LMFDB - Embedded newform 4009.2.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.66661 q^{2} -1.96401 q^{3} +5.11081 q^{4} -2.76483 q^{5} +5.23725 q^{6} +2.50280 q^{7} -8.29531 q^{8} +0.857341 q^{9} +O(q^{10})$	\(q-2.66661 q^{2} -1.96401 q^{3} +5.11081 q^{4} -2.76483 q^{5} +5.23725 q^{6} +2.50280 q^{7} -8.29531 q^{8} +0.857341 q^{9} +7.37272 q^{10} -4.45765 q^{11} -10.0377 q^{12} -2.71084 q^{13} -6.67400 q^{14} +5.43016 q^{15} +11.8987 q^{16} -6.37246 q^{17} -2.28619 q^{18} +1.00000 q^{19} -14.1305 q^{20} -4.91554 q^{21} +11.8868 q^{22} +4.98973 q^{23} +16.2921 q^{24} +2.64428 q^{25} +7.22875 q^{26} +4.20821 q^{27} +12.7914 q^{28} +6.14099 q^{29} -14.4801 q^{30} -4.08470 q^{31} -15.1387 q^{32} +8.75488 q^{33} +16.9929 q^{34} -6.91983 q^{35} +4.38170 q^{36} -0.469337 q^{37} -2.66661 q^{38} +5.32412 q^{39} +22.9351 q^{40} -6.42681 q^{41} +13.1078 q^{42} +9.76238 q^{43} -22.7822 q^{44} -2.37040 q^{45} -13.3057 q^{46} +13.1628 q^{47} -23.3693 q^{48} -0.735969 q^{49} -7.05126 q^{50} +12.5156 q^{51} -13.8546 q^{52} -12.3526 q^{53} -11.2216 q^{54} +12.3246 q^{55} -20.7615 q^{56} -1.96401 q^{57} -16.3756 q^{58} +8.81616 q^{59} +27.7525 q^{60} -10.6230 q^{61} +10.8923 q^{62} +2.14576 q^{63} +16.5715 q^{64} +7.49501 q^{65} -23.3458 q^{66} +5.32202 q^{67} -32.5684 q^{68} -9.79988 q^{69} +18.4525 q^{70} -14.9279 q^{71} -7.11191 q^{72} +2.77902 q^{73} +1.25154 q^{74} -5.19339 q^{75} +5.11081 q^{76} -11.1566 q^{77} -14.1973 q^{78} +4.85685 q^{79} -32.8980 q^{80} -10.8370 q^{81} +17.1378 q^{82} +14.7162 q^{83} -25.1224 q^{84} +17.6188 q^{85} -26.0325 q^{86} -12.0610 q^{87} +36.9776 q^{88} +6.53154 q^{89} +6.32093 q^{90} -6.78470 q^{91} +25.5015 q^{92} +8.02240 q^{93} -35.1001 q^{94} -2.76483 q^{95} +29.7325 q^{96} +8.97717 q^{97} +1.96254 q^{98} -3.82173 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * 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 $

$2$	−2.66661	−1.88558	−0.942789	−	0.333390i	$-0.891807\pi$
$2$	−2.66661	−1.88558	−0.942789	+	0.333390i	$0.891807\pi$
$3$	−1.96401	−1.13392	−0.566961	−	0.823744i	$-0.691881\pi$
$3$	−1.96401	−1.13392	−0.566961	+	0.823744i	$0.691881\pi$
$4$	5.11081	2.55540
$5$	−2.76483	−1.23647	−0.618234	−	0.785994i	$-0.712152\pi$
$5$	−2.76483	−1.23647	−0.618234	+	0.785994i	$0.712152\pi$
$6$	5.23725	2.13810
$7$	2.50280	0.945971	0.472986	−	0.881070i	$-0.343176\pi$
$7$	2.50280	0.945971	0.472986	+	0.881070i	$0.343176\pi$
$8$	−8.29531	−2.93283
$9$	0.857341	0.285780
$10$	7.37272	2.33146
$11$	−4.45765	−1.34403	−0.672016	−	0.740537i	$-0.734571\pi$
$11$	−4.45765	−1.34403	−0.672016	+	0.740537i	$0.734571\pi$
$12$	−10.0377	−2.89763
$13$	−2.71084	−0.751851	−0.375926	−	0.926650i	$-0.622675\pi$
$13$	−2.71084	−0.751851	−0.375926	+	0.926650i	$0.622675\pi$
$14$	−6.67400	−1.78370
$15$	5.43016	1.40206
$16$	11.8987	2.97468
$17$	−6.37246	−1.54555	−0.772774	−	0.634681i	$-0.781132\pi$
$17$	−6.37246	−1.54555	−0.772774	+	0.634681i	$0.781132\pi$
$18$	−2.28619	−0.538861
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−14.1305	−3.15968
$21$	−4.91554	−1.07266
$22$	11.8868	2.53428
$23$	4.98973	1.04043	0.520215	−	0.854035i	$-0.325852\pi$
$23$	4.98973	1.04043	0.520215	+	0.854035i	$0.325852\pi$
$24$	16.2921	3.32561
$25$	2.64428	0.528856
$26$	7.22875	1.41767
$27$	4.20821	0.809870
$28$	12.7914	2.41734
$29$	6.14099	1.14035	0.570177	−	0.821522i	$-0.306875\pi$
$29$	6.14099	1.14035	0.570177	+	0.821522i	$0.306875\pi$
$30$	−14.4801	−2.64369
$31$	−4.08470	−0.733634	−0.366817	−	0.930293i	$-0.619552\pi$
$31$	−4.08470	−0.733634	−0.366817	+	0.930293i	$0.619552\pi$
$32$	−15.1387	−2.67616
$33$	8.75488	1.52403
$34$	16.9929	2.91425
$35$	−6.91983	−1.16966
$36$	4.38170	0.730284
$37$	−0.469337	−0.0771585	−0.0385793	−	0.999256i	$-0.512283\pi$
$37$	−0.469337	−0.0771585	−0.0385793	+	0.999256i	$0.512283\pi$
$38$	−2.66661	−0.432581
$39$	5.32412	0.852541
$40$	22.9351	3.62636
$41$	−6.42681	−1.00370	−0.501849	−	0.864955i	$-0.667347\pi$
$41$	−6.42681	−1.00370	−0.501849	+	0.864955i	$0.667347\pi$
$42$	13.1078	2.02258
$43$	9.76238	1.48875	0.744375	−	0.667762i	$-0.232748\pi$
$43$	9.76238	1.48875	0.744375	+	0.667762i	$0.232748\pi$
$44$	−22.7822	−3.43454
$45$	−2.37040	−0.353358
$46$	−13.3057	−1.96181
$47$	13.1628	1.92000	0.959998	−	0.280006i	$-0.0903366\pi$
$47$	13.1628	1.92000	0.959998	+	0.280006i	$0.0903366\pi$
$48$	−23.3693	−3.37306
$49$	−0.735969	−0.105138
$50$	−7.05126	−0.997198
$51$	12.5156	1.75253
$52$	−13.8546	−1.92128
$53$	−12.3526	−1.69676	−0.848379	−	0.529389i	$-0.822421\pi$
$53$	−12.3526	−1.69676	−0.848379	+	0.529389i	$0.822421\pi$
$54$	−11.2216	−1.52707
$55$	12.3246	1.66185
$56$	−20.7615	−2.77438
$57$	−1.96401	−0.260140
$58$	−16.3756	−2.15023
$59$	8.81616	1.14777	0.573883	−	0.818937i	$-0.305436\pi$
$59$	8.81616	1.14777	0.573883	+	0.818937i	$0.305436\pi$
$60$	27.7525	3.58283
$61$	−10.6230	−1.36013	−0.680065	−	0.733152i	$-0.738048\pi$
$61$	−10.6230	−1.36013	−0.680065	+	0.733152i	$0.738048\pi$
$62$	10.8923	1.38332
$63$	2.14576	0.270340
$64$	16.5715	2.07143
$65$	7.49501	0.929641
$66$	−23.3458	−2.87367
$67$	5.32202	0.650188	0.325094	−	0.945682i	$-0.394604\pi$
$67$	5.32202	0.650188	0.325094	+	0.945682i	$0.394604\pi$
$68$	−32.5684	−3.94950
$69$	−9.79988	−1.17977
$70$	18.4525	2.20549
$71$	−14.9279	−1.77161	−0.885806	−	0.464055i	$-0.846394\pi$
$71$	−14.9279	−1.77161	−0.885806	+	0.464055i	$0.846394\pi$
$72$	−7.11191	−0.838146
$73$	2.77902	0.325259	0.162630	−	0.986687i	$-0.448002\pi$
$73$	2.77902	0.325259	0.162630	+	0.986687i	$0.448002\pi$
