LMFDB - Embedded newform 4015.2.a.c.1.7

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			1.7
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)

$f(q)$	$=$	$q-1.22952 q^{2} -0.451066 q^{3} -0.488282 q^{4} +1.00000 q^{5} +0.554594 q^{6} -2.48868 q^{7} +3.05939 q^{8} -2.79654 q^{9} +O(q^{10})$	\(q-1.22952 q^{2} -0.451066 q^{3} -0.488282 q^{4} +1.00000 q^{5} +0.554594 q^{6} -2.48868 q^{7} +3.05939 q^{8} -2.79654 q^{9} -1.22952 q^{10} -1.00000 q^{11} +0.220247 q^{12} +0.572678 q^{13} +3.05988 q^{14} -0.451066 q^{15} -2.78502 q^{16} -4.98415 q^{17} +3.43840 q^{18} -0.991776 q^{19} -0.488282 q^{20} +1.12256 q^{21} +1.22952 q^{22} +7.38662 q^{23} -1.37999 q^{24} +1.00000 q^{25} -0.704119 q^{26} +2.61462 q^{27} +1.21518 q^{28} +10.2737 q^{29} +0.554594 q^{30} +3.51322 q^{31} -2.69455 q^{32} +0.451066 q^{33} +6.12811 q^{34} -2.48868 q^{35} +1.36550 q^{36} -5.05291 q^{37} +1.21941 q^{38} -0.258315 q^{39} +3.05939 q^{40} +4.09170 q^{41} -1.38021 q^{42} +1.31409 q^{43} +0.488282 q^{44} -2.79654 q^{45} -9.08199 q^{46} +4.64758 q^{47} +1.25623 q^{48} -0.806480 q^{49} -1.22952 q^{50} +2.24818 q^{51} -0.279628 q^{52} +6.59701 q^{53} -3.21473 q^{54} -1.00000 q^{55} -7.61384 q^{56} +0.447356 q^{57} -12.6317 q^{58} -11.0492 q^{59} +0.220247 q^{60} -5.18907 q^{61} -4.31958 q^{62} +6.95969 q^{63} +8.88304 q^{64} +0.572678 q^{65} -0.554594 q^{66} -4.75667 q^{67} +2.43367 q^{68} -3.33185 q^{69} +3.05988 q^{70} -11.2756 q^{71} -8.55571 q^{72} +1.00000 q^{73} +6.21265 q^{74} -0.451066 q^{75} +0.484266 q^{76} +2.48868 q^{77} +0.317604 q^{78} +6.70273 q^{79} -2.78502 q^{80} +7.21025 q^{81} -5.03082 q^{82} -0.880945 q^{83} -0.548124 q^{84} -4.98415 q^{85} -1.61570 q^{86} -4.63412 q^{87} -3.05939 q^{88} -0.942817 q^{89} +3.43840 q^{90} -1.42521 q^{91} -3.60675 q^{92} -1.58469 q^{93} -5.71429 q^{94} -0.991776 q^{95} +1.21542 q^{96} -4.05200 q^{97} +0.991583 q^{98} +2.79654 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) $ 23 * q - 3 * q^2 - 5 * q^3 + 15 * q^4 + 23 * q^5 - 5 * q^6 - 6 * q^8 + 8 * q^9	\( 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 23 q^{11} - 18 q^{12} - 5 q^{13} - 17 q^{14} - 5 q^{15} - q^{16} - 36 q^{17} + q^{18} + 6 q^{19} + 15 q^{20} - 18 q^{21} + 3 q^{22} - 14 q^{23} - 7 q^{24} + 23 q^{25} - 21 q^{26} - 29 q^{27} + 28 q^{28} - 36 q^{29} - 5 q^{30} - 16 q^{31} - 5 q^{32} + 5 q^{33} - 28 q^{34} - 14 q^{36} - 24 q^{37} + q^{38} - 10 q^{39} - 6 q^{40} - 36 q^{41} - 5 q^{42} + 17 q^{43} - 15 q^{44} + 8 q^{45} - 25 q^{46} - 21 q^{47} - 17 q^{48} - 27 q^{49} - 3 q^{50} + 19 q^{51} - 21 q^{52} - 28 q^{53} - 15 q^{54} - 23 q^{55} - 46 q^{56} - 23 q^{57} - 16 q^{58} - 61 q^{59} - 18 q^{60} - 17 q^{61} - 22 q^{62} - 9 q^{63} - 18 q^{64} - 5 q^{65} + 5 q^{66} + 2 q^{67} - 39 q^{68} - 36 q^{69} - 17 q^{70} - 50 q^{71} + 15 q^{72} + 23 q^{73} + 17 q^{74} - 5 q^{75} - 21 q^{76} + 49 q^{78} - 18 q^{79} - q^{80} - 57 q^{81} + 14 q^{82} - 20 q^{83} - 38 q^{84} - 36 q^{85} - 45 q^{86} + 37 q^{87} + 6 q^{88} - 93 q^{89} + q^{90} - 42 q^{91} - 39 q^{92} - 18 q^{93} - 6 q^{94} + 6 q^{95} - 9 q^{96} - 31 q^{97} - 31 q^{98} - 8 q^{99}+O(q^{100}) \) 23 * q - 3 * q^2 - 5 * q^3 + 15 * q^4 + 23 * q^5 - 5 * q^6 - 6 * q^8 + 8 * q^9 - 3 * q^10 - 23 * q^11 - 18 * q^12 - 5 * q^13 - 17 * q^14 - 5 * q^15 - q^16 - 36 * q^17 + q^18 + 6 * q^19 + 15 * q^20 - 18 * q^21 + 3 * q^22 - 14 * q^23 - 7 * q^24 + 23 * q^25 - 21 * q^26 - 29 * q^27 + 28 * q^28 - 36 * q^29 - 5 * q^30 - 16 * q^31 - 5 * q^32 + 5 * q^33 - 28 * q^34 - 14 * q^36 - 24 * q^37 + q^38 - 10 * q^39 - 6 * q^40 - 36 * q^41 - 5 * q^42 + 17 * q^43 - 15 * q^44 + 8 * q^45 - 25 * q^46 - 21 * q^47 - 17 * q^48 - 27 * q^49 - 3 * q^50 + 19 * q^51 - 21 * q^52 - 28 * q^53 - 15 * q^54 - 23 * q^55 - 46 * q^56 - 23 * q^57 - 16 * q^58 - 61 * q^59 - 18 * q^60 - 17 * q^61 - 22 * q^62 - 9 * q^63 - 18 * q^64 - 5 * q^65 + 5 * q^66 + 2 * q^67 - 39 * q^68 - 36 * q^69 - 17 * q^70 - 50 * q^71 + 15 * q^72 + 23 * q^73 + 17 * q^74 - 5 * q^75 - 21 * q^76 + 49 * q^78 - 18 * q^79 - q^80 - 57 * q^81 + 14 * q^82 - 20 * q^83 - 38 * q^84 - 36 * q^85 - 45 * q^86 + 37 * q^87 + 6 * q^88 - 93 * q^89 + q^90 - 42 * q^91 - 39 * q^92 - 18 * q^93 - 6 * q^94 + 6 * q^95 - 9 * q^96 - 31 * q^97 - 31 * 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 $

$2$	−1.22952	−0.869402	−0.434701	−	0.900575i	$-0.643146\pi$
$2$	−1.22952	−0.869402	−0.434701	+	0.900575i	$0.643146\pi$
$3$	−0.451066	−0.260423	−0.130211	−	0.991486i	$-0.541566\pi$
$3$	−0.451066	−0.260423	−0.130211	+	0.991486i	$0.541566\pi$
$4$	−0.488282	−0.244141
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.554594	0.226412
$7$	−2.48868	−0.940632	−0.470316	−	0.882498i	$-0.655860\pi$
$7$	−2.48868	−0.940632	−0.470316	+	0.882498i	$0.655860\pi$
$8$	3.05939	1.08166
$9$	−2.79654	−0.932180
$10$	−1.22952	−0.388808
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.220247	0.0635798
$13$	0.572678	0.158832	0.0794162	−	0.996842i	$-0.474694\pi$
$13$	0.572678	0.158832	0.0794162	+	0.996842i	$0.474694\pi$
$14$	3.05988	0.817787
$15$	−0.451066	−0.116465
$16$	−2.78502	−0.696254
$17$	−4.98415	−1.20883	−0.604417	−	0.796668i	$-0.706594\pi$
$17$	−4.98415	−1.20883	−0.604417	+	0.796668i	$0.706594\pi$
$18$	3.43840	0.810439
$19$	−0.991776	−0.227529	−0.113765	−	0.993508i	$-0.536291\pi$
$19$	−0.991776	−0.227529	−0.113765	+	0.993508i	$0.536291\pi$
$20$	−0.488282	−0.109183
$21$	1.12256	0.244962
$22$	1.22952	0.262134
$23$	7.38662	1.54022	0.770108	−	0.637914i	$-0.220202\pi$
$23$	7.38662	1.54022	0.770108	+	0.637914i	$0.220202\pi$
$24$	−1.37999	−0.281688
$25$	1.00000	0.200000
$26$	−0.704119	−0.138089
$27$	2.61462	0.503184
$28$	1.21518	0.229647
$29$	10.2737	1.90778	0.953891	−	0.300154i	$-0.0970381\pi$
$29$	10.2737	1.90778	0.953891	+	0.300154i	$0.0970381\pi$
$30$	0.554594	0.101255
$31$	3.51322	0.630993	0.315497	−	0.948927i	$-0.397829\pi$
$31$	3.51322	0.630993	0.315497	+	0.948927i	$0.397829\pi$
$32$	−2.69455	−0.476333
$33$	0.451066	0.0785204
$34$	6.12811	1.05096
$35$	−2.48868	−0.420663
$36$	1.36550	0.227583
$37$	−5.05291	−0.830693	−0.415346	−	0.909663i	$-0.636340\pi$
$37$	−5.05291	−0.830693	−0.415346	+	0.909663i	$0.636340\pi$
$38$	1.21941	0.197814
$39$	−0.258315	−0.0413636
$40$	3.05939	0.483732
$41$	4.09170	0.639016	0.319508	−	0.947584i	$-0.396482\pi$
$41$	4.09170	0.639016	0.319508	+	0.947584i	$0.396482\pi$
$42$	−1.38021	−0.212970
$43$	1.31409	0.200396	0.100198	−	0.994967i	$-0.468052\pi$
$43$	1.31409	0.200396	0.100198	+	0.994967i	$0.468052\pi$
$44$	0.488282	0.0736112
$45$	−2.79654	−0.416884
$46$	−9.08199	−1.33907
$47$	4.64758	0.677919	0.338960	−	0.940801i	$-0.389925\pi$
$47$	4.64758	0.677919	0.338960	+	0.940801i	$0.389925\pi$
$48$	1.25623	0.181321
$49$	−0.806480	−0.115211
$50$	−1.22952	−0.173880
$51$	2.24818	0.314808
$52$	−0.279628	−0.0387775
$53$	6.59701	0.906169	0.453084	−	0.891468i	$-0.350324\pi$
$53$	6.59701	0.906169	0.453084	+	0.891468i	$0.350324\pi$
$54$	−3.21473	−0.437469
$55$	−1.00000	−0.134840
$56$	−7.61384	−1.01744
$57$	0.447356	0.0592538
$58$	−12.6317	−1.65863
$59$	−11.0492	−1.43848	−0.719241	−	0.694761i	$-0.755510\pi$
$59$	−11.0492	−1.43848	−0.719241	+	0.694761i	$0.755510\pi$
$60$	0.220247	0.0284338
