LMFDB - Embedded newform 1339.2.a.d.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, 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("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.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:	$10.6919688306$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 9 x^{18} + 13 x^{17} + 106 x^{16} - 351 x^{15} - 279 x^{14} + 2337 x^{13} - 1079 x^{12} + \cdots - 63 $ x^19 - 9x^18 + 13x^17 + 106x^16 - 351x^15 - 279x^14 + 2337x^13 - 1079x^12 - 6826x^11 + 7199x^10 + 9364x^9 - 14841x^8 - 5183x^7 + 14037x^6 - 108x^5 - 6099x^4 + 855x^3 + 1062x^2 - 135x - 63
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.49342$ of defining polynomial
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.49342 q^{2} +2.98467 q^{3} +0.230315 q^{4} -3.68204 q^{5} -4.45737 q^{6} -3.54305 q^{7} +2.64289 q^{8} +5.90824 q^{9} +O(q^{10})$	\(q-1.49342 q^{2} +2.98467 q^{3} +0.230315 q^{4} -3.68204 q^{5} -4.45737 q^{6} -3.54305 q^{7} +2.64289 q^{8} +5.90824 q^{9} +5.49884 q^{10} +3.04962 q^{11} +0.687414 q^{12} +1.00000 q^{13} +5.29128 q^{14} -10.9897 q^{15} -4.40759 q^{16} +7.25478 q^{17} -8.82351 q^{18} -6.13016 q^{19} -0.848029 q^{20} -10.5748 q^{21} -4.55437 q^{22} -7.92349 q^{23} +7.88815 q^{24} +8.55740 q^{25} -1.49342 q^{26} +8.68013 q^{27} -0.816019 q^{28} -4.68559 q^{29} +16.4122 q^{30} +2.66016 q^{31} +1.29661 q^{32} +9.10209 q^{33} -10.8345 q^{34} +13.0457 q^{35} +1.36076 q^{36} +1.40162 q^{37} +9.15493 q^{38} +2.98467 q^{39} -9.73122 q^{40} -7.10414 q^{41} +15.7927 q^{42} -9.16677 q^{43} +0.702373 q^{44} -21.7544 q^{45} +11.8331 q^{46} -0.586766 q^{47} -13.1552 q^{48} +5.55323 q^{49} -12.7798 q^{50} +21.6531 q^{51} +0.230315 q^{52} -12.9298 q^{53} -12.9631 q^{54} -11.2288 q^{55} -9.36390 q^{56} -18.2965 q^{57} +6.99757 q^{58} +0.603268 q^{59} -2.53108 q^{60} +0.0966424 q^{61} -3.97274 q^{62} -20.9332 q^{63} +6.87878 q^{64} -3.68204 q^{65} -13.5933 q^{66} +4.60119 q^{67} +1.67089 q^{68} -23.6490 q^{69} -19.4827 q^{70} -15.0879 q^{71} +15.6148 q^{72} -13.4439 q^{73} -2.09321 q^{74} +25.5410 q^{75} -1.41187 q^{76} -10.8050 q^{77} -4.45737 q^{78} +9.23913 q^{79} +16.2289 q^{80} +8.18259 q^{81} +10.6095 q^{82} -5.64747 q^{83} -2.43554 q^{84} -26.7124 q^{85} +13.6899 q^{86} -13.9849 q^{87} +8.05980 q^{88} -6.20259 q^{89} +32.4885 q^{90} -3.54305 q^{91} -1.82490 q^{92} +7.93968 q^{93} +0.876291 q^{94} +22.5715 q^{95} +3.86996 q^{96} -3.97895 q^{97} -8.29333 q^{98} +18.0179 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9}+O(q^{10}) $ 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9	\( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9} - q^{11} + 18 q^{12} + 19 q^{13} - 8 q^{14} - 11 q^{15} + 21 q^{16} - 16 q^{17} - 10 q^{19} - 20 q^{20} - 33 q^{21} + 2 q^{22} - 14 q^{23} - 9 q^{24} + 23 q^{25} - 9 q^{26} + q^{27} + 10 q^{28} - 22 q^{29} + 28 q^{30} - 3 q^{31} - 47 q^{32} - 25 q^{33} - 35 q^{34} - 3 q^{35} - 33 q^{36} - 19 q^{37} - 15 q^{38} - 2 q^{39} + 8 q^{40} - 52 q^{41} - 3 q^{42} - 2 q^{43} - 54 q^{44} - 40 q^{45} + 33 q^{46} - 24 q^{47} - 8 q^{48} + 7 q^{49} - 10 q^{50} - 7 q^{51} + 17 q^{52} - 11 q^{53} - 23 q^{54} - 29 q^{55} - 17 q^{56} - 34 q^{57} + 18 q^{58} - 52 q^{59} - 71 q^{60} - 25 q^{61} - 22 q^{62} - 19 q^{63} + 48 q^{64} - 18 q^{65} + 9 q^{66} - 2 q^{67} + 18 q^{68} - 26 q^{69} - 12 q^{70} - 44 q^{71} - 6 q^{72} - 39 q^{73} - 4 q^{74} + 17 q^{75} - 10 q^{76} - 18 q^{77} - 8 q^{78} + q^{79} - 9 q^{80} - 13 q^{81} + 47 q^{82} - 27 q^{83} - 24 q^{84} - 26 q^{85} - q^{86} + 31 q^{87} + 19 q^{88} - 86 q^{89} + 48 q^{90} - 8 q^{91} - 19 q^{92} + 32 q^{93} + 45 q^{94} + 17 q^{95} - 25 q^{96} - 20 q^{97} + 39 q^{98} + 50 q^{99}+O(q^{100}) \) 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9 - q^11 + 18 * q^12 + 19 * q^13 - 8 * q^14 - 11 * q^15 + 21 * q^16 - 16 * q^17 - 10 * q^19 - 20 * q^20 - 33 * q^21 + 2 * q^22 - 14 * q^23 - 9 * q^24 + 23 * q^25 - 9 * q^26 + q^27 + 10 * q^28 - 22 * q^29 + 28 * q^30 - 3 * q^31 - 47 * q^32 - 25 * q^33 - 35 * q^34 - 3 * q^35 - 33 * q^36 - 19 * q^37 - 15 * q^38 - 2 * q^39 + 8 * q^40 - 52 * q^41 - 3 * q^42 - 2 * q^43 - 54 * q^44 - 40 * q^45 + 33 * q^46 - 24 * q^47 - 8 * q^48 + 7 * q^49 - 10 * q^50 - 7 * q^51 + 17 * q^52 - 11 * q^53 - 23 * q^54 - 29 * q^55 - 17 * q^56 - 34 * q^57 + 18 * q^58 - 52 * q^59 - 71 * q^60 - 25 * q^61 - 22 * q^62 - 19 * q^63 + 48 * q^64 - 18 * q^65 + 9 * q^66 - 2 * q^67 + 18 * q^68 - 26 * q^69 - 12 * q^70 - 44 * q^71 - 6 * q^72 - 39 * q^73 - 4 * q^74 + 17 * q^75 - 10 * q^76 - 18 * q^77 - 8 * q^78 + q^79 - 9 * q^80 - 13 * q^81 + 47 * q^82 - 27 * q^83 - 24 * q^84 - 26 * q^85 - q^86 + 31 * q^87 + 19 * q^88 - 86 * q^89 + 48 * q^90 - 8 * q^91 - 19 * q^92 + 32 * q^93 + 45 * q^94 + 17 * q^95 - 25 * q^96 - 20 * q^97 + 39 * q^98 + 50 * 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 $

$2$	−1.49342	−1.05601	−0.528005	−	0.849241i	$-0.677060\pi$
$2$	−1.49342	−1.05601	−0.528005	+	0.849241i	$0.677060\pi$
$3$	2.98467	1.72320	0.861599	−	0.507589i	$-0.169463\pi$
$3$	2.98467	1.72320	0.861599	+	0.507589i	$0.169463\pi$
$4$	0.230315	0.115158
$5$	−3.68204	−1.64666	−0.823329	−	0.567565i	$-0.807886\pi$
$5$	−3.68204	−1.64666	−0.823329	+	0.567565i	$0.807886\pi$
$6$	−4.45737	−1.81972
$7$	−3.54305	−1.33915	−0.669574	−	0.742745i	$-0.733523\pi$
$7$	−3.54305	−1.33915	−0.669574	+	0.742745i	$0.733523\pi$
$8$	2.64289	0.934403
$9$	5.90824	1.96941
$10$	5.49884	1.73889
$11$	3.04962	0.919494	0.459747	−	0.888050i	$-0.347940\pi$
$11$	3.04962	0.919494	0.459747	+	0.888050i	$0.347940\pi$
$12$	0.687414	0.198439
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	5.29128	1.41415
$15$	−10.9897	−2.83752
$16$	−4.40759	−1.10190
$17$	7.25478	1.75954	0.879771	−	0.475397i	$-0.157696\pi$
$17$	7.25478	1.75954	0.879771	+	0.475397i	$0.157696\pi$
$18$	−8.82351	−2.07972
$19$	−6.13016	−1.40636	−0.703178	−	0.711014i	$-0.748236\pi$
$19$	−6.13016	−1.40636	−0.703178	+	0.711014i	$0.748236\pi$
$20$	−0.848029	−0.189625
$21$	−10.5748	−2.30762
$22$	−4.55437	−0.970995
$23$	−7.92349	−1.65216	−0.826081	−	0.563551i	$-0.809435\pi$
$23$	−7.92349	−1.65216	−0.826081	+	0.563551i	$0.809435\pi$
$24$	7.88815	1.61016
