LMFDB - Embedded newform 5070.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +0.267949 q^{11} -1.00000 q^{12} +3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +5.73205 q^{19} +1.00000 q^{20} +3.00000 q^{21} -0.267949 q^{22} -3.46410 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -3.00000 q^{28} +1.46410 q^{29} +1.00000 q^{30} +4.92820 q^{31} -1.00000 q^{32} -0.267949 q^{33} +4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -5.92820 q^{37} -5.73205 q^{38} -1.00000 q^{40} +4.00000 q^{41} -3.00000 q^{42} -6.00000 q^{43} +0.267949 q^{44} +1.00000 q^{45} +3.46410 q^{46} +6.46410 q^{47} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +4.00000 q^{51} -0.267949 q^{53} +1.00000 q^{54} +0.267949 q^{55} +3.00000 q^{56} -5.73205 q^{57} -1.46410 q^{58} +11.4641 q^{59} -1.00000 q^{60} -0.535898 q^{61} -4.92820 q^{62} -3.00000 q^{63} +1.00000 q^{64} +0.267949 q^{66} -1.46410 q^{67} -4.00000 q^{68} +3.46410 q^{69} +3.00000 q^{70} -12.9282 q^{71} -1.00000 q^{72} +6.92820 q^{73} +5.92820 q^{74} -1.00000 q^{75} +5.73205 q^{76} -0.803848 q^{77} +3.07180 q^{79} +1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} +9.46410 q^{83} +3.00000 q^{84} -4.00000 q^{85} +6.00000 q^{86} -1.46410 q^{87} -0.267949 q^{88} -14.1244 q^{89} -1.00000 q^{90} -3.46410 q^{92} -4.92820 q^{93} -6.46410 q^{94} +5.73205 q^{95} +1.00000 q^{96} +8.39230 q^{97} -2.00000 q^{98} +0.267949 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} - 2 q^{12} + 6 q^{14} - 2 q^{15} + 2 q^{16} - 8 q^{17} - 2 q^{18} + 8 q^{19} + 2 q^{20} + 6 q^{21} - 4 q^{22} + 2 q^{24} + 2 q^{25} - 2 q^{27} - 6 q^{28} - 4 q^{29} + 2 q^{30} - 4 q^{31} - 2 q^{32} - 4 q^{33} + 8 q^{34} - 6 q^{35} + 2 q^{36} + 2 q^{37} - 8 q^{38} - 2 q^{40} + 8 q^{41} - 6 q^{42} - 12 q^{43} + 4 q^{44} + 2 q^{45} + 6 q^{47} - 2 q^{48} + 4 q^{49} - 2 q^{50} + 8 q^{51} - 4 q^{53} + 2 q^{54} + 4 q^{55} + 6 q^{56} - 8 q^{57} + 4 q^{58} + 16 q^{59} - 2 q^{60} - 8 q^{61} + 4 q^{62} - 6 q^{63} + 2 q^{64} + 4 q^{66} + 4 q^{67} - 8 q^{68} + 6 q^{70} - 12 q^{71} - 2 q^{72} - 2 q^{74} - 2 q^{75} + 8 q^{76} - 12 q^{77} + 20 q^{79} + 2 q^{80} + 2 q^{81} - 8 q^{82} + 12 q^{83} + 6 q^{84} - 8 q^{85} + 12 q^{86} + 4 q^{87} - 4 q^{88} - 4 q^{89} - 2 q^{90} + 4 q^{93} - 6 q^{94} + 8 q^{95} + 2 q^{96} - 4 q^{97} - 4 q^{98} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 4 * q^11 - 2 * q^12 + 6 * q^14 - 2 * q^15 + 2 * q^16 - 8 * q^17 - 2 * q^18 + 8 * q^19 + 2 * q^20 + 6 * q^21 - 4 * q^22 + 2 * q^24 + 2 * q^25 - 2 * q^27 - 6 * q^28 - 4 * q^29 + 2 * q^30 - 4 * q^31 - 2 * q^32 - 4 * q^33 + 8 * q^34 - 6 * q^35 + 2 * q^36 + 2 * q^37 - 8 * q^38 - 2 * q^40 + 8 * q^41 - 6 * q^42 - 12 * q^43 + 4 * q^44 + 2 * q^45 + 6 * q^47 - 2 * q^48 + 4 * q^49 - 2 * q^50 + 8 * q^51 - 4 * q^53 + 2 * q^54 + 4 * q^55 + 6 * q^56 - 8 * q^57 + 4 * q^58 + 16 * q^59 - 2 * q^60 - 8 * q^61 + 4 * q^62 - 6 * q^63 + 2 * q^64 + 4 * q^66 + 4 * q^67 - 8 * q^68 + 6 * q^70 - 12 * q^71 - 2 * q^72 - 2 * q^74 - 2 * q^75 + 8 * q^76 - 12 * q^77 + 20 * q^79 + 2 * q^80 + 2 * q^81 - 8 * q^82 + 12 * q^83 + 6 * q^84 - 8 * q^85 + 12 * q^86 + 4 * q^87 - 4 * q^88 - 4 * q^89 - 2 * q^90 + 4 * q^93 - 6 * q^94 + 8 * q^95 + 2 * q^96 - 4 * q^97 - 4 * q^98 + 4 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0.267949	0.0807897	0.0403949	−	0.999184i	$-0.487138\pi$
$11$	0.267949	0.0807897	0.0403949	+	0.999184i	$0.487138\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	3.00000	0.801784
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−1.00000	−0.235702
$19$	5.73205	1.31502	0.657511	−	0.753445i	$-0.271609\pi$
$19$	5.73205	1.31502	0.657511	+	0.753445i	$0.271609\pi$
$20$	1.00000	0.223607
$21$	3.00000	0.654654
$22$	−0.267949	−0.0571270
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−3.00000	−0.566947
$29$	1.46410	0.271877	0.135938	−	0.990717i	$-0.456595\pi$
$29$	1.46410	0.271877	0.135938	+	0.990717i	$0.456595\pi$
$30$	1.00000	0.182574
$31$	4.92820	0.885131	0.442566	−	0.896736i	$-0.354068\pi$
$31$	4.92820	0.885131	0.442566	+	0.896736i	$0.354068\pi$
$32$	−1.00000	−0.176777
$33$	−0.267949	−0.0466440
$34$	4.00000	0.685994
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	−5.92820	−0.974591	−0.487295	−	0.873237i	$-0.662016\pi$
$37$	−5.92820	−0.974591	−0.487295	+	0.873237i	$0.662016\pi$
$38$	−5.73205	−0.929861
$39$	0	0
$40$	−1.00000	−0.158114
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	−3.00000	−0.462910
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0.267949	0.0403949
$45$	1.00000	0.149071
$46$	3.46410	0.510754
$47$	6.46410	0.942886	0.471443	−	0.881897i	$-0.343733\pi$
$47$	6.46410	0.942886	0.471443	+	0.881897i	$0.343733\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	4.00000	0.560112
$52$	0	0
$53$	−0.267949	−0.0368057	−0.0184028	−	0.999831i	$-0.505858\pi$
$53$	−0.267949	−0.0368057	−0.0184028	+	0.999831i	$0.505858\pi$
$54$	1.00000	0.136083
$55$	0.267949	0.0361303
$56$	3.00000	0.400892
$57$	−5.73205	−0.759229
$58$	−1.46410	−0.192246
$59$	11.4641	1.49250	0.746249	−	0.665666i	$-0.231853\pi$
$59$	11.4641	1.49250	0.746249	+	0.665666i	$0.231853\pi$
$60$	−1.00000	−0.129099
$61$	−0.535898	−0.0686148	−0.0343074	−	0.999411i	$-0.510923\pi$
$61$	−0.535898	−0.0686148	−0.0343074	+	0.999411i	$0.510923\pi$
$62$	−4.92820	−0.625882
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	0.267949	0.0329823
$67$	−1.46410	−0.178868	−0.0894342	−	0.995993i	$-0.528506\pi$
$67$	−1.46410	−0.178868	−0.0894342	+	0.995993i	$0.528506\pi$
$68$	−4.00000	−0.485071
$69$	3.46410	0.417029
$70$	3.00000	0.358569
$71$	−12.9282	−1.53430	−0.767148	−	0.641470i	$-0.778325\pi$
$71$	−12.9282	−1.53430	−0.767148	+	0.641470i	$0.778325\pi$
$72$	−1.00000	−0.117851
