LMFDB - Embedded newform 8047.2.a.c.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, 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("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$1$
Dimension:	$151$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.25853 q^{2} -0.662024 q^{3} +3.10095 q^{4} +2.15399 q^{5} +1.49520 q^{6} +0.474761 q^{7} -2.48652 q^{8} -2.56172 q^{9} +O(q^{10})$	\(q-2.25853 q^{2} -0.662024 q^{3} +3.10095 q^{4} +2.15399 q^{5} +1.49520 q^{6} +0.474761 q^{7} -2.48652 q^{8} -2.56172 q^{9} -4.86485 q^{10} +0.981650 q^{11} -2.05290 q^{12} -1.00000 q^{13} -1.07226 q^{14} -1.42599 q^{15} -0.586027 q^{16} -2.52063 q^{17} +5.78572 q^{18} -3.44260 q^{19} +6.67941 q^{20} -0.314303 q^{21} -2.21708 q^{22} +6.57104 q^{23} +1.64613 q^{24} -0.360327 q^{25} +2.25853 q^{26} +3.68199 q^{27} +1.47221 q^{28} -1.28482 q^{29} +3.22064 q^{30} -6.52214 q^{31} +6.29659 q^{32} -0.649876 q^{33} +5.69291 q^{34} +1.02263 q^{35} -7.94377 q^{36} -4.94265 q^{37} +7.77520 q^{38} +0.662024 q^{39} -5.35593 q^{40} +6.95257 q^{41} +0.709862 q^{42} +5.14965 q^{43} +3.04404 q^{44} -5.51793 q^{45} -14.8409 q^{46} -8.91987 q^{47} +0.387964 q^{48} -6.77460 q^{49} +0.813808 q^{50} +1.66872 q^{51} -3.10095 q^{52} +12.0290 q^{53} -8.31589 q^{54} +2.11447 q^{55} -1.18050 q^{56} +2.27908 q^{57} +2.90179 q^{58} -3.90211 q^{59} -4.42193 q^{60} +12.0861 q^{61} +14.7304 q^{62} -1.21621 q^{63} -13.0490 q^{64} -2.15399 q^{65} +1.46776 q^{66} +13.8725 q^{67} -7.81633 q^{68} -4.35018 q^{69} -2.30964 q^{70} +8.55245 q^{71} +6.36977 q^{72} +5.40608 q^{73} +11.1631 q^{74} +0.238545 q^{75} -10.6753 q^{76} +0.466049 q^{77} -1.49520 q^{78} -5.64662 q^{79} -1.26230 q^{80} +5.24760 q^{81} -15.7026 q^{82} +8.48522 q^{83} -0.974636 q^{84} -5.42941 q^{85} -11.6306 q^{86} +0.850579 q^{87} -2.44089 q^{88} +7.00428 q^{89} +12.4624 q^{90} -0.474761 q^{91} +20.3764 q^{92} +4.31781 q^{93} +20.1458 q^{94} -7.41532 q^{95} -4.16849 q^{96} -10.9730 q^{97} +15.3006 q^{98} -2.51472 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) $ 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9 - 3 * q^10 - 27 * q^11 - 52 * q^12 - 151 * q^13 - 9 * q^14 - 14 * q^15 + 143 * q^16 - 111 * q^17 - 37 * q^18 - 17 * q^19 - 107 * q^20 - 29 * q^21 - 16 * q^22 - 47 * q^23 - 46 * q^24 + 122 * q^25 + 13 * q^26 - 55 * q^27 - 44 * q^28 + 37 * q^29 - 14 * q^30 - 27 * q^31 - 86 * q^32 - 94 * q^33 - 10 * q^34 - 47 * q^35 + 124 * q^36 - 59 * q^37 - 80 * q^38 + 16 * q^39 + 5 * q^40 - 129 * q^41 - 77 * q^42 - 11 * q^43 - 99 * q^44 - 122 * q^45 - 17 * q^46 - 130 * q^47 - 111 * q^48 + 99 * q^49 - 72 * q^50 + 15 * q^51 - 151 * q^52 - 43 * q^53 - 49 * q^54 - 40 * q^55 - 50 * q^56 - 85 * q^57 - 73 * q^58 - 74 * q^59 - 43 * q^60 - 7 * q^61 - 110 * q^62 - 70 * q^63 + 141 * q^64 + 43 * q^65 - 16 * q^66 - 39 * q^67 - 222 * q^68 + 19 * q^69 - 52 * q^70 - 72 * q^71 - 106 * q^72 - 143 * q^73 + 20 * q^74 - 73 * q^75 - 88 * q^76 - 86 * q^77 + 17 * q^78 + 10 * q^79 - 239 * q^80 + 103 * q^81 - 96 * q^82 - 96 * q^83 - 75 * q^84 - 24 * q^85 - 109 * q^86 - 65 * q^87 - 45 * q^88 - 237 * q^89 - 79 * q^90 + 18 * q^91 - 153 * q^92 - 137 * q^93 - 23 * q^94 + 10 * q^95 - 109 * q^96 - 160 * q^97 - 119 * q^98 - 86 * 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 $

$2$	−2.25853	−1.59702	−0.798510	−	0.601982i	$-0.794378\pi$
$2$	−2.25853	−1.59702	−0.798510	+	0.601982i	$0.794378\pi$
$3$	−0.662024	−0.382220	−0.191110	−	0.981569i	$-0.561209\pi$
$3$	−0.662024	−0.382220	−0.191110	+	0.981569i	$0.561209\pi$
$4$	3.10095	1.55047
$5$	2.15399	0.963294	0.481647	−	0.876365i	$-0.340039\pi$
$5$	2.15399	0.963294	0.481647	+	0.876365i	$0.340039\pi$
$6$	1.49520	0.610412
$7$	0.474761	0.179443	0.0897214	−	0.995967i	$-0.471402\pi$
$7$	0.474761	0.179443	0.0897214	+	0.995967i	$0.471402\pi$
$8$	−2.48652	−0.879116
$9$	−2.56172	−0.853908
$10$	−4.86485	−1.53840
$11$	0.981650	0.295979	0.147989	−	0.988989i	$-0.452720\pi$
$11$	0.981650	0.295979	0.147989	+	0.988989i	$0.452720\pi$
$12$	−2.05290	−0.592621
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.07226	−0.286574
$15$	−1.42599	−0.368190
$16$	−0.586027	−0.146507
$17$	−2.52063	−0.611342	−0.305671	−	0.952137i	$-0.598881\pi$
$17$	−2.52063	−0.611342	−0.305671	+	0.952137i	$0.598881\pi$
$18$	5.78572	1.36371
$19$	−3.44260	−0.789786	−0.394893	−	0.918727i	$-0.629218\pi$
$19$	−3.44260	−0.789786	−0.394893	+	0.918727i	$0.629218\pi$
$20$	6.67941	1.49356
$21$	−0.314303	−0.0685865
$22$	−2.21708	−0.472684
$23$	6.57104	1.37016	0.685078	−	0.728470i	$-0.259768\pi$
$23$	6.57104	1.37016	0.685078	+	0.728470i	$0.259768\pi$
$24$	1.64613	0.336015
$25$	−0.360327	−0.0720654
$26$	2.25853	0.442934
$27$	3.68199	0.708600
$28$	1.47221	0.278221
$29$	−1.28482	−0.238584	−0.119292	−	0.992859i	$-0.538063\pi$
$29$	−1.28482	−0.238584	−0.119292	+	0.992859i	$0.538063\pi$
$30$	3.22064	0.588006
$31$	−6.52214	−1.17141	−0.585705	−	0.810524i	$-0.699182\pi$
$31$	−6.52214	−1.17141	−0.585705	+	0.810524i	$0.699182\pi$
$32$	6.29659	1.11309
$33$	−0.649876	−0.113129
$34$	5.69291	0.976326
$35$	1.02263	0.172856
$36$	−7.94377	−1.32396
$37$	−4.94265	−0.812567	−0.406283	−	0.913747i	$-0.633175\pi$
$37$	−4.94265	−0.812567	−0.406283	+	0.913747i	$0.633175\pi$
$38$	7.77520	1.26130
$39$	0.662024	0.106009
$40$	−5.35593	−0.846847
$41$	6.95257	1.08581	0.542904	−	0.839795i	$-0.317325\pi$
$41$	6.95257	1.08581	0.542904	+	0.839795i	$0.317325\pi$
$42$	0.709862	0.109534
$43$	5.14965	0.785314	0.392657	−	0.919685i	$-0.371556\pi$
$43$	5.14965	0.785314	0.392657	+	0.919685i	$0.371556\pi$
$44$	3.04404	0.458907
$45$	−5.51793	−0.822564
$46$	−14.8409	−2.18817
$47$	−8.91987	−1.30110	−0.650549	−	0.759465i	$-0.725461\pi$
$47$	−8.91987	−1.30110	−0.650549	+	0.759465i	$0.725461\pi$
$48$	0.387964	0.0559978
$49$	−6.77460	−0.967800
$50$	0.813808	0.115090
$51$	1.66872	0.233667
$52$	−3.10095	−0.430024
