LMFDB - Embedded newform 1339.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, 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("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.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:	$10.6919688306$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 9 x^{18} + 13 x^{17} + 106 x^{16} - 351 x^{15} - 279 x^{14} + 2337 x^{13} - 1079 x^{12} + \cdots - 63 $ x^19 - 9x^18 + 13x^17 + 106x^16 - 351x^15 - 279x^14 + 2337x^13 - 1079x^12 - 6826x^11 + 7199x^10 + 9364x^9 - 14841x^8 - 5183x^7 + 14037x^6 - 108x^5 - 6099x^4 + 855x^3 + 1062x^2 - 135x - 63
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.78640$ of defining polynomial
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78640 q^{2} -0.0654083 q^{3} +5.76405 q^{4} -3.53830 q^{5} +0.182254 q^{6} +1.33123 q^{7} -10.4882 q^{8} -2.99572 q^{9} +O(q^{10})$	\(q-2.78640 q^{2} -0.0654083 q^{3} +5.76405 q^{4} -3.53830 q^{5} +0.182254 q^{6} +1.33123 q^{7} -10.4882 q^{8} -2.99572 q^{9} +9.85914 q^{10} -1.80499 q^{11} -0.377017 q^{12} +1.00000 q^{13} -3.70933 q^{14} +0.231434 q^{15} +17.6962 q^{16} +7.17950 q^{17} +8.34729 q^{18} +0.918122 q^{19} -20.3949 q^{20} -0.0870732 q^{21} +5.02943 q^{22} -0.637666 q^{23} +0.686013 q^{24} +7.51957 q^{25} -2.78640 q^{26} +0.392170 q^{27} +7.67325 q^{28} +1.02352 q^{29} -0.644869 q^{30} +9.05964 q^{31} -28.3324 q^{32} +0.118061 q^{33} -20.0050 q^{34} -4.71028 q^{35} -17.2675 q^{36} -11.7398 q^{37} -2.55826 q^{38} -0.0654083 q^{39} +37.1103 q^{40} -2.02384 q^{41} +0.242621 q^{42} -5.74053 q^{43} -10.4040 q^{44} +10.5998 q^{45} +1.77680 q^{46} +11.8738 q^{47} -1.15748 q^{48} -5.22784 q^{49} -20.9526 q^{50} -0.469599 q^{51} +5.76405 q^{52} +10.8346 q^{53} -1.09274 q^{54} +6.38659 q^{55} -13.9621 q^{56} -0.0600528 q^{57} -2.85195 q^{58} -3.34094 q^{59} +1.33400 q^{60} -9.78838 q^{61} -25.2438 q^{62} -3.98798 q^{63} +43.5531 q^{64} -3.53830 q^{65} -0.328966 q^{66} -3.38397 q^{67} +41.3830 q^{68} +0.0417087 q^{69} +13.1247 q^{70} -16.2230 q^{71} +31.4196 q^{72} +4.45188 q^{73} +32.7119 q^{74} -0.491842 q^{75} +5.29210 q^{76} -2.40285 q^{77} +0.182254 q^{78} -9.89386 q^{79} -62.6144 q^{80} +8.96151 q^{81} +5.63924 q^{82} +8.80540 q^{83} -0.501894 q^{84} -25.4032 q^{85} +15.9954 q^{86} -0.0669470 q^{87} +18.9310 q^{88} -4.86483 q^{89} -29.5352 q^{90} +1.33123 q^{91} -3.67554 q^{92} -0.592575 q^{93} -33.0853 q^{94} -3.24859 q^{95} +1.85317 q^{96} +6.61086 q^{97} +14.5669 q^{98} +5.40724 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9}+O(q^{10}) $ 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9	\( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9} - q^{11} + 18 q^{12} + 19 q^{13} - 8 q^{14} - 11 q^{15} + 21 q^{16} - 16 q^{17} - 10 q^{19} - 20 q^{20} - 33 q^{21} + 2 q^{22} - 14 q^{23} - 9 q^{24} + 23 q^{25} - 9 q^{26} + q^{27} + 10 q^{28} - 22 q^{29} + 28 q^{30} - 3 q^{31} - 47 q^{32} - 25 q^{33} - 35 q^{34} - 3 q^{35} - 33 q^{36} - 19 q^{37} - 15 q^{38} - 2 q^{39} + 8 q^{40} - 52 q^{41} - 3 q^{42} - 2 q^{43} - 54 q^{44} - 40 q^{45} + 33 q^{46} - 24 q^{47} - 8 q^{48} + 7 q^{49} - 10 q^{50} - 7 q^{51} + 17 q^{52} - 11 q^{53} - 23 q^{54} - 29 q^{55} - 17 q^{56} - 34 q^{57} + 18 q^{58} - 52 q^{59} - 71 q^{60} - 25 q^{61} - 22 q^{62} - 19 q^{63} + 48 q^{64} - 18 q^{65} + 9 q^{66} - 2 q^{67} + 18 q^{68} - 26 q^{69} - 12 q^{70} - 44 q^{71} - 6 q^{72} - 39 q^{73} - 4 q^{74} + 17 q^{75} - 10 q^{76} - 18 q^{77} - 8 q^{78} + q^{79} - 9 q^{80} - 13 q^{81} + 47 q^{82} - 27 q^{83} - 24 q^{84} - 26 q^{85} - q^{86} + 31 q^{87} + 19 q^{88} - 86 q^{89} + 48 q^{90} - 8 q^{91} - 19 q^{92} + 32 q^{93} + 45 q^{94} + 17 q^{95} - 25 q^{96} - 20 q^{97} + 39 q^{98} + 50 q^{99}+O(q^{100}) \) 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9 - q^11 + 18 * q^12 + 19 * q^13 - 8 * q^14 - 11 * q^15 + 21 * q^16 - 16 * q^17 - 10 * q^19 - 20 * q^20 - 33 * q^21 + 2 * q^22 - 14 * q^23 - 9 * q^24 + 23 * q^25 - 9 * q^26 + q^27 + 10 * q^28 - 22 * q^29 + 28 * q^30 - 3 * q^31 - 47 * q^32 - 25 * q^33 - 35 * q^34 - 3 * q^35 - 33 * q^36 - 19 * q^37 - 15 * q^38 - 2 * q^39 + 8 * q^40 - 52 * q^41 - 3 * q^42 - 2 * q^43 - 54 * q^44 - 40 * q^45 + 33 * q^46 - 24 * q^47 - 8 * q^48 + 7 * q^49 - 10 * q^50 - 7 * q^51 + 17 * q^52 - 11 * q^53 - 23 * q^54 - 29 * q^55 - 17 * q^56 - 34 * q^57 + 18 * q^58 - 52 * q^59 - 71 * q^60 - 25 * q^61 - 22 * q^62 - 19 * q^63 + 48 * q^64 - 18 * q^65 + 9 * q^66 - 2 * q^67 + 18 * q^68 - 26 * q^69 - 12 * q^70 - 44 * q^71 - 6 * q^72 - 39 * q^73 - 4 * q^74 + 17 * q^75 - 10 * q^76 - 18 * q^77 - 8 * q^78 + q^79 - 9 * q^80 - 13 * q^81 + 47 * q^82 - 27 * q^83 - 24 * q^84 - 26 * q^85 - q^86 + 31 * q^87 + 19 * q^88 - 86 * q^89 + 48 * q^90 - 8 * q^91 - 19 * q^92 + 32 * q^93 + 45 * q^94 + 17 * q^95 - 25 * q^96 - 20 * q^97 + 39 * q^98 + 50 * 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 $

$2$	−2.78640	−1.97029	−0.985143	−	0.171737i	$-0.945062\pi$
$2$	−2.78640	−1.97029	−0.985143	+	0.171737i	$0.945062\pi$
$3$	−0.0654083	−0.0377635	−0.0188817	−	0.999822i	$-0.506011\pi$
$3$	−0.0654083	−0.0377635	−0.0188817	+	0.999822i	$0.506011\pi$
$4$	5.76405	2.88203
$5$	−3.53830	−1.58238	−0.791188	−	0.611573i	$-0.790537\pi$
$5$	−3.53830	−1.58238	−0.791188	+	0.611573i	$0.790537\pi$
$6$	0.182254	0.0744049
$7$	1.33123	0.503156	0.251578	−	0.967837i	$-0.419051\pi$
$7$	1.33123	0.503156	0.251578	+	0.967837i	$0.419051\pi$
$8$	−10.4882	−3.70813
$9$	−2.99572	−0.998574
$10$	9.85914	3.11773
$11$	−1.80499	−0.544224	−0.272112	−	0.962266i	$-0.587722\pi$
$11$	−1.80499	−0.544224	−0.272112	+	0.962266i	$0.587722\pi$
$12$	−0.377017	−0.108835
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−3.70933	−0.991361
$15$	0.231434	0.0597561
$16$	17.6962	4.42405
$17$	7.17950	1.74129	0.870643	−	0.491916i	$-0.163703\pi$
$17$	7.17950	1.74129	0.870643	+	0.491916i	$0.163703\pi$
$18$	8.34729	1.96748
$19$	0.918122	0.210632	0.105316	−	0.994439i	$-0.466415\pi$
$19$	0.918122	0.210632	0.105316	+	0.994439i	$0.466415\pi$
$20$	−20.3949	−4.56045
$21$	−0.0870732	−0.0190009
$22$	5.02943	1.07228
$23$	−0.637666	−0.132963	−0.0664813	−	0.997788i	$-0.521177\pi$
$23$	−0.637666	−0.132963	−0.0664813	+	0.997788i	$0.521177\pi$
$24$	0.686013	0.140032
