LMFDB - Embedded newform 847.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.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:	$6.76332905120$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7674048.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.276564$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27656 q^{2} -2.57603 q^{3} -0.370384 q^{4} -4.09144 q^{5} +3.28847 q^{6} -1.00000 q^{7} +3.02595 q^{8} +3.63595 q^{9} +O(q^{10})$	\(q-1.27656 q^{2} -2.57603 q^{3} -0.370384 q^{4} -4.09144 q^{5} +3.28847 q^{6} -1.00000 q^{7} +3.02595 q^{8} +3.63595 q^{9} +5.22298 q^{10} +0.954122 q^{12} +4.39091 q^{13} +1.27656 q^{14} +10.5397 q^{15} -3.12205 q^{16} -4.19146 q^{17} -4.64152 q^{18} +1.24880 q^{19} +1.51540 q^{20} +2.57603 q^{21} +4.97180 q^{23} -7.79494 q^{24} +11.7399 q^{25} -5.60527 q^{26} -1.63823 q^{27} +0.370384 q^{28} +1.93542 q^{29} -13.4546 q^{30} -1.56278 q^{31} -2.06640 q^{32} +5.35067 q^{34} +4.09144 q^{35} -1.34670 q^{36} -0.716296 q^{37} -1.59417 q^{38} -11.3111 q^{39} -12.3805 q^{40} -4.80626 q^{41} -3.28847 q^{42} -1.35362 q^{43} -14.8763 q^{45} -6.34682 q^{46} -10.4662 q^{47} +8.04250 q^{48} +1.00000 q^{49} -14.9867 q^{50} +10.7973 q^{51} -1.62632 q^{52} -3.97180 q^{53} +2.09131 q^{54} -3.02595 q^{56} -3.21695 q^{57} -2.47069 q^{58} +13.7588 q^{59} -3.90373 q^{60} +11.7271 q^{61} +1.99499 q^{62} -3.63595 q^{63} +8.88199 q^{64} -17.9651 q^{65} +7.59274 q^{67} +1.55245 q^{68} -12.8075 q^{69} -5.22298 q^{70} +0.218316 q^{71} +11.0022 q^{72} -10.9714 q^{73} +0.914398 q^{74} -30.2423 q^{75} -0.462535 q^{76} +14.4394 q^{78} +4.56248 q^{79} +12.7737 q^{80} -6.68771 q^{81} +6.13550 q^{82} -2.45458 q^{83} -0.954122 q^{84} +17.1491 q^{85} +1.72798 q^{86} -4.98571 q^{87} +4.20456 q^{89} +18.9905 q^{90} -4.39091 q^{91} -1.84148 q^{92} +4.02578 q^{93} +13.3608 q^{94} -5.10938 q^{95} +5.32312 q^{96} -10.9249 q^{97} -1.27656 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) $ 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9	\( 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9} + 8 q^{10} - 14 q^{12} - 4 q^{13} + 4 q^{14} + 2 q^{15} + 8 q^{16} - 22 q^{17} - 24 q^{18} - 6 q^{19} + 2 q^{20} + 2 q^{21} + 2 q^{23} + 20 q^{24} + 4 q^{25} + 6 q^{26} - 2 q^{27} - 4 q^{28} - 12 q^{29} - 20 q^{30} - 2 q^{31} - 8 q^{32} + 24 q^{34} + 4 q^{35} + 18 q^{36} + 14 q^{37} - 22 q^{38} - 20 q^{39} - 18 q^{40} - 26 q^{41} - 6 q^{42} + 4 q^{43} - 36 q^{45} - 12 q^{46} - 16 q^{47} - 24 q^{48} + 6 q^{49} + 4 q^{50} + 4 q^{51} - 12 q^{52} + 4 q^{53} + 32 q^{54} + 12 q^{56} - 20 q^{57} - 2 q^{58} - 4 q^{59} + 24 q^{60} + 8 q^{61} - 20 q^{62} - 8 q^{63} + 26 q^{64} - 24 q^{65} + 6 q^{67} - 12 q^{68} - 14 q^{69} - 8 q^{70} + 22 q^{71} - 16 q^{72} - 14 q^{73} - 44 q^{74} - 20 q^{75} + 30 q^{76} + 32 q^{78} + 28 q^{79} - 4 q^{80} - 6 q^{81} - 4 q^{82} - 22 q^{83} + 14 q^{84} + 24 q^{85} - 30 q^{86} - 22 q^{87} + 22 q^{90} + 4 q^{91} + 10 q^{92} - 50 q^{93} + 38 q^{94} + 24 q^{95} + 62 q^{96} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9 + 8 * q^10 - 14 * q^12 - 4 * q^13 + 4 * q^14 + 2 * q^15 + 8 * q^16 - 22 * q^17 - 24 * q^18 - 6 * q^19 + 2 * q^20 + 2 * q^21 + 2 * q^23 + 20 * q^24 + 4 * q^25 + 6 * q^26 - 2 * q^27 - 4 * q^28 - 12 * q^29 - 20 * q^30 - 2 * q^31 - 8 * q^32 + 24 * q^34 + 4 * q^35 + 18 * q^36 + 14 * q^37 - 22 * q^38 - 20 * q^39 - 18 * q^40 - 26 * q^41 - 6 * q^42 + 4 * q^43 - 36 * q^45 - 12 * q^46 - 16 * q^47 - 24 * q^48 + 6 * q^49 + 4 * q^50 + 4 * q^51 - 12 * q^52 + 4 * q^53 + 32 * q^54 + 12 * q^56 - 20 * q^57 - 2 * q^58 - 4 * q^59 + 24 * q^60 + 8 * q^61 - 20 * q^62 - 8 * q^63 + 26 * q^64 - 24 * q^65 + 6 * q^67 - 12 * q^68 - 14 * q^69 - 8 * q^70 + 22 * q^71 - 16 * q^72 - 14 * q^73 - 44 * q^74 - 20 * q^75 + 30 * q^76 + 32 * q^78 + 28 * q^79 - 4 * q^80 - 6 * q^81 - 4 * q^82 - 22 * q^83 + 14 * q^84 + 24 * q^85 - 30 * q^86 - 22 * q^87 + 22 * q^90 + 4 * q^91 + 10 * q^92 - 50 * q^93 + 38 * q^94 + 24 * q^95 + 62 * q^96 - 4 * q^97 - 4 * q^98

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 $

$2$	−1.27656	−0.902667	−0.451334	−	0.892355i	$-0.649052\pi$
$2$	−1.27656	−0.902667	−0.451334	+	0.892355i	$0.649052\pi$
$3$	−2.57603	−1.48727	−0.743637	−	0.668584i	$-0.766901\pi$
$3$	−2.57603	−1.48727	−0.743637	+	0.668584i	$0.766901\pi$
$4$	−0.370384	−0.185192
$5$	−4.09144	−1.82975	−0.914873	−	0.403741i	$-0.867710\pi$
$5$	−4.09144	−1.82975	−0.914873	+	0.403741i	$0.867710\pi$
$6$	3.28847	1.34251
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.02595	1.06983
$9$	3.63595	1.21198
$10$	5.22298	1.65165
$11$	0	0
$11$	0	0
$12$	0.954122	0.275431
$13$	4.39091	1.21782	0.608909	−	0.793240i	$-0.291607\pi$
$13$	4.39091	1.21782	0.608909	+	0.793240i	$0.291607\pi$
$14$	1.27656	0.341176
$15$	10.5397	2.72133
$16$	−3.12205	−0.780512
$17$	−4.19146	−1.01658	−0.508289	−	0.861186i	$-0.669722\pi$
$17$	−4.19146	−1.01658	−0.508289	+	0.861186i	$0.669722\pi$
$18$	−4.64152	−1.09402
$19$	1.24880	0.286494	0.143247	−	0.989687i	$-0.454246\pi$
$19$	1.24880	0.286494	0.143247	+	0.989687i	$0.454246\pi$
$20$	1.51540	0.338855
$21$	2.57603	0.562137
$22$	0	0
$23$	4.97180	1.03669	0.518346	−	0.855171i	$-0.326548\pi$
$23$	4.97180	1.03669	0.518346	+	0.855171i	$0.326548\pi$
$24$	−7.79494	−1.59114
$25$	11.7399	2.34797
$26$	−5.60527	−1.09928
$27$	−1.63823	−0.315278
$28$	0.370384	0.0699960
$29$	1.93542	0.359399	0.179699	−	0.983722i	$-0.442488\pi$
$29$	1.93542	0.359399	0.179699	+	0.983722i	$0.442488\pi$
$30$	−13.4546	−2.45646
$31$	−1.56278	−0.280684	−0.140342	−	0.990103i	$-0.544820\pi$
$31$	−1.56278	−0.280684	−0.140342	+	0.990103i	$0.544820\pi$
$32$	−2.06640	−0.365292
$33$	0	0
$34$	5.35067	0.917632
$35$	4.09144	0.691579
$36$	−1.34670	−0.224450
$37$	−0.716296	−0.117758	−0.0588792	−	0.998265i	$-0.518753\pi$
$37$	−0.716296	−0.117758	−0.0588792	+	0.998265i	$0.518753\pi$
$38$	−1.59417	−0.258609
$39$	−11.3111	−1.81123
$40$	−12.3805	−1.95752
$41$	−4.80626	−0.750611	−0.375306	−	0.926901i	$-0.622462\pi$
