LMFDB - Embedded newform 4002.2.a.bk.1.1

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 30x^{6} + 52x^{5} + 267x^{4} - 352x^{3} - 632x^{2} + 240x + 288 $ x^8 - 2x^7 - 30x^6 + 52x^5 + 267x^4 - 352x^3 - 632x^2 + 240*x + 288
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.78725$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.78725 q^{5} -1.00000 q^{6} +3.57193 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.78725 q^{5} -1.00000 q^{6} +3.57193 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.78725 q^{10} +2.34220 q^{11} -1.00000 q^{12} +4.66679 q^{13} +3.57193 q^{14} +3.78725 q^{15} +1.00000 q^{16} -1.44505 q^{17} +1.00000 q^{18} -1.40051 q^{19} -3.78725 q^{20} -3.57193 q^{21} +2.34220 q^{22} -1.00000 q^{23} -1.00000 q^{24} +9.34325 q^{25} +4.66679 q^{26} -1.00000 q^{27} +3.57193 q^{28} -1.00000 q^{29} +3.78725 q^{30} +1.30550 q^{31} +1.00000 q^{32} -2.34220 q^{33} -1.44505 q^{34} -13.5278 q^{35} +1.00000 q^{36} +8.01866 q^{37} -1.40051 q^{38} -4.66679 q^{39} -3.78725 q^{40} -7.66636 q^{41} -3.57193 q^{42} -5.48760 q^{43} +2.34220 q^{44} -3.78725 q^{45} -1.00000 q^{46} -6.80634 q^{47} -1.00000 q^{48} +5.75866 q^{49} +9.34325 q^{50} +1.44505 q^{51} +4.66679 q^{52} +0.554378 q^{53} -1.00000 q^{54} -8.87050 q^{55} +3.57193 q^{56} +1.40051 q^{57} -1.00000 q^{58} +3.88380 q^{59} +3.78725 q^{60} -0.342202 q^{61} +1.30550 q^{62} +3.57193 q^{63} +1.00000 q^{64} -17.6743 q^{65} -2.34220 q^{66} +11.2917 q^{67} -1.44505 q^{68} +1.00000 q^{69} -13.5278 q^{70} +15.1708 q^{71} +1.00000 q^{72} +0.445478 q^{73} +8.01866 q^{74} -9.34325 q^{75} -1.40051 q^{76} +8.36617 q^{77} -4.66679 q^{78} -5.56500 q^{79} -3.78725 q^{80} +1.00000 q^{81} -7.66636 q^{82} +8.84556 q^{83} -3.57193 q^{84} +5.47275 q^{85} -5.48760 q^{86} +1.00000 q^{87} +2.34220 q^{88} +7.99564 q^{89} -3.78725 q^{90} +16.6694 q^{91} -1.00000 q^{92} -1.30550 q^{93} -6.80634 q^{94} +5.30409 q^{95} -1.00000 q^{96} -8.60968 q^{97} +5.75866 q^{98} +2.34220 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 - 8 * q^12 + 13 * q^13 + 5 * q^14 - 2 * q^15 + 8 * q^16 + 5 * q^17 + 8 * q^18 + 7 * q^19 + 2 * q^20 - 5 * q^21 + 3 * q^22 - 8 * q^23 - 8 * q^24 + 24 * q^25 + 13 * q^26 - 8 * q^27 + 5 * q^28 - 8 * q^29 - 2 * q^30 + 15 * q^31 + 8 * q^32 - 3 * q^33 + 5 * q^34 + 9 * q^35 + 8 * q^36 + 22 * q^37 + 7 * q^38 - 13 * q^39 + 2 * q^40 - 8 * q^41 - 5 * q^42 - q^43 + 3 * q^44 + 2 * q^45 - 8 * q^46 - 9 * q^47 - 8 * q^48 + 33 * q^49 + 24 * q^50 - 5 * q^51 + 13 * q^52 + 14 * q^53 - 8 * q^54 - 17 * q^55 + 5 * q^56 - 7 * q^57 - 8 * q^58 - 4 * q^59 - 2 * q^60 + 13 * q^61 + 15 * q^62 + 5 * q^63 + 8 * q^64 + 21 * q^65 - 3 * q^66 - 3 * q^67 + 5 * q^68 + 8 * q^69 + 9 * q^70 + 7 * q^71 + 8 * q^72 + 16 * q^73 + 22 * q^74 - 24 * q^75 + 7 * q^76 - 13 * q^78 + 14 * q^79 + 2 * q^80 + 8 * q^81 - 8 * q^82 + 36 * q^83 - 5 * q^84 + 47 * q^85 - q^86 + 8 * q^87 + 3 * q^88 - 12 * q^89 + 2 * q^90 + 20 * q^91 - 8 * q^92 - 15 * q^93 - 9 * q^94 + 7 * q^95 - 8 * q^96 + 10 * q^97 + 33 * q^98 + 3 * 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
$5$	−3.78725	−1.69371	−0.846855	−	0.531825i	$-0.821507\pi$
$5$	−3.78725	−1.69371	−0.846855	+	0.531825i	$0.821507\pi$
$6$	−1.00000	−0.408248
$7$	3.57193	1.35006	0.675031	−	0.737790i	$-0.264130\pi$
$7$	3.57193	1.35006	0.675031	+	0.737790i	$0.264130\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.78725	−1.19763
$11$	2.34220	0.706200	0.353100	−	0.935585i	$-0.385127\pi$
$11$	2.34220	0.706200	0.353100	+	0.935585i	$0.385127\pi$
$12$	−1.00000	−0.288675
$13$	4.66679	1.29434	0.647168	−	0.762347i	$-0.275953\pi$
$13$	4.66679	1.29434	0.647168	+	0.762347i	$0.275953\pi$
$14$	3.57193	0.954638
$15$	3.78725	0.977863
$16$	1.00000	0.250000
$17$	−1.44505	−0.350475	−0.175238	−	0.984526i	$-0.556069\pi$
$17$	−1.44505	−0.350475	−0.175238	+	0.984526i	$0.556069\pi$
$18$	1.00000	0.235702
$19$	−1.40051	−0.321300	−0.160650	−	0.987011i	$-0.551359\pi$
$19$	−1.40051	−0.321300	−0.160650	+	0.987011i	$0.551359\pi$
$20$	−3.78725	−0.846855
$21$	−3.57193	−0.779458
$22$	2.34220	0.499359
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	9.34325	1.86865
$26$	4.66679	0.915234
$27$	−1.00000	−0.192450
$28$	3.57193	0.675031
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	3.78725	0.691454
$31$	1.30550	0.234475	0.117238	−	0.993104i	$-0.462596\pi$
$31$	1.30550	0.234475	0.117238	+	0.993104i	$0.462596\pi$
$32$	1.00000	0.176777
$33$	−2.34220	−0.407725
$34$	−1.44505	−0.247823
$35$	−13.5278	−2.28661
$36$	1.00000	0.166667
$37$	8.01866	1.31826	0.659130	−	0.752029i	$-0.270925\pi$
$37$	8.01866	1.31826	0.659130	+	0.752029i	$0.270925\pi$
$38$	−1.40051	−0.227193
$39$	−4.66679	−0.747285
$40$	−3.78725	−0.598817
$41$	−7.66636	−1.19728	−0.598642	−	0.801016i	$-0.704293\pi$
$41$	−7.66636	−1.19728	−0.598642	+	0.801016i	$0.704293\pi$
$42$	−3.57193	−0.551160
$43$	−5.48760	−0.836851	−0.418425	−	0.908251i	$-0.637418\pi$
$43$	−5.48760	−0.836851	−0.418425	+	0.908251i	$0.637418\pi$
$44$	2.34220	0.353100
$45$	−3.78725	−0.564570
$46$	−1.00000	−0.147442
$47$	−6.80634	−0.992807	−0.496403	−	0.868092i	$-0.665346\pi$
$47$	−6.80634	−0.992807	−0.496403	+	0.868092i	$0.665346\pi$
$48$	−1.00000	−0.144338
$49$	5.75866	0.822666
$50$	9.34325	1.32134
$51$	1.44505	0.202347
$52$	4.66679	0.647168
$53$	0.554378	0.0761497	0.0380748	−	0.999275i	$-0.487877\pi$
$53$	0.554378	0.0761497	0.0380748	+	0.999275i	$0.487877\pi$
$54$	−1.00000	−0.136083
$55$	−8.87050	−1.19610
$56$	3.57193	0.477319
$57$	1.40051	0.185503
$58$	−1.00000	−0.131306
$59$	3.88380	0.505628	0.252814	−	0.967515i	$-0.418644\pi$
$59$	3.88380	0.505628	0.252814	+	0.967515i	$0.418644\pi$
$60$	3.78725	0.488932
$61$	−0.342202	−0.0438145	−0.0219072	−	0.999760i	$-0.506974\pi$
$61$	−0.342202	−0.0438145	−0.0219072	+	0.999760i	$0.506974\pi$