$25$	8.55740	1.71148
$26$	−1.49342	−0.292885
$27$	8.68013	1.67049
$28$	−0.816019	−0.154213
$29$	−4.68559	−0.870091	−0.435046	−	0.900408i	$-0.643268\pi$
$29$	−4.68559	−0.870091	−0.435046	+	0.900408i	$0.643268\pi$
$30$	16.4122	2.99645
$31$	2.66016	0.477778	0.238889	−	0.971047i	$-0.423217\pi$
$31$	2.66016	0.477778	0.238889	+	0.971047i	$0.423217\pi$
$32$	1.29661	0.229211
$33$	9.10209	1.58447
$34$	−10.8345	−1.85810
$35$	13.0457	2.20512
$36$	1.36076	0.226793
$37$	1.40162	0.230424	0.115212	−	0.993341i	$-0.463245\pi$
$37$	1.40162	0.230424	0.115212	+	0.993341i	$0.463245\pi$
$38$	9.15493	1.48513
$39$	2.98467	0.477929
$40$	−9.73122	−1.53864
$41$	−7.10414	−1.10948	−0.554740	−	0.832024i	$-0.687182\pi$
$41$	−7.10414	−1.10948	−0.554740	+	0.832024i	$0.687182\pi$
$42$	15.7927	2.43687
$43$	−9.16677	−1.39792	−0.698960	−	0.715161i	$-0.746353\pi$
$43$	−9.16677	−1.39792	−0.698960	+	0.715161i	$0.746353\pi$
$44$	0.702373	0.105887
$45$	−21.7544	−3.24295
$46$	11.8331	1.74470
$47$	−0.586766	−0.0855887	−0.0427943	−	0.999084i	$-0.513626\pi$
$47$	−0.586766	−0.0855887	−0.0427943	+	0.999084i	$0.513626\pi$
$48$	−13.1552	−1.89879
$49$	5.55323	0.793319
$50$	−12.7798	−1.80734
$51$	21.6531	3.03204
$52$	0.230315	0.0319390
$53$	−12.9298	−1.77604	−0.888020	−	0.459805i	$-0.847919\pi$
$53$	−12.9298	−1.77604	−0.888020	+	0.459805i	$0.847919\pi$
$54$	−12.9631	−1.76406
$55$	−11.2288	−1.51409
$56$	−9.36390	−1.25130
$57$	−18.2965	−2.42343
$58$	6.99757	0.918825
$59$	0.603268	0.0785389	0.0392694	−	0.999229i	$-0.487497\pi$
$59$	0.603268	0.0785389	0.0392694	+	0.999229i	$0.487497\pi$
$60$	−2.53108	−0.326762
$61$	0.0966424	0.0123738	0.00618689	−	0.999981i	$-0.498031\pi$
$61$	0.0966424	0.0123738	0.00618689	+	0.999981i	$0.498031\pi$
$62$	−3.97274	−0.504539
$63$	−20.9332	−2.63734
$64$	6.87878	0.859847
$65$	−3.68204	−0.456701
$66$	−13.5933	−1.67322
$67$	4.60119	0.562125	0.281062	−	0.959689i	$-0.409313\pi$
$67$	4.60119	0.562125	0.281062	+	0.959689i	$0.409313\pi$
$68$	1.67089	0.202625
$69$	−23.6490	−2.84700
$70$	−19.4827	−2.32863
$71$	−15.0879	−1.79060	−0.895299	−	0.445465i	$-0.853038\pi$
$71$	−15.0879	−1.79060	−0.895299	+	0.445465i	$0.853038\pi$
$72$	15.6148	1.84023
$73$	−13.4439	−1.57349	−0.786743	−	0.617281i	$-0.788234\pi$
$73$	−13.4439	−1.57349	−0.786743	+	0.617281i	$0.788234\pi$
$74$	−2.09321	−0.243331
$75$	25.5410	2.94922
$76$	−1.41187	−0.161952
$77$	−10.8050	−1.23134
$78$	−4.45737	−0.504698
$79$	9.23913	1.03948	0.519741	−	0.854324i	$-0.326028\pi$
$79$	9.23913	1.03948	0.519741	+	0.854324i	$0.326028\pi$
$80$	16.2289	1.81445
$81$	8.18259	0.909177
$82$	10.6095	1.17162
$83$	−5.64747	−0.619891	−0.309945	−	0.950754i	$-0.600311\pi$
$83$	−5.64747	−0.619891	−0.309945	+	0.950754i	$0.600311\pi$
$84$	−2.43554	−0.265740
$85$	−26.7124	−2.89736
$86$	13.6899	1.47622
$87$	−13.9849	−1.49934
$88$	8.05980	0.859178
$89$	−6.20259	−0.657473	−0.328736	−	0.944422i	$-0.606623\pi$
$89$	−6.20259	−0.657473	−0.328736	+	0.944422i	$0.606623\pi$
$90$	32.4885	3.42459
$91$	−3.54305	−0.371413
$92$	−1.82490	−0.190259
$93$	7.93968	0.823307
$94$	0.876291	0.0903825
$95$	22.5715	2.31579
$96$	3.86996	0.394976
$97$	−3.97895	−0.404001	−0.202001	−	0.979385i	$-0.564744\pi$
$97$	−3.97895	−0.404001	−0.202001	+	0.979385i	$0.564744\pi$
$98$	−8.29333	−0.837753
$99$	18.0179	1.81086
$100$	1.97090	0.197090
$101$	−2.95113	−0.293648	−0.146824	−	0.989163i	$-0.546905\pi$
$101$	−2.95113	−0.293648	−0.146824	+	0.989163i	$0.546905\pi$
$102$	−32.3373	−3.20187
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	2.64289	0.259157
$105$	38.9370	3.79986
$106$	19.3096	1.87552
$107$	15.1545	1.46504	0.732522	−	0.680744i	$-0.238343\pi$
$107$	15.1545	1.46504	0.732522	+	0.680744i	$0.238343\pi$
$108$	1.99917	0.192370
$109$	12.3339	1.18137	0.590685	−	0.806902i	$-0.298857\pi$
$109$	12.3339	1.18137	0.590685	+	0.806902i	$0.298857\pi$
$110$	16.7694	1.59890
$111$	4.18336	0.397067
$112$	15.6163	1.47560
$113$	−0.538137	−0.0506237	−0.0253118	−	0.999680i	$-0.508058\pi$
$113$	−0.538137	−0.0506237	−0.0253118	+	0.999680i	$0.508058\pi$
$114$	27.3244	2.55917
$115$	29.1746	2.72054
$116$	−1.07916	−0.100198
$117$	5.90824	0.546217
$118$	−0.900935	−0.0829378
$119$	−25.7041	−2.35629
$120$	−29.0445	−2.65138
$121$	−1.69984	−0.154531
$122$	−0.144328	−0.0130668
$123$	−21.2035	−1.91185
$124$	0.612674	0.0550197
$125$	−13.0985	−1.17156
$126$	31.2622	2.78506
$127$	12.8114	1.13683	0.568414	−	0.822743i	$-0.307557\pi$
$127$	12.8114	1.13683	0.568414	+	0.822743i	$0.307557\pi$
$128$	−12.8662	−1.13722
$129$	−27.3598	−2.40889
$130$	5.49884	0.482280
$131$	−15.8473	−1.38458	−0.692292	−	0.721617i	$-0.743399\pi$
$131$	−15.8473	−1.38458	−0.692292	+	0.721617i	$0.743399\pi$
$132$	2.09635	0.182464
$133$	21.7195	1.88332
$134$	−6.87153	−0.593610
$135$	−31.9606	−2.75073
$136$	19.1736	1.64412
$137$	−0.692462	−0.0591610	−0.0295805	−	0.999562i	$-0.509417\pi$
$137$	−0.692462	−0.0591610	−0.0295805	+	0.999562i	$0.509417\pi$
$138$	35.3180	3.00646
$139$	−3.18888	−0.270477	−0.135239	−	0.990813i	$-0.543180\pi$
$139$	−3.18888	−0.270477	−0.135239	+	0.990813i	$0.543180\pi$
$140$	3.00461	0.253936
$141$	−1.75130	−0.147486
$142$	22.5326	1.89089
$143$	3.04962	0.255022
$144$	−26.0411	−2.17009
$145$	17.2525	1.43274
$146$	20.0774	1.66162
$147$	16.5746	1.36705
$148$	0.322814	0.0265351
$149$	−12.3706	−1.01344	−0.506720	−	0.862111i	$-0.669142\pi$
$149$	−12.3706	−1.01344	−0.506720	+	0.862111i	$0.669142\pi$
$150$	−38.1435	−3.11441
$151$	12.4947	1.01680	0.508402	−	0.861120i	$-0.330236\pi$
$151$	12.4947	1.01680	0.508402	+	0.861120i	$0.330236\pi$
$152$	−16.2013	−1.31410
$153$	42.8630	3.46527
$154$	16.1364	1.30031
$155$	−9.79480	−0.786737
$156$	0.687414	0.0550372
$157$	−15.7048	−1.25338	−0.626689	−	0.779269i	$-0.715590\pi$
$157$	−15.7048	−1.25338	−0.626689	+	0.779269i	$0.715590\pi$
$158$	−13.7979	−1.09770
$159$	−38.5911	−3.06047
$160$	−4.77418	−0.377432
$161$	28.0734	2.21249
$162$	−12.2201	−0.960100
$163$	−4.66841	−0.365658	−0.182829	−	0.983145i	$-0.558526\pi$
$163$	−4.66841	−0.365658	−0.182829	+	0.983145i	$0.558526\pi$
$164$	−1.63619	−0.127765
$165$	−33.5142	−2.60908
$166$	8.43407	0.654611
$167$	9.61913	0.744350	0.372175	−	0.928162i	$-0.378612\pi$
