LMFDB - Embedded newform 1849.2.a.o.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,2,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, 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("1849.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1849.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:	$14.7643393337$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 15 x^{16} + 106 x^{15} + 47 x^{14} - 897 x^{13} + 364 x^{12} + 3855 x^{11} + \cdots - 27 $ x^18 - 5x^17 - 15x^16 + 106x^15 + 47x^14 - 897x^13 + 364x^12 + 3855x^11 - 3223x^10 - 8851x^9 + 9909x^8 + 10400x^7 - 14483x^6 - 5118x^5 + 9945x^4 - 27x^3 - 2493x^2 + 513*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.10272$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.10272 q^{2} +0.645986 q^{3} -0.784019 q^{4} +3.32240 q^{5} +0.712338 q^{6} +2.46546 q^{7} -3.06998 q^{8} -2.58270 q^{9} +O(q^{10})$	\(q+1.10272 q^{2} +0.645986 q^{3} -0.784019 q^{4} +3.32240 q^{5} +0.712338 q^{6} +2.46546 q^{7} -3.06998 q^{8} -2.58270 q^{9} +3.66366 q^{10} +1.19972 q^{11} -0.506465 q^{12} +5.76244 q^{13} +2.71871 q^{14} +2.14622 q^{15} -1.81728 q^{16} +1.06784 q^{17} -2.84799 q^{18} +2.15635 q^{19} -2.60482 q^{20} +1.59265 q^{21} +1.32295 q^{22} -5.91845 q^{23} -1.98316 q^{24} +6.03834 q^{25} +6.35433 q^{26} -3.60635 q^{27} -1.93297 q^{28} -2.38034 q^{29} +2.36667 q^{30} +7.64330 q^{31} +4.13602 q^{32} +0.775003 q^{33} +1.17753 q^{34} +8.19126 q^{35} +2.02489 q^{36} -5.66067 q^{37} +2.37784 q^{38} +3.72245 q^{39} -10.1997 q^{40} +9.74127 q^{41} +1.75624 q^{42} -0.940604 q^{44} -8.58077 q^{45} -6.52637 q^{46} +5.29979 q^{47} -1.17393 q^{48} -0.921486 q^{49} +6.65857 q^{50} +0.689810 q^{51} -4.51786 q^{52} -3.67667 q^{53} -3.97677 q^{54} +3.98595 q^{55} -7.56893 q^{56} +1.39297 q^{57} -2.62484 q^{58} -0.938103 q^{59} -1.68268 q^{60} +2.80073 q^{61} +8.42838 q^{62} -6.36756 q^{63} +8.19541 q^{64} +19.1451 q^{65} +0.854608 q^{66} -11.9885 q^{67} -0.837208 q^{68} -3.82323 q^{69} +9.03262 q^{70} +15.1881 q^{71} +7.92885 q^{72} -6.81669 q^{73} -6.24211 q^{74} +3.90068 q^{75} -1.69062 q^{76} +2.95787 q^{77} +4.10481 q^{78} +2.37603 q^{79} -6.03772 q^{80} +5.41846 q^{81} +10.7419 q^{82} -7.13848 q^{83} -1.24867 q^{84} +3.54780 q^{85} -1.53767 q^{87} -3.68312 q^{88} +1.51670 q^{89} -9.46214 q^{90} +14.2071 q^{91} +4.64018 q^{92} +4.93746 q^{93} +5.84416 q^{94} +7.16425 q^{95} +2.67181 q^{96} -10.8065 q^{97} -1.01614 q^{98} -3.09852 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9}+O(q^{10}) $ 18 * q + 5 * q^2 + 5 * q^3 + 19 * q^4 + 11 * q^5 + 4 * q^6 + 6 * q^7 + 12 * q^8 + 15 * q^9	\( 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9} - 7 q^{10} - 2 q^{11} + 16 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 9 q^{16} - 11 q^{17} + 25 q^{18} + 31 q^{19} + 25 q^{20} + 19 q^{22} - 11 q^{23} + 18 q^{24} + 9 q^{25} + 27 q^{26} + 23 q^{27} + 20 q^{28} + 37 q^{29} - 17 q^{30} - 12 q^{31} + 39 q^{32} + 38 q^{33} + 14 q^{34} + 16 q^{35} + 47 q^{36} - 19 q^{37} + 56 q^{38} + 46 q^{39} + 6 q^{40} - 7 q^{41} - q^{42} + 7 q^{44} + 23 q^{45} - 47 q^{46} - q^{47} + 15 q^{48} - 6 q^{49} - 3 q^{50} + 38 q^{51} + 15 q^{52} + 3 q^{53} - 67 q^{54} + 28 q^{55} - 81 q^{56} + 46 q^{57} + 34 q^{58} + 17 q^{59} - 83 q^{60} + 28 q^{61} + 33 q^{62} + 26 q^{63} + 10 q^{64} + 16 q^{65} - 72 q^{66} + 18 q^{67} + 53 q^{68} + 7 q^{69} - 34 q^{70} + 86 q^{71} - 2 q^{72} - 27 q^{73} - 79 q^{74} + 31 q^{75} + 59 q^{76} + 43 q^{77} + 91 q^{78} + 17 q^{79} + 8 q^{80} - 10 q^{81} + 13 q^{82} - 12 q^{83} - 32 q^{84} + 28 q^{85} - 43 q^{87} - 23 q^{88} + 51 q^{89} + 10 q^{90} - 20 q^{91} + 18 q^{92} - 30 q^{93} - 15 q^{94} - q^{95} - 20 q^{96} - 19 q^{97} - 5 q^{98} + 38 q^{99}+O(q^{100}) \) 18 * q + 5 * q^2 + 5 * q^3 + 19 * q^4 + 11 * q^5 + 4 * q^6 + 6 * q^7 + 12 * q^8 + 15 * q^9 - 7 * q^10 - 2 * q^11 + 16 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 9 * q^16 - 11 * q^17 + 25 * q^18 + 31 * q^19 + 25 * q^20 + 19 * q^22 - 11 * q^23 + 18 * q^24 + 9 * q^25 + 27 * q^26 + 23 * q^27 + 20 * q^28 + 37 * q^29 - 17 * q^30 - 12 * q^31 + 39 * q^32 + 38 * q^33 + 14 * q^34 + 16 * q^35 + 47 * q^36 - 19 * q^37 + 56 * q^38 + 46 * q^39 + 6 * q^40 - 7 * q^41 - q^42 + 7 * q^44 + 23 * q^45 - 47 * q^46 - q^47 + 15 * q^48 - 6 * q^49 - 3 * q^50 + 38 * q^51 + 15 * q^52 + 3 * q^53 - 67 * q^54 + 28 * q^55 - 81 * q^56 + 46 * q^57 + 34 * q^58 + 17 * q^59 - 83 * q^60 + 28 * q^61 + 33 * q^62 + 26 * q^63 + 10 * q^64 + 16 * q^65 - 72 * q^66 + 18 * q^67 + 53 * q^68 + 7 * q^69 - 34 * q^70 + 86 * q^71 - 2 * q^72 - 27 * q^73 - 79 * q^74 + 31 * q^75 + 59 * q^76 + 43 * q^77 + 91 * q^78 + 17 * q^79 + 8 * q^80 - 10 * q^81 + 13 * q^82 - 12 * q^83 - 32 * q^84 + 28 * q^85 - 43 * q^87 - 23 * q^88 + 51 * q^89 + 10 * q^90 - 20 * q^91 + 18 * q^92 - 30 * q^93 - 15 * q^94 - q^95 - 20 * q^96 - 19 * q^97 - 5 * q^98 + 38 * 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 $

$2$	1.10272	0.779737	0.389869	−	0.920870i	$-0.372520\pi$
$2$	1.10272	0.779737	0.389869	+	0.920870i	$0.372520\pi$
$3$	0.645986	0.372960	0.186480	−	0.982459i	$-0.440292\pi$
$3$	0.645986	0.372960	0.186480	+	0.982459i	$0.440292\pi$
$4$	−0.784019	−0.392009
$5$	3.32240	1.48582	0.742911	−	0.669390i	$-0.233445\pi$
$5$	3.32240	1.48582	0.742911	+	0.669390i	$0.233445\pi$
$6$	0.712338	0.290811
$7$	2.46546	0.931858	0.465929	−	0.884822i	$-0.345720\pi$
$7$	2.46546	0.931858	0.465929	+	0.884822i	$0.345720\pi$
$8$	−3.06998	−1.08540
$9$	−2.58270	−0.860901
$10$	3.66366	1.15855
$11$	1.19972	0.361730	0.180865	−	0.983508i	$-0.442110\pi$
$11$	1.19972	0.361730	0.180865	+	0.983508i	$0.442110\pi$
$12$	−0.506465	−0.146204
$13$	5.76244	1.59821	0.799107	−	0.601189i	$-0.205306\pi$
$13$	5.76244	1.59821	0.799107	+	0.601189i	$0.205306\pi$
$14$	2.71871	0.726604
$15$	2.14622	0.554152
$16$	−1.81728	−0.454319
$17$	1.06784	0.258990	0.129495	−	0.991580i	$-0.458664\pi$
$17$	1.06784	0.258990	0.129495	+	0.991580i	$0.458664\pi$
$18$	−2.84799	−0.671277
$19$	2.15635	0.494700	0.247350	−	0.968926i	$-0.420440\pi$
$19$	2.15635	0.494700	0.247350	+	0.968926i	$0.420440\pi$
$20$	−2.60482	−0.582456
$21$	1.59265	0.347546
$22$	1.32295	0.282054
$23$	−5.91845	−1.23408	−0.617041	−	0.786931i	$-0.711669\pi$
