LMFDB - Embedded newform 231.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [231,2,Mod(1,231)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(231, 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("231.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 231 = 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	231.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:	$1.84454428669$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
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			1.2
Root			$-0.167449$ of defining polynomial
Character	$\chi$	$=$	231.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +3.80451 q^{5} +0.167449 q^{6} -1.00000 q^{7} +0.665102 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +3.80451 q^{5} +0.167449 q^{6} -1.00000 q^{7} +0.665102 q^{8} +1.00000 q^{9} -0.637062 q^{10} +1.00000 q^{11} +1.97196 q^{12} +3.80451 q^{13} +0.167449 q^{14} -3.80451 q^{15} +3.83255 q^{16} +0.334898 q^{17} -0.167449 q^{18} +8.13941 q^{19} -7.50235 q^{20} +1.00000 q^{21} -0.167449 q^{22} -1.66510 q^{23} -0.665102 q^{24} +9.47431 q^{25} -0.637062 q^{26} -1.00000 q^{27} +1.97196 q^{28} +0.195488 q^{29} +0.637062 q^{30} -9.94392 q^{31} -1.97196 q^{32} -1.00000 q^{33} -0.0560785 q^{34} -3.80451 q^{35} -1.97196 q^{36} -4.47431 q^{37} -1.36294 q^{38} -3.80451 q^{39} +2.53039 q^{40} -6.27882 q^{41} -0.167449 q^{42} +2.33490 q^{43} -1.97196 q^{44} +3.80451 q^{45} +0.278820 q^{46} -12.1394 q^{47} -3.83255 q^{48} +1.00000 q^{49} -1.58647 q^{50} -0.334898 q^{51} -7.50235 q^{52} +7.94392 q^{53} +0.167449 q^{54} +3.80451 q^{55} -0.665102 q^{56} -8.13941 q^{57} -0.0327344 q^{58} +3.74843 q^{59} +7.50235 q^{60} +6.00000 q^{61} +1.66510 q^{62} -1.00000 q^{63} -7.33490 q^{64} +14.4743 q^{65} +0.167449 q^{66} -0.139410 q^{67} -0.660406 q^{68} +1.66510 q^{69} +0.637062 q^{70} +4.66980 q^{71} +0.665102 q^{72} +4.19549 q^{73} +0.749219 q^{74} -9.47431 q^{75} -16.0506 q^{76} -1.00000 q^{77} +0.637062 q^{78} +3.33020 q^{79} +14.5810 q^{80} +1.00000 q^{81} +1.05138 q^{82} -13.9439 q^{83} -1.97196 q^{84} +1.27412 q^{85} -0.390977 q^{86} -0.195488 q^{87} +0.665102 q^{88} -9.88784 q^{89} -0.637062 q^{90} -3.80451 q^{91} +3.28352 q^{92} +9.94392 q^{93} +2.03273 q^{94} +30.9665 q^{95} +1.97196 q^{96} +0.0560785 q^{97} -0.167449 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 6 * q^4 - 3 * q^7 + 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 9 q^{10} + 3 q^{11} - 6 q^{12} + 12 q^{16} + 12 q^{19} - 21 q^{20} + 3 q^{21} - 6 q^{23} - 3 q^{24} + 15 q^{25} + 9 q^{26} - 3 q^{27} - 6 q^{28} + 12 q^{29} - 9 q^{30} - 6 q^{31} + 6 q^{32} - 3 q^{33} - 24 q^{34} + 6 q^{36} - 15 q^{38} + 18 q^{40} + 6 q^{41} + 6 q^{43} + 6 q^{44} - 24 q^{46} - 24 q^{47} - 12 q^{48} + 3 q^{49} - 39 q^{50} - 21 q^{52} - 3 q^{56} - 12 q^{57} - 9 q^{58} - 24 q^{59} + 21 q^{60} + 18 q^{61} + 6 q^{62} - 3 q^{63} - 21 q^{64} + 30 q^{65} + 12 q^{67} - 6 q^{68} + 6 q^{69} - 9 q^{70} + 12 q^{71} + 3 q^{72} + 24 q^{73} + 39 q^{74} - 15 q^{75} - 3 q^{76} - 3 q^{77} - 9 q^{78} + 12 q^{79} + 9 q^{80} + 3 q^{81} + 30 q^{82} - 18 q^{83} + 6 q^{84} - 18 q^{85} - 24 q^{86} - 12 q^{87} + 3 q^{88} + 18 q^{89} + 9 q^{90} - 18 q^{92} + 6 q^{93} + 15 q^{94} + 12 q^{95} - 6 q^{96} + 24 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 6 * q^4 - 3 * q^7 + 3 * q^8 + 3 * q^9 + 9 * q^10 + 3 * q^11 - 6 * q^12 + 12 * q^16 + 12 * q^19 - 21 * q^20 + 3 * q^21 - 6 * q^23 - 3 * q^24 + 15 * q^25 + 9 * q^26 - 3 * q^27 - 6 * q^28 + 12 * q^29 - 9 * q^30 - 6 * q^31 + 6 * q^32 - 3 * q^33 - 24 * q^34 + 6 * q^36 - 15 * q^38 + 18 * q^40 + 6 * q^41 + 6 * q^43 + 6 * q^44 - 24 * q^46 - 24 * q^47 - 12 * q^48 + 3 * q^49 - 39 * q^50 - 21 * q^52 - 3 * q^56 - 12 * q^57 - 9 * q^58 - 24 * q^59 + 21 * q^60 + 18 * q^61 + 6 * q^62 - 3 * q^63 - 21 * q^64 + 30 * q^65 + 12 * q^67 - 6 * q^68 + 6 * q^69 - 9 * q^70 + 12 * q^71 + 3 * q^72 + 24 * q^73 + 39 * q^74 - 15 * q^75 - 3 * q^76 - 3 * q^77 - 9 * q^78 + 12 * q^79 + 9 * q^80 + 3 * q^81 + 30 * q^82 - 18 * q^83 + 6 * q^84 - 18 * q^85 - 24 * q^86 - 12 * q^87 + 3 * q^88 + 18 * q^89 + 9 * q^90 - 18 * q^92 + 6 * q^93 + 15 * q^94 + 12 * q^95 - 6 * q^96 + 24 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.167449	−0.118404	−0.0592022	−	0.998246i	$-0.518856\pi$
$2$	−0.167449	−0.118404	−0.0592022	+	0.998246i	$0.518856\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.97196	−0.985980
$5$	3.80451	1.70143	0.850715	−	0.525628i	$-0.176170\pi$
$5$	3.80451	1.70143	0.850715	+	0.525628i	$0.176170\pi$
$6$	0.167449	0.0683608
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.665102	0.235149
$9$	1.00000	0.333333
$10$	−0.637062	−0.201457
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.97196	0.569256
$13$	3.80451	1.05518	0.527591	−	0.849499i	$-0.323095\pi$
$13$	3.80451	1.05518	0.527591	+	0.849499i	$0.323095\pi$
$14$	0.167449	0.0447527
$15$	−3.80451	−0.982321
$16$	3.83255	0.958138
$17$	0.334898	0.0812248	0.0406124	−	0.999175i	$-0.487069\pi$
$17$	0.334898	0.0812248	0.0406124	+	0.999175i	$0.487069\pi$
$18$	−0.167449	−0.0394682
$19$	8.13941	1.86731	0.933654	−	0.358175i	$-0.116601\pi$
$19$	8.13941	1.86731	0.933654	+	0.358175i	$0.116601\pi$
$20$	−7.50235	−1.67758
$21$	1.00000	0.218218
$22$	−0.167449	−0.0357003
$23$	−1.66510	−0.347198	−0.173599	−	0.984816i	$-0.555540\pi$
$23$	−1.66510	−0.347198	−0.173599	+	0.984816i	$0.555540\pi$
$24$	−0.665102	−0.135763
$25$	9.47431	1.89486
$26$	−0.637062	−0.124938
$27$	−1.00000	−0.192450
$28$	1.97196	0.372666
$29$	0.195488	0.0363013	0.0181506	−	0.999835i	$-0.494222\pi$
$29$	0.195488	0.0363013	0.0181506	+	0.999835i	$0.494222\pi$
$30$	0.637062	0.116311
$31$	−9.94392	−1.78598	−0.892991	−	0.450075i	$-0.851397\pi$
$31$	−9.94392	−1.78598	−0.892991	+	0.450075i	$0.851397\pi$
$32$	−1.97196	−0.348597
$33$	−1.00000	−0.174078
$34$	−0.0560785	−0.00961738
$35$	−3.80451	−0.643080
$36$	−1.97196	−0.328660
$37$	−4.47431	−0.735572	−0.367786	−	0.929911i	$-0.619884\pi$
$37$	−4.47431	−0.735572	−0.367786	+	0.929911i	$0.619884\pi$
$38$	−1.36294	−0.221098
$39$	−3.80451	−0.609209
$40$	2.53039	0.400089
$41$	−6.27882	−0.980587	−0.490293	−	0.871557i	$-0.663110\pi$
