LMFDB - Embedded newform 309.2.a.d.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [309,2,Mod(1,309)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(309, 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("309.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 309 = 3 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	309.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:	$2.46737742246$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 13x^{6} + 11x^{5} + 52x^{4} - 35x^{3} - 59x^{2} + 27x + 1 $ x^8 - x^7 - 13x^6 + 11x^5 + 52x^4 - 35x^3 - 59x^2 + 27x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.27306$ of defining polynomial
Character	$\chi$	$=$	309.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.27306 q^{2} +1.00000 q^{3} -0.379326 q^{4} +1.25442 q^{5} +1.27306 q^{6} +2.55696 q^{7} -3.02902 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.27306 q^{2} +1.00000 q^{3} -0.379326 q^{4} +1.25442 q^{5} +1.27306 q^{6} +2.55696 q^{7} -3.02902 q^{8} +1.00000 q^{9} +1.59695 q^{10} -0.263412 q^{11} -0.379326 q^{12} +2.56998 q^{13} +3.25516 q^{14} +1.25442 q^{15} -3.09746 q^{16} +0.293552 q^{17} +1.27306 q^{18} -5.31376 q^{19} -0.475836 q^{20} +2.55696 q^{21} -0.335338 q^{22} +3.68477 q^{23} -3.02902 q^{24} -3.42642 q^{25} +3.27173 q^{26} +1.00000 q^{27} -0.969922 q^{28} -8.22506 q^{29} +1.59695 q^{30} +3.61056 q^{31} +2.11479 q^{32} -0.263412 q^{33} +0.373708 q^{34} +3.20752 q^{35} -0.379326 q^{36} +0.269739 q^{37} -6.76472 q^{38} +2.56998 q^{39} -3.79968 q^{40} -10.3358 q^{41} +3.25516 q^{42} +3.52503 q^{43} +0.0999190 q^{44} +1.25442 q^{45} +4.69092 q^{46} -11.4483 q^{47} -3.09746 q^{48} -0.461936 q^{49} -4.36203 q^{50} +0.293552 q^{51} -0.974859 q^{52} +4.35871 q^{53} +1.27306 q^{54} -0.330431 q^{55} -7.74509 q^{56} -5.31376 q^{57} -10.4710 q^{58} -9.99270 q^{59} -0.475836 q^{60} +12.7035 q^{61} +4.59645 q^{62} +2.55696 q^{63} +8.88717 q^{64} +3.22384 q^{65} -0.335338 q^{66} +3.51633 q^{67} -0.111352 q^{68} +3.68477 q^{69} +4.08335 q^{70} -0.0409897 q^{71} -3.02902 q^{72} -0.813252 q^{73} +0.343393 q^{74} -3.42642 q^{75} +2.01565 q^{76} -0.673535 q^{77} +3.27173 q^{78} +0.783502 q^{79} -3.88553 q^{80} +1.00000 q^{81} -13.1581 q^{82} +8.45280 q^{83} -0.969922 q^{84} +0.368239 q^{85} +4.48756 q^{86} -8.22506 q^{87} +0.797879 q^{88} -2.01361 q^{89} +1.59695 q^{90} +6.57134 q^{91} -1.39773 q^{92} +3.61056 q^{93} -14.5744 q^{94} -6.66571 q^{95} +2.11479 q^{96} +11.7859 q^{97} -0.588071 q^{98} -0.263412 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 8 q^{3} + 11 q^{4} - q^{5} - q^{6} + 6 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - q^2 + 8 * q^3 + 11 * q^4 - q^5 - q^6 + 6 * q^7 - 3 * q^8 + 8 * q^9	\( 8 q - q^{2} + 8 q^{3} + 11 q^{4} - q^{5} - q^{6} + 6 q^{7} - 3 q^{8} + 8 q^{9} + 3 q^{10} + 6 q^{11} + 11 q^{12} + 9 q^{13} - 6 q^{14} - q^{15} + 9 q^{16} - 4 q^{17} - q^{18} + 16 q^{19} - 23 q^{20} + 6 q^{21} - 4 q^{22} - 11 q^{23} - 3 q^{24} + 15 q^{25} - 14 q^{26} + 8 q^{27} - 5 q^{28} + 3 q^{30} + 17 q^{31} - 12 q^{32} + 6 q^{33} - 8 q^{34} + 4 q^{35} + 11 q^{36} - 6 q^{37} - 3 q^{38} + 9 q^{39} + 3 q^{40} + 12 q^{41} - 6 q^{42} + 9 q^{43} + 8 q^{44} - q^{45} - 30 q^{46} - 6 q^{47} + 9 q^{48} + 18 q^{49} - 36 q^{50} - 4 q^{51} + 23 q^{52} - 16 q^{53} - q^{54} - 10 q^{55} - 13 q^{56} + 16 q^{57} - 22 q^{58} + 11 q^{59} - 23 q^{60} + 5 q^{61} - 25 q^{62} + 6 q^{63} - 35 q^{64} - 41 q^{65} - 4 q^{66} + 5 q^{67} - 19 q^{68} - 11 q^{69} - 48 q^{70} - 10 q^{71} - 3 q^{72} + 14 q^{73} + 4 q^{74} + 15 q^{75} - 12 q^{76} - 40 q^{77} - 14 q^{78} + 14 q^{79} - 19 q^{80} + 8 q^{81} - 13 q^{82} - 23 q^{83} - 5 q^{84} - 4 q^{85} - 3 q^{86} - 30 q^{88} - 14 q^{89} + 3 q^{90} - 6 q^{91} - 21 q^{92} + 17 q^{93} + 22 q^{94} + 6 q^{95} - 12 q^{96} + 3 q^{97} - 18 q^{98} + 6 q^{99}+O(q^{100}) \) 8 * q - q^2 + 8 * q^3 + 11 * q^4 - q^5 - q^6 + 6 * q^7 - 3 * q^8 + 8 * q^9 + 3 * q^10 + 6 * q^11 + 11 * q^12 + 9 * q^13 - 6 * q^14 - q^15 + 9 * q^16 - 4 * q^17 - q^18 + 16 * q^19 - 23 * q^20 + 6 * q^21 - 4 * q^22 - 11 * q^23 - 3 * q^24 + 15 * q^25 - 14 * q^26 + 8 * q^27 - 5 * q^28 + 3 * q^30 + 17 * q^31 - 12 * q^32 + 6 * q^33 - 8 * q^34 + 4 * q^35 + 11 * q^36 - 6 * q^37 - 3 * q^38 + 9 * q^39 + 3 * q^40 + 12 * q^41 - 6 * q^42 + 9 * q^43 + 8 * q^44 - q^45 - 30 * q^46 - 6 * q^47 + 9 * q^48 + 18 * q^49 - 36 * q^50 - 4 * q^51 + 23 * q^52 - 16 * q^53 - q^54 - 10 * q^55 - 13 * q^56 + 16 * q^57 - 22 * q^58 + 11 * q^59 - 23 * q^60 + 5 * q^61 - 25 * q^62 + 6 * q^63 - 35 * q^64 - 41 * q^65 - 4 * q^66 + 5 * q^67 - 19 * q^68 - 11 * q^69 - 48 * q^70 - 10 * q^71 - 3 * q^72 + 14 * q^73 + 4 * q^74 + 15 * q^75 - 12 * q^76 - 40 * q^77 - 14 * q^78 + 14 * q^79 - 19 * q^80 + 8 * q^81 - 13 * q^82 - 23 * q^83 - 5 * q^84 - 4 * q^85 - 3 * q^86 - 30 * q^88 - 14 * q^89 + 3 * q^90 - 6 * q^91 - 21 * q^92 + 17 * q^93 + 22 * q^94 + 6 * q^95 - 12 * q^96 + 3 * q^97 - 18 * q^98 + 6 * 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 $

$2$	1.27306	0.900187	0.450094	−	0.892981i	$-0.351391\pi$
$2$	1.27306	0.900187	0.450094	+	0.892981i	$0.351391\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.379326	−0.189663
$5$	1.25442	0.560996	0.280498	−	0.959855i	$-0.409500\pi$
$5$	1.25442	0.560996	0.280498	+	0.959855i	$0.409500\pi$
$6$	1.27306	0.519723
$7$	2.55696	0.966441	0.483221	−	0.875499i	$-0.339467\pi$
$7$	2.55696	0.966441	0.483221	+	0.875499i	$0.339467\pi$
$8$	−3.02902	−1.07092
$9$	1.00000	0.333333
$10$	1.59695	0.505001
$11$	−0.263412	−0.0794217	−0.0397108	−	0.999211i	$-0.512644\pi$
$11$	−0.263412	−0.0794217	−0.0397108	+	0.999211i	$0.512644\pi$
$12$	−0.379326	−0.109502
$13$	2.56998	0.712784	0.356392	−	0.934337i	$-0.384007\pi$
$13$	2.56998	0.712784	0.356392	+	0.934337i	$0.384007\pi$
$14$	3.25516	0.869978
$15$	1.25442	0.323891
$16$	−3.09746	−0.774365
$17$	0.293552	0.0711968	0.0355984	−	0.999366i	$-0.488666\pi$
$17$	0.293552	0.0711968	0.0355984	+	0.999366i	$0.488666\pi$
$18$	1.27306	0.300062
$19$	−5.31376	−1.21906	−0.609530	−	0.792763i	$-0.708642\pi$
$19$	−5.31376	−1.21906	−0.609530	+	0.792763i	$0.708642\pi$
$20$	−0.475836	−0.106400
$21$	2.55696	0.557975
$22$	−0.335338	−0.0714944
$23$	3.68477	0.768328	0.384164	−	0.923265i	$-0.374490\pi$
$23$	3.68477	0.768328	0.384164	+	0.923265i	$0.374490\pi$
$24$	−3.02902	−0.618296
