LMFDB - Embedded newform 138.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,2,Mod(1,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, 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("138.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	138.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:	$1.10193554789$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	138.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.23607 q^{5} +1.00000 q^{6} +4.47214 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.23607 q^{5} +1.00000 q^{6} +4.47214 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.23607 q^{10} -0.763932 q^{11} +1.00000 q^{12} -4.47214 q^{13} +4.47214 q^{14} -3.23607 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -7.70820 q^{19} -3.23607 q^{20} +4.47214 q^{21} -0.763932 q^{22} +1.00000 q^{23} +1.00000 q^{24} +5.47214 q^{25} -4.47214 q^{26} +1.00000 q^{27} +4.47214 q^{28} +4.47214 q^{29} -3.23607 q^{30} +6.47214 q^{31} +1.00000 q^{32} -0.763932 q^{33} -4.00000 q^{34} -14.4721 q^{35} +1.00000 q^{36} +6.76393 q^{37} -7.70820 q^{38} -4.47214 q^{39} -3.23607 q^{40} -2.00000 q^{41} +4.47214 q^{42} -9.23607 q^{43} -0.763932 q^{44} -3.23607 q^{45} +1.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} +13.0000 q^{49} +5.47214 q^{50} -4.00000 q^{51} -4.47214 q^{52} +0.763932 q^{53} +1.00000 q^{54} +2.47214 q^{55} +4.47214 q^{56} -7.70820 q^{57} +4.47214 q^{58} +8.94427 q^{59} -3.23607 q^{60} +5.23607 q^{61} +6.47214 q^{62} +4.47214 q^{63} +1.00000 q^{64} +14.4721 q^{65} -0.763932 q^{66} -3.70820 q^{67} -4.00000 q^{68} +1.00000 q^{69} -14.4721 q^{70} -8.94427 q^{71} +1.00000 q^{72} +4.47214 q^{73} +6.76393 q^{74} +5.47214 q^{75} -7.70820 q^{76} -3.41641 q^{77} -4.47214 q^{78} -4.47214 q^{79} -3.23607 q^{80} +1.00000 q^{81} -2.00000 q^{82} +8.76393 q^{83} +4.47214 q^{84} +12.9443 q^{85} -9.23607 q^{86} +4.47214 q^{87} -0.763932 q^{88} -1.52786 q^{89} -3.23607 q^{90} -20.0000 q^{91} +1.00000 q^{92} +6.47214 q^{93} +4.00000 q^{94} +24.9443 q^{95} +1.00000 q^{96} -8.47214 q^{97} +13.0000 q^{98} -0.763932 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 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 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 6 q^{11} + 2 q^{12} - 2 q^{15} + 2 q^{16} - 8 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 6 q^{22} + 2 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{27} - 2 q^{30} + 4 q^{31} + 2 q^{32} - 6 q^{33} - 8 q^{34} - 20 q^{35} + 2 q^{36} + 18 q^{37} - 2 q^{38} - 2 q^{40} - 4 q^{41} - 14 q^{43} - 6 q^{44} - 2 q^{45} + 2 q^{46} + 8 q^{47} + 2 q^{48} + 26 q^{49} + 2 q^{50} - 8 q^{51} + 6 q^{53} + 2 q^{54} - 4 q^{55} - 2 q^{57} - 2 q^{60} + 6 q^{61} + 4 q^{62} + 2 q^{64} + 20 q^{65} - 6 q^{66} + 6 q^{67} - 8 q^{68} + 2 q^{69} - 20 q^{70} + 2 q^{72} + 18 q^{74} + 2 q^{75} - 2 q^{76} + 20 q^{77} - 2 q^{80} + 2 q^{81} - 4 q^{82} + 22 q^{83} + 8 q^{85} - 14 q^{86} - 6 q^{88} - 12 q^{89} - 2 q^{90} - 40 q^{91} + 2 q^{92} + 4 q^{93} + 8 q^{94} + 32 q^{95} + 2 q^{96} - 8 q^{97} + 26 q^{98} - 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 6 * q^11 + 2 * q^12 - 2 * q^15 + 2 * q^16 - 8 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 6 * q^22 + 2 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^27 - 2 * q^30 + 4 * q^31 + 2 * q^32 - 6 * q^33 - 8 * q^34 - 20 * q^35 + 2 * q^36 + 18 * q^37 - 2 * q^38 - 2 * q^40 - 4 * q^41 - 14 * q^43 - 6 * q^44 - 2 * q^45 + 2 * q^46 + 8 * q^47 + 2 * q^48 + 26 * q^49 + 2 * q^50 - 8 * q^51 + 6 * q^53 + 2 * q^54 - 4 * q^55 - 2 * q^57 - 2 * q^60 + 6 * q^61 + 4 * q^62 + 2 * q^64 + 20 * q^65 - 6 * q^66 + 6 * q^67 - 8 * q^68 + 2 * q^69 - 20 * q^70 + 2 * q^72 + 18 * q^74 + 2 * q^75 - 2 * q^76 + 20 * q^77 - 2 * q^80 + 2 * q^81 - 4 * q^82 + 22 * q^83 + 8 * q^85 - 14 * q^86 - 6 * q^88 - 12 * q^89 - 2 * q^90 - 40 * q^91 + 2 * q^92 + 4 * q^93 + 8 * q^94 + 32 * q^95 + 2 * q^96 - 8 * q^97 + 26 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	1.00000	0.408248
$7$	4.47214	1.69031	0.845154	−	0.534522i	$-0.179509\pi$
$7$	4.47214	1.69031	0.845154	+	0.534522i	$0.179509\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.23607	−1.02333
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	1.00000	0.288675
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	4.47214	1.19523
$15$	−3.23607	−0.835549
$16$	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
$19$	−7.70820	−1.76838	−0.884192	−	0.467124i	$-0.845290\pi$
$19$	−7.70820	−1.76838	−0.884192	+	0.467124i	$0.845290\pi$
$20$	−3.23607	−0.723607
$21$	4.47214	0.975900
$22$	−0.763932	−0.162871
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	5.47214	1.09443
$26$	−4.47214	−0.877058
$27$	1.00000	0.192450
$28$	4.47214	0.845154
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	−3.23607	−0.590822
$31$	6.47214	1.16243	0.581215	−	0.813750i	$-0.302578\pi$
$31$	6.47214	1.16243	0.581215	+	0.813750i	$0.302578\pi$
$32$	1.00000	0.176777
$33$	−0.763932	−0.132983
$34$	−4.00000	−0.685994
$35$	−14.4721	−2.44624
$36$	1.00000	0.166667
$37$	6.76393	1.11198	0.555992	−	0.831188i	$-0.312339\pi$
$37$	6.76393	1.11198	0.555992	+	0.831188i	$0.312339\pi$
$38$	−7.70820	−1.25044
$39$	−4.47214	−0.716115
$40$	−3.23607	−0.511667
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	4.47214	0.690066
$43$	−9.23607	−1.40849	−0.704244	−	0.709958i	$-0.748714\pi$
$43$	−9.23607	−1.40849	−0.704244	+	0.709958i	$0.748714\pi$
$44$	−0.763932	−0.115167
$45$	−3.23607	−0.482405
$46$	1.00000	0.147442
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	1.00000	0.144338
$49$	13.0000	1.85714
$50$	5.47214	0.773877
$51$	−4.00000	−0.560112
$52$	−4.47214	−0.620174
$53$	0.763932	0.104934	0.0524671	−	0.998623i	$-0.483292\pi$
$53$	0.763932	0.104934	0.0524671	+	0.998623i	$0.483292\pi$
$54$	1.00000	0.136083
$55$	2.47214	0.333343
$56$	4.47214	0.597614
