LMFDB - Embedded newform 354.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,4,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("354.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	354.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:	$20.8866761420$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.39445 q^{5} +6.00000 q^{6} -27.7250 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.39445 q^{5} +6.00000 q^{6} -27.7250 q^{7} +8.00000 q^{8} +9.00000 q^{9} -12.7889 q^{10} -21.9722 q^{11} +12.0000 q^{12} -14.9445 q^{13} -55.4500 q^{14} -19.1833 q^{15} +16.0000 q^{16} -7.09167 q^{17} +18.0000 q^{18} -25.6972 q^{19} -25.5778 q^{20} -83.1749 q^{21} -43.9445 q^{22} -177.369 q^{23} +24.0000 q^{24} -84.1110 q^{25} -29.8890 q^{26} +27.0000 q^{27} -110.900 q^{28} -76.9916 q^{29} -38.3667 q^{30} +234.902 q^{31} +32.0000 q^{32} -65.9167 q^{33} -14.1833 q^{34} +177.286 q^{35} +36.0000 q^{36} -198.836 q^{37} -51.3944 q^{38} -44.8335 q^{39} -51.1556 q^{40} -410.463 q^{41} -166.350 q^{42} -84.5391 q^{43} -87.8890 q^{44} -57.5500 q^{45} -354.738 q^{46} +586.958 q^{47} +48.0000 q^{48} +425.675 q^{49} -168.222 q^{50} -21.2750 q^{51} -59.7779 q^{52} +609.005 q^{53} +54.0000 q^{54} +140.500 q^{55} -221.800 q^{56} -77.0917 q^{57} -153.983 q^{58} +59.0000 q^{59} -76.7334 q^{60} -66.9193 q^{61} +469.805 q^{62} -249.525 q^{63} +64.0000 q^{64} +95.5618 q^{65} -131.833 q^{66} -729.394 q^{67} -28.3667 q^{68} -532.108 q^{69} +354.572 q^{70} -40.4719 q^{71} +72.0000 q^{72} +193.636 q^{73} -397.672 q^{74} -252.333 q^{75} -102.789 q^{76} +609.180 q^{77} -89.6669 q^{78} -76.0059 q^{79} -102.311 q^{80} +81.0000 q^{81} -820.927 q^{82} +536.596 q^{83} -332.700 q^{84} +45.3473 q^{85} -169.078 q^{86} -230.975 q^{87} -175.778 q^{88} -972.719 q^{89} -115.100 q^{90} +414.336 q^{91} -709.477 q^{92} +704.707 q^{93} +1173.92 q^{94} +164.320 q^{95} +96.0000 q^{96} -599.671 q^{97} +851.349 q^{98} -197.750 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 20 q^{5} + 12 q^{6} - 23 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 20 * q^5 + 12 * q^6 - 23 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 20 q^{5} + 12 q^{6} - 23 q^{7} + 16 q^{8} + 18 q^{9} - 40 q^{10} - 80 q^{11} + 24 q^{12} - 102 q^{13} - 46 q^{14} - 60 q^{15} + 32 q^{16} - 25 q^{17} + 36 q^{18} - 55 q^{19} - 80 q^{20} - 69 q^{21} - 160 q^{22} - 5 q^{23} + 48 q^{24} - 24 q^{25} - 204 q^{26} + 54 q^{27} - 92 q^{28} - 35 q^{29} - 120 q^{30} - 53 q^{31} + 64 q^{32} - 240 q^{33} - 50 q^{34} + 113 q^{35} + 72 q^{36} - 221 q^{37} - 110 q^{38} - 306 q^{39} - 160 q^{40} - 89 q^{41} - 138 q^{42} - 508 q^{43} - 320 q^{44} - 180 q^{45} - 10 q^{46} + 579 q^{47} + 96 q^{48} + 105 q^{49} - 48 q^{50} - 75 q^{51} - 408 q^{52} + 432 q^{53} + 108 q^{54} + 930 q^{55} - 184 q^{56} - 165 q^{57} - 70 q^{58} + 118 q^{59} - 240 q^{60} + 151 q^{61} - 106 q^{62} - 207 q^{63} + 128 q^{64} + 1280 q^{65} - 480 q^{66} - 168 q^{67} - 100 q^{68} - 15 q^{69} + 226 q^{70} + 532 q^{71} + 144 q^{72} - 49 q^{73} - 442 q^{74} - 72 q^{75} - 220 q^{76} + 335 q^{77} - 612 q^{78} - 664 q^{79} - 320 q^{80} + 162 q^{81} - 178 q^{82} - 351 q^{83} - 276 q^{84} + 289 q^{85} - 1016 q^{86} - 105 q^{87} - 640 q^{88} - 1401 q^{89} - 360 q^{90} + 3 q^{91} - 20 q^{92} - 159 q^{93} + 1158 q^{94} + 563 q^{95} + 192 q^{96} + 452 q^{97} + 210 q^{98} - 720 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 20 * q^5 + 12 * q^6 - 23 * q^7 + 16 * q^8 + 18 * q^9 - 40 * q^10 - 80 * q^11 + 24 * q^12 - 102 * q^13 - 46 * q^14 - 60 * q^15 + 32 * q^16 - 25 * q^17 + 36 * q^18 - 55 * q^19 - 80 * q^20 - 69 * q^21 - 160 * q^22 - 5 * q^23 + 48 * q^24 - 24 * q^25 - 204 * q^26 + 54 * q^27 - 92 * q^28 - 35 * q^29 - 120 * q^30 - 53 * q^31 + 64 * q^32 - 240 * q^33 - 50 * q^34 + 113 * q^35 + 72 * q^36 - 221 * q^37 - 110 * q^38 - 306 * q^39 - 160 * q^40 - 89 * q^41 - 138 * q^42 - 508 * q^43 - 320 * q^44 - 180 * q^45 - 10 * q^46 + 579 * q^47 + 96 * q^48 + 105 * q^49 - 48 * q^50 - 75 * q^51 - 408 * q^52 + 432 * q^53 + 108 * q^54 + 930 * q^55 - 184 * q^56 - 165 * q^57 - 70 * q^58 + 118 * q^59 - 240 * q^60 + 151 * q^61 - 106 * q^62 - 207 * q^63 + 128 * q^64 + 1280 * q^65 - 480 * q^66 - 168 * q^67 - 100 * q^68 - 15 * q^69 + 226 * q^70 + 532 * q^71 + 144 * q^72 - 49 * q^73 - 442 * q^74 - 72 * q^75 - 220 * q^76 + 335 * q^77 - 612 * q^78 - 664 * q^79 - 320 * q^80 + 162 * q^81 - 178 * q^82 - 351 * q^83 - 276 * q^84 + 289 * q^85 - 1016 * q^86 - 105 * q^87 - 640 * q^88 - 1401 * q^89 - 360 * q^90 + 3 * q^91 - 20 * q^92 - 159 * q^93 + 1158 * q^94 + 563 * q^95 + 192 * q^96 + 452 * q^97 + 210 * q^98 - 720 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−6.39445	−0.571937	−0.285968	−	0.958239i	$-0.592315\pi$
$5$	−6.39445	−0.571937	−0.285968	+	0.958239i	$0.592315\pi$
$6$	6.00000	0.408248
$7$	−27.7250	−1.49701	−0.748504	−	0.663130i	$-0.769228\pi$
$7$	−27.7250	−1.49701	−0.748504	+	0.663130i	$0.769228\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−12.7889	−0.404420
$11$	−21.9722	−0.602262	−0.301131	−	0.953583i	$-0.597364\pi$
$11$	−21.9722	−0.602262	−0.301131	+	0.953583i	$0.597364\pi$
$12$	12.0000	0.288675
$13$	−14.9445	−0.318835	−0.159418	−	0.987211i	$-0.550962\pi$
$13$	−14.9445	−0.318835	−0.159418	+	0.987211i	$0.550962\pi$
$14$	−55.4500	−1.05854
$15$	−19.1833	−0.330208
$16$	16.0000	0.250000
$17$	−7.09167	−0.101175	−0.0505877	−	0.998720i	$-0.516109\pi$
$17$	−7.09167	−0.101175	−0.0505877	+	0.998720i	$0.516109\pi$
$18$	18.0000	0.235702
$19$	−25.6972	−0.310281	−0.155141	−	0.987892i	$-0.549583\pi$
$19$	−25.6972	−0.310281	−0.155141	+	0.987892i	$0.549583\pi$
$20$	−25.5778	−0.285968
$21$	−83.1749	−0.864298
$22$	−43.9445	−0.425863
$23$	−177.369	−1.60800	−0.804001	−	0.594628i	$-0.797299\pi$
$23$	−177.369	−1.60800	−0.804001	+	0.594628i	$0.797299\pi$
$24$	24.0000	0.204124
$25$	−84.1110	−0.672888
$26$	−29.8890	−0.225450
$27$	27.0000	0.192450
$28$	−110.900	−0.748504
$29$	−76.9916	−0.492999	−0.246500	−	0.969143i	$-0.579280\pi$
$29$	−76.9916	−0.492999	−0.246500	+	0.969143i	$0.579280\pi$
$30$	−38.3667	−0.233492
$31$	234.902	1.36096	0.680479	−	0.732767i	$-0.261772\pi$
$31$	234.902	1.36096	0.680479	+	0.732767i	$0.261772\pi$
$32$	32.0000	0.176777
