LMFDB - Embedded newform 8042.2.a.a.1.48

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			1.48
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.24504 q^{3} +1.00000 q^{4} +1.49320 q^{5} +1.24504 q^{6} +1.54540 q^{7} +1.00000 q^{8} -1.44988 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.24504 q^{3} +1.00000 q^{4} +1.49320 q^{5} +1.24504 q^{6} +1.54540 q^{7} +1.00000 q^{8} -1.44988 q^{9} +1.49320 q^{10} -0.809236 q^{11} +1.24504 q^{12} -5.71050 q^{13} +1.54540 q^{14} +1.85909 q^{15} +1.00000 q^{16} -7.43171 q^{17} -1.44988 q^{18} -1.82949 q^{19} +1.49320 q^{20} +1.92409 q^{21} -0.809236 q^{22} -1.30270 q^{23} +1.24504 q^{24} -2.77036 q^{25} -5.71050 q^{26} -5.54027 q^{27} +1.54540 q^{28} -6.08893 q^{29} +1.85909 q^{30} +1.89111 q^{31} +1.00000 q^{32} -1.00753 q^{33} -7.43171 q^{34} +2.30759 q^{35} -1.44988 q^{36} +5.94454 q^{37} -1.82949 q^{38} -7.10980 q^{39} +1.49320 q^{40} +8.50899 q^{41} +1.92409 q^{42} -3.72057 q^{43} -0.809236 q^{44} -2.16496 q^{45} -1.30270 q^{46} +2.58089 q^{47} +1.24504 q^{48} -4.61173 q^{49} -2.77036 q^{50} -9.25276 q^{51} -5.71050 q^{52} +8.00627 q^{53} -5.54027 q^{54} -1.20835 q^{55} +1.54540 q^{56} -2.27778 q^{57} -6.08893 q^{58} -5.06172 q^{59} +1.85909 q^{60} +1.47275 q^{61} +1.89111 q^{62} -2.24065 q^{63} +1.00000 q^{64} -8.52691 q^{65} -1.00753 q^{66} -4.78729 q^{67} -7.43171 q^{68} -1.62191 q^{69} +2.30759 q^{70} -9.05377 q^{71} -1.44988 q^{72} -1.67754 q^{73} +5.94454 q^{74} -3.44921 q^{75} -1.82949 q^{76} -1.25060 q^{77} -7.10980 q^{78} +4.97491 q^{79} +1.49320 q^{80} -2.54821 q^{81} +8.50899 q^{82} -6.70911 q^{83} +1.92409 q^{84} -11.0970 q^{85} -3.72057 q^{86} -7.58096 q^{87} -0.809236 q^{88} +2.58402 q^{89} -2.16496 q^{90} -8.82503 q^{91} -1.30270 q^{92} +2.35450 q^{93} +2.58089 q^{94} -2.73179 q^{95} +1.24504 q^{96} -15.9750 q^{97} -4.61173 q^{98} +1.17329 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.24504	0.718823	0.359412	−	0.933179i	$-0.382977\pi$
$3$	1.24504	0.718823	0.359412	+	0.933179i	$0.382977\pi$
$4$	1.00000	0.500000
$5$	1.49320	0.667778	0.333889	−	0.942612i	$-0.391639\pi$
$5$	1.49320	0.667778	0.333889	+	0.942612i	$0.391639\pi$
$6$	1.24504	0.508285
$7$	1.54540	0.584107	0.292054	−	0.956402i	$-0.405661\pi$
$7$	1.54540	0.584107	0.292054	+	0.956402i	$0.405661\pi$
$8$	1.00000	0.353553
$9$	−1.44988	−0.483293
$10$	1.49320	0.472190
$11$	−0.809236	−0.243994	−0.121997	−	0.992530i	$-0.538930\pi$
$11$	−0.809236	−0.243994	−0.121997	+	0.992530i	$0.538930\pi$
$12$	1.24504	0.359412
$13$	−5.71050	−1.58381	−0.791905	−	0.610645i	$-0.790910\pi$
$13$	−5.71050	−1.58381	−0.791905	+	0.610645i	$0.790910\pi$
$14$	1.54540	0.413026
$15$	1.85909	0.480015
$16$	1.00000	0.250000
$17$	−7.43171	−1.80245	−0.901227	−	0.433348i	$-0.857332\pi$
$17$	−7.43171	−1.80245	−0.901227	+	0.433348i	$0.857332\pi$
$18$	−1.44988	−0.341740
$19$	−1.82949	−0.419714	−0.209857	−	0.977732i	$-0.567300\pi$
$19$	−1.82949	−0.419714	−0.209857	+	0.977732i	$0.567300\pi$
$20$	1.49320	0.333889
$21$	1.92409	0.419870
$22$	−0.809236	−0.172530
$23$	−1.30270	−0.271631	−0.135816	−	0.990734i	$-0.543365\pi$
$23$	−1.30270	−0.271631	−0.135816	+	0.990734i	$0.543365\pi$
$24$	1.24504	0.254142
$25$	−2.77036	−0.554072
$26$	−5.71050	−1.11992
$27$	−5.54027	−1.06623
$28$	1.54540	0.292054
$29$	−6.08893	−1.13069	−0.565343	−	0.824856i	$-0.691256\pi$
$29$	−6.08893	−1.13069	−0.565343	+	0.824856i	$0.691256\pi$
$30$	1.85909	0.339422
$31$	1.89111	0.339653	0.169826	−	0.985474i	$-0.445679\pi$
$31$	1.89111	0.339653	0.169826	+	0.985474i	$0.445679\pi$
$32$	1.00000	0.176777
$33$	−1.00753	−0.175389
$34$	−7.43171	−1.27453
$35$	2.30759	0.390054
$36$	−1.44988	−0.241647
$37$	5.94454	0.977277	0.488638	−	0.872486i	$-0.337494\pi$
$37$	5.94454	0.977277	0.488638	+	0.872486i	$0.337494\pi$
$38$	−1.82949	−0.296782
$39$	−7.10980	−1.13848
$40$	1.49320	0.236095
$41$	8.50899	1.32888	0.664441	−	0.747341i	$-0.268670\pi$
$41$	8.50899	1.32888	0.664441	+	0.747341i	$0.268670\pi$
$42$	1.92409	0.296893
$43$	−3.72057	−0.567381	−0.283691	−	0.958916i	$-0.591559\pi$
$43$	−3.72057	−0.567381	−0.283691	+	0.958916i	$0.591559\pi$
$44$	−0.809236	−0.121997
$45$	−2.16496	−0.322733
$46$	−1.30270	−0.192072
$47$	2.58089	0.376461	0.188230	−	0.982125i	$-0.439725\pi$
$47$	2.58089	0.376461	0.188230	+	0.982125i	$0.439725\pi$
$48$	1.24504	0.179706
$49$	−4.61173	−0.658819
$50$	−2.77036	−0.391788
$51$	−9.25276	−1.29565
$52$	−5.71050	−0.791905
$53$	8.00627	1.09975	0.549873	−	0.835248i	$-0.314676\pi$
$53$	8.00627	1.09975	0.549873	+	0.835248i	$0.314676\pi$
$54$	−5.54027	−0.753935
$55$	−1.20835	−0.162934
$56$	1.54540	0.206513
$57$	−2.27778	−0.301700
$58$	−6.08893	−0.799516
$59$	−5.06172	−0.658980	−0.329490	−	0.944159i	$-0.606877\pi$
$59$	−5.06172	−0.658980	−0.329490	+	0.944159i	$0.606877\pi$
$60$	1.85909	0.240007
$61$	1.47275	0.188566	0.0942831	−	0.995545i	$-0.469944\pi$
$61$	1.47275	0.188566	0.0942831	+	0.995545i	$0.469944\pi$
$62$	1.89111	0.240171
$63$	−2.24065	−0.282295
$64$	1.00000	0.125000
$65$	−8.52691	−1.05763
$66$	−1.00753	−0.124018
$67$	−4.78729	−0.584861	−0.292431	−	0.956287i	$-0.594464\pi$
$67$	−4.78729	−0.584861	−0.292431	+	0.956287i	$0.594464\pi$
$68$	−7.43171	−0.901227
$69$	−1.62191	−0.195255
$70$	2.30759	0.275810
$71$	−9.05377	−1.07448	−0.537242	−	0.843428i	$-0.680534\pi$
$71$	−9.05377	−1.07448	−0.537242	+	0.843428i	$0.680534\pi$
$72$	−1.44988	−0.170870
$73$	−1.67754	−0.196341	−0.0981705	−	0.995170i	$-0.531299\pi$
$73$	−1.67754	−0.196341	−0.0981705	+	0.995170i	$0.531299\pi$
$74$	5.94454	0.691039
$75$	−3.44921	−0.398280
$76$	−1.82949	−0.209857