$57$	−7.70820	−1.02098
$58$	4.47214	0.587220
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	−3.23607	−0.417775
$61$	5.23607	0.670410	0.335205	−	0.942145i	$-0.391194\pi$
$61$	5.23607	0.670410	0.335205	+	0.942145i	$0.391194\pi$
$62$	6.47214	0.821962
$63$	4.47214	0.563436
$64$	1.00000	0.125000
$65$	14.4721	1.79505
$66$	−0.763932	−0.0940335
$67$	−3.70820	−0.453029	−0.226515	−	0.974008i	$-0.572733\pi$
$67$	−3.70820	−0.453029	−0.226515	+	0.974008i	$0.572733\pi$
$68$	−4.00000	−0.485071
$69$	1.00000	0.120386
$70$	−14.4721	−1.72975
$71$	−8.94427	−1.06149	−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427	−1.06149	−0.530745	+	0.847532i	$0.678088\pi$
$72$	1.00000	0.117851
$73$	4.47214	0.523424	0.261712	−	0.965146i	$-0.415713\pi$
$73$	4.47214	0.523424	0.261712	+	0.965146i	$0.415713\pi$
$74$	6.76393	0.786291
$75$	5.47214	0.631868
$76$	−7.70820	−0.884192
$77$	−3.41641	−0.389336
$78$	−4.47214	−0.506370
$79$	−4.47214	−0.503155	−0.251577	−	0.967837i	$-0.580949\pi$
$79$	−4.47214	−0.503155	−0.251577	+	0.967837i	$0.580949\pi$
$80$	−3.23607	−0.361803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	8.76393	0.961967	0.480983	−	0.876730i	$-0.340280\pi$
$83$	8.76393	0.961967	0.480983	+	0.876730i	$0.340280\pi$
$84$	4.47214	0.487950
$85$	12.9443	1.40400
$86$	−9.23607	−0.995951
$87$	4.47214	0.479463
$88$	−0.763932	−0.0814354
$89$	−1.52786	−0.161953	−0.0809766	−	0.996716i	$-0.525804\pi$
$89$	−1.52786	−0.161953	−0.0809766	+	0.996716i	$0.525804\pi$
$90$	−3.23607	−0.341112
$91$	−20.0000	−2.09657
$92$	1.00000	0.104257
$93$	6.47214	0.671129
$94$	4.00000	0.412568
$95$	24.9443	2.55923
$96$	1.00000	0.102062
$97$	−8.47214	−0.860215	−0.430108	−	0.902778i	$-0.641524\pi$
$97$	−8.47214	−0.860215	−0.430108	+	0.902778i	$0.641524\pi$
$98$	13.0000	1.31320
$99$	−0.763932	−0.0767781
$100$	5.47214	0.547214
$101$	−4.47214	−0.444994	−0.222497	−	0.974933i	$-0.571421\pi$
$101$	−4.47214	−0.444994	−0.222497	+	0.974933i	$0.571421\pi$
$102$	−4.00000	−0.396059
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	−4.47214	−0.438529
$105$	−14.4721	−1.41234
$106$	0.763932	0.0741996
$107$	18.6525	1.80320	0.901601	−	0.432568i	$-0.142392\pi$
$107$	18.6525	1.80320	0.901601	+	0.432568i	$0.142392\pi$
$108$	1.00000	0.0962250
$109$	9.23607	0.884655	0.442327	−	0.896854i	$-0.354153\pi$
$109$	9.23607	0.884655	0.442327	+	0.896854i	$0.354153\pi$
$110$	2.47214	0.235709
$111$	6.76393	0.642004
$112$	4.47214	0.422577
$113$	−14.4721	−1.36142	−0.680712	−	0.732551i	$-0.738329\pi$
$113$	−14.4721	−1.36142	−0.680712	+	0.732551i	$0.738329\pi$
$114$	−7.70820	−0.721939
$115$	−3.23607	−0.301765
$116$	4.47214	0.415227
$117$	−4.47214	−0.413449
$118$	8.94427	0.823387
$119$	−17.8885	−1.63984
$120$	−3.23607	−0.295411
$121$	−10.4164	−0.946946
$122$	5.23607	0.474051
$123$	−2.00000	−0.180334
$124$	6.47214	0.581215
$125$	−1.52786	−0.136656
$126$	4.47214	0.398410
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	1.00000	0.0883883
$129$	−9.23607	−0.813190
$130$	14.4721	1.26929
$131$	−18.4721	−1.61392	−0.806959	−	0.590607i	$-0.798888\pi$
$131$	−18.4721	−1.61392	−0.806959	+	0.590607i	$0.798888\pi$
$132$	−0.763932	−0.0664917
$133$	−34.4721	−2.98911
$134$	−3.70820	−0.320340
$135$	−3.23607	−0.278516
$136$	−4.00000	−0.342997
$137$	20.9443	1.78939	0.894695	−	0.446678i	$-0.147393\pi$
$137$	20.9443	1.78939	0.894695	+	0.446678i	$0.147393\pi$
$138$	1.00000	0.0851257
$139$	0.944272	0.0800921	0.0400460	−	0.999198i	$-0.487250\pi$
$139$	0.944272	0.0800921	0.0400460	+	0.999198i	$0.487250\pi$
$140$	−14.4721	−1.22312
$141$	4.00000	0.336861
$142$	−8.94427	−0.750587
$143$	3.41641	0.285694
$144$	1.00000	0.0833333
$145$	−14.4721	−1.20185
$146$	4.47214	0.370117
$147$	13.0000	1.07222
$148$	6.76393	0.555992
$149$	1.70820	0.139942	0.0699708	−	0.997549i	$-0.477709\pi$
$149$	1.70820	0.139942	0.0699708	+	0.997549i	$0.477709\pi$
$150$	5.47214	0.446798
$151$	−5.52786	−0.449851	−0.224926	−	0.974376i	$-0.572214\pi$
$151$	−5.52786	−0.449851	−0.224926	+	0.974376i	$0.572214\pi$
$152$	−7.70820	−0.625218
$153$	−4.00000	−0.323381
$154$	−3.41641	−0.275302
$155$	−20.9443	−1.68228
$156$	−4.47214	−0.358057
$157$	24.6525	1.96748	0.983741	−	0.179594i	$-0.0574783\pi$
$157$	24.6525	1.96748	0.983741	+	0.179594i	$0.0574783\pi$
$158$	−4.47214	−0.355784
$159$	0.763932	0.0605838
$160$	−3.23607	−0.255834
$161$	4.47214	0.352454
$162$	1.00000	0.0785674
$163$	6.47214	0.506937	0.253468	−	0.967344i	$-0.418429\pi$
$163$	6.47214	0.506937	0.253468	+	0.967344i	$0.418429\pi$
$164$	−2.00000	−0.156174
$165$	2.47214	0.192456
$166$	8.76393	0.680213
$167$	−0.944272	−0.0730700	−0.0365350	−	0.999332i	$-0.511632\pi$
$167$	−0.944272	−0.0730700	−0.0365350	+	0.999332i	$0.511632\pi$
$168$	4.47214	0.345033
$169$	7.00000	0.538462
$170$	12.9443	0.992780
$171$	−7.70820	−0.589461
$172$	−9.23607	−0.704244
$173$	9.41641	0.715916	0.357958	−	0.933738i	$-0.383473\pi$
$173$	9.41641	0.715916	0.357958	+	0.933738i	$0.383473\pi$
$174$	4.47214	0.339032
$175$	24.4721	1.84992
$176$	−0.763932	−0.0575835
$177$	8.94427	0.672293
$178$	−1.52786	−0.114518
$179$	−7.41641	−0.554328	−0.277164	−	0.960823i	$-0.589395\pi$
$179$	−7.41641	−0.554328	−0.277164	+	0.960823i	$0.589395\pi$
$180$	−3.23607	−0.241202
$181$	−6.76393	−0.502759	−0.251380	−	0.967889i	$-0.580884\pi$
$181$	−6.76393	−0.502759	−0.251380	+	0.967889i	$0.580884\pi$
$182$	−20.0000	−1.48250
$183$	5.23607	0.387061
$184$	1.00000	0.0737210
$185$	−21.8885	−1.60928
$186$	6.47214	0.474560
$187$	3.05573	0.223457
$188$	4.00000	0.291730
$189$	4.47214	0.325300
$190$	24.9443	1.80965
$191$	−2.47214	−0.178877	−0.0894387	−	0.995992i	$-0.528507\pi$
$191$	−2.47214	−0.178877	−0.0894387	+	0.995992i	$0.528507\pi$
$192$	1.00000	0.0721688
$193$	−11.8885	−0.855756	−0.427878	−	0.903836i	$-0.640739\pi$
$193$	−11.8885	−0.855756	−0.427878	+	0.903836i	$0.640739\pi$
$194$	−8.47214	−0.608264
$195$	14.4721	1.03637
$196$	13.0000	0.928571
