LMFDB - Embedded newform 8045.2.a.e.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, 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("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$0$
Dimension:	$142$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58612 q^{2} +1.58438 q^{3} +4.68803 q^{4} +1.00000 q^{5} -4.09739 q^{6} -2.71567 q^{7} -6.95157 q^{8} -0.489749 q^{9} +O(q^{10})$	\(q-2.58612 q^{2} +1.58438 q^{3} +4.68803 q^{4} +1.00000 q^{5} -4.09739 q^{6} -2.71567 q^{7} -6.95157 q^{8} -0.489749 q^{9} -2.58612 q^{10} -0.283992 q^{11} +7.42760 q^{12} +4.05561 q^{13} +7.02304 q^{14} +1.58438 q^{15} +8.60155 q^{16} +0.100323 q^{17} +1.26655 q^{18} +0.606273 q^{19} +4.68803 q^{20} -4.30264 q^{21} +0.734438 q^{22} -5.64772 q^{23} -11.0139 q^{24} +1.00000 q^{25} -10.4883 q^{26} -5.52908 q^{27} -12.7311 q^{28} -4.42803 q^{29} -4.09739 q^{30} -5.63092 q^{31} -8.34152 q^{32} -0.449950 q^{33} -0.259447 q^{34} -2.71567 q^{35} -2.29596 q^{36} -2.57414 q^{37} -1.56790 q^{38} +6.42561 q^{39} -6.95157 q^{40} -9.66482 q^{41} +11.1271 q^{42} +0.0774746 q^{43} -1.33136 q^{44} -0.489749 q^{45} +14.6057 q^{46} +5.86816 q^{47} +13.6281 q^{48} +0.374840 q^{49} -2.58612 q^{50} +0.158949 q^{51} +19.0128 q^{52} -2.28736 q^{53} +14.2989 q^{54} -0.283992 q^{55} +18.8781 q^{56} +0.960566 q^{57} +11.4514 q^{58} +2.76580 q^{59} +7.42760 q^{60} +7.73021 q^{61} +14.5622 q^{62} +1.33000 q^{63} +4.36909 q^{64} +4.05561 q^{65} +1.16363 q^{66} +11.5707 q^{67} +0.470317 q^{68} -8.94812 q^{69} +7.02304 q^{70} -0.557844 q^{71} +3.40453 q^{72} +0.324794 q^{73} +6.65703 q^{74} +1.58438 q^{75} +2.84223 q^{76} +0.771227 q^{77} -16.6174 q^{78} +13.4457 q^{79} +8.60155 q^{80} -7.29090 q^{81} +24.9944 q^{82} +8.90987 q^{83} -20.1709 q^{84} +0.100323 q^{85} -0.200359 q^{86} -7.01567 q^{87} +1.97419 q^{88} +15.0748 q^{89} +1.26655 q^{90} -11.0137 q^{91} -26.4767 q^{92} -8.92149 q^{93} -15.1758 q^{94} +0.606273 q^{95} -13.2161 q^{96} +4.43169 q^{97} -0.969381 q^{98} +0.139085 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9}+O(q^{10}) $ 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9} + 21 q^{10} + 36 q^{11} + 55 q^{12} + 57 q^{13} + 2 q^{14} + 33 q^{15} + 179 q^{16} + 55 q^{17} + 65 q^{18} + 130 q^{19} + 157 q^{20} + 28 q^{21} + 30 q^{22} + 117 q^{23} + 21 q^{24} + 142 q^{25} + 21 q^{26} + 120 q^{27} + 135 q^{28} + 12 q^{29} + 15 q^{30} + 74 q^{31} + 126 q^{32} + 55 q^{33} + 35 q^{34} + 63 q^{35} + 186 q^{36} + 75 q^{37} + 65 q^{38} + 23 q^{39} + 60 q^{40} + 22 q^{41} + 10 q^{42} + 190 q^{43} + 22 q^{44} + 157 q^{45} + 56 q^{46} + 102 q^{47} + 78 q^{48} + 197 q^{49} + 21 q^{50} + 30 q^{51} + 120 q^{52} + 56 q^{53} - 6 q^{54} + 36 q^{55} + 3 q^{56} + 68 q^{57} + 31 q^{58} + 55 q^{59} + 55 q^{60} + 90 q^{61} + 68 q^{62} + 167 q^{63} + 180 q^{64} + 57 q^{65} + 17 q^{66} + 151 q^{67} + 119 q^{68} + 21 q^{69} + 2 q^{70} + 4 q^{71} + 130 q^{72} + 143 q^{73} - 46 q^{74} + 33 q^{75} + 213 q^{76} + 75 q^{77} - 24 q^{78} + 47 q^{79} + 179 q^{80} + 150 q^{81} + 69 q^{82} + 201 q^{83} - 31 q^{84} + 55 q^{85} - 4 q^{86} + 153 q^{87} + 37 q^{88} + 25 q^{89} + 65 q^{90} + 132 q^{91} + 194 q^{92} + 52 q^{93} + 18 q^{94} + 130 q^{95} + 13 q^{96} + 80 q^{97} + 58 q^{98} + 103 q^{99}+O(q^{100}) \) 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9 + 21 * q^10 + 36 * q^11 + 55 * q^12 + 57 * q^13 + 2 * q^14 + 33 * q^15 + 179 * q^16 + 55 * q^17 + 65 * q^18 + 130 * q^19 + 157 * q^20 + 28 * q^21 + 30 * q^22 + 117 * q^23 + 21 * q^24 + 142 * q^25 + 21 * q^26 + 120 * q^27 + 135 * q^28 + 12 * q^29 + 15 * q^30 + 74 * q^31 + 126 * q^32 + 55 * q^33 + 35 * q^34 + 63 * q^35 + 186 * q^36 + 75 * q^37 + 65 * q^38 + 23 * q^39 + 60 * q^40 + 22 * q^41 + 10 * q^42 + 190 * q^43 + 22 * q^44 + 157 * q^45 + 56 * q^46 + 102 * q^47 + 78 * q^48 + 197 * q^49 + 21 * q^50 + 30 * q^51 + 120 * q^52 + 56 * q^53 - 6 * q^54 + 36 * q^55 + 3 * q^56 + 68 * q^57 + 31 * q^58 + 55 * q^59 + 55 * q^60 + 90 * q^61 + 68 * q^62 + 167 * q^63 + 180 * q^64 + 57 * q^65 + 17 * q^66 + 151 * q^67 + 119 * q^68 + 21 * q^69 + 2 * q^70 + 4 * q^71 + 130 * q^72 + 143 * q^73 - 46 * q^74 + 33 * q^75 + 213 * q^76 + 75 * q^77 - 24 * q^78 + 47 * q^79 + 179 * q^80 + 150 * q^81 + 69 * q^82 + 201 * q^83 - 31 * q^84 + 55 * q^85 - 4 * q^86 + 153 * q^87 + 37 * q^88 + 25 * q^89 + 65 * q^90 + 132 * q^91 + 194 * q^92 + 52 * q^93 + 18 * q^94 + 130 * q^95 + 13 * q^96 + 80 * q^97 + 58 * q^98 + 103 * 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 $

$2$	−2.58612	−1.82866	−0.914332	−	0.404965i	$-0.867284\pi$
$2$	−2.58612	−1.82866	−0.914332	+	0.404965i	$0.867284\pi$
$3$	1.58438	0.914741	0.457370	−	0.889276i	$-0.348791\pi$
$3$	1.58438	0.914741	0.457370	+	0.889276i	$0.348791\pi$
$4$	4.68803	2.34401
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−4.09739	−1.67275
$7$	−2.71567	−1.02643	−0.513213	−	0.858262i	$-0.671545\pi$
$7$	−2.71567	−1.02643	−0.513213	+	0.858262i	$0.671545\pi$
$8$	−6.95157	−2.45775
$9$	−0.489749	−0.163250
$10$	−2.58612	−0.817804
$11$	−0.283992	−0.0856268	−0.0428134	−	0.999083i	$-0.513632\pi$
$11$	−0.283992	−0.0856268	−0.0428134	+	0.999083i	$0.513632\pi$
$12$	7.42760	2.14416
$13$	4.05561	1.12482	0.562411	−	0.826858i	$-0.309874\pi$
$13$	4.05561	1.12482	0.562411	+	0.826858i	$0.309874\pi$
$14$	7.02304	1.87699
$15$	1.58438	0.409084
$16$	8.60155	2.15039
$17$	0.100323	0.0243319	0.0121659	−	0.999926i	$-0.496127\pi$
$17$	0.100323	0.0243319	0.0121659	+	0.999926i	$0.496127\pi$
$18$	1.26655	0.298529
$19$	0.606273	0.139089	0.0695443	−	0.997579i	$-0.477845\pi$
$19$	0.606273	0.139089	0.0695443	+	0.997579i	$0.477845\pi$
$20$	4.68803	1.04827
$21$	−4.30264	−0.938913
$22$	0.734438	0.156583
$23$	−5.64772	−1.17763	−0.588816	−	0.808267i	$-0.700406\pi$
$23$	−5.64772	−1.17763	−0.588816	+	0.808267i	$0.700406\pi$
$24$	−11.0139	−2.24820
$25$	1.00000	0.200000
$26$	−10.4883	−2.05692
$27$	−5.52908	−1.06407
$28$	−12.7311	−2.40595
$29$	−4.42803	−0.822264	−0.411132	−	0.911576i	$-0.634866\pi$
$29$	−4.42803	−0.822264	−0.411132	+	0.911576i	$0.634866\pi$
$30$	−4.09739	−0.748078
$31$	−5.63092	−1.01134	−0.505671	−	0.862726i	$-0.668755\pi$
$31$	−5.63092	−1.01134	−0.505671	+	0.862726i	$0.668755\pi$
