LMFDB - Embedded newform 507.4.b.e.337.4

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:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 17x^{2} + 289 $ x^4 - 17*x^2 + 289
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.4
Root			$3.57071 + 2.06155i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.e.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} -3.05006i q^{5} -12.3693i q^{6} +6.68324i q^{7} -4.12311i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} -3.05006i q^{5} -12.3693i q^{6} +6.68324i q^{7} -4.12311i q^{8} +9.00000 q^{9} +12.5757 q^{10} -32.2500i q^{11} +27.0000 q^{12} -27.5557 q^{14} +9.15018i q^{15} -55.0000 q^{16} +28.8586 q^{17} +37.1080i q^{18} -101.194i q^{19} +27.4505i q^{20} -20.0497i q^{21} +132.970 q^{22} +118.990 q^{23} +12.3693i q^{24} +115.697 q^{25} -27.0000 q^{27} -60.1492i q^{28} +160.111 q^{29} -37.7271 q^{30} -38.0705i q^{31} -259.756i q^{32} +96.7499i q^{33} +118.987i q^{34} +20.3843 q^{35} -81.0000 q^{36} +327.568i q^{37} +417.233 q^{38} -12.5757 q^{40} +56.0846i q^{41} +82.6671 q^{42} -127.879 q^{43} +290.250i q^{44} -27.4505i q^{45} +490.608i q^{46} +517.983i q^{47} +165.000 q^{48} +298.334 q^{49} +477.032i q^{50} -86.5757 q^{51} -695.546 q^{53} -111.324i q^{54} -98.3643 q^{55} +27.5557 q^{56} +303.581i q^{57} +660.156i q^{58} -656.523i q^{59} -82.3516i q^{60} +701.304 q^{61} +156.969 q^{62} +60.1492i q^{63} +631.000 q^{64} -398.910 q^{66} -57.1750i q^{67} -259.727 q^{68} -356.970 q^{69} +84.0465i q^{70} +309.226i q^{71} -37.1080i q^{72} +389.711i q^{73} -1350.60 q^{74} -347.091 q^{75} +910.744i q^{76} +215.534 q^{77} +901.820 q^{79} +167.753i q^{80} +81.0000 q^{81} -231.243 q^{82} +687.095i q^{83} +180.448i q^{84} -88.0203i q^{85} -527.257i q^{86} -480.334 q^{87} -132.970 q^{88} -1070.54i q^{89} +113.181 q^{90} -1070.91 q^{92} +114.211i q^{93} -2135.70 q^{94} -308.647 q^{95} +779.267i q^{96} +1754.48i q^{97} +1230.06i q^{98} -290.250i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 - 36 * q^4 + 36 * q^9	$ 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9} + 136 q^{10} + 108 q^{12} + 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} + 120 q^{25} - 108 q^{27} + 12 q^{29} - 408 q^{30} - 804 q^{35} - 324 q^{36} + 612 q^{38} - 136 q^{40} - 612 q^{42} - 940 q^{43} + 660 q^{48} - 692 q^{49} - 432 q^{51} - 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} + 2524 q^{64} + 204 q^{66} - 1296 q^{68} - 828 q^{69} - 3060 q^{74} - 360 q^{75} + 2976 q^{77} + 8 q^{79} + 324 q^{81} - 68 q^{82} - 36 q^{87} + 68 q^{88} + 1224 q^{90} - 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) $ 4 * q - 12 * q^3 - 36 * q^4 + 36 * q^9 + 136 * q^10 + 108 * q^12 + 204 * q^14 - 220 * q^16 + 144 * q^17 - 68 * q^22 + 276 * q^23 + 120 * q^25 - 108 * q^27 + 12 * q^29 - 408 * q^30 - 804 * q^35 - 324 * q^36 + 612 * q^38 - 136 * q^40 - 612 * q^42 - 940 * q^43 + 660 * q^48 - 692 * q^49 - 432 * q^51 - 2268 * q^53 + 892 * q^55 - 204 * q^56 + 320 * q^61 + 2856 * q^62 + 2524 * q^64 + 204 * q^66 - 1296 * q^68 - 828 * q^69 - 3060 * q^74 - 360 * q^75 + 2976 * q^77 + 8 * q^79 + 324 * q^81 - 68 * q^82 - 36 * q^87 + 68 * q^88 + 1224 * q^90 - 2484 * q^92 - 5372 * q^94 + 108 * 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 $

$2$	4.12311i	1.45774i	0.684653	+	0.728869i	$0.259954\pi$
$2$	4.12311i	1.45774i	−0.684653	+	0.728869i	$0.740046\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−9.00000	−1.12500
$5$	− 3.05006i	− 0.272806i	−0.990653	−	0.136403i	$-0.956446\pi$
$5$	− 3.05006i	− 0.272806i	0.990653	−	0.136403i	$-0.0435541\pi$
$6$	− 12.3693i	− 0.841625i
$7$	6.68324i	0.360861i	0.983588	+	0.180431i	$0.0577491\pi$
$7$	6.68324i	0.360861i	−0.983588	+	0.180431i	$0.942251\pi$
$8$	− 4.12311i	− 0.182217i
$9$	9.00000	0.333333
$10$	12.5757	0.397679
$11$	− 32.2500i	− 0.883975i	−0.897021	−	0.441988i	$-0.854273\pi$
$11$	− 32.2500i	− 0.883975i	0.897021	−	0.441988i	$-0.145727\pi$
$12$	27.0000	0.649519
$13$	0	0
$13$	0	0
$14$	−27.5557	−0.526041
$15$	9.15018i	0.157504i
$16$	−55.0000	−0.859375
$17$	28.8586	0.411720	0.205860	−	0.978582i	$-0.434001\pi$
$17$	28.8586	0.411720	0.205860	+	0.978582i	$0.434001\pi$
$18$	37.1080i	0.485913i
$19$	− 101.194i	− 1.22187i	−0.791682	−	0.610933i	$-0.790794\pi$
$19$	− 101.194i	− 1.22187i	0.791682	−	0.610933i	$-0.209206\pi$
$20$	27.4505i	0.306906i
$21$	− 20.0497i	− 0.208343i
$22$	132.970	1.28860
$23$	118.990	1.07874	0.539372	−	0.842067i	$-0.318662\pi$
$23$	118.990	1.07874	0.539372	+	0.842067i	$0.318662\pi$
$24$	12.3693i	0.105203i
$25$	115.697	0.925577
$26$	0	0
$27$	−27.0000	−0.192450
$28$	− 60.1492i	− 0.405969i
$29$	160.111	1.02524	0.512620	−	0.858616i	$-0.328675\pi$
$29$	160.111	1.02524	0.512620	+	0.858616i	$0.328675\pi$
$30$	−37.7271	−0.229600
$31$	− 38.0705i	− 0.220570i	−0.993900	−	0.110285i	$-0.964824\pi$
$31$	− 38.0705i	− 0.220570i	0.993900	−	0.110285i	$-0.0351763\pi$
$32$	− 259.756i	− 1.43496i
$33$	96.7499i	0.510363i
$34$	118.987i	0.600179i
$35$	20.3843	0.0984449
$36$	−81.0000	−0.375000
$37$	327.568i	1.45545i	0.685866	+	0.727727i	$0.259423\pi$
$37$	327.568i	1.45545i	−0.685866	+	0.727727i	$0.740577\pi$
$38$	417.233	1.78116
$39$	0	0
$40$	−12.5757	−0.0497099
$41$	56.0846i	0.213633i	0.994279	+	0.106816i	$0.0340657\pi$
$41$	56.0846i	0.213633i	−0.994279	+	0.106816i	$0.965934\pi$
$42$	82.6671	0.303710
$43$	−127.879	−0.453519	−0.226759	−	0.973951i	$-0.572813\pi$
$43$	−127.879	−0.453519	−0.226759	+	0.973951i	$0.572813\pi$
$44$	290.250i	0.994472i
$45$	− 27.4505i	− 0.0909352i
$46$	490.608i	1.57253i
$47$	517.983i	1.60757i	0.594923	+	0.803783i	$0.297183\pi$
$47$	517.983i	1.60757i	−0.594923	+	0.803783i	$0.702817\pi$
$48$	165.000	0.496160
$49$	298.334	0.869779
$50$	477.032i	1.34925i
$51$	−86.5757	−0.237706
$52$	0	0
$53$	−695.546	−1.80265	−0.901326	−	0.433141i	$-0.857405\pi$
$53$	−695.546	−1.80265	−0.901326	+	0.433141i	$0.857405\pi$
$54$	− 111.324i	− 0.280542i
$55$	−98.3643	−0.241153
$56$	27.5557	0.0657551
$57$	303.581i	0.705445i
$58$	660.156i	1.49453i
