LMFDB - Embedded newform 1045.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.91899$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0881559 q^{2} -3.20362 q^{3} -1.99223 q^{4} +1.00000 q^{5} -0.282418 q^{6} -1.95185 q^{7} -0.351939 q^{8} +7.26315 q^{9} +O(q^{10})$	\(q+0.0881559 q^{2} -3.20362 q^{3} -1.99223 q^{4} +1.00000 q^{5} -0.282418 q^{6} -1.95185 q^{7} -0.351939 q^{8} +7.26315 q^{9} +0.0881559 q^{10} -1.00000 q^{11} +6.38233 q^{12} +3.20786 q^{13} -0.172067 q^{14} -3.20362 q^{15} +3.95343 q^{16} +0.503983 q^{17} +0.640290 q^{18} +1.00000 q^{19} -1.99223 q^{20} +6.25297 q^{21} -0.0881559 q^{22} +1.05529 q^{23} +1.12748 q^{24} +1.00000 q^{25} +0.282792 q^{26} -13.6575 q^{27} +3.88853 q^{28} -7.91058 q^{29} -0.282418 q^{30} +9.52800 q^{31} +1.05240 q^{32} +3.20362 q^{33} +0.0444291 q^{34} -1.95185 q^{35} -14.4699 q^{36} -5.22650 q^{37} +0.0881559 q^{38} -10.2768 q^{39} -0.351939 q^{40} +8.32622 q^{41} +0.551237 q^{42} -8.04835 q^{43} +1.99223 q^{44} +7.26315 q^{45} +0.0930303 q^{46} -9.39522 q^{47} -12.6653 q^{48} -3.19028 q^{49} +0.0881559 q^{50} -1.61457 q^{51} -6.39079 q^{52} +2.37178 q^{53} -1.20399 q^{54} -1.00000 q^{55} +0.686931 q^{56} -3.20362 q^{57} -0.697365 q^{58} -2.80063 q^{59} +6.38233 q^{60} -1.53640 q^{61} +0.839949 q^{62} -14.1766 q^{63} -7.81409 q^{64} +3.20786 q^{65} +0.282418 q^{66} -9.79354 q^{67} -1.00405 q^{68} -3.38075 q^{69} -0.172067 q^{70} -4.45027 q^{71} -2.55618 q^{72} -1.55991 q^{73} -0.460747 q^{74} -3.20362 q^{75} -1.99223 q^{76} +1.95185 q^{77} -0.905957 q^{78} +3.00849 q^{79} +3.95343 q^{80} +21.9639 q^{81} +0.734005 q^{82} -2.22730 q^{83} -12.4574 q^{84} +0.503983 q^{85} -0.709509 q^{86} +25.3425 q^{87} +0.351939 q^{88} +6.10788 q^{89} +0.640290 q^{90} -6.26126 q^{91} -2.10238 q^{92} -30.5240 q^{93} -0.828244 q^{94} +1.00000 q^{95} -3.37147 q^{96} -4.69521 q^{97} -0.281242 q^{98} -7.26315 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 5 q^{5} + 2 q^{6} - 11 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) $ 5 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 5 * q^5 + 2 * q^6 - 11 * q^7 + 3 * q^8 + 8 * q^9	\( 5 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 5 q^{5} + 2 q^{6} - 11 q^{7} + 3 q^{8} + 8 q^{9} - 3 q^{10} - 5 q^{11} - 7 q^{12} + q^{13} - 7 q^{15} - 3 q^{16} - 3 q^{17} - 7 q^{18} + 5 q^{19} + 5 q^{20} + 11 q^{21} + 3 q^{22} - 8 q^{23} + 9 q^{24} + 5 q^{25} - 16 q^{26} - 10 q^{27} - 22 q^{28} + 11 q^{29} + 2 q^{30} - 5 q^{31} - 2 q^{32} + 7 q^{33} + 4 q^{34} - 11 q^{35} - 3 q^{36} - 9 q^{37} - 3 q^{38} - 8 q^{39} + 3 q^{40} + 15 q^{41} + 11 q^{42} - 13 q^{43} - 5 q^{44} + 8 q^{45} + 18 q^{46} - 20 q^{47} - 20 q^{48} + 20 q^{49} - 3 q^{50} + 24 q^{51} + q^{52} - 5 q^{53} + 17 q^{54} - 5 q^{55} - 7 q^{57} - 33 q^{58} - 17 q^{59} - 7 q^{60} + 3 q^{61} + 14 q^{62} - 22 q^{63} - 17 q^{64} + q^{65} - 2 q^{66} - 28 q^{67} - 25 q^{68} - 2 q^{69} - 6 q^{71} - 26 q^{72} - 16 q^{73} - 21 q^{74} - 7 q^{75} + 5 q^{76} + 11 q^{77} + 29 q^{78} + 3 q^{79} - 3 q^{80} + q^{81} + 2 q^{82} - 33 q^{83} + 33 q^{84} - 3 q^{85} + 10 q^{86} - 3 q^{88} - 16 q^{89} - 7 q^{90} - 22 q^{91} - 19 q^{92} - 26 q^{93} - 10 q^{94} + 5 q^{95} + 5 q^{96} - 14 q^{97} + 10 q^{98} - 8 q^{99}+O(q^{100}) \) 5 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 5 * q^5 + 2 * q^6 - 11 * q^7 + 3 * q^8 + 8 * q^9 - 3 * q^10 - 5 * q^11 - 7 * q^12 + q^13 - 7 * q^15 - 3 * q^16 - 3 * q^17 - 7 * q^18 + 5 * q^19 + 5 * q^20 + 11 * q^21 + 3 * q^22 - 8 * q^23 + 9 * q^24 + 5 * q^25 - 16 * q^26 - 10 * q^27 - 22 * q^28 + 11 * q^29 + 2 * q^30 - 5 * q^31 - 2 * q^32 + 7 * q^33 + 4 * q^34 - 11 * q^35 - 3 * q^36 - 9 * q^37 - 3 * q^38 - 8 * q^39 + 3 * q^40 + 15 * q^41 + 11 * q^42 - 13 * q^43 - 5 * q^44 + 8 * q^45 + 18 * q^46 - 20 * q^47 - 20 * q^48 + 20 * q^49 - 3 * q^50 + 24 * q^51 + q^52 - 5 * q^53 + 17 * q^54 - 5 * q^55 - 7 * q^57 - 33 * q^58 - 17 * q^59 - 7 * q^60 + 3 * q^61 + 14 * q^62 - 22 * q^63 - 17 * q^64 + q^65 - 2 * q^66 - 28 * q^67 - 25 * q^68 - 2 * q^69 - 6 * q^71 - 26 * q^72 - 16 * q^73 - 21 * q^74 - 7 * q^75 + 5 * q^76 + 11 * q^77 + 29 * q^78 + 3 * q^79 - 3 * q^80 + q^81 + 2 * q^82 - 33 * q^83 + 33 * q^84 - 3 * q^85 + 10 * q^86 - 3 * q^88 - 16 * q^89 - 7 * q^90 - 22 * q^91 - 19 * q^92 - 26 * q^93 - 10 * q^94 + 5 * q^95 + 5 * q^96 - 14 * q^97 + 10 * q^98 - 8 * 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 $

$2$	0.0881559	0.0623356	0.0311678	−	0.999514i	$-0.490077\pi$
$2$	0.0881559	0.0623356	0.0311678	+	0.999514i	$0.490077\pi$
$3$	−3.20362	−1.84961	−0.924804	−	0.380443i	$-0.875771\pi$
$3$	−3.20362	−1.84961	−0.924804	+	0.380443i	$0.875771\pi$
$4$	−1.99223	−0.996114
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.282418	−0.115297
$7$	−1.95185	−0.737730	−0.368865	−	0.929483i	$-0.620253\pi$
$7$	−1.95185	−0.737730	−0.368865	+	0.929483i	$0.620253\pi$
$8$	−0.351939	−0.124429
$9$	7.26315	2.42105
$10$	0.0881559	0.0278774
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	6.38233	1.84242
$13$	3.20786	0.889700	0.444850	−	0.895605i	$-0.353257\pi$
$13$	3.20786	0.889700	0.444850	+	0.895605i	$0.353257\pi$
$14$	−0.172067	−0.0459869
$15$	−3.20362	−0.827170
$16$	3.95343	0.988358
$17$	0.503983	0.122234	0.0611169	−	0.998131i	$-0.480534\pi$
$17$	0.503983	0.122234	0.0611169	+	0.998131i	$0.480534\pi$
$18$	0.640290	0.150918
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.99223	−0.445476
$21$	6.25297	1.36451
$22$	−0.0881559	−0.0187949
$23$	1.05529	0.220044	0.110022	−	0.993929i	$-0.464908\pi$
$23$	1.05529	0.220044	0.110022	+	0.993929i	$0.464908\pi$
$24$	1.12748	0.230145
$25$	1.00000	0.200000
$26$	0.282792	0.0554601
$27$	−13.6575	−2.62839
$28$	3.88853	0.734863
$29$	−7.91058	−1.46896	−0.734479	−	0.678631i	$-0.762574\pi$
$29$	−7.91058	−1.46896	−0.734479	+	0.678631i	$0.762574\pi$
$30$	−0.282418	−0.0515622
$31$	9.52800	1.71128	0.855639	−	0.517573i	$-0.173164\pi$
$31$	9.52800	1.71128	0.855639	+	0.517573i	$0.173164\pi$
$32$	1.05240	0.186039
$33$	3.20362	0.557678
$34$	0.0444291	0.00761953
$35$	−1.95185	−0.329923
$36$	−14.4699	−2.41164
$37$	−5.22650	−0.859231	−0.429615	−	0.903012i	$-0.641351\pi$
$37$	−5.22650	−0.859231	−0.429615	+	0.903012i	$0.641351\pi$
$38$	0.0881559	0.0143008