$197$	−14.9443	−1.06474	−0.532368	−	0.846513i	$-0.678698\pi$
$197$	−14.9443	−1.06474	−0.532368	+	0.846513i	$0.678698\pi$
$198$	−0.763932	−0.0542903
$199$	−9.41641	−0.667511	−0.333756	−	0.942660i	$-0.608316\pi$
$199$	−9.41641	−0.667511	−0.333756	+	0.942660i	$0.608316\pi$
$200$	5.47214	0.386938
$201$	−3.70820	−0.261557
$202$	−4.47214	−0.314658
$203$	20.0000	1.40372
$204$	−4.00000	−0.280056
$205$	6.47214	0.452034
$206$	−6.00000	−0.418040
$207$	1.00000	0.0695048
$208$	−4.47214	−0.310087
$209$	5.88854	0.407319
$210$	−14.4721	−0.998672
$211$	3.41641	0.235195	0.117598	−	0.993061i	$-0.462481\pi$
$211$	3.41641	0.235195	0.117598	+	0.993061i	$0.462481\pi$
$212$	0.763932	0.0524671
$213$	−8.94427	−0.612851
$214$	18.6525	1.27506
$215$	29.8885	2.03838
$216$	1.00000	0.0680414
$217$	28.9443	1.96487
$218$	9.23607	0.625545
$219$	4.47214	0.302199
$220$	2.47214	0.166671
$221$	17.8885	1.20331
$222$	6.76393	0.453965
$223$	−7.41641	−0.496639	−0.248320	−	0.968678i	$-0.579878\pi$
$223$	−7.41641	−0.496639	−0.248320	+	0.968678i	$0.579878\pi$
$224$	4.47214	0.298807
$225$	5.47214	0.364809
$226$	−14.4721	−0.962672
$227$	−4.76393	−0.316193	−0.158097	−	0.987424i	$-0.550536\pi$
$227$	−4.76393	−0.316193	−0.158097	+	0.987424i	$0.550536\pi$
$228$	−7.70820	−0.510488
$229$	4.29180	0.283610	0.141805	−	0.989895i	$-0.454709\pi$
$229$	4.29180	0.283610	0.141805	+	0.989895i	$0.454709\pi$
$230$	−3.23607	−0.213380
$231$	−3.41641	−0.224783
$232$	4.47214	0.293610
$233$	−15.8885	−1.04089	−0.520447	−	0.853894i	$-0.674235\pi$
$233$	−15.8885	−1.04089	−0.520447	+	0.853894i	$0.674235\pi$
$234$	−4.47214	−0.292353
$235$	−12.9443	−0.844391
$236$	8.94427	0.582223
$237$	−4.47214	−0.290496
$238$	−17.8885	−1.15954
$239$	12.9443	0.837295	0.418648	−	0.908149i	$-0.362504\pi$
$239$	12.9443	0.837295	0.418648	+	0.908149i	$0.362504\pi$
$240$	−3.23607	−0.208887
$241$	−3.52786	−0.227250	−0.113625	−	0.993524i	$-0.536246\pi$
$241$	−3.52786	−0.227250	−0.113625	+	0.993524i	$0.536246\pi$
$242$	−10.4164	−0.669592
$243$	1.00000	0.0641500
$244$	5.23607	0.335205
$245$	−42.0689	−2.68768
$246$	−2.00000	−0.127515
$247$	34.4721	2.19341
$248$	6.47214	0.410981
$249$	8.76393	0.555392
$250$	−1.52786	−0.0966306
$251$	−6.29180	−0.397135	−0.198567	−	0.980087i	$-0.563629\pi$
$251$	−6.29180	−0.397135	−0.198567	+	0.980087i	$0.563629\pi$
$252$	4.47214	0.281718
$253$	−0.763932	−0.0480280
$254$	−4.00000	−0.250982
$255$	12.9443	0.810602
$256$	1.00000	0.0625000
$257$	23.8885	1.49013	0.745063	−	0.666994i	$-0.232419\pi$
$257$	23.8885	1.49013	0.745063	+	0.666994i	$0.232419\pi$
$258$	−9.23607	−0.575012
$259$	30.2492	1.87960
$260$	14.4721	0.897524
$261$	4.47214	0.276818
$262$	−18.4721	−1.14121
$263$	7.05573	0.435075	0.217537	−	0.976052i	$-0.430198\pi$
$263$	7.05573	0.435075	0.217537	+	0.976052i	$0.430198\pi$
$264$	−0.763932	−0.0470168
$265$	−2.47214	−0.151862
$266$	−34.4721	−2.11362
$267$	−1.52786	−0.0935038
$268$	−3.70820	−0.226515
$269$	−30.9443	−1.88671	−0.943353	−	0.331791i	$-0.892347\pi$
$269$	−30.9443	−1.88671	−0.943353	+	0.331791i	$0.892347\pi$
$270$	−3.23607	−0.196941
$271$	0.944272	0.0573604	0.0286802	−	0.999589i	$-0.490870\pi$
$271$	0.944272	0.0573604	0.0286802	+	0.999589i	$0.490870\pi$
$272$	−4.00000	−0.242536
$273$	−20.0000	−1.21046
$274$	20.9443	1.26529
$275$	−4.18034	−0.252084
$276$	1.00000	0.0601929
$277$	−11.5279	−0.692642	−0.346321	−	0.938116i	$-0.612569\pi$
$277$	−11.5279	−0.692642	−0.346321	+	0.938116i	$0.612569\pi$
$278$	0.944272	0.0566337
$279$	6.47214	0.387477
$280$	−14.4721	−0.864876
$281$	−22.4721	−1.34058	−0.670288	−	0.742101i	$-0.733829\pi$
$281$	−22.4721	−1.34058	−0.670288	+	0.742101i	$0.733829\pi$
$282$	4.00000	0.238197
$283$	−26.1803	−1.55626	−0.778130	−	0.628103i	$-0.783831\pi$
$283$	−26.1803	−1.55626	−0.778130	+	0.628103i	$0.783831\pi$
$284$	−8.94427	−0.530745
$285$	24.9443	1.47757
$286$	3.41641	0.202016
$287$	−8.94427	−0.527964
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	−14.4721	−0.849833
$291$	−8.47214	−0.496645
$292$	4.47214	0.261712
$293$	13.7082	0.800842	0.400421	−	0.916331i	$-0.368864\pi$
$293$	13.7082	0.800842	0.400421	+	0.916331i	$0.368864\pi$
$294$	13.0000	0.758175
$295$	−28.9443	−1.68520
$296$	6.76393	0.393146
$297$	−0.763932	−0.0443278
$298$	1.70820	0.0989536
$299$	−4.47214	−0.258630
$300$	5.47214	0.315934
$301$	−41.3050	−2.38078
$302$	−5.52786	−0.318093
$303$	−4.47214	−0.256917
$304$	−7.70820	−0.442096
$305$	−16.9443	−0.970226
$306$	−4.00000	−0.228665
$307$	−11.4164	−0.651569	−0.325784	−	0.945444i	$-0.605628\pi$
$307$	−11.4164	−0.651569	−0.325784	+	0.945444i	$0.605628\pi$
$308$	−3.41641	−0.194668
$309$	−6.00000	−0.341328
$310$	−20.9443	−1.18955
$311$	−3.05573	−0.173274	−0.0866372	−	0.996240i	$-0.527612\pi$
$311$	−3.05573	−0.173274	−0.0866372	+	0.996240i	$0.527612\pi$
$312$	−4.47214	−0.253185
$313$	24.4721	1.38325	0.691623	−	0.722258i	$-0.256896\pi$
$313$	24.4721	1.38325	0.691623	+	0.722258i	$0.256896\pi$
$314$	24.6525	1.39122
$315$	−14.4721	−0.815412
$316$	−4.47214	−0.251577
$317$	28.4721	1.59915	0.799577	−	0.600563i	$-0.205057\pi$
$317$	28.4721	1.59915	0.799577	+	0.600563i	$0.205057\pi$
$318$	0.763932	0.0428392
$319$	−3.41641	−0.191282
$320$	−3.23607	−0.180902
$321$	18.6525	1.04108
$322$	4.47214	0.249222
$323$	30.8328	1.71558
$324$	1.00000	0.0555556
$325$	−24.4721	−1.35747
$326$	6.47214	0.358458
$327$	9.23607	0.510756
$328$	−2.00000	−0.110432
$329$	17.8885	0.986227
$330$	2.47214	0.136087
$331$	−1.52786	−0.0839790	−0.0419895	−	0.999118i	$-0.513370\pi$
$331$	−1.52786	−0.0839790	−0.0419895	+	0.999118i	$0.513370\pi$
$332$	8.76393	0.480983
$333$	6.76393	0.370661
$334$	−0.944272	−0.0516683
$335$	12.0000	0.655630
$336$	4.47214	0.243975
$337$	15.8885	0.865504	0.432752	−	0.901513i	$-0.357543\pi$
$337$	15.8885	0.865504	0.432752	+	0.901513i	$0.357543\pi$
$338$	7.00000	0.380750
$339$	−14.4721	−0.786019
$340$	12.9443	0.702002
