LMFDB - Embedded newform 8042.2.a.a.1.56

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			1.56
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.85929 q^{3} +1.00000 q^{4} -1.01796 q^{5} +1.85929 q^{6} -1.17174 q^{7} +1.00000 q^{8} +0.456947 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.85929 q^{3} +1.00000 q^{4} -1.01796 q^{5} +1.85929 q^{6} -1.17174 q^{7} +1.00000 q^{8} +0.456947 q^{9} -1.01796 q^{10} +1.08861 q^{11} +1.85929 q^{12} +2.13486 q^{13} -1.17174 q^{14} -1.89269 q^{15} +1.00000 q^{16} -4.74650 q^{17} +0.456947 q^{18} +2.90296 q^{19} -1.01796 q^{20} -2.17860 q^{21} +1.08861 q^{22} -8.82576 q^{23} +1.85929 q^{24} -3.96375 q^{25} +2.13486 q^{26} -4.72826 q^{27} -1.17174 q^{28} +1.01696 q^{29} -1.89269 q^{30} -3.34018 q^{31} +1.00000 q^{32} +2.02403 q^{33} -4.74650 q^{34} +1.19279 q^{35} +0.456947 q^{36} -6.27824 q^{37} +2.90296 q^{38} +3.96931 q^{39} -1.01796 q^{40} +2.79406 q^{41} -2.17860 q^{42} -1.62328 q^{43} +1.08861 q^{44} -0.465156 q^{45} -8.82576 q^{46} -8.69124 q^{47} +1.85929 q^{48} -5.62702 q^{49} -3.96375 q^{50} -8.82510 q^{51} +2.13486 q^{52} -2.38381 q^{53} -4.72826 q^{54} -1.10817 q^{55} -1.17174 q^{56} +5.39744 q^{57} +1.01696 q^{58} +5.06228 q^{59} -1.89269 q^{60} +10.8652 q^{61} -3.34018 q^{62} -0.535424 q^{63} +1.00000 q^{64} -2.17321 q^{65} +2.02403 q^{66} +11.0826 q^{67} -4.74650 q^{68} -16.4096 q^{69} +1.19279 q^{70} +4.35068 q^{71} +0.456947 q^{72} -10.2240 q^{73} -6.27824 q^{74} -7.36974 q^{75} +2.90296 q^{76} -1.27557 q^{77} +3.96931 q^{78} -14.9430 q^{79} -1.01796 q^{80} -10.1620 q^{81} +2.79406 q^{82} +1.00368 q^{83} -2.17860 q^{84} +4.83177 q^{85} -1.62328 q^{86} +1.89081 q^{87} +1.08861 q^{88} +7.32916 q^{89} -0.465156 q^{90} -2.50150 q^{91} -8.82576 q^{92} -6.21036 q^{93} -8.69124 q^{94} -2.95512 q^{95} +1.85929 q^{96} -1.72909 q^{97} -5.62702 q^{98} +0.497436 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.85929	1.07346	0.536730	−	0.843754i	$-0.319659\pi$
$3$	1.85929	1.07346	0.536730	+	0.843754i	$0.319659\pi$
$4$	1.00000	0.500000
$5$	−1.01796	−0.455248	−0.227624	−	0.973749i	$-0.573096\pi$
$5$	−1.01796	−0.455248	−0.227624	+	0.973749i	$0.573096\pi$
$6$	1.85929	0.759051
$7$	−1.17174	−0.442876	−0.221438	−	0.975174i	$-0.571075\pi$
$7$	−1.17174	−0.442876	−0.221438	+	0.975174i	$0.571075\pi$
$8$	1.00000	0.353553
$9$	0.456947	0.152316
$10$	−1.01796	−0.321909
$11$	1.08861	0.328228	0.164114	−	0.986441i	$-0.447524\pi$
$11$	1.08861	0.328228	0.164114	+	0.986441i	$0.447524\pi$
$12$	1.85929	0.536730
$13$	2.13486	0.592103	0.296052	−	0.955172i	$-0.404330\pi$
$13$	2.13486	0.592103	0.296052	+	0.955172i	$0.404330\pi$
$14$	−1.17174	−0.313161
$15$	−1.89269	−0.488690
$16$	1.00000	0.250000
$17$	−4.74650	−1.15119	−0.575597	−	0.817733i	$-0.695230\pi$
$17$	−4.74650	−1.15119	−0.575597	+	0.817733i	$0.695230\pi$
$18$	0.456947	0.107703
$19$	2.90296	0.665985	0.332993	−	0.942929i	$-0.391942\pi$
$19$	2.90296	0.665985	0.332993	+	0.942929i	$0.391942\pi$
$20$	−1.01796	−0.227624
$21$	−2.17860	−0.475410
$22$	1.08861	0.232092
$23$	−8.82576	−1.84030	−0.920149	−	0.391568i	$-0.871933\pi$
$23$	−8.82576	−1.84030	−0.920149	+	0.391568i	$0.871933\pi$
$24$	1.85929	0.379525
$25$	−3.96375	−0.792749
$26$	2.13486	0.418680
$27$	−4.72826	−0.909955
$28$	−1.17174	−0.221438
$29$	1.01696	0.188844	0.0944221	−	0.995532i	$-0.469900\pi$
$29$	1.01696	0.188844	0.0944221	+	0.995532i	$0.469900\pi$
$30$	−1.89269	−0.345556
$31$	−3.34018	−0.599915	−0.299957	−	0.953953i	$-0.596972\pi$
$31$	−3.34018	−0.599915	−0.299957	+	0.953953i	$0.596972\pi$
$32$	1.00000	0.176777
$33$	2.02403	0.352339
$34$	−4.74650	−0.814017
$35$	1.19279	0.201619
$36$	0.456947	0.0761579
$37$	−6.27824	−1.03214	−0.516068	−	0.856547i	$-0.672605\pi$
$37$	−6.27824	−1.03214	−0.516068	+	0.856547i	$0.672605\pi$
$38$	2.90296	0.470923
$39$	3.96931	0.635599
$40$	−1.01796	−0.160954
$41$	2.79406	0.436358	0.218179	−	0.975909i	$-0.429988\pi$
$41$	2.79406	0.436358	0.218179	+	0.975909i	$0.429988\pi$
$42$	−2.17860	−0.336166
$43$	−1.62328	−0.247548	−0.123774	−	0.992310i	$-0.539500\pi$
$43$	−1.62328	−0.247548	−0.123774	+	0.992310i	$0.539500\pi$
$44$	1.08861	0.164114
$45$	−0.465156	−0.0693414
$46$	−8.82576	−1.30129
$47$	−8.69124	−1.26775	−0.633874	−	0.773436i	$-0.718536\pi$
$47$	−8.69124	−1.26775	−0.633874	+	0.773436i	$0.718536\pi$
$48$	1.85929	0.268365
$49$	−5.62702	−0.803860
$50$	−3.96375	−0.560559
$51$	−8.82510	−1.23576
$52$	2.13486	0.296052
$53$	−2.38381	−0.327441	−0.163721	−	0.986507i	$-0.552350\pi$
$53$	−2.38381	−0.327441	−0.163721	+	0.986507i	$0.552350\pi$
$54$	−4.72826	−0.643435
$55$	−1.10817	−0.149425
$56$	−1.17174	−0.156580
$57$	5.39744	0.714909
$58$	1.01696	0.133533
$59$	5.06228	0.659052	0.329526	−	0.944146i	$-0.393111\pi$
$59$	5.06228	0.659052	0.329526	+	0.944146i	$0.393111\pi$
$60$	−1.89269	−0.244345
$61$	10.8652	1.39115	0.695574	−	0.718455i	$-0.255150\pi$
$61$	10.8652	1.39115	0.695574	+	0.718455i	$0.255150\pi$
$62$	−3.34018	−0.424204
$63$	−0.535424	−0.0674571
$64$	1.00000	0.125000
$65$	−2.17321	−0.269554
$66$	2.02403	0.249141
$67$	11.0826	1.35395	0.676977	−	0.736004i	$-0.263289\pi$
$67$	11.0826	1.35395	0.676977	+	0.736004i	$0.263289\pi$
$68$	−4.74650	−0.575597
$69$	−16.4096	−1.97549
$70$	1.19279	0.142566
$71$	4.35068	0.516330	0.258165	−	0.966101i	$-0.416882\pi$
$71$	4.35068	0.516330	0.258165	+	0.966101i	$0.416882\pi$
$72$	0.456947	0.0538517
$73$	−10.2240	−1.19663	−0.598315	−	0.801261i	$-0.704163\pi$
$73$	−10.2240	−1.19663	−0.598315	+	0.801261i	$0.704163\pi$
$74$	−6.27824	−0.729831
$75$	−7.36974	−0.850985
$76$	2.90296	0.332993
