LMFDB - Embedded newform 507.4.b.g.337.6

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:	$6$
Coefficient field:	6.0.158155776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 $ x^6 - 2x^5 + 2x^4 + 24x^3 + 81x^2 + 54*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.6
Root			$-0.376763 + 0.376763i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.g.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.20905i q^{2} +3.00000 q^{3} -9.71610 q^{4} -11.4322i q^{5} +12.6271i q^{6} +11.2543i q^{7} -7.22315i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.20905i q^{2} +3.00000 q^{3} -9.71610 q^{4} -11.4322i q^{5} +12.6271i q^{6} +11.2543i q^{7} -7.22315i q^{8} +9.00000 q^{9} +48.1187 q^{10} -25.8785i q^{11} -29.1483 q^{12} -47.3699 q^{14} -34.2966i q^{15} -47.3262 q^{16} +20.3276 q^{17} +37.8814i q^{18} +154.712i q^{19} +111.076i q^{20} +33.7629i q^{21} +108.924 q^{22} +180.418 q^{23} -21.6695i q^{24} -5.69520 q^{25} +27.0000 q^{27} -109.348i q^{28} -20.4522 q^{29} +144.356 q^{30} +266.424i q^{31} -256.984i q^{32} -77.6355i q^{33} +85.5599i q^{34} +128.661 q^{35} -87.4449 q^{36} -115.984i q^{37} -651.190 q^{38} -82.5765 q^{40} +391.184i q^{41} -142.110 q^{42} -151.407 q^{43} +251.438i q^{44} -102.890i q^{45} +759.390i q^{46} +467.365i q^{47} -141.979 q^{48} +216.341 q^{49} -23.9714i q^{50} +60.9828 q^{51} +79.9842 q^{53} +113.644i q^{54} -295.848 q^{55} +81.2915 q^{56} +464.136i q^{57} -86.0843i q^{58} +873.710i q^{59} +333.229i q^{60} -187.068 q^{61} -1121.39 q^{62} +101.289i q^{63} +703.047 q^{64} +326.772 q^{66} -609.204i q^{67} -197.505 q^{68} +541.255 q^{69} +541.542i q^{70} +248.038i q^{71} -65.0084i q^{72} -852.765i q^{73} +488.181 q^{74} -17.0856 q^{75} -1503.20i q^{76} +291.244 q^{77} -331.221 q^{79} +541.043i q^{80} +81.0000 q^{81} -1646.51 q^{82} -435.432i q^{83} -328.044i q^{84} -232.389i q^{85} -637.281i q^{86} -61.3566 q^{87} -186.924 q^{88} -259.233i q^{89} +433.068 q^{90} -1752.96 q^{92} +799.273i q^{93} -1967.16 q^{94} +1768.70 q^{95} -770.951i q^{96} +1225.17i q^{97} +910.589i q^{98} -232.907i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9}+O(q^{10}) $ 6 * q + 18 * q^3 - 20 * q^4 + 54 * q^9	$ 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9} + 8 q^{10} - 60 q^{12} - 352 q^{14} - 220 q^{16} + 292 q^{17} - 112 q^{22} + 96 q^{23} - 290 q^{25} + 162 q^{27} - 4 q^{29} + 24 q^{30} + 160 q^{35} - 180 q^{36} - 624 q^{38} - 1032 q^{40} - 1056 q^{42} + 520 q^{43} - 660 q^{48} - 894 q^{49} + 876 q^{51} - 1356 q^{53} - 3104 q^{55} - 192 q^{56} + 460 q^{61} - 3904 q^{62} + 1500 q^{64} - 336 q^{66} - 920 q^{68} + 288 q^{69} + 3448 q^{74} - 870 q^{75} - 224 q^{77} - 48 q^{79} + 486 q^{81} - 1128 q^{82} - 12 q^{87} - 3216 q^{88} + 72 q^{90} - 2592 q^{92} - 3840 q^{94} + 8064 q^{95}+O(q^{100}) $ 6 * q + 18 * q^3 - 20 * q^4 + 54 * q^9 + 8 * q^10 - 60 * q^12 - 352 * q^14 - 220 * q^16 + 292 * q^17 - 112 * q^22 + 96 * q^23 - 290 * q^25 + 162 * q^27 - 4 * q^29 + 24 * q^30 + 160 * q^35 - 180 * q^36 - 624 * q^38 - 1032 * q^40 - 1056 * q^42 + 520 * q^43 - 660 * q^48 - 894 * q^49 + 876 * q^51 - 1356 * q^53 - 3104 * q^55 - 192 * q^56 + 460 * q^61 - 3904 * q^62 + 1500 * q^64 - 336 * q^66 - 920 * q^68 + 288 * q^69 + 3448 * q^74 - 870 * q^75 - 224 * q^77 - 48 * q^79 + 486 * q^81 - 1128 * q^82 - 12 * q^87 - 3216 * q^88 + 72 * q^90 - 2592 * q^92 - 3840 * q^94 + 8064 * 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 $

$2$	4.20905i	1.48812i	0.668111	+	0.744062i	$0.267103\pi$
$2$	4.20905i	1.48812i	−0.668111	+	0.744062i	$0.732897\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−9.71610	−1.21451
$5$	− 11.4322i	− 1.02253i	−0.859424	−	0.511264i	$-0.829178\pi$
$5$	− 11.4322i	− 1.02253i	0.859424	−	0.511264i	$-0.170822\pi$
$6$	12.6271i	0.859169i
$7$	11.2543i	0.607675i	0.952724	+	0.303838i	$0.0982680\pi$
$7$	11.2543i	0.607675i	−0.952724	+	0.303838i	$0.901732\pi$
$8$	− 7.22315i	− 0.319221i
$9$	9.00000	0.333333
$10$	48.1187	1.52165
$11$	− 25.8785i	− 0.709333i	−0.934993	−	0.354666i	$-0.884594\pi$
$11$	− 25.8785i	− 0.709333i	0.934993	−	0.354666i	$-0.115406\pi$
$12$	−29.1483	−0.701199
$13$	0	0
$13$	0	0
$14$	−47.3699	−0.904296
$15$	− 34.2966i	− 0.590356i
$16$	−47.3262	−0.739472
$17$	20.3276	0.290010	0.145005	−	0.989431i	$-0.453680\pi$
$17$	20.3276	0.290010	0.145005	+	0.989431i	$0.453680\pi$
$18$	37.8814i	0.496041i
$19$	154.712i	1.86807i	0.357181	+	0.934035i	$0.383738\pi$
$19$	154.712i	1.86807i	−0.357181	+	0.934035i	$0.616262\pi$
$20$	111.076i	1.24187i
$21$	33.7629i	0.350841i
$22$	108.924	1.05558
$23$	180.418	1.63565	0.817823	−	0.575471i	$-0.195181\pi$
$23$	180.418	1.63565	0.817823	+	0.575471i	$0.195181\pi$
$24$	− 21.6695i	− 0.184302i
$25$	−5.69520	−0.0455616
$26$	0	0
$27$	27.0000	0.192450
$28$	− 109.348i	− 0.738029i
$29$	−20.4522	−0.130961	−0.0654806	−	0.997854i	$-0.520858\pi$
$29$	−20.4522	−0.130961	−0.0654806	+	0.997854i	$0.520858\pi$
$30$	144.356	0.878523
$31$	266.424i	1.54359i	0.635873	+	0.771794i	$0.280640\pi$
$31$	266.424i	1.54359i	−0.635873	+	0.771794i	$0.719360\pi$
$32$	− 256.984i	− 1.41965i
$33$	− 77.6355i	− 0.409534i
$34$	85.5599i	0.431571i
$35$	128.661	0.621364
$36$	−87.4449	−0.404837
$37$	− 115.984i	− 0.515340i	−0.966233	−	0.257670i	$-0.917045\pi$
$37$	− 115.984i	− 0.515340i	0.966233	−	0.257670i	$-0.0829548\pi$
$38$	−651.190	−2.77992
$39$	0	0
$40$	−82.5765	−0.326412
$41$	391.184i	1.49006i	0.667029	+	0.745032i	$0.267566\pi$
$41$	391.184i	1.49006i	−0.667029	+	0.745032i	$0.732434\pi$
$42$	−142.110	−0.522095
$43$	−151.407	−0.536963	−0.268482	−	0.963285i	$-0.586522\pi$
$43$	−151.407	−0.536963	−0.268482	+	0.963285i	$0.586522\pi$
$44$	251.438i	0.861494i
$45$	− 102.890i	− 0.340842i
$46$	759.390i	2.43404i
$47$	467.365i	1.45047i	0.688500	+	0.725236i	$0.258269\pi$
$47$	467.365i	1.45047i	−0.688500	+	0.725236i	$0.741731\pi$
$48$	−141.979	−0.426934
$49$	216.341	0.630731
$50$	− 23.9714i	− 0.0678012i
$51$	60.9828	0.167437
$52$	0	0
$53$	79.9842	0.207296	0.103648	−	0.994614i	$-0.466949\pi$
$53$	79.9842	0.207296	0.103648	+	0.994614i	$0.466949\pi$
