LMFDB - Embedded newform 507.4.b.g.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:	$6$
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			337.4
Root			$2.79911 - 2.79911i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.g.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.52644i q^{2} +3.00000 q^{3} +5.66998 q^{4} +19.3400i q^{5} +4.57932i q^{6} -4.84136i q^{7} +20.8664i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+1.52644i q^{2} +3.00000 q^{3} +5.66998 q^{4} +19.3400i q^{5} +4.57932i q^{6} -4.84136i q^{7} +20.8664i q^{8} +9.00000 q^{9} -29.5213 q^{10} +61.0728i q^{11} +17.0099 q^{12} +7.39005 q^{14} +58.0199i q^{15} +13.5085 q^{16} +41.7885 q^{17} +13.7380i q^{18} -107.561i q^{19} +109.657i q^{20} -14.5241i q^{21} -93.2239 q^{22} -28.5138 q^{23} +62.5992i q^{24} -249.034 q^{25} +27.0000 q^{27} -27.4504i q^{28} -89.8886 q^{29} -88.5639 q^{30} +183.108i q^{31} +187.551i q^{32} +183.218i q^{33} +63.7876i q^{34} +93.6318 q^{35} +51.0298 q^{36} -418.029i q^{37} +164.185 q^{38} -403.555 q^{40} -142.674i q^{41} +22.1701 q^{42} +71.0935 q^{43} +346.281i q^{44} +174.060i q^{45} -43.5246i q^{46} -323.711i q^{47} +40.5256 q^{48} +319.561 q^{49} -380.136i q^{50} +125.365 q^{51} -25.1047 q^{53} +41.2139i q^{54} -1181.15 q^{55} +101.022 q^{56} -322.683i q^{57} -137.210i q^{58} +684.508i q^{59} +328.972i q^{60} +308.125 q^{61} -279.503 q^{62} -43.5723i q^{63} -178.217 q^{64} -279.672 q^{66} +672.808i q^{67} +236.940 q^{68} -85.5413 q^{69} +142.923i q^{70} -326.837i q^{71} +187.798i q^{72} -24.3058i q^{73} +638.095 q^{74} -747.103 q^{75} -609.869i q^{76} +295.675 q^{77} +166.810 q^{79} +261.255i q^{80} +81.0000 q^{81} +217.783 q^{82} -201.093i q^{83} -82.3513i q^{84} +808.188i q^{85} +108.520i q^{86} -269.666 q^{87} -1274.37 q^{88} -108.834i q^{89} -265.692 q^{90} -161.673 q^{92} +549.323i q^{93} +494.126 q^{94} +2080.23 q^{95} +562.654i q^{96} +1157.95i q^{97} +487.791i q^{98} +549.655i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.52644i	0.539678i	0.962905	+	0.269839i	$0.0869705\pi$
$2$	1.52644i	0.539678i	−0.962905	+	0.269839i	$0.913030\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	5.66998	0.708748
$5$	19.3400i	1.72982i	0.501928	+	0.864909i	$0.332624\pi$
$5$	19.3400i	1.72982i	−0.501928	+	0.864909i	$0.667376\pi$
$6$	4.57932i	0.311583i
$7$	− 4.84136i	− 0.261409i	−0.991421	−	0.130704i	$-0.958276\pi$
$7$	− 4.84136i	− 0.261409i	0.991421	−	0.130704i	$-0.0417239\pi$
$8$	20.8664i	0.922173i
$9$	9.00000	0.333333
$10$	−29.5213	−0.933545
$11$	61.0728i	1.67401i	0.547194	+	0.837006i	$0.315696\pi$
$11$	61.0728i	1.67401i	−0.547194	+	0.837006i	$0.684304\pi$
$12$	17.0099	0.409196
$13$	0	0
$13$	0	0
$14$	7.39005	0.141077
$15$	58.0199i	0.998711i
$16$	13.5085	0.211071
$17$	41.7885	0.596188	0.298094	−	0.954537i	$-0.403649\pi$
$17$	41.7885	0.596188	0.298094	+	0.954537i	$0.403649\pi$
$18$	13.7380i	0.179893i
$19$	− 107.561i	− 1.29875i	−0.760469	−	0.649374i	$-0.775031\pi$
$19$	− 107.561i	− 1.29875i	0.760469	−	0.649374i	$-0.224969\pi$
$20$	109.657i	1.22601i
$21$	− 14.5241i	− 0.150925i
$22$	−93.2239	−0.903427
$23$	−28.5138	−0.258502	−0.129251	−	0.991612i	$-0.541257\pi$
$23$	−28.5138	−0.258502	−0.129251	+	0.991612i	$0.541257\pi$
$24$	62.5992i	0.532417i
$25$	−249.034	−1.99227
$26$	0	0
$27$	27.0000	0.192450
$28$	− 27.4504i	− 0.185273i
$29$	−89.8886	−0.575583	−0.287791	−	0.957693i	$-0.592921\pi$
$29$	−89.8886	−0.575583	−0.287791	+	0.957693i	$0.592921\pi$
$30$	−88.5639	−0.538983
$31$	183.108i	1.06087i	0.847724	+	0.530437i	$0.177972\pi$
$31$	183.108i	1.06087i	−0.847724	+	0.530437i	$0.822028\pi$
$32$	187.551i	1.03608i
$33$	183.218i	0.966491i
$34$	63.7876i	0.321750i
$35$	93.6318	0.452190
$36$	51.0298	0.236249
$37$	− 418.029i	− 1.85739i	−0.370843	−	0.928696i	$-0.620931\pi$
$37$	− 418.029i	− 1.85739i	0.370843	−	0.928696i	$-0.379069\pi$
$38$	164.185	0.700905
$39$	0	0
$40$	−403.555	−1.59519
$41$	− 142.674i	− 0.543460i	−0.962373	−	0.271730i	$-0.912404\pi$
$41$	− 142.674i	− 0.543460i	0.962373	−	0.271730i	$-0.0875958\pi$
$42$	22.1701	0.0814506
$43$	71.0935	0.252132	0.126066	−	0.992022i	$-0.459765\pi$
$43$	71.0935	0.252132	0.126066	+	0.992022i	$0.459765\pi$
$44$	346.281i	1.18645i
$45$	174.060i	0.576606i
$46$	− 43.5246i	− 0.139508i
$47$	− 323.711i	− 1.00464i	−0.864681	−	0.502321i	$-0.832480\pi$
$47$	− 323.711i	− 1.00464i	0.864681	−	0.502321i	$-0.167520\pi$
$48$	40.5256	0.121862
$49$	319.561	0.931665
$50$	− 380.136i	− 1.07519i
$51$	125.365	0.344209
$52$	0	0
$53$	−25.1047	−0.0650641	−0.0325321	−	0.999471i	$-0.510357\pi$
$53$	−25.1047	−0.0650641	−0.0325321	+	0.999471i	$0.510357\pi$
$54$	41.2139i	0.103861i
$55$	−1181.15	−2.89574
$56$	101.022	0.241064
$57$	− 322.683i	− 0.749832i
$58$	− 137.210i	− 0.310629i
$59$	684.508i	1.51043i	0.655477	+	0.755215i	$0.272467\pi$
$59$	684.508i	1.51043i	−0.655477	+	0.755215i	$0.727533\pi$
$60$	328.972i	0.707834i
$61$	308.125	0.646744	0.323372	−	0.946272i	$-0.395184\pi$
$61$	308.125	0.646744	0.323372	+	0.946272i	$0.395184\pi$
$62$	−279.503	−0.572531
$63$	− 43.5723i	− 0.0871363i
$64$	−178.217	−0.348081
$65$	0	0
$66$	−279.672	−0.521594
$67$	672.808i	1.22681i	0.789767	+	0.613407i	$0.210202\pi$
$67$	672.808i	1.22681i	−0.789767	+	0.613407i	$0.789798\pi$
$68$	236.940	0.422547
$69$	−85.5413	−0.149246
$70$	142.923i	0.244037i
$71$	− 326.837i	− 0.546315i	−0.961969	−	0.273158i	$-0.911932\pi$
$71$	− 326.837i	− 0.546315i	0.961969	−	0.273158i	$-0.0880681\pi$
$72$	187.798i	0.307391i
$73$	− 24.3058i	− 0.0389695i	−0.999810	−	0.0194847i	$-0.993797\pi$
$73$	− 24.3058i	− 0.0389695i	0.999810	−	0.0194847i	$-0.00620258\pi$
$74$	638.095	1.00239
$75$	−747.103	−1.15024
$76$	− 609.869i	− 0.920484i
$77$	295.675	0.437602
$78$	0	0
