LMFDB - Embedded newform 171.4.a.f.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [171,4,Mod(1,171)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("171.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	171.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,0,21,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$10.0893266110$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.20905$ of defining polynomial
Character	$\chi$	$=$	171.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.96257 q^{2} +7.70200 q^{4} -18.1342 q^{5} -25.8362 q^{7} -1.18085 q^{8} -71.8581 q^{10} +8.13420 q^{11} -4.56640 q^{13} -102.378 q^{14} -66.2952 q^{16} +62.5850 q^{17} -19.0000 q^{19} -139.670 q^{20} +32.2324 q^{22} +52.7502 q^{23} +203.849 q^{25} -18.0947 q^{26} -198.990 q^{28} -171.620 q^{29} +168.749 q^{31} -253.253 q^{32} +247.998 q^{34} +468.519 q^{35} -147.534 q^{37} -75.2889 q^{38} +21.4138 q^{40} -308.774 q^{41} -448.950 q^{43} +62.6496 q^{44} +209.027 q^{46} -113.335 q^{47} +324.509 q^{49} +807.768 q^{50} -35.1704 q^{52} -155.402 q^{53} -147.507 q^{55} +30.5086 q^{56} -680.059 q^{58} -182.347 q^{59} +404.080 q^{61} +668.681 q^{62} -473.172 q^{64} +82.8080 q^{65} -106.400 q^{67} +482.030 q^{68} +1856.54 q^{70} -472.079 q^{71} +843.821 q^{73} -584.616 q^{74} -146.338 q^{76} -210.157 q^{77} -591.036 q^{79} +1202.21 q^{80} -1223.54 q^{82} -290.388 q^{83} -1134.93 q^{85} -1779.00 q^{86} -9.60526 q^{88} +964.896 q^{89} +117.978 q^{91} +406.282 q^{92} -449.099 q^{94} +344.550 q^{95} -219.495 q^{97} +1285.89 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 21 q^{4} - 14 q^{5} - 35 q^{7} - 27 q^{8} - 88 q^{10} - 16 q^{11} + 65 q^{13} - 37 q^{14} + 33 q^{16} - 29 q^{17} - 57 q^{19} - 100 q^{20} + 118 q^{22} + 101 q^{23} - 37 q^{25} - 299 q^{26}+ \cdots + 2450 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 21 * q^4 - 14 * q^5 - 35 * q^7 - 27 * q^8 - 88 * q^10 - 16 * q^11 + 65 * q^13 - 37 * q^14 + 33 * q^16 - 29 * q^17 - 57 * q^19 - 100 * q^20 + 118 * q^22 + 101 * q^23 - 37 * q^25 - 299 * q^26 - 493 * q^28 - 377 * q^29 - 140 * q^31 - 579 * q^32 + 329 * q^34 + 438 * q^35 - 290 * q^37 + 57 * q^38 - 72 * q^40 - 956 * q^41 - 570 * q^43 - 110 * q^44 - 239 * q^46 - 66 * q^47 - 98 * q^49 + 1639 * q^50 + 697 * q^52 - 817 * q^53 - 198 * q^55 + 915 * q^56 + 647 * q^58 - 265 * q^59 + 988 * q^61 + 1436 * q^62 - 159 * q^64 + 240 * q^65 - 207 * q^67 + 1529 * q^68 + 2040 * q^70 - 846 * q^71 + 627 * q^73 + 770 * q^74 - 399 * q^76 - 88 * q^77 + 382 * q^79 + 1472 * q^80 + 872 * q^82 + 766 * q^83 - 1242 * q^85 - 2592 * q^86 + 342 * q^88 + 172 * q^89 - 457 * q^91 + 2677 * q^92 - 96 * q^94 + 266 * q^95 - 2450 * q^97 + 2450 * q^98

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 $

$2$	3.96257	1.40098	0.700491	−	0.713661i	$-0.252964\pi$
$2$	3.96257	1.40098	0.700491	+	0.713661i	$0.252964\pi$
$3$	0	0
$3$	0	0
$4$	7.70200	0.962750
$5$	−18.1342	−1.62197	−0.810986	−	0.585065i	$-0.801069\pi$
$5$	−18.1342	−1.62197	−0.810986	+	0.585065i	$0.801069\pi$
$6$	0	0
$7$	−25.8362	−1.39502	−0.697512	−	0.716573i	$-0.745710\pi$
$7$	−25.8362	−1.39502	−0.697512	+	0.716573i	$0.745710\pi$
$8$	−1.18085	−0.0521866
$9$	0	0
$10$	−71.8581	−2.27235
$11$	8.13420	0.222959	0.111480	−	0.993767i	$-0.464441\pi$
$11$	8.13420	0.222959	0.111480	+	0.993767i	$0.464441\pi$
$12$	0	0
$13$	−4.56640	−0.0974224	−0.0487112	−	0.998813i	$-0.515511\pi$
$13$	−4.56640	−0.0974224	−0.0487112	+	0.998813i	$0.515511\pi$
$14$	−102.378	−1.95440
$15$	0	0
$16$	−66.2952	−1.03586
$17$	62.5850	0.892888	0.446444	−	0.894812i	$-0.352690\pi$
$17$	62.5850	0.892888	0.446444	+	0.894812i	$0.352690\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−139.670	−1.56155
$21$	0	0
$22$	32.2324	0.312362
$23$	52.7502	0.478225	0.239113	−	0.970992i	$-0.423144\pi$
$23$	52.7502	0.478225	0.239113	+	0.970992i	$0.423144\pi$
$24$	0	0
$25$	203.849	1.63079
$26$	−18.0947	−0.136487
$27$	0	0
$28$	−198.990	−1.34306
$29$	−171.620	−1.09894	−0.549468	−	0.835515i	$-0.685169\pi$
$29$	−171.620	−1.09894	−0.549468	+	0.835515i	$0.685169\pi$
$30$	0	0
$31$	168.749	0.977685	0.488842	−	0.872372i	$-0.337419\pi$
$31$	168.749	0.977685	0.488842	+	0.872372i	$0.337419\pi$
$32$	−253.253	−1.39904
$33$	0	0
$34$	247.998	1.25092
$35$	468.519	2.26269
$36$	0	0
$37$	−147.534	−0.655528	−0.327764	−	0.944760i	$-0.606295\pi$
$37$	−147.534	−0.655528	−0.327764	+	0.944760i	$0.606295\pi$
$38$	−75.2889	−0.321407
$39$	0	0
$40$	21.4138	0.0846453
$41$	−308.774	−1.17616	−0.588078	−	0.808804i	$-0.700115\pi$
$41$	−308.774	−1.17616	−0.588078	+	0.808804i	$0.700115\pi$
$42$	0	0
$43$	−448.950	−1.59219	−0.796096	−	0.605170i	$-0.793105\pi$
$43$	−448.950	−1.59219	−0.796096	+	0.605170i	$0.793105\pi$
$44$	62.6496	0.214654
$45$	0	0
$46$	209.027	0.669985
$47$	−113.335	−0.351737	−0.175868	−	0.984414i	$-0.556273\pi$
$47$	−113.335	−0.351737	−0.175868	+	0.984414i	$0.556273\pi$
$48$	0	0
$49$	324.509	0.946091
$50$	807.768	2.28471
$51$	0	0
$52$	−35.1704	−0.0937934
$53$	−155.402	−0.402758	−0.201379	−	0.979513i	$-0.564542\pi$
$53$	−155.402	−0.402758	−0.201379	+	0.979513i	$0.564542\pi$
$54$	0	0
$55$	−147.507	−0.361634
$56$	30.5086	0.0728016
$57$	0	0
