LMFDB - Embedded newform 6045.2.a.bg.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 27 x^{14} + 51 x^{13} + 294 x^{12} - 517 x^{11} - 1657 x^{10} + 2678 x^{9} + \cdots - 428 $ x^16 - 2x^15 - 27x^14 + 51x^13 + 294x^12 - 517x^11 - 1657x^10 + 2678x^9 + 5124x^8 - 7583x^7 - 8358x^6 + 11579x^5 + 6027x^4 - 8572x^3 - 780x^2 + 2178*x - 428
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.399270$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.399270 q^{2} -1.00000 q^{3} -1.84058 q^{4} -1.00000 q^{5} +0.399270 q^{6} -2.18061 q^{7} +1.53343 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.399270 q^{2} -1.00000 q^{3} -1.84058 q^{4} -1.00000 q^{5} +0.399270 q^{6} -2.18061 q^{7} +1.53343 q^{8} +1.00000 q^{9} +0.399270 q^{10} +5.45298 q^{11} +1.84058 q^{12} -1.00000 q^{13} +0.870651 q^{14} +1.00000 q^{15} +3.06891 q^{16} +6.93522 q^{17} -0.399270 q^{18} +2.35863 q^{19} +1.84058 q^{20} +2.18061 q^{21} -2.17721 q^{22} +4.58805 q^{23} -1.53343 q^{24} +1.00000 q^{25} +0.399270 q^{26} -1.00000 q^{27} +4.01359 q^{28} -0.133279 q^{29} -0.399270 q^{30} +1.00000 q^{31} -4.29219 q^{32} -5.45298 q^{33} -2.76903 q^{34} +2.18061 q^{35} -1.84058 q^{36} +8.94864 q^{37} -0.941729 q^{38} +1.00000 q^{39} -1.53343 q^{40} -0.596045 q^{41} -0.870651 q^{42} -7.42317 q^{43} -10.0367 q^{44} -1.00000 q^{45} -1.83187 q^{46} +9.95184 q^{47} -3.06891 q^{48} -2.24496 q^{49} -0.399270 q^{50} -6.93522 q^{51} +1.84058 q^{52} -1.10442 q^{53} +0.399270 q^{54} -5.45298 q^{55} -3.34381 q^{56} -2.35863 q^{57} +0.0532144 q^{58} +7.17351 q^{59} -1.84058 q^{60} -7.44977 q^{61} -0.399270 q^{62} -2.18061 q^{63} -4.42408 q^{64} +1.00000 q^{65} +2.17721 q^{66} -14.2012 q^{67} -12.7648 q^{68} -4.58805 q^{69} -0.870651 q^{70} +10.0161 q^{71} +1.53343 q^{72} +1.69655 q^{73} -3.57293 q^{74} -1.00000 q^{75} -4.34125 q^{76} -11.8908 q^{77} -0.399270 q^{78} -0.227163 q^{79} -3.06891 q^{80} +1.00000 q^{81} +0.237983 q^{82} -3.26940 q^{83} -4.01359 q^{84} -6.93522 q^{85} +2.96385 q^{86} +0.133279 q^{87} +8.36176 q^{88} +10.8360 q^{89} +0.399270 q^{90} +2.18061 q^{91} -8.44470 q^{92} -1.00000 q^{93} -3.97347 q^{94} -2.35863 q^{95} +4.29219 q^{96} +6.37910 q^{97} +0.896346 q^{98} +5.45298 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * 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 $

$2$	−0.399270	−0.282327	−0.141163	−	0.989986i	$-0.545084\pi$
$2$	−0.399270	−0.282327	−0.141163	+	0.989986i	$0.545084\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.84058	−0.920292
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.399270	0.163001
$7$	−2.18061	−0.824191	−0.412096	−	0.911141i	$-0.635203\pi$
$7$	−2.18061	−0.824191	−0.412096	+	0.911141i	$0.635203\pi$
$8$	1.53343	0.542150
$9$	1.00000	0.333333
$10$	0.399270	0.126260
$11$	5.45298	1.64413	0.822067	−	0.569391i	$-0.192821\pi$
$11$	5.45298	1.64413	0.822067	+	0.569391i	$0.192821\pi$
$12$	1.84058	0.531331
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0.870651	0.232691
$15$	1.00000	0.258199
$16$	3.06891	0.767228
$17$	6.93522	1.68204	0.841019	−	0.541006i	$-0.181956\pi$
$17$	6.93522	1.68204	0.841019	+	0.541006i	$0.181956\pi$
$18$	−0.399270	−0.0941089
$19$	2.35863	0.541106	0.270553	−	0.962705i	$-0.412793\pi$
$19$	2.35863	0.541106	0.270553	+	0.962705i	$0.412793\pi$
$20$	1.84058	0.411567
$21$	2.18061	0.475847
$22$	−2.17721	−0.464183
$23$	4.58805	0.956675	0.478338	−	0.878176i	$-0.341239\pi$
$23$	4.58805	0.956675	0.478338	+	0.878176i	$0.341239\pi$
$24$	−1.53343	−0.313010
$25$	1.00000	0.200000
$26$	0.399270	0.0783033
$27$	−1.00000	−0.192450
$28$	4.01359	0.758496
$29$	−0.133279	−0.0247493	−0.0123747	−	0.999923i	$-0.503939\pi$
$29$	−0.133279	−0.0247493	−0.0123747	+	0.999923i	$0.503939\pi$
$30$	−0.399270	−0.0728964
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−4.29219	−0.758759
$33$	−5.45298	−0.949241
$34$	−2.76903	−0.474884
$35$	2.18061	0.368590
$36$	−1.84058	−0.306764
$37$	8.94864	1.47115	0.735574	−	0.677444i	$-0.236913\pi$
$37$	8.94864	1.47115	0.735574	+	0.677444i	$0.236913\pi$
$38$	−0.941729	−0.152769
$39$	1.00000	0.160128
$40$	−1.53343	−0.242457
$41$	−0.596045	−0.0930866	−0.0465433	−	0.998916i	$-0.514821\pi$
$41$	−0.596045	−0.0930866	−0.0465433	+	0.998916i	$0.514821\pi$
$42$	−0.870651	−0.134344
$43$	−7.42317	−1.13202	−0.566012	−	0.824397i	$-0.691514\pi$
$43$	−7.42317	−1.13202	−0.566012	+	0.824397i	$0.691514\pi$
$44$	−10.0367	−1.51308
$45$	−1.00000	−0.149071
$46$	−1.83187	−0.270095
$47$	9.95184	1.45163	0.725813	−	0.687892i	$-0.241464\pi$
$47$	9.95184	1.45163	0.725813	+	0.687892i	$0.241464\pi$
$48$	−3.06891	−0.442960
$49$	−2.24496	−0.320709
$50$	−0.399270	−0.0564653
$51$	−6.93522	−0.971125
$52$	1.84058	0.255243
$53$	−1.10442	−0.151704	−0.0758518	−	0.997119i	$-0.524168\pi$
$53$	−1.10442	−0.151704	−0.0758518	+	0.997119i	$0.524168\pi$
$54$	0.399270	0.0543338
$55$	−5.45298	−0.735279
$56$	−3.34381	−0.446835
$57$	−2.35863	−0.312408
$58$	0.0532144	0.00698740
$59$	7.17351	0.933911	0.466956	−	0.884281i	$-0.345351\pi$
$59$	7.17351	0.933911	0.466956	+	0.884281i	$0.345351\pi$
$60$	−1.84058	−0.237618
$61$	−7.44977	−0.953845	−0.476922	−	0.878945i	$-0.658248\pi$
$61$	−7.44977	−0.953845	−0.476922	+	0.878945i	$0.658248\pi$
$62$	−0.399270	−0.0507074
$63$	−2.18061	−0.274730
$64$	−4.42408	−0.553011
$65$	1.00000	0.124035
$66$	2.17721	0.267996
$67$	−14.2012	−1.73496	−0.867479	−	0.497474i	$-0.834261\pi$
$67$	−14.2012	−1.73496	−0.867479	+	0.497474i	$0.834261\pi$
$68$	−12.7648	−1.54797
$69$	−4.58805	−0.552337
$70$	−0.870651	−0.104063
$71$	10.0161	1.18869	0.594347	−	0.804209i	$-0.297411\pi$
$71$	10.0161	1.18869	0.594347	+	0.804209i	$0.297411\pi$