$79$	166.810	0.237565	0.118783	−	0.992920i	$-0.462101\pi$
$79$	166.810	0.237565	0.118783	+	0.992920i	$0.462101\pi$
$80$	261.255i	0.365115i
$81$	81.0000	0.111111
$82$	217.783	0.293294
$83$	− 201.093i	− 0.265938i	−0.991120	−	0.132969i	$-0.957549\pi$
$83$	− 201.093i	− 0.265938i	0.991120	−	0.132969i	$-0.0424511\pi$
$84$	− 82.3513i	− 0.106967i
$85$	808.188i	1.03130i
$86$	108.520i	0.136070i
$87$	−269.666	−0.332313
$88$	−1274.37	−1.54373
$89$	− 108.834i	− 0.129622i	−0.997898	−	0.0648109i	$-0.979356\pi$
$89$	− 108.834i	− 0.129622i	0.997898	−	0.0648109i	$-0.0206444\pi$
$90$	−265.692	−0.311182
$91$	0	0
$92$	−161.673	−0.183212
$93$	549.323i	0.612496i
$94$	494.126	0.542183
$95$	2080.23	2.24660
$96$	562.654i	0.598183i
$97$	1157.95i	1.21208i	0.795434	+	0.606041i	$0.207243\pi$
$97$	1157.95i	1.21208i	−0.795434	+	0.606041i	$0.792757\pi$
$98$	487.791i	0.502799i
$99$	549.655i	0.558004i
$100$	−1412.02	−1.41202
$101$	−1702.75	−1.67752	−0.838761	−	0.544500i	$-0.816719\pi$
$101$	−1702.75	−1.67752	−0.838761	+	0.544500i	$0.816719\pi$
$102$	191.363i	0.185762i
$103$	1455.14	1.39203	0.696015	−	0.718027i	$-0.254955\pi$
$103$	1455.14	1.39203	0.696015	+	0.718027i	$0.254955\pi$
$104$	0	0
$105$	280.895	0.261072
$106$	− 38.3209i	− 0.0351137i
$107$	−822.762	−0.743359	−0.371679	−	0.928361i	$-0.621218\pi$
$107$	−822.762	−0.743359	−0.371679	+	0.928361i	$0.621218\pi$
$108$	153.090	0.136399
$109$	457.264i	0.401816i	0.979610	+	0.200908i	$0.0643893\pi$
$109$	457.264i	0.401816i	−0.979610	+	0.200908i	$0.935611\pi$
$110$	− 1802.95i	− 1.56277i
$111$	− 1254.09i	− 1.07237i
$112$	− 65.3998i	− 0.0551759i
$113$	−381.693	−0.317758	−0.158879	−	0.987298i	$-0.550788\pi$
$113$	−381.693	−0.317758	−0.158879	+	0.987298i	$0.550788\pi$
$114$	492.556	0.404668
$115$	− 551.456i	− 0.447161i
$116$	−509.667	−0.407943
$117$	0	0
$118$	−1044.86	−0.815146
$119$	− 202.313i	− 0.155849i
$120$	−1210.67	−0.920985
$121$	−2398.88	−1.80232
$122$	470.334i	0.349033i
$123$	− 428.021i	− 0.313767i
$124$	1038.22i	0.751892i
$125$	− 2398.82i	− 1.71645i
$126$	66.5104	0.0470255
$127$	1129.09	0.788905	0.394452	−	0.918916i	$-0.370934\pi$
$127$	1129.09	0.788905	0.394452	+	0.918916i	$0.370934\pi$
$128$	1228.37i	0.848232i
$129$	213.281	0.145568
$130$	0	0
$131$	−852.761	−0.568749	−0.284374	−	0.958713i	$-0.591786\pi$
$131$	−852.761	−0.568749	−0.284374	+	0.958713i	$0.591786\pi$
$132$	1038.84i	0.684999i
$133$	−520.742	−0.339504
$134$	−1027.00	−0.662085
$135$	522.179i	0.332904i
$136$	871.975i	0.549789i
$137$	488.903i	0.304889i	0.988312	+	0.152445i	$0.0487146\pi$
$137$	488.903i	0.304889i	−0.988312	+	0.152445i	$0.951285\pi$
$138$	− 130.574i	− 0.0805447i
$139$	407.123	0.248430	0.124215	−	0.992255i	$-0.460359\pi$
$139$	407.123	0.248430	0.124215	+	0.992255i	$0.460359\pi$
$140$	530.890	0.320489
$141$	− 971.134i	− 0.580030i
$142$	498.897	0.294834
$143$	0	0
$144$	121.577	0.0703570
$145$	− 1738.44i	− 0.995654i
$146$	37.1013	0.0210310
$147$	958.684	0.537897
$148$	− 2370.21i	− 1.31642i
$149$	1717.63i	0.944388i	0.881495	+	0.472194i	$0.156538\pi$
$149$	1717.63i	0.944388i	−0.881495	+	0.472194i	$0.843462\pi$
$150$	− 1140.41i	− 0.620759i
$151$	− 1341.79i	− 0.723133i	−0.932346	−	0.361567i	$-0.882242\pi$
$151$	− 1341.79i	− 0.723133i	0.932346	−	0.361567i	$-0.117758\pi$
$152$	2244.41	1.19767
$153$	376.096	0.198729
$154$	451.331i	0.236164i
$155$	−3541.30	−1.83512
$156$	0	0
$157$	−760.546	−0.386612	−0.193306	−	0.981138i	$-0.561921\pi$
$157$	−760.546	−0.386612	−0.193306	+	0.981138i	$0.561921\pi$
$158$	254.626i	0.128209i
$159$	−75.3142	−0.0375648
$160$	−3627.23	−1.79224
$161$	138.046i	0.0675746i
$162$	123.642i	0.0599642i
$163$	− 2712.09i	− 1.30323i	−0.758549	−	0.651616i	$-0.774091\pi$
$163$	− 2712.09i	− 1.30323i	0.758549	−	0.651616i	$-0.225909\pi$
$164$	− 808.957i	− 0.385176i
$165$	−3543.44	−1.67185
$166$	306.957	0.143521
$167$	− 1551.69i	− 0.719004i	−0.933144	−	0.359502i	$-0.882947\pi$
$167$	− 1551.69i	− 0.719004i	0.933144	−	0.359502i	$-0.117053\pi$
$168$	303.065	0.139179
$169$	0	0
$170$	−1233.65	−0.556568
$171$	− 968.050i	− 0.432916i
$172$	403.099	0.178698
$173$	3970.26	1.74482	0.872409	−	0.488777i	$-0.162557\pi$
$173$	3970.26	1.74482	0.872409	+	0.488777i	$0.162557\pi$
$174$	− 411.629i	− 0.179342i
$175$	1205.66i	0.520798i
$176$	825.004i	0.353335i
$177$	2053.52i	0.872047i
$178$	166.128	0.0699540
$179$	2690.95	1.12364	0.561818	−	0.827261i	$-0.310102\pi$
$179$	2690.95	1.12364	0.561818	+	0.827261i	$0.310102\pi$
$180$	986.915i	0.408668i
$181$	4371.10	1.79503	0.897517	−	0.440980i	$-0.145369\pi$
$181$	4371.10	1.79503	0.897517	+	0.440980i	$0.145369\pi$
$182$	0	0
$183$	924.375	0.373398
$184$	− 594.980i	− 0.238383i
$185$	8084.66	3.21295
$186$	−838.508	−0.330551
$187$	2552.14i	0.998026i
$188$	− 1835.44i	− 0.712038i
$189$	− 130.717i	− 0.0503082i
$190$	3175.34i	1.21244i
$191$	1408.47	0.533578	0.266789	−	0.963755i	$-0.414037\pi$
$191$	1408.47	0.533578	0.266789	+	0.963755i	$0.414037\pi$
$192$	−534.652	−0.200964
$193$	4131.69i	1.54096i	0.637463	+	0.770481i	$0.279984\pi$
$193$	4131.69i	1.54096i	−0.637463	+	0.770481i	$0.720016\pi$
$194$	−1767.54	−0.654134
$195$	0	0
$196$	1811.91	0.660316
$197$	− 3401.23i	− 1.23009i	−0.788492	−	0.615045i	$-0.789138\pi$
$197$	− 3401.23i	− 1.23009i	0.788492	−	0.615045i	$-0.210862\pi$
$198$	−839.015	−0.301142
$199$	3520.74	1.25416	0.627081	−	0.778954i	$-0.284249\pi$
$199$	3520.74	1.25416	0.627081	+	0.778954i	$0.284249\pi$
$200$	− 5196.45i	− 1.83722i
$201$	2018.42i	0.708302i
$202$	− 2599.14i	− 0.905321i
$203$	435.183i	0.150463i
$204$	710.820	0.243958
$205$	2759.30	0.940088
$206$	2221.18i	0.751248i
$207$	−256.624	−0.0861672
$208$	0	0
$209$	6569.05	2.17412
$210$	428.770i	0.140895i
$211$	−2245.22	−0.732545	−0.366272	−	0.930508i	$-0.619366\pi$
