LMFDB - Embedded newform 4002.2.a.bc.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.11324.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 4x + 2 $ x^4 - x^3 - 5x^2 + 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.18994$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.07486 q^{5} +1.00000 q^{6} +2.07486 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.07486 q^{5} +1.00000 q^{6} +2.07486 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.07486 q^{10} -5.68076 q^{11} +1.00000 q^{12} -0.699126 q^{13} +2.07486 q^{14} -4.07486 q^{15} +1.00000 q^{16} +4.45474 q^{17} +1.00000 q^{18} +2.45474 q^{19} -4.07486 q^{20} +2.07486 q^{21} -5.68076 q^{22} -1.00000 q^{23} +1.00000 q^{24} +11.6044 q^{25} -0.699126 q^{26} +1.00000 q^{27} +2.07486 q^{28} -1.00000 q^{29} -4.07486 q^{30} -7.63704 q^{31} +1.00000 q^{32} -5.68076 q^{33} +4.45474 q^{34} -8.45474 q^{35} +1.00000 q^{36} -3.98578 q^{37} +2.45474 q^{38} -0.699126 q^{39} -4.07486 q^{40} -5.15386 q^{41} +2.07486 q^{42} -3.09323 q^{43} -5.68076 q^{44} -4.07486 q^{45} -1.00000 q^{46} +1.09323 q^{47} +1.00000 q^{48} -2.69497 q^{49} +11.6044 q^{50} +4.45474 q^{51} -0.699126 q^{52} -11.7698 q^{53} +1.00000 q^{54} +23.1483 q^{55} +2.07486 q^{56} +2.45474 q^{57} -1.00000 q^{58} -7.05649 q^{59} -4.07486 q^{60} +0.566333 q^{61} -7.63704 q^{62} +2.07486 q^{63} +1.00000 q^{64} +2.84884 q^{65} -5.68076 q^{66} -2.37988 q^{67} +4.45474 q^{68} -1.00000 q^{69} -8.45474 q^{70} +13.1850 q^{71} +1.00000 q^{72} -6.75976 q^{73} -3.98578 q^{74} +11.6044 q^{75} +2.45474 q^{76} -11.7867 q^{77} -0.699126 q^{78} +0.848837 q^{79} -4.07486 q^{80} +1.00000 q^{81} -5.15386 q^{82} -9.54796 q^{83} +2.07486 q^{84} -18.1524 q^{85} -3.09323 q^{86} -1.00000 q^{87} -5.68076 q^{88} -8.93791 q^{89} -4.07486 q^{90} -1.45059 q^{91} -1.00000 q^{92} -7.63704 q^{93} +1.09323 q^{94} -10.0027 q^{95} +1.00000 q^{96} -10.5012 q^{97} -2.69497 q^{98} -5.68076 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{9} - 4 q^{10} - 12 q^{11} + 4 q^{12} - 6 q^{13} - 4 q^{14} - 4 q^{15} + 4 q^{16} - 2 q^{17} + 4 q^{18} - 10 q^{19} - 4 q^{20} - 4 q^{21} - 12 q^{22} - 4 q^{23} + 4 q^{24} + 2 q^{25} - 6 q^{26} + 4 q^{27} - 4 q^{28} - 4 q^{29} - 4 q^{30} - 6 q^{31} + 4 q^{32} - 12 q^{33} - 2 q^{34} - 14 q^{35} + 4 q^{36} - 10 q^{37} - 10 q^{38} - 6 q^{39} - 4 q^{40} - 4 q^{41} - 4 q^{42} - 14 q^{43} - 12 q^{44} - 4 q^{45} - 4 q^{46} + 6 q^{47} + 4 q^{48} - 6 q^{49} + 2 q^{50} - 2 q^{51} - 6 q^{52} - 30 q^{53} + 4 q^{54} + 22 q^{55} - 4 q^{56} - 10 q^{57} - 4 q^{58} - 2 q^{59} - 4 q^{60} - 2 q^{61} - 6 q^{62} - 4 q^{63} + 4 q^{64} - 10 q^{65} - 12 q^{66} - 2 q^{67} - 2 q^{68} - 4 q^{69} - 14 q^{70} + 10 q^{71} + 4 q^{72} - 12 q^{73} - 10 q^{74} + 2 q^{75} - 10 q^{76} + 2 q^{77} - 6 q^{78} - 18 q^{79} - 4 q^{80} + 4 q^{81} - 4 q^{82} - 20 q^{83} - 4 q^{84} - 10 q^{85} - 14 q^{86} - 4 q^{87} - 12 q^{88} - 8 q^{89} - 4 q^{90} + 22 q^{91} - 4 q^{92} - 6 q^{93} + 6 q^{94} - 2 q^{95} + 4 q^{96} + 2 q^{97} - 6 q^{98} - 12 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^9 - 4 * q^10 - 12 * q^11 + 4 * q^12 - 6 * q^13 - 4 * q^14 - 4 * q^15 + 4 * q^16 - 2 * q^17 + 4 * q^18 - 10 * q^19 - 4 * q^20 - 4 * q^21 - 12 * q^22 - 4 * q^23 + 4 * q^24 + 2 * q^25 - 6 * q^26 + 4 * q^27 - 4 * q^28 - 4 * q^29 - 4 * q^30 - 6 * q^31 + 4 * q^32 - 12 * q^33 - 2 * q^34 - 14 * q^35 + 4 * q^36 - 10 * q^37 - 10 * q^38 - 6 * q^39 - 4 * q^40 - 4 * q^41 - 4 * q^42 - 14 * q^43 - 12 * q^44 - 4 * q^45 - 4 * q^46 + 6 * q^47 + 4 * q^48 - 6 * q^49 + 2 * q^50 - 2 * q^51 - 6 * q^52 - 30 * q^53 + 4 * q^54 + 22 * q^55 - 4 * q^56 - 10 * q^57 - 4 * q^58 - 2 * q^59 - 4 * q^60 - 2 * q^61 - 6 * q^62 - 4 * q^63 + 4 * q^64 - 10 * q^65 - 12 * q^66 - 2 * q^67 - 2 * q^68 - 4 * q^69 - 14 * q^70 + 10 * q^71 + 4 * q^72 - 12 * q^73 - 10 * q^74 + 2 * q^75 - 10 * q^76 + 2 * q^77 - 6 * q^78 - 18 * q^79 - 4 * q^80 + 4 * q^81 - 4 * q^82 - 20 * q^83 - 4 * q^84 - 10 * q^85 - 14 * q^86 - 4 * q^87 - 12 * q^88 - 8 * q^89 - 4 * q^90 + 22 * q^91 - 4 * q^92 - 6 * q^93 + 6 * q^94 - 2 * q^95 + 4 * q^96 + 2 * q^97 - 6 * q^98 - 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−4.07486	−1.82233	−0.911165	−	0.412041i	$-0.864816\pi$
$5$	−4.07486	−1.82233	−0.911165	+	0.412041i	$0.864816\pi$
$6$	1.00000	0.408248
$7$	2.07486	0.784222	0.392111	−	0.919918i	$-0.371745\pi$
$7$	2.07486	0.784222	0.392111	+	0.919918i	$0.371745\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−4.07486	−1.28858
$11$	−5.68076	−1.71281	−0.856406	−	0.516303i	$-0.827308\pi$
$11$	−5.68076	−1.71281	−0.856406	+	0.516303i	$0.827308\pi$
$12$	1.00000	0.288675
$13$	−0.699126	−0.193903	−0.0969513	−	0.995289i	$-0.530909\pi$
$13$	−0.699126	−0.193903	−0.0969513	+	0.995289i	$0.530909\pi$
$14$	2.07486	0.554529
$15$	−4.07486	−1.05212
$16$	1.00000	0.250000
$17$	4.45474	1.08043	0.540216	−	0.841526i	$-0.318342\pi$
$17$	4.45474	1.08043	0.540216	+	0.841526i	$0.318342\pi$
$18$	1.00000	0.235702
$19$	2.45474	0.563155	0.281578	−	0.959538i	$-0.409142\pi$
$19$	2.45474	0.563155	0.281578	+	0.959538i	$0.409142\pi$
$20$	−4.07486	−0.911165
$21$	2.07486	0.452771
$22$	−5.68076	−1.21114
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	11.6044	2.32089
$26$	−0.699126	−0.137110
$27$	1.00000	0.192450
$28$	2.07486	0.392111
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−4.07486	−0.743963
$31$	−7.63704	−1.37165	−0.685826	−	0.727765i	$-0.740559\pi$
$31$	−7.63704	−1.37165	−0.685826	+	0.727765i	$0.740559\pi$
$32$	1.00000	0.176777
$33$	−5.68076	−0.988893
$34$	4.45474	0.763981
$35$	−8.45474	−1.42911
$36$	1.00000	0.166667
$37$	−3.98578	−0.655258	−0.327629	−	0.944806i	$-0.606250\pi$
$37$	−3.98578	−0.655258	−0.327629	+	0.944806i	$0.606250\pi$
$38$	2.45474	0.398211
$39$	−0.699126	−0.111950
$40$	−4.07486	−0.644291
$41$	−5.15386	−0.804898	−0.402449	−	0.915442i	$-0.631841\pi$
$41$	−5.15386	−0.804898	−0.402449	+	0.915442i	$0.631841\pi$
$42$	2.07486	0.320157
$43$	−3.09323	−0.471713	−0.235856	−	0.971788i	$-0.575789\pi$
$43$	−3.09323	−0.471713	−0.235856	+	0.971788i	$0.575789\pi$
