LMFDB - Embedded newform 4002.2.a.bk.1.6

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			1.6
Root			$2.56063$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.56063 q^{5} -1.00000 q^{6} +3.94138 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.56063 q^{5} -1.00000 q^{6} +3.94138 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.56063 q^{10} +1.29902 q^{11} -1.00000 q^{12} -5.00113 q^{13} +3.94138 q^{14} -2.56063 q^{15} +1.00000 q^{16} +3.85966 q^{17} +1.00000 q^{18} +4.54152 q^{19} +2.56063 q^{20} -3.94138 q^{21} +1.29902 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.55684 q^{25} -5.00113 q^{26} -1.00000 q^{27} +3.94138 q^{28} -1.00000 q^{29} -2.56063 q^{30} +1.25832 q^{31} +1.00000 q^{32} -1.29902 q^{33} +3.85966 q^{34} +10.0924 q^{35} +1.00000 q^{36} +8.85699 q^{37} +4.54152 q^{38} +5.00113 q^{39} +2.56063 q^{40} -7.41434 q^{41} -3.94138 q^{42} -6.81438 q^{43} +1.29902 q^{44} +2.56063 q^{45} -1.00000 q^{46} +8.11911 q^{47} -1.00000 q^{48} +8.53447 q^{49} +1.55684 q^{50} -3.85966 q^{51} -5.00113 q^{52} +11.1325 q^{53} -1.00000 q^{54} +3.32632 q^{55} +3.94138 q^{56} -4.54152 q^{57} -1.00000 q^{58} -6.72787 q^{59} -2.56063 q^{60} +0.700976 q^{61} +1.25832 q^{62} +3.94138 q^{63} +1.00000 q^{64} -12.8061 q^{65} -1.29902 q^{66} -2.80720 q^{67} +3.85966 q^{68} +1.00000 q^{69} +10.0924 q^{70} +7.16777 q^{71} +1.00000 q^{72} -14.2751 q^{73} +8.85699 q^{74} -1.55684 q^{75} +4.54152 q^{76} +5.11995 q^{77} +5.00113 q^{78} +6.58464 q^{79} +2.56063 q^{80} +1.00000 q^{81} -7.41434 q^{82} -2.40118 q^{83} -3.94138 q^{84} +9.88316 q^{85} -6.81438 q^{86} +1.00000 q^{87} +1.29902 q^{88} -17.5887 q^{89} +2.56063 q^{90} -19.7114 q^{91} -1.00000 q^{92} -1.25832 q^{93} +8.11911 q^{94} +11.6292 q^{95} -1.00000 q^{96} -1.23989 q^{97} +8.53447 q^{98} +1.29902 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.56063	1.14515	0.572575	−	0.819853i	$-0.305945\pi$
$5$	2.56063	1.14515	0.572575	+	0.819853i	$0.305945\pi$
$6$	−1.00000	−0.408248
$7$	3.94138	1.48970	0.744851	−	0.667231i	$-0.232521\pi$
$7$	3.94138	1.48970	0.744851	+	0.667231i	$0.232521\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.56063	0.809743
$11$	1.29902	0.391670	0.195835	−	0.980637i	$-0.437258\pi$
$11$	1.29902	0.391670	0.195835	+	0.980637i	$0.437258\pi$
$12$	−1.00000	−0.288675
$13$	−5.00113	−1.38707	−0.693533	−	0.720425i	$-0.743947\pi$
$13$	−5.00113	−1.38707	−0.693533	+	0.720425i	$0.743947\pi$
$14$	3.94138	1.05338
$15$	−2.56063	−0.661152
$16$	1.00000	0.250000
$17$	3.85966	0.936104	0.468052	−	0.883701i	$-0.344956\pi$
$17$	3.85966	0.936104	0.468052	+	0.883701i	$0.344956\pi$
$18$	1.00000	0.235702
$19$	4.54152	1.04190	0.520948	−	0.853588i	$-0.325579\pi$
$19$	4.54152	1.04190	0.520948	+	0.853588i	$0.325579\pi$
$20$	2.56063	0.572575
$21$	−3.94138	−0.860079
$22$	1.29902	0.276953
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	1.55684	0.311367
$26$	−5.00113	−0.980803
$27$	−1.00000	−0.192450
$28$	3.94138	0.744851
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−2.56063	−0.467505
$31$	1.25832	0.226001	0.113001	−	0.993595i	$-0.463954\pi$
$31$	1.25832	0.226001	0.113001	+	0.993595i	$0.463954\pi$
$32$	1.00000	0.176777
$33$	−1.29902	−0.226131
$34$	3.85966	0.661926
$35$	10.0924	1.70593
$36$	1.00000	0.166667
$37$	8.85699	1.45608	0.728040	−	0.685534i	$-0.240431\pi$
$37$	8.85699	1.45608	0.728040	+	0.685534i	$0.240431\pi$
$38$	4.54152	0.736732
$39$	5.00113	0.800822
$40$	2.56063	0.404871
$41$	−7.41434	−1.15793	−0.578963	−	0.815354i	$-0.696542\pi$
$41$	−7.41434	−1.15793	−0.578963	+	0.815354i	$0.696542\pi$
$42$	−3.94138	−0.608168
$43$	−6.81438	−1.03918	−0.519592	−	0.854415i	$-0.673916\pi$
$43$	−6.81438	−1.03918	−0.519592	+	0.854415i	$0.673916\pi$
$44$	1.29902	0.195835
$45$	2.56063	0.381716
$46$	−1.00000	−0.147442
$47$	8.11911	1.18429	0.592147	−	0.805830i	$-0.298280\pi$
$47$	8.11911	1.18429	0.592147	+	0.805830i	$0.298280\pi$
$48$	−1.00000	−0.144338
$49$	8.53447	1.21921
$50$	1.55684	0.220170
$51$	−3.85966	−0.540460
$52$	−5.00113	−0.693533
$53$	11.1325	1.52916	0.764580	−	0.644528i	$-0.222946\pi$
$53$	11.1325	1.52916	0.764580	+	0.644528i	$0.222946\pi$
$54$	−1.00000	−0.136083
$55$	3.32632	0.448521
$56$	3.94138	0.526689
$57$	−4.54152	−0.601539
$58$	−1.00000	−0.131306
$59$	−6.72787	−0.875895	−0.437947	−	0.899001i	$-0.644294\pi$
$59$	−6.72787	−0.875895	−0.437947	+	0.899001i	$0.644294\pi$
$60$	−2.56063	−0.330576
$61$	0.700976	0.0897508	0.0448754	−	0.998993i	$-0.485711\pi$
$61$	0.700976	0.0897508	0.0448754	+	0.998993i	$0.485711\pi$
$62$	1.25832	0.159807
$63$	3.94138	0.496567
$64$	1.00000	0.125000
$65$	−12.8061	−1.58840
$66$	−1.29902	−0.159899
$67$	−2.80720	−0.342954	−0.171477	−	0.985188i	$-0.554854\pi$
$67$	−2.80720	−0.342954	−0.171477	+	0.985188i	$0.554854\pi$
$68$	3.85966	0.468052
$69$	1.00000	0.120386
$70$	10.0924	1.20627
$71$	7.16777	0.850658	0.425329	−	0.905039i	$-0.360158\pi$
$71$	7.16777	0.850658	0.425329	+	0.905039i	$0.360158\pi$
$72$	1.00000	0.117851
$73$	−14.2751	−1.67078	−0.835389	−	0.549659i	$-0.814757\pi$
$73$	−14.2751	−1.67078	−0.835389	+	0.549659i	$0.814757\pi$
$74$	8.85699	1.02960
$75$	−1.55684	−0.179768
$76$	4.54152	0.520948
$77$	5.11995	0.583472
$78$	5.00113	0.566267
$79$	6.58464	0.740830	0.370415	−	0.928866i	$-0.379215\pi$
$79$	6.58464	0.740830	0.370415	+	0.928866i	$0.379215\pi$
$80$	2.56063	0.286287
$81$	1.00000	0.111111
$82$	−7.41434	−0.818777
$83$	−2.40118	−0.263563	−0.131782	−	0.991279i	$-0.542070\pi$
$83$	−2.40118	−0.263563	−0.131782	+	0.991279i	$0.542070\pi$
