LMFDB - Embedded newform 6422.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.16609.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} - x + 9 $ x^4 - 7*x^2 - x + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.42534$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.88230 q^{3} +1.00000 q^{4} -0.456953 q^{5} +1.88230 q^{6} -0.317076 q^{7} -1.00000 q^{8} +0.543047 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.88230 q^{3} +1.00000 q^{4} -0.456953 q^{5} +1.88230 q^{6} -0.317076 q^{7} -1.00000 q^{8} +0.543047 q^{9} +0.456953 q^{10} -4.10827 q^{11} -1.88230 q^{12} +0.317076 q^{14} +0.860122 q^{15} +1.00000 q^{16} +3.24815 q^{17} -0.543047 q^{18} -1.00000 q^{19} -0.456953 q^{20} +0.596831 q^{21} +4.10827 q^{22} +5.02218 q^{23} +1.88230 q^{24} -4.79119 q^{25} +4.62472 q^{27} -0.317076 q^{28} -0.171486 q^{29} -0.860122 q^{30} -10.4754 q^{31} -1.00000 q^{32} +7.73299 q^{33} -3.24815 q^{34} +0.144889 q^{35} +0.543047 q^{36} +6.93236 q^{37} +1.00000 q^{38} +0.456953 q^{40} -2.13616 q^{41} -0.596831 q^{42} +5.84627 q^{43} -4.10827 q^{44} -0.248147 q^{45} -5.02218 q^{46} +8.01846 q^{47} -1.88230 q^{48} -6.89946 q^{49} +4.79119 q^{50} -6.11398 q^{51} -6.31206 q^{53} -4.62472 q^{54} +1.87729 q^{55} +0.317076 q^{56} +1.88230 q^{57} +0.171486 q^{58} +0.913907 q^{59} +0.860122 q^{60} +10.2165 q^{61} +10.4754 q^{62} -0.172187 q^{63} +1.00000 q^{64} -7.73299 q^{66} +13.7014 q^{67} +3.24815 q^{68} -9.45323 q^{69} -0.144889 q^{70} +15.9229 q^{71} -0.543047 q^{72} -6.85069 q^{73} -6.93236 q^{74} +9.01846 q^{75} -1.00000 q^{76} +1.30263 q^{77} +2.73170 q^{79} -0.456953 q^{80} -10.3342 q^{81} +2.13616 q^{82} -13.0944 q^{83} +0.596831 q^{84} -1.48425 q^{85} -5.84627 q^{86} +0.322788 q^{87} +4.10827 q^{88} +5.93608 q^{89} +0.248147 q^{90} +5.02218 q^{92} +19.7178 q^{93} -8.01846 q^{94} +0.456953 q^{95} +1.88230 q^{96} +17.7565 q^{97} +6.89946 q^{98} -2.23098 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 7 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 7 * q^7 - 4 * q^8 + 2 * q^9	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 7 q^{7} - 4 q^{8} + 2 q^{9} + 2 q^{10} - q^{11} + 2 q^{12} + 7 q^{14} + 9 q^{15} + 4 q^{16} - 8 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{20} - 3 q^{21} + q^{22} + 5 q^{23} - 2 q^{24} + 2 q^{25} + 5 q^{27} - 7 q^{28} - 5 q^{29} - 9 q^{30} - 9 q^{31} - 4 q^{32} + 2 q^{33} + 8 q^{34} + 7 q^{35} + 2 q^{36} - 5 q^{37} + 4 q^{38} + 2 q^{40} + 15 q^{41} + 3 q^{42} - 9 q^{43} - q^{44} + 20 q^{45} - 5 q^{46} - q^{47} + 2 q^{48} + 9 q^{49} - 2 q^{50} - 16 q^{51} - 19 q^{53} - 5 q^{54} - 14 q^{55} + 7 q^{56} - 2 q^{57} + 5 q^{58} + 4 q^{59} + 9 q^{60} + 10 q^{61} + 9 q^{62} + 4 q^{64} - 2 q^{66} + 16 q^{67} - 8 q^{68} - 20 q^{69} - 7 q^{70} + 6 q^{71} - 2 q^{72} - 8 q^{73} + 5 q^{74} + 3 q^{75} - 4 q^{76} - 26 q^{77} - 12 q^{79} - 2 q^{80} - 20 q^{81} - 15 q^{82} + q^{83} - 3 q^{84} + q^{85} + 9 q^{86} + 14 q^{87} + q^{88} + 9 q^{89} - 20 q^{90} + 5 q^{92} + 18 q^{93} + q^{94} + 2 q^{95} - 2 q^{96} + 21 q^{97} - 9 q^{98} - 15 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 7 * q^7 - 4 * q^8 + 2 * q^9 + 2 * q^10 - q^11 + 2 * q^12 + 7 * q^14 + 9 * q^15 + 4 * q^16 - 8 * q^17 - 2 * q^18 - 4 * q^19 - 2 * q^20 - 3 * q^21 + q^22 + 5 * q^23 - 2 * q^24 + 2 * q^25 + 5 * q^27 - 7 * q^28 - 5 * q^29 - 9 * q^30 - 9 * q^31 - 4 * q^32 + 2 * q^33 + 8 * q^34 + 7 * q^35 + 2 * q^36 - 5 * q^37 + 4 * q^38 + 2 * q^40 + 15 * q^41 + 3 * q^42 - 9 * q^43 - q^44 + 20 * q^45 - 5 * q^46 - q^47 + 2 * q^48 + 9 * q^49 - 2 * q^50 - 16 * q^51 - 19 * q^53 - 5 * q^54 - 14 * q^55 + 7 * q^56 - 2 * q^57 + 5 * q^58 + 4 * q^59 + 9 * q^60 + 10 * q^61 + 9 * q^62 + 4 * q^64 - 2 * q^66 + 16 * q^67 - 8 * q^68 - 20 * q^69 - 7 * q^70 + 6 * q^71 - 2 * q^72 - 8 * q^73 + 5 * q^74 + 3 * q^75 - 4 * q^76 - 26 * q^77 - 12 * q^79 - 2 * q^80 - 20 * q^81 - 15 * q^82 + q^83 - 3 * q^84 + q^85 + 9 * q^86 + 14 * q^87 + q^88 + 9 * q^89 - 20 * q^90 + 5 * q^92 + 18 * q^93 + q^94 + 2 * q^95 - 2 * q^96 + 21 * q^97 - 9 * q^98 - 15 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.88230	−1.08675	−0.543373	−	0.839492i	$-0.682853\pi$
$3$	−1.88230	−1.08675	−0.543373	+	0.839492i	$0.682853\pi$
$4$	1.00000	0.500000
$5$	−0.456953	−0.204356	−0.102178	−	0.994766i	$-0.532581\pi$
$5$	−0.456953	−0.204356	−0.102178	+	0.994766i	$0.532581\pi$
$6$	1.88230	0.768445
$7$	−0.317076	−0.119843	−0.0599217	−	0.998203i	$-0.519085\pi$
$7$	−0.317076	−0.119843	−0.0599217	+	0.998203i	$0.519085\pi$
$8$	−1.00000	−0.353553
$9$	0.543047	0.181016
$10$	0.456953	0.144501
$11$	−4.10827	−1.23869	−0.619345	−	0.785119i	$-0.712602\pi$
$11$	−4.10827	−1.23869	−0.619345	+	0.785119i	$0.712602\pi$
$12$	−1.88230	−0.543373
$13$	0	0
$13$	0	0
$14$	0.317076	0.0847420
$15$	0.860122	0.222083
$16$	1.00000	0.250000
$17$	3.24815	0.787791	0.393896	−	0.919155i	$-0.371127\pi$
$17$	3.24815	0.787791	0.393896	+	0.919155i	$0.371127\pi$
$18$	−0.543047	−0.127997
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.456953	−0.102178
$21$	0.596831	0.130239
$22$	4.10827	0.875886
$23$	5.02218	1.04720	0.523598	−	0.851965i	$-0.324589\pi$
$23$	5.02218	1.04720	0.523598	+	0.851965i	$0.324589\pi$
$24$	1.88230	0.384223
$25$	−4.79119	−0.958239
$26$	0	0
$27$	4.62472	0.890028
$28$	−0.317076	−0.0599217
$29$	−0.171486	−0.0318441	−0.0159221	−	0.999873i	$-0.505068\pi$
$29$	−0.171486	−0.0318441	−0.0159221	+	0.999873i	$0.505068\pi$
$30$	−0.860122	−0.157036
$31$	−10.4754	−1.88144	−0.940719	−	0.339186i	$-0.889849\pi$
$31$	−10.4754	−1.88144	−0.940719	+	0.339186i	$0.889849\pi$
$32$	−1.00000	−0.176777
$33$	7.73299	1.34614
$34$	−3.24815	−0.557053
$35$	0.144889	0.0244907
$36$	0.543047	0.0905078
$37$	6.93236	1.13967	0.569837	−	0.821758i	$-0.307006\pi$
$37$	6.93236	1.13967	0.569837	+	0.821758i	$0.307006\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0.456953	0.0722507
$41$	−2.13616	−0.333612	−0.166806	−	0.985990i	$-0.553345\pi$
$41$	−2.13616	−0.333612	−0.166806	+	0.985990i	$0.553345\pi$