$73$	6.92820	0.810885	0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820	0.810885	0.405442	+	0.914121i	$0.367117\pi$
$74$	5.92820	0.689140
$75$	−1.00000	−0.115470
$76$	5.73205	0.657511
$77$	−0.803848	−0.0916069
$78$	0	0
$79$	3.07180	0.345604	0.172802	−	0.984957i	$-0.444718\pi$
$79$	3.07180	0.345604	0.172802	+	0.984957i	$0.444718\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−4.00000	−0.441726
$83$	9.46410	1.03882	0.519410	−	0.854525i	$-0.326152\pi$
$83$	9.46410	1.03882	0.519410	+	0.854525i	$0.326152\pi$
$84$	3.00000	0.327327
$85$	−4.00000	−0.433861
$86$	6.00000	0.646997
$87$	−1.46410	−0.156968
$88$	−0.267949	−0.0285635
$89$	−14.1244	−1.49718	−0.748589	−	0.663034i	$-0.769269\pi$
$89$	−14.1244	−1.49718	−0.748589	+	0.663034i	$0.769269\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−3.46410	−0.361158
$93$	−4.92820	−0.511031
$94$	−6.46410	−0.666721
$95$	5.73205	0.588096
$96$	1.00000	0.102062
$97$	8.39230	0.852109	0.426055	−	0.904697i	$-0.359903\pi$
$97$	8.39230	0.852109	0.426055	+	0.904697i	$0.359903\pi$
$98$	−2.00000	−0.202031
$99$	0.267949	0.0269299
$100$	1.00000	0.100000
$101$	−12.3923	−1.23308	−0.616540	−	0.787323i	$-0.711466\pi$
$101$	−12.3923	−1.23308	−0.616540	+	0.787323i	$0.711466\pi$
$102$	−4.00000	−0.396059
$103$	11.5885	1.14184	0.570922	−	0.821004i	$-0.306586\pi$
$103$	11.5885	1.14184	0.570922	+	0.821004i	$0.306586\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0.267949	0.0260255
$107$	−12.9282	−1.24982	−0.624908	−	0.780698i	$-0.714864\pi$
$107$	−12.9282	−1.24982	−0.624908	+	0.780698i	$0.714864\pi$
$108$	−1.00000	−0.0962250
$109$	−10.3923	−0.995402	−0.497701	−	0.867349i	$-0.665822\pi$
$109$	−10.3923	−0.995402	−0.497701	+	0.867349i	$0.665822\pi$
$110$	−0.267949	−0.0255480
$111$	5.92820	0.562680
$112$	−3.00000	−0.283473
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	5.73205	0.536856
$115$	−3.46410	−0.323029
$116$	1.46410	0.135938
$117$	0	0
$118$	−11.4641	−1.05536
$119$	12.0000	1.10004
$120$	1.00000	0.0912871
$121$	−10.9282	−0.993473
$122$	0.535898	0.0485180
$123$	−4.00000	−0.360668
$124$	4.92820	0.442566
$125$	1.00000	0.0894427
$126$	3.00000	0.267261
$127$	12.6603	1.12342	0.561708	−	0.827336i	$-0.310144\pi$
$127$	12.6603	1.12342	0.561708	+	0.827336i	$0.310144\pi$
$128$	−1.00000	−0.0883883
$129$	6.00000	0.528271
$130$	0	0
$131$	14.3205	1.25119	0.625594	−	0.780149i	$-0.284857\pi$
$131$	14.3205	1.25119	0.625594	+	0.780149i	$0.284857\pi$
$132$	−0.267949	−0.0233220
$133$	−17.1962	−1.49110
$134$	1.46410	0.126479
$135$	−1.00000	−0.0860663
$136$	4.00000	0.342997
$137$	−13.4641	−1.15032	−0.575158	−	0.818042i	$-0.695059\pi$
$137$	−13.4641	−1.15032	−0.575158	+	0.818042i	$0.695059\pi$
$138$	−3.46410	−0.294884
$139$	−19.7846	−1.67811	−0.839054	−	0.544048i	$-0.816891\pi$
$139$	−19.7846	−1.67811	−0.839054	+	0.544048i	$0.816891\pi$
$140$	−3.00000	−0.253546
$141$	−6.46410	−0.544376
$142$	12.9282	1.08491
$143$	0	0
$144$	1.00000	0.0833333
$145$	1.46410	0.121587
$146$	−6.92820	−0.573382
$147$	−2.00000	−0.164957
$148$	−5.92820	−0.487295
$149$	18.7846	1.53890	0.769448	−	0.638710i	$-0.220532\pi$
$149$	18.7846	1.53890	0.769448	+	0.638710i	$0.220532\pi$
$150$	1.00000	0.0816497
$151$	−22.7846	−1.85419	−0.927093	−	0.374832i	$-0.877700\pi$
$151$	−22.7846	−1.85419	−0.927093	+	0.374832i	$0.877700\pi$
$152$	−5.73205	−0.464931
$153$	−4.00000	−0.323381
$154$	0.803848	0.0647759
$155$	4.92820	0.395843
$156$	0	0
$157$	21.1962	1.69164	0.845819	−	0.533471i	$-0.179113\pi$
$157$	21.1962	1.69164	0.845819	+	0.533471i	$0.179113\pi$
$158$	−3.07180	−0.244379
$159$	0.267949	0.0212498
$160$	−1.00000	−0.0790569
$161$	10.3923	0.819028
$162$	−1.00000	−0.0785674
$163$	2.92820	0.229355	0.114677	−	0.993403i	$-0.463417\pi$
$163$	2.92820	0.229355	0.114677	+	0.993403i	$0.463417\pi$
$164$	4.00000	0.312348
$165$	−0.267949	−0.0208598
$166$	−9.46410	−0.734557
$167$	−0.464102	−0.0359133	−0.0179566	−	0.999839i	$-0.505716\pi$
$167$	−0.464102	−0.0359133	−0.0179566	+	0.999839i	$0.505716\pi$
$168$	−3.00000	−0.231455
$169$	0	0
$170$	4.00000	0.306786
$171$	5.73205	0.438341
$172$	−6.00000	−0.457496
$173$	2.12436	0.161512	0.0807559	−	0.996734i	$-0.474267\pi$
$173$	2.12436	0.161512	0.0807559	+	0.996734i	$0.474267\pi$
$174$	1.46410	0.110993
$175$	−3.00000	−0.226779
$176$	0.267949	0.0201974
$177$	−11.4641	−0.861695
$178$	14.1244	1.05867
$179$	−9.07180	−0.678058	−0.339029	−	0.940776i	$-0.610098\pi$
$179$	−9.07180	−0.678058	−0.339029	+	0.940776i	$0.610098\pi$
$180$	1.00000	0.0745356
$181$	−16.9282	−1.25826	−0.629132	−	0.777299i	$-0.716589\pi$
$181$	−16.9282	−1.25826	−0.629132	+	0.777299i	$0.716589\pi$
$182$	0	0
$183$	0.535898	0.0396147
$184$	3.46410	0.255377
$185$	−5.92820	−0.435850
$186$	4.92820	0.361353
$187$	−1.07180	−0.0783775
$188$	6.46410	0.471443
$189$	3.00000	0.218218
$190$	−5.73205	−0.415847
$191$	−17.3205	−1.25327	−0.626634	−	0.779314i	$-0.715568\pi$
$191$	−17.3205	−1.25327	−0.626634	+	0.779314i	$0.715568\pi$
$192$	−1.00000	−0.0721688
$193$	−9.85641	−0.709480	−0.354740	−	0.934965i	$-0.615431\pi$
$193$	−9.85641	−0.709480	−0.354740	+	0.934965i	$0.615431\pi$
$194$	−8.39230	−0.602532
$195$	0	0
$196$	2.00000	0.142857
$197$	11.3923	0.811668	0.405834	−	0.913947i	$-0.366981\pi$
$197$	11.3923	0.811668	0.405834	+	0.913947i	$0.366981\pi$
$198$	−0.267949	−0.0190423
$199$	24.9282	1.76711	0.883557	−	0.468324i	$-0.155142\pi$
$199$	24.9282	1.76711	0.883557	+	0.468324i	$0.155142\pi$
$200$	−1.00000	−0.0707107
$201$	1.46410	0.103270
$202$	12.3923	0.871920
$203$	−4.39230	−0.308279
$204$	4.00000	0.280056
$205$	4.00000	0.279372
$206$	−11.5885	−0.807406
$207$	−3.46410	−0.240772
$208$	0	0
$209$	1.53590	0.106240
$210$	−3.00000	−0.207020
$211$	−8.07180	−0.555685	−0.277843	−	0.960627i	$-0.589619\pi$
$211$	−8.07180	−0.555685	−0.277843	+	0.960627i	$0.589619\pi$