$44$	−5.68076	−0.856406
$45$	−4.07486	−0.607444
$46$	−1.00000	−0.147442
$47$	1.09323	0.159463	0.0797317	−	0.996816i	$-0.474594\pi$
$47$	1.09323	0.159463	0.0797317	+	0.996816i	$0.474594\pi$
$48$	1.00000	0.144338
$49$	−2.69497	−0.384996
$50$	11.6044	1.64112
$51$	4.45474	0.623788
$52$	−0.699126	−0.0969513
$53$	−11.7698	−1.61671	−0.808355	−	0.588695i	$-0.799642\pi$
$53$	−11.7698	−1.61671	−0.808355	+	0.588695i	$0.799642\pi$
$54$	1.00000	0.136083
$55$	23.1483	3.12131
$56$	2.07486	0.277264
$57$	2.45474	0.325138
$58$	−1.00000	−0.131306
$59$	−7.05649	−0.918676	−0.459338	−	0.888262i	$-0.651913\pi$
$59$	−7.05649	−0.918676	−0.459338	+	0.888262i	$0.651913\pi$
$60$	−4.07486	−0.526062
$61$	0.566333	0.0725116	0.0362558	−	0.999343i	$-0.488457\pi$
$61$	0.566333	0.0725116	0.0362558	+	0.999343i	$0.488457\pi$
$62$	−7.63704	−0.969905
$63$	2.07486	0.261407
$64$	1.00000	0.125000
$65$	2.84884	0.353355
$66$	−5.68076	−0.699253
$67$	−2.37988	−0.290749	−0.145374	−	0.989377i	$-0.546439\pi$
$67$	−2.37988	−0.290749	−0.145374	+	0.989377i	$0.546439\pi$
$68$	4.45474	0.540216
$69$	−1.00000	−0.120386
$70$	−8.45474	−1.01053
$71$	13.1850	1.56477	0.782386	−	0.622794i	$-0.214003\pi$
$71$	13.1850	1.56477	0.782386	+	0.622794i	$0.214003\pi$
$72$	1.00000	0.117851
$73$	−6.75976	−0.791170	−0.395585	−	0.918429i	$-0.629458\pi$
$73$	−6.75976	−0.791170	−0.395585	+	0.918429i	$0.629458\pi$
$74$	−3.98578	−0.463338
$75$	11.6044	1.33997
$76$	2.45474	0.281578
$77$	−11.7867	−1.34322
$78$	−0.699126	−0.0791604
$79$	0.848837	0.0955017	0.0477508	−	0.998859i	$-0.484795\pi$
$79$	0.848837	0.0955017	0.0477508	+	0.998859i	$0.484795\pi$
$80$	−4.07486	−0.455583
$81$	1.00000	0.111111
$82$	−5.15386	−0.569149
$83$	−9.54796	−1.04803	−0.524013	−	0.851711i	$-0.675566\pi$
$83$	−9.54796	−1.04803	−0.524013	+	0.851711i	$0.675566\pi$
$84$	2.07486	0.226385
$85$	−18.1524	−1.96891
$86$	−3.09323	−0.333551
$87$	−1.00000	−0.107211
$88$	−5.68076	−0.605571
$89$	−8.93791	−0.947417	−0.473708	−	0.880682i	$-0.657085\pi$
$89$	−8.93791	−0.947417	−0.473708	+	0.880682i	$0.657085\pi$
$90$	−4.07486	−0.429528
$91$	−1.45059	−0.152063
$92$	−1.00000	−0.104257
$93$	−7.63704	−0.791924
$94$	1.09323	0.112758
$95$	−10.0027	−1.02626
$96$	1.00000	0.102062
$97$	−10.5012	−1.06623	−0.533115	−	0.846042i	$-0.678979\pi$
$97$	−10.5012	−1.06623	−0.533115	+	0.846042i	$0.678979\pi$
$98$	−2.69497	−0.272233
$99$	−5.68076	−0.570937
$100$	11.6044	1.16044
$101$	5.09593	0.507064	0.253532	−	0.967327i	$-0.418408\pi$
$101$	5.09593	0.507064	0.253532	+	0.967327i	$0.418408\pi$
$102$	4.45474	0.441085
$103$	3.88840	0.383136	0.191568	−	0.981479i	$-0.438643\pi$
$103$	3.88840	0.383136	0.191568	+	0.981479i	$0.438643\pi$
$104$	−0.699126	−0.0685549
$105$	−8.45474	−0.825098
$106$	−11.7698	−1.14319
$107$	−19.4533	−1.88062	−0.940310	−	0.340319i	$-0.889465\pi$
$107$	−19.4533	−1.88062	−0.940310	+	0.340319i	$0.889465\pi$
$108$	1.00000	0.0962250
$109$	−10.9095	−1.04494	−0.522469	−	0.852658i	$-0.674989\pi$
$109$	−10.9095	−1.04494	−0.522469	+	0.852658i	$0.674989\pi$
$110$	23.1483	2.20710
$111$	−3.98578	−0.378314
$112$	2.07486	0.196055
$113$	−17.3642	−1.63349	−0.816744	−	0.577000i	$-0.804223\pi$
$113$	−17.3642	−1.63349	−0.816744	+	0.577000i	$0.804223\pi$
$114$	2.45474	0.229907
$115$	4.07486	0.379982
$116$	−1.00000	−0.0928477
$117$	−0.699126	−0.0646342
$118$	−7.05649	−0.649602
$119$	9.24294	0.847299
$120$	−4.07486	−0.371982
$121$	21.2710	1.93373
$122$	0.566333	0.0512734
$123$	−5.15386	−0.464708
$124$	−7.63704	−0.685826
$125$	−26.9122	−2.40710
$126$	2.07486	0.184843
$127$	9.17506	0.814155	0.407077	−	0.913394i	$-0.366548\pi$
$127$	9.17506	0.814155	0.407077	+	0.913394i	$0.366548\pi$
$128$	1.00000	0.0883883
$129$	−3.09323	−0.272343
$130$	2.84884	0.249860
$131$	−12.8121	−1.11940	−0.559699	−	0.828696i	$-0.689083\pi$
$131$	−12.8121	−1.11940	−0.559699	+	0.828696i	$0.689083\pi$
$132$	−5.68076	−0.494446
$133$	5.09323	0.441639
$134$	−2.37988	−0.205590
$135$	−4.07486	−0.350708
$136$	4.45474	0.381991
$137$	13.6692	1.16784	0.583921	−	0.811811i	$-0.301518\pi$
$137$	13.6692	1.16784	0.583921	+	0.811811i	$0.301518\pi$
$138$	−1.00000	−0.0851257
$139$	3.38995	0.287532	0.143766	−	0.989612i	$-0.454079\pi$
$139$	3.38995	0.287532	0.143766	+	0.989612i	$0.454079\pi$
$140$	−8.45474	−0.714556
$141$	1.09323	0.0920662
$142$	13.1850	1.10646
$143$	3.97156	0.332119
$144$	1.00000	0.0833333
$145$	4.07486	0.338398
$146$	−6.75976	−0.559442
$147$	−2.69497	−0.222278
$148$	−3.98578	−0.327629
$149$	−1.12864	−0.0924618	−0.0462309	−	0.998931i	$-0.514721\pi$
$149$	−1.12864	−0.0924618	−0.0462309	+	0.998931i	$0.514721\pi$
$150$	11.6044	0.947499
$151$	15.1539	1.23320	0.616602	−	0.787275i	$-0.288509\pi$
$151$	15.1539	1.23320	0.616602	+	0.787275i	$0.288509\pi$
$152$	2.45474	0.199105
$153$	4.45474	0.360144
$154$	−11.7867	−0.949803
$155$	31.1198	2.49960
$156$	−0.699126	−0.0559749
$157$	18.6214	1.48615	0.743073	−	0.669210i	$-0.233367\pi$
$157$	18.6214	1.48615	0.743073	+	0.669210i	$0.233367\pi$
$158$	0.848837	0.0675299
$159$	−11.7698	−0.933408
$160$	−4.07486	−0.322146
$161$	−2.07486	−0.163522
$162$	1.00000	0.0785674
$163$	−5.93376	−0.464768	−0.232384	−	0.972624i	$-0.574653\pi$
$163$	−5.93376	−0.464768	−0.232384	+	0.972624i	$0.574653\pi$
$164$	−5.15386	−0.402449
$165$	23.1483	1.80209
$166$	−9.54796	−0.741066
$167$	−4.90947	−0.379907	−0.189953	−	0.981793i	$-0.560834\pi$
$167$	−4.90947	−0.379907	−0.189953	+	0.981793i	$0.560834\pi$
$168$	2.07486	0.160079
$169$	−12.5112	−0.962402
$170$	−18.1524	−1.39223
$171$	2.45474	0.187718
$172$	−3.09323	−0.235856
$173$	17.6030	1.33833	0.669166	−	0.743113i	$-0.266652\pi$
$173$	17.6030	1.33833	0.669166	+	0.743113i	$0.266652\pi$
$174$	−1.00000	−0.0758098
$175$	24.0776	1.82009
$176$	−5.68076	−0.428203
$177$	−7.05649	−0.530398
$178$	−8.93791	−0.669925
$179$	−3.88413	−0.290313	−0.145157	−	0.989409i	$-0.546369\pi$
$179$	−3.88413	−0.290313	−0.145157	+	0.989409i	$0.546369\pi$
$180$	−4.07486	−0.303722
$181$	18.6678	1.38757	0.693783	−	0.720184i	$-0.255943\pi$
$181$	18.6678	1.38757	0.693783	+	0.720184i	$0.255943\pi$