$25$	7.51957	1.50391
$26$	−2.78640	−0.546459
$27$	0.392170	0.0754731
$28$	7.67325	1.45011
$29$	1.02352	0.190064	0.0950318	−	0.995474i	$-0.469705\pi$
$29$	1.02352	0.190064	0.0950318	+	0.995474i	$0.469705\pi$
$30$	−0.644869	−0.117737
$31$	9.05964	1.62716	0.813579	−	0.581454i	$-0.197516\pi$
$31$	9.05964	1.62716	0.813579	+	0.581454i	$0.197516\pi$
$32$	−28.3324	−5.00851
$33$	0.118061	0.0205518
$34$	−20.0050	−3.43083
$35$	−4.71028	−0.796182
$36$	−17.2675	−2.87792
$37$	−11.7398	−1.93002	−0.965009	−	0.262218i	$-0.915546\pi$
$37$	−11.7398	−1.93002	−0.965009	+	0.262218i	$0.915546\pi$
$38$	−2.55826	−0.415004
$39$	−0.0654083	−0.0104737
$40$	37.1103	5.86765
$41$	−2.02384	−0.316071	−0.158036	−	0.987433i	$-0.550516\pi$
$41$	−2.02384	−0.316071	−0.158036	+	0.987433i	$0.550516\pi$
$42$	0.242621	0.0374373
$43$	−5.74053	−0.875423	−0.437711	−	0.899116i	$-0.644211\pi$
$43$	−5.74053	−0.875423	−0.437711	+	0.899116i	$0.644211\pi$
$44$	−10.4040	−1.56847
$45$	10.5998	1.58012
$46$	1.77680	0.261974
$47$	11.8738	1.73198	0.865988	−	0.500065i	$-0.166690\pi$
$47$	11.8738	1.73198	0.865988	+	0.500065i	$0.166690\pi$
$48$	−1.15748	−0.167067
$49$	−5.22784	−0.746834
$50$	−20.9526	−2.96314
$51$	−0.469599	−0.0657570
$52$	5.76405	0.799330
$53$	10.8346	1.48824	0.744121	−	0.668045i	$-0.232869\pi$
$53$	10.8346	1.48824	0.744121	+	0.668045i	$0.232869\pi$
$54$	−1.09274	−0.148704
$55$	6.38659	0.861168
$56$	−13.9621	−1.86577
$57$	−0.0600528	−0.00795418
$58$	−2.85195	−0.374480
$59$	−3.34094	−0.434953	−0.217477	−	0.976066i	$-0.569783\pi$
$59$	−3.34094	−0.434953	−0.217477	+	0.976066i	$0.569783\pi$
$60$	1.33400	0.172219
$61$	−9.78838	−1.25327	−0.626637	−	0.779311i	$-0.715569\pi$
$61$	−9.78838	−1.25327	−0.626637	+	0.779311i	$0.715569\pi$
$62$	−25.2438	−3.20597
$63$	−3.98798	−0.502438
$64$	43.5531	5.44414
$65$	−3.53830	−0.438872
$66$	−0.328966	−0.0404929
$67$	−3.38397	−0.413418	−0.206709	−	0.978402i	$-0.566275\pi$
$67$	−3.38397	−0.413418	−0.206709	+	0.978402i	$0.566275\pi$
$68$	41.3830	5.01843
$69$	0.0417087	0.00502113
$70$	13.1247	1.56871
$71$	−16.2230	−1.92531	−0.962655	−	0.270731i	$-0.912735\pi$
$71$	−16.2230	−1.92531	−0.962655	+	0.270731i	$0.912735\pi$
$72$	31.4196	3.70284
$73$	4.45188	0.521053	0.260527	−	0.965467i	$-0.416104\pi$
$73$	4.45188	0.521053	0.260527	+	0.965467i	$0.416104\pi$
$74$	32.7119	3.80269
$75$	−0.491842	−0.0567931
$76$	5.29210	0.607046
$77$	−2.40285	−0.273830
$78$	0.182254	0.0206362
$79$	−9.89386	−1.11315	−0.556573	−	0.830799i	$-0.687884\pi$
$79$	−9.89386	−1.11315	−0.556573	+	0.830799i	$0.687884\pi$
$80$	−62.6144	−7.00050
$81$	8.96151	0.995724
$82$	5.63924	0.622750
$83$	8.80540	0.966518	0.483259	−	0.875477i	$-0.339453\pi$
$83$	8.80540	0.966518	0.483259	+	0.875477i	$0.339453\pi$
$84$	−0.501894	−0.0547612
$85$	−25.4032	−2.75537
$86$	15.9954	1.72483
$87$	−0.0669470	−0.00717747
$88$	18.9310	2.01805
$89$	−4.86483	−0.515671	−0.257835	−	0.966189i	$-0.583009\pi$
$89$	−4.86483	−0.515671	−0.257835	+	0.966189i	$0.583009\pi$
$90$	−29.5352	−3.11329
$91$	1.33123	0.139550
$92$	−3.67554	−0.383202
$93$	−0.592575	−0.0614472
$94$	−33.0853	−3.41249
$95$	−3.24859	−0.333298
$96$	1.85317	0.189139
$97$	6.61086	0.671231	0.335615	−	0.941999i	$-0.391056\pi$
$97$	6.61086	0.671231	0.335615	+	0.941999i	$0.391056\pi$
$98$	14.5669	1.47148
$99$	5.40724	0.543448
$100$	43.3432	4.33432
$101$	8.78721	0.874360	0.437180	−	0.899374i	$-0.355977\pi$
$101$	8.78721	0.874360	0.437180	+	0.899374i	$0.355977\pi$
$102$	1.30849	0.129560
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−10.4882	−1.02845
$105$	0.308091	0.0300666
$106$	−30.1895	−2.93226
$107$	−7.42412	−0.717717	−0.358858	−	0.933392i	$-0.616834\pi$
$107$	−7.42412	−0.717717	−0.358858	+	0.933392i	$0.616834\pi$
$108$	2.26049	0.217516
$109$	2.14707	0.205652	0.102826	−	0.994699i	$-0.467211\pi$
$109$	2.14707	0.205652	0.102826	+	0.994699i	$0.467211\pi$
$110$	−17.7956	−1.69675
$111$	0.767883	0.0728842
$112$	23.5576	2.22598
$113$	9.34048	0.878679	0.439339	−	0.898321i	$-0.355213\pi$
$113$	9.34048	0.878679	0.439339	+	0.898321i	$0.355213\pi$
$114$	0.167331	0.0156720
$115$	2.25625	0.210397
$116$	5.89964	0.547768
$117$	−2.99572	−0.276955
$118$	9.30920	0.856982
$119$	9.55754	0.876138
$120$	−2.42732	−0.221583
$121$	−7.74202	−0.703820
$122$	27.2744	2.46931
$123$	0.132376	0.0119360
$124$	52.2202	4.68951
$125$	−8.91501	−0.797383
$126$	11.1121	0.989947
$127$	−17.3719	−1.54150	−0.770752	−	0.637135i	$-0.780119\pi$
$127$	−17.3719	−1.54150	−0.770752	+	0.637135i	$0.780119\pi$
$128$	−64.6919	−5.71801
$129$	0.375478	0.0330590
$130$	9.85914	0.864704
$131$	−11.1450	−0.973743	−0.486871	−	0.873474i	$-0.661862\pi$
$131$	−11.1450	−0.973743	−0.486871	+	0.873474i	$0.661862\pi$
$132$	0.680511	0.0592309
$133$	1.22223	0.105981
$134$	9.42911	0.814551
$135$	−1.38762	−0.119427
$136$	−75.2998	−6.45691
$137$	−12.7354	−1.08806	−0.544031	−	0.839065i	$-0.683103\pi$
$137$	−12.7354	−1.08806	−0.544031	+	0.839065i	$0.683103\pi$
$138$	−0.116217	−0.00989307
$139$	−9.77489	−0.829096	−0.414548	−	0.910028i	$-0.636060\pi$
$139$	−9.77489	−0.829096	−0.414548	+	0.910028i	$0.636060\pi$
$140$	−27.1503	−2.29462
$141$	−0.776647	−0.0654055
$142$	45.2037	3.79341
$143$	−1.80499	−0.150941
$144$	−53.0128	−4.41774
$145$	−3.62154	−0.300752
$146$	−12.4047	−1.02662
$147$	0.341944	0.0282031
$148$	−67.6690	−5.56236
$149$	−15.8843	−1.30130	−0.650648	−	0.759379i	$-0.725503\pi$
$149$	−15.8843	−1.30130	−0.650648	+	0.759379i	$0.725503\pi$
$150$	1.37047	0.111899
$151$	−16.4335	−1.33734	−0.668668	−	0.743561i	$-0.733135\pi$
$151$	−16.4335	−1.33734	−0.668668	+	0.743561i	$0.733135\pi$
$152$	−9.62942	−0.781049
$153$	−21.5078	−1.73880
$154$	6.69530	0.539523
$155$	−32.0557	−2.57478
$156$	−0.377017	−0.0301855
$157$	−0.547772	−0.0437170	−0.0218585	−	0.999761i	$-0.506958\pi$
$157$	−0.547772	−0.0437170	−0.0218585	+	0.999761i	$0.506958\pi$
$158$	27.5683	2.19322
$159$	−0.708670	−0.0562012
$160$	100.248	7.92534
$161$	−0.848877	−0.0669009
$162$	−24.9704	−1.96186
$163$	−8.69044	−0.680688	−0.340344	−	0.940301i	$-0.610544\pi$
$163$	−8.69044	−0.680688	−0.340344	+	0.940301i	$0.610544\pi$
$164$	−11.6655	−0.910925
$165$	−0.417736	−0.0325207
$166$	−24.5354	−1.90432
$167$	5.47091	0.423352	0.211676	−	0.977340i	$-0.432108\pi$
