LMFDB - Embedded newform 847.2.a.n.1.3

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:	$0$
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			1.3
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	$-0.486451\pi$
$2$	0.120360	0.0851073	0.0425536	+	0.999094i	$0.486451\pi$
$3$	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	$-0.452162\pi$
$13$	1.07967	0.299448	0.149724	+	0.988728i	$0.452162\pi$
$14$	0.120360	0.0321675
$15$	−7.77355	−2.00712
$16$	3.91329	0.978323
$17$	6.95828	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$17$	6.95828	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$18$	0.560990	0.132227
$19$	7.54411	1.73074	0.865369	−	0.501135i	$-0.167084\pi$
$19$	7.54411	1.73074	0.865369	+	0.501135i	$0.167084\pi$
$20$	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	$-0.463650\pi$
$29$	1.22726	0.227897	0.113949	+	0.993487i	$0.463650\pi$
$30$	−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	$-0.241286\pi$
$41$	9.29986	1.45239	0.726197	+	0.687487i	$0.241286\pi$
$42$	0.333137	0.0514041
$43$	−5.23402	−0.798181	−0.399091	−	0.916912i	$-0.630674\pi$
$43$	−5.23402	−0.798181	−0.399091	+	0.916912i	$0.630674\pi$
$44$	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	$-0.715861\pi$
$61$	−9.79952	−1.25470	−0.627350	+	0.778737i	$0.715861\pi$
$62$	−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	$-0.452000\pi$
$73$	2.56708	0.300454	0.150227	+	0.988652i	$0.452000\pi$
$74$	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	$-0.830969\pi$
$79$	−15.3283	−1.72457	−0.862287	+	0.506420i	$0.830969\pi$
$80$	−10.9906	−1.22878
$81$	−1.25848	−0.139832
$82$	1.11933	0.123609
$83$	2.04602	0.224580	0.112290	−	0.993675i	$-0.464181\pi$
$83$	2.04602	0.224580	0.112290	+	0.993675i	$0.464181\pi$
$84$	−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	$-0.575842\pi$
$101$	−4.74385	−0.472031	−0.236015	+	0.971749i	$0.575842\pi$
$102$	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	$-0.460701\pi$
$107$	2.54774	0.246299	0.123149	+	0.992388i	$0.460701\pi$
$108$	−9.12781	−0.878324
$109$	−3.81522	−0.365432	−0.182716	−	0.983166i	$-0.558489\pi$
$109$	−3.81522	−0.365432	−0.182716	+	0.983166i	$0.558489\pi$
$110$	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	$-0.356089\pi$
$127$	9.84644	0.873730	0.436865	+	0.899527i	$0.356089\pi$
$128$	−3.78208	−0.334291
$129$	−14.4869	−1.27550
$130$	−0.364966	−0.0320097
$131$	16.3782	1.43097	0.715484	−	0.698629i	$-0.246206\pi$
$131$	16.3782	1.43097	0.715484	+	0.698629i	$0.246206\pi$
$132$	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	$-0.627281\pi$
$139$	−9.17942	−0.778588	−0.389294	+	0.921114i	$0.627281\pi$
$140$	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	$-0.342096\pi$
$149$	11.6200	0.951947	0.475973	+	0.879460i	$0.342096\pi$
$150$	0.962037	0.0785500
$151$	17.9150	1.45790	0.728952	−	0.684565i	$-0.240008\pi$
$151$	17.9150	1.45790	0.728952	+	0.684565i	$0.240008\pi$
$152$	−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	$-0.567031\pi$
$167$	−5.40259	−0.418065	−0.209032	+	0.977909i	$0.567031\pi$
$168$	−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	$-0.769795\pi$
$173$	−19.7211	−1.49937	−0.749685	+	0.661795i	$0.769795\pi$
$174$	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	$-0.767015\pi$
$193$	−20.6685	−1.48775	−0.743877	+	0.668317i	$0.767015\pi$
$194$	1.09709	0.0787668
$195$	−8.39290	−0.601028
$196$	−1.98551	−0.141822
$197$	6.27954	0.447399	0.223699	−	0.974658i	$-0.428187\pi$
$197$	6.27954	0.447399	0.223699	+	0.974658i	$0.428187\pi$
$198$	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	$-0.385122\pi$
$211$	10.2586	0.706232	0.353116	+	0.935580i	$0.385122\pi$
$212$	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	$-0.351228\pi$
$227$	13.5764	0.901100	0.450550	+	0.892751i	$0.351228\pi$
$228$	−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	$-0.413397\pi$
$233$	8.20387	0.537453	0.268727	+	0.963216i	$0.413397\pi$
$234$	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	$-0.390984\pi$
$239$	10.3835	0.671655	0.335827	+	0.941924i	$0.390984\pi$
$240$	−30.4202	−1.96361
$241$	−8.20445	−0.528495	−0.264248	−	0.964455i	$-0.585124\pi$
$241$	−8.20445	−0.528495	−0.264248	+	0.964455i	$0.585124\pi$
$242$	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	$-0.481579\pi$