$59$	− 656.523i	− 1.44868i	−0.689444	−	0.724339i	$-0.742145\pi$
$59$	− 656.523i	− 1.44868i	0.689444	−	0.724339i	$-0.257855\pi$
$60$	− 82.3516i	− 0.177192i
$61$	701.304	1.47201	0.736007	−	0.676974i	$-0.236709\pi$
$61$	701.304	1.47201	0.736007	+	0.676974i	$0.236709\pi$
$62$	156.969	0.321533
$63$	60.1492i	0.120287i
$64$	631.000	1.23242
$65$	0	0
$66$	−398.910	−0.743976
$67$	− 57.1750i	− 0.104254i	−0.998640	−	0.0521271i	$-0.983400\pi$
$67$	− 57.1750i	− 0.104254i	0.998640	−	0.0521271i	$-0.0166001\pi$
$68$	−259.727	−0.463184
$69$	−356.970	−0.622814
$70$	84.0465i	0.143507i
$71$	309.226i	0.516879i	0.966027	+	0.258440i	$0.0832083\pi$
$71$	309.226i	0.516879i	−0.966027	+	0.258440i	$0.916792\pi$
$72$	− 37.1080i	− 0.0607391i
$73$	389.711i	0.624826i	0.949946	+	0.312413i	$0.101137\pi$
$73$	389.711i	0.624826i	−0.949946	+	0.312413i	$0.898863\pi$
$74$	−1350.60	−2.12167
$75$	−347.091	−0.534382
$76$	910.744i	1.37460i
$77$	215.534	0.318992
$78$	0	0
$79$	901.820	1.28434	0.642169	−	0.766563i	$-0.278035\pi$
$79$	901.820	1.28434	0.642169	+	0.766563i	$0.278035\pi$
$80$	167.753i	0.234442i
$81$	81.0000	0.111111
$82$	−231.243	−0.311421
$83$	687.095i	0.908657i	0.890834	+	0.454328i	$0.150121\pi$
$83$	687.095i	0.908657i	−0.890834	+	0.454328i	$0.849879\pi$
$84$	180.448i	0.234386i
$85$	− 88.0203i	− 0.112319i
$86$	− 527.257i	− 0.661111i
$87$	−480.334	−0.591922
$88$	−132.970	−0.161076
$89$	− 1070.54i	− 1.27502i	−0.770442	−	0.637510i	$-0.779964\pi$
$89$	− 1070.54i	− 1.27502i	0.770442	−	0.637510i	$-0.220036\pi$
$90$	113.181	0.132560
$91$	0	0
$92$	−1070.91	−1.21359
$93$	114.211i	0.127346i
$94$	−2135.70	−2.34341
$95$	−308.647	−0.333332
$96$	779.267i	0.828475i
$97$	1754.48i	1.83650i	0.395998	+	0.918251i	$0.370399\pi$
$97$	1754.48i	1.83650i	−0.395998	+	0.918251i	$0.629601\pi$
$98$	1230.06i	1.26791i
$99$	− 290.250i	− 0.294658i
$100$	−1041.27	−1.04127
$101$	640.840	0.631346	0.315673	−	0.948868i	$-0.397770\pi$
$101$	640.840	0.631346	0.315673	+	0.948868i	$0.397770\pi$
$102$	− 356.961i	− 0.346514i
$103$	693.153	0.663091	0.331546	−	0.943439i	$-0.392430\pi$
$103$	693.153	0.663091	0.331546	+	0.943439i	$0.392430\pi$
$104$	0	0
$105$	−61.1528	−0.0568372
$106$	− 2867.81i	− 2.62779i
$107$	−405.676	−0.366525	−0.183262	−	0.983064i	$-0.558666\pi$
$107$	−405.676	−0.366525	−0.183262	+	0.983064i	$0.558666\pi$
$108$	243.000	0.216506
$109$	− 479.516i	− 0.421370i	−0.977554	−	0.210685i	$-0.932431\pi$
$109$	− 479.516i	− 0.421370i	0.977554	−	0.210685i	$-0.0675694\pi$
$110$	− 405.566i	− 0.351538i
$111$	− 982.704i	− 0.840307i
$112$	− 367.578i	− 0.310115i
$113$	−1547.20	−1.28804	−0.644021	−	0.765008i	$-0.722735\pi$
$113$	−1547.20	−1.28804	−0.644021	+	0.765008i	$0.722735\pi$
$114$	−1251.70	−1.02835
$115$	− 362.926i	− 0.294288i
$116$	−1441.00	−1.15339
$117$	0	0
$118$	2706.91	2.11179
$119$	192.869i	0.148574i
$120$	37.7271	0.0287000
$121$	290.940	0.218588
$122$	2891.55i	2.14581i
$123$	− 168.254i	− 0.123341i
$124$	342.634i	0.248141i
$125$	− 734.140i	− 0.525308i
$126$	−248.001	−0.175347
$127$	2495.15	1.74338	0.871690	−	0.490059i	$-0.163025\pi$
$127$	2495.15	1.74338	0.871690	+	0.490059i	$0.163025\pi$
$128$	523.634i	0.361587i
$129$	383.636	0.261839
$130$	0	0
$131$	43.7571	0.0291838	0.0145919	−	0.999894i	$-0.495355\pi$
$131$	43.7571	0.0291838	0.0145919	+	0.999894i	$0.495355\pi$
$132$	− 870.749i	− 0.574159i
$133$	676.303	0.440924
$134$	235.739	0.151975
$135$	82.3516i	0.0525015i
$136$	− 118.987i	− 0.0750224i
$137$	206.194i	0.128586i	0.997931	+	0.0642932i	$0.0204793\pi$
$137$	206.194i	0.128586i	−0.997931	+	0.0642932i	$0.979521\pi$
$138$	− 1471.83i	− 0.907899i
$139$	−100.000	−0.0610208	−0.0305104	−	0.999534i	$-0.509713\pi$
$139$	−100.000	−0.0610208	−0.0305104	+	0.999534i	$0.509713\pi$
$140$	−183.459	−0.110751
$141$	− 1553.95i	− 0.928128i
$142$	−1274.97	−0.753474
$143$	0	0
$144$	−495.000	−0.286458
$145$	− 488.349i	− 0.279691i
$146$	−1606.82	−0.910832
$147$	−895.003	−0.502167
$148$	− 2948.11i	− 1.63739i
$149$	380.451i	0.209179i	0.994515	+	0.104590i	$0.0333529\pi$
$149$	380.451i	0.209179i	−0.994515	+	0.104590i	$0.966647\pi$
$150$	− 1431.09i	− 0.778989i
$151$	1517.45i	0.817805i	0.912578	+	0.408902i	$0.134088\pi$
$151$	1517.45i	0.817805i	−0.912578	+	0.408902i	$0.865912\pi$
$152$	−417.233	−0.222645
$153$	259.727	0.137240
$154$	888.671i	0.465007i
$155$	−116.117	−0.0601726
$156$	0	0
$157$	1450.16	0.737166	0.368583	−	0.929595i	$-0.379843\pi$
$157$	1450.16	0.737166	0.368583	+	0.929595i	$0.379843\pi$
$158$	3718.30i	1.87223i
$159$	2086.64	1.04076
$160$	−792.270	−0.391465
$161$	795.239i	0.389277i
$162$	333.972i	0.161971i
$163$	− 2342.36i	− 1.12557i	−0.826603	−	0.562785i	$-0.809730\pi$
$163$	− 2342.36i	− 1.12557i	0.826603	−	0.562785i	$-0.190270\pi$
$164$	− 504.762i	− 0.240337i
$165$	295.093	0.139230
$166$	−2832.97	−1.32458
$167$	40.0731i	0.0185686i	0.999957	+	0.00928428i	$0.00295532\pi$
$167$	40.0731i	0.0185686i	−0.999957	+	0.00928428i	$0.997045\pi$
$168$	−82.6671	−0.0379637
$169$	0	0
$170$	362.917	0.163732
$171$	− 910.744i	− 0.407289i
$172$	1150.91	0.510208
$173$	−1909.54	−0.839189	−0.419594	−	0.907712i	$-0.637828\pi$
$173$	−1909.54	−0.839189	−0.419594	+	0.907712i	$0.637828\pi$
$174$	− 1980.47i	− 0.862868i
$175$	773.232i	0.334005i
$176$	1773.75i	0.759666i
$177$	1969.57i	0.836395i
$178$	4413.94	1.85864
$179$	509.959	0.212939	0.106470	−	0.994316i	$-0.466045\pi$
$179$	509.959	0.212939	0.106470	+	0.994316i	$0.466045\pi$
$180$	247.055i	0.102302i
$181$	2136.88	0.877531	0.438766	−	0.898602i	$-0.355416\pi$
$181$	2136.88	0.877531	0.438766	+	0.898602i	$0.355416\pi$
$182$	0	0
$183$	−2103.91	−0.849867
$184$	− 490.608i	− 0.196566i
$185$	999.101	0.397056
$186$	−470.906	−0.185637
$187$	− 930.688i	− 0.363950i
$188$	− 4661.85i	− 1.80851i
$189$	− 180.448i	− 0.0694478i
$190$	− 1272.58i	− 0.485911i
$191$	4057.74	1.53721	0.768607	−	0.639721i	$-0.220950\pi$
$191$	4057.74	1.53721	0.768607	+	0.639721i	$0.220950\pi$
$192$	−1893.00	−0.711539
$193$	− 873.394i	− 0.325742i	−0.986647	−	0.162871i	$-0.947925\pi$