$84$	−3.94138	−0.430040
$85$	9.88316	1.07198
$86$	−6.81438	−0.734814
$87$	1.00000	0.107211
$88$	1.29902	0.138476
$89$	−17.5887	−1.86439	−0.932197	−	0.361950i	$-0.882111\pi$
$89$	−17.5887	−1.86439	−0.932197	+	0.361950i	$0.882111\pi$
$90$	2.56063	0.269914
$91$	−19.7114	−2.06631
$92$	−1.00000	−0.104257
$93$	−1.25832	−0.130482
$94$	8.11911	0.837422
$95$	11.6292	1.19313
$96$	−1.00000	−0.102062
$97$	−1.23989	−0.125892	−0.0629460	−	0.998017i	$-0.520050\pi$
$97$	−1.23989	−0.125892	−0.0629460	+	0.998017i	$0.520050\pi$
$98$	8.53447	0.862111
$99$	1.29902	0.130557
$100$	1.55684	0.155684
$101$	6.11583	0.608547	0.304274	−	0.952585i	$-0.401586\pi$
$101$	6.11583	0.608547	0.304274	+	0.952585i	$0.401586\pi$
$102$	−3.85966	−0.382163
$103$	−13.6082	−1.34085	−0.670427	−	0.741976i	$-0.733889\pi$
$103$	−13.6082	−1.34085	−0.670427	+	0.741976i	$0.733889\pi$
$104$	−5.00113	−0.490402
$105$	−10.0924	−0.984919
$106$	11.1325	1.08128
$107$	−13.2818	−1.28400	−0.641999	−	0.766705i	$-0.721895\pi$
$107$	−13.2818	−1.28400	−0.641999	+	0.766705i	$0.721895\pi$
$108$	−1.00000	−0.0962250
$109$	7.74423	0.741763	0.370881	−	0.928680i	$-0.379056\pi$
$109$	7.74423	0.741763	0.370881	+	0.928680i	$0.379056\pi$
$110$	3.32632	0.317152
$111$	−8.85699	−0.840669
$112$	3.94138	0.372425
$113$	4.72818	0.444790	0.222395	−	0.974957i	$-0.428613\pi$
$113$	4.72818	0.444790	0.222395	+	0.974957i	$0.428613\pi$
$114$	−4.54152	−0.425352
$115$	−2.56063	−0.238780
$116$	−1.00000	−0.0928477
$117$	−5.00113	−0.462355
$118$	−6.72787	−0.619351
$119$	15.2124	1.39452
$120$	−2.56063	−0.233753
$121$	−9.31254	−0.846594
$122$	0.700976	0.0634634
$123$	7.41434	0.668528
$124$	1.25832	0.113001
$125$	−8.81668	−0.788588
$126$	3.94138	0.351126
$127$	−0.512958	−0.0455176	−0.0227588	−	0.999741i	$-0.507245\pi$
$127$	−0.512958	−0.0455176	−0.0227588	+	0.999741i	$0.507245\pi$
$128$	1.00000	0.0883883
$129$	6.81438	0.599973
$130$	−12.8061	−1.12317
$131$	−0.622867	−0.0544201	−0.0272101	−	0.999630i	$-0.508662\pi$
$131$	−0.622867	−0.0544201	−0.0272101	+	0.999630i	$0.508662\pi$
$132$	−1.29902	−0.113066
$133$	17.8999	1.55211
$134$	−2.80720	−0.242505
$135$	−2.56063	−0.220384
$136$	3.85966	0.330963
$137$	8.58464	0.733436	0.366718	−	0.930332i	$-0.380481\pi$
$137$	8.58464	0.733436	0.366718	+	0.930332i	$0.380481\pi$
$138$	1.00000	0.0851257
$139$	10.9010	0.924609	0.462305	−	0.886721i	$-0.347023\pi$
$139$	10.9010	0.924609	0.462305	+	0.886721i	$0.347023\pi$
$140$	10.0924	0.852965
$141$	−8.11911	−0.683753
$142$	7.16777	0.601506
$143$	−6.49659	−0.543273
$144$	1.00000	0.0833333
$145$	−2.56063	−0.212649
$146$	−14.2751	−1.18142
$147$	−8.53447	−0.703911
$148$	8.85699	0.728040
$149$	5.97382	0.489394	0.244697	−	0.969600i	$-0.421311\pi$
$149$	5.97382	0.489394	0.244697	+	0.969600i	$0.421311\pi$
$150$	−1.55684	−0.127115
$151$	14.3537	1.16809	0.584046	−	0.811721i	$-0.301469\pi$
$151$	14.3537	1.16809	0.584046	+	0.811721i	$0.301469\pi$
$152$	4.54152	0.368366
$153$	3.85966	0.312035
$154$	5.11995	0.412577
$155$	3.22210	0.258805
$156$	5.00113	0.400411
$157$	7.11639	0.567949	0.283975	−	0.958832i	$-0.408347\pi$
$157$	7.11639	0.567949	0.283975	+	0.958832i	$0.408347\pi$
$158$	6.58464	0.523846
$159$	−11.1325	−0.882861
$160$	2.56063	0.202436
$161$	−3.94138	−0.310624
$162$	1.00000	0.0785674
$163$	−6.23792	−0.488592	−0.244296	−	0.969701i	$-0.578557\pi$
$163$	−6.23792	−0.488592	−0.244296	+	0.969701i	$0.578557\pi$
$164$	−7.41434	−0.578963
$165$	−3.32632	−0.258954
$166$	−2.40118	−0.186367
$167$	−3.11626	−0.241143	−0.120572	−	0.992705i	$-0.538473\pi$
$167$	−3.11626	−0.241143	−0.120572	+	0.992705i	$0.538473\pi$
$168$	−3.94138	−0.304084
$169$	12.0113	0.923950
$170$	9.88316	0.758004
$171$	4.54152	0.347299
$172$	−6.81438	−0.519592
$173$	15.3688	1.16847	0.584234	−	0.811585i	$-0.301395\pi$
$173$	15.3688	1.16847	0.584234	+	0.811585i	$0.301395\pi$
$174$	1.00000	0.0758098
$175$	6.13608	0.463844
$176$	1.29902	0.0979176
$177$	6.72787	0.505698
$178$	−17.5887	−1.31833
$179$	−15.2408	−1.13915	−0.569575	−	0.821940i	$-0.692892\pi$
$179$	−15.2408	−1.13915	−0.569575	+	0.821940i	$0.692892\pi$
$180$	2.56063	0.190858
$181$	−6.99922	−0.520248	−0.260124	−	0.965575i	$-0.583763\pi$
$181$	−6.99922	−0.520248	−0.260124	+	0.965575i	$0.583763\pi$
$182$	−19.7114	−1.46110
$183$	−0.700976	−0.0518176
$184$	−1.00000	−0.0737210
$185$	22.6795	1.66743
$186$	−1.25832	−0.0922646
$187$	5.01379	0.366644
$188$	8.11911	0.592147
$189$	−3.94138	−0.286693
$190$	11.6292	0.843668
$191$	−18.9929	−1.37428	−0.687139	−	0.726526i	$-0.741134\pi$
$191$	−18.9929	−1.37428	−0.687139	+	0.726526i	$0.741134\pi$
$192$	−1.00000	−0.0721688
$193$	−13.0004	−0.935788	−0.467894	−	0.883785i	$-0.654987\pi$
$193$	−13.0004	−0.935788	−0.467894	+	0.883785i	$0.654987\pi$
$194$	−1.23989	−0.0890191
$195$	12.8061	0.917061
$196$	8.53447	0.609605
$197$	−9.44605	−0.673003	−0.336502	−	0.941683i	$-0.609244\pi$
$197$	−9.44605	−0.673003	−0.336502	+	0.941683i	$0.609244\pi$
$198$	1.29902	0.0923176
$199$	25.1472	1.78264	0.891320	−	0.453374i	$-0.149780\pi$
$199$	25.1472	1.78264	0.891320	+	0.453374i	$0.149780\pi$
$200$	1.55684	0.110085
$201$	2.80720	0.198005
$202$	6.11583	0.430308
$203$	−3.94138	−0.276631
$204$	−3.85966	−0.270230
$205$	−18.9854	−1.32600
$206$	−13.6082	−0.948127
$207$	−1.00000	−0.0695048
$208$	−5.00113	−0.346766
$209$	5.89954	0.408080
$210$	−10.0924	−0.696443
$211$	12.9638	0.892467	0.446233	−	0.894917i	$-0.352765\pi$
$211$	12.9638	0.892467	0.446233	+	0.894917i	$0.352765\pi$
$212$	11.1325	0.764580
