LMFDB - Embedded newform 4009.2.a.c.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.69837 q^{2} +1.88273 q^{3} +5.28118 q^{4} -2.05530 q^{5} -5.08030 q^{6} +0.784756 q^{7} -8.85381 q^{8} +0.544674 q^{9} +O(q^{10})$	\(q-2.69837 q^{2} +1.88273 q^{3} +5.28118 q^{4} -2.05530 q^{5} -5.08030 q^{6} +0.784756 q^{7} -8.85381 q^{8} +0.544674 q^{9} +5.54594 q^{10} -2.34721 q^{11} +9.94303 q^{12} +4.30855 q^{13} -2.11756 q^{14} -3.86957 q^{15} +13.3285 q^{16} -2.48420 q^{17} -1.46973 q^{18} +1.00000 q^{19} -10.8544 q^{20} +1.47748 q^{21} +6.33362 q^{22} +4.84592 q^{23} -16.6693 q^{24} -0.775762 q^{25} -11.6260 q^{26} -4.62272 q^{27} +4.14444 q^{28} +2.17707 q^{29} +10.4415 q^{30} +0.527020 q^{31} -18.2575 q^{32} -4.41916 q^{33} +6.70329 q^{34} -1.61291 q^{35} +2.87652 q^{36} -5.01494 q^{37} -2.69837 q^{38} +8.11184 q^{39} +18.1972 q^{40} -1.52853 q^{41} -3.98679 q^{42} -4.10155 q^{43} -12.3960 q^{44} -1.11947 q^{45} -13.0761 q^{46} +0.0151453 q^{47} +25.0939 q^{48} -6.38416 q^{49} +2.09329 q^{50} -4.67708 q^{51} +22.7542 q^{52} +2.70200 q^{53} +12.4738 q^{54} +4.82420 q^{55} -6.94809 q^{56} +1.88273 q^{57} -5.87454 q^{58} -10.7088 q^{59} -20.4359 q^{60} +5.59481 q^{61} -1.42209 q^{62} +0.427436 q^{63} +22.6084 q^{64} -8.85534 q^{65} +11.9245 q^{66} +5.88688 q^{67} -13.1195 q^{68} +9.12357 q^{69} +4.35221 q^{70} +13.0097 q^{71} -4.82244 q^{72} +1.58068 q^{73} +13.5321 q^{74} -1.46055 q^{75} +5.28118 q^{76} -1.84199 q^{77} -21.8887 q^{78} +15.7171 q^{79} -27.3939 q^{80} -10.3374 q^{81} +4.12453 q^{82} -13.9489 q^{83} +7.80286 q^{84} +5.10577 q^{85} +11.0675 q^{86} +4.09884 q^{87} +20.7817 q^{88} -11.7529 q^{89} +3.02073 q^{90} +3.38116 q^{91} +25.5922 q^{92} +0.992236 q^{93} -0.0408675 q^{94} -2.05530 q^{95} -34.3739 q^{96} -1.28700 q^{97} +17.2268 q^{98} -1.27846 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.69837	−1.90803	−0.954016	−	0.299755i	$-0.903095\pi$
$2$	−2.69837	−1.90803	−0.954016	+	0.299755i	$0.903095\pi$
$3$	1.88273	1.08699	0.543497	−	0.839411i	$-0.317100\pi$
$3$	1.88273	1.08699	0.543497	+	0.839411i	$0.317100\pi$
$4$	5.28118	2.64059
$5$	−2.05530	−0.919156	−0.459578	−	0.888137i	$-0.651999\pi$
$5$	−2.05530	−0.919156	−0.459578	+	0.888137i	$0.651999\pi$
$6$	−5.08030	−2.07402
$7$	0.784756	0.296610	0.148305	−	0.988942i	$-0.452618\pi$
$7$	0.784756	0.296610	0.148305	+	0.988942i	$0.452618\pi$
$8$	−8.85381	−3.13030
$9$	0.544674	0.181558
$10$	5.54594	1.75378
$11$	−2.34721	−0.707710	−0.353855	−	0.935300i	$-0.615129\pi$
$11$	−2.34721	−0.707710	−0.353855	+	0.935300i	$0.615129\pi$
$12$	9.94303	2.87031
$13$	4.30855	1.19498	0.597488	−	0.801878i	$-0.296165\pi$
$13$	4.30855	1.19498	0.597488	+	0.801878i	$0.296165\pi$
$14$	−2.11756	−0.565942
$15$	−3.86957	−0.999118
$16$	13.3285	3.33212
$17$	−2.48420	−0.602508	−0.301254	−	0.953544i	$-0.597405\pi$
$17$	−2.48420	−0.602508	−0.301254	+	0.953544i	$0.597405\pi$
$18$	−1.46973	−0.346419
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−10.8544	−2.42711
$21$	1.47748	0.322414
$22$	6.33362	1.35033
$23$	4.84592	1.01045	0.505223	−	0.862989i	$-0.331410\pi$
$23$	4.84592	1.01045	0.505223	+	0.862989i	$0.331410\pi$
$24$	−16.6693	−3.40262
$25$	−0.775762	−0.155152
$26$	−11.6260	−2.28005
$27$	−4.62272	−0.889642
$28$	4.14444	0.783225
$29$	2.17707	0.404272	0.202136	−	0.979357i	$-0.435212\pi$
$29$	2.17707	0.404272	0.202136	+	0.979357i	$0.435212\pi$
$30$	10.4415	1.90635
$31$	0.527020	0.0946555	0.0473278	−	0.998879i	$-0.484929\pi$
$31$	0.527020	0.0946555	0.0473278	+	0.998879i	$0.484929\pi$
$32$	−18.2575	−3.22749
$33$	−4.41916	−0.769277
$34$	6.70329	1.14960
$35$	−1.61291	−0.272631
$36$	2.87652	0.479420
$37$	−5.01494	−0.824451	−0.412226	−	0.911082i	$-0.635248\pi$
$37$	−5.01494	−0.824451	−0.412226	+	0.911082i	$0.635248\pi$
$38$	−2.69837	−0.437733
$39$	8.11184	1.29893
$40$	18.1972	2.87723
$41$	−1.52853	−0.238716	−0.119358	−	0.992851i	$-0.538084\pi$
$41$	−1.52853	−0.238716	−0.119358	+	0.992851i	$0.538084\pi$
$42$	−3.98679	−0.615176
$43$	−4.10155	−0.625480	−0.312740	−	0.949839i	$-0.601247\pi$
$43$	−4.10155	−0.625480	−0.312740	+	0.949839i	$0.601247\pi$
$44$	−12.3960	−1.86877
$45$	−1.11947	−0.166880
$46$	−13.0761	−1.92796
$47$	0.0151453	0.00220916	0.00110458	−	0.999999i	$-0.499648\pi$
$47$	0.0151453	0.00220916	0.00110458	+	0.999999i	$0.499648\pi$
$48$	25.0939	3.62200
$49$	−6.38416	−0.912022
$50$	2.09329	0.296036
$51$	−4.67708	−0.654923
$52$	22.7542	3.15544
$53$	2.70200	0.371148	0.185574	−	0.982630i	$-0.440586\pi$
$53$	2.70200	0.371148	0.185574	+	0.982630i	$0.440586\pi$
$54$	12.4738	1.69747
$55$	4.82420	0.650496
$56$	−6.94809	−0.928477
$57$	1.88273	0.249374
$58$	−5.87454	−0.771365
$59$	−10.7088	−1.39417	−0.697084	−	0.716990i	$-0.745519\pi$
$59$	−10.7088	−1.39417	−0.697084	+	0.716990i	$0.745519\pi$
$60$	−20.4359	−2.63826
$61$	5.59481	0.716342	0.358171	−	0.933656i	$-0.383401\pi$
$61$	5.59481	0.716342	0.358171	+	0.933656i	$0.383401\pi$
$62$	−1.42209	−0.180606
$63$	0.427436	0.0538519
$64$	22.6084	2.82605
$65$	−8.85534	−1.09837
$66$	11.9245	1.46781
$67$	5.88688	0.719197	0.359599	−	0.933107i	$-0.382914\pi$
$67$	5.88688	0.719197	0.359599	+	0.933107i	$0.382914\pi$
$68$	−13.1195	−1.59097
$69$	9.12357	1.09835
$70$	4.35221	0.520189
$71$	13.0097	1.54397	0.771985	−	0.635641i	$-0.219264\pi$
$71$	13.0097	1.54397	0.771985	+	0.635641i	$0.219264\pi$
$72$	−4.82244	−0.568330
$73$	1.58068	0.185004	0.0925021	−	0.995712i	$-0.470514\pi$
$73$	1.58068	0.185004	0.0925021	+	0.995712i	$0.470514\pi$
$74$	13.5321	1.57308
$75$	−1.46055	−0.168650
$76$	5.28118	0.605793
