LMFDB - Embedded newform 8031.2.a.d.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8031,2,Mod(1,8031)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8031, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8031 = 3 \cdot 2677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8031.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:	$64.1278578633$
Analytic rank:	$0$
Dimension:	$132$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8031.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.19060 q^{2} +1.00000 q^{3} +2.79873 q^{4} +1.38696 q^{5} -2.19060 q^{6} -0.332475 q^{7} -1.74969 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.19060 q^{2} +1.00000 q^{3} +2.79873 q^{4} +1.38696 q^{5} -2.19060 q^{6} -0.332475 q^{7} -1.74969 q^{8} +1.00000 q^{9} -3.03827 q^{10} +2.25325 q^{11} +2.79873 q^{12} +3.95774 q^{13} +0.728319 q^{14} +1.38696 q^{15} -1.76458 q^{16} -1.12327 q^{17} -2.19060 q^{18} -0.289067 q^{19} +3.88172 q^{20} -0.332475 q^{21} -4.93596 q^{22} -1.35246 q^{23} -1.74969 q^{24} -3.07635 q^{25} -8.66982 q^{26} +1.00000 q^{27} -0.930506 q^{28} -4.13736 q^{29} -3.03827 q^{30} -2.47795 q^{31} +7.36487 q^{32} +2.25325 q^{33} +2.46063 q^{34} -0.461129 q^{35} +2.79873 q^{36} -0.641426 q^{37} +0.633230 q^{38} +3.95774 q^{39} -2.42675 q^{40} -9.86537 q^{41} +0.728319 q^{42} -4.35309 q^{43} +6.30622 q^{44} +1.38696 q^{45} +2.96270 q^{46} +7.85619 q^{47} -1.76458 q^{48} -6.88946 q^{49} +6.73904 q^{50} -1.12327 q^{51} +11.0766 q^{52} +8.37098 q^{53} -2.19060 q^{54} +3.12516 q^{55} +0.581728 q^{56} -0.289067 q^{57} +9.06331 q^{58} -2.51037 q^{59} +3.88172 q^{60} +10.8139 q^{61} +5.42819 q^{62} -0.332475 q^{63} -12.6043 q^{64} +5.48922 q^{65} -4.93596 q^{66} +11.6248 q^{67} -3.14372 q^{68} -1.35246 q^{69} +1.01015 q^{70} -5.67619 q^{71} -1.74969 q^{72} +9.00509 q^{73} +1.40511 q^{74} -3.07635 q^{75} -0.809019 q^{76} -0.749148 q^{77} -8.66982 q^{78} +12.0115 q^{79} -2.44740 q^{80} +1.00000 q^{81} +21.6111 q^{82} +9.23361 q^{83} -0.930506 q^{84} -1.55793 q^{85} +9.53587 q^{86} -4.13736 q^{87} -3.94248 q^{88} -2.07495 q^{89} -3.03827 q^{90} -1.31585 q^{91} -3.78516 q^{92} -2.47795 q^{93} -17.2098 q^{94} -0.400924 q^{95} +7.36487 q^{96} +11.2925 q^{97} +15.0920 q^{98} +2.25325 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 132 q + 4 q^{2} + 132 q^{3} + 156 q^{4} + 20 q^{5} + 4 q^{6} + 44 q^{7} + 9 q^{8} + 132 q^{9}+O(q^{10}) $ 132 * q + 4 * q^2 + 132 * q^3 + 156 * q^4 + 20 * q^5 + 4 * q^6 + 44 * q^7 + 9 * q^8 + 132 * q^9	\( 132 q + 4 q^{2} + 132 q^{3} + 156 q^{4} + 20 q^{5} + 4 q^{6} + 44 q^{7} + 9 q^{8} + 132 q^{9} + 40 q^{10} + 24 q^{11} + 156 q^{12} + 62 q^{13} + 25 q^{14} + 20 q^{15} + 192 q^{16} + 77 q^{17} + 4 q^{18} + 86 q^{19} + 26 q^{20} + 44 q^{21} + 52 q^{22} + 17 q^{23} + 9 q^{24} + 212 q^{25} + 13 q^{26} + 132 q^{27} + 95 q^{28} + 52 q^{29} + 40 q^{30} + 59 q^{31} - 8 q^{32} + 24 q^{33} + 41 q^{34} + 21 q^{35} + 156 q^{36} + 76 q^{37} + 2 q^{38} + 62 q^{39} + 91 q^{40} + 114 q^{41} + 25 q^{42} + 173 q^{43} + 44 q^{44} + 20 q^{45} + 48 q^{46} + 15 q^{47} + 192 q^{48} + 262 q^{49} - 9 q^{50} + 77 q^{51} + 144 q^{52} + 15 q^{53} + 4 q^{54} + 111 q^{55} + 66 q^{56} + 86 q^{57} + 33 q^{58} + 20 q^{59} + 26 q^{60} + 182 q^{61} + 16 q^{62} + 44 q^{63} + 255 q^{64} + 70 q^{65} + 52 q^{66} + 169 q^{67} + 128 q^{68} + 17 q^{69} + 2 q^{70} + 23 q^{71} + 9 q^{72} + 148 q^{73} + 57 q^{74} + 212 q^{75} + 143 q^{76} + 31 q^{77} + 13 q^{78} + 152 q^{79} + 27 q^{80} + 132 q^{81} + 67 q^{82} + 28 q^{83} + 95 q^{84} + 88 q^{85} - 10 q^{86} + 52 q^{87} + 130 q^{88} + 136 q^{89} + 40 q^{90} + 125 q^{91} + 59 q^{93} + 95 q^{94} + 2 q^{95} - 8 q^{96} + 147 q^{97} - 18 q^{98} + 24 q^{99}+O(q^{100}) \) 132 * q + 4 * q^2 + 132 * q^3 + 156 * q^4 + 20 * q^5 + 4 * q^6 + 44 * q^7 + 9 * q^8 + 132 * q^9 + 40 * q^10 + 24 * q^11 + 156 * q^12 + 62 * q^13 + 25 * q^14 + 20 * q^15 + 192 * q^16 + 77 * q^17 + 4 * q^18 + 86 * q^19 + 26 * q^20 + 44 * q^21 + 52 * q^22 + 17 * q^23 + 9 * q^24 + 212 * q^25 + 13 * q^26 + 132 * q^27 + 95 * q^28 + 52 * q^29 + 40 * q^30 + 59 * q^31 - 8 * q^32 + 24 * q^33 + 41 * q^34 + 21 * q^35 + 156 * q^36 + 76 * q^37 + 2 * q^38 + 62 * q^39 + 91 * q^40 + 114 * q^41 + 25 * q^42 + 173 * q^43 + 44 * q^44 + 20 * q^45 + 48 * q^46 + 15 * q^47 + 192 * q^48 + 262 * q^49 - 9 * q^50 + 77 * q^51 + 144 * q^52 + 15 * q^53 + 4 * q^54 + 111 * q^55 + 66 * q^56 + 86 * q^57 + 33 * q^58 + 20 * q^59 + 26 * q^60 + 182 * q^61 + 16 * q^62 + 44 * q^63 + 255 * q^64 + 70 * q^65 + 52 * q^66 + 169 * q^67 + 128 * q^68 + 17 * q^69 + 2 * q^70 + 23 * q^71 + 9 * q^72 + 148 * q^73 + 57 * q^74 + 212 * q^75 + 143 * q^76 + 31 * q^77 + 13 * q^78 + 152 * q^79 + 27 * q^80 + 132 * q^81 + 67 * q^82 + 28 * q^83 + 95 * q^84 + 88 * q^85 - 10 * q^86 + 52 * q^87 + 130 * q^88 + 136 * q^89 + 40 * q^90 + 125 * q^91 + 59 * q^93 + 95 * q^94 + 2 * q^95 - 8 * q^96 + 147 * q^97 - 18 * q^98 + 24 * 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 $

$2$	−2.19060	−1.54899	−0.774494	−	0.632581i	$-0.781995\pi$
$2$	−2.19060	−1.54899	−0.774494	+	0.632581i	$0.781995\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.79873	1.39936
$5$	1.38696	0.620267	0.310133	−	0.950693i	$-0.399626\pi$
$5$	1.38696	0.620267	0.310133	+	0.950693i	$0.399626\pi$
$6$	−2.19060	−0.894309
$7$	−0.332475	−0.125664	−0.0628318	−	0.998024i	$-0.520013\pi$
$7$	−0.332475	−0.125664	−0.0628318	+	0.998024i	$0.520013\pi$
$8$	−1.74969	−0.618609
$9$	1.00000	0.333333
$10$	−3.03827	−0.960786
$11$	2.25325	0.679380	0.339690	−	0.940538i	$-0.389678\pi$
$11$	2.25325	0.679380	0.339690	+	0.940538i	$0.389678\pi$
$12$	2.79873	0.807923
$13$	3.95774	1.09768	0.548840	−	0.835928i	$-0.315070\pi$
$13$	3.95774	1.09768	0.548840	+	0.835928i	$0.315070\pi$
$14$	0.728319	0.194651
$15$	1.38696	0.358111
$16$	−1.76458	−0.441146
$17$	−1.12327	−0.272432	−0.136216	−	0.990679i	$-0.543494\pi$
$17$	−1.12327	−0.272432	−0.136216	+	0.990679i	$0.543494\pi$
$18$	−2.19060	−0.516329
$19$	−0.289067	−0.0663165	−0.0331583	−	0.999450i	$-0.510557\pi$
$19$	−0.289067	−0.0663165	−0.0331583	+	0.999450i	$0.510557\pi$
$20$	3.88172	0.867978
$21$	−0.332475	−0.0725519
$22$	−4.93596	−1.05235
$23$	−1.35246	−0.282007	−0.141004	−	0.990009i	$-0.545033\pi$
$23$	−1.35246	−0.282007	−0.141004	+	0.990009i	$0.545033\pi$
