LMFDB - Embedded newform 4002.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.9561308889$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.23252.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 2 $ x^4 - x^3 - 6*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.95372$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.27661 q^{5} -1.00000 q^{6} -5.23034 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.27661 q^{5} -1.00000 q^{6} -5.23034 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.27661 q^{10} -0.0462763 q^{11} +1.00000 q^{12} -5.72448 q^{13} +5.23034 q^{14} +1.27661 q^{15} +1.00000 q^{16} +5.23034 q^{17} -1.00000 q^{18} +8.21863 q^{19} +1.27661 q^{20} -5.23034 q^{21} +0.0462763 q^{22} +1.00000 q^{23} -1.00000 q^{24} -3.37026 q^{25} +5.72448 q^{26} +1.00000 q^{27} -5.23034 q^{28} -1.00000 q^{29} -1.27661 q^{30} -4.18296 q^{31} -1.00000 q^{32} -0.0462763 q^{33} -5.23034 q^{34} -6.67711 q^{35} +1.00000 q^{36} +6.31965 q^{37} -8.21863 q^{38} -5.72448 q^{39} -1.27661 q^{40} +7.07871 q^{41} +5.23034 q^{42} -11.1691 q^{43} -0.0462763 q^{44} +1.27661 q^{45} -1.00000 q^{46} +2.31118 q^{47} +1.00000 q^{48} +20.3564 q^{49} +3.37026 q^{50} +5.23034 q^{51} -5.72448 q^{52} +1.04304 q^{53} -1.00000 q^{54} -0.0590769 q^{55} +5.23034 q^{56} +8.21863 q^{57} +1.00000 q^{58} +7.63193 q^{59} +1.27661 q^{60} -5.95372 q^{61} +4.18296 q^{62} -5.23034 q^{63} +1.00000 q^{64} -7.30795 q^{65} +0.0462763 q^{66} -9.46391 q^{67} +5.23034 q^{68} +1.00000 q^{69} +6.67711 q^{70} -6.52548 q^{71} -1.00000 q^{72} -2.12389 q^{73} -6.31965 q^{74} -3.37026 q^{75} +8.21863 q^{76} +0.242041 q^{77} +5.72448 q^{78} -12.5846 q^{79} +1.27661 q^{80} +1.00000 q^{81} -7.07871 q^{82} -14.8266 q^{83} -5.23034 q^{84} +6.67711 q^{85} +11.1691 q^{86} -1.00000 q^{87} +0.0462763 q^{88} -16.4059 q^{89} -1.27661 q^{90} +29.9410 q^{91} +1.00000 q^{92} -4.18296 q^{93} -2.31118 q^{94} +10.4920 q^{95} -1.00000 q^{96} -12.4059 q^{97} -20.3564 q^{98} -0.0462763 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} + 3 q^{10} - 11 q^{11} + 4 q^{12} - q^{13} + 2 q^{14} - 3 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 8 q^{19} - 3 q^{20} - 2 q^{21} + 11 q^{22} + 4 q^{23} - 4 q^{24} + 3 q^{25} + q^{26} + 4 q^{27} - 2 q^{28} - 4 q^{29} + 3 q^{30} - 17 q^{31} - 4 q^{32} - 11 q^{33} - 2 q^{34} - 24 q^{35} + 4 q^{36} + 15 q^{37} - 8 q^{38} - q^{39} + 3 q^{40} + q^{41} + 2 q^{42} + 4 q^{43} - 11 q^{44} - 3 q^{45} - 4 q^{46} + 6 q^{47} + 4 q^{48} + 16 q^{49} - 3 q^{50} + 2 q^{51} - q^{52} + 2 q^{53} - 4 q^{54} + 13 q^{55} + 2 q^{56} + 8 q^{57} + 4 q^{58} - 13 q^{59} - 3 q^{60} - 13 q^{61} + 17 q^{62} - 2 q^{63} + 4 q^{64} - 13 q^{65} + 11 q^{66} - 13 q^{67} + 2 q^{68} + 4 q^{69} + 24 q^{70} - 15 q^{71} - 4 q^{72} - 22 q^{73} - 15 q^{74} + 3 q^{75} + 8 q^{76} - 12 q^{77} + q^{78} - 26 q^{79} - 3 q^{80} + 4 q^{81} - q^{82} - 22 q^{83} - 2 q^{84} + 24 q^{85} - 4 q^{86} - 4 q^{87} + 11 q^{88} - 24 q^{89} + 3 q^{90} + 30 q^{91} + 4 q^{92} - 17 q^{93} - 6 q^{94} - 4 q^{95} - 4 q^{96} - 8 q^{97} - 16 q^{98} - 11 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 + 3 * q^10 - 11 * q^11 + 4 * q^12 - q^13 + 2 * q^14 - 3 * q^15 + 4 * q^16 + 2 * q^17 - 4 * q^18 + 8 * q^19 - 3 * q^20 - 2 * q^21 + 11 * q^22 + 4 * q^23 - 4 * q^24 + 3 * q^25 + q^26 + 4 * q^27 - 2 * q^28 - 4 * q^29 + 3 * q^30 - 17 * q^31 - 4 * q^32 - 11 * q^33 - 2 * q^34 - 24 * q^35 + 4 * q^36 + 15 * q^37 - 8 * q^38 - q^39 + 3 * q^40 + q^41 + 2 * q^42 + 4 * q^43 - 11 * q^44 - 3 * q^45 - 4 * q^46 + 6 * q^47 + 4 * q^48 + 16 * q^49 - 3 * q^50 + 2 * q^51 - q^52 + 2 * q^53 - 4 * q^54 + 13 * q^55 + 2 * q^56 + 8 * q^57 + 4 * q^58 - 13 * q^59 - 3 * q^60 - 13 * q^61 + 17 * q^62 - 2 * q^63 + 4 * q^64 - 13 * q^65 + 11 * q^66 - 13 * q^67 + 2 * q^68 + 4 * q^69 + 24 * q^70 - 15 * q^71 - 4 * q^72 - 22 * q^73 - 15 * q^74 + 3 * q^75 + 8 * q^76 - 12 * q^77 + q^78 - 26 * q^79 - 3 * q^80 + 4 * q^81 - q^82 - 22 * q^83 - 2 * q^84 + 24 * q^85 - 4 * q^86 - 4 * q^87 + 11 * q^88 - 24 * q^89 + 3 * q^90 + 30 * q^91 + 4 * q^92 - 17 * q^93 - 6 * q^94 - 4 * q^95 - 4 * q^96 - 8 * q^97 - 16 * q^98 - 11 * 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
$5$	1.27661	0.570918	0.285459	−	0.958391i	$-0.407854\pi$
$5$	1.27661	0.570918	0.285459	+	0.958391i	$0.407854\pi$
$6$	−1.00000	−0.408248
$7$	−5.23034	−1.97688	−0.988441	−	0.151609i	$-0.951555\pi$
$7$	−5.23034	−1.97688	−0.988441	+	0.151609i	$0.951555\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.27661	−0.403700
$11$	−0.0462763	−0.0139528	−0.00697641	−	0.999976i	$-0.502221\pi$
$11$	−0.0462763	−0.0139528	−0.00697641	+	0.999976i	$0.502221\pi$
$12$	1.00000	0.288675
$13$	−5.72448	−1.58769	−0.793843	−	0.608123i	$-0.791923\pi$
$13$	−5.72448	−1.58769	−0.793843	+	0.608123i	$0.791923\pi$
$14$	5.23034	1.39787
$15$	1.27661	0.329620
$16$	1.00000	0.250000
$17$	5.23034	1.26854	0.634271	−	0.773110i	$-0.281300\pi$
$17$	5.23034	1.26854	0.634271	+	0.773110i	$0.281300\pi$
$18$	−1.00000	−0.235702
$19$	8.21863	1.88548	0.942742	−	0.333524i	$-0.108238\pi$
$19$	8.21863	1.88548	0.942742	+	0.333524i	$0.108238\pi$
$20$	1.27661	0.285459
$21$	−5.23034	−1.14135
$22$	0.0462763	0.00986614
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	−3.37026	−0.674052
$26$	5.72448	1.12266
$27$	1.00000	0.192450
$28$	−5.23034	−0.988441
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−1.27661	−0.233076
$31$	−4.18296	−0.751282	−0.375641	−	0.926765i	$-0.622577\pi$
$31$	−4.18296	−0.751282	−0.375641	+	0.926765i	$0.622577\pi$
$32$	−1.00000	−0.176777
$33$	−0.0462763	−0.00805567
$34$	−5.23034	−0.896995
$35$	−6.67711	−1.12864
$36$	1.00000	0.166667
$37$	6.31965	1.03894	0.519472	−	0.854487i	$-0.326129\pi$
$37$	6.31965	1.03894	0.519472	+	0.854487i	$0.326129\pi$
$38$	−8.21863	−1.33324
$39$	−5.72448	−0.916651
$40$	−1.27661	−0.201850
$41$	7.07871	1.10551	0.552754	−	0.833344i	$-0.313577\pi$
$41$	7.07871	1.10551	0.552754	+	0.833344i	$0.313577\pi$
$42$	5.23034	0.807058
$43$	−11.1691	−1.70327	−0.851637	−	0.524132i	$-0.824390\pi$
