LMFDB - Embedded newform 847.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, 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("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$6.76332905120$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7674048.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.70320$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70320 q^{2} -2.80719 q^{3} +5.30727 q^{4} -0.445072 q^{5} +7.58839 q^{6} -1.00000 q^{7} -8.94020 q^{8} +4.88032 q^{9} +O(q^{10})$	\(q-2.70320 q^{2} -2.80719 q^{3} +5.30727 q^{4} -0.445072 q^{5} +7.58839 q^{6} -1.00000 q^{7} -8.94020 q^{8} +4.88032 q^{9} +1.20312 q^{10} -14.8985 q^{12} -0.450933 q^{13} +2.70320 q^{14} +1.24940 q^{15} +13.5526 q^{16} -4.83117 q^{17} -13.1925 q^{18} +1.08644 q^{19} -2.36212 q^{20} +2.80719 q^{21} +4.57222 q^{23} +25.0968 q^{24} -4.80191 q^{25} +1.21896 q^{26} -5.27841 q^{27} -5.30727 q^{28} +1.98431 q^{29} -3.37738 q^{30} +8.25861 q^{31} -18.7549 q^{32} +13.0596 q^{34} +0.445072 q^{35} +25.9012 q^{36} +7.31725 q^{37} -2.93685 q^{38} +1.26586 q^{39} +3.97903 q^{40} -1.77073 q^{41} -7.58839 q^{42} +11.4084 q^{43} -2.17209 q^{45} -12.3596 q^{46} +1.02259 q^{47} -38.0446 q^{48} +1.00000 q^{49} +12.9805 q^{50} +13.5620 q^{51} -2.39322 q^{52} -3.57222 q^{53} +14.2686 q^{54} +8.94020 q^{56} -3.04984 q^{57} -5.36399 q^{58} -14.3996 q^{59} +6.63092 q^{60} -4.92965 q^{61} -22.3246 q^{62} -4.88032 q^{63} +23.5929 q^{64} +0.200698 q^{65} -6.18858 q^{67} -25.6403 q^{68} -12.8351 q^{69} -1.20312 q^{70} -5.92165 q^{71} -43.6310 q^{72} +1.65776 q^{73} -19.7800 q^{74} +13.4799 q^{75} +5.76602 q^{76} -3.42186 q^{78} +3.60833 q^{79} -6.03187 q^{80} +0.176558 q^{81} +4.78663 q^{82} +10.8048 q^{83} +14.8985 q^{84} +2.15022 q^{85} -30.8392 q^{86} -5.57035 q^{87} -5.21170 q^{89} +5.87160 q^{90} +0.450933 q^{91} +24.2660 q^{92} -23.1835 q^{93} -2.76425 q^{94} -0.483543 q^{95} +52.6484 q^{96} -5.30985 q^{97} -2.70320 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) $ 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9	\( 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9} + 8 q^{10} - 14 q^{12} - 4 q^{13} + 4 q^{14} + 2 q^{15} + 8 q^{16} - 22 q^{17} - 24 q^{18} - 6 q^{19} + 2 q^{20} + 2 q^{21} + 2 q^{23} + 20 q^{24} + 4 q^{25} + 6 q^{26} - 2 q^{27} - 4 q^{28} - 12 q^{29} - 20 q^{30} - 2 q^{31} - 8 q^{32} + 24 q^{34} + 4 q^{35} + 18 q^{36} + 14 q^{37} - 22 q^{38} - 20 q^{39} - 18 q^{40} - 26 q^{41} - 6 q^{42} + 4 q^{43} - 36 q^{45} - 12 q^{46} - 16 q^{47} - 24 q^{48} + 6 q^{49} + 4 q^{50} + 4 q^{51} - 12 q^{52} + 4 q^{53} + 32 q^{54} + 12 q^{56} - 20 q^{57} - 2 q^{58} - 4 q^{59} + 24 q^{60} + 8 q^{61} - 20 q^{62} - 8 q^{63} + 26 q^{64} - 24 q^{65} + 6 q^{67} - 12 q^{68} - 14 q^{69} - 8 q^{70} + 22 q^{71} - 16 q^{72} - 14 q^{73} - 44 q^{74} - 20 q^{75} + 30 q^{76} + 32 q^{78} + 28 q^{79} - 4 q^{80} - 6 q^{81} - 4 q^{82} - 22 q^{83} + 14 q^{84} + 24 q^{85} - 30 q^{86} - 22 q^{87} + 22 q^{90} + 4 q^{91} + 10 q^{92} - 50 q^{93} + 38 q^{94} + 24 q^{95} + 62 q^{96} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9 + 8 * q^10 - 14 * q^12 - 4 * q^13 + 4 * q^14 + 2 * q^15 + 8 * q^16 - 22 * q^17 - 24 * q^18 - 6 * q^19 + 2 * q^20 + 2 * q^21 + 2 * q^23 + 20 * q^24 + 4 * q^25 + 6 * q^26 - 2 * q^27 - 4 * q^28 - 12 * q^29 - 20 * q^30 - 2 * q^31 - 8 * q^32 + 24 * q^34 + 4 * q^35 + 18 * q^36 + 14 * q^37 - 22 * q^38 - 20 * q^39 - 18 * q^40 - 26 * q^41 - 6 * q^42 + 4 * q^43 - 36 * q^45 - 12 * q^46 - 16 * q^47 - 24 * q^48 + 6 * q^49 + 4 * q^50 + 4 * q^51 - 12 * q^52 + 4 * q^53 + 32 * q^54 + 12 * q^56 - 20 * q^57 - 2 * q^58 - 4 * q^59 + 24 * q^60 + 8 * q^61 - 20 * q^62 - 8 * q^63 + 26 * q^64 - 24 * q^65 + 6 * q^67 - 12 * q^68 - 14 * q^69 - 8 * q^70 + 22 * q^71 - 16 * q^72 - 14 * q^73 - 44 * q^74 - 20 * q^75 + 30 * q^76 + 32 * q^78 + 28 * q^79 - 4 * q^80 - 6 * q^81 - 4 * q^82 - 22 * q^83 + 14 * q^84 + 24 * q^85 - 30 * q^86 - 22 * q^87 + 22 * q^90 + 4 * q^91 + 10 * q^92 - 50 * q^93 + 38 * q^94 + 24 * q^95 + 62 * q^96 - 4 * q^97 - 4 * q^98

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 $

$2$	−2.70320	−1.91145	−0.955724	−	0.294264i	$-0.904925\pi$
$2$	−2.70320	−1.91145	−0.955724	+	0.294264i	$0.904925\pi$
$3$	−2.80719	−1.62073	−0.810366	−	0.585924i	$-0.800732\pi$
$3$	−2.80719	−1.62073	−0.810366	+	0.585924i	$0.800732\pi$
$4$	5.30727	2.65363
$5$	−0.445072	−0.199042	−0.0995212	−	0.995035i	$-0.531731\pi$
$5$	−0.445072	−0.199042	−0.0995212	+	0.995035i	$0.531731\pi$
$6$	7.58839	3.09795
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−8.94020	−3.16084
$9$	4.88032	1.62677
$10$	1.20312	0.380459
$11$	0	0
$11$	0	0
$12$	−14.8985	−4.30083
$13$	−0.450933	−0.125066	−0.0625332	−	0.998043i	$-0.519918\pi$
$13$	−0.450933	−0.125066	−0.0625332	+	0.998043i	$0.519918\pi$
$14$	2.70320	0.722460
$15$	1.24940	0.322594
$16$	13.5526	3.38814
$17$	−4.83117	−1.17173	−0.585866	−	0.810408i	$-0.699245\pi$
$17$	−4.83117	−1.17173	−0.585866	+	0.810408i	$0.699245\pi$
$18$	−13.1925	−3.10949
$19$	1.08644	0.249246	0.124623	−	0.992204i	$-0.460228\pi$
$19$	1.08644	0.249246	0.124623	+	0.992204i	$0.460228\pi$
$20$	−2.36212	−0.528186
$21$	2.80719	0.612579
$22$	0	0
$23$	4.57222	0.953373	0.476687	−	0.879073i	$-0.341838\pi$
$23$	4.57222	0.953373	0.476687	+	0.879073i	$0.341838\pi$
$24$	25.0968	5.12287
$25$	−4.80191	−0.960382
$26$	1.21896	0.239058
$27$	−5.27841	−1.01583
$28$	−5.30727	−1.00298
$29$	1.98431	0.368478	0.184239	−	0.982881i	$-0.441018\pi$
$29$	1.98431	0.368478	0.184239	+	0.982881i	$0.441018\pi$
$30$	−3.37738	−0.616623
$31$	8.25861	1.48329	0.741645	−	0.670793i	$-0.234046\pi$
$31$	8.25861	1.48329	0.741645	+	0.670793i	$0.234046\pi$
$32$	−18.7549	−3.31542
$33$	0	0
$34$	13.0596	2.23970
$35$	0.445072	0.0752309
$36$	25.9012	4.31686
$37$	7.31725	1.20295	0.601474	−	0.798892i	$-0.294580\pi$
$37$	7.31725	1.20295	0.601474	+	0.798892i	$0.294580\pi$
$38$	−2.93685	−0.476421
$39$	1.26586	0.202699
$40$	3.97903	0.629140