$62$	1.30550	0.165799
$63$	3.57193	0.450020
$64$	1.00000	0.125000
$65$	−17.6743	−2.19223
$66$	−2.34220	−0.288305
$67$	11.2917	1.37950	0.689749	−	0.724049i	$-0.257721\pi$
$67$	11.2917	1.37950	0.689749	+	0.724049i	$0.257721\pi$
$68$	−1.44505	−0.175238
$69$	1.00000	0.120386
$70$	−13.5278	−1.61688
$71$	15.1708	1.80044	0.900221	−	0.435434i	$-0.143405\pi$
$71$	15.1708	1.80044	0.900221	+	0.435434i	$0.143405\pi$
$72$	1.00000	0.117851
$73$	0.445478	0.0521393	0.0260696	−	0.999660i	$-0.491701\pi$
$73$	0.445478	0.0521393	0.0260696	+	0.999660i	$0.491701\pi$
$74$	8.01866	0.932150
$75$	−9.34325	−1.07887
$76$	−1.40051	−0.160650
$77$	8.36617	0.953414
$78$	−4.66679	−0.528410
$79$	−5.56500	−0.626111	−0.313056	−	0.949735i	$-0.601353\pi$
$79$	−5.56500	−0.626111	−0.313056	+	0.949735i	$0.601353\pi$
$80$	−3.78725	−0.423427
$81$	1.00000	0.111111
$82$	−7.66636	−0.846608
$83$	8.84556	0.970926	0.485463	−	0.874257i	$-0.338651\pi$
$83$	8.84556	0.970926	0.485463	+	0.874257i	$0.338651\pi$
$84$	−3.57193	−0.389729
$85$	5.47275	0.593603
$86$	−5.48760	−0.591743
$87$	1.00000	0.107211
$88$	2.34220	0.249680
$89$	7.99564	0.847537	0.423768	−	0.905771i	$-0.360707\pi$
$89$	7.99564	0.847537	0.423768	+	0.905771i	$0.360707\pi$
$90$	−3.78725	−0.399211
$91$	16.6694	1.74743
$92$	−1.00000	−0.104257
$93$	−1.30550	−0.135374
$94$	−6.80634	−0.702020
$95$	5.30409	0.544188
$96$	−1.00000	−0.102062
$97$	−8.60968	−0.874180	−0.437090	−	0.899418i	$-0.643991\pi$
$97$	−8.60968	−0.874180	−0.437090	+	0.899418i	$0.643991\pi$
$98$	5.75866	0.581713
$99$	2.34220	0.235400
$100$	9.34325	0.934325
$101$	−1.37139	−0.136458	−0.0682291	−	0.997670i	$-0.521735\pi$
$101$	−1.37139	−0.136458	−0.0682291	+	0.997670i	$0.521735\pi$
$102$	1.44505	0.143081
$103$	16.0535	1.58180	0.790898	−	0.611948i	$-0.209614\pi$
$103$	16.0535	1.58180	0.790898	+	0.611948i	$0.209614\pi$
$104$	4.66679	0.457617
$105$	13.5278	1.32018
$106$	0.554378	0.0538459
$107$	0.933549	0.0902496	0.0451248	−	0.998981i	$-0.485631\pi$
$107$	0.933549	0.0902496	0.0451248	+	0.998981i	$0.485631\pi$
$108$	−1.00000	−0.0962250
$109$	16.4318	1.57388	0.786938	−	0.617031i	$-0.211665\pi$
$109$	16.4318	1.57388	0.786938	+	0.617031i	$0.211665\pi$
$110$	−8.87050	−0.845769
$111$	−8.01866	−0.761097
$112$	3.57193	0.337515
$113$	5.68368	0.534676	0.267338	−	0.963603i	$-0.413856\pi$
$113$	5.68368	0.534676	0.267338	+	0.963603i	$0.413856\pi$
$114$	1.40051	0.131170
$115$	3.78725	0.353163
$116$	−1.00000	−0.0928477
$117$	4.66679	0.431445
$118$	3.88380	0.357533
$119$	−5.16160	−0.473163
$120$	3.78725	0.345727
$121$	−5.51409	−0.501281
$122$	−0.342202	−0.0309815
$123$	7.66636	0.691253
$124$	1.30550	0.117238
$125$	−16.4490	−1.47124
$126$	3.57193	0.318213
$127$	−11.7171	−1.03972	−0.519860	−	0.854251i	$-0.674016\pi$
$127$	−11.7171	−1.03972	−0.519860	+	0.854251i	$0.674016\pi$
$128$	1.00000	0.0883883
$129$	5.48760	0.483156
$130$	−17.6743	−1.55014
$131$	12.3385	1.07802	0.539009	−	0.842300i	$-0.318799\pi$
$131$	12.3385	1.07802	0.539009	+	0.842300i	$0.318799\pi$
$132$	−2.34220	−0.203863
$133$	−5.00253	−0.433774
$134$	11.2917	0.975452
$135$	3.78725	0.325954
$136$	−1.44505	−0.123912
$137$	−3.56500	−0.304578	−0.152289	−	0.988336i	$-0.548665\pi$
$137$	−3.56500	−0.304578	−0.152289	+	0.988336i	$0.548665\pi$
$138$	1.00000	0.0851257
$139$	3.62041	0.307079	0.153539	−	0.988143i	$-0.450933\pi$
$139$	3.62041	0.307079	0.153539	+	0.988143i	$0.450933\pi$
$140$	−13.5278	−1.14331
$141$	6.80634	0.573197
$142$	15.1708	1.27310
$143$	10.9306	0.914061
$144$	1.00000	0.0833333
$145$	3.78725	0.314514
$146$	0.445478	0.0368680
$147$	−5.75866	−0.474966
$148$	8.01866	0.659130
$149$	−9.41698	−0.771469	−0.385735	−	0.922610i	$-0.626052\pi$
$149$	−9.41698	−0.771469	−0.385735	+	0.922610i	$0.626052\pi$
$150$	−9.34325	−0.762873
$151$	4.85524	0.395114	0.197557	−	0.980291i	$-0.436699\pi$
$151$	4.85524	0.395114	0.197557	+	0.980291i	$0.436699\pi$
$152$	−1.40051	−0.113597
$153$	−1.44505	−0.116825
$154$	8.36617	0.674166
$155$	−4.94426	−0.397132
$156$	−4.66679	−0.373643
$157$	5.30967	0.423758	0.211879	−	0.977296i	$-0.432042\pi$
$157$	5.30967	0.423758	0.211879	+	0.977296i	$0.432042\pi$
$158$	−5.56500	−0.442728
$159$	−0.554378	−0.0439650
$160$	−3.78725	−0.299408
$161$	−3.57193	−0.281507
$162$	1.00000	0.0785674
$163$	−4.22663	−0.331055	−0.165528	−	0.986205i	$-0.552933\pi$
$163$	−4.22663	−0.331055	−0.165528	+	0.986205i	$0.552933\pi$
$164$	−7.66636	−0.598642
$165$	8.87050	0.690568
$166$	8.84556	0.686549
$167$	−17.4722	−1.35204	−0.676021	−	0.736882i	$-0.736297\pi$
$167$	−17.4722	−1.35204	−0.676021	+	0.736882i	$0.736297\pi$
$168$	−3.57193	−0.275580
$169$	8.77897	0.675305
$170$	5.47275	0.419741
$171$	−1.40051	−0.107100
$172$	−5.48760	−0.418425
$173$	6.23022	0.473675	0.236837	−	0.971549i	$-0.423889\pi$
$173$	6.23022	0.473675	0.236837	+	0.971549i	$0.423889\pi$
$174$	1.00000	0.0758098
$175$	33.3734	2.52279
$176$	2.34220	0.176550
$177$	−3.88380	−0.291924
$178$	7.99564	0.599299
$179$	16.0519	1.19978	0.599889	−	0.800083i	$-0.295211\pi$
$179$	16.0519	1.19978	0.599889	+	0.800083i	$0.295211\pi$
$180$	−3.78725	−0.282285
$181$	2.16116	0.160637	0.0803187	−	0.996769i	$-0.474406\pi$
$181$	2.16116	0.160637	0.0803187	+	0.996769i	$0.474406\pi$
$182$	16.6694	1.23562
$183$	0.342202	0.0252963
$184$	−1.00000	−0.0737210
$185$	−30.3687	−2.23275
$186$	−1.30550	−0.0957240
$187$	−3.38459	−0.247506
$188$	−6.80634	−0.496403
$189$	−3.57193	−0.259819
$190$	5.30409	0.384799
$191$	25.9760	1.87956	0.939779	−	0.341782i	$-0.111031\pi$
$191$	25.9760	1.87956	0.939779	+	0.341782i	$0.111031\pi$
$192$	−1.00000	−0.0721688
$193$	21.7463	1.56533	0.782667	−	0.622441i	$-0.213859\pi$
$193$	21.7463	1.56533	0.782667	+	0.622441i	$0.213859\pi$
$194$	−8.60968	−0.618139
$195$	17.6743	1.26568
$196$	5.75866	0.411333
