LMFDB - Embedded newform 8009.2.a.a.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8009,2,Mod(1,8009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8009, 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("8009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8009 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8009.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:	$63.9521869788$
Analytic rank:	$1$
Dimension:	$306$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.55906 q^{2} +0.342269 q^{3} +4.54881 q^{4} -2.51856 q^{5} -0.875888 q^{6} -3.84856 q^{7} -6.52258 q^{8} -2.88285 q^{9} +O(q^{10})$	\(q-2.55906 q^{2} +0.342269 q^{3} +4.54881 q^{4} -2.51856 q^{5} -0.875888 q^{6} -3.84856 q^{7} -6.52258 q^{8} -2.88285 q^{9} +6.44516 q^{10} -2.74183 q^{11} +1.55692 q^{12} -0.0963708 q^{13} +9.84871 q^{14} -0.862024 q^{15} +7.59408 q^{16} -3.25437 q^{17} +7.37741 q^{18} -5.50192 q^{19} -11.4565 q^{20} -1.31724 q^{21} +7.01652 q^{22} +6.26772 q^{23} -2.23247 q^{24} +1.34314 q^{25} +0.246619 q^{26} -2.01352 q^{27} -17.5064 q^{28} -1.15728 q^{29} +2.20598 q^{30} +8.65601 q^{31} -6.38858 q^{32} -0.938442 q^{33} +8.32815 q^{34} +9.69283 q^{35} -13.1136 q^{36} -0.0286873 q^{37} +14.0798 q^{38} -0.0329847 q^{39} +16.4275 q^{40} -5.99425 q^{41} +3.37091 q^{42} -1.30249 q^{43} -12.4721 q^{44} +7.26063 q^{45} -16.0395 q^{46} +11.2903 q^{47} +2.59921 q^{48} +7.81141 q^{49} -3.43719 q^{50} -1.11387 q^{51} -0.438373 q^{52} -2.54827 q^{53} +5.15272 q^{54} +6.90546 q^{55} +25.1025 q^{56} -1.88313 q^{57} +2.96155 q^{58} +14.2740 q^{59} -3.92119 q^{60} -11.6303 q^{61} -22.1513 q^{62} +11.0948 q^{63} +1.16063 q^{64} +0.242716 q^{65} +2.40153 q^{66} -6.99587 q^{67} -14.8035 q^{68} +2.14524 q^{69} -24.8046 q^{70} +7.34007 q^{71} +18.8036 q^{72} -8.41645 q^{73} +0.0734127 q^{74} +0.459715 q^{75} -25.0272 q^{76} +10.5521 q^{77} +0.0844100 q^{78} +7.10354 q^{79} -19.1261 q^{80} +7.95939 q^{81} +15.3397 q^{82} +6.78568 q^{83} -5.99188 q^{84} +8.19633 q^{85} +3.33314 q^{86} -0.396101 q^{87} +17.8838 q^{88} +17.0523 q^{89} -18.5804 q^{90} +0.370889 q^{91} +28.5107 q^{92} +2.96268 q^{93} -28.8926 q^{94} +13.8569 q^{95} -2.18661 q^{96} +1.70654 q^{97} -19.9899 q^{98} +7.90429 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) $ 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9} - 61 q^{10} - 43 q^{11} - 50 q^{12} - 89 q^{13} - 40 q^{14} - 61 q^{15} + 151 q^{16} - 52 q^{17} - 57 q^{18} - 185 q^{19} - 66 q^{20} - 63 q^{21} - 55 q^{22} - 62 q^{23} - 131 q^{24} + 209 q^{25} - 57 q^{26} - 88 q^{27} - 182 q^{28} - 67 q^{29} - 68 q^{30} - 240 q^{31} - 64 q^{32} - 52 q^{33} - 128 q^{34} - 99 q^{35} + 106 q^{36} - 49 q^{37} - 45 q^{38} - 190 q^{39} - 158 q^{40} - 72 q^{41} - 36 q^{42} - 141 q^{43} - 80 q^{44} - 100 q^{45} - 91 q^{46} - 105 q^{47} - 85 q^{48} + 116 q^{49} - 51 q^{50} - 145 q^{51} - 237 q^{52} - 48 q^{53} - 156 q^{54} - 420 q^{55} - 116 q^{56} - 35 q^{57} - 43 q^{58} - 139 q^{59} - 73 q^{60} - 233 q^{61} - 58 q^{62} - 252 q^{63} - 3 q^{64} - 45 q^{65} - 127 q^{66} - 108 q^{67} - 85 q^{68} - 164 q^{69} - 56 q^{70} - 131 q^{71} - 117 q^{72} - 118 q^{73} - 47 q^{74} - 112 q^{75} - 389 q^{76} - 36 q^{77} + 9 q^{78} - 382 q^{79} - 119 q^{80} + 102 q^{81} - 131 q^{82} - 59 q^{83} - 144 q^{84} - 140 q^{85} - 38 q^{86} - 301 q^{87} - 131 q^{88} - 98 q^{89} - 138 q^{90} - 176 q^{91} - 97 q^{92} - 60 q^{93} - 342 q^{94} - 154 q^{95} - 243 q^{96} - 109 q^{97} - 21 q^{98} - 173 q^{99}+O(q^{100}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9 - 61 * q^10 - 43 * q^11 - 50 * q^12 - 89 * q^13 - 40 * q^14 - 61 * q^15 + 151 * q^16 - 52 * q^17 - 57 * q^18 - 185 * q^19 - 66 * q^20 - 63 * q^21 - 55 * q^22 - 62 * q^23 - 131 * q^24 + 209 * q^25 - 57 * q^26 - 88 * q^27 - 182 * q^28 - 67 * q^29 - 68 * q^30 - 240 * q^31 - 64 * q^32 - 52 * q^33 - 128 * q^34 - 99 * q^35 + 106 * q^36 - 49 * q^37 - 45 * q^38 - 190 * q^39 - 158 * q^40 - 72 * q^41 - 36 * q^42 - 141 * q^43 - 80 * q^44 - 100 * q^45 - 91 * q^46 - 105 * q^47 - 85 * q^48 + 116 * q^49 - 51 * q^50 - 145 * q^51 - 237 * q^52 - 48 * q^53 - 156 * q^54 - 420 * q^55 - 116 * q^56 - 35 * q^57 - 43 * q^58 - 139 * q^59 - 73 * q^60 - 233 * q^61 - 58 * q^62 - 252 * q^63 - 3 * q^64 - 45 * q^65 - 127 * q^66 - 108 * q^67 - 85 * q^68 - 164 * q^69 - 56 * q^70 - 131 * q^71 - 117 * q^72 - 118 * q^73 - 47 * q^74 - 112 * q^75 - 389 * q^76 - 36 * q^77 + 9 * q^78 - 382 * q^79 - 119 * q^80 + 102 * q^81 - 131 * q^82 - 59 * q^83 - 144 * q^84 - 140 * q^85 - 38 * q^86 - 301 * q^87 - 131 * q^88 - 98 * q^89 - 138 * q^90 - 176 * q^91 - 97 * q^92 - 60 * q^93 - 342 * q^94 - 154 * q^95 - 243 * q^96 - 109 * q^97 - 21 * q^98 - 173 * 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 $

$2$	−2.55906	−1.80953	−0.904766	−	0.425909i	$-0.859954\pi$
$2$	−2.55906	−1.80953	−0.904766	+	0.425909i	$0.859954\pi$
$3$	0.342269	0.197609	0.0988044	−	0.995107i	$-0.468498\pi$
$3$	0.342269	0.197609	0.0988044	+	0.995107i	$0.468498\pi$
$4$	4.54881	2.27441
$5$	−2.51856	−1.12633	−0.563167	−	0.826343i	$-0.690417\pi$
$5$	−2.51856	−1.12633	−0.563167	+	0.826343i	$0.690417\pi$
$6$	−0.875888	−0.357580
$7$	−3.84856	−1.45462	−0.727309	−	0.686310i	$-0.759229\pi$
$7$	−3.84856	−1.45462	−0.727309	+	0.686310i	$0.759229\pi$
$8$	−6.52258	−2.30608
$9$	−2.88285	−0.960951
$10$	6.44516	2.03814
$11$	−2.74183	−0.826692	−0.413346	−	0.910574i	$-0.635640\pi$
$11$	−2.74183	−0.826692	−0.413346	+	0.910574i	$0.635640\pi$
$12$	1.55692	0.449443
$13$	−0.0963708	−0.0267285	−0.0133642	−	0.999911i	$-0.504254\pi$
$13$	−0.0963708	−0.0267285	−0.0133642	+	0.999911i	$0.504254\pi$
$14$	9.84871	2.63218
$15$	−0.862024	−0.222574
$16$	7.59408	1.89852
$17$	−3.25437	−0.789301	−0.394650	−	0.918831i	$-0.629134\pi$
$17$	−3.25437	−0.789301	−0.394650	+	0.918831i	$0.629134\pi$
$18$	7.37741	1.73887
$19$	−5.50192	−1.26223	−0.631113	−	0.775691i	$-0.717402\pi$
$19$	−5.50192	−1.26223	−0.631113	+	0.775691i	$0.717402\pi$
$20$	−11.4565	−2.56174
$21$	−1.31724	−0.287446
$22$	7.01652	1.49593
$23$	6.26772	1.30691	0.653455	−	0.756965i	$-0.273319\pi$
$23$	6.26772	1.30691	0.653455	+	0.756965i	$0.273319\pi$
$24$	−2.23247	−0.455702
$25$	1.34314	0.268628
$26$	0.246619	0.0483660
$27$	−2.01352	−0.387501
