LMFDB - Embedded newform 8019.2.a.i.1.16

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:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q-1.42916 q^{2} +0.0424985 q^{4} -0.596655 q^{5} -4.06486 q^{7} +2.79758 q^{8} +O(q^{10})$	\(q-1.42916 q^{2} +0.0424985 q^{4} -0.596655 q^{5} -4.06486 q^{7} +2.79758 q^{8} +0.852715 q^{10} -1.00000 q^{11} -6.34217 q^{13} +5.80934 q^{14} -4.08319 q^{16} +6.18090 q^{17} +4.48638 q^{19} -0.0253569 q^{20} +1.42916 q^{22} +3.88339 q^{23} -4.64400 q^{25} +9.06398 q^{26} -0.172750 q^{28} -2.01695 q^{29} -5.51710 q^{31} +0.240367 q^{32} -8.83350 q^{34} +2.42532 q^{35} +1.56041 q^{37} -6.41175 q^{38} -1.66919 q^{40} -0.599885 q^{41} +7.75052 q^{43} -0.0424985 q^{44} -5.54999 q^{46} +0.217861 q^{47} +9.52309 q^{49} +6.63702 q^{50} -0.269533 q^{52} -11.9771 q^{53} +0.596655 q^{55} -11.3718 q^{56} +2.88255 q^{58} -10.6092 q^{59} +7.45640 q^{61} +7.88482 q^{62} +7.82286 q^{64} +3.78409 q^{65} +5.02445 q^{67} +0.262679 q^{68} -3.46617 q^{70} -6.30959 q^{71} +3.74077 q^{73} -2.23007 q^{74} +0.190664 q^{76} +4.06486 q^{77} +13.8856 q^{79} +2.43626 q^{80} +0.857332 q^{82} +11.1892 q^{83} -3.68787 q^{85} -11.0767 q^{86} -2.79758 q^{88} +12.5897 q^{89} +25.7801 q^{91} +0.165038 q^{92} -0.311358 q^{94} -2.67682 q^{95} +1.02661 q^{97} -13.6100 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * 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 $

$2$	−1.42916	−1.01057	−0.505284	−	0.862953i	$-0.668612\pi$
$2$	−1.42916	−1.01057	−0.505284	+	0.862953i	$0.668612\pi$
$3$	0	0
$3$	0	0
$4$	0.0424985	0.0212492
$5$	−0.596655	−0.266832	−0.133416	−	0.991060i	$-0.542595\pi$
$5$	−0.596655	−0.266832	−0.133416	+	0.991060i	$0.542595\pi$
$6$	0	0
$7$	−4.06486	−1.53637	−0.768186	−	0.640226i	$-0.778841\pi$
$7$	−4.06486	−1.53637	−0.768186	+	0.640226i	$0.778841\pi$
$8$	2.79758	0.989095
$9$	0	0
$10$	0.852715	0.269652
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.34217	−1.75900	−0.879501	−	0.475897i	$-0.842124\pi$
$13$	−6.34217	−1.75900	−0.879501	+	0.475897i	$0.842124\pi$
$14$	5.80934	1.55261
$15$	0	0
$16$	−4.08319	−1.02080
$17$	6.18090	1.49909	0.749545	−	0.661954i	$-0.230272\pi$
$17$	6.18090	1.49909	0.749545	+	0.661954i	$0.230272\pi$
$18$	0	0
$19$	4.48638	1.02925	0.514623	−	0.857417i	$-0.327932\pi$
$19$	4.48638	1.02925	0.514623	+	0.857417i	$0.327932\pi$
$20$	−0.0253569	−0.00566998
$21$	0	0
$22$	1.42916	0.304698
$23$	3.88339	0.809743	0.404871	−	0.914374i	$-0.367316\pi$
$23$	3.88339	0.809743	0.404871	+	0.914374i	$0.367316\pi$
$24$	0	0
$25$	−4.64400	−0.928801
$26$	9.06398	1.77759
$27$	0	0
$28$	−0.172750	−0.0326467
$29$	−2.01695	−0.374539	−0.187269	−	0.982309i	$-0.559964\pi$
$29$	−2.01695	−0.374539	−0.187269	+	0.982309i	$0.559964\pi$
$30$	0	0
$31$	−5.51710	−0.990901	−0.495450	−	0.868636i	$-0.664997\pi$
$31$	−5.51710	−0.990901	−0.495450	+	0.868636i	$0.664997\pi$
$32$	0.240367	0.0424913
$33$	0	0
$34$	−8.83350	−1.51493
$35$	2.42532	0.409954
$36$	0	0
$37$	1.56041	0.256529	0.128265	−	0.991740i	$-0.459059\pi$
$37$	1.56041	0.256529	0.128265	+	0.991740i	$0.459059\pi$
$38$	−6.41175	−1.04012
$39$	0	0
$40$	−1.66919	−0.263922
$41$	−0.599885	−0.0936864	−0.0468432	−	0.998902i	$-0.514916\pi$
$41$	−0.599885	−0.0936864	−0.0468432	+	0.998902i	$0.514916\pi$
$42$	0	0
$43$	7.75052	1.18194	0.590972	−	0.806692i	$-0.298745\pi$
$43$	7.75052	1.18194	0.590972	+	0.806692i	$0.298745\pi$
$44$	−0.0424985	−0.00640688
$45$	0	0
$46$	−5.54999	−0.818301
$47$	0.217861	0.0317783	0.0158892	−	0.999874i	$-0.494942\pi$
$47$	0.217861	0.0317783	0.0158892	+	0.999874i	$0.494942\pi$
$48$	0	0
$49$	9.52309	1.36044
$50$	6.63702	0.938617
$51$	0	0
$52$	−0.269533	−0.0373774
$53$	−11.9771	−1.64518	−0.822589	−	0.568637i	$-0.807471\pi$
$53$	−11.9771	−1.64518	−0.822589	+	0.568637i	$0.807471\pi$
$54$	0	0
$55$	0.596655	0.0804529
$56$	−11.3718	−1.51962
$57$	0	0
$58$	2.88255	0.378497
$59$	−10.6092	−1.38120	−0.690601	−	0.723236i	$-0.742654\pi$
$59$	−10.6092	−1.38120	−0.690601	+	0.723236i	$0.742654\pi$
$60$	0	0
$61$	7.45640	0.954695	0.477347	−	0.878715i	$-0.341598\pi$
$61$	7.45640	0.954695	0.477347	+	0.878715i	$0.341598\pi$
$62$	7.88482	1.00137
$63$	0	0
$64$	7.82286	0.977857
$65$	3.78409	0.469358
$66$	0	0
$67$	5.02445	0.613835	0.306917	−	0.951736i	$-0.400702\pi$
$67$	5.02445	0.613835	0.306917	+	0.951736i	$0.400702\pi$
$68$	0.262679	0.0318545
$69$	0	0
$70$	−3.46617	−0.414286
$71$	−6.30959	−0.748810	−0.374405	−	0.927265i	$-0.622153\pi$
$71$	−6.30959	−0.748810	−0.374405	+	0.927265i	$0.622153\pi$
$72$	0	0
$73$	3.74077	0.437824	0.218912	−	0.975745i	$-0.429749\pi$
$73$	3.74077	0.437824	0.218912	+	0.975745i	$0.429749\pi$
$74$	−2.23007	−0.259240
$75$	0	0
$76$	0.190664	0.0218707
$77$	4.06486	0.463234
$78$	0	0
$79$	13.8856	1.56225	0.781125	−	0.624375i	$-0.214646\pi$
$79$	13.8856	1.56225	0.781125	+	0.624375i	$0.214646\pi$
$80$	2.43626	0.272382
