LMFDB - Embedded newform 354.4.a.i.1.1

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:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 492x^{4} + 3376x^{3} + 13255x^{2} - 108942x + 106740 $ x^6 - 2x^5 - 492x^4 + 3376x^3 + 13255x^2 - 108942*x + 106740
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-23.5704$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -20.5704 q^{5} +6.00000 q^{6} +10.6977 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -20.5704 q^{5} +6.00000 q^{6} +10.6977 q^{7} +8.00000 q^{8} +9.00000 q^{9} -41.1409 q^{10} -2.23495 q^{11} +12.0000 q^{12} +85.5511 q^{13} +21.3953 q^{14} -61.7113 q^{15} +16.0000 q^{16} +60.1629 q^{17} +18.0000 q^{18} +0.284714 q^{19} -82.2818 q^{20} +32.0930 q^{21} -4.46989 q^{22} +2.95213 q^{23} +24.0000 q^{24} +298.143 q^{25} +171.102 q^{26} +27.0000 q^{27} +42.7907 q^{28} +66.9477 q^{29} -123.423 q^{30} +111.170 q^{31} +32.0000 q^{32} -6.70484 q^{33} +120.326 q^{34} -220.056 q^{35} +36.0000 q^{36} +153.036 q^{37} +0.569428 q^{38} +256.653 q^{39} -164.564 q^{40} -196.485 q^{41} +64.1860 q^{42} +308.000 q^{43} -8.93979 q^{44} -185.134 q^{45} +5.90427 q^{46} +130.989 q^{47} +48.0000 q^{48} -228.560 q^{49} +596.286 q^{50} +180.489 q^{51} +342.205 q^{52} -297.454 q^{53} +54.0000 q^{54} +45.9738 q^{55} +85.5814 q^{56} +0.854142 q^{57} +133.895 q^{58} -59.0000 q^{59} -246.845 q^{60} +92.3530 q^{61} +222.340 q^{62} +96.2791 q^{63} +64.0000 q^{64} -1759.82 q^{65} -13.4097 q^{66} -484.184 q^{67} +240.652 q^{68} +8.85640 q^{69} -440.112 q^{70} -485.984 q^{71} +72.0000 q^{72} -729.508 q^{73} +306.071 q^{74} +894.429 q^{75} +1.13886 q^{76} -23.9087 q^{77} +513.307 q^{78} -354.288 q^{79} -329.127 q^{80} +81.0000 q^{81} -392.970 q^{82} +1254.88 q^{83} +128.372 q^{84} -1237.58 q^{85} +616.001 q^{86} +200.843 q^{87} -17.8796 q^{88} +873.062 q^{89} -370.268 q^{90} +915.198 q^{91} +11.8085 q^{92} +333.510 q^{93} +261.978 q^{94} -5.85669 q^{95} +96.0000 q^{96} +1214.61 q^{97} -457.120 q^{98} -20.1145 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} + 18 q^{3} + 24 q^{4} + 20 q^{5} + 36 q^{6} + 26 q^{7} + 48 q^{8} + 54 q^{9}+O(q^{10}) $ 6 * q + 12 * q^2 + 18 * q^3 + 24 * q^4 + 20 * q^5 + 36 * q^6 + 26 * q^7 + 48 * q^8 + 54 * q^9	\( 6 q + 12 q^{2} + 18 q^{3} + 24 q^{4} + 20 q^{5} + 36 q^{6} + 26 q^{7} + 48 q^{8} + 54 q^{9} + 40 q^{10} + 63 q^{11} + 72 q^{12} + 93 q^{13} + 52 q^{14} + 60 q^{15} + 96 q^{16} + 230 q^{17} + 108 q^{18} + 89 q^{19} + 80 q^{20} + 78 q^{21} + 126 q^{22} + 81 q^{23} + 144 q^{24} + 304 q^{25} + 186 q^{26} + 162 q^{27} + 104 q^{28} + 131 q^{29} + 120 q^{30} + 51 q^{31} + 192 q^{32} + 189 q^{33} + 460 q^{34} - 87 q^{35} + 216 q^{36} - 16 q^{37} + 178 q^{38} + 279 q^{39} + 160 q^{40} + 176 q^{41} + 156 q^{42} + 375 q^{43} + 252 q^{44} + 180 q^{45} + 162 q^{46} - 255 q^{47} + 288 q^{48} - 290 q^{49} + 608 q^{50} + 690 q^{51} + 372 q^{52} - 256 q^{53} + 324 q^{54} - 184 q^{55} + 208 q^{56} + 267 q^{57} + 262 q^{58} - 354 q^{59} + 240 q^{60} + 39 q^{61} + 102 q^{62} + 234 q^{63} + 384 q^{64} - 954 q^{65} + 378 q^{66} + 86 q^{67} + 920 q^{68} + 243 q^{69} - 174 q^{70} - 895 q^{71} + 432 q^{72} + 155 q^{73} - 32 q^{74} + 912 q^{75} + 356 q^{76} - 498 q^{77} + 558 q^{78} - 565 q^{79} + 320 q^{80} + 486 q^{81} + 352 q^{82} + 140 q^{83} + 312 q^{84} + q^{85} + 750 q^{86} + 393 q^{87} + 504 q^{88} - 1769 q^{89} + 360 q^{90} - 422 q^{91} + 324 q^{92} + 153 q^{93} - 510 q^{94} - 2209 q^{95} + 576 q^{96} + 48 q^{97} - 580 q^{98} + 567 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 + 18 * q^3 + 24 * q^4 + 20 * q^5 + 36 * q^6 + 26 * q^7 + 48 * q^8 + 54 * q^9 + 40 * q^10 + 63 * q^11 + 72 * q^12 + 93 * q^13 + 52 * q^14 + 60 * q^15 + 96 * q^16 + 230 * q^17 + 108 * q^18 + 89 * q^19 + 80 * q^20 + 78 * q^21 + 126 * q^22 + 81 * q^23 + 144 * q^24 + 304 * q^25 + 186 * q^26 + 162 * q^27 + 104 * q^28 + 131 * q^29 + 120 * q^30 + 51 * q^31 + 192 * q^32 + 189 * q^33 + 460 * q^34 - 87 * q^35 + 216 * q^36 - 16 * q^37 + 178 * q^38 + 279 * q^39 + 160 * q^40 + 176 * q^41 + 156 * q^42 + 375 * q^43 + 252 * q^44 + 180 * q^45 + 162 * q^46 - 255 * q^47 + 288 * q^48 - 290 * q^49 + 608 * q^50 + 690 * q^51 + 372 * q^52 - 256 * q^53 + 324 * q^54 - 184 * q^55 + 208 * q^56 + 267 * q^57 + 262 * q^58 - 354 * q^59 + 240 * q^60 + 39 * q^61 + 102 * q^62 + 234 * q^63 + 384 * q^64 - 954 * q^65 + 378 * q^66 + 86 * q^67 + 920 * q^68 + 243 * q^69 - 174 * q^70 - 895 * q^71 + 432 * q^72 + 155 * q^73 - 32 * q^74 + 912 * q^75 + 356 * q^76 - 498 * q^77 + 558 * q^78 - 565 * q^79 + 320 * q^80 + 486 * q^81 + 352 * q^82 + 140 * q^83 + 312 * q^84 + q^85 + 750 * q^86 + 393 * q^87 + 504 * q^88 - 1769 * q^89 + 360 * q^90 - 422 * q^91 + 324 * q^92 + 153 * q^93 - 510 * q^94 - 2209 * q^95 + 576 * q^96 + 48 * q^97 - 580 * q^98 + 567 * 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
$5$	−20.5704	−1.83988	−0.919938	−	0.392064i	$-0.871761\pi$
$5$	−20.5704	−1.83988	−0.919938	+	0.392064i	$0.871761\pi$
$6$	6.00000	0.408248
$7$	10.6977	0.577620	0.288810	−	0.957386i	$-0.406740\pi$
$7$	10.6977	0.577620	0.288810	+	0.957386i	$0.406740\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−41.1409	−1.30099
$11$	−2.23495	−0.0612602	−0.0306301	−	0.999531i	$-0.509751\pi$
$11$	−2.23495	−0.0612602	−0.0306301	+	0.999531i	$0.509751\pi$
$12$	12.0000	0.288675
$13$	85.5511	1.82520	0.912601	−	0.408852i	$-0.134071\pi$
$13$	85.5511	1.82520	0.912601	+	0.408852i	$0.134071\pi$
$14$	21.3953	0.408439
$15$	−61.7113	−1.06225
$16$	16.0000	0.250000
$17$	60.1629	0.858332	0.429166	−	0.903226i	$-0.358808\pi$
$17$	60.1629	0.858332	0.429166	+	0.903226i	$0.358808\pi$
$18$	18.0000	0.235702
$19$	0.284714	0.00343778	0.00171889	−	0.999999i	$-0.499453\pi$
$19$	0.284714	0.00343778	0.00171889	+	0.999999i	$0.499453\pi$
$20$	−82.2818	−0.919938
$21$	32.0930	0.333489
$22$	−4.46989	−0.0433175
$23$	2.95213	0.0267636	0.0133818	−	0.999910i	$-0.495740\pi$
$23$	2.95213	0.0267636	0.0133818	+	0.999910i	$0.495740\pi$
$24$	24.0000	0.204124
$25$	298.143	2.38514
$26$	171.102	1.29061
$27$	27.0000	0.192450
$28$	42.7907	0.288810
$29$	66.9477	0.428686	0.214343	−	0.976758i	$-0.431239\pi$
$29$	66.9477	0.428686	0.214343	+	0.976758i	$0.431239\pi$
$30$	−123.423	−0.751126
