LMFDB - Embedded newform 4015.2.a.b.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4015,2,Mod(1,4015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4015, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4015 = 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4015.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:	$32.0599364115$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	4015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.55677 q^{2} +0.976240 q^{3} +0.423526 q^{4} +1.00000 q^{5} +1.51978 q^{6} -0.0535957 q^{7} -2.45420 q^{8} -2.04696 q^{9} +O(q^{10})$	\(q+1.55677 q^{2} +0.976240 q^{3} +0.423526 q^{4} +1.00000 q^{5} +1.51978 q^{6} -0.0535957 q^{7} -2.45420 q^{8} -2.04696 q^{9} +1.55677 q^{10} +1.00000 q^{11} +0.413463 q^{12} -4.93652 q^{13} -0.0834361 q^{14} +0.976240 q^{15} -4.66768 q^{16} +1.52053 q^{17} -3.18663 q^{18} +2.49796 q^{19} +0.423526 q^{20} -0.0523223 q^{21} +1.55677 q^{22} -4.45847 q^{23} -2.39589 q^{24} +1.00000 q^{25} -7.68501 q^{26} -4.92704 q^{27} -0.0226992 q^{28} -5.20663 q^{29} +1.51978 q^{30} +2.28605 q^{31} -2.35808 q^{32} +0.976240 q^{33} +2.36711 q^{34} -0.0535957 q^{35} -0.866939 q^{36} +1.00995 q^{37} +3.88874 q^{38} -4.81923 q^{39} -2.45420 q^{40} +3.36257 q^{41} -0.0814536 q^{42} -1.86334 q^{43} +0.423526 q^{44} -2.04696 q^{45} -6.94080 q^{46} +1.99164 q^{47} -4.55677 q^{48} -6.99713 q^{49} +1.55677 q^{50} +1.48440 q^{51} -2.09075 q^{52} -10.5893 q^{53} -7.67026 q^{54} +1.00000 q^{55} +0.131535 q^{56} +2.43861 q^{57} -8.10551 q^{58} -8.74388 q^{59} +0.413463 q^{60} -2.17329 q^{61} +3.55884 q^{62} +0.109708 q^{63} +5.66437 q^{64} -4.93652 q^{65} +1.51978 q^{66} -11.3821 q^{67} +0.643983 q^{68} -4.35254 q^{69} -0.0834361 q^{70} -11.8133 q^{71} +5.02364 q^{72} -1.00000 q^{73} +1.57226 q^{74} +0.976240 q^{75} +1.05795 q^{76} -0.0535957 q^{77} -7.50242 q^{78} -5.46451 q^{79} -4.66768 q^{80} +1.33089 q^{81} +5.23474 q^{82} +9.51766 q^{83} -0.0221599 q^{84} +1.52053 q^{85} -2.90078 q^{86} -5.08292 q^{87} -2.45420 q^{88} -6.43943 q^{89} -3.18663 q^{90} +0.264576 q^{91} -1.88828 q^{92} +2.23173 q^{93} +3.10052 q^{94} +2.49796 q^{95} -2.30206 q^{96} +9.27946 q^{97} -10.8929 q^{98} -2.04696 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 5 q^{2} - 3 q^{3} + 15 q^{4} + 23 q^{5} - 15 q^{6} - 10 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) $ 23 * q - 5 * q^2 - 3 * q^3 + 15 * q^4 + 23 * q^5 - 15 * q^6 - 10 * q^7 - 12 * q^8 + 8 * q^9	\( 23 q - 5 q^{2} - 3 q^{3} + 15 q^{4} + 23 q^{5} - 15 q^{6} - 10 q^{7} - 12 q^{8} + 8 q^{9} - 5 q^{10} + 23 q^{11} + 4 q^{12} - 19 q^{13} - 5 q^{14} - 3 q^{15} - q^{16} - 26 q^{17} + 5 q^{18} - 34 q^{19} + 15 q^{20} - 26 q^{21} - 5 q^{22} - 4 q^{23} - 23 q^{24} + 23 q^{25} - 13 q^{26} - 3 q^{27} - 28 q^{28} - 36 q^{29} - 15 q^{30} - 24 q^{31} - 19 q^{32} - 3 q^{33} - 4 q^{34} - 10 q^{35} - 14 q^{36} + 4 q^{37} - 15 q^{38} - 34 q^{39} - 12 q^{40} - 74 q^{41} + 7 q^{42} - 15 q^{43} + 15 q^{44} + 8 q^{45} + 3 q^{46} + q^{47} + 29 q^{48} + q^{49} - 5 q^{50} - 47 q^{51} - 23 q^{52} - 6 q^{53} - 11 q^{54} + 23 q^{55} - 20 q^{56} - 19 q^{57} + 22 q^{58} - 17 q^{59} + 4 q^{60} - 59 q^{61} + 38 q^{62} - 21 q^{63} - 18 q^{64} - 19 q^{65} - 15 q^{66} + 12 q^{67} + 5 q^{68} - 8 q^{69} - 5 q^{70} - 34 q^{71} + 21 q^{72} - 23 q^{73} - 9 q^{74} - 3 q^{75} - 53 q^{76} - 10 q^{77} + 23 q^{78} - 62 q^{79} - q^{80} + 7 q^{81} + 24 q^{82} - 8 q^{83} + 46 q^{84} - 26 q^{85} - 11 q^{86} + q^{87} - 12 q^{88} - 77 q^{89} + 5 q^{90} - 6 q^{91} + 47 q^{92} - 32 q^{94} - 34 q^{95} - 53 q^{96} - 21 q^{97} - 3 q^{98} + 8 q^{99}+O(q^{100}) \) 23 * q - 5 * q^2 - 3 * q^3 + 15 * q^4 + 23 * q^5 - 15 * q^6 - 10 * q^7 - 12 * q^8 + 8 * q^9 - 5 * q^10 + 23 * q^11 + 4 * q^12 - 19 * q^13 - 5 * q^14 - 3 * q^15 - q^16 - 26 * q^17 + 5 * q^18 - 34 * q^19 + 15 * q^20 - 26 * q^21 - 5 * q^22 - 4 * q^23 - 23 * q^24 + 23 * q^25 - 13 * q^26 - 3 * q^27 - 28 * q^28 - 36 * q^29 - 15 * q^30 - 24 * q^31 - 19 * q^32 - 3 * q^33 - 4 * q^34 - 10 * q^35 - 14 * q^36 + 4 * q^37 - 15 * q^38 - 34 * q^39 - 12 * q^40 - 74 * q^41 + 7 * q^42 - 15 * q^43 + 15 * q^44 + 8 * q^45 + 3 * q^46 + q^47 + 29 * q^48 + q^49 - 5 * q^50 - 47 * q^51 - 23 * q^52 - 6 * q^53 - 11 * q^54 + 23 * q^55 - 20 * q^56 - 19 * q^57 + 22 * q^58 - 17 * q^59 + 4 * q^60 - 59 * q^61 + 38 * q^62 - 21 * q^63 - 18 * q^64 - 19 * q^65 - 15 * q^66 + 12 * q^67 + 5 * q^68 - 8 * q^69 - 5 * q^70 - 34 * q^71 + 21 * q^72 - 23 * q^73 - 9 * q^74 - 3 * q^75 - 53 * q^76 - 10 * q^77 + 23 * q^78 - 62 * q^79 - q^80 + 7 * q^81 + 24 * q^82 - 8 * q^83 + 46 * q^84 - 26 * q^85 - 11 * q^86 + q^87 - 12 * q^88 - 77 * q^89 + 5 * q^90 - 6 * q^91 + 47 * q^92 - 32 * q^94 - 34 * q^95 - 53 * q^96 - 21 * q^97 - 3 * q^98 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.55677	1.10080	0.550401	−	0.834901i	$-0.314475\pi$
$2$	1.55677	1.10080	0.550401	+	0.834901i	$0.314475\pi$
$3$	0.976240	0.563633	0.281816	−	0.959468i	$-0.409063\pi$
$3$	0.976240	0.563633	0.281816	+	0.959468i	$0.409063\pi$
$4$	0.423526	0.211763
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.51978	0.620447
$7$	−0.0535957	−0.0202573	−0.0101286	−	0.999949i	$-0.503224\pi$
$7$	−0.0535957	−0.0202573	−0.0101286	+	0.999949i	$0.503224\pi$
$8$	−2.45420	−0.867692
$9$	−2.04696	−0.682318
$10$	1.55677	0.492293
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.413463	0.119357
$13$	−4.93652	−1.36914	−0.684572	−	0.728945i	$-0.740011\pi$
$13$	−4.93652	−1.36914	−0.684572	+	0.728945i	$0.740011\pi$
$14$	−0.0834361	−0.0222992
$15$	0.976240	0.252064
$16$	−4.66768	−1.16692
$17$	1.52053	0.368782	0.184391	−	0.982853i	$-0.440969\pi$
$17$	1.52053	0.368782	0.184391	+	0.982853i	$0.440969\pi$
$18$	−3.18663	−0.751097
$19$	2.49796	0.573071	0.286536	−	0.958070i	$-0.407496\pi$
$19$	2.49796	0.573071	0.286536	+	0.958070i	$0.407496\pi$
$20$	0.423526	0.0947033
$21$	−0.0523223	−0.0114177
$22$	1.55677	0.331904
$23$	−4.45847	−0.929655	−0.464828	−	0.885401i	$-0.653884\pi$
$23$	−4.45847	−0.929655	−0.464828	+	0.885401i	$0.653884\pi$
$24$	−2.39589	−0.489059
$25$	1.00000	0.200000
$26$	−7.68501	−1.50716
$27$	−4.92704	−0.948209
$28$	−0.0226992	−0.00428974
$29$	−5.20663	−0.966847	−0.483423	−	0.875387i	$-0.660607\pi$
$29$	−5.20663	−0.966847	−0.483423	+	0.875387i	$0.660607\pi$
