LMFDB - Embedded newform 8045.2.a.e.1.2

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			1.2
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)

$f(q)$	$=$	$q-2.68924 q^{2} -0.0494506 q^{3} +5.23199 q^{4} +1.00000 q^{5} +0.132984 q^{6} +2.47489 q^{7} -8.69158 q^{8} -2.99755 q^{9} +O(q^{10})$	\(q-2.68924 q^{2} -0.0494506 q^{3} +5.23199 q^{4} +1.00000 q^{5} +0.132984 q^{6} +2.47489 q^{7} -8.69158 q^{8} -2.99755 q^{9} -2.68924 q^{10} -1.00446 q^{11} -0.258725 q^{12} +6.92941 q^{13} -6.65555 q^{14} -0.0494506 q^{15} +12.9097 q^{16} -1.98355 q^{17} +8.06113 q^{18} +2.36368 q^{19} +5.23199 q^{20} -0.122385 q^{21} +2.70123 q^{22} -4.94693 q^{23} +0.429804 q^{24} +1.00000 q^{25} -18.6348 q^{26} +0.296583 q^{27} +12.9486 q^{28} +7.17849 q^{29} +0.132984 q^{30} +6.67857 q^{31} -17.3342 q^{32} +0.0496712 q^{33} +5.33425 q^{34} +2.47489 q^{35} -15.6832 q^{36} +5.58001 q^{37} -6.35648 q^{38} -0.342663 q^{39} -8.69158 q^{40} +4.47396 q^{41} +0.329121 q^{42} +4.01274 q^{43} -5.25533 q^{44} -2.99755 q^{45} +13.3035 q^{46} +2.45993 q^{47} -0.638394 q^{48} -0.874942 q^{49} -2.68924 q^{50} +0.0980879 q^{51} +36.2546 q^{52} -0.842054 q^{53} -0.797580 q^{54} -1.00446 q^{55} -21.5107 q^{56} -0.116885 q^{57} -19.3046 q^{58} +5.61455 q^{59} -0.258725 q^{60} +3.64217 q^{61} -17.9603 q^{62} -7.41860 q^{63} +20.7962 q^{64} +6.92941 q^{65} -0.133578 q^{66} +1.97441 q^{67} -10.3779 q^{68} +0.244629 q^{69} -6.65555 q^{70} -5.46767 q^{71} +26.0535 q^{72} -4.58225 q^{73} -15.0060 q^{74} -0.0494506 q^{75} +12.3667 q^{76} -2.48593 q^{77} +0.921503 q^{78} -14.0834 q^{79} +12.9097 q^{80} +8.97800 q^{81} -12.0315 q^{82} +7.32829 q^{83} -0.640315 q^{84} -1.98355 q^{85} -10.7912 q^{86} -0.354980 q^{87} +8.73036 q^{88} -8.76084 q^{89} +8.06113 q^{90} +17.1495 q^{91} -25.8823 q^{92} -0.330259 q^{93} -6.61532 q^{94} +2.36368 q^{95} +0.857184 q^{96} +4.06353 q^{97} +2.35293 q^{98} +3.01093 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.68924	−1.90158	−0.950788	−	0.309841i	$-0.899724\pi$
$2$	−2.68924	−1.90158	−0.950788	+	0.309841i	$0.899724\pi$
$3$	−0.0494506	−0.0285503	−0.0142752	−	0.999898i	$-0.504544\pi$
$3$	−0.0494506	−0.0285503	−0.0142752	+	0.999898i	$0.504544\pi$
$4$	5.23199	2.61599
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.132984	0.0542906
$7$	2.47489	0.935419	0.467709	−	0.883882i	$-0.345079\pi$
$7$	2.47489	0.935419	0.467709	+	0.883882i	$0.345079\pi$
$8$	−8.69158	−3.07294
$9$	−2.99755	−0.999185
$10$	−2.68924	−0.850411
$11$	−1.00446	−0.302856	−0.151428	−	0.988468i	$-0.548387\pi$
$11$	−1.00446	−0.302856	−0.151428	+	0.988468i	$0.548387\pi$
$12$	−0.258725	−0.0746875
$13$	6.92941	1.92187	0.960937	−	0.276769i	$-0.0892635\pi$
$13$	6.92941	1.92187	0.960937	+	0.276769i	$0.0892635\pi$
$14$	−6.65555	−1.77877
$15$	−0.0494506	−0.0127681
$16$	12.9097	3.22743
$17$	−1.98355	−0.481083	−0.240541	−	0.970639i	$-0.577325\pi$
$17$	−1.98355	−0.481083	−0.240541	+	0.970639i	$0.577325\pi$
$18$	8.06113	1.90003
$19$	2.36368	0.542264	0.271132	−	0.962542i	$-0.412602\pi$
$19$	2.36368	0.542264	0.271132	+	0.962542i	$0.412602\pi$
$20$	5.23199	1.16991
$21$	−0.122385	−0.0267065
$22$	2.70123	0.575905
$23$	−4.94693	−1.03151	−0.515753	−	0.856737i	$-0.672488\pi$
$23$	−4.94693	−1.03151	−0.515753	+	0.856737i	$0.672488\pi$
$24$	0.429804	0.0877333
$25$	1.00000	0.200000
$26$	−18.6348	−3.65459
$27$	0.296583	0.0570773
$28$	12.9486	2.44705
$29$	7.17849	1.33301	0.666506	−	0.745500i	$-0.267789\pi$
$29$	7.17849	1.33301	0.666506	+	0.745500i	$0.267789\pi$
$30$	0.132984	0.0242795
$31$	6.67857	1.19951	0.599753	−	0.800185i	$-0.295265\pi$
$31$	6.67857	1.19951	0.599753	+	0.800185i	$0.295265\pi$
$32$	−17.3342	−3.06428
$33$	0.0496712	0.00864664
$34$	5.33425	0.914816
$35$	2.47489	0.418332
$36$	−15.6832	−2.61386
$37$	5.58001	0.917348	0.458674	−	0.888605i	$-0.348324\pi$
$37$	5.58001	0.917348	0.458674	+	0.888605i	$0.348324\pi$
$38$	−6.35648	−1.03116
$39$	−0.342663	−0.0548701
$40$	−8.69158	−1.37426
$41$	4.47396	0.698715	0.349358	−	0.936989i	$-0.386400\pi$
$41$	4.47396	0.698715	0.349358	+	0.936989i	$0.386400\pi$
$42$	0.329121	0.0507844
$43$	4.01274	0.611937	0.305969	−	0.952042i	$-0.401020\pi$
$43$	4.01274	0.611937	0.305969	+	0.952042i	$0.401020\pi$
$44$	−5.25533	−0.792271
$45$	−2.99755	−0.446849
$46$	13.3035	1.96149
$47$	2.45993	0.358817	0.179409	−	0.983775i	$-0.442582\pi$
$47$	2.45993	0.358817	0.179409	+	0.983775i	$0.442582\pi$
$48$	−0.638394	−0.0921442
$49$	−0.874942	−0.124992
$50$	−2.68924	−0.380315
$51$	0.0980879	0.0137351
$52$	36.2546	5.02761
$53$	−0.842054	−0.115665	−0.0578325	−	0.998326i	$-0.518419\pi$
$53$	−0.842054	−0.115665	−0.0578325	+	0.998326i	$0.518419\pi$
$54$	−0.797580	−0.108537
$55$	−1.00446	−0.135441
$56$	−21.5107	−2.87448
$57$	−0.116885	−0.0154818
$58$	−19.3046	−2.53482
$59$	5.61455	0.730953	0.365476	−	0.930821i	$-0.380906\pi$
$59$	5.61455	0.730953	0.365476	+	0.930821i	$0.380906\pi$
$60$	−0.258725	−0.0334012
$61$	3.64217	0.466332	0.233166	−	0.972437i	$-0.425091\pi$
$61$	3.64217	0.466332	0.233166	+	0.972437i	$0.425091\pi$
$62$	−17.9603	−2.28095
$63$	−7.41860	−0.934656
$64$	20.7962	2.59952
$65$	6.92941	0.859488
$66$	−0.133578	−0.0164423
$67$	1.97441	0.241212	0.120606	−	0.992700i	$-0.461516\pi$
$67$	1.97441	0.241212	0.120606	+	0.992700i	$0.461516\pi$
$68$	−10.3779	−1.25851
$69$	0.244629	0.0294498
$70$	−6.65555	−0.795490
$71$	−5.46767	−0.648893	−0.324446	−	0.945904i	$-0.605178\pi$
$71$	−5.46767	−0.648893	−0.324446	+	0.945904i	$0.605178\pi$
$72$	26.0535	3.07043
$73$	−4.58225	−0.536311	−0.268156	−	0.963376i	$-0.586414\pi$
$73$	−4.58225	−0.536311	−0.268156	+	0.963376i	$0.586414\pi$
