LMFDB - Embedded newform 165.4.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("165.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$9.73531515095$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.23612.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 20x + 26 $ x^3 - x^2 - 20*x + 26
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.59056$ of defining polynomial
Character	$\chi$	$=$	165.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.59056 q^{2} -3.00000 q^{3} +4.89212 q^{4} -5.00000 q^{5} -10.7717 q^{6} -16.1465 q^{7} -11.1590 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+3.59056 q^{2} -3.00000 q^{3} +4.89212 q^{4} -5.00000 q^{5} -10.7717 q^{6} -16.1465 q^{7} -11.1590 q^{8} +9.00000 q^{9} -17.9528 q^{10} +11.0000 q^{11} -14.6764 q^{12} -54.1214 q^{13} -57.9749 q^{14} +15.0000 q^{15} -79.2041 q^{16} -107.010 q^{17} +32.3150 q^{18} +48.7496 q^{19} -24.4606 q^{20} +48.4394 q^{21} +39.4962 q^{22} +11.9498 q^{23} +33.4771 q^{24} +25.0000 q^{25} -194.326 q^{26} -27.0000 q^{27} -78.9905 q^{28} +239.733 q^{29} +53.8584 q^{30} -82.0851 q^{31} -195.115 q^{32} -33.0000 q^{33} -384.224 q^{34} +80.7324 q^{35} +44.0291 q^{36} -21.7573 q^{37} +175.038 q^{38} +162.364 q^{39} +55.7952 q^{40} -124.835 q^{41} +173.925 q^{42} +224.459 q^{43} +53.8133 q^{44} -45.0000 q^{45} +42.9064 q^{46} -186.832 q^{47} +237.612 q^{48} -82.2913 q^{49} +89.7640 q^{50} +321.029 q^{51} -264.768 q^{52} +233.997 q^{53} -96.9451 q^{54} -55.0000 q^{55} +180.179 q^{56} -146.249 q^{57} +860.774 q^{58} +232.936 q^{59} +73.3818 q^{60} +163.849 q^{61} -294.731 q^{62} -145.318 q^{63} -66.9386 q^{64} +270.607 q^{65} -118.488 q^{66} -876.918 q^{67} -523.503 q^{68} -35.8493 q^{69} +289.874 q^{70} -733.141 q^{71} -100.431 q^{72} +1161.97 q^{73} -78.1208 q^{74} -75.0000 q^{75} +238.489 q^{76} -177.611 q^{77} +582.978 q^{78} -588.831 q^{79} +396.021 q^{80} +81.0000 q^{81} -448.226 q^{82} -1161.06 q^{83} +236.971 q^{84} +535.048 q^{85} +805.933 q^{86} -719.198 q^{87} -122.749 q^{88} -1042.16 q^{89} -161.575 q^{90} +873.869 q^{91} +58.4597 q^{92} +246.255 q^{93} -670.831 q^{94} -243.748 q^{95} +585.345 q^{96} +1546.63 q^{97} -295.472 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q - 4 * q^2 - 9 * q^3 + 22 * q^4 - 15 * q^5 + 12 * q^6 - 4 * q^7 - 48 * q^8 + 27 * q^9	\( 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9} + 20 q^{10} + 33 q^{11} - 66 q^{12} - 56 q^{14} + 45 q^{15} + 50 q^{16} - 218 q^{17} - 36 q^{18} + 146 q^{19} - 110 q^{20} + 12 q^{21} - 44 q^{22} - 200 q^{23} + 144 q^{24} + 75 q^{25} - 508 q^{26} - 81 q^{27} - 340 q^{28} + 68 q^{29} - 60 q^{30} - 68 q^{31} - 688 q^{32} - 99 q^{33} - 176 q^{34} + 20 q^{35} + 198 q^{36} - 390 q^{37} - 316 q^{38} + 240 q^{40} - 196 q^{41} + 168 q^{42} - 524 q^{43} + 242 q^{44} - 135 q^{45} + 1160 q^{46} - 60 q^{47} - 150 q^{48} - 157 q^{49} - 100 q^{50} + 654 q^{51} + 1020 q^{52} - 158 q^{53} + 108 q^{54} - 165 q^{55} + 1368 q^{56} - 438 q^{57} + 1092 q^{58} - 1044 q^{59} + 330 q^{60} + 642 q^{61} + 88 q^{62} - 36 q^{63} + 1166 q^{64} + 132 q^{66} - 236 q^{67} + 144 q^{68} + 600 q^{69} + 280 q^{70} - 544 q^{71} - 432 q^{72} + 900 q^{73} + 1536 q^{74} - 225 q^{75} + 1996 q^{76} - 44 q^{77} + 1524 q^{78} - 1586 q^{79} - 250 q^{80} + 243 q^{81} + 380 q^{82} - 1582 q^{83} + 1020 q^{84} + 1090 q^{85} + 3568 q^{86} - 204 q^{87} - 528 q^{88} - 2122 q^{89} + 180 q^{90} - 8 q^{91} - 4128 q^{92} + 204 q^{93} - 2152 q^{94} - 730 q^{95} + 2064 q^{96} + 618 q^{97} + 572 q^{98} + 297 q^{99}+O(q^{100}) \) 3 * q - 4 * q^2 - 9 * q^3 + 22 * q^4 - 15 * q^5 + 12 * q^6 - 4 * q^7 - 48 * q^8 + 27 * q^9 + 20 * q^10 + 33 * q^11 - 66 * q^12 - 56 * q^14 + 45 * q^15 + 50 * q^16 - 218 * q^17 - 36 * q^18 + 146 * q^19 - 110 * q^20 + 12 * q^21 - 44 * q^22 - 200 * q^23 + 144 * q^24 + 75 * q^25 - 508 * q^26 - 81 * q^27 - 340 * q^28 + 68 * q^29 - 60 * q^30 - 68 * q^31 - 688 * q^32 - 99 * q^33 - 176 * q^34 + 20 * q^35 + 198 * q^36 - 390 * q^37 - 316 * q^38 + 240 * q^40 - 196 * q^41 + 168 * q^42 - 524 * q^43 + 242 * q^44 - 135 * q^45 + 1160 * q^46 - 60 * q^47 - 150 * q^48 - 157 * q^49 - 100 * q^50 + 654 * q^51 + 1020 * q^52 - 158 * q^53 + 108 * q^54 - 165 * q^55 + 1368 * q^56 - 438 * q^57 + 1092 * q^58 - 1044 * q^59 + 330 * q^60 + 642 * q^61 + 88 * q^62 - 36 * q^63 + 1166 * q^64 + 132 * q^66 - 236 * q^67 + 144 * q^68 + 600 * q^69 + 280 * q^70 - 544 * q^71 - 432 * q^72 + 900 * q^73 + 1536 * q^74 - 225 * q^75 + 1996 * q^76 - 44 * q^77 + 1524 * q^78 - 1586 * q^79 - 250 * q^80 + 243 * q^81 + 380 * q^82 - 1582 * q^83 + 1020 * q^84 + 1090 * q^85 + 3568 * q^86 - 204 * q^87 - 528 * q^88 - 2122 * q^89 + 180 * q^90 - 8 * q^91 - 4128 * q^92 + 204 * q^93 - 2152 * q^94 - 730 * q^95 + 2064 * q^96 + 618 * q^97 + 572 * q^98 + 297 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	3.59056	1.26945	0.634727	−	0.772736i	$-0.281112\pi$
$2$	3.59056	1.26945	0.634727	+	0.772736i	$0.281112\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.89212	0.611515
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−10.7717	−0.732920
$7$	−16.1465	−0.871828	−0.435914	−	0.899988i	$-0.643575\pi$
$7$	−16.1465	−0.871828	−0.435914	+	0.899988i	$0.643575\pi$
$8$	−11.1590	−0.493164
$9$	9.00000	0.333333
$10$	−17.9528	−0.567717
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−14.6764	−0.353058
$13$	−54.1214	−1.15466	−0.577329	−	0.816511i	$-0.695905\pi$
$13$	−54.1214	−1.15466	−0.577329	+	0.816511i	$0.695905\pi$
$14$	−57.9749	−1.10675
$15$	15.0000	0.258199
$16$	−79.2041	−1.23756
$17$	−107.010	−1.52668	−0.763342	−	0.645995i	$-0.776443\pi$
$17$	−107.010	−1.52668	−0.763342	+	0.645995i	$0.776443\pi$
$18$	32.3150	0.423152
$19$	48.7496	0.588628	0.294314	−	0.955709i	$-0.404909\pi$
$19$	48.7496	0.588628	0.294314	+	0.955709i	$0.404909\pi$
$20$	−24.4606	−0.273478
$21$	48.4394	0.503350
$22$	39.4962	0.382755
$23$	11.9498	0.108335	0.0541674	−	0.998532i	$-0.482750\pi$
$23$	11.9498	0.108335	0.0541674	+	0.998532i	$0.482750\pi$
$24$	33.4771	0.284729
$25$	25.0000	0.200000
$26$	−194.326	−1.46579
$27$	−27.0000	−0.192450
$28$	−78.9905	−0.533136
$29$	239.733	1.53508	0.767538	−	0.641003i	$-0.221481\pi$
$29$	239.733	1.53508	0.767538	+	0.641003i	$0.221481\pi$
