LMFDB - Embedded newform 8042.2.a.a.1.40

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.40
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.391292 q^{3} +1.00000 q^{4} +3.34195 q^{5} +0.391292 q^{6} +0.999654 q^{7} +1.00000 q^{8} -2.84689 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.391292 q^{3} +1.00000 q^{4} +3.34195 q^{5} +0.391292 q^{6} +0.999654 q^{7} +1.00000 q^{8} -2.84689 q^{9} +3.34195 q^{10} -3.10216 q^{11} +0.391292 q^{12} -3.80522 q^{13} +0.999654 q^{14} +1.30768 q^{15} +1.00000 q^{16} -3.58273 q^{17} -2.84689 q^{18} -5.00864 q^{19} +3.34195 q^{20} +0.391156 q^{21} -3.10216 q^{22} -8.59178 q^{23} +0.391292 q^{24} +6.16865 q^{25} -3.80522 q^{26} -2.28784 q^{27} +0.999654 q^{28} +5.44439 q^{29} +1.30768 q^{30} +1.81344 q^{31} +1.00000 q^{32} -1.21385 q^{33} -3.58273 q^{34} +3.34080 q^{35} -2.84689 q^{36} +3.23667 q^{37} -5.00864 q^{38} -1.48895 q^{39} +3.34195 q^{40} -6.06772 q^{41} +0.391156 q^{42} +4.09193 q^{43} -3.10216 q^{44} -9.51417 q^{45} -8.59178 q^{46} -1.12172 q^{47} +0.391292 q^{48} -6.00069 q^{49} +6.16865 q^{50} -1.40189 q^{51} -3.80522 q^{52} -1.09695 q^{53} -2.28784 q^{54} -10.3673 q^{55} +0.999654 q^{56} -1.95984 q^{57} +5.44439 q^{58} +6.63738 q^{59} +1.30768 q^{60} -6.21437 q^{61} +1.81344 q^{62} -2.84591 q^{63} +1.00000 q^{64} -12.7169 q^{65} -1.21385 q^{66} +16.0803 q^{67} -3.58273 q^{68} -3.36189 q^{69} +3.34080 q^{70} -5.85751 q^{71} -2.84689 q^{72} +3.07134 q^{73} +3.23667 q^{74} +2.41374 q^{75} -5.00864 q^{76} -3.10109 q^{77} -1.48895 q^{78} -17.5919 q^{79} +3.34195 q^{80} +7.64546 q^{81} -6.06772 q^{82} -3.75190 q^{83} +0.391156 q^{84} -11.9733 q^{85} +4.09193 q^{86} +2.13035 q^{87} -3.10216 q^{88} +1.81593 q^{89} -9.51417 q^{90} -3.80390 q^{91} -8.59178 q^{92} +0.709583 q^{93} -1.12172 q^{94} -16.7386 q^{95} +0.391292 q^{96} +2.83644 q^{97} -6.00069 q^{98} +8.83151 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	0.391292	0.225912	0.112956	−	0.993600i	$-0.463968\pi$
$3$	0.391292	0.225912	0.112956	+	0.993600i	$0.463968\pi$
$4$	1.00000	0.500000
$5$	3.34195	1.49457	0.747283	−	0.664506i	$-0.231358\pi$
$5$	3.34195	1.49457	0.747283	+	0.664506i	$0.231358\pi$
$6$	0.391292	0.159744
$7$	0.999654	0.377834	0.188917	−	0.981993i	$-0.439502\pi$
$7$	0.999654	0.377834	0.188917	+	0.981993i	$0.439502\pi$
$8$	1.00000	0.353553
$9$	−2.84689	−0.948964
$10$	3.34195	1.05682
$11$	−3.10216	−0.935336	−0.467668	−	0.883904i	$-0.654906\pi$
$11$	−3.10216	−0.935336	−0.467668	+	0.883904i	$0.654906\pi$
$12$	0.391292	0.112956
$13$	−3.80522	−1.05538	−0.527689	−	0.849438i	$-0.676941\pi$
$13$	−3.80522	−1.05538	−0.527689	+	0.849438i	$0.676941\pi$
$14$	0.999654	0.267169
$15$	1.30768	0.337641
$16$	1.00000	0.250000
$17$	−3.58273	−0.868941	−0.434470	−	0.900686i	$-0.643064\pi$
$17$	−3.58273	−0.868941	−0.434470	+	0.900686i	$0.643064\pi$
$18$	−2.84689	−0.671019
$19$	−5.00864	−1.14906	−0.574530	−	0.818483i	$-0.694815\pi$
$19$	−5.00864	−1.14906	−0.574530	+	0.818483i	$0.694815\pi$
$20$	3.34195	0.747283
$21$	0.391156	0.0853573
$22$	−3.10216	−0.661383
$23$	−8.59178	−1.79151	−0.895755	−	0.444547i	$-0.853365\pi$
$23$	−8.59178	−1.79151	−0.895755	+	0.444547i	$0.853365\pi$
$24$	0.391292	0.0798721
$25$	6.16865	1.23373
$26$	−3.80522	−0.746265
$27$	−2.28784	−0.440295
$28$	0.999654	0.188917
$29$	5.44439	1.01100	0.505499	−	0.862827i	$-0.331308\pi$
$29$	5.44439	1.01100	0.505499	+	0.862827i	$0.331308\pi$
$30$	1.30768	0.238748
$31$	1.81344	0.325703	0.162852	−	0.986651i	$-0.447931\pi$
$31$	1.81344	0.325703	0.162852	+	0.986651i	$0.447931\pi$
$32$	1.00000	0.176777
$33$	−1.21385	−0.211304
$34$	−3.58273	−0.614434
$35$	3.34080	0.564698
$36$	−2.84689	−0.474482
$37$	3.23667	0.532105	0.266052	−	0.963959i	$-0.414281\pi$
$37$	3.23667	0.532105	0.266052	+	0.963959i	$0.414281\pi$
$38$	−5.00864	−0.812509
$39$	−1.48895	−0.238423
$40$	3.34195	0.528409
$41$	−6.06772	−0.947619	−0.473809	−	0.880627i	$-0.657121\pi$
$41$	−6.06772	−0.947619	−0.473809	+	0.880627i	$0.657121\pi$
$42$	0.391156	0.0603567
$43$	4.09193	0.624013	0.312007	−	0.950080i	$-0.398999\pi$
$43$	4.09193	0.624013	0.312007	+	0.950080i	$0.398999\pi$
$44$	−3.10216	−0.467668
$45$	−9.51417	−1.41829
$46$	−8.59178	−1.26679
$47$	−1.12172	−0.163619	−0.0818096	−	0.996648i	$-0.526070\pi$
$47$	−1.12172	−0.163619	−0.0818096	+	0.996648i	$0.526070\pi$
$48$	0.391292	0.0564781
$49$	−6.00069	−0.857242
$50$	6.16865	0.872378
$51$	−1.40189	−0.196304
$52$	−3.80522	−0.527689
$53$	−1.09695	−0.150678	−0.0753392	−	0.997158i	$-0.524004\pi$
$53$	−1.09695	−0.150678	−0.0753392	+	0.997158i	$0.524004\pi$
$54$	−2.28784	−0.311336
$55$	−10.3673	−1.39792
$56$	0.999654	0.133584
$57$	−1.95984	−0.259587
$58$	5.44439	0.714884
$59$	6.63738	0.864114	0.432057	−	0.901846i	$-0.357788\pi$
$59$	6.63738	0.864114	0.432057	+	0.901846i	$0.357788\pi$
$60$	1.30768	0.168821
$61$	−6.21437	−0.795668	−0.397834	−	0.917457i	$-0.630238\pi$
$61$	−6.21437	−0.795668	−0.397834	+	0.917457i	$0.630238\pi$
$62$	1.81344	0.230307
$63$	−2.84591	−0.358550
$64$	1.00000	0.125000
$65$	−12.7169	−1.57733
$66$	−1.21385	−0.149414
$67$	16.0803	1.96452	0.982259	−	0.187530i	$-0.0600482\pi$
$67$	16.0803	1.96452	0.982259	+	0.187530i	$0.0600482\pi$
$68$	−3.58273	−0.434470
$69$	−3.36189	−0.404724
$70$	3.34080	0.399301
$71$	−5.85751	−0.695158	−0.347579	−	0.937651i	$-0.612996\pi$
$71$	−5.85751	−0.695158	−0.347579	+	0.937651i	$0.612996\pi$
$72$	−2.84689	−0.335509
$73$	3.07134	0.359473	0.179736	−	0.983715i	$-0.442476\pi$
$73$	3.07134	0.359473	0.179736	+	0.983715i	$0.442476\pi$
$74$	3.23667	0.376255
$75$	2.41374	0.278715
