LMFDB - Embedded newform 786.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [786,2,Mod(1,786)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(786, 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("786.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 786 = 2 \cdot 3 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	786.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.27624159887$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.64268.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 6x + 32 $ x^4 - x^3 - 12x^2 + 6x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.72110$ of defining polynomial
Character	$\chi$	$=$	786.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.31673 q^{5} +1.00000 q^{6} -3.03783 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.31673 q^{5} +1.00000 q^{6} -3.03783 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.31673 q^{10} +5.85421 q^{11} +1.00000 q^{12} -0.412014 q^{13} -3.03783 q^{14} +1.31673 q^{15} +1.00000 q^{16} -1.85421 q^{17} +1.00000 q^{18} +2.81638 q^{19} +1.31673 q^{20} -3.03783 q^{21} +5.85421 q^{22} +3.44219 q^{23} +1.00000 q^{24} -3.26622 q^{25} -0.412014 q^{26} +1.00000 q^{27} -3.03783 q^{28} +3.44984 q^{29} +1.31673 q^{30} -0.904717 q^{31} +1.00000 q^{32} +5.85421 q^{33} -1.85421 q^{34} -4.00000 q^{35} +1.00000 q^{36} +1.72110 q^{37} +2.81638 q^{38} -0.412014 q^{39} +1.31673 q^{40} +1.03783 q^{41} -3.03783 q^{42} -4.07565 q^{43} +5.85421 q^{44} +1.31673 q^{45} +3.44219 q^{46} -4.00000 q^{47} +1.00000 q^{48} +2.22839 q^{49} -3.26622 q^{50} -1.85421 q^{51} -0.412014 q^{52} -10.9495 q^{53} +1.00000 q^{54} +7.70841 q^{55} -3.03783 q^{56} +2.81638 q^{57} +3.44984 q^{58} -7.92986 q^{59} +1.31673 q^{60} +3.85421 q^{61} -0.904717 q^{62} -3.03783 q^{63} +1.00000 q^{64} -0.542511 q^{65} +5.85421 q^{66} -14.1884 q^{67} -1.85421 q^{68} +3.44219 q^{69} -4.00000 q^{70} -11.4422 q^{71} +1.00000 q^{72} +14.1513 q^{73} +1.72110 q^{74} -3.26622 q^{75} +2.81638 q^{76} -17.7841 q^{77} -0.412014 q^{78} -8.12546 q^{79} +1.31673 q^{80} +1.00000 q^{81} +1.03783 q^{82} -2.41966 q^{83} -3.03783 q^{84} -2.44149 q^{85} -4.07565 q^{86} +3.44984 q^{87} +5.85421 q^{88} +5.30405 q^{89} +1.31673 q^{90} +1.25163 q^{91} +3.44219 q^{92} -0.904717 q^{93} -4.00000 q^{94} +3.70841 q^{95} +1.00000 q^{96} +13.1506 q^{97} +2.22839 q^{98} +5.85421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{6} + q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^6 + q^7 + 4 * q^8 + 4 * q^9	$ 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{6} + q^{7} + 4 q^{8} + 4 q^{9} + 5 q^{11} + 4 q^{12} + q^{13} + q^{14} + 4 q^{16} + 11 q^{17} + 4 q^{18} + 6 q^{19} + q^{21} + 5 q^{22} - 2 q^{23} + 4 q^{24} + 8 q^{25} + q^{26} + 4 q^{27} + q^{28} - 2 q^{29} - q^{31} + 4 q^{32} + 5 q^{33} + 11 q^{34} - 16 q^{35} + 4 q^{36} - q^{37} + 6 q^{38} + q^{39} - 9 q^{41} + q^{42} + 10 q^{43} + 5 q^{44} - 2 q^{46} - 16 q^{47} + 4 q^{48} + q^{49} + 8 q^{50} + 11 q^{51} + q^{52} - 28 q^{53} + 4 q^{54} - 6 q^{55} + q^{56} + 6 q^{57} - 2 q^{58} + 13 q^{59} - 3 q^{61} - q^{62} + q^{63} + 4 q^{64} - 18 q^{65} + 5 q^{66} + 9 q^{67} + 11 q^{68} - 2 q^{69} - 16 q^{70} - 30 q^{71} + 4 q^{72} + 4 q^{73} - q^{74} + 8 q^{75} + 6 q^{76} - 8 q^{77} + q^{78} - 22 q^{79} + 4 q^{81} - 9 q^{82} - 7 q^{83} + q^{84} + 6 q^{85} + 10 q^{86} - 2 q^{87} + 5 q^{88} - 13 q^{89} - 16 q^{91} - 2 q^{92} - q^{93} - 16 q^{94} - 22 q^{95} + 4 q^{96} + q^{98} + 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^6 + q^7 + 4 * q^8 + 4 * q^9 + 5 * q^11 + 4 * q^12 + q^13 + q^14 + 4 * q^16 + 11 * q^17 + 4 * q^18 + 6 * q^19 + q^21 + 5 * q^22 - 2 * q^23 + 4 * q^24 + 8 * q^25 + q^26 + 4 * q^27 + q^28 - 2 * q^29 - q^31 + 4 * q^32 + 5 * q^33 + 11 * q^34 - 16 * q^35 + 4 * q^36 - q^37 + 6 * q^38 + q^39 - 9 * q^41 + q^42 + 10 * q^43 + 5 * q^44 - 2 * q^46 - 16 * q^47 + 4 * q^48 + q^49 + 8 * q^50 + 11 * q^51 + q^52 - 28 * q^53 + 4 * q^54 - 6 * q^55 + q^56 + 6 * q^57 - 2 * q^58 + 13 * q^59 - 3 * q^61 - q^62 + q^63 + 4 * q^64 - 18 * q^65 + 5 * q^66 + 9 * q^67 + 11 * q^68 - 2 * q^69 - 16 * q^70 - 30 * q^71 + 4 * q^72 + 4 * q^73 - q^74 + 8 * q^75 + 6 * q^76 - 8 * q^77 + q^78 - 22 * q^79 + 4 * q^81 - 9 * q^82 - 7 * q^83 + q^84 + 6 * q^85 + 10 * q^86 - 2 * q^87 + 5 * q^88 - 13 * q^89 - 16 * q^91 - 2 * q^92 - q^93 - 16 * q^94 - 22 * q^95 + 4 * q^96 + q^98 + 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.31673	0.588860	0.294430	−	0.955673i	$-0.404870\pi$
$5$	1.31673	0.588860	0.294430	+	0.955673i	$0.404870\pi$
$6$	1.00000	0.408248
$7$	−3.03783	−1.14819	−0.574095	−	0.818788i	$-0.694646\pi$
$7$	−3.03783	−1.14819	−0.574095	+	0.818788i	$0.694646\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.31673	0.416387
$11$	5.85421	1.76511	0.882555	−	0.470209i	$-0.155822\pi$
$11$	5.85421	1.76511	0.882555	+	0.470209i	$0.155822\pi$
$12$	1.00000	0.288675
$13$	−0.412014	−0.114272	−0.0571360	−	0.998366i	$-0.518197\pi$
$13$	−0.412014	−0.114272	−0.0571360	+	0.998366i	$0.518197\pi$
$14$	−3.03783	−0.811893
$15$	1.31673	0.339978
$16$	1.00000	0.250000
$17$	−1.85421	−0.449711	−0.224856	−	0.974392i	$-0.572191\pi$
$17$	−1.85421	−0.449711	−0.224856	+	0.974392i	$0.572191\pi$
$18$	1.00000	0.235702
$19$	2.81638	0.646122	0.323061	−	0.946378i	$-0.395288\pi$
$19$	2.81638	0.646122	0.323061	+	0.946378i	$0.395288\pi$
$20$	1.31673	0.294430
$21$	−3.03783	−0.662908
$22$	5.85421	1.24812
$23$	3.44219	0.717747	0.358873	−	0.933386i	$-0.383161\pi$
$23$	3.44219	0.717747	0.358873	+	0.933386i	$0.383161\pi$
$24$	1.00000	0.204124
$25$	−3.26622	−0.653244
$26$	−0.412014	−0.0808025
$27$	1.00000	0.192450
$28$	−3.03783	−0.574095
$29$	3.44984	0.640619	0.320310	−	0.947313i	$-0.396213\pi$
$29$	3.44984	0.640619	0.320310	+	0.947313i	$0.396213\pi$
$30$	1.31673	0.240401
$31$	−0.904717	−0.162492	−0.0812460	−	0.996694i	$-0.525890\pi$
$31$	−0.904717	−0.162492	−0.0812460	+	0.996694i	$0.525890\pi$
$32$	1.00000	0.176777
$33$	5.85421	1.01909
$34$	−1.85421	−0.317994
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	1.72110	0.282947	0.141473	−	0.989942i	$-0.454816\pi$