$167$	5.47091	0.423352	0.211676	+	0.977340i	$0.432108\pi$
$168$	0.913238	0.0704579
$169$	1.00000	0.0769231
$170$	70.7837	5.42886
$171$	−2.75044	−0.210331
$172$	−33.0887	−2.52299
$173$	−4.52994	−0.344405	−0.172202	−	0.985062i	$-0.555088\pi$
$173$	−4.52994	−0.344405	−0.172202	+	0.985062i	$0.555088\pi$
$174$	0.186541	0.0141417
$175$	10.0102	0.756703
$176$	−31.9414	−2.40767
$177$	0.218525	0.0164253
$178$	13.5554	1.01602
$179$	−21.9894	−1.64357	−0.821783	−	0.569800i	$-0.807021\pi$
$179$	−21.9894	−1.64357	−0.821783	+	0.569800i	$0.807021\pi$
$180$	61.0976	4.55395
$181$	−8.32061	−0.618466	−0.309233	−	0.950986i	$-0.600072\pi$
$181$	−8.32061	−0.618466	−0.309233	+	0.950986i	$0.600072\pi$
$182$	−3.70933	−0.274954
$183$	0.640241	0.0473280
$184$	6.68795	0.493042
$185$	41.5391	3.05401
$186$	1.65115	0.121069
$187$	−12.9589	−0.947650
$188$	68.4414	4.99160
$189$	0.522067	0.0379748
$190$	9.05189	0.656693
$191$	−8.31127	−0.601382	−0.300691	−	0.953722i	$-0.597217\pi$
$191$	−8.31127	−0.601382	−0.300691	+	0.953722i	$0.597217\pi$
$192$	−2.84874	−0.205590
$193$	1.00001	0.0719823	0.0359912	−	0.999352i	$-0.488541\pi$
$193$	1.00001	0.0719823	0.0359912	+	0.999352i	$0.488541\pi$
$194$	−18.4205	−1.32252
$195$	0.231434	0.0165734
$196$	−30.1335	−2.15240
$197$	−6.60977	−0.470926	−0.235463	−	0.971883i	$-0.575661\pi$
$197$	−6.60977	−0.470926	−0.235463	+	0.971883i	$0.575661\pi$
$198$	−15.0668	−1.07075
$199$	19.0442	1.35000	0.675002	−	0.737816i	$-0.264143\pi$
$199$	19.0442	1.35000	0.675002	+	0.737816i	$0.264143\pi$
$200$	−78.8666	−5.57671
$201$	0.221340	0.0156121
$202$	−24.4847	−1.72274
$203$	1.36254	0.0956316
$204$	−2.70679	−0.189513
$205$	7.16096	0.500143
$206$	−2.78640	−0.194138
$207$	1.91027	0.132773
$208$	17.6962	1.22701
$209$	−1.65720	−0.114631
$210$	−0.858467	−0.0592398
$211$	15.3899	1.05949	0.529743	−	0.848158i	$-0.322288\pi$
$211$	15.3899	1.05949	0.529743	+	0.848158i	$0.322288\pi$
$212$	62.4510	4.28915
$213$	1.06112	0.0727064
$214$	20.6866	1.41411
$215$	20.3117	1.38525
$216$	−4.11314	−0.279864
$217$	12.0604	0.818714
$218$	−5.98262	−0.405194
$219$	−0.291190	−0.0196768
$220$	36.8126	2.48191
$221$	7.17950	0.482946
$222$	−2.13963	−0.143603
$223$	14.8179	0.992282	0.496141	−	0.868242i	$-0.334750\pi$
$223$	14.8179	0.992282	0.496141	+	0.868242i	$0.334750\pi$
$224$	−37.7168	−2.52006
$225$	−22.5265	−1.50177
$226$	−26.0264	−1.73125
$227$	0.574467	0.0381287	0.0190643	−	0.999818i	$-0.493931\pi$
$227$	0.574467	0.0381287	0.0190643	+	0.999818i	$0.493931\pi$
$228$	−0.346147	−0.0229242
$229$	−12.3210	−0.814193	−0.407096	−	0.913385i	$-0.633459\pi$
$229$	−12.3210	−0.814193	−0.407096	+	0.913385i	$0.633459\pi$
$230$	−6.28684	−0.414542
$231$	0.157166	0.0103408
$232$	−10.7349	−0.704780
$233$	9.74779	0.638599	0.319299	−	0.947654i	$-0.396552\pi$
$233$	9.74779	0.638599	0.319299	+	0.947654i	$0.396552\pi$
$234$	8.34729	0.545680
$235$	−42.0132	−2.74064
$236$	−19.2573	−1.25355
$237$	0.647141	0.0420363
$238$	−26.6312	−1.72624
$239$	1.08544	0.0702116	0.0351058	−	0.999384i	$-0.488823\pi$
$239$	1.08544	0.0702116	0.0351058	+	0.999384i	$0.488823\pi$
$240$	4.09550	0.264364
$241$	1.94784	0.125472	0.0627358	−	0.998030i	$-0.480017\pi$
$241$	1.94784	0.125472	0.0627358	+	0.998030i	$0.480017\pi$
$242$	21.5724	1.38673
$243$	−1.76267	−0.113075
$244$	−56.4207	−3.61197
$245$	18.4977	1.18177
$246$	−0.368853	−0.0235172
$247$	0.918122	0.0584187
$248$	−95.0190	−6.03371
$249$	−0.575946	−0.0364991
$250$	24.8408	1.57107
$251$	2.19956	0.138835	0.0694175	−	0.997588i	$-0.477886\pi$
$251$	2.19956	0.138835	0.0694175	+	0.997588i	$0.477886\pi$
$252$	−22.9869	−1.44804
$253$	1.15098	0.0723615
$254$	48.4051	3.03720
$255$	1.66158	0.104052
$256$	93.1514	5.82197
$257$	−10.2227	−0.637675	−0.318837	−	0.947809i	$-0.603292\pi$
$257$	−10.2227	−0.637675	−0.318837	+	0.947809i	$0.603292\pi$
$258$	−1.04623	−0.0651357
$259$	−15.6284	−0.971099
$260$	−20.3949	−1.26484
$261$	−3.06619	−0.189793
$262$	31.0545	1.91855
$263$	−22.7437	−1.40244	−0.701218	−	0.712947i	$-0.747360\pi$
$263$	−22.7437	−1.40244	−0.701218	+	0.712947i	$0.747360\pi$
$264$	−1.23825	−0.0762088
$265$	−38.3359	−2.35496
$266$	−3.40562	−0.208812
$267$	0.318200	0.0194735
$268$	−19.5054	−1.19148
$269$	−18.3394	−1.11817	−0.559085	−	0.829110i	$-0.688847\pi$
$269$	−18.3394	−1.11817	−0.559085	+	0.829110i	$0.688847\pi$
$270$	3.86646	0.235305
$271$	9.15705	0.556251	0.278126	−	0.960545i	$-0.410287\pi$
$271$	9.15705	0.556251	0.278126	+	0.960545i	$0.410287\pi$
$272$	127.050	7.70352
$273$	−0.0870732	−0.00526991
$274$	35.4861	2.14379
$275$	−13.5727	−0.818467
$276$	0.240411	0.0144710
$277$	22.3100	1.34048	0.670238	−	0.742146i	$-0.266192\pi$
$277$	22.3100	1.34048	0.670238	+	0.742146i	$0.266192\pi$
$278$	27.2368	1.63356
$279$	−27.1401	−1.62484
$280$	49.4022	2.95234
$281$	31.5807	1.88395	0.941973	−	0.335689i	$-0.108969\pi$
$281$	31.5807	1.88395	0.941973	+	0.335689i	$0.108969\pi$
$282$	2.16405	0.128867
$283$	−16.7272	−0.994330	−0.497165	−	0.867656i	$-0.665626\pi$
$283$	−16.7272	−0.994330	−0.497165	+	0.867656i	$0.665626\pi$
$284$	−93.5099	−5.54879
$285$	0.212485	0.0125865
$286$	5.02943	0.297396
$287$	−2.69419	−0.159033
$288$	84.8759	5.00136
$289$	34.5453	2.03207
$290$	10.0911	0.592568
$291$	−0.432405	−0.0253480
$292$	25.6609	1.50169
$293$	−5.92181	−0.345956	−0.172978	−	0.984926i	$-0.555339\pi$
$293$	−5.92181	−0.345956	−0.172978	+	0.984926i	$0.555339\pi$
$294$	−0.952795	−0.0555681
$295$	11.8212	0.688259
$296$	123.129	7.15675
$297$	−0.707862	−0.0410743
$298$	44.2602	2.56392
$299$	−0.637666	−0.0368772
$300$	−2.83501	−0.163679
$301$	−7.64194	−0.440474
$302$	45.7903	2.63493
$303$	−0.574756	−0.0330189
$304$	16.2472	0.931844
$305$	34.6342	1.98315
$306$	59.9294	3.42594
$307$	15.8592	0.905130	0.452565	−	0.891731i	$-0.350509\pi$
$307$	15.8592	0.905130	0.452565	+	0.891731i	$0.350509\pi$
$308$	−13.8501	−0.789184
$309$	−0.0654083	−0.00372095
$310$	89.3202	5.07305
$311$	−10.4271	−0.591269	−0.295635	−	0.955301i	$-0.595531\pi$
$311$	−10.4271	−0.591269	−0.295635	+	0.955301i	$0.595531\pi$
$312$	0.686013	0.0388379
$313$	12.3026	0.695384	0.347692	−	0.937609i	$-0.386966\pi$
$313$	12.3026	0.695384	0.347692	+	0.937609i	$0.386966\pi$
$314$	1.52632	0.0861350
$315$	14.1107	0.795046
