LMFDB - Embedded newform 4002.2.a.bc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.11324.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 4x + 2 $ x^4 - x^3 - 5x^2 + 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.356500$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.380691 q^{5} +1.00000 q^{6} -2.38069 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.380691 q^{5} +1.00000 q^{6} -2.38069 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.380691 q^{10} +1.61010 q^{11} +1.00000 q^{12} -4.89710 q^{13} -2.38069 q^{14} +0.380691 q^{15} +1.00000 q^{16} -3.09369 q^{17} +1.00000 q^{18} -5.09369 q^{19} +0.380691 q^{20} -2.38069 q^{21} +1.61010 q^{22} -1.00000 q^{23} +1.00000 q^{24} -4.85507 q^{25} -4.89710 q^{26} +1.00000 q^{27} -2.38069 q^{28} -1.00000 q^{29} +0.380691 q^{30} -8.59453 q^{31} +1.00000 q^{32} +1.61010 q^{33} -3.09369 q^{34} -0.906309 q^{35} +1.00000 q^{36} +1.94241 q^{37} -5.09369 q^{38} -4.89710 q^{39} +0.380691 q^{40} -1.80341 q^{41} -2.38069 q^{42} -10.1265 q^{43} +1.61010 q^{44} +0.380691 q^{45} -1.00000 q^{46} +8.12651 q^{47} +1.00000 q^{48} -1.33231 q^{49} -4.85507 q^{50} -3.09369 q^{51} -4.89710 q^{52} -5.95162 q^{53} +1.00000 q^{54} +0.612951 q^{55} -2.38069 q^{56} -5.09369 q^{57} -1.00000 q^{58} +8.88789 q^{59} +0.380691 q^{60} +11.5400 q^{61} -8.59453 q^{62} -2.38069 q^{63} +1.00000 q^{64} -1.86428 q^{65} +1.61010 q^{66} +0.713000 q^{67} -3.09369 q^{68} -1.00000 q^{69} -0.906309 q^{70} +13.6273 q^{71} +1.00000 q^{72} -0.574000 q^{73} +1.94241 q^{74} -4.85507 q^{75} -5.09369 q^{76} -3.83315 q^{77} -4.89710 q^{78} -3.86428 q^{79} +0.380691 q^{80} +1.00000 q^{81} -1.80341 q^{82} -9.03282 q^{83} -2.38069 q^{84} -1.17774 q^{85} -10.1265 q^{86} -1.00000 q^{87} +1.61010 q^{88} -5.69743 q^{89} +0.380691 q^{90} +11.6585 q^{91} -1.00000 q^{92} -8.59453 q^{93} +8.12651 q^{94} -1.93912 q^{95} +1.00000 q^{96} +13.3592 q^{97} -1.33231 q^{98} +1.61010 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{9} - 4 q^{10} - 12 q^{11} + 4 q^{12} - 6 q^{13} - 4 q^{14} - 4 q^{15} + 4 q^{16} - 2 q^{17} + 4 q^{18} - 10 q^{19} - 4 q^{20} - 4 q^{21} - 12 q^{22} - 4 q^{23} + 4 q^{24} + 2 q^{25} - 6 q^{26} + 4 q^{27} - 4 q^{28} - 4 q^{29} - 4 q^{30} - 6 q^{31} + 4 q^{32} - 12 q^{33} - 2 q^{34} - 14 q^{35} + 4 q^{36} - 10 q^{37} - 10 q^{38} - 6 q^{39} - 4 q^{40} - 4 q^{41} - 4 q^{42} - 14 q^{43} - 12 q^{44} - 4 q^{45} - 4 q^{46} + 6 q^{47} + 4 q^{48} - 6 q^{49} + 2 q^{50} - 2 q^{51} - 6 q^{52} - 30 q^{53} + 4 q^{54} + 22 q^{55} - 4 q^{56} - 10 q^{57} - 4 q^{58} - 2 q^{59} - 4 q^{60} - 2 q^{61} - 6 q^{62} - 4 q^{63} + 4 q^{64} - 10 q^{65} - 12 q^{66} - 2 q^{67} - 2 q^{68} - 4 q^{69} - 14 q^{70} + 10 q^{71} + 4 q^{72} - 12 q^{73} - 10 q^{74} + 2 q^{75} - 10 q^{76} + 2 q^{77} - 6 q^{78} - 18 q^{79} - 4 q^{80} + 4 q^{81} - 4 q^{82} - 20 q^{83} - 4 q^{84} - 10 q^{85} - 14 q^{86} - 4 q^{87} - 12 q^{88} - 8 q^{89} - 4 q^{90} + 22 q^{91} - 4 q^{92} - 6 q^{93} + 6 q^{94} - 2 q^{95} + 4 q^{96} + 2 q^{97} - 6 q^{98} - 12 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^9 - 4 * q^10 - 12 * q^11 + 4 * q^12 - 6 * q^13 - 4 * q^14 - 4 * q^15 + 4 * q^16 - 2 * q^17 + 4 * q^18 - 10 * q^19 - 4 * q^20 - 4 * q^21 - 12 * q^22 - 4 * q^23 + 4 * q^24 + 2 * q^25 - 6 * q^26 + 4 * q^27 - 4 * q^28 - 4 * q^29 - 4 * q^30 - 6 * q^31 + 4 * q^32 - 12 * q^33 - 2 * q^34 - 14 * q^35 + 4 * q^36 - 10 * q^37 - 10 * q^38 - 6 * q^39 - 4 * q^40 - 4 * q^41 - 4 * q^42 - 14 * q^43 - 12 * q^44 - 4 * q^45 - 4 * q^46 + 6 * q^47 + 4 * q^48 - 6 * q^49 + 2 * q^50 - 2 * q^51 - 6 * q^52 - 30 * q^53 + 4 * q^54 + 22 * q^55 - 4 * q^56 - 10 * q^57 - 4 * q^58 - 2 * q^59 - 4 * q^60 - 2 * q^61 - 6 * q^62 - 4 * q^63 + 4 * q^64 - 10 * q^65 - 12 * q^66 - 2 * q^67 - 2 * q^68 - 4 * q^69 - 14 * q^70 + 10 * q^71 + 4 * q^72 - 12 * q^73 - 10 * q^74 + 2 * q^75 - 10 * q^76 + 2 * q^77 - 6 * q^78 - 18 * q^79 - 4 * q^80 + 4 * q^81 - 4 * q^82 - 20 * q^83 - 4 * q^84 - 10 * q^85 - 14 * q^86 - 4 * q^87 - 12 * q^88 - 8 * q^89 - 4 * q^90 + 22 * q^91 - 4 * q^92 - 6 * q^93 + 6 * q^94 - 2 * q^95 + 4 * q^96 + 2 * q^97 - 6 * q^98 - 12 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0.380691	0.170250	0.0851252	−	0.996370i	$-0.472871\pi$
$5$	0.380691	0.170250	0.0851252	+	0.996370i	$0.472871\pi$
$6$	1.00000	0.408248
$7$	−2.38069	−0.899817	−0.449908	−	0.893075i	$-0.648543\pi$
$7$	−2.38069	−0.899817	−0.449908	+	0.893075i	$0.648543\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0.380691	0.120385
$11$	1.61010	0.485463	0.242732	−	0.970093i	$-0.421957\pi$
$11$	1.61010	0.485463	0.242732	+	0.970093i	$0.421957\pi$
$12$	1.00000	0.288675
$13$	−4.89710	−1.35821	−0.679105	−	0.734041i	$-0.737632\pi$
$13$	−4.89710	−1.35821	−0.679105	+	0.734041i	$0.737632\pi$
$14$	−2.38069	−0.636267
$15$	0.380691	0.0982941
$16$	1.00000	0.250000
$17$	−3.09369	−0.750330	−0.375165	−	0.926958i	$-0.622414\pi$
$17$	−3.09369	−0.750330	−0.375165	+	0.926958i	$0.622414\pi$
$18$	1.00000	0.235702
$19$	−5.09369	−1.16857	−0.584286	−	0.811547i	$-0.698626\pi$
$19$	−5.09369	−1.16857	−0.584286	+	0.811547i	$0.698626\pi$
$20$	0.380691	0.0851252
$21$	−2.38069	−0.519509
$22$	1.61010	0.343274
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	−4.85507	−0.971015
$26$	−4.89710	−0.960400
$27$	1.00000	0.192450
$28$	−2.38069	−0.449908
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0.380691	0.0695044
$31$	−8.59453	−1.54362	−0.771812	−	0.635851i	$-0.780649\pi$
$31$	−8.59453	−1.54362	−0.771812	+	0.635851i	$0.780649\pi$
$32$	1.00000	0.176777
$33$	1.61010	0.280282
$34$	−3.09369	−0.530564
$35$	−0.906309	−0.153194
$36$	1.00000	0.166667
$37$	1.94241	0.319330	0.159665	−	0.987171i	$-0.448959\pi$
$37$	1.94241	0.319330	0.159665	+	0.987171i	$0.448959\pi$
$38$	−5.09369	−0.826306
$39$	−4.89710	−0.784163
$40$	0.380691	0.0601926
$41$	−1.80341	−0.281645	−0.140823	−	0.990035i	$-0.544975\pi$
$41$	−1.80341	−0.281645	−0.140823	+	0.990035i	$0.544975\pi$
$42$	−2.38069	−0.367349
$43$	−10.1265	−1.54428	−0.772139	−	0.635454i	$-0.780813\pi$
$43$	−10.1265	−1.54428	−0.772139	+	0.635454i	$0.780813\pi$
