LMFDB - Embedded newform 4730.2.a.w.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4730,2,Mod(1,4730)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4730, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4730.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4730 = 2 \cdot 5 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4730.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:	$37.7692401561$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 12x^{6} + 7x^{5} + 41x^{4} - 6x^{3} - 28x^{2} + 2x + 4 $ x^8 - x^7 - 12x^6 + 7x^5 + 41x^4 - 6x^3 - 28x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.484643$ of defining polynomial
Character	$\chi$	$=$	4730.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.515357 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.515357 q^{6} -4.64210 q^{7} -1.00000 q^{8} -2.73441 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.515357 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.515357 q^{6} -4.64210 q^{7} -1.00000 q^{8} -2.73441 q^{9} -1.00000 q^{10} +1.00000 q^{11} -0.515357 q^{12} -0.108612 q^{13} +4.64210 q^{14} -0.515357 q^{15} +1.00000 q^{16} +4.19244 q^{17} +2.73441 q^{18} -3.02164 q^{19} +1.00000 q^{20} +2.39234 q^{21} -1.00000 q^{22} -2.10861 q^{23} +0.515357 q^{24} +1.00000 q^{25} +0.108612 q^{26} +2.95527 q^{27} -4.64210 q^{28} +8.06684 q^{29} +0.515357 q^{30} +3.88930 q^{31} -1.00000 q^{32} -0.515357 q^{33} -4.19244 q^{34} -4.64210 q^{35} -2.73441 q^{36} +6.82873 q^{37} +3.02164 q^{38} +0.0559741 q^{39} -1.00000 q^{40} +2.16893 q^{41} -2.39234 q^{42} -1.00000 q^{43} +1.00000 q^{44} -2.73441 q^{45} +2.10861 q^{46} -9.78816 q^{47} -0.515357 q^{48} +14.5491 q^{49} -1.00000 q^{50} -2.16060 q^{51} -0.108612 q^{52} +1.84597 q^{53} -2.95527 q^{54} +1.00000 q^{55} +4.64210 q^{56} +1.55722 q^{57} -8.06684 q^{58} +1.63626 q^{59} -0.515357 q^{60} +8.07848 q^{61} -3.88930 q^{62} +12.6934 q^{63} +1.00000 q^{64} -0.108612 q^{65} +0.515357 q^{66} -3.43337 q^{67} +4.19244 q^{68} +1.08669 q^{69} +4.64210 q^{70} -13.9385 q^{71} +2.73441 q^{72} +7.68332 q^{73} -6.82873 q^{74} -0.515357 q^{75} -3.02164 q^{76} -4.64210 q^{77} -0.0559741 q^{78} +6.50096 q^{79} +1.00000 q^{80} +6.68021 q^{81} -2.16893 q^{82} -6.38996 q^{83} +2.39234 q^{84} +4.19244 q^{85} +1.00000 q^{86} -4.15730 q^{87} -1.00000 q^{88} -10.4488 q^{89} +2.73441 q^{90} +0.504189 q^{91} -2.10861 q^{92} -2.00438 q^{93} +9.78816 q^{94} -3.02164 q^{95} +0.515357 q^{96} -13.7821 q^{97} -14.5491 q^{98} -2.73441 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 7 q^{3} + 8 q^{4} + 8 q^{5} + 7 q^{6} - 6 q^{7} - 8 q^{8} + 7 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - 7 * q^3 + 8 * q^4 + 8 * q^5 + 7 * q^6 - 6 * q^7 - 8 * q^8 + 7 * q^9	\( 8 q - 8 q^{2} - 7 q^{3} + 8 q^{4} + 8 q^{5} + 7 q^{6} - 6 q^{7} - 8 q^{8} + 7 q^{9} - 8 q^{10} + 8 q^{11} - 7 q^{12} - 2 q^{13} + 6 q^{14} - 7 q^{15} + 8 q^{16} - 8 q^{17} - 7 q^{18} + 8 q^{20} + 14 q^{21} - 8 q^{22} - 18 q^{23} + 7 q^{24} + 8 q^{25} + 2 q^{26} - 22 q^{27} - 6 q^{28} + 8 q^{29} + 7 q^{30} - 11 q^{31} - 8 q^{32} - 7 q^{33} + 8 q^{34} - 6 q^{35} + 7 q^{36} - 17 q^{37} - 6 q^{39} - 8 q^{40} + 12 q^{41} - 14 q^{42} - 8 q^{43} + 8 q^{44} + 7 q^{45} + 18 q^{46} - 19 q^{47} - 7 q^{48} - 2 q^{49} - 8 q^{50} - q^{51} - 2 q^{52} - 7 q^{53} + 22 q^{54} + 8 q^{55} + 6 q^{56} - 3 q^{57} - 8 q^{58} + q^{59} - 7 q^{60} + 6 q^{61} + 11 q^{62} - 15 q^{63} + 8 q^{64} - 2 q^{65} + 7 q^{66} - 22 q^{67} - 8 q^{68} + 8 q^{69} + 6 q^{70} - 14 q^{71} - 7 q^{72} - 13 q^{73} + 17 q^{74} - 7 q^{75} - 6 q^{77} + 6 q^{78} - 8 q^{79} + 8 q^{80} + 28 q^{81} - 12 q^{82} - 4 q^{83} + 14 q^{84} - 8 q^{85} + 8 q^{86} - 30 q^{87} - 8 q^{88} + 5 q^{89} - 7 q^{90} - 8 q^{91} - 18 q^{92} + q^{93} + 19 q^{94} + 7 q^{96} - 23 q^{97} + 2 q^{98} + 7 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 7 * q^3 + 8 * q^4 + 8 * q^5 + 7 * q^6 - 6 * q^7 - 8 * q^8 + 7 * q^9 - 8 * q^10 + 8 * q^11 - 7 * q^12 - 2 * q^13 + 6 * q^14 - 7 * q^15 + 8 * q^16 - 8 * q^17 - 7 * q^18 + 8 * q^20 + 14 * q^21 - 8 * q^22 - 18 * q^23 + 7 * q^24 + 8 * q^25 + 2 * q^26 - 22 * q^27 - 6 * q^28 + 8 * q^29 + 7 * q^30 - 11 * q^31 - 8 * q^32 - 7 * q^33 + 8 * q^34 - 6 * q^35 + 7 * q^36 - 17 * q^37 - 6 * q^39 - 8 * q^40 + 12 * q^41 - 14 * q^42 - 8 * q^43 + 8 * q^44 + 7 * q^45 + 18 * q^46 - 19 * q^47 - 7 * q^48 - 2 * q^49 - 8 * q^50 - q^51 - 2 * q^52 - 7 * q^53 + 22 * q^54 + 8 * q^55 + 6 * q^56 - 3 * q^57 - 8 * q^58 + q^59 - 7 * q^60 + 6 * q^61 + 11 * q^62 - 15 * q^63 + 8 * q^64 - 2 * q^65 + 7 * q^66 - 22 * q^67 - 8 * q^68 + 8 * q^69 + 6 * q^70 - 14 * q^71 - 7 * q^72 - 13 * q^73 + 17 * q^74 - 7 * q^75 - 6 * q^77 + 6 * q^78 - 8 * q^79 + 8 * q^80 + 28 * q^81 - 12 * q^82 - 4 * q^83 + 14 * q^84 - 8 * q^85 + 8 * q^86 - 30 * q^87 - 8 * q^88 + 5 * q^89 - 7 * q^90 - 8 * q^91 - 18 * q^92 + q^93 + 19 * q^94 + 7 * q^96 - 23 * q^97 + 2 * q^98 + 7 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.515357	−0.297541	−0.148771	−	0.988872i	$-0.547532\pi$
$3$	−0.515357	−0.297541	−0.148771	+	0.988872i	$0.547532\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.515357	0.210394
$7$	−4.64210	−1.75455	−0.877275	−	0.479988i	$-0.840641\pi$
$7$	−4.64210	−1.75455	−0.877275	+	0.479988i	$0.840641\pi$
$8$	−1.00000	−0.353553
$9$	−2.73441	−0.911469
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.515357	−0.148771
$13$	−0.108612	−0.0301236	−0.0150618	−	0.999887i	$-0.504795\pi$
$13$	−0.108612	−0.0301236	−0.0150618	+	0.999887i	$0.504795\pi$
$14$	4.64210	1.24065
$15$	−0.515357	−0.133065
$16$	1.00000	0.250000
$17$	4.19244	1.01682	0.508409	−	0.861116i	$-0.330234\pi$
$17$	4.19244	1.01682	0.508409	+	0.861116i	$0.330234\pi$
$18$	2.73441	0.644506
$19$	−3.02164	−0.693212	−0.346606	−	0.938011i	$-0.612666\pi$
$19$	−3.02164	−0.693212	−0.346606	+	0.938011i	$0.612666\pi$
$20$	1.00000	0.223607
$21$	2.39234	0.522051
$22$	−1.00000	−0.213201
$23$	−2.10861	−0.439676	−0.219838	−	0.975536i	$-0.570553\pi$
$23$	−2.10861	−0.439676	−0.219838	+	0.975536i	$0.570553\pi$
$24$	0.515357	0.105197
$25$	1.00000	0.200000
$26$	0.108612	0.0213006
$27$	2.95527	0.568741
$28$	−4.64210	−0.877275
$29$	8.06684	1.49798	0.748988	−	0.662584i	$-0.230540\pi$
$29$	8.06684	1.49798	0.748988	+	0.662584i	$0.230540\pi$
