LMFDB - Embedded newform 6045.2.a.bg.1.1

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			1.1
Root			$2.74907$ 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)

$f(q)$	$=$	$q-2.74907 q^{2} -1.00000 q^{3} +5.55740 q^{4} -1.00000 q^{5} +2.74907 q^{6} -0.783225 q^{7} -9.77954 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.74907 q^{2} -1.00000 q^{3} +5.55740 q^{4} -1.00000 q^{5} +2.74907 q^{6} -0.783225 q^{7} -9.77954 q^{8} +1.00000 q^{9} +2.74907 q^{10} -4.40508 q^{11} -5.55740 q^{12} -1.00000 q^{13} +2.15314 q^{14} +1.00000 q^{15} +15.7699 q^{16} -4.56709 q^{17} -2.74907 q^{18} +7.83137 q^{19} -5.55740 q^{20} +0.783225 q^{21} +12.1099 q^{22} +2.11998 q^{23} +9.77954 q^{24} +1.00000 q^{25} +2.74907 q^{26} -1.00000 q^{27} -4.35269 q^{28} +0.715668 q^{29} -2.74907 q^{30} +1.00000 q^{31} -23.7934 q^{32} +4.40508 q^{33} +12.5553 q^{34} +0.783225 q^{35} +5.55740 q^{36} -2.53664 q^{37} -21.5290 q^{38} +1.00000 q^{39} +9.77954 q^{40} +7.47619 q^{41} -2.15314 q^{42} +8.74061 q^{43} -24.4808 q^{44} -1.00000 q^{45} -5.82799 q^{46} -0.973088 q^{47} -15.7699 q^{48} -6.38656 q^{49} -2.74907 q^{50} +4.56709 q^{51} -5.55740 q^{52} +4.11647 q^{53} +2.74907 q^{54} +4.40508 q^{55} +7.65958 q^{56} -7.83137 q^{57} -1.96742 q^{58} -2.81784 q^{59} +5.55740 q^{60} -10.4526 q^{61} -2.74907 q^{62} -0.783225 q^{63} +33.8701 q^{64} +1.00000 q^{65} -12.1099 q^{66} -11.6393 q^{67} -25.3811 q^{68} -2.11998 q^{69} -2.15314 q^{70} -2.44064 q^{71} -9.77954 q^{72} -5.12309 q^{73} +6.97341 q^{74} -1.00000 q^{75} +43.5221 q^{76} +3.45017 q^{77} -2.74907 q^{78} -16.3408 q^{79} -15.7699 q^{80} +1.00000 q^{81} -20.5526 q^{82} -14.8861 q^{83} +4.35269 q^{84} +4.56709 q^{85} -24.0286 q^{86} -0.715668 q^{87} +43.0797 q^{88} +2.18878 q^{89} +2.74907 q^{90} +0.783225 q^{91} +11.7816 q^{92} -1.00000 q^{93} +2.67509 q^{94} -7.83137 q^{95} +23.7934 q^{96} -8.60810 q^{97} +17.5571 q^{98} -4.40508 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.74907	−1.94389	−0.971944	−	0.235213i	$-0.924421\pi$
$2$	−2.74907	−1.94389	−0.971944	+	0.235213i	$0.924421\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.55740	2.77870
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.74907	1.12230
$7$	−0.783225	−0.296031	−0.148016	−	0.988985i	$-0.547289\pi$
$7$	−0.783225	−0.296031	−0.148016	+	0.988985i	$0.547289\pi$
$8$	−9.77954	−3.45759
$9$	1.00000	0.333333
$10$	2.74907	0.869333
$11$	−4.40508	−1.32818	−0.664092	−	0.747651i	$-0.731181\pi$
$11$	−4.40508	−1.32818	−0.664092	+	0.747651i	$0.731181\pi$
$12$	−5.55740	−1.60428
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.15314	0.575451
$15$	1.00000	0.258199
$16$	15.7699	3.94247
$17$	−4.56709	−1.10768	−0.553841	−	0.832623i	$-0.686838\pi$
$17$	−4.56709	−1.10768	−0.553841	+	0.832623i	$0.686838\pi$
$18$	−2.74907	−0.647963
$19$	7.83137	1.79664	0.898320	−	0.439341i	$-0.144788\pi$
$19$	7.83137	1.79664	0.898320	+	0.439341i	$0.144788\pi$
$20$	−5.55740	−1.24267
$21$	0.783225	0.170914
$22$	12.1099	2.58184
$23$	2.11998	0.442047	0.221024	−	0.975268i	$-0.429060\pi$
$23$	2.11998	0.442047	0.221024	+	0.975268i	$0.429060\pi$
$24$	9.77954	1.99624
$25$	1.00000	0.200000
$26$	2.74907	0.539137
$27$	−1.00000	−0.192450
$28$	−4.35269	−0.822582
$29$	0.715668	0.132896	0.0664481	−	0.997790i	$-0.478833\pi$
$29$	0.715668	0.132896	0.0664481	+	0.997790i	$0.478833\pi$
$30$	−2.74907	−0.501910
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−23.7934	−4.20613
$33$	4.40508	0.766827
$34$	12.5553	2.15321
$35$	0.783225	0.132389
$36$	5.55740	0.926233
$37$	−2.53664	−0.417021	−0.208511	−	0.978020i	$-0.566862\pi$
$37$	−2.53664	−0.417021	−0.208511	+	0.978020i	$0.566862\pi$
$38$	−21.5290	−3.49247
$39$	1.00000	0.160128
$40$	9.77954	1.54628
$41$	7.47619	1.16759	0.583793	−	0.811903i	$-0.301568\pi$
$41$	7.47619	1.16759	0.583793	+	0.811903i	$0.301568\pi$
$42$	−2.15314	−0.332237
$43$	8.74061	1.33293	0.666465	−	0.745536i	$-0.267807\pi$
$43$	8.74061	1.33293	0.666465	+	0.745536i	$0.267807\pi$
$44$	−24.4808	−3.69062
$45$	−1.00000	−0.149071
$46$	−5.82799	−0.859291
$47$	−0.973088	−0.141940	−0.0709698	−	0.997478i	$-0.522609\pi$
$47$	−0.973088	−0.141940	−0.0709698	+	0.997478i	$0.522609\pi$
$48$	−15.7699	−2.27619
$49$	−6.38656	−0.912366
$50$	−2.74907	−0.388778
$51$	4.56709	0.639520
$52$	−5.55740	−0.770672
$53$	4.11647	0.565441	0.282720	−	0.959202i	$-0.408763\pi$
$53$	4.11647	0.565441	0.282720	+	0.959202i	$0.408763\pi$
$54$	2.74907	0.374101
$55$	4.40508	0.593982
$56$	7.65958	1.02355
$57$	−7.83137	−1.03729
$58$	−1.96742	−0.258335
$59$	−2.81784	−0.366852	−0.183426	−	0.983034i	$-0.558719\pi$
$59$	−2.81784	−0.366852	−0.183426	+	0.983034i	$0.558719\pi$
$60$	5.55740	0.717457
$61$	−10.4526	−1.33832	−0.669158	−	0.743120i	$-0.733345\pi$
$61$	−10.4526	−1.33832	−0.669158	+	0.743120i	$0.733345\pi$
$62$	−2.74907	−0.349133
$63$	−0.783225	−0.0986771
$64$	33.8701	4.23377
$65$	1.00000	0.124035
$66$	−12.1099	−1.49063
$67$	−11.6393	−1.42197	−0.710984	−	0.703208i	$-0.751750\pi$
$67$	−11.6393	−1.42197	−0.710984	+	0.703208i	$0.751750\pi$
$68$	−25.3811	−3.07791
$69$	−2.11998	−0.255216
$70$	−2.15314	−0.257350
$71$	−2.44064	−0.289650	−0.144825	−	0.989457i	$-0.546262\pi$
$71$	−2.44064	−0.289650	−0.144825	+	0.989457i	$0.546262\pi$
$72$	−9.77954	−1.15253
$73$	−5.12309	−0.599612	−0.299806	−	0.954000i	$-0.596922\pi$
$73$	−5.12309	−0.599612	−0.299806	+	0.954000i	$0.596922\pi$
$74$	6.97341	0.810642
$75$	−1.00000	−0.115470
$76$	43.5221	4.99232
$77$	3.45017	0.393184
$78$	−2.74907	−0.311271
