LMFDB - Embedded newform 8045.2.a.b.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, 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("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$1$
Dimension:	$126$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51572 q^{2} -0.0214177 q^{3} +4.32884 q^{4} -1.00000 q^{5} +0.0538809 q^{6} +1.26771 q^{7} -5.85870 q^{8} -2.99954 q^{9} +O(q^{10})$	\(q-2.51572 q^{2} -0.0214177 q^{3} +4.32884 q^{4} -1.00000 q^{5} +0.0538809 q^{6} +1.26771 q^{7} -5.85870 q^{8} -2.99954 q^{9} +2.51572 q^{10} +3.93794 q^{11} -0.0927137 q^{12} +2.08307 q^{13} -3.18920 q^{14} +0.0214177 q^{15} +6.08116 q^{16} -3.08854 q^{17} +7.54600 q^{18} -6.53090 q^{19} -4.32884 q^{20} -0.0271515 q^{21} -9.90675 q^{22} -1.53462 q^{23} +0.125480 q^{24} +1.00000 q^{25} -5.24041 q^{26} +0.128496 q^{27} +5.48772 q^{28} -0.108241 q^{29} -0.0538809 q^{30} +0.323055 q^{31} -3.58108 q^{32} -0.0843416 q^{33} +7.76990 q^{34} -1.26771 q^{35} -12.9845 q^{36} +6.32518 q^{37} +16.4299 q^{38} -0.0446145 q^{39} +5.85870 q^{40} +9.10657 q^{41} +0.0683054 q^{42} -11.2121 q^{43} +17.0467 q^{44} +2.99954 q^{45} +3.86068 q^{46} +9.05621 q^{47} -0.130244 q^{48} -5.39291 q^{49} -2.51572 q^{50} +0.0661495 q^{51} +9.01725 q^{52} +9.08799 q^{53} -0.323261 q^{54} -3.93794 q^{55} -7.42714 q^{56} +0.139877 q^{57} +0.272303 q^{58} -5.32710 q^{59} +0.0927137 q^{60} +4.83182 q^{61} -0.812717 q^{62} -3.80255 q^{63} -3.15332 q^{64} -2.08307 q^{65} +0.212180 q^{66} -12.5127 q^{67} -13.3698 q^{68} +0.0328681 q^{69} +3.18920 q^{70} +4.09103 q^{71} +17.5734 q^{72} +14.6704 q^{73} -15.9124 q^{74} -0.0214177 q^{75} -28.2712 q^{76} +4.99217 q^{77} +0.112237 q^{78} -8.24167 q^{79} -6.08116 q^{80} +8.99587 q^{81} -22.9096 q^{82} -12.1198 q^{83} -0.117534 q^{84} +3.08854 q^{85} +28.2064 q^{86} +0.00231827 q^{87} -23.0712 q^{88} +8.70002 q^{89} -7.54600 q^{90} +2.64073 q^{91} -6.64314 q^{92} -0.00691910 q^{93} -22.7829 q^{94} +6.53090 q^{95} +0.0766985 q^{96} +8.03392 q^{97} +13.5670 q^{98} -11.8120 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) $ 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.51572	−1.77888	−0.889441	−	0.457051i	$-0.848906\pi$
$2$	−2.51572	−1.77888	−0.889441	+	0.457051i	$0.848906\pi$
$3$	−0.0214177	−0.0123655	−0.00618276	−	0.999981i	$-0.501968\pi$
$3$	−0.0214177	−0.0123655	−0.00618276	+	0.999981i	$0.501968\pi$
$4$	4.32884	2.16442
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.0538809	0.0219968
$7$	1.26771	0.479150	0.239575	−	0.970878i	$-0.422992\pi$
$7$	1.26771	0.479150	0.239575	+	0.970878i	$0.422992\pi$
$8$	−5.85870	−2.07136
$9$	−2.99954	−0.999847
$10$	2.51572	0.795540
$11$	3.93794	1.18733	0.593667	−	0.804711i	$-0.297680\pi$
$11$	3.93794	1.18733	0.593667	+	0.804711i	$0.297680\pi$
$12$	−0.0927137	−0.0267641
$13$	2.08307	0.577738	0.288869	−	0.957369i	$-0.406721\pi$
$13$	2.08307	0.577738	0.288869	+	0.957369i	$0.406721\pi$
$14$	−3.18920	−0.852351
$15$	0.0214177	0.00553002
$16$	6.08116	1.52029
$17$	−3.08854	−0.749082	−0.374541	−	0.927210i	$-0.622200\pi$
$17$	−3.08854	−0.749082	−0.374541	+	0.927210i	$0.622200\pi$
$18$	7.54600	1.77861
$19$	−6.53090	−1.49829	−0.749145	−	0.662406i	$-0.769535\pi$
$19$	−6.53090	−1.49829	−0.749145	+	0.662406i	$0.769535\pi$
$20$	−4.32884	−0.967957
$21$	−0.0271515	−0.00592493
$22$	−9.90675	−2.11213
$23$	−1.53462	−0.319991	−0.159996	−	0.987118i	$-0.551148\pi$
$23$	−1.53462	−0.319991	−0.159996	+	0.987118i	$0.551148\pi$
$24$	0.125480	0.0256135
$25$	1.00000	0.200000
$26$	−5.24041	−1.02773
$27$	0.128496	0.0247291
$28$	5.48772	1.03708
$29$	−0.108241	−0.0200998	−0.0100499	−	0.999949i	$-0.503199\pi$
$29$	−0.108241	−0.0200998	−0.0100499	+	0.999949i	$0.503199\pi$
$30$	−0.0538809	−0.00983726
$31$	0.323055	0.0580225	0.0290112	−	0.999579i	$-0.490764\pi$
$31$	0.323055	0.0580225	0.0290112	+	0.999579i	$0.490764\pi$
$32$	−3.58108	−0.633052
$33$	−0.0843416	−0.0146820
$34$	7.76990	1.33253
$35$	−1.26771	−0.214282
$36$	−12.9845	−2.16409
$37$	6.32518	1.03985	0.519927	−	0.854211i	$-0.325959\pi$
$37$	6.32518	1.03985	0.519927	+	0.854211i	$0.325959\pi$
$38$	16.4299	2.66528
$39$	−0.0446145	−0.00714403
$40$	5.85870	0.926341
$41$	9.10657	1.42221	0.711104	−	0.703087i	$-0.248195\pi$
$41$	9.10657	1.42221	0.711104	+	0.703087i	$0.248195\pi$
$42$	0.0683054	0.0105398
$43$	−11.2121	−1.70982	−0.854911	−	0.518775i	$-0.826388\pi$
$43$	−11.2121	−1.70982	−0.854911	+	0.518775i	$0.826388\pi$
$44$	17.0467	2.56989
$45$	2.99954	0.447145
$46$	3.86068	0.569226
$47$	9.05621	1.32098	0.660492	−	0.750833i	$-0.270348\pi$
$47$	9.05621	1.32098	0.660492	+	0.750833i	$0.270348\pi$
$48$	−0.130244	−0.0187992
$49$	−5.39291	−0.770415
$50$	−2.51572	−0.355776
$51$	0.0661495	0.00926278
$52$	9.01725	1.25047
$53$	9.08799	1.24833	0.624166	−	0.781292i	$-0.285439\pi$
$53$	9.08799	1.24833	0.624166	+	0.781292i	$0.285439\pi$
$54$	−0.323261	−0.0439902
$55$	−3.93794	−0.530992
$56$	−7.42714	−0.992493
$57$	0.139877	0.0185271
$58$	0.272303	0.0357552
$59$	−5.32710	−0.693529	−0.346765	−	0.937952i	$-0.612720\pi$
$59$	−5.32710	−0.693529	−0.346765	+	0.937952i	$0.612720\pi$
$60$	0.0927137	0.0119693
$61$	4.83182	0.618652	0.309326	−	0.950956i	$-0.399897\pi$
$61$	4.83182	0.618652	0.309326	+	0.950956i	$0.399897\pi$
$62$	−0.812717	−0.103215
$63$	−3.80255	−0.479077
$64$	−3.15332	−0.394165
$65$	−2.08307	−0.258372
$66$	0.212180	0.0261175
$67$	−12.5127	−1.52867	−0.764335	−	0.644820i	$-0.776933\pi$
$67$	−12.5127	−1.52867	−0.764335	+	0.644820i	$0.776933\pi$
$68$	−13.3698	−1.62133
$69$	0.0328681	0.00395686
$70$	3.18920	0.381183
$71$	4.09103	0.485516	0.242758	−	0.970087i	$-0.421948\pi$
$71$	4.09103	0.485516	0.242758	+	0.970087i	$0.421948\pi$