$74$	1.25154	0.145488
$75$	−5.19339	−0.599681
$76$	5.11081	0.586250
$77$	−11.1566	−1.27142
$78$	−14.1973	−1.60753
$79$	4.85685	0.546438	0.273219	−	0.961952i	$-0.411912\pi$
$79$	4.85685	0.546438	0.273219	+	0.961952i	$0.411912\pi$
$80$	−32.8980	−3.67811
$81$	−10.8370	−1.20411
$82$	17.1378	1.89255
$83$	14.7162	1.61532	0.807658	−	0.589652i	$-0.200735\pi$
$83$	14.7162	1.61532	0.807658	+	0.589652i	$0.200735\pi$
$84$	−25.1224	−2.74107
$85$	17.6188	1.91102
$86$	−26.0325	−2.80715
$87$	−12.0610	−1.29307
$88$	36.9776	3.94182
$89$	6.53154	0.692342	0.346171	−	0.938171i	$-0.387482\pi$
$89$	6.53154	0.692342	0.346171	+	0.938171i	$0.387482\pi$
$90$	6.32093	0.666285
$91$	−6.78470	−0.711230
$92$	25.5015	2.65872
$93$	8.02240	0.831884
$94$	−35.1001	−3.62030
$95$	−2.76483	−0.283665
$96$	29.7325	3.03456
$97$	8.97717	0.911494	0.455747	−	0.890109i	$-0.349372\pi$
$97$	8.97717	0.911494	0.455747	+	0.890109i	$0.349372\pi$
$98$	1.96254	0.198247
$99$	−3.82173	−0.384098
$100$	13.5144	1.35144
$101$	11.0461	1.09913	0.549565	−	0.835451i	$-0.314793\pi$
$101$	11.0461	1.09913	0.549565	+	0.835451i	$0.314793\pi$
$102$	−33.3742	−3.30454
$103$	−12.7577	−1.25705	−0.628526	−	0.777788i	$-0.716342\pi$
$103$	−12.7577	−1.25705	−0.628526	+	0.777788i	$0.716342\pi$
$104$	22.4872	2.20506
$105$	13.5906	1.32631
$106$	32.9395	3.19937
$107$	5.30078	0.512446	0.256223	−	0.966618i	$-0.417522\pi$
$107$	5.30078	0.512446	0.256223	+	0.966618i	$0.417522\pi$
$108$	21.5073	2.06954
$109$	0.362031	0.0346763	0.0173381	−	0.999850i	$-0.494481\pi$
$109$	0.362031	0.0346763	0.0173381	+	0.999850i	$0.494481\pi$
$110$	−32.8650	−3.13355
$111$	0.921783	0.0874918
$112$	29.7802	2.81397
$113$	−17.6390	−1.65934	−0.829670	−	0.558254i	$-0.811471\pi$
$113$	−17.6390	−1.65934	−0.829670	+	0.558254i	$0.811471\pi$
$114$	5.23725	0.490514
$115$	−13.7957	−1.28646
$116$	31.3854	2.91406
$117$	−2.32411	−0.214864
$118$	−23.5092	−2.16420
$119$	−15.9490	−1.46204
$120$	−45.0448	−4.11201
$121$	8.87064	0.806422
$122$	28.3273	2.56463
$123$	12.6223	1.13812
$124$	−20.8761	−1.87473
$125$	6.51317	0.582556
$126$	−5.72190	−0.509747
$127$	19.9917	1.77397	0.886986	−	0.461796i	$-0.152795\pi$
$127$	19.9917	1.77397	0.886986	+	0.461796i	$0.152795\pi$
$128$	−13.9123	−1.22968
$129$	−19.1734	−1.68813
$130$	−19.9863	−1.75291
$131$	−1.66459	−0.145436	−0.0727181	−	0.997353i	$-0.523167\pi$
$131$	−1.66459	−0.145436	−0.0727181	+	0.997353i	$0.523167\pi$
$132$	44.7445	3.89451
$133$	2.50280	0.217021
$134$	−14.1917	−1.22598
$135$	−11.6350	−1.00138
$136$	52.8615	4.53284
$137$	11.7208	1.00138	0.500690	−	0.865627i	$-0.333080\pi$
$137$	11.7208	1.00138	0.500690	+	0.865627i	$0.333080\pi$
$138$	26.1325	2.22454
$139$	9.15570	0.776577	0.388288	−	0.921538i	$-0.373066\pi$
$139$	9.15570	0.776577	0.388288	+	0.921538i	$0.373066\pi$
$140$	−35.3659	−2.98896
$141$	−25.8520	−2.17713
$142$	39.8068	3.34051
$143$	12.0840	1.01051
$144$	10.2013	0.850106
$145$	−16.9788	−1.41001
$146$	−7.41055	−0.613302
$147$	1.44545	0.119219
$148$	−2.39869	−0.197171
$149$	11.0107	0.902031	0.451015	−	0.892516i	$-0.351062\pi$
$149$	11.0107	0.902031	0.451015	+	0.892516i	$0.351062\pi$
$150$	13.8487	1.13075
$151$	−3.53329	−0.287535	−0.143768	−	0.989611i	$-0.545922\pi$
$151$	−3.53329	−0.287535	−0.143768	+	0.989611i	$0.545922\pi$
$152$	−8.29531	−0.672838
$153$	−5.46337	−0.441687
$154$	29.7504	2.39735
$155$	11.2935	0.907115
$156$	27.2105	2.17859
$157$	23.2044	1.85191	0.925957	−	0.377630i	$-0.123261\pi$
$157$	23.2044	1.85191	0.925957	+	0.377630i	$0.123261\pi$
$158$	−12.9513	−1.03035
$159$	24.2606	1.92399
$160$	41.8558	3.30899
$161$	12.4883	0.984217
$162$	28.8980	2.27044
$163$	11.5374	0.903682	0.451841	−	0.892099i	$-0.350767\pi$
$163$	11.5374	0.903682	0.451841	+	0.892099i	$0.350767\pi$
$164$	−32.8462	−2.56486
$165$	−24.2057	−1.88441
$166$	−39.2424	−3.04580
$167$	−13.6959	−1.05982	−0.529912	−	0.848053i	$-0.677775\pi$
$167$	−13.6959	−1.05982	−0.529912	+	0.848053i	$0.677775\pi$
$168$	40.7759	3.14593
$169$	−5.65135	−0.434719
$170$	−46.9824	−3.60338
$171$	0.857341	0.0655625
$172$	49.8936	3.80435
$173$	−3.42622	−0.260491	−0.130245	−	0.991482i	$-0.541577\pi$
$173$	−3.42622	−0.260491	−0.130245	+	0.991482i	$0.541577\pi$
$174$	32.1619	2.43819
$175$	6.61811	0.500282
$176$	−53.0404	−3.99807
$177$	−17.3150	−1.30148
$178$	−17.4171	−1.30546
$179$	5.32355	0.397901	0.198950	−	0.980010i	$-0.436247\pi$
$179$	5.32355	0.397901	0.198950	+	0.980010i	$0.436247\pi$
$180$	−12.1147	−0.902974
$181$	−14.2684	−1.06056	−0.530280	−	0.847823i	$-0.677913\pi$
$181$	−14.2684	−1.06056	−0.530280	+	0.847823i	$0.677913\pi$
$182$	18.0921	1.34108
$183$	20.8636	1.54228
$184$	−41.3913	−3.05141
$185$	1.29764	0.0954041
$186$	−21.3926	−1.56858
$187$	28.4062	2.07727
$188$	67.2727	4.90637
$189$	10.5323	0.766114
$190$	7.37272	0.534873
$191$	−16.2012	−1.17227	−0.586137	−	0.810212i	$-0.699352\pi$
$191$	−16.2012	−1.17227	−0.586137	+	0.810212i	$0.699352\pi$
$192$	−32.5465	−2.34884
$193$	−24.6621	−1.77522	−0.887610	−	0.460596i	$-0.847636\pi$
$193$	−24.6621	−1.77522	−0.887610	+	0.460596i	$0.847636\pi$
$194$	−23.9386	−1.71869
$195$	−14.7203	−1.05414
$196$	−3.76140	−0.268671
$197$	−16.5289	−1.17763	−0.588816	−	0.808267i	$-0.700406\pi$
$197$	−16.5289	−1.17763	−0.588816	+	0.808267i	$0.700406\pi$
$198$	10.1911	0.724246
$199$	−5.01692	−0.355640	−0.177820	−	0.984063i	$-0.556904\pi$
$199$	−5.01692	−0.355640	−0.177820	+	0.984063i	$0.556904\pi$
$200$	−21.9351	−1.55105
$201$	−10.4525	−0.737263