$72$	1.53343	0.180717
$73$	1.69655	0.198566	0.0992830	−	0.995059i	$-0.468345\pi$
$73$	1.69655	0.198566	0.0992830	+	0.995059i	$0.468345\pi$
$74$	−3.57293	−0.415344
$75$	−1.00000	−0.115470
$76$	−4.34125	−0.497975
$77$	−11.8908	−1.35508
$78$	−0.399270	−0.0452084
$79$	−0.227163	−0.0255579	−0.0127789	−	0.999918i	$-0.504068\pi$
$79$	−0.227163	−0.0255579	−0.0127789	+	0.999918i	$0.504068\pi$
$80$	−3.06891	−0.343115
$81$	1.00000	0.111111
$82$	0.237983	0.0262808
$83$	−3.26940	−0.358863	−0.179432	−	0.983770i	$-0.557426\pi$
$83$	−3.26940	−0.358863	−0.179432	+	0.983770i	$0.557426\pi$
$84$	−4.01359	−0.437918
$85$	−6.93522	−0.752230
$86$	2.96385	0.319600
$87$	0.133279	0.0142890
$88$	8.36176	0.891366
$89$	10.8360	1.14862	0.574309	−	0.818638i	$-0.305271\pi$
$89$	10.8360	1.14862	0.574309	+	0.818638i	$0.305271\pi$
$90$	0.399270	0.0420868
$91$	2.18061	0.228590
$92$	−8.44470	−0.880420
$93$	−1.00000	−0.103695
$94$	−3.97347	−0.409833
$95$	−2.35863	−0.241990
$96$	4.29219	0.438069
$97$	6.37910	0.647699	0.323850	−	0.946109i	$-0.395023\pi$
$97$	6.37910	0.647699	0.323850	+	0.946109i	$0.395023\pi$
$98$	0.896346	0.0905446
$99$	5.45298	0.548045
$100$	−1.84058	−0.184058
$101$	10.5105	1.04584	0.522918	−	0.852383i	$-0.324843\pi$
$101$	10.5105	1.04584	0.522918	+	0.852383i	$0.324843\pi$
$102$	2.76903	0.274174
$103$	−15.8965	−1.56633	−0.783166	−	0.621813i	$-0.786396\pi$
$103$	−15.8965	−1.56633	−0.783166	+	0.621813i	$0.786396\pi$
$104$	−1.53343	−0.150365
$105$	−2.18061	−0.212805
$106$	0.440962	0.0428300
$107$	19.1024	1.84669	0.923347	−	0.383966i	$-0.125442\pi$
$107$	19.1024	1.84669	0.923347	+	0.383966i	$0.125442\pi$
$108$	1.84058	0.177110
$109$	−11.9280	−1.14250	−0.571249	−	0.820777i	$-0.693541\pi$
$109$	−11.9280	−1.14250	−0.571249	+	0.820777i	$0.693541\pi$
$110$	2.17721	0.207589
$111$	−8.94864	−0.849368
$112$	−6.69209	−0.632343
$113$	−18.2665	−1.71837	−0.859185	−	0.511664i	$-0.829029\pi$
$113$	−18.2665	−1.71837	−0.859185	+	0.511664i	$0.829029\pi$
$114$	0.941729	0.0882010
$115$	−4.58805	−0.427838
$116$	0.245312	0.0227766
$117$	−1.00000	−0.0924500
$118$	−2.86417	−0.263668
$119$	−15.1230	−1.38632
$120$	1.53343	0.139982
$121$	18.7349	1.70318
$122$	2.97447	0.269296
$123$	0.596045	0.0537436
$124$	−1.84058	−0.165289
$125$	−1.00000	−0.0894427
$126$	0.870651	0.0775637
$127$	−11.4635	−1.01722	−0.508612	−	0.860996i	$-0.669841\pi$
$127$	−11.4635	−1.01722	−0.508612	+	0.860996i	$0.669841\pi$
$128$	10.3508	0.914888
$129$	7.42317	0.653574
$130$	−0.399270	−0.0350183
$131$	−10.2088	−0.891950	−0.445975	−	0.895045i	$-0.647143\pi$
$131$	−10.2088	−0.891950	−0.445975	+	0.895045i	$0.647143\pi$
$132$	10.0367	0.873579
$133$	−5.14323	−0.445975
$134$	5.67013	0.489825
$135$	1.00000	0.0860663
$136$	10.6347	0.911916
$137$	12.7225	1.08696	0.543479	−	0.839423i	$-0.317107\pi$
$137$	12.7225	1.08696	0.543479	+	0.839423i	$0.317107\pi$
$138$	1.83187	0.155939
$139$	17.3235	1.46936	0.734681	−	0.678413i	$-0.237332\pi$
$139$	17.3235	1.46936	0.734681	+	0.678413i	$0.237332\pi$
$140$	−4.01359	−0.339210
$141$	−9.95184	−0.838096
$142$	−3.99914	−0.335600
$143$	−5.45298	−0.456001
$144$	3.06891	0.255743
$145$	0.133279	0.0110682
$146$	−0.677382	−0.0560605
$147$	2.24496	0.185161
$148$	−16.4707	−1.35389
$149$	10.9926	0.900548	0.450274	−	0.892890i	$-0.351326\pi$
$149$	10.9926	0.900548	0.450274	+	0.892890i	$0.351326\pi$
$150$	0.399270	0.0326003
$151$	−12.6392	−1.02856	−0.514282	−	0.857621i	$-0.671942\pi$
$151$	−12.6392	−1.02856	−0.514282	+	0.857621i	$0.671942\pi$
$152$	3.61679	0.293360
$153$	6.93522	0.560679
$154$	4.74764	0.382575
$155$	−1.00000	−0.0803219
$156$	−1.84058	−0.147365
$157$	−3.56585	−0.284586	−0.142293	−	0.989825i	$-0.545448\pi$
$157$	−3.56585	−0.284586	−0.142293	+	0.989825i	$0.545448\pi$
$158$	0.0906996	0.00721567
$159$	1.10442	0.0875861
$160$	4.29219	0.339327
$161$	−10.0047	−0.788484
$162$	−0.399270	−0.0313696
$163$	8.09081	0.633721	0.316861	−	0.948472i	$-0.397371\pi$
$163$	8.09081	0.633721	0.316861	+	0.948472i	$0.397371\pi$
$164$	1.09707	0.0856668
$165$	5.45298	0.424514
$166$	1.30537	0.101317
$167$	−3.16195	−0.244679	−0.122339	−	0.992488i	$-0.539040\pi$
$167$	−3.16195	−0.244679	−0.122339	+	0.992488i	$0.539040\pi$
$168$	3.34381	0.257980
$169$	1.00000	0.0769231
$170$	2.76903	0.212375
$171$	2.35863	0.180369
$172$	13.6630	1.04179
$173$	1.51121	0.114896	0.0574478	−	0.998349i	$-0.481704\pi$
$173$	1.51121	0.114896	0.0574478	+	0.998349i	$0.481704\pi$
$174$	−0.0532144	−0.00403418
$175$	−2.18061	−0.164838
$176$	16.7347	1.26143
$177$	−7.17351	−0.539194
$178$	−4.32651	−0.324286
$179$	−2.71362	−0.202825	−0.101413	−	0.994844i	$-0.532336\pi$
$179$	−2.71362	−0.202825	−0.101413	+	0.994844i	$0.532336\pi$
$180$	1.84058	0.137189
$181$	11.9030	0.884746	0.442373	−	0.896831i	$-0.354137\pi$
$181$	11.9030	0.884746	0.442373	+	0.896831i	$0.354137\pi$
$182$	−0.870651	−0.0645369
$183$	7.44977	0.550703
$184$	7.03546	0.518661
$185$	−8.94864	−0.657917
$186$	0.399270	0.0292759
$187$	37.8176	2.76550
$188$	−18.3172	−1.33592
$189$	2.18061	0.158616
$190$	0.941729	0.0683202
$191$	15.2973	1.10688	0.553438	−	0.832890i	$-0.313315\pi$
$191$	15.2973	1.10688	0.553438	+	0.832890i	$0.313315\pi$
$192$	4.42408	0.319281
$193$	−22.9289	−1.65046	−0.825229	−	0.564798i	$-0.808954\pi$
$193$	−22.9289	−1.65046	−0.825229	+	0.564798i	$0.808954\pi$
$194$	−2.54698	−0.182863
$195$	−1.00000	−0.0716115
$196$	4.13204	0.295146
$197$	8.94927	0.637609	0.318805	−	0.947820i	$-0.396719\pi$
$197$	8.94927	0.637609	0.318805	+	0.947820i	$0.396719\pi$
$198$	−2.17721	−0.154728
$199$	5.14636	0.364816	0.182408	−	0.983223i	$-0.441611\pi$
