LMFDB - Embedded newform 6021.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6021,2,Mod(1,6021)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6021, 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("6021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6021 = 3^{3} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6021.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:	$48.0779270570$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	6021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69102 q^{2} +5.24161 q^{4} -2.40662 q^{5} +2.96509 q^{7} -8.72327 q^{8} +O(q^{10})$	\(q-2.69102 q^{2} +5.24161 q^{4} -2.40662 q^{5} +2.96509 q^{7} -8.72327 q^{8} +6.47626 q^{10} +0.991862 q^{11} +3.08869 q^{13} -7.97914 q^{14} +12.9913 q^{16} +3.39643 q^{17} -3.05082 q^{19} -12.6145 q^{20} -2.66912 q^{22} +5.70909 q^{23} +0.791796 q^{25} -8.31175 q^{26} +15.5419 q^{28} +3.78412 q^{29} +4.30571 q^{31} -17.5134 q^{32} -9.13988 q^{34} -7.13584 q^{35} +5.83764 q^{37} +8.20982 q^{38} +20.9935 q^{40} +0.306744 q^{41} +0.173316 q^{43} +5.19896 q^{44} -15.3633 q^{46} -3.13156 q^{47} +1.79178 q^{49} -2.13074 q^{50} +16.1897 q^{52} +1.76013 q^{53} -2.38703 q^{55} -25.8653 q^{56} -10.1832 q^{58} +3.38654 q^{59} +5.00170 q^{61} -11.5868 q^{62} +21.1463 q^{64} -7.43329 q^{65} +3.68493 q^{67} +17.8028 q^{68} +19.2027 q^{70} +14.8689 q^{71} -1.26962 q^{73} -15.7092 q^{74} -15.9912 q^{76} +2.94096 q^{77} -0.660487 q^{79} -31.2650 q^{80} -0.825454 q^{82} +6.79682 q^{83} -8.17390 q^{85} -0.466397 q^{86} -8.65227 q^{88} -3.21025 q^{89} +9.15826 q^{91} +29.9249 q^{92} +8.42711 q^{94} +7.34214 q^{95} -7.76179 q^{97} -4.82172 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) $ 40 * q + 46 * q^4 + 16 * q^7	$ 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) $ 40 * q + 46 * q^4 + 16 * q^7 + 22 * q^10 + 14 * q^13 + 50 * q^16 + 64 * q^19 + 12 * q^22 + 40 * q^25 + 48 * q^28 + 54 * q^31 + 32 * q^34 + 24 * q^37 + 40 * q^40 + 24 * q^43 + 52 * q^46 + 64 * q^49 + 18 * q^52 + 36 * q^55 + 8 * q^58 + 58 * q^61 + 120 * q^64 + 52 * q^67 - 30 * q^70 + 50 * q^73 + 112 * q^76 + 60 * q^79 + 50 * q^82 + 38 * q^85 + 16 * q^88 + 118 * q^91 + 44 * q^94 + 38 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.69102	−1.90284	−0.951421	−	0.307893i	$-0.900376\pi$
$2$	−2.69102	−1.90284	−0.951421	+	0.307893i	$0.900376\pi$
$3$	0	0
$3$	0	0
$4$	5.24161	2.62081
$5$	−2.40662	−1.07627	−0.538135	−	0.842858i	$-0.680871\pi$
$5$	−2.40662	−1.07627	−0.538135	+	0.842858i	$0.680871\pi$
$6$	0	0
$7$	2.96509	1.12070	0.560350	−	0.828256i	$-0.310667\pi$
$7$	2.96509	1.12070	0.560350	+	0.828256i	$0.310667\pi$
$8$	−8.72327	−3.08414
$9$	0	0
$10$	6.47626	2.04797
$11$	0.991862	0.299058	0.149529	−	0.988757i	$-0.452224\pi$
$11$	0.991862	0.299058	0.149529	+	0.988757i	$0.452224\pi$
$12$	0	0
$13$	3.08869	0.856649	0.428324	−	0.903625i	$-0.359104\pi$
$13$	3.08869	0.856649	0.428324	+	0.903625i	$0.359104\pi$
$14$	−7.97914	−2.13251
$15$	0	0
$16$	12.9913	3.24782
$17$	3.39643	0.823756	0.411878	−	0.911239i	$-0.364873\pi$
$17$	3.39643	0.823756	0.411878	+	0.911239i	$0.364873\pi$
$18$	0	0
$19$	−3.05082	−0.699905	−0.349953	−	0.936767i	$-0.613802\pi$
$19$	−3.05082	−0.699905	−0.349953	+	0.936767i	$0.613802\pi$
$20$	−12.6145	−2.82070
$21$	0	0
$22$	−2.66912	−0.569059
$23$	5.70909	1.19043	0.595214	−	0.803567i	$-0.297067\pi$
$23$	5.70909	1.19043	0.595214	+	0.803567i	$0.297067\pi$
$24$	0	0
$25$	0.791796	0.158359
$26$	−8.31175	−1.63007
$27$	0	0
$28$	15.5419	2.93714
$29$	3.78412	0.702694	0.351347	−	0.936245i	$-0.385724\pi$
$29$	3.78412	0.702694	0.351347	+	0.936245i	$0.385724\pi$
$30$	0	0
$31$	4.30571	0.773328	0.386664	−	0.922221i	$-0.373627\pi$
$31$	4.30571	0.773328	0.386664	+	0.922221i	$0.373627\pi$
$32$	−17.5134	−3.09595
$33$	0	0
$34$	−9.13988	−1.56748
$35$	−7.13584	−1.20618
$36$	0	0
$37$	5.83764	0.959702	0.479851	−	0.877350i	$-0.340691\pi$
$37$	5.83764	0.959702	0.479851	+	0.877350i	$0.340691\pi$
$38$	8.20982	1.33181
$39$	0	0
$40$	20.9935	3.31937
$41$	0.306744	0.0479053	0.0239526	−	0.999713i	$-0.492375\pi$
$41$	0.306744	0.0479053	0.0239526	+	0.999713i	$0.492375\pi$
$42$	0	0
$43$	0.173316	0.0264304	0.0132152	−	0.999913i	$-0.495793\pi$
$43$	0.173316	0.0264304	0.0132152	+	0.999913i	$0.495793\pi$
$44$	5.19896	0.783772
$45$	0	0
$46$	−15.3633	−2.26520
$47$	−3.13156	−0.456785	−0.228393	−	0.973569i	$-0.573347\pi$
$47$	−3.13156	−0.456785	−0.228393	+	0.973569i	$0.573347\pi$
$48$	0	0
$49$	1.79178	0.255968
$50$	−2.13074	−0.301333
$51$	0	0
$52$	16.1897	2.24511
$53$	1.76013	0.241772	0.120886	−	0.992666i	$-0.461426\pi$
$53$	1.76013	0.241772	0.120886	+	0.992666i	$0.461426\pi$
$54$	0	0
$55$	−2.38703	−0.321867
$56$	−25.8653	−3.45640
$57$	0	0
$58$	−10.1832	−1.33712
$59$	3.38654	0.440890	0.220445	−	0.975399i	$-0.429249\pi$
$59$	3.38654	0.440890	0.220445	+	0.975399i	$0.429249\pi$
$60$	0	0
$61$	5.00170	0.640402	0.320201	−	0.947350i	$-0.396249\pi$
$61$	5.00170	0.640402	0.320201	+	0.947350i	$0.396249\pi$
