LMFDB - Embedded newform 169.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	169.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:	$9.97132279097$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.438447 q^{2} -3.68466 q^{3} -7.80776 q^{4} +17.8078 q^{5} -1.61553 q^{6} -5.43845 q^{7} -6.93087 q^{8} -13.4233 q^{9} +O(q^{10})$	\(q+0.438447 q^{2} -3.68466 q^{3} -7.80776 q^{4} +17.8078 q^{5} -1.61553 q^{6} -5.43845 q^{7} -6.93087 q^{8} -13.4233 q^{9} +7.80776 q^{10} +22.4233 q^{11} +28.7689 q^{12} -2.38447 q^{14} -65.6155 q^{15} +59.4233 q^{16} +67.9848 q^{17} -5.88540 q^{18} +80.8078 q^{19} -139.039 q^{20} +20.0388 q^{21} +9.83143 q^{22} +140.531 q^{23} +25.5379 q^{24} +192.116 q^{25} +148.946 q^{27} +42.4621 q^{28} -106.693 q^{29} -28.7689 q^{30} +276.155 q^{31} +81.5009 q^{32} -82.6222 q^{33} +29.8078 q^{34} -96.8466 q^{35} +104.806 q^{36} +4.29168 q^{37} +35.4299 q^{38} -123.423 q^{40} -227.769 q^{41} +8.78596 q^{42} +27.5294 q^{43} -175.076 q^{44} -239.039 q^{45} +61.6155 q^{46} -318.617 q^{47} -218.955 q^{48} -313.423 q^{49} +84.2329 q^{50} -250.501 q^{51} -67.6562 q^{53} +65.3050 q^{54} +399.309 q^{55} +37.6932 q^{56} -297.749 q^{57} -46.7793 q^{58} +291.115 q^{59} +512.311 q^{60} +663.311 q^{61} +121.080 q^{62} +73.0019 q^{63} -439.652 q^{64} -36.2255 q^{66} +425.101 q^{67} -530.810 q^{68} -517.810 q^{69} -42.4621 q^{70} +152.963 q^{71} +93.0351 q^{72} -117.268 q^{73} +1.88167 q^{74} -707.884 q^{75} -630.928 q^{76} -121.948 q^{77} +202.462 q^{79} +1058.20 q^{80} -186.386 q^{81} -99.8647 q^{82} -336.155 q^{83} -156.458 q^{84} +1210.66 q^{85} +12.0702 q^{86} +393.128 q^{87} -155.413 q^{88} -718.194 q^{89} -104.806 q^{90} -1097.23 q^{92} -1017.54 q^{93} -139.697 q^{94} +1439.01 q^{95} -300.303 q^{96} -759.368 q^{97} -137.420 q^{98} -300.994 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9}+O(q^{10}) $ 2 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 15 * q^5 + 38 * q^6 - 15 * q^7 + 15 * q^8 + 35 * q^9	\( 2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9} - 5 q^{10} - 17 q^{11} + 140 q^{12} - 46 q^{14} - 90 q^{15} + 57 q^{16} + 70 q^{17} + 215 q^{18} + 141 q^{19} - 175 q^{20} - 63 q^{21} - 170 q^{22} + 145 q^{23} + 216 q^{24} + 75 q^{25} + 335 q^{27} - 80 q^{28} + 34 q^{29} - 140 q^{30} + 140 q^{31} - 105 q^{32} - 425 q^{33} + 39 q^{34} - 70 q^{35} + 725 q^{36} + 190 q^{37} + 310 q^{38} - 185 q^{40} - 538 q^{41} - 370 q^{42} + 455 q^{43} - 680 q^{44} - 375 q^{45} + 82 q^{46} - 60 q^{47} - 240 q^{48} - 565 q^{49} - 450 q^{50} - 233 q^{51} + 545 q^{53} + 914 q^{54} + 510 q^{55} - 172 q^{56} + 225 q^{57} + 595 q^{58} + 809 q^{59} + 200 q^{60} + 502 q^{61} - 500 q^{62} - 390 q^{63} - 1271 q^{64} - 1598 q^{66} + 475 q^{67} - 505 q^{68} - 479 q^{69} + 80 q^{70} - 127 q^{71} + 1155 q^{72} - 585 q^{73} + 849 q^{74} - 1725 q^{75} + 140 q^{76} + 255 q^{77} + 240 q^{79} + 1065 q^{80} + 122 q^{81} - 1515 q^{82} - 260 q^{83} - 1220 q^{84} + 1205 q^{85} + 1962 q^{86} + 1615 q^{87} - 1020 q^{88} - 921 q^{89} - 725 q^{90} - 1040 q^{92} - 2200 q^{93} + 1040 q^{94} + 1270 q^{95} - 1920 q^{96} + 415 q^{97} - 1285 q^{98} - 2210 q^{99}+O(q^{100}) \) 2 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 15 * q^5 + 38 * q^6 - 15 * q^7 + 15 * q^8 + 35 * q^9 - 5 * q^10 - 17 * q^11 + 140 * q^12 - 46 * q^14 - 90 * q^15 + 57 * q^16 + 70 * q^17 + 215 * q^18 + 141 * q^19 - 175 * q^20 - 63 * q^21 - 170 * q^22 + 145 * q^23 + 216 * q^24 + 75 * q^25 + 335 * q^27 - 80 * q^28 + 34 * q^29 - 140 * q^30 + 140 * q^31 - 105 * q^32 - 425 * q^33 + 39 * q^34 - 70 * q^35 + 725 * q^36 + 190 * q^37 + 310 * q^38 - 185 * q^40 - 538 * q^41 - 370 * q^42 + 455 * q^43 - 680 * q^44 - 375 * q^45 + 82 * q^46 - 60 * q^47 - 240 * q^48 - 565 * q^49 - 450 * q^50 - 233 * q^51 + 545 * q^53 + 914 * q^54 + 510 * q^55 - 172 * q^56 + 225 * q^57 + 595 * q^58 + 809 * q^59 + 200 * q^60 + 502 * q^61 - 500 * q^62 - 390 * q^63 - 1271 * q^64 - 1598 * q^66 + 475 * q^67 - 505 * q^68 - 479 * q^69 + 80 * q^70 - 127 * q^71 + 1155 * q^72 - 585 * q^73 + 849 * q^74 - 1725 * q^75 + 140 * q^76 + 255 * q^77 + 240 * q^79 + 1065 * q^80 + 122 * q^81 - 1515 * q^82 - 260 * q^83 - 1220 * q^84 + 1205 * q^85 + 1962 * q^86 + 1615 * q^87 - 1020 * q^88 - 921 * q^89 - 725 * q^90 - 1040 * q^92 - 2200 * q^93 + 1040 * q^94 + 1270 * q^95 - 1920 * q^96 + 415 * q^97 - 1285 * q^98 - 2210 * 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 $

$2$	0.438447	0.155014	0.0775072	−	0.996992i	$-0.475304\pi$
$2$	0.438447	0.155014	0.0775072	+	0.996992i	$0.475304\pi$
$3$	−3.68466	−0.709113	−0.354556	−	0.935035i	$-0.615368\pi$
$3$	−3.68466	−0.709113	−0.354556	+	0.935035i	$0.615368\pi$
$4$	−7.80776	−0.975971
$5$	17.8078	1.59277	0.796387	−	0.604787i	$-0.206742\pi$
$5$	17.8078	1.59277	0.796387	+	0.604787i	$0.206742\pi$
$6$	−1.61553	−0.109923
$7$	−5.43845	−0.293649	−0.146824	−	0.989163i	$-0.546905\pi$
$7$	−5.43845	−0.293649	−0.146824	+	0.989163i	$0.546905\pi$
$8$	−6.93087	−0.306304
$9$	−13.4233	−0.497159
$10$	7.80776	0.246903
$11$	22.4233	0.614625	0.307313	−	0.951609i	$-0.400570\pi$
$11$	22.4233	0.614625	0.307313	+	0.951609i	$0.400570\pi$
$12$	28.7689	0.692073
$13$	0	0
$13$	0	0
$14$	−2.38447	−0.0455198
$15$	−65.6155	−1.12946
$16$	59.4233	0.928489
$17$	67.9848	0.969926	0.484963	−	0.874535i	$-0.338833\pi$
$17$	67.9848	0.969926	0.484963	+	0.874535i	$0.338833\pi$
$18$	−5.88540	−0.0770668
$19$	80.8078	0.975714	0.487857	−	0.872923i	$-0.337779\pi$
$19$	80.8078	0.975714	0.487857	+	0.872923i	$0.337779\pi$
$20$	−139.039	−1.55450
$21$	20.0388	0.208230
$22$	9.83143	0.0952758
$23$	140.531	1.27403	0.637017	−	0.770850i	$-0.280168\pi$
$23$	140.531	1.27403	0.637017	+	0.770850i	$0.280168\pi$
$24$	25.5379	0.217204
$25$	192.116	1.53693
$26$	0	0
$27$	148.946	1.06165
$28$	42.4621	0.286592
$29$	−106.693	−0.683187	−0.341594	−	0.939848i	$-0.610967\pi$
$29$	−106.693	−0.683187	−0.341594	+	0.939848i	$0.610967\pi$
$30$	−28.7689	−0.175082
$31$	276.155	1.59997	0.799983	−	0.600023i	$-0.204842\pi$
$31$	276.155	1.59997	0.799983	+	0.600023i	$0.204842\pi$
$32$	81.5009	0.450233
$33$	−82.6222	−0.435839
$34$	29.8078	0.150353
$35$	−96.8466	−0.467716
$36$	104.806	0.485212