$43$	−11.1691	−1.70327	−0.851637	+	0.524132i	$0.824390\pi$
$44$	−0.0462763	−0.00697641
$45$	1.27661	0.190306
$46$	−1.00000	−0.147442
$47$	2.31118	0.337121	0.168560	−	0.985691i	$-0.446088\pi$
$47$	2.31118	0.337121	0.168560	+	0.985691i	$0.446088\pi$
$48$	1.00000	0.144338
$49$	20.3564	2.90806
$50$	3.37026	0.476627
$51$	5.23034	0.732394
$52$	−5.72448	−0.793843
$53$	1.04304	0.143272	0.0716362	−	0.997431i	$-0.477178\pi$
$53$	1.04304	0.143272	0.0716362	+	0.997431i	$0.477178\pi$
$54$	−1.00000	−0.136083
$55$	−0.0590769	−0.00796592
$56$	5.23034	0.698933
$57$	8.21863	1.08858
$58$	1.00000	0.131306
$59$	7.63193	0.993593	0.496796	−	0.867867i	$-0.334510\pi$
$59$	7.63193	0.993593	0.496796	+	0.867867i	$0.334510\pi$
$60$	1.27661	0.164810
$61$	−5.95372	−0.762296	−0.381148	−	0.924514i	$-0.624471\pi$
$61$	−5.95372	−0.762296	−0.381148	+	0.924514i	$0.624471\pi$
$62$	4.18296	0.531237
$63$	−5.23034	−0.658960
$64$	1.00000	0.125000
$65$	−7.30795	−0.906439
$66$	0.0462763	0.00569622
$67$	−9.46391	−1.15620	−0.578100	−	0.815966i	$-0.696206\pi$
$67$	−9.46391	−1.15620	−0.578100	+	0.815966i	$0.696206\pi$
$68$	5.23034	0.634271
$69$	1.00000	0.120386
$70$	6.67711	0.798067
$71$	−6.52548	−0.774432	−0.387216	−	0.921989i	$-0.626563\pi$
$71$	−6.52548	−0.774432	−0.387216	+	0.921989i	$0.626563\pi$
$72$	−1.00000	−0.117851
$73$	−2.12389	−0.248582	−0.124291	−	0.992246i	$-0.539666\pi$
$73$	−2.12389	−0.248582	−0.124291	+	0.992246i	$0.539666\pi$
$74$	−6.31965	−0.734645
$75$	−3.37026	−0.389164
$76$	8.21863	0.942742
$77$	0.242041	0.0275831
$78$	5.72448	0.648170
$79$	−12.5846	−1.41587	−0.707937	−	0.706276i	$-0.750374\pi$
$79$	−12.5846	−1.41587	−0.707937	+	0.706276i	$0.750374\pi$
$80$	1.27661	0.142730
$81$	1.00000	0.111111
$82$	−7.07871	−0.781712
$83$	−14.8266	−1.62743	−0.813715	−	0.581264i	$-0.802559\pi$
$83$	−14.8266	−1.62743	−0.813715	+	0.581264i	$0.802559\pi$
$84$	−5.23034	−0.570676
$85$	6.67711	0.724234
$86$	11.1691	1.20440
$87$	−1.00000	−0.107211
$88$	0.0462763	0.00493307
$89$	−16.4059	−1.73902	−0.869512	−	0.493911i	$-0.835567\pi$
$89$	−16.4059	−1.73902	−0.869512	+	0.493911i	$0.835567\pi$
$90$	−1.27661	−0.134567
$91$	29.9410	3.13867
$92$	1.00000	0.104257
$93$	−4.18296	−0.433753
$94$	−2.31118	−0.238380
$95$	10.4920	1.07646
$96$	−1.00000	−0.102062
$97$	−12.4059	−1.25963	−0.629816	−	0.776745i	$-0.716870\pi$
$97$	−12.4059	−1.25963	−0.629816	+	0.776745i	$0.716870\pi$
$98$	−20.3564	−2.05631
$99$	−0.0462763	−0.00465094
$100$	−3.37026	−0.337026
$101$	−13.2660	−1.32002	−0.660008	−	0.751258i	$-0.729447\pi$
$101$	−13.2660	−1.32002	−0.660008	+	0.751258i	$0.729447\pi$
$102$	−5.23034	−0.517880
$103$	−3.03457	−0.299005	−0.149503	−	0.988761i	$-0.547767\pi$
$103$	−3.03457	−0.299005	−0.149503	+	0.988761i	$0.547767\pi$
$104$	5.72448	0.561332
$105$	−6.67711	−0.651619
$106$	−1.04304	−0.101309
$107$	1.85270	0.179108	0.0895538	−	0.995982i	$-0.471456\pi$
$107$	1.85270	0.179108	0.0895538	+	0.995982i	$0.471456\pi$
$108$	1.00000	0.0962250
$109$	−1.37763	−0.131953	−0.0659766	−	0.997821i	$-0.521016\pi$
$109$	−1.37763	−0.131953	−0.0659766	+	0.997821i	$0.521016\pi$
$110$	0.0590769	0.00563276
$111$	6.31965	0.599835
$112$	−5.23034	−0.494220
$113$	−13.0139	−1.22424	−0.612122	−	0.790763i	$-0.709684\pi$
$113$	−13.0139	−1.22424	−0.612122	+	0.790763i	$0.709684\pi$
$114$	−8.21863	−0.769745
$115$	1.27661	0.119045
$116$	−1.00000	−0.0928477
$117$	−5.72448	−0.529229
$118$	−7.63193	−0.702576
$119$	−27.3564	−2.50776
$120$	−1.27661	−0.116538
$121$	−10.9979	−0.999805
$122$	5.95372	0.539025
$123$	7.07871	0.638265
$124$	−4.18296	−0.375641
$125$	−10.6856	−0.955747
$126$	5.23034	0.465955
$127$	−19.2660	−1.70958	−0.854791	−	0.518973i	$-0.826315\pi$
$127$	−19.2660	−1.70958	−0.854791	+	0.518973i	$0.826315\pi$
$128$	−1.00000	−0.0883883
$129$	−11.1691	−0.983386
$130$	7.30795	0.640949
$131$	−15.7836	−1.37902	−0.689508	−	0.724278i	$-0.742173\pi$
$131$	−15.7836	−1.37902	−0.689508	+	0.724278i	$0.742173\pi$
$132$	−0.0462763	−0.00402783
$133$	−42.9862	−3.72738
$134$	9.46391	0.817557
$135$	1.27661	0.109873
$136$	−5.23034	−0.448498
$137$	21.0452	1.79802	0.899008	−	0.437933i	$-0.144289\pi$
$137$	21.0452	1.79802	0.899008	+	0.437933i	$0.144289\pi$
$138$	−1.00000	−0.0851257
$139$	6.58456	0.558495	0.279247	−	0.960219i	$-0.409915\pi$
$139$	6.58456	0.558495	0.279247	+	0.960219i	$0.409915\pi$
$140$	−6.67711	−0.564319
$141$	2.31118	0.194637
$142$	6.52548	0.547606
$143$	0.264908	0.0221527
$144$	1.00000	0.0833333
$145$	−1.27661	−0.106017
$146$	2.12389	0.175774
$147$	20.3564	1.67897
$148$	6.31965	0.519472
$149$	−4.04628	−0.331484	−0.165742	−	0.986169i	$-0.553002\pi$
$149$	−4.04628	−0.331484	−0.165742	+	0.986169i	$0.553002\pi$
$150$	3.37026	0.275181
$151$	14.8979	1.21238	0.606188	−	0.795321i	$-0.292698\pi$
$151$	14.8979	1.21238	0.606188	+	0.795321i	$0.292698\pi$
$152$	−8.21863	−0.666619
$153$	5.23034	0.422848
$154$	−0.242041	−0.0195042
$155$	−5.34002	−0.428921
$156$	−5.72448	−0.458326
$157$	3.96867	0.316734	0.158367	−	0.987380i	$-0.449377\pi$
$157$	3.96867	0.316734	0.158367	+	0.987380i	$0.449377\pi$
$158$	12.5846	1.00117
$159$	1.04304	0.0827183
$160$	−1.27661	−0.100925
$161$	−5.23034	−0.412208
$162$	−1.00000	−0.0785674
$163$	4.30112	0.336890	0.168445	−	0.985711i	$-0.446126\pi$
$163$	4.30112	0.336890	0.168445	+	0.985711i	$0.446126\pi$
$164$	7.07871	0.552754
$165$	−0.0590769	−0.00459913
$166$	14.8266	1.15077
$167$	−1.77923	−0.137681	−0.0688404	−	0.997628i	$-0.521930\pi$
$167$	−1.77923	−0.137681	−0.0688404	+	0.997628i	$0.521930\pi$
$168$	5.23034	0.403529
$169$	19.7697	1.52075
$170$	−6.67711	−0.512111
$171$	8.21863	0.628494
$172$	−11.1691	−0.851637
$173$	6.30471	0.479338	0.239669	−	0.970855i	$-0.422961\pi$
$173$	6.30471	0.479338	0.239669	+	0.970855i	$0.422961\pi$
$174$	1.00000	0.0758098
$175$	17.6276	1.33252
$176$	−0.0462763	−0.00348821
$177$	7.63193	0.573651
$178$	16.4059	1.22968
$179$	−10.9192	−0.816136	−0.408068	−	0.912952i	$-0.633797\pi$
$179$	−10.9192	−0.816136	−0.408068	+	0.912952i	$0.633797\pi$
$180$	1.27661	0.0951531
$181$	25.2937	1.88007	0.940035	−	0.341079i	$-0.110792\pi$