$28$	−17.5064	−3.30839
$29$	−1.15728	−0.214902	−0.107451	−	0.994210i	$-0.534269\pi$
$29$	−1.15728	−0.214902	−0.107451	+	0.994210i	$0.534269\pi$
$30$	2.20598	0.402754
$31$	8.65601	1.55467	0.777333	−	0.629090i	$-0.216572\pi$
$31$	8.65601	1.55467	0.777333	+	0.629090i	$0.216572\pi$
$32$	−6.38858	−1.12935
$33$	−0.938442	−0.163362
$34$	8.32815	1.42827
$35$	9.69283	1.63839
$36$	−13.1136	−2.18559
$37$	−0.0286873	−0.00471617	−0.00235808	−	0.999997i	$-0.500751\pi$
$37$	−0.0286873	−0.00471617	−0.00235808	+	0.999997i	$0.500751\pi$
$38$	14.0798	2.28404
$39$	−0.0329847	−0.00528178
$40$	16.4275	2.59742
$41$	−5.99425	−0.936144	−0.468072	−	0.883690i	$-0.655051\pi$
$41$	−5.99425	−0.936144	−0.468072	+	0.883690i	$0.655051\pi$
$42$	3.37091	0.520142
$43$	−1.30249	−0.198627	−0.0993136	−	0.995056i	$-0.531665\pi$
$43$	−1.30249	−0.198627	−0.0993136	+	0.995056i	$0.531665\pi$
$44$	−12.4721	−1.88023
$45$	7.26063	1.08235
$46$	−16.0395	−2.36490
$47$	11.2903	1.64686	0.823430	−	0.567418i	$-0.192058\pi$
$47$	11.2903	1.64686	0.823430	+	0.567418i	$0.192058\pi$
$48$	2.59921	0.375164
$49$	7.81141	1.11592
$50$	−3.43719	−0.486091
$51$	−1.11387	−0.155973
$52$	−0.438373	−0.0607914
$53$	−2.54827	−0.350032	−0.175016	−	0.984566i	$-0.555998\pi$
$53$	−2.54827	−0.350032	−0.175016	+	0.984566i	$0.555998\pi$
$54$	5.15272	0.701196
$55$	6.90546	0.931132
$56$	25.1025	3.35447
$57$	−1.88313	−0.249427
$58$	2.96155	0.388871
$59$	14.2740	1.85832	0.929160	−	0.369677i	$-0.120532\pi$
$59$	14.2740	1.85832	0.929160	+	0.369677i	$0.120532\pi$
$60$	−3.92119	−0.506223
$61$	−11.6303	−1.48911	−0.744555	−	0.667561i	$-0.767339\pi$
$61$	−11.6303	−1.48911	−0.744555	+	0.667561i	$0.767339\pi$
$62$	−22.1513	−2.81322
$63$	11.0948	1.39782
$64$	1.16063	0.145079
$65$	0.242716	0.0301052
$66$	2.40153	0.295608
$67$	−6.99587	−0.854682	−0.427341	−	0.904091i	$-0.640550\pi$
$67$	−6.99587	−0.854682	−0.427341	+	0.904091i	$0.640550\pi$
$68$	−14.8035	−1.79519
$69$	2.14524	0.258257
$70$	−24.8046	−2.96471
$71$	7.34007	0.871106	0.435553	−	0.900163i	$-0.356553\pi$
$71$	7.34007	0.871106	0.435553	+	0.900163i	$0.356553\pi$
$72$	18.8036	2.21603
$73$	−8.41645	−0.985071	−0.492536	−	0.870292i	$-0.663930\pi$
$73$	−8.41645	−0.985071	−0.492536	+	0.870292i	$0.663930\pi$
$74$	0.0734127	0.00853405
$75$	0.459715	0.0530833
$76$	−25.0272	−2.87082
$77$	10.5521	1.20252
$78$	0.0844100	0.00955755
$79$	7.10354	0.799210	0.399605	−	0.916687i	$-0.369147\pi$
$79$	7.10354	0.799210	0.399605	+	0.916687i	$0.369147\pi$
$80$	−19.1261	−2.13837
$81$	7.95939	0.884377
$82$	15.3397	1.69398
$83$	6.78568	0.744826	0.372413	−	0.928067i	$-0.378531\pi$
$83$	6.78568	0.744826	0.372413	+	0.928067i	$0.378531\pi$
$84$	−5.99188	−0.653768
$85$	8.19633	0.889016
$86$	3.33314	0.359422
$87$	−0.396101	−0.0424664
$88$	17.8838	1.90642
$89$	17.0523	1.80754	0.903771	−	0.428016i	$-0.140787\pi$
$89$	17.0523	1.80754	0.903771	+	0.428016i	$0.140787\pi$
$90$	−18.5804	−1.95855
$91$	0.370889	0.0388797
$92$	28.5107	2.97244
$93$	2.96268	0.307216
$94$	−28.8926	−2.98005
$95$	13.8569	1.42169
$96$	−2.18661	−0.223170
$97$	1.70654	0.173273	0.0866365	−	0.996240i	$-0.472388\pi$
$97$	1.70654	0.173273	0.0866365	+	0.996240i	$0.472388\pi$
$98$	−19.9899	−2.01929
$99$	7.90429	0.794411
$100$	6.10970	0.610970
$101$	−8.14677	−0.810634	−0.405317	−	0.914176i	$-0.632839\pi$
$101$	−8.14677	−0.810634	−0.405317	+	0.914176i	$0.632839\pi$
$102$	2.85046	0.282238
$103$	−7.77661	−0.766252	−0.383126	−	0.923696i	$-0.625153\pi$
$103$	−7.77661	−0.766252	−0.383126	+	0.923696i	$0.625153\pi$
$104$	0.628586	0.0616380
$105$	3.31755	0.323760
$106$	6.52119	0.633394
$107$	15.2963	1.47874	0.739372	−	0.673297i	$-0.235122\pi$
$107$	15.2963	1.47874	0.739372	+	0.673297i	$0.235122\pi$
$108$	−9.15911	−0.881336
$109$	0.169330	0.0162188	0.00810942	−	0.999967i	$-0.497419\pi$
$109$	0.169330	0.0162188	0.00810942	+	0.999967i	$0.497419\pi$
$110$	−17.6715	−1.68491
$111$	−0.00981877	−0.000931956 0
$112$	−29.2263	−2.76162
$113$	−18.4132	−1.73217	−0.866086	−	0.499895i	$-0.833372\pi$
$113$	−18.4132	−1.73217	−0.866086	+	0.499895i	$0.833372\pi$
$114$	4.81906	0.451347
$115$	−15.7856	−1.47202
$116$	−5.26425	−0.488773
$117$	0.277823	0.0256847
$118$	−36.5282	−3.36269
$119$	12.5246	1.14813
$120$	5.62262	0.513273
$121$	−3.48238	−0.316580
$122$	29.7628	2.69459
$123$	−2.05164	−0.184990
$124$	39.3746	3.53594
$125$	9.21002	0.823769
$126$	−28.3924	−2.52939
$127$	5.97200	0.529930	0.264965	−	0.964258i	$-0.414640\pi$
$127$	5.97200	0.529930	0.264965	+	0.964258i	$0.414640\pi$
$128$	9.80703	0.866827
$129$	−0.445800	−0.0392505
$130$	−0.621125	−0.0544763
$131$	0.135923	0.0118756	0.00593781	−	0.999982i	$-0.498110\pi$
$131$	0.135923	0.0118756	0.00593781	+	0.999982i	$0.498110\pi$
$132$	−4.26880	−0.371551
$133$	21.1745	1.83606
$134$	17.9029	1.54657
$135$	5.07116	0.436456
$136$	21.2269	1.82019
$137$	7.08877	0.605635	0.302817	−	0.953049i	$-0.402073\pi$
$137$	7.08877	0.605635	0.302817	+	0.953049i	$0.402073\pi$
$138$	−5.48982	−0.467324
$139$	2.16168	0.183352	0.0916759	−	0.995789i	$-0.470778\pi$
$139$	2.16168	0.183352	0.0916759	+	0.995789i	$0.470778\pi$
$140$	44.0909	3.72636
$141$	3.86432	0.325434
$142$	−18.7837	−1.57629
$143$	0.264232	0.0220962
$144$	−21.8926	−1.82438
$145$	2.91468	0.242051
$146$	21.5382	1.78252
$147$	2.67360	0.220515
$148$	−0.130493	−0.0107265
$149$	0.959590	0.0786127	0.0393064	−	0.999227i	$-0.487485\pi$
$149$	0.959590	0.0786127	0.0393064	+	0.999227i	$0.487485\pi$
$150$	−1.17644	−0.0960560
$151$	−8.12902	−0.661530	−0.330765	−	0.943713i	$-0.607307\pi$
$151$	−8.12902	−0.661530	−0.330765	+	0.943713i	$0.607307\pi$
$152$	35.8867	2.91080
$153$	9.38187	0.758479
$154$	−27.0035	−2.17600
$155$	−21.8007	−1.75107
$156$	−0.150041	−0.0120129
$157$	10.8874	0.868913	0.434457	−	0.900693i	$-0.356940\pi$
$157$	10.8874	0.868913	0.434457	+	0.900693i	$0.356940\pi$
$158$	−18.1784	−1.44620
$159$	−0.872193	−0.0691694
$160$	16.0900	1.27203
$161$	−24.1217	−1.90106
$162$	−20.3686	−1.60031
$163$	25.4540	1.99371	0.996855	−	0.0792446i	$-0.0252508\pi$
$163$	25.4540	1.99371	0.996855	+	0.0792446i	$0.0252508\pi$
$164$	−27.2667	−2.12917
$165$	2.36352	0.184000
$166$	−17.3650	−1.34779
$167$	10.4948	0.812108	0.406054	−	0.913849i	$-0.366904\pi$