$31$	111.170	0.644087	0.322043	−	0.946725i	$-0.395630\pi$
$31$	111.170	0.644087	0.322043	+	0.946725i	$0.395630\pi$
$32$	32.0000	0.176777
$33$	−6.70484	−0.0353686
$34$	120.326	0.606932
$35$	−220.056	−1.06275
$36$	36.0000	0.166667
$37$	153.036	0.679970	0.339985	−	0.940431i	$-0.389578\pi$
$37$	153.036	0.679970	0.339985	+	0.940431i	$0.389578\pi$
$38$	0.569428	0.00243088
$39$	256.653	1.05378
$40$	−164.564	−0.650494
$41$	−196.485	−0.748435	−0.374218	−	0.927341i	$-0.622089\pi$
$41$	−196.485	−0.748435	−0.374218	+	0.927341i	$0.622089\pi$
$42$	64.1860	0.235812
$43$	308.000	1.09232	0.546158	−	0.837682i	$-0.316090\pi$
$43$	308.000	1.09232	0.546158	+	0.837682i	$0.316090\pi$
$44$	−8.93979	−0.0306301
$45$	−185.134	−0.613292
$46$	5.90427	0.0189247
$47$	130.989	0.406526	0.203263	−	0.979124i	$-0.434845\pi$
$47$	130.989	0.406526	0.203263	+	0.979124i	$0.434845\pi$
$48$	48.0000	0.144338
$49$	−228.560	−0.666355
$50$	596.286	1.68655
$51$	180.489	0.495558
$52$	342.205	0.912601
$53$	−297.454	−0.770915	−0.385457	−	0.922726i	$-0.625956\pi$
$53$	−297.454	−0.770915	−0.385457	+	0.922726i	$0.625956\pi$
$54$	54.0000	0.136083
$55$	45.9738	0.112711
$56$	85.5814	0.204220
$57$	0.854142	0.00198480
$58$	133.895	0.303127
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	−246.845	−0.531126
$61$	92.3530	0.193846	0.0969229	−	0.995292i	$-0.469100\pi$
$61$	92.3530	0.193846	0.0969229	+	0.995292i	$0.469100\pi$
$62$	222.340	0.455438
$63$	96.2791	0.192540
$64$	64.0000	0.125000
$65$	−1759.82	−3.35814
$66$	−13.4097	−0.0250094
$67$	−484.184	−0.882873	−0.441436	−	0.897293i	$-0.645531\pi$
$67$	−484.184	−0.882873	−0.441436	+	0.897293i	$0.645531\pi$
$68$	240.652	0.429166
$69$	8.85640	0.0154520
$70$	−440.112	−0.751477
$71$	−485.984	−0.812333	−0.406166	−	0.913799i	$-0.633135\pi$
$71$	−485.984	−0.812333	−0.406166	+	0.913799i	$0.633135\pi$
$72$	72.0000	0.117851
$73$	−729.508	−1.16962	−0.584812	−	0.811169i	$-0.698832\pi$
$73$	−729.508	−1.16962	−0.584812	+	0.811169i	$0.698832\pi$
$74$	306.071	0.480811
$75$	894.429	1.37706
$76$	1.13886	0.00171889
$77$	−23.9087	−0.0353851
$78$	513.307	0.745135
$79$	−354.288	−0.504564	−0.252282	−	0.967654i	$-0.581181\pi$
$79$	−354.288	−0.504564	−0.252282	+	0.967654i	$0.581181\pi$
$80$	−329.127	−0.459969
$81$	81.0000	0.111111
$82$	−392.970	−0.529224
$83$	1254.88	1.65953	0.829765	−	0.558112i	$-0.188474\pi$
$83$	1254.88	1.65953	0.829765	+	0.558112i	$0.188474\pi$
$84$	128.372	0.166745
$85$	−1237.58	−1.57922
$86$	616.001	0.772384
$87$	200.843	0.247502
$88$	−17.8796	−0.0216587
$89$	873.062	1.03982	0.519912	−	0.854220i	$-0.325965\pi$
$89$	873.062	1.03982	0.519912	+	0.854220i	$0.325965\pi$
$90$	−370.268	−0.433663
$91$	915.198	1.05427
$92$	11.8085	0.0133818
$93$	333.510	0.371864
$94$	261.978	0.287457
$95$	−5.85669	−0.00632509
$96$	96.0000	0.102062
$97$	1214.61	1.27139	0.635695	−	0.771940i	$-0.280713\pi$
$97$	1214.61	1.27139	0.635695	+	0.771940i	$0.280713\pi$
$98$	−457.120	−0.471184
$99$	−20.1145	−0.0204201
$100$	1192.57	1.19257
$101$	−1493.51	−1.47139	−0.735694	−	0.677314i	$-0.763144\pi$
$101$	−1493.51	−1.47139	−0.735694	+	0.677314i	$0.763144\pi$
$102$	360.977	0.350413
$103$	−1251.38	−1.19711	−0.598555	−	0.801082i	$-0.704258\pi$
$103$	−1251.38	−1.19711	−0.598555	+	0.801082i	$0.704258\pi$
$104$	684.409	0.645306
$105$	−660.168	−0.613579
$106$	−594.909	−0.545119
$107$	1533.68	1.38567	0.692834	−	0.721097i	$-0.256362\pi$
$107$	1533.68	1.38567	0.692834	+	0.721097i	$0.256362\pi$
$108$	108.000	0.0962250
$109$	257.128	0.225949	0.112974	−	0.993598i	$-0.463962\pi$
$109$	257.128	0.225949	0.112974	+	0.993598i	$0.463962\pi$
$110$	91.9477	0.0796988
$111$	459.107	0.392581
$112$	171.163	0.144405
$113$	2128.87	1.77227	0.886137	−	0.463423i	$-0.153379\pi$
$113$	2128.87	1.77227	0.886137	+	0.463423i	$0.153379\pi$
$114$	1.70828	0.00140347
$115$	−60.7267	−0.0492417
$116$	267.791	0.214343
$117$	769.960	0.608400
$118$	−118.000	−0.0920575
$119$	643.603	0.495790
$120$	−493.691	−0.375563
$121$	−1326.01	−0.996247
$122$	184.706	0.137070
$123$	−589.456	−0.432109
$124$	444.679	0.322043
$125$	−3561.63	−2.54849
$126$	192.558	0.136146
$127$	1861.39	1.30056	0.650282	−	0.759693i	$-0.274651\pi$
$127$	1861.39	1.30056	0.650282	+	0.759693i	$0.274651\pi$
$128$	128.000	0.0883883
$129$	924.001	0.630649
$130$	−3519.65	−2.37457
$131$	−254.823	−0.169954	−0.0849771	−	0.996383i	$-0.527082\pi$
$131$	−254.823	−0.169954	−0.0849771	+	0.996383i	$0.527082\pi$
$132$	−26.8194	−0.0176843
$133$	3.04578	0.00198573
$134$	−968.368	−0.624285
$135$	−555.402	−0.354084
$136$	481.303	0.303466
$137$	−2391.69	−1.49150	−0.745751	−	0.666225i	$-0.767909\pi$
$137$	−2391.69	−1.49150	−0.745751	+	0.666225i	$0.767909\pi$
$138$	17.7128	0.0109262
$139$	−1981.83	−1.20933	−0.604663	−	0.796481i	$-0.706692\pi$
$139$	−1981.83	−1.20933	−0.604663	+	0.796481i	$0.706692\pi$
$140$	−880.223	−0.531375
$141$	392.967	0.234708
$142$	−971.967	−0.574406
$143$	−191.202	−0.111812
$144$	144.000	0.0833333
$145$	−1377.14	−0.788728
$146$	−1459.02	−0.827049
$147$	−685.679	−0.384720
$148$	612.142	0.339985
$149$	−3117.67	−1.71416	−0.857079	−	0.515184i	$-0.827723\pi$
$149$	−3117.67	−1.71416	−0.857079	+	0.515184i	$0.827723\pi$
$150$	1788.86	0.973731
$151$	−1691.78	−0.911758	−0.455879	−	0.890042i	$-0.650675\pi$
$151$	−1691.78	−0.911758	−0.455879	+	0.890042i	$0.650675\pi$
$152$	2.27771	0.00121544
$153$	541.466	0.286111
$154$	−47.8175	−0.0250210
$155$	−2286.81	−1.18504
$156$	1026.61	0.526890
$157$	1600.30	0.813489	0.406744	−	0.913542i	$-0.366664\pi$
$157$	1600.30	0.813489	0.406744	+	0.913542i	$0.366664\pi$
$158$	−708.576	−0.356780
$159$	−892.363	−0.445088
$160$	−658.254	−0.325247
$161$	31.5810	0.0154592
$162$	162.000	0.0785674
$163$	1610.34	0.773812	0.386906	−	0.922119i	$-0.373544\pi$
$163$	1610.34	0.773812	0.386906	+	0.922119i	$0.373544\pi$
$164$	−785.941	−0.374218
$165$	137.921	0.0650738
$166$	2509.76	1.17347
$167$	−1983.03	−0.918869	−0.459435	−	0.888212i	$-0.651948\pi$
$167$	−1983.03	−0.918869	−0.459435	+	0.888212i	$0.651948\pi$
$168$	256.744	0.117906
$169$	5122.00	2.33136
$170$	−2475.15	−1.11668
$171$	2.56243	0.00114593
$172$	1232.00	0.546158
$173$	3462.71	1.52176	0.760882	−	0.648890i	$-0.224766\pi$
$173$	3462.71	1.52176	0.760882	+	0.648890i	$0.224766\pi$
$174$	401.686	0.175010
$175$	3189.44	1.37771
