LMFDB - Embedded newform 4002.2.a.bi.1.4

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			1.4
Root			$1.98746$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.881455 q^{5} -1.00000 q^{6} +0.253486 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.881455 q^{5} -1.00000 q^{6} +0.253486 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.881455 q^{10} -3.18357 q^{11} +1.00000 q^{12} +2.13413 q^{13} -0.253486 q^{14} -0.881455 q^{15} +1.00000 q^{16} -5.96126 q^{17} -1.00000 q^{18} +3.48434 q^{19} -0.881455 q^{20} +0.253486 q^{21} +3.18357 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.22304 q^{25} -2.13413 q^{26} +1.00000 q^{27} +0.253486 q^{28} +1.00000 q^{29} +0.881455 q^{30} +10.8272 q^{31} -1.00000 q^{32} -3.18357 q^{33} +5.96126 q^{34} -0.223436 q^{35} +1.00000 q^{36} +9.11932 q^{37} -3.48434 q^{38} +2.13413 q^{39} +0.881455 q^{40} -4.89141 q^{41} -0.253486 q^{42} -8.97041 q^{43} -3.18357 q^{44} -0.881455 q^{45} +1.00000 q^{46} -0.153477 q^{47} +1.00000 q^{48} -6.93575 q^{49} +4.22304 q^{50} -5.96126 q^{51} +2.13413 q^{52} -10.3575 q^{53} -1.00000 q^{54} +2.80618 q^{55} -0.253486 q^{56} +3.48434 q^{57} -1.00000 q^{58} +8.73661 q^{59} -0.881455 q^{60} -11.0288 q^{61} -10.8272 q^{62} +0.253486 q^{63} +1.00000 q^{64} -1.88114 q^{65} +3.18357 q^{66} -1.61142 q^{67} -5.96126 q^{68} -1.00000 q^{69} +0.223436 q^{70} +14.9152 q^{71} -1.00000 q^{72} -6.69346 q^{73} -9.11932 q^{74} -4.22304 q^{75} +3.48434 q^{76} -0.806990 q^{77} -2.13413 q^{78} -0.899667 q^{79} -0.881455 q^{80} +1.00000 q^{81} +4.89141 q^{82} -17.0301 q^{83} +0.253486 q^{84} +5.25458 q^{85} +8.97041 q^{86} +1.00000 q^{87} +3.18357 q^{88} -1.38034 q^{89} +0.881455 q^{90} +0.540972 q^{91} -1.00000 q^{92} +10.8272 q^{93} +0.153477 q^{94} -3.07129 q^{95} -1.00000 q^{96} -8.28427 q^{97} +6.93575 q^{98} -3.18357 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.881455	−0.394198	−0.197099	−	0.980384i	$-0.563152\pi$
$5$	−0.881455	−0.394198	−0.197099	+	0.980384i	$0.563152\pi$
$6$	−1.00000	−0.408248
$7$	0.253486	0.0958086	0.0479043	−	0.998852i	$-0.484746\pi$
$7$	0.253486	0.0958086	0.0479043	+	0.998852i	$0.484746\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0.881455	0.278740
$11$	−3.18357	−0.959884	−0.479942	−	0.877300i	$-0.659342\pi$
$11$	−3.18357	−0.959884	−0.479942	+	0.877300i	$0.659342\pi$
$12$	1.00000	0.288675
$13$	2.13413	0.591901	0.295951	−	0.955203i	$-0.404364\pi$
$13$	2.13413	0.591901	0.295951	+	0.955203i	$0.404364\pi$
$14$	−0.253486	−0.0677469
$15$	−0.881455	−0.227591
$16$	1.00000	0.250000
$17$	−5.96126	−1.44582	−0.722910	−	0.690943i	$-0.757196\pi$
$17$	−5.96126	−1.44582	−0.722910	+	0.690943i	$0.757196\pi$
$18$	−1.00000	−0.235702
$19$	3.48434	0.799363	0.399682	−	0.916654i	$-0.369121\pi$
$19$	3.48434	0.799363	0.399682	+	0.916654i	$0.369121\pi$
$20$	−0.881455	−0.197099
$21$	0.253486	0.0553151
$22$	3.18357	0.678740
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	−4.22304	−0.844608
$26$	−2.13413	−0.418538
$27$	1.00000	0.192450
$28$	0.253486	0.0479043
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0.881455	0.160931
$31$	10.8272	1.94461	0.972307	−	0.233707i	$-0.0750856\pi$
$31$	10.8272	1.94461	0.972307	+	0.233707i	$0.0750856\pi$
$32$	−1.00000	−0.176777
$33$	−3.18357	−0.554189
$34$	5.96126	1.02235
$35$	−0.223436	−0.0377676
$36$	1.00000	0.166667
$37$	9.11932	1.49921	0.749603	−	0.661887i	$-0.230244\pi$
$37$	9.11932	1.49921	0.749603	+	0.661887i	$0.230244\pi$
$38$	−3.48434	−0.565235
$39$	2.13413	0.341734
$40$	0.881455	0.139370
$41$	−4.89141	−0.763910	−0.381955	−	0.924181i	$-0.624749\pi$
$41$	−4.89141	−0.763910	−0.381955	+	0.924181i	$0.624749\pi$
$42$	−0.253486	−0.0391137
$43$	−8.97041	−1.36797	−0.683987	−	0.729494i	$-0.739756\pi$
$43$	−8.97041	−1.36797	−0.683987	+	0.729494i	$0.739756\pi$
$44$	−3.18357	−0.479942
$45$	−0.881455	−0.131399
$46$	1.00000	0.147442
$47$	−0.153477	−0.0223869	−0.0111935	−	0.999937i	$-0.503563\pi$
$47$	−0.153477	−0.0223869	−0.0111935	+	0.999937i	$0.503563\pi$
$48$	1.00000	0.144338
$49$	−6.93575	−0.990821
$50$	4.22304	0.597228
$51$	−5.96126	−0.834744
$52$	2.13413	0.295951
$53$	−10.3575	−1.42271	−0.711355	−	0.702833i	$-0.751918\pi$
$53$	−10.3575	−1.42271	−0.711355	+	0.702833i	$0.751918\pi$
$54$	−1.00000	−0.136083
$55$	2.80618	0.378385
$56$	−0.253486	−0.0338734
$57$	3.48434	0.461512
$58$	−1.00000	−0.131306
$59$	8.73661	1.13741	0.568705	−	0.822542i	$-0.307445\pi$
$59$	8.73661	1.13741	0.568705	+	0.822542i	$0.307445\pi$
$60$	−0.881455	−0.113795
$61$	−11.0288	−1.41209	−0.706045	−	0.708167i	$-0.749522\pi$
$61$	−11.0288	−1.41209	−0.706045	+	0.708167i	$0.749522\pi$
$62$	−10.8272	−1.37505
$63$	0.253486	0.0319362
$64$	1.00000	0.125000
$65$	−1.88114	−0.233327
$66$	3.18357	0.391871
$67$	−1.61142	−0.196867	−0.0984333	−	0.995144i	$-0.531383\pi$
$67$	−1.61142	−0.196867	−0.0984333	+	0.995144i	$0.531383\pi$
$68$	−5.96126	−0.722910
$69$	−1.00000	−0.120386
$70$	0.223436	0.0267057
$71$	14.9152	1.77011	0.885054	−	0.465488i	$-0.154121\pi$
$71$	14.9152	1.77011	0.885054	+	0.465488i	$0.154121\pi$
$72$	−1.00000	−0.117851
$73$	−6.69346	−0.783410	−0.391705	−	0.920091i	$-0.628115\pi$
$73$	−6.69346	−0.783410	−0.391705	+	0.920091i	$0.628115\pi$
$74$	−9.11932	−1.06010
$75$	−4.22304	−0.487634
$76$	3.48434	0.399682
$77$	−0.806990	−0.0919651
$78$	−2.13413	−0.241643
$79$	−0.899667	−0.101220	−0.0506102	−	0.998718i	$-0.516117\pi$
$79$	−0.899667	−0.101220	−0.0506102	+	0.998718i	$0.516117\pi$
$80$	−0.881455	−0.0985496
$81$	1.00000	0.111111
$82$	4.89141	0.540166
$83$	−17.0301	−1.86930	−0.934649	−	0.355571i	$-0.884287\pi$