$30$	1.51978	0.277472
$31$	2.28605	0.410586	0.205293	−	0.978701i	$-0.434185\pi$
$31$	2.28605	0.410586	0.205293	+	0.978701i	$0.434185\pi$
$32$	−2.35808	−0.416854
$33$	0.976240	0.169942
$34$	2.36711	0.405956
$35$	−0.0535957	−0.00905933
$36$	−0.866939	−0.144490
$37$	1.00995	0.166035	0.0830176	−	0.996548i	$-0.473544\pi$
$37$	1.00995	0.166035	0.0830176	+	0.996548i	$0.473544\pi$
$38$	3.88874	0.630837
$39$	−4.81923	−0.771694
$40$	−2.45420	−0.388044
$41$	3.36257	0.525145	0.262573	−	0.964912i	$-0.415429\pi$
$41$	3.36257	0.525145	0.262573	+	0.964912i	$0.415429\pi$
$42$	−0.0814536	−0.0125686
$43$	−1.86334	−0.284156	−0.142078	−	0.989855i	$-0.545378\pi$
$43$	−1.86334	−0.284156	−0.142078	+	0.989855i	$0.545378\pi$
$44$	0.423526	0.0638490
$45$	−2.04696	−0.305142
$46$	−6.94080	−1.02337
$47$	1.99164	0.290510	0.145255	−	0.989394i	$-0.453600\pi$
$47$	1.99164	0.290510	0.145255	+	0.989394i	$0.453600\pi$
$48$	−4.55677	−0.657714
$49$	−6.99713	−0.999590
$50$	1.55677	0.220160
$51$	1.48440	0.207858
$52$	−2.09075	−0.289934
$53$	−10.5893	−1.45455	−0.727277	−	0.686344i	$-0.759214\pi$
$53$	−10.5893	−1.45455	−0.727277	+	0.686344i	$0.759214\pi$
$54$	−7.67026	−1.04379
$55$	1.00000	0.134840
$56$	0.131535	0.0175771
$57$	2.43861	0.323002
$58$	−8.10551	−1.06431
$59$	−8.74388	−1.13836	−0.569178	−	0.822214i	$-0.692739\pi$
$59$	−8.74388	−1.13836	−0.569178	+	0.822214i	$0.692739\pi$
$60$	0.413463	0.0533779
$61$	−2.17329	−0.278262	−0.139131	−	0.990274i	$-0.544431\pi$
$61$	−2.17329	−0.278262	−0.139131	+	0.990274i	$0.544431\pi$
$62$	3.55884	0.451974
$63$	0.109708	0.0138219
$64$	5.66437	0.708046
$65$	−4.93652	−0.612300
$66$	1.51978	0.187072
$67$	−11.3821	−1.39055	−0.695275	−	0.718744i	$-0.744718\pi$
$67$	−11.3821	−1.39055	−0.695275	+	0.718744i	$0.744718\pi$
$68$	0.643983	0.0780944
$69$	−4.35254	−0.523984
$70$	−0.0834361	−0.00997252
$71$	−11.8133	−1.40198	−0.700990	−	0.713171i	$-0.747258\pi$
$71$	−11.8133	−1.40198	−0.700990	+	0.713171i	$0.747258\pi$
$72$	5.02364	0.592042
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	1.57226	0.182772
$75$	0.976240	0.112727
$76$	1.05795	0.121355
$77$	−0.0535957	−0.00610780
$78$	−7.50242	−0.849482
$79$	−5.46451	−0.614806	−0.307403	−	0.951579i	$-0.599460\pi$
$79$	−5.46451	−0.614806	−0.307403	+	0.951579i	$0.599460\pi$
$80$	−4.66768	−0.521862
$81$	1.33089	0.147877
$82$	5.23474	0.578081
$83$	9.51766	1.04470	0.522350	−	0.852731i	$-0.325056\pi$
$83$	9.51766	1.04470	0.522350	+	0.852731i	$0.325056\pi$
$84$	−0.0221599	−0.00241784
$85$	1.52053	0.164924
$86$	−2.90078	−0.312799
$87$	−5.08292	−0.544946
$88$	−2.45420	−0.261619
$89$	−6.43943	−0.682578	−0.341289	−	0.939958i	$-0.610863\pi$
$89$	−6.43943	−0.682578	−0.341289	+	0.939958i	$0.610863\pi$
$90$	−3.18663	−0.335901
$91$	0.264576	0.0277351
$92$	−1.88828	−0.196867
$93$	2.23173	0.231420
$94$	3.10052	0.319794
$95$	2.49796	0.256285
$96$	−2.30206	−0.234953
$97$	9.27946	0.942187	0.471093	−	0.882083i	$-0.343859\pi$
$97$	9.27946	0.942187	0.471093	+	0.882083i	$0.343859\pi$
$98$	−10.8929	−1.10035
$99$	−2.04696	−0.205727
$100$	0.423526	0.0423526
$101$	14.4829	1.44110	0.720551	−	0.693402i	$-0.243889\pi$
$101$	14.4829	1.44110	0.720551	+	0.693402i	$0.243889\pi$
$102$	2.31087	0.228810
$103$	13.5360	1.33374	0.666870	−	0.745174i	$-0.267633\pi$
$103$	13.5360	1.33374	0.666870	+	0.745174i	$0.267633\pi$
$104$	12.1152	1.18800
$105$	−0.0523223	−0.00510613
$106$	−16.4851	−1.60117
$107$	−13.3655	−1.29209	−0.646046	−	0.763298i	$-0.723579\pi$
$107$	−13.3655	−1.29209	−0.646046	+	0.763298i	$0.723579\pi$
$108$	−2.08673	−0.200796
$109$	−6.95789	−0.666445	−0.333223	−	0.942848i	$-0.608136\pi$
$109$	−6.95789	−0.666445	−0.333223	+	0.942848i	$0.608136\pi$
$110$	1.55677	0.148432
$111$	0.985956	0.0935828
$112$	0.250167	0.0236386
$113$	−14.9594	−1.40726	−0.703629	−	0.710568i	$-0.748438\pi$
$113$	−14.9594	−1.40726	−0.703629	+	0.710568i	$0.748438\pi$
$114$	3.79635	0.355560
$115$	−4.45847	−0.415754
$116$	−2.20514	−0.204742
$117$	10.1048	0.934192
$118$	−13.6122	−1.25310
$119$	−0.0814937	−0.00747052
$120$	−2.39589	−0.218714
$121$	1.00000	0.0909091
$122$	−3.38331	−0.306311
$123$	3.28268	0.295989
$124$	0.968201	0.0869470
$125$	1.00000	0.0894427
$126$	0.170790	0.0152152
$127$	16.3997	1.45524	0.727620	−	0.685980i	$-0.240626\pi$
$127$	16.3997	1.45524	0.727620	+	0.685980i	$0.240626\pi$
$128$	13.5343	1.19627
$129$	−1.81906	−0.160160
$130$	−7.68501	−0.674020
$131$	−4.51558	−0.394528	−0.197264	−	0.980350i	$-0.563206\pi$
$131$	−4.51558	−0.394528	−0.197264	+	0.980350i	$0.563206\pi$
$132$	0.413463	0.0359874
$133$	−0.133880	−0.0116089
$134$	−17.7194	−1.53072
$135$	−4.92704	−0.424052
$136$	−3.73168	−0.319989
$137$	21.9820	1.87805	0.939025	−	0.343848i	$-0.111731\pi$
$137$	21.9820	1.87805	0.939025	+	0.343848i	$0.111731\pi$
$138$	−6.77589	−0.576802
$139$	−3.04242	−0.258054	−0.129027	−	0.991641i	$-0.541185\pi$
$139$	−3.04242	−0.258054	−0.129027	+	0.991641i	$0.541185\pi$
$140$	−0.0226992	−0.00191843
$141$	1.94432	0.163741
$142$	−18.3905	−1.54330
$143$	−4.93652	−0.412812
$144$	9.55453	0.796211
$145$	−5.20663	−0.432387
$146$	−1.55677	−0.128839
$147$	−6.83088	−0.563401
$148$	0.427741	0.0351601
$149$	−0.888862	−0.0728184	−0.0364092	−	0.999337i	$-0.511592\pi$
$149$	−0.888862	−0.0728184	−0.0364092	+	0.999337i	$0.511592\pi$
$150$	1.51978	0.124089
$151$	−6.48514	−0.527754	−0.263877	−	0.964556i	$-0.585001\pi$
$151$	−6.48514	−0.527754	−0.263877	+	0.964556i	$0.585001\pi$
$152$	−6.13050	−0.497249
$153$	−3.11245	−0.251627
$154$	−0.0834361	−0.00672347
$155$	2.28605	0.183620
$156$	−2.04107	−0.163416
$157$	15.6644	1.25016	0.625078	−	0.780562i	$-0.285067\pi$
$157$	15.6644	1.25016	0.625078	+	0.780562i	$0.285067\pi$
$158$	−8.50698	−0.676779
$159$	−10.3377	−0.819834
$160$	−2.35808	−0.186423
$161$	0.238955	0.0188323
$162$	2.07189	0.162783
$163$	−1.52747	−0.119640	−0.0598202	−	0.998209i	$-0.519053\pi$
$163$	−1.52747	−0.119640	−0.0598202	+	0.998209i	$0.519053\pi$
$164$	1.42414	0.111206
$165$	0.976240	0.0760002
$166$	14.8168	1.15001
$167$	−19.3833	−1.49992	−0.749961	−	0.661482i	$-0.769928\pi$
$167$	−19.3833	−1.49992	−0.749961	+	0.661482i	$0.769928\pi$
$168$	0.128410	0.00990701
$169$	11.3692	0.874556
$170$	2.36711	0.181549
$171$	−5.11321	−0.391017
$172$	−0.789172	−0.0601738
$173$	10.6108	0.806724	0.403362	−	0.915041i	$-0.367842\pi$