$41$	−4.80626	−0.750611	−0.375306	+	0.926901i	$0.622462\pi$
$42$	−3.28847	−0.507422
$43$	−1.35362	−0.206424	−0.103212	−	0.994659i	$-0.532912\pi$
$43$	−1.35362	−0.206424	−0.103212	+	0.994659i	$0.532912\pi$
$44$	0	0
$45$	−14.8763	−2.21762
$46$	−6.34682	−0.935788
$47$	−10.4662	−1.52665	−0.763327	−	0.646013i	$-0.776435\pi$
$47$	−10.4662	−1.52665	−0.763327	+	0.646013i	$0.776435\pi$
$48$	8.04250	1.16083
$49$	1.00000	0.142857
$50$	−14.9867	−2.11944
$51$	10.7973	1.51193
$52$	−1.62632	−0.225530
$53$	−3.97180	−0.545569	−0.272784	−	0.962075i	$-0.587945\pi$
$53$	−3.97180	−0.545569	−0.272784	+	0.962075i	$0.587945\pi$
$54$	2.09131	0.284591
$55$	0	0
$56$	−3.02595	−0.404359
$57$	−3.21695	−0.426095
$58$	−2.47069	−0.324417
$59$	13.7588	1.79125	0.895624	−	0.444811i	$-0.146729\pi$
$59$	13.7588	1.79125	0.895624	+	0.444811i	$0.146729\pi$
$60$	−3.90373	−0.503970
$61$	11.7271	1.50151	0.750753	−	0.660583i	$-0.229691\pi$
$61$	11.7271	1.50151	0.750753	+	0.660583i	$0.229691\pi$
$62$	1.99499	0.253364
$63$	−3.63595	−0.458087
$64$	8.88199	1.11025
$65$	−17.9651	−2.22830
$66$	0	0
$67$	7.59274	0.927600	0.463800	−	0.885940i	$-0.346486\pi$
$67$	7.59274	0.927600	0.463800	+	0.885940i	$0.346486\pi$
$68$	1.55245	0.188262
$69$	−12.8075	−1.54185
$70$	−5.22298	−0.624266
$71$	0.218316	0.0259094	0.0129547	−	0.999916i	$-0.495876\pi$
$71$	0.218316	0.0259094	0.0129547	+	0.999916i	$0.495876\pi$
$72$	11.0022	1.29662
$73$	−10.9714	−1.28411	−0.642054	−	0.766659i	$-0.721918\pi$
$73$	−10.9714	−1.28411	−0.642054	+	0.766659i	$0.721918\pi$
$74$	0.914398	0.106297
$75$	−30.2423	−3.49208
$76$	−0.462535	−0.0530564
$77$	0	0
$78$	14.4394	1.63494
$79$	4.56248	0.513319	0.256659	−	0.966502i	$-0.417378\pi$
$79$	4.56248	0.513319	0.256659	+	0.966502i	$0.417378\pi$
$80$	12.7737	1.42814
$81$	−6.68771	−0.743079
$82$	6.13550	0.677552
$83$	−2.45458	−0.269425	−0.134712	−	0.990885i	$-0.543011\pi$
$83$	−2.45458	−0.269425	−0.134712	+	0.990885i	$0.543011\pi$
$84$	−0.954122	−0.104103
$85$	17.1491	1.86008
$86$	1.72798	0.186333
$87$	−4.98571	−0.534524
$88$	0	0
$89$	4.20456	0.445683	0.222841	−	0.974855i	$-0.428467\pi$
$89$	4.20456	0.445683	0.222841	+	0.974855i	$0.428467\pi$
$90$	18.9905	2.00178
$91$	−4.39091	−0.460292
$92$	−1.84148	−0.191987
$93$	4.02578	0.417454
$94$	13.3608	1.37806
$95$	−5.10938	−0.524211
$96$	5.32312	0.543289
$97$	−10.9249	−1.10925	−0.554627	−	0.832099i	$-0.687139\pi$
$97$	−10.9249	−1.10925	−0.554627	+	0.832099i	$0.687139\pi$
$98$	−1.27656	−0.128952
$99$	0	0
$100$	−4.34826	−0.434826
$101$	−7.36579	−0.732924	−0.366462	−	0.930433i	$-0.619431\pi$
$101$	−7.36579	−0.732924	−0.366462	+	0.930433i	$0.619431\pi$
$102$	−13.7835	−1.36477
$103$	−0.153886	−0.0151629	−0.00758143	−	0.999971i	$-0.502413\pi$
$103$	−0.153886	−0.0151629	−0.00758143	+	0.999971i	$0.502413\pi$
$104$	13.2867	1.30286
$105$	−10.5397	−1.02857
$106$	5.07026	0.492467
$107$	−7.94617	−0.768185	−0.384092	−	0.923295i	$-0.625486\pi$
$107$	−7.94617	−0.768185	−0.384092	+	0.923295i	$0.625486\pi$
$108$	0.606775	0.0583869
$109$	4.91440	0.470714	0.235357	−	0.971909i	$-0.424374\pi$
$109$	4.91440	0.470714	0.235357	+	0.971909i	$0.424374\pi$
$110$	0	0
$111$	1.84520	0.175139
$112$	3.12205	0.295006
$113$	3.04208	0.286175	0.143088	−	0.989710i	$-0.454297\pi$
$113$	3.04208	0.286175	0.143088	+	0.989710i	$0.454297\pi$
$114$	4.10664	0.384622
$115$	−20.3418	−1.89688
$116$	−0.716849	−0.0665578
$117$	15.9651	1.47598
$118$	−17.5640	−1.61690
$119$	4.19146	0.384231
$120$	31.8925	2.91138
$121$	0	0
$122$	−14.9704	−1.35536
$123$	12.3811	1.11636
$124$	0.578830	0.0519804
$125$	−27.5757	−2.46645
$126$	4.64152	0.413500
$127$	10.7954	0.957932	0.478966	−	0.877833i	$-0.341012\pi$
$127$	10.7954	0.957932	0.478966	+	0.877833i	$0.341012\pi$
$128$	−7.20562	−0.636893
$129$	3.48696	0.307010
$130$	22.9336	2.01141
$131$	8.70429	0.760497	0.380249	−	0.924884i	$-0.375838\pi$
$131$	8.70429	0.760497	0.380249	+	0.924884i	$0.375838\pi$
$132$	0	0
$133$	−1.24880	−0.108285
$134$	−9.69261	−0.837314
$135$	6.70272	0.576878
$136$	−12.6831	−1.08757
$137$	11.9977	1.02503	0.512516	−	0.858677i	$-0.328713\pi$
$137$	11.9977	1.02503	0.512516	+	0.858677i	$0.328713\pi$
$138$	16.3496	1.39177
$139$	−8.18967	−0.694639	−0.347319	−	0.937747i	$-0.612908\pi$
$139$	−8.18967	−0.694639	−0.347319	+	0.937747i	$0.612908\pi$
$140$	−1.51540	−0.128075
$141$	26.9613	2.27055
$142$	−0.278695	−0.0233875
$143$	0	0
$144$	−11.3516	−0.945968
$145$	−7.91865	−0.657608
$146$	14.0057	1.15912
$147$	−2.57603	−0.212468
$148$	0.265305	0.0218079
$149$	−12.0816	−0.989766	−0.494883	−	0.868960i	$-0.664789\pi$
$149$	−12.0816	−0.989766	−0.494883	+	0.868960i	$0.664789\pi$
$150$	38.6062	3.15218
$151$	6.94850	0.565461	0.282730	−	0.959199i	$-0.408760\pi$
$151$	6.94850	0.565461	0.282730	+	0.959199i	$0.408760\pi$
$152$	3.77880	0.306501
$153$	−15.2399	−1.23208
$154$	0	0
$155$	6.39402	0.513580
$156$	4.18946	0.335425
$157$	10.4930	0.837436	0.418718	−	0.908116i	$-0.362480\pi$
$157$	10.4930	0.837436	0.418718	+	0.908116i	$0.362480\pi$
$158$	−5.82429	−0.463356
$159$	10.2315	0.811410
$160$	8.45455	0.668391
$161$	−4.97180	−0.391833
$162$	8.53730	0.670753
$163$	−7.81743	−0.612309	−0.306154	−	0.951982i	$-0.599042\pi$
$163$	−7.81743	−0.612309	−0.306154	+	0.951982i	$0.599042\pi$
$164$	1.78016	0.139007
$165$	0	0
$166$	3.13342	0.243201
$167$	−21.3503	−1.65214	−0.826068	−	0.563571i	$-0.809427\pi$
$167$	−21.3503	−1.65214	−0.826068	+	0.563571i	$0.809427\pi$
$168$	7.79494	0.601393
$169$	6.28007	0.483082
$170$	−21.8919	−1.67903
$171$	4.54057	0.347226
$172$	0.501358	0.0382282
$173$	−19.4384	−1.47787	−0.738936	−	0.673776i	$-0.764671\pi$
$173$	−19.4384	−1.47787	−0.738936	+	0.673776i	$0.764671\pi$
$174$	6.36458	0.482497
$175$	−11.7399	−0.887450
$176$	0	0
$177$	−35.4432	−2.66408
$178$	−5.36740	−0.402303
$179$	17.9964	1.34511	0.672557	−	0.740045i	$-0.265196\pi$
$179$	17.9964	1.34511	0.672557	+	0.740045i	$0.265196\pi$
$180$	5.50993	0.410686
$181$	15.1575	1.12665	0.563326	−	0.826235i	$-0.309522\pi$
$181$	15.1575	1.12665	0.563326	+	0.826235i	$0.309522\pi$
