LMFDB - Embedded newform 847.2.a.m.1.4

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			1.4
Root			$0.879640$ 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)

$f(q)$	$=$	$q-0.120360 q^{2} +2.76784 q^{3} -1.98551 q^{4} -2.80853 q^{5} -0.333137 q^{6} -1.00000 q^{7} +0.479696 q^{8} +4.66094 q^{9} +O(q^{10})$	\(q-0.120360 q^{2} +2.76784 q^{3} -1.98551 q^{4} -2.80853 q^{5} -0.333137 q^{6} -1.00000 q^{7} +0.479696 q^{8} +4.66094 q^{9} +0.338034 q^{10} -5.49558 q^{12} -1.07967 q^{13} +0.120360 q^{14} -7.77355 q^{15} +3.91329 q^{16} -6.95828 q^{17} -0.560990 q^{18} -7.54411 q^{19} +5.57636 q^{20} -2.76784 q^{21} +4.82552 q^{23} +1.32772 q^{24} +2.88781 q^{25} +0.129949 q^{26} +4.59720 q^{27} +1.98551 q^{28} -1.22726 q^{29} +0.935623 q^{30} -8.07409 q^{31} -1.43040 q^{32} +0.837498 q^{34} +2.80853 q^{35} -9.25435 q^{36} +1.53525 q^{37} +0.908009 q^{38} -2.98836 q^{39} -1.34724 q^{40} -9.29986 q^{41} +0.333137 q^{42} +5.23402 q^{43} -13.0904 q^{45} -0.580799 q^{46} -1.89387 q^{47} +10.8314 q^{48} +1.00000 q^{49} -0.347577 q^{50} -19.2594 q^{51} +2.14371 q^{52} -3.82552 q^{53} -0.553319 q^{54} -0.479696 q^{56} -20.8809 q^{57} +0.147713 q^{58} -6.66349 q^{59} +15.4345 q^{60} +9.79952 q^{61} +0.971796 q^{62} -4.66094 q^{63} -7.65442 q^{64} +3.03229 q^{65} -2.06100 q^{67} +13.8158 q^{68} +13.3563 q^{69} -0.338034 q^{70} +12.5212 q^{71} +2.23583 q^{72} -2.56708 q^{73} -0.184783 q^{74} +7.99301 q^{75} +14.9789 q^{76} +0.359679 q^{78} +15.3283 q^{79} -10.9906 q^{80} -1.25848 q^{81} +1.11933 q^{82} -2.04602 q^{83} +5.49558 q^{84} +19.5425 q^{85} -0.629967 q^{86} -3.39687 q^{87} +4.76119 q^{89} +1.57555 q^{90} +1.07967 q^{91} -9.58114 q^{92} -22.3478 q^{93} +0.227945 q^{94} +21.1878 q^{95} -3.95910 q^{96} +9.11512 q^{97} -0.120360 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.120360	−0.0851073	−0.0425536	−	0.999094i	$-0.513549\pi$
$2$	−0.120360	−0.0851073	−0.0425536	+	0.999094i	$0.513549\pi$
$3$	2.76784	1.59801	0.799006	−	0.601322i	$-0.205359\pi$
$3$	2.76784	1.59801	0.799006	+	0.601322i	$0.205359\pi$
$4$	−1.98551	−0.992757
$5$	−2.80853	−1.25601	−0.628005	−	0.778209i	$-0.716128\pi$
$5$	−2.80853	−1.25601	−0.628005	+	0.778209i	$0.716128\pi$
$6$	−0.333137	−0.136003
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.479696	0.169598
$9$	4.66094	1.55365
$10$	0.338034	0.106896
$11$	0	0
$11$	0	0
$12$	−5.49558	−1.58644
$13$	−1.07967	−0.299448	−0.149724	−	0.988728i	$-0.547838\pi$
$13$	−1.07967	−0.299448	−0.149724	+	0.988728i	$0.547838\pi$
$14$	0.120360	0.0321675
$15$	−7.77355	−2.00712
$16$	3.91329	0.978323
$17$	−6.95828	−1.68763	−0.843816	−	0.536633i	$-0.819696\pi$
$17$	−6.95828	−1.68763	−0.843816	+	0.536633i	$0.819696\pi$
$18$	−0.560990	−0.132227
$19$	−7.54411	−1.73074	−0.865369	−	0.501135i	$-0.832916\pi$
$19$	−7.54411	−1.73074	−0.865369	+	0.501135i	$0.832916\pi$
$20$	5.57636	1.24691
$21$	−2.76784	−0.603992
$22$	0	0
$23$	4.82552	1.00619	0.503095	−	0.864231i	$-0.332194\pi$
$23$	4.82552	1.00619	0.503095	+	0.864231i	$0.332194\pi$
$24$	1.32772	0.271020
$25$	2.88781	0.577563
$26$	0.129949	0.0254852
$27$	4.59720	0.884732
$28$	1.98551	0.375227
$29$	−1.22726	−0.227897	−0.113949	−	0.993487i	$-0.536350\pi$
$29$	−1.22726	−0.227897	−0.113949	+	0.993487i	$0.536350\pi$
$30$	0.935623	0.170821
$31$	−8.07409	−1.45015	−0.725074	−	0.688671i	$-0.758195\pi$
$31$	−8.07409	−1.45015	−0.725074	+	0.688671i	$0.758195\pi$
$32$	−1.43040	−0.252861
$33$	0	0
$34$	0.837498	0.143630
$35$	2.80853	0.474727
$36$	−9.25435	−1.54239
$37$	1.53525	0.252394	0.126197	−	0.992005i	$-0.459723\pi$
$37$	1.53525	0.252394	0.126197	+	0.992005i	$0.459723\pi$
$38$	0.908009	0.147298
$39$	−2.98836	−0.478521
$40$	−1.34724	−0.213017
$41$	−9.29986	−1.45239	−0.726197	−	0.687487i	$-0.758714\pi$
$41$	−9.29986	−1.45239	−0.726197	+	0.687487i	$0.758714\pi$
$42$	0.333137	0.0514041
$43$	5.23402	0.798181	0.399091	−	0.916912i	$-0.369326\pi$
$43$	5.23402	0.798181	0.399091	+	0.916912i	$0.369326\pi$
$44$	0	0
$45$	−13.0904	−1.95140
$46$	−0.580799	−0.0856341
$47$	−1.89387	−0.276249	−0.138124	−	0.990415i	$-0.544107\pi$
$47$	−1.89387	−0.276249	−0.138124	+	0.990415i	$0.544107\pi$
$48$	10.8314	1.56337
$49$	1.00000	0.142857
$50$	−0.347577	−0.0491548
$51$	−19.2594	−2.69686
$52$	2.14371	0.297279
$53$	−3.82552	−0.525476	−0.262738	−	0.964867i	$-0.584625\pi$
$53$	−3.82552	−0.525476	−0.262738	+	0.964867i	$0.584625\pi$
$54$	−0.553319	−0.0752972
$55$	0	0
$56$	−0.479696	−0.0641021
$57$	−20.8809	−2.76574
$58$	0.147713	0.0193957
$59$	−6.66349	−0.867513	−0.433756	−	0.901030i	$-0.642812\pi$
$59$	−6.66349	−0.867513	−0.433756	+	0.901030i	$0.642812\pi$
$60$	15.4345	1.99258
$61$	9.79952	1.25470	0.627350	−	0.778737i	$-0.284139\pi$
$61$	9.79952	1.25470	0.627350	+	0.778737i	$0.284139\pi$
$62$	0.971796	0.123418
$63$	−4.66094	−0.587223
$64$	−7.65442	−0.956802
$65$	3.03229	0.376110
$66$	0	0
$67$	−2.06100	−0.251792	−0.125896	−	0.992043i	$-0.540181\pi$
$67$	−2.06100	−0.251792	−0.125896	+	0.992043i	$0.540181\pi$
$68$	13.8158	1.67541
$69$	13.3563	1.60791
$70$	−0.338034	−0.0404028
$71$	12.5212	1.48599	0.742996	−	0.669296i	$-0.233404\pi$
$71$	12.5212	1.48599	0.742996	+	0.669296i	$0.233404\pi$
$72$	2.23583	0.263495
$73$	−2.56708	−0.300454	−0.150227	−	0.988652i	$-0.548000\pi$
$73$	−2.56708	−0.300454	−0.150227	+	0.988652i	$0.548000\pi$
$74$	−0.184783	−0.0214806
$75$	7.99301	0.922953
$76$	14.9789	1.71820
$77$	0	0
$78$	0.359679	0.0407257
$79$	15.3283	1.72457	0.862287	−	0.506420i	$-0.169031\pi$
