LMFDB - Embedded newform 8042.2.a.a.1.30

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			1.30
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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.449086 q^{3} +1.00000 q^{4} +0.325053 q^{5} -0.449086 q^{6} +4.49527 q^{7} +1.00000 q^{8} -2.79832 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.449086 q^{3} +1.00000 q^{4} +0.325053 q^{5} -0.449086 q^{6} +4.49527 q^{7} +1.00000 q^{8} -2.79832 q^{9} +0.325053 q^{10} -4.78204 q^{11} -0.449086 q^{12} -2.30266 q^{13} +4.49527 q^{14} -0.145977 q^{15} +1.00000 q^{16} -4.04648 q^{17} -2.79832 q^{18} +3.64818 q^{19} +0.325053 q^{20} -2.01876 q^{21} -4.78204 q^{22} +0.292576 q^{23} -0.449086 q^{24} -4.89434 q^{25} -2.30266 q^{26} +2.60394 q^{27} +4.49527 q^{28} +3.89276 q^{29} -0.145977 q^{30} -4.13030 q^{31} +1.00000 q^{32} +2.14755 q^{33} -4.04648 q^{34} +1.46120 q^{35} -2.79832 q^{36} +3.23926 q^{37} +3.64818 q^{38} +1.03409 q^{39} +0.325053 q^{40} -2.67113 q^{41} -2.01876 q^{42} +1.46775 q^{43} -4.78204 q^{44} -0.909604 q^{45} +0.292576 q^{46} +5.96691 q^{47} -0.449086 q^{48} +13.2074 q^{49} -4.89434 q^{50} +1.81722 q^{51} -2.30266 q^{52} -5.72402 q^{53} +2.60394 q^{54} -1.55442 q^{55} +4.49527 q^{56} -1.63835 q^{57} +3.89276 q^{58} -3.93696 q^{59} -0.145977 q^{60} -5.78940 q^{61} -4.13030 q^{62} -12.5792 q^{63} +1.00000 q^{64} -0.748489 q^{65} +2.14755 q^{66} -14.6678 q^{67} -4.04648 q^{68} -0.131392 q^{69} +1.46120 q^{70} +7.96883 q^{71} -2.79832 q^{72} +3.48592 q^{73} +3.23926 q^{74} +2.19798 q^{75} +3.64818 q^{76} -21.4966 q^{77} +1.03409 q^{78} -15.7497 q^{79} +0.325053 q^{80} +7.22557 q^{81} -2.67113 q^{82} -6.24447 q^{83} -2.01876 q^{84} -1.31532 q^{85} +1.46775 q^{86} -1.74818 q^{87} -4.78204 q^{88} +13.7817 q^{89} -0.909604 q^{90} -10.3511 q^{91} +0.292576 q^{92} +1.85486 q^{93} +5.96691 q^{94} +1.18585 q^{95} -0.449086 q^{96} -16.9554 q^{97} +13.2074 q^{98} +13.3817 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.449086	−0.259280	−0.129640	−	0.991561i	$-0.541382\pi$
$3$	−0.449086	−0.259280	−0.129640	+	0.991561i	$0.541382\pi$
$4$	1.00000	0.500000
$5$	0.325053	0.145368	0.0726841	−	0.997355i	$-0.476844\pi$
$5$	0.325053	0.145368	0.0726841	+	0.997355i	$0.476844\pi$
$6$	−0.449086	−0.183338
$7$	4.49527	1.69905	0.849526	−	0.527547i	$-0.176888\pi$
$7$	4.49527	1.69905	0.849526	+	0.527547i	$0.176888\pi$
$8$	1.00000	0.353553
$9$	−2.79832	−0.932774
$10$	0.325053	0.102791
$11$	−4.78204	−1.44184	−0.720920	−	0.693018i	$-0.756281\pi$
$11$	−4.78204	−1.44184	−0.720920	+	0.693018i	$0.756281\pi$
$12$	−0.449086	−0.129640
$13$	−2.30266	−0.638644	−0.319322	−	0.947646i	$-0.603455\pi$
$13$	−2.30266	−0.638644	−0.319322	+	0.947646i	$0.603455\pi$
$14$	4.49527	1.20141
$15$	−0.145977	−0.0376910
$16$	1.00000	0.250000
$17$	−4.04648	−0.981415	−0.490708	−	0.871324i	$-0.663262\pi$
$17$	−4.04648	−0.981415	−0.490708	+	0.871324i	$0.663262\pi$
$18$	−2.79832	−0.659571
$19$	3.64818	0.836950	0.418475	−	0.908228i	$-0.362565\pi$
$19$	3.64818	0.836950	0.418475	+	0.908228i	$0.362565\pi$
$20$	0.325053	0.0726841
$21$	−2.01876	−0.440530
$22$	−4.78204	−1.01954
$23$	0.292576	0.0610063	0.0305031	−	0.999535i	$-0.490289\pi$
$23$	0.292576	0.0610063	0.0305031	+	0.999535i	$0.490289\pi$
$24$	−0.449086	−0.0916692
$25$	−4.89434	−0.978868
$26$	−2.30266	−0.451590
$27$	2.60394	0.501129
$28$	4.49527	0.849526
$29$	3.89276	0.722867	0.361434	−	0.932398i	$-0.382287\pi$
$29$	3.89276	0.722867	0.361434	+	0.932398i	$0.382287\pi$
$30$	−0.145977	−0.0266516
$31$	−4.13030	−0.741825	−0.370912	−	0.928668i	$-0.620955\pi$
$31$	−4.13030	−0.741825	−0.370912	+	0.928668i	$0.620955\pi$
$32$	1.00000	0.176777
$33$	2.14755	0.373840
$34$	−4.04648	−0.693965
$35$	1.46120	0.246988
$36$	−2.79832	−0.466387
$37$	3.23926	0.532531	0.266266	−	0.963900i	$-0.414210\pi$
$37$	3.23926	0.532531	0.266266	+	0.963900i	$0.414210\pi$
$38$	3.64818	0.591813
$39$	1.03409	0.165587
$40$	0.325053	0.0513954
$41$	−2.67113	−0.417161	−0.208581	−	0.978005i	$-0.566884\pi$
$41$	−2.67113	−0.417161	−0.208581	+	0.978005i	$0.566884\pi$
$42$	−2.01876	−0.311502
$43$	1.46775	0.223830	0.111915	−	0.993718i	$-0.464302\pi$
$43$	1.46775	0.223830	0.111915	+	0.993718i	$0.464302\pi$
$44$	−4.78204	−0.720920
$45$	−0.909604	−0.135596
$46$	0.292576	0.0431380
$47$	5.96691	0.870364	0.435182	−	0.900343i	$-0.356684\pi$
$47$	5.96691	0.870364	0.435182	+	0.900343i	$0.356684\pi$
$48$	−0.449086	−0.0648199
$49$	13.2074	1.88678
$50$	−4.89434	−0.692164
$51$	1.81722	0.254461
$52$	−2.30266	−0.319322
$53$	−5.72402	−0.786254	−0.393127	−	0.919484i	$-0.628607\pi$
$53$	−5.72402	−0.786254	−0.393127	+	0.919484i	$0.628607\pi$
$54$	2.60394	0.354352
$55$	−1.55442	−0.209598
$56$	4.49527	0.600706
$57$	−1.63835	−0.217004
$58$	3.89276	0.511144
$59$	−3.93696	−0.512548	−0.256274	−	0.966604i	$-0.582495\pi$
$59$	−3.93696	−0.512548	−0.256274	+	0.966604i	$0.582495\pi$
$60$	−0.145977	−0.0188455
$61$	−5.78940	−0.741257	−0.370628	−	0.928781i	$-0.620858\pi$
$61$	−5.78940	−0.741257	−0.370628	+	0.928781i	$0.620858\pi$
$62$	−4.13030	−0.524549
$63$	−12.5792	−1.58483
$64$	1.00000	0.125000
$65$	−0.748489	−0.0928386
$66$	2.14755	0.264345
$67$	−14.6678	−1.79196	−0.895980	−	0.444094i	$-0.853526\pi$
$67$	−14.6678	−1.79196	−0.895980	+	0.444094i	$0.853526\pi$
$68$	−4.04648	−0.490708
$69$	−0.131392	−0.0158177
$70$	1.46120	0.174647
$71$	7.96883	0.945726	0.472863	−	0.881136i	$-0.343221\pi$
$71$	7.96883	0.945726	0.472863	+	0.881136i	$0.343221\pi$
$72$	−2.79832	−0.329785
$73$	3.48592	0.407996	0.203998	−	0.978971i	$-0.434606\pi$
$73$	3.48592	0.407996	0.203998	+	0.978971i	$0.434606\pi$