$30$	53.8584	0.327772
$31$	−82.0851	−0.475578	−0.237789	−	0.971317i	$-0.576423\pi$
$31$	−82.0851	−0.475578	−0.237789	+	0.971317i	$0.576423\pi$
$32$	−195.115	−1.07787
$33$	−33.0000	−0.174078
$34$	−384.224	−1.93806
$35$	80.7324	0.389893
$36$	44.0291	0.203838
$37$	−21.7573	−0.0966723	−0.0483361	−	0.998831i	$-0.515392\pi$
$37$	−21.7573	−0.0966723	−0.0483361	+	0.998831i	$0.515392\pi$
$38$	175.038	0.747236
$39$	162.364	0.666643
$40$	55.7952	0.220550
$41$	−124.835	−0.475510	−0.237755	−	0.971325i	$-0.576412\pi$
$41$	−124.835	−0.475510	−0.237755	+	0.971325i	$0.576412\pi$
$42$	173.925	0.638980
$43$	224.459	0.796039	0.398019	−	0.917377i	$-0.369698\pi$
$43$	224.459	0.796039	0.398019	+	0.917377i	$0.369698\pi$
$44$	53.8133	0.184379
$45$	−45.0000	−0.149071
$46$	42.9064	0.137526
$47$	−186.832	−0.579835	−0.289917	−	0.957052i	$-0.593628\pi$
$47$	−186.832	−0.579835	−0.289917	+	0.957052i	$0.593628\pi$
$48$	237.612	0.714508
$49$	−82.2913	−0.239916
$50$	89.7640	0.253891
$51$	321.029	0.881431
$52$	−264.768	−0.706091
$53$	233.997	0.606453	0.303226	−	0.952919i	$-0.401936\pi$
$53$	233.997	0.606453	0.303226	+	0.952919i	$0.401936\pi$
$54$	−96.9451	−0.244307
$55$	−55.0000	−0.134840
$56$	180.179	0.429954
$57$	−146.249	−0.339844
$58$	860.774	1.94871
$59$	232.936	0.513996	0.256998	−	0.966412i	$-0.417267\pi$
$59$	232.936	0.513996	0.256998	+	0.966412i	$0.417267\pi$
$60$	73.3818	0.157892
$61$	163.849	0.343913	0.171957	−	0.985105i	$-0.444991\pi$
$61$	163.849	0.343913	0.171957	+	0.985105i	$0.444991\pi$
$62$	−294.731	−0.603725
$63$	−145.318	−0.290609
$64$	−66.9386	−0.130739
$65$	270.607	0.516379
$66$	−118.488	−0.220984
$67$	−876.918	−1.59899	−0.799497	−	0.600670i	$-0.794900\pi$
$67$	−876.918	−1.59899	−0.799497	+	0.600670i	$0.794900\pi$
$68$	−523.503	−0.933590
$69$	−35.8493	−0.0625471
$70$	289.874	0.494952
$71$	−733.141	−1.22546	−0.612731	−	0.790291i	$-0.709929\pi$
$71$	−733.141	−1.22546	−0.612731	+	0.790291i	$0.709929\pi$
$72$	−100.431	−0.164388
$73$	1161.97	1.86299	0.931496	−	0.363750i	$-0.118504\pi$
$73$	1161.97	1.86299	0.931496	+	0.363750i	$0.118504\pi$
$74$	−78.1208	−0.122721
$75$	−75.0000	−0.115470
$76$	238.489	0.359954
$77$	−177.611	−0.262866
$78$	582.978	0.846272
$79$	−588.831	−0.838591	−0.419296	−	0.907850i	$-0.637723\pi$
$79$	−588.831	−0.838591	−0.419296	+	0.907850i	$0.637723\pi$
$80$	396.021	0.553456
$81$	81.0000	0.111111
$82$	−448.226	−0.603638
$83$	−1161.06	−1.53546	−0.767731	−	0.640772i	$-0.778614\pi$
$83$	−1161.06	−1.53546	−0.767731	+	0.640772i	$0.778614\pi$
$84$	236.971	0.307806
$85$	535.048	0.682754
$86$	805.933	1.01053
$87$	−719.198	−0.886277
$88$	−122.749	−0.148695
$89$	−1042.16	−1.24122	−0.620610	−	0.784120i	$-0.713115\pi$
$89$	−1042.16	−1.24122	−0.620610	+	0.784120i	$0.713115\pi$
$90$	−161.575	−0.189239
$91$	873.869	1.00666
$92$	58.4597	0.0662483
$93$	246.255	0.274575
$94$	−670.831	−0.736074
$95$	−243.748	−0.263242
$96$	585.345	0.622307
$97$	1546.63	1.61893	0.809464	−	0.587169i	$-0.199758\pi$
$97$	1546.63	1.61893	0.809464	+	0.587169i	$0.199758\pi$
$98$	−295.472	−0.304563
$99$	99.0000	0.100504
$100$	122.303	0.122303
$101$	−662.282	−0.652470	−0.326235	−	0.945289i	$-0.605780\pi$
$101$	−662.282	−0.652470	−0.326235	+	0.945289i	$0.605780\pi$
$102$	1152.67	1.11894
$103$	399.592	0.382262	0.191131	−	0.981565i	$-0.438784\pi$
$103$	399.592	0.382262	0.191131	+	0.981565i	$0.438784\pi$
$104$	603.942	0.569436
$105$	−242.197	−0.225105
$106$	840.181	0.769864
$107$	−1591.22	−1.43765	−0.718827	−	0.695189i	$-0.755321\pi$
$107$	−1591.22	−1.43765	−0.718827	+	0.695189i	$0.755321\pi$
$108$	−132.087	−0.117686
$109$	755.128	0.663561	0.331780	−	0.943357i	$-0.392351\pi$
$109$	755.128	0.663561	0.331780	+	0.943357i	$0.392351\pi$
$110$	−197.481	−0.171173
$111$	65.2718	0.0558138
$112$	1278.87	1.07894
$113$	−1145.65	−0.953753	−0.476876	−	0.878970i	$-0.658231\pi$
$113$	−1145.65	−0.953753	−0.476876	+	0.878970i	$0.658231\pi$
$114$	−525.115	−0.431417
$115$	−59.7488	−0.0484488
$116$	1172.80	0.938722
$117$	−487.092	−0.384886
$118$	836.372	0.652494
$119$	1727.83	1.33101
$120$	−167.386	−0.127334
$121$	121.000	0.0909091
$122$	588.309	0.436582
$123$	374.504	0.274536
$124$	−401.570	−0.290823
$125$	−125.000	−0.0894427
$126$	−521.774	−0.368915
$127$	1461.29	1.02101	0.510505	−	0.859875i	$-0.329458\pi$
$127$	1461.29	1.02101	0.510505	+	0.859875i	$0.329458\pi$
$128$	1320.57	0.911900
$129$	−673.377	−0.459593
$130$	971.630	0.655520
$131$	−1524.94	−1.01706	−0.508528	−	0.861045i	$-0.669810\pi$
$131$	−1524.94	−1.01706	−0.508528	+	0.861045i	$0.669810\pi$
$132$	−161.440	−0.106451
$133$	−787.134	−0.513182
$134$	−3148.63	−2.02985
$135$	135.000	0.0860663
$136$	1194.12	0.752906
$137$	−2125.68	−1.32561	−0.662805	−	0.748792i	$-0.730634\pi$
$137$	−2125.68	−1.32561	−0.662805	+	0.748792i	$0.730634\pi$
$138$	−128.719	−0.0794007
$139$	−1774.28	−1.08268	−0.541339	−	0.840805i	$-0.682082\pi$
$139$	−1774.28	−1.08268	−0.541339	+	0.840805i	$0.682082\pi$
$140$	394.952	0.238425
$141$	560.496	0.334768
$142$	−2632.39	−1.55567
$143$	−595.335	−0.348143
$144$	−712.837	−0.412521
$145$	−1198.66	−0.686507
$146$	4172.13	2.36498
$147$	246.874	0.138516
$148$	−106.439	−0.0591165
$149$	1575.78	0.866393	0.433197	−	0.901299i	$-0.357386\pi$
$149$	1575.78	0.866393	0.433197	+	0.901299i	$0.357386\pi$
$150$	−269.292	−0.146584
$151$	420.978	0.226879	0.113439	−	0.993545i	$-0.463813\pi$
$151$	420.978	0.226879	0.113439	+	0.993545i	$0.463813\pi$
$152$	−543.998	−0.290290
$153$	−963.086	−0.508895
$154$	−637.724	−0.333696
$155$	410.425	0.212685
$156$	794.304	0.407662
$157$	−2224.30	−1.13069	−0.565345	−	0.824854i	$-0.691257\pi$
$157$	−2224.30	−1.13069	−0.565345	+	0.824854i	$0.691257\pi$
$158$	−2114.23	−1.06455
$159$	−701.992	−0.350136
$160$	975.574	0.482037
$161$	−192.947	−0.0944492
$162$	290.835	0.141051
$163$	−3093.37	−1.48645	−0.743226	−	0.669040i	$-0.766705\pi$
$163$	−3093.37	−1.48645	−0.743226	+	0.669040i	$0.766705\pi$
$164$	−610.706	−0.290781
$165$	165.000	0.0778499
$166$	−4168.87	−1.94920
$167$	−2416.43	−1.11970	−0.559848	−	0.828595i	$-0.689140\pi$
$167$	−2416.43	−1.11970	−0.559848	+	0.828595i	$0.689140\pi$
$168$	−540.537	−0.248234
$169$	732.122	0.333237
$170$	1921.12	0.866725