$41$	−6.27882	−0.980587	−0.490293	+	0.871557i	$0.663110\pi$
$42$	−0.167449	−0.0258380
$43$	2.33490	0.356069	0.178034	−	0.984024i	$-0.443026\pi$
$43$	2.33490	0.356069	0.178034	+	0.984024i	$0.443026\pi$
$44$	−1.97196	−0.297284
$45$	3.80451	0.567143
$46$	0.278820	0.0411098
$47$	−12.1394	−1.77071	−0.885357	−	0.464911i	$-0.846086\pi$
$47$	−12.1394	−1.77071	−0.885357	+	0.464911i	$0.846086\pi$
$48$	−3.83255	−0.553181
$49$	1.00000	0.142857
$50$	−1.58647	−0.224360
$51$	−0.334898	−0.0468952
$52$	−7.50235	−1.04039
$53$	7.94392	1.09118	0.545591	−	0.838052i	$-0.316305\pi$
$53$	7.94392	1.09118	0.545591	+	0.838052i	$0.316305\pi$
$54$	0.167449	0.0227869
$55$	3.80451	0.513000
$56$	−0.665102	−0.0888779
$57$	−8.13941	−1.07809
$58$	−0.0327344	−0.00429823
$59$	3.74843	0.488004	0.244002	−	0.969775i	$-0.421540\pi$
$59$	3.74843	0.488004	0.244002	+	0.969775i	$0.421540\pi$
$60$	7.50235	0.968549
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	1.66510	0.211468
$63$	−1.00000	−0.125988
$64$	−7.33490	−0.916862
$65$	14.4743	1.79532
$66$	0.167449	0.0206116
$67$	−0.139410	−0.0170316	−0.00851582	−	0.999964i	$-0.502711\pi$
$67$	−0.139410	−0.0170316	−0.00851582	+	0.999964i	$0.502711\pi$
$68$	−0.660406	−0.0800860
$69$	1.66510	0.200455
$70$	0.637062	0.0761435
$71$	4.66980	0.554203	0.277101	−	0.960841i	$-0.410626\pi$
$71$	4.66980	0.554203	0.277101	+	0.960841i	$0.410626\pi$
$72$	0.665102	0.0783830
$73$	4.19549	0.491045	0.245522	−	0.969391i	$-0.421041\pi$
$73$	4.19549	0.491045	0.245522	+	0.969391i	$0.421041\pi$
$74$	0.749219	0.0870950
$75$	−9.47431	−1.09400
$76$	−16.0506	−1.84113
$77$	−1.00000	−0.113961
$78$	0.637062	0.0721331
$79$	3.33020	0.374677	0.187339	−	0.982295i	$-0.440014\pi$
$79$	3.33020	0.374677	0.187339	+	0.982295i	$0.440014\pi$
$80$	14.5810	1.63020
$81$	1.00000	0.111111
$82$	1.05138	0.116106
$83$	−13.9439	−1.53054	−0.765272	−	0.643707i	$-0.777396\pi$
$83$	−13.9439	−1.53054	−0.765272	+	0.643707i	$0.777396\pi$
$84$	−1.97196	−0.215159
$85$	1.27412	0.138198
$86$	−0.390977	−0.0421601
$87$	−0.195488	−0.0209586
$88$	0.665102	0.0709001
$89$	−9.88784	−1.04811	−0.524055	−	0.851685i	$-0.675581\pi$
$89$	−9.88784	−1.04811	−0.524055	+	0.851685i	$0.675581\pi$
$90$	−0.637062	−0.0671523
$91$	−3.80451	−0.398821
$92$	3.28352	0.342330
$93$	9.94392	1.03114
$94$	2.03273	0.209661
$95$	30.9665	3.17709
$96$	1.97196	0.201262
$97$	0.0560785	0.00569391	0.00284695	−	0.999996i	$-0.499094\pi$
$97$	0.0560785	0.00569391	0.00284695	+	0.999996i	$0.499094\pi$
$98$	−0.167449	−0.0169149
$99$	1.00000	0.100504
$100$	−18.6830	−1.86830
$101$	−18.8831	−1.87894	−0.939472	−	0.342626i	$-0.888683\pi$
$101$	−18.8831	−1.87894	−0.939472	+	0.342626i	$0.888683\pi$
$102$	0.0560785	0.00555260
$103$	−8.27882	−0.815736	−0.407868	−	0.913041i	$-0.633728\pi$
$103$	−8.27882	−0.815736	−0.407868	+	0.913041i	$0.633728\pi$
$104$	2.53039	0.248125
$105$	3.80451	0.371282
$106$	−1.33020	−0.129201
$107$	−8.13941	−0.786866	−0.393433	−	0.919353i	$-0.628713\pi$
$107$	−8.13941	−0.786866	−0.393433	+	0.919353i	$0.628713\pi$
$108$	1.97196	0.189752
$109$	11.5529	1.10657	0.553286	−	0.832992i	$-0.313374\pi$
$109$	11.5529	1.10657	0.553286	+	0.832992i	$0.313374\pi$
$110$	−0.637062	−0.0607415
$111$	4.47431	0.424683
$112$	−3.83255	−0.362142
$113$	−1.33020	−0.125135	−0.0625675	−	0.998041i	$-0.519929\pi$
$113$	−1.33020	−0.125135	−0.0625675	+	0.998041i	$0.519929\pi$
$114$	1.36294	0.127651
$115$	−6.33490	−0.590732
$116$	−0.385496	−0.0357924
$117$	3.80451	0.351727
$118$	−0.627672	−0.0577819
$119$	−0.334898	−0.0307001
$120$	−2.53039	−0.230992
$121$	1.00000	0.0909091
$122$	−1.00470	−0.0909608
$123$	6.27882	0.566142
$124$	19.6090	1.76094
$125$	17.0226	1.52254
$126$	0.167449	0.0149176
$127$	14.6137	1.29676	0.648379	−	0.761318i	$-0.275447\pi$
$127$	14.6137	1.29676	0.648379	+	0.761318i	$0.275447\pi$
$128$	5.17214	0.457157
$129$	−2.33490	−0.205576
$130$	−2.42371	−0.212574
$131$	20.5576	1.79613	0.898065	−	0.439863i	$-0.144973\pi$
$131$	20.5576	1.79613	0.898065	+	0.439863i	$0.144973\pi$
$132$	1.97196	0.171637
$133$	−8.13941	−0.705776
$134$	0.0233441	0.00201662
$135$	−3.80451	−0.327440
$136$	0.222741	0.0190999
$137$	−16.2227	−1.38600	−0.693001	−	0.720936i	$-0.743712\pi$
$137$	−16.2227	−1.38600	−0.693001	+	0.720936i	$0.743712\pi$
$138$	−0.278820	−0.0237347
$139$	22.5482	1.91252	0.956259	−	0.292522i	$-0.0944945\pi$
$139$	22.5482	1.91252	0.956259	+	0.292522i	$0.0944945\pi$
$140$	7.50235	0.634064
$141$	12.1394	1.02232
$142$	−0.781954	−0.0656201
$143$	3.80451	0.318149
$144$	3.83255	0.319379
$145$	0.743738	0.0617641
$146$	−0.702531	−0.0581419
$147$	−1.00000	−0.0824786
$148$	8.82316	0.725259
$149$	8.19549	0.671401	0.335700	−	0.941969i	$-0.391027\pi$
$149$	8.19549	0.671401	0.335700	+	0.941969i	$0.391027\pi$
$150$	1.58647	0.129534
$151$	−13.2741	−1.08023	−0.540116	−	0.841590i	$-0.681620\pi$
$151$	−13.2741	−1.08023	−0.540116	+	0.841590i	$0.681620\pi$
$152$	5.41353	0.439096
$153$	0.334898	0.0270749
$154$	0.167449	0.0134934
$155$	−37.8318	−3.03872
$156$	7.50235	0.600669
$157$	−16.9392	−1.35190	−0.675949	−	0.736949i	$-0.736266\pi$
$157$	−16.9392	−1.35190	−0.675949	+	0.736949i	$0.736266\pi$
$158$	−0.557640	−0.0443634
$159$	−7.94392	−0.629994
$160$	−7.50235	−0.593113
$161$	1.66510	0.131228
$162$	−0.167449	−0.0131561
$163$	−6.79982	−0.532603	−0.266301	−	0.963890i	$-0.585802\pi$
$163$	−6.79982	−0.532603	−0.266301	+	0.963890i	$0.585802\pi$
$164$	12.3816	0.966839
$165$	−3.80451	−0.296181
$166$	2.33490	0.181223
$167$	−18.2227	−1.41012	−0.705059	−	0.709149i	$-0.749080\pi$
$167$	−18.2227	−1.41012	−0.705059	+	0.709149i	$0.749080\pi$
$168$	0.665102	0.0513137
$169$	1.47431	0.113408
$170$	−0.213351	−0.0163633
$171$	8.13941	0.622436
$172$	−4.60433	−0.351077
$173$	−1.72118	−0.130859	−0.0654294	−	0.997857i	$-0.520842\pi$
$173$	−1.72118	−0.130859	−0.0654294	+	0.997857i	$0.520842\pi$
$174$	0.0327344	0.00248159
$175$	−9.47431	−0.716190
$176$	3.83255	0.288889
$177$	−3.74843	−0.281749
$178$	1.65571	0.124101
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	−7.50235	−0.559192
$181$	0.725875	0.0539539	0.0269769	−	0.999636i	$-0.491412\pi$