$33$	−65.9167	−0.347716
$34$	−14.1833	−0.0715419
$35$	177.286	0.856194
$36$	36.0000	0.166667
$37$	−198.836	−0.883471	−0.441736	−	0.897145i	$-0.645637\pi$
$37$	−198.836	−0.883471	−0.441736	+	0.897145i	$0.645637\pi$
$38$	−51.3944	−0.219402
$39$	−44.8335	−0.184079
$40$	−51.1556	−0.202210
$41$	−410.463	−1.56350	−0.781751	−	0.623590i	$-0.785673\pi$
$41$	−410.463	−1.56350	−0.781751	+	0.623590i	$0.785673\pi$
$42$	−166.350	−0.611151
$43$	−84.5391	−0.299816	−0.149908	−	0.988700i	$-0.547898\pi$
$43$	−84.5391	−0.299816	−0.149908	+	0.988700i	$0.547898\pi$
$44$	−87.8890	−0.301131
$45$	−57.5500	−0.190646
$46$	−354.738	−1.13703
$47$	586.958	1.82163	0.910815	−	0.412815i	$-0.135454\pi$
$47$	586.958	1.82163	0.910815	+	0.412815i	$0.135454\pi$
$48$	48.0000	0.144338
$49$	425.675	1.24103
$50$	−168.222	−0.475804
$51$	−21.2750	−0.0584137
$52$	−59.7779	−0.159418
$53$	609.005	1.57836	0.789182	−	0.614160i	$-0.210505\pi$
$53$	609.005	1.57836	0.789182	+	0.614160i	$0.210505\pi$
$54$	54.0000	0.136083
$55$	140.500	0.344456
$56$	−221.800	−0.529272
$57$	−77.0917	−0.179141
$58$	−153.983	−0.348603
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	−76.7334	−0.165104
$61$	−66.9193	−0.140461	−0.0702306	−	0.997531i	$-0.522374\pi$
$61$	−66.9193	−0.140461	−0.0702306	+	0.997531i	$0.522374\pi$
$62$	469.805	0.962343
$63$	−249.525	−0.499003
$64$	64.0000	0.125000
$65$	95.5618	0.182353
$66$	−131.833	−0.245872
$67$	−729.394	−1.32999	−0.664997	−	0.746846i	$-0.731567\pi$
$67$	−729.394	−1.32999	−0.664997	+	0.746846i	$0.731567\pi$
$68$	−28.3667	−0.0505877
$69$	−532.108	−0.928380
$70$	354.572	0.605421
$71$	−40.4719	−0.0676497	−0.0338248	−	0.999428i	$-0.510769\pi$
$71$	−40.4719	−0.0676497	−0.0338248	+	0.999428i	$0.510769\pi$
$72$	72.0000	0.117851
$73$	193.636	0.310457	0.155229	−	0.987879i	$-0.450389\pi$
$73$	193.636	0.310457	0.155229	+	0.987879i	$0.450389\pi$
$74$	−397.672	−0.624709
$75$	−252.333	−0.388492
$76$	−102.789	−0.155141
$77$	609.180	0.901591
$78$	−89.6669	−0.130164
$79$	−76.0059	−0.108245	−0.0541223	−	0.998534i	$-0.517236\pi$
$79$	−76.0059	−0.108245	−0.0541223	+	0.998534i	$0.517236\pi$
$80$	−102.311	−0.142984
$81$	81.0000	0.111111
$82$	−820.927	−1.10556
$83$	536.596	0.709628	0.354814	−	0.934937i	$-0.384544\pi$
$83$	536.596	0.709628	0.354814	+	0.934937i	$0.384544\pi$
$84$	−332.700	−0.432149
$85$	45.3473	0.0578660
$86$	−169.078	−0.212002
$87$	−230.975	−0.284633
$88$	−175.778	−0.212932
$89$	−972.719	−1.15852	−0.579259	−	0.815144i	$-0.696658\pi$
$89$	−972.719	−1.15852	−0.579259	+	0.815144i	$0.696658\pi$
$90$	−115.100	−0.134807
$91$	414.336	0.477299
$92$	−709.477	−0.804001
$93$	704.707	0.785750
$94$	1173.92	1.28809
$95$	164.320	0.177461
$96$	96.0000	0.102062
$97$	−599.671	−0.627705	−0.313853	−	0.949472i	$-0.601620\pi$
$97$	−599.671	−0.627705	−0.313853	+	0.949472i	$0.601620\pi$
$98$	851.349	0.877543
$99$	−197.750	−0.200754
$100$	−336.444	−0.336444
$101$	841.016	0.828557	0.414278	−	0.910150i	$-0.364034\pi$
$101$	841.016	0.828557	0.414278	+	0.910150i	$0.364034\pi$
$102$	−42.5500	−0.0413047
$103$	416.705	0.398633	0.199316	−	0.979935i	$-0.436128\pi$
$103$	416.705	0.398633	0.199316	+	0.979935i	$0.436128\pi$
$104$	−119.556	−0.112725
$105$	531.858	0.494324
$106$	1218.01	1.11607
$107$	−260.819	−0.235648	−0.117824	−	0.993034i	$-0.537592\pi$
$107$	−260.819	−0.235648	−0.117824	+	0.993034i	$0.537592\pi$
$108$	108.000	0.0962250
$109$	142.876	0.125551	0.0627755	−	0.998028i	$-0.480005\pi$
$109$	142.876	0.125551	0.0627755	+	0.998028i	$0.480005\pi$
$110$	281.001	0.243567
$111$	−596.508	−0.510072
$112$	−443.600	−0.374252
$113$	−1323.95	−1.10218	−0.551091	−	0.834445i	$-0.685788\pi$
$113$	−1323.95	−1.10218	−0.551091	+	0.834445i	$0.685788\pi$
$114$	−154.183	−0.126672
$115$	1134.18	0.919676
$116$	−307.966	−0.246500
$117$	−134.500	−0.106278
$118$	118.000	0.0920575
$119$	196.616	0.151461
$120$	−153.467	−0.116746
$121$	−848.221	−0.637281
$122$	−133.839	−0.0993211
$123$	−1231.39	−0.902689
$124$	939.610	0.680479
$125$	1337.15	0.956786
$126$	−499.050	−0.352848
$127$	1825.64	1.27558	0.637792	−	0.770209i	$-0.279848\pi$
$127$	1825.64	1.27558	0.637792	+	0.770209i	$0.279848\pi$
$128$	128.000	0.0883883
$129$	−253.617	−0.173099
$130$	191.124	0.128943
$131$	541.793	0.361349	0.180674	−	0.983543i	$-0.442172\pi$
$131$	541.793	0.361349	0.180674	+	0.983543i	$0.442172\pi$
$132$	−263.667	−0.173858
$133$	712.455	0.464494
$134$	−1458.79	−0.940448
$135$	−172.650	−0.110069
$136$	−56.7334	−0.0357709
$137$	1809.32	1.12832	0.564162	−	0.825664i	$-0.309199\pi$
$137$	1809.32	1.12832	0.564162	+	0.825664i	$0.309199\pi$
$138$	−1064.22	−0.656464
$139$	−1874.87	−1.14406	−0.572029	−	0.820234i	$-0.693843\pi$
$139$	−1874.87	−1.14406	−0.572029	+	0.820234i	$0.693843\pi$
$140$	709.144	0.428097
$141$	1760.87	1.05172
$142$	−80.9437	−0.0478355
$143$	328.364	0.192022
$144$	144.000	0.0833333
$145$	492.319	0.281964
$146$	387.272	0.219526
$147$	1277.02	0.716511
$148$	−795.344	−0.441736
$149$	614.885	0.338076	0.169038	−	0.985610i	$-0.445934\pi$
$149$	614.885	0.338076	0.169038	+	0.985610i	$0.445934\pi$
$150$	−504.666	−0.274705
$151$	1185.35	0.638826	0.319413	−	0.947616i	$-0.396514\pi$
$151$	1185.35	0.638826	0.319413	+	0.947616i	$0.396514\pi$
$152$	−205.578	−0.109701
$153$	−63.8251	−0.0337252
$154$	1218.36	0.637521
$155$	−1502.07	−0.778383
$156$	−179.334	−0.0920397
$157$	−2243.13	−1.14026	−0.570131	−	0.821554i	$-0.693108\pi$
$157$	−2243.13	−1.14026	−0.570131	+	0.821554i	$0.693108\pi$
$158$	−152.012	−0.0765405
$159$	1827.02	0.911269
$160$	−204.622	−0.101105
$161$	4917.56	2.40719
$162$	162.000	0.0785674
$163$	−2239.14	−1.07597	−0.537985	−	0.842954i	$-0.680814\pi$
$163$	−2239.14	−1.07597	−0.537985	+	0.842954i	$0.680814\pi$
$164$	−1641.85	−0.781751
$165$	421.501	0.198872
$166$	1073.19	0.501783
$167$	−1106.34	−0.512641	−0.256321	−	0.966592i	$-0.582510\pi$
$167$	−1106.34	−0.512641	−0.256321	+	0.966592i	$0.582510\pi$
$168$	−665.400	−0.305576
$169$	−1973.66	−0.898344
$170$	90.6947	0.0409174
$171$	−231.275	−0.103427
$172$	−338.156	−0.149908
$173$	3363.54	1.47818	0.739090	−	0.673606i	$-0.235256\pi$
$173$	3363.54	1.47818	0.739090	+	0.673606i	$0.235256\pi$
$174$	−461.950	−0.201266
$175$	2331.98	1.00732
$176$	−351.556	−0.150565
$177$	177.000	0.0751646