$213$	−7.16777	−0.491128
$214$	−13.2818	−0.907924
$215$	−17.4491	−1.19002
$216$	−1.00000	−0.0680414
$217$	4.95952	0.336674
$218$	7.74423	0.524505
$219$	14.2751	0.964624
$220$	3.32632	0.224261
$221$	−19.3027	−1.29844
$222$	−8.85699	−0.594443
$223$	9.95293	0.666497	0.333249	−	0.942839i	$-0.391855\pi$
$223$	9.95293	0.666497	0.333249	+	0.942839i	$0.391855\pi$
$224$	3.94138	0.263344
$225$	1.55684	0.103789
$226$	4.72818	0.314514
$227$	7.74988	0.514378	0.257189	−	0.966361i	$-0.417204\pi$
$227$	7.74988	0.514378	0.257189	+	0.966361i	$0.417204\pi$
$228$	−4.54152	−0.300770
$229$	17.1578	1.13382	0.566908	−	0.823781i	$-0.308139\pi$
$229$	17.1578	1.13382	0.566908	+	0.823781i	$0.308139\pi$
$230$	−2.56063	−0.168843
$231$	−5.11995	−0.336868
$232$	−1.00000	−0.0656532
$233$	2.97688	0.195022	0.0975109	−	0.995234i	$-0.468912\pi$
$233$	2.97688	0.195022	0.0975109	+	0.995234i	$0.468912\pi$
$234$	−5.00113	−0.326934
$235$	20.7901	1.35619
$236$	−6.72787	−0.437947
$237$	−6.58464	−0.427719
$238$	15.2124	0.986071
$239$	7.01362	0.453674	0.226837	−	0.973933i	$-0.427162\pi$
$239$	7.01362	0.453674	0.226837	+	0.973933i	$0.427162\pi$
$240$	−2.56063	−0.165288
$241$	−26.3107	−1.69482	−0.847411	−	0.530937i	$-0.821840\pi$
$241$	−26.3107	−1.69482	−0.847411	+	0.530937i	$0.821840\pi$
$242$	−9.31254	−0.598633
$243$	−1.00000	−0.0641500
$244$	0.700976	0.0448754
$245$	21.8536	1.39618
$246$	7.41434	0.472721
$247$	−22.7128	−1.44518
$248$	1.25832	0.0799035
$249$	2.40118	0.152168
$250$	−8.81668	−0.557616
$251$	−30.3765	−1.91735	−0.958674	−	0.284507i	$-0.908170\pi$
$251$	−30.3765	−1.91735	−0.958674	+	0.284507i	$0.908170\pi$
$252$	3.94138	0.248284
$253$	−1.29902	−0.0816689
$254$	−0.512958	−0.0321858
$255$	−9.88316	−0.618907
$256$	1.00000	0.0625000
$257$	13.1573	0.820727	0.410363	−	0.911922i	$-0.365402\pi$
$257$	13.1573	0.820727	0.410363	+	0.911922i	$0.365402\pi$
$258$	6.81438	0.424245
$259$	34.9088	2.16913
$260$	−12.8061	−0.794198
$261$	−1.00000	−0.0618984
$262$	−0.622867	−0.0384809
$263$	−21.8819	−1.34930	−0.674649	−	0.738139i	$-0.735705\pi$
$263$	−21.8819	−1.34930	−0.674649	+	0.738139i	$0.735705\pi$
$264$	−1.29902	−0.0799494
$265$	28.5061	1.75112
$266$	17.8999	1.09751
$267$	17.5887	1.07641
$268$	−2.80720	−0.171477
$269$	2.55644	0.155869	0.0779343	−	0.996958i	$-0.475168\pi$
$269$	2.55644	0.155869	0.0779343	+	0.996958i	$0.475168\pi$
$270$	−2.56063	−0.155835
$271$	−0.612404	−0.0372009	−0.0186005	−	0.999827i	$-0.505921\pi$
$271$	−0.612404	−0.0372009	−0.0186005	+	0.999827i	$0.505921\pi$
$272$	3.85966	0.234026
$273$	19.7114	1.19299
$274$	8.58464	0.518617
$275$	2.02237	0.121953
$276$	1.00000	0.0601929
$277$	11.1060	0.667298	0.333649	−	0.942697i	$-0.391720\pi$
$277$	11.1060	0.667298	0.333649	+	0.942697i	$0.391720\pi$
$278$	10.9010	0.653797
$279$	1.25832	0.0753337
$280$	10.0924	0.603137
$281$	23.8468	1.42258	0.711290	−	0.702899i	$-0.248111\pi$
$281$	23.8468	1.42258	0.711290	+	0.702899i	$0.248111\pi$
$282$	−8.11911	−0.483486
$283$	17.1729	1.02082	0.510411	−	0.859931i	$-0.329493\pi$
$283$	17.1729	1.02082	0.510411	+	0.859931i	$0.329493\pi$
$284$	7.16777	0.425329
$285$	−11.6292	−0.688852
$286$	−6.49659	−0.384152
$287$	−29.2227	−1.72496
$288$	1.00000	0.0589256
$289$	−2.10306	−0.123709
$290$	−2.56063	−0.150365
$291$	1.23989	0.0726838
$292$	−14.2751	−0.835389
$293$	1.70479	0.0995951	0.0497976	−	0.998759i	$-0.484142\pi$
$293$	1.70479	0.0995951	0.0497976	+	0.998759i	$0.484142\pi$
$294$	−8.53447	−0.497740
$295$	−17.2276	−1.00303
$296$	8.85699	0.514802
$297$	−1.29902	−0.0753770
$298$	5.97382	0.346054
$299$	5.00113	0.289223
$300$	−1.55684	−0.0898839
$301$	−26.8581	−1.54807
$302$	14.3537	0.825965
$303$	−6.11583	−0.351345
$304$	4.54152	0.260474
$305$	1.79494	0.102778
$306$	3.85966	0.220642
$307$	13.7302	0.783625	0.391813	−	0.920045i	$-0.371848\pi$
$307$	13.7302	0.783625	0.391813	+	0.920045i	$0.371848\pi$
$308$	5.11995	0.291736
$309$	13.6082	0.774142
$310$	3.22210	0.183003
$311$	4.72949	0.268185	0.134092	−	0.990969i	$-0.457188\pi$
$311$	4.72949	0.268185	0.134092	+	0.990969i	$0.457188\pi$
$312$	5.00113	0.283133
$313$	6.76189	0.382205	0.191102	−	0.981570i	$-0.438794\pi$
$313$	6.76189	0.382205	0.191102	+	0.981570i	$0.438794\pi$
$314$	7.11639	0.401601
$315$	10.0924	0.568643
$316$	6.58464	0.370415
$317$	13.7503	0.772294	0.386147	−	0.922437i	$-0.373806\pi$
$317$	13.7503	0.772294	0.386147	+	0.922437i	$0.373806\pi$
$318$	−11.1325	−0.624277
$319$	−1.29902	−0.0727314
$320$	2.56063	0.143144
$321$	13.2818	0.741317
$322$	−3.94138	−0.219644
$323$	17.5287	0.975323
$324$	1.00000	0.0555556
$325$	−7.78594	−0.431886
$326$	−6.23792	−0.345487
$327$	−7.74423	−0.428257
$328$	−7.41434	−0.409388
$329$	32.0005	1.76424
$330$	−3.32632	−0.183108
$331$	−22.6054	−1.24251	−0.621253	−	0.783610i	$-0.713376\pi$
$331$	−22.6054	−1.24251	−0.621253	+	0.783610i	$0.713376\pi$
$332$	−2.40118	−0.131782
$333$	8.85699	0.485360
$334$	−3.11626	−0.170514
$335$	−7.18820	−0.392734
$336$	−3.94138	−0.215020
$337$	4.69618	0.255817	0.127909	−	0.991786i	$-0.459174\pi$
$337$	4.69618	0.255817	0.127909	+	0.991786i	$0.459174\pi$
$338$	12.0113	0.653331
$339$	−4.72818	−0.256799
$340$	9.88316	0.535989
$341$	1.63459	0.0885180
$342$	4.54152	0.245577
$343$	6.04792	0.326557
$344$	−6.81438	−0.367407
$345$	2.56063	0.137860
$346$	15.3688	0.826231
$347$	−21.1535	−1.13558	−0.567789	−	0.823174i	$-0.692201\pi$
$347$	−21.1535	−1.13558	−0.567789	+	0.823174i	$0.692201\pi$
$348$	1.00000	0.0536056
$349$	25.0918	1.34314	0.671568	−	0.740943i	$-0.265621\pi$
