LMFDB - Embedded newform 4002.2.a.bi.1.2

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:	$8$
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} - 24x^{6} - 3x^{5} + 194x^{4} + 39x^{3} - 607x^{2} - 104x + 600 $ x^8 - 24x^6 - 3x^5 + 194x^4 + 39x^3 - 607x^2 - 104x + 600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.57534$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.69359 q^{5} -1.00000 q^{6} -3.87906 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.69359 q^{5} -1.00000 q^{6} -3.87906 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.69359 q^{10} -6.10949 q^{11} +1.00000 q^{12} +5.79409 q^{13} +3.87906 q^{14} -2.69359 q^{15} +1.00000 q^{16} +8.21209 q^{17} -1.00000 q^{18} +4.11557 q^{19} -2.69359 q^{20} -3.87906 q^{21} +6.10949 q^{22} -1.00000 q^{23} -1.00000 q^{24} +2.25542 q^{25} -5.79409 q^{26} +1.00000 q^{27} -3.87906 q^{28} +1.00000 q^{29} +2.69359 q^{30} +0.978738 q^{31} -1.00000 q^{32} -6.10949 q^{33} -8.21209 q^{34} +10.4486 q^{35} +1.00000 q^{36} -2.93762 q^{37} -4.11557 q^{38} +5.79409 q^{39} +2.69359 q^{40} -10.0259 q^{41} +3.87906 q^{42} -4.09974 q^{43} -6.10949 q^{44} -2.69359 q^{45} +1.00000 q^{46} +6.91570 q^{47} +1.00000 q^{48} +8.04711 q^{49} -2.25542 q^{50} +8.21209 q^{51} +5.79409 q^{52} +4.37463 q^{53} -1.00000 q^{54} +16.4565 q^{55} +3.87906 q^{56} +4.11557 q^{57} -1.00000 q^{58} -6.65937 q^{59} -2.69359 q^{60} +6.57573 q^{61} -0.978738 q^{62} -3.87906 q^{63} +1.00000 q^{64} -15.6069 q^{65} +6.10949 q^{66} +9.19748 q^{67} +8.21209 q^{68} -1.00000 q^{69} -10.4486 q^{70} -11.3406 q^{71} -1.00000 q^{72} +13.1310 q^{73} +2.93762 q^{74} +2.25542 q^{75} +4.11557 q^{76} +23.6991 q^{77} -5.79409 q^{78} -10.9603 q^{79} -2.69359 q^{80} +1.00000 q^{81} +10.0259 q^{82} -2.83411 q^{83} -3.87906 q^{84} -22.1200 q^{85} +4.09974 q^{86} +1.00000 q^{87} +6.10949 q^{88} -12.6111 q^{89} +2.69359 q^{90} -22.4756 q^{91} -1.00000 q^{92} +0.978738 q^{93} -6.91570 q^{94} -11.0856 q^{95} -1.00000 q^{96} +7.30840 q^{97} -8.04711 q^{98} -6.10949 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9} + 3 q^{10} - 7 q^{11} + 8 q^{12} - 3 q^{13} + 6 q^{14} - 3 q^{15} + 8 q^{16} - 12 q^{17} - 8 q^{18} - 4 q^{19} - 3 q^{20} - 6 q^{21} + 7 q^{22} - 8 q^{23} - 8 q^{24} + 13 q^{25} + 3 q^{26} + 8 q^{27} - 6 q^{28} + 8 q^{29} + 3 q^{30} - q^{31} - 8 q^{32} - 7 q^{33} + 12 q^{34} - 6 q^{35} + 8 q^{36} - 11 q^{37} + 4 q^{38} - 3 q^{39} + 3 q^{40} - 17 q^{41} + 6 q^{42} - 10 q^{43} - 7 q^{44} - 3 q^{45} + 8 q^{46} - 26 q^{47} + 8 q^{48} + 10 q^{49} - 13 q^{50} - 12 q^{51} - 3 q^{52} - 2 q^{53} - 8 q^{54} + q^{55} + 6 q^{56} - 4 q^{57} - 8 q^{58} - 25 q^{59} - 3 q^{60} + 3 q^{61} + q^{62} - 6 q^{63} + 8 q^{64} - 25 q^{65} + 7 q^{66} + q^{67} - 12 q^{68} - 8 q^{69} + 6 q^{70} - 27 q^{71} - 8 q^{72} + 16 q^{73} + 11 q^{74} + 13 q^{75} - 4 q^{76} - 16 q^{77} + 3 q^{78} - 6 q^{79} - 3 q^{80} + 8 q^{81} + 17 q^{82} - 44 q^{83} - 6 q^{84} - 20 q^{85} + 10 q^{86} + 8 q^{87} + 7 q^{88} - 52 q^{89} + 3 q^{90} - 18 q^{91} - 8 q^{92} - q^{93} + 26 q^{94} - 56 q^{95} - 8 q^{96} - 4 q^{97} - 10 q^{98} - 7 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9 + 3 * q^10 - 7 * q^11 + 8 * q^12 - 3 * q^13 + 6 * q^14 - 3 * q^15 + 8 * q^16 - 12 * q^17 - 8 * q^18 - 4 * q^19 - 3 * q^20 - 6 * q^21 + 7 * q^22 - 8 * q^23 - 8 * q^24 + 13 * q^25 + 3 * q^26 + 8 * q^27 - 6 * q^28 + 8 * q^29 + 3 * q^30 - q^31 - 8 * q^32 - 7 * q^33 + 12 * q^34 - 6 * q^35 + 8 * q^36 - 11 * q^37 + 4 * q^38 - 3 * q^39 + 3 * q^40 - 17 * q^41 + 6 * q^42 - 10 * q^43 - 7 * q^44 - 3 * q^45 + 8 * q^46 - 26 * q^47 + 8 * q^48 + 10 * q^49 - 13 * q^50 - 12 * q^51 - 3 * q^52 - 2 * q^53 - 8 * q^54 + q^55 + 6 * q^56 - 4 * q^57 - 8 * q^58 - 25 * q^59 - 3 * q^60 + 3 * q^61 + q^62 - 6 * q^63 + 8 * q^64 - 25 * q^65 + 7 * q^66 + q^67 - 12 * q^68 - 8 * q^69 + 6 * q^70 - 27 * q^71 - 8 * q^72 + 16 * q^73 + 11 * q^74 + 13 * q^75 - 4 * q^76 - 16 * q^77 + 3 * q^78 - 6 * q^79 - 3 * q^80 + 8 * q^81 + 17 * q^82 - 44 * q^83 - 6 * q^84 - 20 * q^85 + 10 * q^86 + 8 * q^87 + 7 * q^88 - 52 * q^89 + 3 * q^90 - 18 * q^91 - 8 * q^92 - q^93 + 26 * q^94 - 56 * q^95 - 8 * q^96 - 4 * q^97 - 10 * q^98 - 7 * 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
$5$	−2.69359	−1.20461	−0.602305	−	0.798266i	$-0.705751\pi$
$5$	−2.69359	−1.20461	−0.602305	+	0.798266i	$0.705751\pi$
$6$	−1.00000	−0.408248
$7$	−3.87906	−1.46615	−0.733074	−	0.680149i	$-0.761915\pi$
$7$	−3.87906	−1.46615	−0.733074	+	0.680149i	$0.761915\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.69359	0.851787
$11$	−6.10949	−1.84208	−0.921041	−	0.389466i	$-0.872660\pi$
$11$	−6.10949	−1.84208	−0.921041	+	0.389466i	$0.872660\pi$
$12$	1.00000	0.288675
$13$	5.79409	1.60699	0.803496	−	0.595310i	$-0.202971\pi$
$13$	5.79409	1.60699	0.803496	+	0.595310i	$0.202971\pi$
$14$	3.87906	1.03672
$15$	−2.69359	−0.695481
$16$	1.00000	0.250000
$17$	8.21209	1.99172	0.995862	−	0.0908798i	$-0.0289679\pi$
$17$	8.21209	1.99172	0.995862	+	0.0908798i	$0.0289679\pi$
$18$	−1.00000	−0.235702
$19$	4.11557	0.944175	0.472088	−	0.881552i	$-0.343501\pi$
$19$	4.11557	0.944175	0.472088	+	0.881552i	$0.343501\pi$
$20$	−2.69359	−0.602305
$21$	−3.87906	−0.846481
$22$	6.10949	1.30255
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	2.25542	0.451083
$26$	−5.79409	−1.13631
$27$	1.00000	0.192450
$28$	−3.87906	−0.733074
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	2.69359	0.491780
$31$	0.978738	0.175787	0.0878933	−	0.996130i	$-0.471987\pi$
$31$	0.978738	0.175787	0.0878933	+	0.996130i	$0.471987\pi$
$32$	−1.00000	−0.176777
$33$	−6.10949	−1.06353
$34$	−8.21209	−1.40836
$35$	10.4486	1.76613
$36$	1.00000	0.166667
$37$	−2.93762	−0.482942	−0.241471	−	0.970408i	$-0.577630\pi$
$37$	−2.93762	−0.482942	−0.241471	+	0.970408i	$0.577630\pi$
$38$	−4.11557	−0.667633
$39$	5.79409	0.927797
$40$	2.69359	0.425894
$41$	−10.0259	−1.56578	−0.782888	−	0.622163i	$-0.786254\pi$
$41$	−10.0259	−1.56578	−0.782888	+	0.622163i	$0.786254\pi$
$42$	3.87906	0.598552
$43$	−4.09974	−0.625205	−0.312602	−	0.949884i	$-0.601201\pi$
