LMFDB - Embedded newform 8037.2.a.o.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 28 x^{16} + 25 x^{15} + 322 x^{14} - 247 x^{13} - 1971 x^{12} + 1231 x^{11} + 6953 x^{10} + \cdots + 9 $ x^18 - x^17 - 28x^16 + 25x^15 + 322x^14 - 247x^13 - 1971x^12 + 1231x^11 + 6953x^10 - 3283x^9 - 14235x^8 + 4562x^7 + 15962x^6 - 2882x^5 - 8159x^4 + 606x^3 + 890x^2 - 179x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.146947$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.146947 q^{2} -1.97841 q^{4} -2.45635 q^{5} -1.38215 q^{7} +0.584613 q^{8} +O(q^{10})$	\(q-0.146947 q^{2} -1.97841 q^{4} -2.45635 q^{5} -1.38215 q^{7} +0.584613 q^{8} +0.360953 q^{10} +0.590228 q^{11} +1.64960 q^{13} +0.203103 q^{14} +3.87091 q^{16} -6.16682 q^{17} -1.00000 q^{19} +4.85966 q^{20} -0.0867321 q^{22} -9.25247 q^{23} +1.03366 q^{25} -0.242403 q^{26} +2.73446 q^{28} -3.55186 q^{29} +5.70846 q^{31} -1.73804 q^{32} +0.906193 q^{34} +3.39505 q^{35} +3.94965 q^{37} +0.146947 q^{38} -1.43602 q^{40} +8.87986 q^{41} -9.81538 q^{43} -1.16771 q^{44} +1.35962 q^{46} -1.00000 q^{47} -5.08965 q^{49} -0.151893 q^{50} -3.26358 q^{52} +6.20088 q^{53} -1.44981 q^{55} -0.808025 q^{56} +0.521934 q^{58} -12.4931 q^{59} -7.91648 q^{61} -0.838838 q^{62} -7.48641 q^{64} -4.05199 q^{65} -10.6473 q^{67} +12.2005 q^{68} -0.498892 q^{70} +1.81174 q^{71} -7.95953 q^{73} -0.580387 q^{74} +1.97841 q^{76} -0.815786 q^{77} +2.18796 q^{79} -9.50830 q^{80} -1.30487 q^{82} -11.7404 q^{83} +15.1479 q^{85} +1.44234 q^{86} +0.345055 q^{88} -12.0548 q^{89} -2.28000 q^{91} +18.3052 q^{92} +0.146947 q^{94} +2.45635 q^{95} -15.5094 q^{97} +0.747907 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * 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 $

$2$	−0.146947	−0.103907	−0.0519535	−	0.998650i	$-0.516545\pi$
$2$	−0.146947	−0.103907	−0.0519535	+	0.998650i	$0.516545\pi$
$3$	0	0
$3$	0	0
$4$	−1.97841	−0.989203
$5$	−2.45635	−1.09851	−0.549257	−	0.835654i	$-0.685089\pi$
$5$	−2.45635	−1.09851	−0.549257	+	0.835654i	$0.685089\pi$
$6$	0	0
$7$	−1.38215	−0.522405	−0.261202	−	0.965284i	$-0.584119\pi$
$7$	−1.38215	−0.522405	−0.261202	+	0.965284i	$0.584119\pi$
$8$	0.584613	0.206692
$9$	0	0
$10$	0.360953	0.114143
$11$	0.590228	0.177961	0.0889803	−	0.996033i	$-0.471639\pi$
$11$	0.590228	0.177961	0.0889803	+	0.996033i	$0.471639\pi$
$12$	0	0
$13$	1.64960	0.457516	0.228758	−	0.973483i	$-0.426533\pi$
$13$	1.64960	0.457516	0.228758	+	0.973483i	$0.426533\pi$
$14$	0.203103	0.0542815
$15$	0	0
$16$	3.87091	0.967727
$17$	−6.16682	−1.49567	−0.747837	−	0.663883i	$-0.768907\pi$
$17$	−6.16682	−1.49567	−0.747837	+	0.663883i	$0.768907\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	4.85966	1.08665
$21$	0	0
$22$	−0.0867321	−0.0184913
$23$	−9.25247	−1.92927	−0.964637	−	0.263582i	$-0.915096\pi$
$23$	−9.25247	−1.92927	−0.964637	+	0.263582i	$0.915096\pi$
$24$	0	0
$25$	1.03366	0.206732
$26$	−0.242403	−0.0475391
$27$	0	0
$28$	2.73446	0.516764
$29$	−3.55186	−0.659564	−0.329782	−	0.944057i	$-0.606975\pi$
$29$	−3.55186	−0.659564	−0.329782	+	0.944057i	$0.606975\pi$
$30$	0	0
$31$	5.70846	1.02527	0.512635	−	0.858607i	$-0.328670\pi$
$31$	5.70846	1.02527	0.512635	+	0.858607i	$0.328670\pi$
$32$	−1.73804	−0.307246
$33$	0	0
$34$	0.906193	0.155411
$35$	3.39505	0.573869
$36$	0	0
$37$	3.94965	0.649318	0.324659	−	0.945831i	$-0.394751\pi$
$37$	3.94965	0.649318	0.324659	+	0.945831i	$0.394751\pi$
$38$	0.146947	0.0238379
$39$	0	0
$40$	−1.43602	−0.227054
$41$	8.87986	1.38680	0.693400	−	0.720553i	$-0.256112\pi$
$41$	8.87986	1.38680	0.693400	+	0.720553i	$0.256112\pi$
$42$	0	0
$43$	−9.81538	−1.49683	−0.748416	−	0.663230i	$-0.769185\pi$
$43$	−9.81538	−1.49683	−0.748416	+	0.663230i	$0.769185\pi$
$44$	−1.16771	−0.176039
$45$	0	0
$46$	1.35962	0.200465
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−5.08965	−0.727093
$50$	−0.151893	−0.0214809
$51$	0	0
$52$	−3.26358	−0.452577
$53$	6.20088	0.851756	0.425878	−	0.904781i	$-0.359965\pi$
$53$	6.20088	0.851756	0.425878	+	0.904781i	$0.359965\pi$
$54$	0	0
$55$	−1.44981	−0.195492
$56$	−0.808025	−0.107977
$57$	0	0
$58$	0.521934	0.0685333
$59$	−12.4931	−1.62646	−0.813232	−	0.581940i	$-0.802294\pi$
$59$	−12.4931	−1.62646	−0.813232	+	0.581940i	$0.802294\pi$
$60$	0	0
$61$	−7.91648	−1.01360	−0.506801	−	0.862063i	$-0.669172\pi$
$61$	−7.91648	−1.01360	−0.506801	+	0.862063i	$0.669172\pi$
$62$	−0.838838	−0.106533
$63$	0	0
$64$	−7.48641	−0.935802
$65$	−4.05199	−0.502588
$66$	0	0
$67$	−10.6473	−1.30077	−0.650385	−	0.759604i	$-0.725393\pi$
$67$	−10.6473	−1.30077	−0.650385	+	0.759604i	$0.725393\pi$
$68$	12.2005	1.47952
$69$	0	0
$70$	−0.498892	−0.0596289
$71$	1.81174	0.215014	0.107507	−	0.994204i	$-0.465713\pi$
$71$	1.81174	0.215014	0.107507	+	0.994204i	$0.465713\pi$
$72$	0	0
$73$	−7.95953	−0.931593	−0.465796	−	0.884892i	$-0.654232\pi$
$73$	−7.95953	−0.931593	−0.465796	+	0.884892i	$0.654232\pi$
