LMFDB - Embedded newform 6047.2.a.a.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6047,2,Mod(1,6047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6047, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6047 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6047.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:	$48.2855381023$
Analytic rank:	$1$
Dimension:	$217$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	6047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.65451 q^{2} -2.99510 q^{3} +5.04641 q^{4} -2.08040 q^{5} +7.95052 q^{6} -1.11556 q^{7} -8.08670 q^{8} +5.97065 q^{9} +O(q^{10})$	\(q-2.65451 q^{2} -2.99510 q^{3} +5.04641 q^{4} -2.08040 q^{5} +7.95052 q^{6} -1.11556 q^{7} -8.08670 q^{8} +5.97065 q^{9} +5.52243 q^{10} +3.38425 q^{11} -15.1145 q^{12} -3.23622 q^{13} +2.96127 q^{14} +6.23101 q^{15} +11.3734 q^{16} +4.29870 q^{17} -15.8491 q^{18} +0.715567 q^{19} -10.4985 q^{20} +3.34123 q^{21} -8.98352 q^{22} +0.981717 q^{23} +24.2205 q^{24} -0.671946 q^{25} +8.59058 q^{26} -8.89739 q^{27} -5.62959 q^{28} -2.22960 q^{29} -16.5402 q^{30} +0.355075 q^{31} -14.0173 q^{32} -10.1362 q^{33} -11.4109 q^{34} +2.32082 q^{35} +30.1303 q^{36} +11.1937 q^{37} -1.89948 q^{38} +9.69283 q^{39} +16.8236 q^{40} -1.13870 q^{41} -8.86932 q^{42} -6.13987 q^{43} +17.0783 q^{44} -12.4213 q^{45} -2.60597 q^{46} -2.54799 q^{47} -34.0645 q^{48} -5.75552 q^{49} +1.78369 q^{50} -12.8750 q^{51} -16.3313 q^{52} -12.6824 q^{53} +23.6182 q^{54} -7.04059 q^{55} +9.02124 q^{56} -2.14320 q^{57} +5.91850 q^{58} +12.1366 q^{59} +31.4442 q^{60} +7.65047 q^{61} -0.942549 q^{62} -6.66064 q^{63} +14.4624 q^{64} +6.73263 q^{65} +26.9066 q^{66} -9.28979 q^{67} +21.6930 q^{68} -2.94034 q^{69} -6.16063 q^{70} -3.24012 q^{71} -48.2828 q^{72} -4.55854 q^{73} -29.7138 q^{74} +2.01255 q^{75} +3.61104 q^{76} -3.77535 q^{77} -25.7297 q^{78} -2.45178 q^{79} -23.6612 q^{80} +8.73667 q^{81} +3.02268 q^{82} +7.49212 q^{83} +16.8612 q^{84} -8.94300 q^{85} +16.2983 q^{86} +6.67790 q^{87} -27.3675 q^{88} +11.6238 q^{89} +32.9725 q^{90} +3.61022 q^{91} +4.95414 q^{92} -1.06349 q^{93} +6.76364 q^{94} -1.48866 q^{95} +41.9834 q^{96} -10.6560 q^{97} +15.2781 q^{98} +20.2062 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9}+O(q^{10}) $ 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9	\( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9} - 46 q^{10} - 32 q^{11} - 72 q^{12} - 80 q^{13} - 22 q^{14} - 43 q^{15} + 122 q^{16} - 61 q^{17} - 88 q^{18} - 43 q^{19} - 41 q^{20} - 61 q^{21} - 93 q^{22} - 60 q^{23} - 41 q^{24} + 26 q^{25} - 9 q^{26} - 93 q^{27} - 126 q^{28} - 47 q^{29} - 36 q^{30} - 100 q^{31} - 114 q^{32} - 133 q^{33} - 75 q^{34} - 37 q^{35} + 75 q^{36} - 264 q^{37} - 35 q^{38} - 47 q^{39} - 118 q^{40} - 72 q^{41} - 64 q^{42} - 107 q^{43} - 59 q^{44} - 69 q^{45} - 111 q^{46} - 54 q^{47} - 135 q^{48} + 33 q^{49} - 42 q^{50} - 26 q^{51} - 173 q^{52} - 103 q^{53} - 28 q^{54} - 78 q^{55} - 44 q^{56} - 205 q^{57} - 189 q^{58} - 38 q^{59} - 105 q^{60} - 108 q^{61} - 14 q^{62} - 116 q^{63} + 39 q^{64} - 146 q^{65} + 5 q^{66} - 206 q^{67} - 62 q^{68} - 55 q^{69} - 125 q^{70} - 78 q^{71} - 225 q^{72} - 326 q^{73} + 3 q^{74} - 95 q^{75} - 84 q^{76} - 79 q^{77} - 86 q^{78} - 117 q^{79} - 39 q^{80} + q^{81} - 96 q^{82} - 23 q^{83} - 57 q^{84} - 224 q^{85} - 7 q^{86} - 45 q^{87} - 250 q^{88} - 104 q^{89} - 36 q^{90} - 96 q^{91} - 137 q^{92} - 155 q^{93} - 48 q^{94} - 38 q^{95} - 33 q^{96} - 447 q^{97} - 46 q^{98} - 94 q^{99}+O(q^{100}) \) 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9 - 46 * q^10 - 32 * q^11 - 72 * q^12 - 80 * q^13 - 22 * q^14 - 43 * q^15 + 122 * q^16 - 61 * q^17 - 88 * q^18 - 43 * q^19 - 41 * q^20 - 61 * q^21 - 93 * q^22 - 60 * q^23 - 41 * q^24 + 26 * q^25 - 9 * q^26 - 93 * q^27 - 126 * q^28 - 47 * q^29 - 36 * q^30 - 100 * q^31 - 114 * q^32 - 133 * q^33 - 75 * q^34 - 37 * q^35 + 75 * q^36 - 264 * q^37 - 35 * q^38 - 47 * q^39 - 118 * q^40 - 72 * q^41 - 64 * q^42 - 107 * q^43 - 59 * q^44 - 69 * q^45 - 111 * q^46 - 54 * q^47 - 135 * q^48 + 33 * q^49 - 42 * q^50 - 26 * q^51 - 173 * q^52 - 103 * q^53 - 28 * q^54 - 78 * q^55 - 44 * q^56 - 205 * q^57 - 189 * q^58 - 38 * q^59 - 105 * q^60 - 108 * q^61 - 14 * q^62 - 116 * q^63 + 39 * q^64 - 146 * q^65 + 5 * q^66 - 206 * q^67 - 62 * q^68 - 55 * q^69 - 125 * q^70 - 78 * q^71 - 225 * q^72 - 326 * q^73 + 3 * q^74 - 95 * q^75 - 84 * q^76 - 79 * q^77 - 86 * q^78 - 117 * q^79 - 39 * q^80 + q^81 - 96 * q^82 - 23 * q^83 - 57 * q^84 - 224 * q^85 - 7 * q^86 - 45 * q^87 - 250 * q^88 - 104 * q^89 - 36 * q^90 - 96 * q^91 - 137 * q^92 - 155 * q^93 - 48 * q^94 - 38 * q^95 - 33 * q^96 - 447 * q^97 - 46 * q^98 - 94 * 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 $

$2$	−2.65451	−1.87702	−0.938510	−	0.345253i	$-0.887793\pi$
$2$	−2.65451	−1.87702	−0.938510	+	0.345253i	$0.887793\pi$
$3$	−2.99510	−1.72922	−0.864612	−	0.502440i	$-0.832436\pi$
$3$	−2.99510	−1.72922	−0.864612	+	0.502440i	$0.832436\pi$
$4$	5.04641	2.52320
$5$	−2.08040	−0.930382	−0.465191	−	0.885210i	$-0.654014\pi$
$5$	−2.08040	−0.930382	−0.465191	+	0.885210i	$0.654014\pi$
$6$	7.95052	3.24579
$7$	−1.11556	−0.421644	−0.210822	−	0.977524i	$-0.567614\pi$
$7$	−1.11556	−0.421644	−0.210822	+	0.977524i	$0.567614\pi$
$8$	−8.08670	−2.85908
$9$	5.97065	1.99022
$10$	5.52243	1.74635
$11$	3.38425	1.02039	0.510195	−	0.860059i	$-0.329573\pi$
$11$	3.38425	1.02039	0.510195	+	0.860059i	$0.329573\pi$
$12$	−15.1145	−4.36318
$13$	−3.23622	−0.897567	−0.448784	−	0.893640i	$-0.648143\pi$
$13$	−3.23622	−0.897567	−0.448784	+	0.893640i	$0.648143\pi$
$14$	2.96127	0.791434
$15$	6.23101	1.60884
$16$	11.3734	2.84335
$17$	4.29870	1.04259	0.521293	−	0.853378i	$-0.325450\pi$
$17$	4.29870	1.04259	0.521293	+	0.853378i	$0.325450\pi$
$18$	−15.8491	−3.73567
$19$	0.715567	0.164162	0.0820811	−	0.996626i	$-0.473843\pi$
$19$	0.715567	0.164162	0.0820811	+	0.996626i	$0.473843\pi$
$20$	−10.4985	−2.34754
$21$	3.34123	0.729117
$22$	−8.98352	−1.91529
$23$	0.981717	0.204702	0.102351	−	0.994748i	$-0.467363\pi$
$23$	0.981717	0.204702	0.102351	+	0.994748i	$0.467363\pi$
$24$	24.2205	4.94399
$25$	−0.671946	−0.134389
$26$	8.59058	1.68475