$83$	−17.0301	−1.86930	−0.934649	+	0.355571i	$0.884287\pi$
$84$	0.253486	0.0276575
$85$	5.25458	0.569940
$86$	8.97041	0.967304
$87$	1.00000	0.107211
$88$	3.18357	0.339370
$89$	−1.38034	−0.146316	−0.0731581	−	0.997320i	$-0.523308\pi$
$89$	−1.38034	−0.146316	−0.0731581	+	0.997320i	$0.523308\pi$
$90$	0.881455	0.0929135
$91$	0.540972	0.0567092
$92$	−1.00000	−0.104257
$93$	10.8272	1.12272
$94$	0.153477	0.0158299
$95$	−3.07129	−0.315108
$96$	−1.00000	−0.102062
$97$	−8.28427	−0.841140	−0.420570	−	0.907260i	$-0.638170\pi$
$97$	−8.28427	−0.841140	−0.420570	+	0.907260i	$0.638170\pi$
$98$	6.93575	0.700616
$99$	−3.18357	−0.319961
$100$	−4.22304	−0.422304
$101$	−4.05768	−0.403754	−0.201877	−	0.979411i	$-0.564704\pi$
$101$	−4.05768	−0.403754	−0.201877	+	0.979411i	$0.564704\pi$
$102$	5.96126	0.590253
$103$	−18.6717	−1.83977	−0.919887	−	0.392184i	$-0.871720\pi$
$103$	−18.6717	−1.83977	−0.919887	+	0.392184i	$0.871720\pi$
$104$	−2.13413	−0.209269
$105$	−0.223436	−0.0218051
$106$	10.3575	1.00601
$107$	−14.5325	−1.40491	−0.702453	−	0.711730i	$-0.747912\pi$
$107$	−14.5325	−1.40491	−0.702453	+	0.711730i	$0.747912\pi$
$108$	1.00000	0.0962250
$109$	7.67530	0.735160	0.367580	−	0.929992i	$-0.380186\pi$
$109$	7.67530	0.735160	0.367580	+	0.929992i	$0.380186\pi$
$110$	−2.80618	−0.267558
$111$	9.11932	0.865568
$112$	0.253486	0.0239521
$113$	13.3247	1.25348	0.626742	−	0.779227i	$-0.284388\pi$
$113$	13.3247	1.25348	0.626742	+	0.779227i	$0.284388\pi$
$114$	−3.48434	−0.326339
$115$	0.881455	0.0821961
$116$	1.00000	0.0928477
$117$	2.13413	0.197300
$118$	−8.73661	−0.804270
$119$	−1.51109	−0.138522
$120$	0.881455	0.0804654
$121$	−0.864853	−0.0786230
$122$	11.0288	0.998498
$123$	−4.89141	−0.441043
$124$	10.8272	0.972307
$125$	8.12969	0.727142
$126$	−0.253486	−0.0225823
$127$	11.1416	0.988655	0.494328	−	0.869276i	$-0.335414\pi$
$127$	11.1416	0.988655	0.494328	+	0.869276i	$0.335414\pi$
$128$	−1.00000	−0.0883883
$129$	−8.97041	−0.789801
$130$	1.88114	0.164987
$131$	5.68602	0.496789	0.248395	−	0.968659i	$-0.420097\pi$
$131$	5.68602	0.496789	0.248395	+	0.968659i	$0.420097\pi$
$132$	−3.18357	−0.277095
$133$	0.883231	0.0765858
$134$	1.61142	0.139206
$135$	−0.881455	−0.0758635
$136$	5.96126	0.511174
$137$	−3.95394	−0.337808	−0.168904	−	0.985633i	$-0.554023\pi$
$137$	−3.95394	−0.337808	−0.168904	+	0.985633i	$0.554023\pi$
$138$	1.00000	0.0851257
$139$	−0.573158	−0.0486147	−0.0243073	−	0.999705i	$-0.507738\pi$
$139$	−0.573158	−0.0486147	−0.0243073	+	0.999705i	$0.507738\pi$
$140$	−0.223436	−0.0188838
$141$	−0.153477	−0.0129251
$142$	−14.9152	−1.25166
$143$	−6.79416	−0.568157
$144$	1.00000	0.0833333
$145$	−0.881455	−0.0732008
$146$	6.69346	0.553955
$147$	−6.93575	−0.572051
$148$	9.11932	0.749603
$149$	−8.53366	−0.699105	−0.349553	−	0.936917i	$-0.613666\pi$
$149$	−8.53366	−0.699105	−0.349553	+	0.936917i	$0.613666\pi$
$150$	4.22304	0.344810
$151$	−21.3556	−1.73789	−0.868947	−	0.494905i	$-0.835203\pi$
$151$	−21.3556	−1.73789	−0.868947	+	0.494905i	$0.835203\pi$
$152$	−3.48434	−0.282618
$153$	−5.96126	−0.481940
$154$	0.806990	0.0650291
$155$	−9.54364	−0.766564
$156$	2.13413	0.170867
$157$	−10.6386	−0.849055	−0.424527	−	0.905415i	$-0.639560\pi$
$157$	−10.6386	−0.849055	−0.424527	+	0.905415i	$0.639560\pi$
$158$	0.899667	0.0715737
$159$	−10.3575	−0.821402
$160$	0.881455	0.0696851
$161$	−0.253486	−0.0199775
$162$	−1.00000	−0.0785674
$163$	9.38753	0.735288	0.367644	−	0.929967i	$-0.380164\pi$
$163$	9.38753	0.735288	0.367644	+	0.929967i	$0.380164\pi$
$164$	−4.89141	−0.381955
$165$	2.80618	0.218461
$166$	17.0301	1.32179
$167$	−4.69173	−0.363057	−0.181528	−	0.983386i	$-0.558104\pi$
$167$	−4.69173	−0.363057	−0.181528	+	0.983386i	$0.558104\pi$
$168$	−0.253486	−0.0195568
$169$	−8.44549	−0.649653
$170$	−5.25458	−0.403008
$171$	3.48434	0.266454
$172$	−8.97041	−0.683987
$173$	14.6777	1.11593	0.557963	−	0.829866i	$-0.311583\pi$
$173$	14.6777	1.11593	0.557963	+	0.829866i	$0.311583\pi$
$174$	−1.00000	−0.0758098
$175$	−1.07048	−0.0809206
$176$	−3.18357	−0.239971
$177$	8.73661	0.656683
$178$	1.38034	0.103461
$179$	−22.8412	−1.70723	−0.853614	−	0.520905i	$-0.825594\pi$
$179$	−22.8412	−1.70723	−0.853614	+	0.520905i	$0.825594\pi$
$180$	−0.881455	−0.0656997
$181$	9.66709	0.718549	0.359274	−	0.933232i	$-0.383024\pi$
$181$	9.66709	0.718549	0.359274	+	0.933232i	$0.383024\pi$
$182$	−0.540972	−0.0400995
$183$	−11.0288	−0.815270
$184$	1.00000	0.0737210
$185$	−8.03827	−0.590985
$186$	−10.8272	−0.793885
$187$	18.9781	1.38782
$188$	−0.153477	−0.0111935
$189$	0.253486	0.0184384
$190$	3.07129	0.222815
$191$	−27.4995	−1.98979	−0.994895	−	0.100913i	$-0.967824\pi$
$191$	−27.4995	−1.98979	−0.994895	+	0.100913i	$0.967824\pi$
$192$	1.00000	0.0721688
$193$	−17.1988	−1.23800	−0.618999	−	0.785392i	$-0.712461\pi$
$193$	−17.1988	−1.23800	−0.618999	+	0.785392i	$0.712461\pi$
$194$	8.28427	0.594776
$195$	−1.88114	−0.134711
$196$	−6.93575	−0.495410
$197$	3.77448	0.268921	0.134460	−	0.990919i	$-0.457070\pi$
$197$	3.77448	0.268921	0.134460	+	0.990919i	$0.457070\pi$
$198$	3.18357	0.226247
$199$	14.0528	0.996177	0.498088	−	0.867126i	$-0.334035\pi$
$199$	14.0528	0.996177	0.498088	+	0.867126i	$0.334035\pi$
$200$	4.22304	0.298614
$201$	−1.61142	−0.113661
$202$	4.05768	0.285497
$203$	0.253486	0.0177912
$204$	−5.96126	−0.417372
$205$	4.31155	0.301132
$206$	18.6717	1.30092
$207$	−1.00000	−0.0695048
$208$	2.13413	0.147975
$209$	−11.0927	−0.767296
$210$	0.223436	0.0154186
$211$	−11.4091	−0.785432	−0.392716	−	0.919660i	$-0.628465\pi$