$61$	−5.18907	−0.664392	−0.332196	−	0.943210i	$-0.607790\pi$
$61$	−5.18907	−0.664392	−0.332196	+	0.943210i	$0.607790\pi$
$62$	−4.31958	−0.548587
$63$	6.95969	0.876838
$64$	8.88304	1.11038
$65$	0.572678	0.0710320
$66$	−0.554594	−0.0682658
$67$	−4.75667	−0.581120	−0.290560	−	0.956857i	$-0.593842\pi$
$67$	−4.75667	−0.581120	−0.290560	+	0.956857i	$0.593842\pi$
$68$	2.43367	0.295126
$69$	−3.33185	−0.401107
$70$	3.05988	0.365725
$71$	−11.2756	−1.33816	−0.669082	−	0.743189i	$-0.733312\pi$
$71$	−11.2756	−1.33816	−0.669082	+	0.743189i	$0.733312\pi$
$72$	−8.55571	−1.00830
$73$	1.00000	0.117041
$73$	1.00000	0.117041
$74$	6.21265	0.722205
$75$	−0.451066	−0.0520846
$76$	0.484266	0.0555491
$77$	2.48868	0.283611
$78$	0.317604	0.0359616
$79$	6.70273	0.754117	0.377058	−	0.926190i	$-0.376936\pi$
$79$	6.70273	0.754117	0.377058	+	0.926190i	$0.376936\pi$
$80$	−2.78502	−0.311374
$81$	7.21025	0.801139
$82$	−5.03082	−0.555561
$83$	−0.880945	−0.0966963	−0.0483482	−	0.998831i	$-0.515396\pi$
$83$	−0.880945	−0.0966963	−0.0483482	+	0.998831i	$0.515396\pi$
$84$	−0.548124	−0.0598052
$85$	−4.98415	−0.540607
$86$	−1.61570	−0.174225
$87$	−4.63412	−0.496830
$88$	−3.05939	−0.326132
$89$	−0.942817	−0.0999384	−0.0499692	−	0.998751i	$-0.515912\pi$
$89$	−0.942817	−0.0999384	−0.0499692	+	0.998751i	$0.515912\pi$
$90$	3.43840	0.362439
$91$	−1.42521	−0.149403
$92$	−3.60675	−0.376030
$93$	−1.58469	−0.164325
$94$	−5.71429	−0.589384
$95$	−0.991776	−0.101754
$96$	1.21542	0.124048
$97$	−4.05200	−0.411418	−0.205709	−	0.978613i	$-0.565950\pi$
$97$	−4.05200	−0.411418	−0.205709	+	0.978613i	$0.565950\pi$
$98$	0.991583	0.100165
$99$	2.79654	0.281063
$100$	−0.488282	−0.0488282
$101$	6.06190	0.603181	0.301591	−	0.953437i	$-0.402482\pi$
$101$	6.06190	0.603181	0.301591	+	0.953437i	$0.402482\pi$
$102$	−2.76418	−0.273695
$103$	0.644774	0.0635314	0.0317657	−	0.999495i	$-0.489887\pi$
$103$	0.644774	0.0635314	0.0317657	+	0.999495i	$0.489887\pi$
$104$	1.75205	0.171802
$105$	1.12256	0.109550
$106$	−8.11115	−0.787825
$107$	20.4480	1.97678	0.988391	−	0.151931i	$-0.0485492\pi$
$107$	20.4480	1.97678	0.988391	+	0.151931i	$0.0485492\pi$
$108$	−1.27667	−0.122848
$109$	−1.06638	−0.102141	−0.0510705	−	0.998695i	$-0.516263\pi$
$109$	−1.06638	−0.102141	−0.0510705	+	0.998695i	$0.516263\pi$
$110$	1.22952	0.117230
$111$	2.27919	0.216331
$112$	6.93101	0.654919
$113$	−11.5692	−1.08834	−0.544172	−	0.838974i	$-0.683156\pi$
$113$	−11.5692	−1.08834	−0.544172	+	0.838974i	$0.683156\pi$
$114$	−0.550033	−0.0515153
$115$	7.38662	0.688805
$116$	−5.01647	−0.465767
$117$	−1.60152	−0.148060
$118$	13.5852	1.25062
$119$	12.4040	1.13707
$120$	−1.37999	−0.125975
$121$	1.00000	0.0909091
$122$	6.38006	0.577624
$123$	−1.84562	−0.166414
$124$	−1.71544	−0.154051
$125$	1.00000	0.0894427
$126$	−8.55707	−0.762325
$127$	−13.9899	−1.24140	−0.620701	−	0.784048i	$-0.713152\pi$
$127$	−13.9899	−1.24140	−0.620701	+	0.784048i	$0.713152\pi$
$128$	−5.53277	−0.489032
$129$	−0.592740	−0.0521878
$130$	−0.704119	−0.0617553
$131$	5.31506	0.464379	0.232189	−	0.972671i	$-0.425411\pi$
$131$	5.31506	0.464379	0.232189	+	0.972671i	$0.425411\pi$
$132$	−0.220247	−0.0191700
$133$	2.46821	0.214021
$134$	5.84842	0.505226
$135$	2.61462	0.225031
$136$	−15.2485	−1.30755
$137$	5.56667	0.475593	0.237797	−	0.971315i	$-0.423575\pi$
$137$	5.56667	0.475593	0.237797	+	0.971315i	$0.423575\pi$
$138$	4.09657	0.348723
$139$	−6.18958	−0.524993	−0.262497	−	0.964933i	$-0.584546\pi$
$139$	−6.18958	−0.524993	−0.262497	+	0.964933i	$0.584546\pi$
$140$	1.21518	0.102701
$141$	−2.09636	−0.176546
$142$	13.8635	1.16340
$143$	−0.572678	−0.0478898
$144$	7.78841	0.649034