$77$	−1.27557	−0.145364
$78$	3.96931	0.449436
$79$	−14.9430	−1.68122	−0.840610	−	0.541641i	$-0.817803\pi$
$79$	−14.9430	−1.68122	−0.840610	+	0.541641i	$0.817803\pi$
$80$	−1.01796	−0.113812
$81$	−10.1620	−1.12912
$82$	2.79406	0.308552
$83$	1.00368	0.110169	0.0550844	−	0.998482i	$-0.482457\pi$
$83$	1.00368	0.110169	0.0550844	+	0.998482i	$0.482457\pi$
$84$	−2.17860	−0.237705
$85$	4.83177	0.524079
$86$	−1.62328	−0.175043
$87$	1.89081	0.202717
$88$	1.08861	0.116046
$89$	7.32916	0.776889	0.388445	−	0.921472i	$-0.373012\pi$
$89$	7.32916	0.776889	0.388445	+	0.921472i	$0.373012\pi$
$90$	−0.465156	−0.0490318
$91$	−2.50150	−0.262229
$92$	−8.82576	−0.920149
$93$	−6.21036	−0.643984
$94$	−8.69124	−0.896433
$95$	−2.95512	−0.303188
$96$	1.85929	0.189763
$97$	−1.72909	−0.175562	−0.0877811	−	0.996140i	$-0.527978\pi$
$97$	−1.72909	−0.175562	−0.0877811	+	0.996140i	$0.527978\pi$
$98$	−5.62702	−0.568415
$99$	0.497436	0.0499942
$100$	−3.96375	−0.396375
$101$	−7.01837	−0.698354	−0.349177	−	0.937057i	$-0.613539\pi$
$101$	−7.01837	−0.698354	−0.349177	+	0.937057i	$0.613539\pi$
$102$	−8.82510	−0.873815
$103$	1.63131	0.160738	0.0803691	−	0.996765i	$-0.474390\pi$
$103$	1.63131	0.160738	0.0803691	+	0.996765i	$0.474390\pi$
$104$	2.13486	0.209340
$105$	2.21774	0.216429
$106$	−2.38381	−0.231536
$107$	−11.2451	−1.08711	−0.543553	−	0.839375i	$-0.682921\pi$
$107$	−11.2451	−1.08711	−0.543553	+	0.839375i	$0.682921\pi$
$108$	−4.72826	−0.454977
$109$	−3.18706	−0.305265	−0.152632	−	0.988283i	$-0.548775\pi$
$109$	−3.18706	−0.305265	−0.152632	+	0.988283i	$0.548775\pi$
$110$	−1.10817	−0.105659
$111$	−11.6731	−1.10796
$112$	−1.17174	−0.110719
$113$	11.7222	1.10273	0.551365	−	0.834264i	$-0.314107\pi$
$113$	11.7222	1.10273	0.551365	+	0.834264i	$0.314107\pi$
$114$	5.39744	0.505517
$115$	8.98431	0.837792
$116$	1.01696	0.0944221
$117$	0.975517	0.0901866
$118$	5.06228	0.466020
$119$	5.56166	0.509837
$120$	−1.89269	−0.172778
$121$	−9.81493	−0.892267
$122$	10.8652	0.983690
$123$	5.19495	0.468413
$124$	−3.34018	−0.299957
$125$	9.12478	0.816145
$126$	−0.535424	−0.0476993
$127$	10.4378	0.926207	0.463103	−	0.886304i	$-0.346736\pi$
$127$	10.4378	0.926207	0.463103	+	0.886304i	$0.346736\pi$
$128$	1.00000	0.0883883
$129$	−3.01815	−0.265733
$130$	−2.17321	−0.190603
$131$	4.65907	0.407065	0.203533	−	0.979068i	$-0.434758\pi$
$131$	4.65907	0.407065	0.203533	+	0.979068i	$0.434758\pi$
$132$	2.02403	0.176170
$133$	−3.40152	−0.294949
$134$	11.0826	0.957391
$135$	4.81321	0.414255
$136$	−4.74650	−0.407009
$137$	5.20125	0.444373	0.222187	−	0.975004i	$-0.428681\pi$
$137$	5.20125	0.444373	0.222187	+	0.975004i	$0.428681\pi$
$138$	−16.4096	−1.39688
$139$	7.36123	0.624372	0.312186	−	0.950021i	$-0.398939\pi$
$139$	7.36123	0.624372	0.312186	+	0.950021i	$0.398939\pi$
$140$	1.19279	0.100809
$141$	−16.1595	−1.36088
$142$	4.35068	0.365101
$143$	2.32402	0.194345
$144$	0.456947	0.0380789
$145$	−1.03523	−0.0859709
$146$	−10.2240	−0.846145
$147$	−10.4622	−0.862912
$148$	−6.27824	−0.516068
$149$	2.40536	0.197055	0.0985273	−	0.995134i	$-0.468587\pi$
$149$	2.40536	0.197055	0.0985273	+	0.995134i	$0.468587\pi$
$150$	−7.36974	−0.601737
$151$	10.5609	0.859436	0.429718	−	0.902963i	$-0.358613\pi$
$151$	10.5609	0.859436	0.429718	+	0.902963i	$0.358613\pi$
$152$	2.90296	0.235461
$153$	−2.16890	−0.175345
$154$	−1.27557	−0.102788
$155$	3.40019	0.273110
$156$	3.96931	0.317799
$157$	−9.11624	−0.727555	−0.363778	−	0.931486i	$-0.618513\pi$
$157$	−9.11624	−0.727555	−0.363778	+	0.931486i	$0.618513\pi$
$158$	−14.9430	−1.18880
$159$	−4.43218	−0.351495
$160$	−1.01796	−0.0804772
$161$	10.3415	0.815025
$162$	−10.1620	−0.798405
$163$	−9.68993	−0.758974	−0.379487	−	0.925197i	$-0.623899\pi$
$163$	−9.68993	−0.758974	−0.379487	+	0.925197i	$0.623899\pi$
$164$	2.79406	0.218179
$165$	−2.06040	−0.160402
$166$	1.00368	0.0779011
$167$	3.69806	0.286164	0.143082	−	0.989711i	$-0.454299\pi$
$167$	3.69806	0.286164	0.143082	+	0.989711i	$0.454299\pi$
$168$	−2.17860	−0.168083
$169$	−8.44238	−0.649414
$170$	4.83177	0.370580
$171$	1.32650	0.101440
$172$	−1.62328	−0.123774
$173$	−10.3362	−0.785849	−0.392925	−	0.919571i	$-0.628537\pi$
$173$	−10.3362	−0.785849	−0.392925	+	0.919571i	$0.628537\pi$
$174$	1.89081	0.143342
$175$	4.64449	0.351090
$176$	1.08861	0.0820569
$177$	9.41222	0.707466
$178$	7.32916	0.549344
$179$	9.85570	0.736649	0.368325	−	0.929697i	$-0.379932\pi$
$179$	9.85570	0.736649	0.368325	+	0.929697i	$0.379932\pi$
$180$	−0.465156	−0.0346707
$181$	−19.5838	−1.45565	−0.727826	−	0.685762i	$-0.759469\pi$
$181$	−19.5838	−1.45565	−0.727826	+	0.685762i	$0.759469\pi$
$182$	−2.50150	−0.185424
$183$	20.2015	1.49334
$184$	−8.82576	−0.650644
$185$	6.39103	0.469878
$186$	−6.21036	−0.455366
$187$	−5.16707	−0.377854
$188$	−8.69124	−0.633874
$189$	5.54030	0.402998
$190$	−2.95512	−0.214387
$191$	−21.3071	−1.54173	−0.770863	−	0.637001i	$-0.780175\pi$
$191$	−21.3071	−1.54173	−0.770863	+	0.637001i	$0.780175\pi$
$192$	1.85929	0.134182
$193$	20.0964	1.44657	0.723285	−	0.690549i	$-0.242631\pi$
$193$	20.0964	1.44657	0.723285	+	0.690549i	$0.242631\pi$
$194$	−1.72909	−0.124141
$195$	−4.04062	−0.289355
$196$	−5.62702	−0.401930
$197$	1.00529	0.0716237	0.0358119	−	0.999359i	$-0.488598\pi$
$197$	1.00529	0.0716237	0.0358119	+	0.999359i	$0.488598\pi$
$198$	0.497436	0.0353513
$199$	−19.6146	−1.39044	−0.695222	−	0.718795i	$-0.744694\pi$
$199$	−19.6146	−1.39044	−0.695222	+	0.718795i	$0.744694\pi$
$200$	−3.96375	−0.280279
$201$	20.6057	1.45342
$202$	−7.01837	−0.493811
$203$	−1.19161	−0.0836347
$204$	−8.82510	−0.617880
$205$	−2.84425	−0.198651