$27$	−8.89739	−1.71230
$28$	−5.62959	−1.06389
$29$	−2.22960	−0.414027	−0.207014	−	0.978338i	$-0.566374\pi$
$29$	−2.22960	−0.414027	−0.207014	+	0.978338i	$0.566374\pi$
$30$	−16.5402	−3.01982
$31$	0.355075	0.0637734	0.0318867	−	0.999491i	$-0.489848\pi$
$31$	0.355075	0.0637734	0.0318867	+	0.999491i	$0.489848\pi$
$32$	−14.0173	−2.47794
$33$	−10.1362	−1.76448
$34$	−11.4109	−1.95696
$35$	2.32082	0.392290
$36$	30.1303	5.02172
$37$	11.1937	1.84024	0.920118	−	0.391641i	$-0.128092\pi$
$37$	11.1937	1.84024	0.920118	+	0.391641i	$0.128092\pi$
$38$	−1.89948	−0.308136
$39$	9.69283	1.55209
$40$	16.8236	2.66004
$41$	−1.13870	−0.177834	−0.0889172	−	0.996039i	$-0.528341\pi$
$41$	−1.13870	−0.177834	−0.0889172	+	0.996039i	$0.528341\pi$
$42$	−8.86932	−1.36857
$43$	−6.13987	−0.936321	−0.468160	−	0.883643i	$-0.655083\pi$
$43$	−6.13987	−0.936321	−0.468160	+	0.883643i	$0.655083\pi$
$44$	17.0783	2.57465
$45$	−12.4213	−1.85166
$46$	−2.60597	−0.384230
$47$	−2.54799	−0.371662	−0.185831	−	0.982582i	$-0.559498\pi$
$47$	−2.54799	−0.371662	−0.185831	+	0.982582i	$0.559498\pi$
$48$	−34.0645	−4.91679
$49$	−5.75552	−0.822216
$50$	1.78369	0.252251
$51$	−12.8750	−1.80287
$52$	−16.3313	−2.26474
$53$	−12.6824	−1.74206	−0.871028	−	0.491233i	$-0.836546\pi$
$53$	−12.6824	−1.74206	−0.871028	+	0.491233i	$0.836546\pi$
$54$	23.6182	3.21403
$55$	−7.04059	−0.949353
$56$	9.02124	1.20551
$57$	−2.14320	−0.283873
$58$	5.91850	0.777137
$59$	12.1366	1.58005	0.790027	−	0.613072i	$-0.210067\pi$
$59$	12.1366	1.58005	0.790027	+	0.613072i	$0.210067\pi$
$60$	31.4442	4.05943
$61$	7.65047	0.979543	0.489771	−	0.871851i	$-0.337080\pi$
$61$	7.65047	0.979543	0.489771	+	0.871851i	$0.337080\pi$
$62$	−0.942549	−0.119704
$63$	−6.66064	−0.839162
$64$	14.4624	1.80779
$65$	6.73263	0.835080
$66$	26.9066	3.31197
$67$	−9.28979	−1.13493	−0.567464	−	0.823398i	$-0.692076\pi$
$67$	−9.28979	−1.13493	−0.567464	+	0.823398i	$0.692076\pi$
$68$	21.6930	2.63066
$69$	−2.94034	−0.353976
$70$	−6.16063	−0.736336
$71$	−3.24012	−0.384531	−0.192266	−	0.981343i	$-0.561584\pi$
$71$	−3.24012	−0.384531	−0.192266	+	0.981343i	$0.561584\pi$
$72$	−48.2828	−5.69019
$73$	−4.55854	−0.533537	−0.266769	−	0.963761i	$-0.585956\pi$
$73$	−4.55854	−0.533537	−0.266769	+	0.963761i	$0.585956\pi$
$74$	−29.7138	−3.45416
$75$	2.01255	0.232389
$76$	3.61104	0.414215
$77$	−3.77535	−0.430241
$78$	−25.7297	−2.91331
$79$	−2.45178	−0.275847	−0.137924	−	0.990443i	$-0.544043\pi$
$79$	−2.45178	−0.275847	−0.137924	+	0.990443i	$0.544043\pi$
$80$	−23.6612	−2.64540
$81$	8.73667	0.970742
$82$	3.02268	0.333799
$83$	7.49212	0.822367	0.411184	−	0.911553i	$-0.365115\pi$
$83$	7.49212	0.822367	0.411184	+	0.911553i	$0.365115\pi$
$84$	16.8612	1.83971
$85$	−8.94300	−0.970004
$86$	16.2983	1.75749
$87$	6.67790	0.715946
$88$	−27.3675	−2.91738
$89$	11.6238	1.23212	0.616059	−	0.787700i	$-0.288728\pi$
$89$	11.6238	1.23212	0.616059	+	0.787700i	$0.288728\pi$
$90$	32.9725	3.47560
$91$	3.61022	0.378454
$92$	4.95414	0.516505
$93$	−1.06349	−0.110278
$94$	6.76364	0.697617
$95$	−1.48866	−0.152734
$96$	41.9834	4.28491
$97$	−10.6560	−1.08196	−0.540979	−	0.841036i	$-0.681946\pi$
$97$	−10.6560	−1.08196	−0.540979	+	0.841036i	$0.681946\pi$
$98$	15.2781	1.54332
$99$	20.2062	2.03080
$100$	−3.39091	−0.339091
$101$	4.79542	0.477162	0.238581	−	0.971123i	$-0.423318\pi$
$101$	4.79542	0.477162	0.238581	+	0.971123i	$0.423318\pi$
$102$	34.1769	3.38402
$103$	−15.0541	−1.48332	−0.741662	−	0.670774i	$-0.765962\pi$
$103$	−15.0541	−1.48332	−0.741662	+	0.670774i	$0.765962\pi$
$104$	26.1704	2.56622
$105$	−6.95109	−0.678357
$106$	33.6654	3.26987
$107$	4.83698	0.467609	0.233804	−	0.972284i	$-0.424882\pi$
$107$	4.83698	0.467609	0.233804	+	0.972284i	$0.424882\pi$
$108$	−44.8998	−4.32049
$109$	−10.7199	−1.02678	−0.513390	−	0.858156i	$-0.671610\pi$
$109$	−10.7199	−1.02678	−0.513390	+	0.858156i	$0.671610\pi$
$110$	18.6893	1.78195
$111$	−33.5264	−3.18218
$112$	−12.6878	−1.19888
$113$	−7.17112	−0.674602	−0.337301	−	0.941397i	$-0.609514\pi$
$113$	−7.17112	−0.674602	−0.337301	+	0.941397i	$0.609514\pi$
$114$	5.68913	0.532836
$115$	−2.04236	−0.190451
$116$	−11.2515	−1.04467
$117$	−19.3223	−1.78635
$118$	−32.2167	−2.96579
$119$	−4.79547	−0.439600
$120$	−50.3883	−4.59980
$121$	0.453170	0.0411973
$122$	−20.3082	−1.83862
$123$	3.41051	0.307516
$124$	1.79185	0.160913
$125$	11.7999	1.05542
$126$	17.6807	1.57512
$127$	17.7855	1.57821	0.789106	−	0.614258i	$-0.210544\pi$
$127$	17.7855	1.57821	0.789106	+	0.614258i	$0.210544\pi$
$128$	−10.3557	−0.915325
$129$	18.3895	1.61911
$130$	−17.8718	−1.56746
$131$	−3.25337	−0.284248	−0.142124	−	0.989849i	$-0.545393\pi$
$131$	−3.25337	−0.284248	−0.142124	+	0.989849i	$0.545393\pi$
$132$	−51.1513	−4.45215
$133$	−0.798261	−0.0692180
$134$	24.6598	2.13028
$135$	18.5101	1.59310
$136$	−34.7623	−2.98084
$137$	10.0179	0.855884	0.427942	−	0.903806i	$-0.359239\pi$
$137$	10.0179	0.855884	0.427942	+	0.903806i	$0.359239\pi$
$138$	7.80516	0.664420
$139$	−12.9376	−1.09735	−0.548676	−	0.836035i	$-0.684868\pi$
$139$	−12.9376	−1.09735	−0.548676	+	0.836035i	$0.684868\pi$
$140$	11.7118	0.989827
$141$	7.63148	0.642687
$142$	8.60092	0.721773
$143$	−10.9522	−0.915869
$144$	67.9065	5.65888
$145$	4.63846	0.385203
$146$	12.1007	1.00146
$147$	17.2384	1.42180
$148$	56.4880	4.64329
$149$	18.6786	1.53021	0.765105	−	0.643906i	$-0.222687\pi$
$149$	18.6786	1.53021	0.765105	+	0.643906i	$0.222687\pi$
$150$	−5.34232	−0.436199
$151$	−14.4072	−1.17244	−0.586221	−	0.810151i	$-0.699385\pi$
$151$	−14.4072	−1.17244	−0.586221	+	0.810151i	$0.699385\pi$
$152$	−5.78657	−0.469353
$153$	25.6660	2.07497
$154$	10.0217	0.807572
$155$	−0.738697	−0.0593336
$156$	48.9139	3.91625
$157$	16.7766	1.33892	0.669458	−	0.742850i	$-0.266526\pi$
$157$	16.7766	1.33892	0.669458	+	0.742850i	$0.266526\pi$
$158$	6.50828	0.517771
$159$	37.9850	3.01241
$160$	29.1617	2.30543
$161$	−1.09517	−0.0863114
$162$	−23.1916	−1.82210
$163$	4.31063	0.337635	0.168817	−	0.985647i	$-0.446005\pi$
$163$	4.31063	0.337635	0.168817	+	0.985647i	$0.446005\pi$
$164$	−5.74632	−0.448712
$165$	21.0873	1.64164
$166$	−19.8879	−1.54360
$167$	−14.3337	−1.10917	−0.554587	−	0.832126i	$-0.687124\pi$
$167$	−14.3337	−1.10917	−0.554587	+	0.832126i	$0.687124\pi$