$62$	−11.5868	−1.47152
$63$	0	0
$64$	21.1463	2.64329
$65$	−7.43329	−0.921986
$66$	0	0
$67$	3.68493	0.450186	0.225093	−	0.974337i	$-0.427731\pi$
$67$	3.68493	0.450186	0.225093	+	0.974337i	$0.427731\pi$
$68$	17.8028	2.15890
$69$	0	0
$70$	19.2027	2.29516
$71$	14.8689	1.76461	0.882304	−	0.470680i	$-0.155991\pi$
$71$	14.8689	1.76461	0.882304	+	0.470680i	$0.155991\pi$
$72$	0	0
$73$	−1.26962	−0.148597	−0.0742987	−	0.997236i	$-0.523672\pi$
$73$	−1.26962	−0.148597	−0.0742987	+	0.997236i	$0.523672\pi$
$74$	−15.7092	−1.82616
$75$	0	0
$76$	−15.9912	−1.83432
$77$	2.94096	0.335154
$78$	0	0
$79$	−0.660487	−0.0743106	−0.0371553	−	0.999310i	$-0.511830\pi$
$79$	−0.660487	−0.0743106	−0.0371553	+	0.999310i	$0.511830\pi$
$80$	−31.2650	−3.49554
$81$	0	0
$82$	−0.825454	−0.0911562
$83$	6.79682	0.746048	0.373024	−	0.927822i	$-0.378321\pi$
$83$	6.79682	0.746048	0.373024	+	0.927822i	$0.378321\pi$
$84$	0	0
$85$	−8.17390	−0.886584
$86$	−0.466397	−0.0502929
$87$	0	0
$88$	−8.65227	−0.922336
$89$	−3.21025	−0.340286	−0.170143	−	0.985419i	$-0.554423\pi$
$89$	−3.21025	−0.340286	−0.170143	+	0.985419i	$0.554423\pi$
$90$	0	0
$91$	9.15826	0.960046
$92$	29.9249	3.11988
$93$	0	0
$94$	8.42711	0.869190
$95$	7.34214	0.753288
$96$	0	0
$97$	−7.76179	−0.788091	−0.394045	−	0.919091i	$-0.628925\pi$
$97$	−7.76179	−0.788091	−0.394045	+	0.919091i	$0.628925\pi$
$98$	−4.82172	−0.487067
$99$	0	0
$100$	4.15029	0.415029
$101$	12.2515	1.21907	0.609534	−	0.792760i	$-0.291356\pi$
$101$	12.2515	1.21907	0.609534	+	0.792760i	$0.291356\pi$
$102$	0	0
$103$	−12.3546	−1.21733	−0.608667	−	0.793426i	$-0.708295\pi$
$103$	−12.3546	−1.21733	−0.608667	+	0.793426i	$0.708295\pi$
$104$	−26.9435	−2.64203
$105$	0	0
$106$	−4.73655	−0.460055
$107$	−19.4121	−1.87664	−0.938320	−	0.345767i	$-0.887619\pi$
$107$	−19.4121	−1.87664	−0.938320	+	0.345767i	$0.887619\pi$
$108$	0	0
$109$	−3.68687	−0.353138	−0.176569	−	0.984288i	$-0.556500\pi$
$109$	−3.68687	−0.353138	−0.176569	+	0.984288i	$0.556500\pi$
$110$	6.42356	0.612462
$111$	0	0
$112$	38.5204	3.63984
$113$	−1.30561	−0.122821	−0.0614105	−	0.998113i	$-0.519560\pi$
$113$	−1.30561	−0.122821	−0.0614105	+	0.998113i	$0.519560\pi$
$114$	0	0
$115$	−13.7396	−1.28122
$116$	19.8349	1.84163
$117$	0	0
$118$	−9.11327	−0.838944
$119$	10.0707	0.923183
$120$	0	0
$121$	−10.0162	−0.910565
$122$	−13.4597	−1.21858
$123$	0	0
$124$	22.5689	2.02674
$125$	10.1275	0.905834
$126$	0	0
$127$	16.9571	1.50470	0.752349	−	0.658765i	$-0.228921\pi$
$127$	16.9571	1.50470	0.752349	+	0.658765i	$0.228921\pi$
$128$	−21.8785	−1.93381
$129$	0	0
$130$	20.0032	1.75439
$131$	−5.95130	−0.519967	−0.259984	−	0.965613i	$-0.583717\pi$
$131$	−5.95130	−0.519967	−0.259984	+	0.965613i	$0.583717\pi$
$132$	0	0
$133$	−9.04596	−0.784384
$134$	−9.91625	−0.856633
$135$	0	0
$136$	−29.6280	−2.54058
$137$	7.68045	0.656185	0.328093	−	0.944646i	$-0.393594\pi$
$137$	7.68045	0.656185	0.328093	+	0.944646i	$0.393594\pi$
$138$	0	0
$139$	−10.7583	−0.912504	−0.456252	−	0.889851i	$-0.650808\pi$
$139$	−10.7583	−0.912504	−0.456252	+	0.889851i	$0.650808\pi$
$140$	−37.4033	−3.16116
$141$	0	0
$142$	−40.0124	−3.35777
$143$	3.06356	0.256187
$144$	0	0
$145$	−9.10693	−0.756289
$146$	3.41657	0.282757
$147$	0	0
$148$	30.5986	2.51519
$149$	−7.37112	−0.603866	−0.301933	−	0.953329i	$-0.597632\pi$
$149$	−7.37112	−0.603866	−0.301933	+	0.953329i	$0.597632\pi$
$150$	0	0
$151$	20.2959	1.65165	0.825827	−	0.563923i	$-0.190708\pi$
$151$	20.2959	1.65165	0.825827	+	0.563923i	$0.190708\pi$
$152$	26.6131	2.15861
$153$	0	0
$154$	−7.91420	−0.637745
$155$	−10.3622	−0.832311
$156$	0	0
$157$	−0.790865	−0.0631179	−0.0315590	−	0.999502i	$-0.510047\pi$
$157$	−0.790865	−0.0631179	−0.0315590	+	0.999502i	$0.510047\pi$
$158$	1.77739	0.141401
$159$	0	0
$160$	42.1479	3.33209
$161$	16.9280	1.33411
$162$	0	0
$163$	10.1377	0.794047	0.397023	−	0.917808i	$-0.370043\pi$
$163$	10.1377	0.794047	0.397023	+	0.917808i	$0.370043\pi$
$164$	1.60783	0.125551
$165$	0	0
$166$	−18.2904	−1.41961
$167$	−2.19350	−0.169738	−0.0848692	−	0.996392i	$-0.527047\pi$
$167$	−2.19350	−0.169738	−0.0848692	+	0.996392i	$0.527047\pi$
$168$	0	0
$169$	−3.45999	−0.266153
$170$	21.9962	1.68703
$171$	0	0
$172$	0.908454	0.0692690
$173$	13.6966	1.04133	0.520667	−	0.853760i	$-0.325683\pi$
$173$	13.6966	1.04133	0.520667	+	0.853760i	$0.325683\pi$
$174$	0	0
$175$	2.34775	0.177473
$176$	12.8856	0.971286
$177$	0	0
$178$	8.63887	0.647511
$179$	−11.9178	−0.890776	−0.445388	−	0.895338i	$-0.646934\pi$
$179$	−11.9178	−0.890776	−0.445388	+	0.895338i	$0.646934\pi$
$180$	0	0
$181$	10.8434	0.805981	0.402991	−	0.915204i	$-0.367971\pi$
$181$	10.8434	0.805981	0.402991	+	0.915204i	$0.367971\pi$
$182$	−24.6451	−1.82682
$183$	0	0
$184$	−49.8019	−3.67145
$185$	−14.0489	−1.03290
$186$	0	0
$187$	3.36879	0.246350
$188$	−16.4144	−1.19715
$189$	0	0
$190$	−19.7579	−1.43339
$191$	−9.87036	−0.714194	−0.357097	−	0.934067i	$-0.616233\pi$
$191$	−9.87036	−0.714194	−0.357097	+	0.934067i	$0.616233\pi$
