LMFDB - Embedded newform 1682.2.a.u.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1682,2,Mod(1,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1682.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1682 = 2 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1682.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:	$13.4308376200$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.32836640625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 18x^{6} + 17x^{5} + 95x^{4} - 77x^{3} - 128x^{2} + 51x + 31 $ x^8 - x^7 - 18x^6 + 17x^5 + 95x^4 - 77x^3 - 128x^2 + 51x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.66631$ of defining polynomial
Character	$\chi$	$=$	1682.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.66631 q^{3} +1.00000 q^{4} +1.88957 q^{5} -2.66631 q^{6} -4.43319 q^{7} -1.00000 q^{8} +4.10924 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.66631 q^{3} +1.00000 q^{4} +1.88957 q^{5} -2.66631 q^{6} -4.43319 q^{7} -1.00000 q^{8} +4.10924 q^{9} -1.88957 q^{10} +3.10473 q^{11} +2.66631 q^{12} +4.26344 q^{13} +4.43319 q^{14} +5.03819 q^{15} +1.00000 q^{16} +0.827423 q^{17} -4.10924 q^{18} +3.05075 q^{19} +1.88957 q^{20} -11.8203 q^{21} -3.10473 q^{22} +6.90272 q^{23} -2.66631 q^{24} -1.42953 q^{25} -4.26344 q^{26} +2.95757 q^{27} -4.43319 q^{28} -5.03819 q^{30} -5.36667 q^{31} -1.00000 q^{32} +8.27819 q^{33} -0.827423 q^{34} -8.37681 q^{35} +4.10924 q^{36} +1.91948 q^{37} -3.05075 q^{38} +11.3677 q^{39} -1.88957 q^{40} -6.95185 q^{41} +11.8203 q^{42} -9.61591 q^{43} +3.10473 q^{44} +7.76469 q^{45} -6.90272 q^{46} +7.58301 q^{47} +2.66631 q^{48} +12.6531 q^{49} +1.42953 q^{50} +2.20617 q^{51} +4.26344 q^{52} +2.69817 q^{53} -2.95757 q^{54} +5.86660 q^{55} +4.43319 q^{56} +8.13425 q^{57} +4.06594 q^{59} +5.03819 q^{60} -2.70091 q^{61} +5.36667 q^{62} -18.2170 q^{63} +1.00000 q^{64} +8.05607 q^{65} -8.27819 q^{66} +4.14583 q^{67} +0.827423 q^{68} +18.4048 q^{69} +8.37681 q^{70} +11.7789 q^{71} -4.10924 q^{72} +2.76905 q^{73} -1.91948 q^{74} -3.81157 q^{75} +3.05075 q^{76} -13.7638 q^{77} -11.3677 q^{78} +10.6224 q^{79} +1.88957 q^{80} -4.44189 q^{81} +6.95185 q^{82} +6.98757 q^{83} -11.8203 q^{84} +1.56347 q^{85} +9.61591 q^{86} -3.10473 q^{88} +4.36203 q^{89} -7.76469 q^{90} -18.9006 q^{91} +6.90272 q^{92} -14.3092 q^{93} -7.58301 q^{94} +5.76460 q^{95} -2.66631 q^{96} -10.5299 q^{97} -12.6531 q^{98} +12.7581 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - q^{3} + 8 q^{4} + 5 q^{5} + q^{6} + 7 q^{7} - 8 q^{8} + 13 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - q^3 + 8 * q^4 + 5 * q^5 + q^6 + 7 * q^7 - 8 * q^8 + 13 * q^9	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} + 5 q^{5} + q^{6} + 7 q^{7} - 8 q^{8} + 13 q^{9} - 5 q^{10} + 7 q^{11} - q^{12} + 13 q^{13} - 7 q^{14} + 20 q^{15} + 8 q^{16} - 9 q^{17} - 13 q^{18} + 13 q^{19} + 5 q^{20} - 34 q^{21} - 7 q^{22} + 12 q^{23} + q^{24} + 45 q^{25} - 13 q^{26} + 2 q^{27} + 7 q^{28} - 20 q^{30} - 15 q^{31} - 8 q^{32} - 4 q^{33} + 9 q^{34} + 13 q^{36} - 4 q^{37} - 13 q^{38} + 9 q^{39} - 5 q^{40} - 8 q^{41} + 34 q^{42} - 12 q^{43} + 7 q^{44} + 30 q^{45} - 12 q^{46} + 13 q^{47} - q^{48} + 27 q^{49} - 45 q^{50} - 42 q^{51} + 13 q^{52} + 4 q^{53} - 2 q^{54} + 35 q^{55} - 7 q^{56} - 11 q^{57} + 8 q^{59} + 20 q^{60} - 9 q^{61} + 15 q^{62} + 12 q^{63} + 8 q^{64} - 15 q^{65} + 4 q^{66} + 34 q^{67} - 9 q^{68} - 4 q^{69} - 11 q^{71} - 13 q^{72} + 13 q^{73} + 4 q^{74} + 15 q^{75} + 13 q^{76} + 23 q^{77} - 9 q^{78} + 8 q^{79} + 5 q^{80} + 12 q^{81} + 8 q^{82} + 10 q^{83} - 34 q^{84} + 15 q^{85} + 12 q^{86} - 7 q^{88} - 6 q^{89} - 30 q^{90} + 12 q^{91} + 12 q^{92} + 15 q^{93} - 13 q^{94} + 15 q^{95} + q^{96} + 11 q^{97} - 27 q^{98} + 2 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 + 5 * q^5 + q^6 + 7 * q^7 - 8 * q^8 + 13 * q^9 - 5 * q^10 + 7 * q^11 - q^12 + 13 * q^13 - 7 * q^14 + 20 * q^15 + 8 * q^16 - 9 * q^17 - 13 * q^18 + 13 * q^19 + 5 * q^20 - 34 * q^21 - 7 * q^22 + 12 * q^23 + q^24 + 45 * q^25 - 13 * q^26 + 2 * q^27 + 7 * q^28 - 20 * q^30 - 15 * q^31 - 8 * q^32 - 4 * q^33 + 9 * q^34 + 13 * q^36 - 4 * q^37 - 13 * q^38 + 9 * q^39 - 5 * q^40 - 8 * q^41 + 34 * q^42 - 12 * q^43 + 7 * q^44 + 30 * q^45 - 12 * q^46 + 13 * q^47 - q^48 + 27 * q^49 - 45 * q^50 - 42 * q^51 + 13 * q^52 + 4 * q^53 - 2 * q^54 + 35 * q^55 - 7 * q^56 - 11 * q^57 + 8 * q^59 + 20 * q^60 - 9 * q^61 + 15 * q^62 + 12 * q^63 + 8 * q^64 - 15 * q^65 + 4 * q^66 + 34 * q^67 - 9 * q^68 - 4 * q^69 - 11 * q^71 - 13 * q^72 + 13 * q^73 + 4 * q^74 + 15 * q^75 + 13 * q^76 + 23 * q^77 - 9 * q^78 + 8 * q^79 + 5 * q^80 + 12 * q^81 + 8 * q^82 + 10 * q^83 - 34 * q^84 + 15 * q^85 + 12 * q^86 - 7 * q^88 - 6 * q^89 - 30 * q^90 + 12 * q^91 + 12 * q^92 + 15 * q^93 - 13 * q^94 + 15 * q^95 + q^96 + 11 * q^97 - 27 * q^98 + 2 * 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
$3$	2.66631	1.53940	0.769699	−	0.638407i	$-0.220406\pi$
$3$	2.66631	1.53940	0.769699	+	0.638407i	$0.220406\pi$
$4$	1.00000	0.500000
$5$	1.88957	0.845041	0.422521	−	0.906353i	$-0.361145\pi$
$5$	1.88957	0.845041	0.422521	+	0.906353i	$0.361145\pi$
$6$	−2.66631	−1.08852
$7$	−4.43319	−1.67559	−0.837793	−	0.545988i	$-0.816155\pi$
$7$	−4.43319	−1.67559	−0.837793	+	0.545988i	$0.816155\pi$
$8$	−1.00000	−0.353553
$9$	4.10924	1.36975
$10$	−1.88957	−0.597534
$11$	3.10473	0.936111	0.468056	−	0.883699i	$-0.344955\pi$
$11$	3.10473	0.936111	0.468056	+	0.883699i	$0.344955\pi$
$12$	2.66631	0.769699
$13$	4.26344	1.18247	0.591233	−	0.806501i	$-0.298641\pi$
$13$	4.26344	1.18247	0.591233	+	0.806501i	$0.298641\pi$
$14$	4.43319	1.18482
$15$	5.03819	1.30085
$16$	1.00000	0.250000
$17$	0.827423	0.200680	0.100340	−	0.994953i	$-0.468007\pi$
$17$	0.827423	0.200680	0.100340	+	0.994953i	$0.468007\pi$
$18$	−4.10924	−0.968556
$19$	3.05075	0.699889	0.349945	−	0.936770i	$-0.386200\pi$
$19$	3.05075	0.699889	0.349945	+	0.936770i	$0.386200\pi$
$20$	1.88957	0.422521
$21$	−11.8203	−2.57939
$22$	−3.10473	−0.661931
$23$	6.90272	1.43932	0.719658	−	0.694329i	$-0.244299\pi$
$23$	6.90272	1.43932	0.719658	+	0.694329i	$0.244299\pi$
$24$	−2.66631	−0.544259