$81$	0	0
$82$	0.857332	0.0946765
$83$	11.1892	1.22818	0.614089	−	0.789237i	$-0.289524\pi$
$83$	11.1892	1.22818	0.614089	+	0.789237i	$0.289524\pi$
$84$	0	0
$85$	−3.68787	−0.400005
$86$	−11.0767	−1.19443
$87$	0	0
$88$	−2.79758	−0.298223
$89$	12.5897	1.33451	0.667255	−	0.744829i	$-0.267469\pi$
$89$	12.5897	1.33451	0.667255	+	0.744829i	$0.267469\pi$
$90$	0	0
$91$	25.7801	2.70248
$92$	0.165038	0.0172064
$93$	0	0
$94$	−0.311358	−0.0321142
$95$	−2.67682	−0.274636
$96$	0	0
$97$	1.02661	0.104237	0.0521184	−	0.998641i	$-0.483403\pi$
$97$	1.02661	0.104237	0.0521184	+	0.998641i	$0.483403\pi$
$98$	−13.6100	−1.37482
$99$	0	0
$100$	−0.197363	−0.0197363
$101$	6.37885	0.634720	0.317360	−	0.948305i	$-0.397204\pi$
$101$	6.37885	0.634720	0.317360	+	0.948305i	$0.397204\pi$
$102$	0	0
$103$	6.96632	0.686412	0.343206	−	0.939260i	$-0.388487\pi$
$103$	6.96632	0.686412	0.343206	+	0.939260i	$0.388487\pi$
$104$	−17.7428	−1.73982
$105$	0	0
$106$	17.1172	1.66256
$107$	−3.25990	−0.315147	−0.157573	−	0.987507i	$-0.550367\pi$
$107$	−3.25990	−0.315147	−0.157573	+	0.987507i	$0.550367\pi$
$108$	0	0
$109$	−11.0186	−1.05539	−0.527694	−	0.849435i	$-0.676943\pi$
$109$	−11.0186	−1.05539	−0.527694	+	0.849435i	$0.676943\pi$
$110$	−0.852715	−0.0813032
$111$	0	0
$112$	16.5976	1.56833
$113$	18.5449	1.74455	0.872277	−	0.489012i	$-0.162642\pi$
$113$	18.5449	1.74455	0.872277	+	0.489012i	$0.162642\pi$
$114$	0	0
$115$	−2.31704	−0.216065
$116$	−0.0857174	−0.00795866
$117$	0	0
$118$	15.1623	1.39580
$119$	−25.1245	−2.30316
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.6564	−0.964785
$123$	0	0
$124$	−0.234468	−0.0210559
$125$	5.75414	0.514666
$126$	0	0
$127$	−9.02027	−0.800419	−0.400210	−	0.916424i	$-0.631063\pi$
$127$	−9.02027	−0.800419	−0.400210	+	0.916424i	$0.631063\pi$
$128$	−11.6609	−1.03068
$129$	0	0
$130$	−5.40807	−0.474319
$131$	10.2008	0.891245	0.445622	−	0.895221i	$-0.352982\pi$
$131$	10.2008	0.891245	0.445622	+	0.895221i	$0.352982\pi$
$132$	0	0
$133$	−18.2365	−1.58131
$134$	−7.18075	−0.620322
$135$	0	0
$136$	17.2916	1.48274
$137$	8.82281	0.753783	0.376892	−	0.926257i	$-0.376993\pi$
$137$	8.82281	0.753783	0.376892	+	0.926257i	$0.376993\pi$
$138$	0	0
$139$	8.65064	0.733737	0.366869	−	0.930273i	$-0.380430\pi$
$139$	8.65064	0.733737	0.366869	+	0.930273i	$0.380430\pi$
$140$	0.103072	0.00871120
$141$	0	0
$142$	9.01741	0.756724
$143$	6.34217	0.530359
$144$	0	0
$145$	1.20343	0.0999390
$146$	−5.34616	−0.442451
$147$	0	0
$148$	0.0663148	0.00545105
$149$	12.7642	1.04568	0.522842	−	0.852429i	$-0.324872\pi$
$149$	12.7642	1.04568	0.522842	+	0.852429i	$0.324872\pi$
$150$	0	0
$151$	−23.5804	−1.91894	−0.959472	−	0.281805i	$-0.909067\pi$
$151$	−23.5804	−1.91894	−0.959472	+	0.281805i	$0.909067\pi$
$152$	12.5510	1.01802
$153$	0	0
$154$	−5.80934	−0.468130
$155$	3.29181	0.264404
$156$	0	0
$157$	0.675363	0.0538999	0.0269499	−	0.999637i	$-0.491421\pi$
$157$	0.675363	0.0538999	0.0269499	+	0.999637i	$0.491421\pi$
$158$	−19.8447	−1.57876
$159$	0	0
$160$	−0.143416	−0.0113380
$161$	−15.7854	−1.24407
$162$	0	0
$163$	−21.8433	−1.71090	−0.855448	−	0.517888i	$-0.826718\pi$
$163$	−21.8433	−1.71090	−0.855448	+	0.517888i	$0.826718\pi$
$164$	−0.0254942	−0.00199076
$165$	0	0
$166$	−15.9912	−1.24116
$167$	−10.2608	−0.794007	−0.397004	−	0.917817i	$-0.629950\pi$
$167$	−10.2608	−0.794007	−0.397004	+	0.917817i	$0.629950\pi$
$168$	0	0
$169$	27.2232	2.09409
$170$	5.27055	0.404233
$171$	0	0
$172$	0.329385	0.0251154
$173$	−9.96734	−0.757803	−0.378901	−	0.925437i	$-0.623698\pi$
$173$	−9.96734	−0.757803	−0.378901	+	0.925437i	$0.623698\pi$
$174$	0	0
$175$	18.8772	1.42698
$176$	4.08319	0.307782
$177$	0	0
$178$	−17.9928	−1.34861
$179$	−0.304848	−0.0227854	−0.0113927	−	0.999935i	$-0.503626\pi$
$179$	−0.304848	−0.0227854	−0.0113927	+	0.999935i	$0.503626\pi$
$180$	0	0
$181$	21.5936	1.60504	0.802518	−	0.596627i	$-0.203493\pi$
$181$	21.5936	1.60504	0.802518	+	0.596627i	$0.203493\pi$
$182$	−36.8438	−2.73105
$183$	0	0
$184$	10.8641	0.800913
$185$	−0.931024	−0.0684502
$186$	0	0
$187$	−6.18090	−0.451992
$188$	0.00925876	0.000675264 0
$189$	0	0
$190$	3.82560	0.277538
$191$	2.82063	0.204093	0.102047	−	0.994780i	$-0.467461\pi$
$191$	2.82063	0.204093	0.102047	+	0.994780i	$0.467461\pi$
$192$	0	0
$193$	−23.6800	−1.70452	−0.852261	−	0.523116i	$-0.824769\pi$
$193$	−23.6800	−1.70452	−0.852261	+	0.523116i	$0.824769\pi$
$194$	−1.46719	−0.105338
$195$	0	0
$196$	0.404717	0.0289083
$197$	−6.35788	−0.452980	−0.226490	−	0.974013i	$-0.572725\pi$
$197$	−6.35788	−0.452980	−0.226490	+	0.974013i	$0.572725\pi$
$198$	0	0
$199$	11.1707	0.791870	0.395935	−	0.918279i	$-0.370421\pi$
$199$	11.1707	0.791870	0.395935	+	0.918279i	$0.370421\pi$
$200$	−12.9920	−0.918672
$201$	0	0
$202$	−9.11640	−0.641428
$203$	8.19864	0.575431
$204$	0	0
$205$	0.357925	0.0249985
$206$	−9.95599	−0.693667
$207$	0	0
$208$	25.8963	1.79559