$182$	5.60527	0.415491
$183$	−30.2095	−2.23315
$184$	15.0444	1.10909
$185$	2.93068	0.215468
$186$	−5.13916	−0.376822
$187$	0	0
$188$	3.87652	0.282724
$189$	1.63823	0.119164
$190$	6.52245	0.473188
$191$	−23.7574	−1.71903	−0.859513	−	0.511115i	$-0.829233\pi$
$191$	−23.7574	−1.71903	−0.859513	+	0.511115i	$0.829233\pi$
$192$	−22.8803	−1.65124
$193$	−14.0582	−1.01193	−0.505965	−	0.862554i	$-0.668863\pi$
$193$	−14.0582	−1.01193	−0.505965	+	0.862554i	$0.668863\pi$
$194$	13.9463	1.00129
$195$	46.2788	3.31409
$196$	−0.370384	−0.0264560
$197$	−18.0665	−1.28718	−0.643591	−	0.765369i	$-0.722556\pi$
$197$	−18.0665	−1.28718	−0.643591	+	0.765369i	$0.722556\pi$
$198$	0	0
$199$	1.54374	0.109433	0.0547163	−	0.998502i	$-0.482575\pi$
$199$	1.54374	0.109433	0.0547163	+	0.998502i	$0.482575\pi$
$200$	35.5242	2.51194
$201$	−19.5591	−1.37960
$202$	9.40291	0.661586
$203$	−1.93542	−0.135840
$204$	−3.99917	−0.279998
$205$	19.6645	1.37343
$206$	0.196446	0.0136870
$207$	18.0772	1.25645
$208$	−13.7086	−0.950522
$209$	0	0
$210$	13.4546	0.928454
$211$	22.0836	1.52030	0.760149	−	0.649749i	$-0.225126\pi$
$211$	22.0836	1.52030	0.760149	+	0.649749i	$0.225126\pi$
$212$	1.47109	0.101035
$213$	−0.562390	−0.0385343
$214$	10.1438	0.693415
$215$	5.53824	0.377704
$216$	−4.95720	−0.337295
$217$	1.56278	0.106089
$218$	−6.27354	−0.424898
$219$	28.2628	1.90982
$220$	0	0
$221$	−18.4043	−1.23801
$222$	−2.35552	−0.158092
$223$	−8.57414	−0.574167	−0.287083	−	0.957906i	$-0.592686\pi$
$223$	−8.57414	−0.574167	−0.287083	+	0.957906i	$0.592686\pi$
$224$	2.06640	0.138067
$225$	42.6856	2.84570
$226$	−3.88342	−0.258321
$227$	26.5709	1.76357	0.881786	−	0.471650i	$-0.156341\pi$
$227$	26.5709	1.76357	0.881786	+	0.471650i	$0.156341\pi$
$228$	1.19151	0.0789094
$229$	11.4113	0.754081	0.377040	−	0.926197i	$-0.376942\pi$
$229$	11.4113	0.754081	0.377040	+	0.926197i	$0.376942\pi$
$230$	25.9676	1.71225
$231$	0	0
$232$	5.85648	0.384497
$233$	−26.6130	−1.74348	−0.871738	−	0.489972i	$-0.837007\pi$
$233$	−26.6130	−1.74348	−0.871738	+	0.489972i	$0.837007\pi$
$234$	−20.3805	−1.33232
$235$	42.8218	2.79339
$236$	−5.09606	−0.331725
$237$	−11.7531	−0.763445
$238$	−5.35067	−0.346832
$239$	7.51280	0.485963	0.242981	−	0.970031i	$-0.421875\pi$
$239$	7.51280	0.485963	0.242981	+	0.970031i	$0.421875\pi$
$240$	−32.9054	−2.12403
$241$	−27.4388	−1.76749	−0.883744	−	0.467971i	$-0.844985\pi$
$241$	−27.4388	−1.76749	−0.883744	+	0.467971i	$0.844985\pi$
$242$	0	0
$243$	22.1425	1.42044
$244$	−4.34355	−0.278067
$245$	−4.09144	−0.261392
$246$	−15.8052	−1.00771
$247$	5.48336	0.348898
$248$	−4.72889	−0.300285
$249$	6.32307	0.400708
$250$	35.2022	2.22638
$251$	1.92514	0.121514	0.0607568	−	0.998153i	$-0.480649\pi$
$251$	1.92514	0.121514	0.0607568	+	0.998153i	$0.480649\pi$
$252$	1.34670	0.0848340
$253$	0	0
$254$	−13.7810	−0.864694
$255$	−44.1767	−2.76645
$256$	−8.56553	−0.535346
$257$	10.1604	0.633791	0.316895	−	0.948460i	$-0.397360\pi$
$257$	10.1604	0.633791	0.316895	+	0.948460i	$0.397360\pi$
$258$	−4.45133	−0.277128
$259$	0.716296	0.0445085
$260$	6.65400	0.412663
$261$	7.03709	0.435585
$262$	−11.1116	−0.686476
$263$	4.38774	0.270560	0.135280	−	0.990807i	$-0.456807\pi$
$263$	4.38774	0.270560	0.135280	+	0.990807i	$0.456807\pi$
$264$	0	0
$265$	16.2504	0.998253
$266$	1.59417	0.0977449
$267$	−10.8311	−0.662853
$268$	−2.81223	−0.171784
$269$	0.625379	0.0381300	0.0190650	−	0.999818i	$-0.493931\pi$
$269$	0.625379	0.0381300	0.0190650	+	0.999818i	$0.493931\pi$
$270$	−8.55645	−0.520729
$271$	11.9375	0.725151	0.362575	−	0.931954i	$-0.381898\pi$
$271$	11.9375	0.725151	0.362575	+	0.931954i	$0.381898\pi$
$272$	13.0859	0.793452
$273$	11.3111	0.684581
$274$	−15.3158	−0.925263
$275$	0	0
$276$	4.74371	0.285538
$277$	−28.5571	−1.71583	−0.857915	−	0.513792i	$-0.828240\pi$
$277$	−28.5571	−1.71583	−0.857915	+	0.513792i	$0.828240\pi$
$278$	10.4546	0.627028
$279$	−5.68220	−0.340184
$280$	12.3805	0.739875
$281$	−14.6981	−0.876814	−0.438407	−	0.898777i	$-0.644457\pi$
$281$	−14.6981	−0.876814	−0.438407	+	0.898777i	$0.644457\pi$
$282$	−34.4178	−2.04955
$283$	−22.5396	−1.33984	−0.669921	−	0.742433i	$-0.733672\pi$
$283$	−22.5396	−1.33984	−0.669921	+	0.742433i	$0.733672\pi$
$284$	−0.0808609	−0.00479821
$285$	13.1619	0.779646
$286$	0	0
$287$	4.80626	0.283704
$288$	−7.51333	−0.442727
$289$	0.568350	0.0334323
$290$	10.1087	0.593601
$291$	28.1429	1.64976
$292$	4.06364	0.237807
$293$	−16.8280	−0.983105	−0.491553	−	0.870848i	$-0.663570\pi$
$293$	−16.8280	−0.983105	−0.491553	+	0.870848i	$0.663570\pi$
$294$	3.28847	0.191788
$295$	−56.2934	−3.27753
$296$	−2.16747	−0.125982
$297$	0	0
$298$	15.4230	0.893429
$299$	21.8307	1.26250
$300$	11.2013	0.646705
$301$	1.35362	0.0780211
$302$	−8.87021	−0.510423
$303$	18.9745	1.09006
$304$	−3.89881	−0.223612
$305$	−47.9808	−2.74738
$306$	19.4548	1.11216
$307$	−0.238354	−0.0136036	−0.00680179	−	0.999977i	$-0.502165\pi$
$307$	−0.238354	−0.0136036	−0.00680179	+	0.999977i	$0.502165\pi$
$308$	0	0
$309$	0.396416	0.0225513
$310$	−8.16238	−0.463592
$311$	−0.656523	−0.0372280	−0.0186140	−	0.999827i	$-0.505925\pi$
$311$	−0.656523	−0.0372280	−0.0186140	+	0.999827i	$0.505925\pi$
$312$	−34.2269	−1.93772
$313$	−7.45262	−0.421247	−0.210624	−	0.977567i	$-0.567549\pi$
$313$	−7.45262	−0.421247	−0.210624	+	0.977567i	$0.567549\pi$
$314$	−13.3950	−0.755926
$315$	14.8763	0.838183
$316$	−1.68987	−0.0950625
$317$	17.4412	0.979595	0.489797	−	0.871836i	$-0.337071\pi$
$317$	17.4412	0.979595	0.489797	+	0.871836i	$0.337071\pi$
$318$	−13.0612	−0.732433
$319$	0	0
$320$	−36.3401	−2.03147
$321$	20.4696	1.14250
$322$	6.34682	0.353695
$323$	−5.23429	−0.291244
$324$	2.47702	0.137612
$325$	51.5486	2.85940
$326$	9.97946	0.552711
$327$	−12.6597	−0.700081
$328$	−14.5435	−0.803030
$329$	10.4662	0.577021
$330$	0	0
$331$	−28.6146	−1.57280	−0.786399	−	0.617719i	$-0.788057\pi$
$331$	−28.6146	−1.57280	−0.786399	+	0.617719i	$0.788057\pi$
$332$	0.909136	0.0498953
$333$	−2.60442	−0.142721
$334$	27.2550	1.49133
$335$	−31.0652	−1.69727
