LMFDB - Embedded newform 8042.2.a.a.1.52

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			1.52
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.54806 q^{3} +1.00000 q^{4} -0.496638 q^{5} +1.54806 q^{6} -1.67357 q^{7} +1.00000 q^{8} -0.603508 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.54806 q^{3} +1.00000 q^{4} -0.496638 q^{5} +1.54806 q^{6} -1.67357 q^{7} +1.00000 q^{8} -0.603508 q^{9} -0.496638 q^{10} +0.109322 q^{11} +1.54806 q^{12} +2.67921 q^{13} -1.67357 q^{14} -0.768826 q^{15} +1.00000 q^{16} -6.97252 q^{17} -0.603508 q^{18} -3.72371 q^{19} -0.496638 q^{20} -2.59079 q^{21} +0.109322 q^{22} +3.09611 q^{23} +1.54806 q^{24} -4.75335 q^{25} +2.67921 q^{26} -5.57845 q^{27} -1.67357 q^{28} +5.48849 q^{29} -0.768826 q^{30} +6.47208 q^{31} +1.00000 q^{32} +0.169238 q^{33} -6.97252 q^{34} +0.831160 q^{35} -0.603508 q^{36} -3.64003 q^{37} -3.72371 q^{38} +4.14758 q^{39} -0.496638 q^{40} -1.82844 q^{41} -2.59079 q^{42} +6.81985 q^{43} +0.109322 q^{44} +0.299725 q^{45} +3.09611 q^{46} -7.60435 q^{47} +1.54806 q^{48} -4.19916 q^{49} -4.75335 q^{50} -10.7939 q^{51} +2.67921 q^{52} +13.3705 q^{53} -5.57845 q^{54} -0.0542937 q^{55} -1.67357 q^{56} -5.76453 q^{57} +5.48849 q^{58} -5.16570 q^{59} -0.768826 q^{60} -9.27004 q^{61} +6.47208 q^{62} +1.01001 q^{63} +1.00000 q^{64} -1.33060 q^{65} +0.169238 q^{66} -3.78847 q^{67} -6.97252 q^{68} +4.79297 q^{69} +0.831160 q^{70} -2.99509 q^{71} -0.603508 q^{72} -2.54494 q^{73} -3.64003 q^{74} -7.35848 q^{75} -3.72371 q^{76} -0.182959 q^{77} +4.14758 q^{78} -4.23339 q^{79} -0.496638 q^{80} -6.82526 q^{81} -1.82844 q^{82} -12.6474 q^{83} -2.59079 q^{84} +3.46282 q^{85} +6.81985 q^{86} +8.49651 q^{87} +0.109322 q^{88} -17.2754 q^{89} +0.299725 q^{90} -4.48386 q^{91} +3.09611 q^{92} +10.0192 q^{93} -7.60435 q^{94} +1.84934 q^{95} +1.54806 q^{96} +13.0089 q^{97} -4.19916 q^{98} -0.0659769 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.54806	0.893773	0.446887	−	0.894591i	$-0.352533\pi$
$3$	1.54806	0.893773	0.446887	+	0.894591i	$0.352533\pi$
$4$	1.00000	0.500000
$5$	−0.496638	−0.222103	−0.111052	−	0.993815i	$-0.535422\pi$
$5$	−0.496638	−0.222103	−0.111052	+	0.993815i	$0.535422\pi$
$6$	1.54806	0.631993
$7$	−1.67357	−0.632551	−0.316275	−	0.948667i	$-0.602432\pi$
$7$	−1.67357	−0.632551	−0.316275	+	0.948667i	$0.602432\pi$
$8$	1.00000	0.353553
$9$	−0.603508	−0.201169
$10$	−0.496638	−0.157051
$11$	0.109322	0.0329620	0.0164810	−	0.999864i	$-0.494754\pi$
$11$	0.109322	0.0329620	0.0164810	+	0.999864i	$0.494754\pi$
$12$	1.54806	0.446887
$13$	2.67921	0.743080	0.371540	−	0.928417i	$-0.378830\pi$
$13$	2.67921	0.743080	0.371540	+	0.928417i	$0.378830\pi$
$14$	−1.67357	−0.447281
$15$	−0.768826	−0.198510
$16$	1.00000	0.250000
$17$	−6.97252	−1.69108	−0.845542	−	0.533909i	$-0.820722\pi$
$17$	−6.97252	−1.69108	−0.845542	+	0.533909i	$0.820722\pi$
$18$	−0.603508	−0.142248
$19$	−3.72371	−0.854277	−0.427139	−	0.904186i	$-0.640478\pi$
$19$	−3.72371	−0.854277	−0.427139	+	0.904186i	$0.640478\pi$
$20$	−0.496638	−0.111052
$21$	−2.59079	−0.565357
$22$	0.109322	0.0233076
$23$	3.09611	0.645584	0.322792	−	0.946470i	$-0.395379\pi$
$23$	3.09611	0.645584	0.322792	+	0.946470i	$0.395379\pi$
$24$	1.54806	0.315997
$25$	−4.75335	−0.950670
$26$	2.67921	0.525437
$27$	−5.57845	−1.07357
$28$	−1.67357	−0.316275
$29$	5.48849	1.01919	0.509593	−	0.860416i	$-0.329796\pi$
$29$	5.48849	1.01919	0.509593	+	0.860416i	$0.329796\pi$
$30$	−0.768826	−0.140368
$31$	6.47208	1.16242	0.581210	−	0.813754i	$-0.302580\pi$
$31$	6.47208	1.16242	0.581210	+	0.813754i	$0.302580\pi$
$32$	1.00000	0.176777
$33$	0.169238	0.0294605
$34$	−6.97252	−1.19578
$35$	0.831160	0.140492
$36$	−0.603508	−0.100585
$37$	−3.64003	−0.598417	−0.299208	−	0.954188i	$-0.596723\pi$
$37$	−3.64003	−0.598417	−0.299208	+	0.954188i	$0.596723\pi$
$38$	−3.72371	−0.604065
$39$	4.14758	0.664145
$40$	−0.496638	−0.0785254
$41$	−1.82844	−0.285554	−0.142777	−	0.989755i	$-0.545603\pi$
$41$	−1.82844	−0.285554	−0.142777	+	0.989755i	$0.545603\pi$
$42$	−2.59079	−0.399768
$43$	6.81985	1.04002	0.520008	−	0.854161i	$-0.325929\pi$
$43$	6.81985	1.04002	0.520008	+	0.854161i	$0.325929\pi$
$44$	0.109322	0.0164810
$45$	0.299725	0.0446804
$46$	3.09611	0.456497
$47$	−7.60435	−1.10921	−0.554604	−	0.832114i	$-0.687130\pi$
$47$	−7.60435	−1.10921	−0.554604	+	0.832114i	$0.687130\pi$
$48$	1.54806	0.223443
$49$	−4.19916	−0.599879
$50$	−4.75335	−0.672225
$51$	−10.7939	−1.51145
$52$	2.67921	0.371540
$53$	13.3705	1.83658	0.918288	−	0.395914i	$-0.129572\pi$
$53$	13.3705	1.83658	0.918288	+	0.395914i	$0.129572\pi$
$54$	−5.57845	−0.759131
$55$	−0.0542937	−0.00732097
$56$	−1.67357	−0.223641
$57$	−5.76453	−0.763530
$58$	5.48849	0.720673
$59$	−5.16570	−0.672516	−0.336258	−	0.941770i	$-0.609161\pi$
$59$	−5.16570	−0.672516	−0.336258	+	0.941770i	$0.609161\pi$
$60$	−0.768826	−0.0992551
$61$	−9.27004	−1.18691	−0.593454	−	0.804868i	$-0.702236\pi$
$61$	−9.27004	−1.18691	−0.593454	+	0.804868i	$0.702236\pi$
$62$	6.47208	0.821954
$63$	1.01001	0.127250
$64$	1.00000	0.125000
$65$	−1.33060	−0.165041
$66$	0.169238	0.0208317
$67$	−3.78847	−0.462835	−0.231417	−	0.972855i	$-0.574336\pi$
$67$	−3.78847	−0.462835	−0.231417	+	0.972855i	$0.574336\pi$
$68$	−6.97252	−0.845542
$69$	4.79297	0.577006
$70$	0.831160	0.0993427
$71$	−2.99509	−0.355452	−0.177726	−	0.984080i	$-0.556874\pi$
$71$	−2.99509	−0.355452	−0.177726	+	0.984080i	$0.556874\pi$
$72$	−0.603508	−0.0711240
$73$	−2.54494	−0.297862	−0.148931	−	0.988848i	$-0.547583\pi$
$73$	−2.54494	−0.297862	−0.148931	+	0.988848i	$0.547583\pi$
$74$	−3.64003	−0.423145
