LMFDB - Embedded newform 1045.4.a.b.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + \cdots + 17756160 $ x^20 - 8x^19 - 82x^18 + 700x^17 + 2826x^16 - 25467x^15 - 53768x^14 + 498499x^13 + 623613x^12 - 5673747x^11 - 4539454x^10 + 37893109x^9 + 19879768x^8 - 143049638x^7 - 45064360x^6 + 280461480x^5 + 43097920x^4 - 240447168x^3 - 36068096x^2 + 74703872*x + 17756160
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.12533$ 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)

$f(q)$	$=$	$q+0.125334 q^{2} -3.53244 q^{3} -7.98429 q^{4} +5.00000 q^{5} -0.442735 q^{6} -25.7104 q^{7} -2.00338 q^{8} -14.5219 q^{9} +O(q^{10})$	\(q+0.125334 q^{2} -3.53244 q^{3} -7.98429 q^{4} +5.00000 q^{5} -0.442735 q^{6} -25.7104 q^{7} -2.00338 q^{8} -14.5219 q^{9} +0.626671 q^{10} +11.0000 q^{11} +28.2040 q^{12} -9.72805 q^{13} -3.22239 q^{14} -17.6622 q^{15} +63.6232 q^{16} +102.892 q^{17} -1.82008 q^{18} +19.0000 q^{19} -39.9215 q^{20} +90.8203 q^{21} +1.37868 q^{22} +119.737 q^{23} +7.07681 q^{24} +25.0000 q^{25} -1.21926 q^{26} +146.674 q^{27} +205.279 q^{28} -270.097 q^{29} -2.21368 q^{30} +68.5788 q^{31} +24.0012 q^{32} -38.8568 q^{33} +12.8959 q^{34} -128.552 q^{35} +115.947 q^{36} +383.867 q^{37} +2.38135 q^{38} +34.3638 q^{39} -10.0169 q^{40} -97.2378 q^{41} +11.3829 q^{42} -516.813 q^{43} -87.8272 q^{44} -72.6093 q^{45} +15.0072 q^{46} +102.280 q^{47} -224.745 q^{48} +318.023 q^{49} +3.13335 q^{50} -363.459 q^{51} +77.6716 q^{52} +136.916 q^{53} +18.3832 q^{54} +55.0000 q^{55} +51.5075 q^{56} -67.1164 q^{57} -33.8523 q^{58} +106.238 q^{59} +141.020 q^{60} +359.398 q^{61} +8.59526 q^{62} +373.362 q^{63} -505.978 q^{64} -48.6402 q^{65} -4.87009 q^{66} -843.511 q^{67} -821.519 q^{68} -422.965 q^{69} -16.1119 q^{70} +442.232 q^{71} +29.0928 q^{72} -675.198 q^{73} +48.1116 q^{74} -88.3110 q^{75} -151.702 q^{76} -282.814 q^{77} +4.30695 q^{78} +547.339 q^{79} +318.116 q^{80} -126.025 q^{81} -12.1872 q^{82} -339.235 q^{83} -725.136 q^{84} +514.459 q^{85} -64.7743 q^{86} +954.101 q^{87} -22.0371 q^{88} +1475.24 q^{89} -9.10042 q^{90} +250.112 q^{91} -956.018 q^{92} -242.251 q^{93} +12.8192 q^{94} +95.0000 q^{95} -84.7827 q^{96} -593.234 q^{97} +39.8591 q^{98} -159.741 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) $ 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * 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 $

$2$	0.125334	0.0443123	0.0221561	−	0.999755i	$-0.492947\pi$
$2$	0.125334	0.0443123	0.0221561	+	0.999755i	$0.492947\pi$
$3$	−3.53244	−0.679818	−0.339909	−	0.940458i	$-0.610396\pi$
$3$	−3.53244	−0.679818	−0.339909	+	0.940458i	$0.610396\pi$
$4$	−7.98429	−0.998036
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−0.442735	−0.0301243
$7$	−25.7104	−1.38823	−0.694115	−	0.719865i	$-0.744204\pi$
$7$	−25.7104	−1.38823	−0.694115	+	0.719865i	$0.744204\pi$
$8$	−2.00338	−0.0885376
$9$	−14.5219	−0.537847
$10$	0.626671	0.0198171
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	28.2040	0.678484
$13$	−9.72805	−0.207544	−0.103772	−	0.994601i	$-0.533091\pi$
$13$	−9.72805	−0.207544	−0.103772	+	0.994601i	$0.533091\pi$
$14$	−3.22239	−0.0615156
$15$	−17.6622	−0.304024
$16$	63.6232	0.994113
$17$	102.892	1.46794	0.733969	−	0.679183i	$-0.237666\pi$
$17$	102.892	1.46794	0.733969	+	0.679183i	$0.237666\pi$
$18$	−1.82008	−0.0238332
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−39.9215	−0.446335
$21$	90.8203	0.943744
$22$	1.37868	0.0133607
$23$	119.737	1.08552	0.542760	−	0.839888i	$-0.317379\pi$
$23$	119.737	1.08552	0.542760	+	0.839888i	$0.317379\pi$
$24$	7.07681	0.0601895
$25$	25.0000	0.200000
$26$	−1.21926	−0.00919676
$27$	146.674	1.04546
$28$	205.279	1.38550
$29$	−270.097	−1.72951	−0.864754	−	0.502196i	$-0.832526\pi$
$29$	−270.097	−1.72951	−0.864754	+	0.502196i	$0.832526\pi$
$30$	−2.21368	−0.0134720
$31$	68.5788	0.397326	0.198663	−	0.980068i	$-0.436340\pi$
$31$	68.5788	0.397326	0.198663	+	0.980068i	$0.436340\pi$
$32$	24.0012	0.132589
$33$	−38.8568	−0.204973
$34$	12.8959	0.0650477
$35$	−128.552	−0.620835
$36$	115.947	0.536791
$37$	383.867	1.70560	0.852802	−	0.522234i	$-0.174901\pi$
$37$	383.867	1.70560	0.852802	+	0.522234i	$0.174901\pi$
$38$	2.38135	0.0101659
$39$	34.3638	0.141092
$40$	−10.0169	−0.0395952
$41$	−97.2378	−0.370390	−0.185195	−	0.982702i	$-0.559292\pi$
$41$	−97.2378	−0.370390	−0.185195	+	0.982702i	$0.559292\pi$
$42$	11.3829	0.0418195
$43$	−516.813	−1.83287	−0.916434	−	0.400187i	$-0.868945\pi$
$43$	−516.813	−1.83287	−0.916434	+	0.400187i	$0.868945\pi$
$44$	−87.8272	−0.300919
$45$	−72.6093	−0.240532
$46$	15.0072	0.0481019
$47$	102.280	0.317427	0.158714	−	0.987325i	$-0.449265\pi$
$47$	102.280	0.317427	0.158714	+	0.987325i	$0.449265\pi$
$48$	−224.745	−0.675816
$49$	318.023	0.927180
$50$	3.13335	0.00886246
$51$	−363.459	−0.997932
$52$	77.6716	0.207137
$53$	136.916	0.354845	0.177423	−	0.984135i	$-0.443224\pi$
$53$	136.916	0.354845	0.177423	+	0.984135i	$0.443224\pi$
$54$	18.3832	0.0463266
$55$	55.0000	0.134840
$56$	51.5075	0.122910
$57$	−67.1164	−0.155961
$58$	−33.8523	−0.0766385
$59$	106.238	0.234423	0.117211	−	0.993107i	$-0.462604\pi$
$59$	106.238	0.234423	0.117211	+	0.993107i	$0.462604\pi$
$60$	141.020	0.303427
$61$	359.398	0.754364	0.377182	−	0.926139i	$-0.376893\pi$
$61$	359.398	0.754364	0.377182	+	0.926139i	$0.376893\pi$
$62$	8.59526	0.0176064
$63$	373.362	0.746655
$64$	−505.978	−0.988238
$65$	−48.6402	−0.0928166
$66$	−4.87009	−0.00908282
$67$	−843.511	−1.53808	−0.769040	−	0.639201i	$-0.779265\pi$
$67$	−843.511	−1.53808	−0.769040	+	0.639201i	$0.779265\pi$
$68$	−821.519	−1.46506
$69$	−422.965	−0.737957
$70$	−16.1119	−0.0275106