$171$	438.746	0.196209
$172$	1098.08	0.486789
$173$	−3758.02	−1.65154	−0.825771	−	0.564005i	$-0.809260\pi$
$173$	−3758.02	−1.65154	−0.825771	+	0.564005i	$0.809260\pi$
$174$	−2582.32	−1.12509
$175$	−403.662	−0.174366
$176$	−871.245	−0.373140
$177$	−698.809	−0.296756
$178$	−3741.93	−1.57567
$179$	2533.99	1.05810	0.529049	−	0.848591i	$-0.322549\pi$
$179$	2533.99	1.05810	0.529049	+	0.848591i	$0.322549\pi$
$180$	−220.145	−0.0911592
$181$	−13.8995	−0.00570798	−0.00285399	−	0.999996i	$-0.500908\pi$
$181$	−13.8995	−0.00570798	−0.00285399	+	0.999996i	$0.500908\pi$
$182$	3137.68	1.27791
$183$	−491.547	−0.198558
$184$	−133.348	−0.0534268
$185$	108.786	0.0432332
$186$	884.194	0.348561
$187$	−1177.10	−0.460312
$188$	−914.004	−0.354577
$189$	435.955	0.167783
$190$	−875.192	−0.334174
$191$	3495.39	1.32417	0.662087	−	0.749427i	$-0.269671\pi$
$191$	3495.39	1.32417	0.662087	+	0.749427i	$0.269671\pi$
$192$	200.816	0.0754824
$193$	3469.33	1.29393	0.646963	−	0.762522i	$-0.276039\pi$
$193$	3469.33	1.29393	0.646963	+	0.762522i	$0.276039\pi$
$194$	5553.25	2.05516
$195$	−811.820	−0.298132
$196$	−402.579	−0.146712
$197$	3638.39	1.31586	0.657930	−	0.753079i	$-0.271432\pi$
$197$	3638.39	1.31586	0.657930	+	0.753079i	$0.271432\pi$
$198$	355.465	0.127585
$199$	51.6049	0.0183828	0.00919140	−	0.999958i	$-0.497074\pi$
$199$	51.6049	0.0183828	0.00919140	+	0.999958i	$0.497074\pi$
$200$	−278.976	−0.0986329
$201$	2630.75	0.923179
$202$	−2377.96	−0.828281
$203$	−3870.84	−1.33832
$204$	1570.51	0.539008
$205$	624.173	0.212654
$206$	1434.76	0.485265
$207$	107.548	0.0361116
$208$	4286.63	1.42896
$209$	536.246	0.177478
$210$	−869.623	−0.285761
$211$	2084.57	0.680131	0.340065	−	0.940402i	$-0.389551\pi$
$211$	2084.57	0.680131	0.340065	+	0.940402i	$0.389551\pi$
$212$	1144.74	0.370855
$213$	2199.42	0.707521
$214$	−5713.37	−1.82504
$215$	−1122.29	−0.355999
$216$	301.294	0.0949095
$217$	1325.38	0.414622
$218$	2711.33	0.842361
$219$	−3485.91	−1.07560
$220$	−269.067	−0.0824566
$221$	5791.50	1.76280
$222$	234.362	0.0708531
$223$	1887.36	0.566757	0.283378	−	0.959008i	$-0.408545\pi$
$223$	1887.36	0.566757	0.283378	+	0.959008i	$0.408545\pi$
$224$	3150.42	0.939715
$225$	225.000	0.0666667
$226$	−4113.54	−1.21075
$227$	−1150.73	−0.336462	−0.168231	−	0.985748i	$-0.553806\pi$
$227$	−1150.73	−0.336462	−0.168231	+	0.985748i	$0.553806\pi$
$228$	−715.466	−0.207820
$229$	4106.79	1.18508	0.592542	−	0.805540i	$-0.298125\pi$
$229$	4106.79	1.18508	0.592542	+	0.805540i	$0.298125\pi$
$230$	−214.532	−0.0615035
$231$	532.834	0.151766
$232$	−2675.18	−0.757045
$233$	−5733.58	−1.61210	−0.806050	−	0.591848i	$-0.798399\pi$
$233$	−5733.58	−1.61210	−0.806050	+	0.591848i	$0.798399\pi$
$234$	−1748.93	−0.488596
$235$	934.159	0.259310
$236$	1139.55	0.314316
$237$	1766.49	0.484161
$238$	6203.86	1.68965
$239$	6036.18	1.63367	0.816837	−	0.576868i	$-0.195725\pi$
$239$	6036.18	1.63367	0.816837	+	0.576868i	$0.195725\pi$
$240$	−1188.06	−0.319538
$241$	3720.90	0.994540	0.497270	−	0.867596i	$-0.334336\pi$
$241$	3720.90	0.994540	0.497270	+	0.867596i	$0.334336\pi$
$242$	434.458	0.115405
$243$	−243.000	−0.0641500
$244$	801.568	0.210308
$245$	411.457	0.107294
$246$	1344.68	0.348511
$247$	−2638.39	−0.679664
$248$	915.990	0.234538
$249$	3483.19	0.886500
$250$	−448.820	−0.113543
$251$	−3809.88	−0.958077	−0.479038	−	0.877794i	$-0.659015\pi$
$251$	−3809.88	−0.958077	−0.479038	+	0.877794i	$0.659015\pi$
$252$	−710.914	−0.177712
$253$	131.447	0.0326641
$254$	5246.84	1.29613
$255$	−1605.14	−0.394188
$256$	5277.10	1.28835
$257$	1225.95	0.297559	0.148780	−	0.988870i	$-0.452465\pi$
$257$	1225.95	0.297559	0.148780	+	0.988870i	$0.452465\pi$
$258$	−2417.80	−0.583433
$259$	351.303	0.0842816
$260$	1323.84	0.315773
$261$	2157.59	0.511692
$262$	−5475.38	−1.29111
$263$	7397.68	1.73445	0.867225	−	0.497916i	$-0.165901\pi$
$263$	7397.68	1.73445	0.867225	+	0.497916i	$0.165901\pi$
$264$	368.248	0.0858489
$265$	−1169.99	−0.271214
$266$	−2826.25	−0.651461
$267$	3126.47	0.716618
$268$	−4289.99	−0.977808
$269$	−3214.48	−0.728589	−0.364295	−	0.931284i	$-0.618690\pi$
$269$	−3214.48	−0.728589	−0.364295	+	0.931284i	$0.618690\pi$
$270$	484.726	0.109257
$271$	−7377.37	−1.65367	−0.826833	−	0.562448i	$-0.809860\pi$
$271$	−7377.37	−1.65367	−0.826833	+	0.562448i	$0.809860\pi$
$272$	8475.60	1.88937
$273$	−2621.61	−0.581197
$274$	−7632.36	−1.68280
$275$	275.000	0.0603023
$276$	−175.379	−0.0382485
$277$	−810.606	−0.175829	−0.0879144	−	0.996128i	$-0.528020\pi$
$277$	−810.606	−0.175829	−0.0879144	+	0.996128i	$0.528020\pi$
$278$	−6370.65	−1.37441
$279$	−738.766	−0.158526
$280$	−900.895	−0.192281
$281$	1114.72	0.236651	0.118325	−	0.992975i	$-0.462247\pi$
$281$	1114.72	0.236651	0.118325	+	0.992975i	$0.462247\pi$
$282$	2012.49	0.424972
$283$	2265.40	0.475844	0.237922	−	0.971284i	$-0.423534\pi$
$283$	2265.40	0.475844	0.237922	+	0.971284i	$0.423534\pi$
$284$	−3586.61	−0.749388
$285$	731.244	0.151983
$286$	−2137.59	−0.441951
$287$	2015.64	0.414563
$288$	−1756.03	−0.359289
$289$	6538.04	1.33076
$290$	−4303.87	−0.871490
$291$	−4639.88	−0.934689
$292$	5684.50	1.13925
$293$	3802.06	0.758084	0.379042	−	0.925380i	$-0.376254\pi$
$293$	3802.06	0.758084	0.379042	+	0.925380i	$0.376254\pi$
$294$	886.416	0.175840
$295$	−1164.68	−0.229866
$296$	242.790	0.0476753
$297$	−297.000	−0.0580259
$298$	5657.92	1.09985
$299$	−646.738	−0.125090
$300$	−366.909	−0.0706116
$301$	−3624.22	−0.694009
$302$	1511.55	0.288013
$303$	1986.85	0.376704
$304$	−3861.17	−0.728465
$305$	−819.244	−0.153803
$306$	−3458.02	−0.646019
$307$	−1356.35	−0.252153	−0.126077	−	0.992021i	$-0.540239\pi$
$307$	−1356.35	−0.252153	−0.126077	+	0.992021i	$0.540239\pi$
$308$	−868.895	−0.160746
$309$	−1198.78	−0.220699
$310$	1473.66	0.269994
$311$	−8078.07	−1.47288	−0.736440	−	0.676503i	$-0.763495\pi$
$311$	−8078.07	−1.47288	−0.736440	+	0.676503i	$0.763495\pi$
$312$	−1811.83	−0.328764
$313$	5761.54	1.04045	0.520226	−	0.854029i	$-0.325848\pi$
$313$	5761.54	1.04045	0.520226	+	0.854029i	$0.325848\pi$
$314$	−7986.48	−1.43536
$315$	726.591	0.129964
$316$	−2880.63	−0.512811
$317$	5107.00	0.904851	0.452426	−	0.891802i	$-0.350559\pi$
$317$	5107.00	0.904851	0.452426	+	0.891802i	$0.350559\pi$
$318$	−2520.54	−0.444481
