LMFDB - Embedded newform 165.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,4,Mod(1,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("165.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 165 = 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	165.a (trivial)

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:	yes
Analytic conductor:	$9.73531515095$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1957.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 10 $ x^3 - x^2 - 9*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.91150$ of defining polynomial
Character	$\chi$	$=$	165.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.38835 q^{2} -3.00000 q^{3} +11.2577 q^{4} +5.00000 q^{5} +13.1651 q^{6} -11.7304 q^{7} -14.2958 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.38835 q^{2} -3.00000 q^{3} +11.2577 q^{4} +5.00000 q^{5} +13.1651 q^{6} -11.7304 q^{7} -14.2958 q^{8} +9.00000 q^{9} -21.9418 q^{10} +11.0000 q^{11} -33.7730 q^{12} -72.8298 q^{13} +51.4772 q^{14} -15.0000 q^{15} -27.3264 q^{16} -9.89921 q^{17} -39.4952 q^{18} +0.0238576 q^{19} +56.2883 q^{20} +35.1912 q^{21} -48.2719 q^{22} +73.0456 q^{23} +42.8873 q^{24} +25.0000 q^{25} +319.603 q^{26} -27.0000 q^{27} -132.057 q^{28} +202.097 q^{29} +65.8253 q^{30} +181.642 q^{31} +234.284 q^{32} -33.0000 q^{33} +43.4413 q^{34} -58.6521 q^{35} +101.319 q^{36} +299.887 q^{37} -0.104695 q^{38} +218.489 q^{39} -71.4788 q^{40} +88.5438 q^{41} -154.432 q^{42} -146.114 q^{43} +123.834 q^{44} +45.0000 q^{45} -320.550 q^{46} +185.746 q^{47} +81.9792 q^{48} -205.397 q^{49} -109.709 q^{50} +29.6976 q^{51} -819.893 q^{52} -347.001 q^{53} +118.486 q^{54} +55.0000 q^{55} +167.695 q^{56} -0.0715727 q^{57} -886.874 q^{58} +691.824 q^{59} -168.865 q^{60} +491.854 q^{61} -797.108 q^{62} -105.574 q^{63} -809.510 q^{64} -364.149 q^{65} +144.816 q^{66} +715.379 q^{67} -111.442 q^{68} -219.137 q^{69} +257.386 q^{70} +541.957 q^{71} -128.662 q^{72} -159.702 q^{73} -1316.01 q^{74} -75.0000 q^{75} +0.268580 q^{76} -129.035 q^{77} -958.809 q^{78} -212.257 q^{79} -136.632 q^{80} +81.0000 q^{81} -388.562 q^{82} +413.252 q^{83} +396.171 q^{84} -49.4961 q^{85} +641.202 q^{86} -606.292 q^{87} -157.253 q^{88} -1099.40 q^{89} -197.476 q^{90} +854.323 q^{91} +822.322 q^{92} -544.925 q^{93} -815.121 q^{94} +0.119288 q^{95} -702.852 q^{96} +567.999 q^{97} +901.357 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q + q^2 - 9 * q^3 + 17 * q^4 + 15 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 27 * q^9	\( 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9} + 5 q^{10} + 33 q^{11} - 51 q^{12} - 20 q^{13} + 144 q^{14} - 45 q^{15} + 25 q^{16} + 32 q^{17} + 9 q^{18} + 116 q^{19} + 85 q^{20} - 18 q^{21} + 11 q^{22} + 240 q^{23} + 9 q^{24} + 75 q^{25} + 302 q^{26} - 81 q^{27} + 160 q^{28} + 238 q^{29} - 15 q^{30} + 92 q^{31} + 197 q^{32} - 99 q^{33} + 354 q^{34} + 30 q^{35} + 153 q^{36} - 90 q^{37} + 324 q^{38} + 60 q^{39} - 15 q^{40} - 46 q^{41} - 432 q^{42} - 134 q^{43} + 187 q^{44} + 135 q^{45} - 240 q^{46} - 220 q^{47} - 75 q^{48} - 457 q^{49} + 25 q^{50} - 96 q^{51} - 1530 q^{52} - 798 q^{53} - 27 q^{54} + 165 q^{55} + 688 q^{56} - 348 q^{57} - 978 q^{58} + 1236 q^{59} - 255 q^{60} + 342 q^{61} - 1792 q^{62} + 54 q^{63} - 1919 q^{64} - 100 q^{65} - 33 q^{66} + 764 q^{67} + 1074 q^{68} - 720 q^{69} + 720 q^{70} + 1816 q^{71} - 27 q^{72} + 100 q^{73} - 1874 q^{74} - 225 q^{75} + 396 q^{76} + 66 q^{77} - 906 q^{78} - 96 q^{79} + 125 q^{80} + 243 q^{81} - 910 q^{82} + 858 q^{83} - 480 q^{84} + 160 q^{85} + 188 q^{86} - 714 q^{87} - 33 q^{88} + 838 q^{89} + 45 q^{90} + 332 q^{91} - 688 q^{92} - 276 q^{93} - 3112 q^{94} + 580 q^{95} - 591 q^{96} - 1322 q^{97} + 1017 q^{98} + 297 q^{99}+O(q^{100}) \) 3 * q + q^2 - 9 * q^3 + 17 * q^4 + 15 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 27 * q^9 + 5 * q^10 + 33 * q^11 - 51 * q^12 - 20 * q^13 + 144 * q^14 - 45 * q^15 + 25 * q^16 + 32 * q^17 + 9 * q^18 + 116 * q^19 + 85 * q^20 - 18 * q^21 + 11 * q^22 + 240 * q^23 + 9 * q^24 + 75 * q^25 + 302 * q^26 - 81 * q^27 + 160 * q^28 + 238 * q^29 - 15 * q^30 + 92 * q^31 + 197 * q^32 - 99 * q^33 + 354 * q^34 + 30 * q^35 + 153 * q^36 - 90 * q^37 + 324 * q^38 + 60 * q^39 - 15 * q^40 - 46 * q^41 - 432 * q^42 - 134 * q^43 + 187 * q^44 + 135 * q^45 - 240 * q^46 - 220 * q^47 - 75 * q^48 - 457 * q^49 + 25 * q^50 - 96 * q^51 - 1530 * q^52 - 798 * q^53 - 27 * q^54 + 165 * q^55 + 688 * q^56 - 348 * q^57 - 978 * q^58 + 1236 * q^59 - 255 * q^60 + 342 * q^61 - 1792 * q^62 + 54 * q^63 - 1919 * q^64 - 100 * q^65 - 33 * q^66 + 764 * q^67 + 1074 * q^68 - 720 * q^69 + 720 * q^70 + 1816 * q^71 - 27 * q^72 + 100 * q^73 - 1874 * q^74 - 225 * q^75 + 396 * q^76 + 66 * q^77 - 906 * q^78 - 96 * q^79 + 125 * q^80 + 243 * q^81 - 910 * q^82 + 858 * q^83 - 480 * q^84 + 160 * q^85 + 188 * q^86 - 714 * q^87 - 33 * q^88 + 838 * q^89 + 45 * q^90 + 332 * q^91 - 688 * q^92 - 276 * q^93 - 3112 * q^94 + 580 * q^95 - 591 * q^96 - 1322 * q^97 + 1017 * q^98 + 297 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4.38835	−1.55152	−0.775759	−	0.631029i	$-0.782633\pi$
$2$	−4.38835	−1.55152	−0.775759	+	0.631029i	$0.782633\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	11.2577	1.40721
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	13.1651	0.895769
$7$	−11.7304	−0.633383	−0.316691	−	0.948529i	$-0.602572\pi$
$7$	−11.7304	−0.633383	−0.316691	+	0.948529i	$0.602572\pi$
$8$	−14.2958	−0.631789
$9$	9.00000	0.333333
$10$	−21.9418	−0.693860
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−33.7730	−0.812452
$13$	−72.8298	−1.55380	−0.776898	−	0.629627i	$-0.783208\pi$
$13$	−72.8298	−1.55380	−0.776898	+	0.629627i	$0.783208\pi$
$14$	51.4772	0.982705
$15$	−15.0000	−0.258199
$16$	−27.3264	−0.426975
$17$	−9.89921	−0.141230	−0.0706150	−	0.997504i	$-0.522496\pi$
$17$	−9.89921	−0.141230	−0.0706150	+	0.997504i	$0.522496\pi$
$18$	−39.4952	−0.517173
$19$	0.0238576	0.000288069 0	0.000144034	−	1.00000i	$-0.499954\pi$
$19$	0.0238576	0.000288069 0	0.000144034		1.00000i	$0.499954\pi$
$20$	56.2883	0.629322
$21$	35.1912	0.365684
$22$	−48.2719	−0.467800
$23$	73.0456	0.662220	0.331110	−	0.943592i	$-0.392577\pi$
$23$	73.0456	0.662220	0.331110	+	0.943592i	$0.392577\pi$
$24$	42.8873	0.364764
$25$	25.0000	0.200000
$26$	319.603	2.41074