$79$	−16.3408	−1.83848	−0.919241	−	0.393696i	$-0.871196\pi$
$79$	−16.3408	−1.83848	−0.919241	+	0.393696i	$0.871196\pi$
$80$	−15.7699	−1.76313
$81$	1.00000	0.111111
$82$	−20.5526	−2.26965
$83$	−14.8861	−1.63396	−0.816982	−	0.576663i	$-0.804354\pi$
$83$	−14.8861	−1.63396	−0.816982	+	0.576663i	$0.804354\pi$
$84$	4.35269	0.474918
$85$	4.56709	0.495370
$86$	−24.0286	−2.59107
$87$	−0.715668	−0.0767276
$88$	43.0797	4.59231
$89$	2.18878	0.232010	0.116005	−	0.993249i	$-0.462991\pi$
$89$	2.18878	0.232010	0.116005	+	0.993249i	$0.462991\pi$
$90$	2.74907	0.289778
$91$	0.783225	0.0821043
$92$	11.7816	1.22832
$93$	−1.00000	−0.103695
$94$	2.67509	0.275915
$95$	−7.83137	−0.803482
$96$	23.7934	2.42841
$97$	−8.60810	−0.874020	−0.437010	−	0.899457i	$-0.643963\pi$
$97$	−8.60810	−0.874020	−0.437010	+	0.899457i	$0.643963\pi$
$98$	17.5571	1.77354
$99$	−4.40508	−0.442728
$100$	5.55740	0.555740
$101$	17.2377	1.71522	0.857609	−	0.514303i	$-0.171949\pi$
$101$	17.2377	1.71522	0.857609	+	0.514303i	$0.171949\pi$
$102$	−12.5553	−1.24316
$103$	2.36029	0.232567	0.116283	−	0.993216i	$-0.462902\pi$
$103$	2.36029	0.232567	0.116283	+	0.993216i	$0.462902\pi$
$104$	9.77954	0.958963
$105$	−0.783225	−0.0764349
$106$	−11.3165	−1.09915
$107$	−3.04004	−0.293891	−0.146946	−	0.989145i	$-0.546944\pi$
$107$	−3.04004	−0.293891	−0.146946	+	0.989145i	$0.546944\pi$
$108$	−5.55740	−0.534761
$109$	−6.18979	−0.592874	−0.296437	−	0.955052i	$-0.595799\pi$
$109$	−6.18979	−0.592874	−0.296437	+	0.955052i	$0.595799\pi$
$110$	−12.1099	−1.15463
$111$	2.53664	0.240767
$112$	−12.3514	−1.16709
$113$	18.6617	1.75555	0.877774	−	0.479075i	$-0.159028\pi$
$113$	18.6617	1.75555	0.877774	+	0.479075i	$0.159028\pi$
$114$	21.5290	2.01638
$115$	−2.11998	−0.197690
$116$	3.97725	0.369278
$117$	−1.00000	−0.0924500
$118$	7.74646	0.713119
$119$	3.57706	0.327908
$120$	−9.77954	−0.892746
$121$	8.40477	0.764070
$122$	28.7349	2.60154
$123$	−7.47619	−0.674106
$124$	5.55740	0.499069
$125$	−1.00000	−0.0894427
$126$	2.15314	0.191817
$127$	5.13808	0.455931	0.227965	−	0.973669i	$-0.426793\pi$
$127$	5.13808	0.455931	0.227965	+	0.973669i	$0.426793\pi$
$128$	−45.5246	−4.02384
$129$	−8.74061	−0.769568
$130$	−2.74907	−0.241110
$131$	−9.11354	−0.796254	−0.398127	−	0.917330i	$-0.630340\pi$
$131$	−9.11354	−0.796254	−0.398127	+	0.917330i	$0.630340\pi$
$132$	24.4808	2.13078
$133$	−6.13373	−0.531862
$134$	31.9973	2.76415
$135$	1.00000	0.0860663
$136$	44.6640	3.82991
$137$	−18.7271	−1.59996	−0.799981	−	0.600025i	$-0.795157\pi$
$137$	−18.7271	−1.59996	−0.799981	+	0.600025i	$0.795157\pi$
$138$	5.82799	0.496112
$139$	6.79073	0.575982	0.287991	−	0.957633i	$-0.407013\pi$
$139$	6.79073	0.575982	0.287991	+	0.957633i	$0.407013\pi$
$140$	4.35269	0.367870
$141$	0.973088	0.0819488
$142$	6.70948	0.563047
$143$	4.40508	0.368372
$144$	15.7699	1.31416
$145$	−0.715668	−0.0594330
$146$	14.0837	1.16558
$147$	6.38656	0.526754
$148$	−14.0971	−1.15878
$149$	1.32798	0.108792	0.0543962	−	0.998519i	$-0.482677\pi$
$149$	1.32798	0.108792	0.0543962	+	0.998519i	$0.482677\pi$
$150$	2.74907	0.224461
$151$	7.43162	0.604776	0.302388	−	0.953185i	$-0.402216\pi$
$151$	7.43162	0.604776	0.302388	+	0.953185i	$0.402216\pi$
$152$	−76.5873	−6.21205
$153$	−4.56709	−0.369227
$154$	−9.48477	−0.764305
$155$	−1.00000	−0.0803219
$156$	5.55740	0.444948
$157$	−1.88468	−0.150414	−0.0752071	−	0.997168i	$-0.523962\pi$
$157$	−1.88468	−0.150414	−0.0752071	+	0.997168i	$0.523962\pi$
$158$	44.9220	3.57380
$159$	−4.11647	−0.326457
$160$	23.7934	1.88104
$161$	−1.66043	−0.130860
$162$	−2.74907	−0.215988
$163$	−13.1229	−1.02787	−0.513933	−	0.857830i	$-0.671812\pi$
$163$	−13.1229	−1.02787	−0.513933	+	0.857830i	$0.671812\pi$
$164$	41.5482	3.24437
$165$	−4.40508	−0.342935
$166$	40.9230	3.17624
$167$	2.32211	0.179690	0.0898449	−	0.995956i	$-0.471363\pi$
$167$	2.32211	0.179690	0.0898449	+	0.995956i	$0.471363\pi$
$168$	−7.65958	−0.590950
$169$	1.00000	0.0769231
$170$	−12.5553	−0.962944
$171$	7.83137	0.598880
$172$	48.5750	3.70381
$173$	−7.11584	−0.541007	−0.270504	−	0.962719i	$-0.587190\pi$
$173$	−7.11584	−0.541007	−0.270504	+	0.962719i	$0.587190\pi$
$174$	1.96742	0.149150
$175$	−0.783225	−0.0592062
$176$	−69.4677	−5.23632
$177$	2.81784	0.211802
$178$	−6.01712	−0.451002
$179$	−0.694840	−0.0519348	−0.0259674	−	0.999663i	$-0.508267\pi$
$179$	−0.694840	−0.0519348	−0.0259674	+	0.999663i	$0.508267\pi$
$180$	−5.55740	−0.414224
$181$	20.3815	1.51495	0.757473	−	0.652867i	$-0.226434\pi$
$181$	20.3815	1.51495	0.757473	+	0.652867i	$0.226434\pi$
$182$	−2.15314	−0.159602
$183$	10.4526	0.772677
$184$	−20.7325	−1.52842
$185$	2.53664	0.186497
$186$	2.74907	0.201572
$187$	20.1184	1.47120
$188$	−5.40784	−0.394407
$189$	0.783225	0.0569712
$190$	21.5290	1.56188
$191$	8.96960	0.649018	0.324509	−	0.945883i	$-0.394801\pi$
$191$	8.96960	0.649018	0.324509	+	0.945883i	$0.394801\pi$
$192$	−33.8701	−2.44437
$193$	18.4779	1.33007	0.665035	−	0.746812i	$-0.268416\pi$
$193$	18.4779	1.33007	0.665035	+	0.746812i	$0.268416\pi$
$194$	23.6643	1.69900
$195$	−1.00000	−0.0716115
$196$	−35.4926	−2.53519
$197$	15.4466	1.10053	0.550263	−	0.834992i	$-0.314528\pi$
$197$	15.4466	1.10053	0.550263	+	0.834992i	$0.314528\pi$
$198$	12.1099	0.860613
$199$	−3.54252	−0.251123	−0.125561	−	0.992086i	$-0.540073\pi$
$199$	−3.54252	−0.251123	−0.125561	+	0.992086i	$0.540073\pi$
$200$	−9.77954	−0.691518
$201$	11.6393	0.820973
$202$	−47.3877	−3.33419
$203$	−0.560529	−0.0393414
$204$	25.3811	1.77703
$205$	−7.47619	−0.522160
$206$	−6.48862	−0.452084