$30$	0.515357	0.0940908
$31$	3.88930	0.698538	0.349269	−	0.937022i	$-0.386430\pi$
$31$	3.88930	0.698538	0.349269	+	0.937022i	$0.386430\pi$
$32$	−1.00000	−0.176777
$33$	−0.515357	−0.0897121
$34$	−4.19244	−0.718998
$35$	−4.64210	−0.784659
$36$	−2.73441	−0.455735
$37$	6.82873	1.12264	0.561318	−	0.827600i	$-0.310294\pi$
$37$	6.82873	1.12264	0.561318	+	0.827600i	$0.310294\pi$
$38$	3.02164	0.490175
$39$	0.0559741	0.00896302
$40$	−1.00000	−0.158114
$41$	2.16893	0.338730	0.169365	−	0.985553i	$-0.445828\pi$
$41$	2.16893	0.338730	0.169365	+	0.985553i	$0.445828\pi$
$42$	−2.39234	−0.369146
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	1.00000	0.150756
$45$	−2.73441	−0.407621
$46$	2.10861	0.310898
$47$	−9.78816	−1.42775	−0.713875	−	0.700273i	$-0.753061\pi$
$47$	−9.78816	−1.42775	−0.713875	+	0.700273i	$0.753061\pi$
$48$	−0.515357	−0.0743853
$49$	14.5491	2.07845
$50$	−1.00000	−0.141421
$51$	−2.16060	−0.302545
$52$	−0.108612	−0.0150618
$53$	1.84597	0.253563	0.126781	−	0.991931i	$-0.459535\pi$
$53$	1.84597	0.253563	0.126781	+	0.991931i	$0.459535\pi$
$54$	−2.95527	−0.402161
$55$	1.00000	0.134840
$56$	4.64210	0.620327
$57$	1.55722	0.206259
$58$	−8.06684	−1.05923
$59$	1.63626	0.213023	0.106512	−	0.994311i	$-0.466032\pi$
$59$	1.63626	0.213023	0.106512	+	0.994311i	$0.466032\pi$
$60$	−0.515357	−0.0665323
$61$	8.07848	1.03434	0.517172	−	0.855882i	$-0.326985\pi$
$61$	8.07848	1.03434	0.517172	+	0.855882i	$0.326985\pi$
$62$	−3.88930	−0.493941
$63$	12.6934	1.59922
$64$	1.00000	0.125000
$65$	−0.108612	−0.0134717
$66$	0.515357	0.0634360
$67$	−3.43337	−0.419453	−0.209726	−	0.977760i	$-0.567257\pi$
$67$	−3.43337	−0.419453	−0.209726	+	0.977760i	$0.567257\pi$
$68$	4.19244	0.508409
$69$	1.08669	0.130822
$70$	4.64210	0.554838
$71$	−13.9385	−1.65420	−0.827098	−	0.562058i	$-0.810010\pi$
$71$	−13.9385	−1.65420	−0.827098	+	0.562058i	$0.810010\pi$
$72$	2.73441	0.322253
$73$	7.68332	0.899265	0.449633	−	0.893214i	$-0.351555\pi$
$73$	7.68332	0.899265	0.449633	+	0.893214i	$0.351555\pi$
$74$	−6.82873	−0.793823
$75$	−0.515357	−0.0595083
$76$	−3.02164	−0.346606
$77$	−4.64210	−0.529017
$78$	−0.0559741	−0.00633782
$79$	6.50096	0.731415	0.365707	−	0.930730i	$-0.380827\pi$
$79$	6.50096	0.731415	0.365707	+	0.930730i	$0.380827\pi$
$80$	1.00000	0.111803
$81$	6.68021	0.742245
$82$	−2.16893	−0.239518
$83$	−6.38996	−0.701389	−0.350695	−	0.936490i	$-0.614054\pi$
$83$	−6.38996	−0.701389	−0.350695	+	0.936490i	$0.614054\pi$
$84$	2.39234	0.261026
$85$	4.19244	0.454734
$86$	1.00000	0.107833
$87$	−4.15730	−0.445710
$88$	−1.00000	−0.106600
$89$	−10.4488	−1.10757	−0.553783	−	0.832661i	$-0.686816\pi$
$89$	−10.4488	−1.10757	−0.553783	+	0.832661i	$0.686816\pi$
$90$	2.73441	0.288232
$91$	0.504189	0.0528534
$92$	−2.10861	−0.219838
$93$	−2.00438	−0.207844
$94$	9.78816	1.00957
$95$	−3.02164	−0.310014
$96$	0.515357	0.0525984
$97$	−13.7821	−1.39936	−0.699680	−	0.714456i	$-0.746674\pi$
$97$	−13.7821	−1.39936	−0.699680	+	0.714456i	$0.746674\pi$
$98$	−14.5491	−1.46968
$99$	−2.73441	−0.274818
$100$	1.00000	0.100000
$101$	−8.50862	−0.846640	−0.423320	−	0.905980i	$-0.639135\pi$
$101$	−8.50862	−0.846640	−0.423320	+	0.905980i	$0.639135\pi$
$102$	2.16060	0.213932
$103$	−7.21580	−0.710994	−0.355497	−	0.934677i	$-0.615688\pi$
$103$	−7.21580	−0.710994	−0.355497	+	0.934677i	$0.615688\pi$
$104$	0.108612	0.0106503
$105$	2.39234	0.233468
$106$	−1.84597	−0.179296
$107$	1.17431	0.113525	0.0567623	−	0.998388i	$-0.481922\pi$
$107$	1.17431	0.113525	0.0567623	+	0.998388i	$0.481922\pi$
$108$	2.95527	0.284371
$109$	−2.34821	−0.224918	−0.112459	−	0.993656i	$-0.535873\pi$
$109$	−2.34821	−0.224918	−0.112459	+	0.993656i	$0.535873\pi$
$110$	−1.00000	−0.0953463
$111$	−3.51923	−0.334031
$112$	−4.64210	−0.438638
$113$	−9.55477	−0.898837	−0.449419	−	0.893321i	$-0.648369\pi$
$113$	−9.55477	−0.898837	−0.449419	+	0.893321i	$0.648369\pi$
$114$	−1.55722	−0.145847
$115$	−2.10861	−0.196629
$116$	8.06684	0.748988
$117$	0.296990	0.0274568
$118$	−1.63626	−0.150630
$119$	−19.4618	−1.78406
$120$	0.515357	0.0470454
$121$	1.00000	0.0909091
$122$	−8.07848	−0.731392
$123$	−1.11777	−0.100786
$124$	3.88930	0.349269
$125$	1.00000	0.0894427
$126$	−12.6934	−1.13082
$127$	4.23775	0.376039	0.188020	−	0.982165i	$-0.439793\pi$
$127$	4.23775	0.376039	0.188020	+	0.982165i	$0.439793\pi$
$128$	−1.00000	−0.0883883
$129$	0.515357	0.0453746
$130$	0.108612	0.00952593
$131$	12.4238	1.08547	0.542736	−	0.839903i	$-0.317388\pi$
$131$	12.4238	1.08547	0.542736	+	0.839903i	$0.317388\pi$
$132$	−0.515357	−0.0448560
$133$	14.0268	1.21628
$134$	3.43337	0.296598
$135$	2.95527	0.254349
$136$	−4.19244	−0.359499
$137$	−19.5669	−1.67172	−0.835859	−	0.548945i	$-0.815030\pi$
$137$	−19.5669	−1.67172	−0.835859	+	0.548945i	$0.815030\pi$
$138$	−1.08669	−0.0925050
$139$	−3.28946	−0.279009	−0.139504	−	0.990221i	$-0.544551\pi$
$139$	−3.28946	−0.279009	−0.139504	+	0.990221i	$0.544551\pi$
$140$	−4.64210	−0.392329
$141$	5.04439	0.424815
$142$	13.9385	1.16969
$143$	−0.108612	−0.00908261
$144$	−2.73441	−0.227867
$145$	8.06684	0.669915
$146$	−7.68332	−0.635876
$147$	−7.49799	−0.618424
$148$	6.82873	0.561318
$149$	−11.9446	−0.978542	−0.489271	−	0.872132i	$-0.662737\pi$
$149$	−11.9446	−0.978542	−0.489271	+	0.872132i	$0.662737\pi$
$150$	0.515357	0.0420787
$151$	−5.74924	−0.467866	−0.233933	−	0.972253i	$-0.575160\pi$
$151$	−5.74924	−0.467866	−0.233933	+	0.972253i	$0.575160\pi$
$152$	3.02164	0.245088
$153$	−11.4639	−0.926797
$154$	4.64210	0.374071
$155$	3.88930	0.312396
$156$	0.0559741	0.00448151
$157$	−20.9392	−1.67113	−0.835565	−	0.549392i	$-0.814859\pi$
$157$	−20.9392	−1.67113	−0.835565	+	0.549392i	$0.814859\pi$
$158$	−6.50096	−0.517188
$159$	−0.951331	−0.0754454
$160$	−1.00000	−0.0790569
$161$	9.78840	0.771434
$162$	−6.68021	−0.524847
$163$	−2.46138	−0.192790	−0.0963949	−	0.995343i	$-0.530731\pi$
$163$	−2.46138	−0.192790	−0.0963949	+	0.995343i	$0.530731\pi$
$164$	2.16893	0.169365
$165$	−0.515357	−0.0401205
$166$	6.38996	0.495957
$167$	−15.9201	−1.23193	−0.615967	−	0.787772i	$-0.711235\pi$
$167$	−15.9201	−1.23193	−0.615967	+	0.787772i	$0.711235\pi$
$168$	−2.39234	−0.184573
$169$	−12.9882	−0.999093
$170$	−4.19244	−0.321546
$171$	8.26240	0.631842
$172$	−1.00000	−0.0762493
