LMFDB - Embedded newform 8042.2.a.a.1.18

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			1.18
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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.59531 q^{3} +1.00000 q^{4} +2.52318 q^{5} -1.59531 q^{6} -1.75596 q^{7} +1.00000 q^{8} -0.454973 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.59531 q^{3} +1.00000 q^{4} +2.52318 q^{5} -1.59531 q^{6} -1.75596 q^{7} +1.00000 q^{8} -0.454973 q^{9} +2.52318 q^{10} +0.567436 q^{11} -1.59531 q^{12} -0.804323 q^{13} -1.75596 q^{14} -4.02526 q^{15} +1.00000 q^{16} +5.08505 q^{17} -0.454973 q^{18} +1.51333 q^{19} +2.52318 q^{20} +2.80130 q^{21} +0.567436 q^{22} -3.45744 q^{23} -1.59531 q^{24} +1.36643 q^{25} -0.804323 q^{26} +5.51177 q^{27} -1.75596 q^{28} -6.60941 q^{29} -4.02526 q^{30} -7.81110 q^{31} +1.00000 q^{32} -0.905239 q^{33} +5.08505 q^{34} -4.43059 q^{35} -0.454973 q^{36} -6.06802 q^{37} +1.51333 q^{38} +1.28315 q^{39} +2.52318 q^{40} -7.44313 q^{41} +2.80130 q^{42} +8.13872 q^{43} +0.567436 q^{44} -1.14798 q^{45} -3.45744 q^{46} +0.390686 q^{47} -1.59531 q^{48} -3.91661 q^{49} +1.36643 q^{50} -8.11225 q^{51} -0.804323 q^{52} -0.0978787 q^{53} +5.51177 q^{54} +1.43174 q^{55} -1.75596 q^{56} -2.41424 q^{57} -6.60941 q^{58} -5.89067 q^{59} -4.02526 q^{60} -3.77328 q^{61} -7.81110 q^{62} +0.798914 q^{63} +1.00000 q^{64} -2.02945 q^{65} -0.905239 q^{66} -2.77674 q^{67} +5.08505 q^{68} +5.51570 q^{69} -4.43059 q^{70} +2.79418 q^{71} -0.454973 q^{72} -0.0296022 q^{73} -6.06802 q^{74} -2.17988 q^{75} +1.51333 q^{76} -0.996394 q^{77} +1.28315 q^{78} +10.9999 q^{79} +2.52318 q^{80} -7.42808 q^{81} -7.44313 q^{82} +15.2509 q^{83} +2.80130 q^{84} +12.8305 q^{85} +8.13872 q^{86} +10.5441 q^{87} +0.567436 q^{88} +7.79244 q^{89} -1.14798 q^{90} +1.41236 q^{91} -3.45744 q^{92} +12.4612 q^{93} +0.390686 q^{94} +3.81841 q^{95} -1.59531 q^{96} -14.3668 q^{97} -3.91661 q^{98} -0.258168 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.59531	−0.921055	−0.460527	−	0.887645i	$-0.652340\pi$
$3$	−1.59531	−0.921055	−0.460527	+	0.887645i	$0.652340\pi$
$4$	1.00000	0.500000
$5$	2.52318	1.12840	0.564200	−	0.825638i	$-0.309185\pi$
$5$	2.52318	1.12840	0.564200	+	0.825638i	$0.309185\pi$
$6$	−1.59531	−0.651284
$7$	−1.75596	−0.663690	−0.331845	−	0.943334i	$-0.607671\pi$
$7$	−1.75596	−0.663690	−0.331845	+	0.943334i	$0.607671\pi$
$8$	1.00000	0.353553
$9$	−0.454973	−0.151658
$10$	2.52318	0.797899
$11$	0.567436	0.171088	0.0855442	−	0.996334i	$-0.472737\pi$
$11$	0.567436	0.171088	0.0855442	+	0.996334i	$0.472737\pi$
$12$	−1.59531	−0.460527
$13$	−0.804323	−0.223079	−0.111540	−	0.993760i	$-0.535578\pi$
$13$	−0.804323	−0.223079	−0.111540	+	0.993760i	$0.535578\pi$
$14$	−1.75596	−0.469299
$15$	−4.02526	−1.03932
$16$	1.00000	0.250000
$17$	5.08505	1.23330	0.616652	−	0.787236i	$-0.288488\pi$
$17$	5.08505	1.23330	0.616652	+	0.787236i	$0.288488\pi$
$18$	−0.454973	−0.107238
$19$	1.51333	0.347182	0.173591	−	0.984818i	$-0.444463\pi$
$19$	1.51333	0.347182	0.173591	+	0.984818i	$0.444463\pi$
$20$	2.52318	0.564200
$21$	2.80130	0.611295
$22$	0.567436	0.120978
$23$	−3.45744	−0.720925	−0.360463	−	0.932774i	$-0.617381\pi$
$23$	−3.45744	−0.720925	−0.360463	+	0.932774i	$0.617381\pi$
$24$	−1.59531	−0.325642
$25$	1.36643	0.273285
$26$	−0.804323	−0.157741
$27$	5.51177	1.06074
$28$	−1.75596	−0.331845
$29$	−6.60941	−1.22734	−0.613668	−	0.789564i	$-0.710307\pi$
$29$	−6.60941	−1.22734	−0.613668	+	0.789564i	$0.710307\pi$
$30$	−4.02526	−0.734909
$31$	−7.81110	−1.40292	−0.701458	−	0.712711i	$-0.747467\pi$
$31$	−7.81110	−1.40292	−0.701458	+	0.712711i	$0.747467\pi$
$32$	1.00000	0.176777
$33$	−0.905239	−0.157582
$34$	5.08505	0.872078
$35$	−4.43059	−0.748907
$36$	−0.454973	−0.0758289
$37$	−6.06802	−0.997576	−0.498788	−	0.866724i	$-0.666221\pi$
$37$	−6.06802	−0.997576	−0.498788	+	0.866724i	$0.666221\pi$
$38$	1.51333	0.245495
$39$	1.28315	0.205468
$40$	2.52318	0.398949
$41$	−7.44313	−1.16242	−0.581211	−	0.813753i	$-0.697421\pi$
$41$	−7.44313	−1.16242	−0.581211	+	0.813753i	$0.697421\pi$
$42$	2.80130	0.432251
$43$	8.13872	1.24114	0.620572	−	0.784150i	$-0.286901\pi$
$43$	8.13872	1.24114	0.620572	+	0.784150i	$0.286901\pi$
$44$	0.567436	0.0855442
$45$	−1.14798	−0.171130
$46$	−3.45744	−0.509771
$47$	0.390686	0.0569874	0.0284937	−	0.999594i	$-0.490929\pi$
$47$	0.390686	0.0569874	0.0284937	+	0.999594i	$0.490929\pi$
$48$	−1.59531	−0.230264
$49$	−3.91661	−0.559516
$50$	1.36643	0.193242
$51$	−8.11225	−1.13594
$52$	−0.804323	−0.111540
$53$	−0.0978787	−0.0134447	−0.00672234	−	0.999977i	$-0.502140\pi$
$53$	−0.0978787	−0.0134447	−0.00672234	+	0.999977i	$0.502140\pi$
$54$	5.51177	0.750057
$55$	1.43174	0.193056
$56$	−1.75596	−0.234650
$57$	−2.41424	−0.319774
$58$	−6.60941	−0.867858
$59$	−5.89067	−0.766900	−0.383450	−	0.923562i	$-0.625264\pi$
$59$	−5.89067	−0.766900	−0.383450	+	0.923562i	$0.625264\pi$
$60$	−4.02526	−0.519659
$61$	−3.77328	−0.483119	−0.241559	−	0.970386i	$-0.577659\pi$
$61$	−3.77328	−0.483119	−0.241559	+	0.970386i	$0.577659\pi$
$62$	−7.81110	−0.992011
$63$	0.798914	0.100654
$64$	1.00000	0.125000
$65$	−2.02945	−0.251722
$66$	−0.905239	−0.111427
$67$	−2.77674	−0.339233	−0.169617	−	0.985510i	$-0.554253\pi$
$67$	−2.77674	−0.339233	−0.169617	+	0.985510i	$0.554253\pi$
$68$	5.08505	0.616652
$69$	5.51570	0.664012
$70$	−4.43059	−0.529557
$71$	2.79418	0.331608	0.165804	−	0.986159i	$-0.446978\pi$
$71$	2.79418	0.331608	0.165804	+	0.986159i	$0.446978\pi$
$72$	−0.454973	−0.0536191
$73$	−0.0296022	−0.00346468	−0.00173234	−	0.999998i	$-0.500551\pi$
$73$	−0.0296022	−0.00346468	−0.00173234	+	0.999998i	$0.500551\pi$