$182$	−1.45059	−0.107525
$183$	0.566333	0.0418646
$184$	−1.00000	−0.0737210
$185$	16.2415	1.19410
$186$	−7.63704	−0.559975
$187$	−25.3063	−1.85058
$188$	1.09323	0.0797317
$189$	2.07486	0.150924
$190$	−10.0027	−0.725672
$191$	−5.74863	−0.415956	−0.207978	−	0.978133i	$-0.566688\pi$
$191$	−5.74863	−0.415956	−0.207978	+	0.978133i	$0.566688\pi$
$192$	1.00000	0.0721688
$193$	17.5112	1.26049	0.630243	−	0.776398i	$-0.282955\pi$
$193$	17.5112	1.26049	0.630243	+	0.776398i	$0.282955\pi$
$194$	−10.5012	−0.753939
$195$	2.84884	0.204009
$196$	−2.69497	−0.192498
$197$	−7.49003	−0.533642	−0.266821	−	0.963746i	$-0.585973\pi$
$197$	−7.49003	−0.533642	−0.266821	+	0.963746i	$0.585973\pi$
$198$	−5.68076	−0.403714
$199$	2.23017	0.158093	0.0790463	−	0.996871i	$-0.474813\pi$
$199$	2.23017	0.158093	0.0790463	+	0.996871i	$0.474813\pi$
$200$	11.6044	0.820558
$201$	−2.37988	−0.167864
$202$	5.09593	0.358548
$203$	−2.07486	−0.145626
$204$	4.45474	0.311894
$205$	21.0012	1.46679
$206$	3.88840	0.270918
$207$	−1.00000	−0.0695048
$208$	−0.699126	−0.0484757
$209$	−13.9448	−0.964579
$210$	−8.45474	−0.583432
$211$	−5.51846	−0.379907	−0.189953	−	0.981793i	$-0.560834\pi$
$211$	−5.51846	−0.379907	−0.189953	+	0.981793i	$0.560834\pi$
$212$	−11.7698	−0.808355
$213$	13.1850	0.903421
$214$	−19.4533	−1.32980
$215$	12.6044	0.859616
$216$	1.00000	0.0680414
$217$	−15.8457	−1.07568
$218$	−10.9095	−0.738883
$219$	−6.75976	−0.456782
$220$	23.1483	1.56066
$221$	−3.11442	−0.209499
$222$	−3.98578	−0.267508
$223$	12.6723	0.848602	0.424301	−	0.905521i	$-0.360520\pi$
$223$	12.6723	0.848602	0.424301	+	0.905521i	$0.360520\pi$
$224$	2.07486	0.138632
$225$	11.6044	0.773630
$226$	−17.3642	−1.15505
$227$	−12.9674	−0.860677	−0.430339	−	0.902668i	$-0.641606\pi$
$227$	−12.9674	−0.860677	−0.430339	+	0.902668i	$0.641606\pi$
$228$	2.45474	0.162569
$229$	−10.4717	−0.691987	−0.345993	−	0.938237i	$-0.612458\pi$
$229$	−10.4717	−0.691987	−0.345993	+	0.938237i	$0.612458\pi$
$230$	4.07486	0.268688
$231$	−11.7867	−0.775511
$232$	−1.00000	−0.0656532
$233$	17.0675	1.11813	0.559064	−	0.829124i	$-0.311161\pi$
$233$	17.0675	1.11813	0.559064	+	0.829124i	$0.311161\pi$
$234$	−0.699126	−0.0457033
$235$	−4.45474	−0.290595
$236$	−7.05649	−0.459338
$237$	0.848837	0.0551379
$238$	9.24294	0.599131
$239$	−16.0623	−1.03898	−0.519491	−	0.854476i	$-0.673878\pi$
$239$	−16.0623	−1.03898	−0.519491	+	0.854476i	$0.673878\pi$
$240$	−4.07486	−0.263031
$241$	12.5438	0.808018	0.404009	−	0.914755i	$-0.367616\pi$
$241$	12.5438	0.808018	0.404009	+	0.914755i	$0.367616\pi$
$242$	21.2710	1.36735
$243$	1.00000	0.0641500
$244$	0.566333	0.0362558
$245$	10.9816	0.701591
$246$	−5.15386	−0.328598
$247$	−1.71617	−0.109197
$248$	−7.63704	−0.484952
$249$	−9.54796	−0.605078
$250$	−26.9122	−1.70208
$251$	14.8658	0.938318	0.469159	−	0.883114i	$-0.344557\pi$
$251$	14.8658	0.938318	0.469159	+	0.883114i	$0.344557\pi$
$252$	2.07486	0.130704
$253$	5.68076	0.357146
$254$	9.17506	0.575694
$255$	−18.1524	−1.13675
$256$	1.00000	0.0625000
$257$	−2.69082	−0.167849	−0.0839244	−	0.996472i	$-0.526745\pi$
$257$	−2.69082	−0.167849	−0.0839244	+	0.996472i	$0.526745\pi$
$258$	−3.09323	−0.192576
$259$	−8.26992	−0.513868
$260$	2.84884	0.176677
$261$	−1.00000	−0.0618984
$262$	−12.8121	−0.791534
$263$	18.0196	1.11114	0.555569	−	0.831471i	$-0.312501\pi$
$263$	18.0196	1.11114	0.555569	+	0.831471i	$0.312501\pi$
$264$	−5.68076	−0.349626
$265$	47.9604	2.94618
$266$	5.09323	0.312286
$267$	−8.93791	−0.546991
$268$	−2.37988	−0.145374
$269$	−19.5434	−1.19158	−0.595792	−	0.803139i	$-0.703162\pi$
$269$	−19.5434	−1.19158	−0.595792	+	0.803139i	$0.703162\pi$
$270$	−4.07486	−0.247988
$271$	0.460341	0.0279637	0.0139819	−	0.999902i	$-0.495549\pi$
$271$	0.460341	0.0279637	0.0139819	+	0.999902i	$0.495549\pi$
$272$	4.45474	0.270108
$273$	−1.45059	−0.0877934
$274$	13.6692	0.825789
$275$	−65.9220	−3.97525
$276$	−1.00000	−0.0601929
$277$	3.71617	0.223283	0.111642	−	0.993749i	$-0.464389\pi$
$277$	3.71617	0.223283	0.111642	+	0.993749i	$0.464389\pi$
$278$	3.38995	0.203316
$279$	−7.63704	−0.457217
$280$	−8.45474	−0.505267
$281$	−9.95628	−0.593942	−0.296971	−	0.954887i	$-0.595976\pi$
$281$	−9.95628	−0.593942	−0.296971	+	0.954887i	$0.595976\pi$
$282$	1.09323	0.0651006
$283$	−16.4405	−0.977287	−0.488644	−	0.872483i	$-0.662508\pi$
$283$	−16.4405	−0.977287	−0.488644	+	0.872483i	$0.662508\pi$
$284$	13.1850	0.782386
$285$	−10.0027	−0.592509
$286$	3.97156	0.234843
$287$	−10.6935	−0.631219
$288$	1.00000	0.0589256
$289$	2.84469	0.167334
$290$	4.07486	0.239284
$291$	−10.5012	−0.615589
$292$	−6.75976	−0.395585
$293$	11.7768	0.688008	0.344004	−	0.938968i	$-0.388217\pi$
$293$	11.7768	0.688008	0.344004	+	0.938968i	$0.388217\pi$
$294$	−2.69497	−0.157174
$295$	28.7542	1.67413
$296$	−3.98578	−0.231669
$297$	−5.68076	−0.329631
$298$	−1.12864	−0.0653804
$299$	0.699126	0.0404315
$300$	11.6044	0.669983
$301$	−6.41800	−0.369927
$302$	15.1539	0.872007
$303$	5.09593	0.292753
$304$	2.45474	0.140789
$305$	−2.30773	−0.132140
$306$	4.45474	0.254660
$307$	23.0224	1.31396	0.656980	−	0.753908i	$-0.271834\pi$
$307$	23.0224	1.31396	0.656980	+	0.753908i	$0.271834\pi$
$308$	−11.7867	−0.671612
$309$	3.88840	0.221204
$310$	31.1198	1.76749
$311$	−12.1892	−0.691183	−0.345592	−	0.938385i	$-0.612322\pi$
$311$	−12.1892	−0.691183	−0.345592	+	0.938385i	$0.612322\pi$
$312$	−0.699126	−0.0395802
$313$	20.9307	1.18307	0.591536	−	0.806279i	$-0.298522\pi$
$313$	20.9307	1.18307	0.591536	+	0.806279i	$0.298522\pi$
$314$	18.6214	1.05086
$315$	−8.45474	−0.476370
$316$	0.848837	0.0477508
$317$	1.66547	0.0935423	0.0467712	−	0.998906i	$-0.485107\pi$
$317$	1.66547	0.0935423	0.0467712	+	0.998906i	$0.485107\pi$
$318$	−11.7698	−0.660019
$319$	5.68076	0.318061
$320$	−4.07486	−0.227791
$321$	−19.4533	−1.08578
$322$	−2.07486	−0.115627
$323$	10.9352	0.608451
$324$	1.00000	0.0555556
$325$	−8.11297	−0.450027
$326$	−5.93376	−0.328641
$327$	−10.9095	−0.603295
$328$	−5.15386	−0.284574
$329$	2.26829	0.125055
$330$	23.1483	1.27427