$181$	0.725875	0.0539539	0.0269769	+	0.999636i	$0.491412\pi$
$182$	0.637062	0.0472222
$183$	−6.00000	−0.443533
$184$	−1.10746	−0.0816432
$185$	−17.0226	−1.25152
$186$	−1.66510	−0.122091
$187$	0.334898	0.0244902
$188$	23.9384	1.74589
$189$	1.00000	0.0727393
$190$	−5.18531	−0.376182
$191$	5.27412	0.381622	0.190811	−	0.981627i	$-0.438888\pi$
$191$	5.27412	0.381622	0.190811	+	0.981627i	$0.438888\pi$
$192$	7.33490	0.529351
$193$	−19.8318	−1.42752	−0.713761	−	0.700390i	$-0.753010\pi$
$193$	−19.8318	−1.42752	−0.713761	+	0.700390i	$0.753010\pi$
$194$	−0.00939029	−0.000674184 0
$195$	−14.4743	−1.03653
$196$	−1.97196	−0.140854
$197$	2.66980	0.190215	0.0951076	−	0.995467i	$-0.469680\pi$
$197$	2.66980	0.190215	0.0951076	+	0.995467i	$0.469680\pi$
$198$	−0.167449	−0.0119001
$199$	13.5529	0.960743	0.480371	−	0.877065i	$-0.340502\pi$
$199$	13.5529	0.960743	0.480371	+	0.877065i	$0.340502\pi$
$200$	6.30138	0.445575
$201$	0.139410	0.00983322
$202$	3.16197	0.222475
$203$	−0.195488	−0.0137206
$204$	0.660406	0.0462377
$205$	−23.8878	−1.66840
$206$	1.38628	0.0965868
$207$	−1.66510	−0.115733
$208$	14.5810	1.01101
$209$	8.13941	0.563015
$210$	−0.637062	−0.0439615
$211$	−4.27882	−0.294566	−0.147283	−	0.989094i	$-0.547053\pi$
$211$	−4.27882	−0.294566	−0.147283	+	0.989094i	$0.547053\pi$
$212$	−15.6651	−1.07588
$213$	−4.66980	−0.319969
$214$	1.36294	0.0931685
$215$	8.88315	0.605826
$216$	−0.665102	−0.0452544
$217$	9.94392	0.675037
$218$	−1.93453	−0.131023
$219$	−4.19549	−0.283505
$220$	−7.50235	−0.505808
$221$	1.27412	0.0857069
$222$	−0.749219	−0.0502843
$223$	−10.2694	−0.687692	−0.343846	−	0.939026i	$-0.611730\pi$
$223$	−10.2694	−0.687692	−0.343846	+	0.939026i	$0.611730\pi$
$224$	1.97196	0.131757
$225$	9.47431	0.631621
$226$	0.222741	0.0148165
$227$	−0.390977	−0.0259500	−0.0129750	−	0.999916i	$-0.504130\pi$
$227$	−0.390977	−0.0259500	−0.0129750	+	0.999916i	$0.504130\pi$
$228$	16.0506	1.06298
$229$	7.94392	0.524949	0.262475	−	0.964939i	$-0.415461\pi$
$229$	7.94392	0.524949	0.262475	+	0.964939i	$0.415461\pi$
$230$	1.06077	0.0699453
$231$	1.00000	0.0657952
$232$	0.130020	0.00853621
$233$	26.5576	1.73985	0.869924	−	0.493185i	$-0.164167\pi$
$233$	26.5576	1.73985	0.869924	+	0.493185i	$0.164167\pi$
$234$	−0.637062	−0.0416461
$235$	−46.1845	−3.01275
$236$	−7.39176	−0.481163
$237$	−3.33020	−0.216320
$238$	0.0560785	0.00363503
$239$	10.7998	0.698582	0.349291	−	0.937014i	$-0.386422\pi$
$239$	10.7998	0.698582	0.349291	+	0.937014i	$0.386422\pi$
$240$	−14.5810	−0.941198
$241$	−12.0833	−0.778356	−0.389178	−	0.921163i	$-0.627241\pi$
$241$	−12.0833	−0.778356	−0.389178	+	0.921163i	$0.627241\pi$
$242$	−0.167449	−0.0107640
$243$	−1.00000	−0.0641500
$244$	−11.8318	−0.757451
$245$	3.80451	0.243061
$246$	−1.05138	−0.0670338
$247$	30.9665	1.97035
$248$	−6.61372	−0.419972
$249$	13.9439	0.883660
$250$	−2.85041	−0.180276
$251$	4.80921	0.303554	0.151777	−	0.988415i	$-0.451500\pi$
$251$	4.80921	0.303554	0.151777	+	0.988415i	$0.451500\pi$
$252$	1.97196	0.124222
$253$	−1.66510	−0.104684
$254$	−2.44706	−0.153542
$255$	−1.27412	−0.0797888
$256$	13.8037	0.862733
$257$	−16.7531	−1.04503	−0.522516	−	0.852630i	$-0.675006\pi$
$257$	−16.7531	−1.04503	−0.522516	+	0.852630i	$0.675006\pi$
$258$	0.390977	0.0243412
$259$	4.47431	0.278020
$260$	−28.5428	−1.77015
$261$	0.195488	0.0121004
$262$	−3.44236	−0.212670
$263$	−12.1394	−0.748548	−0.374274	−	0.927318i	$-0.622108\pi$
$263$	−12.1394	−0.748548	−0.374274	+	0.927318i	$0.622108\pi$
$264$	−0.665102	−0.0409342
$265$	30.2227	1.85657
$266$	1.36294	0.0835671
$267$	9.88784	0.605126
$268$	0.274911	0.0167929
$269$	−25.2180	−1.53757	−0.768786	−	0.639506i	$-0.779139\pi$
$269$	−25.2180	−1.53757	−0.768786	+	0.639506i	$0.779139\pi$
$270$	0.637062	0.0387704
$271$	4.13941	0.251451	0.125726	−	0.992065i	$-0.459874\pi$
$271$	4.13941	0.251451	0.125726	+	0.992065i	$0.459874\pi$
$272$	1.28352	0.0778245
$273$	3.80451	0.230260
$274$	2.71648	0.164109
$275$	9.47431	0.571322
$276$	−3.28352	−0.197644
$277$	−18.2788	−1.09827	−0.549134	−	0.835734i	$-0.685042\pi$
$277$	−18.2788	−1.09827	−0.549134	+	0.835734i	$0.685042\pi$
$278$	−3.77569	−0.226451
$279$	−9.94392	−0.595327
$280$	−2.53039	−0.151220
$281$	6.74374	0.402298	0.201149	−	0.979561i	$-0.435533\pi$
$281$	6.74374	0.402298	0.201149	+	0.979561i	$0.435533\pi$
$282$	−2.03273	−0.121048
$283$	23.3575	1.38846	0.694228	−	0.719755i	$-0.255746\pi$
$283$	23.3575	1.38846	0.694228	+	0.719755i	$0.255746\pi$
$284$	−9.20866	−0.546433
$285$	−30.9665	−1.83430
$286$	−0.637062	−0.0376703
$287$	6.27882	0.370627
$288$	−1.97196	−0.116199
$289$	−16.8878	−0.993403
$290$	−0.124538	−0.00731314
$291$	−0.0560785	−0.00328738
$292$	−8.27334	−0.484161
$293$	−14.1667	−0.827625	−0.413813	−	0.910362i	$-0.635803\pi$
$293$	−14.1667	−0.827625	−0.413813	+	0.910362i	$0.635803\pi$
$294$	0.167449	0.00976584
$295$	14.2610	0.830305
$296$	−2.97587	−0.172969
$297$	−1.00000	−0.0580259
$298$	−1.37233	−0.0794968
$299$	−6.33490	−0.366357
$300$	18.6830	1.07866
$301$	−2.33490	−0.134581
$302$	2.22274	0.127904
$303$	18.8831	1.08481
$304$	31.1947	1.78914
$305$	22.8271	1.30707
$306$	−0.0560785	−0.00320579
$307$	−12.5576	−0.716702	−0.358351	−	0.933587i	$-0.616661\pi$
$307$	−12.5576	−0.716702	−0.358351	+	0.933587i	$0.616661\pi$
$308$	1.97196	0.112363
$309$	8.27882	0.470966
$310$	6.33490	0.359798
$311$	9.33959	0.529600	0.264800	−	0.964303i	$-0.414694\pi$
$311$	9.33959	0.529600	0.264800	+	0.964303i	$0.414694\pi$
$312$	−2.53039	−0.143255
$313$	−2.99530	−0.169305	−0.0846523	−	0.996411i	$-0.526978\pi$
$313$	−2.99530	−0.169305	−0.0846523	+	0.996411i	$0.526978\pi$
$314$	2.83646	0.160071
$315$	−3.80451	−0.214360
$316$	−6.56703	−0.369424
$317$	9.93453	0.557979	0.278989	−	0.960294i	$-0.410001\pi$
$317$	9.93453	0.557979	0.278989	+	0.960294i	$0.410001\pi$
$318$	1.33020	0.0745941
$319$	0.195488	0.0109453
$320$	−27.9057	−1.55998
$321$	8.13941	0.454298
$322$	−0.278820	−0.0155380
$323$	2.72588	0.151672
$324$	−1.97196	−0.109553
$325$	36.0451	1.99942
$326$	1.13862	0.0630625
$327$	−11.5529	−0.638879
$328$	−4.17605	−0.230584
$329$	12.1394	0.669267
$330$	0.637062	0.0350691
$331$	22.5482	1.23936	0.619682	−	0.784853i	$-0.287262\pi$