$74$	3.23926	0.376556
$75$	2.19798	0.253801
$76$	3.64818	0.418475
$77$	−21.4966	−2.44976
$78$	1.03409	0.117088
$79$	−15.7497	−1.77198	−0.885988	−	0.463709i	$-0.846518\pi$
$79$	−15.7497	−1.77198	−0.885988	+	0.463709i	$0.846518\pi$
$80$	0.325053	0.0363421
$81$	7.22557	0.802841
$82$	−2.67113	−0.294977
$83$	−6.24447	−0.685419	−0.342710	−	0.939441i	$-0.611345\pi$
$83$	−6.24447	−0.685419	−0.342710	+	0.939441i	$0.611345\pi$
$84$	−2.01876	−0.220265
$85$	−1.31532	−0.142667
$86$	1.46775	0.158271
$87$	−1.74818	−0.187425
$88$	−4.78204	−0.509768
$89$	13.7817	1.46086	0.730428	−	0.682989i	$-0.239321\pi$
$89$	13.7817	1.46086	0.730428	+	0.682989i	$0.239321\pi$
$90$	−0.909604	−0.0958807
$91$	−10.3511	−1.08509
$92$	0.292576	0.0305031
$93$	1.85486	0.192340
$94$	5.96691	0.615440
$95$	1.18585	0.121666
$96$	−0.449086	−0.0458346
$97$	−16.9554	−1.72156	−0.860781	−	0.508975i	$-0.830025\pi$
$97$	−16.9554	−1.72156	−0.860781	+	0.508975i	$0.830025\pi$
$98$	13.2074	1.33415
$99$	13.3817	1.34491
$100$	−4.89434	−0.489434
$101$	−12.6427	−1.25799	−0.628996	−	0.777408i	$-0.716534\pi$
$101$	−12.6427	−1.25799	−0.628996	+	0.777408i	$0.716534\pi$
$102$	1.81722	0.179931
$103$	−15.9938	−1.57591	−0.787956	−	0.615732i	$-0.788860\pi$
$103$	−15.9938	−1.57591	−0.787956	+	0.615732i	$0.788860\pi$
$104$	−2.30266	−0.225795
$105$	−0.656205	−0.0640390
$106$	−5.72402	−0.555966
$107$	−9.68046	−0.935846	−0.467923	−	0.883769i	$-0.654998\pi$
$107$	−9.68046	−0.935846	−0.467923	+	0.883769i	$0.654998\pi$
$108$	2.60394	0.250565
$109$	−5.24399	−0.502283	−0.251141	−	0.967950i	$-0.580806\pi$
$109$	−5.24399	−0.502283	−0.251141	+	0.967950i	$0.580806\pi$
$110$	−1.55442	−0.148208
$111$	−1.45471	−0.138075
$112$	4.49527	0.424763
$113$	4.29034	0.403602	0.201801	−	0.979427i	$-0.435321\pi$
$113$	4.29034	0.403602	0.201801	+	0.979427i	$0.435321\pi$
$114$	−1.63835	−0.153445
$115$	0.0951027	0.00886838
$116$	3.89276	0.361434
$117$	6.44360	0.595711
$118$	−3.93696	−0.362426
$119$	−18.1900	−1.66748
$120$	−0.145977	−0.0133258
$121$	11.8680	1.07890
$122$	−5.78940	−0.524148
$123$	1.19957	0.108161
$124$	−4.13030	−0.370912
$125$	−3.21619	−0.287665
$126$	−12.5792	−1.12065
$127$	−20.2705	−1.79872	−0.899359	−	0.437210i	$-0.855967\pi$
$127$	−20.2705	−1.79872	−0.899359	+	0.437210i	$0.855967\pi$
$128$	1.00000	0.0883883
$129$	−0.659145	−0.0580345
$130$	−0.748489	−0.0656468
$131$	3.66610	0.320309	0.160154	−	0.987092i	$-0.448801\pi$
$131$	3.66610	0.320309	0.160154	+	0.987092i	$0.448801\pi$
$132$	2.14755	0.186920
$133$	16.3996	1.42202
$134$	−14.6678	−1.26711
$135$	0.846420	0.0728483
$136$	−4.04648	−0.346983
$137$	−4.32403	−0.369427	−0.184713	−	0.982792i	$-0.559136\pi$
$137$	−4.32403	−0.369427	−0.184713	+	0.982792i	$0.559136\pi$
$138$	−0.131392	−0.0111848
$139$	−11.3249	−0.960566	−0.480283	−	0.877114i	$-0.659466\pi$
$139$	−11.3249	−0.960566	−0.480283	+	0.877114i	$0.659466\pi$
$140$	1.46120	0.123494
$141$	−2.67966	−0.225668
$142$	7.96883	0.668729
$143$	11.0114	0.920823
$144$	−2.79832	−0.233194
$145$	1.26535	0.105082
$146$	3.48592	0.288496
$147$	−5.93127	−0.489203
$148$	3.23926	0.266266
$149$	−7.97108	−0.653016	−0.326508	−	0.945194i	$-0.605872\pi$
$149$	−7.97108	−0.653016	−0.326508	+	0.945194i	$0.605872\pi$
$150$	2.19798	0.179464
$151$	−3.44572	−0.280408	−0.140204	−	0.990123i	$-0.544776\pi$
$151$	−3.44572	−0.280408	−0.140204	+	0.990123i	$0.544776\pi$
$152$	3.64818	0.295907
$153$	11.3233	0.915438
$154$	−21.4966	−1.73224
$155$	−1.34257	−0.107838
$156$	1.03409	0.0827937
$157$	18.9095	1.50914	0.754572	−	0.656217i	$-0.227844\pi$
$157$	18.9095	1.50914	0.754572	+	0.656217i	$0.227844\pi$
$158$	−15.7497	−1.25298
$159$	2.57057	0.203860
$160$	0.325053	0.0256977
$161$	1.31521	0.103653
$162$	7.22557	0.567695
$163$	−1.94522	−0.152361	−0.0761805	−	0.997094i	$-0.524273\pi$
$163$	−1.94522	−0.152361	−0.0761805	+	0.997094i	$0.524273\pi$
$164$	−2.67113	−0.208581
$165$	0.698067	0.0543445
$166$	−6.24447	−0.484665
$167$	0.880492	0.0681345	0.0340673	−	0.999420i	$-0.489154\pi$
$167$	0.880492	0.0681345	0.0340673	+	0.999420i	$0.489154\pi$
$168$	−2.01876	−0.155751
$169$	−7.69774	−0.592134
$170$	−1.31532	−0.100881
$171$	−10.2088	−0.780685
$172$	1.46775	0.111915
$173$	22.5163	1.71188	0.855941	−	0.517073i	$-0.172978\pi$
$173$	22.5163	1.71188	0.855941	+	0.517073i	$0.172978\pi$
$174$	−1.74818	−0.132529
$175$	−22.0014	−1.66315
$176$	−4.78204	−0.360460
$177$	1.76803	0.132893
$178$	13.7817	1.03298
$179$	15.1462	1.13208	0.566040	−	0.824378i	$-0.308475\pi$
$179$	15.1462	1.13208	0.566040	+	0.824378i	$0.308475\pi$
$180$	−0.909604	−0.0677979
$181$	14.1482	1.05163	0.525814	−	0.850600i	$-0.323761\pi$
$181$	14.1482	1.05163	0.525814	+	0.850600i	$0.323761\pi$
$182$	−10.3511	−0.767274
$183$	2.59994	0.192193
$184$	0.292576	0.0215690
$185$	1.05293	0.0774131
$186$	1.85486	0.136005
$187$	19.3504	1.41504
$188$	5.96691	0.435182
$189$	11.7054	0.851444
$190$	1.18585	0.0860308
$191$	−1.09137	−0.0789685	−0.0394843	−	0.999220i	$-0.512572\pi$
$191$	−1.09137	−0.0789685	−0.0394843	+	0.999220i	$0.512572\pi$
$192$	−0.449086	−0.0324100
$193$	5.43339	0.391104	0.195552	−	0.980693i	$-0.437350\pi$
$193$	5.43339	0.391104	0.195552	+	0.980693i	$0.437350\pi$
$194$	−16.9554	−1.21733
$195$	0.336136	0.0240712
$196$	13.2074	0.943389
$197$	2.33572	0.166413	0.0832065	−	0.996532i	$-0.473484\pi$
$197$	2.33572	0.166413	0.0832065	+	0.996532i	$0.473484\pi$
$198$	13.3817	0.950996
$199$	−8.85424	−0.627660	−0.313830	−	0.949479i	$-0.601612\pi$
$199$	−8.85424	−0.627660	−0.313830	+	0.949479i	$0.601612\pi$
$200$	−4.89434	−0.346082
$201$	6.58711	0.464619
$202$	−12.6427	−0.889535