$176$	−35.7591	−0.0153150
$177$	−177.000	−0.0751646
$178$	1746.12	0.735267
$179$	−3713.11	−1.55045	−0.775226	−	0.631684i	$-0.782364\pi$
$179$	−3713.11	−1.55045	−0.775226	+	0.631684i	$0.782364\pi$
$180$	−740.536	−0.306646
$181$	219.945	0.0903224	0.0451612	−	0.998980i	$-0.485620\pi$
$181$	219.945	0.0903224	0.0451612	+	0.998980i	$0.485620\pi$
$182$	1830.40	0.745484
$183$	277.059	0.111917
$184$	23.6171	0.00946236
$185$	−3148.01	−1.25106
$186$	667.019	0.262947
$187$	−134.461	−0.0525815
$188$	523.956	0.203263
$189$	288.837	0.111163
$190$	−11.7134	−0.00447252
$191$	−3454.10	−1.30853	−0.654267	−	0.756263i	$-0.727023\pi$
$191$	−3454.10	−1.30853	−0.654267	+	0.756263i	$0.727023\pi$
$192$	192.000	0.0721688
$193$	564.088	0.210383	0.105192	−	0.994452i	$-0.466454\pi$
$193$	564.088	0.210383	0.105192	+	0.994452i	$0.466454\pi$
$194$	2429.22	0.899009
$195$	−5279.47	−1.93883
$196$	−914.239	−0.333178
$197$	−748.278	−0.270623	−0.135311	−	0.990803i	$-0.543203\pi$
$197$	−748.278	−0.270623	−0.135311	+	0.990803i	$0.543203\pi$
$198$	−40.2290	−0.0144392
$199$	−338.672	−0.120642	−0.0603212	−	0.998179i	$-0.519213\pi$
$199$	−338.672	−0.120642	−0.0603212	+	0.998179i	$0.519213\pi$
$200$	2385.14	0.843276
$201$	−1452.55	−0.509727
$202$	−2987.03	−1.04043
$203$	716.185	0.247617
$204$	721.955	0.247779
$205$	4041.79	1.37703
$206$	−2502.76	−0.846484
$207$	26.5692	0.00892119
$208$	1368.82	0.456300
$209$	−0.636320	−0.000210599 0
$210$	−1320.34	−0.433866
$211$	1962.10	0.640172	0.320086	−	0.947388i	$-0.396288\pi$
$211$	1962.10	0.640172	0.320086	+	0.947388i	$0.396288\pi$
$212$	−1189.82	−0.385457
$213$	−1457.95	−0.469001
$214$	3067.36	0.979815
$215$	−6335.70	−2.00973
$216$	216.000	0.0680414
$217$	1189.26	0.372038
$218$	514.256	0.159770
$219$	−2188.53	−0.675282
$220$	183.895	0.0563555
$221$	5147.00	1.56663
$222$	918.213	0.277597
$223$	−5094.92	−1.52996	−0.764980	−	0.644055i	$-0.777251\pi$
$223$	−5094.92	−1.52996	−0.764980	+	0.644055i	$0.777251\pi$
$224$	342.326	0.102110
$225$	2683.29	0.795048
$226$	4257.74	1.25319
$227$	5673.58	1.65889	0.829447	−	0.558586i	$-0.188656\pi$
$227$	5673.58	1.65889	0.829447	+	0.558586i	$0.188656\pi$
$228$	3.41657	0.000992402 0
$229$	−3447.28	−0.994772	−0.497386	−	0.867529i	$-0.665707\pi$
$229$	−3447.28	−0.994772	−0.497386	+	0.867529i	$0.665707\pi$
$230$	−121.453	−0.0348191
$231$	−71.7262	−0.0204296
$232$	535.582	0.151563
$233$	−3411.20	−0.959122	−0.479561	−	0.877509i	$-0.659204\pi$
$233$	−3411.20	−0.959122	−0.479561	+	0.877509i	$0.659204\pi$
$234$	1539.92	0.430204
$235$	−2694.50	−0.747957
$236$	−236.000	−0.0650945
$237$	−1062.86	−0.291310
$238$	1287.21	0.350576
$239$	4885.08	1.32213	0.661066	−	0.750328i	$-0.270104\pi$
$239$	4885.08	1.32213	0.661066	+	0.750328i	$0.270104\pi$
$240$	−987.381	−0.265563
$241$	2885.77	0.771322	0.385661	−	0.922641i	$-0.373973\pi$
$241$	2885.77	0.771322	0.385661	+	0.922641i	$0.373973\pi$
$242$	−2652.01	−0.704453
$243$	243.000	0.0641500
$244$	369.412	0.0969229
$245$	4701.57	1.22601
$246$	−1178.91	−0.305547
$247$	24.3576	0.00627465
$248$	889.359	0.227719
$249$	3764.64	0.958131
$250$	−7123.25	−1.80206
$251$	2011.37	0.505803	0.252901	−	0.967492i	$-0.418615\pi$
$251$	2011.37	0.505803	0.252901	+	0.967492i	$0.418615\pi$
$252$	385.116	0.0962700
$253$	−6.59786	−0.00163954
$254$	3722.78	0.919637
$255$	−3712.73	−0.911765
$256$	256.000	0.0625000
$257$	−3671.61	−0.891163	−0.445581	−	0.895241i	$-0.647003\pi$
$257$	−3671.61	−0.891163	−0.445581	+	0.895241i	$0.647003\pi$
$258$	1848.00	0.445936
$259$	1637.12	0.392764
$260$	−7039.30	−1.67907
$261$	602.530	0.142895
$262$	−509.647	−0.120176
$263$	−3963.12	−0.929188	−0.464594	−	0.885524i	$-0.653800\pi$
$263$	−3963.12	−0.929188	−0.464594	+	0.885524i	$0.653800\pi$
$264$	−53.6387	−0.0125047
$265$	6118.77	1.41839
$266$	6.09155	0.00140412
$267$	2619.18	0.600343
$268$	−1936.74	−0.441436
$269$	−407.113	−0.0922755	−0.0461378	−	0.998935i	$-0.514691\pi$
$269$	−407.113	−0.0922755	−0.0461378	+	0.998935i	$0.514691\pi$
$270$	−1110.80	−0.250375
$271$	−5324.46	−1.19350	−0.596749	−	0.802428i	$-0.703541\pi$
$271$	−5324.46	−1.19350	−0.596749	+	0.802428i	$0.703541\pi$
$272$	962.606	0.214583
$273$	2745.59	0.608685
$274$	−4783.38	−1.05465
$275$	−666.333	−0.146114
$276$	35.4256	0.00772598
$277$	1794.59	0.389266	0.194633	−	0.980876i	$-0.437648\pi$
$277$	1794.59	0.389266	0.194633	+	0.980876i	$0.437648\pi$
$278$	−3963.65	−0.855123
$279$	1000.53	0.214696
$280$	−1760.45	−0.375739
$281$	−1375.35	−0.291981	−0.145990	−	0.989286i	$-0.546637\pi$
$281$	−1375.35	−0.291981	−0.145990	+	0.989286i	$0.546637\pi$
$282$	785.934	0.165963
$283$	5607.66	1.17788	0.588941	−	0.808176i	$-0.299545\pi$
$283$	5607.66	1.17788	0.588941	+	0.808176i	$0.299545\pi$
$284$	−1943.93	−0.406166
$285$	−17.5701	−0.00365179
$286$	−382.404	−0.0790631
$287$	−2101.93	−0.432311
$288$	288.000	0.0589256
$289$	−1293.43	−0.263266
$290$	−2754.29	−0.557715
$291$	3643.83	0.734038
$292$	−2918.03	−0.584812
$293$	4607.07	0.918593	0.459297	−	0.888283i	$-0.348101\pi$
$293$	4607.07	0.918593	0.459297	+	0.888283i	$0.348101\pi$
$294$	−1371.36	−0.272038
$295$	1213.66	0.239531
$296$	1224.28	0.240406
$297$	−60.3436	−0.0117895
$298$	−6235.34	−1.21209
$299$	252.558	0.0488489
$300$	3577.72	0.688532
$301$	3294.89	0.630944
$302$	−3383.57	−0.644710
$303$	−4480.54	−0.849506
$304$	4.55542	0.000859446 0
$305$	−1899.74	−0.356652
$306$	1082.93	0.202311
$307$	−1501.61	−0.279158	−0.139579	−	0.990211i	$-0.544575\pi$
$307$	−1501.61	−0.279158	−0.139579	+	0.990211i	$0.544575\pi$
$308$	−95.6349	−0.0176925
$309$	−3754.14	−0.691152
$310$	−4573.62	−0.837950
$311$	7757.92	1.41450	0.707252	−	0.706961i	$-0.249934\pi$
$311$	7757.92	1.41450	0.707252	+	0.706961i	$0.249934\pi$
$312$	2053.23	0.372568
$313$	7752.37	1.39997	0.699984	−	0.714158i	$-0.253190\pi$
$313$	7752.37	1.39997	0.699984	+	0.714158i	$0.253190\pi$
$314$	3200.60	0.575223
$315$	−1980.50	−0.354250
$316$	−1417.15	−0.252282
$317$	−3065.47	−0.543135	−0.271568	−	0.962419i	$-0.587542\pi$
$317$	−3065.47	−0.543135	−0.271568	+	0.962419i	$0.587542\pi$
$318$	−1784.73	−0.314725
$319$	−149.625	−0.0262614
$320$	−1316.51	−0.229984
$321$	4601.04	0.800016
$322$	63.1619	0.0109313
$323$	17.1292	0.00295076
$324$	324.000	0.0555556
$325$	25506.5	4.35337