$77$	−1.84199	−0.209914
$78$	−21.8887	−2.47841
$79$	15.7171	1.76831	0.884157	−	0.467190i	$-0.154733\pi$
$79$	15.7171	1.76831	0.884157	+	0.467190i	$0.154733\pi$
$80$	−27.3939	−3.06274
$81$	−10.3374	−1.14859
$82$	4.12453	0.455478
$83$	−13.9489	−1.53109	−0.765544	−	0.643384i	$-0.777530\pi$
$83$	−13.9489	−1.53109	−0.765544	+	0.643384i	$0.777530\pi$
$84$	7.80286	0.851362
$85$	5.10577	0.553798
$86$	11.0675	1.19344
$87$	4.09884	0.439442
$88$	20.7817	2.21534
$89$	−11.7529	−1.24580	−0.622900	−	0.782301i	$-0.714046\pi$
$89$	−11.7529	−1.24580	−0.622900	+	0.782301i	$0.714046\pi$
$90$	3.02073	0.318413
$91$	3.38116	0.354442
$92$	25.5922	2.66817
$93$	0.992236	0.102890
$94$	−0.0408675	−0.00421516
$95$	−2.05530	−0.210869
$96$	−34.3739	−3.50827
$97$	−1.28700	−0.130675	−0.0653373	−	0.997863i	$-0.520812\pi$
$97$	−1.28700	−0.130675	−0.0653373	+	0.997863i	$0.520812\pi$
$98$	17.2268	1.74017
$99$	−1.27846	−0.128490
$100$	−4.09694	−0.409694
$101$	9.87358	0.982458	0.491229	−	0.871030i	$-0.336548\pi$
$101$	9.87358	0.982458	0.491229	+	0.871030i	$0.336548\pi$
$102$	12.6205	1.24961
$103$	0.332662	0.0327781	0.0163891	−	0.999866i	$-0.494783\pi$
$103$	0.332662	0.0327781	0.0163891	+	0.999866i	$0.494783\pi$
$104$	−38.1471	−3.74063
$105$	−3.03667	−0.296348
$106$	−7.29098	−0.708162
$107$	−0.822910	−0.0795537	−0.0397769	−	0.999209i	$-0.512665\pi$
$107$	−0.822910	−0.0795537	−0.0397769	+	0.999209i	$0.512665\pi$
$108$	−24.4134	−2.34918
$109$	−8.65733	−0.829222	−0.414611	−	0.909999i	$-0.636082\pi$
$109$	−8.65733	−0.829222	−0.414611	+	0.909999i	$0.636082\pi$
$110$	−13.0175	−1.24117
$111$	−9.44178	−0.896174
$112$	10.4596	0.988340
$113$	−7.47827	−0.703496	−0.351748	−	0.936095i	$-0.614413\pi$
$113$	−7.47827	−0.703496	−0.351748	+	0.936095i	$0.614413\pi$
$114$	−5.08030	−0.475813
$115$	−9.95980	−0.928757
$116$	11.4975	1.06752
$117$	2.34676	0.216958
$118$	28.8963	2.66012
$119$	−1.94949	−0.178710
$120$	34.2604	3.12753
$121$	−5.49062	−0.499147
$122$	−15.0968	−1.36680
$123$	−2.87781	−0.259483
$124$	2.78328	0.249946
$125$	11.8709	1.06177
$126$	−1.15338	−0.102751
$127$	−9.92691	−0.880871	−0.440435	−	0.897784i	$-0.645176\pi$
$127$	−9.92691	−0.880871	−0.440435	+	0.897784i	$0.645176\pi$
$128$	−24.4907	−2.16469
$129$	−7.72211	−0.679894
$130$	23.8950	2.09573
$131$	−19.8075	−1.73059	−0.865294	−	0.501265i	$-0.832868\pi$
$131$	−19.8075	−1.73059	−0.865294	+	0.501265i	$0.832868\pi$
$132$	−23.3384	−2.03134
$133$	0.784756	0.0680470
$134$	−15.8850	−1.37225
$135$	9.50105	0.817720
$136$	21.9947	1.88603
$137$	−0.314536	−0.0268726	−0.0134363	−	0.999910i	$-0.504277\pi$
$137$	−0.314536	−0.0268726	−0.0134363	+	0.999910i	$0.504277\pi$
$138$	−24.6187	−2.09569
$139$	1.40641	0.119290	0.0596451	−	0.998220i	$-0.481003\pi$
$139$	1.40641	0.119290	0.0596451	+	0.998220i	$0.481003\pi$
$140$	−8.51804	−0.719906
$141$	0.0285145	0.00240135
$142$	−35.1050	−2.94594
$143$	−10.1131	−0.845697
$144$	7.25967	0.604973
$145$	−4.47453	−0.371589
$146$	−4.26524	−0.352994
$147$	−12.0196	−0.991364
$148$	−26.4848	−2.17704
$149$	−16.7981	−1.37615	−0.688075	−	0.725639i	$-0.741544\pi$
$149$	−16.7981	−1.37615	−0.688075	+	0.725639i	$0.741544\pi$
$150$	3.94110	0.321790
$151$	−9.37824	−0.763190	−0.381595	−	0.924330i	$-0.624625\pi$
$151$	−9.37824	−0.763190	−0.381595	+	0.924330i	$0.624625\pi$
$152$	−8.85381	−0.718139
$153$	−1.35308	−0.109390
$154$	4.97035	0.400522
$155$	−1.08318	−0.0870032
$156$	42.8401	3.42995
$157$	−0.457209	−0.0364892	−0.0182446	−	0.999834i	$-0.505808\pi$
$157$	−0.457209	−0.0364892	−0.0182446	+	0.999834i	$0.505808\pi$
$158$	−42.4105	−3.37400
$159$	5.08713	0.403436
$160$	37.5245	2.96657
$161$	3.80287	0.299708
$162$	27.8940	2.19156
$163$	0.370943	0.0290545	0.0145273	−	0.999894i	$-0.495376\pi$
$163$	0.370943	0.0290545	0.0145273	+	0.999894i	$0.495376\pi$
$164$	−8.07243	−0.630351
$165$	9.08268	0.707085
$166$	37.6392	2.92136
$167$	5.26793	0.407645	0.203822	−	0.979008i	$-0.434663\pi$
$167$	5.26793	0.407645	0.203822	+	0.979008i	$0.434663\pi$
$168$	−13.0814	−1.00925
$169$	5.56361	0.427970
$170$	−13.7772	−1.05667
$171$	0.544674	0.0416523
$172$	−21.6610	−1.65164
$173$	−16.9153	−1.28605	−0.643023	−	0.765847i	$-0.722320\pi$
$173$	−16.9153	−1.28605	−0.643023	+	0.765847i	$0.722320\pi$
$174$	−11.0602	−0.838470
$175$	−0.608784	−0.0460198
$176$	−31.2847	−2.35817
$177$	−20.1618	−1.51545
$178$	31.7135	2.37703
$179$	10.2855	0.768773	0.384386	−	0.923172i	$-0.374413\pi$
$179$	10.2855	0.768773	0.384386	+	0.923172i	$0.374413\pi$
$180$	−5.91210	−0.440662
$181$	9.96371	0.740597	0.370298	−	0.928913i	$-0.379255\pi$
$181$	9.96371	0.740597	0.370298	+	0.928913i	$0.379255\pi$
$182$	−9.12361	−0.676287
$183$	10.5335	0.778660
$184$	−42.9049	−3.16299
$185$	10.3072	0.757799
$186$	−2.67741	−0.196318
$187$	5.83094	0.426401
$188$	0.0799848	0.00583349
$189$	−3.62771	−0.263877
$190$	5.54594	0.402345
$191$	−15.1598	−1.09692	−0.548462	−	0.836175i	$-0.684787\pi$
$191$	−15.1598	−1.09692	−0.548462	+	0.836175i	$0.684787\pi$
$192$	42.5655	3.07190
$193$	−1.98183	−0.142655	−0.0713277	−	0.997453i	$-0.522724\pi$
$193$	−1.98183	−0.142655	−0.0713277	+	0.997453i	$0.522724\pi$
$194$	3.47278	0.249331
$195$	−16.6722	−1.19392
$196$	−33.7159	−2.40828
$197$	−6.30309	−0.449077	−0.224538	−	0.974465i	$-0.572087\pi$
$197$	−6.30309	−0.449077	−0.224538	+	0.974465i	$0.572087\pi$
$198$	3.44976	0.245164
$199$	−6.01077	−0.426092	−0.213046	−	0.977042i	$-0.568338\pi$
$199$	−6.01077	−0.426092	−0.213046	+	0.977042i	$0.568338\pi$
$200$	6.86846	0.485673
$201$	11.0834	0.781764
$202$	−26.6425	−1.87456