$209$	−4.48638	−0.310329
$210$	0	0
$211$	−22.0489	−1.51791	−0.758955	−	0.651143i	$-0.774290\pi$
$211$	−22.0489	−1.51791	−0.758955	+	0.651143i	$0.774290\pi$
$212$	−0.509007	−0.0349587
$213$	0	0
$214$	4.65892	0.318477
$215$	−4.62439	−0.315380
$216$	0	0
$217$	22.4262	1.52239
$218$	15.7473	1.06654
$219$	0	0
$220$	0.0253569	0.00170956
$221$	−39.2004	−2.63690
$222$	0	0
$223$	25.9878	1.74027	0.870136	−	0.492812i	$-0.164031\pi$
$223$	25.9878	1.74027	0.870136	+	0.492812i	$0.164031\pi$
$224$	−0.977059	−0.0652825
$225$	0	0
$226$	−26.5036	−1.76299
$227$	−12.0349	−0.798783	−0.399392	−	0.916780i	$-0.630779\pi$
$227$	−12.0349	−0.798783	−0.399392	+	0.916780i	$0.630779\pi$
$228$	0	0
$229$	6.18769	0.408894	0.204447	−	0.978878i	$-0.434460\pi$
$229$	6.18769	0.408894	0.204447	+	0.978878i	$0.434460\pi$
$230$	3.31143	0.218349
$231$	0	0
$232$	−5.64260	−0.370455
$233$	−19.9236	−1.30524	−0.652621	−	0.757685i	$-0.726330\pi$
$233$	−19.9236	−1.30524	−0.652621	+	0.757685i	$0.726330\pi$
$234$	0	0
$235$	−0.129988	−0.00847947
$236$	−0.450875	−0.0293495
$237$	0	0
$238$	35.9070	2.32750
$239$	−1.06846	−0.0691126	−0.0345563	−	0.999403i	$-0.511002\pi$
$239$	−1.06846	−0.0691126	−0.0345563	+	0.999403i	$0.511002\pi$
$240$	0	0
$241$	21.8800	1.40942	0.704709	−	0.709497i	$-0.251078\pi$
$241$	21.8800	1.40942	0.704709	+	0.709497i	$0.251078\pi$
$242$	−1.42916	−0.0918699
$243$	0	0
$244$	0.316886	0.0202865
$245$	−5.68200	−0.363010
$246$	0	0
$247$	−28.4534	−1.81045
$248$	−15.4345	−0.980095
$249$	0	0
$250$	−8.22359	−0.520105
$251$	−22.5338	−1.42232	−0.711160	−	0.703030i	$-0.751830\pi$
$251$	−22.5338	−1.42232	−0.711160	+	0.703030i	$0.751830\pi$
$252$	0	0
$253$	−3.88339	−0.244147
$254$	12.8914	0.808879
$255$	0	0
$256$	1.01951	0.0637191
$257$	23.4429	1.46233	0.731163	−	0.682203i	$-0.238978\pi$
$257$	23.4429	1.46233	0.731163	+	0.682203i	$0.238978\pi$
$258$	0	0
$259$	−6.34283	−0.394124
$260$	0.160818	0.00997350
$261$	0	0
$262$	−14.5785	−0.900664
$263$	−5.73928	−0.353899	−0.176950	−	0.984220i	$-0.556623\pi$
$263$	−5.73928	−0.353899	−0.176950	+	0.984220i	$0.556623\pi$
$264$	0	0
$265$	7.14618	0.438986
$266$	26.0629	1.59802
$267$	0	0
$268$	0.213531	0.0130435
$269$	4.82711	0.294314	0.147157	−	0.989113i	$-0.452988\pi$
$269$	4.82711	0.294314	0.147157	+	0.989113i	$0.452988\pi$
$270$	0	0
$271$	−8.44366	−0.512916	−0.256458	−	0.966555i	$-0.582555\pi$
$271$	−8.44366	−0.512916	−0.256458	+	0.966555i	$0.582555\pi$
$272$	−25.2378	−1.53027
$273$	0	0
$274$	−12.6092	−0.761750
$275$	4.64400	0.280044
$276$	0	0
$277$	7.50728	0.451069	0.225534	−	0.974235i	$-0.427587\pi$
$277$	7.50728	0.451069	0.225534	+	0.974235i	$0.427587\pi$
$278$	−12.3631	−0.741492
$279$	0	0
$280$	6.78503	0.405483
$281$	−17.9847	−1.07288	−0.536439	−	0.843939i	$-0.680231\pi$
$281$	−17.9847	−1.07288	−0.536439	+	0.843939i	$0.680231\pi$
$282$	0	0
$283$	−16.2464	−0.965746	−0.482873	−	0.875690i	$-0.660407\pi$
$283$	−16.2464	−0.965746	−0.482873	+	0.875690i	$0.660407\pi$
$284$	−0.268148	−0.0159116
$285$	0	0
$286$	−9.06398	−0.535964
$287$	2.43845	0.143937
$288$	0	0
$289$	21.2036	1.24727
$290$	−1.71989	−0.100995
$291$	0	0
$292$	0.158977	0.00930342
$293$	0.0690118	0.00403171	0.00201586	−	0.999998i	$-0.499358\pi$
$293$	0.0690118	0.00403171	0.00201586	+	0.999998i	$0.499358\pi$
$294$	0	0
$295$	6.33004	0.368549
$296$	4.36537	0.253732
$297$	0	0
$298$	−18.2421	−1.05674
$299$	−24.6291	−1.42434
$300$	0	0
$301$	−31.5048	−1.81591
$302$	33.7001	1.93922
$303$	0	0
$304$	−18.3187	−1.05065
$305$	−4.44890	−0.254743
$306$	0	0
$307$	20.1361	1.14923	0.574613	−	0.818425i	$-0.305152\pi$
$307$	20.1361	1.14923	0.574613	+	0.818425i	$0.305152\pi$
$308$	0.172750	0.00984336
$309$	0	0
$310$	−4.70452	−0.267199
$311$	33.6333	1.90717	0.953585	−	0.301125i	$-0.0973622\pi$
$311$	33.6333	1.90717	0.953585	+	0.301125i	$0.0973622\pi$
$312$	0	0
$313$	1.72817	0.0976821	0.0488411	−	0.998807i	$-0.484447\pi$
$313$	1.72817	0.0976821	0.0488411	+	0.998807i	$0.484447\pi$
$314$	−0.965202	−0.0544695
$315$	0	0
$316$	0.590116	0.0331966
$317$	−28.7544	−1.61501	−0.807504	−	0.589862i	$-0.799182\pi$
$317$	−28.7544	−1.61501	−0.807504	+	0.589862i	$0.799182\pi$
$318$	0	0
$319$	2.01695	0.112928
$320$	−4.66755	−0.260924
$321$	0	0
$322$	22.5599	1.25722
$323$	27.7299	1.54293
$324$	0	0
$325$	29.4531	1.63376
$326$	31.2175	1.72898
$327$	0	0
$328$	−1.67823	−0.0926647
$329$	−0.885575	−0.0488233
$330$	0	0
$331$	10.3939	0.571301	0.285651	−	0.958334i	$-0.407790\pi$
$331$	10.3939	0.571301	0.285651	+	0.958334i	$0.407790\pi$
$332$	0.475525	0.0260978
$333$	0	0
$334$	14.6644	0.802399
$335$	−2.99786	−0.163791
$336$	0	0
$337$	−5.80975	−0.316477	−0.158239	−	0.987401i	$-0.550582\pi$
$337$	−5.80975	−0.316477	−0.158239	+	0.987401i	$0.550582\pi$
$338$	−38.9063	−2.11622
$339$	0	0
$340$	−0.156729	−0.00849980
$341$	5.51710	0.298768
$342$	0	0
$343$	−10.2560	−0.553772
$344$	21.6827	1.16905
