LMFDB - Embedded newform 8045.2.a.c.1.13

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:	$1$
Dimension:	$127$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	$q-2.47054 q^{2} -3.37029 q^{3} +4.10358 q^{4} +1.00000 q^{5} +8.32645 q^{6} +4.71800 q^{7} -5.19700 q^{8} +8.35885 q^{9} +O(q^{10})$	\(q-2.47054 q^{2} -3.37029 q^{3} +4.10358 q^{4} +1.00000 q^{5} +8.32645 q^{6} +4.71800 q^{7} -5.19700 q^{8} +8.35885 q^{9} -2.47054 q^{10} -2.32047 q^{11} -13.8303 q^{12} -1.92946 q^{13} -11.6560 q^{14} -3.37029 q^{15} +4.63224 q^{16} -1.96692 q^{17} -20.6509 q^{18} -6.32515 q^{19} +4.10358 q^{20} -15.9010 q^{21} +5.73282 q^{22} -0.107343 q^{23} +17.5154 q^{24} +1.00000 q^{25} +4.76681 q^{26} -18.0609 q^{27} +19.3607 q^{28} +6.39418 q^{29} +8.32645 q^{30} -2.17549 q^{31} -1.05015 q^{32} +7.82066 q^{33} +4.85937 q^{34} +4.71800 q^{35} +34.3013 q^{36} +1.98665 q^{37} +15.6266 q^{38} +6.50283 q^{39} -5.19700 q^{40} +12.5405 q^{41} +39.2842 q^{42} -10.8578 q^{43} -9.52224 q^{44} +8.35885 q^{45} +0.265197 q^{46} -10.5490 q^{47} -15.6120 q^{48} +15.2596 q^{49} -2.47054 q^{50} +6.62910 q^{51} -7.91769 q^{52} +7.65747 q^{53} +44.6202 q^{54} -2.32047 q^{55} -24.5195 q^{56} +21.3176 q^{57} -15.7971 q^{58} -1.49317 q^{59} -13.8303 q^{60} +5.85389 q^{61} +5.37464 q^{62} +39.4371 q^{63} -6.67004 q^{64} -1.92946 q^{65} -19.3213 q^{66} -11.3832 q^{67} -8.07143 q^{68} +0.361779 q^{69} -11.6560 q^{70} +16.3784 q^{71} -43.4409 q^{72} -8.05356 q^{73} -4.90811 q^{74} -3.37029 q^{75} -25.9558 q^{76} -10.9480 q^{77} -16.0655 q^{78} -2.88132 q^{79} +4.63224 q^{80} +35.7939 q^{81} -30.9818 q^{82} -1.77505 q^{83} -65.2513 q^{84} -1.96692 q^{85} +26.8247 q^{86} -21.5503 q^{87} +12.0595 q^{88} -8.61657 q^{89} -20.6509 q^{90} -9.10318 q^{91} -0.440493 q^{92} +7.33202 q^{93} +26.0617 q^{94} -6.32515 q^{95} +3.53931 q^{96} -1.10158 q^{97} -37.6994 q^{98} -19.3965 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9}+O(q^{10}) $ 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9	\( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9} - 20 q^{10} - 32 q^{11} - 65 q^{12} - 49 q^{13} - 4 q^{14} - 31 q^{15} + 98 q^{16} - 53 q^{17} - 60 q^{18} - 126 q^{19} + 116 q^{20} - 24 q^{21} - 46 q^{22} - 121 q^{23} - 51 q^{24} + 127 q^{25} - 9 q^{26} - 112 q^{27} - 123 q^{28} - 6 q^{29} - 17 q^{30} - 54 q^{31} - 119 q^{32} - 57 q^{33} - 53 q^{34} - 63 q^{35} + 133 q^{36} - 61 q^{37} - 27 q^{38} - 33 q^{39} - 57 q^{40} - 8 q^{41} - 46 q^{42} - 208 q^{43} - 54 q^{44} + 122 q^{45} - 34 q^{46} - 116 q^{47} - 106 q^{48} + 110 q^{49} - 20 q^{50} - 54 q^{51} - 142 q^{52} - 60 q^{53} - 62 q^{54} - 32 q^{55} + 33 q^{56} - 56 q^{57} - 87 q^{58} - 53 q^{59} - 65 q^{60} - 76 q^{61} - 84 q^{62} - 215 q^{63} + 67 q^{64} - 49 q^{65} + 9 q^{66} - 145 q^{67} - 133 q^{68} + q^{69} - 4 q^{70} - 4 q^{71} - 167 q^{72} - 155 q^{73} - 14 q^{74} - 31 q^{75} - 199 q^{76} - 97 q^{77} - 24 q^{78} - 73 q^{79} + 98 q^{80} + 127 q^{81} - 69 q^{82} - 225 q^{83} - 59 q^{84} - 53 q^{85} + 30 q^{86} - 179 q^{87} - 119 q^{88} - 25 q^{89} - 60 q^{90} - 160 q^{91} - 188 q^{92} - 44 q^{93} - 32 q^{94} - 126 q^{95} - 43 q^{96} - 72 q^{97} - 111 q^{98} - 141 q^{99}+O(q^{100}) \) 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9 - 20 * q^10 - 32 * q^11 - 65 * q^12 - 49 * q^13 - 4 * q^14 - 31 * q^15 + 98 * q^16 - 53 * q^17 - 60 * q^18 - 126 * q^19 + 116 * q^20 - 24 * q^21 - 46 * q^22 - 121 * q^23 - 51 * q^24 + 127 * q^25 - 9 * q^26 - 112 * q^27 - 123 * q^28 - 6 * q^29 - 17 * q^30 - 54 * q^31 - 119 * q^32 - 57 * q^33 - 53 * q^34 - 63 * q^35 + 133 * q^36 - 61 * q^37 - 27 * q^38 - 33 * q^39 - 57 * q^40 - 8 * q^41 - 46 * q^42 - 208 * q^43 - 54 * q^44 + 122 * q^45 - 34 * q^46 - 116 * q^47 - 106 * q^48 + 110 * q^49 - 20 * q^50 - 54 * q^51 - 142 * q^52 - 60 * q^53 - 62 * q^54 - 32 * q^55 + 33 * q^56 - 56 * q^57 - 87 * q^58 - 53 * q^59 - 65 * q^60 - 76 * q^61 - 84 * q^62 - 215 * q^63 + 67 * q^64 - 49 * q^65 + 9 * q^66 - 145 * q^67 - 133 * q^68 + q^69 - 4 * q^70 - 4 * q^71 - 167 * q^72 - 155 * q^73 - 14 * q^74 - 31 * q^75 - 199 * q^76 - 97 * q^77 - 24 * q^78 - 73 * q^79 + 98 * q^80 + 127 * q^81 - 69 * q^82 - 225 * q^83 - 59 * q^84 - 53 * q^85 + 30 * q^86 - 179 * q^87 - 119 * q^88 - 25 * q^89 - 60 * q^90 - 160 * q^91 - 188 * q^92 - 44 * q^93 - 32 * q^94 - 126 * q^95 - 43 * q^96 - 72 * q^97 - 111 * q^98 - 141 * 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 $

$2$	−2.47054	−1.74694	−0.873469	−	0.486880i	$-0.838135\pi$
$2$	−2.47054	−1.74694	−0.873469	+	0.486880i	$0.838135\pi$
$3$	−3.37029	−1.94584	−0.972919	−	0.231147i	$-0.925752\pi$
$3$	−3.37029	−1.94584	−0.972919	+	0.231147i	$0.925752\pi$
$4$	4.10358	2.05179
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	8.32645	3.39926
$7$	4.71800	1.78324	0.891619	−	0.452787i	$-0.149570\pi$
$7$	4.71800	1.78324	0.891619	+	0.452787i	$0.149570\pi$
$8$	−5.19700	−1.83742
$9$	8.35885	2.78628
$10$	−2.47054	−0.781254
$11$	−2.32047	−0.699648	−0.349824	−	0.936815i	$-0.613759\pi$
$11$	−2.32047	−0.699648	−0.349824	+	0.936815i	$0.613759\pi$
$12$	−13.8303	−3.99245
$13$	−1.92946	−0.535135	−0.267567	−	0.963539i	$-0.586220\pi$
$13$	−1.92946	−0.535135	−0.267567	+	0.963539i	$0.586220\pi$
$14$	−11.6560	−3.11521
$15$	−3.37029	−0.870205
$16$	4.63224	1.15806
$17$	−1.96692	−0.477049	−0.238524	−	0.971136i	$-0.576664\pi$
$17$	−1.96692	−0.477049	−0.238524	+	0.971136i	$0.576664\pi$
$18$	−20.6509	−4.86747
$19$	−6.32515	−1.45109	−0.725545	−	0.688175i	$-0.758412\pi$
$19$	−6.32515	−1.45109	−0.725545	+	0.688175i	$0.758412\pi$
$20$	4.10358	0.917589
$21$	−15.9010	−3.46989
$22$	5.73282	1.22224
$23$	−0.107343	−0.0223827	−0.0111913	−	0.999937i	$-0.503562\pi$
$23$	−0.107343	−0.0223827	−0.0111913	+	0.999937i	$0.503562\pi$
$24$	17.5154	3.57531
$25$	1.00000	0.200000
$26$	4.76681	0.934848
$27$	−18.0609	−3.47582
$28$	19.3607	3.65883
$29$	6.39418	1.18737	0.593685	−	0.804697i	$-0.297672\pi$
$29$	6.39418	1.18737	0.593685	+	0.804697i	$0.297672\pi$
$30$	8.32645	1.52019
$31$	−2.17549	−0.390729	−0.195365	−	0.980731i	$-0.562589\pi$
$31$	−2.17549	−0.390729	−0.195365	+	0.980731i	$0.562589\pi$
$32$	−1.05015	−0.185642