$32$	−8.34152	−1.47459
$33$	−0.449950	−0.0783263
$34$	−0.259447	−0.0444949
$35$	−2.71567	−0.459031
$36$	−2.29596	−0.382660
$37$	−2.57414	−0.423186	−0.211593	−	0.977358i	$-0.567865\pi$
$37$	−2.57414	−0.423186	−0.211593	+	0.977358i	$0.567865\pi$
$38$	−1.56790	−0.254346
$39$	6.42561	1.02892
$40$	−6.95157	−1.09914
$41$	−9.66482	−1.50939	−0.754695	−	0.656075i	$-0.772215\pi$
$41$	−9.66482	−1.50939	−0.754695	+	0.656075i	$0.772215\pi$
$42$	11.1271	1.71696
$43$	0.0774746	0.0118148	0.00590738	−	0.999983i	$-0.498120\pi$
$43$	0.0774746	0.0118148	0.00590738	+	0.999983i	$0.498120\pi$
$44$	−1.33136	−0.200710
$45$	−0.489749	−0.0730075
$46$	14.6057	2.15349
$47$	5.86816	0.855960	0.427980	−	0.903788i	$-0.359225\pi$
$47$	5.86816	0.855960	0.427980	+	0.903788i	$0.359225\pi$
$48$	13.6281	1.96705
$49$	0.374840	0.0535485
$50$	−2.58612	−0.365733
$51$	0.158949	0.0222574
$52$	19.0128	2.63660
$53$	−2.28736	−0.314193	−0.157097	−	0.987583i	$-0.550213\pi$
$53$	−2.28736	−0.314193	−0.157097	+	0.987583i	$0.550213\pi$
$54$	14.2989	1.94583
$55$	−0.283992	−0.0382935
$56$	18.8781	2.52270
$57$	0.960566	0.127230
$58$	11.4514	1.50365
$59$	2.76580	0.360076	0.180038	−	0.983660i	$-0.442378\pi$
$59$	2.76580	0.360076	0.180038	+	0.983660i	$0.442378\pi$
$60$	7.42760	0.958899
$61$	7.73021	0.989752	0.494876	−	0.868964i	$-0.335213\pi$
$61$	7.73021	0.989752	0.494876	+	0.868964i	$0.335213\pi$
$62$	14.5622	1.84941
$63$	1.33000	0.167564
$64$	4.36909	0.546136
$65$	4.05561	0.503036
$66$	1.16363	0.143233
$67$	11.5707	1.41358	0.706791	−	0.707423i	$-0.250142\pi$
$67$	11.5707	1.41358	0.706791	+	0.707423i	$0.250142\pi$
$68$	0.470317	0.0570343
$69$	−8.94812	−1.07723
$70$	7.02304	0.839414
$71$	−0.557844	−0.0662040	−0.0331020	−	0.999452i	$-0.510539\pi$
$71$	−0.557844	−0.0662040	−0.0331020	+	0.999452i	$0.510539\pi$
$72$	3.40453	0.401227
$73$	0.324794	0.0380143	0.0190071	−	0.999819i	$-0.493949\pi$
$73$	0.324794	0.0380143	0.0190071	+	0.999819i	$0.493949\pi$
$74$	6.65703	0.773865
$75$	1.58438	0.182948
$76$	2.84223	0.326026
$77$	0.771227	0.0878895
$78$	−16.6174	−1.88155
$79$	13.4457	1.51275	0.756377	−	0.654135i	$-0.226967\pi$
$79$	13.4457	1.51275	0.756377	+	0.654135i	$0.226967\pi$
$80$	8.60155	0.961682
$81$	−7.29090	−0.810100
$82$	24.9944	2.76017
$83$	8.90987	0.977985	0.488993	−	0.872288i	$-0.337364\pi$
$83$	8.90987	0.977985	0.488993	+	0.872288i	$0.337364\pi$
$84$	−20.1709	−2.20082
$85$	0.100323	0.0108816
$86$	−0.200359	−0.0216052
$87$	−7.01567	−0.752158
$88$	1.97419	0.210449
$89$	15.0748	1.59792	0.798961	−	0.601383i	$-0.205383\pi$
$89$	15.0748	1.59792	0.798961	+	0.601383i	$0.205383\pi$
$90$	1.26655	0.133506
$91$	−11.0137	−1.15455
$92$	−26.4767	−2.76039
$93$	−8.92149	−0.925116
$94$	−15.1758	−1.56526
$95$	0.606273	0.0622023
$96$	−13.2161	−1.34886
$97$	4.43169	0.449970	0.224985	−	0.974362i	$-0.427767\pi$
$97$	4.43169	0.449970	0.224985	+	0.974362i	$0.427767\pi$
$98$	−0.969381	−0.0979223
$99$	0.139085	0.0139786
$100$	4.68803	0.468803
$101$	7.35430	0.731780	0.365890	−	0.930658i	$-0.380765\pi$
$101$	7.35430	0.731780	0.365890	+	0.930658i	$0.380765\pi$
$102$	−0.411063	−0.0407013
$103$	7.94065	0.782415	0.391208	−	0.920302i	$-0.372057\pi$
$103$	7.94065	0.782415	0.391208	+	0.920302i	$0.372057\pi$
$104$	−28.1928	−2.76453
$105$	−4.30264	−0.419894
$106$	5.91539	0.574554
$107$	9.39277	0.908033	0.454017	−	0.890993i	$-0.349991\pi$
$107$	9.39277	0.908033	0.454017	+	0.890993i	$0.349991\pi$
$108$	−25.9205	−2.49420
$109$	−0.134714	−0.0129032	−0.00645161	−	0.999979i	$-0.502054\pi$
$109$	−0.134714	−0.0129032	−0.00645161	+	0.999979i	$0.502054\pi$
$110$	0.734438	0.0700259
$111$	−4.07840	−0.387105
$112$	−23.3589	−2.20721
$113$	8.43620	0.793611	0.396806	−	0.917903i	$-0.370119\pi$
$113$	8.43620	0.793611	0.396806	+	0.917903i	$0.370119\pi$
$114$	−2.48414	−0.232661
$115$	−5.64772	−0.526653
$116$	−20.7587	−1.92740
$117$	−1.98623	−0.183627
$118$	−7.15270	−0.658459
$119$	−0.272444	−0.0249749
$120$	−11.0139	−1.00543
$121$	−10.9193	−0.992668
$122$	−19.9913	−1.80992
$123$	−15.3127	−1.38070
$124$	−26.3979	−2.37060
$125$	1.00000	0.0894427
$126$	−3.43953	−0.306418
$127$	−3.84650	−0.341321	−0.170661	−	0.985330i	$-0.554590\pi$
$127$	−3.84650	−0.341321	−0.170661	+	0.985330i	$0.554590\pi$
$128$	5.38404	0.475886
$129$	0.122749	0.0108074
$130$	−10.4883	−0.919884
$131$	−10.1024	−0.882647	−0.441324	−	0.897348i	$-0.645491\pi$
$131$	−10.1024	−0.882647	−0.441324	+	0.897348i	$0.645491\pi$
$132$	−2.10938	−0.183598
$133$	−1.64644	−0.142764
$134$	−29.9231	−2.58497
$135$	−5.52908	−0.475867
$136$	−0.697402	−0.0598017
$137$	13.0952	1.11880	0.559399	−	0.828898i	$-0.311032\pi$
$137$	13.0952	1.11880	0.559399	+	0.828898i	$0.311032\pi$
$138$	23.1409	1.96989
$139$	20.6458	1.75115	0.875577	−	0.483079i	$-0.160482\pi$
$139$	20.6458	1.75115	0.875577	+	0.483079i	$0.160482\pi$
$140$	−12.7311	−1.07598
$141$	9.29738	0.782981
$142$	1.44265	0.121065
$143$	−1.15176	−0.0963150
$144$	−4.21260	−0.351050
$145$	−4.42803	−0.367728
$146$	−0.839957	−0.0695153
$147$	0.593887	0.0489830
$148$	−12.0676	−0.991953
$149$	−13.9895	−1.14607	−0.573033	−	0.819532i	$-0.694233\pi$
$149$	−13.9895	−1.14607	−0.573033	+	0.819532i	$0.694233\pi$
$150$	−4.09739	−0.334551
$151$	−9.88448	−0.804388	−0.402194	−	0.915555i	$-0.631752\pi$
$151$	−9.88448	−0.804388	−0.402194	+	0.915555i	$0.631752\pi$
$152$	−4.21455	−0.341845
$153$	−0.0491331	−0.00397218
$154$	−1.99449	−0.160720
$155$	−5.63092	−0.452286
$156$	30.1234	2.41181
$157$	22.8634	1.82470	0.912351	−	0.409410i	$-0.134265\pi$
$157$	22.8634	1.82470	0.912351	+	0.409410i	$0.134265\pi$
$158$	−34.7721	−2.76632
$159$	−3.62404	−0.287405
$160$	−8.34152	−0.659455
$161$	15.3373	1.20875
$162$	18.8552	1.48140
$163$	5.18051	0.405769	0.202885	−	0.979203i	$-0.434968\pi$
$163$	5.18051	0.405769	0.202885	+	0.979203i	$0.434968\pi$
$164$	−45.3089	−3.53803
$165$	−0.449950	−0.0350286
$166$	−23.0420	−1.78841
$167$	−17.4085	−1.34711	−0.673556	−	0.739137i	$-0.735234\pi$
$167$	−17.4085	−1.34711	−0.673556	+	0.739137i	$0.735234\pi$
$168$	29.9101	2.30761
$169$	3.44795	0.265227
$170$	−0.259447	−0.0198987
$171$	−0.296922	−0.0227062
$172$	0.363203	0.0276940
$173$	5.22438	0.397202	0.198601	−	0.980080i	$-0.436360\pi$