$168$	−27.0196	−2.08460
$169$	−2.52685	−0.194373
$170$	23.7392	1.82072
$171$	4.27239	0.326718
$172$	−30.9843	−2.36253
$173$	−7.50239	−0.570396	−0.285198	−	0.958469i	$-0.592059\pi$
$173$	−7.50239	−0.570396	−0.285198	+	0.958469i	$0.592059\pi$
$174$	−17.7265	−1.34384
$175$	0.749600	0.0566644
$176$	38.4905	2.90133
$177$	−36.3504	−2.73227
$178$	−30.8554	−2.31271
$179$	−2.38194	−0.178035	−0.0890173	−	0.996030i	$-0.528373\pi$
$179$	−2.38194	−0.178035	−0.0890173	+	0.996030i	$0.528373\pi$
$180$	−62.6830	−4.67211
$181$	−2.47823	−0.184205	−0.0921026	−	0.995750i	$-0.529359\pi$
$181$	−2.47823	−0.184205	−0.0921026	+	0.995750i	$0.529359\pi$
$182$	−9.58335	−0.710365
$183$	−22.9140	−1.69385
$184$	−7.93886	−0.585260
$185$	−23.2874	−1.71212
$186$	2.82303	0.206995
$187$	14.5479	1.06385
$188$	−12.8582	−0.937778
$189$	9.92562	0.721982
$190$	3.95167	0.286684
$191$	−1.92863	−0.139551	−0.0697754	−	0.997563i	$-0.522228\pi$
$191$	−1.92863	−0.139551	−0.0697754	+	0.997563i	$0.522228\pi$
$192$	−43.3162	−3.12608
$193$	10.2727	0.739447	0.369723	−	0.929142i	$-0.379452\pi$
$193$	10.2727	0.739447	0.369723	+	0.929142i	$0.379452\pi$
$194$	28.2866	2.03086
$195$	−20.1649	−1.44404
$196$	−29.0447	−2.07462
$197$	22.7684	1.62218	0.811090	−	0.584922i	$-0.198875\pi$
$197$	22.7684	1.62218	0.811090	+	0.584922i	$0.198875\pi$
$198$	−53.6374	−3.81185
$199$	20.5648	1.45780	0.728900	−	0.684621i	$-0.240032\pi$
$199$	20.5648	1.45780	0.728900	+	0.684621i	$0.240032\pi$
$200$	5.43383	0.384230
$201$	27.8239	1.96255
$202$	−12.7295	−0.895643
$203$	2.48727	0.174572
$204$	−64.9727	−4.54900
$205$	2.36894	0.165454
$206$	39.9612	2.78423
$207$	5.86149	0.407401
$208$	−36.8069	−2.55210
$209$	2.42166	0.167510
$210$	18.4517	1.27329
$211$	−11.0025	−0.757446	−0.378723	−	0.925510i	$-0.623637\pi$
$211$	−11.0025	−0.757446	−0.378723	+	0.925510i	$0.623637\pi$
$212$	−64.0003	−4.39556
$213$	9.70449	0.664941
$214$	−12.8398	−0.877711
$215$	12.7734	0.871136
$216$	71.9506	4.89562
$217$	−0.396109	−0.0268897
$218$	28.4560	1.92728
$219$	13.6533	0.922605
$220$	−35.5297	−2.39541
$221$	−13.9115	−0.935792
$222$	88.9959	5.97302
$223$	22.3094	1.49395	0.746973	−	0.664854i	$-0.231506\pi$
$223$	22.3094	1.49395	0.746973	+	0.664854i	$0.231506\pi$
$224$	15.6373	1.04481
$225$	−4.01195	−0.267464
$226$	19.0358	1.26624
$227$	−5.76706	−0.382773	−0.191387	−	0.981515i	$-0.561298\pi$
$227$	−5.76706	−0.382773	−0.191387	+	0.981515i	$0.561298\pi$
$228$	−10.8154	−0.716270
$229$	−10.0640	−0.665048	−0.332524	−	0.943095i	$-0.607900\pi$
$229$	−10.0640	−0.665048	−0.332524	+	0.943095i	$0.607900\pi$
$230$	5.42146	0.357481
$231$	11.3076	0.743984
$232$	18.0301	1.18374
$233$	−8.36503	−0.548011	−0.274006	−	0.961728i	$-0.588349\pi$
$233$	−8.36503	−0.548011	−0.274006	+	0.961728i	$0.588349\pi$
$234$	51.2913	3.35302
$235$	5.30082	0.345788
$236$	61.2463	3.98679
$237$	7.34335	0.477002
$238$	12.7296	0.825138
$239$	11.2520	0.727832	0.363916	−	0.931432i	$-0.381440\pi$
$239$	11.2520	0.727832	0.363916	+	0.931432i	$0.381440\pi$
$240$	70.8677	4.57449
$241$	−7.91275	−0.509705	−0.254853	−	0.966980i	$-0.582027\pi$
$241$	−7.91275	−0.509705	−0.254853	+	0.966980i	$0.582027\pi$
$242$	−1.20294	−0.0773281
$243$	0.524932	0.0336744
$244$	38.6074	2.47158
$245$	11.9738	0.764975
$246$	−9.05323	−0.577213
$247$	−2.31573	−0.147347
$248$	−2.87139	−0.182333
$249$	−22.4397	−1.42206
$250$	−31.3229	−1.98104
$251$	5.57587	0.351946	0.175973	−	0.984395i	$-0.443693\pi$
$251$	5.57587	0.351946	0.175973	+	0.984395i	$0.443693\pi$
$252$	−33.6123	−2.11738
$253$	3.32238	0.208876
$254$	−47.2118	−2.96233
$255$	26.7852	1.67735
$256$	−1.43539	−0.0897121
$257$	25.4249	1.58596	0.792979	−	0.609248i	$-0.208529\pi$
$257$	25.4249	1.58596	0.792979	+	0.609248i	$0.208529\pi$
$258$	−48.8151	−3.03910
$259$	−12.4873	−0.775924
$260$	33.9756	2.10708
$261$	−13.3122	−0.824003
$262$	8.63608	0.533539
$263$	15.9468	0.983324	0.491662	−	0.870786i	$-0.336390\pi$
$263$	15.9468	0.983324	0.491662	+	0.870786i	$0.336390\pi$
$264$	81.9684	5.04480
$265$	26.3844	1.62078
$266$	2.11899	0.129924
$267$	−34.8144	−2.13061
$268$	−46.8801	−2.86366
$269$	14.3454	0.874657	0.437329	−	0.899302i	$-0.355925\pi$
$269$	14.3454	0.874657	0.437329	+	0.899302i	$0.355925\pi$
$270$	−49.1352	−2.99027
$271$	17.8005	1.08130	0.540651	−	0.841247i	$-0.318178\pi$
$271$	17.8005	1.08130	0.540651	+	0.841247i	$0.318178\pi$
$272$	48.8908	2.96444
$273$	−10.8130	−0.654431
$274$	−26.5925	−1.60651
$275$	−2.27404	−0.137130
$276$	−14.8382	−0.893153
$277$	1.67238	0.100484	0.0502419	−	0.998737i	$-0.484001\pi$
$277$	1.67238	0.100484	0.0502419	+	0.998737i	$0.484001\pi$
$278$	34.3429	2.05975
$279$	2.12003	0.126923
$280$	−18.7678	−1.12159
$281$	−10.7527	−0.641453	−0.320727	−	0.947172i	$-0.603927\pi$
$281$	−10.7527	−0.641453	−0.320727	+	0.947172i	$0.603927\pi$
$282$	−20.2578	−1.20634
$283$	24.5654	1.46026	0.730130	−	0.683308i	$-0.239459\pi$
$283$	24.5654	1.46026	0.730130	+	0.683308i	$0.239459\pi$
$284$	−16.3510	−0.970251
$285$	4.45870	0.264111
$286$	29.0727	1.71910
$287$	1.27029	0.0749828
$288$	−83.6926	−4.93164
$289$	1.47878	0.0869873
$290$	−12.3128	−0.723034
$291$	31.9160	1.87095
$292$	−23.0043	−1.34622
$293$	−3.89252	−0.227403	−0.113702	−	0.993515i	$-0.536271\pi$
$293$	−3.89252	−0.227403	−0.113702	+	0.993515i	$0.536271\pi$
$294$	−45.7594	−2.66874
$295$	−25.2490	−1.47005
$296$	−90.5203	−5.26139
$297$	−30.1110	−1.74722
$298$	−49.5824	−2.87223
$299$	−3.17706	−0.183734
$300$	10.1561	0.586365
$301$	6.84942	0.394794
$302$	38.2440	2.20070
$303$	−14.3628	−0.825120
$304$	8.13842	0.466770
$305$	−15.9160	−0.911349
$306$	−68.1305	−3.89476
$307$	6.84135	0.390457	0.195228	−	0.980758i	$-0.437455\pi$
$307$	6.84135	0.390457	0.195228	+	0.980758i	$0.437455\pi$
$308$	−19.0520	−1.08559
$309$	45.0886	2.56500
$310$	1.96088	0.111370
$311$	26.9202	1.52650	0.763252	−	0.646101i	$-0.223602\pi$
$311$	26.9202	1.52650	0.763252	+	0.646101i	$0.223602\pi$
$312$	−78.3830	−4.43756
$313$	−9.98612	−0.564449	−0.282224	−	0.959348i	$-0.591072\pi$
$313$	−9.98612	−0.564449	−0.282224	+	0.959348i	$0.591072\pi$
$314$	−44.5335	−2.51317
$315$	13.8568	0.780741
$316$	−12.3727	−0.696019
$317$	17.0621	0.958302	0.479151	−	0.877732i	$-0.340945\pi$