$207$	2.11998	0.147349
$208$	−15.7699	−1.09344
$209$	−34.4979	−2.38627
$210$	2.15314	0.148581
$211$	−12.5285	−0.862497	−0.431249	−	0.902233i	$-0.641927\pi$
$211$	−12.5285	−0.862497	−0.431249	+	0.902233i	$0.641927\pi$
$212$	22.8769	1.57119
$213$	2.44064	0.167230
$214$	8.35728	0.571292
$215$	−8.74061	−0.596105
$216$	9.77954	0.665414
$217$	−0.783225	−0.0531688
$218$	17.0162	1.15248
$219$	5.12309	0.346186
$220$	24.4808	1.65050
$221$	4.56709	0.307216
$222$	−6.97341	−0.468024
$223$	10.1265	0.678123	0.339062	−	0.940764i	$-0.389890\pi$
$223$	10.1265	0.678123	0.339062	+	0.940764i	$0.389890\pi$
$224$	18.6356	1.24514
$225$	1.00000	0.0666667
$226$	−51.3024	−3.41259
$227$	−25.6950	−1.70544	−0.852719	−	0.522370i	$-0.825048\pi$
$227$	−25.6950	−1.70544	−0.852719	+	0.522370i	$0.825048\pi$
$228$	−43.5221	−2.88232
$229$	−5.87681	−0.388351	−0.194175	−	0.980967i	$-0.562203\pi$
$229$	−5.87681	−0.388351	−0.194175	+	0.980967i	$0.562203\pi$
$230$	5.82799	0.384286
$231$	−3.45017	−0.227005
$232$	−6.99891	−0.459501
$233$	11.9911	0.785566	0.392783	−	0.919631i	$-0.371512\pi$
$233$	11.9911	0.785566	0.392783	+	0.919631i	$0.371512\pi$
$234$	2.74907	0.179712
$235$	0.973088	0.0634773
$236$	−15.6599	−1.01937
$237$	16.3408	1.06145
$238$	−9.83359	−0.637417
$239$	−27.6384	−1.78778	−0.893889	−	0.448288i	$-0.852034\pi$
$239$	−27.6384	−1.78778	−0.893889	+	0.448288i	$0.852034\pi$
$240$	15.7699	1.01794
$241$	−26.8790	−1.73143	−0.865716	−	0.500536i	$-0.833136\pi$
$241$	−26.8790	−1.73143	−0.865716	+	0.500536i	$0.833136\pi$
$242$	−23.1053	−1.48527
$243$	−1.00000	−0.0641500
$244$	−58.0891	−3.71878
$245$	6.38656	0.408022
$246$	20.5526	1.31039
$247$	−7.83137	−0.498298
$248$	−9.77954	−0.621002
$249$	14.8861	0.943369
$250$	2.74907	0.173867
$251$	11.9413	0.753731	0.376866	−	0.926268i	$-0.377002\pi$
$251$	11.9413	0.753731	0.376866	+	0.926268i	$0.377002\pi$
$252$	−4.35269	−0.274194
$253$	−9.33871	−0.587120
$254$	−14.1250	−0.886278
$255$	−4.56709	−0.286002
$256$	57.4101	3.58813
$257$	−9.88884	−0.616849	−0.308424	−	0.951249i	$-0.599802\pi$
$257$	−9.88884	−0.616849	−0.308424	+	0.951249i	$0.599802\pi$
$258$	24.0286	1.49595
$259$	1.98676	0.123451
$260$	5.55740	0.344655
$261$	0.715668	0.0442987
$262$	25.0538	1.54783
$263$	−10.1386	−0.625174	−0.312587	−	0.949889i	$-0.601196\pi$
$263$	−10.1386	−0.625174	−0.312587	+	0.949889i	$0.601196\pi$
$264$	−43.0797	−2.65137
$265$	−4.11647	−0.252873
$266$	16.8621	1.03388
$267$	−2.18878	−0.133951
$268$	−64.6843	−3.95122
$269$	10.7359	0.654578	0.327289	−	0.944924i	$-0.393865\pi$
$269$	10.7359	0.654578	0.327289	+	0.944924i	$0.393865\pi$
$270$	−2.74907	−0.167303
$271$	1.41918	0.0862088	0.0431044	−	0.999071i	$-0.486275\pi$
$271$	1.41918	0.0862088	0.0431044	+	0.999071i	$0.486275\pi$
$272$	−72.0224	−4.36700
$273$	−0.783225	−0.0474029
$274$	51.4821	3.11015
$275$	−4.40508	−0.265637
$276$	−11.7816	−0.709169
$277$	3.22701	0.193892	0.0969462	−	0.995290i	$-0.469093\pi$
$277$	3.22701	0.193892	0.0969462	+	0.995290i	$0.469093\pi$
$278$	−18.6682	−1.11964
$279$	1.00000	0.0598684
$280$	−7.65958	−0.457748
$281$	−17.8035	−1.06207	−0.531035	−	0.847350i	$-0.678197\pi$
$281$	−17.8035	−1.06207	−0.531035	+	0.847350i	$0.678197\pi$
$282$	−2.67509	−0.159299
$283$	17.0270	1.01215	0.506075	−	0.862490i	$-0.331096\pi$
$283$	17.0270	1.01215	0.506075	+	0.862490i	$0.331096\pi$
$284$	−13.5636	−0.804850
$285$	7.83137	0.463891
$286$	−12.1099	−0.716073
$287$	−5.85554	−0.345642
$288$	−23.7934	−1.40204
$289$	3.85828	0.226958
$290$	1.96742	0.115531
$291$	8.60810	0.504616
$292$	−28.4710	−1.66614
$293$	16.9324	0.989200	0.494600	−	0.869121i	$-0.335315\pi$
$293$	16.9324	0.989200	0.494600	+	0.869121i	$0.335315\pi$
$294$	−17.5571	−1.02395
$295$	2.81784	0.164061
$296$	24.8072	1.44189
$297$	4.40508	0.255609
$298$	−3.65071	−0.211480
$299$	−2.11998	−0.122602
$300$	−5.55740	−0.320857
$301$	−6.84586	−0.394589
$302$	−20.4300	−1.17562
$303$	−17.2377	−0.990281
$304$	123.500	7.08320
$305$	10.4526	0.598513
$306$	12.5553	0.717736
$307$	3.43382	0.195979	0.0979893	−	0.995187i	$-0.468759\pi$
$307$	3.43382	0.195979	0.0979893	+	0.995187i	$0.468759\pi$
$308$	19.1740	1.09254
$309$	−2.36029	−0.134272
$310$	2.74907	0.156137
$311$	18.8217	1.06728	0.533642	−	0.845711i	$-0.320823\pi$
$311$	18.8217	1.06728	0.533642	+	0.845711i	$0.320823\pi$
$312$	−9.77954	−0.553658
$313$	13.4983	0.762971	0.381486	−	0.924375i	$-0.375413\pi$
$313$	13.4983	0.762971	0.381486	+	0.924375i	$0.375413\pi$
$314$	5.18113	0.292388
$315$	0.783225	0.0441297
$316$	−90.8122	−5.10859
$317$	−2.29691	−0.129008	−0.0645038	−	0.997917i	$-0.520546\pi$
$317$	−2.29691	−0.129008	−0.0645038	+	0.997917i	$0.520546\pi$
$318$	11.3165	0.634596
$319$	−3.15258	−0.176510
$320$	−33.8701	−1.89340
$321$	3.04004	0.169678
$322$	4.56463	0.254377
$323$	−35.7666	−1.99010
$324$	5.55740	0.308744
$325$	−1.00000	−0.0554700
$326$	36.0759	1.99806
$327$	6.18979	0.342296
$328$	−73.1138	−4.03703
$329$	0.762147	0.0420185
$330$	12.1099	0.666628
$331$	25.8781	1.42239	0.711194	−	0.702996i	$-0.248155\pi$
$331$	25.8781	1.42239	0.711194	+	0.702996i	$0.248155\pi$
$332$	−82.7281	−4.54029
$333$	−2.53664	−0.139007
$334$	−6.38363	−0.349297
$335$	11.6393	0.635923
$336$	12.3514	0.673822
$337$	18.6957	1.01842	0.509210	−	0.860642i	$-0.329938\pi$
$337$	18.6957	1.01842	0.509210	+	0.860642i	$0.329938\pi$
$338$	−2.74907	−0.149530
$339$	−18.6617	−1.01357
$340$	25.3811	1.37648
$341$	−4.40508	−0.238549
$342$	−21.5290	−1.16416
$343$	10.4847	0.566120
$344$	−85.4792	−4.60873
$345$	2.11998	0.114136
$346$	19.5620	1.05166