$316$	−57.0287	−3.20812
$317$	−8.28607	−0.465392	−0.232696	−	0.972550i	$-0.574755\pi$
$317$	−8.28607	−0.465392	−0.232696	+	0.972550i	$0.574755\pi$
$318$	1.97464	0.110732
$319$	−1.84745	−0.103437
$320$	−154.104	−8.61468
$321$	0.485599	0.0271035
$322$	2.36532	0.131814
$323$	6.59166	0.366770
$324$	51.6546	2.86970
$325$	7.51957	0.417111
$326$	24.2151	1.34115
$327$	−0.140436	−0.00776615
$328$	21.2264	1.17203
$329$	15.8067	0.871454
$330$	1.16398	0.0640751
$331$	−22.5809	−1.24116	−0.620581	−	0.784143i	$-0.713103\pi$
$331$	−22.5809	−1.24116	−0.620581	+	0.784143i	$0.713103\pi$
$332$	50.7548	2.78553
$333$	35.1693	1.92726
$334$	−15.2442	−0.834124
$335$	11.9735	0.654182
$336$	−1.54086	−0.0840610
$337$	−12.3813	−0.674451	−0.337226	−	0.941424i	$-0.609488\pi$
$337$	−12.3813	−0.674451	−0.337226	+	0.941424i	$0.609488\pi$
$338$	−2.78640	−0.151560
$339$	−0.610945	−0.0331820
$340$	−146.426	−7.94104
$341$	−16.3525	−0.885539
$342$	7.66383	0.414413
$343$	−16.2780	−0.878930
$344$	60.2076	3.24618
$345$	−0.147578	−0.00794532
$346$	12.6222	0.678576
$347$	14.6716	0.787611	0.393805	−	0.919194i	$-0.371158\pi$
$347$	14.6716	0.787611	0.393805	+	0.919194i	$0.371158\pi$
$348$	−0.385886	−0.0206856
$349$	−14.1875	−0.759437	−0.379718	−	0.925102i	$-0.623979\pi$
$349$	−14.1875	−0.759437	−0.379718	+	0.925102i	$0.623979\pi$
$350$	−27.8926	−1.49092
$351$	0.392170	0.0209325
$352$	51.1396	2.72575
$353$	−15.0041	−0.798589	−0.399295	−	0.916823i	$-0.630745\pi$
$353$	−15.0041	−0.798589	−0.399295	+	0.916823i	$0.630745\pi$
$354$	−0.608899	−0.0323626
$355$	57.4017	3.04656
$356$	−28.0411	−1.48618
$357$	−0.625142	−0.0330860
$358$	61.2714	3.23830
$359$	−18.3042	−0.966060	−0.483030	−	0.875604i	$-0.660464\pi$
$359$	−18.3042	−0.966060	−0.483030	+	0.875604i	$0.660464\pi$
$360$	−111.172	−5.85929
$361$	−18.1571	−0.955634
$362$	23.1846	1.21856
$363$	0.506392	0.0265787
$364$	7.67325	0.402188
$365$	−15.7521	−0.824502
$366$	−1.78397	−0.0932497
$367$	14.2698	0.744877	0.372438	−	0.928057i	$-0.378522\pi$
$367$	14.2698	0.744877	0.372438	+	0.928057i	$0.378522\pi$
$368$	−11.2843	−0.588233
$369$	6.06287	0.315620
$370$	−115.745	−6.01728
$371$	14.4232	0.748817
$372$	−3.41563	−0.177092
$373$	−19.9115	−1.03098	−0.515490	−	0.856896i	$-0.672390\pi$
$373$	−19.9115	−1.03098	−0.515490	+	0.856896i	$0.672390\pi$
$374$	36.1088	1.86714
$375$	0.583115	0.0301120
$376$	−124.535	−6.42239
$377$	1.02352	0.0527142
$378$	−1.45469	−0.0748211
$379$	9.41319	0.483523	0.241762	−	0.970336i	$-0.422275\pi$
$379$	9.41319	0.483523	0.241762	+	0.970336i	$0.422275\pi$
$380$	−18.7250	−0.960574
$381$	1.13626	0.0582126
$382$	23.1586	1.18489
$383$	0.438820	0.0224226	0.0112113	−	0.999937i	$-0.496431\pi$
$383$	0.438820	0.0224226	0.0112113	+	0.999937i	$0.496431\pi$
$384$	4.23138	0.215932
$385$	8.50199	0.433302
$386$	−2.78643	−0.141826
$387$	17.1970	0.874174
$388$	38.1053	1.93450
$389$	−15.7292	−0.797501	−0.398751	−	0.917059i	$-0.630556\pi$
$389$	−15.7292	−0.797501	−0.398751	+	0.917059i	$0.630556\pi$
$390$	−0.644869	−0.0326542
$391$	−4.57813	−0.231526
$392$	54.8305	2.76936
$393$	0.728975	0.0367719
$394$	18.4175	0.927860
$395$	35.0075	1.76142
$396$	31.1676	1.56623
$397$	−3.16925	−0.159060	−0.0795299	−	0.996832i	$-0.525342\pi$
$397$	−3.16925	−0.159060	−0.0795299	+	0.996832i	$0.525342\pi$
$398$	−53.0647	−2.65989
$399$	−0.0799438	−0.00400219
$400$	133.068	6.65339
$401$	17.8971	0.893739	0.446869	−	0.894599i	$-0.352539\pi$
$401$	17.8971	0.893739	0.446869	+	0.894599i	$0.352539\pi$
$402$	−0.616742	−0.0307603
$403$	9.05964	0.451293
$404$	50.6499	2.51993
$405$	−31.7085	−1.57561
$406$	−3.79659	−0.188422
$407$	21.1903	1.05036
$408$	4.92523	0.243835
$409$	−16.4254	−0.812183	−0.406092	−	0.913832i	$-0.633109\pi$
$409$	−16.4254	−0.812183	−0.406092	+	0.913832i	$0.633109\pi$
$410$	−19.9533	−0.985425
$411$	0.833004	0.0410891
$412$	5.76405	0.283974
$413$	−4.44754	−0.218849
$414$	−5.32279	−0.261601
$415$	−31.1561	−1.52940
$416$	−28.3324	−1.38911
$417$	0.639359	0.0313096
$418$	4.61763	0.225855
$419$	−21.0568	−1.02869	−0.514347	−	0.857582i	$-0.671966\pi$
$419$	−21.0568	−1.02869	−0.514347	+	0.857582i	$0.671966\pi$
$420$	1.77585	0.0866527
$421$	26.3930	1.28631	0.643157	−	0.765734i	$-0.277624\pi$
$421$	26.3930	1.28631	0.643157	+	0.765734i	$0.277624\pi$
$422$	−42.8826	−2.08749
$423$	−35.5707	−1.72951
$424$	−113.635	−5.51859
$425$	53.9868	2.61874
$426$	−2.95670	−0.143252
$427$	−13.0305	−0.630592
$428$	−42.7930	−2.06848
$429$	0.118061	0.00570005
$430$	−56.5967	−2.72933
$431$	−4.16564	−0.200652	−0.100326	−	0.994955i	$-0.531989\pi$
$431$	−4.16564	−0.200652	−0.100326	+	0.994955i	$0.531989\pi$
$432$	6.93991	0.333897
$433$	−26.5117	−1.27407	−0.637035	−	0.770835i	$-0.719839\pi$
$433$	−26.5117	−1.27407	−0.637035	+	0.770835i	$0.719839\pi$
$434$	−33.6052	−1.61310
$435$	0.236879	0.0113575
$436$	12.3758	0.592695
$437$	−0.585455	−0.0280061
$438$	0.811373	0.0387689
$439$	9.49517	0.453180	0.226590	−	0.973990i	$-0.427242\pi$
$439$	9.49517	0.453180	0.226590	+	0.973990i	$0.427242\pi$
$440$	−66.9836	−3.19332
$441$	15.6612	0.745769
$442$	−20.0050	−0.951541
$443$	15.0752	0.716247	0.358123	−	0.933674i	$-0.383417\pi$
$443$	15.0752	0.716247	0.358123	+	0.933674i	$0.383417\pi$
$444$	4.42612	0.210054
$445$	17.2132	0.815985
$446$	−41.2887	−1.95508
$447$	1.03897	0.0491415
$448$	57.9790	2.73925
$449$	−17.2207	−0.812694	−0.406347	−	0.913719i	$-0.633198\pi$
$449$	−17.2207	−0.812694	−0.406347	+	0.913719i	$0.633198\pi$
$450$	62.7681	2.95892
$451$	3.65301	0.172014
$452$	53.8390	2.53237
$453$	1.07488	0.0505025
$454$	−1.60070	−0.0751244
$455$	−4.71028	−0.220821
$456$	0.629844	0.0294951
$457$	5.47975	0.256332	0.128166	−	0.991753i	$-0.459091\pi$
$457$	5.47975	0.256332	0.128166	+	0.991753i	$0.459091\pi$
$458$	34.3312	1.60419
$459$	2.81559	0.131420
$460$	13.0052	0.606369
$461$	−7.11942	−0.331584	−0.165792	−	0.986161i	$-0.553018\pi$
$461$	−7.11942	−0.331584	−0.165792	+	0.986161i	$0.553018\pi$
$462$	−0.437928	−0.0203743
$463$	9.10770	0.423271	0.211635	−	0.977349i	$-0.432121\pi$
$463$	9.10770	0.423271	0.211635	+	0.977349i	$0.432121\pi$
$464$	18.1125	0.840850
$465$	2.09671	0.0972326
$466$	−27.1613	−1.25822
$467$	−11.3038	−0.523075	−0.261538	−	0.965193i	$-0.584230\pi$