$72$	17.5734	2.07105
$73$	14.6704	1.71704	0.858520	−	0.512781i	$-0.171385\pi$
$73$	14.6704	1.71704	0.858520	+	0.512781i	$0.171385\pi$
$74$	−15.9124	−1.84978
$75$	−0.0214177	−0.00247310
$76$	−28.2712	−3.24293
$77$	4.99217	0.568911
$78$	0.112237	0.0127084
$79$	−8.24167	−0.927261	−0.463630	−	0.886029i	$-0.653453\pi$
$79$	−8.24167	−0.927261	−0.463630	+	0.886029i	$0.653453\pi$
$80$	−6.08116	−0.679894
$81$	8.99587	0.999541
$82$	−22.9096	−2.52994
$83$	−12.1198	−1.33032	−0.665158	−	0.746703i	$-0.731636\pi$
$83$	−12.1198	−1.33032	−0.665158	+	0.746703i	$0.731636\pi$
$84$	−0.117534	−0.0128240
$85$	3.08854	0.335000
$86$	28.2064	3.04157
$87$	0.00231827	0.000248544 0
$88$	−23.0712	−2.45940
$89$	8.70002	0.922200	0.461100	−	0.887348i	$-0.347455\pi$
$89$	8.70002	0.922200	0.461100	+	0.887348i	$0.347455\pi$
$90$	−7.54600	−0.795418
$91$	2.64073	0.276823
$92$	−6.64314	−0.692595
$93$	−0.00691910	−0.000717478 0
$94$	−22.7829	−2.34987
$95$	6.53090	0.670056
$96$	0.0766985	0.00782801
$97$	8.03392	0.815721	0.407861	−	0.913044i	$-0.366275\pi$
$97$	8.03392	0.815721	0.407861	+	0.913044i	$0.366275\pi$
$98$	13.5670	1.37048
$99$	−11.8120	−1.18715
$100$	4.32884	0.432884
$101$	−15.5135	−1.54365	−0.771825	−	0.635835i	$-0.780656\pi$
$101$	−15.5135	−1.54365	−0.771825	+	0.635835i	$0.780656\pi$
$102$	−0.166413	−0.0164774
$103$	−6.50485	−0.640942	−0.320471	−	0.947258i	$-0.603841\pi$
$103$	−6.50485	−0.640942	−0.320471	+	0.947258i	$0.603841\pi$
$104$	−12.2041	−1.19671
$105$	0.0271515	0.00264971
$106$	−22.8628	−2.22063
$107$	−13.2032	−1.27640	−0.638200	−	0.769871i	$-0.720321\pi$
$107$	−13.2032	−1.27640	−0.638200	+	0.769871i	$0.720321\pi$
$108$	0.556240	0.0535242
$109$	8.83368	0.846113	0.423057	−	0.906103i	$-0.360957\pi$
$109$	8.83368	0.846113	0.423057	+	0.906103i	$0.360957\pi$
$110$	9.90675	0.944571
$111$	−0.135471	−0.0128583
$112$	7.70915	0.728447
$113$	14.1582	1.33189	0.665946	−	0.746000i	$-0.268028\pi$
$113$	14.1582	1.33189	0.665946	+	0.746000i	$0.268028\pi$
$114$	−0.351890	−0.0329576
$115$	1.53462	0.143104
$116$	−0.468557	−0.0435044
$117$	−6.24824	−0.577650
$118$	13.4015	1.23371
$119$	−3.91538	−0.358923
$120$	−0.125480	−0.0114547
$121$	4.50737	0.409761
$122$	−12.1555	−1.10051
$123$	−0.195042	−0.0175863
$124$	1.39845	0.125585
$125$	−1.00000	−0.0894427
$126$	9.56615	0.852220
$127$	−17.7093	−1.57145	−0.785724	−	0.618577i	$-0.787709\pi$
$127$	−17.7093	−1.57145	−0.785724	+	0.618577i	$0.787709\pi$
$128$	15.0950	1.33423
$129$	0.240136	0.0211428
$130$	5.24041	0.459614
$131$	−2.35056	−0.205370	−0.102685	−	0.994714i	$-0.532743\pi$
$131$	−2.35056	−0.205370	−0.102685	+	0.994714i	$0.532743\pi$
$132$	−0.365101	−0.0317780
$133$	−8.27929	−0.717906
$134$	31.4784	2.71932
$135$	−0.128496	−0.0110592
$136$	18.0948	1.55162
$137$	18.2047	1.55533	0.777664	−	0.628680i	$-0.216404\pi$
$137$	18.2047	1.55533	0.777664	+	0.628680i	$0.216404\pi$
$138$	−0.0826869	−0.00703878
$139$	−7.19896	−0.610608	−0.305304	−	0.952255i	$-0.598758\pi$
$139$	−7.19896	−0.610608	−0.305304	+	0.952255i	$0.598758\pi$
$140$	−5.48772	−0.463797
$141$	−0.193963	−0.0163346
$142$	−10.2919	−0.863675
$143$	8.20299	0.685968
$144$	−18.2407	−1.52006
$145$	0.108241	0.00898891
$146$	−36.9066	−3.05441
$147$	0.115504	0.00952658
$148$	27.3807	2.25068
$149$	−19.0213	−1.55828	−0.779142	−	0.626848i	$-0.784345\pi$
$149$	−19.0213	−1.55828	−0.779142	+	0.626848i	$0.784345\pi$
$150$	0.0538809	0.00439936
$151$	0.316605	0.0257649	0.0128825	−	0.999917i	$-0.495899\pi$
$151$	0.316605	0.0257649	0.0128825	+	0.999917i	$0.495899\pi$
$152$	38.2625	3.10350
$153$	9.26421	0.748967
$154$	−12.5589	−1.01202
$155$	−0.323055	−0.0259484
$156$	−0.193129	−0.0154627
$157$	12.9229	1.03136	0.515680	−	0.856782i	$-0.327540\pi$
$157$	12.9229	1.03136	0.515680	+	0.856782i	$0.327540\pi$
$158$	20.7337	1.64949
$159$	−0.194644	−0.0154363
$160$	3.58108	0.283109
$161$	−1.94546	−0.153324
$162$	−22.6311	−1.77807
$163$	14.1740	1.11020	0.555098	−	0.831785i	$-0.312681\pi$
$163$	14.1740	1.11020	0.555098	+	0.831785i	$0.312681\pi$
$164$	39.4209	3.07825
$165$	0.0843416	0.00656598
$166$	30.4899	2.36647
$167$	16.5749	1.28261	0.641304	−	0.767287i	$-0.278394\pi$
$167$	16.5749	1.28261	0.641304	+	0.767287i	$0.278394\pi$
$168$	0.159072	0.0122727
$169$	−8.66084	−0.666218
$170$	−7.76990	−0.595925
$171$	19.5897	1.49806
$172$	−48.5351	−3.70077
$173$	−19.0261	−1.44653	−0.723265	−	0.690570i	$-0.757360\pi$
$173$	−19.0261	−1.44653	−0.723265	+	0.690570i	$0.757360\pi$
$174$	−0.00583211	−0.000442131 0
$175$	1.26771	0.0958300
$176$	23.9472	1.80509
$177$	0.114094	0.00857584
$178$	−21.8868	−1.64048
$179$	−8.46273	−0.632534	−0.316267	−	0.948670i	$-0.602430\pi$
$179$	−8.46273	−0.632534	−0.316267	+	0.948670i	$0.602430\pi$
$180$	12.9845	0.967809
$181$	−18.9450	−1.40817	−0.704087	−	0.710114i	$-0.748643\pi$
$181$	−18.9450	−1.40817	−0.704087	+	0.710114i	$0.748643\pi$
$182$	−6.64332	−0.492436
$183$	−0.103487	−0.00764995
$184$	8.99090	0.662818
$185$	−6.32518	−0.465036
$186$	0.0174065	0.00127631
$187$	−12.1625	−0.889410
$188$	39.2029	2.85916
$189$	0.162896	0.0118490
$190$	−16.4299	−1.19195
$191$	7.70723	0.557675	0.278838	−	0.960338i	$-0.410051\pi$
$191$	7.70723	0.557675	0.278838	+	0.960338i	$0.410051\pi$
$192$	0.0675369	0.00487406
$193$	22.0565	1.58766	0.793832	−	0.608138i	$-0.208083\pi$
$193$	22.0565	1.58766	0.793832	+	0.608138i	$0.208083\pi$
$194$	−20.2111	−1.45107
$195$	0.0446145	0.00319491
$196$	−23.3450	−1.66750
$197$	8.97138	0.639184	0.319592	−	0.947555i	$-0.396454\pi$
$197$	8.97138	0.639184	0.319592	+	0.947555i	$0.396454\pi$
$198$	29.7157	2.11180
$199$	−3.41567	−0.242130	−0.121065	−	0.992645i	$-0.538631\pi$
$199$	−3.41567	−0.242130	−0.121065	+	0.992645i	$0.538631\pi$