$74$	−0.580387	−0.0674686
$75$	0	0
$76$	1.97841	0.226939
$77$	−0.815786	−0.0929674
$78$	0	0
$79$	2.18796	0.246165	0.123083	−	0.992396i	$-0.460722\pi$
$79$	2.18796	0.246165	0.123083	+	0.992396i	$0.460722\pi$
$80$	−9.50830	−1.06306
$81$	0	0
$82$	−1.30487	−0.144098
$83$	−11.7404	−1.28867	−0.644337	−	0.764742i	$-0.722867\pi$
$83$	−11.7404	−1.28867	−0.644337	+	0.764742i	$0.722867\pi$
$84$	0	0
$85$	15.1479	1.64302
$86$	1.44234	0.155531
$87$	0	0
$88$	0.345055	0.0367830
$89$	−12.0548	−1.27781	−0.638904	−	0.769286i	$-0.720612\pi$
$89$	−12.0548	−1.27781	−0.638904	+	0.769286i	$0.720612\pi$
$90$	0	0
$91$	−2.28000	−0.239009
$92$	18.3052	1.90844
$93$	0	0
$94$	0.146947	0.0151564
$95$	2.45635	0.252016
$96$	0	0
$97$	−15.5094	−1.57474	−0.787372	−	0.616478i	$-0.788559\pi$
$97$	−15.5094	−1.57474	−0.787372	+	0.616478i	$0.788559\pi$
$98$	0.747907	0.0755501
$99$	0	0
$100$	−2.04500	−0.204500
$101$	15.9301	1.58510	0.792552	−	0.609805i	$-0.208752\pi$
$101$	15.9301	1.58510	0.792552	+	0.609805i	$0.208752\pi$
$102$	0	0
$103$	11.7451	1.15728	0.578641	−	0.815582i	$-0.303583\pi$
$103$	11.7451	1.15728	0.578641	+	0.815582i	$0.303583\pi$
$104$	0.964378	0.0945650
$105$	0	0
$106$	−0.911198	−0.0885034
$107$	10.7783	1.04198	0.520989	−	0.853563i	$-0.325563\pi$
$107$	10.7783	1.04198	0.520989	+	0.853563i	$0.325563\pi$
$108$	0	0
$109$	−6.83486	−0.654661	−0.327330	−	0.944910i	$-0.606149\pi$
$109$	−6.83486	−0.654661	−0.327330	+	0.944910i	$0.606149\pi$
$110$	0.213044	0.0203130
$111$	0	0
$112$	−5.35018	−0.505545
$113$	1.13489	0.106762	0.0533809	−	0.998574i	$-0.483000\pi$
$113$	1.13489	0.106762	0.0533809	+	0.998574i	$0.483000\pi$
$114$	0	0
$115$	22.7273	2.11933
$116$	7.02703	0.652443
$117$	0	0
$118$	1.83582	0.169001
$119$	8.52349	0.781347
$120$	0	0
$121$	−10.6516	−0.968330
$122$	1.16330	0.105320
$123$	0	0
$124$	−11.2936	−1.01420
$125$	9.74272	0.871416
$126$	0	0
$127$	4.68977	0.416150	0.208075	−	0.978113i	$-0.433280\pi$
$127$	4.68977	0.416150	0.208075	+	0.978113i	$0.433280\pi$
$128$	4.57619	0.404482
$129$	0	0
$130$	0.595427	0.0522224
$131$	−8.93491	−0.780646	−0.390323	−	0.920678i	$-0.627637\pi$
$131$	−8.93491	−0.780646	−0.390323	+	0.920678i	$0.627637\pi$
$132$	0	0
$133$	1.38215	0.119848
$134$	1.56458	0.135159
$135$	0	0
$136$	−3.60521	−0.309144
$137$	−18.9201	−1.61645	−0.808226	−	0.588872i	$-0.799572\pi$
$137$	−18.9201	−1.61645	−0.808226	+	0.588872i	$0.799572\pi$
$138$	0	0
$139$	22.1644	1.87996	0.939982	−	0.341225i	$-0.110842\pi$
$139$	22.1644	1.87996	0.939982	+	0.341225i	$0.110842\pi$
$140$	−6.71679	−0.567673
$141$	0	0
$142$	−0.266229	−0.0223414
$143$	0.973640	0.0814199
$144$	0	0
$145$	8.72462	0.724540
$146$	1.16963	0.0967990
$147$	0	0
$148$	−7.81400	−0.642307
$149$	−4.45033	−0.364585	−0.182292	−	0.983244i	$-0.558352\pi$
$149$	−4.45033	−0.364585	−0.182292	+	0.983244i	$0.558352\pi$
$150$	0	0
$151$	−17.6106	−1.43313	−0.716565	−	0.697520i	$-0.754287\pi$
$151$	−17.6106	−1.43313	−0.716565	+	0.697520i	$0.754287\pi$
$152$	−0.584613	−0.0474184
$153$	0	0
$154$	0.119877	0.00965996
$155$	−14.0220	−1.12627
$156$	0	0
$157$	−11.3609	−0.906701	−0.453351	−	0.891332i	$-0.649771\pi$
$157$	−11.3609	−0.906701	−0.453351	+	0.891332i	$0.649771\pi$
$158$	−0.321514	−0.0255783
$159$	0	0
$160$	4.26925	0.337513
$161$	12.7883	1.00786
$162$	0	0
$163$	16.3970	1.28431	0.642155	−	0.766575i	$-0.278041\pi$
$163$	16.3970	1.28431	0.642155	+	0.766575i	$0.278041\pi$
$164$	−17.5680	−1.37183
$165$	0	0
$166$	1.72521	0.133902
$167$	20.3238	1.57271	0.786353	−	0.617778i	$-0.211967\pi$
$167$	20.3238	1.57271	0.786353	+	0.617778i	$0.211967\pi$
$168$	0	0
$169$	−10.2788	−0.790679
$170$	−2.22593	−0.170721
$171$	0	0
$172$	19.4188	1.48067
$173$	−19.0218	−1.44620	−0.723102	−	0.690742i	$-0.757284\pi$
$173$	−19.0218	−1.44620	−0.723102	+	0.690742i	$0.757284\pi$
$174$	0	0
$175$	−1.42868	−0.107998
$176$	2.28472	0.172217
$177$	0	0
$178$	1.77141	0.132773
$179$	−19.7651	−1.47731	−0.738655	−	0.674083i	$-0.764539\pi$
$179$	−19.7651	−1.47731	−0.738655	+	0.674083i	$0.764539\pi$
$180$	0	0
$181$	−3.25992	−0.242308	−0.121154	−	0.992634i	$-0.538659\pi$
$181$	−3.25992	−0.242308	−0.121154	+	0.992634i	$0.538659\pi$
$182$	0.335038	0.0248347
$183$	0	0
$184$	−5.40912	−0.398766
$185$	−9.70172	−0.713284
$186$	0	0
$187$	−3.63983	−0.266171
$188$	1.97841	0.144290
$189$	0	0
$190$	−0.360953	−0.0261862
$191$	5.18119	0.374898	0.187449	−	0.982274i	$-0.439978\pi$
$191$	5.18119	0.374898	0.187449	+	0.982274i	$0.439978\pi$
$192$	0	0
$193$	14.3272	1.03130	0.515648	−	0.856800i	$-0.327551\pi$
$193$	14.3272	1.03130	0.515648	+	0.856800i	$0.327551\pi$
$194$	2.27906	0.163627
$195$	0	0
$196$	10.0694	0.719243
$197$	14.5275	1.03504	0.517521	−	0.855671i	$-0.326855\pi$
$197$	14.5275	1.03504	0.517521	+	0.855671i	$0.326855\pi$
$198$	0	0
$199$	−8.31163	−0.589196	−0.294598	−	0.955621i	$-0.595186\pi$
$199$	−8.31163	−0.589196	−0.294598	+	0.955621i	$0.595186\pi$