$212$	−0.267949	−0.0184028
$213$	12.9282	0.885826
$214$	12.9282	0.883754
$215$	−6.00000	−0.409197
$216$	1.00000	0.0680414
$217$	−14.7846	−1.00364
$218$	10.3923	0.703856
$219$	−6.92820	−0.468165
$220$	0.267949	0.0180651
$221$	0	0
$222$	−5.92820	−0.397875
$223$	18.8564	1.26272	0.631359	−	0.775491i	$-0.282497\pi$
$223$	18.8564	1.26272	0.631359	+	0.775491i	$0.282497\pi$
$224$	3.00000	0.200446
$225$	1.00000	0.0666667
$226$	12.0000	0.798228
$227$	6.53590	0.433803	0.216901	−	0.976194i	$-0.430405\pi$
$227$	6.53590	0.433803	0.216901	+	0.976194i	$0.430405\pi$
$228$	−5.73205	−0.379614
$229$	−4.53590	−0.299741	−0.149870	−	0.988706i	$-0.547886\pi$
$229$	−4.53590	−0.299741	−0.149870	+	0.988706i	$0.547886\pi$
$230$	3.46410	0.228416
$231$	0.803848	0.0528893
$232$	−1.46410	−0.0961230
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	6.46410	0.421671
$236$	11.4641	0.746249
$237$	−3.07180	−0.199535
$238$	−12.0000	−0.777844
$239$	−3.46410	−0.224074	−0.112037	−	0.993704i	$-0.535738\pi$
$239$	−3.46410	−0.224074	−0.112037	+	0.993704i	$0.535738\pi$
$240$	−1.00000	−0.0645497
$241$	25.1962	1.62303	0.811513	−	0.584334i	$-0.198644\pi$
$241$	25.1962	1.62303	0.811513	+	0.584334i	$0.198644\pi$
$242$	10.9282	0.702492
$243$	−1.00000	−0.0641500
$244$	−0.535898	−0.0343074
$245$	2.00000	0.127775
$246$	4.00000	0.255031
$247$	0	0
$248$	−4.92820	−0.312941
$249$	−9.46410	−0.599763
$250$	−1.00000	−0.0632456
$251$	−19.5359	−1.23309	−0.616547	−	0.787318i	$-0.711469\pi$
$251$	−19.5359	−1.23309	−0.616547	+	0.787318i	$0.711469\pi$
$252$	−3.00000	−0.188982
$253$	−0.928203	−0.0583556
$254$	−12.6603	−0.794375
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	−14.5359	−0.906724	−0.453362	−	0.891326i	$-0.649776\pi$
$257$	−14.5359	−0.906724	−0.453362	+	0.891326i	$0.649776\pi$
$258$	−6.00000	−0.373544
$259$	17.7846	1.10508
$260$	0	0
$261$	1.46410	0.0906256
$262$	−14.3205	−0.884724
$263$	−24.5167	−1.51176	−0.755881	−	0.654709i	$-0.772791\pi$
$263$	−24.5167	−1.51176	−0.755881	+	0.654709i	$0.772791\pi$
$264$	0.267949	0.0164911
$265$	−0.267949	−0.0164600
$266$	17.1962	1.05436
$267$	14.1244	0.864397
$268$	−1.46410	−0.0894342
$269$	−17.0718	−1.04089	−0.520443	−	0.853896i	$-0.674233\pi$
$269$	−17.0718	−1.04089	−0.520443	+	0.853896i	$0.674233\pi$
$270$	1.00000	0.0608581
$271$	24.3923	1.48173	0.740863	−	0.671656i	$-0.234417\pi$
$271$	24.3923	1.48173	0.740863	+	0.671656i	$0.234417\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	13.4641	0.813396
$275$	0.267949	0.0161579
$276$	3.46410	0.208514
$277$	−11.5885	−0.696283	−0.348141	−	0.937442i	$-0.613187\pi$
$277$	−11.5885	−0.696283	−0.348141	+	0.937442i	$0.613187\pi$
$278$	19.7846	1.18660
$279$	4.92820	0.295044
$280$	3.00000	0.179284
$281$	6.92820	0.413302	0.206651	−	0.978415i	$-0.433744\pi$
$281$	6.92820	0.413302	0.206651	+	0.978415i	$0.433744\pi$
$282$	6.46410	0.384932
$283$	6.39230	0.379983	0.189992	−	0.981786i	$-0.439154\pi$
$283$	6.39230	0.379983	0.189992	+	0.981786i	$0.439154\pi$
$284$	−12.9282	−0.767148
$285$	−5.73205	−0.339537
$286$	0	0
$287$	−12.0000	−0.708338
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	−1.46410	−0.0859750
$291$	−8.39230	−0.491966
$292$	6.92820	0.405442
$293$	−29.2487	−1.70873	−0.854364	−	0.519675i	$-0.826053\pi$
$293$	−29.2487	−1.70873	−0.854364	+	0.519675i	$0.826053\pi$
$294$	2.00000	0.116642
$295$	11.4641	0.667466
$296$	5.92820	0.344570
$297$	−0.267949	−0.0155480
$298$	−18.7846	−1.08816
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	18.0000	1.03750
$302$	22.7846	1.31111
$303$	12.3923	0.711919
$304$	5.73205	0.328756
$305$	−0.535898	−0.0306855
$306$	4.00000	0.228665
$307$	−28.2487	−1.61224	−0.806120	−	0.591753i	$-0.798436\pi$
$307$	−28.2487	−1.61224	−0.806120	+	0.591753i	$0.798436\pi$
$308$	−0.803848	−0.0458035
$309$	−11.5885	−0.659244
$310$	−4.92820	−0.279903
$311$	−18.9282	−1.07332	−0.536660	−	0.843799i	$-0.680314\pi$
$311$	−18.9282	−1.07332	−0.536660	+	0.843799i	$0.680314\pi$
$312$	0	0
$313$	−33.3205	−1.88339	−0.941693	−	0.336473i	$-0.890766\pi$
$313$	−33.3205	−1.88339	−0.941693	+	0.336473i	$0.890766\pi$
$314$	−21.1962	−1.19617
$315$	−3.00000	−0.169031
$316$	3.07180	0.172802
$317$	−30.4641	−1.71103	−0.855517	−	0.517774i	$-0.826761\pi$
$317$	−30.4641	−1.71103	−0.855517	+	0.517774i	$0.826761\pi$
$318$	−0.267949	−0.0150258
$319$	0.392305	0.0219649
$320$	1.00000	0.0559017
$321$	12.9282	0.721582
$322$	−10.3923	−0.579141
$323$	−22.9282	−1.27576
$324$	1.00000	0.0555556
$325$	0	0
$326$	−2.92820	−0.162178
$327$	10.3923	0.574696
$328$	−4.00000	−0.220863
$329$	−19.3923	−1.06913
$330$	0.267949	0.0147501
$331$	14.3923	0.791073	0.395536	−	0.918450i	$-0.370559\pi$
$331$	14.3923	0.791073	0.395536	+	0.918450i	$0.370559\pi$
$332$	9.46410	0.519410
$333$	−5.92820	−0.324864
$334$	0.464102	0.0253945
$335$	−1.46410	−0.0799924
$336$	3.00000	0.163663
$337$	26.3923	1.43768	0.718840	−	0.695175i	$-0.244673\pi$
$337$	26.3923	1.43768	0.718840	+	0.695175i	$0.244673\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	−4.00000	−0.216930
$341$	1.32051	0.0715095
$342$	−5.73205	−0.309954
$343$	15.0000	0.809924
$344$	6.00000	0.323498
$345$	3.46410	0.186501
$346$	−2.12436	−0.114206
$347$	4.39230	0.235791	0.117896	−	0.993026i	$-0.462385\pi$
$347$	4.39230	0.235791	0.117896	+	0.993026i	$0.462385\pi$
$348$	−1.46410	−0.0784841
$349$	−10.5359	−0.563974	−0.281987	−	0.959418i	$-0.590993\pi$
$349$	−10.5359	−0.563974	−0.281987	+	0.959418i	$0.590993\pi$
$350$	3.00000	0.160357
$351$	0	0
$352$	−0.267949	−0.0142817
$353$	−7.60770	−0.404917	−0.202458	−	0.979291i	$-0.564893\pi$
$353$	−7.60770	−0.404917	−0.202458	+	0.979291i	$0.564893\pi$
$354$	11.4641	0.609310
$355$	−12.9282	−0.686158
$356$	−14.1244	−0.748589
$357$	−12.0000	−0.635107