$167$	9.61913	0.744350	0.372175	+	0.928162i	$0.378612\pi$
$168$	−27.9481	−2.15625
$169$	1.00000	0.0769231
$170$	39.8929	3.05965
$171$	−36.2185	−2.76970
$172$	−2.11124	−0.160981
$173$	−16.5827	−1.26076	−0.630380	−	0.776287i	$-0.717101\pi$
$173$	−16.5827	−1.26076	−0.630380	+	0.776287i	$0.717101\pi$
$174$	20.8854	1.58332
$175$	−30.3193	−2.29193
$176$	−13.4414	−1.01319
$177$	1.80056	0.135338
$178$	9.26309	0.694298
$179$	−1.53092	−0.114426	−0.0572132	−	0.998362i	$-0.518221\pi$
$179$	−1.53092	−0.114426	−0.0572132	+	0.998362i	$0.518221\pi$
$180$	−5.01036	−0.373450
$181$	14.3465	1.06636	0.533182	−	0.846001i	$-0.320996\pi$
$181$	14.3465	1.06636	0.533182	+	0.846001i	$0.320996\pi$
$182$	5.29128	0.392216
$183$	0.288445	0.0213225
$184$	−20.9409	−1.54378
$185$	−5.16081	−0.379430
$186$	−11.8573	−0.869420
$187$	22.1243	1.61789
$188$	−0.135141	−0.00985618
$189$	−30.7542	−2.23704
$190$	−33.7088	−2.44549
$191$	6.85965	0.496347	0.248174	−	0.968716i	$-0.420170\pi$
$191$	6.85965	0.496347	0.248174	+	0.968716i	$0.420170\pi$
$192$	20.5309	1.48169
$193$	4.92128	0.354242	0.177121	−	0.984189i	$-0.443322\pi$
$193$	4.92128	0.354242	0.177121	+	0.984189i	$0.443322\pi$
$194$	5.94226	0.426630
$195$	−10.9897	−0.786986
$196$	1.27899	0.0913566
$197$	23.7216	1.69010	0.845048	−	0.534691i	$-0.179572\pi$
$197$	23.7216	1.69010	0.845048	+	0.534691i	$0.179572\pi$
$198$	−26.9083	−1.91229
$199$	8.19645	0.581031	0.290515	−	0.956870i	$-0.406173\pi$
$199$	8.19645	0.581031	0.290515	+	0.956870i	$0.406173\pi$
$200$	22.6163	1.59921
$201$	13.7330	0.968653
$202$	4.40729	0.310096
$203$	16.6013	1.16518
$204$	4.98704	0.349162
$205$	26.1577	1.82693
$206$	−1.49342	−0.104052
$207$	−46.8139	−3.25379
$208$	−4.40759	−0.305611
$209$	−18.6946	−1.29314
$210$	−58.1494	−4.01269
$211$	−9.75498	−0.671560	−0.335780	−	0.941940i	$-0.609000\pi$
$211$	−9.75498	−0.671560	−0.335780	+	0.941940i	$0.609000\pi$
$212$	−2.97792	−0.204524
$213$	−45.0322	−3.08556
$214$	−22.6321	−1.54710
$215$	33.7524	2.30189
$216$	22.9406	1.56091
$217$	−9.42508	−0.639816
$218$	−18.4197	−1.24754
$219$	−40.1255	−2.71143
$220$	−2.58616	−0.174359
$221$	7.25478	0.488009
$222$	−6.24753	−0.419307
$223$	24.3762	1.63235	0.816177	−	0.577802i	$-0.196089\pi$
$223$	24.3762	1.63235	0.816177	+	0.577802i	$0.196089\pi$
$224$	−4.59397	−0.306948
$225$	50.5592	3.37061
$226$	0.803667	0.0534591
$227$	7.31900	0.485779	0.242890	−	0.970054i	$-0.421905\pi$
$227$	7.31900	0.485779	0.242890	+	0.970054i	$0.421905\pi$
$228$	−4.21396	−0.279076
$229$	−6.27594	−0.414726	−0.207363	−	0.978264i	$-0.566488\pi$
$229$	−6.27594	−0.414726	−0.207363	+	0.978264i	$0.566488\pi$
$230$	−43.5700	−2.87292
$231$	−32.2492	−2.12184
$232$	−12.3835	−0.813016
$233$	−15.6430	−1.02481	−0.512405	−	0.858744i	$-0.671245\pi$
$233$	−15.6430	−1.02481	−0.512405	+	0.858744i	$0.671245\pi$
$234$	−8.82351	−0.576811
$235$	2.16050	0.140935
$236$	0.138942	0.00904434
$237$	27.5757	1.79124
$238$	38.3871	2.48827
$239$	−18.3583	−1.18750	−0.593750	−	0.804649i	$-0.702353\pi$
$239$	−18.3583	−1.18750	−0.593750	+	0.804649i	$0.702353\pi$
$240$	48.4379	3.12665
$241$	12.7815	0.823329	0.411664	−	0.911336i	$-0.364948\pi$
$241$	12.7815	0.823329	0.411664	+	0.911336i	$0.364948\pi$
$242$	2.53858	0.163186
$243$	−1.61809	−0.103800
$244$	0.0222582	0.00142493
$245$	−20.4472	−1.30632
$246$	31.6658	2.01894
$247$	−6.13016	−0.390053
$248$	7.03050	0.446437
$249$	−16.8558	−1.06819
$250$	19.5616	1.23718
$251$	8.35465	0.527341	0.263670	−	0.964613i	$-0.415067\pi$
$251$	8.35465	0.527341	0.263670	+	0.964613i	$0.415067\pi$
$252$	−4.82124	−0.303709
$253$	−24.1636	−1.51915
$254$	−19.1328	−1.20050
$255$	−79.7276	−4.99273
$256$	5.45707	0.341067
$257$	−18.6157	−1.16122	−0.580609	−	0.814183i	$-0.697186\pi$
$257$	−18.6157	−1.16122	−0.580609	+	0.814183i	$0.697186\pi$
$258$	40.8597	2.54381
$259$	−4.96601	−0.308573
$260$	−0.848029	−0.0525925
$261$	−27.6836	−1.71357
$262$	23.6667	1.46214
$263$	15.2149	0.938189	0.469095	−	0.883148i	$-0.344580\pi$
$263$	15.2149	0.938189	0.469095	+	0.883148i	$0.344580\pi$
$264$	24.0558	1.48053
$265$	47.6079	2.92453
$266$	−32.4364	−1.98880
$267$	−18.5127	−1.13296
$268$	1.05972	0.0647329
$269$	−18.3385	−1.11812	−0.559059	−	0.829128i	$-0.688838\pi$
$269$	−18.3385	−1.11812	−0.559059	+	0.829128i	$0.688838\pi$
$270$	47.7307	2.90480
$271$	−3.25773	−0.197893	−0.0989465	−	0.995093i	$-0.531547\pi$
$271$	−3.25773	−0.197893	−0.0989465	+	0.995093i	$0.531547\pi$
$272$	−31.9761	−1.93883
$273$	−10.5748	−0.640018
$274$	1.03414	0.0624746
$275$	26.0968	1.57370
$276$	−5.44672	−0.327854
$277$	−22.0075	−1.32230	−0.661152	−	0.750252i	$-0.729932\pi$
$277$	−22.0075	−1.32230	−0.661152	+	0.750252i	$0.729932\pi$
$278$	4.76235	0.285627
$279$	15.7168	0.940943
$280$	34.4782	2.06047
$281$	3.37047	0.201065	0.100533	−	0.994934i	$-0.467945\pi$
$281$	3.37047	0.201065	0.100533	+	0.994934i	$0.467945\pi$
$282$	2.61544	0.155747
$283$	−10.8182	−0.643078	−0.321539	−	0.946896i	$-0.604200\pi$
$283$	−10.8182	−0.643078	−0.321539	+	0.946896i	$0.604200\pi$
$284$	−3.47496	−0.206201
$285$	67.3684	3.99056
$286$	−4.55437	−0.269306
$287$	25.1703	1.48576
$288$	7.66070	0.451411
$289$	35.6318	2.09599
$290$	−25.7653	−1.51299
$291$	−11.8759	−0.696175
$292$	−3.09633	−0.181199
$293$	27.0100	1.57794	0.788971	−	0.614430i	$-0.210614\pi$
$293$	27.0100	1.57794	0.788971	+	0.614430i	$0.210614\pi$
$294$	−24.7528	−1.44361
$295$	−2.22126	−0.129327
$296$	3.70432	0.215309
$297$	26.4711	1.53601
$298$	18.4746	1.07020
$299$	−7.92349	−0.458227
$300$	5.88248	0.339625
$301$	32.4784	1.87202
$302$	−18.6599	−1.07376
$303$	−8.80814	−0.506014
$304$	27.0192	1.54966
$305$	−0.355841	−0.0203754
$306$	−64.0126	−3.65936
$307$	0.368219	0.0210154	0.0105077	−	0.999945i	$-0.496655\pi$
$307$	0.368219	0.0210154	0.0105077	+	0.999945i	$0.496655\pi$
$308$	−2.48854	−0.141798
$309$	2.98467	0.169792
$310$	14.6278	0.830802
$311$	11.9705	0.678786	0.339393	−	0.940645i	$-0.389778\pi$
$311$	11.9705	0.678786	0.339393	+	0.940645i	$0.389778\pi$
$312$	7.88815	0.446578
$313$	25.4029	1.43585	0.717927	−	0.696118i	$-0.245091\pi$
$313$	25.4029	1.43585	0.717927	+	0.696118i	$0.245091\pi$
$314$	23.4539	1.32358
$315$	77.0769	4.34279
$316$	2.12791	0.119704