$76$	−5.00864	−0.574530
$77$	−3.10109	−0.353401
$78$	−1.48895	−0.168590
$79$	−17.5919	−1.97924	−0.989620	−	0.143710i	$-0.954097\pi$
$79$	−17.5919	−1.97924	−0.989620	+	0.143710i	$0.954097\pi$
$80$	3.34195	0.373642
$81$	7.64546	0.849496
$82$	−6.06772	−0.670067
$83$	−3.75190	−0.411824	−0.205912	−	0.978570i	$-0.566016\pi$
$83$	−3.75190	−0.411824	−0.205912	+	0.978570i	$0.566016\pi$
$84$	0.391156	0.0426786
$85$	−11.9733	−1.29869
$86$	4.09193	0.441244
$87$	2.13035	0.228397
$88$	−3.10216	−0.330691
$89$	1.81593	0.192488	0.0962440	−	0.995358i	$-0.469317\pi$
$89$	1.81593	0.192488	0.0962440	+	0.995358i	$0.469317\pi$
$90$	−9.51417	−1.00288
$91$	−3.80390	−0.398757
$92$	−8.59178	−0.895755
$93$	0.709583	0.0735803
$94$	−1.12172	−0.115696
$95$	−16.7386	−1.71735
$96$	0.391292	0.0399360
$97$	2.83644	0.287997	0.143999	−	0.989578i	$-0.454004\pi$
$97$	2.83644	0.287997	0.143999	+	0.989578i	$0.454004\pi$
$98$	−6.00069	−0.606161
$99$	8.83151	0.887600
$100$	6.16865	0.616865
$101$	−14.5089	−1.44369	−0.721846	−	0.692054i	$-0.756706\pi$
$101$	−14.5089	−1.44369	−0.721846	+	0.692054i	$0.756706\pi$
$102$	−1.40189	−0.138808
$103$	4.68676	0.461800	0.230900	−	0.972978i	$-0.425833\pi$
$103$	4.68676	0.461800	0.230900	+	0.972978i	$0.425833\pi$
$104$	−3.80522	−0.373132
$105$	1.30723	0.127572
$106$	−1.09695	−0.106546
$107$	19.5626	1.89118	0.945592	−	0.325354i	$-0.105483\pi$
$107$	19.5626	1.89118	0.945592	+	0.325354i	$0.105483\pi$
$108$	−2.28784	−0.220147
$109$	−18.4062	−1.76299	−0.881497	−	0.472189i	$-0.843464\pi$
$109$	−18.4062	−1.76299	−0.881497	+	0.472189i	$0.843464\pi$
$110$	−10.3673	−0.988480
$111$	1.26648	0.120209
$112$	0.999654	0.0944584
$113$	5.71145	0.537288	0.268644	−	0.963240i	$-0.413424\pi$
$113$	5.71145	0.537288	0.268644	+	0.963240i	$0.413424\pi$
$114$	−1.95984	−0.183556
$115$	−28.7133	−2.67753
$116$	5.44439	0.505499
$117$	10.8330	1.00151
$118$	6.63738	0.611021
$119$	−3.58149	−0.328315
$120$	1.30768	0.119374
$121$	−1.37661	−0.125146
$122$	−6.21437	−0.562622
$123$	−2.37425	−0.214079
$124$	1.81344	0.162852
$125$	3.90556	0.349324
$126$	−2.84591	−0.253533
$127$	−12.0116	−1.06586	−0.532930	−	0.846160i	$-0.678909\pi$
$127$	−12.0116	−1.06586	−0.532930	+	0.846160i	$0.678909\pi$
$128$	1.00000	0.0883883
$129$	1.60114	0.140972
$130$	−12.7169	−1.11534
$131$	6.20516	0.542148	0.271074	−	0.962559i	$-0.412621\pi$
$131$	6.20516	0.542148	0.271074	+	0.962559i	$0.412621\pi$
$132$	−1.21385	−0.105652
$133$	−5.00691	−0.434154
$134$	16.0803	1.38912
$135$	−7.64585	−0.658050
$136$	−3.58273	−0.307217
$137$	−1.35172	−0.115486	−0.0577428	−	0.998331i	$-0.518390\pi$
$137$	−1.35172	−0.115486	−0.0577428	+	0.998331i	$0.518390\pi$
$138$	−3.36189	−0.286183
$139$	−17.9797	−1.52502	−0.762508	−	0.646979i	$-0.776032\pi$
$139$	−17.9797	−1.52502	−0.762508	+	0.646979i	$0.776032\pi$
$140$	3.34080	0.282349
$141$	−0.438918	−0.0369636
$142$	−5.85751	−0.491551
$143$	11.8044	0.987133
$144$	−2.84689	−0.237241
$145$	18.1949	1.51100
$146$	3.07134	0.254185
$147$	−2.34802	−0.193661
$148$	3.23667	0.266052
$149$	−24.3233	−1.99264	−0.996320	−	0.0857115i	$-0.972684\pi$
$149$	−24.3233	−1.99264	−0.996320	+	0.0857115i	$0.972684\pi$
$150$	2.41374	0.197081
$151$	6.59106	0.536373	0.268186	−	0.963367i	$-0.413576\pi$
$151$	6.59106	0.536373	0.268186	+	0.963367i	$0.413576\pi$
$152$	−5.00864	−0.406254
$153$	10.1997	0.824593
$154$	−3.10109	−0.249893
$155$	6.06042	0.486785
$156$	−1.48895	−0.119211
$157$	−0.525903	−0.0419716	−0.0209858	−	0.999780i	$-0.506680\pi$
$157$	−0.525903	−0.0419716	−0.0209858	+	0.999780i	$0.506680\pi$
$158$	−17.5919	−1.39953
$159$	−0.429229	−0.0340401
$160$	3.34195	0.264205
$161$	−8.58881	−0.676893
$162$	7.64546	0.600684
$163$	−3.94493	−0.308991	−0.154495	−	0.987994i	$-0.549375\pi$
$163$	−3.94493	−0.308991	−0.154495	+	0.987994i	$0.549375\pi$
$164$	−6.06772	−0.473809
$165$	−4.05663	−0.315808
$166$	−3.75190	−0.291204
$167$	−14.5358	−1.12482	−0.562408	−	0.826860i	$-0.690125\pi$
$167$	−14.5358	−1.12482	−0.562408	+	0.826860i	$0.690125\pi$
$168$	0.391156	0.0301784
$169$	1.47968	0.113821
$170$	−11.9733	−0.918312
$171$	14.2591	1.09042
$172$	4.09193	0.312007
$173$	−8.75438	−0.665583	−0.332792	−	0.943000i	$-0.607991\pi$
$173$	−8.75438	−0.665583	−0.332792	+	0.943000i	$0.607991\pi$
$174$	2.13035	0.161501
$175$	6.16651	0.466144
$176$	−3.10216	−0.233834
$177$	2.59715	0.195214
$178$	1.81593	0.136110
$179$	−12.1903	−0.911148	−0.455574	−	0.890198i	$-0.650566\pi$
$179$	−12.1903	−0.911148	−0.455574	+	0.890198i	$0.650566\pi$
$180$	−9.51417	−0.709145
$181$	−13.6918	−1.01771	−0.508853	−	0.860854i	$-0.669930\pi$
$181$	−13.6918	−1.01771	−0.508853	+	0.860854i	$0.669930\pi$
$182$	−3.80390	−0.281964
$183$	−2.43163	−0.179751
$184$	−8.59178	−0.633395
$185$	10.8168	0.795266
$186$	0.709583	0.0520292
$187$	11.1142	0.812751
$188$	−1.12172	−0.0818096
$189$	−2.28705	−0.166358
$190$	−16.7386	−1.21435
$191$	23.8815	1.72800	0.864002	−	0.503488i	$-0.167950\pi$
$191$	23.8815	1.72800	0.864002	+	0.503488i	$0.167950\pi$
$192$	0.391292	0.0282390
$193$	−14.4703	−1.04159	−0.520796	−	0.853681i	$-0.674365\pi$
$193$	−14.4703	−1.04159	−0.520796	+	0.853681i	$0.674365\pi$
$194$	2.83644	0.203645
$195$	−4.97600	−0.356339
$196$	−6.00069	−0.428621
$197$	−0.272400	−0.0194077	−0.00970385	−	0.999953i	$-0.503089\pi$
$197$	−0.272400	−0.0194077	−0.00970385	+	0.999953i	$0.503089\pi$
$198$	8.83151	0.627628
$199$	−6.10321	−0.432645	−0.216323	−	0.976322i	$-0.569406\pi$
$199$	−6.10321	−0.432645	−0.216323	+	0.976322i	$0.569406\pi$
$200$	6.16865	0.436189
$201$	6.29208	0.443809
$202$	−14.5089	−1.02084
$203$	5.44251	0.381989