$74$	−15.0060	−1.74441
$75$	−0.0494506	−0.00571006
$76$	12.3667	1.41856
$77$	−2.48593	−0.283298
$78$	0.921503	0.104340
$79$	−14.0834	−1.58451	−0.792253	−	0.610193i	$-0.791092\pi$
$79$	−14.0834	−1.58451	−0.792253	+	0.610193i	$0.791092\pi$
$80$	12.9097	1.44335
$81$	8.97800	0.997555
$82$	−12.0315	−1.32866
$83$	7.32829	0.804385	0.402192	−	0.915555i	$-0.368248\pi$
$83$	7.32829	0.804385	0.402192	+	0.915555i	$0.368248\pi$
$84$	−0.640315	−0.0698640
$85$	−1.98355	−0.215147
$86$	−10.7912	−1.16365
$87$	−0.354980	−0.0380579
$88$	8.73036	0.930659
$89$	−8.76084	−0.928647	−0.464324	−	0.885666i	$-0.653703\pi$
$89$	−8.76084	−0.928647	−0.464324	+	0.885666i	$0.653703\pi$
$90$	8.06113	0.849718
$91$	17.1495	1.79776
$92$	−25.8823	−2.69842
$93$	−0.330259	−0.0342463
$94$	−6.61532	−0.682318
$95$	2.36368	0.242508
$96$	0.857184	0.0874860
$97$	4.06353	0.412589	0.206294	−	0.978490i	$-0.433860\pi$
$97$	4.06353	0.412589	0.206294	+	0.978490i	$0.433860\pi$
$98$	2.35293	0.237681
$99$	3.01093	0.302610
$100$	5.23199	0.523199
$101$	−14.2874	−1.42165	−0.710823	−	0.703371i	$-0.751677\pi$
$101$	−14.2874	−1.42165	−0.710823	+	0.703371i	$0.751677\pi$
$102$	−0.263782	−0.0261183
$103$	10.4902	1.03363	0.516813	−	0.856098i	$-0.327118\pi$
$103$	10.4902	1.03363	0.516813	+	0.856098i	$0.327118\pi$
$104$	−60.2276	−5.90580
$105$	−0.122385	−0.0119435
$106$	2.26448	0.219946
$107$	1.62034	0.156644	0.0783219	−	0.996928i	$-0.475044\pi$
$107$	1.62034	0.156644	0.0783219	+	0.996928i	$0.475044\pi$
$108$	1.55172	0.149314
$109$	3.13274	0.300062	0.150031	−	0.988681i	$-0.452063\pi$
$109$	3.13274	0.300062	0.150031	+	0.988681i	$0.452063\pi$
$110$	2.70123	0.257552
$111$	−0.275935	−0.0261906
$112$	31.9501	3.01900
$113$	9.59153	0.902296	0.451148	−	0.892449i	$-0.351015\pi$
$113$	9.59153	0.902296	0.451148	+	0.892449i	$0.351015\pi$
$114$	0.314332	0.0294399
$115$	−4.94693	−0.461304
$116$	37.5578	3.48715
$117$	−20.7713	−1.92031
$118$	−15.0989	−1.38996
$119$	−4.90907	−0.450014
$120$	0.429804	0.0392355
$121$	−9.99106	−0.908278
$122$	−9.79465	−0.886766
$123$	−0.221240	−0.0199485
$124$	34.9422	3.13790
$125$	1.00000	0.0894427
$126$	19.9504	1.77732
$127$	10.5543	0.936541	0.468271	−	0.883585i	$-0.344877\pi$
$127$	10.5543	0.936541	0.468271	+	0.883585i	$0.344877\pi$
$128$	−21.2575	−1.87892
$129$	−0.198432	−0.0174710
$130$	−18.6348	−1.63438
$131$	−1.68618	−0.147322	−0.0736609	−	0.997283i	$-0.523468\pi$
$131$	−1.68618	−0.147322	−0.0736609	+	0.997283i	$0.523468\pi$
$132$	0.259879	0.0226196
$133$	5.84983	0.507244
$134$	−5.30965	−0.458684
$135$	0.296583	0.0255258
$136$	17.2402	1.47834
$137$	−6.08664	−0.520017	−0.260009	−	0.965606i	$-0.583725\pi$
$137$	−6.08664	−0.520017	−0.260009	+	0.965606i	$0.583725\pi$
$138$	−0.657864	−0.0560011
$139$	−11.5079	−0.976084	−0.488042	−	0.872820i	$-0.662289\pi$
$139$	−11.5079	−0.976084	−0.488042	+	0.872820i	$0.662289\pi$
$140$	12.9486	1.09435
$141$	−0.121645	−0.0102443
$142$	14.7039	1.23392
$143$	−6.96032	−0.582052
$144$	−38.6976	−3.22480
$145$	7.17849	0.596141
$146$	12.3227	1.01984
$147$	0.0432664	0.00356855
$148$	29.1946	2.39978
$149$	−17.3100	−1.41809	−0.709045	−	0.705164i	$-0.750874\pi$
$149$	−17.3100	−1.41809	−0.709045	+	0.705164i	$0.750874\pi$
$150$	0.132984	0.0108581
$151$	−20.9432	−1.70433	−0.852166	−	0.523271i	$-0.824711\pi$
$151$	−20.9432	−1.70433	−0.852166	+	0.523271i	$0.824711\pi$
$152$	−20.5441	−1.66634
$153$	5.94581	0.480691
$154$	6.68524	0.538712
$155$	6.67857	0.536436
$156$	−1.79281	−0.143540
$157$	−3.71117	−0.296184	−0.148092	−	0.988974i	$-0.547313\pi$
$157$	−3.71117	−0.296184	−0.148092	+	0.988974i	$0.547313\pi$
$158$	37.8736	3.01306
$159$	0.0416401	0.00330227
$160$	−17.3342	−1.37039
$161$	−12.2431	−0.964890
$162$	−24.1440	−1.89693
$163$	−15.7553	−1.23405	−0.617025	−	0.786943i	$-0.711662\pi$
$163$	−15.7553	−1.23405	−0.617025	+	0.786943i	$0.711662\pi$
$164$	23.4077	1.82784
$165$	0.0496712	0.00386690
$166$	−19.7075	−1.52960
$167$	7.43824	0.575589	0.287794	−	0.957692i	$-0.407078\pi$
$167$	7.43824	0.575589	0.287794	+	0.957692i	$0.407078\pi$
$168$	1.06372	0.0820674
$169$	35.0168	2.69360
$170$	5.33425	0.409118
$171$	−7.08525	−0.541822
$172$	20.9946	1.60082
$173$	15.2446	1.15903	0.579513	−	0.814963i	$-0.303243\pi$
$173$	15.2446	1.15903	0.579513	+	0.814963i	$0.303243\pi$
$174$	0.954626	0.0723700
$175$	2.47489	0.187084
$176$	−12.9673	−0.977449
$177$	−0.277643	−0.0208689
$178$	23.5600	1.76589
$179$	7.94808	0.594068	0.297034	−	0.954867i	$-0.404003\pi$
$179$	7.94808	0.594068	0.297034	+	0.954867i	$0.404003\pi$
$180$	−15.6832	−1.16895
$181$	−14.2599	−1.05993	−0.529967	−	0.848018i	$-0.677796\pi$
$181$	−14.2599	−1.05993	−0.529967	+	0.848018i	$0.677796\pi$
$182$	−46.1191	−3.41857
$183$	−0.180107	−0.0133139
$184$	42.9967	3.16976
$185$	5.58001	0.410251
$186$	0.888145	0.0651219
$187$	1.99240	0.145699
$188$	12.8703	0.938663
$189$	0.734008	0.0533912
$190$	−6.35648	−0.461148
$191$	20.5585	1.48756	0.743781	−	0.668423i	$-0.233031\pi$
$191$	20.5585	1.48756	0.743781	+	0.668423i	$0.233031\pi$
$192$	−1.02838	−0.0742172
$193$	1.79201	0.128992	0.0644959	−	0.997918i	$-0.479456\pi$
$193$	1.79201	0.128992	0.0644959	+	0.997918i	$0.479456\pi$
$194$	−10.9278	−0.784569
$195$	−0.342663	−0.0245386
$196$	−4.57769	−0.326978
$197$	14.4749	1.03129	0.515646	−	0.856802i	$-0.327552\pi$
$197$	14.4749	1.03129	0.515646	+	0.856802i	$0.327552\pi$
$198$	−8.09709	−0.575435
$199$	17.0155	1.20620	0.603100	−	0.797666i	$-0.293932\pi$
$199$	17.0155	1.20620	0.603100	+	0.797666i	$0.293932\pi$
$200$	−8.69158	−0.614588
$201$	−0.0976356	−0.00688668
$202$	38.4221	2.70337
$203$	17.7659	1.24692