$74$	−6.06802	−0.705393
$75$	−2.17988	−0.251711
$76$	1.51333	0.173591
$77$	−0.996394	−0.113550
$78$	1.28315	0.145288
$79$	10.9999	1.23758	0.618792	−	0.785555i	$-0.287622\pi$
$79$	10.9999	1.23758	0.618792	+	0.785555i	$0.287622\pi$
$80$	2.52318	0.282100
$81$	−7.42808	−0.825342
$82$	−7.44313	−0.821956
$83$	15.2509	1.67401	0.837003	−	0.547199i	$-0.184306\pi$
$83$	15.2509	1.67401	0.837003	+	0.547199i	$0.184306\pi$
$84$	2.80130	0.305647
$85$	12.8305	1.39166
$86$	8.13872	0.877621
$87$	10.5441	1.13044
$88$	0.567436	0.0604889
$89$	7.79244	0.825997	0.412999	−	0.910732i	$-0.364481\pi$
$89$	7.79244	0.825997	0.412999	+	0.910732i	$0.364481\pi$
$90$	−1.14798	−0.121008
$91$	1.41236	0.148055
$92$	−3.45744	−0.360463
$93$	12.4612	1.29216
$94$	0.390686	0.0402962
$95$	3.81841	0.391760
$96$	−1.59531	−0.162821
$97$	−14.3668	−1.45873	−0.729364	−	0.684126i	$-0.760184\pi$
$97$	−14.3668	−1.45873	−0.729364	+	0.684126i	$0.760184\pi$
$98$	−3.91661	−0.395638
$99$	−0.258168	−0.0259469
$100$	1.36643	0.136643
$101$	−9.19530	−0.914966	−0.457483	−	0.889218i	$-0.651249\pi$
$101$	−9.19530	−0.914966	−0.457483	+	0.889218i	$0.651249\pi$
$102$	−8.11225	−0.803232
$103$	4.30984	0.424661	0.212330	−	0.977198i	$-0.431895\pi$
$103$	4.30984	0.424661	0.212330	+	0.977198i	$0.431895\pi$
$104$	−0.804323	−0.0788703
$105$	7.06819	0.689785
$106$	−0.0978787	−0.00950682
$107$	1.97490	0.190921	0.0954605	−	0.995433i	$-0.469568\pi$
$107$	1.97490	0.190921	0.0954605	+	0.995433i	$0.469568\pi$
$108$	5.51177	0.530370
$109$	7.09215	0.679305	0.339652	−	0.940551i	$-0.389691\pi$
$109$	7.09215	0.679305	0.339652	+	0.940551i	$0.389691\pi$
$110$	1.43174	0.136511
$111$	9.68039	0.918822
$112$	−1.75596	−0.165922
$113$	−1.07604	−0.101226	−0.0506128	−	0.998718i	$-0.516117\pi$
$113$	−1.07604	−0.101226	−0.0506128	+	0.998718i	$0.516117\pi$
$114$	−2.41424	−0.226114
$115$	−8.72373	−0.813492
$116$	−6.60941	−0.613668
$117$	0.365945	0.0338317
$118$	−5.89067	−0.542281
$119$	−8.92913	−0.818532
$120$	−4.02526	−0.367454
$121$	−10.6780	−0.970729
$122$	−3.77328	−0.341617
$123$	11.8741	1.07065
$124$	−7.81110	−0.701458
$125$	−9.16815	−0.820025
$126$	0.798914	0.0711729
$127$	−9.46065	−0.839497	−0.419748	−	0.907640i	$-0.637882\pi$
$127$	−9.46065	−0.839497	−0.419748	+	0.907640i	$0.637882\pi$
$128$	1.00000	0.0883883
$129$	−12.9838	−1.14316
$130$	−2.02945	−0.177994
$131$	−13.3410	−1.16561	−0.582804	−	0.812613i	$-0.698044\pi$
$131$	−13.3410	−1.16561	−0.582804	+	0.812613i	$0.698044\pi$
$132$	−0.905239	−0.0787909
$133$	−2.65735	−0.230421
$134$	−2.77674	−0.239874
$135$	13.9072	1.19694
$136$	5.08505	0.436039
$137$	0.778785	0.0665361	0.0332681	−	0.999446i	$-0.489408\pi$
$137$	0.778785	0.0665361	0.0332681	+	0.999446i	$0.489408\pi$
$138$	5.51570	0.469527
$139$	−21.5348	−1.82656	−0.913280	−	0.407333i	$-0.866459\pi$
$139$	−21.5348	−1.82656	−0.913280	+	0.407333i	$0.866459\pi$
$140$	−4.43059	−0.374453
$141$	−0.623267	−0.0524885
$142$	2.79418	0.234483
$143$	−0.456402	−0.0381662
$144$	−0.454973	−0.0379144
$145$	−16.6767	−1.38493
$146$	−0.0296022	−0.00244990
$147$	6.24823	0.515345
$148$	−6.06802	−0.498788
$149$	−0.971434	−0.0795830	−0.0397915	−	0.999208i	$-0.512669\pi$
$149$	−0.971434	−0.0795830	−0.0397915	+	0.999208i	$0.512669\pi$
$150$	−2.17988	−0.177986
$151$	−6.55659	−0.533568	−0.266784	−	0.963756i	$-0.585961\pi$
$151$	−6.55659	−0.533568	−0.266784	+	0.963756i	$0.585961\pi$
$152$	1.51333	0.122747
$153$	−2.31356	−0.187040
$154$	−0.996394	−0.0802917
$155$	−19.7088	−1.58305
$156$	1.28315	0.102734
$157$	15.4574	1.23364	0.616818	−	0.787106i	$-0.288421\pi$
$157$	15.4574	1.23364	0.616818	+	0.787106i	$0.288421\pi$
$158$	10.9999	0.875104
$159$	0.156147	0.0123833
$160$	2.52318	0.199475
$161$	6.07111	0.478471
$162$	−7.42808	−0.583605
$163$	−4.58599	−0.359203	−0.179601	−	0.983739i	$-0.557481\pi$
$163$	−4.58599	−0.359203	−0.179601	+	0.983739i	$0.557481\pi$
$164$	−7.44313	−0.581211
$165$	−2.28408	−0.177815
$166$	15.2509	1.18370
$167$	15.3669	1.18912	0.594562	−	0.804050i	$-0.297325\pi$
$167$	15.3669	1.18912	0.594562	+	0.804050i	$0.297325\pi$
$168$	2.80130	0.216125
$169$	−12.3531	−0.950236
$170$	12.8305	0.984052
$171$	−0.688525	−0.0526529
$172$	8.13872	0.620572
$173$	−1.20405	−0.0915425	−0.0457712	−	0.998952i	$-0.514575\pi$
$173$	−1.20405	−0.0915425	−0.0457712	+	0.998952i	$0.514575\pi$
$174$	10.5441	0.799345
$175$	−2.39938	−0.181376
$176$	0.567436	0.0427721
$177$	9.39748	0.706357
$178$	7.79244	0.584068
$179$	−16.3825	−1.22449	−0.612244	−	0.790669i	$-0.709733\pi$
$179$	−16.3825	−1.22449	−0.612244	+	0.790669i	$0.709733\pi$
$180$	−1.14798	−0.0855652
$181$	−3.77800	−0.280817	−0.140408	−	0.990094i	$-0.544842\pi$
$181$	−3.77800	−0.280817	−0.140408	+	0.990094i	$0.544842\pi$
$182$	1.41236	0.104691
$183$	6.01956	0.444979
$184$	−3.45744	−0.254886
$185$	−15.3107	−1.12566
$186$	12.4612	0.913696
$187$	2.88544	0.211004
$188$	0.390686	0.0284937
$189$	−9.67843	−0.704002
$190$	3.81841	0.277016
$191$	8.80926	0.637415	0.318708	−	0.947853i	$-0.396751\pi$
$191$	8.80926	0.637415	0.318708	+	0.947853i	$0.396751\pi$
$192$	−1.59531	−0.115132
$193$	−3.87984	−0.279277	−0.139639	−	0.990203i	$-0.544594\pi$
$193$	−3.87984	−0.279277	−0.139639	+	0.990203i	$0.544594\pi$
$194$	−14.3668	−1.03148
$195$	3.23761	0.231850
$196$	−3.91661	−0.279758
$197$	−27.5906	−1.96575	−0.982875	−	0.184272i	$-0.941007\pi$
$197$	−27.5906	−1.96575	−0.982875	+	0.184272i	$0.941007\pi$
$198$	−0.258168	−0.0183472
$199$	−10.6045	−0.751734	−0.375867	−	0.926674i	$-0.622655\pi$
$199$	−10.6045	−0.751734	−0.375867	+	0.926674i	$0.622655\pi$
$200$	1.36643	0.0966208
$201$	4.42978	0.312452