$37$	4.29168	0.0190688	0.00953442	−	0.999955i	$-0.496965\pi$
$37$	4.29168	0.0190688	0.00953442	+	0.999955i	$0.496965\pi$
$38$	35.4299	0.151250
$39$	0	0
$40$	−123.423	−0.487873
$41$	−227.769	−0.867598	−0.433799	−	0.901010i	$-0.642827\pi$
$41$	−227.769	−0.867598	−0.433799	+	0.901010i	$0.642827\pi$
$42$	8.78596	0.0322787
$43$	27.5294	0.0976323	0.0488162	−	0.998808i	$-0.484455\pi$
$43$	27.5294	0.0976323	0.0488162	+	0.998808i	$0.484455\pi$
$44$	−175.076	−0.599856
$45$	−239.039	−0.791862
$46$	61.6155	0.197494
$47$	−318.617	−0.988832	−0.494416	−	0.869225i	$-0.664618\pi$
$47$	−318.617	−0.988832	−0.494416	+	0.869225i	$0.664618\pi$
$48$	−218.955	−0.658403
$49$	−313.423	−0.913771
$50$	84.2329	0.238247
$51$	−250.501	−0.687787
$52$	0	0
$53$	−67.6562	−0.175345	−0.0876726	−	0.996149i	$-0.527943\pi$
$53$	−67.6562	−0.175345	−0.0876726	+	0.996149i	$0.527943\pi$
$54$	65.3050	0.164572
$55$	399.309	0.978960
$56$	37.6932	0.0899457
$57$	−297.749	−0.691892
$58$	−46.7793	−0.105904
$59$	291.115	0.642371	0.321186	−	0.947016i	$-0.395919\pi$
$59$	291.115	0.642371	0.321186	+	0.947016i	$0.395919\pi$
$60$	512.311	1.10232
$61$	663.311	1.39227	0.696133	−	0.717913i	$-0.254902\pi$
$61$	663.311	1.39227	0.696133	+	0.717913i	$0.254902\pi$
$62$	121.080	0.248018
$63$	73.0019	0.145990
$64$	−439.652	−0.858696
$65$	0	0
$66$	−36.2255	−0.0675613
$67$	425.101	0.775140	0.387570	−	0.921840i	$-0.373315\pi$
$67$	425.101	0.775140	0.387570	+	0.921840i	$0.373315\pi$
$68$	−530.810	−0.946619
$69$	−517.810	−0.903434
$70$	−42.4621	−0.0725028
$71$	152.963	0.255681	0.127841	−	0.991795i	$-0.459195\pi$
$71$	152.963	0.255681	0.127841	+	0.991795i	$0.459195\pi$
$72$	93.0351	0.152282
$73$	−117.268	−0.188016	−0.0940081	−	0.995571i	$-0.529968\pi$
$73$	−117.268	−0.188016	−0.0940081	+	0.995571i	$0.529968\pi$
$74$	1.88167	0.00295595
$75$	−707.884	−1.08986
$76$	−630.928	−0.952268
$77$	−121.948	−0.180484
$78$	0	0
$79$	202.462	0.288339	0.144169	−	0.989553i	$-0.453949\pi$
$79$	202.462	0.288339	0.144169	+	0.989553i	$0.453949\pi$
$80$	1058.20	1.47887
$81$	−186.386	−0.255674
$82$	−99.8647	−0.134490
$83$	−336.155	−0.444552	−0.222276	−	0.974984i	$-0.571349\pi$
$83$	−336.155	−0.444552	−0.222276	+	0.974984i	$0.571349\pi$
$84$	−156.458	−0.203226
$85$	1210.66	1.54487
$86$	12.0702	0.0151344
$87$	393.128	0.484457
$88$	−155.413	−0.188262
$89$	−718.194	−0.855376	−0.427688	−	0.903927i	$-0.640672\pi$
$89$	−718.194	−0.855376	−0.427688	+	0.903927i	$0.640672\pi$
$90$	−104.806	−0.122750
$91$	0	0
$92$	−1097.23	−1.24342
$93$	−1017.54	−1.13456
$94$	−139.697	−0.153283
$95$	1439.01	1.55409
$96$	−300.303	−0.319266
$97$	−759.368	−0.794868	−0.397434	−	0.917631i	$-0.630099\pi$
$97$	−759.368	−0.794868	−0.397434	+	0.917631i	$0.630099\pi$
$98$	−137.420	−0.141648
$99$	−300.994	−0.305566
$100$	−1500.00	−1.50000
$101$	−348.697	−0.343531	−0.171766	−	0.985138i	$-0.554947\pi$
$101$	−348.697	−0.343531	−0.171766	+	0.985138i	$0.554947\pi$
$102$	−109.831	−0.106617
$103$	−580.303	−0.555136	−0.277568	−	0.960706i	$-0.589528\pi$
$103$	−580.303	−0.555136	−0.277568	+	0.960706i	$0.589528\pi$
$104$	0	0
$105$	356.847	0.331663
$106$	−29.6637	−0.0271810
$107$	571.493	0.516340	0.258170	−	0.966100i	$-0.416881\pi$
$107$	571.493	0.516340	0.258170	+	0.966100i	$0.416881\pi$
$108$	−1162.94	−1.03614
$109$	−176.004	−0.154661	−0.0773307	−	0.997005i	$-0.524640\pi$
$109$	−176.004	−0.154661	−0.0773307	+	0.997005i	$0.524640\pi$
$110$	175.076	0.151753
$111$	−15.8134	−0.0135220
$112$	−323.170	−0.272649
$113$	1264.88	1.05301	0.526505	−	0.850172i	$-0.323502\pi$
$113$	1264.88	1.05301	0.526505	+	0.850172i	$0.323502\pi$
$114$	−130.547	−0.107253
$115$	2502.55	2.02925
$116$	833.035	0.666770
$117$	0	0
$118$	127.638	0.0995768
$119$	−369.732	−0.284817
$120$	454.773	0.345957
$121$	−828.196	−0.622236
$122$	290.827	0.215821
$123$	839.251	0.615225
$124$	−2156.16	−1.56152
$125$	1195.19	0.855211
$126$	32.0075	0.0226306
$127$	2604.11	1.81950	0.909752	−	0.415151i	$-0.136271\pi$
$127$	2604.11	1.81950	0.909752	+	0.415151i	$0.136271\pi$
$128$	−844.772	−0.583344
$129$	−101.436	−0.0692323
$130$	0	0
$131$	2131.70	1.42174	0.710870	−	0.703324i	$-0.248302\pi$
$131$	2131.70	1.42174	0.710870	+	0.703324i	$0.248302\pi$
$132$	645.094	0.425366
$133$	−439.469	−0.286517
$134$	186.384	0.120158
$135$	2652.40	1.69098
$136$	−471.194	−0.297092
$137$	−687.985	−0.429040	−0.214520	−	0.976720i	$-0.568819\pi$
$137$	−687.985	−0.429040	−0.214520	+	0.976720i	$0.568819\pi$
$138$	−227.032	−0.140045
$139$	−679.580	−0.414685	−0.207343	−	0.978268i	$-0.566482\pi$
$139$	−679.580	−0.414685	−0.207343	+	0.978268i	$0.566482\pi$
$140$	756.155	0.456477
$141$	1174.00	0.701194
$142$	67.0662	0.0396343
$143$	0	0
$144$	−797.656	−0.461607
$145$	−1899.97	−1.08816
$146$	−51.4158	−0.0291452
$147$	1154.86	0.647966
$148$	−33.5084	−0.0186106
$149$	1975.46	1.08615	0.543074	−	0.839685i	$-0.317260\pi$
$149$	1975.46	1.08615	0.543074	+	0.839685i	$0.317260\pi$
$150$	−310.370	−0.168944
$151$	−1803.24	−0.971824	−0.485912	−	0.874008i	$-0.661513\pi$
$151$	−1803.24	−0.971824	−0.485912	+	0.874008i	$0.661513\pi$
$152$	−560.068	−0.298865
$153$	−912.580	−0.482208
$154$	−53.4677	−0.0279776
$155$	4917.71	2.54839
$156$	0	0
$157$	−397.168	−0.201894	−0.100947	−	0.994892i	$-0.532187\pi$
$157$	−397.168	−0.201894	−0.100947	+	0.994892i	$0.532187\pi$
$158$	88.7689	0.0446967
$159$	249.290	0.124340
$160$	1451.35	0.717120
$161$	−764.272	−0.374118
$162$	−81.7206	−0.0396332
$163$	−941.393	−0.452365	−0.226183	−	0.974085i	$-0.572625\pi$
$163$	−941.393	−0.452365	−0.226183	+	0.974085i	$0.572625\pi$
$164$	1778.37	0.846750
$165$	−1471.32	−0.694193
$166$	−147.386	−0.0689120
$167$	−3680.43	−1.70539	−0.852696	−	0.522408i	$-0.825034\pi$
$167$	−3680.43	−1.70539	−0.852696	+	0.522408i	$0.825034\pi$
$168$	−138.886	−0.0637817
$169$	0	0
$170$	530.810	0.239478
$171$	−1084.71	−0.485085
$172$	−214.943	−0.0952863
$173$	1422.77	0.625269	0.312634	−	0.949874i	$-0.398789\pi$
$173$	1422.77	0.625269	0.312634	+	0.949874i	$0.398789\pi$
$174$	172.366	0.0750978
$175$	−1044.82	−0.451318
$176$	1332.47	0.570673
$177$	−1072.66	−0.455514
$178$	−314.890	−0.132596
$179$	1167.89	0.487666	0.243833	−	0.969817i	$-0.421595\pi$