$77$	−1.25060	−0.142519
$78$	−7.10980	−0.805026
$79$	4.97491	0.559721	0.279861	−	0.960041i	$-0.409712\pi$
$79$	4.97491	0.559721	0.279861	+	0.960041i	$0.409712\pi$
$80$	1.49320	0.166945
$81$	−2.54821	−0.283135
$82$	8.50899	0.939661
$83$	−6.70911	−0.736420	−0.368210	−	0.929743i	$-0.620029\pi$
$83$	−6.70911	−0.736420	−0.368210	+	0.929743i	$0.620029\pi$
$84$	1.92409	0.209935
$85$	−11.0970	−1.20364
$86$	−3.72057	−0.401199
$87$	−7.58096	−0.812764
$88$	−0.809236	−0.0862649
$89$	2.58402	0.273906	0.136953	−	0.990578i	$-0.456269\pi$
$89$	2.58402	0.273906	0.136953	+	0.990578i	$0.456269\pi$
$90$	−2.16496	−0.228206
$91$	−8.82503	−0.925115
$92$	−1.30270	−0.135816
$93$	2.35450	0.244150
$94$	2.58089	0.266198
$95$	−2.73179	−0.280276
$96$	1.24504	0.127071
$97$	−15.9750	−1.62201	−0.811007	−	0.585036i	$-0.801080\pi$
$97$	−15.9750	−1.62201	−0.811007	+	0.585036i	$0.801080\pi$
$98$	−4.61173	−0.465855
$99$	1.17329	0.117921
$100$	−2.77036	−0.277036
$101$	−6.65878	−0.662573	−0.331286	−	0.943530i	$-0.607483\pi$
$101$	−6.65878	−0.662573	−0.331286	+	0.943530i	$0.607483\pi$
$102$	−9.25276	−0.916160
$103$	15.5124	1.52848	0.764240	−	0.644932i	$-0.223114\pi$
$103$	15.5124	1.52848	0.764240	+	0.644932i	$0.223114\pi$
$104$	−5.71050	−0.559961
$105$	2.87304	0.280380
$106$	8.00627	0.777638
$107$	0.292436	0.0282709	0.0141354	−	0.999900i	$-0.495500\pi$
$107$	0.292436	0.0282709	0.0141354	+	0.999900i	$0.495500\pi$
$108$	−5.54027	−0.533113
$109$	−8.76816	−0.839837	−0.419919	−	0.907562i	$-0.637941\pi$
$109$	−8.76816	−0.839837	−0.419919	+	0.907562i	$0.637941\pi$
$110$	−1.20835	−0.115212
$111$	7.40118	0.702489
$112$	1.54540	0.146027
$113$	1.02156	0.0961000	0.0480500	−	0.998845i	$-0.484699\pi$
$113$	1.02156	0.0961000	0.0480500	+	0.998845i	$0.484699\pi$
$114$	−2.27778	−0.213334
$115$	−1.94518	−0.181389
$116$	−6.08893	−0.565343
$117$	8.27954	0.765444
$118$	−5.06172	−0.465969
$119$	−11.4850	−1.05283
$120$	1.85909	0.169711
$121$	−10.3451	−0.940467
$122$	1.47275	0.133336
$123$	10.5940	0.955231
$124$	1.89111	0.169826
$125$	−11.6027	−1.03778
$126$	−2.24065	−0.199613
$127$	8.64061	0.766730	0.383365	−	0.923597i	$-0.374765\pi$
$127$	8.64061	0.766730	0.383365	+	0.923597i	$0.374765\pi$
$128$	1.00000	0.0883883
$129$	−4.63225	−0.407847
$130$	−8.52691	−0.747860
$131$	−19.2228	−1.67950	−0.839750	−	0.542973i	$-0.817298\pi$
$131$	−19.2228	−1.67950	−0.839750	+	0.542973i	$0.817298\pi$
$132$	−1.00753	−0.0876943
$133$	−2.82730	−0.245158
$134$	−4.78729	−0.413559
$135$	−8.27272	−0.712002
$136$	−7.43171	−0.637264
$137$	0.853884	0.0729522	0.0364761	−	0.999335i	$-0.488387\pi$
$137$	0.853884	0.0729522	0.0364761	+	0.999335i	$0.488387\pi$
$138$	−1.62191	−0.138066
$139$	15.1559	1.28551	0.642755	−	0.766072i	$-0.277791\pi$
$139$	15.1559	1.28551	0.642755	+	0.766072i	$0.277791\pi$
$140$	2.30759	0.195027
$141$	3.21330	0.270609
$142$	−9.05377	−0.759775
$143$	4.62115	0.386440
$144$	−1.44988	−0.120823
$145$	−9.09198	−0.755048
$146$	−1.67754	−0.138834
$147$	−5.74178	−0.473574
$148$	5.94454	0.488638
$149$	18.6462	1.52755	0.763777	−	0.645480i	$-0.223343\pi$
$149$	18.6462	1.52755	0.763777	+	0.645480i	$0.223343\pi$
$150$	−3.44921	−0.281627
$151$	−13.5519	−1.10284	−0.551419	−	0.834229i	$-0.685913\pi$
$151$	−13.5519	−1.10284	−0.551419	+	0.834229i	$0.685913\pi$
$152$	−1.82949	−0.148391
$153$	10.7751	0.871113
$154$	−1.25060	−0.100776
$155$	2.82380	0.226813
$156$	−7.10980	−0.569239
$157$	11.9061	0.950214	0.475107	−	0.879928i	$-0.342409\pi$
$157$	11.9061	0.950214	0.475107	+	0.879928i	$0.342409\pi$
$158$	4.97491	0.395783
$159$	9.96812	0.790523
$160$	1.49320	0.118048
$161$	−2.01319	−0.158662
$162$	−2.54821	−0.200207
$163$	−11.8804	−0.930542	−0.465271	−	0.885168i	$-0.654043\pi$
$163$	−11.8804	−0.930542	−0.465271	+	0.885168i	$0.654043\pi$
$164$	8.50899	0.664441
$165$	−1.50444	−0.117121
$166$	−6.70911	−0.520728
$167$	5.77079	0.446557	0.223279	−	0.974755i	$-0.428324\pi$
$167$	5.77079	0.446557	0.223279	+	0.974755i	$0.428324\pi$
$168$	1.92409	0.148446
$169$	19.6099	1.50845
$170$	−11.0970	−0.851102
$171$	2.65254	0.202845
$172$	−3.72057	−0.283691
$173$	18.0810	1.37467	0.687335	−	0.726341i	$-0.258781\pi$
$173$	18.0810	1.37467	0.687335	+	0.726341i	$0.258781\pi$
$174$	−7.58096	−0.574711
$175$	−4.28132	−0.323638
$176$	−0.809236	−0.0609985
$177$	−6.30204	−0.473690
$178$	2.58402	0.193680
$179$	−10.5814	−0.790889	−0.395445	−	0.918490i	$-0.629409\pi$
$179$	−10.5814	−0.790889	−0.395445	+	0.918490i	$0.629409\pi$
$180$	−2.16496	−0.161366
$181$	−5.90091	−0.438611	−0.219306	−	0.975656i	$-0.570379\pi$
$181$	−5.90091	−0.438611	−0.219306	+	0.975656i	$0.570379\pi$
$182$	−8.82503	−0.654155
$183$	1.83363	0.135546
$184$	−1.30270	−0.0960361
$185$	8.87637	0.652604
$186$	2.35450	0.172640
$187$	6.01401	0.439788
$188$	2.58089	0.188230
$189$	−8.56195	−0.622790
$190$	−2.73179	−0.198185
$191$	7.49007	0.541962	0.270981	−	0.962585i	$-0.412652\pi$
$191$	7.49007	0.541962	0.270981	+	0.962585i	$0.412652\pi$
$192$	1.24504	0.0898529
$193$	5.70990	0.411008	0.205504	−	0.978656i	$-0.434117\pi$
$193$	5.70990	0.411008	0.205504	+	0.978656i	$0.434117\pi$
$194$	−15.9750	−1.14694
$195$	−10.6163	−0.760251
$196$	−4.61173	−0.329409
$197$	14.1787	1.01019	0.505094	−	0.863065i	$-0.331458\pi$
$197$	14.1787	1.01019	0.505094	+	0.863065i	$0.331458\pi$
$198$	1.17329	0.0833824
$199$	9.05908	0.642181	0.321091	−	0.947049i	$-0.395951\pi$
$199$	9.05908	0.642181	0.321091	+	0.947049i	$0.395951\pi$
$200$	−2.77036	−0.195894
$201$	−5.96037	−0.420412
$202$	−6.65878	−0.468510
$203$	−9.40986	−0.660442
$204$	−9.25276	−0.647823
$205$	12.7056	0.887398