$25$	−1.42953	−0.285905
$26$	−4.26344	−0.836130
$27$	2.95757	0.569185
$28$	−4.43319	−0.837793
$29$	0	0
$29$	0	0
$30$	−5.03819	−0.919843
$31$	−5.36667	−0.963882	−0.481941	−	0.876204i	$-0.660068\pi$
$31$	−5.36667	−0.963882	−0.481941	+	0.876204i	$0.660068\pi$
$32$	−1.00000	−0.176777
$33$	8.27819	1.44105
$34$	−0.827423	−0.141902
$35$	−8.37681	−1.41594
$36$	4.10924	0.684873
$37$	1.91948	0.315560	0.157780	−	0.987474i	$-0.449566\pi$
$37$	1.91948	0.315560	0.157780	+	0.987474i	$0.449566\pi$
$38$	−3.05075	−0.494897
$39$	11.3677	1.82029
$40$	−1.88957	−0.298767
$41$	−6.95185	−1.08570	−0.542849	−	0.839831i	$-0.682654\pi$
$41$	−6.95185	−1.08570	−0.542849	+	0.839831i	$0.682654\pi$
$42$	11.8203	1.82391
$43$	−9.61591	−1.46641	−0.733206	−	0.680006i	$-0.761977\pi$
$43$	−9.61591	−1.46641	−0.733206	+	0.680006i	$0.761977\pi$
$44$	3.10473	0.468056
$45$	7.76469	1.15749
$46$	−6.90272	−1.01775
$47$	7.58301	1.10610	0.553048	−	0.833149i	$-0.313465\pi$
$47$	7.58301	1.10610	0.553048	+	0.833149i	$0.313465\pi$
$48$	2.66631	0.384849
$49$	12.6531	1.80759
$50$	1.42953	0.202166
$51$	2.20617	0.308926
$52$	4.26344	0.591233
$53$	2.69817	0.370622	0.185311	−	0.982680i	$-0.440671\pi$
$53$	2.69817	0.370622	0.185311	+	0.982680i	$0.440671\pi$
$54$	−2.95757	−0.402474
$55$	5.86660	0.791053
$56$	4.43319	0.592409
$57$	8.13425	1.07741
$58$	0	0
$59$	4.06594	0.529340	0.264670	−	0.964339i	$-0.414737\pi$
$59$	4.06594	0.529340	0.264670	+	0.964339i	$0.414737\pi$
$60$	5.03819	0.650427
$61$	−2.70091	−0.345816	−0.172908	−	0.984938i	$-0.555316\pi$
$61$	−2.70091	−0.345816	−0.172908	+	0.984938i	$0.555316\pi$
$62$	5.36667	0.681567
$63$	−18.2170	−2.29513
$64$	1.00000	0.125000
$65$	8.05607	0.999232
$66$	−8.27819	−1.01897
$67$	4.14583	0.506494	0.253247	−	0.967402i	$-0.418501\pi$
$67$	4.14583	0.506494	0.253247	+	0.967402i	$0.418501\pi$
$68$	0.827423	0.100340
$69$	18.4048	2.21568
$70$	8.37681	1.00122
$71$	11.7789	1.39790	0.698949	−	0.715171i	$-0.253651\pi$
$71$	11.7789	1.39790	0.698949	+	0.715171i	$0.253651\pi$
$72$	−4.10924	−0.484278
$73$	2.76905	0.324093	0.162046	−	0.986783i	$-0.448191\pi$
$73$	2.76905	0.324093	0.162046	+	0.986783i	$0.448191\pi$
$74$	−1.91948	−0.223135
$75$	−3.81157	−0.440122
$76$	3.05075	0.349945
$77$	−13.7638	−1.56854
$78$	−11.3677	−1.28714
$79$	10.6224	1.19511	0.597554	−	0.801828i	$-0.296139\pi$
$79$	10.6224	1.19511	0.597554	+	0.801828i	$0.296139\pi$
$80$	1.88957	0.211260
$81$	−4.44189	−0.493543
$82$	6.95185	0.767704
$83$	6.98757	0.766985	0.383493	−	0.923544i	$-0.374721\pi$
$83$	6.98757	0.766985	0.383493	+	0.923544i	$0.374721\pi$
$84$	−11.8203	−1.28970
$85$	1.56347	0.169582
$86$	9.61591	1.03691
$87$	0	0
$88$	−3.10473	−0.330965
$89$	4.36203	0.462374	0.231187	−	0.972909i	$-0.425739\pi$
$89$	4.36203	0.462374	0.231187	+	0.972909i	$0.425739\pi$
$90$	−7.76469	−0.818470
$91$	−18.9006	−1.98132
$92$	6.90272	0.719658
$93$	−14.3092	−1.48380
$94$	−7.58301	−0.782128
$95$	5.76460	0.591435
$96$	−2.66631	−0.272130
$97$	−10.5299	−1.06914	−0.534572	−	0.845123i	$-0.679527\pi$
$97$	−10.5299	−1.06914	−0.534572	+	0.845123i	$0.679527\pi$
$98$	−12.6531	−1.27816
$99$	12.7581	1.28223
$100$	−1.42953	−0.142953
$101$	−11.2561	−1.12003	−0.560013	−	0.828484i	$-0.689204\pi$
$101$	−11.2561	−1.12003	−0.560013	+	0.828484i	$0.689204\pi$
$102$	−2.20617	−0.218443
$103$	19.0154	1.87365	0.936823	−	0.349804i	$-0.113752\pi$
$103$	19.0154	1.87365	0.936823	+	0.349804i	$0.113752\pi$
$104$	−4.26344	−0.418065
$105$	−22.3352	−2.17969
$106$	−2.69817	−0.262069
$107$	2.37176	0.229287	0.114643	−	0.993407i	$-0.463427\pi$
$107$	2.37176	0.229287	0.114643	+	0.993407i	$0.463427\pi$
$108$	2.95757	0.284592
$109$	5.72813	0.548655	0.274328	−	0.961636i	$-0.411545\pi$
$109$	5.72813	0.548655	0.274328	+	0.961636i	$0.411545\pi$
$110$	−5.86660	−0.559359
$111$	5.11793	0.485773
$112$	−4.43319	−0.418897
$113$	−12.4938	−1.17532	−0.587658	−	0.809109i	$-0.699950\pi$
$113$	−12.4938	−1.17532	−0.587658	+	0.809109i	$0.699950\pi$
$114$	−8.13425	−0.761843
$115$	13.0432	1.21628
$116$	0	0
$117$	17.5195	1.61968
$118$	−4.06594	−0.374300
$119$	−3.66812	−0.336256
$120$	−5.03819	−0.459921
$121$	−1.36065	−0.123696
$122$	2.70091	0.244529
$123$	−18.5358	−1.67132
$124$	−5.36667	−0.481941
$125$	−12.1490	−1.08664
$126$	18.2170	1.62290
$127$	−10.9754	−0.973909	−0.486955	−	0.873427i	$-0.661892\pi$
$127$	−10.9754	−0.973909	−0.486955	+	0.873427i	$0.661892\pi$
$128$	−1.00000	−0.0883883
$129$	−25.6390	−2.25739
$130$	−8.05607	−0.706564
$131$	3.35552	0.293173	0.146587	−	0.989198i	$-0.453171\pi$
$131$	3.35552	0.293173	0.146587	+	0.989198i	$0.453171\pi$
$132$	8.27819	0.720524
$133$	−13.5245	−1.17273
$134$	−4.14583	−0.358145
$135$	5.58854	0.480985
$136$	−0.827423	−0.0709509
$137$	−13.5265	−1.15564	−0.577822	−	0.816162i	$-0.696097\pi$
$137$	−13.5265	−1.15564	−0.577822	+	0.816162i	$0.696097\pi$
$138$	−18.4048	−1.56672
$139$	−19.8393	−1.68275	−0.841373	−	0.540454i	$-0.818252\pi$
$139$	−19.8393	−1.68275	−0.841373	+	0.540454i	$0.818252\pi$
$140$	−8.37681	−0.707970
$141$	20.2187	1.70272
$142$	−11.7789	−0.988464
$143$	13.2368	1.10692
$144$	4.10924	0.342436
$145$	0	0
$146$	−2.76905	−0.229168
$147$	33.7372	2.78260
$148$	1.91948	0.157780
$149$	6.25711	0.512602	0.256301	−	0.966597i	$-0.417496\pi$
$149$	6.25711	0.512602	0.256301	+	0.966597i	$0.417496\pi$
$150$	3.81157	0.311213
$151$	7.74673	0.630420	0.315210	−	0.949022i	$-0.397925\pi$
$151$	7.74673	0.630420	0.315210	+	0.949022i	$0.397925\pi$
$152$	−3.05075	−0.247448
$153$	3.40008	0.274880
$154$	13.7638	1.10912
$155$	−10.1407	−0.814520
$156$	11.3677	0.910143
$157$	5.26142	0.419907	0.209954	−	0.977711i	$-0.432669\pi$
$157$	5.26142	0.419907	0.209954	+	0.977711i	$0.432669\pi$
$158$	−10.6224	−0.845069
$159$	7.19417	0.570535
$160$	−1.88957	−0.149384
$161$	−30.6010	−2.41170
$162$	4.44189	0.348988
$163$	12.2657	0.960720	0.480360	−	0.877071i	$-0.340506\pi$
$163$	12.2657	0.960720	0.480360	+	0.877071i	$0.340506\pi$
$164$	−6.95185	−0.542849
$165$	15.6422	1.21774
$166$	−6.98757	−0.542341
$167$	11.9027	0.921057	0.460529	−	0.887645i	$-0.347660\pi$
$167$	11.9027	0.921057	0.460529	+	0.887645i	$0.347660\pi$
$168$	11.8203	0.911953
$169$	5.17693	0.398225