$341$	−4.94427	−0.267747
$342$	−7.70820	−0.416812
$343$	26.8328	1.44884
$344$	−9.23607	−0.497975
$345$	−3.23607	−0.174224
$346$	9.41641	0.506229
$347$	21.5279	1.15568	0.577838	−	0.816151i	$-0.303896\pi$
$347$	21.5279	1.15568	0.577838	+	0.816151i	$0.303896\pi$
$348$	4.47214	0.239732
$349$	−31.8885	−1.70695	−0.853477	−	0.521130i	$-0.825511\pi$
$349$	−31.8885	−1.70695	−0.853477	+	0.521130i	$0.825511\pi$
$350$	24.4721	1.30809
$351$	−4.47214	−0.238705
$352$	−0.763932	−0.0407177
$353$	31.8885	1.69726	0.848628	−	0.528990i	$-0.177429\pi$
$353$	31.8885	1.69726	0.848628	+	0.528990i	$0.177429\pi$
$354$	8.94427	0.475383
$355$	28.9443	1.53620
$356$	−1.52786	−0.0809766
$357$	−17.8885	−0.946762
$358$	−7.41641	−0.391969
$359$	−33.3050	−1.75777	−0.878884	−	0.477035i	$-0.841711\pi$
$359$	−33.3050	−1.75777	−0.878884	+	0.477035i	$0.841711\pi$
$360$	−3.23607	−0.170556
$361$	40.4164	2.12718
$362$	−6.76393	−0.355504
$363$	−10.4164	−0.546720
$364$	−20.0000	−1.04828
$365$	−14.4721	−0.757506
$366$	5.23607	0.273694
$367$	−17.4164	−0.909129	−0.454565	−	0.890714i	$-0.650205\pi$
$367$	−17.4164	−0.909129	−0.454565	+	0.890714i	$0.650205\pi$
$368$	1.00000	0.0521286
$369$	−2.00000	−0.104116
$370$	−21.8885	−1.13793
$371$	3.41641	0.177371
$372$	6.47214	0.335565
$373$	13.5967	0.704013	0.352006	−	0.935998i	$-0.385500\pi$
$373$	13.5967	0.704013	0.352006	+	0.935998i	$0.385500\pi$
$374$	3.05573	0.158008
$375$	−1.52786	−0.0788986
$376$	4.00000	0.206284
$377$	−20.0000	−1.03005
$378$	4.47214	0.230022
$379$	0.291796	0.0149886	0.00749428	−	0.999972i	$-0.497614\pi$
$379$	0.291796	0.0149886	0.00749428	+	0.999972i	$0.497614\pi$
$380$	24.9443	1.27961
$381$	−4.00000	−0.204926
$382$	−2.47214	−0.126485
$383$	24.9443	1.27459	0.637296	−	0.770619i	$-0.280053\pi$
$383$	24.9443	1.27459	0.637296	+	0.770619i	$0.280053\pi$
$384$	1.00000	0.0510310
$385$	11.0557	0.563452
$386$	−11.8885	−0.605111
$387$	−9.23607	−0.469496
$388$	−8.47214	−0.430108
$389$	10.2918	0.521815	0.260907	−	0.965364i	$-0.415978\pi$
$389$	10.2918	0.521815	0.260907	+	0.965364i	$0.415978\pi$
$390$	14.4721	0.732825
$391$	−4.00000	−0.202289
$392$	13.0000	0.656599
$393$	−18.4721	−0.931796
$394$	−14.9443	−0.752882
$395$	14.4721	0.728172
$396$	−0.763932	−0.0383890
$397$	9.05573	0.454494	0.227247	−	0.973837i	$-0.427028\pi$
$397$	9.05573	0.454494	0.227247	+	0.973837i	$0.427028\pi$
$398$	−9.41641	−0.472002
$399$	−34.4721	−1.72577
$400$	5.47214	0.273607
$401$	−8.94427	−0.446656	−0.223328	−	0.974743i	$-0.571692\pi$
$401$	−8.94427	−0.446656	−0.223328	+	0.974743i	$0.571692\pi$
$402$	−3.70820	−0.184948
$403$	−28.9443	−1.44182
$404$	−4.47214	−0.222497
$405$	−3.23607	−0.160802
$406$	20.0000	0.992583
$407$	−5.16718	−0.256128
$408$	−4.00000	−0.198030
$409$	−0.111456	−0.00551115	−0.00275558	−	0.999996i	$-0.500877\pi$
$409$	−0.111456	−0.00551115	−0.00275558	+	0.999996i	$0.500877\pi$
$410$	6.47214	0.319636
$411$	20.9443	1.03310
$412$	−6.00000	−0.295599
$413$	40.0000	1.96827
$414$	1.00000	0.0491473
$415$	−28.3607	−1.39217
$416$	−4.47214	−0.219265
$417$	0.944272	0.0462412
$418$	5.88854	0.288018
$419$	−30.0689	−1.46896	−0.734481	−	0.678630i	$-0.762574\pi$
$419$	−30.0689	−1.46896	−0.734481	+	0.678630i	$0.762574\pi$
$420$	−14.4721	−0.706168
$421$	−5.23607	−0.255190	−0.127595	−	0.991826i	$-0.540726\pi$
$421$	−5.23607	−0.255190	−0.127595	+	0.991826i	$0.540726\pi$
$422$	3.41641	0.166308
$423$	4.00000	0.194487
$424$	0.763932	0.0370998
$425$	−21.8885	−1.06175
$426$	−8.94427	−0.433351
$427$	23.4164	1.13320
$428$	18.6525	0.901601
$429$	3.41641	0.164946
$430$	29.8885	1.44135
$431$	3.41641	0.164563	0.0822813	−	0.996609i	$-0.473779\pi$
$431$	3.41641	0.164563	0.0822813	+	0.996609i	$0.473779\pi$
$432$	1.00000	0.0481125
$433$	5.41641	0.260296	0.130148	−	0.991495i	$-0.458455\pi$
$433$	5.41641	0.260296	0.130148	+	0.991495i	$0.458455\pi$
$434$	28.9443	1.38937
$435$	−14.4721	−0.693886
$436$	9.23607	0.442327
$437$	−7.70820	−0.368733
$438$	4.47214	0.213687
$439$	36.9443	1.76325	0.881627	−	0.471947i	$-0.156449\pi$
$439$	36.9443	1.76325	0.881627	+	0.471947i	$0.156449\pi$
$440$	2.47214	0.117854
$441$	13.0000	0.619048
$442$	17.8885	0.850871
$443$	−32.9443	−1.56523	−0.782615	−	0.622506i	$-0.786115\pi$
$443$	−32.9443	−1.56523	−0.782615	+	0.622506i	$0.786115\pi$
$444$	6.76393	0.321002
$445$	4.94427	0.234381
$446$	−7.41641	−0.351177
$447$	1.70820	0.0807953
$448$	4.47214	0.211289
$449$	−19.8885	−0.938598	−0.469299	−	0.883039i	$-0.655493\pi$
$449$	−19.8885	−0.938598	−0.469299	+	0.883039i	$0.655493\pi$
$450$	5.47214	0.257959
$451$	1.52786	0.0719443
$452$	−14.4721	−0.680712
$453$	−5.52786	−0.259722
$454$	−4.76393	−0.223582
$455$	64.7214	3.03418
$456$	−7.70820	−0.360970
$457$	36.4721	1.70609	0.853047	−	0.521834i	$-0.174752\pi$
$457$	36.4721	1.70609	0.853047	+	0.521834i	$0.174752\pi$
$458$	4.29180	0.200542
$459$	−4.00000	−0.186704
$460$	−3.23607	−0.150882
$461$	22.0000	1.02464	0.512321	−	0.858794i	$-0.328786\pi$
$461$	22.0000	1.02464	0.512321	+	0.858794i	$0.328786\pi$
$462$	−3.41641	−0.158946
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	4.47214	0.207614
$465$	−20.9443	−0.971267
$466$	−15.8885	−0.736023
$467$	−40.1803	−1.85932	−0.929662	−	0.368413i	$-0.879901\pi$
$467$	−40.1803	−1.85932	−0.929662	+	0.368413i	$0.879901\pi$
$468$	−4.47214	−0.206725
$469$	−16.5836	−0.765759
$470$	−12.9443	−0.597075
$471$	24.6525	1.13593
$472$	8.94427	0.411693
$473$	7.05573	0.324423
$474$	−4.47214	−0.205412
$475$	−42.1803	−1.93537
$476$	−17.8885	−0.819920
$477$	0.763932	0.0349780
$478$	12.9443	0.592057
$479$	−11.0557	−0.505149	−0.252575	−	0.967577i	$-0.581277\pi$
$479$	−11.0557	−0.505149	−0.252575	+	0.967577i	$0.581277\pi$
$480$	−3.23607	−0.147706
$481$	−30.2492	−1.37925
$482$	−3.52786	−0.160690
$483$	4.47214	0.203489
$484$	−10.4164	−0.473473
$485$	27.4164	1.24491
$486$	1.00000	0.0453609
$487$	−25.8885	−1.17312	−0.586561	−	0.809905i	$-0.699519\pi$