$206$	15.5124	1.08080
$207$	1.88875	0.131277
$208$	−5.71050	−0.395952
$209$	1.48049	0.102408
$210$	2.87304	0.198259
$211$	28.8212	1.98413	0.992067	−	0.125714i	$-0.0401222\pi$
$211$	28.8212	1.98413	0.992067	+	0.125714i	$0.0401222\pi$
$212$	8.00627	0.549873
$213$	−11.2723	−0.772365
$214$	0.292436	0.0199905
$215$	−5.55554	−0.378885
$216$	−5.54027	−0.376968
$217$	2.92252	0.198394
$218$	−8.76816	−0.593855
$219$	−2.08860	−0.141134
$220$	−1.20835	−0.0814669
$221$	42.4388	2.85474
$222$	7.40118	0.496735
$223$	−17.8039	−1.19224	−0.596119	−	0.802896i	$-0.703291\pi$
$223$	−17.8039	−1.19224	−0.596119	+	0.802896i	$0.703291\pi$
$224$	1.54540	0.103257
$225$	4.01669	0.267779
$226$	1.02156	0.0679530
$227$	6.41516	0.425789	0.212895	−	0.977075i	$-0.431711\pi$
$227$	6.41516	0.425789	0.212895	+	0.977075i	$0.431711\pi$
$228$	−2.27778	−0.150850
$229$	8.44276	0.557914	0.278957	−	0.960304i	$-0.410011\pi$
$229$	8.44276	0.557914	0.278957	+	0.960304i	$0.410011\pi$
$230$	−1.94518	−0.128262
$231$	−1.55704	−0.102446
$232$	−6.08893	−0.399758
$233$	−25.3382	−1.65996	−0.829979	−	0.557795i	$-0.811648\pi$
$233$	−25.3382	−1.65996	−0.829979	+	0.557795i	$0.811648\pi$
$234$	8.27954	0.541251
$235$	3.85377	0.251392
$236$	−5.06172	−0.329490
$237$	6.19396	0.402341
$238$	−11.4850	−0.744461
$239$	−15.4257	−0.997803	−0.498902	−	0.866659i	$-0.666263\pi$
$239$	−15.4257	−0.997803	−0.498902	+	0.866659i	$0.666263\pi$
$240$	1.85909	0.120004
$241$	−7.36974	−0.474726	−0.237363	−	0.971421i	$-0.576283\pi$
$241$	−7.36974	−0.474726	−0.237363	+	0.971421i	$0.576283\pi$
$242$	−10.3451	−0.665011
$243$	13.4482	0.862702
$244$	1.47275	0.0942831
$245$	−6.88622	−0.439945
$246$	10.5940	0.675450
$247$	10.4473	0.664746
$248$	1.89111	0.120085
$249$	−8.35310	−0.529356
$250$	−11.6027	−0.733818
$251$	−20.4199	−1.28889	−0.644447	−	0.764649i	$-0.722913\pi$
$251$	−20.4199	−1.28889	−0.644447	+	0.764649i	$0.722913\pi$
$252$	−2.24065	−0.141148
$253$	1.05419	0.0662763
$254$	8.64061	0.542160
$255$	−13.8162	−0.865204
$256$	1.00000	0.0625000
$257$	30.7650	1.91907	0.959533	−	0.281598i	$-0.0908643\pi$
$257$	30.7650	1.91907	0.959533	+	0.281598i	$0.0908643\pi$
$258$	−4.63225	−0.288391
$259$	9.18671	0.570835
$260$	−8.52691	−0.528817
$261$	8.82822	0.546453
$262$	−19.2228	−1.18759
$263$	−0.163551	−0.0100850	−0.00504249	−	0.999987i	$-0.501605\pi$
$263$	−0.163551	−0.0100850	−0.00504249	+	0.999987i	$0.501605\pi$
$264$	−1.00753	−0.0620092
$265$	11.9549	0.734386
$266$	−2.82730	−0.173353
$267$	3.21720	0.196890
$268$	−4.78729	−0.292431
$269$	5.02527	0.306396	0.153198	−	0.988196i	$-0.451043\pi$
$269$	5.02527	0.306396	0.153198	+	0.988196i	$0.451043\pi$
$270$	−8.27272	−0.503462
$271$	−10.0332	−0.609473	−0.304737	−	0.952437i	$-0.598568\pi$
$271$	−10.0332	−0.609473	−0.304737	+	0.952437i	$0.598568\pi$
$272$	−7.43171	−0.450613
$273$	−10.9875	−0.664994
$274$	0.853884	0.0515850
$275$	2.24188	0.135190
$276$	−1.62191	−0.0976274
$277$	−0.368794	−0.0221587	−0.0110793	−	0.999939i	$-0.503527\pi$
$277$	−0.368794	−0.0221587	−0.0110793	+	0.999939i	$0.503527\pi$
$278$	15.1559	0.908993
$279$	−2.74188	−0.164152
$280$	2.30759	0.137905
$281$	−7.94886	−0.474189	−0.237094	−	0.971487i	$-0.576195\pi$
$281$	−7.94886	−0.474189	−0.237094	+	0.971487i	$0.576195\pi$
$282$	3.21330	0.191349
$283$	−11.5941	−0.689199	−0.344599	−	0.938750i	$-0.611985\pi$
$283$	−11.5941	−0.689199	−0.344599	+	0.938750i	$0.611985\pi$
$284$	−9.05377	−0.537242
$285$	−3.40118	−0.201469
$286$	4.62115	0.273254
$287$	13.1498	0.776209
$288$	−1.44988	−0.0854349
$289$	38.2303	2.24884
$290$	−9.09198	−0.533899
$291$	−19.8895	−1.16594
$292$	−1.67754	−0.0981705
$293$	−1.75024	−0.102250	−0.0511251	−	0.998692i	$-0.516281\pi$
$293$	−1.75024	−0.102250	−0.0511251	+	0.998692i	$0.516281\pi$
$294$	−5.74178	−0.334867
$295$	−7.55815	−0.440052
$296$	5.94454	0.345519
$297$	4.48339	0.260153
$298$	18.6462	1.08014
$299$	7.43906	0.430212
$300$	−3.44921	−0.199140
$301$	−5.74977	−0.331411
$302$	−13.5519	−0.779824
$303$	−8.29043	−0.476273
$304$	−1.82949	−0.104928
$305$	2.19911	0.125920
$306$	10.7751	0.615970
$307$	−12.4839	−0.712491	−0.356246	−	0.934392i	$-0.615943\pi$
$307$	−12.4839	−0.712491	−0.356246	+	0.934392i	$0.615943\pi$
$308$	−1.25060	−0.0712593
$309$	19.3135	1.09871
$310$	2.82380	0.160381
$311$	−17.6662	−1.00176	−0.500880	−	0.865517i	$-0.666990\pi$
$311$	−17.6662	−1.00176	−0.500880	+	0.865517i	$0.666990\pi$
$312$	−7.10980	−0.402513
$313$	−21.3900	−1.20903	−0.604516	−	0.796593i	$-0.706634\pi$
$313$	−21.3900	−1.20903	−0.604516	+	0.796593i	$0.706634\pi$
$314$	11.9061	0.671903
$315$	−3.34573	−0.188510
$316$	4.97491	0.279861
$317$	35.3686	1.98650	0.993250	−	0.115996i	$-0.0370061\pi$
$317$	35.3686	1.98650	0.993250	+	0.115996i	$0.0370061\pi$
$318$	9.96812	0.558984
$319$	4.92739	0.275881
$320$	1.49320	0.0834723
$321$	0.364094	0.0203218
$322$	−2.01319	−0.112191
$323$	13.5962	0.756514
$324$	−2.54821	−0.141567
$325$	15.8202	0.877545
$326$	−11.8804	−0.657993
$327$	−10.9167	−0.603695
$328$	8.50899	0.469830
$329$	3.98851	0.219894
$330$	−1.50444	−0.0828168
$331$	−19.5056	−1.07212	−0.536061	−	0.844179i	$-0.680088\pi$
$331$	−19.5056	−1.07212	−0.536061	+	0.844179i	$0.680088\pi$
$332$	−6.70911	−0.368210
$333$	−8.61887	−0.472311
$334$	5.77079	0.315764
$335$	−7.14837	−0.390557
$336$	1.92409	0.104968
$337$	5.51020	0.300160	0.150080	−	0.988674i	$-0.452047\pi$
$337$	5.51020	0.300160	0.150080	+	0.988674i	$0.452047\pi$
$338$	19.6099	1.06664
$339$	1.27188	0.0690790
$340$	−11.0970	−0.601820
$341$	−1.53035	−0.0828732
$342$	2.65254	0.143433
$343$	−17.9448	−0.968928
$344$	−3.72057	−0.200599