$319$	2637.06	0.462843
$320$	334.693	0.0584684
$321$	4773.66	0.830030
$322$	−692.786	−0.119899
$323$	−5216.67	−0.898648
$324$	396.262	0.0679461
$325$	−1353.03	−0.230932
$326$	−11106.9	−1.88698
$327$	−2265.38	−0.383107
$328$	1393.03	0.234504
$329$	3016.68	0.505516
$330$	592.442	0.0988269
$331$	−2780.94	−0.461796	−0.230898	−	0.972978i	$-0.574166\pi$
$331$	−2780.94	−0.461796	−0.230898	+	0.972978i	$0.574166\pi$
$332$	−5680.06	−0.938958
$333$	−195.816	−0.0322241
$334$	−8676.35	−1.42140
$335$	4384.59	0.715092
$336$	−3836.60	−0.622928
$337$	4939.28	0.798397	0.399198	−	0.916865i	$-0.369288\pi$
$337$	4939.28	0.798397	0.399198	+	0.916865i	$0.369288\pi$
$338$	2628.73	0.423029
$339$	3436.96	0.550649
$340$	2617.52	0.417514
$341$	−902.936	−0.143392
$342$	1575.34	0.249079
$343$	6866.96	1.08099
$344$	−2504.74	−0.392578
$345$	179.247	0.0279719
$346$	−13493.4	−2.09656
$347$	−2711.58	−0.419496	−0.209748	−	0.977755i	$-0.567264\pi$
$347$	−2711.58	−0.419496	−0.209748	+	0.977755i	$0.567264\pi$
$348$	−3518.40	−0.541971
$349$	5496.03	0.842967	0.421484	−	0.906836i	$-0.361510\pi$
$349$	5496.03	0.842967	0.421484	+	0.906836i	$0.361510\pi$
$350$	−1449.37	−0.221349
$351$	1461.28	0.222214
$352$	−2146.26	−0.324989
$353$	6372.23	0.960792	0.480396	−	0.877052i	$-0.340493\pi$
$353$	6372.23	0.960792	0.480396	+	0.877052i	$0.340493\pi$
$354$	−2509.12	−0.376718
$355$	3665.71	0.548043
$356$	−5098.36	−0.759024
$357$	−5183.48	−0.768456
$358$	9098.46	1.34321
$359$	630.622	0.0927101	0.0463551	−	0.998925i	$-0.485239\pi$
$359$	630.622	0.0927101	0.0463551	+	0.998925i	$0.485239\pi$
$360$	502.157	0.0735166
$361$	−4482.48	−0.653518
$362$	−49.9071	−0.00724602
$363$	−363.000	−0.0524864
$364$	4275.07	0.615590
$365$	−5809.86	−0.833156
$366$	−1764.93	−0.252061
$367$	−5374.49	−0.764431	−0.382216	−	0.924073i	$-0.624839\pi$
$367$	−5374.49	−0.764431	−0.382216	+	0.924073i	$0.624839\pi$
$368$	−946.471	−0.134071
$369$	−1123.51	−0.158503
$370$	390.604	0.0548825
$371$	−3778.23	−0.528722
$372$	1204.71	0.167907
$373$	7520.12	1.04391	0.521953	−	0.852974i	$-0.325204\pi$
$373$	7520.12	1.04391	0.521953	+	0.852974i	$0.325204\pi$
$374$	−4226.46	−0.584346
$375$	375.000	0.0516398
$376$	2084.86	0.285954
$377$	−12974.7	−1.77249
$378$	1565.32	0.212993
$379$	−12509.1	−1.69538	−0.847690	−	0.530492i	$-0.822007\pi$
$379$	−12509.1	−1.69538	−0.847690	+	0.530492i	$0.822007\pi$
$380$	−1192.44	−0.160977
$381$	−4383.86	−0.589481
$382$	12550.4	1.68098
$383$	−11149.9	−1.48755	−0.743775	−	0.668430i	$-0.766967\pi$
$383$	−11149.9	−1.48755	−0.743775	+	0.668430i	$0.766967\pi$
$384$	−3961.72	−0.526486
$385$	888.056	0.117557
$386$	12456.8	1.64258
$387$	2020.13	0.265346
$388$	7566.28	0.989999
$389$	3194.22	0.416333	0.208166	−	0.978093i	$-0.433250\pi$
$389$	3194.22	0.416333	0.208166	+	0.978093i	$0.433250\pi$
$390$	−2914.89	−0.378465
$391$	−1278.74	−0.165393
$392$	918.292	0.118318
$393$	4574.81	0.587198
$394$	13063.8	1.67042
$395$	2944.16	0.375029
$396$	484.320	0.0614596
$397$	−584.410	−0.0738809	−0.0369404	−	0.999317i	$-0.511761\pi$
$397$	−584.410	−0.0738809	−0.0369404	+	0.999317i	$0.511761\pi$
$398$	185.291	0.0233361
$399$	2361.40	0.296286
$400$	−1980.10	−0.247513
$401$	−6951.24	−0.865657	−0.432829	−	0.901476i	$-0.642484\pi$
$401$	−6951.24	−0.865657	−0.432829	+	0.901476i	$0.642484\pi$
$402$	9445.88	1.17193
$403$	4442.56	0.549130
$404$	−3239.96	−0.398995
$405$	−405.000	−0.0496904
$406$	−13898.5	−1.69894
$407$	−239.330	−0.0291478
$408$	−3582.37	−0.434690
$409$	11754.1	1.42104	0.710519	−	0.703678i	$-0.248460\pi$
$409$	11754.1	1.42104	0.710519	+	0.703678i	$0.248460\pi$
$410$	2241.13	0.269955
$411$	6377.03	0.765342
$412$	1954.85	0.233759
$413$	−3761.10	−0.448116
$414$	386.157	0.0458420
$415$	5805.32	0.686680
$416$	10559.9	1.24457
$417$	5322.83	0.625084
$418$	1925.42	0.225300
$419$	11829.8	1.37929	0.689646	−	0.724146i	$-0.257766\pi$
$419$	11829.8	1.37929	0.689646	+	0.724146i	$0.257766\pi$
$420$	−1184.86	−0.137655
$421$	−2000.07	−0.231538	−0.115769	−	0.993276i	$-0.536933\pi$
$421$	−2000.07	−0.231538	−0.115769	+	0.993276i	$0.536933\pi$
$422$	7484.76	0.863395
$423$	−1681.49	−0.193278
$424$	−2611.18	−0.299081
$425$	−2675.24	−0.305337
$426$	7897.16	0.898166
$427$	−2645.58	−0.299833
$428$	−7784.43	−0.879147
$429$	1786.00	0.201000
$430$	−4029.67	−0.451925
$431$	−8064.11	−0.901240	−0.450620	−	0.892716i	$-0.648797\pi$
$431$	−8064.11	−0.901240	−0.450620	+	0.892716i	$0.648797\pi$
$432$	2138.51	0.238169
$433$	−10710.3	−1.18869	−0.594345	−	0.804210i	$-0.702589\pi$
$433$	−10710.3	−1.18869	−0.594345	+	0.804210i	$0.702589\pi$
$434$	4758.87	0.526344
$435$	3595.99	0.396355
$436$	3694.18	0.405777
$437$	582.546	0.0637688
$438$	−12516.4	−1.36542
$439$	4658.08	0.506419	0.253210	−	0.967411i	$-0.418514\pi$
$439$	4658.08	0.506419	0.253210	+	0.967411i	$0.418514\pi$
$440$	613.747	0.0664983
$441$	−740.622	−0.0799721
$442$	20794.7	2.23779
$443$	2094.73	0.224658	0.112329	−	0.993671i	$-0.464169\pi$
$443$	2094.73	0.224658	0.112329	+	0.993671i	$0.464169\pi$
$444$	319.318	0.0341310
$445$	5210.79	0.555090
$446$	6776.67	0.719472
$447$	−4727.33	−0.500212
$448$	1080.82	0.113982
$449$	−1402.21	−0.147382	−0.0736909	−	0.997281i	$-0.523478\pi$
$449$	−1402.21	−0.147382	−0.0736909	+	0.997281i	$0.523478\pi$
$450$	807.876	0.0846303
$451$	−1373.18	−0.143372
$452$	−5604.67	−0.583234
$453$	−1262.93	−0.130989
$454$	−4131.78	−0.427124
$455$	−4369.35	−0.450194
$456$	1632.00	0.167599
$457$	−4914.19	−0.503011	−0.251506	−	0.967856i	$-0.580926\pi$
$457$	−4914.19	−0.503011	−0.251506	+	0.967856i	$0.580926\pi$
$458$	14745.7	1.50441
$459$	2889.26	0.293810
$460$	−292.298	−0.0296271
$461$	−2214.08	−0.223688	−0.111844	−	0.993726i	$-0.535676\pi$
$461$	−2214.08	−0.223688	−0.111844	+	0.993726i	$0.535676\pi$
$462$	1913.17	0.192660
$463$	−5567.02	−0.558793	−0.279396	−	0.960176i	$-0.590134\pi$
$463$	−5567.02	−0.558793	−0.279396	+	0.960176i	$0.590134\pi$
$464$	−18987.8	−1.89976
$465$	−1231.28	−0.122794
$466$	−20586.8	−2.04649
$467$	497.054	0.0492525	0.0246263	−	0.999697i	$-0.492160\pi$
$467$	497.054	0.0492525	0.0246263	+	0.999697i	$0.492160\pi$
$468$	−2382.91	−0.235364
$469$	14159.1	1.39405
$470$	3354.15	0.329182
$471$	6672.90	0.652804
$472$	−2599.35	−0.253484