$193$	− 873.394i	− 0.325742i	0.986647	−	0.162871i	$-0.0520755\pi$
$194$	−7233.92	−2.67714
$195$	0	0
$196$	−2685.01	−0.978502
$197$	4147.25i	1.49989i	0.661498	+	0.749947i	$0.269921\pi$
$197$	4147.25i	1.49989i	−0.661498	+	0.749947i	$0.730079\pi$
$198$	1196.73	0.429535
$199$	2404.06	0.856379	0.428189	−	0.903689i	$-0.359152\pi$
$199$	2404.06	0.856379	0.428189	+	0.903689i	$0.359152\pi$
$200$	− 477.032i	− 0.168656i
$201$	171.525i	0.0601912i
$202$	2642.25i	0.920337i
$203$	1070.06i	0.369969i
$204$	779.181	0.267420
$205$	171.061	0.0582802
$206$	2857.94i	0.966613i
$207$	1070.91	0.359582
$208$	0	0
$209$	−3263.50	−1.08010
$210$	− 252.140i	− 0.0828538i
$211$	3868.85	1.26229	0.631144	−	0.775665i	$-0.282586\pi$
$211$	3868.85	1.26229	0.631144	+	0.775665i	$0.282586\pi$
$212$	6259.91	2.02798
$213$	− 927.679i	− 0.298420i
$214$	− 1672.64i	− 0.534297i
$215$	390.037i	0.123722i
$216$	111.324i	0.0350677i
$217$	254.434	0.0795950
$218$	1977.09	0.614246
$219$	− 1169.13i	− 0.360743i
$220$	885.279	0.271298
$221$	0	0
$222$	4051.79	1.22495
$223$	2813.55i	0.844885i	0.906390	+	0.422443i	$0.138827\pi$
$223$	2813.55i	0.844885i	−0.906390	+	0.422443i	$0.861173\pi$
$224$	1736.01	0.517822
$225$	1041.27	0.308526
$226$	− 6379.29i	− 1.87763i
$227$	− 4518.37i	− 1.32112i	−0.750772	−	0.660561i	$-0.770318\pi$
$227$	− 4518.37i	− 1.32112i	0.750772	−	0.660561i	$-0.229682\pi$
$228$	− 2732.23i	− 0.793625i
$229$	1305.27i	0.376658i	0.982106	+	0.188329i	$0.0603070\pi$
$229$	1305.27i	0.376658i	−0.982106	+	0.188329i	$0.939693\pi$
$230$	1496.38	0.428994
$231$	−646.603	−0.184170
$232$	− 660.156i	− 0.186816i
$233$	3360.55	0.944879	0.472440	−	0.881363i	$-0.343373\pi$
$233$	3360.55	0.944879	0.472440	+	0.881363i	$0.343373\pi$
$234$	0	0
$235$	1579.88	0.438553
$236$	5908.71i	1.62976i
$237$	−2705.46	−0.741513
$238$	−795.219	−0.216581
$239$	− 4737.17i	− 1.28210i	−0.767499	−	0.641050i	$-0.778499\pi$
$239$	− 4737.17i	− 1.28210i	0.767499	−	0.641050i	$-0.221501\pi$
$240$	− 503.260i	− 0.135355i
$241$	− 4785.28i	− 1.27903i	−0.768778	−	0.639516i	$-0.779135\pi$
$241$	− 4785.28i	− 1.27903i	0.768778	−	0.639516i	$-0.220865\pi$
$242$	1199.58i	0.318643i
$243$	−243.000	−0.0641500
$244$	−6311.74	−1.65601
$245$	− 909.937i	− 0.237281i
$246$	693.729	0.179799
$247$	0	0
$248$	−156.969	−0.0401916
$249$	− 2061.29i	− 0.524613i
$250$	3026.94	0.765762
$251$	3273.86	0.823284	0.411642	−	0.911346i	$-0.364955\pi$
$251$	3273.86	0.823284	0.411642	+	0.911346i	$0.364955\pi$
$252$	− 541.343i	− 0.135323i
$253$	− 3837.42i	− 0.953584i
$254$	10287.8i	2.54139i
$255$	264.061i	0.0648476i
$256$	2889.00	0.705322
$257$	6545.81	1.58878	0.794390	−	0.607408i	$-0.207791\pi$
$257$	6545.81	1.58878	0.794390	+	0.607408i	$0.207791\pi$
$258$	1581.77i	0.381693i
$259$	−2189.22	−0.525217
$260$	0	0
$261$	1441.00	0.341746
$262$	180.415i	0.0425424i
$263$	88.2014	0.0206796	0.0103398	−	0.999947i	$-0.496709\pi$
$263$	88.2014	0.0206796	0.0103398	+	0.999947i	$0.496709\pi$
$264$	398.910	0.0929970
$265$	2121.46i	0.491773i
$266$	2788.47i	0.642752i
$267$	3211.61i	0.736133i
$268$	514.575i	0.117286i
$269$	−4527.60	−1.02622	−0.513109	−	0.858324i	$-0.671506\pi$
$269$	−4527.60	−1.02622	−0.513109	+	0.858324i	$0.671506\pi$
$270$	−339.544	−0.0765334
$271$	− 8321.82i	− 1.86537i	−0.360695	−	0.932684i	$-0.617460\pi$
$271$	− 8321.82i	− 1.86537i	0.360695	−	0.932684i	$-0.382540\pi$
$272$	−1587.22	−0.353821
$273$	0	0
$274$	−850.160	−0.187445
$275$	− 3731.23i	− 0.818187i
$276$	3212.73	0.700665
$277$	2881.31	0.624986	0.312493	−	0.949920i	$-0.398836\pi$
$277$	2881.31	0.624986	0.312493	+	0.949920i	$0.398836\pi$
$278$	− 412.311i	− 0.0889523i
$279$	− 342.634i	− 0.0735232i
$280$	− 84.0465i	− 0.0179384i
$281$	2817.99i	0.598247i	0.954214	+	0.299123i	$0.0966942\pi$
$281$	2817.99i	0.598247i	−0.954214	+	0.299123i	$0.903306\pi$
$282$	6407.10	1.35297
$283$	−264.601	−0.0555792	−0.0277896	−	0.999614i	$-0.508847\pi$
$283$	−264.601	−0.0555792	−0.0277896	+	0.999614i	$0.508847\pi$
$284$	− 2783.04i	− 0.581489i
$285$	925.941	0.192449
$286$	0	0
$287$	−374.827	−0.0770918
$288$	− 2337.80i	− 0.478320i
$289$	−4080.18	−0.830487
$290$	2013.52	0.407716
$291$	− 5263.45i	− 1.06031i
$292$	− 3507.40i	− 0.702929i
$293$	− 4292.52i	− 0.855877i	−0.903808	−	0.427938i	$-0.859240\pi$
$293$	− 4292.52i	− 0.855877i	0.903808	−	0.427938i	$-0.140760\pi$
$294$	− 3690.19i	− 0.732028i
$295$	−2002.43	−0.395208
$296$	1350.60	0.265209
$297$	870.749i	0.170121i
$298$	−1568.64	−0.304929
$299$	0	0
$300$	3123.82	0.601180
$301$	− 854.643i	− 0.163657i
$302$	−6256.62	−1.19214
$303$	−1922.52	−0.364508
$304$	5565.66i	1.05004i
$305$	− 2139.02i	− 0.401573i
$306$	1070.88i	0.200060i
$307$	− 7026.26i	− 1.30622i	−0.757263	−	0.653110i	$-0.773464\pi$
$307$	− 7026.26i	− 1.30622i	0.757263	−	0.653110i	$-0.226536\pi$
$308$	−1939.81	−0.358866
$309$	−2079.46	−0.382836
$310$	− 478.763i	− 0.0877159i
$311$	1133.21	0.206618	0.103309	−	0.994649i	$-0.467057\pi$
$311$	1133.21	0.206618	0.103309	+	0.994649i	$0.467057\pi$
$312$	0	0
$313$	−5285.95	−0.954566	−0.477283	−	0.878750i	$-0.658378\pi$
$313$	−5285.95	−0.954566	−0.477283	+	0.878750i	$0.658378\pi$
$314$	5979.15i	1.07459i
$315$	183.459	0.0328150
$316$	−8116.38	−1.44488
$317$	− 4782.16i	− 0.847296i	−0.905827	−	0.423648i	$-0.860749\pi$
$317$	− 4782.16i	− 0.847296i	0.905827	−	0.423648i	$-0.139251\pi$
$318$	8603.43i	1.51716i
$319$	− 5163.59i	− 0.906286i
$320$	− 1924.59i	− 0.336212i
$321$	1217.03	0.211613
$322$	−3278.85	−0.567464
$323$	− 2920.31i	− 0.503066i
$324$	−729.000	−0.125000
$325$	0	0
$326$	9657.80	1.64079
$327$	1438.55i	0.243278i
$328$	231.243	0.0389276
$329$	−3461.81	−0.580108
$330$	1216.70i	0.202961i
$331$	8669.98i	1.43971i	0.694122	+	0.719857i	$0.255793\pi$
$331$	8669.98i	1.43971i	−0.694122	+	0.719857i	$0.744207\pi$
$332$	− 6183.86i	− 1.02224i
$333$	2948.11i	0.485152i
$334$	−165.226	−0.0270681
$335$	−174.387	−0.0284412
$336$	1102.73i	0.179045i
$337$	−8526.59	−1.37826	−0.689129	−	0.724639i	$-0.742007\pi$
$337$	−8526.59	−1.37826	−0.689129	+	0.724639i	$0.742007\pi$
$338$	0	0
$339$	4641.61	0.743651