$167$	10.4948	0.812108	0.406054	+	0.913849i	$0.366904\pi$
$168$	8.59181	0.662872
$169$	−12.9907	−0.999286
$170$	−20.9749	−1.60870
$171$	15.8612	1.21294
$172$	−5.92476	−0.451759
$173$	11.0008	0.836376	0.418188	−	0.908361i	$-0.362665\pi$
$173$	11.0008	0.836376	0.418188	+	0.908361i	$0.362665\pi$
$174$	1.01365	0.0768444
$175$	−5.16916	−0.390752
$176$	−20.8217	−1.56949
$177$	4.88555	0.367221
$178$	−43.6380	−3.27081
$179$	−12.0951	−0.904030	−0.452015	−	0.892010i	$-0.649295\pi$
$179$	−12.0951	−0.904030	−0.452015	+	0.892010i	$0.649295\pi$
$180$	33.0273	2.46171
$181$	22.0823	1.64137	0.820683	−	0.571384i	$-0.193593\pi$
$181$	22.0823	1.64137	0.820683	+	0.571384i	$0.193593\pi$
$182$	−0.949129	−0.0703541
$183$	−3.98070	−0.294262
$184$	−40.8817	−3.01384
$185$	0.0722507	0.00531198
$186$	−7.58169	−0.555917
$187$	8.92293	0.652509
$188$	51.3575	3.74563
$189$	7.74913	0.563667
$190$	−35.4607	−2.57259
$191$	23.7897	1.72136	0.860682	−	0.509144i	$-0.170038\pi$
$191$	23.7897	1.72136	0.860682	+	0.509144i	$0.170038\pi$
$192$	0.397247	0.0286688
$193$	−23.0117	−1.65642	−0.828211	−	0.560417i	$-0.810641\pi$
$193$	−23.0117	−1.65642	−0.828211	+	0.560417i	$0.810641\pi$
$194$	−4.36715	−0.313543
$195$	0.0830740	0.00594905
$196$	35.5326	2.53805
$197$	−22.9471	−1.63491	−0.817457	−	0.575990i	$-0.804617\pi$
$197$	−22.9471	−1.63491	−0.817457	+	0.575990i	$0.804617\pi$
$198$	−20.2276	−1.43751
$199$	8.22657	0.583166	0.291583	−	0.956546i	$-0.405818\pi$
$199$	8.22657	0.583166	0.291583	+	0.956546i	$0.405818\pi$
$200$	−8.76074	−0.619478
$201$	−2.39447	−0.168893
$202$	20.8481	1.46687
$203$	4.45386	0.312600
$204$	−5.06678	−0.354746
$205$	15.0969	1.05441
$206$	19.9009	1.38656
$207$	−18.0689	−1.25588
$208$	−0.731847	−0.0507445
$209$	15.0853	1.04347
$210$	−8.48983	−0.585854
$211$	−25.6223	−1.76391	−0.881955	−	0.471334i	$-0.843773\pi$
$211$	−25.6223	−1.76391	−0.881955	+	0.471334i	$0.843773\pi$
$212$	−11.5916	−0.796115
$213$	2.51227	0.172138
$214$	−39.1441	−2.67584
$215$	3.28039	0.223721
$216$	13.1333	0.893609
$217$	−33.3132	−2.26145
$218$	−0.433325	−0.0293485
$219$	−2.88069	−0.194659
$220$	31.4116	2.11777
$221$	0.313626	0.0210968
$222$	0.0251269	0.00168640
$223$	28.6474	1.91837	0.959186	−	0.282775i	$-0.0912550\pi$
$223$	28.6474	1.91837	0.959186	+	0.282775i	$0.0912550\pi$
$224$	24.5868	1.64278
$225$	−3.87208	−0.258139
$226$	47.1207	3.13442
$227$	19.5343	1.29654	0.648270	−	0.761411i	$-0.275493\pi$
$227$	19.5343	1.29654	0.648270	+	0.761411i	$0.275493\pi$
$228$	−8.56603	−0.567299
$229$	−5.02591	−0.332121	−0.166061	−	0.986116i	$-0.553105\pi$
$229$	−5.02591	−0.332121	−0.166061	+	0.986116i	$0.553105\pi$
$230$	40.3964	2.66366
$231$	3.61165	0.237629
$232$	7.54845	0.495580
$233$	−28.7824	−1.88560	−0.942798	−	0.333364i	$-0.891816\pi$
$233$	−28.7824	−1.88560	−0.942798	+	0.333364i	$0.891816\pi$
$234$	−0.710967	−0.0464774
$235$	−28.4353	−1.85491
$236$	64.9299	4.22658
$237$	2.43132	0.157931
$238$	−32.0514	−2.07758
$239$	−18.4117	−1.19095	−0.595476	−	0.803373i	$-0.703037\pi$
$239$	−18.4117	−1.19095	−0.595476	+	0.803373i	$0.703037\pi$
$240$	−6.54627	−0.422560
$241$	−4.29187	−0.276463	−0.138232	−	0.990400i	$-0.544142\pi$
$241$	−4.29187	−0.276463	−0.138232	+	0.990400i	$0.544142\pi$
$242$	8.91162	0.572861
$243$	8.76480	0.562262
$244$	−52.9042	−3.38684
$245$	−19.6735	−1.25689
$246$	5.25029	0.334746
$247$	0.530225	0.0337374
$248$	−56.4595	−3.58518
$249$	2.32253	0.147184
$250$	−23.5690	−1.49064
$251$	−17.4117	−1.09902	−0.549508	−	0.835488i	$-0.685185\pi$
$251$	−17.4117	−1.09902	−0.549508	+	0.835488i	$0.685185\pi$
$252$	50.4683	3.17920
$253$	−17.1850	−1.08041
$254$	−15.2827	−0.958925
$255$	2.80535	0.175678
$256$	−27.4181	−1.71363
$257$	−24.2265	−1.51121	−0.755605	−	0.655028i	$-0.772657\pi$
$257$	−24.2265	−1.51121	−0.755605	+	0.655028i	$0.772657\pi$
$258$	1.14083	0.0710250
$259$	0.110405	0.00686022
$260$	1.10407	0.0684714
$261$	3.33627	0.206510
$262$	−0.347835	−0.0214893
$263$	4.90886	0.302693	0.151347	−	0.988481i	$-0.451639\pi$
$263$	4.90886	0.302693	0.151347	+	0.988481i	$0.451639\pi$
$264$	6.12106	0.376725
$265$	6.41797	0.394253
$266$	−54.1868	−3.32241
$267$	5.83647	0.357186
$268$	−31.8229	−1.94389
$269$	−2.61506	−0.159443	−0.0797217	−	0.996817i	$-0.525403\pi$
$269$	−2.61506	−0.159443	−0.0797217	+	0.996817i	$0.525403\pi$
$270$	−12.9774	−0.789781
$271$	−2.72563	−0.165570	−0.0827852	−	0.996567i	$-0.526382\pi$
$271$	−2.72563	−0.165570	−0.0827852	+	0.996567i	$0.526382\pi$
$272$	−24.7139	−1.49850
$273$	0.126944	0.00768298
$274$	−18.1406	−1.09592
$275$	−3.68266	−0.222073
$276$	9.75831	0.587382
$277$	−25.8814	−1.55506	−0.777530	−	0.628846i	$-0.783528\pi$
$277$	−25.8814	−1.55506	−0.777530	+	0.628846i	$0.783528\pi$
$278$	−5.53189	−0.331781
$279$	−24.9540	−1.49396
$280$	−63.2222	−3.77825
$281$	−11.0778	−0.660847	−0.330424	−	0.943833i	$-0.607192\pi$
$281$	−11.0778	−0.660847	−0.330424	+	0.943833i	$0.607192\pi$
$282$	−9.88904	−0.588884
$283$	−15.4745	−0.919863	−0.459932	−	0.887954i	$-0.652126\pi$
$283$	−15.4745	−0.919863	−0.459932	+	0.887954i	$0.652126\pi$
$284$	33.3886	1.98125
$285$	4.74279	0.280938
$286$	−0.676188	−0.0399838
$287$	23.0692	1.36173
$288$	18.4173	1.08525
$289$	−6.40907	−0.377004
$290$	−7.45885	−0.437999
$291$	0.584096	0.0342403
$292$	−38.2849	−2.24045
$293$	15.2164	0.888949	0.444474	−	0.895792i	$-0.353390\pi$
$293$	15.2164	0.888949	0.444474	+	0.895792i	$0.353390\pi$
$294$	−6.84192	−0.399029
$295$	−35.9500	−2.09309
$296$	0.187115	0.0108759
$297$	5.52072	0.320344
$298$	−2.45565	−0.142252
$299$	−0.604025	−0.0349317
$300$	2.09116	0.120733
$301$	5.01269	0.288927
$302$	20.8027	1.19706
$303$	−2.78838	−0.160188
$304$	−41.7820	−2.39636
$305$	29.2917	1.67724
$306$	−24.0088	−1.37249
$307$	2.71396	0.154894	0.0774470	−	0.996996i	$-0.475323\pi$
$307$	2.71396	0.154894	0.0774470	+	0.996996i	$0.475323\pi$
$308$	47.9995	2.73502
$309$	−2.66169	−0.151418
$310$	55.7893	3.16862
$311$	−25.6102	−1.45222	−0.726111	−	0.687577i	$-0.758674\pi$
$311$	−25.6102	−1.45222	−0.726111	+	0.687577i	$0.758674\pi$
$312$	0.215145	0.0121802
$313$	−12.4322	−0.702709	−0.351354	−	0.936243i	$-0.614279\pi$
$313$	−12.4322	−0.702709	−0.351354	+	0.936243i	$0.614279\pi$
$314$	−27.8617	−1.57233
$315$	−27.9430	−1.57441
$316$	32.3127	1.81773