$178$	−1945.44	−0.819195
$179$	1381.23	0.576749	0.288374	−	0.957518i	$-0.406885\pi$
$179$	1381.23	0.576749	0.288374	+	0.957518i	$0.406885\pi$
$180$	−230.200	−0.0953228
$181$	−3462.81	−1.42204	−0.711018	−	0.703174i	$-0.751766\pi$
$181$	−3462.81	−1.42204	−0.711018	+	0.703174i	$0.751766\pi$
$182$	828.671	0.337501
$183$	−200.758	−0.0810953
$184$	−1418.95	−0.568515
$185$	1271.45	0.505290
$186$	1409.41	0.555609
$187$	155.820	0.0609341
$188$	2347.83	0.910815
$189$	−748.574	−0.288099
$190$	328.639	0.125484
$191$	1625.04	0.615623	0.307811	−	0.951447i	$-0.400403\pi$
$191$	1625.04	0.615623	0.307811	+	0.951447i	$0.400403\pi$
$192$	192.000	0.0721688
$193$	3088.78	1.15200	0.575999	−	0.817451i	$-0.304613\pi$
$193$	3088.78	1.15200	0.575999	+	0.817451i	$0.304613\pi$
$194$	−1199.34	−0.443854
$195$	286.685	0.105282
$196$	1702.70	0.620517
$197$	3491.47	1.26273	0.631363	−	0.775487i	$-0.282496\pi$
$197$	3491.47	1.26273	0.631363	+	0.775487i	$0.282496\pi$
$198$	−395.500	−0.141954
$199$	−5264.53	−1.87534	−0.937669	−	0.347529i	$-0.887021\pi$
$199$	−5264.53	−1.87534	−0.937669	+	0.347529i	$0.887021\pi$
$200$	−672.888	−0.237902
$201$	−2188.18	−0.767873
$202$	1682.03	0.585878
$203$	2134.59	0.738024
$204$	−85.1001	−0.0292068
$205$	2624.69	0.894225
$206$	833.410	0.281876
$207$	−1596.32	−0.536001
$208$	−239.112	−0.0797088
$209$	564.626	0.186871
$210$	1063.72	0.349540
$211$	1300.58	0.424339	0.212170	−	0.977233i	$-0.431947\pi$
$211$	1300.58	0.424339	0.212170	+	0.977233i	$0.431947\pi$
$212$	2436.02	0.789182
$213$	−121.416	−0.0390575
$214$	−521.638	−0.166628
$215$	540.581	0.171476
$216$	216.000	0.0680414
$217$	−6512.67	−2.03737
$218$	285.753	0.0887780
$219$	580.908	0.179242
$220$	562.002	0.172228
$221$	105.981	0.0322583
$222$	−1193.02	−0.360676
$223$	3841.78	1.15365	0.576827	−	0.816866i	$-0.304291\pi$
$223$	3841.78	1.15365	0.576827	+	0.816866i	$0.304291\pi$
$224$	−887.199	−0.264636
$225$	−756.999	−0.224296
$226$	−2647.90	−0.779361
$227$	1647.74	0.481781	0.240890	−	0.970552i	$-0.422561\pi$
$227$	1647.74	0.481781	0.240890	+	0.970552i	$0.422561\pi$
$228$	−308.367	−0.0895705
$229$	−5223.43	−1.50731	−0.753655	−	0.657270i	$-0.771711\pi$
$229$	−5223.43	−1.50731	−0.753655	+	0.657270i	$0.771711\pi$
$230$	2268.36	0.650309
$231$	1827.54	0.520534
$232$	−615.933	−0.174302
$233$	1918.67	0.539469	0.269735	−	0.962935i	$-0.413064\pi$
$233$	1918.67	0.539469	0.269735	+	0.962935i	$0.413064\pi$
$234$	−269.001	−0.0751501
$235$	−3753.27	−1.04186
$236$	236.000	0.0650945
$237$	−228.018	−0.0624951
$238$	393.233	0.107099
$239$	−226.258	−0.0612362	−0.0306181	−	0.999531i	$-0.509748\pi$
$239$	−226.258	−0.0612362	−0.0306181	+	0.999531i	$0.509748\pi$
$240$	−306.934	−0.0825520
$241$	−6253.48	−1.67146	−0.835730	−	0.549141i	$-0.814955\pi$
$241$	−6253.48	−1.67146	−0.835730	+	0.549141i	$0.814955\pi$
$242$	−1696.44	−0.450625
$243$	243.000	0.0641500
$244$	−267.677	−0.0702306
$245$	−2721.95	−0.709793
$246$	−2462.78	−0.638297
$247$	384.032	0.0989286
$248$	1879.22	0.481172
$249$	1609.79	0.409704
$250$	2674.30	0.676550
$251$	−1802.44	−0.453263	−0.226632	−	0.973981i	$-0.572771\pi$
$251$	−1802.44	−0.453263	−0.226632	+	0.973981i	$0.572771\pi$
$252$	−998.099	−0.249501
$253$	3897.20	0.968438
$254$	3651.28	0.901974
$255$	136.042	0.0334089
$256$	256.000	0.0625000
$257$	−3109.74	−0.754786	−0.377393	−	0.926053i	$-0.623179\pi$
$257$	−3109.74	−0.754786	−0.377393	+	0.926053i	$0.623179\pi$
$258$	−507.235	−0.122399
$259$	5512.72	1.32256
$260$	382.247	0.0911767
$261$	−692.924	−0.164333
$262$	1083.59	0.255512
$263$	−3680.72	−0.862976	−0.431488	−	0.902119i	$-0.642011\pi$
$263$	−3680.72	−0.862976	−0.431488	+	0.902119i	$0.642011\pi$
$264$	−527.334	−0.122936
$265$	−3894.25	−0.902724
$266$	1424.91	0.328447
$267$	−2918.16	−0.668870
$268$	−2917.57	−0.664997
$269$	−7964.61	−1.80525	−0.902623	−	0.430433i	$-0.858361\pi$
$269$	−7964.61	−1.80525	−0.902623	+	0.430433i	$0.858361\pi$
$270$	−345.300	−0.0778308
$271$	−414.339	−0.0928757	−0.0464379	−	0.998921i	$-0.514787\pi$
$271$	−414.339	−0.0928757	−0.0464379	+	0.998921i	$0.514787\pi$
$272$	−113.467	−0.0252939
$273$	1243.01	0.275568
$274$	3618.63	0.797845
$275$	1848.11	0.405255
$276$	−2128.43	−0.464190
$277$	−353.066	−0.0765836	−0.0382918	−	0.999267i	$-0.512192\pi$
$277$	−353.066	−0.0765836	−0.0382918	+	0.999267i	$0.512192\pi$
$278$	−3749.73	−0.808971
$279$	2114.12	0.453653
$280$	1418.29	0.302710
$281$	−404.140	−0.0857970	−0.0428985	−	0.999079i	$-0.513659\pi$
$281$	−404.140	−0.0857970	−0.0428985	+	0.999079i	$0.513659\pi$
$282$	3521.75	0.743677
$283$	−3198.12	−0.671761	−0.335881	−	0.941905i	$-0.609034\pi$
$283$	−3198.12	−0.671761	−0.335881	+	0.941905i	$0.609034\pi$
$284$	−161.887	−0.0338248
$285$	492.959	0.102457
$286$	656.728	0.135780
$287$	11380.1	2.34058
$288$	288.000	0.0589256
$289$	−4862.71	−0.989764
$290$	984.638	0.199379
$291$	−1799.01	−0.362406
$292$	774.543	0.155229
$293$	4764.55	0.949993	0.474996	−	0.879988i	$-0.342449\pi$
$293$	4764.55	0.949993	0.474996	+	0.879988i	$0.342449\pi$
$294$	2554.05	0.506650
$295$	−377.272	−0.0744598
$296$	−1590.69	−0.312354
$297$	−593.251	−0.115905
$298$	1229.77	0.239056
$299$	2650.69	0.512687
$300$	−1009.33	−0.194246
$301$	2343.84	0.448827
$302$	2370.71	0.451718
$303$	2523.05	0.478367
$304$	−411.156	−0.0775704
$305$	427.912	0.0803350
$306$	−127.650	−0.0238473
$307$	8.88111	0.00165105	0.000825524	−	1.00000i	$-0.499737\pi$
$307$	8.88111	0.00165105	0.000825524		1.00000i	$0.499737\pi$
$308$	2436.72	0.450796
$309$	1250.11	0.230151
$310$	−3004.14	−0.550400
$311$	−5104.58	−0.930721	−0.465360	−	0.885121i	$-0.654075\pi$
$311$	−5104.58	−0.930721	−0.465360	+	0.885121i	$0.654075\pi$
$312$	−358.668	−0.0650819
$313$	692.367	0.125032	0.0625158	−	0.998044i	$-0.480088\pi$
$313$	692.367	0.125032	0.0625158	+	0.998044i	$0.480088\pi$
$314$	−4486.26	−0.806287
$315$	1595.57	0.285398
$316$	−304.023	−0.0541223
$317$	−7661.86	−1.35752	−0.678758	−	0.734362i	$-0.737482\pi$
$317$	−7661.86	−1.35752	−0.678758	+	0.734362i	$0.737482\pi$
$318$	3654.03	0.644364
$319$	1691.68	0.296915
$320$	−409.245	−0.0714921
$321$	−782.458	−0.136051
$322$	9835.12	1.70214
$323$	182.236	0.0313929
$324$	324.000	0.0555556
$325$	1257.00	0.214540
$326$	−4478.28	−0.760826