$25$	−3.42642	−0.685284
$26$	3.27173	0.641639
$27$	1.00000	0.192450
$28$	−0.969922	−0.183298
$29$	−8.22506	−1.52736	−0.763678	−	0.645598i	$-0.776608\pi$
$29$	−8.22506	−1.52736	−0.763678	+	0.645598i	$0.776608\pi$
$30$	1.59695	0.291563
$31$	3.61056	0.648477	0.324238	−	0.945975i	$-0.394892\pi$
$31$	3.61056	0.648477	0.324238	+	0.945975i	$0.394892\pi$
$32$	2.11479	0.373846
$33$	−0.263412	−0.0458541
$34$	0.373708	0.0640904
$35$	3.20752	0.542170
$36$	−0.379326	−0.0632210
$37$	0.269739	0.0443449	0.0221724	−	0.999754i	$-0.492942\pi$
$37$	0.269739	0.0443449	0.0221724	+	0.999754i	$0.492942\pi$
$38$	−6.76472	−1.09738
$39$	2.56998	0.411526
$40$	−3.79968	−0.600781
$41$	−10.3358	−1.61418	−0.807092	−	0.590425i	$-0.798960\pi$
$41$	−10.3358	−1.61418	−0.807092	+	0.590425i	$0.798960\pi$
$42$	3.25516	0.502282
$43$	3.52503	0.537562	0.268781	−	0.963201i	$-0.413379\pi$
$43$	3.52503	0.537562	0.268781	+	0.963201i	$0.413379\pi$
$44$	0.0999190	0.0150633
$45$	1.25442	0.186999
$46$	4.69092	0.691639
$47$	−11.4483	−1.66991	−0.834956	−	0.550317i	$-0.814507\pi$
$47$	−11.4483	−1.66991	−0.834956	+	0.550317i	$0.814507\pi$
$48$	−3.09746	−0.447080
$49$	−0.461936	−0.0659909
$50$	−4.36203	−0.616884
$51$	0.293552	0.0411055
$52$	−0.974859	−0.135189
$53$	4.35871	0.598715	0.299357	−	0.954141i	$-0.403228\pi$
$53$	4.35871	0.598715	0.299357	+	0.954141i	$0.403228\pi$
$54$	1.27306	0.173241
$55$	−0.330431	−0.0445552
$56$	−7.74509	−1.03498
$57$	−5.31376	−0.703825
$58$	−10.4710	−1.37491
$59$	−9.99270	−1.30094	−0.650469	−	0.759533i	$-0.725428\pi$
$59$	−9.99270	−1.30094	−0.650469	+	0.759533i	$0.725428\pi$
$60$	−0.475836	−0.0614301
$61$	12.7035	1.62651	0.813256	−	0.581906i	$-0.197693\pi$
$61$	12.7035	1.62651	0.813256	+	0.581906i	$0.197693\pi$
$62$	4.59645	0.583750
$63$	2.55696	0.322147
$64$	8.88717	1.11090
$65$	3.22384	0.399869
$66$	−0.335338	−0.0412773
$67$	3.51633	0.429588	0.214794	−	0.976659i	$-0.431092\pi$
$67$	3.51633	0.429588	0.214794	+	0.976659i	$0.431092\pi$
$68$	−0.111352	−0.0135034
$69$	3.68477	0.443594
$70$	4.08335	0.488054
$71$	−0.0409897	−0.00486458	−0.00243229	−	0.999997i	$-0.500774\pi$
$71$	−0.0409897	−0.00486458	−0.00243229	+	0.999997i	$0.500774\pi$
$72$	−3.02902	−0.356973
$73$	−0.813252	−0.0951839	−0.0475919	−	0.998867i	$-0.515155\pi$
$73$	−0.813252	−0.0951839	−0.0475919	+	0.998867i	$0.515155\pi$
$74$	0.343393	0.0399187
$75$	−3.42642	−0.395649
$76$	2.01565	0.231211
$77$	−0.673535	−0.0767564
$78$	3.27173	0.370450
$79$	0.783502	0.0881508	0.0440754	−	0.999028i	$-0.485966\pi$
$79$	0.783502	0.0881508	0.0440754	+	0.999028i	$0.485966\pi$
$80$	−3.88553	−0.434416
$81$	1.00000	0.111111
$82$	−13.1581	−1.45307
$83$	8.45280	0.927816	0.463908	−	0.885884i	$-0.346447\pi$
$83$	8.45280	0.927816	0.463908	+	0.885884i	$0.346447\pi$
$84$	−0.969922	−0.105827
$85$	0.368239	0.0399411
$86$	4.48756	0.483906
$87$	−8.22506	−0.881819
$88$	0.797879	0.0850542
$89$	−2.01361	−0.213442	−0.106721	−	0.994289i	$-0.534035\pi$
$89$	−2.01361	−0.213442	−0.106721	+	0.994289i	$0.534035\pi$
$90$	1.59695	0.168334
$91$	6.57134	0.688864
$92$	−1.39773	−0.145723
$93$	3.61056	0.374398
$94$	−14.5744	−1.50323
$95$	−6.66571	−0.683888
$96$	2.11479	0.215840
$97$	11.7859	1.19668	0.598340	−	0.801242i	$-0.295827\pi$
$97$	11.7859	1.19668	0.598340	+	0.801242i	$0.295827\pi$
$98$	−0.588071	−0.0594042
$99$	−0.263412	−0.0264739
$100$	1.29973	0.129973
$101$	2.74529	0.273167	0.136583	−	0.990629i	$-0.456388\pi$
$101$	2.74529	0.273167	0.136583	+	0.990629i	$0.456388\pi$
$102$	0.373708	0.0370026
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−7.78451	−0.763334
$105$	3.20752	0.313022
$106$	5.54889	0.538956
$107$	8.98639	0.868747	0.434374	−	0.900733i	$-0.356970\pi$
$107$	8.98639	0.868747	0.434374	+	0.900733i	$0.356970\pi$
$108$	−0.379326	−0.0365006
$109$	−8.83401	−0.846145	−0.423073	−	0.906096i	$-0.639048\pi$
$109$	−8.83401	−0.846145	−0.423073	+	0.906096i	$0.639048\pi$
$110$	−0.420657	−0.0401081
$111$	0.269739	0.0256025
$112$	−7.92009	−0.748379
$113$	−4.95002	−0.465659	−0.232830	−	0.972518i	$-0.574798\pi$
$113$	−4.95002	−0.465659	−0.232830	+	0.972518i	$0.574798\pi$
$114$	−6.76472	−0.633574
$115$	4.62227	0.431029
$116$	3.11998	0.289683
$117$	2.56998	0.237595
$118$	−12.7213	−1.17109
$119$	0.750601	0.0688075
$120$	−3.79968	−0.346861
$121$	−10.9306	−0.993692
$122$	16.1722	1.46417
$123$	−10.3358	−0.931950
$124$	−1.36958	−0.122992
$125$	−10.5703	−0.945437
$126$	3.25516	0.289993
$127$	20.3717	1.80770	0.903849	−	0.427852i	$-0.140729\pi$
$127$	20.3717	1.80770	0.903849	+	0.427852i	$0.140729\pi$
$128$	7.08429	0.626169
$129$	3.52503	0.310361
$130$	4.10414	0.359957
$131$	2.06488	0.180410	0.0902049	−	0.995923i	$-0.471248\pi$
$131$	2.06488	0.180410	0.0902049	+	0.995923i	$0.471248\pi$
$132$	0.0999190	0.00869683
$133$	−13.5871	−1.17815
$134$	4.47648	0.386709
$135$	1.25442	0.107964
$136$	−0.889174	−0.0762460
$137$	2.22059	0.189718	0.0948588	−	0.995491i	$-0.469760\pi$
$137$	2.22059	0.189718	0.0948588	+	0.995491i	$0.469760\pi$
$138$	4.69092	0.399318
$139$	12.1026	1.02653	0.513264	−	0.858231i	$-0.328436\pi$
$139$	12.1026	1.02653	0.513264	+	0.858231i	$0.328436\pi$
$140$	−1.21669	−0.102829
$141$	−11.4483	−0.964124
$142$	−0.0521822	−0.00437903
$143$	−0.676963	−0.0566105
$144$	−3.09746	−0.258122
$145$	−10.3177	−0.856840
$146$	−1.03532	−0.0856833
$147$	−0.461936	−0.0380999
$148$	−0.102319	−0.00841057
$149$	−3.39846	−0.278413	−0.139207	−	0.990263i	$-0.544455\pi$
$149$	−3.39846	−0.278413	−0.139207	+	0.990263i	$0.544455\pi$
$150$	−4.36203	−0.356158
$151$	23.7639	1.93388	0.966941	−	0.255002i	$-0.0820760\pi$
$151$	23.7639	1.93388	0.966941	+	0.255002i	$0.0820760\pi$
$152$	16.0955	1.30552
$153$	0.293552	0.0237323
$154$	−0.857448	−0.0690951
$155$	4.52918	0.363793
$156$	−0.974859	−0.0780512
$157$	19.3703	1.54592	0.772960	−	0.634455i	$-0.218775\pi$
$157$	19.3703	1.54592	0.772960	+	0.634455i	$0.218775\pi$
$158$	0.997443	0.0793523
$159$	4.35871	0.345668
$160$	2.65285	0.209726
$161$	9.42182	0.742544
$162$	1.27306	0.100021
$163$	6.13272	0.480352	0.240176	−	0.970729i	$-0.422795\pi$
$163$	6.13272	0.480352	0.240176	+	0.970729i	$0.422795\pi$
$164$	3.92065	0.306151
$165$	−0.330431	−0.0257240
$166$	10.7609	0.835208
$167$	4.77970	0.369865	0.184932	−	0.982751i	$-0.440793\pi$
$167$	4.77970	0.369865	0.184932	+	0.982751i	$0.440793\pi$
$168$	−7.74509	−0.597547