$200$	−5.85870	−0.414273
$201$	0.267993	0.0189028
$202$	39.0276	2.74597
$203$	−0.137218	−0.00963082
$204$	0.286350	0.0200485
$205$	−9.10657	−0.636031
$206$	16.3644	1.14016
$207$	4.60317	0.319942
$208$	12.6675	0.878330
$209$	−25.7183	−1.77897
$210$	−0.0683054	−0.00471352
$211$	−21.9784	−1.51305	−0.756526	−	0.653963i	$-0.773105\pi$
$211$	−21.9784	−1.51305	−0.756526	+	0.653963i	$0.773105\pi$
$212$	39.3404	2.70191
$213$	−0.0876204	−0.00600365
$214$	33.2155	2.27056
$215$	11.2121	0.764656
$216$	−0.752821	−0.0512230
$217$	0.409541	0.0278015
$218$	−22.2231	−1.50513
$219$	−0.314206	−0.0212321
$220$	−17.0467	−1.14929
$221$	−6.43364	−0.432773
$222$	0.340806	0.0228734
$223$	−23.4337	−1.56924	−0.784619	−	0.619978i	$-0.787141\pi$
$223$	−23.4337	−1.56924	−0.784619	+	0.619978i	$0.787141\pi$
$224$	−4.53978	−0.303327
$225$	−2.99954	−0.199969
$226$	−35.6180	−2.36928
$227$	9.70141	0.643905	0.321952	−	0.946756i	$-0.395661\pi$
$227$	9.70141	0.643905	0.321952	+	0.946756i	$0.395661\pi$
$228$	0.605504	0.0401005
$229$	−15.4174	−1.01881	−0.509405	−	0.860527i	$-0.670134\pi$
$229$	−15.4174	−1.01881	−0.509405	+	0.860527i	$0.670134\pi$
$230$	−3.86068	−0.254566
$231$	−0.106921	−0.00703487
$232$	0.634150	0.0416340
$233$	18.6783	1.22366	0.611828	−	0.790991i	$-0.290435\pi$
$233$	18.6783	1.22366	0.611828	+	0.790991i	$0.290435\pi$
$234$	15.7188	1.02757
$235$	−9.05621	−0.590762
$236$	−23.0601	−1.50109
$237$	0.176518	0.0114661
$238$	9.85000	0.638481
$239$	4.87905	0.315600	0.157800	−	0.987471i	$-0.449560\pi$
$239$	4.87905	0.315600	0.157800	+	0.987471i	$0.449560\pi$
$240$	0.130244	0.00840724
$241$	−9.37107	−0.603644	−0.301822	−	0.953364i	$-0.597595\pi$
$241$	−9.37107	−0.603644	−0.301822	+	0.953364i	$0.597595\pi$
$242$	−11.3393	−0.728916
$243$	−0.578160	−0.0370890
$244$	20.9162	1.33902
$245$	5.39291	0.344540
$246$	0.490670	0.0312840
$247$	−13.6043	−0.865620
$248$	−1.89268	−0.120186
$249$	0.259577	0.0164500
$250$	2.51572	0.159108
$251$	11.9094	0.751714	0.375857	−	0.926678i	$-0.377348\pi$
$251$	11.9094	0.751714	0.375857	+	0.926678i	$0.377348\pi$
$252$	−16.4606	−1.03692
$253$	−6.04326	−0.379936
$254$	44.5517	2.79542
$255$	−0.0661495	−0.00414244
$256$	−31.6682	−1.97926
$257$	−2.71330	−0.169251	−0.0846255	−	0.996413i	$-0.526969\pi$
$257$	−2.71330	−0.169251	−0.0846255	+	0.996413i	$0.526969\pi$
$258$	−0.604115	−0.0376106
$259$	8.01850	0.498246
$260$	−9.01725	−0.559226
$261$	0.324673	0.0200967
$262$	5.91335	0.365328
$263$	22.3399	1.37754	0.688770	−	0.724980i	$-0.258151\pi$
$263$	22.3399	1.37754	0.688770	+	0.724980i	$0.258151\pi$
$264$	0.494132	0.0304117
$265$	−9.08799	−0.558271
$266$	20.8284	1.27707
$267$	−0.186334	−0.0114035
$268$	−54.1654	−3.30868
$269$	−9.80238	−0.597662	−0.298831	−	0.954306i	$-0.596597\pi$
$269$	−9.80238	−0.597662	−0.298831	+	0.954306i	$0.596597\pi$
$270$	0.323261	0.0196730
$271$	−4.43509	−0.269413	−0.134706	−	0.990886i	$-0.543009\pi$
$271$	−4.43509	−0.269413	−0.134706	+	0.990886i	$0.543009\pi$
$272$	−18.7819	−1.13882
$273$	−0.0565583	−0.00342306
$274$	−45.7978	−2.76674
$275$	3.93794	0.237467
$276$	0.142281	0.00856429
$277$	−12.9549	−0.778385	−0.389192	−	0.921157i	$-0.627246\pi$
$277$	−12.9549	−0.778385	−0.389192	+	0.921157i	$0.627246\pi$
$278$	18.1105	1.08620
$279$	−0.969018	−0.0580136
$280$	7.42714	0.443856
$281$	−30.3181	−1.80863	−0.904314	−	0.426869i	$-0.859617\pi$
$281$	−30.3181	−1.80863	−0.904314	+	0.426869i	$0.859617\pi$
$282$	0.487957	0.0290574
$283$	−13.1938	−0.784290	−0.392145	−	0.919903i	$-0.628267\pi$
$283$	−13.1938	−0.784290	−0.392145	+	0.919903i	$0.628267\pi$
$284$	17.7094	1.05086
$285$	−0.139877	−0.00828558
$286$	−20.6364	−1.22026
$287$	11.5445	0.681451
$288$	10.7416	0.632955
$289$	−7.46090	−0.438876
$290$	−0.272303	−0.0159902
$291$	−0.172068	−0.0100868
$292$	63.5057	3.71639
$293$	−19.5195	−1.14034	−0.570170	−	0.821527i	$-0.693123\pi$
$293$	−19.5195	−1.14034	−0.570170	+	0.821527i	$0.693123\pi$
$294$	−0.290575	−0.0169467
$295$	5.32710	0.310156
$296$	−37.0573	−2.15391
$297$	0.506011	0.0293617
$298$	47.8522	2.77200
$299$	−3.19672	−0.184871
$300$	−0.0927137	−0.00535283
$301$	−14.2136	−0.819261
$302$	−0.796488	−0.0458327
$303$	0.332263	0.0190880
$304$	−39.7154	−2.27783
$305$	−4.83182	−0.276669
$306$	−23.3062	−1.33232
$307$	10.3006	0.587884	0.293942	−	0.955823i	$-0.405033\pi$
$307$	10.3006	0.587884	0.293942	+	0.955823i	$0.405033\pi$
$308$	21.6103	1.23136
$309$	0.139319	0.00792558
$310$	0.812717	0.0461592
$311$	20.1218	1.14101	0.570503	−	0.821296i	$-0.306748\pi$
$311$	20.1218	1.14101	0.570503	+	0.821296i	$0.306748\pi$
$312$	0.261383	0.0147979
$313$	−30.9648	−1.75024	−0.875118	−	0.483910i	$-0.839216\pi$
$313$	−30.9648	−1.75024	−0.875118	+	0.483910i	$0.839216\pi$
$314$	−32.5104	−1.83467
$315$	3.80255	0.214250
$316$	−35.6769	−2.00698
$317$	3.82181	0.214655	0.107327	−	0.994224i	$-0.465771\pi$
$317$	3.82181	0.214655	0.107327	+	0.994224i	$0.465771\pi$
$318$	0.489669	0.0274593
$319$	−0.426246	−0.0238652
$320$	3.15332	0.176276
$321$	0.282782	0.0157833
$322$	4.89423	0.272745
$323$	20.1710	1.12234
$324$	38.9417	2.16343
$325$	2.08307	0.115548
$326$	−35.6578	−1.97491
$327$	−0.189197	−0.0104626
$328$	−53.3527	−2.94591
$329$	11.4807	0.632949
$330$	−0.212180	−0.0116801
$331$	7.37932	0.405604	0.202802	−	0.979220i	$-0.434995\pi$
$331$	7.37932	0.405604	0.202802	+	0.979220i	$0.434995\pi$
$332$	−52.4644	−2.87936
$333$	−18.9726	−1.03969
$334$	−41.6979	−2.28161
$335$	12.5127	0.683642
$336$	−0.165112	−0.00900761
$337$	−8.65898	−0.471685	−0.235842	−	0.971791i	$-0.575785\pi$
$337$	−8.65898	−0.471685	−0.235842	+	0.971791i	$0.575785\pi$
$338$	21.7882	1.18512
$339$	−0.303236	−0.0164695