$358$	9.07180	0.479459
$359$	0.928203	0.0489887	0.0244943	−	0.999700i	$-0.492202\pi$
$359$	0.928203	0.0489887	0.0244943	+	0.999700i	$0.492202\pi$
$360$	−1.00000	−0.0527046
$361$	13.8564	0.729285
$362$	16.9282	0.889727
$363$	10.9282	0.573582
$364$	0	0
$365$	6.92820	0.362639
$366$	−0.535898	−0.0280119
$367$	21.3205	1.11292	0.556461	−	0.830874i	$-0.312159\pi$
$367$	21.3205	1.11292	0.556461	+	0.830874i	$0.312159\pi$
$368$	−3.46410	−0.180579
$369$	4.00000	0.208232
$370$	5.92820	0.308193
$371$	0.803848	0.0417337
$372$	−4.92820	−0.255515
$373$	9.07180	0.469720	0.234860	−	0.972029i	$-0.424537\pi$
$373$	9.07180	0.469720	0.234860	+	0.972029i	$0.424537\pi$
$374$	1.07180	0.0554213
$375$	−1.00000	−0.0516398
$376$	−6.46410	−0.333361
$377$	0	0
$378$	−3.00000	−0.154303
$379$	9.73205	0.499902	0.249951	−	0.968259i	$-0.419586\pi$
$379$	9.73205	0.499902	0.249951	+	0.968259i	$0.419586\pi$
$380$	5.73205	0.294048
$381$	−12.6603	−0.648604
$382$	17.3205	0.886194
$383$	4.78461	0.244482	0.122241	−	0.992500i	$-0.460992\pi$
$383$	4.78461	0.244482	0.122241	+	0.992500i	$0.460992\pi$
$384$	1.00000	0.0510310
$385$	−0.803848	−0.0409679
$386$	9.85641	0.501678
$387$	−6.00000	−0.304997
$388$	8.39230	0.426055
$389$	−17.8564	−0.905356	−0.452678	−	0.891674i	$-0.649531\pi$
$389$	−17.8564	−0.905356	−0.452678	+	0.891674i	$0.649531\pi$
$390$	0	0
$391$	13.8564	0.700749
$392$	−2.00000	−0.101015
$393$	−14.3205	−0.722374
$394$	−11.3923	−0.573936
$395$	3.07180	0.154559
$396$	0.267949	0.0134650
$397$	5.92820	0.297528	0.148764	−	0.988873i	$-0.452471\pi$
$397$	5.92820	0.297528	0.148764	+	0.988873i	$0.452471\pi$
$398$	−24.9282	−1.24954
$399$	17.1962	0.860884
$400$	1.00000	0.0500000
$401$	−22.1244	−1.10484	−0.552419	−	0.833567i	$-0.686295\pi$
$401$	−22.1244	−1.10484	−0.552419	+	0.833567i	$0.686295\pi$
$402$	−1.46410	−0.0730228
$403$	0	0
$404$	−12.3923	−0.616540
$405$	1.00000	0.0496904
$406$	4.39230	0.217986
$407$	−1.58846	−0.0787369
$408$	−4.00000	−0.198030
$409$	−39.0526	−1.93102	−0.965512	−	0.260357i	$-0.916160\pi$
$409$	−39.0526	−1.93102	−0.965512	+	0.260357i	$0.916160\pi$
$410$	−4.00000	−0.197546
$411$	13.4641	0.664135
$412$	11.5885	0.570922
$413$	−34.3923	−1.69233
$414$	3.46410	0.170251
$415$	9.46410	0.464574
$416$	0	0
$417$	19.7846	0.968857
$418$	−1.53590	−0.0751232
$419$	9.85641	0.481517	0.240758	−	0.970585i	$-0.422604\pi$
$419$	9.85641	0.481517	0.240758	+	0.970585i	$0.422604\pi$
$420$	3.00000	0.146385
$421$	−21.8564	−1.06522	−0.532608	−	0.846362i	$-0.678788\pi$
$421$	−21.8564	−1.06522	−0.532608	+	0.846362i	$0.678788\pi$
$422$	8.07180	0.392929
$423$	6.46410	0.314295
$424$	0.267949	0.0130128
$425$	−4.00000	−0.194029
$426$	−12.9282	−0.626373
$427$	1.60770	0.0778018
$428$	−12.9282	−0.624908
$429$	0	0
$430$	6.00000	0.289346
$431$	13.8564	0.667440	0.333720	−	0.942672i	$-0.391696\pi$
$431$	13.8564	0.667440	0.333720	+	0.942672i	$0.391696\pi$
$432$	−1.00000	−0.0481125
$433$	8.78461	0.422161	0.211081	−	0.977469i	$-0.432302\pi$
$433$	8.78461	0.422161	0.211081	+	0.977469i	$0.432302\pi$
$434$	14.7846	0.709684
$435$	−1.46410	−0.0701983
$436$	−10.3923	−0.497701
$437$	−19.8564	−0.949861
$438$	6.92820	0.331042
$439$	13.3205	0.635753	0.317877	−	0.948132i	$-0.397030\pi$
$439$	13.3205	0.635753	0.317877	+	0.948132i	$0.397030\pi$
$440$	−0.267949	−0.0127740
$441$	2.00000	0.0952381
$442$	0	0
$443$	−19.8564	−0.943406	−0.471703	−	0.881757i	$-0.656361\pi$
$443$	−19.8564	−0.943406	−0.471703	+	0.881757i	$0.656361\pi$
$444$	5.92820	0.281340
$445$	−14.1244	−0.669559
$446$	−18.8564	−0.892877
$447$	−18.7846	−0.888482
$448$	−3.00000	−0.141737
$449$	6.12436	0.289026	0.144513	−	0.989503i	$-0.453838\pi$
$449$	6.12436	0.289026	0.144513	+	0.989503i	$0.453838\pi$
$450$	−1.00000	−0.0471405
$451$	1.07180	0.0504689
$452$	−12.0000	−0.564433
$453$	22.7846	1.07051
$454$	−6.53590	−0.306745
$455$	0	0
$456$	5.73205	0.268428
$457$	7.46410	0.349156	0.174578	−	0.984643i	$-0.444144\pi$
$457$	7.46410	0.349156	0.174578	+	0.984643i	$0.444144\pi$
$458$	4.53590	0.211949
$459$	4.00000	0.186704
$460$	−3.46410	−0.161515
$461$	−11.6077	−0.540624	−0.270312	−	0.962773i	$-0.587127\pi$
$461$	−11.6077	−0.540624	−0.270312	+	0.962773i	$0.587127\pi$
$462$	−0.803848	−0.0373984
$463$	−40.7846	−1.89542	−0.947711	−	0.319131i	$-0.896609\pi$
$463$	−40.7846	−1.89542	−0.947711	+	0.319131i	$0.896609\pi$
$464$	1.46410	0.0679692
$465$	−4.92820	−0.228540
$466$	18.0000	0.833834
$467$	−24.3923	−1.12874	−0.564371	−	0.825522i	$-0.690881\pi$
$467$	−24.3923	−1.12874	−0.564371	+	0.825522i	$0.690881\pi$
$468$	0	0
$469$	4.39230	0.202818
$470$	−6.46410	−0.298167
$471$	−21.1962	−0.976667
$472$	−11.4641	−0.527678
$473$	−1.60770	−0.0739219
$474$	3.07180	0.141092
$475$	5.73205	0.263005
$476$	12.0000	0.550019
$477$	−0.267949	−0.0122686
$478$	3.46410	0.158444
$479$	−22.2487	−1.01657	−0.508285	−	0.861189i	$-0.669720\pi$
$479$	−22.2487	−1.01657	−0.508285	+	0.861189i	$0.669720\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−25.1962	−1.14765
$483$	−10.3923	−0.472866
$484$	−10.9282	−0.496737
$485$	8.39230	0.381075
$486$	1.00000	0.0453609
$487$	21.0000	0.951601	0.475800	−	0.879553i	$-0.342158\pi$
$487$	21.0000	0.951601	0.475800	+	0.879553i	$0.342158\pi$
$488$	0.535898	0.0242590
$489$	−2.92820	−0.132418
$490$	−2.00000	−0.0903508
$491$	−5.39230	−0.243351	−0.121676	−	0.992570i	$-0.538827\pi$
$491$	−5.39230	−0.243351	−0.121676	+	0.992570i	$0.538827\pi$
$492$	−4.00000	−0.180334
$493$	−5.85641	−0.263759
$494$	0	0
$495$	0.267949	0.0120434
$496$	4.92820	0.221283
$497$	38.7846	1.73973
$498$	9.46410	0.424097
$499$	−33.3205	−1.49163	−0.745815	−	0.666153i	$-0.767940\pi$
$499$	−33.3205	−1.49163	−0.745815	+	0.666153i	$0.767940\pi$
$500$	1.00000	0.0447214
$501$	0.464102	0.0207345