$326$	3220.67	0.547168
$327$	771.384	0.130452
$328$	−1571.88	−0.264612
$329$	1401.28	0.234817
$330$	275.843	0.0460141
$331$	4787.75	0.795041	0.397521	−	0.917593i	$-0.369871\pi$
$331$	4787.75	0.795041	0.397521	+	0.917593i	$0.369871\pi$
$332$	5019.52	0.829765
$333$	1377.32	0.226657
$334$	−3966.05	−0.649739
$335$	9959.87	1.62438
$336$	513.488	0.0833723
$337$	4927.46	0.796486	0.398243	−	0.917280i	$-0.369620\pi$
$337$	4927.46	0.796486	0.398243	+	0.917280i	$0.369620\pi$
$338$	10244.0	1.64852
$339$	6386.60	1.02322
$340$	−4950.31	−0.789612
$341$	−248.459	−0.0394569
$342$	5.12485	0.000810293 0
$343$	−6114.36	−0.962520
$344$	2464.00	0.386192
$345$	−182.180	−0.0284297
$346$	6925.43	1.07605
$347$	2074.22	0.320893	0.160446	−	0.987045i	$-0.448707\pi$
$347$	2074.22	0.320893	0.160446	+	0.987045i	$0.448707\pi$
$348$	803.373	0.123751
$349$	−4556.88	−0.698923	−0.349461	−	0.936951i	$-0.613635\pi$
$349$	−4556.88	−0.698923	−0.349461	+	0.936951i	$0.613635\pi$
$350$	6378.87	0.974186
$351$	2309.88	0.351260
$352$	−71.5183	−0.0108294
$353$	−5680.79	−0.856539	−0.428269	−	0.903651i	$-0.640877\pi$
$353$	−5680.79	−0.856539	−0.428269	+	0.903651i	$0.640877\pi$
$354$	−354.000	−0.0531494
$355$	9996.89	1.49459
$356$	3492.25	0.519912
$357$	1930.81	0.286244
$358$	−7426.22	−1.09634
$359$	−11834.3	−1.73980	−0.869900	−	0.493228i	$-0.835817\pi$
$359$	−11834.3	−1.73980	−0.869900	+	0.493228i	$0.835817\pi$
$360$	−1481.07	−0.216831
$361$	−6858.92	−0.999988
$362$	439.889	0.0638676
$363$	−3978.02	−0.575184
$364$	3660.79	0.527136
$365$	15006.3	2.15196
$366$	554.118	0.0791372
$367$	5218.71	0.742273	0.371137	−	0.928578i	$-0.378968\pi$
$367$	5218.71	0.742273	0.371137	+	0.928578i	$0.378968\pi$
$368$	47.2341	0.00669090
$369$	−1768.37	−0.249478
$370$	−6296.02	−0.884633
$371$	−3182.07	−0.445296
$372$	1334.04	0.185932
$373$	−9115.29	−1.26534	−0.632670	−	0.774421i	$-0.718041\pi$
$373$	−9115.29	−1.26534	−0.632670	+	0.774421i	$0.718041\pi$
$374$	−268.922	−0.0371808
$375$	−10684.9	−1.47137
$376$	1047.91	0.143728
$377$	5727.46	0.782438
$378$	577.674	0.0786041
$379$	−5005.47	−0.678400	−0.339200	−	0.940714i	$-0.610156\pi$
$379$	−5005.47	−0.678400	−0.339200	+	0.940714i	$0.610156\pi$
$380$	−23.4268	−0.00316255
$381$	5584.17	0.750880
$382$	−6908.21	−0.925274
$383$	4486.35	0.598542	0.299271	−	0.954168i	$-0.403257\pi$
$383$	4486.35	0.598542	0.299271	+	0.954168i	$0.403257\pi$
$384$	384.000	0.0510310
$385$	491.813	0.0651042
$386$	1128.18	0.148763
$387$	2772.00	0.364105
$388$	4858.44	0.635695
$389$	−1874.92	−0.244376	−0.122188	−	0.992507i	$-0.538991\pi$
$389$	−1874.92	−0.244376	−0.122188	+	0.992507i	$0.538991\pi$
$390$	−10558.9	−1.37096
$391$	177.609	0.0229720
$392$	−1828.48	−0.235592
$393$	−764.470	−0.0981232
$394$	−1496.56	−0.191359
$395$	7287.86	0.928335
$396$	−80.4581	−0.0102100
$397$	4124.57	0.521427	0.260713	−	0.965416i	$-0.416042\pi$
$397$	4124.57	0.521427	0.260713	+	0.965416i	$0.416042\pi$
$398$	−677.345	−0.0853071
$399$	9.13733	0.00114646
$400$	4770.29	0.596286
$401$	−5543.46	−0.690342	−0.345171	−	0.938540i	$-0.612179\pi$
$401$	−5543.46	−0.690342	−0.345171	+	0.938540i	$0.612179\pi$
$402$	−2905.10	−0.360431
$403$	9510.71	1.17559
$404$	−5974.05	−0.735694
$405$	−1666.21	−0.204431
$406$	1432.37	0.175092
$407$	−342.026	−0.0416551
$408$	1443.91	0.175206
$409$	14156.7	1.71150	0.855748	−	0.517393i	$-0.173097\pi$
$409$	14156.7	1.71150	0.855748	+	0.517393i	$0.173097\pi$
$410$	8083.58	0.973706
$411$	−7175.07	−0.861119
$412$	−5005.53	−0.598555
$413$	−631.163	−0.0751997
$414$	53.1384	0.00630824
$415$	−25813.4	−3.05333
$416$	2737.64	0.322653
$417$	−5945.48	−0.698205
$418$	−1.27264	−0.000148916 0
$419$	10128.0	1.18088	0.590438	−	0.807083i	$-0.298955\pi$
$419$	10128.0	1.18088	0.590438	+	0.807083i	$0.298955\pi$
$420$	−2640.67	−0.306789
$421$	−5141.33	−0.595185	−0.297593	−	0.954693i	$-0.596184\pi$
$421$	−5141.33	−0.595185	−0.297593	+	0.954693i	$0.596184\pi$
$422$	3924.19	0.452670
$423$	1178.90	0.135509
$424$	−2379.63	−0.272560
$425$	17937.1	2.04724
$426$	−2915.90	−0.331634
$427$	987.963	0.111969
$428$	6134.72	0.692834
$429$	−573.607	−0.0645548
$430$	−12671.4	−1.42109
$431$	−14321.7	−1.60059	−0.800293	−	0.599609i	$-0.795323\pi$
$431$	−14321.7	−1.60059	−0.800293	+	0.599609i	$0.795323\pi$
$432$	432.000	0.0481125
$433$	−9152.49	−1.01580	−0.507899	−	0.861417i	$-0.669578\pi$
$433$	−9152.49	−1.01580	−0.507899	+	0.861417i	$0.669578\pi$
$434$	2378.52	0.263070
$435$	−4131.43	−0.455373
$436$	1028.51	0.112974
$437$	0.840514	9.20074e−5 0
$438$	−4377.05	−0.477497
$439$	−13370.0	−1.45356	−0.726781	−	0.686869i	$-0.758985\pi$
$439$	−13370.0	−1.45356	−0.726781	+	0.686869i	$0.758985\pi$
$440$	367.791	0.0398494
$441$	−2057.04	−0.222118
$442$	10294.0	1.10777
$443$	9301.03	0.997529	0.498764	−	0.866738i	$-0.333787\pi$
$443$	9301.03	0.997529	0.498764	+	0.866738i	$0.333787\pi$
$444$	1836.43	0.196290
$445$	−17959.3	−1.91315
$446$	−10189.8	−1.08184
$447$	−9353.02	−0.989670
$448$	684.651	0.0722025
$449$	−173.955	−0.0182838	−0.00914192	−	0.999958i	$-0.502910\pi$
$449$	−173.955	−0.0182838	−0.00914192	+	0.999958i	$0.502910\pi$
$450$	5366.57	0.562184
$451$	439.134	0.0458492
$452$	8515.47	0.886137
$453$	−5075.35	−0.526404
$454$	11347.2	1.17301
$455$	−18826.0	−1.93973
$456$	6.83314	0.000701734 0
$457$	6334.55	0.648397	0.324199	−	0.945989i	$-0.394905\pi$
$457$	6334.55	0.648397	0.324199	+	0.945989i	$0.394905\pi$
$458$	−6894.56	−0.703410
$459$	1624.40	0.165186
$460$	−242.907	−0.0246208
$461$	16961.2	1.71358	0.856792	−	0.515661i	$-0.172454\pi$
$461$	16961.2	1.71358	0.856792	+	0.515661i	$0.172454\pi$
$462$	−143.452	−0.0144459
$463$	−3211.30	−0.322336	−0.161168	−	0.986927i	$-0.551526\pi$
$463$	−3211.30	−0.322336	−0.161168	+	0.986927i	$0.551526\pi$
$464$	1071.16	0.107171
$465$	−6860.44	−0.684183
$466$	−6822.41	−0.678201
$467$	−6991.32	−0.692762	−0.346381	−	0.938094i	$-0.612590\pi$
$467$	−6991.32	−0.692762	−0.346381	+	0.938094i	$0.612590\pi$
$468$	3079.84	0.304200
$469$	−5179.64	−0.509965
$470$	−5389.00	−0.528885
$471$	4800.89	0.469668
$472$	−472.000	−0.0460287
$473$	−688.364	−0.0669155
$474$	−2125.73	−0.205987
$475$	84.8855	0.00819960
$476$	2574.41	0.247895
$477$	−2677.09	−0.256972
$478$	9770.16	0.934889
$479$	−14165.5	−1.35123	−0.675614	−	0.737255i	$-0.736122\pi$