$145$	10.2737	0.853186
$146$	−1.22952	−0.101756
$147$	0.363775	0.0300037
$148$	2.46724	0.202806
$149$	7.74743	0.634694	0.317347	−	0.948309i	$-0.397208\pi$
$149$	7.74743	0.634694	0.317347	+	0.948309i	$0.397208\pi$
$150$	0.554594	0.0452824
$151$	−19.2811	−1.56907	−0.784537	−	0.620082i	$-0.787099\pi$
$151$	−19.2811	−1.56907	−0.784537	+	0.620082i	$0.787099\pi$
$152$	−3.03423	−0.246109
$153$	13.9384	1.12685
$154$	−3.05988	−0.246572
$155$	3.51322	0.282189
$156$	0.126131	0.0100985
$157$	2.97964	0.237801	0.118900	−	0.992906i	$-0.462063\pi$
$157$	2.97964	0.237801	0.118900	+	0.992906i	$0.462063\pi$
$158$	−8.24114	−0.655630
$159$	−2.97568	−0.235987
$160$	−2.69455	−0.213023
$161$	−18.3829	−1.44878
$162$	−8.86515	−0.696512
$163$	9.66071	0.756685	0.378342	−	0.925666i	$-0.376494\pi$
$163$	9.66071	0.756685	0.378342	+	0.925666i	$0.376494\pi$
$164$	−1.99790	−0.156010
$165$	0.451066	0.0351154
$166$	1.08314	0.0840679
$167$	−17.4971	−1.35397	−0.676983	−	0.735999i	$-0.736713\pi$
$167$	−17.4971	−1.35397	−0.676983	+	0.735999i	$0.736713\pi$
$168$	3.43434	0.264965
$169$	−12.6720	−0.974772
$170$	6.12811	0.470005
$171$	2.77354	0.212098
$172$	−0.641645	−0.0489249
$173$	−14.4750	−1.10051	−0.550257	−	0.834995i	$-0.685470\pi$
$173$	−14.4750	−1.10051	−0.550257	+	0.834995i	$0.685470\pi$
$174$	5.69774	0.431945
$175$	−2.48868	−0.188126
$176$	2.78502	0.209929
$177$	4.98391	0.374613
$178$	1.15921	0.0868866
$179$	−6.47207	−0.483745	−0.241872	−	0.970308i	$-0.577762\pi$
$179$	−6.47207	−0.483745	−0.241872	+	0.970308i	$0.577762\pi$
$180$	1.36550	0.101778
$181$	−21.3801	−1.58917	−0.794584	−	0.607154i	$-0.792311\pi$
$181$	−21.3801	−1.58917	−0.794584	+	0.607154i	$0.792311\pi$
$182$	1.75233	0.129891
$183$	2.34061	0.173023
$184$	22.5985	1.66599
$185$	−5.05291	−0.371497
$186$	1.94841	0.142865
$187$	4.98415	0.364477
$188$	−2.26933	−0.165508
$189$	−6.50695	−0.473311
$190$	1.21941	0.0884652
$191$	24.2829	1.75705	0.878525	−	0.477696i	$-0.158528\pi$
$191$	24.2829	1.75705	0.878525	+	0.477696i	$0.158528\pi$
$192$	−4.00683	−0.289168
$193$	3.10124	0.223232	0.111616	−	0.993751i	$-0.464397\pi$
$193$	3.10124	0.223232	0.111616	+	0.993751i	$0.464397\pi$
$194$	4.98202	0.357688
$195$	−0.258315	−0.0184984
$196$	0.393789	0.0281278
$197$	4.19576	0.298935	0.149468	−	0.988767i	$-0.452244\pi$
$197$	4.19576	0.298935	0.149468	+	0.988767i	$0.452244\pi$
$198$	−3.43840	−0.244356
$199$	13.7506	0.974754	0.487377	−	0.873192i	$-0.337954\pi$
$199$	13.7506	0.974754	0.487377	+	0.873192i	$0.337954\pi$
$200$	3.05939	0.216332
$201$	2.14557	0.151337
$202$	−7.45322	−0.524407
$203$	−25.5680	−1.79452
$204$	−1.09775	−0.0768575
$205$	4.09170	0.285777
$206$	−0.792762	−0.0552343
$207$	−20.6570	−1.43576
$208$	−1.59492	−0.110588
$209$	0.991776	0.0686026
$210$	−1.38021	−0.0952433
$211$	−8.88283	−0.611519	−0.305760	−	0.952109i	$-0.598910\pi$
$211$	−8.88283	−0.611519	−0.305760	+	0.952109i	$0.598910\pi$
$212$	−3.22120	−0.221233
$213$	5.08602	0.348488
$214$	−25.1412	−1.71862
$215$	1.31409	0.0896200
$216$	7.99914	0.544273
$217$	−8.74328	−0.593533
$218$	1.31114	0.0888016
$219$	−0.451066	−0.0304802
$220$	0.488282	0.0329199
$221$	−2.85432	−0.192002
$222$	−2.80231	−0.188079
$223$	−15.1288	−1.01310	−0.506550	−	0.862210i	$-0.669080\pi$
$223$	−15.1288	−1.01310	−0.506550	+	0.862210i	$0.669080\pi$
$224$	6.70586	0.448054
$225$	−2.79654	−0.186436
$226$	14.2246	0.946207
$227$	13.9051	0.922914	0.461457	−	0.887163i	$-0.347327\pi$
$227$	13.9051	0.922914	0.461457	+	0.887163i	$0.347327\pi$
$228$	−0.218436	−0.0144663
$229$	−25.4867	−1.68421	−0.842103	−	0.539316i	$-0.818683\pi$