$487$	−25.8885	−1.17312	−0.586561	+	0.809905i	$0.699519\pi$
$488$	5.23607	0.237026
$489$	6.47214	0.292680
$490$	−42.0689	−1.90048
$491$	−33.3050	−1.50303	−0.751516	−	0.659715i	$-0.770677\pi$
$491$	−33.3050	−1.50303	−0.751516	+	0.659715i	$0.770677\pi$
$492$	−2.00000	−0.0901670
$493$	−17.8885	−0.805659
$494$	34.4721	1.55097
$495$	2.47214	0.111114
$496$	6.47214	0.290607
$497$	−40.0000	−1.79425
$498$	8.76393	0.392721
$499$	27.4164	1.22733	0.613663	−	0.789568i	$-0.289695\pi$
$499$	27.4164	1.22733	0.613663	+	0.789568i	$0.289695\pi$
$500$	−1.52786	−0.0683282
$501$	−0.944272	−0.0421870
$502$	−6.29180	−0.280817
$503$	1.52786	0.0681241	0.0340620	−	0.999420i	$-0.489156\pi$
$503$	1.52786	0.0681241	0.0340620	+	0.999420i	$0.489156\pi$
$504$	4.47214	0.199205
$505$	14.4721	0.644002
$506$	−0.763932	−0.0339609
$507$	7.00000	0.310881
$508$	−4.00000	−0.177471
$509$	−6.58359	−0.291813	−0.145906	−	0.989298i	$-0.546610\pi$
$509$	−6.58359	−0.291813	−0.145906	+	0.989298i	$0.546610\pi$
$510$	12.9443	0.573182
$511$	20.0000	0.884748
$512$	1.00000	0.0441942
$513$	−7.70820	−0.340326
$514$	23.8885	1.05368
$515$	19.4164	0.855589
$516$	−9.23607	−0.406595
$517$	−3.05573	−0.134391
$518$	30.2492	1.32907
$519$	9.41641	0.413334
$520$	14.4721	0.634645
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	4.47214	0.195740
$523$	12.2918	0.537483	0.268741	−	0.963212i	$-0.413392\pi$
$523$	12.2918	0.537483	0.268741	+	0.963212i	$0.413392\pi$
$524$	−18.4721	−0.806959
$525$	24.4721	1.06805
$526$	7.05573	0.307644
$527$	−25.8885	−1.12772
$528$	−0.763932	−0.0332459
$529$	1.00000	0.0434783
$530$	−2.47214	−0.107383
$531$	8.94427	0.388148
$532$	−34.4721	−1.49456
$533$	8.94427	0.387419
$534$	−1.52786	−0.0661171
$535$	−60.3607	−2.60962
$536$	−3.70820	−0.160170
$537$	−7.41641	−0.320042
$538$	−30.9443	−1.33410
$539$	−9.93112	−0.427763
$540$	−3.23607	−0.139258
$541$	43.8885	1.88692	0.943458	−	0.331492i	$-0.107552\pi$
$541$	43.8885	1.88692	0.943458	+	0.331492i	$0.107552\pi$
$542$	0.944272	0.0405600
$543$	−6.76393	−0.290268
$544$	−4.00000	−0.171499
$545$	−29.8885	−1.28028
$546$	−20.0000	−0.855921
$547$	−21.3050	−0.910934	−0.455467	−	0.890253i	$-0.650528\pi$
$547$	−21.3050	−0.910934	−0.455467	+	0.890253i	$0.650528\pi$
$548$	20.9443	0.894695
$549$	5.23607	0.223470
$550$	−4.18034	−0.178250
$551$	−34.4721	−1.46856
$552$	1.00000	0.0425628
$553$	−20.0000	−0.850487
$554$	−11.5279	−0.489772
$555$	−21.8885	−0.929117
$556$	0.944272	0.0400460
$557$	25.1246	1.06456	0.532282	−	0.846567i	$-0.321335\pi$
$557$	25.1246	1.06456	0.532282	+	0.846567i	$0.321335\pi$
$558$	6.47214	0.273987
$559$	41.3050	1.74701
$560$	−14.4721	−0.611559
$561$	3.05573	0.129013
$562$	−22.4721	−0.947930
$563$	−6.65248	−0.280368	−0.140184	−	0.990125i	$-0.544769\pi$
$563$	−6.65248	−0.280368	−0.140184	+	0.990125i	$0.544769\pi$
$564$	4.00000	0.168430
$565$	46.8328	1.97027
$566$	−26.1803	−1.10044
$567$	4.47214	0.187812
$568$	−8.94427	−0.375293
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	24.9443	1.04480
$571$	−27.7082	−1.15955	−0.579776	−	0.814776i	$-0.696860\pi$
$571$	−27.7082	−1.15955	−0.579776	+	0.814776i	$0.696860\pi$
$572$	3.41641	0.142847
$573$	−2.47214	−0.103275
$574$	−8.94427	−0.373327
$575$	5.47214	0.228204
$576$	1.00000	0.0416667
$577$	10.3607	0.431321	0.215660	−	0.976468i	$-0.430810\pi$
$577$	10.3607	0.431321	0.215660	+	0.976468i	$0.430810\pi$
$578$	−1.00000	−0.0415945
$579$	−11.8885	−0.494071
$580$	−14.4721	−0.600923
$581$	39.1935	1.62602
$582$	−8.47214	−0.351181
$583$	−0.583592	−0.0241699
$584$	4.47214	0.185058
$585$	14.4721	0.598349
$586$	13.7082	0.566281
$587$	−2.47214	−0.102036	−0.0510180	−	0.998698i	$-0.516247\pi$
$587$	−2.47214	−0.102036	−0.0510180	+	0.998698i	$0.516247\pi$
$588$	13.0000	0.536111
$589$	−49.8885	−2.05562
$590$	−28.9443	−1.19162
$591$	−14.9443	−0.614725
$592$	6.76393	0.277996
$593$	−37.7771	−1.55132	−0.775660	−	0.631152i	$-0.782583\pi$
$593$	−37.7771	−1.55132	−0.775660	+	0.631152i	$0.782583\pi$
$594$	−0.763932	−0.0313445
$595$	57.8885	2.37320
$596$	1.70820	0.0699708
$597$	−9.41641	−0.385388
$598$	−4.47214	−0.182879
$599$	−1.88854	−0.0771638	−0.0385819	−	0.999255i	$-0.512284\pi$
$599$	−1.88854	−0.0771638	−0.0385819	+	0.999255i	$0.512284\pi$
$600$	5.47214	0.223399
$601$	34.3607	1.40160	0.700801	−	0.713357i	$-0.252826\pi$
$601$	34.3607	1.40160	0.700801	+	0.713357i	$0.252826\pi$
$602$	−41.3050	−1.68346
$603$	−3.70820	−0.151010
$604$	−5.52786	−0.224926
$605$	33.7082	1.37043
$606$	−4.47214	−0.181668
$607$	−26.4721	−1.07447	−0.537235	−	0.843432i	$-0.680531\pi$
$607$	−26.4721	−1.07447	−0.537235	+	0.843432i	$0.680531\pi$
$608$	−7.70820	−0.312609
$609$	20.0000	0.810441
$610$	−16.9443	−0.686054
$611$	−17.8885	−0.723693
$612$	−4.00000	−0.161690
$613$	15.1246	0.610877	0.305439	−	0.952212i	$-0.401197\pi$
$613$	15.1246	0.610877	0.305439	+	0.952212i	$0.401197\pi$
$614$	−11.4164	−0.460729
$615$	6.47214	0.260982
$616$	−3.41641	−0.137651
$617$	−24.3607	−0.980724	−0.490362	−	0.871519i	$-0.663135\pi$
$617$	−24.3607	−0.980724	−0.490362	+	0.871519i	$0.663135\pi$
$618$	−6.00000	−0.241355
$619$	31.7082	1.27446	0.637230	−	0.770674i	$-0.280080\pi$
$619$	31.7082	1.27446	0.637230	+	0.770674i	$0.280080\pi$
$620$	−20.9443	−0.841142
$621$	1.00000	0.0401286
$622$	−3.05573	−0.122524
$623$	−6.83282	−0.273751
$624$	−4.47214	−0.179029
$625$	−22.4164	−0.896656
$626$	24.4721	0.978103
$627$	5.88854	0.235166
$628$	24.6525	0.983741
$629$	−27.0557	−1.07878
$630$	−14.4721	−0.576584
$631$	−6.00000	−0.238856	−0.119428	−	0.992843i	$-0.538106\pi$
$631$	−6.00000	−0.238856	−0.119428	+	0.992843i	$0.538106\pi$
$632$	−4.47214	−0.177892
$633$	3.41641	0.135790
$634$	28.4721	1.13077
$635$	12.9443	0.513678
$636$	0.763932	0.0302919
$637$	−58.1378	−2.30350
$638$	−3.41641	−0.135257
$639$	−8.94427	−0.353830
$640$	−3.23607	−0.127917
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	18.6525	0.736155