$211$	−11.4091	−0.785432	−0.392716	+	0.919660i	$0.628465\pi$
$212$	−10.3575	−0.711355
$213$	14.9152	1.02197
$214$	14.5325	0.993418
$215$	7.90701	0.539254
$216$	−1.00000	−0.0680414
$217$	2.74453	0.186311
$218$	−7.67530	−0.519837
$219$	−6.69346	−0.452302
$220$	2.80618	0.189192
$221$	−12.7221	−0.855782
$222$	−9.11932	−0.612049
$223$	12.8044	0.857443	0.428722	−	0.903437i	$-0.358964\pi$
$223$	12.8044	0.857443	0.428722	+	0.903437i	$0.358964\pi$
$224$	−0.253486	−0.0169367
$225$	−4.22304	−0.281536
$226$	−13.3247	−0.886347
$227$	−8.25395	−0.547834	−0.273917	−	0.961753i	$-0.588319\pi$
$227$	−8.25395	−0.547834	−0.273917	+	0.961753i	$0.588319\pi$
$228$	3.48434	0.230756
$229$	−10.3469	−0.683741	−0.341871	−	0.939747i	$-0.611060\pi$
$229$	−10.3469	−0.683741	−0.341871	+	0.939747i	$0.611060\pi$
$230$	−0.881455	−0.0581214
$231$	−0.806990	−0.0530961
$232$	−1.00000	−0.0656532
$233$	−0.750585	−0.0491725	−0.0245862	−	0.999698i	$-0.507827\pi$
$233$	−0.750585	−0.0491725	−0.0245862	+	0.999698i	$0.507827\pi$
$234$	−2.13413	−0.139513
$235$	0.135283	0.00882489
$236$	8.73661	0.568705
$237$	−0.899667	−0.0584397
$238$	1.51109	0.0979497
$239$	0.641568	0.0414996	0.0207498	−	0.999785i	$-0.493395\pi$
$239$	0.641568	0.0414996	0.0207498	+	0.999785i	$0.493395\pi$
$240$	−0.881455	−0.0568977
$241$	22.2474	1.43308	0.716541	−	0.697545i	$-0.245724\pi$
$241$	22.2474	1.43308	0.716541	+	0.697545i	$0.245724\pi$
$242$	0.864853	0.0555949
$243$	1.00000	0.0641500
$244$	−11.0288	−0.706045
$245$	6.11354	0.390580
$246$	4.89141	0.311865
$247$	7.43604	0.473144
$248$	−10.8272	−0.687525
$249$	−17.0301	−1.07924
$250$	−8.12969	−0.514167
$251$	24.9396	1.57417	0.787087	−	0.616842i	$-0.211588\pi$
$251$	24.9396	1.57417	0.787087	+	0.616842i	$0.211588\pi$
$252$	0.253486	0.0159681
$253$	3.18357	0.200150
$254$	−11.1416	−0.699085
$255$	5.25458	0.329055
$256$	1.00000	0.0625000
$257$	−2.91885	−0.182073	−0.0910366	−	0.995848i	$-0.529018\pi$
$257$	−2.91885	−0.182073	−0.0910366	+	0.995848i	$0.529018\pi$
$258$	8.97041	0.558473
$259$	2.31162	0.143637
$260$	−1.88114	−0.116663
$261$	1.00000	0.0618984
$262$	−5.68602	−0.351283
$263$	15.7584	0.971705	0.485853	−	0.874041i	$-0.338509\pi$
$263$	15.7584	0.971705	0.485853	+	0.874041i	$0.338509\pi$
$264$	3.18357	0.195935
$265$	9.12965	0.560830
$266$	−0.883231	−0.0541544
$267$	−1.38034	−0.0844757
$268$	−1.61142	−0.0984333
$269$	−8.72871	−0.532199	−0.266099	−	0.963946i	$-0.585735\pi$
$269$	−8.72871	−0.532199	−0.266099	+	0.963946i	$0.585735\pi$
$270$	0.881455	0.0536436
$271$	−6.99426	−0.424871	−0.212435	−	0.977175i	$-0.568140\pi$
$271$	−6.99426	−0.424871	−0.212435	+	0.977175i	$0.568140\pi$
$272$	−5.96126	−0.361455
$273$	0.540972	0.0327411
$274$	3.95394	0.238866
$275$	13.4444	0.810725
$276$	−1.00000	−0.0601929
$277$	24.6600	1.48167	0.740837	−	0.671685i	$-0.234429\pi$
$277$	24.6600	1.48167	0.740837	+	0.671685i	$0.234429\pi$
$278$	0.573158	0.0343758
$279$	10.8272	0.648205
$280$	0.223436	0.0133529
$281$	16.1802	0.965230	0.482615	−	0.875832i	$-0.339687\pi$
$281$	16.1802	0.965230	0.482615	+	0.875832i	$0.339687\pi$
$282$	0.153477	0.00913942
$283$	−7.11722	−0.423075	−0.211537	−	0.977370i	$-0.567847\pi$
$283$	−7.11722	−0.423075	−0.211537	+	0.977370i	$0.567847\pi$
$284$	14.9152	0.885054
$285$	−3.07129	−0.181928
$286$	6.79416	0.401747
$287$	−1.23990	−0.0731891
$288$	−1.00000	−0.0589256
$289$	18.5367	1.09039
$290$	0.881455	0.0517608
$291$	−8.28427	−0.485633
$292$	−6.69346	−0.391705
$293$	−23.4459	−1.36973	−0.684863	−	0.728672i	$-0.740138\pi$
$293$	−23.4459	−1.36973	−0.684863	+	0.728672i	$0.740138\pi$
$294$	6.93575	0.404501
$295$	−7.70092	−0.448365
$296$	−9.11932	−0.530050
$297$	−3.18357	−0.184730
$298$	8.53366	0.494342
$299$	−2.13413	−0.123420
$300$	−4.22304	−0.243817
$301$	−2.27387	−0.131064
$302$	21.3556	1.22888
$303$	−4.05768	−0.233107
$304$	3.48434	0.199841
$305$	9.72136	0.556644
$306$	5.96126	0.340783
$307$	−28.0138	−1.59883	−0.799415	−	0.600779i	$-0.794857\pi$
$307$	−28.0138	−1.59883	−0.799415	+	0.600779i	$0.794857\pi$
$308$	−0.806990	−0.0459825
$309$	−18.6717	−1.06219
$310$	9.54364	0.542043
$311$	16.2662	0.922369	0.461185	−	0.887304i	$-0.347425\pi$
$311$	16.2662	0.922369	0.461185	+	0.887304i	$0.347425\pi$
$312$	−2.13413	−0.120821
$313$	21.4754	1.21386	0.606931	−	0.794754i	$-0.292400\pi$
$313$	21.4754	1.21386	0.606931	+	0.794754i	$0.292400\pi$
$314$	10.6386	0.600372
$315$	−0.223436	−0.0125892
$316$	−0.899667	−0.0506102
$317$	−17.9480	−1.00806	−0.504030	−	0.863686i	$-0.668150\pi$
$317$	−17.9480	−1.00806	−0.504030	+	0.863686i	$0.668150\pi$
$318$	10.3575	0.580819
$319$	−3.18357	−0.178246
$320$	−0.881455	−0.0492748
$321$	−14.5325	−0.811123
$322$	0.253486	0.0141262
$323$	−20.7711	−1.15573
$324$	1.00000	0.0555556
$325$	−9.01252	−0.499924
$326$	−9.38753	−0.519927
$327$	7.67530	0.424445
$328$	4.89141	0.270083
$329$	−0.0389042	−0.00214486
$330$	−2.80618	−0.154475
$331$	−29.9727	−1.64745	−0.823725	−	0.566990i	$-0.808108\pi$
$331$	−29.9727	−1.64745	−0.823725	+	0.566990i	$0.808108\pi$
$332$	−17.0301	−0.934649
$333$	9.11932	0.499736
$334$	4.69173	0.256720
$335$	1.42040	0.0776046
$336$	0.253486	0.0138288
$337$	−11.5002	−0.626456	−0.313228	−	0.949678i	$-0.601410\pi$
$337$	−11.5002	−0.626456	−0.313228	+	0.949678i	$0.601410\pi$
$338$	8.44549	0.459374
$339$	13.3247	0.723699
$340$	5.25458	0.284970
$341$	−34.4691	−1.86660
$342$	−3.48434	−0.188412
$343$	−3.53251	−0.190738
$344$	8.97041	0.483652
$345$	0.881455	0.0474559
$346$	−14.6777	−0.789079
$347$	−6.67485	−0.358325	−0.179162	−	0.983819i	$-0.557339\pi$
$347$	−6.67485	−0.358325	−0.179162	+	0.983819i	$0.557339\pi$