$349$	25.0918	1.34314	0.671568	+	0.740943i	$0.265621\pi$
$350$	6.13608	0.327987
$351$	5.00113	0.266941
$352$	1.29902	0.0692382
$353$	27.9371	1.48694	0.743471	−	0.668768i	$-0.233178\pi$
$353$	27.9371	1.48694	0.743471	+	0.668768i	$0.233178\pi$
$354$	6.72787	0.357583
$355$	18.3540	0.974130
$356$	−17.5887	−0.932197
$357$	−15.2124	−0.805124
$358$	−15.2408	−0.805500
$359$	−18.6367	−0.983607	−0.491803	−	0.870706i	$-0.663662\pi$
$359$	−18.6367	−0.983607	−0.491803	+	0.870706i	$0.663662\pi$
$360$	2.56063	0.134957
$361$	1.62541	0.0855477
$362$	−6.99922	−0.367871
$363$	9.31254	0.488781
$364$	−19.7114	−1.03316
$365$	−36.5534	−1.91329
$366$	−0.700976	−0.0366406
$367$	−20.3576	−1.06266	−0.531329	−	0.847166i	$-0.678307\pi$
$367$	−20.3576	−1.06266	−0.531329	+	0.847166i	$0.678307\pi$
$368$	−1.00000	−0.0521286
$369$	−7.41434	−0.385975
$370$	22.6795	1.17905
$371$	43.8772	2.27799
$372$	−1.25832	−0.0652409
$373$	1.36257	0.0705514	0.0352757	−	0.999378i	$-0.488769\pi$
$373$	1.36257	0.0705514	0.0352757	+	0.999378i	$0.488769\pi$
$374$	5.01379	0.259257
$375$	8.81668	0.455291
$376$	8.11911	0.418711
$377$	5.00113	0.257572
$378$	−3.94138	−0.202723
$379$	−15.7149	−0.807221	−0.403610	−	0.914931i	$-0.632245\pi$
$379$	−15.7149	−0.807221	−0.403610	+	0.914931i	$0.632245\pi$
$380$	11.6292	0.596563
$381$	0.512958	0.0262796
$382$	−18.9929	−0.971761
$383$	5.85118	0.298981	0.149491	−	0.988763i	$-0.452237\pi$
$383$	5.85118	0.298981	0.149491	+	0.988763i	$0.452237\pi$
$384$	−1.00000	−0.0510310
$385$	13.1103	0.668163
$386$	−13.0004	−0.661702
$387$	−6.81438	−0.346394
$388$	−1.23989	−0.0629460
$389$	29.1921	1.48010	0.740050	−	0.672552i	$-0.234802\pi$
$389$	29.1921	1.48010	0.740050	+	0.672552i	$0.234802\pi$
$390$	12.8061	0.648460
$391$	−3.85966	−0.195191
$392$	8.53447	0.431056
$393$	0.622867	0.0314195
$394$	−9.44605	−0.475885
$395$	16.8608	0.848361
$396$	1.29902	0.0652784
$397$	28.3701	1.42386	0.711928	−	0.702252i	$-0.247822\pi$
$397$	28.3701	1.42386	0.711928	+	0.702252i	$0.247822\pi$
$398$	25.1472	1.26052
$399$	−17.8999	−0.896113
$400$	1.55684	0.0778418
$401$	11.3449	0.566536	0.283268	−	0.959041i	$-0.408581\pi$
$401$	11.3449	0.566536	0.283268	+	0.959041i	$0.408581\pi$
$402$	2.80720	0.140010
$403$	−6.29304	−0.313478
$404$	6.11583	0.304274
$405$	2.56063	0.127239
$406$	−3.94138	−0.195607
$407$	11.5054	0.570304
$408$	−3.85966	−0.191081
$409$	−38.5468	−1.90602	−0.953008	−	0.302946i	$-0.902030\pi$
$409$	−38.5468	−1.90602	−0.953008	+	0.302946i	$0.902030\pi$
$410$	−18.9854	−0.937622
$411$	−8.58464	−0.423449
$412$	−13.6082	−0.670427
$413$	−26.5171	−1.30482
$414$	−1.00000	−0.0491473
$415$	−6.14853	−0.301819
$416$	−5.00113	−0.245201
$417$	−10.9010	−0.533823
$418$	5.89954	0.288556
$419$	−22.8817	−1.11784	−0.558921	−	0.829221i	$-0.688785\pi$
$419$	−22.8817	−1.11784	−0.558921	+	0.829221i	$0.688785\pi$
$420$	−10.0924	−0.492460
$421$	31.0742	1.51447	0.757233	−	0.653145i	$-0.226551\pi$
$421$	31.0742	1.51447	0.757233	+	0.653145i	$0.226551\pi$
$422$	12.9638	0.631069
$423$	8.11911	0.394765
$424$	11.1325	0.540640
$425$	6.00885	0.291472
$426$	−7.16777	−0.347280
$427$	2.76281	0.133702
$428$	−13.2818	−0.641999
$429$	6.49659	0.313659
$430$	−17.4491	−0.841471
$431$	−31.6300	−1.52356	−0.761781	−	0.647835i	$-0.775675\pi$
$431$	−31.6300	−1.52356	−0.761781	+	0.647835i	$0.775675\pi$
$432$	−1.00000	−0.0481125
$433$	0.715152	0.0343680	0.0171840	−	0.999852i	$-0.494530\pi$
$433$	0.715152	0.0343680	0.0171840	+	0.999852i	$0.494530\pi$
$434$	4.95952	0.238065
$435$	2.56063	0.122773
$436$	7.74423	0.370881
$437$	−4.54152	−0.217250
$438$	14.2751	0.682092
$439$	−10.1649	−0.485145	−0.242572	−	0.970133i	$-0.577991\pi$
$439$	−10.1649	−0.485145	−0.242572	+	0.970133i	$0.577991\pi$
$440$	3.32632	0.158576
$441$	8.53447	0.406403
$442$	−19.3027	−0.918134
$443$	−25.5952	−1.21607	−0.608033	−	0.793912i	$-0.708041\pi$
$443$	−25.5952	−1.21607	−0.608033	+	0.793912i	$0.708041\pi$
$444$	−8.85699	−0.420334
$445$	−45.0381	−2.13501
$446$	9.95293	0.471285
$447$	−5.97382	−0.282552
$448$	3.94138	0.186213
$449$	−30.0523	−1.41826	−0.709129	−	0.705079i	$-0.750911\pi$
$449$	−30.0523	−1.41826	−0.709129	+	0.705079i	$0.750911\pi$
$450$	1.55684	0.0733899
$451$	−9.63141	−0.453525
$452$	4.72818	0.222395
$453$	−14.3537	−0.674398
$454$	7.74988	0.363720
$455$	−50.4736	−2.36624
$456$	−4.54152	−0.212676
$457$	−29.9139	−1.39931	−0.699656	−	0.714480i	$-0.746664\pi$
$457$	−29.9139	−1.39931	−0.699656	+	0.714480i	$0.746664\pi$
$458$	17.1578	0.801729
$459$	−3.85966	−0.180153
$460$	−2.56063	−0.119390
$461$	−8.87221	−0.413220	−0.206610	−	0.978423i	$-0.566243\pi$
$461$	−8.87221	−0.413220	−0.206610	+	0.978423i	$0.566243\pi$
$462$	−5.11995	−0.238201
$463$	−21.9219	−1.01880	−0.509399	−	0.860530i	$-0.670132\pi$
$463$	−21.9219	−1.01880	−0.509399	+	0.860530i	$0.670132\pi$
$464$	−1.00000	−0.0464238
$465$	−3.22210	−0.149421
$466$	2.97688	0.137901
$467$	7.40897	0.342846	0.171423	−	0.985197i	$-0.445163\pi$
$467$	7.40897	0.342846	0.171423	+	0.985197i	$0.445163\pi$
$468$	−5.00113	−0.231178
$469$	−11.0642	−0.510899
$470$	20.7901	0.958974
$471$	−7.11639	−0.327906
$472$	−6.72787	−0.309676
$473$	−8.85204	−0.407017
$474$	−6.58464	−0.302443
$475$	7.07040	0.324412
$476$	15.2124	0.697258
$477$	11.1325	0.509720
$478$	7.01362	0.320796
$479$	1.09797	0.0501673	0.0250837	−	0.999685i	$-0.492015\pi$
$479$	1.09797	0.0501673	0.0250837	+	0.999685i	$0.492015\pi$
$480$	−2.56063	−0.116876
$481$	−44.2950	−2.01968
$482$	−26.3107	−1.19842
$483$	3.94138	0.179339