$327$	428.629	0.0724869
$328$	−3283.71	−0.552782
$329$	−16273.4	−2.72700
$330$	843.002	0.140623
$331$	5639.78	0.936527	0.468263	−	0.883589i	$-0.344880\pi$
$331$	5639.78	0.936527	0.468263	+	0.883589i	$0.344880\pi$
$332$	2146.39	0.354814
$333$	−1789.52	−0.294490
$334$	−2212.68	−0.362492
$335$	4664.07	0.760673
$336$	−1330.80	−0.216075
$337$	−12069.8	−1.95100	−0.975498	−	0.220009i	$-0.929391\pi$
$337$	−12069.8	−1.95100	−0.975498	+	0.220009i	$0.929391\pi$
$338$	−3947.32	−0.635225
$339$	−3971.85	−0.636345
$340$	181.389	0.0289330
$341$	−5161.33	−0.819654
$342$	−462.550	−0.0731340
$343$	−2292.15	−0.360829
$344$	−676.313	−0.106001
$345$	3402.54	0.530975
$346$	6727.08	1.04523
$347$	−5432.23	−0.840397	−0.420198	−	0.907432i	$-0.638039\pi$
$347$	−5432.23	−0.840397	−0.420198	+	0.907432i	$0.638039\pi$
$348$	−923.899	−0.142317
$349$	−7286.27	−1.11755	−0.558775	−	0.829319i	$-0.688729\pi$
$349$	−7286.27	−1.11755	−0.558775	+	0.829319i	$0.688729\pi$
$350$	4663.95	0.712282
$351$	−403.501	−0.0613598
$352$	−703.112	−0.106466
$353$	−7614.91	−1.14816	−0.574080	−	0.818799i	$-0.694640\pi$
$353$	−7614.91	−1.14816	−0.574080	+	0.818799i	$0.694640\pi$
$354$	354.000	0.0531494
$355$	258.795	0.0386913
$356$	−3890.88	−0.579259
$357$	589.849	0.0874458
$358$	2762.46	0.407823
$359$	−6778.72	−0.996566	−0.498283	−	0.867015i	$-0.666036\pi$
$359$	−6778.72	−0.996566	−0.498283	+	0.867015i	$0.666036\pi$
$360$	−460.400	−0.0674034
$361$	−6198.65	−0.903725
$362$	−6925.62	−1.00553
$363$	−2544.66	−0.367934
$364$	1657.34	0.238649
$365$	−1238.19	−0.177562
$366$	−401.516	−0.0573431
$367$	31.1181	0.00442603	0.00221301	−	0.999998i	$-0.499296\pi$
$367$	31.1181	0.00442603	0.00221301	+	0.999998i	$0.499296\pi$
$368$	−2837.91	−0.402000
$369$	−3694.17	−0.521168
$370$	2542.89	0.357294
$371$	−16884.7	−2.36282
$372$	2818.83	0.392875
$373$	−748.691	−0.103930	−0.0519648	−	0.998649i	$-0.516548\pi$
$373$	−748.691	−0.103930	−0.0519648	+	0.998649i	$0.516548\pi$
$374$	311.640	0.0430869
$375$	4011.45	0.552401
$376$	4695.66	0.644044
$377$	1150.60	0.157185
$378$	−1497.15	−0.203717
$379$	5339.65	0.723692	0.361846	−	0.932238i	$-0.382147\pi$
$379$	5339.65	0.723692	0.361846	+	0.932238i	$0.382147\pi$
$380$	657.278	0.0887307
$381$	5476.91	0.736459
$382$	3250.08	0.435311
$383$	−2648.39	−0.353333	−0.176666	−	0.984271i	$-0.556531\pi$
$383$	−2648.39	−0.353333	−0.176666	+	0.984271i	$0.556531\pi$
$384$	384.000	0.0510310
$385$	−3895.37	−0.515653
$386$	6177.57	0.814585
$387$	−760.852	−0.0999387
$388$	−2398.68	−0.313853
$389$	14335.0	1.86842	0.934209	−	0.356725i	$-0.116107\pi$
$389$	14335.0	1.86842	0.934209	+	0.356725i	$0.116107\pi$
$390$	573.371	0.0744455
$391$	1257.84	0.162690
$392$	3405.40	0.438772
$393$	1625.38	0.208625
$394$	6982.94	0.892882
$395$	486.016	0.0619091
$396$	−791.001	−0.100377
$397$	−12970.0	−1.63966	−0.819828	−	0.572610i	$-0.805931\pi$
$397$	−12970.0	−1.63966	−0.819828	+	0.572610i	$0.805931\pi$
$398$	−10529.1	−1.32606
$399$	2137.37	0.268176
$400$	−1345.78	−0.168222
$401$	788.303	0.0981695	0.0490847	−	0.998795i	$-0.484370\pi$
$401$	788.303	0.0981695	0.0490847	+	0.998795i	$0.484370\pi$
$402$	−4376.36	−0.542968
$403$	−3510.50	−0.433921
$404$	3364.06	0.414278
$405$	−517.950	−0.0635485
$406$	4269.18	0.521862
$407$	4368.87	0.532081
$408$	−170.200	−0.0206524
$409$	−3458.72	−0.418149	−0.209074	−	0.977900i	$-0.567045\pi$
$409$	−3458.72	−0.418149	−0.209074	+	0.977900i	$0.567045\pi$
$410$	5249.38	0.632313
$411$	5427.95	0.651438
$412$	1666.82	0.199316
$413$	−1635.77	−0.194894
$414$	−3192.65	−0.379010
$415$	−3431.24	−0.405862
$416$	−478.224	−0.0563626
$417$	−5624.60	−0.660522
$418$	1129.25	0.132138
$419$	8729.81	1.01785	0.508925	−	0.860811i	$-0.330043\pi$
$419$	8729.81	1.01785	0.508925	+	0.860811i	$0.330043\pi$
$420$	2127.43	0.247162
$421$	−2882.93	−0.333742	−0.166871	−	0.985979i	$-0.553366\pi$
$421$	−2882.93	−0.333742	−0.166871	+	0.985979i	$0.553366\pi$
$422$	2601.16	0.300053
$423$	5282.62	0.607210
$424$	4872.04	0.558036
$425$	596.488	0.0680798
$426$	−242.831	−0.0276179
$427$	1855.34	0.210272
$428$	−1043.28	−0.117824
$429$	985.092	0.110864
$430$	1081.16	0.121252
$431$	17573.3	1.96398	0.981989	−	0.188937i	$-0.0605041\pi$
$431$	17573.3	1.96398	0.981989	+	0.188937i	$0.0605041\pi$
$432$	432.000	0.0481125
$433$	−1185.72	−0.131598	−0.0657991	−	0.997833i	$-0.520960\pi$
$433$	−1185.72	−0.131598	−0.0657991	+	0.997833i	$0.520960\pi$
$434$	−13025.3	−1.44064
$435$	1476.96	0.162792
$436$	571.505	0.0627755
$437$	4557.90	0.498933
$438$	1161.82	0.126744
$439$	11927.0	1.29669	0.648344	−	0.761347i	$-0.275462\pi$
$439$	11927.0	1.29669	0.648344	+	0.761347i	$0.275462\pi$
$440$	1124.00	0.121784
$441$	3831.07	0.413678
$442$	211.963	0.0228101
$443$	−766.275	−0.0821824	−0.0410912	−	0.999155i	$-0.513083\pi$
$443$	−766.275	−0.0821824	−0.0410912	+	0.999155i	$0.513083\pi$
$444$	−2386.03	−0.255036
$445$	6220.00	0.662599
$446$	7683.57	0.815757
$447$	1844.65	0.195188
$448$	−1774.40	−0.187126
$449$	9070.01	0.953319	0.476660	−	0.879088i	$-0.341847\pi$
$449$	9070.01	0.953319	0.476660	+	0.879088i	$0.341847\pi$
$450$	−1514.00	−0.158601
$451$	9018.80	0.941638
$452$	−5295.80	−0.551091
$453$	3556.06	0.368826
$454$	3295.48	0.340670
$455$	−2649.45	−0.272985
$456$	−616.733	−0.0633359
$457$	−550.016	−0.0562991	−0.0281495	−	0.999604i	$-0.508961\pi$
$457$	−550.016	−0.0562991	−0.0281495	+	0.999604i	$0.508961\pi$
$458$	−10446.9	−1.06583
$459$	−191.475	−0.0194712
$460$	4536.71	0.459838
$461$	−17481.2	−1.76612	−0.883060	−	0.469260i	$-0.844521\pi$
$461$	−17481.2	−1.76612	−0.883060	+	0.469260i	$0.844521\pi$
$462$	3655.08	0.368073
$463$	−3594.88	−0.360838	−0.180419	−	0.983590i	$-0.557745\pi$
$463$	−3594.88	−0.360838	−0.180419	+	0.983590i	$0.557745\pi$
$464$	−1231.87	−0.123250
$465$	−4506.22	−0.449399
$466$	3837.34	0.381462
$467$	9298.39	0.921366	0.460683	−	0.887565i	$-0.347604\pi$
$467$	9298.39	0.921366	0.460683	+	0.887565i	$0.347604\pi$
$468$	−538.002	−0.0531392
$469$	20222.4	1.99101
$470$	−7506.55	−0.736705
$471$	−6729.38	−0.658330
$472$	472.000	0.0460287
$473$	1857.51	0.180568
$474$	−456.035	−0.0441907
$475$	2161.42	0.208785
$476$	786.466	0.0757303
$477$	5481.05	0.526121
$478$	−452.517	−0.0433005
$479$	−3901.16	−0.372127	−0.186063	−	0.982538i	$-0.559573\pi$