$75$	−7.35848	−0.849684
$76$	−3.72371	−0.427139
$77$	−0.182959	−0.0208501
$78$	4.14758	0.469621
$79$	−4.23339	−0.476293	−0.238147	−	0.971229i	$-0.576540\pi$
$79$	−4.23339	−0.476293	−0.238147	+	0.971229i	$0.576540\pi$
$80$	−0.496638	−0.0555259
$81$	−6.82526	−0.758362
$82$	−1.82844	−0.201917
$83$	−12.6474	−1.38823	−0.694117	−	0.719862i	$-0.744205\pi$
$83$	−12.6474	−1.38823	−0.694117	+	0.719862i	$0.744205\pi$
$84$	−2.59079	−0.282679
$85$	3.46282	0.375596
$86$	6.81985	0.735403
$87$	8.49651	0.910921
$88$	0.109322	0.0116538
$89$	−17.2754	−1.83119	−0.915597	−	0.402097i	$-0.868281\pi$
$89$	−17.2754	−1.83119	−0.915597	+	0.402097i	$0.868281\pi$
$90$	0.299725	0.0315938
$91$	−4.48386	−0.470036
$92$	3.09611	0.322792
$93$	10.0192	1.03894
$94$	−7.60435	−0.784329
$95$	1.84934	0.189738
$96$	1.54806	0.157998
$97$	13.0089	1.32085	0.660427	−	0.750890i	$-0.270375\pi$
$97$	13.0089	1.32085	0.660427	+	0.750890i	$0.270375\pi$
$98$	−4.19916	−0.424179
$99$	−0.0659769	−0.00663093
$100$	−4.75335	−0.475335
$101$	−2.49112	−0.247876	−0.123938	−	0.992290i	$-0.539552\pi$
$101$	−2.49112	−0.247876	−0.123938	+	0.992290i	$0.539552\pi$
$102$	−10.7939	−1.06875
$103$	−5.60126	−0.551909	−0.275954	−	0.961171i	$-0.588994\pi$
$103$	−5.60126	−0.551909	−0.275954	+	0.961171i	$0.588994\pi$
$104$	2.67921	0.262718
$105$	1.28669	0.125568
$106$	13.3705	1.29865
$107$	−1.11608	−0.107896	−0.0539478	−	0.998544i	$-0.517180\pi$
$107$	−1.11608	−0.107896	−0.0539478	+	0.998544i	$0.517180\pi$
$108$	−5.57845	−0.536787
$109$	−13.1380	−1.25839	−0.629194	−	0.777249i	$-0.716615\pi$
$109$	−13.1380	−1.25839	−0.629194	+	0.777249i	$0.716615\pi$
$110$	−0.0542937	−0.00517671
$111$	−5.63499	−0.534849
$112$	−1.67357	−0.158138
$113$	−12.7506	−1.19948	−0.599739	−	0.800196i	$-0.704729\pi$
$113$	−12.7506	−1.19948	−0.599739	+	0.800196i	$0.704729\pi$
$114$	−5.76453	−0.539897
$115$	−1.53765	−0.143386
$116$	5.48849	0.509593
$117$	−1.61692	−0.149485
$118$	−5.16570	−0.475541
$119$	11.6690	1.06970
$120$	−0.768826	−0.0701839
$121$	−10.9880	−0.998914
$122$	−9.27004	−0.839270
$123$	−2.83054	−0.255221
$124$	6.47208	0.581210
$125$	4.84389	0.433251
$126$	1.01001	0.0899792
$127$	−6.92030	−0.614077	−0.307039	−	0.951697i	$-0.599338\pi$
$127$	−6.92030	−0.614077	−0.307039	+	0.951697i	$0.599338\pi$
$128$	1.00000	0.0883883
$129$	10.5575	0.929539
$130$	−1.33060	−0.116701
$131$	−16.0279	−1.40036	−0.700181	−	0.713966i	$-0.746897\pi$
$131$	−16.0279	−1.40036	−0.700181	+	0.713966i	$0.746897\pi$
$132$	0.169238	0.0147303
$133$	6.23190	0.540374
$134$	−3.78847	−0.327274
$135$	2.77047	0.238444
$136$	−6.97252	−0.597888
$137$	−3.33659	−0.285064	−0.142532	−	0.989790i	$-0.545524\pi$
$137$	−3.33659	−0.285064	−0.142532	+	0.989790i	$0.545524\pi$
$138$	4.79297	0.408005
$139$	−15.3049	−1.29815	−0.649073	−	0.760726i	$-0.724843\pi$
$139$	−15.3049	−1.29815	−0.649073	+	0.760726i	$0.724843\pi$
$140$	0.831160	0.0702459
$141$	−11.7720	−0.991381
$142$	−2.99509	−0.251343
$143$	0.292898	0.0244934
$144$	−0.603508	−0.0502923
$145$	−2.72579	−0.226365
$146$	−2.54494	−0.210621
$147$	−6.50055	−0.536156
$148$	−3.64003	−0.299208
$149$	−1.45254	−0.118997	−0.0594983	−	0.998228i	$-0.518950\pi$
$149$	−1.45254	−0.118997	−0.0594983	+	0.998228i	$0.518950\pi$
$150$	−7.35848	−0.600817
$151$	−12.6925	−1.03290	−0.516451	−	0.856317i	$-0.672747\pi$
$151$	−12.6925	−1.03290	−0.516451	+	0.856317i	$0.672747\pi$
$152$	−3.72371	−0.302033
$153$	4.20797	0.340194
$154$	−0.182959	−0.0147433
$155$	−3.21428	−0.258177
$156$	4.14758	0.332072
$157$	19.5303	1.55868	0.779342	−	0.626599i	$-0.215553\pi$
$157$	19.5303	1.55868	0.779342	+	0.626599i	$0.215553\pi$
$158$	−4.23339	−0.336790
$159$	20.6983	1.64148
$160$	−0.496638	−0.0392627
$161$	−5.18157	−0.408365
$162$	−6.82526	−0.536243
$163$	−8.32932	−0.652403	−0.326201	−	0.945300i	$-0.605769\pi$
$163$	−8.32932	−0.652403	−0.326201	+	0.945300i	$0.605769\pi$
$164$	−1.82844	−0.142777
$165$	−0.0840500	−0.00654329
$166$	−12.6474	−0.981630
$167$	20.9582	1.62180	0.810898	−	0.585187i	$-0.198979\pi$
$167$	20.9582	1.62180	0.810898	+	0.585187i	$0.198979\pi$
$168$	−2.59079	−0.199884
$169$	−5.82182	−0.447832
$170$	3.46282	0.265586
$171$	2.24729	0.171854
$172$	6.81985	0.520008
$173$	12.8467	0.976716	0.488358	−	0.872643i	$-0.337596\pi$
$173$	12.8467	0.976716	0.488358	+	0.872643i	$0.337596\pi$
$174$	8.49651	0.644119
$175$	7.95508	0.601347
$176$	0.109322	0.00824049
$177$	−7.99681	−0.601077
$178$	−17.2754	−1.29485
$179$	13.1282	0.981249	0.490625	−	0.871371i	$-0.336769\pi$
$179$	13.1282	0.981249	0.490625	+	0.871371i	$0.336769\pi$
$180$	0.299725	0.0223402
$181$	2.96797	0.220607	0.110304	−	0.993898i	$-0.464818\pi$
$181$	2.96797	0.220607	0.110304	+	0.993898i	$0.464818\pi$
$182$	−4.48386	−0.332366
$183$	−14.3506	−1.06083
$184$	3.09611	0.228248
$185$	1.80778	0.132910
$186$	10.0192	0.734641
$187$	−0.762253	−0.0557415
$188$	−7.60435	−0.554604
$189$	9.33594	0.679090
$190$	1.84934	0.134165
$191$	−20.9099	−1.51299	−0.756493	−	0.654002i	$-0.773089\pi$
$191$	−20.9099	−1.51299	−0.756493	+	0.654002i	$0.773089\pi$
$192$	1.54806	0.111722
$193$	23.6283	1.70080	0.850400	−	0.526136i	$-0.176360\pi$
$193$	23.6283	1.70080	0.850400	+	0.526136i	$0.176360\pi$
$194$	13.0089	0.933985
$195$	−2.05985	−0.147509
$196$	−4.19916	−0.299940
$197$	3.82269	0.272355	0.136178	−	0.990684i	$-0.456518\pi$
$197$	3.82269	0.272355	0.136178	+	0.990684i	$0.456518\pi$
$198$	−0.0659769	−0.00468878
$199$	20.3566	1.44304	0.721521	−	0.692392i	$-0.243443\pi$
$199$	20.3566	1.44304	0.721521	+	0.692392i	$0.243443\pi$
$200$	−4.75335	−0.336113
$201$	−5.86478	−0.413670
$202$	−2.49112	−0.175275
$203$	−9.18538	−0.644687