$44$	1.61010	0.242732
$45$	0.380691	0.0567501
$46$	−1.00000	−0.147442
$47$	8.12651	1.18537	0.592686	−	0.805433i	$-0.298067\pi$
$47$	8.12651	1.18537	0.592686	+	0.805433i	$0.298067\pi$
$48$	1.00000	0.144338
$49$	−1.33231	−0.190330
$50$	−4.85507	−0.686611
$51$	−3.09369	−0.433203
$52$	−4.89710	−0.679105
$53$	−5.95162	−0.817517	−0.408759	−	0.912642i	$-0.634038\pi$
$53$	−5.95162	−0.817517	−0.408759	+	0.912642i	$0.634038\pi$
$54$	1.00000	0.136083
$55$	0.612951	0.0826503
$56$	−2.38069	−0.318133
$57$	−5.09369	−0.674676
$58$	−1.00000	−0.131306
$59$	8.88789	1.15710	0.578552	−	0.815645i	$-0.303618\pi$
$59$	8.88789	1.15710	0.578552	+	0.815645i	$0.303618\pi$
$60$	0.380691	0.0491470
$61$	11.5400	1.47755	0.738774	−	0.673954i	$-0.235405\pi$
$61$	11.5400	1.47755	0.738774	+	0.673954i	$0.235405\pi$
$62$	−8.59453	−1.09151
$63$	−2.38069	−0.299939
$64$	1.00000	0.125000
$65$	−1.86428	−0.231236
$66$	1.61010	0.198190
$67$	0.713000	0.0871068	0.0435534	−	0.999051i	$-0.486132\pi$
$67$	0.713000	0.0871068	0.0435534	+	0.999051i	$0.486132\pi$
$68$	−3.09369	−0.375165
$69$	−1.00000	−0.120386
$70$	−0.906309	−0.108325
$71$	13.6273	1.61727	0.808634	−	0.588312i	$-0.200207\pi$
$71$	13.6273	1.61727	0.808634	+	0.588312i	$0.200207\pi$
$72$	1.00000	0.117851
$73$	−0.574000	−0.0671816	−0.0335908	−	0.999436i	$-0.510694\pi$
$73$	−0.574000	−0.0671816	−0.0335908	+	0.999436i	$0.510694\pi$
$74$	1.94241	0.225800
$75$	−4.85507	−0.560616
$76$	−5.09369	−0.584286
$77$	−3.83315	−0.436828
$78$	−4.89710	−0.554487
$79$	−3.86428	−0.434766	−0.217383	−	0.976086i	$-0.569752\pi$
$79$	−3.86428	−0.434766	−0.217383	+	0.976086i	$0.569752\pi$
$80$	0.380691	0.0425626
$81$	1.00000	0.111111
$82$	−1.80341	−0.199153
$83$	−9.03282	−0.991480	−0.495740	−	0.868471i	$-0.665103\pi$
$83$	−9.03282	−0.991480	−0.495740	+	0.868471i	$0.665103\pi$
$84$	−2.38069	−0.259755
$85$	−1.17774	−0.127744
$86$	−10.1265	−1.09197
$87$	−1.00000	−0.107211
$88$	1.61010	0.171637
$89$	−5.69743	−0.603927	−0.301963	−	0.953320i	$-0.597642\pi$
$89$	−5.69743	−0.603927	−0.301963	+	0.953320i	$0.597642\pi$
$90$	0.380691	0.0401284
$91$	11.6585	1.22214
$92$	−1.00000	−0.104257
$93$	−8.59453	−0.891211
$94$	8.12651	0.838185
$95$	−1.93912	−0.198950
$96$	1.00000	0.102062
$97$	13.3592	1.35642	0.678211	−	0.734868i	$-0.262756\pi$
$97$	13.3592	1.35642	0.678211	+	0.734868i	$0.262756\pi$
$98$	−1.33231	−0.134583
$99$	1.61010	0.161821
$100$	−4.85507	−0.485507
$101$	4.06563	0.404545	0.202273	−	0.979329i	$-0.435167\pi$
$101$	4.06563	0.404545	0.202273	+	0.979329i	$0.435167\pi$
$102$	−3.09369	−0.306321
$103$	−14.6337	−1.44190	−0.720951	−	0.692986i	$-0.756295\pi$
$103$	−14.6337	−1.44190	−0.720951	+	0.692986i	$0.756295\pi$
$104$	−4.89710	−0.480200
$105$	−0.906309	−0.0884467
$106$	−5.95162	−0.578072
$107$	1.71936	0.166217	0.0831083	−	0.996541i	$-0.473515\pi$
$107$	1.71936	0.166217	0.0831083	+	0.996541i	$0.473515\pi$
$108$	1.00000	0.0962250
$109$	4.18738	0.401079	0.200539	−	0.979686i	$-0.435731\pi$
$109$	4.18738	0.401079	0.200539	+	0.979686i	$0.435731\pi$
$110$	0.612951	0.0584426
$111$	1.94241	0.184365
$112$	−2.38069	−0.224954
$113$	5.28107	0.496802	0.248401	−	0.968657i	$-0.420095\pi$
$113$	5.28107	0.496802	0.248401	+	0.968657i	$0.420095\pi$
$114$	−5.09369	−0.477068
$115$	−0.380691	−0.0354997
$116$	−1.00000	−0.0928477
$117$	−4.89710	−0.452737
$118$	8.88789	0.818197
$119$	7.36512	0.675160
$120$	0.380691	0.0347522
$121$	−8.40758	−0.764326
$122$	11.5400	1.04478
$123$	−1.80341	−0.162608
$124$	−8.59453	−0.771812
$125$	−3.75174	−0.335566
$126$	−2.38069	−0.212089
$127$	−19.4732	−1.72797	−0.863984	−	0.503519i	$-0.832038\pi$
$127$	−19.4732	−1.72797	−0.863984	+	0.503519i	$0.832038\pi$
$128$	1.00000	0.0883883
$129$	−10.1265	−0.891589
$130$	−1.86428	−0.163508
$131$	14.8787	1.29996	0.649978	−	0.759953i	$-0.274778\pi$
$131$	14.8787	1.29996	0.649978	+	0.759953i	$0.274778\pi$
$132$	1.61010	0.140141
$133$	12.1265	1.05150
$134$	0.713000	0.0615938
$135$	0.380691	0.0327647
$136$	−3.09369	−0.265282
$137$	−7.61338	−0.650455	−0.325228	−	0.945636i	$-0.605441\pi$
$137$	−7.61338	−0.650455	−0.325228	+	0.945636i	$0.605441\pi$
$138$	−1.00000	−0.0851257
$139$	0.664617	0.0563721	0.0281860	−	0.999603i	$-0.491027\pi$
$139$	0.664617	0.0563721	0.0281860	+	0.999603i	$0.491027\pi$
$140$	−0.906309	−0.0765971
$141$	8.12651	0.684375
$142$	13.6273	1.14358
$143$	−7.88482	−0.659361
$144$	1.00000	0.0833333
$145$	−0.380691	−0.0316147
$146$	−0.574000	−0.0475046
$147$	−1.33231	−0.109887
$148$	1.94241	0.159665
$149$	11.2077	0.918171	0.459086	−	0.888392i	$-0.348177\pi$
$149$	11.2077	0.918171	0.459086	+	0.888392i	$0.348177\pi$
$150$	−4.85507	−0.396415
$151$	11.8034	0.960548	0.480274	−	0.877119i	$-0.340537\pi$
$151$	11.8034	0.960548	0.480274	+	0.877119i	$0.340537\pi$
$152$	−5.09369	−0.413153
$153$	−3.09369	−0.250110
$154$	−3.83315	−0.308884
$155$	−3.27186	−0.262802
$156$	−4.89710	−0.392082
$157$	0.0264580	0.00211158	0.00105579	−	0.999999i	$-0.499664\pi$
$157$	0.0264580	0.00211158	0.00105579	+	0.999999i	$0.499664\pi$
$158$	−3.86428	−0.307426
$159$	−5.95162	−0.471994
$160$	0.380691	0.0300963
$161$	2.38069	0.187625
$162$	1.00000	0.0785674
$163$	2.86736	0.224589	0.112294	−	0.993675i	$-0.464180\pi$
$163$	2.86736	0.224589	0.112294	+	0.993675i	$0.464180\pi$
$164$	−1.80341	−0.140823
$165$	0.612951	0.0477182
$166$	−9.03282	−0.701082
$167$	10.1874	0.788323	0.394162	−	0.919041i	$-0.371035\pi$
$167$	10.1874	0.788323	0.394162	+	0.919041i	$0.371035\pi$
$168$	−2.38069	−0.183674
$169$	10.9816	0.844737
$170$	−1.17774	−0.0903287
$171$	−5.09369	−0.389524
$172$	−10.1265	−0.772139
$173$	−12.4807	−0.948893	−0.474447	−	0.880284i	$-0.657352\pi$
$173$	−12.4807	−0.948893	−0.474447	+	0.880284i	$0.657352\pi$
$174$	−1.00000	−0.0758098
$175$	11.5584	0.873735
$176$	1.61010	0.121366
$177$	8.88789	0.668055
$178$	−5.69743	−0.427041
$179$	−8.52445	−0.637147	−0.318574	−	0.947898i	$-0.603204\pi$
$179$	−8.52445	−0.637147	−0.318574	+	0.947898i	$0.603204\pi$
$180$	0.380691	0.0283751
$181$	−16.2390	−1.20704	−0.603519	−	0.797348i	$-0.706235\pi$
$181$	−16.2390	−1.20704	−0.603519	+	0.797348i	$0.706235\pi$
$182$	11.6585	0.864184