$179$	1167.89	0.487666	0.243833	+	0.969817i	$0.421595\pi$
$180$	1866.36	0.772834
$181$	−1133.96	−0.465673	−0.232836	−	0.972516i	$-0.574801\pi$
$181$	−1133.96	−0.465673	−0.232836	+	0.972516i	$0.574801\pi$
$182$	0	0
$183$	−2444.07	−0.987274
$184$	−974.004	−0.390242
$185$	76.4252	0.0303724
$186$	−446.137	−0.175873
$187$	1524.44	0.596141
$188$	2487.69	0.965071
$189$	−810.035	−0.311753
$190$	630.928	0.240907
$191$	2682.12	1.01608	0.508040	−	0.861333i	$-0.330370\pi$
$191$	2682.12	1.01608	0.508040	+	0.861333i	$0.330370\pi$
$192$	1619.97	0.608913
$193$	1970.67	0.734983	0.367491	−	0.930027i	$-0.380217\pi$
$193$	1970.67	0.734983	0.367491	+	0.930027i	$0.380217\pi$
$194$	−332.943	−0.123216
$195$	0	0
$196$	2447.14	0.891813
$197$	4016.05	1.45244	0.726222	−	0.687460i	$-0.241274\pi$
$197$	4016.05	1.45244	0.726222	+	0.687460i	$0.241274\pi$
$198$	−131.970	−0.0473672
$199$	−4226.06	−1.50541	−0.752707	−	0.658356i	$-0.771252\pi$
$199$	−4226.06	−1.50541	−0.752707	+	0.658356i	$0.771252\pi$
$200$	−1331.53	−0.470768
$201$	−1566.35	−0.549662
$202$	−152.885	−0.0532523
$203$	580.245	0.200617
$204$	1955.85	0.671260
$205$	−4056.06	−1.38189
$206$	−254.432	−0.0860541
$207$	−1886.39	−0.633398
$208$	0	0
$209$	1811.98	0.599699
$210$	156.458	0.0514126
$211$	1364.67	0.445249	0.222625	−	0.974904i	$-0.428538\pi$
$211$	1364.67	0.445249	0.222625	+	0.974904i	$0.428538\pi$
$212$	528.244	0.171132
$213$	−563.617	−0.181307
$214$	250.570	0.0800401
$215$	490.237	0.155506
$216$	−1032.33	−0.325189
$217$	−1501.86	−0.469828
$218$	−77.1683	−0.0239748
$219$	432.093	0.133325
$220$	−3117.71	−0.955436
$221$	0	0
$222$	−6.93332	−0.00209610
$223$	1059.47	0.318149	0.159075	−	0.987267i	$-0.449149\pi$
$223$	1059.47	0.318149	0.159075	+	0.987267i	$0.449149\pi$
$224$	−443.239	−0.132210
$225$	−2578.84	−0.764099
$226$	554.584	0.163232
$227$	−3464.19	−1.01289	−0.506446	−	0.862272i	$-0.669041\pi$
$227$	−3464.19	−1.01289	−0.506446	+	0.862272i	$0.669041\pi$
$228$	2324.75	0.675266
$229$	2324.64	0.670815	0.335407	−	0.942073i	$-0.391126\pi$
$229$	2324.64	0.670815	0.335407	+	0.942073i	$0.391126\pi$
$230$	1097.23	0.314563
$231$	449.336	0.127983
$232$	739.476	0.209263
$233$	−3731.01	−1.04904	−0.524521	−	0.851398i	$-0.675755\pi$
$233$	−3731.01	−1.04904	−0.524521	+	0.851398i	$0.675755\pi$
$234$	0	0
$235$	−5673.86	−1.57499
$236$	−2272.95	−0.626935
$237$	−746.004	−0.204465
$238$	−162.108	−0.0441508
$239$	−6044.47	−1.63592	−0.817958	−	0.575278i	$-0.804894\pi$
$239$	−6044.47	−1.63592	−0.817958	+	0.575278i	$0.804894\pi$
$240$	−3899.09	−1.04869
$241$	5173.96	1.38292	0.691461	−	0.722414i	$-0.256967\pi$
$241$	5173.96	1.38292	0.691461	+	0.722414i	$0.256967\pi$
$242$	−363.120	−0.0964556
$243$	−3334.77	−0.880353
$244$	−5178.97	−1.35881
$245$	−5581.37	−1.45543
$246$	367.967	0.0953688
$247$	0	0
$248$	−1914.00	−0.490076
$249$	1238.62	0.315238
$250$	524.029	0.132570
$251$	5620.73	1.41346	0.706728	−	0.707486i	$-0.250171\pi$
$251$	5620.73	1.41346	0.706728	+	0.707486i	$0.250171\pi$
$252$	−569.981	−0.142482
$253$	3151.17	0.783054
$254$	1141.76	0.282050
$255$	−4460.86	−1.09549
$256$	3146.83	0.768270
$257$	−1674.14	−0.406342	−0.203171	−	0.979143i	$-0.565125\pi$
$257$	−1674.14	−0.406342	−0.203171	+	0.979143i	$0.565125\pi$
$258$	−44.4745	−0.0107320
$259$	−23.3401	−0.00559954
$260$	0	0
$261$	1432.17	0.339653
$262$	934.640	0.220390
$263$	−6309.18	−1.47924	−0.739622	−	0.673023i	$-0.764996\pi$
$263$	−6309.18	−1.47924	−0.739622	+	0.673023i	$0.764996\pi$
$264$	572.644	0.133499
$265$	−1204.81	−0.279285
$266$	−192.684	−0.0444143
$267$	2646.30	0.606558
$268$	−3319.09	−0.756514
$269$	−2482.73	−0.562731	−0.281366	−	0.959601i	$-0.590787\pi$
$269$	−2482.73	−0.562731	−0.281366	+	0.959601i	$0.590787\pi$
$270$	1162.94	0.262126
$271$	−2835.72	−0.635638	−0.317819	−	0.948151i	$-0.602950\pi$
$271$	−2835.72	−0.635638	−0.317819	+	0.948151i	$0.602950\pi$
$272$	4039.88	0.900566
$273$	0	0
$274$	−301.645	−0.0665075
$275$	4307.88	0.944637
$276$	4042.94	0.881725
$277$	−3837.51	−0.832396	−0.416198	−	0.909274i	$-0.636638\pi$
$277$	−3837.51	−0.832396	−0.416198	+	0.909274i	$0.636638\pi$
$278$	−297.960	−0.0642822
$279$	−3706.91	−0.795438
$280$	671.231	0.143263
$281$	9122.13	1.93659	0.968293	−	0.249819i	$-0.0803712\pi$
$281$	9122.13	1.93659	0.968293	+	0.249819i	$0.0803712\pi$
$282$	514.735	0.108695
$283$	2127.85	0.446952	0.223476	−	0.974709i	$-0.428260\pi$
$283$	2127.85	0.446952	0.223476	+	0.974709i	$0.428260\pi$
$284$	−1194.30	−0.249537
$285$	−5302.24	−1.10203
$286$	0	0
$287$	1238.71	0.254769
$288$	−1094.01	−0.223838
$289$	−291.061	−0.0592430
$290$	−833.035	−0.168681
$291$	2798.01	0.563651
$292$	915.601	0.183498
$293$	−8274.77	−1.64989	−0.824944	−	0.565215i	$-0.808793\pi$
$293$	−8274.77	−1.64989	−0.824944	+	0.565215i	$0.808793\pi$
$294$	506.344	0.100444
$295$	5184.10	1.02315
$296$	−29.7450	−0.00584086
$297$	3339.86	0.652520
$298$	866.136	0.168369
$299$	0	0
$300$	5526.99	1.06367
$301$	−149.717	−0.0286696
$302$	−790.625	−0.150647
$303$	1284.83	0.243602
$304$	4801.86	0.905940
$305$	11812.1	2.21757
$306$	−400.118	−0.0747492
$307$	3610.49	0.671211	0.335605	−	0.942003i	$-0.391059\pi$
$307$	3610.49	0.671211	0.335605	+	0.942003i	$0.391059\pi$
$308$	952.140	0.176147
$309$	2138.22	0.393654
$310$	2156.16	0.395037
$311$	3331.06	0.607354	0.303677	−	0.952775i	$-0.401786\pi$
$311$	3331.06	0.607354	0.303677	+	0.952775i	$0.401786\pi$
$312$	0	0
$313$	−358.125	−0.0646724	−0.0323362	−	0.999477i	$-0.510295\pi$
$313$	−358.125	−0.0646724	−0.0323362	+	0.999477i	$0.510295\pi$
$314$	−174.137	−0.0312966
$315$	1300.00	0.232529
$316$	−1580.78	−0.281410
$317$	−3047.46	−0.539944	−0.269972	−	0.962868i	$-0.587014\pi$
$317$	−3047.46	−0.539944	−0.269972	+	0.962868i	$0.587014\pi$
$318$	109.301	0.0192744
$319$	−2392.41	−0.419904
$320$	−7829.23	−1.36771
$321$	−2105.76	−0.366143
$322$	−335.093	−0.0579938
$323$	5493.70	0.946371
$324$	1455.26	0.249530
$325$	0	0
$326$	−412.751	−0.0701232
$327$	648.514	0.109672
$328$	1578.64	0.265749
$329$	1732.78	0.290369
$330$	−645.094	−0.107610
$331$	−7694.77	−1.27777	−0.638887	−	0.769301i	$-0.720605\pi$
$331$	−7694.77	−1.27777	−0.638887	+	0.769301i	$0.720605\pi$
$332$	2624.62	0.433870
$333$	−57.6084	−0.00948025