$199$	5.14636	0.364816	0.182408	+	0.983223i	$0.441611\pi$
$200$	1.53343	0.108430
$201$	14.2012	1.00168
$202$	−4.19654	−0.295267
$203$	0.290629	0.0203982
$204$	12.7648	0.893718
$205$	0.596045	0.0416296
$206$	6.34701	0.442217
$207$	4.58805	0.318892
$208$	−3.06891	−0.212791
$209$	12.8615	0.889650
$210$	0.870651	0.0600806
$211$	27.5730	1.89820	0.949101	−	0.314971i	$-0.101995\pi$
$211$	27.5730	1.89820	0.949101	+	0.314971i	$0.101995\pi$
$212$	2.03277	0.139612
$213$	−10.0161	−0.686293
$214$	−7.62700	−0.521371
$215$	7.42317	0.506256
$216$	−1.53343	−0.104337
$217$	−2.18061	−0.148029
$218$	4.76251	0.322558
$219$	−1.69655	−0.114642
$220$	10.0367	0.676671
$221$	−6.93522	−0.466513
$222$	3.57293	0.239799
$223$	−27.7784	−1.86018	−0.930089	−	0.367335i	$-0.880270\pi$
$223$	−27.7784	−1.86018	−0.930089	+	0.367335i	$0.880270\pi$
$224$	9.35956	0.625362
$225$	1.00000	0.0666667
$226$	7.29328	0.485142
$227$	−25.5062	−1.69291	−0.846453	−	0.532464i	$-0.821266\pi$
$227$	−25.5062	−1.69291	−0.846453	+	0.532464i	$0.821266\pi$
$228$	4.34125	0.287506
$229$	25.9480	1.71469	0.857346	−	0.514741i	$-0.172112\pi$
$229$	25.9480	1.71469	0.857346	+	0.514741i	$0.172112\pi$
$230$	1.83187	0.120790
$231$	11.8908	0.782356
$232$	−0.204374	−0.0134178
$233$	4.12305	0.270110	0.135055	−	0.990838i	$-0.456879\pi$
$233$	4.12305	0.270110	0.135055	+	0.990838i	$0.456879\pi$
$234$	0.399270	0.0261011
$235$	−9.95184	−0.649187
$236$	−13.2034	−0.859471
$237$	0.227163	0.0147558
$238$	6.03815	0.391395
$239$	6.33629	0.409860	0.204930	−	0.978777i	$-0.434303\pi$
$239$	6.33629	0.409860	0.204930	+	0.978777i	$0.434303\pi$
$240$	3.06891	0.198098
$241$	−3.72023	−0.239641	−0.119821	−	0.992796i	$-0.538232\pi$
$241$	−3.72023	−0.239641	−0.119821	+	0.992796i	$0.538232\pi$
$242$	−7.48030	−0.480852
$243$	−1.00000	−0.0641500
$244$	13.7119	0.877815
$245$	2.24496	0.143425
$246$	−0.237983	−0.0151732
$247$	−2.35863	−0.150076
$248$	1.53343	0.0973729
$249$	3.26940	0.207190
$250$	0.399270	0.0252521
$251$	8.51196	0.537270	0.268635	−	0.963242i	$-0.413427\pi$
$251$	8.51196	0.537270	0.268635	+	0.963242i	$0.413427\pi$
$252$	4.01359	0.252832
$253$	25.0185	1.57290
$254$	4.57705	0.287190
$255$	6.93522	0.434300
$256$	4.71541	0.294713
$257$	−6.22975	−0.388601	−0.194301	−	0.980942i	$-0.562244\pi$
$257$	−6.22975	−0.388601	−0.194301	+	0.980942i	$0.562244\pi$
$258$	−2.96385	−0.184521
$259$	−19.5135	−1.21251
$260$	−1.84058	−0.114148
$261$	−0.133279	−0.00824978
$262$	4.07608	0.251821
$263$	10.4571	0.644812	0.322406	−	0.946602i	$-0.395508\pi$
$263$	10.4571	0.644812	0.322406	+	0.946602i	$0.395508\pi$
$264$	−8.36176	−0.514631
$265$	1.10442	0.0678439
$266$	2.05354	0.125911
$267$	−10.8360	−0.663155
$268$	26.1386	1.59667
$269$	−29.1258	−1.77583	−0.887916	−	0.460006i	$-0.847847\pi$
$269$	−29.1258	−1.77583	−0.887916	+	0.460006i	$0.847847\pi$
$270$	−0.399270	−0.0242988
$271$	−10.9240	−0.663584	−0.331792	−	0.943353i	$-0.607653\pi$
$271$	−10.9240	−0.663584	−0.331792	+	0.943353i	$0.607653\pi$
$272$	21.2836	1.29051
$273$	−2.18061	−0.131976
$274$	−5.07972	−0.306877
$275$	5.45298	0.328827
$276$	8.44470	0.508311
$277$	5.76219	0.346217	0.173108	−	0.984903i	$-0.444619\pi$
$277$	5.76219	0.346217	0.173108	+	0.984903i	$0.444619\pi$
$278$	−6.91677	−0.414840
$279$	1.00000	0.0598684
$280$	3.34381	0.199831
$281$	−24.9786	−1.49010	−0.745048	−	0.667010i	$-0.767574\pi$
$281$	−24.9786	−1.49010	−0.745048	+	0.667010i	$0.767574\pi$
$282$	3.97347	0.236617
$283$	6.20341	0.368754	0.184377	−	0.982856i	$-0.440973\pi$
$283$	6.20341	0.368754	0.184377	+	0.982856i	$0.440973\pi$
$284$	−18.4355	−1.09395
$285$	2.35863	0.139713
$286$	2.17721	0.128741
$287$	1.29974	0.0767212
$288$	−4.29219	−0.252920
$289$	31.0973	1.82925
$290$	−0.0532144	−0.00312486
$291$	−6.37910	−0.373949
$292$	−3.12264	−0.182739
$293$	−12.6392	−0.738387	−0.369194	−	0.929353i	$-0.620366\pi$
$293$	−12.6392	−0.738387	−0.369194	+	0.929353i	$0.620366\pi$
$294$	−0.896346	−0.0522760
$295$	−7.17351	−0.417658
$296$	13.7221	0.797582
$297$	−5.45298	−0.316414
$298$	−4.38901	−0.254249
$299$	−4.58805	−0.265334
$300$	1.84058	0.106266
$301$	16.1870	0.933004
$302$	5.04646	0.290391
$303$	−10.5105	−0.603814
$304$	7.23842	0.415152
$305$	7.44977	0.426572
$306$	−2.76903	−0.158295
$307$	23.8542	1.36143	0.680717	−	0.732547i	$-0.261668\pi$
$307$	23.8542	1.36143	0.680717	+	0.732547i	$0.261668\pi$
$308$	21.8860	1.24707
$309$	15.8965	0.904322
$310$	0.399270	0.0226770
$311$	−13.9726	−0.792312	−0.396156	−	0.918183i	$-0.629656\pi$
$311$	−13.9726	−0.792312	−0.396156	+	0.918183i	$0.629656\pi$
$312$	1.53343	0.0868134
$313$	26.2047	1.48118	0.740588	−	0.671959i	$-0.234547\pi$
$313$	26.2047	1.48118	0.740588	+	0.671959i	$0.234547\pi$
$314$	1.42374	0.0803462
$315$	2.18061	0.122863
$316$	0.418113	0.0235207
$317$	−21.0810	−1.18403	−0.592013	−	0.805928i	$-0.701667\pi$
$317$	−21.0810	−1.18403	−0.592013	+	0.805928i	$0.701667\pi$
$318$	−0.440962	−0.0247279
$319$	−0.726769	−0.0406912
$320$	4.42408	0.247314
$321$	−19.1024	−1.06619
$322$	3.99459	0.222610
$323$	16.3576	0.910160
$324$	−1.84058	−0.102255
$325$	−1.00000	−0.0554700
$326$	−3.23042	−0.178916
$327$	11.9280	0.659622
$328$	−0.913994	−0.0504669
$329$	−21.7010	−1.19642
$330$	−2.17721	−0.119851
$331$	30.4715	1.67487	0.837433	−	0.546540i	$-0.184055\pi$
$331$	30.4715	1.67487	0.837433	+	0.546540i	$0.184055\pi$
$332$	6.01761	0.330259
$333$	8.94864	0.490383
$334$	1.26247	0.0690793
$335$	14.2012	0.775897
$336$	6.69209	0.365083
$337$	22.4975	1.22552	0.612758	−	0.790270i	$-0.290060\pi$
$337$	22.4975	1.22552	0.612758	+	0.790270i	$0.290060\pi$
$338$	−0.399270	−0.0217174
$339$	18.2665	0.992102
$340$	12.7648	0.692271