$331$	22.5482	1.23936	0.619682	+	0.784853i	$0.287262\pi$
$332$	27.4969	1.50909
$333$	−4.47431	−0.245191
$334$	3.05138	0.166964
$335$	−0.530387	−0.0289781
$336$	3.83255	0.209083
$337$	13.2835	0.723599	0.361800	−	0.932256i	$-0.382162\pi$
$337$	13.2835	0.723599	0.361800	+	0.932256i	$0.382162\pi$
$338$	−0.246872	−0.0134281
$339$	1.33020	0.0722467
$340$	−2.51252	−0.136261
$341$	−9.94392	−0.538494
$342$	−1.36294	−0.0736992
$343$	−1.00000	−0.0539949
$344$	1.55294	0.0837292
$345$	6.33490	0.341059
$346$	0.288210	0.0154943
$347$	−27.7757	−1.49108	−0.745538	−	0.666463i	$-0.767808\pi$
$347$	−27.7757	−1.49108	−0.745538	+	0.666463i	$0.767808\pi$
$348$	0.385496	0.0206647
$349$	−2.85589	−0.152873	−0.0764363	−	0.997074i	$-0.524354\pi$
$349$	−2.85589	−0.152873	−0.0764363	+	0.997074i	$0.524354\pi$
$350$	1.58647	0.0848001
$351$	−3.80451	−0.203070
$352$	−1.97196	−0.105106
$353$	24.6410	1.31151	0.655753	−	0.754975i	$-0.272351\pi$
$353$	24.6410	1.31151	0.655753	+	0.754975i	$0.272351\pi$
$354$	0.627672	0.0333604
$355$	17.7663	0.942937
$356$	19.4984	1.03342
$357$	0.334898	0.0177247
$358$	2.00939	0.106200
$359$	−11.8878	−0.627416	−0.313708	−	0.949519i	$-0.601571\pi$
$359$	−11.8878	−0.627416	−0.313708	+	0.949519i	$0.601571\pi$
$360$	2.53039	0.133363
$361$	47.2500	2.48684
$362$	−0.121547	−0.00638838
$363$	−1.00000	−0.0524864
$364$	7.50235	0.393230
$365$	15.9618	0.835478
$366$	1.00470	0.0525163
$367$	3.60902	0.188389	0.0941947	−	0.995554i	$-0.469972\pi$
$367$	3.60902	0.188389	0.0941947	+	0.995554i	$0.469972\pi$
$368$	−6.38159	−0.332663
$369$	−6.27882	−0.326862
$370$	2.85041	0.148186
$371$	−7.94392	−0.412428
$372$	−19.6090	−1.01668
$373$	12.3349	0.638677	0.319338	−	0.947641i	$-0.396539\pi$
$373$	12.3349	0.638677	0.319338	+	0.947641i	$0.396539\pi$
$374$	−0.0560785	−0.00289975
$375$	−17.0226	−0.879041
$376$	−8.07394	−0.416382
$377$	0.743738	0.0383045
$378$	−0.167449	−0.00861266
$379$	23.3575	1.19979	0.599896	−	0.800078i	$-0.295209\pi$
$379$	23.3575	1.19979	0.599896	+	0.800078i	$0.295209\pi$
$380$	−61.0647	−3.13255
$381$	−14.6137	−0.748683
$382$	−0.883148	−0.0451858
$383$	−15.2180	−0.777606	−0.388803	−	0.921321i	$-0.627111\pi$
$383$	−15.2180	−0.777606	−0.388803	+	0.921321i	$0.627111\pi$
$384$	−5.17214	−0.263940
$385$	−3.80451	−0.193896
$386$	3.32081	0.169025
$387$	2.33490	0.118690
$388$	−0.110585	−0.00561408
$389$	3.73057	0.189147	0.0945737	−	0.995518i	$-0.469851\pi$
$389$	3.73057	0.189147	0.0945737	+	0.995518i	$0.469851\pi$
$390$	2.42371	0.122729
$391$	−0.557640	−0.0282011
$392$	0.665102	0.0335927
$393$	−20.5576	−1.03700
$394$	−0.447055	−0.0225223
$395$	12.6698	0.637487
$396$	−1.97196	−0.0990948
$397$	20.8925	1.04857	0.524283	−	0.851544i	$-0.324333\pi$
$397$	20.8925	1.04857	0.524283	+	0.851544i	$0.324333\pi$
$398$	−2.26943	−0.113756
$399$	8.13941	0.407480
$400$	36.3108	1.81554
$401$	23.2741	1.16225	0.581127	−	0.813813i	$-0.302612\pi$
$401$	23.2741	1.16225	0.581127	+	0.813813i	$0.302612\pi$
$402$	−0.0233441	−0.00116430
$403$	−37.8318	−1.88453
$404$	37.2368	1.85260
$405$	3.80451	0.189048
$406$	0.0327344	0.00162458
$407$	−4.47431	−0.221783
$408$	−0.222741	−0.0110273
$409$	−23.8972	−1.18164	−0.590821	−	0.806803i	$-0.701196\pi$
$409$	−23.8972	−1.18164	−0.590821	+	0.806803i	$0.701196\pi$
$410$	4.00000	0.197546
$411$	16.2227	0.800209
$412$	16.3255	0.804300
$413$	−3.74843	−0.184448
$414$	0.278820	0.0137033
$415$	−53.0498	−2.60411
$416$	−7.50235	−0.367833
$417$	−22.5482	−1.10419
$418$	−1.36294	−0.0666635
$419$	−30.2967	−1.48009	−0.740045	−	0.672557i	$-0.765196\pi$
$419$	−30.2967	−1.48009	−0.740045	+	0.672557i	$0.765196\pi$
$420$	−7.50235	−0.366077
$421$	−15.5257	−0.756676	−0.378338	−	0.925668i	$-0.623504\pi$
$421$	−15.5257	−0.756676	−0.378338	+	0.925668i	$0.623504\pi$
$422$	0.716485	0.0348779
$423$	−12.1394	−0.590238
$424$	5.28352	0.256590
$425$	3.17293	0.153910
$426$	0.781954	0.0378858
$427$	−6.00000	−0.290360
$428$	16.0506	0.775835
$429$	−3.80451	−0.183684
$430$	−1.48748	−0.0717325
$431$	3.07864	0.148293	0.0741463	−	0.997247i	$-0.476377\pi$
$431$	3.07864	0.148293	0.0741463	+	0.997247i	$0.476377\pi$
$432$	−3.83255	−0.184394
$433$	16.9392	0.814047	0.407024	−	0.913418i	$-0.366567\pi$
$433$	16.9392	0.814047	0.407024	+	0.913418i	$0.366567\pi$
$434$	−1.66510	−0.0799274
$435$	−0.743738	−0.0356595
$436$	−22.7820	−1.09106
$437$	−13.5529	−0.648325
$438$	0.702531	0.0335682
$439$	−11.0786	−0.528754	−0.264377	−	0.964419i	$-0.585166\pi$
$439$	−11.0786	−0.528754	−0.264377	+	0.964419i	$0.585166\pi$
$440$	2.53039	0.120631
$441$	1.00000	0.0476190
$442$	−0.213351	−0.0101481
$443$	−14.5482	−0.691208	−0.345604	−	0.938380i	$-0.612326\pi$
$443$	−14.5482	−0.691208	−0.345604	+	0.938380i	$0.612326\pi$
$444$	−8.82316	−0.418729
$445$	−37.6184	−1.78328
$446$	1.71961	0.0814258
$447$	−8.19549	−0.387633
$448$	7.33490	0.346541
$449$	27.4875	1.29721	0.648607	−	0.761123i	$-0.275352\pi$
$449$	27.4875	1.29721	0.648607	+	0.761123i	$0.275352\pi$
$450$	−1.58647	−0.0747867
$451$	−6.27882	−0.295658
$452$	2.62311	0.123381
$453$	13.2741	0.623673
$454$	0.0654688	0.00307260
$455$	−14.4743	−0.678566
$456$	−5.41353	−0.253512
$457$	−7.94392	−0.371601	−0.185800	−	0.982587i	$-0.559488\pi$
$457$	−7.94392	−0.371601	−0.185800	+	0.982587i	$0.559488\pi$
$458$	−1.33020	−0.0621563
$459$	−0.334898	−0.0156317
$460$	12.4922	0.582450
$461$	8.26943	0.385146	0.192573	−	0.981283i	$-0.438317\pi$
$461$	8.26943	0.385146	0.192573	+	0.981283i	$0.438317\pi$
$462$	−0.167449	−0.00779044
$463$	9.73904	0.452612	0.226306	−	0.974056i	$-0.427335\pi$
$463$	9.73904	0.452612	0.226306	+	0.974056i	$0.427335\pi$
$464$	0.749219	0.0347816
$465$	37.8318	1.75441
$466$	−4.44706	−0.206006
$467$	−40.6970	−1.88323	−0.941617	−	0.336685i	$-0.890694\pi$
$467$	−40.6970	−1.88323	−0.941617	+	0.336685i	$0.890694\pi$
$468$	−7.50235	−0.346796
$469$	0.139410	0.00643735
$470$	7.73356	0.356723
$471$	16.9392	0.780518
$472$	2.49309	0.114754
$473$	2.33490	0.107359
$474$	0.557640	0.0256132
$475$	77.1153	3.53829
$476$	0.660406	0.0302697
$477$	7.94392	0.363727
$478$	−1.80842	−0.0827152
$479$	37.8318	1.72858	0.864289	−	0.502996i	$-0.167769\pi$
$479$	37.8318	1.72858	0.864289	+	0.502996i	$0.167769\pi$