$27$	−27.0000	−0.192450
$28$	−132.057	−0.891301
$29$	202.097	1.29409	0.647043	−	0.762453i	$-0.276005\pi$
$29$	202.097	1.29409	0.647043	+	0.762453i	$0.276005\pi$
$30$	65.8253	0.400600
$31$	181.642	1.05238	0.526191	−	0.850367i	$-0.323620\pi$
$31$	181.642	1.05238	0.526191	+	0.850367i	$0.323620\pi$
$32$	234.284	1.29425
$33$	−33.0000	−0.174078
$34$	43.4413	0.219121
$35$	−58.6521	−0.283257
$36$	101.319	0.469069
$37$	299.887	1.33246	0.666231	−	0.745746i	$-0.267907\pi$
$37$	299.887	1.33246	0.666231	+	0.745746i	$0.267907\pi$
$38$	−0.104695	−0.000446943 0
$39$	218.489	0.897085
$40$	−71.4788	−0.282545
$41$	88.5438	0.337274	0.168637	−	0.985678i	$-0.446063\pi$
$41$	88.5438	0.337274	0.168637	+	0.985678i	$0.446063\pi$
$42$	−154.432	−0.567365
$43$	−146.114	−0.518191	−0.259096	−	0.965852i	$-0.583424\pi$
$43$	−146.114	−0.518191	−0.259096	+	0.965852i	$0.583424\pi$
$44$	123.834	0.424289
$45$	45.0000	0.149071
$46$	−320.550	−1.02745
$47$	185.746	0.576466	0.288233	−	0.957560i	$-0.406932\pi$
$47$	185.746	0.576466	0.288233	+	0.957560i	$0.406932\pi$
$48$	81.9792	0.246514
$49$	−205.397	−0.598826
$50$	−109.709	−0.310304
$51$	29.6976	0.0815392
$52$	−819.893	−2.18651
$53$	−347.001	−0.899325	−0.449662	−	0.893199i	$-0.648456\pi$
$53$	−347.001	−0.899325	−0.449662	+	0.893199i	$0.648456\pi$
$54$	118.486	0.298590
$55$	55.0000	0.134840
$56$	167.695	0.400164
$57$	−0.0715727	−0.000166316 0
$58$	−886.874	−2.00780
$59$	691.824	1.52657	0.763287	−	0.646060i	$-0.223584\pi$
$59$	691.824	1.52657	0.763287	+	0.646060i	$0.223584\pi$
$60$	−168.865	−0.363339
$61$	491.854	1.03238	0.516192	−	0.856473i	$-0.327349\pi$
$61$	491.854	1.03238	0.516192	+	0.856473i	$0.327349\pi$
$62$	−797.108	−1.63279
$63$	−105.574	−0.211128
$64$	−809.510	−1.58107
$65$	−364.149	−0.694879
$66$	144.816	0.270085
$67$	715.379	1.30444	0.652220	−	0.758030i	$-0.273838\pi$
$67$	715.379	1.30444	0.652220	+	0.758030i	$0.273838\pi$
$68$	−111.442	−0.198740
$69$	−219.137	−0.382333
$70$	257.386	0.439479
$71$	541.957	0.905893	0.452947	−	0.891538i	$-0.350373\pi$
$71$	541.957	0.905893	0.452947	+	0.891538i	$0.350373\pi$
$72$	−128.662	−0.210596
$73$	−159.702	−0.256051	−0.128026	−	0.991771i	$-0.540864\pi$
$73$	−159.702	−0.256051	−0.128026	+	0.991771i	$0.540864\pi$
$74$	−1316.01	−2.06734
$75$	−75.0000	−0.115470
$76$	0.268580	0.000405372 0
$77$	−129.035	−0.190972
$78$	−958.809	−1.39184
$79$	−212.257	−0.302288	−0.151144	−	0.988512i	$-0.548296\pi$
$79$	−212.257	−0.302288	−0.151144	+	0.988512i	$0.548296\pi$
$80$	−136.632	−0.190949
$81$	81.0000	0.111111
$82$	−388.562	−0.523286
$83$	413.252	0.546510	0.273255	−	0.961942i	$-0.411900\pi$
$83$	413.252	0.546510	0.273255	+	0.961942i	$0.411900\pi$
$84$	396.171	0.514593
$85$	−49.4961	−0.0631600
$86$	641.202	0.803983
$87$	−606.292	−0.747141
$88$	−157.253	−0.190492
$89$	−1099.40	−1.30939	−0.654697	−	0.755891i	$-0.727204\pi$
$89$	−1099.40	−1.30939	−0.654697	+	0.755891i	$0.727204\pi$
$90$	−197.476	−0.231287
$91$	854.323	0.984148
$92$	822.322	0.931880
$93$	−544.925	−0.607593
$94$	−815.121	−0.894397
$95$	0.119288	0.000128828 0
$96$	−702.852	−0.747235
$97$	567.999	0.594553	0.297276	−	0.954791i	$-0.403922\pi$
$97$	567.999	0.594553	0.297276	+	0.954791i	$0.403922\pi$
$98$	901.357	0.929090
$99$	99.0000	0.100504
$100$	281.441	0.281441
$101$	944.617	0.930623	0.465311	−	0.885147i	$-0.345942\pi$
$101$	944.617	0.930623	0.465311	+	0.885147i	$0.345942\pi$
$102$	−130.324	−0.126510
$103$	1832.42	1.75295	0.876475	−	0.481448i	$-0.159889\pi$
$103$	1832.42	1.75295	0.876475	+	0.481448i	$0.159889\pi$
$104$	1041.16	0.981672
$105$	175.956	0.163539
$106$	1522.76	1.39532
$107$	−706.148	−0.637999	−0.319000	−	0.947755i	$-0.603347\pi$
$107$	−706.148	−0.637999	−0.319000	+	0.947755i	$0.603347\pi$
$108$	−303.957	−0.270817
$109$	−1830.83	−1.60882	−0.804411	−	0.594074i	$-0.797519\pi$
$109$	−1830.83	−1.60882	−0.804411	+	0.594074i	$0.797519\pi$
$110$	−241.360	−0.209207
$111$	−899.660	−0.769297
$112$	320.550	0.270439
$113$	654.436	0.544815	0.272408	−	0.962182i	$-0.412180\pi$
$113$	654.436	0.544815	0.272408	+	0.962182i	$0.412180\pi$
$114$	0.314086	0.000258043 0
$115$	365.228	0.296154
$116$	2275.14	1.82105
$117$	−655.468	−0.517932
$118$	−3035.97	−2.36851
$119$	116.122	0.0894527
$120$	214.436	0.163127
$121$	121.000	0.0909091
$122$	−2158.43	−1.60176
$123$	−265.631	−0.194725
$124$	2044.86	1.48092
$125$	125.000	0.0894427
$126$	463.295	0.327568
$127$	814.399	0.569025	0.284513	−	0.958672i	$-0.408168\pi$
$127$	814.399	0.569025	0.284513	+	0.958672i	$0.408168\pi$
$128$	1678.15	1.15882
$129$	438.343	0.299178
$130$	1598.01	1.07812
$131$	2320.98	1.54798	0.773989	−	0.633199i	$-0.218259\pi$
$131$	2320.98	1.54798	0.773989	+	0.633199i	$0.218259\pi$
$132$	−371.503	−0.244963
$133$	−0.279859	−0.000182458 0
$134$	−3139.34	−2.02386
$135$	−135.000	−0.0860663
$136$	141.517	0.0892277
$137$	2014.15	1.25606	0.628030	−	0.778189i	$-0.283861\pi$
$137$	2014.15	1.25606	0.628030	+	0.778189i	$0.283861\pi$
$138$	961.650	0.593196
$139$	1832.51	1.11821	0.559106	−	0.829096i	$-0.311144\pi$
$139$	1832.51	1.11821	0.559106	+	0.829096i	$0.311144\pi$
$140$	−660.285	−0.398602
$141$	−557.239	−0.332823
$142$	−2378.30	−1.40551
$143$	−801.128	−0.468487
$144$	−245.938	−0.142325
$145$	1010.49	0.578733
$146$	700.830	0.397268
$147$	616.192	0.345733
$148$	3376.02	1.87505
$149$	−3583.75	−1.97042	−0.985208	−	0.171365i	$-0.945182\pi$
$149$	−3583.75	−1.97042	−0.985208	+	0.171365i	$0.945182\pi$
$150$	329.127	0.179154
$151$	−2822.48	−1.52112	−0.760562	−	0.649265i	$-0.775077\pi$
$151$	−2822.48	−1.52112	−0.760562	+	0.649265i	$0.775077\pi$
$152$	−0.341062	−0.000181999 0
$153$	−89.0929	−0.0470767
$154$	566.249	0.296297
$155$	908.209	0.470639
$156$	2459.68	1.26238
$157$	2052.04	1.04313	0.521564	−	0.853212i	$-0.325349\pi$
$157$	2052.04	1.04313	0.521564	+	0.853212i	$0.325349\pi$
$158$	931.459	0.469005
$159$	1041.00	0.519225
$160$	1171.42	0.578806
$161$	−856.855	−0.419439
$162$	−355.457	−0.172391
$163$	−742.552	−0.356817	−0.178409	−	0.983956i	$-0.557095\pi$
$163$	−742.552	−0.356817	−0.178409	+	0.983956i	$0.557095\pi$
$164$	996.796	0.474614
$165$	−165.000	−0.0778499
$166$	−1813.50	−0.847920
$167$	−699.537	−0.324143	−0.162071	−	0.986779i	$-0.551817\pi$
$167$	−699.537	−0.324143	−0.162071	+	0.986779i	$0.551817\pi$
$168$	−503.086	−0.231035
$169$	3107.18	1.41428
$170$	217.206	0.0979939