$467$	−11.3038	−0.523075	−0.261538	+	0.965193i	$0.584230\pi$
$468$	−17.2675	−0.798190
$469$	−4.50483	−0.208014
$470$	117.066	5.39984
$471$	0.0358289	0.00165091
$472$	35.0403	1.61286
$473$	10.3616	0.476426
$474$	−1.80320	−0.0828235
$475$	6.90388	0.316772
$476$	55.0901	2.52505
$477$	−32.4573	−1.48612
$478$	−3.02449	−0.138337
$479$	−31.9615	−1.46036	−0.730178	−	0.683257i	$-0.760563\pi$
$479$	−31.9615	−1.46036	−0.730178	+	0.683257i	$0.760563\pi$
$480$	−6.55708	−0.299289
$481$	−11.7398	−0.535290
$482$	−5.42748	−0.247215
$483$	0.0555236	0.00252641
$484$	−44.6254	−2.02843
$485$	−23.3912	−1.06214
$486$	4.91150	0.222790
$487$	3.48184	0.157777	0.0788885	−	0.996883i	$-0.474863\pi$
$487$	3.48184	0.157777	0.0788885	+	0.996883i	$0.474863\pi$
$488$	102.662	4.64730
$489$	0.568427	0.0257052
$490$	−51.5420	−2.32843
$491$	21.7494	0.981537	0.490769	−	0.871290i	$-0.336716\pi$
$491$	21.7494	0.981537	0.490769	+	0.871290i	$0.336716\pi$
$492$	0.763023	0.0343997
$493$	7.34839	0.330955
$494$	−2.55826	−0.115101
$495$	−19.1324	−0.859940
$496$	160.321	7.19862
$497$	−21.5964	−0.968731
$498$	1.60482	0.0719137
$499$	30.5296	1.36669	0.683346	−	0.730095i	$-0.260524\pi$
$499$	30.5296	1.36669	0.683346	+	0.730095i	$0.260524\pi$
$500$	−51.3866	−2.29808
$501$	−0.357843	−0.0159872
$502$	−6.12887	−0.273545
$503$	43.6072	1.94435	0.972173	−	0.234262i	$-0.0752674\pi$
$503$	43.6072	1.94435	0.972173	+	0.234262i	$0.0752674\pi$
$504$	41.8266	1.86311
$505$	−31.0918	−1.38357
$506$	−3.20710	−0.142573
$507$	−0.0654083	−0.00290488
$508$	−100.132	−4.44265
$509$	−27.1180	−1.20199	−0.600993	−	0.799254i	$-0.705228\pi$
$509$	−27.1180	−1.20199	−0.600993	+	0.799254i	$0.705228\pi$
$510$	−4.62984	−0.205013
$511$	5.92645	0.262171
$512$	−130.174	−5.75293
$513$	0.360060	0.0158970
$514$	28.4846	1.25640
$515$	−3.53830	−0.155916
$516$	2.16428	0.0952769
$517$	−21.4321	−0.942584
$518$	43.5470	1.91334
$519$	0.296296	0.0130059
$520$	37.1103	1.62739
$521$	−22.1817	−0.971797	−0.485898	−	0.874015i	$-0.661508\pi$
$521$	−22.1817	−0.971797	−0.485898	+	0.874015i	$0.661508\pi$
$522$	8.54366	0.373946
$523$	−4.45541	−0.194821	−0.0974107	−	0.995244i	$-0.531056\pi$
$523$	−4.45541	−0.194821	−0.0974107	+	0.995244i	$0.531056\pi$
$524$	−64.2403	−2.80635
$525$	−0.654753	−0.0285758
$526$	63.3732	2.76320
$527$	65.0437	2.83335
$528$	2.08923	0.0909222
$529$	−22.5934	−0.982321
$530$	106.819	4.63994
$531$	10.0085	0.434333
$532$	7.04498	0.305439
$533$	−2.02384	−0.0876624
$534$	−0.886634	−0.0383684
$535$	26.2688	1.13570
$536$	35.4916	1.53301
$537$	1.43829	0.0620668
$538$	51.1009	2.20311
$539$	9.43619	0.406445
$540$	−7.99829	−0.344191
$541$	14.5908	0.627309	0.313654	−	0.949537i	$-0.398447\pi$
$541$	14.5908	0.627309	0.313654	+	0.949537i	$0.398447\pi$
$542$	−25.5153	−1.09597
$543$	0.544237	0.0233555
$544$	−203.412	−8.72124
$545$	−7.59699	−0.325419
$546$	0.242621	0.0103832
$547$	8.16760	0.349221	0.174611	−	0.984638i	$-0.444133\pi$
$547$	8.16760	0.349221	0.174611	+	0.984638i	$0.444133\pi$
$548$	−73.4078	−3.13582
$549$	29.3233	1.25149
$550$	37.8191	1.61261
$551$	0.939720	0.0400334
$552$	−0.437448	−0.0186190
$553$	−13.1710	−0.560086
$554$	−62.1646	−2.64112
$555$	−2.71700	−0.115330
$556$	−56.3430	−2.38947
$557$	−10.9549	−0.464175	−0.232087	−	0.972695i	$-0.574556\pi$
$557$	−10.9549	−0.464175	−0.232087	+	0.972695i	$0.574556\pi$
$558$	75.6234	3.20140
$559$	−5.74053	−0.242799
$560$	−83.3539	−3.52234
$561$	0.847621	0.0357866
$562$	−87.9966	−3.71191
$563$	34.6599	1.46074	0.730371	−	0.683051i	$-0.239347\pi$
$563$	34.6599	1.46074	0.730371	+	0.683051i	$0.239347\pi$
$564$	−4.47663	−0.188500
$565$	−33.0494	−1.39040
$566$	46.6088	1.95911
$567$	11.9298	0.501004
$568$	170.149	7.13930
$569$	8.64713	0.362506	0.181253	−	0.983436i	$-0.441985\pi$
$569$	8.64713	0.362506	0.181253	+	0.983436i	$0.441985\pi$
$570$	−0.592069	−0.0247990
$571$	1.96618	0.0822821	0.0411410	−	0.999153i	$-0.486901\pi$
$571$	1.96618	0.0822821	0.0411410	+	0.999153i	$0.486901\pi$
$572$	−10.4040	−0.435015
$573$	0.543626	0.0227103
$574$	7.50711	0.313341
$575$	−4.79498	−0.199964
$576$	−130.473	−5.43638
$577$	−5.06916	−0.211032	−0.105516	−	0.994418i	$-0.533649\pi$
$577$	−5.06916	−0.211032	−0.105516	+	0.994418i	$0.533649\pi$
$578$	−96.2571	−4.00377
$579$	−0.0654090	−0.00271830
$580$	−20.8747	−0.866775
$581$	11.7220	0.486309
$582$	1.20486	0.0499428
$583$	−19.5563	−0.809937
$584$	−46.6921	−1.93213
$585$	10.5998	0.438246
$586$	16.5006	0.681632
$587$	−17.3636	−0.716672	−0.358336	−	0.933593i	$-0.616656\pi$
$587$	−17.3636	−0.716672	−0.358336	+	0.933593i	$0.616656\pi$
$588$	1.97098	0.0812820
$589$	8.31785	0.342731
$590$	−32.9388	−1.35607
$591$	0.432334	0.0177838
$592$	−207.750	−8.53848
$593$	13.3155	0.546800	0.273400	−	0.961900i	$-0.411852\pi$
$593$	13.3155	0.546800	0.273400	+	0.961900i	$0.411852\pi$
$594$	1.97239	0.0809282
$595$	−33.8174	−1.38638
$596$	−91.5582	−3.75037
$597$	−1.24565	−0.0509809
$598$	1.77680	0.0726586
$599$	−0.954701	−0.0390080	−0.0195040	−	0.999810i	$-0.506209\pi$
$599$	−0.954701	−0.0390080	−0.0195040	+	0.999810i	$0.506209\pi$
$600$	5.15853	0.210596
$601$	21.6586	0.883471	0.441736	−	0.897145i	$-0.354363\pi$
$601$	21.6586	0.883471	0.441736	+	0.897145i	$0.354363\pi$
$602$	21.2935	0.867860
$603$	10.1374	0.412828
$604$	−94.7233	−3.85424
$605$	27.3936	1.11371
$606$	1.60150	0.0650567
$607$	−21.4926	−0.872359	−0.436180	−	0.899860i	$-0.643669\pi$
$607$	−21.4926	−0.872359	−0.436180	+	0.899860i	$0.643669\pi$
$608$	−26.0126	−1.05495
$609$	−0.0891215	−0.00361139
$610$	−96.5050	−3.90737
$611$	11.8738	0.480364
$612$	−123.972	−5.01127
$613$	−43.5689	−1.75973	−0.879866	−	0.475222i	$-0.842368\pi$
$613$	−43.5689	−1.75973	−0.879866	+	0.475222i	$0.842368\pi$
$614$	−44.1900	−1.78337
$615$	−0.468387	−0.0188872
$616$	25.2015	1.01540
$617$	43.4400	1.74883	0.874415	−	0.485179i	$-0.161245\pi$
$617$	43.4400	1.74883	0.874415	+	0.485179i	$0.161245\pi$
$618$	0.182254	0.00733133
$619$	25.6581	1.03129	0.515643	−	0.856804i	$-0.327553\pi$
$619$	25.6581	1.03129	0.515643	+	0.856804i	$0.327553\pi$
$620$	−184.771	−7.42057
$621$	−0.250074	−0.0100351
$622$	29.0542	1.16497
$623$	−6.47618	−0.259463
$624$	−1.15748	−0.0463362
$625$	−6.05389	−0.242155
$626$	−34.2800	−1.37010
$627$	0.108395	0.00432886
