LMFDB - Embedded newform 6044.2.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, 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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$0$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50519 q^{3} -1.30564 q^{5} +3.79113 q^{7} +3.27598 q^{9} +O(q^{10})$	$q-2.50519 q^{3} -1.30564 q^{5} +3.79113 q^{7} +3.27598 q^{9} -0.363607 q^{11} +1.90367 q^{13} +3.27087 q^{15} +3.86448 q^{17} +7.31712 q^{19} -9.49750 q^{21} -2.47630 q^{23} -3.29532 q^{25} -0.691391 q^{27} +4.83219 q^{29} +9.05699 q^{31} +0.910906 q^{33} -4.94983 q^{35} +10.6804 q^{37} -4.76906 q^{39} +4.69290 q^{41} +0.101108 q^{43} -4.27724 q^{45} -12.9937 q^{47} +7.37264 q^{49} -9.68127 q^{51} -6.44907 q^{53} +0.474738 q^{55} -18.3308 q^{57} +10.2369 q^{59} +3.80135 q^{61} +12.4197 q^{63} -2.48550 q^{65} -5.43573 q^{67} +6.20361 q^{69} -5.86368 q^{71} +13.4405 q^{73} +8.25540 q^{75} -1.37848 q^{77} +4.85479 q^{79} -8.09588 q^{81} -5.14749 q^{83} -5.04560 q^{85} -12.1056 q^{87} -4.25481 q^{89} +7.21706 q^{91} -22.6895 q^{93} -9.55349 q^{95} -16.3728 q^{97} -1.19117 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * q^99

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	0
$2$	0	0
$3$	−2.50519	−1.44637	−0.723186	−	0.690653i	$-0.757323\pi$
$3$	−2.50519	−1.44637	−0.723186	+	0.690653i	$0.757323\pi$
$4$	0	0
$5$	−1.30564	−0.583898	−0.291949	−	0.956434i	$-0.594304\pi$
$5$	−1.30564	−0.583898	−0.291949	+	0.956434i	$0.594304\pi$
$6$	0	0
$7$	3.79113	1.43291	0.716456	−	0.697633i	$-0.245763\pi$
$7$	3.79113	1.43291	0.716456	+	0.697633i	$0.245763\pi$
$8$	0	0
$9$	3.27598	1.09199
$10$	0	0
$11$	−0.363607	−0.109632	−0.0548158	−	0.998496i	$-0.517457\pi$
$11$	−0.363607	−0.109632	−0.0548158	+	0.998496i	$0.517457\pi$
$12$	0	0
$13$	1.90367	0.527983	0.263992	−	0.964525i	$-0.414961\pi$
$13$	1.90367	0.527983	0.263992	+	0.964525i	$0.414961\pi$
$14$	0	0
$15$	3.27087	0.844534
$16$	0	0
$17$	3.86448	0.937275	0.468637	−	0.883391i	$-0.344745\pi$
$17$	3.86448	0.937275	0.468637	+	0.883391i	$0.344745\pi$
$18$	0	0
$19$	7.31712	1.67866	0.839331	−	0.543620i	$-0.182947\pi$
$19$	7.31712	1.67866	0.839331	+	0.543620i	$0.182947\pi$
$20$	0	0
$21$	−9.49750	−2.07252
$22$	0	0
$23$	−2.47630	−0.516344	−0.258172	−	0.966099i	$-0.583120\pi$
$23$	−2.47630	−0.516344	−0.258172	+	0.966099i	$0.583120\pi$
$24$	0	0
$25$	−3.29532	−0.659063
$26$	0	0
$27$	−0.691391	−0.133058
$28$	0	0
$29$	4.83219	0.897315	0.448658	−	0.893704i	$-0.351902\pi$
$29$	4.83219	0.897315	0.448658	+	0.893704i	$0.351902\pi$
$30$	0	0
$31$	9.05699	1.62668	0.813342	−	0.581786i	$-0.197646\pi$
$31$	9.05699	1.62668	0.813342	+	0.581786i	$0.197646\pi$
$32$	0	0
$33$	0.910906	0.158568
$34$	0	0
$35$	−4.94983	−0.836674
$36$	0	0
$37$	10.6804	1.75585	0.877923	−	0.478801i	$-0.158929\pi$
$37$	10.6804	1.75585	0.877923	+	0.478801i	$0.158929\pi$
$38$	0	0
$39$	−4.76906	−0.763661
$40$	0	0
$41$	4.69290	0.732908	0.366454	−	0.930436i	$-0.380572\pi$
$41$	4.69290	0.732908	0.366454	+	0.930436i	$0.380572\pi$
$42$	0	0
$43$	0.101108	0.0154189	0.00770945	−	0.999970i	$-0.497546\pi$
$43$	0.101108	0.0154189	0.00770945	+	0.999970i	$0.497546\pi$
$44$	0	0
$45$	−4.27724	−0.637613
$46$	0	0
$47$	−12.9937	−1.89532	−0.947662	−	0.319276i	$-0.896560\pi$
$47$	−12.9937	−1.89532	−0.947662	+	0.319276i	$0.896560\pi$
$48$	0	0
$49$	7.37264	1.05323
$50$	0	0
$51$	−9.68127	−1.35565
$52$	0	0
$53$	−6.44907	−0.885847	−0.442924	−	0.896559i	$-0.646059\pi$
$53$	−6.44907	−0.885847	−0.442924	+	0.896559i	$0.646059\pi$
$54$	0	0
$55$	0.474738	0.0640137
$56$	0	0
$57$	−18.3308	−2.42797
$58$	0	0
$59$	10.2369	1.33273	0.666365	−	0.745626i	$-0.267849\pi$
$59$	10.2369	1.33273	0.666365	+	0.745626i	$0.267849\pi$
$60$	0	0
$61$	3.80135	0.486713	0.243356	−	0.969937i	$-0.421752\pi$
$61$	3.80135	0.486713	0.243356	+	0.969937i	$0.421752\pi$
$62$	0	0
$63$	12.4197	1.56473
$64$	0	0
$65$	−2.48550	−0.308288
$66$	0	0
$67$	−5.43573	−0.664081	−0.332040	−	0.943265i	$-0.607737\pi$
$67$	−5.43573	−0.664081	−0.332040	+	0.943265i	$0.607737\pi$
$68$	0	0
$69$	6.20361	0.746826
$70$	0	0
$71$	−5.86368	−0.695890	−0.347945	−	0.937515i	$-0.613120\pi$
$71$	−5.86368	−0.695890	−0.347945	+	0.937515i	$0.613120\pi$
$72$	0	0
$73$	13.4405	1.57309	0.786543	−	0.617535i	$-0.211869\pi$
$73$	13.4405	1.57309	0.786543	+	0.617535i	$0.211869\pi$
$74$	0	0
$75$	8.25540	0.953251
$76$	0	0
$77$	−1.37848	−0.157092
$78$	0	0
$79$	4.85479	0.546207	0.273103	−	0.961985i	$-0.411950\pi$
$79$	4.85479	0.546207	0.273103	+	0.961985i	$0.411950\pi$