$263$	1.87602	0.115681	0.0578403	+	0.998326i	$0.481579\pi$
$264$	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	$-0.486913\pi$
$271$	1.35328	0.0822056	0.0411028	+	0.999155i	$0.486913\pi$
$272$	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	$-0.631709\pi$
$277$	−13.3835	−0.804138	−0.402069	+	0.915609i	$0.631709\pi$
$278$	−1.10483	−0.0662635
$279$	−37.6328	−2.25302
$280$	1.34724	0.0805129
$281$	2.31887	0.138332	0.0691661	−	0.997605i	$-0.477966\pi$
$281$	2.31887	0.138332	0.0691661	+	0.997605i	$0.477966\pi$
$282$	−0.630916	−0.0375705
$283$	−22.2679	−1.32369	−0.661845	−	0.749641i	$-0.730226\pi$
$283$	−22.2679	−1.32369	−0.661845	+	0.749641i	$0.730226\pi$
$284$	−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	$-0.522704\pi$
$293$	−2.43981	−0.142535	−0.0712675	+	0.997457i	$0.522704\pi$
$294$	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	$-0.581649\pi$
$307$	−8.89055	−0.507410	−0.253705	+	0.967282i	$0.581649\pi$
$308$	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	$-0.655439\pi$
$337$	−17.2248	−0.938297	−0.469148	+	0.883119i	$0.655439\pi$
$338$	−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	$-0.575331\pi$
$347$	−8.73485	−0.468911	−0.234456	+	0.972127i	$0.575331\pi$
$348$	−6.74453	−0.361545
$349$	−29.9851	−1.60506	−0.802532	−	0.596609i	$-0.796514\pi$
$349$	−29.9851	−1.60506	−0.802532	+	0.596609i	$0.796514\pi$
$350$	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	$-0.458132\pi$
$359$	4.96996	0.262304	0.131152	+	0.991362i	$0.458132\pi$
$360$	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	$-0.425333\pi$
$373$	8.97781	0.464853	0.232427	+	0.972614i	$0.425333\pi$
$374$	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	$-0.547257\pi$
$409$	−5.98291	−0.295836	−0.147918	+	0.989000i	$0.547257\pi$
$410$	−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	$-0.145245\pi$
$431$	37.2730	1.79538	0.897689	+	0.440630i	$0.145245\pi$
$432$	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	$-0.510863\pi$
$439$	−1.42974	−0.0682379	−0.0341189	+	0.999418i	$0.510863\pi$
$440$	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	$-0.257797\pi$
$457$	29.4829	1.37915	0.689575	+	0.724214i	$0.257797\pi$
$458$	−0.174878	−0.00817150
$459$	31.9887	1.49310
$460$	26.9089	1.25463
$461$	31.7282	1.47773	0.738866	−	0.673853i	$-0.235362\pi$
$461$	31.7282	1.47773	0.738866	+	0.673853i	$0.235362\pi$
$462$	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	$-0.726242\pi$
$479$	−28.5574	−1.30482	−0.652411	+	0.757866i	$0.726242\pi$
$480$	−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	$-0.363329\pi$
$491$	18.4489	0.832586	0.416293	+	0.909230i	$0.363329\pi$
$492$	−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	$-0.724041\pi$
$503$	−29.0283	−1.29431	−0.647154	+	0.762359i	$0.724041\pi$
$504$	−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	$-0.614038\pi$
$523$	−16.0380	−0.701295	−0.350647	+	0.936508i	$0.614038\pi$
$524$	−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	$-0.405599\pi$
$541$	13.5946	0.584480	0.292240	+	0.956345i	$0.405599\pi$
$542$	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	$-0.560415\pi$
$547$	−8.82486	−0.377324	−0.188662	+	0.982042i	$0.560415\pi$
$548$	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	$-0.244075\pi$
$557$	33.9920	1.44029	0.720145	+	0.693824i	$0.244075\pi$
$558$	−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	$-0.640149\pi$
$563$	−20.2256	−0.852406	−0.426203	+	0.904628i	$0.640149\pi$
$564$	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	$-0.292586\pi$
$569$	28.9330	1.21293	0.606467	+	0.795109i	$0.292586\pi$
$570$	−7.05845	−0.295646
$571$	−7.51312	−0.314414	−0.157207	−	0.987566i	$-0.550249\pi$
$571$	−7.51312	−0.314414	−0.157207	+	0.987566i	$0.550249\pi$
$572$	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	$-0.294637\pi$
$593$	29.2867	1.20266	0.601331	+	0.799000i	$0.294637\pi$
$594$	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	$-0.274738\pi$
$601$	31.8735	1.30015	0.650073	+	0.759872i	$0.274738\pi$
$602$	−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	$-0.634073\pi$
$607$	−20.1463	−0.817714	−0.408857	+	0.912598i	$0.634073\pi$
$608$	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	$-0.525040\pi$