$41$	−1.77073	−0.276542	−0.138271	−	0.990394i	$-0.544154\pi$
$41$	−1.77073	−0.276542	−0.138271	+	0.990394i	$0.544154\pi$
$42$	−7.58839	−1.17091
$43$	11.4084	1.73977	0.869884	−	0.493257i	$-0.164194\pi$
$43$	11.4084	1.73977	0.869884	+	0.493257i	$0.164194\pi$
$44$	0	0
$45$	−2.17209	−0.323797
$46$	−12.3596	−1.82232
$47$	1.02259	0.149159	0.0745797	−	0.997215i	$-0.476238\pi$
$47$	1.02259	0.149159	0.0745797	+	0.997215i	$0.476238\pi$
$48$	−38.0446	−5.49127
$49$	1.00000	0.142857
$50$	12.9805	1.83572
$51$	13.5620	1.89906
$52$	−2.39322	−0.331880
$53$	−3.57222	−0.490682	−0.245341	−	0.969437i	$-0.578900\pi$
$53$	−3.57222	−0.490682	−0.245341	+	0.969437i	$0.578900\pi$
$54$	14.2686	1.94171
$55$	0	0
$56$	8.94020	1.19468
$57$	−3.04984	−0.403961
$58$	−5.36399	−0.704326
$59$	−14.3996	−1.87467	−0.937333	−	0.348435i	$-0.886713\pi$
$59$	−14.3996	−1.87467	−0.937333	+	0.348435i	$0.886713\pi$
$60$	6.63092	0.856048
$61$	−4.92965	−0.631177	−0.315588	−	0.948896i	$-0.602202\pi$
$61$	−4.92965	−0.631177	−0.315588	+	0.948896i	$0.602202\pi$
$62$	−22.3246	−2.83523
$63$	−4.88032	−0.614862
$64$	23.5929	2.94911
$65$	0.200698	0.0248935
$66$	0	0
$67$	−6.18858	−0.756055	−0.378028	−	0.925794i	$-0.623398\pi$
$67$	−6.18858	−0.756055	−0.378028	+	0.925794i	$0.623398\pi$
$68$	−25.6403	−3.10935
$69$	−12.8351	−1.54516
$70$	−1.20312	−0.143800
$71$	−5.92165	−0.702771	−0.351385	−	0.936231i	$-0.614289\pi$
$71$	−5.92165	−0.702771	−0.351385	+	0.936231i	$0.614289\pi$
$72$	−43.6310	−5.14196
$73$	1.65776	0.194027	0.0970133	−	0.995283i	$-0.469071\pi$
$73$	1.65776	0.194027	0.0970133	+	0.995283i	$0.469071\pi$
$74$	−19.7800	−2.29937
$75$	13.4799	1.55652
$76$	5.76602	0.661407
$77$	0	0
$78$	−3.42186	−0.387449
$79$	3.60833	0.405969	0.202984	−	0.979182i	$-0.434936\pi$
$79$	3.60833	0.405969	0.202984	+	0.979182i	$0.434936\pi$
$80$	−6.03187	−0.674384
$81$	0.176558	0.0196175
$82$	4.78663	0.528595
$83$	10.8048	1.18598	0.592992	−	0.805208i	$-0.297947\pi$
$83$	10.8048	1.18598	0.592992	+	0.805208i	$0.297947\pi$
$84$	14.8985	1.62556
$85$	2.15022	0.233224
$86$	−30.8392	−3.32548
$87$	−5.57035	−0.597204
$88$	0	0
$89$	−5.21170	−0.552439	−0.276220	−	0.961095i	$-0.589082\pi$
$89$	−5.21170	−0.552439	−0.276220	+	0.961095i	$0.589082\pi$
$90$	5.87160	0.618921
$91$	0.450933	0.0472706
$92$	24.2660	2.52990
$93$	−23.1835	−2.40402
$94$	−2.76425	−0.285110
$95$	−0.483543	−0.0496105
$96$	52.6484	5.37341
$97$	−5.30985	−0.539133	−0.269567	−	0.962982i	$-0.586880\pi$
$97$	−5.30985	−0.539133	−0.269567	+	0.962982i	$0.586880\pi$
$98$	−2.70320	−0.273064
$99$	0	0
$100$	−25.4850	−2.54850
$101$	−15.5604	−1.54832	−0.774158	−	0.632992i	$-0.781826\pi$
$101$	−15.5604	−1.54832	−0.774158	+	0.632992i	$0.781826\pi$
$102$	−36.6608	−3.62996
$103$	−14.1713	−1.39634	−0.698172	−	0.715930i	$-0.746003\pi$
$103$	−14.1713	−1.39634	−0.698172	+	0.715930i	$0.746003\pi$
$104$	4.03143	0.395314
$105$	−1.24940	−0.121929
$106$	9.65640	0.937913
$107$	11.7551	1.13641	0.568205	−	0.822887i	$-0.307638\pi$
$107$	11.7551	1.13641	0.568205	+	0.822887i	$0.307638\pi$
$108$	−28.0140	−2.69565
$109$	−15.7800	−1.51145	−0.755723	−	0.654891i	$-0.772714\pi$
$109$	−15.7800	−1.51145	−0.755723	+	0.654891i	$0.772714\pi$
$110$	0	0
$111$	−20.5409	−1.94966
$112$	−13.5526	−1.28060
$113$	−5.92347	−0.557233	−0.278617	−	0.960402i	$-0.589876\pi$
$113$	−5.92347	−0.557233	−0.278617	+	0.960402i	$0.589876\pi$
$114$	8.24431	0.772150
$115$	−2.03497	−0.189762
$116$	10.5313	0.977805
$117$	−2.20070	−0.203455
$118$	38.9249	3.58333
$119$	4.83117	0.442873
$120$	−11.1699	−1.01967
$121$	0	0
$122$	13.3258	1.20646
$123$	4.97078	0.448200
$124$	43.8306	3.93611
$125$	4.36256	0.390199
$126$	13.1925	1.17528
$127$	−10.1337	−0.899222	−0.449611	−	0.893224i	$-0.648437\pi$
$127$	−10.1337	−0.899222	−0.449611	+	0.893224i	$0.648437\pi$
$128$	−26.2666	−2.32166
$129$	−32.0256	−2.81970
$130$	−0.542526	−0.0475826
$131$	6.83340	0.597037	0.298519	−	0.954404i	$-0.403508\pi$
$131$	6.83340	0.597037	0.298519	+	0.954404i	$0.403508\pi$
$132$	0	0
$133$	−1.08644	−0.0942061
$134$	16.7289	1.44516
$135$	2.34928	0.202193
$136$	43.1916	3.70365
$137$	14.4673	1.23602	0.618012	−	0.786169i	$-0.287938\pi$
$137$	14.4673	1.23602	0.618012	+	0.786169i	$0.287938\pi$
$138$	34.6958	2.95350
$139$	14.6957	1.24647	0.623235	−	0.782035i	$-0.285818\pi$
$139$	14.6957	1.24647	0.623235	+	0.782035i	$0.285818\pi$
$140$	2.36212	0.199635
$141$	−2.87059	−0.241747
$142$	16.0074	1.34331
$143$	0	0
$144$	66.1409	5.51174
$145$	−0.883163	−0.0733427
$146$	−4.48126	−0.370872
$147$	−2.80719	−0.231533
$148$	38.8346	3.19219
$149$	−1.00140	−0.0820377	−0.0410189	−	0.999158i	$-0.513060\pi$
$149$	−1.00140	−0.0820377	−0.0410189	+	0.999158i	$0.513060\pi$
$150$	−36.4388	−2.97521
$151$	−1.37539	−0.111927	−0.0559636	−	0.998433i	$-0.517823\pi$
$151$	−1.37539	−0.111927	−0.0559636	+	0.998433i	$0.517823\pi$
$152$	−9.71297	−0.787826
$153$	−23.5777	−1.90614
$154$	0	0
$155$	−3.67568	−0.295237
$156$	6.71824	0.537889
$157$	5.37668	0.429106	0.214553	−	0.976712i	$-0.431171\pi$
$157$	5.37668	0.429106	0.214553	+	0.976712i	$0.431171\pi$
$158$	−9.75403	−0.775989
$159$	10.0279	0.795264
$160$	8.34726	0.659909
$161$	−4.57222	−0.360341
$162$	−0.477270	−0.0374979
$163$	−9.42513	−0.738233	−0.369116	−	0.929383i	$-0.620340\pi$
$163$	−9.42513	−0.738233	−0.369116	+	0.929383i	$0.620340\pi$
$164$	−9.39775	−0.733841
$165$	0	0
$166$	−29.2076	−2.26695
$167$	−20.0118	−1.54856	−0.774281	−	0.632842i	$-0.781888\pi$
$167$	−20.0118	−1.54856	−0.774281	+	0.632842i	$0.781888\pi$
$168$	−25.0968	−1.93626
$169$	−12.7967	−0.984358
$170$	−5.81247	−0.445796
$171$	5.30216	0.405466
$172$	60.5476	4.61671
$173$	−10.8465	−0.824647	−0.412324	−	0.911037i	$-0.635283\pi$
$173$	−10.8465	−0.824647	−0.412324	+	0.911037i	$0.635283\pi$
$174$	15.0577	1.14152
$175$	4.80191	0.362990
$176$	0	0
$177$	40.4224	3.03833
$178$	14.0883	1.05596
$179$	16.8484	1.25931	0.629654	−	0.776876i	$-0.283197\pi$
$179$	16.8484	1.25931	0.629654	+	0.776876i	$0.283197\pi$
$180$	−11.5279	−0.859238