$347$	35.7150	1.91728	0.958642	−	0.284615i	$-0.0918657\pi$
$347$	35.7150	1.91728	0.958642	+	0.284615i	$0.0918657\pi$
$348$	−3.97725	−0.213203
$349$	8.74377	0.468043	0.234022	−	0.972231i	$-0.424811\pi$
$349$	8.74377	0.468043	0.234022	+	0.972231i	$0.424811\pi$
$350$	2.15314	0.115090
$351$	1.00000	0.0533761
$352$	104.812	5.58651
$353$	21.2069	1.12873	0.564364	−	0.825526i	$-0.309121\pi$
$353$	21.2069	1.12873	0.564364	+	0.825526i	$0.309121\pi$
$354$	−7.74646	−0.411720
$355$	2.44064	0.129535
$356$	12.1639	0.644687
$357$	−3.57706	−0.189318
$358$	1.91017	0.100955
$359$	−17.4422	−0.920563	−0.460281	−	0.887773i	$-0.652251\pi$
$359$	−17.4422	−0.920563	−0.460281	+	0.887773i	$0.652251\pi$
$360$	9.77954	0.515427
$361$	42.3304	2.22792
$362$	−56.0302	−2.94488
$363$	−8.40477	−0.441136
$364$	4.35269	0.228143
$365$	5.12309	0.268155
$366$	−28.7349	−1.50200
$367$	−17.9394	−0.936430	−0.468215	−	0.883614i	$-0.655103\pi$
$367$	−17.9394	−0.936430	−0.468215	+	0.883614i	$0.655103\pi$
$368$	33.4319	1.74276
$369$	7.47619	0.389195
$370$	−6.97341	−0.362530
$371$	−3.22412	−0.167388
$372$	−5.55740	−0.288138
$373$	−3.73797	−0.193545	−0.0967724	−	0.995307i	$-0.530852\pi$
$373$	−3.73797	−0.193545	−0.0967724	+	0.995307i	$0.530852\pi$
$374$	−55.3070	−2.85985
$375$	1.00000	0.0516398
$376$	9.51636	0.490769
$377$	−0.715668	−0.0368588
$378$	−2.15314	−0.110746
$379$	16.2701	0.835741	0.417871	−	0.908507i	$-0.362777\pi$
$379$	16.2701	0.835741	0.417871	+	0.908507i	$0.362777\pi$
$380$	−43.5221	−2.23263
$381$	−5.13808	−0.263232
$382$	−24.6581	−1.26162
$383$	−14.4400	−0.737849	−0.368924	−	0.929459i	$-0.620274\pi$
$383$	−14.4400	−0.737849	−0.368924	+	0.929459i	$0.620274\pi$
$384$	45.5246	2.32317
$385$	−3.45017	−0.175837
$386$	−50.7972	−2.58551
$387$	8.74061	0.444310
$388$	−47.8386	−2.42864
$389$	−1.71200	−0.0868019	−0.0434010	−	0.999058i	$-0.513819\pi$
$389$	−1.71200	−0.0868019	−0.0434010	+	0.999058i	$0.513819\pi$
$390$	2.74907	0.139205
$391$	−9.68216	−0.489648
$392$	62.4576	3.15459
$393$	9.11354	0.459717
$394$	−42.4638	−2.13930
$395$	16.3408	0.822194
$396$	−24.4808	−1.23021
$397$	−38.1499	−1.91469	−0.957345	−	0.288949i	$-0.906694\pi$
$397$	−38.1499	−1.91469	−0.957345	+	0.288949i	$0.906694\pi$
$398$	9.73864	0.488154
$399$	6.13373	0.307070
$400$	15.7699	0.788494
$401$	−11.9188	−0.595196	−0.297598	−	0.954691i	$-0.596186\pi$
$401$	−11.9188	−0.595196	−0.297598	+	0.954691i	$0.596186\pi$
$402$	−31.9973	−1.59588
$403$	−1.00000	−0.0498135
$404$	95.7969	4.76607
$405$	−1.00000	−0.0496904
$406$	1.54093	0.0764753
$407$	11.1741	0.553880
$408$	−44.6640	−2.21120
$409$	−28.7203	−1.42013	−0.710064	−	0.704137i	$-0.751334\pi$
$409$	−28.7203	−1.42013	−0.710064	+	0.704137i	$0.751334\pi$
$410$	20.5526	1.01502
$411$	18.7271	0.923739
$412$	13.1171	0.646233
$413$	2.20701	0.108600
$414$	−5.82799	−0.286430
$415$	14.8861	0.730731
$416$	23.7934	1.16657
$417$	−6.79073	−0.332543
$418$	94.8371	4.63864
$419$	5.65750	0.276387	0.138194	−	0.990405i	$-0.455870\pi$
$419$	5.65750	0.276387	0.138194	+	0.990405i	$0.455870\pi$
$420$	−4.35269	−0.212390
$421$	−2.90000	−0.141337	−0.0706687	−	0.997500i	$-0.522513\pi$
$421$	−2.90000	−0.141337	−0.0706687	+	0.997500i	$0.522513\pi$
$422$	34.4417	1.67660
$423$	−0.973088	−0.0473132
$424$	−40.2572	−1.95506
$425$	−4.56709	−0.221536
$426$	−6.70948	−0.325075
$427$	8.18672	0.396183
$428$	−16.8947	−0.816636
$429$	−4.40508	−0.212680
$430$	24.0286	1.15876
$431$	−2.27765	−0.109711	−0.0548553	−	0.998494i	$-0.517470\pi$
$431$	−2.27765	−0.109711	−0.0548553	+	0.998494i	$0.517470\pi$
$432$	−15.7699	−0.758729
$433$	−21.3841	−1.02765	−0.513827	−	0.857894i	$-0.671772\pi$
$433$	−21.3841	−1.02765	−0.513827	+	0.857894i	$0.671772\pi$
$434$	2.15314	0.103354
$435$	0.715668	0.0343136
$436$	−34.3991	−1.64742
$437$	16.6024	0.794200
$438$	−14.0837	−0.672947
$439$	23.2650	1.11038	0.555190	−	0.831724i	$-0.312646\pi$
$439$	23.2650	1.11038	0.555190	+	0.831724i	$0.312646\pi$
$440$	−43.0797	−2.05375
$441$	−6.38656	−0.304122
$442$	−12.5553	−0.597192
$443$	−8.51236	−0.404435	−0.202217	−	0.979341i	$-0.564815\pi$
$443$	−8.51236	−0.404435	−0.202217	+	0.979341i	$0.564815\pi$
$444$	14.0971	0.669020
$445$	−2.18878	−0.103758
$446$	−27.8386	−1.31820
$447$	−1.32798	−0.0628113
$448$	−26.5279	−1.25333
$449$	13.5192	0.638009	0.319005	−	0.947753i	$-0.396651\pi$
$449$	13.5192	0.638009	0.319005	+	0.947753i	$0.396651\pi$
$450$	−2.74907	−0.129593
$451$	−32.9333	−1.55077
$452$	103.711	4.87814
$453$	−7.43162	−0.349168
$454$	70.6374	3.31518
$455$	−0.783225	−0.0367182
$456$	76.5873	3.58653
$457$	6.34661	0.296882	0.148441	−	0.988921i	$-0.452575\pi$
$457$	6.34661	0.296882	0.148441	+	0.988921i	$0.452575\pi$
$458$	16.1558	0.754910
$459$	4.56709	0.213173
$460$	−11.7816	−0.549320
$461$	−32.2147	−1.50039	−0.750194	−	0.661218i	$-0.770040\pi$
$461$	−32.2147	−1.50039	−0.750194	+	0.661218i	$0.770040\pi$
$462$	9.48477	0.441272
$463$	22.1389	1.02888	0.514440	−	0.857526i	$-0.328000\pi$
$463$	22.1389	1.02888	0.514440	+	0.857526i	$0.328000\pi$
$464$	11.2860	0.523939
$465$	1.00000	0.0463739
$466$	−32.9645	−1.52705
$467$	22.3377	1.03366	0.516832	−	0.856087i	$-0.327111\pi$
$467$	22.3377	1.03366	0.516832	+	0.856087i	$0.327111\pi$
$468$	−5.55740	−0.256891
$469$	9.11620	0.420947
$470$	−2.67509	−0.123393
$471$	1.88468	0.0868417
$472$	27.5572	1.26842
$473$	−38.5031	−1.77038
$474$	−44.9220	−2.06334
$475$	7.83137	0.359328
$476$	19.8791	0.911158
$477$	4.11647	0.188480
$478$	75.9799	3.47524
$479$	39.7514	1.81629	0.908143	−	0.418660i	$-0.137500\pi$