$173$	5.22438	0.397202	0.198601	+	0.980080i	$0.436360\pi$
$174$	18.1434	1.37545
$175$	−2.71567	−0.205285
$176$	−2.44277	−0.184131
$177$	4.38207	0.329377
$178$	−38.9852	−2.92206
$179$	−7.89176	−0.589858	−0.294929	−	0.955519i	$-0.595296\pi$
$179$	−7.89176	−0.589858	−0.294929	+	0.955519i	$0.595296\pi$
$180$	−2.29596	−0.171131
$181$	18.3256	1.36213	0.681064	−	0.732224i	$-0.261518\pi$
$181$	18.3256	1.36213	0.681064	+	0.732224i	$0.261518\pi$
$182$	28.4827	2.11128
$183$	12.2476	0.905367
$184$	39.2605	2.89433
$185$	−2.57414	−0.189254
$186$	23.0721	1.69173
$187$	−0.0284909	−0.00208346
$188$	27.5101	2.00638
$189$	15.0151	1.09219
$190$	−1.56790	−0.113747
$191$	−19.5636	−1.41557	−0.707786	−	0.706427i	$-0.750306\pi$
$191$	−19.5636	−1.41557	−0.707786	+	0.706427i	$0.750306\pi$
$192$	6.92229	0.499573
$193$	15.4414	1.11149	0.555747	−	0.831351i	$-0.312432\pi$
$193$	15.4414	1.11149	0.555747	+	0.831351i	$0.312432\pi$
$194$	−11.4609	−0.822844
$195$	6.42561	0.460148
$196$	1.75726	0.125518
$197$	7.88375	0.561694	0.280847	−	0.959753i	$-0.409385\pi$
$197$	7.88375	0.561694	0.280847	+	0.959753i	$0.409385\pi$
$198$	−0.359690	−0.0255621
$199$	4.12110	0.292137	0.146069	−	0.989274i	$-0.453338\pi$
$199$	4.12110	0.292137	0.146069	+	0.989274i	$0.453338\pi$
$200$	−6.95157	−0.491550
$201$	18.3323	1.29306
$202$	−19.0191	−1.33818
$203$	12.0250	0.843992
$204$	0.745159	0.0521716
$205$	−9.66482	−0.675020
$206$	−20.5355	−1.43078
$207$	2.76597	0.192248
$208$	34.8845	2.41880
$209$	−0.172177	−0.0119097
$210$	11.1271	0.767846
$211$	18.8259	1.29603	0.648014	−	0.761628i	$-0.275600\pi$
$211$	18.8259	1.29603	0.648014	+	0.761628i	$0.275600\pi$
$212$	−10.7232	−0.736473
$213$	−0.883836	−0.0605594
$214$	−24.2908	−1.66049
$215$	0.0774746	0.00528372
$216$	38.4358	2.61522
$217$	15.2917	1.03807
$218$	0.348386	0.0235957
$219$	0.514596	0.0347732
$220$	−1.33136	−0.0897604
$221$	0.406871	0.0273691
$222$	10.5473	0.707885
$223$	1.15067	0.0770545	0.0385273	−	0.999258i	$-0.487733\pi$
$223$	1.15067	0.0770545	0.0385273	+	0.999258i	$0.487733\pi$
$224$	22.6528	1.51355
$225$	−0.489749	−0.0326500
$226$	−21.8171	−1.45125
$227$	−5.59983	−0.371674	−0.185837	−	0.982581i	$-0.559500\pi$
$227$	−5.59983	−0.371674	−0.185837	+	0.982581i	$0.559500\pi$
$228$	4.50316	0.298229
$229$	−10.4542	−0.690833	−0.345416	−	0.938450i	$-0.612262\pi$
$229$	−10.4542	−0.690833	−0.345416	+	0.938450i	$0.612262\pi$
$230$	14.6057	0.963072
$231$	1.22191	0.0803961
$232$	30.7817	2.02092
$233$	−14.8176	−0.970736	−0.485368	−	0.874310i	$-0.661314\pi$
$233$	−14.8176	−0.970736	−0.485368	+	0.874310i	$0.661314\pi$
$234$	5.13664	0.335792
$235$	5.86816	0.382797
$236$	12.9661	0.844024
$237$	21.3030	1.38378
$238$	0.704573	0.0456707
$239$	19.7122	1.27508	0.637538	−	0.770419i	$-0.279953\pi$
$239$	19.7122	1.27508	0.637538	+	0.770419i	$0.279953\pi$
$240$	13.6281	0.879690
$241$	−3.78625	−0.243894	−0.121947	−	0.992537i	$-0.538914\pi$
$241$	−3.78625	−0.243894	−0.121947	+	0.992537i	$0.538914\pi$
$242$	28.2388	1.81526
$243$	5.03571	0.323041
$244$	36.2395	2.31999
$245$	0.374840	0.0239476
$246$	39.6005	2.52484
$247$	2.45881	0.156450
$248$	39.1437	2.48563
$249$	14.1166	0.894603
$250$	−2.58612	−0.163561
$251$	7.89572	0.498374	0.249187	−	0.968455i	$-0.419837\pi$
$251$	7.89572	0.498374	0.249187	+	0.968455i	$0.419837\pi$
$252$	6.23506	0.392772
$253$	1.60391	0.100837
$254$	9.94751	0.624162
$255$	0.158949	0.00995380
$256$	−22.6620	−1.41637
$257$	21.2223	1.32381	0.661905	−	0.749588i	$-0.269748\pi$
$257$	21.2223	1.32381	0.661905	+	0.749588i	$0.269748\pi$
$258$	−0.317444	−0.0197632
$259$	6.99050	0.434368
$260$	19.0128	1.17912
$261$	2.16862	0.134234
$262$	26.1259	1.61407
$263$	10.0949	0.622478	0.311239	−	0.950332i	$-0.399256\pi$
$263$	10.0949	0.622478	0.311239	+	0.950332i	$0.399256\pi$
$264$	3.12786	0.192506
$265$	−2.28736	−0.140511
$266$	4.25788	0.261068
$267$	23.8841	1.46168
$268$	54.2436	3.31345
$269$	−10.5950	−0.645987	−0.322994	−	0.946401i	$-0.604689\pi$
$269$	−10.5950	−0.645987	−0.322994	+	0.946401i	$0.604689\pi$
$270$	14.2989	0.870202
$271$	28.8745	1.75400	0.877000	−	0.480491i	$-0.159542\pi$
$271$	28.8745	1.75400	0.877000	+	0.480491i	$0.159542\pi$
$272$	0.862933	0.0523230
$273$	−17.4498	−1.05611
$274$	−33.8658	−2.04591
$275$	−0.283992	−0.0171254
$276$	−41.9491	−2.52504
$277$	3.01015	0.180863	0.0904313	−	0.995903i	$-0.471175\pi$
$277$	3.01015	0.180863	0.0904313	+	0.995903i	$0.471175\pi$
$278$	−53.3926	−3.20227
$279$	2.75774	0.165101
$280$	18.8781	1.12818
$281$	−2.83046	−0.168851	−0.0844254	−	0.996430i	$-0.526905\pi$
$281$	−2.83046	−0.168851	−0.0844254	+	0.996430i	$0.526905\pi$
$282$	−24.0442	−1.43181
$283$	−30.4876	−1.81230	−0.906151	−	0.422955i	$-0.860993\pi$
$283$	−30.4876	−1.81230	−0.906151	+	0.422955i	$0.860993\pi$
$284$	−2.61519	−0.155183
$285$	0.960566	0.0568990
$286$	2.97859	0.176128
$287$	26.2464	1.54928
$288$	4.08525	0.240726
$289$	−16.9899	−0.999408
$290$	11.4514	0.672451
$291$	7.02147	0.411606
$292$	1.52264	0.0891060
$293$	−31.9892	−1.86883	−0.934415	−	0.356187i	$-0.884077\pi$
$293$	−31.9892	−1.86883	−0.934415	+	0.356187i	$0.884077\pi$
$294$	−1.53587	−0.0895735
$295$	2.76580	0.161031
$296$	17.8943	1.04008
$297$	1.57021	0.0911130
$298$	36.1786	2.09577
$299$	−22.9049	−1.32463
$300$	7.42760	0.428833
$301$	−0.210395	−0.0121270
$302$	25.5625	1.47095
$303$	11.6520	0.669389
$304$	5.21489	0.299094
$305$	7.73021	0.442631
$306$	0.127064	0.00726378
$307$	31.6932	1.80883	0.904413	−	0.426659i	$-0.140309\pi$
$307$	31.6932	1.80883	0.904413	+	0.426659i	$0.140309\pi$
$308$	3.61553	0.206014
$309$	12.5810	0.715707
$310$	14.5622	0.827079
$311$	−7.95851	−0.451286	−0.225643	−	0.974210i	$-0.572448\pi$
$311$	−7.95851	−0.451286	−0.225643	+	0.974210i	$0.572448\pi$
$312$	−44.6681	−2.52883
$313$	−25.7545	−1.45573	−0.727864	−	0.685721i	$-0.759487\pi$
$313$	−25.7545	−1.45573	−0.727864	+	0.685721i	$0.759487\pi$
$314$	−59.1277	−3.33677
$315$	1.33000	0.0749368
$316$	63.0336	3.54592
$317$	3.11343	0.174868	0.0874339	−	0.996170i	$-0.472133\pi$
$317$	3.11343	0.174868	0.0874339	+	0.996170i	$0.472133\pi$
$318$	9.37221	0.525568
$319$	1.25752	0.0704078
$320$	4.36909	0.244240
$321$	14.8817	0.830615
$322$	−39.6642	−2.21040
$323$	0.0608231	0.00338429
