LMFDB - Embedded newform 1155.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1155,2,Mod(1,1155)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1155, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1155.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1155 = 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1155.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:	$9.22272143346$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*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			1.2
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	1155.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.470683 q^{2} +1.00000 q^{3} -1.77846 q^{4} +1.00000 q^{5} +0.470683 q^{6} -1.00000 q^{7} -1.77846 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.470683 q^{2} +1.00000 q^{3} -1.77846 q^{4} +1.00000 q^{5} +0.470683 q^{6} -1.00000 q^{7} -1.77846 q^{8} +1.00000 q^{9} +0.470683 q^{10} +1.00000 q^{11} -1.77846 q^{12} -0.249141 q^{13} -0.470683 q^{14} +1.00000 q^{15} +2.71982 q^{16} +5.71982 q^{17} +0.470683 q^{18} +1.83709 q^{19} -1.77846 q^{20} -1.00000 q^{21} +0.470683 q^{22} +0.778457 q^{23} -1.77846 q^{24} +1.00000 q^{25} -0.117266 q^{26} +1.00000 q^{27} +1.77846 q^{28} +3.47068 q^{29} +0.470683 q^{30} -3.30777 q^{31} +4.83709 q^{32} +1.00000 q^{33} +2.69223 q^{34} -1.00000 q^{35} -1.77846 q^{36} +10.8647 q^{37} +0.864688 q^{38} -0.249141 q^{39} -1.77846 q^{40} +9.80605 q^{41} -0.470683 q^{42} -5.96896 q^{43} -1.77846 q^{44} +1.00000 q^{45} +0.366407 q^{46} +3.30777 q^{47} +2.71982 q^{48} +1.00000 q^{49} +0.470683 q^{50} +5.71982 q^{51} +0.443086 q^{52} -2.77846 q^{53} +0.470683 q^{54} +1.00000 q^{55} +1.77846 q^{56} +1.83709 q^{57} +1.63359 q^{58} -3.58795 q^{59} -1.77846 q^{60} -7.27674 q^{61} -1.55691 q^{62} -1.00000 q^{63} -3.16291 q^{64} -0.249141 q^{65} +0.470683 q^{66} +5.55691 q^{67} -10.1725 q^{68} +0.778457 q^{69} -0.470683 q^{70} +2.74742 q^{71} -1.77846 q^{72} +4.94137 q^{73} +5.11383 q^{74} +1.00000 q^{75} -3.26719 q^{76} -1.00000 q^{77} -0.117266 q^{78} +2.48367 q^{79} +2.71982 q^{80} +1.00000 q^{81} +4.61555 q^{82} -7.15947 q^{83} +1.77846 q^{84} +5.71982 q^{85} -2.80949 q^{86} +3.47068 q^{87} -1.77846 q^{88} -14.5845 q^{89} +0.470683 q^{90} +0.249141 q^{91} -1.38445 q^{92} -3.30777 q^{93} +1.55691 q^{94} +1.83709 q^{95} +4.83709 q^{96} -14.0242 q^{97} +0.470683 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} + 3 q^{11} + 3 q^{12} + 8 q^{13} - q^{14} + 3 q^{15} - q^{16} + 8 q^{17} + q^{18} - 2 q^{19} + 3 q^{20} - 3 q^{21} + q^{22} - 6 q^{23} + 3 q^{24} + 3 q^{25} - 2 q^{26} + 3 q^{27} - 3 q^{28} + 10 q^{29} + q^{30} - 2 q^{31} + 7 q^{32} + 3 q^{33} + 16 q^{34} - 3 q^{35} + 3 q^{36} + 8 q^{37} - 22 q^{38} + 8 q^{39} + 3 q^{40} + 4 q^{41} - q^{42} + 3 q^{44} + 3 q^{45} - 6 q^{46} + 2 q^{47} - q^{48} + 3 q^{49} + q^{50} + 8 q^{51} + 18 q^{52} + q^{54} + 3 q^{55} - 3 q^{56} - 2 q^{57} + 12 q^{58} - 12 q^{59} + 3 q^{60} + 4 q^{61} + 12 q^{62} - 3 q^{63} - 17 q^{64} + 8 q^{65} + q^{66} + 2 q^{68} - 6 q^{69} - q^{70} - 18 q^{71} + 3 q^{72} + 14 q^{73} - 18 q^{74} + 3 q^{75} - 24 q^{76} - 3 q^{77} - 2 q^{78} + 2 q^{79} - q^{80} + 3 q^{81} - 2 q^{82} + 6 q^{83} - 3 q^{84} + 8 q^{85} - 18 q^{86} + 10 q^{87} + 3 q^{88} - 10 q^{89} + q^{90} - 8 q^{91} - 20 q^{92} - 2 q^{93} - 12 q^{94} - 2 q^{95} + 7 q^{96} + 10 q^{97} + q^{98} + 3 q^{99}+O(q^{100}) \) 3 * q + q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9 + q^10 + 3 * q^11 + 3 * q^12 + 8 * q^13 - q^14 + 3 * q^15 - q^16 + 8 * q^17 + q^18 - 2 * q^19 + 3 * q^20 - 3 * q^21 + q^22 - 6 * q^23 + 3 * q^24 + 3 * q^25 - 2 * q^26 + 3 * q^27 - 3 * q^28 + 10 * q^29 + q^30 - 2 * q^31 + 7 * q^32 + 3 * q^33 + 16 * q^34 - 3 * q^35 + 3 * q^36 + 8 * q^37 - 22 * q^38 + 8 * q^39 + 3 * q^40 + 4 * q^41 - q^42 + 3 * q^44 + 3 * q^45 - 6 * q^46 + 2 * q^47 - q^48 + 3 * q^49 + q^50 + 8 * q^51 + 18 * q^52 + q^54 + 3 * q^55 - 3 * q^56 - 2 * q^57 + 12 * q^58 - 12 * q^59 + 3 * q^60 + 4 * q^61 + 12 * q^62 - 3 * q^63 - 17 * q^64 + 8 * q^65 + q^66 + 2 * q^68 - 6 * q^69 - q^70 - 18 * q^71 + 3 * q^72 + 14 * q^73 - 18 * q^74 + 3 * q^75 - 24 * q^76 - 3 * q^77 - 2 * q^78 + 2 * q^79 - q^80 + 3 * q^81 - 2 * q^82 + 6 * q^83 - 3 * q^84 + 8 * q^85 - 18 * q^86 + 10 * q^87 + 3 * q^88 - 10 * q^89 + q^90 - 8 * q^91 - 20 * q^92 - 2 * q^93 - 12 * q^94 - 2 * q^95 + 7 * q^96 + 10 * q^97 + q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.470683	0.332823	0.166412	−	0.986056i	$-0.446782\pi$
$2$	0.470683	0.332823	0.166412	+	0.986056i	$0.446782\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.77846	−0.889229
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.470683	0.192156
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.77846	−0.628780
$9$	1.00000	0.333333
$10$	0.470683	0.148843
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.77846	−0.513396
$13$	−0.249141	−0.0690992	−0.0345496	−	0.999403i	$-0.511000\pi$
$13$	−0.249141	−0.0690992	−0.0345496	+	0.999403i	$0.511000\pi$
$14$	−0.470683	−0.125795
$15$	1.00000	0.258199
$16$	2.71982	0.679956
$17$	5.71982	1.38726	0.693631	−	0.720331i	$-0.256010\pi$
$17$	5.71982	1.38726	0.693631	+	0.720331i	$0.256010\pi$
$18$	0.470683	0.110941
$19$	1.83709	0.421457	0.210729	−	0.977545i	$-0.432416\pi$
$19$	1.83709	0.421457	0.210729	+	0.977545i	$0.432416\pi$
$20$	−1.77846	−0.397675
$21$	−1.00000	−0.218218
$22$	0.470683	0.100350
$23$	0.778457	0.162320	0.0811598	−	0.996701i	$-0.474138\pi$
$23$	0.778457	0.162320	0.0811598	+	0.996701i	$0.474138\pi$
$24$	−1.77846	−0.363026
$25$	1.00000	0.200000
$26$	−0.117266	−0.0229978
$27$	1.00000	0.192450
$28$	1.77846	0.336097
$29$	3.47068	0.644490	0.322245	−	0.946656i	$-0.395563\pi$
$29$	3.47068	0.644490	0.322245	+	0.946656i	$0.395563\pi$
$30$	0.470683	0.0859346
$31$	−3.30777	−0.594094	−0.297047	−	0.954863i	$-0.596002\pi$
$31$	−3.30777	−0.594094	−0.297047	+	0.954863i	$0.596002\pi$
$32$	4.83709	0.855085
$33$	1.00000	0.174078
$34$	2.69223	0.461713
$35$	−1.00000	−0.169031
$36$	−1.77846	−0.296410
$37$	10.8647	1.78614	0.893072	−	0.449914i	$-0.148545\pi$
$37$	10.8647	1.78614	0.893072	+	0.449914i	$0.148545\pi$
$38$	0.864688	0.140271