$71$	442.232	0.739201	0.369600	−	0.929191i	$-0.379495\pi$
$71$	442.232	0.739201	0.369600	+	0.929191i	$0.379495\pi$
$72$	29.0928	0.0476197
$73$	−675.198	−1.08255	−0.541274	−	0.840846i	$-0.682058\pi$
$73$	−675.198	−1.08255	−0.541274	+	0.840846i	$0.682058\pi$
$74$	48.1116	0.0755793
$75$	−88.3110	−0.135964
$76$	−151.702	−0.228965
$77$	−282.814	−0.418567
$78$	4.30695	0.00625213
$79$	547.339	0.779500	0.389750	−	0.920921i	$-0.372561\pi$
$79$	547.339	0.779500	0.389750	+	0.920921i	$0.372561\pi$
$80$	318.116	0.444581
$81$	−126.025	−0.172874
$82$	−12.1872	−0.0164128
$83$	−339.235	−0.448625	−0.224313	−	0.974517i	$-0.572014\pi$
$83$	−339.235	−0.448625	−0.224313	+	0.974517i	$0.572014\pi$
$84$	−725.136	−0.941891
$85$	514.459	0.656482
$86$	−64.7743	−0.0812186
$87$	954.101	1.17575
$88$	−22.0371	−0.0266951
$89$	1475.24	1.75702	0.878509	−	0.477726i	$-0.158539\pi$
$89$	1475.24	1.75702	0.878509	+	0.477726i	$0.158539\pi$
$90$	−9.10042	−0.0106585
$91$	250.112	0.288119
$92$	−956.018	−1.08339
$93$	−242.251	−0.270110
$94$	12.8192	0.0140659
$95$	95.0000	0.102598
$96$	−84.7827	−0.0901365
$97$	−593.234	−0.620966	−0.310483	−	0.950579i	$-0.600491\pi$
$97$	−593.234	−0.620966	−0.310483	+	0.950579i	$0.600491\pi$
$98$	39.8591	0.0410855
$99$	−159.741	−0.162167
$100$	−199.607	−0.199607
$101$	−943.980	−0.929995	−0.464997	−	0.885312i	$-0.653945\pi$
$101$	−943.980	−0.929995	−0.464997	+	0.885312i	$0.653945\pi$
$102$	−45.5539	−0.0442206
$103$	−345.506	−0.330522	−0.165261	−	0.986250i	$-0.552847\pi$
$103$	−345.506	−0.330522	−0.165261	+	0.986250i	$0.552847\pi$
$104$	19.4889	0.0183755
$105$	454.102	0.422055
$106$	17.1602	0.0157240
$107$	533.270	0.481805	0.240902	−	0.970549i	$-0.422557\pi$
$107$	533.270	0.481805	0.240902	+	0.970549i	$0.422557\pi$
$108$	−1171.08	−1.04340
$109$	−1496.67	−1.31518	−0.657592	−	0.753374i	$-0.728425\pi$
$109$	−1496.67	−1.31518	−0.657592	+	0.753374i	$0.728425\pi$
$110$	6.89338	0.00597507
$111$	−1355.99	−1.15950
$112$	−1635.78	−1.38006
$113$	1127.07	0.938285	0.469143	−	0.883122i	$-0.344563\pi$
$113$	1127.07	0.938285	0.469143	+	0.883122i	$0.344563\pi$
$114$	−8.41197	−0.00691099
$115$	598.687	0.485459
$116$	2156.53	1.72611
$117$	141.269	0.111627
$118$	13.3152	0.0103878
$119$	−2645.39	−2.03783
$120$	35.3840	0.0269176
$121$	121.000	0.0909091
$122$	45.0449	0.0334276
$123$	343.487	0.251798
$124$	−547.553	−0.396546
$125$	125.000	0.0894427
$126$	46.7950	0.0330860
$127$	−2196.46	−1.53468	−0.767338	−	0.641243i	$-0.778419\pi$
$127$	−2196.46	−1.53468	−0.767338	+	0.641243i	$0.778419\pi$
$128$	−255.426	−0.176380
$129$	1825.61	1.24602
$130$	−6.09628	−0.00411292
$131$	−58.1627	−0.0387916	−0.0193958	−	0.999812i	$-0.506174\pi$
$131$	−58.1627	−0.0387916	−0.0193958	+	0.999812i	$0.506174\pi$
$132$	310.244	0.204571
$133$	−488.497	−0.318482
$134$	−105.721	−0.0681558
$135$	733.368	0.467542
$136$	−206.131	−0.129968
$137$	2358.79	1.47099	0.735493	−	0.677533i	$-0.236951\pi$
$137$	2358.79	1.47099	0.735493	+	0.677533i	$0.236951\pi$
$138$	−53.0119	−0.0327005
$139$	654.253	0.399230	0.199615	−	0.979874i	$-0.436031\pi$
$139$	654.253	0.399230	0.199615	+	0.979874i	$0.436031\pi$
$140$	1026.40	0.619616
$141$	−361.298	−0.215793
$142$	55.4267	0.0327557
$143$	−107.009	−0.0625769
$144$	−923.928	−0.534681
$145$	−1350.48	−0.773459
$146$	−84.6254	−0.0479702
$147$	−1123.40	−0.630314
$148$	−3064.91	−1.70226
$149$	1105.02	0.607565	0.303782	−	0.952741i	$-0.401750\pi$
$149$	1105.02	0.607565	0.303782	+	0.952741i	$0.401750\pi$
$150$	−11.0684	−0.00602486
$151$	−1959.96	−1.05629	−0.528144	−	0.849155i	$-0.677112\pi$
$151$	−1959.96	−1.05629	−0.528144	+	0.849155i	$0.677112\pi$
$152$	−38.0642	−0.0203119
$153$	−1494.18	−0.789526
$154$	−35.4462	−0.0185477
$155$	342.894	0.177690
$156$	−274.370	−0.140815
$157$	−319.892	−0.162612	−0.0813062	−	0.996689i	$-0.525909\pi$
$157$	−319.892	−0.162612	−0.0813062	+	0.996689i	$0.525909\pi$
$158$	68.6003	0.0345414
$159$	−483.646	−0.241230
$160$	120.006	0.0592956
$161$	−3078.49	−1.50695
$162$	−15.7952	−0.00766044
$163$	−3224.11	−1.54928	−0.774638	−	0.632404i	$-0.782068\pi$
$163$	−3224.11	−1.54928	−0.774638	+	0.632404i	$0.782068\pi$
$164$	776.375	0.369663
$165$	−194.284	−0.0916667
$166$	−42.5177	−0.0198796
$167$	3743.48	1.73461	0.867303	−	0.497781i	$-0.165852\pi$
$167$	3743.48	1.73461	0.867303	+	0.497781i	$0.165852\pi$
$168$	−181.947	−0.0835568
$169$	−2102.37	−0.956925
$170$	64.4793	0.0290902
$171$	−275.915	−0.123391
$172$	4126.39	1.82927
$173$	−3590.04	−1.57772	−0.788859	−	0.614574i	$-0.789328\pi$
$173$	−3590.04	−1.57772	−0.788859	+	0.614574i	$0.789328\pi$
$174$	119.581	0.0521002
$175$	−642.759	−0.277646
$176$	699.856	0.299736
$177$	−375.278	−0.159365
$178$	184.897	0.0778575
$179$	−2891.99	−1.20758	−0.603792	−	0.797142i	$-0.706344\pi$
$179$	−2891.99	−1.20758	−0.603792	+	0.797142i	$0.706344\pi$
$180$	579.734	0.240060
$181$	−3389.38	−1.39188	−0.695940	−	0.718100i	$-0.745012\pi$
$181$	−3389.38	−1.39188	−0.695940	+	0.718100i	$0.745012\pi$
$182$	31.3475	0.0127672
$183$	−1269.55	−0.512831
$184$	−239.879	−0.0961093
$185$	1919.34	0.762770
$186$	−30.3623	−0.0119692
$187$	1131.81	0.442600
$188$	−816.633	−0.316804
$189$	−3771.03	−1.45133
$190$	11.9067	0.00454635
$191$	−169.556	−0.0642336	−0.0321168	−	0.999484i	$-0.510225\pi$
$191$	−169.556	−0.0642336	−0.0321168	+	0.999484i	$0.510225\pi$
$192$	1787.34	0.671822
$193$	−757.259	−0.282429	−0.141214	−	0.989979i	$-0.545101\pi$
$193$	−757.259	−0.282429	−0.141214	+	0.989979i	$0.545101\pi$
$194$	−74.3524	−0.0275164
$195$	171.819	0.0630985
$196$	−2539.19	−0.925360
$197$	−4592.57	−1.66095	−0.830475	−	0.557055i	$-0.811931\pi$
$197$	−4592.57	−1.66095	−0.830475	+	0.557055i	$0.811931\pi$