$324$	−34.1799	−1.89888
$325$	4.05561	0.224965
$326$	−13.3974	−0.742015
$327$	−0.213437	−0.0118031
$328$	67.1856	3.70971
$329$	−15.9360	−0.878578
$330$	1.16363	0.0640555
$331$	−13.1555	−0.723093	−0.361546	−	0.932354i	$-0.617751\pi$
$331$	−13.1555	−0.723093	−0.361546	+	0.932354i	$0.617751\pi$
$332$	41.7697	2.29241
$333$	1.26068	0.0690850
$334$	45.0205	2.46341
$335$	11.5707	0.632173
$336$	−37.0094	−2.01903
$337$	−7.94519	−0.432802	−0.216401	−	0.976305i	$-0.569432\pi$
$337$	−7.94519	−0.432802	−0.216401	+	0.976305i	$0.569432\pi$
$338$	−8.91681	−0.485010
$339$	13.3661	0.725948
$340$	0.470317	0.0255065
$341$	1.59913	0.0865980
$342$	0.767877	0.0415220
$343$	17.9917	0.971462
$344$	−0.538570	−0.0290377
$345$	−8.94812	−0.481751
$346$	−13.5109	−0.726349
$347$	33.7684	1.81278	0.906392	−	0.422439i	$-0.138826\pi$
$347$	33.7684	1.81278	0.906392	+	0.422439i	$0.138826\pi$
$348$	−32.8896	−1.76307
$349$	−0.107829	−0.00577195	−0.00288597	−	0.999996i	$-0.500919\pi$
$349$	−0.107829	−0.00577195	−0.00288597	+	0.999996i	$0.500919\pi$
$350$	7.02304	0.375397
$351$	−22.4238	−1.19689
$352$	2.36892	0.126264
$353$	12.5011	0.665366	0.332683	−	0.943039i	$-0.392046\pi$
$353$	12.5011	0.665366	0.332683	+	0.943039i	$0.392046\pi$
$354$	−11.3326	−0.602319
$355$	−0.557844	−0.0296073
$356$	70.6709	3.74555
$357$	−0.431653	−0.0228455
$358$	20.4090	1.07865
$359$	10.4447	0.551249	0.275624	−	0.961265i	$-0.411115\pi$
$359$	10.4447	0.551249	0.275624	+	0.961265i	$0.411115\pi$
$360$	3.40453	0.179434
$361$	−18.6324	−0.980654
$362$	−47.3921	−2.49087
$363$	−17.3004	−0.908034
$364$	−51.6324	−2.70627
$365$	0.324794	0.0170005
$366$	−31.6737	−1.65561
$367$	−27.7786	−1.45003	−0.725015	−	0.688733i	$-0.758167\pi$
$367$	−27.7786	−1.45003	−0.725015	+	0.688733i	$0.758167\pi$
$368$	−48.5792	−2.53236
$369$	4.73334	0.246408
$370$	6.65703	0.346083
$371$	6.21171	0.322496
$372$	−41.8242	−2.16848
$373$	23.3417	1.20859	0.604293	−	0.796762i	$-0.293456\pi$
$373$	23.3417	1.20859	0.604293	+	0.796762i	$0.293456\pi$
$374$	0.0736810	0.00380995
$375$	1.58438	0.0818169
$376$	−40.7929	−2.10374
$377$	−17.9583	−0.924901
$378$	−38.8310	−1.99725
$379$	−13.3190	−0.684151	−0.342076	−	0.939672i	$-0.611130\pi$
$379$	−13.3190	−0.684151	−0.342076	+	0.939672i	$0.611130\pi$
$380$	2.84223	0.145803
$381$	−6.09430	−0.312220
$382$	50.5938	2.58861
$383$	−4.61559	−0.235846	−0.117923	−	0.993023i	$-0.537624\pi$
$383$	−4.61559	−0.235846	−0.117923	+	0.993023i	$0.537624\pi$
$384$	8.53035	0.435312
$385$	0.771227	0.0393054
$386$	−39.9333	−2.03255
$387$	−0.0379431	−0.00192876
$388$	20.7759	1.05474
$389$	−33.5487	−1.70099	−0.850493	−	0.525987i	$-0.823696\pi$
$389$	−33.5487	−1.70099	−0.850493	+	0.525987i	$0.823696\pi$
$390$	−16.6174	−0.841455
$391$	−0.566596	−0.0286540
$392$	−2.60572	−0.131609
$393$	−16.0059	−0.807393
$394$	−20.3883	−1.02715
$395$	13.4457	0.676525
$396$	0.652034	0.0327659
$397$	1.03932	0.0521617	0.0260809	−	0.999660i	$-0.491697\pi$
$397$	1.03932	0.0521617	0.0260809	+	0.999660i	$0.491697\pi$
$398$	−10.6577	−0.534221
$399$	−2.60858	−0.130592
$400$	8.60155	0.430077
$401$	2.42934	0.121315	0.0606577	−	0.998159i	$-0.480680\pi$
$401$	2.42934	0.121315	0.0606577	+	0.998159i	$0.480680\pi$
$402$	−47.4095	−2.36457
$403$	−22.8368	−1.13758
$404$	34.4772	1.71530
$405$	−7.29090	−0.362288
$406$	−31.0982	−1.54338
$407$	0.731034	0.0362360
$408$	−1.10495	−0.0547031
$409$	16.1661	0.799362	0.399681	−	0.916654i	$-0.369121\pi$
$409$	16.1661	0.799362	0.399681	+	0.916654i	$0.369121\pi$
$410$	24.9944	1.23439
$411$	20.7477	1.02341
$412$	37.2260	1.83399
$413$	−7.51099	−0.369592
$414$	−7.15313	−0.351557
$415$	8.90987	0.437368
$416$	−33.8299	−1.65865
$417$	32.7107	1.60185
$418$	0.445270	0.0217789
$419$	20.7993	1.01611	0.508057	−	0.861324i	$-0.330364\pi$
$419$	20.7993	1.01611	0.508057	+	0.861324i	$0.330364\pi$
$420$	−20.1709	−0.984239
$421$	−25.9093	−1.26274	−0.631371	−	0.775481i	$-0.717507\pi$
$421$	−25.9093	−1.26274	−0.631371	+	0.775481i	$0.717507\pi$
$422$	−48.6861	−2.37000
$423$	−2.87393	−0.139735
$424$	15.9007	0.772208
$425$	0.100323	0.00486638
$426$	2.28571	0.110743
$427$	−20.9927	−1.01591
$428$	44.0336	2.12844
$429$	−1.82482	−0.0881032
$430$	−0.200359	−0.00966216
$431$	23.6282	1.13813	0.569065	−	0.822293i	$-0.307305\pi$
$431$	23.6282	1.13813	0.569065	+	0.822293i	$0.307305\pi$
$432$	−47.5586	−2.28817
$433$	−13.6203	−0.654552	−0.327276	−	0.944929i	$-0.606131\pi$
$433$	−13.6203	−0.654552	−0.327276	+	0.944929i	$0.606131\pi$
$434$	−39.5462	−1.89828
$435$	−7.01567	−0.336375
$436$	−0.631541	−0.0302453
$437$	−3.42406	−0.163795
$438$	−1.33081	−0.0635885
$439$	10.5845	0.505169	0.252585	−	0.967575i	$-0.418719\pi$
$439$	10.5845	0.505169	0.252585	+	0.967575i	$0.418719\pi$
$440$	1.97419	0.0941158
$441$	−0.183577	−0.00874179
$442$	−1.05222	−0.0500488
$443$	15.8344	0.752313	0.376157	−	0.926556i	$-0.377245\pi$
$443$	15.8344	0.752313	0.376157	+	0.926556i	$0.377245\pi$
$444$	−19.1197	−0.907380
$445$	15.0748	0.714613
$446$	−2.97577	−0.140907
$447$	−22.1647	−1.04835
$448$	−11.8650	−0.560568
$449$	−23.7166	−1.11926	−0.559629	−	0.828744i	$-0.689056\pi$
$449$	−23.7166	−1.11926	−0.559629	+	0.828744i	$0.689056\pi$
$450$	1.26655	0.0597058
$451$	2.74473	0.129244
$452$	39.5492	1.86024
$453$	−15.6607	−0.735806
$454$	14.4818	0.679667
$455$	−11.0137	−0.516329
$456$	−6.67744	−0.312700
$457$	27.0068	1.26333	0.631663	−	0.775243i	$-0.282373\pi$
$457$	27.0068	1.26333	0.631663	+	0.775243i	$0.282373\pi$
$458$	27.0358	1.26330
$459$	−0.554694	−0.0258909
$460$	−26.4767	−1.23448
$461$	10.7210	0.499327	0.249663	−	0.968333i	$-0.419680\pi$
$461$	10.7210	0.499327	0.249663	+	0.968333i	$0.419680\pi$
$462$	−3.16002	−0.147017
$463$	15.7897	0.733812	0.366906	−	0.930258i	$-0.380417\pi$
$463$	15.7897	0.733812	0.366906	+	0.930258i	$0.380417\pi$
$464$	−38.0879	−1.76819
$465$	−8.92149	−0.413724
$466$	38.3202	1.77515
$467$	6.33913	0.293340	0.146670	−	0.989185i	$-0.453144\pi$
$467$	6.33913	0.293340	0.146670	+	0.989185i	$0.453144\pi$
$468$	−9.31150	−0.430424
$469$	−31.4220	−1.45094
$470$	−15.1758	−0.700007
$471$	36.2243	1.66913
$472$	−19.2266	−0.884978
$473$	−0.0220022	−0.00101166
$474$	−55.0921	−2.53047
$475$	0.606273	0.0278177
$476$	−1.27722	−0.0585414
