LMFDB - Embedded newform 507.4.b.j.337.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 97 x^{16} + 3906 x^{14} + 84743 x^{12} + 1077128 x^{10} + 8187552 x^{8} + 36483705 x^{6} + \cdots + 26460736 $ x^18 + 97x^16 + 3906x^14 + 84743x^12 + 1077128x^10 + 8187552x^8 + 36483705x^6 + 88861676x^4 + 98825392x^2 + 26460736
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.9
Root			$-2.05129i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.j.337.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.05129i q^{2} -3.00000 q^{3} +6.89480 q^{4} +17.8886i q^{5} +3.15386i q^{6} +30.1975i q^{7} -15.6587i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.05129i q^{2} -3.00000 q^{3} +6.89480 q^{4} +17.8886i q^{5} +3.15386i q^{6} +30.1975i q^{7} -15.6587i q^{8} +9.00000 q^{9} +18.8061 q^{10} +50.8457i q^{11} -20.6844 q^{12} +31.7462 q^{14} -53.6659i q^{15} +38.6966 q^{16} +2.99137 q^{17} -9.46157i q^{18} +72.7016i q^{19} +123.339i q^{20} -90.5926i q^{21} +53.4533 q^{22} +41.9071 q^{23} +46.9761i q^{24} -195.003 q^{25} -27.0000 q^{27} +208.206i q^{28} -135.233 q^{29} -56.4182 q^{30} -316.820i q^{31} -165.951i q^{32} -152.537i q^{33} -3.14479i q^{34} -540.192 q^{35} +62.0532 q^{36} -261.777i q^{37} +76.4301 q^{38} +280.113 q^{40} +198.911i q^{41} -95.2386 q^{42} +201.351 q^{43} +350.571i q^{44} +160.998i q^{45} -44.0563i q^{46} -97.3687i q^{47} -116.090 q^{48} -568.890 q^{49} +205.004i q^{50} -8.97412 q^{51} -150.458 q^{53} +28.3847i q^{54} -909.560 q^{55} +472.853 q^{56} -218.105i q^{57} +142.168i q^{58} -497.812i q^{59} -370.016i q^{60} +525.066 q^{61} -333.068 q^{62} +271.778i q^{63} +135.112 q^{64} -160.360 q^{66} +777.584i q^{67} +20.6249 q^{68} -125.721 q^{69} +567.896i q^{70} +1012.16i q^{71} -140.928i q^{72} -612.910i q^{73} -275.202 q^{74} +585.010 q^{75} +501.263i q^{76} -1535.41 q^{77} +718.804 q^{79} +692.230i q^{80} +81.0000 q^{81} +209.112 q^{82} +397.730i q^{83} -624.617i q^{84} +53.5116i q^{85} -211.677i q^{86} +405.699 q^{87} +796.176 q^{88} +648.413i q^{89} +169.255 q^{90} +288.941 q^{92} +950.460i q^{93} -102.362 q^{94} -1300.53 q^{95} +497.852i q^{96} +272.412i q^{97} +598.066i q^{98} +457.611i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) $ 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9	$ 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100}) $ 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9 - 396 * q^10 + 192 * q^12 + 196 * q^14 + 64 * q^16 + 268 * q^17 + 548 * q^22 - 452 * q^23 - 1224 * q^25 - 486 * q^27 - 1094 * q^29 + 1188 * q^30 + 276 * q^35 - 576 * q^36 + 832 * q^38 + 2684 * q^40 - 588 * q^42 - 316 * q^43 - 192 * q^48 - 1284 * q^49 - 804 * q^51 + 2798 * q^53 - 2816 * q^55 + 1232 * q^56 + 4184 * q^61 + 586 * q^62 - 4962 * q^64 - 1644 * q^66 - 3158 * q^68 + 1356 * q^69 + 2074 * q^74 + 3672 * q^75 + 3372 * q^77 - 230 * q^79 + 1458 * q^81 + 10294 * q^82 + 3282 * q^87 + 968 * q^88 - 3564 * q^90 + 4174 * q^92 - 936 * q^94 + 444 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.05129i	− 0.371686i	−0.982579	−	0.185843i	$-0.940498\pi$
$2$	− 1.05129i	− 0.371686i	0.982579	−	0.185843i	$-0.0595015\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	6.89480	0.861850
$5$	17.8886i	1.60001i	0.599994	+	0.800004i	$0.295169\pi$
$5$	17.8886i	1.60001i	−0.599994	+	0.800004i	$0.704831\pi$
$6$	3.15386i	0.214593i
$7$	30.1975i	1.63051i	0.579100	+	0.815256i	$0.303404\pi$
$7$	30.1975i	1.63051i	−0.579100	+	0.815256i	$0.696596\pi$
$8$	− 15.6587i	− 0.692023i
$9$	9.00000	0.333333
$10$	18.8061	0.594700
$11$	50.8457i	1.39369i	0.717224	+	0.696843i	$0.245413\pi$
$11$	50.8457i	1.39369i	−0.717224	+	0.696843i	$0.754587\pi$
$12$	−20.6844	−0.497589
$13$	0	0
$13$	0	0
$14$	31.7462	0.606038
$15$	− 53.6659i	− 0.923765i
$16$	38.6966	0.604635
$17$	2.99137	0.0426773	0.0213387	−	0.999772i	$-0.493207\pi$
$17$	2.99137	0.0426773	0.0213387	+	0.999772i	$0.493207\pi$
$18$	− 9.46157i	− 0.123895i
$19$	72.7016i	0.877836i	0.898527	+	0.438918i	$0.144638\pi$
$19$	72.7016i	0.877836i	−0.898527	+	0.438918i	$0.855362\pi$
$20$	123.339i	1.37897i
$21$	− 90.5926i	− 0.941377i
$22$	53.4533	0.518013
$23$	41.9071	0.379923	0.189961	−	0.981792i	$-0.439164\pi$
$23$	41.9071	0.379923	0.189961	+	0.981792i	$0.439164\pi$
$24$	46.9761i	0.399539i
$25$	−195.003	−1.56003
$26$	0	0
$27$	−27.0000	−0.192450
$28$	208.206i	1.40526i
$29$	−135.233	−0.865935	−0.432967	−	0.901410i	$-0.642534\pi$
$29$	−135.233	−0.865935	−0.432967	+	0.901410i	$0.642534\pi$
$30$	−56.4182	−0.343350
$31$	− 316.820i	− 1.83557i	−0.397082	−	0.917783i	$-0.629977\pi$
$31$	− 316.820i	− 1.83557i	0.397082	−	0.917783i	$-0.370023\pi$
$32$	− 165.951i	− 0.916757i
$33$	− 152.537i	− 0.804645i
$34$	− 3.14479i	− 0.0158625i
$35$	−540.192	−2.60883
$36$	62.0532	0.287283
$37$	− 261.777i	− 1.16313i	−0.813500	−	0.581565i	$-0.802441\pi$
$37$	− 261.777i	− 1.16313i	0.813500	−	0.581565i	$-0.197559\pi$
$38$	76.4301	0.326279
$39$	0	0
$40$	280.113	1.10724
$41$	198.911i	0.757674i	0.925463	+	0.378837i	$0.123676\pi$
$41$	198.911i	0.757674i	−0.925463	+	0.378837i	$0.876324\pi$
$42$	−95.2386	−0.349896
$43$	201.351	0.714085	0.357043	−	0.934088i	$-0.383785\pi$
$43$	201.351	0.714085	0.357043	+	0.934088i	$0.383785\pi$
$44$	350.571i	1.20115i
$45$	160.998i	0.533336i
$46$	− 44.0563i	− 0.141212i
$47$	− 97.3687i	− 0.302185i	−0.988520	−	0.151092i	$-0.951721\pi$
$47$	− 97.3687i	− 0.302185i	0.988520	−	0.151092i	$-0.0482791\pi$
$48$	−116.090	−0.349086
$49$	−568.890	−1.65857
$50$	205.004i	0.579839i
$51$	−8.97412	−0.0246398
$52$	0	0
$53$	−150.458	−0.389943	−0.194972	−	0.980809i	$-0.562461\pi$
$53$	−150.458	−0.389943	−0.194972	+	0.980809i	$0.562461\pi$
$54$	28.3847i	0.0715309i
$55$	−909.560	−2.22991
$56$	472.853	1.12835
$57$	− 218.105i	− 0.506819i
$58$	142.168i	0.321855i
$59$	− 497.812i	− 1.09847i	−0.835668	−	0.549234i	$-0.814919\pi$
$59$	− 497.812i	− 1.09847i	0.835668	−	0.549234i	$-0.185081\pi$