$58$	−680.059	−1.53959
$59$	−182.347	−0.402365	−0.201183	−	0.979554i	$-0.564478\pi$
$59$	−182.347	−0.402365	−0.201183	+	0.979554i	$0.564478\pi$
$60$	0	0
$61$	404.080	0.848149	0.424075	−	0.905627i	$-0.360599\pi$
$61$	404.080	0.848149	0.424075	+	0.905627i	$0.360599\pi$
$62$	668.681	1.36972
$63$	0	0
$64$	−473.172	−0.924164
$65$	82.8080	0.158016
$66$	0	0
$67$	−106.400	−0.194013	−0.0970064	−	0.995284i	$-0.530927\pi$
$67$	−106.400	−0.194013	−0.0970064	+	0.995284i	$0.530927\pi$
$68$	482.030	0.859628
$69$	0	0
$70$	1856.54	3.16999
$71$	−472.079	−0.789091	−0.394546	−	0.918876i	$-0.629098\pi$
$71$	−472.079	−0.789091	−0.394546	+	0.918876i	$0.629098\pi$
$72$	0	0
$73$	843.821	1.35290	0.676451	−	0.736488i	$-0.263517\pi$
$73$	843.821	1.35290	0.676451	+	0.736488i	$0.263517\pi$
$74$	−584.616	−0.918382
$75$	0	0
$76$	−146.338	−0.220870
$77$	−210.157	−0.311034
$78$	0	0
$79$	−591.036	−0.841731	−0.420866	−	0.907123i	$-0.638274\pi$
$79$	−591.036	−0.841731	−0.420866	+	0.907123i	$0.638274\pi$
$80$	1202.21	1.68014
$81$	0	0
$82$	−1223.54	−1.64777
$83$	−290.388	−0.384027	−0.192013	−	0.981392i	$-0.561502\pi$
$83$	−290.388	−0.384027	−0.192013	+	0.981392i	$0.561502\pi$
$84$	0	0
$85$	−1134.93	−1.44824
$86$	−1779.00	−2.23063
$87$	0	0
$88$	−9.60526	−0.0116355
$89$	964.896	1.14920	0.574600	−	0.818435i	$-0.305158\pi$
$89$	964.896	1.14920	0.574600	+	0.818435i	$0.305158\pi$
$90$	0	0
$91$	117.978	0.135907
$92$	406.282	0.460411
$93$	0	0
$94$	−449.099	−0.492777
$95$	344.550	0.372106
$96$	0	0
$97$	−219.495	−0.229756	−0.114878	−	0.993380i	$-0.536648\pi$
$97$	−219.495	−0.229756	−0.114878	+	0.993380i	$0.536648\pi$
$98$	1285.89	1.32546
$99$	0	0
$100$	1570.05	1.57005
$101$	−1447.94	−1.42649	−0.713247	−	0.700913i	$-0.752776\pi$
$101$	−1447.94	−1.42649	−0.713247	+	0.700913i	$0.752776\pi$
$102$	0	0
$103$	883.567	0.845247	0.422623	−	0.906305i	$-0.361109\pi$
$103$	883.567	0.845247	0.422623	+	0.906305i	$0.361109\pi$
$104$	5.39223	0.00508415
$105$	0	0
$106$	−615.793	−0.564256
$107$	1307.82	1.18160	0.590801	−	0.806817i	$-0.298812\pi$
$107$	1307.82	1.18160	0.590801	+	0.806817i	$0.298812\pi$
$108$	0	0
$109$	870.507	0.764949	0.382475	−	0.923966i	$-0.375072\pi$
$109$	870.507	0.764949	0.382475	+	0.923966i	$0.375072\pi$
$110$	−584.508	−0.506643
$111$	0	0
$112$	1712.82	1.44505
$113$	1181.41	0.983521	0.491761	−	0.870730i	$-0.336354\pi$
$113$	1181.41	0.983521	0.491761	+	0.870730i	$0.336354\pi$
$114$	0	0
$115$	−956.583	−0.775668
$116$	−1321.82	−1.05800
$117$	0	0
$118$	−722.564	−0.563707
$119$	−1616.96	−1.24560
$120$	0	0
$121$	−1264.83	−0.950289
$122$	1601.20	1.18824
$123$	0	0
$124$	1299.71	0.941266
$125$	−1429.87	−1.02313
$126$	0	0
$127$	887.509	0.620108	0.310054	−	0.950719i	$-0.399653\pi$
$127$	887.509	0.620108	0.310054	+	0.950719i	$0.399653\pi$
$128$	151.044	0.104301
$129$	0	0
$130$	328.133	0.221378
$131$	2344.76	1.56384	0.781920	−	0.623379i	$-0.214241\pi$
$131$	2344.76	1.56384	0.781920	+	0.623379i	$0.214241\pi$
$132$	0	0
$133$	490.888	0.320040
$134$	−421.619	−0.271809
$135$	0	0
$136$	−73.9034	−0.0465968
$137$	−2244.82	−1.39991	−0.699956	−	0.714186i	$-0.746797\pi$
$137$	−2244.82	−1.39991	−0.699956	+	0.714186i	$0.746797\pi$
$138$	0	0
$139$	−296.146	−0.180711	−0.0903554	−	0.995910i	$-0.528800\pi$
$139$	−296.146	−0.180711	−0.0903554	+	0.995910i	$0.528800\pi$
$140$	3608.53	2.17840
$141$	0	0
$142$	−1870.65	−1.10550
$143$	−37.1440	−0.0217212
$144$	0	0
$145$	3112.20	1.78244
$146$	3343.70	1.89539
$147$	0	0
$148$	−1136.31	−0.631109
$149$	−1791.09	−0.984780	−0.492390	−	0.870375i	$-0.663877\pi$
$149$	−1791.09	−0.984780	−0.492390	+	0.870375i	$0.663877\pi$
$150$	0	0
$151$	−2352.65	−1.26792	−0.633960	−	0.773366i	$-0.718571\pi$
$151$	−2352.65	−1.26792	−0.633960	+	0.773366i	$0.718571\pi$
$152$	22.4361	0.0119724
$153$	0	0
$154$	−832.762	−0.435752
$155$	−3060.13	−1.58578
$156$	0	0
$157$	−1438.26	−0.731118	−0.365559	−	0.930788i	$-0.619122\pi$
$157$	−1438.26	−0.731118	−0.365559	+	0.930788i	$0.619122\pi$
$158$	−2342.02	−1.17925
$159$	0	0
$160$	4592.54	2.26920
$161$	−1362.86	−0.667135
$162$	0	0
$163$	127.493	0.0612640	0.0306320	−	0.999531i	$-0.490248\pi$
$163$	127.493	0.0612640	0.0306320	+	0.999531i	$0.490248\pi$
$164$	−2378.18	−1.13234
$165$	0	0
$166$	−1150.68	−0.538014
$167$	−3419.05	−1.58428	−0.792139	−	0.610341i	$-0.791033\pi$
$167$	−3419.05	−1.58428	−0.792139	+	0.610341i	$0.791033\pi$
$168$	0	0
$169$	−2176.15	−0.990509
$170$	−4497.24	−2.02896
$171$	0	0
$172$	−3457.81	−1.53288
$173$	362.598	0.159352	0.0796758	−	0.996821i	$-0.474611\pi$
$173$	362.598	0.159352	0.0796758	+	0.996821i	$0.474611\pi$
$174$	0	0
$175$	−5266.69	−2.27500
$176$	−539.258	−0.230955
$177$	0	0
$178$	3823.47	1.61001
$179$	−2417.89	−1.00962	−0.504809	−	0.863231i	$-0.668437\pi$
$179$	−2417.89	−1.00962	−0.504809	+	0.863231i	$0.668437\pi$
$180$	0	0
$181$	−2444.64	−1.00391	−0.501957	−	0.864892i	$-0.667387\pi$
$181$	−2444.64	−1.00391	−0.501957	+	0.864892i	$0.667387\pi$
$182$	467.498	0.190403
$183$	0	0
$184$	−62.2900	−0.0249570
$185$	2675.42	1.06325
$186$	0	0
$187$	509.079	0.199078
$188$	−872.908	−0.338635
$189$	0	0
$190$	1365.30	0.521314
$191$	1387.66	0.525693	0.262846	−	0.964838i	$-0.415339\pi$
$191$	1387.66	0.525693	0.262846	+	0.964838i	$0.415339\pi$