$80$	0	0
$81$	−8.09588	−0.899543
$82$	0	0
$83$	−5.14749	−0.565010	−0.282505	−	0.959266i	$-0.591165\pi$
$83$	−5.14749	−0.565010	−0.282505	+	0.959266i	$0.591165\pi$
$84$	0	0
$85$	−5.04560	−0.547273
$86$	0	0
$87$	−12.1056	−1.29785
$88$	0	0
$89$	−4.25481	−0.451009	−0.225505	−	0.974242i	$-0.572403\pi$
$89$	−4.25481	−0.451009	−0.225505	+	0.974242i	$0.572403\pi$
$90$	0	0
$91$	7.21706	0.756553
$92$	0	0
$93$	−22.6895	−2.35279
$94$	0	0
$95$	−9.55349	−0.980167
$96$	0	0
$97$	−16.3728	−1.66240	−0.831202	−	0.555971i	$-0.812347\pi$
$97$	−16.3728	−1.66240	−0.831202	+	0.555971i	$0.812347\pi$
$98$	0	0
$99$	−1.19117	−0.119717
$100$	0	0
$101$	−10.9032	−1.08491	−0.542457	−	0.840084i	$-0.682506\pi$
$101$	−10.9032	−1.08491	−0.542457	+	0.840084i	$0.682506\pi$
$102$	0	0
$103$	9.12837	0.899445	0.449722	−	0.893168i	$-0.351523\pi$
$103$	9.12837	0.899445	0.449722	+	0.893168i	$0.351523\pi$
$104$	0	0
$105$	12.4003	1.21014
$106$	0	0
$107$	8.86200	0.856722	0.428361	−	0.903608i	$-0.359091\pi$
$107$	8.86200	0.856722	0.428361	+	0.903608i	$0.359091\pi$
$108$	0	0
$109$	17.0160	1.62984	0.814919	−	0.579575i	$-0.196782\pi$
$109$	17.0160	1.62984	0.814919	+	0.579575i	$0.196782\pi$
$110$	0	0
$111$	−26.7564	−2.53961
$112$	0	0
$113$	7.85554	0.738987	0.369493	−	0.929233i	$-0.379531\pi$
$113$	7.85554	0.738987	0.369493	+	0.929233i	$0.379531\pi$
$114$	0	0
$115$	3.23314	0.301492
$116$	0	0
$117$	6.23639	0.576555
$118$	0	0
$119$	14.6507	1.34303
$120$	0	0
$121$	−10.8678	−0.987981
$122$	0	0
$123$	−11.7566	−1.06006
$124$	0	0
$125$	10.8307	0.968723
$126$	0	0
$127$	−11.0754	−0.982783	−0.491392	−	0.870939i	$-0.663512\pi$
$127$	−11.0754	−0.982783	−0.491392	+	0.870939i	$0.663512\pi$
$128$	0	0
$129$	−0.253296	−0.0223015
$130$	0	0
$131$	1.53234	0.133881	0.0669407	−	0.997757i	$-0.478676\pi$
$131$	1.53234	0.133881	0.0669407	+	0.997757i	$0.478676\pi$
$132$	0	0
$133$	27.7401	2.40537
$134$	0	0
$135$	0.902704	0.0776924
$136$	0	0
$137$	−22.7154	−1.94071	−0.970353	−	0.241693i	$-0.922297\pi$
$137$	−22.7154	−1.94071	−0.970353	+	0.241693i	$0.922297\pi$
$138$	0	0
$139$	−4.29404	−0.364215	−0.182108	−	0.983279i	$-0.558292\pi$
$139$	−4.29404	−0.364215	−0.182108	+	0.983279i	$0.558292\pi$
$140$	0	0
$141$	32.5517	2.74134
$142$	0	0
$143$	−0.692188	−0.0578837
$144$	0	0
$145$	−6.30908	−0.523940
$146$	0	0
$147$	−18.4699	−1.52337
$148$	0	0
$149$	−15.3655	−1.25879	−0.629395	−	0.777085i	$-0.716697\pi$
$149$	−15.3655	−1.25879	−0.629395	+	0.777085i	$0.716697\pi$
$150$	0	0
$151$	1.62007	0.131840	0.0659198	−	0.997825i	$-0.479002\pi$
$151$	1.62007	0.131840	0.0659198	+	0.997825i	$0.479002\pi$
$152$	0	0
$153$	12.6600	1.02350
$154$	0	0
$155$	−11.8251	−0.949817
$156$	0	0
$157$	−6.22084	−0.496477	−0.248238	−	0.968699i	$-0.579852\pi$
$157$	−6.22084	−0.496477	−0.248238	+	0.968699i	$0.579852\pi$
$158$	0	0
$159$	16.1561	1.28127
$160$	0	0
$161$	−9.38797	−0.739875
$162$	0	0
$163$	0.606523	0.0475066	0.0237533	−	0.999718i	$-0.492438\pi$
$163$	0.606523	0.0475066	0.0237533	+	0.999718i	$0.492438\pi$
$164$	0	0
$165$	−1.18931	−0.0925877
$166$	0	0
$167$	7.57408	0.586100	0.293050	−	0.956097i	$-0.405330\pi$
$167$	7.57408	0.586100	0.293050	+	0.956097i	$0.405330\pi$
$168$	0	0
$169$	−9.37604	−0.721234
$170$	0	0
$171$	23.9708	1.83309
$172$	0	0
$173$	−6.51583	−0.495390	−0.247695	−	0.968838i	$-0.579673\pi$
$173$	−6.51583	−0.495390	−0.247695	+	0.968838i	$0.579673\pi$
$174$	0	0
$175$	−12.4930	−0.944379
$176$	0	0
$177$	−25.6454	−1.92762
$178$	0	0
$179$	16.1836	1.20962	0.604811	−	0.796369i	$-0.293249\pi$
$179$	16.1836	1.20962	0.604811	+	0.796369i	$0.293249\pi$
$180$	0	0
$181$	−7.78504	−0.578657	−0.289329	−	0.957230i	$-0.593432\pi$
$181$	−7.78504	−0.578657	−0.289329	+	0.957230i	$0.593432\pi$
$182$	0	0
$183$	−9.52310	−0.703968
$184$	0	0
$185$	−13.9447	−1.02523
$186$	0	0
$187$	−1.40515	−0.102755
$188$	0	0
$189$	−2.62115	−0.190661
$190$	0	0
$191$	−8.48036	−0.613618	−0.306809	−	0.951771i	$-0.599261\pi$
$191$	−8.48036	−0.613618	−0.306809	+	0.951771i	$0.599261\pi$
$192$	0	0
$193$	−14.5606	−1.04809	−0.524047	−	0.851689i	$-0.675578\pi$
$193$	−14.5606	−1.04809	−0.524047	+	0.851689i	$0.675578\pi$
$194$	0	0
$195$	6.22665	0.445900
$196$	0	0
$197$	18.8459	1.34272	0.671359	−	0.741132i	$-0.265711\pi$
$197$	18.8459	1.34272	0.671359	+	0.741132i	$0.265711\pi$
$198$	0	0
$199$	−25.6566	−1.81874	−0.909372	−	0.415983i	$-0.863438\pi$
$199$	−25.6566	−1.81874	−0.909372	+	0.415983i	$0.863438\pi$
$200$	0	0
$201$	13.6176	0.960508
$202$	0	0