$479$	39.7514	1.81629	0.908143	+	0.418660i	$0.137500\pi$
$480$	−23.7934	−1.08602
$481$	2.53664	0.115661
$482$	73.8924	3.36571
$483$	1.66043	0.0755520
$484$	46.7087	2.12312
$485$	8.60810	0.390874
$486$	2.74907	0.124700
$487$	−16.7459	−0.758831	−0.379416	−	0.925226i	$-0.623875\pi$
$487$	−16.7459	−0.758831	−0.379416	+	0.925226i	$0.623875\pi$
$488$	102.221	4.62735
$489$	13.1229	0.593439
$490$	−17.5571	−0.793149
$491$	−22.1498	−0.999609	−0.499804	−	0.866138i	$-0.666595\pi$
$491$	−22.1498	−0.999609	−0.499804	+	0.866138i	$0.666595\pi$
$492$	−41.5482	−1.87314
$493$	−3.26852	−0.147207
$494$	21.5290	0.968636
$495$	4.40508	0.197994
$496$	15.7699	0.708088
$497$	1.91157	0.0857455
$498$	−40.9230	−1.83380
$499$	27.4194	1.22746	0.613731	−	0.789515i	$-0.289668\pi$
$499$	27.4194	1.22746	0.613731	+	0.789515i	$0.289668\pi$
$500$	−5.55740	−0.248534
$501$	−2.32211	−0.103744
$502$	−32.8276	−1.46517
$503$	24.5265	1.09358	0.546792	−	0.837269i	$-0.315849\pi$
$503$	24.5265	1.09358	0.546792	+	0.837269i	$0.315849\pi$
$504$	7.65958	0.341185
$505$	−17.2377	−0.767068
$506$	25.6728	1.14130
$507$	−1.00000	−0.0444116
$508$	28.5544	1.26689
$509$	26.3967	1.17001	0.585007	−	0.811029i	$-0.301092\pi$
$509$	26.3967	1.17001	0.585007	+	0.811029i	$0.301092\pi$
$510$	12.5553	0.555956
$511$	4.01253	0.177504
$512$	−66.7753	−2.95108
$513$	−7.83137	−0.345764
$514$	27.1851	1.19908
$515$	−2.36029	−0.104007
$516$	−48.5750	−2.13840
$517$	4.28654	0.188522
$518$	−5.46175	−0.239975
$519$	7.11584	0.312351
$520$	−9.77954	−0.428861
$521$	6.84316	0.299804	0.149902	−	0.988701i	$-0.452104\pi$
$521$	6.84316	0.299804	0.149902	+	0.988701i	$0.452104\pi$
$522$	−1.96742	−0.0861117
$523$	−8.02843	−0.351059	−0.175529	−	0.984474i	$-0.556164\pi$
$523$	−8.02843	−0.351059	−0.175529	+	0.984474i	$0.556164\pi$
$524$	−50.6476	−2.21255
$525$	0.783225	0.0341827
$526$	27.8718	1.21527
$527$	−4.56709	−0.198945
$528$	69.4677	3.02319
$529$	−18.5057	−0.804594
$530$	11.3165	0.491556
$531$	−2.81784	−0.122284
$532$	−34.0876	−1.47788
$533$	−7.47619	−0.323830
$534$	6.01712	0.260386
$535$	3.04004	0.131432
$536$	113.827	4.91658
$537$	0.694840	0.0299846
$538$	−29.5137	−1.27243
$539$	28.1333	1.21179
$540$	5.55740	0.239152
$541$	−10.0587	−0.432458	−0.216229	−	0.976343i	$-0.569376\pi$
$541$	−10.0587	−0.432458	−0.216229	+	0.976343i	$0.569376\pi$
$542$	−3.90142	−0.167580
$543$	−20.3815	−0.874654
$544$	108.667	4.65905
$545$	6.18979	0.265141
$546$	2.15314	0.0921460
$547$	40.0161	1.71097	0.855483	−	0.517830i	$-0.173260\pi$
$547$	40.0161	1.71097	0.855483	+	0.517830i	$0.173260\pi$
$548$	−104.074	−4.44581
$549$	−10.4526	−0.446105
$550$	12.1099	0.516368
$551$	5.60466	0.238767
$552$	20.7325	0.882433
$553$	12.7985	0.544248
$554$	−8.87129	−0.376905
$555$	−2.53664	−0.107674
$556$	37.7388	1.60048
$557$	−40.1980	−1.70324	−0.851622	−	0.524157i	$-0.824380\pi$
$557$	−40.1980	−1.70324	−0.851622	+	0.524157i	$0.824380\pi$
$558$	−2.74907	−0.116378
$559$	−8.74061	−0.369688
$560$	12.3514	0.521940
$561$	−20.1184	−0.849400
$562$	48.9432	2.06454
$563$	9.15848	0.385984	0.192992	−	0.981200i	$-0.438181\pi$
$563$	9.15848	0.385984	0.192992	+	0.981200i	$0.438181\pi$
$564$	5.40784	0.227711
$565$	−18.6617	−0.785105
$566$	−46.8084	−1.96750
$567$	−0.783225	−0.0328924
$568$	23.8683	1.00149
$569$	−3.71553	−0.155763	−0.0778816	−	0.996963i	$-0.524816\pi$
$569$	−3.71553	−0.155763	−0.0778816	+	0.996963i	$0.524816\pi$
$570$	−21.5290	−0.901751
$571$	−3.58955	−0.150218	−0.0751089	−	0.997175i	$-0.523930\pi$
$571$	−3.58955	−0.150218	−0.0751089	+	0.997175i	$0.523930\pi$
$572$	24.4808	1.02359
$573$	−8.96960	−0.374710
$574$	16.0973	0.671889
$575$	2.11998	0.0884095
$576$	33.8701	1.41126
$577$	−24.5829	−1.02340	−0.511699	−	0.859165i	$-0.670984\pi$
$577$	−24.5829	−1.02340	−0.511699	+	0.859165i	$0.670984\pi$
$578$	−10.6067	−0.441181
$579$	−18.4779	−0.767917
$580$	−3.97725	−0.165146
$581$	11.6592	0.483704
$582$	−23.6643	−0.980916
$583$	−18.1334	−0.751009
$584$	50.1015	2.07321
$585$	1.00000	0.0413449
$586$	−46.5483	−1.92289
$587$	27.0056	1.11464	0.557319	−	0.830298i	$-0.311830\pi$
$587$	27.0056	1.11464	0.557319	+	0.830298i	$0.311830\pi$
$588$	35.4926	1.46369
$589$	7.83137	0.322686
$590$	−7.74646	−0.318917
$591$	−15.4466	−0.635389
$592$	−40.0025	−1.64409
$593$	32.3016	1.32647	0.663235	−	0.748412i	$-0.269183\pi$
$593$	32.3016	1.32647	0.663235	+	0.748412i	$0.269183\pi$
$594$	−12.1099	−0.496875
$595$	−3.57706	−0.146645
$596$	7.38011	0.302301
$597$	3.54252	0.144986
$598$	5.82799	0.238324
$599$	−5.64797	−0.230770	−0.115385	−	0.993321i	$-0.536810\pi$
$599$	−5.64797	−0.230770	−0.115385	+	0.993321i	$0.536810\pi$
$600$	9.77954	0.399248
$601$	−14.9257	−0.608832	−0.304416	−	0.952539i	$-0.598461\pi$
$601$	−14.9257	−0.608832	−0.304416	+	0.952539i	$0.598461\pi$
$602$	18.8198	0.767037
$603$	−11.6393	−0.473989
$604$	41.3005	1.68049
$605$	−8.40477	−0.341703
$606$	47.3877	1.92500
$607$	2.19744	0.0891913	0.0445956	−	0.999005i	$-0.485800\pi$
$607$	2.19744	0.0891913	0.0445956	+	0.999005i	$0.485800\pi$
$608$	−186.335	−7.55690
$609$	0.560529	0.0227138
$610$	−28.7349	−1.16344
$611$	0.973088	0.0393669
$612$	−25.3811	−1.02597
$613$	−8.67956	−0.350564	−0.175282	−	0.984518i	$-0.556084\pi$
$613$	−8.67956	−0.350564	−0.175282	+	0.984518i	$0.556084\pi$
$614$	−9.43983	−0.380961
$615$	7.47619	0.301469
$616$	−33.7411	−1.35947
$617$	33.8561	1.36299	0.681497	−	0.731821i	$-0.261329\pi$
$617$	33.8561	1.36299	0.681497	+	0.731821i	$0.261329\pi$
$618$	6.48862	0.261011
$619$	43.3865	1.74385	0.871926	−	0.489637i	$-0.162871\pi$