$477$	1.12023	0.0512920
$478$	−50.9781	−2.33168
$479$	−19.6611	−0.898340	−0.449170	−	0.893446i	$-0.648280\pi$
$479$	−19.6611	−0.898340	−0.449170	+	0.893446i	$0.648280\pi$
$480$	−13.2161	−0.603230
$481$	−10.4397	−0.476009
$482$	9.79172	0.446000
$483$	24.3001	1.10569
$484$	−51.1902	−2.32683
$485$	4.43169	0.201233
$486$	−13.0229	−0.590733
$487$	35.9181	1.62761	0.813803	−	0.581141i	$-0.197394\pi$
$487$	35.9181	1.62761	0.813803	+	0.581141i	$0.197394\pi$
$488$	−53.7371	−2.43256
$489$	8.20788	0.371173
$490$	−0.969381	−0.0437922
$491$	30.9290	1.39581	0.697903	−	0.716192i	$-0.254117\pi$
$491$	30.9290	1.39581	0.697903	+	0.716192i	$0.254117\pi$
$492$	−71.7864	−3.23638
$493$	−0.444233	−0.0200072
$494$	−6.35877	−0.286095
$495$	0.139085	0.00625140
$496$	−48.4346	−2.17478
$497$	1.51492	0.0679534
$498$	−36.5072	−1.63593
$499$	0.162431	0.00727142	0.00363571	−	0.999993i	$-0.498843\pi$
$499$	0.162431	0.00727142	0.00363571	+	0.999993i	$0.498843\pi$
$500$	4.68803	0.209655
$501$	−27.5817	−1.23226
$502$	−20.4193	−0.911358
$503$	−15.1290	−0.674567	−0.337283	−	0.941403i	$-0.609508\pi$
$503$	−15.1290	−0.674567	−0.337283	+	0.941403i	$0.609508\pi$
$504$	−9.24555	−0.411830
$505$	7.35430	0.327262
$506$	−4.14790	−0.184397
$507$	5.46285	0.242614
$508$	−18.0325	−0.800062
$509$	−12.9442	−0.573742	−0.286871	−	0.957969i	$-0.592615\pi$
$509$	−12.9442	−0.573742	−0.286871	+	0.957969i	$0.592615\pi$
$510$	−0.411063	−0.0182022
$511$	−0.882032	−0.0390188
$512$	47.8385	2.11418
$513$	−3.35213	−0.148000
$514$	−54.8834	−2.42080
$515$	7.94065	0.349907
$516$	0.575450	0.0253328
$517$	−1.66651	−0.0732931
$518$	−18.0783	−0.794314
$519$	8.27738	0.363337
$520$	−28.1928	−1.23634
$521$	−22.8803	−1.00240	−0.501202	−	0.865330i	$-0.667109\pi$
$521$	−22.8803	−1.00240	−0.501202	+	0.865330i	$0.667109\pi$
$522$	−5.60833	−0.245470
$523$	33.5498	1.46703	0.733514	−	0.679674i	$-0.237879\pi$
$523$	33.5498	1.46703	0.733514	+	0.679674i	$0.237879\pi$
$524$	−47.3601	−2.06894
$525$	−4.30264	−0.187783
$526$	−26.1066	−1.13830
$527$	−0.564910	−0.0246079
$528$	−3.87027	−0.168432
$529$	8.89679	0.386817
$530$	5.91539	0.256948
$531$	−1.35455	−0.0587824
$532$	−7.71854	−0.334641
$533$	−39.1967	−1.69780
$534$	−61.7672	−2.67293
$535$	9.39277	0.406085
$536$	−80.4342	−3.47423
$537$	−12.5035	−0.539567
$538$	27.3999	1.18129
$539$	−0.106451	−0.00458519
$540$	−25.9205	−1.11544
$541$	28.8730	1.24135	0.620673	−	0.784070i	$-0.286860\pi$
$541$	28.8730	1.24135	0.620673	+	0.784070i	$0.286860\pi$
$542$	−74.6729	−3.20748
$543$	29.0346	1.24599
$544$	−0.836846	−0.0358795
$545$	−0.134714	−0.00577049
$546$	45.1273	1.93127
$547$	−5.89138	−0.251897	−0.125949	−	0.992037i	$-0.540197\pi$
$547$	−5.89138	−0.251897	−0.125949	+	0.992037i	$0.540197\pi$
$548$	61.3906	2.62248
$549$	−3.78587	−0.161577
$550$	0.734438	0.0313165
$551$	−2.68460	−0.114368
$552$	62.2035	2.64756
$553$	−36.5139	−1.55273
$554$	−7.78462	−0.330737
$555$	−4.07840	−0.173119
$556$	96.7881	4.10473
$557$	4.37267	0.185276	0.0926381	−	0.995700i	$-0.470470\pi$
$557$	4.37267	0.185276	0.0926381	+	0.995700i	$0.470470\pi$
$558$	−7.13185	−0.301915
$559$	0.314206	0.0132895
$560$	−23.3589	−0.987095
$561$	−0.0451404	−0.00190583
$562$	7.31991	0.308772
$563$	−32.1366	−1.35439	−0.677197	−	0.735802i	$-0.736806\pi$
$563$	−32.1366	−1.35439	−0.677197	+	0.735802i	$0.736806\pi$
$564$	43.5864	1.83532
$565$	8.43620	0.354914
$566$	78.8447	3.31409
$567$	19.7996	0.831507
$568$	3.87789	0.162713
$569$	2.86118	0.119947	0.0599736	−	0.998200i	$-0.480898\pi$
$569$	2.86118	0.119947	0.0599736	+	0.998200i	$0.480898\pi$
$570$	−2.48414	−0.104049
$571$	−4.60443	−0.192690	−0.0963448	−	0.995348i	$-0.530715\pi$
$571$	−4.60443	−0.192690	−0.0963448	+	0.995348i	$0.530715\pi$
$572$	−5.39948	−0.225764
$573$	−30.9961	−1.29488
$574$	−67.8764	−2.83311
$575$	−5.64772	−0.235526
$576$	−2.13976	−0.0891566
$577$	8.73831	0.363781	0.181890	−	0.983319i	$-0.441778\pi$
$577$	8.73831	0.363781	0.181890	+	0.983319i	$0.441778\pi$
$578$	43.9380	1.82758
$579$	24.4650	1.01673
$580$	−20.7587	−0.861959
$581$	−24.1962	−1.00383
$582$	−18.1584	−0.752689
$583$	0.649592	0.0269033
$584$	−2.25783	−0.0934296
$585$	−1.98623	−0.0821205
$586$	82.7280	3.41746
$587$	41.5797	1.71618	0.858089	−	0.513501i	$-0.171652\pi$
$587$	41.5797	1.71618	0.858089	+	0.513501i	$0.171652\pi$
$588$	2.78416	0.114817
$589$	−3.41387	−0.140666
$590$	−7.15270	−0.294472
$591$	12.4908	0.513804
$592$	−22.1416	−0.910013
$593$	−5.75601	−0.236371	−0.118186	−	0.992992i	$-0.537708\pi$
$593$	−5.75601	−0.236371	−0.118186	+	0.992992i	$0.537708\pi$
$594$	−4.06076	−0.166615
$595$	−0.272444	−0.0111691
$596$	−65.5833	−2.68640
$597$	6.52938	0.267230
$598$	59.2350	2.42230
$599$	−18.7692	−0.766889	−0.383444	−	0.923564i	$-0.625262\pi$
$599$	−18.7692	−0.766889	−0.383444	+	0.923564i	$0.625262\pi$
$600$	−11.0139	−0.449641
$601$	−2.98396	−0.121718	−0.0608591	−	0.998146i	$-0.519384\pi$
$601$	−2.98396	−0.121718	−0.0608591	+	0.998146i	$0.519384\pi$
$602$	0.544107	0.0221762
$603$	−5.66672	−0.230767
$604$	−46.3387	−1.88550
$605$	−10.9193	−0.443935
$606$	−30.1335	−1.22409
$607$	12.6994	0.515454	0.257727	−	0.966218i	$-0.417027\pi$
$607$	12.6994	0.515454	0.257727	+	0.966218i	$0.417027\pi$
$608$	−5.05724	−0.205098
$609$	19.0522	0.772034
$610$	−19.9913	−0.809423
$611$	23.7990	0.962803
$612$	−0.230337	−0.00931084
$613$	43.0799	1.73998	0.869991	−	0.493068i	$-0.164125\pi$
$613$	43.0799	1.73998	0.869991	+	0.493068i	$0.164125\pi$
$614$	−81.9624	−3.30774
$615$	−15.3127	−0.617468
$616$	−5.36124	−0.216010
$617$	11.3147	0.455514	0.227757	−	0.973718i	$-0.426861\pi$
$617$	11.3147	0.455514	0.227757	+	0.973718i	$0.426861\pi$
$618$	−32.5360	−1.30879
$619$	−15.9971	−0.642978	−0.321489	−	0.946913i	$-0.604183\pi$
$619$	−15.9971	−0.642978	−0.321489	+	0.946913i	$0.604183\pi$
$620$	−26.3979	−1.06016
$621$	31.2267	1.25308
$622$	20.5817	0.825250
$623$	−40.9380	−1.64015
$624$	55.2702	2.21258
$625$	1.00000	0.0400000
$626$	66.6042	2.66204
$627$	−0.272793	−0.0108943
$628$	107.184	4.27713
$629$	−0.258245	−0.0102969
$630$	−3.43953	−0.137034
$631$	25.2521	1.00527	0.502636	−	0.864498i	$-0.332364\pi$
$631$	25.2521	1.00527	0.502636	+	0.864498i	$0.332364\pi$
$632$	−93.4684	−3.71797