$181$	−20.3041	−1.50919	−0.754597	−	0.656188i	$-0.772168\pi$
$181$	−20.3041	−1.50919	−0.754597	+	0.656188i	$0.772168\pi$
$182$	−1.21896	−0.0903554
$183$	13.8385	1.02297
$184$	−40.8765	−3.01346
$185$	−3.25671	−0.239438
$186$	62.6695	4.59515
$187$	0	0
$188$	5.42714	0.395815
$189$	5.27841	0.383948
$190$	1.30711	0.0948279
$191$	2.54435	0.184103	0.0920513	−	0.995754i	$-0.470658\pi$
$191$	2.54435	0.184103	0.0920513	+	0.995754i	$0.470658\pi$
$192$	−66.2298	−4.77972
$193$	−17.5086	−1.26030	−0.630148	−	0.776475i	$-0.717006\pi$
$193$	−17.5086	−1.26030	−0.630148	+	0.776475i	$0.717006\pi$
$194$	14.3536	1.03053
$195$	−0.563397	−0.0403457
$196$	5.30727	0.379091
$197$	2.16558	0.154291	0.0771457	−	0.997020i	$-0.475419\pi$
$197$	2.16558	0.154291	0.0771457	+	0.997020i	$0.475419\pi$
$198$	0	0
$199$	−14.5756	−1.03324	−0.516620	−	0.856215i	$-0.672810\pi$
$199$	−14.5756	−1.03324	−0.516620	+	0.856215i	$0.672810\pi$
$200$	42.9300	3.03561
$201$	17.3725	1.22536
$202$	42.0628	2.95953
$203$	−1.98431	−0.139272
$204$	71.9773	5.03942
$205$	0.788103	0.0550435
$206$	38.3079	2.66904
$207$	22.3139	1.55092
$208$	−6.11130	−0.423743
$209$	0	0
$210$	3.37738	0.233061
$211$	3.63034	0.249923	0.124961	−	0.992162i	$-0.460119\pi$
$211$	3.63034	0.249923	0.124961	+	0.992162i	$0.460119\pi$
$212$	−18.9587	−1.30209
$213$	16.6232	1.13900
$214$	−31.7764	−2.17219
$215$	−5.07757	−0.346287
$216$	47.1901	3.21088
$217$	−8.25861	−0.560631
$218$	42.6563	2.88905
$219$	−4.65366	−0.314465
$220$	0	0
$221$	2.17854	0.146544
$222$	55.5261	3.72667
$223$	−4.84062	−0.324152	−0.162076	−	0.986778i	$-0.551819\pi$
$223$	−4.84062	−0.324152	−0.162076	+	0.986778i	$0.551819\pi$
$224$	18.7549	1.25311
$225$	−23.4349	−1.56232
$226$	16.0123	1.06512
$227$	−22.7306	−1.50868	−0.754342	−	0.656482i	$-0.772044\pi$
$227$	−22.7306	−1.50868	−0.754342	+	0.656482i	$0.772044\pi$
$228$	−16.1863	−1.07196
$229$	−22.6732	−1.49828	−0.749142	−	0.662409i	$-0.769534\pi$
$229$	−22.6732	−1.49828	−0.749142	+	0.662409i	$0.769534\pi$
$230$	5.50092	0.362720
$231$	0	0
$232$	−17.7402	−1.16470
$233$	0.587282	0.0384741	0.0192371	−	0.999815i	$-0.493876\pi$
$233$	0.587282	0.0384741	0.0192371	+	0.999815i	$0.493876\pi$
$234$	5.94892	0.388893
$235$	−0.455124	−0.0296890
$236$	−76.4225	−4.97468
$237$	−10.1293	−0.657967
$238$	−13.0596	−0.846529
$239$	14.9721	0.968465	0.484233	−	0.874939i	$-0.339099\pi$
$239$	14.9721	0.968465	0.484233	+	0.874939i	$0.339099\pi$
$240$	16.9326	1.09300
$241$	18.2746	1.17717	0.588586	−	0.808435i	$-0.299685\pi$
$241$	18.2746	1.17717	0.588586	+	0.808435i	$0.299685\pi$
$242$	0	0
$243$	15.3396	0.984037
$244$	−26.1630	−1.67491
$245$	−0.445072	−0.0284346
$246$	−13.4370	−0.856711
$247$	−0.489911	−0.0311723
$248$	−73.8336	−4.68844
$249$	−30.3312	−1.92216
$250$	−11.7929	−0.745845
$251$	10.5649	0.666850	0.333425	−	0.942777i	$-0.391796\pi$
$251$	10.5649	0.666850	0.333425	+	0.942777i	$0.391796\pi$
$252$	−25.9012	−1.63162
$253$	0	0
$254$	27.3934	1.71882
$255$	−6.03608	−0.377994
$256$	23.8178	1.48861
$257$	−19.4680	−1.21438	−0.607189	−	0.794557i	$-0.707703\pi$
$257$	−19.4680	−1.21438	−0.607189	+	0.794557i	$0.707703\pi$
$258$	86.5715	5.38971
$259$	−7.31725	−0.454672
$260$	1.06516	0.0660583
$261$	9.68408	0.599430
$262$	−18.4720	−1.14121
$263$	−21.1885	−1.30654	−0.653269	−	0.757126i	$-0.726603\pi$
$263$	−21.1885	−1.30654	−0.653269	+	0.757126i	$0.726603\pi$
$264$	0	0
$265$	1.58989	0.0976665
$266$	2.93685	0.180070
$267$	14.6302	0.895356
$268$	−32.8444	−2.00629
$269$	23.1564	1.41187	0.705935	−	0.708276i	$-0.250527\pi$
$269$	23.1564	1.41187	0.705935	+	0.708276i	$0.250527\pi$
$270$	−6.35055	−0.386482
$271$	−26.8232	−1.62939	−0.814696	−	0.579888i	$-0.803096\pi$
$271$	−26.8232	−1.62939	−0.814696	+	0.579888i	$0.803096\pi$
$272$	−65.4748	−3.96999
$273$	−1.26586	−0.0766131
$274$	−39.1079	−2.36260
$275$	0	0
$276$	−68.1193	−4.10030
$277$	14.0508	0.844230	0.422115	−	0.906542i	$-0.361288\pi$
$277$	14.0508	0.844230	0.422115	+	0.906542i	$0.361288\pi$
$278$	−39.7253	−2.38256
$279$	40.3046	2.41298
$280$	−3.97903	−0.237793
$281$	1.74538	0.104120	0.0520602	−	0.998644i	$-0.483421\pi$
$281$	1.74538	0.104120	0.0520602	+	0.998644i	$0.483421\pi$
$282$	7.75977	0.462088
$283$	4.66531	0.277324	0.138662	−	0.990340i	$-0.455720\pi$
$283$	4.66531	0.277324	0.138662	+	0.990340i	$0.455720\pi$
$284$	−31.4278	−1.86490
$285$	1.35740	0.0804053
$286$	0	0
$287$	1.77073	0.104523
$288$	−91.5297	−5.39344
$289$	6.34024	0.372955
$290$	2.38736	0.140191
$291$	14.9057	0.873790
$292$	8.79820	0.514876
$293$	−14.5092	−0.847637	−0.423819	−	0.905747i	$-0.639311\pi$
$293$	−14.5092	−0.847637	−0.423819	+	0.905747i	$0.639311\pi$
$294$	7.58839	0.442564
$295$	6.40886	0.373138
$296$	−65.4177	−3.80232
$297$	0	0
$298$	2.70698	0.156811
$299$	−2.06176	−0.119235
$300$	71.5413	4.13044
$301$	−11.4084	−0.657570
$302$	3.71794	0.213943
$303$	43.6810	2.50941
$304$	14.7240	0.844480
$305$	2.19405	0.125631
$306$	63.7351	3.64349
$307$	−2.35679	−0.134509	−0.0672547	−	0.997736i	$-0.521424\pi$
$307$	−2.35679	−0.134509	−0.0672547	+	0.997736i	$0.521424\pi$
$308$	0	0
$309$	39.7816	2.26310
$310$	9.93608	0.564331
$311$	−20.0050	−1.13438	−0.567190	−	0.823587i	$-0.691969\pi$
$311$	−20.0050	−1.13438	−0.567190	+	0.823587i	$0.691969\pi$
$312$	−11.3170	−0.640699
$313$	18.7530	1.05998	0.529992	−	0.848003i	$-0.322195\pi$
$313$	18.7530	1.05998	0.529992	+	0.848003i	$0.322195\pi$
$314$	−14.5342	−0.820214
$315$	2.17209	0.122384
$316$	19.1504	1.07729
$317$	4.20203	0.236009	0.118005	−	0.993013i	$-0.462350\pi$
$317$	4.20203	0.236009	0.118005	+	0.993013i	$0.462350\pi$
$318$	−27.1074	−1.52011
$319$	0	0
$320$	−10.5005	−0.586999
$321$	−32.9989	−1.84182
$322$	12.3596	0.688774
$323$	−5.24877	−0.292049
$324$	0.937039	0.0520577
$325$	2.16534	0.120112
$326$	25.4780	1.41109
$327$	44.2974	2.44965
$328$	15.8307	0.874103
$329$	−1.02259	−0.0563770
$330$	0	0
$331$	−0.0682694	−0.00375242	−0.00187621	−	0.999998i	$-0.500597\pi$
$331$	−0.0682694	−0.00375242	−0.00187621	+	0.999998i	$0.500597\pi$