$204$	0.513195	0.0359308
$205$	4.47396	0.312475
$206$	−28.2105	−1.96552
$207$	14.8287	1.03067
$208$	89.4569	6.20272
$209$	−2.37422	−0.164228
$210$	0.329121	0.0227115
$211$	22.4323	1.54430	0.772151	−	0.635439i	$-0.219181\pi$
$211$	22.4323	1.54430	0.772151	+	0.635439i	$0.219181\pi$
$212$	−4.40562	−0.302579
$213$	0.270379	0.0185261
$214$	−4.35747	−0.297870
$215$	4.01274	0.273667
$216$	−2.57777	−0.175395
$217$	16.5287	1.12204
$218$	−8.42468	−0.570592
$219$	0.226595	0.0153119
$220$	−5.25533	−0.354314
$221$	−13.7449	−0.924580
$222$	0.742054	0.0498034
$223$	12.6969	0.850245	0.425122	−	0.905136i	$-0.360231\pi$
$223$	12.6969	0.850245	0.425122	+	0.905136i	$0.360231\pi$
$224$	−42.9001	−2.86638
$225$	−2.99755	−0.199837
$226$	−25.7939	−1.71578
$227$	12.2144	0.810701	0.405351	−	0.914161i	$-0.367149\pi$
$227$	12.2144	0.810701	0.405351	+	0.914161i	$0.367149\pi$
$228$	−0.611542	−0.0405003
$229$	−4.33366	−0.286376	−0.143188	−	0.989695i	$-0.545735\pi$
$229$	−4.33366	−0.286376	−0.143188	+	0.989695i	$0.545735\pi$
$230$	13.3035	0.877204
$231$	0.122930	0.00808823
$232$	−62.3924	−4.09626
$233$	−1.69204	−0.110849	−0.0554246	−	0.998463i	$-0.517651\pi$
$233$	−1.69204	−0.110849	−0.0554246	+	0.998463i	$0.517651\pi$
$234$	55.8589	3.65161
$235$	2.45993	0.160468
$236$	29.3753	1.91217
$237$	0.696432	0.0452381
$238$	13.2016	0.855736
$239$	−17.6614	−1.14242	−0.571210	−	0.820804i	$-0.693526\pi$
$239$	−17.6614	−1.14242	−0.571210	+	0.820804i	$0.693526\pi$
$240$	−0.638394	−0.0412082
$241$	−6.75547	−0.435158	−0.217579	−	0.976043i	$-0.569816\pi$
$241$	−6.75547	−0.435158	−0.217579	+	0.976043i	$0.569816\pi$
$242$	26.8683	1.72716
$243$	−1.33371	−0.0855579
$244$	19.0558	1.21992
$245$	−0.874942	−0.0558980
$246$	0.594966	0.0379337
$247$	16.3789	1.04216
$248$	−58.0474	−3.68601
$249$	−0.362388	−0.0229654
$250$	−2.68924	−0.170082
$251$	10.1589	0.641222	0.320611	−	0.947211i	$-0.396112\pi$
$251$	10.1589	0.641222	0.320611	+	0.947211i	$0.396112\pi$
$252$	−38.8141	−2.44506
$253$	4.96900	0.312398
$254$	−28.3830	−1.78091
$255$	0.0980879	0.00614250
$256$	15.5741	0.973380
$257$	12.4885	0.779008	0.389504	−	0.921025i	$-0.372646\pi$
$257$	12.4885	0.779008	0.389504	+	0.921025i	$0.372646\pi$
$258$	0.533631	0.0332224
$259$	13.8099	0.858105
$260$	36.2546	2.24842
$261$	−21.5179	−1.33193
$262$	4.53452	0.280144
$263$	2.19760	0.135510	0.0677551	−	0.997702i	$-0.478416\pi$
$263$	2.19760	0.135510	0.0677551	+	0.997702i	$0.478416\pi$
$264$	−0.431721	−0.0265706
$265$	−0.842054	−0.0517270
$266$	−15.7316	−0.964564
$267$	0.433229	0.0265132
$268$	10.3301	0.631010
$269$	32.2047	1.96355	0.981776	−	0.190041i	$-0.0608619\pi$
$269$	32.2047	1.96355	0.981776	+	0.190041i	$0.0608619\pi$
$270$	−0.797580	−0.0485392
$271$	3.41670	0.207550	0.103775	−	0.994601i	$-0.466908\pi$
$271$	3.41670	0.207550	0.103775	+	0.994601i	$0.466908\pi$
$272$	−25.6072	−1.55266
$273$	−0.848053	−0.0513265
$274$	16.3684	0.988852
$275$	−1.00446	−0.0605713
$276$	1.27989	0.0770406
$277$	22.2196	1.33505	0.667524	−	0.744589i	$-0.267354\pi$
$277$	22.2196	1.33505	0.667524	+	0.744589i	$0.267354\pi$
$278$	30.9474	1.85610
$279$	−20.0194	−1.19853
$280$	−21.5107	−1.28551
$281$	1.42805	0.0851904	0.0425952	−	0.999092i	$-0.486437\pi$
$281$	1.42805	0.0851904	0.0425952	+	0.999092i	$0.486437\pi$
$282$	0.327131	0.0194804
$283$	−12.9922	−0.772309	−0.386154	−	0.922434i	$-0.626197\pi$
$283$	−12.9922	−0.772309	−0.386154	+	0.922434i	$0.626197\pi$
$284$	−28.6068	−1.69750
$285$	−0.116885	−0.00692368
$286$	18.7180	1.10682
$287$	11.0725	0.653591
$288$	51.9601	3.06178
$289$	−13.0655	−0.768559
$290$	−19.3046	−1.13361
$291$	−0.200944	−0.0117795
$292$	−23.9743	−1.40299
$293$	−12.2455	−0.715389	−0.357694	−	0.933839i	$-0.616437\pi$
$293$	−12.2455	−0.715389	−0.357694	+	0.933839i	$0.616437\pi$
$294$	−0.116354	−0.00678588
$295$	5.61455	0.326892
$296$	−48.4991	−2.81896
$297$	−0.297906	−0.0172862
$298$	46.5506	2.69661
$299$	−34.2793	−1.98242
$300$	−0.258725	−0.0149375
$301$	9.93107	0.572417
$302$	56.3212	3.24092
$303$	0.706519	0.0405884
$304$	30.5144	1.75012
$305$	3.64217	0.208550
$306$	−15.9897	−0.914070
$307$	−2.10560	−0.120173	−0.0600865	−	0.998193i	$-0.519138\pi$
$307$	−2.10560	−0.120173	−0.0600865	+	0.998193i	$0.519138\pi$
$308$	−13.0063	−0.741105
$309$	−0.518745	−0.0295104
$310$	−17.9603	−1.02007
$311$	28.5916	1.62128	0.810640	−	0.585545i	$-0.199119\pi$
$311$	28.5916	1.62128	0.810640	+	0.585545i	$0.199119\pi$
$312$	2.97829	0.168612
$313$	−8.06880	−0.456076	−0.228038	−	0.973652i	$-0.573231\pi$
$313$	−8.06880	−0.456076	−0.228038	+	0.973652i	$0.573231\pi$
$314$	9.98022	0.563216
$315$	−7.41860	−0.417991
$316$	−73.6842	−4.14506
$317$	1.01800	0.0571763	0.0285882	−	0.999591i	$-0.490899\pi$
$317$	1.01800	0.0571763	0.0285882	+	0.999591i	$0.490899\pi$
$318$	−0.111980	−0.00627953
$319$	−7.21051	−0.403711
$320$	20.7962	1.16254
$321$	−0.0801266	−0.00447223
$322$	32.9245	1.83481
$323$	−4.68848	−0.260874
$324$	46.9728	2.60960
$325$	6.92941	0.384375
$326$	42.3697	2.34664
$327$	−0.154916	−0.00856687
$328$	−38.8858	−2.14711
$329$	6.08803	0.335644
$330$	−0.133578	−0.00735320
$331$	−10.8435	−0.596010	−0.298005	−	0.954564i	$-0.596321\pi$
$331$	−10.8435	−0.596010	−0.298005	+	0.954564i	$0.596321\pi$
$332$	38.3416	2.10427
$333$	−16.7264	−0.916601
$334$	−20.0032	−1.09453
$335$	1.97441	0.107873
$336$	−1.57995	−0.0861934
$337$	12.6257	0.687767	0.343884	−	0.939012i	$-0.388257\pi$
$337$	12.6257	0.687767	0.343884	+	0.939012i	$0.388257\pi$
$338$	−94.1683	−5.12208
$339$	−0.474307	−0.0257608
$340$	−10.3779	−0.562823
$341$	−6.70836	−0.363278
$342$	19.0539	1.03032
$343$	−19.4896	−1.05234