$24$	−1.74969	−0.357154
$25$	−3.07635	−0.615269
$26$	−8.66982	−1.70029
$27$	1.00000	0.192450
$28$	−0.930506	−0.175849
$29$	−4.13736	−0.768289	−0.384145	−	0.923273i	$-0.625504\pi$
$29$	−4.13736	−0.768289	−0.384145	+	0.923273i	$0.625504\pi$
$30$	−3.03827	−0.554710
$31$	−2.47795	−0.445053	−0.222526	−	0.974927i	$-0.571430\pi$
$31$	−2.47795	−0.445053	−0.222526	+	0.974927i	$0.571430\pi$
$32$	7.36487	1.30194
$33$	2.25325	0.392240
$34$	2.46063	0.421995
$35$	−0.461129	−0.0779450
$36$	2.79873	0.466454
$37$	−0.641426	−0.105450	−0.0527249	−	0.998609i	$-0.516791\pi$
$37$	−0.641426	−0.105450	−0.0527249	+	0.998609i	$0.516791\pi$
$38$	0.633230	0.102723
$39$	3.95774	0.633746
$40$	−2.42675	−0.383702
$41$	−9.86537	−1.54071	−0.770356	−	0.637614i	$-0.779922\pi$
$41$	−9.86537	−1.54071	−0.770356	+	0.637614i	$0.779922\pi$
$42$	0.728319	0.112382
$43$	−4.35309	−0.663840	−0.331920	−	0.943308i	$-0.607696\pi$
$43$	−4.35309	−0.663840	−0.331920	+	0.943308i	$0.607696\pi$
$44$	6.30622	0.950699
$45$	1.38696	0.206756
$46$	2.96270	0.436826
$47$	7.85619	1.14594	0.572972	−	0.819575i	$-0.305790\pi$
$47$	7.85619	1.14594	0.572972	+	0.819575i	$0.305790\pi$
$48$	−1.76458	−0.254696
$49$	−6.88946	−0.984209
$50$	6.73904	0.953044
$51$	−1.12327	−0.157289
$52$	11.0766	1.53605
$53$	8.37098	1.14984	0.574922	−	0.818209i	$-0.305033\pi$
$53$	8.37098	1.14984	0.574922	+	0.818209i	$0.305033\pi$
$54$	−2.19060	−0.298103
$55$	3.12516	0.421397
$56$	0.581728	0.0777366
$57$	−0.289067	−0.0382878
$58$	9.06331	1.19007
$59$	−2.51037	−0.326822	−0.163411	−	0.986558i	$-0.552250\pi$
$59$	−2.51037	−0.326822	−0.163411	+	0.986558i	$0.552250\pi$
$60$	3.88172	0.501128
$61$	10.8139	1.38458	0.692290	−	0.721619i	$-0.256602\pi$
$61$	10.8139	1.38458	0.692290	+	0.721619i	$0.256602\pi$
$62$	5.42819	0.689381
$63$	−0.332475	−0.0418879
$64$	−12.6043	−1.57554
$65$	5.48922	0.680854
$66$	−4.93596	−0.607575
$67$	11.6248	1.42020	0.710099	−	0.704102i	$-0.248650\pi$
$67$	11.6248	1.42020	0.710099	+	0.704102i	$0.248650\pi$
$68$	−3.14372	−0.381232
$69$	−1.35246	−0.162817
$70$	1.01015	0.120736
$71$	−5.67619	−0.673640	−0.336820	−	0.941569i	$-0.609351\pi$
$71$	−5.67619	−0.673640	−0.336820	+	0.941569i	$0.609351\pi$
$72$	−1.74969	−0.206203
$73$	9.00509	1.05397	0.526983	−	0.849876i	$-0.323323\pi$
$73$	9.00509	1.05397	0.526983	+	0.849876i	$0.323323\pi$
$74$	1.40511	0.163340
$75$	−3.07635	−0.355226
$76$	−0.809019	−0.0928009
$77$	−0.749148	−0.0853733
$78$	−8.66982	−0.981664
$79$	12.0115	1.35140	0.675701	−	0.737176i	$-0.263841\pi$
$79$	12.0115	1.35140	0.675701	+	0.737176i	$0.263841\pi$
$80$	−2.44740	−0.273628
$81$	1.00000	0.111111
$82$	21.6111	2.38654
$83$	9.23361	1.01352	0.506760	−	0.862087i	$-0.330843\pi$
$83$	9.23361	1.01352	0.506760	+	0.862087i	$0.330843\pi$
$84$	−0.930506	−0.101527
$85$	−1.55793	−0.168981
$86$	9.53587	1.02828
$87$	−4.13736	−0.443572
$88$	−3.94248	−0.420270
$89$	−2.07495	−0.219944	−0.109972	−	0.993935i	$-0.535076\pi$
$89$	−2.07495	−0.219944	−0.109972	+	0.993935i	$0.535076\pi$
$90$	−3.03827	−0.320262
$91$	−1.31585	−0.137938
$92$	−3.78516	−0.394631
$93$	−2.47795	−0.256951
$94$	−17.2098	−1.77505
$95$	−0.400924	−0.0411339
$96$	7.36487	0.751674
$97$	11.2925	1.14658	0.573291	−	0.819352i	$-0.305667\pi$
$97$	11.2925	1.14658	0.573291	+	0.819352i	$0.305667\pi$
$98$	15.0920	1.52453
$99$	2.25325	0.226460
$100$	−8.60985	−0.860985
$101$	−14.6094	−1.45369	−0.726845	−	0.686802i	$-0.759014\pi$
$101$	−14.6094	−1.45369	−0.726845	+	0.686802i	$0.759014\pi$
$102$	2.46063	0.243639
$103$	9.56869	0.942831	0.471416	−	0.881911i	$-0.343743\pi$
$103$	9.56869	0.942831	0.471416	+	0.881911i	$0.343743\pi$
$104$	−6.92482	−0.679034
$105$	−0.461129	−0.0450015
$106$	−18.3375	−1.78109
$107$	5.04347	0.487570	0.243785	−	0.969829i	$-0.421611\pi$
$107$	5.04347	0.487570	0.243785	+	0.969829i	$0.421611\pi$
$108$	2.79873	0.269308
$109$	7.39484	0.708297	0.354149	−	0.935189i	$-0.384771\pi$
$109$	7.39484	0.708297	0.354149	+	0.935189i	$0.384771\pi$
$110$	−6.84598	−0.652738
$111$	−0.641426	−0.0608815
$112$	0.586679	0.0554360
$113$	16.1070	1.51522	0.757611	−	0.652706i	$-0.226366\pi$
$113$	16.1070	1.51522	0.757611	+	0.652706i	$0.226366\pi$
$114$	0.633230	0.0593074
$115$	−1.87580	−0.174920
$116$	−11.5794	−1.07512
$117$	3.95774	0.365893
$118$	5.49921	0.506243
$119$	0.373458	0.0342349
$120$	−2.42675	−0.221531
$121$	−5.92288	−0.538443
$122$	−23.6890	−2.14470
$123$	−9.86537	−0.889531
$124$	−6.93510	−0.622790
$125$	−11.2016	−1.00190
$126$	0.728319	0.0648838
$127$	11.3561	1.00769	0.503847	−	0.863793i	$-0.331917\pi$
$127$	11.3561	1.00769	0.503847	+	0.863793i	$0.331917\pi$
$128$	12.8813	1.13856
$129$	−4.35309	−0.383268
$130$	−12.0247	−1.05463
$131$	8.91210	0.778653	0.389327	−	0.921100i	$-0.372708\pi$
$131$	8.91210	0.778653	0.389327	+	0.921100i	$0.372708\pi$
$132$	6.30622	0.548886
$133$	0.0961074	0.00833357
$134$	−25.4653	−2.19987
$135$	1.38696	0.119370
$136$	1.96537	0.168529
$137$	10.5445	0.900875	0.450438	−	0.892808i	$-0.351268\pi$
$137$	10.5445	0.900875	0.450438	+	0.892808i	$0.351268\pi$
$138$	2.96270	0.252201
$139$	2.53379	0.214913	0.107457	−	0.994210i	$-0.465729\pi$
$139$	2.53379	0.214913	0.107457	+	0.994210i	$0.465729\pi$
$140$	−1.29057	−0.109073
$141$	7.85619	0.661611
$142$	12.4343	1.04346
$143$	8.91777	0.745741
$144$	−1.76458	−0.147049
$145$	−5.73835	−0.476544
$146$	−19.7266	−1.63258
$147$	−6.88946	−0.568233
$148$	−1.79518	−0.147563
$149$	2.12971	0.174473	0.0872364	−	0.996188i	$-0.472196\pi$
$149$	2.12971	0.174473	0.0872364	+	0.996188i	$0.472196\pi$
$150$	6.73904	0.550240
$151$	11.8872	0.967369	0.483685	−	0.875242i	$-0.339298\pi$
$151$	11.8872	0.967369	0.483685	+	0.875242i	$0.339298\pi$
$152$	0.505777	0.0410240
$153$	−1.12327	−0.0908108
$154$	1.64108	0.132242
$155$	−3.43681	−0.276051
$156$	11.0766	0.886840
$157$	−13.4126	−1.07044	−0.535221	−	0.844712i	$-0.679772\pi$
$157$	−13.4126	−1.07044	−0.535221	+	0.844712i	$0.679772\pi$
$158$	−26.3124	−2.09330
$159$	8.37098	0.663862
$160$	10.2148	0.807549
$161$	0.449658	0.0354381
$162$	−2.19060	−0.172110
$163$	−6.45116	−0.505294	−0.252647	−	0.967559i	$-0.581301\pi$
$163$	−6.45116	−0.505294	−0.252647	+	0.967559i	$0.581301\pi$
$164$	−27.6105	−2.15602
$165$	3.12516	0.243293
$166$	−20.2271	−1.56993
$167$	−18.1213	−1.40227	−0.701134	−	0.713029i	$-0.747323\pi$