$197$	2.10710	0.150125	0.0750624	−	0.997179i	$-0.476084\pi$
$197$	2.10710	0.150125	0.0750624	+	0.997179i	$0.476084\pi$
$198$	2.34220	0.166453
$199$	25.3300	1.79560	0.897798	−	0.440408i	$-0.145166\pi$
$199$	25.3300	1.79560	0.897798	+	0.440408i	$0.145166\pi$
$200$	9.34325	0.660668
$201$	−11.2917	−0.796453
$202$	−1.37139	−0.0964905
$203$	−3.57193	−0.250700
$204$	1.44505	0.101173
$205$	29.0344	2.02785
$206$	16.0535	1.11850
$207$	−1.00000	−0.0695048
$208$	4.66679	0.323584
$209$	−3.28029	−0.226902
$210$	13.5278	0.933505
$211$	−8.16289	−0.561957	−0.280978	−	0.959714i	$-0.590659\pi$
$211$	−8.16289	−0.561957	−0.280978	+	0.959714i	$0.590659\pi$
$212$	0.554378	0.0380748
$213$	−15.1708	−1.03949
$214$	0.933549	0.0638161
$215$	20.7829	1.41738
$216$	−1.00000	−0.0680414
$217$	4.66316	0.316556
$218$	16.4318	1.11290
$219$	−0.445478	−0.0301026
$220$	−8.87050	−0.598049
$221$	−6.74374	−0.453633
$222$	−8.01866	−0.538177
$223$	18.1626	1.21626	0.608128	−	0.793839i	$-0.291921\pi$
$223$	18.1626	1.21626	0.608128	+	0.793839i	$0.291921\pi$
$224$	3.57193	0.238659
$225$	9.34325	0.622883
$226$	5.68368	0.378073
$227$	13.8314	0.918024	0.459012	−	0.888430i	$-0.348204\pi$
$227$	13.8314	0.918024	0.459012	+	0.888430i	$0.348204\pi$
$228$	1.40051	0.0927513
$229$	−14.2401	−0.941010	−0.470505	−	0.882397i	$-0.655928\pi$
$229$	−14.2401	−0.941010	−0.470505	+	0.882397i	$0.655928\pi$
$230$	3.78725	0.249724
$231$	−8.36617	−0.550454
$232$	−1.00000	−0.0656532
$233$	−20.5518	−1.34639	−0.673196	−	0.739464i	$-0.735079\pi$
$233$	−20.5518	−1.34639	−0.673196	+	0.739464i	$0.735079\pi$
$234$	4.66679	0.305078
$235$	25.7773	1.68153
$236$	3.88380	0.252814
$237$	5.56500	0.361486
$238$	−5.16160	−0.334577
$239$	−0.852557	−0.0551474	−0.0275737	−	0.999620i	$-0.508778\pi$
$239$	−0.852557	−0.0551474	−0.0275737	+	0.999620i	$0.508778\pi$
$240$	3.78725	0.244466
$241$	−17.9577	−1.15675	−0.578377	−	0.815769i	$-0.696314\pi$
$241$	−17.9577	−1.15675	−0.578377	+	0.815769i	$0.696314\pi$
$242$	−5.51409	−0.354459
$243$	−1.00000	−0.0641500
$244$	−0.342202	−0.0219072
$245$	−21.8095	−1.39336
$246$	7.66636	0.488789
$247$	−6.53591	−0.415870
$248$	1.30550	0.0828994
$249$	−8.84556	−0.560565
$250$	−16.4490	−1.04032
$251$	5.79440	0.365739	0.182870	−	0.983137i	$-0.441461\pi$
$251$	5.79440	0.365739	0.182870	+	0.983137i	$0.441461\pi$
$252$	3.57193	0.225010
$253$	−2.34220	−0.147253
$254$	−11.7171	−0.735194
$255$	−5.47275	−0.342717
$256$	1.00000	0.0625000
$257$	3.15754	0.196962	0.0984809	−	0.995139i	$-0.468602\pi$
$257$	3.15754	0.196962	0.0984809	+	0.995139i	$0.468602\pi$
$258$	5.48760	0.341643
$259$	28.6421	1.77973
$260$	−17.6743	−1.09611
$261$	−1.00000	−0.0618984
$262$	12.3385	0.762273
$263$	−2.09401	−0.129122	−0.0645611	−	0.997914i	$-0.520565\pi$
$263$	−2.09401	−0.129122	−0.0645611	+	0.997914i	$0.520565\pi$
$264$	−2.34220	−0.144153
$265$	−2.09957	−0.128975
$266$	−5.00253	−0.306725
$267$	−7.99564	−0.489325
$268$	11.2917	0.689749
$269$	14.0144	0.854470	0.427235	−	0.904141i	$-0.359488\pi$
$269$	14.0144	0.854470	0.427235	+	0.904141i	$0.359488\pi$
$270$	3.78725	0.230485
$271$	−26.2382	−1.59386	−0.796929	−	0.604073i	$-0.793543\pi$
$271$	−26.2382	−1.59386	−0.796929	+	0.604073i	$0.793543\pi$
$272$	−1.44505	−0.0876188
$273$	−16.6694	−1.00888
$274$	−3.56500	−0.215370
$275$	21.8838	1.31964
$276$	1.00000	0.0601929
$277$	19.0265	1.14319	0.571595	−	0.820536i	$-0.306325\pi$
$277$	19.0265	1.14319	0.571595	+	0.820536i	$0.306325\pi$
$278$	3.62041	0.217138
$279$	1.30550	0.0781583
$280$	−13.5278	−0.808439
$281$	13.7926	0.822800	0.411400	−	0.911455i	$-0.365040\pi$
$281$	13.7926	0.822800	0.411400	+	0.911455i	$0.365040\pi$
$282$	6.80634	0.405312
$283$	−19.6024	−1.16524	−0.582622	−	0.812743i	$-0.697973\pi$
$283$	−19.6024	−1.16524	−0.582622	+	0.812743i	$0.697973\pi$
$284$	15.1708	0.900221
$285$	−5.30409	−0.314187
$286$	10.9306	0.646338
$287$	−27.3837	−1.61641
$288$	1.00000	0.0589256
$289$	−14.9118	−0.877167
$290$	3.78725	0.222395
$291$	8.60968	0.504708
$292$	0.445478	0.0260696
$293$	−29.3911	−1.71704	−0.858522	−	0.512776i	$-0.828617\pi$
$293$	−29.3911	−1.71704	−0.858522	+	0.512776i	$0.828617\pi$
$294$	−5.75866	−0.335852
$295$	−14.7089	−0.856387
$296$	8.01866	0.466075
$297$	−2.34220	−0.135908
$298$	−9.41698	−0.545511
$299$	−4.66679	−0.269888
$300$	−9.34325	−0.539433
$301$	−19.6013	−1.12980
$302$	4.85524	0.279388
$303$	1.37139	0.0787841
$304$	−1.40051	−0.0803249
$305$	1.29600	0.0742090
$306$	−1.44505	−0.0826078
$307$	−21.9547	−1.25302	−0.626512	−	0.779412i	$-0.715518\pi$
$307$	−21.9547	−1.25302	−0.626512	+	0.779412i	$0.715518\pi$
$308$	8.36617	0.476707
$309$	−16.0535	−0.913251
$310$	−4.94426	−0.280815
$311$	−24.9590	−1.41530	−0.707649	−	0.706564i	$-0.750244\pi$
$311$	−24.9590	−1.41530	−0.707649	+	0.706564i	$0.750244\pi$
$312$	−4.66679	−0.264205
$313$	15.0472	0.850521	0.425260	−	0.905071i	$-0.360183\pi$
$313$	15.0472	0.850521	0.425260	+	0.905071i	$0.360183\pi$
$314$	5.30967	0.299642
$315$	−13.5278	−0.762204
$316$	−5.56500	−0.313056
$317$	32.8887	1.84721	0.923606	−	0.383344i	$-0.125227\pi$
$317$	32.8887	1.84721	0.923606	+	0.383344i	$0.125227\pi$
$318$	−0.554378	−0.0310880
$319$	−2.34220	−0.131138
$320$	−3.78725	−0.211714
$321$	−0.933549	−0.0521056
$322$	−3.57193	−0.199056
$323$	2.02381	0.112608
$324$	1.00000	0.0555556
$325$	43.6030	2.41866
$326$	−4.22663	−0.234091
$327$	−16.4318	−0.908678
$328$	−7.66636	−0.423304
$329$	−24.3117	−1.34035
$330$	8.87050	0.488305
$331$	22.2324	1.22201	0.611003	−	0.791628i	$-0.290766\pi$
$331$	22.2324	1.22201	0.611003	+	0.791628i	$0.290766\pi$
$332$	8.84556	0.485463
$333$	8.01866	0.439420
$334$	−17.4722	−0.956038
$335$	−42.7644	−2.33647
$336$	−3.57193	−0.194865
$337$	7.92729	0.431827	0.215913	−	0.976413i	$-0.430727\pi$
$337$	7.92729	0.431827	0.215913	+	0.976413i	$0.430727\pi$