$33$	7.82066	1.36140
$34$	4.85937	0.833375
$35$	4.71800	0.797488
$36$	34.3013	5.71688
$37$	1.98665	0.326604	0.163302	−	0.986576i	$-0.447786\pi$
$37$	1.98665	0.326604	0.163302	+	0.986576i	$0.447786\pi$
$38$	15.6266	2.53496
$39$	6.50283	1.04129
$40$	−5.19700	−0.821717
$41$	12.5405	1.95849	0.979247	−	0.202670i	$-0.0649619\pi$
$41$	12.5405	1.95849	0.979247	+	0.202670i	$0.0649619\pi$
$42$	39.2842	6.06169
$43$	−10.8578	−1.65580	−0.827901	−	0.560874i	$-0.810465\pi$
$43$	−10.8578	−1.65580	−0.827901	+	0.560874i	$0.810465\pi$
$44$	−9.52224	−1.43553
$45$	8.35885	1.24606
$46$	0.265197	0.0391011
$47$	−10.5490	−1.53872	−0.769362	−	0.638813i	$-0.779426\pi$
$47$	−10.5490	−1.53872	−0.769362	+	0.638813i	$0.779426\pi$
$48$	−15.6120	−2.25340
$49$	15.2596	2.17994
$50$	−2.47054	−0.349388
$51$	6.62910	0.928260
$52$	−7.91769	−1.09799
$53$	7.65747	1.05183	0.525917	−	0.850536i	$-0.323722\pi$
$53$	7.65747	1.05183	0.525917	+	0.850536i	$0.323722\pi$
$54$	44.6202	6.07204
$55$	−2.32047	−0.312892
$56$	−24.5195	−3.27655
$57$	21.3176	2.82358
$58$	−15.7971	−2.07426
$59$	−1.49317	−0.194395	−0.0971973	−	0.995265i	$-0.530988\pi$
$59$	−1.49317	−0.194395	−0.0971973	+	0.995265i	$0.530988\pi$
$60$	−13.8303	−1.78548
$61$	5.85389	0.749514	0.374757	−	0.927123i	$-0.377726\pi$
$61$	5.85389	0.749514	0.374757	+	0.927123i	$0.377726\pi$
$62$	5.37464	0.682580
$63$	39.4371	4.96861
$64$	−6.67004	−0.833754
$65$	−1.92946	−0.239320
$66$	−19.3213	−2.37828
$67$	−11.3832	−1.39068	−0.695339	−	0.718682i	$-0.744746\pi$
$67$	−11.3832	−1.39068	−0.695339	+	0.718682i	$0.744746\pi$
$68$	−8.07143	−0.978805
$69$	0.361779	0.0435530
$70$	−11.6560	−1.39316
$71$	16.3784	1.94375	0.971877	−	0.235488i	$-0.0756688\pi$
$71$	16.3784	1.94375	0.971877	+	0.235488i	$0.0756688\pi$
$72$	−43.4409	−5.11956
$73$	−8.05356	−0.942598	−0.471299	−	0.881974i	$-0.656215\pi$
$73$	−8.05356	−0.942598	−0.471299	+	0.881974i	$0.656215\pi$
$74$	−4.90811	−0.570556
$75$	−3.37029	−0.389168
$76$	−25.9558	−2.97733
$77$	−10.9480	−1.24764
$78$	−16.0655	−1.81906
$79$	−2.88132	−0.324174	−0.162087	−	0.986777i	$-0.551822\pi$
$79$	−2.88132	−0.324174	−0.162087	+	0.986777i	$0.551822\pi$
$80$	4.63224	0.517900
$81$	35.7939	3.97710
$82$	−30.9818	−3.42137
$83$	−1.77505	−0.194837	−0.0974185	−	0.995244i	$-0.531059\pi$
$83$	−1.77505	−0.194837	−0.0974185	+	0.995244i	$0.531059\pi$
$84$	−65.2513	−7.11950
$85$	−1.96692	−0.213343
$86$	26.8247	2.89258
$87$	−21.5503	−2.31043
$88$	12.0595	1.28554
$89$	−8.61657	−0.913354	−0.456677	−	0.889633i	$-0.650961\pi$
$89$	−8.61657	−0.913354	−0.456677	+	0.889633i	$0.650961\pi$
$90$	−20.6509	−2.17680
$91$	−9.10318	−0.954273
$92$	−0.440493	−0.0459246
$93$	7.33202	0.760295
$94$	26.0617	2.68805
$95$	−6.32515	−0.648947
$96$	3.53931	0.361229
$97$	−1.10158	−0.111849	−0.0559243	−	0.998435i	$-0.517811\pi$
$97$	−1.10158	−0.111849	−0.0559243	+	0.998435i	$0.517811\pi$
$98$	−37.6994	−3.80821
$99$	−19.3965	−1.94942
$100$	4.10358	0.410358
$101$	−5.38743	−0.536070	−0.268035	−	0.963409i	$-0.586374\pi$
$101$	−5.38743	−0.536070	−0.268035	+	0.963409i	$0.586374\pi$
$102$	−16.3775	−1.62161
$103$	−2.69468	−0.265514	−0.132757	−	0.991149i	$-0.542383\pi$
$103$	−2.69468	−0.265514	−0.132757	+	0.991149i	$0.542383\pi$
$104$	10.0274	0.983265
$105$	−15.9010	−1.55178
$106$	−18.9181	−1.83749
$107$	15.8007	1.52751	0.763757	−	0.645504i	$-0.223353\pi$
$107$	15.8007	1.52751	0.763757	+	0.645504i	$0.223353\pi$
$108$	−74.1144	−7.13166
$109$	7.97113	0.763496	0.381748	−	0.924266i	$-0.375322\pi$
$109$	7.97113	0.763496	0.381748	+	0.924266i	$0.375322\pi$
$110$	5.73282	0.546603
$111$	−6.69559	−0.635518
$112$	21.8549	2.06510
$113$	1.39308	0.131050	0.0655252	−	0.997851i	$-0.479128\pi$
$113$	1.39308	0.131050	0.0655252	+	0.997851i	$0.479128\pi$
$114$	−52.6661	−4.93263
$115$	−0.107343	−0.0100098
$116$	26.2391	2.43624
$117$	−16.1280	−1.49104
$118$	3.68895	0.339595
$119$	−9.27995	−0.850692
$120$	17.5154	1.59893
$121$	−5.61542	−0.510493
$122$	−14.4623	−1.30935
$123$	−42.2651	−3.81091
$124$	−8.92730	−0.801695
$125$	1.00000	0.0894427
$126$	−97.4311	−8.67985
$127$	18.5468	1.64576	0.822879	−	0.568216i	$-0.192366\pi$
$127$	18.5468	1.64576	0.822879	+	0.568216i	$0.192366\pi$
$128$	18.5789	1.64216
$129$	36.5940	3.22192
$130$	4.76681	0.418077
$131$	15.0846	1.31795	0.658975	−	0.752165i	$-0.270990\pi$
$131$	15.0846	1.31795	0.658975	+	0.752165i	$0.270990\pi$
$132$	32.0927	2.79331
$133$	−29.8421	−2.58764
$134$	28.1227	2.42943
$135$	−18.0609	−1.55443
$136$	10.2221	0.876537
$137$	6.45008	0.551067	0.275534	−	0.961291i	$-0.411145\pi$
$137$	6.45008	0.551067	0.275534	+	0.961291i	$0.411145\pi$
$138$	−0.893790	−0.0760844
$139$	−19.7315	−1.67361	−0.836804	−	0.547503i	$-0.815579\pi$
$139$	−19.7315	−1.67361	−0.836804	+	0.547503i	$0.815579\pi$
$140$	19.3607	1.63628
$141$	35.5530	2.99411
$142$	−40.4635	−3.39562
$143$	4.47725	0.374406
$144$	38.7202	3.22668
$145$	6.39418	0.531008
$146$	19.8967	1.64666
$147$	−51.4291	−4.24180
$148$	8.15239	0.670123
$149$	−2.37459	−0.194534	−0.0972670	−	0.995258i	$-0.531010\pi$
$149$	−2.37459	−0.194534	−0.0972670	+	0.995258i	$0.531010\pi$
$150$	8.32645	0.679852
$151$	20.4959	1.66794	0.833968	−	0.551813i	$-0.186064\pi$
$151$	20.4959	1.66794	0.833968	+	0.551813i	$0.186064\pi$
$152$	32.8718	2.66625
$153$	−16.4412	−1.32919
$154$	27.0475	2.17955
$155$	−2.17549	−0.174739
$156$	26.6849	2.13650
$157$	−20.0213	−1.59787	−0.798936	−	0.601416i	$-0.794603\pi$
$157$	−20.0213	−1.59787	−0.798936	+	0.601416i	$0.794603\pi$
$158$	7.11842	0.566311
$159$	−25.8079	−2.04670
$160$	−1.05015	−0.0830216
$161$	−0.506447	−0.0399136
$162$	−88.4303	−6.94774
$163$	−6.41443	−0.502417	−0.251208	−	0.967933i	$-0.580828\pi$
$163$	−6.41443	−0.502417	−0.251208	+	0.967933i	$0.580828\pi$
$164$	51.4609	4.01842
$165$	7.82066	0.608837
$166$	4.38534	0.340368
$167$	−12.5549	−0.971528	−0.485764	−	0.874090i	$-0.661458\pi$
$167$	−12.5549	−0.971528	−0.485764	+	0.874090i	$0.661458\pi$
$168$	82.6377	6.37563
$169$	−9.27720	−0.713631
$170$	4.85937	0.372697
$171$	−52.8710	−4.04315
$172$	−44.5560	−3.39736
$173$	−8.58623	−0.652799	−0.326400	−	0.945232i	$-0.605836\pi$
$173$	−8.58623	−0.652799	−0.326400	+	0.945232i	$0.605836\pi$