$171$	0.214718	9.60228e−5 0
$172$	−1644.91	−0.729203
$173$	−734.967	−0.322997	−0.161499	−	0.986873i	$-0.551633\pi$
$173$	−734.967	−0.322997	−0.161499	+	0.986873i	$0.551633\pi$
$174$	2660.62	1.15920
$175$	−293.260	−0.126677
$176$	−300.590	−0.128738
$177$	−2075.47	−0.881367
$178$	4824.56	2.03155
$179$	−2687.28	−1.12211	−0.561053	−	0.827780i	$-0.689604\pi$
$179$	−2687.28	−1.12211	−0.561053	+	0.827780i	$0.689604\pi$
$180$	506.595	0.209774
$181$	−3766.56	−1.54677	−0.773387	−	0.633934i	$-0.781439\pi$
$181$	−3766.56	−1.54677	−0.773387	+	0.633934i	$0.781439\pi$
$182$	−3749.07	−1.52692
$183$	−1475.56	−0.596047
$184$	−1044.24	−0.418383
$185$	1499.43	0.595895
$186$	2391.33	0.942691
$187$	−108.891	−0.0425825
$188$	2091.07	0.811207
$189$	316.721	0.121895
$190$	−0.523477	−0.000199879 0
$191$	−1784.15	−0.675899	−0.337950	−	0.941164i	$-0.609733\pi$
$191$	−1784.15	−0.675899	−0.337950	+	0.941164i	$0.609733\pi$
$192$	2428.53	0.912834
$193$	−4463.74	−1.66480	−0.832402	−	0.554173i	$-0.813035\pi$
$193$	−4463.74	−1.66480	−0.832402	+	0.554173i	$0.813035\pi$
$194$	−2492.58	−0.922459
$195$	1092.45	0.401188
$196$	−2312.29	−0.842673
$197$	1166.19	0.421765	0.210883	−	0.977511i	$-0.432366\pi$
$197$	1166.19	0.421765	0.210883	+	0.977511i	$0.432366\pi$
$198$	−434.447	−0.155933
$199$	−1747.63	−0.622542	−0.311271	−	0.950321i	$-0.600755\pi$
$199$	−1747.63	−0.622542	−0.311271	+	0.950321i	$0.600755\pi$
$200$	−357.394	−0.126358
$201$	−2146.14	−0.753119
$202$	−4145.31	−1.44388
$203$	−2370.68	−0.819652
$204$	334.326	0.114743
$205$	442.719	0.150833
$206$	−8041.31	−2.71973
$207$	657.410	0.220740
$208$	1990.18	0.663432
$209$	0.262433	8.68559e−5 0
$210$	−772.158	−0.253733
$211$	−415.081	−0.135428	−0.0677141	−	0.997705i	$-0.521571\pi$
$211$	−415.081	−0.135428	−0.0677141	+	0.997705i	$0.521571\pi$
$212$	−3906.42	−1.26554
$213$	−1625.87	−0.523018
$214$	3098.83	0.989867
$215$	−730.572	−0.231742
$216$	385.986	0.121588
$217$	−2130.73	−0.666560
$218$	8034.32	2.49611
$219$	479.107	0.147831
$220$	619.171	0.189748
$221$	720.957	0.219443
$222$	3948.03	1.19358
$223$	−72.9064	−0.0218932	−0.0109466	−	0.999940i	$-0.503484\pi$
$223$	−72.9064	−0.0218932	−0.0109466	+	0.999940i	$0.503484\pi$
$224$	−2748.25	−0.819755
$225$	225.000	0.0666667
$226$	−2871.90	−0.845291
$227$	2362.35	0.690727	0.345363	−	0.938469i	$-0.387756\pi$
$227$	2362.35	0.690727	0.345363	+	0.938469i	$0.387756\pi$
$228$	−0.805741	−0.000234042 0
$229$	4534.32	1.30846	0.654228	−	0.756297i	$-0.272993\pi$
$229$	4534.32	1.30846	0.654228	+	0.756297i	$0.272993\pi$
$230$	−1602.75	−0.459488
$231$	387.104	0.110258
$232$	−2889.13	−0.817590
$233$	231.076	0.0649713	0.0324856	−	0.999472i	$-0.489658\pi$
$233$	231.076	0.0649713	0.0324856	+	0.999472i	$0.489658\pi$
$234$	2876.43	0.803581
$235$	928.732	0.257803
$236$	7788.32	2.14821
$237$	636.771	0.174526
$238$	−509.584	−0.138787
$239$	4463.18	1.20795	0.603973	−	0.797004i	$-0.293583\pi$
$239$	4463.18	1.20795	0.603973	+	0.797004i	$0.293583\pi$
$240$	409.896	0.110244
$241$	−2097.51	−0.560633	−0.280316	−	0.959908i	$-0.590439\pi$
$241$	−2097.51	−0.560633	−0.280316	+	0.959908i	$0.590439\pi$
$242$	−530.991	−0.141047
$243$	−243.000	−0.0641500
$244$	5537.12	1.45278
$245$	−1026.99	−0.267803
$246$	1165.68	0.302119
$247$	−1.73754	−0.000447600 0
$248$	−2596.71	−0.664883
$249$	−1239.76	−0.315528
$250$	−548.544	−0.138772
$251$	3400.42	0.855110	0.427555	−	0.903989i	$-0.359375\pi$
$251$	3400.42	0.855110	0.427555	+	0.903989i	$0.359375\pi$
$252$	−1188.51	−0.297100
$253$	803.501	0.199667
$254$	−3573.87	−0.882853
$255$	148.488	0.0364655
$256$	−888.218	−0.216850
$257$	4994.11	1.21215	0.606077	−	0.795406i	$-0.292742\pi$
$257$	4994.11	1.21215	0.606077	+	0.795406i	$0.292742\pi$
$258$	−1923.60	−0.464180
$259$	−3517.80	−0.843958
$260$	−4099.46	−0.977838
$261$	1818.88	0.431362
$262$	−10185.3	−2.40172
$263$	6869.81	1.61069	0.805344	−	0.592808i	$-0.201981\pi$
$263$	6869.81	1.61069	0.805344	+	0.592808i	$0.201981\pi$
$264$	471.760	0.109980
$265$	−1735.00	−0.402190
$266$	1.22812	0.000283086 0
$267$	3298.20	0.755979
$268$	8053.49	1.83562
$269$	−2019.60	−0.457758	−0.228879	−	0.973455i	$-0.573506\pi$
$269$	−2019.60	−0.457758	−0.228879	+	0.973455i	$0.573506\pi$
$270$	592.428	0.133533
$271$	5191.61	1.16372	0.581859	−	0.813289i	$-0.302325\pi$
$271$	5191.61	1.16372	0.581859	+	0.813289i	$0.302325\pi$
$272$	270.510	0.0603017
$273$	−2562.97	−0.568198
$274$	−8838.80	−1.94880
$275$	275.000	0.0603023
$276$	−2466.97	−0.538021
$277$	−456.134	−0.0989402	−0.0494701	−	0.998776i	$-0.515753\pi$
$277$	−456.134	−0.0989402	−0.0494701	+	0.998776i	$0.515753\pi$
$278$	−8041.70	−1.73493
$279$	1634.78	0.350794
$280$	838.476	0.178959
$281$	3004.92	0.637930	0.318965	−	0.947767i	$-0.396665\pi$
$281$	3004.92	0.637930	0.318965	+	0.947767i	$0.396665\pi$
$282$	2445.36	0.516380
$283$	−409.156	−0.0859428	−0.0429714	−	0.999076i	$-0.513682\pi$
$283$	−409.156	−0.0859428	−0.0429714	+	0.999076i	$0.513682\pi$
$284$	6101.16	1.27478
$285$	−0.357864	−7.43790e−5 0
$286$	3515.63	0.726866
$287$	−1038.66	−0.213623
$288$	2108.56	0.431416
$289$	−4815.01	−0.980054
$290$	−4434.37	−0.897915
$291$	−1704.00	−0.343265
$292$	−1797.87	−0.360317
$293$	−7410.14	−1.47749	−0.738745	−	0.673985i	$-0.764581\pi$
$293$	−7410.14	−1.47749	−0.738745	+	0.673985i	$0.764581\pi$
$294$	−2704.07	−0.536410
$295$	3459.12	0.682704
$296$	−4287.11	−0.841835
$297$	−297.000	−0.0580259
$298$	15726.7	3.05713
$299$	−5319.89	−1.02895
$300$	−844.324	−0.162490
$301$	1713.98	0.328213
$302$	12386.0	2.36005
$303$	−2833.85	−0.537295
$304$	−0.651941	−0.000122998 0
$305$	2459.27	0.461696
$306$	390.971	0.0730403
$307$	4985.77	0.926883	0.463442	−	0.886127i	$-0.346614\pi$
$307$	4985.77	0.926883	0.463442	+	0.886127i	$0.346614\pi$
$308$	−1452.63	−0.268737
$309$	−5497.26	−1.01207
$310$	−3985.54	−0.730205
$311$	−6238.99	−1.13756	−0.568779	−	0.822491i	$-0.692584\pi$
$311$	−6238.99	−1.13756	−0.568779	+	0.822491i	$0.692584\pi$
$312$	−3123.47	−0.566768
$313$	−9821.30	−1.77359	−0.886794	−	0.462166i	$-0.847072\pi$
$313$	−9821.30	−1.77359	−0.886794	+	0.462166i	$0.847072\pi$
$314$	−9005.10	−1.61843
$315$	−527.869	−0.0944191
$316$	−2389.52	−0.425382
$317$	−1747.26	−0.309577	−0.154789	−	0.987948i	$-0.549470\pi$
$317$	−1747.26	−0.309577	−0.154789	+	0.987948i	$0.549470\pi$
$318$	−4568.29	−0.805588
$319$	2223.07	0.390182