$334$	−1613.68	−0.264360
$335$	7570.10	1.23462
$336$	1190.77	0.193339
$337$	4712.21	0.761693	0.380846	−	0.924638i	$-0.375633\pi$
$337$	4712.21	0.761693	0.380846	+	0.924638i	$0.375633\pi$
$338$	0	0
$339$	−4660.66	−0.746703
$340$	−9452.53	−1.50775
$341$	6192.31	0.983380
$342$	−475.586	−0.0751952
$343$	3569.92	0.561976
$344$	−190.803	−0.0299052
$345$	−9221.03	−1.43897
$346$	623.811	0.0969257
$347$	−5261.98	−0.814058	−0.407029	−	0.913415i	$-0.633435\pi$
$347$	−5261.98	−0.814058	−0.407029	+	0.913415i	$0.633435\pi$
$348$	−3069.45	−0.472815
$349$	−50.3345	−0.00772018	−0.00386009	−	0.999993i	$-0.501229\pi$
$349$	−50.3345	−0.00772018	−0.00386009	+	0.999993i	$0.501229\pi$
$350$	−458.096	−0.0699608
$351$	0	0
$352$	1827.52	0.276725
$353$	−9057.64	−1.36569	−0.682846	−	0.730562i	$-0.739258\pi$
$353$	−9057.64	−1.36569	−0.682846	+	0.730562i	$0.739258\pi$
$354$	−470.304	−0.0706112
$355$	2723.93	0.407243
$356$	5607.49	0.834821
$357$	1362.34	0.201968
$358$	512.059	0.0755953
$359$	−7177.86	−1.05525	−0.527623	−	0.849479i	$-0.676917\pi$
$359$	−7177.86	−1.05525	−0.527623	+	0.849479i	$0.676917\pi$
$360$	1656.75	0.242551
$361$	−329.105	−0.0479815
$362$	−497.183	−0.0721861
$363$	3051.62	0.441235
$364$	0	0
$365$	−2088.28	−0.299467
$366$	−1071.60	−0.153042
$367$	4004.14	0.569522	0.284761	−	0.958599i	$-0.408086\pi$
$367$	4004.14	0.569522	0.284761	+	0.958599i	$0.408086\pi$
$368$	8350.83	1.18293
$369$	3057.41	0.431334
$370$	33.5084	0.00470816
$371$	367.945	0.0514899
$372$	7944.70	1.10729
$373$	−10014.2	−1.39012	−0.695060	−	0.718952i	$-0.744622\pi$
$373$	−10014.2	−1.39012	−0.695060	+	0.718952i	$0.744622\pi$
$374$	668.388	0.0924105
$375$	−4403.88	−0.606441
$376$	2208.30	0.302883
$377$	0	0
$378$	−355.158	−0.0483263
$379$	8169.12	1.10717	0.553587	−	0.832791i	$-0.313258\pi$
$379$	8169.12	1.10717	0.553587	+	0.832791i	$0.313258\pi$
$380$	−11235.4	−1.51675
$381$	−9595.24	−1.29023
$382$	1175.97	0.157507
$383$	7310.25	0.975290	0.487645	−	0.873042i	$-0.337856\pi$
$383$	7310.25	0.975290	0.487645	+	0.873042i	$0.337856\pi$
$384$	3112.70	0.413656
$385$	−2171.62	−0.287470
$386$	864.033	0.113933
$387$	−369.535	−0.0485388
$388$	5928.97	0.775767
$389$	8785.47	1.14509	0.572546	−	0.819872i	$-0.305956\pi$
$389$	8785.47	1.14509	0.572546	+	0.819872i	$0.305956\pi$
$390$	0	0
$391$	9553.99	1.23572
$392$	2172.30	0.279892
$393$	−7854.60	−1.00817
$394$	1760.82	0.225150
$395$	3605.40	0.459259
$396$	2350.09	0.298224
$397$	−11266.8	−1.42434	−0.712171	−	0.702006i	$-0.752288\pi$
$397$	−11266.8	−1.42434	−0.712171	+	0.702006i	$0.752288\pi$
$398$	−1852.90	−0.233361
$399$	1619.29	0.203173
$400$	11416.2	1.42702
$401$	−1576.23	−0.196293	−0.0981464	−	0.995172i	$-0.531291\pi$
$401$	−1576.23	−0.196293	−0.0981464	+	0.995172i	$0.531291\pi$
$402$	−686.763	−0.0852055
$403$	0	0
$404$	2722.54	0.335276
$405$	−3319.12	−0.407231
$406$	254.407	0.0310985
$407$	96.2335	0.0117202
$408$	1736.19	0.210672
$409$	−6755.78	−0.816753	−0.408377	−	0.912814i	$-0.633905\pi$
$409$	−6755.78	−0.816753	−0.408377	+	0.912814i	$0.633905\pi$
$410$	−1778.37	−0.214213
$411$	2534.99	0.304238
$412$	4530.87	0.541796
$413$	−1583.21	−0.188631
$414$	−827.083	−0.0981858
$415$	−5986.17	−0.708072
$416$	0	0
$417$	2504.02	0.294059
$418$	794.456	0.0929620
$419$	10756.2	1.25411	0.627057	−	0.778973i	$-0.284259\pi$
$419$	10756.2	1.25411	0.627057	+	0.778973i	$0.284259\pi$
$420$	−2786.17	−0.323694
$421$	−7886.03	−0.912925	−0.456463	−	0.889743i	$-0.650884\pi$
$421$	−7886.03	−0.912925	−0.456463	+	0.889743i	$0.650884\pi$
$422$	598.335	0.0690201
$423$	4276.89	0.491607
$424$	468.916	0.0537089
$425$	13061.0	1.49071
$426$	−247.116	−0.0281052
$427$	−3607.38	−0.408837
$428$	−4462.08	−0.503932
$429$	0	0
$430$	214.943	0.0241057
$431$	14084.6	1.57409	0.787044	−	0.616897i	$-0.211610\pi$
$431$	14084.6	1.57409	0.787044	+	0.616897i	$0.211610\pi$
$432$	8850.86	0.985735
$433$	1864.14	0.206894	0.103447	−	0.994635i	$-0.467013\pi$
$433$	1864.14	0.206894	0.103447	+	0.994635i	$0.467013\pi$
$434$	−658.485	−0.0728301
$435$	7000.73	0.771630
$436$	1374.20	0.150945
$437$	11356.0	1.24309
$438$	189.450	0.0206673
$439$	6154.49	0.669106	0.334553	−	0.942377i	$-0.391415\pi$
$439$	6154.49	0.669106	0.334553	+	0.942377i	$0.391415\pi$
$440$	−2767.56	−0.299859
$441$	4207.17	0.454289
$442$	0	0
$443$	−14539.3	−1.55933	−0.779663	−	0.626200i	$-0.784609\pi$
$443$	−14539.3	−1.55933	−0.779663	+	0.626200i	$0.784609\pi$
$444$	123.467	0.0131970
$445$	−12789.4	−1.36242
$446$	464.521	0.0493177
$447$	−7278.90	−0.770202
$448$	2391.03	0.252155
$449$	7043.87	0.740358	0.370179	−	0.928960i	$-0.379296\pi$
$449$	7043.87	0.740358	0.370179	+	0.928960i	$0.379296\pi$
$450$	−1130.68	−0.118446
$451$	−5107.33	−0.533248
$452$	−9875.90	−1.02771
$453$	6644.32	0.689133
$454$	−1518.87	−0.157013
$455$	0	0
$456$	2063.66	0.211929
$457$	−14098.9	−1.44314	−0.721572	−	0.692340i	$-0.756580\pi$
$457$	−14098.9	−1.44314	−0.721572	+	0.692340i	$0.756580\pi$
$458$	1019.23	0.103986
$459$	10126.1	1.02973
$460$	−19539.3	−1.98049
$461$	−14449.7	−1.45985	−0.729924	−	0.683529i	$-0.760444\pi$
$461$	−14449.7	−1.45985	−0.729924	+	0.683529i	$0.760444\pi$
$462$	197.010	0.0198393
$463$	−15806.5	−1.58659	−0.793293	−	0.608840i	$-0.791635\pi$
$463$	−15806.5	−1.58659	−0.793293	+	0.608840i	$0.791635\pi$
$464$	−6340.06	−0.634332
$465$	−18120.1	−1.80709
$466$	−1635.85	−0.162617
$467$	−15071.3	−1.49340	−0.746699	−	0.665162i	$-0.768362\pi$
$467$	−15071.3	−1.49340	−0.746699	+	0.665162i	$0.768362\pi$
$468$	0	0
$469$	−2311.89	−0.227619
$470$	−2487.69	−0.244146
$471$	1463.43	0.143166
$472$	−2017.68	−0.196761
$473$	617.299	0.0600073
$474$	−327.083	−0.0316950
$475$	15524.5	1.49961
$476$	2886.78	0.277973
$477$	908.169	0.0871744
$478$	−2650.18	−0.253591
$479$	392.545	0.0374443	0.0187222	−	0.999825i	$-0.494040\pi$
$479$	392.545	0.0374443	0.0187222	+	0.999825i	$0.494040\pi$
$480$	−5347.73	−0.508519
$481$	0	0
$482$	2268.51	0.214373
$483$	2816.08	0.265292
$484$	6466.36	0.607284
$485$	−13522.7	−1.26605
$486$	−1462.12	−0.136467
$487$	−9497.89	−0.883758	−0.441879	−	0.897075i	$-0.645688\pi$
$487$	−9497.89	−0.883758	−0.441879	+	0.897075i	$0.645688\pi$
$488$	−4597.32	−0.426457
$489$	3468.71	0.320778