$174$	53.2408	4.03618
$175$	4.71800	0.356648
$176$	−10.7490	−0.810234
$177$	5.03243	0.378260
$178$	21.2876	1.59557
$179$	−7.22463	−0.539994	−0.269997	−	0.962861i	$-0.587023\pi$
$179$	−7.22463	−0.539994	−0.269997	+	0.962861i	$0.587023\pi$
$180$	34.3013	2.55667
$181$	−6.09346	−0.452923	−0.226462	−	0.974020i	$-0.572716\pi$
$181$	−6.09346	−0.452923	−0.226462	+	0.974020i	$0.572716\pi$
$182$	22.4898	1.66706
$183$	−19.7293	−1.45843
$184$	0.557864	0.0411262
$185$	1.98665	0.146062
$186$	−18.1141	−1.32819
$187$	4.56418	0.333766
$188$	−43.2885	−3.15714
$189$	−85.2113	−6.19821
$190$	15.6266	1.13367
$191$	22.6239	1.63701	0.818503	−	0.574502i	$-0.194804\pi$
$191$	22.6239	1.63701	0.818503	+	0.574502i	$0.194804\pi$
$192$	22.4800	1.62235
$193$	20.6434	1.48594	0.742972	−	0.669323i	$-0.233416\pi$
$193$	20.6434	1.48594	0.742972	+	0.669323i	$0.233416\pi$
$194$	2.72150	0.195392
$195$	6.50283	0.465677
$196$	62.6189	4.47278
$197$	11.6555	0.830419	0.415209	−	0.909726i	$-0.363708\pi$
$197$	11.6555	0.830419	0.415209	+	0.909726i	$0.363708\pi$
$198$	47.9198	3.40551
$199$	7.07925	0.501835	0.250917	−	0.968009i	$-0.419268\pi$
$199$	7.07925	0.501835	0.250917	+	0.968009i	$0.419268\pi$
$200$	−5.19700	−0.367483
$201$	38.3646	2.70603
$202$	13.3099	0.936480
$203$	30.1678	2.11736
$204$	27.2031	1.90460
$205$	12.5405	0.875865
$206$	6.65731	0.463837
$207$	−0.897268	−0.0623644
$208$	−8.93770	−0.619718
$209$	14.6773	1.01525
$210$	39.2842	2.71087
$211$	2.29979	0.158324	0.0791620	−	0.996862i	$-0.474776\pi$
$211$	2.29979	0.158324	0.0791620	+	0.996862i	$0.474776\pi$
$212$	31.4231	2.15815
$213$	−55.1998	−3.78223
$214$	−39.0364	−2.66847
$215$	−10.8578	−0.740497
$216$	93.8624	6.38653
$217$	−10.2640	−0.696763
$218$	−19.6930	−1.33378
$219$	27.1428	1.83414
$220$	−9.52224	−0.641990
$221$	3.79509	0.255286
$222$	16.5418	1.11021
$223$	−3.95020	−0.264525	−0.132262	−	0.991215i	$-0.542224\pi$
$223$	−3.95020	−0.264525	−0.132262	+	0.991215i	$0.542224\pi$
$224$	−4.95461	−0.331044
$225$	8.35885	0.557257
$226$	−3.44168	−0.228937
$227$	−1.32998	−0.0882737	−0.0441368	−	0.999025i	$-0.514054\pi$
$227$	−1.32998	−0.0882737	−0.0441368	+	0.999025i	$0.514054\pi$
$228$	87.4786	5.79341
$229$	−16.7549	−1.10720	−0.553598	−	0.832784i	$-0.686746\pi$
$229$	−16.7549	−1.10720	−0.553598	+	0.832784i	$0.686746\pi$
$230$	0.265197	0.0174865
$231$	36.8979	2.42770
$232$	−33.2306	−2.18169
$233$	−20.3823	−1.33529	−0.667643	−	0.744482i	$-0.732697\pi$
$233$	−20.3823	−1.33529	−0.667643	+	0.744482i	$0.732697\pi$
$234$	39.8450	2.60475
$235$	−10.5490	−0.688138
$236$	−6.12736	−0.398857
$237$	9.71087	0.630789
$238$	22.9265	1.48611
$239$	−17.0961	−1.10586	−0.552929	−	0.833229i	$-0.686490\pi$
$239$	−17.0961	−1.10586	−0.552929	+	0.833229i	$0.686490\pi$
$240$	−15.6120	−1.00775
$241$	−21.6106	−1.39206	−0.696029	−	0.718014i	$-0.745052\pi$
$241$	−21.6106	−1.39206	−0.696029	+	0.718014i	$0.745052\pi$
$242$	13.8731	0.891799
$243$	−66.4531	−4.26297
$244$	24.0219	1.53785
$245$	15.2596	0.974898
$246$	104.418	6.65743
$247$	12.2041	0.776529
$248$	11.3060	0.717932
$249$	5.98243	0.379121
$250$	−2.47054	−0.156251
$251$	1.08885	0.0687275	0.0343638	−	0.999409i	$-0.489060\pi$
$251$	1.08885	0.0687275	0.0343638	+	0.999409i	$0.489060\pi$
$252$	161.833	10.1946
$253$	0.249087	0.0156600
$254$	−45.8206	−2.87504
$255$	6.62910	0.415130
$256$	−32.5599	−2.03500
$257$	17.7117	1.10483	0.552414	−	0.833570i	$-0.313707\pi$
$257$	17.7117	1.10483	0.552414	+	0.833570i	$0.313707\pi$
$258$	−90.4071	−5.62850
$259$	9.37303	0.582412
$260$	−7.91769	−0.491034
$261$	53.4480	3.30835
$262$	−37.2672	−2.30238
$263$	−22.8075	−1.40637	−0.703184	−	0.711008i	$-0.748239\pi$
$263$	−22.8075	−1.40637	−0.703184	+	0.711008i	$0.748239\pi$
$264$	−40.6439	−2.50146
$265$	7.65747	0.470395
$266$	73.7262	4.52044
$267$	29.0403	1.77724
$268$	−46.7119	−2.85338
$269$	−28.3515	−1.72862	−0.864312	−	0.502956i	$-0.832246\pi$
$269$	−28.3515	−1.72862	−0.864312	+	0.502956i	$0.832246\pi$
$270$	44.6202	2.71550
$271$	13.5644	0.823980	0.411990	−	0.911188i	$-0.364834\pi$
$271$	13.5644	0.823980	0.411990	+	0.911188i	$0.364834\pi$
$272$	−9.11125	−0.552451
$273$	30.6804	1.85686
$274$	−15.9352	−0.962681
$275$	−2.32047	−0.139930
$276$	1.48459	0.0893617
$277$	20.8102	1.25036	0.625181	−	0.780480i	$-0.285025\pi$
$277$	20.8102	1.25036	0.625181	+	0.780480i	$0.285025\pi$
$278$	48.7476	2.92369
$279$	−18.1846	−1.08868
$280$	−24.5195	−1.46532
$281$	−2.08716	−0.124510	−0.0622548	−	0.998060i	$-0.519829\pi$
$281$	−2.08716	−0.124510	−0.0622548	+	0.998060i	$0.519829\pi$
$282$	−87.8353	−5.23052
$283$	−14.2344	−0.846146	−0.423073	−	0.906096i	$-0.639049\pi$
$283$	−14.2344	−0.846146	−0.423073	+	0.906096i	$0.639049\pi$
$284$	67.2100	3.98818
$285$	21.3176	1.26275
$286$	−11.0612	−0.654064
$287$	59.1660	3.49246
$288$	−8.77804	−0.517251
$289$	−13.1312	−0.772424
$290$	−15.7971	−0.927638
$291$	3.71265	0.217639
$292$	−33.0485	−1.93401
$293$	15.0615	0.879901	0.439950	−	0.898022i	$-0.354996\pi$
$293$	15.0615	0.879901	0.439950	+	0.898022i	$0.354996\pi$
$294$	127.058	7.41017
$295$	−1.49317	−0.0869359
$296$	−10.3246	−0.600107
$297$	41.9098	2.43185
$298$	5.86653	0.339839
$299$	0.207114	0.0119777
$300$	−13.8303	−0.798491
$301$	−51.2272	−2.95269
$302$	−50.6361	−2.91378
$303$	18.1572	1.04310
$304$	−29.2996	−1.68045
$305$	5.85389	0.335193
$306$	40.6187	2.32202
$307$	−9.18112	−0.523994	−0.261997	−	0.965069i	$-0.584381\pi$
$307$	−9.18112	−0.523994	−0.261997	+	0.965069i	$0.584381\pi$
$308$	−44.9260	−2.55990
$309$	9.08184	0.516648
$310$	5.37464	0.305259
$311$	17.8992	1.01497	0.507486	−	0.861660i	$-0.330575\pi$
$311$	17.8992	1.01497	0.507486	+	0.861660i	$0.330575\pi$
$312$	−33.7952	−1.91328
$313$	14.9847	0.846985	0.423492	−	0.905900i	$-0.360804\pi$
$313$	14.9847	0.846985	0.423492	+	0.905900i	$0.360804\pi$
$314$	49.4634	2.79138
$315$	39.4371	2.22203
$316$	−11.8237	−0.665137
$317$	−7.84290	−0.440501	−0.220251	−	0.975443i	$-0.570687\pi$
$317$	−7.84290	−0.440501	−0.220251	+	0.975443i	$0.570687\pi$
$318$	63.7595	3.57546
$319$	−14.8375	−0.830741
$320$	−6.67004	−0.372866
$321$	−53.2530	−2.97229
$322$	1.25120	0.0697266
$323$	12.4411	0.692241