$331$	16.0918	0.884484	0.442242	−	0.896896i	$-0.354183\pi$
$331$	16.0918	0.884484	0.442242	+	0.896896i	$0.354183\pi$
$332$	−9.54796	−0.524013
$333$	−3.98578	−0.218419
$334$	−4.90947	−0.268635
$335$	9.69767	0.529841
$336$	2.07486	0.113193
$337$	24.2273	1.31974	0.659872	−	0.751378i	$-0.270611\pi$
$337$	24.2273	1.31974	0.659872	+	0.751378i	$0.270611\pi$
$338$	−12.5112	−0.680521
$339$	−17.3642	−0.943095
$340$	−18.1524	−0.984453
$341$	43.3841	2.34938
$342$	2.45474	0.132737
$343$	−20.1157	−1.08614
$344$	−3.09323	−0.166776
$345$	4.07486	0.219383
$346$	17.6030	0.946343
$347$	34.3360	1.84325	0.921625	−	0.388081i	$-0.126862\pi$
$347$	34.3360	1.84325	0.921625	+	0.388081i	$0.126862\pi$
$348$	−1.00000	−0.0536056
$349$	2.63849	0.141235	0.0706175	−	0.997503i	$-0.477503\pi$
$349$	2.63849	0.141235	0.0706175	+	0.997503i	$0.477503\pi$
$350$	24.0776	1.28700
$351$	−0.699126	−0.0373166
$352$	−5.68076	−0.302785
$353$	−0.228845	−0.0121802	−0.00609009	−	0.999981i	$-0.501939\pi$
$353$	−0.228845	−0.0121802	−0.00609009	+	0.999981i	$0.501939\pi$
$354$	−7.05649	−0.375048
$355$	−53.7270	−2.85153
$356$	−8.93791	−0.473708
$357$	9.24294	0.489188
$358$	−3.88413	−0.205282
$359$	−8.14380	−0.429813	−0.214907	−	0.976635i	$-0.568945\pi$
$359$	−8.14380	−0.429813	−0.214907	+	0.976635i	$0.568945\pi$
$360$	−4.07486	−0.214764
$361$	−12.9743	−0.682856
$362$	18.6678	0.981157
$363$	21.2710	1.11644
$364$	−1.45059	−0.0760313
$365$	27.5451	1.44177
$366$	0.566333	0.0296027
$367$	−22.4730	−1.17308	−0.586541	−	0.809919i	$-0.699511\pi$
$367$	−22.4730	−1.17308	−0.586541	+	0.809919i	$0.699511\pi$
$368$	−1.00000	−0.0521286
$369$	−5.15386	−0.268299
$370$	16.2415	0.844355
$371$	−24.4207	−1.26786
$372$	−7.63704	−0.395962
$373$	−26.3048	−1.36201	−0.681006	−	0.732278i	$-0.738457\pi$
$373$	−26.3048	−1.36201	−0.681006	+	0.732278i	$0.738457\pi$
$374$	−25.3063	−1.30856
$375$	−26.9122	−1.38974
$376$	1.09323	0.0563788
$377$	0.699126	0.0360068
$378$	2.07486	0.106719
$379$	−21.9265	−1.12629	−0.563145	−	0.826358i	$-0.690409\pi$
$379$	−21.9265	−1.12629	−0.563145	+	0.826358i	$0.690409\pi$
$380$	−10.0027	−0.513128
$381$	9.17506	0.470053
$382$	−5.74863	−0.294126
$383$	15.1144	0.772311	0.386155	−	0.922434i	$-0.373803\pi$
$383$	15.1144	0.772311	0.386155	+	0.922434i	$0.373803\pi$
$384$	1.00000	0.0510310
$385$	48.0293	2.44780
$386$	17.5112	0.891298
$387$	−3.09323	−0.157238
$388$	−10.5012	−0.533115
$389$	−2.33616	−0.118448	−0.0592241	−	0.998245i	$-0.518863\pi$
$389$	−2.33616	−0.118448	−0.0592241	+	0.998245i	$0.518863\pi$
$390$	2.84884	0.144256
$391$	−4.45474	−0.225286
$392$	−2.69497	−0.136117
$393$	−12.8121	−0.646285
$394$	−7.49003	−0.377342
$395$	−3.45889	−0.174036
$396$	−5.68076	−0.285469
$397$	29.6424	1.48771	0.743856	−	0.668340i	$-0.232995\pi$
$397$	29.6424	1.48771	0.743856	+	0.668340i	$0.232995\pi$
$398$	2.23017	0.111788
$399$	5.09323	0.254980
$400$	11.6044	0.580222
$401$	−34.8942	−1.74253	−0.871266	−	0.490810i	$-0.836701\pi$
$401$	−34.8942	−1.74253	−0.871266	+	0.490810i	$0.836701\pi$
$402$	−2.37988	−0.118698
$403$	5.33925	0.265967
$404$	5.09593	0.253532
$405$	−4.07486	−0.202481
$406$	−2.07486	−0.102973
$407$	22.6423	1.12233
$408$	4.45474	0.220542
$409$	−18.1452	−0.897221	−0.448611	−	0.893727i	$-0.648081\pi$
$409$	−18.1452	−0.897221	−0.448611	+	0.893727i	$0.648081\pi$
$410$	21.0012	1.03718
$411$	13.6692	0.674254
$412$	3.88840	0.191568
$413$	−14.6412	−0.720446
$414$	−1.00000	−0.0491473
$415$	38.9066	1.90985
$416$	−0.699126	−0.0342775
$417$	3.38995	0.166006
$418$	−13.9448	−0.682061
$419$	6.10008	0.298008	0.149004	−	0.988837i	$-0.452393\pi$
$419$	6.10008	0.298008	0.149004	+	0.988837i	$0.452393\pi$
$420$	−8.45474	−0.412549
$421$	−2.14273	−0.104430	−0.0522152	−	0.998636i	$-0.516628\pi$
$421$	−2.14273	−0.104430	−0.0522152	+	0.998636i	$0.516628\pi$
$422$	−5.51846	−0.268635
$423$	1.09323	0.0531545
$424$	−11.7698	−0.571593
$425$	51.6948	2.50756
$426$	13.1850	0.638815
$427$	1.17506	0.0568651
$428$	−19.4533	−0.940310
$429$	3.97156	0.191749
$430$	12.6044	0.607840
$431$	−32.1452	−1.54838	−0.774189	−	0.632954i	$-0.781842\pi$
$431$	−32.1452	−1.54838	−0.774189	+	0.632954i	$0.781842\pi$
$432$	1.00000	0.0481125
$433$	−34.9602	−1.68008	−0.840041	−	0.542523i	$-0.817469\pi$
$433$	−34.9602	−1.68008	−0.840041	+	0.542523i	$0.817469\pi$
$434$	−15.8457	−0.760620
$435$	4.07486	0.195374
$436$	−10.9095	−0.522469
$437$	−2.45474	−0.117426
$438$	−6.75976	−0.322994
$439$	−13.5663	−0.647482	−0.323741	−	0.946146i	$-0.604941\pi$
$439$	−13.5663	−0.647482	−0.323741	+	0.946146i	$0.604941\pi$
$440$	23.1483	1.10355
$441$	−2.69497	−0.128332
$442$	−3.11442	−0.148138
$443$	33.1805	1.57645	0.788226	−	0.615386i	$-0.211000\pi$
$443$	33.1805	1.57645	0.788226	+	0.615386i	$0.211000\pi$
$444$	−3.98578	−0.189157
$445$	36.4207	1.72651
$446$	12.6723	0.600052
$447$	−1.12864	−0.0533829
$448$	2.07486	0.0980277
$449$	−11.1622	−0.526775	−0.263388	−	0.964690i	$-0.584840\pi$
$449$	−11.1622	−0.526775	−0.263388	+	0.964690i	$0.584840\pi$
$450$	11.6044	0.547039
$451$	29.2778	1.37864
$452$	−17.3642	−0.816744
$453$	15.1539	0.711991
$454$	−12.9674	−0.608591
$455$	5.91093	0.277108
$456$	2.45474	0.114954
$457$	−22.4024	−1.04794	−0.523970	−	0.851737i	$-0.675550\pi$
$457$	−22.4024	−1.04794	−0.523970	+	0.851737i	$0.675550\pi$
$458$	−10.4717	−0.489308
$459$	4.45474	0.207929
$460$	4.07486	0.189991
$461$	15.4844	0.721181	0.360591	−	0.932724i	$-0.382575\pi$
$461$	15.4844	0.721181	0.360591	+	0.932724i	$0.382575\pi$
$462$	−11.7867	−0.548369
$463$	−21.0876	−0.980025	−0.490012	−	0.871715i	$-0.663008\pi$
$463$	−21.0876	−0.980025	−0.490012	+	0.871715i	$0.663008\pi$
$464$	−1.00000	−0.0464238
$465$	31.1198	1.44315
$466$	17.0675	0.790636
$467$	22.0086	1.01844	0.509219	−	0.860637i	$-0.329934\pi$
$467$	22.0086	1.01844	0.509219	+	0.860637i	$0.329934\pi$
$468$	−0.699126	−0.0323171
$469$	−4.93791	−0.228012
$470$	−4.45474	−0.205482
$471$	18.6214	0.858027
$472$	−7.05649	−0.324801
$473$	17.5719	0.807955
$474$	0.848837	0.0389884
$475$	28.4859	1.30702
$476$	9.24294	0.423649
$477$	−11.7698	−0.538903
$478$	−16.0623	−0.734671
