LMFDB - Embedded newform 507.4.b.k.337.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + \cdots + 729 $ x^18 + 112x^16 + 5026x^14 + 114847x^12 + 1397921x^10 + 8545747x^8 + 21033277x^6 + 6703200x^4 + 137781x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.5
Root			$-2.37150i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.k.337.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.17344i q^{2} +3.00000 q^{3} -2.07074 q^{4} +6.74147i q^{5} -9.52033i q^{6} +14.1726i q^{7} -18.8162i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.17344i q^{2} +3.00000 q^{3} -2.07074 q^{4} +6.74147i q^{5} -9.52033i q^{6} +14.1726i q^{7} -18.8162i q^{8} +9.00000 q^{9} +21.3937 q^{10} -62.4956i q^{11} -6.21221 q^{12} +44.9761 q^{14} +20.2244i q^{15} -76.2779 q^{16} +58.6172 q^{17} -28.5610i q^{18} +64.1652i q^{19} -13.9598i q^{20} +42.5179i q^{21} -198.326 q^{22} -10.9221 q^{23} -56.4485i q^{24} +79.5526 q^{25} +27.0000 q^{27} -29.3478i q^{28} +216.316 q^{29} +64.1810 q^{30} -38.6271i q^{31} +91.5342i q^{32} -187.487i q^{33} -186.018i q^{34} -95.5445 q^{35} -18.6366 q^{36} -423.770i q^{37} +203.625 q^{38} +126.849 q^{40} -366.126i q^{41} +134.928 q^{42} +128.297 q^{43} +129.412i q^{44} +60.6732i q^{45} +34.6605i q^{46} +93.1169i q^{47} -228.834 q^{48} +142.136 q^{49} -252.455i q^{50} +175.852 q^{51} +131.909 q^{53} -85.6829i q^{54} +421.313 q^{55} +266.675 q^{56} +192.496i q^{57} -686.467i q^{58} -386.729i q^{59} -41.8794i q^{60} -621.077 q^{61} -122.581 q^{62} +127.554i q^{63} -319.745 q^{64} -594.979 q^{66} +865.273i q^{67} -121.381 q^{68} -32.7662 q^{69} +303.205i q^{70} -607.506i q^{71} -169.346i q^{72} -980.958i q^{73} -1344.81 q^{74} +238.658 q^{75} -132.869i q^{76} +885.728 q^{77} +1331.91 q^{79} -514.226i q^{80} +81.0000 q^{81} -1161.88 q^{82} +907.633i q^{83} -88.0434i q^{84} +395.166i q^{85} -407.142i q^{86} +648.949 q^{87} -1175.93 q^{88} +1033.67i q^{89} +192.543 q^{90} +22.6167 q^{92} -115.881i q^{93} +295.501 q^{94} -432.568 q^{95} +274.603i q^{96} +1046.17i q^{97} -451.061i q^{98} -562.461i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * 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 $

$2$	− 3.17344i	− 1.12198i	−0.827822	−	0.560991i	$-0.810420\pi$
$2$	− 3.17344i	− 1.12198i	0.827822	−	0.560991i	$-0.189580\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−2.07074	−0.258842
$5$	6.74147i	0.602975i	0.953470	+	0.301488i	$0.0974832\pi$
$5$	6.74147i	0.602975i	−0.953470	+	0.301488i	$0.902517\pi$
$6$	− 9.52033i	− 0.647776i
$7$	14.1726i	0.765251i	0.923904	+	0.382625i	$0.124980\pi$
$7$	14.1726i	0.765251i	−0.923904	+	0.382625i	$0.875020\pi$
$8$	− 18.8162i	− 0.831565i
$9$	9.00000	0.333333
$10$	21.3937	0.676527
$11$	− 62.4956i	− 1.71301i	−0.516136	−	0.856507i	$-0.672630\pi$
$11$	− 62.4956i	− 1.71301i	0.516136	−	0.856507i	$-0.327370\pi$
$12$	−6.21221	−0.149442
$13$	0	0
$13$	0	0
$14$	44.9761	0.858597
$15$	20.2244i	0.348128i
$16$	−76.2779	−1.19184
$17$	58.6172	0.836280	0.418140	−	0.908383i	$-0.362682\pi$
$17$	58.6172	0.836280	0.418140	+	0.908383i	$0.362682\pi$
$18$	− 28.5610i	− 0.373994i
$19$	64.1652i	0.774764i	0.921919	+	0.387382i	$0.126620\pi$
$19$	64.1652i	0.774764i	−0.921919	+	0.387382i	$0.873380\pi$
$20$	− 13.9598i	− 0.156075i
$21$	42.5179i	0.441818i
$22$	−198.326	−1.92197
$23$	−10.9221	−0.0990177	−0.0495088	−	0.998774i	$-0.515766\pi$
$23$	−10.9221	−0.0990177	−0.0495088	+	0.998774i	$0.515766\pi$
$24$	− 56.4485i	− 0.480105i
$25$	79.5526	0.636421
$26$	0	0
$27$	27.0000	0.192450
$28$	− 29.3478i	− 0.198079i
$29$	216.316	1.38514	0.692568	−	0.721353i	$-0.256479\pi$
$29$	216.316	1.38514	0.692568	+	0.721353i	$0.256479\pi$
$30$	64.1810	0.390593
$31$	− 38.6271i	− 0.223795i	−0.993720	−	0.111897i	$-0.964307\pi$
$31$	− 38.6271i	− 0.223795i	0.993720	−	0.111897i	$-0.0356928\pi$
$32$	91.5342i	0.505660i
$33$	− 187.487i	− 0.989009i
$34$	− 186.018i	− 0.938290i
$35$	−95.5445	−0.461427
$36$	−18.6366	−0.0862807
$37$	− 423.770i	− 1.88290i	−0.337147	−	0.941452i	$-0.609462\pi$
$37$	− 423.770i	− 1.88290i	0.337147	−	0.941452i	$-0.390538\pi$
$38$	203.625	0.869270
$39$	0	0
$40$	126.849	0.501414
$41$	− 366.126i	− 1.39461i	−0.716772	−	0.697307i	$-0.754381\pi$
$41$	− 366.126i	− 1.39461i	0.716772	−	0.697307i	$-0.245619\pi$
$42$	134.928	0.495711
$43$	128.297	0.455001	0.227501	−	0.973778i	$-0.426945\pi$
$43$	128.297	0.455001	0.227501	+	0.973778i	$0.426945\pi$
$44$	129.412i	0.443400i
$45$	60.6732i	0.200992i
$46$	34.6605i	0.111096i
$47$	93.1169i	0.288989i	0.989506	+	0.144495i	$0.0461556\pi$
$47$	93.1169i	0.288989i	−0.989506	+	0.144495i	$0.953844\pi$
$48$	−228.834	−0.688111
$49$	142.136	0.414391
$50$	− 252.455i	− 0.714052i
$51$	175.852	0.482826
$52$	0	0
$53$	131.909	0.341869	0.170934	−	0.985282i	$-0.445321\pi$
$53$	131.909	0.341869	0.170934	+	0.985282i	$0.445321\pi$
$54$	− 85.6829i	− 0.215925i
$55$	421.313	1.03290
$56$	266.675	0.636356
$57$	192.496i	0.447310i
$58$	− 686.467i	− 1.55410i
$59$	− 386.729i	− 0.853353i	−0.904404	−	0.426677i	$-0.859684\pi$
$59$	− 386.729i	− 0.853353i	0.904404	−	0.426677i	$-0.140316\pi$
$60$	− 41.8794i	− 0.0901102i
$61$	−621.077	−1.30362	−0.651810	−	0.758382i	$-0.725990\pi$
$61$	−621.077	−1.30362	−0.651810	+	0.758382i	$0.725990\pi$
$62$	−122.581	−0.251094
$63$	127.554i	0.255084i
$64$	−319.745	−0.624502
$65$	0	0
$66$	−594.979	−1.10965
$67$	865.273i	1.57776i	0.614547	+	0.788880i	$0.289339\pi$
$67$	865.273i	1.57776i	−0.614547	+	0.788880i	$0.710661\pi$
$68$	−121.381	−0.216464
$69$	−32.7662	−0.0571679
$70$	303.205i	0.517713i
$71$	− 607.506i	− 1.01546i	−0.861516	−	0.507730i	$-0.830485\pi$
$71$	− 607.506i	− 1.01546i	0.861516	−	0.507730i	$-0.169515\pi$
$72$	− 169.346i	− 0.277188i