$206$	1.63131	0.113659
$207$	−4.03291	−0.280306
$208$	2.13486	0.148026
$209$	3.16019	0.218595
$210$	2.21774	0.153039
$211$	9.65392	0.664604	0.332302	−	0.943173i	$-0.392175\pi$
$211$	9.65392	0.664604	0.332302	+	0.943173i	$0.392175\pi$
$212$	−2.38381	−0.163721
$213$	8.08915	0.554260
$214$	−11.2451	−0.768700
$215$	1.65244	0.112696
$216$	−4.72826	−0.321718
$217$	3.91383	0.265688
$218$	−3.18706	−0.215855
$219$	−19.0094	−1.28453
$220$	−1.10817	−0.0747125
$221$	−10.1331	−0.681626
$222$	−11.6731	−0.783444
$223$	−22.0427	−1.47609	−0.738043	−	0.674753i	$-0.764250\pi$
$223$	−22.0427	−1.47609	−0.738043	+	0.674753i	$0.764250\pi$
$224$	−1.17174	−0.0782902
$225$	−1.81122	−0.120748
$226$	11.7222	0.779748
$227$	−5.93190	−0.393714	−0.196857	−	0.980432i	$-0.563073\pi$
$227$	−5.93190	−0.393714	−0.196857	+	0.980432i	$0.563073\pi$
$228$	5.39744	0.357454
$229$	2.65940	0.175738	0.0878691	−	0.996132i	$-0.471994\pi$
$229$	2.65940	0.175738	0.0878691	+	0.996132i	$0.471994\pi$
$230$	8.98431	0.592408
$231$	−2.37164	−0.156043
$232$	1.01696	0.0667665
$233$	−5.80761	−0.380469	−0.190235	−	0.981739i	$-0.560925\pi$
$233$	−5.80761	−0.380469	−0.190235	+	0.981739i	$0.560925\pi$
$234$	0.975517	0.0637716
$235$	8.84738	0.577139
$236$	5.06228	0.329526
$237$	−27.7833	−1.80472
$238$	5.56166	0.360509
$239$	22.6822	1.46719	0.733594	−	0.679588i	$-0.237841\pi$
$239$	22.6822	1.46719	0.733594	+	0.679588i	$0.237841\pi$
$240$	−1.89269	−0.122173
$241$	21.0958	1.35890	0.679450	−	0.733721i	$-0.262218\pi$
$241$	21.0958	1.35890	0.679450	+	0.733721i	$0.262218\pi$
$242$	−9.81493	−0.630928
$243$	−4.70935	−0.302105
$244$	10.8652	0.695574
$245$	5.72811	0.365956
$246$	5.19495	0.331218
$247$	6.19742	0.394332
$248$	−3.34018	−0.212102
$249$	1.86614	0.118262
$250$	9.12478	0.577102
$251$	−17.0104	−1.07369	−0.536843	−	0.843682i	$-0.680383\pi$
$251$	−17.0104	−1.07369	−0.536843	+	0.843682i	$0.680383\pi$
$252$	−0.535424	−0.0337285
$253$	−9.60780	−0.604037
$254$	10.4378	0.654927
$255$	8.98364	0.562577
$256$	1.00000	0.0625000
$257$	−10.4635	−0.652693	−0.326347	−	0.945250i	$-0.605818\pi$
$257$	−10.4635	−0.652693	−0.326347	+	0.945250i	$0.605818\pi$
$258$	−3.01815	−0.187902
$259$	7.35647	0.457109
$260$	−2.17321	−0.134777
$261$	0.464696	0.0287639
$262$	4.65907	0.287838
$263$	−0.633366	−0.0390550	−0.0195275	−	0.999809i	$-0.506216\pi$
$263$	−0.633366	−0.0390550	−0.0195275	+	0.999809i	$0.506216\pi$
$264$	2.02403	0.124571
$265$	2.42663	0.149067
$266$	−3.40152	−0.208561
$267$	13.6270	0.833959
$268$	11.0826	0.676977
$269$	11.6238	0.708717	0.354358	−	0.935110i	$-0.384699\pi$
$269$	11.6238	0.708717	0.354358	+	0.935110i	$0.384699\pi$
$270$	4.81321	0.292922
$271$	−7.21081	−0.438026	−0.219013	−	0.975722i	$-0.570284\pi$
$271$	−7.21081	−0.438026	−0.219013	+	0.975722i	$0.570284\pi$
$272$	−4.74650	−0.287799
$273$	−4.65101	−0.281492
$274$	5.20125	0.314219
$275$	−4.31497	−0.260202
$276$	−16.4096	−0.987743
$277$	28.1431	1.69096	0.845479	−	0.534009i	$-0.179315\pi$
$277$	28.1431	1.69096	0.845479	+	0.534009i	$0.179315\pi$
$278$	7.36123	0.441497
$279$	−1.52629	−0.0913765
$280$	1.19279	0.0712829
$281$	12.6556	0.754970	0.377485	−	0.926016i	$-0.376789\pi$
$281$	12.6556	0.754970	0.377485	+	0.926016i	$0.376789\pi$
$282$	−16.1595	−0.962285
$283$	12.1062	0.719641	0.359820	−	0.933022i	$-0.382838\pi$
$283$	12.1062	0.719641	0.359820	+	0.933022i	$0.382838\pi$
$284$	4.35068	0.258165
$285$	−5.49441	−0.325461
$286$	2.32402	0.137422
$287$	−3.27391	−0.193253
$288$	0.456947	0.0269259
$289$	5.52922	0.325248
$290$	−1.03523	−0.0607906
$291$	−3.21487	−0.188459
$292$	−10.2240	−0.598315
$293$	−9.03515	−0.527839	−0.263919	−	0.964545i	$-0.585015\pi$
$293$	−9.03515	−0.527839	−0.263919	+	0.964545i	$0.585015\pi$
$294$	−10.4622	−0.610171
$295$	−5.15322	−0.300032
$296$	−6.27824	−0.364915
$297$	−5.14723	−0.298672
$298$	2.40536	0.139339
$299$	−18.8417	−1.08965
$300$	−7.36974	−0.425492
$301$	1.90207	0.109633
$302$	10.5609	0.607713
$303$	−13.0492	−0.749655
$304$	2.90296	0.166496
$305$	−11.0604	−0.633317
$306$	−2.16890	−0.123988
$307$	−26.9796	−1.53981	−0.769905	−	0.638159i	$-0.779696\pi$
$307$	−26.9796	−1.53981	−0.769905	+	0.638159i	$0.779696\pi$
$308$	−1.27557	−0.0726822
$309$	3.03308	0.172546
$310$	3.40019	0.193118
$311$	17.8100	1.00991	0.504955	−	0.863146i	$-0.331509\pi$
$311$	17.8100	1.00991	0.504955	+	0.863146i	$0.331509\pi$
$312$	3.96931	0.224718
$313$	−9.30565	−0.525986	−0.262993	−	0.964798i	$-0.584710\pi$
$313$	−9.30565	−0.525986	−0.262993	+	0.964798i	$0.584710\pi$
$314$	−9.11624	−0.514459
$315$	0.545043	0.0307097
$316$	−14.9430	−0.840610
$317$	−2.71677	−0.152589	−0.0762944	−	0.997085i	$-0.524309\pi$
$317$	−2.71677	−0.152589	−0.0762944	+	0.997085i	$0.524309\pi$
$318$	−4.43218	−0.248544
$319$	1.10707	0.0619839
$320$	−1.01796	−0.0569060
$321$	−20.9079	−1.16696
$322$	10.3415	0.576310
$323$	−13.7789	−0.766679
$324$	−10.1620	−0.564558
$325$	−8.46204	−0.469389
$326$	−9.68993	−0.536676
$327$	−5.92565	−0.327689
$328$	2.79406	0.154276
$329$	10.1839	0.561456
$330$	−2.06040	−0.113421
$331$	−26.5174	−1.45753	−0.728765	−	0.684764i	$-0.759905\pi$
$331$	−26.5174	−1.45753	−0.728765	+	0.684764i	$0.759905\pi$
$332$	1.00368	0.0550844
$333$	−2.86882	−0.157211
$334$	3.69806	0.202349
$335$	−11.2817	−0.616385
$336$	−2.17860	−0.118853
$337$	−25.6870	−1.39926	−0.699631	−	0.714504i	$-0.746652\pi$
$337$	−25.6870	−1.39926	−0.699631	+	0.714504i	$0.746652\pi$
$338$	−8.44238	−0.459205
$339$	21.7949	1.18374
$340$	4.83177	0.262039
$341$	−3.63615	−0.196909
$342$	1.32650	0.0717290
$343$	14.7956	0.798887
$344$	−1.62328	−0.0875215