$336$	−8.04250	−0.438754
$337$	−1.73053	−0.0942679	−0.0471339	−	0.998889i	$-0.515009\pi$
$337$	−1.73053	−0.0942679	−0.0471339	+	0.998889i	$0.515009\pi$
$338$	−8.01691	−0.436062
$339$	−7.83651	−0.425621
$340$	−6.35176	−0.344472
$341$	0	0
$342$	−5.79633	−0.313430
$343$	−1.00000	−0.0539949
$344$	−4.09597	−0.220840
$345$	52.4012	2.82119
$346$	24.8143	1.33403
$347$	−2.73009	−0.146559	−0.0732796	−	0.997311i	$-0.523347\pi$
$347$	−2.73009	−0.146559	−0.0732796	+	0.997311i	$0.523347\pi$
$348$	1.84663	0.0989896
$349$	−30.9063	−1.65437	−0.827187	−	0.561927i	$-0.810060\pi$
$349$	−30.9063	−1.65437	−0.827187	+	0.561927i	$0.810060\pi$
$350$	14.9867	0.801072
$351$	−7.19332	−0.383951
$352$	0	0
$353$	−23.9543	−1.27496	−0.637480	−	0.770467i	$-0.720023\pi$
$353$	−23.9543	−1.27496	−0.637480	+	0.770467i	$0.720023\pi$
$354$	45.2456	2.40478
$355$	−0.893227	−0.0474076
$356$	−1.55730	−0.0825369
$357$	−10.7973	−0.571456
$358$	−22.9736	−1.21419
$359$	1.51611	0.0800170	0.0400085	−	0.999199i	$-0.487261\pi$
$359$	1.51611	0.0800170	0.0400085	+	0.999199i	$0.487261\pi$
$360$	−45.0148	−2.37249
$361$	−17.4405	−0.917921
$362$	−19.3496	−1.01699
$363$	0	0
$364$	1.62632	0.0852425
$365$	44.8889	2.34959
$366$	38.5644	2.01579
$367$	8.72207	0.455288	0.227644	−	0.973744i	$-0.426898\pi$
$367$	8.72207	0.455288	0.227644	+	0.973744i	$0.426898\pi$
$368$	−15.5222	−0.809151
$369$	−17.4753	−0.909729
$370$	−3.74120	−0.194496
$371$	3.97180	0.206206
$372$	−1.49108	−0.0773091
$373$	36.8111	1.90601	0.953004	−	0.302957i	$-0.0979739\pi$
$373$	36.8111	1.90601	0.953004	+	0.302957i	$0.0979739\pi$
$374$	0	0
$375$	71.0360	3.66828
$376$	−31.6702	−1.63327
$377$	8.49825	0.437682
$378$	−2.09131	−0.107565
$379$	2.59838	0.133470	0.0667349	−	0.997771i	$-0.478742\pi$
$379$	2.59838	0.133470	0.0667349	+	0.997771i	$0.478742\pi$
$380$	1.89243	0.0970798
$381$	−27.8092	−1.42471
$382$	30.3278	1.55171
$383$	19.2392	0.983078	0.491539	−	0.870856i	$-0.336435\pi$
$383$	19.2392	0.983078	0.491539	+	0.870856i	$0.336435\pi$
$384$	18.5619	0.947235
$385$	0	0
$386$	17.9462	0.913435
$387$	−4.92168	−0.250183
$388$	4.04640	0.205425
$389$	−28.3331	−1.43654	−0.718272	−	0.695762i	$-0.755067\pi$
$389$	−28.3331	−1.43654	−0.718272	+	0.695762i	$0.755067\pi$
$390$	−59.0778	−2.99152
$391$	−20.8391	−1.05388
$392$	3.02595	0.152833
$393$	−22.4225	−1.13107
$394$	23.0630	1.16190
$395$	−18.6671	−0.939243
$396$	0	0
$397$	−10.3666	−0.520287	−0.260143	−	0.965570i	$-0.583770\pi$
$397$	−10.3666	−0.520287	−0.260143	+	0.965570i	$0.583770\pi$
$398$	−1.97068	−0.0987812
$399$	3.21695	0.161049
$400$	−36.6524	−1.83262
$401$	−12.7878	−0.638592	−0.319296	−	0.947655i	$-0.603446\pi$
$401$	−12.7878	−0.638592	−0.319296	+	0.947655i	$0.603446\pi$
$402$	24.9685	1.24532
$403$	−6.86203	−0.341822
$404$	2.72817	0.135732
$405$	27.3624	1.35965
$406$	2.47069	0.122618
$407$	0	0
$408$	32.6722	1.61752
$409$	32.0217	1.58337	0.791686	−	0.610928i	$-0.209203\pi$
$409$	32.0217	1.58337	0.791686	+	0.610928i	$0.209203\pi$
$410$	−25.1030	−1.23975
$411$	−30.9065	−1.52450
$412$	0.0569970	0.00280804
$413$	−13.7588	−0.677028
$414$	−23.0767	−1.13416
$415$	10.0427	0.492979
$416$	−9.07338	−0.444859
$417$	21.0969	1.03312
$418$	0	0
$419$	−20.0934	−0.981629	−0.490815	−	0.871264i	$-0.663301\pi$
$419$	−20.0934	−0.981629	−0.490815	+	0.871264i	$0.663301\pi$
$420$	3.90373	0.190483
$421$	−22.4243	−1.09289	−0.546446	−	0.837494i	$-0.684020\pi$
$421$	−22.4243	−1.09289	−0.546446	+	0.837494i	$0.684020\pi$
$422$	−28.1911	−1.37232
$423$	−38.0546	−1.85028
$424$	−12.0185	−0.583668
$425$	−49.2072	−2.38690
$426$	0.717927	0.0347837
$427$	−11.7271	−0.567516
$428$	2.94313	0.142262
$429$	0	0
$430$	−7.06991	−0.340941
$431$	−12.0856	−0.582144	−0.291072	−	0.956701i	$-0.594012\pi$
$431$	−12.0856	−0.582144	−0.291072	+	0.956701i	$0.594012\pi$
$432$	5.11463	0.246078
$433$	−0.302453	−0.0145350	−0.00726749	−	0.999974i	$-0.502313\pi$
$433$	−0.302453	−0.0145350	−0.00726749	+	0.999974i	$0.502313\pi$
$434$	−1.99499	−0.0957626
$435$	20.3987	0.978044
$436$	−1.82022	−0.0871725
$437$	6.20878	0.297006
$438$	−36.0792	−1.72393
$439$	3.52592	0.168283	0.0841416	−	0.996454i	$-0.473185\pi$
$439$	3.52592	0.168283	0.0841416	+	0.996454i	$0.473185\pi$
$440$	0	0
$441$	3.63595	0.173141
$442$	23.4943	1.11751
$443$	−21.8443	−1.03786	−0.518928	−	0.854818i	$-0.673669\pi$
$443$	−21.8443	−1.03786	−0.518928	+	0.854818i	$0.673669\pi$
$444$	−0.683434	−0.0324343
$445$	−17.2027	−0.815487
$446$	10.9454	0.518281
$447$	31.1227	1.47205
$448$	−8.88199	−0.419634
$449$	9.30172	0.438975	0.219488	−	0.975615i	$-0.429561\pi$
$449$	9.30172	0.438975	0.219488	+	0.975615i	$0.429561\pi$
$450$	−54.4908	−2.56872
$451$	0	0
$452$	−1.12674	−0.0529974
$453$	−17.8996	−0.840995
$454$	−33.9194	−1.59192
$455$	17.9651	0.842218
$456$	−9.73431	−0.455851
$457$	11.3165	0.529365	0.264683	−	0.964336i	$-0.414733\pi$
$457$	11.3165	0.529365	0.264683	+	0.964336i	$0.414733\pi$
$458$	−14.5673	−0.680684
$459$	6.86658	0.320505
$460$	7.53429	0.351288
$461$	0.678821	0.0316158	0.0158079	−	0.999875i	$-0.494968\pi$
$461$	0.678821	0.0316158	0.0158079	+	0.999875i	$0.494968\pi$
$462$	0	0
$463$	−13.4936	−0.627103	−0.313551	−	0.949571i	$-0.601519\pi$
$463$	−13.4936	−0.627103	−0.313551	+	0.949571i	$0.601519\pi$
$464$	−6.04247	−0.280515
$465$	−16.4712	−0.763835
$466$	33.9732	1.57378
$467$	29.6568	1.37235	0.686176	−	0.727435i	$-0.259288\pi$
$467$	29.6568	1.37235	0.686176	+	0.727435i	$0.259288\pi$
$468$	−5.91323	−0.273339
$469$	−7.59274	−0.350600
$470$	−54.6648	−2.52150
$471$	−27.0304	−1.24550
$472$	41.6335	1.91634
$473$	0	0
$474$	15.0036	0.689137
$475$	14.6607	0.672680
$476$	−1.55245	−0.0711565
$477$	−14.4413	−0.661220
$478$	−9.59057	−0.438662
$479$	−4.70136	−0.214811	−0.107405	−	0.994215i	$-0.534254\pi$
$479$	−4.70136	−0.214811	−0.107405	+	0.994215i	$0.534254\pi$
$480$	−21.7792	−0.994080
$481$	−3.14519	−0.143408
$482$	35.0274	1.59545
$483$	12.8075	0.582763
$484$	0	0
$485$	44.6985	2.02965
$486$	−28.2663	−1.28218
$487$	32.3738	1.46700	0.733498	−	0.679692i	$-0.237886\pi$