$60$	− 370.016i	− 0.796147i
$61$	525.066	1.10209	0.551047	−	0.834474i	$-0.314228\pi$
$61$	525.066	1.10209	0.551047	+	0.834474i	$0.314228\pi$
$62$	−333.068	−0.682253
$63$	271.778i	0.543504i
$64$	135.112	0.263890
$65$	0	0
$66$	−160.360	−0.299075
$67$	777.584i	1.41787i	0.705276	+	0.708933i	$0.250823\pi$
$67$	777.584i	1.41787i	−0.705276	+	0.708933i	$0.749177\pi$
$68$	20.6249	0.0367814
$69$	−125.721	−0.219349
$70$	567.896i	0.969666i
$71$	1012.16i	1.69185i	0.533303	+	0.845924i	$0.320951\pi$
$71$	1012.16i	1.69185i	−0.533303	+	0.845924i	$0.679049\pi$
$72$	− 140.928i	− 0.230674i
$73$	− 612.910i	− 0.982680i	−0.870968	−	0.491340i	$-0.836507\pi$
$73$	− 612.910i	− 0.982680i	0.870968	−	0.491340i	$-0.163493\pi$
$74$	−275.202	−0.432318
$75$	585.010	0.900682
$76$	501.263i	0.756563i
$77$	−1535.41	−2.27242
$78$	0	0
$79$	718.804	1.02369	0.511846	−	0.859077i	$-0.328962\pi$
$79$	718.804	1.02369	0.511846	+	0.859077i	$0.328962\pi$
$80$	692.230i	0.967421i
$81$	81.0000	0.111111
$82$	209.112	0.281616
$83$	397.730i	0.525982i	0.964798	+	0.262991i	$0.0847089\pi$
$83$	397.730i	0.525982i	−0.964798	+	0.262991i	$0.915291\pi$
$84$	− 624.617i	− 0.811326i
$85$	53.5116i	0.0682841i
$86$	− 211.677i	− 0.265415i
$87$	405.699	0.499948
$88$	796.176	0.964462
$89$	648.413i	0.772265i	0.922443	+	0.386133i	$0.126189\pi$
$89$	648.413i	0.772265i	−0.922443	+	0.386133i	$0.873811\pi$
$90$	169.255	0.198233
$91$	0	0
$92$	288.941	0.327436
$93$	950.460i	1.05976i
$94$	−102.362	−0.112318
$95$	−1300.53	−1.40454
$96$	497.852i	0.529290i
$97$	272.412i	0.285147i	0.989784	+	0.142574i	$0.0455378\pi$
$97$	272.412i	0.285147i	−0.989784	+	0.142574i	$0.954462\pi$
$98$	598.066i	0.616467i
$99$	457.611i	0.464562i
$100$	−1344.51	−1.34451
$101$	−416.057	−0.409893	−0.204947	−	0.978773i	$-0.565702\pi$
$101$	−416.057	−0.409893	−0.204947	+	0.978773i	$0.565702\pi$
$102$	9.43436i	0.00915824i
$103$	−261.509	−0.250168	−0.125084	−	0.992146i	$-0.539920\pi$
$103$	−261.509	−0.250168	−0.125084	+	0.992146i	$0.539920\pi$
$104$	0	0
$105$	1620.58	1.50621
$106$	158.174i	0.144936i
$107$	1041.57	0.941052	0.470526	−	0.882386i	$-0.344064\pi$
$107$	1041.57	0.941052	0.470526	+	0.882386i	$0.344064\pi$
$108$	−186.160	−0.165863
$109$	− 609.644i	− 0.535718i	−0.963458	−	0.267859i	$-0.913684\pi$
$109$	− 609.644i	− 0.535718i	0.963458	−	0.267859i	$-0.0863161\pi$
$110$	956.207i	0.828825i
$111$	785.330i	0.671533i
$112$	1168.54i	0.985865i
$113$	−333.997	−0.278051	−0.139025	−	0.990289i	$-0.544397\pi$
$113$	−333.997	−0.278051	−0.139025	+	0.990289i	$0.544397\pi$
$114$	−229.290	−0.188377
$115$	749.660i	0.607880i
$116$	−932.403	−0.746306
$117$	0	0
$118$	−523.343	−0.408285
$119$	90.3320i	0.0695859i
$120$	−840.338	−0.639266
$121$	−1254.28	−0.942361
$122$	− 551.994i	− 0.409633i
$123$	− 596.732i	− 0.437443i
$124$	− 2184.41i	− 1.58198i
$125$	− 1252.26i	− 0.896048i
$126$	285.716	0.202013
$127$	1110.44	0.775872	0.387936	−	0.921686i	$-0.373188\pi$
$127$	1110.44	0.775872	0.387936	+	0.921686i	$0.373188\pi$
$128$	− 1469.65i	− 1.01484i
$129$	−604.052	−0.412277
$130$	0	0
$131$	−557.489	−0.371817	−0.185909	−	0.982567i	$-0.559523\pi$
$131$	−557.489	−0.371817	−0.185909	+	0.982567i	$0.559523\pi$
$132$	− 1051.71i	− 0.693483i
$133$	−2195.41	−1.43132
$134$	817.463	0.527000
$135$	− 482.993i	− 0.307922i
$136$	− 46.8410i	− 0.0295337i
$137$	1364.89i	0.851170i	0.904918	+	0.425585i	$0.139932\pi$
$137$	1364.89i	0.851170i	−0.904918	+	0.425585i	$0.860068\pi$
$138$	132.169i	0.0815287i
$139$	−1936.99	−1.18197	−0.590985	−	0.806683i	$-0.701261\pi$
$139$	−1936.99	−1.18197	−0.590985	+	0.806683i	$0.701261\pi$
$140$	−3724.52	−2.24842
$141$	292.106i	0.174467i
$142$	1064.07	0.628836
$143$	0	0
$144$	348.270	0.201545
$145$	− 2419.13i	− 1.38550i
$146$	−644.343	−0.365248
$147$	1706.67	0.957577
$148$	− 1804.90i	− 1.00244i
$149$	3499.94i	1.92434i	0.272452	+	0.962169i	$0.412165\pi$
$149$	3499.94i	1.92434i	−0.272452	+	0.962169i	$0.587835\pi$
$150$	− 615.013i	− 0.334770i
$151$	− 672.434i	− 0.362397i	−0.983447	−	0.181198i	$-0.942002\pi$
$151$	− 672.434i	− 0.362397i	0.983447	−	0.181198i	$-0.0579976\pi$
$152$	1138.41	0.607482
$153$	26.9224	0.0142258
$154$	1614.16i	0.844627i
$155$	5667.48	2.93692
$156$	0	0
$157$	1272.02	0.646612	0.323306	−	0.946295i	$-0.395206\pi$
$157$	1272.02	0.646612	0.323306	+	0.946295i	$0.395206\pi$
$158$	− 755.668i	− 0.380492i
$159$	451.374	0.225134
$160$	2968.63	1.46682
$161$	1265.49i	0.619469i
$162$	− 85.1541i	− 0.0412984i
$163$	− 1476.22i	− 0.709366i	−0.934987	−	0.354683i	$-0.884589\pi$
$163$	− 1476.22i	− 0.709366i	0.934987	−	0.354683i	$-0.115411\pi$
$164$	1371.45i	0.653001i
$165$	2728.68	1.28744
$166$	418.127	0.195500
$167$	− 3234.78i	− 1.49889i	−0.662066	−	0.749446i	$-0.730320\pi$
$167$	− 3234.78i	− 1.49889i	0.662066	−	0.749446i	$-0.269680\pi$
$168$	−1418.56	−0.651454
$169$	0	0
$170$	56.2559	0.0253802
$171$	654.314i	0.292612i
$172$	1388.27	0.615434
$173$	−424.597	−0.186598	−0.0932991	−	0.995638i	$-0.529741\pi$
$173$	−424.597	−0.186598	−0.0932991	+	0.995638i	$0.529741\pi$
$174$	− 426.505i	− 0.185823i
$175$	− 5888.62i	− 2.54364i
$176$	1967.56i	0.842671i
$177$	1493.44i	0.634201i
$178$	681.667	0.287040
$179$	4400.04	1.83729	0.918644	−	0.395086i	$-0.129285\pi$
$179$	4400.04	1.83729	0.918644	+	0.395086i	$0.129285\pi$
$180$	1110.05i	0.459656i
$181$	345.151	0.141740	0.0708698	−	0.997486i	$-0.477423\pi$
$181$	345.151	0.141740	0.0708698	+	0.997486i	$0.477423\pi$
$182$	0	0
$183$	−1575.20	−0.636295
$184$	− 656.209i	− 0.262915i
$185$	4682.83	1.86102
$186$	999.205	0.393899
$187$	152.098i	0.0594788i
$188$	− 671.338i	− 0.260438i
$189$	− 815.333i	− 0.313792i
$190$	1367.23i	0.522049i
$191$	−4076.61	−1.54436	−0.772181	−	0.635402i	$-0.780834\pi$
$191$	−4076.61	−1.54436	−0.772181	+	0.635402i	$0.780834\pi$
$192$	−405.335	−0.152357
$193$	− 3221.08i	− 1.20134i	−0.799497	−	0.600670i	$-0.794901\pi$