$203$	17.4990	1.22819
$204$	1.81722	0.127231
$205$	−0.868261	−0.0606420
$206$	−15.9938	−1.11434
$207$	−0.818721	−0.0569051
$208$	−2.30266	−0.159661
$209$	−17.4458	−1.20675
$210$	−0.656205	−0.0452824
$211$	9.73960	0.670502	0.335251	−	0.942129i	$-0.391179\pi$
$211$	9.73960	0.670502	0.335251	+	0.942129i	$0.391179\pi$
$212$	−5.72402	−0.393127
$213$	−3.57868	−0.245207
$214$	−9.68046	−0.661743
$215$	0.477097	0.0325377
$216$	2.60394	0.177176
$217$	−18.5668	−1.26040
$218$	−5.24399	−0.355168
$219$	−1.56547	−0.105785
$220$	−1.55442	−0.104799
$221$	9.31768	0.626775
$222$	−1.45471	−0.0976334
$223$	−27.5688	−1.84614	−0.923071	−	0.384630i	$-0.874329\pi$
$223$	−27.5688	−1.84614	−0.923071	+	0.384630i	$0.874329\pi$
$224$	4.49527	0.300353
$225$	13.6959	0.913063
$226$	4.29034	0.285389
$227$	−3.67615	−0.243994	−0.121997	−	0.992530i	$-0.538930\pi$
$227$	−3.67615	−0.243994	−0.121997	+	0.992530i	$0.538930\pi$
$228$	−1.63835	−0.108502
$229$	−15.3435	−1.01393	−0.506965	−	0.861967i	$-0.669233\pi$
$229$	−15.3435	−1.01393	−0.506965	+	0.861967i	$0.669233\pi$
$230$	0.0951027	0.00627089
$231$	9.65380	0.635174
$232$	3.89276	0.255572
$233$	−16.6933	−1.09361	−0.546807	−	0.837259i	$-0.684157\pi$
$233$	−16.6933	−1.09361	−0.546807	+	0.837259i	$0.684157\pi$
$234$	6.44360	0.421231
$235$	1.93957	0.126523
$236$	−3.93696	−0.256274
$237$	7.07295	0.459437
$238$	−18.1900	−1.17908
$239$	28.7540	1.85994	0.929969	−	0.367637i	$-0.119833\pi$
$239$	28.7540	1.85994	0.929969	+	0.367637i	$0.119833\pi$
$240$	−0.145977	−0.00942276
$241$	−6.08392	−0.391900	−0.195950	−	0.980614i	$-0.562779\pi$
$241$	−6.08392	−0.391900	−0.195950	+	0.980614i	$0.562779\pi$
$242$	11.8680	0.762901
$243$	−11.0567	−0.709290
$244$	−5.78940	−0.370628
$245$	4.29312	0.274278
$246$	1.19957	0.0764817
$247$	−8.40054	−0.534513
$248$	−4.13030	−0.262275
$249$	2.80430	0.177715
$250$	−3.21619	−0.203410
$251$	30.6032	1.93165	0.965827	−	0.259188i	$-0.0834548\pi$
$251$	30.6032	1.93165	0.965827	+	0.259188i	$0.0834548\pi$
$252$	−12.5792	−0.792416
$253$	−1.39911	−0.0879613
$254$	−20.2705	−1.27189
$255$	0.590692	0.0369906
$256$	1.00000	0.0625000
$257$	−8.73603	−0.544939	−0.272469	−	0.962164i	$-0.587840\pi$
$257$	−8.73603	−0.544939	−0.272469	+	0.962164i	$0.587840\pi$
$258$	−0.659145	−0.0410366
$259$	14.5613	0.904798
$260$	−0.748489	−0.0464193
$261$	−10.8932	−0.674272
$262$	3.66610	0.226492
$263$	29.3791	1.81159	0.905797	−	0.423712i	$-0.139273\pi$
$263$	29.3791	1.81159	0.905797	+	0.423712i	$0.139273\pi$
$264$	2.14755	0.132172
$265$	−1.86061	−0.114296
$266$	16.3996	1.00552
$267$	−6.18916	−0.378770
$268$	−14.6678	−0.895980
$269$	−26.7854	−1.63314	−0.816568	−	0.577250i	$-0.804126\pi$
$269$	−26.7854	−1.63314	−0.816568	+	0.577250i	$0.804126\pi$
$270$	0.846420	0.0515115
$271$	−6.41023	−0.389394	−0.194697	−	0.980863i	$-0.562372\pi$
$271$	−6.41023	−0.389394	−0.194697	+	0.980863i	$0.562372\pi$
$272$	−4.04648	−0.245354
$273$	4.64853	0.281342
$274$	−4.32403	−0.261224
$275$	23.4050	1.41137
$276$	−0.131392	−0.00790884
$277$	16.1883	0.972658	0.486329	−	0.873776i	$-0.338336\pi$
$277$	16.1883	0.972658	0.486329	+	0.873776i	$0.338336\pi$
$278$	−11.3249	−0.679223
$279$	11.5579	0.691955
$280$	1.46120	0.0873235
$281$	13.0179	0.776584	0.388292	−	0.921536i	$-0.373065\pi$
$281$	13.0179	0.776584	0.388292	+	0.921536i	$0.373065\pi$
$282$	−2.67966	−0.159571
$283$	−29.0483	−1.72674	−0.863372	−	0.504568i	$-0.831652\pi$
$283$	−29.0483	−1.72674	−0.863372	+	0.504568i	$0.831652\pi$
$284$	7.96883	0.472863
$285$	−0.532550	−0.0315455
$286$	11.0114	0.651120
$287$	−12.0075	−0.708778
$288$	−2.79832	−0.164893
$289$	−0.626016	−0.0368245
$290$	1.26535	0.0743041
$291$	7.61444	0.446366
$292$	3.48592	0.203998
$293$	−32.3273	−1.88858	−0.944292	−	0.329110i	$-0.893251\pi$
$293$	−32.3273	−1.88858	−0.944292	+	0.329110i	$0.893251\pi$
$294$	−5.93127	−0.345919
$295$	−1.27972	−0.0745082
$296$	3.23926	0.188278
$297$	−12.4522	−0.722548
$298$	−7.97108	−0.461752
$299$	−0.673704	−0.0389613
$300$	2.19798	0.126900
$301$	6.59793	0.380298
$302$	−3.44572	−0.198279
$303$	5.67764	0.326172
$304$	3.64818	0.209238
$305$	−1.88186	−0.107755
$306$	11.3233	0.647313
$307$	−22.8472	−1.30396	−0.651980	−	0.758236i	$-0.726062\pi$
$307$	−22.8472	−1.30396	−0.651980	+	0.758236i	$0.726062\pi$
$308$	−21.4966	−1.22488
$309$	7.18257	0.408602
$310$	−1.34257	−0.0762528
$311$	10.3233	0.585380	0.292690	−	0.956207i	$-0.405450\pi$
$311$	10.3233	0.585380	0.292690	+	0.956207i	$0.405450\pi$
$312$	1.03409	0.0585440
$313$	21.2765	1.20262	0.601311	−	0.799015i	$-0.294645\pi$
$313$	21.2765	1.20262	0.601311	+	0.799015i	$0.294645\pi$
$314$	18.9095	1.06713
$315$	−4.08891	−0.230384
$316$	−15.7497	−0.885988
$317$	10.5657	0.593427	0.296713	−	0.954967i	$-0.404109\pi$
$317$	10.5657	0.593427	0.296713	+	0.954967i	$0.404109\pi$
$318$	2.57057	0.144151
$319$	−18.6153	−1.04226
$320$	0.325053	0.0181710
$321$	4.34736	0.242646
$322$	1.31521	0.0732936
$323$	−14.7623	−0.821395
$324$	7.22557	0.401421
$325$	11.2700	0.625148
$326$	−1.94522	−0.107736
$327$	2.35500	0.130232
$328$	−2.67113	−0.147489
$329$	26.8229	1.47879
$330$	0.698067	0.0384274
$331$	7.17242	0.394232	0.197116	−	0.980380i	$-0.436843\pi$
$331$	7.17242	0.394232	0.197116	+	0.980380i	$0.436843\pi$
$332$	−6.24447	−0.342710
$333$	−9.06450	−0.496731
$334$	0.880492	0.0481784
$335$	−4.76783	−0.260494
$336$	−2.01876	−0.110132
$337$	−21.2885	−1.15966	−0.579829	−	0.814739i	$-0.696880\pi$
$337$	−21.2885	−1.15966	−0.579829	+	0.814739i	$0.696880\pi$
$338$	−7.69774	−0.418702
$339$	−1.92673	−0.104646
$340$	−1.31532	−0.0713333
$341$	19.7513	1.06959
$342$	−10.2088	−0.552028