$487$	32.3738	1.46700	0.733498	+	0.679692i	$0.237886\pi$
$488$	35.4857	1.60636
$489$	20.1380	0.910671
$490$	5.22298	0.235950
$491$	6.85594	0.309404	0.154702	−	0.987961i	$-0.450558\pi$
$491$	6.85594	0.309404	0.154702	+	0.987961i	$0.450558\pi$
$492$	−4.58576	−0.206742
$493$	−8.11224	−0.365357
$494$	−6.99986	−0.314939
$495$	0	0
$496$	4.87908	0.219077
$497$	−0.218316	−0.00979282
$498$	−8.07181	−0.361706
$499$	10.9670	0.490952	0.245476	−	0.969403i	$-0.421056\pi$
$499$	10.9670	0.490952	0.245476	+	0.969403i	$0.421056\pi$
$500$	10.2136	0.456766
$501$	54.9991	2.45718
$502$	−2.45756	−0.109686
$503$	26.9334	1.20090	0.600451	−	0.799662i	$-0.294988\pi$
$503$	26.9334	1.20090	0.600451	+	0.799662i	$0.294988\pi$
$504$	−11.0022	−0.490077
$505$	30.1367	1.34106
$506$	0	0
$507$	−16.1777	−0.718475
$508$	−3.99843	−0.177402
$509$	30.3958	1.34727	0.673635	−	0.739064i	$-0.264732\pi$
$509$	30.3958	1.34727	0.673635	+	0.739064i	$0.264732\pi$
$510$	56.3944	2.49718
$511$	10.9714	0.485347
$512$	25.3457	1.12013
$513$	−2.04582	−0.0903252
$514$	−12.9705	−0.572102
$515$	0.629616	0.0277442
$516$	−1.29151	−0.0568558
$517$	0	0
$518$	−0.914398	−0.0401763
$519$	50.0739	2.19800
$520$	−54.3615	−2.38391
$521$	−0.935426	−0.0409818	−0.0204909	−	0.999790i	$-0.506523\pi$
$521$	−0.935426	−0.0409818	−0.0204909	+	0.999790i	$0.506523\pi$
$522$	−8.98330	−0.393188
$523$	−26.1853	−1.14500	−0.572502	−	0.819903i	$-0.694027\pi$
$523$	−26.1853	−1.14500	−0.572502	+	0.819903i	$0.694027\pi$
$524$	−3.22393	−0.140838
$525$	30.2423	1.31988
$526$	−5.60124	−0.244226
$527$	6.55034	0.285337
$528$	0	0
$529$	1.71881	0.0747307
$530$	−20.7446	−0.901090
$531$	50.0265	2.17096
$532$	0.462535	0.0200534
$533$	−21.1038	−0.914109
$534$	13.8266	0.598335
$535$	32.5112	1.40558
$536$	22.9752	0.992378
$537$	−46.3594	−2.00055
$538$	−0.798336	−0.0344187
$539$	0	0
$540$	−2.48258	−0.106833
$541$	−26.7566	−1.15035	−0.575177	−	0.818029i	$-0.695067\pi$
$541$	−26.7566	−1.15035	−0.575177	+	0.818029i	$0.695067\pi$
$542$	−15.2390	−0.654570
$543$	−39.0464	−1.67564
$544$	8.66124	0.371348
$545$	−20.1070	−0.861287
$546$	−14.4394	−0.617948
$547$	−46.4581	−1.98641	−0.993203	−	0.116397i	$-0.962865\pi$
$547$	−46.4581	−1.98641	−0.993203	+	0.116397i	$0.962865\pi$
$548$	−4.44376	−0.189828
$549$	42.6393	1.81980
$550$	0	0
$551$	2.41695	0.102966
$552$	−38.7549	−1.64952
$553$	−4.56248	−0.194016
$554$	36.4550	1.54882
$555$	−7.54953	−0.320460
$556$	3.03332	0.128642
$557$	−35.0974	−1.48712	−0.743562	−	0.668667i	$-0.766865\pi$
$557$	−35.0974	−1.48712	−0.743562	+	0.668667i	$0.766865\pi$
$558$	7.25369	0.307073
$559$	−5.94360	−0.251388
$560$	−12.7737	−0.539786
$561$	0	0
$562$	18.7630	0.791471
$563$	8.94791	0.377109	0.188555	−	0.982063i	$-0.439620\pi$
$563$	8.94791	0.377109	0.188555	+	0.982063i	$0.439620\pi$
$564$	−9.98604	−0.420488
$565$	−12.4465	−0.523628
$566$	28.7733	1.20943
$567$	6.68771	0.280858
$568$	0.660614	0.0277187
$569$	−19.4256	−0.814365	−0.407183	−	0.913347i	$-0.633489\pi$
$569$	−19.4256	−0.814365	−0.407183	+	0.913347i	$0.633489\pi$
$570$	−16.8021	−0.703761
$571$	−16.0171	−0.670295	−0.335148	−	0.942166i	$-0.608786\pi$
$571$	−16.0171	−0.670295	−0.335148	+	0.942166i	$0.608786\pi$
$572$	0	0
$573$	61.1999	2.55666
$574$	−6.13550	−0.256091
$575$	58.3683	2.43412
$576$	32.2945	1.34560
$577$	−32.9192	−1.37044	−0.685222	−	0.728335i	$-0.740295\pi$
$577$	−32.9192	−1.37044	−0.685222	+	0.728335i	$0.740295\pi$
$578$	−0.725535	−0.0301783
$579$	36.2143	1.50502
$580$	2.93294	0.121784
$581$	2.45458	0.101833
$582$	−35.9262	−1.48919
$583$	0	0
$584$	−33.1990	−1.37378
$585$	−65.3203	−2.70066
$586$	21.4821	0.887417
$587$	27.4372	1.13246	0.566228	−	0.824249i	$-0.308402\pi$
$587$	27.4372	1.13246	0.566228	+	0.824249i	$0.308402\pi$
$588$	0.954122	0.0393473
$589$	−1.95160	−0.0804142
$590$	71.8622	2.95852
$591$	46.5398	1.91439
$592$	2.23631	0.0919118
$593$	−14.6132	−0.600092	−0.300046	−	0.953925i	$-0.597002\pi$
$593$	−14.6132	−0.600092	−0.300046	+	0.953925i	$0.597002\pi$
$594$	0	0
$595$	−17.1491	−0.703045
$596$	4.47485	0.183297
$597$	−3.97672	−0.162756
$598$	−27.8683	−1.13962
$599$	25.2765	1.03277	0.516384	−	0.856357i	$-0.327278\pi$
$599$	25.2765	1.03277	0.516384	+	0.856357i	$0.327278\pi$
$600$	−91.5115	−3.73594
$601$	9.30098	0.379395	0.189697	−	0.981843i	$-0.439249\pi$
$601$	9.30098	0.379395	0.189697	+	0.981843i	$0.439249\pi$
$602$	−1.72798	−0.0704271
$603$	27.6068	1.12424
$604$	−2.57361	−0.104719
$605$	0	0
$606$	−24.2222	−0.983960
$607$	2.12671	0.0863207	0.0431603	−	0.999068i	$-0.486257\pi$
$607$	2.12671	0.0863207	0.0431603	+	0.999068i	$0.486257\pi$
$608$	−2.58052	−0.104654
$609$	4.98571	0.202031
$610$	61.2506	2.47997
$611$	−45.9561	−1.85919
$612$	5.64463	0.228171
$613$	40.9448	1.65374	0.826872	−	0.562390i	$-0.190118\pi$
$613$	40.9448	1.65374	0.826872	+	0.562390i	$0.190118\pi$
$614$	0.304274	0.0122795
$615$	−50.6564	−2.04266
$616$	0	0
$617$	−7.53813	−0.303474	−0.151737	−	0.988421i	$-0.548487\pi$
$617$	−7.53813	−0.303474	−0.151737	+	0.988421i	$0.548487\pi$
$618$	−0.506051	−0.0203563
$619$	−19.5055	−0.783994	−0.391997	−	0.919967i	$-0.628216\pi$
$619$	−19.5055	−0.783994	−0.391997	+	0.919967i	$0.628216\pi$
$620$	−2.36824	−0.0951110
$621$	−8.14496	−0.326846
$622$	0.838094	0.0336045
$623$	−4.20456	−0.168452
$624$	35.3139	1.41369
$625$	54.1250	2.16500
$626$	9.51375	0.380246
$627$	0	0
$628$	−3.88645	−0.155086
$629$	3.00233	0.119711
$630$	−18.9905	−0.756600
$631$	−10.6654	−0.424582	−0.212291	−	0.977207i	$-0.568092\pi$
$631$	−10.6654	−0.424582	−0.212291	+	0.977207i	$0.568092\pi$
$632$	13.8058	0.549166
$633$	−56.8881	−2.26110
$634$	−22.2648	−0.884248
$635$	−44.1685	−1.75277
$636$	−3.78958	−0.150267
$637$	4.39091	0.173974
$638$	0	0
$639$	0.793787	0.0314017
$640$	29.4814	1.16535
$641$	0.448349	0.0177087	0.00885436	−	0.999961i	$-0.497182\pi$
$641$	0.448349	0.0177087	0.00885436	+	0.999961i	$0.497182\pi$
$642$	−26.1307	−1.03130
$643$	−31.6440	−1.24792	−0.623958	−	0.781458i	$-0.714476\pi$