$73$	− 980.958i	− 1.57277i	−0.617735	−	0.786387i	$-0.711949\pi$
$73$	− 980.958i	− 1.57277i	0.617735	−	0.786387i	$-0.288051\pi$
$74$	−1344.81	−2.11258
$75$	238.658	0.367438
$76$	− 132.869i	− 0.200541i
$77$	885.728	1.31088
$78$	0	0
$79$	1331.91	1.89685	0.948425	−	0.317003i	$-0.102676\pi$
$79$	1331.91	1.89685	0.948425	+	0.317003i	$0.102676\pi$
$80$	− 514.226i	− 0.718652i
$81$	81.0000	0.111111
$82$	−1161.88	−1.56473
$83$	907.633i	1.20031i	0.799884	+	0.600155i	$0.204894\pi$
$83$	907.633i	1.20031i	−0.799884	+	0.600155i	$0.795106\pi$
$84$	− 88.0434i	− 0.114361i
$85$	395.166i	0.504256i
$86$	− 407.142i	− 0.510503i
$87$	648.949	0.799708
$88$	−1175.93	−1.42448
$89$	1033.67i	1.23110i	0.788096	+	0.615552i	$0.211067\pi$
$89$	1033.67i	1.23110i	−0.788096	+	0.615552i	$0.788933\pi$
$90$	192.543	0.225509
$91$	0	0
$92$	22.6167	0.0256299
$93$	− 115.881i	− 0.129208i
$94$	295.501	0.324240
$95$	−432.568	−0.467163
$96$	274.603i	0.291943i
$97$	1046.17i	1.09508i	0.836781	+	0.547538i	$0.184435\pi$
$97$	1046.17i	1.09508i	−0.836781	+	0.547538i	$0.815565\pi$
$98$	− 451.061i	− 0.464939i
$99$	− 562.461i	− 0.571004i
$100$	−164.732	−0.164732
$101$	−1416.64	−1.39566	−0.697828	−	0.716265i	$-0.745850\pi$
$101$	−1416.64	−1.39566	−0.697828	+	0.716265i	$0.745850\pi$
$102$	− 558.055i	− 0.541722i
$103$	−387.629	−0.370818	−0.185409	−	0.982661i	$-0.559361\pi$
$103$	−387.629	−0.370818	−0.185409	+	0.982661i	$0.559361\pi$
$104$	0	0
$105$	−286.633	−0.266405
$106$	− 418.605i	− 0.383570i
$107$	86.4526	0.0781092	0.0390546	−	0.999237i	$-0.487565\pi$
$107$	86.4526	0.0781092	0.0390546	+	0.999237i	$0.487565\pi$
$108$	−55.9099	−0.0498142
$109$	− 940.072i	− 0.826079i	−0.910713	−	0.413039i	$-0.864467\pi$
$109$	− 940.072i	− 0.826079i	0.910713	−	0.413039i	$-0.135533\pi$
$110$	− 1337.01i	− 1.15890i
$111$	− 1271.31i	− 1.08709i
$112$	− 1081.06i	− 0.912059i
$113$	960.499	0.799612	0.399806	−	0.916600i	$-0.369078\pi$
$113$	960.499	0.799612	0.399806	+	0.916600i	$0.369078\pi$
$114$	610.874	0.501873
$115$	− 73.6307i	− 0.0597052i
$116$	−447.934	−0.358531
$117$	0	0
$118$	−1227.26	−0.957447
$119$	830.760i	0.639964i
$120$	380.546	0.289491
$121$	−2574.71	−1.93441
$122$	1970.95i	1.46264i
$123$	− 1098.38i	− 0.805181i
$124$	79.9866i	0.0579275i
$125$	1378.99i	0.986721i
$126$	404.785	0.286199
$127$	−2022.18	−1.41291	−0.706456	−	0.707757i	$-0.749707\pi$
$127$	−2022.18	−1.41291	−0.706456	+	0.707757i	$0.749707\pi$
$128$	1746.97i	1.20634i
$129$	384.890	0.262695
$130$	0	0
$131$	1857.90	1.23912	0.619561	−	0.784948i	$-0.287310\pi$
$131$	1857.90	1.23912	0.619561	+	0.784948i	$0.287310\pi$
$132$	388.236i	0.255997i
$133$	−909.390	−0.592888
$134$	2745.90	1.77022
$135$	182.020i	0.116043i
$136$	− 1102.95i	− 0.695421i
$137$	− 1894.12i	− 1.18121i	−0.806961	−	0.590604i	$-0.798890\pi$
$137$	− 1894.12i	− 1.18121i	0.806961	−	0.590604i	$-0.201110\pi$
$138$	103.982i	0.0641413i
$139$	−1226.08	−0.748165	−0.374082	−	0.927395i	$-0.622042\pi$
$139$	−1226.08	−0.748165	−0.374082	+	0.927395i	$0.622042\pi$
$140$	197.847	0.119437
$141$	279.351i	0.166848i
$142$	−1927.89	−1.13933
$143$	0	0
$144$	−686.501	−0.397281
$145$	1458.29i	0.835203i
$146$	−3113.01	−1.76462
$147$	426.409	0.239249
$148$	877.517i	0.487375i
$149$	3195.65i	1.75703i	0.477711	+	0.878517i	$0.341467\pi$
$149$	3195.65i	1.75703i	−0.477711	+	0.878517i	$0.658533\pi$
$150$	− 757.366i	− 0.412258i
$151$	− 508.232i	− 0.273903i	−0.990578	−	0.136951i	$-0.956270\pi$
$151$	− 508.232i	− 0.273903i	0.990578	−	0.136951i	$-0.0437304\pi$
$152$	1207.34	0.644267
$153$	527.555	0.278760
$154$	− 2810.81i	− 1.47079i
$155$	260.404	0.134943
$156$	0	0
$157$	−1243.08	−0.631900	−0.315950	−	0.948776i	$-0.602323\pi$
$157$	−1243.08	−0.631900	−0.315950	+	0.948776i	$0.602323\pi$
$158$	− 4226.73i	− 2.12823i
$159$	395.726	0.197378
$160$	−617.075	−0.304901
$161$	− 154.794i	− 0.0757734i
$162$	− 257.049i	− 0.124665i
$163$	33.9996i	0.0163378i	0.999967	+	0.00816888i	$0.00260026\pi$
$163$	33.9996i	0.0163378i	−0.999967	+	0.00816888i	$0.997400\pi$
$164$	758.149i	0.360985i
$165$	1263.94	0.596348
$166$	2880.32	1.34673
$167$	2210.67i	1.02435i	0.858880	+	0.512176i	$0.171161\pi$
$167$	2210.67i	1.02435i	−0.858880	+	0.512176i	$0.828839\pi$
$168$	800.025	0.367400
$169$	0	0
$170$	1254.04	0.565766
$171$	577.487i	0.258255i
$172$	−265.668	−0.117773
$173$	661.307	0.290626	0.145313	−	0.989386i	$-0.453581\pi$
$173$	661.307	0.290626	0.145313	+	0.989386i	$0.453581\pi$
$174$	− 2059.40i	− 0.897258i
$175$	1127.47i	0.487021i
$176$	4767.04i	2.04164i
$177$	− 1160.19i	− 0.492684i
$178$	3280.28	1.38128
$179$	2325.05	0.970850	0.485425	−	0.874278i	$-0.338665\pi$
$179$	2325.05	0.970850	0.485425	+	0.874278i	$0.338665\pi$
$180$	− 125.638i	− 0.0520251i
$181$	−2122.20	−0.871503	−0.435752	−	0.900067i	$-0.643517\pi$
$181$	−2122.20	−0.871503	−0.435752	+	0.900067i	$0.643517\pi$
$182$	0	0
$183$	−1863.23	−0.752645
$184$	205.511i	0.0823397i
$185$	2856.84	1.13534
$186$	−367.743	−0.144969
$187$	− 3663.32i	− 1.43256i
$188$	− 192.820i	− 0.0748025i
$189$	382.661i	0.147273i
$190$	1372.73i	0.524149i
$191$	−2484.37	−0.941166	−0.470583	−	0.882356i	$-0.655956\pi$
$191$	−2484.37	−0.941166	−0.470583	+	0.882356i	$0.655956\pi$
$192$	−959.235	−0.360556
$193$	266.771i	0.0994955i	0.998762	+	0.0497478i	$0.0158417\pi$
$193$	266.771i	0.0994955i	−0.998762	+	0.0497478i	$0.984158\pi$
$194$	3319.95	1.22865
$195$	0	0
$196$	−294.327	−0.107262
$197$	− 1231.03i	− 0.445216i	−0.974908	−	0.222608i	$-0.928543\pi$
$197$	− 1231.03i	− 0.445216i	0.974908	−	0.222608i	$-0.0714569\pi$
$198$	−1784.94	−0.640656
$199$	3246.14	1.15635	0.578173	−	0.815914i	$-0.303766\pi$
$199$	3246.14	1.15635	0.578173	+	0.815914i	$0.303766\pi$
$200$	− 1496.88i	− 0.529225i
$201$	2595.82i	0.910921i