$317$	−17.5708	−0.986877	−0.493438	−	0.869781i	$-0.664260\pi$
$317$	−17.5708	−0.986877	−0.493438	+	0.869781i	$0.664260\pi$
$318$	2.23200	0.125164
$319$	3.17306	0.177657
$320$	−2.92311	−0.163407
$321$	5.23543	0.292213
$322$	61.7290	3.44002
$323$	17.9053	0.996277
$324$	36.2058	2.01143
$325$	−0.129440	−0.00718002
$326$	−65.1384	−3.60768
$327$	0.0579562	0.00320499
$328$	39.0980	2.15882
$329$	−43.4514	−2.39555
$330$	−6.04841	−0.332954
$331$	−13.9198	−0.765102	−0.382551	−	0.923934i	$-0.624954\pi$
$331$	−13.9198	−0.765102	−0.382551	+	0.923934i	$0.624954\pi$
$332$	30.8668	1.69404
$333$	0.0827013	0.00453200
$334$	−26.8568	−1.46954
$335$	17.6195	0.962657
$336$	−10.0032	−0.545721
$337$	22.4773	1.22442	0.612208	−	0.790697i	$-0.290281\pi$
$337$	22.4773	1.22442	0.612208	+	0.790697i	$0.290281\pi$
$338$	33.2441	1.80824
$339$	−6.30228	−0.342293
$340$	37.2836	2.02198
$341$	−23.7333	−1.28523
$342$	−40.5899	−2.19485
$343$	−3.12276	−0.168613
$344$	8.49556	0.458050
$345$	−5.40292	−0.290884
$346$	−28.1518	−1.51345
$347$	24.0019	1.28849	0.644244	−	0.764820i	$-0.277172\pi$
$347$	24.0019	1.28849	0.644244	+	0.764820i	$0.277172\pi$
$348$	−1.80179	−0.0965860
$349$	31.9742	1.71154	0.855769	−	0.517359i	$-0.173085\pi$
$349$	31.9742	1.71154	0.855769	+	0.517359i	$0.173085\pi$
$350$	13.2282	0.707078
$351$	0.194044	0.0103573
$352$	17.5164	0.933626
$353$	19.2339	1.02372	0.511860	−	0.859069i	$-0.328957\pi$
$353$	19.2339	1.02372	0.511860	+	0.859069i	$0.328957\pi$
$354$	−12.5024	−0.664497
$355$	−18.4864	−0.981156
$356$	77.5678	4.11109
$357$	4.28679	0.226881
$358$	30.9521	1.63587
$359$	−12.2724	−0.647714	−0.323857	−	0.946106i	$-0.604980\pi$
$359$	−12.2724	−0.647714	−0.323857	+	0.946106i	$0.604980\pi$
$360$	−47.3581	−2.49599
$361$	11.2711	0.593217
$362$	−56.5101	−2.97010
$363$	−1.19191	−0.0625589
$364$	1.68710	0.0884283
$365$	21.1973	1.10952
$366$	10.1869	0.532476
$367$	33.3439	1.74054	0.870269	−	0.492577i	$-0.163945\pi$
$367$	33.3439	1.74054	0.870269	+	0.492577i	$0.163945\pi$
$368$	47.5975	2.48119
$369$	17.2805	0.899588
$370$	−0.184894	−0.00961220
$371$	9.80717	0.509163
$372$	13.4767	0.698733
$373$	17.9727	0.930590	0.465295	−	0.885156i	$-0.345948\pi$
$373$	17.9727	0.930590	0.465295	+	0.885156i	$0.345948\pi$
$374$	−22.8343	−1.18074
$375$	3.15230	0.162784
$376$	−73.6419	−3.79779
$377$	0.111528	0.00574399
$378$	−19.8305	−1.01997
$379$	−18.7897	−0.965160	−0.482580	−	0.875852i	$-0.660300\pi$
$379$	−18.7897	−0.965160	−0.482580	+	0.875852i	$0.660300\pi$
$380$	63.0325	3.23350
$381$	2.04403	0.104719
$382$	−60.8794	−3.11486
$383$	−3.74770	−0.191499	−0.0957493	−	0.995405i	$-0.530525\pi$
$383$	−3.74770	−0.191499	−0.0957493	+	0.995405i	$0.530525\pi$
$384$	3.35664	0.171293
$385$	−26.5761	−1.35444
$386$	58.8885	2.99735
$387$	3.75487	0.190871
$388$	7.76274	0.394093
$389$	33.6832	1.70781	0.853903	−	0.520432i	$-0.174229\pi$
$389$	33.6832	1.70781	0.853903	+	0.520432i	$0.174229\pi$
$390$	−0.212592	−0.0107650
$391$	−20.3975	−1.03155
$392$	−50.9505	−2.57339
$393$	0.0465220	0.00234673
$394$	58.7231	2.95843
$395$	−17.8907	−0.900178
$396$	35.9551	1.80681
$397$	11.9067	0.597578	0.298789	−	0.954319i	$-0.403417\pi$
$397$	11.9067	0.597578	0.298789	+	0.954319i	$0.403417\pi$
$398$	−21.0523	−1.05526
$399$	7.24736	0.362822
$400$	10.1999	0.509996
$401$	18.2746	0.912592	0.456296	−	0.889828i	$-0.349176\pi$
$401$	18.2746	0.912592	0.456296	+	0.889828i	$0.349176\pi$
$402$	6.12760	0.305617
$403$	−0.834187	−0.0415538
$404$	−37.0581	−1.84371
$405$	−20.0462	−0.996104
$406$	−11.3977	−0.565659
$407$	0.0786557	0.00389882
$408$	7.26530	0.359686
$409$	8.89053	0.439608	0.219804	−	0.975544i	$-0.429458\pi$
$409$	8.89053	0.439608	0.219804	+	0.975544i	$0.429458\pi$
$410$	−38.6339	−1.90799
$411$	2.42626	0.119679
$412$	−35.3744	−1.74277
$413$	−54.9344	−2.70315
$414$	46.2395	2.27255
$415$	−17.0901	−0.838922
$416$	0.615672	0.0301858
$417$	0.739877	0.0362319
$418$	−38.6043	−1.88820
$419$	14.4020	0.703583	0.351791	−	0.936078i	$-0.385573\pi$
$419$	14.4020	0.703583	0.351791	+	0.936078i	$0.385573\pi$
$420$	15.0909	0.736361
$421$	30.1805	1.47091	0.735454	−	0.677575i	$-0.236969\pi$
$421$	30.1805	1.47091	0.735454	+	0.677575i	$0.236969\pi$
$422$	65.5690	3.19185
$423$	−32.5483	−1.58255
$424$	16.6213	0.807201
$425$	−4.37108	−0.212028
$426$	−6.42907	−0.311490
$427$	44.7600	2.16609
$428$	69.5798	3.36327
$429$	0.0904384	0.00436641
$430$	−8.39472	−0.404829
$431$	−0.885200	−0.0426386	−0.0213193	−	0.999773i	$-0.506787\pi$
$431$	−0.885200	−0.0426386	−0.0213193	+	0.999773i	$0.506787\pi$
$432$	−15.2908	−0.735679
$433$	−8.47023	−0.407053	−0.203527	−	0.979069i	$-0.565240\pi$
$433$	−8.47023	−0.407053	−0.203527	+	0.979069i	$0.565240\pi$
$434$	85.2506	4.09216
$435$	0.997603	0.0478314
$436$	0.770249	0.0368882
$437$	−34.4845	−1.64962
$438$	7.37187	0.352241
$439$	−5.22691	−0.249467	−0.124733	−	0.992190i	$-0.539808\pi$
$439$	−5.22691	−0.249467	−0.124733	+	0.992190i	$0.539808\pi$
$440$	−45.0414	−2.14726
$441$	−22.5191	−1.07234
$442$	−0.802590	−0.0381753
$443$	−21.5289	−1.02287	−0.511434	−	0.859322i	$-0.670886\pi$
$443$	−21.5289	−1.02287	−0.511434	+	0.859322i	$0.670886\pi$
$444$	−0.0446637	−0.00211965
$445$	−42.9473	−2.03590
$446$	−73.3106	−3.47136
$447$	0.328438	0.0155346
$448$	−4.46675	−0.211034
$449$	12.6703	0.597949	0.298974	−	0.954261i	$-0.403355\pi$
$449$	12.6703	0.597949	0.298974	+	0.954261i	$0.403355\pi$
$450$	9.90890	0.467110
$451$	16.4352	0.773903
$452$	−83.7584	−3.93966
$453$	−2.78231	−0.130724
$454$	−49.9897	−2.34613
$455$	−0.934106	−0.0437916
$456$	12.2829	0.575199
$457$	28.2936	1.32352	0.661758	−	0.749717i	$-0.269810\pi$
$457$	28.2936	1.32352	0.661758	+	0.749717i	$0.269810\pi$
$458$	12.8616	0.600984
$459$	6.55273	0.305855
$460$	−71.8059	−3.34797
$461$	−3.76726	−0.175459	−0.0877294	−	0.996144i	$-0.527961\pi$
$461$	−3.76726	−0.175459	−0.0877294	+	0.996144i	$0.527961\pi$
$462$	−9.24245	−0.429997
$463$	9.63051	0.447568	0.223784	−	0.974639i	$-0.428159\pi$
$463$	9.63051	0.447568	0.223784	+	0.974639i	$0.428159\pi$
$464$	−8.78847	−0.407995
$465$	−7.46169	−0.346027
$466$	73.6560	3.41205
$467$	24.7201	1.14391	0.571955	−	0.820285i	$-0.306185\pi$
$467$	24.7201	1.14391	0.571955	+	0.820285i	$0.306185\pi$
$468$	1.26376	0.0584175
$469$	26.9240	1.24324