$39$	−0.249141	−0.0398944
$40$	−1.77846	−0.281199
$41$	9.80605	1.53145	0.765724	−	0.643169i	$-0.222381\pi$
$41$	9.80605	1.53145	0.765724	+	0.643169i	$0.222381\pi$
$42$	−0.470683	−0.0726280
$43$	−5.96896	−0.910259	−0.455129	−	0.890425i	$-0.650407\pi$
$43$	−5.96896	−0.910259	−0.455129	+	0.890425i	$0.650407\pi$
$44$	−1.77846	−0.268112
$45$	1.00000	0.149071
$46$	0.366407	0.0540237
$47$	3.30777	0.482488	0.241244	−	0.970464i	$-0.422445\pi$
$47$	3.30777	0.482488	0.241244	+	0.970464i	$0.422445\pi$
$48$	2.71982	0.392573
$49$	1.00000	0.142857
$50$	0.470683	0.0665647
$51$	5.71982	0.800936
$52$	0.443086	0.0614449
$53$	−2.77846	−0.381650	−0.190825	−	0.981624i	$-0.561116\pi$
$53$	−2.77846	−0.381650	−0.190825	+	0.981624i	$0.561116\pi$
$54$	0.470683	0.0640519
$55$	1.00000	0.134840
$56$	1.77846	0.237656
$57$	1.83709	0.243329
$58$	1.63359	0.214501
$59$	−3.58795	−0.467111	−0.233556	−	0.972343i	$-0.575036\pi$
$59$	−3.58795	−0.467111	−0.233556	+	0.972343i	$0.575036\pi$
$60$	−1.77846	−0.229598
$61$	−7.27674	−0.931691	−0.465845	−	0.884866i	$-0.654250\pi$
$61$	−7.27674	−0.931691	−0.465845	+	0.884866i	$0.654250\pi$
$62$	−1.55691	−0.197728
$63$	−1.00000	−0.125988
$64$	−3.16291	−0.395364
$65$	−0.249141	−0.0309021
$66$	0.470683	0.0579371
$67$	5.55691	0.678885	0.339443	−	0.940627i	$-0.389762\pi$
$67$	5.55691	0.678885	0.339443	+	0.940627i	$0.389762\pi$
$68$	−10.1725	−1.23359
$69$	0.778457	0.0937152
$70$	−0.470683	−0.0562574
$71$	2.74742	0.326059	0.163029	−	0.986621i	$-0.447873\pi$
$71$	2.74742	0.326059	0.163029	+	0.986621i	$0.447873\pi$
$72$	−1.77846	−0.209593
$73$	4.94137	0.578343	0.289172	−	0.957277i	$-0.406620\pi$
$73$	4.94137	0.578343	0.289172	+	0.957277i	$0.406620\pi$
$74$	5.11383	0.594470
$75$	1.00000	0.115470
$76$	−3.26719	−0.374772
$77$	−1.00000	−0.113961
$78$	−0.117266	−0.0132778
$79$	2.48367	0.279435	0.139718	−	0.990191i	$-0.455381\pi$
$79$	2.48367	0.279435	0.139718	+	0.990191i	$0.455381\pi$
$80$	2.71982	0.304086
$81$	1.00000	0.111111
$82$	4.61555	0.509702
$83$	−7.15947	−0.785854	−0.392927	−	0.919570i	$-0.628538\pi$
$83$	−7.15947	−0.785854	−0.392927	+	0.919570i	$0.628538\pi$
$84$	1.77846	0.194046
$85$	5.71982	0.620402
$86$	−2.80949	−0.302955
$87$	3.47068	0.372096
$88$	−1.77846	−0.189584
$89$	−14.5845	−1.54596	−0.772978	−	0.634433i	$-0.781234\pi$
$89$	−14.5845	−1.54596	−0.772978	+	0.634433i	$0.781234\pi$
$90$	0.470683	0.0496144
$91$	0.249141	0.0261170
$92$	−1.38445	−0.144339
$93$	−3.30777	−0.343000
$94$	1.55691	0.160583
$95$	1.83709	0.188481
$96$	4.83709	0.493683
$97$	−14.0242	−1.42394	−0.711969	−	0.702211i	$-0.752196\pi$
$97$	−14.0242	−1.42394	−0.711969	+	0.702211i	$0.752196\pi$
$98$	0.470683	0.0475462
$99$	1.00000	0.100504
$100$	−1.77846	−0.177846
$101$	14.9966	1.49221	0.746107	−	0.665826i	$-0.231921\pi$
$101$	14.9966	1.49221	0.746107	+	0.665826i	$0.231921\pi$
$102$	2.69223	0.266570
$103$	2.52932	0.249221	0.124610	−	0.992206i	$-0.460232\pi$
$103$	2.52932	0.249221	0.124610	+	0.992206i	$0.460232\pi$
$104$	0.443086	0.0434481
$105$	−1.00000	−0.0975900
$106$	−1.30777	−0.127022
$107$	8.30434	0.802810	0.401405	−	0.915901i	$-0.368522\pi$
$107$	8.30434	0.802810	0.401405	+	0.915901i	$0.368522\pi$
$108$	−1.77846	−0.171132
$109$	15.3630	1.47151	0.735753	−	0.677250i	$-0.236829\pi$
$109$	15.3630	1.47151	0.735753	+	0.677250i	$0.236829\pi$
$110$	0.470683	0.0448779
$111$	10.8647	1.03123
$112$	−2.71982	−0.256999
$113$	−7.60256	−0.715188	−0.357594	−	0.933877i	$-0.616403\pi$
$113$	−7.60256	−0.715188	−0.357594	+	0.933877i	$0.616403\pi$
$114$	0.864688	0.0809854
$115$	0.778457	0.0725915
$116$	−6.17246	−0.573099
$117$	−0.249141	−0.0230331
$118$	−1.68879	−0.155466
$119$	−5.71982	−0.524335
$120$	−1.77846	−0.162350
$121$	1.00000	0.0909091
$122$	−3.42504	−0.310089
$123$	9.80605	0.884182
$124$	5.88273	0.528285
$125$	1.00000	0.0894427
$126$	−0.470683	−0.0419318
$127$	−3.52588	−0.312871	−0.156436	−	0.987688i	$-0.550000\pi$
$127$	−3.52588	−0.312871	−0.156436	+	0.987688i	$0.550000\pi$
$128$	−11.1629	−0.986671
$129$	−5.96896	−0.525538
$130$	−0.117266	−0.0102849
$131$	−18.4216	−1.60950	−0.804751	−	0.593612i	$-0.797701\pi$
$131$	−18.4216	−1.60950	−0.804751	+	0.593612i	$0.797701\pi$
$132$	−1.77846	−0.154795
$133$	−1.83709	−0.159296
$134$	2.61555	0.225949
$135$	1.00000	0.0860663
$136$	−10.1725	−0.872281
$137$	−15.8207	−1.35165	−0.675825	−	0.737062i	$-0.736213\pi$
$137$	−15.8207	−1.35165	−0.675825	+	0.737062i	$0.736213\pi$
$138$	0.366407	0.0311906
$139$	4.17246	0.353904	0.176952	−	0.984220i	$-0.443376\pi$
$139$	4.17246	0.353904	0.176952	+	0.984220i	$0.443376\pi$
$140$	1.77846	0.150307
$141$	3.30777	0.278565
$142$	1.29317	0.108520
$143$	−0.249141	−0.0208342
$144$	2.71982	0.226652
$145$	3.47068	0.288225
$146$	2.32582	0.192486
$147$	1.00000	0.0824786
$148$	−19.3224	−1.58829
$149$	−9.61211	−0.787455	−0.393727	−	0.919227i	$-0.628815\pi$
$149$	−9.61211	−0.787455	−0.393727	+	0.919227i	$0.628815\pi$
$150$	0.470683	0.0384311
$151$	6.44309	0.524331	0.262165	−	0.965023i	$-0.415563\pi$
$151$	6.44309	0.524331	0.262165	+	0.965023i	$0.415563\pi$
$152$	−3.26719	−0.265004
$153$	5.71982	0.462420
$154$	−0.470683	−0.0379288
$155$	−3.30777	−0.265687
$156$	0.443086	0.0354753
$157$	17.5259	1.39872	0.699359	−	0.714771i	$-0.253469\pi$
$157$	17.5259	1.39872	0.699359	+	0.714771i	$0.253469\pi$
$158$	1.16902	0.0930025
$159$	−2.77846	−0.220346
$160$	4.83709	0.382406
$161$	−0.778457	−0.0613510
$162$	0.470683	0.0369804
$163$	−12.5389	−0.982120	−0.491060	−	0.871126i	$-0.663390\pi$
$163$	−12.5389	−0.982120	−0.491060	+	0.871126i	$0.663390\pi$
$164$	−17.4396	−1.36181
$165$	1.00000	0.0778499
$166$	−3.36984	−0.261551
$167$	13.0586	1.01051	0.505254	−	0.862971i	$-0.331399\pi$
$167$	13.0586	1.01051	0.505254	+	0.862971i	$0.331399\pi$
$168$	1.77846	0.137211
$169$	−12.9379	−0.995225
$170$	2.69223	0.206484
$171$	1.83709	0.140486
$172$	10.6155	0.809428
$173$	−5.93793	−0.451452	−0.225726	−	0.974191i	$-0.572475\pi$
$173$	−5.93793	−0.451452	−0.225726	+	0.974191i	$0.572475\pi$
$174$	1.63359	0.123842
$175$	−1.00000	−0.0755929
$176$	2.71982	0.205014
$177$	−3.58795	−0.269687
$178$	−6.86469	−0.514530
$179$	−10.2491	−0.766057	−0.383028	−	0.923737i	$-0.625119\pi$