$344$	−34.8771	−1.88045
$345$	0.244629	0.0131704
$346$	−40.9963	−2.20398
$347$	−10.1964	−0.547371	−0.273686	−	0.961819i	$-0.588243\pi$
$347$	−10.1964	−0.547371	−0.273686	+	0.961819i	$0.588243\pi$
$348$	−1.85725	−0.0995593
$349$	−34.0020	−1.82009	−0.910043	−	0.414514i	$-0.863952\pi$
$349$	−34.0020	−1.82009	−0.910043	+	0.414514i	$0.863952\pi$
$350$	−6.65555	−0.355754
$351$	2.05514	0.109695
$352$	17.4115	0.928036
$353$	−15.2806	−0.813307	−0.406653	−	0.913583i	$-0.633304\pi$
$353$	−15.2806	−0.813307	−0.406653	+	0.913583i	$0.633304\pi$
$354$	0.746647	0.0396839
$355$	−5.46767	−0.290194
$356$	−45.8366	−2.42934
$357$	0.242756	0.0128480
$358$	−21.3743	−1.12967
$359$	−22.6025	−1.19292	−0.596458	−	0.802644i	$-0.703426\pi$
$359$	−22.6025	−1.19292	−0.596458	+	0.802644i	$0.703426\pi$
$360$	26.0535	1.37314
$361$	−13.4130	−0.705949
$362$	38.3484	2.01554
$363$	0.494064	0.0259316
$364$	89.7260	4.70292
$365$	−4.58225	−0.239846
$366$	0.484351	0.0253175
$367$	−8.26561	−0.431461	−0.215731	−	0.976453i	$-0.569213\pi$
$367$	−8.26561	−0.431461	−0.215731	+	0.976453i	$0.569213\pi$
$368$	−63.8636	−3.32912
$369$	−13.4109	−0.698146
$370$	−15.0060	−0.780123
$371$	−2.08399	−0.108195
$372$	−1.72791	−0.0895881
$373$	−34.6637	−1.79482	−0.897408	−	0.441201i	$-0.854553\pi$
$373$	−34.6637	−1.79482	−0.897408	+	0.441201i	$0.854553\pi$
$374$	−5.35804	−0.277058
$375$	−0.0494506	−0.00255362
$376$	−21.3806	−1.10262
$377$	49.7427	2.56188
$378$	−1.97392	−0.101528
$379$	−6.98106	−0.358593	−0.179296	−	0.983795i	$-0.557382\pi$
$379$	−6.98106	−0.358593	−0.179296	+	0.983795i	$0.557382\pi$
$380$	12.3667	0.634400
$381$	−0.521915	−0.0267385
$382$	−55.2867	−2.82871
$383$	36.7882	1.87979	0.939896	−	0.341461i	$-0.110922\pi$
$383$	36.7882	1.87979	0.939896	+	0.341461i	$0.110922\pi$
$384$	1.05120	0.0536436
$385$	−2.48593	−0.126695
$386$	−4.81914	−0.245288
$387$	−12.0284	−0.611438
$388$	21.2603	1.07933
$389$	−2.87243	−0.145638	−0.0728189	−	0.997345i	$-0.523200\pi$
$389$	−2.87243	−0.145638	−0.0728189	+	0.997345i	$0.523200\pi$
$390$	0.921503	0.0466621
$391$	9.81251	0.496240
$392$	7.60463	0.384092
$393$	0.0833824	0.00420608
$394$	−38.9264	−1.96108
$395$	−14.0834	−0.708613
$396$	15.7531	0.791625
$397$	−5.88385	−0.295302	−0.147651	−	0.989040i	$-0.547171\pi$
$397$	−5.88385	−0.295302	−0.147651	+	0.989040i	$0.547171\pi$
$398$	−45.7588	−2.29368
$399$	−0.289277	−0.0144820
$400$	12.9097	0.645487
$401$	11.5185	0.575207	0.287604	−	0.957750i	$-0.407141\pi$
$401$	11.5185	0.575207	0.287604	+	0.957750i	$0.407141\pi$
$402$	0.262565	0.0130956
$403$	46.2786	2.30530
$404$	−74.7514	−3.71902
$405$	8.97800	0.446120
$406$	−47.7768	−2.37112
$407$	−5.60491	−0.277825
$408$	−0.852539	−0.0422070
$409$	−6.96157	−0.344228	−0.172114	−	0.985077i	$-0.555060\pi$
$409$	−6.96157	−0.344228	−0.172114	+	0.985077i	$0.555060\pi$
$410$	−12.0315	−0.594195
$411$	0.300988	0.0148466
$412$	54.8844	2.70396
$413$	13.8954	0.683747
$414$	−39.8779	−1.95989
$415$	7.32829	0.359732
$416$	−120.116	−5.88915
$417$	0.569071	0.0278675
$418$	6.38484	0.312293
$419$	−4.47915	−0.218821	−0.109410	−	0.993997i	$-0.534896\pi$
$419$	−4.47915	−0.218821	−0.109410	+	0.993997i	$0.534896\pi$
$420$	−0.640315	−0.0312442
$421$	−26.8551	−1.30884	−0.654418	−	0.756133i	$-0.727086\pi$
$421$	−26.8551	−1.30884	−0.654418	+	0.756133i	$0.727086\pi$
$422$	−60.3257	−2.93661
$423$	−7.37376	−0.358525
$424$	7.31879	0.355432
$425$	−1.98355	−0.0962165
$426$	−0.727114	−0.0352288
$427$	9.01395	0.436216
$428$	8.47758	0.409779
$429$	0.344192	0.0166178
$430$	−10.7912	−0.520398
$431$	0.338271	0.0162940	0.00814698	−	0.999967i	$-0.497407\pi$
$431$	0.338271	0.0162940	0.00814698	+	0.999967i	$0.497407\pi$
$432$	3.82880	0.184213
$433$	29.1590	1.40129	0.700647	−	0.713508i	$-0.252895\pi$
$433$	29.1590	1.40129	0.700647	+	0.713508i	$0.252895\pi$
$434$	−44.4496	−2.13365
$435$	−0.354980	−0.0170200
$436$	16.3905	0.784961
$437$	−11.6929	−0.559349
$438$	−0.609367	−0.0291167
$439$	9.91914	0.473415	0.236707	−	0.971581i	$-0.423932\pi$
$439$	9.91914	0.473415	0.236707	+	0.971581i	$0.423932\pi$
$440$	8.73036	0.416203
$441$	2.62269	0.124890
$442$	36.9632	1.75816
$443$	−18.5791	−0.882721	−0.441360	−	0.897330i	$-0.645504\pi$
$443$	−18.5791	−0.882721	−0.441360	+	0.897330i	$0.645504\pi$
$444$	−1.44369	−0.0685144
$445$	−8.76084	−0.415304
$446$	−34.1448	−1.61681
$447$	0.855989	0.0404869
$448$	51.4682	2.43164
$449$	16.5986	0.783334	0.391667	−	0.920107i	$-0.371898\pi$
$449$	16.5986	0.783334	0.391667	+	0.920107i	$0.371898\pi$
$450$	8.06113	0.380005
$451$	−4.49392	−0.211610
$452$	50.1828	2.36040
$453$	1.03565	0.0486592
$454$	−32.8475	−1.54161
$455$	17.1495	0.803981
$456$	1.01592	0.0475747
$457$	−11.0447	−0.516648	−0.258324	−	0.966058i	$-0.583170\pi$
$457$	−11.0447	−0.516648	−0.258324	+	0.966058i	$0.583170\pi$
$458$	11.6542	0.544566
$459$	−0.588288	−0.0274589
$460$	−25.8823	−1.20677
$461$	−27.6059	−1.28573	−0.642867	−	0.765978i	$-0.722255\pi$
$461$	−27.6059	−1.28573	−0.642867	+	0.765978i	$0.722255\pi$
$462$	−0.330589	−0.0153804
$463$	−0.0289373	−0.00134483	−0.000672415	−	1.00000i	$-0.500214\pi$
$463$	−0.0289373	−0.00134483	−0.000672415		1.00000i	$0.500214\pi$
$464$	92.6724	4.30221
$465$	−0.330259	−0.0153154
$466$	4.55029	0.210788
$467$	2.60144	0.120380	0.0601900	−	0.998187i	$-0.480829\pi$
$467$	2.60144	0.120380	0.0601900	+	0.998187i	$0.480829\pi$
$468$	−108.675	−5.02351
$469$	4.88643	0.225634
$470$	−6.61532	−0.305142
$471$	0.183520	0.00845614
$472$	−48.7994	−2.24617
$473$	−4.03064	−0.185329
$474$	−1.87287	−0.0860238
$475$	2.36368	0.108453
$476$	−25.6842	−1.17723
$477$	2.52410	0.115571
$478$	47.4957	2.17240