$345$	16.7044	0.899336
$346$	−10.3362	−0.555679
$347$	−3.03689	−0.163029	−0.0815145	−	0.996672i	$-0.525976\pi$
$347$	−3.03689	−0.163029	−0.0815145	+	0.996672i	$0.525976\pi$
$348$	1.89081	0.101358
$349$	−22.5033	−1.20457	−0.602286	−	0.798280i	$-0.705743\pi$
$349$	−22.5033	−1.20457	−0.602286	+	0.798280i	$0.705743\pi$
$350$	4.64449	0.248258
$351$	−10.0942	−0.538787
$352$	1.08861	0.0580230
$353$	5.22665	0.278186	0.139093	−	0.990279i	$-0.455581\pi$
$353$	5.22665	0.278186	0.139093	+	0.990279i	$0.455581\pi$
$354$	9.41222	0.500254
$355$	−4.42884	−0.235058
$356$	7.32916	0.388445
$357$	10.3407	0.547289
$358$	9.85570	0.520890
$359$	25.0074	1.31984	0.659920	−	0.751336i	$-0.270590\pi$
$359$	25.0074	1.31984	0.659920	+	0.751336i	$0.270590\pi$
$360$	−0.465156	−0.0245159
$361$	−10.5728	−0.556463
$362$	−19.5838	−1.02930
$363$	−18.2488	−0.957812
$364$	−2.50150	−0.131114
$365$	10.4077	0.544763
$366$	20.2015	1.05595
$367$	−32.3541	−1.68887	−0.844436	−	0.535657i	$-0.820064\pi$
$367$	−32.3541	−1.68887	−0.844436	+	0.535657i	$0.820064\pi$
$368$	−8.82576	−0.460075
$369$	1.27674	0.0664642
$370$	6.39103	0.332254
$371$	2.79321	0.145016
$372$	−6.21036	−0.321992
$373$	−17.5399	−0.908184	−0.454092	−	0.890955i	$-0.650036\pi$
$373$	−17.5399	−0.908184	−0.454092	+	0.890955i	$0.650036\pi$
$374$	−5.16707	−0.267183
$375$	16.9656	0.876099
$376$	−8.69124	−0.448217
$377$	2.17106	0.111815
$378$	5.54030	0.284962
$379$	−36.1855	−1.85872	−0.929361	−	0.369172i	$-0.879641\pi$
$379$	−36.1855	−1.85872	−0.929361	+	0.369172i	$0.879641\pi$
$380$	−2.95512	−0.151594
$381$	19.4069	0.994246
$382$	−21.3071	−1.09016
$383$	11.9969	0.613014	0.306507	−	0.951868i	$-0.400840\pi$
$383$	11.9969	0.613014	0.306507	+	0.951868i	$0.400840\pi$
$384$	1.85929	0.0948813
$385$	1.29848	0.0661768
$386$	20.0964	1.02288
$387$	−0.741754	−0.0377055
$388$	−1.72909	−0.0877811
$389$	22.5812	1.14491	0.572456	−	0.819935i	$-0.305991\pi$
$389$	22.5812	1.14491	0.572456	+	0.819935i	$0.305991\pi$
$390$	−4.04062	−0.204605
$391$	41.8914	2.11854
$392$	−5.62702	−0.284208
$393$	8.66255	0.436968
$394$	1.00529	0.0506456
$395$	15.2115	0.765372
$396$	0.497436	0.0249971
$397$	−12.5034	−0.627527	−0.313764	−	0.949501i	$-0.601590\pi$
$397$	−12.5034	−0.627527	−0.313764	+	0.949501i	$0.601590\pi$
$398$	−19.6146	−0.983192
$399$	−6.32440	−0.316616
$400$	−3.96375	−0.198187
$401$	13.2373	0.661040	0.330520	−	0.943799i	$-0.392776\pi$
$401$	13.2373	0.661040	0.330520	+	0.943799i	$0.392776\pi$
$402$	20.6057	1.02772
$403$	−7.13082	−0.355212
$404$	−7.01837	−0.349177
$405$	10.3446	0.514027
$406$	−1.19161	−0.0591386
$407$	−6.83455	−0.338776
$408$	−8.82510	−0.436907
$409$	−24.2845	−1.20079	−0.600396	−	0.799703i	$-0.704990\pi$
$409$	−24.2845	−1.20079	−0.600396	+	0.799703i	$0.704990\pi$
$410$	−2.84425	−0.140468
$411$	9.67062	0.477017
$412$	1.63131	0.0803691
$413$	−5.93168	−0.291879
$414$	−4.03291	−0.198207
$415$	−1.02172	−0.0501541
$416$	2.13486	0.104670
$417$	13.6866	0.670238
$418$	3.16019	0.154570
$419$	9.91496	0.484377	0.242189	−	0.970229i	$-0.422135\pi$
$419$	9.91496	0.484377	0.242189	+	0.970229i	$0.422135\pi$
$420$	2.21774	0.108215
$421$	−12.5062	−0.609513	−0.304756	−	0.952430i	$-0.598575\pi$
$421$	−12.5062	−0.609513	−0.304756	+	0.952430i	$0.598575\pi$
$422$	9.65392	0.469946
$423$	−3.97144	−0.193098
$424$	−2.38381	−0.115768
$425$	18.8139	0.912609
$426$	8.08915	0.391921
$427$	−12.7312	−0.616106
$428$	−11.2451	−0.543553
$429$	4.32103	0.208621
$430$	1.65244	0.0796880
$431$	−12.5414	−0.604096	−0.302048	−	0.953293i	$-0.597670\pi$
$431$	−12.5414	−0.604096	−0.302048	+	0.953293i	$0.597670\pi$
$432$	−4.72826	−0.227489
$433$	−5.30573	−0.254977	−0.127489	−	0.991840i	$-0.540692\pi$
$433$	−5.30573	−0.254977	−0.127489	+	0.991840i	$0.540692\pi$
$434$	3.91383	0.187870
$435$	−1.92478	−0.0922863
$436$	−3.18706	−0.152632
$437$	−25.6209	−1.22561
$438$	−19.0094	−0.908302
$439$	23.5404	1.12352	0.561760	−	0.827300i	$-0.310124\pi$
$439$	23.5404	1.12352	0.561760	+	0.827300i	$0.310124\pi$
$440$	−1.10817	−0.0528297
$441$	−2.57125	−0.122441
$442$	−10.1331	−0.481982
$443$	12.5769	0.597545	0.298773	−	0.954324i	$-0.403423\pi$
$443$	12.5769	0.597545	0.298773	+	0.954324i	$0.403423\pi$
$444$	−11.6731	−0.553979
$445$	−7.46083	−0.353677
$446$	−22.0427	−1.04375
$447$	4.47225	0.211530
$448$	−1.17174	−0.0553596
$449$	22.6864	1.07064	0.535319	−	0.844650i	$-0.320191\pi$
$449$	22.6864	1.07064	0.535319	+	0.844650i	$0.320191\pi$
$450$	−1.81122	−0.0853819
$451$	3.04163	0.143225
$452$	11.7222	0.551365
$453$	19.6358	0.922570
$454$	−5.93190	−0.278398
$455$	2.54644	0.119379
$456$	5.39744	0.252758
$457$	16.9499	0.792884	0.396442	−	0.918060i	$-0.370245\pi$
$457$	16.9499	0.792884	0.396442	+	0.918060i	$0.370245\pi$
$458$	2.65940	0.124266
$459$	22.4427	1.04753
$460$	8.98431	0.418896
$461$	2.17423	0.101264	0.0506320	−	0.998717i	$-0.483876\pi$
$461$	2.17423	0.101264	0.0506320	+	0.998717i	$0.483876\pi$
$462$	−2.37164	−0.110339
$463$	11.3330	0.526690	0.263345	−	0.964702i	$-0.415174\pi$
$463$	11.3330	0.526690	0.263345	+	0.964702i	$0.415174\pi$
$464$	1.01696	0.0472111
$465$	6.32193	0.293173
$466$	−5.80761	−0.269032
$467$	−8.84416	−0.409259	−0.204630	−	0.978839i	$-0.565599\pi$
$467$	−8.84416	−0.409259	−0.204630	+	0.978839i	$0.565599\pi$
$468$	0.975517	0.0450933
$469$	−12.9859	−0.599635
$470$	8.84738	0.408099
$471$	−16.9497	−0.781001
$472$	5.06228	0.233010
$473$	−1.76712	−0.0812522
$474$	−27.7833	−1.27613
$475$	−11.5066	−0.527960
$476$	5.56166	0.254918
$477$	−1.08927	−0.0498745
$478$	22.6822	1.03746
$479$	37.2346	1.70129	0.850646	−	0.525740i	$-0.176211\pi$