$204$	−10.7939	−0.755723
$205$	0.908073	0.0634226
$206$	−5.60126	−0.390258
$207$	−1.86853	−0.129872
$208$	2.67921	0.185770
$209$	−0.407085	−0.0281587
$210$	1.28669	0.0887898
$211$	−15.9642	−1.09902	−0.549511	−	0.835487i	$-0.685186\pi$
$211$	−15.9642	−1.09902	−0.549511	+	0.835487i	$0.685186\pi$
$212$	13.3705	0.918288
$213$	−4.63659	−0.317694
$214$	−1.11608	−0.0762937
$215$	−3.38700	−0.230991
$216$	−5.57845	−0.379565
$217$	−10.8315	−0.735289
$218$	−13.1380	−0.889814
$219$	−3.93972	−0.266221
$220$	−0.0542937	−0.00366048
$221$	−18.6809	−1.25661
$222$	−5.63499	−0.378195
$223$	21.8240	1.46144	0.730720	−	0.682677i	$-0.239185\pi$
$223$	21.8240	1.46144	0.730720	+	0.682677i	$0.239185\pi$
$224$	−1.67357	−0.111820
$225$	2.86868	0.191246
$226$	−12.7506	−0.848159
$227$	−2.11883	−0.140632	−0.0703158	−	0.997525i	$-0.522401\pi$
$227$	−2.11883	−0.140632	−0.0703158	+	0.997525i	$0.522401\pi$
$228$	−5.76453	−0.381765
$229$	20.2165	1.33595	0.667973	−	0.744185i	$-0.267162\pi$
$229$	20.2165	1.33595	0.667973	+	0.744185i	$0.267162\pi$
$230$	−1.53765	−0.101390
$231$	−0.283232	−0.0186353
$232$	5.48849	0.360337
$233$	−6.60893	−0.432966	−0.216483	−	0.976286i	$-0.569459\pi$
$233$	−6.60893	−0.432966	−0.216483	+	0.976286i	$0.569459\pi$
$234$	−1.61692	−0.105702
$235$	3.77661	0.246359
$236$	−5.16570	−0.336258
$237$	−6.55354	−0.425698
$238$	11.6690	0.756390
$239$	2.18395	0.141268	0.0706339	−	0.997502i	$-0.477498\pi$
$239$	2.18395	0.141268	0.0706339	+	0.997502i	$0.477498\pi$
$240$	−0.768826	−0.0496275
$241$	6.63904	0.427658	0.213829	−	0.976871i	$-0.431406\pi$
$241$	6.63904	0.427658	0.213829	+	0.976871i	$0.431406\pi$
$242$	−10.9880	−0.706339
$243$	6.16943	0.395769
$244$	−9.27004	−0.593454
$245$	2.08546	0.133235
$246$	−2.83054	−0.180468
$247$	−9.97660	−0.634796
$248$	6.47208	0.410977
$249$	−19.5790	−1.24077
$250$	4.84389	0.306354
$251$	19.9834	1.26134	0.630671	−	0.776050i	$-0.282780\pi$
$251$	19.9834	1.26134	0.630671	+	0.776050i	$0.282780\pi$
$252$	1.01001	0.0636249
$253$	0.338475	0.0212797
$254$	−6.92030	−0.434218
$255$	5.36066	0.335697
$256$	1.00000	0.0625000
$257$	11.0335	0.688249	0.344125	−	0.938924i	$-0.388176\pi$
$257$	11.0335	0.688249	0.344125	+	0.938924i	$0.388176\pi$
$258$	10.5575	0.657284
$259$	6.09185	0.378529
$260$	−1.33060	−0.0825203
$261$	−3.31234	−0.205029
$262$	−16.0279	−0.990205
$263$	30.9328	1.90740	0.953698	−	0.300767i	$-0.0972426\pi$
$263$	30.9328	1.90740	0.953698	+	0.300767i	$0.0972426\pi$
$264$	0.169238	0.0104159
$265$	−6.64029	−0.407910
$266$	6.23190	0.382102
$267$	−26.7434	−1.63667
$268$	−3.78847	−0.231417
$269$	17.9567	1.09484	0.547420	−	0.836858i	$-0.315610\pi$
$269$	17.9567	1.09484	0.547420	+	0.836858i	$0.315610\pi$
$270$	2.77047	0.168606
$271$	−4.75874	−0.289073	−0.144536	−	0.989499i	$-0.546169\pi$
$271$	−4.75874	−0.289073	−0.144536	+	0.989499i	$0.546169\pi$
$272$	−6.97252	−0.422771
$273$	−6.94128	−0.420105
$274$	−3.33659	−0.201571
$275$	−0.519648	−0.0313360
$276$	4.79297	0.288503
$277$	−0.748360	−0.0449646	−0.0224823	−	0.999747i	$-0.507157\pi$
$277$	−0.748360	−0.0449646	−0.0224823	+	0.999747i	$0.507157\pi$
$278$	−15.3049	−0.917928
$279$	−3.90595	−0.233843
$280$	0.831160	0.0496713
$281$	2.94930	0.175940	0.0879701	−	0.996123i	$-0.471962\pi$
$281$	2.94930	0.175940	0.0879701	+	0.996123i	$0.471962\pi$
$282$	−11.7720	−0.701012
$283$	1.37351	0.0816467	0.0408233	−	0.999166i	$-0.487002\pi$
$283$	1.37351	0.0816467	0.0408233	+	0.999166i	$0.487002\pi$
$284$	−2.99509	−0.177726
$285$	2.86289	0.169583
$286$	0.292898	0.0173194
$287$	3.06003	0.180628
$288$	−0.603508	−0.0355620
$289$	31.6160	1.85976
$290$	−2.72579	−0.160064
$291$	20.1386	1.18054
$292$	−2.54494	−0.148931
$293$	8.62732	0.504013	0.252007	−	0.967726i	$-0.418910\pi$
$293$	8.62732	0.504013	0.252007	+	0.967726i	$0.418910\pi$
$294$	−6.50055	−0.379120
$295$	2.56548	0.149368
$296$	−3.64003	−0.211572
$297$	−0.609850	−0.0353871
$298$	−1.45254	−0.0841433
$299$	8.29514	0.479720
$300$	−7.35848	−0.424842
$301$	−11.4135	−0.657864
$302$	−12.6925	−0.730372
$303$	−3.85641	−0.221545
$304$	−3.72371	−0.213569
$305$	4.60386	0.263616
$306$	4.20797	0.240553
$307$	8.86029	0.505683	0.252842	−	0.967508i	$-0.418635\pi$
$307$	8.86029	0.505683	0.252842	+	0.967508i	$0.418635\pi$
$308$	−0.182959	−0.0104251
$309$	−8.67109	−0.493281
$310$	−3.21428	−0.182559
$311$	−6.95670	−0.394478	−0.197239	−	0.980355i	$-0.563198\pi$
$311$	−6.95670	−0.394478	−0.197239	+	0.980355i	$0.563198\pi$
$312$	4.14758	0.234811
$313$	−2.32055	−0.131165	−0.0655827	−	0.997847i	$-0.520891\pi$
$313$	−2.32055	−0.131165	−0.0655827	+	0.997847i	$0.520891\pi$
$314$	19.5303	1.10216
$315$	−0.501612	−0.0282626
$316$	−4.23339	−0.238147
$317$	−29.8241	−1.67509	−0.837544	−	0.546369i	$-0.816010\pi$
$317$	−29.8241	−1.67509	−0.837544	+	0.546369i	$0.816010\pi$
$318$	20.6983	1.16070
$319$	0.600015	0.0335944
$320$	−0.496638	−0.0277629
$321$	−1.72776	−0.0964343
$322$	−5.18157	−0.288758
$323$	25.9636	1.44465
$324$	−6.82526	−0.379181
$325$	−12.7352	−0.706424
$326$	−8.32932	−0.461319
$327$	−20.3383	−1.12471
$328$	−1.82844	−0.100959
$329$	12.7264	0.701631
$330$	−0.0840500	−0.00462680
$331$	−17.1180	−0.940890	−0.470445	−	0.882429i	$-0.655907\pi$
$331$	−17.1180	−0.940890	−0.470445	+	0.882429i	$0.655907\pi$
$332$	−12.6474	−0.694117
$333$	2.19678	0.120383
$334$	20.9582	1.14678
$335$	1.88150	0.102797
$336$	−2.59079	−0.141339
$337$	−23.2653	−1.26734	−0.633671	−	0.773602i	$-0.718453\pi$
$337$	−23.2653	−1.26734	−0.633671	+	0.773602i	$0.718453\pi$
$338$	−5.82182	−0.316665
$339$	−19.7388	−1.07206
$340$	3.46282	0.187798
$341$	0.707543	0.0383156
$342$	2.24729	0.121519