$183$	11.5400	0.853062
$184$	−1.00000	−0.0737210
$185$	0.739458	0.0543660
$186$	−8.59453	−0.630182
$187$	−4.98115	−0.364258
$188$	8.12651	0.592686
$189$	−2.38069	−0.173170
$190$	−1.93912	−0.140679
$191$	−25.2282	−1.82545	−0.912726	−	0.408573i	$-0.866026\pi$
$191$	−25.2282	−1.82545	−0.912726	+	0.408573i	$0.866026\pi$
$192$	1.00000	0.0721688
$193$	−5.98158	−0.430564	−0.215282	−	0.976552i	$-0.569067\pi$
$193$	−5.98158	−0.430564	−0.215282	+	0.976552i	$0.569067\pi$
$194$	13.3592	0.959135
$195$	−1.86428	−0.133504
$196$	−1.33231	−0.0951649
$197$	−9.29504	−0.662244	−0.331122	−	0.943588i	$-0.607427\pi$
$197$	−9.29504	−0.662244	−0.331122	+	0.943588i	$0.607427\pi$
$198$	1.61010	0.114425
$199$	8.04838	0.570535	0.285267	−	0.958448i	$-0.407918\pi$
$199$	8.04838	0.570535	0.285267	+	0.958448i	$0.407918\pi$
$200$	−4.85507	−0.343306
$201$	0.713000	0.0502911
$202$	4.06563	0.286057
$203$	2.38069	0.167092
$204$	−3.09369	−0.216602
$205$	−0.686542	−0.0479502
$206$	−14.6337	−1.01958
$207$	−1.00000	−0.0695048
$208$	−4.89710	−0.339553
$209$	−8.20135	−0.567299
$210$	−0.906309	−0.0625412
$211$	−19.1799	−1.32040	−0.660198	−	0.751092i	$-0.729528\pi$
$211$	−19.1799	−1.32040	−0.660198	+	0.751092i	$0.729528\pi$
$212$	−5.95162	−0.408759
$213$	13.6273	0.933730
$214$	1.71936	0.117533
$215$	−3.85507	−0.262914
$216$	1.00000	0.0680414
$217$	20.4609	1.38898
$218$	4.18738	0.283605
$219$	−0.574000	−0.0387873
$220$	0.612951	0.0413251
$221$	15.1501	1.01911
$222$	1.94241	0.130366
$223$	22.9833	1.53907	0.769537	−	0.638603i	$-0.220487\pi$
$223$	22.9833	1.53907	0.769537	+	0.638603i	$0.220487\pi$
$224$	−2.38069	−0.159067
$225$	−4.85507	−0.323672
$226$	5.28107	0.351292
$227$	4.44961	0.295331	0.147665	−	0.989037i	$-0.452824\pi$
$227$	4.44961	0.295331	0.147665	+	0.989037i	$0.452824\pi$
$228$	−5.09369	−0.337338
$229$	−0.787841	−0.0520620	−0.0260310	−	0.999661i	$-0.508287\pi$
$229$	−0.787841	−0.0520620	−0.0260310	+	0.999661i	$0.508287\pi$
$230$	−0.380691	−0.0251020
$231$	−3.83315	−0.252203
$232$	−1.00000	−0.0656532
$233$	4.18082	0.273894	0.136947	−	0.990578i	$-0.456271\pi$
$233$	4.18082	0.273894	0.136947	+	0.990578i	$0.456271\pi$
$234$	−4.89710	−0.320133
$235$	3.09369	0.201810
$236$	8.88789	0.578552
$237$	−3.86428	−0.251012
$238$	7.36512	0.477410
$239$	−23.6479	−1.52965	−0.764827	−	0.644236i	$-0.777176\pi$
$239$	−23.6479	−1.52965	−0.764827	+	0.644236i	$0.777176\pi$
$240$	0.380691	0.0245735
$241$	6.46802	0.416642	0.208321	−	0.978060i	$-0.433200\pi$
$241$	6.46802	0.416642	0.208321	+	0.978060i	$0.433200\pi$
$242$	−8.40758	−0.540460
$243$	1.00000	0.0641500
$244$	11.5400	0.738774
$245$	−0.507198	−0.0324037
$246$	−1.80341	−0.114981
$247$	24.9443	1.58717
$248$	−8.59453	−0.545753
$249$	−9.03282	−0.572431
$250$	−3.75174	−0.237281
$251$	8.01725	0.506044	0.253022	−	0.967460i	$-0.418575\pi$
$251$	8.01725	0.506044	0.253022	+	0.967460i	$0.418575\pi$
$252$	−2.38069	−0.149969
$253$	−1.61010	−0.101226
$254$	−19.4732	−1.22186
$255$	−1.17774	−0.0737530
$256$	1.00000	0.0625000
$257$	4.23248	0.264015	0.132007	−	0.991249i	$-0.457858\pi$
$257$	4.23248	0.264015	0.132007	+	0.991249i	$0.457858\pi$
$258$	−10.1265	−0.630449
$259$	−4.62427	−0.287338
$260$	−1.86428	−0.115618
$261$	−1.00000	−0.0618984
$262$	14.8787	0.919208
$263$	7.82066	0.482242	0.241121	−	0.970495i	$-0.422485\pi$
$263$	7.82066	0.482242	0.241121	+	0.970495i	$0.422485\pi$
$264$	1.61010	0.0990948
$265$	−2.26573	−0.139183
$266$	12.1265	0.743524
$267$	−5.69743	−0.348677
$268$	0.713000	0.0435534
$269$	26.1895	1.59680	0.798401	−	0.602126i	$-0.205680\pi$
$269$	26.1895	1.59680	0.798401	+	0.602126i	$0.205680\pi$
$270$	0.380691	0.0231681
$271$	12.0968	0.734826	0.367413	−	0.930058i	$-0.380244\pi$
$271$	12.0968	0.734826	0.367413	+	0.930058i	$0.380244\pi$
$272$	−3.09369	−0.187583
$273$	11.6585	0.705603
$274$	−7.61338	−0.459941
$275$	−7.81715	−0.471392
$276$	−1.00000	−0.0601929
$277$	−22.9443	−1.37859	−0.689295	−	0.724481i	$-0.742080\pi$
$277$	−22.9443	−1.37859	−0.689295	+	0.724481i	$0.742080\pi$
$278$	0.664617	0.0398611
$279$	−8.59453	−0.514541
$280$	−0.906309	−0.0541623
$281$	−18.2046	−1.08600	−0.542999	−	0.839734i	$-0.682711\pi$
$281$	−18.2046	−1.08600	−0.542999	+	0.839734i	$0.682711\pi$
$282$	8.12651	0.483926
$283$	−2.96390	−0.176186	−0.0880928	−	0.996112i	$-0.528077\pi$
$283$	−2.96390	−0.176186	−0.0880928	+	0.996112i	$0.528077\pi$
$284$	13.6273	0.808634
$285$	−1.93912	−0.114864
$286$	−7.88482	−0.466239
$287$	4.29336	0.253429
$288$	1.00000	0.0589256
$289$	−7.42907	−0.437004
$290$	−0.380691	−0.0223550
$291$	13.3592	0.783130
$292$	−0.574000	−0.0335908
$293$	−25.2674	−1.47614	−0.738069	−	0.674725i	$-0.764262\pi$
$293$	−25.2674	−1.47614	−0.738069	+	0.674725i	$0.764262\pi$
$294$	−1.33231	−0.0777018
$295$	3.38354	0.196997
$296$	1.94241	0.112900
$297$	1.61010	0.0934274
$298$	11.2077	0.649245
$299$	4.89710	0.283207
$300$	−4.85507	−0.280308
$301$	24.1081	1.38957
$302$	11.8034	0.679210
$303$	4.06563	0.233564
$304$	−5.09369	−0.292143
$305$	4.39318	0.251553
$306$	−3.09369	−0.176855
$307$	−23.9632	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$307$	−23.9632	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$308$	−3.83315	−0.218414
$309$	−14.6337	−0.832482
$310$	−3.27186	−0.185829
$311$	−18.1921	−1.03158	−0.515791	−	0.856715i	$-0.672502\pi$
$311$	−18.1921	−1.03158	−0.515791	+	0.856715i	$0.672502\pi$
$312$	−4.89710	−0.277244
$313$	−19.4640	−1.10017	−0.550085	−	0.835109i	$-0.685405\pi$
$313$	−19.4640	−1.10017	−0.550085	+	0.835109i	$0.685405\pi$
$314$	0.0264580	0.00149311
$315$	−0.906309	−0.0510647
$316$	−3.86428	−0.217383
$317$	14.4793	0.813241	0.406621	−	0.913597i	$-0.366707\pi$
$317$	14.4793	0.813241	0.406621	+	0.913597i	$0.366707\pi$
$318$	−5.95162	−0.333750
$319$	−1.61010	−0.0901482
$320$	0.380691	0.0212813
$321$	1.71936	0.0959652
$322$	2.38069	0.132671
$323$	15.7583	0.876816
$324$	1.00000	0.0555556
$325$	23.7758	1.31884
$326$	2.86736	0.158808
$327$	4.18738	0.231563
$328$	−1.80341	−0.0995765
$329$	−19.3467	−1.06662
$330$	0.612951	0.0337418
$331$	9.50084	0.522213	0.261107	−	0.965310i	$-0.415913\pi$