$179$	−10.2491	−0.766057	−0.383028	+	0.923737i	$0.625119\pi$
$180$	−1.77846	−0.132558
$181$	14.9966	1.11469	0.557343	−	0.830282i	$-0.311821\pi$
$181$	14.9966	1.11469	0.557343	+	0.830282i	$0.311821\pi$
$182$	0.117266	0.00869236
$183$	−7.27674	−0.537912
$184$	−1.38445	−0.102063
$185$	10.8647	0.798788
$186$	−1.55691	−0.114158
$187$	5.71982	0.418275
$188$	−5.88273	−0.429042
$189$	−1.00000	−0.0727393
$190$	0.864688	0.0627311
$191$	−13.2311	−0.957368	−0.478684	−	0.877987i	$-0.658886\pi$
$191$	−13.2311	−0.957368	−0.478684	+	0.877987i	$0.658886\pi$
$192$	−3.16291	−0.228263
$193$	8.61555	0.620161	0.310080	−	0.950710i	$-0.399644\pi$
$193$	8.61555	0.620161	0.310080	+	0.950710i	$0.399644\pi$
$194$	−6.60094	−0.473920
$195$	−0.249141	−0.0178413
$196$	−1.77846	−0.127033
$197$	−4.94137	−0.352058	−0.176029	−	0.984385i	$-0.556325\pi$
$197$	−4.94137	−0.352058	−0.176029	+	0.984385i	$0.556325\pi$
$198$	0.470683	0.0334500
$199$	6.87930	0.487660	0.243830	−	0.969818i	$-0.421596\pi$
$199$	6.87930	0.487660	0.243830	+	0.969818i	$0.421596\pi$
$200$	−1.77846	−0.125756
$201$	5.55691	0.391955
$202$	7.05863	0.496644
$203$	−3.47068	−0.243594
$204$	−10.1725	−0.712215
$205$	9.80605	0.684885
$206$	1.19051	0.0829466
$207$	0.778457	0.0541065
$208$	−0.677618	−0.0469844
$209$	1.83709	0.127074
$210$	−0.470683	−0.0324802
$211$	−4.17246	−0.287244	−0.143622	−	0.989633i	$-0.545875\pi$
$211$	−4.17246	−0.287244	−0.143622	+	0.989633i	$0.545875\pi$
$212$	4.94137	0.339374
$213$	2.74742	0.188250
$214$	3.90871	0.267194
$215$	−5.96896	−0.407080
$216$	−1.77846	−0.121009
$217$	3.30777	0.224546
$218$	7.23109	0.489752
$219$	4.94137	0.333907
$220$	−1.77846	−0.119904
$221$	−1.42504	−0.0958586
$222$	5.11383	0.343218
$223$	16.8483	1.12824	0.564121	−	0.825692i	$-0.309215\pi$
$223$	16.8483	1.12824	0.564121	+	0.825692i	$0.309215\pi$
$224$	−4.83709	−0.323192
$225$	1.00000	0.0666667
$226$	−3.57840	−0.238031
$227$	−17.1855	−1.14064	−0.570319	−	0.821423i	$-0.693181\pi$
$227$	−17.1855	−1.14064	−0.570319	+	0.821423i	$0.693181\pi$
$228$	−3.26719	−0.216375
$229$	−24.2491	−1.60243	−0.801214	−	0.598378i	$-0.795812\pi$
$229$	−24.2491	−1.60243	−0.801214	+	0.598378i	$0.795812\pi$
$230$	0.366407	0.0241602
$231$	−1.00000	−0.0657952
$232$	−6.17246	−0.405242
$233$	2.45769	0.161009	0.0805044	−	0.996754i	$-0.474347\pi$
$233$	2.45769	0.161009	0.0805044	+	0.996754i	$0.474347\pi$
$234$	−0.117266	−0.00766594
$235$	3.30777	0.215775
$236$	6.38101	0.415369
$237$	2.48367	0.161332
$238$	−2.69223	−0.174511
$239$	−11.3534	−0.734392	−0.367196	−	0.930144i	$-0.619682\pi$
$239$	−11.3534	−0.734392	−0.367196	+	0.930144i	$0.619682\pi$
$240$	2.71982	0.175564
$241$	−3.79145	−0.244229	−0.122114	−	0.992516i	$-0.538967\pi$
$241$	−3.79145	−0.244229	−0.122114	+	0.992516i	$0.538967\pi$
$242$	0.470683	0.0302567
$243$	1.00000	0.0641500
$244$	12.9414	0.828486
$245$	1.00000	0.0638877
$246$	4.61555	0.294277
$247$	−0.457694	−0.0291224
$248$	5.88273	0.373554
$249$	−7.15947	−0.453713
$250$	0.470683	0.0297686
$251$	−12.9966	−0.820336	−0.410168	−	0.912010i	$-0.634530\pi$
$251$	−12.9966	−0.820336	−0.410168	+	0.912010i	$0.634530\pi$
$252$	1.77846	0.112032
$253$	0.778457	0.0489412
$254$	−1.65957	−0.104131
$255$	5.71982	0.358189
$256$	1.07162	0.0669764
$257$	−7.19051	−0.448532	−0.224266	−	0.974528i	$-0.571998\pi$
$257$	−7.19051	−0.448532	−0.224266	+	0.974528i	$0.571998\pi$
$258$	−2.80949	−0.174911
$259$	−10.8647	−0.675099
$260$	0.443086	0.0274790
$261$	3.47068	0.214830
$262$	−8.67074	−0.535680
$263$	−16.2637	−1.00287	−0.501433	−	0.865197i	$-0.667194\pi$
$263$	−16.2637	−1.00287	−0.501433	+	0.865197i	$0.667194\pi$
$264$	−1.77846	−0.109456
$265$	−2.77846	−0.170679
$266$	−0.864688	−0.0530174
$267$	−14.5845	−0.892558
$268$	−9.88273	−0.603684
$269$	−10.6758	−0.650915	−0.325457	−	0.945557i	$-0.605518\pi$
$269$	−10.6758	−0.650915	−0.325457	+	0.945557i	$0.605518\pi$
$270$	0.470683	0.0286449
$271$	−22.0096	−1.33699	−0.668493	−	0.743719i	$-0.733060\pi$
$271$	−22.0096	−1.33699	−0.668493	+	0.743719i	$0.733060\pi$
$272$	15.5569	0.943276
$273$	0.249141	0.0150787
$274$	−7.44652	−0.449861
$275$	1.00000	0.0603023
$276$	−1.38445	−0.0833343
$277$	22.4362	1.34806	0.674031	−	0.738703i	$-0.264562\pi$
$277$	22.4362	1.34806	0.674031	+	0.738703i	$0.264562\pi$
$278$	1.96391	0.117787
$279$	−3.30777	−0.198031
$280$	1.77846	0.106283
$281$	−0.208553	−0.0124412	−0.00622062	−	0.999981i	$-0.501980\pi$
$281$	−0.208553	−0.0124412	−0.00622062	+	0.999981i	$0.501980\pi$
$282$	1.55691	0.0927129
$283$	11.0732	0.658236	0.329118	−	0.944289i	$-0.393249\pi$
$283$	11.0732	0.658236	0.329118	+	0.944289i	$0.393249\pi$
$284$	−4.88617	−0.289941
$285$	1.83709	0.108820
$286$	−0.117266	−0.00693410
$287$	−9.80605	−0.578833
$288$	4.83709	0.285028
$289$	15.7164	0.924493
$290$	1.63359	0.0959279
$291$	−14.0242	−0.822111
$292$	−8.78801	−0.514279
$293$	6.30939	0.368599	0.184299	−	0.982870i	$-0.440998\pi$
$293$	6.30939	0.368599	0.184299	+	0.982870i	$0.440998\pi$
$294$	0.470683	0.0274508
$295$	−3.58795	−0.208899
$296$	−19.3224	−1.12309
$297$	1.00000	0.0580259
$298$	−4.52426	−0.262083
$299$	−0.193945	−0.0112161
$300$	−1.77846	−0.102679
$301$	5.96896	0.344045
$302$	3.03265	0.174510
$303$	14.9966	0.861530
$304$	4.99656	0.286572
$305$	−7.27674	−0.416665
$306$	2.69223	0.153904
$307$	28.1725	1.60789	0.803944	−	0.594705i	$-0.202731\pi$
$307$	28.1725	1.60789	0.803944	+	0.594705i	$0.202731\pi$
$308$	1.77846	0.101337
$309$	2.52932	0.143888
$310$	−1.55691	−0.0884268
$311$	17.7294	1.00534	0.502670	−	0.864478i	$-0.332351\pi$
$311$	17.7294	1.00534	0.502670	+	0.864478i	$0.332351\pi$
$312$	0.443086	0.0250848
$313$	20.9655	1.18504	0.592521	−	0.805555i	$-0.298133\pi$
$313$	20.9655	1.18504	0.592521	+	0.805555i	$0.298133\pi$
$314$	8.24914	0.465526
$315$	−1.00000	−0.0563436
$316$	−4.41711	−0.248482
$317$	−4.67762	−0.262721	−0.131361	−	0.991335i	$-0.541935\pi$
$317$	−4.67762	−0.262721	−0.131361	+	0.991335i	$0.541935\pi$
$318$	−1.30777	−0.0733363
$319$	3.47068	0.194321
$320$	−3.16291	−0.176812
$321$	8.30434	0.463503
$322$	−0.366407	−0.0204191
$323$	10.5078	0.584671
$324$	−1.77846	−0.0988032
$325$	−0.249141	−0.0138198
$326$	−5.90184	−0.326873
$327$	15.3630	0.849574