$192$	0	0
$193$	−3208.03	−1.19647	−0.598237	−	0.801319i	$-0.704132\pi$
$193$	−3208.03	−1.19647	−0.598237	+	0.801319i	$0.704132\pi$
$194$	−869.764	−0.321884
$195$	0	0
$196$	2499.37	0.910849
$197$	3445.36	1.24605	0.623025	−	0.782202i	$-0.285903\pi$
$197$	3445.36	1.24605	0.623025	+	0.782202i	$0.285903\pi$
$198$	0	0
$199$	2025.71	0.721602	0.360801	−	0.932643i	$-0.382503\pi$
$199$	2025.71	0.721602	0.360801	+	0.932643i	$0.382503\pi$
$200$	−240.715	−0.0851056
$201$	0	0
$202$	−5737.59	−1.99849
$203$	4434.02	1.53304
$204$	0	0
$205$	5599.37	1.90769
$206$	3501.20	1.18418
$207$	0	0
$208$	302.730	0.100916
$209$	−154.550	−0.0511504
$210$	0	0
$211$	4309.54	1.40607	0.703036	−	0.711155i	$-0.251827\pi$
$211$	4309.54	1.40607	0.703036	+	0.711155i	$0.251827\pi$
$212$	−1196.91	−0.387755
$213$	0	0
$214$	5182.33	1.65540
$215$	8141.35	2.58249
$216$	0	0
$217$	−4359.84	−1.36389
$218$	3449.45	1.07168
$219$	0	0
$220$	−1136.10	−0.348163
$221$	−285.788	−0.0869873
$222$	0	0
$223$	825.648	0.247935	0.123968	−	0.992286i	$-0.460438\pi$
$223$	825.648	0.247935	0.123968	+	0.992286i	$0.460438\pi$
$224$	6543.09	1.95169
$225$	0	0
$226$	4681.43	1.37790
$227$	1501.19	0.438931	0.219466	−	0.975620i	$-0.429569\pi$
$227$	1501.19	0.438931	0.219466	+	0.975620i	$0.429569\pi$
$228$	0	0
$229$	−5250.40	−1.51509	−0.757547	−	0.652781i	$-0.773602\pi$
$229$	−5250.40	−1.51509	−0.757547	+	0.652781i	$0.773602\pi$
$230$	−3790.53	−1.08670
$231$	0	0
$232$	202.658	0.0573497
$233$	−2139.06	−0.601435	−0.300717	−	0.953713i	$-0.597226\pi$
$233$	−2139.06	−0.601435	−0.300717	+	0.953713i	$0.597226\pi$
$234$	0	0
$235$	2055.24	0.570508
$236$	−1404.44	−0.387377
$237$	0	0
$238$	−6407.32	−1.74506
$239$	−3772.70	−1.02107	−0.510534	−	0.859857i	$-0.670552\pi$
$239$	−3772.70	−1.02107	−0.510534	+	0.859857i	$0.670552\pi$
$240$	0	0
$241$	6415.39	1.71474	0.857369	−	0.514702i	$-0.172097\pi$
$241$	6415.39	1.71474	0.857369	+	0.514702i	$0.172097\pi$
$242$	−5012.00	−1.33134
$243$	0	0
$244$	3112.22	0.816555
$245$	−5884.71	−1.53453
$246$	0	0
$247$	86.7616	0.0223502
$248$	−199.267	−0.0510221
$249$	0	0
$250$	−5665.95	−1.43339
$251$	6277.31	1.57857	0.789283	−	0.614029i	$-0.210452\pi$
$251$	6277.31	1.57857	0.789283	+	0.614029i	$0.210452\pi$
$252$	0	0
$253$	429.081	0.106625
$254$	3516.82	0.868760
$255$	0	0
$256$	4383.90	1.07029
$257$	3183.98	0.772807	0.386404	−	0.922330i	$-0.373717\pi$
$257$	3183.98	0.772807	0.386404	+	0.922330i	$0.373717\pi$
$258$	0	0
$259$	3811.73	0.914476
$260$	637.787	0.152130
$261$	0	0
$262$	9291.31	2.19091
$263$	2624.18	0.615261	0.307630	−	0.951506i	$-0.400464\pi$
$263$	2624.18	0.615261	0.307630	+	0.951506i	$0.400464\pi$
$264$	0	0
$265$	2818.10	0.653261
$266$	1945.18	0.448371
$267$	0	0
$268$	−819.495	−0.186786
$269$	−7444.76	−1.68742	−0.843708	−	0.536803i	$-0.819632\pi$
$269$	−7444.76	−1.68742	−0.843708	+	0.536803i	$0.819632\pi$
$270$	0	0
$271$	−4004.49	−0.897621	−0.448810	−	0.893627i	$-0.648152\pi$
$271$	−4004.49	−0.897621	−0.448810	+	0.893627i	$0.648152\pi$
$272$	−4149.08	−0.924909
$273$	0	0
$274$	−8895.27	−1.96125
$275$	1658.15	0.363601
$276$	0	0
$277$	−5830.66	−1.26473	−0.632365	−	0.774671i	$-0.717916\pi$
$277$	−5830.66	−1.26473	−0.632365	+	0.774671i	$0.717916\pi$
$278$	−1173.50	−0.253172
$279$	0	0
$280$	−553.250	−0.118082
$281$	7504.37	1.59314	0.796572	−	0.604544i	$-0.206645\pi$
$281$	7504.37	1.59314	0.796572	+	0.604544i	$0.206645\pi$
$282$	0	0
$283$	5910.87	1.24157	0.620785	−	0.783980i	$-0.286814\pi$
$283$	5910.87	1.24157	0.620785	+	0.783980i	$0.286814\pi$
$284$	−3635.95	−0.759697
$285$	0	0
$286$	−147.186	−0.0304311
$287$	7977.54	1.64076
$288$	0	0
$289$	−996.118	−0.202752
$290$	12332.3	2.49717
$291$	0	0
$292$	6499.11	1.30251
$293$	−3245.59	−0.647131	−0.323566	−	0.946206i	$-0.604882\pi$
$293$	−3245.59	−0.647131	−0.323566	+	0.946206i	$0.604882\pi$
$294$	0	0
$295$	3306.72	0.652625
$296$	174.216	0.0342098
$297$	0	0
$298$	−7097.35	−1.37966
$299$	−240.878	−0.0465898
$300$	0	0
$301$	11599.2	2.22115
$302$	−9322.54	−1.77633
$303$	0	0
$304$	1259.61	0.237643
$305$	−7327.66	−1.37567
$306$	0	0
$307$	7489.14	1.39227	0.696137	−	0.717909i	$-0.254901\pi$
$307$	7489.14	1.39227	0.696137	+	0.717909i	$0.254901\pi$
$308$	−1618.63	−0.299448
$309$	0	0
$310$	−12126.0	−2.22165
$311$	−2136.71	−0.389588	−0.194794	−	0.980844i	$-0.562404\pi$
$311$	−2136.71	−0.389588	−0.194794	+	0.980844i	$0.562404\pi$
$312$	0	0
$313$	2212.15	0.399483	0.199742	−	0.979849i	$-0.435990\pi$
$313$	2212.15	0.399483	0.199742	+	0.979849i	$0.435990\pi$
$314$	−5699.21	−1.02428
$315$	0	0
$316$	−4552.16	−0.810377
$317$	−429.326	−0.0760674	−0.0380337	−	0.999276i	$-0.512109\pi$
$317$	−429.326	−0.0760674	−0.0380337	+	0.999276i	$0.512109\pi$
$318$	0	0
$319$	−1396.00	−0.245018
$320$	8580.60	1.49897
$321$	0	0
$322$	−5400.45	−0.934644
$323$	−1189.11	−0.204842
$324$	0	0
$325$	−930.857	−0.158876
$326$	505.201	0.0858297
$327$	0	0
$328$	364.615	0.0613796
$329$	2928.15	0.490681
$330$	0	0
$331$	−765.454	−0.127109	−0.0635546	−	0.997978i	$-0.520244\pi$
$331$	−765.454	−0.127109	−0.0635546	+	0.997978i	$0.520244\pi$
$332$	−2236.57	−0.369722
$333$	0	0
$334$	−13548.3	−2.21954
$335$	1929.48	0.314684
$336$	0	0