$53$	12.0290	1.65231	0.826157	−	0.563439i	$-0.190522\pi$
$53$	12.0290	1.65231	0.826157	+	0.563439i	$0.190522\pi$
$54$	−8.31589	−1.13165
$55$	2.11447	0.285114
$56$	−1.18050	−0.157751
$57$	2.27908	0.301872
$58$	2.90179	0.381024
$59$	−3.90211	−0.508011	−0.254006	−	0.967203i	$-0.581748\pi$
$59$	−3.90211	−0.508011	−0.254006	+	0.967203i	$0.581748\pi$
$60$	−4.42193	−0.570868
$61$	12.0861	1.54747	0.773736	−	0.633508i	$-0.218385\pi$
$61$	12.0861	1.54747	0.773736	+	0.633508i	$0.218385\pi$
$62$	14.7304	1.87077
$63$	−1.21621	−0.153228
$64$	−13.0490	−1.63112
$65$	−2.15399	−0.267170
$66$	1.46776	0.180669
$67$	13.8725	1.69479	0.847397	−	0.530959i	$-0.178168\pi$
$67$	13.8725	1.69479	0.847397	+	0.530959i	$0.178168\pi$
$68$	−7.81633	−0.947870
$69$	−4.35018	−0.523701
$70$	−2.30964	−0.276055
$71$	8.55245	1.01499	0.507494	−	0.861655i	$-0.330572\pi$
$71$	8.55245	1.01499	0.507494	+	0.861655i	$0.330572\pi$
$72$	6.36977	0.750684
$73$	5.40608	0.632734	0.316367	−	0.948637i	$-0.397537\pi$
$73$	5.40608	0.632734	0.316367	+	0.948637i	$0.397537\pi$
$74$	11.1631	1.29769
$75$	0.238545	0.0275448
$76$	−10.6753	−1.22454
$77$	0.466049	0.0531112
$78$	−1.49520	−0.169298
$79$	−5.64662	−0.635294	−0.317647	−	0.948209i	$-0.602893\pi$
$79$	−5.64662	−0.635294	−0.317647	+	0.948209i	$0.602893\pi$
$80$	−1.26230	−0.141129
$81$	5.24760	0.583067
$82$	−15.7026	−1.73406
$83$	8.48522	0.931374	0.465687	−	0.884950i	$-0.345807\pi$
$83$	8.48522	0.931374	0.465687	+	0.884950i	$0.345807\pi$
$84$	−0.974636	−0.106342
$85$	−5.42941	−0.588902
$86$	−11.6306	−1.25416
$87$	0.850579	0.0911916
$88$	−2.44089	−0.260200
$89$	7.00428	0.742452	0.371226	−	0.928543i	$-0.378938\pi$
$89$	7.00428	0.742452	0.371226	+	0.928543i	$0.378938\pi$
$90$	12.4624	1.31365
$91$	−0.474761	−0.0497685
$92$	20.3764	2.12439
$93$	4.31781	0.447736
$94$	20.1458	2.07788
$95$	−7.41532	−0.760796
$96$	−4.16849	−0.425445
$97$	−10.9730	−1.11414	−0.557068	−	0.830467i	$-0.688074\pi$
$97$	−10.9730	−1.11414	−0.557068	+	0.830467i	$0.688074\pi$
$98$	15.3006	1.54560
$99$	−2.51472	−0.252739
$100$	−1.11735	−0.111735
$101$	1.58178	0.157393	0.0786965	−	0.996899i	$-0.474924\pi$
$101$	1.58178	0.157393	0.0786965	+	0.996899i	$0.474924\pi$
$102$	−3.76884	−0.373171
$103$	4.25135	0.418898	0.209449	−	0.977820i	$-0.432833\pi$
$103$	4.25135	0.418898	0.209449	+	0.977820i	$0.432833\pi$
$104$	2.48652	0.243823
$105$	−0.677005	−0.0660690
$106$	−27.1679	−2.63878
$107$	−19.7125	−1.90568	−0.952840	−	0.303475i	$-0.901853\pi$
$107$	−19.7125	−1.90568	−0.952840	+	0.303475i	$0.901853\pi$
$108$	11.4177	1.09867
$109$	14.9658	1.43346	0.716730	−	0.697351i	$-0.245638\pi$
$109$	14.9658	1.43346	0.716730	+	0.697351i	$0.245638\pi$
$110$	−4.77558	−0.455333
$111$	3.27215	0.310579
$112$	−0.278223	−0.0262896
$113$	−6.19985	−0.583233	−0.291616	−	0.956535i	$-0.594193\pi$
$113$	−6.19985	−0.583233	−0.291616	+	0.956535i	$0.594193\pi$
$114$	−5.14737	−0.482095
$115$	14.1539	1.31986
$116$	−3.98414	−0.369919
$117$	2.56172	0.236831
$118$	8.81302	0.811304
$119$	−1.19670	−0.109701
$120$	3.54575	0.323682
$121$	−10.0364	−0.912397
$122$	−27.2969	−2.47134
$123$	−4.60277	−0.415017
$124$	−20.2248	−1.81624
$125$	−11.5461	−1.03271
$126$	2.74683	0.244708
$127$	−9.78247	−0.868054	−0.434027	−	0.900900i	$-0.642908\pi$
$127$	−9.78247	−0.868054	−0.434027	+	0.900900i	$0.642908\pi$
$128$	16.8783	1.49184
$129$	−3.40919	−0.300162
$130$	4.86485	0.426675
$131$	4.16052	0.363507	0.181753	−	0.983344i	$-0.441823\pi$
$131$	4.16052	0.363507	0.181753	+	0.983344i	$0.441823\pi$
$132$	−2.01523	−0.175403
$133$	−1.63441	−0.141721
$134$	−31.3314	−2.70662
$135$	7.93098	0.682590
$136$	6.26758	0.537441
$137$	1.61582	0.138049	0.0690245	−	0.997615i	$-0.478011\pi$
$137$	1.61582	0.138049	0.0690245	+	0.997615i	$0.478011\pi$
$138$	9.82501	0.836360
$139$	−20.2130	−1.71445	−0.857224	−	0.514944i	$-0.827813\pi$
$139$	−20.2130	−1.71445	−0.857224	+	0.514944i	$0.827813\pi$
$140$	3.17112	0.268009
$141$	5.90517	0.497305
$142$	−19.3159	−1.62096
$143$	−0.981650	−0.0820897
$144$	1.50124	0.125103
$145$	−2.76748	−0.229827
$146$	−12.2098	−1.01049
$147$	4.48495	0.369912
$148$	−15.3269	−1.25986
$149$	−14.0819	−1.15364	−0.576818	−	0.816873i	$-0.695706\pi$
$149$	−14.0819	−1.15364	−0.576818	+	0.816873i	$0.695706\pi$
$150$	−0.538760	−0.0439896
$151$	−10.8677	−0.884403	−0.442201	−	0.896916i	$-0.645802\pi$
$151$	−10.8677	−0.884403	−0.442201	+	0.896916i	$0.645802\pi$
$152$	8.56007	0.694314
$153$	6.45716	0.522030
$154$	−1.05258	−0.0848197
$155$	−14.0486	−1.12841
$156$	2.05290	0.164364
$157$	−3.98949	−0.318396	−0.159198	−	0.987247i	$-0.550891\pi$
$157$	−3.98949	−0.318396	−0.159198	+	0.987247i	$0.550891\pi$
$158$	12.7530	1.01458
$159$	−7.96351	−0.631547
$160$	13.5628	1.07223
$161$	3.11967	0.245864
$162$	−11.8519	−0.931170
$163$	−6.98936	−0.547449	−0.273724	−	0.961808i	$-0.588256\pi$
$163$	−6.98936	−0.547449	−0.273724	+	0.961808i	$0.588256\pi$
$164$	21.5595	1.68352
$165$	−1.39983	−0.108976
$166$	−19.1641	−1.48742
$167$	4.17963	0.323429	0.161715	−	0.986838i	$-0.448298\pi$
$167$	4.17963	0.323429	0.161715	+	0.986838i	$0.448298\pi$
$168$	0.781519	0.0602955
$169$	1.00000	0.0769231
$170$	12.2625	0.940489
$171$	8.81899	0.674405
$172$	15.9688	1.21761
$173$	−8.13425	−0.618435	−0.309218	−	0.950991i	$-0.600067\pi$
$173$	−8.13425	−0.618435	−0.309218	+	0.950991i	$0.600067\pi$
$174$	−1.92106	−0.145635
$175$	−0.171069	−0.0129316
$176$	−0.575274	−0.0433629
$177$	2.58329	0.194172
$178$	−15.8194	−1.18571
$179$	−11.5761	−0.865240	−0.432620	−	0.901576i	$-0.642411\pi$
$179$	−11.5761	−0.865240	−0.432620	+	0.901576i	$0.642411\pi$
$180$	−17.1108	−1.27536
$181$	−7.11606	−0.528933	−0.264466	−	0.964395i	$-0.585196\pi$
$181$	−7.11606	−0.528933	−0.264466	+	0.964395i	$0.585196\pi$
$182$	1.07226	0.0794812
$183$	−8.00132	−0.591475
$184$	−16.3390	−1.20453
$185$	−10.6464	−0.782741
$186$	−9.75189	−0.715043
$187$	−2.47438	−0.180944