$338$	8.77897	0.477513
$339$	−5.68368	−0.308695
$340$	5.47275	0.296802
$341$	3.05775	0.165586
$342$	−1.40051	−0.0757311
$343$	−4.43397	−0.239412
$344$	−5.48760	−0.295871
$345$	−3.78725	−0.203899
$346$	6.23022	0.334939
$347$	−22.8001	−1.22397	−0.611987	−	0.790868i	$-0.709630\pi$
$347$	−22.8001	−1.22397	−0.611987	+	0.790868i	$0.709630\pi$
$348$	1.00000	0.0536056
$349$	11.4172	0.611149	0.305574	−	0.952168i	$-0.401151\pi$
$349$	11.4172	0.611149	0.305574	+	0.952168i	$0.401151\pi$
$350$	33.3734	1.78388
$351$	−4.66679	−0.249095
$352$	2.34220	0.124840
$353$	−33.5445	−1.78540	−0.892698	−	0.450656i	$-0.851190\pi$
$353$	−33.5445	−1.78540	−0.892698	+	0.450656i	$0.851190\pi$
$354$	−3.88380	−0.206422
$355$	−57.4556	−3.04942
$356$	7.99564	0.423768
$357$	5.16160	0.273181
$358$	16.0519	0.848371
$359$	5.08140	0.268186	0.134093	−	0.990969i	$-0.457188\pi$
$359$	5.08140	0.268186	0.134093	+	0.990969i	$0.457188\pi$
$360$	−3.78725	−0.199606
$361$	−17.0386	−0.896766
$362$	2.16116	0.113588
$363$	5.51409	0.289415
$364$	16.6694	0.873716
$365$	−1.68714	−0.0883088
$366$	0.342202	0.0178872
$367$	25.4876	1.33044	0.665220	−	0.746647i	$-0.268338\pi$
$367$	25.4876	1.33044	0.665220	+	0.746647i	$0.268338\pi$
$368$	−1.00000	−0.0521286
$369$	−7.66636	−0.399095
$370$	−30.3687	−1.57879
$371$	1.98020	0.102807
$372$	−1.30550	−0.0676871
$373$	25.6742	1.32936	0.664680	−	0.747128i	$-0.268568\pi$
$373$	25.6742	1.32936	0.664680	+	0.747128i	$0.268568\pi$
$374$	−3.38459	−0.175013
$375$	16.4490	0.849421
$376$	−6.80634	−0.351010
$377$	−4.66679	−0.240352
$378$	−3.57193	−0.183720
$379$	33.6985	1.73098	0.865488	−	0.500930i	$-0.167009\pi$
$379$	33.6985	1.73098	0.865488	+	0.500930i	$0.167009\pi$
$380$	5.30409	0.272094
$381$	11.7171	0.600283
$382$	25.9760	1.32905
$383$	−18.9508	−0.968339	−0.484169	−	0.874974i	$-0.660878\pi$
$383$	−18.9508	−0.968339	−0.484169	+	0.874974i	$0.660878\pi$
$384$	−1.00000	−0.0510310
$385$	−31.6848	−1.61481
$386$	21.7463	1.10686
$387$	−5.48760	−0.278950
$388$	−8.60968	−0.437090
$389$	−4.03989	−0.204831	−0.102415	−	0.994742i	$-0.532657\pi$
$389$	−4.03989	−0.204831	−0.102415	+	0.994742i	$0.532657\pi$
$390$	17.6743	0.894974
$391$	1.44505	0.0730791
$392$	5.75866	0.290856
$393$	−12.3385	−0.622393
$394$	2.10710	0.106154
$395$	21.0760	1.06045
$396$	2.34220	0.117700
$397$	14.1022	0.707769	0.353885	−	0.935289i	$-0.384861\pi$
$397$	14.1022	0.707769	0.353885	+	0.935289i	$0.384861\pi$
$398$	25.3300	1.26968
$399$	5.00253	0.250440
$400$	9.34325	0.467163
$401$	27.6506	1.38081	0.690403	−	0.723425i	$-0.257433\pi$
$401$	27.6506	1.38081	0.690403	+	0.723425i	$0.257433\pi$
$402$	−11.2917	−0.563178
$403$	6.09251	0.303489
$404$	−1.37139	−0.0682291
$405$	−3.78725	−0.188190
$406$	−3.57193	−0.177272
$407$	18.7813	0.930955
$408$	1.44505	0.0715405
$409$	−18.5570	−0.917587	−0.458793	−	0.888543i	$-0.651718\pi$
$409$	−18.5570	−0.917587	−0.458793	+	0.888543i	$0.651718\pi$
$410$	29.0344	1.43391
$411$	3.56500	0.175848
$412$	16.0535	0.790898
$413$	13.8727	0.682629
$414$	−1.00000	−0.0491473
$415$	−33.5003	−1.64447
$416$	4.66679	0.228808
$417$	−3.62041	−0.177292
$418$	−3.28029	−0.160444
$419$	30.4264	1.48642	0.743212	−	0.669056i	$-0.233301\pi$
$419$	30.4264	1.48642	0.743212	+	0.669056i	$0.233301\pi$
$420$	13.5278	0.660088
$421$	−20.9140	−1.01928	−0.509642	−	0.860386i	$-0.670222\pi$
$421$	−20.9140	−1.01928	−0.509642	+	0.860386i	$0.670222\pi$
$422$	−8.16289	−0.397363
$423$	−6.80634	−0.330936
$424$	0.554378	0.0269230
$425$	−13.5014	−0.654916
$426$	−15.1708	−0.735027
$427$	−1.22232	−0.0591522
$428$	0.933549	0.0451248
$429$	−10.9306	−0.527733
$430$	20.7829	1.00224
$431$	6.74285	0.324792	0.162396	−	0.986726i	$-0.448078\pi$
$431$	6.74285	0.324792	0.162396	+	0.986726i	$0.448078\pi$
$432$	−1.00000	−0.0481125
$433$	−21.7300	−1.04428	−0.522138	−	0.852861i	$-0.674865\pi$
$433$	−21.7300	−1.04428	−0.522138	+	0.852861i	$0.674865\pi$
$434$	4.66316	0.223839
$435$	−3.78725	−0.181585
$436$	16.4318	0.786938
$437$	1.40051	0.0669956
$438$	−0.445478	−0.0212858
$439$	−32.7473	−1.56294	−0.781472	−	0.623941i	$-0.785531\pi$
$439$	−32.7473	−1.56294	−0.781472	+	0.623941i	$0.785531\pi$
$440$	−8.87050	−0.422885
$441$	5.75866	0.274222
$442$	−6.74374	−0.320767
$443$	14.8655	0.706283	0.353142	−	0.935570i	$-0.385113\pi$
$443$	14.8655	0.706283	0.353142	+	0.935570i	$0.385113\pi$
$444$	−8.01866	−0.380549
$445$	−30.2815	−1.43548
$446$	18.1626	0.860023
$447$	9.41698	0.445408
$448$	3.57193	0.168758
$449$	14.4939	0.684008	0.342004	−	0.939698i	$-0.388894\pi$
$449$	14.4939	0.684008	0.342004	+	0.939698i	$0.388894\pi$
$450$	9.34325	0.440445
$451$	−17.9562	−0.845523
$452$	5.68368	0.267338
$453$	−4.85524	−0.228119
$454$	13.8314	0.649141
$455$	−63.1313	−2.95964
$456$	1.40051	0.0655850
$457$	−0.319638	−0.0149520	−0.00747602	−	0.999972i	$-0.502380\pi$
$457$	−0.319638	−0.0149520	−0.00747602	+	0.999972i	$0.502380\pi$
$458$	−14.2401	−0.665395
$459$	1.44505	0.0674490
$460$	3.78725	0.176581
$461$	−4.02268	−0.187355	−0.0936773	−	0.995603i	$-0.529862\pi$
$461$	−4.02268	−0.187355	−0.0936773	+	0.995603i	$0.529862\pi$
$462$	−8.36617	−0.389230
$463$	−16.9775	−0.789010	−0.394505	−	0.918894i	$-0.629084\pi$
$463$	−16.9775	−0.789010	−0.394505	+	0.918894i	$0.629084\pi$
$464$	−1.00000	−0.0464238
$465$	4.94426	0.229285
$466$	−20.5518	−0.952043
$467$	−18.8441	−0.872001	−0.436000	−	0.899947i	$-0.643605\pi$
$467$	−18.8441	−0.872001	−0.436000	+	0.899947i	$0.643605\pi$
$468$	4.66679	0.215723
$469$	40.3330	1.86241
$470$	25.7773	1.18902
$471$	−5.30967	−0.244657
$472$	3.88380	0.178766
$473$	−12.8531	−0.590984
$474$	5.56500	0.255609
$475$	−13.0853	−0.600397
$476$	−5.16160	−0.236582
$477$	0.554378	0.0253832
$478$	−0.852557	−0.0389951
$479$	4.12593	0.188518	0.0942592	−	0.995548i	$-0.469952\pi$