$484$	−9.31254	−0.423297
$485$	−3.17491	−0.144165
$486$	−1.00000	−0.0453609
$487$	13.8183	0.626167	0.313084	−	0.949726i	$-0.398638\pi$
$487$	13.8183	0.626167	0.313084	+	0.949726i	$0.398638\pi$
$488$	0.700976	0.0317317
$489$	6.23792	0.282089
$490$	21.8536	0.987246
$491$	−9.28961	−0.419234	−0.209617	−	0.977784i	$-0.567222\pi$
$491$	−9.28961	−0.419234	−0.209617	+	0.977784i	$0.567222\pi$
$492$	7.41434	0.334264
$493$	−3.85966	−0.173830
$494$	−22.7128	−1.02190
$495$	3.32632	0.149507
$496$	1.25832	0.0565003
$497$	28.2509	1.26723
$498$	2.40118	0.107599
$499$	35.2253	1.57690	0.788451	−	0.615098i	$-0.210883\pi$
$499$	35.2253	1.57690	0.788451	+	0.615098i	$0.210883\pi$
$500$	−8.81668	−0.394294
$501$	3.11626	0.139224
$502$	−30.3765	−1.35577
$503$	−7.45977	−0.332615	−0.166307	−	0.986074i	$-0.553184\pi$
$503$	−7.45977	−0.332615	−0.166307	+	0.986074i	$0.553184\pi$
$504$	3.94138	0.175563
$505$	15.6604	0.696878
$506$	−1.29902	−0.0577487
$507$	−12.0113	−0.533443
$508$	−0.512958	−0.0227588
$509$	−36.3148	−1.60962	−0.804812	−	0.593530i	$-0.797734\pi$
$509$	−36.3148	−1.60962	−0.804812	+	0.593530i	$0.797734\pi$
$510$	−9.88316	−0.437634
$511$	−56.2637	−2.48896
$512$	1.00000	0.0441942
$513$	−4.54152	−0.200513
$514$	13.1573	0.580342
$515$	−34.8455	−1.53548
$516$	6.81438	0.299986
$517$	10.5469	0.463853
$518$	34.9088	1.53380
$519$	−15.3688	−0.674615
$520$	−12.8061	−0.561583
$521$	−33.9926	−1.48924	−0.744621	−	0.667487i	$-0.767370\pi$
$521$	−33.9926	−1.48924	−0.744621	+	0.667487i	$0.767370\pi$
$522$	−1.00000	−0.0437688
$523$	−15.4643	−0.676206	−0.338103	−	0.941109i	$-0.609785\pi$
$523$	−15.4643	−0.676206	−0.338103	+	0.941109i	$0.609785\pi$
$524$	−0.622867	−0.0272101
$525$	−6.13608	−0.267800
$526$	−21.8819	−0.954097
$527$	4.85669	0.211561
$528$	−1.29902	−0.0565328
$529$	1.00000	0.0434783
$530$	28.5061	1.23823
$531$	−6.72787	−0.291965
$532$	17.8999	0.776057
$533$	37.0801	1.60612
$534$	17.5887	0.761136
$535$	−34.0098	−1.47037
$536$	−2.80720	−0.121253
$537$	15.2408	0.657688
$538$	2.55644	0.110216
$539$	11.0865	0.477528
$540$	−2.56063	−0.110192
$541$	−22.4469	−0.965067	−0.482533	−	0.875878i	$-0.660283\pi$
$541$	−22.4469	−0.965067	−0.482533	+	0.875878i	$0.660283\pi$
$542$	−0.612404	−0.0263050
$543$	6.99922	0.300365
$544$	3.85966	0.165481
$545$	19.8301	0.849429
$546$	19.7114	0.843569
$547$	12.5043	0.534646	0.267323	−	0.963607i	$-0.413861\pi$
$547$	12.5043	0.534646	0.267323	+	0.963607i	$0.413861\pi$
$548$	8.58464	0.366718
$549$	0.700976	0.0299169
$550$	2.02237	0.0862340
$551$	−4.54152	−0.193475
$552$	1.00000	0.0425628
$553$	25.9526	1.10362
$554$	11.1060	0.471851
$555$	−22.6795	−0.962691
$556$	10.9010	0.462305
$557$	25.8404	1.09489	0.547446	−	0.836841i	$-0.315600\pi$
$557$	25.8404	1.09489	0.547446	+	0.836841i	$0.315600\pi$
$558$	1.25832	0.0532690
$559$	34.0796	1.44142
$560$	10.0924	0.426483
$561$	−5.01379	−0.211682
$562$	23.8468	1.00592
$563$	−38.6286	−1.62800	−0.814000	−	0.580865i	$-0.802714\pi$
$563$	−38.6286	−1.62800	−0.814000	+	0.580865i	$0.802714\pi$
$564$	−8.11911	−0.341876
$565$	12.1071	0.509351
$566$	17.1729	0.721830
$567$	3.94138	0.165522
$568$	7.16777	0.300753
$569$	26.0031	1.09011	0.545054	−	0.838401i	$-0.316509\pi$
$569$	26.0031	1.09011	0.545054	+	0.838401i	$0.316509\pi$
$570$	−11.6292	−0.487092
$571$	39.9500	1.67186	0.835928	−	0.548840i	$-0.184930\pi$
$571$	39.9500	1.67186	0.835928	+	0.548840i	$0.184930\pi$
$572$	−6.49659	−0.271636
$573$	18.9929	0.793440
$574$	−29.2227	−1.21973
$575$	−1.55684	−0.0649245
$576$	1.00000	0.0416667
$577$	21.8814	0.910934	0.455467	−	0.890253i	$-0.349472\pi$
$577$	21.8814	0.910934	0.455467	+	0.890253i	$0.349472\pi$
$578$	−2.10306	−0.0874756
$579$	13.0004	0.540277
$580$	−2.56063	−0.106324
$581$	−9.46394	−0.392631
$582$	1.23989	0.0513952
$583$	14.4613	0.598927
$584$	−14.2751	−0.590709
$585$	−12.8061	−0.529466
$586$	1.70479	0.0704244
$587$	−7.13618	−0.294542	−0.147271	−	0.989096i	$-0.547049\pi$
$587$	−7.13618	−0.294542	−0.147271	+	0.989096i	$0.547049\pi$
$588$	−8.53447	−0.351956
$589$	5.71469	0.235470
$590$	−17.2276	−0.709249
$591$	9.44605	0.388559
$592$	8.85699	0.364020
$593$	−1.09241	−0.0448598	−0.0224299	−	0.999748i	$-0.507140\pi$
$593$	−1.09241	−0.0448598	−0.0224299	+	0.999748i	$0.507140\pi$
$594$	−1.29902	−0.0532996
$595$	38.9533	1.59693
$596$	5.97382	0.244697
$597$	−25.1472	−1.02921
$598$	5.00113	0.204512
$599$	7.88408	0.322135	0.161067	−	0.986943i	$-0.448506\pi$
$599$	7.88408	0.322135	0.161067	+	0.986943i	$0.448506\pi$
$600$	−1.55684	−0.0635575
$601$	20.8136	0.849004	0.424502	−	0.905427i	$-0.360449\pi$
$601$	20.8136	0.849004	0.424502	+	0.905427i	$0.360449\pi$
$602$	−26.8581	−1.09465
$603$	−2.80720	−0.114318
$604$	14.3537	0.584046
$605$	−23.8460	−0.969477
$606$	−6.11583	−0.248438
$607$	−15.0359	−0.610289	−0.305144	−	0.952306i	$-0.598705\pi$
$607$	−15.0359	−0.610289	−0.305144	+	0.952306i	$0.598705\pi$
$608$	4.54152	0.184183
$609$	3.94138	0.159713
$610$	1.79494	0.0726750
$611$	−40.6048	−1.64269
$612$	3.85966	0.156017
$613$	17.0601	0.689053	0.344526	−	0.938777i	$-0.388040\pi$
$613$	17.0601	0.689053	0.344526	+	0.938777i	$0.388040\pi$
$614$	13.7302	0.554107
$615$	18.9854	0.765565
$616$	5.11995	0.206289
$617$	−33.0478	−1.33045	−0.665226	−	0.746642i	$-0.731665\pi$
$617$	−33.0478	−1.33045	−0.665226	+	0.746642i	$0.731665\pi$
$618$	13.6082	0.547401
$619$	−4.28930	−0.172401	−0.0862007	−	0.996278i	$-0.527473\pi$
$619$	−4.28930	−0.172401	−0.0862007	+	0.996278i	$0.527473\pi$
$620$	3.22210	0.129403
$621$	1.00000	0.0401286