$317$	−6.76219	−0.379802	−0.189901	−	0.981803i	$-0.560817\pi$
$317$	−6.76219	−0.379802	−0.189901	+	0.981803i	$0.560817\pi$
$318$	57.6328	3.23189
$319$	−14.2892	−0.800044
$320$	−25.3279	−1.41587
$321$	45.2312	2.52456
$322$	−41.9254	−2.33641
$323$	−44.4730	−2.47454
$324$	1.88457	0.104699
$325$	8.55740	0.474679
$326$	6.97192	0.386139
$327$	36.8125	2.03574
$328$	−18.7755	−1.03670
$329$	2.07894	0.114616
$330$	50.0510	2.75522
$331$	−22.5399	−1.23891	−0.619453	−	0.785034i	$-0.712646\pi$
$331$	−22.5399	−1.23891	−0.619453	+	0.785034i	$0.712646\pi$
$332$	−1.30070	−0.0713851
$333$	8.28109	0.453801
$334$	−14.3654	−0.786042
$335$	−16.9418	−0.925627
$336$	46.6095	2.54276
$337$	−18.6231	−1.01447	−0.507233	−	0.861809i	$-0.669332\pi$
$337$	−18.6231	−1.01447	−0.507233	+	0.861809i	$0.669332\pi$
$338$	−1.49342	−0.0812316
$339$	−1.60616	−0.0872347
$340$	−6.15226	−0.333653
$341$	8.11246	0.439314
$342$	54.0896	2.92483
$343$	5.12597	0.276777
$344$	−24.2268	−1.30622
$345$	87.0765	4.68804
$346$	24.7650	1.33138
$347$	−2.15260	−0.115558	−0.0577789	−	0.998329i	$-0.518402\pi$
$347$	−2.15260	−0.115558	−0.0577789	+	0.998329i	$0.518402\pi$
$348$	−3.22094	−0.172660
$349$	5.51337	0.295124	0.147562	−	0.989053i	$-0.452857\pi$
$349$	5.51337	0.295124	0.147562	+	0.989053i	$0.452857\pi$
$350$	45.2796	2.42030
$351$	8.68013	0.463311
$352$	3.95417	0.210758
$353$	−12.2580	−0.652427	−0.326213	−	0.945296i	$-0.605773\pi$
$353$	−12.2580	−0.652427	−0.326213	+	0.945296i	$0.605773\pi$
$354$	−2.68899	−0.142918
$355$	55.5541	2.94850
$356$	−1.42855	−0.0757129
$357$	−76.7181	−4.06035
$358$	2.28631	0.120835
$359$	20.0718	1.05935	0.529676	−	0.848200i	$-0.322314\pi$
$359$	20.0718	1.05935	0.529676	+	0.848200i	$0.322314\pi$
$360$	−57.4944	−3.03022
$361$	18.5789	0.977837
$362$	−21.4253	−1.12609
$363$	−5.07345	−0.266287
$364$	−0.816019	−0.0427710
$365$	49.5008	2.59099
$366$	−0.430771	−0.0225168
$367$	−32.3666	−1.68952	−0.844761	−	0.535144i	$-0.820257\pi$
$367$	−32.3666	−1.68952	−0.844761	+	0.535144i	$0.820257\pi$
$368$	34.9235	1.82051
$369$	−41.9730	−2.18503
$370$	7.70728	0.400682
$371$	45.8109	2.37838
$372$	1.82863	0.0948100
$373$	−16.4382	−0.851136	−0.425568	−	0.904927i	$-0.639926\pi$
$373$	−16.4382	−0.851136	−0.425568	+	0.904927i	$0.639926\pi$
$374$	−33.0410	−1.70851
$375$	−39.0946	−2.01884
$376$	−1.55076	−0.0799743
$377$	−4.68559	−0.241320
$378$	45.9290	2.36233
$379$	2.96221	0.152159	0.0760793	−	0.997102i	$-0.475760\pi$
$379$	2.96221	0.152159	0.0760793	+	0.997102i	$0.475760\pi$
$380$	5.19855	0.266680
$381$	38.2378	1.95898
$382$	−10.2444	−0.524148
$383$	23.1553	1.18318	0.591590	−	0.806239i	$-0.298501\pi$
$383$	23.1553	1.18318	0.591590	+	0.806239i	$0.298501\pi$
$384$	−38.4012	−1.95965
$385$	39.7843	2.02759
$386$	−7.34956	−0.374083
$387$	−54.1595	−2.75308
$388$	−0.916413	−0.0465238
$389$	−7.71004	−0.390914	−0.195457	−	0.980712i	$-0.562619\pi$
$389$	−7.71004	−0.390914	−0.195457	+	0.980712i	$0.562619\pi$
$390$	16.4122	0.831065
$391$	−57.4832	−2.90705
$392$	14.6766	0.741279
$393$	−47.2989	−2.38591
$394$	−35.4264	−1.78476
$395$	−34.0188	−1.71167
$396$	4.14979	0.208535
$397$	21.1010	1.05903	0.529515	−	0.848300i	$-0.322374\pi$
$397$	21.1010	1.05903	0.529515	+	0.848300i	$0.322374\pi$
$398$	−12.2408	−0.613574
$399$	64.8255	3.24533
$400$	−37.7175	−1.88587
$401$	−29.3406	−1.46520	−0.732600	−	0.680660i	$-0.761693\pi$
$401$	−29.3406	−1.46520	−0.732600	+	0.680660i	$0.761693\pi$
$402$	−20.5092	−1.02291
$403$	2.66016	0.132512
$404$	−0.679689	−0.0338158
$405$	−30.1286	−1.49710
$406$	−24.7928	−1.23044
$407$	4.27440	0.211874
$408$	57.2268	2.83315
$409$	−5.04116	−0.249269	−0.124635	−	0.992203i	$-0.539776\pi$
$409$	−5.04116	−0.249269	−0.124635	+	0.992203i	$0.539776\pi$
$410$	−39.0645	−1.92926
$411$	−2.06677	−0.101946
$412$	0.230315	0.0113468
$413$	−2.13741	−0.105175
$414$	69.9130	3.43604
$415$	20.7942	1.02075
$416$	1.29661	0.0635717
$417$	−9.51775	−0.466086
$418$	27.9190	1.36556
$419$	−5.98141	−0.292211	−0.146105	−	0.989269i	$-0.546674\pi$
$419$	−5.98141	−0.292211	−0.146105	+	0.989269i	$0.546674\pi$
$420$	8.96777	0.437582
$421$	34.3087	1.67210	0.836051	−	0.548651i	$-0.184859\pi$
$421$	34.3087	1.67210	0.836051	+	0.548651i	$0.184859\pi$
$422$	14.5683	0.709174
$423$	−3.46676	−0.168559
$424$	−34.1720	−1.65954
$425$	62.0821	3.01142
$426$	67.2522	3.25838
$427$	−0.342409	−0.0165703
$428$	3.49032	0.168711
$429$	9.10209	0.439453
$430$	−50.4066	−2.43082
$431$	32.5757	1.56911	0.784557	−	0.620056i	$-0.212890\pi$
$431$	32.5757	1.56911	0.784557	+	0.620056i	$0.212890\pi$
$432$	−38.2584	−1.84071
$433$	11.3434	0.545127	0.272563	−	0.962138i	$-0.412129\pi$
$433$	11.3434	0.545127	0.272563	+	0.962138i	$0.412129\pi$
$434$	14.0756	0.675652
$435$	51.4930	2.46890
$436$	2.84068	0.136044
$437$	48.5723	2.32353
$438$	59.9244	2.86330
$439$	−5.40863	−0.258140	−0.129070	−	0.991635i	$-0.541199\pi$
$439$	−5.40863	−0.258140	−0.129070	+	0.991635i	$0.541199\pi$
$440$	−29.6765	−1.41477
$441$	32.8098	1.56237
$442$	−10.8345	−0.515343
$443$	5.04385	0.239641	0.119820	−	0.992796i	$-0.461768\pi$
$443$	5.04385	0.239641	0.119820	+	0.992796i	$0.461768\pi$
$444$	0.963491	0.0457253
$445$	22.8382	1.08263
$446$	−36.4041	−1.72378
$447$	−36.9222	−1.74636
$448$	−24.3719	−1.15146
$449$	−12.8418	−0.606044	−0.303022	−	0.952984i	$-0.597996\pi$
$449$	−12.8418	−0.606044	−0.303022	+	0.952984i	$0.597996\pi$
$450$	−75.5063	−3.55940
$451$	−21.6649	−1.02016
$452$	−0.123941	−0.00582970
$453$	37.2925	1.75216
$454$	−10.9304	−0.512988
$455$	13.0457	0.611590
$456$	−48.3556	−2.26446
$457$	−18.6026	−0.870194	−0.435097	−	0.900384i	$-0.643286\pi$
$457$	−18.6026	−0.870194	−0.435097	+	0.900384i	$0.643286\pi$
$458$	9.37264	0.437955
$459$	62.9725	2.93930
$460$	6.71935	0.313291
$461$	−39.8116	−1.85421	−0.927105	−	0.374803i	$-0.877710\pi$
$461$	−39.8116	−1.85421	−0.927105	+	0.374803i	$0.877710\pi$
$462$	48.1617	2.24069
$463$	−25.9383	−1.20545	−0.602727	−	0.797948i	$-0.705919\pi$
$463$	−25.9383	−1.20545	−0.602727	+	0.797948i	$0.705919\pi$
$464$	20.6521	0.958750
$465$	−29.2342	−1.35570
$466$	23.3617	1.08221
$467$	3.52204	0.162981	0.0814903	−	0.996674i	$-0.474032\pi$
$467$	3.52204	0.162981	0.0814903	+	0.996674i	$0.474032\pi$