$173$	10.6108	0.806724	0.403362	+	0.915041i	$0.367842\pi$
$174$	−7.91293	−0.599878
$175$	−0.0535957	−0.00405145
$176$	−4.66768	−0.351839
$177$	−8.53613	−0.641615
$178$	−10.0247	−0.751382
$179$	21.1778	1.58290	0.791450	−	0.611234i	$-0.209327\pi$
$179$	21.1778	1.58290	0.791450	+	0.611234i	$0.209327\pi$
$180$	−0.866939	−0.0646178
$181$	−10.0951	−0.750366	−0.375183	−	0.926951i	$-0.622420\pi$
$181$	−10.0951	−0.750366	−0.375183	+	0.926951i	$0.622420\pi$
$182$	0.411884	0.0305309
$183$	−2.12166	−0.156837
$184$	10.9420	0.806654
$185$	1.00995	0.0742532
$186$	3.47429	0.254747
$187$	1.52053	0.111192
$188$	0.843510	0.0615193
$189$	0.264068	0.0192081
$190$	3.88874	0.282119
$191$	−12.2641	−0.887397	−0.443698	−	0.896176i	$-0.646334\pi$
$191$	−12.2641	−0.887397	−0.443698	+	0.896176i	$0.646334\pi$
$192$	5.52978	0.399078
$193$	−21.6728	−1.56004	−0.780021	−	0.625753i	$-0.784792\pi$
$193$	−21.6728	−1.56004	−0.780021	+	0.625753i	$0.784792\pi$
$194$	14.4460	1.03716
$195$	−4.81923	−0.345112
$196$	−2.96347	−0.211676
$197$	10.1002	0.719611	0.359805	−	0.933027i	$-0.382843\pi$
$197$	10.1002	0.719611	0.359805	+	0.933027i	$0.382843\pi$
$198$	−3.18663	−0.226464
$199$	−0.0243400	−0.00172542	−0.000862708	−	1.00000i	$-0.500275\pi$
$199$	−0.0243400	−0.00172542	−0.000862708		1.00000i	$0.500275\pi$
$200$	−2.45420	−0.173538
$201$	−11.1117	−0.783760
$202$	22.5465	1.58637
$203$	0.279053	0.0195857
$204$	0.628682	0.0440166
$205$	3.36257	0.234852
$206$	21.0724	1.46818
$207$	9.12629	0.634321
$208$	23.0421	1.59768
$209$	2.49796	0.172787
$210$	−0.0814536	−0.00562083
$211$	−9.86201	−0.678929	−0.339464	−	0.940619i	$-0.610246\pi$
$211$	−9.86201	−0.678929	−0.339464	+	0.940619i	$0.610246\pi$
$212$	−4.48485	−0.308021
$213$	−11.5326	−0.790201
$214$	−20.8070	−1.42234
$215$	−1.86334	−0.127078
$216$	12.0920	0.822754
$217$	−0.122522	−0.00831736
$218$	−10.8318	−0.733624
$219$	−0.976240	−0.0659682
$220$	0.423526	0.0285541
$221$	−7.50611	−0.504916
$222$	1.53490	0.103016
$223$	18.1057	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$223$	18.1057	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$224$	0.126383	0.00844433
$225$	−2.04696	−0.136464
$226$	−23.2882	−1.54911
$227$	24.2448	1.60918	0.804591	−	0.593829i	$-0.202385\pi$
$227$	24.2448	1.60918	0.804591	+	0.593829i	$0.202385\pi$
$228$	1.03281	0.0683998
$229$	27.1605	1.79481	0.897407	−	0.441203i	$-0.145448\pi$
$229$	27.1605	1.79481	0.897407	+	0.441203i	$0.145448\pi$
$230$	−6.94080	−0.457663
$231$	−0.0523223	−0.00344255
$232$	12.7781	0.838925
$233$	10.7228	0.702477	0.351238	−	0.936286i	$-0.385761\pi$
$233$	10.7228	0.702477	0.351238	+	0.936286i	$0.385761\pi$
$234$	15.7309	1.02836
$235$	1.99164	0.129920
$236$	−3.70326	−0.241062
$237$	−5.33468	−0.346525
$238$	−0.126867	−0.00822355
$239$	−11.6601	−0.754232	−0.377116	−	0.926166i	$-0.623084\pi$
$239$	−11.6601	−0.754232	−0.377116	+	0.926166i	$0.623084\pi$
$240$	−4.55677	−0.294139
$241$	−19.6152	−1.26353	−0.631764	−	0.775161i	$-0.717669\pi$
$241$	−19.6152	−1.26353	−0.631764	+	0.775161i	$0.717669\pi$
$242$	1.55677	0.100073
$243$	16.0804	1.03156
$244$	−0.920447	−0.0589256
$245$	−6.99713	−0.447030
$246$	5.11037	0.325825
$247$	−12.3312	−0.784617
$248$	−5.61043	−0.356262
$249$	9.29153	0.588827
$250$	1.55677	0.0984586
$251$	20.2899	1.28069	0.640344	−	0.768088i	$-0.278792\pi$
$251$	20.2899	1.28069	0.640344	+	0.768088i	$0.278792\pi$
$252$	0.0464642	0.00292697
$253$	−4.45847	−0.280302
$254$	25.5306	1.60193
$255$	1.48440	0.0929567
$256$	9.74099	0.608812
$257$	−14.9921	−0.935184	−0.467592	−	0.883944i	$-0.654878\pi$
$257$	−14.9921	−0.935184	−0.467592	+	0.883944i	$0.654878\pi$
$258$	−2.83186	−0.176304
$259$	−0.0541291	−0.00336342
$260$	−2.09075	−0.129663
$261$	10.6577	0.659697
$262$	−7.02971	−0.434297
$263$	7.76190	0.478619	0.239310	−	0.970943i	$-0.423079\pi$
$263$	7.76190	0.478619	0.239310	+	0.970943i	$0.423079\pi$
$264$	−2.39589	−0.147457
$265$	−10.5893	−0.650496
$266$	−0.208420	−0.0127790
$267$	−6.28643	−0.384723
$268$	−4.82064	−0.294467
$269$	5.36646	0.327199	0.163599	−	0.986527i	$-0.447690\pi$
$269$	5.36646	0.327199	0.163599	+	0.986527i	$0.447690\pi$
$270$	−7.67026	−0.466797
$271$	20.8322	1.26547	0.632734	−	0.774369i	$-0.281933\pi$
$271$	20.8322	1.26547	0.632734	+	0.774369i	$0.281933\pi$
$272$	−7.09733	−0.430339
$273$	0.258290	0.0156324
$274$	34.2209	2.06736
$275$	1.00000	0.0603023
$276$	−1.84341	−0.110960
$277$	3.18827	0.191564	0.0957821	−	0.995402i	$-0.469465\pi$
$277$	3.18827	0.191564	0.0957821	+	0.995402i	$0.469465\pi$
$278$	−4.73634	−0.284067
$279$	−4.67944	−0.280150
$280$	0.131535	0.00786071
$281$	1.08300	0.0646065	0.0323033	−	0.999478i	$-0.489716\pi$
$281$	1.08300	0.0646065	0.0323033	+	0.999478i	$0.489716\pi$
$282$	3.02685	0.180246
$283$	−15.6595	−0.930860	−0.465430	−	0.885085i	$-0.654100\pi$
$283$	−15.6595	−0.930860	−0.465430	+	0.885085i	$0.654100\pi$
$284$	−5.00324	−0.296888
$285$	2.43861	0.144451
$286$	−7.68501	−0.454424
$287$	−0.180219	−0.0106380
$288$	4.82689	0.284427
$289$	−14.6880	−0.864000
$290$	−8.10551	−0.475972
$291$	9.05898	0.531047
$292$	−0.423526	−0.0247850
$293$	−15.2758	−0.892421	−0.446211	−	0.894928i	$-0.647227\pi$
$293$	−15.2758	−0.892421	−0.446211	+	0.894928i	$0.647227\pi$
$294$	−10.6341	−0.620193
$295$	−8.74388	−0.509088
$296$	−2.47863	−0.144067
$297$	−4.92704	−0.285896
$298$	−1.38375	−0.0801586
$299$	22.0093	1.27283
$300$	0.413463	0.0238713
$301$	0.0998668	0.00575623
$302$	−10.0959	−0.580952
$303$	14.1388	0.812252
$304$	−11.6597	−0.668728
$305$	−2.17329	−0.124442
$306$	−4.84536	−0.276991
$307$	−16.9336	−0.966454	−0.483227	−	0.875495i	$-0.660535\pi$
$307$	−16.9336	−0.966454	−0.483227	+	0.875495i	$0.660535\pi$
$308$	−0.0226992	−0.00129341
$309$	13.2144	0.751740
$310$	3.55884	0.202129
$311$	−19.8443	−1.12527	−0.562635	−	0.826706i	$-0.690212\pi$
$311$	−19.8443	−1.12527	−0.562635	+	0.826706i	$0.690212\pi$
$312$	11.8274	0.669593
$313$	6.82543	0.385796	0.192898	−	0.981219i	$-0.438211\pi$
$313$	6.82543	0.385796	0.192898	+	0.981219i	$0.438211\pi$
$314$	24.3859	1.37617
$315$	0.109708	0.00618134
$316$	−2.31436	−0.130193
$317$	5.92449	0.332752	0.166376	−	0.986062i	$-0.446793\pi$
$317$	5.92449	0.332752	0.166376	+	0.986062i	$0.446793\pi$
$318$	−16.0934	−0.902474
$319$	−5.20663	−0.291515
$320$	5.66437	0.316648
$321$	−13.0479	−0.728265