$622$	4.72949	0.189635
$623$	−69.3236	−2.77739
$624$	5.00113	0.200206
$625$	−30.3604	−1.21442
$626$	6.76189	0.270260
$627$	−5.89954	−0.235605
$628$	7.11639	0.283975
$629$	34.1849	1.36304
$630$	10.0924	0.402092
$631$	−17.9790	−0.715733	−0.357867	−	0.933773i	$-0.616496\pi$
$631$	−17.9790	−0.715733	−0.357867	+	0.933773i	$0.616496\pi$
$632$	6.58464	0.261923
$633$	−12.9638	−0.515266
$634$	13.7503	0.546094
$635$	−1.31350	−0.0521245
$636$	−11.1325	−0.441431
$637$	−42.6820	−1.69112
$638$	−1.29902	−0.0514289
$639$	7.16777	0.283553
$640$	2.56063	0.101218
$641$	−15.6214	−0.617009	−0.308505	−	0.951223i	$-0.599829\pi$
$641$	−15.6214	−0.617009	−0.308505	+	0.951223i	$0.599829\pi$
$642$	13.2818	0.524190
$643$	−29.8162	−1.17583	−0.587917	−	0.808921i	$-0.700052\pi$
$643$	−29.8162	−1.17583	−0.587917	+	0.808921i	$0.700052\pi$
$644$	−3.94138	−0.155312
$645$	17.4491	0.687058
$646$	17.5287	0.689658
$647$	−39.7274	−1.56184	−0.780922	−	0.624629i	$-0.785250\pi$
$647$	−39.7274	−1.56184	−0.780922	+	0.624629i	$0.785250\pi$
$648$	1.00000	0.0392837
$649$	−8.73967	−0.343062
$650$	−7.78594	−0.305390
$651$	−4.95952	−0.194379
$652$	−6.23792	−0.244296
$653$	−2.78743	−0.109081	−0.0545403	−	0.998512i	$-0.517369\pi$
$653$	−2.78743	−0.109081	−0.0545403	+	0.998512i	$0.517369\pi$
$654$	−7.74423	−0.302823
$655$	−1.59493	−0.0623192
$656$	−7.41434	−0.289481
$657$	−14.2751	−0.556926
$658$	32.0005	1.24751
$659$	−25.2988	−0.985500	−0.492750	−	0.870171i	$-0.664008\pi$
$659$	−25.2988	−0.985500	−0.492750	+	0.870171i	$0.664008\pi$
$660$	−3.32632	−0.129477
$661$	13.3248	0.518275	0.259138	−	0.965840i	$-0.416562\pi$
$661$	13.3248	0.518275	0.259138	+	0.965840i	$0.416562\pi$
$662$	−22.6054	−0.878585
$663$	19.3027	0.749653
$664$	−2.40118	−0.0931837
$665$	45.8349	1.77740
$666$	8.85699	0.343202
$667$	1.00000	0.0387202
$668$	−3.11626	−0.120572
$669$	−9.95293	−0.384802
$670$	−7.18820	−0.277705
$671$	0.910585	0.0351527
$672$	−3.94138	−0.152042
$673$	23.4342	0.903320	0.451660	−	0.892190i	$-0.350832\pi$
$673$	23.4342	0.903320	0.451660	+	0.892190i	$0.350832\pi$
$674$	4.69618	0.180890
$675$	−1.55684	−0.0599226
$676$	12.0113	0.461975
$677$	−40.3000	−1.54886	−0.774428	−	0.632663i	$-0.781962\pi$
$677$	−40.3000	−1.54886	−0.774428	+	0.632663i	$0.781962\pi$
$678$	−4.72818	−0.181585
$679$	−4.88689	−0.187541
$680$	9.88316	0.379002
$681$	−7.74988	−0.296976
$682$	1.63459	0.0625917
$683$	−6.93127	−0.265218	−0.132609	−	0.991168i	$-0.542335\pi$
$683$	−6.93127	−0.265218	−0.132609	+	0.991168i	$0.542335\pi$
$684$	4.54152	0.173649
$685$	21.9821	0.839893
$686$	6.04792	0.230911
$687$	−17.1578	−0.654609
$688$	−6.81438	−0.259796
$689$	−55.6749	−2.12105
$690$	2.56063	0.0974816
$691$	1.81864	0.0691844	0.0345922	−	0.999402i	$-0.488987\pi$
$691$	1.81864	0.0691844	0.0345922	+	0.999402i	$0.488987\pi$
$692$	15.3688	0.584234
$693$	5.11995	0.194491
$694$	−21.1535	−0.802975
$695$	27.9134	1.05882
$696$	1.00000	0.0379049
$697$	−28.6168	−1.08394
$698$	25.0918	0.949740
$699$	−2.97688	−0.112596
$700$	6.13608	0.231922
$701$	29.9909	1.13274	0.566370	−	0.824151i	$-0.308347\pi$
$701$	29.9909	1.13274	0.566370	+	0.824151i	$0.308347\pi$
$702$	5.00113	0.188756
$703$	40.2242	1.51709
$704$	1.29902	0.0489588
$705$	−20.7901	−0.782999
$706$	27.9371	1.05143
$707$	24.1048	0.906554
$708$	6.72787	0.252849
$709$	−33.6637	−1.26427	−0.632133	−	0.774860i	$-0.717820\pi$
$709$	−33.6637	−1.26427	−0.632133	+	0.774860i	$0.717820\pi$
$710$	18.3540	0.688814
$711$	6.58464	0.246943
$712$	−17.5887	−0.659163
$713$	−1.25832	−0.0471245
$714$	−15.2124	−0.569309
$715$	−16.6354	−0.622128
$716$	−15.2408	−0.569575
$717$	−7.01362	−0.261929
$718$	−18.6367	−0.695515
$719$	−51.0227	−1.90282	−0.951412	−	0.307920i	$-0.900367\pi$
$719$	−51.0227	−1.90282	−0.951412	+	0.307920i	$0.900367\pi$
$720$	2.56063	0.0954291
$721$	−53.6350	−1.99747
$722$	1.62541	0.0604913
$723$	26.3107	0.978506
$724$	−6.99922	−0.260124
$725$	−1.55684	−0.0578194
$726$	9.31254	0.345621
$727$	5.41966	0.201004	0.100502	−	0.994937i	$-0.467955\pi$
$727$	5.41966	0.201004	0.100502	+	0.994937i	$0.467955\pi$
$728$	−19.7114	−0.730552
$729$	1.00000	0.0370370
$730$	−36.5534	−1.35290
$731$	−26.3012	−0.972784
$732$	−0.700976	−0.0259088
$733$	−20.9515	−0.773861	−0.386931	−	0.922109i	$-0.626465\pi$
$733$	−20.9515	−0.773861	−0.386931	+	0.922109i	$0.626465\pi$
$734$	−20.3576	−0.751413
$735$	−21.8536	−0.806083
$736$	−1.00000	−0.0368605
$737$	−3.64662	−0.134325
$738$	−7.41434	−0.272926
$739$	−14.6555	−0.539112	−0.269556	−	0.962985i	$-0.586877\pi$
$739$	−14.6555	−0.539112	−0.269556	+	0.962985i	$0.586877\pi$
$740$	22.6795	0.833715
$741$	22.7128	0.834374
$742$	43.8772	1.61078
$743$	6.30334	0.231247	0.115624	−	0.993293i	$-0.463113\pi$
$743$	6.30334	0.231247	0.115624	+	0.993293i	$0.463113\pi$
$744$	−1.25832	−0.0461323
$745$	15.2967	0.560430
$746$	1.36257	0.0498873
$747$	−2.40118	−0.0878544
$748$	5.01379	0.183322
$749$	−52.3485	−1.91277
$750$	8.81668	0.321940
$751$	−43.0131	−1.56957	−0.784785	−	0.619769i	$-0.787226\pi$
$751$	−43.0131	−1.56957	−0.784785	+	0.619769i	$0.787226\pi$
$752$	8.11911	0.296074
$753$	30.3765	1.10698
$754$	5.00113	0.182131
$755$	36.7547	1.33764
$756$	−3.94138	−0.143347
$757$	45.8442	1.66624	0.833118	−	0.553096i	$-0.186554\pi$
$757$	45.8442	1.66624	0.833118	+	0.553096i	$0.186554\pi$
$758$	−15.7149	−0.570791
$759$	1.29902	0.0471516
$760$	11.6292	0.421834
$761$	−14.8009	−0.536534	−0.268267	−	0.963345i	$-0.586451\pi$
$761$	−14.8009	−0.536534	−0.268267	+	0.963345i	$0.586451\pi$
$762$	0.512958	0.0185825