$181$	25.2937	1.88007	0.940035	+	0.341079i	$0.110792\pi$
$182$	−29.9410	−2.21937
$183$	−5.95372	−0.440112
$184$	−1.00000	−0.0737210
$185$	8.06774	0.593152
$186$	4.18296	0.306710
$187$	−0.242041	−0.0176998
$188$	2.31118	0.168560
$189$	−5.23034	−0.380451
$190$	−10.4920	−0.761170
$191$	−13.0915	−0.947268	−0.473634	−	0.880722i	$-0.657058\pi$
$191$	−13.0915	−0.947268	−0.473634	+	0.880722i	$0.657058\pi$
$192$	1.00000	0.0721688
$193$	−11.0196	−0.793210	−0.396605	−	0.917989i	$-0.629812\pi$
$193$	−11.0196	−0.793210	−0.396605	+	0.917989i	$0.629812\pi$
$194$	12.4059	0.890694
$195$	−7.30795	−0.523333
$196$	20.3564	1.45403
$197$	−21.6042	−1.53923	−0.769617	−	0.638505i	$-0.779553\pi$
$197$	−21.6042	−1.53923	−0.769617	+	0.638505i	$0.779553\pi$
$198$	0.0462763	0.00328871
$199$	9.40269	0.666539	0.333270	−	0.942832i	$-0.391848\pi$
$199$	9.40269	0.666539	0.333270	+	0.942832i	$0.391848\pi$
$200$	3.37026	0.238313
$201$	−9.46391	−0.667533
$202$	13.2660	0.933393
$203$	5.23034	0.367098
$204$	5.23034	0.366197
$205$	9.03676	0.631155
$206$	3.03457	0.211429
$207$	1.00000	0.0695048
$208$	−5.72448	−0.396922
$209$	−0.380328	−0.0263078
$210$	6.67711	0.460764
$211$	16.5576	1.13987	0.569935	−	0.821690i	$-0.306969\pi$
$211$	16.5576	1.13987	0.569935	+	0.821690i	$0.306969\pi$
$212$	1.04304	0.0716362
$213$	−6.52548	−0.447119
$214$	−1.85270	−0.126648
$215$	−14.2586	−0.972431
$216$	−1.00000	−0.0680414
$217$	21.8783	1.48520
$218$	1.37763	0.0933050
$219$	−2.12389	−0.143519
$220$	−0.0590769	−0.00398296
$221$	−29.9410	−2.01405
$222$	−6.31965	−0.424147
$223$	−9.48678	−0.635282	−0.317641	−	0.948211i	$-0.602891\pi$
$223$	−9.48678	−0.635282	−0.317641	+	0.948211i	$0.602891\pi$
$224$	5.23034	0.349467
$225$	−3.37026	−0.224684
$226$	13.0139	0.865672
$227$	−7.74356	−0.513958	−0.256979	−	0.966417i	$-0.582727\pi$
$227$	−7.74356	−0.513958	−0.256979	+	0.966417i	$0.582727\pi$
$228$	8.21863	0.544292
$229$	−15.3479	−1.01422	−0.507110	−	0.861881i	$-0.669286\pi$
$229$	−15.3479	−1.01422	−0.507110	+	0.861881i	$0.669286\pi$
$230$	−1.27661	−0.0841773
$231$	0.242041	0.0159251
$232$	1.00000	0.0656532
$233$	−1.69674	−0.111157	−0.0555786	−	0.998454i	$-0.517700\pi$
$233$	−1.69674	−0.111157	−0.0555786	+	0.998454i	$0.517700\pi$
$234$	5.72448	0.374221
$235$	2.95049	0.192468
$236$	7.63193	0.496796
$237$	−12.5846	−0.817455
$238$	27.3564	1.77325
$239$	0.315518	0.0204092	0.0102046	−	0.999948i	$-0.496752\pi$
$239$	0.315518	0.0204092	0.0102046	+	0.999948i	$0.496752\pi$
$240$	1.27661	0.0824050
$241$	28.0591	1.80745	0.903724	−	0.428116i	$-0.140823\pi$
$241$	28.0591	1.80745	0.903724	+	0.428116i	$0.140823\pi$
$242$	10.9979	0.706969
$243$	1.00000	0.0641500
$244$	−5.95372	−0.381148
$245$	25.9873	1.66026
$246$	−7.07871	−0.451322
$247$	−47.0474	−2.99356
$248$	4.18296	0.265618
$249$	−14.8266	−0.939598
$250$	10.6856	0.675815
$251$	−10.6308	−0.671012	−0.335506	−	0.942038i	$-0.608907\pi$
$251$	−10.6308	−0.671012	−0.335506	+	0.942038i	$0.608907\pi$
$252$	−5.23034	−0.329480
$253$	−0.0462763	−0.00290937
$254$	19.2660	1.20886
$255$	6.67711	0.418137
$256$	1.00000	0.0625000
$257$	−14.4059	−0.898617	−0.449309	−	0.893377i	$-0.648330\pi$
$257$	−14.4059	−0.898617	−0.449309	+	0.893377i	$0.648330\pi$
$258$	11.1691	0.695359
$259$	−33.0539	−2.05387
$260$	−7.30795	−0.453220
$261$	−1.00000	−0.0618984
$262$	15.7836	0.975112
$263$	6.92708	0.427142	0.213571	−	0.976928i	$-0.431491\pi$
$263$	6.92708	0.427142	0.213571	+	0.976928i	$0.431491\pi$
$264$	0.0462763	0.00284811
$265$	1.33156	0.0817968
$266$	42.9862	2.63565
$267$	−16.4059	−1.00403
$268$	−9.46391	−0.578100
$269$	16.8857	1.02954	0.514769	−	0.857329i	$-0.327878\pi$
$269$	16.8857	1.02954	0.514769	+	0.857329i	$0.327878\pi$
$270$	−1.27661	−0.0776922
$271$	−10.5511	−0.640933	−0.320466	−	0.947260i	$-0.603840\pi$
$271$	−10.5511	−0.640933	−0.320466	+	0.947260i	$0.603840\pi$
$272$	5.23034	0.317136
$273$	29.9410	1.81211
$274$	−21.0452	−1.27139
$275$	0.155963	0.00940493
$276$	1.00000	0.0601929
$277$	27.9075	1.67680	0.838400	−	0.545056i	$-0.183492\pi$
$277$	27.9075	1.67680	0.838400	+	0.545056i	$0.183492\pi$
$278$	−6.58456	−0.394916
$279$	−4.18296	−0.250427
$280$	6.67711	0.399034
$281$	−20.1261	−1.20062	−0.600311	−	0.799767i	$-0.704956\pi$
$281$	−20.1261	−1.20062	−0.600311	+	0.799767i	$0.704956\pi$
$282$	−2.31118	−0.137629
$283$	9.74302	0.579162	0.289581	−	0.957153i	$-0.406484\pi$
$283$	9.74302	0.579162	0.289581	+	0.957153i	$0.406484\pi$
$284$	−6.52548	−0.387216
$285$	10.4920	0.621493
$286$	−0.264908	−0.0156643
$287$	−37.0240	−2.18546
$288$	−1.00000	−0.0589256
$289$	10.3564	0.609201
$290$	1.27661	0.0749653
$291$	−12.4059	−0.727248
$292$	−2.12389	−0.124291
$293$	−4.27337	−0.249653	−0.124827	−	0.992179i	$-0.539837\pi$
$293$	−4.27337	−0.249653	−0.124827	+	0.992179i	$0.539837\pi$
$294$	−20.3564	−1.18721
$295$	9.74302	0.567260
$296$	−6.31965	−0.367322
$297$	−0.0462763	−0.00268522
$298$	4.04628	0.234394
$299$	−5.72448	−0.331055
$300$	−3.37026	−0.194582
$301$	58.4182	3.36717
$302$	−14.8979	−0.857280
$303$	−13.2660	−0.762112
$304$	8.21863	0.471371
$305$	−7.60060	−0.435209
$306$	−5.23034	−0.298998
$307$	0.157362	0.00898115	0.00449057	−	0.999990i	$-0.498571\pi$
$307$	0.157362	0.00898115	0.00449057	+	0.999990i	$0.498571\pi$
$308$	0.242041	0.0137915
$309$	−3.03457	−0.172631
$310$	5.34002	0.303293
$311$	27.3420	1.55042	0.775212	−	0.631702i	$-0.217643\pi$
$311$	27.3420	1.55042	0.775212	+	0.631702i	$0.217643\pi$
$312$	5.72448	0.324085
$313$	3.04523	0.172127	0.0860633	−	0.996290i	$-0.472571\pi$
$313$	3.04523	0.172127	0.0860633	+	0.996290i	$0.472571\pi$
$314$	−3.96867	−0.223965
$315$	−6.67711	−0.376213
$316$	−12.5846	−0.707937
$317$	29.8585	1.67702	0.838510	−	0.544886i	$-0.183427\pi$
$317$	29.8585	1.67702	0.838510	+	0.544886i	$0.183427\pi$
$318$	−1.04304	−0.0584907
$319$	0.0462763	0.00259097
$320$	1.27661	0.0713648
$321$	1.85270	0.103408
$322$	5.23034	0.291475
$323$	42.9862	2.39182
$324$	1.00000	0.0555556
$325$	19.2930	1.07018
$326$	−4.30112	−0.238217
$327$	−1.37763	−0.0761832
$328$	−7.07871	−0.390856
$329$	−12.0883	−0.666448
$330$	0.0590769	0.00325207