$79$	15.3283	1.72457	0.862287	+	0.506420i	$0.169031\pi$
$80$	−10.9906	−1.22878
$81$	−1.25848	−0.139832
$82$	1.11933	0.123609
$83$	−2.04602	−0.224580	−0.112290	−	0.993675i	$-0.535819\pi$
$83$	−2.04602	−0.224580	−0.112290	+	0.993675i	$0.535819\pi$
$84$	5.49558	0.599617
$85$	19.5425	2.11968
$86$	−0.629967	−0.0679310
$87$	−3.39687	−0.364183
$88$	0	0
$89$	4.76119	0.504685	0.252342	−	0.967638i	$-0.418799\pi$
$89$	4.76119	0.504685	0.252342	+	0.967638i	$0.418799\pi$
$90$	1.57555	0.166078
$91$	1.07967	0.113181
$92$	−9.58114	−0.998902
$93$	−22.3478	−2.31736
$94$	0.227945	0.0235108
$95$	21.1878	2.17383
$96$	−3.95910	−0.404074
$97$	9.11512	0.925500	0.462750	−	0.886489i	$-0.346863\pi$
$97$	9.11512	0.925500	0.462750	+	0.886489i	$0.346863\pi$
$98$	−0.120360	−0.0121582
$99$	0	0
$100$	−5.73379	−0.573379
$101$	4.74385	0.472031	0.236015	−	0.971749i	$-0.424158\pi$
$101$	4.74385	0.472031	0.236015	+	0.971749i	$0.424158\pi$
$102$	2.31806	0.229522
$103$	0.350901	0.0345753	0.0172876	−	0.999851i	$-0.494497\pi$
$103$	0.350901	0.0345753	0.0172876	+	0.999851i	$0.494497\pi$
$104$	−0.517915	−0.0507858
$105$	7.77355	0.758621
$106$	0.460439	0.0447218
$107$	−2.54774	−0.246299	−0.123149	−	0.992388i	$-0.539299\pi$
$107$	−2.54774	−0.246299	−0.123149	+	0.992388i	$0.539299\pi$
$108$	−9.12781	−0.878324
$109$	3.81522	0.365432	0.182716	−	0.983166i	$-0.441511\pi$
$109$	3.81522	0.365432	0.182716	+	0.983166i	$0.441511\pi$
$110$	0	0
$111$	4.24933	0.403329
$112$	−3.91329	−0.369771
$113$	2.31468	0.217747	0.108873	−	0.994056i	$-0.465276\pi$
$113$	2.31468	0.217747	0.108873	+	0.994056i	$0.465276\pi$
$114$	2.51322	0.235385
$115$	−13.5526	−1.26379
$116$	2.43675	0.226246
$117$	−5.03229	−0.465236
$118$	0.802017	0.0738316
$119$	6.95828	0.637865
$120$	−3.72894	−0.340404
$121$	0	0
$122$	−1.17947	−0.106784
$123$	−25.7405	−2.32094
$124$	16.0312	1.43964
$125$	5.93213	0.530586
$126$	0.560990	0.0499769
$127$	−9.84644	−0.873730	−0.436865	−	0.899527i	$-0.643911\pi$
$127$	−9.84644	−0.873730	−0.436865	+	0.899527i	$0.643911\pi$
$128$	3.78208	0.334291
$129$	14.4869	1.27550
$130$	−0.364966	−0.0320097
$131$	−16.3782	−1.43097	−0.715484	−	0.698629i	$-0.753794\pi$
$131$	−16.3782	−1.43097	−0.715484	+	0.698629i	$0.753794\pi$
$132$	0	0
$133$	7.54411	0.654158
$134$	0.248062	0.0214293
$135$	−12.9114	−1.11123
$136$	−3.33786	−0.286219
$137$	−14.2142	−1.21440	−0.607199	−	0.794550i	$-0.707707\pi$
$137$	−14.2142	−1.21440	−0.607199	+	0.794550i	$0.707707\pi$
$138$	−1.60756	−0.136844
$139$	9.17942	0.778588	0.389294	−	0.921114i	$-0.372719\pi$
$139$	9.17942	0.778588	0.389294	+	0.921114i	$0.372719\pi$
$140$	−5.57636	−0.471289
$141$	−5.24192	−0.441449
$142$	−1.50705	−0.126469
$143$	0	0
$144$	18.2396	1.51997
$145$	3.44680	0.286241
$146$	0.308974	0.0255708
$147$	2.76784	0.228288
$148$	−3.04827	−0.250566
$149$	−11.6200	−0.951947	−0.475973	−	0.879460i	$-0.657904\pi$
$149$	−11.6200	−0.951947	−0.475973	+	0.879460i	$0.657904\pi$
$150$	−0.962037	−0.0785500
$151$	−17.9150	−1.45790	−0.728952	−	0.684565i	$-0.759992\pi$
$151$	−17.9150	−1.45790	−0.728952	+	0.684565i	$0.759992\pi$
$152$	−3.61888	−0.293530
$153$	−32.4321	−2.62198
$154$	0	0
$155$	22.6763	1.82140
$156$	5.93344	0.475055
$157$	7.42805	0.592823	0.296412	−	0.955060i	$-0.404210\pi$
$157$	7.42805	0.592823	0.296412	+	0.955060i	$0.404210\pi$
$158$	−1.84492	−0.146774
$159$	−10.5884	−0.839717
$160$	4.01730	0.317596
$161$	−4.82552	−0.380304
$162$	0.151471	0.0119007
$163$	7.85296	0.615091	0.307546	−	0.951533i	$-0.400492\pi$
$163$	7.85296	0.615091	0.307546	+	0.951533i	$0.400492\pi$
$164$	18.4650	1.44187
$165$	0	0
$166$	0.246259	0.0191134
$167$	5.40259	0.418065	0.209032	−	0.977909i	$-0.432969\pi$
$167$	5.40259	0.418065	0.209032	+	0.977909i	$0.432969\pi$
$168$	−1.32772	−0.102436
$169$	−11.8343	−0.910331
$170$	−2.35214	−0.180401
$171$	−35.1626	−2.68895
$172$	−10.3922	−0.792400
$173$	19.7211	1.49937	0.749685	−	0.661795i	$-0.230205\pi$
$173$	19.7211	1.49937	0.749685	+	0.661795i	$0.230205\pi$
$174$	0.408847	0.0309946
$175$	−2.88781	−0.218298
$176$	0	0
$177$	−18.4435	−1.38630
$178$	−0.573056	−0.0429524
$179$	−23.3292	−1.74370	−0.871851	−	0.489770i	$-0.837081\pi$
$179$	−23.3292	−1.74370	−0.871851	+	0.489770i	$0.837081\pi$
$180$	25.9911	1.93726
$181$	3.08500	0.229306	0.114653	−	0.993406i	$-0.463424\pi$
$181$	3.08500	0.229306	0.114653	+	0.993406i	$0.463424\pi$
$182$	−0.129949	−0.00963250
$183$	27.1235	2.00503
$184$	2.31478	0.170648
$185$	−4.31180	−0.317010
$186$	2.68978	0.197224
$187$	0	0
$188$	3.76030	0.274248
$189$	−4.59720	−0.334397
$190$	−2.55017	−0.185008
$191$	12.5715	0.909640	0.454820	−	0.890583i	$-0.349704\pi$
$191$	12.5715	0.909640	0.454820	+	0.890583i	$0.349704\pi$
$192$	−21.1862	−1.52898
$193$	20.6685	1.48775	0.743877	−	0.668317i	$-0.232985\pi$
$193$	20.6685	1.48775	0.743877	+	0.668317i	$0.232985\pi$
$194$	−1.09709	−0.0787668
$195$	8.39290	0.601028
$196$	−1.98551	−0.141822
$197$	−6.27954	−0.447399	−0.223699	−	0.974658i	$-0.571813\pi$
$197$	−6.27954	−0.447399	−0.223699	+	0.974658i	$0.571813\pi$
$198$	0	0
$199$	−6.86896	−0.486927	−0.243464	−	0.969910i	$-0.578284\pi$
$199$	−6.86896	−0.486927	−0.243464	+	0.969910i	$0.578284\pi$
$200$	1.38527	0.0979536
$201$	−5.70453	−0.402366
$202$	−0.570969	−0.0401733
$203$	1.22726	0.0861370
$204$	38.2398	2.67732
$205$	26.1189	1.82422
$206$	−0.0422344	−0.00294261
$207$	22.4914	1.56326
$208$	−4.22508	−0.292957
$209$	0	0
$210$	−0.935623	−0.0645641
$211$	−10.2586	−0.706232	−0.353116	−	0.935580i	$-0.614878\pi$
$211$	−10.2586	−0.706232	−0.353116	+	0.935580i	$0.614878\pi$