$192$	0	0
$193$	−16.9393	−1.21932	−0.609659	−	0.792664i	$-0.708694\pi$
$193$	−16.9393	−1.21932	−0.609659	+	0.792664i	$0.708694\pi$
$194$	20.8872	1.49961
$195$	0	0
$196$	9.39181	0.670844
$197$	22.5951	1.60983	0.804917	−	0.593387i	$-0.202210\pi$
$197$	22.5951	1.60983	0.804917	+	0.593387i	$0.202210\pi$
$198$	0	0
$199$	−4.99554	−0.354124	−0.177062	−	0.984200i	$-0.556659\pi$
$199$	−4.99554	−0.354124	−0.177062	+	0.984200i	$0.556659\pi$
$200$	−6.90705	−0.488402
$201$	0	0
$202$	−32.9691	−2.31969
$203$	11.2203	0.787509
$204$	0	0
$205$	−0.738214	−0.0515591
$206$	33.2465	2.31639
$207$	0	0
$208$	40.1261	2.78224
$209$	−3.02599	−0.209312
$210$	0	0
$211$	−4.03956	−0.278095	−0.139047	−	0.990286i	$-0.544404\pi$
$211$	−4.03956	−0.278095	−0.139047	+	0.990286i	$0.544404\pi$
$212$	9.22592	0.633639
$213$	0	0
$214$	52.2385	3.57095
$215$	−0.417104	−0.0284463
$216$	0	0
$217$	12.7668	0.866669
$218$	9.92146	0.671966
$219$	0	0
$220$	−12.5119	−0.843551
$221$	10.4905	0.705669
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−51.9288	−3.46964
$225$	0	0
$226$	3.51342	0.233709
$227$	9.65211	0.640633	0.320317	−	0.947311i	$-0.396211\pi$
$227$	9.65211	0.640633	0.320317	+	0.947311i	$0.396211\pi$
$228$	0	0
$229$	−13.7334	−0.907527	−0.453764	−	0.891122i	$-0.649919\pi$
$229$	−13.7334	−0.907527	−0.453764	+	0.891122i	$0.649919\pi$
$230$	36.9736	2.43797
$231$	0	0
$232$	−33.0099	−2.16721
$233$	−25.9508	−1.70009	−0.850047	−	0.526707i	$-0.823427\pi$
$233$	−25.9508	−1.70009	−0.850047	+	0.526707i	$0.823427\pi$
$234$	0	0
$235$	7.53646	0.491625
$236$	17.7509	1.15549
$237$	0	0
$238$	−27.1006	−1.75667
$239$	1.96532	0.127126	0.0635631	−	0.997978i	$-0.479754\pi$
$239$	1.96532	0.127126	0.0635631	+	0.997978i	$0.479754\pi$
$240$	0	0
$241$	25.6552	1.65260	0.826300	−	0.563230i	$-0.190442\pi$
$241$	25.6552	1.65260	0.826300	+	0.563230i	$0.190442\pi$
$242$	26.9539	1.73266
$243$	0	0
$244$	26.2170	1.67837
$245$	−4.31212	−0.275491
$246$	0	0
$247$	−9.42303	−0.599573
$248$	−37.5599	−2.38505
$249$	0	0
$250$	−27.2534	−1.72366
$251$	11.9846	0.756462	0.378231	−	0.925711i	$-0.376532\pi$
$251$	11.9846	0.756462	0.378231	+	0.925711i	$0.376532\pi$
$252$	0	0
$253$	5.66263	0.356007
$254$	−45.6319	−2.86320
$255$	0	0
$256$	16.5830	1.03644
$257$	−27.4096	−1.70976	−0.854882	−	0.518823i	$-0.826370\pi$
$257$	−27.4096	−1.70976	−0.854882	+	0.518823i	$0.826370\pi$
$258$	0	0
$259$	17.3091	1.07554
$260$	−38.9624	−2.41635
$261$	0	0
$262$	16.0151	0.989416
$263$	−16.0236	−0.988060	−0.494030	−	0.869445i	$-0.664477\pi$
$263$	−16.0236	−0.988060	−0.494030	+	0.869445i	$0.664477\pi$
$264$	0	0
$265$	−4.23596	−0.260213
$266$	24.3429	1.49256
$267$	0	0
$268$	19.3150	1.17985
$269$	−13.1263	−0.800327	−0.400163	−	0.916444i	$-0.631047\pi$
$269$	−13.1263	−0.800327	−0.400163	+	0.916444i	$0.631047\pi$
$270$	0	0
$271$	25.1912	1.53026	0.765129	−	0.643877i	$-0.222675\pi$
$271$	25.1912	1.53026	0.765129	+	0.643877i	$0.222675\pi$
$272$	44.1240	2.67541
$273$	0	0
$274$	−20.6683	−1.24862
$275$	0.785352	0.0473585
$276$	0	0
$277$	22.4203	1.34710	0.673551	−	0.739140i	$-0.264768\pi$
$277$	22.4203	1.34710	0.673551	+	0.739140i	$0.264768\pi$
$278$	28.9508	1.73635
$279$	0	0
$280$	62.2478	3.72002
$281$	−9.25912	−0.552352	−0.276176	−	0.961107i	$-0.589067\pi$
$281$	−9.25912	−0.552352	−0.276176	+	0.961107i	$0.589067\pi$
$282$	0	0
$283$	6.05899	0.360170	0.180085	−	0.983651i	$-0.442363\pi$
$283$	6.05899	0.360170	0.180085	+	0.983651i	$0.442363\pi$
$284$	77.9368	4.62470
$285$	0	0
$286$	−8.24410	−0.487484
$287$	0.909523	0.0536875
$288$	0	0
$289$	−5.46425	−0.321427
$290$	24.5070	1.43910
$291$	0	0
$292$	−6.65484	−0.389445
$293$	1.67003	0.0975640	0.0487820	−	0.998809i	$-0.484466\pi$
$293$	1.67003	0.0975640	0.0487820	+	0.998809i	$0.484466\pi$
$294$	0	0
$295$	−8.15010	−0.474517
$296$	−50.9233	−2.95985
$297$	0	0
$298$	19.8359	1.14906
$299$	17.6336	1.01978
$300$	0	0
$301$	0.513897	0.0296205
$302$	−54.6167	−3.14284
$303$	0	0
$304$	−39.6341	−2.27317
$305$	−12.0372	−0.689246
$306$	0	0
$307$	−15.7718	−0.900141	−0.450071	−	0.892993i	$-0.648601\pi$
$307$	−15.7718	−0.900141	−0.450071	+	0.892993i	$0.648601\pi$
$308$	15.4154	0.878374
$309$	0	0
$310$	27.8849	1.58376
$311$	−12.9739	−0.735683	−0.367842	−	0.929888i	$-0.619903\pi$
$311$	−12.9739	−0.735683	−0.367842	+	0.929888i	$0.619903\pi$
$312$	0	0
$313$	26.0654	1.47330	0.736651	−	0.676273i	$-0.236406\pi$
$313$	26.0654	1.47330	0.736651	+	0.676273i	$0.236406\pi$
$314$	2.12824	0.120103
$315$	0	0
$316$	−3.46202	−0.194754
$317$	−15.1994	−0.853684	−0.426842	−	0.904326i	$-0.640374\pi$
$317$	−15.1994	−0.853684	−0.426842	+	0.904326i	$0.640374\pi$
$318$	0	0
$319$	3.75333	0.210146
$320$	−50.8910	−2.84489
$321$	0	0
$322$	−45.5537	−2.53861
$323$	−10.3619	−0.576551
$324$	0	0
$325$	2.44561	0.135658
$326$	−27.2808	−1.51095
$327$	0	0
$328$	−2.67581	−0.147747
$329$	−9.28537	−0.511919
$330$	0	0
$331$	−24.4417	−1.34344	−0.671719	−	0.740806i	$-0.734444\pi$
$331$	−24.4417	−1.34344	−0.671719	+	0.740806i	$0.734444\pi$