$468$	1.36076	0.0629010
$469$	−16.3023	−0.752769
$470$	−3.22654	−0.148829
$471$	−46.8736	−2.15982
$472$	1.59437	0.0733869
$473$	−27.9551	−1.28538
$474$	−41.1822	−1.89156
$475$	−52.4583	−2.40695
$476$	−5.92004	−0.271344
$477$	−76.3922	−3.49776
$478$	27.4167	1.25401
$479$	25.1259	1.14803	0.574016	−	0.818844i	$-0.305385\pi$
$479$	25.1259	1.14803	0.574016	+	0.818844i	$0.305385\pi$
$480$	−14.2493	−0.650390
$481$	1.40162	0.0639083
$482$	−19.0882	−0.869443
$483$	83.7896	3.81256
$484$	−0.391498	−0.0177954
$485$	14.6507	0.665252
$486$	2.41649	0.109614
$487$	−11.1887	−0.507009	−0.253505	−	0.967334i	$-0.581583\pi$
$487$	−11.1887	−0.507009	−0.253505	+	0.967334i	$0.581583\pi$
$488$	0.255415	0.0115621
$489$	−13.9337	−0.630102
$490$	30.5364	1.37949
$491$	−3.08011	−0.139003	−0.0695017	−	0.997582i	$-0.522141\pi$
$491$	−3.08011	−0.139003	−0.0695017	+	0.997582i	$0.522141\pi$
$492$	−4.88348	−0.220164
$493$	−33.9929	−1.53096
$494$	9.15493	0.411900
$495$	−66.3425	−2.98187
$496$	−11.7249	−0.526462
$497$	53.4571	2.39788
$498$	25.1729	1.12802
$499$	−21.0221	−0.941078	−0.470539	−	0.882379i	$-0.655941\pi$
$499$	−21.0221	−0.941078	−0.470539	+	0.882379i	$0.655941\pi$
$500$	−3.01678	−0.134914
$501$	28.7099	1.28266
$502$	−12.4770	−0.556877
$503$	−4.56993	−0.203763	−0.101882	−	0.994797i	$-0.532486\pi$
$503$	−4.56993	−0.203763	−0.101882	+	0.994797i	$0.532486\pi$
$504$	−55.3242	−2.46434
$505$	10.8662	0.483538
$506$	36.0865	1.60424
$507$	2.98467	0.132554
$508$	2.95066	0.130914
$509$	−38.6360	−1.71251	−0.856254	−	0.516554i	$-0.827214\pi$
$509$	−38.6360	−1.71251	−0.856254	+	0.516554i	$0.827214\pi$
$510$	119.067	5.27238
$511$	47.6324	2.10713
$512$	17.5826	0.777048
$513$	−53.2106	−2.34931
$514$	27.8012	1.22626
$515$	−3.68204	−0.162250
$516$	−6.30136	−0.277402
$517$	−1.78941	−0.0786983
$518$	7.41635	0.325856
$519$	−49.4939	−2.17254
$520$	−9.73122	−0.426742
$521$	2.26323	0.0991540	0.0495770	−	0.998770i	$-0.484213\pi$
$521$	2.26323	0.0991540	0.0495770	+	0.998770i	$0.484213\pi$
$522$	41.3433	1.80955
$523$	38.7544	1.69461	0.847306	−	0.531105i	$-0.178223\pi$
$523$	38.7544	1.69461	0.847306	+	0.531105i	$0.178223\pi$
$524$	−3.64987	−0.159445
$525$	−90.4932	−3.94945
$526$	−22.7223	−0.990737
$527$	19.2988	0.840671
$528$	−40.1182	−1.74592
$529$	39.7817	1.72964
$530$	−71.0988	−3.08833
$531$	3.56426	0.154675
$532$	5.00233	0.216878
$533$	−7.10414	−0.307714
$534$	27.6472	1.19641
$535$	−55.7995	−2.41242
$536$	12.1604	0.525251
$537$	−4.56929	−0.197179
$538$	27.3872	1.18074
$539$	16.9352	0.729452
$540$	−7.36100	−0.316767
$541$	6.77155	0.291132	0.145566	−	0.989349i	$-0.453500\pi$
$541$	6.77155	0.291132	0.145566	+	0.989349i	$0.453500\pi$
$542$	4.86517	0.208977
$543$	42.8194	1.83756
$544$	9.40665	0.403307
$545$	−45.4138	−1.94531
$546$	15.7927	0.675866
$547$	21.1423	0.903981	0.451990	−	0.892023i	$-0.350714\pi$
$547$	21.1423	0.903981	0.451990	+	0.892023i	$0.350714\pi$
$548$	−0.159484	−0.00681284
$549$	0.570986	0.0243691
$550$	−38.9736	−1.66184
$551$	28.7234	1.22366
$552$	−62.5017	−2.66025
$553$	−32.7347	−1.39202
$554$	32.8666	1.39637
$555$	−15.4033	−0.653834
$556$	−0.734447	−0.0311475
$557$	16.4611	0.697478	0.348739	−	0.937220i	$-0.386610\pi$
$557$	16.4611	0.697478	0.348739	+	0.937220i	$0.386610\pi$
$558$	−23.4719	−0.993645
$559$	−9.16677	−0.387713
$560$	−57.4999	−2.42981
$561$	66.0337	2.78794
$562$	−5.03354	−0.212327
$563$	−0.988304	−0.0416521	−0.0208260	−	0.999783i	$-0.506630\pi$
$563$	−0.988304	−0.0416521	−0.0208260	+	0.999783i	$0.506630\pi$
$564$	−0.403351	−0.0169842
$565$	1.98144	0.0833599
$566$	16.1562	0.679097
$567$	−28.9914	−1.21752
$568$	−39.8755	−1.67314
$569$	14.1465	0.593053	0.296526	−	0.955025i	$-0.404172\pi$
$569$	14.1465	0.593053	0.296526	+	0.955025i	$0.404172\pi$
$570$	−100.610	−4.21407
$571$	−35.7408	−1.49570	−0.747852	−	0.663865i	$-0.768915\pi$
$571$	−35.7408	−1.49570	−0.747852	+	0.663865i	$0.768915\pi$
$572$	0.702373	0.0293677
$573$	20.4738	0.855305
$574$	−37.5900	−1.56898
$575$	−67.8045	−2.82764
$576$	40.6415	1.69339
$577$	2.07318	0.0863076	0.0431538	−	0.999068i	$-0.486259\pi$
$577$	2.07318	0.0863076	0.0431538	+	0.999068i	$0.486259\pi$
$578$	−53.2135	−2.21339
$579$	14.6884	0.610429
$580$	3.97351	0.164991
$581$	20.0093	0.830126
$582$	17.7357	0.735168
$583$	−39.4308	−1.63306
$584$	−35.5307	−1.47027
$585$	−21.7544	−0.899432
$586$	−40.3374	−1.66632
$587$	−0.649172	−0.0267942	−0.0133971	−	0.999910i	$-0.504265\pi$
$587$	−0.649172	−0.0267942	−0.0133971	+	0.999910i	$0.504265\pi$
$588$	3.81737	0.157426
$589$	−16.3072	−0.671926
$590$	3.31728	0.136570
$591$	70.8012	2.91237
$592$	−6.17775	−0.253904
$593$	−20.2382	−0.831085	−0.415543	−	0.909574i	$-0.636408\pi$
$593$	−20.2382	−0.831085	−0.415543	+	0.909574i	$0.636408\pi$
$594$	−39.5325	−1.62204
$595$	94.6434	3.88000
$596$	−2.84914	−0.116705
$597$	24.4637	1.00123
$598$	11.8331	0.483893
$599$	−3.28275	−0.134129	−0.0670647	−	0.997749i	$-0.521363\pi$
$599$	−3.28275	−0.134129	−0.0670647	+	0.997749i	$0.521363\pi$
$600$	67.5021	2.75576
$601$	4.70993	0.192122	0.0960610	−	0.995375i	$-0.469376\pi$
$601$	4.70993	0.192122	0.0960610	+	0.995375i	$0.469376\pi$
$602$	−48.5040	−1.97687
$603$	27.1849	1.10706
$604$	2.87772	0.117093
$605$	6.25887	0.254459
$606$	13.1543	0.534356
$607$	27.8324	1.12968	0.564841	−	0.825200i	$-0.308938\pi$
$607$	27.8324	1.12968	0.564841	+	0.825200i	$0.308938\pi$
$608$	−7.94845	−0.322352
$609$	49.5493	2.00784
$610$	0.531421	0.0215166
$611$	−0.586766	−0.0237380
$612$	9.87199	0.399052
$613$	−21.8494	−0.882489	−0.441245	−	0.897387i	$-0.645463\pi$
$613$	−21.8494	−0.882489	−0.441245	+	0.897387i	$0.645463\pi$
$614$	−0.549907	−0.0221924
$615$	78.0721	3.14817
$616$	−28.5563	−1.15057
$617$	−9.36189	−0.376896	−0.188448	−	0.982083i	$-0.560346\pi$
$617$	−9.36189	−0.376896	−0.188448	+	0.982083i	$0.560346\pi$
$618$	−4.45737	−0.179302
$619$	15.0541	0.605077	0.302538	−	0.953137i	$-0.402166\pi$
$619$	15.0541	0.605077	0.302538	+	0.953137i	$0.402166\pi$
$620$	−2.25589	−0.0905987
$621$	−68.7770	−2.75992
$622$	−17.8771	−0.716805
$623$	21.9761	0.880454
$624$	−13.1552	−0.526629
$625$	5.44213	0.217685
$626$	−37.9372	−1.51628
$627$	−55.7973	−2.22833
$628$	−3.61705	−0.144336
$629$	10.1684	0.405442