$643$	34.5410	1.36216	0.681082	−	0.732207i	$-0.261510\pi$
$643$	34.5410	1.36216	0.681082	+	0.732207i	$0.261510\pi$
$644$	4.47214	0.176227
$645$	29.8885	1.17686
$646$	30.8328	1.21310
$647$	−4.94427	−0.194379	−0.0971897	−	0.995266i	$-0.530985\pi$
$647$	−4.94427	−0.194379	−0.0971897	+	0.995266i	$0.530985\pi$
$648$	1.00000	0.0392837
$649$	−6.83282	−0.268211
$650$	−24.4721	−0.959876
$651$	28.9443	1.13442
$652$	6.47214	0.253468
$653$	−17.4164	−0.681557	−0.340778	−	0.940144i	$-0.610691\pi$
$653$	−17.4164	−0.681557	−0.340778	+	0.940144i	$0.610691\pi$
$654$	9.23607	0.361159
$655$	59.7771	2.33568
$656$	−2.00000	−0.0780869
$657$	4.47214	0.174475
$658$	17.8885	0.697368
$659$	−1.34752	−0.0524921	−0.0262460	−	0.999656i	$-0.508355\pi$
$659$	−1.34752	−0.0524921	−0.0262460	+	0.999656i	$0.508355\pi$
$660$	2.47214	0.0962278
$661$	26.1803	1.01830	0.509149	−	0.860679i	$-0.329960\pi$
$661$	26.1803	1.01830	0.509149	+	0.860679i	$0.329960\pi$
$662$	−1.52786	−0.0593821
$663$	17.8885	0.694733
$664$	8.76393	0.340107
$665$	111.554	4.32589
$666$	6.76393	0.262097
$667$	4.47214	0.173162
$668$	−0.944272	−0.0365350
$669$	−7.41641	−0.286735
$670$	12.0000	0.463600
$671$	−4.00000	−0.154418
$672$	4.47214	0.172516
$673$	−4.47214	−0.172388	−0.0861941	−	0.996278i	$-0.527471\pi$
$673$	−4.47214	−0.172388	−0.0861941	+	0.996278i	$0.527471\pi$
$674$	15.8885	0.612004
$675$	5.47214	0.210623
$676$	7.00000	0.269231
$677$	−21.1246	−0.811885	−0.405942	−	0.913899i	$-0.633057\pi$
$677$	−21.1246	−0.811885	−0.405942	+	0.913899i	$0.633057\pi$
$678$	−14.4721	−0.555799
$679$	−37.8885	−1.45403
$680$	12.9443	0.496390
$681$	−4.76393	−0.182554
$682$	−4.94427	−0.189326
$683$	5.52786	0.211518	0.105759	−	0.994392i	$-0.466273\pi$
$683$	5.52786	0.211518	0.105759	+	0.994392i	$0.466273\pi$
$684$	−7.70820	−0.294731
$685$	−67.7771	−2.58963
$686$	26.8328	1.02448
$687$	4.29180	0.163742
$688$	−9.23607	−0.352122
$689$	−3.41641	−0.130155
$690$	−3.23607	−0.123195
$691$	43.4164	1.65164	0.825819	−	0.563935i	$-0.190713\pi$
$691$	43.4164	1.65164	0.825819	+	0.563935i	$0.190713\pi$
$692$	9.41641	0.357958
$693$	−3.41641	−0.129779
$694$	21.5279	0.817187
$695$	−3.05573	−0.115910
$696$	4.47214	0.169516
$697$	8.00000	0.303022
$698$	−31.8885	−1.20700
$699$	−15.8885	−0.600960
$700$	24.4721	0.924960
$701$	37.1246	1.40218	0.701089	−	0.713074i	$-0.252698\pi$
$701$	37.1246	1.40218	0.701089	+	0.713074i	$0.252698\pi$
$702$	−4.47214	−0.168790
$703$	−52.1378	−1.96641
$704$	−0.763932	−0.0287918
$705$	−12.9443	−0.487509
$706$	31.8885	1.20014
$707$	−20.0000	−0.752177
$708$	8.94427	0.336146
$709$	−41.0132	−1.54028	−0.770141	−	0.637874i	$-0.779814\pi$
$709$	−41.0132	−1.54028	−0.770141	+	0.637874i	$0.779814\pi$
$710$	28.9443	1.08626
$711$	−4.47214	−0.167718
$712$	−1.52786	−0.0572591
$713$	6.47214	0.242383
$714$	−17.8885	−0.669462
$715$	−11.0557	−0.413461
$716$	−7.41641	−0.277164
$717$	12.9443	0.483413
$718$	−33.3050	−1.24293
$719$	−20.9443	−0.781090	−0.390545	−	0.920584i	$-0.627713\pi$
$719$	−20.9443	−0.781090	−0.390545	+	0.920584i	$0.627713\pi$
$720$	−3.23607	−0.120601
$721$	−26.8328	−0.999306
$722$	40.4164	1.50414
$723$	−3.52786	−0.131203
$724$	−6.76393	−0.251380
$725$	24.4721	0.908872
$726$	−10.4164	−0.386589
$727$	43.3050	1.60609	0.803046	−	0.595917i	$-0.203211\pi$
$727$	43.3050	1.60609	0.803046	+	0.595917i	$0.203211\pi$
$728$	−20.0000	−0.741249
$729$	1.00000	0.0370370
$730$	−14.4721	−0.535638
$731$	36.9443	1.36643
$732$	5.23607	0.193531
$733$	−8.29180	−0.306264	−0.153132	−	0.988206i	$-0.548936\pi$
$733$	−8.29180	−0.306264	−0.153132	+	0.988206i	$0.548936\pi$
$734$	−17.4164	−0.642851
$735$	−42.0689	−1.55173
$736$	1.00000	0.0368605
$737$	2.83282	0.104348
$738$	−2.00000	−0.0736210
$739$	−15.0557	−0.553834	−0.276917	−	0.960894i	$-0.589313\pi$
$739$	−15.0557	−0.553834	−0.276917	+	0.960894i	$0.589313\pi$
$740$	−21.8885	−0.804639
$741$	34.4721	1.26637
$742$	3.41641	0.125420
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	6.47214	0.237280
$745$	−5.52786	−0.202525
$746$	13.5967	0.497812
$747$	8.76393	0.320656
$748$	3.05573	0.111728
$749$	83.4164	3.04797
$750$	−1.52786	−0.0557897
$751$	−18.0000	−0.656829	−0.328415	−	0.944534i	$-0.606514\pi$
$751$	−18.0000	−0.656829	−0.328415	+	0.944534i	$0.606514\pi$
$752$	4.00000	0.145865
$753$	−6.29180	−0.229286
$754$	−20.0000	−0.728357
$755$	17.8885	0.651031
$756$	4.47214	0.162650
$757$	−32.6525	−1.18677	−0.593387	−	0.804917i	$-0.702210\pi$
$757$	−32.6525	−1.18677	−0.593387	+	0.804917i	$0.702210\pi$
$758$	0.291796	0.0105985
$759$	−0.763932	−0.0277290
$760$	24.9443	0.904824
$761$	−22.9443	−0.831729	−0.415865	−	0.909427i	$-0.636521\pi$
$761$	−22.9443	−0.831729	−0.415865	+	0.909427i	$0.636521\pi$
$762$	−4.00000	−0.144905
$763$	41.3050	1.49534
$764$	−2.47214	−0.0894387
$765$	12.9443	0.468001
$766$	24.9443	0.901273
$767$	−40.0000	−1.44432
$768$	1.00000	0.0360844
$769$	3.52786	0.127218	0.0636090	−	0.997975i	$-0.479739\pi$
$769$	3.52786	0.127218	0.0636090	+	0.997975i	$0.479739\pi$
$770$	11.0557	0.398421
$771$	23.8885	0.860325
$772$	−11.8885	−0.427878
$773$	−15.8197	−0.568994	−0.284497	−	0.958677i	$-0.591827\pi$
$773$	−15.8197	−0.568994	−0.284497	+	0.958677i	$0.591827\pi$
$774$	−9.23607	−0.331984
$775$	35.4164	1.27219
$776$	−8.47214	−0.304132
$777$	30.2492	1.08518
$778$	10.2918	0.368979
$779$	15.4164	0.552350
$780$	14.4721	0.518186
$781$	6.83282	0.244497
$782$	−4.00000	−0.143040
$783$	4.47214	0.159821
$784$	13.0000	0.464286
$785$	−79.7771	−2.84737
$786$	−18.4721	−0.658879
$787$	30.7639	1.09662	0.548308	−	0.836277i	$-0.315272\pi$
$787$	30.7639	1.09662	0.548308	+	0.836277i	$0.315272\pi$
$788$	−14.9443	−0.532368
$789$	7.05573	0.251191
$790$	14.4721	0.514895
$791$	−64.7214	−2.30123
$792$	−0.763932	−0.0271451
$793$	−23.4164	−0.831541
$794$	9.05573	0.321376
$795$	−2.47214	−0.0876776
$796$	−9.41641	−0.333756
$797$	−7.59675	−0.269091	−0.134545	−	0.990907i	$-0.542957\pi$