$473$	2469.05	0.240015
$474$	6342.70	0.614620
$475$	1218.74	0.117726
$476$	8452.73	0.813929
$477$	2105.97	0.202151
$478$	21673.3	2.07388
$479$	−9349.28	−0.891815	−0.445908	−	0.895079i	$-0.647119\pi$
$479$	−9349.28	−0.891815	−0.445908	+	0.895079i	$0.647119\pi$
$480$	−2926.72	−0.278304
$481$	1177.53	0.111624
$482$	13360.1	1.26252
$483$	578.840	0.0545303
$484$	591.946	0.0555923
$485$	−7733.13	−0.724007
$486$	−872.506	−0.0814355
$487$	197.750	0.0184003	0.00920013	−	0.999958i	$-0.497071\pi$
$487$	197.750	0.0184003	0.00920013	+	0.999958i	$0.497071\pi$
$488$	−1828.40	−0.169606
$489$	9280.12	0.858204
$490$	1477.36	0.136205
$491$	−9997.05	−0.918861	−0.459430	−	0.888214i	$-0.651946\pi$
$491$	−9997.05	−0.918861	−0.459430	+	0.888214i	$0.651946\pi$
$492$	1832.12	0.167883
$493$	−25653.7	−2.34358
$494$	−9473.31	−0.862803
$495$	−495.000	−0.0449467
$496$	6501.48	0.588558
$497$	11837.6	1.06839
$498$	12506.6	1.12537
$499$	−8714.73	−0.781813	−0.390907	−	0.920430i	$-0.627838\pi$
$499$	−8714.73	−0.781813	−0.390907	+	0.920430i	$0.627838\pi$
$500$	−611.515	−0.0546955
$501$	7249.30	0.646457
$502$	−13679.6	−1.21623
$503$	5978.53	0.529959	0.264979	−	0.964254i	$-0.414635\pi$
$503$	5978.53	0.529959	0.264979	+	0.964254i	$0.414635\pi$
$504$	1621.61	0.143318
$505$	3311.41	0.291794
$506$	471.970	0.0414657
$507$	−2196.36	−0.192394
$508$	7148.79	0.624363
$509$	−8205.79	−0.714569	−0.357284	−	0.933996i	$-0.616297\pi$
$509$	−8205.79	−0.714569	−0.357284	+	0.933996i	$0.616297\pi$
$510$	−5763.36	−0.500404
$511$	−18761.7	−1.62421
$512$	8383.17	0.723608
$513$	−1316.24	−0.113281
$514$	4401.85	0.377738
$515$	−1997.96	−0.170953
$516$	−3294.24	−0.281048
$517$	−2055.15	−0.174827
$518$	1261.38	0.106992
$519$	11274.1	0.953519
$520$	−3019.71	−0.254660
$521$	−5266.06	−0.442822	−0.221411	−	0.975181i	$-0.571066\pi$
$521$	−5266.06	−0.442822	−0.221411	+	0.975181i	$0.571066\pi$
$522$	7746.97	0.649570
$523$	−22398.6	−1.87270	−0.936350	−	0.351068i	$-0.885819\pi$
$523$	−22398.6	−1.87270	−0.936350	+	0.351068i	$0.885819\pi$
$524$	−7460.18	−0.621945
$525$	1210.99	0.100670
$526$	26561.8	2.20181
$527$	8783.89	0.726057
$528$	2613.74	0.215432
$529$	−12024.2	−0.988264
$530$	−4200.90	−0.344294
$531$	2096.43	0.171332
$532$	−3850.75	−0.313818
$533$	6756.22	0.549052
$534$	11225.8	0.909714
$535$	7956.10	0.642938
$536$	9785.56	0.788566
$537$	−7601.98	−0.610893
$538$	−11541.8	−0.924911
$539$	−905.205	−0.0723375
$540$	660.436	0.0526308
$541$	−13030.2	−1.03551	−0.517757	−	0.855528i	$-0.673233\pi$
$541$	−13030.2	−1.03551	−0.517757	+	0.855528i	$0.673233\pi$
$542$	−26488.9	−2.09925
$543$	41.6986	0.00329550
$544$	20879.1	1.64556
$545$	−3775.64	−0.296753
$546$	−9413.04	−0.737804
$547$	10448.9	0.816747	0.408374	−	0.912815i	$-0.366096\pi$
$547$	10448.9	0.816747	0.408374	+	0.912815i	$0.366096\pi$
$548$	−10399.1	−0.810631
$549$	1474.64	0.114638
$550$	987.404	0.0765510
$551$	11686.9	0.903588
$552$	400.044	0.0308460
$553$	9507.55	0.731107
$554$	−2910.53	−0.223207
$555$	−326.359	−0.0249607
$556$	−8679.97	−0.662073
$557$	−11448.7	−0.870911	−0.435456	−	0.900210i	$-0.643413\pi$
$557$	−11448.7	−0.870911	−0.435456	+	0.900210i	$0.643413\pi$
$558$	−2652.58	−0.201242
$559$	−12148.0	−0.919153
$560$	−6394.34	−0.482518
$561$	3531.31	0.265762
$562$	4002.49	0.300418
$563$	−26035.8	−1.94898	−0.974492	−	0.224422i	$-0.927951\pi$
$563$	−26035.8	−1.94898	−0.974492	+	0.224422i	$0.927951\pi$
$564$	2742.01	0.204715
$565$	5728.27	0.426531
$566$	8134.05	0.604063
$567$	−1307.86	−0.0968697
$568$	8181.15	0.604354
$569$	25075.0	1.84745	0.923725	−	0.383057i	$-0.125129\pi$
$569$	25075.0	1.84745	0.923725	+	0.383057i	$0.125129\pi$
$570$	2625.57	0.192935
$571$	19056.6	1.39666	0.698330	−	0.715776i	$-0.253927\pi$
$571$	19056.6	1.39666	0.698330	+	0.715776i	$0.253927\pi$
$572$	−2912.45	−0.212894
$573$	−10486.2	−0.764512
$574$	7237.28	0.526268
$575$	298.744	0.0216669
$576$	−602.447	−0.0435798
$577$	−6482.55	−0.467716	−0.233858	−	0.972271i	$-0.575135\pi$
$577$	−6482.55	−0.467716	−0.233858	+	0.972271i	$0.575135\pi$
$578$	23475.2	1.68934
$579$	−10408.0	−0.747048
$580$	−5864.00	−0.419809
$581$	18747.1	1.33866
$582$	−16659.8	−1.18654
$583$	2573.97	0.182852
$584$	−12966.5	−0.918762
$585$	2435.46	0.172126
$586$	13651.5	0.962353
$587$	24650.3	1.73327	0.866633	−	0.498946i	$-0.166280\pi$
$587$	24650.3	1.73327	0.866633	+	0.498946i	$0.166280\pi$
$588$	1207.74	0.0847045
$589$	−4001.61	−0.279938
$590$	−4181.86	−0.291804
$591$	−10915.2	−0.759712
$592$	1723.27	0.119638
$593$	23163.2	1.60405	0.802024	−	0.597292i	$-0.203757\pi$
$593$	23163.2	1.60405	0.802024	+	0.597292i	$0.203757\pi$
$594$	−1066.40	−0.0736612
$595$	−8639.13	−0.595244
$596$	7708.88	0.529812
$597$	−154.815	−0.0106133
$598$	−2322.15	−0.158796
$599$	5355.44	0.365304	0.182652	−	0.983178i	$-0.441532\pi$
$599$	5355.44	0.365304	0.182652	+	0.983178i	$0.441532\pi$
$600$	836.928	0.0569457
$601$	−20417.6	−1.38578	−0.692889	−	0.721045i	$-0.743662\pi$
$601$	−20417.6	−1.38578	−0.692889	+	0.721045i	$0.743662\pi$
$602$	−13013.0	−0.881012
$603$	−7892.26	−0.532998
$604$	2059.48	0.138740
$605$	−605.000	−0.0406558
$606$	7133.89	0.478208
$607$	−6749.81	−0.451345	−0.225673	−	0.974203i	$-0.572458\pi$
$607$	−6749.81	−0.451345	−0.225673	+	0.974203i	$0.572458\pi$
$608$	−9511.77	−0.634463
$609$	11612.5	0.772681
$610$	−2941.55	−0.195245
$611$	10111.6	0.669511
$612$	−4711.53	−0.311197
$613$	−30321.0	−1.99780	−0.998901	−	0.0468613i	$-0.985078\pi$
$613$	−30321.0	−1.99780	−0.998901	+	0.0468613i	$0.985078\pi$
$614$	−4870.06	−0.320097
$615$	−1872.52	−0.122776
$616$	1981.97	0.129636
$617$	15236.8	0.994182	0.497091	−	0.867699i	$-0.334402\pi$
$617$	15236.8	0.994182	0.497091	+	0.867699i	$0.334402\pi$
$618$	−4304.28	−0.280168
$619$	−20875.4	−1.35550	−0.677749	−	0.735293i	$-0.737044\pi$
$619$	−20875.4	−1.35550	−0.677749	+	0.735293i	$0.737044\pi$
$620$	2007.85	0.130060
$621$	−322.644	−0.0208490
$622$	−29004.8	−1.86975
$623$	16827.2	1.08213
$624$	−12859.9	−0.825013
$625$	625.000	0.0400000
$626$	20687.2	1.32081
$627$	−1608.74	−0.102467
$628$	−10881.5	−0.691434
$629$	2328.24	0.147588
$630$	2608.87	0.164984
$631$	27966.1	1.76437	0.882183	−	0.470907i	$-0.156073\pi$
$631$	27966.1	1.76437	0.882183	+	0.470907i	$0.156073\pi$