$203$	1.70847	0.119911
$204$	−24.7005	−1.72938
$205$	3.14158	0.219417
$206$	−0.897642	−0.0625417
$207$	2.63945	0.183454
$208$	57.4264	3.98180
$209$	−2.34721	−0.162360
$210$	8.19404	0.565442
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	14.2697	0.980049
$213$	24.4938	1.67829
$214$	2.22051	0.151791
$215$	8.42989	0.574914
$216$	40.9287	2.78484
$217$	0.413582	0.0280758
$218$	23.3606	1.58218
$219$	2.97599	0.201099
$220$	25.4775	1.71769
$221$	−10.7033	−0.719983
$222$	25.4774	1.70993
$223$	−24.4009	−1.63400	−0.817002	−	0.576635i	$-0.804366\pi$
$223$	−24.4009	−1.63400	−0.817002	+	0.576635i	$0.804366\pi$
$224$	−14.3277	−0.957307
$225$	−0.422538	−0.0281692
$226$	20.1791	1.34229
$227$	14.3866	0.954873	0.477437	−	0.878666i	$-0.341566\pi$
$227$	14.3866	0.954873	0.477437	+	0.878666i	$0.341566\pi$
$228$	9.94303	0.658493
$229$	6.16069	0.407110	0.203555	−	0.979064i	$-0.434750\pi$
$229$	6.16069	0.407110	0.203555	+	0.979064i	$0.434750\pi$
$230$	26.8752	1.77210
$231$	−3.46796	−0.228175
$232$	−19.2754	−1.26549
$233$	7.71844	0.505652	0.252826	−	0.967512i	$-0.418640\pi$
$233$	7.71844	0.505652	0.252826	+	0.967512i	$0.418640\pi$
$234$	−6.33240	−0.413962
$235$	−0.0311280	−0.00203057
$236$	−56.5551	−3.68142
$237$	29.5911	1.92215
$238$	5.26045	0.340984
$239$	−5.35067	−0.346106	−0.173053	−	0.984913i	$-0.555363\pi$
$239$	−5.35067	−0.346106	−0.173053	+	0.984913i	$0.555363\pi$
$240$	−51.5754	−3.32918
$241$	−12.2947	−0.791974	−0.395987	−	0.918256i	$-0.629597\pi$
$241$	−12.2947	−0.791974	−0.395987	+	0.918256i	$0.629597\pi$
$242$	14.8157	0.952388
$243$	−5.59430	−0.358874
$244$	29.5472	1.89156
$245$	13.1213	0.838291
$246$	7.76537	0.495102
$247$	4.30855	0.274146
$248$	−4.66613	−0.296300
$249$	−26.2620	−1.66428
$250$	−32.0320	−2.02588
$251$	−4.80456	−0.303261	−0.151631	−	0.988437i	$-0.548452\pi$
$251$	−4.80456	−0.303261	−0.151631	+	0.988437i	$0.548452\pi$
$252$	2.25737	0.142201
$253$	−11.3744	−0.715102
$254$	26.7864	1.68073
$255$	9.61279	0.601976
$256$	20.8682	1.30426
$257$	5.29185	0.330096	0.165048	−	0.986285i	$-0.447222\pi$
$257$	5.29185	0.330096	0.165048	+	0.986285i	$0.447222\pi$
$258$	20.8371	1.29726
$259$	−3.93551	−0.244541
$260$	−46.7666	−2.90034
$261$	1.18580	0.0733989
$262$	53.4478	3.30202
$263$	2.89234	0.178350	0.0891748	−	0.996016i	$-0.471577\pi$
$263$	2.89234	0.178350	0.0891748	+	0.996016i	$0.471577\pi$
$264$	39.1264	2.40806
$265$	−5.55340	−0.341143
$266$	−2.11756	−0.129836
$267$	−22.1275	−1.35418
$268$	31.0897	1.89910
$269$	29.8616	1.82069	0.910347	−	0.413846i	$-0.135815\pi$
$269$	29.8616	1.82069	0.910347	+	0.413846i	$0.135815\pi$
$270$	−25.6373	−1.56024
$271$	0.174135	0.0105779	0.00528896	−	0.999986i	$-0.498316\pi$
$271$	0.174135	0.0105779	0.00528896	+	0.999986i	$0.498316\pi$
$272$	−33.1106	−2.00763
$273$	6.36582	0.385277
$274$	0.848732	0.0512738
$275$	1.82088	0.109803
$276$	48.1832	2.90029
$277$	−7.65965	−0.460224	−0.230112	−	0.973164i	$-0.573909\pi$
$277$	−7.65965	−0.460224	−0.230112	+	0.973164i	$0.573909\pi$
$278$	−3.79501	−0.227610
$279$	0.287054	0.0171855
$280$	14.2804	0.853415
$281$	−24.9921	−1.49090	−0.745451	−	0.666561i	$-0.767766\pi$
$281$	−24.9921	−1.49090	−0.745451	+	0.666561i	$0.767766\pi$
$282$	−0.0769424	−0.00458185
$283$	4.40963	0.262125	0.131063	−	0.991374i	$-0.458161\pi$
$283$	4.40963	0.262125	0.131063	+	0.991374i	$0.458161\pi$
$284$	68.7066	4.07699
$285$	−3.86957	−0.229213
$286$	27.2887	1.61362
$287$	−1.19952	−0.0708055
$288$	−9.94437	−0.585977
$289$	−10.8287	−0.636985
$290$	12.0739	0.709005
$291$	−2.42307	−0.142043
$292$	8.34783	0.488520
$293$	−8.08961	−0.472600	−0.236300	−	0.971680i	$-0.575935\pi$
$293$	−8.08961	−0.472600	−0.236300	+	0.971680i	$0.575935\pi$
$294$	32.4334	1.89155
$295$	22.0097	1.28146
$296$	44.4014	2.58078
$297$	10.8505	0.629609
$298$	45.3273	2.62574
$299$	20.8789	1.20746
$300$	−7.71343	−0.445335
$301$	−3.21872	−0.185524
$302$	25.3059	1.45619
$303$	18.5893	1.06793
$304$	13.3285	0.764440
$305$	−11.4990	−0.658430
$306$	3.65111	0.208720
$307$	−26.7688	−1.52777	−0.763887	−	0.645350i	$-0.776711\pi$
$307$	−26.7688	−1.52777	−0.763887	+	0.645350i	$0.776711\pi$
$308$	−9.72786	−0.554296
$309$	0.626312	0.0356296
$310$	2.92282	0.166005
$311$	5.98927	0.339620	0.169810	−	0.985477i	$-0.445685\pi$
$311$	5.98927	0.339620	0.169810	+	0.985477i	$0.445685\pi$
$312$	−71.8207	−4.06605
$313$	1.56947	0.0887115	0.0443557	−	0.999016i	$-0.485876\pi$
$313$	1.56947	0.0887115	0.0443557	+	0.999016i	$0.485876\pi$
$314$	1.23372	0.0696227
$315$	−0.878508	−0.0494983
$316$	83.0049	4.66939
$317$	1.75982	0.0988416	0.0494208	−	0.998778i	$-0.484262\pi$
$317$	1.75982	0.0988416	0.0494208	+	0.998778i	$0.484262\pi$
$318$	−13.7269	−0.769769
$319$	−5.11004	−0.286108
$320$	−46.4669	−2.59758
$321$	−1.54932	−0.0864745
$322$	−10.2615	−0.571853
$323$	−2.48420	−0.138225
$324$	−54.5934	−3.03297
$325$	−3.34241	−0.185404
$326$	−1.00094	−0.0554369
$327$	−16.2994	−0.901360
$328$	13.5333	0.747252
$329$	0.0118853	0.000655260 0
$330$	−24.5084	−1.34914
$331$	−32.2037	−1.77008	−0.885038	−	0.465519i	$-0.845868\pi$
$331$	−32.2037	−1.77008	−0.885038	+	0.465519i	$0.845868\pi$
$332$	−73.6664	−4.04297
$333$	−2.73151	−0.149686
$334$	−14.2148	−0.777800
$335$	−12.0993	−0.661054
$336$	19.6926	1.07432
$337$	31.9035	1.73790	0.868948	−	0.494903i	$-0.164796\pi$
$337$	31.9035	1.73790	0.868948	+	0.494903i	$0.164796\pi$
$338$	−15.0126	−0.816580
$339$	−14.0796	−0.764697
$340$	26.9645	1.46235
$341$	−1.23702	−0.0669886
$342$	−1.46973	−0.0794739
$343$	−10.5033	−0.567125
$344$	36.3143	1.95794
$345$	−18.7516	−1.00955