$479$	−3901.16	−0.372127	−0.186063	+	0.982538i	$0.559573\pi$
$480$	−613.867	−0.0583731
$481$	2971.50	0.281682
$482$	−12507.0	−1.18190
$483$	14752.7	1.38979
$484$	−3392.88	−0.318640
$485$	3834.57	0.359008
$486$	486.000	0.0453609
$487$	9068.83	0.843836	0.421918	−	0.906634i	$-0.361357\pi$
$487$	9068.83	0.843836	0.421918	+	0.906634i	$0.361357\pi$
$488$	−535.354	−0.0496605
$489$	−6717.42	−0.621211
$490$	−5443.91	−0.501899
$491$	−16155.9	−1.48494	−0.742472	−	0.669877i	$-0.766347\pi$
$491$	−16155.9	−1.48494	−0.742472	+	0.669877i	$0.766347\pi$
$492$	−4925.56	−0.451344
$493$	545.999	0.0498794
$494$	768.064	0.0699531
$495$	1264.50	0.114819
$496$	3758.44	0.340240
$497$	1122.08	0.101272
$498$	3219.58	0.289704
$499$	12184.5	1.09309	0.546544	−	0.837430i	$-0.315943\pi$
$499$	12184.5	1.09309	0.546544	+	0.837430i	$0.315943\pi$
$500$	5348.60	0.478393
$501$	−3319.02	−0.295974
$502$	−3604.88	−0.320505
$503$	13361.3	1.18439	0.592197	−	0.805794i	$-0.298261\pi$
$503$	13361.3	1.18439	0.592197	+	0.805794i	$0.298261\pi$
$504$	−1996.20	−0.176424
$505$	−5377.83	−0.473882
$506$	7794.40	0.684789
$507$	−5920.99	−0.518659
$508$	7302.55	0.637792
$509$	6768.26	0.589386	0.294693	−	0.955592i	$-0.404783\pi$
$509$	6768.26	0.589386	0.294693	+	0.955592i	$0.404783\pi$
$510$	272.084	0.0236237
$511$	−5368.55	−0.464757
$512$	512.000	0.0441942
$513$	−693.825	−0.0597137
$514$	−6219.47	−0.533714
$515$	−2664.60	−0.227993
$516$	−1014.47	−0.0865494
$517$	−12896.8	−1.09710
$518$	11025.4	0.935194
$519$	10090.6	0.853428
$520$	764.494	0.0644717
$521$	−12104.0	−1.01782	−0.508911	−	0.860819i	$-0.669952\pi$
$521$	−12104.0	−1.01782	−0.508911	+	0.860819i	$0.669952\pi$
$522$	−1385.85	−0.116201
$523$	21639.9	1.80926	0.904632	−	0.426193i	$-0.140146\pi$
$523$	21639.9	1.80926	0.904632	+	0.426193i	$0.140146\pi$
$524$	2167.17	0.180674
$525$	6995.93	0.581576
$526$	−7361.43	−0.610216
$527$	−1665.85	−0.137696
$528$	−1054.67	−0.0869290
$529$	19292.8	1.58567
$530$	−7788.50	−0.638323
$531$	531.000	0.0433963
$532$	2849.82	0.232247
$533$	6134.17	0.498499
$534$	−5836.31	−0.472963
$535$	1667.79	0.134776
$536$	−5835.15	−0.470224
$537$	4143.69	0.332986
$538$	−15929.2	−1.27650
$539$	−9353.03	−0.747427
$540$	−690.600	−0.0550347
$541$	16839.0	1.33820	0.669098	−	0.743174i	$-0.266681\pi$
$541$	16839.0	1.33820	0.669098	+	0.743174i	$0.266681\pi$
$542$	−828.679	−0.0656731
$543$	−10388.4	−0.821013
$544$	−226.934	−0.0178855
$545$	−913.615	−0.0718073
$546$	2486.01	0.194856
$547$	−20645.2	−1.61376	−0.806880	−	0.590716i	$-0.798845\pi$
$547$	−20645.2	−1.61376	−0.806880	+	0.590716i	$0.798845\pi$
$548$	7237.27	0.564162
$549$	−602.273	−0.0468204
$550$	3696.22	0.286559
$551$	1978.47	0.152969
$552$	−4256.86	−0.328232
$553$	2107.26	0.162043
$554$	−706.131	−0.0541528
$555$	3814.34	0.291729
$556$	−7499.46	−0.572029
$557$	−597.786	−0.0454740	−0.0227370	−	0.999741i	$-0.507238\pi$
$557$	−597.786	−0.0454740	−0.0227370	+	0.999741i	$0.507238\pi$
$558$	4228.24	0.320781
$559$	1263.39	0.0955919
$560$	2836.58	0.214049
$561$	467.460	0.0351803
$562$	−808.279	−0.0606676
$563$	8.84411	0.000662051 0	0.000331025	−	1.00000i	$-0.499895\pi$
$563$	8.84411	0.000662051 0	0.000331025		1.00000i	$0.499895\pi$
$564$	7043.50	0.525859
$565$	8465.92	0.630379
$566$	−6396.24	−0.475007
$567$	−2245.72	−0.166334
$568$	−323.775	−0.0239178
$569$	589.011	0.0433966	0.0216983	−	0.999765i	$-0.493093\pi$
$569$	589.011	0.0433966	0.0216983	+	0.999765i	$0.493093\pi$
$570$	985.918	0.0724483
$571$	1352.32	0.0991121	0.0495560	−	0.998771i	$-0.484219\pi$
$571$	1352.32	0.0991121	0.0495560	+	0.998771i	$0.484219\pi$
$572$	1313.46	0.0960111
$573$	4875.13	0.355430
$574$	22760.2	1.65504
$575$	14918.7	1.08201
$576$	576.000	0.0416667
$577$	16915.2	1.22043	0.610215	−	0.792236i	$-0.291083\pi$
$577$	16915.2	1.22043	0.610215	+	0.792236i	$0.291083\pi$
$578$	−9725.42	−0.699868
$579$	9266.35	0.665106
$580$	1969.28	0.140982
$581$	−14877.1	−1.06232
$582$	−3598.03	−0.256260
$583$	−13381.2	−0.950588
$584$	1549.09	0.109763
$585$	860.056	0.0607845
$586$	9529.10	0.671746
$587$	−13102.4	−0.921285	−0.460643	−	0.887586i	$-0.652381\pi$
$587$	−13102.4	−0.921285	−0.460643	+	0.887586i	$0.652381\pi$
$588$	5108.09	0.358256
$589$	−6036.34	−0.422280
$590$	−754.545	−0.0526511
$591$	10474.4	0.729035
$592$	−3181.38	−0.220868
$593$	−15886.7	−1.10015	−0.550076	−	0.835115i	$-0.685401\pi$
$593$	−15886.7	−1.10015	−0.550076	+	0.835115i	$0.685401\pi$
$594$	−1186.50	−0.0819575
$595$	−1257.25	−0.0866259
$596$	2459.54	0.169038
$597$	−15793.6	−1.08273
$598$	5301.38	0.362525
$599$	10291.1	0.701978	0.350989	−	0.936380i	$-0.385845\pi$
$599$	10291.1	0.701978	0.350989	+	0.936380i	$0.385845\pi$
$600$	−2018.66	−0.137353
$601$	22498.9	1.52704	0.763519	−	0.645786i	$-0.223470\pi$
$601$	22498.9	1.52704	0.763519	+	0.645786i	$0.223470\pi$
$602$	4687.69	0.317369
$603$	−6564.54	−0.443331
$604$	4741.41	0.319413
$605$	5423.90	0.364484
$606$	5046.10	0.338257
$607$	3318.97	0.221932	0.110966	−	0.993824i	$-0.464606\pi$
$607$	3318.97	0.221932	0.110966	+	0.993824i	$0.464606\pi$
$608$	−822.311	−0.0548505
$609$	6403.77	0.426098
$610$	855.824	0.0568054
$611$	−8771.79	−0.580799
$612$	−255.300	−0.0168626
$613$	15216.4	1.00259	0.501293	−	0.865278i	$-0.332858\pi$
$613$	15216.4	1.00259	0.501293	+	0.865278i	$0.332858\pi$
$614$	17.7622	0.00116747
$615$	7874.06	0.516281
$616$	4873.44	0.318761
$617$	13512.3	0.881662	0.440831	−	0.897590i	$-0.354684\pi$
$617$	13512.3	0.881662	0.440831	+	0.897590i	$0.354684\pi$
$618$	2500.23	0.162741
$619$	26392.5	1.71374	0.856868	−	0.515536i	$-0.172407\pi$
$619$	26392.5	1.71374	0.856868	+	0.515536i	$0.172407\pi$
$620$	−6008.29	−0.389191
$621$	−4788.97	−0.309460
$622$	−10209.2	−0.658119
$623$	26968.6	1.73431
$624$	−717.335	−0.0460199
$625$	1963.54	0.125667
$626$	1384.73	0.0884107
$627$	1693.88	0.107890
$628$	−8972.51	−0.570131
$629$	1410.08	0.0893856
$630$	3191.15	0.201807
$631$	−24601.6	−1.55210	−0.776048	−	0.630674i	$-0.782779\pi$
$631$	−24601.6	−1.55210	−0.776048	+	0.630674i	$0.782779\pi$
$632$	−608.047	−0.0382703
$633$	3901.74	0.244992
$634$	−15323.7	−0.959909
$635$	−11673.9	−0.729553
$636$	7308.06	0.455634
$637$	−6361.49	−0.395685
$638$	3383.36	0.209950
$639$	−364.247	−0.0225499