$320$	−4047.55	−0.707078
$321$	2118.44	0.368349
$322$	3760.18	0.650766
$323$	−0.236171	−4.06839e−5 0
$324$	911.870	0.156356
$325$	−1820.74	−0.310759
$326$	3258.58	0.553608
$327$	5492.48	0.928853
$328$	−1265.80	−0.213086
$329$	−2178.88	−0.365124
$330$	724.079	0.120785
$331$	−7209.01	−1.19711	−0.598554	−	0.801082i	$-0.704258\pi$
$331$	−7209.01	−1.19711	−0.598554	+	0.801082i	$0.704258\pi$
$332$	4652.25	0.769053
$333$	2698.98	0.444154
$334$	3069.82	0.502913
$335$	3576.90	0.583363
$336$	−961.650	−0.156138
$337$	53.0763	0.00857938	0.00428969	−	0.999991i	$-0.498635\pi$
$337$	53.0763	0.00857938	0.00428969	+	0.999991i	$0.498635\pi$
$338$	−13635.4	−2.19428
$339$	−1963.31	−0.314549
$340$	−557.210	−0.0888792
$341$	1998.06	0.317305
$342$	−0.942259	−0.000148981 0
$343$	6432.93	1.01267
$344$	2088.82	0.327388
$345$	−1095.68	−0.170984
$346$	3225.30	0.501136
$347$	3326.14	0.514572	0.257286	−	0.966335i	$-0.417172\pi$
$347$	3326.14	0.514572	0.257286	+	0.966335i	$0.417172\pi$
$348$	−6825.42	−1.05138
$349$	12042.6	1.84706	0.923530	−	0.383526i	$-0.125290\pi$
$349$	12042.6	1.84706	0.923530	+	0.383526i	$0.125290\pi$
$350$	1286.93	0.196541
$351$	1966.40	0.299028
$352$	2577.12	0.390231
$353$	10994.7	1.65775	0.828876	−	0.559432i	$-0.188981\pi$
$353$	10994.7	1.65775	0.828876	+	0.559432i	$0.188981\pi$
$354$	9107.91	1.36746
$355$	2709.78	0.405128
$356$	−12376.7	−1.84259
$357$	−348.366	−0.0516455
$358$	11792.8	1.74097
$359$	2385.44	0.350694	0.175347	−	0.984507i	$-0.443895\pi$
$359$	2385.44	0.350694	0.175347	+	0.984507i	$0.443895\pi$
$360$	−643.309	−0.0941816
$361$	−6859.00	−1.00000
$362$	16529.0	2.39985
$363$	−363.000	−0.0524864
$364$	9617.68	1.38490
$365$	−798.511	−0.114510
$366$	6475.29	0.924778
$367$	3227.78	0.459098	0.229549	−	0.973297i	$-0.426275\pi$
$367$	3227.78	0.459098	0.229549	+	0.973297i	$0.426275\pi$
$368$	−1996.07	−0.282751
$369$	796.894	0.112425
$370$	−6580.05	−0.924542
$371$	4070.46	0.569617
$372$	−6134.58	−0.855009
$373$	−5948.20	−0.825700	−0.412850	−	0.910799i	$-0.635467\pi$
$373$	−5948.20	−0.825700	−0.412850	+	0.910799i	$0.635467\pi$
$374$	477.854	0.0660675
$375$	−375.000	−0.0516398
$376$	−2655.39	−0.364205
$377$	−14718.7	−2.01075
$378$	−1389.88	−0.189122
$379$	10332.2	1.40034	0.700170	−	0.713976i	$-0.253107\pi$
$379$	10332.2	1.40034	0.700170	+	0.713976i	$0.253107\pi$
$380$	1.34290	0.000181288 0
$381$	−2443.20	−0.328527
$382$	7829.50	1.04867
$383$	−6686.90	−0.892127	−0.446063	−	0.895001i	$-0.647174\pi$
$383$	−6686.90	−0.892127	−0.446063	+	0.895001i	$0.647174\pi$
$384$	−5034.44	−0.669043
$385$	−645.173	−0.0854053
$386$	19588.5	2.58297
$387$	−1315.03	−0.172730
$388$	6394.34	0.836659
$389$	6099.57	0.795014	0.397507	−	0.917599i	$-0.369875\pi$
$389$	6099.57	0.795014	0.397507	+	0.917599i	$0.369875\pi$
$390$	−4794.04	−0.622451
$391$	−723.094	−0.0935254
$392$	2936.31	0.378332
$393$	−6962.94	−0.893725
$394$	−5117.67	−0.654376
$395$	−1061.28	−0.135187
$396$	1114.51	0.141430
$397$	2640.37	0.333795	0.166897	−	0.985974i	$-0.446625\pi$
$397$	2640.37	0.333795	0.166897	+	0.985974i	$0.446625\pi$
$398$	7669.20	0.965885
$399$	0.839578	0.000105342 0
$400$	−683.160	−0.0853950
$401$	8787.59	1.09434	0.547171	−	0.837021i	$-0.315705\pi$
$401$	8787.59	1.09434	0.547171	+	0.837021i	$0.315705\pi$
$402$	9418.01	1.16848
$403$	−13228.9	−1.63519
$404$	10634.2	1.30958
$405$	405.000	0.0496904
$406$	10403.4	1.27170
$407$	3298.75	0.401752
$408$	−424.550	−0.0515156
$409$	−5870.27	−0.709698	−0.354849	−	0.934924i	$-0.615468\pi$
$409$	−5870.27	−0.709698	−0.354849	+	0.934924i	$0.615468\pi$
$410$	−1942.81	−0.234021
$411$	−6042.44	−0.725187
$412$	20628.8	2.46676
$413$	−8115.38	−0.966905
$414$	−2884.95	−0.342482
$415$	2066.26	0.244407
$416$	−17062.9	−2.01100
$417$	−5497.53	−0.645600
$418$	−1.15165	−0.000134759 0
$419$	14208.8	1.65667	0.828337	−	0.560231i	$-0.189288\pi$
$419$	14208.8	1.65667	0.828337	+	0.560231i	$0.189288\pi$
$420$	1980.85	0.230133
$421$	−12301.9	−1.42413	−0.712063	−	0.702115i	$-0.752239\pi$
$421$	−12301.9	−1.42413	−0.712063	+	0.702115i	$0.752239\pi$
$422$	1821.52	0.210119
$423$	1671.72	0.192155
$424$	4960.64	0.568184
$425$	−247.480	−0.0282460
$426$	7134.89	0.811471
$427$	−5769.65	−0.653895
$428$	−7949.57	−0.897797
$429$	2403.38	0.270481
$430$	3206.01	0.359552
$431$	−5640.08	−0.630331	−0.315166	−	0.949037i	$-0.602060\pi$
$431$	−5640.08	−0.630331	−0.315166	+	0.949037i	$0.602060\pi$
$432$	737.813	0.0821714
$433$	10122.9	1.12350	0.561749	−	0.827308i	$-0.310129\pi$
$433$	10122.9	1.12350	0.561749	+	0.827308i	$0.310129\pi$
$434$	9350.41	1.03418
$435$	−3031.46	−0.334132
$436$	−20610.8	−2.26394
$437$	1.74269	0.000190765 0
$438$	−2102.49	−0.229363
$439$	−10753.2	−1.16907	−0.584534	−	0.811369i	$-0.698723\pi$
$439$	−10753.2	−1.16907	−0.584534	+	0.811369i	$0.698723\pi$
$440$	−786.267	−0.0851905
$441$	−1848.58	−0.199609
$442$	−3163.82	−0.340469
$443$	−10346.6	−1.10967	−0.554834	−	0.831961i	$-0.687218\pi$
$443$	−10346.6	−1.10967	−0.554834	+	0.831961i	$0.687218\pi$
$444$	−10128.1	−1.08256
$445$	−5497.00	−0.585579
$446$	319.939	0.0339676
$447$	10751.2	1.13762
$448$	9495.89	1.00143
$449$	9569.21	1.00579	0.502894	−	0.864348i	$-0.332269\pi$
$449$	9569.21	1.00579	0.502894	+	0.864348i	$0.332269\pi$
$450$	−987.380	−0.103435
$451$	973.982	0.101692
$452$	7367.42	0.766668
$453$	8467.43	0.878222
$454$	−10366.8	−1.07167
$455$	4271.62	0.440124
$456$	1.02319	0.000105077 0
$457$	7446.31	0.762197	0.381098	−	0.924535i	$-0.375546\pi$
$457$	7446.31	0.762197	0.381098	+	0.924535i	$0.375546\pi$
$458$	−19898.2	−2.03009
$459$	267.279	0.0271797
$460$	4111.61	0.416750
$461$	10934.9	1.10475	0.552373	−	0.833597i	$-0.313722\pi$
$461$	10934.9	1.10475	0.552373	+	0.833597i	$0.313722\pi$
$462$	−1698.75	−0.171067
$463$	−9889.15	−0.992630	−0.496315	−	0.868142i	$-0.665314\pi$
$463$	−9889.15	−0.992630	−0.496315	+	0.868142i	$0.665314\pi$
$464$	−5522.59	−0.552543
$465$	−2724.63	−0.271724
$466$	−1014.04	−0.100804
$467$	−100.396	−0.00994808	−0.00497404	−	0.999988i	$-0.501583\pi$
$467$	−100.396	−0.00994808	−0.00497404	+	0.999988i	$0.501583\pi$
$468$	−7379.04	−0.728838
$469$	−8391.69	−0.826210
$470$	−4075.61	−0.399987
$471$	−6156.13	−0.602250
$472$	−9890.15	−0.964473
$473$	−1607.26	−0.156241
$474$	−2794.38	−0.270780
$475$	0.596439	5.76137e−5 0
$476$	1307.26	0.125878
$477$	−3123.01	−0.299775
$478$	−19586.0	−1.87415