$343$	27.9041	1.50668
$344$	1.46775	0.0791357
$345$	−0.0427093	−0.00229939
$346$	22.5163	1.21048
$347$	8.88351	0.476892	0.238446	−	0.971156i	$-0.423362\pi$
$347$	8.88351	0.476892	0.238446	+	0.971156i	$0.423362\pi$
$348$	−1.74818	−0.0937124
$349$	−1.31236	−0.0702492	−0.0351246	−	0.999383i	$-0.511183\pi$
$349$	−1.31236	−0.0702492	−0.0351246	+	0.999383i	$0.511183\pi$
$350$	−22.0014	−1.17602
$351$	−5.99601	−0.320043
$352$	−4.78204	−0.254884
$353$	−31.2347	−1.66245	−0.831227	−	0.555933i	$-0.812361\pi$
$353$	−31.2347	−1.66245	−0.831227	+	0.555933i	$0.812361\pi$
$354$	1.76803	0.0939698
$355$	2.59029	0.137478
$356$	13.7817	0.730428
$357$	8.16887	0.432342
$358$	15.1462	0.800501
$359$	−29.3361	−1.54830	−0.774150	−	0.633002i	$-0.781822\pi$
$359$	−29.3361	−1.54830	−0.774150	+	0.633002i	$0.781822\pi$
$360$	−0.909604	−0.0479403
$361$	−5.69078	−0.299515
$362$	14.1482	0.743613
$363$	−5.32973	−0.279738
$364$	−10.3511	−0.542545
$365$	1.13311	0.0593096
$366$	2.59994	0.135901
$367$	−1.20230	−0.0627596	−0.0313798	−	0.999508i	$-0.509990\pi$
$367$	−1.20230	−0.0627596	−0.0313798	+	0.999508i	$0.509990\pi$
$368$	0.292576	0.0152516
$369$	7.47469	0.389117
$370$	1.05293	0.0547394
$371$	−25.7310	−1.33589
$372$	1.85486	0.0961700
$373$	35.9635	1.86212	0.931059	−	0.364868i	$-0.118886\pi$
$373$	35.9635	1.86212	0.931059	+	0.364868i	$0.118886\pi$
$374$	19.3504	1.00059
$375$	1.44434	0.0745856
$376$	5.96691	0.307720
$377$	−8.96371	−0.461655
$378$	11.7054	0.602062
$379$	−16.7491	−0.860341	−0.430170	−	0.902748i	$-0.641547\pi$
$379$	−16.7491	−0.860341	−0.430170	+	0.902748i	$0.641547\pi$
$380$	1.18585	0.0608330
$381$	9.10320	0.466371
$382$	−1.09137	−0.0558392
$383$	11.4905	0.587136	0.293568	−	0.955938i	$-0.405157\pi$
$383$	11.4905	0.587136	0.293568	+	0.955938i	$0.405157\pi$
$384$	−0.449086	−0.0229173
$385$	−6.98753	−0.356118
$386$	5.43339	0.276552
$387$	−4.10724	−0.208783
$388$	−16.9554	−0.860781
$389$	13.2516	0.671885	0.335943	−	0.941882i	$-0.390945\pi$
$389$	13.2516	0.671885	0.335943	+	0.941882i	$0.390945\pi$
$390$	0.336136	0.0170209
$391$	−1.18390	−0.0598725
$392$	13.2074	0.667076
$393$	−1.64639	−0.0830495
$394$	2.33572	0.117672
$395$	−5.11948	−0.257589
$396$	13.3817	0.672456
$397$	28.2973	1.42020	0.710100	−	0.704101i	$-0.248650\pi$
$397$	28.2973	1.42020	0.710100	+	0.704101i	$0.248650\pi$
$398$	−8.85424	−0.443823
$399$	−7.36480	−0.368701
$400$	−4.89434	−0.244717
$401$	25.7497	1.28588	0.642939	−	0.765918i	$-0.277715\pi$
$401$	25.7497	1.28588	0.642939	+	0.765918i	$0.277715\pi$
$402$	6.58711	0.328535
$403$	9.51070	0.473762
$404$	−12.6427	−0.628996
$405$	2.34870	0.116708
$406$	17.4990	0.868460
$407$	−15.4903	−0.767825
$408$	1.81722	0.0899656
$409$	−9.14618	−0.452249	−0.226125	−	0.974098i	$-0.572606\pi$
$409$	−9.14618	−0.452249	−0.226125	+	0.974098i	$0.572606\pi$
$410$	−0.868261	−0.0428804
$411$	1.94186	0.0957848
$412$	−15.9938	−0.787956
$413$	−17.6977	−0.870846
$414$	−0.818721	−0.0402380
$415$	−2.02978	−0.0996382
$416$	−2.30266	−0.112897
$417$	5.08585	0.249055
$418$	−17.4458	−0.853300
$419$	−18.8306	−0.919934	−0.459967	−	0.887936i	$-0.652139\pi$
$419$	−18.8306	−0.919934	−0.459967	+	0.887936i	$0.652139\pi$
$420$	−0.656205	−0.0320195
$421$	19.1746	0.934515	0.467257	−	0.884121i	$-0.345242\pi$
$421$	19.1746	0.934515	0.467257	+	0.884121i	$0.345242\pi$
$422$	9.73960	0.474116
$423$	−16.6973	−0.811853
$424$	−5.72402	−0.277983
$425$	19.8048	0.960676
$426$	−3.57868	−0.173388
$427$	−26.0249	−1.25943
$428$	−9.68046	−0.467923
$429$	−4.94508	−0.238751
$430$	0.477097	0.0230077
$431$	−19.6430	−0.946170	−0.473085	−	0.881017i	$-0.656860\pi$
$431$	−19.6430	−0.946170	−0.473085	+	0.881017i	$0.656860\pi$
$432$	2.60394	0.125282
$433$	−0.970513	−0.0466399	−0.0233199	−	0.999728i	$-0.507424\pi$
$433$	−0.970513	−0.0466399	−0.0233199	+	0.999728i	$0.507424\pi$
$434$	−18.5668	−0.891236
$435$	−0.568252	−0.0272456
$436$	−5.24399	−0.251141
$437$	1.06737	0.0510592
$438$	−1.56547	−0.0748013
$439$	34.9148	1.66639	0.833196	−	0.552978i	$-0.186509\pi$
$439$	34.9148	1.66639	0.833196	+	0.552978i	$0.186509\pi$
$440$	−1.55442	−0.0741040
$441$	−36.9587	−1.75994
$442$	9.31768	0.443197
$443$	35.2228	1.67348	0.836742	−	0.547597i	$-0.184457\pi$
$443$	35.2228	1.67348	0.836742	+	0.547597i	$0.184457\pi$
$444$	−1.45471	−0.0690373
$445$	4.47979	0.212362
$446$	−27.5688	−1.30542
$447$	3.57970	0.169314
$448$	4.49527	0.212381
$449$	−29.8487	−1.40865	−0.704324	−	0.709878i	$-0.748750\pi$
$449$	−29.8487	−1.40865	−0.704324	+	0.709878i	$0.748750\pi$
$450$	13.6959	0.645633
$451$	12.7735	0.601480
$452$	4.29034	0.201801
$453$	1.54742	0.0727042
$454$	−3.67615	−0.172530
$455$	−3.36466	−0.157738
$456$	−1.63835	−0.0767226
$457$	−23.5252	−1.10046	−0.550232	−	0.835012i	$-0.685461\pi$
$457$	−23.5252	−1.10046	−0.550232	+	0.835012i	$0.685461\pi$
$458$	−15.3435	−0.716957
$459$	−10.5368	−0.491816
$460$	0.0951027	0.00443419
$461$	18.9293	0.881624	0.440812	−	0.897599i	$-0.354690\pi$
$461$	18.9293	0.881624	0.440812	+	0.897599i	$0.354690\pi$
$462$	9.65380	0.449136
$463$	4.77729	0.222019	0.111010	−	0.993819i	$-0.464592\pi$
$463$	4.77729	0.222019	0.111010	+	0.993819i	$0.464592\pi$
$464$	3.89276	0.180717
$465$	0.602929	0.0279601
$466$	−16.6933	−0.773301
$467$	2.92229	0.135228	0.0676138	−	0.997712i	$-0.478461\pi$
$467$	2.92229	0.135228	0.0676138	+	0.997712i	$0.478461\pi$
$468$	6.44360	0.297855
$469$	−65.9358	−3.04463
$470$	1.93957	0.0894655
$471$	−8.49200	−0.391290
$472$	−3.93696	−0.181213
$473$	−7.01884	−0.322727
$474$	7.07295	0.324871
$475$	−17.8554	−0.819264
$476$	−18.1900	−0.833738
$477$	16.0176	0.733398
$478$	28.7540	1.31518