$643$	−31.6440	−1.24792	−0.623958	+	0.781458i	$0.714476\pi$
$644$	1.84148	0.0725643
$645$	−14.2667	−0.561750
$646$	6.68191	0.262896
$647$	−30.1242	−1.18430	−0.592152	−	0.805826i	$-0.701721\pi$
$647$	−30.1242	−1.18430	−0.592152	+	0.805826i	$0.701721\pi$
$648$	−20.2367	−0.794972
$649$	0	0
$650$	−65.8051	−2.58109
$651$	−4.02578	−0.157783
$652$	2.89545	0.113395
$653$	34.6110	1.35443	0.677217	−	0.735784i	$-0.263186\pi$
$653$	34.6110	1.35443	0.677217	+	0.735784i	$0.263186\pi$
$654$	16.1609	0.631940
$655$	−35.6131	−1.39152
$656$	15.0054	0.585861
$657$	−39.8916	−1.55632
$658$	−13.3608	−0.520858
$659$	20.2587	0.789167	0.394584	−	0.918860i	$-0.370889\pi$
$659$	20.2587	0.789167	0.394584	+	0.918860i	$0.370889\pi$
$660$	0	0
$661$	40.6737	1.58202	0.791012	−	0.611801i	$-0.209555\pi$
$661$	40.6737	1.58202	0.791012	+	0.611801i	$0.209555\pi$
$662$	36.5283	1.41971
$663$	47.4102	1.84126
$664$	−7.42742	−0.288240
$665$	5.10938	0.198133
$666$	3.32471	0.128830
$667$	9.62253	0.372586
$668$	7.90781	0.305962
$669$	22.0873	0.853943
$670$	39.6567	1.53207
$671$	0	0
$672$	−5.32312	−0.205344
$673$	−24.7484	−0.953980	−0.476990	−	0.878909i	$-0.658272\pi$
$673$	−24.7484	−0.953980	−0.476990	+	0.878909i	$0.658272\pi$
$674$	2.20913	0.0850925
$675$	−19.2326	−0.740263
$676$	−2.32604	−0.0894630
$677$	11.3821	0.437452	0.218726	−	0.975786i	$-0.429810\pi$
$677$	11.3821	0.437452	0.218726	+	0.975786i	$0.429810\pi$
$678$	10.0038	0.384194
$679$	10.9249	0.419259
$680$	51.8923	1.98998
$681$	−68.4475	−2.62291
$682$	0	0
$683$	−0.212700	−0.00813875	−0.00406938	−	0.999992i	$-0.501295\pi$
$683$	−0.212700	−0.00813875	−0.00406938	+	0.999992i	$0.501295\pi$
$684$	−1.68176	−0.0643035
$685$	−49.0878	−1.87555
$686$	1.27656	0.0487394
$687$	−29.3959	−1.12152
$688$	4.22605	0.161117
$689$	−17.4398	−0.664404
$690$	−66.8935	−2.54659
$691$	36.3804	1.38398	0.691988	−	0.721909i	$-0.256735\pi$
$691$	36.3804	1.38398	0.691988	+	0.721909i	$0.256735\pi$
$692$	7.19966	0.273690
$693$	0	0
$694$	3.48514	0.132294
$695$	33.5075	1.27101
$696$	−15.0865	−0.571852
$697$	20.1452	0.763056
$698$	39.4538	1.49335
$699$	68.5560	2.59303
$700$	4.34826	0.164349
$701$	18.7394	0.707779	0.353889	−	0.935287i	$-0.384859\pi$
$701$	18.7394	0.707779	0.353889	+	0.935287i	$0.384859\pi$
$702$	9.18273	0.346580
$703$	−0.894509	−0.0337371
$704$	0	0
$705$	−110.310	−4.15453
$706$	30.5792	1.15086
$707$	7.36579	0.277019
$708$	13.1276	0.493366
$709$	−12.7596	−0.479195	−0.239598	−	0.970872i	$-0.577015\pi$
$709$	−12.7596	−0.479195	−0.239598	+	0.970872i	$0.577015\pi$
$710$	1.14026	0.0427933
$711$	16.5889	0.622134
$712$	12.7228	0.476807
$713$	−7.76984	−0.290983
$714$	13.7835	0.515835
$715$	0	0
$716$	−6.66558	−0.249105
$717$	−19.3532	−0.722759
$718$	−1.93541	−0.0722287
$719$	−17.1228	−0.638571	−0.319286	−	0.947658i	$-0.603443\pi$
$719$	−17.1228	−0.638571	−0.319286	+	0.947658i	$0.603443\pi$
$720$	46.4444	1.73088
$721$	0.153886	0.00573102
$722$	22.2639	0.828577
$723$	70.6832	2.62874
$724$	−5.61411	−0.208647
$725$	22.7216	0.843858
$726$	0	0
$727$	−8.65786	−0.321102	−0.160551	−	0.987028i	$-0.551327\pi$
$727$	−8.65786	−0.321102	−0.160551	+	0.987028i	$0.551327\pi$
$728$	−13.2867	−0.492436
$729$	−36.9766	−1.36950
$730$	−57.3036	−2.12090
$731$	5.67363	0.209847
$732$	11.1891	0.413562
$733$	−28.0866	−1.03740	−0.518702	−	0.854955i	$-0.673584\pi$
$733$	−28.0866	−1.03740	−0.518702	+	0.854955i	$0.673584\pi$
$734$	−11.1343	−0.410974
$735$	10.5397	0.388762
$736$	−10.2737	−0.378695
$737$	0	0
$738$	22.3084	0.821182
$739$	33.8965	1.24690	0.623451	−	0.781862i	$-0.285730\pi$
$739$	33.8965	1.24690	0.623451	+	0.781862i	$0.285730\pi$
$740$	−1.08548	−0.0399030
$741$	−14.1253	−0.518906
$742$	−5.07026	−0.186135
$743$	5.65321	0.207396	0.103698	−	0.994609i	$-0.466932\pi$
$743$	5.65321	0.207396	0.103698	+	0.994609i	$0.466932\pi$
$744$	12.1818	0.446606
$745$	49.4313	1.81102
$746$	−46.9918	−1.72049
$747$	−8.92472	−0.326538
$748$	0	0
$749$	7.94617	0.290347
$750$	−90.6820	−3.31124
$751$	44.3409	1.61802	0.809012	−	0.587792i	$-0.200003\pi$
$751$	44.3409	1.61802	0.809012	+	0.587792i	$0.200003\pi$
$752$	32.6760	1.19157
$753$	−4.95922	−0.180724
$754$	−10.8486	−0.395081
$755$	−28.4294	−1.03465
$756$	−0.606775	−0.0220682
$757$	−21.3781	−0.777001	−0.388500	−	0.921449i	$-0.627007\pi$
$757$	−21.3781	−0.777001	−0.388500	+	0.921449i	$0.627007\pi$
$758$	−3.31700	−0.120479
$759$	0	0
$760$	−15.4607	−0.560819
$761$	4.58574	0.166233	0.0831164	−	0.996540i	$-0.473513\pi$
$761$	4.58574	0.166233	0.0831164	+	0.996540i	$0.473513\pi$
$762$	35.5002	1.28604
$763$	−4.91440	−0.177913
$764$	8.79936	0.318350
$765$	62.3533	2.25439
$766$	−24.5601	−0.887392
$767$	60.4138	2.18142
$768$	22.0651	0.796206
$769$	−12.8223	−0.462385	−0.231193	−	0.972908i	$-0.574263\pi$
$769$	−12.8223	−0.462385	−0.231193	+	0.972908i	$0.574263\pi$
$770$	0	0
$771$	−26.1736	−0.942621
$772$	5.20692	0.187401
$773$	54.0639	1.94454	0.972272	−	0.233855i	$-0.0751340\pi$
$773$	54.0639	1.94454	0.972272	+	0.233855i	$0.0751340\pi$
$774$	6.28284	0.225832
$775$	−18.3468	−0.659038
$776$	−33.0581	−1.18672
$777$	−1.84520	−0.0661963
$778$	36.1690	1.29672
$779$	−6.00205	−0.215046
$780$	−17.1409	−0.613743
$781$	0	0
$782$	26.6025	0.951302
$783$	−3.17067	−0.113310
$784$	−3.12205	−0.111502
$785$	−42.9316	−1.53229
$786$	28.6238	1.02098
$787$	43.8514	1.56314	0.781568	−	0.623821i	$-0.214420\pi$
$787$	43.8514	1.56314	0.781568	+	0.623821i	$0.214420\pi$
$788$	6.69153	0.238376
$789$	−11.3030	−0.402397
$790$	23.8297	0.847824
$791$	−3.04208	−0.108164
$792$	0	0
$793$	51.4928	1.82856
$794$	13.2337	0.469646
$795$	−41.8615	−1.48468
$796$	−0.571776	−0.0202660
$797$	−35.7697	−1.26703	−0.633514	−	0.773731i	$-0.718388\pi$
$797$	−35.7697	−1.26703	−0.633514	+	0.773731i	$0.718388\pi$
$798$	−4.10664	−0.145373
$799$	43.8687	1.55196
$800$	−24.2593	−0.857694
$801$	15.2876	0.540160
$802$	16.3244	0.576436
$803$	0	0
$804$	7.24440	0.255490
$805$	20.3418	0.716955
$806$	8.75982	0.308551
$807$	−1.61100	−0.0567098
$808$	−22.2885	−0.784107