$640$	−818.489	−0.0505526
$641$	−17003.5	−1.04774	−0.523868	−	0.851799i	$-0.675512\pi$
$641$	−17003.5	−1.04774	−0.523868	+	0.851799i	$0.675512\pi$
$642$	−1564.92	−0.0962029
$643$	3125.47	0.191690	0.0958449	−	0.995396i	$-0.469445\pi$
$643$	3125.47	0.191690	0.0958449	+	0.995396i	$0.469445\pi$
$644$	19670.2	1.20360
$645$	1621.74	0.0990016
$646$	364.473	0.0221981
$647$	2303.24	0.139953	0.0699766	−	0.997549i	$-0.477708\pi$
$647$	2303.24	0.139953	0.0699766	+	0.997549i	$0.477708\pi$
$648$	648.000	0.0392837
$649$	−1296.36	−0.0784078
$650$	2513.99	0.151703
$651$	−19538.0	−1.17627
$652$	−8956.57	−0.537985
$653$	−11851.4	−0.710231	−0.355116	−	0.934822i	$-0.615558\pi$
$653$	−11851.4	−0.710231	−0.355116	+	0.934822i	$0.615558\pi$
$654$	857.258	0.0512560
$655$	−3464.47	−0.206669
$656$	−6567.42	−0.390876
$657$	1742.72	0.103486
$658$	−32546.8	−1.92828
$659$	−11729.2	−0.693333	−0.346667	−	0.937988i	$-0.612686\pi$
$659$	−11729.2	−0.693333	−0.346667	+	0.937988i	$0.612686\pi$
$660$	1686.00	0.0994358
$661$	−18775.1	−1.10479	−0.552396	−	0.833582i	$-0.686286\pi$
$661$	−18775.1	−1.10479	−0.552396	+	0.833582i	$0.686286\pi$
$662$	11279.6	0.662225
$663$	317.944	0.0186243
$664$	4292.77	0.250891
$665$	−4555.76	−0.265661
$666$	−3579.05	−0.208236
$667$	13655.9	0.792744
$668$	−4425.36	−0.256321
$669$	11525.4	0.666062
$670$	9328.14	0.537877
$671$	1470.37	0.0845944
$672$	−2661.60	−0.152788
$673$	−10630.8	−0.608895	−0.304448	−	0.952529i	$-0.598472\pi$
$673$	−10630.8	−0.608895	−0.304448	+	0.952529i	$0.598472\pi$
$674$	−24139.7	−1.37956
$675$	−2271.00	−0.129497
$676$	−7894.65	−0.449172
$677$	12267.4	0.696414	0.348207	−	0.937418i	$-0.386791\pi$
$677$	12267.4	0.696414	0.348207	+	0.937418i	$0.386791\pi$
$678$	−7943.69	−0.449964
$679$	16625.9	0.939680
$680$	362.779	0.0204587
$681$	4943.21	0.278156
$682$	−10322.7	−0.579583
$683$	5972.61	0.334606	0.167303	−	0.985906i	$-0.446494\pi$
$683$	5972.61	0.334606	0.167303	+	0.985906i	$0.446494\pi$
$684$	−925.100	−0.0517136
$685$	−11569.6	−0.645330
$686$	−4584.30	−0.255145
$687$	−15670.3	−0.870246
$688$	−1352.63	−0.0749540
$689$	−9101.27	−0.503238
$690$	6805.07	0.375456
$691$	−5639.88	−0.310494	−0.155247	−	0.987876i	$-0.549617\pi$
$691$	−5639.88	−0.310494	−0.155247	+	0.987876i	$0.549617\pi$
$692$	13454.2	0.739090
$693$	5482.62	0.300530
$694$	−10864.5	−0.594250
$695$	11988.7	0.654329
$696$	−1847.80	−0.100633
$697$	2910.87	0.158188
$698$	−14572.5	−0.790228
$699$	5756.01	0.311463
$700$	9327.91	0.503660
$701$	−14652.1	−0.789448	−0.394724	−	0.918800i	$-0.629160\pi$
$701$	−14652.1	−0.789448	−0.394724	+	0.918800i	$0.629160\pi$
$702$	−807.002	−0.0433879
$703$	5109.53	0.274125
$704$	−1406.22	−0.0752827
$705$	−11259.8	−0.601517
$706$	−15229.8	−0.811872
$707$	−23317.2	−1.24036
$708$	708.000	0.0375823
$709$	−16041.6	−0.849726	−0.424863	−	0.905258i	$-0.639678\pi$
$709$	−16041.6	−0.849726	−0.424863	+	0.905258i	$0.639678\pi$
$710$	517.590	0.0273589
$711$	−684.053	−0.0360815
$712$	−7781.75	−0.409598
$713$	−41664.5	−2.18842
$714$	1179.70	0.0618335
$715$	−2099.71	−0.109825
$716$	5524.92	0.288374
$717$	−678.775	−0.0353547
$718$	−13557.4	−0.704678
$719$	−3908.80	−0.202745	−0.101373	−	0.994849i	$-0.532323\pi$
$719$	−3908.80	−0.202745	−0.101373	+	0.994849i	$0.532323\pi$
$720$	−920.801	−0.0476614
$721$	−11553.1	−0.596756
$722$	−12397.3	−0.639030
$723$	−18760.4	−0.965018
$724$	−13851.2	−0.711018
$725$	6475.84	0.331733
$726$	−5089.32	−0.260169
$727$	−32754.6	−1.67098	−0.835490	−	0.549506i	$-0.814816\pi$
$727$	−32754.6	−1.67098	−0.835490	+	0.549506i	$0.814816\pi$
$728$	3314.68	0.168751
$729$	729.000	0.0370370
$730$	−2476.39	−0.125555
$731$	599.524	0.0303340
$732$	−803.031	−0.0405477
$733$	−34637.4	−1.74538	−0.872688	−	0.488278i	$-0.837625\pi$
$733$	−34637.4	−1.74538	−0.872688	+	0.488278i	$0.837625\pi$
$734$	62.2362	0.00312968
$735$	−8165.86	−0.409799
$736$	−5675.82	−0.284257
$737$	16026.4	0.801005
$738$	−7388.34	−0.368521
$739$	28824.8	1.43483	0.717413	−	0.696648i	$-0.245326\pi$
$739$	28824.8	1.43483	0.717413	+	0.696648i	$0.245326\pi$
$740$	5085.79	0.252645
$741$	1152.10	0.0571164
$742$	−33769.3	−1.67077
$743$	−4164.72	−0.205638	−0.102819	−	0.994700i	$-0.532786\pi$
$743$	−4164.72	−0.205638	−0.102819	+	0.994700i	$0.532786\pi$
$744$	5637.66	0.277805
$745$	−3931.85	−0.193358
$746$	−1497.38	−0.0734893
$747$	4829.37	0.236543
$748$	623.280	0.0304671
$749$	7231.21	0.352767
$750$	8022.90	0.390606
$751$	−980.964	−0.0476643	−0.0238322	−	0.999716i	$-0.507587\pi$
$751$	−980.964	−0.0476643	−0.0238322	+	0.999716i	$0.507587\pi$
$752$	9391.33	0.455408
$753$	−5407.32	−0.261692
$754$	2301.20	0.111147
$755$	−7579.68	−0.365368
$756$	−2994.30	−0.144050
$757$	1349.58	0.0647972	0.0323986	−	0.999475i	$-0.489685\pi$
$757$	1349.58	0.0647972	0.0323986	+	0.999475i	$0.489685\pi$
$758$	10679.3	0.511727
$759$	11691.6	0.559128
$760$	1314.56	0.0627421
$761$	29468.2	1.40371	0.701854	−	0.712321i	$-0.252356\pi$
$761$	29468.2	1.40371	0.701854	+	0.712321i	$0.252356\pi$
$762$	10953.8	0.520755
$763$	−3961.24	−0.187951
$764$	6500.17	0.307811
$765$	408.126	0.0192887
$766$	−5296.78	−0.249844
$767$	−881.725	−0.0415088
$768$	768.000	0.0360844
$769$	27894.2	1.30805	0.654025	−	0.756473i	$-0.273079\pi$
$769$	27894.2	1.30805	0.654025	+	0.756473i	$0.273079\pi$
$770$	−7790.74	−0.364622
$771$	−9329.21	−0.435776
$772$	12355.1	0.575999
$773$	−11084.0	−0.515734	−0.257867	−	0.966180i	$-0.583020\pi$
$773$	−11084.0	−0.515734	−0.257867	+	0.966180i	$0.583020\pi$
$774$	−1521.70	−0.0706673
$775$	−19757.9	−0.915773
$776$	−4797.37	−0.221927
$777$	16538.2	0.763583
$778$	28670.1	1.32117
$779$	10547.8	0.485126
$780$	1146.74	0.0526409
$781$	889.258	0.0407428
$782$	2515.69	0.115039
$783$	−2078.77	−0.0948778
$784$	6810.79	0.310258
$785$	14343.6	0.652158
$786$	3250.76	0.147520
$787$	−11226.7	−0.508499	−0.254250	−	0.967139i	$-0.581828\pi$
$787$	−11226.7	−0.508499	−0.254250	+	0.967139i	$0.581828\pi$
$788$	13965.9	0.631363
$789$	−11042.1	−0.498239
$790$	972.031	0.0437763
$791$	36706.5	1.64998
$792$	−1582.00	−0.0709772
$793$	1000.07	0.0447840
$794$	−25939.9	−1.15941
$795$	−11682.8	−0.521188
$796$	−21058.1	−0.937669
$797$	6642.23	0.295207	0.147603	−	0.989047i	$-0.452844\pi$
$797$	6642.23	0.295207	0.147603	+	0.989047i	$0.452844\pi$