$633$	29.8273	1.18553
$634$	−8.05172	−0.319774
$635$	−3.84650	−0.152644
$636$	−16.9896	−0.673682
$637$	1.52020	0.0602326
$638$	−3.25211	−0.128752
$639$	0.273204	0.0108078
$640$	5.38404	0.212823
$641$	−20.5854	−0.813072	−0.406536	−	0.913635i	$-0.633264\pi$
$641$	−20.5854	−0.813072	−0.406536	+	0.913635i	$0.633264\pi$
$642$	−38.4859	−1.51892
$643$	−23.9571	−0.944776	−0.472388	−	0.881391i	$-0.656608\pi$
$643$	−23.9571	−0.944776	−0.472388	+	0.881391i	$0.656608\pi$
$644$	71.9018	2.83333
$645$	0.122749	0.00483324
$646$	−0.157296	−0.00618873
$647$	−34.8358	−1.36954	−0.684768	−	0.728761i	$-0.740097\pi$
$647$	−34.8358	−1.36954	−0.684768	+	0.728761i	$0.740097\pi$
$648$	50.6832	1.99102
$649$	−0.785465	−0.0308322
$650$	−10.4883	−0.411385
$651$	24.2278	0.949562
$652$	24.2864	0.951128
$653$	8.99329	0.351935	0.175967	−	0.984396i	$-0.443695\pi$
$653$	8.99329	0.351935	0.175967	+	0.984396i	$0.443695\pi$
$654$	0.551974	0.0215839
$655$	−10.1024	−0.394732
$656$	−83.1324	−3.24577
$657$	−0.159068	−0.00620582
$658$	41.2124	1.60663
$659$	2.51340	0.0979083	0.0489542	−	0.998801i	$-0.484411\pi$
$659$	2.51340	0.0979083	0.0489542	+	0.998801i	$0.484411\pi$
$660$	−2.10938	−0.0821075
$661$	40.8236	1.58785	0.793927	−	0.608013i	$-0.208033\pi$
$661$	40.8236	1.58785	0.793927	+	0.608013i	$0.208033\pi$
$662$	34.0218	1.32229
$663$	0.644636	0.0250356
$664$	−61.9376	−2.40364
$665$	−1.64644	−0.0638460
$666$	−3.26028	−0.126333
$667$	25.0083	0.968324
$668$	−81.6116	−3.15765
$669$	1.82309	0.0704849
$670$	−29.9231	−1.15603
$671$	−2.19532	−0.0847493
$672$	35.8905	1.38451
$673$	37.0753	1.42915	0.714574	−	0.699560i	$-0.246621\pi$
$673$	37.0753	1.42915	0.714574	+	0.699560i	$0.246621\pi$
$674$	20.5472	0.791450
$675$	−5.52908	−0.212814
$676$	16.1641	0.621695
$677$	11.8700	0.456201	0.228101	−	0.973638i	$-0.426748\pi$
$677$	11.8700	0.456201	0.228101	+	0.973638i	$0.426748\pi$
$678$	−34.5664	−1.32752
$679$	−12.0350	−0.461861
$680$	−0.697402	−0.0267441
$681$	−8.87224	−0.339985
$682$	−4.13556	−0.158359
$683$	−20.2412	−0.774506	−0.387253	−	0.921973i	$-0.626576\pi$
$683$	−20.2412	−0.774506	−0.387253	+	0.921973i	$0.626576\pi$
$684$	−1.39198	−0.0532236
$685$	13.0952	0.500342
$686$	−46.5288	−1.77648
$687$	−16.5634	−0.631933
$688$	0.666401	0.0254063
$689$	−9.27663	−0.353412
$690$	23.1409	0.880961
$691$	39.5052	1.50285	0.751424	−	0.659820i	$-0.229367\pi$
$691$	39.5052	1.50285	0.751424	+	0.659820i	$0.229367\pi$
$692$	24.4920	0.931047
$693$	−0.377708	−0.0143479
$694$	−87.3292	−3.31497
$695$	20.6458	0.783140
$696$	48.7699	1.84862
$697$	−0.969603	−0.0367263
$698$	0.278859	0.0105550
$699$	−23.4767	−0.887971
$700$	−12.7311	−0.481191
$701$	24.1886	0.913591	0.456796	−	0.889572i	$-0.348997\pi$
$701$	24.1886	0.913591	0.456796	+	0.889572i	$0.348997\pi$
$702$	57.9906	2.18871
$703$	−1.56063	−0.0588603
$704$	−1.24079	−0.0467639
$705$	9.29738	0.350160
$706$	−32.3293	−1.21673
$707$	−19.9718	−0.751118
$708$	20.5433	0.772063
$709$	−2.56745	−0.0964228	−0.0482114	−	0.998837i	$-0.515352\pi$
$709$	−2.56745	−0.0964228	−0.0482114	+	0.998837i	$0.515352\pi$
$710$	1.44265	0.0541418
$711$	−6.58500	−0.246957
$712$	−104.793	−3.92729
$713$	31.8019	1.19099
$714$	1.11631	0.0417768
$715$	−1.15176	−0.0430734
$716$	−36.9968	−1.38263
$717$	31.2315	1.16636
$718$	−27.0112	−1.00805
$719$	−1.83114	−0.0682901	−0.0341450	−	0.999417i	$-0.510871\pi$
$719$	−1.83114	−0.0682901	−0.0341450	+	0.999417i	$0.510871\pi$
$720$	−4.21260	−0.156994
$721$	−21.5641	−0.803091
$722$	48.1857	1.79329
$723$	−5.99885	−0.223100
$724$	85.9107	3.19285
$725$	−4.42803	−0.164453
$726$	44.7409	1.66049
$727$	17.0263	0.631470	0.315735	−	0.948847i	$-0.397749\pi$
$727$	17.0263	0.631470	0.315735	+	0.948847i	$0.397749\pi$
$728$	76.5623	2.83759
$729$	29.8511	1.10560
$730$	−0.839957	−0.0310882
$731$	0.00777248	0.000287476 0
$732$	57.4170	2.12219
$733$	44.6159	1.64793	0.823963	−	0.566644i	$-0.191758\pi$
$733$	44.6159	1.64793	0.823963	+	0.566644i	$0.191758\pi$
$734$	71.8388	2.65162
$735$	0.593887	0.0219059
$736$	47.1106	1.73652
$737$	−3.28597	−0.121040
$738$	−12.2410	−0.450597
$739$	2.67348	0.0983455	0.0491728	−	0.998790i	$-0.484342\pi$
$739$	2.67348	0.0983455	0.0491728	+	0.998790i	$0.484342\pi$
$740$	−12.0676	−0.443615
$741$	3.89568	0.143111
$742$	−16.0642	−0.589736
$743$	−15.5890	−0.571906	−0.285953	−	0.958244i	$-0.592310\pi$
$743$	−15.5890	−0.571906	−0.285953	+	0.958244i	$0.592310\pi$
$744$	62.0184	2.27370
$745$	−13.9895	−0.512537
$746$	−60.3644	−2.21010
$747$	−4.36360	−0.159656
$748$	−0.133566	−0.00488366
$749$	−25.5076	−0.932028
$750$	−4.09739	−0.149616
$751$	10.1812	0.371518	0.185759	−	0.982595i	$-0.440526\pi$
$751$	10.1812	0.371518	0.185759	+	0.982595i	$0.440526\pi$
$752$	50.4753	1.84064
$753$	12.5098	0.455882
$754$	46.4425	1.69133
$755$	−9.88448	−0.359733
$756$	70.3913	2.56011
$757$	−8.45290	−0.307226	−0.153613	−	0.988131i	$-0.549091\pi$
$757$	−8.45290	−0.307226	−0.153613	+	0.988131i	$0.549091\pi$
$758$	34.4446	1.25108
$759$	2.54120	0.0922395
$760$	−4.21455	−0.152878
$761$	8.53310	0.309325	0.154662	−	0.987967i	$-0.450571\pi$
$761$	8.53310	0.309325	0.154662	+	0.987967i	$0.450571\pi$
$762$	15.7606	0.570946
$763$	0.365837	0.0132442
$764$	−91.7146	−3.31812
$765$	−0.0491331	−0.00177641
$766$	11.9365	0.431283
$767$	11.2170	0.405022
$768$	−35.9051	−1.29561
$769$	18.8658	0.680318	0.340159	−	0.940368i	$-0.389519\pi$
$769$	18.8658	0.680318	0.340159	+	0.940368i	$0.389519\pi$
$770$	−1.99449	−0.0718763
$771$	33.6241	1.21094
$772$	72.3896	2.60536
$773$	−38.4746	−1.38383	−0.691917	−	0.721977i	$-0.743234\pi$
$773$	−38.4746	−1.38383	−0.691917	+	0.721977i	$0.743234\pi$
$774$	0.0981256	0.00352705
$775$	−5.63092	−0.202268
$776$	−30.8072	−1.10591
$777$	11.0756	0.397334
$778$	86.7610	3.11053
$779$	−5.85952	−0.209939
$780$	30.1234	1.07859
$781$	0.158423	0.00566883
$782$	1.46529	0.0523986
$783$	24.4829	0.874948
$784$	3.22420	0.115150
$785$	22.8634	0.816031
$786$	41.3933	1.47645
$787$	11.9892	0.427369	0.213685	−	0.976903i	$-0.431454\pi$
$787$	11.9892	0.427369	0.213685	+	0.976903i	$0.431454\pi$
$788$	36.9592	1.31662
$789$	15.9941	0.569406
$790$	−34.7721	−1.23714
$791$	−22.9099	−0.814583
$792$	−0.966858	−0.0343558
$793$	31.3507	1.11330
$794$	−2.68780	−0.0953863