$54$	113.644i	0.286390i
$55$	−295.848	−0.725312
$56$	81.2915	0.193983
$57$	464.136i	1.07853i
$58$	− 86.0843i	− 0.194887i
$59$	873.710i	1.92792i	0.266045	+	0.963960i	$0.414283\pi$
$59$	873.710i	1.92792i	−0.266045	+	0.963960i	$0.585717\pi$
$60$	333.229i	0.716995i
$61$	−187.068	−0.392649	−0.196325	−	0.980539i	$-0.562901\pi$
$61$	−187.068	−0.392649	−0.196325	+	0.980539i	$0.562901\pi$
$62$	−1121.39	−2.29705
$63$	101.289i	0.202558i
$64$	703.047	1.37314
$65$	0	0
$66$	326.772	0.609437
$67$	− 609.204i	− 1.11084i	−0.831571	−	0.555418i	$-0.812558\pi$
$67$	− 609.204i	− 1.11084i	0.831571	−	0.555418i	$-0.187442\pi$
$68$	−197.505	−0.352221
$69$	541.255	0.944340
$70$	541.542i	0.924667i
$71$	248.038i	0.414601i	0.978277	+	0.207301i	$0.0664679\pi$
$71$	248.038i	0.414601i	−0.978277	+	0.207301i	$0.933532\pi$
$72$	− 65.0084i	− 0.106407i
$73$	− 852.765i	− 1.36724i	−0.729838	−	0.683621i	$-0.760404\pi$
$73$	− 852.765i	− 1.36724i	0.729838	−	0.683621i	$-0.239596\pi$
$74$	488.181	0.766890
$75$	−17.0856	−0.0263050
$76$	− 1503.20i	− 2.26880i
$77$	291.244	0.431044
$78$	0	0
$79$	−331.221	−0.471712	−0.235856	−	0.971788i	$-0.575789\pi$
$79$	−331.221	−0.471712	−0.235856	+	0.971788i	$0.575789\pi$
$80$	541.043i	0.756130i
$81$	81.0000	0.111111
$82$	−1646.51	−2.21740
$83$	− 435.432i	− 0.575842i	−0.957654	−	0.287921i	$-0.907036\pi$
$83$	− 435.432i	− 0.575842i	0.957654	−	0.287921i	$-0.0929641\pi$
$84$	− 328.044i	− 0.426101i
$85$	− 232.389i	− 0.296543i
$86$	− 637.281i	− 0.799067i
$87$	−61.3566	−0.0756105
$88$	−186.924	−0.226434
$89$	− 259.233i	− 0.308749i	−0.988012	−	0.154375i	$-0.950664\pi$
$89$	− 259.233i	− 0.308749i	0.988012	−	0.154375i	$-0.0493362\pi$
$90$	433.068	0.507216
$91$	0	0
$92$	−1752.96	−1.98651
$93$	799.273i	0.891191i
$94$	−1967.16	−2.15848
$95$	1768.70	1.91015
$96$	− 770.951i	− 0.819634i
$97$	1225.17i	1.28245i	0.767355	+	0.641223i	$0.221572\pi$
$97$	1225.17i	1.28245i	−0.767355	+	0.641223i	$0.778428\pi$
$98$	910.589i	0.938606i
$99$	− 232.907i	− 0.236444i
$100$	55.3351	0.0553351
$101$	−645.416	−0.635855	−0.317927	−	0.948115i	$-0.602987\pi$
$101$	−645.416	−0.635855	−0.317927	+	0.948115i	$0.602987\pi$
$102$	256.680i	0.249167i
$103$	511.137	0.488969	0.244484	−	0.969653i	$-0.421381\pi$
$103$	511.137	0.488969	0.244484	+	0.969653i	$0.421381\pi$
$104$	0	0
$105$	385.984	0.358745
$106$	336.657i	0.308482i
$107$	608.195	0.549499	0.274750	−	0.961516i	$-0.411405\pi$
$107$	608.195	0.549499	0.274750	+	0.961516i	$0.411405\pi$
$108$	−262.335	−0.233733
$109$	− 1300.04i	− 1.14239i	−0.820813	−	0.571197i	$-0.806479\pi$
$109$	− 1300.04i	− 1.14239i	0.820813	−	0.571197i	$-0.193521\pi$
$110$	− 1245.24i	− 1.07935i
$111$	− 347.951i	− 0.297532i
$112$	− 532.623i	− 0.449359i
$113$	42.1953	0.0351274	0.0175637	−	0.999846i	$-0.494409\pi$
$113$	42.1953	0.0351274	0.0175637	+	0.999846i	$0.494409\pi$
$114$	−1953.57	−1.60499
$115$	− 2062.58i	− 1.67249i
$116$	198.716	0.159054
$117$	0	0
$118$	−3677.49	−2.86899
$119$	228.773i	0.176232i
$120$	−247.729	−0.188454
$121$	661.303	0.496847
$122$	− 787.378i	− 0.584311i
$123$	1173.55i	0.860289i
$124$	− 2588.61i	− 1.87471i
$125$	− 1363.92i	− 0.975939i
$126$	−426.329	−0.301432
$127$	311.018	0.217310	0.108655	−	0.994080i	$-0.465346\pi$
$127$	311.018	0.217310	0.108655	+	0.994080i	$0.465346\pi$
$128$	903.291i	0.623753i
$129$	−454.222	−0.310016
$130$	0	0
$131$	2000.98	1.33456	0.667278	−	0.744809i	$-0.267459\pi$
$131$	2000.98	1.33456	0.667278	+	0.744809i	$0.267459\pi$
$132$	754.314i	0.497384i
$133$	−1741.17	−1.13518
$134$	2564.17	1.65306
$135$	− 308.669i	− 0.196785i
$136$	− 146.829i	− 0.0925773i
$137$	− 1038.53i	− 0.647644i	−0.946118	−	0.323822i	$-0.895032\pi$
$137$	− 1038.53i	− 0.647644i	0.946118	−	0.323822i	$-0.104968\pi$
$138$	2278.17i	1.40529i
$139$	−2858.46	−1.74426	−0.872128	−	0.489277i	$-0.837261\pi$
$139$	−2858.46	−1.74426	−0.872128	+	0.489277i	$0.837261\pi$
$140$	−1250.09	−0.754655
$141$	1402.09i	0.837430i
$142$	−1044.00	−0.616978
$143$	0	0
$144$	−425.936	−0.246491
$145$	233.814i	0.133911i
$146$	3589.33	2.03462
$147$	649.022	0.364153
$148$	1126.91i	0.625887i
$149$	743.479i	0.408780i	0.978890	+	0.204390i	$0.0655211\pi$
$149$	743.479i	0.408780i	−0.978890	+	0.204390i	$0.934479\pi$
$150$	− 71.9141i	− 0.0391451i
$151$	− 2277.24i	− 1.22728i	−0.789586	−	0.613640i	$-0.789705\pi$
$151$	− 2277.24i	− 1.22728i	0.789586	−	0.613640i	$-0.210295\pi$
$152$	1117.51	0.596328
$153$	182.948	0.0966700
$154$	1225.86i	0.641447i
$155$	3045.82	1.57836
$156$	0	0
$157$	3173.51	1.61321	0.806605	−	0.591091i	$-0.201303\pi$
$157$	3173.51	1.61321	0.806605	+	0.591091i	$0.201303\pi$
$158$	− 1394.12i	− 0.701966i
$159$	239.953	0.119682
$160$	−2937.89	−1.45163
$161$	2030.48i	0.993941i
$162$	340.933i	0.165347i
$163$	2314.65i	1.11225i	0.831098	+	0.556126i	$0.187713\pi$
$163$	2314.65i	1.11225i	−0.831098	+	0.556126i	$0.812287\pi$
$164$	− 3800.78i	− 1.80970i
$165$	−887.545	−0.418759
$166$	1832.76	0.856925
$167$	2665.65i	1.23517i	0.786502	+	0.617587i	$0.211890\pi$
$167$	2665.65i	1.23517i	−0.786502	+	0.617587i	$0.788110\pi$
$168$	243.874	0.111996
$169$	0	0
$170$	978.138	0.441293
$171$	1392.41i	0.622690i
$172$	1471.09	0.652148
$173$	165.243	0.0726198	0.0363099	−	0.999341i	$-0.488440\pi$
$173$	165.243	0.0726198	0.0363099	+	0.999341i	$0.488440\pi$
$174$	− 258.253i	− 0.112518i
$175$	− 64.0954i	− 0.0276866i
$176$	1224.73i	0.524532i
$177$	2621.13i	1.11309i
$178$	1091.13	0.459457
$179$	−712.339	−0.297446	−0.148723	−	0.988879i	$-0.547516\pi$
$179$	−712.339	−0.297446	−0.148723	+	0.988879i	$0.547516\pi$
$180$	999.688i	0.413957i
$181$	−2206.53	−0.906133	−0.453066	−	0.891477i	$-0.649670\pi$
$181$	−2206.53	−0.906133	−0.453066	+	0.891477i	$0.649670\pi$
$182$	0	0
$183$	−561.204	−0.226696
$184$	− 1303.19i	− 0.522132i
$185$	−1325.95	−0.526949
$186$	−3364.18	−1.32620
$187$	− 526.048i	− 0.205714i
$188$	− 4540.96i	− 1.76162i
$189$	303.866i	0.116947i
$190$	7444.54i	2.84254i
$191$	1470.64	0.557129	0.278565	−	0.960417i	$-0.410141\pi$
$191$	1470.64	0.557129	0.278565	+	0.960417i	$0.410141\pi$