$479$	17.4675	0.798111	0.399055	−	0.916927i	$-0.369338\pi$
$479$	17.4675	0.798111	0.399055	+	0.916927i	$0.369338\pi$
$480$	−4.07486	−0.185991
$481$	2.78656	0.127056
$482$	12.5438	0.571355
$483$	−2.07486	−0.0944092
$484$	21.2710	0.966863
$485$	42.7907	1.94303
$486$	1.00000	0.0453609
$487$	−0.846137	−0.0383421	−0.0191711	−	0.999816i	$-0.506103\pi$
$487$	−0.846137	−0.0383421	−0.0191711	+	0.999816i	$0.506103\pi$
$488$	0.566333	0.0256367
$489$	−5.93376	−0.268334
$490$	10.9816	0.496099
$491$	19.1767	0.865432	0.432716	−	0.901530i	$-0.357555\pi$
$491$	19.1767	0.865432	0.432716	+	0.901530i	$0.357555\pi$
$492$	−5.15386	−0.232354
$493$	−4.45474	−0.200631
$494$	−1.71617	−0.0772142
$495$	23.1483	1.04044
$496$	−7.63704	−0.342913
$497$	27.3570	1.22713
$498$	−9.54796	−0.427854
$499$	−10.1229	−0.453164	−0.226582	−	0.973992i	$-0.572755\pi$
$499$	−10.1229	−0.453164	−0.226582	+	0.973992i	$0.572755\pi$
$500$	−26.9122	−1.20355
$501$	−4.90947	−0.219339
$502$	14.8658	0.663491
$503$	22.6415	1.00953	0.504767	−	0.863255i	$-0.331578\pi$
$503$	22.6415	1.00953	0.504767	+	0.863255i	$0.331578\pi$
$504$	2.07486	0.0924214
$505$	−20.7652	−0.924038
$506$	5.68076	0.252540
$507$	−12.5112	−0.555643
$508$	9.17506	0.407077
$509$	18.1084	0.802640	0.401320	−	0.915938i	$-0.368552\pi$
$509$	18.1084	0.802640	0.401320	+	0.915938i	$0.368552\pi$
$510$	−18.1524	−0.803802
$511$	−14.0255	−0.620453
$512$	1.00000	0.0441942
$513$	2.45474	0.108379
$514$	−2.69082	−0.118687
$515$	−15.8447	−0.698200
$516$	−3.09323	−0.136172
$517$	−6.21035	−0.273131
$518$	−8.26992	−0.363360
$519$	17.6030	0.772686
$520$	2.84884	0.124930
$521$	−1.44361	−0.0632456	−0.0316228	−	0.999500i	$-0.510068\pi$
$521$	−1.44361	−0.0632456	−0.0316228	+	0.999500i	$0.510068\pi$
$522$	−1.00000	−0.0437688
$523$	−26.7582	−1.17005	−0.585027	−	0.811014i	$-0.698916\pi$
$523$	−26.7582	−1.17005	−0.585027	+	0.811014i	$0.698916\pi$
$524$	−12.8121	−0.559699
$525$	24.0776	1.05083
$526$	18.0196	0.785693
$527$	−34.0210	−1.48198
$528$	−5.68076	−0.247223
$529$	1.00000	0.0434783
$530$	47.9604	2.08326
$531$	−7.05649	−0.306225
$532$	5.09323	0.220819
$533$	3.60320	0.156072
$534$	−8.93791	−0.386781
$535$	79.2693	3.42711
$536$	−2.37988	−0.102795
$537$	−3.88413	−0.167612
$538$	−19.5434	−0.842577
$539$	15.3095	0.659426
$540$	−4.07486	−0.175354
$541$	28.4518	1.22324	0.611620	−	0.791152i	$-0.290518\pi$
$541$	28.4518	1.22324	0.611620	+	0.791152i	$0.290518\pi$
$542$	0.460341	0.0197733
$543$	18.6678	0.801111
$544$	4.45474	0.190995
$545$	44.4545	1.90422
$546$	−1.45059	−0.0620793
$547$	−33.7739	−1.44407	−0.722034	−	0.691858i	$-0.756793\pi$
$547$	−33.7739	−1.44407	−0.722034	+	0.691858i	$0.756793\pi$
$548$	13.6692	0.583921
$549$	0.566333	0.0241705
$550$	−65.9220	−2.81093
$551$	−2.45474	−0.104575
$552$	−1.00000	−0.0425628
$553$	1.76121	0.0748945
$554$	3.71617	0.157885
$555$	16.2415	0.689413
$556$	3.38995	0.143766
$557$	26.8602	1.13810	0.569051	−	0.822303i	$-0.307311\pi$
$557$	26.8602	1.13810	0.569051	+	0.822303i	$0.307311\pi$
$558$	−7.63704	−0.323302
$559$	2.16255	0.0914663
$560$	−8.45474	−0.357278
$561$	−25.3063	−1.06843
$562$	−9.95628	−0.419980
$563$	−15.4758	−0.652228	−0.326114	−	0.945331i	$-0.605739\pi$
$563$	−15.4758	−0.652228	−0.326114	+	0.945331i	$0.605739\pi$
$564$	1.09323	0.0460331
$565$	70.7567	2.97675
$566$	−16.4405	−0.691047
$567$	2.07486	0.0871357
$568$	13.1850	0.553230
$569$	−20.9151	−0.876806	−0.438403	−	0.898779i	$-0.644456\pi$
$569$	−20.9151	−0.876806	−0.438403	+	0.898779i	$0.644456\pi$
$570$	−10.0027	−0.418967
$571$	14.5819	0.610235	0.305117	−	0.952315i	$-0.401304\pi$
$571$	14.5819	0.610235	0.305117	+	0.952315i	$0.401304\pi$
$572$	3.97156	0.166059
$573$	−5.74863	−0.240153
$574$	−10.6935	−0.446339
$575$	−11.6044	−0.483939
$576$	1.00000	0.0416667
$577$	−38.3432	−1.59625	−0.798124	−	0.602493i	$-0.794174\pi$
$577$	−38.3432	−1.59625	−0.798124	+	0.602493i	$0.794174\pi$
$578$	2.84469	0.118323
$579$	17.5112	0.727742
$580$	4.07486	0.169199
$581$	−19.8106	−0.821884
$582$	−10.5012	−0.435287
$583$	66.8615	2.76912
$584$	−6.75976	−0.279721
$585$	2.84884	0.117785
$586$	11.7768	0.486495
$587$	−13.2595	−0.547280	−0.273640	−	0.961832i	$-0.588228\pi$
$587$	−13.2595	−0.547280	−0.273640	+	0.961832i	$0.588228\pi$
$588$	−2.69497	−0.111139
$589$	−18.7469	−0.772453
$590$	28.7542	1.18379
$591$	−7.49003	−0.308099
$592$	−3.98578	−0.163815
$593$	33.6663	1.38251	0.691255	−	0.722611i	$-0.257058\pi$
$593$	33.6663	1.38251	0.691255	+	0.722611i	$0.257058\pi$
$594$	−5.68076	−0.233084
$595$	−37.6636	−1.54406
$596$	−1.12864	−0.0462309
$597$	2.23017	0.0912748
$598$	0.699126	0.0285894
$599$	25.1751	1.02863	0.514313	−	0.857603i	$-0.328047\pi$
$599$	25.1751	1.02863	0.514313	+	0.857603i	$0.328047\pi$
$600$	11.6044	0.473750
$601$	−38.0054	−1.55027	−0.775136	−	0.631794i	$-0.782319\pi$
$601$	−38.0054	−1.55027	−0.775136	+	0.631794i	$0.782319\pi$
$602$	−6.41800	−0.261578
$603$	−2.37988	−0.0969163
$604$	15.1539	0.616602
$605$	−86.6762	−3.52389
$606$	5.09593	0.207008
$607$	38.4813	1.56191	0.780955	−	0.624588i	$-0.214733\pi$
$607$	38.4813	1.56191	0.780955	+	0.624588i	$0.214733\pi$
$608$	2.45474	0.0995527
$609$	−2.07486	−0.0840774
$610$	−2.30773	−0.0934371
$611$	−0.764303	−0.0309204
$612$	4.45474	0.180072
$613$	−28.9516	−1.16934	−0.584672	−	0.811270i	$-0.698777\pi$
$613$	−28.9516	−1.16934	−0.584672	+	0.811270i	$0.698777\pi$
$614$	23.0224	0.929110
$615$	21.0012	0.846852
$616$	−11.7867	−0.474902
$617$	28.3223	1.14021	0.570106	−	0.821571i	$-0.306902\pi$
$617$	28.3223	1.14021	0.570106	+	0.821571i	$0.306902\pi$
$618$	3.88840	0.156415
$619$	20.9690	0.842817	0.421409	−	0.906871i	$-0.361536\pi$
$619$	20.9690	0.842817	0.421409	+	0.906871i	$0.361536\pi$
$620$	31.1198	1.24980
$621$	−1.00000	−0.0401286
$622$	−12.1892	−0.488740
$623$	−18.5449	−0.742985
$624$	−0.699126	−0.0279874
$625$	51.6410	2.06564
$626$	20.9307	0.836558
$627$	−13.9448	−0.556900
$628$	18.6214	0.743073
$629$	−17.7556	−0.707963
$630$	−8.45474	−0.336845
$631$	24.1932	0.963117	0.481559	−	0.876414i	$-0.340071\pi$
$631$	24.1932	0.963117	0.481559	+	0.876414i	$0.340071\pi$