$188$	−27.6600	−2.01732
$189$	1.74807	0.127153
$190$	16.7477	1.21501
$191$	2.39361	0.173196	0.0865978	−	0.996243i	$-0.472400\pi$
$191$	2.39361	0.173196	0.0865978	+	0.996243i	$0.472400\pi$
$192$	8.63873	0.623447
$193$	24.0281	1.72958	0.864792	−	0.502131i	$-0.167450\pi$
$193$	24.0281	1.72958	0.864792	+	0.502131i	$0.167450\pi$
$194$	24.7827	1.77930
$195$	1.42599	0.102117
$196$	−21.0077	−1.50055
$197$	−15.1404	−1.07871	−0.539355	−	0.842078i	$-0.681332\pi$
$197$	−15.1404	−1.07871	−0.539355	+	0.842078i	$0.681332\pi$
$198$	5.67956	0.403629
$199$	23.4609	1.66310	0.831548	−	0.555453i	$-0.187455\pi$
$199$	23.4609	1.66310	0.831548	+	0.555453i	$0.187455\pi$
$200$	0.895958	0.0633538
$201$	−9.18392	−0.647784
$202$	−3.57249	−0.251360
$203$	−0.609980	−0.0428122
$204$	5.17460	0.362295
$205$	14.9758	1.04595
$206$	−9.60179	−0.668988
$207$	−16.8332	−1.16999
$208$	0.586027	0.0406337
$209$	−3.37943	−0.233760
$210$	1.52904	0.105513
$211$	3.97815	0.273867	0.136934	−	0.990580i	$-0.456275\pi$
$211$	3.97815	0.273867	0.136934	+	0.990580i	$0.456275\pi$
$212$	37.3014	2.56187
$213$	−5.66193	−0.387949
$214$	44.5212	3.04341
$215$	11.0923	0.756488
$216$	−9.15534	−0.622942
$217$	−3.09645	−0.210201
$218$	−33.8006	−2.28926
$219$	−3.57896	−0.241843
$220$	6.55684	0.442062
$221$	2.52063	0.169556
$222$	−7.39025	−0.496001
$223$	−5.14423	−0.344483	−0.172242	−	0.985055i	$-0.555101\pi$
$223$	−5.14423	−0.344483	−0.172242	+	0.985055i	$0.555101\pi$
$224$	2.98937	0.199736
$225$	0.923058	0.0615372
$226$	14.0025	0.931435
$227$	−12.6222	−0.837763	−0.418881	−	0.908041i	$-0.637578\pi$
$227$	−12.6222	−0.837763	−0.418881	+	0.908041i	$0.637578\pi$
$228$	7.06731	0.468044
$229$	14.9895	0.990531	0.495266	−	0.868742i	$-0.335071\pi$
$229$	14.9895	0.990531	0.495266	+	0.868742i	$0.335071\pi$
$230$	−31.9671	−2.10785
$231$	−0.308536	−0.0203002
$232$	3.19472	0.209743
$233$	−13.0459	−0.854663	−0.427331	−	0.904095i	$-0.640546\pi$
$233$	−13.0459	−0.854663	−0.427331	+	0.904095i	$0.640546\pi$
$234$	−5.78572	−0.378225
$235$	−19.2133	−1.25334
$236$	−12.1002	−0.787658
$237$	3.73820	0.242822
$238$	2.70277	0.175195
$239$	−11.4463	−0.740398	−0.370199	−	0.928952i	$-0.620711\pi$
$239$	−11.4463	−0.740398	−0.370199	+	0.928952i	$0.620711\pi$
$240$	0.835671	0.0539423
$241$	−26.3185	−1.69532	−0.847661	−	0.530537i	$-0.821990\pi$
$241$	−26.3185	−1.69532	−0.847661	+	0.530537i	$0.821990\pi$
$242$	22.6674	1.45712
$243$	−14.5200	−0.931460
$244$	37.4785	2.39931
$245$	−14.5924	−0.932276
$246$	10.3955	0.662791
$247$	3.44260	0.219047
$248$	16.2174	1.02981
$249$	−5.61742	−0.355989
$250$	26.0772	1.64926
$251$	−9.97546	−0.629645	−0.314823	−	0.949151i	$-0.601945\pi$
$251$	−9.97546	−0.629645	−0.314823	+	0.949151i	$0.601945\pi$
$252$	−3.77139	−0.237575
$253$	6.45046	0.405537
$254$	22.0940	1.38630
$255$	3.59440	0.225090
$256$	−12.0221	−0.751381
$257$	29.1530	1.81852	0.909258	−	0.416233i	$-0.136650\pi$
$257$	29.1530	1.81852	0.909258	+	0.416233i	$0.136650\pi$
$258$	7.69975	0.479365
$259$	−2.34658	−0.145809
$260$	−6.67941	−0.414239
$261$	3.29134	0.203729
$262$	−9.39666	−0.580527
$263$	−4.58259	−0.282575	−0.141287	−	0.989969i	$-0.545124\pi$
$263$	−4.58259	−0.282575	−0.141287	+	0.989969i	$0.545124\pi$
$264$	1.61593	0.0994534
$265$	25.9104	1.59166
$266$	3.69136	0.226332
$267$	−4.63700	−0.283780
$268$	43.0178	2.62773
$269$	−24.1325	−1.47138	−0.735692	−	0.677317i	$-0.763143\pi$
$269$	−24.1325	−1.47138	−0.735692	+	0.677317i	$0.763143\pi$
$270$	−17.9123	−1.09011
$271$	11.7347	0.712830	0.356415	−	0.934328i	$-0.383999\pi$
$271$	11.7347	0.712830	0.356415	+	0.934328i	$0.383999\pi$
$272$	1.47716	0.0895659
$273$	0.314303	0.0190225
$274$	−3.64938	−0.220467
$275$	−0.353715	−0.0213298
$276$	−13.4897	−0.811983
$277$	2.71601	0.163189	0.0815946	−	0.996666i	$-0.473999\pi$
$277$	2.71601	0.163189	0.0815946	+	0.996666i	$0.473999\pi$
$278$	45.6517	2.73801
$279$	16.7079	1.00028
$280$	−2.54279	−0.151961
$281$	8.52935	0.508818	0.254409	−	0.967097i	$-0.418119\pi$
$281$	8.52935	0.508818	0.254409	+	0.967097i	$0.418119\pi$
$282$	−13.3370	−0.794206
$283$	0.00668517	0.000397392 0	0.000198696	−	1.00000i	$-0.499937\pi$
$283$	0.00668517	0.000397392 0	0.000198696		1.00000i	$0.499937\pi$
$284$	26.5207	1.57371
$285$	4.90912	0.290791
$286$	2.21708	0.131099
$287$	3.30081	0.194840
$288$	−16.1301	−0.950477
$289$	−10.6464	−0.626260
$290$	6.25043	0.367038
$291$	7.26436	0.425845
$292$	16.7640	0.981037
$293$	−30.6630	−1.79135	−0.895676	−	0.444707i	$-0.853308\pi$
$293$	−30.6630	−1.79135	−0.895676	+	0.444707i	$0.853308\pi$
$294$	−10.1294	−0.590757
$295$	−8.40511	−0.489364
$296$	12.2900	0.714341
$297$	3.61443	0.209731
$298$	31.8044	1.84238
$299$	−6.57104	−0.380013
$300$	0.739715	0.0427075
$301$	2.44485	0.140919
$302$	24.5451	1.41241
$303$	−1.04718	−0.0601587
$304$	2.01746	0.115709
$305$	26.0334	1.49067
$306$	−14.5837	−0.833693
$307$	−19.5402	−1.11522	−0.557609	−	0.830103i	$-0.688281\pi$
$307$	−19.5402	−1.11522	−0.557609	+	0.830103i	$0.688281\pi$
$308$	1.44519	0.0823475
$309$	−2.81449	−0.160111
$310$	31.7292	1.80210
$311$	−3.93175	−0.222949	−0.111475	−	0.993767i	$-0.535557\pi$
$311$	−3.93175	−0.222949	−0.111475	+	0.993767i	$0.535557\pi$
$312$	−1.64613	−0.0931939
$313$	9.77593	0.552568	0.276284	−	0.961076i	$-0.410897\pi$
$313$	9.77593	0.552568	0.276284	+	0.961076i	$0.410897\pi$
$314$	9.01037	0.508485
$315$	−2.61970	−0.147603
$316$	−17.5099	−0.985007
$317$	−13.0985	−0.735688	−0.367844	−	0.929888i	$-0.619904\pi$
$317$	−13.0985	−0.735688	−0.367844	+	0.929888i	$0.619904\pi$
$318$	17.9858	1.00859
$319$	−1.26124	−0.0706159
$320$	−28.1073	−1.57125
$321$	13.0501	0.728388
$322$	−7.04586	−0.392650
$323$	8.67751	0.482830
$324$	16.2725	0.904030
$325$	0.360327	0.0199873
$326$	15.7856	0.874286
$327$	−9.90769	−0.547897
$328$	−17.2877	−0.954552
$329$	−4.23481	−0.233472
$330$	3.16155	0.174037
$331$	21.2507	1.16805	0.584023	−	0.811737i	$-0.301478\pi$