$479$	4.29873	0.196414	0.0982070	−	0.995166i	$-0.468689\pi$
$479$	4.29873	0.196414	0.0982070	+	0.995166i	$0.468689\pi$
$480$	0.857184	0.0391249
$481$	38.6662	1.76303
$482$	18.1671	0.827487
$483$	0.605428	0.0275479
$484$	−52.2731	−2.37605
$485$	4.06353	0.184515
$486$	3.58667	0.162695
$487$	1.23722	0.0560639	0.0280319	−	0.999607i	$-0.491076\pi$
$487$	1.23722	0.0560639	0.0280319	+	0.999607i	$0.491076\pi$
$488$	−31.6562	−1.43301
$489$	0.779109	0.0352325
$490$	2.35293	0.106294
$491$	−16.7909	−0.757765	−0.378882	−	0.925445i	$-0.623692\pi$
$491$	−16.7909	−0.757765	−0.378882	+	0.925445i	$0.623692\pi$
$492$	−1.15753	−0.0521853
$493$	−14.2389	−0.641289
$494$	−44.0467	−1.98175
$495$	3.01093	0.135331
$496$	86.2186	3.87133
$497$	−13.5319	−0.606987
$498$	0.974548	0.0436705
$499$	−0.602473	−0.0269704	−0.0134852	−	0.999909i	$-0.504293\pi$
$499$	−0.602473	−0.0269704	−0.0134852	+	0.999909i	$0.504293\pi$
$500$	5.23199	0.233982
$501$	−0.367825	−0.0164332
$502$	−27.3196	−1.21933
$503$	24.3855	1.08729	0.543647	−	0.839314i	$-0.317043\pi$
$503$	24.3855	1.08729	0.543647	+	0.839314i	$0.317043\pi$
$504$	64.4794	2.87214
$505$	−14.2874	−0.635780
$506$	−13.3628	−0.594049
$507$	−1.73160	−0.0769030
$508$	55.2199	2.44999
$509$	0.0240053	0.00106402	0.000532008	−	1.00000i	$-0.499831\pi$
$509$	0.0240053	0.00106402	0.000532008		1.00000i	$0.499831\pi$
$510$	−0.263782	−0.0116804
$511$	−11.3405	−0.501676
$512$	0.632621	0.0279582
$513$	0.701025	0.0309510
$514$	−33.5844	−1.48134
$515$	10.4902	0.462252
$516$	−1.03820	−0.0457040
$517$	−2.47090	−0.108670
$518$	−37.1381	−1.63175
$519$	−0.753855	−0.0330906
$520$	−60.2276	−2.64115
$521$	44.2229	1.93744	0.968719	−	0.248159i	$-0.0798256\pi$
$521$	44.2229	1.93744	0.968719	+	0.248159i	$0.0798256\pi$
$522$	57.8667	2.53276
$523$	27.7635	1.21401	0.607007	−	0.794696i	$-0.292370\pi$
$523$	27.7635	1.21401	0.607007	+	0.794696i	$0.292370\pi$
$524$	−8.82205	−0.385393
$525$	−0.122385	−0.00534130
$526$	−5.90988	−0.257683
$527$	−13.2473	−0.577062
$528$	0.641242	0.0279065
$529$	1.47212	0.0640053
$530$	2.26448	0.0983628
$531$	−16.8299	−0.730357
$532$	30.6062	1.32695
$533$	31.0019	1.34284
$534$	−1.16505	−0.0504168
$535$	1.62034	0.0700532
$536$	−17.1607	−0.741230
$537$	−0.393037	−0.0169608
$538$	−86.6059	−3.73385
$539$	0.878846	0.0378546
$540$	1.55172	0.0667753
$541$	37.9912	1.63337	0.816686	−	0.577083i	$-0.195809\pi$
$541$	37.9912	1.63337	0.816686	+	0.577083i	$0.195809\pi$
$542$	−9.18831	−0.394672
$543$	0.705163	0.0302614
$544$	34.3833	1.47417
$545$	3.13274	0.134192
$546$	2.28061	0.0976013
$547$	28.6335	1.22428	0.612140	−	0.790749i	$-0.290309\pi$
$547$	28.6335	1.22428	0.612140	+	0.790749i	$0.290309\pi$
$548$	−31.8453	−1.36036
$549$	−10.9176	−0.465952
$550$	2.70123	0.115181
$551$	16.9676	0.722845
$552$	−2.12621	−0.0904975
$553$	−34.8548	−1.48218
$554$	−59.7538	−2.53869
$555$	−0.275935	−0.0117128
$556$	−60.2090	−2.55343
$557$	19.0376	0.806648	0.403324	−	0.915057i	$-0.367855\pi$
$557$	19.0376	0.806648	0.403324	+	0.915057i	$0.367855\pi$
$558$	53.8368	2.27910
$559$	27.8059	1.17607
$560$	31.9501	1.35014
$561$	−0.0985255	−0.00415975
$562$	−3.84037	−0.161996
$563$	−13.0305	−0.549168	−0.274584	−	0.961563i	$-0.588540\pi$
$563$	−13.0305	−0.549168	−0.274584	+	0.961563i	$0.588540\pi$
$564$	−0.636444	−0.0267991
$565$	9.59153	0.403519
$566$	34.9392	1.46860
$567$	22.2195	0.933132
$568$	47.5227	1.99401
$569$	−10.9312	−0.458261	−0.229131	−	0.973396i	$-0.573588\pi$
$569$	−10.9312	−0.458261	−0.229131	+	0.973396i	$0.573588\pi$
$570$	0.314332	0.0131659
$571$	−19.8024	−0.828704	−0.414352	−	0.910117i	$-0.635992\pi$
$571$	−19.8024	−0.828704	−0.414352	+	0.910117i	$0.635992\pi$
$572$	−36.4163	−1.52264
$573$	−1.01663	−0.0424703
$574$	−29.7767	−1.24285
$575$	−4.94693	−0.206301
$576$	−62.3377	−2.59740
$577$	24.6740	1.02719	0.513595	−	0.858033i	$-0.328313\pi$
$577$	24.6740	1.02719	0.513595	+	0.858033i	$0.328313\pi$
$578$	35.1362	1.46147
$579$	−0.0886161	−0.00368276
$580$	37.5578	1.55950
$581$	18.1367	0.752437
$582$	0.540385	0.0223997
$583$	0.845811	0.0350299
$584$	39.8270	1.64805
$585$	−20.7713	−0.858787
$586$	32.9310	1.36037
$587$	−15.3447	−0.633344	−0.316672	−	0.948535i	$-0.602565\pi$
$587$	−15.3447	−0.633344	−0.316672	+	0.948535i	$0.602565\pi$
$588$	0.226369	0.00933532
$589$	15.7860	0.650450
$590$	−15.0989	−0.621610
$591$	−0.715791	−0.0294437
$592$	72.0365	2.96068
$593$	26.3342	1.08142	0.540708	−	0.841210i	$-0.318156\pi$
$593$	26.3342	1.08142	0.540708	+	0.841210i	$0.318156\pi$
$594$	0.801139	0.0328711
$595$	−4.90907	−0.201252
$596$	−90.5657	−3.70971
$597$	−0.841428	−0.0344374
$598$	92.1852	3.76973
$599$	−42.5965	−1.74045	−0.870224	−	0.492657i	$-0.836026\pi$
$599$	−42.5965	−1.74045	−0.870224	+	0.492657i	$0.836026\pi$
$600$	0.429804	0.0175467
$601$	−30.2640	−1.23449	−0.617247	−	0.786769i	$-0.711752\pi$
$601$	−30.2640	−1.23449	−0.617247	+	0.786769i	$0.711752\pi$
$602$	−26.7070	−1.08850
$603$	−5.91839	−0.241016
$604$	−109.575	−4.45852
$605$	−9.99106	−0.406194
$606$	−1.90000	−0.0771820
$607$	−0.352204	−0.0142955	−0.00714776	−	0.999974i	$-0.502275\pi$
$607$	−0.352204	−0.0142955	−0.00714776	+	0.999974i	$0.502275\pi$
$608$	−40.9723	−1.66165
$609$	−0.878536	−0.0356001
$610$	−9.79465	−0.396574
$611$	17.0458	0.689601
$612$	31.1084	1.25748
$613$	−22.2444	−0.898441	−0.449221	−	0.893421i	$-0.648298\pi$
$613$	−22.2444	−0.898441	−0.449221	+	0.893421i	$0.648298\pi$
$614$	5.66246	0.228518
$615$	−0.221240	−0.00892126
$616$	21.6066	0.870556
$617$	−14.9329	−0.601178	−0.300589	−	0.953754i	$-0.597183\pi$
$617$	−14.9329	−0.601178	−0.300589	+	0.953754i	$0.597183\pi$