$317$	17.0621	0.958302	0.479151	+	0.877732i	$0.340945\pi$
$318$	−100.831	−5.65434
$319$	−7.54555	−0.422469
$320$	−30.0874	−1.68194
$321$	−14.4873	−0.808600
$322$	2.90713	0.162008
$323$	3.07600	0.171153
$324$	44.0888	2.44938
$325$	2.17457	0.120623
$326$	−11.4426	−0.633747
$327$	32.1072	1.77553
$328$	9.20830	0.508443
$329$	2.84244	0.156709
$330$	−55.9764	−3.08140
$331$	12.4753	0.685707	0.342853	−	0.939389i	$-0.388607\pi$
$331$	12.4753	0.685707	0.342853	+	0.939389i	$0.388607\pi$
$332$	37.8083	2.07500
$333$	66.8337	3.66247
$334$	38.0488	2.08194
$335$	19.3265	1.05592
$336$	38.0012	2.07313
$337$	−4.17402	−0.227374	−0.113687	−	0.993517i	$-0.536266\pi$
$337$	−4.17402	−0.227374	−0.113687	+	0.993517i	$0.536266\pi$
$338$	6.70755	0.364843
$339$	21.4782	1.16654
$340$	−45.1300	−2.44752
$341$	1.20166	0.0650738
$342$	−11.3411	−0.613256
$343$	14.2296	0.768326
$344$	49.6513	2.67702
$345$	6.11709	0.329333
$346$	19.9152	1.07065
$347$	−1.54513	−0.0829468	−0.0414734	−	0.999140i	$-0.513205\pi$
$347$	−1.54513	−0.0829468	−0.0414734	+	0.999140i	$0.513205\pi$
$348$	33.6994	1.80648
$349$	15.9650	0.854589	0.427295	−	0.904112i	$-0.359467\pi$
$349$	15.9650	0.854589	0.427295	+	0.904112i	$0.359467\pi$
$350$	−1.98982	−0.106360
$351$	28.7940	1.53691
$352$	−47.4383	−2.52847
$353$	−20.0161	−1.06535	−0.532675	−	0.846320i	$-0.678813\pi$
$353$	−20.0161	−1.06535	−0.532675	+	0.846320i	$0.678813\pi$
$354$	96.4925	5.12852
$355$	6.74074	0.357761
$356$	58.6583	3.10888
$357$	14.3629	0.760167
$358$	6.32288	0.334174
$359$	3.06598	0.161816	0.0809080	−	0.996722i	$-0.474218\pi$
$359$	3.06598	0.161816	0.0809080	+	0.996722i	$0.474218\pi$
$360$	100.447	5.29405
$361$	−18.4880	−0.973051
$362$	6.57847	0.345757
$363$	−1.35729	−0.0712394
$364$	18.2186	0.954915
$365$	9.48358	0.496393
$366$	60.8253	3.17939
$367$	19.9579	1.04179	0.520897	−	0.853620i	$-0.325598\pi$
$367$	19.9579	1.04179	0.520897	+	0.853620i	$0.325598\pi$
$368$	11.1655	0.582040
$369$	−6.79875	−0.353929
$370$	61.8165	3.21369
$371$	14.1480	0.734527
$372$	−5.36679	−0.278255
$373$	−31.2790	−1.61956	−0.809781	−	0.586732i	$-0.800414\pi$
$373$	−31.2790	−1.61956	−0.809781	+	0.586732i	$0.800414\pi$
$374$	−38.6174	−1.99686
$375$	−35.3419	−1.82505
$376$	20.6048	1.06261
$377$	7.21550	0.371617
$378$	−26.3476	−1.35518
$379$	21.2304	1.09053	0.545267	−	0.838262i	$-0.316428\pi$
$379$	21.2304	1.09053	0.545267	+	0.838262i	$0.316428\pi$
$380$	−7.51240	−0.385378
$381$	−53.2695	−2.72908
$382$	5.11956	0.261939
$383$	−7.97632	−0.407571	−0.203785	−	0.979016i	$-0.565324\pi$
$383$	−7.97632	−0.407571	−0.203785	+	0.979016i	$0.565324\pi$
$384$	31.0164	1.58280
$385$	7.85424	0.400289
$386$	−27.2690	−1.38796
$387$	−36.6590	−1.86348
$388$	−53.7747	−2.73000
$389$	−13.0994	−0.664168	−0.332084	−	0.943250i	$-0.607752\pi$
$389$	−13.0994	−0.664168	−0.332084	+	0.943250i	$0.607752\pi$
$390$	53.5279	2.71049
$391$	4.22010	0.213420
$392$	46.5431	2.35078
$393$	9.74417	0.491528
$394$	−60.4388	−3.04486
$395$	5.10069	0.256643
$396$	101.969	5.12411
$397$	−7.38795	−0.370790	−0.185395	−	0.982664i	$-0.559357\pi$
$397$	−7.38795	−0.370790	−0.185395	+	0.982664i	$0.559357\pi$
$398$	−54.5894	−2.73632
$399$	2.39087	0.119693
$400$	−7.64231	−0.382116
$401$	−8.17292	−0.408136	−0.204068	−	0.978957i	$-0.565416\pi$
$401$	−8.17292	−0.408136	−0.204068	+	0.978957i	$0.565416\pi$
$402$	−73.8587	−3.68374
$403$	−1.14910	−0.0572409
$404$	24.1996	1.20398
$405$	−18.1758	−0.903161
$406$	−6.60247	−0.327675
$407$	37.8824	1.87776
$408$	104.117	5.15454
$409$	−21.3014	−1.05329	−0.526643	−	0.850087i	$-0.676549\pi$
$409$	−21.3014	−1.05329	−0.526643	+	0.850087i	$0.676549\pi$
$410$	−6.28837	−0.310560
$411$	−30.0045	−1.48001
$412$	−75.9691	−3.74273
$413$	−13.5392	−0.666220
$414$	−15.5594	−0.764700
$415$	−15.5866	−0.765116
$416$	45.3633	2.22412
$417$	38.7494	1.89757
$418$	−6.42831	−0.314419
$419$	−12.8051	−0.625572	−0.312786	−	0.949824i	$-0.601262\pi$
$419$	−12.8051	−0.625572	−0.312786	+	0.949824i	$0.601262\pi$
$420$	−35.0780	−1.71163
$421$	−33.8202	−1.64830	−0.824148	−	0.566374i	$-0.808346\pi$
$421$	−33.8202	−1.64830	−0.824148	+	0.566374i	$0.808346\pi$
$422$	29.2063	1.42174
$423$	−15.2131	−0.739687
$424$	102.558	4.98068
$425$	−2.88849	−0.140112
$426$	−25.7606	−1.24811
$427$	−8.53460	−0.413018
$428$	24.4094	1.17987
$429$	32.8030	1.58374
$430$	−33.9070	−1.63514
$431$	1.97569	0.0951655	0.0475827	−	0.998867i	$-0.484848\pi$
$431$	1.97569	0.0951655	0.0475827	+	0.998867i	$0.484848\pi$
$432$	−101.194	−4.86868
$433$	30.4554	1.46359	0.731797	−	0.681523i	$-0.238682\pi$
$433$	30.4554	1.46359	0.731797	+	0.681523i	$0.238682\pi$
$434$	1.05147	0.0504724
$435$	−13.8927	−0.666103
$436$	−54.0969	−2.59077
$437$	0.702484	0.0336044
$438$	−36.2428	−1.73175
$439$	36.2092	1.72817	0.864084	−	0.503347i	$-0.167898\pi$
$439$	36.2092	1.72817	0.864084	+	0.503347i	$0.167898\pi$
$440$	56.9352	2.71428
$441$	−34.3641	−1.63639
$442$	36.9283	1.75650
$443$	−22.2613	−1.05767	−0.528834	−	0.848725i	$-0.677371\pi$
$443$	−22.2613	−1.05767	−0.528834	+	0.848725i	$0.677371\pi$
$444$	−169.188	−8.02929
$445$	−24.1821	−1.14634
$446$	−59.2204	−2.80417
$447$	−55.9443	−2.64607
$448$	−16.1337	−0.762245
$449$	26.3429	1.24320	0.621599	−	0.783336i	$-0.286483\pi$
$449$	26.3429	1.24320	0.621599	+	0.783336i	$0.286483\pi$
$450$	10.6498	0.502034
$451$	−3.85364	−0.181461
$452$	−36.1884	−1.70216
$453$	43.1511	2.02741
$454$	15.3087	0.718473
$455$	−7.51069	−0.352106
$456$	17.3314	0.811617
$457$	−41.1676	−1.92574	−0.962870	−	0.269967i	$-0.912987\pi$
$457$	−41.1676	−1.92574	−0.962870	+	0.269967i	$0.912987\pi$
$458$	26.7149	1.24831
$459$	−38.2472	−1.78523
$460$	−10.3066	−0.480547
$461$	−2.21473	−0.103150	−0.0515752	−	0.998669i	$-0.516424\pi$
$461$	−2.21473	−0.103150	−0.0515752	+	0.998669i	$0.516424\pi$
$462$	−30.0160	−1.39647
$463$	−13.0498	−0.606476	−0.303238	−	0.952915i	$-0.598068\pi$
$463$	−13.0498	−0.606476	−0.303238	+	0.952915i	$0.598068\pi$
$464$	−25.3582	−1.17722
$465$	2.21248	0.102601
$466$	22.2050	1.02863
$467$	11.2239	0.519379	0.259689	−	0.965692i	$-0.416380\pi$
$467$	11.2239	0.519379	0.259689	+	0.965692i	$0.416380\pi$
$468$	−97.5084	−4.50733
$469$	10.3634	0.478536
$470$	−14.0711	−0.649050
$471$	−50.2476	−2.31529
$472$	−98.1452	−4.51750