$331$	9.50084	0.522213	0.261107	+	0.965310i	$0.415913\pi$
$332$	−9.03282	−0.495740
$333$	1.94241	0.106443
$334$	10.1874	0.557429
$335$	0.271433	0.0148300
$336$	−2.38069	−0.129877
$337$	2.79705	0.152365	0.0761825	−	0.997094i	$-0.475727\pi$
$337$	2.79705	0.152365	0.0761825	+	0.997094i	$0.475727\pi$
$338$	10.9816	0.597319
$339$	5.28107	0.286829
$340$	−1.17774	−0.0638720
$341$	−13.8380	−0.749372
$342$	−5.09369	−0.275435
$343$	19.8367	1.07108
$344$	−10.1265	−0.545985
$345$	−0.380691	−0.0204957
$346$	−12.4807	−0.670969
$347$	4.17942	0.224363	0.112182	−	0.993688i	$-0.464216\pi$
$347$	4.17942	0.224363	0.112182	+	0.993688i	$0.464216\pi$
$348$	−1.00000	−0.0536056
$349$	17.2202	0.921776	0.460888	−	0.887458i	$-0.347531\pi$
$349$	17.2202	0.921776	0.460888	+	0.887458i	$0.347531\pi$
$350$	11.5584	0.617824
$351$	−4.89710	−0.261388
$352$	1.61010	0.0858186
$353$	36.3002	1.93207	0.966033	−	0.258419i	$-0.0832014\pi$
$353$	36.3002	1.93207	0.966033	+	0.258419i	$0.0832014\pi$
$354$	8.88789	0.472386
$355$	5.18781	0.275341
$356$	−5.69743	−0.301963
$357$	7.36512	0.389804
$358$	−8.52445	−0.450531
$359$	−4.42579	−0.233584	−0.116792	−	0.993156i	$-0.537261\pi$
$359$	−4.42579	−0.233584	−0.116792	+	0.993156i	$0.537261\pi$
$360$	0.380691	0.0200642
$361$	6.94569	0.365563
$362$	−16.2390	−0.853505
$363$	−8.40758	−0.441284
$364$	11.6585	0.611070
$365$	−0.218517	−0.0114377
$366$	11.5400	0.603206
$367$	37.6216	1.96383	0.981917	−	0.189312i	$-0.0606259\pi$
$367$	37.6216	1.96383	0.981917	+	0.189312i	$0.0606259\pi$
$368$	−1.00000	−0.0521286
$369$	−1.80341	−0.0938817
$370$	0.739458	0.0384426
$371$	14.1690	0.735616
$372$	−8.59453	−0.445606
$373$	7.64452	0.395818	0.197909	−	0.980220i	$-0.436585\pi$
$373$	7.64452	0.395818	0.197909	+	0.980220i	$0.436585\pi$
$374$	−4.98115	−0.257569
$375$	−3.75174	−0.193739
$376$	8.12651	0.419093
$377$	4.89710	0.252213
$378$	−2.38069	−0.122450
$379$	24.0288	1.23428	0.617138	−	0.786855i	$-0.288292\pi$
$379$	24.0288	1.23428	0.617138	+	0.786855i	$0.288292\pi$
$380$	−1.93912	−0.0994750
$381$	−19.4732	−0.997643
$382$	−25.2282	−1.29079
$383$	−3.15011	−0.160963	−0.0804816	−	0.996756i	$-0.525646\pi$
$383$	−3.15011	−0.160963	−0.0804816	+	0.996756i	$0.525646\pi$
$384$	1.00000	0.0510310
$385$	−1.45925	−0.0743701
$386$	−5.98158	−0.304454
$387$	−10.1265	−0.514759
$388$	13.3592	0.678211
$389$	−7.49163	−0.379841	−0.189920	−	0.981799i	$-0.560823\pi$
$389$	−7.49163	−0.379841	−0.189920	+	0.981799i	$0.560823\pi$
$390$	−1.86428	−0.0944017
$391$	3.09369	0.156455
$392$	−1.33231	−0.0672917
$393$	14.8787	0.750530
$394$	−9.29504	−0.468277
$395$	−1.47110	−0.0740190
$396$	1.61010	0.0809105
$397$	14.4728	0.726368	0.363184	−	0.931717i	$-0.381690\pi$
$397$	14.4728	0.726368	0.363184	+	0.931717i	$0.381690\pi$
$398$	8.04838	0.403429
$399$	12.1265	0.607085
$400$	−4.85507	−0.242754
$401$	−39.9021	−1.99261	−0.996307	−	0.0858626i	$-0.972635\pi$
$401$	−39.9021	−1.99261	−0.996307	+	0.0858626i	$0.972635\pi$
$402$	0.713000	0.0355612
$403$	42.0883	2.09657
$404$	4.06563	0.202273
$405$	0.380691	0.0189167
$406$	2.38069	0.118152
$407$	3.12747	0.155023
$408$	−3.09369	−0.153161
$409$	35.9837	1.77928	0.889640	−	0.456663i	$-0.150956\pi$
$409$	35.9837	1.77928	0.889640	+	0.456663i	$0.150956\pi$
$410$	−0.686542	−0.0339059
$411$	−7.61338	−0.375540
$412$	−14.6337	−0.720951
$413$	−21.1593	−1.04118
$414$	−1.00000	−0.0491473
$415$	−3.43872	−0.168800
$416$	−4.89710	−0.240100
$417$	0.664617	0.0325464
$418$	−8.20135	−0.401141
$419$	10.6304	0.519330	0.259665	−	0.965699i	$-0.416388\pi$
$419$	10.6304	0.519330	0.259665	+	0.965699i	$0.416388\pi$
$420$	−0.906309	−0.0442233
$421$	−24.4576	−1.19199	−0.595996	−	0.802987i	$-0.703243\pi$
$421$	−24.4576	−1.19199	−0.595996	+	0.802987i	$0.703243\pi$
$422$	−19.1799	−0.933661
$423$	8.12651	0.395124
$424$	−5.95162	−0.289036
$425$	15.0201	0.728582
$426$	13.6273	0.660247
$427$	−27.4732	−1.32952
$428$	1.71936	0.0831083
$429$	−7.88482	−0.380682
$430$	−3.85507	−0.185908
$431$	21.9837	1.05892	0.529459	−	0.848336i	$-0.322395\pi$
$431$	21.9837	1.05892	0.529459	+	0.848336i	$0.322395\pi$
$432$	1.00000	0.0481125
$433$	−13.4572	−0.646712	−0.323356	−	0.946277i	$-0.604811\pi$
$433$	−13.4572	−0.646712	−0.323356	+	0.946277i	$0.604811\pi$
$434$	20.4609	0.982156
$435$	−0.380691	−0.0182528
$436$	4.18738	0.200539
$437$	5.09369	0.243664
$438$	−0.574000	−0.0274268
$439$	39.4951	1.88500	0.942500	−	0.334206i	$-0.108468\pi$
$439$	39.4951	1.88500	0.942500	+	0.334206i	$0.108468\pi$
$440$	0.612951	0.0292213
$441$	−1.33231	−0.0634433
$442$	15.1501	0.720617
$443$	−11.5950	−0.550893	−0.275447	−	0.961316i	$-0.588826\pi$
$443$	−11.5950	−0.550893	−0.275447	+	0.961316i	$0.588826\pi$
$444$	1.94241	0.0921826
$445$	−2.16896	−0.102819
$446$	22.9833	1.08829
$447$	11.2077	0.530106
$448$	−2.38069	−0.112477
$449$	−18.9330	−0.893503	−0.446752	−	0.894658i	$-0.647419\pi$
$449$	−18.9330	−0.893503	−0.446752	+	0.894658i	$0.647419\pi$
$450$	−4.85507	−0.228870
$451$	−2.90367	−0.136728
$452$	5.28107	0.248401
$453$	11.8034	0.554572
$454$	4.44961	0.208830
$455$	4.43828	0.208070
$456$	−5.09369	−0.238534
$457$	−36.3590	−1.70080	−0.850401	−	0.526135i	$-0.823641\pi$
$457$	−36.3590	−1.70080	−0.850401	+	0.526135i	$0.823641\pi$
$458$	−0.787841	−0.0368134
$459$	−3.09369	−0.144401
$460$	−0.380691	−0.0177498
$461$	−1.89542	−0.0882784	−0.0441392	−	0.999025i	$-0.514055\pi$
$461$	−1.89542	−0.0882784	−0.0441392	+	0.999025i	$0.514055\pi$
$462$	−3.83315	−0.178334
$463$	−8.93605	−0.415293	−0.207647	−	0.978204i	$-0.566580\pi$
$463$	−8.93605	−0.415293	−0.207647	+	0.978204i	$0.566580\pi$
$464$	−1.00000	−0.0464238
$465$	−3.27186	−0.151729
$466$	4.18082	0.193673
$467$	8.75195	0.404992	0.202496	−	0.979283i	$-0.435095\pi$
$467$	8.75195	0.404992	0.202496	+	0.979283i	$0.435095\pi$
$468$	−4.89710	−0.226368
$469$	−1.69743	−0.0783802
$470$	3.09369	0.142701
$471$	0.0264580	0.00121912
$472$	8.88789	0.409098
$473$	−16.3047	−0.749690
$474$	−3.86428	−0.177492
$475$	24.7302	1.13470
$476$	7.36512	0.337580
$477$	−5.95162	−0.272506
$478$	−23.6479	−1.08163
$479$	2.22305	0.101574	0.0507869	−	0.998710i	$-0.483827\pi$