$200$	0.604292	0.0427299
$201$	0	0
$202$	−2.34087	−0.164703
$203$	4.90922	0.344559
$204$	0	0
$205$	−21.8120	−1.52342
$206$	−1.72591	−0.120250
$207$	0	0
$208$	6.38544	0.442751
$209$	−0.590228	−0.0408270
$210$	0	0
$211$	9.67227	0.665866	0.332933	−	0.942950i	$-0.391962\pi$
$211$	9.67227	0.665866	0.332933	+	0.942950i	$0.391962\pi$
$212$	−12.2679	−0.842560
$213$	0	0
$214$	−1.58384	−0.108269
$215$	24.1100	1.64429
$216$	0	0
$217$	−7.88996	−0.535605
$218$	1.00436	0.0680238
$219$	0	0
$220$	2.86831	0.193381
$221$	−10.1728	−0.684295
$222$	0	0
$223$	−16.5939	−1.11121	−0.555604	−	0.831447i	$-0.687513\pi$
$223$	−16.5939	−1.11121	−0.555604	+	0.831447i	$0.687513\pi$
$224$	2.40224	0.160507
$225$	0	0
$226$	−0.166769	−0.0110933
$227$	12.0980	0.802976	0.401488	−	0.915864i	$-0.368493\pi$
$227$	12.0980	0.802976	0.401488	+	0.915864i	$0.368493\pi$
$228$	0	0
$229$	−11.2730	−0.744942	−0.372471	−	0.928044i	$-0.621489\pi$
$229$	−11.2730	−0.744942	−0.372471	+	0.928044i	$0.621489\pi$
$230$	−3.33970	−0.220214
$231$	0	0
$232$	−2.07647	−0.136327
$233$	21.3017	1.39552	0.697759	−	0.716333i	$-0.254181\pi$
$233$	21.3017	1.39552	0.697759	+	0.716333i	$0.254181\pi$
$234$	0	0
$235$	2.45635	0.160235
$236$	24.7164	1.60890
$237$	0	0
$238$	−1.25250	−0.0811874
$239$	2.95699	0.191272	0.0956359	−	0.995416i	$-0.469512\pi$
$239$	2.95699	0.191272	0.0956359	+	0.995416i	$0.469512\pi$
$240$	0	0
$241$	7.74125	0.498658	0.249329	−	0.968419i	$-0.419790\pi$
$241$	7.74125	0.498658	0.249329	+	0.968419i	$0.419790\pi$
$242$	1.56522	0.100616
$243$	0	0
$244$	15.6620	1.00266
$245$	12.5020	0.798722
$246$	0	0
$247$	−1.64960	−0.104961
$248$	3.33724	0.211915
$249$	0	0
$250$	−1.43166	−0.0905461
$251$	−3.83815	−0.242262	−0.121131	−	0.992637i	$-0.538652\pi$
$251$	−3.83815	−0.242262	−0.121131	+	0.992637i	$0.538652\pi$
$252$	0	0
$253$	−5.46107	−0.343335
$254$	−0.689146	−0.0432408
$255$	0	0
$256$	14.3004	0.893773
$257$	20.2806	1.26507	0.632534	−	0.774533i	$-0.282015\pi$
$257$	20.2806	1.26507	0.632534	+	0.774533i	$0.282015\pi$
$258$	0	0
$259$	−5.45901	−0.339207
$260$	8.01649	0.497162
$261$	0	0
$262$	1.31295	0.0811146
$263$	9.35950	0.577132	0.288566	−	0.957460i	$-0.406822\pi$
$263$	9.35950	0.577132	0.288566	+	0.957460i	$0.406822\pi$
$264$	0	0
$265$	−15.2315	−0.935666
$266$	−0.203103	−0.0124530
$267$	0	0
$268$	21.0646	1.28673
$269$	−7.20178	−0.439100	−0.219550	−	0.975601i	$-0.570459\pi$
$269$	−7.20178	−0.439100	−0.219550	+	0.975601i	$0.570459\pi$
$270$	0	0
$271$	0.105714	0.00642169	0.00321084	−	0.999995i	$-0.498978\pi$
$271$	0.105714	0.00642169	0.00321084	+	0.999995i	$0.498978\pi$
$272$	−23.8712	−1.44740
$273$	0	0
$274$	2.78024	0.167961
$275$	0.610096	0.0367901
$276$	0	0
$277$	0.755219	0.0453767	0.0226883	−	0.999743i	$-0.492777\pi$
$277$	0.755219	0.0453767	0.0226883	+	0.999743i	$0.492777\pi$
$278$	−3.25699	−0.195341
$279$	0	0
$280$	1.98479	0.118614
$281$	−2.00605	−0.119671	−0.0598353	−	0.998208i	$-0.519058\pi$
$281$	−2.00605	−0.119671	−0.0598353	+	0.998208i	$0.519058\pi$
$282$	0	0
$283$	14.8618	0.883444	0.441722	−	0.897152i	$-0.354368\pi$
$283$	14.8618	0.883444	0.441722	+	0.897152i	$0.354368\pi$
$284$	−3.58436	−0.212692
$285$	0	0
$286$	−0.143073	−0.00846009
$287$	−12.2733	−0.724471
$288$	0	0
$289$	21.0297	1.23704
$290$	−1.28205	−0.0752848
$291$	0	0
$292$	15.7472	0.921535
$293$	6.33548	0.370122	0.185061	−	0.982727i	$-0.440752\pi$
$293$	6.33548	0.370122	0.185061	+	0.982727i	$0.440752\pi$
$294$	0	0
$295$	30.6874	1.78669
$296$	2.30902	0.134209
$297$	0	0
$298$	0.653960	0.0378829
$299$	−15.2629	−0.882674
$300$	0	0
$301$	13.5664	0.781952
$302$	2.58782	0.148912
$303$	0	0
$304$	−3.87091	−0.222012
$305$	19.4457	1.11346
$306$	0	0
$307$	−6.31980	−0.360690	−0.180345	−	0.983603i	$-0.557721\pi$
$307$	−6.31980	−0.360690	−0.180345	+	0.983603i	$0.557721\pi$
$308$	1.61396	0.0919637
$309$	0	0
$310$	2.06048	0.117028
$311$	−15.2912	−0.867084	−0.433542	−	0.901133i	$-0.642736\pi$
$311$	−15.2912	−0.867084	−0.433542	+	0.901133i	$0.642736\pi$
$312$	0	0
$313$	−2.94311	−0.166354	−0.0831772	−	0.996535i	$-0.526507\pi$
$313$	−2.94311	−0.166354	−0.0831772	+	0.996535i	$0.526507\pi$
$314$	1.66945	0.0942126
$315$	0	0
$316$	−4.32868	−0.243508
$317$	29.4585	1.65455	0.827276	−	0.561795i	$-0.189889\pi$
$317$	29.4585	1.65455	0.827276	+	0.561795i	$0.189889\pi$
$318$	0	0
$319$	−2.09641	−0.117376
$320$	18.3893	1.02799
$321$	0	0
$322$	−1.87920	−0.104724
$323$	6.16682	0.343131
$324$	0	0
$325$	1.70512	0.0945833
$326$	−2.40948	−0.133449
$327$	0	0
$328$	5.19128	0.286641
$329$	1.38215	0.0762006
$330$	0	0
$331$	18.0770	0.993602	0.496801	−	0.867865i	$-0.334508\pi$
$331$	18.0770	0.993602	0.496801	+	0.867865i	$0.334508\pi$
$332$	23.2272	1.27476
$333$	0	0
$334$	−2.98652	−0.163415
$335$	26.1534	1.42891
$336$	0	0
$337$	12.7818	0.696272	0.348136	−	0.937444i	$-0.386815\pi$
$337$	12.7818	0.696272	0.348136	+	0.937444i	$0.386815\pi$
$338$	1.51044	0.0821570