$23$	−5.91845	−1.23408	−0.617041	+	0.786931i	$0.711669\pi$
$24$	−1.98316	−0.404812
$25$	6.03834	1.20767
$26$	6.35433	1.24619
$27$	−3.60635	−0.694042
$28$	−1.93297	−0.365297
$29$	−2.38034	−0.442018	−0.221009	−	0.975272i	$-0.570935\pi$
$29$	−2.38034	−0.442018	−0.221009	+	0.975272i	$0.570935\pi$
$30$	2.36667	0.432093
$31$	7.64330	1.37278	0.686389	−	0.727235i	$-0.259195\pi$
$31$	7.64330	1.37278	0.686389	+	0.727235i	$0.259195\pi$
$32$	4.13602	0.731152
$33$	0.775003	0.134911
$34$	1.17753	0.201944
$35$	8.19126	1.38457
$36$	2.02489	0.337481
$37$	−5.66067	−0.930608	−0.465304	−	0.885151i	$-0.654055\pi$
$37$	−5.66067	−0.930608	−0.465304	+	0.885151i	$0.654055\pi$
$38$	2.37784	0.385736
$39$	3.72245	0.596070
$40$	−10.1997	−1.61271
$41$	9.74127	1.52133	0.760666	−	0.649144i	$-0.224873\pi$
$41$	9.74127	1.52133	0.760666	+	0.649144i	$0.224873\pi$
$42$	1.75624	0.270994
$43$	0	0
$43$	0	0
$44$	−0.940604	−0.141801
$45$	−8.58077	−1.27915
$46$	−6.52637	−0.962260
$47$	5.29979	0.773054	0.386527	−	0.922278i	$-0.373675\pi$
$47$	5.29979	0.773054	0.386527	+	0.922278i	$0.373675\pi$
$48$	−1.17393	−0.169443
$49$	−0.921486	−0.131641
$50$	6.65857	0.941663
$51$	0.689810	0.0965928
$52$	−4.51786	−0.626515
$53$	−3.67667	−0.505029	−0.252515	−	0.967593i	$-0.581258\pi$
$53$	−3.67667	−0.505029	−0.252515	+	0.967593i	$0.581258\pi$
$54$	−3.97677	−0.541170
$55$	3.98595	0.537466
$56$	−7.56893	−1.01144
$57$	1.39297	0.184503
$58$	−2.62484	−0.344658
$59$	−0.938103	−0.122131	−0.0610653	−	0.998134i	$-0.519450\pi$
$59$	−0.938103	−0.122131	−0.0610653	+	0.998134i	$0.519450\pi$
$60$	−1.68268	−0.217233
$61$	2.80073	0.358597	0.179298	−	0.983795i	$-0.442617\pi$
$61$	2.80073	0.358597	0.179298	+	0.983795i	$0.442617\pi$
$62$	8.42838	1.07041
$63$	−6.36756	−0.802237
$64$	8.19541	1.02443
$65$	19.1451	2.37466
$66$	0.854608	0.105195
$67$	−11.9885	−1.46463	−0.732314	−	0.680967i	$-0.761560\pi$
$67$	−11.9885	−1.46463	−0.732314	+	0.680967i	$0.761560\pi$
$68$	−0.837208	−0.101526
$69$	−3.82323	−0.460263
$70$	9.03262	1.07960
$71$	15.1881	1.80250	0.901249	−	0.433301i	$-0.142651\pi$
$71$	15.1881	1.80250	0.901249	+	0.433301i	$0.142651\pi$
$72$	7.92885	0.934423
$73$	−6.81669	−0.797833	−0.398917	−	0.916987i	$-0.630614\pi$
$73$	−6.81669	−0.797833	−0.398917	+	0.916987i	$0.630614\pi$
$74$	−6.24211	−0.725630
$75$	3.90068	0.450412
$76$	−1.69062	−0.193927
$77$	2.95787	0.337081
$78$	4.10481	0.464778
$79$	2.37603	0.267325	0.133662	−	0.991027i	$-0.457326\pi$
$79$	2.37603	0.267325	0.133662	+	0.991027i	$0.457326\pi$
$80$	−6.03772	−0.675037
$81$	5.41846	0.602051
$82$	10.7419	1.18624
$83$	−7.13848	−0.783550	−0.391775	−	0.920061i	$-0.628139\pi$
$83$	−7.13848	−0.783550	−0.391775	+	0.920061i	$0.628139\pi$
$84$	−1.24867	−0.136241
$85$	3.54780	0.384813
$86$	0	0
$87$	−1.53767	−0.164855
$88$	−3.68312	−0.392622
$89$	1.51670	0.160770	0.0803849	−	0.996764i	$-0.474385\pi$
$89$	1.51670	0.160770	0.0803849	+	0.996764i	$0.474385\pi$
$90$	−9.46214	−0.997398
$91$	14.2071	1.48931
$92$	4.64018	0.483772
$93$	4.93746	0.511991
$94$	5.84416	0.602780
$95$	7.16425	0.735036
$96$	2.67181	0.272691
$97$	−10.8065	−1.09724	−0.548619	−	0.836072i	$-0.684846\pi$
$97$	−10.8065	−1.09724	−0.548619	+	0.836072i	$0.684846\pi$
$98$	−1.01614	−0.102645
$99$	−3.09852	−0.311413
$100$	−4.73417	−0.473417
$101$	11.9930	1.19335	0.596676	−	0.802482i	$-0.296488\pi$
$101$	11.9930	1.19335	0.596676	+	0.802482i	$0.296488\pi$
$102$	0.760665	0.0753170
$103$	10.4879	1.03340	0.516700	−	0.856166i	$-0.327160\pi$
$103$	10.4879	1.03340	0.516700	+	0.856166i	$0.327160\pi$
$104$	−17.6906	−1.73470
$105$	5.29143	0.516391
$106$	−4.05432	−0.393790
$107$	−9.38278	−0.907068	−0.453534	−	0.891239i	$-0.649837\pi$
$107$	−9.38278	−0.907068	−0.453534	+	0.891239i	$0.649837\pi$
$108$	2.82744	0.272071
$109$	−14.5901	−1.39748	−0.698738	−	0.715377i	$-0.746255\pi$
$109$	−14.5901	−1.39748	−0.698738	+	0.715377i	$0.746255\pi$
$110$	4.39537	0.419082
$111$	−3.65671	−0.347080
$112$	−4.48043	−0.423361
$113$	2.21384	0.208261	0.104130	−	0.994564i	$-0.466794\pi$
$113$	2.21384	0.208261	0.104130	+	0.994564i	$0.466794\pi$
$114$	1.53605	0.143864
$115$	−19.6635	−1.83363
$116$	1.86623	0.173275
$117$	−14.8827	−1.37590
$118$	−1.03446	−0.0952298
$119$	2.63273	0.241342
$120$	−6.58886	−0.601478
$121$	−9.56067	−0.869152
$122$	3.08841	0.279611
$123$	6.29272	0.567396
$124$	−5.99249	−0.538142
$125$	3.44977	0.308556
$126$	−7.02161	−0.625534
$127$	0.129263	0.0114703	0.00573513	−	0.999984i	$-0.498174\pi$
$127$	0.129263	0.0114703	0.00573513	+	0.999984i	$0.498174\pi$
$128$	0.765157	0.0676310
$129$	0	0
$130$	21.1116	1.85161
$131$	2.13352	0.186407	0.0932033	−	0.995647i	$-0.470289\pi$
$131$	2.13352	0.186407	0.0932033	+	0.995647i	$0.470289\pi$
$132$	−0.607617	−0.0528863
$133$	5.31640	0.460990
$134$	−13.2199	−1.14203
$135$	−11.9817	−1.03122
$136$	−3.27825	−0.281108
$137$	−0.618814	−0.0528689	−0.0264344	−	0.999651i	$-0.508415\pi$
$137$	−0.618814	−0.0528689	−0.0264344	+	0.999651i	$0.508415\pi$
$138$	−4.21594	−0.358885
$139$	9.49962	0.805747	0.402874	−	0.915256i	$-0.368011\pi$
$139$	9.49962	0.805747	0.402874	+	0.915256i	$0.368011\pi$
$140$	−6.42210	−0.542766
$141$	3.42359	0.288318
$142$	16.7482	1.40548
$143$	6.91332	0.578121
$144$	4.69348	0.391124
$145$	−7.90844	−0.656760
$146$	−7.51687	−0.622100
$147$	−0.595267	−0.0490968
$148$	4.43807	0.364807
$149$	4.31670	0.353637	0.176819	−	0.984243i	$-0.443419\pi$
$149$	4.31670	0.353637	0.176819	+	0.984243i	$0.443419\pi$
$150$	4.30134	0.351203
$151$	−4.44657	−0.361857	−0.180928	−	0.983496i	$-0.557910\pi$
$151$	−4.44657	−0.361857	−0.180928	+	0.983496i	$0.557910\pi$
$152$	−6.61995	−0.536948
$153$	−2.75792	−0.222964
$154$	3.26169	0.262834
$155$	25.3941	2.03970
$156$	−2.91847	−0.233665
$157$	−14.0116	−1.11825	−0.559124	−	0.829084i	$-0.688862\pi$
$157$	−14.0116	−1.11825	−0.559124	+	0.829084i	$0.688862\pi$
$158$	2.62009	0.208443
$159$	−2.37507	−0.188356
$160$	13.7415	1.08636
$161$	−14.5917	−1.14999
$162$	5.97502	0.469442
$163$	1.90611	0.149298	0.0746489	−	0.997210i	$-0.476216\pi$
$163$	1.90611	0.149298	0.0746489	+	0.997210i	$0.476216\pi$
$164$	−7.63734	−0.596376
$165$	2.57487	0.200453
$166$	−7.87171	−0.610963
$167$	−13.9089	−1.07630	−0.538152	−	0.842848i	$-0.680877\pi$
$167$	−13.9089	−1.07630	−0.538152	+	0.842848i	$0.680877\pi$