$345$	0	0
$346$	14.2449	0.765812
$347$	29.9809	1.60946	0.804730	−	0.593641i	$-0.202310\pi$
$347$	29.9809	1.60946	0.804730	+	0.593641i	$0.202310\pi$
$348$	0	0
$349$	−29.8619	−1.59847	−0.799235	−	0.601019i	$-0.794762\pi$
$349$	−29.8619	−1.59847	−0.799235	+	0.601019i	$0.794762\pi$
$350$	−26.9786	−1.44207
$351$	0	0
$352$	−0.240367	−0.0128116
$353$	−21.4375	−1.14100	−0.570502	−	0.821296i	$-0.693251\pi$
$353$	−21.4375	−1.14100	−0.570502	+	0.821296i	$0.693251\pi$
$354$	0	0
$355$	3.76465	0.199807
$356$	0.535045	0.0283573
$357$	0	0
$358$	0.435676	0.0230262
$359$	−26.2228	−1.38399	−0.691993	−	0.721904i	$-0.743267\pi$
$359$	−26.2228	−1.38399	−0.691993	+	0.721904i	$0.743267\pi$
$360$	0	0
$361$	1.12760	0.0593472
$362$	−30.8607	−1.62200
$363$	0	0
$364$	1.09561	0.0574257
$365$	−2.23195	−0.116826
$366$	0	0
$367$	−0.214465	−0.0111950	−0.00559749	−	0.999984i	$-0.501782\pi$
$367$	−0.214465	−0.0111950	−0.00559749	+	0.999984i	$0.501782\pi$
$368$	−15.8566	−0.826584
$369$	0	0
$370$	1.33058	0.0691737
$371$	48.6851	2.52761
$372$	0	0
$373$	7.72977	0.400232	0.200116	−	0.979772i	$-0.435868\pi$
$373$	7.72977	0.400232	0.200116	+	0.979772i	$0.435868\pi$
$374$	8.83350	0.456770
$375$	0	0
$376$	0.609485	0.0314318
$377$	12.7919	0.658815
$378$	0	0
$379$	−4.79663	−0.246386	−0.123193	−	0.992383i	$-0.539313\pi$
$379$	−4.79663	−0.246386	−0.123193	+	0.992383i	$0.539313\pi$
$380$	−0.113761	−0.00583580
$381$	0	0
$382$	−4.03113	−0.206250
$383$	−7.18453	−0.367112	−0.183556	−	0.983009i	$-0.558761\pi$
$383$	−7.18453	−0.367112	−0.183556	+	0.983009i	$0.558761\pi$
$384$	0	0
$385$	−2.42532	−0.123606
$386$	33.8425	1.72254
$387$	0	0
$388$	0.0436295	0.00221495
$389$	−28.4470	−1.44232	−0.721160	−	0.692769i	$-0.756391\pi$
$389$	−28.4470	−1.44232	−0.721160	+	0.692769i	$0.756391\pi$
$390$	0	0
$391$	24.0029	1.21388
$392$	26.6416	1.34561
$393$	0	0
$394$	9.08642	0.457767
$395$	−8.28490	−0.416858
$396$	0	0
$397$	10.9773	0.550932	0.275466	−	0.961311i	$-0.411168\pi$
$397$	10.9773	0.550932	0.275466	+	0.961311i	$0.411168\pi$
$398$	−15.9647	−0.800239
$399$	0	0
$400$	18.9624	0.948118
$401$	13.1599	0.657172	0.328586	−	0.944474i	$-0.393428\pi$
$401$	13.1599	0.657172	0.328586	+	0.944474i	$0.393428\pi$
$402$	0	0
$403$	34.9904	1.74300
$404$	0.271091	0.0134873
$405$	0	0
$406$	−11.7172	−0.581513
$407$	−1.56041	−0.0773465
$408$	0	0
$409$	−8.69061	−0.429723	−0.214861	−	0.976645i	$-0.568930\pi$
$409$	−8.69061	−0.429723	−0.214861	+	0.976645i	$0.568930\pi$
$410$	−0.511531	−0.0252627
$411$	0	0
$412$	0.296058	0.0145857
$413$	43.1250	2.12204
$414$	0	0
$415$	−6.67611	−0.327717
$416$	−1.52445	−0.0747423
$417$	0	0
$418$	6.41175	0.313609
$419$	−1.71445	−0.0837563	−0.0418781	−	0.999123i	$-0.513334\pi$
$419$	−1.71445	−0.0837563	−0.0418781	+	0.999123i	$0.513334\pi$
$420$	0	0
$421$	−30.0086	−1.46253	−0.731264	−	0.682095i	$-0.761069\pi$
$421$	−30.0086	−1.46253	−0.731264	+	0.682095i	$0.761069\pi$
$422$	31.5114	1.53395
$423$	0	0
$424$	−33.5069	−1.62724
$425$	−28.7041	−1.39236
$426$	0	0
$427$	−30.3092	−1.46677
$428$	−0.138541	−0.00669662
$429$	0	0
$430$	6.60899	0.318714
$431$	−36.7515	−1.77026	−0.885128	−	0.465347i	$-0.845930\pi$
$431$	−36.7515	−1.77026	−0.885128	+	0.465347i	$0.845930\pi$
$432$	0	0
$433$	−12.2726	−0.589783	−0.294891	−	0.955531i	$-0.595283\pi$
$433$	−12.2726	−0.589783	−0.294891	+	0.955531i	$0.595283\pi$
$434$	−32.0507	−1.53848
$435$	0	0
$436$	−0.468272	−0.0224262
$437$	17.4224	0.833425
$438$	0	0
$439$	−36.7694	−1.75491	−0.877453	−	0.479662i	$-0.840759\pi$
$439$	−36.7694	−1.75491	−0.877453	+	0.479662i	$0.840759\pi$
$440$	1.66919	0.0795756
$441$	0	0
$442$	56.0236	2.66477
$443$	16.1145	0.765622	0.382811	−	0.923827i	$-0.374956\pi$
$443$	16.1145	0.765622	0.382811	+	0.923827i	$0.374956\pi$
$444$	0	0
$445$	−7.51173	−0.356090
$446$	−37.1407	−1.75866
$447$	0	0
$448$	−31.7988	−1.50235
$449$	11.2200	0.529502	0.264751	−	0.964317i	$-0.414710\pi$
$449$	11.2200	0.529502	0.264751	+	0.964317i	$0.414710\pi$
$450$	0	0
$451$	0.599885	0.0282475
$452$	0.788128	0.0370704
$453$	0	0
$454$	17.1998	0.807225
$455$	−15.3818	−0.721110
$456$	0	0
$457$	25.1142	1.17479	0.587396	−	0.809300i	$-0.300153\pi$
$457$	25.1142	1.17479	0.587396	+	0.809300i	$0.300153\pi$
$458$	−8.84320	−0.413216
$459$	0	0
$460$	−0.0984708	−0.00459122
$461$	−12.1746	−0.567029	−0.283515	−	0.958968i	$-0.591500\pi$
$461$	−12.1746	−0.567029	−0.283515	+	0.958968i	$0.591500\pi$
$462$	0	0
$463$	16.8222	0.781793	0.390897	−	0.920435i	$-0.372165\pi$
$463$	16.8222	0.781793	0.390897	+	0.920435i	$0.372165\pi$
$464$	8.23561	0.382328
$465$	0	0
$466$	28.4741	1.31904
$467$	−4.71585	−0.218224	−0.109112	−	0.994029i	$-0.534801\pi$
$467$	−4.71585	−0.218224	−0.109112	+	0.994029i	$0.534801\pi$
$468$	0	0
$469$	−20.4237	−0.943079
$470$	0.185773	0.00856909
$471$	0	0
$472$	−29.6802	−1.36614
$473$	−7.75052	−0.356369
$474$	0	0
$475$	−20.8348	−0.955964