$479$	37.2346	1.70129	0.850646	+	0.525740i	$0.176211\pi$
$480$	−1.89269	−0.0863890
$481$	−13.4032	−0.611131
$482$	21.0958	0.960888
$483$	19.2278	0.874896
$484$	−9.81493	−0.446133
$485$	1.76015	0.0799243
$486$	−4.70935	−0.213621
$487$	−12.5485	−0.568628	−0.284314	−	0.958731i	$-0.591766\pi$
$487$	−12.5485	−0.568628	−0.284314	+	0.958731i	$0.591766\pi$
$488$	10.8652	0.491845
$489$	−18.0164	−0.814728
$490$	5.72811	0.258770
$491$	−11.2788	−0.509007	−0.254504	−	0.967072i	$-0.581912\pi$
$491$	−11.2788	−0.509007	−0.254504	+	0.967072i	$0.581912\pi$
$492$	5.19495	0.234207
$493$	−4.82698	−0.217396
$494$	6.19742	0.278835
$495$	−0.506373	−0.0227598
$496$	−3.34018	−0.149979
$497$	−5.09787	−0.228671
$498$	1.86614	0.0836236
$499$	−11.1122	−0.497451	−0.248725	−	0.968574i	$-0.580012\pi$
$499$	−11.1122	−0.497451	−0.248725	+	0.968574i	$0.580012\pi$
$500$	9.12478	0.408073
$501$	6.87575	0.307186
$502$	−17.0104	−0.759211
$503$	0.149364	0.00665981	0.00332990	−	0.999994i	$-0.498940\pi$
$503$	0.149364	0.00665981	0.00332990	+	0.999994i	$0.498940\pi$
$504$	−0.535424	−0.0238497
$505$	7.14445	0.317924
$506$	−9.60780	−0.427119
$507$	−15.6968	−0.697120
$508$	10.4378	0.463103
$509$	24.8102	1.09969	0.549847	−	0.835265i	$-0.314686\pi$
$509$	24.8102	1.09969	0.549847	+	0.835265i	$0.314686\pi$
$510$	8.98364	0.397802
$511$	11.9799	0.529959
$512$	1.00000	0.0441942
$513$	−13.7260	−0.606017
$514$	−10.4635	−0.461524
$515$	−1.66062	−0.0731757
$516$	−3.01815	−0.132867
$517$	−9.46136	−0.416110
$518$	7.35647	0.323225
$519$	−19.2180	−0.843578
$520$	−2.17321	−0.0953016
$521$	4.29864	0.188327	0.0941634	−	0.995557i	$-0.469982\pi$
$521$	4.29864	0.188327	0.0941634	+	0.995557i	$0.469982\pi$
$522$	0.464696	0.0203392
$523$	35.2554	1.54161	0.770804	−	0.637072i	$-0.219855\pi$
$523$	35.2554	1.54161	0.770804	+	0.637072i	$0.219855\pi$
$524$	4.65907	0.203533
$525$	8.63543	0.376881
$526$	−0.633366	−0.0276161
$527$	15.8542	0.690619
$528$	2.02403	0.0880848
$529$	54.8940	2.38670
$530$	2.42663	0.105406
$531$	2.31319	0.100384
$532$	−3.40152	−0.147475
$533$	5.96492	0.258369
$534$	13.6270	0.589698
$535$	11.4471	0.494903
$536$	11.0826	0.478695
$537$	18.3246	0.790763
$538$	11.6238	0.501138
$539$	−6.12562	−0.263849
$540$	4.81321	0.207127
$541$	25.4275	1.09321	0.546606	−	0.837390i	$-0.315919\pi$
$541$	25.4275	1.09321	0.546606	+	0.837390i	$0.315919\pi$
$542$	−7.21081	−0.309731
$543$	−36.4119	−1.56258
$544$	−4.74650	−0.203504
$545$	3.24431	0.138971
$546$	−4.65101	−0.199045
$547$	35.8483	1.53276	0.766381	−	0.642386i	$-0.222055\pi$
$547$	35.8483	1.53276	0.766381	+	0.642386i	$0.222055\pi$
$548$	5.20125	0.222187
$549$	4.96483	0.211894
$550$	−4.31497	−0.183991
$551$	2.95219	0.125767
$552$	−16.4096	−0.698440
$553$	17.5093	0.744573
$554$	28.1431	1.19569
$555$	11.8828	0.504395
$556$	7.36123	0.312186
$557$	−25.1194	−1.06434	−0.532171	−	0.846637i	$-0.678624\pi$
$557$	−25.1194	−1.06434	−0.532171	+	0.846637i	$0.678624\pi$
$558$	−1.52629	−0.0646129
$559$	−3.46548	−0.146574
$560$	1.19279	0.0504046
$561$	−9.60707	−0.405611
$562$	12.6556	0.533844
$563$	−25.2673	−1.06489	−0.532444	−	0.846465i	$-0.678726\pi$
$563$	−25.2673	−1.06489	−0.532444	+	0.846465i	$0.678726\pi$
$564$	−16.1595	−0.680438
$565$	−11.9328	−0.502016
$566$	12.1062	0.508863
$567$	11.9073	0.500059
$568$	4.35068	0.182550
$569$	12.8930	0.540502	0.270251	−	0.962790i	$-0.412893\pi$
$569$	12.8930	0.540502	0.270251	+	0.962790i	$0.412893\pi$
$570$	−5.49441	−0.230135
$571$	6.00241	0.251193	0.125597	−	0.992081i	$-0.459916\pi$
$571$	6.00241	0.251193	0.125597	+	0.992081i	$0.459916\pi$
$572$	2.32402	0.0971723
$573$	−39.6160	−1.65498
$574$	−3.27391	−0.136650
$575$	34.9831	1.45890
$576$	0.456947	0.0190395
$577$	−3.63979	−0.151526	−0.0757631	−	0.997126i	$-0.524139\pi$
$577$	−3.63979	−0.151526	−0.0757631	+	0.997126i	$0.524139\pi$
$578$	5.52922	0.229985
$579$	37.3650	1.55284
$580$	−1.03523	−0.0429855
$581$	−1.17606	−0.0487911
$582$	−3.21487	−0.133261
$583$	−2.59503	−0.107475
$584$	−10.2240	−0.423072
$585$	−0.993043	−0.0410573
$586$	−9.03515	−0.373238
$587$	4.62263	0.190796	0.0953982	−	0.995439i	$-0.469588\pi$
$587$	4.62263	0.190796	0.0953982	+	0.995439i	$0.469588\pi$
$588$	−10.4622	−0.431456
$589$	−9.69643	−0.399535
$590$	−5.15322	−0.212155
$591$	1.86912	0.0768852
$592$	−6.27824	−0.258034
$593$	10.1455	0.416627	0.208313	−	0.978062i	$-0.433203\pi$
$593$	10.1455	0.416627	0.208313	+	0.978062i	$0.433203\pi$
$594$	−5.14723	−0.211193
$595$	−5.66158	−0.232102
$596$	2.40536	0.0985273
$597$	−36.4692	−1.49259
$598$	−18.8417	−0.770496
$599$	−9.50043	−0.388177	−0.194088	−	0.980984i	$-0.562175\pi$
$599$	−9.50043	−0.388177	−0.194088	+	0.980984i	$0.562175\pi$
$600$	−7.36974	−0.300868
$601$	43.7461	1.78444	0.892220	−	0.451601i	$-0.149147\pi$
$601$	43.7461	1.78444	0.892220	+	0.451601i	$0.149147\pi$
$602$	1.90207	0.0775225
$603$	5.06416	0.206229
$604$	10.5609	0.429718
$605$	9.99126	0.406202
$606$	−13.0492	−0.530086
$607$	−27.9703	−1.13528	−0.567639	−	0.823278i	$-0.692143\pi$
$607$	−27.9703	−1.13528	−0.567639	+	0.823278i	$0.692143\pi$
$608$	2.90296	0.117731
$609$	−2.21555	−0.0897784
$610$	−11.0604	−0.447823
$611$	−18.5546	−0.750638
$612$	−2.16890	−0.0876725
$613$	39.0641	1.57778	0.788892	−	0.614531i	$-0.210655\pi$
$613$	39.0641	1.57778	0.788892	+	0.614531i	$0.210655\pi$
$614$	−26.9796	−1.08881
$615$	−5.28828	−0.213244
$616$	−1.27557	−0.0513941
$617$	38.0480	1.53176	0.765878	−	0.642986i	$-0.222305\pi$
$617$	38.0480	1.53176	0.765878	+	0.642986i	$0.222305\pi$
$618$	3.03308	0.122008
$619$	12.9990	0.522476	0.261238	−	0.965274i	$-0.415869\pi$