$502$	19.5359	0.871930
$503$	3.73205	0.166404	0.0832020	−	0.996533i	$-0.473485\pi$
$503$	3.73205	0.166404	0.0832020	+	0.996533i	$0.473485\pi$
$504$	3.00000	0.133631
$505$	−12.3923	−0.551450
$506$	0.928203	0.0412637
$507$	0	0
$508$	12.6603	0.561708
$509$	−29.3205	−1.29961	−0.649804	−	0.760102i	$-0.725149\pi$
$509$	−29.3205	−1.29961	−0.649804	+	0.760102i	$0.725149\pi$
$510$	−4.00000	−0.177123
$511$	−20.7846	−0.919457
$512$	−1.00000	−0.0441942
$513$	−5.73205	−0.253076
$514$	14.5359	0.641151
$515$	11.5885	0.510648
$516$	6.00000	0.264135
$517$	1.73205	0.0761755
$518$	−17.7846	−0.781411
$519$	−2.12436	−0.0932489
$520$	0	0
$521$	−6.60770	−0.289488	−0.144744	−	0.989469i	$-0.546236\pi$
$521$	−6.60770	−0.289488	−0.144744	+	0.989469i	$0.546236\pi$
$522$	−1.46410	−0.0640820
$523$	−41.7128	−1.82397	−0.911987	−	0.410219i	$-0.865452\pi$
$523$	−41.7128	−1.82397	−0.911987	+	0.410219i	$0.865452\pi$
$524$	14.3205	0.625594
$525$	3.00000	0.130931
$526$	24.5167	1.06898
$527$	−19.7128	−0.858704
$528$	−0.267949	−0.0116610
$529$	−11.0000	−0.478261
$530$	0.267949	0.0116390
$531$	11.4641	0.497500
$532$	−17.1962	−0.745548
$533$	0	0
$534$	−14.1244	−0.611221
$535$	−12.9282	−0.558935
$536$	1.46410	0.0632396
$537$	9.07180	0.391477
$538$	17.0718	0.736017
$539$	0.535898	0.0230828
$540$	−1.00000	−0.0430331
$541$	14.7846	0.635640	0.317820	−	0.948151i	$-0.397049\pi$
$541$	14.7846	0.635640	0.317820	+	0.948151i	$0.397049\pi$
$542$	−24.3923	−1.04774
$543$	16.9282	0.726459
$544$	4.00000	0.171499
$545$	−10.3923	−0.445157
$546$	0	0
$547$	−5.32051	−0.227488	−0.113744	−	0.993510i	$-0.536284\pi$
$547$	−5.32051	−0.227488	−0.113744	+	0.993510i	$0.536284\pi$
$548$	−13.4641	−0.575158
$549$	−0.535898	−0.0228716
$550$	−0.267949	−0.0114254
$551$	8.39230	0.357524
$552$	−3.46410	−0.147442
$553$	−9.21539	−0.391878
$554$	11.5885	0.492346
$555$	5.92820	0.251638
$556$	−19.7846	−0.839054
$557$	−22.6077	−0.957919	−0.478959	−	0.877837i	$-0.658986\pi$
$557$	−22.6077	−0.957919	−0.478959	+	0.877837i	$0.658986\pi$
$558$	−4.92820	−0.208627
$559$	0	0
$560$	−3.00000	−0.126773
$561$	1.07180	0.0452513
$562$	−6.92820	−0.292249
$563$	15.3205	0.645682	0.322841	−	0.946453i	$-0.395362\pi$
$563$	15.3205	0.645682	0.322841	+	0.946453i	$0.395362\pi$
$564$	−6.46410	−0.272188
$565$	−12.0000	−0.504844
$566$	−6.39230	−0.268689
$567$	−3.00000	−0.125988
$568$	12.9282	0.542455
$569$	16.3205	0.684191	0.342096	−	0.939665i	$-0.388863\pi$
$569$	16.3205	0.684191	0.342096	+	0.939665i	$0.388863\pi$
$570$	5.73205	0.240089
$571$	−10.8564	−0.454326	−0.227163	−	0.973857i	$-0.572945\pi$
$571$	−10.8564	−0.454326	−0.227163	+	0.973857i	$0.572945\pi$
$572$	0	0
$573$	17.3205	0.723575
$574$	12.0000	0.500870
$575$	−3.46410	−0.144463
$576$	1.00000	0.0416667
$577$	9.32051	0.388018	0.194009	−	0.981000i	$-0.437851\pi$
$577$	9.32051	0.388018	0.194009	+	0.981000i	$0.437851\pi$
$578$	1.00000	0.0415945
$579$	9.85641	0.409618
$580$	1.46410	0.0607935
$581$	−28.3923	−1.17791
$582$	8.39230	0.347872
$583$	−0.0717968	−0.00297352
$584$	−6.92820	−0.286691
$585$	0	0
$586$	29.2487	1.20825
$587$	8.92820	0.368506	0.184253	−	0.982879i	$-0.441013\pi$
$587$	8.92820	0.368506	0.184253	+	0.982879i	$0.441013\pi$
$588$	−2.00000	−0.0824786
$589$	28.2487	1.16397
$590$	−11.4641	−0.471970
$591$	−11.3923	−0.468617
$592$	−5.92820	−0.243648
$593$	−44.7846	−1.83908	−0.919542	−	0.392992i	$-0.871440\pi$
$593$	−44.7846	−1.83908	−0.919542	+	0.392992i	$0.871440\pi$
$594$	0.267949	0.0109941
$595$	12.0000	0.491952
$596$	18.7846	0.769448
$597$	−24.9282	−1.02024
$598$	0	0
$599$	28.9282	1.18197	0.590987	−	0.806681i	$-0.298738\pi$
$599$	28.9282	1.18197	0.590987	+	0.806681i	$0.298738\pi$
$600$	1.00000	0.0408248
$601$	30.7128	1.25280	0.626401	−	0.779501i	$-0.284527\pi$
$601$	30.7128	1.25280	0.626401	+	0.779501i	$0.284527\pi$
$602$	−18.0000	−0.733625
$603$	−1.46410	−0.0596228
$604$	−22.7846	−0.927093
$605$	−10.9282	−0.444295
$606$	−12.3923	−0.503403
$607$	−21.4449	−0.870420	−0.435210	−	0.900329i	$-0.643326\pi$
$607$	−21.4449	−0.870420	−0.435210	+	0.900329i	$0.643326\pi$
$608$	−5.73205	−0.232465
$609$	4.39230	0.177985
$610$	0.535898	0.0216979
$611$	0	0
$612$	−4.00000	−0.161690
$613$	−45.0000	−1.81753	−0.908766	−	0.417305i	$-0.862975\pi$
$613$	−45.0000	−1.81753	−0.908766	+	0.417305i	$0.862975\pi$
$614$	28.2487	1.14003
$615$	−4.00000	−0.161296
$616$	0.803848	0.0323879
$617$	−35.4641	−1.42773	−0.713865	−	0.700283i	$-0.753057\pi$
$617$	−35.4641	−1.42773	−0.713865	+	0.700283i	$0.753057\pi$
$618$	11.5885	0.466156
$619$	2.51666	0.101153	0.0505766	−	0.998720i	$-0.483894\pi$
$619$	2.51666	0.101153	0.0505766	+	0.998720i	$0.483894\pi$
$620$	4.92820	0.197921
$621$	3.46410	0.139010
$622$	18.9282	0.758952
$623$	42.3731	1.69764
$624$	0	0
$625$	1.00000	0.0400000
$626$	33.3205	1.33176
$627$	−1.53590	−0.0613379
$628$	21.1962	0.845819
$629$	23.7128	0.945492
$630$	3.00000	0.119523
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−3.07180	−0.122190
$633$	8.07180	0.320825
$634$	30.4641	1.20988
$635$	12.6603	0.502407
$636$	0.267949	0.0106249
$637$	0	0
$638$	−0.392305	−0.0155315
$639$	−12.9282	−0.511432
$640$	−1.00000	−0.0395285
$641$	−14.4641	−0.571298	−0.285649	−	0.958334i	$-0.592209\pi$
$641$	−14.4641	−0.571298	−0.285649	+	0.958334i	$0.592209\pi$
$642$	−12.9282	−0.510235
$643$	12.7846	0.504176	0.252088	−	0.967704i	$-0.418883\pi$
$643$	12.7846	0.504176	0.252088	+	0.967704i	$0.418883\pi$
$644$	10.3923	0.409514
$645$	6.00000	0.236250
$646$	22.9282	0.902098
$647$	35.7321	1.40477	0.702386	−	0.711796i	$-0.252118\pi$
$647$	35.7321	1.40477	0.702386	+	0.711796i	$0.252118\pi$
$648$	−1.00000	−0.0392837
$649$	3.07180	0.120579
$650$	0	0
$651$	14.7846	0.579455
$652$	2.92820	0.114677
$653$	−1.58846	−0.0621611	−0.0310806	−	0.999517i	$-0.509895\pi$