$632$	6570.79	0.413563
$633$	−6253.70	−0.392674
$634$	18337.0	1.14867
$635$	−7306.44	−0.456610
$636$	−3434.23	−0.214113
$637$	4453.72	0.277022
$638$	9468.52	0.587558
$639$	−6598.27	−0.408487
$640$	−6602.86	−0.407814
$641$	−17992.0	−1.10865	−0.554323	−	0.832301i	$-0.687023\pi$
$641$	−17992.0	−1.10865	−0.554323	+	0.832301i	$0.687023\pi$
$642$	17140.1	1.05369
$643$	−9448.64	−0.579499	−0.289750	−	0.957102i	$-0.593572\pi$
$643$	−9448.64	−0.579499	−0.289750	+	0.957102i	$0.593572\pi$
$644$	−943.918	−0.0577571
$645$	3366.88	0.205536
$646$	−18730.8	−1.14079
$647$	−7429.22	−0.451426	−0.225713	−	0.974194i	$-0.572471\pi$
$647$	−7429.22	−0.451426	−0.225713	+	0.974194i	$0.572471\pi$
$648$	−903.882	−0.0547960
$649$	2562.30	0.154976
$650$	−4858.15	−0.293157
$651$	−3976.15	−0.239382
$652$	−15133.2	−0.908988
$653$	−4488.63	−0.268995	−0.134497	−	0.990914i	$-0.542942\pi$
$653$	−4488.63	−0.268995	−0.134497	+	0.990914i	$0.542942\pi$
$654$	−8134.00	−0.486337
$655$	7624.69	0.454842
$656$	9887.42	0.588474
$657$	10457.7	0.620998
$658$	10831.6	0.641730
$659$	−25326.7	−1.49710	−0.748550	−	0.663078i	$-0.769250\pi$
$659$	−25326.7	−1.49710	−0.748550	+	0.663078i	$0.769250\pi$
$660$	807.200	0.0476064
$661$	15192.3	0.893969	0.446984	−	0.894542i	$-0.352498\pi$
$661$	15192.3	0.893969	0.446984	+	0.894542i	$0.352498\pi$
$662$	−9985.15	−0.586229
$663$	−17374.5	−1.01775
$664$	12956.4	0.757235
$665$	3935.67	0.229502
$666$	−703.087	−0.0409070
$667$	2864.75	0.166302
$668$	−11821.5	−0.684711
$669$	−5662.07	−0.327217
$670$	15743.1	0.907776
$671$	1802.34	0.103694
$672$	−9451.25	−0.542545
$673$	11718.2	0.671180	0.335590	−	0.942008i	$-0.391064\pi$
$673$	11718.2	0.671180	0.335590	+	0.942008i	$0.391064\pi$
$674$	17734.8	1.01353
$675$	−675.000	−0.0384900
$676$	3581.63	0.203779
$677$	−15649.1	−0.888397	−0.444198	−	0.895928i	$-0.646511\pi$
$677$	−15649.1	−0.888397	−0.444198	+	0.895928i	$0.646511\pi$
$678$	12340.6	0.699024
$679$	−24972.6	−1.41143
$680$	−5970.61	−0.336710
$681$	3452.20	0.194257
$682$	−3242.05	−0.182030
$683$	−18162.8	−1.01754	−0.508770	−	0.860903i	$-0.669899\pi$
$683$	−18162.8	−1.01754	−0.508770	+	0.860903i	$0.669899\pi$
$684$	2146.40	0.119985
$685$	10628.4	0.592831
$686$	24656.2	1.37227
$687$	−12320.4	−0.684208
$688$	−17778.1	−0.985149
$689$	−12664.2	−0.700246
$690$	643.595	0.0355091
$691$	29606.7	1.62995	0.814973	−	0.579500i	$-0.196752\pi$
$691$	29606.7	1.62995	0.814973	+	0.579500i	$0.196752\pi$
$692$	−18384.7	−1.00994
$693$	−1598.50	−0.0876220
$694$	−9736.08	−0.532531
$695$	8871.38	0.484188
$696$	8025.55	0.437080
$697$	13358.5	0.725953
$698$	19733.8	1.07011
$699$	17200.7	0.930746
$700$	−1974.76	−0.106627
$701$	−26164.7	−1.40974	−0.704870	−	0.709336i	$-0.748995\pi$
$701$	−26164.7	−1.40974	−0.704870	+	0.709336i	$0.748995\pi$
$702$	5246.80	0.282091
$703$	−1060.66	−0.0569040
$704$	−736.324	−0.0394194
$705$	−2802.48	−0.149713
$706$	22879.9	1.21968
$707$	10693.5	0.568842
$708$	−3418.66	−0.181470
$709$	−14508.9	−0.768535	−0.384268	−	0.923222i	$-0.625546\pi$
$709$	−14508.9	−0.768535	−0.384268	+	0.923222i	$0.625546\pi$
$710$	13161.9	0.695716
$711$	−5299.48	−0.279530
$712$	11629.5	0.612125
$713$	−980.898	−0.0515216
$714$	−18611.6	−0.975520
$715$	2976.67	0.155694
$716$	12396.6	0.647043
$717$	−18108.5	−0.943202
$718$	2264.28	0.117691
$719$	−2545.80	−0.132048	−0.0660239	−	0.997818i	$-0.521031\pi$
$719$	−2545.80	−0.132048	−0.0660239	+	0.997818i	$0.521031\pi$
$720$	3564.19	0.184485
$721$	−6452.01	−0.333267
$722$	−16094.6	−0.829611
$723$	−11162.7	−0.574198
$724$	−67.9982	−0.00349051
$725$	5993.31	0.307015
$726$	−1303.37	−0.0666291
$727$	−18984.8	−0.968511	−0.484255	−	0.874927i	$-0.660909\pi$
$727$	−18984.8	−0.968511	−0.484255	+	0.874927i	$0.660909\pi$
$728$	−9751.54	−0.496451
$729$	729.000	0.0370370
$730$	−20860.6	−1.05765
$731$	−24019.2	−1.21530
$732$	−2404.70	−0.121421
$733$	36740.4	1.85134	0.925672	−	0.378326i	$-0.123500\pi$
$733$	36740.4	1.85134	0.925672	+	0.378326i	$0.123500\pi$
$734$	−19297.4	−0.970411
$735$	−1234.37	−0.0619461
$736$	−2331.58	−0.116770
$737$	−9646.10	−0.482115
$738$	−4034.04	−0.201213
$739$	1755.17	0.0873682	0.0436841	−	0.999045i	$-0.486090\pi$
$739$	1755.17	0.0873682	0.0436841	+	0.999045i	$0.486090\pi$
$740$	532.196	0.0264377
$741$	7915.18	0.392404
$742$	−13566.0	−0.671189
$743$	17972.6	0.887419	0.443709	−	0.896171i	$-0.353662\pi$
$743$	17972.6	0.887419	0.443709	+	0.896171i	$0.353662\pi$
$744$	−2747.97	−0.135411
$745$	−7878.88	−0.387463
$746$	27001.4	1.32519
$747$	−10449.6	−0.511821
$748$	−5758.54	−0.281488
$749$	25692.6	1.25339
$750$	1346.46	0.0655544
$751$	22704.9	1.10321	0.551606	−	0.834105i	$-0.314015\pi$
$751$	22704.9	1.10321	0.551606	+	0.834105i	$0.314015\pi$
$752$	14797.9	0.717583
$753$	11429.6	0.553146
$754$	−46586.3	−2.25010
$755$	−2104.89	−0.101463
$756$	2132.74	0.102602
$757$	39860.3	1.91380	0.956901	−	0.290414i	$-0.0937929\pi$
$757$	39860.3	1.91380	0.956901	+	0.290414i	$0.0937929\pi$
$758$	−44914.6	−2.15221
$759$	−394.342	−0.0188587
$760$	2719.99	0.129822
$761$	19631.4	0.935135	0.467568	−	0.883957i	$-0.345130\pi$
$761$	19631.4	0.935135	0.467568	+	0.883957i	$0.345130\pi$
$762$	−15740.5	−0.748319
$763$	−12192.7	−0.578511
$764$	17099.8	0.809752
$765$	4815.43	0.227585
$766$	−40034.3	−1.88838
$767$	−12606.8	−0.593490
$768$	−15831.3	−0.743832
$769$	35029.9	1.64267	0.821333	−	0.570449i	$-0.193231\pi$
$769$	35029.9	1.64267	0.821333	+	0.570449i	$0.193231\pi$
$770$	3188.62	0.149234
$771$	−3677.86	−0.171796
$772$	16972.4	0.791255
$773$	−26736.9	−1.24406	−0.622032	−	0.782992i	$-0.713693\pi$
$773$	−26736.9	−1.24406	−0.622032	+	0.782992i	$0.713693\pi$
$774$	7253.40	0.336845
$775$	−2052.13	−0.0951156
$776$	−17258.8	−0.798398
$777$	−1053.91	−0.0486600
$778$	11469.0	0.528515
$779$	−6085.64	−0.279898
$780$	−3971.52	−0.182312
$781$	−8064.55	−0.369491
$782$	−4591.39	−0.209959
$783$	−6472.78	−0.295426
$784$	6517.81	0.296912
$785$	11121.5	0.505660
$786$	16426.1	0.745421
$787$	4799.45	0.217385	0.108693	−	0.994075i	$-0.465334\pi$
$787$	4799.45	0.217385	0.108693	+	0.994075i	$0.465334\pi$
$788$	17799.4	0.804668
$789$	−22193.0	−1.00139
$790$	10571.2	0.476083
$791$	18498.3	0.831508
$792$	−1104.74	−0.0495649