$324$	146.883	8.16018
$325$	−1.92946	−0.107027
$326$	15.8471	0.877691
$327$	−26.8650	−1.48564
$328$	−65.1728	−3.59857
$329$	−49.7700	−2.74391
$330$	−19.3213	−1.06360
$331$	−9.16981	−0.504018	−0.252009	−	0.967725i	$-0.581091\pi$
$331$	−9.16981	−0.504018	−0.252009	+	0.967725i	$0.581091\pi$
$332$	−7.28407	−0.399765
$333$	16.6061	0.910011
$334$	31.0174	1.69720
$335$	−11.3832	−0.621930
$336$	−73.6574	−4.01834
$337$	−5.82321	−0.317211	−0.158605	−	0.987342i	$-0.550700\pi$
$337$	−5.82321	−0.317211	−0.158605	+	0.987342i	$0.550700\pi$
$338$	22.9197	1.24667
$339$	−4.69510	−0.255003
$340$	−8.07143	−0.437735
$341$	5.04815	0.273373
$342$	130.620	7.06313
$343$	38.9686	2.10411
$344$	56.4281	3.04240
$345$	0.361779	0.0194775
$346$	21.2127	1.14040
$347$	−15.3826	−0.825780	−0.412890	−	0.910781i	$-0.635481\pi$
$347$	−15.3826	−0.825780	−0.412890	+	0.910781i	$0.635481\pi$
$348$	−88.4333	−4.74052
$349$	25.3440	1.35663	0.678315	−	0.734771i	$-0.262710\pi$
$349$	25.3440	1.35663	0.678315	+	0.734771i	$0.262710\pi$
$350$	−11.6560	−0.623041
$351$	34.8477	1.86003
$352$	2.43684	0.129884
$353$	−17.8587	−0.950522	−0.475261	−	0.879845i	$-0.657646\pi$
$353$	−17.8587	−0.950522	−0.475261	+	0.879845i	$0.657646\pi$
$354$	−12.4328	−0.660797
$355$	16.3784	0.869273
$356$	−35.3588	−1.87401
$357$	31.2761	1.65531
$358$	17.8488	0.943336
$359$	−28.2065	−1.48868	−0.744341	−	0.667799i	$-0.767236\pi$
$359$	−28.2065	−1.48868	−0.744341	+	0.667799i	$0.767236\pi$
$360$	−43.4409	−2.28954
$361$	21.0076	1.10566
$362$	15.0542	0.791229
$363$	18.9256	0.993336
$364$	−37.3557	−1.95797
$365$	−8.05356	−0.421542
$366$	48.7421	2.54779
$367$	−32.3919	−1.69084	−0.845421	−	0.534101i	$-0.820650\pi$
$367$	−32.3919	−1.69084	−0.845421	+	0.534101i	$0.820650\pi$
$368$	−0.497240	−0.0259204
$369$	104.824	5.45692
$370$	−4.90811	−0.255160
$371$	36.1280	1.87567
$372$	30.0876	1.55997
$373$	−0.946753	−0.0490210	−0.0245105	−	0.999700i	$-0.507803\pi$
$373$	−0.946753	−0.0490210	−0.0245105	+	0.999700i	$0.507803\pi$
$374$	−11.2760	−0.583069
$375$	−3.37029	−0.174041
$376$	54.8229	2.82727
$377$	−12.3373	−0.635403
$378$	210.518	10.8279
$379$	−11.5818	−0.594919	−0.297459	−	0.954734i	$-0.596139\pi$
$379$	−11.5818	−0.594919	−0.297459	+	0.954734i	$0.596139\pi$
$380$	−25.9558	−1.33150
$381$	−62.5080	−3.20238
$382$	−55.8933	−2.85975
$383$	−7.99166	−0.408355	−0.204177	−	0.978934i	$-0.565452\pi$
$383$	−7.99166	−0.408355	−0.204177	+	0.978934i	$0.565452\pi$
$384$	−62.6163	−3.19538
$385$	−10.9480	−0.557961
$386$	−51.0004	−2.59585
$387$	−90.7589	−4.61354
$388$	−4.52043	−0.229490
$389$	28.4339	1.44165	0.720827	−	0.693115i	$-0.243762\pi$
$389$	28.4339	1.44165	0.720827	+	0.693115i	$0.243762\pi$
$390$	−16.0655	−0.813509
$391$	0.211136	0.0106776
$392$	−79.3039	−4.00545
$393$	−50.8395	−2.56452
$394$	−28.7954	−1.45069
$395$	−2.88132	−0.144975
$396$	−79.5950	−3.99980
$397$	−6.76203	−0.339377	−0.169688	−	0.985498i	$-0.554276\pi$
$397$	−6.76203	−0.339377	−0.169688	+	0.985498i	$0.554276\pi$
$398$	−17.4896	−0.876674
$399$	100.577	5.03512
$400$	4.63224	0.231612
$401$	−39.4996	−1.97252	−0.986258	−	0.165213i	$-0.947169\pi$
$401$	−39.4996	−1.97252	−0.986258	+	0.165213i	$0.947169\pi$
$402$	−94.7815	−4.72727
$403$	4.19751	0.209093
$404$	−22.1078	−1.09990
$405$	35.7939	1.77861
$406$	−74.5308	−3.69890
$407$	−4.60997	−0.228508
$408$	−34.4514	−1.70560
$409$	12.4392	0.615080	0.307540	−	0.951535i	$-0.400494\pi$
$409$	12.4392	0.615080	0.307540	+	0.951535i	$0.400494\pi$
$410$	−30.9818	−1.53008
$411$	−21.7386	−1.07229
$412$	−11.0578	−0.544780
$413$	−7.04480	−0.346652
$414$	2.21674	0.108947
$415$	−1.77505	−0.0871338
$416$	2.02622	0.0993435
$417$	66.5010	3.25657
$418$	−36.2610	−1.77358
$419$	7.78183	0.380167	0.190084	−	0.981768i	$-0.439124\pi$
$419$	7.78183	0.380167	0.190084	+	0.981768i	$0.439124\pi$
$420$	−65.2513	−3.18394
$421$	14.8936	0.725871	0.362935	−	0.931814i	$-0.381775\pi$
$421$	14.8936	0.725871	0.362935	+	0.931814i	$0.381775\pi$
$422$	−5.68173	−0.276582
$423$	−88.1772	−4.28732
$424$	−39.7959	−1.93266
$425$	−1.96692	−0.0954098
$426$	136.374	6.60732
$427$	27.6187	1.33656
$428$	64.8396	3.13414
$429$	−15.0896	−0.728533
$430$	26.8247	1.29360
$431$	22.7767	1.09711	0.548557	−	0.836113i	$-0.315178\pi$
$431$	22.7767	1.09711	0.548557	+	0.836113i	$0.315178\pi$
$432$	−83.6623	−4.02521
$433$	28.8514	1.38651	0.693255	−	0.720693i	$-0.256176\pi$
$433$	28.8514	1.38651	0.693255	+	0.720693i	$0.256176\pi$
$434$	25.3576	1.21720
$435$	−21.5503	−1.03326
$436$	32.7102	1.56654
$437$	0.678964	0.0324792
$438$	−67.0575	−3.20413
$439$	−34.7805	−1.65998	−0.829991	−	0.557777i	$-0.811655\pi$
$439$	−34.7805	−1.65998	−0.829991	+	0.557777i	$0.811655\pi$
$440$	12.0595	0.574913
$441$	127.552	6.07393
$442$	−9.37594	−0.445968
$443$	7.22733	0.343381	0.171690	−	0.985151i	$-0.445077\pi$
$443$	7.22733	0.343381	0.171690	+	0.985151i	$0.445077\pi$
$444$	−27.4759	−1.30395
$445$	−8.61657	−0.408464
$446$	9.75914	0.462109
$447$	8.00306	0.378532
$448$	−31.4693	−1.48678
$449$	−18.0017	−0.849554	−0.424777	−	0.905298i	$-0.639647\pi$
$449$	−18.0017	−0.849554	−0.424777	+	0.905298i	$0.639647\pi$
$450$	−20.6509	−0.973493
$451$	−29.0998	−1.37026
$452$	5.71664	0.268888
$453$	−69.0772	−3.24553
$454$	3.28577	0.154209
$455$	−9.10318	−0.426764
$456$	−110.787	−5.18810
$457$	−6.52716	−0.305328	−0.152664	−	0.988278i	$-0.548785\pi$
$457$	−6.52716	−0.305328	−0.152664	+	0.988278i	$0.548785\pi$
$458$	41.3938	1.93420
$459$	35.5244	1.65814
$460$	−0.440493	−0.0205381
$461$	0.423507	0.0197247	0.00986234	−	0.999951i	$-0.496861\pi$
$461$	0.423507	0.0197247	0.00986234	+	0.999951i	$0.496861\pi$
$462$	−91.1578	−4.24105
$463$	20.3599	0.946206	0.473103	−	0.881007i	$-0.343134\pi$
$463$	20.3599	0.946206	0.473103	+	0.881007i	$0.343134\pi$
$464$	29.6194	1.37505
$465$	7.33202	0.340014
$466$	50.3553	2.33266
$467$	1.80417	0.0834871	0.0417435	−	0.999128i	$-0.486709\pi$
$467$	1.80417	0.0834871	0.0417435	+	0.999128i	$0.486709\pi$
$468$	−66.1828	−3.05930
$469$	−53.7059	−2.47991
$470$	26.0617	1.20213
$471$	67.4775	3.10920
$472$	7.76002	0.357184
$473$	25.1952	1.15848
$474$	−23.9911	−1.10195
$475$	−6.32515	−0.290218
$476$	−38.0811	−1.74544