$198$	−20.0209	−0.00718599
$199$	3580.07	1.27530	0.637649	−	0.770327i	$-0.279907\pi$
$199$	3580.07	1.27530	0.637649	+	0.770327i	$0.279907\pi$
$200$	−50.0844	−0.0177075
$201$	2979.65	1.04561
$202$	−118.313	−0.0412102
$203$	6944.29	2.40095
$204$	2901.97	0.995972
$205$	−486.189	−0.165643
$206$	−43.3037	−0.0146462
$207$	−1738.81	−0.583843
$208$	−618.930	−0.206322
$209$	209.000	0.0691714
$210$	56.9144	0.0187022
$211$	−420.956	−0.137345	−0.0686726	−	0.997639i	$-0.521876\pi$
$211$	−420.956	−0.137345	−0.0686726	+	0.997639i	$0.521876\pi$
$212$	−1093.17	−0.354148
$213$	−1562.16	−0.502522
$214$	66.8369	0.0213499
$215$	−2584.07	−0.819683
$216$	−293.842	−0.0925622
$217$	−1763.19	−0.551580
$218$	−187.584	−0.0582788
$219$	2385.10	0.735936
$220$	−439.136	−0.134575
$221$	−1000.94	−0.304662
$222$	−169.952	−0.0513802
$223$	−3381.85	−1.01554	−0.507770	−	0.861493i	$-0.669530\pi$
$223$	−3381.85	−1.01554	−0.507770	+	0.861493i	$0.669530\pi$
$224$	−617.079	−0.184064
$225$	−363.047	−0.107569
$226$	141.261	0.0415776
$227$	−3960.69	−1.15806	−0.579031	−	0.815306i	$-0.696569\pi$
$227$	−3960.69	−1.15806	−0.579031	+	0.815306i	$0.696569\pi$
$228$	535.877	0.155655
$229$	2406.95	0.694566	0.347283	−	0.937760i	$-0.387104\pi$
$229$	2406.95	0.694566	0.347283	+	0.937760i	$0.387104\pi$
$230$	75.0359	0.0215118
$231$	999.024	0.284549
$232$	541.106	0.153126
$233$	6725.93	1.89112	0.945558	−	0.325454i	$-0.105517\pi$
$233$	6725.93	1.89112	0.945558	+	0.325454i	$0.105517\pi$
$234$	17.7059	0.00494645
$235$	511.400	0.141958
$236$	−848.231	−0.233963
$237$	−1933.44	−0.529918
$238$	−331.557	−0.0903011
$239$	−5031.24	−1.36169	−0.680845	−	0.732428i	$-0.738387\pi$
$239$	−5031.24	−1.36169	−0.680845	+	0.732428i	$0.738387\pi$
$240$	−1123.73	−0.302234
$241$	6278.97	1.67827	0.839137	−	0.543919i	$-0.183060\pi$
$241$	6278.97	1.67827	0.839137	+	0.543919i	$0.183060\pi$
$242$	15.1654	0.00402839
$243$	−3515.01	−0.927934
$244$	−2869.54	−0.752883
$245$	1590.11	0.414648
$246$	43.0506	0.0111577
$247$	−184.833	−0.0476139
$248$	−137.389	−0.0351783
$249$	1198.33	0.304984
$250$	15.6668	0.00396341
$251$	2093.00	0.526331	0.263165	−	0.964751i	$-0.415233\pi$
$251$	2093.00	0.526331	0.263165	+	0.964751i	$0.415233\pi$
$252$	−2981.03	−0.745189
$253$	1317.11	0.327297
$254$	−275.291	−0.0680050
$255$	−1817.30	−0.446289
$256$	4015.81	0.980422
$257$	6276.57	1.52343	0.761715	−	0.647912i	$-0.224358\pi$
$257$	6276.57	1.52343	0.761715	+	0.647912i	$0.224358\pi$
$258$	228.811	0.0552139
$259$	−9869.36	−2.36777
$260$	388.358	0.0926344
$261$	3922.31	0.930210
$262$	−7.28977	−0.00171894
$263$	−7254.84	−1.70096	−0.850480	−	0.526007i	$-0.823688\pi$
$263$	−7254.84	−1.70096	−0.850480	+	0.526007i	$0.823688\pi$
$264$	77.8449	0.0181478
$265$	684.578	0.158692
$266$	−61.2253	−0.0141127
$267$	−5211.18	−1.19445
$268$	6734.84	1.53506
$269$	2655.77	0.601953	0.300976	−	0.953632i	$-0.402687\pi$
$269$	2655.77	0.601953	0.300976	+	0.953632i	$0.402687\pi$
$270$	91.9160	0.0207179
$271$	4608.23	1.03295	0.516476	−	0.856301i	$-0.327243\pi$
$271$	4608.23	1.03295	0.516476	+	0.856301i	$0.327243\pi$
$272$	6546.32	1.45930
$273$	−883.505	−0.195869
$274$	295.637	0.0651827
$275$	275.000	0.0603023
$276$	3377.08	0.736508
$277$	−4942.12	−1.07200	−0.535998	−	0.844219i	$-0.680065\pi$
$277$	−4942.12	−1.07200	−0.535998	+	0.844219i	$0.680065\pi$
$278$	82.0002	0.0176908
$279$	−995.892	−0.213701
$280$	257.538	0.0549672
$281$	5320.65	1.12955	0.564774	−	0.825245i	$-0.308963\pi$
$281$	5320.65	1.12955	0.564774	+	0.825245i	$0.308963\pi$
$282$	−45.2830	−0.00956228
$283$	−2333.67	−0.490185	−0.245092	−	0.969500i	$-0.578818\pi$
$283$	−2333.67	−0.490185	−0.245092	+	0.969500i	$0.578818\pi$
$284$	−3530.91	−0.737750
$285$	−335.582	−0.0697479
$286$	−13.4118	−0.00277293
$287$	2500.02	0.514186
$288$	−348.542	−0.0713126
$289$	5673.74	1.15484
$290$	−169.262	−0.0342738
$291$	2095.56	0.422144
$292$	5390.98	1.08042
$293$	7092.51	1.41416	0.707080	−	0.707133i	$-0.250012\pi$
$293$	7092.51	1.41416	0.707080	+	0.707133i	$0.250012\pi$
$294$	−140.800	−0.0279307
$295$	531.188	0.104837
$296$	−769.031	−0.151010
$297$	1613.41	0.315217
$298$	138.497	0.0269226
$299$	−1164.81	−0.225293
$300$	705.101	0.135697
$301$	13287.5	2.54444
$302$	−245.650	−0.0468066
$303$	3334.55	0.632228
$304$	1208.84	0.228065
$305$	1796.99	0.337362
$306$	−187.272	−0.0349857
$307$	1696.63	0.315413	0.157706	−	0.987486i	$-0.449590\pi$
$307$	1696.63	0.315413	0.157706	+	0.987486i	$0.449590\pi$
$308$	2258.07	0.417745
$309$	1220.48	0.224695
$310$	42.9763	0.00787384
$311$	6100.43	1.11229	0.556147	−	0.831084i	$-0.312279\pi$
$311$	6100.43	1.11229	0.556147	+	0.831084i	$0.312279\pi$
$312$	−68.8435	−0.0124920
$313$	−5372.09	−0.970122	−0.485061	−	0.874480i	$-0.661203\pi$
$313$	−5372.09	−0.970122	−0.485061	+	0.874480i	$0.661203\pi$
$314$	−40.0934	−0.00720573
$315$	1866.81	0.333914
$316$	−4370.12	−0.777969
$317$	−1580.72	−0.280069	−0.140035	−	0.990147i	$-0.544721\pi$
$317$	−1580.72	−0.280069	−0.140035	+	0.990147i	$0.544721\pi$
$318$	−60.6173	−0.0106895
$319$	−2971.07	−0.521466
$320$	−2529.89	−0.441953
$321$	−1883.74	−0.327540
$322$	−385.840	−0.0667764
$323$	1954.95	0.336768
$324$	1006.22	0.172535
$325$	−243.201	−0.0415089
$326$	−404.092	−0.0686520
$327$	5286.90	0.894086
$328$	194.804	0.0327934
$329$	−2629.66	−0.440662
$330$	−24.3504	−0.00406196
$331$	1954.87	0.324621	0.162311	−	0.986740i	$-0.448105\pi$
$331$	1954.87	0.324621	0.162311	+	0.986740i	$0.448105\pi$
$332$	2708.55	0.447744
$333$	−5574.47	−0.917354
$334$	469.186	0.0768643
$335$	−4217.56	−0.687850
$336$	5778.28	0.938188
$337$	−5720.45	−0.924667	−0.462333	−	0.886706i	$-0.652988\pi$