$43$	−4.09974	−0.625205	−0.312602	+	0.949884i	$0.601201\pi$
$44$	−6.10949	−0.921041
$45$	−2.69359	−0.401536
$46$	1.00000	0.147442
$47$	6.91570	1.00876	0.504379	−	0.863482i	$-0.331721\pi$
$47$	6.91570	1.00876	0.504379	+	0.863482i	$0.331721\pi$
$48$	1.00000	0.144338
$49$	8.04711	1.14959
$50$	−2.25542	−0.318964
$51$	8.21209	1.14992
$52$	5.79409	0.803496
$53$	4.37463	0.600902	0.300451	−	0.953797i	$-0.402863\pi$
$53$	4.37463	0.600902	0.300451	+	0.953797i	$0.402863\pi$
$54$	−1.00000	−0.136083
$55$	16.4565	2.21899
$56$	3.87906	0.518361
$57$	4.11557	0.545120
$58$	−1.00000	−0.131306
$59$	−6.65937	−0.866976	−0.433488	−	0.901159i	$-0.642717\pi$
$59$	−6.65937	−0.866976	−0.433488	+	0.901159i	$0.642717\pi$
$60$	−2.69359	−0.347741
$61$	6.57573	0.841936	0.420968	−	0.907076i	$-0.361690\pi$
$61$	6.57573	0.841936	0.420968	+	0.907076i	$0.361690\pi$
$62$	−0.978738	−0.124300
$63$	−3.87906	−0.488716
$64$	1.00000	0.125000
$65$	−15.6069	−1.93580
$66$	6.10949	0.752027
$67$	9.19748	1.12365	0.561826	−	0.827256i	$-0.310099\pi$
$67$	9.19748	1.12365	0.561826	+	0.827256i	$0.310099\pi$
$68$	8.21209	0.995862
$69$	−1.00000	−0.120386
$70$	−10.4486	−1.24885
$71$	−11.3406	−1.34588	−0.672940	−	0.739697i	$-0.734969\pi$
$71$	−11.3406	−1.34588	−0.672940	+	0.739697i	$0.734969\pi$
$72$	−1.00000	−0.117851
$73$	13.1310	1.53687	0.768433	−	0.639930i	$-0.221037\pi$
$73$	13.1310	1.53687	0.768433	+	0.639930i	$0.221037\pi$
$74$	2.93762	0.341491
$75$	2.25542	0.260433
$76$	4.11557	0.472088
$77$	23.6991	2.70076
$78$	−5.79409	−0.656052
$79$	−10.9603	−1.23312	−0.616562	−	0.787306i	$-0.711475\pi$
$79$	−10.9603	−1.23312	−0.616562	+	0.787306i	$0.711475\pi$
$80$	−2.69359	−0.301152
$81$	1.00000	0.111111
$82$	10.0259	1.10717
$83$	−2.83411	−0.311084	−0.155542	−	0.987829i	$-0.549712\pi$
$83$	−2.83411	−0.311084	−0.155542	+	0.987829i	$0.549712\pi$
$84$	−3.87906	−0.423240
$85$	−22.1200	−2.39925
$86$	4.09974	0.442087
$87$	1.00000	0.107211
$88$	6.10949	0.651274
$89$	−12.6111	−1.33678	−0.668389	−	0.743812i	$-0.733016\pi$
$89$	−12.6111	−1.33678	−0.668389	+	0.743812i	$0.733016\pi$
$90$	2.69359	0.283929
$91$	−22.4756	−2.35609
$92$	−1.00000	−0.104257
$93$	0.978738	0.101490
$94$	−6.91570	−0.713300
$95$	−11.0856	−1.13736
$96$	−1.00000	−0.102062
$97$	7.30840	0.742056	0.371028	−	0.928622i	$-0.379005\pi$
$97$	7.30840	0.742056	0.371028	+	0.928622i	$0.379005\pi$
$98$	−8.04711	−0.812881
$99$	−6.10949	−0.614027
$100$	2.25542	0.225542
$101$	−2.90175	−0.288735	−0.144368	−	0.989524i	$-0.546115\pi$
$101$	−2.90175	−0.288735	−0.144368	+	0.989524i	$0.546115\pi$
$102$	−8.21209	−0.813118
$103$	0.129262	0.0127365	0.00636827	−	0.999980i	$-0.497973\pi$
$103$	0.129262	0.0127365	0.00636827	+	0.999980i	$0.497973\pi$
$104$	−5.79409	−0.568157
$105$	10.4486	1.01968
$106$	−4.37463	−0.424902
$107$	−9.12874	−0.882509	−0.441254	−	0.897382i	$-0.645466\pi$
$107$	−9.12874	−0.882509	−0.441254	+	0.897382i	$0.645466\pi$
$108$	1.00000	0.0962250
$109$	−16.9420	−1.62275	−0.811375	−	0.584527i	$-0.801280\pi$
$109$	−16.9420	−1.62275	−0.811375	+	0.584527i	$0.801280\pi$
$110$	−16.4565	−1.56906
$111$	−2.93762	−0.278827
$112$	−3.87906	−0.366537
$113$	7.59406	0.714389	0.357194	−	0.934030i	$-0.383733\pi$
$113$	7.59406	0.714389	0.357194	+	0.934030i	$0.383733\pi$
$114$	−4.11557	−0.385458
$115$	2.69359	0.251178
$116$	1.00000	0.0928477
$117$	5.79409	0.535664
$118$	6.65937	0.613045
$119$	−31.8552	−2.92016
$120$	2.69359	0.245890
$121$	26.3259	2.39327
$122$	−6.57573	−0.595339
$123$	−10.0259	−0.904001
$124$	0.978738	0.0878933
$125$	7.39278	0.661230
$126$	3.87906	0.345574
$127$	−10.1474	−0.900435	−0.450218	−	0.892919i	$-0.648654\pi$
$127$	−10.1474	−0.900435	−0.450218	+	0.892919i	$0.648654\pi$
$128$	−1.00000	−0.0883883
$129$	−4.09974	−0.360962
$130$	15.6069	1.36882
$131$	10.8105	0.944516	0.472258	−	0.881460i	$-0.343439\pi$
$131$	10.8105	0.944516	0.472258	+	0.881460i	$0.343439\pi$
$132$	−6.10949	−0.531763
$133$	−15.9645	−1.38430
$134$	−9.19748	−0.794541
$135$	−2.69359	−0.231827
$136$	−8.21209	−0.704181
$137$	−20.3271	−1.73666	−0.868329	−	0.495989i	$-0.834805\pi$
$137$	−20.3271	−1.73666	−0.868329	+	0.495989i	$0.834805\pi$
$138$	1.00000	0.0851257
$139$	−12.0316	−1.02050	−0.510252	−	0.860025i	$-0.670448\pi$
$139$	−12.0316	−1.02050	−0.510252	+	0.860025i	$0.670448\pi$
$140$	10.4486	0.883067
$141$	6.91570	0.582407
$142$	11.3406	0.951681
$143$	−35.3990	−2.96021
$144$	1.00000	0.0833333
$145$	−2.69359	−0.223690
$146$	−13.1310	−1.08673
$147$	8.04711	0.663715
$148$	−2.93762	−0.241471
$149$	7.04400	0.577067	0.288533	−	0.957470i	$-0.406832\pi$
$149$	7.04400	0.577067	0.288533	+	0.957470i	$0.406832\pi$
$150$	−2.25542	−0.184154
$151$	17.8844	1.45541	0.727704	−	0.685891i	$-0.240587\pi$
$151$	17.8844	1.45541	0.727704	+	0.685891i	$0.240587\pi$
$152$	−4.11557	−0.333816
$153$	8.21209	0.663908
$154$	−23.6991	−1.90973
$155$	−2.63632	−0.211754
$156$	5.79409	0.463899
$157$	−20.5190	−1.63759	−0.818797	−	0.574083i	$-0.805359\pi$
$157$	−20.5190	−1.63759	−0.818797	+	0.574083i	$0.805359\pi$
$158$	10.9603	0.871951
$159$	4.37463	0.346931
$160$	2.69359	0.212947
$161$	3.87906	0.305713
$162$	−1.00000	−0.0785674
$163$	−8.68240	−0.680058	−0.340029	−	0.940415i	$-0.610437\pi$
$163$	−8.68240	−0.680058	−0.340029	+	0.940415i	$0.610437\pi$
$164$	−10.0259	−0.782888
$165$	16.4565	1.28113
$166$	2.83411	0.219969
$167$	−19.6860	−1.52335	−0.761675	−	0.647959i	$-0.775623\pi$
$167$	−19.6860	−1.52335	−0.761675	+	0.647959i	$0.775623\pi$
$168$	3.87906	0.299276
$169$	20.5715	1.58242
$170$	22.1200	1.69652
$171$	4.11557	0.314725
$172$	−4.09974	−0.312602
$173$	−5.58200	−0.424391	−0.212196	−	0.977227i	$-0.568061\pi$
$173$	−5.58200	−0.424391	−0.212196	+	0.977227i	$0.568061\pi$
$174$	−1.00000	−0.0758098
$175$	−8.74889	−0.661354
$176$	−6.10949	−0.460521
$177$	−6.65937	−0.500549
$178$	12.6111	0.945244
$179$	−6.00438	−0.448789	−0.224394	−	0.974498i	$-0.572040\pi$
$179$	−6.00438	−0.448789	−0.224394	+	0.974498i	$0.572040\pi$
$180$	−2.69359	−0.200768
$181$	−18.3194	−1.36167	−0.680836	−	0.732436i	$-0.738383\pi$