$797$	−7.59675	−0.269091	−0.134545	+	0.990907i	$0.542957\pi$
$798$	−34.4721	−1.22030
$799$	−16.0000	−0.566039
$800$	5.47214	0.193469
$801$	−1.52786	−0.0539844
$802$	−8.94427	−0.315833
$803$	−3.41641	−0.120562
$804$	−3.70820	−0.130778
$805$	−14.4721	−0.510076
$806$	−28.9443	−1.01952
$807$	−30.9443	−1.08929
$808$	−4.47214	−0.157329
$809$	25.0557	0.880912	0.440456	−	0.897774i	$-0.354817\pi$
$809$	25.0557	0.880912	0.440456	+	0.897774i	$0.354817\pi$
$810$	−3.23607	−0.113704
$811$	−21.3050	−0.748118	−0.374059	−	0.927405i	$-0.622034\pi$
$811$	−21.3050	−0.748118	−0.374059	+	0.927405i	$0.622034\pi$
$812$	20.0000	0.701862
$813$	0.944272	0.0331171
$814$	−5.16718	−0.181110
$815$	−20.9443	−0.733646
$816$	−4.00000	−0.140028
$817$	71.1935	2.49075
$818$	−0.111456	−0.00389697
$819$	−20.0000	−0.698857
$820$	6.47214	0.226017
$821$	48.4721	1.69169	0.845845	−	0.533429i	$-0.179097\pi$
$821$	48.4721	1.69169	0.845845	+	0.533429i	$0.179097\pi$
$822$	20.9443	0.730515
$823$	−7.63932	−0.266290	−0.133145	−	0.991097i	$-0.542508\pi$
$823$	−7.63932	−0.266290	−0.133145	+	0.991097i	$0.542508\pi$
$824$	−6.00000	−0.209020
$825$	−4.18034	−0.145541
$826$	40.0000	1.39178
$827$	27.5967	0.959633	0.479816	−	0.877369i	$-0.340703\pi$
$827$	27.5967	0.959633	0.479816	+	0.877369i	$0.340703\pi$
$828$	1.00000	0.0347524
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	−28.3607	−0.984414
$831$	−11.5279	−0.399897
$832$	−4.47214	−0.155043
$833$	−52.0000	−1.80169
$834$	0.944272	0.0326975
$835$	3.05573	0.105748
$836$	5.88854	0.203660
$837$	6.47214	0.223710
$838$	−30.0689	−1.03871
$839$	−10.1115	−0.349086	−0.174543	−	0.984650i	$-0.555845\pi$
$839$	−10.1115	−0.349086	−0.174543	+	0.984650i	$0.555845\pi$
$840$	−14.4721	−0.499336
$841$	−9.00000	−0.310345
$842$	−5.23607	−0.180447
$843$	−22.4721	−0.773981
$844$	3.41641	0.117598
$845$	−22.6525	−0.779269
$846$	4.00000	0.137523
$847$	−46.5836	−1.60063
$848$	0.763932	0.0262335
$849$	−26.1803	−0.898507
$850$	−21.8885	−0.750771
$851$	6.76393	0.231865
$852$	−8.94427	−0.306426
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	23.4164	0.801293
$855$	24.9443	0.853076
$856$	18.6525	0.637528
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	3.41641	0.116634
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	29.8885	1.01919
$861$	−8.94427	−0.304820
$862$	3.41641	0.116363
$863$	−38.8328	−1.32188	−0.660942	−	0.750437i	$-0.729843\pi$
$863$	−38.8328	−1.32188	−0.660942	+	0.750437i	$0.729843\pi$
$864$	1.00000	0.0340207
$865$	−30.4721	−1.03608
$866$	5.41641	0.184057
$867$	−1.00000	−0.0339618
$868$	28.9443	0.982433
$869$	3.41641	0.115894
$870$	−14.4721	−0.490651
$871$	16.5836	0.561914
$872$	9.23607	0.312773
$873$	−8.47214	−0.286738
$874$	−7.70820	−0.260734
$875$	−6.83282	−0.230991
$876$	4.47214	0.151099
$877$	32.2492	1.08898	0.544489	−	0.838768i	$-0.316723\pi$
$877$	32.2492	1.08898	0.544489	+	0.838768i	$0.316723\pi$
$878$	36.9443	1.24681
$879$	13.7082	0.462366
$880$	2.47214	0.0833357
$881$	12.5836	0.423952	0.211976	−	0.977275i	$-0.432010\pi$
$881$	12.5836	0.423952	0.211976	+	0.977275i	$0.432010\pi$
$882$	13.0000	0.437733
$883$	22.4721	0.756248	0.378124	−	0.925755i	$-0.376569\pi$
$883$	22.4721	0.756248	0.378124	+	0.925755i	$0.376569\pi$
$884$	17.8885	0.601657
$885$	−28.9443	−0.972951
$886$	−32.9443	−1.10678
$887$	0.944272	0.0317055	0.0158528	−	0.999874i	$-0.494954\pi$
$887$	0.944272	0.0317055	0.0158528	+	0.999874i	$0.494954\pi$
$888$	6.76393	0.226983
$889$	−17.8885	−0.599963
$890$	4.94427	0.165732
$891$	−0.763932	−0.0255927
$892$	−7.41641	−0.248320
$893$	−30.8328	−1.03178
$894$	1.70820	0.0571309
$895$	24.0000	0.802232
$896$	4.47214	0.149404
$897$	−4.47214	−0.149320
$898$	−19.8885	−0.663689
$899$	28.9443	0.965346
$900$	5.47214	0.182405
$901$	−3.05573	−0.101801
$902$	1.52786	0.0508723
$903$	−41.3050	−1.37454
$904$	−14.4721	−0.481336
$905$	21.8885	0.727600
$906$	−5.52786	−0.183651
$907$	−18.1803	−0.603668	−0.301834	−	0.953360i	$-0.597599\pi$
$907$	−18.1803	−0.603668	−0.301834	+	0.953360i	$0.597599\pi$
$908$	−4.76393	−0.158097
$909$	−4.47214	−0.148331
$910$	64.7214	2.14549
$911$	43.4164	1.43845	0.719225	−	0.694777i	$-0.244497\pi$
$911$	43.4164	1.43845	0.719225	+	0.694777i	$0.244497\pi$
$912$	−7.70820	−0.255244
$913$	−6.69505	−0.221574
$914$	36.4721	1.20639
$915$	−16.9443	−0.560160
$916$	4.29180	0.141805
$917$	−82.6099	−2.72802
$918$	−4.00000	−0.132020
$919$	−7.52786	−0.248321	−0.124161	−	0.992262i	$-0.539624\pi$
$919$	−7.52786	−0.248321	−0.124161	+	0.992262i	$0.539624\pi$
$920$	−3.23607	−0.106690
$921$	−11.4164	−0.376183
$922$	22.0000	0.724531
$923$	40.0000	1.31662
$924$	−3.41641	−0.112392
$925$	37.0132	1.21699
$926$	24.0000	0.788689
$927$	−6.00000	−0.197066
$928$	4.47214	0.146805
$929$	28.8328	0.945974	0.472987	−	0.881069i	$-0.343176\pi$
$929$	28.8328	0.945974	0.472987	+	0.881069i	$0.343176\pi$
$930$	−20.9443	−0.686790
$931$	−100.207	−3.28414
$932$	−15.8885	−0.520447
$933$	−3.05573	−0.100040
$934$	−40.1803	−1.31474
$935$	−9.88854	−0.323390
$936$	−4.47214	−0.146176
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	−16.5836	−0.541473
$939$	24.4721	0.798618
$940$	−12.9443	−0.422196
$941$	15.8197	0.515706	0.257853	−	0.966184i	$-0.416985\pi$
$941$	15.8197	0.515706	0.257853	+	0.966184i	$0.416985\pi$
$942$	24.6525	0.803221
$943$	−2.00000	−0.0651290
$944$	8.94427	0.291111
$945$	−14.4721	−0.470779
$946$	7.05573	0.229402
$947$	51.1935	1.66357	0.831783	−	0.555102i	$-0.187321\pi$
$947$	51.1935	1.66357	0.831783	+	0.555102i	$0.187321\pi$
$948$	−4.47214	−0.145248
$949$	−20.0000	−0.649227
$950$	−42.1803	−1.36851
$951$	28.4721	0.923272
$952$	−17.8885	−0.579771
$953$	23.7771	0.770215	0.385108	−	0.922872i	$-0.374164\pi$
$953$	23.7771	0.770215	0.385108	+	0.922872i	$0.374164\pi$
$954$	0.763932	0.0247332
$955$	8.00000	0.258874
$956$	12.9443	0.418648
$957$	−3.41641	−0.110437
$958$	−11.0557	−0.357194