$169$	−6.39521	−0.491939
$170$	0.468789	0.0359545
$171$	−5.31376	−0.406353
$172$	−1.33713	−0.101956
$173$	−2.71132	−0.206138	−0.103069	−	0.994674i	$-0.532866\pi$
$173$	−2.71132	−0.206138	−0.103069	+	0.994674i	$0.532866\pi$
$174$	−10.4710	−0.793802
$175$	−8.76123	−0.662286
$176$	0.815908	0.0615014
$177$	−9.99270	−0.751097
$178$	−2.56344	−0.192138
$179$	6.79008	0.507514	0.253757	−	0.967268i	$-0.418334\pi$
$179$	6.79008	0.507514	0.253757	+	0.967268i	$0.418334\pi$
$180$	−0.475836	−0.0354667
$181$	−3.15097	−0.234209	−0.117105	−	0.993120i	$-0.537361\pi$
$181$	−3.15097	−0.234209	−0.117105	+	0.993120i	$0.537361\pi$
$182$	8.36569	0.620106
$183$	12.7035	0.939067
$184$	−11.1612	−0.822817
$185$	0.338368	0.0248773
$186$	4.59645	0.337028
$187$	−0.0773251	−0.00565457
$188$	4.34265	0.316720
$189$	2.55696	0.185992
$190$	−8.48583	−0.615627
$191$	−9.68027	−0.700440	−0.350220	−	0.936668i	$-0.613893\pi$
$191$	−9.68027	−0.700440	−0.350220	+	0.936668i	$0.613893\pi$
$192$	8.88717	0.641376
$193$	7.90112	0.568735	0.284367	−	0.958715i	$-0.408216\pi$
$193$	7.90112	0.568735	0.284367	+	0.958715i	$0.408216\pi$
$194$	15.0042	1.07724
$195$	3.22384	0.230864
$196$	0.175224	0.0125160
$197$	−6.63176	−0.472494	−0.236247	−	0.971693i	$-0.575917\pi$
$197$	−6.63176	−0.472494	−0.236247	+	0.971693i	$0.575917\pi$
$198$	−0.335338	−0.0238315
$199$	−16.3709	−1.16050	−0.580251	−	0.814438i	$-0.697045\pi$
$199$	−16.3709	−1.16050	−0.580251	+	0.814438i	$0.697045\pi$
$200$	10.3787	0.733883
$201$	3.51633	0.248022
$202$	3.49492	0.245901
$203$	−21.0312	−1.47610
$204$	−0.111352	−0.00779619
$205$	−12.9655	−0.905551
$206$	−1.27306	−0.0886981
$207$	3.68477	0.256109
$208$	−7.96041	−0.551955
$209$	1.39971	0.0968198
$210$	4.08335	0.281778
$211$	−6.73471	−0.463636	−0.231818	−	0.972759i	$-0.574467\pi$
$211$	−6.73471	−0.463636	−0.231818	+	0.972759i	$0.574467\pi$
$212$	−1.65337	−0.113554
$213$	−0.0409897	−0.00280857
$214$	11.4402	0.782035
$215$	4.42188	0.301570
$216$	−3.02902	−0.206099
$217$	9.23208	0.626715
$218$	−11.2462	−0.761689
$219$	−0.813252	−0.0549544
$220$	0.125341	0.00845048
$221$	0.754422	0.0507479
$222$	0.343393	0.0230471
$223$	9.87916	0.661558	0.330779	−	0.943708i	$-0.392689\pi$
$223$	9.87916	0.661558	0.330779	+	0.943708i	$0.392689\pi$
$224$	5.40745	0.361300
$225$	−3.42642	−0.228428
$226$	−6.30166	−0.419180
$227$	26.4553	1.75590	0.877951	−	0.478751i	$-0.158910\pi$
$227$	26.4553	1.75590	0.877951	+	0.478751i	$0.158910\pi$
$228$	2.01565	0.133489
$229$	−6.59664	−0.435919	−0.217959	−	0.975958i	$-0.569940\pi$
$229$	−6.59664	−0.435919	−0.217959	+	0.975958i	$0.569940\pi$
$230$	5.88441	0.388006
$231$	−0.673535	−0.0443153
$232$	24.9139	1.63567
$233$	−2.80917	−0.184035	−0.0920174	−	0.995757i	$-0.529332\pi$
$233$	−2.80917	−0.184035	−0.0920174	+	0.995757i	$0.529332\pi$
$234$	3.27173	0.213880
$235$	−14.3611	−0.936814
$236$	3.79049	0.246740
$237$	0.783502	0.0508939
$238$	0.955558	0.0619397
$239$	5.51054	0.356447	0.178223	−	0.983990i	$-0.442965\pi$
$239$	5.51054	0.356447	0.178223	+	0.983990i	$0.442965\pi$
$240$	−3.88553	−0.250810
$241$	−3.17113	−0.204270	−0.102135	−	0.994771i	$-0.532567\pi$
$241$	−3.17113	−0.204270	−0.102135	+	0.994771i	$0.532567\pi$
$242$	−13.9153	−0.894509
$243$	1.00000	0.0641500
$244$	−4.81875	−0.308489
$245$	−0.579464	−0.0370206
$246$	−13.1581	−0.838929
$247$	−13.6562	−0.868926
$248$	−10.9365	−0.694466
$249$	8.45280	0.535675
$250$	−13.4566	−0.851071
$251$	30.2544	1.90964	0.954820	−	0.297186i	$-0.0960482\pi$
$251$	30.2544	1.90964	0.954820	+	0.297186i	$0.0960482\pi$
$252$	−0.969922	−0.0610994
$253$	−0.970612	−0.0610219
$254$	25.9344	1.62727
$255$	0.368239	0.0230600
$256$	−8.75563	−0.547227
$257$	−4.05791	−0.253125	−0.126563	−	0.991959i	$-0.540394\pi$
$257$	−4.05791	−0.253125	−0.126563	+	0.991959i	$0.540394\pi$
$258$	4.48756	0.279383
$259$	0.689713	0.0428567
$260$	−1.22289	−0.0758403
$261$	−8.22506	−0.509118
$262$	2.62872	0.162403
$263$	6.35259	0.391717	0.195859	−	0.980632i	$-0.437251\pi$
$263$	6.35259	0.391717	0.195859	+	0.980632i	$0.437251\pi$
$264$	0.797879	0.0491061
$265$	5.46768	0.335877
$266$	−17.2971	−1.06056
$267$	−2.01361	−0.123231
$268$	−1.33383	−0.0814768
$269$	−1.98537	−0.121050	−0.0605250	−	0.998167i	$-0.519277\pi$
$269$	−1.98537	−0.121050	−0.0605250	+	0.998167i	$0.519277\pi$
$270$	1.59695	0.0971876
$271$	−25.9354	−1.57546	−0.787731	−	0.616019i	$-0.788744\pi$
$271$	−25.9354	−1.57546	−0.787731	+	0.616019i	$0.788744\pi$
$272$	−0.909265	−0.0551323
$273$	6.57134	0.397716
$274$	2.82693	0.170781
$275$	0.902559	0.0544264
$276$	−1.39773	−0.0841334
$277$	−27.8754	−1.67487	−0.837437	−	0.546535i	$-0.815947\pi$
$277$	−27.8754	−1.67487	−0.837437	+	0.546535i	$0.815947\pi$
$278$	15.4073	0.924067
$279$	3.61056	0.216159
$280$	−9.71563	−0.580620
$281$	−9.95610	−0.593931	−0.296966	−	0.954888i	$-0.595975\pi$
$281$	−9.95610	−0.593931	−0.296966	+	0.954888i	$0.595975\pi$
$282$	−14.5744	−0.867892
$283$	−17.5111	−1.04093	−0.520463	−	0.853884i	$-0.674241\pi$
$283$	−17.5111	−1.04093	−0.520463	+	0.853884i	$0.674241\pi$
$284$	0.0155484	0.000922631 0
$285$	−6.66571	−0.394843
$286$	−0.861812	−0.0509600
$287$	−26.4283	−1.56001
$288$	2.11479	0.124615
$289$	−16.9138	−0.994931
$290$	−13.1350	−0.771317
$291$	11.7859	0.690904
$292$	0.308487	0.0180529
$293$	−5.46064	−0.319014	−0.159507	−	0.987197i	$-0.550990\pi$
$293$	−5.46064	−0.319014	−0.159507	+	0.987197i	$0.550990\pi$
$294$	−0.588071	−0.0342970
$295$	−12.5351	−0.729821
$296$	−0.817045	−0.0474898
$297$	−0.263412	−0.0152847
$298$	−4.32644	−0.250624
$299$	9.46978	0.547651
$300$	1.29973	0.0750399
$301$	9.01337	0.519522
$302$	30.2528	1.74086
$303$	2.74529	0.157713
$304$	16.4592	0.943998
$305$	15.9355	0.912467
$306$	0.373708	0.0213635
$307$	−16.8318	−0.960641	−0.480321	−	0.877093i	$-0.659480\pi$
$307$	−16.8318	−0.960641	−0.480321	+	0.877093i	$0.659480\pi$
$308$	0.255489	0.0145578
$309$	−1.00000	−0.0568880
$310$	5.76591	0.327482
$311$	−14.9534	−0.847932	−0.423966	−	0.905678i	$-0.639362\pi$
$311$	−14.9534	−0.847932	−0.423966	+	0.905678i	$0.639362\pi$
$312$	−7.78451	−0.440711
$313$	6.59657	0.372860	0.186430	−	0.982468i	$-0.440308\pi$
$313$	6.59657	0.372860	0.186430	+	0.982468i	$0.440308\pi$
$314$	24.6595	1.39162
$315$	3.20752	0.180723
$316$	−0.297202	−0.0167189
$317$	−34.8832	−1.95924	−0.979619	−	0.200865i	$-0.935625\pi$