$628$	−3.15739	−0.125994
$629$	−84.2862	−3.36071
$630$	−39.3181	−1.56647
$631$	−21.4182	−0.852645	−0.426323	−	0.904571i	$-0.640191\pi$
$631$	−21.4182	−0.852645	−0.426323	+	0.904571i	$0.640191\pi$
$632$	103.768	4.12769
$633$	−1.00663	−0.0400099
$634$	23.0883	0.916955
$635$	61.4669	2.43924
$636$	−4.08481	−0.161973
$637$	−5.22784	−0.207135
$638$	5.14774	0.203801
$639$	48.5994	1.92256
$640$	228.899	9.04804
$641$	−33.5530	−1.32526	−0.662632	−	0.748945i	$-0.730561\pi$
$641$	−33.5530	−1.32526	−0.662632	+	0.748945i	$0.730561\pi$
$642$	−1.35308	−0.0534016
$643$	42.4016	1.67216	0.836079	−	0.548610i	$-0.184843\pi$
$643$	42.4016	1.67216	0.836079	+	0.548610i	$0.184843\pi$
$644$	−4.89297	−0.192810
$645$	−1.32856	−0.0523118
$646$	−18.3670	−0.722641
$647$	−6.49974	−0.255531	−0.127766	−	0.991804i	$-0.540781\pi$
$647$	−6.49974	−0.255531	−0.127766	+	0.991804i	$0.540781\pi$
$648$	−93.9899	−3.69227
$649$	6.03035	0.236712
$650$	−20.9526	−0.821828
$651$	−0.788851	−0.0309175
$652$	−50.0922	−1.96176
$653$	−26.3402	−1.03077	−0.515386	−	0.856958i	$-0.672352\pi$
$653$	−26.3402	−1.03077	−0.515386	+	0.856958i	$0.672352\pi$
$654$	0.391313	0.0153015
$655$	39.4344	1.54083
$656$	−35.8143	−1.39831
$657$	−13.3366	−0.520310
$658$	−44.0440	−1.71701
$659$	13.9867	0.544845	0.272422	−	0.962178i	$-0.412175\pi$
$659$	13.9867	0.544845	0.272422	+	0.962178i	$0.412175\pi$
$660$	−2.40785	−0.0937255
$661$	22.4517	0.873269	0.436635	−	0.899639i	$-0.356170\pi$
$661$	22.4517	0.873269	0.436635	+	0.899639i	$0.356170\pi$
$662$	62.9197	2.44544
$663$	−0.469599	−0.0182377
$664$	−92.3525	−3.58397
$665$	−4.32461	−0.167701
$666$	−97.9959	−3.79726
$667$	−0.652667	−0.0252714
$668$	31.5346	1.22011
$669$	−0.969216	−0.0374720
$670$	−33.3630	−1.28893
$671$	17.6679	0.682062
$672$	2.46699	0.0951662
$673$	24.5953	0.948079	0.474040	−	0.880503i	$-0.342795\pi$
$673$	24.5953	0.948079	0.474040	+	0.880503i	$0.342795\pi$
$674$	34.4992	1.32886
$675$	2.94895	0.113505
$676$	5.76405	0.221694
$677$	25.9958	0.999099	0.499550	−	0.866285i	$-0.333499\pi$
$677$	25.9958	0.999099	0.499550	+	0.866285i	$0.333499\pi$
$678$	1.70234	0.0653780
$679$	8.80054	0.337734
$680$	266.433	10.2173
$681$	−0.0375749	−0.00143987
$682$	45.5648	1.74477
$683$	−15.6788	−0.599933	−0.299966	−	0.953950i	$-0.596975\pi$
$683$	−15.6788	−0.599933	−0.299966	+	0.953950i	$0.596975\pi$
$684$	−15.8537	−0.606180
$685$	45.0619	1.72172
$686$	45.3571	1.73174
$687$	0.805894	0.0307468
$688$	−101.585	−3.87291
$689$	10.8346	0.412764
$690$	0.411211	0.0156546
$691$	32.0298	1.21847	0.609235	−	0.792989i	$-0.291476\pi$
$691$	32.0298	1.21847	0.609235	+	0.792989i	$0.291476\pi$
$692$	−26.1108	−0.992584
$693$	7.19826	0.273439
$694$	−40.8809	−1.55182
$695$	34.5865	1.31194
$696$	0.702151	0.0266150
$697$	−14.5302	−0.550370
$698$	39.5320	1.49631
$699$	−0.637586	−0.0241157
$700$	57.6996	2.18084
$701$	−3.25422	−0.122910	−0.0614550	−	0.998110i	$-0.519574\pi$
$701$	−3.25422	−0.122910	−0.0614550	+	0.998110i	$0.519574\pi$
$702$	−1.09274	−0.0412430
$703$	−10.7786	−0.406523
$704$	−78.6129	−2.96283
$705$	2.74801	0.103496
$706$	41.8076	1.57345
$707$	11.6978	0.439939
$708$	1.25959	0.0473383
$709$	−25.9095	−0.973052	−0.486526	−	0.873666i	$-0.661736\pi$
$709$	−25.9095	−0.973052	−0.486526	+	0.873666i	$0.661736\pi$
$710$	−159.944	−6.00260
$711$	29.6393	1.11156
$712$	51.0231	1.91217
$713$	−5.77702	−0.216351
$714$	1.74190	0.0651889
$715$	6.38659	0.238845
$716$	−126.748	−4.73680
$717$	−0.0709971	−0.00265143
$718$	51.0030	1.90341
$719$	−18.7541	−0.699408	−0.349704	−	0.936860i	$-0.613718\pi$
$719$	−18.7541	−0.699408	−0.349704	+	0.936860i	$0.613718\pi$
$720$	187.575	6.99052
$721$	1.33123	0.0495774
$722$	50.5929	1.88287
$723$	−0.127405	−0.00473824
$724$	−47.9604	−1.78244
$725$	7.69646	0.285839
$726$	−1.41101	−0.0523676
$727$	5.53685	0.205350	0.102675	−	0.994715i	$-0.467260\pi$
$727$	5.53685	0.205350	0.102675	+	0.994715i	$0.467260\pi$
$728$	−13.9621	−0.517470
$729$	−26.7692	−0.991454
$730$	43.8917	1.62450
$731$	−41.2141	−1.52436
$732$	3.69038	0.136401
$733$	18.0311	0.665994	0.332997	−	0.942928i	$-0.391940\pi$
$733$	18.0311	0.665994	0.332997	+	0.942928i	$0.391940\pi$
$734$	−39.7614	−1.46762
$735$	−1.20990	−0.0446279
$736$	18.0666	0.665944
$737$	6.10803	0.224992
$738$	−16.8936	−0.621862
$739$	13.0636	0.480553	0.240276	−	0.970705i	$-0.422762\pi$
$739$	13.0636	0.480553	0.240276	+	0.970705i	$0.422762\pi$
$740$	239.433	8.80174
$741$	−0.0600528	−0.00220609
$742$	−40.1890	−1.47538
$743$	−22.0956	−0.810611	−0.405305	−	0.914181i	$-0.632835\pi$
$743$	−22.0956	−0.810611	−0.405305	+	0.914181i	$0.632835\pi$
$744$	6.21503	0.227854
$745$	56.2036	2.05914
$746$	55.4816	2.03133
$747$	−26.3785	−0.965140
$748$	−74.6959	−2.73115
$749$	−9.88318	−0.361123
$750$	−1.62480	−0.0593291
$751$	16.2678	0.593620	0.296810	−	0.954937i	$-0.404077\pi$
$751$	16.2678	0.593620	0.296810	+	0.954937i	$0.404077\pi$
$752$	210.121	7.66234
$753$	−0.143870	−0.00524290
$754$	−2.85195	−0.103862
$755$	58.1465	2.11617
$756$	3.00922	0.109444
$757$	22.8199	0.829404	0.414702	−	0.909957i	$-0.363886\pi$
$757$	22.8199	0.829404	0.414702	+	0.909957i	$0.363886\pi$
$758$	−26.2290	−0.952679
$759$	−0.0752836	−0.00273262
$760$	34.0718	1.23591
$761$	48.1578	1.74572	0.872860	−	0.487970i	$-0.162263\pi$
$761$	48.1578	1.74572	0.872860	+	0.487970i	$0.162263\pi$
$762$	−3.16609	−0.114695
$763$	2.85824	0.103475
$764$	−47.9066	−1.73320
$765$	76.1010	2.75144
$766$	−1.22273	−0.0441790
$767$	−3.34094	−0.120634
$768$	−6.09288	−0.219858
$769$	22.5174	0.811999	0.405999	−	0.913873i	$-0.366923\pi$
$769$	22.5174	0.811999	0.405999	+	0.913873i	$0.366923\pi$
$770$	−23.6900	−0.853728
$771$	0.668650	0.0240808
$772$	5.76411	0.207455
$773$	−44.0155	−1.58313	−0.791563	−	0.611087i	$-0.790732\pi$
$773$	−44.0155	−1.58313	−0.791563	+	0.611087i	$0.790732\pi$
$774$	−47.9179	−1.72237
$775$	68.1246	2.44711
$776$	−69.3358	−2.48901
$777$	1.02222	0.0366721
$778$	43.8279	1.57130
$779$	−1.85813	−0.0665746
$780$	1.33400	0.0477648
$781$	29.2822	1.04780
$782$	12.7565	0.456172
$783$	0.401395	0.0143447
$784$	−92.5128	−3.30403
$785$	1.93818	0.0691767
$786$	−2.03122	−0.0724512
$787$	−3.97011	−0.141519	−0.0707595	−	0.997493i	$-0.522542\pi$
$787$	−3.97011	−0.141519	−0.0707595	+	0.997493i	$0.522542\pi$