$193$	− 3221.08i	− 1.20134i	0.799497	−	0.600670i	$-0.205099\pi$
$194$	286.383	0.105985
$195$	0	0
$196$	−3922.38	−1.42944
$197$	3068.80i	1.10986i	0.831897	+	0.554931i	$0.187255\pi$
$197$	3068.80i	1.10986i	−0.831897	+	0.554931i	$0.812745\pi$
$198$	481.080	0.172671
$199$	−3992.05	−1.42206	−0.711028	−	0.703164i	$-0.751770\pi$
$199$	−3992.05	−1.42206	−0.711028	+	0.703164i	$0.751770\pi$
$200$	3053.50i	1.07957i
$201$	− 2332.75i	− 0.818605i
$202$	437.395i	0.152351i
$203$	− 4083.70i	− 1.41192i
$204$	−61.8747	−0.0212358
$205$	−3558.24	−1.21228
$206$	274.921i	0.0929838i
$207$	377.163	0.126641
$208$	0	0
$209$	−3696.56	−1.22343
$210$	− 1703.69i	− 0.559837i
$211$	2868.29	0.935837	0.467918	−	0.883772i	$-0.345004\pi$
$211$	2868.29	0.935837	0.467918	+	0.883772i	$0.345004\pi$
$212$	−1037.38	−0.336072
$213$	− 3036.48i	− 0.976789i
$214$	− 1094.99i	− 0.349775i
$215$	3601.89i	1.14254i
$216$	422.784i	0.133180i
$217$	9567.18	2.99291
$218$	−640.909	−0.199119
$219$	1838.73i	0.567351i
$220$	−6271.23	−1.92185
$221$	0	0
$222$	825.606	0.249599
$223$	− 3099.47i	− 0.930745i	−0.885115	−	0.465372i	$-0.845920\pi$
$223$	− 3099.47i	− 0.930745i	0.885115	−	0.465372i	$-0.154080\pi$
$224$	5011.30	1.49478
$225$	−1755.03	−0.520009
$226$	351.126i	0.103347i
$227$	− 258.860i	− 0.0756879i	−0.999284	−	0.0378440i	$-0.987951\pi$
$227$	− 258.860i	− 0.0756879i	0.999284	−	0.0378440i	$-0.0120490\pi$
$228$	− 1503.79i	− 0.436802i
$229$	804.986i	0.232292i	0.993232	+	0.116146i	$0.0370541\pi$
$229$	804.986i	0.232292i	−0.993232	+	0.116146i	$0.962946\pi$
$230$	788.107	0.225940
$231$	4606.24	1.31198
$232$	2117.57i	0.599247i
$233$	1717.65	0.482947	0.241474	−	0.970407i	$-0.422369\pi$
$233$	1717.65	0.482947	0.241474	+	0.970407i	$0.422369\pi$
$234$	0	0
$235$	1741.79	0.483498
$236$	− 3432.32i	− 0.946715i
$237$	−2156.41	−0.591029
$238$	94.9647	0.0258641
$239$	2355.26i	0.637445i	0.947848	+	0.318722i	$0.103254\pi$
$239$	2355.26i	0.637445i	−0.947848	+	0.318722i	$0.896746\pi$
$240$	− 2076.69i	− 0.558541i
$241$	2170.07i	0.580027i	0.957023	+	0.290014i	$0.0936598\pi$
$241$	2170.07i	0.580027i	−0.957023	+	0.290014i	$0.906340\pi$
$242$	1318.61i	0.350262i
$243$	−243.000	−0.0641500
$244$	3620.22	0.949840
$245$	− 10176.7i	− 2.65373i
$246$	−627.335	−0.162591
$247$	0	0
$248$	−4960.98	−1.27025
$249$	− 1193.19i	− 0.303676i
$250$	−1316.49	−0.333048
$251$	−2860.82	−0.719416	−0.359708	−	0.933065i	$-0.617124\pi$
$251$	−2860.82	−0.719416	−0.359708	+	0.933065i	$0.617124\pi$
$252$	1873.85i	0.468419i
$253$	2130.79i	0.529493i
$254$	− 1167.39i	− 0.288381i
$255$	− 160.535i	− 0.0394238i
$256$	−464.125	−0.113312
$257$	−2276.69	−0.552593	−0.276296	−	0.961073i	$-0.589107\pi$
$257$	−2276.69	−0.552593	−0.276296	+	0.961073i	$0.589107\pi$
$258$	635.031i	0.153238i
$259$	7905.00	1.89650
$260$	0	0
$261$	−1217.10	−0.288645
$262$	586.080i	0.138199i
$263$	−3933.20	−0.922172	−0.461086	−	0.887355i	$-0.652540\pi$
$263$	−3933.20	−0.922172	−0.461086	+	0.887355i	$0.652540\pi$
$264$	−2388.53	−0.556833
$265$	− 2691.49i	− 0.623912i
$266$	2308.00i	0.532002i
$267$	− 1945.24i	− 0.445868i
$268$	5361.29i	1.22199i
$269$	−1622.92	−0.367848	−0.183924	−	0.982940i	$-0.558880\pi$
$269$	−1622.92	−0.367848	−0.183924	+	0.982940i	$0.558880\pi$
$270$	−507.764	−0.114450
$271$	− 1554.77i	− 0.348508i	−0.984701	−	0.174254i	$-0.944249\pi$
$271$	− 1554.77i	− 0.348508i	0.984701	−	0.174254i	$-0.0557513\pi$
$272$	115.756	0.0258042
$273$	0	0
$274$	1434.89	0.316368
$275$	− 9915.08i	− 2.17419i
$276$	−866.822	−0.189046
$277$	8258.33	1.79132	0.895658	−	0.444743i	$-0.146705\pi$
$277$	8258.33	1.79132	0.895658	+	0.444743i	$0.146705\pi$
$278$	2036.33i	0.439321i
$279$	− 2851.38i	− 0.611855i
$280$	8458.70i	1.80537i
$281$	5023.16i	1.06639i	0.845991	+	0.533197i	$0.179009\pi$
$281$	5023.16i	1.06639i	−0.845991	+	0.533197i	$0.820991\pi$
$282$	307.087	0.0648467
$283$	4804.71	1.00922	0.504612	−	0.863346i	$-0.331636\pi$
$283$	4804.71	1.00922	0.504612	+	0.863346i	$0.331636\pi$
$284$	6978.64i	1.45812i
$285$	3901.60	0.810914
$286$	0	0
$287$	−6006.61	−1.23540
$288$	− 1493.56i	− 0.305586i
$289$	−4904.05	−0.998179
$290$	−2543.20	−0.514971
$291$	− 817.237i	− 0.164630i
$292$	− 4225.89i	− 0.846923i
$293$	− 2496.53i	− 0.497778i	−0.968532	−	0.248889i	$-0.919935\pi$
$293$	− 2496.53i	− 0.497778i	0.968532	−	0.248889i	$-0.0800654\pi$
$294$	− 1794.20i	− 0.355917i
$295$	8905.18	1.75756
$296$	−4099.08	−0.804912
$297$	− 1372.83i	− 0.268215i
$298$	3679.44	0.715249
$299$	0	0
$300$	4033.53	0.776253
$301$	6080.29i	1.16433i
$302$	−706.920	−0.134698
$303$	1248.17	0.236652
$304$	2813.31i	0.530770i
$305$	9392.71i	1.76336i
$306$	− 28.3031i	− 0.00528751i
$307$	1914.68i	0.355950i	0.984035	+	0.177975i	$0.0569545\pi$
$307$	1914.68i	0.355950i	−0.984035	+	0.177975i	$0.943045\pi$
$308$	−10586.4	−1.95849
$309$	784.528	0.144434
$310$	− 5958.14i	− 1.09161i
$311$	−3171.99	−0.578350	−0.289175	−	0.957276i	$-0.593381\pi$
$311$	−3171.99	−0.578350	−0.289175	+	0.957276i	$0.593381\pi$
$312$	0	0
$313$	1240.50	0.224017	0.112009	−	0.993707i	$-0.464272\pi$
$313$	1240.50	0.224017	0.112009	+	0.993707i	$0.464272\pi$
$314$	− 1337.25i	− 0.240336i
$315$	−4861.73	−0.869611
$316$	4956.01	0.882270
$317$	− 9521.41i	− 1.68699i	−0.537137	−	0.843495i	$-0.680494\pi$
$317$	− 9521.41i	− 1.68699i	0.537137	−	0.843495i	$-0.319506\pi$
$318$	− 474.523i	− 0.0836790i
$319$	− 6876.01i	− 1.20684i
$320$	2416.96i	0.422226i
$321$	−3124.72	−0.543317
$322$	1330.39	0.230248
$323$	217.477i	0.0374637i
$324$	558.479	0.0957611
$325$	0	0
$326$	−1551.93	−0.263661
$327$	1828.93i	0.309297i
$328$	3114.68	0.524327
$329$	2940.29	0.492716
$330$	− 2868.62i	− 0.478522i
$331$	1997.12i	0.331637i	0.986156	+	0.165818i	$0.0530266\pi$
$331$	1997.12i	0.331637i	−0.986156	+	0.165818i	$0.946973\pi$
$332$	2742.27i	0.453317i
$333$	− 2355.99i	− 0.387710i
$334$	−3400.68	−0.557116
$335$	−13909.9	−2.26860
$336$	− 3505.63i	− 0.569190i
$337$	−5200.25	−0.840581	−0.420290	−	0.907390i	$-0.638072\pi$