$42$	−0.596831	−0.0920930
$43$	5.84627	0.891548	0.445774	−	0.895146i	$-0.352929\pi$
$43$	5.84627	0.891548	0.445774	+	0.895146i	$0.352929\pi$
$44$	−4.10827	−0.619345
$45$	−0.248147	−0.0369916
$46$	−5.02218	−0.740479
$47$	8.01846	1.16961	0.584806	−	0.811173i	$-0.301171\pi$
$47$	8.01846	1.16961	0.584806	+	0.811173i	$0.301171\pi$
$48$	−1.88230	−0.271686
$49$	−6.89946	−0.985638
$50$	4.79119	0.677577
$51$	−6.11398	−0.856129
$52$	0	0
$53$	−6.31206	−0.867029	−0.433514	−	0.901147i	$-0.642727\pi$
$53$	−6.31206	−0.867029	−0.433514	+	0.901147i	$0.642727\pi$
$54$	−4.62472	−0.629345
$55$	1.87729	0.253133
$56$	0.317076	0.0423710
$57$	1.88230	0.249316
$58$	0.171486	0.0225172
$59$	0.913907	0.118980	0.0594902	−	0.998229i	$-0.481052\pi$
$59$	0.913907	0.118980	0.0594902	+	0.998229i	$0.481052\pi$
$60$	0.860122	0.111041
$61$	10.2165	1.30809	0.654047	−	0.756454i	$-0.273070\pi$
$61$	10.2165	1.30809	0.654047	+	0.756454i	$0.273070\pi$
$62$	10.4754	1.33038
$63$	−0.172187	−0.0216935
$64$	1.00000	0.125000
$65$	0	0
$66$	−7.73299	−0.951865
$67$	13.7014	1.67389	0.836945	−	0.547287i	$-0.184339\pi$
$67$	13.7014	1.67389	0.836945	+	0.547287i	$0.184339\pi$
$68$	3.24815	0.393896
$69$	−9.45323	−1.13804
$70$	−0.144889	−0.0173175
$71$	15.9229	1.88970	0.944852	−	0.327497i	$-0.106205\pi$
$71$	15.9229	1.88970	0.944852	+	0.327497i	$0.106205\pi$
$72$	−0.543047	−0.0639987
$73$	−6.85069	−0.801813	−0.400906	−	0.916119i	$-0.631305\pi$
$73$	−6.85069	−0.801813	−0.400906	+	0.916119i	$0.631305\pi$
$74$	−6.93236	−0.805871
$75$	9.01846	1.04136
$76$	−1.00000	−0.114708
$77$	1.30263	0.148449
$78$	0	0
$79$	2.73170	0.307340	0.153670	−	0.988122i	$-0.450891\pi$
$79$	2.73170	0.307340	0.153670	+	0.988122i	$0.450891\pi$
$80$	−0.456953	−0.0510889
$81$	−10.3342	−1.14825
$82$	2.13616	0.235899
$83$	−13.0944	−1.43730	−0.718649	−	0.695373i	$-0.755239\pi$
$83$	−13.0944	−1.43730	−0.718649	+	0.695373i	$0.755239\pi$
$84$	0.596831	0.0651196
$85$	−1.48425	−0.160990
$86$	−5.84627	−0.630419
$87$	0.322788	0.0346065
$88$	4.10827	0.437943
$89$	5.93608	0.629223	0.314612	−	0.949220i	$-0.398126\pi$
$89$	5.93608	0.629223	0.314612	+	0.949220i	$0.398126\pi$
$90$	0.248147	0.0261570
$91$	0	0
$92$	5.02218	0.523598
$93$	19.7178	2.04465
$94$	−8.01846	−0.827041
$95$	0.456953	0.0468824
$96$	1.88230	0.192111
$97$	17.7565	1.80289	0.901447	−	0.432889i	$-0.142506\pi$
$97$	17.7565	1.80289	0.901447	+	0.432889i	$0.142506\pi$
$98$	6.89946	0.696951
$99$	−2.23098	−0.224222
$100$	−4.79119	−0.479119
$101$	−9.46598	−0.941900	−0.470950	−	0.882160i	$-0.656089\pi$
$101$	−9.46598	−0.941900	−0.470950	+	0.882160i	$0.656089\pi$
$102$	6.11398	0.605374
$103$	6.11899	0.602922	0.301461	−	0.953478i	$-0.402526\pi$
$103$	6.11899	0.602922	0.301461	+	0.953478i	$0.402526\pi$
$104$	0	0
$105$	−0.272724	−0.0266151
$106$	6.31206	0.613082
$107$	−16.1210	−1.55848	−0.779239	−	0.626727i	$-0.784394\pi$
$107$	−16.1210	−1.55848	−0.779239	+	0.626727i	$0.784394\pi$
$108$	4.62472	0.445014
$109$	−12.4704	−1.19445	−0.597224	−	0.802075i	$-0.703730\pi$
$109$	−12.4704	−1.19445	−0.597224	+	0.802075i	$0.703730\pi$
$110$	−1.87729	−0.178992
$111$	−13.0488	−1.23853
$112$	−0.317076	−0.0299608
$113$	−10.6153	−0.998602	−0.499301	−	0.866429i	$-0.666410\pi$
$113$	−10.6153	−0.998602	−0.499301	+	0.866429i	$0.666410\pi$
$114$	−1.88230	−0.176293
$115$	−2.29490	−0.214001
$116$	−0.171486	−0.0159221
$117$	0	0
$118$	−0.913907	−0.0841319
$119$	−1.02991	−0.0944115
$120$	−0.860122	−0.0785181
$121$	5.87788	0.534352
$122$	−10.2165	−0.924962
$123$	4.02089	0.362551
$124$	−10.4754	−0.940719
$125$	4.47412	0.400177
$126$	0.172187	0.0153396
$127$	−1.31265	−0.116479	−0.0582396	−	0.998303i	$-0.518549\pi$
$127$	−1.31265	−0.116479	−0.0582396	+	0.998303i	$0.518549\pi$
$128$	−1.00000	−0.0883883
$129$	−11.0044	−0.968885
$130$	0	0
$131$	10.6526	0.930723	0.465361	−	0.885121i	$-0.345924\pi$
$131$	10.6526	0.930723	0.465361	+	0.885121i	$0.345924\pi$
$132$	7.73299	0.673070
$133$	0.317076	0.0274939
$134$	−13.7014	−1.18362
$135$	−2.11328	−0.181882
$136$	−3.24815	−0.278526
$137$	−0.342972	−0.0293021	−0.0146510	−	0.999893i	$-0.504664\pi$
$137$	−0.342972	−0.0293021	−0.0146510	+	0.999893i	$0.504664\pi$
$138$	9.45323	0.804713
$139$	−1.73870	−0.147475	−0.0737373	−	0.997278i	$-0.523493\pi$
$139$	−1.73870	−0.147475	−0.0737373	+	0.997278i	$0.523493\pi$
$140$	0.144889	0.0122453
$141$	−15.0931	−1.27107
$142$	−15.9229	−1.33622
$143$	0	0
$144$	0.543047	0.0452539
$145$	0.0783611	0.00650753
$146$	6.85069	0.566967
$147$	12.9868	1.07114
$148$	6.93236	0.569837
$149$	−4.77031	−0.390799	−0.195399	−	0.980724i	$-0.562600\pi$
$149$	−4.77031	−0.390799	−0.195399	+	0.980724i	$0.562600\pi$
$150$	−9.01846	−0.736354
$151$	−9.85441	−0.801941	−0.400970	−	0.916091i	$-0.631327\pi$
$151$	−9.85441	−0.801941	−0.400970	+	0.916091i	$0.631327\pi$
$152$	1.00000	0.0811107
$153$	1.76390	0.142603
$154$	−1.30263	−0.104969
$155$	4.78677	0.384483
$156$	0	0
$157$	4.16334	0.332271	0.166136	−	0.986103i	$-0.446871\pi$
$157$	4.16334	0.332271	0.166136	+	0.986103i	$0.446871\pi$
$158$	−2.73170	−0.217322
$159$	11.8812	0.942240
$160$	0.456953	0.0361253
$161$	−1.59241	−0.125499
$162$	10.3342	0.811935
$163$	−3.22969	−0.252969	−0.126484	−	0.991969i	$-0.540369\pi$
$163$	−3.22969	−0.252969	−0.126484	+	0.991969i	$0.540369\pi$
$164$	−2.13616	−0.166806
$165$	−3.53361	−0.275092
$166$	13.0944	1.01632
$167$	10.4603	0.809440	0.404720	−	0.914441i	$-0.367369\pi$
$167$	10.4603	0.809440	0.404720	+	0.914441i	$0.367369\pi$
$168$	−0.596831	−0.0460465
$169$	0	0
$170$	1.48425	0.113837
$171$	−0.543047	−0.0415278
$172$	5.84627	0.445774
$173$	−0.315785	−0.0240087	−0.0120043	−	0.999928i	$-0.503821\pi$
$173$	−0.315785	−0.0240087	−0.0120043	+	0.999928i	$0.503821\pi$
$174$	−0.322788	−0.0244705
$175$	1.51917	0.114839
$176$	−4.10827	−0.309672
$177$	−1.72024	−0.129302
$178$	−5.93608	−0.444928
$179$	24.3749	1.82186	0.910932	−	0.412557i	$-0.135364\pi$
$179$	24.3749	1.82186	0.910932	+	0.412557i	$0.135364\pi$
$180$	−0.248147	−0.0184958