$619$	43.3865	1.74385	0.871926	+	0.489637i	$0.162871\pi$
$620$	−5.55740	−0.223190
$621$	−2.11998	−0.0850721
$622$	−51.7423	−2.07468
$623$	−1.71431	−0.0686823
$624$	15.7699	0.631300
$625$	1.00000	0.0400000
$626$	−37.1079	−1.48313
$627$	34.4979	1.37771
$628$	−10.4739	−0.417956
$629$	11.5851	0.461926
$630$	−2.15314	−0.0857832
$631$	34.9810	1.39257	0.696287	−	0.717764i	$-0.254834\pi$
$631$	34.9810	1.39257	0.696287	+	0.717764i	$0.254834\pi$
$632$	159.805	6.35672
$633$	12.5285	0.497963
$634$	6.31438	0.250776
$635$	−5.13808	−0.203899
$636$	−22.8769	−0.907127
$637$	6.38656	0.253045
$638$	8.66666	0.343116
$639$	−2.44064	−0.0965500
$640$	45.5246	1.79952
$641$	−43.9436	−1.73567	−0.867834	−	0.496855i	$-0.834488\pi$
$641$	−43.9436	−1.73567	−0.867834	+	0.496855i	$0.834488\pi$
$642$	−8.35728	−0.329835
$643$	7.06097	0.278457	0.139229	−	0.990260i	$-0.455538\pi$
$643$	7.06097	0.278457	0.139229	+	0.990260i	$0.455538\pi$
$644$	−9.22764	−0.363620
$645$	8.74061	0.344161
$646$	98.3249	3.86854
$647$	45.7196	1.79742	0.898712	−	0.438539i	$-0.144504\pi$
$647$	45.7196	1.79742	0.898712	+	0.438539i	$0.144504\pi$
$648$	−9.77954	−0.384177
$649$	12.4128	0.487247
$650$	2.74907	0.107827
$651$	0.783225	0.0306970
$652$	−72.9293	−2.85613
$653$	8.43983	0.330276	0.165138	−	0.986270i	$-0.447193\pi$
$653$	8.43983	0.330276	0.165138	+	0.986270i	$0.447193\pi$
$654$	−17.0162	−0.665385
$655$	9.11354	0.356096
$656$	117.899	4.60317
$657$	−5.12309	−0.199871
$658$	−2.09520	−0.0816793
$659$	18.2955	0.712691	0.356346	−	0.934354i	$-0.384023\pi$
$659$	18.2955	0.712691	0.356346	+	0.934354i	$0.384023\pi$
$660$	−24.4808	−0.952914
$661$	31.0471	1.20759	0.603795	−	0.797139i	$-0.293654\pi$
$661$	31.0471	1.20759	0.603795	+	0.797139i	$0.293654\pi$
$662$	−71.1407	−2.76496
$663$	−4.56709	−0.177371
$664$	145.579	5.64958
$665$	6.13373	0.237856
$666$	6.97341	0.270214
$667$	1.51720	0.0587464
$668$	12.9049	0.499304
$669$	−10.1265	−0.391515
$670$	−31.9973	−1.23616
$671$	46.0445	1.77753
$672$	−18.6356	−0.718885
$673$	31.6879	1.22148	0.610738	−	0.791832i	$-0.290873\pi$
$673$	31.6879	1.22148	0.610738	+	0.791832i	$0.290873\pi$
$674$	−51.3958	−1.97969
$675$	−1.00000	−0.0384900
$676$	5.55740	0.213746
$677$	−26.7996	−1.02999	−0.514996	−	0.857193i	$-0.672206\pi$
$677$	−26.7996	−1.02999	−0.514996	+	0.857193i	$0.672206\pi$
$678$	51.3024	1.97026
$679$	6.74208	0.258737
$680$	−44.6640	−1.71279
$681$	25.6950	0.984635
$682$	12.1099	0.463712
$683$	49.1603	1.88107	0.940533	−	0.339701i	$-0.110326\pi$
$683$	49.1603	1.88107	0.940533	+	0.339701i	$0.110326\pi$
$684$	43.5221	1.66411
$685$	18.7271	0.715525
$686$	−28.8232	−1.10047
$687$	5.87681	0.224214
$688$	137.838	5.25504
$689$	−4.11647	−0.156825
$690$	−5.82799	−0.221868
$691$	43.1991	1.64337	0.821687	−	0.569940i	$-0.193033\pi$
$691$	43.1991	1.64337	0.821687	+	0.569940i	$0.193033\pi$
$692$	−39.5455	−1.50330
$693$	3.45017	0.131061
$694$	−98.1832	−3.72698
$695$	−6.79073	−0.257587
$696$	6.99891	0.265293
$697$	−34.1444	−1.29331
$698$	−24.0373	−0.909824
$699$	−11.9911	−0.453547
$700$	−4.35269	−0.164516
$701$	−25.5603	−0.965401	−0.482701	−	0.875785i	$-0.660344\pi$
$701$	−25.5603	−0.965401	−0.482701	+	0.875785i	$0.660344\pi$
$702$	−2.74907	−0.103757
$703$	−19.8654	−0.749237
$704$	−149.201	−5.62322
$705$	−0.973088	−0.0366486
$706$	−58.2992	−2.19412
$707$	−13.5010	−0.507758
$708$	15.6599	0.588534
$709$	44.2020	1.66004	0.830021	−	0.557732i	$-0.188328\pi$
$709$	44.2020	1.66004	0.830021	+	0.557732i	$0.188328\pi$
$710$	−6.70948	−0.251802
$711$	−16.3408	−0.612827
$712$	−21.4053	−0.802197
$713$	2.11998	0.0793941
$714$	9.83359	0.368013
$715$	−4.40508	−0.164741
$716$	−3.86151	−0.144311
$717$	27.6384	1.03217
$718$	47.9498	1.78947
$719$	−2.30631	−0.0860107	−0.0430053	−	0.999075i	$-0.513693\pi$
$719$	−2.30631	−0.0860107	−0.0430053	+	0.999075i	$0.513693\pi$
$720$	−15.7699	−0.587709
$721$	−1.84864	−0.0688470
$722$	−116.369	−4.33082
$723$	26.8790	0.999642
$724$	113.268	4.20958
$725$	0.715668	0.0265792
$726$	23.1053	0.857519
$727$	−33.2691	−1.23388	−0.616942	−	0.787009i	$-0.711629\pi$
$727$	−33.2691	−1.23388	−0.616942	+	0.787009i	$0.711629\pi$
$728$	−7.65958	−0.283883
$729$	1.00000	0.0370370
$730$	−14.0837	−0.521262
$731$	−39.9191	−1.47646
$732$	58.0891	2.14704
$733$	−27.8948	−1.03032	−0.515159	−	0.857095i	$-0.672267\pi$
$733$	−27.8948	−1.03032	−0.515159	+	0.857095i	$0.672267\pi$
$734$	49.3168	1.82032
$735$	−6.38656	−0.235572
$736$	−50.4417	−1.85931
$737$	51.2721	1.88863
$738$	−20.5526	−0.756552
$739$	34.5366	1.27045	0.635225	−	0.772327i	$-0.280907\pi$
$739$	34.5366	1.27045	0.635225	+	0.772327i	$0.280907\pi$
$740$	14.0971	0.518220
$741$	7.83137	0.287693
$742$	8.86335	0.325384
$743$	36.2758	1.33083	0.665416	−	0.746473i	$-0.268254\pi$
$743$	36.2758	1.33083	0.665416	+	0.746473i	$0.268254\pi$
$744$	9.77954	0.358536
$745$	−1.32798	−0.0486534
$746$	10.2760	0.376229
$747$	−14.8861	−0.544655
$748$	111.806	4.08803
$749$	2.38103	0.0870010
$750$	−2.74907	−0.100382
$751$	25.3737	0.925900	0.462950	−	0.886384i	$-0.346791\pi$
$751$	25.3737	0.925900	0.462950	+	0.886384i	$0.346791\pi$
$752$	−15.3455	−0.559592
$753$	−11.9413	−0.435167
$754$	1.96742	0.0716493
$755$	−7.43162	−0.270464
$756$	4.35269	0.158306
$757$	−25.0811	−0.911589	−0.455795	−	0.890085i	$-0.650645\pi$
$757$	−25.0811	−0.911589	−0.455795	+	0.890085i	$0.650645\pi$
$758$	−44.7278	−1.62459
$759$	9.33871	0.338974
$760$	76.5873	2.77811
$761$	10.1978	0.369670	0.184835	−	0.982770i	$-0.440825\pi$
$761$	10.1978	0.369670	0.184835	+	0.982770i	$0.440825\pi$