$202$	−29.4557	−2.07250
$203$	15.3697	1.07874
$204$	63.9647	4.47843
$205$	17.7690	1.24104
$206$	34.0198	2.37027
$207$	4.27790	0.297334
$208$	−32.2556	−2.23652
$209$	−4.45765	−0.308342
$210$	−36.2409	−2.50086
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−63.1317	−4.33590
$213$	29.3185	2.00887
$214$	−14.1351	−0.966256
$215$	−26.9913	−1.84079
$216$	−34.9084	−2.37521
$217$	−10.2232	−0.693996
$218$	−0.965396	−0.0653848
$219$	−5.45802	−0.368819
$220$	62.9889	4.24671
$221$	17.2747	1.16202
$222$	−2.45804	−0.164973
$223$	16.1972	1.08464	0.542322	−	0.840171i	$-0.317545\pi$
$223$	16.1972	1.08464	0.542322	+	0.840171i	$0.317545\pi$
$224$	−37.8891	−2.53157
$225$	2.26705	0.151136
$226$	47.0364	3.12881
$227$	20.2399	1.34337	0.671684	−	0.740838i	$-0.265571\pi$
$227$	20.2399	1.34337	0.671684	+	0.740838i	$0.265571\pi$
$228$	−10.0377	−0.664762
$229$	11.7308	0.775192	0.387596	−	0.921829i	$-0.373305\pi$
$229$	11.7308	0.775192	0.387596	+	0.921829i	$0.373305\pi$
$230$	36.7879	2.42572
$231$	21.9117	1.44169
$232$	−50.9414	−3.34447
$233$	14.8441	0.972472	0.486236	−	0.873828i	$-0.338370\pi$
$233$	14.8441	0.972472	0.486236	+	0.873828i	$0.338370\pi$
$234$	6.19750	0.405143
$235$	−36.3930	−2.37402
$236$	45.0577	2.93300
$237$	−9.53890	−0.619618
$238$	42.5298	2.75680
$239$	−1.39599	−0.0902988	−0.0451494	−	0.998980i	$-0.514376\pi$
$239$	−1.39599	−0.0902988	−0.0451494	+	0.998980i	$0.514376\pi$
$240$	64.6120	4.17069
$241$	−8.97454	−0.578101	−0.289051	−	0.957314i	$-0.593340\pi$
$241$	−8.97454	−0.578101	−0.289051	+	0.957314i	$0.593340\pi$
$242$	−23.6545	−1.52057
$243$	8.65935	0.555498
$244$	−54.2919	−3.47568
$245$	2.03483	0.130000
$246$	−33.6588	−2.14601
$247$	−2.71084	−0.172487
$248$	33.8838	2.15163
$249$	−28.9028	−1.83164
$250$	−17.3681	−1.09845
$251$	19.5062	1.23122	0.615609	−	0.788052i	$-0.288910\pi$
$251$	19.5062	1.23122	0.615609	+	0.788052i	$0.288910\pi$
$252$	10.9665	0.690828
$253$	−22.2425	−1.39837
$254$	−53.3099	−3.34496
$255$	−34.6034	−2.16695
$256$	3.95565	0.247228
$257$	16.4529	1.02630	0.513152	−	0.858298i	$-0.328478\pi$
$257$	16.4529	1.02630	0.513152	+	0.858298i	$0.328478\pi$
$258$	51.1280	3.18309
$259$	−1.17466	−0.0729898
$260$	38.3055	2.37561
$261$	5.26493	0.325891
$262$	4.43882	0.274231
$263$	−3.75134	−0.231317	−0.115659	−	0.993289i	$-0.536898\pi$
$263$	−3.75134	−0.231317	−0.115659	+	0.993289i	$0.536898\pi$
$264$	−72.6244	−4.46972
$265$	34.1528	2.09799
$266$	−6.67400	−0.409209
$267$	−12.8280	−0.785062
$268$	27.1998	1.66149
$269$	14.5866	0.889361	0.444681	−	0.895689i	$-0.353317\pi$
$269$	14.5866	0.889361	0.444681	+	0.895689i	$0.353317\pi$
$270$	31.0259	1.88818
$271$	−11.1714	−0.678612	−0.339306	−	0.940676i	$-0.610192\pi$
$271$	−11.1714	−0.678612	−0.339306	+	0.940676i	$0.610192\pi$
$272$	−75.8242	−4.59752
$273$	13.3252	0.806480
$274$	−31.2549	−1.88818
$275$	−11.7873	−0.710799
$276$	−50.0853	−3.01478
$277$	16.8162	1.01039	0.505195	−	0.863005i	$-0.331421\pi$
$277$	16.8162	1.01039	0.505195	+	0.863005i	$0.331421\pi$
$278$	−24.4147	−1.46430
$279$	−3.50198	−0.209658
$280$	57.4021	3.43043
$281$	−9.10242	−0.543005	−0.271502	−	0.962438i	$-0.587520\pi$
$281$	−9.10242	−0.543005	−0.271502	+	0.962438i	$0.587520\pi$
$282$	68.9371	4.10514
$283$	−15.5475	−0.924203	−0.462102	−	0.886827i	$-0.652904\pi$
$283$	−15.5475	−0.924203	−0.462102	+	0.886827i	$0.652904\pi$
$284$	−76.2935	−4.52718
$285$	5.43016	0.321655
$286$	−32.2232	−1.90540
$287$	−16.0850	−0.949470
$288$	−12.9790	−0.764795
$289$	23.6083	1.38872
$290$	45.2758	2.65869
$291$	−17.6313	−1.03356
$292$	14.2030	0.831169
$293$	−23.7740	−1.38889	−0.694447	−	0.719544i	$-0.744351\pi$
$293$	−23.7740	−1.38889	−0.694447	+	0.719544i	$0.744351\pi$
$294$	−3.85446	−0.224796
$295$	−24.3752	−1.41918
$296$	3.89330	0.226293
$297$	−18.7587	−1.08849
$298$	−29.3612	−1.70085
$299$	−13.5263	−0.782249
$300$	−26.5424	−1.53243
$301$	24.4333	1.40831
$302$	9.42191	0.542170
$303$	−21.6947	−1.24633
$304$	11.8987	0.682439
$305$	29.3707	1.68176
$306$	14.5687	0.832836
$307$	21.0146	1.19937	0.599684	−	0.800237i	$-0.295293\pi$
$307$	21.0146	1.19937	0.599684	+	0.800237i	$0.295293\pi$
$308$	−57.0194	−3.24898
$309$	25.0563	1.42540
$310$	−30.1153	−1.71044
$311$	22.2164	1.25977	0.629887	−	0.776687i	$-0.283101\pi$
$311$	22.2164	1.25977	0.629887	+	0.776687i	$0.283101\pi$
$312$	−44.1652	−2.50036
$313$	6.69435	0.378387	0.189194	−	0.981940i	$-0.439413\pi$
$313$	6.69435	0.378387	0.189194	+	0.981940i	$0.439413\pi$
$314$	−61.8771	−3.49193
$315$	−5.93265	−0.334267
$316$	24.8224	1.39637
$317$	−25.1536	−1.41277	−0.706384	−	0.707829i	$-0.749675\pi$
$317$	−25.1536	−1.41277	−0.706384	+	0.707829i	$0.749675\pi$
$318$	−64.6936	−3.62784
$319$	−27.3744	−1.53267
$320$	−45.8172	−2.56126
$321$	−10.4108	−0.581074
$322$	−33.3015	−1.85582
$323$	−6.37246	−0.354573
$324$	−55.3858	−3.07699
$325$	−7.16821	−0.397621
$326$	−30.7658	−1.70396
$327$	−0.711033	−0.0393202
$328$	53.3124	2.94368
$329$	32.9440	1.81626
$330$	64.5472	3.55321
$331$	12.2936	0.675716	0.337858	−	0.941197i	$-0.390298\pi$
$331$	12.2936	0.675716	0.337858	+	0.941197i	$0.390298\pi$
$332$	75.2118	4.12778
$333$	−0.402382	−0.0220504
$334$	36.5217	1.99838
$335$	−14.7145	−0.803937
$336$	−58.4887	−3.19082
$337$	−21.4655	−1.16930	−0.584649	−	0.811286i	$-0.698768\pi$
$337$	−21.4655	−1.16930	−0.584649	+	0.811286i	$0.698768\pi$
$338$	15.0700	0.819697
$339$	34.6432	1.88156
$340$	90.0461	4.88344
$341$	18.2082	0.986027
$342$	−2.28619	−0.123623
$343$	−19.3616	−1.04543
$344$	−80.9820	−4.36625