$167$	−18.1213	−1.40227	−0.701134	+	0.713029i	$0.747323\pi$
$168$	0.581728	0.0448813
$169$	2.66371	0.204901
$170$	3.41279	0.261749
$171$	−0.289067	−0.0221055
$172$	−12.1831	−0.928953
$173$	−1.82825	−0.138999	−0.0694995	−	0.997582i	$-0.522140\pi$
$173$	−1.82825	−0.138999	−0.0694995	+	0.997582i	$0.522140\pi$
$174$	9.06331	0.687088
$175$	1.02281	0.0773170
$176$	−3.97604	−0.299705
$177$	−2.51037	−0.188691
$178$	4.54538	0.340690
$179$	10.4689	0.782483	0.391242	−	0.920288i	$-0.372046\pi$
$179$	10.4689	0.782483	0.391242	+	0.920288i	$0.372046\pi$
$180$	3.88172	0.289326
$181$	22.2409	1.65316	0.826578	−	0.562822i	$-0.190284\pi$
$181$	22.2409	1.65316	0.826578	+	0.562822i	$0.190284\pi$
$182$	2.88250	0.213665
$183$	10.8139	0.799388
$184$	2.36638	0.174452
$185$	−0.889631	−0.0654070
$186$	5.42819	0.398014
$187$	−2.53100	−0.185085
$188$	21.9873	1.60359
$189$	−0.332475	−0.0241840
$190$	0.878264	0.0637159
$191$	−7.55757	−0.546847	−0.273423	−	0.961894i	$-0.588156\pi$
$191$	−7.55757	−0.546847	−0.273423	+	0.961894i	$0.588156\pi$
$192$	−12.6043	−0.909639
$193$	11.6362	0.837590	0.418795	−	0.908081i	$-0.362453\pi$
$193$	11.6362	0.837590	0.418795	+	0.908081i	$0.362453\pi$
$194$	−24.7374	−1.77604
$195$	5.48922	0.393091
$196$	−19.2817	−1.37727
$197$	9.67824	0.689546	0.344773	−	0.938686i	$-0.387956\pi$
$197$	9.67824	0.689546	0.344773	+	0.938686i	$0.387956\pi$
$198$	−4.93596	−0.350784
$199$	5.15843	0.365672	0.182836	−	0.983143i	$-0.441472\pi$
$199$	5.15843	0.365672	0.182836	+	0.983143i	$0.441472\pi$
$200$	5.38265	0.380611
$201$	11.6248	0.819952
$202$	32.0034	2.25175
$203$	1.37557	0.0965460
$204$	−3.14372	−0.220104
$205$	−13.6829	−0.955653
$206$	−20.9612	−1.46043
$207$	−1.35246	−0.0940024
$208$	−6.98376	−0.484237
$209$	−0.651339	−0.0450541
$210$	1.01015	0.0697069
$211$	12.1539	0.836707	0.418354	−	0.908284i	$-0.362607\pi$
$211$	12.1539	0.836707	0.418354	+	0.908284i	$0.362607\pi$
$212$	23.4281	1.60905
$213$	−5.67619	−0.388926
$214$	−11.0482	−0.755241
$215$	−6.03755	−0.411758
$216$	−1.74969	−0.119051
$217$	0.823855	0.0559269
$218$	−16.1991	−1.09714
$219$	9.00509	0.608508
$220$	8.74647	0.589687
$221$	−4.44560	−0.299044
$222$	1.40511	0.0943046
$223$	2.57474	0.172417	0.0862087	−	0.996277i	$-0.472525\pi$
$223$	2.57474	0.172417	0.0862087	+	0.996277i	$0.472525\pi$
$224$	−2.44863	−0.163606
$225$	−3.07635	−0.205090
$226$	−35.2841	−2.34706
$227$	−16.6946	−1.10806	−0.554029	−	0.832497i	$-0.686910\pi$
$227$	−16.6946	−1.10806	−0.554029	+	0.832497i	$0.686910\pi$
$228$	−0.809019	−0.0535786
$229$	21.3528	1.41103	0.705517	−	0.708693i	$-0.250715\pi$
$229$	21.3528	1.41103	0.705517	+	0.708693i	$0.250715\pi$
$230$	4.10914	0.270949
$231$	−0.749148	−0.0492903
$232$	7.23910	0.475270
$233$	−18.5904	−1.21790	−0.608949	−	0.793209i	$-0.708409\pi$
$233$	−18.5904	−1.21790	−0.608949	+	0.793209i	$0.708409\pi$
$234$	−8.66982	−0.566764
$235$	10.8962	0.710791
$236$	−7.02583	−0.457343
$237$	12.0115	0.780232
$238$	−0.818097	−0.0530294
$239$	22.2792	1.44112	0.720562	−	0.693390i	$-0.243884\pi$
$239$	22.2792	1.44112	0.720562	+	0.693390i	$0.243884\pi$
$240$	−2.44740	−0.157979
$241$	5.51611	0.355324	0.177662	−	0.984092i	$-0.443147\pi$
$241$	5.51611	0.355324	0.177662	+	0.984092i	$0.443147\pi$
$242$	12.9746	0.834042
$243$	1.00000	0.0641500
$244$	30.2652	1.93753
$245$	−9.55540	−0.610472
$246$	21.6111	1.37787
$247$	−1.14405	−0.0727943
$248$	4.33564	0.275314
$249$	9.23361	0.585156
$250$	24.5381	1.55193
$251$	1.62401	0.102506	0.0512532	−	0.998686i	$-0.483678\pi$
$251$	1.62401	0.102506	0.0512532	+	0.998686i	$0.483678\pi$
$252$	−0.930506	−0.0586164
$253$	−3.04743	−0.191590
$254$	−24.8768	−1.56091
$255$	−1.55793	−0.0975611
$256$	−3.00908	−0.188067
$257$	27.5892	1.72097	0.860485	−	0.509476i	$-0.170161\pi$
$257$	27.5892	1.72097	0.860485	+	0.509476i	$0.170161\pi$
$258$	9.53587	0.593678
$259$	0.213258	0.0132512
$260$	15.3628	0.952762
$261$	−4.13736	−0.256096
$262$	−19.5228	−1.20612
$263$	−31.1077	−1.91818	−0.959092	−	0.283095i	$-0.908639\pi$
$263$	−31.1077	−1.91818	−0.959092	+	0.283095i	$0.908639\pi$
$264$	−3.94248	−0.242643
$265$	11.6102	0.713209
$266$	−0.210533	−0.0129086
$267$	−2.07495	−0.126985
$268$	32.5347	1.98737
$269$	19.2487	1.17361	0.586807	−	0.809727i	$-0.300385\pi$
$269$	19.2487	1.17361	0.586807	+	0.809727i	$0.300385\pi$
$270$	−3.03827	−0.184903
$271$	20.0640	1.21880	0.609402	−	0.792862i	$-0.291410\pi$
$271$	20.0640	1.21880	0.609402	+	0.792862i	$0.291410\pi$
$272$	1.98210	0.120182
$273$	−1.31585	−0.0796388
$274$	−23.0987	−1.39544
$275$	−6.93177	−0.418001
$276$	−3.78516	−0.227840
$277$	1.24271	0.0746675	0.0373337	−	0.999303i	$-0.488114\pi$
$277$	1.24271	0.0746675	0.0373337	+	0.999303i	$0.488114\pi$
$278$	−5.55052	−0.332898
$279$	−2.47795	−0.148351
$280$	0.806832	0.0482174
$281$	28.5263	1.70174	0.850869	−	0.525377i	$-0.176076\pi$
$281$	28.5263	1.70174	0.850869	+	0.525377i	$0.176076\pi$
$282$	−17.2098	−1.02483
$283$	−5.96214	−0.354412	−0.177206	−	0.984174i	$-0.556706\pi$
$283$	−5.96214	−0.354412	−0.177206	+	0.984174i	$0.556706\pi$
$284$	−15.8861	−0.942667
$285$	−0.400924	−0.0237487
$286$	−19.5353	−1.15514
$287$	3.27999	0.193612
$288$	7.36487	0.433979
$289$	−15.7383	−0.925781
$290$	12.5704	0.738161
$291$	11.2925	0.661979
$292$	25.2028	1.47488
$293$	−13.1850	−0.770278	−0.385139	−	0.922859i	$-0.625846\pi$
$293$	−13.1850	−0.770278	−0.385139	+	0.922859i	$0.625846\pi$
$294$	15.0920	0.880186
$295$	−3.48178	−0.202717
$296$	1.12230	0.0652322
$297$	2.25325	0.130747
$298$	−4.66535	−0.270256
$299$	−5.35268	−0.309554
$300$	−8.60985	−0.497090
$301$	1.44729	0.0834205
$302$	−26.0402	−1.49844
$303$	−14.6094	−0.839288
$304$	0.510083	0.0292552
$305$	14.9985	0.858809
$306$	2.46063	0.140665
$307$	17.4748	0.997342	0.498671	−	0.866791i	$-0.333822\pi$
$307$	17.4748	0.997342	0.498671	+	0.866791i	$0.333822\pi$
$308$	−2.09666	−0.119468
$309$	9.56869	0.544344
$310$	7.52868	0.427600
$311$	16.5809	0.940219	0.470110	−	0.882608i	$-0.344214\pi$
$311$	16.5809	0.940219	0.470110	+	0.882608i	$0.344214\pi$
$312$	−6.92482	−0.392041
$313$	−27.9767	−1.58134	−0.790668	−	0.612245i	$-0.790267\pi$
$313$	−27.9767	−1.58134	−0.790668	+	0.612245i	$0.790267\pi$
$314$	29.3817	1.65810
$315$	−0.461129	−0.0259817
$316$	33.6170	1.89110
$317$	−16.5326	−0.928564	−0.464282	−	0.885687i	$-0.653688\pi$