$202$	4495.63i	1.56590i
$203$	3065.77i	1.05998i
$204$	−364.142	−0.124976
$205$	2468.23	0.840919
$206$	1230.12i	0.416051i
$207$	−98.2985	−0.0330059
$208$	0	0
$209$	4010.05	1.32718
$210$	909.614i	0.298902i
$211$	330.708	0.107900	0.0539500	−	0.998544i	$-0.482819\pi$
$211$	330.708	0.107900	0.0539500	+	0.998544i	$0.482819\pi$
$212$	−273.148	−0.0884900
$213$	− 1822.52i	− 0.586277i
$214$	− 274.352i	− 0.0876371i
$215$	864.908i	0.274355i
$216$	− 508.037i	− 0.160035i
$217$	547.449	0.171259
$218$	−2983.26	−0.926845
$219$	− 2942.87i	− 0.908041i
$220$	−872.427	−0.267359
$221$	0	0
$222$	−4034.43	−1.21970
$223$	5785.86i	1.73744i	0.495300	+	0.868722i	$0.335058\pi$
$223$	5785.86i	1.73744i	−0.495300	+	0.868722i	$0.664942\pi$
$224$	−1297.28	−0.386957
$225$	715.973	0.212140
$226$	− 3048.09i	− 0.897149i
$227$	2945.35i	0.861189i	0.902545	+	0.430595i	$0.141696\pi$
$227$	2945.35i	0.861189i	−0.902545	+	0.430595i	$0.858304\pi$
$228$	− 398.608i	− 0.115783i
$229$	3541.26i	1.02189i	0.859613	+	0.510945i	$0.170704\pi$
$229$	3541.26i	1.02189i	−0.859613	+	0.510945i	$0.829296\pi$
$230$	−233.663	−0.0669882
$231$	2657.18	0.756840
$232$	− 4070.25i	− 1.15183i
$233$	−2340.76	−0.658148	−0.329074	−	0.944304i	$-0.606737\pi$
$233$	−2340.76	−0.658148	−0.329074	+	0.944304i	$0.606737\pi$
$234$	0	0
$235$	−627.745	−0.174253
$236$	800.814i	0.220884i
$237$	3995.72	1.09515
$238$	2636.37	0.718027
$239$	− 1515.70i	− 0.410218i	−0.978739	−	0.205109i	$-0.934245\pi$
$239$	− 1515.70i	− 0.410218i	0.978739	−	0.205109i	$-0.0657549\pi$
$240$	− 1542.68i	− 0.414914i
$241$	− 2392.47i	− 0.639472i	−0.947507	−	0.319736i	$-0.896406\pi$
$241$	− 2392.47i	− 0.639472i	0.947507	−	0.319736i	$-0.103594\pi$
$242$	8170.68i	2.17038i
$243$	243.000	0.0641500
$244$	1286.09	0.337432
$245$	958.207i	0.249868i
$246$	−3485.64	−0.903398
$247$	0	0
$248$	−726.815	−0.186100
$249$	2722.90i	0.692999i
$250$	4376.13	1.10708
$251$	2198.78	0.552931	0.276465	−	0.961024i	$-0.410837\pi$
$251$	2198.78	0.552931	0.276465	+	0.961024i	$0.410837\pi$
$252$	− 264.130i	− 0.0660263i
$253$	682.581i	0.169619i
$254$	6417.29i	1.58526i
$255$	1185.50i	0.291132i
$256$	2985.94	0.728988
$257$	6194.26	1.50345	0.751727	−	0.659475i	$-0.229221\pi$
$257$	6194.26	1.50345	0.751727	+	0.659475i	$0.229221\pi$
$258$	− 1221.43i	− 0.294739i
$259$	6005.95	1.44089
$260$	0	0
$261$	1946.85	0.461712
$262$	− 5895.92i	− 1.39027i
$263$	−4181.74	−0.980445	−0.490222	−	0.871597i	$-0.663084\pi$
$263$	−4181.74	−0.980445	−0.490222	+	0.871597i	$0.663084\pi$
$264$	−3527.79	−0.822425
$265$	889.258i	0.206139i
$266$	2885.90i	0.665210i
$267$	3101.00i	0.710778i
$268$	− 1791.75i	− 0.408391i
$269$	2767.69	0.627320	0.313660	−	0.949535i	$-0.398445\pi$
$269$	2767.69	0.627320	0.313660	+	0.949535i	$0.398445\pi$
$270$	577.629	0.130198
$271$	7191.36i	1.61197i	0.591935	+	0.805986i	$0.298364\pi$
$271$	7191.36i	1.61197i	−0.591935	+	0.805986i	$0.701636\pi$
$272$	−4471.20	−0.996714
$273$	0	0
$274$	−6010.88	−1.32529
$275$	− 4971.69i	− 1.09020i
$276$	67.8501	0.0147974
$277$	−1317.27	−0.285729	−0.142864	−	0.989742i	$-0.545631\pi$
$277$	−1317.27	−0.285729	−0.142864	+	0.989742i	$0.545631\pi$
$278$	3890.90i	0.839427i
$279$	− 347.644i	− 0.0745983i
$280$	1797.78i	0.383707i
$281$	3948.92i	0.838338i	0.907908	+	0.419169i	$0.137679\pi$
$281$	3948.92i	0.838338i	−0.907908	+	0.419169i	$0.862321\pi$
$282$	886.503	0.187200
$283$	4981.52	1.04636	0.523181	−	0.852221i	$-0.324745\pi$
$283$	4981.52	1.04636	0.523181	+	0.852221i	$0.324745\pi$
$284$	1257.99i	0.262844i
$285$	−1297.70	−0.269717
$286$	0	0
$287$	5188.97	1.06723
$288$	823.808i	0.168553i
$289$	−1477.03	−0.300636
$290$	4627.80	0.937082
$291$	3138.50i	0.632242i
$292$	2031.31i	0.407100i
$293$	3203.02i	0.638644i	0.947646	+	0.319322i	$0.103455\pi$
$293$	3203.02i	0.638644i	−0.947646	+	0.319322i	$0.896545\pi$
$294$	− 1353.18i	− 0.268433i
$295$	2607.12	0.514551
$296$	−7973.74	−1.56576
$297$	− 1687.38i	− 0.329670i
$298$	10141.2	1.97136
$299$	0	0
$300$	−494.197	−0.0951083
$301$	1818.30i	0.348190i
$302$	−1612.84	−0.307314
$303$	−4249.93	−0.805782
$304$	− 4894.39i	− 0.923396i
$305$	− 4186.97i	− 0.786051i
$306$	− 1674.16i	− 0.312763i
$307$	− 4795.67i	− 0.891542i	−0.895147	−	0.445771i	$-0.852930\pi$
$307$	− 4795.67i	− 0.891542i	0.895147	−	0.445771i	$-0.147070\pi$
$308$	−1834.11	−0.339312
$309$	−1162.89	−0.214092
$310$	− 826.376i	− 0.151403i
$311$	−630.213	−0.114907	−0.0574535	−	0.998348i	$-0.518298\pi$
$311$	−630.213	−0.114907	−0.0574535	+	0.998348i	$0.518298\pi$
$312$	0	0
$313$	−9314.73	−1.68211	−0.841054	−	0.540952i	$-0.818064\pi$
$313$	−9314.73	−1.68211	−0.841054	+	0.540952i	$0.818064\pi$
$314$	3944.83i	0.708980i
$315$	−859.900	−0.153809
$316$	−2758.02	−0.490984
$317$	576.333i	0.102114i	0.998696	+	0.0510569i	$0.0162590\pi$
$317$	576.333i	0.102114i	−0.998696	+	0.0510569i	$0.983741\pi$
$318$	− 1255.81i	− 0.221455i
$319$	− 13518.8i	− 2.37275i
$320$	− 2155.55i	− 0.376559i
$321$	259.358	0.0450964
$322$	−491.231	−0.0850163
$323$	3761.18i	0.647919i
$324$	−167.730	−0.0287602
$325$	0	0
$326$	107.896	0.0183307
$327$	− 2820.22i	− 0.476937i
$328$	−6889.08	−1.15971
$329$	−1319.71	−0.221149
$330$	− 4011.03i	− 0.669091i
$331$	1575.95i	0.261699i	0.991402	+	0.130849i	$0.0417704\pi$
$331$	1575.95i	0.261699i	−0.991402	+	0.130849i	$0.958230\pi$
$332$	− 1879.47i	− 0.310691i
$333$	− 3813.93i	− 0.627635i
$334$	7015.44	1.14930
$335$	−5833.22	−0.951351
$336$	− 3243.18i	− 0.526577i
$337$	−9289.32	−1.50155	−0.750774	−	0.660559i	$-0.770319\pi$
$337$	−9289.32	−1.50155	−0.750774	+	0.660559i	$0.770319\pi$
$338$	0	0
$339$	2881.50	0.461656
$340$	− 818.284i	− 0.130523i
$341$	−2414.03	−0.383363
$342$	1832.62	0.289757
$343$	6875.66i	1.08236i
$344$	− 2414.05i	− 0.378363i