$630$	−115.108	−4.58603
$631$	−16.8655	−0.671405	−0.335702	−	0.941968i	$-0.608974\pi$
$631$	−16.8655	−0.671405	−0.335702	+	0.941968i	$0.608974\pi$
$632$	24.4180	0.971296
$633$	−29.1154	−1.15723
$634$	10.0988	0.401075
$635$	−47.1720	−1.87197
$636$	−8.88810	−0.352436
$637$	5.55323	0.220027
$638$	21.3399	0.844854
$639$	−89.1427	−3.52643
$640$	47.3737	1.87261
$641$	19.2739	0.761273	0.380637	−	0.924725i	$-0.375705\pi$
$641$	19.2739	0.761273	0.380637	+	0.924725i	$0.375705\pi$
$642$	−67.5494	−2.66596
$643$	18.8646	0.743949	0.371975	−	0.928243i	$-0.378681\pi$
$643$	18.8646	0.743949	0.371975	+	0.928243i	$0.378681\pi$
$644$	6.46572	0.254785
$645$	100.740	3.96662
$646$	66.4170	2.61314
$647$	5.12413	0.201450	0.100725	−	0.994914i	$-0.467884\pi$
$647$	5.12413	0.201450	0.100725	+	0.994914i	$0.467884\pi$
$648$	21.6257	0.849537
$649$	1.83974	0.0722160
$650$	−12.7798	−0.501266
$651$	−28.1307	−1.10253
$652$	−1.07521	−0.0421083
$653$	17.6246	0.689705	0.344852	−	0.938657i	$-0.387929\pi$
$653$	17.6246	0.689705	0.344852	+	0.938657i	$0.387929\pi$
$654$	−54.9767	−2.14976
$655$	58.3503	2.27994
$656$	31.3121	1.22253
$657$	−79.4296	−3.09884
$658$	−3.10475	−0.121036
$659$	27.9568	1.08904	0.544522	−	0.838747i	$-0.316711\pi$
$659$	27.9568	1.08904	0.544522	+	0.838747i	$0.316711\pi$
$660$	−7.71884	−0.300455
$661$	39.2784	1.52775	0.763877	−	0.645361i	$-0.223293\pi$
$661$	39.2784	1.52775	0.763877	+	0.645361i	$0.223293\pi$
$662$	33.6617	1.30830
$663$	21.6531	0.840937
$664$	−14.9256	−0.579227
$665$	−79.9720	−3.10118
$666$	−12.3672	−0.479219
$667$	37.1262	1.43753
$668$	2.21543	0.0857176
$669$	72.7550	2.81287
$670$	25.3012	0.977472
$671$	0.294722	0.0113776
$672$	−13.7115	−0.528932
$673$	46.9496	1.80977	0.904887	−	0.425652i	$-0.139955\pi$
$673$	46.9496	1.80977	0.904887	+	0.425652i	$0.139955\pi$
$674$	27.8122	1.07129
$675$	74.2794	2.85902
$676$	0.230315	0.00885827
$677$	−26.2892	−1.01038	−0.505188	−	0.863009i	$-0.668577\pi$
$677$	−26.2892	−1.01038	−0.505188	+	0.863009i	$0.668577\pi$
$678$	2.39868	0.0921207
$679$	14.0976	0.541018
$680$	−70.5979	−2.70730
$681$	21.8448	0.837094
$682$	−12.1153	−0.463920
$683$	4.65105	0.177967	0.0889837	−	0.996033i	$-0.471638\pi$
$683$	4.65105	0.177967	0.0889837	+	0.996033i	$0.471638\pi$
$684$	−8.34166	−0.318951
$685$	2.54967	0.0974179
$686$	−7.65525	−0.292279
$687$	−18.7316	−0.714655
$688$	40.4033	1.54036
$689$	−12.9298	−0.492585
$690$	−130.042	−4.95062
$691$	36.3840	1.38411	0.692056	−	0.721844i	$-0.256705\pi$
$691$	36.3840	1.38411	0.692056	+	0.721844i	$0.256705\pi$
$692$	−3.81925	−0.145186
$693$	−63.8383	−2.42502
$694$	3.21475	0.122030
$695$	11.7416	0.445383
$696$	−36.9606	−1.40099
$697$	−51.5390	−1.95218
$698$	−8.23379	−0.311654
$699$	−46.6893	−1.76595
$700$	−6.98300	−0.263933
$701$	−18.3580	−0.693372	−0.346686	−	0.937981i	$-0.612693\pi$
$701$	−18.3580	−0.693372	−0.346686	+	0.937981i	$0.612693\pi$
$702$	−12.9631	−0.489261
$703$	−8.59214	−0.324059
$704$	20.9776	0.790624
$705$	6.44836	0.242859
$706$	18.3064	0.688969
$707$	10.4560	0.393239
$708$	0.414695	0.0155852
$709$	33.8982	1.27307	0.636537	−	0.771246i	$-0.280366\pi$
$709$	33.8982	1.27307	0.636537	+	0.771246i	$0.280366\pi$
$710$	−82.9657	−3.11365
$711$	54.5870	2.04717
$712$	−16.3927	−0.614344
$713$	−21.0777	−0.789367
$714$	114.573	4.28778
$715$	−11.2288	−0.419933
$716$	−0.352594	−0.0131771
$717$	−54.7935	−2.04630
$718$	−29.9758	−1.11869
$719$	2.50297	0.0933449	0.0466725	−	0.998910i	$-0.485138\pi$
$719$	2.50297	0.0933449	0.0466725	+	0.998910i	$0.485138\pi$
$720$	95.8842	3.57339
$721$	−3.54305	−0.131950
$722$	−27.7462	−1.03261
$723$	38.1485	1.41876
$724$	3.30421	0.122800
$725$	−40.0964	−1.48914
$726$	7.57681	0.281202
$727$	−19.4498	−0.721352	−0.360676	−	0.932691i	$-0.617454\pi$
$727$	−19.4498	−0.721352	−0.360676	+	0.932691i	$0.617454\pi$
$728$	−9.36390	−0.347049
$729$	−29.3772	−1.08805
$730$	−73.9257	−2.73611
$731$	−66.5029	−2.45970
$732$	0.0664333	0.00245545
$733$	−9.20990	−0.340175	−0.170088	−	0.985429i	$-0.554405\pi$
$733$	−9.20990	−0.340175	−0.170088	+	0.985429i	$0.554405\pi$
$734$	48.3370	1.78415
$735$	−61.0281	−2.25106
$736$	−10.2737	−0.378694
$737$	14.0319	0.516871
$738$	62.6834	2.30741
$739$	10.8849	0.400409	0.200205	−	0.979754i	$-0.435839\pi$
$739$	10.8849	0.400409	0.200205	+	0.979754i	$0.435839\pi$
$740$	−1.18861	−0.0436942
$741$	−18.2965	−0.672139
$742$	−68.4150	−2.51160
$743$	−8.01709	−0.294118	−0.147059	−	0.989128i	$-0.546981\pi$
$743$	−8.01709	−0.294118	−0.147059	+	0.989128i	$0.546981\pi$
$744$	20.9837	0.769300
$745$	45.5490	1.66879
$746$	24.5491	0.898808
$747$	−33.3666	−1.22082
$748$	5.09556	0.186312
$749$	−53.6933	−1.96191
$750$	58.3849	2.13191
$751$	−8.29462	−0.302675	−0.151337	−	0.988482i	$-0.548358\pi$
$751$	−8.29462	−0.302675	−0.151337	+	0.988482i	$0.548358\pi$
$752$	2.58622	0.0943098
$753$	24.9359	0.908713
$754$	6.99757	0.254836
$755$	−46.0059	−1.67433
$756$	−7.08315	−0.257612
$757$	45.9862	1.67140	0.835699	−	0.549188i	$-0.185063\pi$
$757$	45.9862	1.67140	0.835699	+	0.549188i	$0.185063\pi$
$758$	−4.42384	−0.160681
$759$	−72.1203	−2.61780
$760$	59.6540	2.16388
$761$	−45.3298	−1.64320	−0.821602	−	0.570062i	$-0.806919\pi$
$761$	−45.3298	−1.64320	−0.821602	+	0.570062i	$0.806919\pi$
$762$	−57.1052	−2.06870
$763$	−43.6996	−1.58203
$764$	1.57988	0.0571581
$765$	−157.823	−5.70611
$766$	−34.5806	−1.24945
$767$	0.603268	0.0217828
$768$	16.2875	0.587726
$769$	−11.4863	−0.414208	−0.207104	−	0.978319i	$-0.566404\pi$
$769$	−11.4863	−0.414208	−0.207104	+	0.978319i	$0.566404\pi$
$770$	−59.4148	−2.14116
$771$	−55.5618	−2.00101
$772$	1.13345	0.0407936
$773$	−32.6057	−1.17275	−0.586373	−	0.810041i	$-0.699445\pi$
$773$	−32.6057	−1.17275	−0.586373	+	0.810041i	$0.699445\pi$
$774$	80.8831	2.90728
$775$	22.7640	0.817708
$776$	−10.5159	−0.377500
$777$	−14.8219	−0.531732
$778$	11.5144	0.412809
$779$	43.5495	1.56032
$780$	−2.53108	−0.0906273
$781$	−46.0122	−1.64644
$782$	85.8468	3.06987
$783$	−40.6715	−1.45348
$784$	−24.4763	−0.874155
$785$	57.8256	2.06389
$786$	70.6373	2.51955
$787$	46.6109	1.66150	0.830750	−	0.556645i	$-0.187912\pi$
$787$	46.6109	1.66150	0.830750	+	0.556645i	$0.187912\pi$
$788$	5.46345	0.194627