$332$	35.6263	1.95525
$333$	0	0
$334$	5.90277	0.322985
$335$	−8.86821	−0.484522
$336$	0	0
$337$	−36.4312	−1.98453	−0.992267	−	0.124122i	$-0.960388\pi$
$337$	−36.4312	−1.98453	−0.992267	+	0.124122i	$0.960388\pi$
$338$	9.31091	0.506447
$339$	0	0
$340$	−42.8445	−2.32357
$341$	4.27067	0.231270
$342$	0	0
$343$	−15.4429	−0.833836
$344$	−1.51188	−0.0815150
$345$	0	0
$346$	−36.8579	−1.98149
$347$	−4.10675	−0.220462	−0.110231	−	0.993906i	$-0.535159\pi$
$347$	−4.10675	−0.220462	−0.110231	+	0.993906i	$0.535159\pi$
$348$	0	0
$349$	−9.47028	−0.506932	−0.253466	−	0.967344i	$-0.581571\pi$
$349$	−9.47028	−0.506932	−0.253466	+	0.967344i	$0.581571\pi$
$350$	−6.31785	−0.337703
$351$	0	0
$352$	−17.3708	−0.925869
$353$	−24.8693	−1.32366	−0.661831	−	0.749653i	$-0.730220\pi$
$353$	−24.8693	−1.32366	−0.661831	+	0.749653i	$0.730220\pi$
$354$	0	0
$355$	−35.7836	−1.89920
$356$	−16.8269	−0.891824
$357$	0	0
$358$	32.0710	1.69501
$359$	1.82528	0.0963348	0.0481674	−	0.998839i	$-0.484662\pi$
$359$	1.82528	0.0963348	0.0481674	+	0.998839i	$0.484662\pi$
$360$	0	0
$361$	−9.69252	−0.510132
$362$	−29.1798	−1.53365
$363$	0	0
$364$	48.0041	2.51610
$365$	3.05548	0.159931
$366$	0	0
$367$	8.60522	0.449189	0.224594	−	0.974452i	$-0.427894\pi$
$367$	8.60522	0.449189	0.224594	+	0.974452i	$0.427894\pi$
$368$	74.1685	3.86630
$369$	0	0
$370$	37.8061	1.96544
$371$	5.21895	0.270954
$372$	0	0
$373$	−16.2180	−0.839736	−0.419868	−	0.907585i	$-0.637924\pi$
$373$	−16.2180	−0.839736	−0.419868	+	0.907585i	$0.637924\pi$
$374$	−9.06550	−0.468766
$375$	0	0
$376$	27.3174	1.40879
$377$	11.6880	0.601962
$378$	0	0
$379$	24.1071	1.23830	0.619149	−	0.785274i	$-0.287478\pi$
$379$	24.1071	1.23830	0.619149	+	0.785274i	$0.287478\pi$
$380$	38.4847	1.97422
$381$	0	0
$382$	26.5614	1.35900
$383$	−21.6702	−1.10729	−0.553647	−	0.832752i	$-0.686764\pi$
$383$	−21.6702	−1.10729	−0.553647	+	0.832752i	$0.686764\pi$
$384$	0	0
$385$	−7.07777	−0.360716
$386$	45.5841	2.32017
$387$	0	0
$388$	−40.6843	−2.06543
$389$	29.2097	1.48099	0.740495	−	0.672062i	$-0.234591\pi$
$389$	29.2097	1.48099	0.740495	+	0.672062i	$0.234591\pi$
$390$	0	0
$391$	19.3905	0.980622
$392$	−15.6302	−0.789442
$393$	0	0
$394$	−60.8040	−3.06326
$395$	1.58954	0.0799784
$396$	0	0
$397$	24.9004	1.24971	0.624857	−	0.780739i	$-0.285157\pi$
$397$	24.9004	1.24971	0.624857	+	0.780739i	$0.285157\pi$
$398$	13.4431	0.673842
$399$	0	0
$400$	10.2865	0.514323
$401$	−19.7276	−0.985150	−0.492575	−	0.870270i	$-0.663944\pi$
$401$	−19.7276	−0.985150	−0.492575	+	0.870270i	$0.663944\pi$
$402$	0	0
$403$	13.2990	0.662471
$404$	64.2176	3.19494
$405$	0	0
$406$	−30.1941	−1.49851
$407$	5.79013	0.287006
$408$	0	0
$409$	29.7696	1.47201	0.736006	−	0.676975i	$-0.236709\pi$
$409$	29.7696	1.47201	0.736006	+	0.676975i	$0.236709\pi$
$410$	1.98655	0.0981088
$411$	0	0
$412$	−64.7580	−3.19040
$413$	10.0414	0.494105
$414$	0	0
$415$	−16.3573	−0.802950
$416$	−54.0934	−2.65215
$417$	0	0
$418$	8.14301	0.398288
$419$	23.9872	1.17185	0.585925	−	0.810365i	$-0.300731\pi$
$419$	23.9872	1.17185	0.585925	+	0.810365i	$0.300731\pi$
$420$	0	0
$421$	15.6723	0.763820	0.381910	−	0.924200i	$-0.375266\pi$
$421$	15.6723	0.763820	0.381910	+	0.924200i	$0.375266\pi$
$422$	10.8706	0.529171
$423$	0	0
$424$	−15.3541	−0.745660
$425$	2.68928	0.130449
$426$	0	0
$427$	14.8305	0.717699
$428$	−101.751	−4.91831
$429$	0	0
$430$	1.12244	0.0541287
$431$	26.2181	1.26288	0.631441	−	0.775424i	$-0.282464\pi$
$431$	26.2181	1.26288	0.631441	+	0.775424i	$0.282464\pi$
$432$	0	0
$433$	20.6571	0.992718	0.496359	−	0.868117i	$-0.334670\pi$
$433$	20.6571	0.992718	0.496359	+	0.868117i	$0.334670\pi$
$434$	−34.3559	−1.64913
$435$	0	0
$436$	−19.3252	−0.925507
$437$	−17.4174	−0.833187
$438$	0	0
$439$	−26.0732	−1.24441	−0.622203	−	0.782856i	$-0.713762\pi$
$439$	−26.0732	−1.24441	−0.622203	+	0.782856i	$0.713762\pi$
$440$	20.8227	0.992683
$441$	0	0
$442$	−28.2303	−1.34278
$443$	11.8427	0.562662	0.281331	−	0.959611i	$-0.409224\pi$
$443$	11.8427	0.562662	0.281331	+	0.959611i	$0.409224\pi$
$444$	0	0
$445$	7.72584	0.366240
$446$	−2.69102	−0.127424
$447$	0	0
$448$	62.7008	2.96233
$449$	33.1513	1.56451	0.782254	−	0.622959i	$-0.214070\pi$
$449$	33.1513	1.56451	0.782254	+	0.622959i	$0.214070\pi$
$450$	0	0
$451$	0.304247	0.0143264
$452$	−6.84348	−0.321890
$453$	0	0
$454$	−25.9741	−1.21902
$455$	−22.0404	−1.03327
$456$	0	0
$457$	−32.4683	−1.51880	−0.759401	−	0.650623i	$-0.774508\pi$
$457$	−32.4683	−1.51880	−0.759401	+	0.650623i	$0.774508\pi$
$458$	36.9569	1.72688
$459$	0	0
$460$	−72.0176	−3.35784
$461$	13.5292	0.630120	0.315060	−	0.949072i	$-0.397975\pi$
$461$	13.5292	0.630120	0.315060	+	0.949072i	$0.397975\pi$
$462$	0	0
$463$	25.4931	1.18476	0.592382	−	0.805658i	$-0.298188\pi$
$463$	25.4931	1.18476	0.592382	+	0.805658i	$0.298188\pi$
$464$	49.1607	2.28223
$465$	0	0
$466$	69.8343	3.23501
$467$	1.46956	0.0680030	0.0340015	−	0.999422i	$-0.489175\pi$
$467$	1.46956	0.0680030	0.0340015	+	0.999422i	$0.489175\pi$
$468$	0	0