$212$	7.59562	0.521669
$213$	34.6567	2.37463
$214$	0.306645	0.0209618
$215$	−14.6999	−1.00252
$216$	2.20526	0.150049
$217$	8.07409	0.548105
$218$	−0.459199	−0.0311009
$219$	−7.10527	−0.480130
$220$	0	0
$221$	7.51268	0.505358
$222$	−0.511449	−0.0343262
$223$	−13.5221	−0.905506	−0.452753	−	0.891636i	$-0.649558\pi$
$223$	−13.5221	−0.905506	−0.452753	+	0.891636i	$0.649558\pi$
$224$	1.43040	0.0955723
$225$	13.4599	0.897328
$226$	−0.278595	−0.0185319
$227$	−13.5764	−0.901100	−0.450550	−	0.892751i	$-0.648772\pi$
$227$	−13.5764	−0.901100	−0.450550	+	0.892751i	$0.648772\pi$
$228$	41.4593	2.74571
$229$	−1.45296	−0.0960141	−0.0480070	−	0.998847i	$-0.515287\pi$
$229$	−1.45296	−0.0960141	−0.0480070	+	0.998847i	$0.515287\pi$
$230$	1.63119	0.107557
$231$	0	0
$232$	−0.588713	−0.0386509
$233$	−8.20387	−0.537453	−0.268727	−	0.963216i	$-0.586603\pi$
$233$	−8.20387	−0.537453	−0.268727	+	0.963216i	$0.586603\pi$
$234$	0.605686	0.0395949
$235$	5.31897	0.346971
$236$	13.2305	0.861229
$237$	42.4264	2.75589
$238$	−0.837498	−0.0542870
$239$	−10.3835	−0.671655	−0.335827	−	0.941924i	$-0.609016\pi$
$239$	−10.3835	−0.671655	−0.335827	+	0.941924i	$0.609016\pi$
$240$	−30.4202	−1.96361
$241$	8.20445	0.528495	0.264248	−	0.964455i	$-0.414876\pi$
$241$	8.20445	0.528495	0.264248	+	0.964455i	$0.414876\pi$
$242$	0	0
$243$	−17.2749	−1.10818
$244$	−19.4571	−1.24561
$245$	−2.80853	−0.179430
$246$	3.09813	0.197529
$247$	8.14519	0.518266
$248$	−3.87311	−0.245943
$249$	−5.66306	−0.358882
$250$	−0.713990	−0.0451567
$251$	−16.4452	−1.03801	−0.519005	−	0.854771i	$-0.673698\pi$
$251$	−16.4452	−1.03801	−0.519005	+	0.854771i	$0.673698\pi$
$252$	9.25435	0.582969
$253$	0	0
$254$	1.18512	0.0743608
$255$	54.0906	3.38728
$256$	14.8536	0.928352
$257$	−12.0440	−0.751282	−0.375641	−	0.926765i	$-0.622577\pi$
$257$	−12.0440	−0.751282	−0.375641	+	0.926765i	$0.622577\pi$
$258$	−1.74365	−0.108555
$259$	−1.53525	−0.0953960
$260$	−6.02066	−0.373385
$261$	−5.72020	−0.354071
$262$	1.97127	0.121786
$263$	−1.87602	−0.115681	−0.0578403	−	0.998326i	$-0.518421\pi$
$263$	−1.87602	−0.115681	−0.0578403	+	0.998326i	$0.518421\pi$
$264$	0	0
$265$	10.7441	0.660003
$266$	−0.908009	−0.0556736
$267$	13.1782	0.806493
$268$	4.09215	0.249968
$269$	−14.0290	−0.855362	−0.427681	−	0.903930i	$-0.640669\pi$
$269$	−14.0290	−0.855362	−0.427681	+	0.903930i	$0.640669\pi$
$270$	1.55401	0.0945741
$271$	−1.35328	−0.0822056	−0.0411028	−	0.999155i	$-0.513087\pi$
$271$	−1.35328	−0.0822056	−0.0411028	+	0.999155i	$0.513087\pi$
$272$	−27.2298	−1.65105
$273$	2.98836	0.180864
$274$	1.71081	0.103354
$275$	0	0
$276$	−26.5190	−1.59626
$277$	13.3835	0.804138	0.402069	−	0.915609i	$-0.368291\pi$
$277$	13.3835	0.804138	0.402069	+	0.915609i	$0.368291\pi$
$278$	−1.10483	−0.0662635
$279$	−37.6328	−2.25302
$280$	1.34724	0.0805129
$281$	−2.31887	−0.138332	−0.0691661	−	0.997605i	$-0.522034\pi$
$281$	−2.31887	−0.138332	−0.0691661	+	0.997605i	$0.522034\pi$
$282$	0.630916	0.0375705
$283$	22.2679	1.32369	0.661845	−	0.749641i	$-0.269774\pi$
$283$	22.2679	1.32369	0.661845	+	0.749641i	$0.269774\pi$
$284$	−24.8610	−1.47523
$285$	58.6445	3.47380
$286$	0	0
$287$	9.29986	0.548953
$288$	−6.66698	−0.392856
$289$	31.4177	1.84810
$290$	−0.414857	−0.0243612
$291$	25.2292	1.47896
$292$	5.09698	0.298278
$293$	2.43981	0.142535	0.0712675	−	0.997457i	$-0.477296\pi$
$293$	2.43981	0.142535	0.0712675	+	0.997457i	$0.477296\pi$
$294$	−0.333137	−0.0194289
$295$	18.7146	1.08961
$296$	0.736455	0.0428056
$297$	0	0
$298$	1.39858	0.0810176
$299$	−5.20999	−0.301301
$300$	−15.8702	−0.916268
$301$	−5.23402	−0.301684
$302$	2.15625	0.124078
$303$	13.1302	0.754311
$304$	−29.5223	−1.69322
$305$	−27.5222	−1.57592
$306$	3.90353	0.223150
$307$	8.89055	0.507410	0.253705	−	0.967282i	$-0.418351\pi$
$307$	8.89055	0.507410	0.253705	+	0.967282i	$0.418351\pi$
$308$	0	0
$309$	0.971237	0.0552517
$310$	−2.72931	−0.155015
$311$	−11.8347	−0.671085	−0.335543	−	0.942025i	$-0.608920\pi$
$311$	−11.8347	−0.671085	−0.335543	+	0.942025i	$0.608920\pi$
$312$	−1.43351	−0.0811563
$313$	−29.5406	−1.66973	−0.834865	−	0.550454i	$-0.814455\pi$
$313$	−29.5406	−1.66973	−0.834865	+	0.550454i	$0.814455\pi$
$314$	−0.894039	−0.0504536
$315$	13.0904	0.737558
$316$	−30.4346	−1.71208
$317$	−25.1545	−1.41282	−0.706409	−	0.707804i	$-0.749686\pi$
$317$	−25.1545	−1.41282	−0.706409	+	0.707804i	$0.749686\pi$
$318$	1.27442	0.0714660
$319$	0	0
$320$	21.4976	1.20175
$321$	−7.05172	−0.393589
$322$	0.580799	0.0323667
$323$	52.4941	2.92085
$324$	2.49874	0.138819
$325$	−3.11790	−0.172950
$326$	−0.945181	−0.0523488
$327$	10.5599	0.583964
$328$	−4.46110	−0.246323
$329$	1.89387	0.104412
$330$	0	0
$331$	−9.46333	−0.520152	−0.260076	−	0.965588i	$-0.583748\pi$
$331$	−9.46333	−0.520152	−0.260076	+	0.965588i	$0.583748\pi$
$332$	4.06241	0.222954
$333$	7.15572	0.392131
$334$	−0.650255	−0.0355804
$335$	5.78838	0.316253
$336$	−10.8314	−0.590899
$337$	17.2248	0.938297	0.469148	−	0.883119i	$-0.344561\pi$
$337$	17.2248	0.938297	0.469148	+	0.883119i	$0.344561\pi$
$338$	1.42438	0.0774758
$339$	6.40667	0.347962
$340$	−38.8019	−2.10433
$341$	0	0
$342$	4.23217	0.228850
$343$	−1.00000	−0.0539949
$344$	2.51074	0.135370
$345$	−37.5114	−2.01955
$346$	−2.37363	−0.127607
$347$	8.73485	0.468911	0.234456	−	0.972127i	$-0.424669\pi$
$347$	8.73485	0.468911	0.234456	+	0.972127i	$0.424669\pi$
$348$	6.74453	0.361545
$349$	29.9851	1.60506	0.802532	−	0.596609i	$-0.203486\pi$
$349$	29.9851	1.60506	0.802532	+	0.596609i	$0.203486\pi$