$470$	72.7678	3.35653
$471$	3.72643	0.171705
$472$	−93.1035	−4.28544
$473$	3.57119	0.164204
$474$	−6.22190	−0.285781
$475$	−7.38986	−0.339070
$476$	56.9722	2.61132
$477$	7.34629	0.336363
$478$	47.1167	2.15507
$479$	−24.3291	−1.11163	−0.555813	−	0.831307i	$-0.687593\pi$
$479$	−24.3291	−1.11163	−0.555813	+	0.831307i	$0.687593\pi$
$480$	5.50711	0.251364
$481$	0.00276462	0.000126056 0
$482$	10.9832	0.500270
$483$	−8.25610	−0.375666
$484$	−15.8407	−0.720031
$485$	−4.29803	−0.195163
$486$	−22.4297	−1.01743
$487$	−23.3430	−1.05777	−0.528886	−	0.848693i	$-0.677390\pi$
$487$	−23.3430	−1.05777	−0.528886	+	0.848693i	$0.677390\pi$
$488$	75.8597	3.43401
$489$	8.71210	0.393975
$490$	50.3458	2.27439
$491$	−38.5257	−1.73864	−0.869320	−	0.494250i	$-0.835443\pi$
$491$	−38.5257	−1.73864	−0.869320	+	0.494250i	$0.835443\pi$
$492$	−9.33254	−0.420743
$493$	3.76622	0.169622
$494$	−1.35688	−0.0610489
$495$	−19.9074	−0.894772
$496$	65.7344	2.95156
$497$	−28.2487	−1.26713
$498$	−5.94350	−0.266334
$499$	−14.6824	−0.657275	−0.328638	−	0.944456i	$-0.606589\pi$
$499$	−14.6824	−0.657275	−0.328638	+	0.944456i	$0.606589\pi$
$500$	41.8946	1.87359
$501$	3.59203	0.160480
$502$	44.5576	1.98870
$503$	21.9428	0.978382	0.489191	−	0.872177i	$-0.337292\pi$
$503$	21.9428	0.978382	0.489191	+	0.872177i	$0.337292\pi$
$504$	−72.3669	−3.22348
$505$	20.5181	0.913045
$506$	43.9776	1.95504
$507$	−4.44631	−0.197468
$508$	27.1655	1.20528
$509$	36.9633	1.63837	0.819186	−	0.573528i	$-0.194426\pi$
$509$	36.9633	1.63837	0.819186	+	0.573528i	$0.194426\pi$
$510$	−7.17906	−0.317894
$511$	32.3912	1.43290
$512$	50.5506	2.23404
$513$	11.0782	0.489115
$514$	61.9973	2.73458
$515$	19.5859	0.863056
$516$	−2.02786	−0.0892716
$517$	−30.9561	−1.36145
$518$	−0.282533	−0.0124138
$519$	3.76523	0.165275
$520$	−1.58313	−0.0694249
$521$	−12.4797	−0.546745	−0.273373	−	0.961908i	$-0.588139\pi$
$521$	−12.4797	−0.546745	−0.273373	+	0.961908i	$0.588139\pi$
$522$	−8.53772	−0.373686
$523$	7.62644	0.333481	0.166740	−	0.986001i	$-0.446676\pi$
$523$	7.62644	0.333481	0.166740	+	0.986001i	$0.446676\pi$
$524$	0.618286	0.0270100
$525$	−1.76924	−0.0772160
$526$	−12.5621	−0.547733
$527$	−28.1699	−1.22710
$528$	−7.12660	−0.310145
$529$	16.2843	0.708014
$530$	−16.4240	−0.713413
$531$	−41.1499	−1.78575
$532$	96.3187	4.17594
$533$	0.577671	0.0250217
$534$	−14.9359	−0.646340
$535$	−38.5245	−1.66556
$536$	45.6311	1.97096
$537$	−4.13977	−0.178644
$538$	6.69212	0.288518
$539$	−21.4175	−0.922519
$540$	23.0678	0.992678
$541$	33.6481	1.44664	0.723322	−	0.690511i	$-0.242614\pi$
$541$	33.6481	1.44664	0.723322	+	0.690511i	$0.242614\pi$
$542$	6.97507	0.299605
$543$	7.55809	0.324349
$544$	20.7908	0.891398
$545$	−0.426467	−0.0182678
$546$	−0.324857	−0.0139026
$547$	21.0984	0.902102	0.451051	−	0.892498i	$-0.351049\pi$
$547$	21.0984	0.902102	0.451051	+	0.892498i	$0.351049\pi$
$548$	32.2455	1.37746
$549$	33.5285	1.43096
$550$	9.42417	0.401848
$551$	6.36726	0.271254
$552$	−13.9925	−0.595561
$553$	−27.3384	−1.16255
$554$	66.2321	2.81393
$555$	0.0247292	0.00104969
$556$	9.83310	0.417016
$557$	10.2969	0.436293	0.218147	−	0.975916i	$-0.429999\pi$
$557$	10.2969	0.436293	0.218147	+	0.975916i	$0.429999\pi$
$558$	63.8589	2.70336
$559$	0.125522	0.00530900
$560$	73.6081	3.11051
$561$	3.05404	0.128942
$562$	28.3489	1.19582
$563$	17.8018	0.750256	0.375128	−	0.926973i	$-0.377599\pi$
$563$	17.8018	0.750256	0.375128	+	0.926973i	$0.377599\pi$
$564$	17.5781	0.740170
$565$	46.3748	1.95100
$566$	39.6002	1.66452
$567$	−30.6322	−1.28643
$568$	−47.8762	−2.00884
$569$	−34.2695	−1.43665	−0.718326	−	0.695706i	$-0.755091\pi$
$569$	−34.2695	−1.43665	−0.718326	+	0.695706i	$0.755091\pi$
$570$	−12.1371	−0.508367
$571$	−13.1806	−0.551590	−0.275795	−	0.961216i	$-0.588941\pi$
$571$	−13.1806	−0.551590	−0.275795	+	0.961216i	$0.588941\pi$
$572$	1.20194	0.0502558
$573$	8.14247	0.340157
$574$	−59.0356	−2.46410
$575$	8.41843	0.351073
$576$	−3.34592	−0.139413
$577$	−23.8903	−0.994565	−0.497283	−	0.867589i	$-0.665669\pi$
$577$	−23.8903	−0.994565	−0.497283	+	0.867589i	$0.665669\pi$
$578$	16.4012	0.682201
$579$	−7.87620	−0.327324
$580$	13.2583	0.550522
$581$	−26.1151	−1.08344
$582$	−1.49474	−0.0619589
$583$	6.98692	0.289369
$584$	54.8970	2.27165
$585$	−0.699713	−0.0289296
$586$	−38.9396	−1.60858
$587$	−22.5071	−0.928968	−0.464484	−	0.885582i	$-0.653760\pi$
$587$	−22.5071	−0.928968	−0.464484	+	0.885582i	$0.653760\pi$
$588$	12.1617	0.501540
$589$	−47.6247	−1.96234
$590$	91.9984	3.78751
$591$	−7.85407	−0.323073
$592$	−0.217854	−0.00895373
$593$	36.7290	1.50828	0.754140	−	0.656714i	$-0.228054\pi$
$593$	36.7290	1.50828	0.754140	+	0.656714i	$0.228054\pi$
$594$	−14.1279	−0.579673
$595$	−31.5440	−1.29318
$596$	4.36500	0.178797
$597$	2.81570	0.115239
$598$	1.54574	0.0632100
$599$	−30.4521	−1.24424	−0.622120	−	0.782922i	$-0.713728\pi$
$599$	−30.4521	−1.24424	−0.622120	+	0.782922i	$0.713728\pi$
$600$	−2.99853	−0.122414
$601$	−33.3406	−1.35999	−0.679995	−	0.733217i	$-0.738018\pi$
$601$	−33.3406	−1.35999	−0.679995	+	0.733217i	$0.738018\pi$
$602$	−12.8278	−0.522822
$603$	20.1681	0.821307
$604$	−36.9774	−1.50459
$605$	8.77057	0.356574
$606$	7.13566	0.289866
$607$	−18.7922	−0.762752	−0.381376	−	0.924420i	$-0.624550\pi$
$607$	−18.7922	−0.762752	−0.381376	+	0.924420i	$0.624550\pi$
$608$	35.1494	1.42550
$609$	1.52442	0.0617725
$610$	−74.9593	−3.03501
$611$	−1.08806	−0.0440180
$612$	42.6764	1.72509
$613$	37.8414	1.52840	0.764200	−	0.644980i	$-0.223134\pi$
$613$	37.8414	1.52840	0.764200	+	0.644980i	$0.223134\pi$
$614$	−6.94520	−0.280286
$615$	5.16718	0.208361
$616$	−68.8268	−2.77311
$617$	25.7592	1.03703	0.518514	−	0.855069i	$-0.326485\pi$
$617$	25.7592	1.03703	0.518514	+	0.855069i	$0.326485\pi$
$618$	6.81144	0.273996
$619$	−19.4741	−0.782730	−0.391365	−	0.920235i	$-0.627997\pi$
$619$	−19.4741	−0.782730	−0.391365	+	0.920235i	$0.627997\pi$
$620$	−99.1672	−3.98265
$621$	−12.6202	−0.506429
$622$	65.5382	2.62784
$623$	−65.6269	−2.62929
$624$	−0.250488	−0.0100276
$625$	−29.9117	−1.19647
$626$	31.8148	1.27157
$627$	5.16323	0.206200
$628$	49.5250	1.97626
$629$	0.0933592	0.00372247
$630$	71.5079	2.84894
$631$	−46.3553	−1.84538	−0.922689	−	0.385546i	$-0.874013\pi$