$345$	27.0950	1.45875
$346$	9.13639	0.491176
$347$	−18.4761	−0.991848	−0.495924	−	0.868366i	$-0.665170\pi$
$347$	−18.4761	−0.991848	−0.495924	+	0.868366i	$0.665170\pi$
$348$	−61.6414	−3.30432
$349$	7.13363	0.381854	0.190927	−	0.981604i	$-0.438851\pi$
$349$	7.13363	0.381854	0.190927	+	0.981604i	$0.438851\pi$
$350$	−17.6479	−0.943321
$351$	−11.4078	−0.608902
$352$	67.4829	3.59685
$353$	−32.9457	−1.75352	−0.876760	−	0.480927i	$-0.840300\pi$
$353$	−32.9457	−1.75352	−0.876760	+	0.480927i	$0.840300\pi$
$354$	46.1724	2.45404
$355$	41.2730	2.19054
$356$	33.3814	1.76921
$357$	31.3241	1.65785
$358$	−14.1958	−0.750273
$359$	21.9548	1.15873	0.579365	−	0.815068i	$-0.303301\pi$
$359$	21.9548	1.15873	0.579365	+	0.815068i	$0.303301\pi$
$360$	19.6632	1.03634
$361$	1.00000	0.0526316
$362$	38.0482	1.99977
$363$	−17.4220	−0.914420
$364$	−34.6753	−1.81748
$365$	−7.68350	−0.402173
$366$	−55.6351	−2.90809
$367$	−11.3445	−0.592176	−0.296088	−	0.955161i	$-0.595682\pi$
$367$	−11.3445	−0.592176	−0.296088	+	0.955161i	$0.595682\pi$
$368$	59.3715	3.09495
$369$	−5.50996	−0.286837
$370$	−3.46029	−0.179892
$371$	−30.9161	−1.60508
$372$	41.0009	2.12580
$373$	14.9759	0.775420	0.387710	−	0.921781i	$-0.373266\pi$
$373$	14.9759	0.775420	0.387710	+	0.921781i	$0.373266\pi$
$374$	−75.7482	−3.91685
$375$	−12.7919	−0.660573
$376$	−109.190	−5.63103
$377$	−16.6472	−0.857377
$378$	−28.0856	−1.44457
$379$	−37.3644	−1.91928	−0.959641	−	0.281227i	$-0.909259\pi$
$379$	−37.3644	−1.91928	−0.959641	+	0.281227i	$0.909259\pi$
$380$	−14.1305	−0.724880
$381$	−39.2638	−2.01155
$382$	43.2022	2.21041
$383$	35.5848	1.81830	0.909149	−	0.416471i	$-0.136733\pi$
$383$	35.5848	1.81830	0.909149	+	0.416471i	$0.136733\pi$
$384$	27.3238	1.39436
$385$	30.8462	1.57207
$386$	65.7643	3.34731
$387$	8.36969	0.425455
$388$	45.8806	2.32923
$389$	25.4532	1.29053	0.645264	−	0.763960i	$-0.276747\pi$
$389$	25.4532	1.29053	0.645264	+	0.763960i	$0.276747\pi$
$390$	39.2532	1.98766
$391$	−31.7968	−1.60804
$392$	6.10509	0.308354
$393$	3.26928	0.164913
$394$	44.0760	2.22052
$395$	−13.4283	−0.675653
$396$	−19.5321	−0.981525
$397$	−10.3768	−0.520796	−0.260398	−	0.965501i	$-0.583854\pi$
$397$	−10.3768	−0.520796	−0.260398	+	0.965501i	$0.583854\pi$
$398$	13.3782	0.670587
$399$	−4.91554	−0.246085
$400$	31.4636	1.57318
$401$	−18.2985	−0.913784	−0.456892	−	0.889522i	$-0.651037\pi$
$401$	−18.2985	−0.913784	−0.456892	+	0.889522i	$0.651037\pi$
$402$	27.8728	1.39017
$403$	11.0730	0.551583
$404$	56.4546	2.80872
$405$	29.9624	1.48884
$406$	−40.9850	−2.03405
$407$	2.09214	0.103704
$408$	−103.821	−5.13989
$409$	−29.8158	−1.47429	−0.737147	−	0.675732i	$-0.763828\pi$
$409$	−29.8158	−1.47429	−0.737147	+	0.675732i	$0.763828\pi$
$410$	−47.3830	−2.34008
$411$	−23.0199	−1.13549
$412$	−65.2021	−3.21228
$413$	22.0651	1.08575
$414$	−11.4075	−0.560647
$415$	−40.6878	−1.99729
$416$	41.0385	2.01208
$417$	−17.9819	−0.880578
$418$	11.8868	0.581403
$419$	22.0042	1.07497	0.537487	−	0.843272i	$-0.319374\pi$
$419$	22.0042	1.07497	0.537487	+	0.843272i	$0.319374\pi$
$420$	69.4590	3.38925
$421$	−22.6175	−1.10231	−0.551155	−	0.834403i	$-0.685813\pi$
$421$	−22.6175	−1.10231	−0.551155	+	0.834403i	$0.685813\pi$
$422$	−2.66661	−0.129809
$423$	11.2850	0.548697
$424$	102.469	4.97631
$425$	−16.8506	−0.817372
$426$	−78.1810	−3.78788
$427$	−26.5872	−1.28664
$428$	27.0913	1.30951
$429$	−23.7331	−1.14584
$430$	71.9753	3.47096
$431$	−34.6128	−1.66724	−0.833619	−	0.552340i	$-0.813735\pi$
$431$	−34.6128	−1.66724	−0.833619	+	0.552340i	$0.813735\pi$
$432$	50.0724	2.40911
$433$	−1.38682	−0.0666465	−0.0333233	−	0.999445i	$-0.510609\pi$
$433$	−1.38682	−0.0666465	−0.0333233	+	0.999445i	$0.510609\pi$
$434$	27.2613	1.30858
$435$	33.3465	1.59884
$436$	1.85027	0.0886119
$437$	4.98973	0.238691
$438$	14.5544	0.695437
$439$	10.9221	0.521284	0.260642	−	0.965435i	$-0.416066\pi$
$439$	10.9221	0.521284	0.260642	+	0.965435i	$0.416066\pi$
$440$	−102.237	−4.87394
$441$	−0.630976	−0.0300465
$442$	−46.0649	−2.19108
$443$	−7.67323	−0.364566	−0.182283	−	0.983246i	$-0.558349\pi$
$443$	−7.67323	−0.364566	−0.182283	+	0.983246i	$0.558349\pi$
$444$	4.71106	0.223577
$445$	−18.0586	−0.856059
$446$	−43.1916	−2.04518
$447$	−21.6251	−1.02283
$448$	41.4751	1.95951
$449$	11.4850	0.542008	0.271004	−	0.962578i	$-0.412644\pi$
$449$	11.4850	0.542008	0.271004	+	0.962578i	$0.412644\pi$
$450$	−6.04533	−0.284980
$451$	28.6485	1.34900
$452$	−90.1497	−4.24028
$453$	6.93942	0.326043
$454$	−53.9719	−2.53303
$455$	18.7585	0.879414
$456$	16.2921	0.762947
$457$	−19.2851	−0.902117	−0.451059	−	0.892494i	$-0.648953\pi$
$457$	−19.2851	−0.902117	−0.451059	+	0.892494i	$0.648953\pi$
$458$	−31.2814	−1.46169
$459$	−26.8166	−1.25169
$460$	−70.5074	−3.28742
$461$	21.5088	1.00176	0.500882	−	0.865516i	$-0.333009\pi$
$461$	21.5088	1.00176	0.500882	+	0.865516i	$0.333009\pi$
$462$	−58.4301	−2.71841
$463$	−19.9383	−0.926613	−0.463307	−	0.886198i	$-0.653337\pi$
$463$	−19.9383	−0.926613	−0.463307	+	0.886198i	$0.653337\pi$
$464$	73.0701	3.39219
$465$	−22.1805	−1.02860
$466$	−39.5835	−1.83367
$467$	9.98586	0.462091	0.231045	−	0.972943i	$-0.425785\pi$
$467$	9.98586	0.462091	0.231045	+	0.972943i	$0.425785\pi$
$468$	−11.8781	−0.549065
$469$	13.3200	0.615059
$470$	97.0459	4.47639
$471$	−45.5737	−2.09993
$472$	−73.1327	−3.36621
$473$	−43.5173	−2.00093
$474$	25.4365	1.16834
$475$	2.64428	0.121328
$476$	−81.5124	−3.73611
$477$	−10.5904	−0.484900
$478$	3.72255	0.170265
$479$	−8.11650	−0.370852	−0.185426	−	0.982658i	$-0.559367\pi$