$350$	0.347577	0.0185788
$351$	−4.96348	−0.264931
$352$	0	0
$353$	7.31999	0.389604	0.194802	−	0.980843i	$-0.437594\pi$
$353$	7.31999	0.389604	0.194802	+	0.980843i	$0.437594\pi$
$354$	2.21985	0.117984
$355$	−35.1661	−1.86642
$356$	−9.45340	−0.501029
$357$	19.2594	1.01932
$358$	2.80789	0.148402
$359$	−4.96996	−0.262304	−0.131152	−	0.991362i	$-0.541868\pi$
$359$	−4.96996	−0.262304	−0.131152	+	0.991362i	$0.541868\pi$
$360$	−6.27939	−0.330953
$361$	37.9137	1.99546
$362$	−0.371311	−0.0195156
$363$	0	0
$364$	−2.14371	−0.112361
$365$	7.20971	0.377374
$366$	−3.26458	−0.170642
$367$	−14.2042	−0.741455	−0.370727	−	0.928742i	$-0.620892\pi$
$367$	−14.2042	−0.741455	−0.370727	+	0.928742i	$0.620892\pi$
$368$	18.8837	0.984379
$369$	−43.3460	−2.25650
$370$	0.518967	0.0269798
$371$	3.82552	0.198611
$372$	44.3718	2.30057
$373$	−8.97781	−0.464853	−0.232427	−	0.972614i	$-0.574667\pi$
$373$	−8.97781	−0.464853	−0.232427	+	0.972614i	$0.574667\pi$
$374$	0	0
$375$	16.4192	0.847883
$376$	−0.908480	−0.0468513
$377$	1.32504	0.0682433
$378$	0.553319	0.0284597
$379$	10.7896	0.554223	0.277112	−	0.960838i	$-0.410623\pi$
$379$	10.7896	0.554223	0.277112	+	0.960838i	$0.410623\pi$
$380$	−42.0687	−2.15808
$381$	−27.2534	−1.39623
$382$	−1.51310	−0.0774170
$383$	−23.5100	−1.20130	−0.600652	−	0.799511i	$-0.705092\pi$
$383$	−23.5100	−1.20130	−0.600652	+	0.799511i	$0.705092\pi$
$384$	10.4682	0.534202
$385$	0	0
$386$	−2.48766	−0.126619
$387$	24.3954	1.24009
$388$	−18.0982	−0.918797
$389$	−11.8745	−0.602063	−0.301032	−	0.953614i	$-0.597331\pi$
$389$	−11.8745	−0.602063	−0.301032	+	0.953614i	$0.597331\pi$
$390$	−1.01017	−0.0511519
$391$	−33.5773	−1.69808
$392$	0.479696	0.0242283
$393$	−45.3322	−2.28670
$394$	0.755804	0.0380769
$395$	−43.0501	−2.16608
$396$	0	0
$397$	−23.7264	−1.19079	−0.595397	−	0.803431i	$-0.703005\pi$
$397$	−23.7264	−1.19079	−0.595397	+	0.803431i	$0.703005\pi$
$398$	0.826747	0.0414411
$399$	20.8809	1.04535
$400$	11.3009	0.565043
$401$	−3.80121	−0.189823	−0.0949117	−	0.995486i	$-0.530257\pi$
$401$	−3.80121	−0.189823	−0.0949117	+	0.995486i	$0.530257\pi$
$402$	0.686596	0.0342443
$403$	8.71738	0.434244
$404$	−9.41898	−0.468612
$405$	3.53448	0.175630
$406$	−0.147713	−0.00733089
$407$	0	0
$408$	−9.23866	−0.457382
$409$	5.98291	0.295836	0.147918	−	0.989000i	$-0.452743\pi$
$409$	5.98291	0.295836	0.147918	+	0.989000i	$0.452743\pi$
$410$	−3.14367	−0.155255
$411$	−39.3425	−1.94062
$412$	−0.696718	−0.0343248
$413$	6.66349	0.327889
$414$	−2.70707	−0.133045
$415$	5.74631	0.282075
$416$	1.54436	0.0757185
$417$	25.4072	1.24419
$418$	0	0
$419$	−4.16889	−0.203664	−0.101832	−	0.994802i	$-0.532470\pi$
$419$	−4.16889	−0.203664	−0.101832	+	0.994802i	$0.532470\pi$
$420$	−15.4345	−0.753126
$421$	23.4555	1.14315	0.571575	−	0.820550i	$-0.306333\pi$
$421$	23.4555	1.14315	0.571575	+	0.820550i	$0.306333\pi$
$422$	1.23473	0.0601055
$423$	−8.82719	−0.429192
$424$	−1.83509	−0.0891197
$425$	−20.0942	−0.974713
$426$	−4.17127	−0.202099
$427$	−9.79952	−0.474232
$428$	5.05856	0.244515
$429$	0	0
$430$	1.76928	0.0853221
$431$	−37.2730	−1.79538	−0.897689	−	0.440630i	$-0.854755\pi$
$431$	−37.2730	−1.79538	−0.897689	+	0.440630i	$0.854755\pi$
$432$	17.9902	0.865554
$433$	−22.7863	−1.09504	−0.547520	−	0.836793i	$-0.684428\pi$
$433$	−22.7863	−1.09504	−0.547520	+	0.836793i	$0.684428\pi$
$434$	−0.971796	−0.0466477
$435$	9.54019	0.457417
$436$	−7.57517	−0.362785
$437$	−36.4043	−1.74145
$438$	0.855190	0.0408625
$439$	1.42974	0.0682379	0.0341189	−	0.999418i	$-0.489137\pi$
$439$	1.42974	0.0682379	0.0341189	+	0.999418i	$0.489137\pi$
$440$	0	0
$441$	4.66094	0.221949
$442$	−0.904225	−0.0430096
$443$	−26.4301	−1.25573	−0.627866	−	0.778321i	$-0.716072\pi$
$443$	−26.4301	−1.25573	−0.627866	+	0.778321i	$0.716072\pi$
$444$	−8.43711	−0.400408
$445$	−13.3719	−0.633890
$446$	1.62752	0.0770651
$447$	−32.1623	−1.52122
$448$	7.65442	0.361637
$449$	14.0870	0.664806	0.332403	−	0.943137i	$-0.392141\pi$
$449$	14.0870	0.664806	0.332403	+	0.943137i	$0.392141\pi$
$450$	−1.62003	−0.0763691
$451$	0	0
$452$	−4.59583	−0.216170
$453$	−49.5859	−2.32975
$454$	1.63406	0.0766902
$455$	−3.03229	−0.142156
$456$	−10.0165	−0.469065
$457$	−29.4829	−1.37915	−0.689575	−	0.724214i	$-0.742203\pi$
$457$	−29.4829	−1.37915	−0.689575	+	0.724214i	$0.742203\pi$
$458$	0.174878	0.00817150
$459$	−31.9887	−1.49310
$460$	26.9089	1.25463
$461$	−31.7282	−1.47773	−0.738866	−	0.673853i	$-0.764638\pi$
$461$	−31.7282	−1.47773	−0.738866	+	0.673853i	$0.764638\pi$
$462$	0	0
$463$	−12.7839	−0.594117	−0.297059	−	0.954859i	$-0.596006\pi$
$463$	−12.7839	−0.594117	−0.297059	+	0.954859i	$0.596006\pi$
$464$	−4.80264	−0.222957
$465$	62.7643	2.91062
$466$	0.987417	0.0457412
$467$	29.5768	1.36865	0.684325	−	0.729177i	$-0.260097\pi$
$467$	29.5768	1.36865	0.684325	+	0.729177i	$0.260097\pi$
$468$	9.99168	0.461866
$469$	2.06100	0.0951683
$470$	−0.640191	−0.0295298
$471$	20.5597	0.947339
$472$	−3.19645	−0.147129
$473$	0	0
$474$	−5.10644	−0.234546
$475$	−21.7860	−0.999610
$476$	−13.8158	−0.633245
$477$	−17.8305	−0.816403
$478$	1.24976	0.0571627
$479$	28.5574	1.30482	0.652411	−	0.757866i	$-0.273758\pi$
$479$	28.5574	1.30482	0.652411	+	0.757866i	$0.273758\pi$
$480$	11.1192	0.507522
$481$	−1.65757	−0.0755788
$482$	−0.987487	−0.0449788
$483$	−13.3563	−0.607731
$484$	0	0
$485$	−25.6000	−1.16244
$486$	2.07920	0.0943146
$487$	1.05828	0.0479552	0.0239776	−	0.999712i	$-0.492367\pi$
$487$	1.05828	0.0479552	0.0239776	+	0.999712i	$0.492367\pi$
$488$	4.70079	0.212795