$339$	0	0
$340$	−29.9687	−1.62528
$341$	3.36929	0.182457
$342$	0	0
$343$	16.7097	0.902242
$344$	−5.73821	−0.309383
$345$	0	0
$346$	2.79519	0.150271
$347$	25.6031	1.37445	0.687224	−	0.726446i	$-0.258829\pi$
$347$	25.6031	1.37445	0.687224	+	0.726446i	$0.258829\pi$
$348$	0	0
$349$	17.3527	0.928870	0.464435	−	0.885607i	$-0.346257\pi$
$349$	17.3527	0.928870	0.464435	+	0.885607i	$0.346257\pi$
$350$	0.209939	0.0112217
$351$	0	0
$352$	−1.02584	−0.0546776
$353$	6.05781	0.322425	0.161212	−	0.986920i	$-0.448460\pi$
$353$	6.05781	0.322425	0.161212	+	0.986920i	$0.448460\pi$
$354$	0	0
$355$	−4.45027	−0.236196
$356$	23.8493	1.26401
$357$	0	0
$358$	2.90441	0.153503
$359$	21.1095	1.11411	0.557057	−	0.830474i	$-0.311930\pi$
$359$	21.1095	1.11411	0.557057	+	0.830474i	$0.311930\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0.479034	0.0251775
$363$	0	0
$364$	4.51076	0.236428
$365$	19.5514	1.02337
$366$	0	0
$367$	−6.96937	−0.363798	−0.181899	−	0.983317i	$-0.558224\pi$
$367$	−6.96937	−0.363798	−0.181899	+	0.983317i	$0.558224\pi$
$368$	−35.8155	−1.86701
$369$	0	0
$370$	1.42563	0.0741152
$371$	−8.57056	−0.444961
$372$	0	0
$373$	8.46854	0.438484	0.219242	−	0.975670i	$-0.429642\pi$
$373$	8.46854	0.438484	0.219242	+	0.975670i	$0.429642\pi$
$374$	0.534861	0.0276570
$375$	0	0
$376$	−0.584613	−0.0301491
$377$	−5.85914	−0.301761
$378$	0	0
$379$	−22.6033	−1.16105	−0.580527	−	0.814241i	$-0.697153\pi$
$379$	−22.6033	−1.16105	−0.580527	+	0.814241i	$0.697153\pi$
$380$	−4.85966	−0.249295
$381$	0	0
$382$	−0.761359	−0.0389545
$383$	27.7063	1.41572	0.707862	−	0.706351i	$-0.249660\pi$
$383$	27.7063	1.41572	0.707862	+	0.706351i	$0.249660\pi$
$384$	0	0
$385$	2.00386	0.102126
$386$	−2.10534	−0.107159
$387$	0	0
$388$	30.6840	1.55774
$389$	−37.2371	−1.88800	−0.943998	−	0.329950i	$-0.892968\pi$
$389$	−37.2371	−1.88800	−0.943998	+	0.329950i	$0.892968\pi$
$390$	0	0
$391$	57.0583	2.88556
$392$	−2.97548	−0.150284
$393$	0	0
$394$	−2.13477	−0.107548
$395$	−5.37441	−0.270416
$396$	0	0
$397$	9.52734	0.478163	0.239082	−	0.970999i	$-0.423154\pi$
$397$	9.52734	0.478163	0.239082	+	0.970999i	$0.423154\pi$
$398$	1.22137	0.0612216
$399$	0	0
$400$	4.00120	0.200060
$401$	−17.4385	−0.870837	−0.435418	−	0.900228i	$-0.643400\pi$
$401$	−17.4385	−0.870837	−0.435418	+	0.900228i	$0.643400\pi$
$402$	0	0
$403$	9.41666	0.469077
$404$	−31.5162	−1.56799
$405$	0	0
$406$	−0.721393	−0.0358021
$407$	2.33119	0.115553
$408$	0	0
$409$	31.0879	1.53720	0.768599	−	0.639731i	$-0.220954\pi$
$409$	31.0879	1.53720	0.768599	+	0.639731i	$0.220954\pi$
$410$	3.20521	0.158294
$411$	0	0
$412$	−23.2366	−1.14479
$413$	17.2674	0.849672
$414$	0	0
$415$	28.8385	1.41563
$416$	−2.86707	−0.140570
$417$	0	0
$418$	0.0867321	0.00424220
$419$	−0.668435	−0.0326552	−0.0163276	−	0.999867i	$-0.505197\pi$
$419$	−0.668435	−0.0326552	−0.0163276	+	0.999867i	$0.505197\pi$
$420$	0	0
$421$	−21.3751	−1.04176	−0.520878	−	0.853631i	$-0.674395\pi$
$421$	−21.3751	−1.04176	−0.520878	+	0.853631i	$0.674395\pi$
$422$	−1.42131	−0.0691882
$423$	0	0
$424$	3.62512	0.176051
$425$	−6.37440	−0.309204
$426$	0	0
$427$	10.9418	0.529510
$428$	−21.3239	−1.03073
$429$	0	0
$430$	−3.54289	−0.170853
$431$	−2.03731	−0.0981337	−0.0490669	−	0.998795i	$-0.515625\pi$
$431$	−2.03731	−0.0981337	−0.0490669	+	0.998795i	$0.515625\pi$
$432$	0	0
$433$	24.7071	1.18735	0.593675	−	0.804705i	$-0.297677\pi$
$433$	24.7071	1.18735	0.593675	+	0.804705i	$0.297677\pi$
$434$	1.15940	0.0556531
$435$	0	0
$436$	13.5221	0.647593
$437$	9.25247	0.442606
$438$	0	0
$439$	−24.9950	−1.19294	−0.596472	−	0.802634i	$-0.703431\pi$
$439$	−24.9950	−1.19294	−0.596472	+	0.802634i	$0.703431\pi$
$440$	−0.847577	−0.0404067
$441$	0	0
$442$	1.49486	0.0711030
$443$	−31.4093	−1.49230	−0.746151	−	0.665776i	$-0.768100\pi$
$443$	−31.4093	−1.49230	−0.746151	+	0.665776i	$0.768100\pi$
$444$	0	0
$445$	29.6109	1.40369
$446$	2.43841	0.115462
$447$	0	0
$448$	10.3474	0.488867
$449$	−13.3304	−0.629103	−0.314551	−	0.949240i	$-0.601854\pi$
$449$	−13.3304	−0.629103	−0.314551	+	0.949240i	$0.601854\pi$
$450$	0	0
$451$	5.24114	0.246796
$452$	−2.24528	−0.105609
$453$	0	0
$454$	−1.77777	−0.0834348
$455$	5.60047	0.262554
$456$	0	0
$457$	28.1234	1.31556	0.657780	−	0.753210i	$-0.271496\pi$
$457$	28.1234	1.31556	0.657780	+	0.753210i	$0.271496\pi$
$458$	1.65653	0.0774047
$459$	0	0
$460$	−44.9639	−2.09645
$461$	25.5903	1.19186	0.595930	−	0.803037i	$-0.296784\pi$
$461$	25.5903	1.19186	0.595930	+	0.803037i	$0.296784\pi$
$462$	0	0
$463$	13.0383	0.605939	0.302970	−	0.953000i	$-0.402022\pi$
$463$	13.0383	0.605939	0.302970	+	0.953000i	$0.402022\pi$
$464$	−13.7489	−0.638278
$465$	0	0
$466$	−3.13021	−0.145004
$467$	−11.3327	−0.524416	−0.262208	−	0.965011i	$-0.584451\pi$
$467$	−11.3327	−0.524416	−0.262208	+	0.965011i	$0.584451\pi$
$468$	0	0
$469$	14.7162	0.679529
$470$	−0.360953	−0.0166495
$471$	0	0
$472$	−7.30363	−0.336177