$345$	−2.42183	−0.130387
$346$	18.0810	0.972038
$347$	−7.93020	−0.425715	−0.212858	−	0.977083i	$-0.568277\pi$
$347$	−7.93020	−0.425715	−0.212858	+	0.977083i	$0.568277\pi$
$348$	−7.58096	−0.406382
$349$	8.50671	0.455354	0.227677	−	0.973737i	$-0.426887\pi$
$349$	8.50671	0.455354	0.227677	+	0.973737i	$0.426887\pi$
$350$	−4.28132	−0.228846
$351$	31.6377	1.68870
$352$	−0.809236	−0.0431324
$353$	2.69809	0.143605	0.0718026	−	0.997419i	$-0.477125\pi$
$353$	2.69809	0.143605	0.0718026	+	0.997419i	$0.477125\pi$
$354$	−6.30204	−0.334949
$355$	−13.5191	−0.717518
$356$	2.58402	0.136953
$357$	−14.2992	−0.756796
$358$	−10.5814	−0.559243
$359$	2.00470	0.105804	0.0529020	−	0.998600i	$-0.483153\pi$
$359$	2.00470	0.105804	0.0529020	+	0.998600i	$0.483153\pi$
$360$	−2.16496	−0.114103
$361$	−15.6530	−0.823841
$362$	−5.90091	−0.310145
$363$	−12.8801	−0.676030
$364$	−8.82503	−0.462557
$365$	−2.50490	−0.131112
$366$	1.83363	0.0958453
$367$	8.53313	0.445426	0.222713	−	0.974884i	$-0.428509\pi$
$367$	8.53313	0.445426	0.222713	+	0.974884i	$0.428509\pi$
$368$	−1.30270	−0.0679078
$369$	−12.3370	−0.642239
$370$	8.87637	0.461461
$371$	12.3729	0.642370
$372$	2.35450	0.122075
$373$	−11.1408	−0.576848	−0.288424	−	0.957503i	$-0.593131\pi$
$373$	−11.1408	−0.576848	−0.288424	+	0.957503i	$0.593131\pi$
$374$	6.01401	0.310977
$375$	−14.4458	−0.745977
$376$	2.58089	0.133099
$377$	34.7709	1.79079
$378$	−8.56195	−0.440379
$379$	11.4251	0.586870	0.293435	−	0.955979i	$-0.405202\pi$
$379$	11.4251	0.586870	0.293435	+	0.955979i	$0.405202\pi$
$380$	−2.73179	−0.140138
$381$	10.7579	0.551144
$382$	7.49007	0.383225
$383$	28.6796	1.46546	0.732729	−	0.680520i	$-0.238246\pi$
$383$	28.6796	1.46546	0.732729	+	0.680520i	$0.238246\pi$
$384$	1.24504	0.0635356
$385$	−1.86739	−0.0951708
$386$	5.70990	0.290626
$387$	5.39437	0.274211
$388$	−15.9750	−0.811007
$389$	−17.9847	−0.911860	−0.455930	−	0.890016i	$-0.650693\pi$
$389$	−17.9847	−0.911860	−0.455930	+	0.890016i	$0.650693\pi$
$390$	−10.6163	−0.537579
$391$	9.68126	0.489602
$392$	−4.61173	−0.232928
$393$	−23.9331	−1.20726
$394$	14.1787	0.714310
$395$	7.42853	0.373770
$396$	1.17329	0.0589603
$397$	9.86044	0.494881	0.247441	−	0.968903i	$-0.420410\pi$
$397$	9.86044	0.494881	0.247441	+	0.968903i	$0.420410\pi$
$398$	9.05908	0.454091
$399$	−3.52009	−0.176225
$400$	−2.77036	−0.138518
$401$	−0.946242	−0.0472531	−0.0236265	−	0.999721i	$-0.507521\pi$
$401$	−0.946242	−0.0472531	−0.0236265	+	0.999721i	$0.507521\pi$
$402$	−5.96037	−0.297276
$403$	−10.7992	−0.537945
$404$	−6.65878	−0.331286
$405$	−3.80499	−0.189071
$406$	−9.40986	−0.467003
$407$	−4.81054	−0.238450
$408$	−9.25276	−0.458080
$409$	−24.7334	−1.22299	−0.611493	−	0.791250i	$-0.709431\pi$
$409$	−24.7334	−1.22299	−0.611493	+	0.791250i	$0.709431\pi$
$410$	12.7056	0.627485
$411$	1.06312	0.0524398
$412$	15.5124	0.764240
$413$	−7.82240	−0.384915
$414$	1.88875	0.0928271
$415$	−10.0180	−0.491765
$416$	−5.71050	−0.279981
$417$	18.8697	0.924055
$418$	1.48049	0.0724131
$419$	−0.626119	−0.0305879	−0.0152939	−	0.999883i	$-0.504868\pi$
$419$	−0.626119	−0.0305879	−0.0152939	+	0.999883i	$0.504868\pi$
$420$	2.87304	0.140190
$421$	−20.0392	−0.976651	−0.488325	−	0.872662i	$-0.662392\pi$
$421$	−20.0392	−0.976651	−0.488325	+	0.872662i	$0.662392\pi$
$422$	28.8212	1.40299
$423$	−3.74197	−0.181941
$424$	8.00627	0.388819
$425$	20.5885	0.998690
$426$	−11.2723	−0.546144
$427$	2.27599	0.110143
$428$	0.292436	0.0141354
$429$	5.75351	0.277782
$430$	−5.55554	−0.267912
$431$	6.31218	0.304047	0.152023	−	0.988377i	$-0.451421\pi$
$431$	6.31218	0.304047	0.152023	+	0.988377i	$0.451421\pi$
$432$	−5.54027	−0.266556
$433$	21.9418	1.05445	0.527227	−	0.849724i	$-0.323232\pi$
$433$	21.9418	1.05445	0.527227	+	0.849724i	$0.323232\pi$
$434$	2.92252	0.140286
$435$	−11.3199	−0.542746
$436$	−8.76816	−0.419919
$437$	2.38327	0.114007
$438$	−2.08860	−0.0997971
$439$	21.7197	1.03662	0.518312	−	0.855191i	$-0.326561\pi$
$439$	21.7197	1.03662	0.518312	+	0.855191i	$0.326561\pi$
$440$	−1.20835	−0.0576058
$441$	6.68645	0.318402
$442$	42.4388	2.01861
$443$	1.38439	0.0657743	0.0328871	−	0.999459i	$-0.489530\pi$
$443$	1.38439	0.0657743	0.0328871	+	0.999459i	$0.489530\pi$
$444$	7.40118	0.351245
$445$	3.85845	0.182908
$446$	−17.8039	−0.843040
$447$	23.2152	1.09804
$448$	1.54540	0.0730134
$449$	37.4257	1.76623	0.883114	−	0.469159i	$-0.155443\pi$
$449$	37.4257	1.76623	0.883114	+	0.469159i	$0.155443\pi$
$450$	4.01669	0.189349
$451$	−6.88578	−0.324239
$452$	1.02156	0.0480500
$453$	−16.8726	−0.792746
$454$	6.41516	0.301078
$455$	−13.1775	−0.617771
$456$	−2.27778	−0.106667
$457$	20.5984	0.963554	0.481777	−	0.876294i	$-0.339992\pi$
$457$	20.5984	0.963554	0.481777	+	0.876294i	$0.339992\pi$
$458$	8.44276	0.394504
$459$	41.1737	1.92182
$460$	−1.94518	−0.0906946
$461$	39.6183	1.84521	0.922604	−	0.385749i	$-0.126057\pi$
$461$	39.6183	1.84521	0.922604	+	0.385749i	$0.126057\pi$
$462$	−1.55704	−0.0724401
$463$	−13.0666	−0.607257	−0.303629	−	0.952790i	$-0.598198\pi$
$463$	−13.0666	−0.607257	−0.303629	+	0.952790i	$0.598198\pi$
$464$	−6.08893	−0.282672
$465$	3.51573	0.163038
$466$	−25.3382	−1.17377
$467$	−27.4432	−1.26992	−0.634961	−	0.772544i	$-0.718984\pi$
$467$	−27.4432	−1.26992	−0.634961	+	0.772544i	$0.718984\pi$
$468$	8.27954	0.382722
$469$	−7.39830	−0.341622
$470$	3.85377	0.177761
$471$	14.8236	0.683036
$472$	−5.06172	−0.232985
$473$	3.01082	0.138438
$474$	6.19396	0.284498
$475$	5.06835	0.232552
$476$	−11.4850	−0.526413
$477$	−11.6081	−0.531500
$478$	−15.4257	−0.705554
$479$	5.62038	0.256802	0.128401	−	0.991722i	$-0.459016\pi$