$762$	14.1250	0.511693
$763$	4.84800	0.175509
$764$	49.8476	1.80342
$765$	4.56709	0.165123
$766$	39.6966	1.43430
$767$	2.81784	0.101746
$768$	−57.4101	−2.07161
$769$	28.4991	1.02770	0.513851	−	0.857879i	$-0.328218\pi$
$769$	28.4991	1.02770	0.513851	+	0.857879i	$0.328218\pi$
$770$	9.48477	0.341808
$771$	9.88884	0.356138
$772$	102.689	3.69587
$773$	−10.6415	−0.382747	−0.191374	−	0.981517i	$-0.561294\pi$
$773$	−10.6415	−0.382747	−0.191374	+	0.981517i	$0.561294\pi$
$774$	−24.0286	−0.863689
$775$	1.00000	0.0359211
$776$	84.1833	3.02200
$777$	−1.98676	−0.0712746
$778$	4.70642	0.168733
$779$	58.5489	2.09773
$780$	−5.55740	−0.198987
$781$	10.7512	0.384708
$782$	26.6169	0.951820
$783$	−0.715668	−0.0255759
$784$	−100.715	−3.59697
$785$	1.88468	0.0672673
$786$	−25.0538	−0.893639
$787$	18.0455	0.643254	0.321627	−	0.946866i	$-0.395770\pi$
$787$	18.0455	0.643254	0.321627	+	0.946866i	$0.395770\pi$
$788$	85.8430	3.05803
$789$	10.1386	0.360945
$790$	−44.9220	−1.59825
$791$	−14.6163	−0.519697
$792$	43.0797	1.53077
$793$	10.4526	0.371182
$794$	104.877	3.72194
$795$	4.11647	0.145996
$796$	−19.6872	−0.697794
$797$	5.01520	0.177647	0.0888237	−	0.996047i	$-0.471689\pi$
$797$	5.01520	0.177647	0.0888237	+	0.996047i	$0.471689\pi$
$798$	−16.8621	−0.596910
$799$	4.44418	0.157224
$800$	−23.7934	−0.841225
$801$	2.18878	0.0773368
$802$	32.7656	1.15700
$803$	22.5676	0.796394
$804$	64.6843	2.28124
$805$	1.66043	0.0585223
$806$	2.74907	0.0968319
$807$	−10.7359	−0.377921
$808$	−168.577	−5.93052
$809$	5.20690	0.183065	0.0915324	−	0.995802i	$-0.470824\pi$
$809$	5.20690	0.183065	0.0915324	+	0.995802i	$0.470824\pi$
$810$	2.74907	0.0965926
$811$	−16.3444	−0.573930	−0.286965	−	0.957941i	$-0.592646\pi$
$811$	−16.3444	−0.573930	−0.286965	+	0.957941i	$0.592646\pi$
$812$	−3.11508	−0.109318
$813$	−1.41918	−0.0497727
$814$	−30.7184	−1.07668
$815$	13.1229	0.459676
$816$	72.0224	2.52129
$817$	68.4510	2.39480
$818$	78.9542	2.76057
$819$	0.783225	0.0273681
$820$	−41.5482	−1.45093
$821$	−47.9708	−1.67419	−0.837097	−	0.547055i	$-0.815749\pi$
$821$	−47.9708	−1.67419	−0.837097	+	0.547055i	$0.815749\pi$
$822$	−51.4821	−1.79564
$823$	−19.8270	−0.691125	−0.345563	−	0.938396i	$-0.612312\pi$
$823$	−19.8270	−0.691125	−0.345563	+	0.938396i	$0.612312\pi$
$824$	−23.0826	−0.804121
$825$	4.40508	0.153365
$826$	−6.06722	−0.211106
$827$	20.5005	0.712872	0.356436	−	0.934320i	$-0.383992\pi$
$827$	20.5005	0.712872	0.356436	+	0.934320i	$0.383992\pi$
$828$	11.7816	0.409439
$829$	−46.2217	−1.60535	−0.802673	−	0.596420i	$-0.796589\pi$
$829$	−46.2217	−1.60535	−0.802673	+	0.596420i	$0.796589\pi$
$830$	−40.9230	−1.42046
$831$	−3.22701	−0.111944
$832$	−33.8701	−1.17424
$833$	29.1680	1.01061
$834$	18.6682	0.646427
$835$	−2.32211	−0.0803598
$836$	−191.718	−6.63072
$837$	−1.00000	−0.0345651
$838$	−15.5529	−0.537265
$839$	−2.61497	−0.0902786	−0.0451393	−	0.998981i	$-0.514373\pi$
$839$	−2.61497	−0.0902786	−0.0451393	+	0.998981i	$0.514373\pi$
$840$	7.65958	0.264281
$841$	−28.4878	−0.982339
$842$	7.97231	0.274744
$843$	17.8035	0.613186
$844$	−69.6258	−2.39662
$845$	−1.00000	−0.0344010
$846$	2.67509	0.0919715
$847$	−6.58283	−0.226189
$848$	64.9162	2.22923
$849$	−17.0270	−0.584365
$850$	12.5553	0.430642
$851$	−5.37764	−0.184343
$852$	13.5636	0.464681
$853$	−31.4586	−1.07712	−0.538562	−	0.842586i	$-0.681032\pi$
$853$	−31.4586	−1.07712	−0.538562	+	0.842586i	$0.681032\pi$
$854$	−22.5059	−0.770136
$855$	−7.83137	−0.267827
$856$	29.7302	1.01616
$857$	38.0737	1.30057	0.650286	−	0.759690i	$-0.274649\pi$
$857$	38.0737	1.30057	0.650286	+	0.759690i	$0.274649\pi$
$858$	12.1099	0.413425
$859$	14.2137	0.484965	0.242483	−	0.970156i	$-0.422038\pi$
$859$	14.2137	0.484965	0.242483	+	0.970156i	$0.422038\pi$
$860$	−48.5750	−1.65640
$861$	5.85554	0.199556
$862$	6.26143	0.213265
$863$	29.3108	0.997752	0.498876	−	0.866673i	$-0.333746\pi$
$863$	29.3108	0.997752	0.498876	+	0.866673i	$0.333746\pi$
$864$	23.7934	0.809470
$865$	7.11584	0.241946
$866$	58.7864	1.99764
$867$	−3.85828	−0.131034
$868$	−4.35269	−0.147740
$869$	71.9825	2.44184
$870$	−1.96742	−0.0667019
$871$	11.6393	0.394383
$872$	60.5333	2.04992
$873$	−8.60810	−0.291340
$874$	−45.6412	−1.54384
$875$	0.783225	0.0264778
$876$	28.4710	0.961947
$877$	41.4866	1.40090	0.700451	−	0.713701i	$-0.252982\pi$
$877$	41.4866	1.40090	0.700451	+	0.713701i	$0.252982\pi$
$878$	−63.9572	−2.15845
$879$	−16.9324	−0.571115
$880$	69.4677	2.34175
$881$	−51.8056	−1.74537	−0.872687	−	0.488279i	$-0.837625\pi$
$881$	−51.8056	−1.74537	−0.872687	+	0.488279i	$0.837625\pi$
$882$	17.5571	0.591179
$883$	−22.6348	−0.761720	−0.380860	−	0.924633i	$-0.624372\pi$
$883$	−22.6348	−0.761720	−0.380860	+	0.924633i	$0.624372\pi$
$884$	25.3811	0.853660
$885$	−2.81784	−0.0947208
$886$	23.4011	0.786175
$887$	44.8829	1.50702	0.753510	−	0.657437i	$-0.228359\pi$
$887$	44.8829	1.50702	0.753510	+	0.657437i	$0.228359\pi$
$888$	−24.8072	−0.832475
$889$	−4.02427	−0.134970
$890$	6.01712	0.201694
$891$	−4.40508	−0.147576
$892$	56.2772	1.88430
$893$	−7.62062	−0.255014
$894$	3.65071	0.122098
$895$	0.694840	0.0232260
$896$	35.6560	1.19118
$897$	2.11998	0.0707842
$898$	−37.1652	−1.24022
$899$	0.715668	0.0238689
$900$	5.55740	0.185247
$901$	−18.8003	−0.626328
$902$	90.5359	3.01452
$903$	6.84586	0.227816
$904$	−182.503	−6.06997
$905$	−20.3815	−0.677504
$906$	20.4300	0.678743
$907$	−16.3814	−0.543935	−0.271968	−	0.962306i	$-0.587674\pi$
$907$	−16.3814	−0.543935	−0.271968	+	0.962306i	$0.587674\pi$
$908$	−142.797	−4.73890