$317$	−16.5326	−0.928564	−0.464282	+	0.885687i	$0.653688\pi$
$318$	−18.3375	−1.02831
$319$	−9.32250	−0.521960
$320$	−17.4817	−0.977255
$321$	5.04347	0.281499
$322$	−0.985022	−0.0548931
$323$	0.324700	0.0180668
$324$	2.79873	0.155485
$325$	−12.1754	−0.675368
$326$	14.1319	0.782694
$327$	7.39484	0.408935
$328$	17.2613	0.953098
$329$	−2.61199	−0.144003
$330$	−6.84598	−0.376859
$331$	1.65806	0.0911349	0.0455675	−	0.998961i	$-0.485490\pi$
$331$	1.65806	0.0911349	0.0455675	+	0.998961i	$0.485490\pi$
$332$	25.8424	1.41828
$333$	−0.641426	−0.0351499
$334$	39.6965	2.17210
$335$	16.1232	0.880902
$336$	0.586679	0.0320060
$337$	24.4241	1.33047	0.665233	−	0.746636i	$-0.268332\pi$
$337$	24.4241	1.33047	0.665233	+	0.746636i	$0.268332\pi$
$338$	−5.83512	−0.317389
$339$	16.1070	0.874814
$340$	−4.36021	−0.236466
$341$	−5.58343	−0.302360
$342$	0.633230	0.0342412
$343$	4.61789	0.249343
$344$	7.61655	0.410657
$345$	−1.87580	−0.100990
$346$	4.00495	0.215308
$347$	−25.2577	−1.35590	−0.677951	−	0.735107i	$-0.737132\pi$
$347$	−25.2577	−1.35590	−0.677951	+	0.735107i	$0.737132\pi$
$348$	−11.5794	−0.620718
$349$	24.8145	1.32829	0.664146	−	0.747603i	$-0.268795\pi$
$349$	24.8145	1.32829	0.664146	+	0.747603i	$0.268795\pi$
$350$	−2.24056	−0.119763
$351$	3.95774	0.211249
$352$	16.5949	0.884510
$353$	1.53546	0.0817241	0.0408621	−	0.999165i	$-0.486990\pi$
$353$	1.53546	0.0817241	0.0408621	+	0.999165i	$0.486990\pi$
$354$	5.49921	0.292280
$355$	−7.87264	−0.417836
$356$	−5.80721	−0.307781
$357$	0.373458	0.0197655
$358$	−22.9332	−1.21206
$359$	10.9541	0.578138	0.289069	−	0.957308i	$-0.406654\pi$
$359$	10.9541	0.578138	0.289069	+	0.957308i	$0.406654\pi$
$360$	−2.42675	−0.127901
$361$	−18.9164	−0.995602
$362$	−48.7210	−2.56072
$363$	−5.92288	−0.310870
$364$	−3.68270	−0.193026
$365$	12.4897	0.653740
$366$	−23.6890	−1.23824
$367$	6.68634	0.349024	0.174512	−	0.984655i	$-0.444165\pi$
$367$	6.68634	0.349024	0.174512	+	0.984655i	$0.444165\pi$
$368$	2.38653	0.124406
$369$	−9.86537	−0.513571
$370$	1.94883	0.101315
$371$	−2.78314	−0.144493
$372$	−6.93510	−0.359568
$373$	−29.6013	−1.53270	−0.766348	−	0.642426i	$-0.777928\pi$
$373$	−29.6013	−1.53270	−0.766348	+	0.642426i	$0.777928\pi$
$374$	5.54441	0.286695
$375$	−11.2016	−0.578446
$376$	−13.7459	−0.708891
$377$	−16.3746	−0.843335
$378$	0.728319	0.0374607
$379$	−21.5899	−1.10900	−0.554498	−	0.832185i	$-0.687090\pi$
$379$	−21.5899	−1.10900	−0.554498	+	0.832185i	$0.687090\pi$
$380$	−1.12208	−0.0575613
$381$	11.3561	0.581793
$382$	16.5556	0.847059
$383$	−31.7196	−1.62080	−0.810399	−	0.585878i	$-0.800750\pi$
$383$	−31.7196	−1.62080	−0.810399	+	0.585878i	$0.800750\pi$
$384$	12.8813	0.657345
$385$	−1.03904	−0.0529542
$386$	−25.4902	−1.29742
$387$	−4.35309	−0.221280
$388$	31.6047	1.60448
$389$	11.4296	0.579505	0.289752	−	0.957102i	$-0.406427\pi$
$389$	11.4296	0.579505	0.289752	+	0.957102i	$0.406427\pi$
$390$	−12.0247	−0.608894
$391$	1.51917	0.0768279
$392$	12.0544	0.608840
$393$	8.91210	0.449556
$394$	−21.2012	−1.06810
$395$	16.6595	0.838229
$396$	6.30622	0.316900
$397$	18.3757	0.922251	0.461126	−	0.887335i	$-0.347446\pi$
$397$	18.3757	0.922251	0.461126	+	0.887335i	$0.347446\pi$
$398$	−11.3001	−0.566421
$399$	0.0961074	0.00481139
$400$	5.42847	0.271423
$401$	−3.93933	−0.196721	−0.0983604	−	0.995151i	$-0.531360\pi$
$401$	−3.93933	−0.196721	−0.0983604	+	0.995151i	$0.531360\pi$
$402$	−25.4653	−1.27010
$403$	−9.80708	−0.488525
$404$	−40.8877	−2.03424
$405$	1.38696	0.0689185
$406$	−3.01332	−0.149549
$407$	−1.44529	−0.0716404
$408$	1.96537	0.0973003
$409$	−21.8412	−1.07998	−0.539990	−	0.841672i	$-0.681572\pi$
$409$	−21.8412	−1.07998	−0.539990	+	0.841672i	$0.681572\pi$
$410$	29.9737	1.48029
$411$	10.5445	0.520121
$412$	26.7802	1.31936
$413$	0.834634	0.0410697
$414$	2.96270	0.145609
$415$	12.8066	0.628653
$416$	29.1483	1.42911
$417$	2.53379	0.124080
$418$	1.42682	0.0697882
$419$	36.4261	1.77953	0.889767	−	0.456416i	$-0.150867\pi$
$419$	36.4261	1.77953	0.889767	+	0.456416i	$0.150867\pi$
$420$	−1.29057	−0.0629735
$421$	17.8712	0.870990	0.435495	−	0.900191i	$-0.356573\pi$
$421$	17.8712	0.870990	0.435495	+	0.900191i	$0.356573\pi$
$422$	−26.6243	−1.29605
$423$	7.85619	0.381981
$424$	−14.6466	−0.711303
$425$	3.45556	0.167619
$426$	12.4343	0.602442
$427$	−3.59535	−0.173991
$428$	14.1153	0.682288
$429$	8.91777	0.430554
$430$	13.2259	0.637808
$431$	−6.88231	−0.331509	−0.165755	−	0.986167i	$-0.553006\pi$
$431$	−6.88231	−0.331509	−0.165755	+	0.986167i	$0.553006\pi$
$432$	−1.76458	−0.0848985
$433$	9.07319	0.436030	0.218015	−	0.975945i	$-0.430042\pi$
$433$	9.07319	0.436030	0.218015	+	0.975945i	$0.430042\pi$
$434$	−1.80474	−0.0866302
$435$	−5.73835	−0.275133
$436$	20.6961	0.991165
$437$	0.390951	0.0187017
$438$	−19.7266	−0.942571
$439$	19.0916	0.911192	0.455596	−	0.890187i	$-0.349426\pi$
$439$	19.0916	0.911192	0.455596	+	0.890187i	$0.349426\pi$
$440$	−5.46806	−0.260680
$441$	−6.88946	−0.328070
$442$	9.73854	0.463215
$443$	8.54064	0.405778	0.202889	−	0.979202i	$-0.434967\pi$
$443$	8.54064	0.405778	0.202889	+	0.979202i	$0.434967\pi$
$444$	−1.79518	−0.0851953
$445$	−2.87787	−0.136424
$446$	−5.64022	−0.267072
$447$	2.12971	0.100732
$448$	4.19062	0.197988
$449$	−22.9356	−1.08240	−0.541198	−	0.840895i	$-0.682029\pi$
$449$	−22.9356	−1.08240	−0.541198	+	0.840895i	$0.682029\pi$
$450$	6.73904	0.317681
$451$	−22.2291	−1.04673
$452$	45.0792	2.12035
$453$	11.8872	0.558511
$454$	36.5712	1.71637
$455$	−1.82503	−0.0855586
$456$	0.505777	0.0236852
$457$	−9.18031	−0.429437	−0.214718	−	0.976676i	$-0.568883\pi$
$457$	−9.18031	−0.429437	−0.214718	+	0.976676i	$0.568883\pi$
$458$	−46.7754	−2.18567
$459$	−1.12327	−0.0524297
$460$	−5.24987	−0.244776
$461$	−11.1563	−0.519602	−0.259801	−	0.965662i	$-0.583657\pi$
$461$	−11.1563	−0.519602	−0.259801	+	0.965662i	$0.583657\pi$
$462$	1.64108	0.0763501
$463$	13.7335	0.638250	0.319125	−	0.947713i	$-0.396611\pi$
$463$	13.7335	0.638250	0.319125	+	0.947713i	$0.396611\pi$
$464$	7.30072	0.338928
$465$	−3.43681	−0.159378
$466$	40.7241	1.88651
$467$	−30.0268	−1.38947	−0.694737	−	0.719264i	$-0.744479\pi$
$467$	−30.0268	−1.38947	−0.694737	+	0.719264i	$0.744479\pi$
$468$	11.0766	0.512018
$469$	−3.86496	−0.178467
$470$	−23.8692	−1.10101
$471$	−13.4126	−0.618020
$472$	4.39237	0.202175