$473$	−5.79332	−0.266377
$474$	0	0
$475$	−1.03366	−0.0474276
$476$	−16.8629	−0.772911
$477$	0	0
$478$	−0.434520	−0.0198745
$479$	−3.60682	−0.164800	−0.0823998	−	0.996599i	$-0.526258\pi$
$479$	−3.60682	−0.164800	−0.0823998	+	0.996599i	$0.526258\pi$
$480$	0	0
$481$	6.51533	0.297073
$482$	−1.13755	−0.0518140
$483$	0	0
$484$	21.0733	0.957875
$485$	38.0966	1.72988
$486$	0	0
$487$	42.4678	1.92440	0.962200	−	0.272344i	$-0.0877988\pi$
$487$	42.4678	1.92440	0.962200	+	0.272344i	$0.0877988\pi$
$488$	−4.62808	−0.209503
$489$	0	0
$490$	−1.83712	−0.0829928
$491$	−8.90333	−0.401802	−0.200901	−	0.979612i	$-0.564387\pi$
$491$	−8.90333	−0.401802	−0.200901	+	0.979612i	$0.564387\pi$
$492$	0	0
$493$	21.9037	0.986492
$494$	0.242403	0.0109062
$495$	0	0
$496$	22.0969	0.992180
$497$	−2.50410	−0.112324
$498$	0	0
$499$	−8.22424	−0.368168	−0.184084	−	0.982911i	$-0.558932\pi$
$499$	−8.22424	−0.368168	−0.184084	+	0.982911i	$0.558932\pi$
$500$	−19.2751	−0.862007
$501$	0	0
$502$	0.564003	0.0251727
$503$	−24.4784	−1.09144	−0.545719	−	0.837969i	$-0.683743\pi$
$503$	−24.4784	−1.09144	−0.545719	+	0.837969i	$0.683743\pi$
$504$	0	0
$505$	−39.1299	−1.74126
$506$	0.802486	0.0356749
$507$	0	0
$508$	−9.27827	−0.411657
$509$	−13.1269	−0.581842	−0.290921	−	0.956747i	$-0.593962\pi$
$509$	−13.1269	−0.581842	−0.290921	+	0.956747i	$0.593962\pi$
$510$	0	0
$511$	11.0013	0.486668
$512$	−11.2538	−0.497351
$513$	0	0
$514$	−2.98016	−0.131449
$515$	−28.8502	−1.27129
$516$	0	0
$517$	−0.590228	−0.0259582
$518$	0.802184	0.0352459
$519$	0	0
$520$	−2.36885	−0.103881
$521$	−1.43732	−0.0629702	−0.0314851	−	0.999504i	$-0.510024\pi$
$521$	−1.43732	−0.0629702	−0.0314851	+	0.999504i	$0.510024\pi$
$522$	0	0
$523$	−21.7258	−0.950001	−0.475000	−	0.879986i	$-0.657552\pi$
$523$	−21.7258	−0.950001	−0.475000	+	0.879986i	$0.657552\pi$
$524$	17.6769	0.772218
$525$	0	0
$526$	−1.37535	−0.0599680
$527$	−35.2030	−1.53347
$528$	0	0
$529$	62.6082	2.72210
$530$	2.23822	0.0972222
$531$	0	0
$532$	−2.73446	−0.118554
$533$	14.6482	0.634484
$534$	0	0
$535$	−26.4753	−1.14463
$536$	−6.22454	−0.268859
$537$	0	0
$538$	1.05828	0.0456256
$539$	−3.00406	−0.129394
$540$	0	0
$541$	25.6500	1.10278	0.551391	−	0.834247i	$-0.314097\pi$
$541$	25.6500	1.10278	0.551391	+	0.834247i	$0.314097\pi$
$542$	−0.0155344	−0.000667258 0
$543$	0	0
$544$	10.7182	0.459539
$545$	16.7888	0.719154
$546$	0	0
$547$	21.1680	0.905077	0.452539	−	0.891745i	$-0.350518\pi$
$547$	21.1680	0.905077	0.452539	+	0.891745i	$0.350518\pi$
$548$	37.4316	1.59900
$549$	0	0
$550$	−0.0896515	−0.00382275
$551$	3.55186	0.151314
$552$	0	0
$553$	−3.02410	−0.128598
$554$	−0.110977	−0.00471495
$555$	0	0
$556$	−43.8503	−1.85967
$557$	32.8085	1.39014	0.695071	−	0.718941i	$-0.255373\pi$
$557$	32.8085	1.39014	0.695071	+	0.718941i	$0.255373\pi$
$558$	0	0
$559$	−16.1914	−0.684825
$560$	13.1419	0.555348
$561$	0	0
$562$	0.294782	0.0124346
$563$	−2.48006	−0.104522	−0.0522611	−	0.998633i	$-0.516643\pi$
$563$	−2.48006	−0.104522	−0.0522611	+	0.998633i	$0.516643\pi$
$564$	0	0
$565$	−2.78770	−0.117279
$566$	−2.18390	−0.0917960
$567$	0	0
$568$	1.05917	0.0444417
$569$	1.53075	0.0641724	0.0320862	−	0.999485i	$-0.489785\pi$
$569$	1.53075	0.0641724	0.0320862	+	0.999485i	$0.489785\pi$
$570$	0	0
$571$	−34.0539	−1.42511	−0.712555	−	0.701616i	$-0.752462\pi$
$571$	−34.0539	−1.42511	−0.712555	+	0.701616i	$0.752462\pi$
$572$	−1.92626	−0.0805408
$573$	0	0
$574$	1.80352	0.0752776
$575$	−9.56391	−0.398843
$576$	0	0
$577$	−12.2785	−0.511161	−0.255581	−	0.966788i	$-0.582267\pi$
$577$	−12.2785	−0.511161	−0.255581	+	0.966788i	$0.582267\pi$
$578$	−3.09024	−0.128537
$579$	0	0
$580$	−17.2608	−0.716717
$581$	16.2270	0.673209
$582$	0	0
$583$	3.65993	0.151579
$584$	−4.65325	−0.192553
$585$	0	0
$586$	−0.930977	−0.0384583
$587$	−3.34132	−0.137911	−0.0689556	−	0.997620i	$-0.521967\pi$
$587$	−3.34132	−0.137911	−0.0689556	+	0.997620i	$0.521967\pi$
$588$	0	0
$589$	−5.70846	−0.235213
$590$	−4.50942	−0.185650
$591$	0	0
$592$	15.2887	0.628362
$593$	27.0538	1.11097	0.555484	−	0.831527i	$-0.312533\pi$
$593$	27.0538	1.11097	0.555484	+	0.831527i	$0.312533\pi$
$594$	0	0
$595$	−20.9367	−0.858320
$596$	8.80455	0.360649
$597$	0	0
$598$	2.24283	0.0917160
$599$	−40.9658	−1.67382	−0.836909	−	0.547342i	$-0.815640\pi$
$599$	−40.9658	−1.67382	−0.836909	+	0.547342i	$0.815640\pi$
$600$	0	0
$601$	−11.7134	−0.477799	−0.238900	−	0.971044i	$-0.576787\pi$
$601$	−11.7134	−0.477799	−0.238900	+	0.971044i	$0.576787\pi$
$602$	−1.99353	−0.0812503
$603$	0	0
$604$	34.8409	1.41766
$605$	26.1641	1.06372
$606$	0	0
$607$	19.4936	0.791221	0.395610	−	0.918418i	$-0.370533\pi$
$607$	19.4936	0.791221	0.395610	+	0.918418i	$0.370533\pi$
$608$	1.73804	0.0704870
$609$	0	0
$610$	−2.85747	−0.115696
$611$	−1.64960	−0.0667356
$612$	0	0
$613$	−33.1469	−1.33879	−0.669396	−	0.742906i	$-0.733447\pi$
$613$	−33.1469	−1.33879	−0.669396	+	0.742906i	$0.733447\pi$