$343$	18.7426	1.01201
$344$	6.81985	0.367701
$345$	−2.38037	−0.128155
$346$	12.8467	0.690642
$347$	−19.8891	−1.06771	−0.533853	−	0.845577i	$-0.679256\pi$
$347$	−19.8891	−1.06771	−0.533853	+	0.845577i	$0.679256\pi$
$348$	8.49651	0.455461
$349$	19.0269	1.01848	0.509242	−	0.860623i	$-0.329926\pi$
$349$	19.0269	1.01848	0.509242	+	0.860623i	$0.329926\pi$
$350$	7.95508	0.425217
$351$	−14.9458	−0.797750
$352$	0.109322	0.00582691
$353$	−16.8745	−0.898137	−0.449069	−	0.893497i	$-0.648244\pi$
$353$	−16.8745	−0.898137	−0.449069	+	0.893497i	$0.648244\pi$
$354$	−7.99681	−0.425026
$355$	1.48748	0.0789471
$356$	−17.2754	−0.915597
$357$	18.0643	0.956066
$358$	13.1282	0.693848
$359$	−21.1366	−1.11555	−0.557773	−	0.829994i	$-0.688344\pi$
$359$	−21.1366	−1.11555	−0.557773	+	0.829994i	$0.688344\pi$
$360$	0.299725	0.0157969
$361$	−5.13400	−0.270210
$362$	2.96797	0.155993
$363$	−17.0102	−0.892802
$364$	−4.48386	−0.235018
$365$	1.26391	0.0661563
$366$	−14.3506	−0.750117
$367$	21.0002	1.09620	0.548102	−	0.836412i	$-0.315351\pi$
$367$	21.0002	1.09620	0.548102	+	0.836412i	$0.315351\pi$
$368$	3.09611	0.161396
$369$	1.10348	0.0574447
$370$	1.80778	0.0939819
$371$	−22.3765	−1.16173
$372$	10.0192	0.519470
$373$	−17.4743	−0.904786	−0.452393	−	0.891819i	$-0.649429\pi$
$373$	−17.4743	−0.904786	−0.452393	+	0.891819i	$0.649429\pi$
$374$	−0.762253	−0.0394152
$375$	7.49863	0.387228
$376$	−7.60435	−0.392165
$377$	14.7048	0.757337
$378$	9.33594	0.480189
$379$	−20.5704	−1.05663	−0.528315	−	0.849048i	$-0.677176\pi$
$379$	−20.5704	−1.05663	−0.528315	+	0.849048i	$0.677176\pi$
$380$	1.84934	0.0948690
$381$	−10.7130	−0.548846
$382$	−20.9099	−1.06984
$383$	4.41300	0.225494	0.112747	−	0.993624i	$-0.464035\pi$
$383$	4.41300	0.225494	0.112747	+	0.993624i	$0.464035\pi$
$384$	1.54806	0.0789992
$385$	0.0908645	0.00463089
$386$	23.6283	1.20265
$387$	−4.11583	−0.209219
$388$	13.0089	0.660427
$389$	−3.40758	−0.172771	−0.0863855	−	0.996262i	$-0.527532\pi$
$389$	−3.40758	−0.172771	−0.0863855	+	0.996262i	$0.527532\pi$
$390$	−2.05985	−0.104305
$391$	−21.5877	−1.09174
$392$	−4.19916	−0.212089
$393$	−24.8121	−1.25161
$394$	3.82269	0.192584
$395$	2.10246	0.105786
$396$	−0.0659769	−0.00331547
$397$	−17.8178	−0.894252	−0.447126	−	0.894471i	$-0.647552\pi$
$397$	−17.8178	−0.894252	−0.447126	+	0.894471i	$0.647552\pi$
$398$	20.3566	1.02039
$399$	9.64735	0.482972
$400$	−4.75335	−0.237668
$401$	32.8090	1.63840	0.819202	−	0.573505i	$-0.194417\pi$
$401$	32.8090	1.63840	0.819202	+	0.573505i	$0.194417\pi$
$402$	−5.86478	−0.292509
$403$	17.3401	0.863770
$404$	−2.49112	−0.123938
$405$	3.38968	0.168435
$406$	−9.18538	−0.455863
$407$	−0.397937	−0.0197250
$408$	−10.7939	−0.534377
$409$	28.0993	1.38942	0.694711	−	0.719289i	$-0.255532\pi$
$409$	28.0993	1.38942	0.694711	+	0.719289i	$0.255532\pi$
$410$	0.908073	0.0448465
$411$	−5.16525	−0.254783
$412$	−5.60126	−0.275954
$413$	8.64517	0.425401
$414$	−1.86853	−0.0918331
$415$	6.28119	0.308332
$416$	2.67921	0.131359
$417$	−23.6929	−1.16025
$418$	−0.407085	−0.0199112
$419$	27.4234	1.33972	0.669860	−	0.742487i	$-0.266354\pi$
$419$	27.4234	1.33972	0.669860	+	0.742487i	$0.266354\pi$
$420$	1.28669	0.0627839
$421$	2.73331	0.133213	0.0666067	−	0.997779i	$-0.478783\pi$
$421$	2.73331	0.133213	0.0666067	+	0.997779i	$0.478783\pi$
$422$	−15.9642	−0.777126
$423$	4.58928	0.223139
$424$	13.3705	0.649327
$425$	33.1428	1.60766
$426$	−4.63659	−0.224643
$427$	15.5141	0.750779
$428$	−1.11608	−0.0539478
$429$	0.453424	0.0218915
$430$	−3.38700	−0.163336
$431$	−34.7027	−1.67157	−0.835785	−	0.549056i	$-0.814987\pi$
$431$	−34.7027	−1.67157	−0.835785	+	0.549056i	$0.814987\pi$
$432$	−5.57845	−0.268393
$433$	−15.0807	−0.724730	−0.362365	−	0.932036i	$-0.618031\pi$
$433$	−15.0807	−0.724730	−0.362365	+	0.932036i	$0.618031\pi$
$434$	−10.8315	−0.519928
$435$	−4.21969	−0.202319
$436$	−13.1380	−0.629194
$437$	−11.5290	−0.551508
$438$	−3.93972	−0.188247
$439$	−27.6748	−1.32085	−0.660423	−	0.750893i	$-0.729623\pi$
$439$	−27.6748	−1.32085	−0.660423	+	0.750893i	$0.729623\pi$
$440$	−0.0542937	−0.00258835
$441$	2.53422	0.120677
$442$	−18.6809	−0.888558
$443$	−34.2453	−1.62705	−0.813523	−	0.581533i	$-0.802453\pi$
$443$	−34.2453	−1.62705	−0.813523	+	0.581533i	$0.802453\pi$
$444$	−5.63499	−0.267425
$445$	8.57965	0.406714
$446$	21.8240	1.03339
$447$	−2.24862	−0.106356
$448$	−1.67357	−0.0790689
$449$	−8.89972	−0.420004	−0.210002	−	0.977701i	$-0.567347\pi$
$449$	−8.89972	−0.420004	−0.210002	+	0.977701i	$0.567347\pi$
$450$	2.86868	0.135231
$451$	−0.199890	−0.00941243
$452$	−12.7506	−0.599739
$453$	−19.6488	−0.923180
$454$	−2.11883	−0.0994415
$455$	2.22685	0.104397
$456$	−5.76453	−0.269949
$457$	−22.0721	−1.03249	−0.516245	−	0.856441i	$-0.672671\pi$
$457$	−22.0721	−1.03249	−0.516245	+	0.856441i	$0.672671\pi$
$458$	20.2165	0.944657
$459$	38.8958	1.81550
$460$	−1.53765	−0.0716932
$461$	−33.9867	−1.58292	−0.791459	−	0.611222i	$-0.790678\pi$
$461$	−33.9867	−1.58292	−0.791459	+	0.611222i	$0.790678\pi$
$462$	−0.283232	−0.0131771
$463$	15.0923	0.701397	0.350699	−	0.936488i	$-0.385944\pi$
$463$	15.0923	0.701397	0.350699	+	0.936488i	$0.385944\pi$
$464$	5.48849	0.254797
$465$	−4.97590	−0.230752
$466$	−6.60893	−0.306153
$467$	8.48395	0.392590	0.196295	−	0.980545i	$-0.437109\pi$
$467$	8.48395	0.392590	0.196295	+	0.980545i	$0.437109\pi$
$468$	−1.61692	−0.0747424
$469$	6.34028	0.292767
$470$	3.77661	0.174202
$471$	30.2340	1.39311
$472$	−5.16570	−0.237770
$473$	0.745563	0.0342810
$474$	−6.55354	−0.301014
$475$	17.7001	0.812136
$476$	11.6690	0.534848
$477$	−8.06918	−0.369462
$478$	2.18395	0.0998914