$331$	21.2507	1.16805	0.584023	+	0.811737i	$0.301478\pi$
$332$	26.3122	1.44407
$333$	12.6617	0.693858
$334$	−9.43980	−0.516523
$335$	29.8812	1.63259
$336$	0.184190	0.0100484
$337$	−29.3617	−1.59943	−0.799716	−	0.600379i	$-0.795016\pi$
$337$	−29.3617	−1.59943	−0.799716	+	0.600379i	$0.795016\pi$
$338$	−2.25853	−0.122848
$339$	4.10445	0.222923
$340$	−16.8363	−0.913077
$341$	−6.40246	−0.346713
$342$	−19.9179	−1.07704
$343$	−6.53964	−0.353107
$344$	−12.8047	−0.690382
$345$	−9.37025	−0.504477
$346$	18.3714	0.987653
$347$	26.4946	1.42230	0.711152	−	0.703038i	$-0.248174\pi$
$347$	26.4946	1.42230	0.711152	+	0.703038i	$0.248174\pi$
$348$	2.63760	0.141390
$349$	8.61542	0.461173	0.230587	−	0.973052i	$-0.425936\pi$
$349$	8.61542	0.461173	0.230587	+	0.973052i	$0.425936\pi$
$350$	0.386364	0.0206520
$351$	−3.68199	−0.196530
$352$	6.18105	0.329451
$353$	22.8950	1.21858	0.609290	−	0.792947i	$-0.291455\pi$
$353$	22.8950	1.21858	0.609290	+	0.792947i	$0.291455\pi$
$354$	−5.83443	−0.310097
$355$	18.4219	0.977732
$356$	21.7199	1.15115
$357$	0.792241	0.0419299
$358$	26.1450	1.38181
$359$	−7.33967	−0.387373	−0.193687	−	0.981063i	$-0.562044\pi$
$359$	−7.33967	−0.387373	−0.193687	+	0.981063i	$0.562044\pi$
$360$	13.7204	0.723129
$361$	−7.14852	−0.376238
$362$	16.0718	0.844716
$363$	6.64431	0.348736
$364$	−1.47221	−0.0771646
$365$	11.6446	0.609509
$366$	18.0712	0.944597
$367$	−3.65033	−0.190546	−0.0952729	−	0.995451i	$-0.530372\pi$
$367$	−3.65033	−0.190546	−0.0952729	+	0.995451i	$0.530372\pi$
$368$	−3.85081	−0.200737
$369$	−17.8106	−0.927181
$370$	24.0452	1.25005
$371$	5.71091	0.296496
$372$	13.3893	0.694203
$373$	−7.29087	−0.377507	−0.188754	−	0.982024i	$-0.560445\pi$
$373$	−7.29087	−0.377507	−0.188754	+	0.982024i	$0.560445\pi$
$374$	5.58845	0.288972
$375$	7.64379	0.394724
$376$	22.1794	1.14382
$377$	1.28482	0.0661714
$378$	−3.94806	−0.203066
$379$	−20.6090	−1.05861	−0.529307	−	0.848431i	$-0.677548\pi$
$379$	−20.6090	−1.05861	−0.529307	+	0.848431i	$0.677548\pi$
$380$	−22.9945	−1.17959
$381$	6.47623	0.331787
$382$	−5.40604	−0.276597
$383$	7.33479	0.374790	0.187395	−	0.982285i	$-0.439996\pi$
$383$	7.33479	0.374790	0.187395	+	0.982285i	$0.439996\pi$
$384$	−11.1738	−0.570211
$385$	1.00387	0.0511617
$386$	−54.2682	−2.76218
$387$	−13.1920	−0.670586
$388$	−34.0266	−1.72744
$389$	−24.5127	−1.24284	−0.621422	−	0.783476i	$-0.713445\pi$
$389$	−24.5127	−1.24284	−0.621422	+	0.783476i	$0.713445\pi$
$390$	−3.22064	−0.163084
$391$	−16.5631	−0.837634
$392$	16.8452	0.850809
$393$	−2.75437	−0.138939
$394$	34.1950	1.72272
$395$	−12.1628	−0.611975
$396$	−7.79800	−0.391864
$397$	−15.4405	−0.774937	−0.387468	−	0.921883i	$-0.626650\pi$
$397$	−15.4405	−0.774937	−0.387468	+	0.921883i	$0.626650\pi$
$398$	−52.9870	−2.65600
$399$	1.08202	0.0541687
$400$	0.211161	0.0105581
$401$	4.90811	0.245099	0.122550	−	0.992462i	$-0.460893\pi$
$401$	4.90811	0.245099	0.122550	+	0.992462i	$0.460893\pi$
$402$	20.7421	1.03452
$403$	6.52214	0.324891
$404$	4.90501	0.244033
$405$	11.3033	0.561665
$406$	1.37766	0.0683720
$407$	−4.85196	−0.240503
$408$	−4.14929	−0.205421
$409$	−33.9660	−1.67951	−0.839755	−	0.542965i	$-0.817302\pi$
$409$	−33.9660	−1.67951	−0.839755	+	0.542965i	$0.817302\pi$
$410$	−33.8232	−1.67041
$411$	−1.06971	−0.0527650
$412$	13.1832	0.649490
$413$	−1.85257	−0.0911590
$414$	38.0182	1.86849
$415$	18.2771	0.897187
$416$	−6.29659	−0.308716
$417$	13.3815	0.655296
$418$	7.63253	0.373319
$419$	4.94290	0.241477	0.120738	−	0.992684i	$-0.461474\pi$
$419$	4.94290	0.241477	0.120738	+	0.992684i	$0.461474\pi$
$420$	−2.09936	−0.102438
$421$	27.0318	1.31745	0.658726	−	0.752383i	$-0.271096\pi$
$421$	27.0318	1.31745	0.658726	+	0.752383i	$0.271096\pi$
$422$	−8.98476	−0.437371
$423$	22.8503	1.11102
$424$	−29.9104	−1.45258
$425$	0.908250	0.0440566
$426$	12.7876	0.619562
$427$	5.73803	0.277683
$428$	−61.1274	−2.95470
$429$	0.649876	0.0313763
$430$	−25.0522	−1.20813
$431$	−22.7260	−1.09467	−0.547336	−	0.836913i	$-0.684358\pi$
$431$	−22.7260	−1.09467	−0.547336	+	0.836913i	$0.684358\pi$
$432$	−2.15775	−0.103815
$433$	34.1349	1.64042	0.820210	−	0.572063i	$-0.193857\pi$
$433$	34.1349	1.64042	0.820210	+	0.572063i	$0.193857\pi$
$434$	6.99343	0.335695
$435$	1.83214	0.0878443
$436$	46.4080	2.22254
$437$	−22.6214	−1.08213
$438$	8.08317	0.386229
$439$	17.8950	0.854082	0.427041	−	0.904232i	$-0.359556\pi$
$439$	17.8950	0.854082	0.427041	+	0.904232i	$0.359556\pi$
$440$	−5.25765	−0.250649
$441$	17.3547	0.826413
$442$	−5.69291	−0.270784
$443$	12.1154	0.575619	0.287809	−	0.957688i	$-0.407073\pi$
$443$	12.1154	0.575619	0.287809	+	0.957688i	$0.407073\pi$
$444$	10.1468	0.481544
$445$	15.0871	0.715199
$446$	11.6184	0.550146
$447$	9.32256	0.440942
$448$	−6.19514	−0.292693
$449$	−15.1841	−0.716580	−0.358290	−	0.933610i	$-0.616640\pi$
$449$	−15.1841	−0.716580	−0.358290	+	0.933610i	$0.616640\pi$
$450$	−2.08475	−0.0982761
$451$	6.82499	0.321376
$452$	−19.2254	−0.904287
$453$	7.19469	0.338036
$454$	28.5075	1.33792
$455$	−1.02263	−0.0479416
$456$	−5.66697	−0.265380
$457$	−11.4935	−0.537644	−0.268822	−	0.963190i	$-0.586634\pi$
$457$	−11.4935	−0.537644	−0.268822	+	0.963190i	$0.586634\pi$
$458$	−33.8541	−1.58190
$459$	−9.28094	−0.433197
$460$	43.8906	2.04641
$461$	25.4336	1.18456	0.592279	−	0.805733i	$-0.298228\pi$
$461$	25.4336	1.18456	0.592279	+	0.805733i	$0.298228\pi$
$462$	0.696836	0.0324198
$463$	−2.80744	−0.130473	−0.0652363	−	0.997870i	$-0.520780\pi$
$463$	−2.80744	−0.130473	−0.0652363	+	0.997870i	$0.520780\pi$
$464$	0.752937	0.0349542
$465$	9.30052	0.431301
$466$	29.4644	1.36491
$467$	−42.6785	−1.97492	−0.987462	−	0.157855i	$-0.949542\pi$
$467$	−42.6785	−1.97492	−0.987462	+	0.157855i	$0.949542\pi$
$468$	7.94377	0.367201
$469$	6.58612	0.304119
$470$	43.3938	2.00161
$471$	2.64114	0.121697
$472$	9.70266	0.446601
$473$	5.05515	0.232436