$332$	57.3441	3.14717
$333$	35.7105	1.95692
$334$	54.0959	2.96000
$335$	2.75436	0.150487
$336$	38.0446	2.07551
$337$	−21.7461	−1.18459	−0.592294	−	0.805722i	$-0.701777\pi$
$337$	−21.7461	−1.18459	−0.592294	+	0.805722i	$0.701777\pi$
$338$	34.5919	1.88155
$339$	16.6283	0.903126
$340$	11.4118	0.618892
$341$	0	0
$342$	−14.3328	−0.775028
$343$	−1.00000	−0.0539949
$344$	−101.994	−5.49912
$345$	5.71254	0.307553
$346$	29.3203	1.57627
$347$	−23.9351	−1.28490	−0.642452	−	0.766326i	$-0.722083\pi$
$347$	−23.9351	−1.28490	−0.642452	+	0.766326i	$0.722083\pi$
$348$	−29.5633	−1.58476
$349$	26.4151	1.41397	0.706983	−	0.707231i	$-0.250056\pi$
$349$	26.4151	1.41397	0.706983	+	0.707231i	$0.250056\pi$
$350$	−12.9805	−0.693837
$351$	2.38021	0.127046
$352$	0	0
$353$	−4.44182	−0.236414	−0.118207	−	0.992989i	$-0.537715\pi$
$353$	−4.44182	−0.236414	−0.118207	+	0.992989i	$0.537715\pi$
$354$	−109.270	−5.80761
$355$	2.63556	0.139881
$356$	−27.6599	−1.46597
$357$	−13.5620	−0.717779
$358$	−45.5445	−2.40710
$359$	14.5325	0.766997	0.383499	−	0.923541i	$-0.374719\pi$
$359$	14.5325	0.766997	0.383499	+	0.923541i	$0.374719\pi$
$360$	19.4190	1.02347
$361$	−17.8197	−0.937876
$362$	54.8860	2.88475
$363$	0	0
$364$	2.39322	0.125439
$365$	−0.737825	−0.0386195
$366$	−37.4081	−1.95535
$367$	4.98158	0.260037	0.130018	−	0.991512i	$-0.458496\pi$
$367$	4.98158	0.260037	0.130018	+	0.991512i	$0.458496\pi$
$368$	61.9653	3.23016
$369$	−8.64173	−0.449871
$370$	8.80351	0.457673
$371$	3.57222	0.185460
$372$	−123.041	−6.37938
$373$	13.6638	0.707485	0.353743	−	0.935343i	$-0.384909\pi$
$373$	13.6638	0.707485	0.353743	+	0.935343i	$0.384909\pi$
$374$	0	0
$375$	−12.2465	−0.632408
$376$	−9.14211	−0.471469
$377$	−0.894793	−0.0460842
$378$	−14.2686	−0.733897
$379$	−8.36232	−0.429544	−0.214772	−	0.976664i	$-0.568901\pi$
$379$	−8.36232	−0.429544	−0.214772	+	0.976664i	$0.568901\pi$
$380$	−2.56629	−0.131648
$381$	28.4473	1.45740
$382$	−6.87787	−0.351902
$383$	−1.98679	−0.101520	−0.0507601	−	0.998711i	$-0.516164\pi$
$383$	−1.98679	−0.101520	−0.0507601	+	0.998711i	$0.516164\pi$
$384$	73.7352	3.76278
$385$	0	0
$386$	47.3292	2.40899
$387$	55.6767	2.83021
$388$	−28.1808	−1.43066
$389$	14.8325	0.752039	0.376019	−	0.926612i	$-0.377293\pi$
$389$	14.8325	0.752039	0.376019	+	0.926612i	$0.377293\pi$
$390$	1.52297	0.0771187
$391$	−22.0892	−1.11710
$392$	−8.94020	−0.451548
$393$	−19.1827	−0.967637
$394$	−5.85400	−0.294920
$395$	−1.60597	−0.0808050
$396$	0	0
$397$	11.4630	0.575314	0.287657	−	0.957734i	$-0.407124\pi$
$397$	11.4630	0.575314	0.287657	+	0.957734i	$0.407124\pi$
$398$	39.4008	1.97498
$399$	3.04984	0.152683
$400$	−65.0782	−3.25391
$401$	4.31030	0.215246	0.107623	−	0.994192i	$-0.465676\pi$
$401$	4.31030	0.215246	0.107623	+	0.994192i	$0.465676\pi$
$402$	−46.9613	−2.34222
$403$	−3.72408	−0.185510
$404$	−82.5831	−4.10867
$405$	−0.0785809	−0.00390472
$406$	5.36399	0.266210
$407$	0	0
$408$	−121.247	−6.00263
$409$	9.31202	0.460450	0.230225	−	0.973137i	$-0.426054\pi$
$409$	9.31202	0.460450	0.230225	+	0.973137i	$0.426054\pi$
$410$	−2.13040	−0.105213
$411$	−40.6124	−2.00326
$412$	−75.2111	−3.70538
$413$	14.3996	0.708557
$414$	−60.3188	−2.96451
$415$	−4.80893	−0.236061
$416$	8.45719	0.414648
$417$	−41.2535	−2.02019
$418$	0	0
$419$	−26.3424	−1.28691	−0.643454	−	0.765485i	$-0.722499\pi$
$419$	−26.3424	−1.28691	−0.643454	+	0.765485i	$0.722499\pi$
$420$	−6.63092	−0.323556
$421$	14.1792	0.691053	0.345526	−	0.938409i	$-0.387700\pi$
$421$	14.1792	0.691053	0.345526	+	0.938409i	$0.387700\pi$
$422$	−9.81352	−0.477715
$423$	4.99054	0.242648
$424$	31.9363	1.55097
$425$	23.1989	1.12531
$426$	−44.9358	−2.17715
$427$	4.92965	0.238562
$428$	62.3876	3.01562
$429$	0	0
$430$	13.7257	0.661911
$431$	−7.59531	−0.365853	−0.182927	−	0.983127i	$-0.558557\pi$
$431$	−7.59531	−0.365853	−0.182927	+	0.983127i	$0.558557\pi$
$432$	−71.5361	−3.44178
$433$	22.4008	1.07652	0.538258	−	0.842780i	$-0.319083\pi$
$433$	22.4008	1.07652	0.538258	+	0.842780i	$0.319083\pi$
$434$	22.3246	1.07162
$435$	2.47921	0.118869
$436$	−83.7485	−4.01083
$437$	4.96743	0.237624
$438$	12.5798	0.601084
$439$	−15.4051	−0.735244	−0.367622	−	0.929975i	$-0.619828\pi$
$439$	−15.4051	−0.735244	−0.367622	+	0.929975i	$0.619828\pi$
$440$	0	0
$441$	4.88032	0.232396
$442$	−5.88901	−0.280112
$443$	21.2099	1.00771	0.503857	−	0.863787i	$-0.331914\pi$
$443$	21.2099	1.00771	0.503857	+	0.863787i	$0.331914\pi$
$444$	−109.016	−5.17368
$445$	2.31958	0.109959
$446$	13.0851	0.619599
$447$	2.81112	0.132961
$448$	−23.5929	−1.11466
$449$	−11.6316	−0.548931	−0.274466	−	0.961597i	$-0.588501\pi$
$449$	−11.6316	−0.548931	−0.274466	+	0.961597i	$0.588501\pi$
$450$	63.3490	2.98630
$451$	0	0
$452$	−31.4375	−1.47869
$453$	3.86097	0.181404
$454$	61.4453	2.88377
$455$	−0.200698	−0.00940886
$456$	27.2661	1.27685
$457$	−31.5976	−1.47807	−0.739037	−	0.673665i	$-0.764719\pi$
$457$	−31.5976	−1.47807	−0.739037	+	0.673665i	$0.764719\pi$
$458$	61.2900	2.86389
$459$	25.5009	1.19028
$460$	−10.8001	−0.503558
$461$	8.86324	0.412802	0.206401	−	0.978467i	$-0.433825\pi$
$461$	8.86324	0.412802	0.206401	+	0.978467i	$0.433825\pi$
$462$	0	0
$463$	−25.6132	−1.19035	−0.595173	−	0.803597i	$-0.702917\pi$
$463$	−25.6132	−1.19035	−0.595173	+	0.803597i	$0.702917\pi$
$464$	26.8925	1.24846
$465$	10.3183	0.478501
$466$	−1.58754	−0.0735413
$467$	20.5402	0.950486	0.475243	−	0.879855i	$-0.342360\pi$
$467$	20.5402	0.950486	0.475243	+	0.879855i	$0.342360\pi$
$468$	−11.6797	−0.539894
$469$	6.18858	0.285762
$470$	1.23029	0.0567491
$471$	−15.0934	−0.695466
$472$	128.735	5.92551
$473$	0	0
$474$	27.3814	1.25767
$475$	−5.21698	−0.239371
$476$	25.6403	1.17522
$477$	−17.4336	−0.798228
$478$	−40.4726	−1.85117
$479$	27.5487	1.25873	0.629367	−	0.777108i	$-0.283314\pi$
$479$	27.5487	1.25873	0.629367	+	0.777108i	$0.283314\pi$
$480$	−23.4324	−1.06954
$481$	−3.29959	−0.150448
$482$	−49.3999	−2.25010
$483$	12.8351	0.584017
$484$	0	0
$485$	2.36327	0.107310
$486$	−41.4660	−1.88093