$345$	− 220.892i	− 0.0344708i
$346$	− 2098.62i	− 0.326077i
$347$	−7701.82	−1.19151	−0.595757	−	0.803164i	$-0.703148\pi$
$347$	−7701.82	−1.19151	−0.595757	+	0.803164i	$0.703148\pi$
$348$	−1343.80	−0.206998
$349$	4972.89i	0.762730i	0.924425	+	0.381365i	$0.124546\pi$
$349$	4972.89i	0.762730i	−0.924425	+	0.381365i	$0.875454\pi$
$350$	3577.96	0.546429
$351$	0	0
$352$	5720.49	0.866202
$353$	1575.34i	0.237526i	0.992923	+	0.118763i	$0.0378929\pi$
$353$	1575.34i	0.237526i	−0.992923	+	0.118763i	$0.962107\pi$
$354$	−3681.79	−0.552782
$355$	4095.49	0.612298
$356$	− 2140.45i	− 0.318661i
$357$	2492.28i	0.369483i
$358$	− 7378.41i	− 1.08928i
$359$	− 7567.42i	− 1.11252i	−0.831010	−	0.556258i	$-0.812237\pi$
$359$	− 7567.42i	− 1.11252i	0.831010	−	0.556258i	$-0.187763\pi$
$360$	1141.64	0.167138
$361$	2741.83	0.399741
$362$	6734.69i	0.977810i
$363$	−7724.12	−1.11683
$364$	0	0
$365$	6613.10	0.948344
$366$	5912.86i	0.844454i
$367$	4368.25	0.621310	0.310655	−	0.950523i	$-0.399452\pi$
$367$	4368.25	0.621310	0.310655	+	0.950523i	$0.399452\pi$
$368$	833.112	0.118014
$369$	− 3295.13i	− 0.464872i
$370$	− 9066.01i	− 1.27384i
$371$	1869.49i	0.261615i
$372$	239.960i	0.0334445i
$373$	−801.944	−0.111322	−0.0556610	−	0.998450i	$-0.517727\pi$
$373$	−801.944	−0.111322	−0.0556610	+	0.998450i	$0.517727\pi$
$374$	−11625.3	−1.60730
$375$	4136.96i	0.569684i
$376$	1752.10	0.240313
$377$	0	0
$378$	1214.35	0.165237
$379$	68.0819i	0.00922727i	0.999989	+	0.00461363i	$0.00146857\pi$
$379$	68.0819i	0.00922727i	−0.999989	+	0.00461363i	$0.998531\pi$
$380$	895.734	0.120922
$381$	−6066.55	−0.815745
$382$	7884.01i	1.05597i
$383$	1549.01i	0.206659i	0.994647	+	0.103330i	$0.0329497\pi$
$383$	1549.01i	0.206659i	−0.994647	+	0.103330i	$0.967050\pi$
$384$	5240.90i	0.696480i
$385$	5971.11i	0.790431i
$386$	846.584	0.111632
$387$	1154.67	0.151667
$388$	− 2166.34i	− 0.283451i
$389$	−7300.51	−0.951544	−0.475772	−	0.879569i	$-0.657831\pi$
$389$	−7300.51	−0.951544	−0.475772	+	0.879569i	$0.657831\pi$
$390$	0	0
$391$	−640.220	−0.0828065
$392$	− 2674.46i	− 0.344594i
$393$	5573.69	0.715408
$394$	−3906.61	−0.499524
$395$	8979.00i	1.14375i
$396$	1164.71i	0.147800i
$397$	− 6096.27i	− 0.770688i	−0.922773	−	0.385344i	$-0.874083\pi$
$397$	− 6096.27i	− 0.770688i	0.922773	−	0.385344i	$-0.125917\pi$
$398$	− 10301.4i	− 1.29740i
$399$	−2728.17	−0.342304
$400$	−6068.11	−0.758513
$401$	− 7592.37i	− 0.945498i	−0.881197	−	0.472749i	$-0.843262\pi$
$401$	− 7592.37i	− 0.945498i	0.881197	−	0.472749i	$-0.156738\pi$
$402$	8237.69	1.02204
$403$	0	0
$404$	2933.49	0.361254
$405$	546.059i	0.0669973i
$406$	9729.05	1.18927
$407$	−26483.8	−3.22544
$408$	− 3308.85i	− 0.401502i
$409$	− 7233.86i	− 0.874551i	−0.899328	−	0.437275i	$-0.855943\pi$
$409$	− 7233.86i	− 0.874551i	0.899328	−	0.437275i	$-0.144057\pi$
$410$	− 7832.77i	− 0.943495i
$411$	− 5682.36i	− 0.681971i
$412$	802.678	0.0959832
$413$	5480.97	0.653029
$414$	311.945i	0.0370320i
$415$	−6118.78	−0.723757
$416$	0	0
$417$	−3678.25	−0.431953
$418$	− 12725.6i	− 1.48907i
$419$	−5312.55	−0.619416	−0.309708	−	0.950832i	$-0.600231\pi$
$419$	−5312.55	−0.619416	−0.309708	+	0.950832i	$0.600231\pi$
$420$	593.542	0.0689569
$421$	15028.1i	1.73973i	0.493290	+	0.869865i	$0.335794\pi$
$421$	15028.1i	1.73973i	−0.493290	+	0.869865i	$0.664206\pi$
$422$	− 1049.48i	− 0.121062i
$423$	838.052i	0.0963297i
$424$	− 2482.02i	− 0.284286i
$425$	4663.15	0.532226
$426$	−5783.66	−0.657791
$427$	− 8802.31i	− 0.997596i
$428$	−179.020	−0.0202179
$429$	0	0
$430$	2744.73	0.307821
$431$	− 7154.66i	− 0.799600i	−0.916602	−	0.399800i	$-0.869080\pi$
$431$	− 7154.66i	− 0.799600i	0.916602	−	0.399800i	$-0.130920\pi$
$432$	−2059.50	−0.229370
$433$	9542.58	1.05909	0.529546	−	0.848281i	$-0.322362\pi$
$433$	9542.58	1.05909	0.529546	+	0.848281i	$0.322362\pi$
$434$	− 1737.30i	− 0.192150i
$435$	4374.87i	0.482204i
$436$	1946.64i	0.213824i
$437$	− 700.816i	− 0.0767153i
$438$	−9339.04	−1.01881
$439$	−7070.70	−0.768715	−0.384358	−	0.923184i	$-0.625577\pi$
$439$	−7070.70	−0.768715	−0.384358	+	0.923184i	$0.625577\pi$
$440$	− 7927.49i	− 0.858928i
$441$	1279.23	0.138130
$442$	0	0
$443$	2092.58	0.224428	0.112214	−	0.993684i	$-0.464206\pi$
$443$	2092.58	0.224428	0.112214	+	0.993684i	$0.464206\pi$
$444$	2632.55i	0.281386i
$445$	−6968.42	−0.742326
$446$	18361.1	1.94938
$447$	9586.96i	1.01442i
$448$	− 4531.63i	− 0.477901i
$449$	− 5842.05i	− 0.614038i	−0.951703	−	0.307019i	$-0.900668\pi$
$449$	− 5842.05i	− 0.614038i	0.951703	−	0.307019i	$-0.0993316\pi$
$450$	− 2272.10i	− 0.238017i
$451$	−22881.3	−2.38899
$452$	−1988.94	−0.206973
$453$	− 1524.69i	− 0.158138i
$454$	9346.91	0.966238
$455$	0	0
$456$	3622.03	0.371967
$457$	5954.40i	0.609486i	0.952435	+	0.304743i	$0.0985705\pi$
$457$	5954.40i	0.609486i	−0.952435	+	0.304743i	$0.901429\pi$
$458$	11238.0	1.14654
$459$	1582.66	0.160942
$460$	152.470i	0.0154542i
$461$	− 1865.94i	− 0.188515i	−0.995548	−	0.0942576i	$-0.969952\pi$
$461$	− 1865.94i	− 0.188515i	0.995548	−	0.0942576i	$-0.0300477\pi$
$462$	− 8432.42i	− 0.849160i
$463$	6700.05i	0.672522i	0.941769	+	0.336261i	$0.109162\pi$
$463$	6700.05i	0.672522i	−0.941769	+	0.336261i	$0.890838\pi$
$464$	−16500.2	−1.65086
$465$	781.211	0.0779093
$466$	7428.28i	0.738430i
$467$	16585.8	1.64347	0.821734	−	0.569871i	$-0.193007\pi$
$467$	16585.8	1.64347	0.821734	+	0.569871i	$0.193007\pi$
$468$	0	0
$469$	−12263.2	−1.20738
$470$	1992.11i	0.195509i
$471$	−3729.23	−0.364828
$472$	−7276.77	−0.709619
$473$	− 8017.98i	− 0.779423i
$474$	− 12680.2i	− 1.22873i
$475$	5104.51i	0.493075i
$476$	− 1720.29i	− 0.165649i
$477$	1187.18	0.113956
$478$	−4809.97	−0.460257
$479$	6166.88i	0.588250i	0.955767	+	0.294125i	$0.0950282\pi$