$479$	−14165.5	−1.35123	−0.675614	+	0.737255i	$0.736122\pi$
$480$	−1974.76	−0.187782
$481$	13092.4	1.24108
$482$	5771.53	0.545407
$483$	94.7429	0.00892536
$484$	−5304.02	−0.498124
$485$	−24985.0	−2.33920
$486$	486.000	0.0453609
$487$	−16442.6	−1.52995	−0.764975	−	0.644060i	$-0.777249\pi$
$487$	−16442.6	−1.52995	−0.764975	+	0.644060i	$0.777249\pi$
$488$	738.824	0.0685348
$489$	4831.01	0.446760
$490$	9403.15	0.866920
$491$	−12860.4	−1.18204	−0.591018	−	0.806658i	$-0.701274\pi$
$491$	−12860.4	−1.18204	−0.591018	+	0.806658i	$0.701274\pi$
$492$	−2357.82	−0.216055
$493$	4027.77	0.367955
$494$	48.7152	0.00443684
$495$	413.764	0.0375704
$496$	1778.72	0.161022
$497$	−5198.89	−0.469220
$498$	7529.28	0.677501
$499$	−3243.78	−0.291005	−0.145502	−	0.989358i	$-0.546480\pi$
$499$	−3243.78	−0.291005	−0.145502	+	0.989358i	$0.546480\pi$
$500$	−14246.5	−1.27425
$501$	−5949.08	−0.530509
$502$	4022.74	0.357657
$503$	18521.3	1.64179	0.820897	−	0.571077i	$-0.193474\pi$
$503$	18521.3	1.64179	0.820897	+	0.571077i	$0.193474\pi$
$504$	770.233	0.0680732
$505$	30722.2	2.70717
$506$	−13.1957	−0.00115933
$507$	15366.0	1.34601
$508$	7445.55	0.650282
$509$	6048.84	0.526739	0.263369	−	0.964695i	$-0.415166\pi$
$509$	6048.84	0.526739	0.263369	+	0.964695i	$0.415166\pi$
$510$	−7425.46	−0.644716
$511$	−7804.04	−0.675598
$512$	512.000	0.0441942
$513$	7.68728	0.000661602 0
$514$	−7343.22	−0.630147
$515$	25741.5	2.20253
$516$	3696.00	0.315325
$517$	−292.753	−0.0249038
$518$	3274.25	0.277726
$519$	10388.1	0.878591
$520$	−14078.6	−1.18728
$521$	−4489.66	−0.377535	−0.188767	−	0.982022i	$-0.560449\pi$
$521$	−4489.66	−0.377535	−0.188767	+	0.982022i	$0.560449\pi$
$522$	1205.06	0.101042
$523$	12017.2	1.00473	0.502365	−	0.864656i	$-0.332463\pi$
$523$	12017.2	1.00473	0.502365	+	0.864656i	$0.332463\pi$
$524$	−1019.29	−0.0849771
$525$	9568.31	0.795419
$526$	−7926.24	−0.657035
$527$	6688.30	0.552840
$528$	−107.277	−0.00884214
$529$	−12158.3	−0.999284
$530$	12237.5	1.00295
$531$	−531.000	−0.0433963
$532$	12.1831	0.000992866 0
$533$	−16809.5	−1.36604
$534$	5238.37	0.424506
$535$	−31548.5	−2.54946
$536$	−3873.47	−0.312143
$537$	−11139.3	−0.895154
$538$	−814.226	−0.0652487
$539$	510.819	0.0408210
$540$	−2221.61	−0.177042
$541$	−4726.84	−0.375642	−0.187821	−	0.982203i	$-0.560143\pi$
$541$	−4726.84	−0.375642	−0.187821	+	0.982203i	$0.560143\pi$
$542$	−10648.9	−0.843930
$543$	659.834	0.0521477
$544$	1925.21	0.151733
$545$	−5289.24	−0.415718
$546$	5491.19	0.430405
$547$	−8146.35	−0.636769	−0.318385	−	0.947962i	$-0.603140\pi$
$547$	−8146.35	−0.636769	−0.318385	+	0.947962i	$0.603140\pi$
$548$	−9566.76	−0.745751
$549$	831.177	0.0646153
$550$	−1332.67	−0.103318
$551$	19.0610	0.00147373
$552$	70.8512	0.00546309
$553$	−3790.06	−0.291446
$554$	3589.18	0.275252
$555$	−9444.02	−0.722300
$556$	−7927.31	−0.604663
$557$	16255.2	1.23654	0.618270	−	0.785965i	$-0.287834\pi$
$557$	16255.2	1.23654	0.618270	+	0.785965i	$0.287834\pi$
$558$	2001.06	0.151813
$559$	26349.8	1.99370
$560$	−3520.89	−0.265687
$561$	−403.382	−0.0303580
$562$	−2750.70	−0.206462
$563$	8289.78	0.620555	0.310278	−	0.950646i	$-0.399578\pi$
$563$	8289.78	0.620555	0.310278	+	0.950646i	$0.399578\pi$
$564$	1571.87	0.117354
$565$	−43791.7	−3.26077
$566$	11215.3	0.832888
$567$	866.512	0.0641800
$568$	−3887.87	−0.287203
$569$	−4884.07	−0.359843	−0.179922	−	0.983681i	$-0.557584\pi$
$569$	−4884.07	−0.359843	−0.179922	+	0.983681i	$0.557584\pi$
$570$	−35.1401	−0.00258221
$571$	5925.23	0.434262	0.217131	−	0.976143i	$-0.430330\pi$
$571$	5925.23	0.434262	0.217131	+	0.976143i	$0.430330\pi$
$572$	−764.809	−0.0559061
$573$	−10362.3	−0.755483
$574$	−4203.87	−0.305690
$575$	880.158	0.0638350
$576$	576.000	0.0416667
$577$	7156.64	0.516352	0.258176	−	0.966098i	$-0.416879\pi$
$577$	7156.64	0.516352	0.258176	+	0.966098i	$0.416879\pi$
$578$	−2586.86	−0.186158
$579$	1692.26	0.121465
$580$	−5508.58	−0.394364
$581$	13424.3	0.958578
$582$	7287.66	0.519043
$583$	664.794	0.0472264
$584$	−5836.07	−0.413524
$585$	−15838.4	−1.11938
$586$	9214.14	0.649544
$587$	−18003.9	−1.26593	−0.632964	−	0.774181i	$-0.718162\pi$
$587$	−18003.9	−1.26593	−0.632964	+	0.774181i	$0.718162\pi$
$588$	−2742.72	−0.192360
$589$	31.6516	0.00221423
$590$	2427.31	0.169374
$591$	−2244.84	−0.156244
$592$	2448.57	0.169992
$593$	−8897.47	−0.616147	−0.308074	−	0.951363i	$-0.599684\pi$
$593$	−8897.47	−0.616147	−0.308074	+	0.951363i	$0.599684\pi$
$594$	−120.687	−0.00833645
$595$	−13239.2	−0.912192
$596$	−12470.7	−0.857079
$597$	−1016.02	−0.0696530
$598$	505.117	0.0345414
$599$	−12124.1	−0.827009	−0.413505	−	0.910502i	$-0.635695\pi$
$599$	−12124.1	−0.827009	−0.413505	+	0.910502i	$0.635695\pi$
$600$	7155.43	0.486865
$601$	5354.06	0.363389	0.181694	−	0.983355i	$-0.441842\pi$
$601$	5354.06	0.363389	0.181694	+	0.983355i	$0.441842\pi$
$602$	6589.77	0.446145
$603$	−4357.65	−0.294291
$604$	−6767.14	−0.455879
$605$	27276.5	1.83297
$606$	−8961.08	−0.600692
$607$	3932.68	0.262969	0.131485	−	0.991318i	$-0.458026\pi$
$607$	3932.68	0.262969	0.131485	+	0.991318i	$0.458026\pi$
$608$	9.11085	0.000607720 0
$609$	2148.56	0.142962
$610$	−3799.49	−0.252191
$611$	11206.3	0.741991
$612$	2165.86	0.143055
$613$	−22292.0	−1.46878	−0.734392	−	0.678726i	$-0.762532\pi$
$613$	−22292.0	−1.46878	−0.734392	+	0.678726i	$0.762532\pi$
$614$	−3003.22	−0.197394
$615$	12125.4	0.795027
$616$	−191.270	−0.0125105
$617$	7665.09	0.500138	0.250069	−	0.968228i	$-0.419547\pi$
$617$	7665.09	0.500138	0.250069	+	0.968228i	$0.419547\pi$
$618$	−7508.29	−0.488718
$619$	−24881.3	−1.61561	−0.807807	−	0.589446i	$-0.799346\pi$
$619$	−24881.3	−1.61561	−0.807807	+	0.589446i	$0.799346\pi$
$620$	−9147.25	−0.592520
$621$	79.7076	0.00515065
$622$	15515.8	1.00021
$623$	9339.73	0.600623
$624$	4106.45	0.263445
$625$	35996.3	2.30377
$626$	15504.7	0.989927
$627$	−1.90896	−0.000121589 0
$628$	6401.19	0.406744
$629$	9207.06	0.583640
$630$	−3961.01	−0.250492
$631$	−10193.5	−0.643099	−0.321549	−	0.946893i	$-0.604204\pi$
$631$	−10193.5	−0.643099	−0.321549	+	0.946893i	$0.604204\pi$
$632$	−2834.31	−0.178390
$633$	5886.29	0.369604
$634$	−6130.94	−0.384055
$635$	−38289.6	−2.39287
$636$	−3569.45	−0.222544
$637$	−19553.5	−1.21623
$638$	−299.249	−0.0185696
$639$	−4373.85	−0.270778