$476$	−1.06775	−0.0489404
$477$	0	0
$478$	1.52699	0.0698431
$479$	−20.1551	−0.920909	−0.460454	−	0.887683i	$-0.652314\pi$
$479$	−20.1551	−0.920909	−0.460454	+	0.887683i	$0.652314\pi$
$480$	0	0
$481$	−9.89637	−0.451236
$482$	−31.2701	−1.42431
$483$	0	0
$484$	0.0424985	0.00193175
$485$	−0.612534	−0.0278137
$486$	0	0
$487$	2.11857	0.0960015	0.0480008	−	0.998847i	$-0.484715\pi$
$487$	2.11857	0.0960015	0.0480008	+	0.998847i	$0.484715\pi$
$488$	20.8599	0.944284
$489$	0	0
$490$	8.12048	0.366846
$491$	14.9250	0.673557	0.336778	−	0.941584i	$-0.390663\pi$
$491$	14.9250	0.673557	0.336778	+	0.941584i	$0.390663\pi$
$492$	0	0
$493$	−12.4666	−0.561467
$494$	40.6645	1.82958
$495$	0	0
$496$	22.5274	1.01151
$497$	25.6476	1.15045
$498$	0	0
$499$	22.6494	1.01393	0.506964	−	0.861967i	$-0.330768\pi$
$499$	22.6494	1.01393	0.506964	+	0.861967i	$0.330768\pi$
$500$	0.244542	0.0109363
$501$	0	0
$502$	32.2044	1.43735
$503$	−38.2626	−1.70604	−0.853022	−	0.521875i	$-0.825233\pi$
$503$	−38.2626	−1.70604	−0.853022	+	0.521875i	$0.825233\pi$
$504$	0	0
$505$	−3.80597	−0.169364
$506$	5.54999	0.246727
$507$	0	0
$508$	−0.383347	−0.0170083
$509$	−40.2917	−1.78590	−0.892949	−	0.450158i	$-0.851368\pi$
$509$	−40.2917	−1.78590	−0.892949	+	0.450158i	$0.851368\pi$
$510$	0	0
$511$	−15.2057	−0.672661
$512$	21.8647	0.966291
$513$	0	0
$514$	−33.5036	−1.47778
$515$	−4.15649	−0.183157
$516$	0	0
$517$	−0.217861	−0.00958152
$518$	9.06492	0.398290
$519$	0	0
$520$	10.5863	0.464240
$521$	−39.8556	−1.74610	−0.873052	−	0.487627i	$-0.837863\pi$
$521$	−39.8556	−1.74610	−0.873052	+	0.487627i	$0.837863\pi$
$522$	0	0
$523$	7.23929	0.316552	0.158276	−	0.987395i	$-0.449406\pi$
$523$	7.23929	0.316552	0.158276	+	0.987395i	$0.449406\pi$
$524$	0.433516	0.0189383
$525$	0	0
$526$	8.20235	0.357639
$527$	−34.1007	−1.48545
$528$	0	0
$529$	−7.91928	−0.344316
$530$	−10.2130	−0.443626
$531$	0	0
$532$	−0.775023	−0.0336015
$533$	3.80458	0.164795
$534$	0	0
$535$	1.94504	0.0840912
$536$	14.0563	0.607141
$537$	0	0
$538$	−6.89871	−0.297425
$539$	−9.52309	−0.410188
$540$	0	0
$541$	−26.6604	−1.14622	−0.573109	−	0.819479i	$-0.694263\pi$
$541$	−26.6604	−1.14622	−0.573109	+	0.819479i	$0.694263\pi$
$542$	12.0673	0.518337
$543$	0	0
$544$	1.48569	0.0636983
$545$	6.57428	0.281611
$546$	0	0
$547$	−22.2446	−0.951109	−0.475554	−	0.879686i	$-0.657753\pi$
$547$	−22.2446	−0.951109	−0.475554	+	0.879686i	$0.657753\pi$
$548$	0.374956	0.0160173
$549$	0	0
$550$	−6.63702	−0.283004
$551$	−9.04882	−0.385493
$552$	0	0
$553$	−56.4429	−2.40020
$554$	−10.7291	−0.455836
$555$	0	0
$556$	0.367639	0.0155914
$557$	23.3797	0.990629	0.495314	−	0.868714i	$-0.335053\pi$
$557$	23.3797	0.990629	0.495314	+	0.868714i	$0.335053\pi$
$558$	0	0
$559$	−49.1551	−2.07904
$560$	−9.90304	−0.418480
$561$	0	0
$562$	25.7030	1.08422
$563$	−7.81802	−0.329490	−0.164745	−	0.986336i	$-0.552680\pi$
$563$	−7.81802	−0.329490	−0.164745	+	0.986336i	$0.552680\pi$
$564$	0	0
$565$	−11.0649	−0.465503
$566$	23.2187	0.975953
$567$	0	0
$568$	−17.6516	−0.740645
$569$	−15.2206	−0.638081	−0.319040	−	0.947741i	$-0.603361\pi$
$569$	−15.2206	−0.638081	−0.319040	+	0.947741i	$0.603361\pi$
$570$	0	0
$571$	−1.76252	−0.0737592	−0.0368796	−	0.999320i	$-0.511742\pi$
$571$	−1.76252	−0.0737592	−0.0368796	+	0.999320i	$0.511742\pi$
$572$	0.269533	0.0112697
$573$	0	0
$574$	−3.48494	−0.145458
$575$	−18.0345	−0.752090
$576$	0	0
$577$	14.8221	0.617052	0.308526	−	0.951216i	$-0.400164\pi$
$577$	14.8221	0.617052	0.308526	+	0.951216i	$0.400164\pi$
$578$	−30.3033	−1.26045
$579$	0	0
$580$	0.0511437	0.00212363
$581$	−45.4827	−1.88694
$582$	0	0
$583$	11.9771	0.496040
$584$	10.4651	0.433050
$585$	0	0
$586$	−0.0986289	−0.00407432
$587$	0.950954	0.0392501	0.0196250	−	0.999807i	$-0.493753\pi$
$587$	0.950954	0.0392501	0.0196250	+	0.999807i	$0.493753\pi$
$588$	0	0
$589$	−24.7518	−1.01988
$590$	−9.04664	−0.372444
$591$	0	0
$592$	−6.37144	−0.261864
$593$	12.9027	0.529851	0.264925	−	0.964269i	$-0.414653\pi$
$593$	12.9027	0.529851	0.264925	+	0.964269i	$0.414653\pi$
$594$	0	0
$595$	14.9907	0.614557
$596$	0.542459	0.0222200
$597$	0	0
$598$	35.1990	1.43939
$599$	−16.8104	−0.686853	−0.343426	−	0.939180i	$-0.611588\pi$
$599$	−16.8104	−0.686853	−0.343426	+	0.939180i	$0.611588\pi$
$600$	0	0
$601$	21.2490	0.866765	0.433383	−	0.901210i	$-0.357320\pi$
$601$	21.2490	0.866765	0.433383	+	0.901210i	$0.357320\pi$
$602$	45.0254	1.83510
$603$	0	0
$604$	−1.00213	−0.0407761
$605$	−0.596655	−0.0242575
$606$	0	0
$607$	10.2918	0.417730	0.208865	−	0.977945i	$-0.433023\pi$
$607$	10.2918	0.417730	0.208865	+	0.977945i	$0.433023\pi$
$608$	1.07838	0.0437340
$609$	0	0
$610$	6.35819	0.257436
$611$	−1.38171	−0.0558981
$612$	0	0
$613$	−3.14832	−0.127160	−0.0635798	−	0.997977i	$-0.520252\pi$
$613$	−3.14832	−0.127160	−0.0635798	+	0.997977i	$0.520252\pi$
$614$	−28.7777	−1.16137
$615$	0	0
$616$	11.3718	0.458182