$211$	−2245.22	−0.732545	−0.366272	+	0.930508i	$0.619366\pi$
$212$	−142.343	−0.0461141
$213$	− 980.510i	− 0.315415i
$214$	− 1255.90i	− 0.401174i
$215$	1374.95i	0.436142i
$216$	563.393i	0.177472i
$217$	886.490	0.277322
$218$	−697.986	−0.216851
$219$	− 72.9173i	− 0.0224990i
$220$	−6697.07	−2.05235
$221$	0	0
$222$	1914.29	0.578732
$223$	3431.26i	1.03038i	0.857077	+	0.515188i	$0.172278\pi$
$223$	3431.26i	1.03038i	−0.857077	+	0.515188i	$0.827722\pi$
$224$	908.003	0.270842
$225$	−2241.31	−0.664091
$226$	− 582.631i	− 0.171487i
$227$	− 4757.91i	− 1.39116i	−0.718448	−	0.695581i	$-0.755147\pi$
$227$	− 4757.91i	− 1.39116i	0.718448	−	0.695581i	$-0.244853\pi$
$228$	− 1829.61i	− 0.531442i
$229$	4368.93i	1.26073i	0.776300	+	0.630364i	$0.217094\pi$
$229$	4368.93i	1.26073i	−0.776300	+	0.630364i	$0.782906\pi$
$230$	841.764	0.241323
$231$	887.026	0.252649
$232$	− 1875.65i	− 0.530787i
$233$	3642.00	1.02401	0.512007	−	0.858981i	$-0.328902\pi$
$233$	3642.00	1.02401	0.512007	+	0.858981i	$0.328902\pi$
$234$	0	0
$235$	6260.57	1.73785
$236$	3881.15i	1.07051i
$237$	500.431	0.137158
$238$	308.819	0.0841082
$239$	2236.17i	0.605213i	0.953116	+	0.302606i	$0.0978568\pi$
$239$	2236.17i	0.605213i	−0.953116	+	0.302606i	$0.902143\pi$
$240$	783.764i	0.210799i
$241$	− 6538.78i	− 1.74772i	−0.486181	−	0.873858i	$-0.661610\pi$
$241$	− 6538.78i	− 1.74772i	0.486181	−	0.873858i	$-0.338390\pi$
$242$	− 3661.75i	− 0.972670i
$243$	243.000	0.0641500
$244$	1747.06	0.458378
$245$	6180.30i	1.61161i
$246$	653.348	0.169333
$247$	0	0
$248$	−3820.80	−0.978310
$249$	− 603.280i	− 0.153539i
$250$	3661.65	0.926332
$251$	−2507.12	−0.630470	−0.315235	−	0.949014i	$-0.602083\pi$
$251$	−2507.12	−0.630470	−0.315235	+	0.949014i	$0.602083\pi$
$252$	− 247.054i	− 0.0617577i
$253$	− 1741.42i	− 0.432735i
$254$	1723.49i	0.425755i
$255$	2424.56i	0.595420i
$256$	−3300.77	−0.805853
$257$	808.131	0.196147	0.0980735	−	0.995179i	$-0.468732\pi$
$257$	808.131	0.196147	0.0980735	+	0.995179i	$0.468732\pi$
$258$	325.560i	0.0785600i
$259$	−2023.83	−0.485539
$260$	0	0
$261$	−808.998	−0.191861
$262$	− 1301.69i	− 0.306941i
$263$	2940.70	0.689472	0.344736	−	0.938700i	$-0.387968\pi$
$263$	2940.70	0.689472	0.344736	+	0.938700i	$0.387968\pi$
$264$	−3823.11	−0.891273
$265$	− 485.525i	− 0.112549i
$266$	− 794.881i	− 0.183223i
$267$	− 326.501i	− 0.0748371i
$268$	3814.81i	0.869502i
$269$	7111.50	1.61188	0.805940	−	0.591997i	$-0.201660\pi$
$269$	7111.50	1.61188	0.805940	+	0.591997i	$0.201660\pi$
$270$	−797.075	−0.179661
$271$	− 2034.96i	− 0.456145i	−0.973644	−	0.228072i	$-0.926758\pi$
$271$	− 2034.96i	− 0.456145i	0.973644	−	0.228072i	$-0.0732423\pi$
$272$	564.502	0.125838
$273$	0	0
$274$	−746.281	−0.164542
$275$	− 15209.2i	− 3.33509i
$276$	−485.018	−0.105778
$277$	−2723.20	−0.590689	−0.295345	−	0.955391i	$-0.595434\pi$
$277$	−2723.20	−0.590689	−0.295345	+	0.955391i	$0.595434\pi$
$278$	621.449i	0.134072i
$279$	1647.97i	0.353625i
$280$	1953.76i	0.416998i
$281$	− 3265.56i	− 0.693263i	−0.938001	−	0.346632i	$-0.887325\pi$
$281$	− 3265.56i	− 0.693263i	0.938001	−	0.346632i	$-0.112675\pi$
$282$	1482.38	0.313030
$283$	−1144.02	−0.240299	−0.120150	−	0.992756i	$-0.538337\pi$
$283$	−1144.02	−0.240299	−0.120150	+	0.992756i	$0.538337\pi$
$284$	− 1853.16i	− 0.387200i
$285$	6240.68	1.29707
$286$	0	0
$287$	−690.735	−0.142065
$288$	1687.96i	0.345361i
$289$	−3166.72	−0.644560
$290$	2653.63	0.537333
$291$	3473.85i	0.699796i
$292$	− 137.813i	− 0.0276195i
$293$	1677.35i	0.334444i	0.985919	+	0.167222i	$0.0534796\pi$
$293$	1677.35i	0.334444i	−0.985919	+	0.167222i	$0.946520\pi$
$294$	1463.37i	0.290291i
$295$	−13238.4	−2.61277
$296$	8722.75	1.71284
$297$	1648.96i	0.322164i
$298$	−2621.86	−0.509666
$299$	0	0
$300$	−4236.06	−0.815230
$301$	− 344.190i	− 0.0659095i
$302$	2048.16	0.390259
$303$	−5108.24	−0.968518
$304$	− 1452.99i	− 0.274128i
$305$	5959.13i	1.11875i
$306$	574.088i	0.107250i
$307$	− 7207.70i	− 1.33995i	−0.742383	−	0.669975i	$-0.766305\pi$
$307$	− 7207.70i	− 1.33995i	0.742383	−	0.669975i	$-0.233695\pi$
$308$	1676.47	0.310149
$309$	4365.42	0.803689
$310$	− 5405.57i	− 0.990374i
$311$	−412.963	−0.0752958	−0.0376479	−	0.999291i	$-0.511987\pi$
$311$	−412.963	−0.0752958	−0.0376479	+	0.999291i	$0.511987\pi$
$312$	0	0
$313$	2936.39	0.530270	0.265135	−	0.964211i	$-0.414583\pi$
$313$	2936.39	0.530270	0.265135	+	0.964211i	$0.414583\pi$
$314$	− 1160.93i	− 0.208646i
$315$	842.686	0.150730
$316$	945.812	0.168374
$317$	377.956i	0.0669657i	0.999439	+	0.0334828i	$0.0106599\pi$
$317$	377.956i	0.0669657i	−0.999439	+	0.0334828i	$0.989340\pi$
$318$	− 114.963i	− 0.0202729i
$319$	− 5489.75i	− 0.963533i
$320$	− 3446.71i	− 0.602116i
$321$	−2468.28	−0.429178
$322$	−210.718	−0.0364685
$323$	− 4494.81i	− 0.774298i
$324$	459.269	0.0787497
$325$	0	0
$326$	4139.83	0.703326
$327$	1371.79i	0.231989i
$328$	2977.09	0.501165
$329$	−1567.20	−0.262622
$330$	− 5408.84i	− 0.902263i
$331$	− 4428.17i	− 0.735330i	−0.929958	−	0.367665i	$-0.880157\pi$
$331$	− 4428.17i	− 0.735330i	0.929958	−	0.367665i	$-0.119843\pi$
$332$	− 1140.20i	− 0.188483i
$333$	− 3762.26i	− 0.619130i
$334$	2368.57	0.388031
$335$	−13012.1	−2.12217
$336$	− 196.199i	− 0.0318558i
$337$	1768.76	0.285907	0.142953	−	0.989729i	$-0.454340\pi$
$337$	1768.76	0.285907	0.142953	+	0.989729i	$0.454340\pi$
$338$	0	0
$339$	−1145.08	−0.183458
$340$	4582.41i	0.730930i
$341$	−11182.9	−1.77592
$342$	1477.67	0.233635
$343$	− 3207.70i	− 0.504955i
$344$	1483.47i	0.232509i
$345$	− 1654.37i	− 0.258168i
$346$	6060.36i	0.941639i
$347$	2412.97	0.373300	0.186650	−	0.982426i	$-0.440237\pi$
$347$	2412.97	0.373300	0.186650	+	0.982426i	$0.440237\pi$
$348$	−1529.00	−0.235526
$349$	9967.45i	1.52878i	0.644752	+	0.764392i	$0.276961\pi$