$337$	−5720.45	−0.924667	−0.462333	+	0.886706i	$0.652988\pi$
$338$	−263.498	−0.0424036
$339$	−3981.32	−0.637864
$340$	−4107.59	−0.655193
$341$	754.367	0.119798
$342$	−34.5816	−0.00546772
$343$	642.171	0.101090
$344$	1035.37	0.162278
$345$	−2114.82	−0.330024
$346$	−449.954	−0.0699123
$347$	3923.21	0.606942	0.303471	−	0.952841i	$-0.401854\pi$
$347$	3923.21	0.606942	0.303471	+	0.952841i	$0.401854\pi$
$348$	−7617.82	−1.17344
$349$	−6176.74	−0.947373	−0.473687	−	0.880693i	$-0.657077\pi$
$349$	−6176.74	−0.947373	−0.473687	+	0.880693i	$0.657077\pi$
$350$	−80.5596	−0.0123031
$351$	−1426.85	−0.216979
$352$	264.013	0.0399771
$353$	8516.19	1.28405	0.642027	−	0.766682i	$-0.278094\pi$
$353$	8516.19	1.28405	0.642027	+	0.766682i	$0.278094\pi$
$354$	−47.0351	−0.00706183
$355$	2211.16	0.330581
$356$	−11778.7	−1.75357
$357$	9344.68	1.38536
$358$	−362.465	−0.0535108
$359$	−7022.75	−1.03244	−0.516221	−	0.856455i	$-0.672662\pi$
$359$	−7022.75	−1.03244	−0.516221	+	0.856455i	$0.672662\pi$
$360$	145.464	0.0212962
$361$	361.000	0.0526316
$362$	−424.804	−0.0616774
$363$	−427.425	−0.0618017
$364$	−1996.96	−0.287553
$365$	−3375.99	−0.484130
$366$	−159.118	−0.0227247
$367$	−10224.2	−1.45422	−0.727109	−	0.686522i	$-0.759137\pi$
$367$	−10224.2	−1.45422	−0.727109	+	0.686522i	$0.759137\pi$
$368$	7618.08	1.07913
$369$	1412.07	0.199213
$370$	240.558	0.0338001
$371$	−3520.15	−0.492607
$372$	1934.20	0.269579
$373$	3258.95	0.452392	0.226196	−	0.974082i	$-0.427371\pi$
$373$	3258.95	0.452392	0.226196	+	0.974082i	$0.427371\pi$
$374$	141.854	0.0196126
$375$	−441.555	−0.0608048
$376$	−204.905	−0.0281042
$377$	2627.52	0.358949
$378$	−472.639	−0.0643119
$379$	−7873.96	−1.06717	−0.533586	−	0.845746i	$-0.679156\pi$
$379$	−7873.96	−1.06717	−0.533586	+	0.845746i	$0.679156\pi$
$380$	−758.508	−0.102396
$381$	7758.85	1.04330
$382$	−21.2511	−0.00284634
$383$	−11078.9	−1.47808	−0.739040	−	0.673661i	$-0.764721\pi$
$383$	−11078.9	−1.47808	−0.739040	+	0.673661i	$0.764721\pi$
$384$	902.276	0.119906
$385$	−1414.07	−0.187189
$386$	−94.9104	−0.0125151
$387$	7505.09	0.985802
$388$	4736.55	0.619747
$389$	12688.9	1.65387	0.826934	−	0.562299i	$-0.190083\pi$
$389$	12688.9	1.65387	0.826934	+	0.562299i	$0.190083\pi$
$390$	21.5348	0.00279604
$391$	12320.0	1.59348
$392$	−637.120	−0.0820903
$393$	205.456	0.0263712
$394$	−575.606	−0.0736005
$395$	2736.70	0.348603
$396$	1275.41	0.161848
$397$	−9445.41	−1.19408	−0.597042	−	0.802210i	$-0.703658\pi$
$397$	−9445.41	−1.19408	−0.597042	+	0.802210i	$0.703658\pi$
$398$	448.705	0.0565114
$399$	1725.59	0.216510
$400$	1590.58	0.198823
$401$	−5682.65	−0.707676	−0.353838	−	0.935307i	$-0.615124\pi$
$401$	−5682.65	−0.707676	−0.353838	+	0.935307i	$0.615124\pi$
$402$	373.452	0.0463336
$403$	−667.138	−0.0824628
$404$	7537.01	0.928169
$405$	−630.126	−0.0773116
$406$	870.356	0.106392
$407$	4222.54	0.514259
$408$	728.146	0.0883544
$409$	−7931.06	−0.958840	−0.479420	−	0.877586i	$-0.659153\pi$
$409$	−7931.06	−0.958840	−0.479420	+	0.877586i	$0.659153\pi$
$410$	−60.9360	−0.00734004
$411$	−8332.28	−1.00000
$412$	2758.62	0.329873
$413$	−2731.41	−0.325433
$414$	−217.932	−0.0258714
$415$	−1696.18	−0.200631
$416$	−233.485	−0.0275181
$417$	−2311.11	−0.271404
$418$	26.1948	0.00306515
$419$	−9587.62	−1.11787	−0.558933	−	0.829213i	$-0.688789\pi$
$419$	−9587.62	−1.11787	−0.558933	+	0.829213i	$0.688789\pi$
$420$	−3625.68	−0.421226
$421$	−11410.7	−1.32096	−0.660479	−	0.750845i	$-0.729647\pi$
$421$	−11410.7	−1.32096	−0.660479	+	0.750845i	$0.729647\pi$
$422$	−52.7602	−0.00608608
$423$	−1485.30	−0.170727
$424$	−274.293	−0.0314171
$425$	2572.30	0.293588
$426$	−195.792	−0.0222679
$427$	−9240.26	−1.04723
$428$	−4257.78	−0.480859
$429$	378.001	0.0425410
$430$	−323.872	−0.0363220
$431$	2507.02	0.280183	0.140091	−	0.990139i	$-0.455260\pi$
$431$	2507.02	0.280183	0.140091	+	0.990139i	$0.455260\pi$
$432$	9331.84	1.03930
$433$	−10363.4	−1.15019	−0.575095	−	0.818086i	$-0.695035\pi$
$433$	−10363.4	−1.15019	−0.575095	+	0.818086i	$0.695035\pi$
$434$	−220.987	−0.0244418
$435$	4770.50	0.525812
$436$	11949.9	1.31260
$437$	2275.01	0.249035
$438$	298.934	0.0326110
$439$	−8845.70	−0.961690	−0.480845	−	0.876805i	$-0.659670\pi$
$439$	−8845.70	−0.961690	−0.480845	+	0.876805i	$0.659670\pi$
$440$	−110.186	−0.0119384
$441$	−4618.29	−0.498681
$442$	−125.452	−0.0135003
$443$	−5286.99	−0.567026	−0.283513	−	0.958968i	$-0.591500\pi$
$443$	−5286.99	−0.567026	−0.283513	+	0.958968i	$0.591500\pi$
$444$	10826.6	1.15722
$445$	7376.18	0.785762
$446$	−423.861	−0.0450009
$447$	−3903.43	−0.413034
$448$	13008.9	1.37190
$449$	−2344.36	−0.246408	−0.123204	−	0.992381i	$-0.539317\pi$
$449$	−2344.36	−0.246408	−0.123204	+	0.992381i	$0.539317\pi$
$450$	−45.5021	−0.00476665
$451$	−1069.62	−0.111677
$452$	−8998.89	−0.936443
$453$	6923.45	0.718084
$454$	−496.409	−0.0513164
$455$	1250.56	0.128851
$456$	134.459	0.0138084
$457$	−1705.73	−0.174597	−0.0872985	−	0.996182i	$-0.527823\pi$
$457$	−1705.73	−0.174597	−0.0872985	+	0.996182i	$0.527823\pi$
$458$	301.673	0.0307778
$459$	15091.5	1.53467
$460$	−4780.09	−0.484506
$461$	17250.5	1.74281	0.871404	−	0.490566i	$-0.163210\pi$
$461$	17250.5	1.74281	0.871404	+	0.490566i	$0.163210\pi$
$462$	125.212	0.0126090
$463$	2727.63	0.273788	0.136894	−	0.990586i	$-0.456288\pi$
$463$	2727.63	0.273788	0.136894	+	0.990586i	$0.456288\pi$
$464$	−17184.4	−1.71933
$465$	−1211.25	−0.120797
$466$	842.988	0.0837997
$467$	10864.1	1.07651	0.538254	−	0.842783i	$-0.319084\pi$
$467$	10864.1	1.07651	0.538254	+	0.842783i	$0.319084\pi$
$468$	−1127.94	−0.111408
$469$	21687.0	2.13521
$470$	64.0959	0.00629047
$471$	1130.00	0.110547
$472$	−212.834	−0.0207552