$479$	−4413.85	−0.421032	−0.210516	−	0.977590i	$-0.567514\pi$
$479$	−4413.85	−0.421032	−0.210516	+	0.977590i	$0.567514\pi$
$480$	−3514.26	−0.334174
$481$	−21840.7	−2.07037
$482$	9204.62	0.869832
$483$	2570.56	0.242163
$484$	1362.18	0.127928
$485$	2840.00	0.265892
$486$	1066.37	0.0995299
$487$	−16795.6	−1.56279	−0.781396	−	0.624035i	$-0.785492\pi$
$487$	−16795.6	−1.56279	−0.781396	+	0.624035i	$0.785492\pi$
$488$	−7031.43	−0.652250
$489$	2227.66	0.206008
$490$	4506.78	0.415502
$491$	19003.5	1.74667	0.873336	−	0.487118i	$-0.161952\pi$
$491$	19003.5	1.74667	0.873336	+	0.487118i	$0.161952\pi$
$492$	−2990.39	−0.274018
$493$	−2000.60	−0.182764
$494$	7.62495	0.000694459 0
$495$	495.000	0.0449467
$496$	−4963.61	−0.449340
$497$	−6357.37	−0.573777
$498$	5440.49	0.489547
$499$	−19969.5	−1.79150	−0.895748	−	0.444562i	$-0.853359\pi$
$499$	−19969.5	−1.79150	−0.895748	+	0.444562i	$0.853359\pi$
$500$	1407.21	0.125864
$501$	2098.61	0.187144
$502$	−14922.2	−1.32672
$503$	15076.5	1.33643	0.668217	−	0.743967i	$-0.267058\pi$
$503$	15076.5	1.33643	0.668217	+	0.743967i	$0.267058\pi$
$504$	1509.26	0.133388
$505$	4723.09	0.416187
$506$	−3526.05	−0.309787
$507$	−9321.53	−0.816536
$508$	9168.23	0.800737
$509$	12404.7	1.08022	0.540108	−	0.841596i	$-0.318384\pi$
$509$	12404.7	1.08022	0.540108	+	0.841596i	$0.318384\pi$
$510$	−651.619	−0.0565768
$511$	1873.37	0.162178
$512$	−9527.35	−0.822370
$513$	−0.644154	−5.54388e−5 0
$514$	−21915.9	−1.88068
$515$	9162.11	0.783943
$516$	4934.72	0.421005
$517$	2043.21	0.173811
$518$	15437.3	1.30942
$519$	2204.90	0.186483
$520$	5205.79	0.439017
$521$	−18075.1	−1.51993	−0.759964	−	0.649965i	$-0.774784\pi$
$521$	−18075.1	−1.51993	−0.759964	+	0.649965i	$0.774784\pi$
$522$	−7981.87	−0.669266
$523$	9878.08	0.825886	0.412943	−	0.910757i	$-0.364501\pi$
$523$	9878.08	0.825886	0.412943	+	0.910757i	$0.364501\pi$
$524$	26128.8	2.17833
$525$	879.781	0.0731367
$526$	−30147.2	−2.49901
$527$	−1798.11	−0.148628
$528$	901.771	0.0743268
$529$	−6831.34	−0.561465
$530$	7613.81	0.624005
$531$	6226.42	0.508858
$532$	−3.15056	−0.000256756 0
$533$	−6448.63	−0.524054
$534$	−14473.7	−1.17292
$535$	−3530.74	−0.285322
$536$	−10226.9	−0.824131
$537$	8061.85	0.647848
$538$	8862.71	0.710220
$539$	−2259.37	−0.180553
$540$	−1519.78	−0.121113
$541$	−4740.76	−0.376749	−0.188374	−	0.982097i	$-0.560322\pi$
$541$	−4740.76	−0.376749	−0.188374	+	0.982097i	$0.560322\pi$
$542$	−22782.6	−1.80553
$543$	11299.7	0.893030
$544$	−2319.23	−0.182787
$545$	−9154.14	−0.719487
$546$	11247.2	0.881569
$547$	20441.4	1.59782	0.798912	−	0.601448i	$-0.205409\pi$
$547$	20441.4	1.59782	0.798912	+	0.601448i	$0.205409\pi$
$548$	22674.6	1.76754
$549$	4426.69	0.344128
$550$	−1206.80	−0.0935600
$551$	4.82155	0.000372786 0
$552$	3132.73	0.241554
$553$	2489.86	0.191464
$554$	2001.68	0.153507
$555$	−4498.30	−0.344040
$556$	20629.8	1.57356
$557$	3587.32	0.272890	0.136445	−	0.990648i	$-0.456432\pi$
$557$	3587.32	0.272890	0.136445	+	0.990648i	$0.456432\pi$
$558$	−7173.98	−0.544263
$559$	10641.5	0.805164
$560$	1602.75	0.120944
$561$	326.674	0.0245850
$562$	−13186.6	−0.989759
$563$	8.82275	0.000660452 0	0.000330226	−	1.00000i	$-0.499895\pi$
$563$	8.82275	0.000660452 0	0.000330226		1.00000i	$0.499895\pi$
$564$	−6273.21	−0.468351
$565$	3272.18	0.243649
$566$	1795.52	0.133342
$567$	−950.163	−0.0703759
$568$	−7747.68	−0.572334
$569$	17601.2	1.29681	0.648403	−	0.761297i	$-0.275437\pi$
$569$	17601.2	1.29681	0.648403	+	0.761297i	$0.275437\pi$
$570$	1.57043	0.000115400 0
$571$	10826.5	0.793477	0.396738	−	0.917932i	$-0.370142\pi$
$571$	10826.5	0.793477	0.396738	+	0.917932i	$0.370142\pi$
$572$	−9018.82	−0.659258
$573$	5352.46	0.390231
$574$	4557.99	0.331440
$575$	1826.14	0.132444
$576$	−7285.59	−0.527025
$577$	−20294.7	−1.46426	−0.732131	−	0.681163i	$-0.761474\pi$
$577$	−20294.7	−1.46426	−0.732131	+	0.681163i	$0.761474\pi$
$578$	21130.0	1.52057
$579$	13391.2	0.961175
$580$	11375.7	0.814398
$581$	−4847.62	−0.346150
$582$	7477.75	0.532582
$583$	−3817.01	−0.271157
$584$	2283.07	0.161770
$585$	−3277.34	−0.231626
$586$	32518.3	2.29235
$587$	1338.81	0.0941375	0.0470687	−	0.998892i	$-0.485012\pi$
$587$	1338.81	0.0941375	0.0470687	+	0.998892i	$0.485012\pi$
$588$	6936.88	0.486517
$589$	4.33353	0.000303158 0
$590$	−15179.8	−1.05923
$591$	−3498.58	−0.243506
$592$	−8194.82	−0.568928
$593$	−11326.3	−0.784345	−0.392173	−	0.919892i	$-0.628276\pi$
$593$	−11326.3	−0.784345	−0.392173	+	0.919892i	$0.628276\pi$
$594$	1303.34	0.0900282
$595$	580.609	0.0400045
$596$	−40344.6	−2.77278
$597$	5242.88	0.359425
$598$	23345.6	1.59644
$599$	−13279.5	−0.905822	−0.452911	−	0.891556i	$-0.649614\pi$
$599$	−13279.5	−0.905822	−0.452911	+	0.891556i	$0.649614\pi$
$600$	1072.18	0.0729528
$601$	6423.07	0.435944	0.217972	−	0.975955i	$-0.430056\pi$
$601$	6423.07	0.435944	0.217972	+	0.975955i	$0.430056\pi$
$602$	−7521.56	−0.509229
$603$	6438.41	0.434813
$604$	−31774.5	−2.14054
$605$	605.000	0.0406558
$606$	12435.9	0.833623
$607$	4767.58	0.318798	0.159399	−	0.987214i	$-0.449044\pi$
$607$	4767.58	0.318798	0.159399	+	0.987214i	$0.449044\pi$
$608$	5.58945	0.000372832 0
$609$	7112.05	0.473226
$610$	−10792.1	−0.716330
$611$	−13527.9	−0.895710
$612$	−1002.98	−0.0662467
$613$	26476.3	1.74448	0.872240	−	0.489078i	$-0.162667\pi$
$613$	26476.3	1.74448	0.872240	+	0.489078i	$0.162667\pi$
$614$	−21879.3	−1.43808
$615$	−1328.16	−0.0870837
$616$	1844.65	0.120654
$617$	−23097.5	−1.50708	−0.753542	−	0.657400i	$-0.771656\pi$
$617$	−23097.5	−1.50708	−0.753542	+	0.657400i	$0.771656\pi$
$618$	24123.9	1.57024
$619$	8579.27	0.557076	0.278538	−	0.960425i	$-0.410150\pi$
$619$	8579.27	0.557076	0.278538	+	0.960425i	$0.410150\pi$
$620$	10224.3	0.662287
$621$	−1972.23	−0.127444
$622$	27378.9	1.76494
$623$	12896.4	0.829348
$624$	−5970.53	−0.383033
$625$	625.000	0.0400000
$626$	43099.4	2.75175
$627$	−0.787300	−5.01463e−5 0
$628$	23101.2	1.46790
$629$	−2968.64	−0.188184
$630$	2316.47	0.146493
$631$	17464.3	1.10181	0.550905	−	0.834568i	$-0.314282\pi$
$631$	17464.3	1.10181	0.550905	+	0.834568i	$0.314282\pi$
$632$	3034.37	0.190982
$633$	1245.24	0.0781895
$634$	7667.60	0.480314
$635$	4072.00	0.254476
$636$	11719.2	0.730658
$637$	14959.0	0.930454
$638$	−9755.62	−0.605374
$639$	4877.61	0.301964
$640$	8390.73	0.518239
$641$	13105.0	0.807513	0.403756	−	0.914867i	$-0.367704\pi$