$477$	64.0077	2.93071
$478$	42.2368	1.93186
$479$	20.4180	0.932921	0.466460	−	0.884542i	$-0.345529\pi$
$479$	20.4180	0.932921	0.466460	+	0.884542i	$0.345529\pi$
$480$	3.53931	0.161547
$481$	−3.83316	−0.174777
$482$	53.3898	2.43184
$483$	1.70687	0.0776654
$484$	−23.0434	−1.04743
$485$	−1.10158	−0.0500202
$486$	164.175	7.44714
$487$	−12.5290	−0.567741	−0.283871	−	0.958863i	$-0.591619\pi$
$487$	−12.5290	−0.567741	−0.283871	+	0.958863i	$0.591619\pi$
$488$	−30.4227	−1.37717
$489$	21.6185	0.977622
$490$	−37.6994	−1.70309
$491$	0.865096	0.0390412	0.0195206	−	0.999809i	$-0.493786\pi$
$491$	0.865096	0.0390412	0.0195206	+	0.999809i	$0.493786\pi$
$492$	−173.438	−7.81920
$493$	−12.5769	−0.566434
$494$	−30.1508	−1.35655
$495$	−19.3965	−0.871806
$496$	−10.0774	−0.452487
$497$	77.2732	3.46618
$498$	−14.7799	−0.662301
$499$	−6.46032	−0.289204	−0.144602	−	0.989490i	$-0.546190\pi$
$499$	−6.46032	−0.289204	−0.144602	+	0.989490i	$0.546190\pi$
$500$	4.10358	0.183518
$501$	42.3137	1.89044
$502$	−2.69005	−0.120063
$503$	21.4899	0.958187	0.479093	−	0.877764i	$-0.340966\pi$
$503$	21.4899	0.958187	0.479093	+	0.877764i	$0.340966\pi$
$504$	−204.955	−9.12940
$505$	−5.38743	−0.239738
$506$	−0.615381	−0.0273570
$507$	31.2668	1.38861
$508$	76.1082	3.37676
$509$	−14.6689	−0.650189	−0.325095	−	0.945681i	$-0.605396\pi$
$509$	−14.6689	−0.650189	−0.325095	+	0.945681i	$0.605396\pi$
$510$	−16.3775	−0.725207
$511$	−37.9967	−1.68088
$512$	43.2829	1.91285
$513$	114.238	5.04373
$514$	−43.7576	−1.93007
$515$	−2.69468	−0.118742
$516$	150.167	6.61072
$517$	24.4785	1.07656
$518$	−23.1565	−1.01744
$519$	28.9381	1.27024
$520$	10.0274	0.439730
$521$	−13.8518	−0.606859	−0.303430	−	0.952854i	$-0.598132\pi$
$521$	−13.8518	−0.606859	−0.303430	+	0.952854i	$0.598132\pi$
$522$	−132.046	−5.77948
$523$	−23.9376	−1.04672	−0.523360	−	0.852112i	$-0.675322\pi$
$523$	−23.9376	−1.04672	−0.523360	+	0.852112i	$0.675322\pi$
$524$	61.9010	2.70416
$525$	−15.9010	−0.693978
$526$	56.3468	2.45684
$527$	4.27902	0.186397
$528$	36.2271	1.57658
$529$	−22.9885	−0.999499
$530$	−18.9181	−0.821750
$531$	−12.4812	−0.541639
$532$	−122.460	−5.30930
$533$	−24.1963	−1.04806
$534$	−71.7454	−3.10473
$535$	15.8007	0.683125
$536$	59.1584	2.55525
$537$	24.3491	1.05074
$538$	70.0437	3.01980
$539$	−35.4093	−1.52519
$540$	−74.1144	−3.18938
$541$	−15.5616	−0.669045	−0.334522	−	0.942388i	$-0.608575\pi$
$541$	−15.5616	−0.669045	−0.334522	+	0.942388i	$0.608575\pi$
$542$	−33.5115	−1.43944
$543$	20.5367	0.881315
$544$	2.06556	0.0885602
$545$	7.97113	0.341446
$546$	−75.7972	−3.24382
$547$	−1.80259	−0.0770732	−0.0385366	−	0.999257i	$-0.512270\pi$
$547$	−1.80259	−0.0770732	−0.0385366	+	0.999257i	$0.512270\pi$
$548$	26.4684	1.13068
$549$	48.9318	2.08836
$550$	5.73282	0.244448
$551$	−40.4442	−1.72298
$552$	−1.88016	−0.0800250
$553$	−13.5941	−0.578078
$554$	−51.4125	−2.18431
$555$	−6.69559	−0.284212
$556$	−80.9701	−3.43390
$557$	11.0655	0.468858	0.234429	−	0.972133i	$-0.424678\pi$
$557$	11.0655	0.468858	0.234429	+	0.972133i	$0.424678\pi$
$558$	44.9258	1.90186
$559$	20.9497	0.886078
$560$	21.8549	0.923539
$561$	−15.3826	−0.649455
$562$	5.15642	0.217511
$563$	43.6821	1.84098	0.920491	−	0.390763i	$-0.127789\pi$
$563$	43.6821	1.84098	0.920491	+	0.390763i	$0.127789\pi$
$564$	145.895	6.14328
$565$	1.39308	0.0586075
$566$	35.1666	1.47816
$567$	168.876	7.09211
$568$	−85.1183	−3.57149
$569$	13.5636	0.568616	0.284308	−	0.958733i	$-0.408236\pi$
$569$	13.5636	0.568616	0.284308	+	0.958733i	$0.408236\pi$
$570$	−52.6661	−2.20594
$571$	24.5263	1.02639	0.513196	−	0.858272i	$-0.328461\pi$
$571$	24.5263	1.02639	0.513196	+	0.858272i	$0.328461\pi$
$572$	18.3728	0.768203
$573$	−76.2490	−3.18535
$574$	−146.172	−6.10111
$575$	−0.107343	−0.00447653
$576$	−55.7539	−2.32308
$577$	24.4374	1.01734	0.508671	−	0.860961i	$-0.330137\pi$
$577$	24.4374	1.01734	0.508671	+	0.860961i	$0.330137\pi$
$578$	32.4412	1.34938
$579$	−69.5742	−2.89141
$580$	26.2391	1.08952
$581$	−8.37469	−0.347441
$582$	−9.17225	−0.380202
$583$	−17.7689	−0.735914
$584$	41.8543	1.73194
$585$	−16.1280	−0.666813
$586$	−37.2100	−1.53713
$587$	−5.06696	−0.209136	−0.104568	−	0.994518i	$-0.533346\pi$
$587$	−5.06696	−0.209136	−0.104568	+	0.994518i	$0.533346\pi$
$588$	−211.044	−8.70330
$589$	13.7603	0.566983
$590$	3.68895	0.151872
$591$	−39.2823	−1.61586
$592$	9.20264	0.378226
$593$	−12.2223	−0.501909	−0.250955	−	0.967999i	$-0.580744\pi$
$593$	−12.2223	−0.501909	−0.250955	+	0.967999i	$0.580744\pi$
$594$	−103.540	−4.24829
$595$	−9.27995	−0.380441
$596$	−9.74433	−0.399143
$597$	−23.8591	−0.976489
$598$	−0.511685	−0.0209244
$599$	2.69653	0.110177	0.0550887	−	0.998481i	$-0.482456\pi$
$599$	2.69653	0.110177	0.0550887	+	0.998481i	$0.482456\pi$
$600$	17.5154	0.715063
$601$	11.4249	0.466033	0.233016	−	0.972473i	$-0.425140\pi$
$601$	11.4249	0.466033	0.233016	+	0.972473i	$0.425140\pi$
$602$	126.559	5.15817
$603$	−95.1504	−3.87482
$604$	84.1068	3.42226
$605$	−5.61542	−0.228299
$606$	−44.8582	−1.82224
$607$	18.3517	0.744874	0.372437	−	0.928057i	$-0.378522\pi$
$607$	18.3517	0.744874	0.372437	+	0.928057i	$0.378522\pi$
$608$	6.64235	0.269383
$609$	−101.674	−4.12005
$610$	−14.4623	−0.585561
$611$	20.3538	0.823425
$612$	−67.4679	−2.72723
$613$	16.5406	0.668070	0.334035	−	0.942561i	$-0.391590\pi$
$613$	16.5406	0.668070	0.334035	+	0.942561i	$0.391590\pi$
$614$	22.6823	0.915385
$615$	−42.2651	−1.70429
$616$	56.8966	2.29243
$617$	−9.96314	−0.401101	−0.200550	−	0.979683i	$-0.564273\pi$
$617$	−9.96314	−0.401101	−0.200550	+	0.979683i	$0.564273\pi$
$618$	−22.4371	−0.902552
$619$	6.39256	0.256939	0.128469	−	0.991713i	$-0.458994\pi$
$619$	6.39256	0.256939	0.128469	+	0.991713i	$0.458994\pi$
$620$	−8.92730	−0.358529
$621$	1.93872	0.0777981
$622$	−44.2208	−1.77309
$623$	−40.6530	−1.62873
$624$	30.1226	1.20587
$625$	1.00000	0.0400000
$626$	−37.0203	−1.47963
$627$	−49.4668	−1.97552
$628$	−82.1590	−3.27850
$629$	−3.90759	−0.155806
$630$	−97.4311	−3.88175
$631$	−5.22099	−0.207844	−0.103922	−	0.994585i	$-0.533139\pi$
$631$	−5.22099	−0.207844	−0.103922	+	0.994585i	$0.533139\pi$
$632$	14.9742	0.595642
$633$	−7.75095	−0.308073
$634$	19.3762	0.769528