$617$	11.5801	0.466198	0.233099	−	0.972453i	$-0.425113\pi$
$617$	11.5801	0.466198	0.233099	+	0.972453i	$0.425113\pi$
$618$	0	0
$619$	15.5599	0.625407	0.312703	−	0.949851i	$-0.398765\pi$
$619$	15.5599	0.625407	0.312703	+	0.949851i	$0.398765\pi$
$620$	0.139897	0.00561838
$621$	0	0
$622$	−48.0674	−1.92733
$623$	−51.1755	−2.05030
$624$	0	0
$625$	19.7868	0.791471
$626$	−2.46984	−0.0987145
$627$	0	0
$628$	0.0287019	0.00114533
$629$	9.64472	0.384560
$630$	0	0
$631$	20.8428	0.829740	0.414870	−	0.909881i	$-0.363827\pi$
$631$	20.8428	0.829740	0.414870	+	0.909881i	$0.363827\pi$
$632$	38.8461	1.54521
$633$	0	0
$634$	41.0946	1.63208
$635$	5.38198	0.213578
$636$	0	0
$637$	−60.3971	−2.39302
$638$	−2.88255	−0.114121
$639$	0	0
$640$	6.95750	0.275019
$641$	−27.3992	−1.08220	−0.541102	−	0.840957i	$-0.681993\pi$
$641$	−27.3992	−1.08220	−0.541102	+	0.840957i	$0.681993\pi$
$642$	0	0
$643$	12.1778	0.480246	0.240123	−	0.970742i	$-0.422812\pi$
$643$	12.1778	0.480246	0.240123	+	0.970742i	$0.422812\pi$
$644$	−0.670857	−0.0264355
$645$	0	0
$646$	−39.6304	−1.55924
$647$	15.1721	0.596478	0.298239	−	0.954491i	$-0.403601\pi$
$647$	15.1721	0.596478	0.298239	+	0.954491i	$0.403601\pi$
$648$	0	0
$649$	10.6092	0.416448
$650$	−42.0932	−1.65103
$651$	0	0
$652$	−0.928305	−0.0363552
$653$	11.5738	0.452916	0.226458	−	0.974021i	$-0.427285\pi$
$653$	11.5738	0.452916	0.226458	+	0.974021i	$0.427285\pi$
$654$	0	0
$655$	−6.08633	−0.237813
$656$	2.44945	0.0956348
$657$	0	0
$658$	1.26563	0.0493393
$659$	25.7436	1.00283	0.501413	−	0.865208i	$-0.332814\pi$
$659$	25.7436	1.00283	0.501413	+	0.865208i	$0.332814\pi$
$660$	0	0
$661$	30.4955	1.18614	0.593069	−	0.805151i	$-0.297916\pi$
$661$	30.4955	1.18614	0.593069	+	0.805151i	$0.297916\pi$
$662$	−14.8546	−0.577339
$663$	0	0
$664$	31.3028	1.21478
$665$	10.8809	0.421943
$666$	0	0
$667$	−7.83262	−0.303280
$668$	−0.436070	−0.0168720
$669$	0	0
$670$	4.28443	0.165522
$671$	−7.45640	−0.287851
$672$	0	0
$673$	−26.1987	−1.00989	−0.504944	−	0.863152i	$-0.668487\pi$
$673$	−26.1987	−1.00989	−0.504944	+	0.863152i	$0.668487\pi$
$674$	8.30307	0.319822
$675$	0	0
$676$	1.15694	0.0444978
$677$	−27.5510	−1.05887	−0.529435	−	0.848350i	$-0.677596\pi$
$677$	−27.5510	−1.05887	−0.529435	+	0.848350i	$0.677596\pi$
$678$	0	0
$679$	−4.17304	−0.160146
$680$	−10.3171	−0.395643
$681$	0	0
$682$	−7.88482	−0.301925
$683$	−17.3957	−0.665629	−0.332815	−	0.942992i	$-0.607998\pi$
$683$	−17.3957	−0.665629	−0.332815	+	0.942992i	$0.607998\pi$
$684$	0	0
$685$	−5.26417	−0.201134
$686$	14.6575	0.559625
$687$	0	0
$688$	−31.6469	−1.20652
$689$	75.9607	2.89387
$690$	0	0
$691$	22.6829	0.862898	0.431449	−	0.902137i	$-0.358003\pi$
$691$	22.6829	0.862898	0.431449	+	0.902137i	$0.358003\pi$
$692$	−0.423597	−0.0161027
$693$	0	0
$694$	−42.8475	−1.62647
$695$	−5.16144	−0.195785
$696$	0	0
$697$	−3.70783	−0.140444
$698$	42.6774	1.61536
$699$	0	0
$700$	0.802253	0.0303223
$701$	30.6412	1.15730	0.578651	−	0.815576i	$-0.303580\pi$
$701$	30.6412	1.15730	0.578651	+	0.815576i	$0.303580\pi$
$702$	0	0
$703$	7.00057	0.264032
$704$	−7.82286	−0.294835
$705$	0	0
$706$	30.6376	1.15306
$707$	−25.9291	−0.975166
$708$	0	0
$709$	−24.9048	−0.935319	−0.467659	−	0.883909i	$-0.654903\pi$
$709$	−24.9048	−0.935319	−0.467659	+	0.883909i	$0.654903\pi$
$710$	−5.38028	−0.201918
$711$	0	0
$712$	35.2208	1.31996
$713$	−21.4251	−0.802375
$714$	0	0
$715$	−3.78409	−0.141517
$716$	−0.0129556	−0.000484172 0
$717$	0	0
$718$	37.4766	1.39861
$719$	16.9203	0.631020	0.315510	−	0.948922i	$-0.397824\pi$
$719$	16.9203	0.631020	0.315510	+	0.948922i	$0.397824\pi$
$720$	0	0
$721$	−28.3171	−1.05458
$722$	−1.61152	−0.0599745
$723$	0	0
$724$	0.917693	0.0341058
$725$	9.36674	0.347872
$726$	0	0
$727$	−0.544095	−0.0201794	−0.0100897	−	0.999949i	$-0.503212\pi$
$727$	−0.544095	−0.0201794	−0.0100897	+	0.999949i	$0.503212\pi$
$728$	72.1218	2.67301
$729$	0	0
$730$	3.18981	0.118060
$731$	47.9052	1.77184
$732$	0	0
$733$	−24.6242	−0.909514	−0.454757	−	0.890615i	$-0.650274\pi$
$733$	−24.6242	−0.909514	−0.454757	+	0.890615i	$0.650274\pi$
$734$	0.306505	0.0113133
$735$	0	0
$736$	0.933439	0.0344070
$737$	−5.02445	−0.185078
$738$	0	0
$739$	−48.3127	−1.77721	−0.888606	−	0.458671i	$-0.848326\pi$
$739$	−48.3127	−1.77721	−0.888606	+	0.458671i	$0.848326\pi$
$740$	−0.0395671	−0.00145451
$741$	0	0
$742$	−69.5788	−2.55432
$743$	−2.45342	−0.0900071	−0.0450035	−	0.998987i	$-0.514330\pi$
$743$	−2.45342	−0.0900071	−0.0450035	+	0.998987i	$0.514330\pi$
$744$	0	0
$745$	−7.61582	−0.279022
$746$	−11.0471	−0.404462
$747$	0	0
$748$	−0.262679	−0.00960449
$749$	13.2510	0.484183
$750$	0	0
$751$	44.2321	1.61405	0.807025	−	0.590517i	$-0.201076\pi$
$751$	44.2321	1.61405	0.807025	+	0.590517i	$0.201076\pi$
$752$	−0.889568	−0.0324392
$753$	0	0
$754$	−18.2816	−0.665778
$755$	14.0693	0.512036
$756$	0	0
$757$	41.2499	1.49925	0.749627	−	0.661860i	$-0.230233\pi$