$203$	18.3194	1.28577
$204$	0	0
$205$	−6.12722	−0.427944
$206$	0	0
$207$	−8.11232	−0.563845
$208$	0	0
$209$	−2.66056	−0.184035
$210$	0	0
$211$	5.97393	0.411262	0.205631	−	0.978630i	$-0.434075\pi$
$211$	5.97393	0.411262	0.205631	+	0.978630i	$0.434075\pi$
$212$	0	0
$213$	14.6896	1.00652
$214$	0	0
$215$	−0.132011	−0.00900306
$216$	0	0
$217$	34.3362	2.33089
$218$	0	0
$219$	−33.6709	−2.27527
$220$	0	0
$221$	7.35670	0.494865
$222$	0	0
$223$	8.52259	0.570715	0.285357	−	0.958421i	$-0.407888\pi$
$223$	8.52259	0.570715	0.285357	+	0.958421i	$0.407888\pi$
$224$	0	0
$225$	−10.7954	−0.719694
$226$	0	0
$227$	29.0376	1.92729	0.963645	−	0.267185i	$-0.0860934\pi$
$227$	29.0376	1.92729	0.963645	+	0.267185i	$0.0860934\pi$
$228$	0	0
$229$	10.8023	0.713833	0.356917	−	0.934136i	$-0.383828\pi$
$229$	10.8023	0.713833	0.356917	+	0.934136i	$0.383828\pi$
$230$	0	0
$231$	3.45336	0.227214
$232$	0	0
$233$	3.81216	0.249743	0.124872	−	0.992173i	$-0.460148\pi$
$233$	3.81216	0.249743	0.124872	+	0.992173i	$0.460148\pi$
$234$	0	0
$235$	16.9650	1.10668
$236$	0	0
$237$	−12.1622	−0.790018
$238$	0	0
$239$	12.6567	0.818692	0.409346	−	0.912379i	$-0.365757\pi$
$239$	12.6567	0.818692	0.409346	+	0.912379i	$0.365757\pi$
$240$	0	0
$241$	27.1475	1.74873	0.874363	−	0.485272i	$-0.161279\pi$
$241$	27.1475	1.74873	0.874363	+	0.485272i	$0.161279\pi$
$242$	0	0
$243$	22.3559	1.43413
$244$	0	0
$245$	−9.62598	−0.614981
$246$	0	0
$247$	13.9294	0.886306
$248$	0	0
$249$	12.8954	0.817215
$250$	0	0
$251$	−3.89105	−0.245601	−0.122801	−	0.992431i	$-0.539188\pi$
$251$	−3.89105	−0.245601	−0.122801	+	0.992431i	$0.539188\pi$
$252$	0	0
$253$	0.900400	0.0566077
$254$	0	0
$255$	12.6402	0.791560
$256$	0	0
$257$	0.448767	0.0279933	0.0139967	−	0.999902i	$-0.495545\pi$
$257$	0.448767	0.0279933	0.0139967	+	0.999902i	$0.495545\pi$
$258$	0	0
$259$	40.4907	2.51597
$260$	0	0
$261$	15.8302	0.979863
$262$	0	0
$263$	16.4410	1.01379	0.506897	−	0.862007i	$-0.330792\pi$
$263$	16.4410	1.01379	0.506897	+	0.862007i	$0.330792\pi$
$264$	0	0
$265$	8.42013	0.517244
$266$	0	0
$267$	10.6591	0.652328
$268$	0	0
$269$	−21.5178	−1.31196	−0.655982	−	0.754777i	$-0.727745\pi$
$269$	−21.5178	−1.31196	−0.655982	+	0.754777i	$0.727745\pi$
$270$	0	0
$271$	18.9037	1.14832	0.574158	−	0.818744i	$-0.305329\pi$
$271$	18.9037	1.14832	0.574158	+	0.818744i	$0.305329\pi$
$272$	0	0
$273$	−18.0801	−1.09426
$274$	0	0
$275$	1.19820	0.0722542
$276$	0	0
$277$	−5.37343	−0.322858	−0.161429	−	0.986884i	$-0.551610\pi$
$277$	−5.37343	−0.322858	−0.161429	+	0.986884i	$0.551610\pi$
$278$	0	0
$279$	29.6706	1.77633
$280$	0	0
$281$	−7.77169	−0.463620	−0.231810	−	0.972761i	$-0.574465\pi$
$281$	−7.77169	−0.463620	−0.231810	+	0.972761i	$0.574465\pi$
$282$	0	0
$283$	19.2276	1.14296	0.571480	−	0.820616i	$-0.306370\pi$
$283$	19.2276	1.14296	0.571480	+	0.820616i	$0.306370\pi$
$284$	0	0
$285$	23.9333	1.41769
$286$	0	0
$287$	17.7914	1.05019
$288$	0	0
$289$	−2.06578	−0.121516
$290$	0	0
$291$	41.0169	2.40446
$292$	0	0
$293$	−15.5778	−0.910062	−0.455031	−	0.890476i	$-0.650372\pi$
$293$	−15.5778	−0.910062	−0.455031	+	0.890476i	$0.650372\pi$
$294$	0	0
$295$	−13.3656	−0.778178
$296$	0	0
$297$	0.251395	0.0145874
$298$	0	0
$299$	−4.71406	−0.272621
$300$	0	0
$301$	0.383315	0.0220939
$302$	0	0
$303$	27.3147	1.56919
$304$	0	0
$305$	−4.96317	−0.284191
$306$	0	0
$307$	−16.6888	−0.952482	−0.476241	−	0.879315i	$-0.658001\pi$
$307$	−16.6888	−0.952482	−0.476241	+	0.879315i	$0.658001\pi$
$308$	0	0
$309$	−22.8683	−1.30093
$310$	0	0
$311$	24.3280	1.37952	0.689758	−	0.724040i	$-0.257717\pi$
$311$	24.3280	1.37952	0.689758	+	0.724040i	$0.257717\pi$
$312$	0	0
$313$	−6.65759	−0.376309	−0.188154	−	0.982139i	$-0.560251\pi$
$313$	−6.65759	−0.376309	−0.188154	+	0.982139i	$0.560251\pi$
$314$	0	0
$315$	−16.2156	−0.913643
$316$	0	0
$317$	18.4545	1.03651	0.518255	−	0.855226i	$-0.326582\pi$
$317$	18.4545	1.03651	0.518255	+	0.855226i	$0.326582\pi$
$318$	0	0
$319$	−1.75702	−0.0983742
$320$	0	0
$321$	−22.2010	−1.23914
$322$	0	0
$323$	28.2769	1.57337
$324$	0	0
$325$	−6.27320	−0.347974
$326$	0	0
$327$	−42.6283	−2.35735
$328$	0	0
$329$	−49.2607	−2.71583
$330$	0	0
$331$	−16.5497	−0.909654	−0.454827	−	0.890580i	$-0.650299\pi$
$331$	−16.5497	−0.909654	−0.454827	+	0.890580i	$0.650299\pi$
$332$	0	0
$333$	34.9888	1.91737
$334$	0	0
$335$	7.09709	0.387755
$336$	0	0
$337$	−4.61696	−0.251502	−0.125751	−	0.992062i	$-0.540134\pi$
$337$	−4.61696	−0.251502	−0.125751	+	0.992062i	$0.540134\pi$
$338$	0	0
$339$	−19.6796	−1.06885