$793$	−8867.72	−0.397102
$794$	−2098.36	−0.0937884
$795$	3509.96	0.156585
$796$	252.457	0.0112413
$797$	38438.9	1.70838	0.854189	−	0.519963i	$-0.174054\pi$
$797$	38438.9	1.70838	0.854189	+	0.519963i	$0.174054\pi$
$798$	8478.76	0.376121
$799$	19992.8	0.885224
$800$	−4877.87	−0.215574
$801$	−9379.42	−0.413740
$802$	−24958.9	−1.09891
$803$	12781.7	0.561713
$804$	12870.0	0.564538
$805$	964.733	0.0422390
$806$	15951.3	0.697096
$807$	9643.45	0.420651
$808$	7390.42	0.321775
$809$	9960.91	0.432889	0.216444	−	0.976295i	$-0.430554\pi$
$809$	9960.91	0.432889	0.216444	+	0.976295i	$0.430554\pi$
$810$	−1454.18	−0.0630797
$811$	−34199.9	−1.48079	−0.740396	−	0.672171i	$-0.765362\pi$
$811$	−34199.9	−1.48079	−0.740396	+	0.672171i	$0.765362\pi$
$812$	−18936.6	−0.818404
$813$	22132.1	0.954744
$814$	−859.329	−0.0370018
$815$	15466.9	0.664762
$816$	−25426.8	−1.09083
$817$	10942.3	0.468570
$818$	42203.9	1.80394
$819$	7864.82	0.335555
$820$	3053.53	0.130041
$821$	−31796.4	−1.35165	−0.675823	−	0.737064i	$-0.736212\pi$
$821$	−31796.4	−1.35165	−0.675823	+	0.737064i	$0.736212\pi$
$822$	22897.1	0.971567
$823$	−10783.6	−0.456733	−0.228366	−	0.973575i	$-0.573338\pi$
$823$	−10783.6	−0.456733	−0.228366	+	0.973575i	$0.573338\pi$
$824$	−4459.07	−0.188518
$825$	−825.000	−0.0348155
$826$	−13504.5	−0.568862
$827$	44197.5	1.85840	0.929201	−	0.369576i	$-0.120497\pi$
$827$	44197.5	1.85840	0.929201	+	0.369576i	$0.120497\pi$
$828$	526.137	0.0220828
$829$	22487.7	0.942137	0.471069	−	0.882097i	$-0.343868\pi$
$829$	22487.7	0.942137	0.471069	+	0.882097i	$0.343868\pi$
$830$	20844.4	0.871709
$831$	2431.82	0.101515
$832$	3622.81	0.150959
$833$	8805.96	0.366276
$834$	19111.9	0.793516
$835$	12082.2	0.500743
$836$	2623.38	0.108530
$837$	2216.30	0.0915250
$838$	42475.6	1.75095
$839$	22491.0	0.925477	0.462738	−	0.886495i	$-0.346867\pi$
$839$	22491.0	0.925477	0.462738	+	0.886495i	$0.346867\pi$
$840$	2702.69	0.111014
$841$	33082.7	1.35646
$842$	−7181.37	−0.293927
$843$	−3344.17	−0.136630
$844$	10198.0	0.415910
$845$	−3660.61	−0.149028
$846$	−6037.48	−0.245358
$847$	−1953.72	−0.0792571
$848$	−18533.5	−0.750524
$849$	−6796.19	−0.274729
$850$	−9605.60	−0.387611
$851$	−259.994	−0.0104730
$852$	10759.8	0.432660
$853$	22327.9	0.896241	0.448120	−	0.893973i	$-0.352094\pi$
$853$	22327.9	0.896241	0.448120	+	0.893973i	$0.352094\pi$
$854$	−9499.12	−0.380624
$855$	−2193.73	−0.0877474
$856$	17756.5	0.708999
$857$	−14505.9	−0.578193	−0.289096	−	0.957300i	$-0.593355\pi$
$857$	−14505.9	−0.578193	−0.289096	+	0.957300i	$0.593355\pi$
$858$	6412.76	0.255161
$859$	−8411.45	−0.334104	−0.167052	−	0.985948i	$-0.553425\pi$
$859$	−8411.45	−0.334104	−0.167052	+	0.985948i	$0.553425\pi$
$860$	−5490.40	−0.217699
$861$	−6046.92	−0.239348
$862$	−28954.7	−1.14408
$863$	32499.6	1.28192	0.640961	−	0.767573i	$-0.278536\pi$
$863$	32499.6	1.28192	0.640961	+	0.767573i	$0.278536\pi$
$864$	5268.10	0.207436
$865$	18790.1	0.738592
$866$	−38455.9	−1.50899
$867$	−19614.1	−0.768316
$868$	6483.94	0.253548
$869$	−6477.15	−0.252845
$870$	12911.6	0.503155
$871$	47460.0	1.84629
$872$	−8426.50	−0.327245
$873$	13919.6	0.539643
$874$	2091.67	0.0809516
$875$	2018.31	0.0779786
$876$	−17053.5	−0.657745
$877$	−32183.1	−1.23916	−0.619581	−	0.784933i	$-0.712697\pi$
$877$	−32183.1	−1.23916	−0.619581	+	0.784933i	$0.712697\pi$
$878$	16725.1	0.642876
$879$	−11406.2	−0.437680
$880$	4356.23	0.166873
$881$	−6246.34	−0.238870	−0.119435	−	0.992842i	$-0.538108\pi$
$881$	−6246.34	−0.238870	−0.119435	+	0.992842i	$0.538108\pi$
$882$	−2659.25	−0.101521
$883$	−11801.1	−0.449762	−0.224881	−	0.974386i	$-0.572199\pi$
$883$	−11801.1	−0.449762	−0.224881	+	0.974386i	$0.572199\pi$
$884$	28332.7	1.07798
$885$	3494.05	0.132713
$886$	7521.24	0.285193
$887$	32375.1	1.22553	0.612767	−	0.790264i	$-0.290056\pi$
$887$	32375.1	1.22553	0.612767	+	0.790264i	$0.290056\pi$
$888$	−728.371	−0.0275254
$889$	−23594.6	−0.890145
$890$	18709.7	0.704662
$891$	891.000	0.0335013
$892$	9233.17	0.346580
$893$	−9107.98	−0.341307
$894$	−16973.8	−0.634997
$895$	−12670.0	−0.473196
$896$	−21322.6	−0.795020
$897$	1940.21	0.0722205
$898$	−5034.72	−0.187095
$899$	−19678.5	−0.730049
$900$	1100.73	0.0407677
$901$	−25039.9	−0.925861
$902$	−4930.49	−0.182004
$903$	10872.7	0.400686
$904$	12784.4	0.470357
$905$	69.4977	0.00255269
$906$	−4534.64	−0.166284
$907$	−19592.8	−0.717277	−0.358638	−	0.933477i	$-0.616759\pi$
$907$	−19592.8	−0.717277	−0.358638	+	0.933477i	$0.616759\pi$
$908$	−5629.53	−0.205752
$909$	−5960.54	−0.217490
$910$	−15688.4	−0.571500
$911$	18673.8	0.679133	0.339567	−	0.940582i	$-0.389720\pi$
$911$	18673.8	0.679133	0.339567	+	0.940582i	$0.389720\pi$
$912$	11583.5	0.420579
$913$	−12771.7	−0.462959
$914$	−17644.7	−0.638550
$915$	2457.73	0.0887980
$916$	20090.9	0.724696
$917$	24622.4	0.886698
$918$	10374.1	0.372979
$919$	−4572.90	−0.164142	−0.0820708	−	0.996627i	$-0.526153\pi$
$919$	−4572.90	−0.164142	−0.0820708	+	0.996627i	$0.526153\pi$
$920$	666.739	0.0238932
$921$	4069.05	0.145581
$922$	−7949.78	−0.283961
$923$	39678.6	1.41499
$924$	2606.69	0.0928070
$925$	−543.932	−0.0193345
$926$	−19988.7	−0.709362
$927$	3596.33	0.127421
$928$	−46775.4	−1.65461
$929$	−44222.1	−1.56176	−0.780882	−	0.624679i	$-0.785230\pi$
$929$	−44222.1	−1.56176	−0.780882	+	0.624679i	$0.785230\pi$
$930$	−4420.97	−0.155881
$931$	−4011.67	−0.141221
$932$	−28049.3	−0.985823
$933$	24234.2	0.850367
$934$	1784.70	0.0625238
$935$	5885.52	0.205858
$936$	5435.48	0.189812
$937$	9218.62	0.321408	0.160704	−	0.987003i	$-0.448624\pi$
$937$	9218.62	0.321408	0.160704	+	0.987003i	$0.448624\pi$
$938$	50839.2	1.76968
$939$	−17284.6	−0.600705
$940$	4570.02	0.158572
$941$	−26484.9	−0.917516	−0.458758	−	0.888561i	$-0.651706\pi$
$941$	−26484.9	−0.917516	−0.458758	+	0.888561i	$0.651706\pi$
$942$	23959.4	0.828705
$943$	−1491.75	−0.0515142
$944$	−18449.5	−0.636103
$945$	−2179.77	−0.0750350
$946$	8865.26	0.304688
$947$	−44972.4	−1.54320	−0.771599	−	0.636110i	$-0.780543\pi$
$947$	−44972.4	−1.54320	−0.771599	+	0.636110i	$0.780543\pi$
$948$	8641.90	0.296072
$949$	−62887.5	−2.15112
$950$	4375.96	0.149447
$951$	−15321.0	−0.522416
$952$	−19280.9	−0.656404
$953$	−2052.50	−0.0697659	−0.0348829	−	0.999391i	$-0.511106\pi$