$337$	−3049.81	−0.492978	−0.246489	−	0.969146i	$-0.579277\pi$
$337$	−3049.81	−0.492978	−0.246489	+	0.969146i	$0.579277\pi$
$338$	−8623.15	−1.38768
$339$	0	0
$340$	−8741.22	−1.39429
$341$	1372.64	0.217984
$342$	0	0
$343$	477.732	0.0752045
$344$	530.142	0.0830912
$345$	0	0
$346$	1436.82	0.223249
$347$	5907.00	0.913845	0.456922	−	0.889507i	$-0.348952\pi$
$347$	5907.00	0.913845	0.456922	+	0.889507i	$0.348952\pi$
$348$	0	0
$349$	−12107.4	−1.85700	−0.928502	−	0.371327i	$-0.878903\pi$
$349$	−12107.4	−1.85700	−0.928502	+	0.371327i	$0.878903\pi$
$350$	−20869.6	−3.18723
$351$	0	0
$352$	−2060.01	−0.311929
$353$	2420.40	0.364943	0.182471	−	0.983211i	$-0.441590\pi$
$353$	2420.40	0.364943	0.182471	+	0.983211i	$0.441590\pi$
$354$	0	0
$355$	8560.77	1.27988
$356$	7431.63	1.10639
$357$	0	0
$358$	−9581.07	−1.41446
$359$	1455.80	0.214023	0.107011	−	0.994258i	$-0.465872\pi$
$359$	1455.80	0.214023	0.107011	+	0.994258i	$0.465872\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−9687.06	−1.40647
$363$	0	0
$364$	908.670	0.130844
$365$	−15302.0	−2.19437
$366$	0	0
$367$	8783.80	1.24935	0.624674	−	0.780886i	$-0.285232\pi$
$367$	8783.80	1.24935	0.624674	+	0.780886i	$0.285232\pi$
$368$	−3497.08	−0.495375
$369$	0	0
$370$	10601.6	1.48959
$371$	4015.00	0.561856
$372$	0	0
$373$	−9199.84	−1.27708	−0.638538	−	0.769590i	$-0.720461\pi$
$373$	−9199.84	−1.27708	−0.638538	+	0.769590i	$0.720461\pi$
$374$	2017.26	0.278904
$375$	0	0
$376$	133.832	0.0183560
$377$	783.688	0.107061
$378$	0	0
$379$	−6161.38	−0.835063	−0.417531	−	0.908662i	$-0.637105\pi$
$379$	−6161.38	−0.835063	−0.417531	+	0.908662i	$0.637105\pi$
$380$	2653.72	0.358245
$381$	0	0
$382$	5498.70	0.736486
$383$	−2630.79	−0.350985	−0.175492	−	0.984481i	$-0.556152\pi$
$383$	−2630.79	−0.350985	−0.175492	+	0.984481i	$0.556152\pi$
$384$	0	0
$385$	3811.03	0.504488
$386$	−12712.1	−1.67624
$387$	0	0
$388$	−1690.55	−0.221197
$389$	5866.48	0.764633	0.382317	−	0.924031i	$-0.375126\pi$
$389$	5866.48	0.764633	0.382317	+	0.924031i	$0.375126\pi$
$390$	0	0
$391$	3301.37	0.427001
$392$	−383.196	−0.0493733
$393$	0	0
$394$	13652.5	1.74569
$395$	10718.0	1.36526
$396$	0	0
$397$	−14254.0	−1.80199	−0.900993	−	0.433833i	$-0.857161\pi$
$397$	−14254.0	−1.80199	−0.900993	+	0.433833i	$0.857161\pi$
$398$	8027.03	1.01095
$399$	0	0
$400$	−13514.2	−1.68928
$401$	−9909.27	−1.23403	−0.617014	−	0.786952i	$-0.711658\pi$
$401$	−9909.27	−1.23403	−0.617014	+	0.786952i	$0.711658\pi$
$402$	0	0
$403$	−770.576	−0.0952484
$404$	−11152.1	−1.37336
$405$	0	0
$406$	17570.1	2.14776
$407$	−1200.07	−0.146156
$408$	0	0
$409$	5805.87	0.701912	0.350956	−	0.936392i	$-0.385857\pi$
$409$	5805.87	0.701912	0.350956	+	0.936392i	$0.385857\pi$
$410$	22187.9	2.67264
$411$	0	0
$412$	6805.23	0.813761
$413$	4711.15	0.561309
$414$	0	0
$415$	5265.95	0.622881
$416$	1156.45	0.136298
$417$	0	0
$418$	−612.415	−0.0716608
$419$	−12260.9	−1.42955	−0.714777	−	0.699353i	$-0.753472\pi$
$419$	−12260.9	−1.42955	−0.714777	+	0.699353i	$0.753472\pi$
$420$	0	0
$421$	5837.85	0.675818	0.337909	−	0.941179i	$-0.390280\pi$
$421$	5837.85	0.675818	0.337909	+	0.941179i	$0.390280\pi$
$422$	17076.9	1.96988
$423$	0	0
$424$	183.507	0.0210186
$425$	12757.9	1.45612
$426$	0	0
$427$	−10439.9	−1.18319
$428$	10072.8	1.13759
$429$	0	0
$430$	32260.7	3.61802
$431$	2770.16	0.309591	0.154796	−	0.987946i	$-0.450528\pi$
$431$	2770.16	0.309591	0.154796	+	0.987946i	$0.450528\pi$
$432$	0	0
$433$	5663.00	0.628513	0.314257	−	0.949338i	$-0.398245\pi$
$433$	5663.00	0.628513	0.314257	+	0.949338i	$0.398245\pi$
$434$	−17276.2	−1.91079
$435$	0	0
$436$	6704.65	0.736455
$437$	−1002.25	−0.109712
$438$	0	0
$439$	−8399.20	−0.913148	−0.456574	−	0.889685i	$-0.650924\pi$
$439$	−8399.20	−0.913148	−0.456574	+	0.889685i	$0.650924\pi$
$440$	174.184	0.0188725
$441$	0	0
$442$	−1132.46	−0.121868
$443$	−6154.68	−0.660085	−0.330043	−	0.943966i	$-0.607063\pi$
$443$	−6154.68	−0.660085	−0.330043	+	0.943966i	$0.607063\pi$
$444$	0	0
$445$	−17497.6	−1.86397
$446$	3271.69	0.347352
$447$	0	0
$448$	12225.0	1.28923
$449$	−3445.03	−0.362095	−0.181048	−	0.983474i	$-0.557949\pi$
$449$	−3445.03	−0.362095	−0.181048	+	0.983474i	$0.557949\pi$
$450$	0	0
$451$	−2511.63	−0.262235
$452$	9099.24	0.946885
$453$	0	0
$454$	5948.57	0.614935
$455$	−2139.44	−0.220437
$456$	0	0
$457$	−502.346	−0.0514196	−0.0257098	−	0.999669i	$-0.508185\pi$
$457$	−502.346	−0.0514196	−0.0257098	+	0.999669i	$0.508185\pi$
$458$	−20805.1	−2.12262
$459$	0	0
$460$	−7367.60	−0.746774
$461$	−546.259	−0.0551883	−0.0275942	−	0.999619i	$-0.508785\pi$
$461$	−546.259	−0.0551883	−0.0275942	+	0.999619i	$0.508785\pi$
$462$	0	0
$463$	18540.2	1.86098	0.930490	−	0.366316i	$-0.119381\pi$
$463$	18540.2	1.86098	0.930490	+	0.366316i	$0.119381\pi$
$464$	11377.6	1.13835
$465$	0	0
$466$	−8476.18	−0.842599
$467$	−12475.1	−1.23614	−0.618070	−	0.786123i	$-0.712085\pi$
$467$	−12475.1	−1.23614	−0.618070	+	0.786123i	$0.712085\pi$
$468$	0	0
$469$	2748.98	0.270653
$470$	8144.05	0.799271
$471$	0	0
$472$	215.324	0.0209981
$473$	−3651.85	−0.354994
$474$	0	0
$475$	−3873.13	−0.374130
$476$	−12453.8	−1.19920
$477$	0	0
$478$	−14949.6	−1.43050
$479$	10569.2	1.00818	0.504091	−	0.863651i	$-0.331828\pi$