$340$	792.183i	0.126359i
$341$	−1227.77	−0.194978
$342$	3755.10	0.593720
$343$	4286.19i	0.674731i
$344$	527.257i	0.0826389i
$345$	1088.78i	0.169907i
$346$	− 7873.23i	− 1.22332i
$347$	−12581.5	−1.94643	−0.973213	−	0.229907i	$-0.926158\pi$
$347$	−12581.5	−1.94643	−0.973213	+	0.229907i	$0.926158\pi$
$348$	4323.01	0.665913
$349$	8961.18i	1.37444i	0.726447	+	0.687222i	$0.241170\pi$
$349$	8961.18i	1.37444i	−0.726447	+	0.687222i	$0.758830\pi$
$350$	−3188.12	−0.486892
$351$	0	0
$352$	−8377.11	−1.26847
$353$	− 5357.99i	− 0.807868i	−0.914788	−	0.403934i	$-0.867643\pi$
$353$	− 5357.99i	− 0.807868i	0.914788	−	0.403934i	$-0.132357\pi$
$354$	−8120.74	−1.21924
$355$	943.159	0.141007
$356$	9634.84i	1.43440i
$357$	− 578.606i	− 0.0857790i
$358$	2102.61i	0.310409i
$359$	2705.40i	0.397731i	0.980027	+	0.198866i	$0.0637257\pi$
$359$	2705.40i	0.397731i	−0.980027	+	0.198866i	$0.936274\pi$
$360$	−113.181	−0.0165700
$361$	−3381.19	−0.492957
$362$	8810.59i	1.27921i
$363$	−872.820	−0.126202
$364$	0	0
$365$	1188.64	0.170456
$366$	− 8674.65i	− 1.23888i
$367$	10473.8	1.48972	0.744858	−	0.667223i	$-0.232517\pi$
$367$	10473.8	1.48972	0.744858	+	0.667223i	$0.232517\pi$
$368$	−6544.45	−0.927046
$369$	504.762i	0.0712110i
$370$	4119.40i	0.578804i
$371$	− 4648.50i	− 0.650507i
$372$	− 1027.90i	− 0.143264i
$373$	−12763.0	−1.77170	−0.885850	−	0.463973i	$-0.846424\pi$
$373$	−12763.0	−1.77170	−0.885850	+	0.463973i	$0.846424\pi$
$374$	3837.32	0.530544
$375$	2202.42i	0.303287i
$376$	2135.70	0.292926
$377$	0	0
$378$	744.004	0.101237
$379$	− 2318.02i	− 0.314166i	−0.987585	−	0.157083i	$-0.949791\pi$
$379$	− 2318.02i	− 0.314166i	0.987585	−	0.157083i	$-0.0502090\pi$
$380$	2777.82	0.374998
$381$	−7485.46	−1.00654
$382$	16730.5i	2.24086i
$383$	1983.34i	0.264606i	0.991209	+	0.132303i	$0.0422372\pi$
$383$	1983.34i	0.264606i	−0.991209	+	0.132303i	$0.957763\pi$
$384$	− 1570.90i	− 0.208763i
$385$	− 657.392i	− 0.0870229i
$386$	3601.10	0.474847
$387$	−1150.91	−0.151173
$388$	− 15790.3i	− 2.06607i
$389$	3244.51	0.422887	0.211444	−	0.977390i	$-0.432184\pi$
$389$	3244.51	0.422887	0.211444	+	0.977390i	$0.432184\pi$
$390$	0	0
$391$	3433.88	0.444140
$392$	− 1230.06i	− 0.158489i
$393$	−131.271	−0.0168493
$394$	−17099.5	−2.18645
$395$	− 2750.60i	− 0.350374i
$396$	2612.25i	0.331491i
$397$	− 3759.72i	− 0.475302i	−0.971351	−	0.237651i	$-0.923622\pi$
$397$	− 3759.72i	− 0.475302i	0.971351	−	0.237651i	$-0.0763775\pi$
$398$	9912.20i	1.24838i
$399$	−2028.91	−0.254568
$400$	−6363.34	−0.795418
$401$	1997.55i	0.248760i	0.992235	+	0.124380i	$0.0396941\pi$
$401$	1997.55i	0.248760i	−0.992235	+	0.124380i	$0.960306\pi$
$402$	−707.216	−0.0877431
$403$	0	0
$404$	−5767.56	−0.710264
$405$	− 247.055i	− 0.0303117i
$406$	−4411.98	−0.539318
$407$	10564.1	1.28659
$408$	356.961i	0.0433142i
$409$	− 5195.23i	− 0.628087i	−0.949409	−	0.314044i	$-0.898316\pi$
$409$	− 5195.23i	− 0.628087i	0.949409	−	0.314044i	$-0.101684\pi$
$410$	705.304i	0.0849573i
$411$	− 618.582i	− 0.0742394i
$412$	−6238.38	−0.745977
$413$	4387.70	0.522772
$414$	4415.48i	0.524176i
$415$	2095.68	0.247887
$416$	0	0
$417$	300.000	0.0352304
$418$	− 13455.7i	− 1.57450i
$419$	−6822.11	−0.795422	−0.397711	−	0.917511i	$-0.630195\pi$
$419$	−6822.11	−0.795422	−0.397711	+	0.917511i	$0.630195\pi$
$420$	550.376	0.0639419
$421$	7537.70i	0.872601i	0.899801	+	0.436300i	$0.143712\pi$
$421$	7537.70i	0.872601i	−0.899801	+	0.436300i	$0.856288\pi$
$422$	15951.7i	1.84009i
$423$	4661.85i	0.535855i
$424$	2867.81i	0.328474i
$425$	3338.85	0.381078
$426$	3824.92	0.435019
$427$	4686.99i	0.531192i
$428$	3651.08	0.412340
$429$	0	0
$430$	−1608.16	−0.180355
$431$	− 13404.2i	− 1.49805i	−0.662544	−	0.749023i	$-0.730523\pi$
$431$	− 13404.2i	− 1.49805i	0.662544	−	0.749023i	$-0.269477\pi$
$432$	1485.00	0.165387
$433$	17715.9	1.96622	0.983110	−	0.183014i	$-0.0585852\pi$
$433$	17715.9	1.96622	0.983110	+	0.183014i	$0.0585852\pi$
$434$	1049.06i	0.116029i
$435$	1465.05i	0.161480i
$436$	4315.64i	0.474041i
$437$	− 12041.1i	− 1.31808i
$438$	4820.46	0.525869
$439$	−7163.47	−0.778801	−0.389401	−	0.921068i	$-0.627318\pi$
$439$	−7163.47	−0.778801	−0.389401	+	0.921068i	$0.627318\pi$
$440$	405.566i	0.0439423i
$441$	2685.01	0.289926
$442$	0	0
$443$	−10169.2	−1.09064	−0.545321	−	0.838227i	$-0.683592\pi$
$443$	−10169.2	−1.09064	−0.545321	+	0.838227i	$0.683592\pi$
$444$	8844.33i	0.945346i
$445$	−3265.20	−0.347833
$446$	−11600.6	−1.23162
$447$	− 1141.35i	− 0.120770i
$448$	4217.13i	0.444733i
$449$	17142.5i	1.80179i	0.434037	+	0.900895i	$0.357089\pi$
$449$	17142.5i	1.80179i	−0.434037	+	0.900895i	$0.642911\pi$
$450$	4293.28i	0.449750i
$451$	1808.73	0.188846
$452$	13924.8	1.44905
$453$	− 4552.36i	− 0.472160i
$454$	18629.7	1.92585
$455$	0	0
$456$	1251.70	0.128544
$457$	14091.1i	1.44235i	0.692750	+	0.721177i	$0.256399\pi$
$457$	14091.1i	1.44235i	−0.692750	+	0.721177i	$0.743601\pi$
$458$	−5381.76	−0.549068
$459$	−779.181	−0.0792355
$460$	3266.34i	0.331074i
$461$	− 2922.22i	− 0.295231i	−0.989045	−	0.147616i	$-0.952840\pi$
$461$	− 2922.22i	− 0.295231i	0.989045	−	0.147616i	$-0.0471598\pi$
$462$	− 2666.01i	− 0.268472i
$463$	2072.61i	0.208040i	0.994575	+	0.104020i	$0.0331706\pi$
$463$	2072.61i	0.208040i	−0.994575	+	0.104020i	$0.966829\pi$
$464$	−8806.13	−0.881065
$465$	348.352	0.0347407
$466$	13855.9i	1.37739i
$467$	2664.19	0.263992	0.131996	−	0.991250i	$-0.457861\pi$
$467$	2664.19	0.263992	0.131996	+	0.991250i	$0.457861\pi$
$468$	0	0
$469$	382.114	0.0376213
$470$	6514.01i	0.639295i
$471$	−4350.47	−0.425603
$472$	−2706.91	−0.263974
$473$	4124.08i	0.400899i
$474$	− 11154.9i	− 1.08093i
$475$	− 11707.8i	− 1.13093i
$476$	− 1735.82i	− 0.167145i
$477$	−6259.91	−0.600884
$478$	19531.8	1.86897
$479$	5220.70i	0.497995i	0.968504	+	0.248998i	$0.0801011\pi$
$479$	5220.70i	0.497995i	−0.968504	+	0.248998i	$0.919899\pi$
$480$	2376.81	0.226013
$481$	0	0
$482$	19730.2	1.86449
$483$	− 2385.72i	− 0.224749i
$484$	−2618.46	−0.245911
$485$	5351.28	0.501008
$486$	− 1001.91i	− 0.0935139i
$487$	− 12224.6i	− 1.13747i	−0.822520	−	0.568737i	$-0.807432\pi$