$37$	1.72110	0.282947	0.141473	+	0.989942i	$0.454816\pi$
$38$	2.81638	0.456877
$39$	−0.412014	−0.0659750
$40$	1.31673	0.208193
$41$	1.03783	0.162081	0.0810407	−	0.996711i	$-0.474176\pi$
$41$	1.03783	0.162081	0.0810407	+	0.996711i	$0.474176\pi$
$42$	−3.03783	−0.468747
$43$	−4.07565	−0.621531	−0.310766	−	0.950487i	$-0.600585\pi$
$43$	−4.07565	−0.621531	−0.310766	+	0.950487i	$0.600585\pi$
$44$	5.85421	0.882555
$45$	1.31673	0.196287
$46$	3.44219	0.507524
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	1.00000	0.144338
$49$	2.22839	0.318342
$50$	−3.26622	−0.461913
$51$	−1.85421	−0.259641
$52$	−0.412014	−0.0571360
$53$	−10.9495	−1.50403	−0.752014	−	0.659147i	$-0.770917\pi$
$53$	−10.9495	−1.50403	−0.752014	+	0.659147i	$0.770917\pi$
$54$	1.00000	0.136083
$55$	7.70841	1.03940
$56$	−3.03783	−0.405947
$57$	2.81638	0.373039
$58$	3.44984	0.452986
$59$	−7.92986	−1.03238	−0.516190	−	0.856474i	$-0.672650\pi$
$59$	−7.92986	−1.03238	−0.516190	+	0.856474i	$0.672650\pi$
$60$	1.31673	0.169989
$61$	3.85421	0.493481	0.246740	−	0.969082i	$-0.420641\pi$
$61$	3.85421	0.493481	0.246740	+	0.969082i	$0.420641\pi$
$62$	−0.904717	−0.114899
$63$	−3.03783	−0.382730
$64$	1.00000	0.125000
$65$	−0.542511	−0.0672902
$66$	5.85421	0.720603
$67$	−14.1884	−1.73339	−0.866697	−	0.498836i	$-0.833761\pi$
$67$	−14.1884	−1.73339	−0.866697	+	0.498836i	$0.833761\pi$
$68$	−1.85421	−0.224856
$69$	3.44219	0.414391
$70$	−4.00000	−0.478091
$71$	−11.4422	−1.35794	−0.678969	−	0.734167i	$-0.737573\pi$
$71$	−11.4422	−1.35794	−0.678969	+	0.734167i	$0.737573\pi$
$72$	1.00000	0.117851
$73$	14.1513	1.65629	0.828143	−	0.560517i	$-0.189398\pi$
$73$	14.1513	1.65629	0.828143	+	0.560517i	$0.189398\pi$
$74$	1.72110	0.200073
$75$	−3.26622	−0.377151
$76$	2.81638	0.323061
$77$	−17.7841	−2.02668
$78$	−0.412014	−0.0466514
$79$	−8.12546	−0.914186	−0.457093	−	0.889419i	$-0.651109\pi$
$79$	−8.12546	−0.914186	−0.457093	+	0.889419i	$0.651109\pi$
$80$	1.31673	0.147215
$81$	1.00000	0.111111
$82$	1.03783	0.114609
$83$	−2.41966	−0.265592	−0.132796	−	0.991143i	$-0.542396\pi$
$83$	−2.41966	−0.265592	−0.132796	+	0.991143i	$0.542396\pi$
$84$	−3.03783	−0.331454
$85$	−2.44149	−0.264817
$86$	−4.07565	−0.439489
$87$	3.44984	0.369862
$88$	5.85421	0.624061
$89$	5.30405	0.562228	0.281114	−	0.959674i	$-0.409296\pi$
$89$	5.30405	0.562228	0.281114	+	0.959674i	$0.409296\pi$
$90$	1.31673	0.138796
$91$	1.25163	0.131206
$92$	3.44219	0.358873
$93$	−0.904717	−0.0938148
$94$	−4.00000	−0.412568
$95$	3.70841	0.380475
$96$	1.00000	0.102062
$97$	13.1506	1.33524	0.667621	−	0.744501i	$-0.267313\pi$
$97$	13.1506	1.33524	0.667621	+	0.744501i	$0.267313\pi$
$98$	2.22839	0.225102
$99$	5.85421	0.588370
$100$	−3.26622	−0.326622
$101$	2.68327	0.266995	0.133498	−	0.991049i	$-0.457379\pi$
$101$	2.68327	0.266995	0.133498	+	0.991049i	$0.457379\pi$
$102$	−1.85421	−0.183594
$103$	0.859944	0.0847328	0.0423664	−	0.999102i	$-0.486510\pi$
$103$	0.859944	0.0847328	0.0423664	+	0.999102i	$0.486510\pi$
$104$	−0.412014	−0.0404013
$105$	−4.00000	−0.390360
$106$	−10.9495	−1.06351
$107$	−10.4120	−1.00657	−0.503284	−	0.864121i	$-0.667875\pi$
$107$	−10.4120	−1.00657	−0.503284	+	0.864121i	$0.667875\pi$
$108$	1.00000	0.0962250
$109$	2.77925	0.266204	0.133102	−	0.991102i	$-0.457506\pi$
$109$	2.77925	0.266204	0.133102	+	0.991102i	$0.457506\pi$
$110$	7.70841	0.734968
$111$	1.72110	0.163359
$112$	−3.03783	−0.287048
$113$	17.0125	1.60040	0.800199	−	0.599735i	$-0.204727\pi$
$113$	17.0125	1.60040	0.800199	+	0.599735i	$0.204727\pi$
$114$	2.81638	0.263778
$115$	4.53244	0.422652
$116$	3.44984	0.320310
$117$	−0.412014	−0.0380907
$118$	−7.92986	−0.730003
$119$	5.63276	0.516354
$120$	1.31673	0.120201
$121$	23.2717	2.11561
$122$	3.85421	0.348943
$123$	1.03783	0.0935777
$124$	−0.904717	−0.0812460
$125$	−10.8844	−0.973529
$126$	−3.03783	−0.270631
$127$	−14.2459	−1.26412	−0.632059	−	0.774920i	$-0.717790\pi$
$127$	−14.2459	−1.26412	−0.632059	+	0.774920i	$0.717790\pi$
$128$	1.00000	0.0883883
$129$	−4.07565	−0.358841
$130$	−0.542511	−0.0475814
$131$	1.00000	0.0873704
$131$	1.00000	0.0873704
$132$	5.85421	0.509543
$133$	−8.55567	−0.741871
$134$	−14.1884	−1.22569
$135$	1.31673	0.113326
$136$	−1.85421	−0.158997
$137$	−19.0201	−1.62500	−0.812499	−	0.582963i	$-0.801893\pi$
$137$	−19.0201	−1.62500	−0.812499	+	0.582963i	$0.801893\pi$
$138$	3.44219	0.293019
$139$	9.89898	0.839620	0.419810	−	0.907612i	$-0.362097\pi$
$139$	9.89898	0.839620	0.419810	+	0.907612i	$0.362097\pi$
$140$	−4.00000	−0.338062
$141$	−4.00000	−0.336861
$142$	−11.4422	−0.960207
$143$	−2.41201	−0.201703
$144$	1.00000	0.0833333
$145$	4.54251	0.377235
$146$	14.1513	1.17117
$147$	2.22839	0.183795
$148$	1.72110	0.141473
$149$	−0.161089	−0.0131970	−0.00659848	−	0.999978i	$-0.502100\pi$
$149$	−0.161089	−0.0131970	−0.00659848	+	0.999978i	$0.502100\pi$
$150$	−3.26622	−0.266686
$151$	−13.9375	−1.13422	−0.567109	−	0.823643i	$-0.691938\pi$
$151$	−13.9375	−1.13422	−0.567109	+	0.823643i	$0.691938\pi$
$152$	2.81638	0.228439
$153$	−1.85421	−0.149904
$154$	−17.7841	−1.43308
$155$	−1.19127	−0.0956850
$156$	−0.412014	−0.0329875
$157$	−6.97272	−0.556484	−0.278242	−	0.960511i	$-0.589752\pi$
$157$	−6.97272	−0.556484	−0.278242	+	0.960511i	$0.589752\pi$
$158$	−8.12546	−0.646427
$159$	−10.9495	−0.868351
$160$	1.31673	0.104097
$161$	−10.4568	−0.824110
$162$	1.00000	0.0785674
$163$	−3.48697	−0.273120	−0.136560	−	0.990632i	$-0.543605\pi$
$163$	−3.48697	−0.273120	−0.136560	+	0.990632i	$0.543605\pi$
$164$	1.03783	0.0810407
$165$	7.70841	0.600099
$166$	−2.41966	−0.187802
$167$	−8.76657	−0.678378	−0.339189	−	0.940718i	$-0.610153\pi$
$167$	−8.76657	−0.678378	−0.339189	+	0.940718i	$0.610153\pi$
$168$	−3.03783	−0.234373
$169$	−12.8302	−0.986942
$170$	−2.44149	−0.187254
$171$	2.81638	0.215374
$172$	−4.07565	−0.310766
$173$	−22.7771	−1.73171	−0.865856	−	0.500293i	$-0.833226\pi$
$173$	−22.7771	−1.73171	−0.865856	+	0.500293i	$0.833226\pi$
$174$	3.44984	0.261532
$175$	9.92221	0.750049
$176$	5.85421	0.441277
$177$	−7.92986	−0.596045
$178$	5.30405	0.397555
$179$	19.9746	1.49297	0.746487	−	0.665400i	$-0.231739\pi$