$349$	9967.45i	1.52878i	−0.644752	+	0.764392i	$0.723039\pi$
$350$	−1840.37	−0.281063
$351$	0	0
$352$	−11454.3	−1.73442
$353$	4516.30i	0.680959i	0.940252	+	0.340479i	$0.110589\pi$
$353$	4516.30i	0.680959i	−0.940252	+	0.340479i	$0.889411\pi$
$354$	−3134.58	−0.470625
$355$	6321.01	0.945027
$356$	− 617.084i	− 0.0918691i
$357$	− 606.939i	− 0.0899794i
$358$	4107.57i	0.606401i
$359$	− 12159.8i	− 1.78767i	−0.448400	−	0.893833i	$-0.648006\pi$
$359$	− 12159.8i	− 1.78767i	0.448400	−	0.893833i	$-0.351994\pi$
$360$	−3632.00	−0.531731
$361$	−4710.38	−0.686745
$362$	6672.22i	0.968740i
$363$	−7196.65	−1.04057
$364$	0	0
$365$	470.072	0.0674102
$366$	1411.00i	0.201514i
$367$	−2674.25	−0.380367	−0.190183	−	0.981749i	$-0.560908\pi$
$367$	−2674.25	−0.380367	−0.190183	+	0.981749i	$0.560908\pi$
$368$	−385.180	−0.0545622
$369$	− 1284.06i	− 0.181153i
$370$	12340.7i	1.73396i
$371$	121.541i	0.0170083i
$372$	3114.65i	0.434105i
$373$	9601.74	1.33287	0.666433	−	0.745564i	$-0.267820\pi$
$373$	9601.74	1.33287	0.666433	+	0.745564i	$0.267820\pi$
$374$	−3895.69	−0.538613
$375$	− 7196.45i	− 0.990995i
$376$	6754.69	0.926454
$377$	0	0
$378$	199.531	0.0271502
$379$	9019.65i	1.22245i	0.791457	+	0.611225i	$0.209323\pi$
$379$	9019.65i	1.22245i	−0.791457	+	0.611225i	$0.790677\pi$
$380$	11794.9	1.59227
$381$	3387.28	0.455474
$382$	2149.95i	0.287960i
$383$	− 4015.34i	− 0.535703i	−0.963460	−	0.267852i	$-0.913686\pi$
$383$	− 4015.34i	− 0.535703i	0.963460	−	0.267852i	$-0.0863137\pi$
$384$	3685.12i	0.489727i
$385$	5718.35i	0.756972i
$386$	−6306.78	−0.831623
$387$	639.842	0.0840439
$388$	6565.55i	0.859060i
$389$	2725.35	0.355221	0.177610	−	0.984101i	$-0.443163\pi$
$389$	2725.35	0.355221	0.177610	+	0.984101i	$0.443163\pi$
$390$	0	0
$391$	−1191.55	−0.154115
$392$	6668.09i	0.859157i
$393$	−2558.28	−0.328367
$394$	5191.78	0.663853
$395$	3226.11i	0.410945i
$396$	3116.53i	0.395484i
$397$	4391.59i	0.555182i	0.960699	+	0.277591i	$0.0895361\pi$
$397$	4391.59i	0.555182i	−0.960699	+	0.277591i	$0.910464\pi$
$398$	5374.19i	0.676844i
$399$	−1562.23	−0.196013
$400$	−3364.09	−0.420511
$401$	− 3762.48i	− 0.468552i	−0.972170	−	0.234276i	$-0.924728\pi$
$401$	− 3762.48i	− 0.468552i	0.972170	−	0.234276i	$-0.0752719\pi$
$402$	−3081.00	−0.382255
$403$	0	0
$404$	−9654.55	−1.18894
$405$	1566.54i	0.192202i
$406$	−664.281	−0.0812013
$407$	25530.2	3.10930
$408$	2615.93i	0.317421i
$409$	− 6797.81i	− 0.821833i	−0.911673	−	0.410917i	$-0.865209\pi$
$409$	− 6797.81i	− 0.821833i	0.911673	−	0.410917i	$-0.134791\pi$
$410$	4211.91i	0.507345i
$411$	1466.71i	0.176028i
$412$	8250.61	0.986599
$413$	3313.95	0.394840
$414$	− 391.721i	− 0.0465025i
$415$	3889.14	0.460025
$416$	0	0
$417$	1221.37	0.143431
$418$	10027.3i	1.17332i
$419$	−12594.2	−1.46841	−0.734207	−	0.678925i	$-0.762446\pi$
$419$	−12594.2	−1.46841	−0.734207	+	0.678925i	$0.762446\pi$
$420$	1592.67	0.185034
$421$	6888.04i	0.797393i	0.917083	+	0.398697i	$0.130537\pi$
$421$	6888.04i	0.797393i	−0.917083	+	0.398697i	$0.869463\pi$
$422$	− 3427.19i	− 0.395338i
$423$	− 2913.40i	− 0.334881i
$424$	− 523.845i	− 0.0600004i
$425$	−10406.8	−1.18777
$426$	1496.69	0.170223
$427$	− 1491.74i	− 0.169065i
$428$	−4665.04	−0.526854
$429$	0	0
$430$	−2098.77	−0.235376
$431$	− 7384.53i	− 0.825291i	−0.910892	−	0.412645i	$-0.864605\pi$
$431$	− 7384.53i	− 0.825291i	0.910892	−	0.412645i	$-0.135395\pi$
$432$	364.731	0.0406206
$433$	−9068.33	−1.00646	−0.503229	−	0.864153i	$-0.667855\pi$
$433$	−9068.33	−1.00646	−0.503229	+	0.864153i	$0.667855\pi$
$434$	1353.17i	0.149665i
$435$	− 5215.33i	− 0.574841i
$436$	2592.68i	0.284786i
$437$	3066.97i	0.335728i
$438$	111.304	0.0121422
$439$	−16875.4	−1.83466	−0.917331	−	0.398125i	$-0.869661\pi$
$439$	−16875.4	−1.83466	−0.917331	+	0.398125i	$0.869661\pi$
$440$	− 24646.3i	− 2.67037i
$441$	2876.05	0.310555
$442$	0	0
$443$	−6766.18	−0.725668	−0.362834	−	0.931854i	$-0.618191\pi$
$443$	−6766.18	−0.725668	−0.362834	+	0.931854i	$0.618191\pi$
$444$	− 7110.64i	− 0.760037i
$445$	2104.84	0.224222
$446$	−5237.61	−0.556072
$447$	5152.90i	0.545243i
$448$	862.814i	0.0909914i
$449$	− 140.944i	− 0.0148141i	−0.999973	−	0.00740706i	$-0.997642\pi$
$449$	− 140.944i	− 0.0148141i	0.999973	−	0.00740706i	$-0.00235776\pi$
$450$	− 3421.22i	− 0.358395i
$451$	8713.47	0.909759
$452$	−2164.19	−0.225210
$453$	− 4025.36i	− 0.417501i
$454$	7262.67	0.750779
$455$	0	0
$456$	6733.24	0.691475
$457$	− 17733.1i	− 1.81514i	−0.419898	−	0.907571i	$-0.637934\pi$
$457$	− 17733.1i	− 1.81514i	0.419898	−	0.907571i	$-0.362066\pi$
$458$	−6668.90	−0.680387
$459$	1128.29	0.114736
$460$	− 3126.74i	− 0.316924i
$461$	2293.37i	0.231699i	0.993267	+	0.115849i	$0.0369590\pi$
$461$	2293.37i	0.231699i	−0.993267	+	0.115849i	$0.963041\pi$
$462$	1353.99i	0.136349i
$463$	− 13770.9i	− 1.38226i	−0.722731	−	0.691129i	$-0.757113\pi$
$463$	− 13770.9i	− 1.38226i	0.722731	−	0.691129i	$-0.242887\pi$
$464$	−1214.27	−0.121489
$465$	−10623.9	−1.05951
$466$	5559.29i	0.552638i
$467$	3477.37	0.344568	0.172284	−	0.985047i	$-0.444885\pi$
$467$	3477.37	0.344568	0.172284	+	0.985047i	$0.444885\pi$
$468$	0	0
$469$	3257.31	0.320700
$470$	9556.38i	0.937879i
$471$	−2281.64	−0.223211
$472$	−14283.2	−1.39288
$473$	4341.88i	0.422072i
$474$	763.878i	0.0740213i
$475$	26786.4i	2.58746i
$476$	− 1147.11i	− 0.110458i
$477$	−225.943	−0.0216880
$478$	−3413.38	−0.326620
$479$	3137.39i	0.299271i	0.988741	+	0.149636i	$0.0478101\pi$
$479$	3137.39i	0.299271i	−0.988741	+	0.149636i	$0.952190\pi$
$480$	−10881.7	−1.03475
$481$	0	0
$482$	9981.05	0.943204
$483$	414.137i	0.0390142i
$484$	−13601.6	−1.27739
$485$	−22394.7	−2.09668
$486$	370.925i	0.0346204i
$487$	5996.52i	0.557964i	0.960296	+	0.278982i	$0.0899970\pi$