$618$	1.39503	0.0561162
$619$	22.3663	0.898978	0.449489	−	0.893286i	$-0.351606\pi$
$619$	22.3663	0.898978	0.449489	+	0.893286i	$0.351606\pi$
$620$	34.9422	1.40331
$621$	−1.46717	−0.0588756
$622$	−76.8895	−3.08299
$623$	−21.6821	−0.868674
$624$	−4.42370	−0.177090
$625$	1.00000	0.0400000
$626$	21.6989	0.867263
$627$	0.117407	0.00468877
$628$	−19.4168	−0.774815
$629$	−11.0683	−0.441320
$630$	19.9504	0.794842
$631$	29.5376	1.17587	0.587937	−	0.808907i	$-0.299940\pi$
$631$	29.5376	1.17587	0.587937	+	0.808907i	$0.299940\pi$
$632$	122.407	4.86909
$633$	−1.10929	−0.0440903
$634$	−2.73763	−0.108725
$635$	10.5543	0.418834
$636$	0.217860	0.00863873
$637$	−6.06284	−0.240218
$638$	19.3908	0.767688
$639$	16.3896	0.648364
$640$	−21.2575	−0.840277
$641$	42.0714	1.66172	0.830860	−	0.556482i	$-0.187849\pi$
$641$	42.0714	1.66172	0.830860	+	0.556482i	$0.187849\pi$
$642$	0.215479	0.00850429
$643$	−8.13528	−0.320824	−0.160412	−	0.987050i	$-0.551282\pi$
$643$	−8.13528	−0.320824	−0.160412	+	0.987050i	$0.551282\pi$
$644$	−64.0557	−2.52415
$645$	−0.198432	−0.00781327
$646$	12.6084	0.496072
$647$	−26.5839	−1.04512	−0.522560	−	0.852603i	$-0.675023\pi$
$647$	−26.5839	−1.04512	−0.522560	+	0.852603i	$0.675023\pi$
$648$	−78.0330	−3.06543
$649$	−5.63960	−0.221374
$650$	−18.6348	−0.730918
$651$	−0.817354	−0.0320346
$652$	−82.4316	−3.22827
$653$	−29.3348	−1.14796	−0.573980	−	0.818869i	$-0.694601\pi$
$653$	−29.3348	−1.14796	−0.573980	+	0.818869i	$0.694601\pi$
$654$	0.416605	0.0162906
$655$	−1.68618	−0.0658843
$656$	57.7577	2.25506
$657$	13.7355	0.535874
$658$	−16.3722	−0.638253
$659$	12.8593	0.500928	0.250464	−	0.968126i	$-0.419417\pi$
$659$	12.8593	0.500928	0.250464	+	0.968126i	$0.419417\pi$
$660$	0.259879	0.0101158
$661$	−23.7064	−0.922072	−0.461036	−	0.887381i	$-0.652522\pi$
$661$	−23.7064	−0.922072	−0.461036	+	0.887381i	$0.652522\pi$
$662$	29.1606	1.13336
$663$	0.679692	0.0263970
$664$	−63.6945	−2.47182
$665$	5.84983	0.226846
$666$	44.9812	1.74299
$667$	−35.5115	−1.37501
$668$	38.9168	1.50574
$669$	−0.627867	−0.0242747
$670$	−5.30965	−0.205130
$671$	−3.65842	−0.141232
$672$	2.12143	0.0818361
$673$	4.35263	0.167782	0.0838908	−	0.996475i	$-0.473265\pi$
$673$	4.35263	0.167782	0.0838908	+	0.996475i	$0.473265\pi$
$674$	−33.9536	−1.30784
$675$	0.296583	0.0114155
$676$	183.207	7.04644
$677$	23.8979	0.918470	0.459235	−	0.888315i	$-0.348124\pi$
$677$	23.8979	0.918470	0.459235	+	0.888315i	$0.348124\pi$
$678$	1.27552	0.0489862
$679$	10.0568	0.385943
$680$	17.2402	0.661133
$681$	−0.604012	−0.0231458
$682$	18.0404	0.690802
$683$	29.0361	1.11104	0.555518	−	0.831505i	$-0.312520\pi$
$683$	29.0361	1.11104	0.555518	+	0.831505i	$0.312520\pi$
$684$	−37.0699	−1.41740
$685$	−6.08664	−0.232559
$686$	52.4121	2.00110
$687$	0.214302	0.00817613
$688$	51.8034	1.97499
$689$	−5.83494	−0.222294
$690$	−0.657864	−0.0250445
$691$	33.7044	1.28218	0.641088	−	0.767468i	$-0.278484\pi$
$691$	33.7044	1.28218	0.641088	+	0.767468i	$0.278484\pi$
$692$	79.7596	3.03201
$693$	7.45170	0.283067
$694$	27.4205	1.04087
$695$	−11.5079	−0.436518
$696$	3.08534	0.116950
$697$	−8.87435	−0.336140
$698$	91.4394	3.46103
$699$	0.0836723	0.00316478
$700$	12.9486	0.489410
$701$	15.8217	0.597578	0.298789	−	0.954319i	$-0.403417\pi$
$701$	15.8217	0.597578	0.298789	+	0.954319i	$0.403417\pi$
$702$	−5.52676	−0.208594
$703$	13.1893	0.497445
$704$	−20.8889	−0.787282
$705$	−0.121645	−0.00458141
$706$	41.0933	1.54657
$707$	−35.3596	−1.32983
$708$	−1.45263	−0.0545930
$709$	6.01791	0.226007	0.113004	−	0.993595i	$-0.463953\pi$
$709$	6.01791	0.226007	0.113004	+	0.993595i	$0.463953\pi$
$710$	14.7039	0.551826
$711$	42.2158	1.58321
$712$	76.1456	2.85368
$713$	−33.0384	−1.23730
$714$	−0.652829	−0.0244315
$715$	−6.96032	−0.260301
$716$	41.5843	1.55408
$717$	0.873366	0.0326165
$718$	60.7835	2.26842
$719$	−9.72000	−0.362495	−0.181247	−	0.983438i	$-0.558013\pi$
$719$	−9.72000	−0.362495	−0.181247	+	0.983438i	$0.558013\pi$
$720$	−38.6976	−1.44218
$721$	25.9620	0.966874
$722$	36.0708	1.34242
$723$	0.334062	0.0124239
$724$	−74.6079	−2.77278
$725$	7.17849	0.266602
$726$	−1.32865	−0.0493110
$727$	−15.1257	−0.560981	−0.280491	−	0.959857i	$-0.590497\pi$
$727$	−15.1257	−0.560981	−0.280491	+	0.959857i	$0.590497\pi$
$728$	−149.056	−5.52439
$729$	−26.8680	−0.995113
$730$	12.3227	0.456085
$731$	−7.95949	−0.294392
$732$	−0.942320	−0.0348292
$733$	23.9179	0.883429	0.441715	−	0.897156i	$-0.354370\pi$
$733$	23.9179	0.883429	0.441715	+	0.897156i	$0.354370\pi$
$734$	22.2282	0.820457
$735$	0.0432664	0.00159591
$736$	85.7509	3.16082
$737$	−1.98321	−0.0730527
$738$	36.0652	1.32758
$739$	−27.1525	−0.998820	−0.499410	−	0.866366i	$-0.666450\pi$
$739$	−27.1525	−0.998820	−0.499410	+	0.866366i	$0.666450\pi$
$740$	29.1946	1.07321
$741$	−0.809945	−0.0297541
$742$	5.60434	0.205742
$743$	26.7291	0.980595	0.490298	−	0.871555i	$-0.336888\pi$
$743$	26.7291	0.980595	0.490298	+	0.871555i	$0.336888\pi$
$744$	2.87048	0.105237
$745$	−17.3100	−0.634189
$746$	93.2188	3.41298
$747$	−21.9670	−0.803729
$748$	10.4242	0.381148
$749$	4.01015	0.146528
$750$	0.132984	0.00485590
$751$	−2.11888	−0.0773192	−0.0386596	−	0.999252i	$-0.512309\pi$
$751$	−2.11888	−0.0773192	−0.0386596	+	0.999252i	$0.512309\pi$
$752$	31.7570	1.15806
$753$	−0.502362	−0.0183071
$754$	−133.770	−4.87161
$755$	−20.9432	−0.762201
$756$	3.84032	0.139671
$757$	23.8809	0.867967	0.433984	−	0.900921i	$-0.357108\pi$
$757$	23.8809	0.867967	0.433984	+	0.900921i	$0.357108\pi$
$758$	18.7737	0.681892
$759$	−0.245720	−0.00891907
$760$	−20.5441	−0.745212
$761$	23.4203	0.848986	0.424493	−	0.905431i	$-0.360452\pi$