$340$	13.3698	0.725079
$341$	1.27217	0.0688920
$342$	−49.2821	−2.66487
$343$	−15.7106	−0.848294
$344$	65.6880	3.54166
$345$	−0.0328681	−0.00176956
$346$	47.8644	2.57321
$347$	13.8217	0.741987	0.370993	−	0.928635i	$-0.379017\pi$
$347$	13.8217	0.741987	0.370993	+	0.928635i	$0.379017\pi$
$348$	0.0100354	0.000537954 0
$349$	7.84849	0.420120	0.210060	−	0.977688i	$-0.432634\pi$
$349$	7.84849	0.420120	0.210060	+	0.977688i	$0.432634\pi$
$350$	−3.18920	−0.170470
$351$	0.267666	0.0142870
$352$	−14.1021	−0.751644
$353$	23.3237	1.24140	0.620699	−	0.784049i	$-0.286849\pi$
$353$	23.3237	1.24140	0.620699	+	0.784049i	$0.286849\pi$
$354$	−0.287029	−0.0152554
$355$	−4.09103	−0.217129
$356$	37.6610	1.99603
$357$	0.0838585	0.00443826
$358$	21.2899	1.12520
$359$	−4.09597	−0.216177	−0.108088	−	0.994141i	$-0.534473\pi$
$359$	−4.09597	−0.216177	−0.108088	+	0.994141i	$0.534473\pi$
$360$	−17.5734	−0.926200
$361$	23.6526	1.24487
$362$	47.6604	2.50497
$363$	−0.0965374	−0.00506690
$364$	11.4313	0.599162
$365$	−14.6704	−0.767883
$366$	0.260343	0.0136083
$367$	23.3001	1.21626	0.608129	−	0.793838i	$-0.291920\pi$
$367$	23.3001	1.21626	0.608129	+	0.793838i	$0.291920\pi$
$368$	−9.33229	−0.486479
$369$	−27.3155	−1.42199
$370$	15.9124	0.827245
$371$	11.5209	0.598138
$372$	−0.0299517	−0.00155292
$373$	−3.07530	−0.159233	−0.0796166	−	0.996826i	$-0.525370\pi$
$373$	−3.07530	−0.159233	−0.0796166	+	0.996826i	$0.525370\pi$
$374$	30.5974	1.58215
$375$	0.0214177	0.00110600
$376$	−53.0576	−2.73624
$377$	−0.225473	−0.0116124
$378$	−0.409801	−0.0210779
$379$	−14.9385	−0.767340	−0.383670	−	0.923470i	$-0.625340\pi$
$379$	−14.9385	−0.767340	−0.383670	+	0.923470i	$0.625340\pi$
$380$	28.2712	1.45028
$381$	0.379293	0.0194318
$382$	−19.3892	−0.992038
$383$	−37.6380	−1.92321	−0.961606	−	0.274435i	$-0.911509\pi$
$383$	−37.6380	−1.92321	−0.961606	+	0.274435i	$0.911509\pi$
$384$	−0.323301	−0.0164984
$385$	−4.99217	−0.254425
$386$	−55.4880	−2.82426
$387$	33.6310	1.70956
$388$	34.7776	1.76556
$389$	9.88125	0.500999	0.250500	−	0.968117i	$-0.419405\pi$
$389$	9.88125	0.500999	0.250500	+	0.968117i	$0.419405\pi$
$390$	−0.112237	−0.00568336
$391$	4.73975	0.239700
$392$	31.5954	1.59581
$393$	0.0503436	0.00253950
$394$	−22.5695	−1.13703
$395$	8.24167	0.414684
$396$	−51.1323	−2.56949
$397$	13.3821	0.671629	0.335815	−	0.941928i	$-0.390988\pi$
$397$	13.3821	0.671629	0.335815	+	0.941928i	$0.390988\pi$
$398$	8.59286	0.430721
$399$	0.177323	0.00887727
$400$	6.08116	0.304058
$401$	1.69938	0.0848631	0.0424316	−	0.999099i	$-0.486490\pi$
$401$	1.69938	0.0848631	0.0424316	+	0.999099i	$0.486490\pi$
$402$	−0.674195	−0.0336258
$403$	0.672946	0.0335218
$404$	−67.1554	−3.34110
$405$	−8.99587	−0.447008
$406$	0.345202	0.0171321
$407$	24.9082	1.23465
$408$	−0.387550	−0.0191866
$409$	36.0371	1.78192	0.890959	−	0.454084i	$-0.150033\pi$
$409$	36.0371	1.78192	0.890959	+	0.454084i	$0.150033\pi$
$410$	22.9096	1.13142
$411$	−0.389902	−0.0192324
$412$	−28.1584	−1.38727
$413$	−6.75323	−0.332304
$414$	−11.5803	−0.569139
$415$	12.1198	0.594935
$416$	−7.45963	−0.365738
$417$	0.154185	0.00755047
$418$	64.6999	3.16458
$419$	25.8445	1.26259	0.631294	−	0.775543i	$-0.282524\pi$
$419$	25.8445	1.26259	0.631294	+	0.775543i	$0.282524\pi$
$420$	0.117534	0.00573508
$421$	−16.9095	−0.824118	−0.412059	−	0.911157i	$-0.635190\pi$
$421$	−16.9095	−0.824118	−0.412059	+	0.911157i	$0.635190\pi$
$422$	55.2914	2.69154
$423$	−27.1645	−1.32078
$424$	−53.2438	−2.58575
$425$	−3.08854	−0.149816
$426$	0.220428	0.0106798
$427$	6.12536	0.296427
$428$	−57.1544	−2.76266
$429$	−0.175689	−0.00848235
$430$	−28.2064	−1.36023
$431$	0.684825	0.0329869	0.0164934	−	0.999864i	$-0.494750\pi$
$431$	0.684825	0.0329869	0.0164934	+	0.999864i	$0.494750\pi$
$432$	0.781406	0.0375954
$433$	17.2171	0.827403	0.413702	−	0.910413i	$-0.364236\pi$
$433$	17.2171	0.827403	0.413702	+	0.910413i	$0.364236\pi$
$434$	−1.03029	−0.0494555
$435$	−0.00231827	−0.000111152 0
$436$	38.2396	1.83134
$437$	10.0225	0.479440
$438$	0.790454	0.0377693
$439$	16.7665	0.800222	0.400111	−	0.916467i	$-0.368972\pi$
$439$	16.7665	0.800222	0.400111	+	0.916467i	$0.368972\pi$
$440$	23.0712	1.09988
$441$	16.1762	0.770298
$442$	16.1852	0.769852
$443$	13.2202	0.628111	0.314056	−	0.949405i	$-0.398312\pi$
$443$	13.2202	0.628111	0.314056	+	0.949405i	$0.398312\pi$
$444$	−0.586431	−0.0278308
$445$	−8.70002	−0.412421
$446$	58.9526	2.79149
$447$	0.407392	0.0192690
$448$	−3.99750	−0.188864
$449$	−8.15974	−0.385082	−0.192541	−	0.981289i	$-0.561673\pi$
$449$	−8.15974	−0.385082	−0.192541	+	0.981289i	$0.561673\pi$
$450$	7.54600	0.355722
$451$	35.8611	1.68864
$452$	61.2886	2.88277
$453$	−0.00678094	−0.000318596 0
$454$	−24.4060	−1.14543
$455$	−2.64073	−0.123799
$456$	−0.819495	−0.0383764
$457$	35.4853	1.65993	0.829966	−	0.557814i	$-0.188360\pi$
$457$	35.4853	1.65993	0.829966	+	0.557814i	$0.188360\pi$
$458$	38.7858	1.81234
$459$	−0.396867	−0.0185241
$460$	6.64314	0.309738
$461$	19.6134	0.913485	0.456743	−	0.889599i	$-0.349016\pi$
$461$	19.6134	0.913485	0.456743	+	0.889599i	$0.349016\pi$
$462$	0.268983	0.0125142
$463$	−0.538993	−0.0250491	−0.0125246	−	0.999922i	$-0.503987\pi$
$463$	−0.538993	−0.0250491	−0.0125246	+	0.999922i	$0.503987\pi$
$464$	−0.658229	−0.0305575
$465$	0.00691910	0.000320866 0
$466$	−46.9893	−2.17674
$467$	−37.2477	−1.72362	−0.861810	−	0.507232i	$-0.830669\pi$
$467$	−37.2477	−1.72362	−0.861810	+	0.507232i	$0.830669\pi$
$468$	−27.0476	−1.25028
$469$	−15.8625	−0.732462
$470$	22.7829	1.05090
$471$	−0.276779	−0.0127533
$472$	31.2099	1.43655
$473$	−44.1524	−2.03013
$474$	−0.444069	−0.0203967
$475$	−6.53090	−0.299658
$476$	−16.9491	−0.776859
$477$	−27.2598	−1.24814