$181$	9.22225	0.685484	0.342742	−	0.939429i	$-0.388644\pi$
$181$	9.22225	0.685484	0.342742	+	0.939429i	$0.388644\pi$
$182$	0	0
$183$	−19.2306	−1.42156
$184$	−5.02218	−0.370240
$185$	−3.16777	−0.232899
$186$	−19.7178	−1.44578
$187$	−13.3443	−0.975829
$188$	8.01846	0.584806
$189$	−1.46639	−0.106664
$190$	−0.456953	−0.0331509
$191$	7.79691	0.564164	0.282082	−	0.959390i	$-0.408975\pi$
$191$	7.79691	0.564164	0.282082	+	0.959390i	$0.408975\pi$
$192$	−1.88230	−0.135843
$193$	−23.2578	−1.67413	−0.837065	−	0.547103i	$-0.815730\pi$
$193$	−23.2578	−1.67413	−0.837065	+	0.547103i	$0.815730\pi$
$194$	−17.7565	−1.27484
$195$	0	0
$196$	−6.89946	−0.492819
$197$	16.9209	1.20557	0.602783	−	0.797905i	$-0.294058\pi$
$197$	16.9209	1.20557	0.602783	+	0.797905i	$0.294058\pi$
$198$	2.23098	0.158549
$199$	−10.0362	−0.711448	−0.355724	−	0.934591i	$-0.615766\pi$
$199$	−10.0362	−0.711448	−0.355724	+	0.934591i	$0.615766\pi$
$200$	4.79119	0.338789
$201$	−25.7901	−1.81909
$202$	9.46598	0.666024
$203$	0.0543740	0.00381631
$204$	−6.11398	−0.428064
$205$	0.976124	0.0681755
$206$	−6.11899	−0.426330
$207$	2.72728	0.189559
$208$	0	0
$209$	4.10827	0.284175
$210$	0.272724	0.0188197
$211$	3.27603	0.225532	0.112766	−	0.993622i	$-0.464029\pi$
$211$	3.27603	0.225532	0.112766	+	0.993622i	$0.464029\pi$
$212$	−6.31206	−0.433514
$213$	−29.9717	−2.05363
$214$	16.1210	1.10201
$215$	−2.67147	−0.182193
$216$	−4.62472	−0.314672
$217$	3.32150	0.225478
$218$	12.4704	0.844602
$219$	12.8950	0.871366
$220$	1.87729	0.126567
$221$	0	0
$222$	13.0488	0.875776
$223$	23.6280	1.58225	0.791125	−	0.611655i	$-0.209496\pi$
$223$	23.6280	1.58225	0.791125	+	0.611655i	$0.209496\pi$
$224$	0.317076	0.0211855
$225$	−2.60184	−0.173456
$226$	10.6153	0.706118
$227$	0.443100	0.0294096	0.0147048	−	0.999892i	$-0.495319\pi$
$227$	0.443100	0.0294096	0.0147048	+	0.999892i	$0.495319\pi$
$228$	1.88230	0.124658
$229$	−8.36957	−0.553077	−0.276538	−	0.961003i	$-0.589187\pi$
$229$	−8.36957	−0.553077	−0.276538	+	0.961003i	$0.589187\pi$
$230$	2.29490	0.151321
$231$	−2.45194	−0.161326
$232$	0.171486	0.0112586
$233$	−10.8113	−0.708275	−0.354138	−	0.935193i	$-0.615226\pi$
$233$	−10.8113	−0.708275	−0.354138	+	0.935193i	$0.615226\pi$
$234$	0	0
$235$	−3.66406	−0.239017
$236$	0.913907	0.0594902
$237$	−5.14187	−0.334000
$238$	1.02991	0.0667590
$239$	25.4945	1.64910	0.824550	−	0.565790i	$-0.191429\pi$
$239$	25.4945	1.64910	0.824550	+	0.565790i	$0.191429\pi$
$240$	0.860122	0.0555207
$241$	2.30322	0.148364	0.0741818	−	0.997245i	$-0.476365\pi$
$241$	2.30322	0.148364	0.0741818	+	0.997245i	$0.476365\pi$
$242$	−5.87788	−0.377844
$243$	5.57797	0.357827
$244$	10.2165	0.654047
$245$	3.15273	0.201421
$246$	−4.02089	−0.256362
$247$	0	0
$248$	10.4754	0.665189
$249$	24.6476	1.56198
$250$	−4.47412	−0.282968
$251$	19.2494	1.21501	0.607507	−	0.794314i	$-0.292170\pi$
$251$	19.2494	1.21501	0.607507	+	0.794314i	$0.292170\pi$
$252$	−0.172187	−0.0108468
$253$	−20.6325	−1.29715
$254$	1.31265	0.0823633
$255$	2.79380	0.174955
$256$	1.00000	0.0625000
$257$	−17.3311	−1.08108	−0.540542	−	0.841317i	$-0.681781\pi$
$257$	−17.3311	−1.08108	−0.540542	+	0.841317i	$0.681781\pi$
$258$	11.0044	0.685105
$259$	−2.19808	−0.136582
$260$	0	0
$261$	−0.0931249	−0.00576429
$262$	−10.6526	−0.658120
$263$	9.21083	0.567964	0.283982	−	0.958830i	$-0.408344\pi$
$263$	9.21083	0.567964	0.283982	+	0.958830i	$0.408344\pi$
$264$	−7.73299	−0.475933
$265$	2.88432	0.177182
$266$	−0.317076	−0.0194412
$267$	−11.1735	−0.683806
$268$	13.7014	0.836945
$269$	−13.8736	−0.845886	−0.422943	−	0.906156i	$-0.639003\pi$
$269$	−13.8736	−0.845886	−0.422943	+	0.906156i	$0.639003\pi$
$270$	2.11328	0.128610
$271$	−1.33855	−0.0813112	−0.0406556	−	0.999173i	$-0.512945\pi$
$271$	−1.33855	−0.0813112	−0.0406556	+	0.999173i	$0.512945\pi$
$272$	3.24815	0.196948
$273$	0	0
$274$	0.342972	0.0207197
$275$	19.6835	1.18696
$276$	−9.45323	−0.569018
$277$	−5.41020	−0.325067	−0.162534	−	0.986703i	$-0.551967\pi$
$277$	−5.41020	−0.325067	−0.162534	+	0.986703i	$0.551967\pi$
$278$	1.73870	0.104280
$279$	−5.68864	−0.340570
$280$	−0.144889	−0.00865876
$281$	9.30193	0.554907	0.277453	−	0.960739i	$-0.410510\pi$
$281$	9.30193	0.554907	0.277453	+	0.960739i	$0.410510\pi$
$282$	15.0931	0.898783
$283$	−24.0813	−1.43148	−0.715741	−	0.698366i	$-0.753911\pi$
$283$	−24.0813	−1.43148	−0.715741	+	0.698366i	$0.753911\pi$
$284$	15.9229	0.944852
$285$	−0.860122	−0.0509493
$286$	0	0
$287$	0.677323	0.0399811
$288$	−0.543047	−0.0319993
$289$	−6.44954	−0.379385
$290$	−0.0783611	−0.00460152
$291$	−33.4229	−1.95929
$292$	−6.85069	−0.400906
$293$	−10.0817	−0.588978	−0.294489	−	0.955655i	$-0.595149\pi$
$293$	−10.0817	−0.588978	−0.294489	+	0.955655i	$0.595149\pi$
$294$	−12.9868	−0.757408
$295$	−0.417613	−0.0243143
$296$	−6.93236	−0.402935
$297$	−18.9996	−1.10247
$298$	4.77031	0.276337
$299$	0	0
$300$	9.01846	0.520681
$301$	−1.85371	−0.106846
$302$	9.85441	0.567058
$303$	17.8178	1.02361
$304$	−1.00000	−0.0573539
$305$	−4.66848	−0.267316
$306$	−1.76390	−0.100835
$307$	−27.0672	−1.54481	−0.772404	−	0.635131i	$-0.780946\pi$
$307$	−27.0672	−1.54481	−0.772404	+	0.635131i	$0.780946\pi$
$308$	1.30263	0.0742244
$309$	−11.5178	−0.655223
$310$	−4.78677	−0.271870
$311$	−0.651316	−0.0369327	−0.0184664	−	0.999829i	$-0.505878\pi$
$311$	−0.651316	−0.0369327	−0.0184664	+	0.999829i	$0.505878\pi$
$312$	0	0
$313$	17.3989	0.983446	0.491723	−	0.870752i	$-0.336367\pi$
$313$	17.3989	0.983446	0.491723	+	0.870752i	$0.336367\pi$
$314$	−4.16334	−0.234951
$315$	0.0786814	0.00443319
$316$	2.73170	0.153670
$317$	20.1345	1.13086	0.565432	−	0.824795i	$-0.308710\pi$
$317$	20.1345	1.13086	0.565432	+	0.824795i	$0.308710\pi$
$318$	−11.8812	−0.666264
$319$	0.704510	0.0394450
$320$	−0.456953	−0.0255445
$321$	30.3446	1.69367
$322$	1.59241	0.0887415
$323$	−3.24815	−0.180732
$324$	−10.3342	−0.574124
$325$	0	0
$326$	3.22969	0.178876
$327$	23.4730	1.29806
$328$	2.13616	0.117950
$329$	−2.54246	−0.140170
$330$	3.53361	0.194519
$331$	−6.85813	−0.376957	−0.188478	−	0.982077i	$-0.560356\pi$