$632$	0.848837	0.0337649
$633$	−5.51846	−0.219339
$634$	1.66547	0.0661444
$635$	−37.3870	−1.48366
$636$	−11.7698	−0.466704
$637$	1.88413	0.0746518
$638$	5.68076	0.224903
$639$	13.1850	0.521591
$640$	−4.07486	−0.161073
$641$	13.5139	0.533768	0.266884	−	0.963729i	$-0.414006\pi$
$641$	13.5139	0.533768	0.266884	+	0.963729i	$0.414006\pi$
$642$	−19.4533	−0.767760
$643$	−0.677668	−0.0267246	−0.0133623	−	0.999911i	$-0.504253\pi$
$643$	−0.677668	−0.0267246	−0.0133623	+	0.999911i	$0.504253\pi$
$644$	−2.07486	−0.0817608
$645$	12.6044	0.496300
$646$	10.9352	0.430240
$647$	21.1904	0.833081	0.416540	−	0.909117i	$-0.363242\pi$
$647$	21.1904	0.833081	0.416540	+	0.909117i	$0.363242\pi$
$648$	1.00000	0.0392837
$649$	40.0862	1.57352
$650$	−8.11297	−0.318217
$651$	−15.8457	−0.621044
$652$	−5.93376	−0.232384
$653$	−25.2172	−0.986825	−0.493413	−	0.869795i	$-0.664251\pi$
$653$	−25.2172	−0.986825	−0.493413	+	0.869795i	$0.664251\pi$
$654$	−10.9095	−0.426594
$655$	52.2074	2.03991
$656$	−5.15386	−0.201225
$657$	−6.75976	−0.263723
$658$	2.26829	0.0884270
$659$	35.7468	1.39250	0.696249	−	0.717801i	$-0.254851\pi$
$659$	35.7468	1.39250	0.696249	+	0.717801i	$0.254851\pi$
$660$	23.1483	0.901045
$661$	49.1722	1.91258	0.956288	−	0.292428i	$-0.0944631\pi$
$661$	49.1722	1.91258	0.956288	+	0.292428i	$0.0944631\pi$
$662$	16.0918	0.625425
$663$	−3.11442	−0.120954
$664$	−9.54796	−0.370533
$665$	−20.7542	−0.804812
$666$	−3.98578	−0.154446
$667$	1.00000	0.0387202
$668$	−4.90947	−0.189953
$669$	12.6723	0.489940
$670$	9.69767	0.374654
$671$	−3.21720	−0.124199
$672$	2.07486	0.0800393
$673$	25.7004	0.990677	0.495338	−	0.868700i	$-0.335044\pi$
$673$	25.7004	0.990677	0.495338	+	0.868700i	$0.335044\pi$
$674$	24.2273	0.933199
$675$	11.6044	0.446655
$676$	−12.5112	−0.481201
$677$	−35.1013	−1.34905	−0.674527	−	0.738250i	$-0.735652\pi$
$677$	−35.1013	−1.34905	−0.674527	+	0.738250i	$0.735652\pi$
$678$	−17.3642	−0.666869
$679$	−21.7884	−0.836161
$680$	−18.1524	−0.696113
$681$	−12.9674	−0.496912
$682$	43.3841	1.66126
$683$	−17.3218	−0.662801	−0.331400	−	0.943490i	$-0.607521\pi$
$683$	−17.3218	−0.662801	−0.331400	+	0.943490i	$0.607521\pi$
$684$	2.45474	0.0938592
$685$	−55.7002	−2.12819
$686$	−20.1157	−0.768020
$687$	−10.4717	−0.399519
$688$	−3.09323	−0.117928
$689$	8.22859	0.313484
$690$	4.07486	0.155127
$691$	−41.0665	−1.56224	−0.781121	−	0.624379i	$-0.785352\pi$
$691$	−41.0665	−1.56224	−0.781121	+	0.624379i	$0.785352\pi$
$692$	17.6030	0.669166
$693$	−11.7867	−0.447742
$694$	34.3360	1.30338
$695$	−13.8135	−0.523978
$696$	−1.00000	−0.0379049
$697$	−22.9591	−0.869638
$698$	2.63849	0.0998683
$699$	17.0675	0.645552
$700$	24.0776	0.910046
$701$	−32.8969	−1.24250	−0.621249	−	0.783613i	$-0.713375\pi$
$701$	−32.8969	−1.24250	−0.621249	+	0.783613i	$0.713375\pi$
$702$	−0.699126	−0.0263868
$703$	−9.78405	−0.369012
$704$	−5.68076	−0.214102
$705$	−4.45474	−0.167775
$706$	−0.228845	−0.00861269
$707$	10.5733	0.397650
$708$	−7.05649	−0.265199
$709$	−33.8796	−1.27237	−0.636187	−	0.771535i	$-0.719489\pi$
$709$	−33.8796	−1.27237	−0.636187	+	0.771535i	$0.719489\pi$
$710$	−53.7270	−2.01634
$711$	0.848837	0.0318339
$712$	−8.93791	−0.334962
$713$	7.63704	0.286009
$714$	9.24294	0.345908
$715$	−16.1835	−0.605230
$716$	−3.88413	−0.145157
$717$	−16.0623	−0.599857
$718$	−8.14380	−0.303924
$719$	1.84652	0.0688637	0.0344319	−	0.999407i	$-0.489038\pi$
$719$	1.84652	0.0688637	0.0344319	+	0.999407i	$0.489038\pi$
$720$	−4.07486	−0.151861
$721$	8.06788	0.300463
$722$	−12.9743	−0.482852
$723$	12.5438	0.466509
$724$	18.6678	0.693783
$725$	−11.6044	−0.430978
$726$	21.2710	0.789440
$727$	−8.53092	−0.316394	−0.158197	−	0.987408i	$-0.550568\pi$
$727$	−8.53092	−0.316394	−0.158197	+	0.987408i	$0.550568\pi$
$728$	−1.45059	−0.0537623
$729$	1.00000	0.0370370
$730$	27.5451	1.01949
$731$	−13.7795	−0.509654
$732$	0.566333	0.0209323
$733$	16.0207	0.591737	0.295869	−	0.955229i	$-0.404391\pi$
$733$	16.0207	0.591737	0.295869	+	0.955229i	$0.404391\pi$
$734$	−22.4730	−0.829495
$735$	10.9816	0.405064
$736$	−1.00000	−0.0368605
$737$	13.5195	0.497998
$738$	−5.15386	−0.189716
$739$	−48.8644	−1.79751	−0.898754	−	0.438454i	$-0.855526\pi$
$739$	−48.8644	−1.79751	−0.898754	+	0.438454i	$0.855526\pi$
$740$	16.2415	0.597049
$741$	−1.71617	−0.0630451
$742$	−24.4207	−0.896512
$743$	−5.05970	−0.185622	−0.0928112	−	0.995684i	$-0.529585\pi$
$743$	−5.05970	−0.185622	−0.0928112	+	0.995684i	$0.529585\pi$
$744$	−7.63704	−0.279987
$745$	4.59905	0.168496
$746$	−26.3048	−0.963088
$747$	−9.54796	−0.349342
$748$	−25.3063	−0.925289
$749$	−40.3628	−1.47482
$750$	−26.9122	−0.982694
$751$	−27.6977	−1.01070	−0.505351	−	0.862914i	$-0.668637\pi$
$751$	−27.6977	−1.01070	−0.505351	+	0.862914i	$0.668637\pi$
$752$	1.09323	0.0398658
$753$	14.8658	0.541738
$754$	0.699126	0.0254607
$755$	−61.7498	−2.24731
$756$	2.07486	0.0754618
$757$	22.9320	0.833478	0.416739	−	0.909026i	$-0.363173\pi$
$757$	22.9320	0.833478	0.416739	+	0.909026i	$0.363173\pi$
$758$	−21.9265	−0.796407
$759$	5.68076	0.206198
$760$	−10.0027	−0.362836
$761$	22.0862	0.800623	0.400312	−	0.916379i	$-0.368902\pi$
$761$	22.0862	0.800623	0.400312	+	0.916379i	$0.368902\pi$
$762$	9.17506	0.332377
$763$	−22.6356	−0.819463
$764$	−5.74863	−0.207978
$765$	−18.1524	−0.656302
$766$	15.1144	0.546106
$767$	4.93337	0.178134
$768$	1.00000	0.0360844
$769$	−24.3477	−0.878000	−0.439000	−	0.898487i	$-0.644667\pi$
$769$	−24.3477	−0.878000	−0.439000	+	0.898487i	$0.644667\pi$
$770$	48.0293	1.73086
$771$	−2.69082	−0.0969076
$772$	17.5112	0.630243
$773$	18.9111	0.680185	0.340093	−	0.940392i	$-0.389542\pi$
$773$	18.9111	0.680185	0.340093	+	0.940392i	$0.389542\pi$
$774$	−3.09323	−0.111184
$775$	−88.6236	−3.18345
$776$	−10.5012	−0.376970
$777$	−8.26992	−0.296682
$778$	−2.33616	−0.0837555
$779$	−12.6514	−0.453283
$780$	2.84884	0.102005
$781$	−74.9008	−2.68016
$782$	−4.45474	−0.159301
$783$	−1.00000	−0.0357371
$784$	−2.69497	−0.0962491
$785$	−75.8794	−2.70825
$786$	−12.8121	−0.456992
$787$	−32.3649	−1.15368	−0.576842	−	0.816856i	$-0.695715\pi$