$479$	4.12593	0.188518	0.0942592	+	0.995548i	$0.469952\pi$
$480$	3.78725	0.172863
$481$	37.4214	1.70627
$482$	−17.9577	−0.817949
$483$	3.57193	0.162528
$484$	−5.51409	−0.250640
$485$	32.6070	1.48061
$486$	−1.00000	−0.0453609
$487$	−35.2878	−1.59904	−0.799521	−	0.600638i	$-0.794913\pi$
$487$	−35.2878	−1.59904	−0.799521	+	0.600638i	$0.794913\pi$
$488$	−0.342202	−0.0154908
$489$	4.22663	0.191135
$490$	−21.8095	−0.985252
$491$	−36.5347	−1.64879	−0.824394	−	0.566017i	$-0.808484\pi$
$491$	−36.5347	−1.64879	−0.824394	+	0.566017i	$0.808484\pi$
$492$	7.66636	0.345626
$493$	1.44505	0.0650816
$494$	−6.53591	−0.294064
$495$	−8.87050	−0.398699
$496$	1.30550	0.0586188
$497$	54.1889	2.43071
$498$	−8.84556	−0.396379
$499$	−23.8310	−1.06682	−0.533412	−	0.845856i	$-0.679090\pi$
$499$	−23.8310	−1.06682	−0.533412	+	0.845856i	$0.679090\pi$
$500$	−16.4490	−0.735620
$501$	17.4722	0.780602
$502$	5.79440	0.258617
$503$	1.61618	0.0720621	0.0360311	−	0.999351i	$-0.488528\pi$
$503$	1.61618	0.0720621	0.0360311	+	0.999351i	$0.488528\pi$
$504$	3.57193	0.159106
$505$	5.19378	0.231120
$506$	−2.34220	−0.104124
$507$	−8.77897	−0.389888
$508$	−11.7171	−0.519860
$509$	8.39347	0.372034	0.186017	−	0.982547i	$-0.440442\pi$
$509$	8.39347	0.372034	0.186017	+	0.982547i	$0.440442\pi$
$510$	−5.47275	−0.242337
$511$	1.59122	0.0703912
$512$	1.00000	0.0441942
$513$	1.40051	0.0618342
$514$	3.15754	0.139273
$515$	−60.7985	−2.67910
$516$	5.48760	0.241578
$517$	−15.9418	−0.701121
$518$	28.6421	1.25846
$519$	−6.23022	−0.273476
$520$	−17.6743	−0.775070
$521$	9.58673	0.420002	0.210001	−	0.977701i	$-0.432653\pi$
$521$	9.58673	0.420002	0.210001	+	0.977701i	$0.432653\pi$
$522$	−1.00000	−0.0437688
$523$	6.46947	0.282890	0.141445	−	0.989946i	$-0.454825\pi$
$523$	6.46947	0.282890	0.141445	+	0.989946i	$0.454825\pi$
$524$	12.3385	0.539009
$525$	−33.3734	−1.45654
$526$	−2.09401	−0.0913032
$527$	−1.88651	−0.0821777
$528$	−2.34220	−0.101931
$529$	1.00000	0.0434783
$530$	−2.09957	−0.0911994
$531$	3.88380	0.168543
$532$	−5.00253	−0.216887
$533$	−35.7773	−1.54969
$534$	−7.99564	−0.346005
$535$	−3.53558	−0.152857
$536$	11.2917	0.487726
$537$	−16.0519	−0.692692
$538$	14.0144	0.604202
$539$	13.4879	0.580967
$540$	3.78725	0.162977
$541$	10.3414	0.444611	0.222305	−	0.974977i	$-0.428642\pi$
$541$	10.3414	0.444611	0.222305	+	0.974977i	$0.428642\pi$
$542$	−26.2382	−1.12703
$543$	−2.16116	−0.0927441
$544$	−1.44505	−0.0619559
$545$	−62.2312	−2.66569
$546$	−16.6694	−0.713387
$547$	14.7461	0.630496	0.315248	−	0.949009i	$-0.397912\pi$
$547$	14.7461	0.630496	0.315248	+	0.949009i	$0.397912\pi$
$548$	−3.56500	−0.152289
$549$	−0.342202	−0.0146048
$550$	21.8838	0.933128
$551$	1.40051	0.0596639
$552$	1.00000	0.0425628
$553$	−19.8778	−0.845289
$554$	19.0265	0.808357
$555$	30.3687	1.28908
$556$	3.62041	0.153539
$557$	6.32137	0.267845	0.133923	−	0.990992i	$-0.457243\pi$
$557$	6.32137	0.267845	0.133923	+	0.990992i	$0.457243\pi$
$558$	1.30550	0.0552663
$559$	−25.6095	−1.08317
$560$	−13.5278	−0.571653
$561$	3.38459	0.142898
$562$	13.7926	0.581808
$563$	−5.33019	−0.224641	−0.112320	−	0.993672i	$-0.535828\pi$
$563$	−5.33019	−0.224641	−0.112320	+	0.993672i	$0.535828\pi$
$564$	6.80634	0.286599
$565$	−21.5255	−0.905586
$566$	−19.6024	−0.823952
$567$	3.57193	0.150007
$568$	15.1708	0.636552
$569$	14.1213	0.591994	0.295997	−	0.955189i	$-0.404348\pi$
$569$	14.1213	0.591994	0.295997	+	0.955189i	$0.404348\pi$
$570$	−5.30409	−0.222164
$571$	−31.8561	−1.33314	−0.666569	−	0.745443i	$-0.732238\pi$
$571$	−31.8561	−1.33314	−0.666569	+	0.745443i	$0.732238\pi$
$572$	10.9306	0.457030
$573$	−25.9760	−1.08516
$574$	−27.3837	−1.14297
$575$	−9.34325	−0.389641
$576$	1.00000	0.0416667
$577$	−19.5054	−0.812021	−0.406010	−	0.913868i	$-0.633080\pi$
$577$	−19.5054	−0.812021	−0.406010	+	0.913868i	$0.633080\pi$
$578$	−14.9118	−0.620251
$579$	−21.7463	−0.903746
$580$	3.78725	0.157257
$581$	31.5957	1.31081
$582$	8.60968	0.356883
$583$	1.29847	0.0537769
$584$	0.445478	0.0184340
$585$	−17.6743	−0.730743
$586$	−29.3911	−1.21413
$587$	−1.66385	−0.0686745	−0.0343372	−	0.999410i	$-0.510932\pi$
$587$	−1.66385	−0.0686745	−0.0343372	+	0.999410i	$0.510932\pi$
$588$	−5.75866	−0.237483
$589$	−1.82837	−0.0753368
$590$	−14.7089	−0.605557
$591$	−2.10710	−0.0866745
$592$	8.01866	0.329565
$593$	26.7886	1.10008	0.550038	−	0.835140i	$-0.314613\pi$
$593$	26.7886	1.10008	0.550038	+	0.835140i	$0.314613\pi$
$594$	−2.34220	−0.0961017
$595$	19.5483	0.801401
$596$	−9.41698	−0.385735
$597$	−25.3300	−1.03669
$598$	−4.66679	−0.190839
$599$	−8.79682	−0.359428	−0.179714	−	0.983719i	$-0.557517\pi$
$599$	−8.79682	−0.359428	−0.179714	+	0.983719i	$0.557517\pi$
$600$	−9.34325	−0.381437
$601$	−8.63645	−0.352288	−0.176144	−	0.984364i	$-0.556362\pi$
$601$	−8.63645	−0.352288	−0.176144	+	0.984364i	$0.556362\pi$
$602$	−19.6013	−0.798889
$603$	11.2917	0.459833
$604$	4.85524	0.197557
$605$	20.8832	0.849024
$606$	1.37139	0.0557088
$607$	33.1161	1.34414	0.672070	−	0.740488i	$-0.265406\pi$
$607$	33.1161	1.34414	0.672070	+	0.740488i	$0.265406\pi$
$608$	−1.40051	−0.0567983
$609$	3.57193	0.144742
$610$	1.29600	0.0524737
$611$	−31.7638	−1.28503
$612$	−1.44505	−0.0584125
$613$	0.623316	0.0251755	0.0125878	−	0.999921i	$-0.495993\pi$
$613$	0.623316	0.0251755	0.0125878	+	0.999921i	$0.495993\pi$
$614$	−21.9547	−0.886021
$615$	−29.0344	−1.17078
$616$	8.36617	0.337083
$617$	−37.3965	−1.50553	−0.752763	−	0.658292i	$-0.771279\pi$
$617$	−37.3965	−1.50553	−0.752763	+	0.658292i	$0.771279\pi$
$618$	−16.0535	−0.645766
$619$	−8.70425	−0.349853	−0.174927	−	0.984581i	$-0.555969\pi$
$619$	−8.70425	−0.349853	−0.174927	+	0.984581i	$0.555969\pi$
$620$	−4.94426	−0.198566
$621$	1.00000	0.0401286
$622$	−24.9590	−1.00077
$623$	28.5599	1.14423
$624$	−4.66679	−0.186821
$625$	15.5801	0.623204