$170$	−1.56347	−0.119913
$171$	12.5362	0.958670
$172$	−9.61591	−0.733206
$173$	−5.46319	−0.415359	−0.207679	−	0.978197i	$-0.566591\pi$
$173$	−5.46319	−0.415359	−0.207679	+	0.978197i	$0.566591\pi$
$174$	0	0
$175$	6.33736	0.479059
$176$	3.10473	0.234028
$177$	10.8411	0.814865
$178$	−4.36203	−0.326948
$179$	−15.1536	−1.13263	−0.566317	−	0.824187i	$-0.691632\pi$
$179$	−15.1536	−1.13263	−0.566317	+	0.824187i	$0.691632\pi$
$180$	7.76469	0.578746
$181$	7.37832	0.548426	0.274213	−	0.961669i	$-0.411583\pi$
$181$	7.37832	0.548426	0.274213	+	0.961669i	$0.411583\pi$
$182$	18.9006	1.40101
$183$	−7.20147	−0.532348
$184$	−6.90272	−0.508875
$185$	3.62699	0.266661
$186$	14.3092	1.04920
$187$	2.56892	0.187858
$188$	7.58301	0.553048
$189$	−13.1115	−0.953718
$190$	−5.76460	−0.418208
$191$	0.600815	0.0434734	0.0217367	−	0.999764i	$-0.493080\pi$
$191$	0.600815	0.0434734	0.0217367	+	0.999764i	$0.493080\pi$
$192$	2.66631	0.192425
$193$	10.8017	0.777526	0.388763	−	0.921338i	$-0.372902\pi$
$193$	10.8017	0.777526	0.388763	+	0.921338i	$0.372902\pi$
$194$	10.5299	0.756000
$195$	21.4800	1.53822
$196$	12.6531	0.903795
$197$	−6.04928	−0.430994	−0.215497	−	0.976505i	$-0.569137\pi$
$197$	−6.04928	−0.430994	−0.215497	+	0.976505i	$0.569137\pi$
$198$	−12.7581	−0.906676
$199$	−22.4713	−1.59295	−0.796475	−	0.604672i	$-0.793304\pi$
$199$	−22.4713	−1.59295	−0.796475	+	0.604672i	$0.793304\pi$
$200$	1.42953	0.101083
$201$	11.0541	0.779696
$202$	11.2561	0.791978
$203$	0	0
$204$	2.20617	0.154463
$205$	−13.1360	−0.917459
$206$	−19.0154	−1.32487
$207$	28.3649	1.97150
$208$	4.26344	0.295616
$209$	9.47175	0.655174
$210$	22.3352	1.54128
$211$	−1.77802	−0.122404	−0.0612020	−	0.998125i	$-0.519493\pi$
$211$	−1.77802	−0.122404	−0.0612020	+	0.998125i	$0.519493\pi$
$212$	2.69817	0.185311
$213$	31.4063	2.15192
$214$	−2.37176	−0.162130
$215$	−18.1699	−1.23918
$216$	−2.95757	−0.201237
$217$	23.7914	1.61507
$218$	−5.72813	−0.387958
$219$	7.38316	0.498907
$220$	5.86660	0.395526
$221$	3.52767	0.237297
$222$	−5.11793	−0.343493
$223$	7.10522	0.475801	0.237900	−	0.971290i	$-0.423541\pi$
$223$	7.10522	0.475801	0.237900	+	0.971290i	$0.423541\pi$
$224$	4.43319	0.296205
$225$	−5.87426	−0.391618
$226$	12.4938	0.831074
$227$	2.49774	0.165781	0.0828903	−	0.996559i	$-0.473585\pi$
$227$	2.49774	0.165781	0.0828903	+	0.996559i	$0.473585\pi$
$228$	8.13425	0.538704
$229$	−10.1469	−0.670525	−0.335262	−	0.942125i	$-0.608825\pi$
$229$	−10.1469	−0.670525	−0.335262	+	0.942125i	$0.608825\pi$
$230$	−13.0432	−0.860041
$231$	−36.6987	−2.41460
$232$	0	0
$233$	−18.5754	−1.21691	−0.608457	−	0.793587i	$-0.708211\pi$
$233$	−18.5754	−1.21691	−0.608457	+	0.793587i	$0.708211\pi$
$234$	−17.5195	−1.14528
$235$	14.3286	0.934697
$236$	4.06594	0.264670
$237$	28.3225	1.83975
$238$	3.66812	0.237769
$239$	−13.9983	−0.905472	−0.452736	−	0.891645i	$-0.649552\pi$
$239$	−13.9983	−0.905472	−0.452736	+	0.891645i	$0.649552\pi$
$240$	5.03819	0.325214
$241$	−13.2572	−0.853971	−0.426986	−	0.904258i	$-0.640425\pi$
$241$	−13.2572	−0.853971	−0.426986	+	0.904258i	$0.640425\pi$
$242$	1.36065	0.0874661
$243$	−20.7162	−1.32894
$244$	−2.70091	−0.172908
$245$	23.9090	1.52749
$246$	18.5358	1.18180
$247$	13.0067	0.827595
$248$	5.36667	0.340784
$249$	18.6311	1.18070
$250$	12.1490	0.768373
$251$	−16.6261	−1.04943	−0.524714	−	0.851278i	$-0.675828\pi$
$251$	−16.6261	−1.04943	−0.524714	+	0.851278i	$0.675828\pi$
$252$	−18.2170	−1.14756
$253$	21.4311	1.34736
$254$	10.9754	0.688658
$255$	4.16871	0.261055
$256$	1.00000	0.0625000
$257$	3.28676	0.205023	0.102511	−	0.994732i	$-0.467312\pi$
$257$	3.28676	0.205023	0.102511	+	0.994732i	$0.467312\pi$
$258$	25.6390	1.59622
$259$	−8.50940	−0.528749
$260$	8.05607	0.499616
$261$	0	0
$262$	−3.35552	−0.207305
$263$	−25.2066	−1.55431	−0.777153	−	0.629312i	$-0.783337\pi$
$263$	−25.2066	−1.55431	−0.777153	+	0.629312i	$0.783337\pi$
$264$	−8.27819	−0.509487
$265$	5.09838	0.313191
$266$	13.5245	0.829242
$267$	11.6305	0.711777
$268$	4.14583	0.253247
$269$	−15.3554	−0.936237	−0.468119	−	0.883666i	$-0.655068\pi$
$269$	−15.3554	−0.936237	−0.468119	+	0.883666i	$0.655068\pi$
$270$	−5.58854	−0.340107
$271$	−15.1856	−0.922463	−0.461231	−	0.887280i	$-0.652592\pi$
$271$	−15.1856	−0.922463	−0.461231	+	0.887280i	$0.652592\pi$
$272$	0.827423	0.0501699
$273$	−50.3950	−3.05005
$274$	13.5265	0.817164
$275$	−4.43830	−0.267639
$276$	18.4048	1.10784
$277$	−28.6613	−1.72209	−0.861044	−	0.508531i	$-0.830189\pi$
$277$	−28.6613	−1.72209	−0.861044	+	0.508531i	$0.830189\pi$
$278$	19.8393	1.18988
$279$	−22.0529	−1.32027
$280$	8.37681	0.500610
$281$	−32.3169	−1.92787	−0.963933	−	0.266146i	$-0.914250\pi$
$281$	−32.3169	−1.92787	−0.963933	+	0.266146i	$0.914250\pi$
$282$	−20.2187	−1.20401
$283$	11.2514	0.668828	0.334414	−	0.942426i	$-0.391462\pi$
$283$	11.2514	0.668828	0.334414	+	0.942426i	$0.391462\pi$
$284$	11.7789	0.698949
$285$	15.3702	0.910454
$286$	−13.2368	−0.782710
$287$	30.8189	1.81918
$288$	−4.10924	−0.242139
$289$	−16.3154	−0.959728
$290$	0	0
$291$	−28.0759	−1.64584
$292$	2.76905	0.162046
$293$	−18.6148	−1.08749	−0.543744	−	0.839251i	$-0.682994\pi$
$293$	−18.6148	−1.08749	−0.543744	+	0.839251i	$0.682994\pi$
$294$	−33.7372	−1.96760
$295$	7.68287	0.447314
$296$	−1.91948	−0.111567
$297$	9.18246	0.532820
$298$	−6.25711	−0.362465
$299$	29.4293	1.70194
$300$	−3.81157	−0.220061
$301$	42.6291	2.45710
$302$	−7.74673	−0.445774
$303$	−30.0124	−1.72416
$304$	3.05075	0.174972
$305$	−5.10355	−0.292229
$306$	−3.40008	−0.194369
$307$	−2.45342	−0.140024	−0.0700119	−	0.997546i	$-0.522304\pi$
$307$	−2.45342	−0.140024	−0.0700119	+	0.997546i	$0.522304\pi$
$308$	−13.7638	−0.784268
$309$	50.7011	2.88429
$310$	10.1407	0.575953
$311$	−2.07536	−0.117683	−0.0588414	−	0.998267i	$-0.518741\pi$
$311$	−2.07536	−0.117683	−0.0588414	+	0.998267i	$0.518741\pi$
$312$	−11.3677	−0.643568
$313$	−12.3137	−0.696010	−0.348005	−	0.937493i	$-0.613141\pi$
$313$	−12.3137	−0.696010	−0.348005	+	0.937493i	$0.613141\pi$
$314$	−5.26142	−0.296919
$315$	−34.4223	−1.93948
$316$	10.6224	0.597554
$317$	16.1586	0.907558	0.453779	−	0.891114i	$-0.350075\pi$
$317$	16.1586	0.907558	0.453779	+	0.891114i	$0.350075\pi$