$787$	−32.3649	−1.15368	−0.576842	+	0.816856i	$0.695715\pi$
$788$	−7.49003	−0.266821
$789$	18.0196	0.641515
$790$	−3.45889	−0.123062
$791$	−36.0282	−1.28102
$792$	−5.68076	−0.201857
$793$	−0.395938	−0.0140602
$794$	29.6424	1.05197
$795$	47.9604	1.70098
$796$	2.23017	0.0790463
$797$	16.2133	0.574303	0.287151	−	0.957885i	$-0.407292\pi$
$797$	16.2133	0.574303	0.287151	+	0.957885i	$0.407292\pi$
$798$	5.09323	0.180298
$799$	4.87003	0.172289
$800$	11.6044	0.410279
$801$	−8.93791	−0.315806
$802$	−34.8942	−1.23216
$803$	38.4006	1.35513
$804$	−2.37988	−0.0839320
$805$	8.45474	0.297990
$806$	5.33925	0.188067
$807$	−19.5434	−0.687961
$808$	5.09593	0.179274
$809$	−46.8983	−1.64886	−0.824428	−	0.565967i	$-0.808503\pi$
$809$	−46.8983	−1.64886	−0.824428	+	0.565967i	$0.808503\pi$
$810$	−4.07486	−0.143176
$811$	41.7149	1.46481	0.732404	−	0.680870i	$-0.238398\pi$
$811$	41.7149	1.46481	0.732404	+	0.680870i	$0.238398\pi$
$812$	−2.07486	−0.0728132
$813$	0.460341	0.0161449
$814$	22.6423	0.793611
$815$	24.1792	0.846961
$816$	4.45474	0.155947
$817$	−7.59306	−0.265647
$818$	−18.1452	−0.634431
$819$	−1.45059	−0.0506876
$820$	21.0012	0.733395
$821$	0.830341	0.0289791	0.0144896	−	0.999895i	$-0.495388\pi$
$821$	0.830341	0.0289791	0.0144896	+	0.999895i	$0.495388\pi$
$822$	13.6692	0.476769
$823$	4.76430	0.166073	0.0830366	−	0.996547i	$-0.473538\pi$
$823$	4.76430	0.166073	0.0830366	+	0.996547i	$0.473538\pi$
$824$	3.88840	0.135459
$825$	−65.9220	−2.29511
$826$	−14.6412	−0.509432
$827$	1.87097	0.0650601	0.0325300	−	0.999471i	$-0.489644\pi$
$827$	1.87097	0.0650601	0.0325300	+	0.999471i	$0.489644\pi$
$828$	−1.00000	−0.0347524
$829$	6.35007	0.220547	0.110274	−	0.993901i	$-0.464827\pi$
$829$	6.35007	0.220547	0.110274	+	0.993901i	$0.464827\pi$
$830$	38.9066	1.35047
$831$	3.71617	0.128913
$832$	−0.699126	−0.0242378
$833$	−12.0054	−0.415962
$834$	3.38995	0.117384
$835$	20.0054	0.692315
$836$	−13.9448	−0.482290
$837$	−7.63704	−0.263975
$838$	6.10008	0.210724
$839$	−16.0279	−0.553345	−0.276673	−	0.960964i	$-0.589232\pi$
$839$	−16.0279	−0.553345	−0.276673	+	0.960964i	$0.589232\pi$
$840$	−8.45474	−0.291716
$841$	1.00000	0.0344828
$842$	−2.14273	−0.0738435
$843$	−9.95628	−0.342913
$844$	−5.51846	−0.189953
$845$	50.9814	1.75381
$846$	1.09323	0.0375859
$847$	44.1342	1.51647
$848$	−11.7698	−0.404178
$849$	−16.4405	−0.564237
$850$	51.6948	1.77312
$851$	3.98578	0.136631
$852$	13.1850	0.451711
$853$	20.3418	0.696489	0.348244	−	0.937404i	$-0.386778\pi$
$853$	20.3418	0.696489	0.348244	+	0.937404i	$0.386778\pi$
$854$	1.17506	0.0402097
$855$	−10.0027	−0.342085
$856$	−19.4533	−0.664900
$857$	45.5636	1.55642	0.778211	−	0.628003i	$-0.216128\pi$
$857$	45.5636	1.55642	0.778211	+	0.628003i	$0.216128\pi$
$858$	3.97156	0.135587
$859$	24.2984	0.829049	0.414525	−	0.910038i	$-0.363948\pi$
$859$	24.2984	0.829049	0.414525	+	0.910038i	$0.363948\pi$
$860$	12.6044	0.429808
$861$	−10.6935	−0.364434
$862$	−32.1452	−1.09487
$863$	−5.52141	−0.187951	−0.0939756	−	0.995575i	$-0.529958\pi$
$863$	−5.52141	−0.187951	−0.0939756	+	0.995575i	$0.529958\pi$
$864$	1.00000	0.0340207
$865$	−71.7297	−2.43888
$866$	−34.9602	−1.18800
$867$	2.84469	0.0966106
$868$	−15.8457	−0.537840
$869$	−4.82204	−0.163576
$870$	4.07486	0.138151
$871$	1.66384	0.0563770
$872$	−10.9095	−0.369441
$873$	−10.5012	−0.355410
$874$	−2.45474	−0.0830327
$875$	−55.8389	−1.88770
$876$	−6.75976	−0.228391
$877$	−25.9901	−0.877622	−0.438811	−	0.898579i	$-0.644600\pi$
$877$	−25.9901	−0.877622	−0.438811	+	0.898579i	$0.644600\pi$
$878$	−13.5663	−0.457839
$879$	11.7768	0.397222
$880$	23.1483	0.780328
$881$	54.6464	1.84108	0.920542	−	0.390644i	$-0.127748\pi$
$881$	54.6464	1.84108	0.920542	+	0.390644i	$0.127748\pi$
$882$	−2.69497	−0.0907445
$883$	−18.4335	−0.620338	−0.310169	−	0.950681i	$-0.600386\pi$
$883$	−18.4335	−0.620338	−0.310169	+	0.950681i	$0.600386\pi$
$884$	−3.11442	−0.104749
$885$	28.7542	0.966561
$886$	33.1805	1.11472
$887$	46.9634	1.57688	0.788439	−	0.615113i	$-0.210890\pi$
$887$	46.9634	1.57688	0.788439	+	0.615113i	$0.210890\pi$
$888$	−3.98578	−0.133754
$889$	19.0369	0.638478
$890$	36.4207	1.22082
$891$	−5.68076	−0.190312
$892$	12.6723	0.424301
$893$	2.68358	0.0898027
$894$	−1.12864	−0.0377474
$895$	15.8273	0.529047
$896$	2.07486	0.0693161
$897$	0.699126	0.0233431
$898$	−11.1622	−0.372486
$899$	7.63704	0.254709
$900$	11.6044	0.386815
$901$	−52.4315	−1.74675
$902$	29.2778	0.974845
$903$	−6.41800	−0.213578
$904$	−17.3642	−0.577525
$905$	−76.0685	−2.52860
$906$	15.1539	0.503453
$907$	−9.15280	−0.303914	−0.151957	−	0.988387i	$-0.548557\pi$
$907$	−9.15280	−0.303914	−0.151957	+	0.988387i	$0.548557\pi$
$908$	−12.9674	−0.430339
$909$	5.09593	0.169021
$910$	5.91093	0.195945
$911$	45.0238	1.49170	0.745852	−	0.666112i	$-0.232043\pi$
$911$	45.0238	1.49170	0.745852	+	0.666112i	$0.232043\pi$
$912$	2.45474	0.0812845
$913$	54.2396	1.79507
$914$	−22.4024	−0.741006
$915$	−2.30773	−0.0762911
$916$	−10.4717	−0.345993
$917$	−26.5833	−0.877856
$918$	4.45474	0.147028
$919$	31.5493	1.04072	0.520358	−	0.853948i	$-0.325798\pi$
$919$	31.5493	1.04072	0.520358	+	0.853948i	$0.325798\pi$
$920$	4.07486	0.134344
$921$	23.0224	0.758615
$922$	15.4844	0.509952
$923$	−9.21798	−0.303413
$924$	−11.7867	−0.387756
$925$	−46.2528	−1.52078
$926$	−21.0876	−0.692982
$927$	3.88840	0.127712
$928$	−1.00000	−0.0328266
$929$	−44.8615	−1.47186	−0.735929	−	0.677058i	$-0.763255\pi$
$929$	−44.8615	−1.47186	−0.735929	+	0.677058i	$0.763255\pi$
$930$	31.1198	1.02046
$931$	−6.61545	−0.216813
$932$	17.0675	0.559064
$933$	−12.1892	−0.399055
$934$	22.0086	0.720144
$935$	103.119	3.37237
$936$	−0.699126	−0.0228516
$937$	42.8349	1.39936	0.699678	−	0.714458i	$-0.253327\pi$
$937$	42.8349	1.39936	0.699678	+	0.714458i	$0.253327\pi$
$938$	−4.93791	−0.161229
$939$	20.9307	0.683047
$940$	−4.45474	−0.145298
$941$	8.01238	0.261196	0.130598	−	0.991435i	$-0.458310\pi$
$941$	8.01238	0.261196	0.130598	+	0.991435i	$0.458310\pi$
$942$	18.6214	0.606717
$943$	5.15386	0.167833
$944$	−7.05649	−0.229669