$798$	4274.73	0.189629
$799$	−4162.51	−0.184304
$800$	−2691.55	−0.118951
$801$	−8754.47	−0.386172
$802$	1576.61	0.0694163
$803$	−4254.61	−0.186976
$804$	−8752.72	−0.383936
$805$	−31445.1	−1.37676
$806$	−7020.99	−0.306829
$807$	−23893.8	−1.04226
$808$	6728.13	0.292939
$809$	13667.2	0.593961	0.296981	−	0.954883i	$-0.404020\pi$
$809$	13667.2	0.593961	0.296981	+	0.954883i	$0.404020\pi$
$810$	−1035.90	−0.0449356
$811$	−2259.48	−0.0978312	−0.0489156	−	0.998803i	$-0.515577\pi$
$811$	−2259.48	−0.0978312	−0.0489156	+	0.998803i	$0.515577\pi$
$812$	8538.36	0.369012
$813$	−1243.02	−0.0536218
$814$	8737.75	0.376238
$815$	14318.1	0.615387
$816$	−340.400	−0.0146034
$817$	2172.42	0.0930274
$818$	−6917.45	−0.295676
$819$	3729.02	0.159100
$820$	10498.8	0.447113
$821$	−10141.9	−0.431126	−0.215563	−	0.976490i	$-0.569159\pi$
$821$	−10141.9	−0.431126	−0.215563	+	0.976490i	$0.569159\pi$
$822$	10855.9	0.460636
$823$	5109.68	0.216418	0.108209	−	0.994128i	$-0.465488\pi$
$823$	5109.68	0.216418	0.108209	+	0.994128i	$0.465488\pi$
$824$	3333.64	0.140938
$825$	5544.32	0.233974
$826$	−3271.55	−0.137811
$827$	33107.9	1.39211	0.696054	−	0.717989i	$-0.254937\pi$
$827$	33107.9	1.39211	0.696054	+	0.717989i	$0.254937\pi$
$828$	−6385.29	−0.268000
$829$	−10062.4	−0.421569	−0.210784	−	0.977533i	$-0.567602\pi$
$829$	−10062.4	−0.421569	−0.210784	+	0.977533i	$0.567602\pi$
$830$	−6862.48	−0.286988
$831$	−1059.20	−0.0442156
$832$	−956.447	−0.0398544
$833$	−3018.74	−0.125562
$834$	−11249.2	−0.467060
$835$	7074.43	0.293198
$836$	2258.50	0.0934353
$837$	6342.37	0.261917
$838$	17459.6	0.719729
$839$	−30356.2	−1.24912	−0.624561	−	0.780976i	$-0.714722\pi$
$839$	−30356.2	−1.24912	−0.624561	+	0.780976i	$0.714722\pi$
$840$	4254.86	0.174770
$841$	−18461.3	−0.756952
$842$	−5765.86	−0.235991
$843$	−1212.42	−0.0495349
$844$	5202.32	0.212170
$845$	12620.5	0.513796
$846$	10565.2	0.429362
$847$	23516.9	0.954014
$848$	9744.08	0.394591
$849$	−9594.36	−0.387841
$850$	1192.98	0.0481397
$851$	35267.4	1.42062
$852$	−485.662	−0.0195288
$853$	19340.7	0.776332	0.388166	−	0.921589i	$-0.373109\pi$
$853$	19340.7	0.776332	0.388166	+	0.921589i	$0.373109\pi$
$854$	3710.67	0.148684
$855$	1478.88	0.0591538
$856$	−2086.55	−0.0833142
$857$	33183.6	1.32267	0.661336	−	0.750090i	$-0.269990\pi$
$857$	33183.6	1.32267	0.661336	+	0.750090i	$0.269990\pi$
$858$	1970.18	0.0783927
$859$	2100.80	0.0834440	0.0417220	−	0.999129i	$-0.486716\pi$
$859$	2100.80	0.0834440	0.0417220	+	0.999129i	$0.486716\pi$
$860$	2162.32	0.0857379
$861$	34140.3	1.35133
$862$	35146.5	1.38874
$863$	22225.4	0.876665	0.438332	−	0.898813i	$-0.355569\pi$
$863$	22225.4	0.876665	0.438332	+	0.898813i	$0.355569\pi$
$864$	864.000	0.0340207
$865$	−21508.0	−0.845426
$866$	−2371.44	−0.0930539
$867$	−14588.1	−0.571440
$868$	−26050.7	−1.01868
$869$	1670.02	0.0651916
$870$	2953.91	0.115112
$871$	10900.4	0.424049
$872$	1143.01	0.0443890
$873$	−5397.04	−0.209235
$874$	9115.79	0.352799
$875$	−37072.5	−1.43232
$876$	2323.63	0.0896212
$877$	−7171.13	−0.276114	−0.138057	−	0.990424i	$-0.544086\pi$
$877$	−7171.13	−0.276114	−0.138057	+	0.990424i	$0.544086\pi$
$878$	23854.1	0.916897
$879$	14293.6	0.548479
$880$	2248.01	0.0861139
$881$	6584.17	0.251789	0.125895	−	0.992044i	$-0.459820\pi$
$881$	6584.17	0.251789	0.125895	+	0.992044i	$0.459820\pi$
$882$	7662.14	0.292514
$883$	6314.37	0.240652	0.120326	−	0.992734i	$-0.461606\pi$
$883$	6314.37	0.240652	0.120326	+	0.992734i	$0.461606\pi$
$884$	423.926	0.0161291
$885$	−1131.82	−0.0429894
$886$	−1532.55	−0.0581117
$887$	−22925.5	−0.867826	−0.433913	−	0.900955i	$-0.642868\pi$
$887$	−22925.5	−0.867826	−0.433913	+	0.900955i	$0.642868\pi$
$888$	−4772.06	−0.180338
$889$	−50615.8	−1.90956
$890$	12440.0	0.468528
$891$	−1779.75	−0.0669180
$892$	15367.1	0.576827
$893$	−15083.2	−0.565218
$894$	3689.31	0.138019
$895$	−8832.21	−0.329864
$896$	−3548.80	−0.132318
$897$	7952.08	0.296000
$898$	18140.0	0.674099
$899$	−18085.5	−0.670952
$900$	−3028.00	−0.112148
$901$	−4318.86	−0.159692
$902$	18037.6	0.665839
$903$	7031.53	0.259130
$904$	−10591.6	−0.389680
$905$	22142.7	0.813315
$906$	7112.12	0.260799
$907$	−29181.9	−1.06832	−0.534162	−	0.845382i	$-0.679373\pi$
$907$	−29181.9	−1.06832	−0.534162	+	0.845382i	$0.679373\pi$
$908$	6590.95	0.240890
$909$	7569.14	0.276186
$910$	−5298.90	−0.193029
$911$	17287.5	0.628715	0.314358	−	0.949305i	$-0.398211\pi$
$911$	17287.5	0.628715	0.314358	+	0.949305i	$0.398211\pi$
$912$	−1233.47	−0.0447853
$913$	−11790.2	−0.427382
$914$	−1100.03	−0.0398095
$915$	1283.74	0.0463814
$916$	−20893.7	−0.753655
$917$	−15021.2	−0.540942
$918$	−382.950	−0.0137682
$919$	−28423.3	−1.02024	−0.510118	−	0.860104i	$-0.670398\pi$
$919$	−28423.3	−1.02024	−0.510118	+	0.860104i	$0.670398\pi$
$920$	9073.43	0.325154
$921$	26.6433	0.000953233 0
$922$	−34962.4	−1.24884
$923$	604.831	0.0215691
$924$	7310.16	0.260267
$925$	16724.3	0.594477
$926$	−7189.76	−0.255151
$927$	3750.34	0.132878
$928$	−2463.73	−0.0871508
$929$	43754.4	1.54525	0.772624	−	0.634864i	$-0.218944\pi$
$929$	43754.4	1.54525	0.772624	+	0.634864i	$0.218944\pi$
$930$	−9012.43	−0.317773
$931$	−10938.7	−0.385070
$932$	7674.68	0.269735
$933$	−15313.7	−0.537352
$934$	18596.8	0.651504
$935$	−996.383	−0.0348505
$936$	−1076.00	−0.0375751
$937$	51255.0	1.78701	0.893505	−	0.449053i	$-0.148239\pi$
$937$	51255.0	1.78701	0.893505	+	0.449053i	$0.148239\pi$
$938$	40444.9	1.40786
$939$	2077.10	0.0721870
$940$	−15013.1	−0.520929
$941$	−38032.4	−1.31756	−0.658778	−	0.752337i	$-0.728927\pi$
$941$	−38032.4	−1.31756	−0.658778	+	0.752337i	$0.728927\pi$
$942$	−13458.8	−0.465510
$943$	72803.6	2.51412
$944$	944.000	0.0325472
$945$	4786.72	0.164775
$946$	3715.03	0.127681
$947$	−5149.20	−0.176691	−0.0883456	−	0.996090i	$-0.528158\pi$
$947$	−5149.20	−0.176691	−0.0883456	+	0.996090i	$0.528158\pi$
$948$	−912.070	−0.0312475
$949$	−2893.79	−0.0989846
$950$	4322.84	0.147633
$951$	−22985.6	−0.783763
$952$	1572.93	0.0535494
$953$	−6144.70	−0.208863	−0.104431	−	0.994532i	$-0.533302\pi$
$953$	−6144.70	−0.208863	−0.104431	+	0.994532i	$0.533302\pi$
$954$	10962.1	0.372024
$955$	−10391.2	−0.352097
$956$	−905.034	−0.0306181
$957$	5075.03	0.171424
$958$	−7802.33	−0.263133
$959$	−50163.3	−1.68911