$640$	−2633.02	−0.162624
$641$	227.166	0.0139977	0.00699883	−	0.999976i	$-0.497772\pi$
$641$	227.166	0.0139977	0.00699883	+	0.999976i	$0.497772\pi$
$642$	9202.08	0.565696
$643$	−26553.3	−1.62855	−0.814276	−	0.580478i	$-0.802866\pi$
$643$	−26553.3	−1.62855	−0.814276	+	0.580478i	$0.802866\pi$
$644$	126.324	0.00772959
$645$	−19007.1	−1.16032
$646$	34.2584	0.00208650
$647$	8435.80	0.512590	0.256295	−	0.966599i	$-0.417498\pi$
$647$	8435.80	0.512590	0.256295	+	0.966599i	$0.417498\pi$
$648$	648.000	0.0392837
$649$	131.862	0.00797539
$650$	51012.9	3.07830
$651$	3567.78	0.214796
$652$	6441.35	0.386906
$653$	−23909.5	−1.43285	−0.716424	−	0.697665i	$-0.754222\pi$
$653$	−23909.5	−1.43285	−0.716424	+	0.697665i	$0.754222\pi$
$654$	1542.77	0.0922432
$655$	5241.83	0.312695
$656$	−3143.76	−0.187109
$657$	−6565.58	−0.389874
$658$	2802.55	0.166041
$659$	27651.8	1.63454	0.817271	−	0.576254i	$-0.195486\pi$
$659$	27651.8	1.63454	0.817271	+	0.576254i	$0.195486\pi$
$660$	551.686	0.0325369
$661$	−26137.9	−1.53804	−0.769021	−	0.639223i	$-0.779256\pi$
$661$	−26137.9	−1.53804	−0.769021	+	0.639223i	$0.779256\pi$
$662$	9575.50	0.562179
$663$	15441.0	0.904493
$664$	10039.0	0.586733
$665$	−62.6530	−0.00365350
$666$	2754.64	0.160270
$667$	197.639	0.0114732
$668$	−7932.10	−0.459435
$669$	−15284.8	−0.883322
$670$	19919.7	1.14861
$671$	−206.404	−0.0118750
$672$	1026.98	0.0589531
$673$	27608.8	1.58134	0.790669	−	0.612244i	$-0.209733\pi$
$673$	27608.8	1.58134	0.790669	+	0.612244i	$0.209733\pi$
$674$	9854.92	0.563201
$675$	8049.86	0.459021
$676$	20488.0	1.16568
$677$	32463.5	1.84294	0.921471	−	0.388447i	$-0.126988\pi$
$677$	32463.5	1.84294	0.921471	+	0.388447i	$0.126988\pi$
$678$	12773.2	0.723528
$679$	12993.5	0.734381
$680$	−9900.61	−0.558340
$681$	17020.7	0.957762
$682$	−496.917	−0.0279002
$683$	−32825.1	−1.83897	−0.919485	−	0.393126i	$-0.871394\pi$
$683$	−32825.1	−1.83897	−0.919485	+	0.393126i	$0.871394\pi$
$684$	10.2497	0.000572964 0
$685$	49198.1	2.74418
$686$	−12228.7	−0.680605
$687$	−10341.8	−0.574332
$688$	4928.01	0.273079
$689$	−25447.6	−1.40707
$690$	−364.360	−0.0201028
$691$	−9824.85	−0.540890	−0.270445	−	0.962735i	$-0.587171\pi$
$691$	−9824.85	−0.540890	−0.270445	+	0.962735i	$0.587171\pi$
$692$	13850.9	0.760882
$693$	−215.179	−0.0117950
$694$	4148.43	0.226905
$695$	40767.1	2.22501
$696$	1606.75	0.0875051
$697$	−11821.1	−0.642406
$698$	−9113.75	−0.494213
$699$	−10233.6	−0.553749
$700$	12757.7	0.688853
$701$	8589.00	0.462771	0.231385	−	0.972862i	$-0.425674\pi$
$701$	8589.00	0.462771	0.231385	+	0.972862i	$0.425674\pi$
$702$	4619.76	0.248378
$703$	43.5714	0.00233759
$704$	−143.037	−0.00765752
$705$	−8083.50	−0.431833
$706$	−11361.6	−0.605664
$707$	−15977.1	−0.849903
$708$	−708.000	−0.0375823
$709$	20291.0	1.07482	0.537409	−	0.843322i	$-0.319403\pi$
$709$	20291.0	1.07482	0.537409	+	0.843322i	$0.319403\pi$
$710$	19993.8	1.05684
$711$	−3188.59	−0.168188
$712$	6984.49	0.367633
$713$	328.188	0.0172381
$714$	3861.62	0.202405
$715$	3933.11	0.205720
$716$	−14852.4	−0.775226
$717$	14655.2	0.763333
$718$	−23668.5	−1.23022
$719$	−29601.8	−1.53541	−0.767705	−	0.640803i	$-0.778602\pi$
$719$	−29601.8	−1.53541	−0.767705	+	0.640803i	$0.778602\pi$
$720$	−2962.14	−0.153323
$721$	−13386.9	−0.691475
$722$	−13717.8	−0.707098
$723$	8657.30	0.445323
$724$	879.778	0.0451612
$725$	19960.0	1.02248
$726$	−7956.03	−0.406716
$727$	−13247.0	−0.675799	−0.337899	−	0.941182i	$-0.609716\pi$
$727$	−13247.0	−0.675799	−0.337899	+	0.941182i	$0.609716\pi$
$728$	7321.59	0.372742
$729$	729.000	0.0370370
$730$	30012.6	1.52167
$731$	18530.2	0.937570
$732$	1108.24	0.0559585
$733$	31066.3	1.56543	0.782714	−	0.622381i	$-0.213835\pi$
$733$	31066.3	1.56543	0.782714	+	0.622381i	$0.213835\pi$
$734$	10437.4	0.524866
$735$	14104.7	0.707838
$736$	94.4683	0.00473118
$737$	1082.12	0.0540849
$738$	−3536.73	−0.176408
$739$	−37244.1	−1.85392	−0.926959	−	0.375164i	$-0.877586\pi$
$739$	−37244.1	−1.85392	−0.926959	+	0.375164i	$0.877586\pi$
$740$	−12592.0	−0.625530
$741$	73.0728	0.00362267
$742$	−6364.14	−0.314872
$743$	−26.2210	−0.00129469	−0.000647347	−	1.00000i	$-0.500206\pi$
$743$	−26.2210	−0.00129469	−0.000647347		1.00000i	$0.500206\pi$
$744$	2668.08	0.131474
$745$	64131.9	3.15384
$746$	−18230.6	−0.894731
$747$	11293.9	0.553177
$748$	−537.843	−0.0262908
$749$	16406.8	0.800389
$750$	−21369.8	−1.04042
$751$	13891.3	0.674967	0.337484	−	0.941331i	$-0.390424\pi$
$751$	13891.3	0.674967	0.337484	+	0.941331i	$0.390424\pi$
$752$	2095.82	0.101631
$753$	6034.11	0.292025
$754$	11454.9	0.553267
$755$	34800.7	1.67752
$756$	1155.35	0.0555815
$757$	28719.9	1.37892	0.689460	−	0.724323i	$-0.257848\pi$
$757$	28719.9	1.37892	0.689460	+	0.724323i	$0.257848\pi$
$758$	−10010.9	−0.479701
$759$	−19.7936	−0.000946590 0
$760$	−46.8535	−0.00223626
$761$	19474.3	0.927653	0.463826	−	0.885926i	$-0.346476\pi$
$761$	19474.3	0.927653	0.463826	+	0.885926i	$0.346476\pi$
$762$	11168.3	0.530953
$763$	2750.67	0.130512
$764$	−13816.4	−0.654267
$765$	−11138.2	−0.526408
$766$	8972.69	0.423233
$767$	−5047.52	−0.237621
$768$	768.000	0.0360844
$769$	−1759.29	−0.0824990	−0.0412495	−	0.999149i	$-0.513134\pi$
$769$	−1759.29	−0.0824990	−0.0412495	+	0.999149i	$0.513134\pi$
$770$	983.626	0.0460356
$771$	−11014.8	−0.514513
$772$	2256.35	0.105192
$773$	−3089.39	−0.143748	−0.0718742	−	0.997414i	$-0.522898\pi$
$773$	−3089.39	−0.143748	−0.0718742	+	0.997414i	$0.522898\pi$
$774$	5544.01	0.257461
$775$	33144.5	1.53624
$776$	9716.87	0.449504
$777$	4911.37	0.226763
$778$	−3749.83	−0.172800
$779$	−55.9421	−0.00257296
$780$	−21117.9	−0.969413
$781$	1086.15	0.0497636
$782$	355.218	0.0162437
$783$	1807.59	0.0825006
$784$	−3656.96	−0.166589
$785$	−32918.8	−1.49672
$786$	−1528.94	−0.0693836
$787$	19285.4	0.873506	0.436753	−	0.899582i	$-0.356128\pi$
$787$	19285.4	0.873506	0.436753	+	0.899582i	$0.356128\pi$
$788$	−2993.11	−0.135311
$789$	−11889.4	−0.536467
$790$	14575.7	0.656432
$791$	22773.9	1.02370
$792$	−160.916	−0.00721958
$793$	7900.91	0.353808
$794$	8249.15	0.368704
$795$	18356.3	0.818907
$796$	−1354.69	−0.0603212
$797$	−15080.2	−0.670222	−0.335111	−	0.942179i	$-0.608774\pi$
$797$	−15080.2	−0.670222	−0.335111	+	0.942179i	$0.608774\pi$
$798$	18.2747	0.000810672 0
$799$	7880.67	0.348934