$202$	−9.19530	−0.646979
$203$	11.6058	0.814571
$204$	−8.11225	−0.567971
$205$	−18.7803	−1.31168
$206$	4.30984	0.300280
$207$	1.57304	0.109334
$208$	−0.804323	−0.0557698
$209$	0.858719	0.0593988
$210$	7.06819	0.487751
$211$	25.2722	1.73981	0.869905	−	0.493219i	$-0.164180\pi$
$211$	25.2722	1.73981	0.869905	+	0.493219i	$0.164180\pi$
$212$	−0.0978787	−0.00672234
$213$	−4.45760	−0.305430
$214$	1.97490	0.135002
$215$	20.5354	1.40051
$216$	5.51177	0.375028
$217$	13.7160	0.931100
$218$	7.09215	0.480341
$219$	0.0472249	0.00319116
$220$	1.43174	0.0965280
$221$	−4.09002	−0.275124
$222$	9.68039	0.649705
$223$	−1.40610	−0.0941596	−0.0470798	−	0.998891i	$-0.514992\pi$
$223$	−1.40610	−0.0941596	−0.0470798	+	0.998891i	$0.514992\pi$
$224$	−1.75596	−0.117325
$225$	−0.621687	−0.0414458
$226$	−1.07604	−0.0715772
$227$	17.3271	1.15004	0.575021	−	0.818139i	$-0.304994\pi$
$227$	17.3271	1.15004	0.575021	+	0.818139i	$0.304994\pi$
$228$	−2.41424	−0.159887
$229$	−14.2781	−0.943520	−0.471760	−	0.881727i	$-0.656381\pi$
$229$	−14.2781	−0.943520	−0.471760	+	0.881727i	$0.656381\pi$
$230$	−8.72373	−0.575225
$231$	1.58956	0.104585
$232$	−6.60941	−0.433929
$233$	−2.31363	−0.151571	−0.0757854	−	0.997124i	$-0.524146\pi$
$233$	−2.31363	−0.151571	−0.0757854	+	0.997124i	$0.524146\pi$
$234$	0.365945	0.0239226
$235$	0.985770	0.0643045
$236$	−5.89067	−0.383450
$237$	−17.5483	−1.13988
$238$	−8.92913	−0.578789
$239$	7.31592	0.473227	0.236614	−	0.971604i	$-0.423962\pi$
$239$	7.31592	0.473227	0.236614	+	0.971604i	$0.423962\pi$
$240$	−4.02526	−0.259829
$241$	21.0925	1.35868	0.679342	−	0.733822i	$-0.262265\pi$
$241$	21.0925	1.35868	0.679342	+	0.733822i	$0.262265\pi$
$242$	−10.6780	−0.686409
$243$	−4.68518	−0.300555
$244$	−3.77328	−0.241559
$245$	−9.88231	−0.631357
$246$	11.8741	0.757067
$247$	−1.21721	−0.0774491
$248$	−7.81110	−0.496005
$249$	−24.3300	−1.54185
$250$	−9.16815	−0.579845
$251$	−19.5196	−1.23207	−0.616033	−	0.787720i	$-0.711261\pi$
$251$	−19.5196	−1.23207	−0.616033	+	0.787720i	$0.711261\pi$
$252$	0.798914	0.0503268
$253$	−1.96187	−0.123342
$254$	−9.46065	−0.593614
$255$	−20.4686	−1.28180
$256$	1.00000	0.0625000
$257$	−28.3700	−1.76967	−0.884835	−	0.465905i	$-0.845729\pi$
$257$	−28.3700	−1.76967	−0.884835	+	0.465905i	$0.845729\pi$
$258$	−12.9838	−0.808337
$259$	10.6552	0.662081
$260$	−2.02945	−0.125861
$261$	3.00710	0.186135
$262$	−13.3410	−0.824209
$263$	−8.74935	−0.539508	−0.269754	−	0.962929i	$-0.586942\pi$
$263$	−8.74935	−0.539508	−0.269754	+	0.962929i	$0.586942\pi$
$264$	−0.905239	−0.0557136
$265$	−0.246965	−0.0151710
$266$	−2.65735	−0.162932
$267$	−12.4314	−0.760789
$268$	−2.77674	−0.169617
$269$	−13.9896	−0.852964	−0.426482	−	0.904496i	$-0.640247\pi$
$269$	−13.9896	−0.852964	−0.426482	+	0.904496i	$0.640247\pi$
$270$	13.9072	0.846363
$271$	6.82598	0.414649	0.207324	−	0.978272i	$-0.433524\pi$
$271$	6.82598	0.414649	0.207324	+	0.978272i	$0.433524\pi$
$272$	5.08505	0.308326
$273$	−2.25315	−0.136367
$274$	0.778785	0.0470481
$275$	0.775359	0.0467559
$276$	5.51570	0.332006
$277$	−3.41024	−0.204902	−0.102451	−	0.994738i	$-0.532668\pi$
$277$	−3.41024	−0.204902	−0.102451	+	0.994738i	$0.532668\pi$
$278$	−21.5348	−1.29157
$279$	3.55384	0.212763
$280$	−4.43059	−0.264779
$281$	26.4075	1.57534	0.787671	−	0.616096i	$-0.211287\pi$
$281$	26.4075	1.57534	0.787671	+	0.616096i	$0.211287\pi$
$282$	−0.623267	−0.0371150
$283$	7.34816	0.436803	0.218401	−	0.975859i	$-0.429916\pi$
$283$	7.34816	0.436803	0.218401	+	0.975859i	$0.429916\pi$
$284$	2.79418	0.165804
$285$	−6.09156	−0.360833
$286$	−0.456402	−0.0269876
$287$	13.0698	0.771487
$288$	−0.454973	−0.0268096
$289$	8.85769	0.521041
$290$	−16.6767	−0.979290
$291$	22.9196	1.34357
$292$	−0.0296022	−0.00173234
$293$	−23.5159	−1.37381	−0.686907	−	0.726745i	$-0.741032\pi$
$293$	−23.5159	−1.37381	−0.686907	+	0.726745i	$0.741032\pi$
$294$	6.24823	0.364404
$295$	−14.8632	−0.865370
$296$	−6.06802	−0.352696
$297$	3.12758	0.181480
$298$	−0.971434	−0.0562737
$299$	2.78089	0.160823
$300$	−2.17988	−0.125855
$301$	−14.2913	−0.823734
$302$	−6.55659	−0.377290
$303$	14.6694	0.842734
$304$	1.51333	0.0867955
$305$	−9.52065	−0.545151
$306$	−2.31356	−0.132257
$307$	−3.19291	−0.182229	−0.0911144	−	0.995840i	$-0.529043\pi$
$307$	−3.19291	−0.182229	−0.0911144	+	0.995840i	$0.529043\pi$
$308$	−0.996394	−0.0567748
$309$	−6.87554	−0.391136
$310$	−19.7088	−1.11938
$311$	16.3953	0.929693	0.464847	−	0.885391i	$-0.346109\pi$
$311$	16.3953	0.929693	0.464847	+	0.885391i	$0.346109\pi$
$312$	1.28315	0.0726439
$313$	19.6926	1.11309	0.556546	−	0.830817i	$-0.312126\pi$
$313$	19.6926	1.11309	0.556546	+	0.830817i	$0.312126\pi$
$314$	15.4574	0.872313
$315$	2.01580	0.113578
$316$	10.9999	0.618792
$317$	14.6377	0.822136	0.411068	−	0.911605i	$-0.365156\pi$
$317$	14.6377	0.822136	0.411068	+	0.911605i	$0.365156\pi$
$318$	0.156147	0.00875631
$319$	−3.75042	−0.209983
$320$	2.52318	0.141050
$321$	−3.15059	−0.175849
$322$	6.07111	0.338330
$323$	7.69536	0.428181
$324$	−7.42808	−0.412671
$325$	−1.09905	−0.0609642
$326$	−4.58599	−0.253995
$327$	−11.3142	−0.625677
$328$	−7.44313	−0.410978
$329$	−0.686028	−0.0378219
$330$	−2.28408	−0.125734
$331$	2.56201	0.140821	0.0704105	−	0.997518i	$-0.477569\pi$
$331$	2.56201	0.140821	0.0704105	+	0.997518i	$0.477569\pi$
$332$	15.2509	0.837003
$333$	2.76078	0.151290
$334$	15.3669	0.840838
$335$	−7.00622	−0.382790
$336$	2.80130	0.152824
$337$	−6.73137	−0.366681	−0.183341	−	0.983049i	$-0.558691\pi$
$337$	−6.73137	−0.366681	−0.183341	+	0.983049i	$0.558691\pi$
$338$	−12.3531	−0.671918
$339$	1.71662	0.0932343
$340$	12.8305	0.695830