$788$	−38.0990	−1.35722
$789$	1.48763	0.0529609
$790$	−97.5449	−3.47049
$791$	12.4343	0.442112
$792$	−56.7121	−2.01518
$793$	−9.78838	−0.347596
$794$	8.83080	0.313393
$795$	2.50749	0.0889314
$796$	109.772	3.89075
$797$	−42.9608	−1.52175	−0.760874	−	0.648899i	$-0.775230\pi$
$797$	−42.9608	−1.52175	−0.760874	+	0.648899i	$0.775230\pi$
$798$	0.222756	0.00788547
$799$	85.2482	3.01586
$800$	−213.047	−7.53236
$801$	14.5737	0.514935
$802$	−49.8686	−1.76092
$803$	−8.03559	−0.283570
$804$	1.27581	0.0449945
$805$	3.00358	0.105862
$806$	−25.2438	−0.889175
$807$	1.19955	0.0422260
$808$	−92.1617	−3.24224
$809$	−53.3114	−1.87433	−0.937164	−	0.348888i	$-0.886559\pi$
$809$	−53.3114	−1.87433	−0.937164	+	0.348888i	$0.886559\pi$
$810$	88.3528	3.10440
$811$	−14.0642	−0.493859	−0.246930	−	0.969033i	$-0.579422\pi$
$811$	−14.0642	−0.493859	−0.246930	+	0.969033i	$0.579422\pi$
$812$	7.85376	0.275613
$813$	−0.598947	−0.0210060
$814$	−59.0447	−2.06951
$815$	30.7494	1.07710
$816$	−8.31011	−0.290912
$817$	−5.27050	−0.184392
$818$	45.7678	1.60023
$819$	−3.98798	−0.139351
$820$	41.2762	1.44143
$821$	0.691073	0.0241186	0.0120593	−	0.999927i	$-0.496161\pi$
$821$	0.691073	0.0241186	0.0120593	+	0.999927i	$0.496161\pi$
$822$	−2.32109	−0.0809572
$823$	−54.0564	−1.88429	−0.942143	−	0.335210i	$-0.891193\pi$
$823$	−54.0564	−1.88429	−0.942143	+	0.335210i	$0.891193\pi$
$824$	−10.4882	−0.365373
$825$	0.887770	0.0309082
$826$	12.3926	0.431195
$827$	14.8722	0.517158	0.258579	−	0.965990i	$-0.416746\pi$
$827$	14.8722	0.517158	0.258579	+	0.965990i	$0.416746\pi$
$828$	11.0109	0.382655
$829$	−13.9983	−0.486179	−0.243090	−	0.970004i	$-0.578161\pi$
$829$	−13.9983	−0.486179	−0.243090	+	0.970004i	$0.578161\pi$
$830$	86.8136	3.01335
$831$	−1.45926	−0.0506211
$832$	43.5531	1.50993
$833$	−37.5333	−1.30045
$834$	−1.78151	−0.0616888
$835$	−19.3577	−0.669902
$836$	−9.55218	−0.330369
$837$	3.55292	0.122807
$838$	58.6729	2.02682
$839$	2.40556	0.0830492	0.0415246	−	0.999137i	$-0.486779\pi$
$839$	2.40556	0.0830492	0.0415246	+	0.999137i	$0.486779\pi$
$840$	−3.23131	−0.111491
$841$	−27.9524	−0.963876
$842$	−73.5415	−2.53441
$843$	−2.06564	−0.0711444
$844$	88.7084	3.05347
$845$	−3.53830	−0.121721
$846$	99.1143	3.40762
$847$	−10.3064	−0.354131
$848$	191.730	6.58405
$849$	1.09410	0.0375494
$850$	−150.429	−5.15967
$851$	7.48610	0.256620
$852$	6.11632	0.209542
$853$	16.5536	0.566786	0.283393	−	0.959004i	$-0.408540\pi$
$853$	16.5536	0.566786	0.283393	+	0.959004i	$0.408540\pi$
$854$	36.3084	1.24245
$855$	9.73187	0.332823
$856$	77.8654	2.66139
$857$	−51.3248	−1.75322	−0.876611	−	0.481200i	$-0.840201\pi$
$857$	−51.3248	−1.75322	−0.876611	+	0.481200i	$0.840201\pi$
$858$	−0.328966	−0.0112307
$859$	33.0033	1.12606	0.563030	−	0.826437i	$-0.309636\pi$
$859$	33.0033	1.12606	0.563030	+	0.826437i	$0.309636\pi$
$860$	117.078	3.99232
$861$	0.176222	0.00600564
$862$	11.6072	0.395342
$863$	31.4342	1.07003	0.535016	−	0.844842i	$-0.320305\pi$
$863$	31.4342	1.07003	0.535016	+	0.844842i	$0.320305\pi$
$864$	−11.1111	−0.378008
$865$	16.0283	0.544978
$866$	73.8723	2.51028
$867$	−2.25955	−0.0767382
$868$	69.5168	2.35956
$869$	17.8583	0.605801
$870$	−0.660039	−0.0223774
$871$	−3.38397	−0.114661
$872$	−22.5189	−0.762585
$873$	−19.8043	−0.670274
$874$	1.63132	0.0551801
$875$	−11.8679	−0.401208
$876$	−1.67843	−0.0567090
$877$	−2.35206	−0.0794235	−0.0397118	−	0.999211i	$-0.512644\pi$
$877$	−2.35206	−0.0794235	−0.0397118	+	0.999211i	$0.512644\pi$
$878$	−26.4574	−0.892894
$879$	0.387336	0.0130645
$880$	113.018	3.80984
$881$	−26.3317	−0.887137	−0.443569	−	0.896240i	$-0.646288\pi$
$881$	−26.3317	−0.887137	−0.443569	+	0.896240i	$0.646288\pi$
$882$	−43.6383	−1.46938
$883$	−43.6025	−1.46734	−0.733671	−	0.679505i	$-0.762194\pi$
$883$	−43.6025	−1.46734	−0.733671	+	0.679505i	$0.762194\pi$
$884$	41.3830	1.39186
$885$	−0.773207	−0.0259911
$886$	−42.0057	−1.41121
$887$	−27.3056	−0.916832	−0.458416	−	0.888738i	$-0.651583\pi$
$887$	−27.3056	−0.916832	−0.458416	+	0.888738i	$0.651583\pi$
$888$	−8.05368	−0.270264
$889$	−23.1259	−0.775617
$890$	−47.9630	−1.60772
$891$	−16.1754	−0.541897
$892$	85.4113	2.85978
$893$	10.9016	0.364809
$894$	−2.89498	−0.0968228
$895$	77.8052	2.60074
$896$	−86.1194	−2.87705
$897$	0.0417087	0.00139261
$898$	47.9838	1.60124
$899$	9.27275	0.309264
$900$	−129.844	−4.32814
$901$	77.7868	2.59145
$902$	−10.1788	−0.338916
$903$	0.499846	0.0166338
$904$	−97.9645	−3.25825
$905$	29.4408	0.978646
$906$	−2.99506	−0.0995043
$907$	13.3752	0.444115	0.222057	−	0.975034i	$-0.428723\pi$
$907$	13.3752	0.444115	0.222057	+	0.975034i	$0.428723\pi$
$908$	3.31125	0.109888
$909$	−26.3240	−0.873113
$910$	13.1247	0.435081
$911$	40.5891	1.34478	0.672389	−	0.740198i	$-0.265268\pi$
$911$	40.5891	1.34478	0.672389	+	0.740198i	$0.265268\pi$
$912$	−1.06270	−0.0351897
$913$	−15.8936	−0.526003
$914$	−15.2688	−0.505047
$915$	−2.26537	−0.0748907
$916$	−71.0187	−2.34652
$917$	−14.8365	−0.489944
$918$	−7.84536	−0.258935
$919$	−41.6069	−1.37248	−0.686242	−	0.727373i	$-0.740741\pi$
$919$	−41.6069	−1.37248	−0.686242	+	0.727373i	$0.740741\pi$
$920$	−23.6640	−0.780178
$921$	−1.03732	−0.0341809
$922$	19.8376	0.653316
$923$	−16.2230	−0.533985
$924$	0.905913	0.0298024
$925$	−88.2786	−2.90258
$926$	−25.3777	−0.833964
$927$	−2.99572	−0.0983924
$928$	−28.9989	−0.951935
$929$	0.129863	0.00426068	0.00213034	−	0.999998i	$-0.499322\pi$
$929$	0.129863	0.00426068	0.00213034	+	0.999998i	$0.499322\pi$
$930$	−5.84228	−0.191576
$931$	−4.79979	−0.157307
$932$	56.1868	1.84046
$933$	0.682022	0.0223284
$934$	31.4968	1.03061
$935$	45.8525	1.49954
$936$	31.4196	1.02698
$937$	−9.53006	−0.311333	−0.155667	−	0.987810i	$-0.549753\pi$
$937$	−9.53006	−0.311333	−0.155667	+	0.987810i	$0.549753\pi$
$938$	12.5523	0.409846
$939$	−0.804692	−0.0262601
$940$	−242.166	−7.89859
$941$	40.0575	1.30584	0.652918	−	0.757429i	$-0.273545\pi$
$941$	40.0575	1.30584	0.652918	+	0.757429i	$0.273545\pi$
$942$	−0.0998337	−0.00325276
$943$	1.29054	0.0420256
$944$	−59.1218	−1.92425
$945$	−1.84723	−0.0600904
$946$	−28.8716	−0.938696
$947$	27.9171	0.907183	0.453592	−	0.891210i	$-0.350143\pi$
$947$	27.9171	0.907183	0.453592	+	0.891210i	$0.350143\pi$
$948$	3.73015	0.121150