$800$	9540.57	0.421638
$801$	7857.55	0.346608
$802$	−11086.9	−0.488145
$803$	1630.41	0.0716513
$804$	−5810.21	−0.254863
$805$	−649.634	−0.0284430
$806$	19021.4	0.831266
$807$	−1221.34	−0.0532753
$808$	−11948.1	−0.520214
$809$	27682.3	1.20304	0.601518	−	0.798859i	$-0.294563\pi$
$809$	27682.3	1.20304	0.601518	+	0.798859i	$0.294563\pi$
$810$	−3332.41	−0.144554
$811$	−3429.01	−0.148469	−0.0742347	−	0.997241i	$-0.523651\pi$
$811$	−3429.01	−0.148469	−0.0742347	+	0.997241i	$0.523651\pi$
$812$	2864.74	0.123809
$813$	−15973.4	−0.689066
$814$	−684.052	−0.0294546
$815$	−33125.3	−1.42372
$816$	2887.82	0.123890
$817$	87.6920	0.00375515
$818$	28313.3	1.21021
$819$	8236.78	0.351424
$820$	16167.2	0.688514
$821$	7419.75	0.315409	0.157705	−	0.987486i	$-0.449591\pi$
$821$	7419.75	0.315409	0.157705	+	0.987486i	$0.449591\pi$
$822$	−14350.1	−0.608903
$823$	−15195.1	−0.643583	−0.321792	−	0.946811i	$-0.604285\pi$
$823$	−15195.1	−0.643583	−0.321792	+	0.946811i	$0.604285\pi$
$824$	−10011.1	−0.423242
$825$	−1999.00	−0.0843591
$826$	−1262.33	−0.0531742
$827$	−2323.17	−0.0976837	−0.0488418	−	0.998807i	$-0.515553\pi$
$827$	−2323.17	−0.0976837	−0.0488418	+	0.998807i	$0.515553\pi$
$828$	106.277	0.00446060
$829$	−33317.4	−1.39585	−0.697927	−	0.716169i	$-0.745894\pi$
$829$	−33317.4	−1.39585	−0.697927	+	0.716169i	$0.745894\pi$
$830$	−51626.9	−2.15903
$831$	5383.78	0.224743
$832$	5475.27	0.228150
$833$	−13750.8	−0.571954
$834$	−11891.0	−0.493706
$835$	40791.7	1.69061
$836$	−2.54528	−0.000105300 0
$837$	3001.59	0.123955
$838$	20256.1	0.835006
$839$	−46106.5	−1.89723	−0.948613	−	0.316437i	$-0.897513\pi$
$839$	−46106.5	−1.89723	−0.948613	+	0.316437i	$0.897513\pi$
$840$	−5281.34	−0.216933
$841$	−19907.0	−0.816229
$842$	−10282.7	−0.420859
$843$	−4126.05	−0.168575
$844$	7848.39	0.320086
$845$	−105362.	−4.28941
$846$	2357.80	0.0958190
$847$	−14185.2	−0.575452
$848$	−4759.27	−0.192729
$849$	16823.0	0.680050
$850$	35874.3	1.44762
$851$	451.781	0.0181984
$852$	−5831.80	−0.234500
$853$	1702.35	0.0683320	0.0341660	−	0.999416i	$-0.489123\pi$
$853$	1702.35	0.0683320	0.0341660	+	0.999416i	$0.489123\pi$
$854$	1975.93	0.0791742
$855$	−52.7102	−0.00210836
$856$	12269.4	0.489908
$857$	−2929.81	−0.116780	−0.0583901	−	0.998294i	$-0.518597\pi$
$857$	−2929.81	−0.116780	−0.0583901	+	0.998294i	$0.518597\pi$
$858$	−1147.21	−0.0456471
$859$	22419.5	0.890503	0.445252	−	0.895405i	$-0.353114\pi$
$859$	22419.5	0.890503	0.445252	+	0.895405i	$0.353114\pi$
$860$	−25342.8	−1.00486
$861$	−6305.80	−0.249595
$862$	−28643.4	−1.13179
$863$	−25151.7	−0.992088	−0.496044	−	0.868297i	$-0.665215\pi$
$863$	−25151.7	−0.992088	−0.496044	+	0.868297i	$0.665215\pi$
$864$	864.000	0.0340207
$865$	−71229.5	−2.79986
$866$	−18305.0	−0.718277
$867$	−3880.28	−0.151997
$868$	4757.03	0.186019
$869$	791.815	0.0309097
$870$	−8262.87	−0.321997
$871$	−41422.5	−1.61142
$872$	2057.02	0.0798849
$873$	10931.5	0.423797
$874$	1.68103	6.50590e−5 0
$875$	−38101.1	−1.47206
$876$	−8754.10	−0.337641
$877$	−9059.18	−0.348811	−0.174405	−	0.984674i	$-0.555800\pi$
$877$	−9059.18	−0.348811	−0.174405	+	0.984674i	$0.555800\pi$
$878$	−26739.9	−1.02782
$879$	13821.2	0.530350
$880$	735.581	0.0281778
$881$	41733.0	1.59594	0.797969	−	0.602699i	$-0.205908\pi$
$881$	41733.0	1.59594	0.797969	+	0.602699i	$0.205908\pi$
$882$	−4114.08	−0.157061
$883$	−40917.8	−1.55945	−0.779725	−	0.626122i	$-0.784641\pi$
$883$	−40917.8	−1.55945	−0.779725	+	0.626122i	$0.784641\pi$
$884$	20588.0	0.783314
$885$	3640.97	0.138294
$886$	18602.1	0.705359
$887$	11039.0	0.417874	0.208937	−	0.977929i	$-0.433000\pi$
$887$	11039.0	0.417874	0.208937	+	0.977929i	$0.433000\pi$
$888$	3672.85	0.138798
$889$	19912.5	0.751231
$890$	−35918.5	−1.35280
$891$	−181.031	−0.00680668
$892$	−20379.7	−0.764980
$893$	37.2944	0.00139755
$894$	−18706.0	−0.699802
$895$	76380.3	2.85264
$896$	1369.30	0.0510549
$897$	757.675	0.0282029
$898$	−347.910	−0.0129286
$899$	7442.57	0.276111
$900$	10733.1	0.397524
$901$	−17895.7	−0.661701
$902$	878.268	0.0324203
$903$	9884.66	0.364276
$904$	17030.9	0.626594
$905$	−4524.36	−0.166182
$906$	−10150.7	−0.372224
$907$	1462.44	0.0535387	0.0267693	−	0.999642i	$-0.491478\pi$
$907$	1462.44	0.0535387	0.0267693	+	0.999642i	$0.491478\pi$
$908$	22694.3	0.829447
$909$	−13441.6	−0.490463
$910$	−37652.1	−1.37160
$911$	40422.5	1.47010	0.735048	−	0.678015i	$-0.237160\pi$
$911$	40422.5	1.47010	0.735048	+	0.678015i	$0.237160\pi$
$912$	13.6663	0.000496201 0
$913$	−2804.59	−0.101663
$914$	12669.1	0.458486
$915$	−5699.23	−0.205913
$916$	−13789.1	−0.497386
$917$	−2726.02	−0.0981690
$918$	3248.80	0.116804
$919$	−55076.9	−1.97695	−0.988476	−	0.151376i	$-0.951630\pi$
$919$	−55076.9	−1.97695	−0.988476	+	0.151376i	$0.951630\pi$
$920$	−485.813	−0.0174096
$921$	−4504.83	−0.161172
$922$	33922.4	1.21169
$923$	−41576.4	−1.48267
$924$	−286.905	−0.0102148
$925$	45626.5	1.62183
$926$	−6422.60	−0.227926
$927$	−11262.4	−0.399037
$928$	2142.33	0.0757816
$929$	−22479.1	−0.793881	−0.396940	−	0.917844i	$-0.629928\pi$
$929$	−22479.1	−0.793881	−0.396940	+	0.917844i	$0.629928\pi$
$930$	−13720.9	−0.483791
$931$	−65.0742	−0.00229078
$932$	−13644.8	−0.479561
$933$	23273.7	0.816665
$934$	−13982.6	−0.489857
$935$	2765.92	0.0967435
$936$	6159.68	0.215102
$937$	46979.2	1.63793	0.818966	−	0.573841i	$-0.194547\pi$
$937$	46979.2	1.63793	0.818966	+	0.573841i	$0.194547\pi$
$938$	−10359.3	−0.360600
$939$	23257.1	0.808272
$940$	−10778.0	−0.373978
$941$	53285.4	1.84597	0.922983	−	0.384841i	$-0.125744\pi$
$941$	53285.4	1.84597	0.922983	+	0.384841i	$0.125744\pi$
$942$	9601.79	0.332105
$943$	−580.051	−0.0200308
$944$	−944.000	−0.0325472
$945$	−5941.51	−0.204526
$946$	−1376.73	−0.0473164
$947$	−16349.6	−0.561025	−0.280513	−	0.959850i	$-0.590504\pi$
$947$	−16349.6	−0.561025	−0.280513	+	0.959850i	$0.590504\pi$
$948$	−4251.46	−0.145655
$949$	−62410.3	−2.13480
$950$	169.771	0.00579800
$951$	−9196.40	−0.313579
$952$	5148.82	0.175288
$953$	17556.7	0.596765	0.298383	−	0.954446i	$-0.403553\pi$
$953$	17556.7	0.596765	0.298383	+	0.954446i	$0.403553\pi$
$954$	−5354.18	−0.181706
$955$	71052.4	2.40754
$956$	19540.3	0.661066
$957$	−448.874	−0.0151620
$958$	−28331.0	−0.955463
$959$	−25585.5	−0.861522
$960$	−3949.52	−0.132782
$961$	−17432.3	−0.585152