$631$	−46.3553	−1.84538	−0.922689	+	0.385546i	$0.874013\pi$
$632$	−46.3334	−1.84304
$633$	−8.76970	−0.348564
$634$	44.9649	1.78579
$635$	−15.0408	−0.596878
$636$	−3.96744	−0.157319
$637$	−0.752792	−0.0298267
$638$	−8.12008	−0.321477
$639$	−21.1603	−0.837090
$640$	−24.6996	−0.976337
$641$	6.97099	0.275338	0.137669	−	0.990478i	$-0.456039\pi$
$641$	6.97099	0.275338	0.137669	+	0.990478i	$0.456039\pi$
$642$	−13.3978	−0.528769
$643$	−33.1696	−1.30808	−0.654040	−	0.756460i	$-0.726927\pi$
$643$	−33.1696	−1.30808	−0.654040	+	0.756460i	$0.726927\pi$
$644$	−109.725	−4.32377
$645$	1.12277	0.0442092
$646$	−45.8208	−1.80279
$647$	−49.9654	−1.96434	−0.982171	−	0.187988i	$-0.939803\pi$
$647$	−49.9654	−1.96434	−0.982171	+	0.187988i	$0.939803\pi$
$648$	−51.9158	−2.03944
$649$	−39.1369	−1.53626
$650$	0.331244	0.0129925
$651$	−11.4021	−0.446882
$652$	115.785	4.53451
$653$	13.3018	0.520538	0.260269	−	0.965536i	$-0.416189\pi$
$653$	13.3018	0.520538	0.260269	+	0.965536i	$0.416189\pi$
$654$	−0.148314	−0.00579952
$655$	−0.342329	−0.0133759
$656$	−45.5208	−1.77729
$657$	24.2634	0.946605
$658$	111.195	4.33483
$659$	36.6683	1.42840	0.714198	−	0.699944i	$-0.246792\pi$
$659$	36.6683	1.42840	0.714198	+	0.699944i	$0.246792\pi$
$660$	10.7512	0.418491
$661$	−7.78564	−0.302826	−0.151413	−	0.988471i	$-0.548382\pi$
$661$	−7.78564	−0.302826	−0.151413	+	0.988471i	$0.548382\pi$
$662$	35.6217	1.38448
$663$	0.107344	0.00416891
$664$	−44.2602	−1.71763
$665$	−53.3291	−2.06802
$666$	−0.211638	−0.00820081
$667$	−7.25351	−0.280857
$668$	47.7387	1.84706
$669$	9.80511	0.379087
$670$	−45.0895	−1.74196
$671$	31.8884	1.23104
$672$	8.41530	0.324627
$673$	38.7453	1.49352	0.746761	−	0.665093i	$-0.231608\pi$
$673$	38.7453	1.49352	0.746761	+	0.665093i	$0.231608\pi$
$674$	−57.5209	−2.21562
$675$	−2.70444	−0.104094
$676$	−59.0923	−2.27278
$677$	23.7719	0.913630	0.456815	−	0.889562i	$-0.348990\pi$
$677$	23.7719	0.913630	0.456815	+	0.889562i	$0.348990\pi$
$678$	16.1279	0.619390
$679$	−6.56773	−0.252046
$680$	−53.4612	−2.05014
$681$	6.68599	0.256208
$682$	60.7351	2.32567
$683$	50.7377	1.94142	0.970712	−	0.240247i	$-0.0772285\pi$
$683$	50.7377	1.94142	0.970712	+	0.240247i	$0.0772285\pi$
$684$	72.1497	2.75871
$685$	−17.8535	−0.682147
$686$	7.99134	0.305111
$687$	−1.72021	−0.0656301
$688$	−9.89117	−0.377097
$689$	0.245579	0.00935581
$690$	13.8264	0.526363
$691$	−28.7225	−1.09265	−0.546327	−	0.837572i	$-0.683975\pi$
$691$	−28.7225	−1.09265	−0.546327	+	0.837572i	$0.683975\pi$
$692$	50.0406	1.90226
$693$	−30.4201	−1.15556
$694$	−61.4223	−2.33156
$695$	−5.44433	−0.206515
$696$	2.58360	0.0979310
$697$	19.5075	0.738899
$698$	−81.8239	−3.09708
$699$	−9.85131	−0.372611
$700$	−23.5135	−0.888728
$701$	15.2628	0.576468	0.288234	−	0.957560i	$-0.406932\pi$
$701$	15.2628	0.576468	0.288234	+	0.957560i	$0.406932\pi$
$702$	−0.496572	−0.0187419
$703$	0.157835	0.00595287
$704$	−3.18225	−0.119935
$705$	−9.73251	−0.366548
$706$	−49.2209	−1.85245
$707$	31.3533	1.17916
$708$	22.2235	0.835209
$709$	41.4399	1.55631	0.778155	−	0.628073i	$-0.216156\pi$
$709$	41.4399	1.55631	0.778155	+	0.628073i	$0.216156\pi$
$710$	47.3079	1.77543
$711$	−20.4784	−0.768002
$712$	−111.225	−4.16834
$713$	54.2535	2.03181
$714$	−10.9702	−0.410549
$715$	−0.665485	−0.0248877
$716$	−55.0184	−2.05613
$717$	−6.30174	−0.235343
$718$	31.4059	1.17206
$719$	5.22860	0.194994	0.0974970	−	0.995236i	$-0.468916\pi$
$719$	5.22860	0.194994	0.0974970	+	0.995236i	$0.468916\pi$
$720$	55.1378	2.05486
$721$	29.9288	1.11460
$722$	−28.8435	−1.07345
$723$	−1.46897	−0.0546316
$724$	100.448	3.73313
$725$	−1.55439	−0.0577286
$726$	3.05017	0.113202
$727$	31.5154	1.16884	0.584421	−	0.811451i	$-0.301322\pi$
$727$	31.5154	1.16884	0.584421	+	0.811451i	$0.301322\pi$
$728$	−2.41915	−0.0896597
$729$	−20.8783	−0.773269
$730$	−54.2454	−2.00771
$731$	4.23877	0.156777
$732$	−18.1074	−0.669270
$733$	−35.2900	−1.30347	−0.651733	−	0.758448i	$-0.725958\pi$
$733$	−35.2900	−1.30347	−0.651733	+	0.758448i	$0.725958\pi$
$734$	−85.3292	−3.14956
$735$	−6.73362	−0.248373
$736$	−40.0418	−1.47596
$737$	19.1815	0.706559
$738$	−44.2220	−1.62783
$739$	−17.5613	−0.646002	−0.323001	−	0.946399i	$-0.604692\pi$
$739$	−17.5613	−0.646002	−0.323001	+	0.946399i	$0.604692\pi$
$740$	0.328655	0.0120816
$741$	0.181479	0.00666681
$742$	−25.0972	−0.921347
$743$	31.6897	1.16258	0.581291	−	0.813695i	$-0.302548\pi$
$743$	31.6897	1.16258	0.581291	+	0.813695i	$0.302548\pi$
$744$	−19.3243	−0.708464
$745$	−2.41679	−0.0885442
$746$	−45.9933	−1.68393
$747$	−19.5621	−0.715741
$748$	40.5887	1.48407
$749$	−58.8685	−2.15101
$750$	−8.06694	−0.294563
$751$	−14.5860	−0.532250	−0.266125	−	0.963939i	$-0.585743\pi$
$751$	−14.5860	−0.532250	−0.266125	+	0.963939i	$0.585743\pi$
$752$	85.7394	3.12660
$753$	−5.95947	−0.217175
$754$	−0.285408	−0.0103939
$755$	20.4734	0.745104
$756$	35.2494	1.28201
$757$	6.35650	0.231031	0.115515	−	0.993306i	$-0.463148\pi$
$757$	6.35650	0.231031	0.115515	+	0.993306i	$0.463148\pi$
$758$	48.0840	1.74649
$759$	−5.88189	−0.213499
$760$	−90.3828	−3.27853
$761$	7.40962	0.268599	0.134299	−	0.990941i	$-0.457122\pi$
$761$	7.40962	0.268599	0.134299	+	0.990941i	$0.457122\pi$
$762$	−5.23080	−0.189492
$763$	−0.651675	−0.0235922
$764$	108.215	3.91508
$765$	−23.6288	−0.854301
$766$	9.59061	0.346523
$767$	−1.37560	−0.0496700
$768$	−9.38435	−0.338628
$769$	−31.2680	−1.12755	−0.563777	−	0.825927i	$-0.690652\pi$
$769$	−31.2680	−1.12755	−0.563777	+	0.825927i	$0.690652\pi$
$770$	68.0099	2.45091
$771$	−8.29198	−0.298629
$772$	−104.676	−3.76738
$773$	−41.7723	−1.50244	−0.751222	−	0.660049i	$-0.770535\pi$
$773$	−41.7723	−1.50244	−0.751222	+	0.660049i	$0.770535\pi$
$774$	−9.60896	−0.345387
$775$	11.6262	0.417627
$776$	−11.1311	−0.399582
$777$	0.0377881	0.00135564
$778$	−86.1975	−3.09033
$779$	32.9799	1.18163
$780$	0.377888	0.0135306
$781$	−20.1252	−0.720137
$782$	52.1985	1.86661
$783$	2.33020	0.0832746
$784$	59.3204	2.11859
$785$	−27.4207	−0.978686
$786$	−0.119053	−0.00424648
$787$	4.17593	0.148856	0.0744280	−	0.997226i	$-0.476287\pi$
$787$	4.17593	0.148856	0.0744280	+	0.997226i	$0.476287\pi$
$788$	−104.382	−3.71846
$789$	1.68015	0.0598149
$790$	45.7834	1.62890
$791$	70.8645	2.51965
$792$	−51.5563	−1.83197
$793$	1.12082	0.0398016