$479$	6166.88i	0.588250i	−0.955767	+	0.294125i	$0.904972\pi$
$480$	−1851.23	−0.176034
$481$	0	0
$482$	−7592.38	−0.717476
$483$	− 464.383i	− 0.0437478i
$484$	5331.53	0.500708
$485$	−7052.71	−0.660303
$486$	− 771.146i	− 0.0719751i
$487$	5718.51i	0.532095i	0.963960	+	0.266047i	$0.0857178\pi$
$487$	5718.51i	0.532095i	−0.963960	+	0.266047i	$0.914282\pi$
$488$	11686.3i	1.08405i
$489$	101.999i	0.00943261i
$490$	3040.82	0.280347
$491$	21060.9	1.93578	0.967888	−	0.251383i	$-0.0808854\pi$
$491$	21060.9	1.93578	0.967888	+	0.251383i	$0.0808854\pi$
$492$	2274.45i	0.208415i
$493$	12679.8	1.15836
$494$	0	0
$495$	3791.81	0.344302
$496$	2946.40i	0.266728i
$497$	8609.97	0.777082
$498$	8640.97	0.777532
$499$	− 7863.87i	− 0.705481i	−0.935721	−	0.352741i	$-0.885250\pi$
$499$	− 7863.87i	− 0.705481i	0.935721	−	0.352741i	$-0.114750\pi$
$500$	− 2855.51i	− 0.255405i
$501$	6632.01i	0.591410i
$502$	− 6977.70i	− 0.620378i
$503$	−6504.06	−0.576544	−0.288272	−	0.957549i	$-0.593081\pi$
$503$	−6504.06	−0.576544	−0.288272	+	0.957549i	$0.593081\pi$
$504$	2400.07	0.212119
$505$	− 9550.26i	− 0.841546i
$506$	2166.13	0.190309
$507$	0	0
$508$	4187.41	0.365721
$509$	14799.3i	1.28873i	0.764717	+	0.644367i	$0.222879\pi$
$509$	14799.3i	1.28873i	−0.764717	+	0.644367i	$0.777121\pi$
$510$	3762.11	0.326645
$511$	13902.8	1.20357
$512$	4500.03i	0.388428i
$513$	1732.46i	0.149103i
$514$	− 19657.1i	− 1.68685i
$515$	− 2613.19i	− 0.223594i
$516$	−797.005	−0.0679965
$517$	5819.40	0.495042
$518$	− 19059.5i	− 1.61666i
$519$	1983.92	0.167793
$520$	0	0
$521$	−6633.65	−0.557822	−0.278911	−	0.960317i	$-0.589974\pi$
$521$	−6633.65	−0.557822	−0.278911	+	0.960317i	$0.589974\pi$
$522$	− 6178.20i	− 0.518032i
$523$	4527.41	0.378527	0.189264	−	0.981926i	$-0.439390\pi$
$523$	4527.41	0.378527	0.189264	+	0.981926i	$0.439390\pi$
$524$	−3847.21	−0.320737
$525$	3382.41i	0.281182i
$526$	13270.5i	1.10004i
$527$	− 2264.21i	− 0.187155i
$528$	14301.1i	1.17874i
$529$	−12047.7	−0.990195
$530$	2822.01	0.231284
$531$	− 3480.56i	− 0.284451i
$532$	1883.11	0.153464
$533$	0	0
$534$	9840.83	0.797480
$535$	582.818i	0.0470980i
$536$	16281.1	1.31201
$537$	6975.14	0.560521
$538$	− 8783.11i	− 0.703841i
$539$	− 8882.89i	− 0.709858i
$540$	− 376.915i	− 0.0300367i
$541$	− 8685.42i	− 0.690232i	−0.938560	−	0.345116i	$-0.887840\pi$
$541$	− 8685.42i	− 0.690232i	0.938560	−	0.345116i	$-0.112160\pi$
$542$	22821.4	1.80860
$543$	−6366.61	−0.503162
$544$	5365.48i	0.422873i
$545$	6337.47	0.498105
$546$	0	0
$547$	24656.6	1.92731	0.963657	−	0.267142i	$-0.0860793\pi$
$547$	24656.6	1.92731	0.963657	+	0.267142i	$0.0860793\pi$
$548$	3922.22i	0.305746i
$549$	−5589.70	−0.434540
$550$	−15777.4	−1.22318
$551$	13880.0i	1.07315i
$552$	616.534i	0.0475388i
$553$	18876.6i	1.45157i
$554$	4180.27i	0.320582i
$555$	8570.51	0.655492
$556$	2538.89	0.193656
$557$	− 9215.90i	− 0.701059i	−0.936552	−	0.350530i	$-0.886002\pi$
$557$	− 9215.90i	− 0.701059i	0.936552	−	0.350530i	$-0.113998\pi$
$558$	−1103.23	−0.0836979
$559$	0	0
$560$	7287.93	0.549949
$561$	− 10990.0i	− 0.827088i
$562$	12531.7	0.940600
$563$	19686.7	1.47370	0.736850	−	0.676056i	$-0.236312\pi$
$563$	19686.7	1.47370	0.736850	+	0.676056i	$0.236312\pi$
$564$	− 578.461i	− 0.0431873i
$565$	6475.17i	0.482146i
$566$	− 15808.6i	− 1.17400i
$567$	1147.98i	0.0850279i
$568$	−11430.9	−0.844422
$569$	3559.36	0.262243	0.131121	−	0.991366i	$-0.458142\pi$
$569$	3559.36	0.262243	0.131121	+	0.991366i	$0.458142\pi$
$570$	4118.19i	0.302617i
$571$	710.968	0.0521070	0.0260535	−	0.999661i	$-0.491706\pi$
$571$	710.968	0.0521070	0.0260535	+	0.999661i	$0.491706\pi$
$572$	0	0
$573$	−7453.11	−0.543383
$574$	− 16466.9i	− 1.19741i
$575$	−868.878	−0.0630169
$576$	−2877.70	−0.208167
$577$	− 8041.67i	− 0.580206i	−0.956995	−	0.290103i	$-0.906310\pi$
$577$	− 8041.67i	− 0.580206i	0.956995	−	0.290103i	$-0.0936896\pi$
$578$	4687.26i	0.337308i
$579$	800.314i	0.0574438i
$580$	− 3019.73i	− 0.216186i
$581$	−12863.6	−0.918538
$582$	9959.86	0.709364
$583$	− 8243.72i	− 0.585626i
$584$	−18457.9	−1.30786
$585$	0	0
$586$	10164.6	0.716546
$587$	− 14641.3i	− 1.02949i	−0.857344	−	0.514744i	$-0.827887\pi$
$587$	− 14641.3i	− 1.02949i	0.857344	−	0.514744i	$-0.172113\pi$
$588$	−882.980	−0.0619277
$589$	2478.52	0.173388
$590$	− 8273.56i	− 0.577317i
$591$	− 3693.10i	− 0.257045i
$592$	32324.3i	2.24413i
$593$	− 5735.76i	− 0.397200i	−0.980081	−	0.198600i	$-0.936361\pi$
$593$	− 5735.76i	− 0.397200i	0.980081	−	0.198600i	$-0.0636394\pi$
$594$	−5354.81	−0.369883
$595$	−5600.55	−0.385882
$596$	− 6617.35i	− 0.454794i
$597$	9738.43	0.667617
$598$	0	0
$599$	−16109.3	−1.09884	−0.549422	−	0.835545i	$-0.685152\pi$
$599$	−16109.3	−1.09884	−0.549422	+	0.835545i	$0.685152\pi$
$600$	− 4490.63i	− 0.305548i
$601$	−21005.4	−1.42567	−0.712837	−	0.701330i	$-0.752590\pi$
$601$	−21005.4	−1.42567	−0.712837	+	0.701330i	$0.752590\pi$
$602$	5770.28	0.390663
$603$	7787.46i	0.525920i
$604$	1052.41i	0.0708975i
$605$	− 17357.3i	− 1.16640i
$606$	13486.9i	0.904073i
$607$	7478.16	0.500048	0.250024	−	0.968240i	$-0.419561\pi$
$607$	7478.16	0.500048	0.250024	+	0.968240i	$0.419561\pi$
$608$	−5873.31	−0.391767
$609$	9197.32i	0.611977i
$610$	−13287.1	−0.881934
$611$	0	0
$612$	−1092.43	−0.0721548
$613$	16435.5i	1.08291i	0.840730	+	0.541455i	$0.182126\pi$
$613$	16435.5i	1.08291i	−0.840730	+	0.541455i	$0.817874\pi$
$614$	−15218.8	−1.00029
$615$	7404.68	0.485505
$616$	− 16666.0i	− 1.09009i
$617$	1290.89i	0.0842289i	0.999113	+	0.0421145i	$0.0134094\pi$
$617$	1290.89i	0.0842289i	−0.999113	+	0.0421145i	$0.986591\pi$
$618$	3690.36i	0.240207i
$619$	26719.0i	1.73494i	0.497490	+	0.867470i	$0.334255\pi$
$619$	26719.0i	1.73494i	−0.497490	+	0.867470i	$0.665745\pi$