$635$	18.5468	0.736006
$636$	−105.905	−4.19940
$637$	−29.4427	−1.16656
$638$	36.6567	1.45125
$639$	136.904	5.41585
$640$	18.5789	0.734396
$641$	−26.5527	−1.04877	−0.524385	−	0.851481i	$-0.675705\pi$
$641$	−26.5527	−1.04877	−0.524385	+	0.851481i	$0.675705\pi$
$642$	131.564	5.19241
$643$	25.8388	1.01898	0.509491	−	0.860476i	$-0.329834\pi$
$643$	25.8388	1.01898	0.509491	+	0.860476i	$0.329834\pi$
$644$	−2.07825	−0.0818944
$645$	36.5940	1.44089
$646$	−30.7362	−1.20930
$647$	−38.9726	−1.53217	−0.766085	−	0.642739i	$-0.777798\pi$
$647$	−38.9726	−1.53217	−0.766085	+	0.642739i	$0.777798\pi$
$648$	−186.021	−7.30758
$649$	3.46486	0.136008
$650$	4.76681	0.186970
$651$	34.5925	1.35579
$652$	−26.3222	−1.03086
$653$	−40.1816	−1.57243	−0.786213	−	0.617956i	$-0.787961\pi$
$653$	−40.1816	−1.57243	−0.786213	+	0.617956i	$0.787961\pi$
$654$	66.3712	2.59532
$655$	15.0846	0.589405
$656$	58.0905	2.26805
$657$	−67.3185	−2.62635
$658$	122.959	4.79344
$659$	−19.1565	−0.746231	−0.373115	−	0.927785i	$-0.621710\pi$
$659$	−19.1565	−0.746231	−0.373115	+	0.927785i	$0.621710\pi$
$660$	32.0927	1.24921
$661$	26.8856	1.04573	0.522863	−	0.852416i	$-0.324864\pi$
$661$	26.8856	1.04573	0.522863	+	0.852416i	$0.324864\pi$
$662$	22.6544	0.880489
$663$	−12.7906	−0.496744
$664$	9.22493	0.357997
$665$	−29.8421	−1.15723
$666$	−41.0262	−1.58973
$667$	−0.686374	−0.0265765
$668$	−51.5201	−1.99337
$669$	13.3133	0.514723
$670$	28.1227	1.08647
$671$	−13.5838	−0.524396
$672$	16.6985	0.644157
$673$	34.2121	1.31878	0.659389	−	0.751802i	$-0.270815\pi$
$673$	34.2121	1.31878	0.659389	+	0.751802i	$0.270815\pi$
$674$	14.3865	0.554147
$675$	−18.0609	−0.695164
$676$	−38.0698	−1.46422
$677$	−33.1636	−1.27458	−0.637290	−	0.770624i	$-0.719945\pi$
$677$	−33.1636	−1.27458	−0.637290	+	0.770624i	$0.719945\pi$
$678$	11.5994	0.445474
$679$	−5.19726	−0.199453
$680$	10.2221	0.391999
$681$	4.48241	0.171766
$682$	−12.4717	−0.477565
$683$	13.4910	0.516217	0.258109	−	0.966116i	$-0.416901\pi$
$683$	13.4910	0.516217	0.258109	+	0.966116i	$0.416901\pi$
$684$	−216.961	−8.29570
$685$	6.45008	0.246445
$686$	−96.2737	−3.67575
$687$	56.4690	2.15443
$688$	−50.2960	−1.91752
$689$	−14.7748	−0.562873
$690$	−0.893790	−0.0340260
$691$	0.221823	0.00843853	0.00421926	−	0.999991i	$-0.498657\pi$
$691$	0.221823	0.00843853	0.00421926	+	0.999991i	$0.498657\pi$
$692$	−35.2343	−1.33941
$693$	−91.5126	−3.47628
$694$	38.0033	1.44259
$695$	−19.7315	−0.748460
$696$	111.997	4.24522
$697$	−24.6662	−0.934297
$698$	−62.6133	−2.36995
$699$	68.6941	2.59825
$700$	19.3607	0.731767
$701$	31.9377	1.20627	0.603134	−	0.797640i	$-0.293918\pi$
$701$	31.9377	1.20627	0.603134	+	0.797640i	$0.293918\pi$
$702$	−86.0928	−3.24936
$703$	−12.5659	−0.473931
$704$	15.4776	0.583335
$705$	35.5530	1.33900
$706$	44.1207	1.66050
$707$	−25.4179	−0.955940
$708$	20.6510	0.776111
$709$	−16.2427	−0.610007	−0.305004	−	0.952351i	$-0.598658\pi$
$709$	−16.2427	−0.610007	−0.305004	+	0.952351i	$0.598658\pi$
$710$	−40.4635	−1.51857
$711$	−24.0845	−0.903240
$712$	44.7803	1.67821
$713$	0.233524	0.00874556
$714$	−77.2690	−2.89172
$715$	4.47725	0.167439
$716$	−29.6469	−1.10796
$717$	57.6190	2.15182
$718$	69.6854	2.60064
$719$	1.68587	0.0628722	0.0314361	−	0.999506i	$-0.489992\pi$
$719$	1.68587	0.0628722	0.0314361	+	0.999506i	$0.489992\pi$
$720$	38.7202	1.44302
$721$	−12.7135	−0.473475
$722$	−51.9001	−1.93152
$723$	72.8338	2.70872
$724$	−25.0050	−0.929304
$725$	6.39418	0.237474
$726$	−46.7565	−1.73530
$727$	−21.5296	−0.798490	−0.399245	−	0.916844i	$-0.630728\pi$
$727$	−21.5296	−0.798490	−0.399245	+	0.916844i	$0.630728\pi$
$728$	47.3092	1.75340
$729$	116.584	4.31794
$730$	19.8967	0.736409
$731$	21.3565	0.789899
$732$	−80.9609	−2.99240
$733$	−48.2101	−1.78068	−0.890339	−	0.455297i	$-0.849533\pi$
$733$	−48.2101	−1.78068	−0.890339	+	0.455297i	$0.849533\pi$
$734$	80.0255	2.95380
$735$	−51.4291	−1.89699
$736$	0.112727	0.00415516
$737$	26.4143	0.972985
$738$	−258.972	−9.53290
$739$	−25.4072	−0.934620	−0.467310	−	0.884093i	$-0.654777\pi$
$739$	−25.4072	−0.934620	−0.467310	+	0.884093i	$0.654777\pi$
$740$	8.15239	0.299688
$741$	−41.1314	−1.51100
$742$	−89.2557	−3.27668
$743$	−14.4034	−0.528408	−0.264204	−	0.964467i	$-0.585109\pi$
$743$	−14.4034	−0.528408	−0.264204	+	0.964467i	$0.585109\pi$
$744$	−38.1045	−1.39698
$745$	−2.37459	−0.0869982
$746$	2.33899	0.0856366
$747$	−14.8374	−0.542871
$748$	18.7295	0.684819
$749$	74.5479	2.72392
$750$	8.32645	0.304039
$751$	−17.3064	−0.631519	−0.315760	−	0.948839i	$-0.602259\pi$
$751$	−17.3064	−0.631519	−0.315760	+	0.948839i	$0.602259\pi$
$752$	−48.8653	−1.78193
$753$	−3.66974	−0.133733
$754$	30.4798	1.11001
$755$	20.4959	0.745923
$756$	−349.672	−12.7174
$757$	5.81745	0.211439	0.105719	−	0.994396i	$-0.466285\pi$
$757$	5.81745	0.211439	0.105719	+	0.994396i	$0.466285\pi$
$758$	28.6134	1.03929
$759$	−0.839496	−0.0304718
$760$	32.8718	1.19239
$761$	−25.7721	−0.934239	−0.467119	−	0.884194i	$-0.654708\pi$
$761$	−25.7721	−0.934239	−0.467119	+	0.884194i	$0.654708\pi$
$762$	154.429	5.59436
$763$	37.6078	1.36149
$764$	92.8390	3.35880
$765$	−16.4412	−0.594434
$766$	19.7437	0.713371
$767$	2.88101	0.104027
$768$	109.736	3.95977
$769$	−33.0620	−1.19225	−0.596123	−	0.802893i	$-0.703293\pi$
$769$	−33.0620	−1.19225	−0.596123	+	0.802893i	$0.703293\pi$
$770$	27.0475	0.974723
$771$	−59.6937	−2.14982
$772$	84.7119	3.04885
$773$	0.278308	0.0100100	0.00500502	−	0.999987i	$-0.498407\pi$
$773$	0.278308	0.0100100	0.00500502	+	0.999987i	$0.498407\pi$
$774$	224.224	8.05956
$775$	−2.17549	−0.0781458
$776$	5.72491	0.205512
$777$	−31.5898	−1.13328
$778$	−70.2471	−2.51848
$779$	−79.3205	−2.84195
$780$	26.6849	0.955473
$781$	−38.0055	−1.35994
$782$	−0.521621	−0.0186531
$783$	−115.485	−4.12708
$784$	70.6859	2.52450
$785$	−20.0213	−0.714590
$786$	125.601	4.48005
$787$	34.4457	1.22786	0.613928	−	0.789362i	$-0.289589\pi$
$787$	34.4457	1.22786	0.613928	+	0.789362i	$0.289589\pi$
$788$	47.8292	1.70385
$789$	76.8678	2.73656
$790$	7.11842	0.253262
$791$	6.57258	0.233694
$792$	100.803	3.58189
$793$	−11.2948	−0.401091
$794$	16.7059	0.592870
$795$	−25.8079	−0.915312
$796$	29.0503	1.02966