$179$	19.9746	1.49297	0.746487	+	0.665400i	$0.231739\pi$
$180$	1.31673	0.0981433
$181$	12.8485	0.955019	0.477510	−	0.878627i	$-0.341540\pi$
$181$	12.8485	0.955019	0.477510	+	0.878627i	$0.341540\pi$
$182$	1.25163	0.0927767
$183$	3.85421	0.284911
$184$	3.44219	0.253762
$185$	2.26622	0.166616
$186$	−0.904717	−0.0663371
$187$	−10.8549	−0.793790
$188$	−4.00000	−0.291730
$189$	−3.03783	−0.220969
$190$	3.70841	0.269037
$191$	−6.11781	−0.442670	−0.221335	−	0.975198i	$-0.571041\pi$
$191$	−6.11781	−0.442670	−0.221335	+	0.975198i	$0.571041\pi$
$192$	1.00000	0.0721688
$193$	8.62581	0.620900	0.310450	−	0.950590i	$-0.399520\pi$
$193$	8.62581	0.620900	0.310450	+	0.950590i	$0.399520\pi$
$194$	13.1506	0.944158
$195$	−0.542511	−0.0388500
$196$	2.22839	0.159171
$197$	−18.1128	−1.29048	−0.645241	−	0.763979i	$-0.723243\pi$
$197$	−18.1128	−1.29048	−0.645241	+	0.763979i	$0.723243\pi$
$198$	5.85421	0.416040
$199$	15.9557	1.13107	0.565535	−	0.824724i	$-0.308670\pi$
$199$	15.9557	1.13107	0.565535	+	0.824724i	$0.308670\pi$
$200$	−3.26622	−0.230957
$201$	−14.1884	−1.00077
$202$	2.68327	0.188794
$203$	−10.4800	−0.735553
$204$	−1.85421	−0.129820
$205$	1.36654	0.0954432
$206$	0.859944	0.0599151
$207$	3.44219	0.239249
$208$	−0.412014	−0.0285680
$209$	16.4877	1.14048
$210$	−4.00000	−0.276026
$211$	11.2669	0.775647	0.387823	−	0.921734i	$-0.373227\pi$
$211$	11.2669	0.775647	0.387823	+	0.921734i	$0.373227\pi$
$212$	−10.9495	−0.752014
$213$	−11.4422	−0.784006
$214$	−10.4120	−0.711751
$215$	−5.36654	−0.365995
$216$	1.00000	0.0680414
$217$	2.74837	0.186572
$218$	2.77925	0.188235
$219$	14.1513	0.956257
$220$	7.70841	0.519701
$221$	0.763959	0.0513894
$222$	1.72110	0.115512
$223$	13.4372	0.899819	0.449909	−	0.893074i	$-0.351456\pi$
$223$	13.4372	0.899819	0.449909	+	0.893074i	$0.351456\pi$
$224$	−3.03783	−0.202973
$225$	−3.26622	−0.217748
$226$	17.0125	1.13165
$227$	−0.404366	−0.0268387	−0.0134193	−	0.999910i	$-0.504272\pi$
$227$	−0.404366	−0.0268387	−0.0134193	+	0.999910i	$0.504272\pi$
$228$	2.81638	0.186519
$229$	−7.48193	−0.494420	−0.247210	−	0.968962i	$-0.579514\pi$
$229$	−7.48193	−0.494420	−0.247210	+	0.968962i	$0.579514\pi$
$230$	4.53244	0.298860
$231$	−17.7841	−1.17011
$232$	3.44984	0.226493
$233$	8.93821	0.585562	0.292781	−	0.956180i	$-0.405419\pi$
$233$	8.93821	0.585562	0.292781	+	0.956180i	$0.405419\pi$
$234$	−0.412014	−0.0269342
$235$	−5.26692	−0.343576
$236$	−7.92986	−0.516190
$237$	−8.12546	−0.527805
$238$	5.63276	0.365118
$239$	1.98872	0.128640	0.0643198	−	0.997929i	$-0.479512\pi$
$239$	1.98872	0.128640	0.0643198	+	0.997929i	$0.479512\pi$
$240$	1.31673	0.0849946
$241$	17.8851	1.15208	0.576040	−	0.817422i	$-0.304597\pi$
$241$	17.8851	1.15208	0.576040	+	0.817422i	$0.304597\pi$
$242$	23.2717	1.49596
$243$	1.00000	0.0641500
$244$	3.85421	0.246740
$245$	2.93419	0.187459
$246$	1.03783	0.0661694
$247$	−1.16039	−0.0738337
$248$	−0.904717	−0.0574496
$249$	−2.41966	−0.153340
$250$	−10.8844	−0.688389
$251$	−8.84586	−0.558346	−0.279173	−	0.960241i	$-0.590060\pi$
$251$	−8.84586	−0.558346	−0.279173	+	0.960241i	$0.590060\pi$
$252$	−3.03783	−0.191365
$253$	20.1513	1.26690
$254$	−14.2459	−0.893867
$255$	−2.44149	−0.152892
$256$	1.00000	0.0625000
$257$	15.5317	0.968843	0.484422	−	0.874835i	$-0.339030\pi$
$257$	15.5317	0.968843	0.484422	+	0.874835i	$0.339030\pi$
$258$	−4.07565	−0.253739
$259$	−5.22839	−0.324877
$260$	−0.542511	−0.0336451
$261$	3.44984	0.213540
$262$	1.00000	0.0617802
$263$	5.26048	0.324375	0.162188	−	0.986760i	$-0.448145\pi$
$263$	5.26048	0.324375	0.162188	+	0.986760i	$0.448145\pi$
$264$	5.85421	0.360302
$265$	−14.4175	−0.885662
$266$	−8.55567	−0.524582
$267$	5.30405	0.324602
$268$	−14.1884	−0.866697
$269$	25.9852	1.58434	0.792172	−	0.610297i	$-0.208950\pi$
$269$	25.9852	1.58434	0.792172	+	0.610297i	$0.208950\pi$
$270$	1.31673	0.0801337
$271$	−8.25092	−0.501208	−0.250604	−	0.968090i	$-0.580629\pi$
$271$	−8.25092	−0.501208	−0.250604	+	0.968090i	$0.580629\pi$
$272$	−1.85421	−0.112428
$273$	1.25163	0.0757519
$274$	−19.0201	−1.14905
$275$	−19.1211	−1.15305
$276$	3.44219	0.207196
$277$	8.92916	0.536501	0.268251	−	0.963349i	$-0.413554\pi$
$277$	8.92916	0.536501	0.268251	+	0.963349i	$0.413554\pi$
$278$	9.89898	0.593701
$279$	−0.904717	−0.0541640
$280$	−4.00000	−0.239046
$281$	19.7994	1.18113	0.590565	−	0.806990i	$-0.298905\pi$
$281$	19.7994	1.18113	0.590565	+	0.806990i	$0.298905\pi$
$282$	−4.00000	−0.238197
$283$	−21.5681	−1.28209	−0.641046	−	0.767502i	$-0.721499\pi$
$283$	−21.5681	−1.28209	−0.641046	+	0.767502i	$0.721499\pi$
$284$	−11.4422	−0.678969
$285$	3.70841	0.219667
$286$	−2.41201	−0.142625
$287$	−3.15274	−0.186100
$288$	1.00000	0.0589256
$289$	−13.5619	−0.797760
$290$	4.54251	0.266745
$291$	13.1506	0.770902
$292$	14.1513	0.828143
$293$	−15.1968	−0.887806	−0.443903	−	0.896075i	$-0.646406\pi$
$293$	−15.1968	−0.887806	−0.443903	+	0.896075i	$0.646406\pi$
$294$	2.22839	0.129963
$295$	−10.4415	−0.607927
$296$	1.72110	0.100037
$297$	5.85421	0.339696
$298$	−0.161089	−0.00933165
$299$	−1.41823	−0.0820184
$300$	−3.26622	−0.188575
$301$	12.3811	0.713637
$302$	−13.9375	−0.802014
$303$	2.68327	0.154150
$304$	2.81638	0.161530
$305$	5.07495	0.290591
$306$	−1.85421	−0.105998
$307$	19.7994	1.13001	0.565005	−	0.825088i	$-0.308874\pi$
$307$	19.7994	1.13001	0.565005	+	0.825088i	$0.308874\pi$
$308$	−17.7841	−1.01334
$309$	0.859944	0.0489205
$310$	−1.19127	−0.0676595
$311$	−13.0713	−0.741207	−0.370603	−	0.928791i	$-0.620849\pi$
$311$	−13.0713	−0.741207	−0.370603	+	0.928791i	$0.620849\pi$
$312$	−0.412014	−0.0233257
$313$	6.92364	0.391347	0.195674	−	0.980669i	$-0.437311\pi$
$313$	6.92364	0.391347	0.195674	+	0.980669i	$0.437311\pi$
$314$	−6.97272	−0.393494
$315$	−4.00000	−0.225374
$316$	−8.12546	−0.457093
$317$	−2.02585	−0.113783	−0.0568914	−	0.998380i	$-0.518119\pi$
$317$	−2.02585	−0.113783	−0.0568914	+	0.998380i	$0.518119\pi$
$318$	−10.9495	−0.614017
$319$	20.1961	1.13076
$320$	1.31673	0.0736075
$321$	−10.4120	−0.581142
$322$	−10.4568	−0.582734
$323$	−5.22215	−0.290568
$324$	1.00000	0.0555556
$325$	1.34573	0.0746475
$326$	−3.48697	−0.193125
$327$	2.77925	0.153693