$480$	7.50235	0.342434
$481$	−17.0226	−0.776162
$482$	2.02334	0.0921608
$483$	−1.66510	−0.0757647
$484$	−1.97196	−0.0896346
$485$	0.213351	0.00968778
$486$	0.167449	0.00759565
$487$	40.5576	1.83784	0.918921	−	0.394442i	$-0.129062\pi$
$487$	40.5576	1.83784	0.918921	+	0.394442i	$0.129062\pi$
$488$	3.99061	0.180646
$489$	6.79982	0.307498
$490$	−0.637062	−0.0287795
$491$	31.0786	1.40256	0.701280	−	0.712886i	$-0.252612\pi$
$491$	31.0786	1.40256	0.701280	+	0.712886i	$0.252612\pi$
$492$	−12.3816	−0.558205
$493$	0.0654688	0.00294856
$494$	−5.18531	−0.233298
$495$	3.80451	0.171000
$496$	−38.1106	−1.71122
$497$	−4.66980	−0.209469
$498$	−2.33490	−0.104629
$499$	15.3575	0.687494	0.343747	−	0.939062i	$-0.388304\pi$
$499$	15.3575	0.687494	0.343747	+	0.939062i	$0.388304\pi$
$500$	−33.5678	−1.50120
$501$	18.2227	0.814132
$502$	−0.805298	−0.0359422
$503$	14.8925	0.664025	0.332013	−	0.943275i	$-0.392272\pi$
$503$	14.8925	0.664025	0.332013	+	0.943275i	$0.392272\pi$
$504$	−0.665102	−0.0296260
$505$	−71.8412	−3.19689
$506$	0.278820	0.0123951
$507$	−1.47431	−0.0654763
$508$	−28.8177	−1.27858
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0.213351	0.00944735
$511$	−4.19549	−0.185597
$512$	−12.6557	−0.559309
$513$	−8.13941	−0.359364
$514$	2.80530	0.123736
$515$	−31.4969	−1.38792
$516$	4.60433	0.202694
$517$	−12.1394	−0.533891
$518$	−0.749219	−0.0329188
$519$	1.72118	0.0755514
$520$	9.62689	0.422167
$521$	27.6924	1.21322	0.606612	−	0.794998i	$-0.292528\pi$
$521$	27.6924	1.21322	0.606612	+	0.794998i	$0.292528\pi$
$522$	−0.0327344	−0.00143274
$523$	18.4088	0.804962	0.402481	−	0.915428i	$-0.368148\pi$
$523$	18.4088	0.804962	0.402481	+	0.915428i	$0.368148\pi$
$524$	−40.5389	−1.77095
$525$	9.47431	0.413493
$526$	2.03273	0.0886314
$527$	−3.33020	−0.145066
$528$	−3.83255	−0.166790
$529$	−20.2274	−0.879454
$530$	−5.06077	−0.219826
$531$	3.74843	0.162668
$532$	16.0506	0.695882
$533$	−23.8878	−1.03470
$534$	−1.65571	−0.0716496
$535$	−30.9665	−1.33880
$536$	−0.0927218	−0.00400497
$537$	12.0000	0.517838
$538$	4.22274	0.182055
$539$	1.00000	0.0430730
$540$	7.50235	0.322850
$541$	−4.26943	−0.183557	−0.0917786	−	0.995779i	$-0.529255\pi$
$541$	−4.26943	−0.183557	−0.0917786	+	0.995779i	$0.529255\pi$
$542$	−0.693141	−0.0297729
$543$	−0.725875	−0.0311503
$544$	−0.660406	−0.0283147
$545$	43.9533	1.88275
$546$	−0.637062	−0.0272638
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	31.9906	1.36657
$549$	6.00000	0.256074
$550$	−1.58647	−0.0676471
$551$	1.59116	0.0677857
$552$	1.10746	0.0471367
$553$	−3.33020	−0.141615
$554$	3.06077	0.130040
$555$	17.0226	0.722567
$556$	−44.4643	−1.88570
$557$	−12.4743	−0.528553	−0.264277	−	0.964447i	$-0.585133\pi$
$557$	−12.4743	−0.528553	−0.264277	+	0.964447i	$0.585133\pi$
$558$	1.66510	0.0704894
$559$	8.88315	0.375717
$560$	−14.5810	−0.616159
$561$	−0.334898	−0.0141394
$562$	−1.12923	−0.0476338
$563$	−32.7710	−1.38113	−0.690566	−	0.723269i	$-0.742639\pi$
$563$	−32.7710	−1.38113	−0.690566	+	0.723269i	$0.742639\pi$
$564$	−23.9384	−1.00799
$565$	−5.06077	−0.212908
$566$	−3.91119	−0.164399
$567$	−1.00000	−0.0419961
$568$	3.10589	0.130320
$569$	−29.2180	−1.22488	−0.612442	−	0.790515i	$-0.709813\pi$
$569$	−29.2180	−1.22488	−0.612442	+	0.790515i	$0.709813\pi$
$570$	5.18531	0.217189
$571$	32.4455	1.35780	0.678901	−	0.734230i	$-0.262457\pi$
$571$	32.4455	1.35780	0.678901	+	0.734230i	$0.262457\pi$
$572$	−7.50235	−0.313689
$573$	−5.27412	−0.220330
$574$	−1.05138	−0.0438839
$575$	−15.7757	−0.657892
$576$	−7.33490	−0.305621
$577$	−14.8831	−0.619594	−0.309797	−	0.950803i	$-0.600261\pi$
$577$	−14.8831	−0.619594	−0.309797	+	0.950803i	$0.600261\pi$
$578$	2.82786	0.117623
$579$	19.8318	0.824180
$580$	−1.46662	−0.0608982
$581$	13.9439	0.578491
$582$	0.00939029	0.000389240 0
$583$	7.94392	0.329004
$584$	2.79043	0.115469
$585$	14.4743	0.598439
$586$	2.37220	0.0979945
$587$	−4.53039	−0.186989	−0.0934945	−	0.995620i	$-0.529804\pi$
$587$	−4.53039	−0.186989	−0.0934945	+	0.995620i	$0.529804\pi$
$588$	1.97196	0.0813223
$589$	−80.9377	−3.33498
$590$	−2.38799	−0.0983118
$591$	−2.66980	−0.109821
$592$	−17.1480	−0.704779
$593$	8.28821	0.340356	0.170178	−	0.985413i	$-0.445566\pi$
$593$	8.28821	0.340356	0.170178	+	0.985413i	$0.445566\pi$
$594$	0.167449	0.00687052
$595$	−1.27412	−0.0522340
$596$	−16.1612	−0.661988
$597$	−13.5529	−0.554685
$598$	1.06077	0.0433783
$599$	1.99061	0.0813341	0.0406671	−	0.999173i	$-0.487052\pi$
$599$	1.99061	0.0813341	0.0406671	+	0.999173i	$0.487052\pi$
$600$	−6.30138	−0.257253
$601$	−8.47431	−0.345674	−0.172837	−	0.984950i	$-0.555293\pi$
$601$	−8.47431	−0.345674	−0.172837	+	0.984950i	$0.555293\pi$
$602$	0.390977	0.0159350
$603$	−0.139410	−0.00567721
$604$	26.1761	1.06509
$605$	3.80451	0.154675
$606$	−3.16197	−0.128446
$607$	−22.9665	−0.932181	−0.466090	−	0.884737i	$-0.654338\pi$
$607$	−22.9665	−0.932181	−0.466090	+	0.884737i	$0.654338\pi$
$608$	−16.0506	−0.650938
$609$	0.195488	0.00792159
$610$	−3.82237	−0.154763
$611$	−46.1845	−1.86843
$612$	−0.660406	−0.0266953
$613$	3.55294	0.143502	0.0717510	−	0.997423i	$-0.477141\pi$
$613$	3.55294	0.143502	0.0717510	+	0.997423i	$0.477141\pi$
$614$	2.10277	0.0848608
$615$	23.8878	0.963251
$616$	−0.665102	−0.0267977
$617$	22.6698	0.912652	0.456326	−	0.889813i	$-0.349165\pi$
$617$	22.6698	0.912652	0.456326	+	0.889813i	$0.349165\pi$
$618$	−1.38628	−0.0557644
$619$	8.71648	0.350345	0.175173	−	0.984538i	$-0.443952\pi$
$619$	8.71648	0.350345	0.175173	+	0.984538i	$0.443952\pi$
$620$	74.6028	2.99612
$621$	1.66510	0.0668182
$622$	−1.56391	−0.0627070
$623$	9.88784	0.396148
$624$	−14.5810	−0.583707
$625$	17.3910	0.695639
$626$	0.501561	0.0200464
$627$	−8.13941	−0.325057
$628$	33.4035	1.33294
$629$	−1.49844	−0.0597467
$630$	0.637062	0.0253812
$631$	−11.8878	−0.473248	−0.236624	−	0.971601i	$-0.576041\pi$
$631$	−11.8878	−0.473248	−0.236624	+	0.971601i	$0.576041\pi$
$632$	2.21492	0.0881049
$633$	4.27882	0.170068
$634$	−1.66353	−0.0660672
$635$	55.5981	2.20634
$636$	15.6651	0.621162
$637$	3.80451	0.150740
$638$	−0.0327344	−0.00129597
$639$	4.66980	0.184734
$640$	19.6775	0.777821