$614$	0.928673	0.0374782
$615$	0	0
$616$	−0.476919	−0.0192156
$617$	−2.48057	−0.0998638	−0.0499319	−	0.998753i	$-0.515900\pi$
$617$	−2.48057	−0.0998638	−0.0499319	+	0.998753i	$0.515900\pi$
$618$	0	0
$619$	−29.4934	−1.18544	−0.592721	−	0.805408i	$-0.701946\pi$
$619$	−29.4934	−1.18544	−0.592721	+	0.805408i	$0.701946\pi$
$620$	27.7412	1.11411
$621$	0	0
$622$	2.24699	0.0900960
$623$	16.6616	0.667533
$624$	0	0
$625$	−29.0998	−1.16399
$626$	0.432480	0.0172854
$627$	0	0
$628$	22.4765	0.896912
$629$	−24.3567	−0.971167
$630$	0	0
$631$	3.37887	0.134510	0.0672552	−	0.997736i	$-0.478576\pi$
$631$	3.37887	0.134510	0.0672552	+	0.997736i	$0.478576\pi$
$632$	1.27911	0.0508804
$633$	0	0
$634$	−4.32882	−0.171920
$635$	−11.5197	−0.457146
$636$	0	0
$637$	−8.39588	−0.332657
$638$	0.308060	0.0121962
$639$	0	0
$640$	−11.2407	−0.444329
$641$	−21.3880	−0.844776	−0.422388	−	0.906415i	$-0.638808\pi$
$641$	−21.3880	−0.844776	−0.422388	+	0.906415i	$0.638808\pi$
$642$	0	0
$643$	13.6949	0.540076	0.270038	−	0.962850i	$-0.412964\pi$
$643$	13.6949	0.540076	0.270038	+	0.962850i	$0.412964\pi$
$644$	−25.3005	−0.996980
$645$	0	0
$646$	−0.906193	−0.0356537
$647$	36.0902	1.41885	0.709425	−	0.704781i	$-0.248955\pi$
$647$	36.0902	1.41885	0.709425	+	0.704781i	$0.248955\pi$
$648$	0	0
$649$	−7.37378	−0.289446
$650$	−0.250562	−0.00982786
$651$	0	0
$652$	−32.4399	−1.27044
$653$	25.6200	1.00259	0.501294	−	0.865277i	$-0.332857\pi$
$653$	25.6200	1.00259	0.501294	+	0.865277i	$0.332857\pi$
$654$	0	0
$655$	21.9473	0.857551
$656$	34.3731	1.34204
$657$	0	0
$658$	−0.203103	−0.00791777
$659$	−8.10569	−0.315753	−0.157876	−	0.987459i	$-0.550465\pi$
$659$	−8.10569	−0.315753	−0.157876	+	0.987459i	$0.550465\pi$
$660$	0	0
$661$	−22.2755	−0.866417	−0.433208	−	0.901294i	$-0.642619\pi$
$661$	−22.2755	−0.866417	−0.433208	+	0.901294i	$0.642619\pi$
$662$	−2.65635	−0.103242
$663$	0	0
$664$	−6.86358	−0.266359
$665$	−3.39505	−0.131654
$666$	0	0
$667$	32.8635	1.27248
$668$	−40.2088	−1.55573
$669$	0	0
$670$	−3.84316	−0.148474
$671$	−4.67253	−0.180381
$672$	0	0
$673$	8.02253	0.309246	0.154623	−	0.987974i	$-0.450584\pi$
$673$	8.02253	0.309246	0.154623	+	0.987974i	$0.450584\pi$
$674$	−1.87825	−0.0723475
$675$	0	0
$676$	20.3357	0.782142
$677$	−8.68583	−0.333824	−0.166912	−	0.985972i	$-0.553380\pi$
$677$	−8.68583	−0.333824	−0.166912	+	0.985972i	$0.553380\pi$
$678$	0	0
$679$	21.4364	0.822654
$680$	8.85565	0.339599
$681$	0	0
$682$	−0.495106	−0.0189586
$683$	18.0022	0.688833	0.344417	−	0.938817i	$-0.388077\pi$
$683$	18.0022	0.688833	0.344417	+	0.938817i	$0.388077\pi$
$684$	0	0
$685$	46.4744	1.77569
$686$	−2.45544	−0.0937492
$687$	0	0
$688$	−37.9944	−1.44852
$689$	10.2290	0.389692
$690$	0	0
$691$	−1.51284	−0.0575511	−0.0287756	−	0.999586i	$-0.509161\pi$
$691$	−1.51284	−0.0575511	−0.0287756	+	0.999586i	$0.509161\pi$
$692$	37.6329	1.43059
$693$	0	0
$694$	−3.76229	−0.142815
$695$	−54.4437	−2.06517
$696$	0	0
$697$	−54.7605	−2.07420
$698$	−2.54992	−0.0965160
$699$	0	0
$700$	2.82650	0.106832
$701$	−34.0516	−1.28611	−0.643056	−	0.765819i	$-0.722334\pi$
$701$	−34.0516	−1.28611	−0.643056	+	0.765819i	$0.722334\pi$
$702$	0	0
$703$	−3.94965	−0.148964
$704$	−4.41869	−0.166536
$705$	0	0
$706$	−0.890175	−0.0335022
$707$	−22.0178	−0.828065
$708$	0	0
$709$	37.8263	1.42060	0.710298	−	0.703901i	$-0.248560\pi$
$709$	37.8263	1.42060	0.710298	+	0.703901i	$0.248560\pi$
$710$	0.653952	0.0245424
$711$	0	0
$712$	−7.04741	−0.264113
$713$	−52.8173	−1.97802
$714$	0	0
$715$	−2.39160	−0.0894408
$716$	39.1033	1.46136
$717$	0	0
$718$	−3.10196	−0.115764
$719$	16.5020	0.615421	0.307710	−	0.951480i	$-0.400437\pi$
$719$	16.5020	0.615421	0.307710	+	0.951480i	$0.400437\pi$
$720$	0	0
$721$	−16.2336	−0.604569
$722$	−0.146947	−0.00546879
$723$	0	0
$724$	6.44945	0.239692
$725$	−3.67142	−0.136353
$726$	0	0
$727$	1.57794	0.0585227	0.0292614	−	0.999572i	$-0.490684\pi$
$727$	1.57794	0.0585227	0.0292614	+	0.999572i	$0.490684\pi$
$728$	−1.33292	−0.0494012
$729$	0	0
$730$	−2.87301	−0.106335
$731$	60.5297	2.23877
$732$	0	0
$733$	−18.7510	−0.692582	−0.346291	−	0.938127i	$-0.612559\pi$
$733$	−18.7510	−0.692582	−0.346291	+	0.938127i	$0.612559\pi$
$734$	1.02413	0.0378012
$735$	0	0
$736$	16.0812	0.592761
$737$	−6.28432	−0.231486
$738$	0	0
$739$	44.0706	1.62116	0.810581	−	0.585627i	$-0.199151\pi$
$739$	44.0706	1.62116	0.810581	+	0.585627i	$0.199151\pi$
$740$	19.1939	0.705583
$741$	0	0
$742$	1.25942	0.0462346
$743$	7.06804	0.259301	0.129651	−	0.991560i	$-0.458614\pi$
$743$	7.06804	0.259301	0.129651	+	0.991560i	$0.458614\pi$
$744$	0	0
$745$	10.9316	0.400501
$746$	−1.24442	−0.0455616
$747$	0	0
$748$	7.20107	0.263297
$749$	−14.8973	−0.544334
$750$	0	0
$751$	−2.44334	−0.0891586	−0.0445793	−	0.999006i	$-0.514195\pi$
$751$	−2.44334	−0.0891586	−0.0445793	+	0.999006i	$0.514195\pi$
$752$	−3.87091	−0.141157
$753$	0	0
$754$	0.860982	0.0313551
$755$	43.2578	1.57431