$761$	23.4203	0.848986	0.424493	+	0.905431i	$0.360452\pi$
$762$	1.40355	0.0508454
$763$	7.75318	0.280684
$764$	107.562	3.89145
$765$	5.94581	0.214971
$766$	−98.9323	−3.57457
$767$	38.9056	1.40480
$768$	−0.770147	−0.0277903
$769$	15.7650	0.568502	0.284251	−	0.958750i	$-0.408255\pi$
$769$	15.7650	0.568502	0.284251	+	0.958750i	$0.408255\pi$
$770$	6.68524	0.240919
$771$	−0.617561	−0.0222409
$772$	9.37579	0.337442
$773$	−31.4848	−1.13243	−0.566215	−	0.824257i	$-0.691593\pi$
$773$	−31.4848	−1.13243	−0.566215	+	0.824257i	$0.691593\pi$
$774$	32.3472	1.16270
$775$	6.67857	0.239901
$776$	−35.3185	−1.26786
$777$	−0.682907	−0.0244992
$778$	7.72463	0.276941
$779$	10.5750	0.378888
$780$	−1.79281	−0.0641930
$781$	5.49206	0.196521
$782$	−26.3881	−0.943638
$783$	2.12901	0.0760848
$784$	−11.2953	−0.403403
$785$	−3.71117	−0.132457
$786$	−0.224235	−0.00799819
$787$	22.9210	0.817044	0.408522	−	0.912748i	$-0.366044\pi$
$787$	22.9210	0.817044	0.408522	+	0.912748i	$0.366044\pi$
$788$	75.7324	2.69785
$789$	−0.108673	−0.00386886
$790$	37.8736	1.34748
$791$	23.7379	0.844024
$792$	−26.1697	−0.929900
$793$	25.2381	0.896231
$794$	15.8231	0.561539
$795$	0.0416401	0.00147682
$796$	89.0251	3.15541
$797$	28.6556	1.01503	0.507517	−	0.861642i	$-0.330564\pi$
$797$	28.6556	1.01503	0.507517	+	0.861642i	$0.330564\pi$
$798$	0.777935	0.0275386
$799$	−4.87940	−0.172621
$800$	−17.3342	−0.612855
$801$	26.2611	0.927890
$802$	−30.9760	−1.09380
$803$	4.60269	0.162425
$804$	−0.510828	−0.0180155
$805$	−12.2431	−0.431512
$806$	−124.454	−4.38371
$807$	−1.59254	−0.0560600
$808$	124.180	4.36863
$809$	30.9271	1.08734	0.543669	−	0.839300i	$-0.317035\pi$
$809$	30.9271	1.08734	0.543669	+	0.839300i	$0.317035\pi$
$810$	−24.1440	−0.848332
$811$	−15.4868	−0.543817	−0.271908	−	0.962323i	$-0.587655\pi$
$811$	−15.4868	−0.543817	−0.271908	+	0.962323i	$0.587655\pi$
$812$	92.9512	3.26195
$813$	−0.168958	−0.00592561
$814$	15.0729	0.528305
$815$	−15.7553	−0.551884
$816$	1.26629	0.0443290
$817$	9.48481	0.331832
$818$	18.7213	0.654575
$819$	−51.4066	−1.79629
$820$	23.4077	0.817433
$821$	51.0669	1.78225	0.891124	−	0.453759i	$-0.149917\pi$
$821$	51.0669	1.78225	0.891124	+	0.453759i	$0.149917\pi$
$822$	−0.809428	−0.0282320
$823$	45.3750	1.58167	0.790837	−	0.612027i	$-0.209646\pi$
$823$	45.3750	1.58167	0.790837	+	0.612027i	$0.209646\pi$
$824$	−91.1761	−3.17627
$825$	0.0496712	0.00172933
$826$	−37.3679	−1.30020
$827$	43.3642	1.50792	0.753961	−	0.656919i	$-0.228141\pi$
$827$	43.3642	1.50792	0.753961	+	0.656919i	$0.228141\pi$
$828$	77.5836	2.69622
$829$	−30.0577	−1.04395	−0.521973	−	0.852962i	$-0.674804\pi$
$829$	−30.0577	−1.04395	−0.521973	+	0.852962i	$0.674804\pi$
$830$	−19.7075	−0.684058
$831$	−1.09877	−0.0381160
$832$	144.105	4.99595
$833$	1.73550	0.0601314
$834$	−1.53037	−0.0529922
$835$	7.43824	0.257411
$836$	−12.4219	−0.429620
$837$	1.98075	0.0684647
$838$	12.0455	0.416105
$839$	41.4673	1.43161	0.715805	−	0.698300i	$-0.246060\pi$
$839$	41.4673	1.43161	0.715805	+	0.698300i	$0.246060\pi$
$840$	1.06372	0.0367017
$841$	22.5307	0.776920
$842$	72.2196	2.48885
$843$	−0.0706179	−0.00243221
$844$	117.366	4.03989
$845$	35.0168	1.20461
$846$	19.8298	0.681762
$847$	−24.7267	−0.849620
$848$	−10.8707	−0.373301
$849$	0.642474	0.0220496
$850$	5.33425	0.182963
$851$	−27.6039	−0.946251
$852$	1.41462	0.0484642
$853$	35.3678	1.21097	0.605486	−	0.795856i	$-0.292979\pi$
$853$	35.3678	1.21097	0.605486	+	0.795856i	$0.292979\pi$
$854$	−24.2406	−0.829498
$855$	−7.08525	−0.242310
$856$	−14.0833	−0.481357
$857$	−50.3110	−1.71859	−0.859295	−	0.511480i	$-0.829097\pi$
$857$	−50.3110	−1.71859	−0.859295	+	0.511480i	$0.829097\pi$
$858$	−0.925614	−0.0315999
$859$	−28.0046	−0.955503	−0.477752	−	0.878495i	$-0.658548\pi$
$859$	−28.0046	−0.955503	−0.477752	+	0.878495i	$0.658548\pi$
$860$	20.9946	0.715910
$861$	−0.547544	−0.0186602
$862$	−0.909691	−0.0309842
$863$	−11.6531	−0.396677	−0.198339	−	0.980134i	$-0.563555\pi$
$863$	−11.6531	−0.396677	−0.198339	+	0.980134i	$0.563555\pi$
$864$	−5.14101	−0.174901
$865$	15.2446	0.518332
$866$	−78.4156	−2.66467
$867$	0.646097	0.0219426
$868$	86.4780	2.93525
$869$	14.1462	0.479878
$870$	0.954626	0.0323649
$871$	13.6815	0.463579
$872$	−27.2285	−0.922073
$873$	−12.1806	−0.412252
$874$	31.4451	1.06365
$875$	2.47489	0.0836664
$876$	1.18554	0.0400557
$877$	40.6662	1.37320	0.686600	−	0.727035i	$-0.259102\pi$
$877$	40.6662	1.37320	0.686600	+	0.727035i	$0.259102\pi$
$878$	−26.6749	−0.900235
$879$	0.605546	0.0204246
$880$	−12.9673	−0.437129
$881$	−11.0352	−0.371785	−0.185892	−	0.982570i	$-0.559518\pi$
$881$	−11.0352	−0.371785	−0.185892	+	0.982570i	$0.559518\pi$
$882$	−7.05303	−0.237488
$883$	−34.2866	−1.15384	−0.576918	−	0.816802i	$-0.695745\pi$
$883$	−34.2866	−1.15384	−0.576918	+	0.816802i	$0.695745\pi$
$884$	−71.9130	−2.41870
$885$	−0.277643	−0.00933287
$886$	49.9636	1.67856
$887$	12.4662	0.418573	0.209286	−	0.977854i	$-0.432886\pi$
$887$	12.4662	0.418573	0.209286	+	0.977854i	$0.432886\pi$
$888$	2.39831	0.0804820
$889$	26.1206	0.876058
$890$	23.5600	0.789732
$891$	−9.01805	−0.302116
$892$	66.4298	2.22424
$893$	5.81446	0.194574
$894$	−2.30196	−0.0769889
$895$	7.94808	0.265675
$896$	−52.6099	−1.75757
$897$	1.69513	0.0565988
$898$	−44.6374	−1.48957
$899$	47.9420	1.59896
$900$	−15.6832	−0.522772
$901$	1.67026	0.0556445
$902$	12.0852	0.402393
$903$	−0.491097	−0.0163427
$904$	−83.3656	−2.77270
$905$	−14.2599	−0.474017
$906$	−2.78511	−0.0925292
$907$	25.0785	0.832718	0.416359	−	0.909200i	$-0.363306\pi$
$907$	25.0785	0.832718	0.416359	+	0.909200i	$0.363306\pi$
$908$	63.9059	2.12079
$909$	42.8272	1.42049