$473$	−5684.95	−0.552630
$474$	−242.326	−0.0234819
$475$	475.000	0.0458831
$476$	21121.5	2.03383
$477$	−1988.27	−0.190852
$478$	−630.586	−0.0603396
$479$	6866.02	0.654940	0.327470	−	0.944862i	$-0.393804\pi$
$479$	6866.02	0.654940	0.327470	+	0.944862i	$0.393804\pi$
$480$	−423.914	−0.0403103
$481$	−3734.28	−0.353988
$482$	786.970	0.0743682
$483$	10874.6	1.02445
$484$	−966.099	−0.0907306
$485$	−2966.17	−0.277705
$486$	−440.550	−0.0411189
$487$	6901.59	0.642179	0.321089	−	0.947049i	$-0.395951\pi$
$487$	6901.59	0.642179	0.321089	+	0.947049i	$0.395951\pi$
$488$	−720.010	−0.0667896
$489$	11389.0	1.05323
$490$	199.296	0.0183740
$491$	4355.57	0.400334	0.200167	−	0.979762i	$-0.435851\pi$
$491$	4355.57	0.400334	0.200167	+	0.979762i	$0.435851\pi$
$492$	−2742.50	−0.251304
$493$	−27790.8	−2.53881
$494$	−23.1659	−0.00210988
$495$	−798.703	−0.0725233
$496$	4363.21	0.394987
$497$	−11369.9	−1.02618
$498$	150.191	0.0135145
$499$	−8061.10	−0.723175	−0.361588	−	0.932338i	$-0.617765\pi$
$499$	−8061.10	−0.723175	−0.361588	+	0.932338i	$0.617765\pi$
$500$	−998.036	−0.0892671
$501$	−13223.6	−1.17922
$502$	262.324	0.0233229
$503$	−16001.9	−1.41847	−0.709235	−	0.704973i	$-0.750959\pi$
$503$	−16001.9	−1.41847	−0.709235	+	0.704973i	$0.750959\pi$
$504$	−747.986	−0.0661070
$505$	−4719.90	−0.415906
$506$	165.079	0.0145033
$507$	7426.48	0.650536
$508$	17537.1	1.53166
$509$	19622.9	1.70878	0.854392	−	0.519629i	$-0.173930\pi$
$509$	19622.9	1.70878	0.854392	+	0.519629i	$0.173930\pi$
$510$	−227.769	−0.0197761
$511$	17359.6	1.50282
$512$	2546.72	0.219825
$513$	2786.80	0.239844
$514$	786.668	0.0675067
$515$	−1727.53	−0.147814
$516$	−14576.2	−1.24357
$517$	1125.08	0.0957079
$518$	−1236.97	−0.104921
$519$	12681.6	1.07256
$520$	97.4447	0.00821776
$521$	−2137.19	−0.179716	−0.0898578	−	0.995955i	$-0.528641\pi$
$521$	−2137.19	−0.179716	−0.0898578	+	0.995955i	$0.528641\pi$
$522$	491.599	0.0412198
$523$	−7642.93	−0.639010	−0.319505	−	0.947585i	$-0.603517\pi$
$523$	−7642.93	−0.639010	−0.319505	+	0.947585i	$0.603517\pi$
$524$	464.388	0.0387154
$525$	2270.51	0.188749
$526$	−909.279	−0.0753734
$527$	7056.20	0.583251
$528$	−2472.20	−0.203766
$529$	2170.03	0.178354
$530$	85.8009	0.00703199
$531$	−1542.77	−0.126084
$532$	3900.30	0.317856
$533$	945.934	0.0768723
$534$	−653.139	−0.0529290
$535$	2666.35	0.215470
$536$	1689.87	0.136178
$537$	10215.8	0.820938
$538$	332.859	0.0266739
$539$	3498.25	0.279555
$540$	−5855.42	−0.466624
$541$	16571.4	1.31693	0.658464	−	0.752612i	$-0.271206\pi$
$541$	16571.4	1.31693	0.658464	+	0.752612i	$0.271206\pi$
$542$	577.569	0.0457725
$543$	11972.8	0.946226
$544$	2469.53	0.194632
$545$	−7483.35	−0.588168
$546$	−110.733	−0.00867939
$547$	−23809.6	−1.86111	−0.930553	−	0.366157i	$-0.880673\pi$
$547$	−23809.6	−1.86111	−0.930553	+	0.366157i	$0.880673\pi$
$548$	−18833.3	−1.46810
$549$	−5219.13	−0.405732
$550$	34.4669	0.00267213
$551$	−5131.84	−0.396776
$552$	847.358	0.0653369
$553$	−14072.3	−1.08212
$554$	−619.416	−0.0475026
$555$	−6779.94	−0.518545
$556$	−5223.74	−0.398446
$557$	20509.3	1.56016	0.780079	−	0.625681i	$-0.215179\pi$
$557$	20509.3	1.56016	0.780079	+	0.625681i	$0.215179\pi$
$558$	−124.819	−0.00946957
$559$	5027.59	0.380401
$560$	−8178.88	−0.617180
$561$	−3998.05	−0.300888
$562$	666.858	0.0500529
$563$	−3234.88	−0.242157	−0.121078	−	0.992643i	$-0.538635\pi$
$563$	−3234.88	−0.242157	−0.121078	+	0.992643i	$0.538635\pi$
$564$	2884.71	0.215369
$565$	5635.37	0.419614
$566$	−292.488	−0.0217212
$567$	3240.15	0.239989
$568$	−885.957	−0.0654471
$569$	11485.0	0.846178	0.423089	−	0.906088i	$-0.360946\pi$
$569$	11485.0	0.846178	0.423089	+	0.906088i	$0.360946\pi$
$570$	−42.0598	−0.00309069
$571$	−2279.07	−0.167033	−0.0835167	−	0.996506i	$-0.526615\pi$
$571$	−2279.07	−0.167033	−0.0835167	+	0.996506i	$0.526615\pi$
$572$	854.387	0.0624541
$573$	598.945	0.0436672
$574$	313.338	0.0227848
$575$	2993.43	0.217104
$576$	7347.74	0.531521
$577$	−19851.2	−1.43227	−0.716133	−	0.697963i	$-0.754090\pi$
$577$	−19851.2	−1.43227	−0.716133	+	0.697963i	$0.754090\pi$
$578$	711.113	0.0511737
$579$	2674.97	0.192000
$580$	10782.7	0.771941
$581$	8721.86	0.622795
$582$	262.645	0.0187062
$583$	1506.07	0.106990
$584$	1352.68	0.0958462
$585$	706.347	0.0499211
$586$	888.934	0.0626647
$587$	−26795.8	−1.88412	−0.942062	−	0.335438i	$-0.891116\pi$
$587$	−26795.8	−1.88412	−0.942062	+	0.335438i	$0.891116\pi$
$588$	8969.53	0.629077
$589$	1303.00	0.0911529
$590$	66.5759	0.00464557
$591$	16223.0	1.12914
$592$	24422.9	1.69556
$593$	13198.7	0.914009	0.457004	−	0.889464i	$-0.348922\pi$
$593$	13198.7	0.914009	0.457004	+	0.889464i	$0.348922\pi$
$594$	202.215	0.0139680
$595$	−13226.9	−0.911347
$596$	−8822.84	−0.606372
$597$	−12646.4	−0.866971
$598$	−145.990	−0.00998327
$599$	−13200.6	−0.900440	−0.450220	−	0.892918i	$-0.648654\pi$
$599$	−13200.6	−0.900440	−0.450220	+	0.892918i	$0.648654\pi$
$600$	176.920	0.0120379
$601$	8600.97	0.583762	0.291881	−	0.956455i	$-0.405719\pi$
$601$	8600.97	0.583762	0.291881	+	0.956455i	$0.405719\pi$
$602$	1665.37	0.112750
$603$	12249.4	0.827251
$604$	15648.9	1.05421
$605$	605.000	0.0406558
$606$	417.933	0.0280155
$607$	23932.4	1.60030	0.800152	−	0.599798i	$-0.204752\pi$
$607$	23932.4	1.60030	0.800152	+	0.599798i	$0.204752\pi$
$608$	456.022	0.0304180
$609$	−24530.3	−1.63221
$610$	225.224	0.0149493
$611$	−994.985	−0.0658802
$612$	11930.0	0.787976
$613$	7245.45	0.477391	0.238696	−	0.971094i	$-0.423280\pi$
$613$	7245.45	0.477391	0.238696	+	0.971094i	$0.423280\pi$
$614$	212.646	0.0139767
$615$	1717.43	0.112607
$616$	566.583	0.0370589
$617$	−13028.7	−0.850107	−0.425054	−	0.905168i	$-0.639745\pi$