$641$	−4.89254	−0.193244	−0.0966218	−	0.995321i	$-0.530804\pi$
$641$	−4.89254	−0.193244	−0.0966218	+	0.995321i	$0.530804\pi$
$642$	−1.36294	−0.0537909
$643$	22.7804	0.898371	0.449185	−	0.893439i	$-0.351714\pi$
$643$	22.7804	0.898371	0.449185	+	0.893439i	$0.351714\pi$
$644$	−3.28352	−0.129389
$645$	−8.88315	−0.349774
$646$	−0.456446	−0.0179586
$647$	31.2453	1.22838	0.614190	−	0.789158i	$-0.289483\pi$
$647$	31.2453	1.22838	0.614190	+	0.789158i	$0.289483\pi$
$648$	0.665102	0.0261277
$649$	3.74843	0.147139
$650$	−6.03573	−0.236741
$651$	−9.94392	−0.389733
$652$	13.4090	0.525136
$653$	28.2882	1.10700	0.553502	−	0.832848i	$-0.313291\pi$
$653$	28.2882	1.10700	0.553502	+	0.832848i	$0.313291\pi$
$654$	1.93453	0.0756462
$655$	78.2118	3.05599
$656$	−24.0639	−0.939537
$657$	4.19549	0.163682
$658$	−2.03273	−0.0792442
$659$	30.2967	1.18019	0.590096	−	0.807333i	$-0.299090\pi$
$659$	30.2967	1.18019	0.590096	+	0.807333i	$0.299090\pi$
$660$	7.50235	0.292028
$661$	−35.7196	−1.38933	−0.694666	−	0.719333i	$-0.744448\pi$
$661$	−35.7196	−1.38933	−0.694666	+	0.719333i	$0.744448\pi$
$662$	−3.77569	−0.146746
$663$	−1.27412	−0.0494829
$664$	−9.27412	−0.359906
$665$	−30.9665	−1.20083
$666$	0.749219	0.0290317
$667$	−0.325508	−0.0126037
$668$	35.9345	1.39035
$669$	10.2694	0.397039
$670$	0.0888128	0.00343114
$671$	6.00000	0.231627
$672$	−1.97196	−0.0760700
$673$	34.9377	1.34675	0.673374	−	0.739302i	$-0.264844\pi$
$673$	34.9377	1.34675	0.673374	+	0.739302i	$0.264844\pi$
$674$	−2.22431	−0.0856774
$675$	−9.47431	−0.364666
$676$	−2.90728	−0.111818
$677$	20.0094	0.769023	0.384512	−	0.923120i	$-0.374370\pi$
$677$	20.0094	0.769023	0.384512	+	0.923120i	$0.374370\pi$
$678$	−0.222741	−0.00855433
$679$	−0.0560785	−0.00215209
$680$	0.847422	0.0324972
$681$	0.390977	0.0149823
$682$	1.66510	0.0637600
$683$	25.0498	0.958504	0.479252	−	0.877677i	$-0.340908\pi$
$683$	25.0498	0.958504	0.479252	+	0.877677i	$0.340908\pi$
$684$	−16.0506	−0.613710
$685$	−61.7196	−2.35818
$686$	0.167449	0.00639324
$687$	−7.94392	−0.303080
$688$	8.94862	0.341163
$689$	30.2227	1.15139
$690$	−1.06077	−0.0403830
$691$	−38.8271	−1.47705	−0.738526	−	0.674225i	$-0.764478\pi$
$691$	−38.8271	−1.47705	−0.738526	+	0.674225i	$0.764478\pi$
$692$	3.39410	0.129024
$693$	−1.00000	−0.0379869
$694$	4.65102	0.176550
$695$	85.7851	3.25401
$696$	−0.130020	−0.00492838
$697$	−2.10277	−0.0796480
$698$	0.478217	0.0181008
$699$	−26.5576	−1.00450
$700$	18.6830	0.706150
$701$	−41.7757	−1.57785	−0.788923	−	0.614492i	$-0.789361\pi$
$701$	−41.7757	−1.57785	−0.788923	+	0.614492i	$0.789361\pi$
$702$	0.637062	0.0240444
$703$	−36.4182	−1.37354
$704$	−7.33490	−0.276444
$705$	46.1845	1.73941
$706$	−4.12611	−0.155288
$707$	18.8831	0.710174
$708$	7.39176	0.277799
$709$	−13.4041	−0.503403	−0.251702	−	0.967805i	$-0.580990\pi$
$709$	−13.4041	−0.503403	−0.251702	+	0.967805i	$0.580990\pi$
$710$	−2.97495	−0.111648
$711$	3.33020	0.124892
$712$	−6.57642	−0.246462
$713$	16.5576	0.620088
$714$	−0.0560785	−0.00209868
$715$	14.4743	0.541308
$716$	23.6635	0.884348
$717$	−10.7998	−0.403327
$718$	1.99061	0.0742889
$719$	5.47900	0.204332	0.102166	−	0.994767i	$-0.467423\pi$
$719$	5.47900	0.204332	0.102166	+	0.994767i	$0.467423\pi$
$720$	14.5810	0.543401
$721$	8.27882	0.308319
$722$	−7.91197	−0.294453
$723$	12.0833	0.449384
$724$	−1.43140	−0.0531975
$725$	1.85212	0.0687859
$726$	0.167449	0.00621462
$727$	−32.8831	−1.21957	−0.609784	−	0.792567i	$-0.708744\pi$
$727$	−32.8831	−1.21957	−0.609784	+	0.792567i	$0.708744\pi$
$728$	−2.53039	−0.0937824
$729$	1.00000	0.0370370
$730$	−2.67279	−0.0989243
$731$	0.781954	0.0289216
$732$	11.8318	0.437315
$733$	−35.8972	−1.32589	−0.662947	−	0.748666i	$-0.730695\pi$
$733$	−35.8972	−1.32589	−0.662947	+	0.748666i	$0.730695\pi$
$734$	−0.604328	−0.0223062
$735$	−3.80451	−0.140332
$736$	3.28352	0.121032
$737$	−0.139410	−0.00513523
$738$	1.05138	0.0387020
$739$	29.6651	1.09125	0.545624	−	0.838030i	$-0.316293\pi$
$739$	29.6651	1.09125	0.545624	+	0.838030i	$0.316293\pi$
$740$	33.5678	1.23398
$741$	−30.9665	−1.13758
$742$	1.33020	0.0488333
$743$	−18.7998	−0.689698	−0.344849	−	0.938658i	$-0.612070\pi$
$743$	−18.7998	−0.689698	−0.344849	+	0.938658i	$0.612070\pi$
$744$	6.61372	0.242471
$745$	31.1798	1.14234
$746$	−2.06547	−0.0756222
$747$	−13.9439	−0.510181
$748$	−0.660406	−0.0241469
$749$	8.13941	0.297408
$750$	2.85041	0.104082
$751$	−9.59116	−0.349986	−0.174993	−	0.984570i	$-0.555990\pi$
$751$	−9.59116	−0.349986	−0.174993	+	0.984570i	$0.555990\pi$
$752$	−46.5249	−1.69659
$753$	−4.80921	−0.175257
$754$	−0.124538	−0.00453542
$755$	−50.5016	−1.83794
$756$	−1.97196	−0.0717195
$757$	2.74374	0.0997229	0.0498614	−	0.998756i	$-0.484122\pi$
$757$	2.74374	0.0997229	0.0498614	+	0.998756i	$0.484122\pi$
$758$	−3.91119	−0.142061
$759$	1.66510	0.0604394
$760$	20.5959	0.747090
$761$	6.39098	0.231673	0.115836	−	0.993268i	$-0.463045\pi$
$761$	6.39098	0.231673	0.115836	+	0.993268i	$0.463045\pi$
$762$	2.44706	0.0886475
$763$	−11.5529	−0.418245
$764$	−10.4004	−0.376272
$765$	1.27412	0.0460661
$766$	2.54825	0.0920720
$767$	14.2610	0.514933
$768$	−13.8037	−0.498099
$769$	17.7951	0.641708	0.320854	−	0.947129i	$-0.396030\pi$
$769$	17.7951	0.641708	0.320854	+	0.947129i	$0.396030\pi$
$770$	0.637062	0.0229581
$771$	16.7531	0.603349
$772$	39.1075	1.40751
$773$	−31.5257	−1.13390	−0.566950	−	0.823752i	$-0.691877\pi$
$773$	−31.5257	−1.13390	−0.566950	+	0.823752i	$0.691877\pi$
$774$	−0.390977	−0.0140534
$775$	−94.2118	−3.38419
$776$	0.0372979	0.00133892
$777$	−4.47431	−0.160515
$778$	−0.624681	−0.0223959
$779$	−51.1059	−1.83106
$780$	28.5428	1.02200
$781$	4.66980	0.167098
$782$	0.0933763	0.00333913
$783$	−0.195488	−0.00698619
$784$	3.83255	0.136877
$785$	−64.4455	−2.30016
$786$	3.44236	0.122785
$787$	31.8606	1.13571	0.567854	−	0.823130i	$-0.307774\pi$
$787$	31.8606	1.13571	0.567854	+	0.823130i	$0.307774\pi$
$788$	−5.26473	−0.187548
$789$	12.1394	0.432174
$790$	−2.12155	−0.0754813
$791$	1.33020	0.0472966
$792$	0.665102	0.0236334
$793$	22.8271	0.810613
$794$	−3.49844	−0.124155
$795$	−30.2227	−1.07189
$796$	−26.7259	−0.947274
$797$	43.8590	1.55357	0.776783	−	0.629768i	$-0.216850\pi$