$479$	2.22305	0.101574	0.0507869	+	0.998710i	$0.483827\pi$
$480$	0.380691	0.0173761
$481$	−9.51216	−0.433717
$482$	6.46802	0.294610
$483$	2.38069	0.108325
$484$	−8.40758	−0.382163
$485$	5.08573	0.230931
$486$	1.00000	0.0453609
$487$	−4.19659	−0.190166	−0.0950829	−	0.995469i	$-0.530312\pi$
$487$	−4.19659	−0.190166	−0.0950829	+	0.995469i	$0.530312\pi$
$488$	11.5400	0.522392
$489$	2.86736	0.129666
$490$	−0.507198	−0.0229129
$491$	8.49777	0.383499	0.191749	−	0.981444i	$-0.438584\pi$
$491$	8.49777	0.383499	0.191749	+	0.981444i	$0.438584\pi$
$492$	−1.80341	−0.0813039
$493$	3.09369	0.139333
$494$	24.9443	1.12230
$495$	0.612951	0.0275501
$496$	−8.59453	−0.385906
$497$	−32.4425	−1.45525
$498$	−9.03282	−0.404770
$499$	−7.32478	−0.327902	−0.163951	−	0.986468i	$-0.552424\pi$
$499$	−7.32478	−0.327902	−0.163951	+	0.986468i	$0.552424\pi$
$500$	−3.75174	−0.167783
$501$	10.1874	0.455139
$502$	8.01725	0.357827
$503$	4.78169	0.213205	0.106603	−	0.994302i	$-0.466003\pi$
$503$	4.78169	0.213205	0.106603	+	0.994302i	$0.466003\pi$
$504$	−2.38069	−0.106044
$505$	1.54775	0.0688740
$506$	−1.61010	−0.0715776
$507$	10.9816	0.487709
$508$	−19.4732	−0.863984
$509$	33.7600	1.49639	0.748193	−	0.663481i	$-0.230922\pi$
$509$	33.7600	1.49639	0.748193	+	0.663481i	$0.230922\pi$
$510$	−1.17774	−0.0521513
$511$	1.36652	0.0604512
$512$	1.00000	0.0441942
$513$	−5.09369	−0.224892
$514$	4.23248	0.186687
$515$	−5.57093	−0.245484
$516$	−10.1265	−0.445795
$517$	13.0845	0.575455
$518$	−4.62427	−0.203179
$519$	−12.4807	−0.547844
$520$	−1.86428	−0.0817542
$521$	−19.5605	−0.856963	−0.428482	−	0.903551i	$-0.640951\pi$
$521$	−19.5605	−0.856963	−0.428482	+	0.903551i	$0.640951\pi$
$522$	−1.00000	−0.0437688
$523$	−35.6713	−1.55980	−0.779898	−	0.625907i	$-0.784729\pi$
$523$	−35.6713	−1.55980	−0.779898	+	0.625907i	$0.784729\pi$
$524$	14.8787	0.649978
$525$	11.5584	0.504451
$526$	7.82066	0.340997
$527$	26.5888	1.15823
$528$	1.61010	0.0700706
$529$	1.00000	0.0434783
$530$	−2.26573	−0.0984170
$531$	8.88789	0.385702
$532$	12.1265	0.525751
$533$	8.83147	0.382533
$534$	−5.69743	−0.246552
$535$	0.654545	0.0282984
$536$	0.713000	0.0307969
$537$	−8.52445	−0.367857
$538$	26.1895	1.12911
$539$	−2.14515	−0.0923981
$540$	0.380691	0.0163823
$541$	−6.34502	−0.272794	−0.136397	−	0.990654i	$-0.543552\pi$
$541$	−6.34502	−0.272794	−0.136397	+	0.990654i	$0.543552\pi$
$542$	12.0968	0.519601
$543$	−16.2390	−0.696884
$544$	−3.09369	−0.132641
$545$	1.59410	0.0682838
$546$	11.6585	0.498937
$547$	30.5187	1.30489	0.652444	−	0.757837i	$-0.273744\pi$
$547$	30.5187	1.30489	0.652444	+	0.757837i	$0.273744\pi$
$548$	−7.61338	−0.325228
$549$	11.5400	0.492516
$550$	−7.81715	−0.333324
$551$	5.09369	0.216999
$552$	−1.00000	−0.0425628
$553$	9.19967	0.391210
$554$	−22.9443	−0.974810
$555$	0.739458	0.0313882
$556$	0.664617	0.0281860
$557$	0.826791	0.0350323	0.0175161	−	0.999847i	$-0.494424\pi$
$557$	0.826791	0.0350323	0.0175161	+	0.999847i	$0.494424\pi$
$558$	−8.59453	−0.363836
$559$	49.5905	2.09746
$560$	−0.906309	−0.0382985
$561$	−4.98115	−0.210304
$562$	−18.2046	−0.767916
$563$	−11.3526	−0.478456	−0.239228	−	0.970963i	$-0.576894\pi$
$563$	−11.3526	−0.478456	−0.239228	+	0.970963i	$0.576894\pi$
$564$	8.12651	0.342188
$565$	2.01046	0.0845807
$566$	−2.96390	−0.124582
$567$	−2.38069	−0.0999796
$568$	13.6273	0.571791
$569$	−25.0031	−1.04818	−0.524092	−	0.851662i	$-0.675595\pi$
$569$	−25.0031	−1.04818	−0.524092	+	0.851662i	$0.675595\pi$
$570$	−1.93912	−0.0812210
$571$	−18.9271	−0.792073	−0.396036	−	0.918235i	$-0.629615\pi$
$571$	−18.9271	−0.792073	−0.396036	+	0.918235i	$0.629615\pi$
$572$	−7.88482	−0.329681
$573$	−25.2282	−1.05392
$574$	4.29336	0.179201
$575$	4.85507	0.202471
$576$	1.00000	0.0416667
$577$	−45.3409	−1.88756	−0.943782	−	0.330568i	$-0.892760\pi$
$577$	−45.3409	−1.88756	−0.943782	+	0.330568i	$0.892760\pi$
$578$	−7.42907	−0.309009
$579$	−5.98158	−0.248586
$580$	−0.380691	−0.0158073
$581$	21.5043	0.892151
$582$	13.3592	0.553757
$583$	−9.58269	−0.396875
$584$	−0.574000	−0.0237523
$585$	−1.86428	−0.0770786
$586$	−25.2674	−1.04379
$587$	−33.6243	−1.38782	−0.693911	−	0.720060i	$-0.744114\pi$
$587$	−33.6243	−1.38782	−0.693911	+	0.720060i	$0.744114\pi$
$588$	−1.33231	−0.0549435
$589$	43.7779	1.80384
$590$	3.38354	0.139298
$591$	−9.29504	−0.382347
$592$	1.94241	0.0798325
$593$	−14.8647	−0.610421	−0.305210	−	0.952285i	$-0.598727\pi$
$593$	−14.8647	−0.610421	−0.305210	+	0.952285i	$0.598727\pi$
$594$	1.61010	0.0660632
$595$	2.80384	0.114946
$596$	11.2077	0.459086
$597$	8.04838	0.329398
$598$	4.89710	0.200257
$599$	−3.47321	−0.141912	−0.0709558	−	0.997479i	$-0.522605\pi$
$599$	−3.47321	−0.141912	−0.0709558	+	0.997479i	$0.522605\pi$
$600$	−4.85507	−0.198208
$601$	−21.8782	−0.892432	−0.446216	−	0.894925i	$-0.647229\pi$
$601$	−21.8782	−0.892432	−0.446216	+	0.894925i	$0.647229\pi$
$602$	24.1081	0.982572
$603$	0.713000	0.0290356
$604$	11.8034	0.480274
$605$	−3.20069	−0.130127
$606$	4.06563	0.165155
$607$	−10.4921	−0.425860	−0.212930	−	0.977067i	$-0.568301\pi$
$607$	−10.4921	−0.425860	−0.212930	+	0.977067i	$0.568301\pi$
$608$	−5.09369	−0.206576
$609$	2.38069	0.0964705
$610$	4.39318	0.177875
$611$	−39.7963	−1.60999
$612$	−3.09369	−0.125055
$613$	−20.7053	−0.836278	−0.418139	−	0.908383i	$-0.637317\pi$
$613$	−20.7053	−0.836278	−0.418139	+	0.908383i	$0.637317\pi$
$614$	−23.9632	−0.967075
$615$	−0.686542	−0.0276840
$616$	−3.83315	−0.154442
$617$	3.17159	0.127684	0.0638418	−	0.997960i	$-0.479665\pi$
$617$	3.17159	0.127684	0.0638418	+	0.997960i	$0.479665\pi$
$618$	−14.6337	−0.588654
$619$	21.5214	0.865017	0.432508	−	0.901630i	$-0.357629\pi$
$619$	21.5214	0.865017	0.432508	+	0.901630i	$0.357629\pi$
$620$	−3.27186	−0.131401
$621$	−1.00000	−0.0401286
$622$	−18.1921	−0.729438
$623$	13.5638	0.543423
$624$	−4.89710	−0.196041
$625$	22.8471	0.913885
$626$	−19.4640	−0.777938
$627$	−8.20135	−0.327530
$628$	0.0264580	0.00105579
$629$	−6.00921	−0.239603
$630$	−0.906309	−0.0361082
$631$	−28.2782	−1.12574	−0.562869	−	0.826546i	$-0.690302\pi$
$631$	−28.2782	−1.12574	−0.562869	+	0.826546i	$0.690302\pi$