$328$	1.03783	0.0573044
$329$	12.1513	0.669923
$330$	7.70841	0.424334
$331$	2.66294	0.146368	0.0731842	−	0.997318i	$-0.476684\pi$
$331$	2.66294	0.146368	0.0731842	+	0.997318i	$0.476684\pi$
$332$	−2.41966	−0.132796
$333$	1.72110	0.0943155
$334$	−8.76657	−0.479685
$335$	−18.6823	−1.02073
$336$	−3.03783	−0.165727
$337$	13.2593	0.722279	0.361139	−	0.932512i	$-0.382388\pi$
$337$	13.2593	0.722279	0.361139	+	0.932512i	$0.382388\pi$
$338$	−12.8302	−0.697873
$339$	17.0125	0.923990
$340$	−2.44149	−0.132408
$341$	−5.29640	−0.286816
$342$	2.81638	0.152292
$343$	14.4953	0.782673
$344$	−4.07565	−0.219745
$345$	4.53244	0.244018
$346$	−22.7771	−1.22451
$347$	35.1638	1.88769	0.943845	−	0.330389i	$-0.107180\pi$
$347$	35.1638	1.88769	0.943845	+	0.330389i	$0.107180\pi$
$348$	3.44984	0.184931
$349$	4.64614	0.248702	0.124351	−	0.992238i	$-0.460315\pi$
$349$	4.64614	0.248702	0.124351	+	0.992238i	$0.460315\pi$
$350$	9.92221	0.530365
$351$	−0.412014	−0.0219917
$352$	5.85421	0.312030
$353$	−9.16377	−0.487738	−0.243869	−	0.969808i	$-0.578417\pi$
$353$	−9.16377	−0.487738	−0.243869	+	0.969808i	$0.578417\pi$
$354$	−7.92986	−0.421467
$355$	−15.0663	−0.799635
$356$	5.30405	0.281114
$357$	5.63276	0.298117
$358$	19.9746	1.05569
$359$	−1.07495	−0.0567338	−0.0283669	−	0.999598i	$-0.509031\pi$
$359$	−1.07495	−0.0567338	−0.0283669	+	0.999598i	$0.509031\pi$
$360$	1.31673	0.0693978
$361$	−11.0680	−0.582527
$362$	12.8485	0.675301
$363$	23.2717	1.22145
$364$	1.25163	0.0656031
$365$	18.6335	0.975320
$366$	3.85421	0.201463
$367$	6.56957	0.342929	0.171464	−	0.985190i	$-0.445150\pi$
$367$	6.56957	0.342929	0.171464	+	0.985190i	$0.445150\pi$
$368$	3.44219	0.179437
$369$	1.03783	0.0540271
$370$	2.26622	0.117815
$371$	33.2627	1.72691
$372$	−0.904717	−0.0469074
$373$	25.1917	1.30438	0.652190	−	0.758056i	$-0.273851\pi$
$373$	25.1917	1.30438	0.652190	+	0.758056i	$0.273851\pi$
$374$	−10.8549	−0.561294
$375$	−10.8844	−0.562067
$376$	−4.00000	−0.206284
$377$	−1.42138	−0.0732049
$378$	−3.03783	−0.156249
$379$	−7.69452	−0.395241	−0.197621	−	0.980279i	$-0.563321\pi$
$379$	−7.69452	−0.395241	−0.197621	+	0.980279i	$0.563321\pi$
$380$	3.70841	0.190238
$381$	−14.2459	−0.729839
$382$	−6.11781	−0.313015
$383$	12.0960	0.618076	0.309038	−	0.951050i	$-0.399993\pi$
$383$	12.0960	0.618076	0.309038	+	0.951050i	$0.399993\pi$
$384$	1.00000	0.0510310
$385$	−23.4168	−1.19343
$386$	8.62581	0.439042
$387$	−4.07565	−0.207177
$388$	13.1506	0.667621
$389$	−23.0433	−1.16834	−0.584172	−	0.811630i	$-0.698581\pi$
$389$	−23.0433	−1.16834	−0.584172	+	0.811630i	$0.698581\pi$
$390$	−0.542511	−0.0274711
$391$	−6.38254	−0.322779
$392$	2.22839	0.112551
$393$	1.00000	0.0504433
$394$	−18.1128	−0.912509
$395$	−10.6990	−0.538327
$396$	5.85421	0.294185
$397$	15.2669	0.766225	0.383112	−	0.923702i	$-0.374852\pi$
$397$	15.2669	0.766225	0.383112	+	0.923702i	$0.374852\pi$
$398$	15.9557	0.799787
$399$	−8.55567	−0.428319
$400$	−3.26622	−0.163311
$401$	16.8396	0.840930	0.420465	−	0.907309i	$-0.361867\pi$
$401$	16.8396	0.840930	0.420465	+	0.907309i	$0.361867\pi$
$402$	−14.1884	−0.707655
$403$	0.372756	0.0185683
$404$	2.68327	0.133498
$405$	1.31673	0.0654289
$406$	−10.4800	−0.520115
$407$	10.0757	0.499432
$408$	−1.85421	−0.0917969
$409$	35.3797	1.74941	0.874707	−	0.484652i	$-0.161054\pi$
$409$	35.3797	1.74941	0.874707	+	0.484652i	$0.161054\pi$
$410$	1.36654	0.0674885
$411$	−19.0201	−0.938193
$412$	0.859944	0.0423664
$413$	24.0895	1.18537
$414$	3.44219	0.169175
$415$	−3.18604	−0.156397
$416$	−0.412014	−0.0202006
$417$	9.89898	0.484755
$418$	16.4877	0.806438
$419$	−15.0496	−0.735220	−0.367610	−	0.929980i	$-0.619824\pi$
$419$	−15.0496	−0.735220	−0.367610	+	0.929980i	$0.619824\pi$
$420$	−4.00000	−0.195180
$421$	3.19678	0.155802	0.0779008	−	0.996961i	$-0.475178\pi$
$421$	3.19678	0.155802	0.0779008	+	0.996961i	$0.475178\pi$
$422$	11.2669	0.548465
$423$	−4.00000	−0.194487
$424$	−10.9495	−0.531754
$425$	6.05625	0.293771
$426$	−11.4422	−0.554376
$427$	−11.7084	−0.566610
$428$	−10.4120	−0.503284
$429$	−2.41201	−0.116453
$430$	−5.36654	−0.258797
$431$	−26.8106	−1.29142	−0.645711	−	0.763582i	$-0.723439\pi$
$431$	−26.8106	−1.29142	−0.645711	+	0.763582i	$0.723439\pi$
$432$	1.00000	0.0481125
$433$	−22.4328	−1.07805	−0.539026	−	0.842289i	$-0.681208\pi$
$433$	−22.4328	−1.07805	−0.539026	+	0.842289i	$0.681208\pi$
$434$	2.74837	0.131926
$435$	4.54251	0.217797
$436$	2.77925	0.133102
$437$	9.69452	0.463752
$438$	14.1513	0.676176
$439$	5.11348	0.244053	0.122027	−	0.992527i	$-0.461061\pi$
$439$	5.11348	0.244053	0.122027	+	0.992527i	$0.461061\pi$
$440$	7.70841	0.367484
$441$	2.22839	0.106114
$442$	0.763959	0.0363378
$443$	−28.5049	−1.35431	−0.677155	−	0.735840i	$-0.736787\pi$
$443$	−28.5049	−1.35431	−0.677155	+	0.735840i	$0.736787\pi$
$444$	1.72110	0.0816796
$445$	6.98400	0.331073
$446$	13.4372	0.636268
$447$	−0.161089	−0.00761926
$448$	−3.03783	−0.143524
$449$	17.4884	0.825327	0.412664	−	0.910883i	$-0.364599\pi$
$449$	17.4884	0.825327	0.412664	+	0.910883i	$0.364599\pi$
$450$	−3.26622	−0.153971
$451$	6.07565	0.286091
$452$	17.0125	0.800199
$453$	−13.9375	−0.654841
$454$	−0.404366	−0.0189778
$455$	1.64806	0.0772620
$456$	2.81638	0.131889
$457$	13.8480	0.647780	0.323890	−	0.946095i	$-0.395009\pi$
$457$	13.8480	0.647780	0.323890	+	0.946095i	$0.395009\pi$
$458$	−7.48193	−0.349608
$459$	−1.85421	−0.0865470
$460$	4.53244	0.211326
$461$	33.1791	1.54530	0.772652	−	0.634830i	$-0.218930\pi$
$461$	33.1791	1.54530	0.772652	+	0.634830i	$0.218930\pi$
$462$	−17.7841	−0.827390
$463$	17.3375	0.805744	0.402872	−	0.915256i	$-0.368012\pi$
$463$	17.3375	0.805744	0.402872	+	0.915256i	$0.368012\pi$
$464$	3.44984	0.160155
$465$	−1.19127	−0.0552438
$466$	8.93821	0.414055
$467$	15.6230	0.722945	0.361473	−	0.932383i	$-0.382274\pi$
$467$	15.6230	0.722945	0.361473	+	0.932383i	$0.382274\pi$
$468$	−0.412014	−0.0190453
$469$	43.1020	1.99027
$470$	−5.26692	−0.242945
$471$	−6.97272	−0.321286
$472$	−7.92986	−0.365001
$473$	−23.8597	−1.09707
$474$	−8.12546	−0.373215
$475$	−9.19892	−0.422075
$476$	5.63276	0.258177
$477$	−10.9495	−0.501343
$478$	1.98872	0.0909620