$318$	−7.19417	−0.403429
$319$	0	0
$320$	1.88957	0.105630
$321$	6.32386	0.352963
$322$	30.6010	1.70533
$323$	2.52426	0.140453
$324$	−4.44189	−0.246772
$325$	−6.09470	−0.338073
$326$	−12.2657	−0.679332
$327$	15.2730	0.844599
$328$	6.95185	0.383852
$329$	−33.6169	−1.85336
$330$	−15.6422	−0.861075
$331$	17.8639	0.981888	0.490944	−	0.871191i	$-0.336652\pi$
$331$	17.8639	0.981888	0.490944	+	0.871191i	$0.336652\pi$
$332$	6.98757	0.383493
$333$	7.88759	0.432237
$334$	−11.9027	−0.651286
$335$	7.83384	0.428008
$336$	−11.8203	−0.644848
$337$	22.2689	1.21307	0.606533	−	0.795058i	$-0.292560\pi$
$337$	22.2689	1.21307	0.606533	+	0.795058i	$0.292560\pi$
$338$	−5.17693	−0.281588
$339$	−33.3124	−1.80928
$340$	1.56347	0.0847912
$341$	−16.6621	−0.902301
$342$	−12.5362	−0.677882
$343$	−25.0614	−1.35319
$344$	9.61591	0.518455
$345$	34.7772	1.87234
$346$	5.46319	0.293703
$347$	−23.5162	−1.26242	−0.631208	−	0.775613i	$-0.717441\pi$
$347$	−23.5162	−1.26242	−0.631208	+	0.775613i	$0.717441\pi$
$348$	0	0
$349$	13.8857	0.743284	0.371642	−	0.928376i	$-0.378795\pi$
$349$	13.8857	0.743284	0.371642	+	0.928376i	$0.378795\pi$
$350$	−6.33736	−0.338746
$351$	12.6094	0.673042
$352$	−3.10473	−0.165483
$353$	26.4253	1.40648	0.703240	−	0.710953i	$-0.251736\pi$
$353$	26.4253	1.40648	0.703240	+	0.710953i	$0.251736\pi$
$354$	−10.8411	−0.576196
$355$	22.2571	1.18128
$356$	4.36203	0.231187
$357$	−9.78036	−0.517632
$358$	15.1536	0.800894
$359$	3.48063	0.183701	0.0918503	−	0.995773i	$-0.470722\pi$
$359$	3.48063	0.183701	0.0918503	+	0.995773i	$0.470722\pi$
$360$	−7.76469	−0.409235
$361$	−9.69294	−0.510155
$362$	−7.37832	−0.387796
$363$	−3.62793	−0.190417
$364$	−18.9006	−0.990662
$365$	5.23231	0.273872
$366$	7.20147	0.376427
$367$	5.89712	0.307827	0.153914	−	0.988084i	$-0.450812\pi$
$367$	5.89712	0.307827	0.153914	+	0.988084i	$0.450812\pi$
$368$	6.90272	0.359829
$369$	−28.5668	−1.48713
$370$	−3.62699	−0.188558
$371$	−11.9615	−0.621009
$372$	−14.3092	−0.741899
$373$	28.0886	1.45437	0.727187	−	0.686439i	$-0.240827\pi$
$373$	28.0886	1.45437	0.727187	+	0.686439i	$0.240827\pi$
$374$	−2.56892	−0.132836
$375$	−32.3932	−1.67278
$376$	−7.58301	−0.391064
$377$	0	0
$378$	13.1115	0.674381
$379$	−32.8894	−1.68942	−0.844708	−	0.535227i	$-0.820226\pi$
$379$	−32.8894	−1.68942	−0.844708	+	0.535227i	$0.820226\pi$
$380$	5.76460	0.295718
$381$	−29.2639	−1.49923
$382$	−0.600815	−0.0307404
$383$	−18.9516	−0.968382	−0.484191	−	0.874962i	$-0.660886\pi$
$383$	−18.9516	−0.968382	−0.484191	+	0.874962i	$0.660886\pi$
$384$	−2.66631	−0.136065
$385$	−26.0077	−1.32548
$386$	−10.8017	−0.549794
$387$	−39.5140	−2.00861
$388$	−10.5299	−0.534572
$389$	25.7959	1.30790	0.653951	−	0.756537i	$-0.273110\pi$
$389$	25.7959	1.30790	0.653951	+	0.756537i	$0.273110\pi$
$390$	−21.4800	−1.08768
$391$	5.71147	0.288841
$392$	−12.6531	−0.639080
$393$	8.94688	0.451310
$394$	6.04928	0.304759
$395$	20.0717	1.00992
$396$	12.7581	0.641117
$397$	24.2601	1.21758	0.608790	−	0.793331i	$-0.291655\pi$
$397$	24.2601	1.21758	0.608790	+	0.793331i	$0.291655\pi$
$398$	22.4713	1.12639
$399$	−36.0606	−1.80529
$400$	−1.42953	−0.0714764
$401$	−19.8886	−0.993188	−0.496594	−	0.867983i	$-0.665416\pi$
$401$	−19.8886	−0.993188	−0.496594	+	0.867983i	$0.665416\pi$
$402$	−11.0541	−0.551328
$403$	−22.8805	−1.13976
$404$	−11.2561	−0.560013
$405$	−8.39326	−0.417064
$406$	0	0
$407$	5.95946	0.295400
$408$	−2.20617	−0.109222
$409$	33.3680	1.64994	0.824971	−	0.565176i	$-0.191192\pi$
$409$	33.3680	1.64994	0.824971	+	0.565176i	$0.191192\pi$
$410$	13.1360	0.648741
$411$	−36.0658	−1.77900
$412$	19.0154	0.936823
$413$	−18.0251	−0.886955
$414$	−28.3649	−1.39406
$415$	13.2035	0.648134
$416$	−4.26344	−0.209032
$417$	−52.8978	−2.59042
$418$	−9.47175	−0.463278
$419$	−18.1156	−0.885004	−0.442502	−	0.896767i	$-0.645909\pi$
$419$	−18.1156	−0.885004	−0.442502	+	0.896767i	$0.645909\pi$
$420$	−22.3352	−1.08985
$421$	−13.8875	−0.676834	−0.338417	−	0.940996i	$-0.609892\pi$
$421$	−13.8875	−0.676834	−0.338417	+	0.940996i	$0.609892\pi$
$422$	1.77802	0.0865527
$423$	31.1604	1.51507
$424$	−2.69817	−0.131035
$425$	−1.18282	−0.0573754
$426$	−31.4063	−1.52164
$427$	11.9736	0.579444
$428$	2.37176	0.114643
$429$	35.2936	1.70399
$430$	18.1699	0.876232
$431$	−17.4399	−0.840048	−0.420024	−	0.907513i	$-0.637978\pi$
$431$	−17.4399	−0.840048	−0.420024	+	0.907513i	$0.637978\pi$
$432$	2.95757	0.142296
$433$	−27.5286	−1.32294	−0.661471	−	0.749971i	$-0.730067\pi$
$433$	−27.5286	−1.32294	−0.661471	+	0.749971i	$0.730067\pi$
$434$	−23.7914	−1.14203
$435$	0	0
$436$	5.72813	0.274328
$437$	21.0584	1.00736
$438$	−7.38316	−0.352781
$439$	−2.15260	−0.102738	−0.0513690	−	0.998680i	$-0.516358\pi$
$439$	−2.15260	−0.102738	−0.0513690	+	0.998680i	$0.516358\pi$
$440$	−5.86660	−0.279679
$441$	51.9947	2.47594
$442$	−3.52767	−0.167794
$443$	25.8500	1.22817	0.614086	−	0.789239i	$-0.289525\pi$
$443$	25.8500	1.22817	0.614086	+	0.789239i	$0.289525\pi$
$444$	5.11793	0.242886
$445$	8.24235	0.390725
$446$	−7.10522	−0.336442
$447$	16.6834	0.789099
$448$	−4.43319	−0.209448
$449$	−18.7693	−0.885777	−0.442888	−	0.896577i	$-0.646046\pi$
$449$	−18.7693	−0.885777	−0.442888	+	0.896577i	$0.646046\pi$
$450$	5.87426	0.276915
$451$	−21.5836	−1.01633
$452$	−12.4938	−0.587658
$453$	20.6552	0.970467
$454$	−2.49774	−0.117225
$455$	−35.7140	−1.67430
$456$	−8.13425	−0.380921
$457$	−13.9278	−0.651514	−0.325757	−	0.945453i	$-0.605619\pi$
$457$	−13.9278	−0.651514	−0.325757	+	0.945453i	$0.605619\pi$
$458$	10.1469	0.474133
$459$	2.44716	0.114224
$460$	13.0432	0.608140
$461$	−15.5360	−0.723583	−0.361792	−	0.932259i	$-0.617835\pi$
$461$	−15.5360	−0.723583	−0.361792	+	0.932259i	$0.617835\pi$
$462$	36.6987	1.70738
$463$	23.6809	1.10054	0.550272	−	0.834986i	$-0.314524\pi$
$463$	23.6809	1.10054	0.550272	+	0.834986i	$0.314524\pi$
$464$	0	0
$465$	−27.0383	−1.25387
$466$	18.5754	0.860488
$467$	−30.5564	−1.41398	−0.706990	−	0.707224i	$-0.749947\pi$
$467$	−30.5564	−1.41398	−0.706990	+	0.707224i	$0.749947\pi$
$468$	17.5195	0.809838
$469$	−18.3792	−0.848675
$470$	−14.3286	−0.660931
$471$	14.0286	0.646404
$472$	−4.06594	−0.187150