$341$	−4.43230	−0.240022
$342$	−0.688525	−0.0372312
$343$	19.1691	1.03503
$344$	8.13872	0.438810
$345$	13.9171	0.749270
$346$	−1.20405	−0.0647303
$347$	−23.3312	−1.25249	−0.626243	−	0.779628i	$-0.715408\pi$
$347$	−23.3312	−1.25249	−0.626243	+	0.779628i	$0.715408\pi$
$348$	10.5441	0.565222
$349$	5.26790	0.281984	0.140992	−	0.990011i	$-0.454971\pi$
$349$	5.26790	0.281984	0.140992	+	0.990011i	$0.454971\pi$
$350$	−2.39938	−0.128253
$351$	−4.43324	−0.236629
$352$	0.567436	0.0302444
$353$	−4.57867	−0.243698	−0.121849	−	0.992549i	$-0.538882\pi$
$353$	−4.57867	−0.243698	−0.121849	+	0.992549i	$0.538882\pi$
$354$	9.39748	0.499470
$355$	7.05022	0.374187
$356$	7.79244	0.412999
$357$	14.2448	0.753913
$358$	−16.3825	−0.865843
$359$	1.48423	0.0783346	0.0391673	−	0.999233i	$-0.487529\pi$
$359$	1.48423	0.0783346	0.0391673	+	0.999233i	$0.487529\pi$
$360$	−1.14798	−0.0605038
$361$	−16.7098	−0.879465
$362$	−3.77800	−0.198568
$363$	17.0348	0.894095
$364$	1.41236	0.0740276
$365$	−0.0746917	−0.00390954
$366$	6.01956	0.314648
$367$	5.56617	0.290552	0.145276	−	0.989391i	$-0.453593\pi$
$367$	5.56617	0.290552	0.145276	+	0.989391i	$0.453593\pi$
$368$	−3.45744	−0.180231
$369$	3.38642	0.176290
$370$	−15.3107	−0.795965
$371$	0.171871	0.00892309
$372$	12.4612	0.646081
$373$	−4.97219	−0.257450	−0.128725	−	0.991680i	$-0.541089\pi$
$373$	−4.97219	−0.257450	−0.128725	+	0.991680i	$0.541089\pi$
$374$	2.88544	0.149202
$375$	14.6261	0.755288
$376$	0.390686	0.0201481
$377$	5.31610	0.273793
$378$	−9.67843	−0.497805
$379$	31.7453	1.63065	0.815323	−	0.579006i	$-0.196559\pi$
$379$	31.7453	1.63065	0.815323	+	0.579006i	$0.196559\pi$
$380$	3.81841	0.195880
$381$	15.0927	0.773223
$382$	8.80926	0.450721
$383$	−13.1805	−0.673490	−0.336745	−	0.941596i	$-0.609326\pi$
$383$	−13.1805	−0.673490	−0.336745	+	0.941596i	$0.609326\pi$
$384$	−1.59531	−0.0814105
$385$	−2.51408	−0.128129
$386$	−3.87984	−0.197479
$387$	−3.70290	−0.188229
$388$	−14.3668	−0.729364
$389$	−29.9685	−1.51946	−0.759731	−	0.650238i	$-0.774669\pi$
$389$	−29.9685	−1.51946	−0.759731	+	0.650238i	$0.774669\pi$
$390$	3.23761	0.163943
$391$	−17.5812	−0.889121
$392$	−3.91661	−0.197819
$393$	21.2831	1.07359
$394$	−27.5906	−1.39000
$395$	27.7547	1.39649
$396$	−0.258168	−0.0129734
$397$	−32.2937	−1.62077	−0.810386	−	0.585896i	$-0.800743\pi$
$397$	−32.2937	−1.62077	−0.810386	+	0.585896i	$0.800743\pi$
$398$	−10.6045	−0.531556
$399$	4.23930	0.212231
$400$	1.36643	0.0683213
$401$	−10.9849	−0.548557	−0.274279	−	0.961650i	$-0.588439\pi$
$401$	−10.9849	−0.548557	−0.274279	+	0.961650i	$0.588439\pi$
$402$	4.42978	0.220937
$403$	6.28265	0.312961
$404$	−9.19530	−0.457483
$405$	−18.7424	−0.931316
$406$	11.6058	0.575988
$407$	−3.44321	−0.170674
$408$	−8.11225	−0.401616
$409$	0.863408	0.0426928	0.0213464	−	0.999772i	$-0.493205\pi$
$409$	0.863408	0.0426928	0.0213464	+	0.999772i	$0.493205\pi$
$410$	−18.7803	−0.927495
$411$	−1.24241	−0.0612834
$412$	4.30984	0.212330
$413$	10.3438	0.508984
$414$	1.57304	0.0773107
$415$	38.4808	1.88895
$416$	−0.804323	−0.0394352
$417$	34.3548	1.68236
$418$	0.858719	0.0420013
$419$	10.1254	0.494658	0.247329	−	0.968932i	$-0.420447\pi$
$419$	10.1254	0.494658	0.247329	+	0.968932i	$0.420447\pi$
$420$	7.06819	0.344892
$421$	15.1397	0.737865	0.368933	−	0.929456i	$-0.379723\pi$
$421$	15.1397	0.737865	0.368933	+	0.929456i	$0.379723\pi$
$422$	25.2722	1.23023
$423$	−0.177752	−0.00864258
$424$	−0.0978787	−0.00475341
$425$	6.94833	0.337044
$426$	−4.45760	−0.215971
$427$	6.62572	0.320641
$428$	1.97490	0.0954605
$429$	0.728104	0.0351532
$430$	20.5354	0.990307
$431$	−15.3926	−0.741434	−0.370717	−	0.928746i	$-0.620888\pi$
$431$	−15.3926	−0.741434	−0.370717	+	0.928746i	$0.620888\pi$
$432$	5.51177	0.265185
$433$	−28.6449	−1.37659	−0.688293	−	0.725433i	$-0.741640\pi$
$433$	−28.6449	−1.37659	−0.688293	+	0.725433i	$0.741640\pi$
$434$	13.7160	0.658387
$435$	26.6046	1.27559
$436$	7.09215	0.339652
$437$	−5.23225	−0.250292
$438$	0.0472249	0.00225649
$439$	−14.0733	−0.671684	−0.335842	−	0.941918i	$-0.609021\pi$
$439$	−14.0733	−0.671684	−0.335842	+	0.941918i	$0.609021\pi$
$440$	1.43174	0.0682556
$441$	1.78195	0.0848549
$442$	−4.09002	−0.194542
$443$	−37.5776	−1.78537	−0.892684	−	0.450684i	$-0.851180\pi$
$443$	−37.5776	−1.78537	−0.892684	+	0.450684i	$0.851180\pi$
$444$	9.68039	0.459411
$445$	19.6617	0.932055
$446$	−1.40610	−0.0665809
$447$	1.54974	0.0733003
$448$	−1.75596	−0.0829612
$449$	25.2836	1.19321	0.596604	−	0.802536i	$-0.296516\pi$
$449$	25.2836	1.19321	0.596604	+	0.802536i	$0.296516\pi$
$450$	−0.621687	−0.0293066
$451$	−4.22350	−0.198877
$452$	−1.07604	−0.0506128
$453$	10.4598	0.491445
$454$	17.3271	0.813202
$455$	3.56363	0.167065
$456$	−2.41424	−0.113057
$457$	3.10932	0.145448	0.0727240	−	0.997352i	$-0.476831\pi$
$457$	3.10932	0.145448	0.0727240	+	0.997352i	$0.476831\pi$
$458$	−14.2781	−0.667170
$459$	28.0276	1.30822
$460$	−8.72373	−0.406746
$461$	−30.2967	−1.41106	−0.705529	−	0.708681i	$-0.749291\pi$
$461$	−30.2967	−1.41106	−0.705529	+	0.708681i	$0.749291\pi$
$462$	1.58956	0.0739531
$463$	−30.6325	−1.42361	−0.711807	−	0.702375i	$-0.752123\pi$
$463$	−30.6325	−1.42361	−0.711807	+	0.702375i	$0.752123\pi$
$464$	−6.60941	−0.306834
$465$	31.4417	1.45807
$466$	−2.31363	−0.107177
$467$	−8.25943	−0.382201	−0.191100	−	0.981571i	$-0.561206\pi$
$467$	−8.25943	−0.382201	−0.191100	+	0.981571i	$0.561206\pi$
$468$	0.365945	0.0169158
$469$	4.87584	0.225146
$470$	0.985770	0.0454702
$471$	−24.6594	−1.13625
$472$	−5.89067	−0.271140
$473$	4.61820	0.212345
$474$	−17.5483	−0.806019
$475$	2.06785	0.0948797
$476$	−8.92913	−0.409266
$477$	0.0445322	0.00203899