$945$	−8.45474	−0.275033
$946$	17.5719	0.571310
$947$	34.4529	1.11957	0.559784	−	0.828638i	$-0.310884\pi$
$947$	34.4529	1.11957	0.559784	+	0.828638i	$0.310884\pi$
$948$	0.848837	0.0275690
$949$	4.72593	0.153410
$950$	28.4859	0.924204
$951$	1.66547	0.0540067
$952$	9.24294	0.299565
$953$	−1.13424	−0.0367418	−0.0183709	−	0.999831i	$-0.505848\pi$
$953$	−1.13424	−0.0367418	−0.0183709	+	0.999831i	$0.505848\pi$
$954$	−11.7698	−0.381062
$955$	23.4248	0.758010
$956$	−16.0623	−0.519491
$957$	5.68076	0.183633
$958$	17.4675	0.564349
$959$	28.3617	0.915847
$960$	−4.07486	−0.131515
$961$	27.3243	0.881430
$962$	2.78656	0.0898424
$963$	−19.4533	−0.626873
$964$	12.5438	0.404009
$965$	−71.3557	−2.29702
$966$	−2.07486	−0.0667574
$967$	−39.1842	−1.26008	−0.630040	−	0.776563i	$-0.716961\pi$
$967$	−39.1842	−1.26008	−0.630040	+	0.776563i	$0.716961\pi$
$968$	21.2710	0.683675
$969$	10.9352	0.351290
$970$	42.7907	1.37393
$971$	−53.9067	−1.72995	−0.864974	−	0.501816i	$-0.832665\pi$
$971$	−53.9067	−1.72995	−0.864974	+	0.501816i	$0.832665\pi$
$972$	1.00000	0.0320750
$973$	7.03365	0.225489
$974$	−0.846137	−0.0271120
$975$	−8.11297	−0.259823
$976$	0.566333	0.0181279
$977$	36.7601	1.17606	0.588029	−	0.808839i	$-0.299904\pi$
$977$	36.7601	1.17606	0.588029	+	0.808839i	$0.299904\pi$
$978$	−5.93376	−0.189741
$979$	50.7741	1.62275
$980$	10.9816	0.350795
$981$	−10.9095	−0.348313
$982$	19.1767	0.611953
$983$	−42.3431	−1.35054	−0.675268	−	0.737572i	$-0.735972\pi$
$983$	−42.3431	−1.35054	−0.675268	+	0.737572i	$0.735972\pi$
$984$	−5.15386	−0.164299
$985$	30.5208	0.972473
$986$	−4.45474	−0.141868
$987$	2.26829	0.0722003
$988$	−1.71617	−0.0545987
$989$	3.09323	0.0983589
$990$	23.1483	0.735700
$991$	30.1001	0.956161	0.478080	−	0.878316i	$-0.341333\pi$
$991$	30.1001	0.956161	0.478080	+	0.878316i	$0.341333\pi$
$992$	−7.63704	−0.242476
$993$	16.0918	0.510657
$994$	27.3570	0.867710
$995$	−9.08762	−0.288097
$996$	−9.54796	−0.302539
$997$	0.872937	0.0276462	0.0138231	−	0.999904i	$-0.495600\pi$
$997$	0.872937	0.0276462	0.0138231	+	0.999904i	$0.495600\pi$
$998$	−10.1229	−0.320435
$999$	−3.98578	−0.126105

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.07486 q^{5} +1.00000 q^{6} +2.07486 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.07486 q^{5} +1.00000 q^{6} +2.07486 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.07486 q^{10} -5.68076 q^{11} +1.00000 q^{12} -0.699126 q^{13} +2.07486 q^{14} -4.07486 q^{15} +1.00000 q^{16} +4.45474 q^{17} +1.00000 q^{18} +2.45474 q^{19} -4.07486 q^{20} +2.07486 q^{21} -5.68076 q^{22} -1.00000 q^{23} +1.00000 q^{24} +11.6044 q^{25} -0.699126 q^{26} +1.00000 q^{27} +2.07486 q^{28} -1.00000 q^{29} -4.07486 q^{30} -7.63704 q^{31} +1.00000 q^{32} -5.68076 q^{33} +4.45474 q^{34} -8.45474 q^{35} +1.00000 q^{36} -3.98578 q^{37} +2.45474 q^{38} -0.699126 q^{39} -4.07486 q^{40} -5.15386 q^{41} +2.07486 q^{42} -3.09323 q^{43} -5.68076 q^{44} -4.07486 q^{45} -1.00000 q^{46} +1.09323 q^{47} +1.00000 q^{48} -2.69497 q^{49} +11.6044 q^{50} +4.45474 q^{51} -0.699126 q^{52} -11.7698 q^{53} +1.00000 q^{54} +23.1483 q^{55} +2.07486 q^{56} +2.45474 q^{57} -1.00000 q^{58} -7.05649 q^{59} -4.07486 q^{60} +0.566333 q^{61} -7.63704 q^{62} +2.07486 q^{63} +1.00000 q^{64} +2.84884 q^{65} -5.68076 q^{66} -2.37988 q^{67} +4.45474 q^{68} -1.00000 q^{69} -8.45474 q^{70} +13.1850 q^{71} +1.00000 q^{72} -6.75976 q^{73} -3.98578 q^{74} +11.6044 q^{75} +2.45474 q^{76} -11.7867 q^{77} -0.699126 q^{78} +0.848837 q^{79} -4.07486 q^{80} +1.00000 q^{81} -5.15386 q^{82} -9.54796 q^{83} +2.07486 q^{84} -18.1524 q^{85} -3.09323 q^{86} -1.00000 q^{87} -5.68076 q^{88} -8.93791 q^{89} -4.07486 q^{90} -1.45059 q^{91} -1.00000 q^{92} -7.63704 q^{93} +1.09323 q^{94} -10.0027 q^{95} +1.00000 q^{96} -10.5012 q^{97} -2.69497 q^{98} -5.68076 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{9} - 4 q^{10} - 12 q^{11} + 4 q^{12} - 6 q^{13} - 4 q^{14} - 4 q^{15} + 4 q^{16} - 2 q^{17} + 4 q^{18} - 10 q^{19} - 4 q^{20} - 4 q^{21} - 12 q^{22} - 4 q^{23} + 4 q^{24} + 2 q^{25} - 6 q^{26} + 4 q^{27} - 4 q^{28} - 4 q^{29} - 4 q^{30} - 6 q^{31} + 4 q^{32} - 12 q^{33} - 2 q^{34} - 14 q^{35} + 4 q^{36} - 10 q^{37} - 10 q^{38} - 6 q^{39} - 4 q^{40} - 4 q^{41} - 4 q^{42} - 14 q^{43} - 12 q^{44} - 4 q^{45} - 4 q^{46} + 6 q^{47} + 4 q^{48} - 6 q^{49} + 2 q^{50} - 2 q^{51} - 6 q^{52} - 30 q^{53} + 4 q^{54} + 22 q^{55} - 4 q^{56} - 10 q^{57} - 4 q^{58} - 2 q^{59} - 4 q^{60} - 2 q^{61} - 6 q^{62} - 4 q^{63} + 4 q^{64} - 10 q^{65} - 12 q^{66} - 2 q^{67} - 2 q^{68} - 4 q^{69} - 14 q^{70} + 10 q^{71} + 4 q^{72} - 12 q^{73} - 10 q^{74} + 2 q^{75} - 10 q^{76} + 2 q^{77} - 6 q^{78} - 18 q^{79} - 4 q^{80} + 4 q^{81} - 4 q^{82} - 20 q^{83} - 4 q^{84} - 10 q^{85} - 14 q^{86} - 4 q^{87} - 12 q^{88} - 8 q^{89} - 4 q^{90} + 22 q^{91} - 4 q^{92} - 6 q^{93} + 6 q^{94} - 2 q^{95} + 4 q^{96} + 2 q^{97} - 6 q^{98} - 12 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^9 - 4 * q^10 - 12 * q^11 + 4 * q^12 - 6 * q^13 - 4 * q^14 - 4 * q^15 + 4 * q^16 - 2 * q^17 + 4 * q^18 - 10 * q^19 - 4 * q^20 - 4 * q^21 - 12 * q^22 - 4 * q^23 + 4 * q^24 + 2 * q^25 - 6 * q^26 + 4 * q^27 - 4 * q^28 - 4 * q^29 - 4 * q^30 - 6 * q^31 + 4 * q^32 - 12 * q^33 - 2 * q^34 - 14 * q^35 + 4 * q^36 - 10 * q^37 - 10 * q^38 - 6 * q^39 - 4 * q^40 - 4 * q^41 - 4 * q^42 - 14 * q^43 - 12 * q^44 - 4 * q^45 - 4 * q^46 + 6 * q^47 + 4 * q^48 - 6 * q^49 + 2 * q^50 - 2 * q^51 - 6 * q^52 - 30 * q^53 + 4 * q^54 + 22 * q^55 - 4 * q^56 - 10 * q^57 - 4 * q^58 - 2 * q^59 - 4 * q^60 - 2 * q^61 - 6 * q^62 - 4 * q^63 + 4 * q^64 - 10 * q^65 - 12 * q^66 - 2 * q^67 - 2 * q^68 - 4 * q^69 - 14 * q^70 + 10 * q^71 + 4 * q^72 - 12 * q^73 - 10 * q^74 + 2 * q^75 - 10 * q^76 + 2 * q^77 - 6 * q^78 - 18 * q^79 - 4 * q^80 + 4 * q^81 - 4 * q^82 - 20 * q^83 - 4 * q^84 - 10 * q^85 - 14 * q^86 - 4 * q^87 - 12 * q^88 - 8 * q^89 - 4 * q^90 + 22 * q^91 - 4 * q^92 - 6 * q^93 + 6 * q^94 - 2 * q^95 + 4 * q^96 + 2 * q^97 - 6 * q^98 - 12 * q^99

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