$479$	5.62038	0.256802	0.128401	+	0.991722i	$0.459016\pi$
$480$	1.85909	0.0848554
$481$	−33.9463	−1.54782
$482$	−7.36974	−0.335682
$483$	−2.50650	−0.114050
$484$	−10.3451	−0.470233
$485$	−23.8538	−1.08315
$486$	13.4482	0.610022
$487$	−15.5862	−0.706280	−0.353140	−	0.935570i	$-0.614886\pi$
$487$	−15.5862	−0.706280	−0.353140	+	0.935570i	$0.614886\pi$
$488$	1.47275	0.0666682
$489$	−14.7915	−0.668896
$490$	−6.88622	−0.311088
$491$	−32.4990	−1.46666	−0.733329	−	0.679874i	$-0.762034\pi$
$491$	−32.4990	−1.46666	−0.733329	+	0.679874i	$0.762034\pi$
$492$	10.5940	0.477615
$493$	45.2512	2.03801
$494$	10.4473	0.470047
$495$	1.75196	0.0787448
$496$	1.89111	0.0849132
$497$	−13.9917	−0.627615
$498$	−8.35310	−0.374311
$499$	10.3035	0.461248	0.230624	−	0.973043i	$-0.425923\pi$
$499$	10.3035	0.461248	0.230624	+	0.973043i	$0.425923\pi$
$500$	−11.6027	−0.518888
$501$	7.18486	0.320996
$502$	−20.4199	−0.911386
$503$	−23.7173	−1.05750	−0.528750	−	0.848778i	$-0.677339\pi$
$503$	−23.7173	−1.05750	−0.528750	+	0.848778i	$0.677339\pi$
$504$	−2.24065	−0.0998064
$505$	−9.94287	−0.442452
$506$	1.05419	0.0468644
$507$	24.4150	1.08431
$508$	8.64061	0.383365
$509$	10.5440	0.467356	0.233678	−	0.972314i	$-0.424924\pi$
$509$	10.5440	0.467356	0.233678	+	0.972314i	$0.424924\pi$
$510$	−13.8162	−0.611792
$511$	−2.59247	−0.114684
$512$	1.00000	0.0441942
$513$	10.1359	0.447509
$514$	30.7650	1.35698
$515$	23.1630	1.02069
$516$	−4.63225	−0.203923
$517$	−2.08855	−0.0918541
$518$	9.18671	0.403641
$519$	22.5115	0.988145
$520$	−8.52691	−0.373930
$521$	−6.79966	−0.297898	−0.148949	−	0.988845i	$-0.547589\pi$
$521$	−6.79966	−0.297898	−0.148949	+	0.988845i	$0.547589\pi$
$522$	8.82822	0.386401
$523$	−4.14934	−0.181438	−0.0907191	−	0.995877i	$-0.528917\pi$
$523$	−4.14934	−0.181438	−0.0907191	+	0.995877i	$0.528917\pi$
$524$	−19.2228	−0.839750
$525$	−5.33041	−0.232638
$526$	−0.163551	−0.00713115
$527$	−14.0542	−0.612208
$528$	−1.00753	−0.0438471
$529$	−21.3030	−0.926217
$530$	11.9549	0.519290
$531$	7.33888	0.318480
$532$	−2.82730	−0.122579
$533$	−48.5906	−2.10469
$534$	3.21720	0.139222
$535$	0.436665	0.0188787
$536$	−4.78729	−0.206780
$537$	−13.1742	−0.568510
$538$	5.02527	0.216655
$539$	3.73198	0.160748
$540$	−8.27272	−0.356001
$541$	−4.61984	−0.198622	−0.0993112	−	0.995056i	$-0.531664\pi$
$541$	−4.61984	−0.198622	−0.0993112	+	0.995056i	$0.531664\pi$
$542$	−10.0332	−0.430963
$543$	−7.34686	−0.315284
$544$	−7.43171	−0.318632
$545$	−13.0926	−0.560825
$546$	−10.9875	−0.470222
$547$	−37.4409	−1.60086	−0.800428	−	0.599428i	$-0.795395\pi$
$547$	−37.4409	−1.60086	−0.800428	+	0.599428i	$0.795395\pi$
$548$	0.853884	0.0364761
$549$	−2.13531	−0.0911327
$550$	2.24188	0.0955939
$551$	11.1396	0.474564
$552$	−1.62191	−0.0690330
$553$	7.68824	0.326937
$554$	−0.368794	−0.0156685
$555$	11.0514	0.469107
$556$	15.1559	0.642755
$557$	−17.3774	−0.736303	−0.368152	−	0.929766i	$-0.620009\pi$
$557$	−17.3774	−0.736303	−0.368152	+	0.929766i	$0.620009\pi$
$558$	−2.74188	−0.116073
$559$	21.2463	0.898623
$560$	2.30759	0.0975135
$561$	7.48767	0.316130
$562$	−7.94886	−0.335302
$563$	−28.9385	−1.21961	−0.609807	−	0.792550i	$-0.708753\pi$
$563$	−28.9385	−1.21961	−0.609807	+	0.792550i	$0.708753\pi$
$564$	3.21330	0.135304
$565$	1.52539	0.0641735
$566$	−11.5941	−0.487337
$567$	−3.93802	−0.165381
$568$	−9.05377	−0.379888
$569$	−3.62243	−0.151860	−0.0759300	−	0.997113i	$-0.524193\pi$
$569$	−3.62243	−0.151860	−0.0759300	+	0.997113i	$0.524193\pi$
$570$	−3.40118	−0.142460
$571$	19.7858	0.828010	0.414005	−	0.910275i	$-0.364130\pi$
$571$	19.7858	0.828010	0.414005	+	0.910275i	$0.364130\pi$
$572$	4.62115	0.193220
$573$	9.32543	0.389575
$574$	13.1498	0.548863
$575$	3.60894	0.150503
$576$	−1.44988	−0.0604116
$577$	−27.2356	−1.13383	−0.566916	−	0.823776i	$-0.691863\pi$
$577$	−27.2356	−1.13383	−0.566916	+	0.823776i	$0.691863\pi$
$578$	38.2303	1.59017
$579$	7.10905	0.295442
$580$	−9.09198	−0.377524
$581$	−10.3683	−0.430149
$582$	−19.8895	−0.824445
$583$	−6.47897	−0.268331
$584$	−1.67754	−0.0694170
$585$	12.3630	0.511147
$586$	−1.75024	−0.0723019
$587$	23.3247	0.962711	0.481356	−	0.876525i	$-0.340145\pi$
$587$	23.3247	0.962711	0.481356	+	0.876525i	$0.340145\pi$
$588$	−5.74178	−0.236787
$589$	−3.45976	−0.142557
$590$	−7.55815	−0.311164
$591$	17.6530	0.726146
$592$	5.94454	0.244319
$593$	−34.7001	−1.42496	−0.712482	−	0.701690i	$-0.752429\pi$
$593$	−34.7001	−1.42496	−0.712482	+	0.701690i	$0.752429\pi$
$594$	4.48339	0.183956
$595$	−17.1493	−0.703055
$596$	18.6462	0.763777
$597$	11.2789	0.461615
$598$	7.43906	0.304206
$599$	11.3659	0.464400	0.232200	−	0.972668i	$-0.425408\pi$
$599$	11.3659	0.464400	0.232200	+	0.972668i	$0.425408\pi$
$600$	−3.44921	−0.140813
$601$	20.3354	0.829500	0.414750	−	0.909935i	$-0.363869\pi$
$601$	20.3354	0.829500	0.414750	+	0.909935i	$0.363869\pi$
$602$	−5.74977	−0.234343
$603$	6.94100	0.282659
$604$	−13.5519	−0.551419
$605$	−15.4473	−0.628023
$606$	−8.29043	−0.336776
$607$	22.3286	0.906288	0.453144	−	0.891437i	$-0.350302\pi$
$607$	22.3286	0.906288	0.453144	+	0.891437i	$0.350302\pi$
$608$	−1.82949	−0.0741956
$609$	−11.7156	−0.474741
$610$	2.19911	0.0890392
$611$	−14.7382	−0.596242
$612$	10.7751	0.435557
$613$	−18.7061	−0.755530	−0.377765	−	0.925901i	$-0.623307\pi$
$613$	−18.7061	−0.755530	−0.377765	+	0.925901i	$0.623307\pi$
$614$	−12.4839	−0.503807
$615$	15.8190	0.637882
$616$	−1.25060	−0.0503880
$617$	−21.3833	−0.860861	−0.430430	−	0.902624i	$-0.641638\pi$
$617$	−21.3833	−0.860861	−0.430430	+	0.902624i	$0.641638\pi$
$618$	19.3135	0.776903