$473$	−20.7789	−0.955413
$474$	−19.4930	−0.895342
$475$	−0.480822	−0.0220616
$476$	−24.1999	−1.10920
$477$	−75.7219	−3.46707
$478$	−29.8685	−1.36616
$479$	−18.5816	−0.849016	−0.424508	−	0.905424i	$-0.639553\pi$
$479$	−18.5816	−0.849016	−0.424508	+	0.905424i	$0.639553\pi$
$480$	−87.3422	−3.98661
$481$	−36.2254	−1.65174
$482$	21.0045	0.956727
$483$	3.28014	0.149252
$484$	2.28688	0.103949
$485$	22.1688	1.00663
$486$	−1.39343	−0.0632075
$487$	−36.3379	−1.64663	−0.823313	−	0.567587i	$-0.807877\pi$
$487$	−36.3379	−1.64663	−0.823313	+	0.567587i	$0.807877\pi$
$488$	−61.8671	−2.80059
$489$	−12.9108	−0.583846
$490$	−31.7844	−1.43587
$491$	24.6739	1.11352	0.556759	−	0.830674i	$-0.312045\pi$
$491$	24.6739	1.11352	0.556759	+	0.830674i	$0.312045\pi$
$492$	17.2108	0.775924
$493$	−9.58439	−0.431659
$494$	6.14713	0.276572
$495$	−42.0369	−1.88942
$496$	4.03841	0.181330
$497$	3.61456	0.162135
$498$	59.5663	2.66923
$499$	12.0686	0.540266	0.270133	−	0.962823i	$-0.412932\pi$
$499$	12.0686	0.540266	0.270133	+	0.962823i	$0.412932\pi$
$500$	59.5471	2.66303
$501$	42.9308	1.91801
$502$	−14.8012	−0.660610
$503$	−35.0987	−1.56497	−0.782487	−	0.622666i	$-0.786049\pi$
$503$	−35.0987	−1.56497	−0.782487	+	0.622666i	$0.786049\pi$
$504$	53.8626	2.39923
$505$	−9.97638	−0.443943
$506$	−8.81928	−0.392065
$507$	7.56819	0.336115
$508$	89.7530	3.98215
$509$	−23.0318	−1.02087	−0.510433	−	0.859917i	$-0.670515\pi$
$509$	−23.0318	−1.02087	−0.510433	+	0.859917i	$0.670515\pi$
$510$	−71.1015	−3.14843
$511$	5.08535	0.224963
$512$	24.5217	1.08372
$513$	−6.36668	−0.281096
$514$	−67.4904	−2.97688
$515$	31.3185	1.38006
$516$	92.8011	4.08534
$517$	−8.62303	−0.379240
$518$	33.1477	1.45643
$519$	22.4704	0.986343
$520$	−54.4448	−2.38756
$521$	−20.0094	−0.876626	−0.438313	−	0.898822i	$-0.644424\pi$
$521$	−20.0094	−0.876626	−0.438313	+	0.898822i	$0.644424\pi$
$522$	35.3373	1.54667
$523$	−19.1158	−0.835875	−0.417937	−	0.908476i	$-0.637247\pi$
$523$	−19.1158	−0.835875	−0.417937	+	0.908476i	$0.637247\pi$
$524$	−16.4178	−0.717215
$525$	−2.24513	−0.0979855
$526$	−42.3310	−1.84572
$527$	1.52636	0.0664893
$528$	−115.283	−5.01704
$529$	−22.0362	−0.958097
$530$	−70.0374	−3.04223
$531$	72.4635	3.14465
$532$	−4.02835	−0.174651
$533$	3.68508	0.159618
$534$	92.4151	3.99919
$535$	−10.0628	−0.435055
$536$	75.1238	3.24485
$537$	7.13416	0.307862
$538$	−38.0801	−1.64175
$539$	−19.4781	−0.838982
$540$	93.4095	4.01971
$541$	−12.8024	−0.550417	−0.275208	−	0.961385i	$-0.588747\pi$
$541$	−12.8024	−0.550417	−0.275208	+	0.961385i	$0.588747\pi$
$542$	−47.2515	−2.02963
$543$	7.42255	0.318532
$544$	−60.2563	−2.58347
$545$	22.3016	0.955297
$546$	28.7031	1.22838
$547$	−14.9095	−0.637486	−0.318743	−	0.947841i	$-0.603261\pi$
$547$	−14.9095	−0.637486	−0.318743	+	0.947841i	$0.603261\pi$
$548$	50.5542	2.15957
$549$	45.6783	1.94950
$550$	6.03645	0.257395
$551$	−1.59543	−0.0679676
$552$	23.7777	1.01205
$553$	2.73512	0.116309
$554$	−4.43936	−0.188610
$555$	69.7481	2.96064
$556$	−65.2883	−2.76884
$557$	−18.1673	−0.769774	−0.384887	−	0.922964i	$-0.625760\pi$
$557$	−18.1673	−0.769774	−0.384887	+	0.922964i	$0.625760\pi$
$558$	−5.62763	−0.238236
$559$	19.8700	0.840411
$560$	26.3956	1.11542
$561$	−43.5724	−1.83963
$562$	28.5432	1.20402
$563$	34.3825	1.44905	0.724524	−	0.689249i	$-0.242060\pi$
$563$	34.3825	1.44905	0.724524	+	0.689249i	$0.242060\pi$
$564$	38.5115	1.62163
$565$	14.9188	0.627637
$566$	−65.2090	−2.74094
$567$	−9.74633	−0.409307
$568$	26.2019	1.09941
$569$	2.33082	0.0977129	0.0488565	−	0.998806i	$-0.484442\pi$
$569$	2.33082	0.0977129	0.0488565	+	0.998806i	$0.484442\pi$
$570$	−11.8356	−0.495741
$571$	34.3236	1.43640	0.718200	−	0.695837i	$-0.244966\pi$
$571$	34.3236	1.43640	0.718200	+	0.695837i	$0.244966\pi$
$572$	−55.2693	−2.31092
$573$	5.77645	0.241314
$574$	−3.37199	−0.140744
$575$	−0.659661	−0.0275098
$576$	86.3496	3.59790
$577$	−22.6331	−0.942228	−0.471114	−	0.882072i	$-0.656148\pi$
$577$	−22.6331	−0.942228	−0.471114	+	0.882072i	$0.656148\pi$
$578$	−3.92544	−0.163277
$579$	−30.7679	−1.27867
$580$	23.4076	0.971946
$581$	−8.35795	−0.346746
$582$	−84.7212	−3.51180
$583$	−42.9203	−1.77758
$584$	36.8636	1.52543
$585$	40.1982	1.66199
$586$	10.3327	0.426841
$587$	12.2595	0.506003	0.253001	−	0.967466i	$-0.418582\pi$
$587$	12.2595	0.506003	0.253001	+	0.967466i	$0.418582\pi$
$588$	86.9918	3.58748
$589$	0.254080	0.0104692
$590$	67.0236	2.75932
$591$	−68.1936	−2.80511
$592$	127.311	5.23243
$593$	2.68708	0.110345	0.0551726	−	0.998477i	$-0.482429\pi$
$593$	2.68708	0.110345	0.0551726	+	0.998477i	$0.482429\pi$
$594$	79.9299	3.27956
$595$	9.97649	0.408996
$596$	94.2597	3.86103
$597$	−61.5937	−2.52086
$598$	8.43352	0.344872
$599$	12.6420	0.516537	0.258268	−	0.966073i	$-0.416848\pi$
$599$	12.6420	0.516537	0.258268	+	0.966073i	$0.416848\pi$
$600$	−16.2749	−0.664419
$601$	4.33060	0.176649	0.0883244	−	0.996092i	$-0.471849\pi$
$601$	4.33060	0.176649	0.0883244	+	0.996092i	$0.471849\pi$
$602$	−18.1818	−0.741036
$603$	−55.4661	−2.25875
$604$	−72.7046	−2.95831
$605$	−0.942774	−0.0383292
$606$	38.1261	1.54877
$607$	0.468327	0.0190088	0.00950441	−	0.999955i	$-0.496975\pi$
$607$	0.468327	0.0190088	0.00950441	+	0.999955i	$0.496975\pi$
$608$	−10.0303	−0.406784
$609$	−7.44963	−0.301874
$610$	42.2492	1.71062
$611$	8.24585	0.333591
$612$	129.521	5.23558
$613$	−13.2115	−0.533609	−0.266805	−	0.963751i	$-0.585968\pi$
$613$	−13.2115	−0.533609	−0.266805	+	0.963751i	$0.585968\pi$
$614$	−18.1604	−0.732895
$615$	−7.09522	−0.286107
$616$	30.5302	1.23010
$617$	−12.5231	−0.504163	−0.252081	−	0.967706i	$-0.581115\pi$
$617$	−12.5231	−0.504163	−0.252081	+	0.967706i	$0.581115\pi$
$618$	−119.688	−4.81456
$619$	−21.8618	−0.878701	−0.439351	−	0.898316i	$-0.644791\pi$
$619$	−21.8618	−0.878701	−0.439351	+	0.898316i	$0.644791\pi$
$620$	−3.72777	−0.149711
$621$	−8.73472	−0.350512
$622$	−71.4598	−2.86528
$623$	−12.9671	−0.519515
$624$	110.240	4.41315
$625$	−21.1888	−0.847550
$626$	26.5082	1.05948
$627$	−7.25312	−0.289662
$628$	84.6614	3.37836
$629$	48.1184	1.91861
$630$	−36.7829	−1.46547
$631$	−30.6788	−1.22130	−0.610652	−	0.791899i	$-0.709093\pi$
$631$	−30.6788	−1.22130	−0.610652	+	0.791899i	$0.709093\pi$
$632$	19.8269	0.788670