$617$	−13028.7	−0.850107	−0.425054	+	0.905168i	$0.639745\pi$
$618$	152.968	0.00995675
$619$	21237.7	1.37902	0.689511	−	0.724275i	$-0.257825\pi$
$619$	21237.7	1.37902	0.689511	+	0.724275i	$0.257825\pi$
$620$	−2737.77	−0.177341
$621$	17562.3	1.13486
$622$	764.592	0.0492883
$623$	−37928.8	−2.43914
$624$	2186.33	0.140262
$625$	625.000	0.0400000
$626$	−673.306	−0.0429883
$627$	−738.280	−0.0470240
$628$	2554.11	0.162293
$629$	39496.8	2.50372
$630$	233.975	0.0147965
$631$	25356.1	1.59970	0.799851	−	0.600198i	$-0.204912\pi$
$631$	25356.1	1.59970	0.799851	+	0.600198i	$0.204912\pi$
$632$	−1096.53	−0.0690150
$633$	1487.00	0.0933698
$634$	−198.118	−0.0124105
$635$	−10982.3	−0.686328
$636$	3861.57	0.240757
$637$	−3093.74	−0.192431
$638$	−372.376	−0.0231074
$639$	−6422.03	−0.397577
$640$	−1277.13	−0.0788796
$641$	−28384.6	−1.74902	−0.874511	−	0.485006i	$-0.838818\pi$
$641$	−28384.6	−1.74902	−0.874511	+	0.485006i	$0.838818\pi$
$642$	−236.097	−0.0145140
$643$	30650.4	1.87983	0.939916	−	0.341405i	$-0.110903\pi$
$643$	30650.4	1.87983	0.939916	+	0.341405i	$0.110903\pi$
$644$	24579.6	1.50399
$645$	9128.06	0.557236
$646$	245.021	0.0149230
$647$	−9787.88	−0.594747	−0.297373	−	0.954761i	$-0.596111\pi$
$647$	−9787.88	−0.594747	−0.297373	+	0.954761i	$0.596111\pi$
$648$	252.476	0.0153058
$649$	1168.61	0.0706811
$650$	−30.4814	−0.00183935
$651$	6228.35	0.374974
$652$	25742.3	1.54623
$653$	−27958.6	−1.67551	−0.837753	−	0.546049i	$-0.816131\pi$
$653$	−27958.6	−1.67551	−0.837753	+	0.546049i	$0.816131\pi$
$654$	662.629	0.0396190
$655$	−290.813	−0.0173481
$656$	−6186.58	−0.368210
$657$	9805.14	0.582245
$658$	−329.586	−0.0195267
$659$	1678.42	0.0992137	0.0496069	−	0.998769i	$-0.484203\pi$
$659$	1678.42	0.0992137	0.0496069	+	0.998769i	$0.484203\pi$
$660$	1551.22	0.0914867
$661$	−1305.49	−0.0768193	−0.0384097	−	0.999262i	$-0.512229\pi$
$661$	−1305.49	−0.0768193	−0.0384097	+	0.999262i	$0.512229\pi$
$662$	245.012	0.0143847
$663$	3535.75	0.207115
$664$	679.616	0.0397202
$665$	−2442.48	−0.142429
$666$	−698.671	−0.0406501
$667$	−32340.7	−1.87742
$668$	−29889.0	−1.73120
$669$	11946.2	0.690383
$670$	−528.604	−0.0304802
$671$	3953.38	0.227449
$672$	2179.79	0.125130
$673$	−3471.07	−0.198811	−0.0994056	−	0.995047i	$-0.531694\pi$
$673$	−3471.07	−0.198811	−0.0994056	+	0.995047i	$0.531694\pi$
$674$	−716.967	−0.0409741
$675$	3666.84	0.209091
$676$	16785.9	0.955046
$677$	9507.29	0.539727	0.269863	−	0.962899i	$-0.413021\pi$
$677$	9507.29	0.539727	0.269863	+	0.962899i	$0.413021\pi$
$678$	−498.996	−0.0282652
$679$	15252.3	0.862044
$680$	−1030.66	−0.0581233
$681$	13990.9	0.787271
$682$	94.5479	0.00530854
$683$	−18567.3	−1.04020	−0.520100	−	0.854105i	$-0.674105\pi$
$683$	−18567.3	−1.04020	−0.520100	+	0.854105i	$0.674105\pi$
$684$	2202.99	0.123148
$685$	11793.9	0.657845
$686$	80.4859	0.00447954
$687$	−8502.40	−0.472179
$688$	−32881.3	−1.82208
$689$	−1331.92	−0.0736461
$690$	−265.060	−0.0146241
$691$	−5770.25	−0.317671	−0.158836	−	0.987305i	$-0.550774\pi$
$691$	−5770.25	−0.317671	−0.158836	+	0.987305i	$0.550774\pi$
$692$	28663.9	1.57462
$693$	4106.99	0.225125
$694$	491.712	0.0268950
$695$	3271.26	0.178541
$696$	−1911.42	−0.104098
$697$	−10005.0	−0.543710
$698$	−774.156	−0.0419803
$699$	−23758.9	−1.28562
$700$	5131.98	0.277101
$701$	−30116.5	−1.62266	−0.811331	−	0.584587i	$-0.801256\pi$
$701$	−30116.5	−1.62266	−0.811331	+	0.584587i	$0.801256\pi$
$702$	−178.833	−0.00961482
$703$	7293.48	0.391293
$704$	−5565.76	−0.297965
$705$	−1806.49	−0.0965055
$706$	1067.37	0.0568994
$707$	24270.1	1.29105
$708$	2996.33	0.159052
$709$	−9497.22	−0.503068	−0.251534	−	0.967848i	$-0.580935\pi$
$709$	−9497.22	−0.503068	−0.251534	+	0.967848i	$0.580935\pi$
$710$	277.134	0.0146488
$711$	−7948.39	−0.419251
$712$	−2955.45	−0.155562
$713$	8211.44	0.431306
$714$	1171.21	0.0613884
$715$	−535.043	−0.0279853
$716$	23090.5	1.20521
$717$	17772.5	0.925701
$718$	−880.190	−0.0457499
$719$	−27826.0	−1.44330	−0.721652	−	0.692256i	$-0.756617\pi$
$719$	−27826.0	−1.44330	−0.721652	+	0.692256i	$0.756617\pi$
$720$	−4619.64	−0.239116
$721$	8883.09	0.458840
$722$	45.2456	0.00233223
$723$	−22180.1	−1.14092
$724$	27061.8	1.38915
$725$	−6752.42	−0.345902
$726$	−53.5710	−0.00273857
$727$	−26404.5	−1.34703	−0.673513	−	0.739175i	$-0.735216\pi$
$727$	−26404.5	−1.34703	−0.673513	+	0.739175i	$0.735216\pi$
$728$	−501.068	−0.0255094
$729$	15819.2	0.803701
$730$	−423.127	−0.0214529
$731$	−53175.9	−2.69054
$732$	10136.5	0.511824
$733$	−11073.5	−0.557992	−0.278996	−	0.960292i	$-0.590002\pi$
$733$	−11073.5	−0.557992	−0.278996	+	0.960292i	$0.590002\pi$
$734$	−1281.44	−0.0644398
$735$	−5616.98	−0.281885
$736$	2873.84	0.143928
$737$	−9278.63	−0.463748
$738$	176.981	0.00882759
$739$	−38178.9	−1.90045	−0.950226	−	0.311561i	$-0.899148\pi$
$739$	−38178.9	−1.90045	−0.950226	+	0.311561i	$0.899148\pi$
$740$	−15324.5	−0.761272
$741$	652.911	0.0323688
$742$	−441.195	−0.0218285
$743$	−26799.6	−1.32326	−0.661629	−	0.749832i	$-0.730135\pi$
$743$	−26799.6	−1.32326	−0.661629	+	0.749832i	$0.730135\pi$
$744$	485.319	0.0239149
$745$	5525.12	0.271711
$746$	408.458	0.0200465
$747$	4926.33	0.241292
$748$	−9036.71	−0.441731
$749$	−13710.6	−0.668856
$750$	−55.3419	−0.00269440
$751$	1574.51	0.0765042	0.0382521	−	0.999268i	$-0.487821\pi$
$751$	1574.51	0.0765042	0.0382521	+	0.999268i	$0.487821\pi$
$752$	6507.39	0.315558
$753$	−7393.40	−0.357809
$754$	329.317	0.0159059
$755$	−9799.82	−0.472387
$756$	30109.0	1.44848
$757$	−31280.1	−1.50184	−0.750920	−	0.660393i	$-0.770390\pi$
$757$	−31280.1	−1.50184	−0.750920	+	0.660393i	$0.770390\pi$
$758$	−986.875	−0.0472888
$759$	−4652.61	−0.222502