$797$	46.0249	1.63028	0.815142	−	0.579261i	$-0.196659\pi$
$797$	46.0249	1.63028	0.815142	+	0.579261i	$0.196659\pi$
$798$	−248.479	−8.79605
$799$	20.7490	0.734046
$800$	−1.05015	−0.0371284
$801$	−72.0246	−2.54486
$802$	97.5855	3.44586
$803$	18.6880	0.659487
$804$	157.433	5.55222
$805$	−0.506447	−0.0178499
$806$	−10.3701	−0.365272
$807$	95.5529	3.36362
$808$	27.9985	0.984983
$809$	−35.3567	−1.24307	−0.621537	−	0.783385i	$-0.713492\pi$
$809$	−35.3567	−1.24307	−0.621537	+	0.783385i	$0.713492\pi$
$810$	−88.4303	−3.10712
$811$	−8.02264	−0.281713	−0.140856	−	0.990030i	$-0.544986\pi$
$811$	−8.02264	−0.281713	−0.140856	+	0.990030i	$0.544986\pi$
$812$	123.796	4.34439
$813$	−45.7160	−1.60333
$814$	11.3891	0.399188
$815$	−6.41443	−0.224688
$816$	30.7076	1.07498
$817$	68.6774	2.40272
$818$	−30.7316	−1.07451
$819$	−76.0922	−2.65888
$820$	51.4609	1.79709
$821$	11.4647	0.400119	0.200060	−	0.979784i	$-0.435886\pi$
$821$	11.4647	0.400119	0.200060	+	0.979784i	$0.435886\pi$
$822$	53.7062	1.87322
$823$	15.3121	0.533747	0.266874	−	0.963732i	$-0.414009\pi$
$823$	15.3121	0.533747	0.266874	+	0.963732i	$0.414009\pi$
$824$	14.0042	0.487860
$825$	7.82066	0.272280
$826$	17.4045	0.605579
$827$	−56.1454	−1.95237	−0.976184	−	0.216946i	$-0.930390\pi$
$827$	−56.1454	−1.95237	−0.976184	+	0.216946i	$0.930390\pi$
$828$	−3.68202	−0.127959
$829$	−17.4600	−0.606412	−0.303206	−	0.952925i	$-0.598057\pi$
$829$	−17.4600	−0.606412	−0.303206	+	0.952925i	$0.598057\pi$
$830$	4.38534	0.152217
$831$	−70.1363	−2.43300
$832$	12.8695	0.446171
$833$	−30.0144	−1.03994
$834$	−164.294	−5.68902
$835$	−12.5549	−0.434480
$836$	60.2296	2.08309
$837$	39.2912	1.35810
$838$	−19.2254	−0.664129
$839$	−11.3994	−0.393551	−0.196776	−	0.980449i	$-0.563047\pi$
$839$	−11.3994	−0.393551	−0.196776	+	0.980449i	$0.563047\pi$
$840$	82.6377	2.85127
$841$	11.8856	0.409848
$842$	−36.7954	−1.26805
$843$	7.03434	0.242275
$844$	9.43737	0.324848
$845$	−9.27720	−0.319145
$846$	217.846	7.48968
$847$	−26.4936	−0.910330
$848$	35.4712	1.21809
$849$	47.9740	1.64646
$850$	4.85937	0.166675
$851$	−0.213254	−0.00731026
$852$	−226.517	−7.76035
$853$	20.3689	0.697417	0.348708	−	0.937231i	$-0.386620\pi$
$853$	20.3689	0.697417	0.348708	+	0.937231i	$0.386620\pi$
$854$	−68.2332	−2.33489
$855$	−52.8710	−1.80815
$856$	−82.1163	−2.80668
$857$	−42.1801	−1.44085	−0.720423	−	0.693535i	$-0.756052\pi$
$857$	−42.1801	−1.44085	−0.720423	+	0.693535i	$0.756052\pi$
$858$	37.2795	1.27270
$859$	24.7203	0.843444	0.421722	−	0.906725i	$-0.361426\pi$
$859$	24.7203	0.843444	0.421722	+	0.906725i	$0.361426\pi$
$860$	−44.5560	−1.51935
$861$	−199.407	−6.79576
$862$	−56.2707	−1.91659
$863$	−10.2710	−0.349629	−0.174814	−	0.984601i	$-0.555933\pi$
$863$	−10.2710	−0.349629	−0.174814	+	0.984601i	$0.555933\pi$
$864$	18.9666	0.645258
$865$	−8.58623	−0.291941
$866$	−71.2786	−2.42215
$867$	44.2560	1.50301
$868$	−42.1190	−1.42961
$869$	6.68601	0.226807
$870$	53.2408	1.80503
$871$	21.9634	0.744200
$872$	−41.4260	−1.40286
$873$	−9.20795	−0.311642
$874$	−1.67741	−0.0567392
$875$	4.71800	0.159498
$876$	111.383	3.76328
$877$	−1.97411	−0.0666610	−0.0333305	−	0.999444i	$-0.510611\pi$
$877$	−1.97411	−0.0666610	−0.0333305	+	0.999444i	$0.510611\pi$
$878$	85.9267	2.89989
$879$	−50.7615	−1.71214
$880$	−10.7490	−0.362348
$881$	36.5108	1.23008	0.615040	−	0.788496i	$-0.289140\pi$
$881$	36.5108	1.23008	0.615040	+	0.788496i	$0.289140\pi$
$882$	−315.124	−10.6108
$883$	19.9648	0.671871	0.335935	−	0.941885i	$-0.390948\pi$
$883$	19.9648	0.671871	0.335935	+	0.941885i	$0.390948\pi$
$884$	15.5735	0.523793
$885$	5.03243	0.169163
$886$	−17.8554	−0.599865
$887$	30.5934	1.02723	0.513614	−	0.858022i	$-0.328306\pi$
$887$	30.5934	1.02723	0.513614	+	0.858022i	$0.328306\pi$
$888$	34.7970	1.16771
$889$	87.5037	2.93478
$890$	21.2876	0.713562
$891$	−83.0586	−2.78257
$892$	−16.2100	−0.542750
$893$	66.7238	2.23283
$894$	−19.7719	−0.661271
$895$	−7.22463	−0.241493
$896$	87.6554	2.92836
$897$	−0.698036	−0.0233067
$898$	44.4740	1.48412
$899$	−13.9105	−0.463940
$900$	34.3013	1.14338
$901$	−15.0617	−0.501776
$902$	71.8923	2.39375
$903$	172.651	5.74545
$904$	−7.23986	−0.240794
$905$	−6.09346	−0.202553
$906$	170.658	5.66974
$907$	−15.0631	−0.500163	−0.250081	−	0.968225i	$-0.580457\pi$
$907$	−15.0631	−0.500163	−0.250081	+	0.968225i	$0.580457\pi$
$908$	−5.45767	−0.181119
$909$	−45.0328	−1.49364
$910$	22.4898	0.745530
$911$	30.8961	1.02363	0.511816	−	0.859095i	$-0.328973\pi$
$911$	30.8961	1.02363	0.511816	+	0.859095i	$0.328973\pi$
$912$	98.7482	3.26988
$913$	4.11895	0.136317
$914$	16.1256	0.533388
$915$	−19.7293	−0.652231
$916$	−68.7553	−2.27174
$917$	71.1693	2.35022
$918$	−87.7645	−2.89666
$919$	4.85651	0.160202	0.0801008	−	0.996787i	$-0.474476\pi$
$919$	4.85651	0.160202	0.0801008	+	0.996787i	$0.474476\pi$
$920$	0.557864	0.0183922
$921$	30.9430	1.01961
$922$	−1.04629	−0.0344578
$923$	−31.6013	−1.04017
$924$	151.414	4.98114
$925$	1.98665	0.0653207
$926$	−50.3001	−1.65296
$927$	−22.5244	−0.739798
$928$	−6.71485	−0.220426
$929$	−38.2987	−1.25654	−0.628269	−	0.777996i	$-0.716236\pi$
$929$	−38.2987	−1.25654	−0.628269	+	0.777996i	$0.716236\pi$
$930$	−18.1141	−0.593984
$931$	−96.5190	−3.16328
$932$	−83.6403	−2.73973
$933$	−60.3256	−1.97497
$934$	−4.45728	−0.145847
$935$	4.56418	0.149265
$936$	83.8174	2.73966
$937$	−2.23109	−0.0728865	−0.0364432	−	0.999336i	$-0.511603\pi$
$937$	−2.23109	−0.0728865	−0.0364432	+	0.999336i	$0.511603\pi$
$938$	132.683	4.33225
$939$	−50.5027	−1.64809
$940$	−43.2885	−1.41192
$941$	−49.7175	−1.62074	−0.810372	−	0.585915i	$-0.800735\pi$
$941$	−49.7175	−1.62074	−0.810372	+	0.585915i	$0.800735\pi$
$942$	−166.706	−5.43158
$943$	−1.34614	−0.0438363
$944$	−6.91673	−0.225120
$945$	−85.2113	−2.77193
$946$	−62.2460	−2.02379
$947$	41.2849	1.34158	0.670789	−	0.741648i	$-0.265956\pi$
$947$	41.2849	1.34158	0.670789	+	0.741648i	$0.265956\pi$
$948$	39.8494	1.29425
$949$	15.5390	0.504417
$950$	15.6266	0.506993
$951$	26.4328	0.857144
$952$	48.2279	1.56307
$953$	−22.8806	−0.741174	−0.370587	−	0.928798i	$-0.620844\pi$
$953$	−22.8806	−0.741174	−0.370587	+	0.928798i	$0.620844\pi$
$954$	−158.134	−5.11977