$181$	−18.3194	−1.36167	−0.680836	+	0.732436i	$0.738383\pi$
$182$	22.4756	1.66600
$183$	6.57573	0.486092
$184$	1.00000	0.0737210
$185$	7.91274	0.581756
$186$	−0.978738	−0.0717646
$187$	−50.1717	−3.66892
$188$	6.91570	0.504379
$189$	−3.87906	−0.282160
$190$	11.0856	0.804237
$191$	0.201020	0.0145453	0.00727263	−	0.999974i	$-0.497685\pi$
$191$	0.201020	0.0145453	0.00727263	+	0.999974i	$0.497685\pi$
$192$	1.00000	0.0721688
$193$	−1.07001	−0.0770210	−0.0385105	−	0.999258i	$-0.512261\pi$
$193$	−1.07001	−0.0770210	−0.0385105	+	0.999258i	$0.512261\pi$
$194$	−7.30840	−0.524713
$195$	−15.6069	−1.11763
$196$	8.04711	0.574794
$197$	21.0590	1.50039	0.750196	−	0.661215i	$-0.229959\pi$
$197$	21.0590	1.50039	0.750196	+	0.661215i	$0.229959\pi$
$198$	6.10949	0.434183
$199$	−21.5663	−1.52880	−0.764398	−	0.644745i	$-0.776963\pi$
$199$	−21.5663	−1.52880	−0.764398	+	0.644745i	$0.776963\pi$
$200$	−2.25542	−0.159482
$201$	9.19748	0.648740
$202$	2.90175	0.204167
$203$	−3.87906	−0.272257
$204$	8.21209	0.574961
$205$	27.0055	1.88615
$206$	−0.129262	−0.00900610
$207$	−1.00000	−0.0695048
$208$	5.79409	0.401748
$209$	−25.1440	−1.73925
$210$	−10.4486	−0.721021
$211$	11.9770	0.824531	0.412265	−	0.911064i	$-0.364738\pi$
$211$	11.9770	0.824531	0.412265	+	0.911064i	$0.364738\pi$
$212$	4.37463	0.300451
$213$	−11.3406	−0.777044
$214$	9.12874	0.624028
$215$	11.0430	0.753127
$216$	−1.00000	−0.0680414
$217$	−3.79659	−0.257729
$218$	16.9420	1.14746
$219$	13.1310	0.887311
$220$	16.4565	1.10949
$221$	47.5816	3.20068
$222$	2.93762	0.197160
$223$	−23.2718	−1.55840	−0.779198	−	0.626778i	$-0.784373\pi$
$223$	−23.2718	−1.55840	−0.779198	+	0.626778i	$0.784373\pi$
$224$	3.87906	0.259181
$225$	2.25542	0.150361
$226$	−7.59406	−0.505149
$227$	−9.46965	−0.628523	−0.314261	−	0.949336i	$-0.601757\pi$
$227$	−9.46965	−0.628523	−0.314261	+	0.949336i	$0.601757\pi$
$228$	4.11557	0.272560
$229$	−26.2271	−1.73314	−0.866568	−	0.499059i	$-0.833679\pi$
$229$	−26.2271	−1.73314	−0.866568	+	0.499059i	$0.833679\pi$
$230$	−2.69359	−0.177610
$231$	23.6991	1.55929
$232$	−1.00000	−0.0656532
$233$	−8.94429	−0.585960	−0.292980	−	0.956119i	$-0.594647\pi$
$233$	−8.94429	−0.585960	−0.292980	+	0.956119i	$0.594647\pi$
$234$	−5.79409	−0.378772
$235$	−18.6280	−1.21516
$236$	−6.65937	−0.433488
$237$	−10.9603	−0.711945
$238$	31.8552	2.06487
$239$	1.38468	0.0895675	0.0447838	−	0.998997i	$-0.485740\pi$
$239$	1.38468	0.0895675	0.0447838	+	0.998997i	$0.485740\pi$
$240$	−2.69359	−0.173870
$241$	−15.1982	−0.979001	−0.489501	−	0.872003i	$-0.662821\pi$
$241$	−15.1982	−0.979001	−0.489501	+	0.872003i	$0.662821\pi$
$242$	−26.3259	−1.69229
$243$	1.00000	0.0641500
$244$	6.57573	0.420968
$245$	−21.6756	−1.38480
$246$	10.0259	0.639225
$247$	23.8460	1.51728
$248$	−0.978738	−0.0621500
$249$	−2.83411	−0.179604
$250$	−7.39278	−0.467560
$251$	−18.9951	−1.19896	−0.599480	−	0.800390i	$-0.704626\pi$
$251$	−18.9951	−1.19896	−0.599480	+	0.800390i	$0.704626\pi$
$252$	−3.87906	−0.244358
$253$	6.10949	0.384101
$254$	10.1474	0.636704
$255$	−22.1200	−1.38521
$256$	1.00000	0.0625000
$257$	23.8053	1.48493	0.742466	−	0.669884i	$-0.233656\pi$
$257$	23.8053	1.48493	0.742466	+	0.669884i	$0.233656\pi$
$258$	4.09974	0.255239
$259$	11.3952	0.708064
$260$	−15.6069	−0.967898
$261$	1.00000	0.0618984
$262$	−10.8105	−0.667874
$263$	−16.9253	−1.04366	−0.521831	−	0.853049i	$-0.674751\pi$
$263$	−16.9253	−1.04366	−0.521831	+	0.853049i	$0.674751\pi$
$264$	6.10949	0.376013
$265$	−11.7834	−0.723851
$266$	15.9645	0.978848
$267$	−12.6111	−0.771789
$268$	9.19748	0.561826
$269$	5.96771	0.363858	0.181929	−	0.983312i	$-0.441766\pi$
$269$	5.96771	0.363858	0.181929	+	0.983312i	$0.441766\pi$
$270$	2.69359	0.163927
$271$	14.5399	0.883236	0.441618	−	0.897203i	$-0.354405\pi$
$271$	14.5399	0.883236	0.441618	+	0.897203i	$0.354405\pi$
$272$	8.21209	0.497931
$273$	−22.4756	−1.36029
$274$	20.3271	1.22800
$275$	−13.7794	−0.830932
$276$	−1.00000	−0.0601929
$277$	4.13692	0.248563	0.124282	−	0.992247i	$-0.460337\pi$
$277$	4.13692	0.248563	0.124282	+	0.992247i	$0.460337\pi$
$278$	12.0316	0.721605
$279$	0.978738	0.0585955
$280$	−10.4486	−0.624423
$281$	1.57465	0.0939358	0.0469679	−	0.998896i	$-0.485044\pi$
$281$	1.57465	0.0939358	0.0469679	+	0.998896i	$0.485044\pi$
$282$	−6.91570	−0.411824
$283$	22.2945	1.32527	0.662635	−	0.748942i	$-0.269438\pi$
$283$	22.2945	1.32527	0.662635	+	0.748942i	$0.269438\pi$
$284$	−11.3406	−0.672940
$285$	−11.0856	−0.656656
$286$	35.3990	2.09318
$287$	38.8909	2.29566
$288$	−1.00000	−0.0589256
$289$	50.4384	2.96696
$290$	2.69359	0.158173
$291$	7.30840	0.428426
$292$	13.1310	0.768433
$293$	5.68927	0.332370	0.166185	−	0.986095i	$-0.446855\pi$
$293$	5.68927	0.332370	0.166185	+	0.986095i	$0.446855\pi$
$294$	−8.04711	−0.469317
$295$	17.9376	1.04437
$296$	2.93762	0.170746
$297$	−6.10949	−0.354509
$298$	−7.04400	−0.408048
$299$	−5.79409	−0.335081
$300$	2.25542	0.130216
$301$	15.9031	0.916642
$302$	−17.8844	−1.02913
$303$	−2.90175	−0.166701
$304$	4.11557	0.236044
$305$	−17.7123	−1.01420
$306$	−8.21209	−0.469454
$307$	5.61881	0.320682	0.160341	−	0.987062i	$-0.448741\pi$
$307$	5.61881	0.320682	0.160341	+	0.987062i	$0.448741\pi$
$308$	23.6991	1.35038
$309$	0.129262	0.00735345
$310$	2.63632	0.149733
$311$	3.40536	0.193100	0.0965500	−	0.995328i	$-0.469219\pi$
$311$	3.40536	0.193100	0.0965500	+	0.995328i	$0.469219\pi$
$312$	−5.79409	−0.328026
$313$	5.65089	0.319407	0.159704	−	0.987165i	$-0.448946\pi$
$313$	5.65089	0.319407	0.159704	+	0.987165i	$0.448946\pi$
$314$	20.5190	1.15795
$315$	10.4486	0.588711
$316$	−10.9603	−0.616562
$317$	−18.3224	−1.02909	−0.514544	−	0.857464i	$-0.672039\pi$
$317$	−18.3224	−1.02909	−0.514544	+	0.857464i	$0.672039\pi$
$318$	−4.37463	−0.245317
$319$	−6.10949	−0.342066
$320$	−2.69359	−0.150576
$321$	−9.12874	−0.509517
$322$	−3.87906	−0.216172
$323$	33.7974	1.88054
$324$	1.00000	0.0555556
$325$	13.0681	0.724887
$326$	8.68240	0.480874
$327$	−16.9420	−0.936895
$328$	10.0259	0.553585
$329$	−26.8264	−1.47899
$330$	−16.4565	−0.905898