$168$	−4.88942	−0.377227
$169$	20.2057	1.55429
$170$	3.91221	0.300053
$171$	−5.56920	−0.425888
$172$	0	0
$173$	−9.52832	−0.724424	−0.362212	−	0.932096i	$-0.617978\pi$
$173$	−9.52832	−0.724424	−0.362212	+	0.932096i	$0.617978\pi$
$174$	−1.69561	−0.128544
$175$	14.8873	1.12537
$176$	−2.18023	−0.164341
$177$	−0.606001	−0.0455498
$178$	1.67249	0.125358
$179$	6.42138	0.479957	0.239978	−	0.970778i	$-0.422860\pi$
$179$	6.42138	0.479957	0.239978	+	0.970778i	$0.422860\pi$
$180$	6.72748	0.501437
$181$	−8.63543	−0.641867	−0.320933	−	0.947102i	$-0.603997\pi$
$181$	−8.63543	−0.641867	−0.320933	+	0.947102i	$0.603997\pi$
$182$	15.6664	1.16127
$183$	1.80923	0.133742
$184$	18.1695	1.33948
$185$	−18.8070	−1.38272
$186$	5.44462	0.399219
$187$	1.28111	0.0936842
$188$	−4.15514	−0.303045
$189$	−8.89132	−0.646748
$190$	7.90013	0.573135
$191$	−2.24473	−0.162423	−0.0812114	−	0.996697i	$-0.525879\pi$
$191$	−2.24473	−0.162423	−0.0812114	+	0.996697i	$0.525879\pi$
$192$	5.29412	0.382070
$193$	16.9623	1.22098	0.610488	−	0.792026i	$-0.290973\pi$
$193$	16.9623	1.22098	0.610488	+	0.792026i	$0.290973\pi$
$194$	−11.9165	−0.855558
$195$	12.3675	0.885653
$196$	0.722463	0.0516045
$197$	−5.61835	−0.400291	−0.200145	−	0.979766i	$-0.564141\pi$
$197$	−5.61835	−0.400291	−0.200145	+	0.979766i	$0.564141\pi$
$198$	−3.41679	−0.242821
$199$	18.3778	1.30277	0.651384	−	0.758748i	$-0.274189\pi$
$199$	18.3778	1.30277	0.651384	+	0.758748i	$0.274189\pi$
$200$	−18.5376	−1.31080
$201$	−7.74440	−0.546248
$202$	13.2249	0.930502
$203$	−5.86864	−0.411898
$204$	−0.540824	−0.0378653
$205$	32.3644	2.26043
$206$	11.5651	0.805781
$207$	15.2856	1.06242
$208$	−10.4719	−0.726099
$209$	2.58702	0.178948
$210$	5.83495	0.402649
$211$	−26.5430	−1.82730	−0.913648	−	0.406507i	$-0.866747\pi$
$211$	−26.5430	−1.82730	−0.913648	+	0.406507i	$0.866747\pi$
$212$	2.88258	0.197976
$213$	9.81131	0.672260
$214$	−10.3465	−0.707275
$215$	0	0
$216$	11.0714	0.753314
$217$	18.8443	1.27923
$218$	−16.0887	−1.08966
$219$	−4.40348	−0.297560
$220$	−3.12506	−0.210692
$221$	6.15337	0.413921
$222$	−4.03231	−0.270631
$223$	−18.8357	−1.26133	−0.630665	−	0.776056i	$-0.717218\pi$
$223$	−18.8357	−1.26133	−0.630665	+	0.776056i	$0.717218\pi$
$224$	10.1972	0.681330
$225$	−15.5952	−1.03968
$226$	2.44124	0.162389
$227$	3.05198	0.202567	0.101283	−	0.994858i	$-0.467705\pi$
$227$	3.05198	0.202567	0.101283	+	0.994858i	$0.467705\pi$
$228$	−1.09211	−0.0723271
$229$	−9.93431	−0.656478	−0.328239	−	0.944595i	$-0.606455\pi$
$229$	−9.93431	−0.656478	−0.328239	+	0.944595i	$0.606455\pi$
$230$	−21.6832	−1.42975
$231$	1.91074	0.125718
$232$	7.30760	0.479767
$233$	7.38263	0.483652	0.241826	−	0.970320i	$-0.422254\pi$
$233$	7.38263	0.483652	0.241826	+	0.970320i	$0.422254\pi$
$234$	−16.4113	−1.07284
$235$	17.6080	1.14862
$236$	0.735490	0.0478764
$237$	1.53488	0.0997014
$238$	2.90315	0.188183
$239$	−14.9263	−0.965500	−0.482750	−	0.875758i	$-0.660362\pi$
$239$	−14.9263	−0.965500	−0.482750	+	0.875758i	$0.660362\pi$
$240$	−3.90028	−0.251762
$241$	−3.84888	−0.247928	−0.123964	−	0.992287i	$-0.539561\pi$
$241$	−3.84888	−0.247928	−0.123964	+	0.992287i	$0.539561\pi$
$242$	−10.5427	−0.677710
$243$	14.3193	0.918583
$244$	−2.19583	−0.140573
$245$	−3.06155	−0.195595
$246$	6.93908	0.442420
$247$	12.4258	0.790636
$248$	−23.4648	−1.49002
$249$	−4.61136	−0.292233
$250$	3.80411	0.240593
$251$	14.2354	0.898527	0.449264	−	0.893399i	$-0.351686\pi$
$251$	14.2354	0.898527	0.449264	+	0.893399i	$0.351686\pi$
$252$	4.99229	0.314485
$253$	−7.10049	−0.446404
$254$	0.142541	0.00894379
$255$	2.29183	0.143520
$256$	−15.5471	−0.971692
$257$	6.95341	0.433742	0.216871	−	0.976200i	$-0.430415\pi$
$257$	6.95341	0.433742	0.216871	+	0.976200i	$0.430415\pi$
$258$	0	0
$259$	−13.9562	−0.867194
$260$	−15.0101	−0.930889
$261$	6.14771	0.380534
$262$	2.35267	0.145348
$263$	−7.22756	−0.445671	−0.222835	−	0.974856i	$-0.571531\pi$
$263$	−7.22756	−0.445671	−0.222835	+	0.974856i	$0.571531\pi$
$264$	−2.37924	−0.146432
$265$	−12.2154	−0.750383
$266$	5.86247	0.359451
$267$	0.979766	0.0599607
$268$	9.39921	0.574148
$269$	−3.73112	−0.227490	−0.113745	−	0.993510i	$-0.536285\pi$
$269$	−3.73112	−0.227490	−0.113745	+	0.993510i	$0.536285\pi$
$270$	−13.2124	−0.804083
$271$	14.3435	0.871307	0.435654	−	0.900114i	$-0.356517\pi$
$271$	14.3435	0.871307	0.435654	+	0.900114i	$0.356517\pi$
$272$	−1.94056	−0.117664
$273$	9.17758	0.555452
$274$	−0.682376	−0.0412238
$275$	7.24432	0.436849
$276$	2.99749	0.180428
$277$	−25.6300	−1.53996	−0.769980	−	0.638068i	$-0.779734\pi$
$277$	−25.6300	−1.53996	−0.769980	+	0.638068i	$0.779734\pi$
$278$	10.4754	0.628271
$279$	−19.7404	−1.18183
$280$	−25.1470	−1.50282
$281$	20.5148	1.22381	0.611906	−	0.790930i	$-0.290403\pi$
$281$	20.5148	1.22381	0.611906	+	0.790930i	$0.290403\pi$
$282$	3.77525	0.224813
$283$	−0.352466	−0.0209520	−0.0104760	−	0.999945i	$-0.503335\pi$
$283$	−0.352466	−0.0209520	−0.0104760	+	0.999945i	$0.503335\pi$
$284$	−11.9078	−0.706597
$285$	4.62800	0.274139
$286$	7.62343	0.450783
$287$	24.0168	1.41766
$288$	−10.6821	−0.629450
$289$	−15.8597	−0.932924
$290$	−8.72076	−0.512100
$291$	−6.98088	−0.409226
$292$	5.34441	0.312758
$293$	−26.1068	−1.52518	−0.762588	−	0.646884i	$-0.776072\pi$
$293$	−26.1068	−1.52518	−0.762588	+	0.646884i	$0.776072\pi$
$294$	−0.656410	−0.0382826
$295$	−3.11675	−0.181464
$296$	17.3781	1.01008
$297$	−4.32661	−0.251055
$298$	4.76009	0.275744
$299$	−34.1047	−1.97233
$300$	−3.05821	−0.176566
$301$	0	0
$302$	−4.90330	−0.282153
$303$	7.74733	0.445073
$304$	−3.91868	−0.224752
$305$	9.30514	0.532811
$306$	−3.04120	−0.173854
$307$	−15.3470	−0.875899	−0.437949	−	0.899000i	$-0.644295\pi$
$307$	−15.3470	−0.875899	−0.437949	+	0.899000i	$0.644295\pi$
$308$	−2.31903	−0.132139
$309$	6.77501	0.385417
$310$	28.0025	1.59043
$311$	−32.2165	−1.82683	−0.913415	−	0.407031i	$-0.866564\pi$
$311$	−32.2165	−1.82683	−0.913415	+	0.407031i	$0.866564\pi$
$312$	−11.4279	−0.646975
$313$	−26.2362	−1.48296	−0.741479	−	0.670976i	$-0.765875\pi$
$313$	−26.2362	−1.48296	−0.741479	+	0.670976i	$0.765875\pi$
$314$	−15.4508	−0.871940
$315$	−21.1556	−1.19198
$316$	−1.86285	−0.104794
$317$	−2.08246	−0.116963	−0.0584814	−	0.998288i	$-0.518626\pi$
$317$	−2.08246	−0.116963	−0.0584814	+	0.998288i	$0.518626\pi$