$474$	−8.44282	−0.387792
$475$	1.24046	0.0569162
$476$	−3.71089	−0.170088
$477$	−30.8151	−1.41092
$478$	25.8517	1.18243
$479$	0.788791	0.0360408	0.0180204	−	0.999838i	$-0.494264\pi$
$479$	0.788791	0.0360408	0.0180204	+	0.999838i	$0.494264\pi$
$480$	−8.97889	−0.409829
$481$	4.94265	0.225366
$482$	59.4410	2.70746
$483$	−2.06530	−0.0939742
$484$	−31.1222	−1.41465
$485$	−23.6357	−1.07324
$486$	32.7939	1.48756
$487$	−15.1504	−0.686532	−0.343266	−	0.939238i	$-0.611533\pi$
$487$	−15.1504	−0.686532	−0.343266	+	0.939238i	$0.611533\pi$
$488$	−30.0524	−1.36041
$489$	4.62712	0.209246
$490$	32.9574	1.48886
$491$	−4.39974	−0.198558	−0.0992788	−	0.995060i	$-0.531654\pi$
$491$	−4.39974	−0.198558	−0.0992788	+	0.995060i	$0.531654\pi$
$492$	−14.2729	−0.643473
$493$	3.23855	0.145857
$494$	−7.77520	−0.349823
$495$	−5.41668	−0.243462
$496$	3.82215	0.171620
$497$	4.06037	0.182132
$498$	12.6871	0.568522
$499$	27.9112	1.24948	0.624738	−	0.780835i	$-0.285206\pi$
$499$	27.9112	1.24948	0.624738	+	0.780835i	$0.285206\pi$
$500$	−35.8038	−1.60119
$501$	−2.76701	−0.123621
$502$	22.5298	1.00556
$503$	−17.3958	−0.775641	−0.387821	−	0.921735i	$-0.626772\pi$
$503$	−17.3958	−0.775641	−0.387821	+	0.921735i	$0.626772\pi$
$504$	3.02412	0.134705
$505$	3.40714	0.151616
$506$	−14.5685	−0.647651
$507$	−0.662024	−0.0294015
$508$	−30.3349	−1.34589
$509$	29.3628	1.30148	0.650741	−	0.759300i	$-0.274458\pi$
$509$	29.3628	1.30148	0.650741	+	0.759300i	$0.274458\pi$
$510$	−8.11805	−0.359473
$511$	2.56660	0.113540
$512$	−6.60431	−0.291872
$513$	−12.6756	−0.559642
$514$	−65.8429	−2.90421
$515$	9.15736	0.403522
$516$	−10.5717	−0.465394
$517$	−8.75620	−0.385097
$518$	5.29981	0.232860
$519$	5.38507	0.236378
$520$	5.35593	0.234873
$521$	−0.820022	−0.0359258	−0.0179629	−	0.999839i	$-0.505718\pi$
$521$	−0.820022	−0.0359258	−0.0179629	+	0.999839i	$0.505718\pi$
$522$	−7.43359	−0.325359
$523$	38.8256	1.69773	0.848863	−	0.528614i	$-0.177288\pi$
$523$	38.8256	1.69773	0.848863	+	0.528614i	$0.177288\pi$
$524$	12.9016	0.563607
$525$	0.113252	0.00494271
$526$	10.3499	0.451278
$527$	16.4399	0.716133
$528$	0.380845	0.0165742
$529$	20.1785	0.877327
$530$	−58.5194	−2.54192
$531$	9.99613	0.433795
$532$	−5.06822	−0.219735
$533$	−6.95257	−0.301149
$534$	10.4728	0.453202
$535$	−42.4605	−1.83573
$536$	−34.4942	−1.48992
$537$	7.66367	0.330712
$538$	54.5039	2.34983
$539$	−6.65029	−0.286448
$540$	24.5935	1.05834
$541$	7.46997	0.321159	0.160579	−	0.987023i	$-0.448664\pi$
$541$	7.46997	0.321159	0.160579	+	0.987023i	$0.448664\pi$
$542$	−26.5030	−1.13840
$543$	4.71100	0.202169
$544$	−15.8714	−0.680479
$545$	32.2361	1.38084
$546$	−0.709862	−0.0303793
$547$	−14.9615	−0.639707	−0.319854	−	0.947467i	$-0.603634\pi$
$547$	−14.9615	−0.639707	−0.319854	+	0.947467i	$0.603634\pi$
$548$	5.01058	0.214041
$549$	−30.9614	−1.32140
$550$	0.798875	0.0340641
$551$	4.42310	0.188431
$552$	10.8168	0.460394
$553$	−2.68079	−0.113999
$554$	−6.13418	−0.260616
$555$	7.04819	0.299179
$556$	−62.6795	−2.65820
$557$	10.2430	0.434010	0.217005	−	0.976170i	$-0.430371\pi$
$557$	10.2430	0.434010	0.217005	+	0.976170i	$0.430371\pi$
$558$	−37.7353	−1.59746
$559$	−5.14965	−0.217807
$560$	−0.599289	−0.0253246
$561$	1.63810	0.0691605
$562$	−19.2638	−0.812593
$563$	−33.3942	−1.40740	−0.703699	−	0.710498i	$-0.748470\pi$
$563$	−33.3942	−1.40740	−0.703699	+	0.710498i	$0.748470\pi$
$564$	18.3116	0.771058
$565$	−13.3544	−0.561825
$566$	−0.0150986	−0.000634643 0
$567$	2.49136	0.104627
$568$	−21.2658	−0.892293
$569$	41.0744	1.72193	0.860965	−	0.508665i	$-0.169861\pi$
$569$	41.0744	1.72193	0.860965	+	0.508665i	$0.169861\pi$
$570$	−11.0874	−0.464399
$571$	4.79821	0.200799	0.100399	−	0.994947i	$-0.467988\pi$
$571$	4.79821	0.200799	0.100399	+	0.994947i	$0.467988\pi$
$572$	−3.04404	−0.127278
$573$	−1.58463	−0.0661988
$574$	−7.45496	−0.311164
$575$	−2.36772	−0.0987408
$576$	33.4279	1.39283
$577$	11.1605	0.464618	0.232309	−	0.972642i	$-0.425372\pi$
$577$	11.1605	0.464618	0.232309	+	0.972642i	$0.425372\pi$
$578$	24.0452	1.00015
$579$	−15.9072	−0.661081
$580$	−8.58181	−0.356340
$581$	4.02845	0.167128
$582$	−16.4068	−0.680082
$583$	11.8083	0.489050
$584$	−13.4423	−0.556247
$585$	5.51793	0.228138
$586$	69.2532	2.86082
$587$	−36.7298	−1.51600	−0.758000	−	0.652255i	$-0.773823\pi$
$587$	−36.7298	−1.51600	−0.758000	+	0.652255i	$0.773823\pi$
$588$	13.9076	0.573539
$589$	22.4531	0.925163
$590$	18.9832	0.781524
$591$	10.0233	0.412304
$592$	2.89653	0.119047
$593$	−16.4034	−0.673608	−0.336804	−	0.941575i	$-0.609346\pi$
$593$	−16.4034	−0.673608	−0.336804	+	0.941575i	$0.609346\pi$
$594$	−8.16329	−0.334944
$595$	−2.57767	−0.105674
$596$	−43.6673	−1.78868
$597$	−15.5317	−0.635668
$598$	14.8409	0.606888
$599$	12.5258	0.511789	0.255894	−	0.966705i	$-0.417630\pi$
$599$	12.5258	0.511789	0.255894	+	0.966705i	$0.417630\pi$
$600$	−0.593146	−0.0242151
$601$	−20.5503	−0.838266	−0.419133	−	0.907925i	$-0.637666\pi$
$601$	−20.5503	−0.838266	−0.419133	+	0.907925i	$0.637666\pi$
$602$	−5.52176	−0.225050
$603$	−35.5375	−1.44720
$604$	−33.7002	−1.37124
$605$	−21.6182	−0.878906
$606$	2.36508	0.0960746
$607$	1.73296	0.0703385	0.0351692	−	0.999381i	$-0.488803\pi$
$607$	1.73296	0.0703385	0.0351692	+	0.999381i	$0.488803\pi$
$608$	−21.6766	−0.879103
$609$	0.403822	0.0163637
$610$	−58.7972	−2.38063
$611$	8.91987	0.360859
$612$	20.0233	0.809394
$613$	−26.7138	−1.07896	−0.539480	−	0.841998i	$-0.681379\pi$
$613$	−26.7138	−1.07896	−0.539480	+	0.841998i	$0.681379\pi$
$614$	44.1321	1.78103
$615$	−9.91431	−0.399784
$616$	−1.15884	−0.0466909
$617$	−36.7980	−1.48143	−0.740716	−	0.671818i	$-0.765514\pi$
$617$	−36.7980	−1.48143	−0.740716	+	0.671818i	$0.765514\pi$
$618$	6.35661	0.255700
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−43.5640	−1.74957
$621$	24.1945	0.970893
$622$	8.87997	0.356054
$623$	3.32536	0.133228
$624$	−0.387964	−0.0155310