$322$	0.371997	0.0207306
$323$	3.79822	0.211338
$324$	0.563667	0.0313148
$325$	−4.93652	−0.273829
$326$	−2.37791	−0.131700
$327$	−6.79257	−0.375630
$328$	−8.25243	−0.455664
$329$	−0.106743	−0.00588494
$330$	1.51978	0.0836611
$331$	−23.4635	−1.28967	−0.644835	−	0.764322i	$-0.723074\pi$
$331$	−23.4635	−1.28967	−0.644835	+	0.764322i	$0.723074\pi$
$332$	4.03098	0.221229
$333$	−2.06733	−0.113289
$334$	−30.1752	−1.65112
$335$	−11.3821	−0.621873
$336$	0.244224	0.0133235
$337$	7.69442	0.419142	0.209571	−	0.977793i	$-0.432793\pi$
$337$	7.69442	0.419142	0.209571	+	0.977793i	$0.432793\pi$
$338$	17.6992	0.962712
$339$	−14.6039	−0.793176
$340$	0.643983	0.0349249
$341$	2.28605	0.123796
$342$	−7.96008	−0.430432
$343$	0.750186	0.0405062
$344$	4.57301	0.246560
$345$	−4.35254	−0.234333
$346$	16.5185	0.888042
$347$	−23.6884	−1.27166	−0.635831	−	0.771828i	$-0.719342\pi$
$347$	−23.6884	−1.27166	−0.635831	+	0.771828i	$0.719342\pi$
$348$	−2.15275	−0.115400
$349$	6.18731	0.331199	0.165600	−	0.986193i	$-0.447044\pi$
$349$	6.18731	0.331199	0.165600	+	0.986193i	$0.447044\pi$
$350$	−0.0834361	−0.00445984
$351$	24.3224	1.29824
$352$	−2.35808	−0.125686
$353$	25.6056	1.36285	0.681426	−	0.731887i	$-0.261360\pi$
$353$	25.6056	1.36285	0.681426	+	0.731887i	$0.261360\pi$
$354$	−13.2888	−0.706290
$355$	−11.8133	−0.626984
$356$	−2.72727	−0.144545
$357$	−0.0795574	−0.00421063
$358$	32.9689	1.74246
$359$	5.92428	0.312671	0.156336	−	0.987704i	$-0.450032\pi$
$359$	5.92428	0.312671	0.156336	+	0.987704i	$0.450032\pi$
$360$	5.02364	0.264769
$361$	−12.7602	−0.671589
$362$	−15.7158	−0.826004
$363$	0.976240	0.0512393
$364$	0.112055	0.00587328
$365$	−1.00000	−0.0523424
$366$	−3.30293	−0.172647
$367$	13.3528	0.697012	0.348506	−	0.937307i	$-0.386689\pi$
$367$	13.3528	0.697012	0.348506	+	0.937307i	$0.386689\pi$
$368$	20.8107	1.08483
$369$	−6.88303	−0.358316
$370$	1.57226	0.0817380
$371$	0.567542	0.0294653
$372$	0.945197	0.0490062
$373$	19.9599	1.03349	0.516743	−	0.856141i	$-0.327144\pi$
$373$	19.9599	1.03349	0.516743	+	0.856141i	$0.327144\pi$
$374$	2.36711	0.122400
$375$	0.976240	0.0504128
$376$	−4.88788	−0.252073
$377$	25.7026	1.32375
$378$	0.411093	0.0211443
$379$	−7.32379	−0.376198	−0.188099	−	0.982150i	$-0.560233\pi$
$379$	−7.32379	−0.376198	−0.188099	+	0.982150i	$0.560233\pi$
$380$	1.05795	0.0542718
$381$	16.0101	0.820221
$382$	−19.0923	−0.976847
$383$	−6.49497	−0.331877	−0.165939	−	0.986136i	$-0.553065\pi$
$383$	−6.49497	−0.331877	−0.165939	+	0.986136i	$0.553065\pi$
$384$	13.2127	0.674258
$385$	−0.0535957	−0.00273149
$386$	−33.7395	−1.71730
$387$	3.81416	0.193885
$388$	3.93010	0.199520
$389$	23.7992	1.20667	0.603334	−	0.797488i	$-0.293838\pi$
$389$	23.7992	1.20667	0.603334	+	0.797488i	$0.293838\pi$
$390$	−7.50242	−0.379900
$391$	−6.77923	−0.342840
$392$	17.1724	0.867336
$393$	−4.40829	−0.222369
$394$	15.7237	0.792148
$395$	−5.46451	−0.274950
$396$	−0.866939	−0.0435653
$397$	−6.26313	−0.314337	−0.157169	−	0.987572i	$-0.550237\pi$
$397$	−6.26313	−0.314337	−0.157169	+	0.987572i	$0.550237\pi$
$398$	−0.0378917	−0.00189934
$399$	−0.130699	−0.00654313
$400$	−4.66768	−0.233384
$401$	20.7792	1.03767	0.518833	−	0.854876i	$-0.326367\pi$
$401$	20.7792	1.03767	0.518833	+	0.854876i	$0.326367\pi$
$402$	−17.2984	−0.862763
$403$	−11.2851	−0.562152
$404$	6.13389	0.305172
$405$	1.33089	0.0661325
$406$	0.434421	0.0215599
$407$	1.00995	0.0500615
$408$	−3.64302	−0.180356
$409$	17.1914	0.850061	0.425031	−	0.905179i	$-0.360263\pi$
$409$	17.1914	0.850061	0.425031	+	0.905179i	$0.360263\pi$
$410$	5.23474	0.258525
$411$	21.4597	1.05853
$412$	5.73285	0.282437
$413$	0.468634	0.0230600
$414$	14.2075	0.698261
$415$	9.51766	0.467204
$416$	11.6407	0.570734
$417$	−2.97013	−0.145448
$418$	3.88874	0.190205
$419$	−4.83591	−0.236250	−0.118125	−	0.992999i	$-0.537688\pi$
$419$	−4.83591	−0.236250	−0.118125	+	0.992999i	$0.537688\pi$
$420$	−0.0221599	−0.00108129
$421$	24.3540	1.18694	0.593471	−	0.804855i	$-0.297757\pi$
$421$	24.3540	1.18694	0.593471	+	0.804855i	$0.297757\pi$
$422$	−15.3529	−0.747365
$423$	−4.07679	−0.198220
$424$	25.9883	1.26211
$425$	1.52053	0.0737564
$426$	−17.9536	−0.869854
$427$	0.116479	0.00563682
$428$	−5.66064	−0.273617
$429$	−4.81923	−0.232675
$430$	−2.90078	−0.139888
$431$	−19.3134	−0.930292	−0.465146	−	0.885234i	$-0.653998\pi$
$431$	−19.3134	−0.930292	−0.465146	+	0.885234i	$0.653998\pi$
$432$	22.9978	1.10648
$433$	−38.5272	−1.85150	−0.925749	−	0.378138i	$-0.876565\pi$
$433$	−38.5272	−1.85150	−0.925749	+	0.378138i	$0.876565\pi$
$434$	−0.190739	−0.00915575
$435$	−5.08292	−0.243707
$436$	−2.94685	−0.141128
$437$	−11.1371	−0.532759
$438$	−1.51978	−0.0726179
$439$	−5.89772	−0.281483	−0.140741	−	0.990046i	$-0.544949\pi$
$439$	−5.89772	−0.281483	−0.140741	+	0.990046i	$0.544949\pi$
$440$	−2.45420	−0.117000
$441$	14.3228	0.682038
$442$	−11.6853	−0.555812
$443$	−20.2346	−0.961374	−0.480687	−	0.876892i	$-0.659613\pi$
$443$	−20.2346	−0.961374	−0.480687	+	0.876892i	$0.659613\pi$
$444$	0.417578	0.0198174
$445$	−6.43943	−0.305258
$446$	28.1864	1.33467
$447$	−0.867743	−0.0410428
$448$	−0.303586	−0.0143431
$449$	−22.3951	−1.05689	−0.528444	−	0.848968i	$-0.677224\pi$
$449$	−22.3951	−1.05689	−0.528444	+	0.848968i	$0.677224\pi$
$450$	−3.18663	−0.150219
$451$	3.36257	0.158337
$452$	−6.33568	−0.298005
$453$	−6.33106	−0.297459
$454$	37.7435	1.77139
$455$	0.264576	0.0124035
$456$	−5.98484	−0.280266
$457$	−39.0367	−1.82606	−0.913030	−	0.407892i	$-0.866264\pi$
$457$	−39.0367	−1.82606	−0.913030	+	0.407892i	$0.866264\pi$
$458$	42.2826	1.97573
$459$	−7.49170	−0.349683
$460$	−1.88828	−0.0880415
$461$	23.1229	1.07694	0.538471	−	0.842644i	$-0.319002\pi$
$461$	23.1229	1.07694	0.538471	+	0.842644i	$0.319002\pi$
$462$	−0.0814536	−0.00378957
$463$	0.319092	0.0148295	0.00741474	−	0.999973i	$-0.497640\pi$
$463$	0.319092	0.0148295	0.00741474	+	0.999973i	$0.497640\pi$
$464$	24.3029	1.12823
$465$	2.23173	0.103494
$466$	16.6930	0.773287
$467$	33.9937	1.57304	0.786522	−	0.617563i	$-0.211880\pi$
$467$	33.9937	1.57304	0.786522	+	0.617563i	$0.211880\pi$
$468$	4.27966	0.197827
$469$	0.610034	0.0281688
$470$	3.10052	0.143016
$471$	15.2922	0.704629
$472$	21.4593	0.987743
$473$	−1.86334	−0.0856763
$474$	−8.30486	−0.381455
$475$	2.49796	0.114614
$476$	−0.0345147	−0.00158198
$477$	21.6759	0.992469