$340$	0	0
$341$	−3.29319	−0.178336
$342$	0	0
$343$	1.41272	0.0762796
$344$	0	0
$345$	−8.09964	−0.436070
$346$	0	0
$347$	19.2410	1.03291	0.516456	−	0.856313i	$-0.327251\pi$
$347$	19.2410	1.03291	0.516456	+	0.856313i	$0.327251\pi$
$348$	0	0
$349$	4.98694	0.266945	0.133472	−	0.991053i	$-0.457387\pi$
$349$	4.98694	0.266945	0.133472	+	0.991053i	$0.457387\pi$
$350$	0	0
$351$	−1.31618	−0.0702526
$352$	0	0
$353$	−33.8970	−1.80416	−0.902079	−	0.431572i	$-0.857959\pi$
$353$	−33.8970	−1.80416	−0.902079	+	0.431572i	$0.857959\pi$
$354$	0	0
$355$	7.65582	0.406329
$356$	0	0
$357$	−36.7029	−1.94252
$358$	0	0
$359$	8.74759	0.461680	0.230840	−	0.972992i	$-0.425853\pi$
$359$	8.74759	0.461680	0.230840	+	0.972992i	$0.425853\pi$
$360$	0	0
$361$	34.5403	1.81791
$362$	0	0
$363$	27.2259	1.42899
$364$	0	0
$365$	−17.5483	−0.918522
$366$	0	0
$367$	28.9781	1.51265	0.756323	−	0.654198i	$-0.226994\pi$
$367$	28.9781	1.51265	0.756323	+	0.654198i	$0.226994\pi$
$368$	0	0
$369$	15.3739	0.800332
$370$	0	0
$371$	−24.4492	−1.26934
$372$	0	0
$373$	−17.7879	−0.921022	−0.460511	−	0.887654i	$-0.652334\pi$
$373$	−17.7879	−0.921022	−0.460511	+	0.887654i	$0.652334\pi$
$374$	0	0
$375$	−27.1329	−1.40114
$376$	0	0
$377$	9.19890	0.473767
$378$	0	0
$379$	35.2765	1.81203	0.906017	−	0.423241i	$-0.139108\pi$
$379$	35.2765	1.81203	0.906017	+	0.423241i	$0.139108\pi$
$380$	0	0
$381$	27.7460	1.42147
$382$	0	0
$383$	34.4048	1.75800	0.879001	−	0.476820i	$-0.158211\pi$
$383$	34.4048	1.75800	0.879001	+	0.476820i	$0.158211\pi$
$384$	0	0
$385$	1.79979	0.0917259
$386$	0	0
$387$	0.331230	0.0168374
$388$	0	0
$389$	24.2221	1.22811	0.614055	−	0.789263i	$-0.289537\pi$
$389$	24.2221	1.22811	0.614055	+	0.789263i	$0.289537\pi$
$390$	0	0
$391$	−9.56962	−0.483956
$392$	0	0
$393$	−3.83881	−0.193642
$394$	0	0
$395$	−6.33858	−0.318929
$396$	0	0
$397$	3.96929	0.199213	0.0996065	−	0.995027i	$-0.468242\pi$
$397$	3.96929	0.199213	0.0996065	+	0.995027i	$0.468242\pi$
$398$	0	0
$399$	−69.4943	−3.47907
$400$	0	0
$401$	−1.40991	−0.0704074	−0.0352037	−	0.999380i	$-0.511208\pi$
$401$	−1.40991	−0.0704074	−0.0352037	+	0.999380i	$0.511208\pi$
$402$	0	0
$403$	17.2415	0.858862
$404$	0	0
$405$	10.5703	0.525241
$406$	0	0
$407$	−3.88347	−0.192496
$408$	0	0
$409$	−2.25583	−0.111544	−0.0557718	−	0.998444i	$-0.517762\pi$
$409$	−2.25583	−0.111544	−0.0557718	+	0.998444i	$0.517762\pi$
$410$	0	0
$411$	56.9064	2.80698
$412$	0	0
$413$	38.8093	1.90968
$414$	0	0
$415$	6.72074	0.329908
$416$	0	0
$417$	10.7574	0.526791
$418$	0	0
$419$	−36.6015	−1.78810	−0.894051	−	0.447965i	$-0.852149\pi$
$419$	−36.6015	−1.78810	−0.894051	+	0.447965i	$0.852149\pi$
$420$	0	0
$421$	−10.0546	−0.490030	−0.245015	−	0.969519i	$-0.578793\pi$
$421$	−10.0546	−0.490030	−0.245015	+	0.969519i	$0.578793\pi$
$422$	0	0
$423$	−42.5671	−2.06968
$424$	0	0
$425$	−12.7347	−0.617723
$426$	0	0
$427$	14.4114	0.697416
$428$	0	0
$429$	1.73406	0.0837214
$430$	0	0
$431$	20.6140	0.992940	0.496470	−	0.868054i	$-0.334629\pi$
$431$	20.6140	0.992940	0.496470	+	0.868054i	$0.334629\pi$
$432$	0	0
$433$	32.7836	1.57548	0.787740	−	0.616008i	$-0.211251\pi$
$433$	32.7836	1.57548	0.787740	+	0.616008i	$0.211251\pi$
$434$	0	0
$435$	15.8054	0.757813
$436$	0	0
$437$	−18.1194	−0.866768
$438$	0	0
$439$	−4.18644	−0.199808	−0.0999038	−	0.994997i	$-0.531854\pi$
$439$	−4.18644	−0.199808	−0.0999038	+	0.994997i	$0.531854\pi$
$440$	0	0
$441$	24.1526	1.15013
$442$	0	0
$443$	10.2069	0.484945	0.242473	−	0.970158i	$-0.422042\pi$
$443$	10.2069	0.484945	0.242473	+	0.970158i	$0.422042\pi$
$444$	0	0
$445$	5.55523	0.263343
$446$	0	0
$447$	38.4935	1.82068
$448$	0	0
$449$	28.1229	1.32720	0.663601	−	0.748087i	$-0.269027\pi$
$449$	28.1229	1.32720	0.663601	+	0.748087i	$0.269027\pi$
$450$	0	0
$451$	−1.70637	−0.0803500
$452$	0	0
$453$	−4.05859	−0.190689
$454$	0	0
$455$	−9.42284	−0.441750
$456$	0	0
$457$	21.6481	1.01265	0.506327	−	0.862342i	$-0.331003\pi$
$457$	21.6481	1.01265	0.506327	+	0.862342i	$0.331003\pi$
$458$	0	0
$459$	−2.67187	−0.124712
$460$	0	0
$461$	21.6030	1.00615	0.503077	−	0.864242i	$-0.332201\pi$
$461$	21.6030	1.00615	0.503077	+	0.864242i	$0.332201\pi$
$462$	0	0
$463$	−42.3106	−1.96634	−0.983169	−	0.182698i	$-0.941517\pi$
$463$	−42.3106	−1.96634	−0.983169	+	0.182698i	$0.941517\pi$
$464$	0	0
$465$	29.6242	1.37379
$466$	0	0
$467$	−11.9213	−0.551652	−0.275826	−	0.961208i	$-0.588951\pi$
$467$	−11.9213	−0.551652	−0.275826	+	0.961208i	$0.588951\pi$
$468$	0	0
$469$	−20.6076	−0.951568
$470$	0	0