$346$	45.6436	2.45382
$347$	20.2132	1.08510	0.542550	−	0.840024i	$-0.317459\pi$
$347$	20.2132	1.08510	0.542550	+	0.840024i	$0.317459\pi$
$348$	21.6467	1.16039
$349$	−5.99779	−0.321054	−0.160527	−	0.987031i	$-0.551319\pi$
$349$	−5.99779	−0.321054	−0.160527	+	0.987031i	$0.551319\pi$
$350$	1.64272	0.0878072
$351$	−19.9172	−1.06310
$352$	42.8541	2.28413
$353$	−10.1564	−0.540570	−0.270285	−	0.962780i	$-0.587118\pi$
$353$	−10.1564	−0.540570	−0.270285	+	0.962780i	$0.587118\pi$
$354$	54.4039	2.89153
$355$	−26.7388	−1.41915
$356$	−62.0689	−3.28965
$357$	−3.67037	−0.194257
$358$	−27.7540	−1.46684
$359$	12.5552	0.662640	0.331320	−	0.943518i	$-0.392506\pi$
$359$	12.5552	0.662640	0.331320	+	0.943518i	$0.392506\pi$
$360$	9.91154	0.522384
$361$	1.00000	0.0526316
$362$	−26.8857	−1.41308
$363$	−10.3373	−0.542570
$364$	17.8565	0.935936
$365$	−3.24876	−0.170048
$366$	−28.4233	−1.48571
$367$	−7.53113	−0.393122	−0.196561	−	0.980492i	$-0.562977\pi$
$367$	−7.53113	−0.393122	−0.196561	+	0.980492i	$0.562977\pi$
$368$	64.5888	3.36692
$369$	−0.832549	−0.0433408
$370$	−27.8126	−1.44591
$371$	2.12041	0.110086
$372$	5.24017	0.271690
$373$	−27.5410	−1.42602	−0.713008	−	0.701156i	$-0.752668\pi$
$373$	−27.5410	−1.42602	−0.713008	+	0.701156i	$0.752668\pi$
$374$	−15.7340	−0.813586
$375$	22.3497	1.15413
$376$	−0.134093	−0.00691534
$377$	9.38003	0.483096
$378$	9.78888	0.503486
$379$	7.58220	0.389472	0.194736	−	0.980856i	$-0.437615\pi$
$379$	7.58220	0.389472	0.194736	+	0.980856i	$0.437615\pi$
$380$	−10.8544	−0.556818
$381$	−18.6897	−0.957502
$382$	40.9067	2.09297
$383$	−22.0172	−1.12503	−0.562514	−	0.826787i	$-0.690166\pi$
$383$	−22.0172	−1.12503	−0.562514	+	0.826787i	$0.690166\pi$
$384$	−46.1094	−2.35301
$385$	3.78583	0.192944
$386$	5.34771	0.272191
$387$	−2.23401	−0.113561
$388$	−6.79685	−0.345058
$389$	14.9288	0.756921	0.378460	−	0.925617i	$-0.376454\pi$
$389$	14.9288	0.756921	0.378460	+	0.925617i	$0.376454\pi$
$390$	44.9878	2.27804
$391$	−12.0383	−0.608801
$392$	56.5241	2.85490
$393$	−37.2921	−1.88114
$394$	17.0080	0.856853
$395$	−32.3033	−1.62536
$396$	−6.75179	−0.339290
$397$	33.4358	1.67810	0.839048	−	0.544058i	$-0.183113\pi$
$397$	33.4358	1.67810	0.839048	+	0.544058i	$0.183113\pi$
$398$	16.2192	0.812997
$399$	1.47748	0.0739668
$400$	−10.3397	−0.516986
$401$	−16.8221	−0.840056	−0.420028	−	0.907511i	$-0.637980\pi$
$401$	−16.8221	−0.840056	−0.420028	+	0.907511i	$0.637980\pi$
$402$	−29.9071	−1.49163
$403$	2.27069	0.113111
$404$	52.1441	2.59427
$405$	21.2463	1.05574
$406$	−4.61008	−0.228795
$407$	11.7711	0.583472
$408$	41.4100	2.05010
$409$	−1.73474	−0.0857775	−0.0428888	−	0.999080i	$-0.513656\pi$
$409$	−1.73474	−0.0857775	−0.0428888	+	0.999080i	$0.513656\pi$
$410$	−8.47712	−0.418655
$411$	−0.592186	−0.0292104
$412$	1.75684	0.0865535
$413$	−8.40380	−0.413524
$414$	−7.12220	−0.350037
$415$	28.6690	1.40731
$416$	−78.6632	−3.85678
$417$	2.64789	0.129668
$418$	6.33362	0.309788
$419$	−6.96118	−0.340076	−0.170038	−	0.985438i	$-0.554389\pi$
$419$	−6.96118	−0.340076	−0.170038	+	0.985438i	$0.554389\pi$
$420$	−16.0372	−0.782534
$421$	−22.9133	−1.11673	−0.558363	−	0.829597i	$-0.688570\pi$
$421$	−22.9133	−1.11673	−0.558363	+	0.829597i	$0.688570\pi$
$422$	−2.69837	−0.131354
$423$	0.00824923	0.000401091 0
$424$	−23.9230	−1.16180
$425$	1.92715	0.0934805
$426$	−66.0932	−3.20223
$427$	4.39056	0.212474
$428$	−4.34593	−0.210069
$429$	−19.0402	−0.919268
$430$	−22.7469	−1.09695
$431$	−1.26689	−0.0610240	−0.0305120	−	0.999534i	$-0.509714\pi$
$431$	−1.26689	−0.0610240	−0.0305120	+	0.999534i	$0.509714\pi$
$432$	−61.6138	−2.96439
$433$	−21.7603	−1.04573	−0.522866	−	0.852415i	$-0.675137\pi$
$433$	−21.7603	−1.04573	−0.522866	+	0.852415i	$0.675137\pi$
$434$	−1.11600	−0.0535695
$435$	−8.42433	−0.403916
$436$	−45.7209	−2.18963
$437$	4.84592	0.231812
$438$	−8.03030	−0.383703
$439$	−4.38773	−0.209415	−0.104707	−	0.994503i	$-0.533391\pi$
$439$	−4.38773	−0.209415	−0.104707	+	0.994503i	$0.533391\pi$
$440$	−42.7126	−2.03624
$441$	−3.47728	−0.165585
$442$	28.8814	1.37375
$443$	31.2985	1.48704	0.743518	−	0.668716i	$-0.233156\pi$
$443$	31.2985	1.48704	0.743518	+	0.668716i	$0.233156\pi$
$444$	−49.8637	−2.36643
$445$	24.1556	1.14508
$446$	65.8425	3.11773
$447$	−31.6262	−1.49587
$448$	17.7421	0.838234
$449$	−24.8735	−1.17385	−0.586925	−	0.809641i	$-0.699662\pi$
$449$	−24.8735	−1.17385	−0.586925	+	0.809641i	$0.699662\pi$
$450$	1.14016	0.0537477
$451$	3.58777	0.168942
$452$	−39.4940	−1.85764
$453$	−17.6567	−0.829584
$454$	−38.8204	−1.82193
$455$	−6.94929	−0.325788
$456$	−16.6693	−0.780614
$457$	−27.0787	−1.26669	−0.633345	−	0.773870i	$-0.718319\pi$
$457$	−27.0787	−1.26669	−0.633345	+	0.773870i	$0.718319\pi$
$458$	−16.6238	−0.776778
$459$	11.4838	0.536016
$460$	−52.5995	−2.45246
$461$	−30.9269	−1.44041	−0.720205	−	0.693761i	$-0.755952\pi$
$461$	−30.9269	−1.44041	−0.720205	+	0.693761i	$0.755952\pi$
$462$	9.35783	0.435366
$463$	−0.188386	−0.00875505	−0.00437753	−	0.999990i	$-0.501393\pi$
$463$	−0.188386	−0.00875505	−0.00437753	+	0.999990i	$0.501393\pi$
$464$	29.0171	1.34708
$465$	−2.03934	−0.0945720
$466$	−20.8272	−0.964800
$467$	19.6298	0.908358	0.454179	−	0.890910i	$-0.349933\pi$
$467$	19.6298	0.908358	0.454179	+	0.890910i	$0.349933\pi$
$468$	12.3936	0.572896
$469$	4.61977	0.213321
$470$	0.0839947	0.00387439
$471$	−0.860801	−0.0396636
$472$	94.8137	4.36416
$473$	9.62719	0.442658
$474$	−79.8476	−3.66752
$475$	−0.775762	−0.0355944
$476$	−10.2956	−0.471899
$477$	1.47171	0.0673849
$478$	14.4381	0.660382