$331$	14.7009	0.808032	0.404016	−	0.914752i	$-0.367614\pi$
$331$	14.7009	0.808032	0.404016	+	0.914752i	$0.367614\pi$
$332$	−2.83411	−0.155542
$333$	−2.93762	−0.160981
$334$	19.6860	1.07717
$335$	−24.7742	−1.35356
$336$	−3.87906	−0.211620
$337$	5.79678	0.315771	0.157885	−	0.987457i	$-0.449532\pi$
$337$	5.79678	0.315771	0.157885	+	0.987457i	$0.449532\pi$
$338$	−20.5715	−1.11894
$339$	7.59406	0.412453
$340$	−22.1200	−1.19962
$341$	−5.97960	−0.323813
$342$	−4.11557	−0.222544
$343$	−4.06182	−0.219318
$344$	4.09974	0.221043
$345$	2.69359	0.145018
$346$	5.58200	0.300090
$347$	−8.77365	−0.470994	−0.235497	−	0.971875i	$-0.575672\pi$
$347$	−8.77365	−0.470994	−0.235497	+	0.971875i	$0.575672\pi$
$348$	1.00000	0.0536056
$349$	−15.5806	−0.834010	−0.417005	−	0.908904i	$-0.636920\pi$
$349$	−15.5806	−0.834010	−0.417005	+	0.908904i	$0.636920\pi$
$350$	8.74889	0.467648
$351$	5.79409	0.309266
$352$	6.10949	0.325637
$353$	−19.1232	−1.01783	−0.508914	−	0.860818i	$-0.669953\pi$
$353$	−19.1232	−1.01783	−0.508914	+	0.860818i	$0.669953\pi$
$354$	6.65937	0.353942
$355$	30.5469	1.62126
$356$	−12.6111	−0.668389
$357$	−31.8552	−1.68596
$358$	6.00438	0.317341
$359$	1.16936	0.0617166	0.0308583	−	0.999524i	$-0.490176\pi$
$359$	1.16936	0.0617166	0.0308583	+	0.999524i	$0.490176\pi$
$360$	2.69359	0.141965
$361$	−2.06212	−0.108533
$362$	18.3194	0.962848
$363$	26.3259	1.38175
$364$	−22.4756	−1.17804
$365$	−35.3695	−1.85132
$366$	−6.57573	−0.343719
$367$	−17.0876	−0.891965	−0.445982	−	0.895042i	$-0.647146\pi$
$367$	−17.0876	−0.891965	−0.445982	+	0.895042i	$0.647146\pi$
$368$	−1.00000	−0.0521286
$369$	−10.0259	−0.521925
$370$	−7.91274	−0.411364
$371$	−16.9695	−0.881010
$372$	0.978738	0.0507452
$373$	24.6063	1.27407	0.637034	−	0.770836i	$-0.280161\pi$
$373$	24.6063	1.27407	0.637034	+	0.770836i	$0.280161\pi$
$374$	50.1717	2.59432
$375$	7.39278	0.381762
$376$	−6.91570	−0.356650
$377$	5.79409	0.298411
$378$	3.87906	0.199517
$379$	21.8644	1.12310	0.561549	−	0.827444i	$-0.310206\pi$
$379$	21.8644	1.12310	0.561549	+	0.827444i	$0.310206\pi$
$380$	−11.0856	−0.568681
$381$	−10.1474	−0.519866
$382$	−0.201020	−0.0102851
$383$	−29.1904	−1.49156	−0.745781	−	0.666191i	$-0.767923\pi$
$383$	−29.1904	−1.49156	−0.745781	+	0.666191i	$0.767923\pi$
$384$	−1.00000	−0.0510310
$385$	−63.8356	−3.25336
$386$	1.07001	0.0544620
$387$	−4.09974	−0.208402
$388$	7.30840	0.371028
$389$	−31.5252	−1.59839	−0.799196	−	0.601071i	$-0.794741\pi$
$389$	−31.5252	−1.59839	−0.799196	+	0.601071i	$0.794741\pi$
$390$	15.6069	0.790286
$391$	−8.21209	−0.415303
$392$	−8.04711	−0.406441
$393$	10.8105	0.545317
$394$	−21.0590	−1.06094
$395$	29.5224	1.48543
$396$	−6.10949	−0.307014
$397$	−22.8284	−1.14572	−0.572862	−	0.819652i	$-0.694167\pi$
$397$	−22.8284	−1.14572	−0.572862	+	0.819652i	$0.694167\pi$
$398$	21.5663	1.08102
$399$	−15.9645	−0.799226
$400$	2.25542	0.112771
$401$	5.07367	0.253367	0.126683	−	0.991943i	$-0.459567\pi$
$401$	5.07367	0.253367	0.126683	+	0.991943i	$0.459567\pi$
$402$	−9.19748	−0.458729
$403$	5.67090	0.282488
$404$	−2.90175	−0.144368
$405$	−2.69359	−0.133845
$406$	3.87906	0.192515
$407$	17.9474	0.889618
$408$	−8.21209	−0.406559
$409$	26.0895	1.29004	0.645021	−	0.764165i	$-0.276849\pi$
$409$	26.0895	1.29004	0.645021	+	0.764165i	$0.276849\pi$
$410$	−27.0055	−1.33371
$411$	−20.3271	−1.00266
$412$	0.129262	0.00636827
$413$	25.8321	1.27111
$414$	1.00000	0.0491473
$415$	7.63392	0.374734
$416$	−5.79409	−0.284079
$417$	−12.0316	−0.589188
$418$	25.1440	1.22983
$419$	25.1650	1.22939	0.614696	−	0.788764i	$-0.289279\pi$
$419$	25.1650	1.22939	0.614696	+	0.788764i	$0.289279\pi$
$420$	10.4486	0.509839
$421$	37.1329	1.80974	0.904872	−	0.425683i	$-0.139966\pi$
$421$	37.1329	1.80974	0.904872	+	0.425683i	$0.139966\pi$
$422$	−11.9770	−0.583031
$423$	6.91570	0.336253
$424$	−4.37463	−0.212451
$425$	18.5217	0.898433
$426$	11.3406	0.549453
$427$	−25.5077	−1.23440
$428$	−9.12874	−0.441254
$429$	−35.3990	−1.70908
$430$	−11.0430	−0.532541
$431$	−16.2673	−0.783566	−0.391783	−	0.920058i	$-0.628142\pi$
$431$	−16.2673	−0.783566	−0.391783	+	0.920058i	$0.628142\pi$
$432$	1.00000	0.0481125
$433$	−36.4818	−1.75320	−0.876601	−	0.481219i	$-0.840194\pi$
$433$	−36.4818	−1.75320	−0.876601	+	0.481219i	$0.840194\pi$
$434$	3.79659	0.182242
$435$	−2.69359	−0.129148
$436$	−16.9420	−0.811375
$437$	−4.11557	−0.196874
$438$	−13.1310	−0.627423
$439$	3.36332	0.160523	0.0802613	−	0.996774i	$-0.474425\pi$
$439$	3.36332	0.160523	0.0802613	+	0.996774i	$0.474425\pi$
$440$	−16.4565	−0.784531
$441$	8.04711	0.383196
$442$	−47.5816	−2.26322
$443$	−21.1581	−1.00525	−0.502625	−	0.864504i	$-0.667632\pi$
$443$	−21.1581	−1.00525	−0.502625	+	0.864504i	$0.667632\pi$
$444$	−2.93762	−0.139413
$445$	33.9692	1.61029
$446$	23.2718	1.10195
$447$	7.04400	0.333170
$448$	−3.87906	−0.183268
$449$	−0.616312	−0.0290855	−0.0145428	−	0.999894i	$-0.504629\pi$
$449$	−0.616312	−0.0290855	−0.0145428	+	0.999894i	$0.504629\pi$
$450$	−2.25542	−0.106321
$451$	61.2529	2.88429
$452$	7.59406	0.357194
$453$	17.8844	0.840281
$454$	9.46965	0.444433
$455$	60.5401	2.83816
$456$	−4.11557	−0.192729
$457$	−0.171829	−0.00803781	−0.00401891	−	0.999992i	$-0.501279\pi$
$457$	−0.171829	−0.00803781	−0.00401891	+	0.999992i	$0.501279\pi$
$458$	26.2271	1.22551
$459$	8.21209	0.383307
$460$	2.69359	0.125589
$461$	−23.8342	−1.11007	−0.555034	−	0.831828i	$-0.687295\pi$
$461$	−23.8342	−1.11007	−0.555034	+	0.831828i	$0.687295\pi$
$462$	−23.6991	−1.10258
$463$	14.1466	0.657446	0.328723	−	0.944426i	$-0.393382\pi$
$463$	14.1466	0.657446	0.328723	+	0.944426i	$0.393382\pi$
$464$	1.00000	0.0464238
$465$	−2.63632	−0.122256
$466$	8.94429	0.414336
$467$	7.98594	0.369545	0.184773	−	0.982781i	$-0.440845\pi$
$467$	7.98594	0.369545	0.184773	+	0.982781i	$0.440845\pi$
$468$	5.79409	0.267832
$469$	−35.6776	−1.64744
$470$	18.6280	0.859247
$471$	−20.5190	−0.945466
$472$	6.65937	0.306522
$473$	25.0474	1.15168
$474$	10.9603	0.503421
$475$	9.28231	0.425902
$476$	−31.8552	−1.46008
$477$	4.37463	0.200301
$478$	−1.38468	−0.0633338