$471$	15.5844	0.718091
$472$	0	0
$473$	−0.0367638	−0.00169040
$474$	0	0
$475$	−24.1122	−1.10635
$476$	0	0
$477$	−21.1270	−0.967340
$478$	0	0
$479$	−9.33770	−0.426650	−0.213325	−	0.976981i	$-0.568429\pi$
$479$	−9.33770	−0.426650	−0.213325	+	0.976981i	$0.568429\pi$
$480$	0	0
$481$	20.3320	0.927058
$482$	0	0
$483$	23.5187	1.07014
$484$	0	0
$485$	21.3769	0.970674
$486$	0	0
$487$	−12.8697	−0.583183	−0.291592	−	0.956543i	$-0.594185\pi$
$487$	−12.8697	−0.583183	−0.291592	+	0.956543i	$0.594185\pi$
$488$	0	0
$489$	−1.51946	−0.0687122
$490$	0	0
$491$	6.45494	0.291307	0.145654	−	0.989336i	$-0.453471\pi$
$491$	6.45494	0.291307	0.145654	+	0.989336i	$0.453471\pi$
$492$	0	0
$493$	18.6739	0.841031
$494$	0	0
$495$	1.55523	0.0699026
$496$	0	0
$497$	−22.2299	−0.997149
$498$	0	0
$499$	−7.62074	−0.341151	−0.170576	−	0.985345i	$-0.554563\pi$
$499$	−7.62074	−0.341151	−0.170576	+	0.985345i	$0.554563\pi$
$500$	0	0
$501$	−18.9745	−0.847719
$502$	0	0
$503$	−22.7140	−1.01277	−0.506383	−	0.862309i	$-0.669018\pi$
$503$	−22.7140	−1.01277	−0.506383	+	0.862309i	$0.669018\pi$
$504$	0	0
$505$	14.2357	0.633478
$506$	0	0
$507$	23.4888	1.04317
$508$	0	0
$509$	24.4508	1.08376	0.541882	−	0.840454i	$-0.317712\pi$
$509$	24.4508	1.08376	0.541882	+	0.840454i	$0.317712\pi$
$510$	0	0
$511$	50.9545	2.25409
$512$	0	0
$513$	−5.05899	−0.223360
$514$	0	0
$515$	−11.9183	−0.525184
$516$	0	0
$517$	4.72460	0.207788
$518$	0	0
$519$	16.3234	0.716518
$520$	0	0
$521$	−24.2450	−1.06219	−0.531097	−	0.847311i	$-0.678220\pi$
$521$	−24.2450	−1.06219	−0.531097	+	0.847311i	$0.678220\pi$
$522$	0	0
$523$	−13.4662	−0.588835	−0.294417	−	0.955677i	$-0.595126\pi$
$523$	−13.4662	−0.588835	−0.294417	+	0.955677i	$0.595126\pi$
$524$	0	0
$525$	31.2973	1.36592
$526$	0	0
$527$	35.0006	1.52465
$528$	0	0
$529$	−16.8679	−0.733389
$530$	0	0
$531$	33.5359	1.45533
$532$	0	0
$533$	8.93375	0.386963
$534$	0	0
$535$	−11.5705	−0.500238
$536$	0	0
$537$	−40.5431	−1.74956
$538$	0	0
$539$	−2.68074	−0.115468
$540$	0	0
$541$	−18.7293	−0.805236	−0.402618	−	0.915368i	$-0.631900\pi$
$541$	−18.7293	−0.805236	−0.402618	+	0.915368i	$0.631900\pi$
$542$	0	0
$543$	19.5030	0.836954
$544$	0	0
$545$	−22.2167	−0.951658
$546$	0	0
$547$	40.6607	1.73853	0.869264	−	0.494349i	$-0.164593\pi$
$547$	40.6607	1.73853	0.869264	+	0.494349i	$0.164593\pi$
$548$	0	0
$549$	12.4532	0.531488
$550$	0	0
$551$	35.3577	1.50629
$552$	0	0
$553$	18.4051	0.782665
$554$	0	0
$555$	34.9341	1.48287
$556$	0	0
$557$	−27.6967	−1.17355	−0.586774	−	0.809751i	$-0.699602\pi$
$557$	−27.6967	−1.17355	−0.586774	+	0.809751i	$0.699602\pi$
$558$	0	0
$559$	0.192477	0.00814092
$560$	0	0
$561$	3.52018	0.148622
$562$	0	0
$563$	5.33718	0.224935	0.112468	−	0.993655i	$-0.464125\pi$
$563$	5.33718	0.224935	0.112468	+	0.993655i	$0.464125\pi$
$564$	0	0
$565$	−10.2565	−0.431493
$566$	0	0
$567$	−30.6925	−1.28896
$568$	0	0
$569$	−28.1561	−1.18036	−0.590182	−	0.807270i	$-0.700944\pi$
$569$	−28.1561	−1.18036	−0.590182	+	0.807270i	$0.700944\pi$
$570$	0	0
$571$	34.0326	1.42422	0.712111	−	0.702067i	$-0.247739\pi$
$571$	34.0326	1.42422	0.712111	+	0.702067i	$0.247739\pi$
$572$	0	0
$573$	21.2449	0.887520
$574$	0	0
$575$	8.16019	0.340304
$576$	0	0
$577$	−42.7261	−1.77871	−0.889356	−	0.457216i	$-0.848847\pi$
$577$	−42.7261	−1.77871	−0.889356	+	0.457216i	$0.848847\pi$
$578$	0	0
$579$	36.4771	1.51593
$580$	0	0
$581$	−19.5148	−0.809609
$582$	0	0
$583$	2.34493	0.0971169
$584$	0	0
$585$	−8.14246	−0.336649
$586$	0	0
$587$	−25.8421	−1.06662	−0.533308	−	0.845921i	$-0.679051\pi$
$587$	−25.8421	−1.06662	−0.533308	+	0.845921i	$0.679051\pi$
$588$	0	0
$589$	66.2711	2.73065
$590$	0	0
$591$	−47.2127	−1.94207
$592$	0	0
$593$	22.7519	0.934307	0.467154	−	0.884176i	$-0.345280\pi$
$593$	22.7519	0.934307	0.467154	+	0.884176i	$0.345280\pi$
$594$	0	0
$595$	−19.1285	−0.784193
$596$	0	0
$597$	64.2746	2.63058
$598$	0	0
$599$	−8.33632	−0.340613	−0.170306	−	0.985391i	$-0.554476\pi$
$599$	−8.33632	−0.340613	−0.170306	+	0.985391i	$0.554476\pi$
$600$	0	0
$601$	−5.43201	−0.221576	−0.110788	−	0.993844i	$-0.535337\pi$
$601$	−5.43201	−0.221576	−0.110788	+	0.993844i	$0.535337\pi$
$602$	0	0
$603$	−17.8074	−0.725172
$604$	0	0
$605$	14.1894	0.576880
$606$	0	0
$607$	18.4685	0.749614	0.374807	−	0.927103i	$-0.377709\pi$
$607$	18.4685	0.749614	0.374807	+	0.927103i	$0.377709\pi$
$608$	0	0
$609$	−45.8937	−1.85971
$610$	0	0
$611$	−24.7357	−1.00070
$612$	0	0
$613$	24.5726	0.992480	0.496240	−	0.868185i	$-0.334714\pi$