$633$	32.9537	1.30979
$634$	−45.2914	−1.79875
$635$	−37.0010	−1.46834
$636$	191.688	7.60091
$637$	18.6261	0.737994
$638$	20.0297	0.792983
$639$	−19.3456	−0.765300
$640$	21.5440	0.851601
$641$	28.0285	1.10706	0.553529	−	0.832830i	$-0.313281\pi$
$641$	28.0285	1.10706	0.553529	+	0.832830i	$0.313281\pi$
$642$	38.4565	1.51776
$643$	39.3726	1.55270	0.776352	−	0.630300i	$-0.217068\pi$
$643$	39.3726	1.55270	0.776352	+	0.630300i	$0.217068\pi$
$644$	−5.52667	−0.217781
$645$	−38.2575	−1.50639
$646$	−8.16527	−0.321258
$647$	2.07740	0.0816710	0.0408355	−	0.999166i	$-0.486998\pi$
$647$	2.07740	0.0816710	0.0408355	+	0.999166i	$0.486998\pi$
$648$	−70.6509	−2.77543
$649$	41.0734	1.61227
$650$	−5.77241	−0.226412
$651$	1.18639	0.0464982
$652$	21.7532	0.851921
$653$	−10.1212	−0.396074	−0.198037	−	0.980195i	$-0.563457\pi$
$653$	−10.1212	−0.396074	−0.198037	+	0.980195i	$0.563457\pi$
$654$	−85.2287	−3.33271
$655$	6.76830	0.264459
$656$	−12.9508	−0.505645
$657$	−27.2175	−1.06185
$658$	−7.54528	−0.294146
$659$	−14.1521	−0.551286	−0.275643	−	0.961260i	$-0.588891\pi$
$659$	−14.1521	−0.551286	−0.275643	+	0.961260i	$0.588891\pi$
$660$	106.415	4.14220
$661$	−36.0369	−1.40167	−0.700836	−	0.713323i	$-0.747189\pi$
$661$	−36.0369	−1.40167	−0.700836	+	0.713323i	$0.747189\pi$
$662$	−33.1159	−1.28708
$663$	41.6665	1.61819
$664$	−60.5866	−2.35122
$665$	1.66070	0.0643992
$666$	−177.411	−6.87452
$667$	−2.18884	−0.0847522
$668$	−72.3335	−2.79867
$669$	−66.8189	−2.58337
$670$	−51.3022	−1.98198
$671$	25.8911	0.999516
$672$	−46.8352	−1.80671
$673$	22.0857	0.851340	0.425670	−	0.904878i	$-0.360038\pi$
$673$	22.0857	0.851340	0.425670	+	0.904878i	$0.360038\pi$
$674$	11.0800	0.426785
$675$	5.97857	0.230115
$676$	−12.7515	−0.490443
$677$	14.4705	0.556146	0.278073	−	0.960560i	$-0.410304\pi$
$677$	14.4705	0.556146	0.278073	+	0.960560i	$0.410304\pi$
$678$	−57.0141	−2.18961
$679$	11.8875	0.456201
$680$	72.3194	2.77332
$681$	17.2730	0.661901
$682$	−3.18983	−0.122145
$683$	−0.0799145	−0.00305784	−0.00152892	−	0.999999i	$-0.500487\pi$
$683$	−0.0799145	−0.00305784	−0.00152892	+	0.999999i	$0.500487\pi$
$684$	21.5602	0.824376
$685$	−20.8411	−0.796299
$686$	−37.7726	−1.44216
$687$	30.1427	1.15002
$688$	−69.8311	−2.66229
$689$	41.0430	1.56361
$690$	−16.2378	−0.618164
$691$	6.46607	0.245981	0.122990	−	0.992408i	$-0.460752\pi$
$691$	6.46607	0.245981	0.122990	+	0.992408i	$0.460752\pi$
$692$	−37.8601	−1.43923
$693$	−22.5413	−0.856273
$694$	4.10155	0.155693
$695$	26.9153	1.02096
$696$	−54.0022	−2.04695
$697$	−4.89491	−0.185408
$698$	−42.3793	−1.60408
$699$	25.0541	0.947634
$700$	3.78278	0.142976
$701$	8.66524	0.327282	0.163641	−	0.986520i	$-0.447676\pi$
$701$	8.66524	0.327282	0.163641	+	0.986520i	$0.447676\pi$
$702$	−76.4337	−2.88481
$703$	8.00985	0.302097
$704$	48.9443	1.84466
$705$	−15.8765	−0.597944
$706$	53.1329	1.99968
$707$	−5.34960	−0.201192
$708$	−183.439	−6.89406
$709$	−3.77392	−0.141733	−0.0708663	−	0.997486i	$-0.522576\pi$
$709$	−3.77392	−0.141733	−0.0708663	+	0.997486i	$0.522576\pi$
$710$	−17.8933	−0.671525
$711$	−14.6387	−0.548996
$712$	−93.9980	−3.52273
$713$	0.348583	0.0130545
$714$	−38.1265	−1.42685
$715$	22.7849	0.852108
$716$	−12.0202	−0.449217
$717$	−33.7009	−1.25858
$718$	−8.13866	−0.303732
$719$	−25.8889	−0.965492	−0.482746	−	0.875761i	$-0.660360\pi$
$719$	−25.8889	−0.965492	−0.482746	+	0.875761i	$0.660360\pi$
$720$	−141.273	−5.26492
$721$	16.7938	0.625435
$722$	49.0764	1.82644
$723$	23.6995	0.881394
$724$	−12.5061	−0.464787
$725$	1.49817	0.0556408
$726$	3.60294	0.133718
$727$	−32.9660	−1.22264	−0.611320	−	0.791383i	$-0.709361\pi$
$727$	−32.9660	−1.22264	−0.611320	+	0.791383i	$0.709361\pi$
$728$	−29.1948	−1.08203
$729$	−27.7822	−1.02897
$730$	−25.1742	−0.931740
$731$	−26.3934	−0.976196
$732$	−115.633	−4.27392
$733$	2.09021	0.0772037	0.0386018	−	0.999255i	$-0.487710\pi$
$733$	2.09021	0.0772037	0.0386018	+	0.999255i	$0.487710\pi$
$734$	−52.9784	−1.95547
$735$	−35.8627	−1.32281
$736$	−13.7611	−0.507240
$737$	−31.4390	−1.15807
$738$	18.0473	0.664331
$739$	−7.31887	−0.269229	−0.134614	−	0.990898i	$-0.542980\pi$
$739$	−7.31887	−0.269229	−0.134614	+	0.990898i	$0.542980\pi$
$740$	−117.518	−4.32003
$741$	6.93586	0.254795
$742$	−37.5559	−1.37872
$743$	−27.8176	−1.02053	−0.510264	−	0.860018i	$-0.670452\pi$
$743$	−27.8176	−1.02053	−0.510264	+	0.860018i	$0.670452\pi$
$744$	8.60010	0.315295
$745$	−38.8589	−1.42368
$746$	83.0302	3.03995
$747$	44.7328	1.63669
$748$	73.4145	2.68430
$749$	−5.39597	−0.197164
$750$	93.8154	3.42565
$751$	−10.1157	−0.369127	−0.184563	−	0.982821i	$-0.559087\pi$
$751$	−10.1157	−0.369127	−0.184563	+	0.982821i	$0.559087\pi$
$752$	−28.9792	−1.05676
$753$	−16.7003	−0.608594
$754$	−19.1536	−0.697533
$755$	29.9727	1.09082
$756$	50.0887	1.82171
$757$	50.1035	1.82104	0.910522	−	0.413461i	$-0.135680\pi$
$757$	50.1035	1.82104	0.910522	+	0.413461i	$0.135680\pi$
$758$	−56.3564	−2.04695
$759$	−9.95087	−0.361194
$760$	12.0384	0.436678
$761$	−23.9710	−0.868947	−0.434473	−	0.900685i	$-0.643066\pi$
$761$	−23.9710	−0.868947	−0.434473	+	0.900685i	$0.643066\pi$
$762$	141.404	5.12254
$763$	11.9587	0.432935
$764$	−9.73265	−0.352115
$765$	−53.3955	−1.93052
$766$	21.1732	0.765018
$767$	−39.2768	−1.41820
$768$	4.29915	0.155132
$769$	−41.4527	−1.49482	−0.747412	−	0.664361i	$-0.768704\pi$
$769$	−41.4527	−1.49482	−0.747412	+	0.664361i	$0.768704\pi$
$770$	−20.8491	−0.751350
$771$	−76.1501	−2.74248
$772$	51.8403	1.86577
$773$	−41.1687	−1.48073	−0.740367	−	0.672203i	$-0.765348\pi$
$773$	−41.1687	−1.48073	−0.740367	+	0.672203i	$0.765348\pi$
$774$	97.3115	3.49779
$775$	−0.238591	−0.00857046
$776$	86.1723	3.09341
$777$	37.4008	1.34175
$778$	34.7726	1.24666
$779$	−0.814813	−0.0291937
$780$	−101.760	−3.64361
$781$	−10.9654	−0.392372
$782$	−11.2023	−0.400593
$783$	19.8377	0.708940
$784$	−65.4598	−2.33785
$785$	−34.9019	−1.24570
$786$	−25.8660	−0.922608
$787$	15.5666	0.554891	0.277445	−	0.960741i	$-0.410512\pi$
$787$	15.5666	0.554891	0.277445	+	0.960741i	$0.410512\pi$
$788$	114.898	4.09309
$789$	−47.7624	−1.70039
$790$	−13.5398	−0.481725
$791$	7.99984	0.284442
$792$	−163.401	−5.80621
$793$	−24.7586	−0.879205
$794$	19.6113	0.695981
$795$	−79.0239	−2.80269