$620$	−539.227	−0.0349289
$621$	−294.896	−0.0190560
$622$	1999.94i	0.128924i
$623$	−14649.8	−0.942103
$624$	0	0
$625$	647.683	0.0414517
$626$	29559.8i	1.88729i
$627$	12030.1	0.766248
$628$	2574.08	0.163562
$629$	− 24840.2i	− 1.57463i
$630$	2728.84i	0.172571i
$631$	10697.4i	0.674893i	0.941345	+	0.337447i	$0.109563\pi$
$631$	10697.4i	0.674893i	−0.941345	+	0.337447i	$0.890437\pi$
$632$	− 25061.4i	− 1.57735i
$633$	992.125	0.0622961
$634$	1828.96	0.114570
$635$	− 13632.5i	− 0.851951i
$636$	−819.444	−0.0510897
$637$	0	0
$638$	−42901.2	−2.66219
$639$	− 5467.56i	− 0.338487i
$640$	−11777.1	−0.727393
$641$	−23572.8	−1.45253	−0.726264	−	0.687416i	$-0.758745\pi$
$641$	−23572.8	−1.45253	−0.726264	+	0.687416i	$0.758745\pi$
$642$	− 823.057i	− 0.0505973i
$643$	− 14000.3i	− 0.858661i	−0.903147	−	0.429331i	$-0.858750\pi$
$643$	− 14000.3i	− 0.858661i	0.903147	−	0.429331i	$-0.141250\pi$
$644$	320.538i	0.0196133i
$645$	2594.72i	0.158399i
$646$	11935.9	0.726953
$647$	−614.196	−0.0373207	−0.0186604	−	0.999826i	$-0.505940\pi$
$647$	−614.196	−0.0373207	−0.0186604	+	0.999826i	$0.505940\pi$
$648$	− 1524.11i	− 0.0923962i
$649$	−24168.9	−1.46181
$650$	0	0
$651$	1642.35	0.0988765
$652$	− 70.4042i	− 0.00422890i
$653$	−5333.42	−0.319622	−0.159811	−	0.987148i	$-0.551088\pi$
$653$	−5333.42	−0.319622	−0.159811	+	0.987148i	$0.551088\pi$
$654$	−8949.79	−0.535114
$655$	12524.9i	0.747161i
$656$	27927.3i	1.66216i
$657$	− 8828.62i	− 0.524258i
$658$	4188.03i	0.248125i
$659$	21396.8	1.26480	0.632398	−	0.774644i	$-0.282071\pi$
$659$	21396.8	1.26480	0.632398	+	0.774644i	$0.282071\pi$
$660$	−2617.28	−0.154360
$661$	16107.9i	0.947841i	0.880568	+	0.473921i	$0.157162\pi$
$661$	16107.9i	0.947841i	−0.880568	+	0.473921i	$0.842838\pi$
$662$	5001.20	0.293621
$663$	0	0
$664$	17078.2	0.998136
$665$	− 6130.63i	− 0.357497i
$666$	−12103.3	−0.704194
$667$	−2362.62	−0.137153
$668$	− 4577.72i	− 0.265145i
$669$	17357.6i	1.00311i
$670$	18511.4i	1.06740i
$671$	38814.6i	2.23312i
$672$	−3891.85	−0.223410
$673$	−20329.9	−1.16443	−0.582213	−	0.813036i	$-0.697813\pi$
$673$	−20329.9	−1.16443	−0.582213	+	0.813036i	$0.697813\pi$
$674$	29479.1i	1.68471i
$675$	2147.92	0.122479
$676$	0	0
$677$	−24883.5	−1.41263	−0.706314	−	0.707899i	$-0.749643\pi$
$677$	−24883.5	−1.41263	−0.706314	+	0.707899i	$0.749643\pi$
$678$	− 9144.26i	− 0.517969i
$679$	−14827.0	−0.838007
$680$	7435.51	0.419322
$681$	8836.06i	0.497208i
$682$	7660.78i	0.430127i
$683$	258.953i	0.0145074i	0.999974	+	0.00725369i	$0.00230894\pi$
$683$	258.953i	0.0145074i	−0.999974	+	0.00725369i	$0.997691\pi$
$684$	− 1195.82i	− 0.0668471i
$685$	12769.1	0.712240
$686$	21819.5	1.21439
$687$	10623.8i	0.589989i
$688$	−9786.20	−0.542290
$689$	0	0
$690$	−700.989	−0.0386756
$691$	− 658.193i	− 0.0362357i	−0.999836	−	0.0181178i	$-0.994233\pi$
$691$	− 658.193i	− 0.0362357i	0.999836	−	0.0181178i	$-0.00576740\pi$
$692$	−1369.39	−0.0752262
$693$	7971.55	0.436962
$694$	24441.3i	1.33686i
$695$	− 8265.60i	− 0.451125i
$696$	− 12210.7i	− 0.665010i
$697$	− 21461.2i	− 1.16629i
$698$	15781.2	0.855768
$699$	−7022.29	−0.379982
$700$	− 2334.69i	− 0.126062i
$701$	8222.16	0.443005	0.221503	−	0.975160i	$-0.428904\pi$
$701$	8222.16	0.443005	0.221503	+	0.975160i	$0.428904\pi$
$702$	0	0
$703$	27191.3	1.45881
$704$	19982.7i	1.06978i
$705$	−1883.23	−0.100605
$706$	4999.25	0.266500
$707$	− 20077.6i	− 1.06803i
$708$	2402.44i	0.127527i
$709$	− 6817.51i	− 0.361124i	−0.983564	−	0.180562i	$-0.942208\pi$
$709$	− 6817.51i	− 0.361124i	0.983564	−	0.180562i	$-0.0577917\pi$
$710$	− 12996.8i	− 0.686987i
$711$	11987.2	0.632283
$712$	19449.6	1.02374
$713$	421.888i	0.0221596i
$714$	7909.11	0.414553
$715$	0	0
$716$	−4814.56	−0.251297
$717$	− 4547.09i	− 0.236840i
$718$	−24014.8	−1.24822
$719$	−23385.7	−1.21299	−0.606496	−	0.795087i	$-0.707425\pi$
$719$	−23385.7	−1.21299	−0.606496	+	0.795087i	$0.707425\pi$
$720$	− 4628.03i	− 0.239551i
$721$	− 5493.73i	− 0.283769i
$722$	− 8701.03i	− 0.448502i
$723$	− 7177.42i	− 0.369199i
$724$	4394.52	0.225582
$725$	17208.5	0.881529
$726$	24512.0i	1.25307i
$727$	−20488.6	−1.04523	−0.522613	−	0.852570i	$-0.675043\pi$
$727$	−20488.6	−1.04523	−0.522613	+	0.852570i	$0.675043\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 20986.3i	− 1.06402i
$731$	7520.38	0.380508
$732$	3858.26	0.194816
$733$	− 16993.4i	− 0.856299i	−0.903708	−	0.428149i	$-0.859166\pi$
$733$	− 16993.4i	− 0.856299i	0.903708	−	0.428149i	$-0.140834\pi$
$734$	− 13862.4i	− 0.697099i
$735$	2874.62i	0.144261i
$736$	− 999.742i	− 0.0500693i
$737$	54075.8	2.70272
$738$	−10456.9	−0.521577
$739$	14014.3i	0.697600i	0.937197	+	0.348800i	$0.113411\pi$
$739$	14014.3i	0.697600i	−0.937197	+	0.348800i	$0.886589\pi$
$740$	−5915.75	−0.293875
$741$	0	0
$742$	5932.73	0.293528
$743$	− 141.560i	− 0.00698971i	−0.999994	−	0.00349485i	$-0.998888\pi$
$743$	− 141.560i	− 0.00698971i	0.999994	−	0.00349485i	$-0.00111245\pi$
$744$	−2180.45	−0.107445
$745$	−21543.4	−1.05945
$746$	2544.92i	0.124901i
$747$	8168.70i	0.400103i
$748$	7585.76i	0.370806i
$749$	1225.26i	0.0597731i
$750$	13128.4	0.639175
$751$	−20734.3	−1.00746	−0.503732	−	0.863860i	$-0.668040\pi$
$751$	−20734.3	−1.00746	−0.503732	+	0.863860i	$0.668040\pi$
$752$	− 7102.76i	− 0.344430i
$753$	6596.34	0.319235
$754$	0	0
$755$	3426.23	0.165157
$756$	− 792.391i	− 0.0381203i
$757$	17449.8	0.837812	0.418906	−	0.908030i	$-0.362414\pi$
$757$	17449.8	0.837812	0.418906	+	0.908030i	$0.362414\pi$
$758$	216.054	0.0103528
$759$	2047.74i	0.0979293i
$760$	8139.27i	0.388477i
$761$	424.121i	0.0202028i	0.999949	+	0.0101014i	$0.00321544\pi$
$761$	424.121i	0.0202028i	−0.999949	+	0.0101014i	$0.996785\pi$
$762$	19251.9i	0.915251i
$763$	13323.3	0.632157
$764$	5144.48	0.243613