$619$	12.9990	0.522476	0.261238	+	0.965274i	$0.415869\pi$
$620$	3.40019	0.136555
$621$	41.7305	1.67459
$622$	17.8100	0.714114
$623$	−8.58788	−0.344066
$624$	3.96931	0.158900
$625$	10.5300	0.421201
$626$	−9.30565	−0.371928
$627$	5.87570	0.234653
$628$	−9.11624	−0.363778
$629$	29.7996	1.18819
$630$	0.545043	0.0217150
$631$	−3.82066	−0.152098	−0.0760490	−	0.997104i	$-0.524231\pi$
$631$	−3.82066	−0.152098	−0.0760490	+	0.997104i	$0.524231\pi$
$632$	−14.9430	−0.594401
$633$	17.9494	0.713425
$634$	−2.71677	−0.107897
$635$	−10.6253	−0.421654
$636$	−4.43218	−0.175747
$637$	−12.0129	−0.475968
$638$	1.10707	0.0438292
$639$	1.98803	0.0786452
$640$	−1.01796	−0.0402386
$641$	−28.7496	−1.13554	−0.567771	−	0.823186i	$-0.692194\pi$
$641$	−28.7496	−1.13554	−0.567771	+	0.823186i	$0.692194\pi$
$642$	−20.9079	−0.825168
$643$	−2.28039	−0.0899299	−0.0449649	−	0.998989i	$-0.514318\pi$
$643$	−2.28039	−0.0899299	−0.0449649	+	0.998989i	$0.514318\pi$
$644$	10.3415	0.407512
$645$	3.07237	0.120974
$646$	−13.7789	−0.542124
$647$	−36.4123	−1.43152	−0.715758	−	0.698349i	$-0.753918\pi$
$647$	−36.4123	−1.43152	−0.715758	+	0.698349i	$0.753918\pi$
$648$	−10.1620	−0.399203
$649$	5.51084	0.216319
$650$	−8.46204	−0.331908
$651$	7.27694	0.285206
$652$	−9.68993	−0.379487
$653$	−46.0714	−1.80291	−0.901457	−	0.432868i	$-0.857502\pi$
$653$	−46.0714	−1.80291	−0.901457	+	0.432868i	$0.857502\pi$
$654$	−5.92565	−0.231711
$655$	−4.74277	−0.185315
$656$	2.79406	0.109090
$657$	−4.67183	−0.182265
$658$	10.1839	0.397009
$659$	−9.61878	−0.374695	−0.187347	−	0.982294i	$-0.559989\pi$
$659$	−9.61878	−0.374695	−0.187347	+	0.982294i	$0.559989\pi$
$660$	−2.06040	−0.0802008
$661$	14.5218	0.564832	0.282416	−	0.959292i	$-0.408864\pi$
$661$	14.5218	0.564832	0.282416	+	0.959292i	$0.408864\pi$
$662$	−26.5174	−1.03063
$663$	−18.8403	−0.731698
$664$	1.00368	0.0389505
$665$	3.46263	0.134275
$666$	−2.86882	−0.111165
$667$	−8.97542	−0.347530
$668$	3.69806	0.143082
$669$	−40.9837	−1.58452
$670$	−11.2817	−0.435850
$671$	11.8280	0.456613
$672$	−2.17860	−0.0840414
$673$	−29.3572	−1.13164	−0.565819	−	0.824529i	$-0.691440\pi$
$673$	−29.3572	−1.13164	−0.565819	+	0.824529i	$0.691440\pi$
$674$	−25.6870	−0.989428
$675$	18.7416	0.721366
$676$	−8.44238	−0.324707
$677$	−13.4420	−0.516617	−0.258308	−	0.966063i	$-0.583165\pi$
$677$	−13.4420	−0.516617	−0.258308	+	0.966063i	$0.583165\pi$
$678$	21.7949	0.837028
$679$	2.02604	0.0777524
$680$	4.83177	0.185290
$681$	−11.0291	−0.422636
$682$	−3.63615	−0.139235
$683$	28.0400	1.07292	0.536460	−	0.843926i	$-0.319761\pi$
$683$	28.0400	1.07292	0.536460	+	0.843926i	$0.319761\pi$
$684$	1.32650	0.0507200
$685$	−5.29469	−0.202300
$686$	14.7956	0.564899
$687$	4.94459	0.188648
$688$	−1.62328	−0.0618871
$689$	−5.08909	−0.193879
$690$	16.7044	0.635926
$691$	2.50441	0.0952723	0.0476362	−	0.998865i	$-0.484831\pi$
$691$	2.50441	0.0952723	0.0476362	+	0.998865i	$0.484831\pi$
$692$	−10.3362	−0.392925
$693$	−0.582867	−0.0221413
$694$	−3.03689	−0.115279
$695$	−7.49348	−0.284244
$696$	1.89081	0.0716712
$697$	−13.2620	−0.502333
$698$	−22.5033	−0.851761
$699$	−10.7980	−0.408419
$700$	4.64449	0.175545
$701$	29.3265	1.10765	0.553823	−	0.832635i	$-0.313169\pi$
$701$	29.3265	1.10765	0.553823	+	0.832635i	$0.313169\pi$
$702$	−10.0942	−0.380980
$703$	−18.2255	−0.687388
$704$	1.08861	0.0410285
$705$	16.4498	0.619536
$706$	5.22665	0.196707
$707$	8.22371	0.309284
$708$	9.41222	0.353733
$709$	−12.5552	−0.471521	−0.235760	−	0.971811i	$-0.575758\pi$
$709$	−12.5552	−0.471521	−0.235760	+	0.971811i	$0.575758\pi$
$710$	−4.42884	−0.166211
$711$	−6.82817	−0.256076
$712$	7.32916	0.274672
$713$	29.4797	1.10402
$714$	10.3407	0.386992
$715$	−2.36578	−0.0884750
$716$	9.85570	0.368325
$717$	42.1727	1.57497
$718$	25.0074	0.933268
$719$	−20.3271	−0.758073	−0.379037	−	0.925382i	$-0.623745\pi$
$719$	−20.3271	−0.758073	−0.379037	+	0.925382i	$0.623745\pi$
$720$	−0.465156	−0.0173353
$721$	−1.91148	−0.0711872
$722$	−10.5728	−0.393479
$723$	39.2232	1.45873
$724$	−19.5838	−0.727826
$725$	−4.03096	−0.149706
$726$	−18.2488	−0.677275
$727$	−4.85727	−0.180146	−0.0900732	−	0.995935i	$-0.528710\pi$
$727$	−4.85727	−0.180146	−0.0900732	+	0.995935i	$0.528710\pi$
$728$	−2.50150	−0.0927118
$729$	21.7301	0.804818
$730$	10.4077	0.385205
$731$	7.70490	0.284976
$732$	20.2015	0.746670
$733$	3.93046	0.145175	0.0725875	−	0.997362i	$-0.476874\pi$
$733$	3.93046	0.145175	0.0725875	+	0.997362i	$0.476874\pi$
$734$	−32.3541	−1.19421
$735$	10.6502	0.392839
$736$	−8.82576	−0.325322
$737$	12.0646	0.444405
$738$	1.27674	0.0469973
$739$	10.4497	0.384400	0.192200	−	0.981356i	$-0.438438\pi$
$739$	10.4497	0.384400	0.192200	+	0.981356i	$0.438438\pi$
$740$	6.39103	0.234939
$741$	11.5228	0.423300
$742$	2.79321	0.102542
$743$	50.4517	1.85089	0.925446	−	0.378880i	$-0.123691\pi$
$743$	50.4517	1.85089	0.925446	+	0.378880i	$0.123691\pi$
$744$	−6.21036	−0.227683
$745$	−2.44857	−0.0897086
$746$	−17.5399	−0.642183
$747$	0.458631	0.0167804
$748$	−5.16707	−0.188927
$749$	13.1764	0.481454
$750$	16.9656	0.619496
$751$	−11.9392	−0.435669	−0.217835	−	0.975986i	$-0.569899\pi$
$751$	−11.9392	−0.435669	−0.217835	+	0.975986i	$0.569899\pi$
$752$	−8.69124	−0.316937
$753$	−31.6272	−1.15256
$754$	2.17106	0.0790653
$755$	−10.7507	−0.391257
$756$	5.54030	0.201499
$757$	17.2441	0.626748	0.313374	−	0.949630i	$-0.398541\pi$
$757$	17.2441	0.626748	0.313374	+	0.949630i	$0.398541\pi$
$758$	−36.1855	−1.31431
$759$	−17.8636	−0.648409
$760$	−2.95512	−0.107193
$761$	44.8803	1.62691	0.813454	−	0.581629i	$-0.197584\pi$