$473$	−9.80858	−0.450999
$474$	−26.3124	−1.20857
$475$	0.889270	0.0408025
$476$	1.04521	0.0479070
$477$	8.37098	0.383281
$478$	−48.8049	−2.23228
$479$	−37.6406	−1.71984	−0.859921	−	0.510428i	$-0.829487\pi$
$479$	−37.6406	−1.71984	−0.859921	+	0.510428i	$0.829487\pi$
$480$	10.2148	0.466239
$481$	−2.53860	−0.115750
$482$	−12.0836	−0.550393
$483$	0.449658	0.0204602
$484$	−16.5765	−0.753478
$485$	15.6623	0.711186
$486$	−2.19060	−0.0993676
$487$	−0.988569	−0.0447964	−0.0223982	−	0.999749i	$-0.507130\pi$
$487$	−0.988569	−0.0447964	−0.0223982	+	0.999749i	$0.507130\pi$
$488$	−18.9210	−0.856513
$489$	−6.45116	−0.291732
$490$	20.9320	0.945614
$491$	−22.2378	−1.00358	−0.501790	−	0.864989i	$-0.667325\pi$
$491$	−22.2378	−1.00358	−0.501790	+	0.864989i	$0.667325\pi$
$492$	−27.6105	−1.24478
$493$	4.64737	0.209307
$494$	2.50616	0.112757
$495$	3.12516	0.140466
$496$	4.37255	0.196333
$497$	1.88719	0.0846520
$498$	−20.2271	−0.906400
$499$	37.4820	1.67793	0.838963	−	0.544189i	$-0.183163\pi$
$499$	37.4820	1.67793	0.838963	+	0.544189i	$0.183163\pi$
$500$	−31.3501	−1.40202
$501$	−18.1213	−0.809600
$502$	−3.55755	−0.158781
$503$	−38.7282	−1.72681	−0.863403	−	0.504515i	$-0.831671\pi$
$503$	−38.7282	−1.72681	−0.863403	+	0.504515i	$0.831671\pi$
$504$	0.581728	0.0259122
$505$	−20.2626	−0.901676
$506$	6.67569	0.296771
$507$	2.66371	0.118300
$508$	31.7827	1.41013
$509$	14.1751	0.628300	0.314150	−	0.949373i	$-0.398281\pi$
$509$	14.1751	0.628300	0.314150	+	0.949373i	$0.398281\pi$
$510$	3.41279	0.151121
$511$	−2.99397	−0.132445
$512$	−19.1709	−0.847241
$513$	−0.289067	−0.0127626
$514$	−60.4370	−2.66576
$515$	13.2714	0.584807
$516$	−12.1831	−0.536331
$517$	17.7020	0.778531
$518$	−0.467163	−0.0205260
$519$	−1.82825	−0.0802511
$520$	−9.60444	−0.421182
$521$	−27.0516	−1.18515	−0.592576	−	0.805515i	$-0.701889\pi$
$521$	−27.0516	−1.18515	−0.592576	+	0.805515i	$0.701889\pi$
$522$	9.06331	0.396690
$523$	11.2343	0.491243	0.245621	−	0.969366i	$-0.421008\pi$
$523$	11.2343	0.491243	0.245621	+	0.969366i	$0.421008\pi$
$524$	24.9425	1.08962
$525$	1.02281	0.0446390
$526$	68.1445	2.97124
$527$	2.78340	0.121247
$528$	−3.97604	−0.173035
$529$	−21.1709	−0.920472
$530$	−25.4333	−1.10475
$531$	−2.51037	−0.108941
$532$	0.268978	0.0116617
$533$	−39.0446	−1.69121
$534$	4.54538	0.196698
$535$	6.99508	0.302424
$536$	−20.3398	−0.878547
$537$	10.4689	0.451767
$538$	−42.1662	−1.81791
$539$	−15.5237	−0.668651
$540$	3.88172	0.167043
$541$	18.0139	0.774476	0.387238	−	0.921980i	$-0.373429\pi$
$541$	18.0139	0.774476	0.387238	+	0.921980i	$0.373429\pi$
$542$	−43.9523	−1.88791
$543$	22.2409	0.954450
$544$	−8.27273	−0.354690
$545$	10.2563	0.439333
$546$	2.88250	0.123360
$547$	−5.57944	−0.238560	−0.119280	−	0.992861i	$-0.538059\pi$
$547$	−5.57944	−0.238560	−0.119280	+	0.992861i	$0.538059\pi$
$548$	29.5111	1.26065
$549$	10.8139	0.461527
$550$	15.1847	0.647479
$551$	1.19598	0.0509503
$552$	2.36638	0.100720
$553$	−3.99353	−0.169822
$554$	−2.72229	−0.115659
$555$	−0.889631	−0.0377627
$556$	7.09139	0.300742
$557$	−5.55264	−0.235273	−0.117636	−	0.993057i	$-0.537532\pi$
$557$	−5.55264	−0.235273	−0.117636	+	0.993057i	$0.537532\pi$
$558$	5.42819	0.229794
$559$	−17.2284	−0.728683
$560$	0.813700	0.0343851
$561$	−2.53100	−0.106859
$562$	−62.4898	−2.63597
$563$	−27.6824	−1.16667	−0.583337	−	0.812230i	$-0.698253\pi$
$563$	−27.6824	−1.16667	−0.583337	+	0.812230i	$0.698253\pi$
$564$	21.9873	0.925834
$565$	22.3398	0.939842
$566$	13.0607	0.548980
$567$	−0.332475	−0.0139626
$568$	9.93157	0.416720
$569$	−23.8680	−1.00060	−0.500300	−	0.865852i	$-0.666777\pi$
$569$	−23.8680	−1.00060	−0.500300	+	0.865852i	$0.666777\pi$
$570$	0.878264	0.0367864
$571$	−0.387411	−0.0162127	−0.00810633	−	0.999967i	$-0.502580\pi$
$571$	−0.387411	−0.0162127	−0.00810633	+	0.999967i	$0.502580\pi$
$572$	24.9584	1.04356
$573$	−7.55757	−0.315722
$574$	−7.18514	−0.299902
$575$	4.16063	0.173510
$576$	−12.6043	−0.525180
$577$	26.7220	1.11245	0.556227	−	0.831031i	$-0.312249\pi$
$577$	26.7220	1.11245	0.556227	+	0.831031i	$0.312249\pi$
$578$	34.4762	1.43402
$579$	11.6362	0.483583
$580$	−16.0601	−0.666858
$581$	−3.06994	−0.127363
$582$	−24.7374	−1.02540
$583$	18.8619	0.781180
$584$	−15.7561	−0.651993
$585$	5.48922	0.226951
$586$	28.8831	1.19315
$587$	−36.4741	−1.50545	−0.752724	−	0.658336i	$-0.771261\pi$
$587$	−36.4741	−1.50545	−0.752724	+	0.658336i	$0.771261\pi$
$588$	−19.2817	−0.795165
$589$	0.716293	0.0295143
$590$	7.62718	0.314006
$591$	9.67824	0.398110
$592$	1.13185	0.0465187
$593$	27.3546	1.12332	0.561659	−	0.827369i	$-0.310163\pi$
$593$	27.3546	1.12332	0.561659	+	0.827369i	$0.310163\pi$
$594$	−4.93596	−0.202525
$595$	0.517971	0.0212347
$596$	5.96048	0.244151
$597$	5.15843	0.211121
$598$	11.7256	0.479495
$599$	−26.0131	−1.06287	−0.531434	−	0.847100i	$-0.678347\pi$
$599$	−26.0131	−1.06287	−0.531434	+	0.847100i	$0.678347\pi$
$600$	5.38265	0.219746
$601$	−44.9454	−1.83336	−0.916681	−	0.399619i	$-0.869142\pi$
$601$	−44.9454	−1.83336	−0.916681	+	0.399619i	$0.869142\pi$
$602$	−3.17044	−0.129217
$603$	11.6248	0.473399
$604$	33.2691	1.35370
$605$	−8.21478	−0.333978
$606$	32.0034	1.30005
$607$	25.4018	1.03103	0.515514	−	0.856881i	$-0.327601\pi$
$607$	25.4018	1.03103	0.515514	+	0.856881i	$0.327601\pi$
$608$	−2.12894	−0.0863400
$609$	1.37557	0.0557409
$610$	−32.8556	−1.33028
$611$	31.0928	1.25788
$612$	−3.14372	−0.127077
$613$	29.8640	1.20620	0.603098	−	0.797667i	$-0.293933\pi$
$613$	29.8640	1.20620	0.603098	+	0.797667i	$0.293933\pi$
$614$	−38.2804	−1.54487
$615$	−13.6829	−0.551746
$616$	1.31078	0.0528127
$617$	−15.5686	−0.626768	−0.313384	−	0.949627i	$-0.601463\pi$
$617$	−15.5686	−0.626768	−0.313384	+	0.949627i	$0.601463\pi$
$618$	−20.9612	−0.843182
$619$	18.6373	0.749097	0.374549	−	0.927207i	$-0.377798\pi$
$619$	18.6373	0.749097	0.374549	+	0.927207i	$0.377798\pi$
$620$	−9.61870	−0.386296
$621$	−1.35246	−0.0542723
$622$	−36.3222	−1.45639
$623$	0.689867	0.0276390
$624$	−6.98376	−0.279574
$625$	−0.154367	−0.00617468
$626$	61.2857	2.44947
$627$	−0.651339	−0.0260120
$628$	−37.5382	−1.49794
$629$	0.720493	0.0287279
$630$	1.01015	0.0402453
$631$	−38.1509	−1.51876	−0.759381	−	0.650646i	$-0.774498\pi$
$631$	−38.1509	−1.51876	−0.759381	+	0.650646i	$0.774498\pi$
$632$	−21.0164	−0.835989
$633$	12.1539	0.483073