$962$	26184.7	0.877577
$963$	13803.1	0.461889
$964$	11543.1	0.385661
$965$	−11603.5	−0.387079
$966$	189.486	0.00631119
$967$	27803.5	0.924612	0.462306	−	0.886720i	$-0.347022\pi$
$967$	27803.5	0.924612	0.462306	+	0.886720i	$0.347022\pi$
$968$	−10608.0	−0.352227
$969$	51.3876	0.00170362
$970$	−49970.1	−1.65407
$971$	−11564.0	−0.382189	−0.191095	−	0.981572i	$-0.561204\pi$
$971$	−11564.0	−0.382189	−0.191095	+	0.981572i	$0.561204\pi$
$972$	972.000	0.0320750
$973$	−21200.9	−0.698531
$974$	−32885.2	−1.08184
$975$	76519.4	2.51342
$976$	1477.65	0.0484615
$977$	17409.6	0.570096	0.285048	−	0.958513i	$-0.407990\pi$
$977$	17409.6	0.570096	0.285048	+	0.958513i	$0.407990\pi$
$978$	9662.02	0.315907
$979$	−1951.25	−0.0636998
$980$	18806.3	0.613005
$981$	2314.15	0.0753162
$982$	−25720.7	−0.835826
$983$	10174.2	0.330120	0.165060	−	0.986284i	$-0.447218\pi$
$983$	10174.2	0.330120	0.165060	+	0.986284i	$0.447218\pi$
$984$	−4715.65	−0.152774
$985$	15392.4	0.497912
$986$	8055.54	0.260183
$987$	4203.83	0.135572
$988$	97.4304	0.00313732
$989$	909.258	0.0292343
$990$	827.529	0.0265663
$991$	22501.5	0.721274	0.360637	−	0.932706i	$-0.382559\pi$
$991$	22501.5	0.721274	0.360637	+	0.932706i	$0.382559\pi$
$992$	3557.44	0.113860
$993$	14363.3	0.459017
$994$	−10397.8	−0.331788
$995$	6966.64	0.221967
$996$	15058.6	0.479065
$997$	−15746.2	−0.500187	−0.250093	−	0.968222i	$-0.580461\pi$
$997$	−15746.2	−0.500187	−0.250093	+	0.968222i	$0.580461\pi$
$998$	−6487.56	−0.205772
$999$	4131.96	0.130860

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.4.a.i.1.1	✓	6
3.2	odd	2		1062.4.a.s.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.4.a.i.1.1	✓	6	1.1	even	1	trivial
1062.4.a.s.1.6		6	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -20.5704 q^{5} +6.00000 q^{6} +10.6977 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -20.5704 q^{5} +6.00000 q^{6} +10.6977 q^{7} +8.00000 q^{8} +9.00000 q^{9} -41.1409 q^{10} -2.23495 q^{11} +12.0000 q^{12} +85.5511 q^{13} +21.3953 q^{14} -61.7113 q^{15} +16.0000 q^{16} +60.1629 q^{17} +18.0000 q^{18} +0.284714 q^{19} -82.2818 q^{20} +32.0930 q^{21} -4.46989 q^{22} +2.95213 q^{23} +24.0000 q^{24} +298.143 q^{25} +171.102 q^{26} +27.0000 q^{27} +42.7907 q^{28} +66.9477 q^{29} -123.423 q^{30} +111.170 q^{31} +32.0000 q^{32} -6.70484 q^{33} +120.326 q^{34} -220.056 q^{35} +36.0000 q^{36} +153.036 q^{37} +0.569428 q^{38} +256.653 q^{39} -164.564 q^{40} -196.485 q^{41} +64.1860 q^{42} +308.000 q^{43} -8.93979 q^{44} -185.134 q^{45} +5.90427 q^{46} +130.989 q^{47} +48.0000 q^{48} -228.560 q^{49} +596.286 q^{50} +180.489 q^{51} +342.205 q^{52} -297.454 q^{53} +54.0000 q^{54} +45.9738 q^{55} +85.5814 q^{56} +0.854142 q^{57} +133.895 q^{58} -59.0000 q^{59} -246.845 q^{60} +92.3530 q^{61} +222.340 q^{62} +96.2791 q^{63} +64.0000 q^{64} -1759.82 q^{65} -13.4097 q^{66} -484.184 q^{67} +240.652 q^{68} +8.85640 q^{69} -440.112 q^{70} -485.984 q^{71} +72.0000 q^{72} -729.508 q^{73} +306.071 q^{74} +894.429 q^{75} +1.13886 q^{76} -23.9087 q^{77} +513.307 q^{78} -354.288 q^{79} -329.127 q^{80} +81.0000 q^{81} -392.970 q^{82} +1254.88 q^{83} +128.372 q^{84} -1237.58 q^{85} +616.001 q^{86} +200.843 q^{87} -17.8796 q^{88} +873.062 q^{89} -370.268 q^{90} +915.198 q^{91} +11.8085 q^{92} +333.510 q^{93} +261.978 q^{94} -5.85669 q^{95} +96.0000 q^{96} +1214.61 q^{97} -457.120 q^{98} -20.1145 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} + 18 q^{3} + 24 q^{4} + 20 q^{5} + 36 q^{6} + 26 q^{7} + 48 q^{8} + 54 q^{9}+O(q^{10}) \) 6 * q + 12 * q^2 + 18 * q^3 + 24 * q^4 + 20 * q^5 + 36 * q^6 + 26 * q^7 + 48 * q^8 + 54 * q^9	\( 6 q + 12 q^{2} + 18 q^{3} + 24 q^{4} + 20 q^{5} + 36 q^{6} + 26 q^{7} + 48 q^{8} + 54 q^{9} + 40 q^{10} + 63 q^{11} + 72 q^{12} + 93 q^{13} + 52 q^{14} + 60 q^{15} + 96 q^{16} + 230 q^{17} + 108 q^{18} + 89 q^{19} + 80 q^{20} + 78 q^{21} + 126 q^{22} + 81 q^{23} + 144 q^{24} + 304 q^{25} + 186 q^{26} + 162 q^{27} + 104 q^{28} + 131 q^{29} + 120 q^{30} + 51 q^{31} + 192 q^{32} + 189 q^{33} + 460 q^{34} - 87 q^{35} + 216 q^{36} - 16 q^{37} + 178 q^{38} + 279 q^{39} + 160 q^{40} + 176 q^{41} + 156 q^{42} + 375 q^{43} + 252 q^{44} + 180 q^{45} + 162 q^{46} - 255 q^{47} + 288 q^{48} - 290 q^{49} + 608 q^{50} + 690 q^{51} + 372 q^{52} - 256 q^{53} + 324 q^{54} - 184 q^{55} + 208 q^{56} + 267 q^{57} + 262 q^{58} - 354 q^{59} + 240 q^{60} + 39 q^{61} + 102 q^{62} + 234 q^{63} + 384 q^{64} - 954 q^{65} + 378 q^{66} + 86 q^{67} + 920 q^{68} + 243 q^{69} - 174 q^{70} - 895 q^{71} + 432 q^{72} + 155 q^{73} - 32 q^{74} + 912 q^{75} + 356 q^{76} - 498 q^{77} + 558 q^{78} - 565 q^{79} + 320 q^{80} + 486 q^{81} + 352 q^{82} + 140 q^{83} + 312 q^{84} + q^{85} + 750 q^{86} + 393 q^{87} + 504 q^{88} - 1769 q^{89} + 360 q^{90} - 422 q^{91} + 324 q^{92} + 153 q^{93} - 510 q^{94} - 2209 q^{95} + 576 q^{96} + 48 q^{97} - 580 q^{98} + 567 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 + 18 * q^3 + 24 * q^4 + 20 * q^5 + 36 * q^6 + 26 * q^7 + 48 * q^8 + 54 * q^9 + 40 * q^10 + 63 * q^11 + 72 * q^12 + 93 * q^13 + 52 * q^14 + 60 * q^15 + 96 * q^16 + 230 * q^17 + 108 * q^18 + 89 * q^19 + 80 * q^20 + 78 * q^21 + 126 * q^22 + 81 * q^23 + 144 * q^24 + 304 * q^25 + 186 * q^26 + 162 * q^27 + 104 * q^28 + 131 * q^29 + 120 * q^30 + 51 * q^31 + 192 * q^32 + 189 * q^33 + 460 * q^34 - 87 * q^35 + 216 * q^36 - 16 * q^37 + 178 * q^38 + 279 * q^39 + 160 * q^40 + 176 * q^41 + 156 * q^42 + 375 * q^43 + 252 * q^44 + 180 * q^45 + 162 * q^46 - 255 * q^47 + 288 * q^48 - 290 * q^49 + 608 * q^50 + 690 * q^51 + 372 * q^52 - 256 * q^53 + 324 * q^54 - 184 * q^55 + 208 * q^56 + 267 * q^57 + 262 * q^58 - 354 * q^59 + 240 * q^60 + 39 * q^61 + 102 * q^62 + 234 * q^63 + 384 * q^64 - 954 * q^65 + 378 * q^66 + 86 * q^67 + 920 * q^68 + 243 * q^69 - 174 * q^70 - 895 * q^71 + 432 * q^72 + 155 * q^73 - 32 * q^74 + 912 * q^75 + 356 * q^76 - 498 * q^77 + 558 * q^78 - 565 * q^79 + 320 * q^80 + 486 * q^81 + 352 * q^82 + 140 * q^83 + 312 * q^84 + q^85 + 750 * q^86 + 393 * q^87 + 504 * q^88 - 1769 * q^89 + 360 * q^90 - 422 * q^91 + 324 * q^92 + 153 * q^93 - 510 * q^94 - 2209 * q^95 + 576 * q^96 + 48 * q^97 - 580 * q^98 + 567 * q^99

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