$348$	1.00000	0.0536056
$349$	−18.0270	−0.964962	−0.482481	−	0.875906i	$-0.660264\pi$
$349$	−18.0270	−0.964962	−0.482481	+	0.875906i	$0.660264\pi$
$350$	1.07048	0.0572195
$351$	2.13413	0.113911
$352$	3.18357	0.169685
$353$	−11.5462	−0.614540	−0.307270	−	0.951622i	$-0.599416\pi$
$353$	−11.5462	−0.614540	−0.307270	+	0.951622i	$0.599416\pi$
$354$	−8.73661	−0.464345
$355$	−13.1471	−0.697774
$356$	−1.38034	−0.0731581
$357$	−1.51109	−0.0799756
$358$	22.8412	1.20719
$359$	−10.2128	−0.539013	−0.269507	−	0.962999i	$-0.586861\pi$
$359$	−10.2128	−0.539013	−0.269507	+	0.962999i	$0.586861\pi$
$360$	0.881455	0.0464567
$361$	−6.85936	−0.361019
$362$	−9.66709	−0.508091
$363$	−0.864853	−0.0453930
$364$	0.540972	0.0283546
$365$	5.89998	0.308819
$366$	11.0288	0.576483
$367$	−25.3240	−1.32190	−0.660952	−	0.750428i	$-0.729847\pi$
$367$	−25.3240	−1.32190	−0.660952	+	0.750428i	$0.729847\pi$
$368$	−1.00000	−0.0521286
$369$	−4.89141	−0.254637
$370$	8.03827	0.417890
$371$	−2.62547	−0.136308
$372$	10.8272	0.561362
$373$	22.4619	1.16304	0.581518	−	0.813534i	$-0.302459\pi$
$373$	22.4619	1.16304	0.581518	+	0.813534i	$0.302459\pi$
$374$	−18.9781	−0.981336
$375$	8.12969	0.419815
$376$	0.153477	0.00791497
$377$	2.13413	0.109913
$378$	−0.253486	−0.0130379
$379$	16.2654	0.835497	0.417749	−	0.908563i	$-0.362819\pi$
$379$	16.2654	0.835497	0.417749	+	0.908563i	$0.362819\pi$
$380$	−3.07129	−0.157554
$381$	11.1416	0.570800
$382$	27.4995	1.40699
$383$	32.4150	1.65633	0.828165	−	0.560485i	$-0.189385\pi$
$383$	32.4150	1.65633	0.828165	+	0.560485i	$0.189385\pi$
$384$	−1.00000	−0.0510310
$385$	0.711325	0.0362525
$386$	17.1988	0.875397
$387$	−8.97041	−0.455992
$388$	−8.28427	−0.420570
$389$	−1.34268	−0.0680765	−0.0340382	−	0.999421i	$-0.510837\pi$
$389$	−1.34268	−0.0680765	−0.0340382	+	0.999421i	$0.510837\pi$
$390$	1.88114	0.0952552
$391$	5.96126	0.301474
$392$	6.93575	0.350308
$393$	5.68602	0.286822
$394$	−3.77448	−0.190156
$395$	0.793016	0.0399009
$396$	−3.18357	−0.159981
$397$	−15.0931	−0.757501	−0.378751	−	0.925499i	$-0.623646\pi$
$397$	−15.0931	−0.757501	−0.378751	+	0.925499i	$0.623646\pi$
$398$	−14.0528	−0.704403
$399$	0.883231	0.0442168
$400$	−4.22304	−0.211152
$401$	−3.73433	−0.186484	−0.0932418	−	0.995643i	$-0.529723\pi$
$401$	−3.73433	−0.186484	−0.0932418	+	0.995643i	$0.529723\pi$
$402$	1.61142	0.0803705
$403$	23.1066	1.15102
$404$	−4.05768	−0.201877
$405$	−0.881455	−0.0437998
$406$	−0.253486	−0.0125803
$407$	−29.0320	−1.43906
$408$	5.96126	0.295127
$409$	−13.7129	−0.678060	−0.339030	−	0.940776i	$-0.610099\pi$
$409$	−13.7129	−0.678060	−0.339030	+	0.940776i	$0.610099\pi$
$410$	−4.31155	−0.212933
$411$	−3.95394	−0.195034
$412$	−18.6717	−0.919887
$413$	2.21460	0.108974
$414$	1.00000	0.0491473
$415$	15.0113	0.736875
$416$	−2.13413	−0.104634
$417$	−0.573158	−0.0280677
$418$	11.0927	0.542560
$419$	4.28997	0.209579	0.104789	−	0.994494i	$-0.466583\pi$
$419$	4.28997	0.209579	0.104789	+	0.994494i	$0.466583\pi$
$420$	−0.223436	−0.0109026
$421$	3.82044	0.186197	0.0930984	−	0.995657i	$-0.470323\pi$
$421$	3.82044	0.186197	0.0930984	+	0.995657i	$0.470323\pi$
$422$	11.4091	0.555384
$423$	−0.153477	−0.00746230
$424$	10.3575	0.503004
$425$	25.1746	1.22115
$426$	−14.9152	−0.722644
$427$	−2.79563	−0.135290
$428$	−14.5325	−0.702453
$429$	−6.79416	−0.328025
$430$	−7.90701	−0.381310
$431$	−17.1289	−0.825069	−0.412535	−	0.910942i	$-0.635356\pi$
$431$	−17.1289	−0.825069	−0.412535	+	0.910942i	$0.635356\pi$
$432$	1.00000	0.0481125
$433$	25.0686	1.20472	0.602361	−	0.798224i	$-0.294227\pi$
$433$	25.0686	1.20472	0.602361	+	0.798224i	$0.294227\pi$
$434$	−2.74453	−0.131742
$435$	−0.881455	−0.0422625
$436$	7.67530	0.367580
$437$	−3.48434	−0.166679
$438$	6.69346	0.319826
$439$	10.7003	0.510696	0.255348	−	0.966849i	$-0.417810\pi$
$439$	10.7003	0.510696	0.255348	+	0.966849i	$0.417810\pi$
$440$	−2.80618	−0.133779
$441$	−6.93575	−0.330274
$442$	12.7221	0.605129
$443$	−30.6408	−1.45579	−0.727894	−	0.685689i	$-0.759501\pi$
$443$	−30.6408	−1.45579	−0.727894	+	0.685689i	$0.759501\pi$
$444$	9.11932	0.432784
$445$	1.21671	0.0576776
$446$	−12.8044	−0.606304
$447$	−8.53366	−0.403628
$448$	0.253486	0.0119761
$449$	15.5745	0.735004	0.367502	−	0.930023i	$-0.380213\pi$
$449$	15.5745	0.735004	0.367502	+	0.930023i	$0.380213\pi$
$450$	4.22304	0.199076
$451$	15.5722	0.733265
$452$	13.3247	0.626742
$453$	−21.3556	−1.00337
$454$	8.25395	0.387377
$455$	−0.476842	−0.0223547
$456$	−3.48434	−0.163169
$457$	4.21570	0.197202	0.0986012	−	0.995127i	$-0.468563\pi$
$457$	4.21570	0.197202	0.0986012	+	0.995127i	$0.468563\pi$
$458$	10.3469	0.483478
$459$	−5.96126	−0.278248
$460$	0.881455	0.0410980
$461$	3.39202	0.157982	0.0789909	−	0.996875i	$-0.474830\pi$
$461$	3.39202	0.157982	0.0789909	+	0.996875i	$0.474830\pi$
$462$	0.806990	0.0375446
$463$	7.79237	0.362142	0.181071	−	0.983470i	$-0.442044\pi$
$463$	7.79237	0.362142	0.181071	+	0.983470i	$0.442044\pi$
$464$	1.00000	0.0464238
$465$	−9.54364	−0.442576
$466$	0.750585	0.0347702
$467$	−15.9168	−0.736541	−0.368271	−	0.929719i	$-0.620050\pi$
$467$	−15.9168	−0.736541	−0.368271	+	0.929719i	$0.620050\pi$
$468$	2.13413	0.0986502
$469$	−0.408473	−0.0188615
$470$	−0.135283	−0.00624014
$471$	−10.6386	−0.490202
$472$	−8.73661	−0.402135
$473$	28.5580	1.31310
$474$	0.899667	0.0413231
$475$	−14.7145	−0.675148
$476$	−1.51109	−0.0692609
$477$	−10.3575	−0.474237
$478$	−0.641568	−0.0293446
$479$	−23.7269	−1.08411	−0.542056	−	0.840343i	$-0.682354\pi$
$479$	−23.7269	−1.08411	−0.542056	+	0.840343i	$0.682354\pi$
$480$	0.881455	0.0402327