$39$	−10.2768	−1.64560
$40$	−0.351939	−0.0556464
$41$	8.32622	1.30034	0.650168	−	0.759790i	$-0.274698\pi$
$41$	8.32622	1.30034	0.650168	+	0.759790i	$0.274698\pi$
$42$	0.551237	0.0850577
$43$	−8.04835	−1.22736	−0.613681	−	0.789554i	$-0.710312\pi$
$43$	−8.04835	−1.22736	−0.613681	+	0.789554i	$0.710312\pi$
$44$	1.99223	0.300340
$45$	7.26315	1.08273
$46$	0.0930303	0.0137166
$47$	−9.39522	−1.37043	−0.685216	−	0.728339i	$-0.740292\pi$
$47$	−9.39522	−1.37043	−0.685216	+	0.728339i	$0.740292\pi$
$48$	−12.6653	−1.82808
$49$	−3.19028	−0.455755
$50$	0.0881559	0.0124671
$51$	−1.61457	−0.226085
$52$	−6.39079	−0.886243
$53$	2.37178	0.325789	0.162895	−	0.986643i	$-0.447917\pi$
$53$	2.37178	0.325789	0.162895	+	0.986643i	$0.447917\pi$
$54$	−1.20399	−0.163842
$55$	−1.00000	−0.134840
$56$	0.686931	0.0917950
$57$	−3.20362	−0.424329
$58$	−0.697365	−0.0915685
$59$	−2.80063	−0.364611	−0.182305	−	0.983242i	$-0.558356\pi$
$59$	−2.80063	−0.364611	−0.182305	+	0.983242i	$0.558356\pi$
$60$	6.38233	0.823956
$61$	−1.53640	−0.196715	−0.0983577	−	0.995151i	$-0.531359\pi$
$61$	−1.53640	−0.196715	−0.0983577	+	0.995151i	$0.531359\pi$
$62$	0.839949	0.106674
$63$	−14.1766	−1.78608
$64$	−7.81409	−0.976761
$65$	3.20786	0.397886
$66$	0.282418	0.0347632
$67$	−9.79354	−1.19647	−0.598236	−	0.801320i	$-0.704131\pi$
$67$	−9.79354	−1.19647	−0.598236	+	0.801320i	$0.704131\pi$
$68$	−1.00405	−0.121759
$69$	−3.38075	−0.406995
$70$	−0.172067	−0.0205659
$71$	−4.45027	−0.528150	−0.264075	−	0.964502i	$-0.585067\pi$
$71$	−4.45027	−0.528150	−0.264075	+	0.964502i	$0.585067\pi$
$72$	−2.55618	−0.301249
$73$	−1.55991	−0.182573	−0.0912866	−	0.995825i	$-0.529098\pi$
$73$	−1.55991	−0.182573	−0.0912866	+	0.995825i	$0.529098\pi$
$74$	−0.460747	−0.0535607
$75$	−3.20362	−0.369922
$76$	−1.99223	−0.228524
$77$	1.95185	0.222434
$78$	−0.905957	−0.102579
$79$	3.00849	0.338482	0.169241	−	0.985575i	$-0.445868\pi$
$79$	3.00849	0.338482	0.169241	+	0.985575i	$0.445868\pi$
$80$	3.95343	0.442007
$81$	21.9639	2.44044
$82$	0.734005	0.0810573
$83$	−2.22730	−0.244478	−0.122239	−	0.992501i	$-0.539008\pi$
$83$	−2.22730	−0.244478	−0.122239	+	0.992501i	$0.539008\pi$
$84$	−12.4574	−1.35921
$85$	0.503983	0.0546647
$86$	−0.709509	−0.0765084
$87$	25.3425	2.71700
$88$	0.351939	0.0375168
$89$	6.10788	0.647434	0.323717	−	0.946154i	$-0.395067\pi$
$89$	6.10788	0.647434	0.323717	+	0.946154i	$0.395067\pi$
$90$	0.640290	0.0674925
$91$	−6.26126	−0.656358
$92$	−2.10238	−0.219189
$93$	−30.5240	−3.16519
$94$	−0.828244	−0.0854268
$95$	1.00000	0.102598
$96$	−3.37147	−0.344099
$97$	−4.69521	−0.476726	−0.238363	−	0.971176i	$-0.576611\pi$
$97$	−4.69521	−0.476726	−0.238363	+	0.971176i	$0.576611\pi$
$98$	−0.281242	−0.0284098
$99$	−7.26315	−0.729974
$100$	−1.99223	−0.199223
$101$	0.160574	0.0159777	0.00798885	−	0.999968i	$-0.497457\pi$
$101$	0.160574	0.0159777	0.00798885	+	0.999968i	$0.497457\pi$
$102$	−0.142334	−0.0140931
$103$	−12.1749	−1.19962	−0.599812	−	0.800141i	$-0.704758\pi$
$103$	−12.1749	−1.19962	−0.599812	+	0.800141i	$0.704758\pi$
$104$	−1.12897	−0.110705
$105$	6.25297	0.610228
$106$	0.209087	0.0203083
$107$	16.2082	1.56690	0.783452	−	0.621452i	$-0.213457\pi$
$107$	16.2082	1.56690	0.783452	+	0.621452i	$0.213457\pi$
$108$	27.2089	2.61817
$109$	17.2276	1.65010	0.825050	−	0.565059i	$-0.191147\pi$
$109$	17.2276	1.65010	0.825050	+	0.565059i	$0.191147\pi$
$110$	−0.0881559	−0.00840534
$111$	16.7437	1.58924
$112$	−7.71650	−0.729141
$113$	−9.74136	−0.916390	−0.458195	−	0.888852i	$-0.651504\pi$
$113$	−9.74136	−0.916390	−0.458195	+	0.888852i	$0.651504\pi$
$114$	−0.282418	−0.0264508
$115$	1.05529	0.0984065
$116$	15.7597	1.46325
$117$	23.2992	2.15401
$118$	−0.246892	−0.0227283
$119$	−0.983699	−0.0901756
$120$	1.12748	0.102924
$121$	1.00000	0.0909091
$122$	−0.135442	−0.0122624
$123$	−26.6740	−2.40511
$124$	−18.9819	−1.70463
$125$	1.00000	0.0894427
$126$	−1.24975	−0.111337
$127$	−6.14600	−0.545370	−0.272685	−	0.962103i	$-0.587912\pi$
$127$	−6.14600	−0.545370	−0.272685	+	0.962103i	$0.587912\pi$
$128$	−2.79365	−0.246926
$129$	25.7838	2.27014
$130$	0.282792	0.0248025
$131$	−13.4596	−1.17597	−0.587987	−	0.808871i	$-0.700079\pi$
$131$	−13.4596	−1.17597	−0.587987	+	0.808871i	$0.700079\pi$
$132$	−6.38233	−0.555511
$133$	−1.95185	−0.169247
$134$	−0.863359	−0.0745828
$135$	−13.6575	−1.17545
$136$	−0.177371	−0.0152095
$137$	7.45968	0.637324	0.318662	−	0.947868i	$-0.396767\pi$
$137$	7.45968	0.637324	0.318662	+	0.947868i	$0.396767\pi$
$138$	−0.298033	−0.0253703
$139$	−18.3550	−1.55685	−0.778425	−	0.627738i	$-0.783981\pi$
$139$	−18.3550	−1.55685	−0.778425	+	0.627738i	$0.783981\pi$
$140$	3.88853	0.328641
$141$	30.0987	2.53476
$142$	−0.392318	−0.0329226
$143$	−3.20786	−0.268255
$144$	28.7144	2.39286
$145$	−7.91058	−0.656938
$146$	−0.137515	−0.0113808
$147$	10.2204	0.842968
$148$	10.4124	0.855892
$149$	−19.2387	−1.57610	−0.788048	−	0.615614i	$-0.788908\pi$
$149$	−19.2387	−1.57610	−0.788048	+	0.615614i	$0.788908\pi$
$150$	−0.282418	−0.0230593
$151$	7.76410	0.631833	0.315917	−	0.948787i	$-0.397688\pi$
$151$	7.76410	0.631833	0.315917	+	0.948787i	$0.397688\pi$
$152$	−0.351939	−0.0285460
$153$	3.66051	0.295934
$154$	0.172067	0.0138656
$155$	9.52800	0.765307
$156$	20.4736	1.63920
$157$	−4.82520	−0.385093	−0.192546	−	0.981288i	$-0.561675\pi$
$157$	−4.82520	−0.385093	−0.192546	+	0.981288i	$0.561675\pi$
$158$	0.265216	0.0210995
$159$	−7.59828	−0.602583
$160$	1.05240	0.0831992
$161$	−2.05977	−0.162333
$162$	1.93625	0.152126
$163$	16.0681	1.25855	0.629276	−	0.777182i	$-0.283352\pi$
$163$	16.0681	1.25855	0.629276	+	0.777182i	$0.283352\pi$
$164$	−16.5877	−1.29528
$165$	3.20362	0.249401
$166$	−0.196350	−0.0152397
$167$	−19.8521	−1.53620	−0.768102	−	0.640327i	$-0.778799\pi$
$167$	−19.8521	−1.53620	−0.768102	+	0.640327i	$0.778799\pi$
$168$	−2.20066	−0.169785
$169$	−2.70963	−0.208433
$170$	0.0444291	0.00340756
$171$	7.26315	0.555427
$172$	16.0341	1.22259
$173$	−23.1362	−1.75901	−0.879507	−	0.475886i	$-0.842127\pi$
$173$	−23.1362	−1.75901	−0.879507	+	0.475886i	$0.842127\pi$
$174$	2.23409	0.169366
$175$	−1.95185	−0.147546
$176$	−3.95343	−0.298001
$177$	8.97214	0.674387
$178$	0.538446	0.0403582
$179$	4.66377	0.348586	0.174293	−	0.984694i	$-0.444236\pi$