$653$	−1.58846	−0.0621611	−0.0310806	+	0.999517i	$0.509895\pi$
$654$	−10.3923	−0.406371
$655$	14.3205	0.559549
$656$	4.00000	0.156174
$657$	6.92820	0.270295
$658$	19.3923	0.755991
$659$	−39.7128	−1.54699	−0.773496	−	0.633801i	$-0.781494\pi$
$659$	−39.7128	−1.54699	−0.773496	+	0.633801i	$0.781494\pi$
$660$	−0.267949	−0.0104299
$661$	−21.6077	−0.840442	−0.420221	−	0.907422i	$-0.638047\pi$
$661$	−21.6077	−0.840442	−0.420221	+	0.907422i	$0.638047\pi$
$662$	−14.3923	−0.559373
$663$	0	0
$664$	−9.46410	−0.367278
$665$	−17.1962	−0.666838
$666$	5.92820	0.229713
$667$	−5.07180	−0.196381
$668$	−0.464102	−0.0179566
$669$	−18.8564	−0.729031
$670$	1.46410	0.0565632
$671$	−0.143594	−0.00554337
$672$	−3.00000	−0.115728
$673$	16.7846	0.646999	0.323500	−	0.946228i	$-0.395141\pi$
$673$	16.7846	0.646999	0.323500	+	0.946228i	$0.395141\pi$
$674$	−26.3923	−1.01659
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	10.9282	0.420005	0.210002	−	0.977701i	$-0.432653\pi$
$677$	10.9282	0.420005	0.210002	+	0.977701i	$0.432653\pi$
$678$	−12.0000	−0.460857
$679$	−25.1769	−0.966201
$680$	4.00000	0.153393
$681$	−6.53590	−0.250456
$682$	−1.32051	−0.0505649
$683$	−41.3205	−1.58109	−0.790543	−	0.612407i	$-0.790201\pi$
$683$	−41.3205	−1.58109	−0.790543	+	0.612407i	$0.790201\pi$
$684$	5.73205	0.219170
$685$	−13.4641	−0.514437
$686$	−15.0000	−0.572703
$687$	4.53590	0.173055
$688$	−6.00000	−0.228748
$689$	0	0
$690$	−3.46410	−0.131876
$691$	−15.0526	−0.572626	−0.286313	−	0.958136i	$-0.592430\pi$
$691$	−15.0526	−0.572626	−0.286313	+	0.958136i	$0.592430\pi$
$692$	2.12436	0.0807559
$693$	−0.803848	−0.0305356
$694$	−4.39230	−0.166730
$695$	−19.7846	−0.750473
$696$	1.46410	0.0554966
$697$	−16.0000	−0.606043
$698$	10.5359	0.398790
$699$	18.0000	0.680823
$700$	−3.00000	−0.113389
$701$	4.39230	0.165895	0.0829475	−	0.996554i	$-0.473567\pi$
$701$	4.39230	0.165895	0.0829475	+	0.996554i	$0.473567\pi$
$702$	0	0
$703$	−33.9808	−1.28161
$704$	0.267949	0.0100987
$705$	−6.46410	−0.243452
$706$	7.60770	0.286319
$707$	37.1769	1.39818
$708$	−11.4641	−0.430847
$709$	27.3205	1.02604	0.513022	−	0.858376i	$-0.328526\pi$
$709$	27.3205	1.02604	0.513022	+	0.858376i	$0.328526\pi$
$710$	12.9282	0.485187
$711$	3.07180	0.115201
$712$	14.1244	0.529333
$713$	−17.0718	−0.639344
$714$	12.0000	0.449089
$715$	0	0
$716$	−9.07180	−0.339029
$717$	3.46410	0.129369
$718$	−0.928203	−0.0346402
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	1.00000	0.0372678
$721$	−34.7654	−1.29473
$722$	−13.8564	−0.515682
$723$	−25.1962	−0.937055
$724$	−16.9282	−0.629132
$725$	1.46410	0.0543754
$726$	−10.9282	−0.405584
$727$	4.66025	0.172839	0.0864196	−	0.996259i	$-0.472457\pi$
$727$	4.66025	0.172839	0.0864196	+	0.996259i	$0.472457\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−6.92820	−0.256424
$731$	24.0000	0.887672
$732$	0.535898	0.0198074
$733$	−20.8564	−0.770349	−0.385174	−	0.922844i	$-0.625859\pi$
$733$	−20.8564	−0.770349	−0.385174	+	0.922844i	$0.625859\pi$
$734$	−21.3205	−0.786954
$735$	−2.00000	−0.0737711
$736$	3.46410	0.127688
$737$	−0.392305	−0.0144507
$738$	−4.00000	−0.147242
$739$	35.8372	1.31829	0.659146	−	0.752015i	$-0.270918\pi$
$739$	35.8372	1.31829	0.659146	+	0.752015i	$0.270918\pi$
$740$	−5.92820	−0.217925
$741$	0	0
$742$	−0.803848	−0.0295102
$743$	10.9282	0.400917	0.200458	−	0.979702i	$-0.435757\pi$
$743$	10.9282	0.400917	0.200458	+	0.979702i	$0.435757\pi$
$744$	4.92820	0.180677
$745$	18.7846	0.688215
$746$	−9.07180	−0.332142
$747$	9.46410	0.346273
$748$	−1.07180	−0.0391888
$749$	38.7846	1.41716
$750$	1.00000	0.0365148
$751$	21.1769	0.772757	0.386378	−	0.922340i	$-0.373726\pi$
$751$	21.1769	0.772757	0.386378	+	0.922340i	$0.373726\pi$
$752$	6.46410	0.235722
$753$	19.5359	0.711928
$754$	0	0
$755$	−22.7846	−0.829217
$756$	3.00000	0.109109
$757$	21.7321	0.789865	0.394932	−	0.918710i	$-0.370768\pi$
$757$	21.7321	0.789865	0.394932	+	0.918710i	$0.370768\pi$
$758$	−9.73205	−0.353484
$759$	0.928203	0.0336916
$760$	−5.73205	−0.207923
$761$	−7.98076	−0.289302	−0.144651	−	0.989483i	$-0.546206\pi$
$761$	−7.98076	−0.289302	−0.144651	+	0.989483i	$0.546206\pi$
$762$	12.6603	0.458633
$763$	31.1769	1.12868
$764$	−17.3205	−0.626634
$765$	−4.00000	−0.144620
$766$	−4.78461	−0.172875
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−15.7128	−0.566619	−0.283309	−	0.959029i	$-0.591432\pi$
$769$	−15.7128	−0.566619	−0.283309	+	0.959029i	$0.591432\pi$
$770$	0.803848	0.0289687
$771$	14.5359	0.523498
$772$	−9.85641	−0.354740
$773$	−32.6077	−1.17282	−0.586409	−	0.810015i	$-0.699459\pi$
$773$	−32.6077	−1.17282	−0.586409	+	0.810015i	$0.699459\pi$
$774$	6.00000	0.215666
$775$	4.92820	0.177026
$776$	−8.39230	−0.301266
$777$	−17.7846	−0.638019
$778$	17.8564	0.640183
$779$	22.9282	0.821488
$780$	0	0
$781$	−3.46410	−0.123955
$782$	−13.8564	−0.495504
$783$	−1.46410	−0.0523227
$784$	2.00000	0.0714286
$785$	21.1962	0.756523
$786$	14.3205	0.510796
$787$	−1.46410	−0.0521896	−0.0260948	−	0.999659i	$-0.508307\pi$
$787$	−1.46410	−0.0521896	−0.0260948	+	0.999659i	$0.508307\pi$
$788$	11.3923	0.405834
$789$	24.5167	0.872816
$790$	−3.07180	−0.109290
$791$	36.0000	1.28001
$792$	−0.267949	−0.00952116
$793$	0	0
$794$	−5.92820	−0.210384
$795$	0.267949	0.00950318
$796$	24.9282	0.883557
$797$	10.1436	0.359305	0.179652	−	0.983730i	$-0.442503\pi$
$797$	10.1436	0.359305	0.179652	+	0.983730i	$0.442503\pi$
$798$	−17.1962	−0.608737
$799$	−25.8564	−0.914734
$800$	−1.00000	−0.0353553
$801$	−14.1244	−0.499060
$802$	22.1244	0.781238
$803$	1.85641	0.0655112
$804$	1.46410	0.0516349
$805$	10.3923	0.366281
$806$	0	0
$807$	17.0718	0.600956
$808$	12.3923	0.435960
$809$	−6.78461	−0.238534	−0.119267	−	0.992862i	$-0.538054\pi$
$809$	−6.78461	−0.238534	−0.119267	+	0.992862i	$0.538054\pi$