$481$	19.4618	0.887383
$482$	−22.2474	−1.01334
$483$	−0.253486	−0.0115340
$484$	−0.864853	−0.0393115
$485$	7.30221	0.331576
$486$	−1.00000	−0.0453609
$487$	42.2746	1.91564	0.957822	−	0.287363i	$-0.0927787\pi$
$487$	42.2746	1.91564	0.957822	+	0.287363i	$0.0927787\pi$
$488$	11.0288	0.499249
$489$	9.38753	0.424519
$490$	−6.11354	−0.276182
$491$	5.73789	0.258947	0.129474	−	0.991583i	$-0.458671\pi$
$491$	5.73789	0.258947	0.129474	+	0.991583i	$0.458671\pi$
$492$	−4.89141	−0.220522
$493$	−5.96126	−0.268482
$494$	−7.43604	−0.334563
$495$	2.80618	0.126128
$496$	10.8272	0.486154
$497$	3.78079	0.169592
$498$	17.0301	0.763138
$499$	28.6996	1.28477	0.642384	−	0.766383i	$-0.277945\pi$
$499$	28.6996	1.28477	0.642384	+	0.766383i	$0.277945\pi$
$500$	8.12969	0.363571
$501$	−4.69173	−0.209611
$502$	−24.9396	−1.11311
$503$	−38.1855	−1.70261	−0.851304	−	0.524672i	$-0.824188\pi$
$503$	−38.1855	−1.70261	−0.851304	+	0.524672i	$0.824188\pi$
$504$	−0.253486	−0.0112911
$505$	3.57666	0.159159
$506$	−3.18357	−0.141527
$507$	−8.44549	−0.375077
$508$	11.1416	0.494328
$509$	−7.92003	−0.351049	−0.175525	−	0.984475i	$-0.556162\pi$
$509$	−7.92003	−0.351049	−0.175525	+	0.984475i	$0.556162\pi$
$510$	−5.25458	−0.232677
$511$	−1.69670	−0.0750574
$512$	−1.00000	−0.0441942
$513$	3.48434	0.153837
$514$	2.91885	0.128745
$515$	16.4582	0.725236
$516$	−8.97041	−0.394900
$517$	0.488605	0.0214888
$518$	−2.31162	−0.101567
$519$	14.6777	0.644281
$520$	1.88114	0.0824934
$521$	24.5778	1.07677	0.538387	−	0.842697i	$-0.319034\pi$
$521$	24.5778	1.07677	0.538387	+	0.842697i	$0.319034\pi$
$522$	−1.00000	−0.0437688
$523$	−39.1768	−1.71308	−0.856541	−	0.516080i	$-0.827391\pi$
$523$	−39.1768	−1.71308	−0.856541	+	0.516080i	$0.827391\pi$
$524$	5.68602	0.248395
$525$	−1.07048	−0.0467196
$526$	−15.7584	−0.687099
$527$	−64.5435	−2.81156
$528$	−3.18357	−0.138547
$529$	1.00000	0.0434783
$530$	−9.12965	−0.396567
$531$	8.73661	0.379136
$532$	0.883231	0.0382929
$533$	−10.4389	−0.452159
$534$	1.38034	0.0597334
$535$	12.8097	0.553812
$536$	1.61142	0.0696029
$537$	−22.8412	−0.985669
$538$	8.72871	0.376321
$539$	22.0805	0.951073
$540$	−0.881455	−0.0379318
$541$	19.8576	0.853747	0.426873	−	0.904311i	$-0.359615\pi$
$541$	19.8576	0.853747	0.426873	+	0.904311i	$0.359615\pi$
$542$	6.99426	0.300429
$543$	9.66709	0.414854
$544$	5.96126	0.255587
$545$	−6.76543	−0.289799
$546$	−0.540972	−0.0231514
$547$	−3.59030	−0.153510	−0.0767550	−	0.997050i	$-0.524456\pi$
$547$	−3.59030	−0.153510	−0.0767550	+	0.997050i	$0.524456\pi$
$548$	−3.95394	−0.168904
$549$	−11.0288	−0.470696
$550$	−13.4444	−0.573269
$551$	3.48434	0.148438
$552$	1.00000	0.0425628
$553$	−0.228053	−0.00969779
$554$	−24.6600	−1.04770
$555$	−8.03827	−0.341205
$556$	−0.573158	−0.0243073
$557$	18.2916	0.775039	0.387519	−	0.921862i	$-0.373332\pi$
$557$	18.2916	0.775039	0.387519	+	0.921862i	$0.373332\pi$
$558$	−10.8272	−0.458350
$559$	−19.1440	−0.809706
$560$	−0.223436	−0.00944190
$561$	18.9781	0.801257
$562$	−16.1802	−0.682521
$563$	−24.4357	−1.02984	−0.514922	−	0.857237i	$-0.672179\pi$
$563$	−24.4357	−1.02984	−0.514922	+	0.857237i	$0.672179\pi$
$564$	−0.153477	−0.00646254
$565$	−11.7451	−0.494121
$566$	7.11722	0.299159
$567$	0.253486	0.0106454
$568$	−14.9152	−0.625828
$569$	10.6690	0.447268	0.223634	−	0.974673i	$-0.428208\pi$
$569$	10.6690	0.447268	0.223634	+	0.974673i	$0.428208\pi$
$570$	3.07129	0.128642
$571$	−19.0717	−0.798127	−0.399063	−	0.916923i	$-0.630665\pi$
$571$	−19.0717	−0.798127	−0.399063	+	0.916923i	$0.630665\pi$
$572$	−6.79416	−0.284078
$573$	−27.4995	−1.14881
$574$	1.23990	0.0517525
$575$	4.22304	0.176113
$576$	1.00000	0.0416667
$577$	2.54472	0.105938	0.0529690	−	0.998596i	$-0.483132\pi$
$577$	2.54472	0.105938	0.0529690	+	0.998596i	$0.483132\pi$
$578$	−18.5367	−0.771024
$579$	−17.1988	−0.714758
$580$	−0.881455	−0.0366004
$581$	−4.31689	−0.179095
$582$	8.28427	0.343394
$583$	32.9738	1.36564
$584$	6.69346	0.276977
$585$	−1.88114	−0.0777755
$586$	23.4459	0.968542
$587$	31.6230	1.30522	0.652610	−	0.757694i	$-0.273674\pi$
$587$	31.6230	1.30522	0.652610	+	0.757694i	$0.273674\pi$
$588$	−6.93575	−0.286025
$589$	37.7255	1.55445
$590$	7.70092	0.317042
$591$	3.77448	0.155261
$592$	9.11932	0.374802
$593$	−5.74003	−0.235715	−0.117857	−	0.993031i	$-0.537603\pi$
$593$	−5.74003	−0.235715	−0.117857	+	0.993031i	$0.537603\pi$
$594$	3.18357	0.130624
$595$	1.33196	0.0546051
$596$	−8.53366	−0.349553
$597$	14.0528	0.575143
$598$	2.13413	0.0872711
$599$	23.3081	0.952345	0.476172	−	0.879352i	$-0.342024\pi$
$599$	23.3081	0.952345	0.476172	+	0.879352i	$0.342024\pi$
$600$	4.22304	0.172405
$601$	−48.5410	−1.98003	−0.990013	−	0.140973i	$-0.954977\pi$
$601$	−48.5410	−1.98003	−0.990013	+	0.140973i	$0.954977\pi$
$602$	2.27387	0.0926760
$603$	−1.61142	−0.0656222
$604$	−21.3556	−0.868947
$605$	0.762329	0.0309931
$606$	4.05768	0.164832
$607$	−22.6699	−0.920143	−0.460071	−	0.887882i	$-0.652176\pi$
$607$	−22.6699	−0.920143	−0.460071	+	0.887882i	$0.652176\pi$
$608$	−3.48434	−0.141309
$609$	0.253486	0.0102718
$610$	−9.72136	−0.393606
$611$	−0.327540	−0.0132508
$612$	−5.96126	−0.240970
$613$	−6.68866	−0.270153	−0.135076	−	0.990835i	$-0.543128\pi$
$613$	−6.68866	−0.270153	−0.135076	+	0.990835i	$0.543128\pi$
$614$	28.0138	1.13054
$615$	4.31155	0.173859
$616$	0.806990	0.0325146
$617$	−25.8809	−1.04192	−0.520962	−	0.853580i	$-0.674427\pi$
$617$	−25.8809	−1.04192	−0.520962	+	0.853580i	$0.674427\pi$
$618$	18.6717	0.751085
$619$	9.15349	0.367910	0.183955	−	0.982935i	$-0.441110\pi$
$619$	9.15349	0.367910	0.183955	+	0.982935i	$0.441110\pi$