$179$	4.66377	0.348586	0.174293	+	0.984694i	$0.444236\pi$
$180$	−14.4699	−1.07852
$181$	23.7883	1.76817	0.884084	−	0.467328i	$-0.154783\pi$
$181$	23.7883	1.76817	0.884084	+	0.467328i	$0.154783\pi$
$182$	−0.551967	−0.0409145
$183$	4.92202	0.363846
$184$	−0.371398	−0.0273798
$185$	−5.22650	−0.384260
$186$	−2.69087	−0.197304
$187$	−0.503983	−0.0368549
$188$	18.7174	1.36511
$189$	26.6574	1.93904
$190$	0.0881559	0.00639550
$191$	−4.12335	−0.298355	−0.149178	−	0.988810i	$-0.547663\pi$
$191$	−4.12335	−0.298355	−0.149178	+	0.988810i	$0.547663\pi$
$192$	25.0333	1.80663
$193$	16.6564	1.19895	0.599477	−	0.800392i	$-0.295375\pi$
$193$	16.6564	1.19895	0.599477	+	0.800392i	$0.295375\pi$
$194$	−0.413910	−0.0297170
$195$	−10.2768	−0.735933
$196$	6.35578	0.453984
$197$	12.7284	0.906862	0.453431	−	0.891291i	$-0.350200\pi$
$197$	12.7284	0.906862	0.453431	+	0.891291i	$0.350200\pi$
$198$	−0.640290	−0.0455034
$199$	−17.0542	−1.20894	−0.604471	−	0.796627i	$-0.706615\pi$
$199$	−17.0542	−1.20894	−0.604471	+	0.796627i	$0.706615\pi$
$200$	−0.351939	−0.0248858
$201$	31.3747	2.21300
$202$	0.0141555	0.000995980 0
$203$	15.4403	1.08369
$204$	3.21659	0.225206
$205$	8.32622	0.581528
$206$	−1.07329	−0.0747794
$207$	7.66475	0.532737
$208$	12.6821	0.879342
$209$	−1.00000	−0.0691714
$210$	0.551237	0.0380389
$211$	−15.0057	−1.03304	−0.516518	−	0.856276i	$-0.672772\pi$
$211$	−15.0057	−1.03304	−0.516518	+	0.856276i	$0.672772\pi$
$212$	−4.72513	−0.324523
$213$	14.2570	0.976871
$214$	1.42885	0.0976740
$215$	−8.04835	−0.548893
$216$	4.80660	0.327048
$217$	−18.5972	−1.26246
$218$	1.51871	0.102860
$219$	4.99734	0.337689
$220$	1.99223	0.134316
$221$	1.61671	0.108752
$222$	1.47605	0.0990663
$223$	4.21760	0.282432	0.141216	−	0.989979i	$-0.454899\pi$
$223$	4.21760	0.282432	0.141216	+	0.989979i	$0.454899\pi$
$224$	−2.05412	−0.137246
$225$	7.26315	0.484210
$226$	−0.858759	−0.0571238
$227$	18.5091	1.22849	0.614245	−	0.789115i	$-0.289461\pi$
$227$	18.5091	1.22849	0.614245	+	0.789115i	$0.289461\pi$
$228$	6.38233	0.422680
$229$	10.0978	0.667282	0.333641	−	0.942700i	$-0.391723\pi$
$229$	10.0978	0.667282	0.333641	+	0.942700i	$0.391723\pi$
$230$	0.0930303	0.00613424
$231$	−6.25297	−0.411416
$232$	2.78404	0.182781
$233$	−5.28276	−0.346085	−0.173043	−	0.984914i	$-0.555360\pi$
$233$	−5.28276	−0.346085	−0.173043	+	0.984914i	$0.555360\pi$
$234$	2.05396	0.134272
$235$	−9.39522	−0.612876
$236$	5.57949	0.363194
$237$	−9.63804	−0.626058
$238$	−0.0867189	−0.00562115
$239$	6.01161	0.388859	0.194429	−	0.980917i	$-0.437714\pi$
$239$	6.01161	0.388859	0.194429	+	0.980917i	$0.437714\pi$
$240$	−12.6653	−0.817540
$241$	−24.1747	−1.55723	−0.778613	−	0.627504i	$-0.784076\pi$
$241$	−24.1747	−1.55723	−0.778613	+	0.627504i	$0.784076\pi$
$242$	0.0881559	0.00566688
$243$	−29.3915	−1.88546
$244$	3.06085	0.195951
$245$	−3.19028	−0.203820
$246$	−2.35147	−0.149924
$247$	3.20786	0.204111
$248$	−3.35327	−0.212933
$249$	7.13543	0.452189
$250$	0.0881559	0.00557547
$251$	−21.5063	−1.35747	−0.678734	−	0.734385i	$-0.737471\pi$
$251$	−21.5063	−1.35747	−0.678734	+	0.734385i	$0.737471\pi$
$252$	28.2430	1.77914
$253$	−1.05529	−0.0663457
$254$	−0.541807	−0.0339960
$255$	−1.61457	−0.101108
$256$	15.3819	0.961369
$257$	−8.04652	−0.501928	−0.250964	−	0.967996i	$-0.580748\pi$
$257$	−8.04652	−0.501928	−0.250964	+	0.967996i	$0.580748\pi$
$258$	2.27300	0.141511
$259$	10.2013	0.633880
$260$	−6.39079	−0.396340
$261$	−57.4558	−3.55642
$262$	−1.18655	−0.0733050
$263$	−19.5827	−1.20752	−0.603762	−	0.797165i	$-0.706332\pi$
$263$	−19.5827	−1.20752	−0.603762	+	0.797165i	$0.706332\pi$
$264$	−1.12748	−0.0693913
$265$	2.37178	0.145697
$266$	−0.172067	−0.0105501
$267$	−19.5673	−1.19750
$268$	19.5110	1.19182
$269$	−27.2287	−1.66017	−0.830083	−	0.557640i	$-0.811707\pi$
$269$	−27.2287	−1.66017	−0.830083	+	0.557640i	$0.811707\pi$
$270$	−1.20399	−0.0732725
$271$	16.4952	1.00201	0.501006	−	0.865444i	$-0.332963\pi$
$271$	16.4952	1.00201	0.501006	+	0.865444i	$0.332963\pi$
$272$	1.99246	0.120811
$273$	20.0587	1.21401
$274$	0.657615	0.0397280
$275$	−1.00000	−0.0603023
$276$	6.73523	0.405413
$277$	−14.0482	−0.844077	−0.422039	−	0.906578i	$-0.638685\pi$
$277$	−14.0482	−0.844077	−0.422039	+	0.906578i	$0.638685\pi$
$278$	−1.61810	−0.0970472
$279$	69.2033	4.14309
$280$	0.686931	0.0410520
$281$	−4.14119	−0.247043	−0.123521	−	0.992342i	$-0.539419\pi$
$281$	−4.14119	−0.247043	−0.123521	+	0.992342i	$0.539419\pi$
$282$	2.65337	0.158006
$283$	4.15687	0.247100	0.123550	−	0.992338i	$-0.460572\pi$
$283$	4.15687	0.247100	0.123550	+	0.992338i	$0.460572\pi$
$284$	8.86596	0.526098
$285$	−3.20362	−0.189766
$286$	−0.282792	−0.0167218
$287$	−16.2515	−0.959297
$288$	7.64371	0.450410
$289$	−16.7460	−0.985059
$290$	−0.697365	−0.0409507
$291$	15.0416	0.881757
$292$	3.10769	0.181864
$293$	−5.34865	−0.312471	−0.156236	−	0.987720i	$-0.549936\pi$
$293$	−5.34865	−0.312471	−0.156236	+	0.987720i	$0.549936\pi$
$294$	0.900993	0.0525470
$295$	−2.80063	−0.163059
$296$	1.83941	0.106913
$297$	13.6575	0.792489
$298$	−1.69601	−0.0982470
$299$	3.38523	0.195773
$300$	6.38233	0.368484
$301$	15.7092	0.905461
$302$	0.684451	0.0393858
$303$	−0.514417	−0.0295525
$304$	3.95343	0.226745
$305$	−1.53640	−0.0879738
$306$	0.322695	0.0184473
$307$	−24.7343	−1.41166	−0.705830	−	0.708381i	$-0.749426\pi$
$307$	−24.7343	−1.41166	−0.705830	+	0.708381i	$0.749426\pi$
$308$	−3.88853	−0.221570
$309$	39.0036	2.21884
$310$	0.839949	0.0477059
$311$	17.9991	1.02063	0.510317	−	0.859986i	$-0.329528\pi$
$311$	17.9991	1.02063	0.510317	+	0.859986i	$0.329528\pi$
$312$	3.61679	0.204760
$313$	−34.3706	−1.94274	−0.971372	−	0.237564i	$-0.923651\pi$
$313$	−34.3706	−1.94274	−0.971372	+	0.237564i	$0.923651\pi$
$314$	−0.425370	−0.0240050
$315$	−14.1766	−0.798760
$316$	−5.99360	−0.337166
$317$	−13.1196	−0.736870	−0.368435	−	0.929654i	$-0.620106\pi$
$317$	−13.1196	−0.736870	−0.368435	+	0.929654i	$0.620106\pi$
$318$	−0.669833	−0.0375624
$319$	7.91058	0.442908
$320$	−7.81409	−0.436821
$321$	−51.9248	−2.89816
$322$	−0.181581	−0.0101191
$323$	0.503983	0.0280424
$324$	−43.7572	−2.43095
$325$	3.20786	0.177940
$326$	1.41650	0.0784527
$327$	−55.1905	−3.05204
$328$	−2.93032	−0.161800
$329$	18.3380	1.01101
$330$	0.282418	0.0155466