$487$	−12.1013	−0.548364	−0.274182	−	0.961678i	$-0.588407\pi$
$487$	−12.1013	−0.548364	−0.274182	+	0.961678i	$0.588407\pi$
$488$	44.0720	1.99505
$489$	26.4581	1.19648
$490$	1.20312	0.0543513
$491$	39.3867	1.77750	0.888749	−	0.458394i	$-0.151575\pi$
$491$	39.3867	1.77750	0.888749	+	0.458394i	$0.151575\pi$
$492$	26.3813	1.18936
$493$	−9.58656	−0.431757
$494$	1.32432	0.0595842
$495$	0	0
$496$	111.925	5.02560
$497$	5.92165	0.265622
$498$	81.9912	3.67411
$499$	−34.7832	−1.55711	−0.778555	−	0.627576i	$-0.784047\pi$
$499$	−34.7832	−1.55711	−0.778555	+	0.627576i	$0.784047\pi$
$500$	23.1533	1.03545
$501$	56.1770	2.50981
$502$	−28.5590	−1.27465
$503$	−3.23224	−0.144119	−0.0720593	−	0.997400i	$-0.522957\pi$
$503$	−3.23224	−0.144119	−0.0720593	+	0.997400i	$0.522957\pi$
$504$	43.6310	1.94348
$505$	6.92550	0.308181
$506$	0	0
$507$	35.9227	1.59538
$508$	−53.7824	−2.38621
$509$	−13.2217	−0.586042	−0.293021	−	0.956106i	$-0.594661\pi$
$509$	−13.2217	−0.586042	−0.293021	+	0.956106i	$0.594661\pi$
$510$	16.3167	0.722516
$511$	−1.65776	−0.0733351
$512$	−11.8512	−0.523752
$513$	−5.73467	−0.253192
$514$	52.6257	2.32122
$515$	6.30727	0.277931
$516$	−169.969	−7.48245
$517$	0	0
$518$	19.7800	0.869082
$519$	30.4483	1.33653
$520$	−1.79428	−0.0786843
$521$	7.30239	0.319924	0.159962	−	0.987123i	$-0.448863\pi$
$521$	7.30239	0.319924	0.159962	+	0.987123i	$0.448863\pi$
$522$	−26.1780	−1.14578
$523$	−8.38007	−0.366435	−0.183217	−	0.983072i	$-0.558651\pi$
$523$	−8.38007	−0.366435	−0.183217	+	0.983072i	$0.558651\pi$
$524$	36.2667	1.58432
$525$	−13.4799	−0.588310
$526$	57.2767	2.49738
$527$	−39.8988	−1.73802
$528$	0	0
$529$	−2.09483	−0.0910794
$530$	−4.29780	−0.186684
$531$	−70.2746	−3.04966
$532$	−5.76602	−0.249989
$533$	0.798481	0.0345861
$534$	−39.5484	−1.71143
$535$	−5.23188	−0.226194
$536$	55.3271	2.38977
$537$	−47.2967	−2.04100
$538$	−62.5963	−2.69872
$539$	0	0
$540$	12.4682	0.536548
$541$	3.79768	0.163275	0.0816375	−	0.996662i	$-0.473985\pi$
$541$	3.79768	0.163275	0.0816375	+	0.996662i	$0.473985\pi$
$542$	72.5083	3.11450
$543$	56.9976	2.44600
$544$	90.6080	3.88478
$545$	7.02323	0.300842
$546$	3.42186	0.146442
$547$	−9.08985	−0.388654	−0.194327	−	0.980937i	$-0.562252\pi$
$547$	−9.08985	−0.388654	−0.194327	+	0.980937i	$0.562252\pi$
$548$	76.7818	3.27996
$549$	−24.0583	−1.02678
$550$	0	0
$551$	2.15583	0.0918416
$552$	114.748	4.88401
$553$	−3.60833	−0.153442
$554$	−37.9820	−1.61370
$555$	9.14219	0.388065
$556$	77.9939	3.30768
$557$	−18.2372	−0.772736	−0.386368	−	0.922345i	$-0.626270\pi$
$557$	−18.2372	−0.772736	−0.386368	+	0.922345i	$0.626270\pi$
$558$	−108.951	−4.61228
$559$	−5.14444	−0.217586
$560$	6.03187	0.254893
$561$	0	0
$562$	−4.71809	−0.199021
$563$	15.5744	0.656384	0.328192	−	0.944611i	$-0.393561\pi$
$563$	15.5744	0.656384	0.328192	+	0.944611i	$0.393561\pi$
$564$	−15.2350	−0.641509
$565$	2.63637	0.110913
$566$	−12.6112	−0.530090
$567$	−0.176558	−0.00741473
$568$	52.9407	2.22134
$569$	−24.4254	−1.02397	−0.511984	−	0.858995i	$-0.671089\pi$
$569$	−24.4254	−1.02397	−0.511984	+	0.858995i	$0.671089\pi$
$570$	−3.66931	−0.153691
$571$	14.9541	0.625808	0.312904	−	0.949785i	$-0.398698\pi$
$571$	14.9541	0.625808	0.312904	+	0.949785i	$0.398698\pi$
$572$	0	0
$573$	−7.14247	−0.298381
$574$	−4.78663	−0.199790
$575$	−21.9554	−0.915603
$576$	115.141	4.79754
$577$	−21.4682	−0.893734	−0.446867	−	0.894600i	$-0.647460\pi$
$577$	−21.4682	−0.893734	−0.446867	+	0.894600i	$0.647460\pi$
$578$	−17.1389	−0.712885
$579$	49.1500	2.04260
$580$	−4.68718	−0.194625
$581$	−10.8048	−0.448260
$582$	−40.2932	−1.67021
$583$	0	0
$584$	−14.8207	−0.613286
$585$	0.979470	0.0404961
$586$	39.2212	1.62021
$587$	−20.6760	−0.853391	−0.426695	−	0.904395i	$-0.640322\pi$
$587$	−20.6760	−0.853391	−0.426695	+	0.904395i	$0.640322\pi$
$588$	−14.8985	−0.614404
$589$	8.97246	0.369704
$590$	−17.3244	−0.713234
$591$	−6.07921	−0.250065
$592$	99.1675	4.07576
$593$	−3.30237	−0.135612	−0.0678060	−	0.997699i	$-0.521600\pi$
$593$	−3.30237	−0.135612	−0.0678060	+	0.997699i	$0.521600\pi$
$594$	0	0
$595$	−2.15022	−0.0881505
$596$	−5.31469	−0.217698
$597$	40.9166	1.67461
$598$	5.57335	0.227911
$599$	38.9212	1.59028	0.795140	−	0.606426i	$-0.207398\pi$
$599$	38.9212	1.59028	0.795140	+	0.606426i	$0.207398\pi$
$600$	−120.513	−4.91991
$601$	−44.3783	−1.81023	−0.905115	−	0.425168i	$-0.860215\pi$
$601$	−44.3783	−1.81023	−0.905115	+	0.425168i	$0.860215\pi$
$602$	30.8392	1.25691
$603$	−30.2022	−1.22993
$604$	−7.29954	−0.297014
$605$	0	0
$606$	−118.078	−4.79660
$607$	18.9059	0.767365	0.383683	−	0.923465i	$-0.374656\pi$
$607$	18.9059	0.767365	0.383683	+	0.923465i	$0.374656\pi$
$608$	−20.3760	−0.826355
$609$	5.57035	0.225722
$610$	−5.93095	−0.240137
$611$	−0.461118	−0.0186548
$612$	−125.133	−5.05820
$613$	9.68116	0.391018	0.195509	−	0.980702i	$-0.437364\pi$
$613$	9.68116	0.391018	0.195509	+	0.980702i	$0.437364\pi$
$614$	6.37088	0.257108
$615$	−2.21236	−0.0892108
$616$	0	0
$617$	−26.2775	−1.05789	−0.528947	−	0.848655i	$-0.677413\pi$
$617$	−26.2775	−1.05789	−0.528947	+	0.848655i	$0.677413\pi$
$618$	−107.538	−4.32580
$619$	−24.7954	−0.996610	−0.498305	−	0.867002i	$-0.666044\pi$
$619$	−24.7954	−0.996610	−0.498305	+	0.867002i	$0.666044\pi$
$620$	−19.5078	−0.783452
$621$	−24.1341	−0.968466
$622$	54.0774	2.16831
$623$	5.21170	0.208802
$624$	17.1556	0.686773
$625$	22.0679	0.882716
$626$	−50.6931	−2.02611
$627$	0	0
$628$	28.5355	1.13869
$629$	−35.3509	−1.40953
$630$	−5.87160	−0.233930
$631$	−11.8107	−0.470178	−0.235089	−	0.971974i	$-0.575538\pi$
$631$	−11.8107	−0.470178	−0.235089	+	0.971974i	$0.575538\pi$
$632$	−32.2592	−1.28320
$633$	−10.1911	−0.405058
$634$	−11.3589	−0.451120
$635$	4.51024	0.178983
$636$	53.2207	2.11034
$637$	−0.450933	−0.0178666
$638$	0	0
$639$	−28.8995	−1.14325
$640$	11.6905	0.462108
$641$	−22.5182	−0.889416	−0.444708	−	0.895676i	$-0.646693\pi$
$641$	−22.5182	−0.889416	−0.444708	+	0.895676i	$0.646693\pi$
$642$	89.2024	3.52054