$794$	−30.4699	−1.08134
$795$	2.19667	0.0779078
$796$	37.4211	1.32636
$797$	−20.7369	−0.734538	−0.367269	−	0.930115i	$-0.619707\pi$
$797$	−20.7369	−0.734538	−0.367269	+	0.930115i	$0.619707\pi$
$798$	−18.5465	−0.656537
$799$	−36.7428	−1.29987
$800$	−8.58076	−0.303376
$801$	−49.1593	−1.73696
$802$	−46.7660	−1.65136
$803$	23.0765	0.814351
$804$	−10.8920	−0.384131
$805$	60.7519	2.14122
$806$	2.13474	0.0751930
$807$	−0.895055	−0.0315074
$808$	53.1380	1.86939
$809$	−5.80006	−0.203919	−0.101960	−	0.994789i	$-0.532511\pi$
$809$	−5.80006	−0.203919	−0.101960	+	0.994789i	$0.532511\pi$
$810$	51.2995	1.80248
$811$	25.9154	0.910014	0.455007	−	0.890488i	$-0.349637\pi$
$811$	25.9154	0.910014	0.455007	+	0.890488i	$0.349637\pi$
$812$	20.2598	0.710979
$813$	−0.932898	−0.0327182
$814$	−0.201285	−0.00705504
$815$	−64.1074	−2.24558
$816$	−8.45881	−0.296117
$817$	7.16617	0.250713
$818$	−22.7514	−0.795485
$819$	−1.06922	−0.0373615
$820$	68.6728	2.39816
$821$	29.3951	1.02590	0.512948	−	0.858420i	$-0.328553\pi$
$821$	29.3951	1.02590	0.512948	+	0.858420i	$0.328553\pi$
$822$	−6.20897	−0.216563
$823$	−21.3147	−0.742983	−0.371492	−	0.928436i	$-0.621154\pi$
$823$	−21.3147	−0.742983	−0.371492	+	0.928436i	$0.621154\pi$
$824$	50.7236	1.76704
$825$	−1.26046	−0.0438836
$826$	140.581	4.89143
$827$	37.1030	1.29020	0.645099	−	0.764099i	$-0.276816\pi$
$827$	37.1030	1.29020	0.645099	+	0.764099i	$0.276816\pi$
$828$	−82.1921	−2.85637
$829$	−14.3494	−0.498375	−0.249187	−	0.968455i	$-0.580163\pi$
$829$	−14.3494	−0.498375	−0.249187	+	0.968455i	$0.580163\pi$
$830$	43.7348	1.51806
$831$	−8.85838	−0.307294
$832$	−0.111851	−0.00387773
$833$	−25.4212	−0.880793
$834$	−1.89339	−0.0655628
$835$	−26.4317	−0.914705
$836$	68.6203	2.37328
$837$	−17.4290	−0.602435
$838$	−36.8556	−1.27316
$839$	−29.5595	−1.02051	−0.510254	−	0.860024i	$-0.670449\pi$
$839$	−29.5595	−1.02051	−0.510254	+	0.860024i	$0.670449\pi$
$840$	−21.6390	−0.746616
$841$	−27.6607	−0.953817
$842$	−77.2338	−2.66165
$843$	−3.79159	−0.130589
$844$	−116.551	−4.01185
$845$	32.7179	1.12553
$846$	83.2932	2.86368
$847$	13.4021	0.460503
$848$	−19.3518	−0.664542
$849$	−5.29643	−0.181773
$850$	11.1859	0.383672
$851$	−0.179804	−0.00616360
$852$	11.4279	0.391512
$853$	35.8796	1.22849	0.614247	−	0.789114i	$-0.289460\pi$
$853$	35.8796	1.22849	0.614247	+	0.789114i	$0.289460\pi$
$854$	−114.544	−3.91961
$855$	−39.9474	−1.36617
$856$	−99.7710	−3.41010
$857$	21.4382	0.732315	0.366158	−	0.930553i	$-0.380673\pi$
$857$	21.4382	0.732315	0.366158	+	0.930553i	$0.380673\pi$
$858$	−0.231438	−0.00790116
$859$	−5.60662	−0.191295	−0.0956477	−	0.995415i	$-0.530492\pi$
$859$	−5.60662	−0.191295	−0.0956477	+	0.995415i	$0.530492\pi$
$860$	14.9219	0.508831
$861$	7.89587	0.269091
$862$	2.26529	0.0771559
$863$	−40.1106	−1.36538	−0.682690	−	0.730708i	$-0.739190\pi$
$863$	−40.1106	−1.36538	−0.682690	+	0.730708i	$0.739190\pi$
$864$	12.8635	0.437625
$865$	−27.7062	−0.942038
$866$	21.6759	0.736576
$867$	−2.19362	−0.0744994
$868$	−151.535	−5.14345
$869$	−19.4767	−0.660701
$870$	−2.55293	−0.0865525
$871$	0.674198	0.0228443
$872$	−1.10447	−0.0374019
$873$	−4.91971	−0.166507
$874$	88.2481	2.98504
$875$	−35.4453	−1.19827
$876$	−13.1037	−0.442733
$877$	−56.4061	−1.90470	−0.952349	−	0.305009i	$-0.901340\pi$
$877$	−56.4061	−1.90470	−0.952349	+	0.305009i	$0.901340\pi$
$878$	13.3760	0.451418
$879$	5.20808	0.175664
$880$	52.4406	1.76777
$881$	−28.1784	−0.949353	−0.474676	−	0.880160i	$-0.657435\pi$
$881$	−28.1784	−0.949353	−0.474676	+	0.880160i	$0.657435\pi$
$882$	57.6279	1.94043
$883$	18.5424	0.624000	0.312000	−	0.950082i	$-0.399001\pi$
$883$	18.5424	0.624000	0.312000	+	0.950082i	$0.399001\pi$
$884$	1.42663	0.0479827
$885$	−12.3046	−0.413613
$886$	55.0938	1.85091
$887$	52.7020	1.76956	0.884780	−	0.466009i	$-0.154309\pi$
$887$	52.7020	1.76956	0.884780	+	0.466009i	$0.154309\pi$
$888$	0.0640437	0.00214917
$889$	−22.9836	−0.770845
$890$	109.905	3.68402
$891$	−21.8233	−0.731108
$892$	130.312	4.36316
$893$	−62.1183	−2.07871
$894$	−0.840493	−0.0281103
$895$	30.4622	1.01824
$896$	−37.7429	−1.26090
$897$	−0.206739	−0.00690281
$898$	−32.4241	−1.08201
$899$	−10.0174	−0.334100
$900$	−17.6134	−0.587112
$901$	8.29302	0.276280
$902$	−42.0587	−1.40040
$903$	1.71569	0.0570945
$904$	120.102	3.99453
$905$	−55.6156	−1.84873
$906$	7.12011	0.236550
$907$	37.6567	1.25037	0.625185	−	0.780476i	$-0.285023\pi$
$907$	37.6567	1.25037	0.625185	+	0.780476i	$0.285023\pi$
$908$	88.8581	2.94886
$909$	23.4859	0.778979
$910$	2.39044	0.0792422
$911$	31.3160	1.03754	0.518772	−	0.854913i	$-0.326389\pi$
$911$	31.3160	1.03754	0.518772	+	0.854913i	$0.326389\pi$
$912$	−14.3007	−0.473542
$913$	−18.6052	−0.615742
$914$	−72.4051	−2.39495
$915$	10.0256	0.331437
$916$	−22.8619	−0.755379
$917$	−0.523106	−0.0172745
$918$	−16.7689	−0.553455
$919$	−43.1500	−1.42339	−0.711694	−	0.702489i	$-0.752072\pi$
$919$	−43.1500	−1.42339	−0.711694	+	0.702489i	$0.752072\pi$
$920$	102.963	3.39459
$921$	0.928904	0.0306084
$922$	9.64066	0.317498
$923$	−0.707368	−0.0232833
$924$	16.4287	0.540465
$925$	−0.0385311	−0.00126690
$926$	−24.6451	−0.809889
$927$	22.4188	0.736331
$928$	7.39337	0.242699
$929$	1.30498	0.0428152	0.0214076	−	0.999771i	$-0.493185\pi$
$929$	1.30498	0.0428152	0.0214076	+	0.999771i	$0.493185\pi$
$930$	19.0949	0.626148
$931$	−42.9777	−1.40854
$932$	−130.926	−4.28861
$933$	−8.76558	−0.286972
$934$	−63.2604	−2.06994
$935$	−22.4729	−0.734943
$936$	−1.81212	−0.0592310
$937$	−22.8761	−0.747329	−0.373665	−	0.927564i	$-0.621899\pi$
$937$	−22.8761	−0.747329	−0.373665	+	0.927564i	$0.621899\pi$
$938$	−68.9003	−2.24968
$939$	−4.25515	−0.138862
$940$	−129.347	−4.21883
$941$	−2.69182	−0.0877508	−0.0438754	−	0.999037i	$-0.513970\pi$
$941$	−2.69182	−0.0877508	−0.0438754	+	0.999037i	$0.513970\pi$
$942$	−9.53618	−0.310706
$943$	−37.5703	−1.22346
$944$	108.398	3.52806
$945$	−19.5167	−0.634877
$946$	−9.13891	−0.297132
$947$	25.0192	0.813013	0.406507	−	0.913648i	$-0.366747\pi$
$947$	25.0192	0.813013	0.406507	+	0.913648i	$0.366747\pi$
$948$	11.0596	0.359199
$949$	0.811101	0.0263294
$950$	18.9111	0.613558
$951$	−6.01395	−0.195016
$952$	−81.6929	−2.64768
$953$	−34.0123	−1.10177	−0.550883	−	0.834582i	$-0.685709\pi$