$318$	−2.61903	−0.146868
$319$	−2.85574	−0.159891
$320$	27.2284	1.52211
$321$	−6.06114	−0.338300
$322$	−16.0905	−0.896690
$323$	2.30264	0.128122
$324$	−4.24818	−0.236010
$325$	34.7955	1.93011
$326$	2.10189	0.116413
$327$	−9.42499	−0.521203
$328$	−29.9055	−1.65126
$329$	13.0665	0.720377
$330$	2.83935	0.156301
$331$	−5.37704	−0.295549	−0.147774	−	0.989021i	$-0.547211\pi$
$331$	−5.37704	−0.295549	−0.147774	+	0.989021i	$0.547211\pi$
$332$	5.59670	0.307159
$333$	14.6198	0.801161
$334$	−15.3376	−0.839234
$335$	−39.8306	−2.17618
$336$	−2.89429	−0.157897
$337$	−12.2282	−0.666111	−0.333055	−	0.942907i	$-0.608080\pi$
$337$	−12.2282	−0.666111	−0.333055	+	0.942907i	$0.608080\pi$
$338$	22.2811	1.21193
$339$	1.43011	0.0776730
$340$	−2.78154	−0.150850
$341$	9.16983	0.496574
$342$	−6.14125	−0.332081
$343$	−19.5301	−1.05453
$344$	0	0
$345$	−12.7023	−0.683869
$346$	−10.5070	−0.564861
$347$	−2.69955	−0.144920	−0.0724598	−	0.997371i	$-0.523085\pi$
$347$	−2.69955	−0.144920	−0.0724598	+	0.997371i	$0.523085\pi$
$348$	1.20556	0.0646247
$349$	7.52972	0.403057	0.201528	−	0.979483i	$-0.435409\pi$
$349$	7.52972	0.403057	0.201528	+	0.979483i	$0.435409\pi$
$350$	16.4165	0.877496
$351$	−20.7814	−1.10923
$352$	4.96207	0.264479
$353$	−5.50147	−0.292814	−0.146407	−	0.989224i	$-0.546771\pi$
$353$	−5.50147	−0.292814	−0.146407	+	0.989224i	$0.546771\pi$
$354$	−0.668247	−0.0355169
$355$	50.4610	2.67819
$356$	−1.18912	−0.0630233
$357$	1.70070	0.0900107
$358$	7.08096	0.374240
$359$	−21.2278	−1.12036	−0.560181	−	0.828370i	$-0.689268\pi$
$359$	−21.2278	−1.12036	−0.560181	+	0.828370i	$0.689268\pi$
$360$	26.3428	1.38839
$361$	−14.3502	−0.755272
$362$	−9.52242	−0.500487
$363$	−6.17605	−0.324159
$364$	−11.1386	−0.583823
$365$	−22.6478	−1.18544
$366$	1.99507	0.104284
$367$	12.0464	0.628817	0.314408	−	0.949288i	$-0.398194\pi$
$367$	12.0464	0.628817	0.314408	+	0.949288i	$0.398194\pi$
$368$	10.7555	0.560667
$369$	−25.1588	−1.30972
$370$	−20.7388	−1.07816
$371$	−9.06469	−0.470615
$372$	−3.87106	−0.200705
$373$	5.89356	0.305157	0.152579	−	0.988291i	$-0.451242\pi$
$373$	5.89356	0.305157	0.152579	+	0.988291i	$0.451242\pi$
$374$	1.41270	0.0730491
$375$	2.22850	0.115079
$376$	−16.2703	−0.839075
$377$	−13.7166	−0.706439
$378$	−9.80459	−0.504294
$379$	17.9864	0.923898	0.461949	−	0.886906i	$-0.347150\pi$
$379$	17.9864	0.923898	0.461949	+	0.886906i	$0.347150\pi$
$380$	−5.61691	−0.288141
$381$	0.0835022	0.00427795
$382$	−2.47529	−0.126647
$383$	36.1531	1.84734	0.923669	−	0.383191i	$-0.125175\pi$
$383$	36.1531	1.84734	0.923669	+	0.383191i	$0.125175\pi$
$384$	0.494281	0.0252237
$385$	9.82723	0.500842
$386$	18.7046	0.952040
$387$	0	0
$388$	8.47254	0.430128
$389$	23.3210	1.18242	0.591210	−	0.806518i	$-0.298651\pi$
$389$	23.3210	1.18242	0.591210	+	0.806518i	$0.298651\pi$
$390$	13.6378	0.690577
$391$	−6.31997	−0.319615
$392$	2.82895	0.142883
$393$	1.37822	0.0695222
$394$	−6.19544	−0.312122
$395$	7.89413	0.397197
$396$	2.42930	0.122077
$397$	−12.6958	−0.637185	−0.318592	−	0.947892i	$-0.603210\pi$
$397$	−12.6958	−0.637185	−0.318592	+	0.947892i	$0.603210\pi$
$398$	20.2655	1.01582
$399$	3.43432	0.171931
$400$	−10.9733	−0.548666
$401$	3.86298	0.192908	0.0964541	−	0.995337i	$-0.469250\pi$
$401$	3.86298	0.192908	0.0964541	+	0.995337i	$0.469250\pi$
$402$	−8.53987	−0.425930
$403$	44.0441	2.19399
$404$	−9.40277	−0.467805
$405$	18.0023	0.894541
$406$	−6.47144	−0.321172
$407$	−6.79122	−0.336629
$408$	−2.11770	−0.104842
$409$	−13.1901	−0.652210	−0.326105	−	0.945334i	$-0.605736\pi$
$409$	−13.1901	−0.652210	−0.326105	+	0.945334i	$0.605736\pi$
$410$	35.6887	1.76254
$411$	−0.399745	−0.0197180
$412$	−8.22269	−0.405103
$413$	−2.31286	−0.113808
$414$	16.8557	0.828411
$415$	−23.7169	−1.16422
$416$	23.8336	1.16854
$417$	6.13662	0.300511
$418$	2.85274	0.139532
$419$	−19.1538	−0.935723	−0.467861	−	0.883802i	$-0.654975\pi$
$419$	−19.1538	−0.935723	−0.467861	+	0.883802i	$0.654975\pi$
$420$	−4.14858	−0.202430
$421$	30.4813	1.48557	0.742783	−	0.669533i	$-0.233506\pi$
$421$	30.4813	1.48557	0.742783	+	0.669533i	$0.233506\pi$
$422$	−29.2694	−1.42481
$423$	−13.6878	−0.665523
$424$	11.2873	0.548160
$425$	6.44799	0.312773
$426$	10.8191	0.524186
$427$	6.90510	0.334161
$428$	7.35628	0.355579
$429$	4.46591	0.215616
$430$	0	0
$431$	−35.9315	−1.73076	−0.865380	−	0.501117i	$-0.832923\pi$
$431$	−35.9315	−1.73076	−0.865380	+	0.501117i	$0.832923\pi$
$432$	6.55373	0.315316
$433$	5.85599	0.281421	0.140711	−	0.990051i	$-0.455061\pi$
$433$	5.85599	0.281421	0.140711	+	0.990051i	$0.455061\pi$
$434$	20.7799	0.997466
$435$	−5.10874	−0.244945
$436$	11.4389	0.547824
$437$	−12.7622	−0.610501
$438$	−4.85579	−0.232019
$439$	−19.4349	−0.927577	−0.463789	−	0.885946i	$-0.653510\pi$
$439$	−19.4349	−0.927577	−0.463789	+	0.885946i	$0.653510\pi$
$440$	−12.2368	−0.583366
$441$	2.37993	0.113330
$442$	6.78542	0.322749
$443$	10.2326	0.486166	0.243083	−	0.970005i	$-0.421841\pi$
$443$	10.2326	0.486166	0.243083	+	0.970005i	$0.421841\pi$
$444$	2.86693	0.136058
$445$	5.03908	0.238875
$446$	−20.7704	−0.983506
$447$	2.78852	0.131893
$448$	20.2055	0.954619
$449$	−13.6961	−0.646361	−0.323181	−	0.946337i	$-0.604752\pi$
$449$	−13.6961	−0.646361	−0.323181	+	0.946337i	$0.604752\pi$
$450$	−17.1971	−0.810679
$451$	11.6868	0.550311
$452$	−1.73569	−0.0816402
$453$	−2.87242	−0.134958
$454$	3.36546	0.157949
$455$	47.2016	2.21285
$456$	−4.27639	−0.200260
$457$	−9.65530	−0.451656	−0.225828	−	0.974167i	$-0.572509\pi$
$457$	−9.65530	−0.451656	−0.225828	+	0.974167i	$0.572509\pi$
$458$	−10.9547	−0.511880
$459$	−3.85101	−0.179750
$460$	15.4165	0.718799
$461$	6.72828	0.313367	0.156684	−	0.987649i	$-0.449920\pi$
$461$	6.72828	0.313367	0.156684	+	0.987649i	$0.449920\pi$
$462$	2.10700	0.0980267
$463$	−5.65537	−0.262827	−0.131414	−	0.991328i	$-0.541952\pi$
$463$	−5.65537	−0.262827	−0.131414	+	0.991328i	$0.541952\pi$
$464$	4.32574	0.200817
$465$	16.4042	0.760727
$466$	8.14094	0.377122
$467$	10.6620	0.493379	0.246690	−	0.969095i	$-0.420657\pi$
$467$	10.6620	0.493379	0.246690	+	0.969095i	$0.420657\pi$
$468$	11.6683	0.539367
$469$	−29.5572	−1.36483
$470$	19.4166	0.895623
$471$	−9.05130	−0.417062
$472$	2.87996	0.132561
$473$	0	0
$474$	1.69254	0.0777409