$479$	10569.2	1.00818	0.504091	+	0.863651i	$0.331828\pi$
$480$	0	0
$481$	673.701	0.0638631
$482$	25421.5	2.40232
$483$	0	0
$484$	−9741.76	−0.914891
$485$	3980.36	0.372657
$486$	0	0
$487$	−11227.9	−1.04473	−0.522366	−	0.852721i	$-0.674951\pi$
$487$	−11227.9	−1.04473	−0.522366	+	0.852721i	$0.674951\pi$
$488$	−477.157	−0.0442621
$489$	0	0
$490$	−23318.6	−2.14985
$491$	536.840	0.0493427	0.0246713	−	0.999696i	$-0.492146\pi$
$491$	536.840	0.0493427	0.0246713	+	0.999696i	$0.492146\pi$
$492$	0	0
$493$	−10740.9	−0.981226
$494$	343.799	0.0313123
$495$	0	0
$496$	−11187.3	−1.01275
$497$	12196.7	1.10080
$498$	0	0
$499$	1319.91	0.118412	0.0592058	−	0.998246i	$-0.481143\pi$
$499$	1319.91	0.118412	0.0592058	+	0.998246i	$0.481143\pi$
$500$	−11012.8	−0.985018
$501$	0	0
$502$	24874.3	2.21154
$503$	−1749.27	−0.155062	−0.0775310	−	0.996990i	$-0.524704\pi$
$503$	−1749.27	−0.155062	−0.0775310	+	0.996990i	$0.524704\pi$
$504$	0	0
$505$	26257.3	2.31373
$506$	1700.26	0.149379
$507$	0	0
$508$	6835.59	0.597009
$509$	−1882.19	−0.163903	−0.0819516	−	0.996636i	$-0.526115\pi$
$509$	−1882.19	−0.163903	−0.0819516	+	0.996636i	$0.526115\pi$
$510$	0	0
$511$	−21801.1	−1.88733
$512$	16163.2	1.39515
$513$	0	0
$514$	12616.8	1.08269
$515$	−16022.8	−1.37097
$516$	0	0
$517$	−921.891	−0.0784231
$518$	15104.3	1.28116
$519$	0	0
$520$	−97.7837	−0.00824635
$521$	3238.50	0.272325	0.136163	−	0.990686i	$-0.456523\pi$
$521$	3238.50	0.272325	0.136163	+	0.990686i	$0.456523\pi$
$522$	0	0
$523$	99.0144	0.00827839	0.00413919	−	0.999991i	$-0.498682\pi$
$523$	99.0144	0.00827839	0.00413919	+	0.999991i	$0.498682\pi$
$524$	18059.4	1.50559
$525$	0	0
$526$	10398.5	0.861969
$527$	10561.2	0.872963
$528$	0	0
$529$	−9384.42	−0.771301
$530$	11166.9	0.915207
$531$	0	0
$532$	3780.82	0.308119
$533$	1409.98	0.114584
$534$	0	0
$535$	−23716.2	−1.91653
$536$	125.643	0.0101249
$537$	0	0
$538$	−29500.4	−2.36404
$539$	2639.62	0.210940
$540$	0	0
$541$	17183.7	1.36559	0.682794	−	0.730611i	$-0.260765\pi$
$541$	17183.7	1.36559	0.682794	+	0.730611i	$0.260765\pi$
$542$	−15868.1	−1.25755
$543$	0	0
$544$	−15849.8	−1.24918
$545$	−15786.0	−1.24073
$546$	0	0
$547$	−1965.86	−0.153664	−0.0768319	−	0.997044i	$-0.524480\pi$
$547$	−1965.86	−0.153664	−0.0768319	+	0.997044i	$0.524480\pi$
$548$	−17289.6	−1.34777
$549$	0	0
$550$	6570.54	0.509398
$551$	3260.79	0.252113
$552$	0	0
$553$	15270.1	1.17423
$554$	−23104.4	−1.77186
$555$	0	0
$556$	−2280.92	−0.173979
$557$	−6039.93	−0.459461	−0.229731	−	0.973254i	$-0.573785\pi$
$557$	−6039.93	−0.459461	−0.229731	+	0.973254i	$0.573785\pi$
$558$	0	0
$559$	2050.09	0.155115
$560$	−31060.5	−2.34384
$561$	0	0
$562$	29736.6	2.23197
$563$	−5260.06	−0.393757	−0.196878	−	0.980428i	$-0.563080\pi$
$563$	−5260.06	−0.393757	−0.196878	+	0.980428i	$0.563080\pi$
$564$	0	0
$565$	−21424.0	−1.59524
$566$	23422.3	1.73942
$567$	0	0
$568$	557.454	0.0411800
$569$	−20567.4	−1.51534	−0.757672	−	0.652635i	$-0.773663\pi$
$569$	−20567.4	−1.51534	−0.757672	+	0.652635i	$0.773663\pi$
$570$	0	0
$571$	11462.4	0.840080	0.420040	−	0.907506i	$-0.362016\pi$
$571$	11462.4	0.840080	0.420040	+	0.907506i	$0.362016\pi$
$572$	−286.083	−0.0209121
$573$	0	0
$574$	31611.6	2.29868
$575$	10753.1	0.779886
$576$	0	0
$577$	−27029.6	−1.95019	−0.975094	−	0.221790i	$-0.928810\pi$
$577$	−27029.6	−1.95019	−0.975094	+	0.221790i	$0.928810\pi$
$578$	−3947.19	−0.284051
$579$	0	0
$580$	23970.2	1.71605
$581$	7502.52	0.535726
$582$	0	0
$583$	−1264.07	−0.0897986
$584$	−996.425	−0.0706034
$585$	0	0
$586$	−12860.9	−0.906619
$587$	−15200.4	−1.06881	−0.534403	−	0.845230i	$-0.679464\pi$
$587$	−15200.4	−1.06881	−0.534403	+	0.845230i	$0.679464\pi$
$588$	0	0
$589$	−3206.23	−0.224296
$590$	13103.1	0.914316
$591$	0	0
$592$	9780.83	0.679036
$593$	19026.7	1.31759	0.658796	−	0.752322i	$-0.271066\pi$
$593$	19026.7	1.31759	0.658796	+	0.752322i	$0.271066\pi$
$594$	0	0
$595$	29322.2	2.02033
$596$	−13795.0	−0.948096
$597$	0	0
$598$	−954.499	−0.0652715
$599$	−3927.31	−0.267889	−0.133945	−	0.990989i	$-0.542764\pi$
$599$	−3927.31	−0.267889	−0.133945	+	0.990989i	$0.542764\pi$
$600$	0	0
$601$	13718.1	0.931069	0.465534	−	0.885030i	$-0.345862\pi$
$601$	13718.1	0.931069	0.465534	+	0.885030i	$0.345862\pi$
$602$	45962.6	3.11178
$603$	0	0
$604$	−18120.1	−1.22069
$605$	22936.8	1.54134
$606$	0	0
$607$	26461.5	1.76942	0.884712	−	0.466138i	$-0.154355\pi$
$607$	26461.5	1.76942	0.884712	+	0.466138i	$0.154355\pi$
$608$	4811.81	0.320961
$609$	0	0
$610$	−29036.4	−1.92729
$611$	517.534	0.0342671
$612$	0	0
$613$	−233.384	−0.0153773	−0.00768865	−	0.999970i	$-0.502447\pi$
$613$	−233.384	−0.0153773	−0.00768865	+	0.999970i	$0.502447\pi$
$614$	29676.3	1.95055
$615$	0	0
$616$	248.163	0.0162318
$617$	−4202.77	−0.274225	−0.137113	−	0.990555i	$-0.543782\pi$
$617$	−4202.77	−0.274225	−0.137113	+	0.990555i	$0.543782\pi$
$618$	0	0
$619$	−23009.4	−1.49407	−0.747033	−	0.664787i	$-0.768522\pi$
$619$	−23009.4	−1.49407	−0.747033	+	0.664787i	$0.768522\pi$
$620$	−23569.1	−1.52671
$621$	0	0
$622$	−8466.89	−0.545806
$623$	−24929.2	−1.60316
$624$	0	0
$625$	448.344	0.0286940
$626$	8765.82	0.559668
$627$	0	0
$628$	−11077.5	−0.703884
$629$	−9233.45	−0.585313
$630$	0	0
$631$	18819.3	1.18730	0.593650	−	0.804723i	$-0.297686\pi$