$634$	36.2163	1.43833
$635$	15.7505	0.625039
$636$	23.4281	0.928984
$637$	−27.2667	−1.08035
$638$	20.4219	0.808510
$639$	−5.67619	−0.224547
$640$	17.8658	0.706208
$641$	−29.3747	−1.16023	−0.580115	−	0.814534i	$-0.696993\pi$
$641$	−29.3747	−1.16023	−0.580115	+	0.814534i	$0.696993\pi$
$642$	−11.0482	−0.436038
$643$	−13.5473	−0.534253	−0.267126	−	0.963661i	$-0.586074\pi$
$643$	−13.5473	−0.534253	−0.267126	+	0.963661i	$0.586074\pi$
$644$	1.25847	0.0495907
$645$	−6.03755	−0.237728
$646$	−0.711287	−0.0279852
$647$	48.9846	1.92578	0.962892	−	0.269887i	$-0.0869865\pi$
$647$	48.9846	1.92578	0.962892	+	0.269887i	$0.0869865\pi$
$648$	−1.74969	−0.0687343
$649$	−5.65648	−0.222036
$650$	26.6714	1.04614
$651$	0.823855	0.0322894
$652$	−18.0550	−0.707090
$653$	28.1686	1.10232	0.551161	−	0.834399i	$-0.314185\pi$
$653$	28.1686	1.10232	0.551161	+	0.834399i	$0.314185\pi$
$654$	−16.1991	−0.633436
$655$	12.3607	0.482973
$656$	17.4083	0.679679
$657$	9.00509	0.351322
$658$	5.72182	0.223060
$659$	−30.6209	−1.19282	−0.596410	−	0.802680i	$-0.703407\pi$
$659$	−30.6209	−1.19282	−0.596410	+	0.802680i	$0.703407\pi$
$660$	8.74647	0.340456
$661$	23.4389	0.911667	0.455834	−	0.890065i	$-0.349341\pi$
$661$	23.4389	0.911667	0.455834	+	0.890065i	$0.349341\pi$
$662$	−3.63213	−0.141167
$663$	−4.44560	−0.172653
$664$	−16.1560	−0.626973
$665$	0.133297	0.00516904
$666$	1.40511	0.0544468
$667$	5.59562	0.216663
$668$	−50.7166	−1.96228
$669$	2.57474	0.0995452
$670$	−35.3194	−1.36451
$671$	24.3664	0.940655
$672$	−2.44863	−0.0944581
$673$	45.0361	1.73602	0.868008	−	0.496551i	$-0.165400\pi$
$673$	45.0361	1.73602	0.868008	+	0.496551i	$0.165400\pi$
$674$	−53.5034	−2.06088
$675$	−3.07635	−0.118409
$676$	7.45500	0.286731
$677$	24.8050	0.953334	0.476667	−	0.879084i	$-0.341845\pi$
$677$	24.8050	0.953334	0.476667	+	0.879084i	$0.341845\pi$
$678$	−35.2841	−1.35508
$679$	−3.75448	−0.144084
$680$	2.72589	0.104533
$681$	−16.6946	−0.639738
$682$	12.2311	0.468352
$683$	4.16000	0.159178	0.0795890	−	0.996828i	$-0.474639\pi$
$683$	4.16000	0.159178	0.0795890	+	0.996828i	$0.474639\pi$
$684$	−0.809019	−0.0309336
$685$	14.6248	0.558783
$686$	−10.1160	−0.386229
$687$	21.3528	0.814660
$688$	7.68139	0.292850
$689$	33.1302	1.26216
$690$	4.10914	0.156432
$691$	−32.2696	−1.22759	−0.613797	−	0.789464i	$-0.710359\pi$
$691$	−32.2696	−1.22759	−0.613797	+	0.789464i	$0.710359\pi$
$692$	−5.11676	−0.194510
$693$	−0.749148	−0.0284578
$694$	55.3294	2.10028
$695$	3.51426	0.133304
$696$	7.23910	0.274398
$697$	11.0815	0.419740
$698$	−54.3587	−2.05751
$699$	−18.5904	−0.703154
$700$	2.86256	0.108195
$701$	40.1102	1.51494	0.757470	−	0.652870i	$-0.226435\pi$
$701$	40.1102	1.51494	0.757470	+	0.652870i	$0.226435\pi$
$702$	−8.66982	−0.327221
$703$	0.185415	0.00699306
$704$	−28.4007	−1.07039
$705$	10.8962	0.410375
$706$	−3.36357	−0.126590
$707$	4.85726	0.182676
$708$	−7.02583	−0.264047
$709$	24.7688	0.930210	0.465105	−	0.885255i	$-0.346017\pi$
$709$	24.7688	0.930210	0.465105	+	0.885255i	$0.346017\pi$
$710$	17.2458	0.647223
$711$	12.0115	0.450467
$712$	3.63051	0.136059
$713$	3.35132	0.125508
$714$	−0.818097	−0.0306165
$715$	12.3686	0.462559
$716$	29.2996	1.09498
$717$	22.2792	0.832034
$718$	−23.9961	−0.895528
$719$	−45.2696	−1.68827	−0.844135	−	0.536130i	$-0.819886\pi$
$719$	−45.2696	−1.68827	−0.844135	+	0.536130i	$0.819886\pi$
$720$	−2.44740	−0.0912093
$721$	−3.18135	−0.118480
$722$	41.4383	1.54218
$723$	5.51611	0.205146
$724$	62.2463	2.31337
$725$	12.7280	0.472705
$726$	12.9746	0.481534
$727$	−6.72811	−0.249532	−0.124766	−	0.992186i	$-0.539818\pi$
$727$	−6.72811	−0.249532	−0.124766	+	0.992186i	$0.539818\pi$
$728$	2.30233	0.0853299
$729$	1.00000	0.0370370
$730$	−27.3599	−1.01264
$731$	4.88968	0.180852
$732$	30.2652	1.11863
$733$	−24.0904	−0.889798	−0.444899	−	0.895581i	$-0.646761\pi$
$733$	−24.0904	−0.889798	−0.444899	+	0.895581i	$0.646761\pi$
$734$	−14.6471	−0.540634
$735$	−9.55540	−0.352456
$736$	−9.96069	−0.367156
$737$	26.1936	0.964854
$738$	21.6111	0.795515
$739$	9.02819	0.332107	0.166054	−	0.986117i	$-0.446898\pi$
$739$	9.02819	0.332107	0.166054	+	0.986117i	$0.446898\pi$
$740$	−2.48983	−0.0915281
$741$	−1.14405	−0.0420278
$742$	6.09675	0.223819
$743$	13.6177	0.499586	0.249793	−	0.968299i	$-0.419637\pi$
$743$	13.6177	0.499586	0.249793	+	0.968299i	$0.419637\pi$
$744$	4.33564	0.158952
$745$	2.95382	0.108220
$746$	64.8445	2.37413
$747$	9.23361	0.337840
$748$	−7.08358	−0.259001
$749$	−1.67683	−0.0612699
$750$	24.5381	0.896006
$751$	15.9126	0.580658	0.290329	−	0.956927i	$-0.406235\pi$
$751$	15.9126	0.580658	0.290329	+	0.956927i	$0.406235\pi$
$752$	−13.8629	−0.505528
$753$	1.62401	0.0591821
$754$	35.8702	1.30632
$755$	16.4871	0.600027
$756$	−0.930506	−0.0338422
$757$	−10.1407	−0.368571	−0.184286	−	0.982873i	$-0.558997\pi$
$757$	−10.1407	−0.368571	−0.184286	+	0.982873i	$0.558997\pi$
$758$	47.2947	1.71782
$759$	−3.04743	−0.110615
$760$	0.701492	0.0254458
$761$	−9.83506	−0.356521	−0.178260	−	0.983983i	$-0.557047\pi$
$761$	−9.83506	−0.356521	−0.178260	+	0.983983i	$0.557047\pi$
$762$	−24.8768	−0.901190
$763$	−2.45860	−0.0890072
$764$	−21.1516	−0.765237
$765$	−1.55793	−0.0563269
$766$	69.4850	2.51060
$767$	−9.93539	−0.358746
$768$	−3.00908	−0.108581
$769$	18.5127	0.667584	0.333792	−	0.942647i	$-0.391672\pi$
$769$	18.5127	0.667584	0.333792	+	0.942647i	$0.391672\pi$
$770$	2.27611	0.0820255
$771$	27.5892	0.993602
$772$	32.5665	1.17209
$773$	40.6002	1.46029	0.730145	−	0.683293i	$-0.239453\pi$
$773$	40.6002	1.46029	0.730145	+	0.683293i	$0.239453\pi$
$774$	9.53587	0.342760
$775$	7.62303	0.273827
$776$	−19.7584	−0.709285
$777$	0.213258	0.00765059
$778$	−25.0377	−0.897646
$779$	2.85175	0.102175
$780$	15.3628	0.550078
$781$	−12.7899	−0.457657
$782$	−3.32790	−0.119006
$783$	−4.13736	−0.147857
$784$	12.1570	0.434179
$785$	−18.6027	−0.663960
$786$	−19.5228	−0.696356
$787$	53.0054	1.88944	0.944719	−	0.327881i	$-0.106334\pi$
$787$	53.0054	1.88944	0.944719	+	0.327881i	$0.106334\pi$
$788$	27.0868	0.964926
$789$	−31.1077	−1.10746
$790$	−36.4943	−1.29841
$791$	−5.35518	−0.190408
$792$	−3.94248	−0.140090
$793$	42.7987	1.51983
$794$	−40.2539	−1.42856
$795$	11.6102	0.411772
$796$	14.4370	0.511707
$797$	−28.8675	−1.02254	−0.511270	−	0.859420i	$-0.670825\pi$
$797$	−28.8675	−1.02254	−0.511270	+	0.859420i	$0.670825\pi$