$331$	−0.911287	−0.0500889	−0.0250444	−	0.999686i	$-0.507973\pi$
$331$	−0.911287	−0.0500889	−0.0250444	+	0.999686i	$0.507973\pi$
$332$	4.43730	0.243528
$333$	−37.9608	−2.08024
$334$	−1.75008	−0.0957603
$335$	−9.79354	−0.535078
$336$	24.7207	1.34863
$337$	12.9894	0.707577	0.353789	−	0.935325i	$-0.384893\pi$
$337$	12.9894	0.707577	0.353789	+	0.935325i	$0.384893\pi$
$338$	−0.238870	−0.0129928
$339$	31.2076	1.69496
$340$	−1.00405	−0.0544522
$341$	−9.52800	−0.515970
$342$	0.640290	0.0346229
$343$	19.8899	1.07395
$344$	2.83252	0.152719
$345$	−3.38075	−0.182014
$346$	−2.03959	−0.109649
$347$	−8.50005	−0.456307	−0.228153	−	0.973625i	$-0.573269\pi$
$347$	−8.50005	−0.456307	−0.228153	+	0.973625i	$0.573269\pi$
$348$	−50.4880	−2.70644
$349$	−22.9281	−1.22731	−0.613657	−	0.789573i	$-0.710302\pi$
$349$	−22.9281	−1.22731	−0.613657	+	0.789573i	$0.710302\pi$
$350$	−0.172067	−0.00919737
$351$	−43.8114	−2.33848
$352$	−1.05240	−0.0560929
$353$	−32.5037	−1.73000	−0.864998	−	0.501775i	$-0.832681\pi$
$353$	−32.5037	−1.73000	−0.864998	+	0.501775i	$0.832681\pi$
$354$	0.790947	0.0420384
$355$	−4.45027	−0.236196
$356$	−12.1683	−0.644919
$357$	3.15139	0.166789
$358$	0.411139	0.0217293
$359$	−7.05497	−0.372347	−0.186174	−	0.982517i	$-0.559609\pi$
$359$	−7.05497	−0.372347	−0.186174	+	0.982517i	$0.559609\pi$
$360$	−2.55618	−0.134723
$361$	1.00000	0.0526316
$362$	2.09708	0.110220
$363$	−3.20362	−0.168146
$364$	12.4739	0.653808
$365$	−1.55991	−0.0816492
$366$	0.433905	0.0226806
$367$	14.8301	0.774127	0.387063	−	0.922053i	$-0.373489\pi$
$367$	14.8301	0.774127	0.387063	+	0.922053i	$0.373489\pi$
$368$	4.17203	0.217482
$369$	60.4746	3.14818
$370$	−0.460747	−0.0239531
$371$	−4.62936	−0.240344
$372$	60.8109	3.15290
$373$	9.69631	0.502056	0.251028	−	0.967980i	$-0.419231\pi$
$373$	9.69631	0.502056	0.251028	+	0.967980i	$0.419231\pi$
$374$	−0.0444291	−0.00229737
$375$	−3.20362	−0.165434
$376$	3.30654	0.170522
$377$	−25.3761	−1.30693
$378$	2.35001	0.120871
$379$	1.03558	0.0531941	0.0265970	−	0.999646i	$-0.491533\pi$
$379$	1.03558	0.0531941	0.0265970	+	0.999646i	$0.491533\pi$
$380$	−1.99223	−0.102199
$381$	19.6894	1.00872
$382$	−0.363498	−0.0185982
$383$	1.35607	0.0692922	0.0346461	−	0.999400i	$-0.488970\pi$
$383$	1.35607	0.0692922	0.0346461	+	0.999400i	$0.488970\pi$
$384$	8.94978	0.456716
$385$	1.95185	0.0994754
$386$	1.46836	0.0747376
$387$	−58.4564	−2.97150
$388$	9.35393	0.474874
$389$	−34.2838	−1.73826	−0.869129	−	0.494585i	$-0.835320\pi$
$389$	−34.2838	−1.73826	−0.869129	+	0.494585i	$0.835320\pi$
$390$	−0.905957	−0.0458749
$391$	0.531850	0.0268968
$392$	1.12278	0.0567092
$393$	43.1195	2.17509
$394$	1.12208	0.0565298
$395$	3.00849	0.151374
$396$	14.4699	0.727138
$397$	−35.4936	−1.78137	−0.890687	−	0.454617i	$-0.849776\pi$
$397$	−35.4936	−1.78137	−0.890687	+	0.454617i	$0.849776\pi$
$398$	−1.50343	−0.0753601
$399$	6.25297	0.313040
$400$	3.95343	0.197672
$401$	28.4914	1.42279	0.711396	−	0.702791i	$-0.248063\pi$
$401$	28.4914	1.42279	0.711396	+	0.702791i	$0.248063\pi$
$402$	2.76587	0.137949
$403$	30.5645	1.52253
$404$	−0.319900	−0.0159156
$405$	21.9639	1.09140
$406$	1.36115	0.0675528
$407$	5.22650	0.259068
$408$	0.568229	0.0281315
$409$	31.1180	1.53869	0.769343	−	0.638836i	$-0.220584\pi$
$409$	31.1180	1.53869	0.769343	+	0.638836i	$0.220584\pi$
$410$	0.734005	0.0362499
$411$	−23.8980	−1.17880
$412$	24.2551	1.19496
$413$	5.46641	0.268984
$414$	0.675693	0.0332085
$415$	−2.22730	−0.109334
$416$	3.37594	0.165519
$417$	58.8023	2.87956
$418$	−0.0881559	−0.00431185
$419$	−23.8806	−1.16664	−0.583322	−	0.812241i	$-0.698247\pi$
$419$	−23.8806	−1.16664	−0.583322	+	0.812241i	$0.698247\pi$
$420$	−12.4574	−0.607857
$421$	−1.13683	−0.0554059	−0.0277029	−	0.999616i	$-0.508819\pi$
$421$	−1.13683	−0.0554059	−0.0277029	+	0.999616i	$0.508819\pi$
$422$	−1.32284	−0.0643950
$423$	−68.2389	−3.31789
$424$	−0.834722	−0.0405377
$425$	0.503983	0.0244468
$426$	1.25684	0.0608939
$427$	2.99881	0.145123
$428$	−32.2904	−1.56082
$429$	10.2768	0.496166
$430$	−0.709509	−0.0342156
$431$	−0.598886	−0.0288473	−0.0144237	−	0.999896i	$-0.504591\pi$
$431$	−0.598886	−0.0288473	−0.0144237	+	0.999896i	$0.504591\pi$
$432$	−53.9940	−2.59779
$433$	32.3456	1.55443	0.777216	−	0.629234i	$-0.216631\pi$
$433$	32.3456	1.55443	0.777216	+	0.629234i	$0.216631\pi$
$434$	−1.63945	−0.0786963
$435$	25.3425	1.21508
$436$	−34.3212	−1.64369
$437$	1.05529	0.0504815
$438$	0.440545	0.0210500
$439$	9.36506	0.446970	0.223485	−	0.974707i	$-0.428257\pi$
$439$	9.36506	0.446970	0.223485	+	0.974707i	$0.428257\pi$
$440$	0.351939	0.0167780
$441$	−23.1715	−1.10341
$442$	0.142522	0.00677910
$443$	10.5557	0.501517	0.250759	−	0.968050i	$-0.419320\pi$
$443$	10.5557	0.501517	0.250759	+	0.968050i	$0.419320\pi$
$444$	−33.3572	−1.58306
$445$	6.10788	0.289541
$446$	0.371807	0.0176056
$447$	61.6334	2.91516
$448$	15.2519	0.720586
$449$	−13.4399	−0.634269	−0.317135	−	0.948381i	$-0.602721\pi$
$449$	−13.4399	−0.634269	−0.317135	+	0.948381i	$0.602721\pi$
$450$	0.640290	0.0301836
$451$	−8.32622	−0.392066
$452$	19.4070	0.912829
$453$	−24.8732	−1.16864
$454$	1.63168	0.0765787
$455$	−6.26126	−0.293532
$456$	1.12748	0.0527989
$457$	3.88536	0.181750	0.0908748	−	0.995862i	$-0.471034\pi$
$457$	3.88536	0.181750	0.0908748	+	0.995862i	$0.471034\pi$
$458$	0.890182	0.0415955
$459$	−6.88315	−0.321278
$460$	−2.10238	−0.0980242
$461$	15.7252	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$461$	15.7252	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$462$	−0.551237	−0.0256459
$463$	35.4757	1.64869	0.824347	−	0.566084i	$-0.191542\pi$
$463$	35.4757	1.64869	0.824347	+	0.566084i	$0.191542\pi$
$464$	−31.2740	−1.45186
$465$	−30.5240	−1.41552
$466$	−0.465707	−0.0215734
$467$	13.5528	0.627148	0.313574	−	0.949564i	$-0.398474\pi$
$467$	13.5528	0.627148	0.313574	+	0.949564i	$0.398474\pi$
$468$	−46.4173	−2.14564
$469$	19.1155	0.882673
$470$	−0.828244	−0.0382040
$471$	15.4581	0.712271
$472$	0.985649	0.0453682
$473$	8.04835	0.370063
$474$	−0.849651	−0.0390257
$475$	1.00000	0.0458831
$476$	1.95975	0.0898252
$477$	17.2266	0.788752
$478$	0.529959	0.0242398
$479$	39.8445	1.82054	0.910271	−	0.414012i	$-0.135873\pi$
$479$	39.8445	1.82054	0.910271	+	0.414012i	$0.135873\pi$