$756$	0	0
$757$	4.22047	0.153396	0.0766978	−	0.997054i	$-0.475562\pi$
$757$	4.22047	0.153396	0.0766978	+	0.997054i	$0.475562\pi$
$758$	3.32148	0.120641
$759$	0	0
$760$	1.43602	0.0520898
$761$	−11.7303	−0.425222	−0.212611	−	0.977137i	$-0.568197\pi$
$761$	−11.7303	−0.425222	−0.212611	+	0.977137i	$0.568197\pi$
$762$	0	0
$763$	9.44682	0.341998
$764$	−10.2505	−0.370850
$765$	0	0
$766$	−4.07134	−0.147104
$767$	−20.6086	−0.744133
$768$	0	0
$769$	−47.5161	−1.71347	−0.856737	−	0.515754i	$-0.827512\pi$
$769$	−47.5161	−1.71347	−0.856737	+	0.515754i	$0.827512\pi$
$770$	−0.294460	−0.0106116
$771$	0	0
$772$	−28.3451	−1.02016
$773$	42.4550	1.52700	0.763501	−	0.645807i	$-0.223479\pi$
$773$	42.4550	1.52700	0.763501	+	0.645807i	$0.223479\pi$
$774$	0	0
$775$	5.90060	0.211956
$776$	−9.06702	−0.325487
$777$	0	0
$778$	5.47187	0.196176
$779$	−8.87986	−0.318154
$780$	0	0
$781$	1.06934	0.0382640
$782$	−8.38453	−0.299830
$783$	0	0
$784$	−19.7016	−0.703628
$785$	27.9064	0.996024
$786$	0	0
$787$	44.7048	1.59355	0.796777	−	0.604274i	$-0.206537\pi$
$787$	44.7048	1.59355	0.796777	+	0.604274i	$0.206537\pi$
$788$	−28.7413	−1.02387
$789$	0	0
$790$	0.789751	0.0280981
$791$	−1.56860	−0.0557729
$792$	0	0
$793$	−13.0590	−0.463739
$794$	−1.40001	−0.0496845
$795$	0	0
$796$	16.4438	0.582835
$797$	−32.2009	−1.14061	−0.570307	−	0.821432i	$-0.693176\pi$
$797$	−32.2009	−1.14061	−0.570307	+	0.821432i	$0.693176\pi$
$798$	0	0
$799$	6.16682	0.218166
$800$	−1.79655	−0.0635175
$801$	0	0
$802$	2.56253	0.0904860
$803$	−4.69794	−0.165787
$804$	0	0
$805$	−31.4126	−1.10715
$806$	−1.38375	−0.0487404
$807$	0	0
$808$	9.31294	0.327628
$809$	29.3904	1.03331	0.516656	−	0.856193i	$-0.327177\pi$
$809$	29.3904	1.03331	0.516656	+	0.856193i	$0.327177\pi$
$810$	0	0
$811$	−22.3059	−0.783264	−0.391632	−	0.920122i	$-0.628089\pi$
$811$	−22.3059	−0.783264	−0.391632	+	0.920122i	$0.628089\pi$
$812$	−9.71242	−0.340839
$813$	0	0
$814$	−0.342561	−0.0120068
$815$	−40.2767	−1.41083
$816$	0	0
$817$	9.81538	0.343397
$818$	−4.56826	−0.159726
$819$	0	0
$820$	43.1531	1.50697
$821$	−23.9644	−0.836365	−0.418183	−	0.908363i	$-0.637333\pi$
$821$	−23.9644	−0.836365	−0.418183	+	0.908363i	$0.637333\pi$
$822$	0	0
$823$	−57.1018	−1.99044	−0.995221	−	0.0976461i	$-0.968869\pi$
$823$	−57.1018	−1.99044	−0.995221	+	0.0976461i	$0.968869\pi$
$824$	6.86636	0.239201
$825$	0	0
$826$	−2.53738	−0.0882868
$827$	46.3140	1.61049	0.805247	−	0.592939i	$-0.202033\pi$
$827$	46.3140	1.61049	0.805247	+	0.592939i	$0.202033\pi$
$828$	0	0
$829$	−28.8520	−1.00207	−0.501035	−	0.865427i	$-0.667047\pi$
$829$	−28.8520	−1.00207	−0.501035	+	0.865427i	$0.667047\pi$
$830$	−4.23772	−0.147093
$831$	0	0
$832$	−12.3496	−0.428145
$833$	31.3870	1.08749
$834$	0	0
$835$	−49.9225	−1.72764
$836$	1.16771	0.0403862
$837$	0	0
$838$	0.0982243	0.00339310
$839$	−11.4661	−0.395853	−0.197926	−	0.980217i	$-0.563421\pi$
$839$	−11.4661	−0.395853	−0.197926	+	0.980217i	$0.563421\pi$
$840$	0	0
$841$	−16.3843	−0.564975
$842$	3.14099	0.108246
$843$	0	0
$844$	−19.1357	−0.658677
$845$	25.2484	0.868571
$846$	0	0
$847$	14.7222	0.505860
$848$	24.0030	0.824267
$849$	0	0
$850$	0.936696	0.0321284
$851$	−36.5440	−1.25271
$852$	0	0
$853$	−12.2391	−0.419057	−0.209529	−	0.977803i	$-0.567193\pi$
$853$	−12.2391	−0.419057	−0.209529	+	0.977803i	$0.567193\pi$
$854$	−1.60786	−0.0550198
$855$	0	0
$856$	6.30114	0.215369
$857$	21.4663	0.733274	0.366637	−	0.930364i	$-0.380509\pi$
$857$	21.4663	0.733274	0.366637	+	0.930364i	$0.380509\pi$
$858$	0	0
$859$	−26.7028	−0.911087	−0.455544	−	0.890213i	$-0.650555\pi$
$859$	−26.7028	−0.911087	−0.455544	+	0.890213i	$0.650555\pi$
$860$	−47.6994	−1.62654
$861$	0	0
$862$	0.299376	0.0101968
$863$	−25.6903	−0.874507	−0.437253	−	0.899338i	$-0.644049\pi$
$863$	−25.6903	−0.874507	−0.437253	+	0.899338i	$0.644049\pi$
$864$	0	0
$865$	46.7243	1.58867
$866$	−3.63063	−0.123374
$867$	0	0
$868$	15.6095	0.529823
$869$	1.29140	0.0438077
$870$	0	0
$871$	−17.5637	−0.595124
$872$	−3.99575	−0.135313
$873$	0	0
$874$	−1.35962	−0.0459898
$875$	−13.4659	−0.455232
$876$	0	0
$877$	6.74633	0.227807	0.113904	−	0.993492i	$-0.463664\pi$
$877$	6.74633	0.227807	0.113904	+	0.993492i	$0.463664\pi$
$878$	3.67292	0.123955
$879$	0	0
$880$	−5.61207	−0.189183
$881$	54.3353	1.83060	0.915301	−	0.402770i	$-0.131952\pi$
$881$	54.3353	1.83060	0.915301	+	0.402770i	$0.131952\pi$
$882$	0	0
$883$	41.1611	1.38518	0.692591	−	0.721330i	$-0.256469\pi$
$883$	41.1611	1.38518	0.692591	+	0.721330i	$0.256469\pi$
$884$	20.1259	0.676907
$885$	0	0
$886$	4.61550	0.155061
$887$	−31.1578	−1.04618	−0.523089	−	0.852278i	$-0.675220\pi$
$887$	−31.1578	−1.04618	−0.523089	+	0.852278i	$0.675220\pi$
$888$	0	0
$889$	−6.48198	−0.217398
$890$	−4.35122	−0.145853
$891$	0	0
$892$	32.8294	1.09921
$893$	1.00000	0.0334637
$894$	0	0
$895$	48.5499	1.62285
$896$	−6.32499	−0.211303
$897$	0	0
$898$	1.95886	0.0653681