$632$	−3.86428	−0.153713
$633$	−19.1799	−0.762331
$634$	14.4793	0.575048
$635$	−7.41328	−0.294187
$636$	−5.95162	−0.235997
$637$	6.52445	0.258508
$638$	−1.61010	−0.0637444
$639$	13.6273	0.539090
$640$	0.380691	0.0150481
$641$	−18.0425	−0.712634	−0.356317	−	0.934365i	$-0.615968\pi$
$641$	−18.0425	−0.712634	−0.356317	+	0.934365i	$0.615968\pi$
$642$	1.71936	0.0678576
$643$	38.2067	1.50673	0.753363	−	0.657604i	$-0.228430\pi$
$643$	38.2067	1.50673	0.753363	+	0.657604i	$0.228430\pi$
$644$	2.38069	0.0938124
$645$	−3.85507	−0.151793
$646$	15.7583	0.620002
$647$	5.50560	0.216447	0.108224	−	0.994127i	$-0.465484\pi$
$647$	5.50560	0.216447	0.108224	+	0.994127i	$0.465484\pi$
$648$	1.00000	0.0392837
$649$	14.3104	0.561732
$650$	23.7758	0.932563
$651$	20.4609	0.801927
$652$	2.86736	0.112294
$653$	−3.41943	−0.133813	−0.0669064	−	0.997759i	$-0.521313\pi$
$653$	−3.41943	−0.133813	−0.0669064	+	0.997759i	$0.521313\pi$
$654$	4.18738	0.163740
$655$	5.66419	0.221318
$656$	−1.80341	−0.0704113
$657$	−0.574000	−0.0223939
$658$	−19.3467	−0.754213
$659$	1.94505	0.0757684	0.0378842	−	0.999282i	$-0.487938\pi$
$659$	1.94505	0.0757684	0.0378842	+	0.999282i	$0.487938\pi$
$660$	0.612951	0.0238591
$661$	−6.72454	−0.261554	−0.130777	−	0.991412i	$-0.541747\pi$
$661$	−6.72454	−0.261554	−0.130777	+	0.991412i	$0.541747\pi$
$662$	9.50084	0.369261
$663$	15.1501	0.588382
$664$	−9.03282	−0.350541
$665$	4.61646	0.179019
$666$	1.94241	0.0752668
$667$	1.00000	0.0387202
$668$	10.1874	0.394162
$669$	22.9833	0.888584
$670$	0.271433	0.0104864
$671$	18.5806	0.717295
$672$	−2.38069	−0.0918372
$673$	8.21056	0.316494	0.158247	−	0.987400i	$-0.449416\pi$
$673$	8.21056	0.316494	0.158247	+	0.987400i	$0.449416\pi$
$674$	2.79705	0.107738
$675$	−4.85507	−0.186872
$676$	10.9816	0.422369
$677$	−17.9439	−0.689639	−0.344820	−	0.938669i	$-0.612060\pi$
$677$	−17.9439	−0.689639	−0.344820	+	0.938669i	$0.612060\pi$
$678$	5.28107	0.202818
$679$	−31.8041	−1.22053
$680$	−1.17774	−0.0451643
$681$	4.44961	0.170509
$682$	−13.8380	−0.529886
$683$	−45.2722	−1.73229	−0.866146	−	0.499792i	$-0.833410\pi$
$683$	−45.2722	−1.73229	−0.866146	+	0.499792i	$0.833410\pi$
$684$	−5.09369	−0.194762
$685$	−2.89835	−0.110740
$686$	19.8367	0.757367
$687$	−0.787841	−0.0300580
$688$	−10.1265	−0.386069
$689$	29.1457	1.11036
$690$	−0.380691	−0.0144927
$691$	38.5454	1.46634	0.733169	−	0.680047i	$-0.238041\pi$
$691$	38.5454	1.46634	0.733169	+	0.680047i	$0.238041\pi$
$692$	−12.4807	−0.474447
$693$	−3.83315	−0.145609
$694$	4.17942	0.158649
$695$	0.253014	0.00959737
$696$	−1.00000	−0.0379049
$697$	5.57919	0.211327
$698$	17.2202	0.651794
$699$	4.18082	0.158133
$700$	11.5584	0.436868
$701$	−29.8412	−1.12709	−0.563543	−	0.826087i	$-0.690562\pi$
$701$	−29.8412	−1.12709	−0.563543	+	0.826087i	$0.690562\pi$
$702$	−4.89710	−0.184829
$703$	−9.89402	−0.373160
$704$	1.61010	0.0606829
$705$	3.09369	0.116515
$706$	36.3002	1.36618
$707$	−9.67901	−0.364017
$708$	8.88789	0.334027
$709$	6.69786	0.251544	0.125772	−	0.992059i	$-0.459859\pi$
$709$	6.69786	0.251544	0.125772	+	0.992059i	$0.459859\pi$
$710$	5.18781	0.194695
$711$	−3.86428	−0.144922
$712$	−5.69743	−0.213520
$713$	8.59453	0.321868
$714$	7.36512	0.275633
$715$	−3.00168	−0.112257
$716$	−8.52445	−0.318574
$717$	−23.6479	−0.883146
$718$	−4.42579	−0.165169
$719$	44.8541	1.67278	0.836388	−	0.548138i	$-0.184663\pi$
$719$	44.8541	1.67278	0.836388	+	0.548138i	$0.184663\pi$
$720$	0.380691	0.0141875
$721$	34.8383	1.29745
$722$	6.94569	0.258492
$723$	6.46802	0.240548
$724$	−16.2390	−0.603519
$725$	4.85507	0.180313
$726$	−8.40758	−0.312035
$727$	−38.8742	−1.44177	−0.720883	−	0.693057i	$-0.756263\pi$
$727$	−38.8742	−1.44177	−0.720883	+	0.693057i	$0.756263\pi$
$728$	11.6585	0.432092
$729$	1.00000	0.0370370
$730$	−0.218517	−0.00808768
$731$	31.3283	1.15872
$732$	11.5400	0.426531
$733$	−20.2112	−0.746518	−0.373259	−	0.927727i	$-0.621760\pi$
$733$	−20.2112	−0.746518	−0.373259	+	0.927727i	$0.621760\pi$
$734$	37.6216	1.38864
$735$	−0.507198	−0.0187083
$736$	−1.00000	−0.0368605
$737$	1.14800	0.0422871
$738$	−1.80341	−0.0663844
$739$	0.331361	0.0121893	0.00609465	−	0.999981i	$-0.498060\pi$
$739$	0.331361	0.0121893	0.00609465	+	0.999981i	$0.498060\pi$
$740$	0.739458	0.0271830
$741$	24.9443	0.916352
$742$	14.1690	0.520159
$743$	8.01419	0.294012	0.147006	−	0.989136i	$-0.453036\pi$
$743$	8.01419	0.294012	0.147006	+	0.989136i	$0.453036\pi$
$744$	−8.59453	−0.315091
$745$	4.26668	0.156319
$746$	7.64452	0.279886
$747$	−9.03282	−0.330493
$748$	−4.98115	−0.182129
$749$	−4.09326	−0.149564
$750$	−3.75174	−0.136994
$751$	−18.2714	−0.666734	−0.333367	−	0.942797i	$-0.608185\pi$
$751$	−18.2714	−0.666734	−0.333367	+	0.942797i	$0.608185\pi$
$752$	8.12651	0.296343
$753$	8.01725	0.292165
$754$	4.89710	0.178342
$755$	4.49346	0.163534
$756$	−2.38069	−0.0865849
$757$	24.8846	0.904446	0.452223	−	0.891905i	$-0.350631\pi$
$757$	24.8846	0.904446	0.452223	+	0.891905i	$0.350631\pi$
$758$	24.0288	0.872765
$759$	−1.61010	−0.0584429
$760$	−1.93912	−0.0703394
$761$	−3.68962	−0.133748	−0.0668742	−	0.997761i	$-0.521303\pi$
$761$	−3.68962	−0.133748	−0.0668742	+	0.997761i	$0.521303\pi$
$762$	−19.4732	−0.705440
$763$	−9.96887	−0.360897
$764$	−25.2282	−0.912726
$765$	−1.17774	−0.0425813
$766$	−3.15011	−0.113818
$767$	−43.5249	−1.57159
$768$	1.00000	0.0360844
$769$	−43.4949	−1.56847	−0.784233	−	0.620466i	$-0.786943\pi$
$769$	−43.4949	−1.56847	−0.784233	+	0.620466i	$0.786943\pi$
$770$	−1.45925	−0.0525876
$771$	4.23248	0.152429
$772$	−5.98158	−0.215282
$773$	21.7836	0.783501	0.391751	−	0.920071i	$-0.371870\pi$
$773$	21.7836	0.783501	0.391751	+	0.920071i	$0.371870\pi$
$774$	−10.1265	−0.363990
$775$	41.7271	1.49888
$776$	13.3592	0.479567
$777$	−4.62427	−0.165895
$778$	−7.49163	−0.268588
$779$	9.18600	0.329123
$780$	−1.86428	−0.0667521
$781$	21.9414	0.785124
$782$	3.09369	0.110630
$783$	−1.00000	−0.0357371
$784$	−1.33231	−0.0475824
$785$	0.0100723	0.000359497 0
$786$	14.8787	0.530705
$787$	−24.9988	−0.891112	−0.445556	−	0.895254i	$-0.646994\pi$
$787$	−24.9988	−0.891112	−0.445556	+	0.895254i	$0.646994\pi$