$757$	41.2499	1.49925	0.749627	+	0.661860i	$0.230233\pi$
$758$	6.85515	0.248990
$759$	0	0
$760$	−7.48863	−0.271641
$761$	−17.5187	−0.635053	−0.317526	−	0.948249i	$-0.602852\pi$
$761$	−17.5187	−0.635053	−0.317526	+	0.948249i	$0.602852\pi$
$762$	0	0
$763$	44.7889	1.62147
$764$	0.119872	0.00433683
$765$	0	0
$766$	10.2678	0.370992
$767$	67.2855	2.42954
$768$	0	0
$769$	−35.5094	−1.28050	−0.640250	−	0.768166i	$-0.721169\pi$
$769$	−35.5094	−1.28050	−0.640250	+	0.768166i	$0.721169\pi$
$770$	3.46617	0.124912
$771$	0	0
$772$	−1.00636	−0.0362198
$773$	32.5461	1.17060	0.585300	−	0.810817i	$-0.300977\pi$
$773$	32.5461	1.17060	0.585300	+	0.810817i	$0.300977\pi$
$774$	0	0
$775$	25.6214	0.920349
$776$	2.87203	0.103100
$777$	0	0
$778$	40.6553	1.45756
$779$	−2.69131	−0.0964263
$780$	0	0
$781$	6.30959	0.225775
$782$	−34.3039	−1.22671
$783$	0	0
$784$	−38.8846	−1.38874
$785$	−0.402959	−0.0143822
$786$	0	0
$787$	−46.6462	−1.66276	−0.831379	−	0.555706i	$-0.812448\pi$
$787$	−46.6462	−1.66276	−0.831379	+	0.555706i	$0.812448\pi$
$788$	−0.270200	−0.00962547
$789$	0	0
$790$	11.8404	0.421264
$791$	−75.3823	−2.68029
$792$	0	0
$793$	−47.2898	−1.67931
$794$	−15.6882	−0.556755
$795$	0	0
$796$	0.474737	0.0168266
$797$	−42.6835	−1.51193	−0.755963	−	0.654614i	$-0.772831\pi$
$797$	−42.6835	−1.51193	−0.755963	+	0.654614i	$0.772831\pi$
$798$	0	0
$799$	1.34658	0.0476385
$800$	−1.11627	−0.0394660
$801$	0	0
$802$	−18.8075	−0.664117
$803$	−3.74077	−0.132009
$804$	0	0
$805$	9.41846	0.331957
$806$	−50.0069	−1.76142
$807$	0	0
$808$	17.8454	0.627798
$809$	−24.4720	−0.860390	−0.430195	−	0.902736i	$-0.641555\pi$
$809$	−24.4720	−0.860390	−0.430195	+	0.902736i	$0.641555\pi$
$810$	0	0
$811$	−48.6616	−1.70874	−0.854369	−	0.519666i	$-0.826056\pi$
$811$	−48.6616	−1.70874	−0.854369	+	0.519666i	$0.826056\pi$
$812$	0.348429	0.0122275
$813$	0	0
$814$	2.23007	0.0781639
$815$	13.0329	0.456522
$816$	0	0
$817$	34.7718	1.21651
$818$	12.4203	0.434265
$819$	0	0
$820$	0.0152112	0.000531199 0
$821$	−37.7271	−1.31668	−0.658342	−	0.752719i	$-0.728742\pi$
$821$	−37.7271	−1.31668	−0.658342	+	0.752719i	$0.728742\pi$
$822$	0	0
$823$	9.07286	0.316260	0.158130	−	0.987418i	$-0.449453\pi$
$823$	9.07286	0.316260	0.158130	+	0.987418i	$0.449453\pi$
$824$	19.4889	0.678927
$825$	0	0
$826$	−61.6325	−2.14447
$827$	45.3448	1.57679	0.788397	−	0.615167i	$-0.210911\pi$
$827$	45.3448	1.57679	0.788397	+	0.615167i	$0.210911\pi$
$828$	0	0
$829$	−3.75929	−0.130566	−0.0652828	−	0.997867i	$-0.520795\pi$
$829$	−3.75929	−0.130566	−0.0652828	+	0.997867i	$0.520795\pi$
$830$	9.54123	0.331181
$831$	0	0
$832$	−49.6139	−1.72005
$833$	58.8613	2.03942
$834$	0	0
$835$	6.12218	0.211867
$836$	−0.190664	−0.00659426
$837$	0	0
$838$	2.45022	0.0846415
$839$	16.5315	0.570732	0.285366	−	0.958419i	$-0.407885\pi$
$839$	16.5315	0.570732	0.285366	+	0.958419i	$0.407885\pi$
$840$	0	0
$841$	−24.9319	−0.859721
$842$	42.8870	1.47798
$843$	0	0
$844$	−0.937044	−0.0322544
$845$	−16.2428	−0.558771
$846$	0	0
$847$	−4.06486	−0.139670
$848$	48.9047	1.67939
$849$	0	0
$850$	41.0228	1.40707
$851$	6.05967	0.207723
$852$	0	0
$853$	46.9881	1.60884	0.804422	−	0.594059i	$-0.202475\pi$
$853$	46.9881	1.60884	0.804422	+	0.594059i	$0.202475\pi$
$854$	43.3168	1.48227
$855$	0	0
$856$	−9.11985	−0.311710
$857$	37.1509	1.26905	0.634525	−	0.772903i	$-0.281196\pi$
$857$	37.1509	1.26905	0.634525	+	0.772903i	$0.281196\pi$
$858$	0	0
$859$	−24.0690	−0.821225	−0.410612	−	0.911810i	$-0.634685\pi$
$859$	−24.0690	−0.821225	−0.410612	+	0.911810i	$0.634685\pi$
$860$	−0.196529	−0.00670159
$861$	0	0
$862$	52.5238	1.78897
$863$	−16.5239	−0.562479	−0.281240	−	0.959638i	$-0.590746\pi$
$863$	−16.5239	−0.562479	−0.281240	+	0.959638i	$0.590746\pi$
$864$	0	0
$865$	5.94706	0.202206
$866$	17.5395	0.596016
$867$	0	0
$868$	0.953081	0.0323497
$869$	−13.8856	−0.471036
$870$	0	0
$871$	−31.8660	−1.07974
$872$	−30.8253	−1.04388
$873$	0	0
$874$	−24.8993	−0.842233
$875$	−23.3898	−0.790719
$876$	0	0
$877$	−33.2051	−1.12126	−0.560628	−	0.828068i	$-0.689440\pi$
$877$	−33.2051	−1.12126	−0.560628	+	0.828068i	$0.689440\pi$
$878$	52.5493	1.77345
$879$	0	0
$880$	−2.43626	−0.0821262
$881$	−22.9268	−0.772424	−0.386212	−	0.922410i	$-0.626217\pi$
$881$	−22.9268	−0.772424	−0.386212	+	0.922410i	$0.626217\pi$
$882$	0	0
$883$	−3.69800	−0.124448	−0.0622238	−	0.998062i	$-0.519819\pi$
$883$	−3.69800	−0.124448	−0.0622238	+	0.998062i	$0.519819\pi$
$884$	−1.66596	−0.0560321
$885$	0	0
$886$	−23.0302	−0.773714
$887$	16.6905	0.560413	0.280206	−	0.959940i	$-0.409597\pi$
$887$	16.6905	0.560413	0.280206	+	0.959940i	$0.409597\pi$
$888$	0	0
$889$	36.6661	1.22974
$890$	10.7355	0.359854
$891$	0	0
$892$	1.10444	0.0369794
$893$	0.977408	0.0327077
$894$	0	0
$895$	0.181889	0.00607988
$896$	47.3997	1.58351
$897$	0	0
$898$	−16.0351	−0.535098
$899$	11.1277	0.371131
$900$	0	0