$631$	18819.3	1.18730	0.593650	+	0.804723i	$0.297686\pi$
$632$	697.924	0.0439271
$633$	0	0
$634$	−1701.24	−0.106569
$635$	−16094.3	−1.00580
$636$	0	0
$637$	−1481.84	−0.0921705
$638$	−5531.74	−0.343266
$639$	0	0
$640$	−2739.06	−0.169173
$641$	−25253.7	−1.55610	−0.778052	−	0.628200i	$-0.783792\pi$
$641$	−25253.7	−1.55610	−0.778052	+	0.628200i	$0.783792\pi$
$642$	0	0
$643$	11712.1	0.718324	0.359162	−	0.933275i	$-0.383063\pi$
$643$	11712.1	0.718324	0.359162	+	0.933275i	$0.383063\pi$
$644$	−10496.8	−0.642284
$645$	0	0
$646$	−4711.96	−0.286981
$647$	26533.3	1.61226	0.806131	−	0.591737i	$-0.201558\pi$
$647$	26533.3	1.61226	0.806131	+	0.591737i	$0.201558\pi$
$648$	0	0
$649$	−1483.25	−0.0897111
$650$	−3688.59	−0.222582
$651$	0	0
$652$	981.952	0.0589819
$653$	−27898.9	−1.67193	−0.835964	−	0.548785i	$-0.815091\pi$
$653$	−27898.9	−1.67193	−0.835964	+	0.548785i	$0.815091\pi$
$654$	0	0
$655$	−42520.4	−2.53650
$656$	20470.2	1.21834
$657$	0	0
$658$	11603.0	0.687436
$659$	1274.66	0.0753468	0.0376734	−	0.999290i	$-0.488005\pi$
$659$	1274.66	0.0753468	0.0376734	+	0.999290i	$0.488005\pi$
$660$	0	0
$661$	−5049.52	−0.297131	−0.148565	−	0.988903i	$-0.547466\pi$
$661$	−5049.52	−0.297131	−0.148565	+	0.988903i	$0.547466\pi$
$662$	−3033.17	−0.178078
$663$	0	0
$664$	342.904	0.0200411
$665$	−8901.86	−0.519097
$666$	0	0
$667$	−9053.01	−0.525538
$668$	−26333.6	−1.52526
$669$	0	0
$670$	7645.73	0.440866
$671$	3286.86	0.189103
$672$	0	0
$673$	8398.64	0.481045	0.240523	−	0.970644i	$-0.422681\pi$
$673$	8398.64	0.481045	0.240523	+	0.970644i	$0.422681\pi$
$674$	−12085.1	−0.690653
$675$	0	0
$676$	−16760.7	−0.953612
$677$	−9875.31	−0.560619	−0.280309	−	0.959910i	$-0.590437\pi$
$677$	−9875.31	−0.560619	−0.280309	+	0.959910i	$0.590437\pi$
$678$	0	0
$679$	5670.91	0.320515
$680$	1340.18	0.0755787
$681$	0	0
$682$	5439.18	0.305392
$683$	8653.78	0.484814	0.242407	−	0.970175i	$-0.422063\pi$
$683$	8653.78	0.484814	0.242407	+	0.970175i	$0.422063\pi$
$684$	0	0
$685$	40708.0	2.27062
$686$	1893.05	0.105360
$687$	0	0
$688$	29763.2	1.64929
$689$	709.629	0.0392376
$690$	0	0
$691$	2916.50	0.160563	0.0802813	−	0.996772i	$-0.474418\pi$
$691$	2916.50	0.160563	0.0802813	+	0.996772i	$0.474418\pi$
$692$	2792.73	0.153416
$693$	0	0
$694$	23406.9	1.28028
$695$	5370.37	0.293108
$696$	0	0
$697$	−19324.6	−1.05017
$698$	−47976.5	−2.60163
$699$	0	0
$700$	−40564.0	−2.19025
$701$	9070.78	0.488729	0.244364	−	0.969683i	$-0.421421\pi$
$701$	9070.78	0.488729	0.244364	+	0.969683i	$0.421421\pi$
$702$	0	0
$703$	2803.16	0.150388
$704$	−3848.88	−0.206051
$705$	0	0
$706$	9591.00	0.511278
$707$	37409.4	1.98999
$708$	0	0
$709$	−5957.11	−0.315549	−0.157774	−	0.987475i	$-0.550432\pi$
$709$	−5957.11	−0.315549	−0.157774	+	0.987475i	$0.550432\pi$
$710$	33922.7	1.79309
$711$	0	0
$712$	−1139.40	−0.0599729
$713$	8901.55	0.467553
$714$	0	0
$715$	673.577	0.0352312
$716$	−18622.6	−0.972010
$717$	0	0
$718$	5768.71	0.299842
$719$	31140.8	1.61524	0.807620	−	0.589703i	$-0.200755\pi$
$719$	31140.8	1.61524	0.807620	+	0.589703i	$0.200755\pi$
$720$	0	0
$721$	−22828.0	−1.17914
$722$	1430.49	0.0737359
$723$	0	0
$724$	−18828.6	−0.966519
$725$	−34984.7	−1.79214
$726$	0	0
$727$	14969.7	0.763682	0.381841	−	0.924228i	$-0.375290\pi$
$727$	14969.7	0.763682	0.381841	+	0.924228i	$0.375290\pi$
$728$	−139.315	−0.00709251
$729$	0	0
$730$	−60635.4	−3.07427
$731$	−28097.5	−1.42165
$732$	0	0
$733$	−12414.1	−0.625545	−0.312772	−	0.949828i	$-0.601258\pi$
$733$	−12414.1	−0.625545	−0.312772	+	0.949828i	$0.601258\pi$
$734$	34806.5	1.75031
$735$	0	0
$736$	−13359.1	−0.669055
$737$	−865.481	−0.0432570
$738$	0	0
$739$	1324.11	0.0659111	0.0329555	−	0.999457i	$-0.489508\pi$
$739$	1324.11	0.0659111	0.0329555	+	0.999457i	$0.489508\pi$
$740$	20606.1	1.02364
$741$	0	0
$742$	15909.8	0.787150
$743$	4391.55	0.216838	0.108419	−	0.994105i	$-0.465421\pi$
$743$	4391.55	0.216838	0.108419	+	0.994105i	$0.465421\pi$
$744$	0	0
$745$	32480.1	1.59729
$746$	−36455.0	−1.78916
$747$	0	0
$748$	3920.92	0.191662
$749$	−33789.0	−1.64836
$750$	0	0
$751$	−31947.5	−1.55230	−0.776152	−	0.630546i	$-0.782831\pi$
$751$	−31947.5	−1.55230	−0.776152	+	0.630546i	$0.782831\pi$
$752$	7513.58	0.364351
$753$	0	0
$754$	3105.42	0.149990
$755$	42663.4	2.05653
$756$	0	0
$757$	18569.8	0.891585	0.445793	−	0.895136i	$-0.352922\pi$
$757$	18569.8	0.891585	0.445793	+	0.895136i	$0.352922\pi$
$758$	−24414.9	−1.16991
$759$	0	0
$760$	−406.861	−0.0194190
$761$	−5507.32	−0.262339	−0.131170	−	0.991360i	$-0.541873\pi$
$761$	−5507.32	−0.262339	−0.131170	+	0.991360i	$0.541873\pi$
$762$	0	0
$763$	−22490.6	−1.06712
$764$	10687.7	0.506111
$765$	0	0
$766$	−10424.7	−0.491723
$767$	832.669	0.0391994
$768$	0	0
$769$	−14977.9	−0.702362	−0.351181	−	0.936308i	$-0.614220\pi$
$769$	−14977.9	−0.702362	−0.351181	+	0.936308i	$0.614220\pi$
$770$	15101.5	0.706778
$771$	0	0
$772$	−24708.3	−1.15190
$773$	−19545.6	−0.909450	−0.454725	−	0.890632i	$-0.650262\pi$
$773$	−19545.6	−0.909450	−0.454725	+	0.890632i	$0.650262\pi$
$774$	0	0
$775$	34399.4	1.59440
$776$	259.190	0.0119902
$777$	0	0
$778$	23246.4	1.07124
$779$	5866.70	0.269829
$780$	0	0
$781$	−3839.98	−0.175935
$782$	13081.9	0.598221