$173$	23.1251	1.75817	0.879083	−	0.476669i	$-0.158156\pi$
$173$	23.1251	1.75817	0.879083	+	0.476669i	$0.158156\pi$
$174$	4.15730	0.315164
$175$	−4.64210	−0.350910
$176$	1.00000	0.0753778
$177$	−0.843259	−0.0633832
$178$	10.4488	0.783168
$179$	7.74932	0.579212	0.289606	−	0.957146i	$-0.406476\pi$
$179$	7.74932	0.579212	0.289606	+	0.957146i	$0.406476\pi$
$180$	−2.73441	−0.203811
$181$	0.948206	0.0704796	0.0352398	−	0.999379i	$-0.488780\pi$
$181$	0.948206	0.0704796	0.0352398	+	0.999379i	$0.488780\pi$
$182$	−0.504189	−0.0373730
$183$	−4.16330	−0.307760
$184$	2.10861	0.155449
$185$	6.82873	0.502058
$186$	2.00438	0.146968
$187$	4.19244	0.306582
$188$	−9.78816	−0.713875
$189$	−13.7186	−0.997885
$190$	3.02164	0.219213
$191$	13.9647	1.01045	0.505224	−	0.862988i	$-0.331410\pi$
$191$	13.9647	1.01045	0.505224	+	0.862988i	$0.331410\pi$
$192$	−0.515357	−0.0371927
$193$	−17.6616	−1.27131	−0.635655	−	0.771974i	$-0.719270\pi$
$193$	−17.6616	−1.27131	−0.635655	+	0.771974i	$0.719270\pi$
$194$	13.7821	0.989497
$195$	0.0559741	0.00400839
$196$	14.5491	1.03922
$197$	−20.8737	−1.48719	−0.743594	−	0.668632i	$-0.766880\pi$
$197$	−20.8737	−1.48719	−0.743594	+	0.668632i	$0.766880\pi$
$198$	2.73441	0.194326
$199$	−24.9553	−1.76903	−0.884516	−	0.466510i	$-0.845511\pi$
$199$	−24.9553	−1.76903	−0.884516	+	0.466510i	$0.845511\pi$
$200$	−1.00000	−0.0707107
$201$	1.76941	0.124805
$202$	8.50862	0.598665
$203$	−37.4471	−2.62827
$204$	−2.16060	−0.151273
$205$	2.16893	0.151485
$206$	7.21580	0.502749
$207$	5.76581	0.400751
$208$	−0.108612	−0.00753091
$209$	−3.02164	−0.209011
$210$	−2.39234	−0.165087
$211$	0.968285	0.0666595	0.0333297	−	0.999444i	$-0.489389\pi$
$211$	0.968285	0.0666595	0.0333297	+	0.999444i	$0.489389\pi$
$212$	1.84597	0.126781
$213$	7.18330	0.492192
$214$	−1.17431	−0.0802740
$215$	−1.00000	−0.0681994
$216$	−2.95527	−0.201080
$217$	−18.0545	−1.22562
$218$	2.34821	0.159041
$219$	−3.95965	−0.267569
$220$	1.00000	0.0674200
$221$	−0.455351	−0.0306302
$222$	3.51923	0.236195
$223$	19.7291	1.32116	0.660579	−	0.750757i	$-0.270311\pi$
$223$	19.7291	1.32116	0.660579	+	0.750757i	$0.270311\pi$
$224$	4.64210	0.310164
$225$	−2.73441	−0.182294
$226$	9.55477	0.635574
$227$	−13.8242	−0.917543	−0.458772	−	0.888554i	$-0.651710\pi$
$227$	−13.8242	−0.917543	−0.458772	+	0.888554i	$0.651710\pi$
$228$	1.55722	0.103130
$229$	24.4201	1.61372	0.806862	−	0.590740i	$-0.201164\pi$
$229$	24.4201	1.61372	0.806862	+	0.590740i	$0.201164\pi$
$230$	2.10861	0.139038
$231$	2.39234	0.157404
$232$	−8.06684	−0.529614
$233$	17.3137	1.13426	0.567130	−	0.823629i	$-0.308054\pi$
$233$	17.3137	1.13426	0.567130	+	0.823629i	$0.308054\pi$
$234$	−0.296990	−0.0194149
$235$	−9.78816	−0.638509
$236$	1.63626	0.106512
$237$	−3.35031	−0.217626
$238$	19.4618	1.26152
$239$	15.6399	1.01166	0.505831	−	0.862632i	$-0.331186\pi$
$239$	15.6399	1.01166	0.505831	+	0.862632i	$0.331186\pi$
$240$	−0.515357	−0.0332661
$241$	−4.82493	−0.310801	−0.155401	−	0.987852i	$-0.549667\pi$
$241$	−4.82493	−0.310801	−0.155401	+	0.987852i	$0.549667\pi$
$242$	−1.00000	−0.0642824
$243$	−12.3085	−0.789590
$244$	8.07848	0.517172
$245$	14.5491	0.929510
$246$	1.11777	0.0712666
$247$	0.328187	0.0208821
$248$	−3.88930	−0.246971
$249$	3.29311	0.208692
$250$	−1.00000	−0.0632456
$251$	29.3009	1.84946	0.924729	−	0.380625i	$-0.124291\pi$
$251$	29.3009	1.84946	0.924729	+	0.380625i	$0.124291\pi$
$252$	12.6934	0.799609
$253$	−2.10861	−0.132567
$254$	−4.23775	−0.265900
$255$	−2.16060	−0.135302
$256$	1.00000	0.0625000
$257$	16.6058	1.03584	0.517921	−	0.855428i	$-0.326706\pi$
$257$	16.6058	1.03584	0.517921	+	0.855428i	$0.326706\pi$
$258$	−0.515357	−0.0320847
$259$	−31.6997	−1.96972
$260$	−0.108612	−0.00673585
$261$	−22.0580	−1.36536
$262$	−12.4238	−0.767545
$263$	−7.44987	−0.459379	−0.229689	−	0.973264i	$-0.573771\pi$
$263$	−7.44987	−0.459379	−0.229689	+	0.973264i	$0.573771\pi$
$264$	0.515357	0.0317180
$265$	1.84597	0.113397
$266$	−14.0268	−0.860037
$267$	5.38484	0.329547
$268$	−3.43337	−0.209726
$269$	−20.2166	−1.23263	−0.616314	−	0.787500i	$-0.711375\pi$
$269$	−20.2166	−1.23263	−0.616314	+	0.787500i	$0.711375\pi$
$270$	−2.95527	−0.179852
$271$	13.9835	0.849436	0.424718	−	0.905326i	$-0.360373\pi$
$271$	13.9835	0.849436	0.424718	+	0.905326i	$0.360373\pi$
$272$	4.19244	0.254204
$273$	−0.259837	−0.0157261
$274$	19.5669	1.18208
$275$	1.00000	0.0603023
$276$	1.08669	0.0654109
$277$	−31.3576	−1.88410	−0.942048	−	0.335477i	$-0.891102\pi$
$277$	−31.3576	−1.88410	−0.942048	+	0.335477i	$0.891102\pi$
$278$	3.28946	0.197289
$279$	−10.6349	−0.636696
$280$	4.64210	0.277419
$281$	22.2132	1.32513	0.662563	−	0.749006i	$-0.269468\pi$
$281$	22.2132	1.32513	0.662563	+	0.749006i	$0.269468\pi$
$282$	−5.04439	−0.300389
$283$	8.84763	0.525937	0.262968	−	0.964804i	$-0.415299\pi$
$283$	8.84763	0.525937	0.262968	+	0.964804i	$0.415299\pi$
$284$	−13.9385	−0.827098
$285$	1.55722	0.0922420
$286$	0.108612	0.00642238
$287$	−10.0684	−0.594319
$288$	2.73441	0.161127
$289$	0.576587	0.0339169
$290$	−8.06684	−0.473701
$291$	7.10270	0.416368
$292$	7.68332	0.449633
$293$	10.6467	0.621984	0.310992	−	0.950413i	$-0.399339\pi$
$293$	10.6467	0.621984	0.310992	+	0.950413i	$0.399339\pi$
$294$	7.49799	0.437292
$295$	1.63626	0.0952669
$296$	−6.82873	−0.396912
$297$	2.95527	0.171482
$298$	11.9446	0.691934
$299$	0.229021	0.0132446
$300$	−0.515357	−0.0297541
$301$	4.64210	0.267566
$302$	5.74924	0.330832
$303$	4.38498	0.251910
$304$	−3.02164	−0.173303
$305$	8.07848	0.462573
$306$	11.4639	0.655345
$307$	5.80506	0.331312	0.165656	−	0.986184i	$-0.447026\pi$
$307$	5.80506	0.331312	0.165656	+	0.986184i	$0.447026\pi$
$308$	−4.64210	−0.264508
$309$	3.71871	0.211550
$310$	−3.88930	−0.220897
$311$	23.6920	1.34345	0.671724	−	0.740801i	$-0.265554\pi$
$311$	23.6920	1.34345	0.671724	+	0.740801i	$0.265554\pi$
$312$	−0.0559741	−0.00316891
$313$	−8.81800	−0.498423	−0.249212	−	0.968449i	$-0.580171\pi$
$313$	−8.81800	−0.498423	−0.249212	+	0.968449i	$0.580171\pi$
$314$	20.9392	1.18167
$315$	12.6934	0.715192
$316$	6.50096	0.365707
$317$	8.44230	0.474167	0.237083	−	0.971489i	$-0.423809\pi$
$317$	8.44230	0.474167	0.237083	+	0.971489i	$0.423809\pi$
$318$	0.951331	0.0533480
$319$	8.06684	0.451657
$320$	1.00000	0.0559017
$321$	−0.605187	−0.0337783