$960$	−1227.73	−0.0412760
$961$	25388.2	0.852209
$962$	5943.00	0.199179
$963$	−2347.37	−0.0785494
$964$	−25013.9	−0.835730
$965$	−19751.1	−0.658870
$966$	29505.4	0.982732
$967$	39476.8	1.31281	0.656406	−	0.754408i	$-0.272076\pi$
$967$	39476.8	1.31281	0.656406	+	0.754408i	$0.272076\pi$
$968$	−6785.76	−0.225313
$969$	546.709	0.0181247
$970$	7669.13	0.253857
$971$	−17595.3	−0.581523	−0.290762	−	0.956796i	$-0.593909\pi$
$971$	−17595.3	−0.581523	−0.290762	+	0.956796i	$0.593909\pi$
$972$	972.000	0.0320750
$973$	51980.6	1.71266
$974$	18137.7	0.596682
$975$	3770.99	0.123865
$976$	−1070.71	−0.0351153
$977$	−2258.07	−0.0739428	−0.0369714	−	0.999316i	$-0.511771\pi$
$977$	−2258.07	−0.0739428	−0.0369714	+	0.999316i	$0.511771\pi$
$978$	−13434.8	−0.439263
$979$	21372.8	0.697731
$980$	−10887.8	−0.354896
$981$	1285.89	0.0418503
$982$	−32311.9	−1.05001
$983$	16949.9	0.549966	0.274983	−	0.961449i	$-0.411328\pi$
$983$	16949.9	0.549966	0.274983	+	0.961449i	$0.411328\pi$
$984$	−9851.12	−0.319149
$985$	−22326.0	−0.722200
$986$	1092.00	0.0352701
$987$	−48820.2	−1.57443
$988$	1536.13	0.0494643
$989$	14994.6	0.482105
$990$	2529.01	0.0811890
$991$	51108.7	1.63827	0.819133	−	0.573603i	$-0.194455\pi$
$991$	51108.7	1.63827	0.819133	+	0.573603i	$0.194455\pi$
$992$	7516.88	0.240586
$993$	16919.3	0.540704
$994$	2244.16	0.0716102
$995$	33663.7	1.07258
$996$	6439.16	0.204852
$997$	−31520.5	−1.00127	−0.500634	−	0.865659i	$-0.666900\pi$
$997$	−31520.5	−1.00127	−0.500634	+	0.865659i	$0.666900\pi$
$998$	24368.9	0.772930
$999$	−5368.57	−0.170024

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.4.a.c.1.2	✓	2
3.2	odd	2		1062.4.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.4.a.c.1.2	✓	2	1.1	even	1	trivial
1062.4.a.f.1.1		2	3.2	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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.39445 q^{5} +6.00000 q^{6} -27.7250 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.39445 q^{5} +6.00000 q^{6} -27.7250 q^{7} +8.00000 q^{8} +9.00000 q^{9} -12.7889 q^{10} -21.9722 q^{11} +12.0000 q^{12} -14.9445 q^{13} -55.4500 q^{14} -19.1833 q^{15} +16.0000 q^{16} -7.09167 q^{17} +18.0000 q^{18} -25.6972 q^{19} -25.5778 q^{20} -83.1749 q^{21} -43.9445 q^{22} -177.369 q^{23} +24.0000 q^{24} -84.1110 q^{25} -29.8890 q^{26} +27.0000 q^{27} -110.900 q^{28} -76.9916 q^{29} -38.3667 q^{30} +234.902 q^{31} +32.0000 q^{32} -65.9167 q^{33} -14.1833 q^{34} +177.286 q^{35} +36.0000 q^{36} -198.836 q^{37} -51.3944 q^{38} -44.8335 q^{39} -51.1556 q^{40} -410.463 q^{41} -166.350 q^{42} -84.5391 q^{43} -87.8890 q^{44} -57.5500 q^{45} -354.738 q^{46} +586.958 q^{47} +48.0000 q^{48} +425.675 q^{49} -168.222 q^{50} -21.2750 q^{51} -59.7779 q^{52} +609.005 q^{53} +54.0000 q^{54} +140.500 q^{55} -221.800 q^{56} -77.0917 q^{57} -153.983 q^{58} +59.0000 q^{59} -76.7334 q^{60} -66.9193 q^{61} +469.805 q^{62} -249.525 q^{63} +64.0000 q^{64} +95.5618 q^{65} -131.833 q^{66} -729.394 q^{67} -28.3667 q^{68} -532.108 q^{69} +354.572 q^{70} -40.4719 q^{71} +72.0000 q^{72} +193.636 q^{73} -397.672 q^{74} -252.333 q^{75} -102.789 q^{76} +609.180 q^{77} -89.6669 q^{78} -76.0059 q^{79} -102.311 q^{80} +81.0000 q^{81} -820.927 q^{82} +536.596 q^{83} -332.700 q^{84} +45.3473 q^{85} -169.078 q^{86} -230.975 q^{87} -175.778 q^{88} -972.719 q^{89} -115.100 q^{90} +414.336 q^{91} -709.477 q^{92} +704.707 q^{93} +1173.92 q^{94} +164.320 q^{95} +96.0000 q^{96} -599.671 q^{97} +851.349 q^{98} -197.750 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 20 q^{5} + 12 q^{6} - 23 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 20 * q^5 + 12 * q^6 - 23 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 20 q^{5} + 12 q^{6} - 23 q^{7} + 16 q^{8} + 18 q^{9} - 40 q^{10} - 80 q^{11} + 24 q^{12} - 102 q^{13} - 46 q^{14} - 60 q^{15} + 32 q^{16} - 25 q^{17} + 36 q^{18} - 55 q^{19} - 80 q^{20} - 69 q^{21} - 160 q^{22} - 5 q^{23} + 48 q^{24} - 24 q^{25} - 204 q^{26} + 54 q^{27} - 92 q^{28} - 35 q^{29} - 120 q^{30} - 53 q^{31} + 64 q^{32} - 240 q^{33} - 50 q^{34} + 113 q^{35} + 72 q^{36} - 221 q^{37} - 110 q^{38} - 306 q^{39} - 160 q^{40} - 89 q^{41} - 138 q^{42} - 508 q^{43} - 320 q^{44} - 180 q^{45} - 10 q^{46} + 579 q^{47} + 96 q^{48} + 105 q^{49} - 48 q^{50} - 75 q^{51} - 408 q^{52} + 432 q^{53} + 108 q^{54} + 930 q^{55} - 184 q^{56} - 165 q^{57} - 70 q^{58} + 118 q^{59} - 240 q^{60} + 151 q^{61} - 106 q^{62} - 207 q^{63} + 128 q^{64} + 1280 q^{65} - 480 q^{66} - 168 q^{67} - 100 q^{68} - 15 q^{69} + 226 q^{70} + 532 q^{71} + 144 q^{72} - 49 q^{73} - 442 q^{74} - 72 q^{75} - 220 q^{76} + 335 q^{77} - 612 q^{78} - 664 q^{79} - 320 q^{80} + 162 q^{81} - 178 q^{82} - 351 q^{83} - 276 q^{84} + 289 q^{85} - 1016 q^{86} - 105 q^{87} - 640 q^{88} - 1401 q^{89} - 360 q^{90} + 3 q^{91} - 20 q^{92} - 159 q^{93} + 1158 q^{94} + 563 q^{95} + 192 q^{96} + 452 q^{97} + 210 q^{98} - 720 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 20 * q^5 + 12 * q^6 - 23 * q^7 + 16 * q^8 + 18 * q^9 - 40 * q^10 - 80 * q^11 + 24 * q^12 - 102 * q^13 - 46 * q^14 - 60 * q^15 + 32 * q^16 - 25 * q^17 + 36 * q^18 - 55 * q^19 - 80 * q^20 - 69 * q^21 - 160 * q^22 - 5 * q^23 + 48 * q^24 - 24 * q^25 - 204 * q^26 + 54 * q^27 - 92 * q^28 - 35 * q^29 - 120 * q^30 - 53 * q^31 + 64 * q^32 - 240 * q^33 - 50 * q^34 + 113 * q^35 + 72 * q^36 - 221 * q^37 - 110 * q^38 - 306 * q^39 - 160 * q^40 - 89 * q^41 - 138 * q^42 - 508 * q^43 - 320 * q^44 - 180 * q^45 - 10 * q^46 + 579 * q^47 + 96 * q^48 + 105 * q^49 - 48 * q^50 - 75 * q^51 - 408 * q^52 + 432 * q^53 + 108 * q^54 + 930 * q^55 - 184 * q^56 - 165 * q^57 - 70 * q^58 + 118 * q^59 - 240 * q^60 + 151 * q^61 - 106 * q^62 - 207 * q^63 + 128 * q^64 + 1280 * q^65 - 480 * q^66 - 168 * q^67 - 100 * q^68 - 15 * q^69 + 226 * q^70 + 532 * q^71 + 144 * q^72 - 49 * q^73 - 442 * q^74 - 72 * q^75 - 220 * q^76 + 335 * q^77 - 612 * q^78 - 664 * q^79 - 320 * q^80 + 162 * q^81 - 178 * q^82 - 351 * q^83 - 276 * q^84 + 289 * q^85 - 1016 * q^86 - 105 * q^87 - 640 * q^88 - 1401 * q^89 - 360 * q^90 + 3 * q^91 - 20 * q^92 - 159 * q^93 + 1158 * q^94 + 563 * q^95 + 192 * q^96 + 452 * q^97 + 210 * q^98 - 720 * q^99

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