$619$	−13.0502	−0.524532	−0.262266	−	0.964996i	$-0.584470\pi$
$619$	−13.0502	−0.524532	−0.262266	+	0.964996i	$0.584470\pi$
$620$	2.82380	0.113406
$621$	7.21729	0.289620
$622$	−17.6662	−0.708351
$623$	3.99335	0.159990
$624$	−7.10980	−0.284620
$625$	−3.47329	−0.138932
$626$	−21.3900	−0.854915
$627$	1.84327	0.0736129
$628$	11.9061	0.475107
$629$	−44.1781	−1.76150
$630$	−3.34573	−0.133297
$631$	32.4217	1.29069	0.645343	−	0.763893i	$-0.276714\pi$
$631$	32.4217	1.29069	0.645343	+	0.763893i	$0.276714\pi$
$632$	4.97491	0.197891
$633$	35.8835	1.42624
$634$	35.3686	1.40467
$635$	12.9021	0.512006
$636$	9.96812	0.395262
$637$	26.3353	1.04344
$638$	4.92739	0.195077
$639$	13.1269	0.519291
$640$	1.49320	0.0590238
$641$	−19.7327	−0.779394	−0.389697	−	0.920943i	$-0.627420\pi$
$641$	−19.7327	−0.779394	−0.389697	+	0.920943i	$0.627420\pi$
$642$	0.364094	0.0143697
$643$	7.97236	0.314399	0.157200	−	0.987567i	$-0.449753\pi$
$643$	7.97236	0.314399	0.157200	+	0.987567i	$0.449753\pi$
$644$	−2.01319	−0.0793308
$645$	−6.91686	−0.272351
$646$	13.5962	0.534936
$647$	−15.6505	−0.615284	−0.307642	−	0.951502i	$-0.599540\pi$
$647$	−15.6505	−0.615284	−0.307642	+	0.951502i	$0.599540\pi$
$648$	−2.54821	−0.100103
$649$	4.09613	0.160787
$650$	15.8202	0.620518
$651$	3.63865	0.142610
$652$	−11.8804	−0.465271
$653$	−6.42947	−0.251605	−0.125802	−	0.992055i	$-0.540151\pi$
$653$	−6.42947	−0.251605	−0.125802	+	0.992055i	$0.540151\pi$
$654$	−10.9167	−0.426877
$655$	−28.7034	−1.12153
$656$	8.50899	0.332220
$657$	2.43223	0.0948902
$658$	3.98851	0.155488
$659$	18.7856	0.731783	0.365891	−	0.930658i	$-0.380764\pi$
$659$	18.7856	0.731783	0.365891	+	0.930658i	$0.380764\pi$
$660$	−1.50444	−0.0585603
$661$	−40.7320	−1.58429	−0.792146	−	0.610331i	$-0.791036\pi$
$661$	−40.7320	−1.58429	−0.792146	+	0.610331i	$0.791036\pi$
$662$	−19.5056	−0.758105
$663$	52.8379	2.05206
$664$	−6.70911	−0.260364
$665$	−4.22171	−0.163711
$666$	−8.61887	−0.333974
$667$	7.93203	0.307130
$668$	5.77079	0.223279
$669$	−22.1666	−0.857008
$670$	−7.14837	−0.276166
$671$	−1.19180	−0.0460090
$672$	1.92409	0.0742232
$673$	15.4724	0.596415	0.298208	−	0.954501i	$-0.403611\pi$
$673$	15.4724	0.596415	0.298208	+	0.954501i	$0.403611\pi$
$674$	5.51020	0.212245
$675$	15.3486	0.590766
$676$	19.6099	0.754226
$677$	8.11671	0.311950	0.155975	−	0.987761i	$-0.450148\pi$
$677$	8.11671	0.311950	0.155975	+	0.987761i	$0.450148\pi$
$678$	1.27188	0.0488462
$679$	−24.6878	−0.947430
$680$	−11.0970	−0.425551
$681$	7.98712	0.306067
$682$	−1.53035	−0.0586002
$683$	4.39436	0.168145	0.0840727	−	0.996460i	$-0.473207\pi$
$683$	4.39436	0.168145	0.0840727	+	0.996460i	$0.473207\pi$
$684$	2.65254	0.101422
$685$	1.27502	0.0487159
$686$	−17.9448	−0.685136
$687$	10.5116	0.401041
$688$	−3.72057	−0.141845
$689$	−45.7199	−1.74179
$690$	−2.42183	−0.0921974
$691$	−0.326367	−0.0124156	−0.00620779	−	0.999981i	$-0.501976\pi$
$691$	−0.326367	−0.0124156	−0.00620779	+	0.999981i	$0.501976\pi$
$692$	18.0810	0.687335
$693$	1.81321	0.0688783
$694$	−7.93020	−0.301026
$695$	22.6308	0.858436
$696$	−7.58096	−0.287355
$697$	−63.2363	−2.39525
$698$	8.50671	0.321984
$699$	−31.5470	−1.19322
$700$	−4.28132	−0.161819
$701$	−6.36153	−0.240272	−0.120136	−	0.992757i	$-0.538333\pi$
$701$	−6.36153	−0.240272	−0.120136	+	0.992757i	$0.538333\pi$
$702$	31.6377	1.19409
$703$	−10.8755	−0.410176
$704$	−0.809236	−0.0304992
$705$	4.79809	0.180707
$706$	2.69809	0.101544
$707$	−10.2905	−0.387014
$708$	−6.30204	−0.236845
$709$	−31.0357	−1.16557	−0.582785	−	0.812627i	$-0.698037\pi$
$709$	−31.0357	−1.16557	−0.582785	+	0.812627i	$0.698037\pi$
$710$	−13.5191	−0.507362
$711$	−7.21302	−0.270509
$712$	2.58402	0.0968402
$713$	−2.46354	−0.0922602
$714$	−14.2992	−0.535136
$715$	6.90029	0.258056
$716$	−10.5814	−0.395445
$717$	−19.2056	−0.717244
$718$	2.00470	0.0748147
$719$	14.1588	0.528033	0.264017	−	0.964518i	$-0.414953\pi$
$719$	14.1588	0.528033	0.264017	+	0.964518i	$0.414953\pi$
$720$	−2.16496	−0.0806831
$721$	23.9729	0.892797
$722$	−15.6530	−0.582543
$723$	−9.17561	−0.341244
$724$	−5.90091	−0.219306
$725$	16.8685	0.626482
$726$	−12.8801	−0.478025
$727$	3.68369	0.136620	0.0683102	−	0.997664i	$-0.478239\pi$
$727$	3.68369	0.136620	0.0683102	+	0.997664i	$0.478239\pi$
$728$	−8.82503	−0.327077
$729$	24.3882	0.903265
$730$	−2.50490	−0.0927103
$731$	27.6502	1.02268
$732$	1.83363	0.0677729
$733$	34.3595	1.26910	0.634548	−	0.772883i	$-0.281186\pi$
$733$	34.3595	1.26910	0.634548	+	0.772883i	$0.281186\pi$
$734$	8.53313	0.314964
$735$	−8.57361	−0.316242
$736$	−1.30270	−0.0480180
$737$	3.87405	0.142703
$738$	−12.3370	−0.454132
$739$	10.8209	0.398054	0.199027	−	0.979994i	$-0.436222\pi$
$739$	10.8209	0.398054	0.199027	+	0.979994i	$0.436222\pi$
$740$	8.87637	0.326302
$741$	13.0073	0.477835
$742$	12.3729	0.454224
$743$	−52.3779	−1.92156	−0.960780	−	0.277312i	$-0.910557\pi$
$743$	−52.3779	−1.92156	−0.960780	+	0.277312i	$0.910557\pi$
$744$	2.35450	0.0863202
$745$	27.8424	1.02007
$746$	−11.1408	−0.407893
$747$	9.72740	0.355907
$748$	6.01401	0.219894
$749$	0.451932	0.0165132
$750$	−14.4458	−0.527486
$751$	27.7143	1.01131	0.505654	−	0.862736i	$-0.331251\pi$
$751$	27.7143	1.01131	0.505654	+	0.862736i	$0.331251\pi$
$752$	2.58089	0.0941152
$753$	−25.4236	−0.926488
$754$	34.7709	1.26628
$755$	−20.2357	−0.736451
$756$	−8.56195	−0.311395
$757$	−6.25685	−0.227409	−0.113705	−	0.993515i	$-0.536272\pi$
$757$	−6.25685	−0.227409	−0.113705	+	0.993515i	$0.536272\pi$
$758$	11.4251	0.414980
$759$	1.31251	0.0476410
$760$	−2.73179	−0.0990924
$761$	8.24871	0.299016	0.149508	−	0.988761i	$-0.452231\pi$