$643$	13.8901	0.547774	0.273887	−	0.961762i	$-0.411691\pi$
$643$	13.8901	0.547774	0.273887	+	0.961762i	$0.411691\pi$
$644$	−24.2660	−0.956214
$645$	14.2537	0.561239
$646$	14.1885	0.558237
$647$	50.8116	1.99761	0.998805	−	0.0488792i	$-0.0155649\pi$
$647$	50.8116	1.99761	0.998805	+	0.0488792i	$0.0155649\pi$
$648$	−1.57846	−0.0620078
$649$	0	0
$650$	−5.85334	−0.229587
$651$	23.1835	0.908632
$652$	−50.0217	−1.95900
$653$	−4.81411	−0.188391	−0.0941953	−	0.995554i	$-0.530028\pi$
$653$	−4.81411	−0.188391	−0.0941953	+	0.995554i	$0.530028\pi$
$654$	−119.744	−4.68238
$655$	−3.04136	−0.118836
$656$	−23.9979	−0.936962
$657$	8.09042	0.315637
$658$	2.76425	0.107762
$659$	28.0409	1.09232	0.546159	−	0.837681i	$-0.316089\pi$
$659$	28.0409	1.09232	0.546159	+	0.837681i	$0.316089\pi$
$660$	0	0
$661$	−41.7390	−1.62346	−0.811730	−	0.584033i	$-0.801474\pi$
$661$	−41.7390	−1.62346	−0.811730	+	0.584033i	$0.801474\pi$
$662$	0.184545	0.00717256
$663$	−6.11557	−0.237509
$664$	−96.5973	−3.74870
$665$	0.483543	0.0187510
$666$	−96.5325	−3.74056
$667$	9.07271	0.351297
$668$	−106.208	−4.10932
$669$	13.5885	0.525363
$670$	−7.44559	−0.287648
$671$	0	0
$672$	−52.6484	−2.03096
$673$	28.7747	1.10918	0.554592	−	0.832123i	$-0.312874\pi$
$673$	28.7747	1.10918	0.554592	+	0.832123i	$0.312874\pi$
$674$	58.7841	2.26428
$675$	25.3465	0.975586
$676$	−67.9153	−2.61213
$677$	12.2862	0.472195	0.236098	−	0.971729i	$-0.424131\pi$
$677$	12.2862	0.472195	0.236098	+	0.971729i	$0.424131\pi$
$678$	−44.9496	−1.72628
$679$	5.30985	0.203773
$680$	−19.2234	−0.737184
$681$	63.8092	2.44517
$682$	0	0
$683$	30.5940	1.17065	0.585323	−	0.810800i	$-0.300968\pi$
$683$	30.5940	1.17065	0.585323	+	0.810800i	$0.300968\pi$
$684$	28.1400	1.07596
$685$	−6.43899	−0.246021
$686$	2.70320	0.103209
$687$	63.6479	2.42832
$688$	154.613	5.89458
$689$	1.61083	0.0613678
$690$	−15.4421	−0.587871
$691$	−17.1121	−0.650976	−0.325488	−	0.945546i	$-0.605529\pi$
$691$	−17.1121	−0.650976	−0.325488	+	0.945546i	$0.605529\pi$
$692$	−57.5655	−2.18831
$693$	0	0
$694$	64.7013	2.45603
$695$	−6.54063	−0.248100
$696$	49.8000	1.88766
$697$	8.55471	0.324033
$698$	−71.4051	−2.70272
$699$	−1.64861	−0.0623563
$700$	25.4850	0.963244
$701$	−13.5936	−0.513423	−0.256712	−	0.966488i	$-0.582639\pi$
$701$	−13.5936	−0.513423	−0.256712	+	0.966488i	$0.582639\pi$
$702$	−6.43418	−0.242842
$703$	7.94974	0.299830
$704$	0	0
$705$	1.27762	0.0481180
$706$	12.0071	0.451893
$707$	15.5604	0.585208
$708$	214.532	8.06262
$709$	29.5995	1.11163	0.555816	−	0.831305i	$-0.312406\pi$
$709$	29.5995	1.11163	0.555816	+	0.831305i	$0.312406\pi$
$710$	−7.12444	−0.267376
$711$	17.6098	0.660419
$712$	46.5936	1.74617
$713$	37.7601	1.41413
$714$	36.6608	1.37200
$715$	0	0
$716$	89.4190	3.34174
$717$	−42.0296	−1.56962
$718$	−39.2842	−1.46608
$719$	45.5407	1.69838	0.849190	−	0.528087i	$-0.177091\pi$
$719$	45.5407	1.69838	0.849190	+	0.528087i	$0.177091\pi$
$720$	−29.4375	−1.09707
$721$	14.1713	0.527768
$722$	48.1700	1.79270
$723$	−51.3004	−1.90788
$724$	−107.759	−4.00485
$725$	−9.52850	−0.353880
$726$	0	0
$727$	−43.8796	−1.62740	−0.813702	−	0.581283i	$-0.802551\pi$
$727$	−43.8796	−1.62740	−0.813702	+	0.581283i	$0.802551\pi$
$728$	−4.03143	−0.149415
$729$	−43.5909	−1.61448
$730$	1.99448	0.0738192
$731$	−55.1161	−2.03854
$732$	73.4445	2.71459
$733$	−14.0878	−0.520345	−0.260172	−	0.965562i	$-0.583779\pi$
$733$	−14.0878	−0.520345	−0.260172	+	0.965562i	$0.583779\pi$
$734$	−13.4662	−0.497046
$735$	1.24940	0.0460849
$736$	−85.7513	−3.16083
$737$	0	0
$738$	23.3603	0.859904
$739$	39.4975	1.45294	0.726470	−	0.687199i	$-0.241160\pi$
$739$	39.4975	1.45294	0.726470	+	0.687199i	$0.241160\pi$
$740$	−17.2842	−0.635380
$741$	1.37527	0.0505219
$742$	−9.65640	−0.354498
$743$	−18.6162	−0.682962	−0.341481	−	0.939889i	$-0.610928\pi$
$743$	−18.6162	−0.682962	−0.341481	+	0.939889i	$0.610928\pi$
$744$	207.265	7.59870
$745$	0.445694	0.0163290
$746$	−36.9360	−1.35232
$747$	52.7310	1.92933
$748$	0	0
$749$	−11.7551	−0.429523
$750$	33.1048	1.20882
$751$	−15.8060	−0.576768	−0.288384	−	0.957515i	$-0.593118\pi$
$751$	−15.8060	−0.576768	−0.288384	+	0.957515i	$0.593118\pi$
$752$	13.8587	0.505373
$753$	−29.6576	−1.08078
$754$	2.41880	0.0880875
$755$	0.612146	0.0222783
$756$	28.0140	1.01886
$757$	−3.61112	−0.131248	−0.0656241	−	0.997844i	$-0.520904\pi$
$757$	−3.61112	−0.131248	−0.0656241	+	0.997844i	$0.520904\pi$
$758$	22.6050	0.821051
$759$	0	0
$760$	4.32297	0.156811
$761$	−39.5819	−1.43484	−0.717422	−	0.696639i	$-0.754678\pi$
$761$	−39.5819	−1.43484	−0.717422	+	0.696639i	$0.754678\pi$
$762$	−76.8986	−2.78574
$763$	15.7800	0.571273
$764$	13.5035	0.488541
$765$	10.4938	0.379403
$766$	5.37068	0.194051
$767$	6.49325	0.234458
$768$	−66.8612	−2.41265
$769$	10.5472	0.380341	0.190171	−	0.981751i	$-0.439096\pi$
$769$	10.5472	0.380341	0.190171	+	0.981751i	$0.439096\pi$
$770$	0	0
$771$	54.6503	1.96818
$772$	−92.9228	−3.34437
$773$	41.5632	1.49492	0.747462	−	0.664305i	$-0.231272\pi$
$773$	41.5632	1.49492	0.747462	+	0.664305i	$0.231272\pi$
$774$	−150.505	−5.40979
$775$	−39.6571	−1.42452
$776$	47.4711	1.70411
$777$	20.5409	0.736901
$778$	−40.0952	−1.43748
$779$	−1.92379	−0.0689269
$780$	−2.99010	−0.107063
$781$	0	0
$782$	59.7114	2.13527
$783$	−10.4740	−0.374311
$784$	13.5526	0.484020
$785$	−2.39301	−0.0854103
$786$	51.8545	1.84959
$787$	−11.5474	−0.411620	−0.205810	−	0.978592i	$-0.565983\pi$
$787$	−11.5474	−0.411620	−0.205810	+	0.978592i	$0.565983\pi$
$788$	11.4933	0.409433
$789$	59.4801	2.11755
$790$	4.34125	0.154455
$791$	5.92347	0.210614
$792$	0	0
$793$	2.22294	0.0789390
$794$	−30.9869	−1.09968
$795$	−4.46314	−0.158291
$796$	−77.3569	−2.74184
$797$	42.3155	1.49889	0.749447	−	0.662065i	$-0.230320\pi$
$797$	42.3155	1.49889	0.749447	+	0.662065i	$0.230320\pi$
$798$	−8.24431	−0.291845
$799$	−4.94029	−0.174775
$800$	90.0591	3.18407
$801$	−25.4348	−0.898693
$802$	−11.6516	−0.411431
$803$	0	0
$804$	92.2006	3.25167
$805$	2.03497	0.0717232
$806$	10.0669	0.354592
$807$	−65.0044	−2.28826