$322$	−9.78840	−0.545486
$323$	−12.6681	−0.704870
$324$	6.68021	0.371123
$325$	−0.108612	−0.00602473
$326$	2.46138	0.136323
$327$	1.21017	0.0669224
$328$	−2.16893	−0.119759
$329$	45.4376	2.50506
$330$	0.515357	0.0283695
$331$	−29.9604	−1.64677	−0.823387	−	0.567481i	$-0.807918\pi$
$331$	−29.9604	−1.64677	−0.823387	+	0.567481i	$0.807918\pi$
$332$	−6.38996	−0.350695
$333$	−18.6725	−1.02325
$334$	15.9201	0.871109
$335$	−3.43337	−0.187585
$336$	2.39234	0.130513
$337$	23.5427	1.28245	0.641226	−	0.767352i	$-0.278426\pi$
$337$	23.5427	1.28245	0.641226	+	0.767352i	$0.278426\pi$
$338$	12.9882	0.706465
$339$	4.92412	0.267441
$340$	4.19244	0.227367
$341$	3.88930	0.210617
$342$	−8.26240	−0.446779
$343$	−35.0438	−1.89219
$344$	1.00000	0.0539164
$345$	1.08669	0.0585053
$346$	−23.1251	−1.24321
$347$	−7.25581	−0.389512	−0.194756	−	0.980852i	$-0.562392\pi$
$347$	−7.25581	−0.389512	−0.194756	+	0.980852i	$0.562392\pi$
$348$	−4.15730	−0.222855
$349$	21.1451	1.13187	0.565936	−	0.824449i	$-0.308515\pi$
$349$	21.1451	1.13187	0.565936	+	0.824449i	$0.308515\pi$
$350$	4.64210	0.248131
$351$	−0.320978	−0.0171325
$352$	−1.00000	−0.0533002
$353$	−23.2194	−1.23584	−0.617922	−	0.786239i	$-0.712025\pi$
$353$	−23.2194	−1.23584	−0.617922	+	0.786239i	$0.712025\pi$
$354$	0.843259	0.0448187
$355$	−13.9385	−0.739779
$356$	−10.4488	−0.553783
$357$	10.0297	0.530831
$358$	−7.74932	−0.409564
$359$	−9.60819	−0.507101	−0.253550	−	0.967322i	$-0.581598\pi$
$359$	−9.60819	−0.507101	−0.253550	+	0.967322i	$0.581598\pi$
$360$	2.73441	0.144116
$361$	−9.86968	−0.519457
$362$	−0.948206	−0.0498366
$363$	−0.515357	−0.0270492
$364$	0.504189	0.0264267
$365$	7.68332	0.402164
$366$	4.16330	0.217619
$367$	−10.2861	−0.536929	−0.268464	−	0.963290i	$-0.586516\pi$
$367$	−10.2861	−0.536929	−0.268464	+	0.963290i	$0.586516\pi$
$368$	−2.10861	−0.109919
$369$	−5.93074	−0.308742
$370$	−6.82873	−0.355009
$371$	−8.56916	−0.444889
$372$	−2.00438	−0.103922
$373$	16.3654	0.847368	0.423684	−	0.905810i	$-0.360737\pi$
$373$	16.3654	0.847368	0.423684	+	0.905810i	$0.360737\pi$
$374$	−4.19244	−0.216786
$375$	−0.515357	−0.0266129
$376$	9.78816	0.504786
$377$	−0.876158	−0.0451245
$378$	13.7186	0.705611
$379$	16.9114	0.868680	0.434340	−	0.900749i	$-0.356982\pi$
$379$	16.9114	0.868680	0.434340	+	0.900749i	$0.356982\pi$
$380$	−3.02164	−0.155007
$381$	−2.18395	−0.111887
$382$	−13.9647	−0.714495
$383$	−30.1347	−1.53981	−0.769907	−	0.638156i	$-0.779697\pi$
$383$	−30.1347	−1.53981	−0.769907	+	0.638156i	$0.779697\pi$
$384$	0.515357	0.0262992
$385$	−4.64210	−0.236584
$386$	17.6616	0.898951
$387$	2.73441	0.138998
$388$	−13.7821	−0.699680
$389$	2.46692	0.125078	0.0625390	−	0.998043i	$-0.480080\pi$
$389$	2.46692	0.125078	0.0625390	+	0.998043i	$0.480080\pi$
$390$	−0.0559741	−0.00283436
$391$	−8.84024	−0.447070
$392$	−14.5491	−0.734842
$393$	−6.40269	−0.322973
$394$	20.8737	1.05160
$395$	6.50096	0.327099
$396$	−2.73441	−0.137409
$397$	−16.0944	−0.807756	−0.403878	−	0.914813i	$-0.632338\pi$
$397$	−16.0944	−0.807756	−0.403878	+	0.914813i	$0.632338\pi$
$398$	24.9553	1.25089
$399$	−7.22879	−0.361892
$400$	1.00000	0.0500000
$401$	−14.3654	−0.717373	−0.358687	−	0.933458i	$-0.616775\pi$
$401$	−14.3654	−0.717373	−0.358687	+	0.933458i	$0.616775\pi$
$402$	−1.76941	−0.0882502
$403$	−0.422425	−0.0210425
$404$	−8.50862	−0.423320
$405$	6.68021	0.331942
$406$	37.4471	1.85847
$407$	6.82873	0.338487
$408$	2.16060	0.106966
$409$	−34.1092	−1.68659	−0.843295	−	0.537452i	$-0.819387\pi$
$409$	−34.1092	−1.68659	−0.843295	+	0.537452i	$0.819387\pi$
$410$	−2.16893	−0.107116
$411$	10.0840	0.497405
$412$	−7.21580	−0.355497
$413$	−7.59570	−0.373760
$414$	−5.76581	−0.283374
$415$	−6.38996	−0.313671
$416$	0.108612	0.00532515
$417$	1.69525	0.0830166
$418$	3.02164	0.147793
$419$	−9.23855	−0.451333	−0.225666	−	0.974205i	$-0.572456\pi$
$419$	−9.23855	−0.451333	−0.225666	+	0.974205i	$0.572456\pi$
$420$	2.39234	0.116734
$421$	−19.8056	−0.965265	−0.482633	−	0.875823i	$-0.660319\pi$
$421$	−19.8056	−0.965265	−0.482633	+	0.875823i	$0.660319\pi$
$422$	−0.968285	−0.0471354
$423$	26.7648	1.30135
$424$	−1.84597	−0.0896480
$425$	4.19244	0.203363
$426$	−7.18330	−0.348032
$427$	−37.5012	−1.81481
$428$	1.17431	0.0567623
$429$	0.0559741	0.00270245
$430$	1.00000	0.0482243
$431$	14.1192	0.680099	0.340050	−	0.940407i	$-0.389556\pi$
$431$	14.1192	0.680099	0.340050	+	0.940407i	$0.389556\pi$
$432$	2.95527	0.142185
$433$	20.3340	0.977191	0.488596	−	0.872510i	$-0.337509\pi$
$433$	20.3340	0.977191	0.488596	+	0.872510i	$0.337509\pi$
$434$	18.0545	0.866645
$435$	−4.15730	−0.199327
$436$	−2.34821	−0.112459
$437$	6.37147	0.304789
$438$	3.95965	0.189200
$439$	−34.3522	−1.63954	−0.819772	−	0.572691i	$-0.805900\pi$
$439$	−34.3522	−1.63954	−0.819772	+	0.572691i	$0.805900\pi$
$440$	−1.00000	−0.0476731
$441$	−39.7832	−1.89444
$442$	0.455351	0.0216588
$443$	14.2974	0.679290	0.339645	−	0.940554i	$-0.389693\pi$
$443$	14.2974	0.679290	0.339645	+	0.940554i	$0.389693\pi$
$444$	−3.51923	−0.167015
$445$	−10.4488	−0.495319
$446$	−19.7291	−0.934199
$447$	6.15575	0.291157
$448$	−4.64210	−0.219319
$449$	−18.5194	−0.873984	−0.436992	−	0.899465i	$-0.643956\pi$
$449$	−18.5194	−0.873984	−0.436992	+	0.899465i	$0.643956\pi$
$450$	2.73441	0.128901
$451$	2.16893	0.102131
$452$	−9.55477	−0.449419
$453$	2.96291	0.139210
$454$	13.8242	0.648801
$455$	0.504189	0.0236368
$456$	−1.55722	−0.0729237
$457$	−30.4296	−1.42343	−0.711717	−	0.702466i	$-0.752082\pi$
$457$	−30.4296	−1.42343	−0.711717	+	0.702466i	$0.752082\pi$
$458$	−24.4201	−1.14108
$459$	12.3898	0.578306
$460$	−2.10861	−0.0983146
$461$	5.19765	0.242079	0.121039	−	0.992648i	$-0.461377\pi$
$461$	5.19765	0.242079	0.121039	+	0.992648i	$0.461377\pi$
$462$	−2.39234	−0.111302
$463$	−33.4985	−1.55681	−0.778405	−	0.627763i	$-0.783971\pi$
$463$	−33.4985	−1.55681	−0.778405	+	0.627763i	$0.783971\pi$
$464$	8.06684	0.374494
$465$	−2.00438	−0.0929507
$466$	−17.3137	−0.802042
$467$	6.88311	0.318512	0.159256	−	0.987237i	$-0.449090\pi$
$467$	6.88311	0.318512	0.159256	+	0.987237i	$0.449090\pi$
$468$	0.296990	0.0137284
$469$	15.9381	0.735951
$470$	9.78816	0.451494
$471$	10.7912	0.497230
$472$	−1.63626	−0.0753151
$473$	−1.00000	−0.0459800
$474$	3.35031	0.153885
$475$	−3.02164	−0.138642