$949$	4.45188	0.144514
$950$	−19.2370	−0.624131
$951$	0.541978	0.0175748
$952$	−100.241	−3.24883
$953$	−19.9989	−0.647827	−0.323913	−	0.946087i	$-0.604999\pi$
$953$	−19.9989	−0.647827	−0.323913	+	0.946087i	$0.604999\pi$
$954$	90.4393	2.92808
$955$	29.4078	0.951613
$956$	6.25656	0.202351
$957$	0.120838	0.00390615
$958$	89.0575	2.87732
$959$	−16.9538	−0.547465
$960$	10.0797	0.325320
$961$	51.0770	1.64764
$962$	32.7119	1.05468
$963$	22.2406	0.716693
$964$	11.2275	0.361612
$965$	−3.53834	−0.113903
$966$	−0.154711	−0.00497775
$967$	29.9183	0.962107	0.481053	−	0.876691i	$-0.340254\pi$
$967$	29.9183	0.962107	0.481053	+	0.876691i	$0.340254\pi$
$968$	81.1996	2.60985
$969$	−0.431149	−0.0138505
$970$	65.1773	2.09272
$971$	−17.7000	−0.568021	−0.284011	−	0.958821i	$-0.591665\pi$
$971$	−17.7000	−0.568021	−0.284011	+	0.958821i	$0.591665\pi$
$972$	−10.1601	−0.325885
$973$	−13.0126	−0.417164
$974$	−9.70180	−0.310866
$975$	−0.491842	−0.0157516
$976$	−173.217	−5.54454
$977$	−32.4158	−1.03707	−0.518536	−	0.855056i	$-0.673523\pi$
$977$	−32.4158	−1.03707	−0.518536	+	0.855056i	$0.673523\pi$
$978$	−1.58387	−0.0506465
$979$	8.78096	0.280641
$980$	106.622	3.40590
$981$	−6.43204	−0.205359
$982$	−60.6027	−1.93391
$983$	15.3075	0.488234	0.244117	−	0.969746i	$-0.421502\pi$
$983$	15.3075	0.488234	0.244117	+	0.969746i	$0.421502\pi$
$984$	−1.38838	−0.0442600
$985$	23.3873	0.745183
$986$	−20.4756	−0.652076
$987$	−1.03389	−0.0329092
$988$	5.29210	0.168364
$989$	3.66054	0.116398
$990$	53.3107	1.69433
$991$	41.6210	1.32213	0.661067	−	0.750327i	$-0.270104\pi$
$991$	41.6210	1.32213	0.661067	+	0.750327i	$0.270104\pi$
$992$	−256.681	−8.14963
$993$	1.47698	0.0468706
$994$	60.1763	1.90868
$995$	−67.3840	−2.13622
$996$	−3.31978	−0.105191
$997$	32.7830	1.03825	0.519124	−	0.854699i	$-0.326258\pi$
$997$	32.7830	1.03825	0.519124	+	0.854699i	$0.326258\pi$
$998$	−85.0677	−2.69277
$999$	−4.60401	−0.145664

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.d.1.1	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.d.1.1	✓	19	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78640 q^{2} -0.0654083 q^{3} +5.76405 q^{4} -3.53830 q^{5} +0.182254 q^{6} +1.33123 q^{7} -10.4882 q^{8} -2.99572 q^{9} +O(q^{10})\)	\(q-2.78640 q^{2} -0.0654083 q^{3} +5.76405 q^{4} -3.53830 q^{5} +0.182254 q^{6} +1.33123 q^{7} -10.4882 q^{8} -2.99572 q^{9} +9.85914 q^{10} -1.80499 q^{11} -0.377017 q^{12} +1.00000 q^{13} -3.70933 q^{14} +0.231434 q^{15} +17.6962 q^{16} +7.17950 q^{17} +8.34729 q^{18} +0.918122 q^{19} -20.3949 q^{20} -0.0870732 q^{21} +5.02943 q^{22} -0.637666 q^{23} +0.686013 q^{24} +7.51957 q^{25} -2.78640 q^{26} +0.392170 q^{27} +7.67325 q^{28} +1.02352 q^{29} -0.644869 q^{30} +9.05964 q^{31} -28.3324 q^{32} +0.118061 q^{33} -20.0050 q^{34} -4.71028 q^{35} -17.2675 q^{36} -11.7398 q^{37} -2.55826 q^{38} -0.0654083 q^{39} +37.1103 q^{40} -2.02384 q^{41} +0.242621 q^{42} -5.74053 q^{43} -10.4040 q^{44} +10.5998 q^{45} +1.77680 q^{46} +11.8738 q^{47} -1.15748 q^{48} -5.22784 q^{49} -20.9526 q^{50} -0.469599 q^{51} +5.76405 q^{52} +10.8346 q^{53} -1.09274 q^{54} +6.38659 q^{55} -13.9621 q^{56} -0.0600528 q^{57} -2.85195 q^{58} -3.34094 q^{59} +1.33400 q^{60} -9.78838 q^{61} -25.2438 q^{62} -3.98798 q^{63} +43.5531 q^{64} -3.53830 q^{65} -0.328966 q^{66} -3.38397 q^{67} +41.3830 q^{68} +0.0417087 q^{69} +13.1247 q^{70} -16.2230 q^{71} +31.4196 q^{72} +4.45188 q^{73} +32.7119 q^{74} -0.491842 q^{75} +5.29210 q^{76} -2.40285 q^{77} +0.182254 q^{78} -9.89386 q^{79} -62.6144 q^{80} +8.96151 q^{81} +5.63924 q^{82} +8.80540 q^{83} -0.501894 q^{84} -25.4032 q^{85} +15.9954 q^{86} -0.0669470 q^{87} +18.9310 q^{88} -4.86483 q^{89} -29.5352 q^{90} +1.33123 q^{91} -3.67554 q^{92} -0.592575 q^{93} -33.0853 q^{94} -3.24859 q^{95} +1.85317 q^{96} +6.61086 q^{97} +14.5669 q^{98} +5.40724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9}+O(q^{10}) \) 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9	\( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9} - q^{11} + 18 q^{12} + 19 q^{13} - 8 q^{14} - 11 q^{15} + 21 q^{16} - 16 q^{17} - 10 q^{19} - 20 q^{20} - 33 q^{21} + 2 q^{22} - 14 q^{23} - 9 q^{24} + 23 q^{25} - 9 q^{26} + q^{27} + 10 q^{28} - 22 q^{29} + 28 q^{30} - 3 q^{31} - 47 q^{32} - 25 q^{33} - 35 q^{34} - 3 q^{35} - 33 q^{36} - 19 q^{37} - 15 q^{38} - 2 q^{39} + 8 q^{40} - 52 q^{41} - 3 q^{42} - 2 q^{43} - 54 q^{44} - 40 q^{45} + 33 q^{46} - 24 q^{47} - 8 q^{48} + 7 q^{49} - 10 q^{50} - 7 q^{51} + 17 q^{52} - 11 q^{53} - 23 q^{54} - 29 q^{55} - 17 q^{56} - 34 q^{57} + 18 q^{58} - 52 q^{59} - 71 q^{60} - 25 q^{61} - 22 q^{62} - 19 q^{63} + 48 q^{64} - 18 q^{65} + 9 q^{66} - 2 q^{67} + 18 q^{68} - 26 q^{69} - 12 q^{70} - 44 q^{71} - 6 q^{72} - 39 q^{73} - 4 q^{74} + 17 q^{75} - 10 q^{76} - 18 q^{77} - 8 q^{78} + q^{79} - 9 q^{80} - 13 q^{81} + 47 q^{82} - 27 q^{83} - 24 q^{84} - 26 q^{85} - q^{86} + 31 q^{87} + 19 q^{88} - 86 q^{89} + 48 q^{90} - 8 q^{91} - 19 q^{92} + 32 q^{93} + 45 q^{94} + 17 q^{95} - 25 q^{96} - 20 q^{97} + 39 q^{98} + 50 q^{99}+O(q^{100}) \) 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9 - q^11 + 18 * q^12 + 19 * q^13 - 8 * q^14 - 11 * q^15 + 21 * q^16 - 16 * q^17 - 10 * q^19 - 20 * q^20 - 33 * q^21 + 2 * q^22 - 14 * q^23 - 9 * q^24 + 23 * q^25 - 9 * q^26 + q^27 + 10 * q^28 - 22 * q^29 + 28 * q^30 - 3 * q^31 - 47 * q^32 - 25 * q^33 - 35 * q^34 - 3 * q^35 - 33 * q^36 - 19 * q^37 - 15 * q^38 - 2 * q^39 + 8 * q^40 - 52 * q^41 - 3 * q^42 - 2 * q^43 - 54 * q^44 - 40 * q^45 + 33 * q^46 - 24 * q^47 - 8 * q^48 + 7 * q^49 - 10 * q^50 - 7 * q^51 + 17 * q^52 - 11 * q^53 - 23 * q^54 - 29 * q^55 - 17 * q^56 - 34 * q^57 + 18 * q^58 - 52 * q^59 - 71 * q^60 - 25 * q^61 - 22 * q^62 - 19 * q^63 + 48 * q^64 - 18 * q^65 + 9 * q^66 - 2 * q^67 + 18 * q^68 - 26 * q^69 - 12 * q^70 - 44 * q^71 - 6 * q^72 - 39 * q^73 - 4 * q^74 + 17 * q^75 - 10 * q^76 - 18 * q^77 - 8 * q^78 + q^79 - 9 * q^80 - 13 * q^81 + 47 * q^82 - 27 * q^83 - 24 * q^84 - 26 * q^85 - q^86 + 31 * q^87 + 19 * q^88 - 86 * q^89 + 48 * q^90 - 8 * q^91 - 19 * q^92 + 32 * q^93 + 45 * q^94 + 17 * q^95 - 25 * q^96 - 20 * q^97 + 39 * q^98 + 50 * q^99

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