$480$	−3.37147	−0.153886
$481$	−16.7659	−0.764458
$482$	−2.13114	−0.0970707
$483$	6.59872	0.300252
$484$	−1.99223	−0.0905558
$485$	−4.69521	−0.213198
$486$	−2.59103	−0.117532
$487$	35.9781	1.63032	0.815161	−	0.579234i	$-0.196648\pi$
$487$	35.9781	1.63032	0.815161	+	0.579234i	$0.196648\pi$
$488$	0.540717	0.0244771
$489$	−51.4761	−2.32783
$490$	−0.281242	−0.0127052
$491$	−37.4470	−1.68996	−0.844980	−	0.534798i	$-0.820388\pi$
$491$	−37.4470	−1.68996	−0.844980	+	0.534798i	$0.820388\pi$
$492$	53.1407	2.39577
$493$	−3.98680	−0.179557
$494$	0.282792	0.0127234
$495$	−7.26315	−0.326454
$496$	37.6683	1.69136
$497$	8.68626	0.389632
$498$	0.629030	0.0281875
$499$	28.2460	1.26446	0.632232	−	0.774779i	$-0.282139\pi$
$499$	28.2460	1.26446	0.632232	+	0.774779i	$0.282139\pi$
$500$	−1.99223	−0.0890952
$501$	63.5986	2.84138
$502$	−1.89591	−0.0846186
$503$	−15.8937	−0.708667	−0.354333	−	0.935119i	$-0.615292\pi$
$503$	−15.8937	−0.708667	−0.354333	+	0.935119i	$0.615292\pi$
$504$	4.98929	0.222240
$505$	0.160574	0.00714545
$506$	−0.0930303	−0.00413570
$507$	8.68062	0.385520
$508$	12.2442	0.543251
$509$	1.13300	0.0502195	0.0251097	−	0.999685i	$-0.492006\pi$
$509$	1.13300	0.0502195	0.0251097	+	0.999685i	$0.492006\pi$
$510$	−0.142334	−0.00630265
$511$	3.04470	0.134690
$512$	6.94330	0.306854
$513$	−13.6575	−0.602994
$514$	−0.709348	−0.0312880
$515$	−12.1749	−0.536488
$516$	−51.3672	−2.26132
$517$	9.39522	0.413201
$518$	0.899308	0.0395133
$519$	74.1195	3.25349
$520$	−1.12897	−0.0495086
$521$	−35.7749	−1.56732	−0.783662	−	0.621187i	$-0.786651\pi$
$521$	−35.7749	−1.56732	−0.783662	+	0.621187i	$0.786651\pi$
$522$	−5.06507	−0.221692
$523$	−11.7973	−0.515862	−0.257931	−	0.966163i	$-0.583041\pi$
$523$	−11.7973	−0.515862	−0.257931	+	0.966163i	$0.583041\pi$
$524$	26.8147	1.17140
$525$	6.25297	0.272902
$526$	−1.72633	−0.0752718
$527$	4.80195	0.209176
$528$	12.6653	0.551185
$529$	−21.8864	−0.951581
$530$	0.209087	0.00908214
$531$	−20.3414	−0.882741
$532$	3.88853	0.168589
$533$	26.7093	1.15691
$534$	−1.72497	−0.0746469
$535$	16.2082	0.700741
$536$	3.44673	0.148876
$537$	−14.9409	−0.644748
$538$	−2.40037	−0.103488
$539$	3.19028	0.137415
$540$	27.2089	1.17088
$541$	−29.1292	−1.25236	−0.626181	−	0.779678i	$-0.715383\pi$
$541$	−29.1292	−1.25236	−0.626181	+	0.779678i	$0.715383\pi$
$542$	1.45415	0.0624611
$543$	−76.2085	−3.27042
$544$	0.530390	0.0227403
$545$	17.2276	0.737947
$546$	1.76829	0.0756758
$547$	25.3904	1.08561	0.542807	−	0.839857i	$-0.317361\pi$
$547$	25.3904	1.08561	0.542807	+	0.839857i	$0.317361\pi$
$548$	−14.8614	−0.634847
$549$	−11.1591	−0.476258
$550$	−0.0881559	−0.00375898
$551$	−7.91058	−0.337002
$552$	1.18982	0.0506420
$553$	−5.87212	−0.249708
$554$	−1.23844	−0.0526161
$555$	16.7437	0.710730
$556$	36.5673	1.55080
$557$	−2.80423	−0.118819	−0.0594096	−	0.998234i	$-0.518922\pi$
$557$	−2.80423	−0.118819	−0.0594096	+	0.998234i	$0.518922\pi$
$558$	6.10068	0.258262
$559$	−25.8180	−1.09198
$560$	−7.71650	−0.326082
$561$	1.61457	0.0681671
$562$	−0.365070	−0.0153996
$563$	14.6693	0.618239	0.309119	−	0.951023i	$-0.399966\pi$
$563$	14.6693	0.618239	0.309119	+	0.951023i	$0.399966\pi$
$564$	−59.9634	−2.52491
$565$	−9.74136	−0.409822
$566$	0.366453	0.0154031
$567$	−42.8703	−1.80038
$568$	1.56622	0.0657172
$569$	−38.3728	−1.60867	−0.804336	−	0.594175i	$-0.797479\pi$
$569$	−38.3728	−1.60867	−0.804336	+	0.594175i	$0.797479\pi$
$570$	−0.282418	−0.0118292
$571$	25.0213	1.04711	0.523554	−	0.851993i	$-0.324606\pi$
$571$	25.0213	1.04711	0.523554	+	0.851993i	$0.324606\pi$
$572$	6.39079	0.267212
$573$	13.2096	0.551840
$574$	−1.43267	−0.0597984
$575$	1.05529	0.0440087
$576$	−56.7549	−2.36479
$577$	−28.9574	−1.20551	−0.602756	−	0.797925i	$-0.705931\pi$
$577$	−28.9574	−1.20551	−0.602756	+	0.797925i	$0.705931\pi$
$578$	−1.47626	−0.0614043
$579$	−53.3607	−2.21760
$580$	15.7597	0.654386
$581$	4.34736	0.180359
$582$	1.32601	0.0549649
$583$	−2.37178	−0.0982292
$584$	0.548991	0.0227174
$585$	23.2992	0.963303
$586$	−0.471515	−0.0194781
$587$	−45.1372	−1.86301	−0.931506	−	0.363726i	$-0.881504\pi$
$587$	−45.1372	−1.86301	−0.931506	+	0.363726i	$0.881504\pi$
$588$	−20.3615	−0.839693
$589$	9.52800	0.392594
$590$	−0.246892	−0.0101644
$591$	−40.7769	−1.67734
$592$	−20.6626	−0.849227
$593$	8.57775	0.352246	0.176123	−	0.984368i	$-0.443644\pi$
$593$	8.57775	0.352246	0.176123	+	0.984368i	$0.443644\pi$
$594$	1.20399	0.0494003
$595$	−0.983699	−0.0403277
$596$	38.3279	1.56997
$597$	54.6351	2.23607
$598$	0.298428	0.0122036
$599$	6.04473	0.246981	0.123490	−	0.992346i	$-0.460591\pi$
$599$	6.04473	0.246981	0.123490	+	0.992346i	$0.460591\pi$
$600$	1.12748	0.0460290
$601$	22.5599	0.920237	0.460118	−	0.887858i	$-0.347807\pi$
$601$	22.5599	0.920237	0.460118	+	0.887858i	$0.347807\pi$
$602$	1.38486	0.0564425
$603$	−71.1320	−2.89672
$604$	−15.4679	−0.629378
$605$	1.00000	0.0406558
$606$	−0.0453489	−0.00184217
$607$	38.8854	1.57831	0.789155	−	0.614194i	$-0.210519\pi$
$607$	38.8854	1.57831	0.789155	+	0.614194i	$0.210519\pi$
$608$	1.05240	0.0426803
$609$	−49.4647	−2.00441
$610$	−0.135442	−0.00548390
$611$	−30.1385	−1.21927
$612$	−7.29257	−0.294785
$613$	−26.0557	−1.05238	−0.526191	−	0.850367i	$-0.676380\pi$
$613$	−26.0557	−1.05238	−0.526191	+	0.850367i	$0.676380\pi$
$614$	−2.18047	−0.0879967
$615$	−26.6740	−1.07560
$616$	−0.686931	−0.0276772
$617$	1.39432	0.0561331	0.0280665	−	0.999606i	$-0.491065\pi$
$617$	1.39432	0.0561331	0.0280665	+	0.999606i	$0.491065\pi$
$618$	3.43840	0.138313
$619$	9.82493	0.394897	0.197449	−	0.980313i	$-0.436734\pi$
$619$	9.82493	0.394897	0.197449	+	0.980313i	$0.436734\pi$
$620$	−18.9819	−0.762333
$621$	−14.4127	−0.578360
$622$	1.58673	0.0636219
$623$	−11.9217	−0.477632
$624$	−40.6284	−1.62644
$625$	1.00000	0.0400000
$626$	−3.02998	−0.121102
$627$	3.20362	0.127940
$628$	9.61290	0.383596
$629$	−2.63407	−0.105027
$630$	−1.24975	−0.0497912
$631$	−39.0163	−1.55321	−0.776607	−	0.629986i	$-0.783061\pi$
$631$	−39.0163	−1.55321	−0.776607	+	0.629986i	$0.783061\pi$
$632$	−1.05880	−0.0421169
$633$	48.0726	1.91071
$634$	−1.15657	−0.0459333
$635$	−6.14600	−0.243897
$636$	15.1375	0.600241
$637$	−10.2340	−0.405485
$638$	0.697365	0.0276089
$639$	−32.3230	−1.27868