$479$	0.338785	0.0154795	0.00773973	−	0.999970i	$-0.497536\pi$
$479$	0.338785	0.0154795	0.00773973	+	0.999970i	$0.497536\pi$
$480$	70.6485	3.22465
$481$	−21.6071	−0.985200
$482$	33.1757	1.51111
$483$	7.15978	0.325781
$484$	−28.9969	−1.31804
$485$	2.64516	0.120110
$486$	15.0955	0.684744
$487$	−15.5325	−0.703846	−0.351923	−	0.936029i	$-0.614472\pi$
$487$	−15.5325	−0.703846	−0.351923	+	0.936029i	$0.614472\pi$
$488$	−49.5354	−2.24236
$489$	0.698386	0.0315821
$490$	−35.4061	−1.59949
$491$	−3.32182	−0.149912	−0.0749558	−	0.997187i	$-0.523882\pi$
$491$	−3.32182	−0.149912	−0.0749558	+	0.997187i	$0.523882\pi$
$492$	−15.1982	−0.685188
$493$	−5.40829	−0.243577
$494$	−11.6260	−0.523080
$495$	2.62762	0.118103
$496$	7.02437	0.315403
$497$	10.2095	0.457957
$498$	70.8644	3.17551
$499$	−21.0866	−0.943965	−0.471982	−	0.881608i	$-0.656461\pi$
$499$	−21.0866	−0.943965	−0.471982	+	0.881608i	$0.656461\pi$
$500$	62.6923	2.80368
$501$	9.91810	0.443108
$502$	12.9645	0.578633
$503$	8.59351	0.383165	0.191583	−	0.981476i	$-0.438638\pi$
$503$	8.59351	0.383165	0.191583	+	0.981476i	$0.438638\pi$
$504$	−3.78444	−0.168572
$505$	−20.2931	−0.903032
$506$	30.6923	1.36444
$507$	10.4748	0.465201
$508$	−52.4258	−2.32602
$509$	−22.5424	−0.999175	−0.499588	−	0.866263i	$-0.666515\pi$
$509$	−22.5424	−0.999175	−0.499588	+	0.866263i	$0.666515\pi$
$510$	−25.9388	−1.14859
$511$	1.24045	0.0548741
$512$	−7.32850	−0.323877
$513$	−4.62272	−0.204098
$514$	−14.2793	−0.629835
$515$	−0.683718	−0.0301282
$516$	−40.7818	−1.79532
$517$	−0.0355491	−0.00156345
$518$	10.6194	0.466591
$519$	−31.8469	−1.39792
$520$	78.4036	3.43822
$521$	21.6596	0.948923	0.474462	−	0.880276i	$-0.342643\pi$
$521$	21.6596	0.948923	0.474462	+	0.880276i	$0.342643\pi$
$522$	−3.19971	−0.140047
$523$	10.0467	0.439312	0.219656	−	0.975577i	$-0.429507\pi$
$523$	10.0467	0.439312	0.219656	+	0.975577i	$0.429507\pi$
$524$	−104.607	−4.56977
$525$	−1.14618	−0.0500233
$526$	−7.80460	−0.340297
$527$	−1.30922	−0.0570307
$528$	−58.9007	−2.56332
$529$	0.482982	0.0209992
$530$	14.9851	0.650912
$531$	−5.83281	−0.253122
$532$	4.14444	0.179684
$533$	−6.58574	−0.285260
$534$	59.7080	2.58382
$535$	1.69132	0.0731223
$536$	−52.1214	−2.25130
$537$	19.3648	0.835652
$538$	−80.5775	−3.47394
$539$	14.9849	0.645447
$540$	50.1767	2.15926
$541$	−20.2534	−0.870761	−0.435381	−	0.900246i	$-0.643386\pi$
$541$	−20.2534	−0.870761	−0.435381	+	0.900246i	$0.643386\pi$
$542$	−0.469879	−0.0201830
$543$	18.7590	0.805025
$544$	45.3552	1.94459
$545$	17.7934	0.762184
$546$	−17.1773	−0.735121
$547$	−33.0050	−1.41119	−0.705596	−	0.708614i	$-0.749321\pi$
$547$	−33.0050	−1.41119	−0.705596	+	0.708614i	$0.749321\pi$
$548$	−1.66112	−0.0709594
$549$	3.04735	0.130058
$550$	−4.91339	−0.209508
$551$	2.17707	0.0927465
$552$	−80.7784	−3.43816
$553$	12.3341	0.524500
$554$	20.6685	0.878121
$555$	19.4057	0.823724
$556$	7.42751	0.314996
$557$	35.5025	1.50429	0.752145	−	0.658997i	$-0.229019\pi$
$557$	35.5025	1.50429	0.752145	+	0.658997i	$0.229019\pi$
$558$	−0.774576	−0.0327904
$559$	−17.6717	−0.747434
$560$	−21.4976	−0.908438
$561$	10.9781	0.463495
$562$	67.4377	2.84469
$563$	−14.9566	−0.630345	−0.315172	−	0.949034i	$-0.602062\pi$
$563$	−14.9566	−0.630345	−0.315172	+	0.949034i	$0.602062\pi$
$564$	0.150590	0.00634098
$565$	15.3700	0.646623
$566$	−11.8988	−0.500144
$567$	−8.11230	−0.340685
$568$	−115.186	−4.83308
$569$	−20.6636	−0.866265	−0.433133	−	0.901330i	$-0.642592\pi$
$569$	−20.6636	−0.866265	−0.433133	+	0.901330i	$0.642592\pi$
$570$	10.4415	0.437347
$571$	−12.3457	−0.516651	−0.258325	−	0.966058i	$-0.583171\pi$
$571$	−12.3457	−0.516651	−0.258325	+	0.966058i	$0.583171\pi$
$572$	−53.4089	−2.23314
$573$	−28.5418	−1.19235
$574$	3.23675	0.135099
$575$	−3.75929	−0.156773
$576$	12.3142	0.513091
$577$	40.6935	1.69409	0.847047	−	0.531518i	$-0.178378\pi$
$577$	40.6935	1.69409	0.847047	+	0.531518i	$0.178378\pi$
$578$	29.2199	1.21539
$579$	−3.73126	−0.155066
$580$	−23.6308	−0.981215
$581$	−10.9465	−0.454136
$582$	6.53832	0.271022
$583$	−6.34215	−0.262665
$584$	−13.9950	−0.579118
$585$	−4.82327	−0.199418
$586$	21.8287	0.901736
$587$	2.28351	0.0942505	0.0471253	−	0.998889i	$-0.484994\pi$
$587$	2.28351	0.0942505	0.0471253	+	0.998889i	$0.484994\pi$
$588$	−63.4779	−2.61778
$589$	0.527020	0.0217155
$590$	−59.3903	−2.44506
$591$	−11.8670	−0.488144
$592$	−66.8415	−2.74717
$593$	−47.7074	−1.95911	−0.979554	−	0.201183i	$-0.935521\pi$
$593$	−47.7074	−1.95911	−0.979554	+	0.201183i	$0.935521\pi$
$594$	−29.2786	−1.20131
$595$	4.00679	0.164262
$596$	−88.7135	−3.63385
$597$	−11.3167	−0.463160
$598$	−56.3389	−2.30387
$599$	−16.4601	−0.672542	−0.336271	−	0.941765i	$-0.609166\pi$
$599$	−16.4601	−0.672542	−0.336271	+	0.941765i	$0.609166\pi$
$600$	12.9315	0.527924
$601$	1.97498	0.0805611	0.0402806	−	0.999188i	$-0.487175\pi$
$601$	1.97498	0.0805611	0.0402806	+	0.999188i	$0.487175\pi$
$602$	8.68527	0.353985
$603$	3.20643	0.130576
$604$	−49.5281	−2.01527
$605$	11.2848	0.458794
$606$	−50.1607	−2.03764
$607$	12.9583	0.525959	0.262980	−	0.964801i	$-0.415295\pi$
$607$	12.9583	0.525959	0.262980	+	0.964801i	$0.415295\pi$
$608$	−18.2575	−0.740438
$609$	3.21659	0.130343
$610$	31.0285	1.25631
$611$	0.0652541	0.00263990
$612$	−7.14586	−0.288854
$613$	−44.0322	−1.77844	−0.889222	−	0.457475i	$-0.848754\pi$
$613$	−44.0322	−1.77844	−0.889222	+	0.457475i	$0.848754\pi$
$614$	72.2319	2.91504
$615$	5.91474	0.238505
$616$	16.3086	0.657092
$617$	45.2637	1.82225	0.911124	−	0.412132i	$-0.135216\pi$
$617$	45.2637	1.82225	0.911124	+	0.412132i	$0.135216\pi$
$618$	−1.69002	−0.0679825