$487$	5996.52i	0.557964i	−0.960296	+	0.278982i	$0.910003\pi$
$488$	6429.46i	0.596410i
$489$	− 8136.26i	− 0.752422i
$490$	−9433.86	−0.869752
$491$	9401.49	0.864121	0.432060	−	0.901845i	$-0.357787\pi$
$491$	9401.49	0.864121	0.432060	+	0.901845i	$0.357787\pi$
$492$	− 2426.87i	− 0.222382i
$493$	−3756.31	−0.343156
$494$	0	0
$495$	−10630.3	−0.965246
$496$	2473.52i	0.223920i
$497$	−1582.34	−0.142812
$498$	920.871	0.0828619
$499$	− 5052.33i	− 0.453253i	−0.973982	−	0.226626i	$-0.927230\pi$
$499$	− 5052.33i	− 0.453253i	0.973982	−	0.226626i	$-0.0727697\pi$
$500$	− 13601.2i	− 1.21653i
$501$	− 4655.08i	− 0.415117i
$502$	− 3826.96i	− 0.340251i
$503$	8184.02	0.725462	0.362731	−	0.931894i	$-0.381844\pi$
$503$	8184.02	0.725462	0.362731	+	0.931894i	$0.381844\pi$
$504$	909.196	0.0803548
$505$	− 32931.1i	− 2.90181i
$506$	2658.17	0.233537
$507$	0	0
$508$	6401.94	0.559134
$509$	6039.12i	0.525892i	0.964811	+	0.262946i	$0.0846941\pi$
$509$	6039.12i	0.525892i	−0.964811	+	0.262946i	$0.915306\pi$
$510$	−3700.95	−0.321335
$511$	−117.673	−0.0101870
$512$	4788.54i	0.413331i
$513$	− 2904.15i	− 0.249944i
$514$	1233.56i	0.105856i
$515$	28142.3i	2.40796i
$516$	1209.30	0.103171
$517$	19770.0	1.68178
$518$	− 3089.25i	− 0.262035i
$519$	11910.8	1.00737
$520$	0	0
$521$	−14602.5	−1.22792	−0.613960	−	0.789337i	$-0.710424\pi$
$521$	−14602.5	−1.22792	−0.613960	+	0.789337i	$0.710424\pi$
$522$	− 1234.89i	− 0.103543i
$523$	8910.70	0.745005	0.372502	−	0.928031i	$-0.378500\pi$
$523$	8910.70	0.745005	0.372502	+	0.928031i	$0.378500\pi$
$524$	−4835.14	−0.403099
$525$	3616.99i	0.300683i
$526$	4488.80i	0.372093i
$527$	7651.79i	0.632481i
$528$	2475.01i	0.203998i
$529$	−11354.0	−0.933177
$530$	741.124	0.0607403
$531$	6160.57i	0.503477i
$532$	−2952.60	−0.240623
$533$	0	0
$534$	498.384	0.0403879
$535$	− 15912.2i	− 1.28588i
$536$	−14039.1	−1.13134
$537$	8072.84	0.648731
$538$	10855.3i	0.869897i
$539$	19516.5i	1.55962i
$540$	2960.75i	0.235945i
$541$	13313.6i	1.05803i	0.848611	+	0.529017i	$0.177439\pi$
$541$	13313.6i	1.05803i	−0.848611	+	0.529017i	$0.822561\pi$
$542$	3106.25	0.246171
$543$	13113.3	1.03636
$544$	7837.48i	0.617701i
$545$	−8843.47	−0.695069
$546$	0	0
$547$	−4116.94	−0.321806	−0.160903	−	0.986970i	$-0.551441\pi$
$547$	−4116.94	−0.321806	−0.160903	+	0.986970i	$0.551441\pi$
$548$	2772.07i	0.216090i
$549$	2773.12	0.215581
$550$	23215.9	1.79987
$551$	9668.52i	0.747537i
$552$	− 1784.94i	− 0.137631i
$553$	− 807.590i	− 0.0621016i
$554$	− 4156.79i	− 0.318782i
$555$	24254.0	1.85500
$556$	2308.38	0.176074
$557$	6888.37i	0.524003i	0.965067	+	0.262002i	$0.0843826\pi$
$557$	6888.37i	0.524003i	−0.965067	+	0.262002i	$0.915617\pi$
$558$	−2515.53	−0.190844
$559$	0	0
$560$	1264.83	0.0954443
$561$	7656.42i	0.576211i
$562$	4984.68	0.374139
$563$	−10537.1	−0.788782	−0.394391	−	0.918943i	$-0.629045\pi$
$563$	−10537.1	−0.788782	−0.394391	+	0.918943i	$0.629045\pi$
$564$	− 5506.31i	− 0.411095i
$565$	− 7381.92i	− 0.549664i
$566$	− 1746.27i	− 0.129684i
$567$	− 392.150i	− 0.0290454i
$568$	6819.91	0.503798
$569$	−26930.1	−1.98413	−0.992065	−	0.125722i	$-0.959875\pi$
$569$	−26930.1	−1.98413	−0.992065	+	0.125722i	$0.959875\pi$
$570$	9526.02i	0.700002i
$571$	3125.60	0.229076	0.114538	−	0.993419i	$-0.463461\pi$
$571$	3125.60	0.229076	0.114538	+	0.993419i	$0.463461\pi$
$572$	0	0
$573$	4225.42	0.308062
$574$	− 1054.36i	− 0.0766696i
$575$	7100.91	0.515006
$576$	−1603.96	−0.116027
$577$	4787.13i	0.345391i	0.984975	+	0.172696i	$0.0552477\pi$
$577$	4787.13i	0.345391i	−0.984975	+	0.172696i	$0.944752\pi$
$578$	− 4833.81i	− 0.347855i
$579$	12395.1i	0.889675i
$580$	− 9856.94i	− 0.705668i
$581$	−973.566	−0.0695186
$582$	−5302.62	−0.377664
$583$	− 1533.22i	− 0.108918i
$584$	507.174	0.0359366
$585$	0	0
$586$	−2560.38	−0.180492
$587$	18380.8i	1.29243i	0.763156	+	0.646214i	$0.223649\pi$
$587$	18380.8i	1.29243i	−0.763156	+	0.646214i	$0.776351\pi$
$588$	5435.72	0.381233
$589$	19695.3	1.37781
$590$	− 20207.6i	− 1.41005i
$591$	− 10203.7i	− 0.710193i
$592$	− 5646.96i	− 0.392042i
$593$	13831.7i	0.957843i	0.877858	+	0.478922i	$0.158972\pi$
$593$	13831.7i	0.957843i	−0.877858	+	0.478922i	$0.841028\pi$
$594$	−2517.05	−0.173865
$595$	3912.73	0.269590
$596$	9738.94i	0.669333i
$597$	10562.2	0.724091
$598$	0	0
$599$	12248.6	0.835502	0.417751	−	0.908562i	$-0.362818\pi$
$599$	12248.6	0.835502	0.417751	+	0.908562i	$0.362818\pi$
$600$	− 15589.3i	− 1.06072i
$601$	9719.56	0.659682	0.329841	−	0.944036i	$-0.393005\pi$
$601$	9719.56	0.659682	0.329841	+	0.944036i	$0.393005\pi$
$602$	525.385	0.0355699
$603$	6055.27i	0.408938i
$604$	− 7607.91i	− 0.512519i
$605$	− 46394.3i	− 3.11768i
$606$	− 7797.42i	− 0.522688i
$607$	1607.83	0.107512	0.0537560	−	0.998554i	$-0.482881\pi$
$607$	1607.83	0.107512	0.0537560	+	0.998554i	$0.482881\pi$
$608$	20173.2	1.34561
$609$	1305.55i	0.0868696i
$610$	−9096.25	−0.603764
$611$	0	0
$612$	2132.46	0.140849
$613$	14731.1i	0.970610i	0.874345	+	0.485305i	$0.161291\pi$
$613$	14731.1i	0.970610i	−0.874345	+	0.485305i	$0.838709\pi$
$614$	11002.1	0.723142
$615$	8277.91	0.542760
$616$	6169.68i	0.403545i
$617$	27951.8i	1.82382i	0.410392	+	0.911909i	$0.365392\pi$
$617$	27951.8i	1.82382i	−0.410392	+	0.911909i	$0.634608\pi$
$618$	6663.55i	0.433733i
$619$	− 16200.2i	− 1.05192i	−0.850509	−	0.525961i	$-0.823706\pi$
$619$	− 16200.2i	− 1.05192i	0.850509	−	0.525961i	$-0.176294\pi$
$620$	−20079.1	−1.30064
$621$	−769.872	−0.0497486
$622$	− 630.364i	− 0.0406355i
$623$	−526.903	−0.0338843
$624$	0	0
$625$	15263.8	0.976880
$626$	4482.22i	0.286175i
$627$	19707.2	1.25523
$628$	−4312.28	−0.274011
$629$	− 17468.8i	− 1.10735i
$630$	1286.31i	0.0813457i
$631$	− 12731.8i	− 0.803239i	−0.915807	−	0.401619i	$-0.868447\pi$
$631$	− 12731.8i	− 0.803239i	0.915807	−	0.401619i	$-0.131553\pi$