$192$	2109.14	0.792782
$193$	− 369.560i	− 0.137832i	−0.997622	−	0.0689158i	$-0.978046\pi$
$193$	− 369.560i	− 0.137832i	0.997622	−	0.0689158i	$-0.0219540\pi$
$194$	−5156.80	−1.90844
$195$	0	0
$196$	−2101.99	−0.766031
$197$	− 4273.41i	− 1.54552i	−0.634697	−	0.772761i	$-0.718875\pi$
$197$	− 4273.41i	− 1.54552i	0.634697	−	0.772761i	$-0.281125\pi$
$198$	980.315	0.351858
$199$	−4154.31	−1.47985	−0.739927	−	0.672687i	$-0.765140\pi$
$199$	−4154.31	−1.47985	−0.739927	+	0.672687i	$0.765140\pi$
$200$	41.1373i	0.0145442i
$201$	− 1827.61i	− 0.641342i
$202$	− 2716.59i	− 0.946230i
$203$	− 230.175i	− 0.0795819i
$204$	−592.515	−0.203355
$205$	4472.09	1.52363
$206$	2151.40i	0.727646i
$207$	1623.77	0.545215
$208$	0	0
$209$	4003.71	1.32508
$210$	1624.63i	0.533857i
$211$	1231.59	0.401830	0.200915	−	0.979609i	$-0.435608\pi$
$211$	1231.59	0.401830	0.200915	+	0.979609i	$0.435608\pi$
$212$	−777.134	−0.251763
$213$	744.114i	0.239370i
$214$	2559.92i	0.817723i
$215$	1730.92i	0.549059i
$216$	− 195.025i	− 0.0614341i
$217$	−2998.42	−0.938000
$218$	5471.92	1.70002
$219$	− 2558.30i	− 0.789377i
$220$	2874.49	0.880901
$221$	0	0
$222$	1464.54	0.442764
$223$	− 2187.24i	− 0.656809i	−0.944537	−	0.328404i	$-0.893489\pi$
$223$	− 2187.24i	− 0.656809i	0.944537	−	0.328404i	$-0.106511\pi$
$224$	2892.17	0.862684
$225$	−51.2568	−0.0151872
$226$	177.602i	0.0522739i
$227$	4138.67i	1.21010i	0.796187	+	0.605051i	$0.206847\pi$
$227$	4138.67i	1.21010i	−0.796187	+	0.605051i	$0.793153\pi$
$228$	− 4509.59i	− 1.30989i
$229$	835.354i	0.241056i	0.992710	+	0.120528i	$0.0384587\pi$
$229$	835.354i	0.241056i	−0.992710	+	0.120528i	$0.961541\pi$
$230$	8681.50	2.48887
$231$	873.733	0.248863
$232$	147.729i	0.0418056i
$233$	−3685.51	−1.03625	−0.518124	−	0.855305i	$-0.673370\pi$
$233$	−3685.51	−1.03625	−0.518124	+	0.855305i	$0.673370\pi$
$234$	0	0
$235$	5343.01	1.48315
$236$	− 8489.05i	− 2.34148i
$237$	−993.662	−0.272343
$238$	−962.917	−0.262255
$239$	3026.21i	0.819034i	0.912303	+	0.409517i	$0.134303\pi$
$239$	3026.21i	0.819034i	−0.912303	+	0.409517i	$0.865697\pi$
$240$	1623.13i	0.436552i
$241$	− 3265.58i	− 0.872839i	−0.899743	−	0.436420i	$-0.856246\pi$
$241$	− 3265.58i	− 0.872839i	0.899743	−	0.436420i	$-0.143754\pi$
$242$	2783.46i	0.739370i
$243$	243.000	0.0641500
$244$	1817.57	0.476877
$245$	− 2473.25i	− 0.644940i
$246$	−4939.53	−1.28022
$247$	0	0
$248$	1924.42	0.492746
$249$	− 1306.30i	− 0.332463i
$250$	5740.79	1.45232
$251$	6363.16	1.60016	0.800078	−	0.599897i	$-0.204792\pi$
$251$	6363.16	1.60016	0.800078	+	0.599897i	$0.204792\pi$
$252$	− 984.131i	− 0.246010i
$253$	− 4668.96i	− 1.16022i
$254$	1309.09i	0.323385i
$255$	− 697.168i	− 0.171209i
$256$	1822.38	0.444917
$257$	6085.36	1.47702	0.738511	−	0.674242i	$-0.235529\pi$
$257$	6085.36	1.47702	0.738511	+	0.674242i	$0.235529\pi$
$258$	− 1911.84i	− 0.461342i
$259$	1305.31	0.313159
$260$	0	0
$261$	−184.070	−0.0436538
$262$	8422.24i	1.98598i
$263$	123.227	0.0288916	0.0144458	−	0.999896i	$-0.495402\pi$
$263$	123.227	0.0288916	0.0144458	+	0.999896i	$0.495402\pi$
$264$	−560.773	−0.130732
$265$	− 914.395i	− 0.211965i
$266$	− 7328.69i	− 1.68929i
$267$	− 777.700i	− 0.178256i
$268$	5919.08i	1.34913i
$269$	−1935.79	−0.438763	−0.219381	−	0.975639i	$-0.570404\pi$
$269$	−1935.79	−0.438763	−0.219381	+	0.975639i	$0.570404\pi$
$270$	1299.20	0.292841
$271$	4612.69i	1.03395i	0.856000	+	0.516976i	$0.172942\pi$
$271$	4612.69i	1.03395i	−0.856000	+	0.516976i	$0.827058\pi$
$272$	−962.028	−0.214454
$273$	0	0
$274$	4371.20	0.963774
$275$	147.383i	0.0323183i
$276$	−5258.89	−1.14691
$277$	5834.30	1.26552	0.632761	−	0.774347i	$-0.281922\pi$
$277$	5834.30	1.26552	0.632761	+	0.774347i	$0.281922\pi$
$278$	− 12031.4i	− 2.59567i
$279$	2397.82i	0.514529i
$280$	− 929.341i	− 0.198353i
$281$	− 4691.91i	− 0.996071i	−0.867157	−	0.498036i	$-0.834055\pi$
$281$	− 4691.91i	− 0.996071i	0.867157	−	0.498036i	$-0.165945\pi$
$282$	−5901.49	−1.24620
$283$	−3465.60	−0.727945	−0.363973	−	0.931410i	$-0.618580\pi$
$283$	−3465.60	−0.727945	−0.363973	+	0.931410i	$0.618580\pi$
$284$	− 2409.96i	− 0.503539i
$285$	5306.09	1.10283
$286$	0	0
$287$	−4402.50	−0.905475
$288$	− 2312.85i	− 0.473216i
$289$	−4499.79	−0.915894
$290$	−984.133	−0.199277
$291$	3675.51i	0.740420i
$292$	8285.55i	1.66053i
$293$	− 2677.31i	− 0.533822i	−0.963721	−	0.266911i	$-0.913997\pi$
$293$	− 2677.31i	− 0.533822i	0.963721	−	0.266911i	$-0.0860030\pi$
$294$	2731.77i	0.541904i
$295$	9988.43	1.97135
$296$	−837.767	−0.164507
$297$	− 698.720i	− 0.136511i
$298$	−3129.34	−0.608315
$299$	0	0
$300$	166.005	0.0319477
$301$	− 1703.98i	− 0.326299i
$302$	9585.02	1.82634
$303$	−1936.25	−0.367111
$304$	− 7321.93i	− 1.38139i
$305$	2138.60i	0.401494i
$306$	770.039i	0.143857i
$307$	− 471.915i	− 0.0877316i	−0.999037	−	0.0438658i	$-0.986033\pi$
$307$	− 471.915i	− 0.0877316i	0.999037	−	0.0438658i	$-0.0139674\pi$
$308$	−2829.76	−0.523508
$309$	1533.41	0.282306
$310$	12820.0i	2.34880i
$311$	1518.52	0.276872	0.138436	−	0.990371i	$-0.455793\pi$
$311$	1518.52	0.276872	0.138436	+	0.990371i	$0.455793\pi$
$312$	0	0
$313$	4049.86	0.731348	0.365674	−	0.930743i	$-0.380839\pi$
$313$	4049.86	0.731348	0.365674	+	0.930743i	$0.380839\pi$
$314$	13357.5i	2.40066i
$315$	1157.95	0.207121
$316$	3218.17	0.572900
$317$	3253.96i	0.576532i	0.957550	+	0.288266i	$0.0930787\pi$
$317$	3253.96i	0.576532i	−0.957550	+	0.288266i	$0.906921\pi$
$318$	1009.97i	0.178102i
$319$	529.272i	0.0928951i
$320$	− 8037.37i	− 1.40407i
$321$	1824.59	0.317254
$322$	−8546.40	−1.47911
$323$	3144.92i	0.541759i
$324$	−787.004	−0.134946
$325$	0	0
$326$	−9742.46	−1.65517
$327$	− 3900.11i	− 0.659561i
$328$	2825.58	0.475660
$329$	−5259.86	−0.881415
$330$	− 3735.72i	− 0.623165i
$331$	− 3422.45i	− 0.568322i	−0.958777	−	0.284161i	$-0.908285\pi$
$331$	− 3422.45i	− 0.568322i	0.958777	−	0.284161i	$-0.0917150\pi$
$332$	4230.71i	0.699368i
$333$	− 1043.85i	− 0.171780i
$334$	−11219.8	−1.83809
$335$	−6964.54	−1.13586
$336$	− 1597.87i	− 0.259437i
$337$	9301.67	1.50354	0.751772	−	0.659423i	$-0.229199\pi$