$809$	30.7068	1.07959	0.539797	−	0.841795i	$-0.318501\pi$
$809$	30.7068	1.07959	0.539797	+	0.841795i	$0.318501\pi$
$810$	−34.9298	−1.22731
$811$	−17.0984	−0.600405	−0.300202	−	0.953876i	$-0.597054\pi$
$811$	−17.0984	−0.600405	−0.300202	+	0.953876i	$0.597054\pi$
$812$	0.716849	0.0251565
$813$	−30.7514	−1.07850
$814$	0	0
$815$	31.9845	1.12037
$816$	−33.7098	−1.18008
$817$	−1.69039	−0.0591394
$818$	−40.8778	−1.42926
$819$	−15.9651	−0.557867
$820$	−7.28342	−0.254348
$821$	12.3822	0.432142	0.216071	−	0.976378i	$-0.430676\pi$
$821$	12.3822	0.432142	0.216071	+	0.976378i	$0.430676\pi$
$822$	39.4541	1.37612
$823$	46.1384	1.60829	0.804143	−	0.594436i	$-0.202625\pi$
$823$	46.1384	1.60829	0.804143	+	0.594436i	$0.202625\pi$
$824$	−0.465652	−0.0162217
$825$	0	0
$826$	17.5640	0.611131
$827$	−46.7103	−1.62428	−0.812138	−	0.583466i	$-0.801696\pi$
$827$	−46.7103	−1.62428	−0.812138	+	0.583466i	$0.801696\pi$
$828$	−6.69552	−0.232685
$829$	4.02555	0.139813	0.0699065	−	0.997554i	$-0.477730\pi$
$829$	4.02555	0.139813	0.0699065	+	0.997554i	$0.477730\pi$
$830$	−12.8202	−0.444996
$831$	73.5641	2.55191
$832$	39.0000	1.35208
$833$	−4.19146	−0.145226
$834$	−26.9315	−0.932562
$835$	87.3534	3.02299
$836$	0	0
$837$	2.56020	0.0884933
$838$	25.6506	0.886084
$839$	−14.3021	−0.493765	−0.246882	−	0.969045i	$-0.579406\pi$
$839$	−14.3021	−0.493765	−0.246882	+	0.969045i	$0.579406\pi$
$840$	−31.8925	−1.10040
$841$	−25.2541	−0.870833
$842$	28.6260	0.986518
$843$	37.8628	1.30406
$844$	−8.17941	−0.281547
$845$	−25.6945	−0.883918
$846$	48.5792	1.67019
$847$	0	0
$848$	12.4002	0.425823
$849$	58.0628	1.99271
$850$	62.8161	2.15457
$851$	−3.56128	−0.122079
$852$	0.208300	0.00713625
$853$	32.3007	1.10595	0.552977	−	0.833197i	$-0.313492\pi$
$853$	32.3007	1.10595	0.552977	+	0.833197i	$0.313492\pi$
$854$	14.9704	0.512278
$855$	−18.5775	−0.635336
$856$	−24.0447	−0.821830
$857$	−16.2318	−0.554468	−0.277234	−	0.960802i	$-0.589418\pi$
$857$	−16.2318	−0.554468	−0.277234	+	0.960802i	$0.589418\pi$
$858$	0	0
$859$	−28.2893	−0.965220	−0.482610	−	0.875835i	$-0.660311\pi$
$859$	−28.2893	−0.965220	−0.482610	+	0.875835i	$0.660311\pi$
$860$	−2.05127	−0.0699479
$861$	−12.3811	−0.421946
$862$	15.4281	0.525482
$863$	31.9865	1.08883	0.544417	−	0.838815i	$-0.316751\pi$
$863$	31.9865	1.08883	0.544417	+	0.838815i	$0.316751\pi$
$864$	3.38524	0.115168
$865$	79.5309	2.70413
$866$	0.386101	0.0131203
$867$	−1.46409	−0.0497230
$868$	−0.578830	−0.0196468
$869$	0	0
$870$	−26.0403	−0.882848
$871$	33.3390	1.12965
$872$	14.8707	0.503586
$873$	−39.7223	−1.34440
$874$	−7.92590	−0.268098
$875$	27.5757	0.932229
$876$	−10.4681	−0.353684
$877$	−45.7968	−1.54645	−0.773224	−	0.634133i	$-0.781357\pi$
$877$	−45.7968	−1.54645	−0.773224	+	0.634133i	$0.781357\pi$
$878$	−4.50107	−0.151904
$879$	43.3496	1.46215
$880$	0	0
$881$	26.7142	0.900025	0.450012	−	0.893022i	$-0.351420\pi$
$881$	26.7142	0.900025	0.450012	+	0.893022i	$0.351420\pi$
$882$	−4.64152	−0.156288
$883$	−28.4869	−0.958662	−0.479331	−	0.877634i	$-0.659121\pi$
$883$	−28.4869	−0.958662	−0.479331	+	0.877634i	$0.659121\pi$
$884$	6.81667	0.229269
$885$	145.014	4.87459
$886$	27.8857	0.936838
$887$	23.6139	0.792879	0.396439	−	0.918061i	$-0.370246\pi$
$887$	23.6139	0.792879	0.396439	+	0.918061i	$0.370246\pi$
$888$	5.58349	0.187370
$889$	−10.7954	−0.362064
$890$	21.9604	0.736113
$891$	0	0
$892$	3.17572	0.106331
$893$	−13.0702	−0.437377
$894$	−39.7301	−1.32877
$895$	−73.6312	−2.46122
$896$	7.20562	0.240723
$897$	−56.2367	−1.87769
$898$	−11.8742	−0.396249
$899$	−3.02464	−0.100877
$900$	−15.8101	−0.527002
$901$	16.6477	0.554614
$902$	0	0
$903$	−3.48696	−0.116039
$904$	9.20519	0.306160
$905$	−62.0161	−2.06149
$906$	22.8500	0.759139
$907$	23.9434	0.795029	0.397514	−	0.917596i	$-0.369873\pi$
$907$	23.9434	0.795029	0.397514	+	0.917596i	$0.369873\pi$
$908$	−9.84143	−0.326599
$909$	−26.7817	−0.888292
$910$	−22.9336	−0.760242
$911$	18.2388	0.604280	0.302140	−	0.953264i	$-0.402299\pi$
$911$	18.2388	0.604280	0.302140	+	0.953264i	$0.402299\pi$
$912$	10.0435	0.332572
$913$	0	0
$914$	−14.4463	−0.477841
$915$	123.600	4.08610
$916$	−4.22657	−0.139650
$917$	−8.70429	−0.287441
$918$	−8.76563	−0.289309
$919$	−28.4024	−0.936909	−0.468455	−	0.883488i	$-0.655189\pi$
$919$	−28.4024	−0.936909	−0.468455	+	0.883488i	$0.655189\pi$
$920$	−61.5533	−2.02935
$921$	0.614008	0.0202323
$922$	−0.866558	−0.0285386
$923$	0.958607	0.0315529
$924$	0	0
$925$	−8.40922	−0.276493
$926$	17.2255	0.566065
$927$	−0.559523	−0.0183771
$928$	−3.99936	−0.131285
$929$	3.38963	0.111210	0.0556051	−	0.998453i	$-0.482291\pi$
$929$	3.38963	0.111210	0.0556051	+	0.998453i	$0.482291\pi$
$930$	21.0266	0.689488
$931$	1.24880	0.0409277
$932$	9.85704	0.322878
$933$	1.69123	0.0553682
$934$	−37.8588	−1.23878
$935$	0	0
$936$	48.3096	1.57905
$937$	−45.3871	−1.48273	−0.741367	−	0.671100i	$-0.765822\pi$
$937$	−45.3871	−1.48273	−0.741367	+	0.671100i	$0.765822\pi$
$938$	9.69261	0.316475
$939$	19.1982	0.626510
$940$	−15.8605	−0.517313
$941$	−21.5279	−0.701789	−0.350894	−	0.936415i	$-0.614122\pi$
$941$	−21.5279	−0.701789	−0.350894	+	0.936415i	$0.614122\pi$
$942$	34.5061	1.12427
$943$	−23.8958	−0.778153
$944$	−42.9558	−1.39809
$945$	−6.70272	−0.218039
$946$	0	0
$947$	−11.2673	−0.366140	−0.183070	−	0.983100i	$-0.558603\pi$
$947$	−11.2673	−0.366140	−0.183070	+	0.983100i	$0.558603\pi$
$948$	4.35316	0.141384
$949$	−48.1745	−1.56381
$950$	−18.7153	−0.607206
$951$	−44.9291	−1.45693
$952$	12.6831	0.411063
$953$	23.8696	0.773213	0.386607	−	0.922245i	$-0.373647\pi$
$953$	23.8696	0.773213	0.386607	+	0.922245i	$0.373647\pi$
$954$	18.4352	0.596862
$955$	97.2019	3.14538
$956$	−2.78262	−0.0899964
$957$	0	0
$958$	6.00159	0.193902
$959$	−11.9977	−0.387426
$960$	93.6133	3.02136
$961$	−28.5577	−0.921217
$962$	4.01504	0.129450
$963$	−28.8919	−0.931027
$964$	10.1629	0.327325
$965$	57.5181	1.85157
$966$	−16.3496	−0.526041
$967$	−20.5165	−0.659765	−0.329882	−	0.944022i	$-0.607009\pi$
$967$	−20.5165	−0.659765	−0.329882	+	0.944022i	$0.607009\pi$
$968$	0	0