$625$	−23.0685	−0.922741
$626$	−22.0792	−0.882463
$627$	2.23726	0.0893476
$628$	−12.3712	−0.493664
$629$	12.4586	0.496757
$630$	5.91666	0.235725
$631$	34.4527	1.37154	0.685770	−	0.727818i	$-0.259465\pi$
$631$	34.4527	1.37154	0.685770	+	0.727818i	$0.259465\pi$
$632$	14.0404	0.558498
$633$	−2.63363	−0.104677
$634$	29.5834	1.17491
$635$	−21.0713	−0.836191
$636$	−24.6944	−0.979197
$637$	6.77460	0.268420
$638$	2.84855	0.112775
$639$	−21.9090	−0.866707
$640$	36.3556	1.43708
$641$	−31.9726	−1.26284	−0.631422	−	0.775440i	$-0.717528\pi$
$641$	−31.9726	−1.26284	−0.631422	+	0.775440i	$0.717528\pi$
$642$	−29.4741	−1.16325
$643$	−42.8443	−1.68962	−0.844808	−	0.535070i	$-0.820285\pi$
$643$	−42.8443	−1.68962	−0.844808	+	0.535070i	$0.820285\pi$
$644$	9.67393	0.381206
$645$	−7.34336	−0.289145
$646$	−19.5984	−0.771089
$647$	25.7157	1.01099	0.505494	−	0.862830i	$-0.331310\pi$
$647$	25.7157	1.01099	0.505494	+	0.862830i	$0.331310\pi$
$648$	−13.0483	−0.512584
$649$	−3.83051	−0.150361
$650$	−0.813808	−0.0319202
$651$	2.04993	0.0803430
$652$	−21.6736	−0.848804
$653$	−6.30137	−0.246592	−0.123296	−	0.992370i	$-0.539346\pi$
$653$	−6.30137	−0.246592	−0.123296	+	0.992370i	$0.539346\pi$
$654$	22.3768	0.875002
$655$	8.96173	0.350164
$656$	−4.07439	−0.159078
$657$	−13.8489	−0.540297
$658$	9.56443	0.372860
$659$	22.3576	0.870929	0.435464	−	0.900206i	$-0.356584\pi$
$659$	22.3576	0.870929	0.435464	+	0.900206i	$0.356584\pi$
$660$	−4.34079	−0.168965
$661$	1.14552	0.0445557	0.0222779	−	0.999752i	$-0.492908\pi$
$661$	1.14552	0.0445557	0.0222779	+	0.999752i	$0.492908\pi$
$662$	−47.9953	−1.86539
$663$	−1.66872	−0.0648076
$664$	−21.0986	−0.818786
$665$	−3.52050	−0.136519
$666$	−28.5968	−1.10810
$667$	−8.44257	−0.326898
$668$	12.9608	0.501468
$669$	3.40560	0.131668
$670$	−67.4875	−2.60727
$671$	11.8644	0.458019
$672$	−1.97904	−0.0763430
$673$	−13.0556	−0.503256	−0.251628	−	0.967824i	$-0.580966\pi$
$673$	−13.0556	−0.503256	−0.251628	+	0.967824i	$0.580966\pi$
$674$	66.3141	2.55432
$675$	−1.32672	−0.0510655
$676$	3.10095	0.119267
$677$	−22.5148	−0.865313	−0.432656	−	0.901559i	$-0.642424\pi$
$677$	−22.5148	−0.865313	−0.432656	+	0.901559i	$0.642424\pi$
$678$	−9.27001	−0.356013
$679$	−5.20953	−0.199923
$680$	13.5003	0.517713
$681$	8.35618	0.320209
$682$	14.4601	0.553707
$683$	−21.5390	−0.824166	−0.412083	−	0.911146i	$-0.635199\pi$
$683$	−21.5390	−0.824166	−0.412083	+	0.911146i	$0.635199\pi$
$684$	27.3472	1.04565
$685$	3.48047	0.132982
$686$	14.7700	0.563920
$687$	−9.92338	−0.378601
$688$	−3.01784	−0.115054
$689$	−12.0290	−0.458270
$690$	21.1630	0.805661
$691$	−33.4654	−1.27308	−0.636542	−	0.771242i	$-0.719636\pi$
$691$	−33.4654	−1.27308	−0.636542	+	0.771242i	$0.719636\pi$
$692$	−25.2239	−0.958867
$693$	−1.19389	−0.0453521
$694$	−59.8387	−2.27145
$695$	−43.5387	−1.65152
$696$	−2.11498	−0.0801680
$697$	−17.5248	−0.663801
$698$	−19.4582	−0.736503
$699$	8.63667	0.326669
$700$	−0.530476	−0.0200501
$701$	23.3959	0.883651	0.441826	−	0.897101i	$-0.354331\pi$
$701$	23.3959	0.883651	0.441826	+	0.897101i	$0.354331\pi$
$702$	8.31589	0.313863
$703$	17.0156	0.641754
$704$	−12.8095	−0.482777
$705$	12.7197	0.479051
$706$	−51.7091	−1.94610
$707$	0.750967	0.0282430
$708$	8.01064	0.301058
$709$	−10.3250	−0.387762	−0.193881	−	0.981025i	$-0.562108\pi$
$709$	−10.3250	−0.387762	−0.193881	+	0.981025i	$0.562108\pi$
$710$	−41.6063	−1.56146
$711$	14.4651	0.542483
$712$	−17.4162	−0.652701
$713$	−42.8572	−1.60501
$714$	−1.78930	−0.0669628
$715$	−2.11447	−0.0790765
$716$	−35.8969	−1.34153
$717$	7.57771	0.282995
$718$	16.5769	0.618643
$719$	21.4513	0.799998	0.399999	−	0.916516i	$-0.369010\pi$
$719$	21.4513	0.799998	0.399999	+	0.916516i	$0.369010\pi$
$720$	3.23366	0.120511
$721$	2.01837	0.0751682
$722$	16.1451	0.600860
$723$	17.4235	0.647986
$724$	−22.0665	−0.820096
$725$	0.462954	0.0171937
$726$	−15.0064	−0.556938
$727$	38.3306	1.42160	0.710802	−	0.703392i	$-0.248332\pi$
$727$	38.3306	1.42160	0.710802	+	0.703392i	$0.248332\pi$
$728$	1.18050	0.0437522
$729$	−6.13021	−0.227045
$730$	−26.2998	−0.973398
$731$	−12.9804	−0.480096
$732$	−24.8117	−0.917065
$733$	−51.3784	−1.89770	−0.948852	−	0.315720i	$-0.897754\pi$
$733$	−51.3784	−1.89770	−0.948852	+	0.315720i	$0.897754\pi$
$734$	8.24437	0.304305
$735$	9.66053	0.356334
$736$	41.3751	1.52511
$737$	13.6179	0.501623
$738$	40.2256	1.48073
$739$	34.1376	1.25577	0.627886	−	0.778305i	$-0.283920\pi$
$739$	34.1376	1.25577	0.627886	+	0.778305i	$0.283920\pi$
$740$	−33.0140	−1.21362
$741$	−2.27908	−0.0837242
$742$	−12.8983	−0.473510
$743$	23.6929	0.869210	0.434605	−	0.900621i	$-0.356888\pi$
$743$	23.6929	0.869210	0.434605	+	0.900621i	$0.356888\pi$
$744$	−10.7363	−0.393612
$745$	−30.3323	−1.11129
$746$	16.4666	0.602886
$747$	−21.7368	−0.795308
$748$	−7.67291	−0.280549
$749$	−9.35872	−0.341960
$750$	−17.2637	−0.630381
$751$	20.8709	0.761591	0.380795	−	0.924659i	$-0.375650\pi$
$751$	20.8709	0.761591	0.380795	+	0.924659i	$0.375650\pi$
$752$	5.22729	0.190620
$753$	6.60399	0.240663
$754$	−2.90179	−0.105677
$755$	−23.4090	−0.851940
$756$	5.42066	0.197147
$757$	1.07500	0.0390716	0.0195358	−	0.999809i	$-0.493781\pi$
$757$	1.07500	0.0390716	0.0195358	+	0.999809i	$0.493781\pi$
$758$	46.5460	1.69063
$759$	−4.27036	−0.155004
$760$	18.4383	0.668828
$761$	−33.7167	−1.22223	−0.611115	−	0.791542i	$-0.709279\pi$
$761$	−33.7167	−1.22223	−0.611115	+	0.791542i	$0.709279\pi$
$762$	−14.6267	−0.529871
$763$	7.10516	0.257224
$764$	7.42246	0.268535
$765$	13.9087	0.502868
$766$	−16.5658	−0.598547
$767$	3.90211	0.140897
$768$	7.95891	0.287193
$769$	4.70756	0.169759	0.0848794	−	0.996391i	$-0.472949\pi$
$769$	4.70756	0.169759	0.0848794	+	0.996391i	$0.472949\pi$
$770$	−2.26726	−0.0817063
$771$	−19.3000	−0.695072
$772$	74.5100	2.68167
$773$	−45.3372	−1.63066	−0.815332	−	0.578993i	$-0.803446\pi$
$773$	−45.3372	−1.63066	−0.815332	+	0.578993i	$0.803446\pi$