$337$	9301.67	1.50354	0.751772	+	0.659423i	$0.229199\pi$
$338$	0	0
$339$	126.586	0.0202808
$340$	2257.92i	0.360155i
$341$	6894.66	1.09492
$342$	−5860.71	−0.926640
$343$	6294.99i	0.990955i
$344$	1093.64i	0.171410i
$345$	− 6187.74i	− 0.965613i
$346$	695.518i	0.108067i
$347$	216.898	0.0335554	0.0167777	−	0.999859i	$-0.494659\pi$
$347$	216.898	0.0335554	0.0167777	+	0.999859i	$0.494659\pi$
$348$	596.147	0.0918299
$349$	4809.84i	0.737721i	0.929485	+	0.368861i	$0.120252\pi$
$349$	4809.84i	0.737721i	−0.929485	+	0.368861i	$0.879748\pi$
$350$	269.781	0.0412011
$351$	0	0
$352$	−6650.35	−1.00700
$353$	− 2859.64i	− 0.431170i	−0.976485	−	0.215585i	$-0.930834\pi$
$353$	− 2859.64i	− 0.431170i	0.976485	−	0.215585i	$-0.0691659\pi$
$354$	−11032.5	−1.65641
$355$	2835.62	0.423941
$356$	2518.74i	0.374980i
$357$	686.319i	0.101747i
$358$	− 2998.27i	− 0.442636i
$359$	− 3686.04i	− 0.541899i	−0.962594	−	0.270949i	$-0.912662\pi$
$359$	− 3686.04i	− 0.541899i	0.962594	−	0.270949i	$-0.0873376\pi$
$360$	−743.188	−0.108804
$361$	−17076.8	−2.48969
$362$	− 9287.39i	− 1.34844i
$363$	1983.91	0.286855
$364$	0	0
$365$	−9748.98	−1.39804
$366$	− 2362.14i	− 0.337352i
$367$	−3470.59	−0.493633	−0.246816	−	0.969062i	$-0.579384\pi$
$367$	−3470.59	−0.493633	−0.246816	+	0.969062i	$0.579384\pi$
$368$	−8538.52	−1.20951
$369$	3520.65i	0.496688i
$370$	− 5580.98i	− 0.784166i
$371$	900.166i	0.125968i
$372$	− 7765.82i	− 1.08236i
$373$	−11963.4	−1.66070	−0.830352	−	0.557240i	$-0.811860\pi$
$373$	−11963.4	−1.66070	−0.830352	+	0.557240i	$0.811860\pi$
$374$	2214.16	0.306127
$375$	− 4091.75i	− 0.563459i
$376$	3375.85	0.463021
$377$	0	0
$378$	−1278.99	−0.174032
$379$	345.604i	0.0468403i	0.999726	+	0.0234202i	$0.00745555\pi$
$379$	345.604i	0.0468403i	−0.999726	+	0.0234202i	$0.992544\pi$
$380$	−17184.8	−2.31990
$381$	933.055	0.125464
$382$	6189.99i	0.829078i
$383$	− 3386.40i	− 0.451793i	−0.974151	−	0.225897i	$-0.927469\pi$
$383$	− 3386.40i	− 0.451793i	0.974151	−	0.225897i	$-0.0725312\pi$
$384$	2709.87i	0.360124i
$385$	− 3329.56i	− 0.440754i
$386$	1555.49	0.205110
$387$	−1362.67	−0.178988
$388$	− 11903.9i	− 1.55755i
$389$	1629.88	0.212438	0.106219	−	0.994343i	$-0.466126\pi$
$389$	1629.88	0.212438	0.106219	+	0.994343i	$0.466126\pi$
$390$	0	0
$391$	3667.47	0.474353
$392$	− 1562.66i	− 0.201343i
$393$	6002.95	0.770506
$394$	17987.0	2.29993
$395$	3786.58i	0.482338i
$396$	2262.94i	0.287165i
$397$	− 7938.94i	− 1.00364i	−0.864973	−	0.501819i	$-0.832664\pi$
$397$	− 7938.94i	− 1.00364i	0.864973	−	0.501819i	$-0.167336\pi$
$398$	− 17485.7i	− 2.20221i
$399$	−5223.52	−0.655396
$400$	269.532	0.0336915
$401$	− 214.402i	− 0.0267001i	−0.999911	−	0.0133500i	$-0.995750\pi$
$401$	− 214.402i	− 0.0267001i	0.999911	−	0.0133500i	$-0.00424958\pi$
$402$	7692.51	0.954396
$403$	0	0
$404$	6270.93	0.772253
$405$	− 926.008i	− 0.113614i
$406$	968.819	0.118428
$407$	−3001.48	−0.365548
$408$	− 440.488i	− 0.0534495i
$409$	− 4783.73i	− 0.578338i	−0.957278	−	0.289169i	$-0.906621\pi$
$409$	− 4783.73i	− 0.578338i	0.957278	−	0.289169i	$-0.0933789\pi$
$410$	18823.2i	2.26735i
$411$	− 3115.58i	− 0.373917i
$412$	−4966.25	−0.593859
$413$	−9832.99	−1.17155
$414$	6834.51i	0.811347i
$415$	−4977.95	−0.588815
$416$	0	0
$417$	−8575.39	−1.00705
$418$	16851.8i	1.97189i
$419$	−9903.67	−1.15472	−0.577358	−	0.816491i	$-0.695916\pi$
$419$	−9903.67	−1.15472	−0.577358	+	0.816491i	$0.695916\pi$
$420$	−3750.26	−0.435700
$421$	− 12120.6i	− 1.40314i	−0.712598	−	0.701572i	$-0.752482\pi$
$421$	− 12120.6i	− 1.40314i	0.712598	−	0.701572i	$-0.247518\pi$
$422$	5183.82i	0.597973i
$423$	4206.28i	0.483491i
$424$	− 577.738i	− 0.0661732i
$425$	−115.770	−0.0132133
$426$	−3132.01	−0.356213
$427$	− 2105.32i	− 0.238603i
$428$	−5909.28	−0.667374
$429$	0	0
$430$	−7285.53	−0.817068
$431$	− 13672.6i	− 1.52805i	−0.645189	−	0.764023i	$-0.723221\pi$
$431$	− 13672.6i	− 1.52805i	0.645189	−	0.764023i	$-0.276779\pi$
$432$	−1277.81	−0.142311
$433$	−7113.10	−0.789455	−0.394727	−	0.918798i	$-0.629161\pi$
$433$	−7113.10	−0.789455	−0.394727	+	0.918798i	$0.629161\pi$
$434$	− 12620.5i	− 1.39586i
$435$	701.441i	0.0773138i
$436$	12631.3i	1.38745i
$437$	27912.9i	3.05550i
$438$	10768.0	1.17469
$439$	6022.04	0.654707	0.327353	−	0.944902i	$-0.393843\pi$
$439$	6022.04	0.654707	0.327353	+	0.944902i	$0.393843\pi$
$440$	2136.96i	0.231535i
$441$	1947.07	0.210244
$442$	0	0
$443$	−12994.4	−1.39364	−0.696821	−	0.717245i	$-0.745403\pi$
$443$	−12994.4	−1.39364	−0.696821	+	0.717245i	$0.745403\pi$
$444$	3380.72i	0.361356i
$445$	−2963.61	−0.315704
$446$	9206.20	0.977413
$447$	2230.44i	0.236009i
$448$	7912.30i	0.834422i
$449$	− 10984.3i	− 1.15452i	−0.816560	−	0.577260i	$-0.804122\pi$
$449$	− 10984.3i	− 1.15452i	0.816560	−	0.577260i	$-0.195878\pi$
$450$	− 215.742i	− 0.0226004i
$451$	10123.2	1.05695
$452$	−409.973	−0.0426627
$453$	− 6831.72i	− 0.708570i
$454$	−17419.9	−1.80078
$455$	0	0
$456$	3352.52	0.344290
$457$	9834.10i	1.00661i	0.864109	+	0.503304i	$0.167882\pi$
$457$	9834.10i	1.00661i	−0.864109	+	0.503304i	$0.832118\pi$
$458$	−3516.05	−0.358721
$459$	548.845	0.0558124
$460$	20040.2i	2.03126i
$461$	3401.42i	0.343644i	0.985128	+	0.171822i	$0.0549653\pi$
$461$	3401.42i	0.343644i	−0.985128	+	0.171822i	$0.945035\pi$
$462$	3677.59i	0.370339i
$463$	− 1739.42i	− 0.174596i	−0.996182	−	0.0872979i	$-0.972177\pi$
$463$	− 1739.42i	− 0.174596i	0.996182	−	0.0872979i	$-0.0278232\pi$
$464$	967.925	0.0968422
$465$	9137.45	0.911267
$466$	− 15512.5i	− 1.54207i
$467$	7958.82	0.788630	0.394315	−	0.918975i	$-0.370982\pi$
$467$	7958.82	0.788630	0.394315	+	0.918975i	$0.370982\pi$
$468$	0	0
$469$	6856.16	0.675028
$470$	22489.0i	2.20711i
$471$	9520.54	0.931387
$472$	6310.94	0.615433
$473$	3918.20i	0.380886i
$474$	− 4182.37i	− 0.405280i
$475$	− 881.114i	− 0.0851122i
$476$	− 2222.78i	− 0.214036i
$477$	719.858	0.0690986
$478$	−12737.5	−1.21882
$479$	− 8431.98i	− 0.804315i	−0.915570	−	0.402158i	$-0.868260\pi$
$479$	− 8431.98i	− 0.804315i	0.915570	−	0.402158i	$-0.131740\pi$
$480$	−8813.66	−0.838097
$481$	0	0
$482$	13745.0	1.29889
$483$	6091.45i	0.573852i