$331$	−5.37982	−0.295702	−0.147851	−	0.989010i	$-0.547236\pi$
$331$	−5.37982	−0.295702	−0.147851	+	0.989010i	$0.547236\pi$
$332$	−14.8266	−0.813715
$333$	6.31965	0.346315
$334$	1.77923	0.0973550
$335$	−12.0817	−0.660096
$336$	−5.23034	−0.285338
$337$	27.8007	1.51440	0.757200	−	0.653183i	$-0.226567\pi$
$337$	27.8007	1.51440	0.757200	+	0.653183i	$0.226567\pi$
$338$	−19.7697	−1.07533
$339$	−13.0139	−0.706818
$340$	6.67711	0.362117
$341$	0.193572	0.0104825
$342$	−8.21863	−0.444413
$343$	−69.8585	−3.77201
$344$	11.1691	0.602198
$345$	1.27661	0.0687305
$346$	−6.30471	−0.338943
$347$	3.08304	0.165506	0.0827531	−	0.996570i	$-0.473629\pi$
$347$	3.08304	0.165506	0.0827531	+	0.996570i	$0.473629\pi$
$348$	−1.00000	−0.0536056
$349$	−18.7441	−1.00335	−0.501675	−	0.865056i	$-0.667283\pi$
$349$	−18.7441	−1.00335	−0.501675	+	0.865056i	$0.667283\pi$
$350$	−17.6276	−0.942235
$351$	−5.72448	−0.305550
$352$	0.0462763	0.00246653
$353$	8.21863	0.437434	0.218717	−	0.975788i	$-0.429813\pi$
$353$	8.21863	0.437434	0.218717	+	0.975788i	$0.429813\pi$
$354$	−7.63193	−0.405633
$355$	−8.33051	−0.442138
$356$	−16.4059	−0.869512
$357$	−27.3564	−1.44786
$358$	10.9192	0.577095
$359$	17.9089	0.945195	0.472598	−	0.881278i	$-0.343316\pi$
$359$	17.9089	0.945195	0.472598	+	0.881278i	$0.343316\pi$
$360$	−1.27661	−0.0672834
$361$	48.5459	2.55505
$362$	−25.2937	−1.32941
$363$	−10.9979	−0.577238
$364$	29.9410	1.56933
$365$	−2.71138	−0.141920
$366$	5.95372	0.311206
$367$	−6.43726	−0.336022	−0.168011	−	0.985785i	$-0.553734\pi$
$367$	−6.43726	−0.336022	−0.168011	+	0.985785i	$0.553734\pi$
$368$	1.00000	0.0521286
$369$	7.07871	0.368503
$370$	−8.06774	−0.419422
$371$	−5.45544	−0.283232
$372$	−4.18296	−0.216877
$373$	7.85270	0.406598	0.203299	−	0.979117i	$-0.434834\pi$
$373$	7.85270	0.406598	0.203299	+	0.979117i	$0.434834\pi$
$374$	0.242041	0.0125156
$375$	−10.6856	−0.551801
$376$	−2.31118	−0.119190
$377$	5.72448	0.294826
$378$	5.23034	0.269019
$379$	21.4432	1.10146	0.550732	−	0.834682i	$-0.314348\pi$
$379$	21.4432	1.10146	0.550732	+	0.834682i	$0.314348\pi$
$380$	10.4920	0.538229
$381$	−19.2660	−0.987027
$382$	13.0915	0.669820
$383$	−30.9293	−1.58041	−0.790206	−	0.612841i	$-0.790026\pi$
$383$	−30.9293	−1.58041	−0.790206	+	0.612841i	$0.790026\pi$
$384$	−1.00000	−0.0510310
$385$	0.308992	0.0157477
$386$	11.0196	0.560884
$387$	−11.1691	−0.567758
$388$	−12.4059	−0.629816
$389$	5.88977	0.298623	0.149312	−	0.988790i	$-0.452294\pi$
$389$	5.88977	0.298623	0.149312	+	0.988790i	$0.452294\pi$
$390$	7.30795	0.370052
$391$	5.23034	0.264509
$392$	−20.3564	−1.02815
$393$	−15.7836	−0.796175
$394$	21.6042	1.08840
$395$	−16.0656	−0.808348
$396$	−0.0462763	−0.00232547
$397$	−9.54152	−0.478875	−0.239438	−	0.970912i	$-0.576963\pi$
$397$	−9.54152	−0.478875	−0.239438	+	0.970912i	$0.576963\pi$
$398$	−9.40269	−0.471314
$399$	−42.9862	−2.15200
$400$	−3.37026	−0.168513
$401$	34.2920	1.71246	0.856229	−	0.516596i	$-0.172801\pi$
$401$	34.2920	1.71246	0.856229	+	0.516596i	$0.172801\pi$
$402$	9.46391	0.472017
$403$	23.9453	1.19280
$404$	−13.2660	−0.660008
$405$	1.27661	0.0634354
$406$	−5.23034	−0.259577
$407$	−0.292450	−0.0144962
$408$	−5.23034	−0.258940
$409$	−14.4942	−0.716692	−0.358346	−	0.933589i	$-0.616659\pi$
$409$	−14.4942	−0.716692	−0.358346	+	0.933589i	$0.616659\pi$
$410$	−9.03676	−0.446294
$411$	21.0452	1.03808
$412$	−3.03457	−0.149503
$413$	−39.9176	−1.96421
$414$	−1.00000	−0.0491473
$415$	−18.9278	−0.929130
$416$	5.72448	0.280666
$417$	6.58456	0.322447
$418$	0.380328	0.0186024
$419$	−7.11292	−0.347489	−0.173745	−	0.984791i	$-0.555587\pi$
$419$	−7.11292	−0.347489	−0.173745	+	0.984791i	$0.555587\pi$
$420$	−6.67711	−0.325810
$421$	−14.1958	−0.691859	−0.345930	−	0.938260i	$-0.612436\pi$
$421$	−14.1958	−0.691859	−0.345930	+	0.938260i	$0.612436\pi$
$422$	−16.5576	−0.806009
$423$	2.31118	0.112374
$424$	−1.04304	−0.0506544
$425$	−17.6276	−0.855064
$426$	6.52548	0.316161
$427$	31.1400	1.50697
$428$	1.85270	0.0895538
$429$	0.264908	0.0127899
$430$	14.2586	0.687612
$431$	1.26022	0.0607027	0.0303513	−	0.999539i	$-0.490337\pi$
$431$	1.26022	0.0607027	0.0303513	+	0.999539i	$0.490337\pi$
$432$	1.00000	0.0481125
$433$	1.34411	0.0645936	0.0322968	−	0.999478i	$-0.489718\pi$
$433$	1.34411	0.0645936	0.0322968	+	0.999478i	$0.489718\pi$
$434$	−21.8783	−1.05019
$435$	−1.27661	−0.0612089
$436$	−1.37763	−0.0659766
$437$	8.21863	0.393150
$438$	2.12389	0.101483
$439$	−26.8893	−1.28335	−0.641677	−	0.766975i	$-0.721761\pi$
$439$	−26.8893	−1.28335	−0.641677	+	0.766975i	$0.721761\pi$
$440$	0.0590769	0.00281638
$441$	20.3564	0.969353
$442$	29.9410	1.42415
$443$	21.9410	1.04245	0.521224	−	0.853420i	$-0.325476\pi$
$443$	21.9410	1.04245	0.521224	+	0.853420i	$0.325476\pi$
$444$	6.31965	0.299917
$445$	−20.9440	−0.992841
$446$	9.48678	0.449212
$447$	−4.04628	−0.191382
$448$	−5.23034	−0.247110
$449$	11.2682	0.531779	0.265890	−	0.964004i	$-0.414334\pi$
$449$	11.2682	0.531779	0.265890	+	0.964004i	$0.414334\pi$
$450$	3.37026	0.158876
$451$	−0.327576	−0.0154250
$452$	−13.0139	−0.612122
$453$	14.8979	0.699966
$454$	7.74356	0.363423
$455$	38.2230	1.79192
$456$	−8.21863	−0.384873
$457$	19.1392	0.895296	0.447648	−	0.894210i	$-0.352262\pi$
$457$	19.1392	0.895296	0.447648	+	0.894210i	$0.352262\pi$
$458$	15.3479	0.717163
$459$	5.23034	0.244131
$460$	1.27661	0.0595224
$461$	−39.2739	−1.82917	−0.914585	−	0.404395i	$-0.867482\pi$
$461$	−39.2739	−1.82917	−0.914585	+	0.404395i	$0.867482\pi$
$462$	−0.242041	−0.0112607
$463$	−8.97390	−0.417052	−0.208526	−	0.978017i	$-0.566867\pi$
$463$	−8.97390	−0.417052	−0.208526	+	0.978017i	$0.566867\pi$
$464$	−1.00000	−0.0464238
$465$	−5.34002	−0.247638
$466$	1.69674	0.0786000
$467$	−23.5666	−1.09053	−0.545266	−	0.838263i	$-0.683571\pi$
$467$	−23.5666	−1.09053	−0.545266	+	0.838263i	$0.683571\pi$
$468$	−5.72448	−0.264614
$469$	49.4994	2.28567
$470$	−2.95049	−0.136096
$471$	3.96867	0.182866
$472$	−7.63193	−0.351288
$473$	0.516865	0.0237655
$474$	12.5846	0.578028
$475$	−27.6989	−1.27091
$476$	−27.3564	−1.25388
$477$	1.04304	0.0477575