$626$	15.0472	0.601409
$627$	3.28029	0.131002
$628$	5.30967	0.211879
$629$	−11.5873	−0.462017
$630$	−13.5278	−0.538959
$631$	−39.7002	−1.58044	−0.790220	−	0.612823i	$-0.790034\pi$
$631$	−39.7002	−1.58044	−0.790220	+	0.612823i	$0.790034\pi$
$632$	−5.56500	−0.221364
$633$	8.16289	0.324446
$634$	32.8887	1.30618
$635$	44.3754	1.76098
$636$	−0.554378	−0.0219825
$637$	26.8745	1.06481
$638$	−2.34220	−0.0927287
$639$	15.1708	0.600147
$640$	−3.78725	−0.149704
$641$	−13.3533	−0.527422	−0.263711	−	0.964602i	$-0.584946\pi$
$641$	−13.3533	−0.527422	−0.263711	+	0.964602i	$0.584946\pi$
$642$	−0.933549	−0.0368442
$643$	28.8141	1.13632	0.568158	−	0.822920i	$-0.307656\pi$
$643$	28.8141	1.13632	0.568158	+	0.822920i	$0.307656\pi$
$644$	−3.57193	−0.140754
$645$	−20.7829	−0.818326
$646$	2.02381	0.0796256
$647$	−16.3657	−0.643401	−0.321701	−	0.946841i	$-0.604254\pi$
$647$	−16.3657	−0.643401	−0.321701	+	0.946841i	$0.604254\pi$
$648$	1.00000	0.0392837
$649$	9.09665	0.357075
$650$	43.6030	1.71025
$651$	−4.66316	−0.182763
$652$	−4.22663	−0.165528
$653$	1.63973	0.0641678	0.0320839	−	0.999485i	$-0.489786\pi$
$653$	1.63973	0.0641678	0.0320839	+	0.999485i	$0.489786\pi$
$654$	−16.4318	−0.642533
$655$	−46.7289	−1.82585
$656$	−7.66636	−0.299321
$657$	0.445478	0.0173798
$658$	−24.3117	−0.947771
$659$	29.7986	1.16079	0.580395	−	0.814335i	$-0.302898\pi$
$659$	29.7986	1.16079	0.580395	+	0.814335i	$0.302898\pi$
$660$	8.87050	0.345284
$661$	−26.7407	−1.04009	−0.520046	−	0.854138i	$-0.674085\pi$
$661$	−26.7407	−1.04009	−0.520046	+	0.854138i	$0.674085\pi$
$662$	22.2324	0.864088
$663$	6.74374	0.261905
$664$	8.84556	0.343274
$665$	18.9458	0.734688
$666$	8.01866	0.310717
$667$	1.00000	0.0387202
$668$	−17.4722	−0.676021
$669$	−18.1626	−0.702206
$670$	−42.7644	−1.65213
$671$	−0.801506	−0.0309418
$672$	−3.57193	−0.137790
$673$	35.6902	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$673$	35.6902	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$674$	7.92729	0.305348
$675$	−9.34325	−0.359622
$676$	8.77897	0.337653
$677$	5.16173	0.198381	0.0991907	−	0.995068i	$-0.468375\pi$
$677$	5.16173	0.198381	0.0991907	+	0.995068i	$0.468375\pi$
$678$	−5.68368	−0.218281
$679$	−30.7531	−1.18020
$680$	5.47275	0.209870
$681$	−13.8314	−0.530021
$682$	3.05775	0.117087
$683$	−4.90914	−0.187843	−0.0939216	−	0.995580i	$-0.529940\pi$
$683$	−4.90914	−0.187843	−0.0939216	+	0.995580i	$0.529940\pi$
$684$	−1.40051	−0.0535500
$685$	13.5015	0.515867
$686$	−4.43397	−0.169290
$687$	14.2401	0.543293
$688$	−5.48760	−0.209213
$689$	2.58717	0.0985633
$690$	−3.78725	−0.144178
$691$	−14.1463	−0.538151	−0.269076	−	0.963119i	$-0.586718\pi$
$691$	−14.1463	−0.538151	−0.269076	+	0.963119i	$0.586718\pi$
$692$	6.23022	0.236837
$693$	8.36617	0.317805
$694$	−22.8001	−0.865481
$695$	−13.7114	−0.520102
$696$	1.00000	0.0379049
$697$	11.0783	0.419619
$698$	11.4172	0.432148
$699$	20.5518	0.777340
$700$	33.3734	1.26140
$701$	22.2077	0.838773	0.419386	−	0.907808i	$-0.362245\pi$
$701$	22.2077	0.838773	0.419386	+	0.907808i	$0.362245\pi$
$702$	−4.66679	−0.176137
$703$	−11.2302	−0.423556
$704$	2.34220	0.0882751
$705$	−25.7773	−0.970829
$706$	−33.5445	−1.26247
$707$	−4.89849	−0.184227
$708$	−3.88380	−0.145962
$709$	−6.65174	−0.249812	−0.124906	−	0.992169i	$-0.539863\pi$
$709$	−6.65174	−0.249812	−0.124906	+	0.992169i	$0.539863\pi$
$710$	−57.4556	−2.15627
$711$	−5.56500	−0.208704
$712$	7.99564	0.299649
$713$	−1.30550	−0.0488914
$714$	5.16160	0.193168
$715$	−41.3968	−1.54815
$716$	16.0519	0.599889
$717$	0.852557	0.0318393
$718$	5.08140	0.189636
$719$	3.65677	0.136374	0.0681872	−	0.997673i	$-0.478278\pi$
$719$	3.65677	0.136374	0.0681872	+	0.997673i	$0.478278\pi$
$720$	−3.78725	−0.141142
$721$	57.3419	2.13552
$722$	−17.0386	−0.634110
$723$	17.9577	0.667853
$724$	2.16116	0.0803187
$725$	−9.34325	−0.347000
$726$	5.51409	0.204647
$727$	−31.7252	−1.17662	−0.588311	−	0.808635i	$-0.700207\pi$
$727$	−31.7252	−1.17662	−0.588311	+	0.808635i	$0.700207\pi$
$728$	16.6694	0.617811
$729$	1.00000	0.0370370
$730$	−1.68714	−0.0624437
$731$	7.92983	0.293295
$732$	0.342202	0.0126482
$733$	−7.63448	−0.281986	−0.140993	−	0.990011i	$-0.545029\pi$
$733$	−7.63448	−0.281986	−0.140993	+	0.990011i	$0.545029\pi$
$734$	25.4876	0.940764
$735$	21.8095	0.804455
$736$	−1.00000	−0.0368605
$737$	26.4474	0.974202
$738$	−7.66636	−0.282203
$739$	5.35462	0.196973	0.0984864	−	0.995138i	$-0.468600\pi$
$739$	5.35462	0.196973	0.0984864	+	0.995138i	$0.468600\pi$
$740$	−30.3687	−1.11637
$741$	6.53591	0.240103
$742$	1.98020	0.0726953
$743$	37.1859	1.36422	0.682109	−	0.731250i	$-0.261063\pi$
$743$	37.1859	1.36422	0.682109	+	0.731250i	$0.261063\pi$
$744$	−1.30550	−0.0478620
$745$	35.6645	1.30664
$746$	25.6742	0.940000
$747$	8.84556	0.323642
$748$	−3.38459	−0.123753
$749$	3.33457	0.121842
$750$	16.4490	0.600632
$751$	17.2349	0.628909	0.314454	−	0.949273i	$-0.398178\pi$
$751$	17.2349	0.628909	0.314454	+	0.949273i	$0.398178\pi$
$752$	−6.80634	−0.248202
$753$	−5.79440	−0.211160
$754$	−4.66679	−0.169955
$755$	−18.3880	−0.669208
$756$	−3.57193	−0.129910
$757$	−24.7280	−0.898755	−0.449377	−	0.893342i	$-0.648354\pi$
$757$	−24.7280	−0.898755	−0.449377	+	0.893342i	$0.648354\pi$
$758$	33.6985	1.22398
$759$	2.34220	0.0850165
$760$	5.30409	0.192400
$761$	32.5123	1.17857	0.589285	−	0.807925i	$-0.299410\pi$
$761$	32.5123	1.17857	0.589285	+	0.807925i	$0.299410\pi$
$762$	11.7171	0.424464
$763$	58.6930	2.12483
$764$	25.9760	0.939779
$765$	5.47275	0.197868
$766$	−18.9508	−0.684719
$767$	18.1249	0.654452
$768$	−1.00000	−0.0360844
$769$	9.17525	0.330868	0.165434	−	0.986221i	$-0.447098\pi$
$769$	9.17525	0.330868	0.165434	+	0.986221i	$0.447098\pi$
$770$	−31.6848	−1.14184
$771$	−3.15754	−0.113716
$772$	21.7463	0.782667