$478$	−18.1521	−0.830259
$479$	−18.5610	−0.848075	−0.424038	−	0.905645i	$-0.639388\pi$
$479$	−18.5610	−0.848075	−0.424038	+	0.905645i	$0.639388\pi$
$480$	−2.30206	−0.105074
$481$	−4.98565	−0.227326
$482$	−30.5363	−1.39089
$483$	0.233277	0.0106145
$484$	0.423526	0.0192512
$485$	9.27946	0.421359
$486$	25.0334	1.13554
$487$	38.0898	1.72601	0.863007	−	0.505192i	$-0.168578\pi$
$487$	38.0898	1.72601	0.863007	+	0.505192i	$0.168578\pi$
$488$	5.33370	0.241445
$489$	−1.49118	−0.0674333
$490$	−10.8929	−0.492091
$491$	7.72591	0.348666	0.174333	−	0.984687i	$-0.444223\pi$
$491$	7.72591	0.348666	0.174333	+	0.984687i	$0.444223\pi$
$492$	1.39030	0.0626795
$493$	−7.91682	−0.356556
$494$	−19.1969	−0.863707
$495$	−2.04696	−0.0920038
$496$	−10.6705	−0.479121
$497$	0.633142	0.0284003
$498$	14.4647	0.648181
$499$	−14.7979	−0.662444	−0.331222	−	0.943553i	$-0.607461\pi$
$499$	−14.7979	−0.662444	−0.331222	+	0.943553i	$0.607461\pi$
$500$	0.423526	0.0189407
$501$	−18.9227	−0.845405
$502$	31.5867	1.40978
$503$	−38.2839	−1.70700	−0.853498	−	0.521097i	$-0.825523\pi$
$503$	−38.2839	−1.70700	−0.853498	+	0.521097i	$0.825523\pi$
$504$	−0.269246	−0.0119932
$505$	14.4829	0.644481
$506$	−6.94080	−0.308556
$507$	11.0991	0.492928
$508$	6.94571	0.308166
$509$	−22.7132	−1.00674	−0.503372	−	0.864070i	$-0.667907\pi$
$509$	−22.7132	−1.00674	−0.503372	+	0.864070i	$0.667907\pi$
$510$	2.31087	0.102327
$511$	0.0535957	0.00237093
$512$	−11.9041	−0.526091
$513$	−12.3075	−0.543391
$514$	−23.3393	−1.02945
$515$	13.5360	0.596467
$516$	−0.770421	−0.0339159
$517$	1.99164	0.0875921
$518$	−0.0842664	−0.00370246
$519$	10.3587	0.454696
$520$	12.1152	0.531288
$521$	12.4638	0.546050	0.273025	−	0.962007i	$-0.411976\pi$
$521$	12.4638	0.546050	0.273025	+	0.962007i	$0.411976\pi$
$522$	16.5916	0.726196
$523$	−9.18186	−0.401495	−0.200747	−	0.979643i	$-0.564337\pi$
$523$	−9.18186	−0.401495	−0.200747	+	0.979643i	$0.564337\pi$
$524$	−1.91247	−0.0835465
$525$	−0.0523223	−0.00228353
$526$	12.0835	0.526865
$527$	3.47600	0.151417
$528$	−4.55677	−0.198308
$529$	−3.12205	−0.135741
$530$	−16.4851	−0.716067
$531$	17.8983	0.776721
$532$	−0.0567016	−0.00245833
$533$	−16.5994	−0.719000
$534$	−9.78651	−0.423504
$535$	−13.3655	−0.577841
$536$	27.9341	1.20657
$537$	20.6746	0.892174
$538$	8.35433	0.360181
$539$	−6.99713	−0.301388
$540$	−2.08673	−0.0897986
$541$	−32.2301	−1.38568	−0.692841	−	0.721090i	$-0.743641\pi$
$541$	−32.2301	−1.38568	−0.692841	+	0.721090i	$0.743641\pi$
$542$	32.4309	1.39303
$543$	−9.85528	−0.422931
$544$	−3.58553	−0.153728
$545$	−6.95789	−0.298043
$546$	0.402097	0.0172082
$547$	−38.3599	−1.64015	−0.820076	−	0.572254i	$-0.806069\pi$
$547$	−38.3599	−1.64015	−0.820076	+	0.572254i	$0.806069\pi$
$548$	9.30996	0.397702
$549$	4.44863	0.189863
$550$	1.55677	0.0663808
$551$	−13.0060	−0.554072
$552$	10.6820	0.454657
$553$	0.292874	0.0124543
$554$	4.96339	0.210874
$555$	0.985956	0.0418515
$556$	−1.28854	−0.0546464
$557$	−15.0739	−0.638701	−0.319351	−	0.947637i	$-0.603465\pi$
$557$	−15.0739	−0.638701	−0.319351	+	0.947637i	$0.603465\pi$
$558$	−7.28480	−0.308390
$559$	9.19839	0.389051
$560$	0.250167	0.0105715
$561$	1.48440	0.0626714
$562$	1.68598	0.0711190
$563$	13.7824	0.580859	0.290430	−	0.956896i	$-0.406202\pi$
$563$	13.7824	0.580859	0.290430	+	0.956896i	$0.406202\pi$
$564$	0.823469	0.0346743
$565$	−14.9594	−0.629345
$566$	−24.3782	−1.02469
$567$	−0.0713300	−0.00299558
$568$	28.9922	1.21649
$569$	−41.2472	−1.72917	−0.864585	−	0.502486i	$-0.832419\pi$
$569$	−41.2472	−1.72917	−0.864585	+	0.502486i	$0.832419\pi$
$570$	3.79635	0.159011
$571$	−16.6070	−0.694980	−0.347490	−	0.937684i	$-0.612966\pi$
$571$	−16.6070	−0.694980	−0.347490	+	0.937684i	$0.612966\pi$
$572$	−2.09075	−0.0874185
$573$	−11.9727	−0.500166
$574$	−0.280560	−0.0117103
$575$	−4.45847	−0.185931
$576$	−11.5947	−0.483113
$577$	20.8946	0.869852	0.434926	−	0.900466i	$-0.356774\pi$
$577$	20.8946	0.869852	0.434926	+	0.900466i	$0.356774\pi$
$578$	−22.8658	−0.951092
$579$	−21.1579	−0.879291
$580$	−2.20514	−0.0915636
$581$	−0.510106	−0.0211628
$582$	14.1027	0.584577
$583$	−10.5893	−0.438565
$584$	2.45420	0.101556
$585$	10.1048	0.417783
$586$	−23.7809	−0.982378
$587$	−20.6396	−0.851889	−0.425944	−	0.904749i	$-0.640058\pi$
$587$	−20.6396	−0.851889	−0.425944	+	0.904749i	$0.640058\pi$
$588$	−2.89306	−0.119308
$589$	5.71045	0.235295
$590$	−13.6122	−0.560405
$591$	9.86024	0.405596
$592$	−4.71413	−0.193750
$593$	−9.12425	−0.374688	−0.187344	−	0.982294i	$-0.559988\pi$
$593$	−9.12425	−0.374688	−0.187344	+	0.982294i	$0.559988\pi$
$594$	−7.67026	−0.314715
$595$	−0.0814937	−0.00334092
$596$	−0.376456	−0.0154203
$597$	−0.0237617	−0.000972501 0
$598$	34.2634	1.40113
$599$	30.6994	1.25434	0.627172	−	0.778881i	$-0.284212\pi$
$599$	30.6994	1.25434	0.627172	+	0.778881i	$0.284212\pi$
$600$	−2.39589	−0.0978119
$601$	9.62783	0.392727	0.196364	−	0.980531i	$-0.437087\pi$
$601$	9.62783	0.392727	0.196364	+	0.980531i	$0.437087\pi$
$602$	0.155469	0.00633646
$603$	23.2987	0.948798
$604$	−2.74663	−0.111759
$605$	1.00000	0.0406558
$606$	22.0108	0.894128
$607$	30.8786	1.25332	0.626661	−	0.779292i	$-0.284421\pi$
$607$	30.8786	1.25332	0.626661	+	0.779292i	$0.284421\pi$
$608$	−5.89040	−0.238887
$609$	0.272423	0.0110391
$610$	−3.38331	−0.136986
$611$	−9.83175	−0.397750
$612$	−1.31820	−0.0532853
$613$	5.13252	0.207300	0.103650	−	0.994614i	$-0.466948\pi$
$613$	5.13252	0.207300	0.103650	+	0.994614i	$0.466948\pi$
$614$	−26.3617	−1.06387
$615$	3.28268	0.132370
$616$	0.131535	0.00529969
$617$	−22.3006	−0.897787	−0.448894	−	0.893585i	$-0.648182\pi$
$617$	−22.3006	−0.897787	−0.448894	+	0.893585i	$0.648182\pi$
$618$	20.5717	0.827516
$619$	39.4581	1.58595	0.792977	−	0.609251i	$-0.208530\pi$
$619$	39.4581	1.58595	0.792977	+	0.609251i	$0.208530\pi$
$620$	0.968201	0.0388839
$621$	21.9671	0.881508
$622$	−30.8930	−1.23870
$623$	0.345126	0.0138272
$624$	22.4946	0.900505
$625$	1.00000	0.0400000
$626$	10.6256	0.424685
$627$	2.43861	0.0973886
$628$	6.63429	0.264737
$629$	1.53566	0.0612308
$630$	0.170790	0.00680443
$631$	10.8762	0.432976	0.216488	−	0.976285i	$-0.430540\pi$
$631$	10.8762	0.432976	0.216488	+	0.976285i	$0.430540\pi$
$632$	13.4110	0.533462
$633$	−9.62769	−0.382666
$634$	9.22305	0.366294
$635$	16.3997	0.650803
$636$	−4.37829	−0.173611