$765$	3556.49i	0.168085i
$766$	4915.68	0.231868
$767$	0	0
$768$	8957.81	0.420882
$769$	38060.7i	1.78479i	0.451252	+	0.892396i	$0.350977\pi$
$769$	38060.7i	1.78479i	−0.451252	+	0.892396i	$0.649023\pi$
$770$	18949.0	0.886849
$771$	18582.8	0.868019
$772$	− 552.413i	− 0.0257536i
$773$	− 16683.3i	− 0.776268i	−0.921603	−	0.388134i	$-0.873120\pi$
$773$	− 16683.3i	− 0.776268i	0.921603	−	0.388134i	$-0.126880\pi$
$774$	− 3664.28i	− 0.170168i
$775$	− 3072.89i	− 0.142428i
$776$	19684.9	0.910627
$777$	18017.8	0.831900
$778$	23167.7i	1.06761i
$779$	23492.5	1.08050
$780$	0	0
$781$	−37966.5	−1.73950
$782$	2031.70i	0.0929073i
$783$	5840.54	0.266569
$784$	−10841.9	−0.493889
$785$	− 8380.17i	− 0.381020i
$786$	− 17687.8i	− 0.802674i
$787$	39105.2i	1.77122i	0.464432	+	0.885609i	$0.346259\pi$
$787$	39105.2i	1.77122i	−0.464432	+	0.885609i	$0.653741\pi$
$788$	2549.14i	0.115240i
$789$	−12545.2	−0.566060
$790$	28494.3	1.28327
$791$	13612.8i	0.611903i
$792$	−10583.4	−0.474827
$793$	0	0
$794$	−19346.2	−0.864697
$795$	2667.78i	0.119014i
$796$	−6721.90	−0.299311
$797$	4346.93	0.193195	0.0965974	−	0.995324i	$-0.469204\pi$
$797$	4346.93	0.193195	0.0965974	+	0.995324i	$0.469204\pi$
$798$	8657.69i	0.384059i
$799$	5458.25i	0.241676i
$800$	7281.78i	0.321812i
$801$	9302.99i	0.410368i
$802$	−24093.9	−1.06083
$803$	−61305.6	−2.69418
$804$	− 5375.26i	− 0.235784i
$805$	1043.54	0.0456895
$806$	0	0
$807$	8303.07	0.362183
$808$	26655.8i	1.16058i
$809$	−23030.2	−1.00086	−0.500432	−	0.865776i	$-0.666826\pi$
$809$	−23030.2	−1.00086	−0.500432	+	0.865776i	$0.666826\pi$
$810$	1732.89	0.0751697
$811$	7898.11i	0.341973i	0.985273	+	0.170987i	$0.0546955\pi$
$811$	7898.11i	0.341973i	−0.985273	+	0.170987i	$0.945305\pi$
$812$	− 6348.41i	− 0.274366i
$813$	21574.1i	0.930672i
$814$	84044.8i	3.61888i
$815$	−229.207	−0.00985127
$816$	−13413.6	−0.575453
$817$	8232.18i	0.352518i
$818$	−22956.2	−0.981230
$819$	0	0
$820$	−5111.04	−0.217665
$821$	− 3939.61i	− 0.167471i	−0.996488	−	0.0837353i	$-0.973315\pi$
$821$	− 3939.61i	− 0.167471i	0.996488	−	0.0837353i	$-0.0266850\pi$
$822$	−18032.6	−0.765159
$823$	17599.8	0.745430	0.372715	−	0.927946i	$-0.378427\pi$
$823$	17599.8	0.745430	0.372715	+	0.927946i	$0.378427\pi$
$824$	7293.70i	0.308359i
$825$	− 14915.1i	− 0.629425i
$826$	− 17393.6i	− 0.732687i
$827$	− 12510.6i	− 0.526042i	−0.964790	−	0.263021i	$-0.915281\pi$
$827$	− 12510.6i	− 0.526042i	0.964790	−	0.263021i	$-0.0847189\pi$
$828$	203.550	0.00854331
$829$	−28630.8	−1.19950	−0.599752	−	0.800186i	$-0.704734\pi$
$829$	−28630.8	−1.19950	−0.599752	+	0.800186i	$0.704734\pi$
$830$	19417.6i	0.812042i
$831$	−3951.80	−0.164966
$832$	0	0
$833$	8331.62	0.346547
$834$	11672.7i	0.484643i
$835$	−14903.2	−0.617660
$836$	−8303.75	−0.343530
$837$	− 1042.93i	− 0.0430693i
$838$	16859.1i	0.694973i
$839$	− 21250.3i	− 0.874426i	−0.899358	−	0.437213i	$-0.855966\pi$
$839$	− 21250.3i	− 0.874426i	0.899358	−	0.437213i	$-0.144034\pi$
$840$	5393.34i	0.221533i
$841$	22403.7	0.918600
$842$	47690.9	1.95194
$843$	11846.8i	0.484015i
$844$	−684.810	−0.0279291
$845$	0	0
$846$	2659.51	0.108080
$847$	− 36490.4i	− 1.48031i
$848$	−10061.7	−0.407454
$849$	14944.6	0.604118
$850$	− 14798.2i	− 0.597147i
$851$	4628.45i	0.186441i
$852$	3773.96i	0.151753i
$853$	− 34721.1i	− 1.39370i	−0.717215	−	0.696852i	$-0.754584\pi$
$853$	− 34721.1i	− 1.39370i	0.717215	−	0.696852i	$-0.245416\pi$
$854$	−27933.6	−1.11928
$855$	−3893.11	−0.155721
$856$	− 1626.71i	− 0.0649529i
$857$	−4898.06	−0.195233	−0.0976163	−	0.995224i	$-0.531122\pi$
$857$	−4898.06	−0.195233	−0.0976163	+	0.995224i	$0.531122\pi$
$858$	0	0
$859$	−12564.2	−0.499051	−0.249526	−	0.968368i	$-0.580275\pi$
$859$	−12564.2	−0.499051	−0.249526	+	0.968368i	$0.580275\pi$
$860$	− 1791.00i	− 0.0710145i
$861$	15566.9	0.616166
$862$	−22704.9	−0.897137
$863$	− 22826.6i	− 0.900380i	−0.892933	−	0.450190i	$-0.851356\pi$
$863$	− 22826.6i	− 0.900380i	0.892933	−	0.450190i	$-0.148644\pi$
$864$	2471.42i	0.0973143i
$865$	4458.18i	0.175240i
$866$	− 30282.8i	− 1.18828i
$867$	−4431.08	−0.173572
$868$	−1133.62	−0.0443291
$869$	− 83238.3i	− 3.24933i
$870$	13883.4	0.541024
$871$	0	0
$872$	−17688.6	−0.686939
$873$	9415.51i	0.365025i
$874$	−2224.00	−0.0860731
$875$	−19543.9	−0.755089
$876$	6093.92i	0.235039i
$877$	25212.7i	0.970776i	0.874299	+	0.485388i	$0.161322\pi$
$877$	25212.7i	0.970776i	−0.874299	+	0.485388i	$0.838678\pi$
$878$	22438.5i	0.862484i
$879$	9609.07i	0.368721i
$880$	−32136.9	−1.23106
$881$	18026.2	0.689352	0.344676	−	0.938722i	$-0.387989\pi$
$881$	18026.2	0.689352	0.344676	+	0.938722i	$0.387989\pi$
$882$	− 4059.55i	− 0.154980i
$883$	18833.1	0.717764	0.358882	−	0.933383i	$-0.383158\pi$
$883$	18833.1	0.717764	0.358882	+	0.933383i	$0.383158\pi$
$884$	0	0
$885$	7821.37	0.297076
$886$	− 6640.68i	− 0.251803i
$887$	38451.4	1.45555	0.727775	−	0.685816i	$-0.240555\pi$
$887$	38451.4	1.45555	0.727775	+	0.685816i	$0.240555\pi$
$888$	−23921.2	−0.903991
$889$	− 28659.7i	− 1.08123i
$890$	22113.9i	0.832876i
$891$	− 5062.15i	− 0.190335i
$892$	− 11981.0i	− 0.449723i
$893$	−5974.86	−0.223898
$894$	30423.7	1.13816
$895$	15674.2i	0.585399i
$896$	−24759.1	−0.923152
$897$	0	0
$898$	−18539.4	−0.688940
$899$	− 8355.68i	− 0.309986i
$900$	−1482.59	−0.0549108
$901$	7732.11	0.285898
$902$	72612.3i	2.68041i
$903$	5454.91i	0.201028i
$904$	− 18072.9i	− 0.664929i
$905$	− 14306.8i	− 0.525495i
$906$	−4838.53	−0.177428
$907$	5531.31	0.202496	0.101248	−	0.994861i	$-0.467716\pi$
$907$	5531.31	0.202496	0.101248	+	0.994861i	$0.467716\pi$
$908$	− 6099.05i	− 0.222912i
$909$	−12749.8	−0.465219
$910$	0	0
$911$	15695.2	0.570806	0.285403	−	0.958408i	$-0.407872\pi$
$911$	15695.2	0.570806	0.285403	+	0.958408i	$0.407872\pi$