$473$	−29.8548	−1.37273
$474$	−28.3225	−1.30090
$475$	−4.36113	−0.200102
$476$	−3.66812	−0.168128
$477$	11.0874	0.507658
$478$	13.9983	0.640265
$479$	−13.8099	−0.630991	−0.315496	−	0.948927i	$-0.602171\pi$
$479$	−13.8099	−0.630991	−0.315496	+	0.948927i	$0.602171\pi$
$480$	−5.03819	−0.229961
$481$	8.18358	0.373139
$482$	13.2572	0.603849
$483$	−81.5919	−3.71256
$484$	−1.36065	−0.0618479
$485$	−19.8969	−0.903471
$486$	20.7162	0.939706
$487$	10.0549	0.455633	0.227817	−	0.973704i	$-0.426841\pi$
$487$	10.0549	0.455633	0.227817	+	0.973704i	$0.426841\pi$
$488$	2.70091	0.122264
$489$	32.7041	1.47893
$490$	−23.9090	−1.08010
$491$	33.0650	1.49220	0.746102	−	0.665832i	$-0.231923\pi$
$491$	33.0650	1.49220	0.746102	+	0.665832i	$0.231923\pi$
$492$	−18.5358	−0.835660
$493$	0	0
$494$	−13.0067	−0.585198
$495$	24.1073	1.08354
$496$	−5.36667	−0.240970
$497$	−52.2181	−2.34230
$498$	−18.6311	−0.834878
$499$	26.1139	1.16902	0.584510	−	0.811387i	$-0.301287\pi$
$499$	26.1139	1.16902	0.584510	+	0.811387i	$0.301287\pi$
$500$	−12.1490	−0.543322
$501$	31.7363	1.41787
$502$	16.6261	0.742058
$503$	7.95167	0.354547	0.177274	−	0.984162i	$-0.443272\pi$
$503$	7.95167	0.354547	0.177274	+	0.984162i	$0.443272\pi$
$504$	18.2170	0.811450
$505$	−21.2692	−0.946468
$506$	−21.4311	−0.952727
$507$	13.8033	0.613027
$508$	−10.9754	−0.486955
$509$	18.8623	0.836058	0.418029	−	0.908434i	$-0.362721\pi$
$509$	18.8623	0.836058	0.418029	+	0.908434i	$0.362721\pi$
$510$	−4.16871	−0.184594
$511$	−12.2757	−0.543045
$512$	−1.00000	−0.0441942
$513$	9.02280	0.398366
$514$	−3.28676	−0.144973
$515$	35.9310	1.58331
$516$	−25.6390	−1.12870
$517$	23.5432	1.03543
$518$	8.50940	0.373882
$519$	−14.5666	−0.639402
$520$	−8.05607	−0.353282
$521$	40.9436	1.79377	0.896885	−	0.442264i	$-0.145824\pi$
$521$	40.9436	1.79377	0.896885	+	0.442264i	$0.145824\pi$
$522$	0	0
$523$	29.9303	1.30876	0.654380	−	0.756166i	$-0.272930\pi$
$523$	29.9303	1.30876	0.654380	+	0.756166i	$0.272930\pi$
$524$	3.35552	0.146587
$525$	16.8974	0.737463
$526$	25.2066	1.09906
$527$	−4.44050	−0.193431
$528$	8.27819	0.360262
$529$	24.6475	1.07163
$530$	−5.09838	−0.221459
$531$	16.7079	0.725061
$532$	−13.5245	−0.586363
$533$	−29.6388	−1.28380
$534$	−11.6305	−0.503303
$535$	4.48161	0.193757
$536$	−4.14583	−0.179073
$537$	−40.4043	−1.74358
$538$	15.3554	0.662020
$539$	39.2845	1.69211
$540$	5.58854	0.240492
$541$	−15.7616	−0.677644	−0.338822	−	0.940851i	$-0.610028\pi$
$541$	−15.7616	−0.677644	−0.338822	+	0.940851i	$0.610028\pi$
$542$	15.1856	0.652280
$543$	19.6729	0.844246
$544$	−0.827423	−0.0354755
$545$	10.8237	0.463636
$546$	50.3950	2.15671
$547$	34.0781	1.45707	0.728536	−	0.685007i	$-0.240201\pi$
$547$	34.0781	1.45707	0.728536	+	0.685007i	$0.240201\pi$
$548$	−13.5265	−0.577822
$549$	−11.0987	−0.473680
$550$	4.43830	0.189250
$551$	0	0
$552$	−18.4048	−0.783361
$553$	−47.0909	−2.00251
$554$	28.6613	1.21770
$555$	9.67069	0.410498
$556$	−19.8393	−0.841373
$557$	−18.1371	−0.768492	−0.384246	−	0.923231i	$-0.625539\pi$
$557$	−18.1371	−0.768492	−0.384246	+	0.923231i	$0.625539\pi$
$558$	22.0529	0.933574
$559$	−40.9969	−1.73398
$560$	−8.37681	−0.353985
$561$	6.84956	0.289189
$562$	32.3169	1.36321
$563$	−27.5452	−1.16089	−0.580446	−	0.814299i	$-0.697122\pi$
$563$	−27.5452	−1.16089	−0.580446	+	0.814299i	$0.697122\pi$
$564$	20.2187	0.851361
$565$	−23.6079	−0.993191
$566$	−11.2514	−0.472933
$567$	19.6917	0.826975
$568$	−11.7789	−0.494232
$569$	−30.9794	−1.29872	−0.649361	−	0.760480i	$-0.724964\pi$
$569$	−30.9794	−1.29872	−0.649361	+	0.760480i	$0.724964\pi$
$570$	−15.3702	−0.643788
$571$	8.54543	0.357615	0.178808	−	0.983884i	$-0.442776\pi$
$571$	8.54543	0.357615	0.178808	+	0.983884i	$0.442776\pi$
$572$	13.2368	0.553460
$573$	1.60196	0.0669229
$574$	−30.8189	−1.28635
$575$	−9.86762	−0.411508
$576$	4.10924	0.171218
$577$	30.3618	1.26398	0.631989	−	0.774977i	$-0.282239\pi$
$577$	30.3618	1.26398	0.631989	+	0.774977i	$0.282239\pi$
$578$	16.3154	0.678630
$579$	28.8008	1.19692
$580$	0	0
$581$	−30.9772	−1.28515
$582$	28.0759	1.16378
$583$	8.37708	0.346943
$584$	−2.76905	−0.114584
$585$	33.1043	1.36869
$586$	18.6148	0.768970
$587$	38.5718	1.59203	0.796015	−	0.605277i	$-0.206938\pi$
$587$	38.5718	1.59203	0.796015	+	0.605277i	$0.206938\pi$
$588$	33.7372	1.39130
$589$	−16.3723	−0.674611
$590$	−7.68287	−0.316299
$591$	−16.1293	−0.663471
$592$	1.91948	0.0788901
$593$	−43.1108	−1.77035	−0.885174	−	0.465261i	$-0.845961\pi$
$593$	−43.1108	−1.77035	−0.885174	+	0.465261i	$0.845961\pi$
$594$	−9.18246	−0.376761
$595$	−6.93117	−0.284150
$596$	6.25711	0.256301
$597$	−59.9156	−2.45218
$598$	−29.4293	−1.20345
$599$	27.5732	1.12661	0.563305	−	0.826249i	$-0.309530\pi$
$599$	27.5732	1.12661	0.563305	+	0.826249i	$0.309530\pi$
$600$	3.81157	0.155607
$601$	7.94225	0.323971	0.161986	−	0.986793i	$-0.448210\pi$
$601$	7.94225	0.323971	0.161986	+	0.986793i	$0.448210\pi$
$602$	−42.6291	−1.73743
$603$	17.0362	0.693768
$604$	7.74673	0.315210
$605$	−2.57105	−0.104528
$606$	30.0124	1.21917
$607$	−39.1665	−1.58972	−0.794859	−	0.606794i	$-0.792455\pi$
$607$	−39.1665	−1.58972	−0.794859	+	0.606794i	$0.792455\pi$
$608$	−3.05075	−0.123724
$609$	0	0
$610$	5.10355	0.206637
$611$	32.3297	1.30792
$612$	3.40008	0.137440
$613$	21.8055	0.880715	0.440358	−	0.897823i	$-0.354852\pi$
$613$	21.8055	0.880715	0.440358	+	0.897823i	$0.354852\pi$
$614$	2.45342	0.0990118
$615$	−35.0247	−1.41233
$616$	13.7638	0.554561
$617$	−23.6828	−0.953432	−0.476716	−	0.879057i	$-0.658173\pi$
$617$	−23.6828	−0.953432	−0.476716	+	0.879057i	$0.658173\pi$
$618$	−50.7011	−2.03950
$619$	27.5854	1.10875	0.554375	−	0.832267i	$-0.312957\pi$
$619$	27.5854	1.10875	0.554375	+	0.832267i	$0.312957\pi$
$620$	−10.1407	−0.407260
$621$	20.4153	0.819237
$622$	2.07536	0.0832143
$623$	−19.3377	−0.774748
$624$	11.3677	0.455071
$625$	−15.8088	−0.632353
$626$	12.3137	0.492153
$627$	25.2547	1.00857
$628$	5.26142	0.209954
$629$	1.58822	0.0633265
$630$	34.4223	1.37142
$631$	10.2968	0.409909	0.204954	−	0.978772i	$-0.434295\pi$
$631$	10.2968	0.409909	0.204954	+	0.978772i	$0.434295\pi$
$632$	−10.6224	−0.422535
$633$	−4.74076	−0.188428
$634$	−16.1586	−0.641741