$969$	13.4837	0.433159
$970$	−57.0605	−1.83210
$971$	−34.7645	−1.11565	−0.557824	−	0.829960i	$-0.688363\pi$
$971$	−34.7645	−1.11565	−0.557824	+	0.829960i	$0.688363\pi$
$972$	−8.20122	−0.263054
$973$	8.18967	0.262549
$974$	−41.3272	−1.32421
$975$	−132.791	−4.25272
$976$	−36.6127	−1.17194
$977$	5.78294	0.185013	0.0925064	−	0.995712i	$-0.470512\pi$
$977$	5.78294	0.185013	0.0925064	+	0.995712i	$0.470512\pi$
$978$	−25.7074	−0.822032
$979$	0	0
$980$	1.51540	0.0484078
$981$	17.8685	0.570498
$982$	−8.75204	−0.279289
$983$	−17.2321	−0.549618	−0.274809	−	0.961499i	$-0.588615\pi$
$983$	−17.2321	−0.549618	−0.274809	+	0.961499i	$0.588615\pi$
$984$	37.4645	1.19432
$985$	73.9178	2.35522
$986$	10.3558	0.329796
$987$	−26.9613	−0.858188
$988$	−2.03095	−0.0646131
$989$	−6.72991	−0.213999
$990$	0	0
$991$	−24.0077	−0.762630	−0.381315	−	0.924445i	$-0.624529\pi$
$991$	−24.0077	−0.762630	−0.381315	+	0.924445i	$0.624529\pi$
$992$	3.22933	0.102531
$993$	73.7121	2.33918
$994$	0.278695	0.00883966
$995$	−6.31610	−0.200234
$996$	−2.34197	−0.0742080
$997$	−36.0335	−1.14119	−0.570596	−	0.821231i	$-0.693288\pi$
$997$	−36.0335	−1.14119	−0.570596	+	0.821231i	$0.693288\pi$
$998$	−14.0001	−0.443166
$999$	1.17346	0.0371266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.m.1.3	✓	6
3.2	odd	2		7623.2.a.cs.1.4		6
7.6	odd	2		5929.2.a.bj.1.3		6
11.2	odd	10		847.2.f.y.323.3		24
11.3	even	5		847.2.f.z.372.3		24
11.4	even	5		847.2.f.z.148.3		24
11.5	even	5		847.2.f.z.729.4		24
11.6	odd	10		847.2.f.y.729.3		24
11.7	odd	10		847.2.f.y.148.4		24
11.8	odd	10		847.2.f.y.372.4		24
11.9	even	5		847.2.f.z.323.4		24
11.10	odd	2		847.2.a.n.1.4	yes	6
33.32	even	2		7623.2.a.cp.1.3		6
77.76	even	2		5929.2.a.bm.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.m.1.3	✓	6	1.1	even	1	trivial
847.2.a.n.1.4	yes	6	11.10	odd	2
847.2.f.y.148.4		24	11.7	odd	10
847.2.f.y.323.3		24	11.2	odd	10
847.2.f.y.372.4		24	11.8	odd	10
847.2.f.y.729.3		24	11.6	odd	10
847.2.f.z.148.3		24	11.4	even	5
847.2.f.z.323.4		24	11.9	even	5
847.2.f.z.372.3		24	11.3	even	5
847.2.f.z.729.4		24	11.5	even	5
5929.2.a.bj.1.3		6	7.6	odd	2
5929.2.a.bm.1.4		6	77.76	even	2
7623.2.a.cp.1.3		6	33.32	even	2
7623.2.a.cs.1.4		6	3.2	odd	2

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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q-1.27656 q^{2} -2.57603 q^{3} -0.370384 q^{4} -4.09144 q^{5} +3.28847 q^{6} -1.00000 q^{7} +3.02595 q^{8} +3.63595 q^{9} +O(q^{10})\)	\(q-1.27656 q^{2} -2.57603 q^{3} -0.370384 q^{4} -4.09144 q^{5} +3.28847 q^{6} -1.00000 q^{7} +3.02595 q^{8} +3.63595 q^{9} +5.22298 q^{10} +0.954122 q^{12} +4.39091 q^{13} +1.27656 q^{14} +10.5397 q^{15} -3.12205 q^{16} -4.19146 q^{17} -4.64152 q^{18} +1.24880 q^{19} +1.51540 q^{20} +2.57603 q^{21} +4.97180 q^{23} -7.79494 q^{24} +11.7399 q^{25} -5.60527 q^{26} -1.63823 q^{27} +0.370384 q^{28} +1.93542 q^{29} -13.4546 q^{30} -1.56278 q^{31} -2.06640 q^{32} +5.35067 q^{34} +4.09144 q^{35} -1.34670 q^{36} -0.716296 q^{37} -1.59417 q^{38} -11.3111 q^{39} -12.3805 q^{40} -4.80626 q^{41} -3.28847 q^{42} -1.35362 q^{43} -14.8763 q^{45} -6.34682 q^{46} -10.4662 q^{47} +8.04250 q^{48} +1.00000 q^{49} -14.9867 q^{50} +10.7973 q^{51} -1.62632 q^{52} -3.97180 q^{53} +2.09131 q^{54} -3.02595 q^{56} -3.21695 q^{57} -2.47069 q^{58} +13.7588 q^{59} -3.90373 q^{60} +11.7271 q^{61} +1.99499 q^{62} -3.63595 q^{63} +8.88199 q^{64} -17.9651 q^{65} +7.59274 q^{67} +1.55245 q^{68} -12.8075 q^{69} -5.22298 q^{70} +0.218316 q^{71} +11.0022 q^{72} -10.9714 q^{73} +0.914398 q^{74} -30.2423 q^{75} -0.462535 q^{76} +14.4394 q^{78} +4.56248 q^{79} +12.7737 q^{80} -6.68771 q^{81} +6.13550 q^{82} -2.45458 q^{83} -0.954122 q^{84} +17.1491 q^{85} +1.72798 q^{86} -4.98571 q^{87} +4.20456 q^{89} +18.9905 q^{90} -4.39091 q^{91} -1.84148 q^{92} +4.02578 q^{93} +13.3608 q^{94} -5.10938 q^{95} +5.32312 q^{96} -10.9249 q^{97} -1.27656 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9	\( 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9} + 8 q^{10} - 14 q^{12} - 4 q^{13} + 4 q^{14} + 2 q^{15} + 8 q^{16} - 22 q^{17} - 24 q^{18} - 6 q^{19} + 2 q^{20} + 2 q^{21} + 2 q^{23} + 20 q^{24} + 4 q^{25} + 6 q^{26} - 2 q^{27} - 4 q^{28} - 12 q^{29} - 20 q^{30} - 2 q^{31} - 8 q^{32} + 24 q^{34} + 4 q^{35} + 18 q^{36} + 14 q^{37} - 22 q^{38} - 20 q^{39} - 18 q^{40} - 26 q^{41} - 6 q^{42} + 4 q^{43} - 36 q^{45} - 12 q^{46} - 16 q^{47} - 24 q^{48} + 6 q^{49} + 4 q^{50} + 4 q^{51} - 12 q^{52} + 4 q^{53} + 32 q^{54} + 12 q^{56} - 20 q^{57} - 2 q^{58} - 4 q^{59} + 24 q^{60} + 8 q^{61} - 20 q^{62} - 8 q^{63} + 26 q^{64} - 24 q^{65} + 6 q^{67} - 12 q^{68} - 14 q^{69} - 8 q^{70} + 22 q^{71} - 16 q^{72} - 14 q^{73} - 44 q^{74} - 20 q^{75} + 30 q^{76} + 32 q^{78} + 28 q^{79} - 4 q^{80} - 6 q^{81} - 4 q^{82} - 22 q^{83} + 14 q^{84} + 24 q^{85} - 30 q^{86} - 22 q^{87} + 22 q^{90} + 4 q^{91} + 10 q^{92} - 50 q^{93} + 38 q^{94} + 24 q^{95} + 62 q^{96} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9 + 8 * q^10 - 14 * q^12 - 4 * q^13 + 4 * q^14 + 2 * q^15 + 8 * q^16 - 22 * q^17 - 24 * q^18 - 6 * q^19 + 2 * q^20 + 2 * q^21 + 2 * q^23 + 20 * q^24 + 4 * q^25 + 6 * q^26 - 2 * q^27 - 4 * q^28 - 12 * q^29 - 20 * q^30 - 2 * q^31 - 8 * q^32 + 24 * q^34 + 4 * q^35 + 18 * q^36 + 14 * q^37 - 22 * q^38 - 20 * q^39 - 18 * q^40 - 26 * q^41 - 6 * q^42 + 4 * q^43 - 36 * q^45 - 12 * q^46 - 16 * q^47 - 24 * q^48 + 6 * q^49 + 4 * q^50 + 4 * q^51 - 12 * q^52 + 4 * q^53 + 32 * q^54 + 12 * q^56 - 20 * q^57 - 2 * q^58 - 4 * q^59 + 24 * q^60 + 8 * q^61 - 20 * q^62 - 8 * q^63 + 26 * q^64 - 24 * q^65 + 6 * q^67 - 12 * q^68 - 14 * q^69 - 8 * q^70 + 22 * q^71 - 16 * q^72 - 14 * q^73 - 44 * q^74 - 20 * q^75 + 30 * q^76 + 32 * q^78 + 28 * q^79 - 4 * q^80 - 6 * q^81 - 4 * q^82 - 22 * q^83 + 14 * q^84 + 24 * q^85 - 30 * q^86 - 22 * q^87 + 22 * q^90 + 4 * q^91 + 10 * q^92 - 50 * q^93 + 38 * q^94 + 24 * q^95 + 62 * q^96 - 4 * q^97 - 4 * q^98

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