$632$	3480.73i	0.219076i
$633$	−6735.65	−0.422935
$634$	−576.927	−0.0361399
$635$	21836.6i	1.36466i
$636$	−427.030	−0.0266240
$637$	0	0
$638$	8379.77	0.519997
$639$	− 2941.53i	− 0.182105i
$640$	−23756.7	−1.46729
$641$	11556.9	0.712119	0.356059	−	0.934463i	$-0.384120\pi$
$641$	11556.9	0.712119	0.356059	+	0.934463i	$0.384120\pi$
$642$	− 3767.69i	− 0.231618i
$643$	− 9181.25i	− 0.563100i	−0.959547	−	0.281550i	$-0.909152\pi$
$643$	− 9181.25i	− 0.563100i	0.959547	−	0.281550i	$-0.0908485\pi$
$644$	782.716i	0.0478934i
$645$	4124.84i	0.251807i
$646$	6861.06	0.417871
$647$	−5244.11	−0.318651	−0.159326	−	0.987226i	$-0.550932\pi$
$647$	−5244.11	−0.318651	−0.159326	+	0.987226i	$0.550932\pi$
$648$	1690.18i	0.102464i
$649$	−41804.8	−2.52848
$650$	0	0
$651$	2659.47	0.160112
$652$	− 15377.5i	− 0.923663i
$653$	−16421.4	−0.984106	−0.492053	−	0.870565i	$-0.663753\pi$
$653$	−16421.4	−0.984106	−0.492053	+	0.870565i	$0.663753\pi$
$654$	−2093.96	−0.125199
$655$	− 16492.4i	− 0.983832i
$656$	− 1927.31i	− 0.114709i
$657$	− 218.752i	− 0.0129898i
$658$	− 2392.24i	− 0.141732i
$659$	1838.11	0.108653	0.0543266	−	0.998523i	$-0.482699\pi$
$659$	1838.11	0.108653	0.0543266	+	0.998523i	$0.482699\pi$
$660$	−20091.2	−1.18492
$661$	5500.93i	0.323694i	0.986816	+	0.161847i	$0.0517450\pi$
$661$	5500.93i	0.323694i	−0.986816	+	0.161847i	$0.948255\pi$
$662$	6759.34	0.396841
$663$	0	0
$664$	4196.10	0.245241
$665$	− 10071.1i	− 0.587281i
$666$	5742.86	0.334131
$667$	2563.07	0.148789
$668$	− 8798.08i	− 0.509593i
$669$	10293.8i	0.594888i
$670$	− 19862.2i	− 1.14529i
$671$	18818.0i	1.08266i
$672$	2724.01	0.156370
$673$	25986.7	1.48843	0.744216	−	0.667939i	$-0.232823\pi$
$673$	25986.7	1.48843	0.744216	+	0.667939i	$0.232823\pi$
$674$	2699.91i	0.154298i
$675$	−6723.92	−0.383413
$676$	0	0
$677$	−11691.3	−0.663714	−0.331857	−	0.943330i	$-0.607675\pi$
$677$	−11691.3	−0.663714	−0.331857	+	0.943330i	$0.607675\pi$
$678$	− 1747.89i	− 0.0990080i
$679$	5606.05	0.316849
$680$	−16864.0	−0.951035
$681$	− 14273.7i	− 0.803188i
$682$	− 17070.0i	− 0.958423i
$683$	11111.5i	0.622501i	0.950328	+	0.311251i	$0.100748\pi$
$683$	11111.5i	0.622501i	−0.950328	+	0.311251i	$0.899252\pi$
$684$	− 5488.82i	− 0.306828i
$685$	−9455.37	−0.527403
$686$	4896.36	0.272513
$687$	13106.8i	0.727882i
$688$	960.370	0.0532177
$689$	0	0
$690$	2525.29	0.139328
$691$	− 7542.55i	− 0.415242i	−0.978209	−	0.207621i	$-0.933428\pi$
$691$	− 7542.55i	− 0.415242i	0.978209	−	0.207621i	$-0.0665721\pi$
$692$	22511.3	1.23664
$693$	2661.08	0.145867
$694$	3683.26i	0.201462i
$695$	7873.75i	0.429738i
$696$	− 5626.96i	− 0.306450i
$697$	− 5962.11i	− 0.324005i
$698$	−15214.7	−0.825051
$699$	10926.0	0.591215
$700$	6836.10i	0.369114i
$701$	8231.17	0.443491	0.221745	−	0.975105i	$-0.428825\pi$
$701$	8231.17	0.443491	0.221745	+	0.975105i	$0.428825\pi$
$702$	0	0
$703$	−44963.6	−2.41228
$704$	− 10884.2i	− 0.582691i
$705$	18781.7	1.00335
$706$	−6893.86	−0.367498
$707$	8243.61i	0.438519i
$708$	11643.4i	0.618061i
$709$	− 28044.6i	− 1.48553i	−0.669554	−	0.742764i	$-0.733515\pi$
$709$	− 28044.6i	− 1.48553i	0.669554	−	0.742764i	$-0.266485\pi$
$710$	9648.64i	0.510010i
$711$	1501.29	0.0791884
$712$	2270.96	0.119534
$713$	− 5221.09i	− 0.274238i
$714$	926.457	0.0485599
$715$	0	0
$716$	15257.6	0.796374
$717$	6708.51i	0.349420i
$718$	18561.3	0.964764
$719$	−29686.4	−1.53980	−0.769901	−	0.638163i	$-0.779694\pi$
$719$	−29686.4	−1.53980	−0.769901	+	0.638163i	$0.779694\pi$
$720$	2351.29i	0.121705i
$721$	− 7044.86i	− 0.363889i
$722$	− 7190.12i	− 0.370621i
$723$	− 19616.3i	− 1.00904i
$724$	24784.0	1.27223
$725$	22385.3	1.14672
$726$	− 10985.3i	− 0.561572i
$727$	27654.5	1.41080	0.705398	−	0.708812i	$-0.250768\pi$
$727$	27654.5	1.41080	0.705398	+	0.708812i	$0.250768\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	717.537i	0.0363798i
$731$	2970.89	0.150318
$732$	5241.19	0.264645
$733$	− 13077.5i	− 0.658975i	−0.944160	−	0.329488i	$-0.893124\pi$
$733$	− 13077.5i	− 0.658975i	0.944160	−	0.329488i	$-0.106876\pi$
$734$	− 4082.08i	− 0.205276i
$735$	18540.9i	0.930465i
$736$	− 5347.79i	− 0.267829i
$737$	−41090.3	−2.05370
$738$	1960.04	0.0977645
$739$	4218.33i	0.209978i	0.994473	+	0.104989i	$0.0334808\pi$
$739$	4218.33i	0.209978i	−0.994473	+	0.104989i	$0.966519\pi$
$740$	45839.9	2.27717
$741$	0	0
$742$	−185.525	−0.00917903
$743$	7725.54i	0.381457i	0.981643	+	0.190728i	$0.0610850\pi$
$743$	7725.54i	0.381457i	−0.981643	+	0.190728i	$0.938915\pi$
$744$	−11462.4	−0.564828
$745$	−33218.9	−1.63362
$746$	14656.5i	0.719319i
$747$	− 1809.84i	− 0.0886460i
$748$	14470.6i	0.707349i
$749$	3983.29i	0.194321i
$750$	10984.9	0.534818
$751$	−7506.12	−0.364717	−0.182358	−	0.983232i	$-0.558373\pi$
$751$	−7506.12	−0.364717	−0.182358	+	0.983232i	$0.558373\pi$
$752$	− 4372.87i	− 0.212051i
$753$	−7521.35	−0.364002
$754$	0	0
$755$	25950.1	1.25089
$756$	− 741.162i	− 0.0356558i
$757$	−2741.62	−0.131632	−0.0658162	−	0.997832i	$-0.520965\pi$
$757$	−2741.62	−0.131632	−0.0658162	+	0.997832i	$0.520965\pi$
$758$	−13768.0	−0.659729
$759$	− 5224.25i	− 0.249839i
$760$	43406.9i	2.07175i
$761$	− 29740.0i	− 1.41666i	−0.705883	−	0.708328i	$-0.749450\pi$
$761$	− 29740.0i	− 1.41666i	0.705883	−	0.708328i	$-0.250550\pi$
$762$	5170.48i	0.245809i
$763$	2213.78	0.105038
$764$	7986.01	0.378172
$765$	7273.69i	0.343766i
$766$	6129.18	0.289107
$767$	0	0
$768$	−9902.32	−0.465259
$769$	− 19896.3i	− 0.933004i	−0.884520	−	0.466502i	$-0.845514\pi$
$769$	− 19896.3i	− 0.933004i	0.884520	−	0.466502i	$-0.154486\pi$
$770$	−8728.72	−0.408521
$771$	2424.39	0.113246
$772$	23426.6i	1.09215i
$773$	− 13601.3i	− 0.632866i	−0.948615	−	0.316433i	$-0.897515\pi$
$773$	− 13601.3i	− 0.632866i	0.948615	−	0.316433i	$-0.102485\pi$