$331$	−6.85813	−0.376957	−0.188478	+	0.982077i	$0.560356\pi$
$332$	−13.0944	−0.718649
$333$	3.76460	0.206299
$334$	−10.4603	−0.572360
$335$	−6.26089	−0.342069
$336$	0.596831	0.0325598
$337$	−23.7299	−1.29265	−0.646324	−	0.763063i	$-0.723694\pi$
$337$	−23.7299	−1.29265	−0.646324	+	0.763063i	$0.723694\pi$
$338$	0	0
$339$	19.9811	1.08523
$340$	−1.48425	−0.0804948
$341$	43.0358	2.33052
$342$	0.543047	0.0293646
$343$	4.40718	0.237965
$344$	−5.84627	−0.315210
$345$	4.31969	0.232564
$346$	0.315785	0.0169767
$347$	−1.97152	−0.105837	−0.0529184	−	0.998599i	$-0.516852\pi$
$347$	−1.97152	−0.105837	−0.0529184	+	0.998599i	$0.516852\pi$
$348$	0.322788	0.0173032
$349$	−18.7330	−1.00275	−0.501377	−	0.865229i	$-0.667173\pi$
$349$	−18.7330	−1.00275	−0.501377	+	0.865229i	$0.667173\pi$
$350$	−1.51917	−0.0812031
$351$	0	0
$352$	4.10827	0.218971
$353$	5.73911	0.305462	0.152731	−	0.988268i	$-0.451193\pi$
$353$	5.73911	0.305462	0.152731	+	0.988268i	$0.451193\pi$
$354$	1.72024	0.0914300
$355$	−7.27603	−0.386172
$356$	5.93608	0.314612
$357$	1.93859	0.102601
$358$	−24.3749	−1.28825
$359$	−14.6596	−0.773706	−0.386853	−	0.922141i	$-0.626438\pi$
$359$	−14.6596	−0.773706	−0.386853	+	0.922141i	$0.626438\pi$
$360$	0.248147	0.0130785
$361$	1.00000	0.0526316
$362$	−9.22225	−0.484711
$363$	−11.0639	−0.580705
$364$	0	0
$365$	3.13045	0.163855
$366$	19.2306	1.00520
$367$	−9.78806	−0.510933	−0.255466	−	0.966818i	$-0.582229\pi$
$367$	−9.78806	−0.510933	−0.255466	+	0.966818i	$0.582229\pi$
$368$	5.02218	0.261799
$369$	−1.16003	−0.0603889
$370$	3.16777	0.164684
$371$	2.00140	0.103908
$372$	19.7178	1.02232
$373$	−0.243726	−0.0126196	−0.00630982	−	0.999980i	$-0.502008\pi$
$373$	−0.243726	−0.0126196	−0.00630982	+	0.999980i	$0.502008\pi$
$374$	13.3443	0.690015
$375$	−8.42162	−0.434891
$376$	−8.01846	−0.413520
$377$	0	0
$378$	1.46639	0.0754227
$379$	−29.1485	−1.49726	−0.748629	−	0.662989i	$-0.769287\pi$
$379$	−29.1485	−1.49726	−0.748629	+	0.662989i	$0.769287\pi$
$380$	0.456953	0.0234412
$381$	2.47081	0.126583
$382$	−7.79691	−0.398924
$383$	25.5774	1.30694	0.653471	−	0.756951i	$-0.273312\pi$
$383$	25.5774	1.30694	0.653471	+	0.756951i	$0.273312\pi$
$384$	1.88230	0.0960556
$385$	−0.595242	−0.0303363
$386$	23.2578	1.18379
$387$	3.17480	0.161384
$388$	17.7565	0.901447
$389$	−18.0532	−0.915333	−0.457667	−	0.889124i	$-0.651315\pi$
$389$	−18.0532	−0.915333	−0.457667	+	0.889124i	$0.651315\pi$
$390$	0	0
$391$	16.3128	0.824972
$392$	6.89946	0.348476
$393$	−20.0514	−1.01146
$394$	−16.9209	−0.852465
$395$	−1.24826	−0.0628067
$396$	−2.23098	−0.112111
$397$	−33.0692	−1.65970	−0.829849	−	0.557988i	$-0.811574\pi$
$397$	−33.0692	−1.65970	−0.829849	+	0.557988i	$0.811574\pi$
$398$	10.0362	0.503070
$399$	−0.596831	−0.0298789
$400$	−4.79119	−0.239560
$401$	29.6590	1.48110	0.740551	−	0.672000i	$-0.234565\pi$
$401$	29.6590	1.48110	0.740551	+	0.672000i	$0.234565\pi$
$402$	25.7901	1.28629
$403$	0	0
$404$	−9.46598	−0.470950
$405$	4.72227	0.234651
$406$	−0.0543740	−0.00269854
$407$	−28.4800	−1.41170
$408$	6.11398	0.302687
$409$	−13.6038	−0.672666	−0.336333	−	0.941743i	$-0.609187\pi$
$409$	−13.6038	−0.672666	−0.336333	+	0.941743i	$0.609187\pi$
$410$	−0.976124	−0.0482073
$411$	0.645575	0.0318439
$412$	6.11899	0.301461
$413$	−0.289778	−0.0142590
$414$	−2.72728	−0.134038
$415$	5.98354	0.293720
$416$	0	0
$417$	3.27275	0.160267
$418$	−4.10827	−0.200942
$419$	−14.2793	−0.697592	−0.348796	−	0.937199i	$-0.613409\pi$
$419$	−14.2793	−0.697592	−0.348796	+	0.937199i	$0.613409\pi$
$420$	−0.272724	−0.0133076
$421$	9.09571	0.443298	0.221649	−	0.975127i	$-0.428856\pi$
$421$	9.09571	0.443298	0.221649	+	0.975127i	$0.428856\pi$
$422$	−3.27603	−0.159475
$423$	4.35440	0.211718
$424$	6.31206	0.306541
$425$	−15.5625	−0.754892
$426$	29.9717	1.45213
$427$	−3.23942	−0.156766
$428$	−16.1210	−0.779239
$429$	0	0
$430$	2.67147	0.128830
$431$	18.5064	0.891423	0.445712	−	0.895177i	$-0.352951\pi$
$431$	18.5064	0.891423	0.445712	+	0.895177i	$0.352951\pi$
$432$	4.62472	0.222507
$433$	−33.7742	−1.62308	−0.811542	−	0.584293i	$-0.801372\pi$
$433$	−33.7742	−1.62308	−0.811542	+	0.584293i	$0.801372\pi$
$434$	−3.32150	−0.159437
$435$	−0.147499	−0.00707203
$436$	−12.4704	−0.597224
$437$	−5.02218	−0.240243
$438$	−12.8950	−0.616149
$439$	−33.2926	−1.58897	−0.794485	−	0.607284i	$-0.792259\pi$
$439$	−33.2926	−1.58897	−0.794485	+	0.607284i	$0.792259\pi$
$440$	−1.87729	−0.0894962
$441$	−3.74673	−0.178416
$442$	0	0
$443$	−21.7110	−1.03152	−0.515760	−	0.856733i	$-0.672490\pi$
$443$	−21.7110	−1.03152	−0.515760	+	0.856733i	$0.672490\pi$
$444$	−13.0488	−0.619267
$445$	−2.71251	−0.128585
$446$	−23.6280	−1.11882
$447$	8.97914	0.424699
$448$	−0.317076	−0.0149804
$449$	29.3730	1.38620	0.693099	−	0.720843i	$-0.256245\pi$
$449$	29.3730	1.38620	0.693099	+	0.720843i	$0.256245\pi$
$450$	2.60184	0.122652
$451$	8.77591	0.413241
$452$	−10.6153	−0.499301
$453$	18.5489	0.871505
$454$	−0.443100	−0.0207957
$455$	0	0
$456$	−1.88230	−0.0881467
$457$	−25.5761	−1.19640	−0.598200	−	0.801347i	$-0.704117\pi$
$457$	−25.5761	−1.19640	−0.598200	+	0.801347i	$0.704117\pi$
$458$	8.36957	0.391084
$459$	15.0218	0.701156
$460$	−2.29490	−0.107000
$461$	−6.83294	−0.318242	−0.159121	−	0.987259i	$-0.550866\pi$
$461$	−6.83294	−0.318242	−0.159121	+	0.987259i	$0.550866\pi$
$462$	2.45194	0.114075
$463$	12.7128	0.590815	0.295408	−	0.955371i	$-0.404545\pi$
$463$	12.7128	0.590815	0.295408	+	0.955371i	$0.404545\pi$
$464$	−0.171486	−0.00796104
$465$	−9.01013	−0.417835
$466$	10.8113	0.500826
$467$	−19.9065	−0.921163	−0.460581	−	0.887617i	$-0.652359\pi$
$467$	−19.9065	−0.921163	−0.460581	+	0.887617i	$0.652359\pi$
$468$	0	0
$469$	−4.34437	−0.200605
$470$	3.66406	0.169010
$471$	−7.83666	−0.361094
$472$	−0.913907	−0.0420660
$473$	−24.0180	−1.10435
$474$	5.14187	0.236174
$475$	4.79119	0.219835
$476$	−1.02991	−0.0472058
$477$	−3.42775	−0.156946
$478$	−25.4945	−1.16609
$479$	−32.2579	−1.47390	−0.736951	−	0.675947i	$-0.763735\pi$