$204$	−1.40189	−0.0981522
$205$	−20.2780	−1.41628
$206$	4.68676	0.326542
$207$	24.4599	1.70008
$208$	−3.80522	−0.263844
$209$	15.5376	1.07476
$210$	1.30723	0.0902071
$211$	−12.6393	−0.870128	−0.435064	−	0.900399i	$-0.643274\pi$
$211$	−12.6393	−0.870128	−0.435064	+	0.900399i	$0.643274\pi$
$212$	−1.09695	−0.0753392
$213$	−2.29199	−0.157045
$214$	19.5626	1.33727
$215$	13.6750	0.932630
$216$	−2.28784	−0.155668
$217$	1.81281	0.123062
$218$	−18.4062	−1.24663
$219$	1.20179	0.0812093
$220$	−10.3673	−0.698961
$221$	13.6331	0.917060
$222$	1.26648	0.0850006
$223$	27.8049	1.86196	0.930978	−	0.365074i	$-0.118956\pi$
$223$	27.8049	1.86196	0.930978	+	0.365074i	$0.118956\pi$
$224$	0.999654	0.0667922
$225$	−17.5615	−1.17076
$226$	5.71145	0.379920
$227$	17.5510	1.16490	0.582451	−	0.812865i	$-0.302094\pi$
$227$	17.5510	1.16490	0.582451	+	0.812865i	$0.302094\pi$
$228$	−1.95984	−0.129794
$229$	10.1502	0.670746	0.335373	−	0.942085i	$-0.391138\pi$
$229$	10.1502	0.670746	0.335373	+	0.942085i	$0.391138\pi$
$230$	−28.7133	−1.89330
$231$	−1.21343	−0.0798378
$232$	5.44439	0.357442
$233$	14.9890	0.981962	0.490981	−	0.871170i	$-0.336638\pi$
$233$	14.9890	0.981962	0.490981	+	0.871170i	$0.336638\pi$
$234$	10.8330	0.708178
$235$	−3.74872	−0.244540
$236$	6.63738	0.432057
$237$	−6.88355	−0.447135
$238$	−3.58149	−0.232154
$239$	19.6731	1.27255	0.636274	−	0.771463i	$-0.280475\pi$
$239$	19.6731	1.27255	0.636274	+	0.771463i	$0.280475\pi$
$240$	1.30768	0.0844103
$241$	23.2762	1.49935	0.749677	−	0.661804i	$-0.230209\pi$
$241$	23.2762	1.49935	0.749677	+	0.661804i	$0.230209\pi$
$242$	−1.37661	−0.0884918
$243$	9.85512	0.632206
$244$	−6.21437	−0.397834
$245$	−20.0540	−1.28120
$246$	−2.37425	−0.151377
$247$	19.0590	1.21269
$248$	1.81344	0.115153
$249$	−1.46809	−0.0930362
$250$	3.90556	0.247009
$251$	8.88267	0.560669	0.280334	−	0.959902i	$-0.409555\pi$
$251$	8.88267	0.560669	0.280334	+	0.959902i	$0.409555\pi$
$252$	−2.84591	−0.179275
$253$	26.6531	1.67566
$254$	−12.0116	−0.753676
$255$	−4.68506	−0.293390
$256$	1.00000	0.0625000
$257$	−9.59562	−0.598558	−0.299279	−	0.954166i	$-0.596746\pi$
$257$	−9.59562	−0.598558	−0.299279	+	0.954166i	$0.596746\pi$
$258$	1.60114	0.0996825
$259$	3.23555	0.201047
$260$	−12.7169	−0.788666
$261$	−15.4996	−0.959401
$262$	6.20516	0.383356
$263$	−12.2294	−0.754097	−0.377049	−	0.926193i	$-0.623061\pi$
$263$	−12.2294	−0.754097	−0.377049	+	0.926193i	$0.623061\pi$
$264$	−1.21385	−0.0747072
$265$	−3.66597	−0.225199
$266$	−5.00691	−0.306993
$267$	0.710558	0.0434854
$268$	16.0803	0.982259
$269$	17.1343	1.04470	0.522349	−	0.852732i	$-0.325056\pi$
$269$	17.1343	1.04470	0.522349	+	0.852732i	$0.325056\pi$
$270$	−7.64585	−0.465312
$271$	24.5519	1.49142	0.745710	−	0.666271i	$-0.232111\pi$
$271$	24.5519	1.49142	0.745710	+	0.666271i	$0.232111\pi$
$272$	−3.58273	−0.217235
$273$	−1.48843	−0.0900842
$274$	−1.35172	−0.0816606
$275$	−19.1361	−1.15395
$276$	−3.36189	−0.202362
$277$	21.2812	1.27866	0.639332	−	0.768931i	$-0.279211\pi$
$277$	21.2812	1.27866	0.639332	+	0.768931i	$0.279211\pi$
$278$	−17.9797	−1.07835
$279$	−5.16266	−0.309080
$280$	3.34080	0.199651
$281$	2.08298	0.124260	0.0621301	−	0.998068i	$-0.480211\pi$
$281$	2.08298	0.124260	0.0621301	+	0.998068i	$0.480211\pi$
$282$	−0.438918	−0.0261372
$283$	−3.45367	−0.205299	−0.102650	−	0.994718i	$-0.532732\pi$
$283$	−3.45367	−0.205299	−0.102650	+	0.994718i	$0.532732\pi$
$284$	−5.85751	−0.347579
$285$	−6.54969	−0.387970
$286$	11.8044	0.698008
$287$	−6.06562	−0.358042
$288$	−2.84689	−0.167755
$289$	−4.16402	−0.244942
$290$	18.1949	1.06844
$291$	1.10988	0.0650621
$292$	3.07134	0.179736
$293$	5.35457	0.312817	0.156409	−	0.987692i	$-0.450008\pi$
$293$	5.35457	0.312817	0.156409	+	0.987692i	$0.450008\pi$
$294$	−2.34802	−0.136939
$295$	22.1818	1.29148
$296$	3.23667	0.188127
$297$	7.09724	0.411824
$298$	−24.3233	−1.40901
$299$	32.6936	1.89072
$300$	2.41374	0.139357
$301$	4.09051	0.235773
$302$	6.59106	0.379273
$303$	−5.67722	−0.326148
$304$	−5.00864	−0.287265
$305$	−20.7681	−1.18918
$306$	10.1997	0.583075
$307$	6.80196	0.388209	0.194104	−	0.980981i	$-0.437820\pi$
$307$	6.80196	0.388209	0.194104	+	0.980981i	$0.437820\pi$
$308$	−3.10109	−0.176701
$309$	1.83389	0.104326
$310$	6.06042	0.344209
$311$	8.28656	0.469888	0.234944	−	0.972009i	$-0.424509\pi$
$311$	8.28656	0.469888	0.234944	+	0.972009i	$0.424509\pi$
$312$	−1.48895	−0.0842952
$313$	27.1892	1.53682	0.768411	−	0.639957i	$-0.221048\pi$
$313$	27.1892	1.53682	0.768411	+	0.639957i	$0.221048\pi$
$314$	−0.525903	−0.0296784
$315$	−9.51088	−0.535877
$316$	−17.5919	−0.989620
$317$	−3.39106	−0.190461	−0.0952305	−	0.995455i	$-0.530359\pi$
$317$	−3.39106	−0.190461	−0.0952305	+	0.995455i	$0.530359\pi$
$318$	−0.429229	−0.0240700
$319$	−16.8894	−0.945623
$320$	3.34195	0.186821
$321$	7.65467	0.427242
$322$	−8.58881	−0.478636
$323$	17.9446	0.998466
$324$	7.64546	0.424748
$325$	−23.4730	−1.30205
$326$	−3.94493	−0.218489
$327$	−7.20220	−0.398282
$328$	−6.06772	−0.335034
$329$	−1.12133	−0.0618208
$330$	−4.05663	−0.223310
$331$	−2.69147	−0.147937	−0.0739684	−	0.997261i	$-0.523566\pi$
$331$	−2.69147	−0.147937	−0.0739684	+	0.997261i	$0.523566\pi$
$332$	−3.75190	−0.205912
$333$	−9.21443	−0.504948
$334$	−14.5358	−0.795365
$335$	53.7395	2.93610
$336$	0.391156	0.0213393
$337$	−0.0378310	−0.00206079	−0.00103039	−	0.999999i	$-0.500328\pi$
$337$	−0.0378310	−0.00206079	−0.00103039	+	0.999999i	$0.500328\pi$
$338$	1.47968	0.0804839
$339$	2.23484	0.121380
$340$	−11.9733	−0.649345
$341$	−5.62557	−0.304642
$342$	14.2591	0.771041
$343$	−12.9962	−0.701728