$808$	139.113	4.89397
$809$	17.9814	0.632194	0.316097	−	0.948727i	$-0.397628\pi$
$809$	17.9814	0.632194	0.316097	+	0.948727i	$0.397628\pi$
$810$	0.212420	0.00746367
$811$	−31.1200	−1.09277	−0.546386	−	0.837533i	$-0.683997\pi$
$811$	−31.1200	−1.09277	−0.546386	+	0.837533i	$0.683997\pi$
$812$	−10.5313	−0.369576
$813$	75.2978	2.64081
$814$	0	0
$815$	4.19486	0.146940
$816$	183.800	6.43430
$817$	12.3945	0.433630
$818$	−25.1722	−0.880126
$819$	2.20070	0.0768986
$820$	4.18268	0.146065
$821$	−48.8675	−1.70549	−0.852744	−	0.522329i	$-0.825063\pi$
$821$	−48.8675	−1.70549	−0.852744	+	0.522329i	$0.825063\pi$
$822$	109.783	3.82914
$823$	29.5821	1.03117	0.515584	−	0.856839i	$-0.327575\pi$
$823$	29.5821	1.03117	0.515584	+	0.856839i	$0.327575\pi$
$824$	126.695	4.41361
$825$	0	0
$826$	−38.9249	−1.35437
$827$	−25.3924	−0.882981	−0.441490	−	0.897266i	$-0.645550\pi$
$827$	−25.3924	−0.882981	−0.441490	+	0.897266i	$0.645550\pi$
$828$	118.426	4.11558
$829$	7.09580	0.246447	0.123224	−	0.992379i	$-0.460677\pi$
$829$	7.09580	0.246447	0.123224	+	0.992379i	$0.460677\pi$
$830$	12.9995	0.451219
$831$	−39.4432	−1.36827
$832$	−10.6388	−0.368835
$833$	−4.83117	−0.167390
$834$	111.516	3.86150
$835$	8.90671	0.308230
$836$	0	0
$837$	−43.5923	−1.50677
$838$	71.2086	2.45986
$839$	−23.7285	−0.819197	−0.409599	−	0.912266i	$-0.634331\pi$
$839$	−23.7285	−0.819197	−0.409599	+	0.912266i	$0.634331\pi$
$840$	11.1699	0.385398
$841$	−25.0625	−0.864224
$842$	−38.3292	−1.32091
$843$	−4.89960	−0.168751
$844$	19.2672	0.663204
$845$	5.69544	0.195929
$846$	−13.4904	−0.463810
$847$	0	0
$848$	−48.4127	−1.66250
$849$	−13.0964	−0.449467
$850$	−62.7111	−2.15097
$851$	33.4561	1.14686
$852$	88.2238	3.02250
$853$	38.6542	1.32350	0.661748	−	0.749726i	$-0.269815\pi$
$853$	38.6542	1.32350	0.661748	+	0.749726i	$0.269815\pi$
$854$	−13.3258	−0.456000
$855$	−2.35985	−0.0807050
$856$	−105.093	−3.59201
$857$	−34.0838	−1.16428	−0.582139	−	0.813089i	$-0.697784\pi$
$857$	−34.0838	−1.16428	−0.582139	+	0.813089i	$0.697784\pi$
$858$	0	0
$859$	56.7283	1.93555	0.967773	−	0.251825i	$-0.0810308\pi$
$859$	56.7283	1.93555	0.967773	+	0.251825i	$0.0810308\pi$
$860$	−26.9480	−0.918920
$861$	−4.97078	−0.169404
$862$	20.5316	0.699310
$863$	24.3930	0.830346	0.415173	−	0.909742i	$-0.363721\pi$
$863$	24.3930	0.830346	0.415173	+	0.909742i	$0.363721\pi$
$864$	98.9959	3.36791
$865$	4.82750	0.164140
$866$	−60.5539	−2.05770
$867$	−17.7983	−0.604461
$868$	−43.8306	−1.48771
$869$	0	0
$870$	−6.70178	−0.227212
$871$	2.79064	0.0945571
$872$	141.076	4.77744
$873$	−25.9137	−0.877047
$874$	−13.4279	−0.454207
$875$	−4.36256	−0.147481
$876$	−24.6982	−0.834475
$877$	−30.4950	−1.02974	−0.514871	−	0.857267i	$-0.672160\pi$
$877$	−30.4950	−1.02974	−0.514871	+	0.857267i	$0.672160\pi$
$878$	41.6429	1.40538
$879$	40.7301	1.37379
$880$	0	0
$881$	2.10056	0.0707697	0.0353848	−	0.999374i	$-0.488734\pi$
$881$	2.10056	0.0707697	0.0353848	+	0.999374i	$0.488734\pi$
$882$	−13.1925	−0.444213
$883$	45.1955	1.52095	0.760475	−	0.649367i	$-0.224966\pi$
$883$	45.1955	1.52095	0.760475	+	0.649367i	$0.224966\pi$
$884$	11.5621	0.388875
$885$	−17.9909	−0.604757
$886$	−57.3346	−1.92619
$887$	−5.18758	−0.174182	−0.0870910	−	0.996200i	$-0.527757\pi$
$887$	−5.18758	−0.174182	−0.0870910	+	0.996200i	$0.527757\pi$
$888$	183.640	6.16255
$889$	10.1337	0.339874
$890$	−6.27029	−0.210181
$891$	0	0
$892$	−25.6905	−0.860181
$893$	1.11098	0.0371774
$894$	−7.59900	−0.254148
$895$	−7.49875	−0.250656
$896$	26.2666	0.877504
$897$	5.78777	0.193248
$898$	31.4426	1.04925
$899$	16.3877	0.546559
$900$	−124.375	−4.14584
$901$	17.2580	0.574947
$902$	0	0
$903$	32.0256	1.06575
$904$	52.9570	1.76132
$905$	9.03681	0.300394
$906$	−10.4370	−0.346745
$907$	−38.6473	−1.28326	−0.641631	−	0.767013i	$-0.721742\pi$
$907$	−38.6473	−1.28326	−0.641631	+	0.767013i	$0.721742\pi$
$908$	−120.637	−4.00350
$909$	−75.9396	−2.51876
$910$	0.542526	0.0179846
$911$	−18.3990	−0.609585	−0.304793	−	0.952419i	$-0.598587\pi$
$911$	−18.3990	−0.609585	−0.304793	+	0.952419i	$0.598587\pi$
$912$	−41.3331	−1.36868
$913$	0	0
$914$	85.4145	2.82526
$915$	−6.15912	−0.203614
$916$	−120.333	−3.97590
$917$	−6.83340	−0.225659
$918$	−68.9340	−2.27516
$919$	−17.6533	−0.582330	−0.291165	−	0.956673i	$-0.594043\pi$
$919$	−17.6533	−0.582330	−0.291165	+	0.956673i	$0.594043\pi$
$920$	18.1930	0.599806
$921$	6.61597	0.218004
$922$	−23.9591	−0.789050
$923$	2.67027	0.0878930
$924$	0	0
$925$	−35.1368	−1.15529
$926$	69.2375	2.27529
$927$	−69.1606	−2.27153
$928$	−37.2155	−1.22166
$929$	−13.4484	−0.441226	−0.220613	−	0.975361i	$-0.570806\pi$
$929$	−13.4484	−0.441226	−0.220613	+	0.975361i	$0.570806\pi$
$930$	−27.8925	−0.914630
$931$	1.08644	0.0356066
$932$	3.11687	0.102096
$933$	56.1579	1.83853
$934$	−55.5241	−1.81680
$935$	0	0
$936$	19.6747	0.643087
$937$	8.59580	0.280813	0.140406	−	0.990094i	$-0.455159\pi$
$937$	8.59580	0.280813	0.140406	+	0.990094i	$0.455159\pi$
$938$	−16.7289	−0.546219
$939$	−52.6434	−1.71795
$940$	−2.41547	−0.0787839
$941$	33.6975	1.09851	0.549254	−	0.835655i	$-0.314912\pi$
$941$	33.6975	1.09851	0.549254	+	0.835655i	$0.314912\pi$
$942$	40.8004	1.32935
$943$	−8.09617	−0.263647
$944$	−195.151	−6.35163
$945$	−2.34928	−0.0764220
$946$	0	0
$947$	−15.3289	−0.498121	−0.249061	−	0.968488i	$-0.580122\pi$
$947$	−15.3289	−0.498121	−0.249061	+	0.968488i	$0.580122\pi$
$948$	−53.7588	−1.74600
$949$	−0.747541	−0.0242662
$950$	14.1025	0.457546
$951$	−11.7959	−0.382508
$952$	−43.1916	−1.39985
$953$	−44.9376	−1.45567	−0.727835	−	0.685752i	$-0.759473\pi$
$953$	−44.9376	−1.45567	−0.727835	+	0.685752i	$0.759473\pi$
$954$	47.1263	1.52577
$955$	−1.13242	−0.0366442
$956$	79.4610	2.56995
$957$	0	0
$958$	−74.4697	−2.40601
$959$	−14.4673	−0.467173
$960$	29.4770	0.951368
$961$	37.2046	1.20015
$962$	8.91944	0.287574
$963$	57.3687	1.84868
$964$	96.9883	3.12378
$965$	7.79259	0.250852
$966$	−34.6958	−1.11632
$967$	−33.9453	−1.09161	−0.545804	−	0.837913i	$-0.683776\pi$