$959$	93.6656	3.02462
$960$	−3.23607	−0.104444
$961$	10.8885	0.351243
$962$	−30.2492	−0.975274
$963$	18.6525	0.601068
$964$	−3.52786	−0.113625
$965$	38.4721	1.23846
$966$	4.47214	0.143889
$967$	−11.4164	−0.367127	−0.183563	−	0.983008i	$-0.558763\pi$
$967$	−11.4164	−0.367127	−0.183563	+	0.983008i	$0.558763\pi$
$968$	−10.4164	−0.334796
$969$	30.8328	0.990493
$970$	27.4164	0.880288
$971$	21.1246	0.677921	0.338961	−	0.940801i	$-0.389925\pi$
$971$	21.1246	0.677921	0.338961	+	0.940801i	$0.389925\pi$
$972$	1.00000	0.0320750
$973$	4.22291	0.135380
$974$	−25.8885	−0.829522
$975$	−24.4721	−0.783736
$976$	5.23607	0.167602
$977$	54.8328	1.75426	0.877129	−	0.480256i	$-0.159456\pi$
$977$	54.8328	1.75426	0.877129	+	0.480256i	$0.159456\pi$
$978$	6.47214	0.206956
$979$	1.16718	0.0373034
$980$	−42.0689	−1.34384
$981$	9.23607	0.294885
$982$	−33.3050	−1.06280
$983$	27.4164	0.874448	0.437224	−	0.899353i	$-0.355962\pi$
$983$	27.4164	0.874448	0.437224	+	0.899353i	$0.355962\pi$
$984$	−2.00000	−0.0637577
$985$	48.3607	1.54090
$986$	−17.8885	−0.569687
$987$	17.8885	0.569399
$988$	34.4721	1.09670
$989$	−9.23607	−0.293690
$990$	2.47214	0.0785696
$991$	−1.52786	−0.0485342	−0.0242671	−	0.999706i	$-0.507725\pi$
$991$	−1.52786	−0.0485342	−0.0242671	+	0.999706i	$0.507725\pi$
$992$	6.47214	0.205491
$993$	−1.52786	−0.0484853
$994$	−40.0000	−1.26872
$995$	30.4721	0.966032
$996$	8.76393	0.277696
$997$	22.9443	0.726652	0.363326	−	0.931662i	$-0.381641\pi$
$997$	22.9443	0.726652	0.363326	+	0.931662i	$0.381641\pi$
$998$	27.4164	0.867851
$999$	6.76393	0.214001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.2.a.d.1.1	✓	2
3.2	odd	2		414.2.a.f.1.2		2
4.3	odd	2		1104.2.a.j.1.1		2
5.2	odd	4		3450.2.d.x.2899.4		4
5.3	odd	4		3450.2.d.x.2899.1		4
5.4	even	2		3450.2.a.be.1.1		2
7.6	odd	2		6762.2.a.cb.1.2		2
8.3	odd	2		4416.2.a.bl.1.2		2
8.5	even	2		4416.2.a.bh.1.2		2
12.11	even	2		3312.2.a.bc.1.2		2
23.22	odd	2		3174.2.a.s.1.2		2
69.68	even	2		9522.2.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.a.d.1.1	✓	2	1.1	even	1	trivial
414.2.a.f.1.2		2	3.2	odd	2
1104.2.a.j.1.1		2	4.3	odd	2
3174.2.a.s.1.2		2	23.22	odd	2
3312.2.a.bc.1.2		2	12.11	even	2
3450.2.a.be.1.1		2	5.4	even	2
3450.2.d.x.2899.1		4	5.3	odd	4
3450.2.d.x.2899.4		4	5.2	odd	4
4416.2.a.bh.1.2		2	8.5	even	2
4416.2.a.bl.1.2		2	8.3	odd	2
6762.2.a.cb.1.2		2	7.6	odd	2
9522.2.a.q.1.1		2	69.68	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.23607 q^{5} +1.00000 q^{6} +4.47214 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.23607 q^{5} +1.00000 q^{6} +4.47214 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.23607 q^{10} -0.763932 q^{11} +1.00000 q^{12} -4.47214 q^{13} +4.47214 q^{14} -3.23607 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -7.70820 q^{19} -3.23607 q^{20} +4.47214 q^{21} -0.763932 q^{22} +1.00000 q^{23} +1.00000 q^{24} +5.47214 q^{25} -4.47214 q^{26} +1.00000 q^{27} +4.47214 q^{28} +4.47214 q^{29} -3.23607 q^{30} +6.47214 q^{31} +1.00000 q^{32} -0.763932 q^{33} -4.00000 q^{34} -14.4721 q^{35} +1.00000 q^{36} +6.76393 q^{37} -7.70820 q^{38} -4.47214 q^{39} -3.23607 q^{40} -2.00000 q^{41} +4.47214 q^{42} -9.23607 q^{43} -0.763932 q^{44} -3.23607 q^{45} +1.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} +13.0000 q^{49} +5.47214 q^{50} -4.00000 q^{51} -4.47214 q^{52} +0.763932 q^{53} +1.00000 q^{54} +2.47214 q^{55} +4.47214 q^{56} -7.70820 q^{57} +4.47214 q^{58} +8.94427 q^{59} -3.23607 q^{60} +5.23607 q^{61} +6.47214 q^{62} +4.47214 q^{63} +1.00000 q^{64} +14.4721 q^{65} -0.763932 q^{66} -3.70820 q^{67} -4.00000 q^{68} +1.00000 q^{69} -14.4721 q^{70} -8.94427 q^{71} +1.00000 q^{72} +4.47214 q^{73} +6.76393 q^{74} +5.47214 q^{75} -7.70820 q^{76} -3.41641 q^{77} -4.47214 q^{78} -4.47214 q^{79} -3.23607 q^{80} +1.00000 q^{81} -2.00000 q^{82} +8.76393 q^{83} +4.47214 q^{84} +12.9443 q^{85} -9.23607 q^{86} +4.47214 q^{87} -0.763932 q^{88} -1.52786 q^{89} -3.23607 q^{90} -20.0000 q^{91} +1.00000 q^{92} +6.47214 q^{93} +4.00000 q^{94} +24.9443 q^{95} +1.00000 q^{96} -8.47214 q^{97} +13.0000 q^{98} -0.763932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 6 q^{11} + 2 q^{12} - 2 q^{15} + 2 q^{16} - 8 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 6 q^{22} + 2 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{27} - 2 q^{30} + 4 q^{31} + 2 q^{32} - 6 q^{33} - 8 q^{34} - 20 q^{35} + 2 q^{36} + 18 q^{37} - 2 q^{38} - 2 q^{40} - 4 q^{41} - 14 q^{43} - 6 q^{44} - 2 q^{45} + 2 q^{46} + 8 q^{47} + 2 q^{48} + 26 q^{49} + 2 q^{50} - 8 q^{51} + 6 q^{53} + 2 q^{54} - 4 q^{55} - 2 q^{57} - 2 q^{60} + 6 q^{61} + 4 q^{62} + 2 q^{64} + 20 q^{65} - 6 q^{66} + 6 q^{67} - 8 q^{68} + 2 q^{69} - 20 q^{70} + 2 q^{72} + 18 q^{74} + 2 q^{75} - 2 q^{76} + 20 q^{77} - 2 q^{80} + 2 q^{81} - 4 q^{82} + 22 q^{83} + 8 q^{85} - 14 q^{86} - 6 q^{88} - 12 q^{89} - 2 q^{90} - 40 q^{91} + 2 q^{92} + 4 q^{93} + 8 q^{94} + 32 q^{95} + 2 q^{96} - 8 q^{97} + 26 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 6 * q^11 + 2 * q^12 - 2 * q^15 + 2 * q^16 - 8 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 6 * q^22 + 2 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^27 - 2 * q^30 + 4 * q^31 + 2 * q^32 - 6 * q^33 - 8 * q^34 - 20 * q^35 + 2 * q^36 + 18 * q^37 - 2 * q^38 - 2 * q^40 - 4 * q^41 - 14 * q^43 - 6 * q^44 - 2 * q^45 + 2 * q^46 + 8 * q^47 + 2 * q^48 + 26 * q^49 + 2 * q^50 - 8 * q^51 + 6 * q^53 + 2 * q^54 - 4 * q^55 - 2 * q^57 - 2 * q^60 + 6 * q^61 + 4 * q^62 + 2 * q^64 + 20 * q^65 - 6 * q^66 + 6 * q^67 - 8 * q^68 + 2 * q^69 - 20 * q^70 + 2 * q^72 + 18 * q^74 + 2 * q^75 - 2 * q^76 + 20 * q^77 - 2 * q^80 + 2 * q^81 - 4 * q^82 + 22 * q^83 + 8 * q^85 - 14 * q^86 - 6 * q^88 - 12 * q^89 - 2 * q^90 - 40 * q^91 + 2 * q^92 + 4 * q^93 + 8 * q^94 + 32 * q^95 + 2 * q^96 - 8 * q^97 + 26 * q^98 - 6 * q^99

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