$478$	−0.315518	−0.0144314
$479$	−9.70376	−0.443376	−0.221688	−	0.975118i	$-0.571157\pi$
$479$	−9.70376	−0.443376	−0.221688	+	0.975118i	$0.571157\pi$
$480$	−1.27661	−0.0582691
$481$	−36.1767	−1.64952
$482$	−28.0591	−1.27806
$483$	−5.23034	−0.237989
$484$	−10.9979	−0.499903
$485$	−15.8376	−0.719147
$486$	−1.00000	−0.0453609
$487$	−25.3856	−1.15033	−0.575165	−	0.818038i	$-0.695062\pi$
$487$	−25.3856	−1.15033	−0.575165	+	0.818038i	$0.695062\pi$
$488$	5.95372	0.269512
$489$	4.30112	0.194503
$490$	−25.9873	−1.17398
$491$	−8.39726	−0.378963	−0.189482	−	0.981884i	$-0.560681\pi$
$491$	−8.39726	−0.378963	−0.189482	+	0.981884i	$0.560681\pi$
$492$	7.07871	0.319133
$493$	−5.23034	−0.235562
$494$	47.0474	2.11676
$495$	−0.0590769	−0.00265531
$496$	−4.18296	−0.187821
$497$	34.1305	1.53096
$498$	14.8266	0.664396
$499$	8.00867	0.358517	0.179259	−	0.983802i	$-0.442630\pi$
$499$	8.00867	0.358517	0.179259	+	0.983802i	$0.442630\pi$
$500$	−10.6856	−0.477874
$501$	−1.77923	−0.0794900
$502$	10.6308	0.474477
$503$	−4.32812	−0.192981	−0.0964906	−	0.995334i	$-0.530762\pi$
$503$	−4.32812	−0.192981	−0.0964906	+	0.995334i	$0.530762\pi$
$504$	5.23034	0.232978
$505$	−16.9355	−0.753622
$506$	0.0462763	0.00205723
$507$	19.7697	0.878004
$508$	−19.2660	−0.854791
$509$	3.76015	0.166666	0.0833329	−	0.996522i	$-0.473444\pi$
$509$	3.76015	0.166666	0.0833329	+	0.996522i	$0.473444\pi$
$510$	−6.67711	−0.295667
$511$	11.1086	0.491417
$512$	−1.00000	−0.0441942
$513$	8.21863	0.362861
$514$	14.4059	0.635418
$515$	−3.87397	−0.170708
$516$	−11.1691	−0.491693
$517$	−0.106953	−0.00470379
$518$	33.0539	1.45230
$519$	6.30471	0.276746
$520$	7.30795	0.320475
$521$	16.1261	0.706496	0.353248	−	0.935530i	$-0.385077\pi$
$521$	16.1261	0.706496	0.353248	+	0.935530i	$0.385077\pi$
$522$	1.00000	0.0437688
$523$	4.47561	0.195705	0.0978525	−	0.995201i	$-0.468803\pi$
$523$	4.47561	0.195705	0.0978525	+	0.995201i	$0.468803\pi$
$524$	−15.7836	−0.689508
$525$	17.6276	0.769331
$526$	−6.92708	−0.302035
$527$	−21.8783	−0.953034
$528$	−0.0462763	−0.00201392
$529$	1.00000	0.0434783
$530$	−1.33156	−0.0578391
$531$	7.63193	0.331198
$532$	−42.9862	−1.86369
$533$	−40.5219	−1.75520
$534$	16.4059	0.709954
$535$	2.36519	0.102256
$536$	9.46391	0.408779
$537$	−10.9192	−0.471196
$538$	−16.8857	−0.727993
$539$	−0.942019	−0.0405756
$540$	1.27661	0.0549366
$541$	−27.2703	−1.17244	−0.586222	−	0.810151i	$-0.699385\pi$
$541$	−27.2703	−1.17244	−0.586222	+	0.810151i	$0.699385\pi$
$542$	10.5511	0.453208
$543$	25.2937	1.08546
$544$	−5.23034	−0.224249
$545$	−1.75870	−0.0753345
$546$	−29.9410	−1.28136
$547$	−16.6736	−0.712910	−0.356455	−	0.934312i	$-0.616015\pi$
$547$	−16.6736	−0.712910	−0.356455	+	0.934312i	$0.616015\pi$
$548$	21.0452	0.899008
$549$	−5.95372	−0.254099
$550$	−0.155963	−0.00665029
$551$	−8.21863	−0.350125
$552$	−1.00000	−0.0425628
$553$	65.8215	2.79901
$554$	−27.9075	−1.18568
$555$	8.06774	0.342457
$556$	6.58456	0.279247
$557$	−30.3989	−1.28804	−0.644022	−	0.765007i	$-0.722735\pi$
$557$	−30.3989	−1.28804	−0.644022	+	0.765007i	$0.722735\pi$
$558$	4.18296	0.177079
$559$	63.9374	2.70427
$560$	−6.67711	−0.282159
$561$	−0.242041	−0.0102190
$562$	20.1261	0.848967
$563$	−18.1054	−0.763052	−0.381526	−	0.924358i	$-0.624601\pi$
$563$	−18.1054	−0.763052	−0.381526	+	0.924358i	$0.624601\pi$
$564$	2.31118	0.0973184
$565$	−16.6137	−0.698944
$566$	−9.74302	−0.409530
$567$	−5.23034	−0.219653
$568$	6.52548	0.273803
$569$	17.4154	0.730093	0.365047	−	0.930989i	$-0.381053\pi$
$569$	17.4154	0.730093	0.365047	+	0.930989i	$0.381053\pi$
$570$	−10.4920	−0.439462
$571$	12.4669	0.521725	0.260863	−	0.965376i	$-0.415993\pi$
$571$	12.4669	0.521725	0.260863	+	0.965376i	$0.415993\pi$
$572$	0.264908	0.0110764
$573$	−13.0915	−0.546906
$574$	37.0240	1.54535
$575$	−3.37026	−0.140550
$576$	1.00000	0.0416667
$577$	−0.816482	−0.0339906	−0.0169953	−	0.999856i	$-0.505410\pi$
$577$	−0.816482	−0.0339906	−0.0169953	+	0.999856i	$0.505410\pi$
$578$	−10.3564	−0.430770
$579$	−11.0196	−0.457960
$580$	−1.27661	−0.0530084
$581$	77.5481	3.21724
$582$	12.4059	0.514242
$583$	−0.0482680	−0.00199905
$584$	2.12389	0.0878871
$585$	−7.30795	−0.302146
$586$	4.27337	0.176531
$587$	−35.5671	−1.46801	−0.734006	−	0.679143i	$-0.762352\pi$
$587$	−35.5671	−1.46801	−0.734006	+	0.679143i	$0.762352\pi$
$588$	20.3564	0.839484
$589$	−34.3782	−1.41653
$590$	−9.74302	−0.401114
$591$	−21.6042	−0.888678
$592$	6.31965	0.259736
$593$	0.405187	0.0166390	0.00831951	−	0.999965i	$-0.497352\pi$
$593$	0.405187	0.0166390	0.00831951	+	0.999965i	$0.497352\pi$
$594$	0.0462763	0.00189874
$595$	−34.9235	−1.43173
$596$	−4.04628	−0.165742
$597$	9.40269	0.384827
$598$	5.72448	0.234092
$599$	5.62186	0.229703	0.114852	−	0.993383i	$-0.463361\pi$
$599$	5.62186	0.229703	0.114852	+	0.993383i	$0.463361\pi$
$600$	3.37026	0.137590
$601$	−27.4656	−1.12034	−0.560172	−	0.828376i	$-0.689265\pi$
$601$	−27.4656	−1.12034	−0.560172	+	0.828376i	$0.689265\pi$
$602$	−58.4182	−2.38095
$603$	−9.46391	−0.385400
$604$	14.8979	0.606188
$605$	−14.0400	−0.570807
$606$	13.2660	0.538895
$607$	−31.6554	−1.28485	−0.642426	−	0.766347i	$-0.722072\pi$
$607$	−31.6554	−1.28485	−0.642426	+	0.766347i	$0.722072\pi$
$608$	−8.21863	−0.333310
$609$	5.23034	0.211944
$610$	7.60060	0.307739
$611$	−13.2303	−0.535242
$612$	5.23034	0.211424
$613$	12.9448	0.522834	0.261417	−	0.965226i	$-0.415810\pi$
$613$	12.9448	0.522834	0.261417	+	0.965226i	$0.415810\pi$
$614$	−0.157362	−0.00635063
$615$	9.03676	0.364397
$616$	−0.242041	−0.00975209
$617$	27.1385	1.09256	0.546278	−	0.837604i	$-0.316044\pi$
$617$	27.1385	1.09256	0.546278	+	0.837604i	$0.316044\pi$
$618$	3.03457	0.122068
$619$	10.2122	0.410461	0.205231	−	0.978714i	$-0.434206\pi$
$619$	10.2122	0.410461	0.205231	+	0.978714i	$0.434206\pi$
$620$	−5.34002	−0.214460
$621$	1.00000	0.0401286
$622$	−27.3420	−1.09631
$623$	85.8085	3.43785
$624$	−5.72448	−0.229163
$625$	3.20996	0.128399
$626$	−3.04523	−0.121712
$627$	−0.380328	−0.0151888
$628$	3.96867	0.158367
$629$	33.0539	1.31795
$630$	6.67711	0.266022
$631$	−21.2532	−0.846077	−0.423038	−	0.906112i	$-0.639036\pi$