$810$	−1.00000	−0.0351364
$811$	−23.5885	−0.828303	−0.414151	−	0.910208i	$-0.635922\pi$
$811$	−23.5885	−0.828303	−0.414151	+	0.910208i	$0.635922\pi$
$812$	−4.39230	−0.154140
$813$	−24.3923	−0.855475
$814$	1.58846	0.0556754
$815$	2.92820	0.102570
$816$	4.00000	0.140028
$817$	−34.3923	−1.20323
$818$	39.0526	1.36544
$819$	0	0
$820$	4.00000	0.139686
$821$	14.6795	0.512318	0.256159	−	0.966635i	$-0.417543\pi$
$821$	14.6795	0.512318	0.256159	+	0.966635i	$0.417543\pi$
$822$	−13.4641	−0.469614
$823$	8.41154	0.293208	0.146604	−	0.989195i	$-0.453166\pi$
$823$	8.41154	0.293208	0.146604	+	0.989195i	$0.453166\pi$
$824$	−11.5885	−0.403703
$825$	−0.267949	−0.00932879
$826$	34.3923	1.19666
$827$	46.4974	1.61687	0.808437	−	0.588583i	$-0.200314\pi$
$827$	46.4974	1.61687	0.808437	+	0.588583i	$0.200314\pi$
$828$	−3.46410	−0.120386
$829$	4.67949	0.162525	0.0812627	−	0.996693i	$-0.474105\pi$
$829$	4.67949	0.162525	0.0812627	+	0.996693i	$0.474105\pi$
$830$	−9.46410	−0.328504
$831$	11.5885	0.401999
$832$	0	0
$833$	−8.00000	−0.277184
$834$	−19.7846	−0.685085
$835$	−0.464102	−0.0160609
$836$	1.53590	0.0531202
$837$	−4.92820	−0.170344
$838$	−9.85641	−0.340484
$839$	−19.1769	−0.662061	−0.331030	−	0.943620i	$-0.607396\pi$
$839$	−19.1769	−0.662061	−0.331030	+	0.943620i	$0.607396\pi$
$840$	−3.00000	−0.103510
$841$	−26.8564	−0.926083
$842$	21.8564	0.753222
$843$	−6.92820	−0.238620
$844$	−8.07180	−0.277843
$845$	0	0
$846$	−6.46410	−0.222240
$847$	32.7846	1.12649
$848$	−0.267949	−0.00920141
$849$	−6.39230	−0.219383
$850$	4.00000	0.137199
$851$	20.5359	0.703962
$852$	12.9282	0.442913
$853$	−24.6410	−0.843692	−0.421846	−	0.906667i	$-0.638618\pi$
$853$	−24.6410	−0.843692	−0.421846	+	0.906667i	$0.638618\pi$
$854$	−1.60770	−0.0550142
$855$	5.73205	0.196032
$856$	12.9282	0.441877
$857$	28.9282	0.988169	0.494084	−	0.869414i	$-0.335503\pi$
$857$	28.9282	0.988169	0.494084	+	0.869414i	$0.335503\pi$
$858$	0	0
$859$	−6.07180	−0.207167	−0.103584	−	0.994621i	$-0.533031\pi$
$859$	−6.07180	−0.207167	−0.103584	+	0.994621i	$0.533031\pi$
$860$	−6.00000	−0.204598
$861$	12.0000	0.408959
$862$	−13.8564	−0.471951
$863$	−2.92820	−0.0996772	−0.0498386	−	0.998757i	$-0.515871\pi$
$863$	−2.92820	−0.0996772	−0.0498386	+	0.998757i	$0.515871\pi$
$864$	1.00000	0.0340207
$865$	2.12436	0.0722303
$866$	−8.78461	−0.298513
$867$	1.00000	0.0339618
$868$	−14.7846	−0.501822
$869$	0.823085	0.0279213
$870$	1.46410	0.0496377
$871$	0	0
$872$	10.3923	0.351928
$873$	8.39230	0.284036
$874$	19.8564	0.671653
$875$	−3.00000	−0.101419
$876$	−6.92820	−0.234082
$877$	21.7128	0.733190	0.366595	−	0.930381i	$-0.380524\pi$
$877$	21.7128	0.733190	0.366595	+	0.930381i	$0.380524\pi$
$878$	−13.3205	−0.449545
$879$	29.2487	0.986535
$880$	0.267949	0.00903257
$881$	−39.1051	−1.31748	−0.658742	−	0.752369i	$-0.728911\pi$
$881$	−39.1051	−1.31748	−0.658742	+	0.752369i	$0.728911\pi$
$882$	−2.00000	−0.0673435
$883$	13.3205	0.448271	0.224135	−	0.974558i	$-0.428044\pi$
$883$	13.3205	0.448271	0.224135	+	0.974558i	$0.428044\pi$
$884$	0	0
$885$	−11.4641	−0.385362
$886$	19.8564	0.667089
$887$	55.7321	1.87130	0.935650	−	0.352930i	$-0.114815\pi$
$887$	55.7321	1.87130	0.935650	+	0.352930i	$0.114815\pi$
$888$	−5.92820	−0.198937
$889$	−37.9808	−1.27383
$890$	14.1244	0.473449
$891$	0.267949	0.00897664
$892$	18.8564	0.631359
$893$	37.0526	1.23992
$894$	18.7846	0.628251
$895$	−9.07180	−0.303237
$896$	3.00000	0.100223
$897$	0	0
$898$	−6.12436	−0.204372
$899$	7.21539	0.240647
$900$	1.00000	0.0333333
$901$	1.07180	0.0357067
$902$	−1.07180	−0.0356869
$903$	−18.0000	−0.599002
$904$	12.0000	0.399114
$905$	−16.9282	−0.562713
$906$	−22.7846	−0.756968
$907$	−20.9282	−0.694910	−0.347455	−	0.937697i	$-0.612954\pi$
$907$	−20.9282	−0.694910	−0.347455	+	0.937697i	$0.612954\pi$
$908$	6.53590	0.216901
$909$	−12.3923	−0.411027
$910$	0	0
$911$	12.7846	0.423573	0.211787	−	0.977316i	$-0.432072\pi$
$911$	12.7846	0.423573	0.211787	+	0.977316i	$0.432072\pi$
$912$	−5.73205	−0.189807
$913$	2.53590	0.0839260
$914$	−7.46410	−0.246891
$915$	0.535898	0.0177163
$916$	−4.53590	−0.149870
$917$	−42.9615	−1.41871
$918$	−4.00000	−0.132020
$919$	−47.9615	−1.58210	−0.791052	−	0.611748i	$-0.790466\pi$
$919$	−47.9615	−1.58210	−0.791052	+	0.611748i	$0.790466\pi$
$920$	3.46410	0.114208
$921$	28.2487	0.930827
$922$	11.6077	0.382279
$923$	0	0
$924$	0.803848	0.0264446
$925$	−5.92820	−0.194918
$926$	40.7846	1.34027
$927$	11.5885	0.380615
$928$	−1.46410	−0.0480615
$929$	57.8564	1.89821	0.949104	−	0.314964i	$-0.101993\pi$
$929$	57.8564	1.89821	0.949104	+	0.314964i	$0.101993\pi$
$930$	4.92820	0.161602
$931$	11.4641	0.375721
$932$	−18.0000	−0.589610
$933$	18.9282	0.619682
$934$	24.3923	0.798141
$935$	−1.07180	−0.0350515
$936$	0	0
$937$	−21.7128	−0.709327	−0.354663	−	0.934994i	$-0.615405\pi$
$937$	−21.7128	−0.709327	−0.354663	+	0.934994i	$0.615405\pi$
$938$	−4.39230	−0.143414
$939$	33.3205	1.08737
$940$	6.46410	0.210836
$941$	60.4974	1.97216	0.986080	−	0.166273i	$-0.0531732\pi$
$941$	60.4974	1.97216	0.986080	+	0.166273i	$0.0531732\pi$
$942$	21.1962	0.690608
$943$	−13.8564	−0.451227
$944$	11.4641	0.373125
$945$	3.00000	0.0975900
$946$	1.60770	0.0522707
$947$	34.3923	1.11760	0.558800	−	0.829303i	$-0.311262\pi$
$947$	34.3923	1.11760	0.558800	+	0.829303i	$0.311262\pi$
$948$	−3.07180	−0.0997673
$949$	0	0
$950$	−5.73205	−0.185972
$951$	30.4641	0.987866
$952$	−12.0000	−0.388922
$953$	53.3205	1.72722	0.863610	−	0.504160i	$-0.168198\pi$
$953$	53.3205	1.72722	0.863610	+	0.504160i	$0.168198\pi$
$954$	0.267949	0.00867518
$955$	−17.3205	−0.560478
$956$	−3.46410	−0.112037
$957$	−0.392305	−0.0126814
$958$	22.2487	0.718823
$959$	40.3923	1.30434
$960$	−1.00000	−0.0322749
$961$	−6.71281	−0.216542
$962$	0	0