$967$	−33.9453	−1.09161	−0.545804	+	0.837913i	$0.683776\pi$
$968$	0	0
$969$	14.7343	0.473334
$970$	−6.38837	−0.205118
$971$	−55.4600	−1.77980	−0.889899	−	0.456158i	$-0.849225\pi$
$971$	−55.4600	−1.77980	−0.889899	+	0.456158i	$0.849225\pi$
$972$	81.4114	2.61127
$973$	−14.6957	−0.471121
$974$	32.7123	1.04817
$975$	−6.07852	−0.194669
$976$	−66.8094	−2.13852
$977$	−37.9269	−1.21339	−0.606694	−	0.794936i	$-0.707505\pi$
$977$	−37.9269	−1.21339	−0.606694	+	0.794936i	$0.707505\pi$
$978$	−71.5215	−2.28701
$979$	0	0
$980$	−2.36212	−0.0754551
$981$	−77.0113	−2.45878
$982$	−106.470	−3.39760
$983$	31.5337	1.00577	0.502884	−	0.864354i	$-0.332272\pi$
$983$	31.5337	1.00577	0.502884	+	0.864354i	$0.332272\pi$
$984$	−44.4397	−1.41669
$985$	−0.963841	−0.0307105
$986$	25.9144	0.825281
$987$	2.87059	0.0913719
$988$	−2.60009	−0.0827198
$989$	52.1618	1.65865
$990$	0	0
$991$	39.8173	1.26484	0.632419	−	0.774627i	$-0.282062\pi$
$991$	39.8173	1.26484	0.632419	+	0.774627i	$0.282062\pi$
$992$	−154.889	−4.91773
$993$	0.191645	0.00608167
$994$	−16.0074	−0.507723
$995$	6.48721	0.205659
$996$	−160.976	−5.10072
$997$	18.7190	0.592836	0.296418	−	0.955058i	$-0.404208\pi$
$997$	18.7190	0.592836	0.296418	+	0.955058i	$0.404208\pi$
$998$	94.0258	2.97634
$999$	−38.6235	−1.22199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.m.1.1	✓	6
3.2	odd	2		7623.2.a.cs.1.6		6
7.6	odd	2		5929.2.a.bj.1.1		6
11.2	odd	10		847.2.f.y.323.1		24
11.3	even	5		847.2.f.z.372.1		24
11.4	even	5		847.2.f.z.148.1		24
11.5	even	5		847.2.f.z.729.6		24
11.6	odd	10		847.2.f.y.729.1		24
11.7	odd	10		847.2.f.y.148.6		24
11.8	odd	10		847.2.f.y.372.6		24
11.9	even	5		847.2.f.z.323.6		24
11.10	odd	2		847.2.a.n.1.6	yes	6
33.32	even	2		7623.2.a.cp.1.1		6
77.76	even	2		5929.2.a.bm.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.m.1.1	✓	6	1.1	even	1	trivial
847.2.a.n.1.6	yes	6	11.10	odd	2
847.2.f.y.148.6		24	11.7	odd	10
847.2.f.y.323.1		24	11.2	odd	10
847.2.f.y.372.6		24	11.8	odd	10
847.2.f.y.729.1		24	11.6	odd	10
847.2.f.z.148.1		24	11.4	even	5
847.2.f.z.323.6		24	11.9	even	5
847.2.f.z.372.1		24	11.3	even	5
847.2.f.z.729.6		24	11.5	even	5
5929.2.a.bj.1.1		6	7.6	odd	2
5929.2.a.bm.1.6		6	77.76	even	2
7623.2.a.cp.1.1		6	33.32	even	2
7623.2.a.cs.1.6		6	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70320 q^{2} -2.80719 q^{3} +5.30727 q^{4} -0.445072 q^{5} +7.58839 q^{6} -1.00000 q^{7} -8.94020 q^{8} +4.88032 q^{9} +O(q^{10})\)	\(q-2.70320 q^{2} -2.80719 q^{3} +5.30727 q^{4} -0.445072 q^{5} +7.58839 q^{6} -1.00000 q^{7} -8.94020 q^{8} +4.88032 q^{9} +1.20312 q^{10} -14.8985 q^{12} -0.450933 q^{13} +2.70320 q^{14} +1.24940 q^{15} +13.5526 q^{16} -4.83117 q^{17} -13.1925 q^{18} +1.08644 q^{19} -2.36212 q^{20} +2.80719 q^{21} +4.57222 q^{23} +25.0968 q^{24} -4.80191 q^{25} +1.21896 q^{26} -5.27841 q^{27} -5.30727 q^{28} +1.98431 q^{29} -3.37738 q^{30} +8.25861 q^{31} -18.7549 q^{32} +13.0596 q^{34} +0.445072 q^{35} +25.9012 q^{36} +7.31725 q^{37} -2.93685 q^{38} +1.26586 q^{39} +3.97903 q^{40} -1.77073 q^{41} -7.58839 q^{42} +11.4084 q^{43} -2.17209 q^{45} -12.3596 q^{46} +1.02259 q^{47} -38.0446 q^{48} +1.00000 q^{49} +12.9805 q^{50} +13.5620 q^{51} -2.39322 q^{52} -3.57222 q^{53} +14.2686 q^{54} +8.94020 q^{56} -3.04984 q^{57} -5.36399 q^{58} -14.3996 q^{59} +6.63092 q^{60} -4.92965 q^{61} -22.3246 q^{62} -4.88032 q^{63} +23.5929 q^{64} +0.200698 q^{65} -6.18858 q^{67} -25.6403 q^{68} -12.8351 q^{69} -1.20312 q^{70} -5.92165 q^{71} -43.6310 q^{72} +1.65776 q^{73} -19.7800 q^{74} +13.4799 q^{75} +5.76602 q^{76} -3.42186 q^{78} +3.60833 q^{79} -6.03187 q^{80} +0.176558 q^{81} +4.78663 q^{82} +10.8048 q^{83} +14.8985 q^{84} +2.15022 q^{85} -30.8392 q^{86} -5.57035 q^{87} -5.21170 q^{89} +5.87160 q^{90} +0.450933 q^{91} +24.2660 q^{92} -23.1835 q^{93} -2.76425 q^{94} -0.483543 q^{95} +52.6484 q^{96} -5.30985 q^{97} -2.70320 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9	\( 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9} + 8 q^{10} - 14 q^{12} - 4 q^{13} + 4 q^{14} + 2 q^{15} + 8 q^{16} - 22 q^{17} - 24 q^{18} - 6 q^{19} + 2 q^{20} + 2 q^{21} + 2 q^{23} + 20 q^{24} + 4 q^{25} + 6 q^{26} - 2 q^{27} - 4 q^{28} - 12 q^{29} - 20 q^{30} - 2 q^{31} - 8 q^{32} + 24 q^{34} + 4 q^{35} + 18 q^{36} + 14 q^{37} - 22 q^{38} - 20 q^{39} - 18 q^{40} - 26 q^{41} - 6 q^{42} + 4 q^{43} - 36 q^{45} - 12 q^{46} - 16 q^{47} - 24 q^{48} + 6 q^{49} + 4 q^{50} + 4 q^{51} - 12 q^{52} + 4 q^{53} + 32 q^{54} + 12 q^{56} - 20 q^{57} - 2 q^{58} - 4 q^{59} + 24 q^{60} + 8 q^{61} - 20 q^{62} - 8 q^{63} + 26 q^{64} - 24 q^{65} + 6 q^{67} - 12 q^{68} - 14 q^{69} - 8 q^{70} + 22 q^{71} - 16 q^{72} - 14 q^{73} - 44 q^{74} - 20 q^{75} + 30 q^{76} + 32 q^{78} + 28 q^{79} - 4 q^{80} - 6 q^{81} - 4 q^{82} - 22 q^{83} + 14 q^{84} + 24 q^{85} - 30 q^{86} - 22 q^{87} + 22 q^{90} + 4 q^{91} + 10 q^{92} - 50 q^{93} + 38 q^{94} + 24 q^{95} + 62 q^{96} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9 + 8 * q^10 - 14 * q^12 - 4 * q^13 + 4 * q^14 + 2 * q^15 + 8 * q^16 - 22 * q^17 - 24 * q^18 - 6 * q^19 + 2 * q^20 + 2 * q^21 + 2 * q^23 + 20 * q^24 + 4 * q^25 + 6 * q^26 - 2 * q^27 - 4 * q^28 - 12 * q^29 - 20 * q^30 - 2 * q^31 - 8 * q^32 + 24 * q^34 + 4 * q^35 + 18 * q^36 + 14 * q^37 - 22 * q^38 - 20 * q^39 - 18 * q^40 - 26 * q^41 - 6 * q^42 + 4 * q^43 - 36 * q^45 - 12 * q^46 - 16 * q^47 - 24 * q^48 + 6 * q^49 + 4 * q^50 + 4 * q^51 - 12 * q^52 + 4 * q^53 + 32 * q^54 + 12 * q^56 - 20 * q^57 - 2 * q^58 - 4 * q^59 + 24 * q^60 + 8 * q^61 - 20 * q^62 - 8 * q^63 + 26 * q^64 - 24 * q^65 + 6 * q^67 - 12 * q^68 - 14 * q^69 - 8 * q^70 + 22 * q^71 - 16 * q^72 - 14 * q^73 - 44 * q^74 - 20 * q^75 + 30 * q^76 + 32 * q^78 + 28 * q^79 - 4 * q^80 - 6 * q^81 - 4 * q^82 - 22 * q^83 + 14 * q^84 + 24 * q^85 - 30 * q^86 - 22 * q^87 + 22 * q^90 + 4 * q^91 + 10 * q^92 - 50 * q^93 + 38 * q^94 + 24 * q^95 + 62 * q^96 - 4 * q^97 - 4 * q^98

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