$631$	−21.2532	−0.846077	−0.423038	+	0.906112i	$0.639036\pi$
$632$	12.5846	0.500587
$633$	16.5576	0.658104
$634$	−29.8585	−1.18583
$635$	−24.5952	−0.976031
$636$	1.04304	0.0413592
$637$	−116.530	−4.61709
$638$	−0.0462763	−0.00183210
$639$	−6.52548	−0.258144
$640$	−1.27661	−0.0504625
$641$	25.5181	1.00790	0.503952	−	0.863731i	$-0.331879\pi$
$641$	25.5181	1.00790	0.503952	+	0.863731i	$0.331879\pi$
$642$	−1.85270	−0.0731204
$643$	1.66321	0.0655908	0.0327954	−	0.999462i	$-0.489559\pi$
$643$	1.66321	0.0655908	0.0327954	+	0.999462i	$0.489559\pi$
$644$	−5.23034	−0.206104
$645$	−14.2586	−0.561433
$646$	−42.9862	−1.69127
$647$	−3.16333	−0.124363	−0.0621817	−	0.998065i	$-0.519806\pi$
$647$	−3.16333	−0.124363	−0.0621817	+	0.998065i	$0.519806\pi$
$648$	−1.00000	−0.0392837
$649$	−0.353177	−0.0138634
$650$	−19.2930	−0.756734
$651$	21.8783	0.857478
$652$	4.30112	0.168445
$653$	8.19089	0.320534	0.160267	−	0.987074i	$-0.448764\pi$
$653$	8.19089	0.320534	0.160267	+	0.987074i	$0.448764\pi$
$654$	1.37763	0.0538697
$655$	−20.1495	−0.787306
$656$	7.07871	0.276377
$657$	−2.12389	−0.0828607
$658$	12.0883	0.471250
$659$	36.5432	1.42352	0.711761	−	0.702422i	$-0.247898\pi$
$659$	36.5432	1.42352	0.711761	+	0.702422i	$0.247898\pi$
$660$	−0.0590769	−0.00229956
$661$	−15.5894	−0.606359	−0.303180	−	0.952933i	$-0.598048\pi$
$661$	−15.5894	−0.606359	−0.303180	+	0.952933i	$0.598048\pi$
$662$	5.37982	0.209093
$663$	−29.9410	−1.16281
$664$	14.8266	0.575384
$665$	−54.8767	−2.12803
$666$	−6.31965	−0.244882
$667$	−1.00000	−0.0387202
$668$	−1.77923	−0.0688404
$669$	−9.48678	−0.366780
$670$	12.0817	0.466758
$671$	0.275516	0.0106362
$672$	5.23034	0.201765
$673$	−36.4651	−1.40562	−0.702812	−	0.711375i	$-0.748073\pi$
$673$	−36.4651	−1.40562	−0.702812	+	0.711375i	$0.748073\pi$
$674$	−27.8007	−1.07084
$675$	−3.37026	−0.129721
$676$	19.7697	0.760374
$677$	31.8149	1.22275	0.611373	−	0.791343i	$-0.290618\pi$
$677$	31.8149	1.22275	0.611373	+	0.791343i	$0.290618\pi$
$678$	13.0139	0.499796
$679$	64.8872	2.49014
$680$	−6.67711	−0.256056
$681$	−7.74356	−0.296734
$682$	−0.193572	−0.00741226
$683$	−49.1202	−1.87953	−0.939765	−	0.341820i	$-0.888957\pi$
$683$	−49.1202	−1.87953	−0.939765	+	0.341820i	$0.888957\pi$
$684$	8.21863	0.314247
$685$	26.8666	1.02652
$686$	69.8585	2.66721
$687$	−15.3479	−0.585561
$688$	−11.1691	−0.425819
$689$	−5.97086	−0.227472
$690$	−1.27661	−0.0485998
$691$	28.7776	1.09475	0.547376	−	0.836887i	$-0.315627\pi$
$691$	28.7776	1.09475	0.547376	+	0.836887i	$0.315627\pi$
$692$	6.30471	0.239669
$693$	0.242041	0.00919436
$694$	−3.08304	−0.117031
$695$	8.40593	0.318855
$696$	1.00000	0.0379049
$697$	37.0240	1.40238
$698$	18.7441	0.709475
$699$	−1.69674	−0.0641766
$700$	17.6276	0.666261
$701$	12.1388	0.458477	0.229239	−	0.973370i	$-0.426376\pi$
$701$	12.1388	0.458477	0.229239	+	0.973370i	$0.426376\pi$
$702$	5.72448	0.216057
$703$	51.9389	1.95891
$704$	−0.0462763	−0.00174410
$705$	2.95049	0.111122
$706$	−8.21863	−0.309312
$707$	69.3857	2.60952
$708$	7.63193	0.286826
$709$	−15.2887	−0.574180	−0.287090	−	0.957904i	$-0.592688\pi$
$709$	−15.2887	−0.574180	−0.287090	+	0.957904i	$0.592688\pi$
$710$	8.33051	0.312638
$711$	−12.5846	−0.471958
$712$	16.4059	0.614838
$713$	−4.18296	−0.156653
$714$	27.3564	1.02379
$715$	0.338185	0.0126474
$716$	−10.9192	−0.408068
$717$	0.315518	0.0117832
$718$	−17.9089	−0.668354
$719$	24.3170	0.906870	0.453435	−	0.891289i	$-0.350198\pi$
$719$	24.3170	0.906870	0.453435	+	0.891289i	$0.350198\pi$
$720$	1.27661	0.0475765
$721$	15.8718	0.591098
$722$	−48.5459	−1.80669
$723$	28.0591	1.04353
$724$	25.2937	0.940035
$725$	3.37026	0.125168
$726$	10.9979	0.408169
$727$	32.5489	1.20717	0.603587	−	0.797297i	$-0.293738\pi$
$727$	32.5489	1.20717	0.603587	+	0.797297i	$0.293738\pi$
$728$	−29.9410	−1.10969
$729$	1.00000	0.0370370
$730$	2.71138	0.100353
$731$	−58.4182	−2.16068
$732$	−5.95372	−0.220056
$733$	−8.97555	−0.331519	−0.165760	−	0.986166i	$-0.553008\pi$
$733$	−8.97555	−0.331519	−0.165760	+	0.986166i	$0.553008\pi$
$734$	6.43726	0.237604
$735$	25.9873	0.958554
$736$	−1.00000	−0.0368605
$737$	0.437955	0.0161323
$738$	−7.07871	−0.260571
$739$	−50.5868	−1.86087	−0.930433	−	0.366463i	$-0.880569\pi$
$739$	−50.5868	−1.86087	−0.930433	+	0.366463i	$0.880569\pi$
$740$	8.06774	0.296576
$741$	−47.0474	−1.72833
$742$	5.45544	0.200276
$743$	−20.9064	−0.766982	−0.383491	−	0.923545i	$-0.625278\pi$
$743$	−20.9064	−0.766982	−0.383491	+	0.923545i	$0.625278\pi$
$744$	4.18296	0.153355
$745$	−5.16553	−0.189250
$746$	−7.85270	−0.287508
$747$	−14.8266	−0.542477
$748$	−0.242041	−0.00884988
$749$	−9.69027	−0.354075
$750$	10.6856	0.390182
$751$	−29.4760	−1.07560	−0.537798	−	0.843074i	$-0.680744\pi$
$751$	−29.4760	−1.07560	−0.537798	+	0.843074i	$0.680744\pi$
$752$	2.31118	0.0842802
$753$	−10.6308	−0.387409
$754$	−5.72448	−0.208473
$755$	19.0189	0.692168
$756$	−5.23034	−0.190225
$757$	−6.71118	−0.243922	−0.121961	−	0.992535i	$-0.538918\pi$
$757$	−6.71118	−0.243922	−0.121961	+	0.992535i	$0.538918\pi$
$758$	−21.4432	−0.778853
$759$	−0.0462763	−0.00167972
$760$	−10.4920	−0.380585
$761$	−21.3564	−0.774169	−0.387085	−	0.922044i	$-0.626518\pi$
$761$	−21.3564	−0.774169	−0.387085	+	0.922044i	$0.626518\pi$
$762$	19.2660	0.697934
$763$	7.20548	0.260856
$764$	−13.0915	−0.473634
$765$	6.67711	0.241411
$766$	30.9293	1.11752
$767$	−43.6889	−1.57751
$768$	1.00000	0.0360844
$769$	−11.9159	−0.429699	−0.214849	−	0.976647i	$-0.568926\pi$
$769$	−11.9159	−0.429699	−0.214849	+	0.976647i	$0.568926\pi$
$770$	−0.308992	−0.0111353
$771$	−14.4059	−0.518817
$772$	−11.0196	−0.396605
$773$	−8.94256	−0.321642	−0.160821	−	0.986984i	$-0.551414\pi$
$773$	−8.94256	−0.321642	−0.160821	+	0.986984i	$0.551414\pi$
$774$	11.1691	0.401466
$775$	14.0977	0.506404
$776$	12.4059	0.445347
$777$	−33.0539	−1.18580
$778$	−5.88977	−0.211159
$779$	58.1773	2.08442
$780$	−7.30795	−0.261666
$781$	0.301975	0.0108055
$782$	−5.23034	−0.187036
$783$	−1.00000	−0.0357371
$784$	20.3564	0.727015
$785$	5.06645	0.180829
$786$	15.7836	0.562981