$344$	4.09193	0.220622
$345$	−11.2353	−0.604888
$346$	−8.75438	−0.470639
$347$	−8.93677	−0.479751	−0.239876	−	0.970804i	$-0.577107\pi$
$347$	−8.93677	−0.479751	−0.239876	+	0.970804i	$0.577107\pi$
$348$	2.13035	0.114199
$349$	−9.14320	−0.489424	−0.244712	−	0.969596i	$-0.578693\pi$
$349$	−9.14320	−0.489424	−0.244712	+	0.969596i	$0.578693\pi$
$350$	6.16651	0.329614
$351$	8.70573	0.464677
$352$	−3.10216	−0.165346
$353$	−10.2998	−0.548204	−0.274102	−	0.961701i	$-0.588381\pi$
$353$	−10.2998	−0.548204	−0.274102	+	0.961701i	$0.588381\pi$
$354$	2.59715	0.138037
$355$	−19.5755	−1.03896
$356$	1.81593	0.0962440
$357$	−1.40141	−0.0741704
$358$	−12.1903	−0.644279
$359$	−8.20472	−0.433028	−0.216514	−	0.976279i	$-0.569469\pi$
$359$	−8.20472	−0.433028	−0.216514	+	0.976279i	$0.569469\pi$
$360$	−9.51417	−0.501441
$361$	6.08648	0.320341
$362$	−13.6918	−0.719626
$363$	−0.538656	−0.0282721
$364$	−3.80390	−0.199379
$365$	10.2643	0.537256
$366$	−2.43163	−0.127103
$367$	8.48297	0.442807	0.221404	−	0.975182i	$-0.428936\pi$
$367$	8.48297	0.442807	0.221404	+	0.975182i	$0.428936\pi$
$368$	−8.59178	−0.447878
$369$	17.2741	0.899255
$370$	10.8168	0.562338
$371$	−1.09658	−0.0569313
$372$	0.709583	0.0367902
$373$	−13.1048	−0.678539	−0.339269	−	0.940689i	$-0.610180\pi$
$373$	−13.1048	−0.678539	−0.339269	+	0.940689i	$0.610180\pi$
$374$	11.1142	0.574702
$375$	1.52821	0.0789166
$376$	−1.12172	−0.0578481
$377$	−20.7171	−1.06698
$378$	−2.28705	−0.117633
$379$	−22.6951	−1.16577	−0.582884	−	0.812555i	$-0.698076\pi$
$379$	−22.6951	−1.16577	−0.582884	+	0.812555i	$0.698076\pi$
$380$	−16.7386	−0.858674
$381$	−4.70005	−0.240791
$382$	23.8815	1.22188
$383$	19.3666	0.989586	0.494793	−	0.869011i	$-0.335244\pi$
$383$	19.3666	0.989586	0.494793	+	0.869011i	$0.335244\pi$
$384$	0.391292	0.0199680
$385$	−10.3637	−0.528182
$386$	−14.4703	−0.736517
$387$	−11.6493	−0.592166
$388$	2.83644	0.143999
$389$	−8.79513	−0.445931	−0.222965	−	0.974826i	$-0.571574\pi$
$389$	−8.79513	−0.445931	−0.222965	+	0.974826i	$0.571574\pi$
$390$	−4.97600	−0.251970
$391$	30.7821	1.55672
$392$	−6.00069	−0.303081
$393$	2.42803	0.122478
$394$	−0.272400	−0.0137233
$395$	−58.7912	−2.95811
$396$	8.83151	0.443800
$397$	−8.24618	−0.413864	−0.206932	−	0.978355i	$-0.566348\pi$
$397$	−8.24618	−0.413864	−0.206932	+	0.978355i	$0.566348\pi$
$398$	−6.10321	−0.305926
$399$	−1.95916	−0.0980807
$400$	6.16865	0.308432
$401$	2.76783	0.138219	0.0691093	−	0.997609i	$-0.477984\pi$
$401$	2.76783	0.138219	0.0691093	+	0.997609i	$0.477984\pi$
$402$	6.29208	0.313820
$403$	−6.90053	−0.343740
$404$	−14.5089	−0.721846
$405$	25.5508	1.26963
$406$	5.44251	0.270107
$407$	−10.0407	−0.497697
$408$	−1.40189	−0.0694041
$409$	17.7203	0.876214	0.438107	−	0.898923i	$-0.355649\pi$
$409$	17.7203	0.876214	0.438107	+	0.898923i	$0.355649\pi$
$410$	−20.2780	−1.00146
$411$	−0.528918	−0.0260896
$412$	4.68676	0.230900
$413$	6.63509	0.326491
$414$	24.4599	1.20214
$415$	−12.5387	−0.615499
$416$	−3.80522	−0.186566
$417$	−7.03530	−0.344520
$418$	15.5376	0.759969
$419$	3.46322	0.169189	0.0845946	−	0.996415i	$-0.473040\pi$
$419$	3.46322	0.169189	0.0845946	+	0.996415i	$0.473040\pi$
$420$	1.30723	0.0637861
$421$	−8.92509	−0.434982	−0.217491	−	0.976062i	$-0.569787\pi$
$421$	−8.92509	−0.434982	−0.217491	+	0.976062i	$0.569787\pi$
$422$	−12.6393	−0.615274
$423$	3.19340	0.155269
$424$	−1.09695	−0.0532728
$425$	−22.1006	−1.07204
$426$	−2.29199	−0.111048
$427$	−6.21221	−0.300630
$428$	19.5626	0.945592
$429$	4.61896	0.223005
$430$	13.6750	0.659469
$431$	4.59463	0.221316	0.110658	−	0.993859i	$-0.464704\pi$
$431$	4.59463	0.221316	0.110658	+	0.993859i	$0.464704\pi$
$432$	−2.28784	−0.110074
$433$	8.66098	0.416220	0.208110	−	0.978105i	$-0.433269\pi$
$433$	8.66098	0.416220	0.208110	+	0.978105i	$0.433269\pi$
$434$	1.81281	0.0870177
$435$	7.11951	0.341355
$436$	−18.4062	−0.881497
$437$	43.0332	2.05856
$438$	1.20179	0.0574236
$439$	−39.7386	−1.89662	−0.948311	−	0.317342i	$-0.897210\pi$
$439$	−39.7386	−1.89662	−0.948311	+	0.317342i	$0.897210\pi$
$440$	−10.3673	−0.494240
$441$	17.0833	0.813491
$442$	13.6331	0.648459
$443$	−29.9026	−1.42071	−0.710357	−	0.703842i	$-0.751466\pi$
$443$	−29.9026	−1.42071	−0.710357	+	0.703842i	$0.751466\pi$
$444$	1.26648	0.0601045
$445$	6.06875	0.287686
$446$	27.8049	1.31660
$447$	−9.51749	−0.450162
$448$	0.999654	0.0472292
$449$	19.1519	0.903835	0.451918	−	0.892060i	$-0.350740\pi$
$449$	19.1519	0.903835	0.451918	+	0.892060i	$0.350740\pi$
$450$	−17.5615	−0.827855
$451$	18.8230	0.886342
$452$	5.71145	0.268644
$453$	2.57903	0.121173
$454$	17.5510	0.823711
$455$	−12.7125	−0.595969
$456$	−1.95984	−0.0917779
$457$	−34.4246	−1.61031	−0.805157	−	0.593061i	$-0.797919\pi$
$457$	−34.4246	−1.61031	−0.805157	+	0.593061i	$0.797919\pi$
$458$	10.1502	0.474289
$459$	8.19672	0.382590
$460$	−28.7133	−1.33877
$461$	21.6982	1.01059	0.505293	−	0.862948i	$-0.331384\pi$
$461$	21.6982	1.01059	0.505293	+	0.862948i	$0.331384\pi$
$462$	−1.21343	−0.0564538
$463$	11.0215	0.512215	0.256107	−	0.966648i	$-0.417560\pi$
$463$	11.0215	0.512215	0.256107	+	0.966648i	$0.417560\pi$
$464$	5.44439	0.252750
$465$	2.37139	0.109971
$466$	14.9890	0.694352
$467$	−6.20454	−0.287112	−0.143556	−	0.989642i	$-0.545854\pi$
$467$	−6.20454	−0.287112	−0.143556	+	0.989642i	$0.545854\pi$
$468$	10.8330	0.500757
$469$	16.0747	0.742261
$470$	−3.74872	−0.172916
$471$	−0.205781	−0.00948191
$472$	6.63738	0.305510
$473$	−12.6938	−0.583662
$474$	−6.88355	−0.316172
$475$	−30.8965	−1.41763
$476$	−3.58149	−0.164157
$477$	3.12291	0.142988
$478$	19.6731	0.899827