$479$	−8.11650	−0.370852	−0.185426	+	0.982658i	$0.559367\pi$
$480$	−82.2053	−3.75214
$481$	1.27230	0.0580118
$482$	23.9316	1.09005
$483$	−24.5272	−1.11603
$484$	45.3361	2.06073
$485$	−24.8203	−1.12703
$486$	−23.0911	−1.04743
$487$	2.92797	0.132679	0.0663394	−	0.997797i	$-0.478868\pi$
$487$	2.92797	0.132679	0.0663394	+	0.997797i	$0.478868\pi$
$488$	88.1207	3.98904
$489$	−22.6597	−1.02471
$490$	−5.42609	−0.245126
$491$	−29.3198	−1.32318	−0.661591	−	0.749865i	$-0.730119\pi$
$491$	−29.3198	−1.32318	−0.661591	+	0.749865i	$0.730119\pi$
$492$	64.5103	2.90835
$493$	−39.1332	−1.76247
$494$	7.22875	0.325237
$495$	10.5664	0.474925
$496$	−48.6028	−2.18233
$497$	−37.3615	−1.67589
$498$	77.0726	3.45370
$499$	−38.6208	−1.72890	−0.864452	−	0.502716i	$-0.832334\pi$
$499$	−38.6208	−1.72890	−0.864452	+	0.502716i	$0.832334\pi$
$500$	33.2876	1.48866
$501$	26.8990	1.20176
$502$	−52.0153	−2.32156
$503$	−29.4445	−1.31287	−0.656434	−	0.754384i	$-0.727936\pi$
$503$	−29.4445	−1.31287	−0.656434	+	0.754384i	$0.727936\pi$
$504$	−17.7997	−0.792862
$505$	−30.5406	−1.35904
$506$	59.3120	2.63674
$507$	11.0993	0.492938
$508$	102.173	4.53321
$509$	−24.6909	−1.09441	−0.547203	−	0.837000i	$-0.684307\pi$
$509$	−24.6909	−1.09441	−0.547203	+	0.837000i	$0.684307\pi$
$510$	92.2739	4.08596
$511$	6.95533	0.307686
$512$	17.2763	0.763514
$513$	4.20821	0.185797
$514$	−43.8735	−1.93518
$515$	35.2728	1.55431
$516$	−97.9917	−4.31384
$517$	−58.6753	−2.58054
$518$	3.13236	0.137628
$519$	6.72914	0.295376
$520$	−62.1734	−2.72648
$521$	−38.6480	−1.69320	−0.846600	−	0.532230i	$-0.821354\pi$
$521$	−38.6480	−1.69320	−0.846600	+	0.532230i	$0.821354\pi$
$522$	−14.0395	−0.614492
$523$	17.7956	0.778147	0.389074	−	0.921207i	$-0.372795\pi$
$523$	17.7956	0.778147	0.389074	+	0.921207i	$0.372795\pi$
$524$	−8.50742	−0.371648
$525$	−12.9980	−0.567281
$526$	10.0033	0.436167
$527$	26.0296	1.13387
$528$	104.172	4.53350
$529$	1.89738	0.0824948
$530$	−91.0721	−3.95592
$531$	7.55845	0.328009
$532$	12.7914	0.554575
$533$	17.4220	0.754632
$534$	34.2073	1.48030
$535$	−14.6557	−0.633623
$536$	−44.1478	−1.90689
$537$	−10.4555	−0.451189
$538$	−38.8968	−1.67696
$539$	3.28069	0.141309
$540$	−59.4641	−2.55893
$541$	−27.1497	−1.16726	−0.583628	−	0.812021i	$-0.698367\pi$
$541$	−27.1497	−1.16726	−0.583628	+	0.812021i	$0.698367\pi$
$542$	29.7896	1.27957
$543$	28.0233	1.20259
$544$	96.4706	4.13614
$545$	−1.00095	−0.0428762
$546$	−35.5332	−1.52068
$547$	18.5459	0.792964	0.396482	−	0.918043i	$-0.370231\pi$
$547$	18.5459	0.792964	0.396482	+	0.918043i	$0.370231\pi$
$548$	59.9030	2.55893
$549$	−9.10749	−0.388698
$550$	31.4320	1.34027
$551$	6.14099	0.261615
$552$	81.2931	3.46006
$553$	12.1557	0.516914
$554$	−44.8423	−1.90517
$555$	−2.54857	−0.108181
$556$	46.7930	1.98447
$557$	−10.0251	−0.424777	−0.212389	−	0.977185i	$-0.568124\pi$
$557$	−10.0251	−0.424777	−0.212389	+	0.977185i	$0.568124\pi$
$558$	9.33841	0.395327
$559$	−26.4642	−1.11932
$560$	−82.3372	−3.47938
$561$	−55.7901	−2.35546
$562$	24.2726	1.02388
$563$	14.3200	0.603518	0.301759	−	0.953384i	$-0.402426\pi$
$563$	14.3200	0.603518	0.301759	+	0.953384i	$0.402426\pi$
$564$	−132.124	−5.56344
$565$	48.7689	2.05172
$566$	41.4591	1.74266
$567$	−27.1229	−1.13905
$568$	123.831	5.19585
$569$	−1.68496	−0.0706374	−0.0353187	−	0.999376i	$-0.511245\pi$
$569$	−1.68496	−0.0706374	−0.0353187	+	0.999376i	$0.511245\pi$
$570$	−14.4801	−0.606505
$571$	−11.7480	−0.491640	−0.245820	−	0.969316i	$-0.579057\pi$
$571$	−11.7480	−0.491640	−0.245820	+	0.969316i	$0.579057\pi$
$572$	61.7588	2.58227
$573$	31.8193	1.32927
$574$	42.8925	1.79030
$575$	13.1942	0.550237
$576$	14.2074	0.591974
$577$	−1.85353	−0.0771635	−0.0385818	−	0.999255i	$-0.512284\pi$
$577$	−1.85353	−0.0771635	−0.0385818	+	0.999255i	$0.512284\pi$
$578$	−62.9540	−2.61854
$579$	48.4367	2.01296
$580$	−86.7754	−3.60315
$581$	36.8318	1.52804
$582$	47.0157	1.94886
$583$	55.0635	2.28050
$584$	−23.0528	−0.953932
$585$	6.42577	0.265673
$586$	63.3960	2.61887
$587$	18.8302	0.777207	0.388603	−	0.921405i	$-0.372958\pi$
$587$	18.8302	0.777207	0.388603	+	0.921405i	$0.372958\pi$
$588$	7.38743	0.304652
$589$	−4.08470	−0.168307
$590$	64.9990	2.67597
$591$	32.4629	1.33534
$592$	−5.58452	−0.229522
$593$	5.19699	0.213415	0.106707	−	0.994290i	$-0.465969\pi$
$593$	5.19699	0.213415	0.106707	+	0.994290i	$0.465969\pi$
$594$	50.0222	2.05243
$595$	44.0963	1.80777
$596$	56.2735	2.30505
$597$	9.85328	0.403268
$598$	36.0695	1.47499
$599$	7.90423	0.322958	0.161479	−	0.986876i	$-0.448374\pi$
$599$	7.90423	0.322958	0.161479	+	0.986876i	$0.448374\pi$
$600$	43.0808	1.75877
$601$	−36.0806	−1.47176	−0.735879	−	0.677114i	$-0.763231\pi$
$601$	−36.0806	−1.47176	−0.735879	+	0.677114i	$0.763231\pi$
$602$	−65.1542	−2.65549
$603$	4.56278	0.185811
$604$	−18.0580	−0.734768
$605$	−24.5258	−0.997116
$606$	57.8513	2.35005
$607$	18.0646	0.733218	0.366609	−	0.930375i	$-0.380519\pi$
$607$	18.0646	0.733218	0.366609	+	0.930375i	$0.380519\pi$
$608$	−15.1387	−0.613954
$609$	−30.1863	−1.22321
$610$	−78.3201	−3.17109
$611$	−35.6823	−1.44355
$612$	−27.9222	−1.12869
$613$	−22.8101	−0.921290	−0.460645	−	0.887584i	$-0.652382\pi$
$613$	−22.8101	−0.921290	−0.460645	+	0.887584i	$0.652382\pi$
$614$	−56.0378	−2.26150
$615$	−34.8986	−1.40725
$616$	92.5477	3.72885
$617$	13.0552	0.525581	0.262791	−	0.964853i	$-0.415357\pi$
$617$	13.0552	0.525581	0.262791	+	0.964853i	$0.415357\pi$
$618$	−66.8152	−2.68770
$619$	16.9702	0.682091	0.341045	−	0.940047i	$-0.389219\pi$