$795$	−3.62404	−0.128532
$796$	19.3199	0.684774
$797$	−16.5403	−0.585889	−0.292944	−	0.956130i	$-0.594635\pi$
$797$	−16.5403	−0.585889	−0.292944	+	0.956130i	$0.594635\pi$
$798$	6.74609	0.238809
$799$	0.588712	0.0208271
$800$	−8.34152	−0.294917
$801$	−7.38286	−0.260860
$802$	−6.28257	−0.221845
$803$	−0.0922389	−0.00325504
$804$	85.9423	3.03095
$805$	15.3373	0.540570
$806$	59.0587	2.08025
$807$	−16.7864	−0.590911
$808$	−51.1239	−1.79853
$809$	−44.5219	−1.56531	−0.782654	−	0.622457i	$-0.786134\pi$
$809$	−44.5219	−1.56531	−0.782654	+	0.622457i	$0.786134\pi$
$810$	18.8552	0.662502
$811$	−32.1464	−1.12881	−0.564407	−	0.825497i	$-0.690895\pi$
$811$	−32.1464	−1.12881	−0.564407	+	0.825497i	$0.690895\pi$
$812$	56.3737	1.97833
$813$	45.7481	1.60445
$814$	−1.89054	−0.0662635
$815$	5.18051	0.181465
$816$	1.36721	0.0478620
$817$	0.0469708	0.00164330
$818$	−41.8075	−1.46177
$819$	5.39394	0.188479
$820$	−45.3089	−1.58226
$821$	11.9175	0.415923	0.207962	−	0.978137i	$-0.433317\pi$
$821$	11.9175	0.415923	0.207962	+	0.978137i	$0.433317\pi$
$822$	−53.6562	−1.87147
$823$	−1.71981	−0.0599489	−0.0299744	−	0.999551i	$-0.509543\pi$
$823$	−1.71981	−0.0599489	−0.0299744	+	0.999551i	$0.509543\pi$
$824$	−55.2000	−1.92298
$825$	−0.449950	−0.0156653
$826$	19.4243	0.675859
$827$	1.31794	0.0458294	0.0229147	−	0.999737i	$-0.492705\pi$
$827$	1.31794	0.0458294	0.0229147	+	0.999737i	$0.492705\pi$
$828$	12.9669	0.450632
$829$	31.1541	1.08203	0.541013	−	0.841015i	$-0.318041\pi$
$829$	31.1541	1.08203	0.541013	+	0.841015i	$0.318041\pi$
$830$	−23.0420	−0.799800
$831$	4.76922	0.165442
$832$	17.7193	0.614307
$833$	0.0376050	0.00130294
$834$	−84.5939	−2.92925
$835$	−17.4085	−0.602446
$836$	−0.807169	−0.0279165
$837$	31.1338	1.07614
$838$	−53.7896	−1.85813
$839$	17.5850	0.607100	0.303550	−	0.952815i	$-0.401828\pi$
$839$	17.5850	0.607100	0.303550	+	0.952815i	$0.401828\pi$
$840$	29.9101	1.03200
$841$	−9.39257	−0.323882
$842$	67.0046	2.30913
$843$	−4.48451	−0.154455
$844$	88.2563	3.03791
$845$	3.44795	0.118613
$846$	7.43233	0.255529
$847$	29.6533	1.01890
$848$	−19.6748	−0.675637
$849$	−48.3039	−1.65779
$850$	−0.259447	−0.00889897
$851$	14.5380	0.498357
$852$	−4.14345	−0.141952
$853$	0.547500	0.0187460	0.00937302	−	0.999956i	$-0.497016\pi$
$853$	0.547500	0.0187460	0.00937302	+	0.999956i	$0.497016\pi$
$854$	54.2896	1.85775
$855$	−0.296922	−0.0101545
$856$	−65.2945	−2.23172
$857$	30.8289	1.05309	0.526547	−	0.850146i	$-0.323486\pi$
$857$	30.8289	1.05309	0.526547	+	0.850146i	$0.323486\pi$
$858$	4.71921	0.161111
$859$	28.5579	0.974382	0.487191	−	0.873295i	$-0.338022\pi$
$859$	28.5579	0.974382	0.487191	+	0.873295i	$0.338022\pi$
$860$	0.363203	0.0123851
$861$	41.5842	1.41719
$862$	−61.1053	−2.08126
$863$	36.2450	1.23380	0.616898	−	0.787043i	$-0.288389\pi$
$863$	36.2450	1.23380	0.616898	+	0.787043i	$0.288389\pi$
$864$	46.1209	1.56907
$865$	5.22438	0.177634
$866$	35.2239	1.19696
$867$	−26.9185	−0.914199
$868$	71.6878	2.43324
$869$	−3.81846	−0.129532
$870$	18.1434	0.615118
$871$	46.9260	1.59003
$872$	0.936470	0.0317129
$873$	−2.17042	−0.0734575
$874$	8.85505	0.299527
$875$	−2.71567	−0.0918063
$876$	2.41244	0.0815088
$877$	−42.1313	−1.42267	−0.711336	−	0.702852i	$-0.751910\pi$
$877$	−42.1313	−1.42267	−0.711336	+	0.702852i	$0.751910\pi$
$878$	−27.3727	−0.923785
$879$	−50.6830	−1.70949
$880$	−2.44277	−0.0823458
$881$	−53.1343	−1.79014	−0.895069	−	0.445927i	$-0.852874\pi$
$881$	−53.1343	−1.79014	−0.895069	+	0.445927i	$0.852874\pi$
$882$	0.474754	0.0159858
$883$	39.8040	1.33951	0.669756	−	0.742581i	$-0.266399\pi$
$883$	39.8040	1.33951	0.669756	+	0.742581i	$0.266399\pi$
$884$	1.90742	0.0641535
$885$	4.38207	0.147302
$886$	−40.9496	−1.37573
$887$	−0.484767	−0.0162769	−0.00813844	−	0.999967i	$-0.502591\pi$
$887$	−0.484767	−0.0162769	−0.00813844	+	0.999967i	$0.502591\pi$
$888$	28.3513	0.951408
$889$	10.4458	0.350341
$890$	−38.9852	−1.30679
$891$	2.07056	0.0693662
$892$	5.39437	0.180617
$893$	3.55771	0.119054
$894$	57.3206	1.91709
$895$	−7.89176	−0.263792
$896$	−14.6212	−0.488462
$897$	−36.2901	−1.21169
$898$	61.3341	2.04675
$899$	24.9338	0.831590
$900$	−2.29596	−0.0765320
$901$	−0.229475	−0.00764491
$902$	−7.09821	−0.236344
$903$	−0.333345	−0.0110930
$904$	−58.6449	−1.95050
$905$	18.3256	0.609162
$906$	40.5006	1.34554
$907$	−34.4364	−1.14344	−0.571721	−	0.820448i	$-0.693724\pi$
$907$	−34.4364	−1.14344	−0.571721	+	0.820448i	$0.693724\pi$
$908$	−26.2522	−0.871209
$909$	−3.60176	−0.119463
$910$	28.4827	0.944192
$911$	27.5755	0.913618	0.456809	−	0.889565i	$-0.348992\pi$
$911$	27.5755	0.913618	0.456809	+	0.889565i	$0.348992\pi$
$912$	8.26235	0.273594
$913$	−2.53033	−0.0837418
$914$	−69.8429	−2.31020
$915$	12.2476	0.404892
$916$	−49.0096	−1.61932
$917$	27.4346	0.905971
$918$	1.43451	0.0473457
$919$	30.2830	0.998943	0.499471	−	0.866330i	$-0.333528\pi$
$919$	30.2830	0.998943	0.499471	+	0.866330i	$0.333528\pi$
$920$	39.2605	1.29438
$921$	50.2140	1.65461
$922$	−27.7258	−0.913101
$923$	−2.26240	−0.0744677
$924$	5.72837	0.188450
$925$	−2.57414	−0.0846371
$926$	−40.8342	−1.34190
$927$	−3.88893	−0.127729
$928$	36.9365	1.21250
$929$	−23.8717	−0.783205	−0.391603	−	0.920134i	$-0.628079\pi$
$929$	−23.8717	−0.783205	−0.391603	+	0.920134i	$0.628079\pi$
$930$	23.0721	0.756563
$931$	0.227255	0.00744799
$932$	−69.4655	−2.27542
$933$	−12.6093	−0.412809
$934$	−16.3938	−0.536421
$935$	−0.0284909	−0.000931753 0
$936$	13.8074	0.451310
$937$	3.57375	0.116749	0.0583746	−	0.998295i	$-0.481408\pi$
$937$	3.57375	0.116749	0.0583746	+	0.998295i	$0.481408\pi$
$938$	81.2612	2.65327
$939$	−40.8048	−1.33161
$940$	27.5101	0.897281
$941$	−8.37229	−0.272929	−0.136464	−	0.990645i	$-0.543574\pi$
$941$	−8.37229	−0.272929	−0.136464	+	0.990645i	$0.543574\pi$
$942$	−93.6805	−3.05228
$943$	54.5842	1.77751
$944$	23.7902	0.774304
$945$	15.0151	0.488442
$946$	0.0569003	0.00184999
$947$	−14.4568	−0.469783	−0.234891	−	0.972022i	$-0.575473\pi$
$947$	−14.4568	−0.469783	−0.234891	+	0.972022i	$0.575473\pi$
$948$	99.8690	3.24360
$949$	1.31724	0.0427593
$950$	−1.56790	−0.0508693
$951$	4.93285	0.159959
$952$	1.89391	0.0613820
$953$	−27.6081	−0.894315	−0.447158	−	0.894455i	$-0.647564\pi$