$761$	44.8803	1.62691	0.813454	+	0.581629i	$0.197584\pi$
$762$	19.4069	0.703038
$763$	3.73440	0.135194
$764$	−21.3071	−0.770863
$765$	2.20786	0.0798254
$766$	11.9969	0.433467
$767$	10.8072	0.390227
$768$	1.85929	0.0670912
$769$	26.2433	0.946358	0.473179	−	0.880966i	$-0.343106\pi$
$769$	26.2433	0.946358	0.473179	+	0.880966i	$0.343106\pi$
$770$	1.29848	0.0467941
$771$	−19.4546	−0.700640
$772$	20.0964	0.723285
$773$	−25.9802	−0.934444	−0.467222	−	0.884140i	$-0.654745\pi$
$773$	−25.9802	−0.934444	−0.467222	+	0.884140i	$0.654745\pi$
$774$	−0.741754	−0.0266618
$775$	13.2396	0.475582
$776$	−1.72909	−0.0620706
$777$	13.6778	0.490688
$778$	22.5812	0.809575
$779$	8.11104	0.290608
$780$	−4.04062	−0.144677
$781$	4.73618	0.169474
$782$	41.8914	1.49803
$783$	−4.80844	−0.171840
$784$	−5.62702	−0.200965
$785$	9.28002	0.331218
$786$	8.66255	0.308983
$787$	6.08412	0.216876	0.108438	−	0.994103i	$-0.465415\pi$
$787$	6.08412	0.216876	0.108438	+	0.994103i	$0.465415\pi$
$788$	1.00529	0.0358119
$789$	−1.17761	−0.0419240
$790$	15.2115	0.541199
$791$	−13.7354	−0.488373
$792$	0.497436	0.0176756
$793$	23.1957	0.823703
$794$	−12.5034	−0.443729
$795$	4.51181	0.160017
$796$	−19.6146	−0.695222
$797$	−40.1485	−1.42213	−0.711066	−	0.703126i	$-0.751787\pi$
$797$	−40.1485	−1.42213	−0.711066	+	0.703126i	$0.751787\pi$
$798$	−6.32440	−0.223881
$799$	41.2529	1.45942
$800$	−3.96375	−0.140140
$801$	3.34904	0.118332
$802$	13.2373	0.467426
$803$	−11.1299	−0.392767
$804$	20.6057	0.726708
$805$	−10.5273	−0.371038
$806$	−7.13082	−0.251172
$807$	21.6120	0.760779
$808$	−7.01837	−0.246905
$809$	28.1590	0.990018	0.495009	−	0.868888i	$-0.335165\pi$
$809$	28.1590	0.990018	0.495009	+	0.868888i	$0.335165\pi$
$810$	10.3446	0.363472
$811$	1.76064	0.0618246	0.0309123	−	0.999522i	$-0.490159\pi$
$811$	1.76064	0.0618246	0.0309123	+	0.999522i	$0.490159\pi$
$812$	−1.19161	−0.0418173
$813$	−13.4070	−0.470203
$814$	−6.83455	−0.239551
$815$	9.86401	0.345521
$816$	−8.82510	−0.308940
$817$	−4.71233	−0.164864
$818$	−24.2845	−0.849088
$819$	−1.14305	−0.0399415
$820$	−2.84425	−0.0993256
$821$	17.0853	0.596280	0.298140	−	0.954522i	$-0.403634\pi$
$821$	17.0853	0.596280	0.298140	+	0.954522i	$0.403634\pi$
$822$	9.67062	0.337302
$823$	−0.327744	−0.0114244	−0.00571222	−	0.999984i	$-0.501818\pi$
$823$	−0.327744	−0.0114244	−0.00571222	+	0.999984i	$0.501818\pi$
$824$	1.63131	0.0568295
$825$	−8.02276	−0.279317
$826$	−5.93168	−0.206389
$827$	36.3786	1.26501	0.632504	−	0.774557i	$-0.282027\pi$
$827$	36.3786	1.26501	0.632504	+	0.774557i	$0.282027\pi$
$828$	−4.03291	−0.140153
$829$	18.7717	0.651968	0.325984	−	0.945375i	$-0.394304\pi$
$829$	18.7717	0.651968	0.325984	+	0.945375i	$0.394304\pi$
$830$	−1.02172	−0.0354643
$831$	52.3262	1.81517
$832$	2.13486	0.0740129
$833$	26.7086	0.925400
$834$	13.6866	0.473930
$835$	−3.76449	−0.130276
$836$	3.16019	0.109297
$837$	15.7933	0.545896
$838$	9.91496	0.342507
$839$	35.7894	1.23559	0.617794	−	0.786340i	$-0.288026\pi$
$839$	35.7894	1.23559	0.617794	+	0.786340i	$0.288026\pi$
$840$	2.21774	0.0765193
$841$	−27.9658	−0.964338
$842$	−12.5062	−0.430991
$843$	23.5304	0.810429
$844$	9.65392	0.332302
$845$	8.59405	0.295644
$846$	−3.97144	−0.136541
$847$	11.5006	0.395164
$848$	−2.38381	−0.0818603
$849$	22.5090	0.772505
$850$	18.8139	0.645312
$851$	55.4103	1.89944
$852$	8.08915	0.277130
$853$	23.6529	0.809861	0.404930	−	0.914347i	$-0.367296\pi$
$853$	23.6529	0.809861	0.404930	+	0.914347i	$0.367296\pi$
$854$	−12.7312	−0.435653
$855$	−1.35033	−0.0461804
$856$	−11.2451	−0.384350
$857$	8.17100	0.279116	0.139558	−	0.990214i	$-0.455432\pi$
$857$	8.17100	0.279116	0.139558	+	0.990214i	$0.455432\pi$
$858$	4.32103	0.147517
$859$	35.2837	1.20387	0.601933	−	0.798547i	$-0.294398\pi$
$859$	35.2837	1.20387	0.601933	+	0.798547i	$0.294398\pi$
$860$	1.65244	0.0563479
$861$	−6.08714	−0.207449
$862$	−12.5414	−0.427161
$863$	−23.0554	−0.784816	−0.392408	−	0.919791i	$-0.628358\pi$
$863$	−23.0554	−0.784816	−0.392408	+	0.919791i	$0.628358\pi$
$864$	−4.72826	−0.160859
$865$	10.5219	0.357756
$866$	−5.30573	−0.180296
$867$	10.2804	0.349141
$868$	3.91383	0.132844
$869$	−16.2671	−0.551823
$870$	−1.92478	−0.0652563
$871$	23.6598	0.801681
$872$	−3.18706	−0.107927
$873$	−0.790102	−0.0267409
$874$	−25.6209	−0.866639
$875$	−10.6919	−0.361452
$876$	−19.0094	−0.642267
$877$	−33.5325	−1.13231	−0.566156	−	0.824298i	$-0.691570\pi$
$877$	−33.5325	−1.13231	−0.566156	+	0.824298i	$0.691570\pi$
$878$	23.5404	0.794449
$879$	−16.7989	−0.566614
$880$	−1.10817	−0.0373562
$881$	−47.1311	−1.58789	−0.793944	−	0.607991i	$-0.791975\pi$
$881$	−47.1311	−1.58789	−0.793944	+	0.607991i	$0.791975\pi$
$882$	−2.57125	−0.0865786
$883$	−5.78288	−0.194609	−0.0973047	−	0.995255i	$-0.531022\pi$
$883$	−5.78288	−0.194609	−0.0973047	+	0.995255i	$0.531022\pi$
$884$	−10.1331	−0.340813
$885$	−9.58131	−0.322072
$886$	12.5769	0.422528
$887$	−53.7488	−1.80471	−0.902355	−	0.430994i	$-0.858163\pi$
$887$	−53.7488	−1.80471	−0.902355	+	0.430994i	$0.858163\pi$
$888$	−11.6731	−0.391722
$889$	−12.2304	−0.410195
$890$	−7.46083	−0.250087
$891$	−11.0625	−0.370607
$892$	−22.0427	−0.738043
$893$	−25.2304	−0.844302
$894$	4.47225	0.149574
$895$	−10.0328	−0.335358
$896$	−1.17174	−0.0391451
$897$	−35.0322	−1.16969
$898$	22.6864	0.757055
$899$	−3.39682	−0.113290
$900$	−1.81122	−0.0603741
$901$	11.3147	0.376949
$902$	3.04163	0.101275
$903$	3.53649	0.117687
$904$	11.7222	0.389874
$905$	19.9356	0.662682
$906$	19.6358	0.652356
$907$	−6.89356	−0.228897	−0.114448	−	0.993429i	$-0.536510\pi$
$907$	−6.89356	−0.228897	−0.114448	+	0.993429i	$0.536510\pi$