$484$	−6425.29	−0.603427
$485$	14006.4	1.31133
$486$	1022.80i	0.0954632i
$487$	− 11684.7i	− 1.08723i	−0.839334	−	0.543617i	$-0.817055\pi$
$487$	− 11684.7i	− 1.08723i	0.839334	−	0.543617i	$-0.182945\pi$
$488$	1351.22i	0.125342i
$489$	6943.94i	0.642159i
$490$	10410.0	0.959750
$491$	3954.70	0.363489	0.181745	−	0.983346i	$-0.441826\pi$
$491$	3954.70	0.363489	0.181745	+	0.983346i	$0.441826\pi$
$492$	− 11402.3i	− 1.04483i
$493$	−415.744	−0.0379801
$494$	0	0
$495$	−2662.63	−0.241771
$496$	− 12608.8i	− 1.14144i
$497$	−2791.49	−0.251943
$498$	5498.27	0.494746
$499$	− 5690.37i	− 0.510493i	−0.966876	−	0.255246i	$-0.917843\pi$
$499$	− 5690.37i	− 0.510493i	0.966876	−	0.255246i	$-0.0821566\pi$
$500$	13251.9i	1.18529i
$501$	7996.95i	0.713128i
$502$	26782.8i	2.38123i
$503$	10859.1	0.962595	0.481298	−	0.876557i	$-0.340166\pi$
$503$	10859.1	0.962595	0.481298	+	0.876557i	$0.340166\pi$
$504$	731.623	0.0646609
$505$	7378.53i	0.650178i
$506$	19651.9	1.72655
$507$	0	0
$508$	−3021.88	−0.263926
$509$	− 18558.6i	− 1.61610i	−0.589115	−	0.808049i	$-0.700524\pi$
$509$	− 18558.6i	− 1.61610i	0.589115	−	0.808049i	$-0.299476\pi$
$510$	2934.41	0.254780
$511$	9597.27	0.830838
$512$	14896.8i	1.28584i
$513$	4177.22i	0.359510i
$514$	25613.6i	2.19799i
$515$	− 5843.42i	− 0.499984i
$516$	4413.27	0.376518
$517$	12094.7	1.02887
$518$	5494.13i	0.466020i
$519$	495.730	0.0419271
$520$	0	0
$521$	17297.5	1.45454	0.727271	−	0.686350i	$-0.240788\pi$
$521$	17297.5	1.45454	0.727271	+	0.686350i	$0.240788\pi$
$522$	− 774.759i	− 0.0649622i
$523$	−5016.11	−0.419386	−0.209693	−	0.977767i	$-0.567247\pi$
$523$	−5016.11	−0.419386	−0.209693	+	0.977767i	$0.567247\pi$
$524$	−19441.8	−1.62084
$525$	− 192.286i	− 0.0159849i
$526$	518.667i	0.0429942i
$527$	5415.77i	0.447656i
$528$	3674.19i	0.302839i
$529$	20383.8	1.67533
$530$	3848.73	0.315431
$531$	7863.39i	0.642640i
$532$	16917.4	1.37869
$533$	0	0
$534$	3273.38	0.265268
$535$	− 6953.01i	− 0.561878i
$536$	−4400.37	−0.354603
$537$	−2137.02	−0.171730
$538$	− 8147.83i	− 0.652933i
$539$	− 5598.57i	− 0.447398i
$540$	2999.06i	0.238998i
$541$	− 17642.3i	− 1.40204i	−0.713144	−	0.701018i	$-0.752729\pi$
$541$	− 17642.3i	− 1.40204i	0.713144	−	0.701018i	$-0.247271\pi$
$542$	−19415.0	−1.53865
$543$	−6619.59	−0.523156
$544$	− 5223.86i	− 0.411712i
$545$	−14862.3	−1.16813
$546$	0	0
$547$	−18414.9	−1.43943	−0.719713	−	0.694271i	$-0.755727\pi$
$547$	−18414.9	−1.43943	−0.719713	+	0.694271i	$0.755727\pi$
$548$	10090.4i	0.786571i
$549$	−1683.61	−0.130883
$550$	−620.343	−0.0480937
$551$	− 3164.20i	− 0.244645i
$552$	− 3909.57i	− 0.301453i
$553$	− 3727.66i	− 0.286648i
$554$	24556.9i	1.88325i
$555$	−3977.84	−0.304234
$556$	27773.1	2.11842
$557$	− 8179.15i	− 0.622193i	−0.950378	−	0.311096i	$-0.899304\pi$
$557$	− 8179.15i	− 0.622193i	0.950378	−	0.311096i	$-0.100696\pi$
$558$	−10092.5	−0.765683
$559$	0	0
$560$	−6089.05	−0.459481
$561$	− 1578.14i	− 0.118769i
$562$	19748.5	1.48228
$563$	1880.07	0.140738	0.0703690	−	0.997521i	$-0.477582\pi$
$563$	1880.07	0.140738	0.0703690	+	0.997521i	$0.477582\pi$
$564$	− 13622.9i	− 1.01707i
$565$	− 482.385i	− 0.0359187i
$566$	− 14586.9i	− 1.08327i
$567$	911.598i	0.0675194i
$568$	1791.62	0.132350
$569$	−10118.3	−0.745485	−0.372743	−	0.927935i	$-0.621583\pi$
$569$	−10118.3	−0.745485	−0.372743	+	0.927935i	$0.621583\pi$
$570$	22333.6i	1.64114i
$571$	−23428.9	−1.71711	−0.858555	−	0.512721i	$-0.828638\pi$
$571$	−23428.9	−1.71711	−0.858555	+	0.512721i	$0.828638\pi$
$572$	0	0
$573$	4411.92	0.321659
$574$	− 18530.3i	− 1.34746i
$575$	−1027.52	−0.0745225
$576$	6327.42	0.457713
$577$	20508.1i	1.47966i	0.672793	+	0.739831i	$0.265094\pi$
$577$	20508.1i	1.47966i	−0.672793	+	0.739831i	$0.734906\pi$
$578$	− 18939.8i	− 1.36296i
$579$	− 1108.68i	− 0.0795771i
$580$	− 2271.76i	− 0.162637i
$581$	4900.49	0.349925
$582$	−15470.4	−1.10184
$583$	− 2069.87i	− 0.147042i
$584$	−6159.65	−0.436452
$585$	0	0
$586$	11268.9	0.794394
$587$	− 5968.43i	− 0.419665i	−0.977737	−	0.209833i	$-0.932708\pi$
$587$	− 5968.43i	− 0.419665i	0.977737	−	0.209833i	$-0.0672919\pi$
$588$	−6305.97	−0.442268
$589$	−41219.0	−2.88353
$590$	42041.8i	2.93361i
$591$	− 12820.2i	− 0.892308i
$592$	5489.06i	0.381080i
$593$	14659.5i	1.01517i	0.861602	+	0.507584i	$0.169461\pi$
$593$	14659.5i	1.01517i	−0.861602	+	0.507584i	$0.830539\pi$
$594$	2940.95	0.203146
$595$	2615.38	0.180202
$596$	− 7223.72i	− 0.496468i
$597$	−12462.9	−0.854394
$598$	0	0
$599$	23635.9	1.61225	0.806125	−	0.591746i	$-0.201561\pi$
$599$	23635.9	1.61225	0.806125	+	0.591746i	$0.201561\pi$
$600$	123.412i	0.00839711i
$601$	−11527.0	−0.782356	−0.391178	−	0.920315i	$-0.627932\pi$
$601$	−11527.0	−0.782356	−0.391178	+	0.920315i	$0.627932\pi$
$602$	7172.15	0.485573
$603$	− 5482.83i	− 0.370279i
$604$	22125.9i	1.49055i
$605$	− 7560.15i	− 0.508039i
$606$	− 8149.77i	− 0.546306i
$607$	−5098.56	−0.340930	−0.170465	−	0.985364i	$-0.554527\pi$
$607$	−5098.56	−0.340930	−0.170465	+	0.985364i	$0.554527\pi$
$608$	39758.4	2.65200
$609$	− 690.525i	− 0.0459466i
$610$	−9001.47	−0.597473
$611$	0	0
$612$	−1777.55	−0.117407
$613$	1516.39i	0.0999128i	0.998751	+	0.0499564i	$0.0159082\pi$
$613$	1516.39i	0.0999128i	−0.998751	+	0.0499564i	$0.984092\pi$
$614$	1986.31	0.130556
$615$	13416.3	0.879668
$616$	− 2103.70i	− 0.137598i
$617$	18539.3i	1.20966i	0.796353	+	0.604832i	$0.206760\pi$
$617$	18539.3i	1.20966i	−0.796353	+	0.604832i	$0.793240\pi$
$618$	6454.20i	0.420107i
$619$	− 25684.9i	− 1.66779i	−0.551920	−	0.833897i	$-0.686105\pi$
$619$	− 25684.9i	− 1.66779i	0.551920	−	0.833897i	$-0.313895\pi$
$620$	−29593.5	−1.91694
$621$	4871.30	0.314780
$622$	6391.51i	0.412020i
$623$	2917.49	0.187619
$624$	0	0
$625$	−16304.5	−1.04349
$626$	17046.1i	1.08834i
$627$	12011.1	0.765038
$628$	−30834.2	−1.95926
$629$	− 2357.67i	− 0.149454i
$630$	4873.88i	0.308222i
$631$	22410.9i	1.41389i	0.707269	+	0.706945i	$0.249927\pi$
$631$	22410.9i	1.41389i	−0.707269	+	0.706945i	$0.750073\pi$
$632$	2392.46i	0.150580i
$633$	3694.77	0.231997
$634$	−13696.1	−0.857950
$635$	− 3555.62i	− 0.222206i