$479$	22.0216	1.00619	0.503097	−	0.864230i	$-0.332194\pi$
$479$	22.0216	1.00619	0.503097	+	0.864230i	$0.332194\pi$
$480$	−0.145977	−0.00666290
$481$	−7.45893	−0.340098
$482$	−6.08392	−0.277115
$483$	−0.590641	−0.0268751
$484$	11.8680	0.539452
$485$	−5.51142	−0.250261
$486$	−11.0567	−0.501543
$487$	38.9861	1.76663	0.883315	−	0.468779i	$-0.155306\pi$
$487$	38.9861	1.76663	0.883315	+	0.468779i	$0.155306\pi$
$488$	−5.78940	−0.262074
$489$	0.873569	0.0395041
$490$	4.29312	0.193944
$491$	−13.1476	−0.593343	−0.296671	−	0.954980i	$-0.595877\pi$
$491$	−13.1476	−0.593343	−0.296671	+	0.954980i	$0.595877\pi$
$492$	1.19957	0.0540807
$493$	−15.7520	−0.709433
$494$	−8.40054	−0.377958
$495$	4.34977	0.195507
$496$	−4.13030	−0.185456
$497$	35.8220	1.60684
$498$	2.80430	0.125664
$499$	10.6328	0.475990	0.237995	−	0.971266i	$-0.423510\pi$
$499$	10.6328	0.475990	0.237995	+	0.971266i	$0.423510\pi$
$500$	−3.21619	−0.143832
$501$	−0.395416	−0.0176659
$502$	30.6032	1.36589
$503$	21.0591	0.938977	0.469489	−	0.882939i	$-0.344438\pi$
$503$	21.0591	0.938977	0.469489	+	0.882939i	$0.344438\pi$
$504$	−12.5792	−0.560323
$505$	−4.10954	−0.182872
$506$	−1.39911	−0.0621981
$507$	3.45694	0.153528
$508$	−20.2705	−0.899359
$509$	40.1675	1.78039	0.890195	−	0.455579i	$-0.150568\pi$
$509$	40.1675	1.78039	0.890195	+	0.455579i	$0.150568\pi$
$510$	0.590692	0.0261563
$511$	15.6701	0.693206
$512$	1.00000	0.0441942
$513$	9.49965	0.419420
$514$	−8.73603	−0.385330
$515$	−5.19882	−0.229088
$516$	−0.659145	−0.0290172
$517$	−28.5340	−1.25493
$518$	14.5613	0.639789
$519$	−10.1117	−0.443856
$520$	−0.748489	−0.0328234
$521$	12.4188	0.544076	0.272038	−	0.962286i	$-0.412302\pi$
$521$	12.4188	0.544076	0.272038	+	0.962286i	$0.412302\pi$
$522$	−10.8932	−0.476782
$523$	13.3861	0.585332	0.292666	−	0.956215i	$-0.405458\pi$
$523$	13.3861	0.585332	0.292666	+	0.956215i	$0.405458\pi$
$524$	3.66610	0.160154
$525$	9.88050	0.431220
$526$	29.3791	1.28099
$527$	16.7132	0.728038
$528$	2.14755	0.0934600
$529$	−22.9144	−0.996278
$530$	−1.86061	−0.0808198
$531$	11.0169	0.478092
$532$	16.3996	0.711011
$533$	6.15073	0.266417
$534$	−6.18916	−0.267831
$535$	−3.14667	−0.136042
$536$	−14.6678	−0.633554
$537$	−6.80194	−0.293525
$538$	−26.7854	−1.15480
$539$	−63.1586	−2.72043
$540$	0.846420	0.0364241
$541$	−36.3328	−1.56207	−0.781035	−	0.624488i	$-0.785308\pi$
$541$	−36.3328	−1.56207	−0.781035	+	0.624488i	$0.785308\pi$
$542$	−6.41023	−0.275343
$543$	−6.35375	−0.272666
$544$	−4.04648	−0.173491
$545$	−1.70457	−0.0730160
$546$	4.64853	0.198939
$547$	−16.7078	−0.714375	−0.357187	−	0.934033i	$-0.616264\pi$
$547$	−16.7078	−0.714375	−0.357187	+	0.934033i	$0.616264\pi$
$548$	−4.32403	−0.184713
$549$	16.2006	0.691425
$550$	23.4050	0.997991
$551$	14.2015	0.605004
$552$	−0.131392	−0.00559240
$553$	−70.7990	−3.01068
$554$	16.1883	0.687773
$555$	−0.472857	−0.0200717
$556$	−11.3249	−0.480283
$557$	15.3274	0.649444	0.324722	−	0.945810i	$-0.394729\pi$
$557$	15.3274	0.649444	0.324722	+	0.945810i	$0.394729\pi$
$558$	11.5579	0.489286
$559$	−3.37973	−0.142948
$560$	1.46120	0.0617471
$561$	−8.69000	−0.366892
$562$	13.0179	0.549127
$563$	15.3118	0.645315	0.322657	−	0.946516i	$-0.395424\pi$
$563$	15.3118	0.645315	0.322657	+	0.946516i	$0.395424\pi$
$564$	−2.67966	−0.112834
$565$	1.39459	0.0586709
$566$	−29.0483	−1.22099
$567$	32.4809	1.36407
$568$	7.96883	0.334364
$569$	16.7654	0.702843	0.351421	−	0.936217i	$-0.385698\pi$
$569$	16.7654	0.702843	0.351421	+	0.936217i	$0.385698\pi$
$570$	−0.532550	−0.0223061
$571$	18.7406	0.784269	0.392135	−	0.919908i	$-0.371737\pi$
$571$	18.7406	0.784269	0.392135	+	0.919908i	$0.371737\pi$
$572$	11.0114	0.460412
$573$	0.490117	0.0204749
$574$	−12.0075	−0.501182
$575$	−1.43197	−0.0597171
$576$	−2.79832	−0.116597
$577$	12.1605	0.506249	0.253124	−	0.967434i	$-0.418542\pi$
$577$	12.1605	0.506249	0.253124	+	0.967434i	$0.418542\pi$
$578$	−0.626016	−0.0260388
$579$	−2.44006	−0.101405
$580$	1.26535	0.0525410
$581$	−28.0706	−1.16456
$582$	7.61444	0.315629
$583$	27.3725	1.13365
$584$	3.48592	0.144248
$585$	2.09451	0.0865974
$586$	−32.3273	−1.33543
$587$	29.8057	1.23021	0.615106	−	0.788444i	$-0.289113\pi$
$587$	29.8057	1.23021	0.615106	+	0.788444i	$0.289113\pi$
$588$	−5.93127	−0.244602
$589$	−15.0681	−0.620870
$590$	−1.27972	−0.0526853
$591$	−1.04894	−0.0431475
$592$	3.23926	0.133133
$593$	13.2698	0.544924	0.272462	−	0.962166i	$-0.412162\pi$
$593$	13.2698	0.544924	0.272462	+	0.962166i	$0.412162\pi$
$594$	−12.4522	−0.510919
$595$	−5.91272	−0.242398
$596$	−7.97108	−0.326508
$597$	3.97631	0.162740
$598$	−0.673704	−0.0275498
$599$	−44.0409	−1.79946	−0.899731	−	0.436446i	$-0.856237\pi$
$599$	−44.0409	−1.79946	−0.899731	+	0.436446i	$0.856237\pi$
$600$	2.19798	0.0897321
$601$	−37.7561	−1.54011	−0.770053	−	0.637980i	$-0.779770\pi$
$601$	−37.7561	−1.54011	−0.770053	+	0.637980i	$0.779770\pi$
$602$	6.59793	0.268911
$603$	41.0453	1.67149
$604$	−3.44572	−0.140204
$605$	3.85772	0.156839
$606$	5.67764	0.230638
$607$	−47.3971	−1.92379	−0.961895	−	0.273420i	$-0.911845\pi$
$607$	−47.3971	−1.92379	−0.961895	+	0.273420i	$0.911845\pi$
$608$	3.64818	0.147953
$609$	−7.85855	−0.318444
$610$	−1.88186	−0.0761945
$611$	−13.7398	−0.555853
$612$	11.3233	0.457719
$613$	−15.0110	−0.606290	−0.303145	−	0.952944i	$-0.598037\pi$
$613$	−15.0110	−0.606290	−0.303145	+	0.952944i	$0.598037\pi$
$614$	−22.8472	−0.922039
$615$	0.389924	0.0157232
$616$	−21.4966	−0.866122
$617$	11.1324	0.448175	0.224088	−	0.974569i	$-0.428060\pi$
$617$	11.1324	0.448175	0.224088	+	0.974569i	$0.428060\pi$
$618$	7.18257	0.288925
$619$	−12.3982	−0.498325	−0.249162	−	0.968462i	$-0.580155\pi$