$478$	7.31592	0.334622
$479$	2.56371	0.117139	0.0585694	−	0.998283i	$-0.481346\pi$
$479$	2.56371	0.117139	0.0585694	+	0.998283i	$0.481346\pi$
$480$	−4.02526	−0.183727
$481$	4.88064	0.222538
$482$	21.0925	0.960735
$483$	−9.68533	−0.440698
$484$	−10.6780	−0.485364
$485$	−36.2500	−1.64603
$486$	−4.68518	−0.212524
$487$	−0.185623	−0.00841138	−0.00420569	−	0.999991i	$-0.501339\pi$
$487$	−0.185623	−0.00841138	−0.00420569	+	0.999991i	$0.501339\pi$
$488$	−3.77328	−0.170808
$489$	7.31610	0.330845
$490$	−9.88231	−0.446437
$491$	2.45074	0.110600	0.0553002	−	0.998470i	$-0.482388\pi$
$491$	2.45074	0.110600	0.0553002	+	0.998470i	$0.482388\pi$
$492$	11.8741	0.535327
$493$	−33.6092	−1.51368
$494$	−1.21721	−0.0547647
$495$	−0.651404	−0.0292784
$496$	−7.81110	−0.350729
$497$	−4.90647	−0.220085
$498$	−24.3300	−1.09025
$499$	23.6939	1.06068	0.530342	−	0.847784i	$-0.322063\pi$
$499$	23.6939	1.06068	0.530342	+	0.847784i	$0.322063\pi$
$500$	−9.16815	−0.410012
$501$	−24.5150	−1.09525
$502$	−19.5196	−0.871202
$503$	−33.2403	−1.48211	−0.741056	−	0.671443i	$-0.765675\pi$
$503$	−33.2403	−1.48211	−0.741056	+	0.671443i	$0.765675\pi$
$504$	0.798914	0.0355864
$505$	−23.2014	−1.03245
$506$	−1.96187	−0.0872159
$507$	19.7070	0.875219
$508$	−9.46065	−0.419748
$509$	3.07485	0.136290	0.0681452	−	0.997675i	$-0.478292\pi$
$509$	3.07485	0.136290	0.0681452	+	0.997675i	$0.478292\pi$
$510$	−20.4686	−0.906366
$511$	0.0519803	0.00229947
$512$	1.00000	0.0441942
$513$	8.34113	0.368270
$514$	−28.3700	−1.25135
$515$	10.8745	0.479187
$516$	−12.9838	−0.571581
$517$	0.221689	0.00974988
$518$	10.6552	0.468162
$519$	1.92084	0.0843157
$520$	−2.02945	−0.0889972
$521$	−23.3857	−1.02455	−0.512274	−	0.858822i	$-0.671197\pi$
$521$	−23.3857	−1.02455	−0.512274	+	0.858822i	$0.671197\pi$
$522$	3.00710	0.131617
$523$	−15.8681	−0.693862	−0.346931	−	0.937891i	$-0.612776\pi$
$523$	−15.8681	−0.693862	−0.346931	+	0.937891i	$0.612776\pi$
$524$	−13.3410	−0.582804
$525$	3.82777	0.167058
$526$	−8.74935	−0.381490
$527$	−39.7198	−1.73022
$528$	−0.905239	−0.0393955
$529$	−11.0461	−0.480267
$530$	−0.246965	−0.0107275
$531$	2.68010	0.116306
$532$	−2.65735	−0.115211
$533$	5.98668	0.259312
$534$	−12.4314	−0.537959
$535$	4.98303	0.215435
$536$	−2.77674	−0.119937
$537$	26.1353	1.12782
$538$	−13.9896	−0.603137
$539$	−2.22243	−0.0957267
$540$	13.9072	0.598469
$541$	−2.93630	−0.126241	−0.0631207	−	0.998006i	$-0.520105\pi$
$541$	−2.93630	−0.126241	−0.0631207	+	0.998006i	$0.520105\pi$
$542$	6.82598	0.293201
$543$	6.02710	0.258648
$544$	5.08505	0.218020
$545$	17.8948	0.766527
$546$	−2.25315	−0.0964260
$547$	31.3660	1.34112	0.670558	−	0.741858i	$-0.266055\pi$
$547$	31.3660	1.34112	0.670558	+	0.741858i	$0.266055\pi$
$548$	0.778785	0.0332681
$549$	1.71674	0.0732687
$550$	0.775359	0.0330614
$551$	−10.0022	−0.426109
$552$	5.51570	0.234764
$553$	−19.3153	−0.821372
$554$	−3.41024	−0.144887
$555$	24.4253	1.03680
$556$	−21.5348	−0.913280
$557$	37.1507	1.57413	0.787064	−	0.616872i	$-0.211600\pi$
$557$	37.1507	1.57413	0.787064	+	0.616872i	$0.211600\pi$
$558$	3.55384	0.150446
$559$	−6.54616	−0.276873
$560$	−4.43059	−0.187227
$561$	−4.60318	−0.194346
$562$	26.4075	1.11394
$563$	−17.8471	−0.752166	−0.376083	−	0.926586i	$-0.622729\pi$
$563$	−17.8471	−0.752166	−0.376083	+	0.926586i	$0.622729\pi$
$564$	−0.623267	−0.0262443
$565$	−2.71505	−0.114223
$566$	7.34816	0.308866
$567$	13.0434	0.547771
$568$	2.79418	0.117241
$569$	−40.2938	−1.68921	−0.844603	−	0.535394i	$-0.820163\pi$
$569$	−40.2938	−1.68921	−0.844603	+	0.535394i	$0.820163\pi$
$570$	−6.09156	−0.255147
$571$	22.7850	0.953522	0.476761	−	0.879033i	$-0.341811\pi$
$571$	22.7850	0.953522	0.476761	+	0.879033i	$0.341811\pi$
$572$	−0.456402	−0.0190831
$573$	−14.0535	−0.587095
$574$	13.0698	0.545524
$575$	−4.72433	−0.197018
$576$	−0.454973	−0.0189572
$577$	40.0111	1.66568	0.832841	−	0.553512i	$-0.186713\pi$
$577$	40.0111	1.66568	0.832841	+	0.553512i	$0.186713\pi$
$578$	8.85769	0.368431
$579$	6.18956	0.257229
$580$	−16.6767	−0.692463
$581$	−26.7800	−1.11102
$582$	22.9196	0.950046
$583$	−0.0555399	−0.00230023
$584$	−0.0296022	−0.00122495
$585$	0.923345	0.0381756
$586$	−23.5159	−0.971433
$587$	−36.7041	−1.51494	−0.757471	−	0.652869i	$-0.773565\pi$
$587$	−36.7041	−1.51494	−0.757471	+	0.652869i	$0.773565\pi$
$588$	6.24823	0.257672
$589$	−11.8208	−0.487067
$590$	−14.8632	−0.611909
$591$	44.0157	1.81056
$592$	−6.06802	−0.249394
$593$	33.1841	1.36271	0.681354	−	0.731954i	$-0.261392\pi$
$593$	33.1841	1.36271	0.681354	+	0.731954i	$0.261392\pi$
$594$	3.12758	0.128326
$595$	−22.5298	−0.923631
$596$	−0.971434	−0.0397915
$597$	16.9175	0.692388
$598$	2.78089	0.113719
$599$	14.0006	0.572050	0.286025	−	0.958222i	$-0.407666\pi$
$599$	14.0006	0.572050	0.286025	+	0.958222i	$0.407666\pi$
$600$	−2.17988	−0.0889931
$601$	22.5328	0.919134	0.459567	−	0.888143i	$-0.348005\pi$
$601$	22.5328	0.919134	0.459567	+	0.888143i	$0.348005\pi$
$602$	−14.2913	−0.582468
$603$	1.26334	0.0514473
$604$	−6.55659	−0.266784
$605$	−26.9425	−1.09537
$606$	14.6694	0.595903
$607$	12.2100	0.495587	0.247794	−	0.968813i	$-0.420295\pi$
$607$	12.2100	0.495587	0.247794	+	0.968813i	$0.420295\pi$
$608$	1.51333	0.0613737
$609$	−18.5150	−0.750264
$610$	−9.52065	−0.385480
$611$	−0.314238	−0.0127127
$612$	−2.31356	−0.0935201
$613$	−31.2474	−1.26207	−0.631036	−	0.775754i	$-0.717370\pi$
$613$	−31.2474	−1.26207	−0.631036	+	0.775754i	$0.717370\pi$
$614$	−3.19291	−0.128855
$615$	29.9605	1.20813
$616$	−0.996394	−0.0401458
$617$	24.2178	0.974973	0.487486	−	0.873131i	$-0.337914\pi$
$617$	24.2178	0.974973	0.487486	+	0.873131i	$0.337914\pi$