$797$	43.8590	1.55357	0.776783	+	0.629768i	$0.216850\pi$
$798$	−1.36294	−0.0482475
$799$	−4.06547	−0.143826
$800$	−18.6830	−0.660543
$801$	−9.88784	−0.349370
$802$	−3.89723	−0.137616
$803$	4.19549	0.148056
$804$	−0.274911	−0.00969536
$805$	6.33490	0.223276
$806$	6.33490	0.223137
$807$	25.2180	0.887717
$808$	−12.5592	−0.441832
$809$	6.74374	0.237097	0.118549	−	0.992948i	$-0.462176\pi$
$809$	6.74374	0.237097	0.118549	+	0.992948i	$0.462176\pi$
$810$	−0.637062	−0.0223841
$811$	3.97275	0.139502	0.0697510	−	0.997564i	$-0.477780\pi$
$811$	3.97275	0.139502	0.0697510	+	0.997564i	$0.477780\pi$
$812$	0.385496	0.0135282
$813$	−4.13941	−0.145175
$814$	0.749219	0.0262601
$815$	−25.8700	−0.906186
$816$	−1.28352	−0.0449320
$817$	19.0047	0.664890
$818$	4.00157	0.139912
$819$	−3.80451	−0.132940
$820$	47.1059	1.64501
$821$	49.5896	1.73069	0.865344	−	0.501178i	$-0.167100\pi$
$821$	49.5896	1.73069	0.865344	+	0.501178i	$0.167100\pi$
$822$	−2.71648	−0.0947483
$823$	23.1908	0.808380	0.404190	−	0.914675i	$-0.367553\pi$
$823$	23.1908	0.808380	0.404190	+	0.914675i	$0.367553\pi$
$824$	−5.50626	−0.191820
$825$	−9.47431	−0.329853
$826$	0.627672	0.0218395
$827$	27.7484	0.964908	0.482454	−	0.875921i	$-0.339746\pi$
$827$	27.7484	0.964908	0.482454	+	0.875921i	$0.339746\pi$
$828$	3.28352	0.114110
$829$	0.269430	0.00935768	0.00467884	−	0.999989i	$-0.498511\pi$
$829$	0.269430	0.00935768	0.00467884	+	0.999989i	$0.498511\pi$
$830$	8.88315	0.308339
$831$	18.2788	0.634085
$832$	−27.9057	−0.967456
$833$	0.334898	0.0116035
$834$	3.77569	0.130741
$835$	−69.3286	−2.39922
$836$	−16.0506	−0.555122
$837$	9.94392	0.343712
$838$	5.07316	0.175249
$839$	−27.3575	−0.944484	−0.472242	−	0.881469i	$-0.656555\pi$
$839$	−27.3575	−0.944484	−0.472242	+	0.881469i	$0.656555\pi$
$840$	2.53039	0.0873066
$841$	−28.9618	−0.998682
$842$	2.59976	0.0895938
$843$	−6.74374	−0.232267
$844$	8.43767	0.290436
$845$	5.60902	0.192956
$846$	2.03273	0.0698868
$847$	−1.00000	−0.0343604
$848$	30.4455	1.04550
$849$	−23.3575	−0.801626
$850$	−0.531305	−0.0182236
$851$	7.45018	0.255389
$852$	9.20866	0.315483
$853$	−28.5482	−0.977473	−0.488737	−	0.872431i	$-0.662542\pi$
$853$	−28.5482	−0.977473	−0.488737	+	0.872431i	$0.662542\pi$
$854$	1.00470	0.0343800
$855$	30.9665	1.05903
$856$	−5.41353	−0.185031
$857$	−39.8972	−1.36286	−0.681432	−	0.731882i	$-0.738642\pi$
$857$	−39.8972	−1.36286	−0.681432	+	0.731882i	$0.738642\pi$
$858$	0.637062	0.0217490
$859$	20.5576	0.701418	0.350709	−	0.936485i	$-0.385941\pi$
$859$	20.5576	0.701418	0.350709	+	0.936485i	$0.385941\pi$
$860$	−17.5172	−0.597332
$861$	−6.27882	−0.213982
$862$	−0.515515	−0.0175585
$863$	−9.66510	−0.329004	−0.164502	−	0.986377i	$-0.552602\pi$
$863$	−9.66510	−0.329004	−0.164502	+	0.986377i	$0.552602\pi$
$864$	1.97196	0.0670875
$865$	−6.54825	−0.222647
$866$	−2.83646	−0.0963868
$867$	16.8878	0.573541
$868$	−19.6090	−0.665574
$869$	3.33020	0.112969
$870$	0.124538	0.00422224
$871$	−0.530387	−0.0179715
$872$	7.68388	0.260209
$873$	0.0560785	0.00189797
$874$	2.26943	0.0767646
$875$	−17.0226	−0.575467
$876$	8.27334	0.279530
$877$	−25.7212	−0.868543	−0.434271	−	0.900782i	$-0.642994\pi$
$877$	−25.7212	−0.868543	−0.434271	+	0.900782i	$0.642994\pi$
$878$	1.85511	0.0626069
$879$	14.1667	0.477830
$880$	14.5810	0.491525
$881$	−14.6316	−0.492950	−0.246475	−	0.969149i	$-0.579272\pi$
$881$	−14.6316	−0.492950	−0.246475	+	0.969149i	$0.579272\pi$
$882$	−0.167449	−0.00563831
$883$	18.6877	0.628890	0.314445	−	0.949276i	$-0.398182\pi$
$883$	18.6877	0.628890	0.314445	+	0.949276i	$0.398182\pi$
$884$	−2.51252	−0.0845053
$885$	−14.2610	−0.479377
$886$	2.43609	0.0818421
$887$	38.6137	1.29652	0.648261	−	0.761418i	$-0.275497\pi$
$887$	38.6137	1.29652	0.648261	+	0.761418i	$0.275497\pi$
$888$	2.97587	0.0998636
$889$	−14.6137	−0.490128
$890$	6.29917	0.211149
$891$	1.00000	0.0335013
$892$	20.2509	0.678051
$893$	−98.8076	−3.30647
$894$	1.37233	0.0458975
$895$	−45.6541	−1.52605
$896$	−5.17214	−0.172789
$897$	6.33490	0.211516
$898$	−4.60276	−0.153596
$899$	−1.94392	−0.0648334
$900$	−18.6830	−0.622765
$901$	2.66041	0.0886310
$902$	1.05138	0.0350072
$903$	2.33490	0.0777006
$904$	−0.884720	−0.0294254
$905$	2.76160	0.0917987
$906$	−2.22274	−0.0738456
$907$	49.0965	1.63022	0.815111	−	0.579304i	$-0.196676\pi$
$907$	49.0965	1.63022	0.815111	+	0.579304i	$0.196676\pi$
$908$	0.770991	0.0255862
$909$	−18.8831	−0.626314
$910$	2.42371	0.0803452
$911$	29.3753	0.973248	0.486624	−	0.873612i	$-0.338228\pi$
$911$	29.3753	0.973248	0.486624	+	0.873612i	$0.338228\pi$
$912$	−31.1947	−1.03296
$913$	−13.9439	−0.461476
$914$	1.33020	0.0439992
$915$	−22.8271	−0.754640
$916$	−15.6651	−0.517590
$917$	−20.5576	−0.678873
$918$	0.0560785	0.00185087
$919$	−27.0047	−0.890803	−0.445401	−	0.895331i	$-0.646939\pi$
$919$	−27.0047	−0.890803	−0.445401	+	0.895331i	$0.646939\pi$
$920$	−4.21335	−0.138910
$921$	12.5576	0.413788
$922$	−1.38471	−0.0456030
$923$	17.7663	0.584785
$924$	−1.97196	−0.0648727
$925$	−42.3910	−1.39381
$926$	−1.63079	−0.0535912
$927$	−8.27882	−0.271912
$928$	−0.385496	−0.0126545
$929$	57.8496	1.89798	0.948992	−	0.315299i	$-0.102105\pi$
$929$	57.8496	1.89798	0.948992	+	0.315299i	$0.102105\pi$
$930$	−6.33490	−0.207730
$931$	8.13941	0.266758
$932$	−52.3706	−1.71546
$933$	−9.33959	−0.305765
$934$	6.81469	0.222983
$935$	1.27412	0.0416683
$936$	2.53039	0.0827083
$937$	25.2180	0.823838	0.411919	−	0.911221i	$-0.364859\pi$
$937$	25.2180	0.823838	0.411919	+	0.911221i	$0.364859\pi$
$938$	−0.0233441	−0.000762211 0
$939$	2.99530	0.0977481
$940$	91.0741	2.97051
$941$	−34.2788	−1.11746	−0.558729	−	0.829350i	$-0.688711\pi$
$941$	−34.2788	−1.11746	−0.558729	+	0.829350i	$0.688711\pi$
$942$	−2.83646	−0.0924169
$943$	10.4549	0.340458
$944$	14.3661	0.467575
$945$	3.80451	0.123761
$946$	−0.390977	−0.0127118
$947$	−18.1573	−0.590032	−0.295016	−	0.955492i	$-0.595325\pi$
$947$	−18.1573	−0.590032	−0.295016	+	0.955492i	$0.595325\pi$
$948$	6.56703	0.213287
$949$	15.9618	0.518141
$950$	−12.9129	−0.418950
$951$	−9.93453	−0.322149
$952$	−0.222741	−0.00721909
$953$	−19.8045	−0.641531	−0.320766	−	0.947159i	$-0.603940\pi$