$955$	22.6239	0.732091
$956$	−70.1555	−2.26899
$957$	50.0067	1.61649
$958$	−50.4435	−1.62975
$959$	30.4315	0.982684
$960$	22.4800	0.725537
$961$	−26.2673	−0.847331
$962$	9.46998	0.305325
$963$	132.076	4.25609
$964$	−88.6807	−2.85621
$965$	20.6434	0.664534
$966$	−4.21690	−0.135677
$967$	−1.60201	−0.0515173	−0.0257586	−	0.999668i	$-0.508200\pi$
$967$	−1.60201	−0.0515173	−0.0257586	+	0.999668i	$0.508200\pi$
$968$	29.1833	0.937987
$969$	−41.9301	−1.34699
$970$	2.72150	0.0873822
$971$	4.22377	0.135547	0.0677737	−	0.997701i	$-0.478410\pi$
$971$	4.22377	0.135547	0.0677737	+	0.997701i	$0.478410\pi$
$972$	−272.696	−8.74672
$973$	−93.0935	−2.98444
$974$	30.9533	0.991809
$975$	6.50283	0.208257
$976$	27.1166	0.867982
$977$	−58.8045	−1.88132	−0.940661	−	0.339349i	$-0.889793\pi$
$977$	−58.8045	−1.88132	−0.940661	+	0.339349i	$0.889793\pi$
$978$	−53.4094	−1.70784
$979$	19.9945	0.639026
$980$	62.6189	2.00029
$981$	66.6295	2.12732
$982$	−2.13726	−0.0682026
$983$	55.3675	1.76595	0.882974	−	0.469422i	$-0.155538\pi$
$983$	55.3675	1.76595	0.882974	+	0.469422i	$0.155538\pi$
$984$	219.651	7.00223
$985$	11.6555	0.371374
$986$	31.0717	0.989524
$987$	167.739	5.33920
$988$	50.0806	1.59328
$989$	1.16552	0.0370613
$990$	47.9198	1.52299
$991$	−43.5828	−1.38445	−0.692226	−	0.721681i	$-0.743370\pi$
$991$	−43.5828	−1.38445	−0.692226	+	0.721681i	$0.743370\pi$
$992$	2.28459	0.0725357
$993$	30.9049	0.980738
$994$	−190.907	−6.05520
$995$	7.07925	0.224427
$996$	24.5494	0.777878
$997$	−46.5426	−1.47402	−0.737010	−	0.675882i	$-0.763763\pi$
$997$	−46.5426	−1.47402	−0.737010	+	0.675882i	$0.763763\pi$
$998$	15.9605	0.505221
$999$	−35.8807	−1.13522

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.c.1.13	✓	127

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.c.1.13	✓	127	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)

\(f(q)\)	\(=\)	\(q-2.47054 q^{2} -3.37029 q^{3} +4.10358 q^{4} +1.00000 q^{5} +8.32645 q^{6} +4.71800 q^{7} -5.19700 q^{8} +8.35885 q^{9} +O(q^{10})\)	\(q-2.47054 q^{2} -3.37029 q^{3} +4.10358 q^{4} +1.00000 q^{5} +8.32645 q^{6} +4.71800 q^{7} -5.19700 q^{8} +8.35885 q^{9} -2.47054 q^{10} -2.32047 q^{11} -13.8303 q^{12} -1.92946 q^{13} -11.6560 q^{14} -3.37029 q^{15} +4.63224 q^{16} -1.96692 q^{17} -20.6509 q^{18} -6.32515 q^{19} +4.10358 q^{20} -15.9010 q^{21} +5.73282 q^{22} -0.107343 q^{23} +17.5154 q^{24} +1.00000 q^{25} +4.76681 q^{26} -18.0609 q^{27} +19.3607 q^{28} +6.39418 q^{29} +8.32645 q^{30} -2.17549 q^{31} -1.05015 q^{32} +7.82066 q^{33} +4.85937 q^{34} +4.71800 q^{35} +34.3013 q^{36} +1.98665 q^{37} +15.6266 q^{38} +6.50283 q^{39} -5.19700 q^{40} +12.5405 q^{41} +39.2842 q^{42} -10.8578 q^{43} -9.52224 q^{44} +8.35885 q^{45} +0.265197 q^{46} -10.5490 q^{47} -15.6120 q^{48} +15.2596 q^{49} -2.47054 q^{50} +6.62910 q^{51} -7.91769 q^{52} +7.65747 q^{53} +44.6202 q^{54} -2.32047 q^{55} -24.5195 q^{56} +21.3176 q^{57} -15.7971 q^{58} -1.49317 q^{59} -13.8303 q^{60} +5.85389 q^{61} +5.37464 q^{62} +39.4371 q^{63} -6.67004 q^{64} -1.92946 q^{65} -19.3213 q^{66} -11.3832 q^{67} -8.07143 q^{68} +0.361779 q^{69} -11.6560 q^{70} +16.3784 q^{71} -43.4409 q^{72} -8.05356 q^{73} -4.90811 q^{74} -3.37029 q^{75} -25.9558 q^{76} -10.9480 q^{77} -16.0655 q^{78} -2.88132 q^{79} +4.63224 q^{80} +35.7939 q^{81} -30.9818 q^{82} -1.77505 q^{83} -65.2513 q^{84} -1.96692 q^{85} +26.8247 q^{86} -21.5503 q^{87} +12.0595 q^{88} -8.61657 q^{89} -20.6509 q^{90} -9.10318 q^{91} -0.440493 q^{92} +7.33202 q^{93} +26.0617 q^{94} -6.32515 q^{95} +3.53931 q^{96} -1.10158 q^{97} -37.6994 q^{98} -19.3965 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9}+O(q^{10}) \) 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9	\( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9} - 20 q^{10} - 32 q^{11} - 65 q^{12} - 49 q^{13} - 4 q^{14} - 31 q^{15} + 98 q^{16} - 53 q^{17} - 60 q^{18} - 126 q^{19} + 116 q^{20} - 24 q^{21} - 46 q^{22} - 121 q^{23} - 51 q^{24} + 127 q^{25} - 9 q^{26} - 112 q^{27} - 123 q^{28} - 6 q^{29} - 17 q^{30} - 54 q^{31} - 119 q^{32} - 57 q^{33} - 53 q^{34} - 63 q^{35} + 133 q^{36} - 61 q^{37} - 27 q^{38} - 33 q^{39} - 57 q^{40} - 8 q^{41} - 46 q^{42} - 208 q^{43} - 54 q^{44} + 122 q^{45} - 34 q^{46} - 116 q^{47} - 106 q^{48} + 110 q^{49} - 20 q^{50} - 54 q^{51} - 142 q^{52} - 60 q^{53} - 62 q^{54} - 32 q^{55} + 33 q^{56} - 56 q^{57} - 87 q^{58} - 53 q^{59} - 65 q^{60} - 76 q^{61} - 84 q^{62} - 215 q^{63} + 67 q^{64} - 49 q^{65} + 9 q^{66} - 145 q^{67} - 133 q^{68} + q^{69} - 4 q^{70} - 4 q^{71} - 167 q^{72} - 155 q^{73} - 14 q^{74} - 31 q^{75} - 199 q^{76} - 97 q^{77} - 24 q^{78} - 73 q^{79} + 98 q^{80} + 127 q^{81} - 69 q^{82} - 225 q^{83} - 59 q^{84} - 53 q^{85} + 30 q^{86} - 179 q^{87} - 119 q^{88} - 25 q^{89} - 60 q^{90} - 160 q^{91} - 188 q^{92} - 44 q^{93} - 32 q^{94} - 126 q^{95} - 43 q^{96} - 72 q^{97} - 111 q^{98} - 141 q^{99}+O(q^{100}) \) 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9 - 20 * q^10 - 32 * q^11 - 65 * q^12 - 49 * q^13 - 4 * q^14 - 31 * q^15 + 98 * q^16 - 53 * q^17 - 60 * q^18 - 126 * q^19 + 116 * q^20 - 24 * q^21 - 46 * q^22 - 121 * q^23 - 51 * q^24 + 127 * q^25 - 9 * q^26 - 112 * q^27 - 123 * q^28 - 6 * q^29 - 17 * q^30 - 54 * q^31 - 119 * q^32 - 57 * q^33 - 53 * q^34 - 63 * q^35 + 133 * q^36 - 61 * q^37 - 27 * q^38 - 33 * q^39 - 57 * q^40 - 8 * q^41 - 46 * q^42 - 208 * q^43 - 54 * q^44 + 122 * q^45 - 34 * q^46 - 116 * q^47 - 106 * q^48 + 110 * q^49 - 20 * q^50 - 54 * q^51 - 142 * q^52 - 60 * q^53 - 62 * q^54 - 32 * q^55 + 33 * q^56 - 56 * q^57 - 87 * q^58 - 53 * q^59 - 65 * q^60 - 76 * q^61 - 84 * q^62 - 215 * q^63 + 67 * q^64 - 49 * q^65 + 9 * q^66 - 145 * q^67 - 133 * q^68 + q^69 - 4 * q^70 - 4 * q^71 - 167 * q^72 - 155 * q^73 - 14 * q^74 - 31 * q^75 - 199 * q^76 - 97 * q^77 - 24 * q^78 - 73 * q^79 + 98 * q^80 + 127 * q^81 - 69 * q^82 - 225 * q^83 - 59 * q^84 - 53 * q^85 + 30 * q^86 - 179 * q^87 - 119 * q^88 - 25 * q^89 - 60 * q^90 - 160 * q^91 - 188 * q^92 - 44 * q^93 - 32 * q^94 - 126 * q^95 - 43 * q^96 - 72 * q^97 - 111 * q^98 - 141 * q^99

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