$479$	29.2118	1.33472	0.667361	−	0.744734i	$-0.267424\pi$
$479$	29.2118	1.33472	0.667361	+	0.744734i	$0.267424\pi$
$480$	−0.768826	−0.0350920
$481$	−9.75241	−0.444672
$482$	6.63904	0.302400
$483$	−8.02138	−0.364986
$484$	−10.9880	−0.499457
$485$	−6.46073	−0.293366
$486$	6.16943	0.279851
$487$	2.06779	0.0937004	0.0468502	−	0.998902i	$-0.485082\pi$
$487$	2.06779	0.0937004	0.0468502	+	0.998902i	$0.485082\pi$
$488$	−9.27004	−0.419635
$489$	−12.8943	−0.583100
$490$	2.08546	0.0942116
$491$	27.4299	1.23790	0.618948	−	0.785432i	$-0.287559\pi$
$491$	27.4299	1.23790	0.618948	+	0.785432i	$0.287559\pi$
$492$	−2.83054	−0.127610
$493$	−38.2686	−1.72353
$494$	−9.97660	−0.448869
$495$	0.0327667	0.00147275
$496$	6.47208	0.290605
$497$	5.01250	0.224842
$498$	−19.5790	−0.877355
$499$	−15.8123	−0.707856	−0.353928	−	0.935273i	$-0.615154\pi$
$499$	−15.8123	−0.707856	−0.353928	+	0.935273i	$0.615154\pi$
$500$	4.84389	0.216625
$501$	32.4446	1.44952
$502$	19.9834	0.891903
$503$	42.2236	1.88266	0.941328	−	0.337493i	$-0.109579\pi$
$503$	42.2236	1.88266	0.941328	+	0.337493i	$0.109579\pi$
$504$	1.01001	0.0449896
$505$	1.23719	0.0550541
$506$	0.338475	0.0150470
$507$	−9.01254	−0.400261
$508$	−6.92030	−0.307039
$509$	3.87671	0.171832	0.0859161	−	0.996302i	$-0.472618\pi$
$509$	3.87671	0.171832	0.0859161	+	0.996302i	$0.472618\pi$
$510$	5.36066	0.237374
$511$	4.25914	0.188413
$512$	1.00000	0.0441942
$513$	20.7725	0.917129
$514$	11.0335	0.486666
$515$	2.78180	0.122581
$516$	10.5575	0.464770
$517$	−0.831327	−0.0365617
$518$	6.09185	0.267661
$519$	19.8875	0.872962
$520$	−1.33060	−0.0583507
$521$	29.0993	1.27486	0.637431	−	0.770507i	$-0.279997\pi$
$521$	29.0993	1.27486	0.637431	+	0.770507i	$0.279997\pi$
$522$	−3.31234	−0.144977
$523$	−16.4221	−0.718087	−0.359044	−	0.933321i	$-0.616897\pi$
$523$	−16.4221	−0.718087	−0.359044	+	0.933321i	$0.616897\pi$
$524$	−16.0279	−0.700181
$525$	12.3149	0.537468
$526$	30.9328	1.34873
$527$	−45.1267	−1.96575
$528$	0.169238	0.00736513
$529$	−13.4141	−0.583221
$530$	−6.64029	−0.288436
$531$	3.11754	0.135290
$532$	6.23190	0.270187
$533$	−4.89878	−0.212190
$534$	−26.7434	−1.15730
$535$	0.554289	0.0239640
$536$	−3.78847	−0.163637
$537$	20.3233	0.877015
$538$	17.9567	0.774169
$539$	−0.459062	−0.0197732
$540$	2.77047	0.119222
$541$	−18.5887	−0.799190	−0.399595	−	0.916692i	$-0.630849\pi$
$541$	−18.5887	−0.799190	−0.399595	+	0.916692i	$0.630849\pi$
$542$	−4.75874	−0.204405
$543$	4.59460	0.197173
$544$	−6.97252	−0.298944
$545$	6.52481	0.279492
$546$	−6.94128	−0.297059
$547$	7.15417	0.305890	0.152945	−	0.988235i	$-0.451124\pi$
$547$	7.15417	0.305890	0.152945	+	0.988235i	$0.451124\pi$
$548$	−3.33659	−0.142532
$549$	5.59454	0.238769
$550$	−0.519648	−0.0221579
$551$	−20.4375	−0.870668
$552$	4.79297	0.204002
$553$	7.08488	0.301280
$554$	−0.748360	−0.0317948
$555$	2.79855	0.118792
$556$	−15.3049	−0.649073
$557$	34.3462	1.45530	0.727648	−	0.685951i	$-0.240613\pi$
$557$	34.3462	1.45530	0.727648	+	0.685951i	$0.240613\pi$
$558$	−3.90595	−0.165352
$559$	18.2718	0.772815
$560$	0.831160	0.0351229
$561$	−1.18001	−0.0498202
$562$	2.94930	0.124409
$563$	27.1861	1.14576	0.572879	−	0.819640i	$-0.305826\pi$
$563$	27.1861	1.14576	0.572879	+	0.819640i	$0.305826\pi$
$564$	−11.7720	−0.495691
$565$	6.33245	0.266408
$566$	1.37351	0.0577329
$567$	11.4226	0.479702
$568$	−2.99509	−0.125671
$569$	18.6728	0.782803	0.391401	−	0.920220i	$-0.371990\pi$
$569$	18.6728	0.782803	0.391401	+	0.920220i	$0.371990\pi$
$570$	2.86289	0.119913
$571$	−0.599376	−0.0250831	−0.0125415	−	0.999921i	$-0.503992\pi$
$571$	−0.599376	−0.0250831	−0.0125415	+	0.999921i	$0.503992\pi$
$572$	0.292898	0.0122467
$573$	−32.3698	−1.35227
$574$	3.06003	0.127723
$575$	−14.7169	−0.613737
$576$	−0.603508	−0.0251461
$577$	15.5598	0.647763	0.323881	−	0.946098i	$-0.395012\pi$
$577$	15.5598	0.647763	0.323881	+	0.946098i	$0.395012\pi$
$578$	31.6160	1.31505
$579$	36.5780	1.52013
$580$	−2.72579	−0.113182
$581$	21.1664	0.878129
$582$	20.1386	0.834771
$583$	1.46169	0.0605371
$584$	−2.54494	−0.105310
$585$	0.803027	0.0332011
$586$	8.62732	0.356391
$587$	13.9406	0.575389	0.287695	−	0.957722i	$-0.407111\pi$
$587$	13.9406	0.575389	0.287695	+	0.957722i	$0.407111\pi$
$588$	−6.50055	−0.268078
$589$	−24.1001	−0.993028
$590$	2.56548	0.105619
$591$	5.91775	0.243424
$592$	−3.64003	−0.149604
$593$	−32.2792	−1.32555	−0.662775	−	0.748819i	$-0.730621\pi$
$593$	−32.2792	−1.32555	−0.662775	+	0.748819i	$0.730621\pi$
$594$	−0.609850	−0.0250224
$595$	−5.79528	−0.237583
$596$	−1.45254	−0.0594983
$597$	31.5133	1.28975
$598$	8.29514	0.339214
$599$	29.0687	1.18772	0.593858	−	0.804570i	$-0.297604\pi$
$599$	29.0687	1.18772	0.593858	+	0.804570i	$0.297604\pi$
$600$	−7.35848	−0.300409
$601$	−34.9794	−1.42684	−0.713420	−	0.700737i	$-0.752855\pi$
$601$	−34.9794	−1.42684	−0.713420	+	0.700737i	$0.752855\pi$
$602$	−11.4135	−0.465180
$603$	2.28637	0.0931081
$604$	−12.6925	−0.516451
$605$	5.45709	0.221862
$606$	−3.85641	−0.156656
$607$	17.1376	0.695592	0.347796	−	0.937570i	$-0.386930\pi$
$607$	17.1376	0.695592	0.347796	+	0.937570i	$0.386930\pi$
$608$	−3.72371	−0.151016
$609$	−14.2195	−0.576204
$610$	4.60386	0.186405
$611$	−20.3737	−0.824231
$612$	4.20797	0.170097
$613$	14.3300	0.578784	0.289392	−	0.957211i	$-0.406547\pi$
$613$	14.3300	0.578784	0.289392	+	0.957211i	$0.406547\pi$
$614$	8.86029	0.357572
$615$	1.40575	0.0566854
$616$	−0.182959	−0.00737163
$617$	−43.7250	−1.76030	−0.880152	−	0.474692i	$-0.842559\pi$
$617$	−43.7250	−1.76030	−0.880152	+	0.474692i	$0.842559\pi$
$618$	−8.67109	−0.348803
$619$	0.483327	0.0194265	0.00971327	−	0.999953i	$-0.496908\pi$