$636$	−2331.40	−0.145356
$637$	0	0
$638$	−2227.73	−0.138239
$639$	2232.34i	0.138200i
$640$	10326.6	0.637804
$641$	−6827.81	−0.420721	−0.210361	−	0.977624i	$-0.567464\pi$
$641$	−6827.81	−0.420721	−0.210361	+	0.977624i	$0.567464\pi$
$642$	7679.77i	0.472113i
$643$	− 23264.3i	− 1.42684i	−0.700738	−	0.713418i	$-0.747146\pi$
$643$	− 23264.3i	− 1.42684i	0.700738	−	0.713418i	$-0.252854\pi$
$644$	− 19728.4i	− 1.20715i
$645$	5192.76i	0.316999i
$646$	−13237.1	−0.806204
$647$	−14745.9	−0.896014	−0.448007	−	0.894030i	$-0.647866\pi$
$647$	−14745.9	−0.896014	−0.448007	+	0.894030i	$0.647866\pi$
$648$	− 585.075i	− 0.0354690i
$649$	22610.3	1.36754
$650$	0	0
$651$	−8995.26	−0.541554
$652$	− 22489.3i	− 1.35084i
$653$	10909.0	0.653755	0.326878	−	0.945067i	$-0.394003\pi$
$653$	10909.0	0.653755	0.326878	+	0.945067i	$0.394003\pi$
$654$	16415.8	0.981509
$655$	− 22875.7i	− 1.36462i
$656$	− 18513.2i	− 1.10186i
$657$	− 7674.89i	− 0.455747i
$658$	− 22139.0i	− 1.31166i
$659$	−4182.99	−0.247263	−0.123631	−	0.992328i	$-0.539454\pi$
$659$	−4182.99	−0.247263	−0.123631	+	0.992328i	$0.539454\pi$
$660$	8623.47	0.508588
$661$	− 2224.23i	− 0.130881i	−0.997856	−	0.0654406i	$-0.979155\pi$
$661$	− 2224.23i	− 0.130881i	0.997856	−	0.0654406i	$-0.0208453\pi$
$662$	14405.2	0.845734
$663$	0	0
$664$	−3145.19	−0.183821
$665$	19905.4i	1.16075i
$666$	4393.63	0.255630
$667$	−3689.95	−0.214206
$668$	− 25899.7i	− 1.50013i
$669$	− 6561.72i	− 0.379209i
$670$	− 29314.1i	− 1.69030i
$671$	4841.04i	0.278519i
$672$	8676.51	0.498071
$673$	24152.5	1.38337	0.691687	−	0.722197i	$-0.256868\pi$
$673$	24152.5	1.38337	0.691687	+	0.722197i	$0.256868\pi$
$674$	39151.2i	2.23746i
$675$	−153.770	−0.00876833
$676$	0	0
$677$	−15310.7	−0.869187	−0.434593	−	0.900627i	$-0.643108\pi$
$677$	−15310.7	−0.869187	−0.434593	+	0.900627i	$0.643108\pi$
$678$	532.806i	0.0301804i
$679$	−13788.4	−0.779310
$680$	−1678.58	−0.0946628
$681$	12416.0i	0.698652i
$682$	29020.0i	1.62937i
$683$	− 11399.6i	− 0.638646i	−0.947646	−	0.319323i	$-0.896545\pi$
$683$	− 11399.6i	− 0.638646i	0.947646	−	0.319323i	$-0.103455\pi$
$684$	− 13528.8i	− 0.756265i
$685$	−11872.6	−0.662233
$686$	−26495.9	−1.47466
$687$	2506.06i	0.139174i
$688$	7165.54	0.397069
$689$	0	0
$690$	26044.5	1.43695
$691$	− 3323.23i	− 0.182955i	−0.995807	−	0.0914773i	$-0.970841\pi$
$691$	− 3323.23i	− 0.182955i	0.995807	−	0.0914773i	$-0.0291589\pi$
$692$	−1605.52	−0.0881976
$693$	2621.20	0.143681
$694$	912.936i	0.0499345i
$695$	32678.5i	1.78355i
$696$	443.188i	0.0241365i
$697$	7951.82i	0.432133i
$698$	−20244.8	−1.09782
$699$	−11056.5	−0.598279
$700$	622.758i	0.0336257i
$701$	12670.4	0.682673	0.341336	−	0.939941i	$-0.389120\pi$
$701$	12670.4	0.682673	0.341336	+	0.939941i	$0.389120\pi$
$702$	0	0
$703$	17944.0	0.962692
$704$	− 18193.8i	− 0.974012i
$705$	16029.0	0.856295
$706$	12036.4	0.641635
$707$	− 7263.71i	− 0.386393i
$708$	− 25467.2i	− 1.35186i
$709$	− 13075.2i	− 0.692594i	−0.938125	−	0.346297i	$-0.887439\pi$
$709$	− 13075.2i	− 0.692594i	0.938125	−	0.346297i	$-0.112561\pi$
$710$	11935.3i	0.630877i
$711$	−2980.99	−0.157237
$712$	−1872.48	−0.0985592
$713$	48067.8i	2.52476i
$714$	−2888.75	−0.151413
$715$	0	0
$716$	6921.16	0.361251
$717$	9078.62i	0.472869i
$718$	15514.7	0.806412
$719$	2988.41	0.155005	0.0775026	−	0.996992i	$-0.475305\pi$
$719$	2988.41	0.155005	0.0775026	+	0.996992i	$0.475305\pi$
$720$	4869.38i	0.252043i
$721$	5752.48i	0.297134i
$722$	− 71877.0i	− 3.70496i
$723$	− 9796.73i	− 0.503934i
$724$	21438.9	1.10051
$725$	116.479	0.00596680
$726$	8350.37i	0.426875i
$727$	5507.46	0.280963	0.140482	−	0.990083i	$-0.455135\pi$
$727$	5507.46	0.280963	0.140482	+	0.990083i	$0.455135\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 41033.9i	− 2.08046i
$731$	−3077.75	−0.155725
$732$	5452.71	0.275325
$733$	− 36585.2i	− 1.84353i	−0.387751	−	0.921764i	$-0.626748\pi$
$733$	− 36585.2i	− 1.84353i	0.387751	−	0.921764i	$-0.373252\pi$
$734$	− 14607.9i	− 0.734587i
$735$	− 7419.75i	− 0.372356i
$736$	− 46364.6i	− 2.32204i
$737$	−15765.3	−0.787953
$738$	−14818.6	−0.739133
$739$	− 6425.89i	− 0.319865i	−0.987128	−	0.159933i	$-0.948872\pi$
$739$	− 6425.89i	− 0.319865i	0.987128	−	0.159933i	$-0.0511277\pi$
$740$	12883.0	0.639986
$741$	0	0
$742$	−3788.84	−0.187457
$743$	20411.0i	1.00782i	0.863757	+	0.503908i	$0.168105\pi$
$743$	20411.0i	1.00782i	−0.863757	+	0.503908i	$0.831895\pi$
$744$	5773.27	0.284487
$745$	8499.60	0.417988
$746$	− 50354.6i	− 2.47133i
$747$	− 3918.89i	− 0.191947i
$748$	5111.13i	0.249842i
$749$	6844.81i	0.333917i
$750$	17222.4	0.838496
$751$	24259.5	1.17875	0.589375	−	0.807860i	$-0.299374\pi$
$751$	24259.5	1.17875	0.589375	+	0.807860i	$0.299374\pi$
$752$	− 22118.6i	− 1.07258i
$753$	19089.5	0.923850
$754$	0	0
$755$	−26033.9	−1.25493
$756$	− 2952.39i	− 0.142034i
$757$	9295.39	0.446297	0.223148	−	0.974785i	$-0.428367\pi$
$757$	9295.39	0.446297	0.223148	+	0.974785i	$0.428367\pi$
$758$	−1454.66	−0.0697042
$759$	− 14006.9i	− 0.669851i
$760$	− 12775.6i	− 0.609761i
$761$	21974.7i	1.04676i	0.852101	+	0.523378i	$0.175328\pi$
$761$	21974.7i	1.04676i	−0.852101	+	0.523378i	$0.824672\pi$
$762$	3927.27i	0.186706i
$763$	14631.0	0.694204
$764$	−14288.9	−0.676641
$765$	− 2091.50i	− 0.0988476i
$766$	14253.5	0.672324
$767$	0	0
$768$	5467.13	0.256873
$769$	22987.4i	1.07795i	0.842320	+	0.538977i	$0.181189\pi$
$769$	22987.4i	1.07795i	−0.842320	+	0.538977i	$0.818811\pi$
$770$	14014.3	0.655897
$771$	18256.1	0.852759
$772$	3590.68i	0.167398i
$773$	31970.9i	1.48760i	0.668404	+	0.743799i	$0.266978\pi$
$773$	31970.9i	1.48760i	−0.668404	+	0.743799i	$0.733022\pi$
$774$	− 5735.53i	− 0.266356i
$775$	− 1517.34i	− 0.0703283i
$776$	8849.59	0.409384
$777$	3915.94	0.180803
$778$	6860.25i	0.316133i
$779$	−60520.8	−2.78354
$780$	0	0
$781$	6418.85	0.294090
$782$	15436.6i	0.705896i
$783$	−552.209	−0.0252035
$784$	−10238.6	−0.466408
$785$	− 36280.2i	− 1.64955i
$786$	25266.7i	1.14661i
$787$	− 6087.26i	− 0.275715i	−0.990452	−	0.137857i	$-0.955978\pi$
$787$	− 6087.26i	− 0.275715i	0.990452	−	0.137857i	$-0.0440215\pi$
$788$	41520.9i	1.87706i