$490$	−2447.14	−0.225613
$491$	−1893.82	−0.174067	−0.0870337	−	0.996205i	$-0.527739\pi$
$491$	−1893.82	−0.174067	−0.0870337	+	0.996205i	$0.527739\pi$
$492$	−6552.67	−0.600442
$493$	−7253.52	−0.662641
$494$	0	0
$495$	−5360.04	−0.486699
$496$	16410.1	1.48555
$497$	−831.881	−0.0750804
$498$	543.068	0.0488664
$499$	13370.1	1.19945	0.599727	−	0.800205i	$-0.295276\pi$
$499$	13370.1	1.19945	0.599727	+	0.800205i	$0.295276\pi$
$500$	−9331.79	−0.834661
$501$	13561.1	1.20932
$502$	2464.39	0.219106
$503$	5554.71	0.492391	0.246195	−	0.969220i	$-0.420820\pi$
$503$	5554.71	0.492391	0.246195	+	0.969220i	$0.420820\pi$
$504$	−505.966	−0.0447173
$505$	−6209.51	−0.547168
$506$	1381.62	0.121385
$507$	0	0
$508$	−20332.3	−1.77578
$509$	−2197.55	−0.191365	−0.0956824	−	0.995412i	$-0.530503\pi$
$509$	−2197.55	−0.191365	−0.0956824	+	0.995412i	$0.530503\pi$
$510$	−1955.85	−0.169817
$511$	637.756	0.0552107
$512$	8137.89	0.702437
$513$	12036.0	1.03587
$514$	−734.022	−0.0629890
$515$	−10333.9	−0.884206
$516$	791.991	0.0675687
$517$	−7144.45	−0.607761
$518$	−10.2334	−0.000868010 0
$519$	−5242.44	−0.443386
$520$	0	0
$521$	17005.2	1.42997	0.714983	−	0.699142i	$-0.246435\pi$
$521$	17005.2	1.42997	0.714983	+	0.699142i	$0.246435\pi$
$522$	627.932	0.0526511
$523$	−14486.2	−1.21116	−0.605581	−	0.795783i	$-0.707059\pi$
$523$	−14486.2	−1.21116	−0.605581	+	0.795783i	$0.707059\pi$
$524$	−16643.8	−1.38758
$525$	3849.79	0.320035
$526$	−2766.24	−0.229304
$527$	18774.4	1.55185
$528$	−4909.68	−0.404671
$529$	7582.03	0.623163
$530$	−528.244	−0.0432933
$531$	−3907.72	−0.319361
$532$	3431.27	0.279632
$533$	0	0
$534$	1160.26	0.0940252
$535$	10177.0	0.822413
$536$	−2946.32	−0.237429
$537$	−4303.28	−0.345810
$538$	−1088.55	−0.0872315
$539$	−7027.98	−0.561626
$540$	−20709.3	−1.65034
$541$	15266.7	1.21325	0.606623	−	0.794990i	$-0.292524\pi$
$541$	15266.7	1.21325	0.606623	+	0.794990i	$0.292524\pi$
$542$	−1243.31	−0.0985330
$543$	4178.27	0.330215
$544$	5540.83	0.436693
$545$	−3134.23	−0.246341
$546$	0	0
$547$	15260.5	1.19286	0.596430	−	0.802665i	$-0.296586\pi$
$547$	15260.5	1.19286	0.596430	+	0.802665i	$0.296586\pi$
$548$	5371.62	0.418731
$549$	−8903.81	−0.692177
$550$	1888.78	0.146432
$551$	−8621.64	−0.666595
$552$	3588.87	0.276726
$553$	−1101.08	−0.0846703
$554$	−1682.55	−0.129033
$555$	−281.601	−0.0215374
$556$	5306.00	0.404721
$557$	10442.1	0.794337	0.397169	−	0.917746i	$-0.369993\pi$
$557$	10442.1	0.794337	0.397169	+	0.917746i	$0.369993\pi$
$558$	−1625.29	−0.123304
$559$	0	0
$560$	−5754.94	−0.434269
$561$	−5617.06	−0.422731
$562$	3999.57	0.300199
$563$	7145.26	0.534879	0.267440	−	0.963575i	$-0.413822\pi$
$563$	7145.26	0.534879	0.267440	+	0.963575i	$0.413822\pi$
$564$	−9166.29	−0.684344
$565$	22524.7	1.67721
$566$	932.950	0.0692841
$567$	1013.65	0.0750783
$568$	−1060.17	−0.0783162
$569$	4438.86	0.327042	0.163521	−	0.986540i	$-0.447715\pi$
$569$	4438.86	0.327042	0.163521	+	0.986540i	$0.447715\pi$
$570$	−2324.75	−0.170830
$571$	10117.3	0.741497	0.370748	−	0.928733i	$-0.379101\pi$
$571$	10117.3	0.741497	0.370748	+	0.928733i	$0.379101\pi$
$572$	0	0
$573$	−9882.70	−0.720516
$574$	543.109	0.0394929
$575$	26998.4	1.95810
$576$	5901.58	0.426909
$577$	−3105.60	−0.224069	−0.112035	−	0.993704i	$-0.535737\pi$
$577$	−3105.60	−0.224069	−0.112035	+	0.993704i	$0.535737\pi$
$578$	−127.615	−0.00918352
$579$	−7261.23	−0.521186
$580$	14834.5	1.06202
$581$	1828.16	0.130542
$582$	1226.78	0.0873741
$583$	−1517.08	−0.107772
$584$	812.769	0.0575901
$585$	0	0
$586$	−3628.05	−0.255757
$587$	−19662.3	−1.38254	−0.691270	−	0.722597i	$-0.742948\pi$
$587$	−19662.3	−1.38254	−0.691270	+	0.722597i	$0.742948\pi$
$588$	−9016.86	−0.632396
$589$	22315.5	1.56111
$590$	2272.95	0.158603
$591$	−14797.8	−1.02995
$592$	255.026	0.0177052
$593$	−6395.51	−0.442888	−0.221444	−	0.975173i	$-0.571077\pi$
$593$	−6395.51	−0.442888	−0.221444	+	0.975173i	$0.571077\pi$
$594$	1464.35	0.101150
$595$	−6584.10	−0.453650
$596$	−15423.9	−1.06005
$597$	15571.6	1.06751
$598$	0	0
$599$	8878.48	0.605618	0.302809	−	0.953051i	$-0.402076\pi$
$599$	8878.48	0.605618	0.302809	+	0.953051i	$0.402076\pi$
$600$	4906.25	0.333828
$601$	19100.6	1.29639	0.648194	−	0.761475i	$-0.275525\pi$
$601$	19100.6	1.29639	0.648194	+	0.761475i	$0.275525\pi$
$602$	−65.6430	−0.00444420
$603$	−5706.26	−0.385368
$604$	14079.3	0.948472
$605$	−14748.3	−0.991082
$606$	563.330	0.0377619
$607$	16595.8	1.10972	0.554861	−	0.831943i	$-0.312771\pi$
$607$	16595.8	1.10972	0.554861	+	0.831943i	$0.312771\pi$
$608$	6585.91	0.439299
$609$	−2138.01	−0.142260
$610$	5178.97	0.343755
$611$	0	0
$612$	7125.21	0.470620
$613$	−16469.2	−1.08513	−0.542564	−	0.840015i	$-0.682546\pi$
$613$	−16469.2	−1.08513	−0.542564	+	0.840015i	$0.682546\pi$
$614$	1583.01	0.104047
$615$	14945.2	0.979915
$616$	845.205	0.0552829
$617$	−10116.0	−0.660055	−0.330027	−	0.943971i	$-0.607058\pi$
$617$	−10116.0	−0.660055	−0.330027	+	0.943971i	$0.607058\pi$
$618$	937.496	0.0610220
$619$	−18854.8	−1.22430	−0.612148	−	0.790743i	$-0.709694\pi$
$619$	−18854.8	−1.22430	−0.612148	+	0.790743i	$0.709694\pi$
$620$	−38396.3	−2.48715
$621$	20931.6	1.35258
$622$	1460.49	0.0941487
$623$	3905.86	0.251180
$624$	0	0
$625$	−2730.82	−0.174773
$626$	−157.019	−0.0100252
$627$	−6676.51	−0.425254
$628$	3100.99	0.197043
$629$	291.769	0.0184954
$630$	569.981	0.0360454
$631$	−18946.2	−1.19531	−0.597653	−	0.801755i	$-0.703900\pi$
$631$	−18946.2	−1.19531	−0.597653	+	0.801755i	$0.703900\pi$
$632$	−1403.24	−0.0883194
$633$	−5028.33	−0.315732
$634$	−1336.15	−0.0836991
$635$	46373.3	2.89806
$636$	−1946.40	−0.121352
$637$	0	0
$638$	−1048.95	−0.0650912
$639$	−2053.27	−0.127114
$640$	−15043.5	−0.929135
$641$	−23586.9	−1.45340	−0.726698	−	0.686957i	$-0.758946\pi$
$641$	−23586.9	−1.45340	−0.726698	+	0.686957i	$0.758946\pi$
$642$	−923.263	−0.0567575
$643$	27153.0	1.66534	0.832669	−	0.553772i	$-0.186812\pi$
$643$	27153.0	1.66534	0.832669	+	0.553772i	$0.186812\pi$
$644$	5967.25	0.365128
$645$	−1806.35	−0.110272
$646$	2408.70	0.146701
$647$	6856.72	0.416639	0.208319	−	0.978061i	$-0.433201\pi$
$647$	6856.72	0.416639	0.208319	+	0.978061i	$0.433201\pi$
$648$	1291.82	0.0783140
$649$	6527.75	0.394817
$650$	0	0