$640$	−2.79365	−0.110429
$641$	24.6377	0.973130	0.486565	−	0.873644i	$-0.338250\pi$
$641$	24.6377	0.973130	0.486565	+	0.873644i	$0.338250\pi$
$642$	−4.57748	−0.180659
$643$	31.8076	1.25437	0.627185	−	0.778871i	$-0.284207\pi$
$643$	31.8076	1.25437	0.627185	+	0.778871i	$0.284207\pi$
$644$	4.10354	0.161702
$645$	25.7838	1.01524
$646$	0.0444291	0.00174804
$647$	18.5631	0.729790	0.364895	−	0.931049i	$-0.381105\pi$
$647$	18.5631	0.729790	0.364895	+	0.931049i	$0.381105\pi$
$648$	−7.72996	−0.303661
$649$	2.80063	0.109934
$650$	0.282792	0.0110920
$651$	59.5783	2.33506
$652$	−32.0114	−1.25366
$653$	11.2747	0.441212	0.220606	−	0.975363i	$-0.429196\pi$
$653$	11.2747	0.441212	0.220606	+	0.975363i	$0.429196\pi$
$654$	−4.86537	−0.190251
$655$	−13.4596	−0.525911
$656$	32.9171	1.28520
$657$	−11.3298	−0.442019
$658$	1.61661	0.0630219
$659$	−39.1971	−1.52690	−0.763451	−	0.645866i	$-0.776497\pi$
$659$	−39.1971	−1.52690	−0.763451	+	0.645866i	$0.776497\pi$
$660$	−6.38233	−0.248432
$661$	−42.1179	−1.63820	−0.819099	−	0.573652i	$-0.805526\pi$
$661$	−42.1179	−1.63820	−0.819099	+	0.573652i	$0.805526\pi$
$662$	−0.0803354	−0.00312232
$663$	−5.17931	−0.201148
$664$	0.783874	0.0304202
$665$	−1.95185	−0.0756895
$666$	−3.34647	−0.129673
$667$	−8.34798	−0.323235
$668$	39.5500	1.53024
$669$	−13.5116	−0.522388
$670$	−0.863359	−0.0333545
$671$	1.53640	0.0593119
$672$	6.58060	0.253852
$673$	25.5042	0.983116	0.491558	−	0.870845i	$-0.336428\pi$
$673$	25.5042	0.983116	0.491558	+	0.870845i	$0.336428\pi$
$674$	1.14509	0.0441073
$675$	−13.6575	−0.525678
$676$	5.39821	0.207623
$677$	48.5446	1.86572	0.932861	−	0.360237i	$-0.117304\pi$
$677$	48.5446	1.86572	0.932861	+	0.360237i	$0.117304\pi$
$678$	2.75113	0.105657
$679$	9.16434	0.351695
$680$	−0.177371	−0.00680187
$681$	−59.2960	−2.27223
$682$	−0.839949	−0.0321633
$683$	22.0305	0.842975	0.421488	−	0.906834i	$-0.361508\pi$
$683$	22.0305	0.842975	0.421488	+	0.906834i	$0.361508\pi$
$684$	−14.4699	−0.553269
$685$	7.45968	0.285020
$686$	1.75341	0.0669456
$687$	−32.3495	−1.23421
$688$	−31.8186	−1.21307
$689$	7.60834	0.289855
$690$	−0.298033	−0.0113459
$691$	−23.7336	−0.902870	−0.451435	−	0.892304i	$-0.649088\pi$
$691$	−23.7336	−0.902870	−0.451435	+	0.892304i	$0.649088\pi$
$692$	46.0926	1.75218
$693$	14.1766	0.538524
$694$	−0.749330	−0.0284442
$695$	−18.3550	−0.696244
$696$	−8.91899	−0.338074
$697$	4.19627	0.158945
$698$	−2.02125	−0.0765054
$699$	16.9239	0.640122
$700$	3.88853	0.146973
$701$	3.46600	0.130909	0.0654546	−	0.997856i	$-0.479150\pi$
$701$	3.46600	0.130909	0.0654546	+	0.997856i	$0.479150\pi$
$702$	−3.86223	−0.145771
$703$	−5.22650	−0.197121
$704$	7.81409	0.294505
$705$	30.0987	1.13358
$706$	−2.86539	−0.107840
$707$	−0.313416	−0.0117872
$708$	−17.8746	−0.671767
$709$	11.4552	0.430208	0.215104	−	0.976591i	$-0.430991\pi$
$709$	11.4552	0.430208	0.215104	+	0.976591i	$0.430991\pi$
$710$	−0.392318	−0.0147234
$711$	21.8511	0.819481
$712$	−2.14960	−0.0805597
$713$	10.0548	0.376556
$714$	0.277814	0.0103969
$715$	−3.20786	−0.119967
$716$	−9.29129	−0.347232
$717$	−19.2589	−0.719237
$718$	−0.621938	−0.0232105
$719$	12.8364	0.478718	0.239359	−	0.970931i	$-0.423063\pi$
$719$	12.8364	0.478718	0.239359	+	0.970931i	$0.423063\pi$
$720$	28.7144	1.07012
$721$	23.7635	0.884998
$722$	0.0881559	0.00328082
$723$	77.4463	2.88026
$724$	−47.3917	−1.76130
$725$	−7.91058	−0.293792
$726$	−0.282418	−0.0104815
$727$	−22.3762	−0.829887	−0.414943	−	0.909847i	$-0.636199\pi$
$727$	−22.3762	−0.829887	−0.414943	+	0.909847i	$0.636199\pi$
$728$	2.20358	0.0816701
$729$	28.2672	1.04693
$730$	−0.137515	−0.00508966
$731$	−4.05623	−0.150025
$732$	−9.80579	−0.362433
$733$	35.6014	1.31497	0.657483	−	0.753469i	$-0.271621\pi$
$733$	35.6014	1.31497	0.657483	+	0.753469i	$0.271621\pi$
$734$	1.30736	0.0482557
$735$	10.2204	0.376987
$736$	1.11059	0.0409367
$737$	9.79354	0.360750
$738$	5.33119	0.196244
$739$	24.3156	0.894465	0.447233	−	0.894418i	$-0.352410\pi$
$739$	24.3156	0.894465	0.447233	+	0.894418i	$0.352410\pi$
$740$	10.4124	0.382766
$741$	−10.2768	−0.377526
$742$	−0.408106	−0.0149820
$743$	34.2814	1.25766	0.628831	−	0.777542i	$-0.283534\pi$
$743$	34.2814	1.25766	0.628831	+	0.777542i	$0.283534\pi$
$744$	10.7426	0.393842
$745$	−19.2387	−0.704852
$746$	0.854787	0.0312960
$747$	−16.1773	−0.591895
$748$	1.00405	0.0367117
$749$	−31.6359	−1.15595
$750$	−0.282418	−0.0103124
$751$	32.2464	1.17669	0.588345	−	0.808610i	$-0.299780\pi$
$751$	32.2464	1.17669	0.588345	+	0.808610i	$0.299780\pi$
$752$	−37.1433	−1.35448
$753$	68.8980	2.51078
$754$	−2.23705	−0.0814685
$755$	7.76410	0.282565
$756$	−53.1076	−1.93151
$757$	2.94119	0.106899	0.0534497	−	0.998571i	$-0.482978\pi$
$757$	2.94119	0.106899	0.0534497	+	0.998571i	$0.482978\pi$
$758$	0.0912923	0.00331589
$759$	3.38075	0.122714
$760$	−0.351939	−0.0127662
$761$	26.6626	0.966519	0.483260	−	0.875477i	$-0.339453\pi$
$761$	26.6626	0.966519	0.483260	+	0.875477i	$0.339453\pi$
$762$	1.73574	0.0628792
$763$	−33.6256	−1.21733
$764$	8.21466	0.297196
$765$	3.66051	0.132346
$766$	0.119546	0.00431937
$767$	−8.98403	−0.324394
$768$	−49.2777	−1.77816
$769$	−7.23123	−0.260765	−0.130382	−	0.991464i	$-0.541621\pi$
$769$	−7.23123	−0.260765	−0.130382	+	0.991464i	$0.541621\pi$
$770$	0.172067	0.00620087
$771$	25.7779	0.928370
$772$	−33.1834	−1.19430
$773$	−8.64003	−0.310760	−0.155380	−	0.987855i	$-0.549660\pi$
$773$	−8.64003	−0.310760	−0.155380	+	0.987855i	$0.549660\pi$
$774$	−5.15327	−0.185231
$775$	9.52800	0.342256
$776$	1.65243	0.0593186
$777$	−32.6811	−1.17243
$778$	−3.02232	−0.108355
$779$	8.32622	0.298318
$780$	20.4736	0.733074
$781$	4.45027	0.159243
$782$	0.0468857	0.00167663
$783$	108.039	3.86099
$784$	−12.6126	−0.450449
$785$	−4.82520	−0.172219
$786$	3.80124	0.135586
$787$	16.4120	0.585025	0.292512	−	0.956262i	$-0.405509\pi$
$787$	16.4120	0.585025	0.292512	+	0.956262i	$0.405509\pi$
$788$	−25.3579	−0.903338
$789$	62.7356	2.23345
$790$	0.265216	0.00943597
$791$	19.0137	0.676048
$792$	2.55618	0.0908300
$793$	−4.92854	−0.175018
$794$	−3.12897	−0.111043
$795$	−7.59828	−0.269483
$796$	33.9759	1.20424
$797$	−54.6736	−1.93664	−0.968319	−	0.249716i	$-0.919663\pi$
$797$	−54.6736	−1.93664	−0.968319	+	0.249716i	$0.919663\pi$