$789$	369.680	0.0166805
$790$	−15937.9	−0.717779
$791$	474.878i	0.0213460i
$792$	−1682.32	−0.0754780
$793$	0	0
$794$	33415.4	1.49354
$795$	− 2743.19i	− 0.122378i
$796$	40363.6	1.79730
$797$	−23080.0	−1.02577	−0.512883	−	0.858458i	$-0.671423\pi$
$797$	−23080.0	−1.02577	−0.512883	+	0.858458i	$0.671423\pi$
$798$	− 21986.1i	− 0.975311i
$799$	9500.41i	0.420651i
$800$	1463.57i	0.0646813i
$801$	− 2333.10i	− 0.102916i
$802$	902.429	0.0397330
$803$	−22068.3	−0.969829
$804$	17757.3i	0.778918i
$805$	23212.9	1.01633
$806$	0	0
$807$	−5807.37	−0.253320
$808$	4661.94i	0.202978i
$809$	−32377.8	−1.40710	−0.703550	−	0.710646i	$-0.748403\pi$
$809$	−32377.8	−1.40710	−0.703550	+	0.710646i	$0.748403\pi$
$810$	3897.61	0.169072
$811$	26352.8i	1.14103i	0.821288	+	0.570513i	$0.193256\pi$
$811$	26352.8i	1.14103i	−0.821288	+	0.570513i	$0.806744\pi$
$812$	2236.40i	0.0966532i
$813$	13838.1i	0.596952i
$814$	− 12633.4i	− 0.543980i
$815$	26461.5	1.13731
$816$	−2886.08	−0.123815
$817$	− 23424.5i	− 1.00308i
$818$	20135.0	0.860638
$819$	0	0
$820$	−43451.3	−1.85047
$821$	35355.3i	1.50294i	0.659770	+	0.751468i	$0.270654\pi$
$821$	35355.3i	1.50294i	−0.659770	+	0.751468i	$0.729346\pi$
$822$	13113.6	0.556435
$823$	12663.3	0.536347	0.268173	−	0.963371i	$-0.413580\pi$
$823$	12663.3	0.536347	0.268173	+	0.963371i	$0.413580\pi$
$824$	− 3692.02i	− 0.156089i
$825$	442.149i	0.0186590i
$826$	− 41387.6i	− 1.74341i
$827$	16295.2i	0.685176i	0.939486	+	0.342588i	$0.111303\pi$
$827$	16295.2i	0.685176i	−0.939486	+	0.342588i	$0.888697\pi$
$828$	−15776.7	−0.662170
$829$	−13638.9	−0.571411	−0.285705	−	0.958318i	$-0.592228\pi$
$829$	−13638.9	−0.571411	−0.285705	+	0.958318i	$0.592228\pi$
$830$	− 20952.4i	− 0.876229i
$831$	17502.9	0.730649
$832$	0	0
$833$	4397.69	0.182918
$834$	− 36094.2i	− 1.49861i
$835$	30474.2	1.26300
$836$	−38900.5	−1.60933
$837$	7193.46i	0.297064i
$838$	− 41685.0i	− 1.71836i
$839$	1890.31i	0.0777838i	0.999243	+	0.0388919i	$0.0123828\pi$
$839$	1890.31i	0.0777838i	−0.999243	+	0.0388919i	$0.987617\pi$
$840$	− 2788.02i	− 0.114519i
$841$	−23970.7	−0.982849
$842$	51016.4	2.08805
$843$	− 14075.7i	− 0.575082i
$844$	−11966.3	−0.488028
$845$	0	0
$846$	−17704.5	−0.719494
$847$	7442.50i	0.301921i
$848$	−3785.35	−0.153289
$849$	−10396.8	−0.420279
$850$	− 487.280i	− 0.0196630i
$851$	− 20925.6i	− 0.842914i
$852$	− 7229.89i	− 0.290718i
$853$	− 1620.21i	− 0.0650351i	−0.999471	−	0.0325175i	$-0.989648\pi$
$853$	− 1620.21i	− 0.0650351i	0.999471	−	0.0325175i	$-0.0103525\pi$
$854$	8861.39	0.355071
$855$	15918.3	0.636718
$856$	− 4393.08i	− 0.175412i
$857$	14508.4	0.578292	0.289146	−	0.957285i	$-0.406629\pi$
$857$	14508.4	0.578292	0.289146	+	0.957285i	$0.406629\pi$
$858$	0	0
$859$	29639.8	1.17730	0.588648	−	0.808389i	$-0.299660\pi$
$859$	29639.8	1.17730	0.588648	+	0.808389i	$0.299660\pi$
$860$	− 16817.8i	− 0.666839i
$861$	−13207.5	−0.522776
$862$	57548.8	2.27392
$863$	21528.8i	0.849186i	0.905384	+	0.424593i	$0.139583\pi$
$863$	21528.8i	0.849186i	−0.905384	+	0.424593i	$0.860417\pi$
$864$	− 6938.56i	− 0.273211i
$865$	− 1889.10i	− 0.0742557i
$866$	− 29939.4i	− 1.17481i
$867$	−13499.4	−0.528792
$868$	29132.9	1.13921
$869$	8571.50i	0.334601i
$870$	−2952.40	−0.115053
$871$	0	0
$872$	−9390.36	−0.364676
$873$	11026.5i	0.427482i
$874$	−117487.	−4.54696
$875$	15349.9	0.593054
$876$	24856.7i	0.958708i
$877$	14865.3i	0.572366i	0.958175	+	0.286183i	$0.0923866\pi$
$877$	14865.3i	0.572366i	−0.958175	+	0.286183i	$0.907613\pi$
$878$	25347.1i	0.974285i
$879$	− 8031.92i	− 0.308202i
$880$	14001.4	0.536348
$881$	21336.0	0.815921	0.407961	−	0.913000i	$-0.366240\pi$
$881$	21336.0	0.815921	0.407961	+	0.913000i	$0.366240\pi$
$882$	8195.30i	0.312869i
$883$	−37538.2	−1.43065	−0.715323	−	0.698794i	$-0.753720\pi$
$883$	−37538.2	−1.43065	−0.715323	+	0.698794i	$0.753720\pi$
$884$	0	0
$885$	29965.3	1.13816
$886$	− 54694.1i	− 2.07391i
$887$	34575.0	1.30881	0.654406	−	0.756144i	$-0.272919\pi$
$887$	34575.0	1.30881	0.654406	+	0.756144i	$0.272919\pi$
$888$	−2513.30	−0.0949784
$889$	3500.29i	0.132054i
$890$	− 12474.0i	− 0.469807i
$891$	− 2096.16i	− 0.0788148i
$892$	21251.4i	0.797703i
$893$	−72306.9	−2.70958
$894$	−9388.02	−0.351211
$895$	8143.61i	0.304146i
$896$	−10165.9	−0.379039
$897$	0	0
$898$	46233.3	1.71807
$899$	− 5448.96i	− 0.202150i
$900$	498.016	0.0184450
$901$	1625.89	0.0601178
$902$	42609.2i	1.57287i
$903$	− 5111.95i	− 0.188389i
$904$	− 304.783i	− 0.0112134i
$905$	25225.5i	0.926545i
$906$	28755.0	1.05444
$907$	10424.8	0.381641	0.190820	−	0.981625i	$-0.438885\pi$
$907$	10424.8	0.381641	0.190820	+	0.981625i	$0.438885\pi$
$908$	− 40211.7i	− 1.46968i
$909$	−5808.75	−0.211952
$910$	0	0
$911$	10961.8	0.398661	0.199331	−	0.979932i	$-0.436123\pi$
$911$	10961.8	0.398661	0.199331	+	0.979932i	$0.436123\pi$
$912$	− 21965.8i	− 0.797543i
$913$	−11268.3	−0.408464
$914$	−41392.2	−1.49796
$915$	6415.80i	0.231803i
$916$	− 8116.38i	− 0.292765i
$917$	22519.7i	0.810976i
$918$	2310.12i	0.0830558i
$919$	−10779.2	−0.386914	−0.193457	−	0.981109i	$-0.561970\pi$
$919$	−10779.2	−0.386914	−0.193457	+	0.981109i	$0.561970\pi$
$920$	−14898.3	−0.533895
$921$	− 1415.74i	− 0.0506519i
$922$	−14316.7	−0.511384
$923$	0	0
$924$	−8489.28	−0.302248
$925$	660.549i	0.0234797i
$926$	7321.32	0.259820
$927$	4600.23	0.162990
$928$	5255.88i	0.185919i
$929$	5429.07i	0.191735i	0.995394	+	0.0958675i	$0.0305625\pi$
$929$	5429.07i	0.191735i	−0.995394	+	0.0958675i	$0.969437\pi$
$930$	38460.0i	1.35608i
$931$	33470.5i	1.17825i
$932$	35808.8	1.25854
$933$	4555.55	0.159852
$934$	33499.1i	1.17358i
$935$	−6013.88	−0.210348
$936$	0	0
$937$	−21300.1	−0.742631	−0.371315	−	0.928507i	$-0.621093\pi$
$937$	−21300.1	−0.742631	−0.371315	+	0.928507i	$0.621093\pi$
$938$	28857.9i	1.00453i
$939$	12149.6	0.422244
$940$	−51913.2	−1.80130
$941$	26851.2i	0.930207i	0.885256	+	0.465103i	$0.153983\pi$
$941$	26851.2i	0.930207i	−0.885256	+	0.465103i	$0.846017\pi$
$942$	40072.4i	1.38602i
$943$	70576.7i	2.43722i
$944$	− 41349.4i	− 1.42564i
$945$	3473.86	0.119582
$946$	−16491.9	−0.566805