$788$	−9.29504	−0.331122
$789$	7.82066	0.278423
$790$	−1.47110	−0.0523394
$791$	−12.5726	−0.447030
$792$	1.61010	0.0572124
$793$	−56.5126	−2.00682
$794$	14.4728	0.513620
$795$	−2.26573	−0.0803571
$796$	8.04838	0.285267
$797$	24.1669	0.856034	0.428017	−	0.903771i	$-0.359212\pi$
$797$	24.1669	0.856034	0.428017	+	0.903771i	$0.359212\pi$
$798$	12.1265	0.429274
$799$	−25.1409	−0.889421
$800$	−4.85507	−0.171653
$801$	−5.69743	−0.201309
$802$	−39.9021	−1.40899
$803$	−0.924197	−0.0326142
$804$	0.713000	0.0251456
$805$	0.906309	0.0319432
$806$	42.0883	1.48250
$807$	26.1895	0.921914
$808$	4.06563	0.143028
$809$	6.56830	0.230929	0.115465	−	0.993312i	$-0.463164\pi$
$809$	6.56830	0.230929	0.115465	+	0.993312i	$0.463164\pi$
$810$	0.380691	0.0133761
$811$	5.77533	0.202799	0.101400	−	0.994846i	$-0.467668\pi$
$811$	5.77533	0.202799	0.101400	+	0.994846i	$0.467668\pi$
$812$	2.38069	0.0835459
$813$	12.0968	0.424252
$814$	3.12747	0.109618
$815$	1.09158	0.0382363
$816$	−3.09369	−0.108301
$817$	51.5813	1.80460
$818$	35.9837	1.25814
$819$	11.6585	0.407380
$820$	−0.686542	−0.0239751
$821$	13.3515	0.465969	0.232985	−	0.972480i	$-0.425151\pi$
$821$	13.3515	0.465969	0.232985	+	0.972480i	$0.425151\pi$
$822$	−7.61338	−0.265547
$823$	43.7963	1.52664	0.763322	−	0.646019i	$-0.223567\pi$
$823$	43.7963	1.52664	0.763322	+	0.646019i	$0.223567\pi$
$824$	−14.6337	−0.509789
$825$	−7.81715	−0.272158
$826$	−21.1593	−0.736227
$827$	−25.4498	−0.884977	−0.442488	−	0.896774i	$-0.645904\pi$
$827$	−25.4498	−0.884977	−0.442488	+	0.896774i	$0.645904\pi$
$828$	−1.00000	−0.0347524
$829$	41.8117	1.45218	0.726090	−	0.687600i	$-0.241336\pi$
$829$	41.8117	1.45218	0.726090	+	0.687600i	$0.241336\pi$
$830$	−3.43872	−0.119360
$831$	−22.9443	−0.795929
$832$	−4.89710	−0.169776
$833$	4.12175	0.142810
$834$	0.664617	0.0230138
$835$	3.87825	0.134212
$836$	−8.20135	−0.283650
$837$	−8.59453	−0.297070
$838$	10.6304	0.367222
$839$	−16.9502	−0.585187	−0.292594	−	0.956237i	$-0.594518\pi$
$839$	−16.9502	−0.585187	−0.292594	+	0.956237i	$0.594518\pi$
$840$	−0.906309	−0.0312706
$841$	1.00000	0.0344828
$842$	−24.4576	−0.842866
$843$	−18.2046	−0.627001
$844$	−19.1799	−0.660198
$845$	4.18059	0.143817
$846$	8.12651	0.279395
$847$	20.0159	0.687753
$848$	−5.95162	−0.204379
$849$	−2.96390	−0.101721
$850$	15.0201	0.515185
$851$	−1.94241	−0.0665849
$852$	13.6273	0.466865
$853$	44.6821	1.52989	0.764943	−	0.644098i	$-0.222767\pi$
$853$	44.6821	1.52989	0.764943	+	0.644098i	$0.222767\pi$
$854$	−27.4732	−0.940114
$855$	−1.93912	−0.0663167
$856$	1.71936	0.0587664
$857$	0.565738	0.0193253	0.00966263	−	0.999953i	$-0.496924\pi$
$857$	0.565738	0.0193253	0.00966263	+	0.999953i	$0.496924\pi$
$858$	−7.88482	−0.269183
$859$	32.5091	1.10920	0.554598	−	0.832118i	$-0.312872\pi$
$859$	32.5091	1.10920	0.554598	+	0.832118i	$0.312872\pi$
$860$	−3.85507	−0.131457
$861$	4.29336	0.146317
$862$	21.9837	0.748768
$863$	46.3269	1.57699	0.788493	−	0.615044i	$-0.210862\pi$
$863$	46.3269	1.57699	0.788493	+	0.615044i	$0.210862\pi$
$864$	1.00000	0.0340207
$865$	−4.75131	−0.161549
$866$	−13.4572	−0.457295
$867$	−7.42907	−0.252305
$868$	20.4609	0.694489
$869$	−6.22188	−0.211063
$870$	−0.380691	−0.0129066
$871$	−3.49163	−0.118309
$872$	4.18738	0.141803
$873$	13.3592	0.452140
$874$	5.09369	0.172297
$875$	8.93174	0.301948
$876$	−0.574000	−0.0193937
$877$	3.10056	0.104698	0.0523492	−	0.998629i	$-0.483329\pi$
$877$	3.10056	0.104698	0.0523492	+	0.998629i	$0.483329\pi$
$878$	39.4951	1.33290
$879$	−25.2674	−0.852249
$880$	0.612951	0.0206626
$881$	9.72536	0.327656	0.163828	−	0.986489i	$-0.447616\pi$
$881$	9.72536	0.327656	0.163828	+	0.986489i	$0.447616\pi$
$882$	−1.33231	−0.0448612
$883$	−36.1829	−1.21765	−0.608826	−	0.793304i	$-0.708359\pi$
$883$	−36.1829	−1.21765	−0.608826	+	0.793304i	$0.708359\pi$
$884$	15.1501	0.509553
$885$	3.38354	0.113737
$886$	−11.5950	−0.389540
$887$	28.3309	0.951259	0.475630	−	0.879646i	$-0.342220\pi$
$887$	28.3309	0.951259	0.475630	+	0.879646i	$0.342220\pi$
$888$	1.94241	0.0651829
$889$	46.3597	1.55485
$890$	−2.16896	−0.0727038
$891$	1.61010	0.0539404
$892$	22.9833	0.769537
$893$	−41.3939	−1.38519
$894$	11.2077	0.374842
$895$	−3.24518	−0.108475
$896$	−2.38069	−0.0795333
$897$	4.89710	0.163509
$898$	−18.9330	−0.631802
$899$	8.59453	0.286644
$900$	−4.85507	−0.161836
$901$	18.4125	0.613408
$902$	−2.90367	−0.0966815
$903$	24.1081	0.802267
$904$	5.28107	0.175646
$905$	−6.18207	−0.205499
$906$	11.8034	0.392142
$907$	−31.8353	−1.05707	−0.528536	−	0.848911i	$-0.677259\pi$
$907$	−31.8353	−1.05707	−0.528536	+	0.848911i	$0.677259\pi$
$908$	4.44961	0.147665
$909$	4.06563	0.134848
$910$	4.43828	0.147128
$911$	40.3854	1.33803	0.669015	−	0.743249i	$-0.266716\pi$
$911$	40.3854	1.33803	0.669015	+	0.743249i	$0.266716\pi$
$912$	−5.09369	−0.168669
$913$	−14.5437	−0.481327
$914$	−36.3590	−1.20265
$915$	4.39318	0.145234
$916$	−0.787841	−0.0260310
$917$	−35.4215	−1.16972
$918$	−3.09369	−0.102107
$919$	−19.3767	−0.639177	−0.319589	−	0.947556i	$-0.603545\pi$
$919$	−19.3767	−0.639177	−0.319589	+	0.947556i	$0.603545\pi$
$920$	−0.380691	−0.0125510
$921$	−23.9632	−0.789613
$922$	−1.89542	−0.0624223
$923$	−66.7345	−2.19659
$924$	−3.83315	−0.126101
$925$	−9.43053	−0.310074
$926$	−8.93605	−0.293657
$927$	−14.6337	−0.480634
$928$	−1.00000	−0.0328266
$929$	31.5827	1.03619	0.518097	−	0.855322i	$-0.326641\pi$
$929$	31.5827	1.03619	0.518097	+	0.855322i	$0.326641\pi$
$930$	−3.27186	−0.107289
$931$	6.78637	0.222414
$932$	4.18082	0.136947
$933$	−18.1921	−0.595584
$934$	8.75195	0.286373
$935$	−1.89628	−0.0620150
$936$	−4.89710	−0.160067
$937$	7.81568	0.255327	0.127664	−	0.991818i	$-0.459252\pi$
$937$	7.81568	0.255327	0.127664	+	0.991818i	$0.459252\pi$
$938$	−1.69743	−0.0554231
$939$	−19.4640	−0.635184
$940$	3.09369	0.100905
$941$	−39.3408	−1.28247	−0.641236	−	0.767343i	$-0.721578\pi$
$941$	−39.3408	−1.28247	−0.641236	+	0.767343i	$0.721578\pi$
$942$	0.0264580	0.000862048 0
$943$	1.80341	0.0587270
$944$	8.88789	0.289276
$945$	−0.906309	−0.0294822
$946$	−16.3047	−0.530111
$947$	−26.3769	−0.857133	−0.428567	−	0.903510i	$-0.640981\pi$