$909$	17.2377	0.571739
$910$	2.15314	0.0713760
$911$	18.5842	0.615723	0.307861	−	0.951431i	$-0.400387\pi$
$911$	18.5842	0.615723	0.307861	+	0.951431i	$0.400387\pi$
$912$	−123.500	−4.08949
$913$	65.5746	2.17020
$914$	−17.4473	−0.577105
$915$	−10.4526	−0.345552
$916$	−32.6598	−1.07911
$917$	7.13796	0.235716
$918$	−12.5553	−0.414385
$919$	−12.2008	−0.402467	−0.201234	−	0.979543i	$-0.564495\pi$
$919$	−12.2008	−0.402467	−0.201234	+	0.979543i	$0.564495\pi$
$920$	20.7325	0.683530
$921$	−3.43382	−0.113148
$922$	88.5605	2.91659
$923$	2.44064	0.0803345
$924$	−19.1740	−0.630778
$925$	−2.53664	−0.0834042
$926$	−60.8613	−2.00003
$927$	2.36029	0.0775222
$928$	−17.0282	−0.558978
$929$	39.0952	1.28267	0.641336	−	0.767260i	$-0.278380\pi$
$929$	39.0952	1.28267	0.641336	+	0.767260i	$0.278380\pi$
$930$	−2.74907	−0.0901456
$931$	−50.0155	−1.63919
$932$	66.6395	2.18285
$933$	−18.8217	−0.616196
$934$	−61.4079	−2.00933
$935$	−20.1184	−0.657942
$936$	9.77954	0.319654
$937$	8.15066	0.266271	0.133135	−	0.991098i	$-0.457496\pi$
$937$	8.15066	0.266271	0.133135	+	0.991098i	$0.457496\pi$
$938$	−25.0611	−0.818273
$939$	−13.4983	−0.440502
$940$	5.40784	0.176384
$941$	25.0877	0.817835	0.408918	−	0.912571i	$-0.365906\pi$
$941$	25.0877	0.817835	0.408918	+	0.912571i	$0.365906\pi$
$942$	−5.18113	−0.168810
$943$	15.8494	0.516128
$944$	−44.4371	−1.44630
$945$	−0.783225	−0.0254783
$946$	105.848	3.44141
$947$	38.8951	1.26392	0.631961	−	0.775000i	$-0.282250\pi$
$947$	38.8951	1.26392	0.631961	+	0.775000i	$0.282250\pi$
$948$	90.8122	2.94944
$949$	5.12309	0.166302
$950$	−21.5290	−0.698493
$951$	2.29691	0.0744825
$952$	−34.9820	−1.13377
$953$	−23.4256	−0.758830	−0.379415	−	0.925227i	$-0.623875\pi$
$953$	−23.4256	−0.758830	−0.379415	+	0.925227i	$0.623875\pi$
$954$	−11.3165	−0.366384
$955$	−8.96960	−0.290249
$956$	−153.598	−4.96770
$957$	3.15258	0.101908
$958$	−109.279	−3.53066
$959$	14.6675	0.473639
$960$	33.8701	1.09315
$961$	1.00000	0.0322581
$962$	−6.97341	−0.224832
$963$	−3.04004	−0.0979638
$964$	−149.378	−4.81113
$965$	−18.4779	−0.594826
$966$	−4.56463	−0.146865
$967$	20.7783	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$967$	20.7783	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$968$	−82.1949	−2.64184
$969$	35.7666	1.14899
$970$	−23.6643	−0.759814
$971$	−25.9403	−0.832463	−0.416231	−	0.909259i	$-0.636649\pi$
$971$	−25.9403	−0.832463	−0.416231	+	0.909259i	$0.636649\pi$
$972$	−5.55740	−0.178254
$973$	−5.31867	−0.170509
$974$	46.0358	1.47508
$975$	1.00000	0.0320256
$976$	−164.836	−5.27627
$977$	−16.7029	−0.534373	−0.267186	−	0.963645i	$-0.586094\pi$
$977$	−16.7029	−0.534373	−0.267186	+	0.963645i	$0.586094\pi$
$978$	−36.0759	−1.15358
$979$	−9.64177	−0.308152
$980$	35.4926	1.13377
$981$	−6.18979	−0.197625
$982$	60.8915	1.94313
$983$	32.7290	1.04389	0.521947	−	0.852978i	$-0.325206\pi$
$983$	32.7290	1.04389	0.521947	+	0.852978i	$0.325206\pi$
$984$	73.1138	2.33078
$985$	−15.4466	−0.492170
$986$	8.98539	0.286153
$987$	−0.762147	−0.0242594
$988$	−43.5221	−1.38462
$989$	18.5300	0.589218
$990$	−12.1099	−0.384878
$991$	−18.9894	−0.603218	−0.301609	−	0.953432i	$-0.597524\pi$
$991$	−18.9894	−0.603218	−0.301609	+	0.953432i	$0.597524\pi$
$992$	−23.7934	−0.755443
$993$	−25.8781	−0.821216
$994$	−5.25503	−0.166680
$995$	3.54252	0.112305
$996$	82.7281	2.62134
$997$	−33.0138	−1.04556	−0.522778	−	0.852469i	$-0.675105\pi$
$997$	−33.0138	−1.04556	−0.522778	+	0.852469i	$0.675105\pi$
$998$	−75.3780	−2.38605
$999$	2.53664	0.0802557

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bg.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-2.74907 q^{2} -1.00000 q^{3} +5.55740 q^{4} -1.00000 q^{5} +2.74907 q^{6} -0.783225 q^{7} -9.77954 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.74907 q^{2} -1.00000 q^{3} +5.55740 q^{4} -1.00000 q^{5} +2.74907 q^{6} -0.783225 q^{7} -9.77954 q^{8} +1.00000 q^{9} +2.74907 q^{10} -4.40508 q^{11} -5.55740 q^{12} -1.00000 q^{13} +2.15314 q^{14} +1.00000 q^{15} +15.7699 q^{16} -4.56709 q^{17} -2.74907 q^{18} +7.83137 q^{19} -5.55740 q^{20} +0.783225 q^{21} +12.1099 q^{22} +2.11998 q^{23} +9.77954 q^{24} +1.00000 q^{25} +2.74907 q^{26} -1.00000 q^{27} -4.35269 q^{28} +0.715668 q^{29} -2.74907 q^{30} +1.00000 q^{31} -23.7934 q^{32} +4.40508 q^{33} +12.5553 q^{34} +0.783225 q^{35} +5.55740 q^{36} -2.53664 q^{37} -21.5290 q^{38} +1.00000 q^{39} +9.77954 q^{40} +7.47619 q^{41} -2.15314 q^{42} +8.74061 q^{43} -24.4808 q^{44} -1.00000 q^{45} -5.82799 q^{46} -0.973088 q^{47} -15.7699 q^{48} -6.38656 q^{49} -2.74907 q^{50} +4.56709 q^{51} -5.55740 q^{52} +4.11647 q^{53} +2.74907 q^{54} +4.40508 q^{55} +7.65958 q^{56} -7.83137 q^{57} -1.96742 q^{58} -2.81784 q^{59} +5.55740 q^{60} -10.4526 q^{61} -2.74907 q^{62} -0.783225 q^{63} +33.8701 q^{64} +1.00000 q^{65} -12.1099 q^{66} -11.6393 q^{67} -25.3811 q^{68} -2.11998 q^{69} -2.15314 q^{70} -2.44064 q^{71} -9.77954 q^{72} -5.12309 q^{73} +6.97341 q^{74} -1.00000 q^{75} +43.5221 q^{76} +3.45017 q^{77} -2.74907 q^{78} -16.3408 q^{79} -15.7699 q^{80} +1.00000 q^{81} -20.5526 q^{82} -14.8861 q^{83} +4.35269 q^{84} +4.56709 q^{85} -24.0286 q^{86} -0.715668 q^{87} +43.0797 q^{88} +2.18878 q^{89} +2.74907 q^{90} +0.783225 q^{91} +11.7816 q^{92} -1.00000 q^{93} +2.67509 q^{94} -7.83137 q^{95} +23.7934 q^{96} -8.60810 q^{97} +17.5571 q^{98} -4.40508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) 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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