$760$	−190.321	−0.00908376
$761$	16627.9	0.792065	0.396032	−	0.918237i	$-0.370387\pi$
$761$	16627.9	0.792065	0.396032	+	0.918237i	$0.370387\pi$
$762$	972.448	0.0462311
$763$	38479.9	1.82578
$764$	1353.78	0.0641074
$765$	−7470.91	−0.353087
$766$	−1388.56	−0.0654971
$767$	−1033.48	−0.0486531
$768$	−14185.6	−0.666509
$769$	−31219.2	−1.46397	−0.731986	−	0.681320i	$-0.761406\pi$
$769$	−31219.2	−1.46397	−0.731986	+	0.681320i	$0.761406\pi$
$770$	−177.231	−0.00829477
$771$	−22171.6	−1.03566
$772$	6046.18	0.281874
$773$	10636.8	0.494929	0.247464	−	0.968897i	$-0.420403\pi$
$773$	10636.8	0.494929	0.247464	+	0.968897i	$0.420403\pi$
$774$	940.644	0.0436831
$775$	1714.47	0.0794653
$776$	1188.47	0.0549789
$777$	34862.9	1.60965
$778$	1590.36	0.0732867
$779$	−1847.52	−0.0849733
$780$	−1371.85	−0.0629746
$781$	4864.55	0.222877
$782$	1544.12	0.0706106
$783$	−39616.1	−1.80813
$784$	20233.6	0.921722
$785$	−1599.46	−0.0727225
$786$	25.7507	0.00116857
$787$	−28834.0	−1.30600	−0.652999	−	0.757359i	$-0.726489\pi$
$787$	−28834.0	−1.30600	−0.652999	+	0.757359i	$0.726489\pi$
$788$	36668.4	1.65769
$789$	25627.3	1.15634
$790$	343.001	0.0154474
$791$	−28977.5	−1.30256
$792$	320.020	0.0143579
$793$	−3496.24	−0.156564
$794$	−1183.83	−0.0529126
$795$	−2418.23	−0.107881
$796$	−28584.3	−1.27279
$797$	6310.51	0.280464	0.140232	−	0.990119i	$-0.455215\pi$
$797$	6310.51	0.280464	0.140232	+	0.990119i	$0.455215\pi$
$798$	216.275	0.00959404
$799$	10523.8	0.465963
$800$	600.029	0.0265178
$801$	−21423.2	−0.945007
$802$	−712.230	−0.0313588
$803$	−7427.18	−0.326400
$804$	−23790.4	−1.04356
$805$	−15392.5	−0.673929
$806$	−83.6151	−0.00365412
$807$	−9381.36	−0.409219
$808$	1891.15	0.0823395
$809$	−10140.1	−0.440676	−0.220338	−	0.975424i	$-0.570716\pi$
$809$	−10140.1	−0.440676	−0.220338	+	0.975424i	$0.570716\pi$
$810$	−78.9762	−0.00342585
$811$	−24725.2	−1.07055	−0.535277	−	0.844676i	$-0.679793\pi$
$811$	−24725.2	−1.07055	−0.535277	+	0.844676i	$0.679793\pi$
$812$	−55445.2	−2.39624
$813$	−16278.3	−0.702220
$814$	529.228	0.0227880
$815$	−16120.6	−0.692858
$816$	−23124.5	−0.992057
$817$	−9819.45	−0.420489
$818$	−994.032	−0.0424884
$819$	−3632.09	−0.154964
$820$	3881.87	0.165318
$821$	−18803.4	−0.799321	−0.399661	−	0.916663i	$-0.630872\pi$
$821$	−18803.4	−0.799321	−0.399661	+	0.916663i	$0.630872\pi$
$822$	−1044.32	−0.0443124
$823$	−23857.0	−1.01045	−0.505227	−	0.862986i	$-0.668591\pi$
$823$	−23857.0	−1.01045	−0.505227	+	0.862986i	$0.668591\pi$
$824$	692.179	0.0292636
$825$	−971.421	−0.0409946
$826$	−342.338	−0.0144207
$827$	17955.1	0.754968	0.377484	−	0.926016i	$-0.376789\pi$
$827$	17955.1	0.754968	0.377484	+	0.926016i	$0.376789\pi$
$828$	13883.2	0.582697
$829$	30829.8	1.29163	0.645817	−	0.763492i	$-0.276517\pi$
$829$	30829.8	1.29163	0.645817	+	0.763492i	$0.276517\pi$
$830$	−212.589	−0.00889043
$831$	17457.7	0.728763
$832$	4922.18	0.205103
$833$	32722.0	1.36104
$834$	−289.661	−0.0120265
$835$	18717.4	0.775739
$836$	−1668.72	−0.0690356
$837$	10058.7	0.415388
$838$	−1201.66	−0.0495352
$839$	9429.22	0.388001	0.194000	−	0.981001i	$-0.437854\pi$
$839$	9429.22	0.388001	0.194000	+	0.981001i	$0.437854\pi$
$840$	−909.737	−0.0373677
$841$	48563.3	1.99120
$842$	−1430.15	−0.0585347
$843$	−18794.9	−0.767888
$844$	3361.04	0.137075
$845$	−10511.8	−0.427950
$846$	−186.158	−0.00756531
$847$	−3110.95	−0.126203
$848$	8711.01	0.352756
$849$	8243.55	0.333237
$850$	322.397	0.0130095
$851$	45963.2	1.85147
$852$	12472.7	0.501536
$853$	−36213.6	−1.45361	−0.726804	−	0.686844i	$-0.758995\pi$
$853$	−36213.6	−1.45361	−0.726804	+	0.686844i	$0.758995\pi$
$854$	−1158.12	−0.0464052
$855$	−1379.58	−0.0551819
$856$	−1068.34	−0.0426578
$857$	−4685.07	−0.186743	−0.0933715	−	0.995631i	$-0.529764\pi$
$857$	−4685.07	−0.186743	−0.0933715	+	0.995631i	$0.529764\pi$
$858$	47.3765	0.00188509
$859$	11160.2	0.443286	0.221643	−	0.975128i	$-0.428858\pi$
$859$	11160.2	0.443286	0.221643	+	0.975128i	$0.428858\pi$
$860$	20631.9	0.818074
$861$	−8831.17	−0.349553
$862$	314.215	0.0124156
$863$	−19855.4	−0.783183	−0.391592	−	0.920139i	$-0.628075\pi$
$863$	−19855.4	−0.783183	−0.391592	+	0.920139i	$0.628075\pi$
$864$	3520.34	0.138616
$865$	−17950.2	−0.705577
$866$	−1298.89	−0.0509676
$867$	−20042.2	−0.785083
$868$	14077.8	0.550497
$869$	6020.73	0.235028
$870$	597.907	0.0232999
$871$	8205.72	0.319220
$872$	2998.39	0.116443
$873$	8614.86	0.333985
$874$	285.136	0.0110353
$875$	−3213.80	−0.124167
$876$	−19043.3	−0.734491
$877$	38958.0	1.50002	0.750010	−	0.661426i	$-0.230049\pi$
$877$	38958.0	1.50002	0.750010	+	0.661426i	$0.230049\pi$
$878$	−1108.67	−0.0426147
$879$	−25053.9	−0.961373
$880$	3499.28	0.134046
$881$	39408.9	1.50706	0.753530	−	0.657414i	$-0.228350\pi$
$881$	39408.9	1.50706	0.753530	+	0.657414i	$0.228350\pi$
$882$	−578.829	−0.0220977
$883$	10512.4	0.400646	0.200323	−	0.979730i	$-0.435801\pi$
$883$	10512.4	0.400646	0.200323	+	0.979730i	$0.435801\pi$
$884$	7991.78	0.304064
$885$	−1876.39	−0.0712702
$886$	−662.641	−0.0251262
$887$	−2331.49	−0.0882569	−0.0441284	−	0.999026i	$-0.514051\pi$
$887$	−2331.49	−0.0882569	−0.0441284	+	0.999026i	$0.514051\pi$
$888$	2716.55	0.102659
$889$	56471.7	2.13048
$890$	924.486	0.0348189
$891$	−1386.28	−0.0521235
$892$	27001.7	1.01355
$893$	1943.32	0.0728228
$894$	−489.233	−0.0183025
$895$	−14460.0	−0.540048
$896$	6567.09	0.244856
$897$	4114.62	0.153159
$898$	−293.828	−0.0109189
$899$	−18522.9	−0.687179
$900$	2898.67	0.107358
$901$	14087.5	0.520891
$902$	−134.059	−0.00494865
$903$	−46937.2	−1.72976
$904$	−2257.96	−0.0830735
$905$	−16946.9	−0.622468
$906$	867.745	0.0318200
$907$	28156.7	1.03079	0.515396	−	0.856952i	$-0.327645\pi$