$341$	5.45298	0.295295
$342$	−0.941729	−0.0509229
$343$	20.1596	1.08852
$344$	−11.3829	−0.613726
$345$	4.58805	0.247013
$346$	−0.603383	−0.0324381
$347$	14.3798	0.771950	0.385975	−	0.922509i	$-0.373865\pi$
$347$	14.3798	0.771950	0.385975	+	0.922509i	$0.373865\pi$
$348$	−0.245312	−0.0131501
$349$	21.3754	1.14420	0.572100	−	0.820184i	$-0.306129\pi$
$349$	21.3754	1.14420	0.572100	+	0.820184i	$0.306129\pi$
$350$	0.870651	0.0465382
$351$	1.00000	0.0533761
$352$	−23.4052	−1.24750
$353$	7.82433	0.416447	0.208223	−	0.978081i	$-0.433232\pi$
$353$	7.82433	0.416447	0.208223	+	0.978081i	$0.433232\pi$
$354$	2.86417	0.152229
$355$	−10.0161	−0.531600
$356$	−19.9446	−1.05706
$357$	15.1230	0.800393
$358$	1.08347	0.0572630
$359$	−9.78424	−0.516392	−0.258196	−	0.966092i	$-0.583128\pi$
$359$	−9.78424	−0.516392	−0.258196	+	0.966092i	$0.583128\pi$
$360$	−1.53343	−0.0808189
$361$	−13.4369	−0.707205
$362$	−4.75253	−0.249787
$363$	−18.7349	−0.983329
$364$	−4.01359	−0.210369
$365$	−1.69655	−0.0888015
$366$	−2.97447	−0.155478
$367$	−24.2107	−1.26379	−0.631895	−	0.775054i	$-0.717723\pi$
$367$	−24.2107	−1.26379	−0.631895	+	0.775054i	$0.717723\pi$
$368$	14.0803	0.733988
$369$	−0.596045	−0.0310289
$370$	3.57293	0.185748
$371$	2.40830	0.125033
$372$	1.84058	0.0954298
$373$	−19.6533	−1.01761	−0.508804	−	0.860882i	$-0.669912\pi$
$373$	−19.6533	−1.01761	−0.508804	+	0.860882i	$0.669912\pi$
$374$	−15.0994	−0.780773
$375$	1.00000	0.0516398
$376$	15.2605	0.786998
$377$	0.133279	0.00686423
$378$	−0.870651	−0.0447814
$379$	0.865726	0.0444693	0.0222347	−	0.999753i	$-0.492922\pi$
$379$	0.865726	0.0444693	0.0222347	+	0.999753i	$0.492922\pi$
$380$	4.34125	0.222701
$381$	11.4635	0.587295
$382$	−6.10777	−0.312501
$383$	−11.1735	−0.570941	−0.285471	−	0.958387i	$-0.592150\pi$
$383$	−11.1735	−0.570941	−0.285471	+	0.958387i	$0.592150\pi$
$384$	−10.3508	−0.528211
$385$	11.8908	0.606011
$386$	9.15483	0.465969
$387$	−7.42317	−0.377341
$388$	−11.7413	−0.596072
$389$	−10.5587	−0.535347	−0.267673	−	0.963510i	$-0.586255\pi$
$389$	−10.5587	−0.535347	−0.267673	+	0.963510i	$0.586255\pi$
$390$	0.399270	0.0202178
$391$	31.8192	1.60916
$392$	−3.44249	−0.173872
$393$	10.2088	0.514968
$394$	−3.57318	−0.180014
$395$	0.227163	0.0114298
$396$	−10.0367	−0.504361
$397$	−23.5819	−1.18354	−0.591772	−	0.806106i	$-0.701571\pi$
$397$	−23.5819	−1.18354	−0.591772	+	0.806106i	$0.701571\pi$
$398$	−2.05479	−0.102997
$399$	5.14323	0.257484
$400$	3.06891	0.153446
$401$	11.9806	0.598284	0.299142	−	0.954209i	$-0.403300\pi$
$401$	11.9806	0.598284	0.299142	+	0.954209i	$0.403300\pi$
$402$	−5.67013	−0.282800
$403$	−1.00000	−0.0498135
$404$	−19.3455	−0.962474
$405$	−1.00000	−0.0496904
$406$	−0.116040	−0.00575895
$407$	48.7967	2.41876
$408$	−10.6347	−0.526495
$409$	18.7741	0.928321	0.464160	−	0.885751i	$-0.346356\pi$
$409$	18.7741	0.928321	0.464160	+	0.885751i	$0.346356\pi$
$410$	−0.237983	−0.0117531
$411$	−12.7225	−0.627555
$412$	29.2589	1.44148
$413$	−15.6426	−0.769721
$414$	−1.83187	−0.0900317
$415$	3.26940	0.160489
$416$	4.29219	0.210442
$417$	−17.3235	−0.848337
$418$	−5.13522	−0.251172
$419$	−6.84423	−0.334362	−0.167181	−	0.985926i	$-0.553467\pi$
$419$	−6.84423	−0.334362	−0.167181	+	0.985926i	$0.553467\pi$
$420$	4.01359	0.195843
$421$	−4.52655	−0.220611	−0.110305	−	0.993898i	$-0.535183\pi$
$421$	−4.52655	−0.220611	−0.110305	+	0.993898i	$0.535183\pi$
$422$	−11.0091	−0.535913
$423$	9.95184	0.483875
$424$	−1.69355	−0.0822460
$425$	6.93522	0.336408
$426$	3.99914	0.193759
$427$	16.2450	0.786151
$428$	−35.1595	−1.69950
$429$	5.45298	0.263272
$430$	−2.96385	−0.142930
$431$	−7.20195	−0.346906	−0.173453	−	0.984842i	$-0.555492\pi$
$431$	−7.20195	−0.346906	−0.173453	+	0.984842i	$0.555492\pi$
$432$	−3.06891	−0.147653
$433$	−10.1480	−0.487683	−0.243842	−	0.969815i	$-0.578408\pi$
$433$	−10.1480	−0.487683	−0.243842	+	0.969815i	$0.578408\pi$
$434$	0.870651	0.0417926
$435$	−0.133279	−0.00639025
$436$	21.9545	1.05143
$437$	10.8215	0.517663
$438$	0.677382	0.0323665
$439$	−25.0861	−1.19729	−0.598647	−	0.801013i	$-0.704295\pi$
$439$	−25.0861	−1.19729	−0.598647	+	0.801013i	$0.704295\pi$
$440$	−8.36176	−0.398631
$441$	−2.24496	−0.106903
$442$	2.76903	0.131709
$443$	34.5758	1.64274	0.821372	−	0.570392i	$-0.193209\pi$
$443$	34.5758	1.64274	0.821372	+	0.570392i	$0.193209\pi$
$444$	16.4707	0.781666
$445$	−10.8360	−0.513678
$446$	11.0911	0.525178
$447$	−10.9926	−0.519932
$448$	9.64718	0.455787
$449$	−22.6760	−1.07015	−0.535074	−	0.844805i	$-0.679716\pi$
$449$	−22.6760	−1.07015	−0.535074	+	0.844805i	$0.679716\pi$
$450$	−0.399270	−0.0188218
$451$	−3.25022	−0.153047
$452$	33.6211	1.58140
$453$	12.6392	0.593842
$454$	10.1839	0.477952
$455$	−2.18061	−0.102228
$456$	−3.61679	−0.169372
$457$	−19.1582	−0.896185	−0.448093	−	0.893987i	$-0.647896\pi$
$457$	−19.1582	−0.896185	−0.448093	+	0.893987i	$0.647896\pi$
$458$	−10.3603	−0.484103
$459$	−6.93522	−0.323708
$460$	8.44470	0.393736
$461$	23.0357	1.07288	0.536439	−	0.843939i	$-0.319769\pi$
$461$	23.0357	1.07288	0.536439	+	0.843939i	$0.319769\pi$
$462$	−4.74764	−0.220880
$463$	5.49005	0.255144	0.127572	−	0.991829i	$-0.459282\pi$
$463$	5.49005	0.255144	0.127572	+	0.991829i	$0.459282\pi$
$464$	−0.409023	−0.0189884
$465$	1.00000	0.0463739
$466$	−1.64621	−0.0762593
$467$	10.9027	0.504517	0.252258	−	0.967660i	$-0.418827\pi$
$467$	10.9027	0.504517	0.252258	+	0.967660i	$0.418827\pi$
$468$	1.84058	0.0850810
$469$	30.9673	1.42994
$470$	3.97347	0.183283
$471$	3.56585	0.164306
$472$	11.0001	0.506320
$473$	−40.4784	−1.86120
$474$	−0.0906996	−0.00416597
$475$	2.35863	0.108221
$476$	27.8351	1.27582
$477$	−1.10442	−0.0505679