$469$	10.9262	0.504524
$470$	−20.2808	−0.935484
$471$	0	0
$472$	−29.5417	−1.35977
$473$	0.171905	0.00790421
$474$	0	0
$475$	−2.41563	−0.110836
$476$	52.7869	2.41948
$477$	0	0
$478$	−5.28874	−0.241901
$479$	−16.0473	−0.733219	−0.366610	−	0.930375i	$-0.619482\pi$
$479$	−16.0473	−0.733219	−0.366610	+	0.930375i	$0.619482\pi$
$480$	0	0
$481$	18.0307	0.822127
$482$	−69.0389	−3.14464
$483$	0	0
$484$	−52.5011	−2.38641
$485$	18.6796	0.848199
$486$	0	0
$487$	−23.3411	−1.05769	−0.528843	−	0.848720i	$-0.677374\pi$
$487$	−23.3411	−1.05769	−0.528843	+	0.848720i	$0.677374\pi$
$488$	−43.6312	−1.97509
$489$	0	0
$490$	11.6040	0.524216
$491$	−34.0225	−1.53541	−0.767706	−	0.640802i	$-0.778602\pi$
$491$	−34.0225	−1.53541	−0.767706	+	0.640802i	$0.778602\pi$
$492$	0	0
$493$	12.8525	0.578848
$494$	25.3576	1.14089
$495$	0	0
$496$	55.9368	2.51163
$497$	44.0875	1.97760
$498$	0	0
$499$	4.97713	0.222807	0.111403	−	0.993775i	$-0.464465\pi$
$499$	4.97713	0.222807	0.111403	+	0.993775i	$0.464465\pi$
$500$	53.0846	2.37402
$501$	0	0
$502$	−32.2509	−1.43943
$503$	−13.8951	−0.619551	−0.309776	−	0.950810i	$-0.600254\pi$
$503$	−13.8951	−0.619551	−0.309776	+	0.950810i	$0.600254\pi$
$504$	0	0
$505$	−29.4846	−1.31205
$506$	−15.2383	−0.677424
$507$	0	0
$508$	88.8825	3.94352
$509$	18.0045	0.798037	0.399019	−	0.916943i	$-0.369351\pi$
$509$	18.0045	0.798037	0.399019	+	0.916943i	$0.369351\pi$
$510$	0	0
$511$	−3.76453	−0.166533
$512$	−0.868258	−0.0383720
$513$	0	0
$514$	73.7599	3.25341
$515$	29.7327	1.31018
$516$	0	0
$517$	−3.10608	−0.136605
$518$	−46.5793	−2.04658
$519$	0	0
$520$	64.8426	2.84353
$521$	36.0883	1.58106	0.790528	−	0.612426i	$-0.209806\pi$
$521$	36.0883	1.58106	0.790528	+	0.612426i	$0.209806\pi$
$522$	0	0
$523$	−10.2584	−0.448568	−0.224284	−	0.974524i	$-0.572004\pi$
$523$	−10.2584	−0.448568	−0.224284	+	0.974524i	$0.572004\pi$
$524$	−31.1944	−1.36273
$525$	0	0
$526$	43.1200	1.88012
$527$	14.6241	0.637034
$528$	0	0
$529$	9.59375	0.417119
$530$	11.3991	0.495144
$531$	0	0
$532$	−47.4154	−2.05572
$533$	0.947436	0.0410380
$534$	0	0
$535$	46.7175	2.01977
$536$	−32.1446	−1.38844
$537$	0	0
$538$	35.3233	1.52289
$539$	1.77720	0.0765493
$540$	0	0
$541$	−21.9925	−0.945531	−0.472766	−	0.881188i	$-0.656744\pi$
$541$	−21.9925	−0.945531	−0.472766	+	0.881188i	$0.656744\pi$
$542$	−67.7902	−2.91184
$543$	0	0
$544$	−59.4829	−2.55031
$545$	8.87288	0.380072
$546$	0	0
$547$	−9.54780	−0.408234	−0.204117	−	0.978946i	$-0.565432\pi$
$547$	−9.54780	−0.408234	−0.204117	+	0.978946i	$0.565432\pi$
$548$	40.2580	1.71974
$549$	0	0
$550$	−2.11340	−0.0901158
$551$	−11.5447	−0.491820
$552$	0	0
$553$	−1.95841	−0.0832799
$554$	−60.3335	−2.56332
$555$	0	0
$556$	−56.3907	−2.39150
$557$	−38.8306	−1.64530	−0.822652	−	0.568545i	$-0.807506\pi$
$557$	−38.8306	−1.64530	−0.822652	+	0.568545i	$0.807506\pi$
$558$	0	0
$559$	0.535319	0.0226416
$560$	−92.7038	−3.91745
$561$	0	0
$562$	24.9165	1.05104
$563$	15.5898	0.657032	0.328516	−	0.944498i	$-0.393451\pi$
$563$	15.5898	0.657032	0.328516	+	0.944498i	$0.393451\pi$
$564$	0	0
$565$	3.14209	0.132189
$566$	−16.3049	−0.685346
$567$	0	0
$568$	−129.705	−5.44230
$569$	40.5343	1.69929	0.849644	−	0.527357i	$-0.176817\pi$
$569$	40.5343	1.69929	0.849644	+	0.527357i	$0.176817\pi$
$570$	0	0
$571$	−24.0528	−1.00658	−0.503289	−	0.864118i	$-0.667877\pi$
$571$	−24.0528	−1.00658	−0.503289	+	0.864118i	$0.667877\pi$
$572$	16.0580	0.671418
$573$	0	0
$574$	−2.44755	−0.102159
$575$	4.52044	0.188515
$576$	0	0
$577$	18.5393	0.771801	0.385900	−	0.922540i	$-0.373891\pi$
$577$	18.5393	0.771801	0.385900	+	0.922540i	$0.373891\pi$
$578$	14.7044	0.611624
$579$	0	0
$580$	−47.7350	−1.98209
$581$	20.1532	0.836096
$582$	0	0
$583$	1.74581	0.0723039
$584$	11.0752	0.458295
$585$	0	0
$586$	−4.49409	−0.185649
$587$	−29.7740	−1.22890	−0.614452	−	0.788954i	$-0.710623\pi$
$587$	−29.7740	−1.22890	−0.614452	+	0.788954i	$0.710623\pi$
$588$	0	0
$589$	−13.1359	−0.541257
$590$	21.9321	0.902931
$591$	0	0
$592$	75.8385	3.11694
$593$	22.7875	0.935770	0.467885	−	0.883789i	$-0.345016\pi$
$593$	22.7875	0.935770	0.467885	+	0.883789i	$0.345016\pi$
$594$	0	0
$595$	−24.2364	−0.993595
$596$	−38.6366	−1.58262
$597$	0	0
$598$	−47.4525	−1.94048
$599$	41.0061	1.67546	0.837732	−	0.546081i	$-0.183881\pi$
$599$	41.0061	1.67546	0.837732	+	0.546081i	$0.183881\pi$
$600$	0	0
$601$	21.9352	0.894756	0.447378	−	0.894345i	$-0.352358\pi$
$601$	21.9352	0.894756	0.447378	+	0.894345i	$0.352358\pi$
$602$	−1.38291	−0.0563632
$603$	0	0
$604$	106.383	4.32867
$605$	24.1052	0.980014
$606$	0	0
$607$	25.1168	1.01946	0.509729	−	0.860335i	$-0.329746\pi$
$607$	25.1168	1.01946	0.509729	+	0.860335i	$0.329746\pi$
$608$	53.4301	2.16688
$609$	0	0
$610$	32.3923	1.31153
$611$	−9.67243	−0.391305
$612$	0	0
$613$	−12.6473	−0.510821	−0.255411	−	0.966833i	$-0.582211\pi$
$613$	−12.6473	−0.510821	−0.255411	+	0.966833i	$0.582211\pi$
$614$	42.4422	1.71283
$615$	0	0
$616$	−25.6548	−1.03366
$617$	10.9184	0.439559	0.219780	−	0.975550i	$-0.429466\pi$