$489$	21.7357	0.982924
$490$	0.338034	0.0152708
$491$	−18.4489	−0.832586	−0.416293	−	0.909230i	$-0.636671\pi$
$491$	−18.4489	−0.832586	−0.416293	+	0.909230i	$0.636671\pi$
$492$	51.1081	2.30413
$493$	8.53965	0.384606
$494$	−0.980354	−0.0441082
$495$	0	0
$496$	−31.5962	−1.41871
$497$	−12.5212	−0.561652
$498$	0.681606	0.0305435
$499$	31.7293	1.42040	0.710199	−	0.704001i	$-0.248605\pi$
$499$	31.7293	1.42040	0.710199	+	0.704001i	$0.248605\pi$
$500$	−11.7783	−0.526742
$501$	14.9535	0.668073
$502$	1.97934	0.0883423
$503$	29.0283	1.29431	0.647154	−	0.762359i	$-0.275959\pi$
$503$	29.0283	1.29431	0.647154	+	0.762359i	$0.275959\pi$
$504$	−2.23583	−0.0995919
$505$	−13.3232	−0.592876
$506$	0	0
$507$	−32.7555	−1.45472
$508$	19.5502	0.867401
$509$	2.88831	0.128022	0.0640110	−	0.997949i	$-0.479611\pi$
$509$	2.88831	0.128022	0.0640110	+	0.997949i	$0.479611\pi$
$510$	−6.51033	−0.288282
$511$	2.56708	0.113561
$512$	−9.35193	−0.413301
$513$	−34.6818	−1.53124
$514$	1.44961	0.0639396
$515$	−0.985513	−0.0434269
$516$	−28.7640	−1.26626
$517$	0	0
$518$	0.184783	0.00811890
$519$	54.5849	2.39601
$520$	1.45458	0.0637875
$521$	31.6708	1.38752	0.693762	−	0.720204i	$-0.255952\pi$
$521$	31.6708	1.38752	0.693762	+	0.720204i	$0.255952\pi$
$522$	0.688482	0.0301340
$523$	16.0380	0.701295	0.350647	−	0.936508i	$-0.385962\pi$
$523$	16.0380	0.701295	0.350647	+	0.936508i	$0.385962\pi$
$524$	32.5191	1.42060
$525$	−7.99301	−0.348843
$526$	0.225798	0.00984526
$527$	56.1818	2.44732
$528$	0	0
$529$	0.285644	0.0124193
$530$	−1.29316	−0.0561711
$531$	−31.0581	−1.34781
$532$	−14.9789	−0.649419
$533$	10.0408	0.434916
$534$	−1.58613	−0.0686384
$535$	7.15538	0.309354
$536$	−0.988655	−0.0427034
$537$	−64.5714	−2.78646
$538$	1.68853	0.0727976
$539$	0	0
$540$	25.6357	1.10318
$541$	−13.5946	−0.584480	−0.292240	−	0.956345i	$-0.594401\pi$
$541$	−13.5946	−0.584480	−0.292240	+	0.956345i	$0.594401\pi$
$542$	0.162880	0.00699630
$543$	8.53879	0.366435
$544$	9.95310	0.426735
$545$	−10.7151	−0.458986
$546$	−0.359679	−0.0153929
$547$	8.82486	0.377324	0.188662	−	0.982042i	$-0.439585\pi$
$547$	8.82486	0.377324	0.188662	+	0.982042i	$0.439585\pi$
$548$	28.2224	1.20560
$549$	45.6749	1.94936
$550$	0	0
$551$	9.25862	0.394430
$552$	6.40695	0.272698
$553$	−15.3283	−0.651828
$554$	−1.61084	−0.0684380
$555$	−11.9344	−0.506586
$556$	−18.2259	−0.772949
$557$	−33.9920	−1.44029	−0.720145	−	0.693824i	$-0.755925\pi$
$557$	−33.9920	−1.44029	−0.720145	+	0.693824i	$0.755925\pi$
$558$	4.52948	0.191748
$559$	−5.65104	−0.239014
$560$	10.9906	0.464437
$561$	0	0
$562$	0.279099	0.0117731
$563$	20.2256	0.852406	0.426203	−	0.904628i	$-0.359851\pi$
$563$	20.2256	0.852406	0.426203	+	0.904628i	$0.359851\pi$
$564$	10.4079	0.438251
$565$	−6.50084	−0.273492
$566$	−2.68016	−0.112656
$567$	1.25848	0.0528513
$568$	6.00637	0.252022
$569$	−28.9330	−1.21293	−0.606467	−	0.795109i	$-0.707414\pi$
$569$	−28.9330	−1.21293	−0.606467	+	0.795109i	$0.707414\pi$
$570$	−7.05845	−0.295646
$571$	7.51312	0.314414	0.157207	−	0.987566i	$-0.449751\pi$
$571$	7.51312	0.314414	0.157207	+	0.987566i	$0.449751\pi$
$572$	0	0
$573$	34.7958	1.45362
$574$	−1.11933	−0.0467199
$575$	13.9352	0.581138
$576$	−35.6768	−1.48653
$577$	34.3748	1.43104	0.715521	−	0.698592i	$-0.246190\pi$
$577$	34.3748	1.43104	0.715521	+	0.698592i	$0.246190\pi$
$578$	−3.78143	−0.157287
$579$	57.2072	2.37745
$580$	−6.84367	−0.284168
$581$	2.04602	0.0848834
$582$	−3.03658	−0.125870
$583$	0	0
$584$	−1.23142	−0.0509565
$585$	14.1333	0.584341
$586$	−0.293655	−0.0121308
$587$	42.0392	1.73514	0.867571	−	0.497313i	$-0.165680\pi$
$587$	42.0392	1.73514	0.867571	+	0.497313i	$0.165680\pi$
$588$	−5.49558	−0.226634
$589$	60.9118	2.50983
$590$	−2.25249	−0.0927333
$591$	−17.3808	−0.714949
$592$	6.00789	0.246923
$593$	−29.2867	−1.20266	−0.601331	−	0.799000i	$-0.705363\pi$
$593$	−29.2867	−1.20266	−0.601331	+	0.799000i	$0.705363\pi$
$594$	0	0
$595$	−19.5425	−0.801165
$596$	23.0716	0.945051
$597$	−19.0122	−0.778116
$598$	0.627074	0.0256430
$599$	−15.1577	−0.619327	−0.309663	−	0.950846i	$-0.600216\pi$
$599$	−15.1577	−0.619327	−0.309663	+	0.950846i	$0.600216\pi$
$600$	3.83421	0.156531
$601$	−31.8735	−1.30015	−0.650073	−	0.759872i	$-0.725262\pi$
$601$	−31.8735	−1.30015	−0.650073	+	0.759872i	$0.725262\pi$
$602$	0.629967	0.0256755
$603$	−9.60621	−0.391195
$604$	35.5705	1.44734
$605$	0	0
$606$	−1.58035	−0.0641974
$607$	20.1463	0.817714	0.408857	−	0.912598i	$-0.365927\pi$
$607$	20.1463	0.817714	0.408857	+	0.912598i	$0.365927\pi$
$608$	10.7911	0.437635
$609$	3.39687	0.137648
$610$	3.31257	0.134122
$611$	2.04476	0.0827220
$612$	64.3944	2.60299
$613$	3.89132	0.157169	0.0785844	−	0.996907i	$-0.474960\pi$
$613$	3.89132	0.157169	0.0785844	+	0.996907i	$0.474960\pi$
$614$	−1.07007	−0.0431843
$615$	72.2929	2.91513
$616$	0	0
$617$	28.6122	1.15189	0.575943	−	0.817490i	$-0.304635\pi$
$617$	28.6122	1.15189	0.575943	+	0.817490i	$0.304635\pi$
$618$	−0.116898	−0.00470233
$619$	−26.1546	−1.05124	−0.525621	−	0.850719i	$-0.676167\pi$
$619$	−26.1546	−1.05124	−0.525621	+	0.850719i	$0.676167\pi$
$620$	−45.0240	−1.80821
$621$	22.1839	0.890209
$622$	1.42443	0.0571143
$623$	−4.76119	−0.190753
$624$	−11.6943	−0.468148
$625$	−31.0996	−1.24398
$626$	3.55550	0.142106
$627$	0	0
$628$	−14.7485	−0.588529
$629$	−10.6827	−0.425948
$630$	−1.57555	−0.0627716
$631$	−5.68272	−0.226225	−0.113113	−	0.993582i	$-0.536082\pi$
$631$	−5.68272	−0.226225	−0.113113	+	0.993582i	$0.536082\pi$
$632$	7.35295	0.292485
$633$	−28.3942	−1.12857