$619$	16.9702	0.682091	0.341045	+	0.940047i	$0.389219\pi$
$620$	57.7189	2.31805
$621$	20.9978	0.842613
$622$	−59.2424	−2.37540
$623$	16.3472	0.654935
$624$	63.3503	2.53604
$625$	−31.2292	−1.24917
$626$	−17.8512	−0.713478
$627$	8.75488	0.349636
$628$	118.593	4.73239
$629$	2.99083	0.119252
$630$	15.8201	0.630286
$631$	31.4802	1.25321	0.626604	−	0.779338i	$-0.284444\pi$
$631$	31.4802	1.25321	0.626604	+	0.779338i	$0.284444\pi$
$632$	−40.2890	−1.60261
$633$	−1.96401	−0.0780624
$634$	67.0749	2.66388
$635$	−55.2735	−2.19346
$636$	123.991	4.91658
$637$	1.99509	0.0790485
$638$	72.9968	2.88997
$639$	−12.7983	−0.506292
$640$	38.4650	1.52046
$641$	−13.3209	−0.526143	−0.263072	−	0.964776i	$-0.584736\pi$
$641$	−13.3209	−0.526143	−0.263072	+	0.964776i	$0.584736\pi$
$642$	27.7615	1.09566
$643$	−10.8218	−0.426770	−0.213385	−	0.976968i	$-0.568449\pi$
$643$	−10.8218	−0.426770	−0.213385	+	0.976968i	$0.568449\pi$
$644$	63.8254	2.51507
$645$	53.0112	2.08732
$646$	16.9929	0.668575
$647$	−31.2529	−1.22868	−0.614340	−	0.789042i	$-0.710578\pi$
$647$	−31.2529	−1.22868	−0.614340	+	0.789042i	$0.710578\pi$
$648$	89.8962	3.53146
$649$	−39.2993	−1.54263
$650$	19.1148	0.749745
$651$	20.0785	0.786938
$652$	58.9656	2.30927
$653$	31.7739	1.24341	0.621704	−	0.783253i	$-0.286441\pi$
$653$	31.7739	1.24341	0.621704	+	0.783253i	$0.286441\pi$
$654$	1.89605	0.0741413
$655$	4.60232	0.179827
$656$	−76.4709	−2.98569
$657$	2.38256	0.0929527
$658$	−87.8488	−3.42470
$659$	3.28492	0.127962	0.0639811	−	0.997951i	$-0.479620\pi$
$659$	3.28492	0.127962	0.0639811	+	0.997951i	$0.479620\pi$
$660$	−123.711	−4.81544
$661$	6.07036	0.236110	0.118055	−	0.993007i	$-0.462334\pi$
$661$	6.07036	0.236110	0.118055	+	0.993007i	$0.462334\pi$
$662$	−32.7822	−1.27412
$663$	−33.9277	−1.31764
$664$	−122.076	−4.73745
$665$	−6.91983	−0.268339
$666$	1.07300	0.0415777
$667$	30.6419	1.18646
$668$	−69.9973	−2.70828
$669$	−31.8114	−1.22990
$670$	39.2378	1.51589
$671$	47.3534	1.82806
$672$	74.4147	2.87061
$673$	26.0860	1.00554	0.502771	−	0.864420i	$-0.332314\pi$
$673$	26.0860	1.00554	0.502771	+	0.864420i	$0.332314\pi$
$674$	57.2400	2.20480
$675$	11.1277	0.428304
$676$	−28.8830	−1.11088
$677$	33.8848	1.30230	0.651149	−	0.758950i	$-0.274287\pi$
$677$	33.8848	1.30230	0.651149	+	0.758950i	$0.274287\pi$
$678$	−92.3800	−3.54783
$679$	22.4681	0.862247
$680$	−146.153	−5.60471
$681$	−39.7514	−1.52328
$682$	−48.5540	−1.85923
$683$	−25.7275	−0.984437	−0.492219	−	0.870472i	$-0.663814\pi$
$683$	−25.7275	−0.984437	−0.492219	+	0.870472i	$0.663814\pi$
$684$	4.38170	0.167539
$685$	−32.4061	−1.23817
$686$	51.6299	1.97124
$687$	−23.0394	−0.879008
$688$	116.160	4.42856
$689$	33.4859	1.27571
$690$	−72.2518	−2.75058
$691$	−39.1024	−1.48753	−0.743763	−	0.668443i	$-0.766961\pi$
$691$	−39.1024	−1.48753	−0.743763	+	0.668443i	$0.766961\pi$
$692$	−17.5108	−0.665659
$693$	−9.56503	−0.363346
$694$	49.2685	1.87021
$695$	−25.3140	−0.960213
$696$	100.050	3.79237
$697$	40.9546	1.55127
$698$	−19.0226	−0.720016
$699$	−29.1541	−1.10271
$700$	33.8239	1.27842
$701$	−0.268417	−0.0101380	−0.00506898	−	0.999987i	$-0.501614\pi$
$701$	−0.268417	−0.0101380	−0.00506898	+	0.999987i	$0.501614\pi$
$702$	30.4201	1.14813
$703$	−0.469337	−0.0177014
$704$	−73.8697	−2.78407
$705$	71.4762	2.69195
$706$	87.8532	3.30640
$707$	27.6463	1.03975
$708$	−88.4938	−3.32580
$709$	−38.8234	−1.45804	−0.729022	−	0.684490i	$-0.760025\pi$
$709$	−38.8234	−1.45804	−0.729022	+	0.684490i	$0.760025\pi$
$710$	−110.059	−4.13044
$711$	4.16397	0.156161
$712$	−54.1811	−2.03052
$713$	−20.3815	−0.763294
$714$	−83.5291	−3.12600
$715$	−33.4101	−1.24947
$716$	27.2076	1.01680
$717$	2.74173	0.102392
$718$	−58.5449	−2.18488
$719$	38.1679	1.42342	0.711711	−	0.702472i	$-0.247920\pi$
$719$	38.1679	1.42342	0.711711	+	0.702472i	$0.247920\pi$
$720$	−28.2048	−1.05113
$721$	−31.9300	−1.18914
$722$	−2.66661	−0.0992409
$723$	17.6261	0.655522
$724$	−72.9229	−2.71016
$725$	16.2385	0.603082
$726$	46.4578	1.72421
$727$	18.3995	0.682400	0.341200	−	0.939991i	$-0.389167\pi$
$727$	18.3995	0.682400	0.341200	+	0.939991i	$0.389167\pi$
$728$	56.2812	2.08592
$729$	15.5039	0.574219
$730$	20.4889	0.758328
$731$	−62.2104	−2.30093
$732$	106.630	3.94115
$733$	26.0551	0.962365	0.481183	−	0.876620i	$-0.340207\pi$
$733$	26.0551	0.962365	0.481183	+	0.876620i	$0.340207\pi$
$734$	30.2512	1.11659
$735$	−3.99643	−0.147410
$736$	−75.5378	−2.78436
$737$	−23.7237	−0.873874
$738$	14.6929	0.540854
$739$	−27.1452	−0.998552	−0.499276	−	0.866443i	$-0.666401\pi$
$739$	−27.1452	−0.998552	−0.499276	+	0.866443i	$0.666401\pi$
$740$	6.63197	0.243796
$741$	5.32412	0.195586
$742$	82.4412	3.02651
$743$	23.5174	0.862771	0.431385	−	0.902168i	$-0.358025\pi$
$743$	23.5174	0.862771	0.431385	+	0.902168i	$0.358025\pi$
$744$	−66.5483	−2.43978
$745$	−30.4427	−1.11533
$746$	−39.9348	−1.46212
$747$	12.6168	0.461625
$748$	145.179	5.30826
$749$	13.2668	0.484759
$750$	34.1111	1.24556
$751$	14.5246	0.530011	0.265005	−	0.964247i	$-0.414626\pi$
$751$	14.5246	0.530011	0.265005	+	0.964247i	$0.414626\pi$
$752$	156.621	5.71138
$753$	−38.3103	−1.39611
$754$	44.3917	1.61665
$755$	9.76894	0.355528
$756$	53.8287	1.95773
$757$	−0.249201	−0.00905735	−0.00452867	−	0.999990i	$-0.501442\pi$
$757$	−0.249201	−0.00905735	−0.00452867	+	0.999990i	$0.501442\pi$
$758$	99.6364	3.61896
$759$	43.6844	1.58564
$760$	22.9351	0.831944
$761$	−6.35173	−0.230250	−0.115125	−	0.993351i	$-0.536727\pi$