$475$	13.0208	0.597433
$476$	−2.06411	−0.0946082
$477$	9.49574	0.434780
$478$	−16.4594	−0.752836
$479$	−10.8213	−0.494438	−0.247219	−	0.968960i	$-0.579517\pi$
$479$	−10.8213	−0.494438	−0.247219	+	0.968960i	$0.579517\pi$
$480$	8.87682	0.405170
$481$	−32.6193	−1.48731
$482$	−4.24422	−0.193319
$483$	−9.42605	−0.428900
$484$	7.49575	0.340716
$485$	−35.9037	−1.63030
$486$	15.7901	0.716253
$487$	19.3365	0.876218	0.438109	−	0.898922i	$-0.355648\pi$
$487$	19.3365	0.876218	0.438109	+	0.898922i	$0.355648\pi$
$488$	−8.59819	−0.389222
$489$	1.23132	0.0556821
$490$	−3.37601	−0.152513
$491$	31.6494	1.42832	0.714158	−	0.699984i	$-0.246810\pi$
$491$	31.6494	1.42832	0.714158	+	0.699984i	$0.246810\pi$
$492$	−4.93361	−0.222424
$493$	−2.54183	−0.114478
$494$	13.7021	0.616489
$495$	−10.2945	−0.462705
$496$	−13.8900	−0.623679
$497$	37.4458	1.67967
$498$	−5.08501	−0.227865
$499$	9.91455	0.443836	0.221918	−	0.975065i	$-0.428768\pi$
$499$	9.91455	0.443836	0.221918	+	0.975065i	$0.428768\pi$
$500$	−2.70468	−0.120957
$501$	−8.98495	−0.401418
$502$	15.6975	0.700615
$503$	−7.71375	−0.343939	−0.171970	−	0.985102i	$-0.555013\pi$
$503$	−7.71375	−0.343939	−0.171970	+	0.985102i	$0.555013\pi$
$504$	19.5483	0.870750
$505$	39.8457	1.77311
$506$	−7.82982	−0.348078
$507$	13.0526	0.579686
$508$	−0.101345	−0.00449645
$509$	−37.2349	−1.65041	−0.825204	−	0.564836i	$-0.808940\pi$
$509$	−37.2349	−1.65041	−0.825204	+	0.564836i	$0.808940\pi$
$510$	2.52723	0.111908
$511$	−16.8063	−0.743467
$512$	−18.6743	−0.825295
$513$	−7.77654	−0.343342
$514$	7.66763	0.338205
$515$	34.8449	1.53545
$516$	0	0
$517$	6.35828	0.279637
$518$	−15.3897	−0.676184
$519$	−6.15516	−0.270181
$520$	−58.7752	−2.57746
$521$	4.86692	0.213223	0.106612	−	0.994301i	$-0.466000\pi$
$521$	4.86692	0.213223	0.106612	+	0.994301i	$0.466000\pi$
$522$	6.77917	0.296716
$523$	8.10948	0.354603	0.177301	−	0.984157i	$-0.443263\pi$
$523$	8.10948	0.354603	0.177301	+	0.984157i	$0.443263\pi$
$524$	−1.67272	−0.0730731
$525$	9.61698	0.419720
$526$	−7.96994	−0.347506
$527$	8.16183	0.355535
$528$	−1.40839	−0.0612925
$529$	12.0281	0.522959
$530$	−13.4701	−0.585102
$531$	2.42284	0.105142
$532$	−4.16816	−0.180713
$533$	56.1335	2.43141
$534$	1.08040	0.0467536
$535$	−31.1733	−1.34774
$536$	36.8044	1.58971
$537$	4.14812	0.179005
$538$	−4.11436	−0.177383
$539$	−1.10553	−0.0476184
$540$	9.39390	0.404249
$541$	16.3366	0.702367	0.351184	−	0.936307i	$-0.385779\pi$
$541$	16.3366	0.702367	0.351184	+	0.936307i	$0.385779\pi$
$542$	15.8168	0.679391
$543$	−5.57836	−0.239391
$544$	4.41662	0.189361
$545$	−48.4741	−2.07640
$546$	10.1203	0.433107
$547$	26.7843	1.14521	0.572607	−	0.819830i	$-0.305932\pi$
$547$	26.7843	1.14521	0.572607	+	0.819830i	$0.305932\pi$
$548$	0.485162	0.0207251
$549$	−7.23345	−0.308716
$550$	7.98842	0.340628
$551$	−5.13284	−0.218666
$552$	11.7373	0.499571
$553$	5.85802	0.249109
$554$	−28.2626	−1.20076
$555$	−12.1491	−0.515698
$556$	−7.44788	−0.315860
$557$	16.0227	0.678903	0.339452	−	0.940624i	$-0.389759\pi$
$557$	16.0227	0.678903	0.339452	+	0.940624i	$0.389759\pi$
$558$	−21.7680	−0.921513
$559$	0	0
$560$	−14.8858	−0.629039
$561$	0.827580	0.0349405
$562$	22.6220	0.954252
$563$	4.84969	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$563$	4.84969	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$564$	−2.68416	−0.113024
$565$	7.35527	0.309439
$566$	−0.388670	−0.0163370
$567$	13.3590	0.561026
$568$	−46.6272	−1.95644
$569$	28.0180	1.17457	0.587287	−	0.809379i	$-0.300196\pi$
$569$	28.0180	1.17457	0.587287	+	0.809379i	$0.300196\pi$
$570$	5.10337	0.213757
$571$	−5.31633	−0.222481	−0.111241	−	0.993793i	$-0.535482\pi$
$571$	−5.31633	−0.222481	−0.111241	+	0.993793i	$0.535482\pi$
$572$	−5.42018	−0.226629
$573$	−1.45006	−0.0605772
$574$	26.4836	1.10541
$575$	−35.7376	−1.49036
$576$	−21.1663	−0.881929
$577$	−0.578860	−0.0240983	−0.0120491	−	0.999927i	$-0.503835\pi$
$577$	−0.578860	−0.0240983	−0.0120491	+	0.999927i	$0.503835\pi$
$578$	−17.4887	−0.727436
$579$	10.9574	0.455375
$580$	6.20037	0.257456
$581$	−17.5997	−0.730157
$582$	−7.69792	−0.319089
$583$	−4.41098	−0.182684
$584$	20.9271	0.865970
$585$	−49.4462	−2.04435
$586$	−28.7884	−1.18924
$587$	35.3888	1.46065	0.730326	−	0.683098i	$-0.239368\pi$
$587$	35.3888	1.46065	0.730326	+	0.683098i	$0.239368\pi$
$588$	0.466701	0.0192464
$589$	16.4816	0.679113
$590$	−3.43689	−0.141495
$591$	−3.62937	−0.149292
$592$	10.2870	0.422793
$593$	14.0549	0.577166	0.288583	−	0.957455i	$-0.406816\pi$
$593$	14.0549	0.577166	0.288583	+	0.957455i	$0.406816\pi$
$594$	−4.77102	−0.195757
$595$	8.74696	0.358591
$596$	−3.38437	−0.138629
$597$	11.8718	0.485881
$598$	−37.6078	−1.53790
$599$	11.1278	0.454669	0.227335	−	0.973817i	$-0.426999\pi$
$599$	11.1278	0.454669	0.227335	+	0.973817i	$0.426999\pi$
$600$	−11.9750	−0.488878
$601$	−0.417393	−0.0170258	−0.00851291	−	0.999964i	$-0.502710\pi$
$601$	−0.417393	−0.0170258	−0.00851291	+	0.999964i	$0.502710\pi$
$602$	0	0
$603$	30.9627	1.26090
$604$	3.48619	0.141851
$605$	−31.7644	−1.29140
$606$	8.54310	0.347040
$607$	0.478432	0.0194189	0.00970947	−	0.999953i	$-0.496909\pi$
$607$	0.478432	0.0194189	0.00970947	+	0.999953i	$0.496909\pi$
$608$	8.91870	0.361701
$609$	−3.79106	−0.153621
$610$	10.2609	0.415453
$611$	30.5397	1.23551
$612$	2.16226	0.0874042
$613$	5.76451	0.232826	0.116413	−	0.993201i	$-0.462860\pi$
$613$	5.76451	0.232826	0.116413	+	0.993201i	$0.462860\pi$
$614$	−16.9234	−0.682971
$615$	20.9069	0.843049
$616$	−9.08060	−0.365868
$617$	−13.8423	−0.557271	−0.278635	−	0.960397i	$-0.589882\pi$
$617$	−13.8423	−0.557271	−0.278635	+	0.960397i	$0.589882\pi$
$618$	7.47091	0.300524
$619$	−32.1146	−1.29079	−0.645397	−	0.763847i	$-0.723308\pi$
$619$	−32.1146	−1.29079	−0.645397	+	0.763847i	$0.723308\pi$
$620$	−19.9094	−0.799583
$621$	21.3440	0.856504
$622$	−35.5256	−1.42445
$623$	3.73937	0.149815
$624$	−6.76473	−0.270806
$625$	−18.7302	−0.749207
$626$	−28.9311	−1.15632
$627$	1.67118	0.0667403
$628$	10.9854	0.438364
$629$	−6.04470	−0.241018
$630$	−23.3286	−0.929433
$631$	48.7336	1.94006	0.970028	−	0.242993i	$-0.0781294\pi$
$631$	48.7336	1.94006	0.970028	+	0.242993i	$0.0781294\pi$
$632$	−7.29437	−0.290155
$633$	−17.1464	−0.681508
$634$	−2.29636	−0.0912003
$635$	0.429464	0.0170428