$317$	−34.8832	−1.95924	−0.979619	+	0.200865i	$0.935625\pi$
$318$	5.54889	0.311166
$319$	2.16658	0.121305
$320$	11.1483	0.623208
$321$	8.98639	0.501572
$322$	11.9945	0.668428
$323$	−1.55986	−0.0867932
$324$	−0.379326	−0.0210737
$325$	−8.80582	−0.488459
$326$	7.80730	0.432407
$327$	−8.83401	−0.488522
$328$	31.3074	1.72866
$329$	−29.2730	−1.61387
$330$	−0.420657	−0.0231564
$331$	29.0160	1.59487	0.797433	−	0.603408i	$-0.206191\pi$
$331$	29.0160	1.59487	0.797433	+	0.603408i	$0.206191\pi$
$332$	−3.20637	−0.175972
$333$	0.269739	0.0147816
$334$	6.08484	0.332947
$335$	4.41097	0.240997
$336$	−7.92009	−0.432077
$337$	17.6065	0.959086	0.479543	−	0.877518i	$-0.340802\pi$
$337$	17.6065	0.959086	0.479543	+	0.877518i	$0.340802\pi$
$338$	−8.14147	−0.442838
$339$	−4.95002	−0.268848
$340$	−0.139682	−0.00757535
$341$	−0.951066	−0.0515031
$342$	−6.76472	−0.365794
$343$	−19.0799	−1.03022
$344$	−10.6774	−0.575685
$345$	4.62227	0.248854
$346$	−3.45167	−0.185563
$347$	−12.1889	−0.654334	−0.327167	−	0.944966i	$-0.606094\pi$
$347$	−12.1889	−0.654334	−0.327167	+	0.944966i	$0.606094\pi$
$348$	3.11998	0.167248
$349$	−30.8182	−1.64966	−0.824831	−	0.565379i	$-0.808730\pi$
$349$	−30.8182	−1.64966	−0.824831	+	0.565379i	$0.808730\pi$
$350$	−11.1535	−0.596182
$351$	2.56998	0.137175
$352$	−0.557061	−0.0296915
$353$	−13.3102	−0.708428	−0.354214	−	0.935164i	$-0.615252\pi$
$353$	−13.3102	−0.708428	−0.354214	+	0.935164i	$0.615252\pi$
$354$	−12.7213	−0.676128
$355$	−0.0514185	−0.00272901
$356$	0.763814	0.0404821
$357$	0.750601	0.0397260
$358$	8.64416	0.456858
$359$	−7.37338	−0.389152	−0.194576	−	0.980887i	$-0.562333\pi$
$359$	−7.37338	−0.389152	−0.194576	+	0.980887i	$0.562333\pi$
$360$	−3.79968	−0.200260
$361$	9.23605	0.486108
$362$	−4.01136	−0.210832
$363$	−10.9306	−0.573708
$364$	−2.49268	−0.130652
$365$	−1.02016	−0.0533978
$366$	16.1722	0.845336
$367$	11.4010	0.595126	0.297563	−	0.954702i	$-0.403826\pi$
$367$	11.4010	0.595126	0.297563	+	0.954702i	$0.403826\pi$
$368$	−11.4134	−0.594966
$369$	−10.3358	−0.538062
$370$	0.430761	0.0223942
$371$	11.1451	0.578623
$372$	−1.36958	−0.0710094
$373$	−27.7872	−1.43876	−0.719382	−	0.694614i	$-0.755575\pi$
$373$	−27.7872	−1.43876	−0.719382	+	0.694614i	$0.755575\pi$
$374$	−0.0984392	−0.00509017
$375$	−10.5703	−0.545848
$376$	34.6772	1.78834
$377$	−21.1382	−1.08867
$378$	3.25516	0.167427
$379$	3.71150	0.190647	0.0953235	−	0.995446i	$-0.469611\pi$
$379$	3.71150	0.190647	0.0953235	+	0.995446i	$0.469611\pi$
$380$	2.52848	0.129708
$381$	20.3717	1.04367
$382$	−12.3235	−0.630527
$383$	8.15093	0.416493	0.208247	−	0.978076i	$-0.433224\pi$
$383$	8.15093	0.416493	0.208247	+	0.978076i	$0.433224\pi$
$384$	7.08429	0.361519
$385$	−0.844899	−0.0430600
$386$	10.0586	0.511968
$387$	3.52503	0.179187
$388$	−4.47071	−0.226966
$389$	13.9731	0.708465	0.354232	−	0.935157i	$-0.384742\pi$
$389$	13.9731	0.708465	0.354232	+	0.935157i	$0.384742\pi$
$390$	4.10414	0.207821
$391$	1.08167	0.0547025
$392$	1.39921	0.0706709
$393$	2.06488	0.104160
$394$	−8.44261	−0.425333
$395$	0.982844	0.0494523
$396$	0.0999190	0.00502112
$397$	36.7059	1.84222	0.921109	−	0.389305i	$-0.127285\pi$
$397$	36.7059	1.84222	0.921109	+	0.389305i	$0.127285\pi$
$398$	−20.8411	−1.04467
$399$	−13.5871	−0.680205
$400$	10.6132	0.530660
$401$	−36.1907	−1.80728	−0.903638	−	0.428298i	$-0.859113\pi$
$401$	−36.1907	−1.80728	−0.903638	+	0.428298i	$0.859113\pi$
$402$	4.47648	0.223267
$403$	9.27907	0.462224
$404$	−1.04136	−0.0518096
$405$	1.25442	0.0623329
$406$	−26.7739	−1.32877
$407$	−0.0710525	−0.00352194
$408$	−0.889174	−0.0440207
$409$	−1.40554	−0.0694993	−0.0347496	−	0.999396i	$-0.511063\pi$
$409$	−1.40554	−0.0694993	−0.0347496	+	0.999396i	$0.511063\pi$
$410$	−16.5058	−0.815165
$411$	2.22059	0.109534
$412$	0.379326	0.0186880
$413$	−25.5510	−1.25728
$414$	4.69092	0.230546
$415$	10.6034	0.520501
$416$	5.43497	0.266471
$417$	12.1026	0.592666
$418$	1.78191	0.0871560
$419$	24.8083	1.21196	0.605982	−	0.795478i	$-0.292780\pi$
$419$	24.8083	1.21196	0.605982	+	0.795478i	$0.292780\pi$
$420$	−1.21669	−0.0593686
$421$	26.9710	1.31449	0.657243	−	0.753679i	$-0.271723\pi$
$421$	26.9710	1.31449	0.657243	+	0.753679i	$0.271723\pi$
$422$	−8.57367	−0.417360
$423$	−11.4483	−0.556637
$424$	−13.2026	−0.641175
$425$	−1.00583	−0.0487900
$426$	−0.0521822	−0.00252824
$427$	32.4823	1.57193
$428$	−3.40877	−0.164769
$429$	−0.676963	−0.0326841
$430$	5.62931	0.271469
$431$	19.8881	0.957974	0.478987	−	0.877822i	$-0.341004\pi$
$431$	19.8881	0.957974	0.478987	+	0.877822i	$0.341004\pi$
$432$	−3.09746	−0.149027
$433$	18.6204	0.894837	0.447419	−	0.894325i	$-0.352343\pi$
$433$	18.6204	0.894837	0.447419	+	0.894325i	$0.352343\pi$
$434$	11.7530	0.564161
$435$	−10.3177	−0.494697
$436$	3.35097	0.160482
$437$	−19.5800	−0.936638
$438$	−1.03532	−0.0494693
$439$	28.3051	1.35093	0.675465	−	0.737392i	$-0.263943\pi$
$439$	28.3051	1.35093	0.675465	+	0.737392i	$0.263943\pi$
$440$	1.00088	0.0477151
$441$	−0.461936	−0.0219970
$442$	0.960422	0.0456826
$443$	−14.8788	−0.706913	−0.353456	−	0.935451i	$-0.614994\pi$
$443$	−14.8788	−0.706913	−0.353456	+	0.935451i	$0.614994\pi$
$444$	−0.102319	−0.00485585
$445$	−2.52592	−0.119740
$446$	12.5767	0.595526
$447$	−3.39846	−0.160742
$448$	22.7242	1.07362
$449$	−14.7581	−0.696477	−0.348238	−	0.937406i	$-0.613220\pi$
$449$	−14.7581	−0.696477	−0.348238	+	0.937406i	$0.613220\pi$
$450$	−4.36203	−0.205628
$451$	2.72258	0.128201
$452$	1.87767	0.0883183
$453$	23.7639	1.11653
$454$	33.6791	1.58064
$455$	8.24325	0.386450
$456$	16.0955	0.753740
$457$	6.00491	0.280898	0.140449	−	0.990088i	$-0.455145\pi$
$457$	6.00491	0.280898	0.140449	+	0.990088i	$0.455145\pi$
$458$	−8.39790	−0.392408
$459$	0.293552	0.0137018
$460$	−1.75335	−0.0817501
$461$	40.1228	1.86871	0.934353	−	0.356348i	$-0.115978\pi$
$461$	40.1228	1.86871	0.934353	+	0.356348i	$0.115978\pi$
$462$	−0.857448	−0.0398921
$463$	−35.2182	−1.63673	−0.818365	−	0.574699i	$-0.805119\pi$
$463$	−35.2182	−1.63673	−0.818365	+	0.574699i	$0.805119\pi$
$464$	25.4768	1.18273
$465$	4.52918	0.210036
$466$	−3.57623	−0.165666
$467$	38.1236	1.76415	0.882074	−	0.471110i	$-0.156146\pi$
$467$	38.1236	1.76415	0.882074	+	0.471110i	$0.156146\pi$
$468$	−0.974859	−0.0450629
$469$	8.99112	0.415171
$470$	−18.2825	−0.843308
$471$	19.3703	0.892537
$472$	30.2681	1.39320