$619$	4.60886	0.185246	0.0926228	−	0.995701i	$-0.470475\pi$
$619$	4.60886	0.185246	0.0926228	+	0.995701i	$0.470475\pi$
$620$	−5.72047	−0.229740
$621$	−22.4013	−0.898935
$622$	−16.1612	−0.648006
$623$	−9.22313	−0.369517
$624$	108.118	4.32820
$625$	−20.5194	−0.820775
$626$	−4.23499	−0.169264
$627$	−4.41916	−0.176484
$628$	−2.41460	−0.0963531
$629$	12.4581	0.496738
$630$	2.37054	0.0944444
$631$	−21.0880	−0.839502	−0.419751	−	0.907639i	$-0.637883\pi$
$631$	−21.0880	−0.839502	−0.419751	+	0.907639i	$0.637883\pi$
$632$	−139.156	−5.53535
$633$	1.88273	0.0748318
$634$	−4.74865	−0.188593
$635$	20.4027	0.809658
$636$	26.8660	1.06531
$637$	−27.5065	−1.08985
$638$	13.7888	0.545903
$639$	7.08605	0.280320
$640$	50.3356	1.98969
$641$	0.752111	0.0297066	0.0148533	−	0.999890i	$-0.495272\pi$
$641$	0.752111	0.0297066	0.0148533	+	0.999890i	$0.495272\pi$
$642$	4.18063	0.164996
$643$	−3.12613	−0.123283	−0.0616413	−	0.998098i	$-0.519633\pi$
$643$	−3.12613	−0.123283	−0.0616413	+	0.998098i	$0.519633\pi$
$644$	20.0836	0.791406
$645$	15.8712	0.624928
$646$	6.70329	0.263737
$647$	30.5402	1.20066	0.600329	−	0.799753i	$-0.295036\pi$
$647$	30.5402	1.20066	0.600329	+	0.799753i	$0.295036\pi$
$648$	91.5250	3.59544
$649$	25.1358	0.986666
$650$	9.01905	0.353756
$651$	0.778663	0.0305182
$652$	1.95902	0.0767210
$653$	3.93799	0.154105	0.0770527	−	0.997027i	$-0.475449\pi$
$653$	3.93799	0.154105	0.0770527	+	0.997027i	$0.475449\pi$
$654$	43.9818	1.71982
$655$	40.7102	1.59068
$656$	−20.3729	−0.795430
$657$	0.860954	0.0335890
$658$	−0.0320710	−0.00125026
$659$	−8.54748	−0.332963	−0.166481	−	0.986045i	$-0.553241\pi$
$659$	−8.54748	−0.332963	−0.166481	+	0.986045i	$0.553241\pi$
$660$	47.9672	1.86712
$661$	−4.42306	−0.172037	−0.0860185	−	0.996294i	$-0.527414\pi$
$661$	−4.42306	−0.172037	−0.0860185	+	0.996294i	$0.527414\pi$
$662$	86.8974	3.37736
$663$	−20.1515	−0.782618
$664$	123.501	4.79276
$665$	−1.61291	−0.0625458
$666$	7.37061	0.285605
$667$	10.5499	0.408495
$668$	27.8209	1.07642
$669$	−45.9403	−1.77615
$670$	32.6483	1.26131
$671$	−13.1322	−0.506962
$672$	−26.9751	−1.04059
$673$	−49.9502	−1.92544	−0.962720	−	0.270499i	$-0.912811\pi$
$673$	−49.9502	−1.92544	−0.962720	+	0.270499i	$0.912811\pi$
$674$	−86.0874	−3.31596
$675$	3.58613	0.138030
$676$	29.3824	1.13009
$677$	−19.4525	−0.747619	−0.373810	−	0.927505i	$-0.621949\pi$
$677$	−19.4525	−0.747619	−0.373810	+	0.927505i	$0.621949\pi$
$678$	37.9918	1.45907
$679$	−1.00998	−0.0387594
$680$	−45.2055	−1.73355
$681$	27.0861	1.03794
$682$	3.33794	0.127816
$683$	41.2759	1.57938	0.789689	−	0.613507i	$-0.210242\pi$
$683$	41.2759	1.57938	0.789689	+	0.613507i	$0.210242\pi$
$684$	2.87652	0.109986
$685$	0.646463	0.0247001
$686$	28.3418	1.08209
$687$	11.5989	0.442526
$688$	−54.6674	−2.08417
$689$	11.6417	0.443513
$690$	50.5987	1.92626
$691$	28.2809	1.07586	0.537929	−	0.842990i	$-0.319207\pi$
$691$	28.2809	1.07586	0.537929	+	0.842990i	$0.319207\pi$
$692$	−89.3326	−3.39592
$693$	−1.00328	−0.0381115
$694$	−54.5425	−2.07041
$695$	−2.89059	−0.109646
$696$	−36.2904	−1.37558
$697$	3.79717	0.143828
$698$	16.1842	0.612582
$699$	14.5317	0.549641
$700$	−3.21510	−0.121519
$701$	−29.7240	−1.12266	−0.561331	−	0.827592i	$-0.689710\pi$
$701$	−29.7240	−1.12266	−0.561331	+	0.827592i	$0.689710\pi$
$702$	53.7439	2.02843
$703$	−5.01494	−0.189142
$704$	−53.0665	−2.00002
$705$	−0.0586056	−0.00220721
$706$	27.4056	1.03142
$707$	7.74835	0.291407
$708$	−106.478	−4.00169
$709$	43.1070	1.61892	0.809458	−	0.587178i	$-0.199761\pi$
$709$	43.1070	1.61892	0.809458	+	0.587178i	$0.199761\pi$
$710$	72.1511	2.70778
$711$	8.56071	0.321052
$712$	104.058	3.89972
$713$	2.55390	0.0956442
$714$	9.90400	0.370648
$715$	20.7853	0.777327
$716$	54.3194	2.03001
$717$	−10.0739	−0.376215
$718$	−33.8786	−1.26434
$719$	−3.88139	−0.144751	−0.0723757	−	0.997377i	$-0.523058\pi$
$719$	−3.88139	−0.144751	−0.0723757	+	0.997377i	$0.523058\pi$
$720$	−14.9208	−0.556064
$721$	0.261058	0.00972232
$722$	−2.69837	−0.100423
$723$	−23.1477	−0.860872
$724$	52.6201	1.95561
$725$	−1.68889	−0.0627239
$726$	27.8939	1.03524
$727$	25.2017	0.934678	0.467339	−	0.884078i	$-0.345213\pi$
$727$	25.2017	0.934678	0.467339	+	0.884078i	$0.345213\pi$
$728$	−29.9362	−1.10951
$729$	20.4795	0.758500
$730$	8.76634	0.324457
$731$	10.1891	0.376857
$732$	55.6294	2.05612
$733$	19.7762	0.730450	0.365225	−	0.930919i	$-0.380992\pi$
$733$	19.7762	0.730450	0.365225	+	0.930919i	$0.380992\pi$
$734$	20.3217	0.750089
$735$	24.7039	0.911218
$736$	−88.4743	−3.26121
$737$	−13.8177	−0.508983
$738$	2.24652	0.0826956
$739$	7.24558	0.266533	0.133267	−	0.991080i	$-0.457453\pi$
$739$	7.24558	0.266533	0.133267	+	0.991080i	$0.457453\pi$
$740$	54.4341	2.00104
$741$	8.11184	0.297996
$742$	−5.72164	−0.210048
$743$	−12.7573	−0.468019	−0.234009	−	0.972234i	$-0.575185\pi$
$743$	−12.7573	−0.468019	−0.234009	+	0.972234i	$0.575185\pi$
$744$	−8.78507	−0.322076
$745$	34.5250	1.26490
$746$	74.3156	2.72089
$747$	−7.59759	−0.277981
$748$	30.7942	1.12595
$749$	−0.645784	−0.0235964
$750$	−60.3077	−2.20212
$751$	13.8163	0.504164	0.252082	−	0.967706i	$-0.418885\pi$
$751$	13.8163	0.504164	0.252082	+	0.967706i	$0.418885\pi$
$752$	0.201863	0.00736120
$753$	−9.04570	−0.329644
$754$	−25.3108	−0.921763
$755$	19.2750	0.701491
$756$	−19.1586	−0.696790
$757$	−0.164360	−0.00597378	−0.00298689	−	0.999996i	$-0.500951\pi$
$757$	−0.164360	−0.00597378	−0.00298689	+	0.999996i	$0.500951\pi$
$758$	−20.4596	−0.743125
$759$	−21.4149	−0.777312
$760$	18.1972	0.660082
$761$	17.6990	0.641588	0.320794	−	0.947149i	$-0.396050\pi$