$635$	−20.7388	−0.822993
$636$	7.19417	0.285267
$637$	53.9459	2.13741
$638$	0	0
$639$	48.4023	1.91476
$640$	−1.88957	−0.0746918
$641$	−32.6454	−1.28941	−0.644707	−	0.764430i	$-0.723021\pi$
$641$	−32.6454	−1.28941	−0.644707	+	0.764430i	$0.723021\pi$
$642$	−6.32386	−0.249583
$643$	9.43433	0.372053	0.186027	−	0.982545i	$-0.440439\pi$
$643$	9.43433	0.372053	0.186027	+	0.982545i	$0.440439\pi$
$644$	−30.6010	−1.20585
$645$	−48.4468	−1.90759
$646$	−2.52426	−0.0993156
$647$	25.9592	1.02056	0.510280	−	0.860008i	$-0.329542\pi$
$647$	25.9592	1.02056	0.510280	+	0.860008i	$0.329542\pi$
$648$	4.44189	0.174494
$649$	12.6236	0.495521
$650$	6.09470	0.239054
$651$	63.4354	2.48623
$652$	12.2657	0.480360
$653$	21.3148	0.834113	0.417057	−	0.908881i	$-0.363062\pi$
$653$	21.3148	0.834113	0.417057	+	0.908881i	$0.363062\pi$
$654$	−15.2730	−0.597221
$655$	6.34049	0.247744
$656$	−6.95185	−0.271424
$657$	11.3787	0.443924
$658$	33.6169	1.31052
$659$	−17.8815	−0.696563	−0.348281	−	0.937390i	$-0.613235\pi$
$659$	−17.8815	−0.696563	−0.348281	+	0.937390i	$0.613235\pi$
$660$	15.6422	0.608872
$661$	32.7118	1.27234	0.636171	−	0.771548i	$-0.280517\pi$
$661$	32.7118	1.27234	0.636171	+	0.771548i	$0.280517\pi$
$662$	−17.8639	−0.694300
$663$	9.40588	0.365294
$664$	−6.98757	−0.271170
$665$	−25.5555	−0.991001
$666$	−7.88759	−0.305638
$667$	0	0
$668$	11.9027	0.460529
$669$	18.9448	0.732447
$670$	−7.83384	−0.302648
$671$	−8.38559	−0.323722
$672$	11.8203	0.455977
$673$	33.5616	1.29371	0.646853	−	0.762615i	$-0.276085\pi$
$673$	33.5616	1.29371	0.646853	+	0.762615i	$0.276085\pi$
$674$	−22.2689	−0.857767
$675$	−4.22793	−0.162733
$676$	5.17693	0.199113
$677$	40.2847	1.54827	0.774133	−	0.633023i	$-0.218186\pi$
$677$	40.2847	1.54827	0.774133	+	0.633023i	$0.218186\pi$
$678$	33.3124	1.27935
$679$	46.6808	1.79144
$680$	−1.56347	−0.0599565
$681$	6.65975	0.255202
$682$	16.6621	0.638023
$683$	−37.6295	−1.43985	−0.719927	−	0.694050i	$-0.755825\pi$
$683$	−37.6295	−1.43985	−0.719927	+	0.694050i	$0.755825\pi$
$684$	12.5362	0.479335
$685$	−25.5592	−0.976568
$686$	25.0614	0.956848
$687$	−27.0548	−1.03220
$688$	−9.61591	−0.366603
$689$	11.5035	0.438248
$690$	−34.7772	−1.32394
$691$	−10.7763	−0.409951	−0.204975	−	0.978767i	$-0.565711\pi$
$691$	−10.7763	−0.409951	−0.204975	+	0.978767i	$0.565711\pi$
$692$	−5.46319	−0.207679
$693$	−56.5589	−2.14849
$694$	23.5162	0.892663
$695$	−37.4877	−1.42199
$696$	0	0
$697$	−5.75212	−0.217877
$698$	−13.8857	−0.525581
$699$	−49.5278	−1.87331
$700$	6.33736	0.239530
$701$	−7.52690	−0.284287	−0.142144	−	0.989846i	$-0.545399\pi$
$701$	−7.52690	−0.284287	−0.142144	+	0.989846i	$0.545399\pi$
$702$	−12.6094	−0.475912
$703$	5.85584	0.220857
$704$	3.10473	0.117014
$705$	38.2047	1.43887
$706$	−26.4253	−0.994531
$707$	49.9005	1.87670
$708$	10.8411	0.407432
$709$	−2.63508	−0.0989625	−0.0494813	−	0.998775i	$-0.515757\pi$
$709$	−2.63508	−0.0989625	−0.0494813	+	0.998775i	$0.515757\pi$
$710$	−22.2571	−0.835292
$711$	43.6498	1.63699
$712$	−4.36203	−0.163474
$713$	−37.0446	−1.38733
$714$	9.78036	0.366021
$715$	25.0119	0.935393
$716$	−15.1536	−0.566317
$717$	−37.3238	−1.39388
$718$	−3.48063	−0.129896
$719$	−42.3268	−1.57852	−0.789261	−	0.614058i	$-0.789536\pi$
$719$	−42.3268	−1.57852	−0.789261	+	0.614058i	$0.789536\pi$
$720$	7.76469	0.289373
$721$	−84.2989	−3.13946
$722$	9.69294	0.360734
$723$	−35.3479	−1.31460
$724$	7.37832	0.274213
$725$	0	0
$726$	3.62793	0.134645
$727$	0.246181	0.00913035	0.00456517	−	0.999990i	$-0.498547\pi$
$727$	0.246181	0.00913035	0.00456517	+	0.999990i	$0.498547\pi$
$728$	18.9006	0.700504
$729$	−41.9102	−1.55223
$730$	−5.23231	−0.193656
$731$	−7.95643	−0.294279
$732$	−7.20147	−0.266174
$733$	−23.9116	−0.883194	−0.441597	−	0.897213i	$-0.645588\pi$
$733$	−23.9116	−0.883194	−0.441597	+	0.897213i	$0.645588\pi$
$734$	−5.89712	−0.217667
$735$	63.7488	2.35141
$736$	−6.90272	−0.254437
$737$	12.8717	0.474135
$738$	28.5668	1.05156
$739$	−7.66404	−0.281926	−0.140963	−	0.990015i	$-0.545020\pi$
$739$	−7.66404	−0.281926	−0.140963	+	0.990015i	$0.545020\pi$
$740$	3.62699	0.133331
$741$	34.6799	1.27400
$742$	11.9615	0.439120
$743$	−18.9783	−0.696247	−0.348124	−	0.937449i	$-0.613181\pi$
$743$	−18.9783	−0.696247	−0.348124	+	0.937449i	$0.613181\pi$
$744$	14.3092	0.524602
$745$	11.8232	0.433170
$746$	−28.0886	−1.02840
$747$	28.7136	1.05057
$748$	2.56892	0.0939292
$749$	−10.5145	−0.384190
$750$	32.3932	1.18283
$751$	−2.12763	−0.0776383	−0.0388191	−	0.999246i	$-0.512360\pi$
$751$	−2.12763	−0.0776383	−0.0388191	+	0.999246i	$0.512360\pi$
$752$	7.58301	0.276524
$753$	−44.3304	−1.61549
$754$	0	0
$755$	14.6380	0.532731
$756$	−13.1115	−0.476859
$757$	−33.7765	−1.22763	−0.613814	−	0.789451i	$-0.710365\pi$
$757$	−33.7765	−1.22763	−0.613814	+	0.789451i	$0.710365\pi$
$758$	32.8894	1.19460
$759$	57.1420	2.07412
$760$	−5.76460	−0.209104
$761$	−0.0142514	−0.000516613 0	−0.000258307	−	1.00000i	$-0.500082\pi$
$761$	−0.0142514	−0.000516613 0	−0.000258307		1.00000i	$0.500082\pi$
$762$	29.2639	1.06012
$763$	−25.3939	−0.919319
$764$	0.600815	0.0217367
$765$	6.42468	0.232285
$766$	18.9516	0.684749
$767$	17.3349	0.625926
$768$	2.66631	0.0962124
$769$	32.0874	1.15710	0.578550	−	0.815647i	$-0.303619\pi$
$769$	32.0874	1.15710	0.578550	+	0.815647i	$0.303619\pi$
$770$	26.0077	0.937254
$771$	8.76355	0.315611
$772$	10.8017	0.388763
$773$	0.560306	0.0201528	0.0100764	−	0.999949i	$-0.496793\pi$
$773$	0.560306	0.0201528	0.0100764	+	0.999949i	$0.496793\pi$
$774$	39.5140	1.42030
$775$	7.67180	0.275579
$776$	10.5299	0.378000
$777$	−22.6887	−0.813954
$778$	−25.7959	−0.924827
$779$	−21.2083	−0.759868
$780$	21.4800	0.769108
$781$	36.5703	1.30859
$782$	−5.71147	−0.204242
$783$	0	0
$784$	12.6531	0.451897
$785$	9.94182	0.354839
$786$	−8.94688	−0.319125
$787$	20.0186	0.713587	0.356794	−	0.934183i	$-0.383870\pi$
$787$	20.0186	0.713587	0.356794	+	0.934183i	$0.383870\pi$
$788$	−6.04928	−0.215497
$789$	−67.2087	−2.39269
$790$	−20.0717	−0.714118
$791$	55.3873	1.96934
$792$	−12.7581	−0.453338
$793$	−11.5152	−0.408915
$794$	−24.2601	−0.860959
$795$	13.5939	0.482125
$796$	−22.4713	−0.796475
$797$	45.1359	1.59880	0.799398	−	0.600801i	$-0.205152\pi$