$636$	1.86210	0.0738372
$637$	−5.31001	−0.210390
$638$	−3.14907	−0.124673
$639$	−39.2264	−1.55177
$640$	2.54216	0.100488
$641$	39.4573	1.55847	0.779235	−	0.626732i	$-0.215608\pi$
$641$	39.4573	1.55847	0.779235	+	0.626732i	$0.215608\pi$
$642$	−6.68371	−0.263785
$643$	31.7285	1.25125	0.625625	−	0.780124i	$-0.284844\pi$
$643$	31.7285	1.25125	0.625625	+	0.780124i	$0.284844\pi$
$644$	11.4402	0.450807
$645$	0	0
$646$	2.53915	0.0999017
$647$	33.6328	1.32224	0.661120	−	0.750280i	$-0.270081\pi$
$647$	33.6328	1.32224	0.661120	+	0.750280i	$0.270081\pi$
$648$	−16.6346	−0.653467
$649$	−1.12546	−0.0441783
$650$	38.3696	1.50498
$651$	12.1731	0.477103
$652$	−1.49442	−0.0585262
$653$	−34.6455	−1.35578	−0.677892	−	0.735161i	$-0.737107\pi$
$653$	−34.6455	−1.35578	−0.677892	+	0.735161i	$0.737107\pi$
$654$	−10.3931	−0.406401
$655$	7.08841	0.276967
$656$	−17.7026	−0.691170
$657$	17.6055	0.686855
$658$	14.4086	0.561705
$659$	22.3153	0.869279	0.434640	−	0.900604i	$-0.356876\pi$
$659$	22.3153	0.869279	0.434640	+	0.900604i	$0.356876\pi$
$660$	−2.01875	−0.0785796
$661$	−19.7276	−0.767315	−0.383657	−	0.923475i	$-0.625336\pi$
$661$	−19.7276	−0.767315	−0.383657	+	0.923475i	$0.625336\pi$
$662$	−5.92935	−0.230451
$663$	3.97499	0.154376
$664$	21.9150	0.850467
$665$	17.6632	0.684949
$666$	16.1215	0.624695
$667$	14.0879	0.545487
$668$	10.9048	0.421921
$669$	−12.1676	−0.470425
$670$	−43.9218	−1.69685
$671$	3.36010	0.129715
$672$	6.58725	0.254109
$673$	−17.2378	−0.664467	−0.332234	−	0.943197i	$-0.607802\pi$
$673$	−17.2378	−0.664467	−0.332234	+	0.943197i	$0.607802\pi$
$674$	−13.4842	−0.519392
$675$	−21.7763	−0.838171
$676$	−15.8417	−0.609295
$677$	−41.4079	−1.59143	−0.795717	−	0.605669i	$-0.792906\pi$
$677$	−41.4079	−1.59143	−0.795717	+	0.605669i	$0.792906\pi$
$678$	1.57701	0.0605645
$679$	−26.6432	−1.02247
$680$	−10.8917	−0.417676
$681$	1.97153	0.0755494
$682$	10.1117	0.387197
$683$	3.69145	0.141249	0.0706246	−	0.997503i	$-0.477501\pi$
$683$	3.69145	0.141249	0.0706246	+	0.997503i	$0.477501\pi$
$684$	4.36636	0.166952
$685$	−2.05595	−0.0785537
$686$	−21.5362	−0.822255
$687$	−6.41742	−0.244840
$688$	0	0
$689$	−21.1866	−0.807144
$690$	−14.0070	−0.533239
$691$	48.5650	1.84750	0.923750	−	0.382997i	$-0.125108\pi$
$691$	48.5650	1.84750	0.923750	+	0.382997i	$0.125108\pi$
$692$	7.47038	0.283981
$693$	−7.63930	−0.290193
$694$	−2.97684	−0.112999
$695$	31.5615	1.19720
$696$	4.72060	0.178934
$697$	10.4021	0.394009
$698$	8.30314	0.314278
$699$	4.76907	0.180383
$700$	−11.6719	−0.441157
$701$	−41.5732	−1.57020	−0.785099	−	0.619371i	$-0.787388\pi$
$701$	−41.5732	−1.57020	−0.785099	+	0.619371i	$0.787388\pi$
$702$	−22.9159	−0.864905
$703$	−12.2064	−0.460372
$704$	9.83221	0.370565
$705$	11.3745	0.428390
$706$	−6.06656	−0.228318
$707$	29.5684	1.11203
$708$	0.475116	0.0178560
$709$	−22.2700	−0.836368	−0.418184	−	0.908362i	$-0.637333\pi$
$709$	−22.2700	−0.836368	−0.418184	+	0.908362i	$0.637333\pi$
$710$	55.6441	2.08829
$711$	−6.13659	−0.230140
$712$	−4.65624	−0.174500
$713$	−45.2365	−1.69412
$714$	1.87539	0.0701848
$715$	22.9688	0.858985
$716$	−5.03449	−0.188148
$717$	−9.64215	−0.360093
$718$	−23.4082	−0.873588
$719$	−13.6665	−0.509673	−0.254837	−	0.966984i	$-0.582022\pi$
$719$	−13.6665	−0.509673	−0.254837	+	0.966984i	$0.582022\pi$
$720$	15.5936	0.581140
$721$	25.8575	0.962982
$722$	−15.8241	−0.588914
$723$	−2.48632	−0.0924673
$724$	6.77034	0.251618
$725$	−14.3733	−0.533811
$726$	−6.81043	−0.252759
$727$	28.7751	1.06721	0.533604	−	0.845734i	$-0.320837\pi$
$727$	28.7751	1.06721	0.533604	+	0.845734i	$0.320837\pi$
$728$	−43.6155	−1.61650
$729$	−7.00533	−0.259457
$730$	−24.9740	−0.924331
$731$	0	0
$732$	−1.41847	−0.0524282
$733$	−42.0159	−1.55189	−0.775947	−	0.630798i	$-0.782728\pi$
$733$	−42.0159	−1.55189	−0.775947	+	0.630798i	$0.782728\pi$
$734$	13.2837	0.490312
$735$	−1.97771	−0.0729491
$736$	−24.4788	−0.902302
$737$	−14.3829	−0.529799
$738$	−27.7430	−1.02123
$739$	47.2291	1.73735	0.868675	−	0.495383i	$-0.164972\pi$
$739$	47.2291	1.73735	0.868675	+	0.495383i	$0.164972\pi$
$740$	14.7450	0.542039
$741$	8.02690	0.294876
$742$	−9.99577	−0.366956
$743$	41.8495	1.53531	0.767654	−	0.640865i	$-0.221424\pi$
$743$	41.8495	1.53531	0.767654	+	0.640865i	$0.221424\pi$
$744$	−15.1579	−0.555716
$745$	14.3418	0.525442
$746$	6.49892	0.237942
$747$	18.4366	0.674559
$748$	−1.00442	−0.0367251
$749$	−23.1329	−0.845258
$750$	2.45740	0.0897316
$751$	−1.27352	−0.0464715	−0.0232357	−	0.999730i	$-0.507397\pi$
$751$	−1.27352	−0.0464715	−0.0232357	+	0.999730i	$0.507397\pi$
$752$	−9.63119	−0.351213
$753$	9.19583	0.335115
$754$	−15.1255	−0.550837
$755$	−14.7733	−0.537655
$756$	6.97096	0.253531
$757$	28.5021	1.03593	0.517964	−	0.855402i	$-0.326690\pi$
$757$	28.5021	1.03593	0.517964	+	0.855402i	$0.326690\pi$
$758$	19.8339	0.720398
$759$	−4.58682	−0.166491
$760$	−21.9941	−0.797810
$761$	23.8329	0.863941	0.431970	−	0.901888i	$-0.357819\pi$
$761$	23.8329	0.863941	0.431970	+	0.901888i	$0.357819\pi$
$762$	0.0920792	0.00333568
$763$	−35.9713	−1.30225
$764$	1.75991	0.0636713
$765$	−9.16290	−0.331285
$766$	39.8666	1.44044
$767$	−5.40576	−0.195191
$768$	−10.0432	−0.362402
$769$	45.9291	1.65625	0.828124	−	0.560546i	$-0.189409\pi$
$769$	45.9291	1.65625	0.828124	+	0.560546i	$0.189409\pi$
$770$	10.8366	0.390525
$771$	4.49180	0.161768
$772$	−13.2988	−0.478634
$773$	−18.2262	−0.655551	−0.327776	−	0.944756i	$-0.606299\pi$
$773$	−18.2262	−0.655551	−0.327776	+	0.944756i	$0.606299\pi$
$774$	0	0
$775$	46.1528	1.65786
$776$	33.1759	1.19095
$777$	−9.01549	−0.323429
$778$	25.7164	0.921977
$779$	21.0056	0.752603
$780$	−9.69634	−0.347185
$781$	18.2215	0.652017
$782$	−6.96913	−0.249215
$783$	8.58433	0.306779
$784$	1.67460	0.0598070
$785$	−46.5522	−1.66152
$786$	1.51979	0.0542090
$787$	−18.9232	−0.674538	−0.337269	−	0.941408i	$-0.609503\pi$
$787$	−18.9232	−0.674538	−0.337269	+	0.941408i	$0.609503\pi$
$788$	4.40489	0.156918
$789$	−4.66890	−0.166217
$790$	8.70498	0.309709
$791$	5.45815	0.194069
$792$	9.51241	0.338009
$793$	16.1390	0.573114
$794$	−13.9999	−0.496837
$795$	−7.89094	−0.279863
$796$	−14.4086	−0.510698
$797$	22.7545	0.806004	0.403002	−	0.915199i	$-0.367967\pi$
$797$	22.7545	0.806004	0.403002	+	0.915199i	$0.367967\pi$
$798$	3.78707	0.134061