$641$	13105.0	0.807513	0.403756	+	0.914867i	$0.367704\pi$
$642$	−9296.49	−0.571500
$643$	2748.22	0.168553	0.0842763	−	0.996442i	$-0.473142\pi$
$643$	2748.22	0.168553	0.0842763	+	0.996442i	$0.473142\pi$
$644$	−9646.18	−0.590237
$645$	2191.72	0.133796
$646$	1.03640	6.31219e−5 0
$647$	−2332.01	−0.141701	−0.0708506	−	0.997487i	$-0.522571\pi$
$647$	−2332.01	−0.141701	−0.0708506	+	0.997487i	$0.522571\pi$
$648$	−1157.96	−0.0701988
$649$	7610.06	0.460279
$650$	7990.07	0.482148
$651$	6392.20	0.384839
$652$	−8359.40	−0.502116
$653$	−4241.59	−0.254190	−0.127095	−	0.991891i	$-0.540565\pi$
$653$	−4241.59	−0.254190	−0.127095	+	0.991891i	$0.540565\pi$
$654$	−24103.0	−1.44113
$655$	11604.9	0.692277
$656$	−2419.58	−0.144007
$657$	−1437.32	−0.0853504
$658$	9561.71	0.566496
$659$	32050.8	1.89457	0.947285	−	0.320391i	$-0.103814\pi$
$659$	32050.8	1.89457	0.947285	+	0.320391i	$0.103814\pi$
$660$	−1857.51	−0.109551
$661$	−27446.2	−1.61503	−0.807513	−	0.589850i	$-0.799187\pi$
$661$	−27446.2	−1.61503	−0.807513	+	0.589850i	$0.799187\pi$
$662$	31635.7	1.85734
$663$	−2162.87	−0.126695
$664$	−5907.76	−0.345279
$665$	−1.39930	−8.15975e−5 0
$666$	−11844.1	−0.689113
$667$	14762.3	0.856970
$668$	−7875.15	−0.456136
$669$	218.719	0.0126400
$670$	−15696.7	−0.905098
$671$	5410.39	0.311276
$672$	8244.74	0.473286
$673$	24175.8	1.38471	0.692354	−	0.721558i	$-0.256574\pi$
$673$	24175.8	1.38471	0.692354	+	0.721558i	$0.256574\pi$
$674$	−232.918	−0.0133111
$675$	−675.000	−0.0384900
$676$	34979.5	1.99019
$677$	−30852.6	−1.75150	−0.875748	−	0.482769i	$-0.839631\pi$
$677$	−30852.6	−1.75150	−0.875748	+	0.482769i	$0.839631\pi$
$678$	8615.69	0.488029
$679$	−6662.87	−0.376579
$680$	707.584	0.0399038
$681$	−7087.06	−0.398791
$682$	−8768.19	−0.492304
$683$	−15562.9	−0.871883	−0.435941	−	0.899975i	$-0.643585\pi$
$683$	−15562.9	−0.871883	−0.435941	+	0.899975i	$0.643585\pi$
$684$	2.41722	0.000135124 0
$685$	10070.7	0.561727
$686$	−28230.0	−1.57117
$687$	−13603.0	−0.755438
$688$	3992.78	0.221255
$689$	25272.0	1.39737
$690$	4808.25	0.265285
$691$	19228.0	1.05856	0.529281	−	0.848446i	$-0.322462\pi$
$691$	19228.0	1.05856	0.529281	+	0.848446i	$0.322462\pi$
$692$	−8274.01	−0.454524
$693$	−1161.31	−0.0636574
$694$	−14596.3	−0.798367
$695$	9162.55	0.500080
$696$	8667.40	0.472036
$697$	−876.514	−0.0476332
$698$	−52847.1	−2.86575
$699$	−693.228	−0.0375112
$700$	−3301.42	−0.178260
$701$	15681.1	0.844891	0.422446	−	0.906388i	$-0.361172\pi$
$701$	15681.1	0.844891	0.422446	+	0.906388i	$0.361172\pi$
$702$	−8629.28	−0.463948
$703$	7.15457	0.000383840 0
$704$	−8904.61	−0.476712
$705$	−2786.20	−0.148843
$706$	−48248.5	−2.57203
$707$	−11080.7	−0.589440
$708$	−23365.0	−1.24027
$709$	21226.9	1.12439	0.562196	−	0.827004i	$-0.309957\pi$
$709$	21226.9	1.12439	0.562196	+	0.827004i	$0.309957\pi$
$710$	−11891.5	−0.628563
$711$	−1910.31	−0.100763
$712$	15716.8	0.827262
$713$	13268.1	0.696908
$714$	1528.75	0.0801290
$715$	−4005.64	−0.209514
$716$	−30252.5	−1.57904
$717$	−13389.5	−0.697408
$718$	−10468.2	−0.544107
$719$	−17928.4	−0.929928	−0.464964	−	0.885330i	$-0.653933\pi$
$719$	−17928.4	−0.929928	−0.464964	+	0.885330i	$0.653933\pi$
$720$	−1229.69	−0.0636497
$721$	−21495.1	−1.11029
$722$	30099.7	1.55152
$723$	6292.53	0.323682
$724$	−42402.6	−2.17663
$725$	5052.43	0.258817
$726$	1592.97	0.0814336
$727$	−11876.7	−0.605890	−0.302945	−	0.953008i	$-0.597970\pi$
$727$	−11876.7	−0.605890	−0.302945	+	0.953008i	$0.597970\pi$
$728$	−12213.2	−0.621774
$729$	729.000	0.0370370
$730$	3504.15	0.177664
$731$	1446.42	0.0731842
$732$	−16611.4	−0.838762
$733$	−25344.9	−1.27713	−0.638563	−	0.769569i	$-0.720471\pi$
$733$	−25344.9	−1.27713	−0.638563	+	0.769569i	$0.720471\pi$
$734$	−14164.7	−0.712299
$735$	3080.96	0.154616
$736$	17113.4	0.857077
$737$	7869.17	0.393303
$738$	−3497.05	−0.174429
$739$	−12511.6	−0.622795	−0.311398	−	0.950280i	$-0.600797\pi$
$739$	−12511.6	−0.622795	−0.311398	+	0.950280i	$0.600797\pi$
$740$	16880.1	0.838548
$741$	5.21263	0.000258422 0
$742$	−17862.6	−0.883771
$743$	−32707.2	−1.61495	−0.807477	−	0.589899i	$-0.799168\pi$
$743$	−32707.2	−1.61495	−0.807477	+	0.589899i	$0.799168\pi$
$744$	7790.12	0.383871
$745$	−17918.7	−0.881196
$746$	26102.8	1.28109
$747$	3719.27	0.182170
$748$	−1225.86	−0.0599224
$749$	8283.41	0.404098
$750$	1645.63	0.0801200
$751$	20409.8	0.991694	0.495847	−	0.868410i	$-0.334858\pi$
$751$	20409.8	0.991694	0.495847	+	0.868410i	$0.334858\pi$
$752$	−5075.78	−0.246137
$753$	−10201.3	−0.493698
$754$	64590.9	3.11971
$755$	−14112.4	−0.680268
$756$	3565.54	0.171531
$757$	5746.29	0.275895	0.137948	−	0.990440i	$-0.455949\pi$
$757$	5746.29	0.275895	0.137948	+	0.990440i	$0.455949\pi$
$758$	−45341.3	−2.17265
$759$	−2410.50	−0.115278
$760$	−1.70531	−8.13923e−5 0
$761$	−2595.93	−0.123656	−0.0618282	−	0.998087i	$-0.519693\pi$
$761$	−2595.93	−0.123656	−0.0618282	+	0.998087i	$0.519693\pi$
$762$	10721.6	0.509715
$763$	21476.4	1.01900
$764$	−20085.4	−0.951130
$765$	−445.465	−0.0210533
$766$	29344.5	1.38415
$767$	−50385.4	−2.37198
$768$	2664.66	0.125199
$769$	12449.7	0.583807	0.291903	−	0.956448i	$-0.405711\pi$
$769$	12449.7	0.583807	0.291903	+	0.956448i	$0.405711\pi$
$770$	2831.25	0.132508
$771$	−14982.3	−0.699838
$772$	−50251.3	−2.34272
$773$	21032.1	0.978621	0.489310	−	0.872110i	$-0.337249\pi$
$773$	21032.1	0.978621	0.489310	+	0.872110i	$0.337249\pi$
$774$	5770.81	0.267994
$775$	4541.04	0.210476
$776$	−8119.98	−0.375632
$777$	10553.4	0.487260
$778$	−26767.1	−1.23348
$779$	2.11244	9.71579e−5 0
$780$	12298.4	0.564555
$781$	5961.52	0.273137
$782$	3173.19	0.145106
$783$	−5456.63	−0.249047
$784$	5612.77	0.255684
$785$	10260.2	0.466501
$786$	30555.9	1.38663
$787$	31784.6	1.43964	0.719821	−	0.694160i	$-0.244224\pi$
$787$	31784.6	1.43964	0.719821	+	0.694160i	$0.244224\pi$
$788$	13128.6	0.593511
$789$	−20609.4	−0.929931
$790$	4657.29	0.209746
$791$	−7676.80	−0.345077
$792$	−1415.28	−0.0634972
$793$	−35821.6	−1.60411
$794$	−11586.9	−0.517888
$795$	5205.01	0.232205
$796$	−19674.2	−0.876046
$797$	30740.5	1.36623	0.683115	−	0.730311i	$-0.260625\pi$
$797$	30740.5	1.36623	0.683115	+	0.730311i	$0.260625\pi$
$798$	−3.68436	−0.000163440 0
$799$	−1838.74	−0.0814143
$800$	5857.10	0.258850
$801$	−9894.60	−0.436465
$802$	−38563.0	−1.69789
$803$	−1756.72	−0.0772023
$804$	−24160.5	−1.05979
$805$	−4284.27	−0.187579