$798$	0.551237	0.0195136
$799$	−4.73503	−0.167513
$800$	1.05240	0.0372078
$801$	44.3625	1.56747
$802$	2.51168	0.0886907
$803$	1.55991	0.0550479
$804$	−62.5057	−2.20440
$805$	−2.05977	−0.0725974
$806$	2.69444	0.0949076
$807$	87.2304	3.07066
$808$	−0.0565122	−0.00198809
$809$	−20.6416	−0.725719	−0.362859	−	0.931844i	$-0.618199\pi$
$809$	−20.6416	−0.725719	−0.362859	+	0.931844i	$0.618199\pi$
$810$	1.93625	0.0680329
$811$	0.728429	0.0255786	0.0127893	−	0.999918i	$-0.495929\pi$
$811$	0.728429	0.0255786	0.0127893	+	0.999918i	$0.495929\pi$
$812$	−30.7605	−1.07948
$813$	−52.8443	−1.85333
$814$	0.460747	0.0161492
$815$	16.0681	0.562842
$816$	−6.38309	−0.223453
$817$	−8.04835	−0.281576
$818$	2.74324	0.0959150
$819$	−45.4765	−1.58908
$820$	−16.5877	−0.579269
$821$	7.88685	0.275253	0.137626	−	0.990484i	$-0.456053\pi$
$821$	7.88685	0.275253	0.137626	+	0.990484i	$0.456053\pi$
$822$	−2.10675	−0.0734812
$823$	21.3819	0.745327	0.372663	−	0.927967i	$-0.378445\pi$
$823$	21.3819	0.745327	0.372663	+	0.927967i	$0.378445\pi$
$824$	4.28480	0.149268
$825$	3.20362	0.111536
$826$	0.481896	0.0167673
$827$	−18.0869	−0.628943	−0.314471	−	0.949267i	$-0.601827\pi$
$827$	−18.0869	−0.628943	−0.314471	+	0.949267i	$0.601827\pi$
$828$	−15.2699	−0.530667
$829$	38.7216	1.34486	0.672429	−	0.740162i	$-0.265251\pi$
$829$	38.7216	1.34486	0.672429	+	0.740162i	$0.265251\pi$
$830$	−0.196350	−0.00681541
$831$	45.0052	1.56121
$832$	−25.0665	−0.869025
$833$	−1.60785	−0.0557087
$834$	5.18377	0.179499
$835$	−19.8521	−0.687012
$836$	1.99223	0.0689027
$837$	−130.129	−4.49790
$838$	−2.10522	−0.0727235
$839$	11.1336	0.384375	0.192187	−	0.981358i	$-0.438442\pi$
$839$	11.1336	0.384375	0.192187	+	0.981358i	$0.438442\pi$
$840$	−2.20066	−0.0759301
$841$	33.5774	1.15784
$842$	−0.100219	−0.00345376
$843$	13.2668	0.456932
$844$	29.8948	1.02902
$845$	−2.70963	−0.0932142
$846$	−6.01566	−0.206823
$847$	−1.95185	−0.0670663
$848$	9.37668	0.321996
$849$	−13.3170	−0.457039
$850$	0.0444291	0.00152391
$851$	−5.51548	−0.189068
$852$	−28.4031	−0.973075
$853$	−37.4831	−1.28340	−0.641698	−	0.766957i	$-0.721770\pi$
$853$	−37.4831	−1.28340	−0.641698	+	0.766957i	$0.721770\pi$
$854$	0.264363	0.00904632
$855$	7.26315	0.248395
$856$	−5.70429	−0.194968
$857$	38.7649	1.32418	0.662091	−	0.749423i	$-0.269669\pi$
$857$	38.7649	1.32418	0.662091	+	0.749423i	$0.269669\pi$
$858$	0.905957	0.0309288
$859$	14.9492	0.510060	0.255030	−	0.966933i	$-0.417915\pi$
$859$	14.9492	0.510060	0.255030	+	0.966933i	$0.417915\pi$
$860$	16.0341	0.546760
$861$	52.0636	1.77432
$862$	−0.0527954	−0.00179822
$863$	−45.1615	−1.53732	−0.768658	−	0.639660i	$-0.779075\pi$
$863$	−45.1615	−1.53732	−0.768658	+	0.639660i	$0.779075\pi$
$864$	−14.3731	−0.488983
$865$	−23.1362	−0.786655
$866$	2.85146	0.0968965
$867$	53.6478	1.82197
$868$	37.0499	1.25756
$869$	−3.00849	−0.102056
$870$	2.23409	0.0757427
$871$	−31.4163	−1.06450
$872$	−6.06304	−0.205320
$873$	−34.1020	−1.15418
$874$	0.0930303	0.00314680
$875$	−1.95185	−0.0659845
$876$	−9.95584	−0.336377
$877$	13.1538	0.444171	0.222086	−	0.975027i	$-0.428714\pi$
$877$	13.1538	0.444171	0.222086	+	0.975027i	$0.428714\pi$
$878$	0.825585	0.0278622
$879$	17.1350	0.577949
$880$	−3.95343	−0.133270
$881$	−19.8386	−0.668380	−0.334190	−	0.942506i	$-0.608463\pi$
$881$	−19.8386	−0.668380	−0.334190	+	0.942506i	$0.608463\pi$
$882$	−2.04271	−0.0687815
$883$	18.2493	0.614138	0.307069	−	0.951687i	$-0.400652\pi$
$883$	18.2493	0.614138	0.307069	+	0.951687i	$0.400652\pi$
$884$	−3.22085	−0.108329
$885$	8.97214	0.301595
$886$	0.930549	0.0312624
$887$	−19.6448	−0.659606	−0.329803	−	0.944050i	$-0.606982\pi$
$887$	−19.6448	−0.659606	−0.329803	+	0.944050i	$0.606982\pi$
$888$	−5.89275	−0.197748
$889$	11.9961	0.402335
$890$	0.538446	0.0180488
$891$	−21.9639	−0.735819
$892$	−8.40243	−0.281334
$893$	−9.39522	−0.314399
$894$	5.43335	0.181718
$895$	4.66377	0.155892
$896$	5.45278	0.182165
$897$	−10.8450	−0.362103
$898$	−1.18481	−0.0395376
$899$	−75.3720	−2.51380
$900$	−14.4699	−0.482329
$901$	1.19534	0.0398225
$902$	−0.734005	−0.0244397
$903$	−50.3261	−1.67475
$904$	3.42836	0.114026
$905$	23.7883	0.790749
$906$	−2.19272	−0.0728482
$907$	−36.7829	−1.22136	−0.610678	−	0.791879i	$-0.709103\pi$
$907$	−36.7829	−1.22136	−0.610678	+	0.791879i	$0.709103\pi$
$908$	−36.8743	−1.22372
$909$	1.16627	0.0386828
$910$	−0.551967	−0.0182975
$911$	−35.8447	−1.18759	−0.593794	−	0.804617i	$-0.702370\pi$
$911$	−35.8447	−1.18759	−0.593794	+	0.804617i	$0.702370\pi$
$912$	−12.6653	−0.419389
$913$	2.22730	0.0737130
$914$	0.342518	0.0113295
$915$	4.92202	0.162717
$916$	−20.1171	−0.664689
$917$	26.2712	0.867550
$918$	−0.606791	−0.0200271
$919$	−2.43716	−0.0803943	−0.0401972	−	0.999192i	$-0.512799\pi$
$919$	−2.43716	−0.0803943	−0.0401972	+	0.999192i	$0.512799\pi$
$920$	−0.371398	−0.0122446
$921$	79.2391	2.61102
$922$	1.38627	0.0456544
$923$	−14.2759	−0.469895
$924$	12.4574	0.409817
$925$	−5.22650	−0.171846
$926$	3.12739	0.102772
$927$	−88.4279	−2.90435
$928$	−8.32506	−0.273284
$929$	50.7417	1.66478	0.832391	−	0.554189i	$-0.186971\pi$
$929$	50.7417	1.66478	0.832391	+	0.554189i	$0.186971\pi$
$930$	−2.69087	−0.0882372
$931$	−3.19028	−0.104557
$932$	10.5245	0.344740
$933$	−57.6621	−1.88777
$934$	1.19476	0.0390937
$935$	−0.503983	−0.0164820
$936$	−8.19988	−0.268021
$937$	−34.9861	−1.14295	−0.571473	−	0.820621i	$-0.693628\pi$
$937$	−34.9861	−1.14295	−0.571473	+	0.820621i	$0.693628\pi$
$938$	1.68515	0.0550220
$939$	110.110	3.59331
$940$	18.7174	0.610495
$941$	−38.7579	−1.26347	−0.631736	−	0.775183i	$-0.717658\pi$
$941$	−38.7579	−1.26347	−0.631736	+	0.775183i	$0.717658\pi$
$942$	1.36272	0.0443999
$943$	8.78660	0.286131
$944$	−11.0721	−0.360366
$945$	26.6574	0.867165
$946$	0.709509	0.0230681
$947$	33.8957	1.10146	0.550732	−	0.834682i	$-0.314349\pi$
$947$	33.8957	1.10146	0.550732	+	0.834682i	$0.314349\pi$
$948$	19.2012	0.623626
$949$	−5.00396	−0.162435
$950$	0.0881559	0.00286016
$951$	42.0301	1.36292
$952$	0.346202	0.0112205
$953$	56.6907	1.83639	0.918195	−	0.396128i	$-0.129646\pi$
$953$	56.6907	1.83639	0.918195	+	0.396128i	$0.129646\pi$
$954$	1.51863	0.0491674
$955$	−4.12335	−0.133428
$956$	−11.9765	−0.387348
$957$	−25.3425	−0.819206
$958$	3.51253	0.113485