$953$	−27.6081	−0.894315	−0.447158	+	0.894455i	$0.647564\pi$
$954$	−2.89706	−0.0937958
$955$	−19.5636	−0.633063
$956$	92.4113	2.98879
$957$	1.99239	0.0644049
$958$	50.8461	1.64276
$959$	−35.5622	−1.14836
$960$	6.92229	0.223416
$961$	0.707212	0.0228133
$962$	26.9983	0.870461
$963$	−4.60010	−0.148236
$964$	−17.7501	−0.571691
$965$	15.4414	0.497076
$966$	−62.8431	−2.02194
$967$	20.6505	0.664074	0.332037	−	0.943266i	$-0.392264\pi$
$967$	20.6505	0.664074	0.332037	+	0.943266i	$0.392264\pi$
$968$	75.9066	2.43973
$969$	0.0963668	0.00309575
$970$	−11.4609	−0.367987
$971$	0.277710	0.00891214	0.00445607	−	0.999990i	$-0.498582\pi$
$971$	0.277710	0.00891214	0.00445607	+	0.999990i	$0.498582\pi$
$972$	23.6075	0.757212
$973$	−56.0671	−1.79743
$974$	−92.8886	−2.97634
$975$	6.42561	0.205784
$976$	66.4918	2.12835
$977$	−14.9804	−0.479264	−0.239632	−	0.970864i	$-0.577027\pi$
$977$	−14.9804	−0.479264	−0.239632	+	0.970864i	$0.577027\pi$
$978$	−21.2266	−0.678752
$979$	−4.28111	−0.136825
$980$	1.75726	0.0561336
$981$	0.0659759	0.00210645
$982$	−79.9862	−2.55246
$983$	−3.82062	−0.121859	−0.0609295	−	0.998142i	$-0.519406\pi$
$983$	−3.82062	−0.121859	−0.0609295	+	0.998142i	$0.519406\pi$
$984$	106.447	3.39342
$985$	7.88375	0.251197
$986$	1.14884	0.0365865
$987$	−25.2486	−0.803671
$988$	11.5270	0.366721
$989$	−0.437555	−0.0139134
$990$	−0.359690	−0.0114317
$991$	23.5800	0.749043	0.374521	−	0.927218i	$-0.377807\pi$
$991$	23.5800	0.749043	0.374521	+	0.927218i	$0.377807\pi$
$992$	46.9704	1.49131
$993$	−20.8433	−0.661442
$994$	−3.91777	−0.124264
$995$	4.12110	0.130648
$996$	66.1790	2.09696
$997$	−32.9184	−1.04253	−0.521267	−	0.853393i	$-0.674541\pi$
$997$	−32.9184	−1.04253	−0.521267	+	0.853393i	$0.674541\pi$
$998$	−0.420067	−0.0132970
$999$	14.2326	0.450300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.e.1.8	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.e.1.8	✓	142	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58612 q^{2} +1.58438 q^{3} +4.68803 q^{4} +1.00000 q^{5} -4.09739 q^{6} -2.71567 q^{7} -6.95157 q^{8} -0.489749 q^{9} +O(q^{10})\)	\(q-2.58612 q^{2} +1.58438 q^{3} +4.68803 q^{4} +1.00000 q^{5} -4.09739 q^{6} -2.71567 q^{7} -6.95157 q^{8} -0.489749 q^{9} -2.58612 q^{10} -0.283992 q^{11} +7.42760 q^{12} +4.05561 q^{13} +7.02304 q^{14} +1.58438 q^{15} +8.60155 q^{16} +0.100323 q^{17} +1.26655 q^{18} +0.606273 q^{19} +4.68803 q^{20} -4.30264 q^{21} +0.734438 q^{22} -5.64772 q^{23} -11.0139 q^{24} +1.00000 q^{25} -10.4883 q^{26} -5.52908 q^{27} -12.7311 q^{28} -4.42803 q^{29} -4.09739 q^{30} -5.63092 q^{31} -8.34152 q^{32} -0.449950 q^{33} -0.259447 q^{34} -2.71567 q^{35} -2.29596 q^{36} -2.57414 q^{37} -1.56790 q^{38} +6.42561 q^{39} -6.95157 q^{40} -9.66482 q^{41} +11.1271 q^{42} +0.0774746 q^{43} -1.33136 q^{44} -0.489749 q^{45} +14.6057 q^{46} +5.86816 q^{47} +13.6281 q^{48} +0.374840 q^{49} -2.58612 q^{50} +0.158949 q^{51} +19.0128 q^{52} -2.28736 q^{53} +14.2989 q^{54} -0.283992 q^{55} +18.8781 q^{56} +0.960566 q^{57} +11.4514 q^{58} +2.76580 q^{59} +7.42760 q^{60} +7.73021 q^{61} +14.5622 q^{62} +1.33000 q^{63} +4.36909 q^{64} +4.05561 q^{65} +1.16363 q^{66} +11.5707 q^{67} +0.470317 q^{68} -8.94812 q^{69} +7.02304 q^{70} -0.557844 q^{71} +3.40453 q^{72} +0.324794 q^{73} +6.65703 q^{74} +1.58438 q^{75} +2.84223 q^{76} +0.771227 q^{77} -16.6174 q^{78} +13.4457 q^{79} +8.60155 q^{80} -7.29090 q^{81} +24.9944 q^{82} +8.90987 q^{83} -20.1709 q^{84} +0.100323 q^{85} -0.200359 q^{86} -7.01567 q^{87} +1.97419 q^{88} +15.0748 q^{89} +1.26655 q^{90} -11.0137 q^{91} -26.4767 q^{92} -8.92149 q^{93} -15.1758 q^{94} +0.606273 q^{95} -13.2161 q^{96} +4.43169 q^{97} -0.969381 q^{98} +0.139085 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9}+O(q^{10}) \) 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9} + 21 q^{10} + 36 q^{11} + 55 q^{12} + 57 q^{13} + 2 q^{14} + 33 q^{15} + 179 q^{16} + 55 q^{17} + 65 q^{18} + 130 q^{19} + 157 q^{20} + 28 q^{21} + 30 q^{22} + 117 q^{23} + 21 q^{24} + 142 q^{25} + 21 q^{26} + 120 q^{27} + 135 q^{28} + 12 q^{29} + 15 q^{30} + 74 q^{31} + 126 q^{32} + 55 q^{33} + 35 q^{34} + 63 q^{35} + 186 q^{36} + 75 q^{37} + 65 q^{38} + 23 q^{39} + 60 q^{40} + 22 q^{41} + 10 q^{42} + 190 q^{43} + 22 q^{44} + 157 q^{45} + 56 q^{46} + 102 q^{47} + 78 q^{48} + 197 q^{49} + 21 q^{50} + 30 q^{51} + 120 q^{52} + 56 q^{53} - 6 q^{54} + 36 q^{55} + 3 q^{56} + 68 q^{57} + 31 q^{58} + 55 q^{59} + 55 q^{60} + 90 q^{61} + 68 q^{62} + 167 q^{63} + 180 q^{64} + 57 q^{65} + 17 q^{66} + 151 q^{67} + 119 q^{68} + 21 q^{69} + 2 q^{70} + 4 q^{71} + 130 q^{72} + 143 q^{73} - 46 q^{74} + 33 q^{75} + 213 q^{76} + 75 q^{77} - 24 q^{78} + 47 q^{79} + 179 q^{80} + 150 q^{81} + 69 q^{82} + 201 q^{83} - 31 q^{84} + 55 q^{85} - 4 q^{86} + 153 q^{87} + 37 q^{88} + 25 q^{89} + 65 q^{90} + 132 q^{91} + 194 q^{92} + 52 q^{93} + 18 q^{94} + 130 q^{95} + 13 q^{96} + 80 q^{97} + 58 q^{98} + 103 q^{99}+O(q^{100}) \) 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9 + 21 * q^10 + 36 * q^11 + 55 * q^12 + 57 * q^13 + 2 * q^14 + 33 * q^15 + 179 * q^16 + 55 * q^17 + 65 * q^18 + 130 * q^19 + 157 * q^20 + 28 * q^21 + 30 * q^22 + 117 * q^23 + 21 * q^24 + 142 * q^25 + 21 * q^26 + 120 * q^27 + 135 * q^28 + 12 * q^29 + 15 * q^30 + 74 * q^31 + 126 * q^32 + 55 * q^33 + 35 * q^34 + 63 * q^35 + 186 * q^36 + 75 * q^37 + 65 * q^38 + 23 * q^39 + 60 * q^40 + 22 * q^41 + 10 * q^42 + 190 * q^43 + 22 * q^44 + 157 * q^45 + 56 * q^46 + 102 * q^47 + 78 * q^48 + 197 * q^49 + 21 * q^50 + 30 * q^51 + 120 * q^52 + 56 * q^53 - 6 * q^54 + 36 * q^55 + 3 * q^56 + 68 * q^57 + 31 * q^58 + 55 * q^59 + 55 * q^60 + 90 * q^61 + 68 * q^62 + 167 * q^63 + 180 * q^64 + 57 * q^65 + 17 * q^66 + 151 * q^67 + 119 * q^68 + 21 * q^69 + 2 * q^70 + 4 * q^71 + 130 * q^72 + 143 * q^73 - 46 * q^74 + 33 * q^75 + 213 * q^76 + 75 * q^77 - 24 * q^78 + 47 * q^79 + 179 * q^80 + 150 * q^81 + 69 * q^82 + 201 * q^83 - 31 * q^84 + 55 * q^85 - 4 * q^86 + 153 * q^87 + 37 * q^88 + 25 * q^89 + 65 * q^90 + 132 * q^91 + 194 * q^92 + 52 * q^93 + 18 * q^94 + 130 * q^95 + 13 * q^96 + 80 * q^97 + 58 * q^98 + 103 * q^99

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