$620$	−9.54364	−0.383282
$621$	−1.00000	−0.0401286
$622$	−16.2662	−0.652214
$623$	−0.349898	−0.0140183
$624$	2.13413	0.0854336
$625$	13.9492	0.557969
$626$	−21.4754	−0.858330
$627$	−11.0927	−0.442998
$628$	−10.6386	−0.424527
$629$	−54.3627	−2.16758
$630$	0.223436	0.00890191
$631$	−37.4528	−1.49097	−0.745487	−	0.666521i	$-0.767783\pi$
$631$	−37.4528	−1.49097	−0.745487	+	0.666521i	$0.767783\pi$
$632$	0.899667	0.0357868
$633$	−11.4091	−0.453470
$634$	17.9480	0.712806
$635$	−9.82079	−0.389726
$636$	−10.3575	−0.410701
$637$	−14.8018	−0.586468
$638$	3.18357	0.126039
$639$	14.9152	0.590036
$640$	0.881455	0.0348426
$641$	−33.0307	−1.30463	−0.652317	−	0.757946i	$-0.726203\pi$
$641$	−33.0307	−1.30463	−0.652317	+	0.757946i	$0.726203\pi$
$642$	14.5325	0.573550
$643$	38.4780	1.51742	0.758711	−	0.651427i	$-0.225829\pi$
$643$	38.4780	1.51742	0.758711	+	0.651427i	$0.225829\pi$
$644$	−0.253486	−0.00998873
$645$	7.90701	0.311338
$646$	20.7711	0.817228
$647$	−7.07033	−0.277963	−0.138982	−	0.990295i	$-0.544383\pi$
$647$	−7.07033	−0.277963	−0.138982	+	0.990295i	$0.544383\pi$
$648$	−1.00000	−0.0392837
$649$	−27.8136	−1.09178
$650$	9.01252	0.353500
$651$	2.74453	0.107567
$652$	9.38753	0.367644
$653$	26.7795	1.04796	0.523981	−	0.851730i	$-0.324446\pi$
$653$	26.7795	1.04796	0.523981	+	0.851730i	$0.324446\pi$
$654$	−7.67530	−0.300128
$655$	−5.01196	−0.195834
$656$	−4.89141	−0.190977
$657$	−6.69346	−0.261137
$658$	0.0389042	0.00151664
$659$	30.2854	1.17975	0.589875	−	0.807495i	$-0.299177\pi$
$659$	30.2854	1.17975	0.589875	+	0.807495i	$0.299177\pi$
$660$	2.80618	0.109230
$661$	−13.3236	−0.518229	−0.259114	−	0.965847i	$-0.583431\pi$
$661$	−13.3236	−0.518229	−0.259114	+	0.965847i	$0.583431\pi$
$662$	29.9727	1.16492
$663$	−12.7221	−0.494086
$664$	17.0301	0.660897
$665$	−0.778528	−0.0301900
$666$	−9.11932	−0.353366
$667$	−1.00000	−0.0387202
$668$	−4.69173	−0.181528
$669$	12.8044	0.495045
$670$	−1.42040	−0.0548747
$671$	35.1109	1.35544
$672$	−0.253486	−0.00977842
$673$	−19.8937	−0.766847	−0.383423	−	0.923573i	$-0.625255\pi$
$673$	−19.8937	−0.766847	−0.383423	+	0.923573i	$0.625255\pi$
$674$	11.5002	0.442972
$675$	−4.22304	−0.162545
$676$	−8.44549	−0.324826
$677$	1.97615	0.0759497	0.0379748	−	0.999279i	$-0.487909\pi$
$677$	1.97615	0.0759497	0.0379748	+	0.999279i	$0.487909\pi$
$678$	−13.3247	−0.511733
$679$	−2.09994	−0.0805884
$680$	−5.25458	−0.201504
$681$	−8.25395	−0.316292
$682$	34.4691	1.31989
$683$	45.3942	1.73696	0.868480	−	0.495724i	$-0.165097\pi$
$683$	45.3942	1.73696	0.868480	+	0.495724i	$0.165097\pi$
$684$	3.48434	0.133227
$685$	3.48522	0.133163
$686$	3.53251	0.134872
$687$	−10.3469	−0.394758
$688$	−8.97041	−0.341994
$689$	−22.1042	−0.842104
$690$	−0.881455	−0.0335564
$691$	36.2352	1.37845	0.689226	−	0.724546i	$-0.257951\pi$
$691$	36.2352	1.37845	0.689226	+	0.724546i	$0.257951\pi$
$692$	14.6777	0.557963
$693$	−0.806990	−0.0306550
$694$	6.67485	0.253374
$695$	0.505213	0.0191638
$696$	−1.00000	−0.0379049
$697$	29.1590	1.10448
$698$	18.0270	0.682331
$699$	−0.750585	−0.0283897
$700$	−1.07048	−0.0404603
$701$	48.7426	1.84098	0.920491	−	0.390764i	$-0.127789\pi$
$701$	48.7426	1.84098	0.920491	+	0.390764i	$0.127789\pi$
$702$	−2.13413	−0.0805476
$703$	31.7748	1.19841
$704$	−3.18357	−0.119985
$705$	0.135283	0.00509505
$706$	11.5462	0.434546
$707$	−1.02856	−0.0386831
$708$	8.73661	0.328342
$709$	−29.3973	−1.10404	−0.552020	−	0.833831i	$-0.686143\pi$
$709$	−29.3973	−1.10404	−0.552020	+	0.833831i	$0.686143\pi$
$710$	13.1471	0.493401
$711$	−0.899667	−0.0337401
$712$	1.38034	0.0517306
$713$	−10.8272	−0.405480
$714$	1.51109	0.0565513
$715$	5.98875	0.223966
$716$	−22.8412	−0.853614
$717$	0.641568	0.0239598
$718$	10.2128	0.381140
$719$	−46.5292	−1.73525	−0.867623	−	0.497222i	$-0.834353\pi$
$719$	−46.5292	−1.73525	−0.867623	+	0.497222i	$0.834353\pi$
$720$	−0.881455	−0.0328499
$721$	−4.73300	−0.176266
$722$	6.85936	0.255279
$723$	22.2474	0.827390
$724$	9.66709	0.359274
$725$	−4.22304	−0.156840
$726$	0.864853	0.0320977
$727$	−3.50443	−0.129972	−0.0649861	−	0.997886i	$-0.520700\pi$
$727$	−3.50443	−0.129972	−0.0649861	+	0.997886i	$0.520700\pi$
$728$	−0.540972	−0.0200497
$729$	1.00000	0.0370370
$730$	−5.89998	−0.218368
$731$	53.4750	1.97784
$732$	−11.0288	−0.407635
$733$	−7.00328	−0.258672	−0.129336	−	0.991601i	$-0.541285\pi$
$733$	−7.00328	−0.258672	−0.129336	+	0.991601i	$0.541285\pi$
$734$	25.3240	0.934727
$735$	6.11354	0.225501
$736$	1.00000	0.0368605
$737$	5.13009	0.188969
$738$	4.89141	0.180055
$739$	23.8077	0.875781	0.437891	−	0.899028i	$-0.355726\pi$
$739$	23.8077	0.875781	0.437891	+	0.899028i	$0.355726\pi$
$740$	−8.03827	−0.295493
$741$	7.43604	0.273170
$742$	2.62547	0.0963842
$743$	37.8081	1.38705	0.693523	−	0.720435i	$-0.256058\pi$
$743$	37.8081	1.38705	0.693523	+	0.720435i	$0.256058\pi$
$744$	−10.8272	−0.396943
$745$	7.52204	0.275586
$746$	−22.4619	−0.822390
$747$	−17.0301	−0.623100
$748$	18.9781	0.693909
$749$	−3.68377	−0.134602
$750$	−8.12969	−0.296854
$751$	−0.298238	−0.0108829	−0.00544144	−	0.999985i	$-0.501732\pi$
$751$	−0.298238	−0.0108829	−0.00544144	+	0.999985i	$0.501732\pi$
$752$	−0.153477	−0.00559673
$753$	24.9396	0.908850
$754$	−2.13413	−0.0777205
$755$	18.8240	0.685075
$756$	0.253486	0.00921918
$757$	−7.78348	−0.282896	−0.141448	−	0.989946i	$-0.545176\pi$
$757$	−7.78348	−0.282896	−0.141448	+	0.989946i	$0.545176\pi$
$758$	−16.2654	−0.590786
$759$	3.18357	0.115556
$760$	3.07129	0.111407
$761$	−25.4291	−0.921806	−0.460903	−	0.887451i	$-0.652474\pi$
$761$	−25.4291	−0.921806	−0.460903	+	0.887451i	$0.652474\pi$