$619$	−12.3982	−0.498325	−0.249162	+	0.968462i	$0.580155\pi$
$620$	−1.34257	−0.0539189
$621$	0.761851	0.0305720
$622$	10.3233	0.413926
$623$	61.9524	2.48207
$624$	1.03409	0.0413969
$625$	23.4263	0.937051
$626$	21.2765	0.850382
$627$	7.83464	0.312885
$628$	18.9095	0.754572
$629$	−13.1076	−0.522634
$630$	−4.08891	−0.162906
$631$	3.09971	0.123397	0.0616987	−	0.998095i	$-0.480348\pi$
$631$	3.09971	0.123397	0.0616987	+	0.998095i	$0.480348\pi$
$632$	−15.7497	−0.626488
$633$	−4.37391	−0.173847
$634$	10.5657	0.419616
$635$	−6.58900	−0.261477
$636$	2.57057	0.101930
$637$	−30.4123	−1.20498
$638$	−18.6153	−0.736989
$639$	−22.2993	−0.882148
$640$	0.325053	0.0128489
$641$	23.2866	0.919765	0.459882	−	0.887980i	$-0.347892\pi$
$641$	23.2866	0.919765	0.459882	+	0.887980i	$0.347892\pi$
$642$	4.34736	0.171577
$643$	−3.06101	−0.120715	−0.0603573	−	0.998177i	$-0.519224\pi$
$643$	−3.06101	−0.120715	−0.0603573	+	0.998177i	$0.519224\pi$
$644$	1.31521	0.0518264
$645$	−0.214257	−0.00843637
$646$	−14.7623	−0.580814
$647$	35.1714	1.38273	0.691364	−	0.722506i	$-0.257010\pi$
$647$	35.1714	1.38273	0.691364	+	0.722506i	$0.257010\pi$
$648$	7.22557	0.283847
$649$	18.8267	0.739013
$650$	11.2700	0.442047
$651$	8.33809	0.326796
$652$	−1.94522	−0.0761805
$653$	−4.45039	−0.174157	−0.0870786	−	0.996201i	$-0.527753\pi$
$653$	−4.45039	−0.174157	−0.0870786	+	0.996201i	$0.527753\pi$
$654$	2.35500	0.0920877
$655$	1.19168	0.0465627
$656$	−2.67113	−0.104290
$657$	−9.75472	−0.380568
$658$	26.8229	1.04566
$659$	−21.1150	−0.822523	−0.411262	−	0.911517i	$-0.634912\pi$
$659$	−21.1150	−0.822523	−0.411262	+	0.911517i	$0.634912\pi$
$660$	0.698067	0.0271722
$661$	−7.33249	−0.285201	−0.142600	−	0.989780i	$-0.545546\pi$
$661$	−7.33249	−0.285201	−0.142600	+	0.989780i	$0.545546\pi$
$662$	7.17242	0.278764
$663$	−4.18444	−0.162510
$664$	−6.24447	−0.242332
$665$	5.33073	0.206717
$666$	−9.06450	−0.351242
$667$	1.13893	0.0440994
$668$	0.880492	0.0340673
$669$	12.3807	0.478667
$670$	−4.76783	−0.184197
$671$	27.6852	1.06877
$672$	−2.01876	−0.0778754
$673$	−26.8237	−1.03398	−0.516989	−	0.855992i	$-0.672947\pi$
$673$	−26.8237	−1.03398	−0.516989	+	0.855992i	$0.672947\pi$
$674$	−21.2885	−0.820001
$675$	−12.7446	−0.490539
$676$	−7.69774	−0.296067
$677$	−20.1850	−0.775772	−0.387886	−	0.921707i	$-0.626795\pi$
$677$	−20.1850	−0.775772	−0.387886	+	0.921707i	$0.626795\pi$
$678$	−1.92673	−0.0739957
$679$	−76.2192	−2.92502
$680$	−1.31532	−0.0504403
$681$	1.65090	0.0632628
$682$	19.7513	0.756316
$683$	−3.83997	−0.146932	−0.0734661	−	0.997298i	$-0.523406\pi$
$683$	−3.83997	−0.146932	−0.0734661	+	0.997298i	$0.523406\pi$
$684$	−10.2088	−0.390343
$685$	−1.40554	−0.0537029
$686$	27.9041	1.06538
$687$	6.89057	0.262892
$688$	1.46775	0.0559574
$689$	13.1805	0.502137
$690$	−0.0427093	−0.00162591
$691$	21.3812	0.813378	0.406689	−	0.913567i	$-0.366683\pi$
$691$	21.3812	0.813378	0.406689	+	0.913567i	$0.366683\pi$
$692$	22.5163	0.855941
$693$	60.1543	2.28507
$694$	8.88351	0.337213
$695$	−3.68120	−0.139636
$696$	−1.74818	−0.0662647
$697$	10.8087	0.409408
$698$	−1.31236	−0.0496737
$699$	7.49671	0.283552
$700$	−22.0014	−0.831574
$701$	−17.4602	−0.659464	−0.329732	−	0.944075i	$-0.606958\pi$
$701$	−17.4602	−0.659464	−0.329732	+	0.944075i	$0.606958\pi$
$702$	−5.99601	−0.226305
$703$	11.8174	0.445702
$704$	−4.78204	−0.180230
$705$	−0.871031	−0.0328049
$706$	−31.2347	−1.17553
$707$	−56.8322	−2.13739
$708$	1.76803	0.0664467
$709$	−16.0245	−0.601813	−0.300906	−	0.953654i	$-0.597289\pi$
$709$	−16.0245	−0.601813	−0.300906	+	0.953654i	$0.597289\pi$
$710$	2.59029	0.0972120
$711$	44.0726	1.65285
$712$	13.7817	0.516491
$713$	−1.20843	−0.0452560
$714$	8.16887	0.305712
$715$	3.57931	0.133858
$716$	15.1462	0.566040
$717$	−12.9130	−0.482244
$718$	−29.3361	−1.09481
$719$	−0.798548	−0.0297808	−0.0148904	−	0.999889i	$-0.504740\pi$
$719$	−0.798548	−0.0297808	−0.0148904	+	0.999889i	$0.504740\pi$
$720$	−0.909604	−0.0338989
$721$	−71.8962	−2.67756
$722$	−5.69078	−0.211789
$723$	2.73220	0.101612
$724$	14.1482	0.525814
$725$	−19.0525	−0.707591
$726$	−5.32973	−0.197805
$727$	−4.25005	−0.157625	−0.0788127	−	0.996889i	$-0.525113\pi$
$727$	−4.25005	−0.157625	−0.0788127	+	0.996889i	$0.525113\pi$
$728$	−10.3511	−0.383637
$729$	−16.7113	−0.618937
$730$	1.13311	0.0419382
$731$	−5.93922	−0.219670
$732$	2.59994	0.0960964
$733$	8.14363	0.300792	0.150396	−	0.988626i	$-0.451945\pi$
$733$	8.14363	0.300792	0.150396	+	0.988626i	$0.451945\pi$
$734$	−1.20230	−0.0443777
$735$	−1.92798	−0.0711146
$736$	0.292576	0.0107845
$737$	70.1422	2.58372
$738$	7.47469	0.275147
$739$	−46.0612	−1.69439	−0.847193	−	0.531285i	$-0.821710\pi$
$739$	−46.0612	−1.69439	−0.847193	+	0.531285i	$0.821710\pi$
$740$	1.05293	0.0387066
$741$	3.77256	0.138588
$742$	−25.7310	−0.944615
$743$	−14.0282	−0.514643	−0.257322	−	0.966326i	$-0.582840\pi$
$743$	−14.0282	−0.514643	−0.257322	+	0.966326i	$0.582840\pi$
$744$	1.85486	0.0680025
$745$	−2.59103	−0.0949278
$746$	35.9635	1.31672
$747$	17.4740	0.639341
$748$	19.3504	0.707522
$749$	−43.5163	−1.59005
$750$	1.44434	0.0527400
$751$	−34.6862	−1.26572	−0.632858	−	0.774268i	$-0.718118\pi$
$751$	−34.6862	−1.26572	−0.632858	+	0.774268i	$0.718118\pi$
$752$	5.96691	0.217591
$753$	−13.7434	−0.500839
$754$	−8.96371	−0.326439
$755$	−1.12004	−0.0407625
$756$	11.7054	0.425722
$757$	−21.2081	−0.770821	−0.385411	−	0.922745i	$-0.625940\pi$
$757$	−21.2081	−0.770821	−0.385411	+	0.922745i	$0.625940\pi$
$758$	−16.7491	−0.608353
$759$	0.628320	0.0228066
$760$	1.18585	0.0430154
$761$	−49.1150	−1.78042	−0.890210	−	0.455551i	$-0.849442\pi$