$328$	−17.4396	−0.962943
$329$	−3.30777	−0.182363
$330$	0.470683	0.0259103
$331$	−26.0096	−1.42961	−0.714807	−	0.699322i	$-0.753486\pi$
$331$	−26.0096	−1.42961	−0.714807	+	0.699322i	$0.753486\pi$
$332$	12.7328	0.698804
$333$	10.8647	0.595381
$334$	6.14648	0.336321
$335$	5.55691	0.303607
$336$	−2.71982	−0.148379
$337$	12.9655	0.706277	0.353139	−	0.935571i	$-0.385114\pi$
$337$	12.9655	0.706277	0.353139	+	0.935571i	$0.385114\pi$
$338$	−6.08967	−0.331234
$339$	−7.60256	−0.412914
$340$	−10.1725	−0.551679
$341$	−3.30777	−0.179126
$342$	0.864688	0.0467570
$343$	−1.00000	−0.0539949
$344$	10.6155	0.572352
$345$	0.778457	0.0419107
$346$	−2.79488	−0.150254
$347$	1.49484	0.0802474	0.0401237	−	0.999195i	$-0.487225\pi$
$347$	1.49484	0.0802474	0.0401237	+	0.999195i	$0.487225\pi$
$348$	−6.17246	−0.330879
$349$	−6.45264	−0.345402	−0.172701	−	0.984974i	$-0.555249\pi$
$349$	−6.45264	−0.345402	−0.172701	+	0.984974i	$0.555249\pi$
$350$	−0.470683	−0.0251591
$351$	−0.249141	−0.0132981
$352$	4.83709	0.257818
$353$	33.6121	1.78899	0.894496	−	0.447076i	$-0.147535\pi$
$353$	33.6121	1.78899	0.894496	+	0.447076i	$0.147535\pi$
$354$	−1.68879	−0.0897581
$355$	2.74742	0.145818
$356$	25.9379	1.37471
$357$	−5.71982	−0.302725
$358$	−4.82410	−0.254962
$359$	−13.3794	−0.706138	−0.353069	−	0.935597i	$-0.614862\pi$
$359$	−13.3794	−0.706138	−0.353069	+	0.935597i	$0.614862\pi$
$360$	−1.77846	−0.0937329
$361$	−15.6251	−0.822374
$362$	7.05863	0.370994
$363$	1.00000	0.0524864
$364$	−0.443086	−0.0232240
$365$	4.94137	0.258643
$366$	−3.42504	−0.179030
$367$	−13.2553	−0.691919	−0.345959	−	0.938250i	$-0.612446\pi$
$367$	−13.2553	−0.691919	−0.345959	+	0.938250i	$0.612446\pi$
$368$	2.11727	0.110370
$369$	9.80605	0.510483
$370$	5.11383	0.265855
$371$	2.77846	0.144250
$372$	5.88273	0.305006
$373$	−21.6983	−1.12350	−0.561749	−	0.827308i	$-0.689871\pi$
$373$	−21.6983	−1.12350	−0.561749	+	0.827308i	$0.689871\pi$
$374$	2.69223	0.139212
$375$	1.00000	0.0516398
$376$	−5.88273	−0.303379
$377$	−0.864688	−0.0445337
$378$	−0.470683	−0.0242093
$379$	−10.8337	−0.556487	−0.278244	−	0.960511i	$-0.589752\pi$
$379$	−10.8337	−0.556487	−0.278244	+	0.960511i	$0.589752\pi$
$380$	−3.26719	−0.167603
$381$	−3.52588	−0.180636
$382$	−6.22766	−0.318635
$383$	14.5941	0.745722	0.372861	−	0.927887i	$-0.378377\pi$
$383$	14.5941	0.745722	0.372861	+	0.927887i	$0.378377\pi$
$384$	−11.1629	−0.569655
$385$	−1.00000	−0.0509647
$386$	4.05520	0.206404
$387$	−5.96896	−0.303420
$388$	24.9414	1.26621
$389$	−35.6267	−1.80635	−0.903173	−	0.429276i	$-0.858769\pi$
$389$	−35.6267	−1.80635	−0.903173	+	0.429276i	$0.858769\pi$
$390$	−0.117266	−0.00593801
$391$	4.45264	0.225180
$392$	−1.77846	−0.0898256
$393$	−18.4216	−0.929247
$394$	−2.32582	−0.117173
$395$	2.48367	0.124967
$396$	−1.77846	−0.0893708
$397$	19.7914	0.993304	0.496652	−	0.867950i	$-0.334562\pi$
$397$	19.7914	0.993304	0.496652	+	0.867950i	$0.334562\pi$
$398$	3.23797	0.162305
$399$	−1.83709	−0.0919695
$400$	2.71982	0.135991
$401$	−4.80949	−0.240175	−0.120087	−	0.992763i	$-0.538317\pi$
$401$	−4.80949	−0.240175	−0.120087	+	0.992763i	$0.538317\pi$
$402$	2.61555	0.130452
$403$	0.824101	0.0410514
$404$	−26.6707	−1.32692
$405$	1.00000	0.0496904
$406$	−1.63359	−0.0810739
$407$	10.8647	0.538543
$408$	−10.1725	−0.503612
$409$	−23.9018	−1.18187	−0.590935	−	0.806719i	$-0.701241\pi$
$409$	−23.9018	−1.18187	−0.590935	+	0.806719i	$0.701241\pi$
$410$	4.61555	0.227946
$411$	−15.8207	−0.780376
$412$	−4.49828	−0.221614
$413$	3.58795	0.176551
$414$	0.366407	0.0180079
$415$	−7.15947	−0.351445
$416$	−1.20512	−0.0590856
$417$	4.17246	0.204326
$418$	0.864688	0.0422933
$419$	7.52588	0.367663	0.183832	−	0.982958i	$-0.441150\pi$
$419$	7.52588	0.367663	0.183832	+	0.982958i	$0.441150\pi$
$420$	1.77846	0.0867798
$421$	−25.8302	−1.25889	−0.629444	−	0.777046i	$-0.716717\pi$
$421$	−25.8302	−1.25889	−0.629444	+	0.777046i	$0.716717\pi$
$422$	−1.96391	−0.0956016
$423$	3.30777	0.160829
$424$	4.94137	0.239974
$425$	5.71982	0.277452
$426$	1.29317	0.0626541
$427$	7.27674	0.352146
$428$	−14.7689	−0.713882
$429$	−0.249141	−0.0120286
$430$	−2.80949	−0.135486
$431$	14.1465	0.681412	0.340706	−	0.940170i	$-0.389334\pi$
$431$	14.1465	0.681412	0.340706	+	0.940170i	$0.389334\pi$
$432$	2.71982	0.130858
$433$	30.8432	1.48223	0.741115	−	0.671378i	$-0.234297\pi$
$433$	30.8432	1.48223	0.741115	+	0.671378i	$0.234297\pi$
$434$	1.55691	0.0747343
$435$	3.47068	0.166407
$436$	−27.3224	−1.30850
$437$	1.43010	0.0684108
$438$	2.32582	0.111132
$439$	14.1008	0.672996	0.336498	−	0.941684i	$-0.390757\pi$
$439$	14.1008	0.672996	0.336498	+	0.941684i	$0.390757\pi$
$440$	−1.77846	−0.0847846
$441$	1.00000	0.0476190
$442$	−0.670743	−0.0319040
$443$	−27.0518	−1.28527	−0.642634	−	0.766173i	$-0.722158\pi$
$443$	−27.0518	−1.28527	−0.642634	+	0.766173i	$0.722158\pi$
$444$	−19.3224	−0.917000
$445$	−14.5845	−0.691372
$446$	7.93020	0.375506
$447$	−9.61211	−0.454637
$448$	3.16291	0.149433
$449$	−18.9560	−0.894588	−0.447294	−	0.894387i	$-0.647612\pi$
$449$	−18.9560	−0.894588	−0.447294	+	0.894387i	$0.647612\pi$
$450$	0.470683	0.0221882
$451$	9.80605	0.461749
$452$	13.5208	0.635966
$453$	6.44309	0.302723
$454$	−8.08891	−0.379631
$455$	0.249141	0.0116799
$456$	−3.26719	−0.153000
$457$	−19.2553	−0.900723	−0.450361	−	0.892846i	$-0.648705\pi$
$457$	−19.2553	−0.900723	−0.450361	+	0.892846i	$0.648705\pi$
$458$	−11.4137	−0.533326
$459$	5.71982	0.266979
$460$	−1.38445	−0.0645504
$461$	19.4948	0.907965	0.453983	−	0.891011i	$-0.350003\pi$
$461$	19.4948	0.907965	0.453983	+	0.891011i	$0.350003\pi$
$462$	−0.470683	−0.0218982
$463$	−20.2829	−0.942624	−0.471312	−	0.881967i	$-0.656219\pi$
$463$	−20.2829	−0.942624	−0.471312	+	0.881967i	$0.656219\pi$
$464$	9.43965	0.438225
$465$	−3.30777	−0.153394
$466$	1.15680	0.0535875
$467$	−40.4983	−1.87404	−0.937018	−	0.349280i	$-0.886426\pi$
$467$	−40.4983	−1.87404	−0.937018	+	0.349280i	$0.886426\pi$
$468$	0.443086	0.0204816
$469$	−5.55691	−0.256594
$470$	1.55691	0.0718151
$471$	17.5259	0.807550
$472$	6.38101	0.293710
$473$	−5.96896	−0.274453
$474$	1.16902	0.0536950
$475$	1.83709	0.0842915
$476$	10.1725	0.466254
$477$	−2.77846	−0.127217
$478$	−5.34387	−0.244423