$479$	−13.8460	−0.632642	−0.316321	−	0.948652i	$-0.602448\pi$
$479$	−13.8460	−0.632642	−0.316321	+	0.948652i	$0.602448\pi$
$480$	1.30768	0.0596871
$481$	−12.3162	−0.561571
$482$	23.2762	1.06020
$483$	−3.36073	−0.152919
$484$	−1.37661	−0.0625732
$485$	9.47926	0.430431
$486$	9.85512	0.447038
$487$	35.5745	1.61203	0.806017	−	0.591893i	$-0.201619\pi$
$487$	35.5745	1.61203	0.806017	+	0.591893i	$0.201619\pi$
$488$	−6.21437	−0.281311
$489$	−1.54362	−0.0698048
$490$	−20.0540	−0.905949
$491$	13.6267	0.614965	0.307483	−	0.951554i	$-0.400513\pi$
$491$	13.6267	0.614965	0.307483	+	0.951554i	$0.400513\pi$
$492$	−2.37425	−0.107039
$493$	−19.5058	−0.878497
$494$	19.0590	0.857503
$495$	29.5145	1.32658
$496$	1.81344	0.0814258
$497$	−5.85548	−0.262654
$498$	−1.46809	−0.0657865
$499$	4.09970	0.183528	0.0917640	−	0.995781i	$-0.470749\pi$
$499$	4.09970	0.183528	0.0917640	+	0.995781i	$0.470749\pi$
$500$	3.90556	0.174662
$501$	−5.68775	−0.254110
$502$	8.88267	0.396453
$503$	−36.6239	−1.63298	−0.816489	−	0.577360i	$-0.804083\pi$
$503$	−36.6239	−1.63298	−0.816489	+	0.577360i	$0.804083\pi$
$504$	−2.84591	−0.126767
$505$	−48.4881	−2.15769
$506$	26.6531	1.18487
$507$	0.578986	0.0257137
$508$	−12.0116	−0.532930
$509$	−14.2329	−0.630861	−0.315430	−	0.948949i	$-0.602149\pi$
$509$	−14.2329	−0.630861	−0.315430	+	0.948949i	$0.602149\pi$
$510$	−4.68506	−0.207458
$511$	3.07027	0.135821
$512$	1.00000	0.0441942
$513$	11.4590	0.505926
$514$	−9.59562	−0.423245
$515$	15.6629	0.690190
$516$	1.60114	0.0704862
$517$	3.47974	0.153039
$518$	3.23555	0.142162
$519$	−3.42552	−0.150364
$520$	−12.7169	−0.557671
$521$	21.2393	0.930512	0.465256	−	0.885176i	$-0.345962\pi$
$521$	21.2393	0.930512	0.465256	+	0.885176i	$0.345962\pi$
$522$	−15.4996	−0.678399
$523$	0.682918	0.0298619	0.0149309	−	0.999889i	$-0.495247\pi$
$523$	0.682918	0.0298619	0.0149309	+	0.999889i	$0.495247\pi$
$524$	6.20516	0.271074
$525$	2.41290	0.105308
$526$	−12.2294	−0.533227
$527$	−6.49706	−0.283017
$528$	−1.21385	−0.0528260
$529$	50.8188	2.20951
$530$	−3.66597	−0.159240
$531$	−18.8959	−0.820012
$532$	−5.00691	−0.217077
$533$	23.0890	1.00010
$534$	0.710558	0.0307488
$535$	65.3772	2.82650
$536$	16.0803	0.694562
$537$	−4.76998	−0.205840
$538$	17.1343	0.738713
$539$	18.6151	0.801809
$540$	−7.64585	−0.329025
$541$	15.7745	0.678198	0.339099	−	0.940751i	$-0.389878\pi$
$541$	15.7745	0.678198	0.339099	+	0.940751i	$0.389878\pi$
$542$	24.5519	1.05459
$543$	−5.35750	−0.229912
$544$	−3.58273	−0.153608
$545$	−61.5127	−2.63491
$546$	−1.48843	−0.0636991
$547$	28.6217	1.22378	0.611888	−	0.790944i	$-0.290410\pi$
$547$	28.6217	1.22378	0.611888	+	0.790944i	$0.290410\pi$
$548$	−1.35172	−0.0577428
$549$	17.6916	0.755060
$550$	−19.1361	−0.815967
$551$	−27.2690	−1.16170
$552$	−3.36189	−0.143092
$553$	−17.5858	−0.747823
$554$	21.2812	0.904152
$555$	4.23252	0.179660
$556$	−17.9797	−0.762508
$557$	34.0496	1.44273	0.721365	−	0.692555i	$-0.243515\pi$
$557$	34.0496	1.44273	0.721365	+	0.692555i	$0.243515\pi$
$558$	−5.16266	−0.218553
$559$	−15.5707	−0.658570
$560$	3.34080	0.141174
$561$	4.34890	0.183611
$562$	2.08298	0.0878652
$563$	−34.8378	−1.46824	−0.734120	−	0.679020i	$-0.762405\pi$
$563$	−34.8378	−1.46824	−0.734120	+	0.679020i	$0.762405\pi$
$564$	−0.438918	−0.0184818
$565$	19.0874	0.803013
$566$	−3.45367	−0.145168
$567$	7.64281	0.320968
$568$	−5.85751	−0.245776
$569$	−6.43171	−0.269631	−0.134816	−	0.990871i	$-0.543044\pi$
$569$	−6.43171	−0.269631	−0.134816	+	0.990871i	$0.543044\pi$
$570$	−6.54969	−0.274336
$571$	−16.8884	−0.706757	−0.353379	−	0.935480i	$-0.614967\pi$
$571$	−16.8884	−0.706757	−0.353379	+	0.935480i	$0.614967\pi$
$572$	11.8044	0.493566
$573$	9.34463	0.390378
$574$	−6.06562	−0.253174
$575$	−52.9997	−2.21024
$576$	−2.84689	−0.118620
$577$	−20.6121	−0.858091	−0.429046	−	0.903283i	$-0.641150\pi$
$577$	−20.6121	−0.858091	−0.429046	+	0.903283i	$0.641150\pi$
$578$	−4.16402	−0.173200
$579$	−5.66209	−0.235308
$580$	18.1949	0.755502
$581$	−3.75060	−0.155601
$582$	1.10988	0.0460059
$583$	3.40293	0.140935
$584$	3.07134	0.127093
$585$	36.2035	1.49683
$586$	5.35457	0.221195
$587$	−29.7371	−1.22738	−0.613690	−	0.789547i	$-0.710316\pi$
$587$	−29.7371	−1.22738	−0.613690	+	0.789547i	$0.710316\pi$
$588$	−2.34802	−0.0968307
$589$	−9.08286	−0.374253
$590$	22.1818	0.913211
$591$	−0.106588	−0.00438444
$592$	3.23667	0.133026
$593$	11.3864	0.467584	0.233792	−	0.972287i	$-0.424887\pi$
$593$	11.3864	0.467584	0.233792	+	0.972287i	$0.424887\pi$
$594$	7.09724	0.291203
$595$	−11.9692	−0.490689
$596$	−24.3233	−0.996320
$597$	−2.38814	−0.0977399
$598$	32.6936	1.33694
$599$	−14.7183	−0.601371	−0.300686	−	0.953723i	$-0.597216\pi$
$599$	−14.7183	−0.601371	−0.300686	+	0.953723i	$0.597216\pi$
$600$	2.41374	0.0985405
$601$	7.43221	0.303166	0.151583	−	0.988445i	$-0.451563\pi$
$601$	7.43221	0.303166	0.151583	+	0.988445i	$0.451563\pi$
$602$	4.09051	0.166717
$603$	−45.7788	−1.86426
$604$	6.59106	0.268186
$605$	−4.60056	−0.187039
$606$	−5.67722	−0.230621
$607$	−0.767275	−0.0311427	−0.0155714	−	0.999879i	$-0.504957\pi$
$607$	−0.767275	−0.0311427	−0.0155714	+	0.999879i	$0.504957\pi$
$608$	−5.00864	−0.203127
$609$	2.12961	0.0862961
$610$	−20.7681	−0.840876
$611$	4.26838	0.172680
$612$	10.1997	0.412296
$613$	−21.4341	−0.865715	−0.432857	−	0.901462i	$-0.642495\pi$
$613$	−21.4341	−0.865715	−0.432857	+	0.901462i	$0.642495\pi$
$614$	6.80196	0.274505
$615$	−7.93462	−0.319955
$616$	−3.10109	−0.124946
$617$	36.8435	1.48326	0.741631	−	0.670808i	$-0.234053\pi$
$617$	36.8435	1.48326	0.741631	+	0.670808i	$0.234053\pi$
$618$	1.83389	0.0737698