$478$	−2.52989	−0.115715
$479$	22.2235	1.01542	0.507708	−	0.861529i	$-0.330493\pi$
$479$	22.2235	1.01542	0.507708	+	0.861529i	$0.330493\pi$
$480$	−4.29219	−0.195911
$481$	−8.94864	−0.408023
$482$	1.48538	0.0676571
$483$	10.0047	0.455231
$484$	−34.4832	−1.56742
$485$	−6.37910	−0.289660
$486$	0.399270	0.0181113
$487$	35.7620	1.62053	0.810265	−	0.586064i	$-0.199323\pi$
$487$	35.7620	1.62053	0.810265	+	0.586064i	$0.199323\pi$
$488$	−11.4237	−0.517127
$489$	−8.09081	−0.365879
$490$	−0.896346	−0.0404928
$491$	23.7947	1.07384	0.536919	−	0.843634i	$-0.319588\pi$
$491$	23.7947	1.07384	0.536919	+	0.843634i	$0.319588\pi$
$492$	−1.09707	−0.0494598
$493$	−0.924321	−0.0416293
$494$	0.941729	0.0423704
$495$	−5.45298	−0.245093
$496$	3.06891	0.137798
$497$	−21.8412	−0.979711
$498$	−1.30537	−0.0584952
$499$	5.67814	0.254188	0.127094	−	0.991891i	$-0.459435\pi$
$499$	5.67814	0.254188	0.127094	+	0.991891i	$0.459435\pi$
$500$	1.84058	0.0823134
$501$	3.16195	0.141265
$502$	−3.39857	−0.151686
$503$	−7.18908	−0.320545	−0.160273	−	0.987073i	$-0.551237\pi$
$503$	−7.18908	−0.320545	−0.160273	+	0.987073i	$0.551237\pi$
$504$	−3.34381	−0.148945
$505$	−10.5105	−0.467712
$506$	−9.98916	−0.444072
$507$	−1.00000	−0.0444116
$508$	21.0996	0.936143
$509$	32.3907	1.43569	0.717845	−	0.696203i	$-0.245128\pi$
$509$	32.3907	1.43569	0.717845	+	0.696203i	$0.245128\pi$
$510$	−2.76903	−0.122615
$511$	−3.69950	−0.163656
$512$	−22.5843	−0.998094
$513$	−2.35863	−0.104136
$514$	2.48735	0.109712
$515$	15.8965	0.700485
$516$	−13.6630	−0.601479
$517$	54.2671	2.38667
$518$	7.79114	0.342323
$519$	−1.51121	−0.0663350
$520$	1.53343	0.0672454
$521$	15.6278	0.684665	0.342332	−	0.939579i	$-0.388783\pi$
$521$	15.6278	0.684665	0.342332	+	0.939579i	$0.388783\pi$
$522$	0.0532144	0.00232913
$523$	−14.4996	−0.634022	−0.317011	−	0.948422i	$-0.602679\pi$
$523$	−14.4996	−0.634022	−0.317011	+	0.948422i	$0.602679\pi$
$524$	18.7902	0.820854
$525$	2.18061	0.0951694
$526$	−4.17520	−0.182048
$527$	6.93522	0.302103
$528$	−16.7347	−0.728285
$529$	−1.94976	−0.0847722
$530$	−0.440962	−0.0191541
$531$	7.17351	0.311304
$532$	9.46655	0.410427
$533$	0.596045	0.0258176
$534$	4.32651	0.187226
$535$	−19.1024	−0.825867
$536$	−21.7766	−0.940606
$537$	2.71362	0.117101
$538$	11.6291	0.501365
$539$	−12.2417	−0.527288
$540$	−1.84058	−0.0792061
$541$	−20.6013	−0.885721	−0.442861	−	0.896591i	$-0.646036\pi$
$541$	−20.6013	−0.885721	−0.442861	+	0.896591i	$0.646036\pi$
$542$	4.36162	0.187348
$543$	−11.9030	−0.510808
$544$	−29.7673	−1.27626
$545$	11.9280	0.510941
$546$	0.870651	0.0372604
$547$	−36.9941	−1.58175	−0.790877	−	0.611975i	$-0.790375\pi$
$547$	−36.9941	−1.58175	−0.790877	+	0.611975i	$0.790375\pi$
$548$	−23.4168	−1.00032
$549$	−7.44977	−0.317948
$550$	−2.17721	−0.0928366
$551$	−0.314356	−0.0133920
$552$	−7.03546	−0.299449
$553$	0.495354	0.0210646
$554$	−2.30067	−0.0977462
$555$	8.94864	0.379849
$556$	−31.8854	−1.35224
$557$	−2.79186	−0.118295	−0.0591474	−	0.998249i	$-0.518838\pi$
$557$	−2.79186	−0.118295	−0.0591474	+	0.998249i	$0.518838\pi$
$558$	−0.399270	−0.0169025
$559$	7.42317	0.313967
$560$	6.69209	0.282792
$561$	−37.8176	−1.59666
$562$	9.97320	0.420694
$563$	5.02698	0.211862	0.105931	−	0.994373i	$-0.466218\pi$
$563$	5.02698	0.211862	0.105931	+	0.994373i	$0.466218\pi$
$564$	18.3172	0.771293
$565$	18.2665	0.768479
$566$	−2.47684	−0.104109
$567$	−2.18061	−0.0915768
$568$	15.3590	0.644450
$569$	−31.7826	−1.33239	−0.666197	−	0.745776i	$-0.732079\pi$
$569$	−31.7826	−1.33239	−0.666197	+	0.745776i	$0.732079\pi$
$570$	−0.941729	−0.0394447
$571$	−3.71231	−0.155355	−0.0776776	−	0.996979i	$-0.524750\pi$
$571$	−3.71231	−0.155355	−0.0776776	+	0.996979i	$0.524750\pi$
$572$	10.0367	0.419654
$573$	−15.2973	−0.639056
$574$	−0.518947	−0.0216604
$575$	4.58805	0.191335
$576$	−4.42408	−0.184337
$577$	23.6347	0.983926	0.491963	−	0.870616i	$-0.336280\pi$
$577$	23.6347	0.983926	0.491963	+	0.870616i	$0.336280\pi$
$578$	−12.4162	−0.516446
$579$	22.9289	0.952893
$580$	−0.245312	−0.0101860
$581$	7.12927	0.295772
$582$	2.54698	0.105576
$583$	−6.02237	−0.249421
$584$	2.60154	0.107653
$585$	1.00000	0.0413449
$586$	5.04644	0.208466
$587$	−16.6300	−0.686395	−0.343198	−	0.939263i	$-0.611510\pi$
$587$	−16.6300	−0.686395	−0.343198	+	0.939263i	$0.611510\pi$
$588$	−4.13204	−0.170402
$589$	2.35863	0.0971855
$590$	2.86417	0.117916
$591$	−8.94927	−0.368124
$592$	27.4626	1.12871
$593$	9.96468	0.409200	0.204600	−	0.978846i	$-0.434411\pi$
$593$	9.96468	0.409200	0.204600	+	0.978846i	$0.434411\pi$
$594$	2.17721	0.0893320
$595$	15.1230	0.619982
$596$	−20.2328	−0.828767
$597$	−5.14636	−0.210626
$598$	1.83187	0.0749109
$599$	33.9240	1.38610	0.693049	−	0.720891i	$-0.256267\pi$
$599$	33.9240	1.38610	0.693049	+	0.720891i	$0.256267\pi$
$600$	−1.53343	−0.0626020
$601$	−30.2338	−1.23326	−0.616632	−	0.787251i	$-0.711503\pi$
$601$	−30.2338	−1.23326	−0.616632	+	0.787251i	$0.711503\pi$
$602$	−6.46299	−0.263412
$603$	−14.2012	−0.578319
$604$	23.2635	0.946579
$605$	−18.7349	−0.761684
$606$	4.19654	0.170473
$607$	−17.1655	−0.696727	−0.348364	−	0.937359i	$-0.613263\pi$
$607$	−17.1655	−0.696727	−0.348364	+	0.937359i	$0.613263\pi$
$608$	−10.1237	−0.410569
$609$	−0.290629	−0.0117769
$610$	−2.97447	−0.120433
$611$	−9.95184	−0.402608
$612$	−12.7648	−0.515988
$613$	−4.71382	−0.190390	−0.0951948	−	0.995459i	$-0.530347\pi$
$613$	−4.71382	−0.190390	−0.0951948	+	0.995459i	$0.530347\pi$
$614$	−9.52429	−0.384369
$615$	−0.596045	−0.0240349
$616$	−18.2337	−0.734656
$617$	−25.6095	−1.03100	−0.515500	−	0.856889i	$-0.672394\pi$
$617$	−25.6095	−1.03100	−0.515500	+	0.856889i	$0.672394\pi$