$762$	−11.1416	−0.403617
$763$	1.94558	0.0704346
$764$	−27.4995	−0.994895
$765$	5.25458	0.189980
$766$	−32.4150	−1.17120
$767$	18.6451	0.673234
$768$	1.00000	0.0360844
$769$	44.7351	1.61319	0.806595	−	0.591105i	$-0.201308\pi$
$769$	44.7351	1.61319	0.806595	+	0.591105i	$0.201308\pi$
$770$	−0.711325	−0.0256344
$771$	−2.91885	−0.105120
$772$	−17.1988	−0.618999
$773$	−1.36385	−0.0490542	−0.0245271	−	0.999699i	$-0.507808\pi$
$773$	−1.36385	−0.0490542	−0.0245271	+	0.999699i	$0.507808\pi$
$774$	8.97041	0.322435
$775$	−45.7235	−1.64244
$776$	8.28427	0.297388
$777$	2.31162	0.0829288
$778$	1.34268	0.0481373
$779$	−17.0433	−0.610641
$780$	−1.88114	−0.0673556
$781$	−47.4836	−1.69910
$782$	−5.96126	−0.213174
$783$	1.00000	0.0357371
$784$	−6.93575	−0.247705
$785$	9.37747	0.334696
$786$	−5.68602	−0.202813
$787$	17.5793	0.626635	0.313318	−	0.949648i	$-0.398560\pi$
$787$	17.5793	0.626635	0.313318	+	0.949648i	$0.398560\pi$
$788$	3.77448	0.134460
$789$	15.7584	0.561014
$790$	−0.793016	−0.0282142
$791$	3.37762	0.120094
$792$	3.18357	0.113123
$793$	−23.5368	−0.835818
$794$	15.0931	0.535634
$795$	9.12965	0.323795
$796$	14.0528	0.498088
$797$	10.0135	0.354695	0.177348	−	0.984148i	$-0.443248\pi$
$797$	10.0135	0.354695	0.177348	+	0.984148i	$0.443248\pi$
$798$	−0.883231	−0.0312660
$799$	0.914917	0.0323674
$800$	4.22304	0.149307
$801$	−1.38034	−0.0487721
$802$	3.73433	0.131864
$803$	21.3091	0.751983
$804$	−1.61142	−0.0568305
$805$	0.223436	0.00787509
$806$	−23.1066	−0.813894
$807$	−8.72871	−0.307265
$808$	4.05768	0.142749
$809$	0.434048	0.0152603	0.00763016	−	0.999971i	$-0.497571\pi$
$809$	0.434048	0.0152603	0.00763016	+	0.999971i	$0.497571\pi$
$810$	0.881455	0.0309712
$811$	48.5306	1.70414	0.852071	−	0.523427i	$-0.175347\pi$
$811$	48.5306	1.70414	0.852071	+	0.523427i	$0.175347\pi$
$812$	0.253486	0.00889560
$813$	−6.99426	−0.245299
$814$	29.0320	1.01757
$815$	−8.27469	−0.289850
$816$	−5.96126	−0.208686
$817$	−31.2560	−1.09351
$818$	13.7129	0.479461
$819$	0.540972	0.0189031
$820$	4.31155	0.150566
$821$	43.4070	1.51491	0.757457	−	0.652885i	$-0.226442\pi$
$821$	43.4070	1.51491	0.757457	+	0.652885i	$0.226442\pi$
$822$	3.95394	0.137910
$823$	−12.9220	−0.450432	−0.225216	−	0.974309i	$-0.572309\pi$
$823$	−12.9220	−0.450432	−0.225216	+	0.974309i	$0.572309\pi$
$824$	18.6717	0.650458
$825$	13.4444	0.468072
$826$	−2.21460	−0.0770559
$827$	−12.7840	−0.444543	−0.222271	−	0.974985i	$-0.571347\pi$
$827$	−12.7840	−0.444543	−0.222271	+	0.974985i	$0.571347\pi$
$828$	−1.00000	−0.0347524
$829$	−39.8884	−1.38538	−0.692691	−	0.721234i	$-0.743575\pi$
$829$	−39.8884	−1.38538	−0.692691	+	0.721234i	$0.743575\pi$
$830$	−15.0113	−0.521049
$831$	24.6600	0.855445
$832$	2.13413	0.0739877
$833$	41.3458	1.43255
$834$	0.573158	0.0198468
$835$	4.13555	0.143116
$836$	−11.0927	−0.383648
$837$	10.8272	0.374241
$838$	−4.28997	−0.148195
$839$	−18.8594	−0.651100	−0.325550	−	0.945525i	$-0.605549\pi$
$839$	−18.8594	−0.651100	−0.325550	+	0.945525i	$0.605549\pi$
$840$	0.223436	0.00770928
$841$	1.00000	0.0344828
$842$	−3.82044	−0.131661
$843$	16.1802	0.557276
$844$	−11.4091	−0.392716
$845$	7.44431	0.256092
$846$	0.153477	0.00527665
$847$	−0.219228	−0.00753276
$848$	−10.3575	−0.355677
$849$	−7.11722	−0.244262
$850$	−25.1746	−0.863483
$851$	−9.11932	−0.312606
$852$	14.9152	0.510986
$853$	21.1304	0.723491	0.361745	−	0.932277i	$-0.382181\pi$
$853$	21.1304	0.723491	0.361745	+	0.932277i	$0.382181\pi$
$854$	2.79563	0.0956647
$855$	−3.07129	−0.105036
$856$	14.5325	0.496709
$857$	48.7817	1.66635	0.833176	−	0.553008i	$-0.186520\pi$
$857$	48.7817	1.66635	0.833176	+	0.553008i	$0.186520\pi$
$858$	6.79416	0.231949
$859$	−29.0239	−0.990282	−0.495141	−	0.868813i	$-0.664884\pi$
$859$	−29.0239	−0.990282	−0.495141	+	0.868813i	$0.664884\pi$
$860$	7.90701	0.269627
$861$	−1.23990	−0.0422557
$862$	17.1289	0.583412
$863$	−23.5677	−0.802255	−0.401127	−	0.916022i	$-0.631382\pi$
$863$	−23.5677	−0.802255	−0.401127	+	0.916022i	$0.631382\pi$
$864$	−1.00000	−0.0340207
$865$	−12.9377	−0.439897
$866$	−25.0686	−0.851867
$867$	18.5367	0.629538
$868$	2.74453	0.0931553
$869$	2.86416	0.0971599
$870$	0.881455	0.0298841
$871$	−3.43899	−0.116526
$872$	−7.67530	−0.259918
$873$	−8.28427	−0.280380
$874$	3.48434	0.117860
$875$	2.06076	0.0696664
$876$	−6.69346	−0.226151
$877$	−0.695364	−0.0234808	−0.0117404	−	0.999931i	$-0.503737\pi$
$877$	−0.695364	−0.0234808	−0.0117404	+	0.999931i	$0.503737\pi$
$878$	−10.7003	−0.361117
$879$	−23.4459	−0.790811
$880$	2.80618	0.0945962
$881$	42.6237	1.43603	0.718014	−	0.696029i	$-0.245051\pi$
$881$	42.6237	1.43603	0.718014	+	0.696029i	$0.245051\pi$
$882$	6.93575	0.233539
$883$	−11.8756	−0.399647	−0.199824	−	0.979832i	$-0.564037\pi$
$883$	−11.8756	−0.399647	−0.199824	+	0.979832i	$0.564037\pi$
$884$	−12.7221	−0.427891
$885$	−7.70092	−0.258864
$886$	30.6408	1.02940
$887$	30.6042	1.02759	0.513795	−	0.857913i	$-0.328239\pi$
$887$	30.6042	1.02759	0.513795	+	0.857913i	$0.328239\pi$
$888$	−9.11932	−0.306024
$889$	2.82423	0.0947216
$890$	−1.21671	−0.0407843
$891$	−3.18357	−0.106654
$892$	12.8044	0.428722
$893$	−0.534766	−0.0178953
$894$	8.53366	0.285408
$895$	20.1334	0.672987
$896$	−0.253486	−0.00846836
$897$	−2.13413	−0.0712566
$898$	−15.5745	−0.519726
$899$	10.8272	0.361106
$900$	−4.22304	−0.140768
$901$	61.7437	2.05698
$902$	−15.5722	−0.518496
$903$	−2.27387	−0.0756697
$904$	−13.3247	−0.443173
$905$	−8.52110	−0.283251
$906$	21.3556	0.709492
$907$	−19.3729	−0.643266	−0.321633	−	0.946864i	$-0.604232\pi$
$907$	−19.3729	−0.643266	−0.321633	+	0.946864i	$0.604232\pi$