$619$	9.32066	0.374629	0.187314	−	0.982300i	$-0.440022\pi$
$619$	9.32066	0.374629	0.187314	+	0.982300i	$0.440022\pi$
$620$	6.06042	0.243392
$621$	19.6566	0.788793
$622$	8.28656	0.332261
$623$	1.81530	0.0727285
$624$	−1.48895	−0.0596057
$625$	−17.7910	−0.711641
$626$	27.1892	1.08670
$627$	6.07973	0.242801
$628$	−0.525903	−0.0209858
$629$	−11.5961	−0.462367
$630$	−9.51088	−0.378923
$631$	9.70398	0.386309	0.193155	−	0.981168i	$-0.438128\pi$
$631$	9.70398	0.386309	0.193155	+	0.981168i	$0.438128\pi$
$632$	−17.5919	−0.699767
$633$	−4.94567	−0.196573
$634$	−3.39106	−0.134676
$635$	−40.1423	−1.59300
$636$	−0.429229	−0.0170200
$637$	22.8339	0.904714
$638$	−16.8894	−0.668657
$639$	16.6757	0.659680
$640$	3.34195	0.132102
$641$	−35.5658	−1.40476	−0.702381	−	0.711801i	$-0.747880\pi$
$641$	−35.5658	−1.40476	−0.702381	+	0.711801i	$0.747880\pi$
$642$	7.65467	0.302106
$643$	21.7781	0.858844	0.429422	−	0.903104i	$-0.358717\pi$
$643$	21.7781	0.858844	0.429422	+	0.903104i	$0.358717\pi$
$644$	−8.58881	−0.338447
$645$	5.35093	0.210693
$646$	17.9446	0.706022
$647$	24.1745	0.950399	0.475200	−	0.879878i	$-0.342376\pi$
$647$	24.1745	0.950399	0.475200	+	0.879878i	$0.342376\pi$
$648$	7.64546	0.300342
$649$	−20.5902	−0.808237
$650$	−23.4730	−0.920688
$651$	0.709338	0.0278011
$652$	−3.94493	−0.154495
$653$	13.4313	0.525609	0.262804	−	0.964849i	$-0.415353\pi$
$653$	13.4313	0.525609	0.262804	+	0.964849i	$0.415353\pi$
$654$	−7.20220	−0.281628
$655$	20.7374	0.810276
$656$	−6.06772	−0.236905
$657$	−8.74376	−0.341126
$658$	−1.12133	−0.0437139
$659$	48.2452	1.87937	0.939683	−	0.342045i	$-0.111120\pi$
$659$	48.2452	1.87937	0.939683	+	0.342045i	$0.111120\pi$
$660$	−4.05663	−0.157904
$661$	−13.0011	−0.505683	−0.252842	−	0.967508i	$-0.581365\pi$
$661$	−13.0011	−0.505683	−0.252842	+	0.967508i	$0.581365\pi$
$662$	−2.69147	−0.104607
$663$	5.33451	0.207175
$664$	−3.75190	−0.145602
$665$	−16.7328	−0.648872
$666$	−9.21443	−0.357052
$667$	−46.7770	−1.81121
$668$	−14.5358	−0.562408
$669$	10.8798	0.420639
$670$	53.7395	2.07614
$671$	19.2780	0.744217
$672$	0.391156	0.0150892
$673$	28.7279	1.10738	0.553689	−	0.832724i	$-0.313220\pi$
$673$	28.7279	1.10738	0.553689	+	0.832724i	$0.313220\pi$
$674$	−0.0378310	−0.00145720
$675$	−14.1129	−0.543205
$676$	1.47968	0.0569107
$677$	−9.51052	−0.365519	−0.182759	−	0.983158i	$-0.558503\pi$
$677$	−9.51052	−0.365519	−0.182759	+	0.983158i	$0.558503\pi$
$678$	2.23484	0.0858286
$679$	2.83546	0.108815
$680$	−11.9733	−0.459156
$681$	6.86757	0.263166
$682$	−5.62557	−0.215414
$683$	10.2268	0.391317	0.195659	−	0.980672i	$-0.437316\pi$
$683$	10.2268	0.391317	0.195659	+	0.980672i	$0.437316\pi$
$684$	14.2591	0.545209
$685$	−4.51740	−0.172601
$686$	−12.9962	−0.496197
$687$	3.97170	0.151530
$688$	4.09193	0.156003
$689$	4.17415	0.159023
$690$	−11.2353	−0.427720
$691$	−28.9382	−1.10086	−0.550431	−	0.834881i	$-0.685536\pi$
$691$	−28.9382	−1.10086	−0.550431	+	0.834881i	$0.685536\pi$
$692$	−8.75438	−0.332792
$693$	8.82845	0.335365
$694$	−8.93677	−0.339235
$695$	−60.0872	−2.27924
$696$	2.13035	0.0807505
$697$	21.7390	0.823424
$698$	−9.14320	−0.346075
$699$	5.86507	0.221837
$700$	6.16651	0.233072
$701$	−16.2192	−0.612592	−0.306296	−	0.951936i	$-0.599090\pi$
$701$	−16.2192	−0.612592	−0.306296	+	0.951936i	$0.599090\pi$
$702$	8.70573	0.328577
$703$	−16.2113	−0.611421
$704$	−3.10216	−0.116917
$705$	−1.46684	−0.0552446
$706$	−10.2998	−0.387639
$707$	−14.5039	−0.545475
$708$	2.59715	0.0976070
$709$	15.7997	0.593369	0.296685	−	0.954976i	$-0.404119\pi$
$709$	15.7997	0.593369	0.296685	+	0.954976i	$0.404119\pi$
$710$	−19.5755	−0.734656
$711$	50.0821	1.87823
$712$	1.81593	0.0680548
$713$	−15.5807	−0.583501
$714$	−1.40141	−0.0524464
$715$	39.4497	1.47534
$716$	−12.1903	−0.455574
$717$	7.69793	0.287484
$718$	−8.20472	−0.306197
$719$	30.8289	1.14972	0.574862	−	0.818250i	$-0.305056\pi$
$719$	30.8289	1.14972	0.574862	+	0.818250i	$0.305056\pi$
$720$	−9.51417	−0.354572
$721$	4.68513	0.174483
$722$	6.08648	0.226515
$723$	9.10779	0.338722
$724$	−13.6918	−0.508853
$725$	33.5845	1.24730
$726$	−0.538656	−0.0199914
$727$	−32.7967	−1.21636	−0.608182	−	0.793798i	$-0.708101\pi$
$727$	−32.7967	−1.21636	−0.608182	+	0.793798i	$0.708101\pi$
$728$	−3.80390	−0.140982
$729$	−19.0802	−0.706672
$730$	10.2643	0.379897
$731$	−14.6603	−0.542230
$732$	−2.43163	−0.0898756
$733$	−39.3860	−1.45476	−0.727378	−	0.686237i	$-0.759261\pi$
$733$	−39.3860	−1.45476	−0.727378	+	0.686237i	$0.759261\pi$
$734$	8.48297	0.313112
$735$	−7.84697	−0.289440
$736$	−8.59178	−0.316697
$737$	−49.8836	−1.83748
$738$	17.2741	0.635870
$739$	24.0072	0.883121	0.441560	−	0.897232i	$-0.354425\pi$
$739$	24.0072	0.883121	0.441560	+	0.897232i	$0.354425\pi$
$740$	10.8168	0.397633
$741$	7.45761	0.273962
$742$	−1.09658	−0.0402565
$743$	19.9144	0.730588	0.365294	−	0.930892i	$-0.380968\pi$
$743$	19.9144	0.730588	0.365294	+	0.930892i	$0.380968\pi$
$744$	0.709583	0.0260146
$745$	−81.2872	−2.97813
$746$	−13.1048	−0.479799
$747$	10.6812	0.390806
$748$	11.1142	0.406376
$749$	19.5558	0.714553
$750$	1.52821	0.0558025
$751$	−9.22265	−0.336539	−0.168270	−	0.985741i	$-0.553818\pi$
$751$	−9.22265	−0.336539	−0.168270	+	0.985741i	$0.553818\pi$
$752$	−1.12172	−0.0409048
$753$	3.47571	0.126662
$754$	−20.7171	−0.754472
$755$	22.0270	0.801644
$756$	−2.28705	−0.0831791
$757$	−11.5239	−0.418843	−0.209421	−	0.977826i	$-0.567158\pi$
$757$	−11.5239	−0.418843	−0.209421	+	0.977826i	$0.567158\pi$
$758$	−22.6951	−0.824322
$759$	10.4291	0.378553
$760$	−16.7386	−0.607174
$761$	−26.2384	−0.951143	−0.475571	−	0.879677i	$-0.657759\pi$