$473$	−0.928534	−0.0426941
$474$	0.997443	0.0458141
$475$	18.2072	0.835402
$476$	−0.284723	−0.0130502
$477$	4.35871	0.199572
$478$	7.01523	0.320869
$479$	−2.53398	−0.115781	−0.0578904	−	0.998323i	$-0.518437\pi$
$479$	−2.53398	−0.115781	−0.0578904	+	0.998323i	$0.518437\pi$
$480$	2.65285	0.121085
$481$	0.693224	0.0316083
$482$	−4.03703	−0.183882
$483$	9.42182	0.428708
$484$	4.14626	0.188467
$485$	14.7846	0.671333
$486$	1.27306	0.0577470
$487$	−27.4844	−1.24544	−0.622718	−	0.782446i	$-0.713972\pi$
$487$	−27.4844	−1.24544	−0.622718	+	0.782446i	$0.713972\pi$
$488$	−38.4790	−1.74186
$489$	6.13272	0.277331
$490$	−0.737691	−0.0333255
$491$	23.4158	1.05674	0.528369	−	0.849015i	$-0.322804\pi$
$491$	23.4158	1.05674	0.528369	+	0.849015i	$0.322804\pi$
$492$	3.92065	0.176756
$493$	−2.41448	−0.108743
$494$	−17.3852	−0.782196
$495$	−0.330431	−0.0148517
$496$	−11.1836	−0.502158
$497$	−0.104809	−0.00470133
$498$	10.7609	0.482207
$499$	−29.8193	−1.33490	−0.667449	−	0.744656i	$-0.732614\pi$
$499$	−29.8193	−1.33490	−0.667449	+	0.744656i	$0.732614\pi$
$500$	4.00959	0.179314
$501$	4.77970	0.213541
$502$	38.5155	1.71903
$503$	25.2475	1.12573	0.562865	−	0.826549i	$-0.309699\pi$
$503$	25.2475	1.12573	0.562865	+	0.826549i	$0.309699\pi$
$504$	−7.74509	−0.344994
$505$	3.44376	0.153246
$506$	−1.23564	−0.0549311
$507$	−6.39521	−0.284021
$508$	−7.72752	−0.342853
$509$	31.3946	1.39154	0.695772	−	0.718263i	$-0.255063\pi$
$509$	31.3946	1.39154	0.695772	+	0.718263i	$0.255063\pi$
$510$	0.468789	0.0207583
$511$	−2.07945	−0.0919897
$512$	−25.3150	−1.11878
$513$	−5.31376	−0.234608
$514$	−5.16594	−0.227860
$515$	−1.25442	−0.0552766
$516$	−1.33713	−0.0588641
$517$	3.01563	0.132627
$518$	0.878044	0.0385791
$519$	−2.71132	−0.119014
$520$	−9.76508	−0.428227
$521$	10.5563	0.462479	0.231240	−	0.972897i	$-0.425722\pi$
$521$	10.5563	0.462479	0.231240	+	0.972897i	$0.425722\pi$
$522$	−10.4710	−0.458302
$523$	20.6556	0.903205	0.451603	−	0.892219i	$-0.350852\pi$
$523$	20.6556	0.903205	0.451603	+	0.892219i	$0.350852\pi$
$524$	−0.783264	−0.0342170
$525$	−8.76123	−0.382371
$526$	8.08720	0.352619
$527$	1.05989	0.0461694
$528$	0.815908	0.0355078
$529$	−9.42247	−0.409673
$530$	6.96066	0.302352
$531$	−9.99270	−0.433646
$532$	5.15394	0.223451
$533$	−26.5628	−1.15056
$534$	−2.56344	−0.110931
$535$	11.2728	0.487364
$536$	−10.6510	−0.460054
$537$	6.79008	0.293013
$538$	−2.52748	−0.108968
$539$	0.121680	0.00524111
$540$	−0.475836	−0.0204767
$541$	−6.49326	−0.279167	−0.139584	−	0.990210i	$-0.544576\pi$
$541$	−6.49326	−0.279167	−0.139584	+	0.990210i	$0.544576\pi$
$542$	−33.0172	−1.41821
$543$	−3.15097	−0.135221
$544$	0.620801	0.0266166
$545$	−11.0816	−0.474684
$546$	8.36569	0.358019
$547$	−12.2400	−0.523344	−0.261672	−	0.965157i	$-0.584274\pi$
$547$	−12.2400	−0.523344	−0.261672	+	0.965157i	$0.584274\pi$
$548$	−0.842326	−0.0359824
$549$	12.7035	0.542171
$550$	1.14901	0.0489939
$551$	43.7060	1.86194
$552$	−11.1612	−0.475054
$553$	2.00339	0.0851926
$554$	−35.4870	−1.50770
$555$	0.338368	0.0143629
$556$	−4.59082	−0.194694
$557$	−31.3177	−1.32697	−0.663486	−	0.748189i	$-0.730924\pi$
$557$	−31.3177	−1.32697	−0.663486	+	0.748189i	$0.730924\pi$
$558$	4.59645	0.194583
$559$	9.05925	0.383165
$560$	−9.93516	−0.419837
$561$	−0.0773251	−0.00326467
$562$	−12.6747	−0.534649
$563$	−38.7773	−1.63427	−0.817135	−	0.576446i	$-0.804439\pi$
$563$	−38.7773	−1.63427	−0.817135	+	0.576446i	$0.804439\pi$
$564$	4.34265	0.182859
$565$	−6.20943	−0.261233
$566$	−22.2926	−0.937028
$567$	2.55696	0.107382
$568$	0.124158	0.00520957
$569$	21.8182	0.914668	0.457334	−	0.889295i	$-0.348804\pi$
$569$	21.8182	0.914668	0.457334	+	0.889295i	$0.348804\pi$
$570$	−8.48583	−0.355432
$571$	−11.5086	−0.481619	−0.240810	−	0.970572i	$-0.577413\pi$
$571$	−11.5086	−0.481619	−0.240810	+	0.970572i	$0.577413\pi$
$572$	0.256790	0.0107369
$573$	−9.68027	−0.404399
$574$	−33.6448	−1.40431
$575$	−12.6256	−0.526522
$576$	8.88717	0.370299
$577$	20.8550	0.868204	0.434102	−	0.900864i	$-0.357066\pi$
$577$	20.8550	0.868204	0.434102	+	0.900864i	$0.357066\pi$
$578$	−21.5323	−0.895624
$579$	7.90112	0.328359
$580$	3.91378	0.162511
$581$	21.6135	0.896679
$582$	15.0042	0.621943
$583$	−1.14814	−0.0475510
$584$	2.46335	0.101934
$585$	3.22384	0.133290
$586$	−6.95171	−0.287173
$587$	44.1721	1.82318	0.911588	−	0.411105i	$-0.134857\pi$
$587$	44.1721	1.82318	0.911588	+	0.411105i	$0.134857\pi$
$588$	0.175224	0.00722613
$589$	−19.1857	−0.790532
$590$	−15.9579	−0.656976
$591$	−6.63176	−0.272794
$592$	−0.835506	−0.0343391
$593$	48.4720	1.99051	0.995254	−	0.0973121i	$-0.0310245\pi$
$593$	48.4720	1.99051	0.995254	+	0.0973121i	$0.0310245\pi$
$594$	−0.335338	−0.0137591
$595$	0.941573	0.0386007
$596$	1.28913	0.0528046
$597$	−16.3709	−0.670016
$598$	12.0556	0.492989
$599$	7.96629	0.325494	0.162747	−	0.986668i	$-0.447965\pi$
$599$	7.96629	0.325494	0.162747	+	0.986668i	$0.447965\pi$
$600$	10.3787	0.423708
$601$	37.1883	1.51694	0.758471	−	0.651706i	$-0.225947\pi$
$601$	37.1883	1.51694	0.758471	+	0.651706i	$0.225947\pi$
$602$	11.4745	0.467667
$603$	3.51633	0.143196
$604$	−9.01427	−0.366786
$605$	−13.7116	−0.557457
$606$	3.49492	0.141971
$607$	−47.7514	−1.93817	−0.969085	−	0.246726i	$-0.920645\pi$
$607$	−47.7514	−1.93817	−0.969085	+	0.246726i	$0.920645\pi$
$608$	−11.2375	−0.455741
$609$	−21.0312	−0.852226
$610$	20.2869	0.821391
$611$	−29.4220	−1.19029
$612$	−0.111352	−0.00450113
$613$	34.4227	1.39032	0.695161	−	0.718854i	$-0.255333\pi$
$613$	34.4227	1.39032	0.695161	+	0.718854i	$0.255333\pi$
$614$	−21.4278	−0.864757
$615$	−12.9655	−0.522820
$616$	2.04015	0.0821999
$617$	−24.1875	−0.973751	−0.486875	−	0.873471i	$-0.661863\pi$
$617$	−24.1875	−0.973751	−0.486875	+	0.873471i	$0.661863\pi$
$618$	−1.27306	−0.0512099
$619$	−19.7687	−0.794569	−0.397285	−	0.917695i	$-0.630047\pi$
$619$	−19.7687	−0.794569	−0.397285	+	0.917695i	$0.630047\pi$
$620$	−1.71804	−0.0689980
$621$	3.68477	0.147865
$622$	−19.0366	−0.763297
$623$	−5.14873	−0.206279
$624$	−7.96041	−0.318671
$625$	3.87243	0.154897
$626$	8.39781	0.335644
$627$	1.39971	0.0558990
$628$	−7.34766	−0.293204
$629$	0.0791824	0.00315721
$630$	4.08335	0.162685
$631$	1.03744	0.0412998	0.0206499	−	0.999787i	$-0.493426\pi$
$631$	1.03744	0.0412998	0.0206499	+	0.999787i	$0.493426\pi$
$632$	−2.37324	−0.0944024
$633$	−6.73471	−0.267681