$763$	30.5229	1.10500
$764$	−18.9929	−0.687139
$765$	9.88316	0.357326
$766$	5.85118	0.211412
$767$	33.6470	1.21492
$768$	−1.00000	−0.0360844
$769$	46.8413	1.68914	0.844571	−	0.535444i	$-0.179856\pi$
$769$	46.8413	1.68914	0.844571	+	0.535444i	$0.179856\pi$
$770$	13.1103	0.472462
$771$	−13.1573	−0.473847
$772$	−13.0004	−0.467894
$773$	−25.3371	−0.911312	−0.455656	−	0.890156i	$-0.650595\pi$
$773$	−25.3371	−0.911312	−0.455656	+	0.890156i	$0.650595\pi$
$774$	−6.81438	−0.244938
$775$	1.95900	0.0703693
$776$	−1.23989	−0.0445095
$777$	−34.9088	−1.25235
$778$	29.1921	1.04659
$779$	−33.6724	−1.20644
$780$	12.8061	0.458531
$781$	9.31111	0.333178
$782$	−3.85966	−0.138021
$783$	1.00000	0.0357371
$784$	8.53447	0.304802
$785$	18.2224	0.650387
$786$	0.622867	0.0222169
$787$	15.1179	0.538894	0.269447	−	0.963015i	$-0.413159\pi$
$787$	15.1179	0.538894	0.269447	+	0.963015i	$0.413159\pi$
$788$	−9.44605	−0.336502
$789$	21.8819	0.779017
$790$	16.8608	0.599882
$791$	18.6355	0.662604
$792$	1.29902	0.0461588
$793$	−3.50568	−0.124490
$794$	28.3701	1.00682
$795$	−28.5061	−1.01101
$796$	25.1472	0.891320
$797$	−32.3613	−1.14630	−0.573149	−	0.819451i	$-0.694278\pi$
$797$	−32.3613	−1.14630	−0.573149	+	0.819451i	$0.694278\pi$
$798$	−17.8999	−0.633648
$799$	31.3370	1.10862
$800$	1.55684	0.0550424
$801$	−17.5887	−0.621465
$802$	11.3449	0.400601
$803$	−18.5437	−0.654394
$804$	2.80720	0.0990023
$805$	−10.0924	−0.355711
$806$	−6.29304	−0.221663
$807$	−2.55644	−0.0899908
$808$	6.11583	0.215154
$809$	12.6962	0.446374	0.223187	−	0.974776i	$-0.428354\pi$
$809$	12.6962	0.446374	0.223187	+	0.974776i	$0.428354\pi$
$810$	2.56063	0.0899714
$811$	13.2176	0.464132	0.232066	−	0.972700i	$-0.425452\pi$
$811$	13.2176	0.464132	0.232066	+	0.972700i	$0.425452\pi$
$812$	−3.94138	−0.138315
$813$	0.612404	0.0214780
$814$	11.5054	0.403266
$815$	−15.9730	−0.559511
$816$	−3.85966	−0.135115
$817$	−30.9476	−1.08272
$818$	−38.5468	−1.34776
$819$	−19.7114	−0.688771
$820$	−18.9854	−0.662999
$821$	−5.44701	−0.190102	−0.0950509	−	0.995472i	$-0.530301\pi$
$821$	−5.44701	−0.190102	−0.0950509	+	0.995472i	$0.530301\pi$
$822$	−8.58464	−0.299424
$823$	22.8393	0.796129	0.398065	−	0.917357i	$-0.369682\pi$
$823$	22.8393	0.796129	0.398065	+	0.917357i	$0.369682\pi$
$824$	−13.6082	−0.474063
$825$	−2.02237	−0.0704098
$826$	−26.5171	−0.922648
$827$	−49.9389	−1.73655	−0.868274	−	0.496086i	$-0.834770\pi$
$827$	−49.9389	−1.73655	−0.868274	+	0.496086i	$0.834770\pi$
$828$	−1.00000	−0.0347524
$829$	−17.8918	−0.621408	−0.310704	−	0.950507i	$-0.600565\pi$
$829$	−17.8918	−0.621408	−0.310704	+	0.950507i	$0.600565\pi$
$830$	−6.14853	−0.213419
$831$	−11.1060	−0.385264
$832$	−5.00113	−0.173383
$833$	32.9401	1.14131
$834$	−10.9010	−0.377470
$835$	−7.97960	−0.276145
$836$	5.89954	0.204040
$837$	−1.25832	−0.0434940
$838$	−22.8817	−0.790434
$839$	−41.3746	−1.42841	−0.714206	−	0.699936i	$-0.753212\pi$
$839$	−41.3746	−1.42841	−0.714206	+	0.699936i	$0.753212\pi$
$840$	−10.0924	−0.348222
$841$	1.00000	0.0344828
$842$	31.0742	1.07089
$843$	−23.8468	−0.821327
$844$	12.9638	0.446233
$845$	30.7566	1.05806
$846$	8.11911	0.279141
$847$	−36.7042	−1.26117
$848$	11.1325	0.382290
$849$	−17.1729	−0.589372
$850$	6.00885	0.206102
$851$	−8.85699	−0.303614
$852$	−7.16777	−0.245564
$853$	16.4431	0.563000	0.281500	−	0.959561i	$-0.409168\pi$
$853$	16.4431	0.563000	0.281500	+	0.959561i	$0.409168\pi$
$854$	2.76281	0.0945415
$855$	11.6292	0.397709
$856$	−13.2818	−0.453962
$857$	32.7125	1.11744	0.558718	−	0.829358i	$-0.311293\pi$
$857$	32.7125	1.11744	0.558718	+	0.829358i	$0.311293\pi$
$858$	6.49659	0.221790
$859$	−20.1346	−0.686984	−0.343492	−	0.939156i	$-0.611610\pi$
$859$	−20.1346	−0.686984	−0.343492	+	0.939156i	$0.611610\pi$
$860$	−17.4491	−0.595010
$861$	29.2227	0.995908
$862$	−31.6300	−1.07732
$863$	18.0833	0.615561	0.307781	−	0.951457i	$-0.400414\pi$
$863$	18.0833	0.615561	0.307781	+	0.951457i	$0.400414\pi$
$864$	−1.00000	−0.0340207
$865$	39.3538	1.33807
$866$	0.715152	0.0243018
$867$	2.10306	0.0714236
$868$	4.95952	0.168337
$869$	8.55361	0.290161
$870$	2.56063	0.0868135
$871$	14.0392	0.475700
$872$	7.74423	0.262253
$873$	−1.23989	−0.0419640
$874$	−4.54152	−0.153619
$875$	−34.7499	−1.17476
$876$	14.2751	0.482312
$877$	43.6504	1.47397	0.736984	−	0.675910i	$-0.236249\pi$
$877$	43.6504	1.47397	0.736984	+	0.675910i	$0.236249\pi$
$878$	−10.1649	−0.343049
$879$	−1.70479	−0.0575013
$880$	3.32632	0.112130
$881$	12.2510	0.412745	0.206373	−	0.978473i	$-0.433834\pi$
$881$	12.2510	0.412745	0.206373	+	0.978473i	$0.433834\pi$
$882$	8.53447	0.287370
$883$	32.3120	1.08738	0.543692	−	0.839285i	$-0.317026\pi$
$883$	32.3120	1.08738	0.543692	+	0.839285i	$0.317026\pi$
$884$	−19.3027	−0.649219
$885$	17.2276	0.579100
$886$	−25.5952	−0.859889
$887$	−8.74866	−0.293751	−0.146876	−	0.989155i	$-0.546922\pi$
$887$	−8.74866	−0.293751	−0.146876	+	0.989155i	$0.546922\pi$
$888$	−8.85699	−0.297221
$889$	−2.02176	−0.0678077
$890$	−45.0381	−1.50968
$891$	1.29902	0.0435189
$892$	9.95293	0.333249
$893$	36.8731	1.23391
$894$	−5.97382	−0.199794
$895$	−39.0260	−1.30450
$896$	3.94138	0.131672
$897$	−5.00113	−0.166983
$898$	−30.0523	−1.00286
$899$	−1.25832	−0.0419674
$900$	1.55684	0.0518945
$901$	42.9675	1.43145
$902$	−9.63141	−0.320691
$903$	26.8581	0.893780
$904$	4.72818	0.157257
$905$	−17.9224	−0.595762
$906$	−14.3537	−0.476871
$907$	−12.3112	−0.408786	−0.204393	−	0.978889i	$-0.565522\pi$
$907$	−12.3112	−0.408786	−0.204393	+	0.978889i	$0.565522\pi$
$908$	7.74988	0.257189