$229$	−25.4867	−1.68421	−0.842103	+	0.539316i	$0.818683\pi$
$230$	−9.08199	−0.598849
$231$	−1.12256	−0.0738588
$232$	31.4313	2.06357
$233$	25.3625	1.66155	0.830775	−	0.556609i	$-0.187898\pi$
$233$	25.3625	1.66155	0.830775	+	0.556609i	$0.187898\pi$
$234$	1.96910	0.128724
$235$	4.64758	0.303175
$236$	5.39512	0.351192
$237$	−3.02337	−0.196389
$238$	−15.2509	−0.988569
$239$	−19.0891	−1.23477	−0.617384	−	0.786662i	$-0.711808\pi$
$239$	−19.0891	−1.23477	−0.617384	+	0.786662i	$0.711808\pi$
$240$	1.25623	0.0810890
$241$	−15.7769	−1.01628	−0.508140	−	0.861274i	$-0.669667\pi$
$241$	−15.7769	−1.01628	−0.508140	+	0.861274i	$0.669667\pi$
$242$	−1.22952	−0.0790365
$243$	−11.0962	−0.711819
$244$	2.53373	0.162205
$245$	−0.806480	−0.0515241
$246$	2.26923	0.144681
$247$	−0.567969	−0.0361390
$248$	10.7483	0.682519
$249$	0.397364	0.0251819
$250$	−1.22952	−0.0777616
$251$	−13.8898	−0.876716	−0.438358	−	0.898800i	$-0.644440\pi$
$251$	−13.8898	−0.876716	−0.438358	+	0.898800i	$0.644440\pi$
$252$	−3.39829	−0.214072
$253$	−7.38662	−0.464393
$254$	17.2008	1.07928
$255$	2.24818	0.140787
$256$	−10.9634	−0.685214
$257$	18.5290	1.15581	0.577904	−	0.816105i	$-0.303871\pi$
$257$	18.5290	1.15581	0.577904	+	0.816105i	$0.303871\pi$
$258$	0.728785	0.0453722
$259$	12.5751	0.781376
$260$	−0.279628	−0.0173418
$261$	−28.7309	−1.77840
$262$	−6.53497	−0.403732
$263$	−26.1453	−1.61219	−0.806093	−	0.591789i	$-0.798422\pi$
$263$	−26.1453	−1.61219	−0.806093	+	0.591789i	$0.798422\pi$
$264$	1.37999	0.0849323
$265$	6.59701	0.405251
$266$	−3.03471	−0.186070
$267$	0.425272	0.0260263
$268$	2.32259	0.141875
$269$	−15.2905	−0.932279	−0.466139	−	0.884711i	$-0.654356\pi$
$269$	−15.2905	−0.932279	−0.466139	+	0.884711i	$0.654356\pi$
$270$	−3.21473	−0.195642
$271$	−5.07426	−0.308239	−0.154120	−	0.988052i	$-0.549254\pi$
$271$	−5.07426	−0.308239	−0.154120	+	0.988052i	$0.549254\pi$
$272$	13.8810	0.841657
$273$	0.642864	0.0389079
$274$	−6.84434	−0.413481
$275$	−1.00000	−0.0603023
$276$	1.62688	0.0979267
$277$	18.8452	1.13230	0.566149	−	0.824303i	$-0.308433\pi$
$277$	18.8452	1.13230	0.566149	+	0.824303i	$0.308433\pi$
$278$	7.61021	0.456430
$279$	−9.82487	−0.588199
$280$	−7.61384	−0.455014
$281$	−30.2589	−1.80509	−0.902547	−	0.430591i	$-0.858305\pi$
$281$	−30.2589	−1.80509	−0.902547	+	0.430591i	$0.858305\pi$
$282$	2.57752	0.153489
$283$	16.2315	0.964861	0.482431	−	0.875934i	$-0.339754\pi$
$283$	16.2315	0.964861	0.482431	+	0.875934i	$0.339754\pi$
$284$	5.50565	0.326700
$285$	0.447356	0.0264991
$286$	0.704119	0.0416354
$287$	−10.1829	−0.601079
$288$	7.53541	0.444028
$289$	7.84179	0.461282
$290$	−12.6317	−0.741761
$291$	1.82772	0.107143
$292$	−0.488282	−0.0285745
$293$	−26.8030	−1.56585	−0.782925	−	0.622116i	$-0.786273\pi$
$293$	−26.8030	−1.56585	−0.782925	+	0.622116i	$0.786273\pi$
$294$	−0.447269	−0.0260853
$295$	−11.0492	−0.643309
$296$	−15.4588	−0.898525
$297$	−2.61462	−0.151716
$298$	−9.52562	−0.551804
$299$	4.23015	0.244636
$300$	0.220247	0.0127160
$301$	−3.27034	−0.188499
$302$	23.7065	1.36416
$303$	−2.73431	−0.157082
$304$	2.76211	0.158418
$305$	−5.18907	−0.297125
$306$	−17.1375	−0.979687
$307$	4.47454	0.255375	0.127688	−	0.991814i	$-0.459245\pi$
$307$	4.47454	0.255375	0.127688	+	0.991814i	$0.459245\pi$
$308$	−1.21518	−0.0692411
$309$	−0.290835	−0.0165450
$310$	−4.31958	−0.245335
$311$	16.2293	0.920281	0.460141	−	0.887846i	$-0.347799\pi$
$311$	16.2293	0.920281	0.460141	+	0.887846i	$0.347799\pi$
$312$	−0.790288	−0.0447412
$313$	33.7435	1.90730	0.953648	−	0.300923i	$-0.0972947\pi$
$313$	33.7435	1.90730	0.953648	+	0.300923i	$0.0972947\pi$
$314$	−3.66352	−0.206744