$337$	−5200.25	−0.840581	−0.420290	+	0.907390i	$0.638072\pi$
$338$	0	0
$339$	1001.99	0.160533
$340$	368.952i	0.0588506i
$341$	16108.9	2.55820
$342$	687.871	0.108760
$343$	− 6821.32i	− 1.07381i
$344$	− 3152.88i	− 0.494163i
$345$	− 2248.98i	− 0.350960i
$346$	446.372i	0.0693558i
$347$	4622.17	0.715075	0.357538	−	0.933899i	$-0.383616\pi$
$347$	4622.17	0.715075	0.357538	+	0.933899i	$0.383616\pi$
$348$	2797.21	0.430880
$349$	9870.06i	1.51385i	0.653504	+	0.756923i	$0.273298\pi$
$349$	9870.06i	1.51385i	−0.653504	+	0.756923i	$0.726702\pi$
$350$	−6190.62	−0.945436
$351$	0	0
$352$	8437.87	1.27767
$353$	8962.46i	1.35134i	0.737203	+	0.675671i	$0.236146\pi$
$353$	8962.46i	1.35134i	−0.737203	+	0.675671i	$0.763854\pi$
$354$	1570.03	0.235723
$355$	−18106.2	−2.70697
$356$	4470.68i	0.665577i
$357$	− 270.996i	− 0.0401754i
$358$	− 4625.70i	− 0.682893i
$359$	− 9174.16i	− 1.34873i	−0.738399	−	0.674365i	$-0.764418\pi$
$359$	− 9174.16i	− 1.34873i	0.738399	−	0.674365i	$-0.235582\pi$
$360$	2521.01	0.369081
$361$	1573.48	0.229404
$362$	− 362.852i	− 0.0526826i
$363$	3762.85	0.544072
$364$	0	0
$365$	10964.1	1.57230
$366$	1655.98i	0.236502i
$367$	10360.7	1.47364	0.736818	−	0.676092i	$-0.236328\pi$
$367$	10360.7	1.47364	0.736818	+	0.676092i	$0.236328\pi$
$368$	1621.66	0.229715
$369$	1790.20i	0.252558i
$370$	− 4922.99i	− 0.691713i
$371$	− 4543.46i	− 0.635807i
$372$	6553.23i	0.913358i
$373$	−5480.60	−0.760790	−0.380395	−	0.924824i	$-0.624212\pi$
$373$	−5480.60	−0.760790	−0.380395	+	0.924824i	$0.624212\pi$
$374$	159.899	0.0221074
$375$	3756.79i	0.517333i
$376$	−1524.67	−0.209119
$377$	0	0
$378$	−857.148	−0.116632
$379$	− 4262.83i	− 0.577748i	−0.957367	−	0.288874i	$-0.906719\pi$
$379$	− 4262.83i	− 0.577748i	0.957367	−	0.288874i	$-0.0932809\pi$
$380$	−8966.91	−1.21051
$381$	−3331.33	−0.447950
$382$	4285.68i	0.574017i
$383$	901.981i	0.120337i	0.998188	+	0.0601685i	$0.0191638\pi$
$383$	901.981i	0.120337i	−0.998188	+	0.0601685i	$0.980836\pi$
$384$	4408.94i	0.585919i
$385$	− 27466.4i	− 3.63590i
$386$	−3386.28	−0.446520
$387$	1812.15	0.238028
$388$	1878.23i	0.245754i
$389$	1093.46	0.142521	0.0712606	−	0.997458i	$-0.477298\pi$
$389$	1093.46	0.142521	0.0712606	+	0.997458i	$0.477298\pi$
$390$	0	0
$391$	125.360	0.0162141
$392$	8908.07i	1.14777i
$393$	1672.47	0.214669
$394$	3226.18	0.412519
$395$	12858.4i	1.63792i
$396$	3155.14i	0.400383i
$397$	2587.62i	0.327126i	0.986533	+	0.163563i	$0.0522987\pi$
$397$	2587.62i	0.327126i	−0.986533	+	0.163563i	$0.947701\pi$
$398$	4196.79i	0.528558i
$399$	6586.22	0.826374
$400$	−7545.98	−0.943247
$401$	422.775i	0.0526494i	0.999653	+	0.0263247i	$0.00838037\pi$
$401$	422.775i	0.0526494i	−0.999653	+	0.0263247i	$0.991620\pi$
$402$	−2452.39	−0.304264
$403$	0	0
$404$	−2868.63	−0.353267
$405$	1448.98i	0.177779i
$406$	−4293.13	−0.524789
$407$	13310.2	1.62104
$408$	140.523i	0.0170513i
$409$	3028.09i	0.366087i	0.983105	+	0.183044i	$0.0585949\pi$
$409$	3028.09i	0.366087i	−0.983105	+	0.183044i	$0.941405\pi$
$410$	3740.73i	0.450588i
$411$	− 4094.66i	− 0.491423i
$412$	−1803.05	−0.215607
$413$	15032.7	1.79107
$414$	− 396.506i	− 0.0470706i
$415$	−7114.84	−0.841576
$416$	0	0
$417$	5810.98	0.682410
$418$	3886.14i	0.454730i
$419$	−9629.51	−1.12275	−0.561375	−	0.827562i	$-0.689727\pi$
$419$	−9629.51	−1.12275	−0.561375	+	0.827562i	$0.689727\pi$
$420$	11173.6	1.29813
$421$	8288.54i	0.959521i	0.877399	+	0.479761i	$0.159277\pi$
$421$	8288.54i	0.959521i	−0.877399	+	0.479761i	$0.840723\pi$
$422$	− 3015.40i	− 0.347837i
$423$	− 876.319i	− 0.100728i
$424$	2355.97i	0.269850i
$425$	−583.328	−0.0665778
$426$	−3192.21	−0.363058
$427$	15855.7i	1.79698i
$428$	7181.43	0.811046
$429$	0	0
$430$	3786.61	0.424667
$431$	3873.84i	0.432938i	0.976289	+	0.216469i	$0.0694541\pi$
$431$	3873.84i	0.432938i	−0.976289	+	0.216469i	$0.930546\pi$
$432$	−1044.81	−0.116362
$433$	−3016.87	−0.334830	−0.167415	−	0.985886i	$-0.553542\pi$
$433$	−3016.87	−0.334830	−0.167415	+	0.985886i	$0.553542\pi$
$434$	− 10057.8i	− 1.11242i
$435$	7257.40i	0.799921i
$436$	− 4203.37i	− 0.461709i
$437$	3046.71i	0.333510i
$438$	1933.03	0.210876
$439$	2862.77	0.311235	0.155618	−	0.987817i	$-0.450263\pi$
$439$	2862.77	0.311235	0.155618	+	0.987817i	$0.450263\pi$
$440$	14242.5i	1.54315i
$441$	−5120.01	−0.552857
$442$	0	0
$443$	13067.0	1.40143	0.700713	−	0.713443i	$-0.252865\pi$
$443$	13067.0	1.40143	0.700713	+	0.713443i	$0.252865\pi$
$444$	5414.69i	0.578761i
$445$	−11599.2	−1.23563
$446$	−3258.43	−0.345944
$447$	− 10499.8i	− 1.11102i
$448$	4080.04i	0.430276i
$449$	9527.11i	1.00136i	0.865631	+	0.500682i	$0.166917\pi$
$449$	9527.11i	1.00136i	−0.865631	+	0.500682i	$0.833083\pi$
$450$	1845.04i	0.193280i
$451$	−10113.7	−1.05596
$452$	−2302.84	−0.239638
$453$	2017.30i	0.209230i
$454$	−272.136	−0.0281321
$455$	0	0
$456$	−3415.23	−0.350730
$457$	6223.04i	0.636983i	0.947926	+	0.318492i	$0.103176\pi$
$457$	6223.04i	0.636983i	−0.947926	+	0.318492i	$0.896824\pi$
$458$	846.270	0.0863397
$459$	−80.7671	−0.00821325
$460$	5168.76i	0.523901i
$461$	17482.6i	1.76626i	0.469132	+	0.883128i	$0.344567\pi$
$461$	17482.6i	1.76626i	−0.469132	+	0.883128i	$0.655433\pi$
$462$	− 4842.47i	− 0.487645i
$463$	3280.46i	0.329278i	0.986354	+	0.164639i	$0.0526460\pi$
$463$	3280.46i	0.329278i	−0.986354	+	0.164639i	$0.947354\pi$
$464$	−5233.06	−0.523575
$465$	−17002.4	−1.69563
$466$	− 1805.74i	− 0.179504i
$467$	−8459.42	−0.838234	−0.419117	−	0.907932i	$-0.637660\pi$
$467$	−8459.42	−0.838234	−0.419117	+	0.907932i	$0.637660\pi$
$468$	0	0
$469$	−23481.1	−2.31185
$470$	− 1831.12i	− 0.179709i
$471$	−3816.05	−0.373321
$472$	−7795.09	−0.760165
$473$	10237.8i	0.995211i
$474$	2267.00i	0.219677i
$475$	− 14177.0i	− 1.36945i
$476$	622.821i	0.0599726i
$477$	−1354.12	−0.129981
$478$	2476.05	0.236929
$479$	− 17593.9i	− 1.67826i	−0.543933	−	0.839129i	$-0.683065\pi$
$479$	− 17593.9i	− 1.67826i	0.543933	−	0.839129i	$-0.316935\pi$