$959$	−14.5602	−0.470173
$960$	25.0333	0.807947
$961$	59.7827	1.92847
$962$	−1.47801	−0.0476530
$963$	117.723	3.79356
$964$	48.1614	1.55118
$965$	16.6564	0.536189
$966$	0.581716	0.0187164
$967$	45.0600	1.44903	0.724517	−	0.689257i	$-0.242063\pi$
$967$	45.0600	1.44903	0.724517	+	0.689257i	$0.242063\pi$
$968$	−0.351939	−0.0113117
$969$	−1.61457	−0.0518674
$970$	−0.413910	−0.0132899
$971$	−24.1262	−0.774247	−0.387124	−	0.922028i	$-0.626531\pi$
$971$	−24.1262	−0.774247	−0.387124	+	0.922028i	$0.626531\pi$
$972$	58.5546	1.87814
$973$	35.8262	1.14853
$974$	3.17168	0.101627
$975$	−10.2768	−0.329119
$976$	−6.07404	−0.194425
$977$	−12.1551	−0.388875	−0.194437	−	0.980915i	$-0.562288\pi$
$977$	−12.1551	−0.388875	−0.194437	+	0.980915i	$0.562288\pi$
$978$	−4.53792	−0.145107
$979$	−6.10788	−0.195209
$980$	6.35578	0.203028
$981$	125.126	3.99498
$982$	−3.30118	−0.105345
$983$	−26.5655	−0.847308	−0.423654	−	0.905824i	$-0.639253\pi$
$983$	−26.5655	−0.847308	−0.423654	+	0.905824i	$0.639253\pi$
$984$	9.38761	0.299266
$985$	12.7284	0.405561
$986$	−0.351460	−0.0111928
$987$	−58.7480	−1.86997
$988$	−6.39079	−0.203318
$989$	−8.49336	−0.270073
$990$	−0.640290	−0.0203498
$991$	6.91424	0.219638	0.109819	−	0.993952i	$-0.464973\pi$
$991$	6.91424	0.219638	0.109819	+	0.993952i	$0.464973\pi$
$992$	10.0272	0.318365
$993$	2.91941	0.0926448
$994$	0.765745	0.0242880
$995$	−17.0542	−0.540655
$996$	−14.2154	−0.450432
$997$	42.2221	1.33719	0.668593	−	0.743628i	$-0.266897\pi$
$997$	42.2221	1.33719	0.668593	+	0.743628i	$0.266897\pi$
$998$	2.49005	0.0788212
$999$	71.3809	2.25839

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.d.1.4	✓	5
3.2	odd	2		9405.2.a.v.1.2		5
5.4	even	2		5225.2.a.j.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.d.1.4	✓	5	1.1	even	1	trivial
5225.2.a.j.1.2		5	5.4	even	2
9405.2.a.v.1.2		5	3.2	odd	2

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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0881559 q^{2} -3.20362 q^{3} -1.99223 q^{4} +1.00000 q^{5} -0.282418 q^{6} -1.95185 q^{7} -0.351939 q^{8} +7.26315 q^{9} +O(q^{10})\)	\(q+0.0881559 q^{2} -3.20362 q^{3} -1.99223 q^{4} +1.00000 q^{5} -0.282418 q^{6} -1.95185 q^{7} -0.351939 q^{8} +7.26315 q^{9} +0.0881559 q^{10} -1.00000 q^{11} +6.38233 q^{12} +3.20786 q^{13} -0.172067 q^{14} -3.20362 q^{15} +3.95343 q^{16} +0.503983 q^{17} +0.640290 q^{18} +1.00000 q^{19} -1.99223 q^{20} +6.25297 q^{21} -0.0881559 q^{22} +1.05529 q^{23} +1.12748 q^{24} +1.00000 q^{25} +0.282792 q^{26} -13.6575 q^{27} +3.88853 q^{28} -7.91058 q^{29} -0.282418 q^{30} +9.52800 q^{31} +1.05240 q^{32} +3.20362 q^{33} +0.0444291 q^{34} -1.95185 q^{35} -14.4699 q^{36} -5.22650 q^{37} +0.0881559 q^{38} -10.2768 q^{39} -0.351939 q^{40} +8.32622 q^{41} +0.551237 q^{42} -8.04835 q^{43} +1.99223 q^{44} +7.26315 q^{45} +0.0930303 q^{46} -9.39522 q^{47} -12.6653 q^{48} -3.19028 q^{49} +0.0881559 q^{50} -1.61457 q^{51} -6.39079 q^{52} +2.37178 q^{53} -1.20399 q^{54} -1.00000 q^{55} +0.686931 q^{56} -3.20362 q^{57} -0.697365 q^{58} -2.80063 q^{59} +6.38233 q^{60} -1.53640 q^{61} +0.839949 q^{62} -14.1766 q^{63} -7.81409 q^{64} +3.20786 q^{65} +0.282418 q^{66} -9.79354 q^{67} -1.00405 q^{68} -3.38075 q^{69} -0.172067 q^{70} -4.45027 q^{71} -2.55618 q^{72} -1.55991 q^{73} -0.460747 q^{74} -3.20362 q^{75} -1.99223 q^{76} +1.95185 q^{77} -0.905957 q^{78} +3.00849 q^{79} +3.95343 q^{80} +21.9639 q^{81} +0.734005 q^{82} -2.22730 q^{83} -12.4574 q^{84} +0.503983 q^{85} -0.709509 q^{86} +25.3425 q^{87} +0.351939 q^{88} +6.10788 q^{89} +0.640290 q^{90} -6.26126 q^{91} -2.10238 q^{92} -30.5240 q^{93} -0.828244 q^{94} +1.00000 q^{95} -3.37147 q^{96} -4.69521 q^{97} -0.281242 q^{98} -7.26315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 5 q^{5} + 2 q^{6} - 11 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) 5 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 5 * q^5 + 2 * q^6 - 11 * q^7 + 3 * q^8 + 8 * q^9	\( 5 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 5 q^{5} + 2 q^{6} - 11 q^{7} + 3 q^{8} + 8 q^{9} - 3 q^{10} - 5 q^{11} - 7 q^{12} + q^{13} - 7 q^{15} - 3 q^{16} - 3 q^{17} - 7 q^{18} + 5 q^{19} + 5 q^{20} + 11 q^{21} + 3 q^{22} - 8 q^{23} + 9 q^{24} + 5 q^{25} - 16 q^{26} - 10 q^{27} - 22 q^{28} + 11 q^{29} + 2 q^{30} - 5 q^{31} - 2 q^{32} + 7 q^{33} + 4 q^{34} - 11 q^{35} - 3 q^{36} - 9 q^{37} - 3 q^{38} - 8 q^{39} + 3 q^{40} + 15 q^{41} + 11 q^{42} - 13 q^{43} - 5 q^{44} + 8 q^{45} + 18 q^{46} - 20 q^{47} - 20 q^{48} + 20 q^{49} - 3 q^{50} + 24 q^{51} + q^{52} - 5 q^{53} + 17 q^{54} - 5 q^{55} - 7 q^{57} - 33 q^{58} - 17 q^{59} - 7 q^{60} + 3 q^{61} + 14 q^{62} - 22 q^{63} - 17 q^{64} + q^{65} - 2 q^{66} - 28 q^{67} - 25 q^{68} - 2 q^{69} - 6 q^{71} - 26 q^{72} - 16 q^{73} - 21 q^{74} - 7 q^{75} + 5 q^{76} + 11 q^{77} + 29 q^{78} + 3 q^{79} - 3 q^{80} + q^{81} + 2 q^{82} - 33 q^{83} + 33 q^{84} - 3 q^{85} + 10 q^{86} - 3 q^{88} - 16 q^{89} - 7 q^{90} - 22 q^{91} - 19 q^{92} - 26 q^{93} - 10 q^{94} + 5 q^{95} + 5 q^{96} - 14 q^{97} + 10 q^{98} - 8 q^{99}+O(q^{100}) \) 5 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 5 * q^5 + 2 * q^6 - 11 * q^7 + 3 * q^8 + 8 * q^9 - 3 * q^10 - 5 * q^11 - 7 * q^12 + q^13 - 7 * q^15 - 3 * q^16 - 3 * q^17 - 7 * q^18 + 5 * q^19 + 5 * q^20 + 11 * q^21 + 3 * q^22 - 8 * q^23 + 9 * q^24 + 5 * q^25 - 16 * q^26 - 10 * q^27 - 22 * q^28 + 11 * q^29 + 2 * q^30 - 5 * q^31 - 2 * q^32 + 7 * q^33 + 4 * q^34 - 11 * q^35 - 3 * q^36 - 9 * q^37 - 3 * q^38 - 8 * q^39 + 3 * q^40 + 15 * q^41 + 11 * q^42 - 13 * q^43 - 5 * q^44 + 8 * q^45 + 18 * q^46 - 20 * q^47 - 20 * q^48 + 20 * q^49 - 3 * q^50 + 24 * q^51 + q^52 - 5 * q^53 + 17 * q^54 - 5 * q^55 - 7 * q^57 - 33 * q^58 - 17 * q^59 - 7 * q^60 + 3 * q^61 + 14 * q^62 - 22 * q^63 - 17 * q^64 + q^65 - 2 * q^66 - 28 * q^67 - 25 * q^68 - 2 * q^69 - 6 * q^71 - 26 * q^72 - 16 * q^73 - 21 * q^74 - 7 * q^75 + 5 * q^76 + 11 * q^77 + 29 * q^78 + 3 * q^79 - 3 * q^80 + q^81 + 2 * q^82 - 33 * q^83 + 33 * q^84 - 3 * q^85 + 10 * q^86 - 3 * q^88 - 16 * q^89 - 7 * q^90 - 22 * q^91 - 19 * q^92 - 26 * q^93 - 10 * q^94 + 5 * q^95 + 5 * q^96 - 14 * q^97 + 10 * q^98 - 8 * q^99

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