$478$	−12.2743	−0.561414
$479$	−10.1785	−0.465070	−0.232535	−	0.972588i	$-0.574702\pi$
$479$	−10.1785	−0.465070	−0.232535	+	0.972588i	$0.574702\pi$
$480$	−0.0766985	−0.00350079
$481$	13.1758	0.600763
$482$	23.5750	1.07381
$483$	0.0416673	0.00189593
$484$	19.5117	0.886894
$485$	−8.03392	−0.364802
$486$	1.45449	0.0659769
$487$	4.21503	0.191001	0.0955006	−	0.995429i	$-0.469555\pi$
$487$	4.21503	0.191001	0.0955006	+	0.995429i	$0.469555\pi$
$488$	−28.3082	−1.28145
$489$	−0.303575	−0.0137281
$490$	−13.5670	−0.612896
$491$	15.5155	0.700204	0.350102	−	0.936712i	$-0.386147\pi$
$491$	15.5155	0.700204	0.350102	+	0.936712i	$0.386147\pi$
$492$	−0.844304	−0.0380642
$493$	0.334306	0.0150564
$494$	34.2245	1.53984
$495$	11.8120	0.530910
$496$	1.96455	0.0882110
$497$	5.18624	0.232635
$498$	−0.653023	−0.0292627
$499$	−9.02280	−0.403916	−0.201958	−	0.979394i	$-0.564730\pi$
$499$	−9.02280	−0.403916	−0.201958	+	0.979394i	$0.564730\pi$
$500$	−4.32884	−0.193591
$501$	−0.354997	−0.0158601
$502$	−29.9607	−1.33721
$503$	−36.0803	−1.60874	−0.804370	−	0.594128i	$-0.797497\pi$
$503$	−36.0803	−1.60874	−0.804370	+	0.594128i	$0.797497\pi$
$504$	22.2780	0.992341
$505$	15.5135	0.690341
$506$	15.2031	0.675862
$507$	0.185495	0.00823813
$508$	−76.6608	−3.40127
$509$	−23.0572	−1.02199	−0.510995	−	0.859584i	$-0.670723\pi$
$509$	−23.0572	−1.02199	−0.510995	+	0.859584i	$0.670723\pi$
$510$	0.166413	0.00736891
$511$	18.5978	0.822719
$512$	49.4782	2.18665
$513$	−0.839196	−0.0370514
$514$	6.82590	0.301077
$515$	6.50485	0.286638
$516$	1.03951	0.0457619
$517$	35.6628	1.56845
$518$	−20.1723	−0.886320
$519$	0.407496	0.0178871
$520$	12.2041	0.535183
$521$	−18.6325	−0.816306	−0.408153	−	0.912914i	$-0.633827\pi$
$521$	−18.6325	−0.816306	−0.408153	+	0.912914i	$0.633827\pi$
$522$	−0.816785	−0.0357497
$523$	−8.50789	−0.372024	−0.186012	−	0.982547i	$-0.559556\pi$
$523$	−8.50789	−0.372024	−0.186012	+	0.982547i	$0.559556\pi$
$524$	−10.1752	−0.444506
$525$	−0.0271515	−0.00118499
$526$	−56.2010	−2.45048
$527$	−0.997771	−0.0434636
$528$	−0.512894	−0.0223209
$529$	−20.6449	−0.897606
$530$	22.8628	0.993097
$531$	15.9789	0.693423
$532$	−35.8397	−1.55385
$533$	18.9696	0.821664
$534$	0.468765	0.0202854
$535$	13.2032	0.570823
$536$	73.3081	3.16643
$537$	0.181252	0.00782161
$538$	24.6600	1.06317
$539$	−21.2369	−0.914740
$540$	−0.556240	−0.0239367
$541$	7.38012	0.317296	0.158648	−	0.987335i	$-0.449287\pi$
$541$	7.38012	0.317296	0.158648	+	0.987335i	$0.449287\pi$
$542$	11.1574	0.479253
$543$	0.405759	0.0174128
$544$	11.0603	0.474208
$545$	−8.83368	−0.378393
$546$	0.142285	0.00608922
$547$	6.81094	0.291215	0.145607	−	0.989342i	$-0.453486\pi$
$547$	6.81094	0.291215	0.145607	+	0.989342i	$0.453486\pi$
$548$	78.8050	3.36638
$549$	−14.4933	−0.618557
$550$	−9.90675	−0.422425
$551$	0.706909	0.0301153
$552$	−0.192564	−0.00819608
$553$	−10.4481	−0.444297
$554$	32.5909	1.38465
$555$	0.135471	0.00575041
$556$	−31.1631	−1.32161
$557$	−39.4581	−1.67189	−0.835946	−	0.548811i	$-0.815081\pi$
$557$	−39.4581	−1.67189	−0.835946	+	0.548811i	$0.815081\pi$
$558$	2.43778	0.103199
$559$	−23.3554	−0.987830
$560$	−7.70915	−0.325771
$561$	0.260493	0.0109980
$562$	76.2718	3.21733
$563$	−36.9578	−1.55758	−0.778792	−	0.627282i	$-0.784167\pi$
$563$	−36.9578	−1.55758	−0.778792	+	0.627282i	$0.784167\pi$
$564$	−0.839635	−0.0353550
$565$	−14.1582	−0.595640
$566$	33.1919	1.39516
$567$	11.4042	0.478930
$568$	−23.9681	−1.00568
$569$	−1.88975	−0.0792224	−0.0396112	−	0.999215i	$-0.512612\pi$
$569$	−1.88975	−0.0792224	−0.0396112	+	0.999215i	$0.512612\pi$
$570$	0.351890	0.0147391
$571$	−21.9563	−0.918842	−0.459421	−	0.888219i	$-0.651943\pi$
$571$	−21.9563	−0.918842	−0.459421	+	0.888219i	$0.651943\pi$
$572$	35.5094	1.48472
$573$	−0.165071	−0.00689594
$574$	−29.0427	−1.21222
$575$	−1.53462	−0.0639983
$576$	9.45852	0.394105
$577$	−8.33988	−0.347194	−0.173597	−	0.984817i	$-0.555539\pi$
$577$	−8.33988	−0.347194	−0.173597	+	0.984817i	$0.555539\pi$
$578$	18.7695	0.780709
$579$	−0.472400	−0.0196323
$580$	0.468557	0.0194558
$581$	−15.3643	−0.637421
$582$	0.432875	0.0179432
$583$	35.7879	1.48219
$584$	−85.9494	−3.55661
$585$	6.24824	0.258333
$586$	49.1055	2.02853
$587$	1.84695	0.0762316	0.0381158	−	0.999273i	$-0.487864\pi$
$587$	1.84695	0.0762316	0.0381158	+	0.999273i	$0.487864\pi$
$588$	0.499996	0.0206195
$589$	−2.10984	−0.0869345
$590$	−13.4015	−0.551730
$591$	−0.192146	−0.00790384
$592$	38.4644	1.58088
$593$	−39.3393	−1.61547	−0.807736	−	0.589545i	$-0.799307\pi$
$593$	−39.3393	−1.61547	−0.807736	+	0.589545i	$0.799307\pi$
$594$	−1.27298	−0.0522310
$595$	3.91538	0.160515
$596$	−82.3400	−3.37278
$597$	0.0731557	0.00299406
$598$	8.04205	0.328864
$599$	−39.8306	−1.62744	−0.813718	−	0.581260i	$-0.802560\pi$
$599$	−39.8306	−1.62744	−0.813718	+	0.581260i	$0.802560\pi$
$600$	0.125480	0.00512269
$601$	29.9928	1.22343	0.611716	−	0.791077i	$-0.290479\pi$
$601$	29.9928	1.22343	0.611716	+	0.791077i	$0.290479\pi$
$602$	35.7575	1.45737
$603$	37.5324	1.52844
$604$	1.37053	0.0557661
$605$	−4.50737	−0.183251
$606$	−0.835880	−0.0339553
$607$	−6.62954	−0.269085	−0.134542	−	0.990908i	$-0.542956\pi$
$607$	−6.62954	−0.269085	−0.134542	+	0.990908i	$0.542956\pi$
$608$	23.3877	0.948495
$609$	0.00293889	0.000119090 0
$610$	12.1555	0.492162
$611$	18.8647	0.763183
$612$	40.1033	1.62108
$613$	−15.5642	−0.628632	−0.314316	−	0.949318i	$-0.601775\pi$
$613$	−15.5642	−0.628632	−0.314316	+	0.949318i	$0.601775\pi$
$614$	−25.9133	−1.04578
$615$	0.195042	0.00786485
$616$	−29.2476	−1.17842
$617$	44.4051	1.78768	0.893841	−	0.448383i	$-0.148000\pi$
$617$	44.4051	1.78768	0.893841	+	0.448383i	$0.148000\pi$
$618$	−0.350487	−0.0140987