$634$	3.02759	0.120241
$635$	27.6540	1.09741
$636$	21.0235	0.833635
$637$	−1.07967	−0.0427782
$638$	0	0
$639$	58.3605	2.30870
$640$	−10.6221	−0.419874
$641$	−20.7292	−0.818756	−0.409378	−	0.912365i	$-0.634254\pi$
$641$	−20.7292	−0.818756	−0.409378	+	0.912365i	$0.634254\pi$
$642$	0.848744	0.0334973
$643$	21.2756	0.839029	0.419515	−	0.907749i	$-0.362200\pi$
$643$	21.2756	0.839029	0.419515	+	0.907749i	$0.362200\pi$
$644$	9.58114	0.377550
$645$	−40.6869	−1.60205
$646$	−6.31818	−0.248586
$647$	−17.2718	−0.679023	−0.339512	−	0.940602i	$-0.610262\pi$
$647$	−17.2718	−0.679023	−0.339512	+	0.940602i	$0.610262\pi$
$648$	−0.603690	−0.0237152
$649$	0	0
$650$	0.375270	0.0147193
$651$	22.3478	0.875878
$652$	−15.5922	−0.610636
$653$	41.0788	1.60754	0.803768	−	0.594942i	$-0.202825\pi$
$653$	41.0788	1.60754	0.803768	+	0.594942i	$0.202825\pi$
$654$	−1.27099	−0.0496996
$655$	45.9985	1.79731
$656$	−36.3930	−1.42091
$657$	−11.9650	−0.466799
$658$	−0.227945	−0.00888624
$659$	−6.00410	−0.233887	−0.116943	−	0.993139i	$-0.537310\pi$
$659$	−6.00410	−0.233887	−0.116943	+	0.993139i	$0.537310\pi$
$660$	0	0
$661$	−1.83502	−0.0713739	−0.0356870	−	0.999363i	$-0.511362\pi$
$661$	−1.83502	−0.0713739	−0.0356870	+	0.999363i	$0.511362\pi$
$662$	1.13901	0.0442687
$663$	20.7939	0.807568
$664$	−0.981469	−0.0380884
$665$	−21.1878	−0.821629
$666$	−0.861261	−0.0333732
$667$	−5.92218	−0.229308
$668$	−10.7269	−0.415037
$669$	−37.4270	−1.44701
$670$	−0.696689	−0.0269154
$671$	0	0
$672$	3.95910	0.152726
$673$	−44.4403	−1.71305	−0.856524	−	0.516107i	$-0.827381\pi$
$673$	−44.4403	−1.71305	−0.856524	+	0.516107i	$0.827381\pi$
$674$	−2.07318	−0.0798559
$675$	13.2759	0.510989
$676$	23.4972	0.903737
$677$	−15.5471	−0.597525	−0.298762	−	0.954327i	$-0.596574\pi$
$677$	−15.5471	−0.597525	−0.298762	+	0.954327i	$0.596574\pi$
$678$	−0.771106	−0.0296141
$679$	−9.11512	−0.349806
$680$	9.37447	0.359494
$681$	−37.5774	−1.43997
$682$	0	0
$683$	−36.8979	−1.41186	−0.705930	−	0.708282i	$-0.749471\pi$
$683$	−36.8979	−1.41186	−0.705930	+	0.708282i	$0.749471\pi$
$684$	69.8159	2.66948
$685$	39.9208	1.52530
$686$	0.120360	0.00459536
$687$	−4.02155	−0.153432
$688$	20.4823	0.780879
$689$	4.13032	0.157352
$690$	4.51487	0.171878
$691$	−18.6726	−0.710339	−0.355169	−	0.934802i	$-0.615577\pi$
$691$	−18.6726	−0.710339	−0.355169	+	0.934802i	$0.615577\pi$
$692$	−39.1566	−1.48851
$693$	0	0
$694$	−1.05133	−0.0399078
$695$	−25.7806	−0.977915
$696$	−1.62946	−0.0617647
$697$	64.7110	2.45111
$698$	−3.60900	−0.136603
$699$	−22.7070	−0.858857
$700$	5.73379	0.216717
$701$	26.1328	0.987022	0.493511	−	0.869739i	$-0.335713\pi$
$701$	26.1328	0.987022	0.493511	+	0.869739i	$0.335713\pi$
$702$	0.597404	0.0225476
$703$	−11.5821	−0.436828
$704$	0	0
$705$	14.7221	0.554465
$706$	−0.881033	−0.0331581
$707$	−4.74385	−0.178411
$708$	36.6198	1.37626
$709$	−16.4449	−0.617602	−0.308801	−	0.951127i	$-0.599928\pi$
$709$	−16.4449	−0.617602	−0.308801	+	0.951127i	$0.599928\pi$
$710$	4.23259	0.158846
$711$	71.4445	2.67938
$712$	2.28392	0.0855936
$713$	−38.9617	−1.45913
$714$	−2.31806	−0.0867513
$715$	0	0
$716$	46.3204	1.73107
$717$	−28.7400	−1.07331
$718$	0.598184	0.0223240
$719$	−9.34913	−0.348664	−0.174332	−	0.984687i	$-0.555777\pi$
$719$	−9.34913	−0.348664	−0.174332	+	0.984687i	$0.555777\pi$
$720$	−51.2264	−1.90909
$721$	−0.350901	−0.0130682
$722$	−4.56328	−0.169828
$723$	22.7086	0.844542
$724$	−6.12531	−0.227646
$725$	−3.54411	−0.131625
$726$	0	0
$727$	27.7523	1.02928	0.514638	−	0.857408i	$-0.327927\pi$
$727$	27.7523	1.02928	0.514638	+	0.857408i	$0.327927\pi$
$728$	0.517915	0.0191952
$729$	−44.0387	−1.63106
$730$	−0.867760	−0.0321173
$731$	−36.4198	−1.34704
$732$	−53.8541	−1.99050
$733$	−1.45630	−0.0537896	−0.0268948	−	0.999638i	$-0.508562\pi$
$733$	−1.45630	−0.0537896	−0.0268948	+	0.999638i	$0.508562\pi$
$734$	1.70962	0.0631032
$735$	−7.77355	−0.286732
$736$	−6.90240	−0.254426
$737$	0	0
$738$	5.21712	0.192045
$739$	27.5966	1.01516	0.507579	−	0.861605i	$-0.330541\pi$
$739$	27.5966	1.01516	0.507579	+	0.861605i	$0.330541\pi$
$740$	8.56113	0.314713
$741$	22.5446	0.828195
$742$	−0.460439	−0.0169033
$743$	−27.9773	−1.02639	−0.513194	−	0.858273i	$-0.671538\pi$
$743$	−27.9773	−1.02639	−0.513194	+	0.858273i	$0.671538\pi$
$744$	−10.7201	−0.393019
$745$	32.6350	1.19566
$746$	1.08057	0.0395624
$747$	−9.53638	−0.348918
$748$	0	0
$749$	2.54774	0.0930922
$750$	−1.97621	−0.0721610
$751$	34.0957	1.24417	0.622085	−	0.782950i	$-0.286286\pi$
$751$	34.0957	1.24417	0.622085	+	0.782950i	$0.286286\pi$
$752$	−7.41125	−0.270260
$753$	−45.5176	−1.65875
$754$	−0.159482	−0.00580800
$755$	50.3148	1.83114
$756$	9.12781	0.331975
$757$	−39.7629	−1.44521	−0.722604	−	0.691262i	$-0.757055\pi$
$757$	−39.7629	−1.44521	−0.722604	+	0.691262i	$0.757055\pi$
$758$	−1.29863	−0.0471684
$759$	0	0
$760$	10.1637	0.368677
$761$	3.82415	0.138625	0.0693127	−	0.997595i	$-0.477919\pi$
$761$	3.82415	0.138625	0.0693127	+	0.997595i	$0.477919\pi$
$762$	3.28021	0.118830
$763$	−3.81522	−0.138120
$764$	−24.9608	−0.903051
$765$	91.0864	3.29324
$766$	2.82966	0.102240
$767$	7.19440	0.259775
$768$	41.1125	1.48352
$769$	19.6583	0.708897	0.354449	−	0.935075i	$-0.384669\pi$
$769$	19.6583	0.708897	0.354449	+	0.935075i	$0.384669\pi$
$770$	0	0
$771$	−33.3358	−1.20056
$772$	−41.0376	−1.47698
$773$	−17.5966	−0.632905	−0.316452	−	0.948608i	$-0.602492\pi$
$773$	−17.5966	−0.632905	−0.316452	+	0.948608i	$0.602492\pi$
$774$	−2.93623	−0.105541
$775$	−23.3165	−0.837552
$776$	4.37249	0.156963