$487$	− 12224.6i	− 1.13747i	0.822520	−	0.568737i	$-0.192568\pi$
$488$	− 2891.55i	− 0.268226i
$489$	7027.08i	0.649848i
$490$	3751.77	0.345893
$491$	−19653.2	−1.80639	−0.903195	−	0.429231i	$-0.858784\pi$
$491$	−19653.2	−1.80639	−0.903195	+	0.429231i	$0.858784\pi$
$492$	1514.29i	0.138759i
$493$	4620.59	0.422111
$494$	0	0
$495$	−885.279	−0.0803845
$496$	2093.88i	0.189552i
$497$	−2066.63	−0.186522
$498$	8498.90	0.764749
$499$	− 11713.6i	− 1.05084i	−0.850842	−	0.525422i	$-0.823907\pi$
$499$	− 11713.6i	− 1.05084i	0.850842	−	0.525422i	$-0.176093\pi$
$500$	6607.26i	0.590972i
$501$	− 120.219i	− 0.0107206i
$502$	13498.5i	1.20013i
$503$	13003.3	1.15266	0.576332	−	0.817216i	$-0.304483\pi$
$503$	13003.3	1.15266	0.576332	+	0.817216i	$0.304483\pi$
$504$	248.001	0.0219184
$505$	− 1954.60i	− 0.172235i
$506$	15822.1	1.39008
$507$	0	0
$508$	−22456.4	−1.96130
$509$	− 5328.93i	− 0.464049i	−0.972710	−	0.232024i	$-0.925465\pi$
$509$	− 5328.93i	− 0.464049i	0.972710	−	0.232024i	$-0.0745349\pi$
$510$	−1088.75	−0.0945308
$511$	−2604.54	−0.225475
$512$	16100.7i	1.38976i
$513$	2732.23i	0.235148i
$514$	26989.1i	2.31603i
$515$	− 2114.16i	− 0.180895i
$516$	−3452.72	−0.294569
$517$	16704.9	1.42105
$518$	− 9026.37i	− 0.765629i
$519$	5728.62	0.484506
$520$	0	0
$521$	−11700.3	−0.983876	−0.491938	−	0.870630i	$-0.663711\pi$
$521$	−11700.3	−0.983876	−0.491938	+	0.870630i	$0.663711\pi$
$522$	5941.41i	0.498177i
$523$	−4535.04	−0.379165	−0.189583	−	0.981865i	$-0.560714\pi$
$523$	−4535.04	−0.379165	−0.189583	+	0.981865i	$0.560714\pi$
$524$	−393.814	−0.0328318
$525$	− 2319.70i	− 0.192838i
$526$	363.664i	0.0301454i
$527$	− 1098.66i	− 0.0908128i
$528$	− 5321.24i	− 0.438594i
$529$	1991.62	0.163690
$530$	−8746.98	−0.716877
$531$	− 5908.71i	− 0.482893i
$532$	−6086.73	−0.496040
$533$	0	0
$534$	−13241.8	−1.07309
$535$	1237.33i	0.0999900i
$536$	−235.739	−0.0189969
$537$	−1529.88	−0.122940
$538$	− 18667.8i	− 1.49596i
$539$	− 9621.27i	− 0.768863i
$540$	− 741.164i	− 0.0590641i
$541$	5184.89i	0.412044i	0.978547	+	0.206022i	$0.0660519\pi$
$541$	5184.89i	0.412044i	−0.978547	+	0.206022i	$0.933948\pi$
$542$	34311.7	2.71922
$543$	−6410.64	−0.506643
$544$	− 7496.18i	− 0.590801i
$545$	−1462.55	−0.114952
$546$	0	0
$547$	5609.12	0.438443	0.219222	−	0.975675i	$-0.429648\pi$
$547$	5609.12	0.438443	0.219222	+	0.975675i	$0.429648\pi$
$548$	− 1855.75i	− 0.144660i
$549$	6311.74	0.490671
$550$	15384.2	1.19270
$551$	− 16202.3i	− 1.25271i
$552$	1471.83i	0.113487i
$553$	6027.08i	0.463468i
$554$	11879.9i	0.911066i
$555$	−2997.30	−0.229241
$556$	900.000	0.0686484
$557$	20150.5i	1.53286i	0.642326	+	0.766432i	$0.277970\pi$
$557$	20150.5i	1.53286i	−0.642326	+	0.766432i	$0.722030\pi$
$558$	1412.72	0.107178
$559$	0	0
$560$	−1121.14	−0.0846011
$561$	2792.06i	0.210127i
$562$	−11618.9	−0.872087
$563$	−16292.2	−1.21960	−0.609800	−	0.792556i	$-0.708750\pi$
$563$	−16292.2	−1.21960	−0.609800	+	0.792556i	$0.708750\pi$
$564$	13985.5i	1.04414i
$565$	4719.06i	0.351385i
$566$	− 1090.98i	− 0.0810200i
$567$	541.343i	0.0400957i
$568$	1274.97	0.0941843
$569$	−10460.5	−0.770700	−0.385350	−	0.922770i	$-0.625919\pi$
$569$	−10460.5	−0.770700	−0.385350	+	0.922770i	$0.625919\pi$
$570$	3817.75i	0.280541i
$571$	−2225.96	−0.163141	−0.0815705	−	0.996668i	$-0.525994\pi$
$571$	−2225.96	−0.163141	−0.0815705	+	0.996668i	$0.525994\pi$
$572$	0	0
$573$	−12173.2	−0.887511
$574$	− 1545.45i	− 0.112380i
$575$	13766.8	0.998461
$576$	5679.00	0.410807
$577$	4686.23i	0.338112i	0.985606	+	0.169056i	$0.0540718\pi$
$577$	4686.23i	0.338112i	−0.985606	+	0.169056i	$0.945928\pi$
$578$	− 16823.0i	− 1.21063i
$579$	2620.18i	0.188067i
$580$	4395.14i	0.314652i
$581$	−4592.03	−0.327899
$582$	21701.8	1.54565
$583$	22431.3i	1.59350i
$584$	1606.82	0.113854
$585$	0	0
$586$	17698.5	1.24764
$587$	12090.6i	0.850138i	0.905161	+	0.425069i	$0.139750\pi$
$587$	12090.6i	0.850138i	−0.905161	+	0.425069i	$0.860250\pi$
$588$	8055.03	0.564938
$589$	−3852.50	−0.269507
$590$	− 8256.25i	− 0.576109i
$591$	− 12441.7i	− 0.865964i
$592$	− 18016.2i	− 1.25078i
$593$	− 6135.97i	− 0.424914i	−0.977170	−	0.212457i	$-0.931853\pi$
$593$	− 6135.97i	− 0.424914i	0.977170	−	0.212457i	$-0.0681466\pi$
$594$	−3590.19	−0.247992
$595$	588.261	0.0405317
$596$	− 3424.06i	− 0.235327i
$597$	−7212.18	−0.494431
$598$	0	0
$599$	−6198.80	−0.422831	−0.211416	−	0.977396i	$-0.567807\pi$
$599$	−6198.80	−0.422831	−0.211416	+	0.977396i	$0.567807\pi$
$600$	1431.09i	0.0973737i
$601$	18345.4	1.24513	0.622565	−	0.782568i	$-0.286091\pi$
$601$	18345.4	1.24513	0.622565	+	0.782568i	$0.286091\pi$
$602$	3523.79	0.238569
$603$	− 514.575i	− 0.0347514i
$604$	− 13657.1i	− 0.920030i
$605$	− 887.384i	− 0.0596319i
$606$	− 7926.75i	− 0.531357i
$607$	10388.1	0.694631	0.347315	−	0.937748i	$-0.387093\pi$
$607$	10388.1	0.694631	0.347315	+	0.937748i	$0.387093\pi$
$608$	−26285.7	−1.75333
$609$	− 3210.19i	− 0.213602i
$610$	8819.40	0.585389
$611$	0	0
$612$	−2337.54	−0.154395
$613$	− 804.480i	− 0.0530060i	−0.999649	−	0.0265030i	$-0.991563\pi$
$613$	− 804.480i	− 0.0530060i	0.999649	−	0.0265030i	$-0.00843715\pi$
$614$	28970.0	1.90413
$615$	−513.184	−0.0336481
$616$	− 888.671i	− 0.0581259i
$617$	15218.3i	0.992973i	0.868044	+	0.496486i	$0.165377\pi$
$617$	15218.3i	0.992973i	−0.868044	+	0.496486i	$0.834623\pi$
$618$	− 8573.83i	− 0.558074i
$619$	11462.5i	0.744291i	0.928174	+	0.372145i	$0.121378\pi$
$619$	11462.5i	0.744291i	−0.928174	+	0.372145i	$0.878622\pi$
$620$	1045.05	0.0676942
$621$	−3212.73	−0.207605
$622$	4672.33i	0.301195i
$623$	7154.66	0.460105
$624$	0	0
$625$	12223.0	0.782270
$626$	− 21794.5i	− 1.39151i
$627$	9790.49	0.623596
$628$	−13051.4	−0.829311
$629$	9453.14i	0.599239i
$630$	756.419i	0.0478356i
$631$	4468.68i	0.281926i	0.990015	+	0.140963i	$0.0450199\pi$
$631$	4468.68i	0.281926i	−0.990015	+	0.140963i	$0.954980\pi$
$632$	− 3718.30i	− 0.234028i
$633$	−11606.6	−0.728783
$634$	19717.3	1.23514
$635$	− 7610.37i	− 0.475603i
$636$	−18779.7	−1.17086
$637$	0	0
$638$	21290.0	1.32113
$639$	2783.04i	0.172293i