$761$	−26.2384	−0.951143	−0.475571	+	0.879677i	$0.657759\pi$
$762$	−4.70005	−0.170265
$763$	−18.3998	−0.666119
$764$	23.8815	0.864002
$765$	34.0867	1.23241
$766$	19.3666	0.699743
$767$	−25.2567	−0.911966
$768$	0.391292	0.0141195
$769$	−14.5177	−0.523521	−0.261760	−	0.965133i	$-0.584303\pi$
$769$	−14.5177	−0.523521	−0.261760	+	0.965133i	$0.584303\pi$
$770$	−10.3637	−0.373481
$771$	−3.75469	−0.135222
$772$	−14.4703	−0.520796
$773$	−33.6089	−1.20883	−0.604414	−	0.796670i	$-0.706593\pi$
$773$	−33.6089	−1.20883	−0.604414	+	0.796670i	$0.706593\pi$
$774$	−11.6493	−0.418725
$775$	11.1865	0.401829
$776$	2.83644	0.101822
$777$	1.26604	0.0454190
$778$	−8.79513	−0.315321
$779$	30.3910	1.08887
$780$	−4.97600	−0.178169
$781$	18.1709	0.650207
$782$	30.7821	1.10076
$783$	−12.4559	−0.445137
$784$	−6.00069	−0.214310
$785$	−1.75754	−0.0627294
$786$	2.42803	0.0866049
$787$	20.2253	0.720953	0.360477	−	0.932768i	$-0.382614\pi$
$787$	20.2253	0.720953	0.360477	+	0.932768i	$0.382614\pi$
$788$	−0.272400	−0.00970385
$789$	−4.78526	−0.170360
$790$	−58.7912	−2.09170
$791$	5.70947	0.203006
$792$	8.83151	0.313814
$793$	23.6470	0.839730
$794$	−8.24618	−0.292646
$795$	−1.43446	−0.0508752
$796$	−6.10321	−0.216323
$797$	2.56335	0.0907986	0.0453993	−	0.998969i	$-0.485544\pi$
$797$	2.56335	0.0907986	0.0453993	+	0.998969i	$0.485544\pi$
$798$	−1.95916	−0.0693535
$799$	4.01881	0.142175
$800$	6.16865	0.218095
$801$	−5.16975	−0.182664
$802$	2.76783	0.0977354
$803$	−9.52777	−0.336228
$804$	6.29208	0.221904
$805$	−28.7034	−1.01166
$806$	−6.90053	−0.243061
$807$	6.70451	0.236010
$808$	−14.5089	−0.510422
$809$	−46.9987	−1.65239	−0.826193	−	0.563388i	$-0.809498\pi$
$809$	−46.9987	−1.65239	−0.826193	+	0.563388i	$0.809498\pi$
$810$	25.5508	0.897762
$811$	40.1411	1.40955	0.704773	−	0.709433i	$-0.251049\pi$
$811$	40.1411	1.40955	0.704773	+	0.709433i	$0.251049\pi$
$812$	5.44251	0.190995
$813$	9.60694	0.336930
$814$	−10.0407	−0.351925
$815$	−13.1838	−0.461807
$816$	−1.40189	−0.0490761
$817$	−20.4950	−0.717029
$818$	17.7203	0.619577
$819$	10.8293	0.378406
$820$	−20.2780	−0.708139
$821$	2.90195	0.101279	0.0506394	−	0.998717i	$-0.483874\pi$
$821$	2.90195	0.101279	0.0506394	+	0.998717i	$0.483874\pi$
$822$	−0.528918	−0.0184481
$823$	40.2033	1.40140	0.700700	−	0.713456i	$-0.252871\pi$
$823$	40.2033	1.40140	0.700700	+	0.713456i	$0.252871\pi$
$824$	4.68676	0.163271
$825$	−7.48781	−0.260692
$826$	6.63509	0.230864
$827$	−21.1528	−0.735554	−0.367777	−	0.929914i	$-0.619881\pi$
$827$	−21.1528	−0.735554	−0.367777	+	0.929914i	$0.619881\pi$
$828$	24.4599	0.850039
$829$	24.4187	0.848095	0.424047	−	0.905640i	$-0.360609\pi$
$829$	24.4187	0.848095	0.424047	+	0.905640i	$0.360609\pi$
$830$	−12.5387	−0.435223
$831$	8.32716	0.288866
$832$	−3.80522	−0.131922
$833$	21.4989	0.744892
$834$	−7.03530	−0.243612
$835$	−48.5780	−1.68111
$836$	15.5376	0.537379
$837$	−4.14886	−0.143405
$838$	3.46322	0.119635
$839$	−2.47258	−0.0853630	−0.0426815	−	0.999089i	$-0.513590\pi$
$839$	−2.47258	−0.0853630	−0.0426815	+	0.999089i	$0.513590\pi$
$840$	1.30723	0.0451036
$841$	0.641408	0.0221175
$842$	−8.92509	−0.307579
$843$	0.815052	0.0280719
$844$	−12.6393	−0.435064
$845$	4.94502	0.170114
$846$	3.19340	0.109792
$847$	−1.37613	−0.0472845
$848$	−1.09695	−0.0376696
$849$	−1.35139	−0.0463796
$850$	−22.1006	−0.758045
$851$	−27.8087	−0.953271
$852$	−2.29199	−0.0785224
$853$	−51.7178	−1.77078	−0.885392	−	0.464846i	$-0.846110\pi$
$853$	−51.7178	−1.77078	−0.885392	+	0.464846i	$0.846110\pi$
$854$	−6.21221	−0.212578
$855$	47.6531	1.62970
$856$	19.5626	0.668635
$857$	42.3160	1.44549	0.722743	−	0.691117i	$-0.242881\pi$
$857$	42.3160	1.44549	0.722743	+	0.691117i	$0.242881\pi$
$858$	4.61896	0.157689
$859$	31.3123	1.06836	0.534180	−	0.845371i	$-0.320620\pi$
$859$	31.3123	1.06836	0.534180	+	0.845371i	$0.320620\pi$
$860$	13.6750	0.466315
$861$	−2.37343	−0.0808861
$862$	4.59463	0.156494
$863$	29.3650	0.999596	0.499798	−	0.866142i	$-0.333408\pi$
$863$	29.3650	0.999596	0.499798	+	0.866142i	$0.333408\pi$
$864$	−2.28784	−0.0778339
$865$	−29.2567	−0.994759
$866$	8.66098	0.294312
$867$	−1.62935	−0.0553355
$868$	1.81281	0.0615308
$869$	54.5728	1.85125
$870$	7.11951	0.241374
$871$	−61.1889	−2.07331
$872$	−18.4062	−0.623313
$873$	−8.07505	−0.273299
$874$	43.0332	1.45562
$875$	3.90421	0.131986
$876$	1.20179	0.0406046
$877$	23.2947	0.786606	0.393303	−	0.919409i	$-0.371332\pi$
$877$	23.2947	0.786606	0.393303	+	0.919409i	$0.371332\pi$
$878$	−39.7386	−1.34111
$879$	2.09520	0.0706693
$880$	−10.3673	−0.349481
$881$	−39.4400	−1.32877	−0.664383	−	0.747392i	$-0.731306\pi$
$881$	−39.4400	−1.32877	−0.664383	+	0.747392i	$0.731306\pi$
$882$	17.0833	0.575225
$883$	−34.0297	−1.14519	−0.572596	−	0.819838i	$-0.694063\pi$
$883$	−34.0297	−1.14519	−0.572596	+	0.819838i	$0.694063\pi$
$884$	13.6331	0.458530
$885$	8.67956	0.291760
$886$	−29.9026	−1.00460
$887$	−10.0118	−0.336162	−0.168081	−	0.985773i	$-0.553757\pi$
$887$	−10.0118	−0.336162	−0.168081	+	0.985773i	$0.553757\pi$
$888$	1.26648	0.0425003
$889$	−12.0075	−0.402717
$890$	6.06875	0.203425
$891$	−23.7174	−0.794564
$892$	27.8049	0.930978
$893$	5.61827	0.188008
$894$	−9.51749	−0.318313
$895$	−40.7395	−1.36177
$896$	0.999654	0.0333961
$897$	12.7927	0.427137
$898$	19.1519	0.639108
$899$	9.87307	0.329285
$900$	−17.5615	−0.585382
$901$	3.93010	0.130931
$902$	18.8230	0.626738
$903$	1.60058	0.0532641
$904$	5.71145	0.189960
$905$	−45.7574	−1.52103
$906$	2.57903	0.0856824
$907$	−24.8818	−0.826187	−0.413093	−	0.910689i	$-0.635552\pi$