$799$	5.65934	0.200213
$800$	24.9747	0.882989
$801$	−3.91718	−0.138407
$802$	4.25977	0.150418
$803$	−8.17813	−0.288600
$804$	6.07175	0.214134
$805$	−48.4795	−1.70868
$806$	48.5681	1.71074
$807$	−2.41025	−0.0848448
$808$	−36.8184	−1.29527
$809$	17.9955	0.632689	0.316344	−	0.948644i	$-0.397544\pi$
$809$	17.9955	0.632689	0.316344	+	0.948644i	$0.397544\pi$
$810$	19.8514	0.697507
$811$	10.9852	0.385742	0.192871	−	0.981224i	$-0.438220\pi$
$811$	10.9852	0.385742	0.192871	+	0.981224i	$0.438220\pi$
$812$	4.60113	0.161468
$813$	9.26571	0.324963
$814$	−7.48879	−0.262482
$815$	6.33285	0.221830
$816$	−1.25358	−0.0438839
$817$	0	0
$818$	−14.5450	−0.508552
$819$	−36.6927	−1.28215
$820$	−25.3743	−0.886109
$821$	41.3967	1.44476	0.722378	−	0.691498i	$-0.243049\pi$
$821$	41.3967	1.44476	0.722378	+	0.691498i	$0.243049\pi$
$822$	−0.440805	−0.0153748
$823$	−29.9932	−1.04550	−0.522748	−	0.852487i	$-0.675093\pi$
$823$	−29.9932	−1.04550	−0.522748	+	0.852487i	$0.675093\pi$
$824$	−32.1976	−1.12165
$825$	4.67973	0.162927
$826$	−2.55043	−0.0887406
$827$	6.93693	0.241221	0.120610	−	0.992700i	$-0.461515\pi$
$827$	6.93693	0.241221	0.120610	+	0.992700i	$0.461515\pi$
$828$	−11.9842	−0.416480
$829$	28.9735	1.00629	0.503145	−	0.864202i	$-0.332176\pi$
$829$	28.9735	1.00629	0.503145	+	0.864202i	$0.332176\pi$
$830$	−26.1530	−0.907783
$831$	−16.5566	−0.574344
$832$	47.2255	1.63725
$833$	−0.984002	−0.0340936
$834$	6.76694	0.234320
$835$	−46.2109	−1.59920
$836$	−2.02827	−0.0701492
$837$	−27.5644	−0.952764
$838$	−21.1212	−0.729618
$839$	50.4742	1.74256	0.871281	−	0.490784i	$-0.163290\pi$
$839$	50.4742	1.74256	0.871281	+	0.490784i	$0.163290\pi$
$840$	−16.2446	−0.560492
$841$	−23.3340	−0.804620
$842$	33.6121	1.15835
$843$	13.2523	0.456433
$844$	20.8102	0.716317
$845$	67.1314	2.30939
$846$	−15.0937	−0.518933
$847$	−23.5715	−0.809926
$848$	6.68152	0.229444
$849$	−0.227688	−0.00781424
$850$	7.11029	0.243881
$851$	33.5024	1.14845
$852$	−7.69225	−0.263532
$853$	22.8353	0.781865	0.390933	−	0.920419i	$-0.372153\pi$
$853$	22.8353	0.781865	0.390933	+	0.920419i	$0.372153\pi$
$854$	7.61436	0.260558
$855$	−18.5031	−0.632793
$856$	28.8050	0.984533
$857$	37.7947	1.29104	0.645520	−	0.763743i	$-0.276641\pi$
$857$	37.7947	1.29104	0.645520	+	0.763743i	$0.276641\pi$
$858$	4.92462	0.168124
$859$	33.4445	1.14111	0.570556	−	0.821258i	$-0.306728\pi$
$859$	33.4445	1.14111	0.570556	+	0.821258i	$0.306728\pi$
$860$	0	0
$861$	15.5145	0.528732
$862$	−39.6222	−1.34954
$863$	−38.7873	−1.32033	−0.660167	−	0.751119i	$-0.729515\pi$
$863$	−38.7873	−1.32033	−0.660167	+	0.751119i	$0.729515\pi$
$864$	−14.9159	−0.507450
$865$	−31.6569	−1.07637
$866$	6.45749	0.219435
$867$	−10.2451	−0.347943
$868$	−14.7743	−0.501472
$869$	2.85058	0.0966992
$870$	−5.63348	−0.190993
$871$	−69.0830	−2.34079
$872$	44.7913	1.51682
$873$	27.9101	0.944614
$874$	−14.0731	−0.476030
$875$	8.50527	0.287531
$876$	3.45242	0.116646
$877$	−34.2891	−1.15786	−0.578930	−	0.815378i	$-0.696529\pi$
$877$	−34.2891	−1.15786	−0.578930	+	0.815378i	$0.696529\pi$
$878$	−21.4312	−0.723267
$879$	−16.8646	−0.568830
$880$	−7.24358	−0.244181
$881$	8.93202	0.300927	0.150464	−	0.988616i	$-0.451923\pi$
$881$	8.93202	0.300927	0.150464	+	0.988616i	$0.451923\pi$
$882$	2.62438	0.0883675
$883$	−3.65426	−0.122976	−0.0614879	−	0.998108i	$-0.519585\pi$
$883$	−3.65426	−0.122976	−0.0614879	+	0.998108i	$0.519585\pi$
$884$	−4.82436	−0.162261
$885$	−2.01338	−0.0676789
$886$	11.2837	0.379082
$887$	31.0771	1.04347	0.521734	−	0.853108i	$-0.325286\pi$
$887$	31.0771	1.04347	0.521734	+	0.853108i	$0.325286\pi$
$888$	11.2260	0.376721
$889$	0.318694	0.0106887
$890$	5.55667	0.186260
$891$	6.50064	0.217780
$892$	14.7675	0.494453
$893$	11.4282	0.382430
$894$	3.07495	0.102842
$895$	21.3344	0.713130
$896$	1.88647	0.0630225
$897$	−22.0312	−0.735599
$898$	−15.1030	−0.503992
$899$	−18.1937	−0.606792
$900$	12.2270	0.407565
$901$	−3.92610	−0.130797
$902$	12.8872	0.429098
$903$	0	0
$904$	−6.79645	−0.226047
$905$	−28.6903	−0.953699
$906$	−3.16746	−0.105232
$907$	−58.4331	−1.94024	−0.970120	−	0.242627i	$-0.921991\pi$
$907$	−58.4331	−1.94024	−0.970120	+	0.242627i	$0.921991\pi$
$908$	−2.39281	−0.0794082
$909$	−30.9745	−1.02736
$910$	52.0499	1.72544
$911$	52.9065	1.75287	0.876435	−	0.481520i	$-0.159915\pi$
$911$	52.9065	1.75287	0.876435	+	0.481520i	$0.159915\pi$
$912$	−2.53141	−0.0838234
$913$	−8.56419	−0.283433
$914$	−10.6470	−0.352173
$915$	6.01099	0.198717
$916$	7.78869	0.257345
$917$	5.26012	0.173704
$918$	−4.24656	−0.140157
$919$	−9.06296	−0.298959	−0.149480	−	0.988765i	$-0.547760\pi$
$919$	−9.06296	−0.298959	−0.149480	+	0.988765i	$0.547760\pi$
$920$	60.3664	1.99022
$921$	−9.91394	−0.326675
$922$	7.41937	0.244344
$923$	87.5206	2.88078
$924$	−1.49806	−0.0492825
$925$	−34.1810	−1.12386
$926$	−6.23626	−0.204936
$927$	−27.0870	−0.889655
$928$	−9.84514	−0.323182
$929$	16.3701	0.537084	0.268542	−	0.963268i	$-0.413458\pi$
$929$	16.3701	0.537084	0.268542	+	0.963268i	$0.413458\pi$
$930$	18.0892	0.593168
$931$	−1.98705	−0.0651228
$932$	−5.78812	−0.189596
$933$	−20.8114	−0.681334
$934$	11.7572	0.384706
$935$	4.25637	0.139198
$936$	45.6895	1.49341
$937$	−4.99550	−0.163196	−0.0815980	−	0.996665i	$-0.526002\pi$
$937$	−4.99550	−0.163196	−0.0815980	+	0.996665i	$0.526002\pi$
$938$	−32.5932	−1.06421
$939$	−16.9482	−0.553084
$940$	−13.8050	−0.450270
$941$	−2.05385	−0.0669536	−0.0334768	−	0.999439i	$-0.510658\pi$
$941$	−2.05385	−0.0669536	−0.0334768	+	0.999439i	$0.510658\pi$
$942$	−9.98101	−0.325199
$943$	−57.6532	−1.87745
$944$	1.70479	0.0554863
$945$	−29.5405	−0.960953
$946$	0	0
$947$	44.9310	1.46006	0.730031	−	0.683414i	$-0.239505\pi$
$947$	44.9310	1.46006	0.730031	+	0.683414i	$0.239505\pi$
$948$	−1.20338	−0.0390839
$949$	−39.2808	−1.27511
$950$	14.3582	0.465841
$951$	−1.34524	−0.0436224
$952$	−8.08242	−0.261953
$953$	60.2932	1.95309	0.976545	−	0.215314i	$-0.0690776\pi$
$953$	60.2932	1.95309	0.976545	+	0.215314i	$0.0690776\pi$
$954$	10.4711	0.339014
$955$	−7.45788	−0.241331
$956$	11.7025	0.378485
$957$	−1.84477	−0.0596330
$958$	−11.9328	−0.385532
$959$	−1.52566	−0.0492663
$960$	17.5892	0.567688
$961$	27.4200	0.884517
$962$	−35.9698	−1.15971
$963$	24.2329	0.780895
$964$	3.01760	0.0971902
$965$	56.3556	1.81415