$806$	58053.2	2.53702
$807$	6058.79	0.264287
$808$	−13504.0	−0.587958
$809$	821.592	0.0357054	0.0178527	−	0.999841i	$-0.494317\pi$
$809$	821.592	0.0357054	0.0178527	+	0.999841i	$0.494317\pi$
$810$	−1777.28	−0.0770955
$811$	35151.8	1.52201	0.761003	−	0.648749i	$-0.224707\pi$
$811$	35151.8	1.52201	0.761003	+	0.648749i	$0.224707\pi$
$812$	−26688.3	−1.15342
$813$	−15574.8	−0.671873
$814$	−14476.1	−0.623326
$815$	−3712.76	−0.159573
$816$	−811.529	−0.0348152
$817$	−3.48593	−0.000149275 0
$818$	25760.8	1.10111
$819$	7688.91	0.328049
$820$	4983.98	0.212254
$821$	5435.02	0.231040	0.115520	−	0.993305i	$-0.463147\pi$
$821$	5435.02	0.231040	0.115520	+	0.993305i	$0.463147\pi$
$822$	26516.4	1.12514
$823$	36291.6	1.53711	0.768557	−	0.639781i	$-0.220975\pi$
$823$	36291.6	1.53711	0.768557	+	0.639781i	$0.220975\pi$
$824$	−26195.9	−1.10749
$825$	−825.000	−0.0348155
$826$	35613.2	1.50017
$827$	−18370.7	−0.772443	−0.386222	−	0.922406i	$-0.626220\pi$
$827$	−18370.7	−0.772443	−0.386222	+	0.922406i	$0.626220\pi$
$828$	7400.90	0.310627
$829$	33861.4	1.41864	0.709322	−	0.704885i	$-0.249001\pi$
$829$	33861.4	1.41864	0.709322	+	0.704885i	$0.249001\pi$
$830$	−9067.49	−0.379201
$831$	1368.40	0.0571231
$832$	58956.5	2.45667
$833$	2033.27	0.0845723
$834$	24125.1	1.00166
$835$	−3497.69	−0.144961
$836$	2.95438	0.000122224 0
$837$	−4904.33	−0.202531
$838$	−62353.3	−2.57036
$839$	−22131.3	−0.910676	−0.455338	−	0.890319i	$-0.650482\pi$
$839$	−22131.3	−0.910676	−0.455338	+	0.890319i	$0.650482\pi$
$840$	−2515.43	−0.103322
$841$	16454.3	0.674660
$842$	53985.0	2.20956
$843$	−9014.75	−0.368309
$844$	−4672.84	−0.190576
$845$	15535.9	0.632486
$846$	−7336.09	−0.298132
$847$	−1419.38	−0.0575803
$848$	9482.28	0.383989
$849$	1227.47	0.0496191
$850$	1086.03	0.0438242
$851$	21905.4	0.882383
$852$	−18303.5	−0.735994
$853$	−34402.7	−1.38092	−0.690460	−	0.723370i	$-0.742592\pi$
$853$	−34402.7	−1.38092	−0.690460	+	0.723370i	$0.742592\pi$
$854$	25319.3	1.01453
$855$	1.07359	4.29427e−5 0
$856$	10094.9	0.403081
$857$	−26140.0	−1.04192	−0.520960	−	0.853581i	$-0.674426\pi$
$857$	−26140.0	−1.04192	−0.520960	+	0.853581i	$0.674426\pi$
$858$	−10546.9	−0.419656
$859$	38993.9	1.54884	0.774421	−	0.632671i	$-0.218042\pi$
$859$	38993.9	1.54884	0.774421	+	0.632671i	$0.218042\pi$
$860$	−8224.53	−0.326109
$861$	3115.97	0.123335
$862$	24750.7	0.977970
$863$	−43742.0	−1.72537	−0.862685	−	0.505741i	$-0.831219\pi$
$863$	−43742.0	−1.72537	−0.862685	+	0.505741i	$0.831219\pi$
$864$	−6325.67	−0.249078
$865$	−3674.84	−0.144449
$866$	−44422.8	−1.74313
$867$	14445.0	0.565834
$868$	−23987.1	−0.937988
$869$	−2334.83	−0.0911433
$870$	13303.1	0.518411
$871$	−52100.9	−2.02683
$872$	26173.1	1.01644
$873$	5111.99	0.198184
$874$	−7.64754	−0.000295975 0
$875$	−1466.30	−0.0566515
$876$	5393.62	0.208029
$877$	−3327.46	−0.128119	−0.0640596	−	0.997946i	$-0.520405\pi$
$877$	−3327.46	−0.128119	−0.0640596	+	0.997946i	$0.520405\pi$
$878$	47188.7	1.81383
$879$	22230.4	0.853030
$880$	−1502.95	−0.0575733
$881$	−2068.21	−0.0790915	−0.0395458	−	0.999218i	$-0.512591\pi$
$881$	−2068.21	−0.0790915	−0.0395458	+	0.999218i	$0.512591\pi$
$882$	8112.21	0.309697
$883$	−24764.3	−0.943812	−0.471906	−	0.881649i	$-0.656434\pi$
$883$	−24764.3	−0.943812	−0.471906	+	0.881649i	$0.656434\pi$
$884$	8116.29	0.308801
$885$	−10377.4	−0.394160
$886$	45404.7	1.72167
$887$	−34771.6	−1.31625	−0.658125	−	0.752908i	$-0.728650\pi$
$887$	−34771.6	−1.31625	−0.658125	+	0.752908i	$0.728650\pi$
$888$	12861.3	0.486034
$889$	−9553.24	−0.360411
$890$	24122.8	0.908536
$891$	891.000	0.0335013
$892$	−820.755	−0.0308082
$893$	4.43146	0.000166062 0
$894$	−47180.2	−1.76504
$895$	−13436.4	−0.501821
$896$	−19685.3	−0.733975
$897$	15959.7	0.594067
$898$	−41993.1	−1.56050
$899$	36709.3	1.36187
$900$	2532.97	0.0938138
$901$	3435.03	0.127012
$902$	−4274.18	−0.157777
$903$	−5141.94	−0.189494
$904$	−9355.66	−0.344209
$905$	−18832.8	−0.691738
$906$	−37158.1	−1.36258
$907$	−12892.0	−0.471965	−0.235983	−	0.971757i	$-0.575831\pi$
$907$	−12892.0	−0.471965	−0.235983	+	0.971757i	$0.575831\pi$
$908$	26594.6	0.971995
$909$	8501.55	0.310208
$910$	−18745.4	−0.682860
$911$	−20530.9	−0.746673	−0.373337	−	0.927696i	$-0.621786\pi$
$911$	−20530.9	−0.746673	−0.373337	+	0.927696i	$0.621786\pi$
$912$	1.95582	7.10130e−5 0
$913$	4545.78	0.164779
$914$	−32677.1	−1.18256
$915$	−7377.81	−0.266561
$916$	51045.9	1.84127
$917$	−27226.1	−0.980462
$918$	−1172.91	−0.0421699
$919$	−19937.9	−0.715658	−0.357829	−	0.933787i	$-0.616483\pi$
$919$	−19937.9	−0.715658	−0.357829	+	0.933787i	$0.616483\pi$
$920$	−5221.21	−0.187107
$921$	−14957.3	−0.535136
$922$	−47986.1	−1.71403
$923$	−39470.6	−1.40757
$924$	4357.88	0.155156
$925$	7497.17	0.266492
$926$	43397.1	1.54008
$927$	16491.8	0.584317
$928$	47348.1	1.67487
$929$	−55075.7	−1.94508	−0.972538	−	0.232746i	$-0.925229\pi$
$929$	−55075.7	−1.94508	−0.972538	+	0.232746i	$0.925229\pi$
$930$	11956.6	0.421584
$931$	−4.90028	−0.000172503 0
$932$	2601.38	0.0914280
$933$	18717.0	0.656769
$934$	440.571	0.0154346
$935$	−544.457	−0.0190435
$936$	9370.41	0.327224
$937$	−29820.6	−1.03970	−0.519848	−	0.854259i	$-0.674011\pi$
$937$	−29820.6	−1.03970	−0.519848	+	0.854259i	$0.674011\pi$
$938$	36825.7	1.28188
$939$	29463.9	1.02398
$940$	10455.3	0.362783
$941$	32262.4	1.11767	0.558833	−	0.829280i	$-0.311249\pi$
$941$	32262.4	1.11767	0.558833	+	0.829280i	$0.311249\pi$
$942$	27015.3	0.934401
$943$	6467.73	0.223349
$944$	−18905.1	−0.651808
$945$	1583.61	0.0545129
$946$	7053.22	0.242410
$947$	−20302.5	−0.696666	−0.348333	−	0.937371i	$-0.613252\pi$
$947$	−20302.5	−0.696666	−0.348333	+	0.937371i	$0.613252\pi$
$948$	7168.55	0.245594
$949$	11631.1	0.397851
$950$	−2.61739	−8.93887e−5 0
$951$	5241.78	0.178734
$952$	−1660.05	−0.0565153
$953$	7243.13	0.246199	0.123100	−	0.992394i	$-0.460717\pi$
$953$	7243.13	0.246199	0.123100	+	0.992394i	$0.460717\pi$
$954$	13704.9	0.465106
$955$	−8920.76	−0.302271
$956$	50245.0	1.69983
$957$	−6669.21	−0.225272
$958$	19369.6	0.653238
$959$	−23626.8	−0.795567
$960$	12142.7	0.408232
$961$	3202.72	0.107506
$962$	95844.7	3.21222
$963$	−6355.33	−0.212666
$964$	−23613.1	−0.788927
$965$	−22318.7	−0.744523
$966$	−11280.5	−0.375720
$967$	−35402.4	−1.17732	−0.588658	−	0.808382i	$-0.700344\pi$
$967$	−35402.4	−1.17732	−0.588658	+	0.808382i	$0.700344\pi$
$968$	−1729.79	−0.0574354