$476$	−19.4618	−0.892028
$477$	−5.04762	−0.231115
$478$	−15.6399	−0.715354
$479$	−41.7556	−1.90786	−0.953931	−	0.300026i	$-0.903005\pi$
$479$	−41.7556	−1.90786	−0.953931	+	0.300026i	$0.903005\pi$
$480$	0.515357	0.0235227
$481$	−0.741684	−0.0338179
$482$	4.82493	0.219770
$483$	−5.04452	−0.229533
$484$	1.00000	0.0454545
$485$	−13.7821	−0.625813
$486$	12.3085	0.558324
$487$	−17.2846	−0.783239	−0.391619	−	0.920127i	$-0.628085\pi$
$487$	−17.2846	−0.783239	−0.391619	+	0.920127i	$0.628085\pi$
$488$	−8.07848	−0.365696
$489$	1.26849	0.0573629
$490$	−14.5491	−0.657263
$491$	1.38700	0.0625944	0.0312972	−	0.999510i	$-0.490036\pi$
$491$	1.38700	0.0625944	0.0312972	+	0.999510i	$0.490036\pi$
$492$	−1.11777	−0.0503931
$493$	33.8198	1.52317
$494$	−0.328187	−0.0147658
$495$	−2.73441	−0.122902
$496$	3.88930	0.174635
$497$	64.7040	2.90237
$498$	−3.29311	−0.147568
$499$	8.42570	0.377186	0.188593	−	0.982055i	$-0.439607\pi$
$499$	8.42570	0.377186	0.188593	+	0.982055i	$0.439607\pi$
$500$	1.00000	0.0447214
$501$	8.20453	0.366551
$502$	−29.3009	−1.30776
$503$	−39.0067	−1.73922	−0.869612	−	0.493735i	$-0.835631\pi$
$503$	−39.0067	−1.73922	−0.869612	+	0.493735i	$0.835631\pi$
$504$	−12.6934	−0.565409
$505$	−8.50862	−0.378629
$506$	2.10861	0.0937392
$507$	6.69356	0.297271
$508$	4.23775	0.188020
$509$	−7.93266	−0.351609	−0.175804	−	0.984425i	$-0.556253\pi$
$509$	−7.93266	−0.351609	−0.175804	+	0.984425i	$0.556253\pi$
$510$	2.16060	0.0956732
$511$	−35.6668	−1.57781
$512$	−1.00000	−0.0441942
$513$	−8.92975	−0.394258
$514$	−16.6058	−0.732452
$515$	−7.21580	−0.317966
$516$	0.515357	0.0226873
$517$	−9.78816	−0.430483
$518$	31.6997	1.39280
$519$	−11.9177	−0.523127
$520$	0.108612	0.00476296
$521$	−12.4595	−0.545862	−0.272931	−	0.962034i	$-0.587993\pi$
$521$	−12.4595	−0.545862	−0.272931	+	0.962034i	$0.587993\pi$
$522$	22.0580	0.965454
$523$	−27.6000	−1.20686	−0.603431	−	0.797415i	$-0.706200\pi$
$523$	−27.6000	−1.20686	−0.603431	+	0.797415i	$0.706200\pi$
$524$	12.4238	0.542736
$525$	2.39234	0.104410
$526$	7.44987	0.324830
$527$	16.3057	0.710286
$528$	−0.515357	−0.0224280
$529$	−18.5538	−0.806685
$530$	−1.84597	−0.0801836
$531$	−4.47421	−0.194164
$532$	14.0268	0.608138
$533$	−0.235573	−0.0102038
$534$	−5.38484	−0.233025
$535$	1.17431	0.0507697
$536$	3.43337	0.148299
$537$	−3.99367	−0.172339
$538$	20.2166	0.871600
$539$	14.5491	0.626675
$540$	2.95527	0.127174
$541$	43.0961	1.85285	0.926423	−	0.376485i	$-0.122867\pi$
$541$	43.0961	1.85285	0.926423	+	0.376485i	$0.122867\pi$
$542$	−13.9835	−0.600642
$543$	−0.488665	−0.0209706
$544$	−4.19244	−0.179750
$545$	−2.34821	−0.100586
$546$	0.259837	0.0111200
$547$	−9.47881	−0.405285	−0.202642	−	0.979253i	$-0.564953\pi$
$547$	−9.47881	−0.405285	−0.202642	+	0.979253i	$0.564953\pi$
$548$	−19.5669	−0.835859
$549$	−22.0899	−0.942773
$550$	−1.00000	−0.0426401
$551$	−24.3751	−1.03841
$552$	−1.08669	−0.0462525
$553$	−30.1781	−1.28330
$554$	31.3576	1.33226
$555$	−3.51923	−0.149383
$556$	−3.28946	−0.139504
$557$	6.96985	0.295322	0.147661	−	0.989038i	$-0.452826\pi$
$557$	6.96985	0.295322	0.147661	+	0.989038i	$0.452826\pi$
$558$	10.6349	0.450212
$559$	0.108612	0.00459381
$560$	−4.64210	−0.196165
$561$	−2.16060	−0.0912208
$562$	−22.2132	−0.937006
$563$	9.30571	0.392189	0.196094	−	0.980585i	$-0.437174\pi$
$563$	9.30571	0.392189	0.196094	+	0.980585i	$0.437174\pi$
$564$	5.04439	0.212407
$565$	−9.55477	−0.401972
$566$	−8.84763	−0.371893
$567$	−31.0102	−1.30231
$568$	13.9385	0.584846
$569$	−6.44699	−0.270272	−0.135136	−	0.990827i	$-0.543147\pi$
$569$	−6.44699	−0.270272	−0.135136	+	0.990827i	$0.543147\pi$
$570$	−1.55722	−0.0652249
$571$	−11.1962	−0.468545	−0.234273	−	0.972171i	$-0.575271\pi$
$571$	−11.1962	−0.468545	−0.234273	+	0.972171i	$0.575271\pi$
$572$	−0.108612	−0.00454131
$573$	−7.19679	−0.300650
$574$	10.0684	0.420247
$575$	−2.10861	−0.0879352
$576$	−2.73441	−0.113934
$577$	3.83486	0.159647	0.0798236	−	0.996809i	$-0.474564\pi$
$577$	3.83486	0.159647	0.0798236	+	0.996809i	$0.474564\pi$
$578$	−0.576587	−0.0239829
$579$	9.10202	0.378267
$580$	8.06684	0.334957
$581$	29.6628	1.23062
$582$	−7.10270	−0.294416
$583$	1.84597	0.0764521
$584$	−7.68332	−0.317938
$585$	0.296990	0.0122790
$586$	−10.6467	−0.439809
$587$	−29.6034	−1.22186	−0.610932	−	0.791683i	$-0.709205\pi$
$587$	−29.6034	−1.22186	−0.610932	+	0.791683i	$0.709205\pi$
$588$	−7.49799	−0.309212
$589$	−11.7521	−0.484235
$590$	−1.63626	−0.0673639
$591$	10.7574	0.442500
$592$	6.82873	0.280659
$593$	7.69040	0.315807	0.157904	−	0.987455i	$-0.449526\pi$
$593$	7.69040	0.315807	0.157904	+	0.987455i	$0.449526\pi$
$594$	−2.95527	−0.121256
$595$	−19.4618	−0.797854
$596$	−11.9446	−0.489271
$597$	12.8609	0.526360
$598$	−0.229021	−0.00936537
$599$	14.3870	0.587837	0.293918	−	0.955831i	$-0.405041\pi$
$599$	14.3870	0.587837	0.293918	+	0.955831i	$0.405041\pi$
$600$	0.515357	0.0210394
$601$	−38.4395	−1.56798	−0.783989	−	0.620775i	$-0.786818\pi$
$601$	−38.4395	−1.56798	−0.783989	+	0.620775i	$0.786818\pi$
$602$	−4.64210	−0.189198
$603$	9.38823	0.382318
$604$	−5.74924	−0.233933
$605$	1.00000	0.0406558
$606$	−4.38498	−0.178128
$607$	0.771069	0.0312967	0.0156484	−	0.999878i	$-0.495019\pi$
$607$	0.771069	0.0312967	0.0156484	+	0.999878i	$0.495019\pi$
$608$	3.02164	0.122544
$609$	19.2986	0.782020
$610$	−8.07848	−0.327088
$611$	1.06311	0.0430090
$612$	−11.4639	−0.463399
$613$	39.7389	1.60504	0.802519	−	0.596627i	$-0.203493\pi$
$613$	39.7389	1.60504	0.802519	+	0.596627i	$0.203493\pi$
$614$	−5.80506	−0.234273
$615$	−1.11777	−0.0450730
$616$	4.64210	0.187036
$617$	−12.5833	−0.506585	−0.253292	−	0.967390i	$-0.581513\pi$
$617$	−12.5833	−0.506585	−0.253292	+	0.967390i	$0.581513\pi$
$618$	−3.71871	−0.149588
$619$	1.55022	0.0623088	0.0311544	−	0.999515i	$-0.490082\pi$
$619$	1.55022	0.0623088	0.0311544	+	0.999515i	$0.490082\pi$
$620$	3.88930	0.156198
$621$	−6.23151	−0.250062
$622$	−23.6920	−0.949962
$623$	48.5042	1.94328
$624$	0.0559741	0.00224076
$625$	1.00000	0.0400000
$626$	8.81800	0.352438
$627$	1.55722	0.0621895
$628$	−20.9392	−0.835565
$629$	28.6291	1.14152
$630$	−12.6934	−0.505717
$631$	−29.4029	−1.17051	−0.585255	−	0.810850i	$-0.699005\pi$
$631$	−29.4029	−1.17051	−0.585255	+	0.810850i	$0.699005\pi$
$632$	−6.50096	−0.258594