$908$	−5.93190	−0.196857
$909$	−3.20702	−0.106370
$910$	2.54644	0.0844137
$911$	1.81816	0.0602384	0.0301192	−	0.999546i	$-0.490411\pi$
$911$	1.81816	0.0602384	0.0301192	+	0.999546i	$0.490411\pi$
$912$	5.39744	0.178727
$913$	1.09262	0.0361604
$914$	16.9499	0.560654
$915$	−20.5645	−0.679840
$916$	2.65940	0.0878691
$917$	−5.45923	−0.180280
$918$	22.4427	0.740719
$919$	−21.0281	−0.693653	−0.346827	−	0.937929i	$-0.612741\pi$
$919$	−21.0281	−0.693653	−0.346827	+	0.937929i	$0.612741\pi$
$920$	8.98431	0.296204
$921$	−50.1629	−1.65292
$922$	2.17423	0.0716044
$923$	9.28808	0.305721
$924$	−2.37164	−0.0780214
$925$	24.8854	0.818226
$926$	11.3330	0.372426
$927$	0.745425	0.0244830
$928$	1.01696	0.0333833
$929$	4.70128	0.154244	0.0771220	−	0.997022i	$-0.475427\pi$
$929$	4.70128	0.154244	0.0771220	+	0.997022i	$0.475427\pi$
$930$	6.32193	0.207304
$931$	−16.3350	−0.535359
$932$	−5.80761	−0.190235
$933$	33.1138	1.08410
$934$	−8.84416	−0.289390
$935$	5.25990	0.172017
$936$	0.975517	0.0318858
$937$	26.3110	0.859542	0.429771	−	0.902938i	$-0.358594\pi$
$937$	26.3110	0.859542	0.429771	+	0.902938i	$0.358594\pi$
$938$	−12.9859	−0.424006
$939$	−17.3019	−0.564625
$940$	8.84738	0.288570
$941$	−17.8248	−0.581072	−0.290536	−	0.956864i	$-0.593834\pi$
$941$	−17.8248	−0.581072	−0.290536	+	0.956864i	$0.593834\pi$
$942$	−16.9497	−0.552251
$943$	−24.6597	−0.803030
$944$	5.06228	0.164763
$945$	−5.63983	−0.183464
$946$	−1.76712	−0.0574540
$947$	−11.6189	−0.377563	−0.188781	−	0.982019i	$-0.560454\pi$
$947$	−11.6189	−0.377563	−0.188781	+	0.982019i	$0.560454\pi$
$948$	−27.7833	−0.902361
$949$	−21.8268	−0.708528
$950$	−11.5066	−0.373324
$951$	−5.05125	−0.163798
$952$	5.56166	0.180255
$953$	−49.3575	−1.59885	−0.799423	−	0.600768i	$-0.794861\pi$
$953$	−49.3575	−1.59885	−0.799423	+	0.600768i	$0.794861\pi$
$954$	−1.08927	−0.0352666
$955$	21.6899	0.701867
$956$	22.6822	0.733594
$957$	2.05836	0.0665372
$958$	37.2346	1.20299
$959$	−6.09452	−0.196802
$960$	−1.89269	−0.0610863
$961$	−19.8432	−0.640102
$962$	−13.4032	−0.432135
$963$	−5.13842	−0.165583
$964$	21.0958	0.679450
$965$	−20.4574	−0.658548
$966$	19.2278	0.618645
$967$	16.9472	0.544986	0.272493	−	0.962158i	$-0.412152\pi$
$967$	16.9472	0.544986	0.272493	+	0.962158i	$0.412152\pi$
$968$	−9.81493	−0.315464
$969$	−25.6189	−0.822999
$970$	1.76015	0.0565150
$971$	32.5146	1.04344	0.521721	−	0.853116i	$-0.325290\pi$
$971$	32.5146	1.04344	0.521721	+	0.853116i	$0.325290\pi$
$972$	−4.70935	−0.151053
$973$	−8.62546	−0.276519
$974$	−12.5485	−0.402081
$975$	−15.7334	−0.503871
$976$	10.8652	0.347787
$977$	−53.4599	−1.71034	−0.855168	−	0.518352i	$-0.826546\pi$
$977$	−53.4599	−1.71034	−0.855168	+	0.518352i	$0.826546\pi$
$978$	−18.0164	−0.576100
$979$	7.97858	0.254997
$980$	5.72811	0.182978
$981$	−1.45632	−0.0464966
$982$	−11.2788	−0.359923
$983$	28.5055	0.909186	0.454593	−	0.890699i	$-0.349785\pi$
$983$	28.5055	0.909186	0.454593	+	0.890699i	$0.349785\pi$
$984$	5.19495	0.165609
$985$	−1.02335	−0.0326065
$986$	−4.82698	−0.153722
$987$	18.9348	0.602700
$988$	6.19742	0.197166
$989$	14.3267	0.455563
$990$	−0.506373	−0.0160936
$991$	5.24681	0.166670	0.0833352	−	0.996522i	$-0.473443\pi$
$991$	5.24681	0.166670	0.0833352	+	0.996522i	$0.473443\pi$
$992$	−3.34018	−0.106051
$993$	−49.3035	−1.56460
$994$	−5.09787	−0.161694
$995$	19.9670	0.632997
$996$	1.86614	0.0591308
$997$	40.3302	1.27727	0.638635	−	0.769510i	$-0.279500\pi$
$997$	40.3302	1.27727	0.638635	+	0.769510i	$0.279500\pi$
$998$	−11.1122	−0.351751
$999$	29.6852	0.939198

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.56	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.56	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.85929 q^{3} +1.00000 q^{4} -1.01796 q^{5} +1.85929 q^{6} -1.17174 q^{7} +1.00000 q^{8} +0.456947 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.85929 q^{3} +1.00000 q^{4} -1.01796 q^{5} +1.85929 q^{6} -1.17174 q^{7} +1.00000 q^{8} +0.456947 q^{9} -1.01796 q^{10} +1.08861 q^{11} +1.85929 q^{12} +2.13486 q^{13} -1.17174 q^{14} -1.89269 q^{15} +1.00000 q^{16} -4.74650 q^{17} +0.456947 q^{18} +2.90296 q^{19} -1.01796 q^{20} -2.17860 q^{21} +1.08861 q^{22} -8.82576 q^{23} +1.85929 q^{24} -3.96375 q^{25} +2.13486 q^{26} -4.72826 q^{27} -1.17174 q^{28} +1.01696 q^{29} -1.89269 q^{30} -3.34018 q^{31} +1.00000 q^{32} +2.02403 q^{33} -4.74650 q^{34} +1.19279 q^{35} +0.456947 q^{36} -6.27824 q^{37} +2.90296 q^{38} +3.96931 q^{39} -1.01796 q^{40} +2.79406 q^{41} -2.17860 q^{42} -1.62328 q^{43} +1.08861 q^{44} -0.465156 q^{45} -8.82576 q^{46} -8.69124 q^{47} +1.85929 q^{48} -5.62702 q^{49} -3.96375 q^{50} -8.82510 q^{51} +2.13486 q^{52} -2.38381 q^{53} -4.72826 q^{54} -1.10817 q^{55} -1.17174 q^{56} +5.39744 q^{57} +1.01696 q^{58} +5.06228 q^{59} -1.89269 q^{60} +10.8652 q^{61} -3.34018 q^{62} -0.535424 q^{63} +1.00000 q^{64} -2.17321 q^{65} +2.02403 q^{66} +11.0826 q^{67} -4.74650 q^{68} -16.4096 q^{69} +1.19279 q^{70} +4.35068 q^{71} +0.456947 q^{72} -10.2240 q^{73} -6.27824 q^{74} -7.36974 q^{75} +2.90296 q^{76} -1.27557 q^{77} +3.96931 q^{78} -14.9430 q^{79} -1.01796 q^{80} -10.1620 q^{81} +2.79406 q^{82} +1.00368 q^{83} -2.17860 q^{84} +4.83177 q^{85} -1.62328 q^{86} +1.89081 q^{87} +1.08861 q^{88} +7.32916 q^{89} -0.465156 q^{90} -2.50150 q^{91} -8.82576 q^{92} -6.21036 q^{93} -8.69124 q^{94} -2.95512 q^{95} +1.85929 q^{96} -1.72909 q^{97} -5.62702 q^{98} +0.497436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 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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