$479$	1.84873	0.0844706	0.0422353	−	0.999108i	$-0.486552\pi$
$479$	1.84873	0.0844706	0.0422353	+	0.999108i	$0.486552\pi$
$480$	2.69359	0.122945
$481$	−17.0208	−0.776083
$482$	15.1982	0.692258
$483$	3.87906	0.176503
$484$	26.3259	1.19663
$485$	−19.6858	−0.893887
$486$	−1.00000	−0.0453609
$487$	−16.2702	−0.737274	−0.368637	−	0.929573i	$-0.620175\pi$
$487$	−16.2702	−0.737274	−0.368637	+	0.929573i	$0.620175\pi$
$488$	−6.57573	−0.297669
$489$	−8.68240	−0.392632
$490$	21.6756	0.979204
$491$	20.0640	0.905476	0.452738	−	0.891643i	$-0.350447\pi$
$491$	20.0640	0.905476	0.452738	+	0.891643i	$0.350447\pi$
$492$	−10.0259	−0.452000
$493$	8.21209	0.369854
$494$	−23.8460	−1.07288
$495$	16.4565	0.739663
$496$	0.978738	0.0439467
$497$	43.9908	1.97326
$498$	2.83411	0.126999
$499$	19.8251	0.887494	0.443747	−	0.896152i	$-0.353649\pi$
$499$	19.8251	0.887494	0.443747	+	0.896152i	$0.353649\pi$
$500$	7.39278	0.330615
$501$	−19.6860	−0.879507
$502$	18.9951	0.847792
$503$	−17.9815	−0.801757	−0.400879	−	0.916131i	$-0.631295\pi$
$503$	−17.9815	−0.801757	−0.400879	+	0.916131i	$0.631295\pi$
$504$	3.87906	0.172787
$505$	7.81613	0.347813
$506$	−6.10949	−0.271600
$507$	20.5715	0.913612
$508$	−10.1474	−0.450218
$509$	−26.2941	−1.16546	−0.582732	−	0.812665i	$-0.698016\pi$
$509$	−26.2941	−1.16546	−0.582732	+	0.812665i	$0.698016\pi$
$510$	22.1200	0.979489
$511$	−50.9359	−2.25327
$512$	−1.00000	−0.0441942
$513$	4.11557	0.181707
$514$	−23.8053	−1.05001
$515$	−0.348178	−0.0153426
$516$	−4.09974	−0.180481
$517$	−42.2514	−1.85821
$518$	−11.3952	−0.500677
$519$	−5.58200	−0.245022
$520$	15.6069	0.684408
$521$	−9.00336	−0.394444	−0.197222	−	0.980359i	$-0.563192\pi$
$521$	−9.00336	−0.394444	−0.197222	+	0.980359i	$0.563192\pi$
$522$	−1.00000	−0.0437688
$523$	10.7877	0.471713	0.235857	−	0.971788i	$-0.424210\pi$
$523$	10.7877	0.471713	0.235857	+	0.971788i	$0.424210\pi$
$524$	10.8105	0.472258
$525$	−8.74889	−0.381833
$526$	16.9253	0.737980
$527$	8.03749	0.350118
$528$	−6.10949	−0.265882
$529$	1.00000	0.0434783
$530$	11.7834	0.511840
$531$	−6.65937	−0.288992
$532$	−15.9645	−0.692150
$533$	−58.0907	−2.51619
$534$	12.6111	0.545737
$535$	24.5891	1.06308
$536$	−9.19748	−0.397271
$537$	−6.00438	−0.259108
$538$	−5.96771	−0.257286
$539$	−49.1638	−2.11764
$540$	−2.69359	−0.115914
$541$	0.355319	0.0152764	0.00763819	−	0.999971i	$-0.497569\pi$
$541$	0.355319	0.0152764	0.00763819	+	0.999971i	$0.497569\pi$
$542$	−14.5399	−0.624542
$543$	−18.3194	−0.786162
$544$	−8.21209	−0.352090
$545$	45.6348	1.95478
$546$	22.4756	0.961868
$547$	−22.3481	−0.955534	−0.477767	−	0.878487i	$-0.658554\pi$
$547$	−22.3481	−0.955534	−0.477767	+	0.878487i	$0.658554\pi$
$548$	−20.3271	−0.868329
$549$	6.57573	0.280645
$550$	13.7794	0.587558
$551$	4.11557	0.175329
$552$	1.00000	0.0425628
$553$	42.5155	1.80794
$554$	−4.13692	−0.175761
$555$	7.91274	0.335877
$556$	−12.0316	−0.510252
$557$	31.1040	1.31792	0.658959	−	0.752179i	$-0.270997\pi$
$557$	31.1040	1.31792	0.658959	+	0.752179i	$0.270997\pi$
$558$	−0.978738	−0.0414333
$559$	−23.7543	−1.00470
$560$	10.4486	0.441534
$561$	−50.1717	−2.11825
$562$	−1.57465	−0.0664226
$563$	15.0566	0.634560	0.317280	−	0.948332i	$-0.397230\pi$
$563$	15.0566	0.634560	0.317280	+	0.948332i	$0.397230\pi$
$564$	6.91570	0.291203
$565$	−20.4553	−0.860559
$566$	−22.2945	−0.937108
$567$	−3.87906	−0.162905
$568$	11.3406	0.475841
$569$	7.40783	0.310553	0.155276	−	0.987871i	$-0.450373\pi$
$569$	7.40783	0.310553	0.155276	+	0.987871i	$0.450373\pi$
$570$	11.0856	0.464326
$571$	−32.5460	−1.36201	−0.681004	−	0.732280i	$-0.738456\pi$
$571$	−32.5460	−1.36201	−0.681004	+	0.732280i	$0.738456\pi$
$572$	−35.3990	−1.48011
$573$	0.201020	0.00839772
$574$	−38.8909	−1.62327
$575$	−2.25542	−0.0940573
$576$	1.00000	0.0416667
$577$	21.5402	0.896732	0.448366	−	0.893850i	$-0.352006\pi$
$577$	21.5402	0.896732	0.448366	+	0.893850i	$0.352006\pi$
$578$	−50.4384	−2.09796
$579$	−1.07001	−0.0444681
$580$	−2.69359	−0.111845
$581$	10.9937	0.456094
$582$	−7.30840	−0.302943
$583$	−26.7268	−1.10691
$584$	−13.1310	−0.543364
$585$	−15.6069	−0.645266
$586$	−5.68927	−0.235021
$587$	0.670750	0.0276848	0.0138424	−	0.999904i	$-0.495594\pi$
$587$	0.670750	0.0276848	0.0138424	+	0.999904i	$0.495594\pi$
$588$	8.04711	0.331857
$589$	4.02806	0.165973
$590$	−17.9376	−0.738479
$591$	21.0590	0.866252
$592$	−2.93762	−0.120735
$593$	34.9286	1.43434	0.717172	−	0.696896i	$-0.245436\pi$
$593$	34.9286	1.43434	0.717172	+	0.696896i	$0.245436\pi$
$594$	6.10949	0.250676
$595$	85.8047	3.51765
$596$	7.04400	0.288533
$597$	−21.5663	−0.882650
$598$	5.79409	0.236938
$599$	−25.7392	−1.05168	−0.525838	−	0.850585i	$-0.676248\pi$
$599$	−25.7392	−1.05168	−0.525838	+	0.850585i	$0.676248\pi$
$600$	−2.25542	−0.0920769
$601$	−1.08614	−0.0443046	−0.0221523	−	0.999755i	$-0.507052\pi$
$601$	−1.08614	−0.0443046	−0.0221523	+	0.999755i	$0.507052\pi$
$602$	−15.9031	−0.648164
$603$	9.19748	0.374550
$604$	17.8844	0.727704
$605$	−70.9112	−2.88295
$606$	2.90175	0.117876
$607$	13.8535	0.562298	0.281149	−	0.959664i	$-0.409285\pi$
$607$	13.8535	0.562298	0.281149	+	0.959664i	$0.409285\pi$
$608$	−4.11557	−0.166908
$609$	−3.87906	−0.157187
$610$	17.7123	0.717150
$611$	40.0702	1.62107
$612$	8.21209	0.331954
$613$	−12.6971	−0.512830	−0.256415	−	0.966567i	$-0.582541\pi$
$613$	−12.6971	−0.512830	−0.256415	+	0.966567i	$0.582541\pi$
$614$	−5.61881	−0.226757
$615$	27.0055	1.08897
$616$	−23.6991	−0.954864
$617$	−14.2698	−0.574480	−0.287240	−	0.957859i	$-0.592738\pi$
$617$	−14.2698	−0.574480	−0.287240	+	0.957859i	$0.592738\pi$
$618$	−0.129262	−0.00519967
$619$	13.9787	0.561853	0.280926	−	0.959729i	$-0.409358\pi$
$619$	13.9787	0.561853	0.280926	+	0.959729i	$0.409358\pi$
$620$	−2.63632	−0.105877
$621$	−1.00000	−0.0401286
$622$	−3.40536	−0.136542
$623$	48.9194	1.95991
$624$	5.79409	0.231949
$625$	−31.1902	−1.24761
$626$	−5.65089	−0.225855
$627$	−25.1440	−1.00416
$628$	−20.5190	−0.818797
$629$	−24.1240	−0.961886
$630$	−10.4486	−0.416282
$631$	3.72145	0.148149	0.0740743	−	0.997253i	$-0.476400\pi$
$631$	3.72145	0.148149	0.0740743	+	0.997253i	$0.476400\pi$