$480$	−8905.90	−0.846868
$481$	0	0
$482$	2281.36	0.215588
$483$	− 3796.47i	− 0.357651i
$484$	−8648.02	−0.812174
$485$	−4873.08	−0.456238
$486$	255.462i	0.0238436i
$487$	12736.9i	1.18515i	0.805517	+	0.592573i	$0.201888\pi$
$487$	12736.9i	1.18515i	−0.805517	+	0.592573i	$0.798112\pi$
$488$	− 8221.84i	− 0.762675i
$489$	4428.67i	0.409553i
$490$	−10698.6	−0.986352
$491$	5458.09	0.501671	0.250835	−	0.968030i	$-0.419295\pi$
$491$	5458.09	0.501671	0.250835	+	0.968030i	$0.419295\pi$
$492$	− 4114.35i	− 0.377010i
$493$	−404.532	−0.0369558
$494$	0	0
$495$	−8186.04	−0.743303
$496$	− 12259.9i	− 1.10985i
$497$	−30564.7	−2.75858
$498$	−1254.38	−0.112872
$499$	− 18109.7i	− 1.62466i	−0.583201	−	0.812328i	$-0.698200\pi$
$499$	− 18109.7i	− 1.62466i	0.583201	−	0.812328i	$-0.301800\pi$
$500$	− 8634.12i	− 0.772259i
$501$	9704.35i	0.865386i
$502$	3007.54i	0.267396i
$503$	19414.1	1.72094	0.860469	−	0.509503i	$-0.170171\pi$
$503$	19414.1	1.72094	0.860469	+	0.509503i	$0.170171\pi$
$504$	4255.68	0.376117
$505$	− 7442.70i	− 0.655833i
$506$	2240.07	0.196805
$507$	0	0
$508$	7656.27	0.668686
$509$	16124.1i	1.40410i	0.712126	+	0.702051i	$0.247732\pi$
$509$	16124.1i	1.40410i	−0.712126	+	0.702051i	$0.752268\pi$
$510$	−168.768	−0.0146533
$511$	18508.3	1.60227
$512$	− 11269.2i	− 0.972724i
$513$	− 1962.94i	− 0.168940i
$514$	2393.46i	0.205391i
$515$	− 4678.05i	− 0.400271i
$516$	−4164.81	−0.355321
$517$	4950.78	0.421151
$518$	− 8310.41i	− 0.704901i
$519$	1273.79	0.107733
$520$	0	0
$521$	18662.7	1.56934	0.784672	−	0.619911i	$-0.212831\pi$
$521$	18662.7	1.56934	0.784672	+	0.619911i	$0.212831\pi$
$522$	1279.51i	0.107285i
$523$	−2290.01	−0.191463	−0.0957314	−	0.995407i	$-0.530519\pi$
$523$	−2290.01	−0.191463	−0.0957314	+	0.995407i	$0.530519\pi$
$524$	−3843.77	−0.320450
$525$	17665.9i	1.46857i
$526$	4134.91i	0.342758i
$527$	− 947.727i	− 0.0783370i
$528$	− 5902.67i	− 0.486517i
$529$	−10410.8	−0.855659
$530$	−2829.52	−0.231899
$531$	− 4480.31i	− 0.366156i
$532$	−15136.9	−1.23359
$533$	0	0
$534$	−2045.00	−0.165723
$535$	18632.3i	1.50569i
$536$	12175.9	0.981195
$537$	−13200.1	−1.06076
$538$	1706.15i	0.136724i
$539$	− 28925.6i	− 2.31153i
$540$	− 3330.14i	− 0.265382i
$541$	11740.6i	0.933029i	0.884514	+	0.466514i	$0.154490\pi$
$541$	11740.6i	0.933029i	−0.884514	+	0.466514i	$0.845510\pi$
$542$	−1634.51	−0.129535
$543$	−1035.45	−0.0818334
$544$	− 496.420i	− 0.0391247i
$545$	10905.7	0.857153
$546$	0	0
$547$	312.599	0.0244347	0.0122174	−	0.999925i	$-0.496111\pi$
$547$	312.599	0.0244347	0.0122174	+	0.999925i	$0.496111\pi$
$548$	9410.63i	0.733581i
$549$	4725.59	0.367365
$550$	−10423.6	−0.808114
$551$	− 9831.64i	− 0.760149i
$552$	1968.63i	0.151794i
$553$	21706.1i	1.66914i
$554$	− 8681.86i	− 0.665806i
$555$	−14048.5	−1.07446
$556$	−13355.2	−1.01868
$557$	7567.62i	0.575674i	0.957679	+	0.287837i	$0.0929361\pi$
$557$	7567.62i	0.575674i	−0.957679	+	0.287837i	$0.907064\pi$
$558$	−2997.61	−0.227418
$559$	0	0
$560$	−20903.6	−1.57739
$561$	− 456.295i	− 0.0343401i
$562$	5280.78	0.396363
$563$	−13068.3	−0.978261	−0.489131	−	0.872211i	$-0.662686\pi$
$563$	−13068.3	−0.978261	−0.489131	+	0.872211i	$0.662686\pi$
$564$	2014.01i	0.150364i
$565$	− 5974.74i	− 0.444884i
$566$	− 5051.12i	− 0.375114i
$567$	2446.00i	0.181168i
$568$	15849.1	1.17080
$569$	5959.19	0.439055	0.219528	−	0.975606i	$-0.429548\pi$
$569$	5959.19	0.439055	0.219528	+	0.975606i	$0.429548\pi$
$570$	− 4101.69i	− 0.301405i
$571$	5460.53	0.400203	0.200102	−	0.979775i	$-0.435873\pi$
$571$	5460.53	0.400203	0.200102	+	0.979775i	$0.435873\pi$
$572$	0	0
$573$	12229.8	0.891638
$574$	6314.66i	0.459179i
$575$	−8172.02	−0.592690
$576$	1216.00	0.0879633
$577$	8560.26i	0.617623i	0.951123	+	0.308811i	$0.0999312\pi$
$577$	8560.26i	0.617623i	−0.951123	+	0.308811i	$0.900069\pi$
$578$	5155.56i	0.371009i
$579$	9663.24i	0.693594i
$580$	− 16679.4i	− 1.19410i
$581$	−12010.4	−0.857620
$582$	−859.149	−0.0611905
$583$	− 7650.14i	− 0.543458i
$584$	−9597.36	−0.680037
$585$	0	0
$586$	−2624.57	−0.185017
$587$	− 25603.9i	− 1.80032i	−0.435564	−	0.900158i	$-0.643451\pi$
$587$	− 25603.9i	− 1.80032i	0.435564	−	0.900158i	$-0.356549\pi$
$588$	11767.1	0.825287
$589$	23033.3	1.61133
$590$	− 9361.89i	− 0.653259i
$591$	− 9206.39i	− 0.640779i
$592$	− 10129.9i	− 0.703269i
$593$	− 16311.5i	− 1.12956i	−0.825240	−	0.564782i	$-0.808960\pi$
$593$	− 16311.5i	− 1.12956i	0.825240	−	0.564782i	$-0.191040\pi$
$594$	−1443.24	−0.0996916
$595$	−1615.92	−0.111338
$596$	24131.4i	1.65849i
$597$	11976.2	0.821024
$598$	0	0
$599$	6549.91	0.446782	0.223391	−	0.974729i	$-0.428287\pi$
$599$	6549.91	0.446782	0.223391	+	0.974729i	$0.428287\pi$
$600$	− 9160.49i	− 0.623292i
$601$	11467.3	0.778306	0.389153	−	0.921173i	$-0.372768\pi$
$601$	11467.3	0.778306	0.389153	+	0.921173i	$0.372768\pi$
$602$	6392.12	0.432763
$603$	6998.26i	0.472622i
$604$	− 4636.30i	− 0.312332i
$605$	− 22437.4i	− 1.50779i
$606$	− 1312.18i	− 0.0879601i
$607$	17594.3	1.17649	0.588245	−	0.808682i	$-0.299819\pi$
$607$	17594.3	1.17649	0.588245	+	0.808682i	$0.299819\pi$
$608$	12064.9	0.804762
$609$	12251.1i	0.815171i
$610$	9874.42	0.655416
$611$	0	0
$612$	185.624	0.0122605
$613$	− 12783.2i	− 0.842265i	−0.906999	−	0.421132i	$-0.861633\pi$
$613$	− 12783.2i	− 0.842265i	0.906999	−	0.421132i	$-0.138367\pi$
$614$	2012.87	0.132301
$615$	10674.7	0.699913
$616$	24042.5i	1.57257i
$617$	− 9767.34i	− 0.637307i	−0.947871	−	0.318653i	$-0.896769\pi$
$617$	− 9767.34i	− 0.637307i	0.947871	−	0.318653i	$-0.103231\pi$
$618$	− 824.763i	− 0.0536842i
$619$	24677.0i	1.60235i	0.598433	+	0.801173i	$0.295790\pi$
$619$	24677.0i	1.60235i	−0.598433	+	0.801173i	$0.704210\pi$
$620$	39076.1	2.53119
$621$	−1131.49	−0.0731162
$622$	3334.66i	0.214964i
$623$	−19580.5	−1.25919
$624$	0	0
$625$	−1974.11	−0.126343
$626$	− 1304.12i	− 0.0832639i
$627$	11089.7	0.706346
$628$	8770.30	0.557282
$629$	− 783.071i	− 0.0496393i