$613$	24.5726	0.992480	0.496240	+	0.868185i	$0.334714\pi$
$614$	0	0
$615$	15.3499	0.618966
$616$	0	0
$617$	−10.6795	−0.429941	−0.214971	−	0.976621i	$-0.568966\pi$
$617$	−10.6795	−0.429941	−0.214971	+	0.976621i	$0.568966\pi$
$618$	0	0
$619$	3.35056	0.134670	0.0673352	−	0.997730i	$-0.478550\pi$
$619$	3.35056	0.134670	0.0673352	+	0.997730i	$0.478550\pi$
$620$	0	0
$621$	1.71209	0.0687039
$622$	0	0
$623$	−16.1305	−0.646256
$624$	0	0
$625$	2.33570	0.0934279
$626$	0	0
$627$	6.66521	0.266183
$628$	0	0
$629$	41.2742	1.64571
$630$	0	0
$631$	1.54404	0.0614672	0.0307336	−	0.999528i	$-0.490216\pi$
$631$	1.54404	0.0614672	0.0307336	+	0.999528i	$0.490216\pi$
$632$	0	0
$633$	−14.9658	−0.594839
$634$	0	0
$635$	14.4604	0.573845
$636$	0	0
$637$	14.0351	0.556090
$638$	0	0
$639$	−19.2093	−0.759908
$640$	0	0
$641$	−21.5504	−0.851191	−0.425595	−	0.904914i	$-0.639935\pi$
$641$	−21.5504	−0.851191	−0.425595	+	0.904914i	$0.639935\pi$
$642$	0	0
$643$	16.7833	0.661868	0.330934	−	0.943654i	$-0.392636\pi$
$643$	16.7833	0.661868	0.330934	+	0.943654i	$0.392636\pi$
$644$	0	0
$645$	0.330712	0.0130218
$646$	0	0
$647$	15.0708	0.592492	0.296246	−	0.955112i	$-0.404265\pi$
$647$	15.0708	0.592492	0.296246	+	0.955112i	$0.404265\pi$
$648$	0	0
$649$	−3.72221	−0.146109
$650$	0	0
$651$	−86.0187	−3.37134
$652$	0	0
$653$	−16.3740	−0.640762	−0.320381	−	0.947289i	$-0.603811\pi$
$653$	−16.3740	−0.640762	−0.320381	+	0.947289i	$0.603811\pi$
$654$	0	0
$655$	−2.00068	−0.0781730
$656$	0	0
$657$	44.0307	1.71780
$658$	0	0
$659$	16.7976	0.654342	0.327171	−	0.944965i	$-0.393905\pi$
$659$	16.7976	0.654342	0.327171	+	0.944965i	$0.393905\pi$
$660$	0	0
$661$	35.3544	1.37513	0.687563	−	0.726125i	$-0.258681\pi$
$661$	35.3544	1.37513	0.687563	+	0.726125i	$0.258681\pi$
$662$	0	0
$663$	−18.4299	−0.715760
$664$	0	0
$665$	−36.2185	−1.40449
$666$	0	0
$667$	−11.9660	−0.463323
$668$	0	0
$669$	−21.3507	−0.825466
$670$	0	0
$671$	−1.38220	−0.0533591
$672$	0	0
$673$	36.7227	1.41556	0.707778	−	0.706435i	$-0.249698\pi$
$673$	36.7227	1.41556	0.707778	+	0.706435i	$0.249698\pi$
$674$	0	0
$675$	2.27835	0.0876938
$676$	0	0
$677$	−22.7767	−0.875379	−0.437690	−	0.899126i	$-0.644203\pi$
$677$	−22.7767	−0.875379	−0.437690	+	0.899126i	$0.644203\pi$
$678$	0	0
$679$	−62.0713	−2.38208
$680$	0	0
$681$	−72.7446	−2.78758
$682$	0	0
$683$	−2.63670	−0.100890	−0.0504452	−	0.998727i	$-0.516064\pi$
$683$	−2.63670	−0.100890	−0.0504452	+	0.998727i	$0.516064\pi$
$684$	0	0
$685$	29.6580	1.13317
$686$	0	0
$687$	−27.0617	−1.03247
$688$	0	0
$689$	−12.2769	−0.467713
$690$	0	0
$691$	−13.6817	−0.520475	−0.260237	−	0.965545i	$-0.583801\pi$
$691$	−13.6817	−0.520475	−0.260237	+	0.965545i	$0.583801\pi$
$692$	0	0
$693$	−4.51588	−0.171544
$694$	0	0
$695$	5.60644	0.212665
$696$	0	0
$697$	18.1356	0.686936
$698$	0	0
$699$	−9.55020	−0.361222
$700$	0	0
$701$	36.1017	1.36354	0.681771	−	0.731566i	$-0.261210\pi$
$701$	36.1017	1.36354	0.681771	+	0.731566i	$0.261210\pi$
$702$	0	0
$703$	78.1498	2.94747
$704$	0	0
$705$	−42.5006	−1.60066
$706$	0	0
$707$	−41.3356	−1.55458
$708$	0	0
$709$	−3.14146	−0.117980	−0.0589900	−	0.998259i	$-0.518788\pi$
$709$	−3.14146	−0.117980	−0.0589900	+	0.998259i	$0.518788\pi$
$710$	0	0
$711$	15.9042	0.596455
$712$	0	0
$713$	−22.4278	−0.839929
$714$	0	0
$715$	0.903746	0.0337982
$716$	0	0
$717$	−31.7074	−1.18413
$718$	0	0
$719$	−29.1814	−1.08828	−0.544142	−	0.838993i	$-0.683145\pi$
$719$	−29.1814	−1.08828	−0.544142	+	0.838993i	$0.683145\pi$
$720$	0	0
$721$	34.6068	1.28882
$722$	0	0
$723$	−68.0098	−2.52931
$724$	0	0
$725$	−15.9236	−0.591387
$726$	0	0
$727$	−50.6424	−1.87822	−0.939112	−	0.343611i	$-0.888350\pi$
$727$	−50.6424	−1.87822	−0.939112	+	0.343611i	$0.888350\pi$
$728$	0	0
$729$	−31.7182	−1.17475
$730$	0	0
$731$	0.390732	0.0144517
$732$	0	0
$733$	−21.6083	−0.798120	−0.399060	−	0.916925i	$-0.630664\pi$
$733$	−21.6083	−0.798120	−0.399060	+	0.916925i	$0.630664\pi$
$734$	0	0
$735$	24.1149	0.889492
$736$	0	0
$737$	1.97647	0.0728043
$738$	0	0
$739$	−8.85134	−0.325602	−0.162801	−	0.986659i	$-0.552053\pi$
$739$	−8.85134	−0.325602	−0.162801	+	0.986659i	$0.552053\pi$
$740$	0	0
$741$	−34.8958	−1.28193
$742$	0	0
$743$	−25.9068	−0.950428	−0.475214	−	0.879870i	$-0.657629\pi$
$743$	−25.9068	−0.950428	−0.475214	+	0.879870i	$0.657629\pi$
$744$	0	0
$745$	20.0617	0.735005
$746$	0	0
$747$	−16.8631	−0.616988
$748$	0	0
$749$	33.5970	1.22761
$750$	0	0
$751$	−26.8156	−0.978516	−0.489258	−	0.872139i	$-0.662732\pi$
$751$	−26.8156	−0.978516	−0.489258	+	0.872139i	$0.662732\pi$