$479$	−32.2579	−1.47390	−0.736951	+	0.675947i	$0.763735\pi$
$480$	−0.860122	−0.0392590
$481$	0	0
$482$	−2.30322	−0.104909
$483$	2.99739	0.136386
$484$	5.87788	0.267176
$485$	−8.11387	−0.368432
$486$	−5.57797	−0.253022
$487$	−35.1123	−1.59109	−0.795545	−	0.605895i	$-0.792815\pi$
$487$	−35.1123	−1.59109	−0.795545	+	0.605895i	$0.792815\pi$
$488$	−10.2165	−0.462481
$489$	6.07924	0.274913
$490$	−3.15273	−0.142426
$491$	−31.2908	−1.41213	−0.706067	−	0.708145i	$-0.749532\pi$
$491$	−31.2908	−1.41213	−0.706067	+	0.708145i	$0.749532\pi$
$492$	4.02089	0.181276
$493$	−0.557012	−0.0250865
$494$	0	0
$495$	1.01945	0.0458211
$496$	−10.4754	−0.470360
$497$	−5.04877	−0.226468
$498$	−24.6476	−1.10449
$499$	11.0689	0.495513	0.247757	−	0.968822i	$-0.420307\pi$
$499$	11.0689	0.495513	0.247757	+	0.968822i	$0.420307\pi$
$500$	4.47412	0.200089
$501$	−19.6893	−0.879655
$502$	−19.2494	−0.859144
$503$	−28.7975	−1.28402	−0.642008	−	0.766698i	$-0.721899\pi$
$503$	−28.7975	−1.28402	−0.642008	+	0.766698i	$0.721899\pi$
$504$	0.172187	0.00766981
$505$	4.32551	0.192483
$506$	20.6325	0.917224
$507$	0	0
$508$	−1.31265	−0.0582396
$509$	−27.5916	−1.22298	−0.611488	−	0.791254i	$-0.709429\pi$
$509$	−27.5916	−1.22298	−0.611488	+	0.791254i	$0.709429\pi$
$510$	−2.79380	−0.123712
$511$	2.17219	0.0960919
$512$	−1.00000	−0.0441942
$513$	−4.62472	−0.204186
$514$	17.3311	0.764442
$515$	−2.79609	−0.123211
$516$	−11.0044	−0.484443
$517$	−32.9420	−1.44879
$518$	2.19808	0.0965782
$519$	0.594401	0.0260913
$520$	0	0
$521$	1.18663	0.0519872	0.0259936	−	0.999662i	$-0.491725\pi$
$521$	1.18663	0.0519872	0.0259936	+	0.999662i	$0.491725\pi$
$522$	0.0931249	0.00407597
$523$	−40.8338	−1.78554	−0.892770	−	0.450513i	$-0.851241\pi$
$523$	−40.8338	−1.78554	−0.892770	+	0.450513i	$0.851241\pi$
$524$	10.6526	0.465361
$525$	−2.85953	−0.124800
$526$	−9.21083	−0.401611
$527$	−34.0257	−1.48218
$528$	7.73299	0.336535
$529$	2.22225	0.0966196
$530$	−2.88432	−0.125287
$531$	0.496294	0.0215373
$532$	0.317076	0.0137470
$533$	0	0
$534$	11.1735	0.483524
$535$	7.36655	0.318484
$536$	−13.7014	−0.591809
$537$	−45.8808	−1.97990
$538$	13.8736	0.598132
$539$	28.3449	1.22090
$540$	−2.11328	−0.0909411
$541$	−32.4292	−1.39424	−0.697121	−	0.716953i	$-0.745536\pi$
$541$	−32.4292	−1.39424	−0.697121	+	0.716953i	$0.745536\pi$
$542$	1.33855	0.0574957
$543$	−17.3590	−0.744947
$544$	−3.24815	−0.139263
$545$	5.69839	0.244092
$546$	0	0
$547$	−33.4939	−1.43210	−0.716048	−	0.698051i	$-0.754051\pi$
$547$	−33.4939	−1.43210	−0.716048	+	0.698051i	$0.754051\pi$
$548$	−0.342972	−0.0146510
$549$	5.54806	0.236785
$550$	−19.6835	−0.839308
$551$	0.171486	0.00730555
$552$	9.45323	0.402356
$553$	−0.866155	−0.0368326
$554$	5.41020	0.229857
$555$	5.96268	0.253102
$556$	−1.73870	−0.0737373
$557$	−42.8307	−1.81480	−0.907398	−	0.420273i	$-0.861934\pi$
$557$	−42.8307	−1.81480	−0.907398	+	0.420273i	$0.861934\pi$
$558$	5.68864	0.240819
$559$	0	0
$560$	0.144889	0.00612267
$561$	25.1179	1.06048
$562$	−9.30193	−0.392378
$563$	25.4305	1.07177	0.535885	−	0.844291i	$-0.319978\pi$
$563$	25.4305	1.07177	0.535885	+	0.844291i	$0.319978\pi$
$564$	−15.0931	−0.635535
$565$	4.85069	0.204070
$566$	24.0813	1.01221
$567$	3.27674	0.137610
$568$	−15.9229	−0.668111
$569$	1.68292	0.0705519	0.0352759	−	0.999378i	$-0.488769\pi$
$569$	1.68292	0.0705519	0.0352759	+	0.999378i	$0.488769\pi$
$570$	0.860122	0.0360266
$571$	19.7058	0.824663	0.412332	−	0.911034i	$-0.364714\pi$
$571$	19.7058	0.824663	0.412332	+	0.911034i	$0.364714\pi$
$572$	0	0
$573$	−14.6761	−0.613103
$574$	−0.677323	−0.0282709
$575$	−24.0622	−1.00346
$576$	0.543047	0.0226269
$577$	21.3939	0.890640	0.445320	−	0.895371i	$-0.353090\pi$
$577$	21.3939	0.890640	0.445320	+	0.895371i	$0.353090\pi$
$578$	6.44954	0.268266
$579$	43.7780	1.81935
$580$	0.0783611	0.00325377
$581$	4.15192	0.172251
$582$	33.4229	1.38543
$583$	25.9317	1.07398
$584$	6.85069	0.283484
$585$	0	0
$586$	10.0817	0.416470
$587$	27.0207	1.11526	0.557631	−	0.830089i	$-0.311710\pi$
$587$	27.0207	1.11526	0.557631	+	0.830089i	$0.311710\pi$
$588$	12.9868	0.535569
$589$	10.4754	0.431632
$590$	0.417613	0.0171928
$591$	−31.8503	−1.31014
$592$	6.93236	0.284918
$593$	4.56835	0.187600	0.0937999	−	0.995591i	$-0.470099\pi$
$593$	4.56835	0.187600	0.0937999	+	0.995591i	$0.470099\pi$
$594$	18.9996	0.779563
$595$	0.470620	0.0192935
$596$	−4.77031	−0.195399
$597$	18.8911	0.773163
$598$	0	0
$599$	17.2394	0.704383	0.352192	−	0.935928i	$-0.385437\pi$
$599$	17.2394	0.704383	0.352192	+	0.935928i	$0.385437\pi$
$600$	−9.01846	−0.368177
$601$	−10.9438	−0.446406	−0.223203	−	0.974772i	$-0.571651\pi$
$601$	−10.9438	−0.446406	−0.223203	+	0.974772i	$0.571651\pi$
$602$	1.85371	0.0755516
$603$	7.44049	0.303000
$604$	−9.85441	−0.400970
$605$	−2.68592	−0.109198
$606$	−17.8178	−0.723798
$607$	44.5687	1.80899	0.904494	−	0.426486i	$-0.140249\pi$
$607$	44.5687	1.80899	0.904494	+	0.426486i	$0.140249\pi$
$608$	1.00000	0.0405554
$609$	−0.102348	−0.00414735
$610$	4.66848	0.189021
$611$	0	0
$612$	1.76390	0.0713013
$613$	−8.59440	−0.347125	−0.173562	−	0.984823i	$-0.555528\pi$
$613$	−8.59440	−0.347125	−0.173562	+	0.984823i	$0.555528\pi$
$614$	27.0672	1.09234
$615$	−1.83736	−0.0740894
$616$	−1.30263	−0.0524845
$617$	5.71884	0.230232	0.115116	−	0.993352i	$-0.463276\pi$
$617$	5.71884	0.230232	0.115116	+	0.993352i	$0.463276\pi$
$618$	11.5178	0.463313
$619$	−31.8667	−1.28083	−0.640416	−	0.768028i	$-0.721238\pi$
$619$	−31.8667	−1.28083	−0.640416	+	0.768028i	$0.721238\pi$
$620$	4.78677	0.192241
$621$	23.2262	0.932033
$622$	0.651316	0.0261154
$623$	−1.88219	−0.0754082
$624$	0	0
$625$	21.9115	0.876460
$626$	−17.3989	−0.695401
$627$	−7.73299	−0.308826
$628$	4.16334	0.166136
$629$	22.5173	0.897825
$630$	−0.0786814	−0.00313474
$631$	−36.7238	−1.46195	−0.730977	−	0.682402i	$-0.760935\pi$
$631$	−36.7238	−1.46195	−0.730977	+	0.682402i	$0.760935\pi$
$632$	−2.73170	−0.108661
$633$	−6.16647	−0.245095
$634$	−20.1345	−0.799641
$635$	0.599822	0.0238032
$636$	11.8812	0.471120