$479$	21.3665	0.976262	0.488131	−	0.872770i	$-0.337679\pi$
$479$	21.3665	0.976262	0.488131	+	0.872770i	$0.337679\pi$
$480$	1.31673	0.0601003
$481$	−0.709115	−0.0323329
$482$	17.8851	0.814643
$483$	−10.4568	−0.475800
$484$	23.2717	1.05781
$485$	17.3158	0.786270
$486$	1.00000	0.0453609
$487$	−25.6692	−1.16318	−0.581590	−	0.813482i	$-0.697569\pi$
$487$	−25.6692	−1.16318	−0.581590	+	0.813482i	$0.697569\pi$
$488$	3.85421	0.174472
$489$	−3.48697	−0.157686
$490$	2.93419	0.132553
$491$	5.59423	0.252464	0.126232	−	0.992001i	$-0.459712\pi$
$491$	5.59423	0.252464	0.126232	+	0.992001i	$0.459712\pi$
$492$	1.03783	0.0467889
$493$	−6.39672	−0.288094
$494$	−1.16039	−0.0522083
$495$	7.70841	0.346467
$496$	−0.904717	−0.0406230
$497$	34.7594	1.55917
$498$	−2.41966	−0.108428
$499$	−3.83649	−0.171745	−0.0858724	−	0.996306i	$-0.527368\pi$
$499$	−3.83649	−0.171745	−0.0858724	+	0.996306i	$0.527368\pi$
$500$	−10.8844	−0.486765
$501$	−8.76657	−0.391662
$502$	−8.84586	−0.394810
$503$	15.8597	0.707150	0.353575	−	0.935406i	$-0.384966\pi$
$503$	15.8597	0.707150	0.353575	+	0.935406i	$0.384966\pi$
$504$	−3.03783	−0.135316
$505$	3.53314	0.157223
$506$	20.1513	0.895835
$507$	−12.8302	−0.569811
$508$	−14.2459	−0.632059
$509$	−10.2641	−0.454948	−0.227474	−	0.973784i	$-0.573047\pi$
$509$	−10.2641	−0.454948	−0.227474	+	0.973784i	$0.573047\pi$
$510$	−2.44149	−0.108111
$511$	−42.9892	−1.90173
$512$	1.00000	0.0441942
$513$	2.81638	0.124346
$514$	15.5317	0.685076
$515$	1.13231	0.0498957
$516$	−4.07565	−0.179421
$517$	−23.4168	−1.02987
$518$	−5.22839	−0.229722
$519$	−22.7771	−0.999805
$520$	−0.542511	−0.0237907
$521$	18.4106	0.806583	0.403292	−	0.915071i	$-0.367866\pi$
$521$	18.4106	0.806583	0.403292	+	0.915071i	$0.367866\pi$
$522$	3.44984	0.150995
$523$	33.7393	1.47532	0.737658	−	0.675174i	$-0.235932\pi$
$523$	33.7393	1.47532	0.737658	+	0.675174i	$0.235932\pi$
$524$	1.00000	0.0436852
$525$	9.92221	0.433041
$526$	5.26048	0.229368
$527$	1.67753	0.0730744
$528$	5.85421	0.254772
$529$	−11.1513	−0.484839
$530$	−14.4175	−0.626257
$531$	−7.92986	−0.344127
$532$	−8.55567	−0.370936
$533$	−0.427599	−0.0185214
$534$	5.30405	0.229529
$535$	−13.7098	−0.592727
$536$	−14.1884	−0.612847
$537$	19.9746	0.861969
$538$	25.9852	1.12030
$539$	13.0455	0.561908
$540$	1.31673	0.0566631
$541$	−27.5794	−1.18573	−0.592866	−	0.805301i	$-0.702004\pi$
$541$	−27.5794	−1.18573	−0.592866	+	0.805301i	$0.702004\pi$
$542$	−8.25092	−0.354408
$543$	12.8485	0.551381
$544$	−1.85421	−0.0794985
$545$	3.65953	0.156757
$546$	1.25163	0.0535647
$547$	−8.94799	−0.382589	−0.191294	−	0.981533i	$-0.561268\pi$
$547$	−8.94799	−0.382589	−0.191294	+	0.981533i	$0.561268\pi$
$548$	−19.0201	−0.812499
$549$	3.85421	0.164494
$550$	−19.1211	−0.815328
$551$	9.71606	0.413918
$552$	3.44219	0.146509
$553$	24.6837	1.04966
$554$	8.92916	0.379364
$555$	2.26622	0.0961957
$556$	9.89898	0.419810
$557$	36.7700	1.55799	0.778996	−	0.627028i	$-0.215729\pi$
$557$	36.7700	1.55799	0.778996	+	0.627028i	$0.215729\pi$
$558$	−0.904717	−0.0382997
$559$	1.67923	0.0710237
$560$	−4.00000	−0.169031
$561$	−10.8549	−0.458295
$562$	19.7994	0.835186
$563$	19.2430	0.810994	0.405497	−	0.914096i	$-0.367098\pi$
$563$	19.2430	0.810994	0.405497	+	0.914096i	$0.367098\pi$
$564$	−4.00000	−0.168430
$565$	22.4008	0.942410
$566$	−21.5681	−0.906576
$567$	−3.03783	−0.127577
$568$	−11.4422	−0.480104
$569$	15.4793	0.648927	0.324463	−	0.945898i	$-0.394816\pi$
$569$	15.4793	0.648927	0.324463	+	0.945898i	$0.394816\pi$
$570$	3.70841	0.155328
$571$	−32.2040	−1.34770	−0.673848	−	0.738870i	$-0.735360\pi$
$571$	−32.2040	−1.34770	−0.673848	+	0.738870i	$0.735360\pi$
$572$	−2.41201	−0.100851
$573$	−6.11781	−0.255575
$574$	−3.15274	−0.131593
$575$	−11.2430	−0.468864
$576$	1.00000	0.0416667
$577$	−36.5008	−1.51955	−0.759775	−	0.650186i	$-0.774691\pi$
$577$	−36.5008	−1.51955	−0.759775	+	0.650186i	$0.774691\pi$
$578$	−13.5619	−0.564101
$579$	8.62581	0.358477
$580$	4.54251	0.188618
$581$	7.35051	0.304951
$582$	13.1506	0.545110
$583$	−64.1006	−2.65477
$584$	14.1513	0.585585
$585$	−0.542511	−0.0224301
$586$	−15.1968	−0.627773
$587$	−18.1822	−0.750459	−0.375230	−	0.926932i	$-0.622436\pi$
$587$	−18.1822	−0.750459	−0.375230	+	0.926932i	$0.622436\pi$
$588$	2.22839	0.0918974
$589$	−2.54803	−0.104990
$590$	−10.4415	−0.429869
$591$	−18.1128	−0.745060
$592$	1.72110	0.0707366
$593$	−18.7285	−0.769088	−0.384544	−	0.923107i	$-0.625641\pi$
$593$	−18.7285	−0.769088	−0.384544	+	0.923107i	$0.625641\pi$
$594$	5.85421	0.240201
$595$	7.41683	0.304060
$596$	−0.161089	−0.00659848
$597$	15.9557	0.653024
$598$	−1.41823	−0.0579958
$599$	36.7707	1.50241	0.751205	−	0.660069i	$-0.229473\pi$
$599$	36.7707	1.50241	0.751205	+	0.660069i	$0.229473\pi$
$600$	−3.26622	−0.133343
$601$	−27.5844	−1.12519	−0.562597	−	0.826732i	$-0.690198\pi$
$601$	−27.5844	−1.12519	−0.562597	+	0.826732i	$0.690198\pi$
$602$	12.3811	0.504617
$603$	−14.1884	−0.577798
$604$	−13.9375	−0.567109
$605$	30.6426	1.24580
$606$	2.68327	0.109000
$607$	5.21571	0.211699	0.105850	−	0.994382i	$-0.466244\pi$
$607$	5.21571	0.211699	0.105850	+	0.994382i	$0.466244\pi$
$608$	2.81638	0.114219
$609$	−10.4800	−0.424672
$610$	5.07495	0.205479
$611$	1.64806	0.0666732
$612$	−1.85421	−0.0749519
$613$	−8.40194	−0.339351	−0.169676	−	0.985500i	$-0.554272\pi$
$613$	−8.40194	−0.339351	−0.169676	+	0.985500i	$0.554272\pi$
$614$	19.7994	0.799037
$615$	1.36654	0.0551042
$616$	−17.7841	−0.716540
$617$	−16.5775	−0.667385	−0.333693	−	0.942682i	$-0.608295\pi$
$617$	−16.5775	−0.667385	−0.333693	+	0.942682i	$0.608295\pi$
$618$	0.859944	0.0345920
$619$	19.1444	0.769477	0.384738	−	0.923026i	$-0.374292\pi$
$619$	19.1444	0.769477	0.384738	+	0.923026i	$0.374292\pi$
$620$	−1.19127	−0.0478425
$621$	3.44219	0.138130
$622$	−13.0713	−0.524112
$623$	−16.1128	−0.645545
$624$	−0.412014	−0.0164938
$625$	1.99930	0.0799719
$626$	6.92364	0.276724
$627$	16.4877	0.658454
$628$	−6.97272	−0.278242
$629$	−3.19127	−0.127244
$630$	−4.00000	−0.159364
$631$	−38.3033	−1.52483	−0.762416	−	0.647087i	$-0.775987\pi$
$631$	−38.3033	−1.52483	−0.762416	+	0.647087i	$0.775987\pi$