$640$	1597.12	0.0986430
$641$	−6142.36	−0.378484	−0.189242	−	0.981930i	$-0.560603\pi$
$641$	−6142.36	−0.378484	−0.189242	+	0.981930i	$0.560603\pi$
$642$	5017.93i	0.308477i
$643$	20738.2i	1.27190i	0.771729	+	0.635951i	$0.219392\pi$
$643$	20738.2i	1.27190i	−0.771729	+	0.635951i	$0.780608\pi$
$644$	− 7157.15i	− 0.437937i
$645$	− 1170.11i	− 0.0714312i
$646$	12040.7	0.733339
$647$	−852.757	−0.0518166	−0.0259083	−	0.999664i	$-0.508248\pi$
$647$	−852.757	−0.0518166	−0.0259083	+	0.999664i	$0.508248\pi$
$648$	− 333.972i	− 0.0202464i
$649$	−21172.8	−1.28060
$650$	0	0
$651$	−763.303	−0.0459542
$652$	21081.3i	1.26627i
$653$	−7345.75	−0.440217	−0.220108	−	0.975475i	$-0.570641\pi$
$653$	−7345.75	−0.440217	−0.220108	+	0.975475i	$0.570641\pi$
$654$	−5931.28	−0.354635
$655$	− 133.462i	− 0.00796151i
$656$	− 3084.65i	− 0.183591i
$657$	3507.40i	0.208275i
$658$	− 14273.4i	− 0.845645i
$659$	12540.7	0.741297	0.370648	−	0.928773i	$-0.379136\pi$
$659$	12540.7	0.741297	0.370648	+	0.928773i	$0.379136\pi$
$660$	−2655.84	−0.156634
$661$	2242.95i	0.131983i	0.997820	+	0.0659915i	$0.0210210\pi$
$661$	2242.95i	0.131983i	−0.997820	+	0.0659915i	$0.978979\pi$
$662$	−35747.3	−2.09873
$663$	0	0
$664$	2832.97	0.165573
$665$	− 2062.76i	− 0.120287i
$666$	−12155.4	−0.707224
$667$	19051.7	1.10597
$668$	− 360.658i	− 0.0208896i
$669$	− 8440.66i	− 0.487795i
$670$	− 719.016i	− 0.0414597i
$671$	− 22617.0i	− 1.30122i
$672$	−5208.03	−0.298964
$673$	4776.46	0.273579	0.136790	−	0.990600i	$-0.456322\pi$
$673$	4776.46	0.273579	0.136790	+	0.990600i	$0.456322\pi$
$674$	− 35156.0i	− 2.00914i
$675$	−3123.82	−0.178127
$676$	0	0
$677$	−7933.57	−0.450387	−0.225193	−	0.974314i	$-0.572301\pi$
$677$	−7933.57	−0.450387	−0.225193	+	0.974314i	$0.572301\pi$
$678$	19137.9i	1.08405i
$679$	−11725.6	−0.662723
$680$	−362.917	−0.0204665
$681$	13555.1i	0.762750i
$682$	− 5062.23i	− 0.284227i
$683$	− 23573.8i	− 1.32068i	−0.750967	−	0.660340i	$-0.770412\pi$
$683$	− 23573.8i	− 1.32068i	0.750967	−	0.660340i	$-0.229588\pi$
$684$	8196.70i	0.458200i
$685$	628.904	0.0350791
$686$	−17672.4	−0.983581
$687$	− 3915.81i	− 0.217463i
$688$	7033.32	0.389743
$689$	0	0
$690$	−4489.15	−0.247680
$691$	− 12543.8i	− 0.690575i	−0.938497	−	0.345288i	$-0.887781\pi$
$691$	− 12543.8i	− 0.690575i	0.938497	−	0.345288i	$-0.112219\pi$
$692$	17185.9	0.944087
$693$	1939.81	0.106331
$694$	− 51874.8i	− 2.83738i
$695$	305.006i	0.0166468i
$696$	1980.47i	0.107858i
$697$	1618.52i	0.0879568i
$698$	−36947.9	−2.00358
$699$	−10081.6	−0.545526
$700$	− 6959.09i	− 0.375755i
$701$	581.786	0.0313463	0.0156731	−	0.999877i	$-0.495011\pi$
$701$	581.786	0.0313463	0.0156731	+	0.999877i	$0.495011\pi$
$702$	0	0
$703$	33147.9	1.77837
$704$	− 20349.7i	− 1.08943i
$705$	−4739.64	−0.253199
$706$	22091.6	1.17766
$707$	4282.89i	0.227828i
$708$	− 17726.1i	− 0.940944i
$709$	20742.0i	1.09871i	0.835590	+	0.549353i	$0.185126\pi$
$709$	20742.0i	1.09871i	−0.835590	+	0.549353i	$0.814874\pi$
$710$	3888.74i	0.205552i
$711$	8116.38	0.428113
$712$	−4413.94	−0.232331
$713$	− 4530.01i	− 0.237938i
$714$	2385.66	0.125043
$715$	0	0
$716$	−4589.63	−0.239556
$717$	14211.5i	0.740221i
$718$	−11154.6	−0.579788
$719$	−25350.2	−1.31489	−0.657443	−	0.753504i	$-0.728362\pi$
$719$	−25350.2	−1.31489	−0.657443	+	0.753504i	$0.728362\pi$
$720$	1509.78i	0.0781474i
$721$	4632.51i	0.239284i
$722$	− 13941.0i	− 0.718602i
$723$	14355.8i	0.738449i
$724$	−19231.9	−0.987223
$725$	18524.4	0.948938
$726$	− 3598.73i	− 0.183969i
$727$	−33428.2	−1.70534	−0.852672	−	0.522447i	$-0.825019\pi$
$727$	−33428.2	−1.70534	−0.852672	+	0.522447i	$0.825019\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	4900.90i	0.248480i
$731$	−3690.39	−0.186722
$732$	18935.2	0.956101
$733$	− 3842.67i	− 0.193632i	−0.995302	−	0.0968160i	$-0.969134\pi$
$733$	− 3842.67i	− 0.193632i	0.995302	−	0.0968160i	$-0.0308658\pi$
$734$	43184.4i	2.17162i
$735$	2729.81i	0.136994i
$736$	− 30908.3i	− 1.54796i
$737$	−1843.89	−0.0921582
$738$	−2081.19	−0.103807
$739$	− 29029.6i	− 1.44502i	−0.691359	−	0.722511i	$-0.742988\pi$
$739$	− 29029.6i	− 1.44502i	0.691359	−	0.722511i	$-0.257012\pi$
$740$	−8991.91	−0.446688
$741$	0	0
$742$	19166.3	0.948269
$743$	− 34996.7i	− 1.72800i	−0.503492	−	0.864000i	$-0.667952\pi$
$743$	− 34996.7i	− 1.72800i	0.503492	−	0.864000i	$-0.332048\pi$
$744$	470.906	0.0232046
$745$	1160.40	0.0570653
$746$	− 52623.2i	− 2.58267i
$747$	6183.86i	0.302886i
$748$	8376.19i	0.409444i
$749$	− 2711.23i	− 0.132265i
$750$	−9080.82	−0.442113
$751$	−10454.1	−0.507957	−0.253979	−	0.967210i	$-0.581739\pi$
$751$	−10454.1	−0.507957	−0.253979	+	0.967210i	$0.581739\pi$
$752$	− 28489.1i	− 1.38150i
$753$	−9821.58	−0.475323
$754$	0	0
$755$	4628.32	0.223102
$756$	1624.03i	0.0781287i
$757$	−28130.4	−1.35062	−0.675308	−	0.737536i	$-0.735989\pi$
$757$	−28130.4	−1.35062	−0.675308	+	0.737536i	$0.735989\pi$
$758$	9557.45	0.457971
$759$	11512.3i	0.550552i
$760$	1272.58i	0.0607388i
$761$	− 21087.0i	− 1.00447i	−0.864731	−	0.502236i	$-0.832511\pi$
$761$	− 21087.0i	− 1.00447i	0.864731	−	0.502236i	$-0.167489\pi$
$762$	− 30863.4i	− 1.46727i
$763$	3204.72	0.152056
$764$	−36519.7	−1.72937
$765$	− 792.183i	− 0.0374398i
$766$	−8177.54	−0.385727
$767$	0	0
$768$	−8667.00	−0.407218
$769$	19527.9i	0.915728i	0.889022	+	0.457864i	$0.151385\pi$
$769$	19527.9i	0.915728i	−0.889022	+	0.457864i	$0.848615\pi$
$770$	2710.50	0.126857
$771$	−19637.4	−0.917283
$772$	7860.55i	0.366460i
$773$	− 29352.0i	− 1.36574i	−0.730540	−	0.682870i	$-0.760731\pi$
$773$	− 29352.0i	− 1.36574i	0.730540	−	0.682870i	$-0.239269\pi$
$774$	− 4745.31i	− 0.220370i
$775$	− 4404.64i	− 0.204154i
$776$	7233.92	0.334643
$777$	6567.65	0.303234
$778$	13377.5i	0.616459i
$779$	5675.42	0.261031
$780$	0	0
$781$	9972.54	0.456908
$782$	14158.3i	0.647440i
$783$	−4323.01	−0.197307
$784$	−16408.4	−0.747467
$785$	− 4423.06i	− 0.201103i
$786$	− 541.246i	− 0.0245618i
$787$	− 4463.12i	− 0.202151i	−0.994879	−	0.101076i	$-0.967772\pi$
$787$	− 4463.12i	− 0.202151i	0.994879	−	0.101076i	$-0.0322284\pi$