$637$	0	0
$638$	−0.704510	−0.0278918
$639$	8.64689	0.342066
$640$	0.456953	0.0180627
$641$	3.00483	0.118684	0.0593418	−	0.998238i	$-0.481100\pi$
$641$	3.00483	0.118684	0.0593418	+	0.998238i	$0.481100\pi$
$642$	−30.3446	−1.19760
$643$	14.7958	0.583489	0.291745	−	0.956496i	$-0.405764\pi$
$643$	14.7958	0.583489	0.291745	+	0.956496i	$0.405764\pi$
$644$	−1.59241	−0.0627497
$645$	5.02851	0.197997
$646$	3.24815	0.127797
$647$	43.2238	1.69930	0.849650	−	0.527346i	$-0.176813\pi$
$647$	43.2238	1.69930	0.849650	+	0.527346i	$0.176813\pi$
$648$	10.3342	0.405967
$649$	−3.75457	−0.147380
$650$	0	0
$651$	−6.25205	−0.245037
$652$	−3.22969	−0.126484
$653$	−41.2789	−1.61537	−0.807684	−	0.589615i	$-0.799279\pi$
$653$	−41.2789	−1.61537	−0.807684	+	0.589615i	$0.799279\pi$
$654$	−23.4730	−0.917867
$655$	−4.86774	−0.190198
$656$	−2.13616	−0.0834029
$657$	−3.72024	−0.145141
$658$	2.54246	0.0991153
$659$	20.8401	0.811817	0.405908	−	0.913914i	$-0.366955\pi$
$659$	20.8401	0.811817	0.405908	+	0.913914i	$0.366955\pi$
$660$	−3.53361	−0.137546
$661$	−17.6013	−0.684609	−0.342305	−	0.939589i	$-0.611208\pi$
$661$	−17.6013	−0.684609	−0.342305	+	0.939589i	$0.611208\pi$
$662$	6.85813	0.266549
$663$	0	0
$664$	13.0944	0.508162
$665$	−0.144889	−0.00561854
$666$	−3.76460	−0.145875
$667$	−0.861233	−0.0333471
$668$	10.4603	0.404720
$669$	−44.4750	−1.71950
$670$	6.26089	0.241879
$671$	−41.9723	−1.62032
$672$	−0.596831	−0.0230233
$673$	−19.5033	−0.751797	−0.375898	−	0.926661i	$-0.622666\pi$
$673$	−19.5033	−0.751797	−0.375898	+	0.926661i	$0.622666\pi$
$674$	23.7299	0.914040
$675$	−22.1579	−0.852859
$676$	0	0
$677$	50.7428	1.95020	0.975102	−	0.221758i	$-0.0711795\pi$
$677$	50.7428	1.95020	0.975102	+	0.221758i	$0.0711795\pi$
$678$	−19.9811	−0.767371
$679$	−5.63014	−0.216065
$680$	1.48425	0.0569184
$681$	−0.834046	−0.0319607
$682$	−43.0358	−1.64793
$683$	−18.0180	−0.689441	−0.344721	−	0.938705i	$-0.612026\pi$
$683$	−18.0180	−0.689441	−0.344721	+	0.938705i	$0.612026\pi$
$684$	−0.543047	−0.0207639
$685$	0.156722	0.00598804
$686$	−4.40718	−0.168267
$687$	15.7540	0.601054
$688$	5.84627	0.222887
$689$	0	0
$690$	−4.31969	−0.164448
$691$	2.39061	0.0909429	0.0454715	−	0.998966i	$-0.485521\pi$
$691$	2.39061	0.0909429	0.0454715	+	0.998966i	$0.485521\pi$
$692$	−0.315785	−0.0120043
$693$	0.707390	0.0268715
$694$	1.97152	0.0748380
$695$	0.794505	0.0301373
$696$	−0.322788	−0.0122352
$697$	−6.93855	−0.262816
$698$	18.7330	0.709054
$699$	20.3502	0.769715
$700$	1.51917	0.0574193
$701$	−27.5661	−1.04116	−0.520579	−	0.853814i	$-0.674284\pi$
$701$	−27.5661	−1.04116	−0.520579	+	0.853814i	$0.674284\pi$
$702$	0	0
$703$	−6.93236	−0.261459
$704$	−4.10827	−0.154836
$705$	6.89685	0.259751
$706$	−5.73911	−0.215994
$707$	3.00143	0.112880
$708$	−1.72024	−0.0646508
$709$	40.4611	1.51955	0.759775	−	0.650186i	$-0.225309\pi$
$709$	40.4611	1.51955	0.759775	+	0.650186i	$0.225309\pi$
$710$	7.27603	0.273065
$711$	1.48344	0.0556333
$712$	−5.93608	−0.222464
$713$	−52.6093	−1.97024
$714$	−1.93859	−0.0725501
$715$	0	0
$716$	24.3749	0.910932
$717$	−47.9882	−1.79215
$718$	14.6596	0.547093
$719$	−19.2217	−0.716850	−0.358425	−	0.933559i	$-0.616686\pi$
$719$	−19.2217	−0.716850	−0.358425	+	0.933559i	$0.616686\pi$
$720$	−0.248147	−0.00924789
$721$	−1.94018	−0.0722562
$722$	−1.00000	−0.0372161
$723$	−4.33535	−0.161233
$724$	9.22225	0.342742
$725$	0.821622	0.0305143
$726$	11.0639	0.410620
$727$	17.5006	0.649061	0.324531	−	0.945875i	$-0.394794\pi$
$727$	17.5006	0.649061	0.324531	+	0.945875i	$0.394794\pi$
$728$	0	0
$729$	20.5033	0.759382
$730$	−3.13045	−0.115863
$731$	18.9895	0.702354
$732$	−19.2306	−0.710782
$733$	−23.6673	−0.874173	−0.437087	−	0.899419i	$-0.643990\pi$
$733$	−23.6673	−0.874173	−0.437087	+	0.899419i	$0.643990\pi$
$734$	9.78806	0.361284
$735$	−5.93438	−0.218893
$736$	−5.02218	−0.185120
$737$	−56.2890	−2.07343
$738$	1.16003	0.0427014
$739$	−25.1437	−0.924928	−0.462464	−	0.886638i	$-0.653035\pi$
$739$	−25.1437	−0.924928	−0.462464	+	0.886638i	$0.653035\pi$
$740$	−3.16777	−0.116449
$741$	0	0
$742$	−2.00140	−0.0734738
$743$	0.356826	0.0130907	0.00654533	−	0.999979i	$-0.497917\pi$
$743$	0.356826	0.0130907	0.00654533	+	0.999979i	$0.497917\pi$
$744$	−19.7178	−0.722891
$745$	2.17981	0.0798620
$746$	0.243726	0.00892343
$747$	−7.11088	−0.260173
$748$	−13.3443	−0.487915
$749$	5.11158	0.186773
$750$	8.42162	0.307514
$751$	7.52175	0.274473	0.137236	−	0.990538i	$-0.456178\pi$
$751$	7.52175	0.274473	0.137236	+	0.990538i	$0.456178\pi$
$752$	8.01846	0.292403
$753$	−36.2332	−1.32041
$754$	0	0
$755$	4.50301	0.163881
$756$	−1.46639	−0.0533319
$757$	32.7497	1.19031	0.595155	−	0.803611i	$-0.297091\pi$
$757$	32.7497	1.19031	0.595155	+	0.803611i	$0.297091\pi$
$758$	29.1485	1.05872
$759$	38.8364	1.40967
$760$	−0.456953	−0.0165754
$761$	25.2964	0.916993	0.458496	−	0.888696i	$-0.348388\pi$
$761$	25.2964	0.916993	0.458496	+	0.888696i	$0.348388\pi$
$762$	−2.47081	−0.0895079
$763$	3.95406	0.143147
$764$	7.79691	0.282082
$765$	−0.806018	−0.0291416
$766$	−25.5774	−0.924148
$767$	0	0
$768$	−1.88230	−0.0679216
$769$	−37.2191	−1.34216	−0.671078	−	0.741386i	$-0.734169\pi$
$769$	−37.2191	−1.34216	−0.671078	+	0.741386i	$0.734169\pi$
$770$	0.595242	0.0214510
$771$	32.6223	1.17486
$772$	−23.2578	−0.837065
$773$	−10.5599	−0.379814	−0.189907	−	0.981802i	$-0.560819\pi$
$773$	−10.5599	−0.379814	−0.189907	+	0.981802i	$0.560819\pi$
$774$	−3.17480	−0.114116
$775$	50.1897	1.80287
$776$	−17.7565	−0.637420
$777$	4.13745	0.148430
$778$	18.0532	0.647238
$779$	2.13616	0.0765358
$780$	0	0
$781$	−65.4157	−2.34076
$782$	−16.3128	−0.583343
$783$	−0.793074	−0.0283422
$784$	−6.89946	−0.246409
$785$	−1.90245	−0.0679015
$786$	20.0514	0.715209
$787$	16.8850	0.601886	0.300943	−	0.953642i	$-0.402699\pi$
$787$	16.8850	0.601886	0.300943	+	0.953642i	$0.402699\pi$
$788$	16.9209	0.602783
$789$	−17.3375	−0.617232
$790$	1.24826	0.0444110
$791$	3.36585	0.119676
$792$	2.23098	0.0792745
$793$	0	0
$794$	33.0692	1.17358
$795$	−5.42915	−0.192552