$774$	976.680i	0.0453566i
$775$	− 45600.1i	− 2.11355i
$776$	−24162.2	−1.11775
$777$	−6071.48	−0.280326
$778$	4160.09i	0.191705i
$779$	−15346.1	−0.705818
$780$	0	0
$781$	19960.8	0.914539
$782$	− 1818.83i	− 0.0831727i
$783$	−2426.99	−0.110771
$784$	4316.81	0.196648
$785$	− 14708.9i	− 0.668770i
$786$	− 3905.06i	− 0.177212i
$787$	1498.29i	0.0678631i	0.999424	+	0.0339315i	$0.0108028\pi$
$787$	1498.29i	0.0678631i	−0.999424	+	0.0339315i	$0.989197\pi$
$788$	− 19284.9i	− 0.871824i
$789$	8822.09	0.398067
$790$	−4924.46	−0.221778
$791$	1847.91i	0.0830647i
$792$	−11469.3	−0.514577
$793$	0	0
$794$	−6703.49	−0.299620
$795$	− 1456.57i	− 0.0649803i
$796$	19962.5	0.888885
$797$	3713.30	0.165034	0.0825168	−	0.996590i	$-0.473704\pi$
$797$	3713.30	0.165034	0.0825168	+	0.996590i	$0.473704\pi$
$798$	− 2384.64i	− 0.105784i
$799$	− 13527.4i	− 0.598955i
$800$	− 46706.7i	− 2.06416i
$801$	− 979.502i	− 0.0432072i
$802$	5743.20	0.252867
$803$	1484.42	0.0652354
$804$	11444.4i	0.502007i
$805$	−2669.80	−0.116892
$806$	0	0
$807$	21334.5	0.930620
$808$	− 35530.2i	− 1.54697i
$809$	−34527.6	−1.50053	−0.750263	−	0.661139i	$-0.770073\pi$
$809$	−34527.6	−1.50053	−0.750263	+	0.661139i	$0.770073\pi$
$810$	−2391.22	−0.103727
$811$	− 37279.2i	− 1.61412i	−0.590471	−	0.807059i	$-0.701058\pi$
$811$	− 37279.2i	− 1.61412i	0.590471	−	0.807059i	$-0.298942\pi$
$812$	2467.48i	0.106640i
$813$	− 6104.89i	− 0.263355i
$814$	38970.3i	1.67802i
$815$	52451.6	2.25436
$816$	1693.51	0.0726526
$817$	− 7646.90i	− 0.327455i
$818$	10376.4	0.443525
$819$	0	0
$820$	15645.2	0.666285
$821$	13877.9i	0.589943i	0.955506	+	0.294972i	$0.0953102\pi$
$821$	13877.9i	0.589943i	−0.955506	+	0.294972i	$0.904690\pi$
$822$	−2238.84	−0.0949983
$823$	−18945.1	−0.802410	−0.401205	−	0.915988i	$-0.631408\pi$
$823$	−18945.1	−0.802410	−0.401205	+	0.915988i	$0.631408\pi$
$824$	30363.5i	1.28369i
$825$	− 45627.6i	− 1.92552i
$826$	5058.55i	0.213086i
$827$	7804.75i	0.328171i	0.986446	+	0.164086i	$0.0524673\pi$
$827$	7804.75i	0.328171i	−0.986446	+	0.164086i	$0.947533\pi$
$828$	−1455.05	−0.0610708
$829$	−5784.85	−0.242360	−0.121180	−	0.992631i	$-0.538668\pi$
$829$	−5784.85	−0.242360	−0.121180	+	0.992631i	$0.538668\pi$
$830$	5936.54i	0.248265i
$831$	−8169.59	−0.341035
$832$	0	0
$833$	13354.0	0.555448
$834$	1864.35i	0.0774065i
$835$	30009.7	1.24375
$836$	37246.4	1.54090
$837$	4943.91i	0.204165i
$838$	− 19224.3i	− 0.792471i
$839$	5011.42i	0.206214i	0.994670	+	0.103107i	$0.0328784\pi$
$839$	5011.42i	0.206214i	−0.994670	+	0.103107i	$0.967122\pi$
$840$	5861.27i	0.240754i
$841$	−16309.0	−0.668704
$842$	−10514.2	−0.430336
$843$	− 9796.68i	− 0.400256i
$844$	−12730.3	−0.519190
$845$	0	0
$846$	4447.13	0.180728
$847$	11613.9i	0.471142i
$848$	−339.128	−0.0137332
$849$	−3432.05	−0.138737
$850$	− 15885.3i	− 0.641013i
$851$	11919.6i	0.480138i
$852$	− 5559.48i	− 0.223550i
$853$	22059.0i	0.885446i	0.896659	+	0.442723i	$0.145987\pi$
$853$	22059.0i	0.885446i	−0.896659	+	0.442723i	$0.854013\pi$
$854$	2277.06	0.0912404
$855$	18722.0	0.748866
$856$	− 17168.1i	− 0.685506i
$857$	−13956.2	−0.556283	−0.278141	−	0.960540i	$-0.589718\pi$
$857$	−13956.2	−0.556283	−0.278141	+	0.960540i	$0.589718\pi$
$858$	0	0
$859$	12498.5	0.496442	0.248221	−	0.968703i	$-0.420154\pi$
$859$	12498.5	0.496442	0.248221	+	0.968703i	$0.420154\pi$
$860$	7795.92i	0.309115i
$861$	−2072.20	−0.0820215
$862$	11272.0	0.445391
$863$	− 38631.2i	− 1.52378i	−0.647707	−	0.761890i	$-0.724272\pi$
$863$	− 38631.2i	− 1.52378i	0.647707	−	0.761890i	$-0.275728\pi$
$864$	5063.88i	0.199394i
$865$	76784.7i	3.01822i
$866$	− 13842.3i	− 0.543163i
$867$	−9500.17	−0.372137
$868$	5026.38	0.196551
$869$	10187.6i	0.397687i
$870$	7960.89	0.310229
$871$	0	0
$872$	−9541.46	−0.370544
$873$	10421.5i	0.404027i
$874$	−4681.55	−0.181185
$875$	−11613.5	−0.448696
$876$	− 413.440i	− 0.0159462i
$877$	− 856.756i	− 0.0329881i	−0.999864	−	0.0164941i	$-0.994750\pi$
$877$	− 856.756i	− 0.0329881i	0.999864	−	0.0164941i	$-0.00525046\pi$
$878$	− 25759.2i	− 0.990127i
$879$	5032.06i	0.193091i
$880$	−15955.6	−0.611206
$881$	−33638.6	−1.28640	−0.643198	−	0.765700i	$-0.722393\pi$
$881$	−33638.6	−1.28640	−0.643198	+	0.765700i	$0.722393\pi$
$882$	4390.12i	0.167600i
$883$	31109.1	1.18562	0.592811	−	0.805342i	$-0.298018\pi$
$883$	31109.1	1.18562	0.592811	+	0.805342i	$0.298018\pi$
$884$	0	0
$885$	−39715.1	−1.50848
$886$	− 10328.2i	− 0.391627i
$887$	26080.3	0.987248	0.493624	−	0.869675i	$-0.335672\pi$
$887$	26080.3	0.987248	0.493624	+	0.869675i	$0.335672\pi$
$888$	26168.3	0.988907
$889$	− 5466.35i	− 0.206227i
$890$	3212.91i	0.121008i
$891$	4946.89i	0.186001i
$892$	19455.2i	0.730277i
$893$	−34818.8	−1.30478
$894$	−7865.58	−0.294256
$895$	52042.8i	1.94369i
$896$	5946.99	0.221736
$897$	0	0
$898$	215.142	0.00799486
$899$	− 16459.3i	− 0.610621i
$900$	−12708.2	−0.470673
$901$	−1049.09	−0.0387905
$902$	13300.6i	0.490977i
$903$	− 1032.57i	− 0.0380529i
$904$	− 7964.56i	− 0.293028i
$905$	84536.9i	3.10508i
$906$	6144.47	0.225316
$907$	−20169.0	−0.738369	−0.369184	−	0.929356i	$-0.620363\pi$
$907$	−20169.0	−0.738369	−0.369184	+	0.929356i	$0.620363\pi$
$908$	− 26977.3i	− 0.985983i
$909$	−15324.7	−0.559174
$910$	0	0
$911$	19982.2	0.726716	0.363358	−	0.931650i	$-0.381630\pi$
$911$	19982.2	0.726716	0.363358	+	0.931650i	$0.381630\pi$
$912$	− 4358.98i	− 0.158268i
$913$	12281.3	0.445184
$914$	27068.5	0.979593
$915$	17877.4i	0.645910i
$916$	24771.7i	0.893538i
$917$	4128.52i	0.148676i
$918$	1722.27i	0.0619207i
$919$	38513.1	1.38241	0.691203	−	0.722661i	$-0.257081\pi$
$919$	38513.1	1.38241	0.691203	+	0.722661i	$0.257081\pi$
$920$	11506.9	0.412360
$921$	− 21623.1i	− 0.773621i
$922$	−3500.70	−0.125043
$923$	0	0
$924$	5029.42	0.179065