$619$	−14.7452	−0.592660	−0.296330	−	0.955086i	$-0.595763\pi$
$619$	−14.7452	−0.592660	−0.296330	+	0.955086i	$0.595763\pi$
$620$	−1.39845	−0.0561633
$621$	−0.197194	−0.00791311
$622$	−50.6209	−2.02971
$623$	11.0291	0.441872
$624$	−0.271308	−0.0108610
$625$	1.00000	0.0400000
$626$	77.8988	3.11346
$627$	0.550826	0.0219979
$628$	55.9411	2.23229
$629$	−19.5356	−0.778935
$630$	−9.56615	−0.381125
$631$	−42.3246	−1.68492	−0.842459	−	0.538761i	$-0.818893\pi$
$631$	−42.3246	−1.68492	−0.842459	+	0.538761i	$0.818893\pi$
$632$	48.2855	1.92069
$633$	0.470726	0.0187097
$634$	−9.61461	−0.381845
$635$	17.7093	0.702773
$636$	−0.842581	−0.0334105
$637$	−11.2338	−0.445099
$638$	1.07231	0.0424533
$639$	−12.2712	−0.485442
$640$	−15.0950	−0.596684
$641$	−13.7514	−0.543147	−0.271574	−	0.962418i	$-0.587544\pi$
$641$	−13.7514	−0.543147	−0.271574	+	0.962418i	$0.587544\pi$
$642$	−0.711399	−0.0280767
$643$	42.9626	1.69428	0.847139	−	0.531372i	$-0.178323\pi$
$643$	42.9626	1.69428	0.847139	+	0.531372i	$0.178323\pi$
$644$	−8.42158	−0.331857
$645$	−0.240136	−0.00945536
$646$	−50.7444	−1.99651
$647$	−32.6448	−1.28340	−0.641699	−	0.766957i	$-0.721770\pi$
$647$	−32.6448	−1.28340	−0.641699	+	0.766957i	$0.721770\pi$
$648$	−52.7041	−2.07041
$649$	−20.9778	−0.823451
$650$	−5.24041	−0.205546
$651$	−0.00877143	−0.000343779 0
$652$	61.3570	2.40293
$653$	−12.5443	−0.490895	−0.245448	−	0.969410i	$-0.578935\pi$
$653$	−12.5443	−0.490895	−0.245448	+	0.969410i	$0.578935\pi$
$654$	0.475966	0.0186118
$655$	2.35056	0.0918440
$656$	55.3785	2.16217
$657$	−44.0044	−1.71678
$658$	−28.8821	−1.12594
$659$	−11.2640	−0.438783	−0.219391	−	0.975637i	$-0.570407\pi$
$659$	−11.2640	−0.438783	−0.219391	+	0.975637i	$0.570407\pi$
$660$	0.365101	0.0142115
$661$	−8.20495	−0.319136	−0.159568	−	0.987187i	$-0.551010\pi$
$661$	−8.20495	−0.319136	−0.159568	+	0.987187i	$0.551010\pi$
$662$	−18.5643	−0.721522
$663$	0.137794	0.00535146
$664$	71.0060	2.75557
$665$	8.27929	0.321057
$666$	47.7298	1.84949
$667$	0.166109	0.00643176
$668$	71.7502	2.77610
$669$	0.501896	0.0194044
$670$	−31.4784	−1.21612
$671$	19.0274	0.734546
$672$	0.0972316	0.00375079
$673$	−23.3084	−0.898473	−0.449237	−	0.893413i	$-0.648304\pi$
$673$	−23.3084	−0.898473	−0.449237	+	0.893413i	$0.648304\pi$
$674$	21.7836	0.839071
$675$	0.128496	0.00494583
$676$	−37.4914	−1.44198
$677$	22.2055	0.853428	0.426714	−	0.904387i	$-0.359671\pi$
$677$	22.2055	0.853428	0.426714	+	0.904387i	$0.359671\pi$
$678$	0.762856	0.0292973
$679$	10.1847	0.390853
$680$	−18.0948	−0.693906
$681$	−0.207782	−0.00796221
$682$	−3.20043	−0.122551
$683$	26.6312	1.01902	0.509508	−	0.860466i	$-0.329828\pi$
$683$	26.6312	1.01902	0.509508	+	0.860466i	$0.329828\pi$
$684$	84.8006	3.24243
$685$	−18.2047	−0.695564
$686$	39.5235	1.50901
$687$	0.330205	0.0125981
$688$	−68.1823	−2.59942
$689$	18.9309	0.721209
$690$	0.0826869	0.00314784
$691$	−10.9495	−0.416539	−0.208270	−	0.978071i	$-0.566783\pi$
$691$	−10.9495	−0.416539	−0.208270	+	0.978071i	$0.566783\pi$
$692$	−82.3611	−3.13090
$693$	−14.9742	−0.568824
$694$	−34.7715	−1.31991
$695$	7.19896	0.273072
$696$	−0.0135820	−0.000514826 0
$697$	−28.1261	−1.06535
$698$	−19.7446	−0.747344
$699$	−0.400046	−0.0151311
$700$	5.48772	0.207416
$701$	−28.5516	−1.07838	−0.539189	−	0.842184i	$-0.681269\pi$
$701$	−28.5516	−1.07838	−0.539189	+	0.842184i	$0.681269\pi$
$702$	−0.673373	−0.0254148
$703$	−41.3091	−1.55800
$704$	−12.4176	−0.468006
$705$	0.193963	0.00730508
$706$	−58.6760	−2.20830
$707$	−19.6666	−0.739640
$708$	0.493895	0.0185617
$709$	36.8908	1.38546	0.692732	−	0.721195i	$-0.256407\pi$
$709$	36.8908	1.38546	0.692732	+	0.721195i	$0.256407\pi$
$710$	10.2919	0.386247
$711$	24.7212	0.927119
$712$	−50.9708	−1.91021
$713$	−0.495769	−0.0185667
$714$	−0.210964	−0.00789514
$715$	−8.20299	−0.306774
$716$	−36.6338	−1.36907
$717$	−0.104498	−0.00390255
$718$	10.3043	0.384553
$719$	−31.0765	−1.15896	−0.579478	−	0.814988i	$-0.696744\pi$
$719$	−31.0765	−1.15896	−0.579478	+	0.814988i	$0.696744\pi$
$720$	18.2407	0.679790
$721$	−8.24628	−0.307107
$722$	−59.5033	−2.21448
$723$	0.200707	0.00746436
$724$	−82.0100	−3.04788
$725$	−0.108241	−0.00401996
$726$	0.242861	0.00901342
$727$	−29.8411	−1.10674	−0.553372	−	0.832934i	$-0.686659\pi$
$727$	−29.8411	−1.10674	−0.553372	+	0.832934i	$0.686659\pi$
$728$	−15.4712	−0.573401
$729$	−26.9752	−0.999083
$730$	36.9066	1.36597
$731$	34.6289	1.28080
$732$	−0.447976	−0.0165577
$733$	51.3143	1.89534	0.947668	−	0.319257i	$-0.103433\pi$
$733$	51.3143	1.89534	0.947668	+	0.319257i	$0.103433\pi$
$734$	−58.6166	−2.16358
$735$	−0.115504	−0.00426042
$736$	5.49561	0.202571
$737$	−49.2743	−1.81504
$738$	68.7182	2.52955
$739$	−2.51638	−0.0925665	−0.0462833	−	0.998928i	$-0.514738\pi$
$739$	−2.51638	−0.0925665	−0.0462833	+	0.998928i	$0.514738\pi$
$740$	−27.3807	−1.00653
$741$	0.291372	0.0107038
$742$	−28.9835	−1.06402
$743$	−23.9465	−0.878511	−0.439255	−	0.898362i	$-0.644758\pi$
$743$	−23.9465	−0.878511	−0.439255	+	0.898362i	$0.644758\pi$
$744$	0.0405369	0.00148616
$745$	19.0213	0.696886
$746$	7.73659	0.283257
$747$	36.3537	1.33011
$748$	−52.6495	−1.92506
$749$	−16.7378	−0.611587
$750$	−0.0538809	−0.00196745
$751$	−0.514508	−0.0187747	−0.00938733	−	0.999956i	$-0.502988\pi$
$751$	−0.514508	−0.0187747	−0.00938733	+	0.999956i	$0.502988\pi$
$752$	55.0723	2.00828
$753$	−0.255072	−0.00929533
$754$	0.567226	0.0206571
$755$	−0.316605	−0.0115224
$756$	0.705151	0.0256461
$757$	27.1889	0.988198	0.494099	−	0.869406i	$-0.335498\pi$
$757$	27.1889	0.988198	0.494099	+	0.869406i	$0.335498\pi$
$758$	37.5811	1.36501
$759$	0.129433	0.00469811
$760$	−38.2625	−1.38793
$761$	28.3709	1.02845	0.514223	−	0.857657i	$-0.328080\pi$