$634$	−44.4083	−1.76368
$635$	25.5548	1.01411
$636$	−1.65337	−0.0655605
$637$	−1.18717	−0.0470372
$638$	2.75818	0.109197
$639$	−0.0409897	−0.00162153
$640$	8.88671	0.351278
$641$	−34.4425	−1.36040	−0.680199	−	0.733028i	$-0.738106\pi$
$641$	−34.4425	−1.36040	−0.680199	+	0.733028i	$0.738106\pi$
$642$	11.4402	0.451508
$643$	−26.9604	−1.06321	−0.531606	−	0.846991i	$-0.678411\pi$
$643$	−26.9604	−1.06321	−0.531606	+	0.846991i	$0.678411\pi$
$644$	−3.57394	−0.140833
$645$	4.42188	0.174111
$646$	−1.98580	−0.0781301
$647$	−28.2016	−1.10872	−0.554359	−	0.832278i	$-0.687036\pi$
$647$	−28.2016	−1.10872	−0.554359	+	0.832278i	$0.687036\pi$
$648$	−3.02902	−0.118991
$649$	2.63220	0.103323
$650$	−11.2103	−0.439705
$651$	9.23208	0.361834
$652$	−2.32630	−0.0911049
$653$	−25.4710	−0.996757	−0.498379	−	0.866959i	$-0.666071\pi$
$653$	−25.4710	−0.996757	−0.498379	+	0.866959i	$0.666071\pi$
$654$	−11.2462	−0.439761
$655$	2.59024	0.101209
$656$	32.0148	1.24997
$657$	−0.813252	−0.0317280
$658$	−37.2662	−1.45279
$659$	46.2987	1.80354	0.901771	−	0.432213i	$-0.142267\pi$
$659$	46.2987	1.80354	0.901771	+	0.432213i	$0.142267\pi$
$660$	0.125341	0.00487889
$661$	−33.0137	−1.28408	−0.642041	−	0.766670i	$-0.721912\pi$
$661$	−33.0137	−1.28408	−0.642041	+	0.766670i	$0.721912\pi$
$662$	36.9391	1.43568
$663$	0.754422	0.0292993
$664$	−25.6037	−0.993616
$665$	−17.0440	−0.660938
$666$	0.343393	0.0133062
$667$	−30.3075	−1.17351
$668$	−1.81307	−0.0701496
$669$	9.87916	0.381951
$670$	5.61541	0.216942
$671$	−3.34624	−0.129180
$672$	5.40745	0.208597
$673$	26.9204	1.03771	0.518853	−	0.854863i	$-0.326359\pi$
$673$	26.9204	1.03771	0.518853	+	0.854863i	$0.326359\pi$
$674$	22.4141	0.863357
$675$	−3.42642	−0.131883
$676$	2.42587	0.0933027
$677$	−19.4416	−0.747203	−0.373601	−	0.927589i	$-0.621877\pi$
$677$	−19.4416	−0.747203	−0.373601	+	0.927589i	$0.621877\pi$
$678$	−6.30166	−0.242014
$679$	30.1362	1.15652
$680$	−1.11540	−0.0427737
$681$	26.4553	1.01377
$682$	−1.21076	−0.0463624
$683$	−0.863761	−0.0330509	−0.0165254	−	0.999863i	$-0.505260\pi$
$683$	−0.863761	−0.0330509	−0.0165254	+	0.999863i	$0.505260\pi$
$684$	2.01565	0.0770702
$685$	2.78556	0.106431
$686$	−24.2898	−0.927389
$687$	−6.59664	−0.251678
$688$	−10.9186	−0.416269
$689$	11.2018	0.426754
$690$	5.88441	0.224016
$691$	21.1991	0.806450	0.403225	−	0.915101i	$-0.367889\pi$
$691$	21.1991	0.806450	0.403225	+	0.915101i	$0.367889\pi$
$692$	1.02848	0.0390968
$693$	−0.673535	−0.0255855
$694$	−15.5172	−0.589023
$695$	15.1818	0.575878
$696$	24.9139	0.944357
$697$	−3.03410	−0.114925
$698$	−39.2334	−1.48500
$699$	−2.80917	−0.106253
$700$	3.32336	0.125611
$701$	−1.56912	−0.0592647	−0.0296323	−	0.999561i	$-0.509434\pi$
$701$	−1.56912	−0.0592647	−0.0296323	+	0.999561i	$0.509434\pi$
$702$	3.27173	0.123483
$703$	−1.43333	−0.0540591
$704$	−2.34099	−0.0882293
$705$	−14.3611	−0.540870
$706$	−16.9446	−0.637718
$707$	7.01962	0.264000
$708$	3.79049	0.142455
$709$	−26.9726	−1.01298	−0.506489	−	0.862246i	$-0.669057\pi$
$709$	−26.9726	−1.01298	−0.506489	+	0.862246i	$0.669057\pi$
$710$	−0.0654587	−0.00245662
$711$	0.783502	0.0293836
$712$	6.09926	0.228579
$713$	13.3041	0.498242
$714$	0.955558	0.0357609
$715$	−0.849199	−0.0317582
$716$	−2.57565	−0.0962566
$717$	5.51054	0.205795
$718$	−9.38673	−0.350310
$719$	−18.7596	−0.699614	−0.349807	−	0.936822i	$-0.613753\pi$
$719$	−18.7596	−0.699614	−0.349807	+	0.936822i	$0.613753\pi$
$720$	−3.88553	−0.144805
$721$	−2.55696	−0.0952263
$722$	11.7580	0.437588
$723$	−3.17113	−0.117936
$724$	1.19524	0.0444208
$725$	28.1825	1.04667
$726$	−13.9153	−0.516445
$727$	−21.5057	−0.797601	−0.398800	−	0.917038i	$-0.630573\pi$
$727$	−21.5057	−0.797601	−0.398800	+	0.917038i	$0.630573\pi$
$728$	−19.9047	−0.737718
$729$	1.00000	0.0370370
$730$	−1.29873	−0.0480680
$731$	1.03478	0.0382727
$732$	−4.81875	−0.178106
$733$	−46.0387	−1.70048	−0.850240	−	0.526396i	$-0.823543\pi$
$733$	−46.0387	−1.70048	−0.850240	+	0.526396i	$0.823543\pi$
$734$	14.5141	0.535725
$735$	−0.579464	−0.0213739
$736$	7.79252	0.287236
$737$	−0.926242	−0.0341186
$738$	−13.1581	−0.484356
$739$	43.6526	1.60579	0.802893	−	0.596123i	$-0.203293\pi$
$739$	43.6526	1.60579	0.802893	+	0.596123i	$0.203293\pi$
$740$	−0.128352	−0.00471830
$741$	−13.6562	−0.501675
$742$	14.1883	0.520869
$743$	37.8923	1.39013	0.695066	−	0.718946i	$-0.255375\pi$
$743$	37.8923	1.39013	0.695066	+	0.718946i	$0.255375\pi$
$744$	−10.9365	−0.400950
$745$	−4.26312	−0.156189
$746$	−35.3746	−1.29516
$747$	8.45280	0.309272
$748$	0.0293314	0.00107246
$749$	22.9779	0.839594
$750$	−13.4566	−0.491366
$751$	5.41830	0.197716	0.0988582	−	0.995102i	$-0.468481\pi$
$751$	5.41830	0.197716	0.0988582	+	0.995102i	$0.468481\pi$
$752$	35.4608	1.29312
$753$	30.2544	1.10253
$754$	−26.9102	−0.980010
$755$	29.8101	1.08490
$756$	−0.969922	−0.0352757
$757$	22.4762	0.816912	0.408456	−	0.912778i	$-0.366067\pi$
$757$	22.4762	0.816912	0.408456	+	0.912778i	$0.366067\pi$
$758$	4.72495	0.171618
$759$	−0.970612	−0.0352310
$760$	20.1906	0.732389
$761$	−29.3780	−1.06495	−0.532476	−	0.846445i	$-0.678738\pi$
$761$	−29.3780	−1.06495	−0.532476	+	0.846445i	$0.678738\pi$
$762$	25.9344	0.939503
$763$	−22.5883	−0.817750
$764$	3.67198	0.132847
$765$	0.368239	0.0133137
$766$	10.3766	0.374922
$767$	−25.6810	−0.927288
$768$	−8.75563	−0.315942
$769$	34.0082	1.22637	0.613184	−	0.789940i	$-0.289889\pi$
$769$	34.0082	1.22637	0.613184	+	0.789940i	$0.289889\pi$
$770$	−1.07560	−0.0387621
$771$	−4.05791	−0.146142
$772$	−2.99710	−0.107868
$773$	39.3389	1.41492	0.707461	−	0.706753i	$-0.249841\pi$
$773$	39.3389	1.41492	0.707461	+	0.706753i	$0.249841\pi$
$774$	4.48756	0.161302
$775$	−12.3713	−0.444390
$776$	−35.6998	−1.28155
$777$	0.689713	0.0247433
$778$	17.7886	0.637751
$779$	54.9221	1.96779
$780$	−1.22289	−0.0437864
$781$	0.0107972	0.000386353 0
$782$	1.37703	0.0492425
$783$	−8.22506	−0.293940
$784$	1.43083	0.0511010
$785$	24.2986	0.867255
$786$	2.62872	0.0937632
$787$	5.45784	0.194551	0.0972755	−	0.995257i	$-0.468987\pi$
$787$	5.45784	0.194551	0.0972755	+	0.995257i	$0.468987\pi$
$788$	2.51560	0.0896145
$789$	6.35259	0.226158
$790$	1.25122	0.0445163
$791$	−12.6570	−0.450032
$792$	0.797879	0.0283514
$793$	32.6476	1.15935
$794$	46.7287	1.65834
$795$	5.46768	0.193918
$796$	6.20990	0.220104
$797$	42.8590	1.51814	0.759071	−	0.651007i	$-0.225653\pi$