$787$	18.6074	0.663283	0.331642	−	0.943405i	$-0.392398\pi$
$787$	18.6074	0.663283	0.331642	+	0.943405i	$0.392398\pi$
$788$	−21.6042	−0.769617
$789$	6.92708	0.246610
$790$	16.0656	0.571588
$791$	68.0671	2.42019
$792$	0.0462763	0.00164436
$793$	34.0820	1.21029
$794$	9.54152	0.338616
$795$	1.33156	0.0472254
$796$	9.40269	0.333270
$797$	−17.3600	−0.614921	−0.307461	−	0.951561i	$-0.599479\pi$
$797$	−17.3600	−0.614921	−0.307461	+	0.951561i	$0.599479\pi$
$798$	42.9862	1.52170
$799$	12.0883	0.427652
$800$	3.37026	0.119157
$801$	−16.4059	−0.579675
$802$	−34.2920	−1.21089
$803$	0.0982856	0.00346842
$804$	−9.46391	−0.333766
$805$	−6.67711	−0.235337
$806$	−23.9453	−0.843437
$807$	16.8857	0.594404
$808$	13.2660	0.466696
$809$	35.7667	1.25749	0.628745	−	0.777612i	$-0.283569\pi$
$809$	35.7667	1.25749	0.628745	+	0.777612i	$0.283569\pi$
$810$	−1.27661	−0.0448556
$811$	34.2857	1.20393	0.601967	−	0.798521i	$-0.294384\pi$
$811$	34.2857	1.20393	0.601967	+	0.798521i	$0.294384\pi$
$812$	5.23034	0.183549
$813$	−10.5511	−0.370043
$814$	0.292450	0.0102504
$815$	5.49086	0.192336
$816$	5.23034	0.183098
$817$	−91.7949	−3.21150
$818$	14.4942	0.506778
$819$	29.9410	1.04622
$820$	9.03676	0.315577
$821$	4.50067	0.157075	0.0785373	−	0.996911i	$-0.474975\pi$
$821$	4.50067	0.157075	0.0785373	+	0.996911i	$0.474975\pi$
$822$	−21.0452	−0.734037
$823$	−8.71133	−0.303658	−0.151829	−	0.988407i	$-0.548516\pi$
$823$	−8.71133	−0.303658	−0.151829	+	0.988407i	$0.548516\pi$
$824$	3.03457	0.105714
$825$	0.155963	0.00542994
$826$	39.9176	1.38891
$827$	−50.5683	−1.75843	−0.879215	−	0.476424i	$-0.841933\pi$
$827$	−50.5683	−1.75843	−0.879215	+	0.476424i	$0.841933\pi$
$828$	1.00000	0.0347524
$829$	30.3973	1.05574	0.527870	−	0.849325i	$-0.322991\pi$
$829$	30.3973	1.05574	0.527870	+	0.849325i	$0.322991\pi$
$830$	18.9278	0.656994
$831$	27.9075	0.968100
$832$	−5.72448	−0.198461
$833$	106.471	3.68900
$834$	−6.58456	−0.228005
$835$	−2.27138	−0.0786045
$836$	−0.380328	−0.0131539
$837$	−4.18296	−0.144584
$838$	7.11292	0.245712
$839$	35.0937	1.21157	0.605785	−	0.795629i	$-0.292859\pi$
$839$	35.0937	1.21157	0.605785	+	0.795629i	$0.292859\pi$
$840$	6.67711	0.230382
$841$	1.00000	0.0344828
$842$	14.1958	0.489218
$843$	−20.1261	−0.693179
$844$	16.5576	0.569935
$845$	25.2383	0.868223
$846$	−2.31118	−0.0794601
$847$	57.5225	1.97650
$848$	1.04304	0.0358181
$849$	9.74302	0.334379
$850$	17.6276	0.604622
$851$	6.31965	0.216635
$852$	−6.52548	−0.223559
$853$	−19.1208	−0.654685	−0.327343	−	0.944906i	$-0.606153\pi$
$853$	−19.1208	−0.654685	−0.327343	+	0.944906i	$0.606153\pi$
$854$	−31.1400	−1.06559
$855$	10.4920	0.358819
$856$	−1.85270	−0.0633241
$857$	−47.9424	−1.63768	−0.818841	−	0.574020i	$-0.805383\pi$
$857$	−47.9424	−1.63768	−0.818841	+	0.574020i	$0.805383\pi$
$858$	−0.264908	−0.00904380
$859$	6.54456	0.223297	0.111649	−	0.993748i	$-0.464387\pi$
$859$	6.54456	0.223297	0.111649	+	0.993748i	$0.464387\pi$
$860$	−14.2586	−0.486215
$861$	−37.0240	−1.26178
$862$	−1.26022	−0.0429233
$863$	43.7670	1.48985	0.744923	−	0.667150i	$-0.232486\pi$
$863$	43.7670	1.48985	0.744923	+	0.667150i	$0.232486\pi$
$864$	−1.00000	−0.0340207
$865$	8.04867	0.273663
$866$	−1.34411	−0.0456746
$867$	10.3564	0.351722
$868$	21.8783	0.742598
$869$	0.582367	0.0197554
$870$	1.27661	0.0432812
$871$	54.1760	1.83568
$872$	1.37763	0.0466525
$873$	−12.4059	−0.419877
$874$	−8.21863	−0.277999
$875$	55.8892	1.88940
$876$	−2.12389	−0.0717595
$877$	−3.52786	−0.119127	−0.0595637	−	0.998225i	$-0.518971\pi$
$877$	−3.52786	−0.119127	−0.0595637	+	0.998225i	$0.518971\pi$
$878$	26.8893	0.907469
$879$	−4.27337	−0.144137
$880$	−0.0590769	−0.00199148
$881$	53.9867	1.81886	0.909429	−	0.415859i	$-0.136519\pi$
$881$	53.9867	1.81886	0.909429	+	0.415859i	$0.136519\pi$
$882$	−20.3564	−0.685436
$883$	9.13340	0.307363	0.153682	−	0.988120i	$-0.450887\pi$
$883$	9.13340	0.307363	0.153682	+	0.988120i	$0.450887\pi$
$884$	−29.9410	−1.00702
$885$	9.74302	0.327508
$886$	−21.9410	−0.737121
$887$	−53.6947	−1.80289	−0.901445	−	0.432894i	$-0.857493\pi$
$887$	−53.6947	−1.80289	−0.901445	+	0.432894i	$0.857493\pi$
$888$	−6.31965	−0.212074
$889$	100.768	3.37964
$890$	20.9440	0.702045
$891$	−0.0462763	−0.00155031
$892$	−9.48678	−0.317641
$893$	18.9948	0.635636
$894$	4.04628	0.135328
$895$	−13.9395	−0.465947
$896$	5.23034	0.174733
$897$	−5.72448	−0.191135
$898$	−11.2682	−0.376025
$899$	4.18296	0.139510
$900$	−3.37026	−0.112342
$901$	5.45544	0.181747
$902$	0.327576	0.0109071
$903$	58.4182	1.94404
$904$	13.0139	0.432836
$905$	32.2903	1.07337
$906$	−14.8979	−0.494951
$907$	−36.1161	−1.19921	−0.599607	−	0.800295i	$-0.704676\pi$
$907$	−36.1161	−1.19921	−0.599607	+	0.800295i	$0.704676\pi$
$908$	−7.74356	−0.256979
$909$	−13.2660	−0.440006
$910$	−38.2230	−1.26708
$911$	−55.3685	−1.83444	−0.917221	−	0.398380i	$-0.869573\pi$
$911$	−55.3685	−1.83444	−0.917221	+	0.398380i	$0.869573\pi$
$912$	8.21863	0.272146
$913$	0.686120	0.0227073
$914$	−19.1392	−0.633070
$915$	−7.60060	−0.251268
$916$	−15.3479	−0.507110
$917$	82.5533	2.72615
$918$	−5.23034	−0.172627
$919$	24.5938	0.811274	0.405637	−	0.914034i	$-0.367050\pi$
$919$	24.5938	0.811274	0.405637	+	0.914034i	$0.367050\pi$
$920$	−1.27661	−0.0420887
$921$	0.157362	0.00518527
$922$	39.2739	1.29342
$923$	37.3550	1.22956
$924$	0.242041	0.00796255
$925$	−21.2989	−0.700303
$926$	8.97390	0.294900
$927$	−3.03457	−0.0996684
$928$	1.00000	0.0328266
$929$	−29.1944	−0.957836	−0.478918	−	0.877860i	$-0.658971\pi$
$929$	−29.1944	−0.957836	−0.478918	+	0.877860i	$0.658971\pi$
$930$	5.34002	0.175106
$931$	167.302	5.48310
$932$	−1.69674	−0.0555786
$933$	27.3420	0.895137
$934$	23.5666	0.771122
$935$	−0.308992	−0.0101051
$936$	5.72448	0.187111
$937$	28.2143	0.921723	0.460861	−	0.887472i	$-0.347541\pi$
$937$	28.2143	0.921723	0.460861	+	0.887472i	$0.347541\pi$
$938$	−49.4994	−1.61621
$939$	3.04523	0.0993774
$940$	2.95049	0.0962342
$941$	−19.2331	−0.626980	−0.313490	−	0.949591i	$-0.601498\pi$
$941$	−19.2331	−0.626980	−0.313490	+	0.949591i	$0.601498\pi$
$942$	−3.96867	−0.129306
$943$	7.07871	0.230514