$632$	−8.12546	−0.323214
$633$	11.2669	0.447820
$634$	−2.02585	−0.0804566
$635$	−18.7580	−0.744388
$636$	−10.9495	−0.434176
$637$	−0.918129	−0.0363776
$638$	20.1961	0.799571
$639$	−11.4422	−0.452646
$640$	1.31673	0.0520483
$641$	−14.3200	−0.565608	−0.282804	−	0.959178i	$-0.591265\pi$
$641$	−14.3200	−0.565608	−0.282804	+	0.959178i	$0.591265\pi$
$642$	−10.4120	−0.410929
$643$	36.5213	1.44026	0.720129	−	0.693840i	$-0.244083\pi$
$643$	36.5213	1.44026	0.720129	+	0.693840i	$0.244083\pi$
$644$	−10.4568	−0.412055
$645$	−5.36654	−0.211307
$646$	−5.22215	−0.205463
$647$	20.6009	0.809906	0.404953	−	0.914338i	$-0.367288\pi$
$647$	20.6009	0.809906	0.404953	+	0.914338i	$0.367288\pi$
$648$	1.00000	0.0392837
$649$	−46.4230	−1.82226
$650$	1.34573	0.0527838
$651$	2.74837	0.107717
$652$	−3.48697	−0.136560
$653$	28.0455	1.09751	0.548753	−	0.835984i	$-0.315103\pi$
$653$	28.0455	1.09751	0.548753	+	0.835984i	$0.315103\pi$
$654$	2.77925	0.108677
$655$	1.31673	0.0514489
$656$	1.03783	0.0405203
$657$	14.1513	0.552095
$658$	12.1513	0.473707
$659$	12.8410	0.500215	0.250108	−	0.968218i	$-0.419534\pi$
$659$	12.8410	0.500215	0.250108	+	0.968218i	$0.419534\pi$
$660$	7.70841	0.300050
$661$	30.3489	1.18043	0.590217	−	0.807244i	$-0.299042\pi$
$661$	30.3489	1.18043	0.590217	+	0.807244i	$0.299042\pi$
$662$	2.66294	0.103498
$663$	0.763959	0.0296697
$664$	−2.41966	−0.0939011
$665$	−11.2655	−0.436858
$666$	1.72110	0.0666911
$667$	11.8750	0.459803
$668$	−8.76657	−0.339189
$669$	13.4372	0.519511
$670$	−18.6823	−0.721762
$671$	22.5633	0.871047
$672$	−3.03783	−0.117187
$673$	33.6830	1.29839	0.649193	−	0.760624i	$-0.275107\pi$
$673$	33.6830	1.29839	0.649193	+	0.760624i	$0.275107\pi$
$674$	13.2593	0.510728
$675$	−3.26622	−0.125717
$676$	−12.8302	−0.493471
$677$	−38.6969	−1.48724	−0.743622	−	0.668601i	$-0.766894\pi$
$677$	−38.6969	−1.48724	−0.743622	+	0.668601i	$0.766894\pi$
$678$	17.0125	0.653360
$679$	−39.9493	−1.53311
$680$	−2.44149	−0.0936269
$681$	−0.404366	−0.0154953
$682$	−5.29640	−0.202810
$683$	−16.1961	−0.619726	−0.309863	−	0.950781i	$-0.600283\pi$
$683$	−16.1961	−0.619726	−0.309863	+	0.950781i	$0.600283\pi$
$684$	2.81638	0.107687
$685$	−25.0444	−0.956896
$686$	14.4953	0.553434
$687$	−7.48193	−0.285453
$688$	−4.07565	−0.155383
$689$	4.51134	0.171868
$690$	4.53244	0.172547
$691$	−41.8597	−1.59242	−0.796209	−	0.605021i	$-0.793165\pi$
$691$	−41.8597	−1.59242	−0.796209	+	0.605021i	$0.793165\pi$
$692$	−22.7771	−0.865856
$693$	−17.7841	−0.675561
$694$	35.1638	1.33480
$695$	13.0343	0.494419
$696$	3.44984	0.130766
$697$	−1.92435	−0.0728898
$698$	4.64614	0.175859
$699$	8.93821	0.338074
$700$	9.92221	0.375024
$701$	−17.4934	−0.660717	−0.330358	−	0.943856i	$-0.607170\pi$
$701$	−17.4934	−0.660717	−0.330358	+	0.943856i	$0.607170\pi$
$702$	−0.412014	−0.0155505
$703$	4.84726	0.182818
$704$	5.85421	0.220639
$705$	−5.26692	−0.198364
$706$	−9.16377	−0.344883
$707$	−8.15131	−0.306562
$708$	−7.92986	−0.298022
$709$	52.2274	1.96144	0.980721	−	0.195411i	$-0.0626042\pi$
$709$	52.2274	1.96144	0.980721	+	0.195411i	$0.0626042\pi$
$710$	−15.0663	−0.565428
$711$	−8.12546	−0.304729
$712$	5.30405	0.198778
$713$	−3.11421	−0.116628
$714$	5.63276	0.210801
$715$	−3.17597	−0.118775
$716$	19.9746	0.746487
$717$	1.98872	0.0742702
$718$	−1.07495	−0.0401169
$719$	−4.26361	−0.159006	−0.0795029	−	0.996835i	$-0.525333\pi$
$719$	−4.26361	−0.159006	−0.0795029	+	0.996835i	$0.525333\pi$
$720$	1.31673	0.0490717
$721$	−2.61236	−0.0972894
$722$	−11.0680	−0.411909
$723$	17.8851	0.665154
$724$	12.8485	0.477510
$725$	−11.2679	−0.418481
$726$	23.2717	0.863695
$727$	−50.0455	−1.85609	−0.928043	−	0.372473i	$-0.878510\pi$
$727$	−50.0455	−1.85609	−0.928043	+	0.372473i	$0.878510\pi$
$728$	1.25163	0.0463884
$729$	1.00000	0.0370370
$730$	18.6335	0.689655
$731$	7.55710	0.279510
$732$	3.85421	0.142456
$733$	49.3816	1.82395	0.911975	−	0.410245i	$-0.134557\pi$
$733$	49.3816	1.82395	0.911975	+	0.410245i	$0.134557\pi$
$734$	6.56957	0.242487
$735$	2.93419	0.108229
$736$	3.44219	0.126881
$737$	−83.0620	−3.05963
$738$	1.03783	0.0382029
$739$	−33.4633	−1.23097	−0.615484	−	0.788150i	$-0.711039\pi$
$739$	−33.4633	−1.23097	−0.615484	+	0.788150i	$0.711039\pi$
$740$	2.26622	0.0833079
$741$	−1.16039	−0.0426279
$742$	33.2627	1.22111
$743$	37.5667	1.37819	0.689095	−	0.724671i	$-0.258008\pi$
$743$	37.5667	1.37819	0.689095	+	0.724671i	$0.258008\pi$
$744$	−0.904717	−0.0331685
$745$	−0.212111	−0.00777116
$746$	25.1917	0.922335
$747$	−2.41966	−0.0885308
$748$	−10.8549	−0.396895
$749$	31.6299	1.15573
$750$	−10.8844	−0.397442
$751$	37.6627	1.37433	0.687166	−	0.726500i	$-0.258854\pi$
$751$	37.6627	1.37433	0.687166	+	0.726500i	$0.258854\pi$
$752$	−4.00000	−0.145865
$753$	−8.84586	−0.322361
$754$	−1.42138	−0.0517637
$755$	−18.3519	−0.667896
$756$	−3.03783	−0.110485
$757$	−1.25134	−0.0454806	−0.0227403	−	0.999741i	$-0.507239\pi$
$757$	−1.25134	−0.0454806	−0.0227403	+	0.999741i	$0.507239\pi$
$758$	−7.69452	−0.279478
$759$	20.1513	0.731446
$760$	3.70841	0.134518
$761$	31.5133	1.14236	0.571178	−	0.820826i	$-0.306487\pi$
$761$	31.5133	1.14236	0.571178	+	0.820826i	$0.306487\pi$
$762$	−14.2459	−0.516074
$763$	−8.44290	−0.305653
$764$	−6.11781	−0.221335
$765$	−2.44149	−0.0882723
$766$	12.0960	0.437046
$767$	3.26721	0.117972
$768$	1.00000	0.0360844
$769$	−24.6761	−0.889843	−0.444921	−	0.895570i	$-0.646768\pi$
$769$	−24.6761	−0.889843	−0.444921	+	0.895570i	$0.646768\pi$
$770$	−23.4168	−0.843884
$771$	15.5317	0.559362
$772$	8.62581	0.310450
$773$	−50.5372	−1.81770	−0.908849	−	0.417126i	$-0.863038\pi$
$773$	−50.5372	−1.81770	−0.908849	+	0.417126i	$0.863038\pi$
$774$	−4.07565	−0.146496
$775$	2.95500	0.106147
$776$	13.1506	0.472079
$777$	−5.22839	−0.187568
$778$	−23.0433	−0.826144
$779$	2.92291	0.104724
$780$	−0.542511	−0.0194250
$781$	−66.9850	−2.39691
$782$	−6.38254	−0.228239
$783$	3.44984	0.123287
$784$	2.22839	0.0795855
$785$	−9.18120	−0.327691
$786$	1.00000	0.0356688
$787$	2.35194	0.0838378	0.0419189	−	0.999121i	$-0.486653\pi$
$787$	2.35194	0.0838378	0.0419189	+	0.999121i	$0.486653\pi$
$788$	−18.1128	−0.645241
$789$	5.26048	0.187278