$618$	−6.87554	−0.276575
$619$	23.2082	0.932814	0.466407	−	0.884570i	$-0.345548\pi$
$619$	23.2082	0.932814	0.466407	+	0.884570i	$0.345548\pi$
$620$	−19.7088	−0.791524
$621$	−19.0566	−0.764714
$622$	16.3953	0.657392
$623$	−13.6832	−0.548206
$624$	1.28315	0.0513670
$625$	−29.9650	−1.19860
$626$	19.6926	0.787075
$627$	−1.36993	−0.0547096
$628$	15.4574	0.616818
$629$	−30.8561	−1.23032
$630$	2.01580	0.0803114
$631$	−6.98747	−0.278167	−0.139083	−	0.990281i	$-0.544416\pi$
$631$	−6.98747	−0.278167	−0.139083	+	0.990281i	$0.544416\pi$
$632$	10.9999	0.437552
$633$	−40.3171	−1.60246
$634$	14.6377	0.581338
$635$	−23.8709	−0.947288
$636$	0.156147	0.00619164
$637$	3.15022	0.124816
$638$	−3.75042	−0.148480
$639$	−1.27128	−0.0502910
$640$	2.52318	0.0997374
$641$	−7.54540	−0.298025	−0.149013	−	0.988835i	$-0.547609\pi$
$641$	−7.54540	−0.298025	−0.149013	+	0.988835i	$0.547609\pi$
$642$	−3.15059	−0.124344
$643$	16.3074	0.643102	0.321551	−	0.946892i	$-0.395796\pi$
$643$	16.3074	0.643102	0.321551	+	0.946892i	$0.395796\pi$
$644$	6.07111	0.239235
$645$	−32.7605	−1.28994
$646$	7.69536	0.302770
$647$	−0.219016	−0.00861040	−0.00430520	−	0.999991i	$-0.501370\pi$
$647$	−0.219016	−0.00861040	−0.00430520	+	0.999991i	$0.501370\pi$
$648$	−7.42808	−0.291803
$649$	−3.34258	−0.131208
$650$	−1.09905	−0.0431082
$651$	−21.8813	−0.857595
$652$	−4.58599	−0.179601
$653$	−10.0239	−0.392266	−0.196133	−	0.980577i	$-0.562838\pi$
$653$	−10.0239	−0.392266	−0.196133	+	0.980577i	$0.562838\pi$
$654$	−11.3142	−0.442421
$655$	−33.6617	−1.31527
$656$	−7.44313	−0.290605
$657$	0.0134682	0.000525445 0
$658$	−0.686028	−0.0267442
$659$	−20.6979	−0.806275	−0.403138	−	0.915139i	$-0.632080\pi$
$659$	−20.6979	−0.806275	−0.403138	+	0.915139i	$0.632080\pi$
$660$	−2.28408	−0.0889076
$661$	−15.9330	−0.619723	−0.309861	−	0.950782i	$-0.600283\pi$
$661$	−15.9330	−0.619723	−0.309861	+	0.950782i	$0.600283\pi$
$662$	2.56201	0.0995755
$663$	6.52486	0.253405
$664$	15.2509	0.591850
$665$	−6.70496	−0.260007
$666$	2.76078	0.106978
$667$	22.8516	0.884818
$668$	15.3669	0.594562
$669$	2.24318	0.0867262
$670$	−7.00622	−0.270674
$671$	−2.14109	−0.0826560
$672$	2.80130	0.108063
$673$	−17.3756	−0.669781	−0.334890	−	0.942257i	$-0.608699\pi$
$673$	−17.3756	−0.669781	−0.334890	+	0.942257i	$0.608699\pi$
$674$	−6.73137	−0.259283
$675$	7.53142	0.289884
$676$	−12.3531	−0.475118
$677$	42.7772	1.64406	0.822030	−	0.569444i	$-0.192841\pi$
$677$	42.7772	1.64406	0.822030	+	0.569444i	$0.192841\pi$
$678$	1.71662	0.0659266
$679$	25.2275	0.968143
$680$	12.8305	0.492026
$681$	−27.6422	−1.05925
$682$	−4.43230	−0.169722
$683$	44.0039	1.68376	0.841882	−	0.539662i	$-0.181448\pi$
$683$	44.0039	1.68376	0.841882	+	0.539662i	$0.181448\pi$
$684$	−0.688525	−0.0263264
$685$	1.96501	0.0750793
$686$	19.1691	0.731880
$687$	22.7780	0.869034
$688$	8.13872	0.310286
$689$	0.0787261	0.00299923
$690$	13.9171	0.529814
$691$	40.1056	1.52569	0.762845	−	0.646581i	$-0.223802\pi$
$691$	40.1056	1.52569	0.762845	+	0.646581i	$0.223802\pi$
$692$	−1.20405	−0.0457712
$693$	0.453332	0.0172207
$694$	−23.3312	−0.885641
$695$	−54.3362	−2.06109
$696$	10.5441	0.399672
$697$	−37.8487	−1.43362
$698$	5.26790	0.199393
$699$	3.69096	0.139605
$700$	−2.39938	−0.0906882
$701$	34.0934	1.28769	0.643845	−	0.765156i	$-0.277338\pi$
$701$	34.0934	1.28769	0.643845	+	0.765156i	$0.277338\pi$
$702$	−4.43324	−0.167322
$703$	−9.18292	−0.346340
$704$	0.567436	0.0213860
$705$	−1.57261	−0.0592280
$706$	−4.57867	−0.172320
$707$	16.1466	0.607254
$708$	9.39748	0.353179
$709$	−25.4063	−0.954154	−0.477077	−	0.878862i	$-0.658304\pi$
$709$	−25.4063	−0.954154	−0.477077	+	0.878862i	$0.658304\pi$
$710$	7.05022	0.264590
$711$	−5.00465	−0.187689
$712$	7.79244	0.292034
$713$	27.0064	1.01140
$714$	14.2448	0.533097
$715$	−1.15158	−0.0430667
$716$	−16.3825	−0.612244
$717$	−11.6712	−0.435868
$718$	1.48423	0.0553909
$719$	−29.0124	−1.08198	−0.540990	−	0.841029i	$-0.681950\pi$
$719$	−29.0124	−1.08198	−0.540990	+	0.841029i	$0.681950\pi$
$720$	−1.14798	−0.0427826
$721$	−7.56789	−0.281843
$722$	−16.7098	−0.621875
$723$	−33.6491	−1.25142
$724$	−3.77800	−0.140408
$725$	−9.03126	−0.335413
$726$	17.0348	0.632220
$727$	−19.1273	−0.709394	−0.354697	−	0.934981i	$-0.615416\pi$
$727$	−19.1273	−0.709394	−0.354697	+	0.934981i	$0.615416\pi$
$728$	1.41236	0.0523454
$729$	29.7586	1.10217
$730$	−0.0746917	−0.00276446
$731$	41.3858	1.53071
$732$	6.01956	0.222489
$733$	51.7685	1.91211	0.956056	−	0.293183i	$-0.0947145\pi$
$733$	51.7685	1.91211	0.956056	+	0.293183i	$0.0947145\pi$
$734$	5.56617	0.205451
$735$	15.7654	0.581515
$736$	−3.45744	−0.127443
$737$	−1.57562	−0.0580388
$738$	3.38642	0.124656
$739$	3.86051	0.142011	0.0710055	−	0.997476i	$-0.477379\pi$
$739$	3.86051	0.142011	0.0710055	+	0.997476i	$0.477379\pi$
$740$	−15.3107	−0.562832
$741$	1.94183	0.0713348
$742$	0.171871	0.00630958
$743$	4.73400	0.173674	0.0868369	−	0.996223i	$-0.472324\pi$
$743$	4.73400	0.173674	0.0868369	+	0.996223i	$0.472324\pi$
$744$	12.4612	0.456848
$745$	−2.45110	−0.0898014
$746$	−4.97219	−0.182045
$747$	−6.93876	−0.253876
$748$	2.88544	0.105502
$749$	−3.46785	−0.126712
$750$	14.6261	0.534069
$751$	32.0930	1.17109	0.585545	−	0.810640i	$-0.300881\pi$
$751$	32.0930	1.17109	0.585545	+	0.810640i	$0.300881\pi$
$752$	0.390686	0.0142468
$753$	31.1399	1.13480
$754$	5.31610	0.193601
$755$	−16.5434	−0.602078
$756$	−9.67843	−0.352001
$757$	16.4083	0.596371	0.298186	−	0.954508i	$-0.403619\pi$
$757$	16.4083	0.596371	0.298186	+	0.954508i	$0.403619\pi$
$758$	31.7453	1.15304
$759$	3.12980	0.113605
$760$	3.81841	0.138508
$761$	−36.0755	−1.30774	−0.653869	−	0.756608i	$-0.726855\pi$