$944$	7.63193	0.248398
$945$	−6.67711	−0.217206
$946$	−0.516865	−0.0168047
$947$	2.51322	0.0816688	0.0408344	−	0.999166i	$-0.486998\pi$
$947$	2.51322	0.0816688	0.0408344	+	0.999166i	$0.486998\pi$
$948$	−12.5846	−0.408727
$949$	12.1582	0.394670
$950$	27.6989	0.898672
$951$	29.8585	0.968228
$952$	27.3564	0.886627
$953$	41.9061	1.35747	0.678736	−	0.734382i	$-0.262528\pi$
$953$	41.9061	1.35747	0.678736	+	0.734382i	$0.262528\pi$
$954$	−1.04304	−0.0337696
$955$	−16.7128	−0.540813
$956$	0.315518	0.0102046
$957$	0.0462763	0.00149590
$958$	9.70376	0.313514
$959$	−110.074	−3.55446
$960$	1.27661	0.0412025
$961$	−13.5028	−0.435575
$962$	36.1767	1.16638
$963$	1.85270	0.0597026
$964$	28.0591	0.903724
$965$	−14.0678	−0.452858
$966$	5.23034	0.168283
$967$	−26.6116	−0.855772	−0.427886	−	0.903833i	$-0.640741\pi$
$967$	−26.6116	−0.855772	−0.427886	+	0.903833i	$0.640741\pi$
$968$	10.9979	0.353485
$969$	42.9862	1.38092
$970$	15.8376	0.508513
$971$	3.71168	0.119114	0.0595568	−	0.998225i	$-0.481031\pi$
$971$	3.71168	0.119114	0.0595568	+	0.998225i	$0.481031\pi$
$972$	1.00000	0.0320750
$973$	−34.4395	−1.10408
$974$	25.3856	0.813406
$975$	19.2930	0.617871
$976$	−5.95372	−0.190574
$977$	25.7520	0.823880	0.411940	−	0.911211i	$-0.364851\pi$
$977$	25.7520	0.823880	0.411940	+	0.911211i	$0.364851\pi$
$978$	−4.30112	−0.137535
$979$	0.759205	0.0242643
$980$	25.9873	0.830132
$981$	−1.37763	−0.0439844
$982$	8.39726	0.267967
$983$	11.1992	0.357199	0.178600	−	0.983922i	$-0.442843\pi$
$983$	11.1992	0.357199	0.178600	+	0.983922i	$0.442843\pi$
$984$	−7.07871	−0.225661
$985$	−27.5802	−0.878777
$986$	5.23034	0.166568
$987$	−12.0883	−0.384774
$988$	−47.0474	−1.49678
$989$	−11.1691	−0.355157
$990$	0.0590769	0.00187759
$991$	−37.7659	−1.19967	−0.599836	−	0.800123i	$-0.704768\pi$
$991$	−37.7659	−1.19967	−0.599836	+	0.800123i	$0.704768\pi$
$992$	4.18296	0.132809
$993$	−5.37982	−0.170724
$994$	−34.1305	−1.08255
$995$	12.0036	0.380539
$996$	−14.8266	−0.469799
$997$	15.1634	0.480229	0.240115	−	0.970745i	$-0.422815\pi$
$997$	15.1634	0.480229	0.240115	+	0.970745i	$0.422815\pi$
$998$	−8.00867	−0.253510
$999$	6.31965	0.199945

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.27661 q^{5} -1.00000 q^{6} -5.23034 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.27661 q^{5} -1.00000 q^{6} -5.23034 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.27661 q^{10} -0.0462763 q^{11} +1.00000 q^{12} -5.72448 q^{13} +5.23034 q^{14} +1.27661 q^{15} +1.00000 q^{16} +5.23034 q^{17} -1.00000 q^{18} +8.21863 q^{19} +1.27661 q^{20} -5.23034 q^{21} +0.0462763 q^{22} +1.00000 q^{23} -1.00000 q^{24} -3.37026 q^{25} +5.72448 q^{26} +1.00000 q^{27} -5.23034 q^{28} -1.00000 q^{29} -1.27661 q^{30} -4.18296 q^{31} -1.00000 q^{32} -0.0462763 q^{33} -5.23034 q^{34} -6.67711 q^{35} +1.00000 q^{36} +6.31965 q^{37} -8.21863 q^{38} -5.72448 q^{39} -1.27661 q^{40} +7.07871 q^{41} +5.23034 q^{42} -11.1691 q^{43} -0.0462763 q^{44} +1.27661 q^{45} -1.00000 q^{46} +2.31118 q^{47} +1.00000 q^{48} +20.3564 q^{49} +3.37026 q^{50} +5.23034 q^{51} -5.72448 q^{52} +1.04304 q^{53} -1.00000 q^{54} -0.0590769 q^{55} +5.23034 q^{56} +8.21863 q^{57} +1.00000 q^{58} +7.63193 q^{59} +1.27661 q^{60} -5.95372 q^{61} +4.18296 q^{62} -5.23034 q^{63} +1.00000 q^{64} -7.30795 q^{65} +0.0462763 q^{66} -9.46391 q^{67} +5.23034 q^{68} +1.00000 q^{69} +6.67711 q^{70} -6.52548 q^{71} -1.00000 q^{72} -2.12389 q^{73} -6.31965 q^{74} -3.37026 q^{75} +8.21863 q^{76} +0.242041 q^{77} +5.72448 q^{78} -12.5846 q^{79} +1.27661 q^{80} +1.00000 q^{81} -7.07871 q^{82} -14.8266 q^{83} -5.23034 q^{84} +6.67711 q^{85} +11.1691 q^{86} -1.00000 q^{87} +0.0462763 q^{88} -16.4059 q^{89} -1.27661 q^{90} +29.9410 q^{91} +1.00000 q^{92} -4.18296 q^{93} -2.31118 q^{94} +10.4920 q^{95} -1.00000 q^{96} -12.4059 q^{97} -20.3564 q^{98} -0.0462763 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} + 3 q^{10} - 11 q^{11} + 4 q^{12} - q^{13} + 2 q^{14} - 3 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 8 q^{19} - 3 q^{20} - 2 q^{21} + 11 q^{22} + 4 q^{23} - 4 q^{24} + 3 q^{25} + q^{26} + 4 q^{27} - 2 q^{28} - 4 q^{29} + 3 q^{30} - 17 q^{31} - 4 q^{32} - 11 q^{33} - 2 q^{34} - 24 q^{35} + 4 q^{36} + 15 q^{37} - 8 q^{38} - q^{39} + 3 q^{40} + q^{41} + 2 q^{42} + 4 q^{43} - 11 q^{44} - 3 q^{45} - 4 q^{46} + 6 q^{47} + 4 q^{48} + 16 q^{49} - 3 q^{50} + 2 q^{51} - q^{52} + 2 q^{53} - 4 q^{54} + 13 q^{55} + 2 q^{56} + 8 q^{57} + 4 q^{58} - 13 q^{59} - 3 q^{60} - 13 q^{61} + 17 q^{62} - 2 q^{63} + 4 q^{64} - 13 q^{65} + 11 q^{66} - 13 q^{67} + 2 q^{68} + 4 q^{69} + 24 q^{70} - 15 q^{71} - 4 q^{72} - 22 q^{73} - 15 q^{74} + 3 q^{75} + 8 q^{76} - 12 q^{77} + q^{78} - 26 q^{79} - 3 q^{80} + 4 q^{81} - q^{82} - 22 q^{83} - 2 q^{84} + 24 q^{85} - 4 q^{86} - 4 q^{87} + 11 q^{88} - 24 q^{89} + 3 q^{90} + 30 q^{91} + 4 q^{92} - 17 q^{93} - 6 q^{94} - 4 q^{95} - 4 q^{96} - 8 q^{97} - 16 q^{98} - 11 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 + 3 * q^10 - 11 * q^11 + 4 * q^12 - q^13 + 2 * q^14 - 3 * q^15 + 4 * q^16 + 2 * q^17 - 4 * q^18 + 8 * q^19 - 3 * q^20 - 2 * q^21 + 11 * q^22 + 4 * q^23 - 4 * q^24 + 3 * q^25 + q^26 + 4 * q^27 - 2 * q^28 - 4 * q^29 + 3 * q^30 - 17 * q^31 - 4 * q^32 - 11 * q^33 - 2 * q^34 - 24 * q^35 + 4 * q^36 + 15 * q^37 - 8 * q^38 - q^39 + 3 * q^40 + q^41 + 2 * q^42 + 4 * q^43 - 11 * q^44 - 3 * q^45 - 4 * q^46 + 6 * q^47 + 4 * q^48 + 16 * q^49 - 3 * q^50 + 2 * q^51 - q^52 + 2 * q^53 - 4 * q^54 + 13 * q^55 + 2 * q^56 + 8 * q^57 + 4 * q^58 - 13 * q^59 - 3 * q^60 - 13 * q^61 + 17 * q^62 - 2 * q^63 + 4 * q^64 - 13 * q^65 + 11 * q^66 - 13 * q^67 + 2 * q^68 + 4 * q^69 + 24 * q^70 - 15 * q^71 - 4 * q^72 - 22 * q^73 - 15 * q^74 + 3 * q^75 + 8 * q^76 - 12 * q^77 + q^78 - 26 * q^79 - 3 * q^80 + 4 * q^81 - q^82 - 22 * q^83 - 2 * q^84 + 24 * q^85 - 4 * q^86 - 4 * q^87 + 11 * q^88 - 24 * q^89 + 3 * q^90 + 30 * q^91 + 4 * q^92 - 17 * q^93 - 6 * q^94 - 4 * q^95 - 4 * q^96 - 8 * q^97 - 16 * q^98 - 11 * q^99

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