$479$	14.1579	0.646889	0.323444	−	0.946247i	$-0.395159\pi$
$479$	14.1579	0.646889	0.323444	+	0.946247i	$0.395159\pi$
$480$	4.83709	0.220782
$481$	−2.70683	−0.123421
$482$	−1.78457	−0.0812850
$483$	−0.778457	−0.0354210
$484$	−1.77846	−0.0808390
$485$	−14.0242	−0.636804
$486$	0.470683	0.0213506
$487$	29.5861	1.34068	0.670338	−	0.742056i	$-0.266149\pi$
$487$	29.5861	1.34068	0.670338	+	0.742056i	$0.266149\pi$
$488$	12.9414	0.585828
$489$	−12.5389	−0.567027
$490$	0.470683	0.0212633
$491$	35.0759	1.58295	0.791477	−	0.611199i	$-0.209312\pi$
$491$	35.0759	1.58295	0.791477	+	0.611199i	$0.209312\pi$
$492$	−17.4396	−0.786240
$493$	19.8517	0.894076
$494$	−0.215429	−0.00969260
$495$	1.00000	0.0449467
$496$	−8.99656	−0.403958
$497$	−2.74742	−0.123239
$498$	−3.36984	−0.151006
$499$	−7.89229	−0.353307	−0.176654	−	0.984273i	$-0.556527\pi$
$499$	−7.89229	−0.353307	−0.176654	+	0.984273i	$0.556527\pi$
$500$	−1.77846	−0.0795350
$501$	13.0586	0.583417
$502$	−6.11727	−0.273027
$503$	−26.1199	−1.16463	−0.582315	−	0.812963i	$-0.697853\pi$
$503$	−26.1199	−1.16463	−0.582315	+	0.812963i	$0.697853\pi$
$504$	1.77846	0.0792188
$505$	14.9966	0.667338
$506$	0.366407	0.0162888
$507$	−12.9379	−0.574594
$508$	6.27062	0.278214
$509$	24.9655	1.10658	0.553289	−	0.832990i	$-0.313373\pi$
$509$	24.9655	1.10658	0.553289	+	0.832990i	$0.313373\pi$
$510$	2.69223	0.119214
$511$	−4.94137	−0.218593
$512$	22.8302	1.00896
$513$	1.83709	0.0811095
$514$	−3.38445	−0.149282
$515$	2.52932	0.111455
$516$	10.6155	0.467323
$517$	3.30777	0.145476
$518$	−5.11383	−0.224689
$519$	−5.93793	−0.260646
$520$	0.443086	0.0194306
$521$	1.41549	0.0620137	0.0310068	−	0.999519i	$-0.490129\pi$
$521$	1.41549	0.0620137	0.0310068	+	0.999519i	$0.490129\pi$
$522$	1.63359	0.0715004
$523$	33.8398	1.47971	0.739855	−	0.672767i	$-0.234894\pi$
$523$	33.8398	1.47971	0.739855	+	0.672767i	$0.234894\pi$
$524$	32.7620	1.43122
$525$	−1.00000	−0.0436436
$526$	−7.65508	−0.333777
$527$	−18.9199	−0.824163
$528$	2.71982	0.118365
$529$	−22.3940	−0.973652
$530$	−1.30777	−0.0568061
$531$	−3.58795	−0.155704
$532$	3.26719	0.141650
$533$	−2.44309	−0.105822
$534$	−6.86469	−0.297064
$535$	8.30434	0.359028
$536$	−9.88273	−0.426869
$537$	−10.2491	−0.442283
$538$	−5.02492	−0.216640
$539$	1.00000	0.0430730
$540$	−1.77846	−0.0765326
$541$	31.3630	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$541$	31.3630	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$542$	−10.3595	−0.444980
$543$	14.9966	0.643564
$544$	27.6673	1.18623
$545$	15.3630	0.658077
$546$	0.117266	0.00501854
$547$	43.9000	1.87703	0.938515	−	0.345240i	$-0.112202\pi$
$547$	43.9000	1.87703	0.938515	+	0.345240i	$0.112202\pi$
$548$	28.1364	1.20193
$549$	−7.27674	−0.310564
$550$	0.470683	0.0200700
$551$	6.37596	0.271625
$552$	−1.38445	−0.0589262
$553$	−2.48367	−0.105617
$554$	10.5604	0.448666
$555$	10.8647	0.461180
$556$	−7.42054	−0.314701
$557$	37.1430	1.57380	0.786901	−	0.617080i	$-0.211684\pi$
$557$	37.1430	1.57380	0.786901	+	0.617080i	$0.211684\pi$
$558$	−1.55691	−0.0659094
$559$	1.48711	0.0628981
$560$	−2.71982	−0.114934
$561$	5.71982	0.241491
$562$	−0.0981625	−0.00414074
$563$	18.4431	0.777283	0.388642	−	0.921389i	$-0.372944\pi$
$563$	18.4431	0.777283	0.388642	+	0.921389i	$0.372944\pi$
$564$	−5.88273	−0.247708
$565$	−7.60256	−0.319842
$566$	5.21199	0.219076
$567$	−1.00000	−0.0419961
$568$	−4.88617	−0.205019
$569$	39.9881	1.67639	0.838194	−	0.545373i	$-0.183612\pi$
$569$	39.9881	1.67639	0.838194	+	0.545373i	$0.183612\pi$
$570$	0.864688	0.0362178
$571$	−21.0180	−0.879578	−0.439789	−	0.898101i	$-0.644947\pi$
$571$	−21.0180	−0.879578	−0.439789	+	0.898101i	$0.644947\pi$
$572$	0.443086	0.0185263
$573$	−13.2311	−0.552737
$574$	−4.61555	−0.192649
$575$	0.778457	0.0324639
$576$	−3.16291	−0.131788
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	7.39744	0.307693
$579$	8.61555	0.358050
$580$	−6.17246	−0.256298
$581$	7.15947	0.297025
$582$	−6.60094	−0.273618
$583$	−2.77846	−0.115072
$584$	−8.78801	−0.363650
$585$	−0.249141	−0.0103007
$586$	2.96973	0.122678
$587$	−5.01805	−0.207117	−0.103558	−	0.994623i	$-0.533023\pi$
$587$	−5.01805	−0.207117	−0.103558	+	0.994623i	$0.533023\pi$
$588$	−1.77846	−0.0733423
$589$	−6.07668	−0.250385
$590$	−1.68879	−0.0695263
$591$	−4.94137	−0.203261
$592$	29.5500	1.21450
$593$	−20.3189	−0.834399	−0.417200	−	0.908815i	$-0.636988\pi$
$593$	−20.3189	−0.834399	−0.417200	+	0.908815i	$0.636988\pi$
$594$	0.470683	0.0193124
$595$	−5.71982	−0.234490
$596$	17.0947	0.700227
$597$	6.87930	0.281551
$598$	−0.0912868	−0.00373299
$599$	−15.0518	−0.614998	−0.307499	−	0.951548i	$-0.599492\pi$
$599$	−15.0518	−0.614998	−0.307499	+	0.951548i	$0.599492\pi$
$600$	−1.77846	−0.0726052
$601$	−33.9836	−1.38622	−0.693109	−	0.720832i	$-0.743760\pi$
$601$	−33.9836	−1.38622	−0.693109	+	0.720832i	$0.743760\pi$
$602$	2.80949	0.114506
$603$	5.55691	0.226295
$604$	−11.4588	−0.466250
$605$	1.00000	0.0406558
$606$	7.05863	0.286737
$607$	16.6415	0.675459	0.337729	−	0.941243i	$-0.390341\pi$
$607$	16.6415	0.675459	0.337729	+	0.941243i	$0.390341\pi$
$608$	8.88617	0.360382
$609$	−3.47068	−0.140639
$610$	−3.42504	−0.138676
$611$	−0.824101	−0.0333395
$612$	−10.1725	−0.411197
$613$	13.0034	0.525204	0.262602	−	0.964904i	$-0.415419\pi$
$613$	13.0034	0.525204	0.262602	+	0.964904i	$0.415419\pi$
$614$	13.2603	0.535143
$615$	9.80605	0.395418
$616$	1.77846	0.0716561
$617$	8.97058	0.361142	0.180571	−	0.983562i	$-0.442205\pi$
$617$	8.97058	0.361142	0.180571	+	0.983562i	$0.442205\pi$
$618$	1.19051	0.0478892
$619$	23.6742	0.951546	0.475773	−	0.879568i	$-0.342168\pi$
$619$	23.6742	0.951546	0.475773	+	0.879568i	$0.342168\pi$
$620$	5.88273	0.236256
$621$	0.778457	0.0312384
$622$	8.34492	0.334601
$623$	14.5845	0.584316
$624$	−0.677618	−0.0271264
$625$	1.00000	0.0400000
$626$	9.86813	0.394410
$627$	1.83709	0.0733663
$628$	−31.1690	−1.24378
$629$	62.1441	2.47785
$630$	−0.470683	−0.0187525
$631$	−1.60256	−0.0637968	−0.0318984	−	0.999491i	$-0.510155\pi$
$631$	−1.60256	−0.0637968	−0.0318984	+	0.999491i	$0.510155\pi$
$632$	−4.41711	−0.175703
$633$	−4.17246	−0.165840
$634$	−2.20168	−0.0874398
$635$	−3.52588	−0.139920
$636$	4.94137	0.195938