$773$	15.0278	0.540513	0.270256	−	0.962788i	$-0.412892\pi$
$773$	15.0278	0.540513	0.270256	+	0.962788i	$0.412892\pi$
$774$	−5.48760	−0.197248
$775$	12.1976	0.438152
$776$	−8.60968	−0.309069
$777$	−28.6421	−1.02753
$778$	−4.03989	−0.144837
$779$	10.7368	0.384687
$780$	17.6743	0.632842
$781$	35.5331	1.27147
$782$	1.44505	0.0516748
$783$	1.00000	0.0357371
$784$	5.75866	0.205666
$785$	−20.1090	−0.717722
$786$	−12.3385	−0.440099
$787$	−12.4048	−0.442185	−0.221092	−	0.975253i	$-0.570962\pi$
$787$	−12.4048	−0.442185	−0.221092	+	0.975253i	$0.570962\pi$
$788$	2.10710	0.0750624
$789$	2.09401	0.0745488
$790$	21.0760	0.749852
$791$	20.3017	0.721846
$792$	2.34220	0.0832265
$793$	−1.59699	−0.0567107
$794$	14.1022	0.500469
$795$	2.09957	0.0744640
$796$	25.3300	0.897798
$797$	−30.0605	−1.06480	−0.532399	−	0.846494i	$-0.678709\pi$
$797$	−30.0605	−1.06480	−0.532399	+	0.846494i	$0.678709\pi$
$798$	5.00253	0.177088
$799$	9.83548	0.347954
$800$	9.34325	0.330334
$801$	7.99564	0.282512
$802$	27.6506	0.976378
$803$	1.04340	0.0368208
$804$	−11.2917	−0.398227
$805$	13.5278	0.476791
$806$	6.09251	0.214599
$807$	−14.0144	−0.493329
$808$	−1.37139	−0.0482452
$809$	34.2036	1.20253	0.601267	−	0.799048i	$-0.294663\pi$
$809$	34.2036	1.20253	0.601267	+	0.799048i	$0.294663\pi$
$810$	−3.78725	−0.133070
$811$	−21.5193	−0.755645	−0.377823	−	0.925878i	$-0.623327\pi$
$811$	−21.5193	−0.755645	−0.377823	+	0.925878i	$0.623327\pi$
$812$	−3.57193	−0.125350
$813$	26.2382	0.920214
$814$	18.7813	0.658285
$815$	16.0073	0.560711
$816$	1.44505	0.0505867
$817$	7.68545	0.268880
$818$	−18.5570	−0.648832
$819$	16.6694	0.582478
$820$	29.0344	1.01393
$821$	−31.6614	−1.10499	−0.552496	−	0.833516i	$-0.686324\pi$
$821$	−31.6614	−1.10499	−0.552496	+	0.833516i	$0.686324\pi$
$822$	3.56500	0.124344
$823$	−31.0191	−1.08126	−0.540628	−	0.841261i	$-0.681813\pi$
$823$	−31.0191	−1.08126	−0.540628	+	0.841261i	$0.681813\pi$
$824$	16.0535	0.559250
$825$	−21.8838	−0.761896
$826$	13.8727	0.482691
$827$	21.5883	0.750700	0.375350	−	0.926883i	$-0.377522\pi$
$827$	21.5883	0.750700	0.375350	+	0.926883i	$0.377522\pi$
$828$	−1.00000	−0.0347524
$829$	−9.74644	−0.338508	−0.169254	−	0.985572i	$-0.554136\pi$
$829$	−9.74644	−0.338508	−0.169254	+	0.985572i	$0.554136\pi$
$830$	−33.5003	−1.16281
$831$	−19.0265	−0.660021
$832$	4.66679	0.161792
$833$	−8.32153	−0.288324
$834$	−3.62041	−0.125364
$835$	66.1717	2.28997
$836$	−3.28029	−0.113451
$837$	−1.30550	−0.0451247
$838$	30.4264	1.05106
$839$	−21.5434	−0.743760	−0.371880	−	0.928281i	$-0.621287\pi$
$839$	−21.5434	−0.743760	−0.371880	+	0.928281i	$0.621287\pi$
$840$	13.5278	0.466753
$841$	1.00000	0.0344828
$842$	−20.9140	−0.720743
$843$	−13.7926	−0.475044
$844$	−8.16289	−0.280978
$845$	−33.2481	−1.14377
$846$	−6.80634	−0.234007
$847$	−19.6959	−0.676760
$848$	0.554378	0.0190374
$849$	19.6024	0.672754
$850$	−13.5014	−0.463095
$851$	−8.01866	−0.274876
$852$	−15.1708	−0.519743
$853$	30.6327	1.04885	0.524423	−	0.851458i	$-0.324281\pi$
$853$	30.6327	1.04885	0.524423	+	0.851458i	$0.324281\pi$
$854$	−1.22232	−0.0418270
$855$	5.30409	0.181396
$856$	0.933549	0.0319080
$857$	19.0855	0.651948	0.325974	−	0.945379i	$-0.394308\pi$
$857$	19.0855	0.651948	0.325974	+	0.945379i	$0.394308\pi$
$858$	−10.9306	−0.373164
$859$	53.8779	1.83829	0.919145	−	0.393918i	$-0.128881\pi$
$859$	53.8779	1.83829	0.919145	+	0.393918i	$0.128881\pi$
$860$	20.7829	0.708691
$861$	27.3837	0.933234
$862$	6.74285	0.229662
$863$	−15.9030	−0.541344	−0.270672	−	0.962672i	$-0.587246\pi$
$863$	−15.9030	−0.541344	−0.270672	+	0.962672i	$0.587246\pi$
$864$	−1.00000	−0.0340207
$865$	−23.5954	−0.802267
$866$	−21.7300	−0.738414
$867$	14.9118	0.506433
$868$	4.66316	0.158278
$869$	−13.0344	−0.442160
$870$	−3.78725	−0.128400
$871$	52.6959	1.78553
$872$	16.4318	0.556450
$873$	−8.60968	−0.291393
$874$	1.40051	0.0473731
$875$	−58.7545	−1.98627
$876$	−0.445478	−0.0150513
$877$	52.5752	1.77534	0.887668	−	0.460483i	$-0.152324\pi$
$877$	52.5752	1.77534	0.887668	+	0.460483i	$0.152324\pi$
$878$	−32.7473	−1.10517
$879$	29.3911	0.991336
$880$	−8.87050	−0.299025
$881$	10.3143	0.347498	0.173749	−	0.984790i	$-0.444412\pi$
$881$	10.3143	0.347498	0.173749	+	0.984790i	$0.444412\pi$
$882$	5.75866	0.193904
$883$	29.2224	0.983412	0.491706	−	0.870761i	$-0.336373\pi$
$883$	29.2224	0.983412	0.491706	+	0.870761i	$0.336373\pi$
$884$	−6.74374	−0.226816
$885$	14.7089	0.494435
$886$	14.8655	0.499418
$887$	−39.7584	−1.33495	−0.667477	−	0.744630i	$-0.732626\pi$
$887$	−39.7584	−1.33495	−0.667477	+	0.744630i	$0.732626\pi$
$888$	−8.01866	−0.269089
$889$	−41.8525	−1.40369
$890$	−30.2815	−1.01504
$891$	2.34220	0.0784667
$892$	18.1626	0.608128
$893$	9.53237	0.318989
$894$	9.41698	0.314951
$895$	−60.7927	−2.03208
$896$	3.57193	0.119330
$897$	4.66679	0.155820
$898$	14.4939	0.483667
$899$	−1.30550	−0.0435409
$900$	9.34325	0.311442
$901$	−0.801102	−0.0266886
$902$	−17.9562	−0.597875
$903$	19.6013	0.652290
$904$	5.68368	0.189037
$905$	−8.18483	−0.272073
$906$	−4.85524	−0.161305
$907$	−50.4594	−1.67548	−0.837738	−	0.546073i	$-0.816122\pi$
$907$	−50.4594	−1.67548	−0.837738	+	0.546073i	$0.816122\pi$
$908$	13.8314	0.459012
$909$	−1.37139	−0.0454860
$910$	−63.1313	−2.09278
$911$	−44.7516	−1.48269	−0.741344	−	0.671125i	$-0.765811\pi$
$911$	−44.7516	−1.48269	−0.741344	+	0.671125i	$0.765811\pi$
$912$	1.40051	0.0463756
$913$	20.7181	0.685669
$914$	−0.319638	−0.0105727
$915$	−1.29600	−0.0428446
$916$	−14.2401	−0.470505
$917$	44.0721	1.45539
$918$	1.44505	0.0476936
$919$	−29.9321	−0.987368	−0.493684	−	0.869641i	$-0.664350\pi$
$919$	−29.9321	−0.987368	−0.493684	+	0.869641i	$0.664350\pi$
$920$	3.78725	0.124862
$921$	21.9547	0.723433
$922$	−4.02268	−0.132480
$923$	70.7990	2.33038
$924$	−8.36617	−0.275227
$925$	74.9204	2.46337
$926$	−16.9775	−0.557914