$315$	6.95969	0.392134
$316$	−3.27282	−0.184111
$317$	12.3289	0.692462	0.346231	−	0.938149i	$-0.387461\pi$
$317$	12.3289	0.692462	0.346231	+	0.938149i	$0.387461\pi$
$318$	3.65866	0.205168
$319$	−10.2737	−0.575218
$320$	8.88304	0.496577
$321$	−9.22339	−0.514799
$322$	22.6021	1.25957
$323$	4.94317	0.275045
$324$	−3.52063	−0.195591
$325$	0.572678	0.0317665
$326$	−11.8780	−0.657863
$327$	0.481009	0.0265999
$328$	12.5181	0.691197
$329$	−11.5663	−0.637672
$330$	−0.554594	−0.0305294
$331$	1.70974	0.0939757	0.0469879	−	0.998895i	$-0.485038\pi$
$331$	1.70974	0.0939757	0.0469879	+	0.998895i	$0.485038\pi$
$332$	0.430149	0.0236075
$333$	14.1307	0.774355
$334$	21.5130	1.17714
$335$	−4.75667	−0.259885
$336$	−3.12634	−0.170556
$337$	6.86508	0.373965	0.186982	−	0.982363i	$-0.440129\pi$
$337$	6.86508	0.373965	0.186982	+	0.982363i	$0.440129\pi$
$338$	15.5805	0.847469
$339$	5.21849	0.283429
$340$	2.43367	0.131984
$341$	−3.51322	−0.190252
$342$	−3.41012	−0.184398
$343$	19.4278	1.04900
$344$	4.02031	0.216760
$345$	−3.33185	−0.179381
$346$	17.7973	0.956789
$347$	−8.81811	−0.473381	−0.236691	−	0.971585i	$-0.576063\pi$
$347$	−8.81811	−0.473381	−0.236691	+	0.971585i	$0.576063\pi$
$348$	2.26276	0.121296
$349$	−31.6523	−1.69431	−0.847153	−	0.531349i	$-0.821685\pi$
$349$	−31.6523	−1.69431	−0.847153	+	0.531349i	$0.821685\pi$
$350$	3.05988	0.163557
$351$	1.49734	0.0799219
$352$	2.69455	0.143620
$353$	−11.3821	−0.605810	−0.302905	−	0.953021i	$-0.597957\pi$
$353$	−11.3821	−0.605810	−0.302905	+	0.953021i	$0.597957\pi$
$354$	−6.12781	−0.325690
$355$	−11.2756	−0.598445
$356$	0.460360	0.0243991
$357$	−5.59500	−0.296119
$358$	7.95753	0.420569
$359$	2.80407	0.147993	0.0739967	−	0.997258i	$-0.476425\pi$
$359$	2.80407	0.147993	0.0739967	+	0.997258i	$0.476425\pi$
$360$	−8.55571	−0.450925
$361$	−18.0164	−0.948231
$362$	26.2872	1.38163
$363$	−0.451066	−0.0236748
$364$	0.695905	0.0364753
$365$	1.00000	0.0523424
$366$	−2.87783	−0.150426
$367$	−18.2530	−0.952800	−0.476400	−	0.879229i	$-0.658059\pi$
$367$	−18.2530	−0.952800	−0.476400	+	0.879229i	$0.658059\pi$
$368$	−20.5719	−1.07238
$369$	−11.4426	−0.595678
$370$	6.21265	0.322980
$371$	−16.4178	−0.852371
$372$	0.773777	0.0401185
$373$	20.0016	1.03565	0.517823	−	0.855488i	$-0.326743\pi$
$373$	20.0016	1.03565	0.517823	+	0.855488i	$0.326743\pi$
$374$	−6.12811	−0.316877
$375$	−0.451066	−0.0232929
$376$	14.2188	0.733277
$377$	5.88354	0.303017
$378$	8.00042	0.411497
$379$	−8.49988	−0.436609	−0.218305	−	0.975881i	$-0.570053\pi$
$379$	−8.49988	−0.436609	−0.218305	+	0.975881i	$0.570053\pi$
$380$	0.484266	0.0248423
$381$	6.31036	0.323289
$382$	−29.8563	−1.52758
$383$	−16.6701	−0.851803	−0.425902	−	0.904770i	$-0.640043\pi$
$383$	−16.6701	−0.851803	−0.425902	+	0.904770i	$0.640043\pi$
$384$	2.49564	0.127355
$385$	2.48868	0.126835
$386$	−3.81303	−0.194078
$387$	−3.67490	−0.186806
$388$	1.97852	0.100444
$389$	−31.4756	−1.59587	−0.797937	−	0.602741i	$-0.794075\pi$
$389$	−31.4756	−1.59587	−0.797937	+	0.602741i	$0.794075\pi$
$390$	0.317604	0.0160825
$391$	−36.8160	−1.86187
$392$	−2.46734	−0.124619
$393$	−2.39744	−0.120935
$394$	−5.15876	−0.259895
$395$	6.70273	0.337251
$396$	−1.36550	−0.0686189
$397$	−30.8841	−1.55003	−0.775014	−	0.631943i	$-0.782257\pi$
$397$	−30.8841	−1.55003	−0.775014	+	0.631943i	$0.782257\pi$
$398$	−16.9066	−0.847453
$399$	−1.11333	−0.0557360
$400$	−2.78502	−0.139251
$401$	−27.0272	−1.34968	−0.674838	−	0.737966i	$-0.735786\pi$
$401$	−27.0272	−1.34968	−0.674838	+	0.737966i	$0.735786\pi$
$402$	−2.63802	−0.131572
$403$	2.01195	0.100222
$404$	−2.95991	−0.147261