$619$	0.483327	0.0194265	0.00971327	+	0.999953i	$0.496908\pi$
$620$	−3.21428	−0.129089
$621$	−17.2715	−0.693082
$622$	−6.95670	−0.278938
$623$	28.9117	1.15832
$624$	4.14758	0.166036
$625$	21.3611	0.854444
$626$	−2.32055	−0.0927480
$627$	−0.630193	−0.0251675
$628$	19.5303	0.779342
$629$	25.3802	1.01197
$630$	−0.501612	−0.0199847
$631$	42.3618	1.68640	0.843198	−	0.537604i	$-0.180670\pi$
$631$	42.3618	1.68640	0.843198	+	0.537604i	$0.180670\pi$
$632$	−4.23339	−0.168395
$633$	−24.7136	−0.982277
$634$	−29.8241	−1.18447
$635$	3.43689	0.136389
$636$	20.6983	0.820741
$637$	−11.2504	−0.445758
$638$	0.600015	0.0237548
$639$	1.80756	0.0715060
$640$	−0.496638	−0.0196314
$641$	18.6705	0.737439	0.368719	−	0.929541i	$-0.379796\pi$
$641$	18.6705	0.737439	0.368719	+	0.929541i	$0.379796\pi$
$642$	−1.72776	−0.0681893
$643$	−43.4246	−1.71250	−0.856249	−	0.516564i	$-0.827211\pi$
$643$	−43.4246	−1.71250	−0.856249	+	0.516564i	$0.827211\pi$
$644$	−5.18157	−0.204182
$645$	−5.24328	−0.206454
$646$	25.9636	1.02152
$647$	−5.92796	−0.233052	−0.116526	−	0.993188i	$-0.537176\pi$
$647$	−5.92796	−0.233052	−0.116526	+	0.993188i	$0.537176\pi$
$648$	−6.82526	−0.268121
$649$	−0.564727	−0.0221675
$650$	−12.7352	−0.499517
$651$	−16.7678	−0.657182
$652$	−8.32932	−0.326201
$653$	27.5601	1.07851	0.539256	−	0.842142i	$-0.318706\pi$
$653$	27.5601	1.07851	0.539256	+	0.842142i	$0.318706\pi$
$654$	−20.3383	−0.795292
$655$	7.96006	0.311025
$656$	−1.82844	−0.0713886
$657$	1.53589	0.0599207
$658$	12.7264	0.496128
$659$	23.5830	0.918663	0.459331	−	0.888265i	$-0.348089\pi$
$659$	23.5830	0.918663	0.459331	+	0.888265i	$0.348089\pi$
$660$	−0.0840500	−0.00327164
$661$	16.9239	0.658263	0.329131	−	0.944284i	$-0.393244\pi$
$661$	16.9239	0.658263	0.329131	+	0.944284i	$0.393244\pi$
$662$	−17.1180	−0.665310
$663$	−28.9191	−1.12312
$664$	−12.6474	−0.490815
$665$	−3.09500	−0.120019
$666$	2.19678	0.0851237
$667$	16.9930	0.657970
$668$	20.9582	0.810898
$669$	33.7848	1.30620
$670$	1.88150	0.0726886
$671$	−1.01342	−0.0391228
$672$	−2.59079	−0.0999420
$673$	−25.0958	−0.967373	−0.483687	−	0.875241i	$-0.660703\pi$
$673$	−25.0958	−0.967373	−0.483687	+	0.875241i	$0.660703\pi$
$674$	−23.2653	−0.896147
$675$	26.5163	1.02061
$676$	−5.82182	−0.223916
$677$	18.1366	0.697048	0.348524	−	0.937300i	$-0.386683\pi$
$677$	18.1366	0.697048	0.348524	+	0.937300i	$0.386683\pi$
$678$	−19.7388	−0.758062
$679$	−21.7714	−0.835508
$680$	3.46282	0.132793
$681$	−3.28007	−0.125693
$682$	0.707543	0.0270932
$683$	−29.5800	−1.13185	−0.565924	−	0.824458i	$-0.691480\pi$
$683$	−29.5800	−1.13185	−0.565924	+	0.824458i	$0.691480\pi$
$684$	2.24729	0.0859271
$685$	1.65708	0.0633138
$686$	18.7426	0.715596
$687$	31.2964	1.19403
$688$	6.81985	0.260004
$689$	35.8223	1.36472
$690$	−2.38037	−0.0906193
$691$	44.1793	1.68066	0.840331	−	0.542074i	$-0.182361\pi$
$691$	44.1793	1.68066	0.840331	+	0.542074i	$0.182361\pi$
$692$	12.8467	0.488358
$693$	0.110417	0.00419440
$694$	−19.8891	−0.754982
$695$	7.60101	0.288323
$696$	8.49651	0.322059
$697$	12.7488	0.482896
$698$	19.0269	0.720178
$699$	−10.2310	−0.386973
$700$	7.95508	0.300674
$701$	0.122199	0.00461539	0.00230770	−	0.999997i	$-0.499265\pi$
$701$	0.122199	0.00461539	0.00230770	+	0.999997i	$0.499265\pi$
$702$	−14.9458	−0.564095
$703$	13.5544	0.511214
$704$	0.109322	0.00412025
$705$	5.84643	0.220189
$706$	−16.8745	−0.635079
$707$	4.16907	0.156794
$708$	−7.99681	−0.300539
$709$	10.1571	0.381459	0.190730	−	0.981643i	$-0.438915\pi$
$709$	10.1571	0.381459	0.190730	+	0.981643i	$0.438915\pi$
$710$	1.48748	0.0558240
$711$	2.55488	0.0958155
$712$	−17.2754	−0.647425
$713$	20.0383	0.750439
$714$	18.0643	0.676041
$715$	−0.145464	−0.00544006
$716$	13.1282	0.490625
$717$	3.38088	0.126261
$718$	−21.1366	−0.788810
$719$	−17.4014	−0.648963	−0.324481	−	0.945892i	$-0.605190\pi$
$719$	−17.4014	−0.648963	−0.324481	+	0.945892i	$0.605190\pi$
$720$	0.299725	0.0111701
$721$	9.37412	0.349110
$722$	−5.13400	−0.191068
$723$	10.2776	0.382229
$724$	2.96797	0.110304
$725$	−26.0887	−0.968910
$726$	−17.0102	−0.631307
$727$	15.2520	0.565667	0.282833	−	0.959169i	$-0.408726\pi$
$727$	15.2520	0.565667	0.282833	+	0.959169i	$0.408726\pi$
$728$	−4.48386	−0.166183
$729$	30.0264	1.11209
$730$	1.26391	0.0467795
$731$	−47.5515	−1.75876
$732$	−14.3506	−0.530413
$733$	29.0075	1.07142	0.535708	−	0.844403i	$-0.320045\pi$
$733$	29.0075	1.07142	0.535708	+	0.844403i	$0.320045\pi$
$734$	21.0002	0.775133
$735$	3.22842	0.119082
$736$	3.09611	0.114124
$737$	−0.414165	−0.0152560
$738$	1.10348	0.0406195
$739$	−47.5983	−1.75093	−0.875465	−	0.483281i	$-0.839445\pi$
$739$	−47.5983	−1.75093	−0.875465	+	0.483281i	$0.839445\pi$
$740$	1.80778	0.0664552
$741$	−15.4444	−0.567364
$742$	−22.3765	−0.821465
$743$	18.2885	0.670939	0.335469	−	0.942051i	$-0.391105\pi$
$743$	18.2885	0.670939	0.335469	+	0.942051i	$0.391105\pi$
$744$	10.0192	0.367320
$745$	0.721386	0.0264296
$746$	−17.4743	−0.639780
$747$	7.63281	0.279270
$748$	−0.762253	−0.0278707
$749$	1.86784	0.0682495
$750$	7.49863	0.273811
$751$	44.6304	1.62858	0.814292	−	0.580455i	$-0.197125\pi$
$751$	44.6304	1.62858	0.814292	+	0.580455i	$0.197125\pi$
$752$	−7.60435	−0.277302
$753$	30.9355	1.12735
$754$	14.7048	0.535518
$755$	6.30359	0.229411
$756$	9.33594	0.339545
$757$	16.0040	0.581676	0.290838	−	0.956772i	$-0.406066\pi$
$757$	16.0040	0.581676	0.290838	+	0.956772i	$0.406066\pi$
$758$	−20.5704	−0.747151
$759$	0.523979	0.0190193
$760$	1.84934	0.0670825
$761$	10.3963	0.376866	0.188433	−	0.982086i	$-0.439659\pi$
$761$	10.3963	0.376866	0.188433	+	0.982086i	$0.439659\pi$