$797$	45.1359	1.59880	0.799398	+	0.600801i	$0.205152\pi$
$798$	36.0606	1.27653
$799$	6.27436	0.221971
$800$	1.42953	0.0505414
$801$	17.9246	0.633335
$802$	19.8886	0.702290
$803$	8.59715	0.303387
$804$	11.0541	0.389848
$805$	−57.8227	−2.03798
$806$	22.8805	0.805930
$807$	−40.9424	−1.44124
$808$	11.2561	0.395989
$809$	30.5309	1.07341	0.536705	−	0.843770i	$-0.319669\pi$
$809$	30.5309	1.07341	0.536705	+	0.843770i	$0.319669\pi$
$810$	8.39326	0.294909
$811$	−7.41012	−0.260204	−0.130102	−	0.991501i	$-0.541531\pi$
$811$	−7.41012	−0.260204	−0.130102	+	0.991501i	$0.541531\pi$
$812$	0	0
$813$	−40.4897	−1.42004
$814$	−5.95946	−0.208879
$815$	23.1768	0.811848
$816$	2.20617	0.0772314
$817$	−29.3357	−1.02633
$818$	−33.3680	−1.16668
$819$	−77.6671	−2.71391
$820$	−13.1360	−0.458729
$821$	53.2997	1.86017	0.930086	−	0.367342i	$-0.119732\pi$
$821$	53.2997	1.86017	0.930086	+	0.367342i	$0.119732\pi$
$822$	36.0658	1.25794
$823$	−3.45715	−0.120509	−0.0602544	−	0.998183i	$-0.519191\pi$
$823$	−3.45715	−0.120509	−0.0602544	+	0.998183i	$0.519191\pi$
$824$	−19.0154	−0.662434
$825$	−11.8339	−0.412003
$826$	18.0251	0.627172
$827$	54.3681	1.89056	0.945281	−	0.326257i	$-0.105787\pi$
$827$	54.3681	1.89056	0.945281	+	0.326257i	$0.105787\pi$
$828$	28.3649	0.985748
$829$	−50.9113	−1.76822	−0.884111	−	0.467277i	$-0.845235\pi$
$829$	−50.9113	−1.76822	−0.884111	+	0.467277i	$0.845235\pi$
$830$	−13.2035	−0.458300
$831$	−76.4199	−2.65098
$832$	4.26344	0.147808
$833$	10.4695	0.362746
$834$	52.8978	1.83170
$835$	22.4909	0.778331
$836$	9.47175	0.327587
$837$	−15.8723	−0.548627
$838$	18.1156	0.625792
$839$	29.2351	1.00931	0.504654	−	0.863322i	$-0.331620\pi$
$839$	29.2351	1.00931	0.504654	+	0.863322i	$0.331620\pi$
$840$	22.3352	0.770638
$841$	0	0
$842$	13.8875	0.478594
$843$	−86.1671	−2.96775
$844$	−1.77802	−0.0612020
$845$	9.78217	0.336517
$846$	−31.1604	−1.07132
$847$	6.03203	0.207263
$848$	2.69817	0.0926555
$849$	29.9998	1.02959
$850$	1.18282	0.0405705
$851$	13.2496	0.454191
$852$	31.4063	1.07596
$853$	−43.7365	−1.49751	−0.748754	−	0.662848i	$-0.769348\pi$
$853$	−43.7365	−1.49751	−0.748754	+	0.662848i	$0.769348\pi$
$854$	−11.9736	−0.409729
$855$	23.6881	0.810116
$856$	−2.37176	−0.0810651
$857$	12.0143	0.410401	0.205200	−	0.978720i	$-0.434215\pi$
$857$	12.0143	0.410401	0.205200	+	0.978720i	$0.434215\pi$
$858$	−35.2936	−1.20490
$859$	−2.06483	−0.0704512	−0.0352256	−	0.999379i	$-0.511215\pi$
$859$	−2.06483	−0.0704512	−0.0352256	+	0.999379i	$0.511215\pi$
$860$	−18.1699	−0.619590
$861$	82.1728	2.80044
$862$	17.4399	0.594004
$863$	−18.1710	−0.618548	−0.309274	−	0.950973i	$-0.600086\pi$
$863$	−18.1710	−0.618548	−0.309274	+	0.950973i	$0.600086\pi$
$864$	−2.95757	−0.100619
$865$	−10.3231	−0.350995
$866$	27.5286	0.935461
$867$	−43.5019	−1.47740
$868$	23.7914	0.807534
$869$	32.9795	1.11875
$870$	0	0
$871$	17.6755	0.598912
$872$	−5.72813	−0.193979
$873$	−43.2697	−1.46446
$874$	−21.0584	−0.712312
$875$	53.8589	1.82076
$876$	7.38316	0.249454
$877$	7.00172	0.236431	0.118216	−	0.992988i	$-0.462283\pi$
$877$	7.00172	0.236431	0.118216	+	0.992988i	$0.462283\pi$
$878$	2.15260	0.0726468
$879$	−49.6329	−1.67408
$880$	5.86660	0.197763
$881$	22.2633	0.750070	0.375035	−	0.927011i	$-0.377631\pi$
$881$	22.2633	0.750070	0.375035	+	0.927011i	$0.377631\pi$
$882$	−51.9947	−1.75075
$883$	57.9963	1.95173	0.975865	−	0.218375i	$-0.0700755\pi$
$883$	57.9963	1.95173	0.975865	+	0.218375i	$0.0700755\pi$
$884$	3.52767	0.118648
$885$	20.4850	0.688594
$886$	−25.8500	−0.868449
$887$	−29.1105	−0.977434	−0.488717	−	0.872442i	$-0.662535\pi$
$887$	−29.1105	−0.977434	−0.488717	+	0.872442i	$0.662535\pi$
$888$	−5.11793	−0.171747
$889$	48.6560	1.63187
$890$	−8.24235	−0.276284
$891$	−13.7909	−0.462011
$892$	7.10522	0.237900
$893$	23.1339	0.774145
$894$	−16.6834	−0.557977
$895$	−28.6338	−0.957123
$896$	4.43319	0.148102
$897$	78.4678	2.61996
$898$	18.7693	0.626339
$899$	0	0
$900$	−5.87426	−0.195809
$901$	2.23253	0.0743763
$902$	21.5836	0.718656
$903$	113.663	3.78246
$904$	12.4938	0.415537
$905$	13.9418	0.463442
$906$	−20.6552	−0.686223
$907$	−35.0625	−1.16423	−0.582116	−	0.813106i	$-0.697775\pi$
$907$	−35.0625	−1.16423	−0.582116	+	0.813106i	$0.697775\pi$
$908$	2.49774	0.0828903
$909$	−46.2540	−1.53415
$910$	35.7140	1.18391
$911$	53.8713	1.78483	0.892417	−	0.451211i	$-0.149008\pi$
$911$	53.8713	1.78483	0.892417	+	0.451211i	$0.149008\pi$
$912$	8.13425	0.269352
$913$	21.6945	0.717984
$914$	13.9278	0.460690
$915$	−13.6077	−0.449856
$916$	−10.1469	−0.335262
$917$	−14.8757	−0.491237
$918$	−2.44716	−0.0807684
$919$	−18.1545	−0.598861	−0.299430	−	0.954118i	$-0.596797\pi$
$919$	−18.1545	−0.598861	−0.299430	+	0.954118i	$0.596797\pi$
$920$	−13.0432	−0.430020
$921$	−6.54158	−0.215552
$922$	15.5360	0.511650
$923$	50.2187	1.65297
$924$	−36.6987	−1.20730
$925$	−2.74395	−0.0902204
$926$	−23.6809	−0.778202
$927$	78.1389	2.56642
$928$	0	0
$929$	−8.58992	−0.281826	−0.140913	−	0.990022i	$-0.545004\pi$
$929$	−8.58992	−0.281826	−0.140913	+	0.990022i	$0.545004\pi$
$930$	27.0383	0.886620
$931$	38.6015	1.26511
$932$	−18.5754	−0.608457
$933$	−5.53356	−0.181161
$934$	30.5564	0.999835
$935$	4.85416	0.158748
$936$	−17.5195	−0.572642
$937$	28.5141	0.931516	0.465758	−	0.884912i	$-0.345782\pi$
$937$	28.5141	0.931516	0.465758	+	0.884912i	$0.345782\pi$
$938$	18.3792	0.600104
$939$	−32.8321	−1.07144
$940$	14.3286	0.467349
$941$	54.2110	1.76723	0.883614	−	0.468216i	$-0.155103\pi$
$941$	54.2110	1.76723	0.883614	+	0.468216i	$0.155103\pi$
$942$	−14.0286	−0.457077
$943$	−47.9867	−1.56266
$944$	4.06594	0.132335
$945$	−24.7750	−0.805931
$946$	29.8548	0.970663
$947$	5.02058	0.163147	0.0815735	−	0.996667i	$-0.474005\pi$
$947$	5.02058	0.163147	0.0815735	+	0.996667i	$0.474005\pi$
$948$	28.3225	0.919874
$949$	11.8057	0.383228
$950$	4.36113	0.141494
$951$	43.0840	1.39709
$952$	3.66812	0.118884
$953$	3.02848	0.0981022	0.0490511	−	0.998796i	$-0.484380\pi$
$953$	3.02848	0.0981022	0.0490511	+	0.998796i	$0.484380\pi$
$954$	−11.0874	−0.358968
$955$	1.13528	0.0367368
$956$	−13.9983	−0.452736
$957$	0	0
$958$	13.8099	0.446178
$959$	59.9654	1.93638