$796$	−10.0362	−0.355724
$797$	4.47895	0.158653	0.0793263	−	0.996849i	$-0.474723\pi$
$797$	4.47895	0.158653	0.0793263	+	0.996849i	$0.474723\pi$
$798$	0.596831	0.0211276
$799$	26.0451	0.921410
$800$	4.79119	0.169394
$801$	3.22357	0.113899
$802$	−29.6590	−1.04730
$803$	28.1445	0.993197
$804$	−25.7901	−0.909546
$805$	0.727657	0.0256465
$806$	0	0
$807$	26.1142	0.919263
$808$	9.46598	0.333012
$809$	33.7003	1.18484	0.592419	−	0.805630i	$-0.298173\pi$
$809$	33.7003	1.18484	0.592419	+	0.805630i	$0.298173\pi$
$810$	−4.72227	−0.165923
$811$	11.1493	0.391505	0.195753	−	0.980653i	$-0.437285\pi$
$811$	11.1493	0.391505	0.195753	+	0.980653i	$0.437285\pi$
$812$	0.0543740	0.00190815
$813$	2.51955	0.0883645
$814$	28.4800	0.998224
$815$	1.47582	0.0516957
$816$	−6.11398	−0.214032
$817$	−5.84627	−0.204535
$818$	13.6038	0.475647
$819$	0	0
$820$	0.976124	0.0340877
$821$	29.1958	1.01894	0.509471	−	0.860488i	$-0.329841\pi$
$821$	29.1958	1.01894	0.509471	+	0.860488i	$0.329841\pi$
$822$	−0.645575	−0.0225170
$823$	45.2365	1.57685	0.788423	−	0.615134i	$-0.210898\pi$
$823$	45.2365	1.57685	0.788423	+	0.615134i	$0.210898\pi$
$824$	−6.11899	−0.213165
$825$	−37.0502	−1.28992
$826$	0.289778	0.0100826
$827$	−35.0052	−1.21725	−0.608625	−	0.793458i	$-0.708279\pi$
$827$	−35.0052	−1.21725	−0.608625	+	0.793458i	$0.708279\pi$
$828$	2.72728	0.0947794
$829$	30.4108	1.05621	0.528105	−	0.849179i	$-0.322903\pi$
$829$	30.4108	1.05621	0.528105	+	0.849179i	$0.322903\pi$
$830$	−5.98354	−0.207692
$831$	10.1836	0.353266
$832$	0	0
$833$	−22.4105	−0.776477
$834$	−3.27275	−0.113326
$835$	−4.77985	−0.165414
$836$	4.10827	0.142087
$837$	−48.4458	−1.67453
$838$	14.2793	0.493272
$839$	−0.485571	−0.0167638	−0.00838188	−	0.999965i	$-0.502668\pi$
$839$	−0.485571	−0.0167638	−0.00838188	+	0.999965i	$0.502668\pi$
$840$	0.272724	0.00940987
$841$	−28.9706	−0.998986
$842$	−9.09571	−0.313459
$843$	−17.5090	−0.603042
$844$	3.27603	0.112766
$845$	0	0
$846$	−4.35440	−0.149707
$847$	−1.86373	−0.0640386
$848$	−6.31206	−0.216757
$849$	45.3281	1.55566
$850$	15.5625	0.533789
$851$	34.8155	1.19346
$852$	−29.9717	−1.02681
$853$	−13.2837	−0.454824	−0.227412	−	0.973799i	$-0.573026\pi$
$853$	−13.2837	−0.454824	−0.227412	+	0.973799i	$0.573026\pi$
$854$	3.23942	0.110851
$855$	0.248147	0.00848645
$856$	16.1210	0.551005
$857$	9.58637	0.327464	0.163732	−	0.986505i	$-0.447647\pi$
$857$	9.58637	0.327464	0.163732	+	0.986505i	$0.447647\pi$
$858$	0	0
$859$	−37.0953	−1.26568	−0.632838	−	0.774285i	$-0.718110\pi$
$859$	−37.0953	−1.26568	−0.632838	+	0.774285i	$0.718110\pi$
$860$	−2.67147	−0.0910964
$861$	−1.27492	−0.0434493
$862$	−18.5064	−0.630331
$863$	15.6822	0.533829	0.266915	−	0.963720i	$-0.413996\pi$
$863$	15.6822	0.533829	0.266915	+	0.963720i	$0.413996\pi$
$864$	−4.62472	−0.157336
$865$	0.144299	0.00490631
$866$	33.7742	1.14769
$867$	12.1400	0.412295
$868$	3.32150	0.112739
$869$	−11.2225	−0.380699
$870$	0.147499	0.00500068
$871$	0	0
$872$	12.4704	0.422301
$873$	9.64258	0.326352
$874$	5.02218	0.169878
$875$	−1.41863	−0.0479586
$876$	12.8950	0.435683
$877$	8.47784	0.286276	0.143138	−	0.989703i	$-0.454281\pi$
$877$	8.47784	0.286276	0.143138	+	0.989703i	$0.454281\pi$
$878$	33.2926	1.12357
$879$	18.9767	0.640069
$880$	1.87729	0.0632833
$881$	−25.0991	−0.845611	−0.422806	−	0.906220i	$-0.638955\pi$
$881$	−25.0991	−0.845611	−0.422806	+	0.906220i	$0.638955\pi$
$882$	3.74673	0.126159
$883$	−47.4667	−1.59738	−0.798692	−	0.601741i	$-0.794474\pi$
$883$	−47.4667	−1.59738	−0.798692	+	0.601741i	$0.794474\pi$
$884$	0	0
$885$	0.786071	0.0264235
$886$	21.7110	0.729395
$887$	19.4282	0.652337	0.326168	−	0.945312i	$-0.394242\pi$
$887$	19.4282	0.652337	0.326168	+	0.945312i	$0.394242\pi$
$888$	13.0488	0.437888
$889$	0.416211	0.0139593
$890$	2.71251	0.0909236
$891$	42.4558	1.42232
$892$	23.6280	0.791125
$893$	−8.01846	−0.268327
$894$	−8.97914	−0.300307
$895$	−11.1382	−0.372308
$896$	0.317076	0.0105928
$897$	0	0
$898$	−29.3730	−0.980189
$899$	1.79639	0.0599128
$900$	−2.60184	−0.0867281
$901$	−20.5025	−0.683038
$902$	−8.77591	−0.292206
$903$	3.48923	0.116114
$904$	10.6153	0.353059
$905$	−4.21414	−0.140083
$906$	−18.5489	−0.616247
$907$	−58.1927	−1.93226	−0.966128	−	0.258064i	$-0.916915\pi$
$907$	−58.1927	−1.93226	−0.966128	+	0.258064i	$0.916915\pi$
$908$	0.443100	0.0147048
$909$	−5.14047	−0.170499
$910$	0	0
$911$	22.6622	0.750833	0.375416	−	0.926856i	$-0.377500\pi$
$911$	22.6622	0.750833	0.375416	+	0.926856i	$0.377500\pi$
$912$	1.88230	0.0623291
$913$	53.7954	1.78037
$914$	25.5761	0.845983
$915$	8.78747	0.290505
$916$	−8.36957	−0.276538
$917$	−3.37768	−0.111541
$918$	−15.0218	−0.495792
$919$	11.3312	0.373782	0.186891	−	0.982381i	$-0.440159\pi$
$919$	11.3312	0.373782	0.186891	+	0.982381i	$0.440159\pi$
$920$	2.29490	0.0756606
$921$	50.9486	1.67881
$922$	6.83294	0.225031
$923$	0	0
$924$	−2.45194	−0.0806630
$925$	−33.2143	−1.09208
$926$	−12.7128	−0.417770
$927$	3.32290	0.109138
$928$	0.171486	0.00562930
$929$	−16.7217	−0.548620	−0.274310	−	0.961641i	$-0.588449\pi$
$929$	−16.7217	−0.548620	−0.274310	+	0.961641i	$0.588449\pi$
$930$	9.01013	0.295454
$931$	6.89946	0.226121
$932$	−10.8113	−0.354138
$933$	1.22597	0.0401365
$934$	19.9065	0.651360
$935$	6.09770	0.199416
$936$	0	0
$937$	−34.4728	−1.12618	−0.563089	−	0.826396i	$-0.690387\pi$
$937$	−34.4728	−1.12618	−0.563089	+	0.826396i	$0.690387\pi$
$938$	4.34437	0.141849
$939$	−32.7500	−1.06875
$940$	−3.66406	−0.119508
$941$	−24.8489	−0.810050	−0.405025	−	0.914306i	$-0.632737\pi$
$941$	−24.8489	−0.810050	−0.405025	+	0.914306i	$0.632737\pi$
$942$	7.83666	0.255332
$943$	−10.7282	−0.349357
$944$	0.913907	0.0297451
$945$	0.670070	0.0217974
$946$	24.0180	0.780894
$947$	−30.4728	−0.990234	−0.495117	−	0.868826i	$-0.664875\pi$
$947$	−30.4728	−0.990234	−0.495117	+	0.868826i	$0.664875\pi$
$948$	−5.14187	−0.167000
$949$	0	0
$950$	−4.79119	−0.155447
$951$	−37.8991	−1.22896
$952$	1.02991	0.0333795
$953$	44.5883	1.44436	0.722179	−	0.691706i	$-0.243141\pi$