$798$	−0.210533	−0.00745278
$799$	−8.82461	−0.312192
$800$	−22.6569	−0.801042
$801$	−2.07495	−0.0733146
$802$	8.62949	0.304718
$803$	20.2907	0.716043
$804$	32.5347	1.14741
$805$	0.623658	0.0219810
$806$	21.4834	0.756720
$807$	19.2487	0.677586
$808$	25.5619	0.899265
$809$	38.5206	1.35431	0.677157	−	0.735839i	$-0.263212\pi$
$809$	38.5206	1.35431	0.677157	+	0.735839i	$0.263212\pi$
$810$	−3.03827	−0.106754
$811$	3.67439	0.129025	0.0645127	−	0.997917i	$-0.479451\pi$
$811$	3.67439	0.129025	0.0645127	+	0.997917i	$0.479451\pi$
$812$	3.84984	0.135103
$813$	20.0640	0.703676
$814$	3.16605	0.110970
$815$	−8.94749	−0.313417
$816$	1.98210	0.0693874
$817$	1.25833	0.0440235
$818$	47.8454	1.67288
$819$	−1.31585	−0.0459795
$820$	−38.2946	−1.33731
$821$	48.5616	1.69481	0.847405	−	0.530947i	$-0.178164\pi$
$821$	48.5616	1.69481	0.847405	+	0.530947i	$0.178164\pi$
$822$	−23.0987	−0.805660
$823$	−14.2285	−0.495974	−0.247987	−	0.968763i	$-0.579769\pi$
$823$	−14.2285	−0.495974	−0.247987	+	0.968763i	$0.579769\pi$
$824$	−16.7422	−0.583244
$825$	−6.93177	−0.241333
$826$	−1.82835	−0.0636164
$827$	49.8635	1.73392	0.866961	−	0.498375i	$-0.166070\pi$
$827$	49.8635	1.73392	0.866961	+	0.498375i	$0.166070\pi$
$828$	−3.78516	−0.131544
$829$	10.6588	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$829$	10.6588	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$830$	−28.0542	−0.973776
$831$	1.24271	0.0431093
$832$	−49.8846	−1.72944
$833$	7.73871	0.268130
$834$	−5.55052	−0.192199
$835$	−25.1335	−0.869781
$836$	−1.82292	−0.0630470
$837$	−2.47795	−0.0856504
$838$	−79.7951	−2.75648
$839$	20.2325	0.698504	0.349252	−	0.937029i	$-0.386436\pi$
$839$	20.2325	0.698504	0.349252	+	0.937029i	$0.386436\pi$
$840$	0.806832	0.0278384
$841$	−11.8822	−0.409732
$842$	−39.1487	−1.34915
$843$	28.5263	0.982499
$844$	34.0154	1.17086
$845$	3.69446	0.127093
$846$	−17.2098	−0.591684
$847$	1.96921	0.0676627
$848$	−14.7713	−0.507248
$849$	−5.96214	−0.204620
$850$	−7.56975	−0.259640
$851$	0.867503	0.0297376
$852$	−15.8861	−0.544249
$853$	23.5573	0.806586	0.403293	−	0.915071i	$-0.367865\pi$
$853$	23.5573	0.806586	0.403293	+	0.915071i	$0.367865\pi$
$854$	7.87598	0.269510
$855$	−0.400924	−0.0137113
$856$	−8.82450	−0.301615
$857$	−19.3776	−0.661925	−0.330963	−	0.943644i	$-0.607373\pi$
$857$	−19.3776	−0.661925	−0.330963	+	0.943644i	$0.607373\pi$
$858$	−19.5353	−0.666923
$859$	0.970475	0.0331122	0.0165561	−	0.999863i	$-0.494730\pi$
$859$	0.970475	0.0331122	0.0165561	+	0.999863i	$0.494730\pi$
$860$	−16.8975	−0.576199
$861$	3.27999	0.111782
$862$	15.0764	0.513504
$863$	−30.2606	−1.03008	−0.515041	−	0.857165i	$-0.672223\pi$
$863$	−30.2606	−1.03008	−0.515041	+	0.857165i	$0.672223\pi$
$864$	7.36487	0.250558
$865$	−2.53570	−0.0862164
$866$	−19.8757	−0.675405
$867$	−15.7383	−0.534500
$868$	2.30575	0.0782621
$869$	27.0649	0.918115
$870$	12.5704	0.426178
$871$	46.0080	1.55892
$872$	−12.9387	−0.438159
$873$	11.2925	0.382194
$874$	−0.856418	−0.0289688
$875$	3.72423	0.125902
$876$	25.2028	0.851524
$877$	12.1507	0.410301	0.205151	−	0.978730i	$-0.434232\pi$
$877$	12.1507	0.410301	0.205151	+	0.978730i	$0.434232\pi$
$878$	−41.8220	−1.41143
$879$	−13.1850	−0.444720
$880$	−5.51461	−0.185897
$881$	−17.1719	−0.578536	−0.289268	−	0.957248i	$-0.593412\pi$
$881$	−17.1719	−0.578536	−0.289268	+	0.957248i	$0.593412\pi$
$882$	15.0920	0.508176
$883$	−48.4360	−1.63000	−0.815001	−	0.579460i	$-0.803264\pi$
$883$	−48.4360	−1.63000	−0.815001	+	0.579460i	$0.803264\pi$
$884$	−12.4420	−0.418471
$885$	−3.48178	−0.117039
$886$	−18.7091	−0.628545
$887$	4.08983	0.137323	0.0686616	−	0.997640i	$-0.478127\pi$
$887$	4.08983	0.137323	0.0686616	+	0.997640i	$0.478127\pi$
$888$	1.12230	0.0376618
$889$	−3.77563	−0.126631
$890$	6.30425	0.211319
$891$	2.25325	0.0754866
$892$	7.20599	0.241275
$893$	−2.27097	−0.0759950
$894$	−4.66535	−0.156033
$895$	14.5199	0.485348
$896$	−4.28270	−0.143075
$897$	−5.35268	−0.178721
$898$	50.2426	1.67662
$899$	10.2522	0.341929
$900$	−8.60985	−0.286995
$901$	−9.40286	−0.313255
$902$	48.6951	1.62137
$903$	1.44729	0.0481629
$904$	−28.1823	−0.937330
$905$	30.8473	1.02540
$906$	−26.0402	−0.865127
$907$	−23.7684	−0.789216	−0.394608	−	0.918850i	$-0.629120\pi$
$907$	−23.7684	−0.789216	−0.394608	+	0.918850i	$0.629120\pi$
$908$	−46.7236	−1.55058
$909$	−14.6094	−0.484563
$910$	3.99790	0.132529
$911$	−1.37735	−0.0456337	−0.0228169	−	0.999740i	$-0.507263\pi$
$911$	−1.37735	−0.0456337	−0.0228169	+	0.999740i	$0.507263\pi$
$912$	0.510083	0.0168905
$913$	20.8056	0.688565
$914$	20.1104	0.665192
$915$	14.9985	0.495834
$916$	59.7607	1.97455
$917$	−2.96305	−0.0978484
$918$	2.46063	0.0812129
$919$	−33.6608	−1.11037	−0.555183	−	0.831728i	$-0.687352\pi$
$919$	−33.6608	−1.11037	−0.555183	+	0.831728i	$0.687352\pi$
$920$	3.28208	0.108207
$921$	17.4748	0.575816
$922$	24.4390	0.804857
$923$	−22.4649	−0.739441
$924$	−2.09666	−0.0689750
$925$	1.97325	0.0648800
$926$	−30.0846	−0.988641
$927$	9.56869	0.314277
$928$	−30.4712	−1.00027
$929$	6.20130	0.203458	0.101729	−	0.994812i	$-0.467563\pi$
$929$	6.20130	0.203458	0.101729	+	0.994812i	$0.467563\pi$
$930$	7.52868	0.246875
$931$	1.99152	0.0652693
$932$	−52.0295	−1.70428
$933$	16.5809	0.542836
$934$	65.7766	2.15228
$935$	−3.51039	−0.114802
$936$	−6.92482	−0.226345
$937$	−43.2396	−1.41258	−0.706288	−	0.707925i	$-0.749632\pi$
$937$	−43.2396	−1.41258	−0.706288	+	0.707925i	$0.749632\pi$
$938$	8.46658	0.276444
$939$	−27.9767	−0.912985
$940$	30.4955	0.994655
$941$	38.8272	1.26573	0.632866	−	0.774261i	$-0.281878\pi$
$941$	38.8272	1.26573	0.632866	+	0.774261i	$0.281878\pi$
$942$	29.3817	0.957306
$943$	13.3425	0.434492
$944$	4.42975	0.144176
$945$	−0.461129	−0.0150005
$946$	21.4867	0.698592
$947$	−47.4220	−1.54101	−0.770504	−	0.637435i	$-0.779996\pi$
$947$	−47.4220	−1.54101	−0.770504	+	0.637435i	$0.779996\pi$
$948$	33.6170	1.09183
$949$	35.6398	1.15692
$950$	−1.94803	−0.0632026
$951$	−16.5326	−0.536107
$952$	−0.653436	−0.0211780
$953$	17.7973	0.576510	0.288255	−	0.957554i	$-0.406925\pi$
$953$	17.7973	0.576510	0.288255	+	0.957554i	$0.406925\pi$
$954$	−18.3375	−0.593698
$955$	−10.4820	−0.339191
$956$	62.3535	2.01666
$957$	−9.32250	−0.301354
$958$	82.4554	2.66401
$959$	−3.50577	−0.113207
$960$	−17.4817	−0.564219
$961$	−24.8598	−0.801928