$637$	34.5415	1.36858
$638$	−8.10551	−0.320900
$639$	24.1813	0.956596
$640$	13.5343	0.534989
$641$	−34.8639	−1.37704	−0.688520	−	0.725217i	$-0.741739\pi$
$641$	−34.8639	−1.37704	−0.688520	+	0.725217i	$0.741739\pi$
$642$	−20.3126	−0.801675
$643$	−8.91951	−0.351751	−0.175876	−	0.984412i	$-0.556276\pi$
$643$	−8.91951	−0.351751	−0.175876	+	0.984412i	$0.556276\pi$
$644$	0.101204	0.00398798
$645$	−1.81906	−0.0716255
$646$	5.91294	0.232642
$647$	48.5231	1.90764	0.953820	−	0.300380i	$-0.0971135\pi$
$647$	48.5231	1.90764	0.953820	+	0.300380i	$0.0971135\pi$
$648$	−3.26628	−0.128311
$649$	−8.74388	−0.343227
$650$	−7.68501	−0.301431
$651$	−0.119611	−0.00468793
$652$	−0.646923	−0.0253354
$653$	5.62648	0.220181	0.110091	−	0.993922i	$-0.464886\pi$
$653$	5.62648	0.220181	0.110091	+	0.993922i	$0.464886\pi$
$654$	−10.5745	−0.413494
$655$	−4.51558	−0.176438
$656$	−15.6954	−0.612802
$657$	2.04696	0.0798593
$658$	−0.166174	−0.00647815
$659$	−34.9993	−1.36338	−0.681690	−	0.731641i	$-0.738755\pi$
$659$	−34.9993	−1.36338	−0.681690	+	0.731641i	$0.738755\pi$
$660$	0.413463	0.0160940
$661$	21.2739	0.827460	0.413730	−	0.910400i	$-0.364226\pi$
$661$	21.2739	0.827460	0.413730	+	0.910400i	$0.364226\pi$
$662$	−36.5272	−1.41967
$663$	−7.32777	−0.284587
$664$	−23.3583	−0.906477
$665$	−0.133880	−0.00519164
$666$	−3.21835	−0.124708
$667$	23.2136	0.898834
$668$	−8.20932	−0.317628
$669$	17.6755	0.683376
$670$	−17.7194	−0.684559
$671$	−2.17329	−0.0838991
$672$	0.123380	0.00475950
$673$	−25.0427	−0.965326	−0.482663	−	0.875806i	$-0.660330\pi$
$673$	−25.0427	−0.965326	−0.482663	+	0.875806i	$0.660330\pi$
$674$	11.9784	0.461392
$675$	−4.92704	−0.189642
$676$	4.81516	0.185199
$677$	−6.34323	−0.243790	−0.121895	−	0.992543i	$-0.538897\pi$
$677$	−6.34323	−0.243790	−0.121895	+	0.992543i	$0.538897\pi$
$678$	−22.7349	−0.873129
$679$	−0.497339	−0.0190861
$680$	−3.73168	−0.143104
$681$	23.6687	0.906987
$682$	3.55884	0.136275
$683$	−37.0705	−1.41846	−0.709232	−	0.704975i	$-0.750958\pi$
$683$	−37.0705	−1.41846	−0.709232	+	0.704975i	$0.750958\pi$
$684$	−2.16558	−0.0828030
$685$	21.9820	0.839890
$686$	1.16787	0.0445893
$687$	26.5151	1.01162
$688$	8.69745	0.331587
$689$	52.2744	1.99149
$690$	−6.77589	−0.257954
$691$	6.98103	0.265571	0.132785	−	0.991145i	$-0.457608\pi$
$691$	6.98103	0.265571	0.132785	+	0.991145i	$0.457608\pi$
$692$	4.49395	0.170834
$693$	0.109708	0.00416746
$694$	−36.8774	−1.39985
$695$	−3.04242	−0.115405
$696$	12.4745	0.472846
$697$	5.11288	0.193664
$698$	9.63221	0.364585
$699$	10.4681	0.395939
$700$	−0.0226992	−0.000857948 0
$701$	−9.20230	−0.347566	−0.173783	−	0.984784i	$-0.555599\pi$
$701$	−9.20230	−0.347566	−0.173783	+	0.984784i	$0.555599\pi$
$702$	37.8644	1.42910
$703$	2.52282	0.0951500
$704$	5.66437	0.213484
$705$	1.94432	0.0732272
$706$	39.8620	1.50023
$707$	−0.776221	−0.0291928
$708$	−3.61527	−0.135870
$709$	0.721624	0.0271012	0.0135506	−	0.999908i	$-0.495687\pi$
$709$	0.721624	0.0271012	0.0135506	+	0.999908i	$0.495687\pi$
$710$	−18.3905	−0.690185
$711$	11.1856	0.419493
$712$	15.8037	0.592267
$713$	−10.1923	−0.381704
$714$	−0.123852	−0.00463506
$715$	−4.93652	−0.184615
$716$	8.96934	0.335200
$717$	−11.3831	−0.425109
$718$	9.22272	0.344189
$719$	−6.92253	−0.258167	−0.129083	−	0.991634i	$-0.541203\pi$
$719$	−6.92253	−0.258167	−0.129083	+	0.991634i	$0.541203\pi$
$720$	9.55453	0.356076
$721$	−0.725471	−0.0270179
$722$	−19.8647	−0.739286
$723$	−19.1492	−0.712165
$724$	−4.27556	−0.158900
$725$	−5.20663	−0.193369
$726$	1.51978	0.0564043
$727$	−3.25052	−0.120555	−0.0602775	−	0.998182i	$-0.519199\pi$
$727$	−3.25052	−0.120555	−0.0602775	+	0.998182i	$0.519199\pi$
$728$	−0.649324	−0.0240655
$729$	11.7057	0.433543
$730$	−1.55677	−0.0576186
$731$	−2.83325	−0.104792
$732$	−0.898577	−0.0332124
$733$	11.8546	0.437860	0.218930	−	0.975741i	$-0.429743\pi$
$733$	11.8546	0.437860	0.218930	+	0.975741i	$0.429743\pi$
$734$	20.7873	0.767272
$735$	−6.83088	−0.251961
$736$	10.5134	0.387531
$737$	−11.3821	−0.419267
$738$	−10.7153	−0.394435
$739$	42.6502	1.56891	0.784457	−	0.620183i	$-0.212942\pi$
$739$	42.6502	1.56891	0.784457	+	0.620183i	$0.212942\pi$
$740$	0.427741	0.0157241
$741$	−12.0382	−0.442236
$742$	0.883531	0.0324354
$743$	0.0830793	0.00304788	0.00152394	−	0.999999i	$-0.499515\pi$
$743$	0.0830793	0.00304788	0.00152394	+	0.999999i	$0.499515\pi$
$744$	−5.47712	−0.200801
$745$	−0.888862	−0.0325654
$746$	31.0730	1.13766
$747$	−19.4822	−0.712818
$748$	0.643983	0.0235464
$749$	0.716334	0.0261743
$750$	1.51978	0.0554945
$751$	−5.85526	−0.213662	−0.106831	−	0.994277i	$-0.534070\pi$
$751$	−5.85526	−0.213662	−0.106831	+	0.994277i	$0.534070\pi$
$752$	−9.29632	−0.339002
$753$	19.8078	0.721837
$754$	40.0130	1.45719
$755$	−6.48514	−0.236019
$756$	0.111840	0.00406757
$757$	26.2209	0.953013	0.476507	−	0.879171i	$-0.341903\pi$
$757$	26.2209	0.953013	0.476507	+	0.879171i	$0.341903\pi$
$758$	−11.4014	−0.414119
$759$	−4.35254	−0.157987
$760$	−6.13050	−0.222377
$761$	21.8041	0.790396	0.395198	−	0.918596i	$-0.370676\pi$
$761$	21.8041	0.790396	0.395198	+	0.918596i	$0.370676\pi$
$762$	24.9240	0.902900
$763$	0.372913	0.0135004
$764$	−5.19415	−0.187918
$765$	−3.11245	−0.112531
$766$	−10.1112	−0.365331
$767$	43.1643	1.55857
$768$	9.50954	0.343146
$769$	9.59373	0.345959	0.172979	−	0.984925i	$-0.444661\pi$
$769$	9.59373	0.345959	0.172979	+	0.984925i	$0.444661\pi$
$770$	−0.0834361	−0.00300683
$771$	−14.6359	−0.527100
$772$	−9.17900	−0.330359
$773$	1.03690	0.0372949	0.0186474	−	0.999826i	$-0.494064\pi$
$773$	1.03690	0.0372949	0.0186474	+	0.999826i	$0.494064\pi$
$774$	5.93777	0.213429
$775$	2.28605	0.0821172
$776$	−22.7737	−0.817528
$777$	−0.0528430	−0.00189573
$778$	37.0499	1.32830
$779$	8.39957	0.300946
$780$	−2.04107	−0.0730820
$781$	−11.8133	−0.422713
$782$	−10.5537	−0.377399
$783$	25.6533	0.916773
$784$	32.6603	1.16644
$785$	15.6644	0.559087
$786$	−6.86269	−0.244784
$787$	−3.59130	−0.128016	−0.0640079	−	0.997949i	$-0.520388\pi$
$787$	−3.59130	−0.128016	−0.0640079	+	0.997949i	$0.520388\pi$
$788$	4.27771	0.152387
$789$	7.57748	0.269765
$790$	−8.50698	−0.302665
$791$	0.801757	0.0285072
$792$	5.02364	0.178507
$793$	10.7285	0.380980
$794$	−9.75024	−0.346023
$795$	−10.3377	−0.366641
$796$	−0.0103086	−0.000365380 0
$797$	37.6517	1.33369	0.666847	−	0.745195i	$-0.267644\pi$