$405$	7.21025	0.358280
$406$	31.4363	1.56016
$407$	5.05291	0.250463
$408$	6.87806	0.340515
$409$	−21.1040	−1.04353	−0.521764	−	0.853090i	$-0.674726\pi$
$409$	−21.1040	−1.04353	−0.521764	+	0.853090i	$0.674726\pi$
$410$	−5.03082	−0.248455
$411$	−2.51094	−0.123855
$412$	−0.314831	−0.0155106
$413$	27.4979	1.35308
$414$	25.3981	1.24825
$415$	−0.880945	−0.0432439
$416$	−1.54311	−0.0756571
$417$	2.79191	0.136720
$418$	−1.21941	−0.0596432
$419$	13.9771	0.682827	0.341414	−	0.939913i	$-0.389094\pi$
$419$	13.9771	0.682827	0.341414	+	0.939913i	$0.389094\pi$
$420$	−0.548124	−0.0267457
$421$	−25.5353	−1.24451	−0.622257	−	0.782813i	$-0.713784\pi$
$421$	−25.5353	−1.24451	−0.622257	+	0.782813i	$0.713784\pi$
$422$	10.9216	0.531656
$423$	−12.9971	−0.631943
$424$	20.1828	0.980165
$425$	−4.98415	−0.241767
$426$	−6.25336	−0.302976
$427$	12.9139	0.624949
$428$	−9.98438	−0.482613
$429$	0.258315	0.0124716
$430$	−1.61570	−0.0779158
$431$	23.4263	1.12841	0.564204	−	0.825636i	$-0.309183\pi$
$431$	23.4263	1.12841	0.564204	+	0.825636i	$0.309183\pi$
$432$	−7.28176	−0.350344
$433$	−20.8368	−1.00135	−0.500677	−	0.865634i	$-0.666915\pi$
$433$	−20.8368	−1.00135	−0.500677	+	0.865634i	$0.666915\pi$
$434$	10.7500	0.516018
$435$	−4.63412	−0.222189
$436$	0.520695	0.0249368
$437$	−7.32587	−0.350444
$438$	0.554594	0.0264995
$439$	20.4221	0.974694	0.487347	−	0.873208i	$-0.337965\pi$
$439$	20.4221	0.974694	0.487347	+	0.873208i	$0.337965\pi$
$440$	−3.05939	−0.145851
$441$	2.25535	0.107398
$442$	3.50944	0.166927
$443$	−14.4506	−0.686567	−0.343284	−	0.939232i	$-0.611539\pi$
$443$	−14.4506	−0.686567	−0.343284	+	0.939232i	$0.611539\pi$
$444$	−1.11289	−0.0528153
$445$	−0.942817	−0.0446938
$446$	18.6012	0.880791
$447$	−3.49460	−0.165289
$448$	−22.1070	−1.04446
$449$	12.5222	0.590958	0.295479	−	0.955349i	$-0.404521\pi$
$449$	12.5222	0.590958	0.295479	+	0.955349i	$0.404521\pi$
$450$	3.43840	0.162088
$451$	−4.09170	−0.192671
$452$	5.64905	0.265709
$453$	8.69704	0.408623
$454$	−17.0966	−0.802383
$455$	−1.42521	−0.0668150
$456$	1.36864	0.0640923
$457$	−6.99225	−0.327084	−0.163542	−	0.986536i	$-0.552292\pi$
$457$	−6.99225	−0.327084	−0.163542	+	0.986536i	$0.552292\pi$
$458$	31.3364	1.46425
$459$	−13.0317	−0.608266
$460$	−3.60675	−0.168166
$461$	−9.49334	−0.442149	−0.221074	−	0.975257i	$-0.570956\pi$
$461$	−9.49334	−0.442149	−0.221074	+	0.975257i	$0.570956\pi$
$462$	1.38021	0.0642130
$463$	39.6283	1.84168	0.920842	−	0.389935i	$-0.127503\pi$
$463$	39.6283	1.84168	0.920842	+	0.389935i	$0.127503\pi$
$464$	−28.6125	−1.32830
$465$	−1.58469	−0.0734884
$466$	−31.1836	−1.44455
$467$	6.44487	0.298233	0.149116	−	0.988820i	$-0.452357\pi$
$467$	6.44487	0.298233	0.149116	+	0.988820i	$0.452357\pi$
$468$	0.781992	0.0361476
$469$	11.8378	0.546620
$470$	−5.71429	−0.263581
$471$	−1.34401	−0.0619288
$472$	−33.8038	−1.55595
$473$	−1.31409	−0.0604218
$474$	3.71730	0.170741
$475$	−0.991776	−0.0455058
$476$	−6.05662	−0.277605
$477$	−18.4488	−0.844713
$478$	23.4704	1.07351
$479$	−21.9796	−1.00427	−0.502136	−	0.864789i	$-0.667452\pi$
$479$	−21.9796	−1.00427	−0.502136	+	0.864789i	$0.667452\pi$
$480$	1.21542	0.0554760
$481$	−2.89369	−0.131941
$482$	19.3980	0.883556
$483$	8.29190	0.377294
$484$	−0.488282	−0.0221946
$485$	−4.05200	−0.183992
$486$	13.6429	0.618856
$487$	−24.6963	−1.11910	−0.559548	−	0.828798i	$-0.689025\pi$
$487$	−24.6963	−1.11910	−0.559548	+	0.828798i	$0.689025\pi$
$488$	−15.8754	−0.718645
$489$	−4.35761	−0.197058
$490$	0.991583	0.0447952
$491$	−24.0094	−1.08353	−0.541764	−	0.840531i	$-0.682243\pi$
$491$	−24.0094	−1.08353	−0.541764	+	0.840531i	$0.682243\pi$