$761$	−6.35173	−0.230250	−0.115125	+	0.993351i	$0.536727\pi$
$762$	104.701	3.79293
$763$	0.906093	0.0328028
$764$	−82.8010	−2.99563
$765$	15.1053	0.546133
$766$	−94.8908	−3.42854
$767$	−23.8992	−0.862949
$768$	−7.76893	−0.280337
$769$	34.3504	1.23871	0.619353	−	0.785112i	$-0.287395\pi$
$769$	34.3504	1.23871	0.619353	+	0.785112i	$0.287395\pi$
$770$	−82.2547	−2.96425
$771$	−32.3137	−1.16375
$772$	−126.043	−4.53640
$773$	−51.6954	−1.85935	−0.929677	−	0.368375i	$-0.879914\pi$
$773$	−51.6954	−1.85935	−0.929677	+	0.368375i	$0.879914\pi$
$774$	−22.3187	−0.802229
$775$	−10.8011	−0.387986
$776$	−74.4684	−2.67326
$777$	2.30704	0.0827647
$778$	−67.8737	−2.43339
$779$	−6.42681	−0.230264
$780$	−75.2325	−2.69376
$781$	66.5432	2.38110
$782$	84.7898	3.03208
$783$	25.8426	0.923538
$784$	−8.75710	−0.312754
$785$	−64.1562	−2.28983
$786$	−8.71790	−0.310957
$787$	−24.5017	−0.873392	−0.436696	−	0.899609i	$-0.643851\pi$
$787$	−24.5017	−0.873392	−0.436696	+	0.899609i	$0.643851\pi$
$788$	−84.4758	−3.00933
$789$	7.36767	0.262296
$790$	35.8082	1.27400
$791$	−44.1470	−1.56969
$792$	31.7024	1.12650
$793$	28.7971	1.02262
$794$	27.6708	0.982001
$795$	−67.0764	−2.37896
$796$	−25.6405	−0.908803
$797$	−1.86395	−0.0660244	−0.0330122	−	0.999455i	$-0.510510\pi$
$797$	−1.86395	−0.0660244	−0.0330122	+	0.999455i	$0.510510\pi$
$798$	13.1078	0.464012
$799$	−83.8796	−2.96745
$800$	−40.0308	−1.41530
$801$	5.59976	0.197858
$802$	48.7950	1.72301
$803$	−12.3879	−0.437159
$804$	−53.4207	−1.88400
$805$	−34.5280	−1.21695
$806$	−29.5273	−1.04005
$807$	−28.6483	−1.00847
$808$	−91.6310	−3.22357
$809$	−12.5263	−0.440403	−0.220201	−	0.975454i	$-0.570671\pi$
$809$	−12.5263	−0.440403	−0.220201	+	0.975454i	$0.570671\pi$
$810$	−79.8981	−2.80733
$811$	−32.0391	−1.12505	−0.562523	−	0.826782i	$-0.690169\pi$
$811$	−32.0391	−1.12505	−0.562523	+	0.826782i	$0.690169\pi$
$812$	78.5516	2.75662
$813$	21.9407	0.769493
$814$	−5.57892	−0.195541
$815$	−31.8990	−1.11737
$816$	148.920	5.21323
$817$	9.76238	0.341542
$818$	79.5070	2.77990
$819$	−5.81680	−0.203255
$820$	90.8140	3.17136
$821$	31.4612	1.09800	0.549001	−	0.835822i	$-0.315008\pi$
$821$	31.4612	1.09800	0.549001	+	0.835822i	$0.315008\pi$
$822$	61.3850	2.14105
$823$	33.4667	1.16658	0.583288	−	0.812265i	$-0.301766\pi$
$823$	33.4667	1.16658	0.583288	+	0.812265i	$0.301766\pi$
$824$	105.829	3.68673
$825$	23.1503	0.805991
$826$	−58.8391	−2.04727
$827$	1.92766	0.0670314	0.0335157	−	0.999438i	$-0.489330\pi$
$827$	1.92766	0.0670314	0.0335157	+	0.999438i	$0.489330\pi$
$828$	21.8635	0.759810
$829$	−34.9913	−1.21530	−0.607649	−	0.794206i	$-0.707887\pi$
$829$	−34.9913	−1.21530	−0.607649	+	0.794206i	$0.707887\pi$
$830$	108.499	3.76604
$831$	−33.0273	−1.14570
$832$	−44.9225	−1.55741
$833$	4.68993	0.162497
$834$	47.9507	1.66040
$835$	37.8669	1.31044
$836$	−22.7822	−0.787939
$837$	−17.1893	−0.594148
$838$	−58.6765	−2.02695
$839$	6.36356	0.219694	0.109847	−	0.993948i	$-0.464964\pi$
$839$	6.36356	0.219694	0.109847	+	0.993948i	$0.464964\pi$
$840$	−112.738	−3.88984
$841$	8.71181	0.300407
$842$	60.3121	2.07849
$843$	17.8773	0.615725
$844$	5.11081	0.175921
$845$	15.6250	0.537517
$846$	−30.0928	−1.03461
$847$	22.2015	0.762852
$848$	−146.980	−5.04732
$849$	30.5355	1.04797
$850$	44.9339	1.54122
$851$	−2.34186	−0.0802781
$852$	149.841	5.13348
$853$	−11.9539	−0.409292	−0.204646	−	0.978836i	$-0.565604\pi$
$853$	−11.9539	−0.409292	−0.204646	+	0.978836i	$0.565604\pi$
$854$	70.8976	2.42607
$855$	−2.37040	−0.0810660
$856$	−43.9716	−1.50292
$857$	−11.8226	−0.403854	−0.201927	−	0.979401i	$-0.564720\pi$
$857$	−11.8226	−0.403854	−0.201927	+	0.979401i	$0.564720\pi$
$858$	63.2868	2.16058
$859$	17.6982	0.603855	0.301928	−	0.953331i	$-0.402370\pi$
$859$	17.6982	0.603855	0.301928	+	0.953331i	$0.402370\pi$
$860$	−137.947	−4.70397
$861$	31.5912	1.07663
$862$	92.2987	3.14371
$863$	−12.6011	−0.428945	−0.214472	−	0.976730i	$-0.568803\pi$
$863$	−12.6011	−0.428945	−0.214472	+	0.976730i	$0.568803\pi$
$864$	−63.7067	−2.16734
$865$	9.47291	0.322089
$866$	3.69812	0.125667
$867$	−46.3669	−1.57470
$868$	−52.2488	−1.77344
$869$	−21.6501	−0.734430
$870$	−88.9222	−3.01475
$871$	−14.4271	−0.488845
$872$	−3.00316	−0.101700
$873$	7.69650	0.260487
$874$	−13.3057	−0.450071
$875$	16.3012	0.551081
$876$	−27.8949	−0.942481
$877$	−18.1793	−0.613871	−0.306935	−	0.951730i	$-0.599304\pi$
$877$	−18.1793	−0.613871	−0.306935	+	0.951730i	$0.599304\pi$
$878$	−29.1250	−0.982922
$879$	46.6924	1.57490
$880$	146.648	4.94349
$881$	−12.2132	−0.411472	−0.205736	−	0.978608i	$-0.565959\pi$
$881$	−12.2132	−0.411472	−0.205736	+	0.978608i	$0.565959\pi$
$882$	1.68257	0.0566550
$883$	−11.7768	−0.396322	−0.198161	−	0.980169i	$-0.563497\pi$
$883$	−11.7768	−0.396322	−0.198161	+	0.980169i	$0.563497\pi$
$884$	88.2877	2.96944
$885$	47.8731	1.60924
$886$	20.4615	0.687418
$887$	48.1650	1.61722	0.808611	−	0.588343i	$-0.200219\pi$
$887$	48.1650	1.61722	0.808611	+	0.588343i	$0.200219\pi$
$888$	−7.64648	−0.256599
$889$	50.0352	1.67813
$890$	48.1552	1.61417
$891$	48.3075	1.61836
$892$	82.7807	2.77170
$893$	13.1628	0.440477
$894$	57.6657	1.92863
$895$	−14.7187	−0.491992
$896$	−34.8197	−1.16324
$897$	26.5659	0.887010
$898$	−30.6259	−1.02200
$899$	−25.0841	−0.836602
$900$	11.5864	0.386215
$901$	78.7164	2.62242
$902$	−76.3943	−2.54365
$903$	−47.9873	−1.59692
$904$	146.321	4.86657
$905$	39.4496	1.31135
$906$	−18.5047	−0.614779
$907$	4.70486	0.156222	0.0781112	−	0.996945i	$-0.475111\pi$
$907$	4.70486	0.156222	0.0781112	+	0.996945i	$0.475111\pi$