$790$	−10.6990	−0.380655
$791$	−51.6809	−1.83756
$792$	5.85421	0.208020
$793$	−1.58799	−0.0563910
$794$	15.2669	0.541803
$795$	−14.4175	−0.511337
$796$	15.9557	0.565535
$797$	−54.3309	−1.92450	−0.962249	−	0.272170i	$-0.912259\pi$
$797$	−54.3309	−1.92450	−0.962249	+	0.272170i	$0.912259\pi$
$798$	−8.55567	−0.302868
$799$	7.41683	0.262388
$800$	−3.26622	−0.115478
$801$	5.30405	0.187409
$802$	16.8396	0.594627
$803$	82.8447	2.92353
$804$	−14.1884	−0.500387
$805$	−13.7688	−0.485285
$806$	0.372756	0.0131298
$807$	25.9852	0.914722
$808$	2.68327	0.0943971
$809$	−18.4932	−0.650186	−0.325093	−	0.945682i	$-0.605396\pi$
$809$	−18.4932	−0.650186	−0.325093	+	0.945682i	$0.605396\pi$
$810$	1.31673	0.0462652
$811$	−7.70841	−0.270679	−0.135340	−	0.990799i	$-0.543213\pi$
$811$	−7.70841	−0.270679	−0.135340	+	0.990799i	$0.543213\pi$
$812$	−10.4800	−0.367777
$813$	−8.25092	−0.289373
$814$	10.0757	0.353151
$815$	−4.59139	−0.160830
$816$	−1.85421	−0.0649102
$817$	−11.4786	−0.401585
$818$	35.3797	1.23702
$819$	1.25163	0.0437354
$820$	1.36654	0.0477216
$821$	−46.0622	−1.60758	−0.803792	−	0.594911i	$-0.797187\pi$
$821$	−46.0622	−1.60758	−0.803792	+	0.594911i	$0.797187\pi$
$822$	−19.0201	−0.663402
$823$	17.9543	0.625848	0.312924	−	0.949778i	$-0.398692\pi$
$823$	17.9543	0.625848	0.312924	+	0.949778i	$0.398692\pi$
$824$	0.859944	0.0299576
$825$	−19.1211	−0.665712
$826$	24.0895	0.838182
$827$	19.2978	0.671050	0.335525	−	0.942031i	$-0.391086\pi$
$827$	19.2978	0.671050	0.335525	+	0.942031i	$0.391086\pi$
$828$	3.44219	0.119624
$829$	−33.6689	−1.16937	−0.584684	−	0.811261i	$-0.698782\pi$
$829$	−33.6689	−1.16937	−0.584684	+	0.811261i	$0.698782\pi$
$830$	−3.18604	−0.110589
$831$	8.92916	0.309749
$832$	−0.412014	−0.0142840
$833$	−4.13190	−0.143162
$834$	9.89898	0.342774
$835$	−11.5432	−0.399469
$836$	16.4877	0.570238
$837$	−0.904717	−0.0312716
$838$	−15.0496	−0.519879
$839$	−51.7922	−1.78807	−0.894033	−	0.448002i	$-0.852136\pi$
$839$	−51.7922	−1.78807	−0.894033	+	0.448002i	$0.852136\pi$
$840$	−4.00000	−0.138013
$841$	−17.0986	−0.589607
$842$	3.19678	0.110168
$843$	19.7994	0.681926
$844$	11.2669	0.387823
$845$	−16.8940	−0.581170
$846$	−4.00000	−0.137523
$847$	−70.6955	−2.42913
$848$	−10.9495	−0.376007
$849$	−21.5681	−0.740216
$850$	6.05625	0.207728
$851$	5.92435	0.203084
$852$	−11.4422	−0.392003
$853$	−6.16612	−0.211124	−0.105562	−	0.994413i	$-0.533664\pi$
$853$	−6.16612	−0.211124	−0.105562	+	0.994413i	$0.533664\pi$
$854$	−11.7084	−0.400654
$855$	3.70841	0.126825
$856$	−10.4120	−0.355875
$857$	−54.9235	−1.87615	−0.938075	−	0.346431i	$-0.887393\pi$
$857$	−54.9235	−1.87615	−0.938075	+	0.346431i	$0.887393\pi$
$858$	−2.41201	−0.0823448
$859$	46.9281	1.60117	0.800584	−	0.599221i	$-0.204523\pi$
$859$	46.9281	1.60117	0.800584	+	0.599221i	$0.204523\pi$
$860$	−5.36654	−0.182997
$861$	−3.15274	−0.107445
$862$	−26.8106	−0.913173
$863$	−49.4503	−1.68331	−0.841654	−	0.540017i	$-0.818418\pi$
$863$	−49.4503	−1.68331	−0.841654	+	0.540017i	$0.818418\pi$
$864$	1.00000	0.0340207
$865$	−29.9913	−1.01974
$866$	−22.4328	−0.762298
$867$	−13.5619	−0.460587
$868$	2.74837	0.0932859
$869$	−47.5681	−1.61364
$870$	4.54251	0.154006
$871$	5.84583	0.198078
$872$	2.77925	0.0941174
$873$	13.1506	0.445081
$874$	9.69452	0.327922
$875$	33.0649	1.11780
$876$	14.1513	0.478128
$877$	34.7397	1.17308	0.586538	−	0.809922i	$-0.300490\pi$
$877$	34.7397	1.17308	0.586538	+	0.809922i	$0.300490\pi$
$878$	5.11348	0.172572
$879$	−15.1968	−0.512575
$880$	7.70841	0.259851
$881$	−24.8729	−0.837990	−0.418995	−	0.907989i	$-0.637617\pi$
$881$	−24.8729	−0.837990	−0.418995	+	0.907989i	$0.637617\pi$
$882$	2.22839	0.0750339
$883$	31.1037	1.04672	0.523361	−	0.852111i	$-0.324678\pi$
$883$	31.1037	1.04672	0.523361	+	0.852111i	$0.324678\pi$
$884$	0.763959	0.0256947
$885$	−10.4415	−0.350987
$886$	−28.5049	−0.957642
$887$	−47.5634	−1.59702	−0.798511	−	0.601980i	$-0.794379\pi$
$887$	−47.5634	−1.59702	−0.798511	+	0.601980i	$0.794379\pi$
$888$	1.72110	0.0577562
$889$	43.2765	1.45145
$890$	6.98400	0.234104
$891$	5.85421	0.196123
$892$	13.4372	0.449909
$893$	−11.2655	−0.376986
$894$	−0.161089	−0.00538763
$895$	26.3012	0.879153
$896$	−3.03783	−0.101487
$897$	−1.41823	−0.0473534
$898$	17.4884	0.583594
$899$	−3.12113	−0.104095
$900$	−3.26622	−0.108874
$901$	20.3026	0.676378
$902$	6.07565	0.202297
$903$	12.3811	0.412018
$904$	17.0125	0.565826
$905$	16.9180	0.562372
$906$	−13.9375	−0.463043
$907$	−50.9739	−1.69256	−0.846281	−	0.532737i	$-0.821163\pi$
$907$	−50.9739	−1.69256	−0.846281	+	0.532737i	$0.821163\pi$
$908$	−0.404366	−0.0134193
$909$	2.68327	0.0889984
$910$	1.64806	0.0546325
$911$	20.2552	0.671086	0.335543	−	0.942025i	$-0.391080\pi$
$911$	20.2552	0.671086	0.335543	+	0.942025i	$0.391080\pi$
$912$	2.81638	0.0932596
$913$	−14.1652	−0.468800
$914$	13.8480	0.458050
$915$	5.07495	0.167773
$916$	−7.48193	−0.247210
$917$	−3.03783	−0.100318
$918$	−1.85421	−0.0611979
$919$	35.4358	1.16892	0.584459	−	0.811423i	$-0.301307\pi$
$919$	35.4358	1.16892	0.584459	+	0.811423i	$0.301307\pi$
$920$	4.53244	0.149430
$921$	19.7994	0.652411
$922$	33.1791	1.09269
$923$	4.71434	0.155174
$924$	−17.7841	−0.585053
$925$	−5.62148	−0.184833
$926$	17.3375	0.569747
$927$	0.859944	0.0282443
$928$	3.44984	0.113247
$929$	49.8854	1.63669	0.818343	−	0.574730i	$-0.194893\pi$
$929$	49.8854	1.63669	0.818343	+	0.574730i	$0.194893\pi$
$930$	−1.19127	−0.0390632
$931$	6.27600	0.205688
$932$	8.93821	0.292781
$933$	−13.0713	−0.427936
$934$	15.6230	0.511199
$935$	−14.2930	−0.467431
$936$	−0.412014	−0.0134671
$937$	−1.20032	−0.0392128	−0.0196064	−	0.999808i	$-0.506241\pi$
$937$	−1.20032	−0.0392128	−0.0196064	+	0.999808i	$0.506241\pi$
$938$	43.1020	1.40733
$939$	6.92364	0.225945
$940$	−5.26692	−0.171788
$941$	11.7903	0.384353	0.192177	−	0.981360i	$-0.438445\pi$
$941$	11.7903	0.384353	0.192177	+	0.981360i	$0.438445\pi$
$942$	−6.97272	−0.227184
$943$	3.57240	0.116333
$944$	−7.92986	−0.258095
$945$	−4.00000	−0.130120
$946$	−23.8597	−0.775746
$947$	17.0867	0.555243	0.277622	−	0.960690i	$-0.410454\pi$
$947$	17.0867	0.555243	0.277622	+	0.960690i	$0.410454\pi$