$632$	10.9603	0.435975
$633$	11.9770	0.476043
$634$	18.3224	0.727675
$635$	27.3329	1.08467
$636$	4.37463	0.173465
$637$	46.6257	1.84738
$638$	6.10949	0.241877
$639$	−11.3406	−0.448627
$640$	2.69359	0.106473
$641$	14.6050	0.576864	0.288432	−	0.957500i	$-0.406866\pi$
$641$	14.6050	0.576864	0.288432	+	0.957500i	$0.406866\pi$
$642$	9.12874	0.360283
$643$	24.8412	0.979640	0.489820	−	0.871824i	$-0.337063\pi$
$643$	24.8412	0.979640	0.489820	+	0.871824i	$0.337063\pi$
$644$	3.87906	0.152856
$645$	11.0430	0.434818
$646$	−33.7974	−1.32974
$647$	−19.9968	−0.786155	−0.393078	−	0.919505i	$-0.628590\pi$
$647$	−19.9968	−0.786155	−0.393078	+	0.919505i	$0.628590\pi$
$648$	−1.00000	−0.0392837
$649$	40.6854	1.59704
$650$	−13.0681	−0.512572
$651$	−3.79659	−0.148800
$652$	−8.68240	−0.340029
$653$	3.47528	0.135998	0.0679991	−	0.997685i	$-0.478339\pi$
$653$	3.47528	0.135998	0.0679991	+	0.997685i	$0.478339\pi$
$654$	16.9420	0.662485
$655$	−29.1190	−1.13777
$656$	−10.0259	−0.391444
$657$	13.1310	0.512289
$658$	26.8264	1.04580
$659$	−9.64628	−0.375766	−0.187883	−	0.982191i	$-0.560163\pi$
$659$	−9.64628	−0.375766	−0.187883	+	0.982191i	$0.560163\pi$
$660$	16.4565	0.640567
$661$	10.1012	0.392890	0.196445	−	0.980515i	$-0.437060\pi$
$661$	10.1012	0.392890	0.196445	+	0.980515i	$0.437060\pi$
$662$	−14.7009	−0.571365
$663$	47.5816	1.84792
$664$	2.83411	0.109985
$665$	43.0019	1.66754
$666$	2.93762	0.113830
$667$	−1.00000	−0.0387202
$668$	−19.6860	−0.761675
$669$	−23.2718	−0.899740
$670$	24.7742	0.957112
$671$	−40.1744	−1.55092
$672$	3.87906	0.149638
$673$	−43.3794	−1.67215	−0.836076	−	0.548613i	$-0.815156\pi$
$673$	−43.3794	−1.67215	−0.836076	+	0.548613i	$0.815156\pi$
$674$	−5.79678	−0.223284
$675$	2.25542	0.0868110
$676$	20.5715	0.791211
$677$	23.7539	0.912937	0.456469	−	0.889739i	$-0.349114\pi$
$677$	23.7539	0.912937	0.456469	+	0.889739i	$0.349114\pi$
$678$	−7.59406	−0.291648
$679$	−28.3497	−1.08796
$680$	22.1200	0.848262
$681$	−9.46965	−0.362878
$682$	5.97960	0.228971
$683$	26.6592	1.02009	0.510043	−	0.860149i	$-0.329629\pi$
$683$	26.6592	1.02009	0.510043	+	0.860149i	$0.329629\pi$
$684$	4.11557	0.157363
$685$	54.7527	2.09199
$686$	4.06182	0.155081
$687$	−26.2271	−1.00063
$688$	−4.09974	−0.156301
$689$	25.3470	0.965644
$690$	−2.69359	−0.102543
$691$	8.93840	0.340033	0.170016	−	0.985441i	$-0.445618\pi$
$691$	8.93840	0.340033	0.170016	+	0.985441i	$0.445618\pi$
$692$	−5.58200	−0.212196
$693$	23.6991	0.900255
$694$	8.77365	0.333043
$695$	32.4081	1.22931
$696$	−1.00000	−0.0379049
$697$	−82.3332	−3.11859
$698$	15.5806	0.589734
$699$	−8.94429	−0.338304
$700$	−8.74889	−0.330677
$701$	−46.8814	−1.77069	−0.885344	−	0.464937i	$-0.846077\pi$
$701$	−46.8814	−1.77069	−0.885344	+	0.464937i	$0.846077\pi$
$702$	−5.79409	−0.218684
$703$	−12.0900	−0.455982
$704$	−6.10949	−0.230260
$705$	−18.6280	−0.701572
$706$	19.1232	0.719713
$707$	11.2561	0.423328
$708$	−6.65937	−0.250275
$709$	−29.5638	−1.11029	−0.555147	−	0.831752i	$-0.687338\pi$
$709$	−29.5638	−1.11029	−0.555147	+	0.831752i	$0.687338\pi$
$710$	−30.5469	−1.14640
$711$	−10.9603	−0.411042
$712$	12.6111	0.472622
$713$	−0.978738	−0.0366540
$714$	31.8552	1.19215
$715$	95.3502	3.56590
$716$	−6.00438	−0.224394
$717$	1.38468	0.0517118
$718$	−1.16936	−0.0436402
$719$	10.6018	0.395380	0.197690	−	0.980265i	$-0.436656\pi$
$719$	10.6018	0.395380	0.197690	+	0.980265i	$0.436656\pi$
$720$	−2.69359	−0.100384
$721$	−0.501415	−0.0186737
$722$	2.06212	0.0767443
$723$	−15.1982	−0.565227
$724$	−18.3194	−0.680836
$725$	2.25542	0.0837640
$726$	−26.3259	−0.977047
$727$	−34.9434	−1.29598	−0.647990	−	0.761649i	$-0.724390\pi$
$727$	−34.9434	−1.29598	−0.647990	+	0.761649i	$0.724390\pi$
$728$	22.4756	0.833002
$729$	1.00000	0.0370370
$730$	35.3695	1.30908
$731$	−33.6674	−1.24524
$732$	6.57573	0.243046
$733$	−42.0432	−1.55290	−0.776451	−	0.630177i	$-0.782982\pi$
$733$	−42.0432	−1.55290	−0.776451	+	0.630177i	$0.782982\pi$
$734$	17.0876	0.630714
$735$	−21.6756	−0.799517
$736$	1.00000	0.0368605
$737$	−56.1920	−2.06986
$738$	10.0259	0.369057
$739$	−31.4012	−1.15511	−0.577556	−	0.816351i	$-0.695994\pi$
$739$	−31.4012	−1.15511	−0.577556	+	0.816351i	$0.695994\pi$
$740$	7.91274	0.290878
$741$	23.8460	0.876003
$742$	16.9695	0.622968
$743$	3.55239	0.130325	0.0651624	−	0.997875i	$-0.479243\pi$
$743$	3.55239	0.130325	0.0651624	+	0.997875i	$0.479243\pi$
$744$	−0.978738	−0.0358823
$745$	−18.9736	−0.695140
$746$	−24.6063	−0.900902
$747$	−2.83411	−0.103695
$748$	−50.1717	−1.83446
$749$	35.4110	1.29389
$750$	−7.39278	−0.269946
$751$	−22.3675	−0.816203	−0.408101	−	0.912937i	$-0.633809\pi$
$751$	−22.3675	−0.816203	−0.408101	+	0.912937i	$0.633809\pi$
$752$	6.91570	0.252189
$753$	−18.9951	−0.692219
$754$	−5.79409	−0.211008
$755$	−48.1731	−1.75320
$756$	−3.87906	−0.141080
$757$	11.3877	0.413894	0.206947	−	0.978352i	$-0.433647\pi$
$757$	11.3877	0.413894	0.206947	+	0.978352i	$0.433647\pi$
$758$	−21.8644	−0.794150
$759$	6.10949	0.221761
$760$	11.0856	0.402118
$761$	−25.9942	−0.942290	−0.471145	−	0.882056i	$-0.656159\pi$
$761$	−25.9942	−0.942290	−0.471145	+	0.882056i	$0.656159\pi$
$762$	10.1474	0.367601
$763$	65.7190	2.37919
$764$	0.201020	0.00727263
$765$	−22.1200	−0.799750
$766$	29.1904	1.05469
$767$	−38.5850	−1.39322
$768$	1.00000	0.0360844
$769$	−5.74922	−0.207322	−0.103661	−	0.994613i	$-0.533056\pi$
$769$	−5.74922	−0.207322	−0.103661	+	0.994613i	$0.533056\pi$
$770$	63.8356	2.30048
$771$	23.8053	0.857326
$772$	−1.07001	−0.0385105
$773$	23.9262	0.860564	0.430282	−	0.902694i	$-0.358414\pi$
$773$	23.9262	0.860564	0.430282	+	0.902694i	$0.358414\pi$
$774$	4.09974	0.147362
$775$	2.20746	0.0792944
$776$	−7.30840	−0.262356
$777$	11.3952	0.408801
$778$	31.5252	1.13023
$779$	−41.2621	−1.47837
$780$	−15.6069	−0.558816
$781$	69.2853	2.47922
$782$	8.21209	0.293664
$783$	1.00000	0.0357371
$784$	8.04711	0.287397
$785$	55.2697	1.97266
$786$	−10.8105	−0.385597
$787$	−19.3170	−0.688577	−0.344288	−	0.938864i	$-0.611880\pi$
$787$	−19.3170	−0.688577	−0.344288	+	0.938864i	$0.611880\pi$