$962$	5.56105	0.179295
$963$	5.04347	0.162523
$964$	15.4381	0.497227
$965$	16.1389	0.519529
$966$	−0.985022	−0.0316926
$967$	−44.7630	−1.43948	−0.719741	−	0.694243i	$-0.755739\pi$
$967$	−44.7630	−1.43948	−0.719741	+	0.694243i	$0.755739\pi$
$968$	10.3632	0.333086
$969$	0.324700	0.0104309
$970$	−34.3097	−1.10162
$971$	27.9050	0.895514	0.447757	−	0.894155i	$-0.352223\pi$
$971$	27.9050	0.895514	0.447757	+	0.894155i	$0.352223\pi$
$972$	2.79873	0.0897692
$973$	−0.842422	−0.0270068
$974$	2.16556	0.0693890
$975$	−12.1754	−0.389924
$976$	−19.0820	−0.610801
$977$	32.1666	1.02910	0.514551	−	0.857460i	$-0.327959\pi$
$977$	32.1666	1.02910	0.514551	+	0.857460i	$0.327959\pi$
$978$	14.1319	0.451889
$979$	−4.67537	−0.149425
$980$	−26.7429	−0.854272
$981$	7.39484	0.236099
$982$	48.7142	1.55453
$983$	−31.6431	−1.00926	−0.504629	−	0.863336i	$-0.668371\pi$
$983$	−31.6431	−1.00926	−0.504629	+	0.863336i	$0.668371\pi$
$984$	17.2613	0.550272
$985$	13.4233	0.427703
$986$	−10.1805	−0.324214
$987$	−2.61199	−0.0831404
$988$	−3.20189	−0.101866
$989$	5.88737	0.187208
$990$	−6.84598	−0.217579
$991$	−3.39712	−0.107913	−0.0539565	−	0.998543i	$-0.517183\pi$
$991$	−3.39712	−0.107913	−0.0539565	+	0.998543i	$0.517183\pi$
$992$	−18.2498	−0.579431
$993$	1.65806	0.0526168
$994$	−4.13408	−0.131125
$995$	7.15453	0.226814
$996$	25.8424	0.818846
$997$	−57.3477	−1.81622	−0.908110	−	0.418732i	$-0.862475\pi$
$997$	−57.3477	−1.81622	−0.908110	+	0.418732i	$0.862475\pi$
$998$	−82.1081	−2.59909
$999$	−0.641426	−0.0202938

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8031.2.a.d.1.19	✓	132

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8031.2.a.d.1.19	✓	132	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:	\( N \)	\(=\)	\( 8031 = 3 \cdot 2677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.19060 q^{2} +1.00000 q^{3} +2.79873 q^{4} +1.38696 q^{5} -2.19060 q^{6} -0.332475 q^{7} -1.74969 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.19060 q^{2} +1.00000 q^{3} +2.79873 q^{4} +1.38696 q^{5} -2.19060 q^{6} -0.332475 q^{7} -1.74969 q^{8} +1.00000 q^{9} -3.03827 q^{10} +2.25325 q^{11} +2.79873 q^{12} +3.95774 q^{13} +0.728319 q^{14} +1.38696 q^{15} -1.76458 q^{16} -1.12327 q^{17} -2.19060 q^{18} -0.289067 q^{19} +3.88172 q^{20} -0.332475 q^{21} -4.93596 q^{22} -1.35246 q^{23} -1.74969 q^{24} -3.07635 q^{25} -8.66982 q^{26} +1.00000 q^{27} -0.930506 q^{28} -4.13736 q^{29} -3.03827 q^{30} -2.47795 q^{31} +7.36487 q^{32} +2.25325 q^{33} +2.46063 q^{34} -0.461129 q^{35} +2.79873 q^{36} -0.641426 q^{37} +0.633230 q^{38} +3.95774 q^{39} -2.42675 q^{40} -9.86537 q^{41} +0.728319 q^{42} -4.35309 q^{43} +6.30622 q^{44} +1.38696 q^{45} +2.96270 q^{46} +7.85619 q^{47} -1.76458 q^{48} -6.88946 q^{49} +6.73904 q^{50} -1.12327 q^{51} +11.0766 q^{52} +8.37098 q^{53} -2.19060 q^{54} +3.12516 q^{55} +0.581728 q^{56} -0.289067 q^{57} +9.06331 q^{58} -2.51037 q^{59} +3.88172 q^{60} +10.8139 q^{61} +5.42819 q^{62} -0.332475 q^{63} -12.6043 q^{64} +5.48922 q^{65} -4.93596 q^{66} +11.6248 q^{67} -3.14372 q^{68} -1.35246 q^{69} +1.01015 q^{70} -5.67619 q^{71} -1.74969 q^{72} +9.00509 q^{73} +1.40511 q^{74} -3.07635 q^{75} -0.809019 q^{76} -0.749148 q^{77} -8.66982 q^{78} +12.0115 q^{79} -2.44740 q^{80} +1.00000 q^{81} +21.6111 q^{82} +9.23361 q^{83} -0.930506 q^{84} -1.55793 q^{85} +9.53587 q^{86} -4.13736 q^{87} -3.94248 q^{88} -2.07495 q^{89} -3.03827 q^{90} -1.31585 q^{91} -3.78516 q^{92} -2.47795 q^{93} -17.2098 q^{94} -0.400924 q^{95} +7.36487 q^{96} +11.2925 q^{97} +15.0920 q^{98} +2.25325 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 4 q^{2} + 132 q^{3} + 156 q^{4} + 20 q^{5} + 4 q^{6} + 44 q^{7} + 9 q^{8} + 132 q^{9}+O(q^{10}) \) 132 * q + 4 * q^2 + 132 * q^3 + 156 * q^4 + 20 * q^5 + 4 * q^6 + 44 * q^7 + 9 * q^8 + 132 * q^9	\( 132 q + 4 q^{2} + 132 q^{3} + 156 q^{4} + 20 q^{5} + 4 q^{6} + 44 q^{7} + 9 q^{8} + 132 q^{9} + 40 q^{10} + 24 q^{11} + 156 q^{12} + 62 q^{13} + 25 q^{14} + 20 q^{15} + 192 q^{16} + 77 q^{17} + 4 q^{18} + 86 q^{19} + 26 q^{20} + 44 q^{21} + 52 q^{22} + 17 q^{23} + 9 q^{24} + 212 q^{25} + 13 q^{26} + 132 q^{27} + 95 q^{28} + 52 q^{29} + 40 q^{30} + 59 q^{31} - 8 q^{32} + 24 q^{33} + 41 q^{34} + 21 q^{35} + 156 q^{36} + 76 q^{37} + 2 q^{38} + 62 q^{39} + 91 q^{40} + 114 q^{41} + 25 q^{42} + 173 q^{43} + 44 q^{44} + 20 q^{45} + 48 q^{46} + 15 q^{47} + 192 q^{48} + 262 q^{49} - 9 q^{50} + 77 q^{51} + 144 q^{52} + 15 q^{53} + 4 q^{54} + 111 q^{55} + 66 q^{56} + 86 q^{57} + 33 q^{58} + 20 q^{59} + 26 q^{60} + 182 q^{61} + 16 q^{62} + 44 q^{63} + 255 q^{64} + 70 q^{65} + 52 q^{66} + 169 q^{67} + 128 q^{68} + 17 q^{69} + 2 q^{70} + 23 q^{71} + 9 q^{72} + 148 q^{73} + 57 q^{74} + 212 q^{75} + 143 q^{76} + 31 q^{77} + 13 q^{78} + 152 q^{79} + 27 q^{80} + 132 q^{81} + 67 q^{82} + 28 q^{83} + 95 q^{84} + 88 q^{85} - 10 q^{86} + 52 q^{87} + 130 q^{88} + 136 q^{89} + 40 q^{90} + 125 q^{91} + 59 q^{93} + 95 q^{94} + 2 q^{95} - 8 q^{96} + 147 q^{97} - 18 q^{98} + 24 q^{99}+O(q^{100}) \) 132 * q + 4 * q^2 + 132 * q^3 + 156 * q^4 + 20 * q^5 + 4 * q^6 + 44 * q^7 + 9 * q^8 + 132 * q^9 + 40 * q^10 + 24 * q^11 + 156 * q^12 + 62 * q^13 + 25 * q^14 + 20 * q^15 + 192 * q^16 + 77 * q^17 + 4 * q^18 + 86 * q^19 + 26 * q^20 + 44 * q^21 + 52 * q^22 + 17 * q^23 + 9 * q^24 + 212 * q^25 + 13 * q^26 + 132 * q^27 + 95 * q^28 + 52 * q^29 + 40 * q^30 + 59 * q^31 - 8 * q^32 + 24 * q^33 + 41 * q^34 + 21 * q^35 + 156 * q^36 + 76 * q^37 + 2 * q^38 + 62 * q^39 + 91 * q^40 + 114 * q^41 + 25 * q^42 + 173 * q^43 + 44 * q^44 + 20 * q^45 + 48 * q^46 + 15 * q^47 + 192 * q^48 + 262 * q^49 - 9 * q^50 + 77 * q^51 + 144 * q^52 + 15 * q^53 + 4 * q^54 + 111 * q^55 + 66 * q^56 + 86 * q^57 + 33 * q^58 + 20 * q^59 + 26 * q^60 + 182 * q^61 + 16 * q^62 + 44 * q^63 + 255 * q^64 + 70 * q^65 + 52 * q^66 + 169 * q^67 + 128 * q^68 + 17 * q^69 + 2 * q^70 + 23 * q^71 + 9 * q^72 + 148 * q^73 + 57 * q^74 + 212 * q^75 + 143 * q^76 + 31 * q^77 + 13 * q^78 + 152 * q^79 + 27 * q^80 + 132 * q^81 + 67 * q^82 + 28 * q^83 + 95 * q^84 + 88 * q^85 - 10 * q^86 + 52 * q^87 + 130 * q^88 + 136 * q^89 + 40 * q^90 + 125 * q^91 + 59 * q^93 + 95 * q^94 + 2 * q^95 - 8 * q^96 + 147 * q^97 - 18 * q^98 + 24 * q^99

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