$761$	8.24871	0.299016	0.149508	+	0.988761i	$0.452231\pi$
$762$	10.7579	0.389717
$763$	−13.5503	−0.490555
$764$	7.49007	0.270981
$765$	16.0893	0.581711
$766$	28.6796	1.03624
$767$	28.9050	1.04370
$768$	1.24504	0.0449265
$769$	−10.1922	−0.367540	−0.183770	−	0.982969i	$-0.558830\pi$
$769$	−10.1922	−0.367540	−0.183770	+	0.982969i	$0.558830\pi$
$770$	−1.86739	−0.0672960
$771$	38.3036	1.37947
$772$	5.70990	0.205504
$773$	−1.75388	−0.0630828	−0.0315414	−	0.999502i	$-0.510042\pi$
$773$	−1.75388	−0.0630828	−0.0315414	+	0.999502i	$0.510042\pi$
$774$	5.39437	0.193897
$775$	−5.23905	−0.188192
$776$	−15.9750	−0.573469
$777$	11.4378	0.410329
$778$	−17.9847	−0.644782
$779$	−15.5671	−0.557749
$780$	−10.6163	−0.380126
$781$	7.32664	0.262168
$782$	9.68126	0.346201
$783$	33.7343	1.20557
$784$	−4.61173	−0.164705
$785$	17.7782	0.634532
$786$	−23.9331	−0.853664
$787$	15.0187	0.535360	0.267680	−	0.963508i	$-0.413743\pi$
$787$	15.0187	0.535360	0.267680	+	0.963508i	$0.413743\pi$
$788$	14.1787	0.505094
$789$	−0.203627	−0.00724931
$790$	7.42853	0.264295
$791$	1.57872	0.0561327
$792$	1.17329	0.0416912
$793$	−8.41014	−0.298653
$794$	9.86044	0.349934
$795$	14.8844	0.527894
$796$	9.05908	0.321091
$797$	31.5717	1.11833	0.559163	−	0.829057i	$-0.311122\pi$
$797$	31.5717	1.11833	0.559163	+	0.829057i	$0.311122\pi$
$798$	−3.52009	−0.124610
$799$	−19.1804	−0.678553
$800$	−2.77036	−0.0979471
$801$	−3.74652	−0.132377
$802$	−0.946242	−0.0334130
$803$	1.35752	0.0479060
$804$	−5.96037	−0.210206
$805$	−3.00609	−0.105951
$806$	−10.7992	−0.380385
$807$	6.25665	0.220245
$808$	−6.65878	−0.234255
$809$	−47.9108	−1.68446	−0.842228	−	0.539122i	$-0.818756\pi$
$809$	−47.9108	−1.68446	−0.842228	+	0.539122i	$0.818756\pi$
$810$	−3.80499	−0.133694
$811$	−42.9656	−1.50873	−0.754364	−	0.656457i	$-0.772054\pi$
$811$	−42.9656	−1.50873	−0.754364	+	0.656457i	$0.772054\pi$
$812$	−9.40986	−0.330221
$813$	−12.4917	−0.438104
$814$	−4.81054	−0.168609
$815$	−17.7397	−0.621396
$816$	−9.25276	−0.323911
$817$	6.80674	0.238138
$818$	−24.7334	−0.864782
$819$	12.7952	0.447101
$820$	12.7056	0.443699
$821$	−5.98172	−0.208764	−0.104382	−	0.994537i	$-0.533286\pi$
$821$	−5.98172	−0.208764	−0.104382	+	0.994537i	$0.533286\pi$
$822$	1.06312	0.0370805
$823$	5.92032	0.206369	0.103185	−	0.994662i	$-0.467097\pi$
$823$	5.92032	0.206369	0.103185	+	0.994662i	$0.467097\pi$
$824$	15.5124	0.540399
$825$	2.79122	0.0971779
$826$	−7.82240	−0.272176
$827$	3.68494	0.128138	0.0640690	−	0.997945i	$-0.479592\pi$
$827$	3.68494	0.128138	0.0640690	+	0.997945i	$0.479592\pi$
$828$	1.88875	0.0656387
$829$	7.70444	0.267586	0.133793	−	0.991009i	$-0.457284\pi$
$829$	7.70444	0.267586	0.133793	+	0.991009i	$0.457284\pi$
$830$	−10.0180	−0.347731
$831$	−0.459162	−0.0159282
$832$	−5.71050	−0.197976
$833$	34.2730	1.18749
$834$	18.8697	0.653405
$835$	8.61693	0.298201
$836$	1.48049	0.0512038
$837$	−10.4772	−0.362147
$838$	−0.626119	−0.0216289
$839$	−42.2768	−1.45956	−0.729778	−	0.683684i	$-0.760376\pi$
$839$	−42.2768	−1.45956	−0.729778	+	0.683684i	$0.760376\pi$
$840$	2.87304	0.0991293
$841$	8.07511	0.278452
$842$	−20.0392	−0.690596
$843$	−9.89663	−0.340858
$844$	28.8212	0.992067
$845$	29.2814	1.00731
$846$	−3.74197	−0.128652
$847$	−15.9874	−0.549334
$848$	8.00627	0.274936
$849$	−14.4351	−0.495412
$850$	20.5885	0.706180
$851$	−7.74393	−0.265459
$852$	−11.2723	−0.386182
$853$	3.80741	0.130363	0.0651816	−	0.997873i	$-0.479237\pi$
$853$	3.80741	0.130363	0.0651816	+	0.997873i	$0.479237\pi$
$854$	2.27599	0.0778828
$855$	3.96076	0.135455
$856$	0.292436	0.00999526
$857$	20.6926	0.706845	0.353423	−	0.935464i	$-0.385018\pi$
$857$	20.6926	0.706845	0.353423	+	0.935464i	$0.385018\pi$
$858$	5.75351	0.196421
$859$	−5.24946	−0.179109	−0.0895546	−	0.995982i	$-0.528544\pi$
$859$	−5.24946	−0.179109	−0.0895546	+	0.995982i	$0.528544\pi$
$860$	−5.55554	−0.189442
$861$	16.3720	0.557957
$862$	6.31218	0.214994
$863$	−0.922640	−0.0314070	−0.0157035	−	0.999877i	$-0.504999\pi$
$863$	−0.922640	−0.0314070	−0.0157035	+	0.999877i	$0.504999\pi$
$864$	−5.54027	−0.188484
$865$	26.9984	0.917974
$866$	21.9418	0.745612
$867$	47.5982	1.61652
$868$	2.92252	0.0991969
$869$	−4.02588	−0.136569
$870$	−11.3199	−0.383779
$871$	27.3379	0.926308
$872$	−8.76816	−0.296927
$873$	23.1618	0.783908
$874$	2.38327	0.0806153
$875$	−17.9308	−0.606172
$876$	−2.08860	−0.0705672
$877$	20.0590	0.677343	0.338672	−	0.940905i	$-0.390022\pi$
$877$	20.0590	0.677343	0.338672	+	0.940905i	$0.390022\pi$
$878$	21.7197	0.733004
$879$	−2.17912	−0.0734999
$880$	−1.20835	−0.0407335
$881$	−19.8512	−0.668804	−0.334402	−	0.942430i	$-0.608534\pi$
$881$	−19.8512	−0.668804	−0.334402	+	0.942430i	$0.608534\pi$
$882$	6.68645	0.225145
$883$	12.6933	0.427165	0.213582	−	0.976925i	$-0.431487\pi$
$883$	12.6933	0.427165	0.213582	+	0.976925i	$0.431487\pi$
$884$	42.4388	1.42737
$885$	−9.41018	−0.316320
$886$	1.38439	0.0465094
$887$	25.3978	0.852773	0.426387	−	0.904541i	$-0.359786\pi$
$887$	25.3978	0.852773	0.426387	+	0.904541i	$0.359786\pi$
$888$	7.40118	0.248367
$889$	13.3532	0.447853
$890$	3.85845	0.129336
$891$	2.06211	0.0690832
$892$	−17.8039	−0.596119
$893$	−4.72170	−0.158006
$894$	23.2152	0.776433
$895$	−15.8001	−0.528139
$896$	1.54540	0.0516283
$897$	9.26191	0.309246
$898$	37.4257	1.24891
$899$	−11.5148	−0.384041
$900$	4.01669	0.133890
$901$	−59.5003	−1.98224
$902$	−6.88578	−0.229272
$903$	−7.15869	−0.238226
$904$	1.02156	0.0339765
$905$	−8.81122	−0.292895
$906$	−16.8726	−0.560556
$907$	−39.3716	−1.30731	−0.653656	−	0.756792i	$-0.726766\pi$
$907$	−39.3716	−1.30731	−0.653656	+	0.756792i	$0.726766\pi$