$633$	−0.499012	−0.0198340
$634$	−8.44230	−0.335286
$635$	4.23775	0.168170
$636$	−0.951331	−0.0377227
$637$	−1.58021	−0.0626103
$638$	−8.06684	−0.319369
$639$	38.1135	1.50775
$640$	−1.00000	−0.0395285
$641$	−29.5181	−1.16589	−0.582947	−	0.812510i	$-0.698101\pi$
$641$	−29.5181	−1.16589	−0.582947	+	0.812510i	$0.698101\pi$
$642$	0.605187	0.0238848
$643$	42.6204	1.68079	0.840393	−	0.541978i	$-0.182324\pi$
$643$	42.6204	1.68079	0.840393	+	0.541978i	$0.182324\pi$
$644$	9.78840	0.385717
$645$	0.515357	0.0202922
$646$	12.6681	0.498418
$647$	14.4992	0.570023	0.285012	−	0.958524i	$-0.408003\pi$
$647$	14.4992	0.570023	0.285012	+	0.958524i	$0.408003\pi$
$648$	−6.68021	−0.262423
$649$	1.63626	0.0642289
$650$	0.108612	0.00426012
$651$	9.30452	0.364673
$652$	−2.46138	−0.0963949
$653$	22.6007	0.884434	0.442217	−	0.896908i	$-0.354192\pi$
$653$	22.6007	0.884434	0.442217	+	0.896908i	$0.354192\pi$
$654$	−1.21017	−0.0473213
$655$	12.4238	0.485438
$656$	2.16893	0.0846826
$657$	−21.0093	−0.819652
$658$	−45.4376	−1.77134
$659$	12.6779	0.493862	0.246931	−	0.969033i	$-0.420578\pi$
$659$	12.6779	0.493862	0.246931	+	0.969033i	$0.420578\pi$
$660$	−0.515357	−0.0200602
$661$	−0.952140	−0.0370340	−0.0185170	−	0.999829i	$-0.505894\pi$
$661$	−0.952140	−0.0370340	−0.0185170	+	0.999829i	$0.505894\pi$
$662$	29.9604	1.16444
$663$	0.234668	0.00911376
$664$	6.38996	0.247978
$665$	14.0268	0.543935
$666$	18.6725	0.723546
$667$	−17.0098	−0.658624
$668$	−15.9201	−0.615967
$669$	−10.1675	−0.393099
$670$	3.43337	0.132643
$671$	8.07848	0.311866
$672$	−2.39234	−0.0922865
$673$	14.3018	0.551295	0.275648	−	0.961259i	$-0.411108\pi$
$673$	14.3018	0.551295	0.275648	+	0.961259i	$0.411108\pi$
$674$	−23.5427	−0.906830
$675$	2.95527	0.113748
$676$	−12.9882	−0.499546
$677$	29.2082	1.12256	0.561281	−	0.827625i	$-0.310309\pi$
$677$	29.2082	1.12256	0.561281	+	0.827625i	$0.310309\pi$
$678$	−4.92412	−0.189110
$679$	63.9779	2.45525
$680$	−4.19244	−0.160773
$681$	7.12439	0.273007
$682$	−3.88930	−0.148929
$683$	−24.3572	−0.932001	−0.466001	−	0.884784i	$-0.654306\pi$
$683$	−24.3572	−0.932001	−0.466001	+	0.884784i	$0.654306\pi$
$684$	8.26240	0.315921
$685$	−19.5669	−0.747615
$686$	35.0438	1.33798
$687$	−12.5851	−0.480150
$688$	−1.00000	−0.0381246
$689$	−0.200494	−0.00763823
$690$	−1.08669	−0.0413695
$691$	41.0692	1.56234	0.781172	−	0.624316i	$-0.214622\pi$
$691$	41.0692	1.56234	0.781172	+	0.624316i	$0.214622\pi$
$692$	23.1251	0.879083
$693$	12.6934	0.482182
$694$	7.25581	0.275427
$695$	−3.28946	−0.124776
$696$	4.15730	0.157582
$697$	9.09312	0.344427
$698$	−21.1451	−0.800354
$699$	−8.92274	−0.337489
$700$	−4.64210	−0.175455
$701$	−21.1868	−0.800214	−0.400107	−	0.916468i	$-0.631027\pi$
$701$	−21.1868	−0.800214	−0.400107	+	0.916468i	$0.631027\pi$
$702$	0.320978	0.0121145
$703$	−20.6340	−0.778225
$704$	1.00000	0.0376889
$705$	5.04439	0.189983
$706$	23.2194	0.873874
$707$	39.4979	1.48547
$708$	−0.843259	−0.0316916
$709$	39.7164	1.49158	0.745791	−	0.666180i	$-0.232072\pi$
$709$	39.7164	1.49158	0.745791	+	0.666180i	$0.232072\pi$
$710$	13.9385	0.523103
$711$	−17.7763	−0.666662
$712$	10.4488	0.391584
$713$	−8.20102	−0.307131
$714$	−10.0297	−0.375354
$715$	−0.108612	−0.00406187
$716$	7.74932	0.289606
$717$	−8.06014	−0.301012
$718$	9.60819	0.358574
$719$	−22.2000	−0.827921	−0.413961	−	0.910295i	$-0.635855\pi$
$719$	−22.2000	−0.827921	−0.413961	+	0.910295i	$0.635855\pi$
$720$	−2.73441	−0.101905
$721$	33.4965	1.24747
$722$	9.86968	0.367311
$723$	2.48656	0.0924762
$724$	0.948206	0.0352398
$725$	8.06684	0.299595
$726$	0.515357	0.0191267
$727$	−30.3674	−1.12627	−0.563133	−	0.826367i	$-0.690404\pi$
$727$	−30.3674	−1.12627	−0.563133	+	0.826367i	$0.690404\pi$
$728$	−0.504189	−0.0186865
$729$	−13.6974	−0.507310
$730$	−7.68332	−0.284373
$731$	−4.19244	−0.155063
$732$	−4.16330	−0.153880
$733$	9.20083	0.339841	0.169920	−	0.985458i	$-0.445649\pi$
$733$	9.20083	0.339841	0.169920	+	0.985458i	$0.445649\pi$
$734$	10.2861	0.379666
$735$	−7.49799	−0.276568
$736$	2.10861	0.0777245
$737$	−3.43337	−0.126470
$738$	5.93074	0.218314
$739$	−8.70249	−0.320126	−0.160063	−	0.987107i	$-0.551170\pi$
$739$	−8.70249	−0.320126	−0.160063	+	0.987107i	$0.551170\pi$
$740$	6.82873	0.251029
$741$	−0.169134	−0.00621328
$742$	8.56916	0.314584
$743$	−14.4927	−0.531684	−0.265842	−	0.964017i	$-0.585650\pi$
$743$	−14.4927	−0.531684	−0.265842	+	0.964017i	$0.585650\pi$
$744$	2.00438	0.0734840
$745$	−11.9446	−0.437617
$746$	−16.3654	−0.599180
$747$	17.4727	0.639294
$748$	4.19244	0.153291
$749$	−5.45126	−0.199185
$750$	0.515357	0.0188182
$751$	7.81010	0.284995	0.142497	−	0.989795i	$-0.454487\pi$
$751$	7.81010	0.284995	0.142497	+	0.989795i	$0.454487\pi$
$752$	−9.78816	−0.356937
$753$	−15.1004	−0.550291
$754$	0.876158	0.0319078
$755$	−5.74924	−0.209236
$756$	−13.7186	−0.498942
$757$	−10.1738	−0.369772	−0.184886	−	0.982760i	$-0.559192\pi$
$757$	−10.1738	−0.369772	−0.184886	+	0.982760i	$0.559192\pi$
$758$	−16.9114	−0.614250
$759$	1.08669	0.0394443
$760$	3.02164	0.109606
$761$	15.3951	0.558072	0.279036	−	0.960281i	$-0.409985\pi$
$761$	15.3951	0.558072	0.279036	+	0.960281i	$0.409985\pi$
$762$	2.18395	0.0791162
$763$	10.9006	0.394630
$764$	13.9647	0.505224
$765$	−11.4639	−0.414476
$766$	30.1347	1.08881
$767$	−0.177718	−0.00641703
$768$	−0.515357	−0.0185963
$769$	−7.01853	−0.253095	−0.126547	−	0.991961i	$-0.540390\pi$
$769$	−7.01853	−0.253095	−0.126547	+	0.991961i	$0.540390\pi$
$770$	4.64210	0.167290
$771$	−8.55792	−0.308206
$772$	−17.6616	−0.635655
$773$	14.4099	0.518288	0.259144	−	0.965839i	$-0.416560\pi$
$773$	14.4099	0.518288	0.259144	+	0.965839i	$0.416560\pi$
$774$	−2.73441	−0.0982862
$775$	3.88930	0.139708
$776$	13.7821	0.494749
$777$	16.3366	0.586074
$778$	−2.46692	−0.0884435
$779$	−6.55373	−0.234812
$780$	0.0559741	0.00200419
$781$	−13.9385	−0.498759
$782$	8.84024	0.316126
$783$	23.8397	0.851960
$784$	14.5491	0.519612
$785$	−20.9392	−0.747352
$786$	6.40269	0.228376
$787$	−42.3946	−1.51120	−0.755602	−	0.655031i	$-0.772655\pi$
$787$	−42.3946	−1.51120	−0.755602	+	0.655031i	$0.772655\pi$
$788$	−20.8737	−0.743594
$789$	3.83934	0.136684
$790$	−6.50096	−0.231294
$791$	44.3542	1.57705
$792$	2.73441	0.0971629
$793$	−0.877423	−0.0311582
$794$	16.0944	0.571170
$795$	−0.951331	−0.0337402