$796$	103.778	3.67832
$797$	21.4327	0.759185	0.379592	−	0.925154i	$-0.376064\pi$
$797$	21.4327	0.759185	0.379592	+	0.925154i	$0.376064\pi$
$798$	−6.34659	−0.224667
$799$	−10.9530	−0.387490
$800$	9.41891	0.333009
$801$	69.4015	2.45218
$802$	21.6951	0.766080
$803$	−15.4273	−0.544417
$804$	140.411	4.95190
$805$	2.27839	0.0803026
$806$	3.05030	0.107442
$807$	−42.9661	−1.51248
$808$	−38.7791	−1.36425
$809$	−7.93616	−0.279020	−0.139510	−	0.990221i	$-0.544553\pi$
$809$	−7.93616	−0.279020	−0.139510	+	0.990221i	$0.544553\pi$
$810$	48.2477	1.69525
$811$	19.6150	0.688774	0.344387	−	0.938828i	$-0.388087\pi$
$811$	19.6150	0.688774	0.344387	+	0.938828i	$0.388087\pi$
$812$	12.5518	0.440480
$813$	−53.3143	−1.86981
$814$	−100.559	−3.52459
$815$	−8.96783	−0.314129
$816$	−146.433	−5.12618
$817$	−4.39348	−0.153709
$818$	56.5446	1.97704
$819$	21.5553	0.753204
$820$	11.9546	0.417474
$821$	26.2538	0.916263	0.458132	−	0.888884i	$-0.348519\pi$
$821$	26.2538	0.916263	0.458132	+	0.888884i	$0.348519\pi$
$822$	79.6472	2.77802
$823$	−27.3948	−0.954921	−0.477461	−	0.878653i	$-0.658443\pi$
$823$	−27.3948	−0.954921	−0.477461	+	0.878653i	$0.658443\pi$
$824$	121.738	4.24095
$825$	6.81098	0.237128
$826$	35.9399	1.25051
$827$	18.4701	0.642267	0.321134	−	0.947034i	$-0.395936\pi$
$827$	18.4701	0.642267	0.321134	+	0.947034i	$0.395936\pi$
$828$	29.5794	1.02796
$829$	25.6304	0.890180	0.445090	−	0.895486i	$-0.353172\pi$
$829$	25.6304	0.890180	0.445090	+	0.895486i	$0.353172\pi$
$830$	41.3747	1.43614
$831$	−5.00897	−0.173759
$832$	−46.8034	−1.62262
$833$	−24.7412	−0.857232
$834$	−102.861	−3.56177
$835$	29.8197	1.03195
$836$	12.2207	0.422661
$837$	−3.15924	−0.109199
$838$	33.9913	1.17421
$839$	−20.4166	−0.704858	−0.352429	−	0.935838i	$-0.614644\pi$
$839$	−20.4166	−0.704858	−0.352429	+	0.935838i	$0.614644\pi$
$840$	56.2114	1.93948
$841$	−24.0289	−0.828582
$842$	89.7760	3.09388
$843$	32.2055	1.10922
$844$	−55.5233	−1.91119
$845$	5.25686	0.180841
$846$	40.3833	1.38841
$847$	−0.505541	−0.0173706
$848$	−144.242	−4.95327
$849$	−73.5759	−2.52512
$850$	7.66752	0.262994
$851$	10.9891	0.376700
$852$	48.9728	1.67778
$853$	5.62072	0.192450	0.0962249	−	0.995360i	$-0.469323\pi$
$853$	5.62072	0.192450	0.0962249	+	0.995360i	$0.469323\pi$
$854$	22.6551	0.775243
$855$	−8.88828	−0.303973
$856$	−39.1153	−1.33693
$857$	−18.7527	−0.640581	−0.320290	−	0.947319i	$-0.603781\pi$
$857$	−18.7527	−0.640581	−0.320290	+	0.947319i	$0.603781\pi$
$858$	−87.0757	−2.97272
$859$	−6.75647	−0.230528	−0.115264	−	0.993335i	$-0.536771\pi$
$859$	−6.75647	−0.230528	−0.115264	+	0.993335i	$0.536771\pi$
$860$	64.4596	2.19805
$861$	−3.80465	−0.129662
$862$	−5.24447	−0.178627
$863$	4.92266	0.167569	0.0837847	−	0.996484i	$-0.473299\pi$
$863$	4.92266	0.167569	0.0837847	+	0.996484i	$0.473299\pi$
$864$	124.718	4.24299
$865$	15.6080	0.530687
$866$	−80.8441	−2.74719
$867$	−4.42911	−0.150421
$868$	−1.99893	−0.0678480
$869$	−8.29746	−0.281472
$870$	36.8782	1.25029
$871$	30.0639	1.01867
$872$	86.6886	2.93565
$873$	−63.6235	−2.15333
$874$	−1.86475	−0.0630761
$875$	−13.1636	−0.445009
$876$	68.9002	2.32792
$877$	33.4684	1.13015	0.565074	−	0.825040i	$-0.308848\pi$
$877$	33.4684	1.13015	0.565074	+	0.825040i	$0.308848\pi$
$878$	−96.1174	−3.24381
$879$	11.6585	0.393232
$880$	−80.0754	−2.69934
$881$	−13.2892	−0.447723	−0.223862	−	0.974621i	$-0.571866\pi$
$881$	−13.2892	−0.447723	−0.223862	+	0.974621i	$0.571866\pi$
$882$	91.2198	3.07153
$883$	−5.26124	−0.177055	−0.0885274	−	0.996074i	$-0.528216\pi$
$883$	−5.26124	−0.177055	−0.0885274	+	0.996074i	$0.528216\pi$
$884$	−70.2033	−2.36119
$885$	75.6234	2.54205
$886$	59.0929	1.98526
$887$	−31.6616	−1.06309	−0.531547	−	0.847029i	$-0.678389\pi$
$887$	−31.6616	−1.06309	−0.531547	+	0.847029i	$0.678389\pi$
$888$	271.118	9.09811
$889$	−19.8409	−0.665443
$890$	64.1915	2.15170
$891$	29.5671	0.990536
$892$	112.582	3.76953
$893$	−1.82325	−0.0610128
$894$	148.505	4.96673
$895$	4.95538	0.165640
$896$	11.5525	0.385941
$897$	9.51561	0.317717
$898$	−69.9274	−2.33351
$899$	−0.791677	−0.0264039
$900$	−20.2459	−0.674865
$901$	−54.5176	−1.81624
$902$	10.2295	0.340605
$903$	−20.5147	−0.682687
$904$	57.9907	1.92874
$905$	5.15570	0.171381
$906$	−114.545	−3.80550
$907$	7.43424	0.246850	0.123425	−	0.992354i	$-0.460612\pi$
$907$	7.43424	0.246850	0.123425	+	0.992354i	$0.460612\pi$
$908$	−29.1029	−0.965815
$909$	28.6318	0.949655
$910$	19.9372	0.660911
$911$	−9.25748	−0.306714	−0.153357	−	0.988171i	$-0.549008\pi$
$911$	−9.25748	−0.306714	−0.153357	+	0.988171i	$0.549008\pi$
$912$	−24.3754	−0.807151
$913$	25.3552	0.839136
$914$	109.280	3.61465
$915$	47.6701	1.57593
$916$	−50.7870	−1.67805
$917$	3.62934	0.119851
$918$	101.527	3.35090
$919$	−2.32860	−0.0768134	−0.0384067	−	0.999262i	$-0.512228\pi$
$919$	−2.32860	−0.0768134	−0.0384067	+	0.999262i	$0.512228\pi$
$920$	16.5160	0.544516
$921$	−20.4906	−0.675187
$922$	5.87903	0.193615
$923$	10.4858	0.345143
$924$	57.0626	1.87722
$925$	−7.52158	−0.247308
$926$	34.6408	1.13837
$927$	−89.8827	−2.95213
$928$	31.2531	1.02593
$929$	−1.00084	−0.0328366	−0.0164183	−	0.999865i	$-0.505226\pi$
$929$	−1.00084	−0.0328366	−0.0164183	+	0.999865i	$0.505226\pi$
$930$	−5.87303	−0.192584
$931$	−4.11845	−0.134977
$932$	−42.2133	−1.38274
$933$	−80.6287	−2.63967
$934$	−29.7938	−0.974884
$935$	−30.2654	−0.989783
$936$	156.254	5.10732
$937$	23.7432	0.775658	0.387829	−	0.921731i	$-0.373225\pi$
$937$	23.7432	0.775658	0.387829	+	0.921731i	$0.373225\pi$
$938$	−27.5096	−0.898221
$939$	29.9095	0.976059
$940$	26.7501	0.872492
$941$	3.67235	0.119715	0.0598576	−	0.998207i	$-0.480935\pi$
$941$	3.67235	0.119715	0.0598576	+	0.998207i	$0.480935\pi$
$942$	133.383	4.34584
$943$	−1.11788	−0.0364031
$944$	138.035	4.49264
$945$	−20.6492	−0.671719
$946$	55.1576	1.79333
$947$	43.5506	1.41520	0.707602	−	0.706611i	$-0.249777\pi$
$947$	43.5506	1.41520	0.707602	+	0.706611i	$0.249777\pi$
$948$	37.0575	1.20357
$949$	14.7525	0.478885
$950$	1.27635	0.0414101
$951$	−51.1027	−1.65712
$952$	38.7796	1.25685
$953$	−14.9357	−0.483815	−0.241907	−	0.970299i	$-0.577773\pi$
$953$	−14.9357	−0.483815	−0.241907	+	0.970299i	$0.577773\pi$
$954$	201.004	6.50775
$955$	4.01232	0.129836
$956$	56.7822	1.83647