$637$	−0.249141	−0.00987131
$638$	1.63359	0.0646746
$639$	2.74742	0.108686
$640$	−11.1629	−0.441253
$641$	−12.8793	−0.508702	−0.254351	−	0.967112i	$-0.581862\pi$
$641$	−12.8793	−0.508702	−0.254351	+	0.967112i	$0.581862\pi$
$642$	3.90871	0.154265
$643$	−21.7535	−0.857876	−0.428938	−	0.903334i	$-0.641112\pi$
$643$	−21.7535	−0.857876	−0.428938	+	0.903334i	$0.641112\pi$
$644$	1.38445	0.0545551
$645$	−5.96896	−0.235028
$646$	4.94586	0.194592
$647$	46.0223	1.80932	0.904662	−	0.426129i	$-0.140123\pi$
$647$	46.0223	1.80932	0.904662	+	0.426129i	$0.140123\pi$
$648$	−1.77846	−0.0698644
$649$	−3.58795	−0.140839
$650$	−0.117266	−0.00459956
$651$	3.30777	0.129642
$652$	22.2998	0.873329
$653$	−0.224981	−0.00880418	−0.00440209	−	0.999990i	$-0.501401\pi$
$653$	−0.224981	−0.00880418	−0.00440209	+	0.999990i	$0.501401\pi$
$654$	7.23109	0.282758
$655$	−18.4216	−0.719792
$656$	26.6707	1.04132
$657$	4.94137	0.192781
$658$	−1.55691	−0.0606948
$659$	−39.4017	−1.53487	−0.767437	−	0.641125i	$-0.778468\pi$
$659$	−39.4017	−1.53487	−0.767437	+	0.641125i	$0.778468\pi$
$660$	−1.77846	−0.0692263
$661$	33.0923	1.28714	0.643572	−	0.765386i	$-0.277452\pi$
$661$	33.0923	1.28714	0.643572	+	0.765386i	$0.277452\pi$
$662$	−12.2423	−0.475809
$663$	−1.42504	−0.0553440
$664$	12.7328	0.494129
$665$	−1.83709	−0.0712393
$666$	5.11383	0.198157
$667$	2.70178	0.104613
$668$	−23.2242	−0.898572
$669$	16.8483	0.651391
$670$	2.61555	0.101047
$671$	−7.27674	−0.280915
$672$	−4.83709	−0.186595
$673$	15.7344	0.606518	0.303259	−	0.952908i	$-0.401925\pi$
$673$	15.7344	0.606518	0.303259	+	0.952908i	$0.401925\pi$
$674$	6.10266	0.235066
$675$	1.00000	0.0384900
$676$	23.0096	0.884983
$677$	3.83709	0.147471	0.0737357	−	0.997278i	$-0.476508\pi$
$677$	3.83709	0.147471	0.0737357	+	0.997278i	$0.476508\pi$
$678$	−3.57840	−0.137428
$679$	14.0242	0.538198
$680$	−10.1725	−0.390096
$681$	−17.1855	−0.658548
$682$	−1.55691	−0.0596173
$683$	−11.0034	−0.421035	−0.210517	−	0.977590i	$-0.567515\pi$
$683$	−11.0034	−0.421035	−0.210517	+	0.977590i	$0.567515\pi$
$684$	−3.26719	−0.124924
$685$	−15.8207	−0.604476
$686$	−0.470683	−0.0179708
$687$	−24.2491	−0.925162
$688$	−16.2345	−0.618936
$689$	0.692226	0.0263717
$690$	0.366407	0.0139489
$691$	−16.6707	−0.634185	−0.317092	−	0.948395i	$-0.602707\pi$
$691$	−16.6707	−0.634185	−0.317092	+	0.948395i	$0.602707\pi$
$692$	10.5604	0.401444
$693$	−1.00000	−0.0379869
$694$	0.703598	0.0267082
$695$	4.17246	0.158270
$696$	−6.17246	−0.233967
$697$	56.0889	2.12452
$698$	−3.03715	−0.114958
$699$	2.45769	0.0929585
$700$	1.77846	0.0672194
$701$	17.5259	0.661943	0.330972	−	0.943641i	$-0.392624\pi$
$701$	17.5259	0.661943	0.330972	+	0.943641i	$0.392624\pi$
$702$	−0.117266	−0.00442593
$703$	19.9594	0.752784
$704$	−3.16291	−0.119207
$705$	3.30777	0.124578
$706$	15.8207	0.595418
$707$	−14.9966	−0.564004
$708$	6.38101	0.239813
$709$	42.7976	1.60730	0.803648	−	0.595105i	$-0.202889\pi$
$709$	42.7976	1.60730	0.803648	+	0.595105i	$0.202889\pi$
$710$	1.29317	0.0485316
$711$	2.48367	0.0931450
$712$	25.9379	0.972065
$713$	−2.57496	−0.0964330
$714$	−2.69223	−0.100754
$715$	−0.249141	−0.00931733
$716$	18.2277	0.681200
$717$	−11.3534	−0.424001
$718$	−6.29746	−0.235019
$719$	−17.2362	−0.642800	−0.321400	−	0.946944i	$-0.604153\pi$
$719$	−17.2362	−0.642800	−0.321400	+	0.946944i	$0.604153\pi$
$720$	2.71982	0.101362
$721$	−2.52932	−0.0941967
$722$	−7.35448	−0.273705
$723$	−3.79145	−0.141005
$724$	−26.6707	−0.991210
$725$	3.47068	0.128898
$726$	0.470683	0.0174687
$727$	−30.2035	−1.12019	−0.560093	−	0.828430i	$-0.689235\pi$
$727$	−30.2035	−1.12019	−0.560093	+	0.828430i	$0.689235\pi$
$728$	−0.443086	−0.0164219
$729$	1.00000	0.0370370
$730$	2.32582	0.0860824
$731$	−34.1414	−1.26277
$732$	12.9414	0.478327
$733$	27.0258	0.998220	0.499110	−	0.866539i	$-0.333660\pi$
$733$	27.0258	0.998220	0.499110	+	0.866539i	$0.333660\pi$
$734$	−6.23903	−0.230287
$735$	1.00000	0.0368856
$736$	3.76547	0.138797
$737$	5.55691	0.204692
$738$	4.61555	0.169901
$739$	22.8172	0.839345	0.419673	−	0.907676i	$-0.362145\pi$
$739$	22.8172	0.839345	0.419673	+	0.907676i	$0.362145\pi$
$740$	−19.3224	−0.710305
$741$	−0.457694	−0.0168138
$742$	1.30777	0.0480099
$743$	−38.4001	−1.40876	−0.704382	−	0.709821i	$-0.748776\pi$
$743$	−38.4001	−1.40876	−0.704382	+	0.709821i	$0.748776\pi$
$744$	5.88273	0.215671
$745$	−9.61211	−0.352160
$746$	−10.2130	−0.373926
$747$	−7.15947	−0.261951
$748$	−10.1725	−0.371942
$749$	−8.30434	−0.303434
$750$	0.470683	0.0171869
$751$	−23.6578	−0.863284	−0.431642	−	0.902045i	$-0.642066\pi$
$751$	−23.6578	−0.863284	−0.431642	+	0.902045i	$0.642066\pi$
$752$	8.99656	0.328071
$753$	−12.9966	−0.473621
$754$	−0.406994	−0.0148219
$755$	6.44309	0.234488
$756$	1.77846	0.0646819
$757$	−29.8466	−1.08479	−0.542397	−	0.840122i	$-0.682483\pi$
$757$	−29.8466	−1.08479	−0.542397	+	0.840122i	$0.682483\pi$
$758$	−5.09922	−0.185212
$759$	0.778457	0.0282562
$760$	−3.26719	−0.118513
$761$	−0.117266	−0.00425090	−0.00212545	−	0.999998i	$-0.500677\pi$
$761$	−0.117266	−0.00425090	−0.00212545	+	0.999998i	$0.500677\pi$
$762$	−1.65957	−0.0601200
$763$	−15.3630	−0.556177
$764$	23.5309	0.851319
$765$	5.71982	0.206801
$766$	6.86918	0.248194
$767$	0.893904	0.0322770
$768$	1.07162	0.0386689
$769$	12.4267	0.448117	0.224058	−	0.974576i	$-0.428069\pi$
$769$	12.4267	0.448117	0.224058	+	0.974576i	$0.428069\pi$
$770$	−0.470683	−0.0169623
$771$	−7.19051	−0.258960
$772$	−15.3224	−0.551465
$773$	47.7294	1.71671	0.858353	−	0.513059i	$-0.171488\pi$
$773$	47.7294	1.71671	0.858353	+	0.513059i	$0.171488\pi$
$774$	−2.80949	−0.100985
$775$	−3.30777	−0.118819
$776$	24.9414	0.895343
$777$	−10.8647	−0.389769
$778$	−16.7689	−0.601194
$779$	18.0146	0.645440
$780$	0.443086	0.0158650
$781$	2.74742	0.0983105
$782$	2.09578	0.0749450
$783$	3.47068	0.124032
$784$	2.71982	0.0971366
$785$	17.5259	0.625525
$786$	−8.67074	−0.309275
$787$	−24.8862	−0.887096	−0.443548	−	0.896251i	$-0.646280\pi$
$787$	−24.8862	−0.887096	−0.443548	+	0.896251i	$0.646280\pi$
$788$	8.78801	0.313060
$789$	−16.2637	−0.579005
$790$	1.16902	0.0415920
$791$	7.60256	0.270316
$792$	−1.77846	−0.0631947
$793$	1.81293	0.0643790
$794$	9.31551	0.330595