$774$	29.7944	1.07094
$775$	2.35010	0.0844181
$776$	27.2844	0.979454
$777$	1.55349	0.0557312
$778$	55.3626	1.98485
$779$	−23.9349	−0.857556
$780$	4.42193	0.158330
$781$	8.39551	0.300415
$782$	37.4083	1.33772
$783$	−4.73069	−0.169061
$784$	3.97010	0.141789
$785$	−8.59332	−0.306709
$786$	6.22081	0.221889
$787$	−29.0639	−1.03602	−0.518008	−	0.855376i	$-0.673326\pi$
$787$	−29.0639	−1.03602	−0.518008	+	0.855376i	$0.673326\pi$
$788$	−46.9496	−1.67251
$789$	3.03379	0.108006
$790$	27.4699	0.977336
$791$	−2.94345	−0.104657
$792$	6.25289	0.222187
$793$	−12.0861	−0.429192
$794$	34.8728	1.23759
$795$	−17.1533	−0.608365
$796$	72.7509	2.57859
$797$	−21.9539	−0.777648	−0.388824	−	0.921312i	$-0.627119\pi$
$797$	−21.9539	−0.777648	−0.388824	+	0.921312i	$0.627119\pi$
$798$	−2.44377	−0.0865085
$799$	22.4837	0.795416
$800$	−2.26883	−0.0802153
$801$	−17.9430	−0.633986
$802$	−11.0851	−0.391428
$803$	5.30688	0.187276
$804$	−28.4788	−1.00437
$805$	6.71974	0.236840
$806$	−14.7304	−0.518857
$807$	15.9763	0.562392
$808$	−3.93312	−0.138367
$809$	40.8994	1.43795	0.718974	−	0.695037i	$-0.244612\pi$
$809$	40.8994	1.43795	0.718974	+	0.695037i	$0.244612\pi$
$810$	−25.5288	−0.896990
$811$	1.97070	0.0692006	0.0346003	−	0.999401i	$-0.488984\pi$
$811$	1.97070	0.0692006	0.0346003	+	0.999401i	$0.488984\pi$
$812$	−1.89152	−0.0663792
$813$	−7.76862	−0.272457
$814$	10.9583	0.384087
$815$	−15.0550	−0.527354
$816$	−0.977914	−0.0342338
$817$	−17.7282	−0.620230
$818$	76.7131	2.68221
$819$	1.21621	0.0424977
$820$	46.4390	1.62172
$821$	−40.7386	−1.42179	−0.710893	−	0.703300i	$-0.751709\pi$
$821$	−40.7386	−1.42179	−0.710893	+	0.703300i	$0.751709\pi$
$822$	2.41598	0.0842668
$823$	−52.8130	−1.84094	−0.920472	−	0.390808i	$-0.872195\pi$
$823$	−52.8130	−1.84094	−0.920472	+	0.390808i	$0.872195\pi$
$824$	−10.5710	−0.368260
$825$	0.234168	0.00815267
$826$	4.18408	0.145583
$827$	15.5154	0.539524	0.269762	−	0.962927i	$-0.413055\pi$
$827$	15.5154	0.539524	0.269762	+	0.962927i	$0.413055\pi$
$828$	−52.1988	−1.81403
$829$	30.0095	1.04227	0.521136	−	0.853474i	$-0.325509\pi$
$829$	30.0095	1.04227	0.521136	+	0.853474i	$0.325509\pi$
$830$	−41.2793	−1.43282
$831$	−1.79806	−0.0623741
$832$	13.0490	0.452392
$833$	17.0763	0.591657
$834$	−30.2225	−1.04652
$835$	9.00287	0.311557
$836$	−10.4794	−0.362438
$837$	−24.0145	−0.830062
$838$	−11.1637	−0.385643
$839$	−31.4134	−1.08451	−0.542256	−	0.840213i	$-0.682430\pi$
$839$	−31.4134	−1.08451	−0.542256	+	0.840213i	$0.682430\pi$
$840$	1.68338	0.0580823
$841$	−27.3492	−0.943078
$842$	−61.0521	−2.10400
$843$	−5.64663	−0.194480
$844$	12.3360	0.424623
$845$	2.15399	0.0740995
$846$	−51.6079	−1.77432
$847$	−4.76487	−0.163723
$848$	−7.04934	−0.242075
$849$	−0.00442574	−0.000151891 0
$850$	−2.05131	−0.0703593
$851$	−32.4783	−1.11334
$852$	−17.5573	−0.601504
$853$	−32.3065	−1.10616	−0.553078	−	0.833130i	$-0.686547\pi$
$853$	−32.3065	−1.10616	−0.553078	+	0.833130i	$0.686547\pi$
$854$	−12.9595	−0.443465
$855$	18.9960	0.649650
$856$	49.0154	1.67531
$857$	−17.4756	−0.596957	−0.298478	−	0.954416i	$-0.596479\pi$
$857$	−17.4756	−0.596957	−0.298478	+	0.954416i	$0.596479\pi$
$858$	−1.46776	−0.0501086
$859$	−20.5405	−0.700833	−0.350417	−	0.936594i	$-0.613960\pi$
$859$	−20.5405	−0.700833	−0.350417	+	0.936594i	$0.613960\pi$
$860$	34.3966	1.17291
$861$	−2.18521	−0.0744718
$862$	51.3272	1.74821
$863$	−47.4772	−1.61614	−0.808071	−	0.589085i	$-0.799488\pi$
$863$	−47.4772	−1.61614	−0.808071	+	0.589085i	$0.799488\pi$
$864$	23.1840	0.788736
$865$	−17.5211	−0.595735
$866$	−77.0947	−2.61978
$867$	7.04819	0.239369
$868$	−9.60194	−0.325911
$869$	−5.54301	−0.188034
$870$	−4.13794	−0.140289
$871$	−13.8725	−0.470051
$872$	−37.2126	−1.26018
$873$	28.1097	0.951369
$874$	51.0911	1.72818
$875$	−5.48163	−0.185313
$876$	−11.0981	−0.374972
$877$	−12.6571	−0.427399	−0.213700	−	0.976899i	$-0.568551\pi$
$877$	−12.6571	−0.427399	−0.213700	+	0.976899i	$0.568551\pi$
$878$	−40.4164	−1.36399
$879$	20.2996	0.684690
$880$	−1.23913	−0.0417712
$881$	−44.6197	−1.50328	−0.751638	−	0.659575i	$-0.770736\pi$
$881$	−44.6197	−1.50328	−0.751638	+	0.659575i	$0.770736\pi$
$882$	−39.1960	−1.31980
$883$	−13.5376	−0.455576	−0.227788	−	0.973711i	$-0.573149\pi$
$883$	−13.5376	−0.455576	−0.227788	+	0.973711i	$0.573149\pi$
$884$	7.81633	0.262892
$885$	5.56438	0.187045
$886$	−27.3629	−0.919274
$887$	5.56424	0.186829	0.0934145	−	0.995627i	$-0.470222\pi$
$887$	5.56424	0.186829	0.0934145	+	0.995627i	$0.470222\pi$
$888$	−8.13626	−0.273035
$889$	−4.64433	−0.155766
$890$	−34.0747	−1.14219
$891$	5.15131	0.172575
$892$	−15.9520	−0.534112
$893$	30.7075	1.02759
$894$	−21.0553	−0.704193
$895$	−24.9348	−0.833480
$896$	8.01314	0.267700
$897$	4.35018	0.145248
$898$	34.2936	1.14439
$899$	8.37975	0.279480
$900$	2.86235	0.0954117
$901$	−30.3207	−1.01013
$902$	−15.4144	−0.513244
$903$	−1.61855	−0.0538620
$904$	15.4160	0.512729
$905$	−15.3279	−0.509518
$906$	−16.2494	−0.539850
$907$	52.0694	1.72894	0.864468	−	0.502688i	$-0.167656\pi$
$907$	52.0694	1.72894	0.864468	+	0.502688i	$0.167656\pi$
$908$	−39.1407	−1.29893
$909$	−4.05208	−0.134399
$910$	2.30964	0.0765637
$911$	4.60804	0.152671	0.0763356	−	0.997082i	$-0.475678\pi$
$911$	4.60804	0.152671	0.0763356	+	0.997082i	$0.475678\pi$
$912$	−1.33560	−0.0442263
$913$	8.32952	0.275667
$914$	25.9584	0.858627
$915$	−17.2348	−0.569764
$916$	46.4815	1.53579
$917$	1.97525	0.0652286
$918$	20.9613	0.691825
$919$	−19.9747	−0.658903	−0.329452	−	0.944172i	$-0.606864\pi$
$919$	−19.9747	−0.658903	−0.329452	+	0.944172i	$0.606864\pi$
$920$	−35.1940	−1.16031
$921$	12.9361	0.426259
$922$	−57.4424	−1.89176
$923$	−8.55245	−0.281507
$924$	−0.956752	−0.0314748
$925$	1.78097	0.0585579
$926$	6.34067	0.208367
$927$	−10.8908	−0.357700
$928$	−8.08996	−0.265566
$929$	−40.2393	−1.32021	−0.660105	−	0.751173i	$-0.729488\pi$