$797$	42.8590	1.51814	0.759071	+	0.651007i	$0.225653\pi$
$798$	−17.2971	−0.612312
$799$	−3.36068	−0.118892
$800$	−7.24616	−0.256190
$801$	−2.01361	−0.0711474
$802$	−46.0728	−1.62689
$803$	0.214220	0.00755967
$804$	−1.33383	−0.0470407
$805$	11.8190	0.416564
$806$	11.8128	0.416088
$807$	−1.98537	−0.0698882
$808$	−8.31554	−0.292540
$809$	−16.9790	−0.596950	−0.298475	−	0.954417i	$-0.596478\pi$
$809$	−16.9790	−0.596950	−0.298475	+	0.954417i	$0.596478\pi$
$810$	1.59695	0.0561113
$811$	39.2430	1.37801	0.689003	−	0.724758i	$-0.258049\pi$
$811$	39.2430	1.37801	0.689003	+	0.724758i	$0.258049\pi$
$812$	7.97767	0.279961
$813$	−25.9354	−0.909594
$814$	−0.0904539	−0.00317041
$815$	7.69304	0.269475
$816$	−0.909265	−0.0318306
$817$	−18.7312	−0.655320
$818$	−1.78933	−0.0625624
$819$	6.57134	0.229621
$820$	4.91816	0.171749
$821$	41.3371	1.44267	0.721337	−	0.692584i	$-0.243528\pi$
$821$	41.3371	1.44267	0.721337	+	0.692584i	$0.243528\pi$
$822$	2.82693	0.0986007
$823$	−6.66487	−0.232323	−0.116161	−	0.993230i	$-0.537059\pi$
$823$	−6.66487	−0.232323	−0.116161	+	0.993230i	$0.537059\pi$
$824$	3.02902	0.105521
$825$	0.902559	0.0314231
$826$	−32.5278	−1.13179
$827$	−36.3819	−1.26512	−0.632561	−	0.774511i	$-0.717996\pi$
$827$	−36.3819	−1.26512	−0.632561	+	0.774511i	$0.717996\pi$
$828$	−1.39773	−0.0485744
$829$	−41.8998	−1.45524	−0.727620	−	0.685981i	$-0.759373\pi$
$829$	−41.8998	−1.45524	−0.727620	+	0.685981i	$0.759373\pi$
$830$	13.4987	0.468548
$831$	−27.8754	−0.966988
$832$	22.8398	0.791829
$833$	−0.135602	−0.00469834
$834$	15.4073	0.533510
$835$	5.99578	0.207493
$836$	−0.530945	−0.0183631
$837$	3.61056	0.124799
$838$	31.5824	1.09099
$839$	22.3434	0.771379	0.385689	−	0.922629i	$-0.373964\pi$
$839$	22.3434	0.771379	0.385689	+	0.922629i	$0.373964\pi$
$840$	−9.71563	−0.335221
$841$	38.6516	1.33281
$842$	34.3356	1.18328
$843$	−9.95610	−0.342906
$844$	2.55465	0.0879346
$845$	−8.02231	−0.275976
$846$	−14.5744	−0.501078
$847$	−27.9492	−0.960345
$848$	−13.5009	−0.463624
$849$	−17.5111	−0.600979
$850$	−1.28048	−0.0439201
$851$	0.993927	0.0340714
$852$	0.0155484	0.000532681 0
$853$	14.7622	0.505447	0.252724	−	0.967538i	$-0.418674\pi$
$853$	14.7622	0.505447	0.252724	+	0.967538i	$0.418674\pi$
$854$	41.3518	1.41503
$855$	−6.66571	−0.227963
$856$	−27.2199	−0.930358
$857$	−44.8558	−1.53224	−0.766122	−	0.642695i	$-0.777816\pi$
$857$	−44.8558	−1.53224	−0.766122	+	0.642695i	$0.777816\pi$
$858$	−0.861812	−0.0294218
$859$	−35.4111	−1.20821	−0.604106	−	0.796904i	$-0.706470\pi$
$859$	−35.4111	−1.20821	−0.604106	+	0.796904i	$0.706470\pi$
$860$	−1.67733	−0.0571966
$861$	−26.4283	−0.900675
$862$	25.3186	0.862356
$863$	−38.7935	−1.32055	−0.660273	−	0.751026i	$-0.729559\pi$
$863$	−38.7935	−1.32055	−0.660273	+	0.751026i	$0.729559\pi$
$864$	2.11479	0.0719467
$865$	−3.40115	−0.115643
$866$	23.7048	0.805521
$867$	−16.9138	−0.574424
$868$	−3.50197	−0.118865
$869$	−0.206384	−0.00700109
$870$	−13.1350	−0.445320
$871$	9.03688	0.306203
$872$	26.7584	0.906153
$873$	11.7859	0.398893
$874$	−24.9264	−0.843149
$875$	−27.0279	−0.913710
$876$	0.308487	0.0104228
$877$	25.7890	0.870833	0.435416	−	0.900229i	$-0.356601\pi$
$877$	25.7890	0.870833	0.435416	+	0.900229i	$0.356601\pi$
$878$	36.0340	1.21609
$879$	−5.46064	−0.184183
$880$	1.02350	0.0345020
$881$	−3.96753	−0.133669	−0.0668347	−	0.997764i	$-0.521290\pi$
$881$	−3.96753	−0.133669	−0.0668347	+	0.997764i	$0.521290\pi$
$882$	−0.588071	−0.0198014
$883$	−32.2023	−1.08369	−0.541847	−	0.840477i	$-0.682275\pi$
$883$	−32.2023	−1.08369	−0.541847	+	0.840477i	$0.682275\pi$
$884$	−0.286172	−0.00962500
$885$	−12.5351	−0.421362
$886$	−18.9416	−0.636354
$887$	−9.09894	−0.305512	−0.152756	−	0.988264i	$-0.548815\pi$
$887$	−9.09894	−0.305512	−0.152756	+	0.988264i	$0.548815\pi$
$888$	−0.817045	−0.0274182
$889$	52.0898	1.74703
$890$	−3.21564	−0.107789
$891$	−0.263412	−0.00882463
$892$	−3.74742	−0.125473
$893$	60.8337	2.03572
$894$	−4.32644	−0.144698
$895$	8.51764	0.284713
$896$	18.1143	0.605156
$897$	9.46978	0.316187
$898$	−18.7879	−0.626960
$899$	−29.6971	−0.990454
$900$	1.29973	0.0433243
$901$	1.27951	0.0426266
$902$	3.46600	0.115405
$903$	9.01337	0.299946
$904$	14.9937	0.498683
$905$	−3.95265	−0.131391
$906$	30.2528	1.00508
$907$	7.41765	0.246299	0.123150	−	0.992388i	$-0.460701\pi$
$907$	7.41765	0.246299	0.123150	+	0.992388i	$0.460701\pi$
$908$	−10.0352	−0.333029
$909$	2.74529	0.0910556
$910$	10.4941	0.347877
$911$	−56.1794	−1.86131	−0.930653	−	0.365902i	$-0.880761\pi$
$911$	−56.1794	−1.86131	−0.930653	+	0.365902i	$0.880761\pi$
$912$	16.4592	0.545017
$913$	−2.22657	−0.0736887
$914$	7.64459	0.252861
$915$	15.9355	0.526813
$916$	2.50228	0.0826776
$917$	5.27983	0.174356
$918$	0.373708	0.0123342
$919$	−11.4294	−0.377022	−0.188511	−	0.982071i	$-0.560366\pi$
$919$	−11.4294	−0.377022	−0.188511	+	0.982071i	$0.560366\pi$
$920$	−14.0009	−0.461597
$921$	−16.8318	−0.554626
$922$	51.0787	1.68219
$923$	−0.105343	−0.00346739
$924$	0.255489	0.00840498
$925$	−0.924239	−0.0303888
$926$	−44.8348	−1.47336
$927$	−1.00000	−0.0328443
$928$	−17.3943	−0.570995
$929$	46.5203	1.52628	0.763140	−	0.646233i	$-0.223656\pi$
$929$	46.5203	1.52628	0.763140	+	0.646233i	$0.223656\pi$
$930$	5.76591	0.189072
$931$	2.45462	0.0804469
$932$	1.06559	0.0349046
$933$	−14.9534	−0.489554
$934$	48.5335	1.58806
$935$	−0.0969985	−0.00317219
$936$	−7.78451	−0.254445
$937$	−28.6109	−0.934676	−0.467338	−	0.884079i	$-0.654787\pi$
$937$	−28.6109	−0.934676	−0.467338	+	0.884079i	$0.654787\pi$
$938$	11.4462	0.373732
$939$	6.59657	0.215271
$940$	5.44753	0.177679
$941$	3.88733	0.126723	0.0633617	−	0.997991i	$-0.479818\pi$
$941$	3.88733	0.126723	0.0633617	+	0.997991i	$0.479818\pi$
$942$	24.6595	0.803451
$943$	−38.0851	−1.24022
$944$	30.9520	1.00740
$945$	3.20752	0.104341
$946$	−1.18208	−0.0384326
$947$	−16.1937	−0.526226	−0.263113	−	0.964765i	$-0.584749\pi$
$947$	−16.1937	−0.526226	−0.263113	+	0.964765i	$0.584749\pi$
$948$	−0.297202	−0.00965269
$949$	−2.09004	−0.0678455
$950$	23.1788	0.752018
$951$	−34.8832	−1.13117
$952$	−2.27358	−0.0736873
$953$	−4.46552	−0.144652	−0.0723262	−	0.997381i	$-0.523042\pi$
$953$	−4.46552	−0.144652	−0.0723262	+	0.997381i	$0.523042\pi$
$954$	5.54889	0.179652
$955$	−12.1432	−0.392944
$956$	−2.09029	−0.0676048
$957$	2.16658	0.0700356
$958$	−3.22591	−0.104224