$788$	21.0590	0.750196
$789$	−16.9253	−0.602558
$790$	−29.5224	−1.05036
$791$	−29.4578	−1.04740
$792$	6.10949	0.217091
$793$	38.1004	1.35298
$794$	22.8284	0.810149
$795$	−11.7834	−0.417916
$796$	−21.5663	−0.764398
$797$	47.9515	1.69853	0.849264	−	0.527968i	$-0.177046\pi$
$797$	47.9515	1.69853	0.849264	+	0.527968i	$0.177046\pi$
$798$	15.9645	0.565138
$799$	56.7923	2.00917
$800$	−2.25542	−0.0797410
$801$	−12.6111	−0.445592
$802$	−5.07367	−0.179157
$803$	−80.2238	−2.83103
$804$	9.19748	0.324370
$805$	−10.4486	−0.368264
$806$	−5.67090	−0.199749
$807$	5.96771	0.210073
$808$	2.90175	0.102083
$809$	25.5392	0.897910	0.448955	−	0.893554i	$-0.351796\pi$
$809$	25.5392	0.897910	0.448955	+	0.893554i	$0.351796\pi$
$810$	2.69359	0.0946430
$811$	−20.6947	−0.726688	−0.363344	−	0.931655i	$-0.618365\pi$
$811$	−20.6947	−0.726688	−0.363344	+	0.931655i	$0.618365\pi$
$812$	−3.87906	−0.136128
$813$	14.5399	0.509936
$814$	−17.9474	−0.629055
$815$	23.3868	0.819204
$816$	8.21209	0.287481
$817$	−16.8728	−0.590303
$818$	−26.0895	−0.912198
$819$	−22.4756	−0.785362
$820$	27.0055	0.943074
$821$	41.2737	1.44046	0.720232	−	0.693734i	$-0.244035\pi$
$821$	41.2737	1.44046	0.720232	+	0.693734i	$0.244035\pi$
$822$	20.3271	0.708987
$823$	29.3465	1.02296	0.511478	−	0.859297i	$-0.329098\pi$
$823$	29.3465	1.02296	0.511478	+	0.859297i	$0.329098\pi$
$824$	−0.129262	−0.00450305
$825$	−13.7794	−0.479739
$826$	−25.8321	−0.898814
$827$	−1.95279	−0.0679051	−0.0339526	−	0.999423i	$-0.510810\pi$
$827$	−1.95279	−0.0679051	−0.0339526	+	0.999423i	$0.510810\pi$
$828$	−1.00000	−0.0347524
$829$	−33.8615	−1.17606	−0.588030	−	0.808839i	$-0.700096\pi$
$829$	−33.8615	−1.17606	−0.588030	+	0.808839i	$0.700096\pi$
$830$	−7.63392	−0.264977
$831$	4.13692	0.143508
$832$	5.79409	0.200874
$833$	66.0836	2.28966
$834$	12.0316	0.416619
$835$	53.0260	1.83504
$836$	−25.1440	−0.869624
$837$	0.978738	0.0338301
$838$	−25.1650	−0.869311
$839$	−44.8191	−1.54733	−0.773664	−	0.633596i	$-0.781578\pi$
$839$	−44.8191	−1.54733	−0.773664	+	0.633596i	$0.781578\pi$
$840$	−10.4486	−0.360511
$841$	1.00000	0.0344828
$842$	−37.1329	−1.27968
$843$	1.57465	0.0542339
$844$	11.9770	0.412265
$845$	−55.4111	−1.90620
$846$	−6.91570	−0.237767
$847$	−102.120	−3.50888
$848$	4.37463	0.150225
$849$	22.2945	0.765146
$850$	−18.5217	−0.635288
$851$	2.93762	0.100700
$852$	−11.3406	−0.388522
$853$	3.24373	0.111063	0.0555316	−	0.998457i	$-0.482315\pi$
$853$	3.24373	0.111063	0.0555316	+	0.998457i	$0.482315\pi$
$854$	25.5077	0.872854
$855$	−11.0856	−0.379121
$856$	9.12874	0.312014
$857$	−20.4104	−0.697207	−0.348603	−	0.937270i	$-0.613344\pi$
$857$	−20.4104	−0.697207	−0.348603	+	0.937270i	$0.613344\pi$
$858$	35.3990	1.20850
$859$	18.7964	0.641324	0.320662	−	0.947194i	$-0.396095\pi$
$859$	18.7964	0.641324	0.320662	+	0.947194i	$0.396095\pi$
$860$	11.0430	0.376564
$861$	38.8909	1.32540
$862$	16.2673	0.554065
$863$	−14.5564	−0.495505	−0.247752	−	0.968823i	$-0.579692\pi$
$863$	−14.5564	−0.495505	−0.247752	+	0.968823i	$0.579692\pi$
$864$	−1.00000	−0.0340207
$865$	15.0356	0.511226
$866$	36.4818	1.23970
$867$	50.4384	1.71298
$868$	−3.79659	−0.128865
$869$	66.9616	2.27152
$870$	2.69359	0.0913212
$871$	53.2910	1.80570
$872$	16.9420	0.573728
$873$	7.30840	0.247352
$874$	4.11557	0.139211
$875$	−28.6770	−0.969461
$876$	13.1310	0.443655
$877$	−16.1896	−0.546684	−0.273342	−	0.961917i	$-0.588129\pi$
$877$	−16.1896	−0.546684	−0.273342	+	0.961917i	$0.588129\pi$
$878$	−3.36332	−0.113507
$879$	5.68927	0.191894
$880$	16.4565	0.554747
$881$	−13.2295	−0.445712	−0.222856	−	0.974851i	$-0.571538\pi$
$881$	−13.2295	−0.445712	−0.222856	+	0.974851i	$0.571538\pi$
$882$	−8.04711	−0.270960
$883$	−5.94164	−0.199952	−0.0999760	−	0.994990i	$-0.531877\pi$
$883$	−5.94164	−0.199952	−0.0999760	+	0.994990i	$0.531877\pi$
$884$	47.5816	1.60034
$885$	17.9376	0.602966
$886$	21.1581	0.710819
$887$	47.6316	1.59931	0.799656	−	0.600458i	$-0.205015\pi$
$887$	47.6316	1.59931	0.799656	+	0.600458i	$0.205015\pi$
$888$	2.93762	0.0985801
$889$	39.3623	1.32017
$890$	−33.9692	−1.13865
$891$	−6.10949	−0.204676
$892$	−23.2718	−0.779198
$893$	28.4620	0.952444
$894$	−7.04400	−0.235586
$895$	16.1733	0.540615
$896$	3.87906	0.129590
$897$	−5.79409	−0.193459
$898$	0.616312	0.0205666
$899$	0.978738	0.0326428
$900$	2.25542	0.0751805
$901$	35.9248	1.19683
$902$	−61.2529	−2.03950
$903$	15.9031	0.529224
$904$	−7.59406	−0.252575
$905$	49.3450	1.64028
$906$	−17.8844	−0.594168
$907$	−52.9568	−1.75840	−0.879200	−	0.476453i	$-0.841922\pi$
$907$	−52.9568	−1.75840	−0.879200	+	0.476453i	$0.841922\pi$
$908$	−9.46965	−0.314261
$909$	−2.90175	−0.0962451
$910$	−60.5401	−2.00688
$911$	37.1947	1.23232	0.616158	−	0.787623i	$-0.288688\pi$
$911$	37.1947	1.23232	0.616158	+	0.787623i	$0.288688\pi$
$912$	4.11557	0.136280
$913$	17.3150	0.573042
$914$	0.171829	0.00568359
$915$	−17.7123	−0.585551
$916$	−26.2271	−0.866568
$917$	−41.9345	−1.38480
$918$	−8.21209	−0.271039
$919$	−19.5776	−0.645807	−0.322903	−	0.946432i	$-0.604659\pi$
$919$	−19.5776	−0.645807	−0.322903	+	0.946432i	$0.604659\pi$
$920$	−2.69359	−0.0888050
$921$	5.61881	0.185146
$922$	23.8342	0.784936
$923$	−65.7084	−2.16282
$924$	23.6991	0.779643
$925$	−6.62555	−0.217847
$926$	−14.1466	−0.464885
$927$	0.129262	0.00424552
$928$	−1.00000	−0.0328266
$929$	−20.1892	−0.662387	−0.331194	−	0.943563i	$-0.607451\pi$
$929$	−20.1892	−0.662387	−0.331194	+	0.943563i	$0.607451\pi$
$930$	2.63632	0.0864483
$931$	33.1184	1.08541
$932$	−8.94429	−0.292980
$933$	3.40536	0.111486
$934$	−7.98594	−0.261308
$935$	135.142	4.41961
$936$	−5.79409	−0.189386
$937$	−26.2446	−0.857373	−0.428686	−	0.903453i	$-0.641023\pi$
$937$	−26.2446	−0.857373	−0.428686	+	0.903453i	$0.641023\pi$
$938$	35.6776	1.16491
$939$	5.65089	0.184410
$940$	−18.6280	−0.607579
$941$	−38.3900	−1.25148	−0.625740	−	0.780032i	$-0.715203\pi$
$941$	−38.3900	−1.25148	−0.625740	+	0.780032i	$0.715203\pi$
$942$	20.5190	0.668545
$943$	10.0259	0.326487
$944$	−6.65937	−0.216744
$945$	10.4486	0.339893
$946$	−25.0474	−0.814360