$969$	0.708513	2.34889e−5 0
$970$	−12462.9	−0.412536
$971$	−4236.86	−0.140028	−0.0700140	−	0.997546i	$-0.522304\pi$
$971$	−4236.86	−0.140028	−0.0700140	+	0.997546i	$0.522304\pi$
$972$	−2735.61	−0.0902724
$973$	−21496.1	−0.708256
$974$	73704.9	2.42470
$975$	5462.23	0.179417
$976$	−13440.6	−0.440802
$977$	31833.5	1.04242	0.521210	−	0.853429i	$-0.325481\pi$
$977$	31833.5	1.04242	0.521210	+	0.853429i	$0.325481\pi$
$978$	−9775.75	−0.319626
$979$	−12093.4	−0.394797
$980$	−11561.5	−0.376855
$981$	−16477.5	−0.536274
$982$	−83394.1	−2.70999
$983$	−36137.8	−1.17255	−0.586274	−	0.810113i	$-0.699406\pi$
$983$	−36137.8	−1.17255	−0.586274	+	0.810113i	$0.699406\pi$
$984$	3797.40	0.123025
$985$	5830.96	0.188619
$986$	8779.36	0.283562
$987$	6536.65	0.210804
$988$	−19.5607	−0.000629866 0
$989$	−10673.0	−0.343157
$990$	−2172.24	−0.0697355
$991$	13795.5	0.442207	0.221103	−	0.975250i	$-0.429034\pi$
$991$	13795.5	0.442207	0.221103	+	0.975250i	$0.429034\pi$
$992$	42555.8	1.36204
$993$	21627.0	0.691151
$994$	27898.4	0.890225
$995$	−8738.13	−0.278409
$996$	−13956.8	−0.444013
$997$	52392.7	1.66429	0.832144	−	0.554560i	$-0.187113\pi$
$997$	52392.7	1.66429	0.832144	+	0.554560i	$0.187113\pi$
$998$	87633.2	2.77954
$999$	−8096.94	−0.256432

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.4.a.g.1.1	✓	3
3.2	odd	2		495.4.a.i.1.3		3
5.2	odd	4		825.4.c.m.199.2		6
5.3	odd	4		825.4.c.m.199.5		6
5.4	even	2		825.4.a.p.1.3		3
11.10	odd	2		1815.4.a.q.1.3		3
15.14	odd	2		2475.4.a.z.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.g.1.1	✓	3	1.1	even	1	trivial
495.4.a.i.1.3		3	3.2	odd	2
825.4.a.p.1.3		3	5.4	even	2
825.4.c.m.199.2		6	5.2	odd	4
825.4.c.m.199.5		6	5.3	odd	4
1815.4.a.q.1.3		3	11.10	odd	2
2475.4.a.z.1.1		3	15.14	odd	2

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:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	165.a (trivial)

\(f(q)\)	\(=\)	\(q-4.38835 q^{2} -3.00000 q^{3} +11.2577 q^{4} +5.00000 q^{5} +13.1651 q^{6} -11.7304 q^{7} -14.2958 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.38835 q^{2} -3.00000 q^{3} +11.2577 q^{4} +5.00000 q^{5} +13.1651 q^{6} -11.7304 q^{7} -14.2958 q^{8} +9.00000 q^{9} -21.9418 q^{10} +11.0000 q^{11} -33.7730 q^{12} -72.8298 q^{13} +51.4772 q^{14} -15.0000 q^{15} -27.3264 q^{16} -9.89921 q^{17} -39.4952 q^{18} +0.0238576 q^{19} +56.2883 q^{20} +35.1912 q^{21} -48.2719 q^{22} +73.0456 q^{23} +42.8873 q^{24} +25.0000 q^{25} +319.603 q^{26} -27.0000 q^{27} -132.057 q^{28} +202.097 q^{29} +65.8253 q^{30} +181.642 q^{31} +234.284 q^{32} -33.0000 q^{33} +43.4413 q^{34} -58.6521 q^{35} +101.319 q^{36} +299.887 q^{37} -0.104695 q^{38} +218.489 q^{39} -71.4788 q^{40} +88.5438 q^{41} -154.432 q^{42} -146.114 q^{43} +123.834 q^{44} +45.0000 q^{45} -320.550 q^{46} +185.746 q^{47} +81.9792 q^{48} -205.397 q^{49} -109.709 q^{50} +29.6976 q^{51} -819.893 q^{52} -347.001 q^{53} +118.486 q^{54} +55.0000 q^{55} +167.695 q^{56} -0.0715727 q^{57} -886.874 q^{58} +691.824 q^{59} -168.865 q^{60} +491.854 q^{61} -797.108 q^{62} -105.574 q^{63} -809.510 q^{64} -364.149 q^{65} +144.816 q^{66} +715.379 q^{67} -111.442 q^{68} -219.137 q^{69} +257.386 q^{70} +541.957 q^{71} -128.662 q^{72} -159.702 q^{73} -1316.01 q^{74} -75.0000 q^{75} +0.268580 q^{76} -129.035 q^{77} -958.809 q^{78} -212.257 q^{79} -136.632 q^{80} +81.0000 q^{81} -388.562 q^{82} +413.252 q^{83} +396.171 q^{84} -49.4961 q^{85} +641.202 q^{86} -606.292 q^{87} -157.253 q^{88} -1099.40 q^{89} -197.476 q^{90} +854.323 q^{91} +822.322 q^{92} -544.925 q^{93} -815.121 q^{94} +0.119288 q^{95} -702.852 q^{96} +567.999 q^{97} +901.357 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q + q^2 - 9 * q^3 + 17 * q^4 + 15 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 27 * q^9	\( 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9} + 5 q^{10} + 33 q^{11} - 51 q^{12} - 20 q^{13} + 144 q^{14} - 45 q^{15} + 25 q^{16} + 32 q^{17} + 9 q^{18} + 116 q^{19} + 85 q^{20} - 18 q^{21} + 11 q^{22} + 240 q^{23} + 9 q^{24} + 75 q^{25} + 302 q^{26} - 81 q^{27} + 160 q^{28} + 238 q^{29} - 15 q^{30} + 92 q^{31} + 197 q^{32} - 99 q^{33} + 354 q^{34} + 30 q^{35} + 153 q^{36} - 90 q^{37} + 324 q^{38} + 60 q^{39} - 15 q^{40} - 46 q^{41} - 432 q^{42} - 134 q^{43} + 187 q^{44} + 135 q^{45} - 240 q^{46} - 220 q^{47} - 75 q^{48} - 457 q^{49} + 25 q^{50} - 96 q^{51} - 1530 q^{52} - 798 q^{53} - 27 q^{54} + 165 q^{55} + 688 q^{56} - 348 q^{57} - 978 q^{58} + 1236 q^{59} - 255 q^{60} + 342 q^{61} - 1792 q^{62} + 54 q^{63} - 1919 q^{64} - 100 q^{65} - 33 q^{66} + 764 q^{67} + 1074 q^{68} - 720 q^{69} + 720 q^{70} + 1816 q^{71} - 27 q^{72} + 100 q^{73} - 1874 q^{74} - 225 q^{75} + 396 q^{76} + 66 q^{77} - 906 q^{78} - 96 q^{79} + 125 q^{80} + 243 q^{81} - 910 q^{82} + 858 q^{83} - 480 q^{84} + 160 q^{85} + 188 q^{86} - 714 q^{87} - 33 q^{88} + 838 q^{89} + 45 q^{90} + 332 q^{91} - 688 q^{92} - 276 q^{93} - 3112 q^{94} + 580 q^{95} - 591 q^{96} - 1322 q^{97} + 1017 q^{98} + 297 q^{99}+O(q^{100}) \) 3 * q + q^2 - 9 * q^3 + 17 * q^4 + 15 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 27 * q^9 + 5 * q^10 + 33 * q^11 - 51 * q^12 - 20 * q^13 + 144 * q^14 - 45 * q^15 + 25 * q^16 + 32 * q^17 + 9 * q^18 + 116 * q^19 + 85 * q^20 - 18 * q^21 + 11 * q^22 + 240 * q^23 + 9 * q^24 + 75 * q^25 + 302 * q^26 - 81 * q^27 + 160 * q^28 + 238 * q^29 - 15 * q^30 + 92 * q^31 + 197 * q^32 - 99 * q^33 + 354 * q^34 + 30 * q^35 + 153 * q^36 - 90 * q^37 + 324 * q^38 + 60 * q^39 - 15 * q^40 - 46 * q^41 - 432 * q^42 - 134 * q^43 + 187 * q^44 + 135 * q^45 - 240 * q^46 - 220 * q^47 - 75 * q^48 - 457 * q^49 + 25 * q^50 - 96 * q^51 - 1530 * q^52 - 798 * q^53 - 27 * q^54 + 165 * q^55 + 688 * q^56 - 348 * q^57 - 978 * q^58 + 1236 * q^59 - 255 * q^60 + 342 * q^61 - 1792 * q^62 + 54 * q^63 - 1919 * q^64 - 100 * q^65 - 33 * q^66 + 764 * q^67 + 1074 * q^68 - 720 * q^69 + 720 * q^70 + 1816 * q^71 - 27 * q^72 + 100 * q^73 - 1874 * q^74 - 225 * q^75 + 396 * q^76 + 66 * q^77 - 906 * q^78 - 96 * q^79 + 125 * q^80 + 243 * q^81 - 910 * q^82 + 858 * q^83 - 480 * q^84 + 160 * q^85 + 188 * q^86 - 714 * q^87 - 33 * q^88 + 838 * q^89 + 45 * q^90 + 332 * q^91 - 688 * q^92 - 276 * q^93 - 3112 * q^94 + 580 * q^95 - 591 * q^96 - 1322 * q^97 + 1017 * q^98 + 297 * q^99

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