$761$	28.3709	1.02845	0.514223	+	0.857657i	$0.328080\pi$
$762$	−0.954194	−0.0345668
$763$	11.1986	0.405415
$764$	33.3633	1.20704
$765$	−9.26421	−0.334948
$766$	94.6866	3.42116
$767$	−11.0967	−0.400679
$768$	0.678260	0.0244746
$769$	5.38533	0.194200	0.0970999	−	0.995275i	$-0.469043\pi$
$769$	5.38533	0.194200	0.0970999	+	0.995275i	$0.469043\pi$
$770$	12.5589	0.452591
$771$	0.0581126	0.00209287
$772$	95.4791	3.43637
$773$	21.4638	0.772000	0.386000	−	0.922499i	$-0.373856\pi$
$773$	21.4638	0.772000	0.386000	+	0.922499i	$0.373856\pi$
$774$	−84.6061	−3.04110
$775$	0.323055	0.0116045
$776$	−47.0683	−1.68965
$777$	−0.171738	−0.00616106
$778$	−24.8584	−0.891219
$779$	−59.4741	−2.13088
$780$	0.193129	0.00691512
$781$	16.1102	0.576469
$782$	−11.9239	−0.426397
$783$	−0.0139085	−0.000497051 0
$784$	−32.7951	−1.17125
$785$	−12.9229	−0.461238
$786$	−0.126650	−0.00451747
$787$	−40.8320	−1.45550	−0.727751	−	0.685841i	$-0.759435\pi$
$787$	−40.8320	−1.45550	−0.727751	+	0.685841i	$0.759435\pi$
$788$	38.8356	1.38346
$789$	−0.478470	−0.0170340
$790$	−20.7337	−0.737673
$791$	17.9485	0.638176
$792$	69.2030	2.45902
$793$	10.0650	0.357419
$794$	−33.6656	−1.19475
$795$	0.194644	0.00690330
$796$	−14.7859	−0.524071
$797$	6.37040	0.225651	0.112826	−	0.993615i	$-0.464010\pi$
$797$	6.37040	0.225651	0.112826	+	0.993615i	$0.464010\pi$
$798$	−0.446096	−0.0157916
$799$	−27.9705	−0.989525
$800$	−3.58108	−0.126610
$801$	−26.0961	−0.922059
$802$	−4.27517	−0.150961
$803$	57.7711	2.03870
$804$	1.16010	0.0409135
$805$	1.94546	0.0685685
$806$	−1.69294	−0.0596313
$807$	0.209944	0.00739039
$808$	90.8888	3.19746
$809$	48.8112	1.71611	0.858056	−	0.513557i	$-0.171672\pi$
$809$	48.8112	1.71611	0.858056	+	0.513557i	$0.171672\pi$
$810$	22.6311	0.795175
$811$	−37.5599	−1.31890	−0.659452	−	0.751746i	$-0.729212\pi$
$811$	−37.5599	−1.31890	−0.659452	+	0.751746i	$0.729212\pi$
$812$	−0.593995	−0.0208451
$813$	0.0949894	0.00333142
$814$	−62.6619	−2.19630
$815$	−14.1740	−0.496494
$816$	0.402265	0.0140821
$817$	73.2247	2.56181
$818$	−90.6591	−3.16982
$819$	−7.92097	−0.276781
$820$	−39.4209	−1.37664
$821$	12.7298	0.444273	0.222136	−	0.975016i	$-0.428697\pi$
$821$	12.7298	0.444273	0.222136	+	0.975016i	$0.428697\pi$
$822$	0.980883	0.0342122
$823$	−49.3245	−1.71934	−0.859671	−	0.510848i	$-0.829332\pi$
$823$	−49.3245	−1.71934	−0.859671	+	0.510848i	$0.829332\pi$
$824$	38.1100	1.32762
$825$	−0.0843416	−0.00293640
$826$	16.9892	0.591130
$827$	23.4532	0.815546	0.407773	−	0.913083i	$-0.366305\pi$
$827$	23.4532	0.815546	0.407773	+	0.913083i	$0.366305\pi$
$828$	19.9264	0.692489
$829$	−20.6405	−0.716873	−0.358437	−	0.933554i	$-0.616690\pi$
$829$	−20.6405	−0.716873	−0.358437	+	0.933554i	$0.616690\pi$
$830$	−30.4899	−1.05832
$831$	0.277464	0.00962512
$832$	−6.56858	−0.227724
$833$	16.6562	0.577104
$834$	−0.387886	−0.0134314
$835$	−16.5749	−0.573600
$836$	−111.330	−3.85044
$837$	0.0415114	0.00143485
$838$	−65.0176	−2.24599
$839$	33.7685	1.16582	0.582910	−	0.812537i	$-0.301914\pi$
$839$	33.7685	1.16582	0.582910	+	0.812537i	$0.301914\pi$
$840$	−0.159072	−0.00548851
$841$	−28.9883	−0.999596
$842$	42.5395	1.46601
$843$	0.649344	0.0223646
$844$	−95.1407	−3.27488
$845$	8.66084	0.297942
$846$	68.3382	2.34951
$847$	5.71404	0.196337
$848$	55.2655	1.89782
$849$	0.282581	0.00969815
$850$	7.76990	0.266506
$851$	−9.70677	−0.332744
$852$	−0.379294	−0.0129944
$853$	−45.2096	−1.54795	−0.773974	−	0.633218i	$-0.781734\pi$
$853$	−45.2096	−1.54795	−0.773974	+	0.633218i	$0.781734\pi$
$854$	−15.4097	−0.527308
$855$	−19.5897	−0.669953
$856$	77.3534	2.64389
$857$	−12.3968	−0.423467	−0.211734	−	0.977327i	$-0.567911\pi$
$857$	−12.3968	−0.423467	−0.211734	+	0.977327i	$0.567911\pi$
$858$	0.441984	0.0150891
$859$	−36.2245	−1.23597	−0.617983	−	0.786192i	$-0.712050\pi$
$859$	−36.2245	−1.23597	−0.617983	+	0.786192i	$0.712050\pi$
$860$	48.5351	1.65503
$861$	−0.247257	−0.00842649
$862$	−1.72283	−0.0586797
$863$	−9.86760	−0.335897	−0.167948	−	0.985796i	$-0.553714\pi$
$863$	−9.86760	−0.335897	−0.167948	+	0.985796i	$0.553714\pi$
$864$	−0.460156	−0.0156548
$865$	19.0261	0.646908
$866$	−43.3135	−1.47185
$867$	0.159795	0.00542693
$868$	1.77284	0.0601740
$869$	−32.4552	−1.10097
$870$	0.00583211	0.000197727 0
$871$	−26.0648	−0.883171
$872$	−51.7539	−1.75261
$873$	−24.0981	−0.815597
$874$	−25.2137	−0.852866
$875$	−1.26771	−0.0428565
$876$	−1.36015	−0.0459551
$877$	−28.3506	−0.957330	−0.478665	−	0.877998i	$-0.658879\pi$
$877$	−28.3506	−0.957330	−0.478665	+	0.877998i	$0.658879\pi$
$878$	−42.1798	−1.42350
$879$	0.418062	0.0141009
$880$	−23.9472	−0.807261
$881$	−13.4698	−0.453810	−0.226905	−	0.973917i	$-0.572861\pi$
$881$	−13.4698	−0.453810	−0.226905	+	0.973917i	$0.572861\pi$
$882$	−40.6949	−1.37027
$883$	6.75618	0.227364	0.113682	−	0.993517i	$-0.463736\pi$
$883$	6.75618	0.227364	0.113682	+	0.993517i	$0.463736\pi$
$884$	−27.8502	−0.936703
$885$	−0.114094	−0.00383523
$886$	−33.2583	−1.11734
$887$	−14.7753	−0.496106	−0.248053	−	0.968746i	$-0.579791\pi$
$887$	−14.7753	−0.496106	−0.248053	+	0.968746i	$0.579791\pi$
$888$	0.793682	0.0266342
$889$	−22.4503	−0.752959
$890$	21.8868	0.733647
$891$	35.4252	1.18679
$892$	−101.441	−3.39649
$893$	−59.1452	−1.97922
$894$	−1.02488	−0.0342772
$895$	8.46273	0.282878
$896$	19.1361	0.639294
$897$	0.0684664	0.00228603
$898$	20.5276	0.685015
$899$	−0.0349678	−0.00116624
$900$	−12.9845	−0.432818
$901$	−28.0686	−0.935102
$902$	−90.2165	−3.00388
$903$	0.304424	0.0101306
$904$	−82.9486	−2.75883
$905$	18.9450	0.629754
$906$	0.0170589	0.000566745 0
$907$	−47.8168	−1.58773	−0.793865	−	0.608094i	$-0.791934\pi$
$907$	−47.8168	−1.58773	−0.793865	+	0.608094i	$0.791934\pi$