$901$	−74.0291	−2.46627
$902$	−0.857332	−0.0285460
$903$	0	0
$904$	51.8808	1.72553
$905$	−12.8839	−0.428275
$906$	0	0
$907$	−46.1885	−1.53366	−0.766831	−	0.641849i	$-0.778168\pi$
$907$	−46.1885	−1.53366	−0.766831	+	0.641849i	$0.778168\pi$
$908$	−0.511464	−0.0169735
$909$	0	0
$910$	21.9830	0.728731
$911$	2.92679	0.0969687	0.0484844	−	0.998824i	$-0.484561\pi$
$911$	2.92679	0.0969687	0.0484844	+	0.998824i	$0.484561\pi$
$912$	0	0
$913$	−11.1892	−0.370310
$914$	−35.8922	−1.18721
$915$	0	0
$916$	0.262967	0.00868868
$917$	−41.4647	−1.36928
$918$	0	0
$919$	34.8677	1.15018	0.575090	−	0.818090i	$-0.304967\pi$
$919$	34.8677	1.15018	0.575090	+	0.818090i	$0.304967\pi$
$920$	−6.48212	−0.213709
$921$	0	0
$922$	17.3995	0.573022
$923$	40.0165	1.31716
$924$	0	0
$925$	−7.24653	−0.238264
$926$	−24.0416	−0.790056
$927$	0	0
$928$	−0.484809	−0.0159146
$929$	−26.1809	−0.858968	−0.429484	−	0.903074i	$-0.641305\pi$
$929$	−26.1809	−0.858968	−0.429484	+	0.903074i	$0.641305\pi$
$930$	0	0
$931$	42.7242	1.40023
$932$	−0.846724	−0.0277354
$933$	0	0
$934$	6.73971	0.220530
$935$	3.68787	0.120606
$936$	0	0
$937$	7.16731	0.234146	0.117073	−	0.993123i	$-0.462649\pi$
$937$	7.16731	0.234146	0.117073	+	0.993123i	$0.462649\pi$
$938$	29.1887	0.953046
$939$	0	0
$940$	−0.00552428	−0.000180182 0
$941$	−41.0527	−1.33828	−0.669139	−	0.743137i	$-0.733337\pi$
$941$	−41.0527	−1.33828	−0.669139	+	0.743137i	$0.733337\pi$
$942$	0	0
$943$	−2.32959	−0.0758619
$944$	43.3195	1.40993
$945$	0	0
$946$	11.0767	0.360136
$947$	−30.2940	−0.984422	−0.492211	−	0.870476i	$-0.663811\pi$
$947$	−30.2940	−0.984422	−0.492211	+	0.870476i	$0.663811\pi$
$948$	0	0
$949$	−23.7246	−0.770134
$950$	29.7762	0.966068
$951$	0	0
$952$	−70.2879	−2.27804
$953$	−55.5618	−1.79982	−0.899911	−	0.436073i	$-0.856369\pi$
$953$	−55.5618	−1.79982	−0.899911	+	0.436073i	$0.856369\pi$
$954$	0	0
$955$	−1.68294	−0.0544587
$956$	−0.0454077	−0.00146859
$957$	0	0
$958$	28.8048	0.930642
$959$	−35.8635	−1.15809
$960$	0	0
$961$	−0.561597	−0.0181160
$962$	14.1435	0.456005
$963$	0	0
$964$	0.929868	0.0299490
$965$	14.1288	0.454822
$966$	0	0
$967$	14.3202	0.460507	0.230253	−	0.973131i	$-0.426045\pi$
$967$	14.3202	0.460507	0.230253	+	0.973131i	$0.426045\pi$
$968$	2.79758	0.0899177
$969$	0	0
$970$	0.875408	0.0281077
$971$	−55.2850	−1.77418	−0.887091	−	0.461595i	$-0.847277\pi$
$971$	−55.2850	−1.77418	−0.887091	+	0.461595i	$0.847277\pi$
$972$	0	0
$973$	−35.1636	−1.12729
$974$	−3.02777	−0.0970162
$975$	0	0
$976$	−30.4459	−0.974550
$977$	−18.5889	−0.594711	−0.297355	−	0.954767i	$-0.596105\pi$
$977$	−18.5889	−0.594711	−0.297355	+	0.954767i	$0.596105\pi$
$978$	0	0
$979$	−12.5897	−0.402370
$980$	−0.241476	−0.00771367
$981$	0	0
$982$	−21.3302	−0.680675
$983$	36.7977	1.17366	0.586832	−	0.809709i	$-0.300375\pi$
$983$	36.7977	1.17366	0.586832	+	0.809709i	$0.300375\pi$
$984$	0	0
$985$	3.79346	0.120870
$986$	17.8168	0.567401
$987$	0	0
$988$	−1.20923	−0.0384706
$989$	30.0983	0.957070
$990$	0	0
$991$	36.3254	1.15391	0.576957	−	0.816775i	$-0.304240\pi$
$991$	36.3254	1.15391	0.576957	+	0.816775i	$0.304240\pi$
$992$	−1.32613	−0.0421047
$993$	0	0
$994$	−36.6545	−1.16261
$995$	−6.66505	−0.211296
$996$	0	0
$997$	−22.3356	−0.707375	−0.353687	−	0.935364i	$-0.615072\pi$
$997$	−22.3356	−0.707375	−0.353687	+	0.935364i	$0.615072\pi$
$998$	−32.3697	−1.02464
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.i.1.16	✓	48
3.2	odd	2		8019.2.a.j.1.33	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8019.2.a.i.1.16	✓	48	1.1	even	1	trivial
8019.2.a.j.1.33	yes	48	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.42916 q^{2} +0.0424985 q^{4} -0.596655 q^{5} -4.06486 q^{7} +2.79758 q^{8} +O(q^{10})\)	\(q-1.42916 q^{2} +0.0424985 q^{4} -0.596655 q^{5} -4.06486 q^{7} +2.79758 q^{8} +0.852715 q^{10} -1.00000 q^{11} -6.34217 q^{13} +5.80934 q^{14} -4.08319 q^{16} +6.18090 q^{17} +4.48638 q^{19} -0.0253569 q^{20} +1.42916 q^{22} +3.88339 q^{23} -4.64400 q^{25} +9.06398 q^{26} -0.172750 q^{28} -2.01695 q^{29} -5.51710 q^{31} +0.240367 q^{32} -8.83350 q^{34} +2.42532 q^{35} +1.56041 q^{37} -6.41175 q^{38} -1.66919 q^{40} -0.599885 q^{41} +7.75052 q^{43} -0.0424985 q^{44} -5.54999 q^{46} +0.217861 q^{47} +9.52309 q^{49} +6.63702 q^{50} -0.269533 q^{52} -11.9771 q^{53} +0.596655 q^{55} -11.3718 q^{56} +2.88255 q^{58} -10.6092 q^{59} +7.45640 q^{61} +7.88482 q^{62} +7.82286 q^{64} +3.78409 q^{65} +5.02445 q^{67} +0.262679 q^{68} -3.46617 q^{70} -6.30959 q^{71} +3.74077 q^{73} -2.23007 q^{74} +0.190664 q^{76} +4.06486 q^{77} +13.8856 q^{79} +2.43626 q^{80} +0.857332 q^{82} +11.1892 q^{83} -3.68787 q^{85} -11.0767 q^{86} -2.79758 q^{88} +12.5897 q^{89} +25.7801 q^{91} +0.165038 q^{92} -0.311358 q^{94} -2.67682 q^{95} +1.02661 q^{97} -13.6100 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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