$796$	−24.9553	−0.884516
$797$	36.7130	1.30044	0.650220	−	0.759746i	$-0.274677\pi$
$797$	36.7130	1.30044	0.650220	+	0.759746i	$0.274677\pi$
$798$	7.22879	0.255897
$799$	−41.0363	−1.45176
$800$	−1.00000	−0.0353553
$801$	28.5712	1.00951
$802$	14.3654	0.507260
$803$	7.68332	0.271139
$804$	1.76941	0.0624023
$805$	9.78840	0.344996
$806$	0.422425	0.0148793
$807$	10.4188	0.366758
$808$	8.50862	0.299332
$809$	48.9521	1.72107	0.860533	−	0.509395i	$-0.170131\pi$
$809$	48.9521	1.72107	0.860533	+	0.509395i	$0.170131\pi$
$810$	−6.68021	−0.234719
$811$	−15.0448	−0.528293	−0.264147	−	0.964483i	$-0.585090\pi$
$811$	−15.0448	−0.528293	−0.264147	+	0.964483i	$0.585090\pi$
$812$	−37.4471	−1.31414
$813$	−7.20648	−0.252742
$814$	−6.82873	−0.239347
$815$	−2.46138	−0.0862182
$816$	−2.16060	−0.0756363
$817$	3.02164	0.105714
$818$	34.1092	1.19260
$819$	−1.37866	−0.0481743
$820$	2.16893	0.0757424
$821$	−4.58575	−0.160044	−0.0800219	−	0.996793i	$-0.525499\pi$
$821$	−4.58575	−0.160044	−0.0800219	+	0.996793i	$0.525499\pi$
$822$	−10.0840	−0.351718
$823$	−54.9163	−1.91426	−0.957131	−	0.289656i	$-0.906459\pi$
$823$	−54.9163	−1.91426	−0.957131	+	0.289656i	$0.906459\pi$
$824$	7.21580	0.251374
$825$	−0.515357	−0.0179424
$826$	7.59570	0.264288
$827$	−12.9128	−0.449021	−0.224510	−	0.974472i	$-0.572078\pi$
$827$	−12.9128	−0.449021	−0.224510	+	0.974472i	$0.572078\pi$
$828$	5.76581	0.200376
$829$	−0.368814	−0.0128094	−0.00640472	−	0.999979i	$-0.502039\pi$
$829$	−0.368814	−0.0128094	−0.00640472	+	0.999979i	$0.502039\pi$
$830$	6.38996	0.221799
$831$	16.1604	0.560597
$832$	−0.108612	−0.00376545
$833$	60.9964	2.11340
$834$	−1.69525	−0.0587016
$835$	−15.9201	−0.550938
$836$	−3.02164	−0.104506
$837$	11.4939	0.397288
$838$	9.23855	0.319141
$839$	27.9683	0.965573	0.482786	−	0.875738i	$-0.339625\pi$
$839$	27.9683	0.965573	0.482786	+	0.875738i	$0.339625\pi$
$840$	−2.39234	−0.0825436
$841$	36.0740	1.24393
$842$	19.8056	0.682546
$843$	−11.4477	−0.394280
$844$	0.968285	0.0333297
$845$	−12.9882	−0.446808
$846$	−26.7648	−0.920193
$847$	−4.64210	−0.159505
$848$	1.84597	0.0633907
$849$	−4.55968	−0.156488
$850$	−4.19244	−0.143800
$851$	−14.3991	−0.493596
$852$	7.18330	0.246096
$853$	23.8544	0.816757	0.408379	−	0.912813i	$-0.366094\pi$
$853$	23.8544	0.816757	0.408379	+	0.912813i	$0.366094\pi$
$854$	37.5012	1.28326
$855$	8.26240	0.282568
$856$	−1.17431	−0.0401370
$857$	−39.4611	−1.34797	−0.673983	−	0.738747i	$-0.735418\pi$
$857$	−39.4611	−1.34797	−0.673983	+	0.738747i	$0.735418\pi$
$858$	−0.0559741	−0.00191092
$859$	−51.4843	−1.75662	−0.878311	−	0.478090i	$-0.841329\pi$
$859$	−51.4843	−1.75662	−0.878311	+	0.478090i	$0.841329\pi$
$860$	−1.00000	−0.0340997
$861$	5.18882	0.176835
$862$	−14.1192	−0.480903
$863$	16.9377	0.576567	0.288284	−	0.957545i	$-0.406915\pi$
$863$	16.9377	0.576567	0.288284	+	0.957545i	$0.406915\pi$
$864$	−2.95527	−0.100540
$865$	23.1251	0.786275
$866$	−20.3340	−0.690979
$867$	−0.297148	−0.0100917
$868$	−18.0545	−0.612810
$869$	6.50096	0.220530
$870$	4.15730	0.140946
$871$	0.372906	0.0126354
$872$	2.34821	0.0795205
$873$	37.6859	1.27547
$874$	−6.37147	−0.215518
$875$	−4.64210	−0.156932
$876$	−3.95965	−0.133784
$877$	10.4830	0.353986	0.176993	−	0.984212i	$-0.443363\pi$
$877$	10.4830	0.353986	0.176993	+	0.984212i	$0.443363\pi$
$878$	34.3522	1.15933
$879$	−5.48682	−0.185066
$880$	1.00000	0.0337100
$881$	−28.0986	−0.946667	−0.473334	−	0.880883i	$-0.656950\pi$
$881$	−28.0986	−0.946667	−0.473334	+	0.880883i	$0.656950\pi$
$882$	39.7832	1.33957
$883$	46.9823	1.58108	0.790541	−	0.612410i	$-0.209800\pi$
$883$	46.9823	1.58108	0.790541	+	0.612410i	$0.209800\pi$
$884$	−0.455351	−0.0153151
$885$	−0.843259	−0.0283458
$886$	−14.2974	−0.480330
$887$	9.00177	0.302250	0.151125	−	0.988515i	$-0.451710\pi$
$887$	9.00177	0.302250	0.151125	+	0.988515i	$0.451710\pi$
$888$	3.51923	0.118098
$889$	−19.6721	−0.659780
$890$	10.4488	0.350243
$891$	6.68021	0.223795
$892$	19.7291	0.660579
$893$	29.5763	0.989733
$894$	−6.15575	−0.205879
$895$	7.74932	0.259031
$896$	4.64210	0.155082
$897$	−0.118028	−0.00394083
$898$	18.5194	0.618000
$899$	31.3744	1.04639
$900$	−2.73441	−0.0911469
$901$	7.73911	0.257827
$902$	−2.16893	−0.0722175
$903$	−2.39234	−0.0796121
$904$	9.55477	0.317787
$905$	0.948206	0.0315194
$906$	−2.96291	−0.0984361
$907$	47.9218	1.59122	0.795608	−	0.605811i	$-0.207151\pi$
$907$	47.9218	1.59122	0.795608	+	0.605811i	$0.207151\pi$
$908$	−13.8242	−0.458772
$909$	23.2660	0.771686
$910$	−0.504189	−0.0167137
$911$	−10.4414	−0.345939	−0.172969	−	0.984927i	$-0.555336\pi$
$911$	−10.4414	−0.345939	−0.172969	+	0.984927i	$0.555336\pi$
$912$	1.55722	0.0515648
$913$	−6.38996	−0.211477
$914$	30.4296	1.00652
$915$	−4.16330	−0.137635
$916$	24.4201	0.806862
$917$	−57.6725	−1.90452
$918$	−12.3898	−0.408924
$919$	21.8478	0.720693	0.360346	−	0.932819i	$-0.382659\pi$
$919$	21.8478	0.720693	0.360346	+	0.932819i	$0.382659\pi$
$920$	2.10861	0.0695189
$921$	−2.99168	−0.0985792
$922$	−5.19765	−0.171176
$923$	1.51389	0.0498304
$924$	2.39234	0.0787022
$925$	6.82873	0.224527
$926$	33.4985	1.10083
$927$	19.7309	0.648049
$928$	−8.06684	−0.264807
$929$	−49.9346	−1.63830	−0.819151	−	0.573578i	$-0.805555\pi$
$929$	−49.9346	−1.63830	−0.819151	+	0.573578i	$0.805555\pi$
$930$	2.00438	0.0657261
$931$	−43.9622	−1.44080
$932$	17.3137	0.567130
$933$	−12.2098	−0.399732
$934$	−6.88311	−0.225222
$935$	4.19244	0.137108
$936$	−0.296990	−0.00970743
$937$	0.289219	0.00944838	0.00472419	−	0.999989i	$-0.498496\pi$
$937$	0.289219	0.00944838	0.00472419	+	0.999989i	$0.498496\pi$
$938$	−15.9381	−0.520396
$939$	4.54442	0.148301
$940$	−9.78816	−0.319255
$941$	24.3719	0.794502	0.397251	−	0.917710i	$-0.369964\pi$
$941$	24.3719	0.794502	0.397251	+	0.917710i	$0.369964\pi$
$942$	−10.7912	−0.351595
$943$	−4.57344	−0.148932
$944$	1.63626	0.0532558
$945$	−13.7186	−0.446268
$946$	1.00000	0.0325128
$947$	28.6203	0.930033	0.465017	−	0.885302i	$-0.346048\pi$
$947$	28.6203	0.930033	0.465017	+	0.885302i	$0.346048\pi$
$948$	−3.35031	−0.108813
$949$	−0.834503	−0.0270891
$950$	3.02164	0.0980350
$951$	−4.35079	−0.141084
$952$	19.4618	0.630759
$953$	38.7967	1.25675	0.628374	−	0.777911i	$-0.283721\pi$
$953$	38.7967	1.25675	0.628374	+	0.777911i	$0.283721\pi$