$929$	−40.2393	−1.32021	−0.660105	+	0.751173i	$0.729488\pi$
$930$	−21.0055	−0.688797
$931$	23.3222	0.764355
$932$	−40.4545	−1.32513
$933$	2.60291	0.0852155
$934$	96.3905	3.15399
$935$	−5.32978	−0.174303
$936$	−6.36977	−0.208202
$937$	42.7360	1.39613	0.698063	−	0.716036i	$-0.254046\pi$
$937$	42.7360	1.39613	0.698063	+	0.716036i	$0.254046\pi$
$938$	−14.8749	−0.485683
$939$	−6.47190	−0.211203
$940$	−59.5795	−1.94327
$941$	48.8849	1.59360	0.796801	−	0.604242i	$-0.206524\pi$
$941$	48.8849	1.59360	0.796801	+	0.604242i	$0.206524\pi$
$942$	−5.96508	−0.194353
$943$	45.6856	1.48773
$944$	2.28674	0.0744272
$945$	3.76532	0.122486
$946$	−11.4172	−0.371205
$947$	−37.2706	−1.21113	−0.605566	−	0.795795i	$-0.707053\pi$
$947$	−37.2706	−1.21113	−0.605566	+	0.795795i	$0.707053\pi$
$948$	11.5919	0.376489
$949$	−5.40608	−0.175489
$950$	−2.80161	−0.0908963
$951$	8.67155	0.281194
$952$	2.97560	0.0964399
$953$	38.9383	1.26133	0.630667	−	0.776053i	$-0.282781\pi$
$953$	38.9383	1.26133	0.630667	+	0.776053i	$0.282781\pi$
$954$	69.5967	2.25328
$955$	5.15582	0.166838
$956$	−35.4943	−1.14797
$957$	0.834971	0.0269908
$958$	−1.78151	−0.0575578
$959$	0.767129	0.0247719
$960$	18.6077	0.600562
$961$	11.5383	0.372202
$962$	−11.1631	−0.359913
$963$	50.4980	1.62727
$964$	−81.6122	−2.62855
$965$	51.7564	1.66610
$966$	4.66453	0.150079
$967$	−23.1829	−0.745511	−0.372755	−	0.927930i	$-0.621587\pi$
$967$	−23.1829	−0.745511	−0.372755	+	0.927930i	$0.621587\pi$
$968$	24.9556	0.802103
$969$	−5.74472	−0.184547
$970$	53.3818	1.71399
$971$	31.2595	1.00317	0.501583	−	0.865110i	$-0.332751\pi$
$971$	31.2595	1.00317	0.501583	+	0.865110i	$0.332751\pi$
$972$	−45.0258	−1.44420
$973$	−9.59636	−0.307645
$974$	34.2177	1.09641
$975$	−0.238545	−0.00763955
$976$	−7.08281	−0.226715
$977$	15.1754	0.485503	0.242752	−	0.970088i	$-0.421950\pi$
$977$	15.1754	0.485503	0.242752	+	0.970088i	$0.421950\pi$
$978$	−10.4505	−0.334169
$979$	6.87575	0.219750
$980$	−45.2503	−1.44547
$981$	−38.3382	−1.22404
$982$	9.93694	0.317100
$983$	−12.1188	−0.386528	−0.193264	−	0.981147i	$-0.561907\pi$
$983$	−12.1188	−0.386528	−0.193264	+	0.981147i	$0.561907\pi$
$984$	11.4448	0.364848
$985$	−32.6123	−1.03911
$986$	−7.31434	−0.232936
$987$	2.80354	0.0892378
$988$	10.6753	0.339627
$989$	33.8385	1.07600
$990$	12.2337	0.388813
$991$	−15.4043	−0.489334	−0.244667	−	0.969607i	$-0.578679\pi$
$991$	−15.4043	−0.489334	−0.244667	+	0.969607i	$0.578679\pi$
$992$	−41.0672	−1.30389
$993$	−14.0685	−0.446450
$994$	−9.17045	−0.290869
$995$	50.5345	1.60205
$996$	−17.4193	−0.551952
$997$	−40.0519	−1.26846	−0.634229	−	0.773145i	$-0.718682\pi$
$997$	−40.0519	−1.26846	−0.634229	+	0.773145i	$0.718682\pi$
$998$	−63.0381	−1.99544
$999$	−18.1988	−0.575785

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.c.1.20	✓	151

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.c.1.20	✓	151	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.25853 q^{2} -0.662024 q^{3} +3.10095 q^{4} +2.15399 q^{5} +1.49520 q^{6} +0.474761 q^{7} -2.48652 q^{8} -2.56172 q^{9} +O(q^{10})\)	\(q-2.25853 q^{2} -0.662024 q^{3} +3.10095 q^{4} +2.15399 q^{5} +1.49520 q^{6} +0.474761 q^{7} -2.48652 q^{8} -2.56172 q^{9} -4.86485 q^{10} +0.981650 q^{11} -2.05290 q^{12} -1.00000 q^{13} -1.07226 q^{14} -1.42599 q^{15} -0.586027 q^{16} -2.52063 q^{17} +5.78572 q^{18} -3.44260 q^{19} +6.67941 q^{20} -0.314303 q^{21} -2.21708 q^{22} +6.57104 q^{23} +1.64613 q^{24} -0.360327 q^{25} +2.25853 q^{26} +3.68199 q^{27} +1.47221 q^{28} -1.28482 q^{29} +3.22064 q^{30} -6.52214 q^{31} +6.29659 q^{32} -0.649876 q^{33} +5.69291 q^{34} +1.02263 q^{35} -7.94377 q^{36} -4.94265 q^{37} +7.77520 q^{38} +0.662024 q^{39} -5.35593 q^{40} +6.95257 q^{41} +0.709862 q^{42} +5.14965 q^{43} +3.04404 q^{44} -5.51793 q^{45} -14.8409 q^{46} -8.91987 q^{47} +0.387964 q^{48} -6.77460 q^{49} +0.813808 q^{50} +1.66872 q^{51} -3.10095 q^{52} +12.0290 q^{53} -8.31589 q^{54} +2.11447 q^{55} -1.18050 q^{56} +2.27908 q^{57} +2.90179 q^{58} -3.90211 q^{59} -4.42193 q^{60} +12.0861 q^{61} +14.7304 q^{62} -1.21621 q^{63} -13.0490 q^{64} -2.15399 q^{65} +1.46776 q^{66} +13.8725 q^{67} -7.81633 q^{68} -4.35018 q^{69} -2.30964 q^{70} +8.55245 q^{71} +6.36977 q^{72} +5.40608 q^{73} +11.1631 q^{74} +0.238545 q^{75} -10.6753 q^{76} +0.466049 q^{77} -1.49520 q^{78} -5.64662 q^{79} -1.26230 q^{80} +5.24760 q^{81} -15.7026 q^{82} +8.48522 q^{83} -0.974636 q^{84} -5.42941 q^{85} -11.6306 q^{86} +0.850579 q^{87} -2.44089 q^{88} +7.00428 q^{89} +12.4624 q^{90} -0.474761 q^{91} +20.3764 q^{92} +4.31781 q^{93} +20.1458 q^{94} -7.41532 q^{95} -4.16849 q^{96} -10.9730 q^{97} +15.3006 q^{98} -2.51472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9 - 3 * q^10 - 27 * q^11 - 52 * q^12 - 151 * q^13 - 9 * q^14 - 14 * q^15 + 143 * q^16 - 111 * q^17 - 37 * q^18 - 17 * q^19 - 107 * q^20 - 29 * q^21 - 16 * q^22 - 47 * q^23 - 46 * q^24 + 122 * q^25 + 13 * q^26 - 55 * q^27 - 44 * q^28 + 37 * q^29 - 14 * q^30 - 27 * q^31 - 86 * q^32 - 94 * q^33 - 10 * q^34 - 47 * q^35 + 124 * q^36 - 59 * q^37 - 80 * q^38 + 16 * q^39 + 5 * q^40 - 129 * q^41 - 77 * q^42 - 11 * q^43 - 99 * q^44 - 122 * q^45 - 17 * q^46 - 130 * q^47 - 111 * q^48 + 99 * q^49 - 72 * q^50 + 15 * q^51 - 151 * q^52 - 43 * q^53 - 49 * q^54 - 40 * q^55 - 50 * q^56 - 85 * q^57 - 73 * q^58 - 74 * q^59 - 43 * q^60 - 7 * q^61 - 110 * q^62 - 70 * q^63 + 141 * q^64 + 43 * q^65 - 16 * q^66 - 39 * q^67 - 222 * q^68 + 19 * q^69 - 52 * q^70 - 72 * q^71 - 106 * q^72 - 143 * q^73 + 20 * q^74 - 73 * q^75 - 88 * q^76 - 86 * q^77 + 17 * q^78 + 10 * q^79 - 239 * q^80 + 103 * q^81 - 96 * q^82 - 96 * q^83 - 75 * q^84 - 24 * q^85 - 109 * q^86 - 65 * q^87 - 45 * q^88 - 237 * q^89 - 79 * q^90 + 18 * q^91 - 153 * q^92 - 137 * q^93 - 23 * q^94 + 10 * q^95 - 109 * q^96 - 160 * q^97 - 119 * q^98 - 86 * q^99

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