$630$	5111.07i	0.323222i
$631$	22666.9i	1.43004i	0.699103	+	0.715021i	$0.253583\pi$
$631$	22666.9i	1.43004i	−0.699103	+	0.715021i	$0.746417\pi$
$632$	− 11255.5i	− 0.708419i
$633$	−8604.88	−0.540306
$634$	−10009.7	−0.627030
$635$	19864.3i	1.24140i
$636$	3112.13	0.194032
$637$	0	0
$638$	−7228.64	−0.448565
$639$	9109.44i	0.563950i
$640$	26290.0	1.62375
$641$	10770.3	0.663653	0.331826	−	0.943341i	$-0.392335\pi$
$641$	10770.3	0.663653	0.331826	+	0.943341i	$0.392335\pi$
$642$	3284.97i	0.201943i
$643$	21220.8i	1.30151i	0.759289	+	0.650753i	$0.225547\pi$
$643$	21220.8i	1.30151i	−0.759289	+	0.650753i	$0.774453\pi$
$644$	8725.29i	0.533889i
$645$	− 10805.7i	− 0.659647i
$646$	228.631	0.0139247
$647$	19430.5	1.18067	0.590334	−	0.807159i	$-0.298996\pi$
$647$	19430.5	1.18067	0.590334	+	0.807159i	$0.298996\pi$
$648$	− 1268.35i	− 0.0768914i
$649$	25311.6	1.53092
$650$	0	0
$651$	−28701.5	−1.72796
$652$	− 10178.3i	− 0.611367i
$653$	28307.3	1.69640	0.848201	−	0.529674i	$-0.177686\pi$
$653$	28307.3	1.69640	0.848201	+	0.529674i	$0.177686\pi$
$654$	1922.73	0.114961
$655$	− 9972.72i	− 0.594910i
$656$	7697.17i	0.458116i
$657$	− 5516.19i	− 0.327560i
$658$	− 3091.09i	− 0.183136i
$659$	25525.5	1.50885	0.754425	−	0.656387i	$-0.227916\pi$
$659$	25525.5	1.50885	0.754425	+	0.656387i	$0.227916\pi$
$660$	18813.7	1.10958
$661$	− 26065.8i	− 1.53380i	−0.641765	−	0.766901i	$-0.721798\pi$
$661$	− 26065.8i	− 1.53380i	0.641765	−	0.766901i	$-0.278202\pi$
$662$	2099.55	0.123265
$663$	0	0
$664$	6227.92	0.363991
$665$	− 39272.8i	− 2.29013i
$666$	−2476.82	−0.144106
$667$	−5667.21	−0.328988
$668$	− 22303.2i	− 1.29182i
$669$	9298.42i	0.537366i
$670$	14623.3i	0.843205i
$671$	26697.3i	1.53597i
$672$	−15033.9	−0.863014
$673$	−27351.3	−1.56659	−0.783295	−	0.621650i	$-0.786463\pi$
$673$	−27351.3	−1.56659	−0.783295	+	0.621650i	$0.786463\pi$
$674$	5466.95i	0.312432i
$675$	5265.09	0.300227
$676$	0	0
$677$	−20998.8	−1.19209	−0.596047	−	0.802949i	$-0.703263\pi$
$677$	−20998.8	−1.19209	−0.596047	+	0.802949i	$0.703263\pi$
$678$	− 1053.38i	− 0.0596677i
$679$	−8226.17	−0.464936
$680$	837.921	0.0472541
$681$	776.581i	0.0436985i
$682$	− 16935.1i	− 0.950847i
$683$	− 21227.9i	− 1.18926i	−0.804000	−	0.594629i	$-0.797299\pi$
$683$	− 21227.9i	− 1.18926i	0.804000	−	0.594629i	$-0.202701\pi$
$684$	4511.36i	0.252188i
$685$	−24416.0	−1.36188
$686$	−7171.15	−0.399119
$687$	− 2414.96i	− 0.134114i
$688$	7791.59	0.431761
$689$	0	0
$690$	−2364.32	−0.130447
$691$	11371.3i	0.626025i	0.949749	+	0.313012i	$0.101338\pi$
$691$	11371.3i	0.626025i	−0.949749	+	0.313012i	$0.898662\pi$
$692$	−2927.51	−0.160820
$693$	−13818.7	−0.757474
$694$	− 4859.22i	− 0.265783i
$695$	− 34650.2i	− 1.89116i
$696$	− 6352.71i	− 0.345975i
$697$	595.016i	0.0323355i
$698$	10376.3	0.562675
$699$	−5152.94	−0.278830
$700$	− 40600.8i	− 2.19224i
$701$	8695.98	0.468534	0.234267	−	0.972172i	$-0.424731\pi$
$701$	8695.98	0.468534	0.234267	+	0.972172i	$0.424731\pi$
$702$	0	0
$703$	19031.6	1.02104
$704$	6869.84i	0.367780i
$705$	−5225.38	−0.279148
$706$	9422.11	0.502274
$707$	− 12563.9i	− 0.668336i
$708$	10296.9i	0.546586i
$709$	9428.79i	0.499444i	0.968318	+	0.249722i	$0.0803392\pi$
$709$	9428.79i	0.499444i	−0.968318	+	0.249722i	$0.919661\pi$
$710$	19034.7i	1.00614i
$711$	6469.23	0.341231
$712$	10153.3	0.534425
$713$	− 13277.0i	− 0.697374i
$714$	−284.894	−0.0149326
$715$	0	0
$716$	30337.4	1.58347
$717$	− 7065.79i	− 0.368029i
$718$	−9644.66	−0.501303
$719$	−790.174	−0.0409854	−0.0204927	−	0.999790i	$-0.506523\pi$
$719$	−790.174	−0.0409854	−0.0204927	+	0.999790i	$0.506523\pi$
$720$	6230.07i	0.322474i
$721$	− 7896.93i	− 0.407902i
$722$	− 1654.18i	− 0.0852663i
$723$	− 6510.21i	− 0.334879i
$724$	2379.75	0.122158
$725$	26370.9	1.35088
$726$	− 3955.83i	− 0.202224i
$727$	1591.88	0.0812101	0.0406050	−	0.999175i	$-0.487071\pi$
$727$	1591.88	0.0812101	0.0406050	+	0.999175i	$0.487071\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 11526.4i	− 0.584400i
$731$	602.314	0.0304752
$732$	−10860.7	−0.548391
$733$	− 2695.20i	− 0.135811i	−0.997692	−	0.0679056i	$-0.978368\pi$
$733$	− 2695.20i	− 0.135811i	0.997692	−	0.0679056i	$-0.0216317\pi$
$734$	− 10892.1i	− 0.547729i
$735$	30530.0i	1.53213i
$736$	− 6954.50i	− 0.348297i
$737$	−39536.8	−1.97606
$738$	1882.01	0.0938721
$739$	1979.17i	0.0985181i	0.998786	+	0.0492590i	$0.0156860\pi$
$739$	1979.17i	0.0985181i	−0.998786	+	0.0492590i	$0.984314\pi$
$740$	32287.1	1.60392
$741$	0	0
$742$	−4776.47	−0.236320
$743$	− 19324.2i	− 0.954156i	−0.878861	−	0.477078i	$-0.841696\pi$
$743$	− 19324.2i	− 0.954156i	0.878861	−	0.477078i	$-0.158304\pi$
$744$	14883.0	0.733381
$745$	−62609.2	−3.07896
$746$	5761.67i	0.282775i
$747$	3579.57i	0.175327i
$748$	1048.69i	0.0512618i
$749$	31452.9i	1.53440i
$750$	3949.46	0.192285
$751$	−13898.2	−0.675303	−0.337652	−	0.941271i	$-0.609633\pi$
$751$	−13898.2	−0.675303	−0.337652	+	0.941271i	$0.609633\pi$
$752$	− 3767.84i	− 0.182712i
$753$	8582.46	0.415355
$754$	0	0
$755$	12028.9	0.579838
$756$	− 5621.56i	− 0.270442i
$757$	16195.7	0.777601	0.388800	−	0.921322i	$-0.372890\pi$
$757$	16195.7	0.777601	0.388800	+	0.921322i	$0.372890\pi$
$758$	−4481.45	−0.214741
$759$	− 6392.38i	− 0.305703i
$760$	20364.6i	0.971977i
$761$	− 22912.0i	− 1.09141i	−0.837978	−	0.545704i	$-0.816262\pi$
$761$	− 22912.0i	− 1.09141i	0.837978	−	0.545704i	$-0.183738\pi$
$762$	3502.17i	0.166497i
$763$	18409.7	0.873495
$764$	−28107.4	−1.33101
$765$	481.604i	0.0227614i
$766$	948.240	0.0447275
$767$	0	0
$768$	1392.37	0.0654205
$769$	− 17213.9i	− 0.807215i	−0.914932	−	0.403607i	$-0.867756\pi$
$769$	− 17213.9i	− 0.807215i	0.914932	−	0.403607i	$-0.132244\pi$
$770$	−28875.1	−1.35141
$771$	6830.08	0.319039
$772$	− 22208.7i	− 1.03537i
$773$	− 19732.2i	− 0.918136i	−0.888401	−	0.459068i	$-0.848183\pi$
$773$	− 19732.2i	− 0.918136i	0.888401	−	0.459068i	$-0.151817\pi$
$774$	− 1905.09i	− 0.0884717i