$795$	−2.77846	−0.0985417
$796$	−12.2345	−0.433642
$797$	−10.2637	−0.363561	−0.181780	−	0.983339i	$-0.558186\pi$
$797$	−10.2637	−0.363561	−0.181780	+	0.983339i	$0.558186\pi$
$798$	−0.864688	−0.0306096
$799$	18.9199	0.669337
$800$	4.83709	0.171017
$801$	−14.5845	−0.515318
$802$	−2.26375	−0.0799357
$803$	4.94137	0.174377
$804$	−9.88273	−0.348537
$805$	−0.778457	−0.0274370
$806$	0.387890	0.0136629
$807$	−10.6758	−0.375806
$808$	−26.6707	−0.938273
$809$	52.6018	1.84938	0.924690	−	0.380720i	$-0.124324\pi$
$809$	52.6018	1.84938	0.924690	+	0.380720i	$0.124324\pi$
$810$	0.470683	0.0165381
$811$	−11.5309	−0.404906	−0.202453	−	0.979292i	$-0.564891\pi$
$811$	−11.5309	−0.404906	−0.202453	+	0.979292i	$0.564891\pi$
$812$	6.17246	0.216611
$813$	−22.0096	−0.771909
$814$	5.11383	0.179240
$815$	−12.5389	−0.439217
$816$	15.5569	0.544601
$817$	−10.9655	−0.383635
$818$	−11.2502	−0.393354
$819$	0.249141	0.00870567
$820$	−17.4396	−0.609019
$821$	−53.7467	−1.87577	−0.937886	−	0.346944i	$-0.887219\pi$
$821$	−53.7467	−1.87577	−0.937886	+	0.346944i	$0.887219\pi$
$822$	−7.44652	−0.259727
$823$	−16.2751	−0.567315	−0.283658	−	0.958926i	$-0.591548\pi$
$823$	−16.2751	−0.567315	−0.283658	+	0.958926i	$0.591548\pi$
$824$	−4.49828	−0.156705
$825$	1.00000	0.0348155
$826$	1.68879	0.0587605
$827$	−3.26719	−0.113611	−0.0568056	−	0.998385i	$-0.518092\pi$
$827$	−3.26719	−0.113611	−0.0568056	+	0.998385i	$0.518092\pi$
$828$	−1.38445	−0.0481131
$829$	−33.6121	−1.16740	−0.583698	−	0.811971i	$-0.698395\pi$
$829$	−33.6121	−1.16740	−0.583698	+	0.811971i	$0.698395\pi$
$830$	−3.36984	−0.116969
$831$	22.4362	0.778304
$832$	0.788009	0.0273193
$833$	5.71982	0.198180
$834$	1.96391	0.0680046
$835$	13.0586	0.451913
$836$	−3.26719	−0.112998
$837$	−3.30777	−0.114333
$838$	3.54231	0.122367
$839$	0.584512	0.0201796	0.0100898	−	0.999949i	$-0.496788\pi$
$839$	0.584512	0.0201796	0.0100898	+	0.999949i	$0.496788\pi$
$840$	1.77846	0.0613626
$841$	−16.9544	−0.584633
$842$	−12.1579	−0.418987
$843$	−0.208553	−0.00718295
$844$	7.42054	0.255426
$845$	−12.9379	−0.445078
$846$	1.55691	0.0535278
$847$	−1.00000	−0.0343604
$848$	−7.55691	−0.259506
$849$	11.0732	0.380033
$850$	2.69223	0.0923426
$851$	8.45769	0.289926
$852$	−4.88617	−0.167397
$853$	6.53887	0.223887	0.111943	−	0.993715i	$-0.464292\pi$
$853$	6.53887	0.223887	0.111943	+	0.993715i	$0.464292\pi$
$854$	3.42504	0.117202
$855$	1.83709	0.0628272
$856$	−14.7689	−0.504791
$857$	21.5829	0.737258	0.368629	−	0.929577i	$-0.379827\pi$
$857$	21.5829	0.737258	0.368629	+	0.929577i	$0.379827\pi$
$858$	−0.117266	−0.00400341
$859$	−48.2614	−1.64666	−0.823328	−	0.567565i	$-0.807885\pi$
$859$	−48.2614	−1.64666	−0.823328	+	0.567565i	$0.807885\pi$
$860$	10.6155	0.361987
$861$	−9.80605	−0.334189
$862$	6.65851	0.226790
$863$	−22.0096	−0.749214	−0.374607	−	0.927184i	$-0.622222\pi$
$863$	−22.0096	−0.749214	−0.374607	+	0.927184i	$0.622222\pi$
$864$	4.83709	0.164561
$865$	−5.93793	−0.201896
$866$	14.5174	0.493321
$867$	15.7164	0.533756
$868$	−5.88273	−0.199673
$869$	2.48367	0.0842528
$870$	1.63359	0.0553840
$871$	−1.38445	−0.0469104
$872$	−27.3224	−0.925253
$873$	−14.0242	−0.474646
$874$	0.673122	0.0227687
$875$	−1.00000	−0.0338062
$876$	−8.78801	−0.296919
$877$	−29.3104	−0.989744	−0.494872	−	0.868966i	$-0.664785\pi$
$877$	−29.3104	−0.989744	−0.494872	+	0.868966i	$0.664785\pi$
$878$	6.63703	0.223989
$879$	6.30939	0.212811
$880$	2.71982	0.0916852
$881$	0.375959	0.0126664	0.00633319	−	0.999980i	$-0.497984\pi$
$881$	0.375959	0.0126664	0.00633319	+	0.999980i	$0.497984\pi$
$882$	0.470683	0.0158487
$883$	49.7156	1.67306	0.836532	−	0.547918i	$-0.184579\pi$
$883$	49.7156	1.67306	0.836532	+	0.547918i	$0.184579\pi$
$884$	2.53437	0.0852402
$885$	−3.58795	−0.120608
$886$	−12.7328	−0.427767
$887$	−39.3027	−1.31966	−0.659828	−	0.751417i	$-0.729371\pi$
$887$	−39.3027	−1.31966	−0.659828	+	0.751417i	$0.729371\pi$
$888$	−19.3224	−0.648417
$889$	3.52588	0.118254
$890$	−6.86469	−0.230105
$891$	1.00000	0.0335013
$892$	−29.9639	−1.00327
$893$	6.07668	0.203348
$894$	−4.52426	−0.151314
$895$	−10.2491	−0.342591
$896$	11.1629	0.372927
$897$	−0.193945	−0.00647564
$898$	−8.92226	−0.297740
$899$	−11.4802	−0.382887
$900$	−1.77846	−0.0592819
$901$	−15.8923	−0.529449
$902$	4.61555	0.153681
$903$	5.96896	0.198635
$904$	13.5208	0.449696
$905$	14.9966	0.498503
$906$	3.03265	0.100753
$907$	−33.4250	−1.10986	−0.554930	−	0.831897i	$-0.687255\pi$
$907$	−33.4250	−1.10986	−0.554930	+	0.831897i	$0.687255\pi$
$908$	30.5636	1.01429
$909$	14.9966	0.497405
$910$	0.117266	0.00388734
$911$	−5.49484	−0.182052	−0.0910261	−	0.995849i	$-0.529015\pi$
$911$	−5.49484	−0.182052	−0.0910261	+	0.995849i	$0.529015\pi$
$912$	4.99656	0.165453
$913$	−7.15947	−0.236944
$914$	−9.06313	−0.299782
$915$	−7.27674	−0.240562
$916$	43.1261	1.42493
$917$	18.4216	0.608335
$918$	2.69223	0.0888567
$919$	32.9053	1.08544	0.542722	−	0.839912i	$-0.317394\pi$
$919$	32.9053	1.08544	0.542722	+	0.839912i	$0.317394\pi$
$920$	−1.38445	−0.0456441
$921$	28.1725	0.928314
$922$	9.17590	0.302192
$923$	−0.684494	−0.0225304
$924$	1.77846	0.0585069
$925$	10.8647	0.357229
$926$	−9.54680	−0.313727
$927$	2.52932	0.0830737
$928$	16.7880	0.551093
$929$	−18.6707	−0.612567	−0.306284	−	0.951940i	$-0.599086\pi$
$929$	−18.6707	−0.612567	−0.306284	+	0.951940i	$0.599086\pi$
$930$	−1.55691	−0.0510532
$931$	1.83709	0.0602082
$932$	−4.37090	−0.143174
$933$	17.7294	0.580434
$934$	−19.0619	−0.623723
$935$	5.71982	0.187058
$936$	0.443086	0.0144827
$937$	−44.9491	−1.46842	−0.734212	−	0.678921i	$-0.762448\pi$
$937$	−44.9491	−1.46842	−0.734212	+	0.678921i	$0.762448\pi$
$938$	−2.61555	−0.0854006
$939$	20.9655	0.684184
$940$	−5.88273	−0.191874
$941$	32.8286	1.07018	0.535091	−	0.844795i	$-0.320277\pi$
$941$	32.8286	1.07018	0.535091	+	0.844795i	$0.320277\pi$
$942$	8.24914	0.268772
$943$	7.63359	0.248584
$944$	−9.75859	−0.317615
$945$	−1.00000	−0.0325300
$946$	−2.80949	−0.0913445
$947$	−41.8371	−1.35952	−0.679761	−	0.733433i	$-0.737917\pi$
$947$	−41.8371	−1.35952	−0.679761	+	0.733433i	$0.737917\pi$
$948$	−4.41711	−0.143461
$949$	−1.23109	−0.0399630
$950$	0.864688	0.0280542
$951$	−4.67762	−0.151682