$777$	−4.24933	−0.152444
$778$	1.42922	0.0512400
$779$	70.1592	2.51371
$780$	−16.6642	−0.596675
$781$	0	0
$782$	4.04136	0.144519
$783$	−5.64198	−0.201628
$784$	3.91329	0.139760
$785$	−20.8619	−0.744592
$786$	5.45617	0.194615
$787$	54.0967	1.92834	0.964169	−	0.265287i	$-0.0854667\pi$
$787$	54.0967	1.92834	0.964169	+	0.265287i	$0.0854667\pi$
$788$	12.4681	0.444158
$789$	−5.19253	−0.184859
$790$	5.18150	0.184349
$791$	−2.31468	−0.0823006
$792$	0	0
$793$	−10.5803	−0.375717
$794$	2.85571	0.101345
$795$	29.7379	1.05469
$796$	13.6384	0.483401
$797$	36.5244	1.29376	0.646881	−	0.762591i	$-0.276073\pi$
$797$	36.5244	1.29376	0.646881	+	0.762591i	$0.276073\pi$
$798$	−2.51322	−0.0889671
$799$	13.1781	0.466206
$800$	−4.13072	−0.146043
$801$	22.1916	0.784101
$802$	0.457513	0.0161554
$803$	0	0
$804$	11.3264	0.399452
$805$	13.5526	0.477666
$806$	−1.04922	−0.0369573
$807$	−38.8300	−1.36688
$808$	2.27561	0.0800555
$809$	−46.9354	−1.65016	−0.825081	−	0.565015i	$-0.808870\pi$
$809$	−46.9354	−1.65016	−0.825081	+	0.565015i	$0.808870\pi$
$810$	−0.425410	−0.0149474
$811$	12.6615	0.444605	0.222303	−	0.974978i	$-0.428643\pi$
$811$	12.6615	0.444605	0.222303	+	0.974978i	$0.428643\pi$
$812$	−2.43675	−0.0855131
$813$	−3.74565	−0.131366
$814$	0	0
$815$	−22.0552	−0.772561
$816$	−75.3677	−2.63840
$817$	−39.4861	−1.38144
$818$	−0.720103	−0.0251778
$819$	5.03229	0.175843
$820$	−51.8594	−1.81101
$821$	49.2938	1.72037	0.860183	−	0.509986i	$-0.170349\pi$
$821$	49.2938	1.72037	0.860183	+	0.509986i	$0.170349\pi$
$822$	4.73526	0.165161
$823$	19.2259	0.670173	0.335087	−	0.942187i	$-0.391234\pi$
$823$	19.2259	0.670173	0.335087	+	0.942187i	$0.391234\pi$
$824$	0.168326	0.00586390
$825$	0	0
$826$	−0.802017	−0.0279057
$827$	−43.6419	−1.51758	−0.758789	−	0.651337i	$-0.774208\pi$
$827$	−43.6419	−1.51758	−0.758789	+	0.651337i	$0.774208\pi$
$828$	−44.6571	−1.55194
$829$	−36.7072	−1.27489	−0.637447	−	0.770494i	$-0.720010\pi$
$829$	−36.7072	−1.27489	−0.637447	+	0.770494i	$0.720010\pi$
$830$	−0.691625	−0.0240067
$831$	37.0434	1.28502
$832$	8.26428	0.286512
$833$	−6.95828	−0.241090
$834$	−3.05800	−0.105890
$835$	−15.1733	−0.525094
$836$	0	0
$837$	−37.1182	−1.28299
$838$	0.501768	0.0173333
$839$	40.4545	1.39665	0.698323	−	0.715783i	$-0.253930\pi$
$839$	40.4545	1.39665	0.698323	+	0.715783i	$0.253930\pi$
$840$	3.72894	0.128661
$841$	−27.4938	−0.948063
$842$	−2.82310	−0.0972904
$843$	−6.41826	−0.221057
$844$	20.3686	0.701117
$845$	33.2369	1.14339
$846$	1.06244	0.0365274
$847$	0	0
$848$	−14.9704	−0.514085
$849$	61.6340	2.11527
$850$	2.41854	0.0829552
$851$	7.40840	0.253957
$852$	−68.8113	−2.35743
$853$	−25.4948	−0.872927	−0.436463	−	0.899722i	$-0.643769\pi$
$853$	−25.4948	−0.872927	−0.436463	+	0.899722i	$0.643769\pi$
$854$	1.17947	0.0403606
$855$	98.7552	3.37735
$856$	−1.22214	−0.0417718
$857$	−49.4756	−1.69005	−0.845027	−	0.534723i	$-0.820416\pi$
$857$	−49.4756	−1.69005	−0.845027	+	0.534723i	$0.820416\pi$
$858$	0	0
$859$	21.6779	0.739641	0.369821	−	0.929103i	$-0.379419\pi$
$859$	21.6779	0.739641	0.369821	+	0.929103i	$0.379419\pi$
$860$	29.1868	0.995262
$861$	25.7405	0.877234
$862$	4.48618	0.152800
$863$	13.6398	0.464306	0.232153	−	0.972679i	$-0.425423\pi$
$863$	13.6398	0.464306	0.232153	+	0.972679i	$0.425423\pi$
$864$	−6.57582	−0.223714
$865$	−55.3873	−1.88322
$866$	2.74256	0.0931959
$867$	86.9592	2.95329
$868$	−16.0312	−0.544135
$869$	0	0
$870$	−1.14826	−0.0389295
$871$	2.22521	0.0753985
$872$	1.83014	0.0619765
$873$	42.4850	1.43790
$874$	4.38161	0.148210
$875$	−5.93213	−0.200542
$876$	14.1076	0.476652
$877$	−29.1290	−0.983618	−0.491809	−	0.870703i	$-0.663664\pi$
$877$	−29.1290	−0.983618	−0.491809	+	0.870703i	$0.663664\pi$
$878$	−0.172084	−0.00580754
$879$	6.75299	0.227773
$880$	0	0
$881$	−48.8256	−1.64498	−0.822488	−	0.568783i	$-0.807414\pi$
$881$	−48.8256	−1.64498	−0.822488	+	0.568783i	$0.807414\pi$
$882$	−0.560990	−0.0188895
$883$	−24.2131	−0.814837	−0.407419	−	0.913242i	$-0.633571\pi$
$883$	−24.2131	−0.814837	−0.407419	+	0.913242i	$0.633571\pi$
$884$	−14.9165	−0.501697
$885$	51.7990	1.74120
$886$	3.18113	0.106872
$887$	−6.64749	−0.223201	−0.111600	−	0.993753i	$-0.535598\pi$
$887$	−6.64749	−0.223201	−0.111600	+	0.993753i	$0.535598\pi$
$888$	2.03839	0.0684038
$889$	9.84644	0.330239
$890$	1.60944	0.0539486
$891$	0	0
$892$	26.8483	0.898947
$893$	14.2875	0.478114
$894$	3.87105	0.129467
$895$	65.5205	2.19011
$896$	−3.78208	−0.126350
$897$	−14.4204	−0.481484
$898$	−1.69551	−0.0565799
$899$	9.90903	0.330485
$900$	−26.7248	−0.890828
$901$	26.6191	0.886809
$902$	0	0
$903$	−14.4869	−0.482095
$904$	1.11034	0.0369295
$905$	−8.66431	−0.288011
$906$	5.96815	0.198279
$907$	−8.37806	−0.278189	−0.139094	−	0.990279i	$-0.544419\pi$
$907$	−8.37806	−0.278189	−0.139094	+	0.990279i	$0.544419\pi$
$908$	26.9562	0.894573
$909$	22.1108	0.733368
$910$	0.364966	0.0120985
$911$	−18.7868	−0.622434	−0.311217	−	0.950339i	$-0.600737\pi$
$911$	−18.7868	−0.622434	−0.311217	+	0.950339i	$0.600737\pi$
$912$	−81.7130	−2.70579
$913$	0	0
$914$	3.54856	0.117376
$915$	−76.1771	−2.51834
$916$	2.88486	0.0953186
$917$	16.3782	0.540855
$918$	3.85015	0.127074
$919$	−18.0039	−0.593894	−0.296947	−	0.954894i	$-0.595968\pi$
$919$	−18.0039	−0.593894	−0.296947	+	0.954894i	$0.595968\pi$
$920$	−6.50113	−0.214336
$921$	24.6076	0.810848
$922$	3.81881	0.125766
$923$	−13.5188	−0.444977
$924$	0	0
$925$	4.43353	0.145773
$926$	1.53867	0.0505637
$927$	1.63553	0.0537177
$928$	1.75547	0.0576262
$929$	−10.1225	−0.332109	−0.166054	−	0.986117i	$-0.553103\pi$