$762$	−10.7130	−0.388093
$763$	21.9873	0.795994
$764$	−20.9099	−0.756493
$765$	−2.08984	−0.0755582
$766$	4.41300	0.159448
$767$	−13.8400	−0.499733
$768$	1.54806	0.0558608
$769$	35.7964	1.29085	0.645426	−	0.763823i	$-0.276680\pi$
$769$	35.7964	1.29085	0.645426	+	0.763823i	$0.276680\pi$
$770$	0.0908645	0.00327453
$771$	17.0805	0.615139
$772$	23.6283	0.850400
$773$	−37.6493	−1.35415	−0.677075	−	0.735914i	$-0.736753\pi$
$773$	−37.6493	−1.35415	−0.677075	+	0.735914i	$0.736753\pi$
$774$	−4.11583	−0.147940
$775$	−30.7640	−1.10508
$776$	13.0089	0.466993
$777$	9.43056	0.338319
$778$	−3.40758	−0.122167
$779$	6.80858	0.243943
$780$	−2.05985	−0.0737544
$781$	−0.327431	−0.0117164
$782$	−21.5877	−0.771974
$783$	−30.6172	−1.09417
$784$	−4.19916	−0.149970
$785$	−9.69947	−0.346189
$786$	−24.8121	−0.885019
$787$	−1.79156	−0.0638623	−0.0319312	−	0.999490i	$-0.510166\pi$
$787$	−1.79156	−0.0638623	−0.0319312	+	0.999490i	$0.510166\pi$
$788$	3.82269	0.136178
$789$	47.8858	1.70478
$790$	2.10246	0.0748022
$791$	21.3391	0.758731
$792$	−0.0659769	−0.00234439
$793$	−24.8364	−0.881967
$794$	−17.8178	−0.632332
$795$	−10.2796	−0.364579
$796$	20.3566	0.721521
$797$	−22.6692	−0.802983	−0.401492	−	0.915863i	$-0.631508\pi$
$797$	−22.6692	−0.802983	−0.401492	+	0.915863i	$0.631508\pi$
$798$	9.64735	0.341513
$799$	53.0215	1.87577
$800$	−4.75335	−0.168056
$801$	10.4259	0.368380
$802$	32.8090	1.15853
$803$	−0.278219	−0.00981813
$804$	−5.86478	−0.206835
$805$	2.57337	0.0906992
$806$	17.3401	0.610778
$807$	27.7981	0.978539
$808$	−2.49112	−0.0876374
$809$	51.4373	1.80844	0.904220	−	0.427068i	$-0.140453\pi$
$809$	51.4373	1.80844	0.904220	+	0.427068i	$0.140453\pi$
$810$	3.38968	0.119101
$811$	20.5757	0.722509	0.361255	−	0.932467i	$-0.382349\pi$
$811$	20.5757	0.722509	0.361255	+	0.932467i	$0.382349\pi$
$812$	−9.18538	−0.322344
$813$	−7.36681	−0.258365
$814$	−0.397937	−0.0139477
$815$	4.13666	0.144901
$816$	−10.7939	−0.377861
$817$	−25.3951	−0.888463
$818$	28.0993	0.982470
$819$	2.70604	0.0945567
$820$	0.908073	0.0317113
$821$	−18.4576	−0.644174	−0.322087	−	0.946710i	$-0.604384\pi$
$821$	−18.4576	−0.644174	−0.322087	+	0.946710i	$0.604384\pi$
$822$	−5.16525	−0.180159
$823$	−26.9930	−0.940916	−0.470458	−	0.882422i	$-0.655911\pi$
$823$	−26.9930	−0.940916	−0.470458	+	0.882422i	$0.655911\pi$
$824$	−5.60126	−0.195129
$825$	−0.804447	−0.0280072
$826$	8.64517	0.300804
$827$	−2.83173	−0.0984688	−0.0492344	−	0.998787i	$-0.515678\pi$
$827$	−2.83173	−0.0984688	−0.0492344	+	0.998787i	$0.515678\pi$
$828$	−1.86853	−0.0649358
$829$	−27.8464	−0.967144	−0.483572	−	0.875304i	$-0.660661\pi$
$829$	−27.8464	−0.967144	−0.483572	+	0.875304i	$0.660661\pi$
$830$	6.28119	0.218023
$831$	−1.15851	−0.0401881
$832$	2.67921	0.0928850
$833$	29.2787	1.01445
$834$	−23.6929	−0.820419
$835$	−10.4087	−0.360207
$836$	−0.407085	−0.0140793
$837$	−36.1041	−1.24794
$838$	27.4234	0.947326
$839$	−43.3979	−1.49826	−0.749131	−	0.662422i	$-0.769529\pi$
$839$	−43.3979	−1.49826	−0.749131	+	0.662422i	$0.769529\pi$
$840$	1.28669	0.0443949
$841$	1.12347	0.0387405
$842$	2.73331	0.0941961
$843$	4.56569	0.157251
$844$	−15.9642	−0.549511
$845$	2.89134	0.0994651
$846$	4.58928	0.157783
$847$	18.3893	0.631864
$848$	13.3705	0.459144
$849$	2.12628	0.0729736
$850$	33.1428	1.13679
$851$	−11.2699	−0.386328
$852$	−4.63659	−0.158847
$853$	2.26611	0.0775900	0.0387950	−	0.999247i	$-0.487648\pi$
$853$	2.26611	0.0775900	0.0387950	+	0.999247i	$0.487648\pi$
$854$	15.5141	0.530881
$855$	−1.11609	−0.0381694
$856$	−1.11608	−0.0381469
$857$	−34.9512	−1.19391	−0.596956	−	0.802274i	$-0.703623\pi$
$857$	−34.9512	−1.19391	−0.596956	+	0.802274i	$0.703623\pi$
$858$	0.453424	0.0154796
$859$	10.0928	0.344361	0.172181	−	0.985065i	$-0.444919\pi$
$859$	10.0928	0.344361	0.172181	+	0.985065i	$0.444919\pi$
$860$	−3.38700	−0.115496
$861$	4.73711	0.161440
$862$	−34.7027	−1.18198
$863$	−30.6726	−1.04411	−0.522054	−	0.852913i	$-0.674834\pi$
$863$	−30.6726	−1.04411	−0.522054	+	0.852913i	$0.674834\pi$
$864$	−5.57845	−0.189783
$865$	−6.38016	−0.216932
$866$	−15.0807	−0.512462
$867$	48.9435	1.66221
$868$	−10.8315	−0.367645
$869$	−0.462804	−0.0156996
$870$	−4.21969	−0.143061
$871$	−10.1501	−0.343923
$872$	−13.1380	−0.444907
$873$	−7.85098	−0.265715
$874$	−11.5290	−0.389975
$875$	−8.10660	−0.274053
$876$	−3.93972	−0.133111
$877$	4.26708	0.144089	0.0720446	−	0.997401i	$-0.477048\pi$
$877$	4.26708	0.144089	0.0720446	+	0.997401i	$0.477048\pi$
$878$	−27.6748	−0.933980
$879$	13.3556	0.450474
$880$	−0.0542937	−0.00183024
$881$	−27.8569	−0.938524	−0.469262	−	0.883059i	$-0.655480\pi$
$881$	−27.8569	−0.938524	−0.469262	+	0.883059i	$0.655480\pi$
$882$	2.53422	0.0853317
$883$	−29.7137	−0.999946	−0.499973	−	0.866041i	$-0.666657\pi$
$883$	−29.7137	−0.999946	−0.499973	+	0.866041i	$0.666657\pi$
$884$	−18.6809	−0.628305
$885$	3.97152	0.133501
$886$	−34.2453	−1.15049
$887$	−24.6191	−0.826628	−0.413314	−	0.910589i	$-0.635629\pi$
$887$	−24.6191	−0.826628	−0.413314	+	0.910589i	$0.635629\pi$
$888$	−5.63499	−0.189098
$889$	11.5816	0.388435
$890$	8.57965	0.287591
$891$	−0.746154	−0.0249971
$892$	21.8240	0.730720
$893$	28.3164	0.947572
$894$	−2.24862	−0.0752050
$895$	−6.51998	−0.217939
$896$	−1.67357	−0.0559101
$897$	12.8414	0.428761
$898$	−8.89972	−0.296988
$899$	35.5219	1.18472
$900$	2.86868	0.0956228
$901$	−93.2258	−3.10580
$902$	−0.199890	−0.00665559
$903$	−17.6688	−0.587981
$904$	−12.7506	−0.424080
$905$	−1.47401	−0.0489977
$906$	−19.6488	−0.652787
$907$	−48.1803	−1.59980	−0.799899	−	0.600134i	$-0.795114\pi$
$907$	−48.1803	−1.59980	−0.799899	+	0.600134i	$0.795114\pi$