$651$	5533.83	0.333161
$652$	7350.17	0.441495
$653$	−8073.89	−0.483853	−0.241926	−	0.970295i	$-0.577779\pi$
$653$	−8073.89	−0.483853	−0.241926	+	0.970295i	$0.577779\pi$
$654$	284.339	0.0170008
$655$	37960.9	2.26451
$656$	−13534.8	−0.805555
$657$	1574.12	0.0934739
$658$	759.734	0.0450114
$659$	5305.73	0.313629	0.156815	−	0.987628i	$-0.449877\pi$
$659$	5305.73	0.313629	0.156815	+	0.987628i	$0.449877\pi$
$660$	11487.7	0.677512
$661$	−25848.3	−1.52100	−0.760502	−	0.649336i	$-0.775047\pi$
$661$	−25848.3	−1.52100	−0.760502	+	0.649336i	$0.775047\pi$
$662$	−3373.75	−0.198073
$663$	0	0
$664$	2329.85	0.136168
$665$	−7825.96	−0.456357
$666$	−25.2583	−0.00146958
$667$	−14993.7	−0.870404
$668$	28735.9	1.66441
$669$	−3903.78	−0.225604
$670$	3319.09	0.191385
$671$	14873.6	0.855722
$672$	1633.18	0.0937521
$673$	−14529.1	−0.832177	−0.416089	−	0.909324i	$-0.636599\pi$
$673$	−14529.1	−0.832177	−0.416089	+	0.909324i	$0.636599\pi$
$674$	2066.06	0.118073
$675$	28615.0	1.63169
$676$	0	0
$677$	12058.1	0.684535	0.342267	−	0.939603i	$-0.388805\pi$
$677$	12058.1	0.684535	0.342267	+	0.939603i	$0.388805\pi$
$678$	−2043.45	−0.115750
$679$	4129.78	0.233412
$680$	−8390.91	−0.473201
$681$	12764.4	0.718255
$682$	2715.00	0.152438
$683$	−30028.8	−1.68231	−0.841156	−	0.540792i	$-0.818125\pi$
$683$	−30028.8	−1.68231	−0.841156	+	0.540792i	$0.818125\pi$
$684$	8469.13	0.473429
$685$	−12251.5	−0.683364
$686$	1565.22	0.0871144
$687$	−8565.50	−0.475683
$688$	1635.89	0.0906505
$689$	0	0
$690$	−4042.94	−0.223061
$691$	−449.696	−0.0247572	−0.0123786	−	0.999923i	$-0.503940\pi$
$691$	−449.696	−0.0247572	−0.0123786	+	0.999923i	$0.503940\pi$
$692$	−11108.7	−0.610244
$693$	1636.94	0.0897291
$694$	−2307.10	−0.126191
$695$	−12101.8	−0.660500
$696$	−2724.72	−0.148391
$697$	−15484.8	−0.841506
$698$	−22.0690	−0.00119674
$699$	13747.5	0.743889
$700$	8157.67	0.440473
$701$	−26986.0	−1.45399	−0.726994	−	0.686644i	$-0.759083\pi$
$701$	−26986.0	−1.45399	−0.726994	+	0.686644i	$0.759083\pi$
$702$	0	0
$703$	346.801	0.0186057
$704$	−9858.46	−0.527776
$705$	20906.2	1.11684
$706$	−3971.29	−0.211702
$707$	1896.37	0.100877
$708$	8375.06	0.444568
$709$	9098.87	0.481968	0.240984	−	0.970529i	$-0.422530\pi$
$709$	9098.87	0.481968	0.240984	+	0.970529i	$0.422530\pi$
$710$	1194.30	0.0631285
$711$	−2717.71	−0.143350
$712$	4977.71	0.262005
$713$	38808.4	2.03841
$714$	597.312	0.0313079
$715$	0	0
$716$	−9118.62	−0.475948
$717$	22271.8	1.16005
$718$	−3147.11	−0.163578
$719$	−6293.55	−0.326439	−0.163220	−	0.986590i	$-0.552188\pi$
$719$	−6293.55	−0.326439	−0.163220	+	0.986590i	$0.552188\pi$
$720$	−14204.5	−0.735235
$721$	3155.95	0.163015
$722$	−144.295	−0.00743783
$723$	−19064.3	−0.980648
$724$	8853.72	0.454483
$725$	−20497.5	−1.05001
$726$	1337.97	0.0683979
$727$	18070.7	0.921878	0.460939	−	0.887432i	$-0.347513\pi$
$727$	18070.7	0.921878	0.460939	+	0.887432i	$0.347513\pi$
$728$	0	0
$729$	17319.9	0.879944
$730$	−915.601	−0.0464218
$731$	1871.58	0.0946962
$732$	19082.7	0.963550
$733$	−34771.5	−1.75214	−0.876068	−	0.482188i	$-0.839842\pi$
$733$	−34771.5	−1.75214	−0.876068	+	0.482188i	$0.839842\pi$
$734$	1755.61	0.0882842
$735$	20565.4	1.03206
$736$	11453.4	0.573613
$737$	9532.17	0.476421
$738$	1340.51	0.0668631
$739$	−23631.5	−1.17632	−0.588158	−	0.808746i	$-0.700147\pi$
$739$	−23631.5	−1.17632	−0.588158	+	0.808746i	$0.700147\pi$
$740$	−596.710	−0.0296425
$741$	0	0
$742$	161.324	0.00798167
$743$	−32502.8	−1.60486	−0.802431	−	0.596745i	$-0.796460\pi$
$743$	−32502.8	−1.60486	−0.802431	+	0.596745i	$0.796460\pi$
$744$	7052.42	0.347519
$745$	35178.6	1.72999
$746$	−4390.69	−0.215489
$747$	4512.31	0.221013
$748$	−11902.5	−0.581816
$749$	−3108.04	−0.151622
$750$	−1930.87	−0.0940072
$751$	2020.86	0.0981920	0.0490960	−	0.998794i	$-0.484366\pi$
$751$	2020.86	0.0981920	0.0490960	+	0.998794i	$0.484366\pi$
$752$	−18933.3	−0.918120
$753$	−20710.5	−1.00230
$754$	0	0
$755$	−32111.6	−1.54790
$756$	6324.56	0.304262
$757$	12568.2	0.603434	0.301717	−	0.953398i	$-0.402440\pi$
$757$	12568.2	0.603434	0.301717	+	0.953398i	$0.402440\pi$
$758$	3581.73	0.171628
$759$	−11611.0	−0.555273
$760$	−9973.56	−0.476025
$761$	8704.81	0.414651	0.207325	−	0.978272i	$-0.433524\pi$
$761$	8704.81	0.414651	0.207325	+	0.978272i	$0.433524\pi$
$762$	−4207.01	−0.200005
$763$	957.187	0.0454161
$764$	−20941.4	−0.991665
$765$	−16251.0	−0.768048
$766$	3205.16	0.151184
$767$	0	0
$768$	−11595.0	−0.544790
$769$	21915.9	1.02771	0.513853	−	0.857878i	$-0.328218\pi$
$769$	21915.9	1.02771	0.513853	+	0.857878i	$0.328218\pi$
$770$	−952.140	−0.0445620
$771$	6168.63	0.288143
$772$	−15386.5	−0.717322
$773$	23077.5	1.07379	0.536896	−	0.843649i	$-0.319597\pi$
$773$	23077.5	1.07379	0.536896	+	0.843649i	$0.319597\pi$
$774$	−162.022	−0.00752422
$775$	53054.0	2.45904
$776$	5263.08	0.243471
$777$	86.0001	0.00397070
$778$	3851.96	0.177506
$779$	−18405.5	−0.846528
$780$	0	0
$781$	3429.94	0.157148
$782$	4188.92	0.191554
$783$	−15891.5	−0.725309
$784$	−18624.6	−0.848426
$785$	−7072.67	−0.321572
$786$	−3443.83	−0.156282
$787$	16522.4	0.748362	0.374181	−	0.927356i	$-0.377924\pi$
$787$	16522.4	0.748362	0.374181	+	0.927356i	$0.377924\pi$
$788$	−31356.4	−1.41754
$789$	23247.2	1.04895
$790$	1580.78	0.0711918
$791$	−6879.00	−0.309215
$792$	2086.15	0.0935962
$793$	0	0
$794$	−4939.89	−0.220794
$795$	4439.30	0.198045
$796$	32996.1	1.46924
$797$	−11719.4	−0.520855	−0.260427	−	0.965493i	$-0.583863\pi$
$797$	−11719.4	−0.520855	−0.260427	+	0.965493i	$0.583863\pi$
$798$	709.974	0.0314948
$799$	−21661.2	−0.959095
$800$	15657.7	0.691978
$801$	9640.53	0.425258
$802$	−691.096	−0.0304282
$803$	−2629.53	−0.115559
$804$	12229.7	0.536454
$805$	−13610.0	−0.595886
$806$	0	0
$807$	9148.01	0.399040
$808$	2416.77	0.105225
$809$	−24096.0	−1.04718	−0.523592	−	0.851969i	$-0.675408\pi$
$809$	−24096.0	−1.04718	−0.523592	+	0.851969i	$0.675408\pi$
$810$	−1455.26	−0.0631267
$811$	−16622.6	−0.719729	−0.359864	−	0.933005i	$-0.617177\pi$
$811$	−16622.6	−0.719729	−0.359864	+	0.933005i	$0.617177\pi$
$812$	−4530.42	−0.195796
$813$	10448.7	0.450739
$814$	42.1933	0.00181680
$815$	−16764.1	−0.720516
$816$	−14885.6	−0.638603
$817$	2224.59	0.0952613
$818$	−2962.05	−0.126609