$752$	0	0
$753$	9.74783	0.355231
$754$	0	0
$755$	−2.11522	−0.0769809
$756$	0	0
$757$	−53.5837	−1.94753	−0.973767	−	0.227548i	$-0.926929\pi$
$757$	−53.5837	−1.94753	−0.973767	+	0.227548i	$0.926929\pi$
$758$	0	0
$759$	−2.25568	−0.0818758
$760$	0	0
$761$	33.7648	1.22397	0.611987	−	0.790868i	$-0.290370\pi$
$761$	33.7648	1.22397	0.611987	+	0.790868i	$0.290370\pi$
$762$	0	0
$763$	64.5098	2.33541
$764$	0	0
$765$	−16.5293	−0.597619
$766$	0	0
$767$	19.4877	0.703659
$768$	0	0
$769$	−3.26570	−0.117764	−0.0588820	−	0.998265i	$-0.518754\pi$
$769$	−3.26570	−0.117764	−0.0588820	+	0.998265i	$0.518754\pi$
$770$	0	0
$771$	−1.12425	−0.0404888
$772$	0	0
$773$	37.6645	1.35470	0.677349	−	0.735662i	$-0.263129\pi$
$773$	37.6645	1.35470	0.677349	+	0.735662i	$0.263129\pi$
$774$	0	0
$775$	−29.8457	−1.07209
$776$	0	0
$777$	−101.437	−3.63903
$778$	0	0
$779$	34.3385	1.23031
$780$	0	0
$781$	2.13207	0.0762916
$782$	0	0
$783$	−3.34093	−0.119395
$784$	0	0
$785$	8.12214	0.289892
$786$	0	0
$787$	−10.2241	−0.364449	−0.182225	−	0.983257i	$-0.558330\pi$
$787$	−10.2241	−0.364449	−0.182225	+	0.983257i	$0.558330\pi$
$788$	0	0
$789$	−41.1878	−1.46632
$790$	0	0
$791$	29.7813	1.05890
$792$	0	0
$793$	7.23652	0.256976
$794$	0	0
$795$	−21.0940	−0.748128
$796$	0	0
$797$	−54.5643	−1.93277	−0.966384	−	0.257103i	$-0.917232\pi$
$797$	−54.5643	−1.93277	−0.966384	+	0.257103i	$0.917232\pi$
$798$	0	0
$799$	−50.2139	−1.77644
$800$	0	0
$801$	−13.9387	−0.492500
$802$	0	0
$803$	−4.88705	−0.172460
$804$	0	0
$805$	12.2573	0.432012
$806$	0	0
$807$	53.9062	1.89759
$808$	0	0
$809$	45.9092	1.61408	0.807041	−	0.590495i	$-0.201068\pi$
$809$	45.9092	1.61408	0.807041	+	0.590495i	$0.201068\pi$
$810$	0	0
$811$	18.4696	0.648555	0.324278	−	0.945962i	$-0.394879\pi$
$811$	18.4696	0.648555	0.324278	+	0.945962i	$0.394879\pi$
$812$	0	0
$813$	−47.3573	−1.66089
$814$	0	0
$815$	−0.791898	−0.0277390
$816$	0	0
$817$	0.739823	0.0258831
$818$	0	0
$819$	23.6430	0.826152
$820$	0	0
$821$	−50.7998	−1.77293	−0.886463	−	0.462799i	$-0.846845\pi$
$821$	−50.7998	−1.77293	−0.886463	+	0.462799i	$0.846845\pi$
$822$	0	0
$823$	55.2692	1.92656	0.963282	−	0.268491i	$-0.0865248\pi$
$823$	55.2692	1.92656	0.963282	+	0.268491i	$0.0865248\pi$
$824$	0	0
$825$	−3.00172	−0.104507
$826$	0	0
$827$	15.6534	0.544322	0.272161	−	0.962252i	$-0.412262\pi$
$827$	15.6534	0.544322	0.272161	+	0.962252i	$0.412262\pi$
$828$	0	0
$829$	−17.8686	−0.620601	−0.310301	−	0.950638i	$-0.600430\pi$
$829$	−17.8686	−0.620601	−0.310301	+	0.950638i	$0.600430\pi$
$830$	0	0
$831$	13.4615	0.466973
$832$	0	0
$833$	28.4914	0.987169
$834$	0	0
$835$	−9.88898	−0.342222
$836$	0	0
$837$	−6.26192	−0.216444
$838$	0	0
$839$	3.71012	0.128088	0.0640438	−	0.997947i	$-0.479600\pi$
$839$	3.71012	0.128088	0.0640438	+	0.997947i	$0.479600\pi$
$840$	0	0
$841$	−5.64995	−0.194826
$842$	0	0
$843$	19.4696	0.670568
$844$	0	0
$845$	12.2417	0.421127
$846$	0	0
$847$	−41.2012	−1.41569
$848$	0	0
$849$	−48.1687	−1.65315
$850$	0	0
$851$	−26.4479	−0.906621
$852$	0	0
$853$	−16.4140	−0.562004	−0.281002	−	0.959707i	$-0.590667\pi$
$853$	−16.4140	−0.562004	−0.281002	+	0.959707i	$0.590667\pi$
$854$	0	0
$855$	−31.2971	−1.07034
$856$	0	0
$857$	−38.0827	−1.30088	−0.650440	−	0.759557i	$-0.725416\pi$
$857$	−38.0827	−1.30088	−0.650440	+	0.759557i	$0.725416\pi$
$858$	0	0
$859$	−32.2033	−1.09876	−0.549382	−	0.835571i	$-0.685137\pi$
$859$	−32.2033	−1.09876	−0.549382	+	0.835571i	$0.685137\pi$
$860$	0	0
$861$	−44.5708	−1.51897
$862$	0	0
$863$	−44.9612	−1.53050	−0.765248	−	0.643736i	$-0.777384\pi$
$863$	−44.9612	−1.53050	−0.765248	+	0.643736i	$0.777384\pi$
$864$	0	0
$865$	8.50730	0.289257
$866$	0	0
$867$	5.17517	0.175758
$868$	0	0
$869$	−1.76524	−0.0598816
$870$	0	0
$871$	−10.3479	−0.350624
$872$	0	0
$873$	−53.6369	−1.81534
$874$	0	0
$875$	41.0604	1.38809
$876$	0	0
$877$	−45.2458	−1.52784	−0.763922	−	0.645309i	$-0.776729\pi$
$877$	−45.2458	−1.52784	−0.763922	+	0.645309i	$0.776729\pi$
$878$	0	0
$879$	39.0253	1.31629
$880$	0	0
$881$	−54.2104	−1.82640	−0.913198	−	0.407516i	$-0.866395\pi$
$881$	−54.2104	−1.82640	−0.913198	+	0.407516i	$0.866395\pi$
$882$	0	0
$883$	57.9733	1.95096	0.975478	−	0.220098i	$-0.0706377\pi$
$883$	57.9733	1.95096	0.975478	+	0.220098i	$0.0706377\pi$
$884$	0	0
$885$	33.4835	1.12554
$886$	0	0
$887$	−6.65363	−0.223407	−0.111704	−	0.993742i	$-0.535631\pi$
$887$	−6.65363	−0.223407	−0.111704	+	0.993742i	$0.535631\pi$
$888$	0	0
$889$	−41.9883	−1.40824
$890$	0	0
$891$	2.94372	0.0986184
$892$	0	0