$947$	8021.68i	0.275258i	0.990484	+	0.137629i	$0.0439482\pi$
$947$	8021.68i	0.275258i	−0.990484	+	0.137629i	$0.956052\pi$
$948$	9654.52	0.330764
$949$	0	0
$950$	3708.65	0.126658
$951$	9761.88i	0.332861i
$952$	1652.46	0.0562569
$953$	−35715.0	−1.21398	−0.606990	−	0.794709i	$-0.707623\pi$
$953$	−35715.0	−1.21398	−0.606990	+	0.794709i	$0.707623\pi$
$954$	3029.92i	0.102827i
$955$	− 16812.6i	− 0.569680i
$956$	− 29402.9i	− 0.994726i
$957$	1587.82i	0.0536330i
$958$	35490.6	1.19692
$959$	11687.9	0.393557
$960$	− 24112.1i	− 0.810641i
$961$	−41190.9	−1.38266
$962$	0	0
$963$	5473.76	0.183166
$964$	31728.7i	1.06007i
$965$	−4224.88	−0.140936
$966$	−25639.2	−0.853963
$967$	53338.8i	1.77380i	0.461965	+	0.886898i	$0.347145\pi$
$967$	53338.8i	1.77380i	−0.461965	+	0.886898i	$0.652855\pi$
$968$	− 4776.69i	− 0.158604i
$969$	9434.77i	0.312785i
$970$	58953.6i	1.95143i
$971$	23112.9	0.763882	0.381941	−	0.924187i	$-0.375256\pi$
$971$	23112.9	0.763882	0.381941	+	0.924187i	$0.375256\pi$
$972$	−2361.01	−0.0779110
$973$	− 32170.0i	− 1.05994i
$974$	49181.3	1.61794
$975$	0	0
$976$	8853.22	0.290353
$977$	− 52874.6i	− 1.73143i	−0.500538	−	0.865715i	$-0.666864\pi$
$977$	− 52874.6i	− 1.73143i	0.500538	−	0.865715i	$-0.333136\pi$
$978$	−29227.4	−0.955612
$979$	−6708.57	−0.219006
$980$	24030.4i	0.783287i
$981$	− 11700.3i	− 0.380798i
$982$	16645.5i	0.540917i
$983$	− 45173.1i	− 1.46572i	−0.680381	−	0.732858i	$-0.738186\pi$
$983$	− 45173.1i	− 1.46572i	0.680381	−	0.732858i	$-0.261814\pi$
$984$	8476.73	0.274622
$985$	−48854.5	−1.58034
$986$	− 1749.89i	− 0.0565190i
$987$	−15779.6	−0.508885
$988$	0	0
$989$	−27316.7	−0.878281
$990$	− 11207.2i	− 0.359785i
$991$	60485.6	1.93884	0.969418	−	0.245414i	$-0.0789239\pi$
$991$	60485.6	1.93884	0.969418	+	0.245414i	$0.0789239\pi$
$992$	68466.7	2.19135
$993$	− 10267.3i	− 0.328121i
$994$	− 11749.5i	− 0.374922i
$995$	47492.8i	1.51319i
$996$	12692.1i	0.403780i
$997$	−18108.1	−0.575214	−0.287607	−	0.957749i	$-0.592860\pi$
$997$	−18108.1	−0.575214	−0.287607	+	0.957749i	$0.592860\pi$
$998$	23951.1	0.759677
$999$	− 3131.56i	− 0.0991773i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.g.337.6		6
13.5	odd	4		39.4.a.c.1.3	✓	3
13.8	odd	4		507.4.a.h.1.1		3
13.12	even	2	inner	507.4.b.g.337.1		6
39.5	even	4		117.4.a.f.1.1		3
39.8	even	4		1521.4.a.u.1.3		3
52.31	even	4		624.4.a.t.1.1		3
65.44	odd	4		975.4.a.l.1.1		3
91.83	even	4		1911.4.a.k.1.3		3
104.5	odd	4		2496.4.a.bl.1.3		3
104.83	even	4		2496.4.a.bp.1.3		3
156.83	odd	4		1872.4.a.bk.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.3	✓	3	13.5	odd	4
117.4.a.f.1.1		3	39.5	even	4
507.4.a.h.1.1		3	13.8	odd	4
507.4.b.g.337.1		6	13.12	even	2	inner
507.4.b.g.337.6		6	1.1	even	1	trivial
624.4.a.t.1.1		3	52.31	even	4
975.4.a.l.1.1		3	65.44	odd	4
1521.4.a.u.1.3		3	39.8	even	4
1872.4.a.bk.1.3		3	156.83	odd	4
1911.4.a.k.1.3		3	91.83	even	4
2496.4.a.bl.1.3		3	104.5	odd	4
2496.4.a.bp.1.3		3	104.83	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)

\(f(q)\)	\(=\)	\(q+4.20905i q^{2} +3.00000 q^{3} -9.71610 q^{4} -11.4322i q^{5} +12.6271i q^{6} +11.2543i q^{7} -7.22315i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.20905i q^{2} +3.00000 q^{3} -9.71610 q^{4} -11.4322i q^{5} +12.6271i q^{6} +11.2543i q^{7} -7.22315i q^{8} +9.00000 q^{9} +48.1187 q^{10} -25.8785i q^{11} -29.1483 q^{12} -47.3699 q^{14} -34.2966i q^{15} -47.3262 q^{16} +20.3276 q^{17} +37.8814i q^{18} +154.712i q^{19} +111.076i q^{20} +33.7629i q^{21} +108.924 q^{22} +180.418 q^{23} -21.6695i q^{24} -5.69520 q^{25} +27.0000 q^{27} -109.348i q^{28} -20.4522 q^{29} +144.356 q^{30} +266.424i q^{31} -256.984i q^{32} -77.6355i q^{33} +85.5599i q^{34} +128.661 q^{35} -87.4449 q^{36} -115.984i q^{37} -651.190 q^{38} -82.5765 q^{40} +391.184i q^{41} -142.110 q^{42} -151.407 q^{43} +251.438i q^{44} -102.890i q^{45} +759.390i q^{46} +467.365i q^{47} -141.979 q^{48} +216.341 q^{49} -23.9714i q^{50} +60.9828 q^{51} +79.9842 q^{53} +113.644i q^{54} -295.848 q^{55} +81.2915 q^{56} +464.136i q^{57} -86.0843i q^{58} +873.710i q^{59} +333.229i q^{60} -187.068 q^{61} -1121.39 q^{62} +101.289i q^{63} +703.047 q^{64} +326.772 q^{66} -609.204i q^{67} -197.505 q^{68} +541.255 q^{69} +541.542i q^{70} +248.038i q^{71} -65.0084i q^{72} -852.765i q^{73} +488.181 q^{74} -17.0856 q^{75} -1503.20i q^{76} +291.244 q^{77} -331.221 q^{79} +541.043i q^{80} +81.0000 q^{81} -1646.51 q^{82} -435.432i q^{83} -328.044i q^{84} -232.389i q^{85} -637.281i q^{86} -61.3566 q^{87} -186.924 q^{88} -259.233i q^{89} +433.068 q^{90} -1752.96 q^{92} +799.273i q^{93} -1967.16 q^{94} +1768.70 q^{95} -770.951i q^{96} +1225.17i q^{97} +910.589i q^{98} -232.907i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9}+O(q^{10}) \) 6 * q + 18 * q^3 - 20 * q^4 + 54 * q^9	\( 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9} + 8 q^{10} - 60 q^{12} - 352 q^{14} - 220 q^{16} + 292 q^{17} - 112 q^{22} + 96 q^{23} - 290 q^{25} + 162 q^{27} - 4 q^{29} + 24 q^{30} + 160 q^{35} - 180 q^{36} - 624 q^{38} - 1032 q^{40} - 1056 q^{42} + 520 q^{43} - 660 q^{48} - 894 q^{49} + 876 q^{51} - 1356 q^{53} - 3104 q^{55} - 192 q^{56} + 460 q^{61} - 3904 q^{62} + 1500 q^{64} - 336 q^{66} - 920 q^{68} + 288 q^{69} + 3448 q^{74} - 870 q^{75} - 224 q^{77} - 48 q^{79} + 486 q^{81} - 1128 q^{82} - 12 q^{87} - 3216 q^{88} + 72 q^{90} - 2592 q^{92} - 3840 q^{94} + 8064 q^{95}+O(q^{100}) \) 6 * q + 18 * q^3 - 20 * q^4 + 54 * q^9 + 8 * q^10 - 60 * q^12 - 352 * q^14 - 220 * q^16 + 292 * q^17 - 112 * q^22 + 96 * q^23 - 290 * q^25 + 162 * q^27 - 4 * q^29 + 24 * q^30 + 160 * q^35 - 180 * q^36 - 624 * q^38 - 1032 * q^40 - 1056 * q^42 + 520 * q^43 - 660 * q^48 - 894 * q^49 + 876 * q^51 - 1356 * q^53 - 3104 * q^55 - 192 * q^56 + 460 * q^61 - 3904 * q^62 + 1500 * q^64 - 336 * q^66 - 920 * q^68 + 288 * q^69 + 3448 * q^74 - 870 * q^75 - 224 * q^77 - 48 * q^79 + 486 * q^81 - 1128 * q^82 - 12 * q^87 - 3216 * q^88 + 72 * q^90 - 2592 * q^92 - 3840 * q^94 + 8064 * q^95

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