$957$	22.5997	0.730544
$958$	49.3251	1.59362
$959$	−11.1756	−0.360878
$960$	90.1150	2.90845
$961$	−30.8739	−0.995933
$962$	96.1605	3.10034
$963$	28.8799	0.930642
$964$	−39.9310	−1.28609
$965$	−21.3713	−0.687968
$966$	−8.70717	−0.280148
$967$	−2.72744	−0.0877084	−0.0438542	−	0.999038i	$-0.513964\pi$
$967$	−2.72744	−0.0877084	−0.0438542	+	0.999038i	$0.513964\pi$
$968$	−3.66465	−0.117786
$969$	−9.21295	−0.295962
$970$	−58.8473	−1.88947
$971$	−24.2522	−0.778289	−0.389145	−	0.921177i	$-0.627229\pi$
$971$	−24.2522	−0.778289	−0.389145	+	0.921177i	$0.627229\pi$
$972$	2.64902	0.0849673
$973$	14.4327	0.462692
$974$	96.4591	3.09075
$975$	−6.51306	−0.208585
$976$	87.0119	2.78518
$977$	−18.6446	−0.596493	−0.298246	−	0.954489i	$-0.596402\pi$
$977$	−18.6446	−0.596493	−0.298246	+	0.954489i	$0.596402\pi$
$978$	34.2718	1.09589
$979$	39.3378	1.25724
$980$	60.4244	1.93019
$981$	−64.0047	−2.04351
$982$	−65.4970	−2.09009
$983$	47.0685	1.50125	0.750626	−	0.660727i	$-0.229752\pi$
$983$	47.0685	1.50125	0.750626	+	0.660727i	$0.229752\pi$
$984$	−27.5798	−0.879212
$985$	−47.3673	−1.50925
$986$	25.4418	0.810233
$987$	−8.51341	−0.270985
$988$	−11.6861	−0.371785
$989$	−6.02761	−0.191667
$990$	111.587	3.54647
$991$	−10.6374	−0.337907	−0.168953	−	0.985624i	$-0.554039\pi$
$991$	−10.6374	−0.337907	−0.168953	+	0.985624i	$0.554039\pi$
$992$	−4.97721	−0.158027
$993$	−37.3649	−1.18574
$994$	−9.59488	−0.304331
$995$	−42.7829	−1.35631
$996$	−113.240	−3.58814
$997$	13.9469	0.441704	0.220852	−	0.975307i	$-0.429116\pi$
$997$	13.9469	0.441704	0.220852	+	0.975307i	$0.429116\pi$
$998$	−32.0362	−1.01409
$999$	−99.5949	−3.15104

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6047.2.a.a.1.8	✓	217

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6047.2.a.a.1.8	✓	217	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:	\( N \)	\(=\)	\( 6047 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.65451 q^{2} -2.99510 q^{3} +5.04641 q^{4} -2.08040 q^{5} +7.95052 q^{6} -1.11556 q^{7} -8.08670 q^{8} +5.97065 q^{9} +O(q^{10})\)	\(q-2.65451 q^{2} -2.99510 q^{3} +5.04641 q^{4} -2.08040 q^{5} +7.95052 q^{6} -1.11556 q^{7} -8.08670 q^{8} +5.97065 q^{9} +5.52243 q^{10} +3.38425 q^{11} -15.1145 q^{12} -3.23622 q^{13} +2.96127 q^{14} +6.23101 q^{15} +11.3734 q^{16} +4.29870 q^{17} -15.8491 q^{18} +0.715567 q^{19} -10.4985 q^{20} +3.34123 q^{21} -8.98352 q^{22} +0.981717 q^{23} +24.2205 q^{24} -0.671946 q^{25} +8.59058 q^{26} -8.89739 q^{27} -5.62959 q^{28} -2.22960 q^{29} -16.5402 q^{30} +0.355075 q^{31} -14.0173 q^{32} -10.1362 q^{33} -11.4109 q^{34} +2.32082 q^{35} +30.1303 q^{36} +11.1937 q^{37} -1.89948 q^{38} +9.69283 q^{39} +16.8236 q^{40} -1.13870 q^{41} -8.86932 q^{42} -6.13987 q^{43} +17.0783 q^{44} -12.4213 q^{45} -2.60597 q^{46} -2.54799 q^{47} -34.0645 q^{48} -5.75552 q^{49} +1.78369 q^{50} -12.8750 q^{51} -16.3313 q^{52} -12.6824 q^{53} +23.6182 q^{54} -7.04059 q^{55} +9.02124 q^{56} -2.14320 q^{57} +5.91850 q^{58} +12.1366 q^{59} +31.4442 q^{60} +7.65047 q^{61} -0.942549 q^{62} -6.66064 q^{63} +14.4624 q^{64} +6.73263 q^{65} +26.9066 q^{66} -9.28979 q^{67} +21.6930 q^{68} -2.94034 q^{69} -6.16063 q^{70} -3.24012 q^{71} -48.2828 q^{72} -4.55854 q^{73} -29.7138 q^{74} +2.01255 q^{75} +3.61104 q^{76} -3.77535 q^{77} -25.7297 q^{78} -2.45178 q^{79} -23.6612 q^{80} +8.73667 q^{81} +3.02268 q^{82} +7.49212 q^{83} +16.8612 q^{84} -8.94300 q^{85} +16.2983 q^{86} +6.67790 q^{87} -27.3675 q^{88} +11.6238 q^{89} +32.9725 q^{90} +3.61022 q^{91} +4.95414 q^{92} -1.06349 q^{93} +6.76364 q^{94} -1.48866 q^{95} +41.9834 q^{96} -10.6560 q^{97} +15.2781 q^{98} +20.2062 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9}+O(q^{10}) \) 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9	\( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9} - 46 q^{10} - 32 q^{11} - 72 q^{12} - 80 q^{13} - 22 q^{14} - 43 q^{15} + 122 q^{16} - 61 q^{17} - 88 q^{18} - 43 q^{19} - 41 q^{20} - 61 q^{21} - 93 q^{22} - 60 q^{23} - 41 q^{24} + 26 q^{25} - 9 q^{26} - 93 q^{27} - 126 q^{28} - 47 q^{29} - 36 q^{30} - 100 q^{31} - 114 q^{32} - 133 q^{33} - 75 q^{34} - 37 q^{35} + 75 q^{36} - 264 q^{37} - 35 q^{38} - 47 q^{39} - 118 q^{40} - 72 q^{41} - 64 q^{42} - 107 q^{43} - 59 q^{44} - 69 q^{45} - 111 q^{46} - 54 q^{47} - 135 q^{48} + 33 q^{49} - 42 q^{50} - 26 q^{51} - 173 q^{52} - 103 q^{53} - 28 q^{54} - 78 q^{55} - 44 q^{56} - 205 q^{57} - 189 q^{58} - 38 q^{59} - 105 q^{60} - 108 q^{61} - 14 q^{62} - 116 q^{63} + 39 q^{64} - 146 q^{65} + 5 q^{66} - 206 q^{67} - 62 q^{68} - 55 q^{69} - 125 q^{70} - 78 q^{71} - 225 q^{72} - 326 q^{73} + 3 q^{74} - 95 q^{75} - 84 q^{76} - 79 q^{77} - 86 q^{78} - 117 q^{79} - 39 q^{80} + q^{81} - 96 q^{82} - 23 q^{83} - 57 q^{84} - 224 q^{85} - 7 q^{86} - 45 q^{87} - 250 q^{88} - 104 q^{89} - 36 q^{90} - 96 q^{91} - 137 q^{92} - 155 q^{93} - 48 q^{94} - 38 q^{95} - 33 q^{96} - 447 q^{97} - 46 q^{98} - 94 q^{99}+O(q^{100}) \) 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9 - 46 * q^10 - 32 * q^11 - 72 * q^12 - 80 * q^13 - 22 * q^14 - 43 * q^15 + 122 * q^16 - 61 * q^17 - 88 * q^18 - 43 * q^19 - 41 * q^20 - 61 * q^21 - 93 * q^22 - 60 * q^23 - 41 * q^24 + 26 * q^25 - 9 * q^26 - 93 * q^27 - 126 * q^28 - 47 * q^29 - 36 * q^30 - 100 * q^31 - 114 * q^32 - 133 * q^33 - 75 * q^34 - 37 * q^35 + 75 * q^36 - 264 * q^37 - 35 * q^38 - 47 * q^39 - 118 * q^40 - 72 * q^41 - 64 * q^42 - 107 * q^43 - 59 * q^44 - 69 * q^45 - 111 * q^46 - 54 * q^47 - 135 * q^48 + 33 * q^49 - 42 * q^50 - 26 * q^51 - 173 * q^52 - 103 * q^53 - 28 * q^54 - 78 * q^55 - 44 * q^56 - 205 * q^57 - 189 * q^58 - 38 * q^59 - 105 * q^60 - 108 * q^61 - 14 * q^62 - 116 * q^63 + 39 * q^64 - 146 * q^65 + 5 * q^66 - 206 * q^67 - 62 * q^68 - 55 * q^69 - 125 * q^70 - 78 * q^71 - 225 * q^72 - 326 * q^73 + 3 * q^74 - 95 * q^75 - 84 * q^76 - 79 * q^77 - 86 * q^78 - 117 * q^79 - 39 * q^80 + q^81 - 96 * q^82 - 23 * q^83 - 57 * q^84 - 224 * q^85 - 7 * q^86 - 45 * q^87 - 250 * q^88 - 104 * q^89 - 36 * q^90 - 96 * q^91 - 137 * q^92 - 155 * q^93 - 48 * q^94 - 38 * q^95 - 33 * q^96 - 447 * q^97 - 46 * q^98 - 94 * q^99

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