$954$	5.04762	0.163423
$955$	13.9647	0.451886
$956$	15.6399	0.505831
$957$	−4.15730	−0.134387
$958$	41.7556	1.34906
$959$	90.8318	2.93311
$960$	−0.515357	−0.0166331
$961$	−15.8734	−0.512044
$962$	0.741684	0.0239128
$963$	−3.21103	−0.103474
$964$	−4.82493	−0.155401
$965$	−17.6616	−0.568547
$966$	5.04452	0.162305
$967$	33.3917	1.07381	0.536903	−	0.843644i	$-0.319594\pi$
$967$	33.3917	1.07381	0.536903	+	0.843644i	$0.319594\pi$
$968$	−1.00000	−0.0321412
$969$	6.52857	0.209728
$970$	13.7821	0.442517
$971$	−38.3504	−1.23072	−0.615361	−	0.788245i	$-0.710990\pi$
$971$	−38.3504	−1.23072	−0.615361	+	0.788245i	$0.710990\pi$
$972$	−12.3085	−0.394795
$973$	15.2700	0.489535
$974$	17.2846	0.553833
$975$	0.0559741	0.00179260
$976$	8.07848	0.258586
$977$	32.1864	1.02973	0.514867	−	0.857270i	$-0.327841\pi$
$977$	32.1864	1.02973	0.514867	+	0.857270i	$0.327841\pi$
$978$	−1.26849	−0.0405617
$979$	−10.4488	−0.333944
$980$	14.5491	0.464755
$981$	6.42097	0.205006
$982$	−1.38700	−0.0442609
$983$	−26.1017	−0.832516	−0.416258	−	0.909247i	$-0.636659\pi$
$983$	−26.1017	−0.832516	−0.416258	+	0.909247i	$0.636659\pi$
$984$	1.11777	0.0356333
$985$	−20.8737	−0.665090
$986$	−33.8198	−1.07704
$987$	−23.4166	−0.745359
$988$	0.328187	0.0104410
$989$	2.10861	0.0670500
$990$	2.73441	0.0869052
$991$	2.66903	0.0847844	0.0423922	−	0.999101i	$-0.486502\pi$
$991$	2.66903	0.0847844	0.0423922	+	0.999101i	$0.486502\pi$
$992$	−3.88930	−0.123485
$993$	15.4403	0.489983
$994$	−64.7040	−2.05228
$995$	−24.9553	−0.791135
$996$	3.29311	0.104346
$997$	−15.0638	−0.477076	−0.238538	−	0.971133i	$-0.576668\pi$
$997$	−15.0638	−0.477076	−0.238538	+	0.971133i	$0.576668\pi$
$998$	−8.42570	−0.266711
$999$	20.1807	0.638489

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4730.2.a.w.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4730.2.a.w.1.5	✓	8	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:	\( N \)	\(=\)	\( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4730.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.515357 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.515357 q^{6} -4.64210 q^{7} -1.00000 q^{8} -2.73441 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.515357 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.515357 q^{6} -4.64210 q^{7} -1.00000 q^{8} -2.73441 q^{9} -1.00000 q^{10} +1.00000 q^{11} -0.515357 q^{12} -0.108612 q^{13} +4.64210 q^{14} -0.515357 q^{15} +1.00000 q^{16} +4.19244 q^{17} +2.73441 q^{18} -3.02164 q^{19} +1.00000 q^{20} +2.39234 q^{21} -1.00000 q^{22} -2.10861 q^{23} +0.515357 q^{24} +1.00000 q^{25} +0.108612 q^{26} +2.95527 q^{27} -4.64210 q^{28} +8.06684 q^{29} +0.515357 q^{30} +3.88930 q^{31} -1.00000 q^{32} -0.515357 q^{33} -4.19244 q^{34} -4.64210 q^{35} -2.73441 q^{36} +6.82873 q^{37} +3.02164 q^{38} +0.0559741 q^{39} -1.00000 q^{40} +2.16893 q^{41} -2.39234 q^{42} -1.00000 q^{43} +1.00000 q^{44} -2.73441 q^{45} +2.10861 q^{46} -9.78816 q^{47} -0.515357 q^{48} +14.5491 q^{49} -1.00000 q^{50} -2.16060 q^{51} -0.108612 q^{52} +1.84597 q^{53} -2.95527 q^{54} +1.00000 q^{55} +4.64210 q^{56} +1.55722 q^{57} -8.06684 q^{58} +1.63626 q^{59} -0.515357 q^{60} +8.07848 q^{61} -3.88930 q^{62} +12.6934 q^{63} +1.00000 q^{64} -0.108612 q^{65} +0.515357 q^{66} -3.43337 q^{67} +4.19244 q^{68} +1.08669 q^{69} +4.64210 q^{70} -13.9385 q^{71} +2.73441 q^{72} +7.68332 q^{73} -6.82873 q^{74} -0.515357 q^{75} -3.02164 q^{76} -4.64210 q^{77} -0.0559741 q^{78} +6.50096 q^{79} +1.00000 q^{80} +6.68021 q^{81} -2.16893 q^{82} -6.38996 q^{83} +2.39234 q^{84} +4.19244 q^{85} +1.00000 q^{86} -4.15730 q^{87} -1.00000 q^{88} -10.4488 q^{89} +2.73441 q^{90} +0.504189 q^{91} -2.10861 q^{92} -2.00438 q^{93} +9.78816 q^{94} -3.02164 q^{95} +0.515357 q^{96} -13.7821 q^{97} -14.5491 q^{98} -2.73441 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 7 q^{3} + 8 q^{4} + 8 q^{5} + 7 q^{6} - 6 q^{7} - 8 q^{8} + 7 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - 7 * q^3 + 8 * q^4 + 8 * q^5 + 7 * q^6 - 6 * q^7 - 8 * q^8 + 7 * q^9	\( 8 q - 8 q^{2} - 7 q^{3} + 8 q^{4} + 8 q^{5} + 7 q^{6} - 6 q^{7} - 8 q^{8} + 7 q^{9} - 8 q^{10} + 8 q^{11} - 7 q^{12} - 2 q^{13} + 6 q^{14} - 7 q^{15} + 8 q^{16} - 8 q^{17} - 7 q^{18} + 8 q^{20} + 14 q^{21} - 8 q^{22} - 18 q^{23} + 7 q^{24} + 8 q^{25} + 2 q^{26} - 22 q^{27} - 6 q^{28} + 8 q^{29} + 7 q^{30} - 11 q^{31} - 8 q^{32} - 7 q^{33} + 8 q^{34} - 6 q^{35} + 7 q^{36} - 17 q^{37} - 6 q^{39} - 8 q^{40} + 12 q^{41} - 14 q^{42} - 8 q^{43} + 8 q^{44} + 7 q^{45} + 18 q^{46} - 19 q^{47} - 7 q^{48} - 2 q^{49} - 8 q^{50} - q^{51} - 2 q^{52} - 7 q^{53} + 22 q^{54} + 8 q^{55} + 6 q^{56} - 3 q^{57} - 8 q^{58} + q^{59} - 7 q^{60} + 6 q^{61} + 11 q^{62} - 15 q^{63} + 8 q^{64} - 2 q^{65} + 7 q^{66} - 22 q^{67} - 8 q^{68} + 8 q^{69} + 6 q^{70} - 14 q^{71} - 7 q^{72} - 13 q^{73} + 17 q^{74} - 7 q^{75} - 6 q^{77} + 6 q^{78} - 8 q^{79} + 8 q^{80} + 28 q^{81} - 12 q^{82} - 4 q^{83} + 14 q^{84} - 8 q^{85} + 8 q^{86} - 30 q^{87} - 8 q^{88} + 5 q^{89} - 7 q^{90} - 8 q^{91} - 18 q^{92} + q^{93} + 19 q^{94} + 7 q^{96} - 23 q^{97} + 2 q^{98} + 7 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 7 * q^3 + 8 * q^4 + 8 * q^5 + 7 * q^6 - 6 * q^7 - 8 * q^8 + 7 * q^9 - 8 * q^10 + 8 * q^11 - 7 * q^12 - 2 * q^13 + 6 * q^14 - 7 * q^15 + 8 * q^16 - 8 * q^17 - 7 * q^18 + 8 * q^20 + 14 * q^21 - 8 * q^22 - 18 * q^23 + 7 * q^24 + 8 * q^25 + 2 * q^26 - 22 * q^27 - 6 * q^28 + 8 * q^29 + 7 * q^30 - 11 * q^31 - 8 * q^32 - 7 * q^33 + 8 * q^34 - 6 * q^35 + 7 * q^36 - 17 * q^37 - 6 * q^39 - 8 * q^40 + 12 * q^41 - 14 * q^42 - 8 * q^43 + 8 * q^44 + 7 * q^45 + 18 * q^46 - 19 * q^47 - 7 * q^48 - 2 * q^49 - 8 * q^50 - q^51 - 2 * q^52 - 7 * q^53 + 22 * q^54 + 8 * q^55 + 6 * q^56 - 3 * q^57 - 8 * q^58 + q^59 - 7 * q^60 + 6 * q^61 + 11 * q^62 - 15 * q^63 + 8 * q^64 - 2 * q^65 + 7 * q^66 - 22 * q^67 - 8 * q^68 + 8 * q^69 + 6 * q^70 - 14 * q^71 - 7 * q^72 - 13 * q^73 + 17 * q^74 - 7 * q^75 - 6 * q^77 + 6 * q^78 - 8 * q^79 + 8 * q^80 + 28 * q^81 - 12 * q^82 - 4 * q^83 + 14 * q^84 - 8 * q^85 + 8 * q^86 - 30 * q^87 - 8 * q^88 + 5 * q^89 - 7 * q^90 - 8 * q^91 - 18 * q^92 + q^93 + 19 * q^94 + 7 * q^96 - 23 * q^97 + 2 * q^98 + 7 * q^99

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