$893$	−95.0764	−3.18161
$894$	0	0
$895$	−21.1299	−0.706296
$896$	0	0
$897$	11.8096	0.394312
$898$	0	0
$899$	43.7651	1.45965
$900$	0	0
$901$	−24.9223	−0.830282
$902$	0	0
$903$	−0.960278	−0.0319560
$904$	0	0
$905$	10.1644	0.337877
$906$	0	0
$907$	−14.1099	−0.468511	−0.234256	−	0.972175i	$-0.575265\pi$
$907$	−14.1099	−0.468511	−0.234256	+	0.972175i	$0.575265\pi$
$908$	0	0
$909$	−35.7188	−1.18472
$910$	0	0
$911$	13.9469	0.462080	0.231040	−	0.972944i	$-0.425787\pi$
$911$	13.9469	0.462080	0.231040	+	0.972944i	$0.425787\pi$
$912$	0	0
$913$	1.87166	0.0619430
$914$	0	0
$915$	12.4337	0.411045
$916$	0	0
$917$	5.80930	0.191840
$918$	0	0
$919$	12.6817	0.418330	0.209165	−	0.977880i	$-0.432925\pi$
$919$	12.6817	0.418330	0.209165	+	0.977880i	$0.432925\pi$
$920$	0	0
$921$	41.8087	1.37764
$922$	0	0
$923$	−11.1625	−0.367419
$924$	0	0
$925$	−35.1953	−1.15721
$926$	0	0
$927$	29.9044	0.982188
$928$	0	0
$929$	−1.03009	−0.0337962	−0.0168981	−	0.999857i	$-0.505379\pi$
$929$	−1.03009	−0.0337962	−0.0168981	+	0.999857i	$0.505379\pi$
$930$	0	0
$931$	53.9465	1.76802
$932$	0	0
$933$	−60.9464	−1.99529
$934$	0	0
$935$	1.83462	0.0599984
$936$	0	0
$937$	−50.3278	−1.64414	−0.822068	−	0.569389i	$-0.807180\pi$
$937$	−50.3278	−1.64414	−0.822068	+	0.569389i	$0.807180\pi$
$938$	0	0
$939$	16.6785	0.544283
$940$	0	0
$941$	32.9666	1.07468	0.537341	−	0.843365i	$-0.319429\pi$
$941$	32.9666	1.07468	0.537341	+	0.843365i	$0.319429\pi$
$942$	0	0
$943$	−11.6210	−0.378433
$944$	0	0
$945$	3.42227	0.111326
$946$	0	0
$947$	5.32202	0.172942	0.0864711	−	0.996254i	$-0.472441\pi$
$947$	5.32202	0.172942	0.0864711	+	0.996254i	$0.472441\pi$
$948$	0	0
$949$	25.5862	0.830564
$950$	0	0
$951$	−46.2322	−1.49918
$952$	0	0
$953$	36.9298	1.19627	0.598136	−	0.801395i	$-0.295908\pi$
$953$	36.9298	1.19627	0.598136	+	0.801395i	$0.295908\pi$
$954$	0	0
$955$	11.0723	0.358290
$956$	0	0
$957$	4.40167	0.142286
$958$	0	0
$959$	−86.1168	−2.78086
$960$	0	0
$961$	51.0291	1.64610
$962$	0	0
$963$	29.0318	0.935535
$964$	0	0
$965$	19.0108	0.611980
$966$	0	0
$967$	−26.7250	−0.859419	−0.429709	−	0.902967i	$-0.641384\pi$
$967$	−26.7250	−0.859419	−0.429709	+	0.902967i	$0.641384\pi$
$968$	0	0
$969$	−70.8390	−2.27568
$970$	0	0
$971$	32.3008	1.03658	0.518291	−	0.855204i	$-0.326568\pi$
$971$	32.3008	1.03658	0.518291	+	0.855204i	$0.326568\pi$
$972$	0	0
$973$	−16.2792	−0.521888
$974$	0	0
$975$	15.7156	0.503301
$976$	0	0
$977$	−16.3219	−0.522185	−0.261092	−	0.965314i	$-0.584083\pi$
$977$	−16.3219	−0.522185	−0.261092	+	0.965314i	$0.584083\pi$
$978$	0	0
$979$	1.54708	0.0494449
$980$	0	0
$981$	55.7441	1.77977
$982$	0	0
$983$	−27.6565	−0.882106	−0.441053	−	0.897481i	$-0.645395\pi$
$983$	−27.6565	−0.882106	−0.441053	+	0.897481i	$0.645395\pi$
$984$	0	0
$985$	−24.6059	−0.784010
$986$	0	0
$987$	123.407	3.92810
$988$	0	0
$989$	−0.250375	−0.00796146
$990$	0	0
$991$	−2.56742	−0.0815568	−0.0407784	−	0.999168i	$-0.512984\pi$
$991$	−2.56742	−0.0815568	−0.0407784	+	0.999168i	$0.512984\pi$
$992$	0	0
$993$	41.4602	1.31570
$994$	0	0
$995$	33.4981	1.06196
$996$	0	0
$997$	47.8210	1.51451	0.757253	−	0.653122i	$-0.226541\pi$
$997$	47.8210	1.51451	0.757253	+	0.653122i	$0.226541\pi$
$998$	0	0
$999$	−7.38433	−0.233630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.b.1.9	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.b.1.9	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.50519 q^{3} -1.30564 q^{5} +3.79113 q^{7} +3.27598 q^{9} +O(q^{10})\)	\(q-2.50519 q^{3} -1.30564 q^{5} +3.79113 q^{7} +3.27598 q^{9} -0.363607 q^{11} +1.90367 q^{13} +3.27087 q^{15} +3.86448 q^{17} +7.31712 q^{19} -9.49750 q^{21} -2.47630 q^{23} -3.29532 q^{25} -0.691391 q^{27} +4.83219 q^{29} +9.05699 q^{31} +0.910906 q^{33} -4.94983 q^{35} +10.6804 q^{37} -4.76906 q^{39} +4.69290 q^{41} +0.101108 q^{43} -4.27724 q^{45} -12.9937 q^{47} +7.37264 q^{49} -9.68127 q^{51} -6.44907 q^{53} +0.474738 q^{55} -18.3308 q^{57} +10.2369 q^{59} +3.80135 q^{61} +12.4197 q^{63} -2.48550 q^{65} -5.43573 q^{67} +6.20361 q^{69} -5.86368 q^{71} +13.4405 q^{73} +8.25540 q^{75} -1.37848 q^{77} +4.85479 q^{79} -8.09588 q^{81} -5.14749 q^{83} -5.04560 q^{85} -12.1056 q^{87} -4.25481 q^{89} +7.21706 q^{91} -22.6895 q^{93} -9.55349 q^{95} -16.3728 q^{97} -1.19117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * q^99

Introduction

L-functions

Modular forms

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Database

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