$761$	−36.0755	−1.30774	−0.653869	+	0.756608i	$0.726855\pi$
$762$	15.0927	0.546751
$763$	−12.4535	−0.450848
$764$	8.80926	0.318708
$765$	−5.83752	−0.211056
$766$	−13.1805	−0.476229
$767$	4.73800	0.171079
$768$	−1.59531	−0.0575659
$769$	29.3025	1.05668	0.528338	−	0.849034i	$-0.322815\pi$
$769$	29.3025	1.05668	0.528338	+	0.849034i	$0.322815\pi$
$770$	−2.51408	−0.0906011
$771$	45.2590	1.62996
$772$	−3.87984	−0.139639
$773$	8.29702	0.298423	0.149212	−	0.988805i	$-0.452326\pi$
$773$	8.29702	0.298423	0.149212	+	0.988805i	$0.452326\pi$
$774$	−3.70290	−0.133098
$775$	−10.6733	−0.383396
$776$	−14.3668	−0.515738
$777$	−16.9984	−0.609813
$778$	−29.9685	−1.07442
$779$	−11.2639	−0.403572
$780$	3.23761	0.115925
$781$	1.58552	0.0567344
$782$	−17.5812	−0.628703
$783$	−36.4295	−1.30189
$784$	−3.91661	−0.139879
$785$	39.0018	1.39203
$786$	21.2831	0.759142
$787$	13.1369	0.468282	0.234141	−	0.972203i	$-0.424772\pi$
$787$	13.1369	0.468282	0.234141	+	0.972203i	$0.424772\pi$
$788$	−27.5906	−0.982875
$789$	13.9580	0.496916
$790$	27.7547	0.987467
$791$	1.88948	0.0671823
$792$	−0.258168	−0.00917361
$793$	3.03493	0.107774
$794$	−32.2937	−1.14606
$795$	0.393987	0.0139733
$796$	−10.6045	−0.375867
$797$	−49.4952	−1.75321	−0.876604	−	0.481212i	$-0.840197\pi$
$797$	−49.4952	−1.75321	−0.876604	+	0.481212i	$0.840197\pi$
$798$	4.23930	0.150070
$799$	1.98666	0.0702828
$800$	1.36643	0.0483104
$801$	−3.54535	−0.125269
$802$	−10.9849	−0.387889
$803$	−0.0167974	−0.000592766 0
$804$	4.42978	0.156226
$805$	15.3185	0.539906
$806$	6.28265	0.221297
$807$	22.3179	0.785627
$808$	−9.19530	−0.323489
$809$	50.6723	1.78154	0.890772	−	0.454450i	$-0.150164\pi$
$809$	50.6723	1.78154	0.890772	+	0.454450i	$0.150164\pi$
$810$	−18.7424	−0.658540
$811$	−16.6077	−0.583176	−0.291588	−	0.956544i	$-0.594184\pi$
$811$	−16.6077	−0.583176	−0.291588	+	0.956544i	$0.594184\pi$
$812$	11.6058	0.407285
$813$	−10.8896	−0.381914
$814$	−3.44321	−0.120684
$815$	−11.5713	−0.405324
$816$	−8.11225	−0.283985
$817$	12.3166	0.430903
$818$	0.863408	0.0301883
$819$	−0.642585	−0.0224537
$820$	−18.7803	−0.655838
$821$	18.1437	0.633219	0.316610	−	0.948556i	$-0.397455\pi$
$821$	18.1437	0.633219	0.316610	+	0.948556i	$0.397455\pi$
$822$	−1.24241	−0.0433339
$823$	−44.7089	−1.55845	−0.779227	−	0.626742i	$-0.784388\pi$
$823$	−44.7089	−1.55845	−0.779227	+	0.626742i	$0.784388\pi$
$824$	4.30984	0.150140
$825$	−1.23694	−0.0430648
$826$	10.3438	0.359906
$827$	4.91716	0.170987	0.0854933	−	0.996339i	$-0.472753\pi$
$827$	4.91716	0.170987	0.0854933	+	0.996339i	$0.472753\pi$
$828$	1.57304	0.0546669
$829$	14.5908	0.506759	0.253379	−	0.967367i	$-0.418458\pi$
$829$	14.5908	0.506759	0.253379	+	0.967367i	$0.418458\pi$
$830$	38.4808	1.33569
$831$	5.44041	0.188726
$832$	−0.804323	−0.0278849
$833$	−19.9162	−0.690054
$834$	34.3548	1.18961
$835$	38.7733	1.34181
$836$	0.858719	0.0296994
$837$	−43.0530	−1.48813
$838$	10.1254	0.349776
$839$	−22.6055	−0.780428	−0.390214	−	0.920724i	$-0.627599\pi$
$839$	−22.6055	−0.780428	−0.390214	+	0.920724i	$0.627599\pi$
$840$	7.06819	0.243876
$841$	14.6843	0.506355
$842$	15.1397	0.521749
$843$	−42.1283	−1.45098
$844$	25.2722	0.869905
$845$	−31.1690	−1.07225
$846$	−0.177752	−0.00611123
$847$	18.7501	0.644263
$848$	−0.0978787	−0.00336117
$849$	−11.7226	−0.402320
$850$	6.94833	0.238326
$851$	20.9798	0.719178
$852$	−4.45760	−0.152715
$853$	23.9858	0.821258	0.410629	−	0.911803i	$-0.365309\pi$
$853$	23.9858	0.821258	0.410629	+	0.911803i	$0.365309\pi$
$854$	6.62572	0.226727
$855$	−1.73727	−0.0594134
$856$	1.97490	0.0675008
$857$	38.3542	1.31015	0.655077	−	0.755562i	$-0.272636\pi$
$857$	38.3542	1.31015	0.655077	+	0.755562i	$0.272636\pi$
$858$	0.728104	0.0248571
$859$	−47.7816	−1.63029	−0.815144	−	0.579259i	$-0.803342\pi$
$859$	−47.7816	−1.63029	−0.815144	+	0.579259i	$0.803342\pi$
$860$	20.5354	0.700253
$861$	−20.8505	−0.710582
$862$	−15.3926	−0.524273
$863$	25.9928	0.884807	0.442403	−	0.896816i	$-0.354126\pi$
$863$	25.9928	0.884807	0.442403	+	0.896816i	$0.354126\pi$
$864$	5.51177	0.187514
$865$	−3.03804	−0.103296
$866$	−28.6449	−0.973393
$867$	−14.1308	−0.479907
$868$	13.7160	0.465550
$869$	6.24173	0.211736
$870$	26.6046	0.901980
$871$	2.23340	0.0756758
$872$	7.09215	0.240171
$873$	6.53651	0.221227
$874$	−5.23225	−0.176983
$875$	16.0989	0.544242
$876$	0.0472249	0.00159558
$877$	7.05799	0.238331	0.119166	−	0.992874i	$-0.461978\pi$
$877$	7.05799	0.238331	0.119166	+	0.992874i	$0.461978\pi$
$878$	−14.0733	−0.474952
$879$	37.5153	1.26536
$880$	1.43174	0.0482640
$881$	−11.3977	−0.383999	−0.192000	−	0.981395i	$-0.561497\pi$
$881$	−11.3977	−0.383999	−0.192000	+	0.981395i	$0.561497\pi$
$882$	1.78195	0.0600015
$883$	−8.56646	−0.288284	−0.144142	−	0.989557i	$-0.546042\pi$
$883$	−8.56646	−0.288284	−0.144142	+	0.989557i	$0.546042\pi$
$884$	−4.09002	−0.137562
$885$	23.7115	0.797053
$886$	−37.5776	−1.26245
$887$	−15.9017	−0.533928	−0.266964	−	0.963706i	$-0.586021\pi$
$887$	−15.9017	−0.533928	−0.266964	+	0.963706i	$0.586021\pi$
$888$	9.68039	0.324853
$889$	16.6125	0.557165
$890$	19.6617	0.659062
$891$	−4.21496	−0.141206
$892$	−1.40610	−0.0470798
$893$	0.591237	0.0197850
$894$	1.54974	0.0518311
$895$	−41.3360	−1.38171
$896$	−1.75596	−0.0586624
$897$	−4.43640	−0.148127
$898$	25.2836	0.843726
$899$	51.6268	1.72185
$900$	−0.621687	−0.0207229
$901$	−0.497718	−0.0165814
$902$	−4.22350	−0.140627
$903$	22.7990	0.758704
$904$	−1.07604	−0.0357886
$905$	−9.53258	−0.316874
$906$	10.4598	0.347504
$907$	−26.6818	−0.885956	−0.442978	−	0.896532i	$-0.646078\pi$
$907$	−26.6818	−0.885956	−0.442978	+	0.896532i	$0.646078\pi$