$952$	10.1725	0.329691
$953$	12.7689	0.413625	0.206813	−	0.978381i	$-0.433691\pi$
$953$	12.7689	0.413625	0.206813	+	0.978381i	$0.433691\pi$
$954$	−1.30777	−0.0423407
$955$	−13.2311	−0.428148
$956$	20.1916	0.653042
$957$	3.47068	0.112191
$958$	6.66387	0.215300
$959$	15.8207	0.510876
$960$	−3.16291	−0.102082
$961$	−20.0586	−0.647053
$962$	−1.27406	−0.0410774
$963$	8.30434	0.267603
$964$	6.74293	0.217175
$965$	8.61555	0.277344
$966$	−0.366407	−0.0117889
$967$	−24.6657	−0.793195	−0.396598	−	0.917993i	$-0.629809\pi$
$967$	−24.6657	−0.793195	−0.396598	+	0.917993i	$0.629809\pi$
$968$	−1.77846	−0.0571618
$969$	10.5078	0.337560
$970$	−6.60094	−0.211943
$971$	−23.1380	−0.742533	−0.371267	−	0.928526i	$-0.621076\pi$
$971$	−23.1380	−0.742533	−0.371267	+	0.928526i	$0.621076\pi$
$972$	−1.77846	−0.0570440
$973$	−4.17246	−0.133763
$974$	13.9257	0.446208
$975$	−0.249141	−0.00797888
$976$	−19.7914	−0.633509
$977$	−33.2406	−1.06346	−0.531731	−	0.846913i	$-0.678458\pi$
$977$	−33.2406	−1.06346	−0.531731	+	0.846913i	$0.678458\pi$
$978$	−5.90184	−0.188720
$979$	−14.5845	−0.466123
$980$	−1.77846	−0.0568107
$981$	15.3630	0.490502
$982$	16.5097	0.526844
$983$	2.94137	0.0938150	0.0469075	−	0.998899i	$-0.485063\pi$
$983$	2.94137	0.0938150	0.0469075	+	0.998899i	$0.485063\pi$
$984$	−17.4396	−0.555956
$985$	−4.94137	−0.157445
$986$	9.34387	0.297569
$987$	−3.30777	−0.105288
$988$	0.813989	0.0258964
$989$	−4.64658	−0.147753
$990$	0.470683	0.0149593
$991$	14.2733	0.453406	0.226703	−	0.973964i	$-0.427205\pi$
$991$	14.2733	0.453406	0.226703	+	0.973964i	$0.427205\pi$
$992$	−16.0000	−0.508001
$993$	−26.0096	−0.825388
$994$	−1.29317	−0.0410167
$995$	6.87930	0.218088
$996$	12.7328	0.403455
$997$	3.23109	0.102330	0.0511649	−	0.998690i	$-0.483707\pi$
$997$	3.23109	0.102330	0.0511649	+	0.998690i	$0.483707\pi$
$998$	−3.71477	−0.117589
$999$	10.8647	0.343744

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1155.2.a.t.1.2	✓	3
3.2	odd	2		3465.2.a.bb.1.2		3
5.4	even	2		5775.2.a.bq.1.2		3
7.6	odd	2		8085.2.a.bl.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.t.1.2	✓	3	1.1	even	1	trivial
3465.2.a.bb.1.2		3	3.2	odd	2
5775.2.a.bq.1.2		3	5.4	even	2
8085.2.a.bl.1.2		3	7.6	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:	\( N \)	\(=\)	\( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1155.a (trivial)

\(f(q)\)	\(=\)	\(q+0.470683 q^{2} +1.00000 q^{3} -1.77846 q^{4} +1.00000 q^{5} +0.470683 q^{6} -1.00000 q^{7} -1.77846 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.470683 q^{2} +1.00000 q^{3} -1.77846 q^{4} +1.00000 q^{5} +0.470683 q^{6} -1.00000 q^{7} -1.77846 q^{8} +1.00000 q^{9} +0.470683 q^{10} +1.00000 q^{11} -1.77846 q^{12} -0.249141 q^{13} -0.470683 q^{14} +1.00000 q^{15} +2.71982 q^{16} +5.71982 q^{17} +0.470683 q^{18} +1.83709 q^{19} -1.77846 q^{20} -1.00000 q^{21} +0.470683 q^{22} +0.778457 q^{23} -1.77846 q^{24} +1.00000 q^{25} -0.117266 q^{26} +1.00000 q^{27} +1.77846 q^{28} +3.47068 q^{29} +0.470683 q^{30} -3.30777 q^{31} +4.83709 q^{32} +1.00000 q^{33} +2.69223 q^{34} -1.00000 q^{35} -1.77846 q^{36} +10.8647 q^{37} +0.864688 q^{38} -0.249141 q^{39} -1.77846 q^{40} +9.80605 q^{41} -0.470683 q^{42} -5.96896 q^{43} -1.77846 q^{44} +1.00000 q^{45} +0.366407 q^{46} +3.30777 q^{47} +2.71982 q^{48} +1.00000 q^{49} +0.470683 q^{50} +5.71982 q^{51} +0.443086 q^{52} -2.77846 q^{53} +0.470683 q^{54} +1.00000 q^{55} +1.77846 q^{56} +1.83709 q^{57} +1.63359 q^{58} -3.58795 q^{59} -1.77846 q^{60} -7.27674 q^{61} -1.55691 q^{62} -1.00000 q^{63} -3.16291 q^{64} -0.249141 q^{65} +0.470683 q^{66} +5.55691 q^{67} -10.1725 q^{68} +0.778457 q^{69} -0.470683 q^{70} +2.74742 q^{71} -1.77846 q^{72} +4.94137 q^{73} +5.11383 q^{74} +1.00000 q^{75} -3.26719 q^{76} -1.00000 q^{77} -0.117266 q^{78} +2.48367 q^{79} +2.71982 q^{80} +1.00000 q^{81} +4.61555 q^{82} -7.15947 q^{83} +1.77846 q^{84} +5.71982 q^{85} -2.80949 q^{86} +3.47068 q^{87} -1.77846 q^{88} -14.5845 q^{89} +0.470683 q^{90} +0.249141 q^{91} -1.38445 q^{92} -3.30777 q^{93} +1.55691 q^{94} +1.83709 q^{95} +4.83709 q^{96} -14.0242 q^{97} +0.470683 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} + 3 q^{11} + 3 q^{12} + 8 q^{13} - q^{14} + 3 q^{15} - q^{16} + 8 q^{17} + q^{18} - 2 q^{19} + 3 q^{20} - 3 q^{21} + q^{22} - 6 q^{23} + 3 q^{24} + 3 q^{25} - 2 q^{26} + 3 q^{27} - 3 q^{28} + 10 q^{29} + q^{30} - 2 q^{31} + 7 q^{32} + 3 q^{33} + 16 q^{34} - 3 q^{35} + 3 q^{36} + 8 q^{37} - 22 q^{38} + 8 q^{39} + 3 q^{40} + 4 q^{41} - q^{42} + 3 q^{44} + 3 q^{45} - 6 q^{46} + 2 q^{47} - q^{48} + 3 q^{49} + q^{50} + 8 q^{51} + 18 q^{52} + q^{54} + 3 q^{55} - 3 q^{56} - 2 q^{57} + 12 q^{58} - 12 q^{59} + 3 q^{60} + 4 q^{61} + 12 q^{62} - 3 q^{63} - 17 q^{64} + 8 q^{65} + q^{66} + 2 q^{68} - 6 q^{69} - q^{70} - 18 q^{71} + 3 q^{72} + 14 q^{73} - 18 q^{74} + 3 q^{75} - 24 q^{76} - 3 q^{77} - 2 q^{78} + 2 q^{79} - q^{80} + 3 q^{81} - 2 q^{82} + 6 q^{83} - 3 q^{84} + 8 q^{85} - 18 q^{86} + 10 q^{87} + 3 q^{88} - 10 q^{89} + q^{90} - 8 q^{91} - 20 q^{92} - 2 q^{93} - 12 q^{94} - 2 q^{95} + 7 q^{96} + 10 q^{97} + q^{98} + 3 q^{99}+O(q^{100}) \) 3 * q + q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9 + q^10 + 3 * q^11 + 3 * q^12 + 8 * q^13 - q^14 + 3 * q^15 - q^16 + 8 * q^17 + q^18 - 2 * q^19 + 3 * q^20 - 3 * q^21 + q^22 - 6 * q^23 + 3 * q^24 + 3 * q^25 - 2 * q^26 + 3 * q^27 - 3 * q^28 + 10 * q^29 + q^30 - 2 * q^31 + 7 * q^32 + 3 * q^33 + 16 * q^34 - 3 * q^35 + 3 * q^36 + 8 * q^37 - 22 * q^38 + 8 * q^39 + 3 * q^40 + 4 * q^41 - q^42 + 3 * q^44 + 3 * q^45 - 6 * q^46 + 2 * q^47 - q^48 + 3 * q^49 + q^50 + 8 * q^51 + 18 * q^52 + q^54 + 3 * q^55 - 3 * q^56 - 2 * q^57 + 12 * q^58 - 12 * q^59 + 3 * q^60 + 4 * q^61 + 12 * q^62 - 3 * q^63 - 17 * q^64 + 8 * q^65 + q^66 + 2 * q^68 - 6 * q^69 - q^70 - 18 * q^71 + 3 * q^72 + 14 * q^73 - 18 * q^74 + 3 * q^75 - 24 * q^76 - 3 * q^77 - 2 * q^78 + 2 * q^79 - q^80 + 3 * q^81 - 2 * q^82 + 6 * q^83 - 3 * q^84 + 8 * q^85 - 18 * q^86 + 10 * q^87 + 3 * q^88 - 10 * q^89 + q^90 - 8 * q^91 - 20 * q^92 - 2 * q^93 - 12 * q^94 - 2 * q^95 + 7 * q^96 + 10 * q^97 + q^98 + 3 * q^99

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