$908$	103.442	3.43285
$909$	9.47030	0.314110
$910$	−50.0217	−1.65820
$911$	−9.79420	−0.324496	−0.162248	−	0.986750i	$-0.551874\pi$
$911$	−9.79420	−0.324496	−0.162248	+	0.986750i	$0.551874\pi$
$912$	−23.3693	−0.773833
$913$	−65.5998	−2.17104
$914$	51.4257	1.70101
$915$	−57.6843	−1.90698
$916$	59.9538	1.98093
$917$	−4.16615	−0.137579
$918$	71.5095	2.36016
$919$	13.5870	0.448195	0.224097	−	0.974567i	$-0.428057\pi$
$919$	13.5870	0.448195	0.224097	+	0.974567i	$0.428057\pi$
$920$	114.440	3.77297
$921$	−41.2729	−1.35999
$922$	−57.3555	−1.88890
$923$	40.4671	1.33199
$924$	111.987	3.68409
$925$	−1.24106	−0.0408057
$926$	53.1678	1.74720
$927$	−10.9377	−0.359241
$928$	−92.9665	−3.05177
$929$	26.2198	0.860245	0.430122	−	0.902771i	$-0.358470\pi$
$929$	26.2198	0.860245	0.430122	+	0.902771i	$0.358470\pi$
$930$	59.1469	1.93950
$931$	−0.735969	−0.0241204
$932$	75.8655	2.48506
$933$	−43.6332	−1.42849
$934$	−26.6284	−0.871308
$935$	−78.5383	−2.56848
$936$	19.2792	0.630162
$937$	−42.6930	−1.39472	−0.697361	−	0.716721i	$-0.745642\pi$
$937$	−42.6930	−1.39472	−0.697361	+	0.716721i	$0.745642\pi$
$938$	−35.5192	−1.15974
$939$	−13.1478	−0.429062
$940$	−185.997	−6.06657
$941$	−13.6886	−0.446235	−0.223118	−	0.974792i	$-0.571623\pi$
$941$	−13.6886	−0.446235	−0.223118	+	0.974792i	$0.571623\pi$
$942$	121.527	3.95957
$943$	−32.0680	−1.04428
$944$	104.901	3.41424
$945$	−29.1201	−0.947276
$946$	116.044	3.77290
$947$	−55.3072	−1.79724	−0.898622	−	0.438724i	$-0.855430\pi$
$947$	−55.3072	−1.79724	−0.898622	+	0.438724i	$0.855430\pi$
$948$	−48.7515	−1.58337
$949$	−7.53347	−0.244547
$950$	−7.05126	−0.228773
$951$	49.4020	1.60197
$952$	132.302	4.28794
$953$	0.319776	0.0103586	0.00517928	−	0.999987i	$-0.498351\pi$
$953$	0.319776	0.0103586	0.00517928	+	0.999987i	$0.498351\pi$
$954$	28.2404	0.914317
$955$	44.7934	1.44948
$956$	−7.13461	−0.230750
$957$	53.7636	1.73793
$958$	21.6435	0.699271
$959$	29.3350	0.947276
$960$	89.9856	2.90427
$961$	−14.3152	−0.461782
$962$	−3.39272	−0.109386
$963$	4.54458	0.146447
$964$	−45.8672	−1.47728
$965$	68.1866	2.19500
$966$	65.4044	2.10435
$967$	−50.8609	−1.63558	−0.817788	−	0.575519i	$-0.804800\pi$
$967$	−50.8609	−1.63558	−0.817788	+	0.575519i	$0.804800\pi$
$968$	−73.5847	−2.36510
$969$	12.5156	0.402059
$970$	66.1862	2.12511
$971$	17.9892	0.577300	0.288650	−	0.957435i	$-0.406794\pi$
$971$	17.9892	0.577300	0.288650	+	0.957435i	$0.406794\pi$
$972$	44.2563	1.41952
$973$	22.9149	0.734619
$974$	−7.80775	−0.250176
$975$	14.0784	0.450871
$976$	−126.400	−4.04596
$977$	47.0123	1.50406	0.752028	−	0.659131i	$-0.229076\pi$
$977$	47.0123	1.50406	0.752028	+	0.659131i	$0.229076\pi$
$978$	60.4245	1.93216
$979$	−29.1153	−0.930529
$980$	10.3996	0.332204
$981$	0.310384	0.00990980
$982$	78.1844	2.49496
$983$	−26.6548	−0.850155	−0.425077	−	0.905157i	$-0.639753\pi$
$983$	−26.6548	−0.850155	−0.425077	+	0.905157i	$0.639753\pi$
$984$	−104.706	−3.33791
$985$	45.6995	1.45611
$986$	104.353	3.32328
$987$	−64.7024	−2.05950
$988$	−13.8546	−0.440773
$989$	48.7116	1.54894
$990$	−28.1765	−0.895508
$991$	3.41726	0.108553	0.0542765	−	0.998526i	$-0.482715\pi$
$991$	3.41726	0.108553	0.0542765	+	0.998526i	$0.482715\pi$
$992$	61.8369	1.96332
$993$	−24.1447	−0.766210
$994$	99.6287	3.16003
$995$	13.8709	0.439738
$996$	−147.717	−4.68059
$997$	15.0759	0.477459	0.238730	−	0.971086i	$-0.423269\pi$
$997$	15.0759	0.477459	0.238730	+	0.971086i	$0.423269\pi$
$998$	102.987	3.25998
$999$	−1.97507	−0.0624884

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.5	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.5	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.66661 q^{2} -1.96401 q^{3} +5.11081 q^{4} -2.76483 q^{5} +5.23725 q^{6} +2.50280 q^{7} -8.29531 q^{8} +0.857341 q^{9} +O(q^{10})\)	\(q-2.66661 q^{2} -1.96401 q^{3} +5.11081 q^{4} -2.76483 q^{5} +5.23725 q^{6} +2.50280 q^{7} -8.29531 q^{8} +0.857341 q^{9} +7.37272 q^{10} -4.45765 q^{11} -10.0377 q^{12} -2.71084 q^{13} -6.67400 q^{14} +5.43016 q^{15} +11.8987 q^{16} -6.37246 q^{17} -2.28619 q^{18} +1.00000 q^{19} -14.1305 q^{20} -4.91554 q^{21} +11.8868 q^{22} +4.98973 q^{23} +16.2921 q^{24} +2.64428 q^{25} +7.22875 q^{26} +4.20821 q^{27} +12.7914 q^{28} +6.14099 q^{29} -14.4801 q^{30} -4.08470 q^{31} -15.1387 q^{32} +8.75488 q^{33} +16.9929 q^{34} -6.91983 q^{35} +4.38170 q^{36} -0.469337 q^{37} -2.66661 q^{38} +5.32412 q^{39} +22.9351 q^{40} -6.42681 q^{41} +13.1078 q^{42} +9.76238 q^{43} -22.7822 q^{44} -2.37040 q^{45} -13.3057 q^{46} +13.1628 q^{47} -23.3693 q^{48} -0.735969 q^{49} -7.05126 q^{50} +12.5156 q^{51} -13.8546 q^{52} -12.3526 q^{53} -11.2216 q^{54} +12.3246 q^{55} -20.7615 q^{56} -1.96401 q^{57} -16.3756 q^{58} +8.81616 q^{59} +27.7525 q^{60} -10.6230 q^{61} +10.8923 q^{62} +2.14576 q^{63} +16.5715 q^{64} +7.49501 q^{65} -23.3458 q^{66} +5.32202 q^{67} -32.5684 q^{68} -9.79988 q^{69} +18.4525 q^{70} -14.9279 q^{71} -7.11191 q^{72} +2.77902 q^{73} +1.25154 q^{74} -5.19339 q^{75} +5.11081 q^{76} -11.1566 q^{77} -14.1973 q^{78} +4.85685 q^{79} -32.8980 q^{80} -10.8370 q^{81} +17.1378 q^{82} +14.7162 q^{83} -25.1224 q^{84} +17.6188 q^{85} -26.0325 q^{86} -12.0610 q^{87} +36.9776 q^{88} +6.53154 q^{89} +6.32093 q^{90} -6.78470 q^{91} +25.5015 q^{92} +8.02240 q^{93} -35.1001 q^{94} -2.76483 q^{95} +29.7325 q^{96} +8.97717 q^{97} +1.96254 q^{98} -3.82173 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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