$797$	37.6517	1.33369	0.666847	+	0.745195i	$0.267644\pi$
$798$	−0.203468	−0.00720268
$799$	3.02834	0.107135
$800$	−2.35808	−0.0833708
$801$	13.1812	0.465735
$802$	32.3484	1.14226
$803$	−1.00000	−0.0352892
$804$	−4.70610	−0.165971
$805$	0.238955	0.00842205
$806$	−17.5683	−0.618817
$807$	5.23896	0.184420
$808$	−35.5440	−1.25043
$809$	25.4432	0.894536	0.447268	−	0.894400i	$-0.352397\pi$
$809$	25.4432	0.894536	0.447268	+	0.894400i	$0.352397\pi$
$810$	2.07189	0.0727987
$811$	16.8415	0.591386	0.295693	−	0.955283i	$-0.404449\pi$
$811$	16.8415	0.591386	0.295693	+	0.955283i	$0.404449\pi$
$812$	0.118186	0.00414752
$813$	20.3373	0.713259
$814$	1.57226	0.0551077
$815$	−1.52747	−0.0535049
$816$	−6.92870	−0.242553
$817$	−4.65454	−0.162842
$818$	26.7631	0.935749
$819$	−0.541576	−0.0189242
$820$	1.42414	0.0497330
$821$	13.9882	0.488191	0.244096	−	0.969751i	$-0.421509\pi$
$821$	13.9882	0.488191	0.244096	+	0.969751i	$0.421509\pi$
$822$	33.4078	1.16523
$823$	36.2651	1.26412	0.632061	−	0.774919i	$-0.282209\pi$
$823$	36.2651	1.26412	0.632061	+	0.774919i	$0.282209\pi$
$824$	−33.2201	−1.15728
$825$	0.976240	0.0339883
$826$	0.729555	0.0253845
$827$	6.15565	0.214053	0.107027	−	0.994256i	$-0.465867\pi$
$827$	6.15565	0.214053	0.107027	+	0.994256i	$0.465867\pi$
$828$	3.86522	0.134326
$829$	−30.2844	−1.05182	−0.525910	−	0.850540i	$-0.676275\pi$
$829$	−30.2844	−1.05182	−0.525910	+	0.850540i	$0.676275\pi$
$830$	14.8168	0.514298
$831$	3.11251	0.107972
$832$	−27.9623	−0.969417
$833$	−10.6393	−0.368631
$834$	−4.62380	−0.160109
$835$	−19.3833	−0.670786
$836$	1.05795	0.0365900
$837$	−11.2634	−0.389322
$838$	−7.52840	−0.260064
$839$	32.0407	1.10617	0.553085	−	0.833125i	$-0.313451\pi$
$839$	32.0407	1.10617	0.553085	+	0.833125i	$0.313451\pi$
$840$	0.128410	0.00443055
$841$	−1.89100	−0.0652071
$842$	37.9136	1.30659
$843$	1.05727	0.0364144
$844$	−4.17682	−0.143772
$845$	11.3692	0.391113
$846$	−6.34662	−0.218201
$847$	−0.0535957	−0.00184157
$848$	49.4275	1.69735
$849$	−15.2874	−0.524663
$850$	2.36711	0.0811911
$851$	−4.50284	−0.154355
$852$	−4.88436	−0.167335
$853$	−7.56456	−0.259005	−0.129503	−	0.991579i	$-0.541338\pi$
$853$	−7.56456	−0.259005	−0.129503	+	0.991579i	$0.541338\pi$
$854$	0.181331	0.00620502
$855$	−5.11321	−0.174868
$856$	32.8017	1.12114
$857$	−45.4820	−1.55363	−0.776817	−	0.629726i	$-0.783167\pi$
$857$	−45.4820	−1.55363	−0.776817	+	0.629726i	$0.783167\pi$
$858$	−7.50242	−0.256128
$859$	−2.03959	−0.0695899	−0.0347950	−	0.999394i	$-0.511078\pi$
$859$	−2.03959	−0.0695899	−0.0347950	+	0.999394i	$0.511078\pi$
$860$	−0.789172	−0.0269105
$861$	−0.175937	−0.00599593
$862$	−30.0664	−1.02407
$863$	−4.11507	−0.140079	−0.0700393	−	0.997544i	$-0.522312\pi$
$863$	−4.11507	−0.140079	−0.0700393	+	0.997544i	$0.522312\pi$
$864$	11.6184	0.395265
$865$	10.6108	0.360778
$866$	−59.9779	−2.03813
$867$	−14.3390	−0.486978
$868$	−0.0518914	−0.00176131
$869$	−5.46451	−0.185371
$870$	−7.91293	−0.268273
$871$	56.1882	1.90386
$872$	17.0761	0.578269
$873$	−18.9946	−0.642871
$874$	−17.3378	−0.586461
$875$	−0.0535957	−0.00181187
$876$	−0.413463	−0.0139696
$877$	−41.5549	−1.40321	−0.701605	−	0.712566i	$-0.747533\pi$
$877$	−41.5549	−1.40321	−0.701605	+	0.712566i	$0.747533\pi$
$878$	−9.18138	−0.309857
$879$	−14.9128	−0.502998
$880$	−4.66768	−0.157347
$881$	−13.8451	−0.466452	−0.233226	−	0.972423i	$-0.574928\pi$
$881$	−13.8451	−0.466452	−0.233226	+	0.972423i	$0.574928\pi$
$882$	22.2973	0.750789
$883$	−1.84116	−0.0619599	−0.0309799	−	0.999520i	$-0.509863\pi$
$883$	−1.84116	−0.0619599	−0.0309799	+	0.999520i	$0.509863\pi$
$884$	−3.17904	−0.106923
$885$	−8.53613	−0.286939
$886$	−31.5006	−1.05828
$887$	−12.6937	−0.426213	−0.213106	−	0.977029i	$-0.568358\pi$
$887$	−12.6937	−0.426213	−0.213106	+	0.977029i	$0.568358\pi$
$888$	−2.41974	−0.0812011
$889$	−0.878955	−0.0294792
$890$	−10.0247	−0.336028
$891$	1.33089	0.0445865
$892$	7.66825	0.256752
$893$	4.97503	0.166483
$894$	−1.35087	−0.0451800
$895$	21.1778	0.707894
$896$	−0.725379	−0.0242332
$897$	21.4864	0.717410
$898$	−34.8639	−1.16342
$899$	−11.9026	−0.396974
$900$	−0.866939	−0.0288980
$901$	−16.1013	−0.536413
$902$	5.23474	0.174298
$903$	0.0974940	0.00324440
$904$	36.7133	1.22107
$905$	−10.0951	−0.335574
$906$	−9.85599	−0.327443
$907$	−7.25728	−0.240974	−0.120487	−	0.992715i	$-0.538446\pi$
$907$	−7.25728	−0.240974	−0.120487	+	0.992715i	$0.538446\pi$
$908$	10.2683	0.340765
$909$	−29.6458	−0.983291
$910$	0.411884	0.0136538
$911$	−51.8736	−1.71865	−0.859325	−	0.511430i	$-0.829116\pi$
$911$	−51.8736	−1.71865	−0.859325	+	0.511430i	$0.829116\pi$
$912$	−11.3826	−0.376917
$913$	9.51766	0.314989
$914$	−60.7711	−2.01013
$915$	−2.12166	−0.0701398
$916$	11.5032	0.380076
$917$	0.242016	0.00799206
$918$	−11.6628	−0.384931
$919$	−55.9215	−1.84468	−0.922341	−	0.386377i	$-0.873726\pi$
$919$	−55.9215	−1.84468	−0.922341	+	0.386377i	$0.873726\pi$
$920$	10.9420	0.360747
$921$	−16.5313	−0.544725
$922$	35.9970	1.18550
$923$	58.3165	1.91951
$924$	−0.0221599	−0.000729006 0
$925$	1.00995	0.0332070
$926$	0.496753	0.0163243
$927$	−27.7076	−0.910036
$928$	12.2777	0.403034
$929$	−6.56464	−0.215379	−0.107689	−	0.994185i	$-0.534345\pi$
$929$	−6.56464	−0.215379	−0.107689	+	0.994185i	$0.534345\pi$
$930$	3.47429	0.113926
$931$	−17.4785	−0.572836
$932$	4.54140	0.148759
$933$	−19.3729	−0.634239
$934$	52.9204	1.73161
$935$	1.52053	0.0497266
$936$	−24.7993	−0.810591
$937$	−36.2937	−1.18566	−0.592832	−	0.805326i	$-0.701990\pi$
$937$	−36.2937	−1.18566	−0.592832	+	0.805326i	$0.701990\pi$
$938$	0.949682	0.0310082
$939$	6.66326	0.217447
$940$	0.843510	0.0275123
$941$	30.4869	0.993846	0.496923	−	0.867795i	$-0.334463\pi$
$941$	30.4869	0.993846	0.496923	+	0.867795i	$0.334463\pi$
$942$	23.8065	0.775656
$943$	−14.9919	−0.488204
$944$	40.8136	1.32837
$945$	0.264068	0.00859014
$946$	−2.90078	−0.0943125
$947$	13.1412	0.427031	0.213516	−	0.976940i	$-0.431509\pi$
$947$	13.1412	0.427031	0.213516	+	0.976940i	$0.431509\pi$
$948$	−2.25938	−0.0733811
$949$	4.93652	0.160246
$950$	3.88874	0.126167
$951$	5.78372	0.187550
$952$	0.200002	0.00648211
$953$	−11.2879	−0.365651	−0.182826	−	0.983145i	$-0.558524\pi$
$953$	−11.2879	−0.365651	−0.182826	+	0.983145i	$0.558524\pi$
$954$	33.7443	1.09251
$955$	−12.2641	−0.396856
$956$	−4.93837	−0.159718