$927$	16.0535	0.527266
$928$	−1.00000	−0.0328266
$929$	−22.1434	−0.726501	−0.363251	−	0.931691i	$-0.618333\pi$
$929$	−22.1434	−0.726501	−0.363251	+	0.931691i	$0.618333\pi$
$930$	4.94426	0.162129
$931$	−8.06508	−0.264322
$932$	−20.5518	−0.673196
$933$	24.9590	0.817123
$934$	−18.8441	−0.616598
$935$	12.8183	0.419203
$936$	4.66679	0.152539
$937$	−8.47194	−0.276766	−0.138383	−	0.990379i	$-0.544191\pi$
$937$	−8.47194	−0.276766	−0.138383	+	0.990379i	$0.544191\pi$
$938$	40.3330	1.31692
$939$	−15.0472	−0.491048
$940$	25.7773	0.840763
$941$	−37.8965	−1.23539	−0.617696	−	0.786417i	$-0.711934\pi$
$941$	−37.8965	−1.23539	−0.617696	+	0.786417i	$0.711934\pi$
$942$	−5.30967	−0.172998
$943$	7.66636	0.249651
$944$	3.88380	0.126407
$945$	13.5278	0.440059
$946$	−12.8531	−0.417889
$947$	−37.5575	−1.22045	−0.610227	−	0.792227i	$-0.708922\pi$
$947$	−37.5575	−1.22045	−0.610227	+	0.792227i	$0.708922\pi$
$948$	5.56500	0.180743
$949$	2.07896	0.0674857
$950$	−13.0853	−0.424545
$951$	−32.8887	−1.06649
$952$	−5.16160	−0.167288
$953$	31.6997	1.02685	0.513427	−	0.858133i	$-0.328376\pi$
$953$	31.6997	1.02685	0.513427	+	0.858133i	$0.328376\pi$
$954$	0.554378	0.0179486
$955$	−98.3776	−3.18342
$956$	−0.852557	−0.0275737
$957$	2.34220	0.0757126
$958$	4.12593	0.133303
$959$	−12.7339	−0.411200
$960$	3.78725	0.122233
$961$	−29.2957	−0.945021
$962$	37.4214	1.20652
$963$	0.933549	0.0300832
$964$	−17.9577	−0.578377
$965$	−82.3587	−2.65122
$966$	3.57193	0.114925
$967$	5.54508	0.178318	0.0891588	−	0.996017i	$-0.471582\pi$
$967$	5.54508	0.178318	0.0891588	+	0.996017i	$0.471582\pi$
$968$	−5.51409	−0.177230
$969$	−2.02381	−0.0650140
$970$	32.6070	1.04695
$971$	−26.5045	−0.850570	−0.425285	−	0.905060i	$-0.639826\pi$
$971$	−26.5045	−0.850570	−0.425285	+	0.905060i	$0.639826\pi$
$972$	−1.00000	−0.0320750
$973$	12.9318	0.414575
$974$	−35.2878	−1.13069
$975$	−43.6030	−1.39641
$976$	−0.342202	−0.0109536
$977$	−44.4577	−1.42233	−0.711165	−	0.703026i	$-0.751832\pi$
$977$	−44.4577	−1.42233	−0.711165	+	0.703026i	$0.751832\pi$
$978$	4.22663	0.135153
$979$	18.7274	0.598531
$980$	−21.8095	−0.696678
$981$	16.4318	0.524626
$982$	−36.5347	−1.16587
$983$	15.1071	0.481842	0.240921	−	0.970545i	$-0.422551\pi$
$983$	15.1071	0.481842	0.240921	+	0.970545i	$0.422551\pi$
$984$	7.66636	0.244395
$985$	−7.98011	−0.254268
$986$	1.44505	0.0460197
$987$	24.3117	0.773851
$988$	−6.53591	−0.207935
$989$	5.48760	0.174495
$990$	−8.87050	−0.281923
$991$	1.20588	0.0383062	0.0191531	−	0.999817i	$-0.493903\pi$
$991$	1.20588	0.0383062	0.0191531	+	0.999817i	$0.493903\pi$
$992$	1.30550	0.0414497
$993$	−22.2324	−0.705525
$994$	54.1889	1.71877
$995$	−95.9310	−3.04122
$996$	−8.84556	−0.280282
$997$	30.3431	0.960975	0.480488	−	0.877002i	$-0.340460\pi$
$997$	30.3431	0.960975	0.480488	+	0.877002i	$0.340460\pi$
$998$	−23.8310	−0.754358
$999$	−8.01866	−0.253699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bk.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bk.1.1	✓	8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.78725 q^{5} -1.00000 q^{6} +3.57193 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.78725 q^{5} -1.00000 q^{6} +3.57193 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.78725 q^{10} +2.34220 q^{11} -1.00000 q^{12} +4.66679 q^{13} +3.57193 q^{14} +3.78725 q^{15} +1.00000 q^{16} -1.44505 q^{17} +1.00000 q^{18} -1.40051 q^{19} -3.78725 q^{20} -3.57193 q^{21} +2.34220 q^{22} -1.00000 q^{23} -1.00000 q^{24} +9.34325 q^{25} +4.66679 q^{26} -1.00000 q^{27} +3.57193 q^{28} -1.00000 q^{29} +3.78725 q^{30} +1.30550 q^{31} +1.00000 q^{32} -2.34220 q^{33} -1.44505 q^{34} -13.5278 q^{35} +1.00000 q^{36} +8.01866 q^{37} -1.40051 q^{38} -4.66679 q^{39} -3.78725 q^{40} -7.66636 q^{41} -3.57193 q^{42} -5.48760 q^{43} +2.34220 q^{44} -3.78725 q^{45} -1.00000 q^{46} -6.80634 q^{47} -1.00000 q^{48} +5.75866 q^{49} +9.34325 q^{50} +1.44505 q^{51} +4.66679 q^{52} +0.554378 q^{53} -1.00000 q^{54} -8.87050 q^{55} +3.57193 q^{56} +1.40051 q^{57} -1.00000 q^{58} +3.88380 q^{59} +3.78725 q^{60} -0.342202 q^{61} +1.30550 q^{62} +3.57193 q^{63} +1.00000 q^{64} -17.6743 q^{65} -2.34220 q^{66} +11.2917 q^{67} -1.44505 q^{68} +1.00000 q^{69} -13.5278 q^{70} +15.1708 q^{71} +1.00000 q^{72} +0.445478 q^{73} +8.01866 q^{74} -9.34325 q^{75} -1.40051 q^{76} +8.36617 q^{77} -4.66679 q^{78} -5.56500 q^{79} -3.78725 q^{80} +1.00000 q^{81} -7.66636 q^{82} +8.84556 q^{83} -3.57193 q^{84} +5.47275 q^{85} -5.48760 q^{86} +1.00000 q^{87} +2.34220 q^{88} +7.99564 q^{89} -3.78725 q^{90} +16.6694 q^{91} -1.00000 q^{92} -1.30550 q^{93} -6.80634 q^{94} +5.30409 q^{95} -1.00000 q^{96} -8.60968 q^{97} +5.75866 q^{98} +2.34220 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 - 8 * q^12 + 13 * q^13 + 5 * q^14 - 2 * q^15 + 8 * q^16 + 5 * q^17 + 8 * q^18 + 7 * q^19 + 2 * q^20 - 5 * q^21 + 3 * q^22 - 8 * q^23 - 8 * q^24 + 24 * q^25 + 13 * q^26 - 8 * q^27 + 5 * q^28 - 8 * q^29 - 2 * q^30 + 15 * q^31 + 8 * q^32 - 3 * q^33 + 5 * q^34 + 9 * q^35 + 8 * q^36 + 22 * q^37 + 7 * q^38 - 13 * q^39 + 2 * q^40 - 8 * q^41 - 5 * q^42 - q^43 + 3 * q^44 + 2 * q^45 - 8 * q^46 - 9 * q^47 - 8 * q^48 + 33 * q^49 + 24 * q^50 - 5 * q^51 + 13 * q^52 + 14 * q^53 - 8 * q^54 - 17 * q^55 + 5 * q^56 - 7 * q^57 - 8 * q^58 - 4 * q^59 - 2 * q^60 + 13 * q^61 + 15 * q^62 + 5 * q^63 + 8 * q^64 + 21 * q^65 - 3 * q^66 - 3 * q^67 + 5 * q^68 + 8 * q^69 + 9 * q^70 + 7 * q^71 + 8 * q^72 + 16 * q^73 + 22 * q^74 - 24 * q^75 + 7 * q^76 - 13 * q^78 + 14 * q^79 + 2 * q^80 + 8 * q^81 - 8 * q^82 + 36 * q^83 - 5 * q^84 + 47 * q^85 - q^86 + 8 * q^87 + 3 * q^88 - 12 * q^89 + 2 * q^90 + 20 * q^91 - 8 * q^92 - 15 * q^93 - 9 * q^94 + 7 * q^95 - 8 * q^96 + 10 * q^97 + 33 * q^98 + 3 * q^99

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