$617$	10.9184	0.439559	0.219780	+	0.975550i	$0.429466\pi$
$618$	0	0
$619$	8.78609	0.353143	0.176571	−	0.984288i	$-0.443499\pi$
$619$	8.78609	0.353143	0.176571	+	0.984288i	$0.443499\pi$
$620$	−54.3146	−2.18133
$621$	0	0
$622$	34.9131	1.39989
$623$	−9.51870	−0.381359
$624$	0	0
$625$	−28.3320	−1.13328
$626$	−70.1426	−2.80346
$627$	0	0
$628$	−4.14541	−0.165420
$629$	19.8271	0.790560
$630$	0	0
$631$	22.5745	0.898676	0.449338	−	0.893362i	$-0.351660\pi$
$631$	22.5745	0.898676	0.449338	+	0.893362i	$0.351660\pi$
$632$	5.76161	0.229184
$633$	0	0
$634$	40.9020	1.62443
$635$	−40.8092	−1.61946
$636$	0	0
$637$	5.53425	0.219275
$638$	−10.1003	−0.399875
$639$	0	0
$640$	52.6532	2.08130
$641$	35.5211	1.40300	0.701500	−	0.712669i	$-0.252514\pi$
$641$	35.5211	1.40300	0.701500	+	0.712669i	$0.252514\pi$
$642$	0	0
$643$	20.6377	0.813871	0.406935	−	0.913457i	$-0.366597\pi$
$643$	20.6377	0.813871	0.406935	+	0.913457i	$0.366597\pi$
$644$	88.7300	3.49645
$645$	0	0
$646$	27.8841	1.09709
$647$	−8.21380	−0.322918	−0.161459	−	0.986879i	$-0.551620\pi$
$647$	−8.21380	−0.322918	−0.161459	+	0.986879i	$0.551620\pi$
$648$	0	0
$649$	3.35898	0.131852
$650$	−6.58121	−0.258136
$651$	0	0
$652$	53.1380	2.08104
$653$	12.9658	0.507393	0.253696	−	0.967284i	$-0.418354\pi$
$653$	12.9658	0.507393	0.253696	+	0.967284i	$0.418354\pi$
$654$	0	0
$655$	14.3225	0.559626
$656$	3.98500	0.155588
$657$	0	0
$658$	24.9872	0.974101
$659$	36.7840	1.43290	0.716451	−	0.697637i	$-0.245765\pi$
$659$	36.7840	1.43290	0.716451	+	0.697637i	$0.245765\pi$
$660$	0	0
$661$	45.1219	1.75504	0.877520	−	0.479541i	$-0.159197\pi$
$661$	45.1219	1.75504	0.877520	+	0.479541i	$0.159197\pi$
$662$	65.7733	2.55635
$663$	0	0
$664$	−59.2905	−2.30092
$665$	21.7701	0.844210
$666$	0	0
$667$	21.6039	0.836507
$668$	−11.4975	−0.444851
$669$	0	0
$670$	23.8646	0.921969
$671$	4.96100	0.191517
$672$	0	0
$673$	−7.28742	−0.280909	−0.140455	−	0.990087i	$-0.544856\pi$
$673$	−7.28742	−0.280909	−0.140455	+	0.990087i	$0.544856\pi$
$674$	98.0373	3.77625
$675$	0	0
$676$	−18.1359	−0.697535
$677$	7.06282	0.271446	0.135723	−	0.990747i	$-0.456664\pi$
$677$	7.06282	0.271446	0.135723	+	0.990747i	$0.456664\pi$
$678$	0	0
$679$	−23.0144	−0.883213
$680$	71.3031	2.73435
$681$	0	0
$682$	−11.4925	−0.440070
$683$	−5.60040	−0.214293	−0.107147	−	0.994243i	$-0.534171\pi$
$683$	−5.60040	−0.214293	−0.107147	+	0.994243i	$0.534171\pi$
$684$	0	0
$685$	−18.4839	−0.706233
$686$	41.5571	1.58666
$687$	0	0
$688$	2.25159	0.0858412
$689$	5.43650	0.207114
$690$	0	0
$691$	23.3133	0.886880	0.443440	−	0.896304i	$-0.353758\pi$
$691$	23.3133	0.886880	0.443440	+	0.896304i	$0.353758\pi$
$692$	71.7924	2.72914
$693$	0	0
$694$	11.0514	0.419504
$695$	25.8910	0.982102
$696$	0	0
$697$	1.04183	0.0394623
$698$	25.4848	0.964612
$699$	0	0
$700$	12.3060	0.465123
$701$	31.2166	1.17903	0.589517	−	0.807756i	$-0.299318\pi$
$701$	31.2166	1.17903	0.589517	+	0.807756i	$0.299318\pi$
$702$	0	0
$703$	−17.8096	−0.671700
$704$	20.9742	0.790496
$705$	0	0
$706$	66.9240	2.51872
$707$	36.3268	1.36621
$708$	0	0
$709$	48.7849	1.83215	0.916077	−	0.401003i	$-0.131338\pi$
$709$	48.7849	1.83215	0.916077	+	0.401003i	$0.131338\pi$
$710$	96.2946	3.61387
$711$	0	0
$712$	28.0039	1.04949
$713$	24.5817	0.920592
$714$	0	0
$715$	−7.37280	−0.275727
$716$	−62.4684	−2.33455
$717$	0	0
$718$	−4.91188	−0.183310
$719$	−6.50222	−0.242492	−0.121246	−	0.992622i	$-0.538689\pi$
$719$	−6.50222	−0.242492	−0.121246	+	0.992622i	$0.538689\pi$
$720$	0	0
$721$	−36.6325	−1.36427
$722$	26.0828	0.970701
$723$	0	0
$724$	56.8367	2.11232
$725$	2.99626	0.111278
$726$	0	0
$727$	−16.5706	−0.614571	−0.307285	−	0.951617i	$-0.599421\pi$
$727$	−16.5706	−0.614571	−0.307285	+	0.951617i	$0.599421\pi$
$728$	−79.8899	−2.96092
$729$	0	0
$730$	−8.22237	−0.304323
$731$	0.588655	0.0217722
$732$	0	0
$733$	10.5189	0.388525	0.194262	−	0.980950i	$-0.437769\pi$
$733$	10.5189	0.388525	0.194262	+	0.980950i	$0.437769\pi$
$734$	−23.1569	−0.854735
$735$	0	0
$736$	−99.9854	−3.68551
$737$	3.65494	0.134632
$738$	0	0
$739$	50.1496	1.84478	0.922391	−	0.386258i	$-0.126233\pi$
$739$	50.1496	1.84478	0.922391	+	0.386258i	$0.126233\pi$
$740$	−73.6392	−2.70703
$741$	0	0
$742$	−14.0443	−0.515583
$743$	19.1896	0.703999	0.352000	−	0.936000i	$-0.385502\pi$
$743$	19.1896	0.703999	0.352000	+	0.936000i	$0.385502\pi$
$744$	0	0
$745$	17.7394	0.649923
$746$	43.6431	1.59789
$747$	0	0
$748$	17.6579	0.645637
$749$	−57.5588	−2.10315
$750$	0	0
$751$	−6.55763	−0.239291	−0.119646	−	0.992817i	$-0.538176\pi$
$751$	−6.55763	−0.239291	−0.119646	+	0.992817i	$0.538176\pi$
$752$	−40.6830	−1.48356
$753$	0	0
$754$	−31.4527	−1.14544
$755$	−48.8444	−1.77763
$756$	0	0
$757$	6.36183	0.231225	0.115612	−	0.993294i	$-0.463117\pi$
$757$	6.36183	0.231225	0.115612	+	0.993294i	$0.463117\pi$
$758$	−64.8728	−2.35629
$759$	0	0
$760$	−64.0475	−2.32325
$761$	−5.56318	−0.201665	−0.100833	−	0.994903i	$-0.532151\pi$
$761$	−5.56318	−0.201665	−0.100833	+	0.994903i	$0.532151\pi$