$899$	−20.2756	−0.676231
$900$	0	0
$901$	−38.2397	−1.27395
$902$	−0.770168	−0.0256438
$903$	0	0
$904$	0.663475	0.0220668
$905$	8.00750	0.266178
$906$	0	0
$907$	−1.86091	−0.0617906	−0.0308953	−	0.999523i	$-0.509836\pi$
$907$	−1.86091	−0.0617906	−0.0308953	+	0.999523i	$0.509836\pi$
$908$	−23.9349	−0.794306
$909$	0	0
$910$	−0.822971	−0.0272812
$911$	−19.0897	−0.632470	−0.316235	−	0.948681i	$-0.602419\pi$
$911$	−19.0897	−0.632470	−0.316235	+	0.948681i	$0.602419\pi$
$912$	0	0
$913$	−6.92951	−0.229333
$914$	−4.13265	−0.136696
$915$	0	0
$916$	22.3026	0.736899
$917$	12.3494	0.407813
$918$	0	0
$919$	−29.9755	−0.988801	−0.494400	−	0.869234i	$-0.664612\pi$
$919$	−29.9755	−0.988801	−0.494400	+	0.869234i	$0.664612\pi$
$920$	13.2867	0.438049
$921$	0	0
$922$	−3.76041	−0.123843
$923$	2.98864	0.0983723
$924$	0	0
$925$	4.08259	0.134235
$926$	−1.91593	−0.0629613
$927$	0	0
$928$	6.17329	0.202648
$929$	25.8162	0.847004	0.423502	−	0.905895i	$-0.360801\pi$
$929$	25.8162	0.847004	0.423502	+	0.905895i	$0.360801\pi$
$930$	0	0
$931$	5.08965	0.166807
$932$	−42.1433	−1.38045
$933$	0	0
$934$	1.66531	0.0544905
$935$	8.94070	0.292392
$936$	0	0
$937$	38.4599	1.25643	0.628215	−	0.778040i	$-0.283786\pi$
$937$	38.4599	1.25643	0.628215	+	0.778040i	$0.283786\pi$
$938$	−2.16249	−0.0706078
$939$	0	0
$940$	−4.85966	−0.158505
$941$	−9.93570	−0.323895	−0.161947	−	0.986799i	$-0.551777\pi$
$941$	−9.93570	−0.323895	−0.161947	+	0.986799i	$0.551777\pi$
$942$	0	0
$943$	−82.1606	−2.67552
$944$	−48.3596	−1.57397
$945$	0	0
$946$	0.851309	0.0276784
$947$	14.2058	0.461626	0.230813	−	0.972998i	$-0.425861\pi$
$947$	14.2058	0.461626	0.230813	+	0.972998i	$0.425861\pi$
$948$	0	0
$949$	−13.1300	−0.426219
$950$	0.151893	0.00492806
$951$	0	0
$952$	4.98294	0.161498
$953$	−28.7407	−0.931004	−0.465502	−	0.885047i	$-0.654126\pi$
$953$	−28.7407	−0.931004	−0.465502	+	0.885047i	$0.654126\pi$
$954$	0	0
$955$	−12.7268	−0.411830
$956$	−5.85013	−0.189207
$957$	0	0
$958$	0.530010	0.0171238
$959$	26.1505	0.844442
$960$	0	0
$961$	1.58648	0.0511767
$962$	−0.957406	−0.0308680
$963$	0	0
$964$	−15.3153	−0.493274
$965$	−35.1927	−1.13289
$966$	0	0
$967$	−4.00109	−0.128666	−0.0643331	−	0.997928i	$-0.520492\pi$
$967$	−4.00109	−0.128666	−0.0643331	+	0.997928i	$0.520492\pi$
$968$	−6.22709	−0.200146
$969$	0	0
$970$	−5.59817	−0.179746
$971$	−14.0988	−0.452453	−0.226226	−	0.974075i	$-0.572639\pi$
$971$	−14.0988	−0.452453	−0.226226	+	0.974075i	$0.572639\pi$
$972$	0	0
$973$	−30.6347	−0.982102
$974$	−6.24050	−0.199959
$975$	0	0
$976$	−30.6440	−0.980889
$977$	−22.1813	−0.709643	−0.354822	−	0.934934i	$-0.615458\pi$
$977$	−22.1813	−0.709643	−0.354822	+	0.934934i	$0.615458\pi$
$978$	0	0
$979$	−7.11510	−0.227399
$980$	−24.7340	−0.790098
$981$	0	0
$982$	1.30831	0.0417500
$983$	−54.0831	−1.72498	−0.862492	−	0.506071i	$-0.831097\pi$
$983$	−54.0831	−1.72498	−0.862492	+	0.506071i	$0.831097\pi$
$984$	0	0
$985$	−35.6846	−1.13701
$986$	−3.21867	−0.102503
$987$	0	0
$988$	3.26358	0.103828
$989$	90.8166	2.88780
$990$	0	0
$991$	45.7163	1.45223	0.726114	−	0.687575i	$-0.241325\pi$
$991$	45.7163	1.45223	0.726114	+	0.687575i	$0.241325\pi$
$992$	−9.92155	−0.315009
$993$	0	0
$994$	0.367969	0.0116713
$995$	20.4163	0.647240
$996$	0	0
$997$	−27.3488	−0.866145	−0.433072	−	0.901359i	$-0.642571\pi$
$997$	−27.3488	−0.866145	−0.433072	+	0.901359i	$0.642571\pi$
$998$	1.20852	0.0382552
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.o.1.9		18
3.2	odd	2		893.2.a.c.1.10	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.c.1.10	✓	18	3.2	odd	2
8037.2.a.o.1.9		18	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-0.146947 q^{2} -1.97841 q^{4} -2.45635 q^{5} -1.38215 q^{7} +0.584613 q^{8} +O(q^{10})\)	\(q-0.146947 q^{2} -1.97841 q^{4} -2.45635 q^{5} -1.38215 q^{7} +0.584613 q^{8} +0.360953 q^{10} +0.590228 q^{11} +1.64960 q^{13} +0.203103 q^{14} +3.87091 q^{16} -6.16682 q^{17} -1.00000 q^{19} +4.85966 q^{20} -0.0867321 q^{22} -9.25247 q^{23} +1.03366 q^{25} -0.242403 q^{26} +2.73446 q^{28} -3.55186 q^{29} +5.70846 q^{31} -1.73804 q^{32} +0.906193 q^{34} +3.39505 q^{35} +3.94965 q^{37} +0.146947 q^{38} -1.43602 q^{40} +8.87986 q^{41} -9.81538 q^{43} -1.16771 q^{44} +1.35962 q^{46} -1.00000 q^{47} -5.08965 q^{49} -0.151893 q^{50} -3.26358 q^{52} +6.20088 q^{53} -1.44981 q^{55} -0.808025 q^{56} +0.521934 q^{58} -12.4931 q^{59} -7.91648 q^{61} -0.838838 q^{62} -7.48641 q^{64} -4.05199 q^{65} -10.6473 q^{67} +12.2005 q^{68} -0.498892 q^{70} +1.81174 q^{71} -7.95953 q^{73} -0.580387 q^{74} +1.97841 q^{76} -0.815786 q^{77} +2.18796 q^{79} -9.50830 q^{80} -1.30487 q^{82} -11.7404 q^{83} +15.1479 q^{85} +1.44234 q^{86} +0.345055 q^{88} -12.0548 q^{89} -2.28000 q^{91} +18.3052 q^{92} +0.146947 q^{94} +2.45635 q^{95} -15.5094 q^{97} +0.747907 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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