$908$	−2.11883	−0.0703158
$909$	1.50341	0.0498650
$910$	2.22685	0.0738195
$911$	−9.66772	−0.320306	−0.160153	−	0.987092i	$-0.551199\pi$
$911$	−9.66772	−0.320306	−0.160153	+	0.987092i	$0.551199\pi$
$912$	−5.76453	−0.190883
$913$	−1.38265	−0.0457590
$914$	−22.0721	−0.730081
$915$	7.12706	0.235613
$916$	20.2165	0.667973
$917$	26.8238	0.885800
$918$	38.8958	1.28375
$919$	−21.7629	−0.717891	−0.358946	−	0.933358i	$-0.616864\pi$
$919$	−21.7629	−0.717891	−0.358946	+	0.933358i	$0.616864\pi$
$920$	−1.53765	−0.0506948
$921$	13.7163	0.451966
$922$	−33.9867	−1.11929
$923$	−8.02449	−0.264129
$924$	−0.283232	−0.00931764
$925$	17.3023	0.568897
$926$	15.0923	0.495963
$927$	3.38040	0.111027
$928$	5.48849	0.180168
$929$	−32.6767	−1.07209	−0.536044	−	0.844190i	$-0.680082\pi$
$929$	−32.6767	−1.07209	−0.536044	+	0.844190i	$0.680082\pi$
$930$	−4.97590	−0.163166
$931$	15.6364	0.512463
$932$	−6.60893	−0.216483
$933$	−10.7694	−0.352574
$934$	8.48395	0.277603
$935$	0.378564	0.0123804
$936$	−1.61692	−0.0528508
$937$	−12.5418	−0.409723	−0.204861	−	0.978791i	$-0.565674\pi$
$937$	−12.5418	−0.409723	−0.204861	+	0.978791i	$0.565674\pi$
$938$	6.34028	0.207017
$939$	−3.59236	−0.117232
$940$	3.77661	0.123180
$941$	13.5864	0.442904	0.221452	−	0.975171i	$-0.428920\pi$
$941$	13.5864	0.442904	0.221452	+	0.975171i	$0.428920\pi$
$942$	30.2340	0.985078
$943$	−5.66105	−0.184349
$944$	−5.16570	−0.168129
$945$	−4.63659	−0.150828
$946$	0.745563	0.0242403
$947$	30.6948	0.997446	0.498723	−	0.866761i	$-0.333802\pi$
$947$	30.6948	0.997446	0.498723	+	0.866761i	$0.333802\pi$
$948$	−6.55354	−0.212849
$949$	−6.81843	−0.221335
$950$	17.7001	0.574267
$951$	−46.1695	−1.49715
$952$	11.6690	0.378195
$953$	−44.5778	−1.44402	−0.722008	−	0.691885i	$-0.756781\pi$
$953$	−44.5778	−1.44402	−0.722008	+	0.691885i	$0.756781\pi$
$954$	−8.06918	−0.261249
$955$	10.3847	0.336039
$956$	2.18395	0.0706339
$957$	0.928860	0.0300258
$958$	29.2118	0.943791
$959$	5.58403	0.180318
$960$	−0.768826	−0.0248138
$961$	10.8878	0.351218
$962$	−9.75241	−0.314430
$963$	0.673563	0.0217053
$964$	6.63904	0.213829
$965$	−11.7347	−0.377754
$966$	−8.02138	−0.258084
$967$	−26.4882	−0.851802	−0.425901	−	0.904770i	$-0.640043\pi$
$967$	−26.4882	−0.851802	−0.425901	+	0.904770i	$0.640043\pi$
$968$	−10.9880	−0.353169
$969$	40.1933	1.29119
$970$	−6.46073	−0.207441
$971$	43.9373	1.41001	0.705007	−	0.709201i	$-0.250944\pi$
$971$	43.9373	1.41001	0.705007	+	0.709201i	$0.250944\pi$
$972$	6.16943	0.197885
$973$	25.6139	0.821143
$974$	2.06779	0.0662562
$975$	−19.7149	−0.631383
$976$	−9.27004	−0.296727
$977$	52.0422	1.66498	0.832488	−	0.554043i	$-0.186916\pi$
$977$	52.0422	1.66498	0.832488	+	0.554043i	$0.186916\pi$
$978$	−12.8943	−0.412314
$979$	−1.88860	−0.0603598
$980$	2.08546	0.0666176
$981$	7.92885	0.253149
$982$	27.4299	0.875324
$983$	22.5687	0.719829	0.359914	−	0.932985i	$-0.382806\pi$
$983$	22.5687	0.719829	0.359914	+	0.932985i	$0.382806\pi$
$984$	−2.83054	−0.0902342
$985$	−1.89849	−0.0604910
$986$	−38.2686	−1.21872
$987$	19.7013	0.627099
$988$	−9.97660	−0.317398
$989$	21.1150	0.671418
$990$	0.0327667	0.00104139
$991$	−22.5648	−0.716793	−0.358397	−	0.933569i	$-0.616676\pi$
$991$	−22.5648	−0.716793	−0.358397	+	0.933569i	$0.616676\pi$
$992$	6.47208	0.205489
$993$	−26.4997	−0.840942
$994$	5.01250	0.158987
$995$	−10.1099	−0.320505
$996$	−19.5790	−0.620384
$997$	−32.4603	−1.02803	−0.514015	−	0.857781i	$-0.671842\pi$
$997$	−32.4603	−1.02803	−0.514015	+	0.857781i	$0.671842\pi$
$998$	−15.8123	−0.500530
$999$	20.3057	0.642444

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.52	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.52	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.54806 q^{3} +1.00000 q^{4} -0.496638 q^{5} +1.54806 q^{6} -1.67357 q^{7} +1.00000 q^{8} -0.603508 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.54806 q^{3} +1.00000 q^{4} -0.496638 q^{5} +1.54806 q^{6} -1.67357 q^{7} +1.00000 q^{8} -0.603508 q^{9} -0.496638 q^{10} +0.109322 q^{11} +1.54806 q^{12} +2.67921 q^{13} -1.67357 q^{14} -0.768826 q^{15} +1.00000 q^{16} -6.97252 q^{17} -0.603508 q^{18} -3.72371 q^{19} -0.496638 q^{20} -2.59079 q^{21} +0.109322 q^{22} +3.09611 q^{23} +1.54806 q^{24} -4.75335 q^{25} +2.67921 q^{26} -5.57845 q^{27} -1.67357 q^{28} +5.48849 q^{29} -0.768826 q^{30} +6.47208 q^{31} +1.00000 q^{32} +0.169238 q^{33} -6.97252 q^{34} +0.831160 q^{35} -0.603508 q^{36} -3.64003 q^{37} -3.72371 q^{38} +4.14758 q^{39} -0.496638 q^{40} -1.82844 q^{41} -2.59079 q^{42} +6.81985 q^{43} +0.109322 q^{44} +0.299725 q^{45} +3.09611 q^{46} -7.60435 q^{47} +1.54806 q^{48} -4.19916 q^{49} -4.75335 q^{50} -10.7939 q^{51} +2.67921 q^{52} +13.3705 q^{53} -5.57845 q^{54} -0.0542937 q^{55} -1.67357 q^{56} -5.76453 q^{57} +5.48849 q^{58} -5.16570 q^{59} -0.768826 q^{60} -9.27004 q^{61} +6.47208 q^{62} +1.01001 q^{63} +1.00000 q^{64} -1.33060 q^{65} +0.169238 q^{66} -3.78847 q^{67} -6.97252 q^{68} +4.79297 q^{69} +0.831160 q^{70} -2.99509 q^{71} -0.603508 q^{72} -2.54494 q^{73} -3.64003 q^{74} -7.35848 q^{75} -3.72371 q^{76} -0.182959 q^{77} +4.14758 q^{78} -4.23339 q^{79} -0.496638 q^{80} -6.82526 q^{81} -1.82844 q^{82} -12.6474 q^{83} -2.59079 q^{84} +3.46282 q^{85} +6.81985 q^{86} +8.49651 q^{87} +0.109322 q^{88} -17.2754 q^{89} +0.299725 q^{90} -4.48386 q^{91} +3.09611 q^{92} +10.0192 q^{93} -7.60435 q^{94} +1.84934 q^{95} +1.54806 q^{96} +13.0089 q^{97} -4.19916 q^{98} -0.0659769 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 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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