$783$	0	0
$784$	−21513.4	−0.980020
$785$	26081.7	1.18585
$786$	0	0
$787$	−4274.62	−0.193613	−0.0968067	−	0.995303i	$-0.530863\pi$
$787$	−4274.62	−0.193613	−0.0968067	+	0.995303i	$0.530863\pi$
$788$	26536.2	1.19963
$789$	0	0
$790$	42470.7	1.91271
$791$	−30523.2	−1.37204
$792$	0	0
$793$	−1845.19	−0.0826287
$794$	−56482.6	−2.52455
$795$	0	0
$796$	15602.0	0.694722
$797$	25450.6	1.13112	0.565562	−	0.824706i	$-0.308659\pi$
$797$	25450.6	1.13112	0.565562	+	0.824706i	$0.308659\pi$
$798$	0	0
$799$	−7093.08	−0.314062
$800$	−51625.4	−2.28154
$801$	0	0
$802$	−39266.2	−1.72885
$803$	6863.81	0.301642
$804$	0	0
$805$	24714.5	1.08207
$806$	−3053.46	−0.133441
$807$	0	0
$808$	1709.80	0.0744439
$809$	−4002.04	−0.173924	−0.0869619	−	0.996212i	$-0.527716\pi$
$809$	−4002.04	−0.173924	−0.0869619	+	0.996212i	$0.527716\pi$
$810$	0	0
$811$	−37915.1	−1.64165	−0.820826	−	0.571179i	$-0.806486\pi$
$811$	−37915.1	−1.64165	−0.820826	+	0.571179i	$0.806486\pi$
$812$	34150.8	1.47593
$813$	0	0
$814$	−4755.39	−0.204762
$815$	−2311.99	−0.0993685
$816$	0	0
$817$	8530.05	0.365274
$818$	23006.2	0.983366
$819$	0	0
$820$	43126.3	1.83663
$821$	−4739.43	−0.201470	−0.100735	−	0.994913i	$-0.532119\pi$
$821$	−4739.43	−0.201470	−0.100735	+	0.994913i	$0.532119\pi$
$822$	0	0
$823$	20752.2	0.878952	0.439476	−	0.898254i	$-0.355164\pi$
$823$	20752.2	0.878952	0.439476	+	0.898254i	$0.355164\pi$
$824$	−1043.36	−0.0441106
$825$	0	0
$826$	18668.3	0.786384
$827$	−34264.8	−1.44075	−0.720377	−	0.693583i	$-0.756031\pi$
$827$	−34264.8	−1.44075	−0.720377	+	0.693583i	$0.756031\pi$
$828$	0	0
$829$	−39707.5	−1.66357	−0.831784	−	0.555100i	$-0.812680\pi$
$829$	−39707.5	−1.66357	−0.831784	+	0.555100i	$0.812680\pi$
$830$	20866.7	0.872645
$831$	0	0
$832$	2160.69	0.0900343
$833$	20309.4	0.844753
$834$	0	0
$835$	62001.8	2.56965
$836$	−1190.34	−0.0492450
$837$	0	0
$838$	−48584.6	−2.00278
$839$	4524.04	0.186159	0.0930794	−	0.995659i	$-0.470329\pi$
$839$	4524.04	0.186159	0.0930794	+	0.995659i	$0.470329\pi$
$840$	0	0
$841$	5064.59	0.207659
$842$	23132.9	0.946809
$843$	0	0
$844$	33192.1	1.35370
$845$	39462.7	1.60658
$846$	0	0
$847$	32678.5	1.32568
$848$	10302.4	0.417201
$849$	0	0
$850$	50554.1	2.03999
$851$	−7782.47	−0.313490
$852$	0	0
$853$	−7595.54	−0.304884	−0.152442	−	0.988312i	$-0.548714\pi$
$853$	−7595.54	−0.304884	−0.152442	+	0.988312i	$0.548714\pi$
$854$	−41368.8	−1.65762
$855$	0	0
$856$	−1544.34	−0.0616639
$857$	19528.9	0.778405	0.389203	−	0.921152i	$-0.372751\pi$
$857$	19528.9	0.778405	0.389203	+	0.921152i	$0.372751\pi$
$858$	0	0
$859$	25980.8	1.03196	0.515979	−	0.856601i	$-0.327428\pi$
$859$	25980.8	1.03196	0.515979	+	0.856601i	$0.327428\pi$
$860$	62704.7	2.48629
$861$	0	0
$862$	10977.0	0.433732
$863$	48294.6	1.90494	0.952472	−	0.304625i	$-0.0985311\pi$
$863$	48294.6	1.90494	0.952472	+	0.304625i	$0.0985311\pi$
$864$	0	0
$865$	−6575.43	−0.258464
$866$	22440.1	0.880536
$867$	0	0
$868$	−33579.4	−1.31309
$869$	−4807.61	−0.187672
$870$	0	0
$871$	485.866	0.0189012
$872$	−1027.94	−0.0399201
$873$	0	0
$874$	−3971.51	−0.153705
$875$	36942.3	1.42729
$876$	0	0
$877$	44377.5	1.70869	0.854346	−	0.519705i	$-0.173958\pi$
$877$	44377.5	1.70869	0.854346	+	0.519705i	$0.173958\pi$
$878$	−33282.5	−1.27930
$879$	0	0
$880$	9779.02	0.374603
$881$	12139.4	0.464231	0.232116	−	0.972688i	$-0.425435\pi$
$881$	12139.4	0.464231	0.232116	+	0.972688i	$0.425435\pi$
$882$	0	0
$883$	5048.07	0.192391	0.0961954	−	0.995362i	$-0.469333\pi$
$883$	5048.07	0.192391	0.0961954	+	0.995362i	$0.469333\pi$
$884$	−2201.14	−0.0837470
$885$	0	0
$886$	−24388.4	−0.924767
$887$	−20373.4	−0.771221	−0.385610	−	0.922662i	$-0.626009\pi$
$887$	−20373.4	−0.771221	−0.385610	+	0.922662i	$0.626009\pi$
$888$	0	0
$889$	−22929.9	−0.865065
$890$	−69335.6	−2.61139
$891$	0	0
$892$	6359.14	0.238699
$893$	2153.37	0.0806940
$894$	0	0
$895$	43846.5	1.63757
$896$	−3902.40	−0.145502
$897$	0	0
$898$	−13651.2	−0.507289
$899$	−28960.8	−1.07441
$900$	0	0
$901$	−9725.85	−0.359617
$902$	−9952.52	−0.367386
$903$	0	0
$904$	−1395.07	−0.0513267
$905$	44331.6	1.62832
$906$	0	0
$907$	−7456.13	−0.272962	−0.136481	−	0.990643i	$-0.543579\pi$
$907$	−7456.13	−0.272962	−0.136481	+	0.990643i	$0.543579\pi$
$908$	11562.2	0.422581
$909$	0	0
$910$	−8477.71	−0.308828
$911$	10653.2	0.387440	0.193720	−	0.981057i	$-0.437945\pi$
$911$	10653.2	0.387440	0.193720	+	0.981057i	$0.437945\pi$
$912$	0	0
$913$	−2362.07	−0.0856224
$914$	−1990.59	−0.0720380
$915$	0	0
$916$	−40438.6	−1.45866
$917$	−60579.8	−2.18159
$918$	0	0
$919$	−12569.7	−0.451183	−0.225591	−	0.974222i	$-0.572431\pi$
$919$	−12569.7	−0.451183	−0.225591	+	0.974222i	$0.572431\pi$
$920$	1129.58	0.0404795
$921$	0	0
$922$	−2164.59	−0.0773179
$923$	2155.70	0.0768752
$924$	0	0
$925$	−30074.8	−1.06903
$926$	73466.8	2.60720
$927$	0	0
$928$	43463.4	1.53745
$929$	−4920.06	−0.173759	−0.0868795	−	0.996219i	$-0.527689\pi$
$929$	−4920.06	−0.173759	−0.0868795	+	0.996219i	$0.527689\pi$
$930$	0	0
$931$	−6165.67	−0.217048
$932$	−16475.0	−0.579031
$933$	0	0
$934$	−49433.4	−1.73181
$935$	−9231.74	−0.322898
$936$	0	0
$937$	1991.87	0.0694465	0.0347233	−	0.999397i	$-0.488945\pi$
$937$	1991.87	0.0694465	0.0347233	+	0.999397i	$0.488945\pi$