$788$	− 37325.2i	− 1.68738i
$789$	−264.604	−0.0119394
$790$	11341.0	0.510754
$791$	− 10340.3i	− 0.464804i
$792$	−1196.73	−0.0536919
$793$	0	0
$794$	15501.7	0.692866
$795$	− 6364.37i	− 0.283926i
$796$	−21636.6	−0.963426
$797$	34785.5	1.54600	0.773002	−	0.634404i	$-0.218754\pi$
$797$	34785.5	1.54600	0.773002	+	0.634404i	$0.218754\pi$
$798$	− 8365.40i	− 0.371093i
$799$	14948.2i	0.661866i
$800$	− 30053.0i	− 1.32817i
$801$	− 9634.84i	− 0.425007i
$802$	−8236.09	−0.362627
$803$	12568.2	0.552330
$804$	− 1543.72i	− 0.0677152i
$805$	2425.53	0.106197
$806$	0	0
$807$	13582.8	0.592487
$808$	− 2642.25i	− 0.115042i
$809$	−10620.0	−0.461530	−0.230765	−	0.973010i	$-0.574123\pi$
$809$	−10620.0	−0.461530	−0.230765	+	0.973010i	$0.574123\pi$
$810$	1018.63	0.0441866
$811$	5497.87i	0.238047i	0.992891	+	0.119024i	$0.0379764\pi$
$811$	5497.87i	0.238047i	−0.992891	+	0.119024i	$0.962024\pi$
$812$	− 9630.57i	− 0.416215i
$813$	24965.5i	1.07697i
$814$	43556.7i	1.87551i
$815$	−7144.34	−0.307062
$816$	4761.66	0.204279
$817$	12940.5i	0.554139i
$818$	21420.5	0.915587
$819$	0	0
$820$	−1539.55	−0.0655653
$821$	21305.3i	0.905678i	0.891592	+	0.452839i	$0.149589\pi$
$821$	21305.3i	0.905678i	−0.891592	+	0.452839i	$0.850411\pi$
$822$	2550.48	0.108222
$823$	−17342.6	−0.734537	−0.367268	−	0.930115i	$-0.619707\pi$
$823$	−17342.6	−0.734537	−0.367268	+	0.930115i	$0.619707\pi$
$824$	− 2857.94i	− 0.120827i
$825$	11193.7i	0.472381i
$826$	18091.0i	0.762064i
$827$	− 5129.96i	− 0.215703i	−0.994167	−	0.107851i	$-0.965603\pi$
$827$	− 5129.96i	− 0.215703i	0.994167	−	0.107851i	$-0.0343971\pi$
$828$	−9638.19	−0.404529
$829$	8471.81	0.354931	0.177466	−	0.984127i	$-0.443210\pi$
$829$	8471.81	0.354931	0.177466	+	0.984127i	$0.443210\pi$
$830$	8640.72i	0.361354i
$831$	−8643.93	−0.360836
$832$	0	0
$833$	8609.50	0.358105
$834$	1236.93i	0.0513566i
$835$	122.225	0.00506561
$836$	29371.5	1.21511
$837$	1027.90i	0.0424486i
$838$	− 28128.3i	− 1.15952i
$839$	19155.0i	0.788207i	0.919066	+	0.394103i	$0.128945\pi$
$839$	19155.0i	0.788207i	−0.919066	+	0.394103i	$0.871055\pi$
$840$	252.140i	0.0103567i
$841$	1246.67	0.0511160
$842$	−31078.7	−1.27202
$843$	− 8453.98i	− 0.345398i
$844$	−34819.7	−1.42007
$845$	0	0
$846$	−19221.3	−0.781136
$847$	1944.42i	0.0788797i
$848$	38255.0	1.54915
$849$	793.804	0.0320887
$850$	13766.4i	0.555512i
$851$	38977.3i	1.57006i
$852$	8349.11i	0.335723i
$853$	18075.1i	0.725532i	0.931880	+	0.362766i	$0.118168\pi$
$853$	18075.1i	0.725532i	−0.931880	+	0.362766i	$0.881832\pi$
$854$	−19324.9	−0.774339
$855$	−2777.82	−0.111111
$856$	1672.64i	0.0667871i
$857$	21054.6	0.839219	0.419609	−	0.907705i	$-0.362167\pi$
$857$	21054.6	0.839219	0.419609	+	0.907705i	$0.362167\pi$
$858$	0	0
$859$	920.322	0.0365553	0.0182776	−	0.999833i	$-0.494182\pi$
$859$	920.322	0.0365553	0.0182776	+	0.999833i	$0.494182\pi$
$860$	− 3510.33i	− 0.139188i
$861$	1124.48	0.0445090
$862$	55267.0	2.18376
$863$	− 19427.5i	− 0.766304i	−0.923685	−	0.383152i	$-0.874839\pi$
$863$	− 19427.5i	− 0.766304i	0.923685	−	0.383152i	$-0.125161\pi$
$864$	7013.40i	0.276158i
$865$	5824.21i	0.228935i
$866$	73044.7i	2.86623i
$867$	12240.5	0.479482
$868$	−2289.91	−0.0895444
$869$	− 29083.7i	− 1.13532i
$870$	−6040.55	−0.235395
$871$	0	0
$872$	−1977.09	−0.0767808
$873$	15790.3i	0.612168i
$874$	49646.5	1.92142
$875$	4906.44	0.189563
$876$	10522.2i	0.405836i
$877$	14872.2i	0.572632i	0.958135	+	0.286316i	$0.0924306\pi$
$877$	14872.2i	0.572632i	−0.958135	+	0.286316i	$0.907569\pi$
$878$	− 29535.7i	− 1.13529i
$879$	12877.6i	0.494141i
$880$	5410.04	0.207241
$881$	12940.6	0.494870	0.247435	−	0.968905i	$-0.420412\pi$
$881$	12940.6	0.494870	0.247435	+	0.968905i	$0.420412\pi$
$882$	11070.6i	0.422637i
$883$	25585.5	0.975108	0.487554	−	0.873093i	$-0.337889\pi$
$883$	25585.5	0.975108	0.487554	+	0.873093i	$0.337889\pi$
$884$	0	0
$885$	6007.30	0.228173
$886$	− 41928.8i	− 1.58987i
$887$	3716.46	0.140684	0.0703418	−	0.997523i	$-0.477591\pi$
$887$	3716.46	0.140684	0.0703418	+	0.997523i	$0.477591\pi$
$888$	−4051.79	−0.153118
$889$	16675.7i	0.629118i
$890$	− 13462.8i	− 0.507049i
$891$	− 2612.25i	− 0.0982195i
$892$	− 25322.0i	− 0.950496i
$893$	52416.7	1.96423
$894$	4705.92	0.176051
$895$	− 1555.40i	− 0.0580910i
$896$	−3499.58	−0.130483
$897$	0	0
$898$	−70680.3	−2.62654
$899$	− 6095.52i	− 0.226137i
$900$	−9371.47	−0.347091
$901$	−20072.5	−0.742187
$902$	7457.57i	0.275288i
$903$	2563.93i	0.0944876i
$904$	6379.29i	0.234703i
$905$	− 6517.61i	− 0.239395i
$906$	18769.8	0.688285
$907$	12960.4	0.474469	0.237235	−	0.971452i	$-0.423759\pi$
$907$	12960.4	0.474469	0.237235	+	0.971452i	$0.423759\pi$
$908$	40665.3i	1.48626i
$909$	5767.56	0.210449
$910$	0	0
$911$	−36607.1	−1.33134	−0.665668	−	0.746248i	$-0.731853\pi$
$911$	−36607.1	−1.33134	−0.665668	+	0.746248i	$0.731853\pi$
$912$	− 16697.0i	− 0.606242i
$913$	22158.8	0.803230
$914$	−58099.3	−2.10258
$915$	6417.06i	0.231849i
$916$	− 11747.4i	− 0.423740i
$917$	292.440i	0.0105313i
$918$	− 3212.65i	− 0.115505i
$919$	20356.3	0.730676	0.365338	−	0.930875i	$-0.380953\pi$
$919$	20356.3	0.730676	0.365338	+	0.930875i	$0.380953\pi$
$920$	−1496.38	−0.0536243
$921$	21078.8i	0.754147i
$922$	12048.6	0.430370
$923$	0	0
$924$	5819.43	0.207192
$925$	37898.7i	1.34714i
$926$	−8545.61	−0.303268
$927$	6238.38	0.221030
$928$	− 41589.8i	− 1.47118i
$929$	− 45069.7i	− 1.59170i	−0.605495	−	0.795849i	$-0.707025\pi$
$929$	− 45069.7i	− 1.59170i	0.605495	−	0.795849i	$-0.292975\pi$
$930$	1436.29i	0.0506428i
$931$	− 30189.6i	− 1.06275i
$932$	−30244.9	−1.06299
$933$	−3399.62	−0.119291
$934$	10984.7i	0.384831i
$935$	−2838.65	−0.0992876
$936$	0	0
$937$	−6771.10	−0.236075	−0.118037	−	0.993009i	$-0.537660\pi$
$937$	−6771.10	−0.236075	−0.118037	+	0.993009i	$0.537660\pi$
$938$	1575.50i	0.0548420i
$939$	15857.8	0.551119
$940$	−14218.9	−0.493372
$941$	36690.7i	1.27108i	0.772070	+	0.635538i	$0.219222\pi$
$941$	36690.7i	1.27108i	−0.772070	+	0.635538i	$0.780778\pi$
$942$	− 17937.4i	− 0.620417i
$943$	6673.51i	0.230455i
$944$	36108.8i	1.24496i