$761$	17.6990	0.641588	0.320794	+	0.947149i	$0.396050\pi$
$762$	50.4316	1.82695
$763$	−6.79390	−0.245956
$764$	−80.0616	−2.89653
$765$	2.78098	0.100547
$766$	59.4106	2.14659
$767$	−46.1394	−1.66600
$768$	39.2891	1.41772
$769$	6.47900	0.233639	0.116819	−	0.993153i	$-0.462730\pi$
$769$	6.47900	0.233639	0.116819	+	0.993153i	$0.462730\pi$
$770$	−10.2155	−0.368143
$771$	9.96312	0.358813
$772$	−10.4664	−0.376694
$773$	−25.6099	−0.921123	−0.460562	−	0.887628i	$-0.652352\pi$
$773$	−25.6099	−0.921123	−0.460562	+	0.887628i	$0.652352\pi$
$774$	6.02817	0.216678
$775$	−0.408842	−0.0146860
$776$	11.3948	0.409050
$777$	−7.40950	−0.265814
$778$	−40.2834	−1.44423
$779$	−1.52853	−0.0547652
$780$	−88.0490	−3.15266
$781$	−30.5365	−1.09268
$782$	32.4836	1.16161
$783$	−10.0640	−0.359658
$784$	−85.0911	−3.03897
$785$	0.939699	0.0335393
$786$	100.628	3.58928
$787$	50.9896	1.81758	0.908791	−	0.417251i	$-0.137006\pi$
$787$	50.9896	1.81758	0.908791	+	0.417251i	$0.137006\pi$
$788$	−33.2877	−1.18583
$789$	5.44550	0.193865
$790$	87.1662	3.10123
$791$	−5.86862	−0.208664
$792$	11.3193	0.402213
$793$	24.1055	0.856012
$794$	−90.2220	−3.20186
$795$	−10.4556	−0.370820
$796$	−31.7439	−1.12513
$797$	19.1130	0.677016	0.338508	−	0.940964i	$-0.390078\pi$
$797$	19.1130	0.677016	0.338508	+	0.940964i	$0.390078\pi$
$798$	−3.98679	−0.141131
$799$	−0.0376239	−0.00133104
$800$	14.1635	0.500754
$801$	−6.40148	−0.226185
$802$	45.3922	1.60285
$803$	−3.71018	−0.130929
$804$	58.5335	2.06432
$805$	−7.81602	−0.275479
$806$	−6.12715	−0.215820
$807$	56.2213	1.97909
$808$	−87.4188	−3.07538
$809$	55.5916	1.95450	0.977248	−	0.212098i	$-0.0680297\pi$
$809$	55.5916	1.95450	0.977248	+	0.212098i	$0.0680297\pi$
$810$	−57.3303	−2.01438
$811$	−3.61666	−0.126998	−0.0634991	−	0.997982i	$-0.520226\pi$
$811$	−3.61666	−0.126998	−0.0634991	+	0.997982i	$0.520226\pi$
$812$	9.02275	0.316636
$813$	0.327849	0.0114982
$814$	−31.7628	−1.11328
$815$	−0.762397	−0.0267056
$816$	−62.3384	−2.18228
$817$	−4.10155	−0.143495
$818$	4.68097	0.163666
$819$	1.84163	0.0643518
$820$	16.5912	0.579390
$821$	14.3160	0.499630	0.249815	−	0.968294i	$-0.419630\pi$
$821$	14.3160	0.499630	0.249815	+	0.968294i	$0.419630\pi$
$822$	1.59793	0.0557343
$823$	−30.5654	−1.06544	−0.532721	−	0.846291i	$-0.678830\pi$
$823$	−30.5654	−1.06544	−0.532721	+	0.846291i	$0.678830\pi$
$824$	−2.94532	−0.102605
$825$	3.42822	0.119355
$826$	22.6765	0.789017
$827$	−38.8642	−1.35144	−0.675720	−	0.737158i	$-0.736167\pi$
$827$	−38.8642	−1.35144	−0.675720	+	0.737158i	$0.736167\pi$
$828$	13.9394	0.484427
$829$	−49.9048	−1.73326	−0.866632	−	0.498948i	$-0.833720\pi$
$829$	−49.9048	−1.73326	−0.866632	+	0.498948i	$0.833720\pi$
$830$	−77.3596	−2.68519
$831$	−14.4210	−0.500261
$832$	97.4093	3.37706
$833$	15.8595	0.549501
$834$	−7.14498	−0.247411
$835$	−10.8272	−0.374689
$836$	−12.3960	−0.428725
$837$	−2.43626	−0.0842095
$838$	18.7838	0.648876
$839$	9.56997	0.330392	0.165196	−	0.986261i	$-0.447174\pi$
$839$	9.56997	0.330392	0.165196	+	0.986261i	$0.447174\pi$
$840$	26.8861	0.927658
$841$	−24.2604	−0.836564
$842$	61.8284	2.13075
$843$	−47.0533	−1.62060
$844$	5.28118	0.181786
$845$	−11.4349	−0.393371
$846$	−0.0222594	−0.000765295 0
$847$	−4.30880	−0.148052
$848$	36.0135	1.23671
$849$	8.30215	0.284929
$850$	−5.20016	−0.178364
$851$	−24.3020	−0.833063
$852$	129.356	4.43166
$853$	27.3445	0.936258	0.468129	−	0.883660i	$-0.344928\pi$
$853$	27.3445	0.936258	0.468129	+	0.883660i	$0.344928\pi$
$854$	−11.8473	−0.405408
$855$	−1.11947	−0.0382849
$856$	7.28589	0.249027
$857$	31.3943	1.07241	0.536205	−	0.844088i	$-0.319857\pi$
$857$	31.3943	1.07241	0.536205	+	0.844088i	$0.319857\pi$
$858$	51.3773	1.75399
$859$	18.3779	0.627047	0.313523	−	0.949581i	$-0.398491\pi$
$859$	18.3779	0.627047	0.313523	+	0.949581i	$0.398491\pi$
$860$	44.5197	1.51811
$861$	−2.25838	−0.0769653
$862$	3.41854	0.116436
$863$	23.8521	0.811934	0.405967	−	0.913888i	$-0.366935\pi$
$863$	23.8521	0.811934	0.405967	+	0.913888i	$0.366935\pi$
$864$	84.3991	2.87132
$865$	34.7659	1.18208
$866$	58.7171	1.99529
$867$	−20.3876	−0.692399
$868$	2.18420	0.0741366
$869$	−36.8913	−1.25145
$870$	22.7319	0.770684
$871$	25.3639	0.859424
$872$	76.6504	2.59571
$873$	−0.700993	−0.0237250
$874$	−13.0761	−0.442305
$875$	9.31576	0.314930
$876$	15.7167	0.531019
$877$	46.3708	1.56583	0.782915	−	0.622129i	$-0.213732\pi$
$877$	46.3708	1.56583	0.782915	+	0.622129i	$0.213732\pi$
$878$	11.8397	0.399570
$879$	−15.2306	−0.513714
$880$	64.2993	2.16753
$881$	−18.8495	−0.635054	−0.317527	−	0.948249i	$-0.602852\pi$
$881$	−18.8495	−0.635054	−0.317527	+	0.948249i	$0.602852\pi$
$882$	9.38298	0.315942
$883$	50.3381	1.69401	0.847006	−	0.531583i	$-0.178403\pi$
$883$	50.3381	1.69401	0.847006	+	0.531583i	$0.178403\pi$
$884$	−56.5261	−1.90118
$885$	41.4384	1.39294
$886$	−84.4548	−2.83731
$887$	0.424913	0.0142672	0.00713360	−	0.999975i	$-0.497729\pi$
$887$	0.424913	0.0142672	0.00713360	+	0.999975i	$0.497729\pi$
$888$	83.5958	2.80529
$889$	−7.79021	−0.261275
$890$	−65.1806	−2.18486
$891$	24.2639	0.812872
$892$	−128.865	−4.31473
$893$	0.0151453	0.000506817 0
$894$	85.3391	2.85417
$895$	−21.1397	−0.706622
$896$	−19.2192	−0.642070
$897$	39.3094	1.31250
$898$	67.1177	2.23975
$899$	1.14736	0.0382666
$900$	−2.23150	−0.0743832
$901$	−6.71231	−0.223619
$902$	−9.68112	−0.322346
$903$	−6.05998	−0.201663
$904$	66.2112	2.20215
$905$	−20.4784	−0.680724
$906$	47.6442	1.58287
$907$	46.9168	1.55785	0.778923	−	0.627120i	$-0.215766\pi$
$907$	46.9168	1.55785	0.778923	+	0.627120i	$0.215766\pi$