$618$	−6.34701	−0.255314
$619$	18.3667	0.738220	0.369110	−	0.929386i	$-0.379663\pi$
$619$	18.3667	0.738220	0.369110	+	0.929386i	$0.379663\pi$
$620$	1.84058	0.0739196
$621$	−4.58805	−0.184112
$622$	5.57884	0.223691
$623$	−23.6291	−0.946681
$624$	3.06891	0.122855
$625$	1.00000	0.0400000
$626$	−10.4627	−0.418175
$627$	−12.8615	−0.513640
$628$	6.56325	0.261902
$629$	62.0608	2.47453
$630$	−0.870651	−0.0346876
$631$	21.1924	0.843657	0.421828	−	0.906676i	$-0.361388\pi$
$631$	21.1924	0.843657	0.421828	+	0.906676i	$0.361388\pi$
$632$	−0.348339	−0.0138562
$633$	−27.5730	−1.09593
$634$	8.41701	0.334282
$635$	11.4635	0.454917
$636$	−2.03277	−0.0806048
$637$	2.24496	0.0889486
$638$	0.290177	0.0114882
$639$	10.0161	0.396231
$640$	−10.3508	−0.409150
$641$	33.3225	1.31616	0.658080	−	0.752948i	$-0.271369\pi$
$641$	33.3225	1.31616	0.658080	+	0.752948i	$0.271369\pi$
$642$	7.62700	0.301014
$643$	−0.655272	−0.0258414	−0.0129207	−	0.999917i	$-0.504113\pi$
$643$	−0.655272	−0.0258414	−0.0129207	+	0.999917i	$0.504113\pi$
$644$	18.4145	0.725635
$645$	−7.42317	−0.292287
$646$	−6.53110	−0.256963
$647$	44.2406	1.73928	0.869639	−	0.493688i	$-0.164351\pi$
$647$	44.2406	1.73928	0.869639	+	0.493688i	$0.164351\pi$
$648$	1.53343	0.0602388
$649$	39.1170	1.53548
$650$	0.399270	0.0156607
$651$	2.18061	0.0854647
$652$	−14.8918	−0.583208
$653$	−42.7015	−1.67104	−0.835520	−	0.549460i	$-0.814833\pi$
$653$	−42.7015	−1.67104	−0.835520	+	0.549460i	$0.814833\pi$
$654$	−4.76251	−0.186229
$655$	10.2088	0.398892
$656$	−1.82921	−0.0714187
$657$	1.69655	0.0661887
$658$	8.66458	0.337780
$659$	−36.9425	−1.43908	−0.719538	−	0.694453i	$-0.755646\pi$
$659$	−36.9425	−1.43908	−0.719538	+	0.694453i	$0.755646\pi$
$660$	−10.0367	−0.390676
$661$	−8.69689	−0.338270	−0.169135	−	0.985593i	$-0.554097\pi$
$661$	−8.69689	−0.338270	−0.169135	+	0.985593i	$0.554097\pi$
$662$	−12.1664	−0.472859
$663$	6.93522	0.269342
$664$	−5.01340	−0.194558
$665$	5.14323	0.199446
$666$	−3.57293	−0.138448
$667$	−0.611492	−0.0236771
$668$	5.81982	0.225176
$669$	27.7784	1.07397
$670$	−5.67013	−0.219056
$671$	−40.6234	−1.56825
$672$	−9.35956	−0.361053
$673$	22.5425	0.868950	0.434475	−	0.900684i	$-0.356934\pi$
$673$	22.5425	0.868950	0.434475	+	0.900684i	$0.356934\pi$
$674$	−8.98258	−0.345996
$675$	−1.00000	−0.0384900
$676$	−1.84058	−0.0707917
$677$	−25.0503	−0.962762	−0.481381	−	0.876512i	$-0.659865\pi$
$677$	−25.0503	−0.962762	−0.481381	+	0.876512i	$0.659865\pi$
$678$	−7.29328	−0.280097
$679$	−13.9103	−0.533828
$680$	−10.6347	−0.407821
$681$	25.5062	0.977399
$682$	−2.17721	−0.0833697
$683$	31.6992	1.21294	0.606468	−	0.795108i	$-0.292586\pi$
$683$	31.6992	1.21294	0.606468	+	0.795108i	$0.292586\pi$
$684$	−4.34125	−0.165992
$685$	−12.7225	−0.486102
$686$	−8.04913	−0.307317
$687$	−25.9480	−0.989977
$688$	−22.7811	−0.868520
$689$	1.10442	0.0420750
$690$	−1.83187	−0.0697382
$691$	39.6020	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$691$	39.6020	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$692$	−2.78152	−0.105737
$693$	−11.8908	−0.451694
$694$	−5.74144	−0.217942
$695$	−17.3235	−0.657119
$696$	0.204374	0.00774679
$697$	−4.13370	−0.156575
$698$	−8.53457	−0.323038
$699$	−4.12305	−0.155948
$700$	4.01359	0.151699
$701$	11.6282	0.439192	0.219596	−	0.975591i	$-0.429526\pi$
$701$	11.6282	0.439192	0.219596	+	0.975591i	$0.429526\pi$
$702$	−0.399270	−0.0150695
$703$	21.1065	0.796047
$704$	−24.1244	−0.909223
$705$	9.95184	0.374808
$706$	−3.12402	−0.117574
$707$	−22.9193	−0.861969
$708$	13.2034	0.496216
$709$	33.0904	1.24274	0.621368	−	0.783519i	$-0.286577\pi$
$709$	33.0904	1.24274	0.621368	+	0.783519i	$0.286577\pi$
$710$	3.99914	0.150085
$711$	−0.227163	−0.00851929
$712$	16.6163	0.622723
$713$	4.58805	0.171824
$714$	−6.03815	−0.225972
$715$	5.45298	0.203930
$716$	4.99464	0.186658
$717$	−6.33629	−0.236633
$718$	3.90656	0.145791
$719$	3.61128	0.134678	0.0673390	−	0.997730i	$-0.478549\pi$
$719$	3.61128	0.134678	0.0673390	+	0.997730i	$0.478549\pi$
$720$	−3.06891	−0.114372
$721$	34.6640	1.29096
$722$	5.36495	0.199663
$723$	3.72023	0.138357
$724$	−21.9085	−0.814224
$725$	−0.133279	−0.00494987
$726$	7.48030	0.277620
$727$	9.19661	0.341083	0.170542	−	0.985350i	$-0.445448\pi$
$727$	9.19661	0.341083	0.170542	+	0.985350i	$0.445448\pi$
$728$	3.34381	0.123930
$729$	1.00000	0.0370370
$730$	0.677382	0.0250710
$731$	−51.4813	−1.90411
$732$	−13.7119	−0.506807
$733$	−20.2113	−0.746523	−0.373261	−	0.927726i	$-0.621761\pi$
$733$	−20.2113	−0.746523	−0.373261	+	0.927726i	$0.621761\pi$
$734$	9.66662	0.356801
$735$	−2.24496	−0.0828066
$736$	−19.6928	−0.725886
$737$	−77.4390	−2.85250
$738$	0.237983	0.00876028
$739$	−44.3611	−1.63185	−0.815924	−	0.578159i	$-0.803771\pi$
$739$	−44.3611	−1.63185	−0.815924	+	0.578159i	$0.803771\pi$
$740$	16.4707	0.605476
$741$	2.35863	0.0866463
$742$	−0.961563	−0.0353001
$743$	−42.1402	−1.54597	−0.772987	−	0.634422i	$-0.781238\pi$
$743$	−42.1402	−1.54597	−0.772987	+	0.634422i	$0.781238\pi$
$744$	−1.53343	−0.0562183
$745$	−10.9926	−0.402737
$746$	7.84697	0.287298
$747$	−3.26940	−0.119621
$748$	−69.6064	−2.54506
$749$	−41.6547	−1.52203
$750$	−0.399270	−0.0145793
$751$	−3.76085	−0.137235	−0.0686176	−	0.997643i	$-0.521859\pi$
$751$	−3.76085	−0.137235	−0.0686176	+	0.997643i	$0.521859\pi$
$752$	30.5413	1.11373
$753$	−8.51196	−0.310193
$754$	−0.0532144	−0.00193796
$755$	12.6392	0.459988
$756$	−4.01359	−0.145973
$757$	44.0250	1.60012	0.800058	−	0.599922i	$-0.204802\pi$
$757$	44.0250	1.60012	0.800058	+	0.599922i	$0.204802\pi$
$758$	−0.345659	−0.0125549
$759$	−25.0185	−0.908116
$760$	−3.61679	−0.131195