$775$	61781.0i	2.86353i
$776$	4265.62	0.197328
$777$	−23715.0	−1.09494
$778$	− 1149.54i	− 0.0529731i
$779$	−14461.1	−0.665113
$780$	0	0
$781$	−51463.9	−2.35791
$782$	− 131.789i	− 0.00602654i
$783$	3651.29	0.166649
$784$	−22014.1	−1.00283
$785$	22754.7i	1.03458i
$786$	− 1758.24i	− 0.0797892i
$787$	− 34771.8i	− 1.57494i	−0.616350	−	0.787472i	$-0.711389\pi$
$787$	− 34771.8i	− 1.57494i	0.616350	−	0.787472i	$-0.288611\pi$
$788$	21158.7i	0.956534i
$789$	11799.6	0.532416
$790$	13517.9	0.608790
$791$	− 10085.9i	− 0.453365i
$792$	7165.59	0.321487
$793$	0	0
$794$	2720.33	0.121588
$795$	8074.46i	0.360216i
$796$	−27524.4	−1.22560
$797$	16429.2	0.730178	0.365089	−	0.930973i	$-0.381039\pi$
$797$	16429.2	0.730178	0.365089	+	0.930973i	$0.381039\pi$
$798$	− 6924.00i	− 0.307151i
$799$	− 291.266i	− 0.0128964i
$800$	32360.9i	1.43017i
$801$	5835.71i	0.257422i
$802$	444.458	0.0195690
$803$	31163.8	1.36955
$804$	− 16083.9i	− 0.705515i
$805$	−22637.9	−0.991156
$806$	0	0
$807$	4868.76	0.212377
$808$	6514.91i	0.283656i
$809$	−1805.75	−0.0784755	−0.0392378	−	0.999230i	$-0.512493\pi$
$809$	−1805.75	−0.0784755	−0.0392378	+	0.999230i	$0.512493\pi$
$810$	1523.29	0.0660778
$811$	− 8758.70i	− 0.379235i	−0.981858	−	0.189618i	$-0.939275\pi$
$811$	− 8758.70i	− 0.379235i	0.981858	−	0.189618i	$-0.0607248\pi$
$812$	− 28156.3i	− 1.21686i
$813$	4664.31i	0.201211i
$814$	− 13992.8i	− 0.602516i
$815$	26407.6	1.13499
$816$	−347.268	−0.0148981
$817$	14638.5i	0.626850i
$818$	3183.39	0.136069
$819$	0	0
$820$	−24533.4	−1.04481
$821$	− 10480.4i	− 0.445518i	−0.974874	−	0.222759i	$-0.928494\pi$
$821$	− 10480.4i	− 0.445518i	0.974874	−	0.222759i	$-0.0715063\pi$
$822$	−4304.66	−0.182655
$823$	38187.3	1.61741	0.808703	−	0.588217i	$-0.200170\pi$
$823$	38187.3	1.61741	0.808703	+	0.588217i	$0.200170\pi$
$824$	4094.89i	0.173122i
$825$	29745.2i	1.25527i
$826$	− 15803.7i	− 0.665714i
$827$	− 10529.0i	− 0.442720i	−0.975192	−	0.221360i	$-0.928950\pi$
$827$	− 10529.0i	− 0.442720i	0.975192	−	0.221360i	$-0.0710495\pi$
$828$	2600.47	0.109145
$829$	−9131.19	−0.382557	−0.191278	−	0.981536i	$-0.561263\pi$
$829$	−9131.19	−0.382557	−0.191278	+	0.981536i	$0.561263\pi$
$830$	7479.73i	0.312801i
$831$	−24775.0	−1.03422
$832$	0	0
$833$	−1701.76	−0.0707834
$834$	− 6109.00i	− 0.253642i
$835$	57865.8	2.39824
$836$	−25487.0	−1.05441
$837$	8554.14i	0.353255i
$838$	10123.4i	0.417310i
$839$	40623.1i	1.67159i	0.549041	+	0.835796i	$0.314993\pi$
$839$	40623.1i	1.67159i	−0.549041	+	0.835796i	$0.685007\pi$
$840$	− 25376.1i	− 1.04233i
$841$	−6101.07	−0.250157
$842$	8713.62	0.356640
$843$	− 15069.5i	− 0.615683i
$844$	19776.3	0.806551
$845$	0	0
$846$	−921.261	−0.0374393
$847$	− 37876.2i	− 1.53653i
$848$	−5822.22	−0.235773
$849$	−14414.1	−0.582675
$850$	613.244i	0.0247460i
$851$	− 10970.3i	− 0.441900i
$852$	− 20935.9i	− 0.841846i
$853$	− 40704.0i	− 1.63385i	−0.576742	−	0.816927i	$-0.695676\pi$
$853$	− 40704.0i	− 1.63385i	0.576742	−	0.816927i	$-0.304324\pi$
$854$	16668.8	0.667911
$855$	−11704.8	−0.468182
$856$	− 16309.7i	− 0.651229i
$857$	−45984.1	−1.83289	−0.916445	−	0.400160i	$-0.868955\pi$
$857$	−45984.1	−1.83289	−0.916445	+	0.400160i	$0.868955\pi$
$858$	0	0
$859$	7787.52	0.309321	0.154660	−	0.987968i	$-0.450572\pi$
$859$	7787.52	0.309321	0.154660	+	0.987968i	$0.450572\pi$
$860$	24834.3i	0.984700i
$861$	18019.8	0.713257
$862$	4072.51	0.160917
$863$	34878.2i	1.37575i	0.725831	+	0.687873i	$0.241455\pi$
$863$	34878.2i	1.37575i	−0.725831	+	0.687873i	$0.758545\pi$
$864$	4480.67i	0.176430i
$865$	− 7595.46i	− 0.298559i
$866$	3171.59i	0.124452i
$867$	14712.2	0.576299
$868$	65963.8	2.57944
$869$	36548.1i	1.42671i
$870$	7629.59	0.297319
$871$	0	0
$872$	−9546.22	−0.370729
$873$	2451.71i	0.0950491i
$874$	3202.96	0.123961
$875$	37815.3	1.46102
$876$	12677.7i	0.488971i
$877$	35625.1i	1.37169i	0.727747	+	0.685846i	$0.240568\pi$
$877$	35625.1i	1.37169i	−0.727747	+	0.685846i	$0.759432\pi$
$878$	− 3009.58i	− 0.115682i
$879$	7489.59i	0.287392i
$880$	−35196.9	−1.34828
$881$	−15602.6	−0.596669	−0.298335	−	0.954461i	$-0.596431\pi$
$881$	−15602.6	−0.596669	−0.298335	+	0.954461i	$0.596431\pi$
$882$	5382.59i	0.205489i
$883$	25341.6	0.965812	0.482906	−	0.875672i	$-0.339581\pi$
$883$	25341.6	0.965812	0.482906	+	0.875672i	$0.339581\pi$
$884$	0	0
$885$	−26715.6	−1.01473
$886$	− 13737.1i	− 0.520890i
$887$	−5091.34	−0.192729	−0.0963644	−	0.995346i	$-0.530721\pi$
$887$	−5091.34	−0.192729	−0.0963644	+	0.995346i	$0.530721\pi$
$888$	12297.2	0.464716
$889$	33532.6i	1.26507i
$890$	12194.1i	0.459266i
$891$	4118.50i	0.154854i
$892$	− 21370.2i	− 0.802162i
$893$	7078.86	0.265269
$894$	−11038.3	−0.412949
$895$	78710.8i	2.93968i
$896$	44379.7	1.65471
$897$	0	0
$898$	10015.7	0.372192
$899$	42844.5i	1.58948i
$900$	−12100.6	−0.448170
$901$	−450.076	−0.0166417
$902$	10632.4i	0.392485i
$903$	− 18240.9i	− 0.672223i
$904$	5229.95i	0.192418i
$905$	6174.28i	0.226785i
$906$	2120.76	0.0777677
$907$	551.828	0.0202019	0.0101010	−	0.999949i	$-0.496785\pi$
$907$	551.828	0.0202019	0.0101010	+	0.999949i	$0.496785\pi$
$908$	− 1784.79i	− 0.0652316i
$909$	−3744.51	−0.136631
$910$	0	0
$911$	37661.8	1.36969	0.684847	−	0.728687i	$-0.259869\pi$
$911$	37661.8	1.36969	0.684847	+	0.728687i	$0.259869\pi$
$912$	− 8439.92i	− 0.306440i
$913$	−20222.8	−0.733054
$914$	6542.19	0.236757
$915$	− 28178.1i	− 1.01808i
$916$	5550.22i	0.200201i
$917$	− 16834.8i	− 0.606252i
$918$	84.9092i	0.00305275i
$919$	16265.4	0.583836	0.291918	−	0.956443i	$-0.405707\pi$
$919$	16265.4	0.583836	0.291918	+	0.956443i	$0.405707\pi$
$920$	11738.7	0.420667
$921$	− 5744.04i	− 0.205508i
$922$	18379.2	0.656492
$923$	0	0
$924$	31759.1	1.13073
$925$	51047.3i	1.81451i
$926$	3448.70	0.122388
$927$	−2353.58	−0.0833893
$928$	22442.0i	0.793852i
$929$	− 10133.6i	− 0.357881i	−0.983860	−	0.178940i	$-0.942733\pi$
$929$	− 10133.6i	− 0.357881i	0.983860	−	0.178940i	$-0.0572669\pi$