$960$	5.03819	0.162607
$961$	−2.19888	−0.0709316
$962$	−8.18358	−0.263849
$963$	9.74612	0.314064
$964$	−13.2572	−0.426986
$965$	20.4106	0.657042
$966$	81.5919	2.62518
$967$	37.9484	1.22034	0.610169	−	0.792271i	$-0.291101\pi$
$967$	37.9484	1.22034	0.610169	+	0.792271i	$0.291101\pi$
$968$	1.36065	0.0437330
$969$	6.73047	0.216214
$970$	19.8969	0.638851
$971$	55.6567	1.78611	0.893055	−	0.449948i	$-0.148557\pi$
$971$	55.6567	1.78611	0.893055	+	0.449948i	$0.148557\pi$
$972$	−20.7162	−0.664472
$973$	87.9513	2.81959
$974$	−10.0549	−0.322181
$975$	−16.2504	−0.520429
$976$	−2.70091	−0.0864540
$977$	−42.1194	−1.34752	−0.673760	−	0.738950i	$-0.735322\pi$
$977$	−42.1194	−1.34752	−0.673760	+	0.738950i	$0.735322\pi$
$978$	−32.7041	−1.04576
$979$	13.5429	0.432834
$980$	23.9090	0.763744
$981$	23.5382	0.751518
$982$	−33.0650	−1.05515
$983$	12.6576	0.403716	0.201858	−	0.979415i	$-0.435302\pi$
$983$	12.6576	0.403716	0.201858	+	0.979415i	$0.435302\pi$
$984$	18.5358	0.590901
$985$	−11.4305	−0.364207
$986$	0	0
$987$	−89.6333	−2.85306
$988$	13.0067	0.413798
$989$	−66.3759	−2.11063
$990$	−24.1073	−0.766179
$991$	−53.8727	−1.71132	−0.855662	−	0.517535i	$-0.826850\pi$
$991$	−53.8727	−1.71132	−0.855662	+	0.517535i	$0.826850\pi$
$992$	5.36667	0.170392
$993$	47.6308	1.51152
$994$	52.2181	1.65626
$995$	−42.4611	−1.34611
$996$	18.6311	0.590348
$997$	11.1555	0.353297	0.176649	−	0.984274i	$-0.443474\pi$
$997$	11.1555	0.353297	0.176649	+	0.984274i	$0.443474\pi$
$998$	−26.1139	−0.826622
$999$	5.67699	0.179612

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.2.a.u.1.7	✓	8
29.12	odd	4		1682.2.b.k.1681.7		16
29.17	odd	4		1682.2.b.k.1681.10		16
29.28	even	2		1682.2.a.v.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1682.2.a.u.1.7	✓	8	1.1	even	1	trivial
1682.2.a.v.1.2	yes	8	29.28	even	2
1682.2.b.k.1681.7		16	29.12	odd	4
1682.2.b.k.1681.10		16	29.17	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:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1682.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.66631 q^{3} +1.00000 q^{4} +1.88957 q^{5} -2.66631 q^{6} -4.43319 q^{7} -1.00000 q^{8} +4.10924 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.66631 q^{3} +1.00000 q^{4} +1.88957 q^{5} -2.66631 q^{6} -4.43319 q^{7} -1.00000 q^{8} +4.10924 q^{9} -1.88957 q^{10} +3.10473 q^{11} +2.66631 q^{12} +4.26344 q^{13} +4.43319 q^{14} +5.03819 q^{15} +1.00000 q^{16} +0.827423 q^{17} -4.10924 q^{18} +3.05075 q^{19} +1.88957 q^{20} -11.8203 q^{21} -3.10473 q^{22} +6.90272 q^{23} -2.66631 q^{24} -1.42953 q^{25} -4.26344 q^{26} +2.95757 q^{27} -4.43319 q^{28} -5.03819 q^{30} -5.36667 q^{31} -1.00000 q^{32} +8.27819 q^{33} -0.827423 q^{34} -8.37681 q^{35} +4.10924 q^{36} +1.91948 q^{37} -3.05075 q^{38} +11.3677 q^{39} -1.88957 q^{40} -6.95185 q^{41} +11.8203 q^{42} -9.61591 q^{43} +3.10473 q^{44} +7.76469 q^{45} -6.90272 q^{46} +7.58301 q^{47} +2.66631 q^{48} +12.6531 q^{49} +1.42953 q^{50} +2.20617 q^{51} +4.26344 q^{52} +2.69817 q^{53} -2.95757 q^{54} +5.86660 q^{55} +4.43319 q^{56} +8.13425 q^{57} +4.06594 q^{59} +5.03819 q^{60} -2.70091 q^{61} +5.36667 q^{62} -18.2170 q^{63} +1.00000 q^{64} +8.05607 q^{65} -8.27819 q^{66} +4.14583 q^{67} +0.827423 q^{68} +18.4048 q^{69} +8.37681 q^{70} +11.7789 q^{71} -4.10924 q^{72} +2.76905 q^{73} -1.91948 q^{74} -3.81157 q^{75} +3.05075 q^{76} -13.7638 q^{77} -11.3677 q^{78} +10.6224 q^{79} +1.88957 q^{80} -4.44189 q^{81} +6.95185 q^{82} +6.98757 q^{83} -11.8203 q^{84} +1.56347 q^{85} +9.61591 q^{86} -3.10473 q^{88} +4.36203 q^{89} -7.76469 q^{90} -18.9006 q^{91} +6.90272 q^{92} -14.3092 q^{93} -7.58301 q^{94} +5.76460 q^{95} -2.66631 q^{96} -10.5299 q^{97} -12.6531 q^{98} +12.7581 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} + 5 q^{5} + q^{6} + 7 q^{7} - 8 q^{8} + 13 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 + 5 * q^5 + q^6 + 7 * q^7 - 8 * q^8 + 13 * q^9	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} + 5 q^{5} + q^{6} + 7 q^{7} - 8 q^{8} + 13 q^{9} - 5 q^{10} + 7 q^{11} - q^{12} + 13 q^{13} - 7 q^{14} + 20 q^{15} + 8 q^{16} - 9 q^{17} - 13 q^{18} + 13 q^{19} + 5 q^{20} - 34 q^{21} - 7 q^{22} + 12 q^{23} + q^{24} + 45 q^{25} - 13 q^{26} + 2 q^{27} + 7 q^{28} - 20 q^{30} - 15 q^{31} - 8 q^{32} - 4 q^{33} + 9 q^{34} + 13 q^{36} - 4 q^{37} - 13 q^{38} + 9 q^{39} - 5 q^{40} - 8 q^{41} + 34 q^{42} - 12 q^{43} + 7 q^{44} + 30 q^{45} - 12 q^{46} + 13 q^{47} - q^{48} + 27 q^{49} - 45 q^{50} - 42 q^{51} + 13 q^{52} + 4 q^{53} - 2 q^{54} + 35 q^{55} - 7 q^{56} - 11 q^{57} + 8 q^{59} + 20 q^{60} - 9 q^{61} + 15 q^{62} + 12 q^{63} + 8 q^{64} - 15 q^{65} + 4 q^{66} + 34 q^{67} - 9 q^{68} - 4 q^{69} - 11 q^{71} - 13 q^{72} + 13 q^{73} + 4 q^{74} + 15 q^{75} + 13 q^{76} + 23 q^{77} - 9 q^{78} + 8 q^{79} + 5 q^{80} + 12 q^{81} + 8 q^{82} + 10 q^{83} - 34 q^{84} + 15 q^{85} + 12 q^{86} - 7 q^{88} - 6 q^{89} - 30 q^{90} + 12 q^{91} + 12 q^{92} + 15 q^{93} - 13 q^{94} + 15 q^{95} + q^{96} + 11 q^{97} - 27 q^{98} + 2 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 + 5 * q^5 + q^6 + 7 * q^7 - 8 * q^8 + 13 * q^9 - 5 * q^10 + 7 * q^11 - q^12 + 13 * q^13 - 7 * q^14 + 20 * q^15 + 8 * q^16 - 9 * q^17 - 13 * q^18 + 13 * q^19 + 5 * q^20 - 34 * q^21 - 7 * q^22 + 12 * q^23 + q^24 + 45 * q^25 - 13 * q^26 + 2 * q^27 + 7 * q^28 - 20 * q^30 - 15 * q^31 - 8 * q^32 - 4 * q^33 + 9 * q^34 + 13 * q^36 - 4 * q^37 - 13 * q^38 + 9 * q^39 - 5 * q^40 - 8 * q^41 + 34 * q^42 - 12 * q^43 + 7 * q^44 + 30 * q^45 - 12 * q^46 + 13 * q^47 - q^48 + 27 * q^49 - 45 * q^50 - 42 * q^51 + 13 * q^52 + 4 * q^53 - 2 * q^54 + 35 * q^55 - 7 * q^56 - 11 * q^57 + 8 * q^59 + 20 * q^60 - 9 * q^61 + 15 * q^62 + 12 * q^63 + 8 * q^64 - 15 * q^65 + 4 * q^66 + 34 * q^67 - 9 * q^68 - 4 * q^69 - 11 * q^71 - 13 * q^72 + 13 * q^73 + 4 * q^74 + 15 * q^75 + 13 * q^76 + 23 * q^77 - 9 * q^78 + 8 * q^79 + 5 * q^80 + 12 * q^81 + 8 * q^82 + 10 * q^83 - 34 * q^84 + 15 * q^85 + 12 * q^86 - 7 * q^88 - 6 * q^89 - 30 * q^90 + 12 * q^91 + 12 * q^92 + 15 * q^93 - 13 * q^94 + 15 * q^95 + q^96 + 11 * q^97 - 27 * q^98 + 2 * q^99

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