$761$	−49.1150	−1.78042	−0.890210	+	0.455551i	$0.849442\pi$
$762$	9.10320	0.329774
$763$	−23.5731	−0.853404
$764$	−1.09137	−0.0394843
$765$	3.68069	0.133076
$766$	11.4905	0.415168
$767$	9.06549	0.327336
$768$	−0.449086	−0.0162050
$769$	40.4551	1.45885	0.729424	−	0.684062i	$-0.239788\pi$
$769$	40.4551	1.45885	0.729424	+	0.684062i	$0.239788\pi$
$770$	−6.98753	−0.251813
$771$	3.92323	0.141292
$772$	5.43339	0.195552
$773$	2.42341	0.0871640	0.0435820	−	0.999050i	$-0.486123\pi$
$773$	2.42341	0.0871640	0.0435820	+	0.999050i	$0.486123\pi$
$774$	−4.10724	−0.147632
$775$	20.2151	0.726148
$776$	−16.9554	−0.608664
$777$	−6.53929	−0.234596
$778$	13.2516	0.475095
$779$	−9.74478	−0.349143
$780$	0.336136	0.0120356
$781$	−38.1073	−1.36359
$782$	−1.18390	−0.0423362
$783$	10.1365	0.362250
$784$	13.2074	0.471694
$785$	6.14660	0.219382
$786$	−1.64639	−0.0587249
$787$	−35.7985	−1.27608	−0.638039	−	0.770004i	$-0.720254\pi$
$787$	−35.7985	−1.27608	−0.638039	+	0.770004i	$0.720254\pi$
$788$	2.33572	0.0832065
$789$	−13.1937	−0.469709
$790$	−5.11948	−0.182143
$791$	19.2862	0.685740
$792$	13.3817	0.475498
$793$	13.3310	0.473399
$794$	28.2973	1.00423
$795$	0.835574	0.0296347
$796$	−8.85424	−0.313830
$797$	−12.2431	−0.433672	−0.216836	−	0.976208i	$-0.569574\pi$
$797$	−12.2431	−0.433672	−0.216836	+	0.976208i	$0.569574\pi$
$798$	−7.36480	−0.260711
$799$	−24.1450	−0.854188
$800$	−4.89434	−0.173041
$801$	−38.5656	−1.36265
$802$	25.7497	0.909253
$803$	−16.6698	−0.588265
$804$	6.58711	0.232309
$805$	0.427512	0.0150678
$806$	9.51070	0.335000
$807$	12.0289	0.423439
$808$	−12.6427	−0.444767
$809$	−27.0591	−0.951349	−0.475674	−	0.879621i	$-0.657796\pi$
$809$	−27.0591	−0.951349	−0.475674	+	0.879621i	$0.657796\pi$
$810$	2.34870	0.0825248
$811$	−35.5244	−1.24743	−0.623716	−	0.781651i	$-0.714378\pi$
$811$	−35.5244	−1.24743	−0.623716	+	0.781651i	$0.714378\pi$
$812$	17.4990	0.614094
$813$	2.87874	0.100962
$814$	−15.4903	−0.542934
$815$	−0.632299	−0.0221485
$816$	1.81722	0.0636153
$817$	5.35461	0.187334
$818$	−9.14618	−0.319789
$819$	28.9657	1.01214
$820$	−0.868261	−0.0303210
$821$	34.0776	1.18932	0.594658	−	0.803979i	$-0.297288\pi$
$821$	34.0776	1.18932	0.594658	+	0.803979i	$0.297288\pi$
$822$	1.94186	0.0677301
$823$	14.3792	0.501226	0.250613	−	0.968087i	$-0.419368\pi$
$823$	14.3792	0.501226	0.250613	+	0.968087i	$0.419368\pi$
$824$	−15.9938	−0.557169
$825$	−10.5108	−0.365940
$826$	−17.6977	−0.615781
$827$	−47.7543	−1.66058	−0.830290	−	0.557332i	$-0.811825\pi$
$827$	−47.7543	−1.66058	−0.830290	+	0.557332i	$0.811825\pi$
$828$	−0.818721	−0.0284525
$829$	49.7301	1.72720	0.863599	−	0.504179i	$-0.168205\pi$
$829$	49.7301	1.72720	0.863599	+	0.504179i	$0.168205\pi$
$830$	−2.02978	−0.0704549
$831$	−7.26991	−0.252191
$832$	−2.30266	−0.0798305
$833$	−53.4436	−1.85171
$834$	5.08585	0.176109
$835$	0.286207	0.00990460
$836$	−17.4458	−0.603374
$837$	−10.7551	−0.371750
$838$	−18.8306	−0.650492
$839$	46.3152	1.59898	0.799489	−	0.600681i	$-0.205104\pi$
$839$	46.3152	1.59898	0.799489	+	0.600681i	$0.205104\pi$
$840$	−0.656205	−0.0226412
$841$	−13.8464	−0.477463
$842$	19.1746	0.660802
$843$	−5.84616	−0.201352
$844$	9.73960	0.335251
$845$	−2.50218	−0.0860774
$846$	−16.6973	−0.574067
$847$	53.3496	1.83312
$848$	−5.72402	−0.196564
$849$	13.0452	0.447710
$850$	19.8048	0.679300
$851$	0.947729	0.0324877
$852$	−3.57868	−0.122604
$853$	20.6746	0.707886	0.353943	−	0.935267i	$-0.384841\pi$
$853$	20.6746	0.707886	0.353943	+	0.935267i	$0.384841\pi$
$854$	−26.0249	−0.890554
$855$	−3.31840	−0.113487
$856$	−9.68046	−0.330871
$857$	−24.2634	−0.828821	−0.414410	−	0.910090i	$-0.636012\pi$
$857$	−24.2634	−0.828821	−0.414410	+	0.910090i	$0.636012\pi$
$858$	−4.94508	−0.168822
$859$	11.1773	0.381365	0.190682	−	0.981652i	$-0.438930\pi$
$859$	11.1773	0.381365	0.190682	+	0.981652i	$0.438930\pi$
$860$	0.477097	0.0162689
$861$	5.39238	0.183772
$862$	−19.6430	−0.669043
$863$	28.1538	0.958365	0.479183	−	0.877715i	$-0.340933\pi$
$863$	28.1538	0.958365	0.479183	+	0.877715i	$0.340933\pi$
$864$	2.60394	0.0885879
$865$	7.31900	0.248853
$866$	−0.970513	−0.0329794
$867$	0.281135	0.00954784
$868$	−18.5668	−0.630199
$869$	75.3156	2.55491
$870$	−0.568252	−0.0192656
$871$	33.7751	1.14443
$872$	−5.24399	−0.177584
$873$	47.4467	1.60583
$874$	1.06737	0.0361043
$875$	−14.4576	−0.488757
$876$	−1.56547	−0.0528925
$877$	−29.8366	−1.00751	−0.503756	−	0.863846i	$-0.668049\pi$
$877$	−29.8366	−1.00751	−0.503756	+	0.863846i	$0.668049\pi$
$878$	34.9148	1.17832
$879$	14.5177	0.489671
$880$	−1.55442	−0.0523995
$881$	44.5025	1.49933	0.749663	−	0.661820i	$-0.230215\pi$
$881$	44.5025	1.49933	0.749663	+	0.661820i	$0.230215\pi$
$882$	−36.9587	−1.24446
$883$	47.0307	1.58271	0.791355	−	0.611357i	$-0.209376\pi$
$883$	47.0307	1.58271	0.791355	+	0.611357i	$0.209376\pi$
$884$	9.31768	0.313388
$885$	0.574704	0.0193185
$886$	35.2228	1.18333
$887$	3.47515	0.116684	0.0583421	−	0.998297i	$-0.481419\pi$
$887$	3.47515	0.116684	0.0583421	+	0.998297i	$0.481419\pi$
$888$	−1.45471	−0.0488167
$889$	−91.1215	−3.05612
$890$	4.47979	0.150163
$891$	−34.5530	−1.15757
$892$	−27.5688	−0.923071
$893$	21.7684	0.728451
$894$	3.57970	0.119723
$895$	4.92332	0.164568
$896$	4.49527	0.150176
$897$	0.302551	0.0101019
$898$	−29.8487	−0.996065
$899$	−16.0783	−0.536240
$900$	13.6959	0.456531
$901$	23.1621	0.771642
$902$	12.7735	0.425310
$903$	−2.96303	−0.0986036
$904$	4.29034	0.142695
$905$	4.59892	0.152873
$906$	1.54742	0.0514096
$907$	−9.53714	−0.316675	−0.158338	−	0.987385i	$-0.550613\pi$
$907$	−9.53714	−0.316675	−0.158338	+	0.987385i	$0.550613\pi$