$907$	28156.7	1.03079	0.515396	+	0.856952i	$0.327645\pi$
$908$	31623.3	1.15579
$909$	13708.3	0.500195
$910$	156.738	0.00570967
$911$	763.112	0.0277531	0.0138765	−	0.999904i	$-0.495583\pi$
$911$	763.112	0.0277531	0.0138765	+	0.999904i	$0.495583\pi$
$912$	−4270.16	−0.155043
$913$	−3731.59	−0.135266
$914$	−213.787	−0.00773680
$915$	−6347.76	−0.229345
$916$	−19217.8	−0.693202
$917$	1495.38	0.0538516
$918$	1891.48	0.0680046
$919$	−48015.9	−1.72350	−0.861751	−	0.507331i	$-0.830632\pi$
$919$	−48015.9	−1.72350	−0.861751	+	0.507331i	$0.830632\pi$
$920$	−1199.39	−0.0429814
$921$	−5993.24	−0.214423
$922$	2162.07	0.0772278
$923$	−4302.05	−0.153417
$924$	−7976.50	−0.283991
$925$	9596.68	0.341121
$926$	341.865	0.0121322
$927$	5017.40	0.177770
$928$	−6482.64	−0.229314
$929$	32040.4	1.13155	0.565776	−	0.824559i	$-0.308577\pi$
$929$	32040.4	1.13155	0.565776	+	0.824559i	$0.308577\pi$
$930$	−151.811	−0.00535278
$931$	6042.43	0.212710
$932$	−53701.7	−1.88740
$933$	−21549.4	−0.756158
$934$	1361.64	0.0477025
$935$	5659.05	0.197937
$936$	−283.016	−0.00988319
$937$	−15676.1	−0.546548	−0.273274	−	0.961936i	$-0.588107\pi$
$937$	−15676.1	−0.546548	−0.273274	+	0.961936i	$0.588107\pi$
$938$	2718.12	0.0946159
$939$	18976.6	0.659507
$940$	−4083.17	−0.141679
$941$	−43353.1	−1.50188	−0.750940	−	0.660370i	$-0.770399\pi$
$941$	−43353.1	−1.50188	−0.750940	+	0.660370i	$0.770399\pi$
$942$	141.627	0.00489859
$943$	−11643.0	−0.402066
$944$	6759.18	0.233043
$945$	−18855.1	−0.649056
$946$	−712.518	−0.0244883
$947$	−13011.0	−0.446464	−0.223232	−	0.974765i	$-0.571661\pi$
$947$	−13011.0	−0.446464	−0.223232	+	0.974765i	$0.571661\pi$
$948$	15437.2	0.528878
$949$	6568.36	0.224677
$950$	59.5337	0.00203319
$951$	5583.79	0.190396
$952$	5299.71	0.180425
$953$	−11013.6	−0.374362	−0.187181	−	0.982325i	$-0.559935\pi$
$953$	−11013.6	−0.374362	−0.187181	+	0.982325i	$0.559935\pi$
$954$	−249.198	−0.00845711
$955$	−847.778	−0.0287261
$956$	40170.9	1.35902
$957$	10495.1	0.354502
$958$	860.546	0.0290219
$959$	−60645.3	−2.04206
$960$	8936.68	0.300448
$961$	−25087.9	−0.842132
$962$	−468.032	−0.0156860
$963$	−7744.07	−0.259137
$964$	−50133.2	−1.67498
$965$	−3786.30	−0.126306
$966$	1362.96	0.0453959
$967$	19072.1	0.634248	0.317124	−	0.948384i	$-0.397283\pi$
$967$	19072.1	0.634248	0.317124	+	0.948384i	$0.397283\pi$
$968$	−242.409	−0.00804887
$969$	−6905.73	−0.228941
$970$	−371.762	−0.0123057
$971$	17693.9	0.584782	0.292391	−	0.956299i	$-0.405549\pi$
$971$	17693.9	0.584782	0.292391	+	0.956299i	$0.405549\pi$
$972$	28064.9	0.926112
$973$	−16821.1	−0.554223
$974$	865.005	0.0284564
$975$	859.094	0.0282185
$976$	22866.1	0.749924
$977$	−29209.9	−0.956506	−0.478253	−	0.878222i	$-0.658730\pi$
$977$	−29209.9	−0.956506	−0.478253	+	0.878222i	$0.658730\pi$
$978$	1427.43	0.0466709
$979$	16227.6	0.529761
$980$	−12695.9	−0.413833
$981$	21734.4	0.707367
$982$	545.902	0.0177397
$983$	−37877.2	−1.22899	−0.614495	−	0.788921i	$-0.710640\pi$
$983$	−37877.2	−1.22899	−0.614495	+	0.788921i	$0.710640\pi$
$984$	−688.133	−0.0222936
$985$	−22962.9	−0.742800
$986$	−3483.13	−0.112501
$987$	9289.11	0.299570
$988$	1475.76	0.0475204
$989$	−61881.8	−1.98961
$990$	−100.105	−0.00321367
$991$	−26723.9	−0.856622	−0.428311	−	0.903631i	$-0.640891\pi$
$991$	−26723.9	−0.856622	−0.428311	+	0.903631i	$0.640891\pi$
$992$	1645.97	0.0526811
$993$	−6905.48	−0.220683
$994$	−1425.04	−0.0454724
$995$	17900.3	0.570331
$996$	−9567.80	−0.304385
$997$	−3400.09	−0.108006	−0.0540029	−	0.998541i	$-0.517198\pi$
$997$	−3400.09	−0.108006	−0.0540029	+	0.998541i	$0.517198\pi$
$998$	−1010.33	−0.0320456
$999$	56303.1	1.78314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.b.1.12	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.b.1.12	✓	20	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+0.125334 q^{2} -3.53244 q^{3} -7.98429 q^{4} +5.00000 q^{5} -0.442735 q^{6} -25.7104 q^{7} -2.00338 q^{8} -14.5219 q^{9} +O(q^{10})\)	\(q+0.125334 q^{2} -3.53244 q^{3} -7.98429 q^{4} +5.00000 q^{5} -0.442735 q^{6} -25.7104 q^{7} -2.00338 q^{8} -14.5219 q^{9} +0.626671 q^{10} +11.0000 q^{11} +28.2040 q^{12} -9.72805 q^{13} -3.22239 q^{14} -17.6622 q^{15} +63.6232 q^{16} +102.892 q^{17} -1.82008 q^{18} +19.0000 q^{19} -39.9215 q^{20} +90.8203 q^{21} +1.37868 q^{22} +119.737 q^{23} +7.07681 q^{24} +25.0000 q^{25} -1.21926 q^{26} +146.674 q^{27} +205.279 q^{28} -270.097 q^{29} -2.21368 q^{30} +68.5788 q^{31} +24.0012 q^{32} -38.8568 q^{33} +12.8959 q^{34} -128.552 q^{35} +115.947 q^{36} +383.867 q^{37} +2.38135 q^{38} +34.3638 q^{39} -10.0169 q^{40} -97.2378 q^{41} +11.3829 q^{42} -516.813 q^{43} -87.8272 q^{44} -72.6093 q^{45} +15.0072 q^{46} +102.280 q^{47} -224.745 q^{48} +318.023 q^{49} +3.13335 q^{50} -363.459 q^{51} +77.6716 q^{52} +136.916 q^{53} +18.3832 q^{54} +55.0000 q^{55} +51.5075 q^{56} -67.1164 q^{57} -33.8523 q^{58} +106.238 q^{59} +141.020 q^{60} +359.398 q^{61} +8.59526 q^{62} +373.362 q^{63} -505.978 q^{64} -48.6402 q^{65} -4.87009 q^{66} -843.511 q^{67} -821.519 q^{68} -422.965 q^{69} -16.1119 q^{70} +442.232 q^{71} +29.0928 q^{72} -675.198 q^{73} +48.1116 q^{74} -88.3110 q^{75} -151.702 q^{76} -282.814 q^{77} +4.30695 q^{78} +547.339 q^{79} +318.116 q^{80} -126.025 q^{81} -12.1872 q^{82} -339.235 q^{83} -725.136 q^{84} +514.459 q^{85} -64.7743 q^{86} +954.101 q^{87} -22.0371 q^{88} +1475.24 q^{89} -9.10042 q^{90} +250.112 q^{91} -956.018 q^{92} -242.251 q^{93} +12.8192 q^{94} +95.0000 q^{95} -84.7827 q^{96} -593.234 q^{97} +39.8591 q^{98} -159.741 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * q^99

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