$613$	−3.89132	−0.157169	−0.0785844	+	0.996907i	$0.525040\pi$
$614$	−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	$-0.462690\pi$
$659$	6.00410	0.233887	0.116943	+	0.993139i	$0.462690\pi$
$660$	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	$-0.172619\pi$
$673$	44.4403	1.71305	0.856524	+	0.516107i	$0.172619\pi$
$674$	−2.07318	−0.0798559
$675$	13.2759	0.510989
$676$	23.4972	0.903737
$677$	15.5471	0.597525	0.298762	−	0.954327i	$-0.403426\pi$
$677$	15.5471	0.597525	0.298762	+	0.954327i	$0.403426\pi$
$678$	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	$-0.664287\pi$
$701$	−26.1328	−0.987022	−0.493511	+	0.869739i	$0.664287\pi$
$702$	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	$-0.491438\pi$
$733$	1.45630	0.0537896	0.0268948	+	0.999638i	$0.491438\pi$
$734$	−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	$-0.669459\pi$
$739$	−27.5966	−1.01516	−0.507579	+	0.861605i	$0.669459\pi$
$740$	8.56113	0.314713
$741$	22.5446	0.828195
$742$	−0.460439	−0.0169033
$743$	27.9773	1.02639	0.513194	−	0.858273i	$-0.328462\pi$
$743$	27.9773	1.02639	0.513194	+	0.858273i	$0.328462\pi$
$744$	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	$-0.522081\pi$
$761$	−3.82415	−0.138625	−0.0693127	+	0.997595i	$0.522081\pi$
$762$	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	$-0.615331\pi$
$769$	−19.6583	−0.708897	−0.354449	+	0.935075i	$0.615331\pi$
$770$	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	$-0.914533\pi$
$787$	−54.0967	−1.92834	−0.964169	+	0.265287i	$0.914533\pi$
$788$	−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	$-0.191130\pi$
$809$	46.9354	1.65016	0.825081	+	0.565015i	$0.191130\pi$
$810$	0.425410	0.0149474
$811$	−12.6615	−0.444605	−0.222303	−	0.974978i	$-0.571357\pi$
$811$	−12.6615	−0.444605	−0.222303	+	0.974978i	$0.571357\pi$
$812$	−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	$-0.829651\pi$
$821$	−49.2938	−1.72037	−0.860183	+	0.509986i	$0.829651\pi$
$822$	−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	$-0.225792\pi$
$827$	43.6419	1.51758	0.758789	+	0.651337i	$0.225792\pi$
$828$	−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	$-0.356231\pi$
$853$	25.4948	0.872927	0.436463	+	0.899722i	$0.356231\pi$
$854$	−1.17947	−0.0403606
$855$	−98.7552	−3.37735
$856$	−1.22214	−0.0417718
$857$	49.4756	1.69005	0.845027	−	0.534723i	$-0.179584\pi$
$857$	49.4756	1.69005	0.845027	+	0.534723i	$0.179584\pi$
$858$	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	$-0.336336\pi$
$877$	29.1290	0.983618	0.491809	+	0.870703i	$0.336336\pi$
$878$	−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	$-0.464402\pi$
$887$	6.64749	0.223201	0.111600	+	0.993753i	$0.464402\pi$
$888$	−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	$-0.404032\pi$
$919$	18.0039	0.593894	0.296947	+	0.954894i	$0.404032\pi$
$920$	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	$-0.652441\pi$
$937$	−28.2113	−0.921622	−0.460811	+	0.887498i	$0.652441\pi$
$938$	−0.248062	−0.00809952
$939$	−81.7635	−2.66825
$940$	−10.5609	−0.344458
$941$	−20.1501	−0.656875	−0.328437	−	0.944526i	$-0.606522\pi$
$941$	−20.1501	−0.656875	−0.328437	+	0.944526i	$0.606522\pi$
$942$	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	$-0.657683\pi$
$953$	−29.3495	−0.950723	−0.475362	+	0.879790i	$0.657683\pi$
$954$	−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	$-0.814672\pi$
$967$	−51.9463	−1.67048	−0.835240	+	0.549885i	$0.814672\pi$
$968$	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	$-0.583762\pi$
$997$	−16.4267	−0.520240	−0.260120	+	0.965576i	$0.583762\pi$
$998$	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.n.1.3	yes	6
3.2	odd	2		7623.2.a.cp.1.4		6
7.6	odd	2		5929.2.a.bm.1.3		6
11.2	odd	10		847.2.f.z.323.3		24
11.3	even	5		847.2.f.y.372.3		24
11.4	even	5		847.2.f.y.148.3		24
11.5	even	5		847.2.f.y.729.4		24
11.6	odd	10		847.2.f.z.729.3		24
11.7	odd	10		847.2.f.z.148.4		24
11.8	odd	10		847.2.f.z.372.4		24
11.9	even	5		847.2.f.y.323.4		24
11.10	odd	2		847.2.a.m.1.4	✓	6
33.32	even	2		7623.2.a.cs.1.3		6
77.76	even	2		5929.2.a.bj.1.4		6

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
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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