$945$	−550.376	−0.0189457
$946$	−17004.0	−0.584406
$947$	50861.2i	1.74527i	0.488375	+	0.872634i	$0.337590\pi$
$947$	50861.2i	1.74527i	−0.488375	+	0.872634i	$0.662410\pi$
$948$	24349.1	0.834202
$949$	0	0
$950$	48272.6	1.64860
$951$	14346.5i	0.489187i
$952$	795.219	0.0270727
$953$	−11855.6	−0.402980	−0.201490	−	0.979491i	$-0.564578\pi$
$953$	−11855.6	−0.402980	−0.201490	+	0.979491i	$0.564578\pi$
$954$	− 25810.3i	− 0.875931i
$955$	− 12376.4i	− 0.419361i
$956$	42634.5i	1.44236i
$957$	15490.8i	0.523245i
$958$	−21525.5	−0.725947
$959$	−1378.04	−0.0464019
$960$	5773.76i	0.194112i
$961$	28341.6	0.951349
$962$	0	0
$963$	−3651.08	−0.122175
$964$	43067.5i	1.43891i
$965$	−2663.90	−0.0888643
$966$	9836.56	0.327625
$967$	40661.7i	1.35221i	0.736803	+	0.676107i	$0.236334\pi$
$967$	40661.7i	1.35221i	−0.736803	+	0.676107i	$0.763666\pi$
$968$	− 1199.58i	− 0.0398304i
$969$	8760.93i	0.290445i
$970$	22063.9i	0.730339i
$971$	57318.3	1.89437	0.947184	−	0.320690i	$-0.103915\pi$
$971$	57318.3	1.89437	0.947184	+	0.320690i	$0.103915\pi$
$972$	2187.00	0.0721688
$973$	− 668.324i	− 0.0220200i
$974$	50403.3	1.65814
$975$	0	0
$976$	−38571.7	−1.26501
$977$	− 3026.73i	− 0.0991134i	−0.998771	−	0.0495567i	$-0.984219\pi$
$977$	− 3026.73i	− 0.0991134i	0.998771	−	0.0495567i	$-0.0157808\pi$
$978$	−28973.4	−0.947308
$979$	−34524.8	−1.12709
$980$	8189.43i	0.266941i
$981$	− 4315.64i	− 0.140457i
$982$	− 81032.3i	− 2.63324i
$983$	− 33942.5i	− 1.10132i	−0.834730	−	0.550659i	$-0.814376\pi$
$983$	− 33942.5i	− 1.10132i	0.834730	−	0.550659i	$-0.185624\pi$
$984$	−693.729	−0.0224749
$985$	12649.3	0.409179
$986$	19051.2i	0.615327i
$987$	10385.4	0.334925
$988$	0	0
$989$	−15216.3	−0.489231
$990$	− 3650.10i	− 0.117179i
$991$	21637.5	0.693580	0.346790	−	0.937943i	$-0.387272\pi$
$991$	21637.5	0.693580	0.346790	+	0.937943i	$0.387272\pi$
$992$	−9889.02	−0.316509
$993$	− 26009.9i	− 0.831219i
$994$	− 8520.95i	− 0.271900i
$995$	− 7332.53i	− 0.233625i
$996$	18551.6i	0.590190i
$997$	19624.6	0.623388	0.311694	−	0.950182i	$-0.399104\pi$
$997$	19624.6	0.623388	0.311694	+	0.950182i	$0.399104\pi$
$998$	48296.3	1.53186
$999$	− 8844.33i	− 0.280102i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.e.337.4		4
13.5	odd	4		507.4.a.k.1.4		4
13.8	odd	4		507.4.a.k.1.1		4
13.9	even	3		39.4.j.b.10.2	yes	4
13.10	even	6		39.4.j.b.4.2	✓	4
13.12	even	2	inner	507.4.b.e.337.1		4
39.5	even	4		1521.4.a.z.1.1		4
39.8	even	4		1521.4.a.z.1.4		4
39.23	odd	6		117.4.q.d.82.1		4
39.35	odd	6		117.4.q.d.10.1		4
52.23	odd	6		624.4.bv.c.433.2		4
52.35	odd	6		624.4.bv.c.49.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.b.4.2	✓	4	13.10	even	6
39.4.j.b.10.2	yes	4	13.9	even	3
117.4.q.d.10.1		4	39.35	odd	6
117.4.q.d.82.1		4	39.23	odd	6
507.4.a.k.1.1		4	13.8	odd	4
507.4.a.k.1.4		4	13.5	odd	4
507.4.b.e.337.1		4	13.12	even	2	inner
507.4.b.e.337.4		4	1.1	even	1	trivial
624.4.bv.c.49.1		4	52.35	odd	6
624.4.bv.c.433.2		4	52.23	odd	6
1521.4.a.z.1.1		4	39.5	even	4
1521.4.a.z.1.4		4	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)

\(f(q)\)	\(=\)	\(q+4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} -3.05006i q^{5} -12.3693i q^{6} +6.68324i q^{7} -4.12311i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} -3.05006i q^{5} -12.3693i q^{6} +6.68324i q^{7} -4.12311i q^{8} +9.00000 q^{9} +12.5757 q^{10} -32.2500i q^{11} +27.0000 q^{12} -27.5557 q^{14} +9.15018i q^{15} -55.0000 q^{16} +28.8586 q^{17} +37.1080i q^{18} -101.194i q^{19} +27.4505i q^{20} -20.0497i q^{21} +132.970 q^{22} +118.990 q^{23} +12.3693i q^{24} +115.697 q^{25} -27.0000 q^{27} -60.1492i q^{28} +160.111 q^{29} -37.7271 q^{30} -38.0705i q^{31} -259.756i q^{32} +96.7499i q^{33} +118.987i q^{34} +20.3843 q^{35} -81.0000 q^{36} +327.568i q^{37} +417.233 q^{38} -12.5757 q^{40} +56.0846i q^{41} +82.6671 q^{42} -127.879 q^{43} +290.250i q^{44} -27.4505i q^{45} +490.608i q^{46} +517.983i q^{47} +165.000 q^{48} +298.334 q^{49} +477.032i q^{50} -86.5757 q^{51} -695.546 q^{53} -111.324i q^{54} -98.3643 q^{55} +27.5557 q^{56} +303.581i q^{57} +660.156i q^{58} -656.523i q^{59} -82.3516i q^{60} +701.304 q^{61} +156.969 q^{62} +60.1492i q^{63} +631.000 q^{64} -398.910 q^{66} -57.1750i q^{67} -259.727 q^{68} -356.970 q^{69} +84.0465i q^{70} +309.226i q^{71} -37.1080i q^{72} +389.711i q^{73} -1350.60 q^{74} -347.091 q^{75} +910.744i q^{76} +215.534 q^{77} +901.820 q^{79} +167.753i q^{80} +81.0000 q^{81} -231.243 q^{82} +687.095i q^{83} +180.448i q^{84} -88.0203i q^{85} -527.257i q^{86} -480.334 q^{87} -132.970 q^{88} -1070.54i q^{89} +113.181 q^{90} -1070.91 q^{92} +114.211i q^{93} -2135.70 q^{94} -308.647 q^{95} +779.267i q^{96} +1754.48i q^{97} +1230.06i q^{98} -290.250i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 - 36 * q^4 + 36 * q^9	\( 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9} + 136 q^{10} + 108 q^{12} + 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} + 120 q^{25} - 108 q^{27} + 12 q^{29} - 408 q^{30} - 804 q^{35} - 324 q^{36} + 612 q^{38} - 136 q^{40} - 612 q^{42} - 940 q^{43} + 660 q^{48} - 692 q^{49} - 432 q^{51} - 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} + 2524 q^{64} + 204 q^{66} - 1296 q^{68} - 828 q^{69} - 3060 q^{74} - 360 q^{75} + 2976 q^{77} + 8 q^{79} + 324 q^{81} - 68 q^{82} - 36 q^{87} + 68 q^{88} + 1224 q^{90} - 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) \) 4 * q - 12 * q^3 - 36 * q^4 + 36 * q^9 + 136 * q^10 + 108 * q^12 + 204 * q^14 - 220 * q^16 + 144 * q^17 - 68 * q^22 + 276 * q^23 + 120 * q^25 - 108 * q^27 + 12 * q^29 - 408 * q^30 - 804 * q^35 - 324 * q^36 + 612 * q^38 - 136 * q^40 - 612 * q^42 - 940 * q^43 + 660 * q^48 - 692 * q^49 - 432 * q^51 - 2268 * q^53 + 892 * q^55 - 204 * q^56 + 320 * q^61 + 2856 * q^62 + 2524 * q^64 + 204 * q^66 - 1296 * q^68 - 828 * q^69 - 3060 * q^74 - 360 * q^75 + 2976 * q^77 + 8 * q^79 + 324 * q^81 - 68 * q^82 - 36 * q^87 + 68 * q^88 + 1224 * q^90 - 2484 * q^92 - 5372 * q^94 + 108 * q^95

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