$963$	−12.9282	−0.416606
$964$	25.1962	0.811513
$965$	−9.85641	−0.317289
$966$	10.3923	0.334367
$967$	1.14359	0.0367755	0.0183877	−	0.999831i	$-0.494147\pi$
$967$	1.14359	0.0367755	0.0183877	+	0.999831i	$0.494147\pi$
$968$	10.9282	0.351246
$969$	22.9282	0.736560
$970$	−8.39230	−0.269461
$971$	−41.3923	−1.32834	−0.664171	−	0.747581i	$-0.731215\pi$
$971$	−41.3923	−1.32834	−0.664171	+	0.747581i	$0.731215\pi$
$972$	−1.00000	−0.0320750
$973$	59.3538	1.90280
$974$	−21.0000	−0.672883
$975$	0	0
$976$	−0.535898	−0.0171537
$977$	5.46410	0.174812	0.0874060	−	0.996173i	$-0.472142\pi$
$977$	5.46410	0.174812	0.0874060	+	0.996173i	$0.472142\pi$
$978$	2.92820	0.0936336
$979$	−3.78461	−0.120957
$980$	2.00000	0.0638877
$981$	−10.3923	−0.331801
$982$	5.39230	0.172075
$983$	46.1769	1.47281	0.736407	−	0.676538i	$-0.236521\pi$
$983$	46.1769	1.47281	0.736407	+	0.676538i	$0.236521\pi$
$984$	4.00000	0.127515
$985$	11.3923	0.362989
$986$	5.85641	0.186506
$987$	19.3923	0.617264
$988$	0	0
$989$	20.7846	0.660912
$990$	−0.267949	−0.00851598
$991$	39.1769	1.24450	0.622248	−	0.782820i	$-0.286220\pi$
$991$	39.1769	1.24450	0.622248	+	0.782820i	$0.286220\pi$
$992$	−4.92820	−0.156471
$993$	−14.3923	−0.456726
$994$	−38.7846	−1.23017
$995$	24.9282	0.790277
$996$	−9.46410	−0.299882
$997$	61.9808	1.96295	0.981475	−	0.191589i	$-0.0613641\pi$
$997$	61.9808	1.96295	0.981475	+	0.191589i	$0.0613641\pi$
$998$	33.3205	1.05474
$999$	5.92820	0.187560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.ba.1.1		2
13.5	odd	4		5070.2.b.p.1351.3		4
13.6	odd	12		390.2.bb.a.361.2	yes	4
13.8	odd	4		5070.2.b.p.1351.2		4
13.11	odd	12		390.2.bb.a.121.2	✓	4
13.12	even	2		5070.2.a.be.1.2		2
39.11	even	12		1170.2.bs.d.901.1		4
39.32	even	12		1170.2.bs.d.361.1		4
65.19	odd	12		1950.2.bc.a.751.1		4
65.24	odd	12		1950.2.bc.a.901.1		4
65.32	even	12		1950.2.y.e.49.2		4
65.37	even	12		1950.2.y.d.199.1		4
65.58	even	12		1950.2.y.d.49.1		4
65.63	even	12		1950.2.y.e.199.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.a.121.2	✓	4	13.11	odd	12
390.2.bb.a.361.2	yes	4	13.6	odd	12
1170.2.bs.d.361.1		4	39.32	even	12
1170.2.bs.d.901.1		4	39.11	even	12
1950.2.y.d.49.1		4	65.58	even	12
1950.2.y.d.199.1		4	65.37	even	12
1950.2.y.e.49.2		4	65.32	even	12
1950.2.y.e.199.2		4	65.63	even	12
1950.2.bc.a.751.1		4	65.19	odd	12
1950.2.bc.a.901.1		4	65.24	odd	12
5070.2.a.ba.1.1		2	1.1	even	1	trivial
5070.2.a.be.1.2		2	13.12	even	2
5070.2.b.p.1351.2		4	13.8	odd	4
5070.2.b.p.1351.3		4	13.5	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +0.267949 q^{11} -1.00000 q^{12} +3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +5.73205 q^{19} +1.00000 q^{20} +3.00000 q^{21} -0.267949 q^{22} -3.46410 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -3.00000 q^{28} +1.46410 q^{29} +1.00000 q^{30} +4.92820 q^{31} -1.00000 q^{32} -0.267949 q^{33} +4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -5.92820 q^{37} -5.73205 q^{38} -1.00000 q^{40} +4.00000 q^{41} -3.00000 q^{42} -6.00000 q^{43} +0.267949 q^{44} +1.00000 q^{45} +3.46410 q^{46} +6.46410 q^{47} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +4.00000 q^{51} -0.267949 q^{53} +1.00000 q^{54} +0.267949 q^{55} +3.00000 q^{56} -5.73205 q^{57} -1.46410 q^{58} +11.4641 q^{59} -1.00000 q^{60} -0.535898 q^{61} -4.92820 q^{62} -3.00000 q^{63} +1.00000 q^{64} +0.267949 q^{66} -1.46410 q^{67} -4.00000 q^{68} +3.46410 q^{69} +3.00000 q^{70} -12.9282 q^{71} -1.00000 q^{72} +6.92820 q^{73} +5.92820 q^{74} -1.00000 q^{75} +5.73205 q^{76} -0.803848 q^{77} +3.07180 q^{79} +1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} +9.46410 q^{83} +3.00000 q^{84} -4.00000 q^{85} +6.00000 q^{86} -1.46410 q^{87} -0.267949 q^{88} -14.1244 q^{89} -1.00000 q^{90} -3.46410 q^{92} -4.92820 q^{93} -6.46410 q^{94} +5.73205 q^{95} +1.00000 q^{96} +8.39230 q^{97} -2.00000 q^{98} +0.267949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} - 2 q^{12} + 6 q^{14} - 2 q^{15} + 2 q^{16} - 8 q^{17} - 2 q^{18} + 8 q^{19} + 2 q^{20} + 6 q^{21} - 4 q^{22} + 2 q^{24} + 2 q^{25} - 2 q^{27} - 6 q^{28} - 4 q^{29} + 2 q^{30} - 4 q^{31} - 2 q^{32} - 4 q^{33} + 8 q^{34} - 6 q^{35} + 2 q^{36} + 2 q^{37} - 8 q^{38} - 2 q^{40} + 8 q^{41} - 6 q^{42} - 12 q^{43} + 4 q^{44} + 2 q^{45} + 6 q^{47} - 2 q^{48} + 4 q^{49} - 2 q^{50} + 8 q^{51} - 4 q^{53} + 2 q^{54} + 4 q^{55} + 6 q^{56} - 8 q^{57} + 4 q^{58} + 16 q^{59} - 2 q^{60} - 8 q^{61} + 4 q^{62} - 6 q^{63} + 2 q^{64} + 4 q^{66} + 4 q^{67} - 8 q^{68} + 6 q^{70} - 12 q^{71} - 2 q^{72} - 2 q^{74} - 2 q^{75} + 8 q^{76} - 12 q^{77} + 20 q^{79} + 2 q^{80} + 2 q^{81} - 8 q^{82} + 12 q^{83} + 6 q^{84} - 8 q^{85} + 12 q^{86} + 4 q^{87} - 4 q^{88} - 4 q^{89} - 2 q^{90} + 4 q^{93} - 6 q^{94} + 8 q^{95} + 2 q^{96} - 4 q^{97} - 4 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 4 * q^11 - 2 * q^12 + 6 * q^14 - 2 * q^15 + 2 * q^16 - 8 * q^17 - 2 * q^18 + 8 * q^19 + 2 * q^20 + 6 * q^21 - 4 * q^22 + 2 * q^24 + 2 * q^25 - 2 * q^27 - 6 * q^28 - 4 * q^29 + 2 * q^30 - 4 * q^31 - 2 * q^32 - 4 * q^33 + 8 * q^34 - 6 * q^35 + 2 * q^36 + 2 * q^37 - 8 * q^38 - 2 * q^40 + 8 * q^41 - 6 * q^42 - 12 * q^43 + 4 * q^44 + 2 * q^45 + 6 * q^47 - 2 * q^48 + 4 * q^49 - 2 * q^50 + 8 * q^51 - 4 * q^53 + 2 * q^54 + 4 * q^55 + 6 * q^56 - 8 * q^57 + 4 * q^58 + 16 * q^59 - 2 * q^60 - 8 * q^61 + 4 * q^62 - 6 * q^63 + 2 * q^64 + 4 * q^66 + 4 * q^67 - 8 * q^68 + 6 * q^70 - 12 * q^71 - 2 * q^72 - 2 * q^74 - 2 * q^75 + 8 * q^76 - 12 * q^77 + 20 * q^79 + 2 * q^80 + 2 * q^81 - 8 * q^82 + 12 * q^83 + 6 * q^84 - 8 * q^85 + 12 * q^86 + 4 * q^87 - 4 * q^88 - 4 * q^89 - 2 * q^90 + 4 * q^93 - 6 * q^94 + 8 * q^95 + 2 * q^96 - 4 * q^97 - 4 * q^98 + 4 * q^99

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