$761$	40.5448	1.46975	0.734875	−	0.678203i	$-0.237241\pi$
$761$	40.5448	1.46975	0.734875	+	0.678203i	$0.237241\pi$
$762$	−4.57705	−0.165809
$763$	26.0103	0.941637
$764$	−28.1560	−1.01865
$765$	−6.93522	−0.250743
$766$	4.46126	0.161192
$767$	−7.17351	−0.259020
$768$	−4.71541	−0.170153
$769$	−10.8568	−0.391506	−0.195753	−	0.980653i	$-0.562715\pi$
$769$	−10.8568	−0.391506	−0.195753	+	0.980653i	$0.562715\pi$
$770$	−4.74764	−0.171093
$771$	6.22975	0.224359
$772$	42.2026	1.51890
$773$	25.5851	0.920231	0.460116	−	0.887859i	$-0.347808\pi$
$773$	25.5851	0.920231	0.460116	+	0.887859i	$0.347808\pi$
$774$	2.96385	0.106533
$775$	1.00000	0.0359211
$776$	9.78191	0.351150
$777$	19.5135	0.700042
$778$	4.21577	0.151143
$779$	−1.40585	−0.0503697
$780$	1.84058	0.0659035
$781$	54.6176	1.95437
$782$	−12.7044	−0.454310
$783$	0.133279	0.00476301
$784$	−6.88959	−0.246057
$785$	3.56585	0.127271
$786$	−4.07608	−0.145389
$787$	−8.24729	−0.293984	−0.146992	−	0.989138i	$-0.546959\pi$
$787$	−8.24729	−0.293984	−0.146992	+	0.989138i	$0.546959\pi$
$788$	−16.4719	−0.586786
$789$	−10.4571	−0.372282
$790$	−0.0906996	−0.00322695
$791$	39.8321	1.41627
$792$	8.36176	0.297122
$793$	7.44977	0.264549
$794$	9.41556	0.334146
$795$	−1.10442	−0.0391697
$796$	−9.47230	−0.335737
$797$	20.0547	0.710373	0.355186	−	0.934796i	$-0.384417\pi$
$797$	20.0547	0.710373	0.355186	+	0.934796i	$0.384417\pi$
$798$	−2.05354	−0.0726945
$799$	69.0182	2.44169
$800$	−4.29219	−0.151752
$801$	10.8360	0.382873
$802$	−4.78350	−0.168911
$803$	9.25124	0.326469
$804$	−26.1386	−0.921836
$805$	10.0047	0.352621
$806$	0.399270	0.0140637
$807$	29.1258	1.02528
$808$	16.1172	0.567000
$809$	17.2910	0.607920	0.303960	−	0.952685i	$-0.401691\pi$
$809$	17.2910	0.607920	0.303960	+	0.952685i	$0.401691\pi$
$810$	0.399270	0.0140289
$811$	23.0428	0.809141	0.404570	−	0.914507i	$-0.367421\pi$
$811$	23.0428	0.809141	0.404570	+	0.914507i	$0.367421\pi$
$812$	−0.534928	−0.0187723
$813$	10.9240	0.383121
$814$	−19.4831	−0.682882
$815$	−8.09081	−0.283409
$816$	−21.2836	−0.745075
$817$	−17.5085	−0.612544
$818$	−7.49595	−0.262090
$819$	2.18061	0.0761965
$820$	−1.09707	−0.0383114
$821$	7.72608	0.269642	0.134821	−	0.990870i	$-0.456954\pi$
$821$	7.72608	0.269642	0.134821	+	0.990870i	$0.456954\pi$
$822$	5.07972	0.177176
$823$	41.8081	1.45734	0.728669	−	0.684866i	$-0.240139\pi$
$823$	41.8081	1.45734	0.728669	+	0.684866i	$0.240139\pi$
$824$	−24.3762	−0.849186
$825$	−5.45298	−0.189848
$826$	6.24562	0.217313
$827$	15.3913	0.535209	0.267605	−	0.963529i	$-0.413768\pi$
$827$	15.3913	0.535209	0.267605	+	0.963529i	$0.413768\pi$
$828$	−8.44470	−0.293473
$829$	−6.12659	−0.212785	−0.106393	−	0.994324i	$-0.533930\pi$
$829$	−6.12659	−0.212785	−0.106393	+	0.994324i	$0.533930\pi$
$830$	−1.30537	−0.0453102
$831$	−5.76219	−0.199888
$832$	4.42408	0.153378
$833$	−15.5693	−0.539444
$834$	6.91677	0.239508
$835$	3.16195	0.109424
$836$	−23.6727	−0.818738
$837$	−1.00000	−0.0345651
$838$	2.73270	0.0943995
$839$	−27.8081	−0.960042	−0.480021	−	0.877257i	$-0.659371\pi$
$839$	−27.8081	−0.960042	−0.480021	+	0.877257i	$0.659371\pi$
$840$	−3.34381	−0.115372
$841$	−28.9822	−0.999387
$842$	1.80732	0.0622842
$843$	24.9786	0.860308
$844$	−50.7504	−1.74690
$845$	−1.00000	−0.0344010
$846$	−3.97347	−0.136611
$847$	−40.8535	−1.40374
$848$	−3.38937	−0.116391
$849$	−6.20341	−0.212900
$850$	−2.76903	−0.0949768
$851$	41.0569	1.40741
$852$	18.4355	0.631590
$853$	−34.5896	−1.18433	−0.592163	−	0.805818i	$-0.701726\pi$
$853$	−34.5896	−1.18433	−0.592163	+	0.805818i	$0.701726\pi$
$854$	−6.48614	−0.221951
$855$	−2.35863	−0.0806633
$856$	29.2921	1.00118
$857$	32.6433	1.11507	0.557536	−	0.830153i	$-0.311747\pi$
$857$	32.6433	1.11507	0.557536	+	0.830153i	$0.311747\pi$
$858$	−2.17721	−0.0743287
$859$	−53.5756	−1.82798	−0.913989	−	0.405740i	$-0.867014\pi$
$859$	−53.5756	−1.82798	−0.913989	+	0.405740i	$0.867014\pi$
$860$	−13.6630	−0.465903
$861$	−1.29974	−0.0442950
$862$	2.87552	0.0979407
$863$	−27.6132	−0.939965	−0.469983	−	0.882676i	$-0.655740\pi$
$863$	−27.6132	−0.939965	−0.469983	+	0.882676i	$0.655740\pi$
$864$	4.29219	0.146023
$865$	−1.51121	−0.0513828
$866$	4.05181	0.137686
$867$	−31.0973	−1.05612
$868$	4.01359	0.136230
$869$	−1.23872	−0.0420206
$870$	0.0532144	0.00180414
$871$	14.2012	0.481191
$872$	−18.2908	−0.619405
$873$	6.37910	0.215900
$874$	−4.32070	−0.146150
$875$	2.18061	0.0737179
$876$	3.12264	0.105504
$877$	−18.4387	−0.622632	−0.311316	−	0.950306i	$-0.600770\pi$
$877$	−18.4387	−0.622632	−0.311316	+	0.950306i	$0.600770\pi$
$878$	10.0161	0.338028
$879$	12.6392	0.426308
$880$	−16.7347	−0.564127
$881$	32.2724	1.08728	0.543642	−	0.839317i	$-0.317045\pi$
$881$	32.2724	1.08728	0.543642	+	0.839317i	$0.317045\pi$
$882$	0.896346	0.0301815
$883$	24.5112	0.824867	0.412434	−	0.910988i	$-0.364679\pi$
$883$	24.5112	0.824867	0.412434	+	0.910988i	$0.364679\pi$
$884$	12.7648	0.429328
$885$	7.17351	0.241135
$886$	−13.8051	−0.463791
$887$	−2.41132	−0.0809640	−0.0404820	−	0.999180i	$-0.512889\pi$
$887$	−2.41132	−0.0809640	−0.0404820	+	0.999180i	$0.512889\pi$
$888$	−13.7221	−0.460484
$889$	24.9975	0.838388
$890$	4.32651	0.145025
$891$	5.45298	0.182682
$892$	51.1284	1.71191
$893$	23.4727	0.785483
$894$	4.38901	0.146791
$895$	2.71362	0.0907063
$896$	−22.5710	−0.754043
$897$	4.58805	0.153191
$898$	9.05387	0.302132
$899$	−0.133279	−0.00444511
$900$	−1.84058	−0.0613528
$901$	−7.65939	−0.255171
$902$	1.29772	0.0432092
$903$	−16.1870	−0.538670
$904$	−28.0105	−0.931614
$905$	−11.9030	−0.395670
$906$	−5.04646	−0.167657
$907$	5.01838	0.166633	0.0833164	−	0.996523i	$-0.473449\pi$
$907$	5.01838	0.166633	0.0833164	+	0.996523i	$0.473449\pi$