$789$	45.4113	1.61669
$790$	50.8045	1.80754
$791$	1.90665	0.0677927
$792$	47.6192	1.69208
$793$	0.0966424	0.00343187
$794$	−31.5128	−1.11835
$795$	142.094	5.03955
$796$	1.88777	0.0669101
$797$	16.8821	0.597994	0.298997	−	0.954254i	$-0.403348\pi$
$797$	16.8821	0.597994	0.298997	+	0.954254i	$0.403348\pi$
$798$	−96.8119	−3.42711
$799$	−4.25686	−0.150597
$800$	11.0956	0.392290
$801$	−36.6464	−1.29484
$802$	43.8179	1.54727
$803$	−40.9986	−1.44681
$804$	3.16292	0.111548
$805$	−103.367	−3.64321
$806$	−3.97274	−0.139934
$807$	−54.7343	−1.92674
$808$	−7.79951	−0.274386
$809$	39.0685	1.37358	0.686788	−	0.726858i	$-0.259020\pi$
$809$	39.0685	1.37358	0.686788	+	0.726858i	$0.259020\pi$
$810$	44.9948	1.58096
$811$	−0.239598	−0.00841344	−0.00420672	−	0.999991i	$-0.501339\pi$
$811$	−0.239598	−0.00841344	−0.00420672	+	0.999991i	$0.501339\pi$
$812$	3.82353	0.134179
$813$	−9.72324	−0.341009
$814$	−6.38349	−0.223741
$815$	17.1893	0.602114
$816$	−95.4379	−3.34100
$817$	56.1938	1.96597
$818$	7.52859	0.263231
$819$	−20.9332	−0.731466
$820$	6.02451	0.210385
$821$	10.0397	0.350388	0.175194	−	0.984534i	$-0.443945\pi$
$821$	10.0397	0.350388	0.175194	+	0.984534i	$0.443945\pi$
$822$	3.08656	0.107656
$823$	−4.69700	−0.163727	−0.0818635	−	0.996644i	$-0.526087\pi$
$823$	−4.69700	−0.163727	−0.0818635	+	0.996644i	$0.526087\pi$
$824$	2.64289	0.0920694
$825$	77.8903	2.71179
$826$	3.19206	0.111066
$827$	29.5991	1.02926	0.514631	−	0.857412i	$-0.327929\pi$
$827$	29.5991	1.02926	0.514631	+	0.857412i	$0.327929\pi$
$828$	−10.7819	−0.374698
$829$	−30.7580	−1.06827	−0.534135	−	0.845399i	$-0.679362\pi$
$829$	−30.7580	−1.06827	−0.534135	+	0.845399i	$0.679362\pi$
$830$	−31.0546	−1.07792
$831$	−65.6851	−2.27859
$832$	6.87878	0.238479
$833$	40.2875	1.39588
$834$	14.2140	0.492192
$835$	−35.4180	−1.22569
$836$	−4.30566	−0.148914
$837$	23.0905	0.798125
$838$	8.93277	0.308578
$839$	−45.3092	−1.56425	−0.782123	−	0.623124i	$-0.785863\pi$
$839$	−45.3092	−1.56425	−0.782123	+	0.623124i	$0.785863\pi$
$840$	102.906	3.55060
$841$	−7.04529	−0.242941
$842$	−51.2374	−1.76576
$843$	10.0597	0.346476
$844$	−2.24672	−0.0773352
$845$	−3.68204	−0.126666
$846$	5.17734	0.178001
$847$	6.02262	0.206940
$848$	56.9890	1.95701
$849$	−32.2889	−1.10815
$850$	−92.7149	−3.18009
$851$	−11.1057	−0.380699
$852$	−10.3716	−0.355325
$853$	30.3573	1.03941	0.519707	−	0.854345i	$-0.326041\pi$
$853$	30.3573	1.03941	0.519707	+	0.854345i	$0.326041\pi$
$854$	0.511362	0.0174984
$855$	133.358	4.56074
$856$	40.0517	1.36894
$857$	8.34845	0.285177	0.142589	−	0.989782i	$-0.454457\pi$
$857$	8.34845	0.285177	0.142589	+	0.989782i	$0.454457\pi$
$858$	−13.5933	−0.464067
$859$	5.44674	0.185840	0.0929201	−	0.995674i	$-0.470380\pi$
$859$	5.44674	0.185840	0.0929201	+	0.995674i	$0.470380\pi$
$860$	7.77368	0.265080
$861$	75.1251	2.56026
$862$	−48.6493	−1.65700
$863$	−35.6345	−1.21301	−0.606506	−	0.795079i	$-0.707429\pi$
$863$	−35.6345	−1.21301	−0.606506	+	0.795079i	$0.707429\pi$
$864$	11.2548	0.382895
$865$	61.0581	2.07604
$866$	−16.9404	−0.575659
$867$	106.349	3.61181
$868$	−2.17074	−0.0736796
$869$	28.1758	0.955798
$870$	−76.9009	−2.60718
$871$	4.60119	0.155905
$872$	32.5971	1.10388
$873$	−23.5086	−0.795646
$874$	−72.5390	−2.45367
$875$	46.4087	1.56890
$876$	−9.24150	−0.312241
$877$	25.4233	0.858483	0.429242	−	0.903190i	$-0.358781\pi$
$877$	25.4233	0.858483	0.429242	+	0.903190i	$0.358781\pi$
$878$	8.07738	0.272598
$879$	80.6159	2.71911
$880$	49.4919	1.66837
$881$	−18.0906	−0.609488	−0.304744	−	0.952434i	$-0.598571\pi$
$881$	−18.0906	−0.609488	−0.304744	+	0.952434i	$0.598571\pi$
$882$	−48.9990	−1.64988
$883$	9.89337	0.332939	0.166469	−	0.986047i	$-0.446763\pi$
$883$	9.89337	0.332939	0.166469	+	0.986047i	$0.446763\pi$
$884$	1.67089	0.0561980
$885$	−6.62971	−0.222855
$886$	−7.53261	−0.253063
$887$	15.2841	0.513190	0.256595	−	0.966519i	$-0.417399\pi$
$887$	15.2841	0.513190	0.256595	+	0.966519i	$0.417399\pi$
$888$	11.0562	0.371021
$889$	−45.3915	−1.52238
$890$	−34.1070	−1.14327
$891$	24.9538	0.835983
$892$	5.61421	0.187978
$893$	3.59697	0.120368
$894$	55.1404	1.84417
$895$	5.63691	0.188421
$896$	45.5855	1.52290
$897$	−23.6490	−0.789617
$898$	19.1783	0.639989
$899$	−12.4644	−0.415711
$900$	11.6445	0.388151
$901$	−93.8026	−3.12502
$902$	32.3549	1.07730
$903$	96.9371	3.22586
$904$	−1.42224	−0.0473029
$905$	−52.8242	−1.75594
$906$	−55.6935	−1.85029
$907$	−54.4358	−1.80751	−0.903756	−	0.428048i	$-0.859201\pi$
$907$	−54.4358	−1.80751	−0.903756	+	0.428048i	$0.859201\pi$
$908$	1.68568	0.0559411
$909$	−17.4360	−0.578315
$910$	−19.4827	−0.645845
$911$	−17.1588	−0.568497	−0.284249	−	0.958751i	$-0.591744\pi$
$911$	−17.1588	−0.568497	−0.284249	+	0.958751i	$0.591744\pi$
$912$	80.6434	2.67037
$913$	−17.2226	−0.569986
$914$	27.7816	0.918934
$915$	−1.06207	−0.0351108
$916$	−1.44544	−0.0477588
$917$	56.1478	1.85416
$918$	−94.0446	−3.10393
$919$	−5.79399	−0.191126	−0.0955630	−	0.995423i	$-0.530465\pi$
$919$	−5.79399	−0.191126	−0.0955630	+	0.995423i	$0.530465\pi$
$920$	77.1052	2.54208
$921$	1.09901	0.0362136
$922$	59.4555	1.95806
$923$	−15.0879	−0.496623
$924$	−7.42748	−0.244346
$925$	11.9942	0.394367
$926$	38.7368	1.27297
$927$	5.90824	0.194052
$928$	−6.07539	−0.199435
$929$	−3.44206	−0.112930	−0.0564652	−	0.998405i	$-0.517983\pi$
$929$	−3.44206	−0.112930	−0.0564652	+	0.998405i	$0.517983\pi$
$930$	43.6591	1.43164
$931$	−34.0422	−1.11569
$932$	−3.60283	−0.118015
$933$	35.7281	1.16968
$934$	−5.25990	−0.172109
$935$	−81.4625	−2.66411
$936$	15.6148	0.510387
$937$	34.1460	1.11550	0.557751	−	0.830008i	$-0.311664\pi$
$937$	34.1460	1.11550	0.557751	+	0.830008i	$0.311664\pi$
$938$	24.3462	0.794932
$939$	75.8191	2.47426
$940$	0.497595	0.0162297
$941$	2.96470	0.0966464	0.0483232	−	0.998832i	$-0.484612\pi$
$941$	2.96470	0.0966464	0.0483232	+	0.998832i	$0.484612\pi$
$942$	70.0021	2.28079
$943$	56.2896	1.83304
$944$	−2.65896	−0.0865417
$945$	113.238	3.68363
$946$	41.7489	1.35737
$947$	13.3301	0.433170	0.216585	−	0.976264i	$-0.430508\pi$
$947$	13.3301	0.433170	0.216585	+	0.976264i	$0.430508\pi$
$948$	6.35110	0.206274
$949$	−13.4439	−0.436406
$950$	78.3424	2.54176
$951$	−20.1829	−0.654475
$952$	−67.9331	−2.20172