$819$	0	0
$820$	31668.7	1.34868
$821$	38005.5	1.61559	0.807797	−	0.589461i	$-0.200660\pi$
$821$	38005.5	1.61559	0.807797	+	0.589461i	$0.200660\pi$
$822$	1111.46	0.0471613
$823$	−15859.5	−0.671722	−0.335861	−	0.941912i	$-0.609027\pi$
$823$	−15859.5	−0.671722	−0.335861	+	0.941912i	$0.609027\pi$
$824$	4022.01	0.170040
$825$	−15873.1	−0.669854
$826$	−694.155	−0.0292406
$827$	−12201.0	−0.513023	−0.256512	−	0.966541i	$-0.582573\pi$
$827$	−12201.0	−0.513023	−0.256512	+	0.966541i	$0.582573\pi$
$828$	14728.5	0.618177
$829$	−5431.41	−0.227552	−0.113776	−	0.993506i	$-0.536295\pi$
$829$	−5431.41	−0.227552	−0.113776	+	0.993506i	$0.536295\pi$
$830$	−2624.62	−0.109761
$831$	14139.9	0.590263
$832$	0	0
$833$	−21308.0	−0.886290
$834$	1097.88	0.0455834
$835$	−65540.3	−2.71630
$836$	−14147.5	−0.585288
$837$	41132.2	1.69861
$838$	4716.02	0.194406
$839$	7960.90	0.327582	0.163791	−	0.986495i	$-0.447628\pi$
$839$	7960.90	0.327582	0.163791	+	0.986495i	$0.447628\pi$
$840$	−2473.26	−0.101590
$841$	−13005.6	−0.533255
$842$	−3457.61	−0.141517
$843$	−33611.9	−1.37326
$844$	−10655.0	−0.434550
$845$	0	0
$846$	1875.19	0.0762062
$847$	4504.10	0.182719
$848$	−4020.35	−0.162806
$849$	−7840.40	−0.316940
$850$	5726.56	0.231082
$851$	603.115	0.0242944
$852$	4400.59	0.176950
$853$	−13576.7	−0.544969	−0.272485	−	0.962160i	$-0.587845\pi$
$853$	−13576.7	−0.544969	−0.272485	+	0.962160i	$0.587845\pi$
$854$	−1581.65	−0.0633756
$855$	−19316.2	−0.772631
$856$	−3960.95	−0.158157
$857$	−31223.9	−1.24456	−0.622281	−	0.782794i	$-0.713794\pi$
$857$	−31223.9	−1.24456	−0.622281	+	0.782794i	$0.713794\pi$
$858$	0	0
$859$	−11815.8	−0.469323	−0.234661	−	0.972077i	$-0.575398\pi$
$859$	−11815.8	−0.469323	−0.234661	+	0.972077i	$0.575398\pi$
$860$	−3827.65	−0.151770
$861$	−4564.22	−0.180660
$862$	6175.36	0.244006
$863$	1790.84	0.0706384	0.0353192	−	0.999376i	$-0.488755\pi$
$863$	1790.84	0.0706384	0.0353192	+	0.999376i	$0.488755\pi$
$864$	12139.2	0.477992
$865$	25336.4	0.995912
$866$	817.328	0.0320715
$867$	1072.46	0.0420100
$868$	11726.1	0.458538
$869$	4539.87	0.177220
$870$	3069.45	0.119614
$871$	0	0
$872$	1219.86	0.0473734
$873$	10193.2	0.395176
$874$	4979.01	0.192698
$875$	−6500.00	−0.251132
$876$	−3373.68	−0.130121
$877$	−43542.5	−1.67654	−0.838270	−	0.545255i	$-0.816433\pi$
$877$	−43542.5	−1.67654	−0.838270	+	0.545255i	$0.816433\pi$
$878$	2698.42	0.103721
$879$	30489.7	1.16996
$880$	23728.2	0.908953
$881$	−1020.04	−0.0390080	−0.0195040	−	0.999810i	$-0.506209\pi$
$881$	−1020.04	−0.0390080	−0.0195040	+	0.999810i	$0.506209\pi$
$882$	1844.62	0.0704214
$883$	−34781.9	−1.32560	−0.662800	−	0.748797i	$-0.730632\pi$
$883$	−34781.9	−1.32560	−0.662800	+	0.748797i	$0.730632\pi$
$884$	0	0
$885$	−19101.6	−0.725531
$886$	−6374.70	−0.241718
$887$	49785.1	1.88458	0.942288	−	0.334802i	$-0.108670\pi$
$887$	49785.1	1.88458	0.942288	+	0.334802i	$0.108670\pi$
$888$	109.600	0.00414183
$889$	−14162.3	−0.534295
$890$	−5607.49	−0.211195
$891$	−4179.40	−0.157144
$892$	−8272.08	−0.310504
$893$	−25746.8	−0.964818
$894$	−3191.41	−0.119392
$895$	20797.5	0.776743
$896$	4594.25	0.171298
$897$	0	0
$898$	3088.36	0.114766
$899$	−29463.9	−1.09308
$900$	20134.9	0.745738
$901$	−4599.60	−0.170072
$902$	−2239.29	−0.0826611
$903$	551.656	0.0203300
$904$	−8766.74	−0.322541
$905$	−20193.3	−0.741712
$906$	2913.18	0.106826
$907$	17389.9	0.636627	0.318314	−	0.947985i	$-0.396884\pi$
$907$	17389.9	0.636627	0.318314	+	0.947985i	$0.396884\pi$
$908$	27047.6	0.988553
$909$	4680.66	0.170790
$910$	0	0
$911$	20419.5	0.742621	0.371311	−	0.928509i	$-0.378909\pi$
$911$	20419.5	0.742621	0.371311	+	0.928509i	$0.378909\pi$
$912$	−17693.2	−0.642414
$913$	−7537.71	−0.273233
$914$	−6181.60	−0.223708
$915$	−43523.5	−1.57250
$916$	−18150.2	−0.654695
$917$	−11593.2	−0.417492
$918$	4439.75	0.159623
$919$	−33231.8	−1.19283	−0.596417	−	0.802674i	$-0.703410\pi$
$919$	−33231.8	−1.19283	−0.596417	+	0.802674i	$0.703410\pi$
$920$	−17344.8	−0.621567
$921$	−13303.4	−0.475964
$922$	−6335.43	−0.226297
$923$	0	0
$924$	−3508.31	−0.124908
$925$	824.502	0.0293075
$926$	−6930.31	−0.245944
$927$	7789.58	0.275991
$928$	−8695.59	−0.307594
$929$	25222.8	0.890780	0.445390	−	0.895337i	$-0.353065\pi$
$929$	25222.8	0.890780	0.445390	+	0.895337i	$0.353065\pi$
$930$	−7944.70	−0.280126
$931$	−25327.0	−0.891579
$932$	29130.9	1.02383
$933$	−12273.8	−0.430683
$934$	−6607.97	−0.231498
$935$	27146.9	0.949519
$936$	0	0
$937$	−26979.4	−0.940639	−0.470319	−	0.882496i	$-0.655861\pi$
$937$	−26979.4	−0.940639	−0.470319	+	0.882496i	$0.655861\pi$
$938$	−1013.64	−0.0352842
$939$	1319.57	0.0458600
$940$	44300.2	1.53714
$941$	−7641.67	−0.264730	−0.132365	−	0.991201i	$-0.542257\pi$
$941$	−7641.67	−0.264730	−0.132365	+	0.991201i	$0.542257\pi$
$942$	641.635	0.0221928
$943$	−32008.7	−1.10535
$944$	17299.0	0.596434
$945$	−14424.9	−0.496553
$946$	270.653	0.00930200
$947$	2869.32	0.0984587	0.0492293	−	0.998788i	$-0.484323\pi$
$947$	2869.32	0.0984587	0.0492293	+	0.998788i	$0.484323\pi$
$948$	5824.62	0.199552
$949$	0	0
$950$	6806.67	0.232461
$951$	11228.8	0.382881
$952$	2562.56	0.0872407
$953$	12313.6	0.418548	0.209274	−	0.977857i	$-0.432890\pi$
$953$	12313.6	0.418548	0.209274	+	0.977857i	$0.432890\pi$
$954$	398.184	0.0135133
$955$	47762.6	1.61839
$956$	47193.8	1.59661
$957$	8815.22	0.297759
$958$	172.110	0.00580442
$959$	3741.57	0.125987
$960$	28848.0	0.969861
$961$	46470.7	1.55989
$962$	0	0
$963$	−7671.32	−0.256703
$964$	−40397.1	−1.34969
$965$	35093.2	1.17066
$966$	1234.70	0.0411241
$967$	17838.0	0.593207	0.296603	−	0.955001i	$-0.404146\pi$
$967$	17838.0	0.593207	0.296603	+	0.955001i	$0.404146\pi$
$968$	5740.12	0.190593
$969$	−20242.4	−0.671084
$970$	−5928.97	−0.196255
$971$	41525.3	1.37241	0.686206	−	0.727408i	$-0.259275\pi$
$971$	41525.3	1.37241	0.686206	+	0.727408i	$0.259275\pi$
$972$	26037.1	0.859199
$973$	3695.86	0.121772
$974$	−4164.32	−0.136995
$975$	0	0
$976$	39416.1	1.29270
$977$	31654.4	1.03655	0.518277	−	0.855213i	$-0.326574\pi$
$977$	31654.4	1.03655	0.518277	+	0.855213i	$0.326574\pi$
$978$	1520.85	0.0497253
$979$	−16104.3	−0.525735
$980$	43578.0	1.42046
$981$	2362.55	0.0768913
$982$	−830.342	−0.0269830