$953$	−2052.50	−0.0697659	−0.0348829	+	0.999391i	$0.511106\pi$
$954$	7561.63	0.256621
$955$	−17476.9	−0.592189
$956$	29529.7	0.999016
$957$	−7911.18	−0.267223
$958$	−33569.2	−1.13212
$959$	34322.2	1.15570
$960$	−1004.08	−0.0337568
$961$	−23053.0	−0.773826
$962$	4228.00	0.141701
$963$	−14321.0	−0.479218
$964$	18203.1	0.608176
$965$	−17346.6	−0.578661
$966$	2078.36	0.0692237
$967$	14950.9	0.497195	0.248598	−	0.968607i	$-0.420030\pi$
$967$	14950.9	0.497195	0.248598	+	0.968607i	$0.420030\pi$
$968$	−1350.24	−0.0448331
$969$	15650.0	0.518835
$970$	−27766.3	−0.919094
$971$	26751.9	0.884151	0.442076	−	0.896978i	$-0.354242\pi$
$971$	26751.9	0.884151	0.442076	+	0.896978i	$0.354242\pi$
$972$	−1188.78	−0.0392287
$973$	28648.3	0.943908
$974$	710.034	0.0233583
$975$	4059.10	0.133329
$976$	−12977.5	−0.425615
$977$	−56893.7	−1.86304	−0.931520	−	0.363690i	$-0.881517\pi$
$977$	−56893.7	−1.86304	−0.931520	+	0.363690i	$0.881517\pi$
$978$	33320.8	1.08945
$979$	−11463.7	−0.374242
$980$	2012.89	0.0656118
$981$	6796.15	0.221187
$982$	−35895.0	−1.16645
$983$	−34633.1	−1.12373	−0.561863	−	0.827230i	$-0.689915\pi$
$983$	−34633.1	−1.12373	−0.561863	+	0.827230i	$0.689915\pi$
$984$	−4179.10	−0.135391
$985$	−18191.9	−0.588470
$986$	−92111.0	−2.97506
$987$	−9050.03	−0.291860
$988$	−12907.3	−0.415625
$989$	2682.23	0.0862386
$990$	−1777.33	−0.0570577
$991$	1961.79	0.0628843	0.0314422	−	0.999506i	$-0.489990\pi$
$991$	1961.79	0.0628843	0.0314422	+	0.999506i	$0.489990\pi$
$992$	16016.0	0.512610
$993$	8342.83	0.266618
$994$	42503.8	1.35628
$995$	−258.025	−0.00822103
$996$	17040.2	0.542108
$997$	13096.5	0.416017	0.208009	−	0.978127i	$-0.433302\pi$
$997$	13096.5	0.416017	0.208009	+	0.978127i	$0.433302\pi$
$998$	−31290.8	−0.992476
$999$	587.447	0.0186046

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.59056 q^{2} -3.00000 q^{3} +4.89212 q^{4} -5.00000 q^{5} -10.7717 q^{6} -16.1465 q^{7} -11.1590 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.59056 q^{2} -3.00000 q^{3} +4.89212 q^{4} -5.00000 q^{5} -10.7717 q^{6} -16.1465 q^{7} -11.1590 q^{8} +9.00000 q^{9} -17.9528 q^{10} +11.0000 q^{11} -14.6764 q^{12} -54.1214 q^{13} -57.9749 q^{14} +15.0000 q^{15} -79.2041 q^{16} -107.010 q^{17} +32.3150 q^{18} +48.7496 q^{19} -24.4606 q^{20} +48.4394 q^{21} +39.4962 q^{22} +11.9498 q^{23} +33.4771 q^{24} +25.0000 q^{25} -194.326 q^{26} -27.0000 q^{27} -78.9905 q^{28} +239.733 q^{29} +53.8584 q^{30} -82.0851 q^{31} -195.115 q^{32} -33.0000 q^{33} -384.224 q^{34} +80.7324 q^{35} +44.0291 q^{36} -21.7573 q^{37} +175.038 q^{38} +162.364 q^{39} +55.7952 q^{40} -124.835 q^{41} +173.925 q^{42} +224.459 q^{43} +53.8133 q^{44} -45.0000 q^{45} +42.9064 q^{46} -186.832 q^{47} +237.612 q^{48} -82.2913 q^{49} +89.7640 q^{50} +321.029 q^{51} -264.768 q^{52} +233.997 q^{53} -96.9451 q^{54} -55.0000 q^{55} +180.179 q^{56} -146.249 q^{57} +860.774 q^{58} +232.936 q^{59} +73.3818 q^{60} +163.849 q^{61} -294.731 q^{62} -145.318 q^{63} -66.9386 q^{64} +270.607 q^{65} -118.488 q^{66} -876.918 q^{67} -523.503 q^{68} -35.8493 q^{69} +289.874 q^{70} -733.141 q^{71} -100.431 q^{72} +1161.97 q^{73} -78.1208 q^{74} -75.0000 q^{75} +238.489 q^{76} -177.611 q^{77} +582.978 q^{78} -588.831 q^{79} +396.021 q^{80} +81.0000 q^{81} -448.226 q^{82} -1161.06 q^{83} +236.971 q^{84} +535.048 q^{85} +805.933 q^{86} -719.198 q^{87} -122.749 q^{88} -1042.16 q^{89} -161.575 q^{90} +873.869 q^{91} +58.4597 q^{92} +246.255 q^{93} -670.831 q^{94} -243.748 q^{95} +585.345 q^{96} +1546.63 q^{97} -295.472 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q - 4 * q^2 - 9 * q^3 + 22 * q^4 - 15 * q^5 + 12 * q^6 - 4 * q^7 - 48 * q^8 + 27 * q^9	\( 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9} + 20 q^{10} + 33 q^{11} - 66 q^{12} - 56 q^{14} + 45 q^{15} + 50 q^{16} - 218 q^{17} - 36 q^{18} + 146 q^{19} - 110 q^{20} + 12 q^{21} - 44 q^{22} - 200 q^{23} + 144 q^{24} + 75 q^{25} - 508 q^{26} - 81 q^{27} - 340 q^{28} + 68 q^{29} - 60 q^{30} - 68 q^{31} - 688 q^{32} - 99 q^{33} - 176 q^{34} + 20 q^{35} + 198 q^{36} - 390 q^{37} - 316 q^{38} + 240 q^{40} - 196 q^{41} + 168 q^{42} - 524 q^{43} + 242 q^{44} - 135 q^{45} + 1160 q^{46} - 60 q^{47} - 150 q^{48} - 157 q^{49} - 100 q^{50} + 654 q^{51} + 1020 q^{52} - 158 q^{53} + 108 q^{54} - 165 q^{55} + 1368 q^{56} - 438 q^{57} + 1092 q^{58} - 1044 q^{59} + 330 q^{60} + 642 q^{61} + 88 q^{62} - 36 q^{63} + 1166 q^{64} + 132 q^{66} - 236 q^{67} + 144 q^{68} + 600 q^{69} + 280 q^{70} - 544 q^{71} - 432 q^{72} + 900 q^{73} + 1536 q^{74} - 225 q^{75} + 1996 q^{76} - 44 q^{77} + 1524 q^{78} - 1586 q^{79} - 250 q^{80} + 243 q^{81} + 380 q^{82} - 1582 q^{83} + 1020 q^{84} + 1090 q^{85} + 3568 q^{86} - 204 q^{87} - 528 q^{88} - 2122 q^{89} + 180 q^{90} - 8 q^{91} - 4128 q^{92} + 204 q^{93} - 2152 q^{94} - 730 q^{95} + 2064 q^{96} + 618 q^{97} + 572 q^{98} + 297 q^{99}+O(q^{100}) \) 3 * q - 4 * q^2 - 9 * q^3 + 22 * q^4 - 15 * q^5 + 12 * q^6 - 4 * q^7 - 48 * q^8 + 27 * q^9 + 20 * q^10 + 33 * q^11 - 66 * q^12 - 56 * q^14 + 45 * q^15 + 50 * q^16 - 218 * q^17 - 36 * q^18 + 146 * q^19 - 110 * q^20 + 12 * q^21 - 44 * q^22 - 200 * q^23 + 144 * q^24 + 75 * q^25 - 508 * q^26 - 81 * q^27 - 340 * q^28 + 68 * q^29 - 60 * q^30 - 68 * q^31 - 688 * q^32 - 99 * q^33 - 176 * q^34 + 20 * q^35 + 198 * q^36 - 390 * q^37 - 316 * q^38 + 240 * q^40 - 196 * q^41 + 168 * q^42 - 524 * q^43 + 242 * q^44 - 135 * q^45 + 1160 * q^46 - 60 * q^47 - 150 * q^48 - 157 * q^49 - 100 * q^50 + 654 * q^51 + 1020 * q^52 - 158 * q^53 + 108 * q^54 - 165 * q^55 + 1368 * q^56 - 438 * q^57 + 1092 * q^58 - 1044 * q^59 + 330 * q^60 + 642 * q^61 + 88 * q^62 - 36 * q^63 + 1166 * q^64 + 132 * q^66 - 236 * q^67 + 144 * q^68 + 600 * q^69 + 280 * q^70 - 544 * q^71 - 432 * q^72 + 900 * q^73 + 1536 * q^74 - 225 * q^75 + 1996 * q^76 - 44 * q^77 + 1524 * q^78 - 1586 * q^79 - 250 * q^80 + 243 * q^81 + 380 * q^82 - 1582 * q^83 + 1020 * q^84 + 1090 * q^85 + 3568 * q^86 - 204 * q^87 - 528 * q^88 - 2122 * q^89 + 180 * q^90 - 8 * q^91 - 4128 * q^92 + 204 * q^93 - 2152 * q^94 - 730 * q^95 + 2064 * q^96 + 618 * q^97 + 572 * q^98 + 297 * q^99

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