$925$	104103.i	3.70043i
$926$	21020.4	0.745975
$927$	13096.3	0.464010
$928$	− 16858.7i	− 0.596352i
$929$	23218.9i	0.820009i	0.912083	+	0.410005i	$0.134473\pi$
$929$	23218.9i	0.820009i	−0.912083	+	0.410005i	$0.865527\pi$
$930$	− 16216.7i	− 0.571793i
$931$	− 34372.3i	− 1.21000i
$932$	20650.1	0.725768
$933$	−1238.89	−0.0434721
$934$	5307.99i	0.185956i
$935$	−49358.3	−1.72640
$936$	0	0
$937$	−11112.9	−0.387452	−0.193726	−	0.981056i	$-0.562057\pi$
$937$	−11112.9	−0.387452	−0.193726	+	0.981056i	$0.562057\pi$
$938$	4972.08i	0.173075i
$939$	8809.17	0.306152
$940$	35497.3	1.23170
$941$	45570.4i	1.57869i	0.613947	+	0.789347i	$0.289581\pi$
$941$	45570.4i	1.57869i	−0.613947	+	0.789347i	$0.710419\pi$
$942$	− 3482.78i	− 0.120462i
$943$	4068.16i	0.140485i
$944$	9246.71i	0.318808i
$945$	2528.06	0.0870240
$946$	−6627.62	−0.227783
$947$	− 34903.9i	− 1.19770i	−0.800860	−	0.598852i	$-0.795624\pi$
$947$	− 34903.9i	− 1.19770i	0.800860	−	0.598852i	$-0.204376\pi$
$948$	2837.44	0.0972106
$949$	0	0
$950$	−40887.8	−1.39640
$951$	1133.87i	0.0386626i
$952$	4221.55	0.143720
$953$	2886.52	0.0981151	0.0490575	−	0.998796i	$-0.484378\pi$
$953$	2886.52	0.0981151	0.0490575	+	0.998796i	$0.484378\pi$
$954$	− 344.888i	− 0.0117046i
$955$	27239.8i	0.922994i
$956$	12679.0i	0.428943i
$957$	− 16469.2i	− 0.556296i
$958$	−4789.03	−0.161510
$959$	2366.96	0.0797008
$960$	− 10340.1i	− 0.347632i
$961$	−3737.42	−0.125455
$962$	0	0
$963$	−7404.85	−0.247786
$964$	− 37074.8i	− 1.23869i
$965$	−79906.8	−2.66559
$966$	−632.155	−0.0210551
$967$	11593.8i	0.385556i	0.981242	+	0.192778i	$0.0617498\pi$
$967$	11593.8i	0.385556i	−0.981242	+	0.192778i	$0.938250\pi$
$968$	− 50056.1i	− 1.66205i
$969$	− 13484.4i	− 0.447041i
$970$	− 34184.1i	− 1.13153i
$971$	−4952.12	−0.163667	−0.0818337	−	0.996646i	$-0.526078\pi$
$971$	−4952.12	−0.163667	−0.0818337	+	0.996646i	$0.526078\pi$
$972$	1377.81	0.0454662
$973$	− 1971.03i	− 0.0649418i
$974$	−9153.33	−0.301121
$975$	0	0
$976$	4162.32	0.136509
$977$	− 19650.1i	− 0.643462i	−0.946831	−	0.321731i	$-0.895735\pi$
$977$	− 19650.1i	− 0.643462i	0.946831	−	0.321731i	$-0.104265\pi$
$978$	12419.5	0.406065
$979$	6646.77	0.216988
$980$	35042.2i	1.14223i
$981$	4115.38i	0.133939i
$982$	14350.8i	0.466347i
$983$	− 56818.4i	− 1.84357i	−0.387707	−	0.921783i	$-0.626733\pi$
$983$	− 56818.4i	− 1.84357i	0.387707	−	0.921783i	$-0.373267\pi$
$984$	8931.26	0.289348
$985$	65779.7	2.12783
$986$	− 5733.78i	− 0.185193i
$987$	−4701.61	−0.151625
$988$	0	0
$989$	−2027.15	−0.0651764
$990$	− 16226.5i	− 0.520922i
$991$	−19120.4	−0.612897	−0.306448	−	0.951887i	$-0.599141\pi$
$991$	−19120.4	−0.612897	−0.306448	+	0.951887i	$0.599141\pi$
$992$	−34342.1	−1.09915
$993$	− 13284.5i	− 0.424543i
$994$	− 2415.34i	− 0.0770723i
$995$	68090.9i	2.16947i
$996$	− 3420.59i	− 0.108821i
$997$	38887.9	1.23530	0.617650	−	0.786453i	$-0.288085\pi$
$997$	38887.9	1.23530	0.617650	+	0.786453i	$0.288085\pi$
$998$	7712.07	0.244611
$999$	− 11286.8i	− 0.357455i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.g.337.4		6
13.5	odd	4		39.4.a.c.1.2	✓	3
13.8	odd	4		507.4.a.h.1.2		3
13.12	even	2	inner	507.4.b.g.337.3		6
39.5	even	4		117.4.a.f.1.2		3
39.8	even	4		1521.4.a.u.1.2		3
52.31	even	4		624.4.a.t.1.3		3
65.44	odd	4		975.4.a.l.1.2		3
91.83	even	4		1911.4.a.k.1.2		3
104.5	odd	4		2496.4.a.bl.1.1		3
104.83	even	4		2496.4.a.bp.1.1		3
156.83	odd	4		1872.4.a.bk.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.2	✓	3	13.5	odd	4
117.4.a.f.1.2		3	39.5	even	4
507.4.a.h.1.2		3	13.8	odd	4
507.4.b.g.337.3		6	13.12	even	2	inner
507.4.b.g.337.4		6	1.1	even	1	trivial
624.4.a.t.1.3		3	52.31	even	4
975.4.a.l.1.2		3	65.44	odd	4
1521.4.a.u.1.2		3	39.8	even	4
1872.4.a.bk.1.1		3	156.83	odd	4
1911.4.a.k.1.2		3	91.83	even	4
2496.4.a.bl.1.1		3	104.5	odd	4
2496.4.a.bp.1.1		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)

\(f(q)\)	\(=\)	\(q+1.52644i q^{2} +3.00000 q^{3} +5.66998 q^{4} +19.3400i q^{5} +4.57932i q^{6} -4.84136i q^{7} +20.8664i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.52644i q^{2} +3.00000 q^{3} +5.66998 q^{4} +19.3400i q^{5} +4.57932i q^{6} -4.84136i q^{7} +20.8664i q^{8} +9.00000 q^{9} -29.5213 q^{10} +61.0728i q^{11} +17.0099 q^{12} +7.39005 q^{14} +58.0199i q^{15} +13.5085 q^{16} +41.7885 q^{17} +13.7380i q^{18} -107.561i q^{19} +109.657i q^{20} -14.5241i q^{21} -93.2239 q^{22} -28.5138 q^{23} +62.5992i q^{24} -249.034 q^{25} +27.0000 q^{27} -27.4504i q^{28} -89.8886 q^{29} -88.5639 q^{30} +183.108i q^{31} +187.551i q^{32} +183.218i q^{33} +63.7876i q^{34} +93.6318 q^{35} +51.0298 q^{36} -418.029i q^{37} +164.185 q^{38} -403.555 q^{40} -142.674i q^{41} +22.1701 q^{42} +71.0935 q^{43} +346.281i q^{44} +174.060i q^{45} -43.5246i q^{46} -323.711i q^{47} +40.5256 q^{48} +319.561 q^{49} -380.136i q^{50} +125.365 q^{51} -25.1047 q^{53} +41.2139i q^{54} -1181.15 q^{55} +101.022 q^{56} -322.683i q^{57} -137.210i q^{58} +684.508i q^{59} +328.972i q^{60} +308.125 q^{61} -279.503 q^{62} -43.5723i q^{63} -178.217 q^{64} -279.672 q^{66} +672.808i q^{67} +236.940 q^{68} -85.5413 q^{69} +142.923i q^{70} -326.837i q^{71} +187.798i q^{72} -24.3058i q^{73} +638.095 q^{74} -747.103 q^{75} -609.869i q^{76} +295.675 q^{77} +166.810 q^{79} +261.255i q^{80} +81.0000 q^{81} +217.783 q^{82} -201.093i q^{83} -82.3513i q^{84} +808.188i q^{85} +108.520i q^{86} -269.666 q^{87} -1274.37 q^{88} -108.834i q^{89} -265.692 q^{90} -161.673 q^{92} +549.323i q^{93} +494.126 q^{94} +2080.23 q^{95} +562.654i q^{96} +1157.95i q^{97} +487.791i q^{98} +549.655i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9}+O(q^{10}) \) 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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