$947$	−26.3769	−0.857133	−0.428567	+	0.903510i	$0.640981\pi$
$948$	−3.86428	−0.125506
$949$	2.81094	0.0912468
$950$	24.7302	0.802355
$951$	14.4793	0.469525
$952$	7.36512	0.238705
$953$	−7.98275	−0.258587	−0.129293	−	0.991606i	$-0.541271\pi$
$953$	−7.98275	−0.258587	−0.129293	+	0.991606i	$0.541271\pi$
$954$	−5.95162	−0.192691
$955$	−9.60417	−0.310784
$956$	−23.6479	−0.764827
$957$	−1.61010	−0.0520471
$958$	2.22305	0.0718235
$959$	18.1251	0.585290
$960$	0.380691	0.0122868
$961$	42.8660	1.38277
$962$	−9.51216	−0.306684
$963$	1.71936	0.0554055
$964$	6.46802	0.208321
$965$	−2.27714	−0.0733036
$966$	2.38069	0.0765975
$967$	39.6877	1.27627	0.638135	−	0.769924i	$-0.279706\pi$
$967$	39.6877	1.27627	0.638135	+	0.769924i	$0.279706\pi$
$968$	−8.40758	−0.270230
$969$	15.7583	0.506230
$970$	5.08573	0.163293
$971$	17.1617	0.550744	0.275372	−	0.961338i	$-0.411199\pi$
$971$	17.1617	0.550744	0.275372	+	0.961338i	$0.411199\pi$
$972$	1.00000	0.0320750
$973$	−1.58225	−0.0507245
$974$	−4.19659	−0.134467
$975$	23.7758	0.761434
$976$	11.5400	0.369387
$977$	6.19637	0.198239	0.0991197	−	0.995076i	$-0.468397\pi$
$977$	6.19637	0.198239	0.0991197	+	0.995076i	$0.468397\pi$
$978$	2.86736	0.0916880
$979$	−9.17343	−0.293184
$980$	−0.507198	−0.0162019
$981$	4.18738	0.133693
$982$	8.49777	0.271175
$983$	−16.2726	−0.519015	−0.259508	−	0.965741i	$-0.583560\pi$
$983$	−16.2726	−0.519015	−0.259508	+	0.965741i	$0.583560\pi$
$984$	−1.80341	−0.0574905
$985$	−3.53854	−0.112747
$986$	3.09369	0.0985232
$987$	−19.3467	−0.615812
$988$	24.9443	0.793584
$989$	10.1265	0.322004
$990$	0.612951	0.0194809
$991$	34.6304	1.10007	0.550036	−	0.835141i	$-0.314614\pi$
$991$	34.6304	1.10007	0.550036	+	0.835141i	$0.314614\pi$
$992$	−8.59453	−0.272877
$993$	9.50084	0.301500
$994$	−32.4425	−1.02901
$995$	3.06395	0.0971337
$996$	−9.03282	−0.286216
$997$	−1.88957	−0.0598433	−0.0299217	−	0.999552i	$-0.509526\pi$
$997$	−1.88957	−0.0598433	−0.0299217	+	0.999552i	$0.509526\pi$
$998$	−7.32478	−0.231862
$999$	1.94241	0.0614551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bc.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bc.1.3	✓	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.380691 q^{5} +1.00000 q^{6} -2.38069 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.380691 q^{5} +1.00000 q^{6} -2.38069 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.380691 q^{10} +1.61010 q^{11} +1.00000 q^{12} -4.89710 q^{13} -2.38069 q^{14} +0.380691 q^{15} +1.00000 q^{16} -3.09369 q^{17} +1.00000 q^{18} -5.09369 q^{19} +0.380691 q^{20} -2.38069 q^{21} +1.61010 q^{22} -1.00000 q^{23} +1.00000 q^{24} -4.85507 q^{25} -4.89710 q^{26} +1.00000 q^{27} -2.38069 q^{28} -1.00000 q^{29} +0.380691 q^{30} -8.59453 q^{31} +1.00000 q^{32} +1.61010 q^{33} -3.09369 q^{34} -0.906309 q^{35} +1.00000 q^{36} +1.94241 q^{37} -5.09369 q^{38} -4.89710 q^{39} +0.380691 q^{40} -1.80341 q^{41} -2.38069 q^{42} -10.1265 q^{43} +1.61010 q^{44} +0.380691 q^{45} -1.00000 q^{46} +8.12651 q^{47} +1.00000 q^{48} -1.33231 q^{49} -4.85507 q^{50} -3.09369 q^{51} -4.89710 q^{52} -5.95162 q^{53} +1.00000 q^{54} +0.612951 q^{55} -2.38069 q^{56} -5.09369 q^{57} -1.00000 q^{58} +8.88789 q^{59} +0.380691 q^{60} +11.5400 q^{61} -8.59453 q^{62} -2.38069 q^{63} +1.00000 q^{64} -1.86428 q^{65} +1.61010 q^{66} +0.713000 q^{67} -3.09369 q^{68} -1.00000 q^{69} -0.906309 q^{70} +13.6273 q^{71} +1.00000 q^{72} -0.574000 q^{73} +1.94241 q^{74} -4.85507 q^{75} -5.09369 q^{76} -3.83315 q^{77} -4.89710 q^{78} -3.86428 q^{79} +0.380691 q^{80} +1.00000 q^{81} -1.80341 q^{82} -9.03282 q^{83} -2.38069 q^{84} -1.17774 q^{85} -10.1265 q^{86} -1.00000 q^{87} +1.61010 q^{88} -5.69743 q^{89} +0.380691 q^{90} +11.6585 q^{91} -1.00000 q^{92} -8.59453 q^{93} +8.12651 q^{94} -1.93912 q^{95} +1.00000 q^{96} +13.3592 q^{97} -1.33231 q^{98} +1.61010 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{9} - 4 q^{10} - 12 q^{11} + 4 q^{12} - 6 q^{13} - 4 q^{14} - 4 q^{15} + 4 q^{16} - 2 q^{17} + 4 q^{18} - 10 q^{19} - 4 q^{20} - 4 q^{21} - 12 q^{22} - 4 q^{23} + 4 q^{24} + 2 q^{25} - 6 q^{26} + 4 q^{27} - 4 q^{28} - 4 q^{29} - 4 q^{30} - 6 q^{31} + 4 q^{32} - 12 q^{33} - 2 q^{34} - 14 q^{35} + 4 q^{36} - 10 q^{37} - 10 q^{38} - 6 q^{39} - 4 q^{40} - 4 q^{41} - 4 q^{42} - 14 q^{43} - 12 q^{44} - 4 q^{45} - 4 q^{46} + 6 q^{47} + 4 q^{48} - 6 q^{49} + 2 q^{50} - 2 q^{51} - 6 q^{52} - 30 q^{53} + 4 q^{54} + 22 q^{55} - 4 q^{56} - 10 q^{57} - 4 q^{58} - 2 q^{59} - 4 q^{60} - 2 q^{61} - 6 q^{62} - 4 q^{63} + 4 q^{64} - 10 q^{65} - 12 q^{66} - 2 q^{67} - 2 q^{68} - 4 q^{69} - 14 q^{70} + 10 q^{71} + 4 q^{72} - 12 q^{73} - 10 q^{74} + 2 q^{75} - 10 q^{76} + 2 q^{77} - 6 q^{78} - 18 q^{79} - 4 q^{80} + 4 q^{81} - 4 q^{82} - 20 q^{83} - 4 q^{84} - 10 q^{85} - 14 q^{86} - 4 q^{87} - 12 q^{88} - 8 q^{89} - 4 q^{90} + 22 q^{91} - 4 q^{92} - 6 q^{93} + 6 q^{94} - 2 q^{95} + 4 q^{96} + 2 q^{97} - 6 q^{98} - 12 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^9 - 4 * q^10 - 12 * q^11 + 4 * q^12 - 6 * q^13 - 4 * q^14 - 4 * q^15 + 4 * q^16 - 2 * q^17 + 4 * q^18 - 10 * q^19 - 4 * q^20 - 4 * q^21 - 12 * q^22 - 4 * q^23 + 4 * q^24 + 2 * q^25 - 6 * q^26 + 4 * q^27 - 4 * q^28 - 4 * q^29 - 4 * q^30 - 6 * q^31 + 4 * q^32 - 12 * q^33 - 2 * q^34 - 14 * q^35 + 4 * q^36 - 10 * q^37 - 10 * q^38 - 6 * q^39 - 4 * q^40 - 4 * q^41 - 4 * q^42 - 14 * q^43 - 12 * q^44 - 4 * q^45 - 4 * q^46 + 6 * q^47 + 4 * q^48 - 6 * q^49 + 2 * q^50 - 2 * q^51 - 6 * q^52 - 30 * q^53 + 4 * q^54 + 22 * q^55 - 4 * q^56 - 10 * q^57 - 4 * q^58 - 2 * q^59 - 4 * q^60 - 2 * q^61 - 6 * q^62 - 4 * q^63 + 4 * q^64 - 10 * q^65 - 12 * q^66 - 2 * q^67 - 2 * q^68 - 4 * q^69 - 14 * q^70 + 10 * q^71 + 4 * q^72 - 12 * q^73 - 10 * q^74 + 2 * q^75 - 10 * q^76 + 2 * q^77 - 6 * q^78 - 18 * q^79 - 4 * q^80 + 4 * q^81 - 4 * q^82 - 20 * q^83 - 4 * q^84 - 10 * q^85 - 14 * q^86 - 4 * q^87 - 12 * q^88 - 8 * q^89 - 4 * q^90 + 22 * q^91 - 4 * q^92 - 6 * q^93 + 6 * q^94 - 2 * q^95 + 4 * q^96 + 2 * q^97 - 6 * q^98 - 12 * q^99

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