$908$	6.41516	0.212895
$909$	9.65442	0.320217
$910$	−13.1775	−0.436830
$911$	37.4052	1.23929	0.619644	−	0.784883i	$-0.287277\pi$
$911$	37.4052	1.23929	0.619644	+	0.784883i	$0.287277\pi$
$912$	−2.27778	−0.0754250
$913$	5.42925	0.179682
$914$	20.5984	0.681335
$915$	2.73797	0.0905145
$916$	8.44276	0.278957
$917$	−29.7069	−0.981008
$918$	41.1737	1.35893
$919$	2.53717	0.0836934	0.0418467	−	0.999124i	$-0.486676\pi$
$919$	2.53717	0.0836934	0.0418467	+	0.999124i	$0.486676\pi$
$920$	−1.94518	−0.0641308
$921$	−15.5429	−0.512155
$922$	39.6183	1.30476
$923$	51.7016	1.70178
$924$	−1.55704	−0.0512229
$925$	−16.4685	−0.541482
$926$	−13.0666	−0.429396
$927$	−22.4911	−0.738704
$928$	−6.08893	−0.199879
$929$	37.3011	1.22381	0.611904	−	0.790932i	$-0.290404\pi$
$929$	37.3011	1.22381	0.611904	+	0.790932i	$0.290404\pi$
$930$	3.51573	0.115285
$931$	8.43711	0.276515
$932$	−25.3382	−0.829979
$933$	−21.9951	−0.720088
$934$	−27.4432	−0.897971
$935$	8.98010	0.293681
$936$	8.27954	0.270625
$937$	−50.6870	−1.65587	−0.827936	−	0.560822i	$-0.810485\pi$
$937$	−50.6870	−1.65587	−0.827936	+	0.560822i	$0.810485\pi$
$938$	−7.39830	−0.241563
$939$	−26.6313	−0.869081
$940$	3.85377	0.125696
$941$	28.3360	0.923727	0.461864	−	0.886951i	$-0.347181\pi$
$941$	28.3360	0.923727	0.461864	+	0.886951i	$0.347181\pi$
$942$	14.8236	0.482979
$943$	−11.0846	−0.360965
$944$	−5.06172	−0.164745
$945$	−12.7847	−0.415886
$946$	3.01082	0.0978901
$947$	−52.4304	−1.70376	−0.851880	−	0.523736i	$-0.824538\pi$
$947$	−52.4304	−1.70376	−0.851880	+	0.523736i	$0.824538\pi$
$948$	6.19396	0.201170
$949$	9.57959	0.310967
$950$	5.06835	0.164439
$951$	44.0353	1.42794
$952$	−11.4850	−0.372230
$953$	−17.7948	−0.576432	−0.288216	−	0.957565i	$-0.593062\pi$
$953$	−17.7948	−0.576432	−0.288216	+	0.957565i	$0.593062\pi$
$954$	−11.6081	−0.375827
$955$	11.1842	0.361911
$956$	−15.4257	−0.498902
$957$	6.13478	0.198309
$958$	5.62038	0.181586
$959$	1.31959	0.0426119
$960$	1.85909	0.0600018
$961$	−27.4237	−0.884636
$962$	−33.9463	−1.09447
$963$	−0.423997	−0.0136631
$964$	−7.36974	−0.237363
$965$	8.52601	0.274462
$966$	−2.50650	−0.0806453
$967$	−47.8331	−1.53821	−0.769105	−	0.639123i	$-0.779298\pi$
$967$	−47.8331	−1.53821	−0.769105	+	0.639123i	$0.779298\pi$
$968$	−10.3451	−0.332505
$969$	16.9278	0.543800
$970$	−23.8538	−0.765900
$971$	30.4896	0.978458	0.489229	−	0.872155i	$-0.337278\pi$
$971$	30.4896	0.978458	0.489229	+	0.872155i	$0.337278\pi$
$972$	13.4482	0.431351
$973$	23.4220	0.750876
$974$	−15.5862	−0.499416
$975$	19.6967	0.630800
$976$	1.47275	0.0471416
$977$	−24.1960	−0.774100	−0.387050	−	0.922059i	$-0.626506\pi$
$977$	−24.1960	−0.774100	−0.387050	+	0.922059i	$0.626506\pi$
$978$	−14.7915	−0.472981
$979$	−2.09108	−0.0668313
$980$	−6.88622	−0.219972
$981$	12.7128	0.405888
$982$	−32.4990	−1.03708
$983$	−16.4288	−0.523998	−0.261999	−	0.965068i	$-0.584382\pi$
$983$	−16.4288	−0.523998	−0.261999	+	0.965068i	$0.584382\pi$
$984$	10.5940	0.337725
$985$	21.1715	0.674581
$986$	45.2512	1.44109
$987$	4.96585	0.158065
$988$	10.4473	0.332373
$989$	4.84677	0.154118
$990$	1.75196	0.0556810
$991$	20.7775	0.660020	0.330010	−	0.943977i	$-0.392948\pi$
$991$	20.7775	0.660020	0.330010	+	0.943977i	$0.392948\pi$
$992$	1.89111	0.0600427
$993$	−24.2852	−0.770667
$994$	−13.9917	−0.443790
$995$	13.5270	0.428835
$996$	−8.35310	−0.264678
$997$	52.2637	1.65521	0.827604	−	0.561312i	$-0.189703\pi$
$997$	52.2637	1.65521	0.827604	+	0.561312i	$0.189703\pi$
$998$	10.3035	0.326152
$999$	−32.9344	−1.04200

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.48	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.48	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.24504 q^{3} +1.00000 q^{4} +1.49320 q^{5} +1.24504 q^{6} +1.54540 q^{7} +1.00000 q^{8} -1.44988 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.24504 q^{3} +1.00000 q^{4} +1.49320 q^{5} +1.24504 q^{6} +1.54540 q^{7} +1.00000 q^{8} -1.44988 q^{9} +1.49320 q^{10} -0.809236 q^{11} +1.24504 q^{12} -5.71050 q^{13} +1.54540 q^{14} +1.85909 q^{15} +1.00000 q^{16} -7.43171 q^{17} -1.44988 q^{18} -1.82949 q^{19} +1.49320 q^{20} +1.92409 q^{21} -0.809236 q^{22} -1.30270 q^{23} +1.24504 q^{24} -2.77036 q^{25} -5.71050 q^{26} -5.54027 q^{27} +1.54540 q^{28} -6.08893 q^{29} +1.85909 q^{30} +1.89111 q^{31} +1.00000 q^{32} -1.00753 q^{33} -7.43171 q^{34} +2.30759 q^{35} -1.44988 q^{36} +5.94454 q^{37} -1.82949 q^{38} -7.10980 q^{39} +1.49320 q^{40} +8.50899 q^{41} +1.92409 q^{42} -3.72057 q^{43} -0.809236 q^{44} -2.16496 q^{45} -1.30270 q^{46} +2.58089 q^{47} +1.24504 q^{48} -4.61173 q^{49} -2.77036 q^{50} -9.25276 q^{51} -5.71050 q^{52} +8.00627 q^{53} -5.54027 q^{54} -1.20835 q^{55} +1.54540 q^{56} -2.27778 q^{57} -6.08893 q^{58} -5.06172 q^{59} +1.85909 q^{60} +1.47275 q^{61} +1.89111 q^{62} -2.24065 q^{63} +1.00000 q^{64} -8.52691 q^{65} -1.00753 q^{66} -4.78729 q^{67} -7.43171 q^{68} -1.62191 q^{69} +2.30759 q^{70} -9.05377 q^{71} -1.44988 q^{72} -1.67754 q^{73} +5.94454 q^{74} -3.44921 q^{75} -1.82949 q^{76} -1.25060 q^{77} -7.10980 q^{78} +4.97491 q^{79} +1.49320 q^{80} -2.54821 q^{81} +8.50899 q^{82} -6.70911 q^{83} +1.92409 q^{84} -11.0970 q^{85} -3.72057 q^{86} -7.58096 q^{87} -0.809236 q^{88} +2.58402 q^{89} -2.16496 q^{90} -8.82503 q^{91} -1.30270 q^{92} +2.35450 q^{93} +2.58089 q^{94} -2.73179 q^{95} +1.24504 q^{96} -15.9750 q^{97} -4.61173 q^{98} +1.17329 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 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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