$908$	75.9783	2.52143
$909$	5.37788	0.178373
$910$	18.7517	0.621613
$911$	9.58750	0.317648	0.158824	−	0.987307i	$-0.449230\pi$
$911$	9.58750	0.317648	0.158824	+	0.987307i	$0.449230\pi$
$912$	25.0939	0.830943
$913$	32.7409	1.08357
$914$	73.0683	2.41689
$915$	−21.6495	−0.715710
$916$	32.5357	1.07501
$917$	−15.5440	−0.513310
$918$	−30.9874	−1.02274
$919$	−0.449869	−0.0148398	−0.00741990	−	0.999972i	$-0.502362\pi$
$919$	−0.449869	−0.0148398	−0.00741990	+	0.999972i	$0.502362\pi$
$920$	88.1823	2.90728
$921$	−50.3984	−1.66068
$922$	83.4521	2.74835
$923$	56.0530	1.84501
$924$	−18.3149	−0.602517
$925$	3.89040	0.127916
$926$	0.508335	0.0167049
$927$	0.181192	0.00595113
$928$	−39.7478	−1.30479
$929$	29.5656	0.970016	0.485008	−	0.874510i	$-0.338817\pi$
$929$	29.5656	0.970016	0.485008	+	0.874510i	$0.338817\pi$
$930$	5.50288	0.180446
$931$	−6.38416	−0.209232
$932$	40.7624	1.33522
$933$	11.2762	0.369165
$934$	−52.9684	−1.73318
$935$	−11.9843	−0.391929
$936$	−20.7777	−0.679142
$937$	24.6765	0.806147	0.403074	−	0.915168i	$-0.367942\pi$
$937$	24.6765	0.806147	0.403074	+	0.915168i	$0.367942\pi$
$938$	−12.4658	−0.407024
$939$	2.95488	0.0964289
$940$	−0.164392	−0.00536189
$941$	−40.6728	−1.32589	−0.662947	−	0.748666i	$-0.730695\pi$
$941$	−40.6728	−1.32589	−0.662947	+	0.748666i	$0.730695\pi$
$942$	2.32276	0.0756795
$943$	−7.40713	−0.241209
$944$	−142.732	−4.64553
$945$	7.45601	0.242544
$946$	−25.9777	−0.844607
$947$	−10.1414	−0.329553	−0.164776	−	0.986331i	$-0.552690\pi$
$947$	−10.1414	−0.329553	−0.164776	+	0.986331i	$0.552690\pi$
$948$	156.276	5.07560
$949$	6.81043	0.221076
$950$	2.09329	0.0679153
$951$	3.31327	0.107440
$952$	17.2605	0.559415
$953$	35.0584	1.13565	0.567826	−	0.823148i	$-0.307784\pi$
$953$	35.0584	1.13565	0.567826	+	0.823148i	$0.307784\pi$
$954$	−3.97121	−0.128573
$955$	31.1579	1.00824
$956$	−28.2578	−0.913924
$957$	−9.62084	−0.310997
$958$	−0.914164	−0.0295353
$959$	−0.246834	−0.00797068
$960$	−87.4846	−2.82355
$961$	−30.7223	−0.991040
$962$	58.3039	1.87979
$963$	−0.448218	−0.0144436
$964$	−64.9307	−2.09128
$965$	4.07325	0.131123
$966$	−19.3197	−0.621601
$967$	−19.9634	−0.641980	−0.320990	−	0.947083i	$-0.604016\pi$
$967$	−19.9634	−0.641980	−0.320990	+	0.947083i	$0.604016\pi$
$968$	48.6129	1.56248
$969$	−4.67708	−0.150250
$970$	−7.13760	−0.229174
$971$	−25.1142	−0.805954	−0.402977	−	0.915210i	$-0.632024\pi$
$971$	−25.1142	−0.805954	−0.402977	+	0.915210i	$0.632024\pi$
$972$	−29.5445	−0.947639
$973$	1.10369	0.0353827
$974$	41.9125	1.34296
$975$	−6.29286	−0.201533
$976$	74.5703	2.38694
$977$	9.79900	0.313498	0.156749	−	0.987638i	$-0.449899\pi$
$977$	9.79900	0.313498	0.156749	+	0.987638i	$0.449899\pi$
$978$	−1.88450	−0.0602597
$979$	27.5864	0.881665
$980$	69.2960	2.21358
$981$	−4.71542	−0.150552
$982$	8.96349	0.286036
$983$	−15.5172	−0.494921	−0.247460	−	0.968898i	$-0.579596\pi$
$983$	−15.5172	−0.494921	−0.247460	+	0.968898i	$0.579596\pi$
$984$	25.4796	0.812259
$985$	12.9547	0.412771
$986$	14.5935	0.464753
$987$	0.0223769	0.000712265 0
$988$	22.7542	0.723908
$989$	−19.8758	−0.632013
$990$	−7.09028	−0.225344
$991$	35.7120	1.13443	0.567215	−	0.823570i	$-0.308021\pi$
$991$	35.7120	1.13443	0.567215	+	0.823570i	$0.308021\pi$
$992$	−9.62204	−0.305500
$993$	−60.6309	−1.92406
$994$	−27.5489	−0.873797
$995$	12.3539	0.391645
$996$	−138.694	−4.39469
$997$	29.2230	0.925500	0.462750	−	0.886489i	$-0.346863\pi$
$997$	29.2230	0.925500	0.462750	+	0.886489i	$0.346863\pi$
$998$	56.8993	1.80112
$999$	23.1827	0.733467

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.3	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-2.69837 q^{2} +1.88273 q^{3} +5.28118 q^{4} -2.05530 q^{5} -5.08030 q^{6} +0.784756 q^{7} -8.85381 q^{8} +0.544674 q^{9} +O(q^{10})\)	\(q-2.69837 q^{2} +1.88273 q^{3} +5.28118 q^{4} -2.05530 q^{5} -5.08030 q^{6} +0.784756 q^{7} -8.85381 q^{8} +0.544674 q^{9} +5.54594 q^{10} -2.34721 q^{11} +9.94303 q^{12} +4.30855 q^{13} -2.11756 q^{14} -3.86957 q^{15} +13.3285 q^{16} -2.48420 q^{17} -1.46973 q^{18} +1.00000 q^{19} -10.8544 q^{20} +1.47748 q^{21} +6.33362 q^{22} +4.84592 q^{23} -16.6693 q^{24} -0.775762 q^{25} -11.6260 q^{26} -4.62272 q^{27} +4.14444 q^{28} +2.17707 q^{29} +10.4415 q^{30} +0.527020 q^{31} -18.2575 q^{32} -4.41916 q^{33} +6.70329 q^{34} -1.61291 q^{35} +2.87652 q^{36} -5.01494 q^{37} -2.69837 q^{38} +8.11184 q^{39} +18.1972 q^{40} -1.52853 q^{41} -3.98679 q^{42} -4.10155 q^{43} -12.3960 q^{44} -1.11947 q^{45} -13.0761 q^{46} +0.0151453 q^{47} +25.0939 q^{48} -6.38416 q^{49} +2.09329 q^{50} -4.67708 q^{51} +22.7542 q^{52} +2.70200 q^{53} +12.4738 q^{54} +4.82420 q^{55} -6.94809 q^{56} +1.88273 q^{57} -5.87454 q^{58} -10.7088 q^{59} -20.4359 q^{60} +5.59481 q^{61} -1.42209 q^{62} +0.427436 q^{63} +22.6084 q^{64} -8.85534 q^{65} +11.9245 q^{66} +5.88688 q^{67} -13.1195 q^{68} +9.12357 q^{69} +4.35221 q^{70} +13.0097 q^{71} -4.82244 q^{72} +1.58068 q^{73} +13.5321 q^{74} -1.46055 q^{75} +5.28118 q^{76} -1.84199 q^{77} -21.8887 q^{78} +15.7171 q^{79} -27.3939 q^{80} -10.3374 q^{81} +4.12453 q^{82} -13.9489 q^{83} +7.80286 q^{84} +5.10577 q^{85} +11.0675 q^{86} +4.09884 q^{87} +20.7817 q^{88} -11.7529 q^{89} +3.02073 q^{90} +3.38116 q^{91} +25.5922 q^{92} +0.992236 q^{93} -0.0408675 q^{94} -2.05530 q^{95} -34.3739 q^{96} -1.28700 q^{97} +17.2268 q^{98} -1.27846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 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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