$957$	−5.08292	−0.164308
$958$	−28.8952	−0.933562
$959$	−1.17814	−0.0380442
$960$	5.52978	0.178473
$961$	−25.7740	−0.831419
$962$	−7.76150	−0.250241
$963$	27.3586	0.881618
$964$	−8.30756	−0.267568
$965$	−21.6728	−0.697672
$966$	0.363159	0.0116844
$967$	3.23231	0.103944	0.0519720	−	0.998649i	$-0.483449\pi$
$967$	3.23231	0.103944	0.0519720	+	0.998649i	$0.483449\pi$
$968$	−2.45420	−0.0788811
$969$	3.70797	0.119117
$970$	14.4460	0.463832
$971$	−6.31370	−0.202616	−0.101308	−	0.994855i	$-0.532303\pi$
$971$	−6.31370	−0.202616	−0.101308	+	0.994855i	$0.532303\pi$
$972$	6.81047	0.218446
$973$	0.163060	0.00522748
$974$	59.2970	1.90000
$975$	−4.81923	−0.154339
$976$	10.1442	0.324709
$977$	20.5190	0.656461	0.328230	−	0.944598i	$-0.393548\pi$
$977$	20.5190	0.656461	0.328230	+	0.944598i	$0.393548\pi$
$978$	−2.32141	−0.0742306
$979$	−6.43943	−0.205805
$980$	−2.96347	−0.0946645
$981$	14.2425	0.454728
$982$	12.0275	0.383812
$983$	−3.04390	−0.0970854	−0.0485427	−	0.998821i	$-0.515458\pi$
$983$	−3.04390	−0.0970854	−0.0485427	+	0.998821i	$0.515458\pi$
$984$	−8.05636	−0.256827
$985$	10.1002	0.321820
$986$	−12.3247	−0.392497
$987$	−0.104207	−0.00331694
$988$	−5.22260	−0.166153
$989$	8.30763	0.264167
$990$	−3.18663	−0.101278
$991$	21.9614	0.697626	0.348813	−	0.937192i	$-0.386585\pi$
$991$	21.9614	0.697626	0.348813	+	0.937192i	$0.386585\pi$
$992$	−5.39069	−0.171155
$993$	−22.9060	−0.726900
$994$	0.985654	0.0312631
$995$	−0.0243400	−0.000771630 0
$996$	3.93520	0.124692
$997$	4.20590	0.133202	0.0666011	−	0.997780i	$-0.478785\pi$
$997$	4.20590	0.133202	0.0666011	+	0.997780i	$0.478785\pi$
$998$	−23.0369	−0.729219
$999$	−4.97608	−0.157436

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.b.1.19	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.b.1.19	✓	23	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4015 = 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4015.a (trivial)

\(f(q)\)	\(=\)	\(q+1.55677 q^{2} +0.976240 q^{3} +0.423526 q^{4} +1.00000 q^{5} +1.51978 q^{6} -0.0535957 q^{7} -2.45420 q^{8} -2.04696 q^{9} +O(q^{10})\)	\(q+1.55677 q^{2} +0.976240 q^{3} +0.423526 q^{4} +1.00000 q^{5} +1.51978 q^{6} -0.0535957 q^{7} -2.45420 q^{8} -2.04696 q^{9} +1.55677 q^{10} +1.00000 q^{11} +0.413463 q^{12} -4.93652 q^{13} -0.0834361 q^{14} +0.976240 q^{15} -4.66768 q^{16} +1.52053 q^{17} -3.18663 q^{18} +2.49796 q^{19} +0.423526 q^{20} -0.0523223 q^{21} +1.55677 q^{22} -4.45847 q^{23} -2.39589 q^{24} +1.00000 q^{25} -7.68501 q^{26} -4.92704 q^{27} -0.0226992 q^{28} -5.20663 q^{29} +1.51978 q^{30} +2.28605 q^{31} -2.35808 q^{32} +0.976240 q^{33} +2.36711 q^{34} -0.0535957 q^{35} -0.866939 q^{36} +1.00995 q^{37} +3.88874 q^{38} -4.81923 q^{39} -2.45420 q^{40} +3.36257 q^{41} -0.0814536 q^{42} -1.86334 q^{43} +0.423526 q^{44} -2.04696 q^{45} -6.94080 q^{46} +1.99164 q^{47} -4.55677 q^{48} -6.99713 q^{49} +1.55677 q^{50} +1.48440 q^{51} -2.09075 q^{52} -10.5893 q^{53} -7.67026 q^{54} +1.00000 q^{55} +0.131535 q^{56} +2.43861 q^{57} -8.10551 q^{58} -8.74388 q^{59} +0.413463 q^{60} -2.17329 q^{61} +3.55884 q^{62} +0.109708 q^{63} +5.66437 q^{64} -4.93652 q^{65} +1.51978 q^{66} -11.3821 q^{67} +0.643983 q^{68} -4.35254 q^{69} -0.0834361 q^{70} -11.8133 q^{71} +5.02364 q^{72} -1.00000 q^{73} +1.57226 q^{74} +0.976240 q^{75} +1.05795 q^{76} -0.0535957 q^{77} -7.50242 q^{78} -5.46451 q^{79} -4.66768 q^{80} +1.33089 q^{81} +5.23474 q^{82} +9.51766 q^{83} -0.0221599 q^{84} +1.52053 q^{85} -2.90078 q^{86} -5.08292 q^{87} -2.45420 q^{88} -6.43943 q^{89} -3.18663 q^{90} +0.264576 q^{91} -1.88828 q^{92} +2.23173 q^{93} +3.10052 q^{94} +2.49796 q^{95} -2.30206 q^{96} +9.27946 q^{97} -10.8929 q^{98} -2.04696 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 5 q^{2} - 3 q^{3} + 15 q^{4} + 23 q^{5} - 15 q^{6} - 10 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) 23 * q - 5 * q^2 - 3 * q^3 + 15 * q^4 + 23 * q^5 - 15 * q^6 - 10 * q^7 - 12 * q^8 + 8 * q^9	\( 23 q - 5 q^{2} - 3 q^{3} + 15 q^{4} + 23 q^{5} - 15 q^{6} - 10 q^{7} - 12 q^{8} + 8 q^{9} - 5 q^{10} + 23 q^{11} + 4 q^{12} - 19 q^{13} - 5 q^{14} - 3 q^{15} - q^{16} - 26 q^{17} + 5 q^{18} - 34 q^{19} + 15 q^{20} - 26 q^{21} - 5 q^{22} - 4 q^{23} - 23 q^{24} + 23 q^{25} - 13 q^{26} - 3 q^{27} - 28 q^{28} - 36 q^{29} - 15 q^{30} - 24 q^{31} - 19 q^{32} - 3 q^{33} - 4 q^{34} - 10 q^{35} - 14 q^{36} + 4 q^{37} - 15 q^{38} - 34 q^{39} - 12 q^{40} - 74 q^{41} + 7 q^{42} - 15 q^{43} + 15 q^{44} + 8 q^{45} + 3 q^{46} + q^{47} + 29 q^{48} + q^{49} - 5 q^{50} - 47 q^{51} - 23 q^{52} - 6 q^{53} - 11 q^{54} + 23 q^{55} - 20 q^{56} - 19 q^{57} + 22 q^{58} - 17 q^{59} + 4 q^{60} - 59 q^{61} + 38 q^{62} - 21 q^{63} - 18 q^{64} - 19 q^{65} - 15 q^{66} + 12 q^{67} + 5 q^{68} - 8 q^{69} - 5 q^{70} - 34 q^{71} + 21 q^{72} - 23 q^{73} - 9 q^{74} - 3 q^{75} - 53 q^{76} - 10 q^{77} + 23 q^{78} - 62 q^{79} - q^{80} + 7 q^{81} + 24 q^{82} - 8 q^{83} + 46 q^{84} - 26 q^{85} - 11 q^{86} + q^{87} - 12 q^{88} - 77 q^{89} + 5 q^{90} - 6 q^{91} + 47 q^{92} - 32 q^{94} - 34 q^{95} - 53 q^{96} - 21 q^{97} - 3 q^{98} + 8 q^{99}+O(q^{100}) \) 23 * q - 5 * q^2 - 3 * q^3 + 15 * q^4 + 23 * q^5 - 15 * q^6 - 10 * q^7 - 12 * q^8 + 8 * q^9 - 5 * q^10 + 23 * q^11 + 4 * q^12 - 19 * q^13 - 5 * q^14 - 3 * q^15 - q^16 - 26 * q^17 + 5 * q^18 - 34 * q^19 + 15 * q^20 - 26 * q^21 - 5 * q^22 - 4 * q^23 - 23 * q^24 + 23 * q^25 - 13 * q^26 - 3 * q^27 - 28 * q^28 - 36 * q^29 - 15 * q^30 - 24 * q^31 - 19 * q^32 - 3 * q^33 - 4 * q^34 - 10 * q^35 - 14 * q^36 + 4 * q^37 - 15 * q^38 - 34 * q^39 - 12 * q^40 - 74 * q^41 + 7 * q^42 - 15 * q^43 + 15 * q^44 + 8 * q^45 + 3 * q^46 + q^47 + 29 * q^48 + q^49 - 5 * q^50 - 47 * q^51 - 23 * q^52 - 6 * q^53 - 11 * q^54 + 23 * q^55 - 20 * q^56 - 19 * q^57 + 22 * q^58 - 17 * q^59 + 4 * q^60 - 59 * q^61 + 38 * q^62 - 21 * q^63 - 18 * q^64 - 19 * q^65 - 15 * q^66 + 12 * q^67 + 5 * q^68 - 8 * q^69 - 5 * q^70 - 34 * q^71 + 21 * q^72 - 23 * q^73 - 9 * q^74 - 3 * q^75 - 53 * q^76 - 10 * q^77 + 23 * q^78 - 62 * q^79 - q^80 + 7 * q^81 + 24 * q^82 - 8 * q^83 + 46 * q^84 - 26 * q^85 - 11 * q^86 + q^87 - 12 * q^88 - 77 * q^89 + 5 * q^90 - 6 * q^91 + 47 * q^92 - 32 * q^94 - 34 * q^95 - 53 * q^96 - 21 * q^97 - 3 * q^98 + 8 * q^99

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