$953$	−19.8045	−0.641531	−0.320766	+	0.947159i	$0.603940\pi$
$954$	−1.33020	−0.0430669
$955$	20.0655	0.649303
$956$	−21.2968	−0.688788
$957$	−0.195488	−0.00631924
$958$	−6.33490	−0.204671
$959$	16.2227	0.523860
$960$	27.9057	0.900653
$961$	67.8816	2.18973
$962$	2.85041	0.0919010
$963$	−8.13941	−0.262289
$964$	23.8279	0.767444
$965$	−75.4502	−2.42883
$966$	0.278820	0.00897088
$967$	−0.278820	−0.00896624	−0.00448312	−	0.999990i	$-0.501427\pi$
$967$	−0.278820	−0.00896624	−0.00448312	+	0.999990i	$0.501427\pi$
$968$	0.665102	0.0213772
$969$	−2.72588	−0.0875677
$970$	−0.0357255	−0.00114708
$971$	25.3481	0.813458	0.406729	−	0.913549i	$-0.366669\pi$
$971$	25.3481	0.813458	0.406729	+	0.913549i	$0.366669\pi$
$972$	1.97196	0.0632507
$973$	−22.5482	−0.722864
$974$	−6.79134	−0.217609
$975$	−36.0451	−1.15437
$976$	22.9953	0.736062
$977$	6.71648	0.214879	0.107440	−	0.994212i	$-0.465735\pi$
$977$	6.71648	0.214879	0.107440	+	0.994212i	$0.465735\pi$
$978$	−1.13862	−0.0364092
$979$	−9.88784	−0.316017
$980$	−7.50235	−0.239654
$981$	11.5529	0.368857
$982$	−5.20409	−0.166069
$983$	9.99061	0.318651	0.159325	−	0.987226i	$-0.449068\pi$
$983$	9.99061	0.318651	0.159325	+	0.987226i	$0.449068\pi$
$984$	4.17605	0.133128
$985$	10.1573	0.323638
$986$	−0.0109627	−0.000349123 0
$987$	−12.1394	−0.386402
$988$	−61.0647	−1.94273
$989$	−3.88784	−0.123626
$990$	−0.637062	−0.0202472
$991$	−26.7064	−0.848358	−0.424179	−	0.905578i	$-0.639437\pi$
$991$	−26.7064	−0.848358	−0.424179	+	0.905578i	$0.639437\pi$
$992$	19.6090	0.622587
$993$	−22.5482	−0.715547
$994$	0.781954	0.0248021
$995$	51.5623	1.63464
$996$	−27.4969	−0.871272
$997$	54.3333	1.72075	0.860377	−	0.509658i	$-0.170228\pi$
$997$	54.3333	1.72075	0.860377	+	0.509658i	$0.170228\pi$
$998$	−2.57159	−0.0814024
$999$	4.47431	0.141561

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.2.a.d.1.2	✓	3
3.2	odd	2		693.2.a.m.1.2		3
4.3	odd	2		3696.2.a.bp.1.3		3
5.4	even	2		5775.2.a.bw.1.2		3
7.6	odd	2		1617.2.a.s.1.2		3
11.10	odd	2		2541.2.a.bi.1.2		3
21.20	even	2		4851.2.a.bp.1.2		3
33.32	even	2		7623.2.a.cb.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.d.1.2	✓	3	1.1	even	1	trivial
693.2.a.m.1.2		3	3.2	odd	2
1617.2.a.s.1.2		3	7.6	odd	2
2541.2.a.bi.1.2		3	11.10	odd	2
3696.2.a.bp.1.3		3	4.3	odd	2
4851.2.a.bp.1.2		3	21.20	even	2
5775.2.a.bw.1.2		3	5.4	even	2
7623.2.a.cb.1.2		3	33.32	even	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:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	231.a (trivial)

\(f(q)\)	\(=\)	\(q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +3.80451 q^{5} +0.167449 q^{6} -1.00000 q^{7} +0.665102 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +3.80451 q^{5} +0.167449 q^{6} -1.00000 q^{7} +0.665102 q^{8} +1.00000 q^{9} -0.637062 q^{10} +1.00000 q^{11} +1.97196 q^{12} +3.80451 q^{13} +0.167449 q^{14} -3.80451 q^{15} +3.83255 q^{16} +0.334898 q^{17} -0.167449 q^{18} +8.13941 q^{19} -7.50235 q^{20} +1.00000 q^{21} -0.167449 q^{22} -1.66510 q^{23} -0.665102 q^{24} +9.47431 q^{25} -0.637062 q^{26} -1.00000 q^{27} +1.97196 q^{28} +0.195488 q^{29} +0.637062 q^{30} -9.94392 q^{31} -1.97196 q^{32} -1.00000 q^{33} -0.0560785 q^{34} -3.80451 q^{35} -1.97196 q^{36} -4.47431 q^{37} -1.36294 q^{38} -3.80451 q^{39} +2.53039 q^{40} -6.27882 q^{41} -0.167449 q^{42} +2.33490 q^{43} -1.97196 q^{44} +3.80451 q^{45} +0.278820 q^{46} -12.1394 q^{47} -3.83255 q^{48} +1.00000 q^{49} -1.58647 q^{50} -0.334898 q^{51} -7.50235 q^{52} +7.94392 q^{53} +0.167449 q^{54} +3.80451 q^{55} -0.665102 q^{56} -8.13941 q^{57} -0.0327344 q^{58} +3.74843 q^{59} +7.50235 q^{60} +6.00000 q^{61} +1.66510 q^{62} -1.00000 q^{63} -7.33490 q^{64} +14.4743 q^{65} +0.167449 q^{66} -0.139410 q^{67} -0.660406 q^{68} +1.66510 q^{69} +0.637062 q^{70} +4.66980 q^{71} +0.665102 q^{72} +4.19549 q^{73} +0.749219 q^{74} -9.47431 q^{75} -16.0506 q^{76} -1.00000 q^{77} +0.637062 q^{78} +3.33020 q^{79} +14.5810 q^{80} +1.00000 q^{81} +1.05138 q^{82} -13.9439 q^{83} -1.97196 q^{84} +1.27412 q^{85} -0.390977 q^{86} -0.195488 q^{87} +0.665102 q^{88} -9.88784 q^{89} -0.637062 q^{90} -3.80451 q^{91} +3.28352 q^{92} +9.94392 q^{93} +2.03273 q^{94} +30.9665 q^{95} +1.97196 q^{96} +0.0560785 q^{97} -0.167449 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 6 * q^4 - 3 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 9 q^{10} + 3 q^{11} - 6 q^{12} + 12 q^{16} + 12 q^{19} - 21 q^{20} + 3 q^{21} - 6 q^{23} - 3 q^{24} + 15 q^{25} + 9 q^{26} - 3 q^{27} - 6 q^{28} + 12 q^{29} - 9 q^{30} - 6 q^{31} + 6 q^{32} - 3 q^{33} - 24 q^{34} + 6 q^{36} - 15 q^{38} + 18 q^{40} + 6 q^{41} + 6 q^{43} + 6 q^{44} - 24 q^{46} - 24 q^{47} - 12 q^{48} + 3 q^{49} - 39 q^{50} - 21 q^{52} - 3 q^{56} - 12 q^{57} - 9 q^{58} - 24 q^{59} + 21 q^{60} + 18 q^{61} + 6 q^{62} - 3 q^{63} - 21 q^{64} + 30 q^{65} + 12 q^{67} - 6 q^{68} + 6 q^{69} - 9 q^{70} + 12 q^{71} + 3 q^{72} + 24 q^{73} + 39 q^{74} - 15 q^{75} - 3 q^{76} - 3 q^{77} - 9 q^{78} + 12 q^{79} + 9 q^{80} + 3 q^{81} + 30 q^{82} - 18 q^{83} + 6 q^{84} - 18 q^{85} - 24 q^{86} - 12 q^{87} + 3 q^{88} + 18 q^{89} + 9 q^{90} - 18 q^{92} + 6 q^{93} + 15 q^{94} + 12 q^{95} - 6 q^{96} + 24 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 6 * q^4 - 3 * q^7 + 3 * q^8 + 3 * q^9 + 9 * q^10 + 3 * q^11 - 6 * q^12 + 12 * q^16 + 12 * q^19 - 21 * q^20 + 3 * q^21 - 6 * q^23 - 3 * q^24 + 15 * q^25 + 9 * q^26 - 3 * q^27 - 6 * q^28 + 12 * q^29 - 9 * q^30 - 6 * q^31 + 6 * q^32 - 3 * q^33 - 24 * q^34 + 6 * q^36 - 15 * q^38 + 18 * q^40 + 6 * q^41 + 6 * q^43 + 6 * q^44 - 24 * q^46 - 24 * q^47 - 12 * q^48 + 3 * q^49 - 39 * q^50 - 21 * q^52 - 3 * q^56 - 12 * q^57 - 9 * q^58 - 24 * q^59 + 21 * q^60 + 18 * q^61 + 6 * q^62 - 3 * q^63 - 21 * q^64 + 30 * q^65 + 12 * q^67 - 6 * q^68 + 6 * q^69 - 9 * q^70 + 12 * q^71 + 3 * q^72 + 24 * q^73 + 39 * q^74 - 15 * q^75 - 3 * q^76 - 3 * q^77 - 9 * q^78 + 12 * q^79 + 9 * q^80 + 3 * q^81 + 30 * q^82 - 18 * q^83 + 6 * q^84 - 18 * q^85 - 24 * q^86 - 12 * q^87 + 3 * q^88 + 18 * q^89 + 9 * q^90 - 18 * q^92 + 6 * q^93 + 15 * q^94 + 12 * q^95 - 6 * q^96 + 24 * q^97 + 3 * q^99

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