$938$	10893.0	0.379179
$939$	0	0
$940$	15829.5	0.549256
$941$	7640.33	0.264684	0.132342	−	0.991204i	$-0.457750\pi$
$941$	7640.33	0.264684	0.132342	+	0.991204i	$0.457750\pi$
$942$	0	0
$943$	−16287.9	−0.562467
$944$	12088.7	0.416795
$945$	0	0
$946$	−14470.7	−0.497340
$947$	6521.15	0.223769	0.111884	−	0.993721i	$-0.464311\pi$
$947$	6521.15	0.223769	0.111884	+	0.993721i	$0.464311\pi$
$948$	0	0
$949$	−3853.22	−0.131803
$950$	−15347.6	−0.524149
$951$	0	0
$952$	1909.38	0.0650037
$953$	35757.9	1.21544	0.607719	−	0.794152i	$-0.292085\pi$
$953$	35757.9	1.21544	0.607719	+	0.794152i	$0.292085\pi$
$954$	0	0
$955$	−25164.1	−0.852659
$956$	−29057.3	−0.983034
$957$	0	0
$958$	41881.3	1.41244
$959$	57997.6	1.95291
$960$	0	0
$961$	−1314.75	−0.0441323
$962$	2669.59	0.0894710
$963$	0	0
$964$	49411.4	1.65086
$965$	58175.1	1.94065
$966$	0	0
$967$	−6342.30	−0.210915	−0.105457	−	0.994424i	$-0.533631\pi$
$967$	−6342.30	−0.210915	−0.105457	+	0.994424i	$0.533631\pi$
$968$	1493.58	0.0495924
$969$	0	0
$970$	15772.5	0.522086
$971$	−30351.2	−1.00311	−0.501553	−	0.865127i	$-0.667238\pi$
$971$	−30351.2	−1.00311	−0.501553	+	0.865127i	$0.667238\pi$
$972$	0	0
$973$	7651.29	0.252096
$974$	−44491.4	−1.46365
$975$	0	0
$976$	−26788.5	−0.878566
$977$	−39843.6	−1.30472	−0.652359	−	0.757910i	$-0.726220\pi$
$977$	−39843.6	−1.30472	−0.652359	+	0.757910i	$0.726220\pi$
$978$	0	0
$979$	7848.66	0.256225
$980$	−45324.1	−1.47737
$981$	0	0
$982$	2127.27	0.0691282
$983$	−24068.7	−0.780949	−0.390475	−	0.920614i	$-0.627689\pi$
$983$	−24068.7	−0.780949	−0.390475	+	0.920614i	$0.627689\pi$
$984$	0	0
$985$	−62478.9	−2.02106
$986$	−42561.5	−1.37468
$987$	0	0
$988$	668.238	0.0215177
$989$	−23682.2	−0.761426
$990$	0	0
$991$	3235.83	0.103723	0.0518615	−	0.998654i	$-0.483485\pi$
$991$	3235.83	0.103723	0.0518615	+	0.998654i	$0.483485\pi$
$992$	−42736.2	−1.36782
$993$	0	0
$994$	48330.4	1.54220
$995$	−36734.6	−1.17042
$996$	0	0
$997$	19444.5	0.617665	0.308833	−	0.951116i	$-0.400062\pi$
$997$	19444.5	0.617665	0.308833	+	0.951116i	$0.400062\pi$
$998$	5230.25	0.165892
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.4.a.f.1.3		3
3.2	odd	2		19.4.a.b.1.1	✓	3
12.11	even	2		304.4.a.i.1.1		3
15.2	even	4		475.4.b.f.324.2		6
15.8	even	4		475.4.b.f.324.5		6
15.14	odd	2		475.4.a.f.1.3		3
21.20	even	2		931.4.a.c.1.1		3
24.5	odd	2		1216.4.a.s.1.1		3
24.11	even	2		1216.4.a.u.1.3		3
33.32	even	2		2299.4.a.h.1.3		3
57.56	even	2		361.4.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.b.1.1	✓	3	3.2	odd	2
171.4.a.f.1.3		3	1.1	even	1	trivial
304.4.a.i.1.1		3	12.11	even	2
361.4.a.i.1.3		3	57.56	even	2
475.4.a.f.1.3		3	15.14	odd	2
475.4.b.f.324.2		6	15.2	even	4
475.4.b.f.324.5		6	15.8	even	4
931.4.a.c.1.1		3	21.20	even	2
1216.4.a.s.1.1		3	24.5	odd	2
1216.4.a.u.1.3		3	24.11	even	2
2299.4.a.h.1.3		3	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

\(f(q)\)	\(=\)	\(q+3.96257 q^{2} +7.70200 q^{4} -18.1342 q^{5} -25.8362 q^{7} -1.18085 q^{8} -71.8581 q^{10} +8.13420 q^{11} -4.56640 q^{13} -102.378 q^{14} -66.2952 q^{16} +62.5850 q^{17} -19.0000 q^{19} -139.670 q^{20} +32.2324 q^{22} +52.7502 q^{23} +203.849 q^{25} -18.0947 q^{26} -198.990 q^{28} -171.620 q^{29} +168.749 q^{31} -253.253 q^{32} +247.998 q^{34} +468.519 q^{35} -147.534 q^{37} -75.2889 q^{38} +21.4138 q^{40} -308.774 q^{41} -448.950 q^{43} +62.6496 q^{44} +209.027 q^{46} -113.335 q^{47} +324.509 q^{49} +807.768 q^{50} -35.1704 q^{52} -155.402 q^{53} -147.507 q^{55} +30.5086 q^{56} -680.059 q^{58} -182.347 q^{59} +404.080 q^{61} +668.681 q^{62} -473.172 q^{64} +82.8080 q^{65} -106.400 q^{67} +482.030 q^{68} +1856.54 q^{70} -472.079 q^{71} +843.821 q^{73} -584.616 q^{74} -146.338 q^{76} -210.157 q^{77} -591.036 q^{79} +1202.21 q^{80} -1223.54 q^{82} -290.388 q^{83} -1134.93 q^{85} -1779.00 q^{86} -9.60526 q^{88} +964.896 q^{89} +117.978 q^{91} +406.282 q^{92} -449.099 q^{94} +344.550 q^{95} -219.495 q^{97} +1285.89 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 21 q^{4} - 14 q^{5} - 35 q^{7} - 27 q^{8} - 88 q^{10} - 16 q^{11} + 65 q^{13} - 37 q^{14} + 33 q^{16} - 29 q^{17} - 57 q^{19} - 100 q^{20} + 118 q^{22} + 101 q^{23} - 37 q^{25} - 299 q^{26}+ \cdots + 2450 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 21 * q^4 - 14 * q^5 - 35 * q^7 - 27 * q^8 - 88 * q^10 - 16 * q^11 + 65 * q^13 - 37 * q^14 + 33 * q^16 - 29 * q^17 - 57 * q^19 - 100 * q^20 + 118 * q^22 + 101 * q^23 - 37 * q^25 - 299 * q^26 - 493 * q^28 - 377 * q^29 - 140 * q^31 - 579 * q^32 + 329 * q^34 + 438 * q^35 - 290 * q^37 + 57 * q^38 - 72 * q^40 - 956 * q^41 - 570 * q^43 - 110 * q^44 - 239 * q^46 - 66 * q^47 - 98 * q^49 + 1639 * q^50 + 697 * q^52 - 817 * q^53 - 198 * q^55 + 915 * q^56 + 647 * q^58 - 265 * q^59 + 988 * q^61 + 1436 * q^62 - 159 * q^64 + 240 * q^65 - 207 * q^67 + 1529 * q^68 + 2040 * q^70 - 846 * q^71 + 627 * q^73 + 770 * q^74 - 399 * q^76 - 88 * q^77 + 382 * q^79 + 1472 * q^80 + 872 * q^82 + 766 * q^83 - 1242 * q^85 - 2592 * q^86 + 342 * q^88 + 172 * q^89 - 457 * q^91 + 2677 * q^92 - 96 * q^94 + 266 * q^95 - 2450 * q^97 + 2450 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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