$959$	5.67796	0.183351
$960$	11.1483	0.359809
$961$	−17.9638	−0.579478
$962$	0.882514	0.0284534
$963$	8.98639	0.289582
$964$	1.20289	0.0387425
$965$	9.91136	0.319058
$966$	11.9945	0.385917
$967$	−33.7167	−1.08426	−0.542128	−	0.840296i	$-0.682381\pi$
$967$	−33.7167	−1.08426	−0.542128	+	0.840296i	$0.682381\pi$
$968$	33.1090	1.06416
$969$	−1.55986	−0.0501101
$970$	18.8216	0.604325
$971$	6.32560	0.202998	0.101499	−	0.994836i	$-0.467636\pi$
$971$	6.32560	0.202998	0.101499	+	0.994836i	$0.467636\pi$
$972$	−0.379326	−0.0121669
$973$	30.9459	0.992079
$974$	−34.9892	−1.12113
$975$	−8.80582	−0.282012
$976$	−39.3485	−1.25951
$977$	−38.0919	−1.21867	−0.609333	−	0.792914i	$-0.708563\pi$
$977$	−38.0919	−1.21867	−0.609333	+	0.792914i	$0.708563\pi$
$978$	7.80730	0.249650
$979$	0.530409	0.0169519
$980$	0.219806	0.00702144
$981$	−8.83401	−0.282048
$982$	29.8096	0.951262
$983$	−35.7252	−1.13946	−0.569729	−	0.821833i	$-0.692952\pi$
$983$	−35.7252	−1.13946	−0.569729	+	0.821833i	$0.692952\pi$
$984$	31.3074	0.998043
$985$	−8.31905	−0.265067
$986$	−3.07377	−0.0978889
$987$	−29.2730	−0.931769
$988$	5.18017	0.164803
$989$	12.9889	0.413024
$990$	−0.420657	−0.0133694
$991$	4.29051	0.136292	0.0681462	−	0.997675i	$-0.478292\pi$
$991$	4.29051	0.136292	0.0681462	+	0.997675i	$0.478292\pi$
$992$	7.63559	0.242430
$993$	29.0160	0.920796
$994$	−0.133428	−0.00423208
$995$	−20.5361	−0.651037
$996$	−3.20637	−0.101598
$997$	25.9878	0.823042	0.411521	−	0.911400i	$-0.364998\pi$
$997$	25.9878	0.823042	0.411521	+	0.911400i	$0.364998\pi$
$998$	−37.9617	−1.20166
$999$	0.269739	0.00853417

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	309.2.a.d.1.6	✓	8
3.2	odd	2		927.2.a.g.1.3		8
4.3	odd	2		4944.2.a.bf.1.6		8
5.4	even	2		7725.2.a.z.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
309.2.a.d.1.6	✓	8	1.1	even	1	trivial
927.2.a.g.1.3		8	3.2	odd	2
4944.2.a.bf.1.6		8	4.3	odd	2
7725.2.a.z.1.3		8	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 309 = 3 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	309.a (trivial)

\(f(q)\)	\(=\)	\(q+1.27306 q^{2} +1.00000 q^{3} -0.379326 q^{4} +1.25442 q^{5} +1.27306 q^{6} +2.55696 q^{7} -3.02902 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.27306 q^{2} +1.00000 q^{3} -0.379326 q^{4} +1.25442 q^{5} +1.27306 q^{6} +2.55696 q^{7} -3.02902 q^{8} +1.00000 q^{9} +1.59695 q^{10} -0.263412 q^{11} -0.379326 q^{12} +2.56998 q^{13} +3.25516 q^{14} +1.25442 q^{15} -3.09746 q^{16} +0.293552 q^{17} +1.27306 q^{18} -5.31376 q^{19} -0.475836 q^{20} +2.55696 q^{21} -0.335338 q^{22} +3.68477 q^{23} -3.02902 q^{24} -3.42642 q^{25} +3.27173 q^{26} +1.00000 q^{27} -0.969922 q^{28} -8.22506 q^{29} +1.59695 q^{30} +3.61056 q^{31} +2.11479 q^{32} -0.263412 q^{33} +0.373708 q^{34} +3.20752 q^{35} -0.379326 q^{36} +0.269739 q^{37} -6.76472 q^{38} +2.56998 q^{39} -3.79968 q^{40} -10.3358 q^{41} +3.25516 q^{42} +3.52503 q^{43} +0.0999190 q^{44} +1.25442 q^{45} +4.69092 q^{46} -11.4483 q^{47} -3.09746 q^{48} -0.461936 q^{49} -4.36203 q^{50} +0.293552 q^{51} -0.974859 q^{52} +4.35871 q^{53} +1.27306 q^{54} -0.330431 q^{55} -7.74509 q^{56} -5.31376 q^{57} -10.4710 q^{58} -9.99270 q^{59} -0.475836 q^{60} +12.7035 q^{61} +4.59645 q^{62} +2.55696 q^{63} +8.88717 q^{64} +3.22384 q^{65} -0.335338 q^{66} +3.51633 q^{67} -0.111352 q^{68} +3.68477 q^{69} +4.08335 q^{70} -0.0409897 q^{71} -3.02902 q^{72} -0.813252 q^{73} +0.343393 q^{74} -3.42642 q^{75} +2.01565 q^{76} -0.673535 q^{77} +3.27173 q^{78} +0.783502 q^{79} -3.88553 q^{80} +1.00000 q^{81} -13.1581 q^{82} +8.45280 q^{83} -0.969922 q^{84} +0.368239 q^{85} +4.48756 q^{86} -8.22506 q^{87} +0.797879 q^{88} -2.01361 q^{89} +1.59695 q^{90} +6.57134 q^{91} -1.39773 q^{92} +3.61056 q^{93} -14.5744 q^{94} -6.66571 q^{95} +2.11479 q^{96} +11.7859 q^{97} -0.588071 q^{98} -0.263412 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 8 q^{3} + 11 q^{4} - q^{5} - q^{6} + 6 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - q^2 + 8 * q^3 + 11 * q^4 - q^5 - q^6 + 6 * q^7 - 3 * q^8 + 8 * q^9	\( 8 q - q^{2} + 8 q^{3} + 11 q^{4} - q^{5} - q^{6} + 6 q^{7} - 3 q^{8} + 8 q^{9} + 3 q^{10} + 6 q^{11} + 11 q^{12} + 9 q^{13} - 6 q^{14} - q^{15} + 9 q^{16} - 4 q^{17} - q^{18} + 16 q^{19} - 23 q^{20} + 6 q^{21} - 4 q^{22} - 11 q^{23} - 3 q^{24} + 15 q^{25} - 14 q^{26} + 8 q^{27} - 5 q^{28} + 3 q^{30} + 17 q^{31} - 12 q^{32} + 6 q^{33} - 8 q^{34} + 4 q^{35} + 11 q^{36} - 6 q^{37} - 3 q^{38} + 9 q^{39} + 3 q^{40} + 12 q^{41} - 6 q^{42} + 9 q^{43} + 8 q^{44} - q^{45} - 30 q^{46} - 6 q^{47} + 9 q^{48} + 18 q^{49} - 36 q^{50} - 4 q^{51} + 23 q^{52} - 16 q^{53} - q^{54} - 10 q^{55} - 13 q^{56} + 16 q^{57} - 22 q^{58} + 11 q^{59} - 23 q^{60} + 5 q^{61} - 25 q^{62} + 6 q^{63} - 35 q^{64} - 41 q^{65} - 4 q^{66} + 5 q^{67} - 19 q^{68} - 11 q^{69} - 48 q^{70} - 10 q^{71} - 3 q^{72} + 14 q^{73} + 4 q^{74} + 15 q^{75} - 12 q^{76} - 40 q^{77} - 14 q^{78} + 14 q^{79} - 19 q^{80} + 8 q^{81} - 13 q^{82} - 23 q^{83} - 5 q^{84} - 4 q^{85} - 3 q^{86} - 30 q^{88} - 14 q^{89} + 3 q^{90} - 6 q^{91} - 21 q^{92} + 17 q^{93} + 22 q^{94} + 6 q^{95} - 12 q^{96} + 3 q^{97} - 18 q^{98} + 6 q^{99}+O(q^{100}) \) 8 * q - q^2 + 8 * q^3 + 11 * q^4 - q^5 - q^6 + 6 * q^7 - 3 * q^8 + 8 * q^9 + 3 * q^10 + 6 * q^11 + 11 * q^12 + 9 * q^13 - 6 * q^14 - q^15 + 9 * q^16 - 4 * q^17 - q^18 + 16 * q^19 - 23 * q^20 + 6 * q^21 - 4 * q^22 - 11 * q^23 - 3 * q^24 + 15 * q^25 - 14 * q^26 + 8 * q^27 - 5 * q^28 + 3 * q^30 + 17 * q^31 - 12 * q^32 + 6 * q^33 - 8 * q^34 + 4 * q^35 + 11 * q^36 - 6 * q^37 - 3 * q^38 + 9 * q^39 + 3 * q^40 + 12 * q^41 - 6 * q^42 + 9 * q^43 + 8 * q^44 - q^45 - 30 * q^46 - 6 * q^47 + 9 * q^48 + 18 * q^49 - 36 * q^50 - 4 * q^51 + 23 * q^52 - 16 * q^53 - q^54 - 10 * q^55 - 13 * q^56 + 16 * q^57 - 22 * q^58 + 11 * q^59 - 23 * q^60 + 5 * q^61 - 25 * q^62 + 6 * q^63 - 35 * q^64 - 41 * q^65 - 4 * q^66 + 5 * q^67 - 19 * q^68 - 11 * q^69 - 48 * q^70 - 10 * q^71 - 3 * q^72 + 14 * q^73 + 4 * q^74 + 15 * q^75 - 12 * q^76 - 40 * q^77 - 14 * q^78 + 14 * q^79 - 19 * q^80 + 8 * q^81 - 13 * q^82 - 23 * q^83 - 5 * q^84 - 4 * q^85 - 3 * q^86 - 30 * q^88 - 14 * q^89 + 3 * q^90 - 6 * q^91 - 21 * q^92 + 17 * q^93 + 22 * q^94 + 6 * q^95 - 12 * q^96 + 3 * q^97 - 18 * q^98 + 6 * q^99

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