$908$	46.9463	1.55797
$909$	10.5105	0.348612
$910$	0.870651	0.0288618
$911$	36.0725	1.19513	0.597567	−	0.801819i	$-0.296134\pi$
$911$	36.0725	1.19513	0.597567	+	0.801819i	$0.296134\pi$
$912$	−7.23842	−0.239688
$913$	−17.8280	−0.590020
$914$	7.64932	0.253017
$915$	−7.44977	−0.246282
$916$	−47.7594	−1.57802
$917$	22.2614	0.735138
$918$	2.76903	0.0913915
$919$	4.51275	0.148862	0.0744310	−	0.997226i	$-0.476286\pi$
$919$	4.51275	0.148862	0.0744310	+	0.997226i	$0.476286\pi$
$920$	−7.03546	−0.231952
$921$	−23.8542	−0.786024
$922$	−9.19746	−0.302902
$923$	−10.0161	−0.329684
$924$	−21.8860	−0.719996
$925$	8.94864	0.294230
$926$	−2.19201	−0.0720340
$927$	−15.8965	−0.522110
$928$	0.572059	0.0187788
$929$	21.2705	0.697863	0.348932	−	0.937148i	$-0.386545\pi$
$929$	21.2705	0.697863	0.348932	+	0.937148i	$0.386545\pi$
$930$	−0.399270	−0.0130926
$931$	−5.29502	−0.173537
$932$	−7.58882	−0.248580
$933$	13.9726	0.457442
$934$	−4.35312	−0.142438
$935$	−37.8176	−1.23677
$936$	−1.53343	−0.0501217
$937$	−29.2739	−0.956336	−0.478168	−	0.878268i	$-0.658699\pi$
$937$	−29.2739	−0.956336	−0.478168	+	0.878268i	$0.658699\pi$
$938$	−12.3643	−0.403709
$939$	−26.2047	−0.855157
$940$	18.3172	0.597441
$941$	−31.7802	−1.03600	−0.518002	−	0.855379i	$-0.673324\pi$
$941$	−31.7802	−1.03600	−0.518002	+	0.855379i	$0.673324\pi$
$942$	−1.42374	−0.0463879
$943$	−2.73469	−0.0890537
$944$	22.0149	0.716523
$945$	−2.18061	−0.0709351
$946$	16.1618	0.525466
$947$	−34.4852	−1.12062	−0.560309	−	0.828284i	$-0.689317\pi$
$947$	−34.4852	−1.12062	−0.560309	+	0.828284i	$0.689317\pi$
$948$	−0.418113	−0.0135797
$949$	−1.69655	−0.0550723
$950$	−0.941729	−0.0305537
$951$	21.0810	0.683598
$952$	−23.1900	−0.751593
$953$	24.5436	0.795044	0.397522	−	0.917593i	$-0.369870\pi$
$953$	24.5436	0.795044	0.397522	+	0.917593i	$0.369870\pi$
$954$	0.440962	0.0142767
$955$	−15.2973	−0.495010
$956$	−11.6625	−0.377191
$957$	0.726769	0.0234931
$958$	−8.87317	−0.286679
$959$	−27.7428	−0.895861
$960$	−4.42408	−0.142787
$961$	1.00000	0.0322581
$962$	3.57293	0.115196
$963$	19.1024	0.615565
$964$	6.84740	0.220540
$965$	22.9289	0.738108
$966$	−3.99459	−0.128524
$967$	28.8626	0.928159	0.464080	−	0.885793i	$-0.346385\pi$
$967$	28.8626	0.928159	0.464080	+	0.885793i	$0.346385\pi$
$968$	28.7287	0.923376
$969$	−16.3576	−0.525481
$970$	2.54698	0.0817787
$971$	1.89293	0.0607470	0.0303735	−	0.999539i	$-0.490330\pi$
$971$	1.89293	0.0607470	0.0303735	+	0.999539i	$0.490330\pi$
$972$	1.84058	0.0590367
$973$	−37.7758	−1.21104
$974$	−14.2787	−0.457519
$975$	1.00000	0.0320256
$976$	−22.8627	−0.731817
$977$	−25.7376	−0.823419	−0.411710	−	0.911315i	$-0.635068\pi$
$977$	−25.7376	−0.823419	−0.411710	+	0.911315i	$0.635068\pi$
$978$	3.23042	0.103297
$979$	59.0887	1.88848
$980$	−4.13204	−0.131993
$981$	−11.9280	−0.380833
$982$	−9.50050	−0.303173
$983$	37.5451	1.19750	0.598751	−	0.800935i	$-0.295664\pi$
$983$	37.5451	1.19750	0.598751	+	0.800935i	$0.295664\pi$
$984$	0.913994	0.0291371
$985$	−8.94927	−0.285147
$986$	0.369054	0.0117531
$987$	21.7010	0.690752
$988$	4.34125	0.138113
$989$	−34.0579	−1.08298
$990$	2.17721	0.0691963
$991$	37.5419	1.19256	0.596279	−	0.802777i	$-0.296645\pi$
$991$	37.5419	1.19256	0.596279	+	0.802777i	$0.296645\pi$
$992$	−4.29219	−0.136277
$993$	−30.4715	−0.966985
$994$	8.72054	0.276599
$995$	−5.14636	−0.163150
$996$	−6.01761	−0.190675
$997$	29.2979	0.927874	0.463937	−	0.885868i	$-0.346436\pi$
$997$	29.2979	0.927874	0.463937	+	0.885868i	$0.346436\pi$
$998$	−2.26711	−0.0717641
$999$	−8.94864	−0.283123

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.bg.1.8	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bg.1.8	✓	16	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-0.399270 q^{2} -1.00000 q^{3} -1.84058 q^{4} -1.00000 q^{5} +0.399270 q^{6} -2.18061 q^{7} +1.53343 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.399270 q^{2} -1.00000 q^{3} -1.84058 q^{4} -1.00000 q^{5} +0.399270 q^{6} -2.18061 q^{7} +1.53343 q^{8} +1.00000 q^{9} +0.399270 q^{10} +5.45298 q^{11} +1.84058 q^{12} -1.00000 q^{13} +0.870651 q^{14} +1.00000 q^{15} +3.06891 q^{16} +6.93522 q^{17} -0.399270 q^{18} +2.35863 q^{19} +1.84058 q^{20} +2.18061 q^{21} -2.17721 q^{22} +4.58805 q^{23} -1.53343 q^{24} +1.00000 q^{25} +0.399270 q^{26} -1.00000 q^{27} +4.01359 q^{28} -0.133279 q^{29} -0.399270 q^{30} +1.00000 q^{31} -4.29219 q^{32} -5.45298 q^{33} -2.76903 q^{34} +2.18061 q^{35} -1.84058 q^{36} +8.94864 q^{37} -0.941729 q^{38} +1.00000 q^{39} -1.53343 q^{40} -0.596045 q^{41} -0.870651 q^{42} -7.42317 q^{43} -10.0367 q^{44} -1.00000 q^{45} -1.83187 q^{46} +9.95184 q^{47} -3.06891 q^{48} -2.24496 q^{49} -0.399270 q^{50} -6.93522 q^{51} +1.84058 q^{52} -1.10442 q^{53} +0.399270 q^{54} -5.45298 q^{55} -3.34381 q^{56} -2.35863 q^{57} +0.0532144 q^{58} +7.17351 q^{59} -1.84058 q^{60} -7.44977 q^{61} -0.399270 q^{62} -2.18061 q^{63} -4.42408 q^{64} +1.00000 q^{65} +2.17721 q^{66} -14.2012 q^{67} -12.7648 q^{68} -4.58805 q^{69} -0.870651 q^{70} +10.0161 q^{71} +1.53343 q^{72} +1.69655 q^{73} -3.57293 q^{74} -1.00000 q^{75} -4.34125 q^{76} -11.8908 q^{77} -0.399270 q^{78} -0.227163 q^{79} -3.06891 q^{80} +1.00000 q^{81} +0.237983 q^{82} -3.26940 q^{83} -4.01359 q^{84} -6.93522 q^{85} +2.96385 q^{86} +0.133279 q^{87} +8.36176 q^{88} +10.8360 q^{89} +0.399270 q^{90} +2.18061 q^{91} -8.44470 q^{92} -1.00000 q^{93} -3.97347 q^{94} -2.35863 q^{95} +4.29219 q^{96} +6.37910 q^{97} +0.896346 q^{98} +5.45298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * q^99

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