$929$	−10.1225	−0.332109	−0.166054	+	0.986117i	$0.553103\pi$
$930$	−7.55430	−0.247715
$931$	−7.54411	−0.247248
$932$	16.2889	0.533560
$933$	−32.7566	−1.07240
$934$	−3.55986	−0.116482
$935$	0	0
$936$	−2.41397	−0.0789031
$937$	28.2113	0.921622	0.460811	−	0.887498i	$-0.347559\pi$
$937$	28.2113	0.921622	0.460811	+	0.887498i	$0.347559\pi$
$938$	−0.248062	−0.00809952
$939$	−81.7635	−2.66825
$940$	−10.5609	−0.344458
$941$	20.1501	0.656875	0.328437	−	0.944526i	$-0.393478\pi$
$941$	20.1501	0.656875	0.328437	+	0.944526i	$0.393478\pi$
$942$	−2.47456	−0.0806255
$943$	−44.8766	−1.46138
$944$	−26.0762	−0.848707
$945$	12.9114	0.420007
$946$	0	0
$947$	0.125141	0.00406653	0.00203326	−	0.999998i	$-0.499353\pi$
$947$	0.125141	0.00406653	0.00203326	+	0.999998i	$0.499353\pi$
$948$	−84.2382	−2.73593
$949$	2.77161	0.0899703
$950$	2.62216	0.0850741
$951$	−69.6236	−2.25770
$952$	3.33786	0.108181
$953$	29.3495	0.950723	0.475362	−	0.879790i	$-0.342317\pi$
$953$	29.3495	0.950723	0.475362	+	0.879790i	$0.342317\pi$
$954$	2.14608	0.0694818
$955$	−35.3073	−1.14252
$956$	20.6166	0.666790
$957$	0	0
$958$	−3.43717	−0.111050
$959$	14.2142	0.458999
$960$	59.5020	1.92042
$961$	34.1909	1.10293
$962$	0.199505	0.00643231
$963$	−11.8748	−0.382661
$964$	−16.2900	−0.524667
$965$	−58.0481	−1.86863
$966$	1.60756	0.0517224
$967$	51.9463	1.67048	0.835240	−	0.549885i	$-0.185328\pi$
$967$	51.9463	1.67048	0.835240	+	0.549885i	$0.185328\pi$
$968$	0	0
$969$	145.295	4.66756
$970$	3.08122	0.0989320
$971$	−25.0066	−0.802500	−0.401250	−	0.915969i	$-0.631424\pi$
$971$	−25.0066	−0.802500	−0.401250	+	0.915969i	$0.631424\pi$
$972$	34.2995	1.10016
$973$	−9.17942	−0.294279
$974$	−0.127374	−0.00408134
$975$	−8.62984	−0.276376
$976$	38.3484	1.22750
$977$	−17.4156	−0.557176	−0.278588	−	0.960411i	$-0.589866\pi$
$977$	−17.4156	−0.557176	−0.278588	+	0.960411i	$0.589866\pi$
$978$	−2.61611	−0.0836540
$979$	0	0
$980$	5.57636	0.178130
$981$	17.7825	0.567751
$982$	2.22051	0.0708592
$983$	−42.3349	−1.35028	−0.675138	−	0.737692i	$-0.735916\pi$
$983$	−42.3349	−1.35028	−0.675138	+	0.737692i	$0.735916\pi$
$984$	−12.3476	−0.393628
$985$	17.6362	0.561937
$986$	−1.02783	−0.0327328
$987$	5.24192	0.166852
$988$	−16.1724	−0.514512
$989$	25.2569	0.803122
$990$	0	0
$991$	55.6263	1.76703	0.883513	−	0.468406i	$-0.155172\pi$
$991$	55.6263	1.76703	0.883513	+	0.468406i	$0.155172\pi$
$992$	11.5491	0.366685
$993$	−26.1930	−0.831209
$994$	1.50705	0.0478007
$995$	19.2916	0.611586
$996$	11.2441	0.356283
$997$	16.4267	0.520240	0.260120	−	0.965576i	$-0.416238\pi$
$997$	16.4267	0.520240	0.260120	+	0.965576i	$0.416238\pi$
$998$	−3.81894	−0.120886
$999$	7.05787	0.223301

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.m.1.4	✓	6
3.2	odd	2		7623.2.a.cs.1.3		6
7.6	odd	2		5929.2.a.bj.1.4		6
11.2	odd	10		847.2.f.y.323.4		24
11.3	even	5		847.2.f.z.372.4		24
11.4	even	5		847.2.f.z.148.4		24
11.5	even	5		847.2.f.z.729.3		24
11.6	odd	10		847.2.f.y.729.4		24
11.7	odd	10		847.2.f.y.148.3		24
11.8	odd	10		847.2.f.y.372.3		24
11.9	even	5		847.2.f.z.323.3		24
11.10	odd	2		847.2.a.n.1.3	yes	6
33.32	even	2		7623.2.a.cp.1.4		6
77.76	even	2		5929.2.a.bm.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.m.1.4	✓	6	1.1	even	1	trivial
847.2.a.n.1.3	yes	6	11.10	odd	2
847.2.f.y.148.3		24	11.7	odd	10
847.2.f.y.323.4		24	11.2	odd	10
847.2.f.y.372.3		24	11.8	odd	10
847.2.f.y.729.4		24	11.6	odd	10
847.2.f.z.148.4		24	11.4	even	5
847.2.f.z.323.3		24	11.9	even	5
847.2.f.z.372.4		24	11.3	even	5
847.2.f.z.729.3		24	11.5	even	5
5929.2.a.bj.1.4		6	7.6	odd	2
5929.2.a.bm.1.3		6	77.76	even	2
7623.2.a.cp.1.4		6	33.32	even	2
7623.2.a.cs.1.3		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)

\(f(q)\)	\(=\)	\(q-0.120360 q^{2} +2.76784 q^{3} -1.98551 q^{4} -2.80853 q^{5} -0.333137 q^{6} -1.00000 q^{7} +0.479696 q^{8} +4.66094 q^{9} +O(q^{10})\)	\(q-0.120360 q^{2} +2.76784 q^{3} -1.98551 q^{4} -2.80853 q^{5} -0.333137 q^{6} -1.00000 q^{7} +0.479696 q^{8} +4.66094 q^{9} +0.338034 q^{10} -5.49558 q^{12} -1.07967 q^{13} +0.120360 q^{14} -7.77355 q^{15} +3.91329 q^{16} -6.95828 q^{17} -0.560990 q^{18} -7.54411 q^{19} +5.57636 q^{20} -2.76784 q^{21} +4.82552 q^{23} +1.32772 q^{24} +2.88781 q^{25} +0.129949 q^{26} +4.59720 q^{27} +1.98551 q^{28} -1.22726 q^{29} +0.935623 q^{30} -8.07409 q^{31} -1.43040 q^{32} +0.837498 q^{34} +2.80853 q^{35} -9.25435 q^{36} +1.53525 q^{37} +0.908009 q^{38} -2.98836 q^{39} -1.34724 q^{40} -9.29986 q^{41} +0.333137 q^{42} +5.23402 q^{43} -13.0904 q^{45} -0.580799 q^{46} -1.89387 q^{47} +10.8314 q^{48} +1.00000 q^{49} -0.347577 q^{50} -19.2594 q^{51} +2.14371 q^{52} -3.82552 q^{53} -0.553319 q^{54} -0.479696 q^{56} -20.8809 q^{57} +0.147713 q^{58} -6.66349 q^{59} +15.4345 q^{60} +9.79952 q^{61} +0.971796 q^{62} -4.66094 q^{63} -7.65442 q^{64} +3.03229 q^{65} -2.06100 q^{67} +13.8158 q^{68} +13.3563 q^{69} -0.338034 q^{70} +12.5212 q^{71} +2.23583 q^{72} -2.56708 q^{73} -0.184783 q^{74} +7.99301 q^{75} +14.9789 q^{76} +0.359679 q^{78} +15.3283 q^{79} -10.9906 q^{80} -1.25848 q^{81} +1.11933 q^{82} -2.04602 q^{83} +5.49558 q^{84} +19.5425 q^{85} -0.629967 q^{86} -3.39687 q^{87} +4.76119 q^{89} +1.57555 q^{90} +1.07967 q^{91} -9.58114 q^{92} -22.3478 q^{93} +0.227945 q^{94} +21.1878 q^{95} -3.95910 q^{96} +9.11512 q^{97} -0.120360 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) 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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