$966$	−10.3942	−0.334429
$967$	42.4414	1.36482	0.682412	−	0.730967i	$-0.260931\pi$
$967$	42.4414	1.36482	0.682412	+	0.730967i	$0.260931\pi$
$968$	29.3511	0.943379
$969$	1.48747	0.0477845
$970$	−39.5915	−1.27121
$971$	−2.50735	−0.0804648	−0.0402324	−	0.999190i	$-0.512810\pi$
$971$	−2.50735	−0.0804648	−0.0402324	+	0.999190i	$0.512810\pi$
$972$	−11.2266	−0.360093
$973$	23.4210	0.750842
$974$	21.3226	0.683220
$975$	22.4774	0.719854
$976$	−5.08970	−0.162917
$977$	15.9228	0.509416	0.254708	−	0.967018i	$-0.418021\pi$
$977$	15.9228	0.509416	0.254708	+	0.967018i	$0.418021\pi$
$978$	1.35779	0.0434174
$979$	1.81962	0.0581552
$980$	2.40031	0.0766751
$981$	37.6818	1.20309
$982$	34.9003	1.11371
$983$	36.4572	1.16280	0.581402	−	0.813617i	$-0.302505\pi$
$983$	36.4572	1.16280	0.581402	+	0.813617i	$0.302505\pi$
$984$	−19.3185	−0.615852
$985$	−18.6664	−0.594760
$986$	−2.80291	−0.0892629
$987$	8.44074	0.268672
$988$	−9.74208	−0.309937
$989$	0	0
$990$	−11.3519	−0.360788
$991$	50.1585	1.59334	0.796668	−	0.604417i	$-0.206594\pi$
$991$	50.1585	1.59334	0.796668	+	0.604417i	$0.206594\pi$
$992$	31.6129	1.00371
$993$	−3.47349	−0.110228
$994$	41.2920	1.30970
$995$	61.0584	1.93568
$996$	3.61539	0.114558
$997$	−38.3514	−1.21460	−0.607301	−	0.794472i	$-0.707748\pi$
$997$	−38.3514	−1.21460	−0.607301	+	0.794472i	$0.707748\pi$
$998$	10.9329	0.346076
$999$	20.4143	0.645881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.o.1.12		18
43.30	odd	42		43.2.g.a.40.3	yes	36
43.33	odd	42		43.2.g.a.14.3	✓	36
43.42	odd	2		1849.2.a.n.1.7		18
129.116	even	42		387.2.y.c.298.1		36
129.119	even	42		387.2.y.c.100.1		36
172.119	even	42		688.2.bg.c.401.2		36
172.159	even	42		688.2.bg.c.513.2		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.14.3	✓	36	43.33	odd	42
43.2.g.a.40.3	yes	36	43.30	odd	42
387.2.y.c.100.1		36	129.119	even	42
387.2.y.c.298.1		36	129.116	even	42
688.2.bg.c.401.2		36	172.119	even	42
688.2.bg.c.513.2		36	172.159	even	42
1849.2.a.n.1.7		18	43.42	odd	2
1849.2.a.o.1.12		18	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+1.10272 q^{2} +0.645986 q^{3} -0.784019 q^{4} +3.32240 q^{5} +0.712338 q^{6} +2.46546 q^{7} -3.06998 q^{8} -2.58270 q^{9} +O(q^{10})\)	\(q+1.10272 q^{2} +0.645986 q^{3} -0.784019 q^{4} +3.32240 q^{5} +0.712338 q^{6} +2.46546 q^{7} -3.06998 q^{8} -2.58270 q^{9} +3.66366 q^{10} +1.19972 q^{11} -0.506465 q^{12} +5.76244 q^{13} +2.71871 q^{14} +2.14622 q^{15} -1.81728 q^{16} +1.06784 q^{17} -2.84799 q^{18} +2.15635 q^{19} -2.60482 q^{20} +1.59265 q^{21} +1.32295 q^{22} -5.91845 q^{23} -1.98316 q^{24} +6.03834 q^{25} +6.35433 q^{26} -3.60635 q^{27} -1.93297 q^{28} -2.38034 q^{29} +2.36667 q^{30} +7.64330 q^{31} +4.13602 q^{32} +0.775003 q^{33} +1.17753 q^{34} +8.19126 q^{35} +2.02489 q^{36} -5.66067 q^{37} +2.37784 q^{38} +3.72245 q^{39} -10.1997 q^{40} +9.74127 q^{41} +1.75624 q^{42} -0.940604 q^{44} -8.58077 q^{45} -6.52637 q^{46} +5.29979 q^{47} -1.17393 q^{48} -0.921486 q^{49} +6.65857 q^{50} +0.689810 q^{51} -4.51786 q^{52} -3.67667 q^{53} -3.97677 q^{54} +3.98595 q^{55} -7.56893 q^{56} +1.39297 q^{57} -2.62484 q^{58} -0.938103 q^{59} -1.68268 q^{60} +2.80073 q^{61} +8.42838 q^{62} -6.36756 q^{63} +8.19541 q^{64} +19.1451 q^{65} +0.854608 q^{66} -11.9885 q^{67} -0.837208 q^{68} -3.82323 q^{69} +9.03262 q^{70} +15.1881 q^{71} +7.92885 q^{72} -6.81669 q^{73} -6.24211 q^{74} +3.90068 q^{75} -1.69062 q^{76} +2.95787 q^{77} +4.10481 q^{78} +2.37603 q^{79} -6.03772 q^{80} +5.41846 q^{81} +10.7419 q^{82} -7.13848 q^{83} -1.24867 q^{84} +3.54780 q^{85} -1.53767 q^{87} -3.68312 q^{88} +1.51670 q^{89} -9.46214 q^{90} +14.2071 q^{91} +4.64018 q^{92} +4.93746 q^{93} +5.84416 q^{94} +7.16425 q^{95} +2.67181 q^{96} -10.8065 q^{97} -1.01614 q^{98} -3.09852 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9}+O(q^{10}) \) 18 * q + 5 * q^2 + 5 * q^3 + 19 * q^4 + 11 * q^5 + 4 * q^6 + 6 * q^7 + 12 * q^8 + 15 * q^9	\( 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9} - 7 q^{10} - 2 q^{11} + 16 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 9 q^{16} - 11 q^{17} + 25 q^{18} + 31 q^{19} + 25 q^{20} + 19 q^{22} - 11 q^{23} + 18 q^{24} + 9 q^{25} + 27 q^{26} + 23 q^{27} + 20 q^{28} + 37 q^{29} - 17 q^{30} - 12 q^{31} + 39 q^{32} + 38 q^{33} + 14 q^{34} + 16 q^{35} + 47 q^{36} - 19 q^{37} + 56 q^{38} + 46 q^{39} + 6 q^{40} - 7 q^{41} - q^{42} + 7 q^{44} + 23 q^{45} - 47 q^{46} - q^{47} + 15 q^{48} - 6 q^{49} - 3 q^{50} + 38 q^{51} + 15 q^{52} + 3 q^{53} - 67 q^{54} + 28 q^{55} - 81 q^{56} + 46 q^{57} + 34 q^{58} + 17 q^{59} - 83 q^{60} + 28 q^{61} + 33 q^{62} + 26 q^{63} + 10 q^{64} + 16 q^{65} - 72 q^{66} + 18 q^{67} + 53 q^{68} + 7 q^{69} - 34 q^{70} + 86 q^{71} - 2 q^{72} - 27 q^{73} - 79 q^{74} + 31 q^{75} + 59 q^{76} + 43 q^{77} + 91 q^{78} + 17 q^{79} + 8 q^{80} - 10 q^{81} + 13 q^{82} - 12 q^{83} - 32 q^{84} + 28 q^{85} - 43 q^{87} - 23 q^{88} + 51 q^{89} + 10 q^{90} - 20 q^{91} + 18 q^{92} - 30 q^{93} - 15 q^{94} - q^{95} - 20 q^{96} - 19 q^{97} - 5 q^{98} + 38 q^{99}+O(q^{100}) \) 18 * q + 5 * q^2 + 5 * q^3 + 19 * q^4 + 11 * q^5 + 4 * q^6 + 6 * q^7 + 12 * q^8 + 15 * q^9 - 7 * q^10 - 2 * q^11 + 16 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 9 * q^16 - 11 * q^17 + 25 * q^18 + 31 * q^19 + 25 * q^20 + 19 * q^22 - 11 * q^23 + 18 * q^24 + 9 * q^25 + 27 * q^26 + 23 * q^27 + 20 * q^28 + 37 * q^29 - 17 * q^30 - 12 * q^31 + 39 * q^32 + 38 * q^33 + 14 * q^34 + 16 * q^35 + 47 * q^36 - 19 * q^37 + 56 * q^38 + 46 * q^39 + 6 * q^40 - 7 * q^41 - q^42 + 7 * q^44 + 23 * q^45 - 47 * q^46 - q^47 + 15 * q^48 - 6 * q^49 - 3 * q^50 + 38 * q^51 + 15 * q^52 + 3 * q^53 - 67 * q^54 + 28 * q^55 - 81 * q^56 + 46 * q^57 + 34 * q^58 + 17 * q^59 - 83 * q^60 + 28 * q^61 + 33 * q^62 + 26 * q^63 + 10 * q^64 + 16 * q^65 - 72 * q^66 + 18 * q^67 + 53 * q^68 + 7 * q^69 - 34 * q^70 + 86 * q^71 - 2 * q^72 - 27 * q^73 - 79 * q^74 + 31 * q^75 + 59 * q^76 + 43 * q^77 + 91 * q^78 + 17 * q^79 + 8 * q^80 - 10 * q^81 + 13 * q^82 - 12 * q^83 - 32 * q^84 + 28 * q^85 - 43 * q^87 - 23 * q^88 + 51 * q^89 + 10 * q^90 - 20 * q^91 + 18 * q^92 - 30 * q^93 - 15 * q^94 - q^95 - 20 * q^96 - 19 * q^97 - 5 * q^98 + 38 * q^99

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