$762$	0	0
$763$	−10.9319	−0.395762
$764$	−51.7366	−1.87177
$765$	0	0
$766$	58.3150	2.10700
$767$	10.4600	0.377688
$768$	0	0
$769$	28.1817	1.01626	0.508128	−	0.861281i	$-0.330338\pi$
$769$	28.1817	1.01626	0.508128	+	0.861281i	$0.330338\pi$
$770$	19.0464	0.686386
$771$	0	0
$772$	−88.7893	−3.19560
$773$	27.9505	1.00531	0.502654	−	0.864488i	$-0.332357\pi$
$773$	27.9505	1.00531	0.502654	+	0.864488i	$0.332357\pi$
$774$	0	0
$775$	3.40925	0.122464
$776$	67.7082	2.43058
$777$	0	0
$778$	−78.6040	−2.81809
$779$	−0.935818	−0.0335292
$780$	0	0
$781$	14.7478	0.527719
$782$	−52.1804	−1.86597
$783$	0	0
$784$	23.2775	0.831340
$785$	1.90331	0.0679320
$786$	0	0
$787$	15.9779	0.569552	0.284776	−	0.958594i	$-0.408081\pi$
$787$	15.9779	0.569552	0.284776	+	0.958594i	$0.408081\pi$
$788$	118.435	4.21907
$789$	0	0
$790$	−4.27749	−0.152186
$791$	−3.87124	−0.137646
$792$	0	0
$793$	15.4487	0.548600
$794$	−67.0076	−2.37801
$795$	0	0
$796$	−26.1847	−0.928091
$797$	−15.8466	−0.561315	−0.280657	−	0.959808i	$-0.590552\pi$
$797$	−15.8466	−0.561315	−0.280657	+	0.959808i	$0.590552\pi$
$798$	0	0
$799$	−10.6361	−0.376279
$800$	−13.8670	−0.490273
$801$	0	0
$802$	53.0875	1.87458
$803$	−1.25928	−0.0444392
$804$	0	0
$805$	−40.7392	−1.43587
$806$	−35.7880	−1.26058
$807$	0	0
$808$	−106.873	−3.75978
$809$	29.5904	1.04034	0.520171	−	0.854062i	$-0.325868\pi$
$809$	29.5904	1.04034	0.520171	+	0.854062i	$0.325868\pi$
$810$	0	0
$811$	−11.9521	−0.419695	−0.209848	−	0.977734i	$-0.567297\pi$
$811$	−11.9521	−0.419695	−0.209848	+	0.977734i	$0.567297\pi$
$812$	58.8124	2.06391
$813$	0	0
$814$	−15.5814	−0.546127
$815$	−24.3976	−0.854610
$816$	0	0
$817$	−0.528754	−0.0184988
$818$	−80.1107	−2.80101
$819$	0	0
$820$	−3.86943	−0.135126
$821$	30.6320	1.06906	0.534531	−	0.845149i	$-0.320488\pi$
$821$	30.6320	1.06906	0.534531	+	0.845149i	$0.320488\pi$
$822$	0	0
$823$	−2.60758	−0.0908944	−0.0454472	−	0.998967i	$-0.514471\pi$
$823$	−2.60758	−0.0908944	−0.0454472	+	0.998967i	$0.514471\pi$
$824$	107.772	3.75443
$825$	0	0
$826$	−27.0217	−0.940205
$827$	−24.4375	−0.849776	−0.424888	−	0.905246i	$-0.639686\pi$
$827$	−24.4375	−0.849776	−0.424888	+	0.905246i	$0.639686\pi$
$828$	0	0
$829$	11.9529	0.415142	0.207571	−	0.978220i	$-0.433444\pi$
$829$	11.9529	0.415142	0.207571	+	0.978220i	$0.433444\pi$
$830$	44.0180	1.52789
$831$	0	0
$832$	65.3144	2.26437
$833$	6.08565	0.210855
$834$	0	0
$835$	5.27892	0.182684
$836$	−15.8611	−0.548567
$837$	0	0
$838$	−64.5501	−2.22985
$839$	43.7774	1.51136	0.755681	−	0.654939i	$-0.227306\pi$
$839$	43.7774	1.51136	0.755681	+	0.654939i	$0.227306\pi$
$840$	0	0
$841$	−14.6804	−0.506221
$842$	−42.1745	−1.45343
$843$	0	0
$844$	−21.1738	−0.728833
$845$	8.32685	0.286452
$846$	0	0
$847$	−29.6990	−1.02047
$848$	22.8664	0.785234
$849$	0	0
$850$	−7.23692	−0.248224
$851$	33.3276	1.14246
$852$	0	0
$853$	−13.6152	−0.466175	−0.233088	−	0.972456i	$-0.574883\pi$
$853$	−13.6152	−0.466175	−0.233088	+	0.972456i	$0.574883\pi$
$854$	−39.9093	−1.36567
$855$	0	0
$856$	169.337	5.78782
$857$	−12.2815	−0.419529	−0.209764	−	0.977752i	$-0.567270\pi$
$857$	−12.2815	−0.419529	−0.209764	+	0.977752i	$0.567270\pi$
$858$	0	0
$859$	34.9788	1.19346	0.596731	−	0.802441i	$-0.296466\pi$
$859$	34.9788	1.19346	0.596731	+	0.802441i	$0.296466\pi$
$860$	−2.18630	−0.0745522
$861$	0	0
$862$	−70.5536	−2.40307
$863$	−28.2944	−0.963151	−0.481575	−	0.876405i	$-0.659935\pi$
$863$	−28.2944	−0.963151	−0.481575	+	0.876405i	$0.659935\pi$
$864$	0	0
$865$	−32.9625	−1.12076
$866$	−55.5888	−1.88899
$867$	0	0
$868$	66.9188	2.27137
$869$	−0.655112	−0.0222232
$870$	0	0
$871$	11.3816	0.385651
$872$	32.1615	1.08913
$873$	0	0
$874$	46.8707	1.58542
$875$	30.0291	1.01517
$876$	0	0
$877$	32.4285	1.09503	0.547516	−	0.836795i	$-0.315574\pi$
$877$	32.4285	1.09503	0.547516	+	0.836795i	$0.315574\pi$
$878$	70.1636	2.36791
$879$	0	0
$880$	−31.0106	−1.04537
$881$	−17.7623	−0.598426	−0.299213	−	0.954186i	$-0.596724\pi$
$881$	−17.7623	−0.598426	−0.299213	+	0.954186i	$0.596724\pi$
$882$	0	0
$883$	−49.9118	−1.67967	−0.839834	−	0.542844i	$-0.817348\pi$
$883$	−49.9118	−1.67967	−0.839834	+	0.542844i	$0.817348\pi$
$884$	54.9873	1.84942
$885$	0	0
$886$	−31.8689	−1.07066
$887$	−27.5344	−0.924513	−0.462257	−	0.886746i	$-0.652960\pi$
$887$	−27.5344	−0.924513	−0.462257	+	0.886746i	$0.652960\pi$
$888$	0	0
$889$	50.2793	1.68632
$890$	−20.7904	−0.696897
$891$	0	0
$892$	5.24161	0.175502
$893$	9.55382	0.319706
$894$	0	0
$895$	28.6815	0.958717
$896$	−64.8718	−2.16722
$897$	0	0
$898$	−89.2111	−2.97701
$899$	16.2933	0.543413
$900$	0	0
$901$	5.97816	0.199161
$902$	−0.818737	−0.0272610
$903$	0	0
$904$	11.3891	0.378797
$905$	−26.0958	−0.867454
$906$	0	0
$907$	−50.3462	−1.67172	−0.835859	−	0.548945i	$-0.815030\pi$
$907$	−50.3462	−1.67172	−0.835859	+	0.548945i	$0.815030\pi$
$908$	50.5927	1.67898
$909$	0	0
$910$	59.3113	1.96615
$911$	−51.4849	−1.70577	−0.852885	−	0.522098i	$-0.825149\pi$
$911$	−51.4849	−1.70577	−0.852885	+	0.522098i	$0.825149\pi$