$908$	−8.25395	−0.273917
$909$	−4.05768	−0.134585
$910$	0.476842	0.0158072
$911$	−1.01847	−0.0337436	−0.0168718	−	0.999858i	$-0.505371\pi$
$911$	−1.01847	−0.0337436	−0.0168718	+	0.999858i	$0.505371\pi$
$912$	3.48434	0.115378
$913$	54.2167	1.79431
$914$	−4.21570	−0.139443
$915$	9.72136	0.321378
$916$	−10.3469	−0.341871
$917$	1.44132	0.0475967
$918$	5.96126	0.196751
$919$	56.5668	1.86597	0.932983	−	0.359920i	$-0.117196\pi$
$919$	56.5668	1.86597	0.932983	+	0.359920i	$0.117196\pi$
$920$	−0.881455	−0.0290607
$921$	−28.0138	−0.923085
$922$	−3.39202	−0.111710
$923$	31.8310	1.04773
$924$	−0.806990	−0.0265480
$925$	−38.5112	−1.26624
$926$	−7.79237	−0.256073
$927$	−18.6717	−0.613258
$928$	−1.00000	−0.0328266
$929$	−11.6951	−0.383704	−0.191852	−	0.981424i	$-0.561449\pi$
$929$	−11.6951	−0.383704	−0.191852	+	0.981424i	$0.561449\pi$
$930$	9.54364	0.312948
$931$	−24.1665	−0.792025
$932$	−0.750585	−0.0245862
$933$	16.2662	0.532530
$934$	15.9168	0.520813
$935$	−16.7284	−0.547076
$936$	−2.13413	−0.0697563
$937$	−40.6021	−1.32641	−0.663206	−	0.748437i	$-0.730805\pi$
$937$	−40.6021	−1.32641	−0.663206	+	0.748437i	$0.730805\pi$
$938$	0.408473	0.0133371
$939$	21.4754	0.700824
$940$	0.135283	0.00441244
$941$	7.85274	0.255992	0.127996	−	0.991775i	$-0.459146\pi$
$941$	7.85274	0.255992	0.127996	+	0.991775i	$0.459146\pi$
$942$	10.6386	0.346625
$943$	4.89141	0.159286
$944$	8.73661	0.284352
$945$	−0.223436	−0.00726838
$946$	−28.5580	−0.928500
$947$	8.09518	0.263058	0.131529	−	0.991312i	$-0.458011\pi$
$947$	8.09518	0.263058	0.131529	+	0.991312i	$0.458011\pi$
$948$	−0.899667	−0.0292198
$949$	−14.2847	−0.463702
$950$	14.7145	0.477402
$951$	−17.9480	−0.582004
$952$	1.51109	0.0489749
$953$	22.5870	0.731665	0.365833	−	0.930681i	$-0.380784\pi$
$953$	22.5870	0.731665	0.365833	+	0.930681i	$0.380784\pi$
$954$	10.3575	0.335336
$955$	24.2395	0.784372
$956$	0.641568	0.0207498
$957$	−3.18357	−0.102910
$958$	23.7269	0.766582
$959$	−1.00227	−0.0323649
$960$	−0.881455	−0.0284488
$961$	86.2273	2.78152
$962$	−19.4618	−0.627474
$963$	−14.5325	−0.468302
$964$	22.2474	0.716541
$965$	15.1600	0.488017
$966$	0.253486	0.00815577
$967$	0.714315	0.0229708	0.0114854	−	0.999934i	$-0.496344\pi$
$967$	0.714315	0.0229708	0.0114854	+	0.999934i	$0.496344\pi$
$968$	0.864853	0.0277974
$969$	−20.7711	−0.667264
$970$	−7.30221	−0.234460
$971$	8.91387	0.286060	0.143030	−	0.989718i	$-0.454315\pi$
$971$	8.91387	0.286060	0.143030	+	0.989718i	$0.454315\pi$
$972$	1.00000	0.0320750
$973$	−0.145287	−0.00465770
$974$	−42.2746	−1.35456
$975$	−9.01252	−0.288631
$976$	−11.0288	−0.353022
$977$	−10.6634	−0.341153	−0.170577	−	0.985344i	$-0.554563\pi$
$977$	−10.6634	−0.341153	−0.170577	+	0.985344i	$0.554563\pi$
$978$	−9.38753	−0.300180
$979$	4.39443	0.140447
$980$	6.11354	0.195290
$981$	7.67530	0.245053
$982$	−5.73789	−0.183103
$983$	−25.2280	−0.804650	−0.402325	−	0.915497i	$-0.631798\pi$
$983$	−25.2280	−0.804650	−0.402325	+	0.915497i	$0.631798\pi$
$984$	4.89141	0.155932
$985$	−3.32703	−0.106008
$986$	5.96126	0.189845
$987$	−0.0389042	−0.00123833
$988$	7.43604	0.236572
$989$	8.97041	0.285242
$990$	−2.80618	−0.0891861
$991$	11.0858	0.352151	0.176076	−	0.984377i	$-0.443660\pi$
$991$	11.0858	0.352151	0.176076	+	0.984377i	$0.443660\pi$
$992$	−10.8272	−0.343762
$993$	−29.9727	−0.951156
$994$	−3.78079	−0.119919
$995$	−12.3869	−0.392691
$996$	−17.0301	−0.539620
$997$	27.6455	0.875542	0.437771	−	0.899087i	$-0.355768\pi$
$997$	27.6455	0.875542	0.437771	+	0.899087i	$0.355768\pi$
$998$	−28.6996	−0.908469
$999$	9.11932	0.288523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bi.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bi.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.881455 q^{5} -1.00000 q^{6} +0.253486 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.881455 q^{5} -1.00000 q^{6} +0.253486 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.881455 q^{10} -3.18357 q^{11} +1.00000 q^{12} +2.13413 q^{13} -0.253486 q^{14} -0.881455 q^{15} +1.00000 q^{16} -5.96126 q^{17} -1.00000 q^{18} +3.48434 q^{19} -0.881455 q^{20} +0.253486 q^{21} +3.18357 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.22304 q^{25} -2.13413 q^{26} +1.00000 q^{27} +0.253486 q^{28} +1.00000 q^{29} +0.881455 q^{30} +10.8272 q^{31} -1.00000 q^{32} -3.18357 q^{33} +5.96126 q^{34} -0.223436 q^{35} +1.00000 q^{36} +9.11932 q^{37} -3.48434 q^{38} +2.13413 q^{39} +0.881455 q^{40} -4.89141 q^{41} -0.253486 q^{42} -8.97041 q^{43} -3.18357 q^{44} -0.881455 q^{45} +1.00000 q^{46} -0.153477 q^{47} +1.00000 q^{48} -6.93575 q^{49} +4.22304 q^{50} -5.96126 q^{51} +2.13413 q^{52} -10.3575 q^{53} -1.00000 q^{54} +2.80618 q^{55} -0.253486 q^{56} +3.48434 q^{57} -1.00000 q^{58} +8.73661 q^{59} -0.881455 q^{60} -11.0288 q^{61} -10.8272 q^{62} +0.253486 q^{63} +1.00000 q^{64} -1.88114 q^{65} +3.18357 q^{66} -1.61142 q^{67} -5.96126 q^{68} -1.00000 q^{69} +0.223436 q^{70} +14.9152 q^{71} -1.00000 q^{72} -6.69346 q^{73} -9.11932 q^{74} -4.22304 q^{75} +3.48434 q^{76} -0.806990 q^{77} -2.13413 q^{78} -0.899667 q^{79} -0.881455 q^{80} +1.00000 q^{81} +4.89141 q^{82} -17.0301 q^{83} +0.253486 q^{84} +5.25458 q^{85} +8.97041 q^{86} +1.00000 q^{87} +3.18357 q^{88} -1.38034 q^{89} +0.881455 q^{90} +0.540972 q^{91} -1.00000 q^{92} +10.8272 q^{93} +0.153477 q^{94} -3.07129 q^{95} -1.00000 q^{96} -8.28427 q^{97} +6.93575 q^{98} -3.18357 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 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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