$912$	− 14683.2i	− 0.533123i
$913$	56723.1	2.05615
$914$	18895.9	0.683832
$915$	− 12560.9i	− 0.453827i
$916$	− 7333.01i	− 0.264508i
$917$	26331.3i	0.948240i
$918$	− 5022.49i	− 0.180574i
$919$	−51503.5	−1.84869	−0.924344	−	0.381561i	$-0.875386\pi$
$919$	−51503.5	−1.84869	−0.924344	+	0.381561i	$0.875386\pi$
$920$	−1385.45	−0.0496488
$921$	− 14387.0i	− 0.514732i
$922$	−5921.45	−0.211510
$923$	0	0
$924$	−5502.33	−0.195902
$925$	− 33712.0i	− 1.19832i
$926$	21262.2	0.754557
$927$	−3488.66	−0.123606
$928$	19800.3i	0.700407i
$929$	− 42927.0i	− 1.51603i	−0.652238	−	0.758014i	$-0.726170\pi$
$929$	− 42927.0i	− 1.51603i	0.652238	−	0.758014i	$-0.273830\pi$
$930$	− 2479.13i	− 0.0874127i
$931$	9120.20i	0.321055i
$932$	4847.10	0.170356
$933$	−1890.64	−0.0663416
$934$	− 52634.1i	− 1.84394i
$935$	24696.2	0.863797
$936$	0	0
$937$	44206.6	1.54127	0.770633	−	0.637280i	$-0.219940\pi$
$937$	44206.6	1.54127	0.770633	+	0.637280i	$0.219940\pi$
$938$	38916.6i	1.35466i
$939$	−27944.2	−0.971165
$940$	1299.89	0.0451041
$941$	− 44067.7i	− 1.52664i	−0.646022	−	0.763319i	$-0.723568\pi$
$941$	− 44067.7i	− 1.52664i	0.646022	−	0.763319i	$-0.276432\pi$
$942$	11834.5i	0.409330i
$943$	3998.85i	0.138092i
$944$	29498.9i	1.01706i
$945$	−2579.70	−0.0888017
$946$	−25444.6	−0.874498
$947$	44402.5i	1.52364i	0.647789	+	0.761820i	$0.275694\pi$
$947$	44402.5i	1.52364i	−0.647789	+	0.761820i	$0.724306\pi$
$948$	−8274.07	−0.283470
$949$	0	0
$950$	16198.9	0.553221
$951$	1729.00i	0.0589554i
$952$	15631.7	0.532172
$953$	10361.7	0.352202	0.176101	−	0.984372i	$-0.443651\pi$
$953$	10361.7	0.352202	0.176101	+	0.984372i	$0.443651\pi$
$954$	− 3767.44i	− 0.127857i
$955$	− 16748.3i	− 0.567500i
$956$	3138.60i	0.106182i
$957$	− 40556.5i	− 1.36991i
$958$	19570.2	0.660006
$959$	26844.7	0.903920
$960$	− 6466.65i	− 0.217407i
$961$	28298.9	0.949916
$962$	0	0
$963$	778.073	0.0260364
$964$	4954.18i	0.165522i
$965$	−1798.43	−0.0599933
$966$	−1473.69	−0.0490842
$967$	8432.54i	0.280426i	0.990121	+	0.140213i	$0.0447788\pi$
$967$	8432.54i	0.280426i	−0.990121	+	0.140213i	$0.955221\pi$
$968$	48446.1i	1.60859i
$969$	11283.5i	0.374076i
$970$	22381.4i	0.740848i
$971$	−36917.1	−1.22011	−0.610055	−	0.792359i	$-0.708853\pi$
$971$	−36917.1	−1.22011	−0.610055	+	0.792359i	$0.708853\pi$
$972$	−503.189	−0.0166047
$973$	− 17376.8i	− 0.572534i
$974$	18147.4	0.597001
$975$	0	0
$976$	47374.5	1.55371
$977$	− 38306.9i	− 1.25440i	−0.778859	−	0.627199i	$-0.784201\pi$
$977$	− 38306.9i	− 1.25440i	0.778859	−	0.627199i	$-0.215799\pi$
$978$	323.687	0.0105832
$979$	64599.6	2.10890
$980$	− 1984.19i	− 0.0646763i
$981$	− 8460.65i	− 0.275360i
$982$	− 66835.6i	− 2.17190i
$983$	18810.9i	0.610350i	0.952296	+	0.305175i	$0.0987149\pi$
$983$	18810.9i	0.610350i	−0.952296	+	0.305175i	$0.901285\pi$
$984$	−20667.3	−0.669561
$985$	8298.97	0.268454
$986$	− 40238.8i	− 1.29966i
$987$	−3959.14	−0.127681
$988$	0	0
$989$	−1401.26	−0.0450532
$990$	− 12033.1i	− 0.386300i
$991$	−6492.09	−0.208101	−0.104051	−	0.994572i	$-0.533180\pi$
$991$	−6492.09	−0.208101	−0.104051	+	0.994572i	$0.533180\pi$
$992$	3535.71	0.113164
$993$	4727.86i	0.151092i
$994$	− 27323.2i	− 0.871872i
$995$	21883.8i	0.697249i
$996$	− 5638.41i	− 0.179377i
$997$	48552.1	1.54229	0.771143	−	0.636662i	$-0.219685\pi$
$997$	48552.1	1.54229	0.771143	+	0.636662i	$0.219685\pi$
$998$	−24955.5	−0.791537
$999$	− 11441.8i	− 0.362365i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.k.337.5		18
13.5	odd	4		507.4.a.p.1.2	yes	9
13.8	odd	4		507.4.a.o.1.8	✓	9
13.12	even	2	inner	507.4.b.k.337.14		18
39.5	even	4		1521.4.a.bf.1.8		9
39.8	even	4		1521.4.a.bi.1.2		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.8	✓	9	13.8	odd	4
507.4.a.p.1.2	yes	9	13.5	odd	4
507.4.b.k.337.5		18	1.1	even	1	trivial
507.4.b.k.337.14		18	13.12	even	2	inner
1521.4.a.bf.1.8		9	39.5	even	4
1521.4.a.bi.1.2		9	39.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.17344i q^{2} +3.00000 q^{3} -2.07074 q^{4} +6.74147i q^{5} -9.52033i q^{6} +14.1726i q^{7} -18.8162i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.17344i q^{2} +3.00000 q^{3} -2.07074 q^{4} +6.74147i q^{5} -9.52033i q^{6} +14.1726i q^{7} -18.8162i q^{8} +9.00000 q^{9} +21.3937 q^{10} -62.4956i q^{11} -6.21221 q^{12} +44.9761 q^{14} +20.2244i q^{15} -76.2779 q^{16} +58.6172 q^{17} -28.5610i q^{18} +64.1652i q^{19} -13.9598i q^{20} +42.5179i q^{21} -198.326 q^{22} -10.9221 q^{23} -56.4485i q^{24} +79.5526 q^{25} +27.0000 q^{27} -29.3478i q^{28} +216.316 q^{29} +64.1810 q^{30} -38.6271i q^{31} +91.5342i q^{32} -187.487i q^{33} -186.018i q^{34} -95.5445 q^{35} -18.6366 q^{36} -423.770i q^{37} +203.625 q^{38} +126.849 q^{40} -366.126i q^{41} +134.928 q^{42} +128.297 q^{43} +129.412i q^{44} +60.6732i q^{45} +34.6605i q^{46} +93.1169i q^{47} -228.834 q^{48} +142.136 q^{49} -252.455i q^{50} +175.852 q^{51} +131.909 q^{53} -85.6829i q^{54} +421.313 q^{55} +266.675 q^{56} +192.496i q^{57} -686.467i q^{58} -386.729i q^{59} -41.8794i q^{60} -621.077 q^{61} -122.581 q^{62} +127.554i q^{63} -319.745 q^{64} -594.979 q^{66} +865.273i q^{67} -121.381 q^{68} -32.7662 q^{69} +303.205i q^{70} -607.506i q^{71} -169.346i q^{72} -980.958i q^{73} -1344.81 q^{74} +238.658 q^{75} -132.869i q^{76} +885.728 q^{77} +1331.91 q^{79} -514.226i q^{80} +81.0000 q^{81} -1161.88 q^{82} +907.633i q^{83} -88.0434i q^{84} +395.166i q^{85} -407.142i q^{86} +648.949 q^{87} -1175.93 q^{88} +1033.67i q^{89} +192.543 q^{90} +22.6167 q^{92} -115.881i q^{93} +295.501 q^{94} -432.568 q^{95} +274.603i q^{96} +1046.17i q^{97} -451.061i q^{98} -562.461i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

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