$912$	0	0
$913$	6.74151	0.223111
$914$	87.3729	2.89004
$915$	0	0
$916$	−71.9851	−2.37845
$917$	−17.6462	−0.582727
$918$	0	0
$919$	29.2289	0.964172	0.482086	−	0.876124i	$-0.339879\pi$
$919$	29.2289	0.964172	0.482086	+	0.876124i	$0.339879\pi$
$920$	119.854	3.95147
$921$	0	0
$922$	−36.4075	−1.19902
$923$	45.9253	1.51165
$924$	0	0
$925$	4.62222	0.151978
$926$	−68.6025	−2.25442
$927$	0	0
$928$	−66.2727	−2.17551
$929$	−9.00567	−0.295466	−0.147733	−	0.989027i	$-0.547198\pi$
$929$	−9.00567	−0.295466	−0.147733	+	0.989027i	$0.547198\pi$
$930$	0	0
$931$	−5.46639	−0.179154
$932$	−136.024	−4.45562
$933$	0	0
$934$	−3.95461	−0.129399
$935$	−8.10738	−0.265140
$936$	0	0
$937$	−16.9246	−0.552902	−0.276451	−	0.961028i	$-0.589158\pi$
$937$	−16.9246	−0.552902	−0.276451	+	0.961028i	$0.589158\pi$
$938$	−29.4026	−0.960029
$939$	0	0
$940$	39.5032	1.28845
$941$	3.83374	0.124976	0.0624881	−	0.998046i	$-0.480096\pi$
$941$	3.83374	0.124976	0.0624881	+	0.998046i	$0.480096\pi$
$942$	0	0
$943$	1.75123	0.0570278
$944$	43.9955	1.43193
$945$	0	0
$946$	−0.462601	−0.0150405
$947$	41.2148	1.33930	0.669651	−	0.742676i	$-0.266444\pi$
$947$	41.2148	1.33930	0.669651	+	0.742676i	$0.266444\pi$
$948$	0	0
$949$	−3.92145	−0.127296
$950$	6.50051	0.210904
$951$	0	0
$952$	−87.8497	−2.84723
$953$	30.0633	0.973845	0.486922	−	0.873445i	$-0.338119\pi$
$953$	30.0633	0.973845	0.486922	+	0.873445i	$0.338119\pi$
$954$	0	0
$955$	23.7542	0.768666
$956$	10.3015	0.333173
$957$	0	0
$958$	43.1836	1.39520
$959$	22.7733	0.735387
$960$	0	0
$961$	−12.4609	−0.401963
$962$	−48.5210	−1.56438
$963$	0	0
$964$	134.475	4.33115
$965$	40.7664	1.31232
$966$	0	0
$967$	38.9783	1.25346	0.626729	−	0.779237i	$-0.284393\pi$
$967$	38.9783	1.25346	0.626729	+	0.779237i	$0.284393\pi$
$968$	87.3741	2.80831
$969$	0	0
$970$	−50.2674	−1.61399
$971$	32.7177	1.04996	0.524981	−	0.851114i	$-0.324073\pi$
$971$	32.7177	1.04996	0.524981	+	0.851114i	$0.324073\pi$
$972$	0	0
$973$	−31.8993	−1.02264
$974$	62.8115	2.01261
$975$	0	0
$976$	64.9786	2.07991
$977$	−26.2707	−0.840476	−0.420238	−	0.907414i	$-0.638053\pi$
$977$	−26.2707	−0.840476	−0.420238	+	0.907414i	$0.638053\pi$
$978$	0	0
$979$	−3.18413	−0.101765
$980$	−22.6025	−0.722009
$981$	0	0
$982$	91.5553	2.92165
$983$	−3.45166	−0.110091	−0.0550454	−	0.998484i	$-0.517530\pi$
$983$	−3.45166	−0.110091	−0.0550454	+	0.998484i	$0.517530\pi$
$984$	0	0
$985$	−54.3777	−1.73262
$986$	−34.5864	−1.10146
$987$	0	0
$988$	−49.3919	−1.57137
$989$	0.989475	0.0314635
$990$	0	0
$991$	54.8023	1.74085	0.870427	−	0.492298i	$-0.163843\pi$
$991$	54.8023	1.74085	0.870427	+	0.492298i	$0.163843\pi$
$992$	−75.4075	−2.39419
$993$	0	0
$994$	−118.641	−3.76305
$995$	12.0223	0.381134
$996$	0	0
$997$	14.1082	0.446813	0.223406	−	0.974725i	$-0.428282\pi$
$997$	14.1082	0.446813	0.223406	+	0.974725i	$0.428282\pi$
$998$	−13.3936	−0.423966
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.t.1.2	✓	40
3.2	odd	2	inner	6021.2.a.t.1.39	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.t.1.2	✓	40	1.1	even	1	trivial
6021.2.a.t.1.39	yes	40	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6021 = 3^{3} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6021.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69102 q^{2} +5.24161 q^{4} -2.40662 q^{5} +2.96509 q^{7} -8.72327 q^{8} +O(q^{10})\)	\(q-2.69102 q^{2} +5.24161 q^{4} -2.40662 q^{5} +2.96509 q^{7} -8.72327 q^{8} +6.47626 q^{10} +0.991862 q^{11} +3.08869 q^{13} -7.97914 q^{14} +12.9913 q^{16} +3.39643 q^{17} -3.05082 q^{19} -12.6145 q^{20} -2.66912 q^{22} +5.70909 q^{23} +0.791796 q^{25} -8.31175 q^{26} +15.5419 q^{28} +3.78412 q^{29} +4.30571 q^{31} -17.5134 q^{32} -9.13988 q^{34} -7.13584 q^{35} +5.83764 q^{37} +8.20982 q^{38} +20.9935 q^{40} +0.306744 q^{41} +0.173316 q^{43} +5.19896 q^{44} -15.3633 q^{46} -3.13156 q^{47} +1.79178 q^{49} -2.13074 q^{50} +16.1897 q^{52} +1.76013 q^{53} -2.38703 q^{55} -25.8653 q^{56} -10.1832 q^{58} +3.38654 q^{59} +5.00170 q^{61} -11.5868 q^{62} +21.1463 q^{64} -7.43329 q^{65} +3.68493 q^{67} +17.8028 q^{68} +19.2027 q^{70} +14.8689 q^{71} -1.26962 q^{73} -15.7092 q^{74} -15.9912 q^{76} +2.94096 q^{77} -0.660487 q^{79} -31.2650 q^{80} -0.825454 q^{82} +6.79682 q^{83} -8.17390 q^{85} -0.466397 q^{86} -8.65227 q^{88} -3.21025 q^{89} +9.15826 q^{91} +29.9249 q^{92} +8.42711 q^{94} +7.34214 q^{95} -7.76179 q^{97} -4.82172 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) \) 40 * q + 46 * q^4 + 16 * q^7	\( 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) \) 40 * q + 46 * q^4 + 16 * q^7 + 22 * q^10 + 14 * q^13 + 50 * q^16 + 64 * q^19 + 12 * q^22 + 40 * q^25 + 48 * q^28 + 54 * q^31 + 32 * q^34 + 24 * q^37 + 40 * q^40 + 24 * q^43 + 52 * q^46 + 64 * q^49 + 18 * q^52 + 36 * q^55 + 8 * q^58 + 58 * q^61 + 120 * q^64 + 52 * q^67 - 30 * q^70 + 50 * q^73 + 112 * q^76 + 60 * q^79 + 50 * q^82 + 38 * q^85 + 16 * q^88 + 118 * q^91 + 44 * q^94 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more