$930$	17874.4i	0.630242i
$931$	− 41359.2i	− 1.45595i
$932$	11842.8	0.416228
$933$	9515.96	0.333911
$934$	8893.26i	0.311559i
$935$	−2720.83	−0.0951666
$936$	0	0
$937$	−45619.6	−1.59053	−0.795265	−	0.606261i	$-0.792669\pi$
$937$	−45619.6	−1.59053	−0.795265	+	0.606261i	$0.792669\pi$
$938$	24685.3i	0.859280i
$939$	−3721.51	−0.129336
$940$	12009.3	0.416703
$941$	− 26998.8i	− 0.935321i	−0.883908	−	0.467660i	$-0.845097\pi$
$941$	− 26998.8i	− 0.935321i	0.883908	−	0.467660i	$-0.154903\pi$
$942$	4011.76i	0.138758i
$943$	8335.76i	0.287858i
$944$	− 19263.7i	− 0.664173i
$945$	14585.2	0.502070
$946$	10762.9	0.369905
$947$	− 16343.4i	− 0.560812i	−0.959882	−	0.280406i	$-0.909531\pi$
$947$	− 16343.4i	− 0.560812i	0.959882	−	0.280406i	$-0.0904690\pi$
$948$	−14868.0	−0.509379
$949$	0	0
$950$	−14904.1	−0.509004
$951$	28564.2i	0.973984i
$952$	1414.48	0.0481550
$953$	19617.5	0.666812	0.333406	−	0.942783i	$-0.391802\pi$
$953$	19617.5	0.666812	0.333406	+	0.942783i	$0.391802\pi$
$954$	1423.57i	0.0483121i
$955$	− 72925.0i	− 2.47099i
$956$	16239.1i	0.549382i
$957$	20628.0i	0.696770i
$958$	−18496.2	−0.623784
$959$	−41216.2	−1.38784
$960$	− 7250.89i	− 0.243772i
$961$	−70583.9	−2.36930
$962$	0	0
$963$	9374.15	0.313684
$964$	14962.2i	0.499896i
$965$	57620.8	1.92215
$966$	−3991.17	−0.132934
$967$	42185.0i	1.40287i	0.712732	+	0.701436i	$0.247457\pi$
$967$	42185.0i	1.40287i	−0.712732	+	0.701436i	$0.752543\pi$
$968$	19640.4i	0.652135i
$969$	− 652.432i	− 0.0216297i
$970$	5123.00i	0.169577i
$971$	−42711.7	−1.41162	−0.705811	−	0.708400i	$-0.749417\pi$
$971$	−42711.7	−1.41162	−0.705811	+	0.708400i	$0.749417\pi$
$972$	−1675.44	−0.0552877
$973$	− 58492.4i	− 1.92722i
$974$	13390.2	0.440502
$975$	0	0
$976$	20318.3	0.666365
$977$	− 30169.0i	− 0.987915i	−0.869486	−	0.493957i	$-0.835550\pi$
$977$	− 30169.0i	− 0.987915i	0.869486	−	0.493957i	$-0.164450\pi$
$978$	4655.79	0.152225
$979$	−32969.0	−1.07630
$980$	− 70166.1i	− 2.28712i
$981$	− 5486.79i	− 0.178573i
$982$	− 5738.01i	− 0.186464i
$983$	− 23434.7i	− 0.760378i	−0.924909	−	0.380189i	$-0.875859\pi$
$983$	− 23434.7i	− 0.760378i	0.924909	−	0.380189i	$-0.124141\pi$
$984$	−9344.04	−0.302721
$985$	−54896.6	−1.77579
$986$	425.278i	0.0137359i
$987$	−8820.88	−0.284470
$988$	0	0
$989$	8438.01	0.271297
$990$	8605.86i	0.276275i
$991$	−41688.8	−1.33632	−0.668158	−	0.744019i	$-0.732917\pi$
$991$	−41688.8	−1.33632	−0.668158	+	0.744019i	$0.732917\pi$
$992$	−52576.5	−1.68277
$993$	− 5991.37i	− 0.191471i
$994$	32132.2i	1.02532i
$995$	− 71412.4i	− 2.27530i
$996$	− 8226.80i	− 0.261723i
$997$	−46249.7	−1.46915	−0.734574	−	0.678528i	$-0.762618\pi$
$997$	−46249.7	−1.46915	−0.734574	+	0.678528i	$0.762618\pi$
$998$	−19038.5	−0.603861
$999$	7067.97i	0.223844i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.j.337.9		18
13.5	odd	4		507.4.a.q.1.4	yes	9
13.8	odd	4		507.4.a.n.1.6	✓	9
13.12	even	2	inner	507.4.b.j.337.10		18
39.5	even	4		1521.4.a.be.1.6		9
39.8	even	4		1521.4.a.bj.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.6	✓	9	13.8	odd	4
507.4.a.q.1.4	yes	9	13.5	odd	4
507.4.b.j.337.9		18	1.1	even	1	trivial
507.4.b.j.337.10		18	13.12	even	2	inner
1521.4.a.be.1.6		9	39.5	even	4
1521.4.a.bj.1.4		9	39.8	even	4

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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.05129i q^{2} -3.00000 q^{3} +6.89480 q^{4} +17.8886i q^{5} +3.15386i q^{6} +30.1975i q^{7} -15.6587i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.05129i q^{2} -3.00000 q^{3} +6.89480 q^{4} +17.8886i q^{5} +3.15386i q^{6} +30.1975i q^{7} -15.6587i q^{8} +9.00000 q^{9} +18.8061 q^{10} +50.8457i q^{11} -20.6844 q^{12} +31.7462 q^{14} -53.6659i q^{15} +38.6966 q^{16} +2.99137 q^{17} -9.46157i q^{18} +72.7016i q^{19} +123.339i q^{20} -90.5926i q^{21} +53.4533 q^{22} +41.9071 q^{23} +46.9761i q^{24} -195.003 q^{25} -27.0000 q^{27} +208.206i q^{28} -135.233 q^{29} -56.4182 q^{30} -316.820i q^{31} -165.951i q^{32} -152.537i q^{33} -3.14479i q^{34} -540.192 q^{35} +62.0532 q^{36} -261.777i q^{37} +76.4301 q^{38} +280.113 q^{40} +198.911i q^{41} -95.2386 q^{42} +201.351 q^{43} +350.571i q^{44} +160.998i q^{45} -44.0563i q^{46} -97.3687i q^{47} -116.090 q^{48} -568.890 q^{49} +205.004i q^{50} -8.97412 q^{51} -150.458 q^{53} +28.3847i q^{54} -909.560 q^{55} +472.853 q^{56} -218.105i q^{57} +142.168i q^{58} -497.812i q^{59} -370.016i q^{60} +525.066 q^{61} -333.068 q^{62} +271.778i q^{63} +135.112 q^{64} -160.360 q^{66} +777.584i q^{67} +20.6249 q^{68} -125.721 q^{69} +567.896i q^{70} +1012.16i q^{71} -140.928i q^{72} -612.910i q^{73} -275.202 q^{74} +585.010 q^{75} +501.263i q^{76} -1535.41 q^{77} +718.804 q^{79} +692.230i q^{80} +81.0000 q^{81} +209.112 q^{82} +397.730i q^{83} -624.617i q^{84} +53.5116i q^{85} -211.677i q^{86} +405.699 q^{87} +796.176 q^{88} +648.413i q^{89} +169.255 q^{90} +288.941 q^{92} +950.460i q^{93} -102.362 q^{94} -1300.53 q^{95} +497.852i q^{96} +272.412i q^{97} +598.066i q^{98} +457.611i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) \) 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9	\( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100}) \) 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9 - 396 * q^10 + 192 * q^12 + 196 * q^14 + 64 * q^16 + 268 * q^17 + 548 * q^22 - 452 * q^23 - 1224 * q^25 - 486 * q^27 - 1094 * q^29 + 1188 * q^30 + 276 * q^35 - 576 * q^36 + 832 * q^38 + 2684 * q^40 - 588 * q^42 - 316 * q^43 - 192 * q^48 - 1284 * q^49 - 804 * q^51 + 2798 * q^53 - 2816 * q^55 + 1232 * q^56 + 4184 * q^61 + 586 * q^62 - 4962 * q^64 - 1644 * q^66 - 3158 * q^68 + 1356 * q^69 + 2074 * q^74 + 3672 * q^75 + 3372 * q^77 - 230 * q^79 + 1458 * q^81 + 10294 * q^82 + 3282 * q^87 + 968 * q^88 - 3564 * q^90 + 4174 * q^92 - 936 * q^94 + 444 * q^95

Introduction

L-functions

Modular forms

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