$908$	41.9958	1.39368
$909$	46.5333	1.54341
$910$	6.64332	0.220224
$911$	−54.4120	−1.80275	−0.901374	−	0.433041i	$-0.857441\pi$
$911$	−54.4120	−1.80275	−0.901374	+	0.433041i	$0.857441\pi$
$912$	0.850612	0.0281666
$913$	−47.7268	−1.57953
$914$	−89.2710	−2.95282
$915$	0.103487	0.00342116
$916$	−66.7393	−2.20513
$917$	−2.97983	−0.0984028
$918$	0.998404	0.0329523
$919$	−44.0111	−1.45179	−0.725895	−	0.687805i	$-0.758574\pi$
$919$	−44.0111	−1.45179	−0.725895	+	0.687805i	$0.758574\pi$
$920$	−8.99090	−0.296421
$921$	−0.220614	−0.00726949
$922$	−49.3417	−1.62498
$923$	8.52188	0.280501
$924$	−0.462843	−0.0152264
$925$	6.32518	0.207971
$926$	1.35595	0.0445594
$927$	19.5116	0.640844
$928$	0.387619	0.0127242
$929$	60.5207	1.98562	0.992809	−	0.119706i	$-0.0381952\pi$
$929$	60.5207	1.98562	0.992809	+	0.119706i	$0.0381952\pi$
$930$	−0.0174065	−0.000570782 0
$931$	35.2205	1.15431
$932$	80.8553	2.64850
$933$	−0.430964	−0.0141091
$934$	93.7048	3.06611
$935$	12.1625	0.397756
$936$	36.6066	1.19652
$937$	−3.40446	−0.111219	−0.0556095	−	0.998453i	$-0.517710\pi$
$937$	−3.40446	−0.111219	−0.0556095	+	0.998453i	$0.517710\pi$
$938$	39.9056	1.30296
$939$	0.663195	0.0216426
$940$	−39.2029	−1.27866
$941$	−43.9118	−1.43148	−0.715742	−	0.698364i	$-0.753912\pi$
$941$	−43.9118	−1.43148	−0.715742	+	0.698364i	$0.753912\pi$
$942$	0.696297	0.0226866
$943$	−13.9752	−0.455094
$944$	−32.3949	−1.05437
$945$	−0.162896	−0.00529902
$946$	111.075	3.61136
$947$	−56.8336	−1.84684	−0.923422	−	0.383786i	$-0.874620\pi$
$947$	−56.8336	−1.84684	−0.923422	+	0.383786i	$0.874620\pi$
$948$	0.764116	0.0248173
$949$	30.5594	0.992000
$950$	16.4299	0.533056
$951$	−0.0818545	−0.00265431
$952$	22.9390	0.743459
$953$	−7.08751	−0.229587	−0.114793	−	0.993389i	$-0.536621\pi$
$953$	−7.08751	−0.229587	−0.114793	+	0.993389i	$0.536621\pi$
$954$	68.5780	2.22029
$955$	−7.70723	−0.249400
$956$	21.1206	0.683090
$957$	0.00912920	0.000295105 0
$958$	25.6063	0.827304
$959$	23.0782	0.745235
$960$	−0.0675369	−0.00217974
$961$	−30.8956	−0.996633
$962$	−33.1465	−1.06869
$963$	39.6035	1.27620
$964$	−40.5658	−1.30654
$965$	−22.0565	−0.710025
$966$	−0.104823	−0.00337263
$967$	−21.1696	−0.680768	−0.340384	−	0.940287i	$-0.610557\pi$
$967$	−21.1696	−0.680768	−0.340384	+	0.940287i	$0.610557\pi$
$968$	−26.4073	−0.848763
$969$	−0.432015	−0.0138783
$970$	20.2111	0.648939
$971$	48.4383	1.55446	0.777230	−	0.629217i	$-0.216624\pi$
$971$	48.4383	1.55446	0.777230	+	0.629217i	$0.216624\pi$
$972$	−2.50276	−0.0802761
$973$	−9.12620	−0.292573
$974$	−10.6038	−0.339769
$975$	−0.0446145	−0.00142881
$976$	29.3831	0.940530
$977$	−21.9998	−0.703836	−0.351918	−	0.936031i	$-0.614470\pi$
$977$	−21.9998	−0.703836	−0.351918	+	0.936031i	$0.614470\pi$
$978$	0.763709	0.0244207
$979$	34.2602	1.09496
$980$	23.3450	0.745729
$981$	−26.4970	−0.845984
$982$	−39.0326	−1.24558
$983$	5.81550	0.185486	0.0927428	−	0.995690i	$-0.470437\pi$
$983$	5.81550	0.185486	0.0927428	+	0.995690i	$0.470437\pi$
$984$	1.14269	0.0364277
$985$	−8.97138	−0.285852
$986$	−0.841021	−0.0267836
$987$	−0.245889	−0.00782674
$988$	−58.8907	−1.87356
$989$	17.2063	0.547128
$990$	−29.7157	−0.944427
$991$	12.3809	0.393293	0.196647	−	0.980474i	$-0.436995\pi$
$991$	12.3809	0.393293	0.196647	+	0.980474i	$0.436995\pi$
$992$	−1.15689	−0.0367312
$993$	−0.158048	−0.00501550
$994$	−13.0471	−0.413830
$995$	3.41567	0.108284
$996$	1.12367	0.0356048
$997$	40.4347	1.28058	0.640290	−	0.768133i	$-0.278814\pi$
$997$	40.4347	1.28058	0.640290	+	0.768133i	$0.278814\pi$
$998$	22.6988	0.718518
$999$	0.812762	0.0257147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.b.1.8	✓	126

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.b.1.8	✓	126	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51572 q^{2} -0.0214177 q^{3} +4.32884 q^{4} -1.00000 q^{5} +0.0538809 q^{6} +1.26771 q^{7} -5.85870 q^{8} -2.99954 q^{9} +O(q^{10})\)	\(q-2.51572 q^{2} -0.0214177 q^{3} +4.32884 q^{4} -1.00000 q^{5} +0.0538809 q^{6} +1.26771 q^{7} -5.85870 q^{8} -2.99954 q^{9} +2.51572 q^{10} +3.93794 q^{11} -0.0927137 q^{12} +2.08307 q^{13} -3.18920 q^{14} +0.0214177 q^{15} +6.08116 q^{16} -3.08854 q^{17} +7.54600 q^{18} -6.53090 q^{19} -4.32884 q^{20} -0.0271515 q^{21} -9.90675 q^{22} -1.53462 q^{23} +0.125480 q^{24} +1.00000 q^{25} -5.24041 q^{26} +0.128496 q^{27} +5.48772 q^{28} -0.108241 q^{29} -0.0538809 q^{30} +0.323055 q^{31} -3.58108 q^{32} -0.0843416 q^{33} +7.76990 q^{34} -1.26771 q^{35} -12.9845 q^{36} +6.32518 q^{37} +16.4299 q^{38} -0.0446145 q^{39} +5.85870 q^{40} +9.10657 q^{41} +0.0683054 q^{42} -11.2121 q^{43} +17.0467 q^{44} +2.99954 q^{45} +3.86068 q^{46} +9.05621 q^{47} -0.130244 q^{48} -5.39291 q^{49} -2.51572 q^{50} +0.0661495 q^{51} +9.01725 q^{52} +9.08799 q^{53} -0.323261 q^{54} -3.93794 q^{55} -7.42714 q^{56} +0.139877 q^{57} +0.272303 q^{58} -5.32710 q^{59} +0.0927137 q^{60} +4.83182 q^{61} -0.812717 q^{62} -3.80255 q^{63} -3.15332 q^{64} -2.08307 q^{65} +0.212180 q^{66} -12.5127 q^{67} -13.3698 q^{68} +0.0328681 q^{69} +3.18920 q^{70} +4.09103 q^{71} +17.5734 q^{72} +14.6704 q^{73} -15.9124 q^{74} -0.0214177 q^{75} -28.2712 q^{76} +4.99217 q^{77} +0.112237 q^{78} -8.24167 q^{79} -6.08116 q^{80} +8.99587 q^{81} -22.9096 q^{82} -12.1198 q^{83} -0.117534 q^{84} +3.08854 q^{85} +28.2064 q^{86} +0.00231827 q^{87} -23.0712 q^{88} +8.70002 q^{89} -7.54600 q^{90} +2.64073 q^{91} -6.64314 q^{92} -0.00691910 q^{93} -22.7829 q^{94} +6.53090 q^{95} +0.0766985 q^{96} +8.03392 q^{97} +13.5670 q^{98} -11.8120 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * q^99

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