$953$	−22.7877	−0.738164	−0.369082	−	0.929397i	$-0.620328\pi$
$953$	−22.7877	−0.738164	−0.369082	+	0.929397i	$0.620328\pi$
$954$	114.086	3.69367
$955$	−25.2575	−0.817314
$956$	−4.22820	−0.136750
$957$	−42.6486	−1.37863
$958$	−37.5236	−1.21233
$959$	2.45343	0.0792254
$960$	−75.5954	−2.43983
$961$	−23.9236	−0.771728
$962$	−2.09321	−0.0674878
$963$	89.5366	2.88528
$964$	2.94377	0.0948125
$965$	−18.1203	−0.583314
$966$	−125.133	−4.02610
$967$	9.30824	0.299333	0.149666	−	0.988737i	$-0.452180\pi$
$967$	9.30824	0.299333	0.149666	+	0.988737i	$0.452180\pi$
$968$	−4.49248	−0.144394
$969$	−132.737	−4.26413
$970$	−21.8796	−0.702513
$971$	4.13699	0.132762	0.0663812	−	0.997794i	$-0.478855\pi$
$971$	4.13699	0.132762	0.0663812	+	0.997794i	$0.478855\pi$
$972$	−0.372670	−0.0119534
$973$	11.2984	0.362209
$974$	16.7095	0.535407
$975$	25.5410	0.817967
$976$	−0.425959	−0.0136346
$977$	21.6159	0.691555	0.345778	−	0.938317i	$-0.387615\pi$
$977$	21.6159	0.691555	0.345778	+	0.938317i	$0.387615\pi$
$978$	20.8089	0.665394
$979$	−18.9155	−0.604542
$980$	−4.70930	−0.150433
$981$	72.8715	2.32661
$982$	4.59991	0.146789
$983$	−0.452535	−0.0144336	−0.00721681	−	0.999974i	$-0.502297\pi$
$983$	−0.452535	−0.0144336	−0.00721681	+	0.999974i	$0.502297\pi$
$984$	−56.0385	−1.78644
$985$	−87.3439	−2.78301
$986$	50.7658	1.61671
$987$	6.20496	0.197506
$988$	−1.41187	−0.0449175
$989$	72.6328	2.30959
$990$	99.0775	3.14889
$991$	−28.4918	−0.905070	−0.452535	−	0.891747i	$-0.649480\pi$
$991$	−28.4918	−0.905070	−0.452535	+	0.891747i	$0.649480\pi$
$992$	3.44919	0.109512
$993$	−67.2742	−2.13488
$994$	−79.8341	−2.53218
$995$	−30.1796	−0.956759
$996$	−3.88215	−0.123011
$997$	54.6511	1.73082	0.865408	−	0.501068i	$-0.167059\pi$
$997$	54.6511	1.73082	0.865408	+	0.501068i	$0.167059\pi$
$998$	31.3949	0.993788
$999$	12.1662	0.384922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.d.1.7	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.d.1.7	✓	19	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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q-1.49342 q^{2} +2.98467 q^{3} +0.230315 q^{4} -3.68204 q^{5} -4.45737 q^{6} -3.54305 q^{7} +2.64289 q^{8} +5.90824 q^{9} +O(q^{10})\)	\(q-1.49342 q^{2} +2.98467 q^{3} +0.230315 q^{4} -3.68204 q^{5} -4.45737 q^{6} -3.54305 q^{7} +2.64289 q^{8} +5.90824 q^{9} +5.49884 q^{10} +3.04962 q^{11} +0.687414 q^{12} +1.00000 q^{13} +5.29128 q^{14} -10.9897 q^{15} -4.40759 q^{16} +7.25478 q^{17} -8.82351 q^{18} -6.13016 q^{19} -0.848029 q^{20} -10.5748 q^{21} -4.55437 q^{22} -7.92349 q^{23} +7.88815 q^{24} +8.55740 q^{25} -1.49342 q^{26} +8.68013 q^{27} -0.816019 q^{28} -4.68559 q^{29} +16.4122 q^{30} +2.66016 q^{31} +1.29661 q^{32} +9.10209 q^{33} -10.8345 q^{34} +13.0457 q^{35} +1.36076 q^{36} +1.40162 q^{37} +9.15493 q^{38} +2.98467 q^{39} -9.73122 q^{40} -7.10414 q^{41} +15.7927 q^{42} -9.16677 q^{43} +0.702373 q^{44} -21.7544 q^{45} +11.8331 q^{46} -0.586766 q^{47} -13.1552 q^{48} +5.55323 q^{49} -12.7798 q^{50} +21.6531 q^{51} +0.230315 q^{52} -12.9298 q^{53} -12.9631 q^{54} -11.2288 q^{55} -9.36390 q^{56} -18.2965 q^{57} +6.99757 q^{58} +0.603268 q^{59} -2.53108 q^{60} +0.0966424 q^{61} -3.97274 q^{62} -20.9332 q^{63} +6.87878 q^{64} -3.68204 q^{65} -13.5933 q^{66} +4.60119 q^{67} +1.67089 q^{68} -23.6490 q^{69} -19.4827 q^{70} -15.0879 q^{71} +15.6148 q^{72} -13.4439 q^{73} -2.09321 q^{74} +25.5410 q^{75} -1.41187 q^{76} -10.8050 q^{77} -4.45737 q^{78} +9.23913 q^{79} +16.2289 q^{80} +8.18259 q^{81} +10.6095 q^{82} -5.64747 q^{83} -2.43554 q^{84} -26.7124 q^{85} +13.6899 q^{86} -13.9849 q^{87} +8.05980 q^{88} -6.20259 q^{89} +32.4885 q^{90} -3.54305 q^{91} -1.82490 q^{92} +7.93968 q^{93} +0.876291 q^{94} +22.5715 q^{95} +3.86996 q^{96} -3.97895 q^{97} -8.29333 q^{98} +18.0179 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9}+O(q^{10}) \) 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9	\( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9} - q^{11} + 18 q^{12} + 19 q^{13} - 8 q^{14} - 11 q^{15} + 21 q^{16} - 16 q^{17} - 10 q^{19} - 20 q^{20} - 33 q^{21} + 2 q^{22} - 14 q^{23} - 9 q^{24} + 23 q^{25} - 9 q^{26} + q^{27} + 10 q^{28} - 22 q^{29} + 28 q^{30} - 3 q^{31} - 47 q^{32} - 25 q^{33} - 35 q^{34} - 3 q^{35} - 33 q^{36} - 19 q^{37} - 15 q^{38} - 2 q^{39} + 8 q^{40} - 52 q^{41} - 3 q^{42} - 2 q^{43} - 54 q^{44} - 40 q^{45} + 33 q^{46} - 24 q^{47} - 8 q^{48} + 7 q^{49} - 10 q^{50} - 7 q^{51} + 17 q^{52} - 11 q^{53} - 23 q^{54} - 29 q^{55} - 17 q^{56} - 34 q^{57} + 18 q^{58} - 52 q^{59} - 71 q^{60} - 25 q^{61} - 22 q^{62} - 19 q^{63} + 48 q^{64} - 18 q^{65} + 9 q^{66} - 2 q^{67} + 18 q^{68} - 26 q^{69} - 12 q^{70} - 44 q^{71} - 6 q^{72} - 39 q^{73} - 4 q^{74} + 17 q^{75} - 10 q^{76} - 18 q^{77} - 8 q^{78} + q^{79} - 9 q^{80} - 13 q^{81} + 47 q^{82} - 27 q^{83} - 24 q^{84} - 26 q^{85} - q^{86} + 31 q^{87} + 19 q^{88} - 86 q^{89} + 48 q^{90} - 8 q^{91} - 19 q^{92} + 32 q^{93} + 45 q^{94} + 17 q^{95} - 25 q^{96} - 20 q^{97} + 39 q^{98} + 50 q^{99}+O(q^{100}) \) 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9 - q^11 + 18 * q^12 + 19 * q^13 - 8 * q^14 - 11 * q^15 + 21 * q^16 - 16 * q^17 - 10 * q^19 - 20 * q^20 - 33 * q^21 + 2 * q^22 - 14 * q^23 - 9 * q^24 + 23 * q^25 - 9 * q^26 + q^27 + 10 * q^28 - 22 * q^29 + 28 * q^30 - 3 * q^31 - 47 * q^32 - 25 * q^33 - 35 * q^34 - 3 * q^35 - 33 * q^36 - 19 * q^37 - 15 * q^38 - 2 * q^39 + 8 * q^40 - 52 * q^41 - 3 * q^42 - 2 * q^43 - 54 * q^44 - 40 * q^45 + 33 * q^46 - 24 * q^47 - 8 * q^48 + 7 * q^49 - 10 * q^50 - 7 * q^51 + 17 * q^52 - 11 * q^53 - 23 * q^54 - 29 * q^55 - 17 * q^56 - 34 * q^57 + 18 * q^58 - 52 * q^59 - 71 * q^60 - 25 * q^61 - 22 * q^62 - 19 * q^63 + 48 * q^64 - 18 * q^65 + 9 * q^66 - 2 * q^67 + 18 * q^68 - 26 * q^69 - 12 * q^70 - 44 * q^71 - 6 * q^72 - 39 * q^73 - 4 * q^74 + 17 * q^75 - 10 * q^76 - 18 * q^77 - 8 * q^78 + q^79 - 9 * q^80 - 13 * q^81 + 47 * q^82 - 27 * q^83 - 24 * q^84 - 26 * q^85 - q^86 + 31 * q^87 + 19 * q^88 - 86 * q^89 + 48 * q^90 - 8 * q^91 - 19 * q^92 + 32 * q^93 + 45 * q^94 + 17 * q^95 - 25 * q^96 - 20 * q^97 + 39 * q^98 + 50 * q^99

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