$947$	44.9550	1.46084	0.730420	−	0.682998i	$-0.239324\pi$
$947$	44.9550	1.46084	0.730420	+	0.682998i	$0.239324\pi$
$948$	−10.9603	−0.355972
$949$	76.0822	2.46973
$950$	−9.28231	−0.301158
$951$	−18.3224	−0.594144
$952$	31.8552	1.03243
$953$	6.40871	0.207598	0.103799	−	0.994598i	$-0.466900\pi$
$953$	6.40871	0.207598	0.103799	+	0.994598i	$0.466900\pi$
$954$	−4.37463	−0.141634
$955$	−0.541464	−0.0175214
$956$	1.38468	0.0447838
$957$	−6.10949	−0.197492
$958$	−1.84873	−0.0597298
$959$	78.8499	2.54620
$960$	−2.69359	−0.0869352
$961$	−30.0421	−0.969099
$962$	17.0208	0.548774
$963$	−9.12874	−0.294170
$964$	−15.1982	−0.489501
$965$	2.88216	0.0927802
$966$	−3.87906	−0.124807
$967$	55.1239	1.77266	0.886332	−	0.463050i	$-0.153245\pi$
$967$	55.1239	1.77266	0.886332	+	0.463050i	$0.153245\pi$
$968$	−26.3259	−0.846147
$969$	33.7974	1.08573
$970$	19.6858	0.632073
$971$	50.6889	1.62668	0.813342	−	0.581787i	$-0.197646\pi$
$971$	50.6889	1.62668	0.813342	+	0.581787i	$0.197646\pi$
$972$	1.00000	0.0320750
$973$	46.6712	1.49621
$974$	16.2702	0.521331
$975$	13.0681	0.418514
$976$	6.57573	0.210484
$977$	57.9315	1.85339	0.926696	−	0.375812i	$-0.122636\pi$
$977$	57.9315	1.85339	0.926696	+	0.375812i	$0.122636\pi$
$978$	8.68240	0.277633
$979$	77.0477	2.46245
$980$	−21.6756	−0.692402
$981$	−16.9420	−0.540916
$982$	−20.0640	−0.640269
$983$	−56.4320	−1.79990	−0.899950	−	0.435993i	$-0.856397\pi$
$983$	−56.4320	−1.79990	−0.899950	+	0.435993i	$0.856397\pi$
$984$	10.0259	0.319613
$985$	−56.7243	−1.80739
$986$	−8.21209	−0.261526
$987$	−26.8264	−0.853894
$988$	23.8460	0.758641
$989$	4.09974	0.130364
$990$	−16.4565	−0.523021
$991$	49.3675	1.56821	0.784105	−	0.620628i	$-0.213122\pi$
$991$	49.3675	1.56821	0.784105	+	0.620628i	$0.213122\pi$
$992$	−0.978738	−0.0310750
$993$	14.7009	0.466517
$994$	−43.9908	−1.39530
$995$	58.0907	1.84160
$996$	−2.83411	−0.0898021
$997$	−39.9362	−1.26479	−0.632395	−	0.774646i	$-0.717928\pi$
$997$	−39.9362	−1.26479	−0.632395	+	0.774646i	$0.717928\pi$
$998$	−19.8251	−0.627553
$999$	−2.93762	−0.0929422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bi.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bi.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.69359 q^{5} -1.00000 q^{6} -3.87906 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.69359 q^{5} -1.00000 q^{6} -3.87906 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.69359 q^{10} -6.10949 q^{11} +1.00000 q^{12} +5.79409 q^{13} +3.87906 q^{14} -2.69359 q^{15} +1.00000 q^{16} +8.21209 q^{17} -1.00000 q^{18} +4.11557 q^{19} -2.69359 q^{20} -3.87906 q^{21} +6.10949 q^{22} -1.00000 q^{23} -1.00000 q^{24} +2.25542 q^{25} -5.79409 q^{26} +1.00000 q^{27} -3.87906 q^{28} +1.00000 q^{29} +2.69359 q^{30} +0.978738 q^{31} -1.00000 q^{32} -6.10949 q^{33} -8.21209 q^{34} +10.4486 q^{35} +1.00000 q^{36} -2.93762 q^{37} -4.11557 q^{38} +5.79409 q^{39} +2.69359 q^{40} -10.0259 q^{41} +3.87906 q^{42} -4.09974 q^{43} -6.10949 q^{44} -2.69359 q^{45} +1.00000 q^{46} +6.91570 q^{47} +1.00000 q^{48} +8.04711 q^{49} -2.25542 q^{50} +8.21209 q^{51} +5.79409 q^{52} +4.37463 q^{53} -1.00000 q^{54} +16.4565 q^{55} +3.87906 q^{56} +4.11557 q^{57} -1.00000 q^{58} -6.65937 q^{59} -2.69359 q^{60} +6.57573 q^{61} -0.978738 q^{62} -3.87906 q^{63} +1.00000 q^{64} -15.6069 q^{65} +6.10949 q^{66} +9.19748 q^{67} +8.21209 q^{68} -1.00000 q^{69} -10.4486 q^{70} -11.3406 q^{71} -1.00000 q^{72} +13.1310 q^{73} +2.93762 q^{74} +2.25542 q^{75} +4.11557 q^{76} +23.6991 q^{77} -5.79409 q^{78} -10.9603 q^{79} -2.69359 q^{80} +1.00000 q^{81} +10.0259 q^{82} -2.83411 q^{83} -3.87906 q^{84} -22.1200 q^{85} +4.09974 q^{86} +1.00000 q^{87} +6.10949 q^{88} -12.6111 q^{89} +2.69359 q^{90} -22.4756 q^{91} -1.00000 q^{92} +0.978738 q^{93} -6.91570 q^{94} -11.0856 q^{95} -1.00000 q^{96} +7.30840 q^{97} -8.04711 q^{98} -6.10949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9} + 3 q^{10} - 7 q^{11} + 8 q^{12} - 3 q^{13} + 6 q^{14} - 3 q^{15} + 8 q^{16} - 12 q^{17} - 8 q^{18} - 4 q^{19} - 3 q^{20} - 6 q^{21} + 7 q^{22} - 8 q^{23} - 8 q^{24} + 13 q^{25} + 3 q^{26} + 8 q^{27} - 6 q^{28} + 8 q^{29} + 3 q^{30} - q^{31} - 8 q^{32} - 7 q^{33} + 12 q^{34} - 6 q^{35} + 8 q^{36} - 11 q^{37} + 4 q^{38} - 3 q^{39} + 3 q^{40} - 17 q^{41} + 6 q^{42} - 10 q^{43} - 7 q^{44} - 3 q^{45} + 8 q^{46} - 26 q^{47} + 8 q^{48} + 10 q^{49} - 13 q^{50} - 12 q^{51} - 3 q^{52} - 2 q^{53} - 8 q^{54} + q^{55} + 6 q^{56} - 4 q^{57} - 8 q^{58} - 25 q^{59} - 3 q^{60} + 3 q^{61} + q^{62} - 6 q^{63} + 8 q^{64} - 25 q^{65} + 7 q^{66} + q^{67} - 12 q^{68} - 8 q^{69} + 6 q^{70} - 27 q^{71} - 8 q^{72} + 16 q^{73} + 11 q^{74} + 13 q^{75} - 4 q^{76} - 16 q^{77} + 3 q^{78} - 6 q^{79} - 3 q^{80} + 8 q^{81} + 17 q^{82} - 44 q^{83} - 6 q^{84} - 20 q^{85} + 10 q^{86} + 8 q^{87} + 7 q^{88} - 52 q^{89} + 3 q^{90} - 18 q^{91} - 8 q^{92} - q^{93} + 26 q^{94} - 56 q^{95} - 8 q^{96} - 4 q^{97} - 10 q^{98} - 7 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9 + 3 * q^10 - 7 * q^11 + 8 * q^12 - 3 * q^13 + 6 * q^14 - 3 * q^15 + 8 * q^16 - 12 * q^17 - 8 * q^18 - 4 * q^19 - 3 * q^20 - 6 * q^21 + 7 * q^22 - 8 * q^23 - 8 * q^24 + 13 * q^25 + 3 * q^26 + 8 * q^27 - 6 * q^28 + 8 * q^29 + 3 * q^30 - q^31 - 8 * q^32 - 7 * q^33 + 12 * q^34 - 6 * q^35 + 8 * q^36 - 11 * q^37 + 4 * q^38 - 3 * q^39 + 3 * q^40 - 17 * q^41 + 6 * q^42 - 10 * q^43 - 7 * q^44 - 3 * q^45 + 8 * q^46 - 26 * q^47 + 8 * q^48 + 10 * q^49 - 13 * q^50 - 12 * q^51 - 3 * q^52 - 2 * q^53 - 8 * q^54 + q^55 + 6 * q^56 - 4 * q^57 - 8 * q^58 - 25 * q^59 - 3 * q^60 + 3 * q^61 + q^62 - 6 * q^63 + 8 * q^64 - 25 * q^65 + 7 * q^66 + q^67 - 12 * q^68 - 8 * q^69 + 6 * q^70 - 27 * q^71 - 8 * q^72 + 16 * q^73 + 11 * q^74 + 13 * q^75 - 4 * q^76 - 16 * q^77 + 3 * q^78 - 6 * q^79 - 3 * q^80 + 8 * q^81 + 17 * q^82 - 44 * q^83 - 6 * q^84 - 20 * q^85 + 10 * q^86 + 8 * q^87 + 7 * q^88 - 52 * q^89 + 3 * q^90 - 18 * q^91 - 8 * q^92 - q^93 + 26 * q^94 - 56 * q^95 - 8 * q^96 - 4 * q^97 - 10 * q^98 - 7 * q^99

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