$983$	39913.2	1.29505	0.647525	−	0.762045i	$-0.275804\pi$
$983$	39913.2	1.29505	0.647525	+	0.762045i	$0.275804\pi$
$984$	−5816.74	−0.188446
$985$	71516.8	2.31342
$986$	−3180.28	−0.102719
$987$	−6384.72	−0.205905
$988$	0	0
$989$	3868.74	0.124387
$990$	−2350.09	−0.0754453
$991$	−2700.94	−0.0865773	−0.0432887	−	0.999063i	$-0.513784\pi$
$991$	−2700.94	−0.0865773	−0.0432887	+	0.999063i	$0.513784\pi$
$992$	22506.9	0.720358
$993$	28352.6	0.906085
$994$	−364.736	−0.0116386
$995$	−75256.6	−2.39778
$996$	−9670.83	−0.307663
$997$	−9729.08	−0.309050	−0.154525	−	0.987989i	$-0.549385\pi$
$997$	−9729.08	−0.309050	−0.154525	+	0.987989i	$0.549385\pi$
$998$	5862.08	0.185933
$999$	639.228	0.0202445

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.j.1.1		2
3.2	odd	2		1521.4.a.l.1.2		2
13.2	odd	12		169.4.e.g.147.3		8
13.3	even	3		169.4.c.f.22.2		4
13.4	even	6		13.4.c.b.3.1	✓	4
13.5	odd	4		169.4.b.e.168.2		4
13.6	odd	12		169.4.e.g.23.2		8
13.7	odd	12		169.4.e.g.23.3		8
13.8	odd	4		169.4.b.e.168.3		4
13.9	even	3		169.4.c.f.146.2		4
13.10	even	6		13.4.c.b.9.1	yes	4
13.11	odd	12		169.4.e.g.147.2		8
13.12	even	2		169.4.a.f.1.2		2
39.17	odd	6		117.4.g.d.55.2		4
39.23	odd	6		117.4.g.d.100.2		4
39.38	odd	2		1521.4.a.t.1.1		2
52.23	odd	6		208.4.i.e.113.1		4
52.43	odd	6		208.4.i.e.81.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.c.b.3.1	✓	4	13.4	even	6
13.4.c.b.9.1	yes	4	13.10	even	6
117.4.g.d.55.2		4	39.17	odd	6
117.4.g.d.100.2		4	39.23	odd	6
169.4.a.f.1.2		2	13.12	even	2
169.4.a.j.1.1		2	1.1	even	1	trivial
169.4.b.e.168.2		4	13.5	odd	4
169.4.b.e.168.3		4	13.8	odd	4
169.4.c.f.22.2		4	13.3	even	3
169.4.c.f.146.2		4	13.9	even	3
169.4.e.g.23.2		8	13.6	odd	12
169.4.e.g.23.3		8	13.7	odd	12
169.4.e.g.147.2		8	13.11	odd	12
169.4.e.g.147.3		8	13.2	odd	12
208.4.i.e.81.1		4	52.43	odd	6
208.4.i.e.113.1		4	52.23	odd	6
1521.4.a.l.1.2		2	3.2	odd	2
1521.4.a.t.1.1		2	39.38	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q+0.438447 q^{2} -3.68466 q^{3} -7.80776 q^{4} +17.8078 q^{5} -1.61553 q^{6} -5.43845 q^{7} -6.93087 q^{8} -13.4233 q^{9} +O(q^{10})\)	\(q+0.438447 q^{2} -3.68466 q^{3} -7.80776 q^{4} +17.8078 q^{5} -1.61553 q^{6} -5.43845 q^{7} -6.93087 q^{8} -13.4233 q^{9} +7.80776 q^{10} +22.4233 q^{11} +28.7689 q^{12} -2.38447 q^{14} -65.6155 q^{15} +59.4233 q^{16} +67.9848 q^{17} -5.88540 q^{18} +80.8078 q^{19} -139.039 q^{20} +20.0388 q^{21} +9.83143 q^{22} +140.531 q^{23} +25.5379 q^{24} +192.116 q^{25} +148.946 q^{27} +42.4621 q^{28} -106.693 q^{29} -28.7689 q^{30} +276.155 q^{31} +81.5009 q^{32} -82.6222 q^{33} +29.8078 q^{34} -96.8466 q^{35} +104.806 q^{36} +4.29168 q^{37} +35.4299 q^{38} -123.423 q^{40} -227.769 q^{41} +8.78596 q^{42} +27.5294 q^{43} -175.076 q^{44} -239.039 q^{45} +61.6155 q^{46} -318.617 q^{47} -218.955 q^{48} -313.423 q^{49} +84.2329 q^{50} -250.501 q^{51} -67.6562 q^{53} +65.3050 q^{54} +399.309 q^{55} +37.6932 q^{56} -297.749 q^{57} -46.7793 q^{58} +291.115 q^{59} +512.311 q^{60} +663.311 q^{61} +121.080 q^{62} +73.0019 q^{63} -439.652 q^{64} -36.2255 q^{66} +425.101 q^{67} -530.810 q^{68} -517.810 q^{69} -42.4621 q^{70} +152.963 q^{71} +93.0351 q^{72} -117.268 q^{73} +1.88167 q^{74} -707.884 q^{75} -630.928 q^{76} -121.948 q^{77} +202.462 q^{79} +1058.20 q^{80} -186.386 q^{81} -99.8647 q^{82} -336.155 q^{83} -156.458 q^{84} +1210.66 q^{85} +12.0702 q^{86} +393.128 q^{87} -155.413 q^{88} -718.194 q^{89} -104.806 q^{90} -1097.23 q^{92} -1017.54 q^{93} -139.697 q^{94} +1439.01 q^{95} -300.303 q^{96} -759.368 q^{97} -137.420 q^{98} -300.994 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9}+O(q^{10}) \) 2 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 15 * q^5 + 38 * q^6 - 15 * q^7 + 15 * q^8 + 35 * q^9	\( 2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9} - 5 q^{10} - 17 q^{11} + 140 q^{12} - 46 q^{14} - 90 q^{15} + 57 q^{16} + 70 q^{17} + 215 q^{18} + 141 q^{19} - 175 q^{20} - 63 q^{21} - 170 q^{22} + 145 q^{23} + 216 q^{24} + 75 q^{25} + 335 q^{27} - 80 q^{28} + 34 q^{29} - 140 q^{30} + 140 q^{31} - 105 q^{32} - 425 q^{33} + 39 q^{34} - 70 q^{35} + 725 q^{36} + 190 q^{37} + 310 q^{38} - 185 q^{40} - 538 q^{41} - 370 q^{42} + 455 q^{43} - 680 q^{44} - 375 q^{45} + 82 q^{46} - 60 q^{47} - 240 q^{48} - 565 q^{49} - 450 q^{50} - 233 q^{51} + 545 q^{53} + 914 q^{54} + 510 q^{55} - 172 q^{56} + 225 q^{57} + 595 q^{58} + 809 q^{59} + 200 q^{60} + 502 q^{61} - 500 q^{62} - 390 q^{63} - 1271 q^{64} - 1598 q^{66} + 475 q^{67} - 505 q^{68} - 479 q^{69} + 80 q^{70} - 127 q^{71} + 1155 q^{72} - 585 q^{73} + 849 q^{74} - 1725 q^{75} + 140 q^{76} + 255 q^{77} + 240 q^{79} + 1065 q^{80} + 122 q^{81} - 1515 q^{82} - 260 q^{83} - 1220 q^{84} + 1205 q^{85} + 1962 q^{86} + 1615 q^{87} - 1020 q^{88} - 921 q^{89} - 725 q^{90} - 1040 q^{92} - 2200 q^{93} + 1040 q^{94} + 1270 q^{95} - 1920 q^{96} + 415 q^{97} - 1285 q^{98} - 2210 q^{99}+O(q^{100}) \) 2 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 15 * q^5 + 38 * q^6 - 15 * q^7 + 15 * q^8 + 35 * q^9 - 5 * q^10 - 17 * q^11 + 140 * q^12 - 46 * q^14 - 90 * q^15 + 57 * q^16 + 70 * q^17 + 215 * q^18 + 141 * q^19 - 175 * q^20 - 63 * q^21 - 170 * q^22 + 145 * q^23 + 216 * q^24 + 75 * q^25 + 335 * q^27 - 80 * q^28 + 34 * q^29 - 140 * q^30 + 140 * q^31 - 105 * q^32 - 425 * q^33 + 39 * q^34 - 70 * q^35 + 725 * q^36 + 190 * q^37 + 310 * q^38 - 185 * q^40 - 538 * q^41 - 370 * q^42 + 455 * q^43 - 680 * q^44 - 375 * q^45 + 82 * q^46 - 60 * q^47 - 240 * q^48 - 565 * q^49 - 450 * q^50 - 233 * q^51 + 545 * q^53 + 914 * q^54 + 510 * q^55 - 172 * q^56 + 225 * q^57 + 595 * q^58 + 809 * q^59 + 200 * q^60 + 502 * q^61 - 500 * q^62 - 390 * q^63 - 1271 * q^64 - 1598 * q^66 + 475 * q^67 - 505 * q^68 - 479 * q^69 + 80 * q^70 - 127 * q^71 + 1155 * q^72 - 585 * q^73 + 849 * q^74 - 1725 * q^75 + 140 * q^76 + 255 * q^77 + 240 * q^79 + 1065 * q^80 + 122 * q^81 - 1515 * q^82 - 260 * q^83 - 1220 * q^84 + 1205 * q^85 + 1962 * q^86 + 1615 * q^87 - 1020 * q^88 - 921 * q^89 - 725 * q^90 - 1040 * q^92 - 2200 * q^93 + 1040 * q^94 + 1270 * q^95 - 1920 * q^96 + 415 * q^97 - 1285 * q^98 - 2210 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more