$953$	44.5883	1.44436	0.722179	+	0.691706i	$0.243141\pi$
$954$	3.42775	0.110977
$955$	−3.56282	−0.115290
$956$	25.4945	0.824550
$957$	−1.32610	−0.0428667
$958$	32.2579	1.04221
$959$	0.108748	0.00351166
$960$	0.860122	0.0277603
$961$	78.7342	2.53981
$962$	0	0
$963$	−8.75446	−0.282109
$964$	2.30322	0.0741818
$965$	10.6277	0.342118
$966$	−2.99739	−0.0964394
$967$	31.8537	1.02435	0.512173	−	0.858882i	$-0.328841\pi$
$967$	31.8537	1.02435	0.512173	+	0.858882i	$0.328841\pi$
$968$	−5.87788	−0.188922
$969$	6.11398	0.196409
$970$	8.11387	0.260521
$971$	13.0736	0.419553	0.209777	−	0.977749i	$-0.432726\pi$
$971$	13.0736	0.419553	0.209777	+	0.977749i	$0.432726\pi$
$972$	5.57797	0.178913
$973$	0.551299	0.0176739
$974$	35.1123	1.12507
$975$	0	0
$976$	10.2165	0.327023
$977$	−11.5857	−0.370658	−0.185329	−	0.982676i	$-0.559335\pi$
$977$	−11.5857	−0.370658	−0.185329	+	0.982676i	$0.559335\pi$
$978$	−6.07924	−0.194393
$979$	−24.3870	−0.779413
$980$	3.15273	0.100710
$981$	−6.77201	−0.216214
$982$	31.2908	0.998530
$983$	−58.0725	−1.85223	−0.926113	−	0.377246i	$-0.876871\pi$
$983$	−58.0725	−1.85223	−0.926113	+	0.377246i	$0.876871\pi$
$984$	−4.02089	−0.128181
$985$	−7.73208	−0.246364
$986$	0.557012	0.0177389
$987$	4.78566	0.152329
$988$	0	0
$989$	29.3610	0.933625
$990$	−1.01945	−0.0324004
$991$	−9.65703	−0.306766	−0.153383	−	0.988167i	$-0.549017\pi$
$991$	−9.65703	−0.306766	−0.153383	+	0.988167i	$0.549017\pi$
$992$	10.4754	0.332595
$993$	12.9090	0.409656
$994$	5.04877	0.160137
$995$	4.58608	0.145388
$996$	24.6476	0.780989
$997$	−12.7140	−0.402657	−0.201328	−	0.979524i	$-0.564526\pi$
$997$	−12.7140	−0.402657	−0.201328	+	0.979524i	$0.564526\pi$
$998$	−11.0689	−0.350381
$999$	32.0602	1.01434

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.ba.1.1		4
13.12	even	2		494.2.a.h.1.1	✓	4
39.38	odd	2		4446.2.a.bl.1.3		4
52.51	odd	2		3952.2.a.u.1.4		4
247.246	odd	2		9386.2.a.bf.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.a.h.1.1	✓	4	13.12	even	2
3952.2.a.u.1.4		4	52.51	odd	2
4446.2.a.bl.1.3		4	39.38	odd	2
6422.2.a.ba.1.1		4	1.1	even	1	trivial
9386.2.a.bf.1.4		4	247.246	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.88230 q^{3} +1.00000 q^{4} -0.456953 q^{5} +1.88230 q^{6} -0.317076 q^{7} -1.00000 q^{8} +0.543047 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.88230 q^{3} +1.00000 q^{4} -0.456953 q^{5} +1.88230 q^{6} -0.317076 q^{7} -1.00000 q^{8} +0.543047 q^{9} +0.456953 q^{10} -4.10827 q^{11} -1.88230 q^{12} +0.317076 q^{14} +0.860122 q^{15} +1.00000 q^{16} +3.24815 q^{17} -0.543047 q^{18} -1.00000 q^{19} -0.456953 q^{20} +0.596831 q^{21} +4.10827 q^{22} +5.02218 q^{23} +1.88230 q^{24} -4.79119 q^{25} +4.62472 q^{27} -0.317076 q^{28} -0.171486 q^{29} -0.860122 q^{30} -10.4754 q^{31} -1.00000 q^{32} +7.73299 q^{33} -3.24815 q^{34} +0.144889 q^{35} +0.543047 q^{36} +6.93236 q^{37} +1.00000 q^{38} +0.456953 q^{40} -2.13616 q^{41} -0.596831 q^{42} +5.84627 q^{43} -4.10827 q^{44} -0.248147 q^{45} -5.02218 q^{46} +8.01846 q^{47} -1.88230 q^{48} -6.89946 q^{49} +4.79119 q^{50} -6.11398 q^{51} -6.31206 q^{53} -4.62472 q^{54} +1.87729 q^{55} +0.317076 q^{56} +1.88230 q^{57} +0.171486 q^{58} +0.913907 q^{59} +0.860122 q^{60} +10.2165 q^{61} +10.4754 q^{62} -0.172187 q^{63} +1.00000 q^{64} -7.73299 q^{66} +13.7014 q^{67} +3.24815 q^{68} -9.45323 q^{69} -0.144889 q^{70} +15.9229 q^{71} -0.543047 q^{72} -6.85069 q^{73} -6.93236 q^{74} +9.01846 q^{75} -1.00000 q^{76} +1.30263 q^{77} +2.73170 q^{79} -0.456953 q^{80} -10.3342 q^{81} +2.13616 q^{82} -13.0944 q^{83} +0.596831 q^{84} -1.48425 q^{85} -5.84627 q^{86} +0.322788 q^{87} +4.10827 q^{88} +5.93608 q^{89} +0.248147 q^{90} +5.02218 q^{92} +19.7178 q^{93} -8.01846 q^{94} +0.456953 q^{95} +1.88230 q^{96} +17.7565 q^{97} +6.89946 q^{98} -2.23098 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 7 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 7 * q^7 - 4 * q^8 + 2 * q^9	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 7 q^{7} - 4 q^{8} + 2 q^{9} + 2 q^{10} - q^{11} + 2 q^{12} + 7 q^{14} + 9 q^{15} + 4 q^{16} - 8 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{20} - 3 q^{21} + q^{22} + 5 q^{23} - 2 q^{24} + 2 q^{25} + 5 q^{27} - 7 q^{28} - 5 q^{29} - 9 q^{30} - 9 q^{31} - 4 q^{32} + 2 q^{33} + 8 q^{34} + 7 q^{35} + 2 q^{36} - 5 q^{37} + 4 q^{38} + 2 q^{40} + 15 q^{41} + 3 q^{42} - 9 q^{43} - q^{44} + 20 q^{45} - 5 q^{46} - q^{47} + 2 q^{48} + 9 q^{49} - 2 q^{50} - 16 q^{51} - 19 q^{53} - 5 q^{54} - 14 q^{55} + 7 q^{56} - 2 q^{57} + 5 q^{58} + 4 q^{59} + 9 q^{60} + 10 q^{61} + 9 q^{62} + 4 q^{64} - 2 q^{66} + 16 q^{67} - 8 q^{68} - 20 q^{69} - 7 q^{70} + 6 q^{71} - 2 q^{72} - 8 q^{73} + 5 q^{74} + 3 q^{75} - 4 q^{76} - 26 q^{77} - 12 q^{79} - 2 q^{80} - 20 q^{81} - 15 q^{82} + q^{83} - 3 q^{84} + q^{85} + 9 q^{86} + 14 q^{87} + q^{88} + 9 q^{89} - 20 q^{90} + 5 q^{92} + 18 q^{93} + q^{94} + 2 q^{95} - 2 q^{96} + 21 q^{97} - 9 q^{98} - 15 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 7 * q^7 - 4 * q^8 + 2 * q^9 + 2 * q^10 - q^11 + 2 * q^12 + 7 * q^14 + 9 * q^15 + 4 * q^16 - 8 * q^17 - 2 * q^18 - 4 * q^19 - 2 * q^20 - 3 * q^21 + q^22 + 5 * q^23 - 2 * q^24 + 2 * q^25 + 5 * q^27 - 7 * q^28 - 5 * q^29 - 9 * q^30 - 9 * q^31 - 4 * q^32 + 2 * q^33 + 8 * q^34 + 7 * q^35 + 2 * q^36 - 5 * q^37 + 4 * q^38 + 2 * q^40 + 15 * q^41 + 3 * q^42 - 9 * q^43 - q^44 + 20 * q^45 - 5 * q^46 - q^47 + 2 * q^48 + 9 * q^49 - 2 * q^50 - 16 * q^51 - 19 * q^53 - 5 * q^54 - 14 * q^55 + 7 * q^56 - 2 * q^57 + 5 * q^58 + 4 * q^59 + 9 * q^60 + 10 * q^61 + 9 * q^62 + 4 * q^64 - 2 * q^66 + 16 * q^67 - 8 * q^68 - 20 * q^69 - 7 * q^70 + 6 * q^71 - 2 * q^72 - 8 * q^73 + 5 * q^74 + 3 * q^75 - 4 * q^76 - 26 * q^77 - 12 * q^79 - 2 * q^80 - 20 * q^81 - 15 * q^82 + q^83 - 3 * q^84 + q^85 + 9 * q^86 + 14 * q^87 + q^88 + 9 * q^89 - 20 * q^90 + 5 * q^92 + 18 * q^93 + q^94 + 2 * q^95 - 2 * q^96 + 21 * q^97 - 9 * q^98 - 15 * q^99

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