$908$	−3.67615	−0.121997
$909$	35.3782	1.17342
$910$	−3.36466	−0.111537
$911$	21.0900	0.698743	0.349371	−	0.936984i	$-0.386395\pi$
$911$	21.0900	0.698743	0.349371	+	0.936984i	$0.386395\pi$
$912$	−1.63835	−0.0542510
$913$	29.8613	0.988266
$914$	−23.5252	−0.778145
$915$	0.845118	0.0279387
$916$	−15.3435	−0.506965
$917$	16.4801	0.544221
$918$	−10.5368	−0.347766
$919$	5.42346	0.178904	0.0894518	−	0.995991i	$-0.471489\pi$
$919$	5.42346	0.178904	0.0894518	+	0.995991i	$0.471489\pi$
$920$	0.0951027	0.00313544
$921$	10.2604	0.338090
$922$	18.9293	0.623403
$923$	−18.3495	−0.603982
$924$	9.65380	0.317587
$925$	−15.8540	−0.521278
$926$	4.77729	0.156991
$927$	44.7557	1.46997
$928$	3.89276	0.127786
$929$	−13.4232	−0.440401	−0.220200	−	0.975455i	$-0.570671\pi$
$929$	−13.4232	−0.440401	−0.220200	+	0.975455i	$0.570671\pi$
$930$	0.602929	0.0197708
$931$	48.1831	1.57914
$932$	−16.6933	−0.546807
$933$	−4.63604	−0.151777
$934$	2.92229	0.0956203
$935$	6.28992	0.205703
$936$	6.44360	0.210616
$937$	−31.2313	−1.02028	−0.510140	−	0.860091i	$-0.670407\pi$
$937$	−31.2313	−1.02028	−0.510140	+	0.860091i	$0.670407\pi$
$938$	−65.9358	−2.15288
$939$	−9.55499	−0.311815
$940$	1.93957	0.0632616
$941$	−26.5807	−0.866506	−0.433253	−	0.901272i	$-0.642634\pi$
$941$	−26.5807	−0.866506	−0.433253	+	0.901272i	$0.642634\pi$
$942$	−8.49200	−0.276684
$943$	−0.781509	−0.0254494
$944$	−3.93696	−0.128137
$945$	3.80489	0.123773
$946$	−7.01884	−0.228202
$947$	28.4179	0.923456	0.461728	−	0.887022i	$-0.347230\pi$
$947$	28.4179	0.923456	0.461728	+	0.887022i	$0.347230\pi$
$948$	7.07295	0.229719
$949$	−8.02690	−0.260564
$950$	−17.8554	−0.579307
$951$	−4.74489	−0.153864
$952$	−18.1900	−0.589541
$953$	−41.1841	−1.33408	−0.667042	−	0.745020i	$-0.732440\pi$
$953$	−41.1841	−1.33408	−0.667042	+	0.745020i	$0.732440\pi$
$954$	16.0176	0.518590
$955$	−0.354752	−0.0114795
$956$	28.7540	0.929969
$957$	8.35988	0.270237
$958$	22.0216	0.711487
$959$	−19.4377	−0.627675
$960$	−0.145977	−0.00471138
$961$	−13.9406	−0.449696
$962$	−7.45893	−0.240486
$963$	27.0891	0.872933
$964$	−6.08392	−0.195950
$965$	1.76614	0.0568541
$966$	−0.590641	−0.0190035
$967$	56.0114	1.80121	0.900603	−	0.434643i	$-0.143126\pi$
$967$	56.0114	1.80121	0.900603	+	0.434643i	$0.143126\pi$
$968$	11.8680	0.381450
$969$	6.62953	0.212971
$970$	−5.51142	−0.176961
$971$	−15.0352	−0.482501	−0.241251	−	0.970463i	$-0.577558\pi$
$971$	−15.0352	−0.482501	−0.241251	+	0.970463i	$0.577558\pi$
$972$	−11.0567	−0.354645
$973$	−50.9085	−1.63205
$974$	38.9861	1.24920
$975$	−5.06121	−0.162088
$976$	−5.78940	−0.185314
$977$	2.29535	0.0734347	0.0367173	−	0.999326i	$-0.488310\pi$
$977$	2.29535	0.0734347	0.0367173	+	0.999326i	$0.488310\pi$
$978$	0.873569	0.0279336
$979$	−65.9047	−2.10632
$980$	4.29312	0.137139
$981$	14.6744	0.468516
$982$	−13.1476	−0.419557
$983$	11.9114	0.379914	0.189957	−	0.981792i	$-0.439165\pi$
$983$	11.9114	0.379914	0.189957	+	0.981792i	$0.439165\pi$
$984$	1.19957	0.0382408
$985$	0.759233	0.0241912
$986$	−15.7520	−0.501645
$987$	−12.0458	−0.383421
$988$	−8.40054	−0.267257
$989$	0.429428	0.0136550
$990$	4.34977	0.138245
$991$	61.0647	1.93978	0.969892	−	0.243534i	$-0.0783068\pi$
$991$	61.0647	1.93978	0.969892	+	0.243534i	$0.0783068\pi$
$992$	−4.13030	−0.131137
$993$	−3.22103	−0.102216
$994$	35.8220	1.13621
$995$	−2.87810	−0.0912419
$996$	2.80430	0.0888577
$997$	55.3042	1.75150	0.875751	−	0.482763i	$-0.160367\pi$
$997$	55.3042	1.75150	0.875751	+	0.482763i	$0.160367\pi$
$998$	10.6328	0.336576
$999$	8.43485	0.266867

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.30	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.30	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.449086 q^{3} +1.00000 q^{4} +0.325053 q^{5} -0.449086 q^{6} +4.49527 q^{7} +1.00000 q^{8} -2.79832 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.449086 q^{3} +1.00000 q^{4} +0.325053 q^{5} -0.449086 q^{6} +4.49527 q^{7} +1.00000 q^{8} -2.79832 q^{9} +0.325053 q^{10} -4.78204 q^{11} -0.449086 q^{12} -2.30266 q^{13} +4.49527 q^{14} -0.145977 q^{15} +1.00000 q^{16} -4.04648 q^{17} -2.79832 q^{18} +3.64818 q^{19} +0.325053 q^{20} -2.01876 q^{21} -4.78204 q^{22} +0.292576 q^{23} -0.449086 q^{24} -4.89434 q^{25} -2.30266 q^{26} +2.60394 q^{27} +4.49527 q^{28} +3.89276 q^{29} -0.145977 q^{30} -4.13030 q^{31} +1.00000 q^{32} +2.14755 q^{33} -4.04648 q^{34} +1.46120 q^{35} -2.79832 q^{36} +3.23926 q^{37} +3.64818 q^{38} +1.03409 q^{39} +0.325053 q^{40} -2.67113 q^{41} -2.01876 q^{42} +1.46775 q^{43} -4.78204 q^{44} -0.909604 q^{45} +0.292576 q^{46} +5.96691 q^{47} -0.449086 q^{48} +13.2074 q^{49} -4.89434 q^{50} +1.81722 q^{51} -2.30266 q^{52} -5.72402 q^{53} +2.60394 q^{54} -1.55442 q^{55} +4.49527 q^{56} -1.63835 q^{57} +3.89276 q^{58} -3.93696 q^{59} -0.145977 q^{60} -5.78940 q^{61} -4.13030 q^{62} -12.5792 q^{63} +1.00000 q^{64} -0.748489 q^{65} +2.14755 q^{66} -14.6678 q^{67} -4.04648 q^{68} -0.131392 q^{69} +1.46120 q^{70} +7.96883 q^{71} -2.79832 q^{72} +3.48592 q^{73} +3.23926 q^{74} +2.19798 q^{75} +3.64818 q^{76} -21.4966 q^{77} +1.03409 q^{78} -15.7497 q^{79} +0.325053 q^{80} +7.22557 q^{81} -2.67113 q^{82} -6.24447 q^{83} -2.01876 q^{84} -1.31532 q^{85} +1.46775 q^{86} -1.74818 q^{87} -4.78204 q^{88} +13.7817 q^{89} -0.909604 q^{90} -10.3511 q^{91} +0.292576 q^{92} +1.85486 q^{93} +5.96691 q^{94} +1.18585 q^{95} -0.449086 q^{96} -16.9554 q^{97} +13.2074 q^{98} +13.3817 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 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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