$910$	−46.1191	−1.52883
$911$	40.1353	1.32974	0.664871	−	0.746959i	$-0.268487\pi$
$911$	40.1353	1.32974	0.664871	+	0.746959i	$0.268487\pi$
$912$	−1.50896	−0.0499665
$913$	−7.36098	−0.243613
$914$	29.7017	0.982445
$915$	−0.180107	−0.00595417
$916$	−22.6737	−0.749159
$917$	−4.17309	−0.137808
$918$	1.58204	0.0522152
$919$	16.4331	0.542079	0.271040	−	0.962568i	$-0.412633\pi$
$919$	16.4331	0.542079	0.271040	+	0.962568i	$0.412633\pi$
$920$	42.9967	1.41756
$921$	0.104123	0.00343098
$922$	74.2387	2.44492
$923$	−37.8877	−1.24709
$924$	0.643171	0.0211588
$925$	5.58001	0.183470
$926$	0.0778192	0.00255730
$927$	−31.4448	−1.03278
$928$	−124.433	−4.08472
$929$	−12.5497	−0.411741	−0.205871	−	0.978579i	$-0.566003\pi$
$929$	−12.5497	−0.411741	−0.205871	+	0.978579i	$0.566003\pi$
$930$	0.888145	0.0291234
$931$	−2.06808	−0.0677786
$932$	−8.85273	−0.289981
$933$	−1.41387	−0.0462880
$934$	−6.99587	−0.228912
$935$	1.99240	0.0651586
$936$	180.535	5.90098
$937$	14.2207	0.464570	0.232285	−	0.972648i	$-0.425380\pi$
$937$	14.2207	0.464570	0.232285	+	0.972648i	$0.425380\pi$
$938$	−13.1408	−0.429061
$939$	0.399007	0.0130211
$940$	12.8703	0.419783
$941$	−30.0654	−0.980106	−0.490053	−	0.871693i	$-0.663023\pi$
$941$	−30.0654	−0.980106	−0.490053	+	0.871693i	$0.663023\pi$
$942$	−0.493528	−0.0160800
$943$	−22.1324	−0.720729
$944$	72.4824	2.35910
$945$	0.734008	0.0238773
$946$	10.8393	0.352417
$947$	−40.0784	−1.30237	−0.651187	−	0.758918i	$-0.725728\pi$
$947$	−40.0784	−1.30237	−0.651187	+	0.758918i	$0.725728\pi$
$948$	3.64373	0.118343
$949$	−31.7523	−1.03072
$950$	−6.35648	−0.206231
$951$	−0.0503405	−0.00163240
$952$	42.6676	1.38286
$953$	9.91223	0.321089	0.160544	−	0.987029i	$-0.448675\pi$
$953$	9.91223	0.321089	0.160544	+	0.987029i	$0.448675\pi$
$954$	−6.78791	−0.219767
$955$	20.5585	0.665258
$956$	−92.4043	−2.98857
$957$	0.356564	0.0115261
$958$	−11.5603	−0.373496
$959$	−15.0637	−0.486434
$960$	−1.02838	−0.0331909
$961$	13.6033	0.438816
$962$	−103.983	−3.35253
$963$	−4.85705	−0.156516
$964$	−35.3445	−1.13837
$965$	1.79201	0.0576869
$966$	−1.62814	−0.0523845
$967$	47.3503	1.52268	0.761341	−	0.648352i	$-0.224541\pi$
$967$	47.3503	1.52268	0.761341	+	0.648352i	$0.224541\pi$
$968$	86.8381	2.79108
$969$	0.231848	0.00744803
$970$	−10.9278	−0.350870
$971$	−5.05840	−0.162332	−0.0811659	−	0.996701i	$-0.525864\pi$
$971$	−5.05840	−0.162332	−0.0811659	+	0.996701i	$0.525864\pi$
$972$	−6.97798	−0.223819
$973$	−28.4806	−0.913048
$974$	−3.32718	−0.106610
$975$	−0.342663	−0.0109740
$976$	47.0195	1.50506
$977$	−35.4567	−1.13436	−0.567181	−	0.823593i	$-0.691966\pi$
$977$	−35.4567	−1.13436	−0.567181	+	0.823593i	$0.691966\pi$
$978$	−2.09521	−0.0669973
$979$	8.79992	0.281247
$980$	−4.57769	−0.146229
$981$	−9.39057	−0.299818
$982$	45.1548	1.44095
$983$	−24.3241	−0.775818	−0.387909	−	0.921698i	$-0.626802\pi$
$983$	−24.3241	−0.775818	−0.387909	+	0.921698i	$0.626802\pi$
$984$	1.92293	0.0613006
$985$	14.4749	0.461208
$986$	38.2918	1.21946
$987$	−0.301057	−0.00958274
$988$	85.6941	2.72629
$989$	−19.8507	−0.631217
$990$	−8.09709	−0.257342
$991$	−1.91384	−0.0607950	−0.0303975	−	0.999538i	$-0.509677\pi$
$991$	−1.91384	−0.0607950	−0.0303975	+	0.999538i	$0.509677\pi$
$992$	−115.767	−3.67562
$993$	0.536215	0.0170163
$994$	36.3903	1.15423
$995$	17.0155	0.539429
$996$	−1.89601	−0.0600774
$997$	40.3460	1.27777	0.638885	−	0.769302i	$-0.279396\pi$
$997$	40.3460	1.27777	0.638885	+	0.769302i	$0.279396\pi$
$998$	1.62019	0.0512863
$999$	1.65493	0.0523598

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.e.1.2	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.e.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-2.68924 q^{2} -0.0494506 q^{3} +5.23199 q^{4} +1.00000 q^{5} +0.132984 q^{6} +2.47489 q^{7} -8.69158 q^{8} -2.99755 q^{9} +O(q^{10})\)	\(q-2.68924 q^{2} -0.0494506 q^{3} +5.23199 q^{4} +1.00000 q^{5} +0.132984 q^{6} +2.47489 q^{7} -8.69158 q^{8} -2.99755 q^{9} -2.68924 q^{10} -1.00446 q^{11} -0.258725 q^{12} +6.92941 q^{13} -6.65555 q^{14} -0.0494506 q^{15} +12.9097 q^{16} -1.98355 q^{17} +8.06113 q^{18} +2.36368 q^{19} +5.23199 q^{20} -0.122385 q^{21} +2.70123 q^{22} -4.94693 q^{23} +0.429804 q^{24} +1.00000 q^{25} -18.6348 q^{26} +0.296583 q^{27} +12.9486 q^{28} +7.17849 q^{29} +0.132984 q^{30} +6.67857 q^{31} -17.3342 q^{32} +0.0496712 q^{33} +5.33425 q^{34} +2.47489 q^{35} -15.6832 q^{36} +5.58001 q^{37} -6.35648 q^{38} -0.342663 q^{39} -8.69158 q^{40} +4.47396 q^{41} +0.329121 q^{42} +4.01274 q^{43} -5.25533 q^{44} -2.99755 q^{45} +13.3035 q^{46} +2.45993 q^{47} -0.638394 q^{48} -0.874942 q^{49} -2.68924 q^{50} +0.0980879 q^{51} +36.2546 q^{52} -0.842054 q^{53} -0.797580 q^{54} -1.00446 q^{55} -21.5107 q^{56} -0.116885 q^{57} -19.3046 q^{58} +5.61455 q^{59} -0.258725 q^{60} +3.64217 q^{61} -17.9603 q^{62} -7.41860 q^{63} +20.7962 q^{64} +6.92941 q^{65} -0.133578 q^{66} +1.97441 q^{67} -10.3779 q^{68} +0.244629 q^{69} -6.65555 q^{70} -5.46767 q^{71} +26.0535 q^{72} -4.58225 q^{73} -15.0060 q^{74} -0.0494506 q^{75} +12.3667 q^{76} -2.48593 q^{77} +0.921503 q^{78} -14.0834 q^{79} +12.9097 q^{80} +8.97800 q^{81} -12.0315 q^{82} +7.32829 q^{83} -0.640315 q^{84} -1.98355 q^{85} -10.7912 q^{86} -0.354980 q^{87} +8.73036 q^{88} -8.76084 q^{89} +8.06113 q^{90} +17.1495 q^{91} -25.8823 q^{92} -0.330259 q^{93} -6.61532 q^{94} +2.36368 q^{95} +0.857184 q^{96} +4.06353 q^{97} +2.35293 q^{98} +3.01093 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9}+O(q^{10}) \) 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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