$909$	6.11583	0.202849
$910$	−50.4736	−1.67318
$911$	−24.5520	−0.813445	−0.406722	−	0.913552i	$-0.633328\pi$
$911$	−24.5520	−0.813445	−0.406722	+	0.913552i	$0.633328\pi$
$912$	−4.54152	−0.150385
$913$	−3.11919	−0.103230
$914$	−29.9139	−0.989463
$915$	−1.79494	−0.0593389
$916$	17.1578	0.566908
$917$	−2.45495	−0.0810698
$918$	−3.85966	−0.127388
$919$	44.5887	1.47085	0.735423	−	0.677608i	$-0.236983\pi$
$919$	44.5887	1.47085	0.735423	+	0.677608i	$0.236983\pi$
$920$	−2.56063	−0.0844215
$921$	−13.7302	−0.452426
$922$	−8.87221	−0.292191
$923$	−35.8470	−1.17992
$924$	−5.11995	−0.168434
$925$	13.7889	0.453376
$926$	−21.9219	−0.720399
$927$	−13.6082	−0.446951
$928$	−1.00000	−0.0328266
$929$	−18.6937	−0.613321	−0.306660	−	0.951819i	$-0.599212\pi$
$929$	−18.6937	−0.613321	−0.306660	+	0.951819i	$0.599212\pi$
$930$	−3.22210	−0.105657
$931$	38.7595	1.27029
$932$	2.97688	0.0975109
$933$	−4.72949	−0.154837
$934$	7.40897	0.242429
$935$	12.8385	0.419863
$936$	−5.00113	−0.163467
$937$	−22.5104	−0.735382	−0.367691	−	0.929948i	$-0.619852\pi$
$937$	−22.5104	−0.735382	−0.367691	+	0.929948i	$0.619852\pi$
$938$	−11.0642	−0.361260
$939$	−6.76189	−0.220666
$940$	20.7901	0.678097
$941$	28.9708	0.944422	0.472211	−	0.881486i	$-0.343456\pi$
$941$	28.9708	0.944422	0.472211	+	0.881486i	$0.343456\pi$
$942$	−7.11639	−0.231864
$943$	7.41434	0.241444
$944$	−6.72787	−0.218974
$945$	−10.0924	−0.328306
$946$	−8.85204	−0.287805
$947$	−33.4560	−1.08717	−0.543587	−	0.839353i	$-0.682934\pi$
$947$	−33.4560	−1.08717	−0.543587	+	0.839353i	$0.682934\pi$
$948$	−6.58464	−0.213859
$949$	71.3918	2.31748
$950$	7.07040	0.229394
$951$	−13.7503	−0.445884
$952$	15.2124	0.493036
$953$	−45.1049	−1.46109	−0.730546	−	0.682864i	$-0.760734\pi$
$953$	−45.1049	−1.46109	−0.730546	+	0.682864i	$0.760734\pi$
$954$	11.1325	0.360427
$955$	−48.6338	−1.57375
$956$	7.01362	0.226837
$957$	1.29902	0.0419915
$958$	1.09797	0.0354737
$959$	33.8353	1.09260
$960$	−2.56063	−0.0826440
$961$	−29.4166	−0.948923
$962$	−44.2950	−1.42813
$963$	−13.2818	−0.427999
$964$	−26.3107	−0.847411
$965$	−33.2892	−1.07162
$966$	3.94138	0.126812
$967$	2.42352	0.0779352	0.0389676	−	0.999240i	$-0.487593\pi$
$967$	2.42352	0.0779352	0.0389676	+	0.999240i	$0.487593\pi$
$968$	−9.31254	−0.299316
$969$	−17.5287	−0.563103
$970$	−3.17491	−0.101940
$971$	−8.96661	−0.287752	−0.143876	−	0.989596i	$-0.545957\pi$
$971$	−8.96661	−0.287752	−0.143876	+	0.989596i	$0.545957\pi$
$972$	−1.00000	−0.0320750
$973$	42.9649	1.37739
$974$	13.8183	0.442767
$975$	7.78594	0.249350
$976$	0.700976	0.0224377
$977$	52.0927	1.66659	0.833297	−	0.552826i	$-0.186451\pi$
$977$	52.0927	1.66659	0.833297	+	0.552826i	$0.186451\pi$
$978$	6.23792	0.199467
$979$	−22.8481	−0.730228
$980$	21.8536	0.698089
$981$	7.74423	0.247254
$982$	−9.28961	−0.296443
$983$	−18.1481	−0.578833	−0.289417	−	0.957203i	$-0.593461\pi$
$983$	−18.1481	−0.578833	−0.289417	+	0.957203i	$0.593461\pi$
$984$	7.41434	0.236361
$985$	−24.1879	−0.770689
$986$	−3.85966	−0.122916
$987$	−32.0005	−1.01859
$988$	−22.7128	−0.722589
$989$	6.81438	0.216685
$990$	3.32632	0.105717
$991$	−43.4064	−1.37885	−0.689425	−	0.724357i	$-0.742137\pi$
$991$	−43.4064	−1.37885	−0.689425	+	0.724357i	$0.742137\pi$
$992$	1.25832	0.0399518
$993$	22.6054	0.717361
$994$	28.2509	0.896064
$995$	64.3928	2.04139
$996$	2.40118	0.0760842
$997$	8.58828	0.271994	0.135997	−	0.990709i	$-0.456576\pi$
$997$	8.58828	0.271994	0.135997	+	0.990709i	$0.456576\pi$
$998$	35.2253	1.11504
$999$	−8.85699	−0.280223

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bk.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bk.1.6	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.56063 q^{5} -1.00000 q^{6} +3.94138 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.56063 q^{5} -1.00000 q^{6} +3.94138 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.56063 q^{10} +1.29902 q^{11} -1.00000 q^{12} -5.00113 q^{13} +3.94138 q^{14} -2.56063 q^{15} +1.00000 q^{16} +3.85966 q^{17} +1.00000 q^{18} +4.54152 q^{19} +2.56063 q^{20} -3.94138 q^{21} +1.29902 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.55684 q^{25} -5.00113 q^{26} -1.00000 q^{27} +3.94138 q^{28} -1.00000 q^{29} -2.56063 q^{30} +1.25832 q^{31} +1.00000 q^{32} -1.29902 q^{33} +3.85966 q^{34} +10.0924 q^{35} +1.00000 q^{36} +8.85699 q^{37} +4.54152 q^{38} +5.00113 q^{39} +2.56063 q^{40} -7.41434 q^{41} -3.94138 q^{42} -6.81438 q^{43} +1.29902 q^{44} +2.56063 q^{45} -1.00000 q^{46} +8.11911 q^{47} -1.00000 q^{48} +8.53447 q^{49} +1.55684 q^{50} -3.85966 q^{51} -5.00113 q^{52} +11.1325 q^{53} -1.00000 q^{54} +3.32632 q^{55} +3.94138 q^{56} -4.54152 q^{57} -1.00000 q^{58} -6.72787 q^{59} -2.56063 q^{60} +0.700976 q^{61} +1.25832 q^{62} +3.94138 q^{63} +1.00000 q^{64} -12.8061 q^{65} -1.29902 q^{66} -2.80720 q^{67} +3.85966 q^{68} +1.00000 q^{69} +10.0924 q^{70} +7.16777 q^{71} +1.00000 q^{72} -14.2751 q^{73} +8.85699 q^{74} -1.55684 q^{75} +4.54152 q^{76} +5.11995 q^{77} +5.00113 q^{78} +6.58464 q^{79} +2.56063 q^{80} +1.00000 q^{81} -7.41434 q^{82} -2.40118 q^{83} -3.94138 q^{84} +9.88316 q^{85} -6.81438 q^{86} +1.00000 q^{87} +1.29902 q^{88} -17.5887 q^{89} +2.56063 q^{90} -19.7114 q^{91} -1.00000 q^{92} -1.25832 q^{93} +8.11911 q^{94} +11.6292 q^{95} -1.00000 q^{96} -1.23989 q^{97} +8.53447 q^{98} +1.29902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 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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