$908$	17.3271	0.575021
$909$	4.18361	0.138762
$910$	3.56363	0.118133
$911$	52.2869	1.73234	0.866172	−	0.499746i	$-0.166573\pi$
$911$	52.2869	1.73234	0.866172	+	0.499746i	$0.166573\pi$
$912$	−2.41424	−0.0799435
$913$	8.65392	0.286403
$914$	3.10932	0.102847
$915$	15.1884	0.502114
$916$	−14.2781	−0.471760
$917$	23.4262	0.773602
$918$	28.0276	0.925048
$919$	20.2470	0.667886	0.333943	−	0.942593i	$-0.391621\pi$
$919$	20.2470	0.667886	0.333943	+	0.942593i	$0.391621\pi$
$920$	−8.72373	−0.287613
$921$	5.09369	0.167843
$922$	−30.2967	−0.997769
$923$	−2.24742	−0.0739749
$924$	1.58956	0.0522927
$925$	−8.29149	−0.272623
$926$	−30.6325	−1.00665
$927$	−1.96086	−0.0644031
$928$	−6.60941	−0.216965
$929$	21.5852	0.708188	0.354094	−	0.935210i	$-0.384789\pi$
$929$	21.5852	0.708188	0.354094	+	0.935210i	$0.384789\pi$
$930$	31.4417	1.03101
$931$	−5.92713	−0.194254
$932$	−2.31363	−0.0757854
$933$	−26.1557	−0.856298
$934$	−8.25943	−0.270257
$935$	7.28047	0.238097
$936$	0.365945	0.0119613
$937$	26.7059	0.872444	0.436222	−	0.899839i	$-0.356316\pi$
$937$	26.7059	0.872444	0.436222	+	0.899839i	$0.356316\pi$
$938$	4.87584	0.159202
$939$	−31.4159	−1.02522
$940$	0.985770	0.0321523
$941$	2.96519	0.0966625	0.0483313	−	0.998831i	$-0.484610\pi$
$941$	2.96519	0.0966625	0.0483313	+	0.998831i	$0.484610\pi$
$942$	−24.6594	−0.803448
$943$	25.7341	0.838019
$944$	−5.89067	−0.191725
$945$	−24.4204	−0.794396
$946$	4.61820	0.150151
$947$	52.3832	1.70223	0.851113	−	0.524983i	$-0.175928\pi$
$947$	52.3832	1.70223	0.851113	+	0.524983i	$0.175928\pi$
$948$	−17.5483	−0.569941
$949$	0.0238098	0.000772897 0
$950$	2.06785	0.0670901
$951$	−23.3518	−0.757233
$952$	−8.92913	−0.289395
$953$	19.3951	0.628267	0.314134	−	0.949379i	$-0.398286\pi$
$953$	19.3951	0.628267	0.314134	+	0.949379i	$0.398286\pi$
$954$	0.0445322	0.00144178
$955$	22.2273	0.719259
$956$	7.31592	0.236614
$957$	5.98309	0.193406
$958$	2.56371	0.0828296
$959$	−1.36751	−0.0441593
$960$	−4.02526	−0.129915
$961$	30.0133	0.968171
$962$	4.88064	0.157358
$963$	−0.898528	−0.0289547
$964$	21.0925	0.679342
$965$	−9.78953	−0.315136
$966$	−9.68533	−0.311620
$967$	−6.75666	−0.217280	−0.108640	−	0.994081i	$-0.534650\pi$
$967$	−6.75666	−0.217280	−0.108640	+	0.994081i	$0.534650\pi$
$968$	−10.6780	−0.343204
$969$	−12.2765	−0.394379
$970$	−36.2500	−1.16392
$971$	−36.5801	−1.17391	−0.586956	−	0.809619i	$-0.699674\pi$
$971$	−36.5801	−1.17391	−0.586956	+	0.809619i	$0.699674\pi$
$972$	−4.68518	−0.150277
$973$	37.8142	1.21227
$974$	−0.185623	−0.00594774
$975$	1.75332	0.0561513
$976$	−3.77328	−0.120780
$977$	33.1882	1.06179	0.530893	−	0.847439i	$-0.321857\pi$
$977$	33.1882	1.06179	0.530893	+	0.847439i	$0.321857\pi$
$978$	7.31610	0.233943
$979$	4.42171	0.141319
$980$	−9.88231	−0.315679
$981$	−3.22674	−0.103022
$982$	2.45074	0.0782062
$983$	−12.3856	−0.395040	−0.197520	−	0.980299i	$-0.563289\pi$
$983$	−12.3856	−0.395040	−0.197520	+	0.980299i	$0.563289\pi$
$984$	11.8741	0.378533
$985$	−69.6160	−2.21815
$986$	−33.6092	−1.07033
$987$	1.09443	0.0348361
$988$	−1.21721	−0.0387245
$989$	−28.1391	−0.894772
$990$	−0.651404	−0.0207030
$991$	−42.2842	−1.34320	−0.671601	−	0.740913i	$-0.734393\pi$
$991$	−42.2842	−1.34320	−0.671601	+	0.740913i	$0.734393\pi$
$992$	−7.81110	−0.248003
$993$	−4.08722	−0.129704
$994$	−4.90647	−0.155624
$995$	−26.7571	−0.848256
$996$	−24.3300	−0.770925
$997$	10.0661	0.318796	0.159398	−	0.987214i	$-0.449045\pi$
$997$	10.0661	0.318796	0.159398	+	0.987214i	$0.449045\pi$
$998$	23.6939	0.750017
$999$	−33.4455	−1.05817

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.18	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.18	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.59531 q^{3} +1.00000 q^{4} +2.52318 q^{5} -1.59531 q^{6} -1.75596 q^{7} +1.00000 q^{8} -0.454973 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.59531 q^{3} +1.00000 q^{4} +2.52318 q^{5} -1.59531 q^{6} -1.75596 q^{7} +1.00000 q^{8} -0.454973 q^{9} +2.52318 q^{10} +0.567436 q^{11} -1.59531 q^{12} -0.804323 q^{13} -1.75596 q^{14} -4.02526 q^{15} +1.00000 q^{16} +5.08505 q^{17} -0.454973 q^{18} +1.51333 q^{19} +2.52318 q^{20} +2.80130 q^{21} +0.567436 q^{22} -3.45744 q^{23} -1.59531 q^{24} +1.36643 q^{25} -0.804323 q^{26} +5.51177 q^{27} -1.75596 q^{28} -6.60941 q^{29} -4.02526 q^{30} -7.81110 q^{31} +1.00000 q^{32} -0.905239 q^{33} +5.08505 q^{34} -4.43059 q^{35} -0.454973 q^{36} -6.06802 q^{37} +1.51333 q^{38} +1.28315 q^{39} +2.52318 q^{40} -7.44313 q^{41} +2.80130 q^{42} +8.13872 q^{43} +0.567436 q^{44} -1.14798 q^{45} -3.45744 q^{46} +0.390686 q^{47} -1.59531 q^{48} -3.91661 q^{49} +1.36643 q^{50} -8.11225 q^{51} -0.804323 q^{52} -0.0978787 q^{53} +5.51177 q^{54} +1.43174 q^{55} -1.75596 q^{56} -2.41424 q^{57} -6.60941 q^{58} -5.89067 q^{59} -4.02526 q^{60} -3.77328 q^{61} -7.81110 q^{62} +0.798914 q^{63} +1.00000 q^{64} -2.02945 q^{65} -0.905239 q^{66} -2.77674 q^{67} +5.08505 q^{68} +5.51570 q^{69} -4.43059 q^{70} +2.79418 q^{71} -0.454973 q^{72} -0.0296022 q^{73} -6.06802 q^{74} -2.17988 q^{75} +1.51333 q^{76} -0.996394 q^{77} +1.28315 q^{78} +10.9999 q^{79} +2.52318 q^{80} -7.42808 q^{81} -7.44313 q^{82} +15.2509 q^{83} +2.80130 q^{84} +12.8305 q^{85} +8.13872 q^{86} +10.5441 q^{87} +0.567436 q^{88} +7.79244 q^{89} -1.14798 q^{90} +1.41236 q^{91} -3.45744 q^{92} +12.4612 q^{93} +0.390686 q^{94} +3.81841 q^{95} -1.59531 q^{96} -14.3668 q^{97} -3.91661 q^{98} -0.258168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 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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