$953$	−34.0123	−1.10177	−0.550883	+	0.834582i	$0.685709\pi$
$954$	−18.7996	−0.608660
$955$	−59.9158	−1.93883
$956$	−83.7513	−2.70871
$957$	1.08604	0.0351067
$958$	62.2598	2.01152
$959$	−27.2816	−0.880968
$960$	−1.00049	−0.0322907
$961$	43.9265	1.41698
$962$	−0.00707484	−0.000228102 0
$963$	−44.0968	−1.42100
$964$	−19.5229	−0.628790
$965$	57.9564	1.86568
$966$	21.1279	0.679779
$967$	50.4546	1.62251	0.811255	−	0.584692i	$-0.198785\pi$
$967$	50.4546	1.62251	0.811255	+	0.584692i	$0.198785\pi$
$968$	22.7141	0.730058
$969$	6.12842	0.196873
$970$	10.9989	0.353154
$971$	27.1584	0.871555	0.435777	−	0.900054i	$-0.356474\pi$
$971$	27.1584	0.871555	0.435777	+	0.900054i	$0.356474\pi$
$972$	39.8694	1.27881
$973$	−8.31937	−0.266707
$974$	59.7363	1.91407
$975$	−0.0443031	−0.00141884
$976$	−88.3216	−2.82711
$977$	−33.4936	−1.07155	−0.535777	−	0.844360i	$-0.679981\pi$
$977$	−33.4936	−1.07155	−0.535777	+	0.844360i	$0.679981\pi$
$978$	−22.2948	−0.712910
$979$	−46.7545	−1.49428
$980$	−89.4911	−2.85869
$981$	−0.488152	−0.0155855
$982$	98.5897	3.14612
$983$	−28.9031	−0.921867	−0.460933	−	0.887435i	$-0.652485\pi$
$983$	−28.9031	−0.921867	−0.460933	+	0.887435i	$0.652485\pi$
$984$	13.3820	0.426603
$985$	57.7936	1.84146
$986$	−9.63800	−0.306936
$987$	−14.8721	−0.473383
$988$	2.41189	0.0767325
$989$	−8.16361	−0.259588
$990$	50.9444	1.61912
$991$	−57.7591	−1.83478	−0.917389	−	0.397992i	$-0.869707\pi$
$991$	−57.7591	−1.83478	−0.917389	+	0.397992i	$0.869707\pi$
$992$	−55.2996	−1.75576
$993$	−4.76431	−0.151191
$994$	72.2902	2.29291
$995$	−20.7191	−0.656840
$996$	10.5647	0.334757
$997$	−35.8873	−1.13656	−0.568281	−	0.822834i	$-0.692392\pi$
$997$	−35.8873	−1.13656	−0.568281	+	0.822834i	$0.692392\pi$
$998$	37.5733	1.18936
$999$	0.0577624	0.00182752

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.a.1.17	✓	306

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.a.1.17	✓	306	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:	\( N \)	\(=\)	\( 8009 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.55906 q^{2} +0.342269 q^{3} +4.54881 q^{4} -2.51856 q^{5} -0.875888 q^{6} -3.84856 q^{7} -6.52258 q^{8} -2.88285 q^{9} +O(q^{10})\)	\(q-2.55906 q^{2} +0.342269 q^{3} +4.54881 q^{4} -2.51856 q^{5} -0.875888 q^{6} -3.84856 q^{7} -6.52258 q^{8} -2.88285 q^{9} +6.44516 q^{10} -2.74183 q^{11} +1.55692 q^{12} -0.0963708 q^{13} +9.84871 q^{14} -0.862024 q^{15} +7.59408 q^{16} -3.25437 q^{17} +7.37741 q^{18} -5.50192 q^{19} -11.4565 q^{20} -1.31724 q^{21} +7.01652 q^{22} +6.26772 q^{23} -2.23247 q^{24} +1.34314 q^{25} +0.246619 q^{26} -2.01352 q^{27} -17.5064 q^{28} -1.15728 q^{29} +2.20598 q^{30} +8.65601 q^{31} -6.38858 q^{32} -0.938442 q^{33} +8.32815 q^{34} +9.69283 q^{35} -13.1136 q^{36} -0.0286873 q^{37} +14.0798 q^{38} -0.0329847 q^{39} +16.4275 q^{40} -5.99425 q^{41} +3.37091 q^{42} -1.30249 q^{43} -12.4721 q^{44} +7.26063 q^{45} -16.0395 q^{46} +11.2903 q^{47} +2.59921 q^{48} +7.81141 q^{49} -3.43719 q^{50} -1.11387 q^{51} -0.438373 q^{52} -2.54827 q^{53} +5.15272 q^{54} +6.90546 q^{55} +25.1025 q^{56} -1.88313 q^{57} +2.96155 q^{58} +14.2740 q^{59} -3.92119 q^{60} -11.6303 q^{61} -22.1513 q^{62} +11.0948 q^{63} +1.16063 q^{64} +0.242716 q^{65} +2.40153 q^{66} -6.99587 q^{67} -14.8035 q^{68} +2.14524 q^{69} -24.8046 q^{70} +7.34007 q^{71} +18.8036 q^{72} -8.41645 q^{73} +0.0734127 q^{74} +0.459715 q^{75} -25.0272 q^{76} +10.5521 q^{77} +0.0844100 q^{78} +7.10354 q^{79} -19.1261 q^{80} +7.95939 q^{81} +15.3397 q^{82} +6.78568 q^{83} -5.99188 q^{84} +8.19633 q^{85} +3.33314 q^{86} -0.396101 q^{87} +17.8838 q^{88} +17.0523 q^{89} -18.5804 q^{90} +0.370889 q^{91} +28.5107 q^{92} +2.96268 q^{93} -28.8926 q^{94} +13.8569 q^{95} -2.18661 q^{96} +1.70654 q^{97} -19.9899 q^{98} +7.90429 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9} - 61 q^{10} - 43 q^{11} - 50 q^{12} - 89 q^{13} - 40 q^{14} - 61 q^{15} + 151 q^{16} - 52 q^{17} - 57 q^{18} - 185 q^{19} - 66 q^{20} - 63 q^{21} - 55 q^{22} - 62 q^{23} - 131 q^{24} + 209 q^{25} - 57 q^{26} - 88 q^{27} - 182 q^{28} - 67 q^{29} - 68 q^{30} - 240 q^{31} - 64 q^{32} - 52 q^{33} - 128 q^{34} - 99 q^{35} + 106 q^{36} - 49 q^{37} - 45 q^{38} - 190 q^{39} - 158 q^{40} - 72 q^{41} - 36 q^{42} - 141 q^{43} - 80 q^{44} - 100 q^{45} - 91 q^{46} - 105 q^{47} - 85 q^{48} + 116 q^{49} - 51 q^{50} - 145 q^{51} - 237 q^{52} - 48 q^{53} - 156 q^{54} - 420 q^{55} - 116 q^{56} - 35 q^{57} - 43 q^{58} - 139 q^{59} - 73 q^{60} - 233 q^{61} - 58 q^{62} - 252 q^{63} - 3 q^{64} - 45 q^{65} - 127 q^{66} - 108 q^{67} - 85 q^{68} - 164 q^{69} - 56 q^{70} - 131 q^{71} - 117 q^{72} - 118 q^{73} - 47 q^{74} - 112 q^{75} - 389 q^{76} - 36 q^{77} + 9 q^{78} - 382 q^{79} - 119 q^{80} + 102 q^{81} - 131 q^{82} - 59 q^{83} - 144 q^{84} - 140 q^{85} - 38 q^{86} - 301 q^{87} - 131 q^{88} - 98 q^{89} - 138 q^{90} - 176 q^{91} - 97 q^{92} - 60 q^{93} - 342 q^{94} - 154 q^{95} - 243 q^{96} - 109 q^{97} - 21 q^{98} - 173 q^{99}+O(q^{100}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9 - 61 * q^10 - 43 * q^11 - 50 * q^12 - 89 * q^13 - 40 * q^14 - 61 * q^15 + 151 * q^16 - 52 * q^17 - 57 * q^18 - 185 * q^19 - 66 * q^20 - 63 * q^21 - 55 * q^22 - 62 * q^23 - 131 * q^24 + 209 * q^25 - 57 * q^26 - 88 * q^27 - 182 * q^28 - 67 * q^29 - 68 * q^30 - 240 * q^31 - 64 * q^32 - 52 * q^33 - 128 * q^34 - 99 * q^35 + 106 * q^36 - 49 * q^37 - 45 * q^38 - 190 * q^39 - 158 * q^40 - 72 * q^41 - 36 * q^42 - 141 * q^43 - 80 * q^44 - 100 * q^45 - 91 * q^46 - 105 * q^47 - 85 * q^48 + 116 * q^49 - 51 * q^50 - 145 * q^51 - 237 * q^52 - 48 * q^53 - 156 * q^54 - 420 * q^55 - 116 * q^56 - 35 * q^57 - 43 * q^58 - 139 * q^59 - 73 * q^60 - 233 * q^61 - 58 * q^62 - 252 * q^63 - 3 * q^64 - 45 * q^65 - 127 * q^66 - 108 * q^67 - 85 * q^68 - 164 * q^69 - 56 * q^70 - 131 * q^71 - 117 * q^72 - 118 * q^73 - 47 * q^74 - 112 * q^75 - 389 * q^76 - 36 * q^77 + 9 * q^78 - 382 * q^79 - 119 * q^80 + 102 * q^81 - 131 * q^82 - 59 * q^83 - 144 * q^84 - 140 * q^85 - 38 * q^86 - 301 * q^87 - 131 * q^88 - 98 * q^89 - 138 * q^90 - 176 * q^91 - 97 * q^92 - 60 * q^93 - 342 * q^94 - 154 * q^95 - 243 * q^96 - 109 * q^97 - 21 * q^98 - 173 * q^99

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