$948$	−8.12546	−0.263903
$949$	−5.83053	−0.189267
$950$	−9.19892	−0.298452
$951$	−2.02585	−0.0656926
$952$	5.63276	0.182559
$953$	42.7559	1.38500	0.692499	−	0.721419i	$-0.256510\pi$
$953$	42.7559	1.38500	0.692499	+	0.721419i	$0.256510\pi$
$954$	−10.9495	−0.354503
$955$	−8.05551	−0.260670
$956$	1.98872	0.0643198
$957$	20.1961	0.652847
$958$	21.3665	0.690322
$959$	57.7798	1.86581
$960$	1.31673	0.0424973
$961$	−30.1815	−0.973596
$962$	−0.709115	−0.0228628
$963$	−10.4120	−0.335522
$964$	17.8851	0.576040
$965$	11.3579	0.365623
$966$	−10.4568	−0.336442
$967$	5.53677	0.178051	0.0890253	−	0.996029i	$-0.471625\pi$
$967$	5.53677	0.178051	0.0890253	+	0.996029i	$0.471625\pi$
$968$	23.2717	0.747982
$969$	−5.22215	−0.167760
$970$	17.3158	0.555977
$971$	−35.8322	−1.14991	−0.574955	−	0.818185i	$-0.694981\pi$
$971$	−35.8322	−1.14991	−0.574955	+	0.818185i	$0.694981\pi$
$972$	1.00000	0.0320750
$973$	−30.0714	−0.964044
$974$	−25.6692	−0.822493
$975$	1.34573	0.0430978
$976$	3.85421	0.123370
$977$	−18.0021	−0.575939	−0.287970	−	0.957640i	$-0.592980\pi$
$977$	−18.0021	−0.575939	−0.287970	+	0.957640i	$0.592980\pi$
$978$	−3.48697	−0.111501
$979$	31.0510	0.992394
$980$	2.93419	0.0937294
$981$	2.77925	0.0887348
$982$	5.59423	0.178519
$983$	17.8736	0.570080	0.285040	−	0.958516i	$-0.407993\pi$
$983$	17.8736	0.570080	0.285040	+	0.958516i	$0.407993\pi$
$984$	1.03783	0.0330847
$985$	−23.8497	−0.759913
$986$	−6.39672	−0.203713
$987$	12.1513	0.386780
$988$	−1.16039	−0.0369168
$989$	−14.0292	−0.446102
$990$	7.70841	0.244989
$991$	−44.3922	−1.41016	−0.705082	−	0.709126i	$-0.749090\pi$
$991$	−44.3922	−1.41016	−0.705082	+	0.709126i	$0.749090\pi$
$992$	−0.904717	−0.0287248
$993$	2.66294	0.0845058
$994$	34.7594	1.10250
$995$	21.0094	0.666042
$996$	−2.41966	−0.0766699
$997$	17.6636	0.559413	0.279707	−	0.960086i	$-0.409763\pi$
$997$	17.6636	0.559413	0.279707	+	0.960086i	$0.409763\pi$
$998$	−3.83649	−0.121442
$999$	1.72110	0.0544531

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	786.2.a.p.1.3	✓	4
3.2	odd	2		2358.2.a.bf.1.2		4
4.3	odd	2		6288.2.a.ba.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.2.a.p.1.3	✓	4	1.1	even	1	trivial
2358.2.a.bf.1.2		4	3.2	odd	2
6288.2.a.ba.1.3		4	4.3	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:	\( N \)	\(=\)	\( 786 = 2 \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	786.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.31673 q^{5} +1.00000 q^{6} -3.03783 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.31673 q^{5} +1.00000 q^{6} -3.03783 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.31673 q^{10} +5.85421 q^{11} +1.00000 q^{12} -0.412014 q^{13} -3.03783 q^{14} +1.31673 q^{15} +1.00000 q^{16} -1.85421 q^{17} +1.00000 q^{18} +2.81638 q^{19} +1.31673 q^{20} -3.03783 q^{21} +5.85421 q^{22} +3.44219 q^{23} +1.00000 q^{24} -3.26622 q^{25} -0.412014 q^{26} +1.00000 q^{27} -3.03783 q^{28} +3.44984 q^{29} +1.31673 q^{30} -0.904717 q^{31} +1.00000 q^{32} +5.85421 q^{33} -1.85421 q^{34} -4.00000 q^{35} +1.00000 q^{36} +1.72110 q^{37} +2.81638 q^{38} -0.412014 q^{39} +1.31673 q^{40} +1.03783 q^{41} -3.03783 q^{42} -4.07565 q^{43} +5.85421 q^{44} +1.31673 q^{45} +3.44219 q^{46} -4.00000 q^{47} +1.00000 q^{48} +2.22839 q^{49} -3.26622 q^{50} -1.85421 q^{51} -0.412014 q^{52} -10.9495 q^{53} +1.00000 q^{54} +7.70841 q^{55} -3.03783 q^{56} +2.81638 q^{57} +3.44984 q^{58} -7.92986 q^{59} +1.31673 q^{60} +3.85421 q^{61} -0.904717 q^{62} -3.03783 q^{63} +1.00000 q^{64} -0.542511 q^{65} +5.85421 q^{66} -14.1884 q^{67} -1.85421 q^{68} +3.44219 q^{69} -4.00000 q^{70} -11.4422 q^{71} +1.00000 q^{72} +14.1513 q^{73} +1.72110 q^{74} -3.26622 q^{75} +2.81638 q^{76} -17.7841 q^{77} -0.412014 q^{78} -8.12546 q^{79} +1.31673 q^{80} +1.00000 q^{81} +1.03783 q^{82} -2.41966 q^{83} -3.03783 q^{84} -2.44149 q^{85} -4.07565 q^{86} +3.44984 q^{87} +5.85421 q^{88} +5.30405 q^{89} +1.31673 q^{90} +1.25163 q^{91} +3.44219 q^{92} -0.904717 q^{93} -4.00000 q^{94} +3.70841 q^{95} +1.00000 q^{96} +13.1506 q^{97} +2.22839 q^{98} +5.85421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{6} + q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^6 + q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{6} + q^{7} + 4 q^{8} + 4 q^{9} + 5 q^{11} + 4 q^{12} + q^{13} + q^{14} + 4 q^{16} + 11 q^{17} + 4 q^{18} + 6 q^{19} + q^{21} + 5 q^{22} - 2 q^{23} + 4 q^{24} + 8 q^{25} + q^{26} + 4 q^{27} + q^{28} - 2 q^{29} - q^{31} + 4 q^{32} + 5 q^{33} + 11 q^{34} - 16 q^{35} + 4 q^{36} - q^{37} + 6 q^{38} + q^{39} - 9 q^{41} + q^{42} + 10 q^{43} + 5 q^{44} - 2 q^{46} - 16 q^{47} + 4 q^{48} + q^{49} + 8 q^{50} + 11 q^{51} + q^{52} - 28 q^{53} + 4 q^{54} - 6 q^{55} + q^{56} + 6 q^{57} - 2 q^{58} + 13 q^{59} - 3 q^{61} - q^{62} + q^{63} + 4 q^{64} - 18 q^{65} + 5 q^{66} + 9 q^{67} + 11 q^{68} - 2 q^{69} - 16 q^{70} - 30 q^{71} + 4 q^{72} + 4 q^{73} - q^{74} + 8 q^{75} + 6 q^{76} - 8 q^{77} + q^{78} - 22 q^{79} + 4 q^{81} - 9 q^{82} - 7 q^{83} + q^{84} + 6 q^{85} + 10 q^{86} - 2 q^{87} + 5 q^{88} - 13 q^{89} - 16 q^{91} - 2 q^{92} - q^{93} - 16 q^{94} - 22 q^{95} + 4 q^{96} + q^{98} + 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^6 + q^7 + 4 * q^8 + 4 * q^9 + 5 * q^11 + 4 * q^12 + q^13 + q^14 + 4 * q^16 + 11 * q^17 + 4 * q^18 + 6 * q^19 + q^21 + 5 * q^22 - 2 * q^23 + 4 * q^24 + 8 * q^25 + q^26 + 4 * q^27 + q^28 - 2 * q^29 - q^31 + 4 * q^32 + 5 * q^33 + 11 * q^34 - 16 * q^35 + 4 * q^36 - q^37 + 6 * q^38 + q^39 - 9 * q^41 + q^42 + 10 * q^43 + 5 * q^44 - 2 * q^46 - 16 * q^47 + 4 * q^48 + q^49 + 8 * q^50 + 11 * q^51 + q^52 - 28 * q^53 + 4 * q^54 - 6 * q^55 + q^56 + 6 * q^57 - 2 * q^58 + 13 * q^59 - 3 * q^61 - q^62 + q^63 + 4 * q^64 - 18 * q^65 + 5 * q^66 + 9 * q^67 + 11 * q^68 - 2 * q^69 - 16 * q^70 - 30 * q^71 + 4 * q^72 + 4 * q^73 - q^74 + 8 * q^75 + 6 * q^76 - 8 * q^77 + q^78 - 22 * q^79 + 4 * q^81 - 9 * q^82 - 7 * q^83 + q^84 + 6 * q^85 + 10 * q^86 - 2 * q^87 + 5 * q^88 - 13 * q^89 - 16 * q^91 - 2 * q^92 - q^93 - 16 * q^94 - 22 * q^95 + 4 * q^96 + q^98 + 5 * q^99

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