$907$	−24.8818	−0.826187	−0.413093	+	0.910689i	$0.635552\pi$
$908$	17.5510	0.582451
$909$	41.3053	1.37001
$910$	−12.7125	−0.421414
$911$	−36.0788	−1.19534	−0.597672	−	0.801741i	$-0.703907\pi$
$911$	−36.0788	−1.19534	−0.597672	+	0.801741i	$0.703907\pi$
$912$	−1.95984	−0.0648968
$913$	11.6390	0.385194
$914$	−34.4246	−1.13866
$915$	−8.12639	−0.268650
$916$	10.1502	0.335373
$917$	6.20302	0.204842
$918$	8.19672	0.270532
$919$	−52.7914	−1.74143	−0.870713	−	0.491791i	$-0.836342\pi$
$919$	−52.7914	−1.74143	−0.870713	+	0.491791i	$0.836342\pi$
$920$	−28.7133	−0.946651
$921$	2.66155	0.0877011
$922$	21.6982	0.714593
$923$	22.2891	0.733655
$924$	−1.21343	−0.0399189
$925$	19.9658	0.656473
$926$	11.0215	0.362191
$927$	−13.3427	−0.438231
$928$	5.44439	0.178721
$929$	44.0477	1.44516	0.722578	−	0.691289i	$-0.242957\pi$
$929$	44.0477	1.44516	0.722578	+	0.691289i	$0.242957\pi$
$930$	2.37139	0.0777610
$931$	30.0553	0.985023
$932$	14.9890	0.490981
$933$	3.24246	0.106153
$934$	−6.20454	−0.203019
$935$	37.1432	1.21471
$936$	10.8330	0.354089
$937$	23.3850	0.763955	0.381977	−	0.924172i	$-0.375243\pi$
$937$	23.3850	0.763955	0.381977	+	0.924172i	$0.375243\pi$
$938$	16.0747	0.524858
$939$	10.6389	0.347187
$940$	−3.74872	−0.122270
$941$	−23.8542	−0.777623	−0.388812	−	0.921317i	$-0.627114\pi$
$941$	−23.8542	−0.777623	−0.388812	+	0.921317i	$0.627114\pi$
$942$	−0.205781	−0.00670472
$943$	52.1325	1.69767
$944$	6.63738	0.216028
$945$	−7.64321	−0.248633
$946$	−12.6938	−0.412712
$947$	−24.7931	−0.805667	−0.402833	−	0.915273i	$-0.631975\pi$
$947$	−24.7931	−0.805667	−0.402833	+	0.915273i	$0.631975\pi$
$948$	−6.88355	−0.223567
$949$	−11.6871	−0.379379
$950$	−30.8965	−1.00242
$951$	−1.32689	−0.0430275
$952$	−3.58149	−0.116077
$953$	−7.86953	−0.254919	−0.127460	−	0.991844i	$-0.540682\pi$
$953$	−7.86953	−0.254919	−0.127460	+	0.991844i	$0.540682\pi$
$954$	3.12291	0.101108
$955$	79.8108	2.58262
$956$	19.6731	0.636274
$957$	−6.60867	−0.213628
$958$	−13.8460	−0.447345
$959$	−1.35126	−0.0436343
$960$	1.30768	0.0422051
$961$	−27.7114	−0.893918
$962$	−12.3162	−0.397091
$963$	−55.6925	−1.79467
$964$	23.2762	0.749677
$965$	−48.3589	−1.55673
$966$	−3.36073	−0.108130
$967$	8.80577	0.283175	0.141587	−	0.989926i	$-0.454779\pi$
$967$	8.80577	0.283175	0.141587	+	0.989926i	$0.454779\pi$
$968$	−1.37661	−0.0442459
$969$	7.02158	0.225566
$970$	9.47926	0.304361
$971$	57.2602	1.83757	0.918784	−	0.394760i	$-0.129172\pi$
$971$	57.2602	1.83757	0.918784	+	0.394760i	$0.129172\pi$
$972$	9.85512	0.316103
$973$	−17.9735	−0.576202
$974$	35.5745	1.13988
$975$	−9.18481	−0.294149
$976$	−6.21437	−0.198917
$977$	−15.5160	−0.496400	−0.248200	−	0.968709i	$-0.579839\pi$
$977$	−15.5160	−0.496400	−0.248200	+	0.968709i	$0.579839\pi$
$978$	−1.54362	−0.0493595
$979$	−5.63330	−0.180041
$980$	−20.0540	−0.640602
$981$	52.4005	1.67302
$982$	13.6267	0.434846
$983$	−11.0504	−0.352452	−0.176226	−	0.984350i	$-0.556389\pi$
$983$	−11.0504	−0.352452	−0.176226	+	0.984350i	$0.556389\pi$
$984$	−2.37425	−0.0756883
$985$	−0.910348	−0.0290061
$986$	−19.5058	−0.621191
$987$	−0.438766	−0.0139661
$988$	19.0590	0.606346
$989$	−35.1570	−1.11793
$990$	29.5145	0.938032
$991$	16.0330	0.509305	0.254653	−	0.967033i	$-0.418039\pi$
$991$	16.0330	0.509305	0.254653	+	0.967033i	$0.418039\pi$
$992$	1.81344	0.0575767
$993$	−1.05315	−0.0334208
$994$	−5.85548	−0.185725
$995$	−20.3966	−0.646617
$996$	−1.46809	−0.0465181
$997$	35.6018	1.12752	0.563760	−	0.825939i	$-0.309354\pi$
$997$	35.6018	1.12752	0.563760	+	0.825939i	$0.309354\pi$
$998$	4.09970	0.129774
$999$	−7.40497	−0.234283

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.40	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.40	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.391292 q^{3} +1.00000 q^{4} +3.34195 q^{5} +0.391292 q^{6} +0.999654 q^{7} +1.00000 q^{8} -2.84689 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.391292 q^{3} +1.00000 q^{4} +3.34195 q^{5} +0.391292 q^{6} +0.999654 q^{7} +1.00000 q^{8} -2.84689 q^{9} +3.34195 q^{10} -3.10216 q^{11} +0.391292 q^{12} -3.80522 q^{13} +0.999654 q^{14} +1.30768 q^{15} +1.00000 q^{16} -3.58273 q^{17} -2.84689 q^{18} -5.00864 q^{19} +3.34195 q^{20} +0.391156 q^{21} -3.10216 q^{22} -8.59178 q^{23} +0.391292 q^{24} +6.16865 q^{25} -3.80522 q^{26} -2.28784 q^{27} +0.999654 q^{28} +5.44439 q^{29} +1.30768 q^{30} +1.81344 q^{31} +1.00000 q^{32} -1.21385 q^{33} -3.58273 q^{34} +3.34080 q^{35} -2.84689 q^{36} +3.23667 q^{37} -5.00864 q^{38} -1.48895 q^{39} +3.34195 q^{40} -6.06772 q^{41} +0.391156 q^{42} +4.09193 q^{43} -3.10216 q^{44} -9.51417 q^{45} -8.59178 q^{46} -1.12172 q^{47} +0.391292 q^{48} -6.00069 q^{49} +6.16865 q^{50} -1.40189 q^{51} -3.80522 q^{52} -1.09695 q^{53} -2.28784 q^{54} -10.3673 q^{55} +0.999654 q^{56} -1.95984 q^{57} +5.44439 q^{58} +6.63738 q^{59} +1.30768 q^{60} -6.21437 q^{61} +1.81344 q^{62} -2.84591 q^{63} +1.00000 q^{64} -12.7169 q^{65} -1.21385 q^{66} +16.0803 q^{67} -3.58273 q^{68} -3.36189 q^{69} +3.34080 q^{70} -5.85751 q^{71} -2.84689 q^{72} +3.07134 q^{73} +3.23667 q^{74} +2.41374 q^{75} -5.00864 q^{76} -3.10109 q^{77} -1.48895 q^{78} -17.5919 q^{79} +3.34195 q^{80} +7.64546 q^{81} -6.06772 q^{82} -3.75190 q^{83} +0.391156 q^{84} -11.9733 q^{85} +4.09193 q^{86} +2.13035 q^{87} -3.10216 q^{88} +1.81593 q^{89} -9.51417 q^{90} -3.80390 q^{91} -8.59178 q^{92} +0.709583 q^{93} -1.12172 q^{94} -16.7386 q^{95} +0.391292 q^{96} +2.83644 q^{97} -6.00069 q^{98} +8.83151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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