$492$	0.901184	0.0406285
$493$	−51.2058	−2.30619
$494$	0.698329	0.0314193
$495$	2.79654	0.125695
$496$	−9.78439	−0.439332
$497$	28.0613	1.25872
$498$	−0.488567	−0.0218932
$499$	12.0231	0.538228	0.269114	−	0.963108i	$-0.413269\pi$
$499$	12.0231	0.538228	0.269114	+	0.963108i	$0.413269\pi$
$500$	−0.488282	−0.0218366
$501$	7.89233	0.352603
$502$	17.0778	0.762219
$503$	3.67841	0.164012	0.0820061	−	0.996632i	$-0.473867\pi$
$503$	3.67841	0.164012	0.0820061	+	0.996632i	$0.473867\pi$
$504$	21.2924	0.948439
$505$	6.06190	0.269751
$506$	9.08199	0.403744
$507$	5.71592	0.253853
$508$	6.83100	0.303077
$509$	−38.5706	−1.70961	−0.854805	−	0.518949i	$-0.826323\pi$
$509$	−38.5706	−1.70961	−0.854805	+	0.518949i	$0.826323\pi$
$510$	−2.76418	−0.122400
$511$	−2.48868	−0.110093
$512$	24.5453	1.08476
$513$	−2.59312	−0.114489
$514$	−22.7818	−1.00486
$515$	0.644774	0.0284121
$516$	0.289424	0.0127412
$517$	−4.64758	−0.204400
$518$	−15.4613	−0.679330
$519$	6.52918	0.286599
$520$	1.75205	0.0768323
$521$	5.46668	0.239500	0.119750	−	0.992804i	$-0.461791\pi$
$521$	5.46668	0.239500	0.119750	+	0.992804i	$0.461791\pi$
$522$	35.3252	1.54614
$523$	−23.8950	−1.04486	−0.522429	−	0.852683i	$-0.674974\pi$
$523$	−23.8950	−1.04486	−0.522429	+	0.852683i	$0.674974\pi$
$524$	−2.59524	−0.113374
$525$	1.12256	0.0489924
$526$	32.1461	1.40164
$527$	−17.5104	−0.762767
$528$	−1.25623	−0.0546702
$529$	31.5621	1.37226
$530$	−8.11115	−0.352326
$531$	30.8995	1.34092
$532$	−1.20518	−0.0522513
$533$	2.34323	0.101496
$534$	−0.522881	−0.0226273
$535$	20.4480	0.884044
$536$	−14.5525	−0.628573
$537$	2.91933	0.125978
$538$	18.8000	0.810525
$539$	0.806480	0.0347376
$540$	−1.27667	−0.0549392
$541$	0.184811	0.00794567	0.00397283	−	0.999992i	$-0.498735\pi$
$541$	0.184811	0.00794567	0.00397283	+	0.999992i	$0.498735\pi$
$542$	6.23890	0.267984
$543$	9.64381	0.413856
$544$	13.4300	0.575808
$545$	−1.06638	−0.0456789
$546$	−0.790414	−0.0338266
$547$	22.1015	0.944993	0.472497	−	0.881332i	$-0.343353\pi$
$547$	22.1015	0.944993	0.472497	+	0.881332i	$0.343353\pi$
$548$	−2.71811	−0.116112
$549$	14.5114	0.619333
$550$	1.22952	0.0524269
$551$	−10.1892	−0.434076
$552$	−10.1934	−0.433861
$553$	−16.6810	−0.709346
$554$	−23.1705	−0.984422
$555$	2.27919	0.0967463
$556$	3.02226	0.128172
$557$	−26.6867	−1.13075	−0.565376	−	0.824833i	$-0.691269\pi$
$557$	−26.6867	−1.13075	−0.565376	+	0.824833i	$0.691269\pi$
$558$	12.0799	0.511382
$559$	0.752549	0.0318294
$560$	6.93101	0.292889
$561$	−2.24818	−0.0949183
$562$	37.2039	1.56935
$563$	38.0393	1.60317	0.801583	−	0.597883i	$-0.203991\pi$
$563$	38.0393	1.60317	0.801583	+	0.597883i	$0.203991\pi$
$564$	1.02362	0.0431020
$565$	−11.5692	−0.486722
$566$	−19.9569	−0.838852
$567$	−17.9440	−0.753577
$568$	−34.4964	−1.44744
$569$	18.9394	0.793980	0.396990	−	0.917823i	$-0.370055\pi$
$569$	18.9394	0.793980	0.396990	+	0.917823i	$0.370055\pi$
$570$	−0.550033	−0.0230384
$571$	3.66077	0.153198	0.0765992	−	0.997062i	$-0.475594\pi$
$571$	3.66077	0.153198	0.0765992	+	0.997062i	$0.475594\pi$
$572$	0.279628	0.0116918
$573$	−10.9532	−0.457576
$574$	12.5201	0.522579
$575$	7.38662	0.308043
$576$	−24.8418	−1.03507
$577$	−35.6762	−1.48522	−0.742610	−	0.669724i	$-0.766412\pi$
$577$	−35.6762	−1.48522	−0.742610	+	0.669724i	$0.766412\pi$
$578$	−9.64164	−0.401039
$579$	−1.39886	−0.0581347
$580$	−5.01647	−0.208297
$581$	2.19239	0.0909556
$582$	−2.24722	−0.0931501
$583$	−6.59701	−0.273220
$584$	3.05939	0.126598
$585$	−1.60152	−0.0662146
$586$	32.9549	1.36135
$587$	7.58097	0.312900	0.156450	−	0.987686i	$-0.449995\pi$
$587$	7.58097	0.312900	0.156450	+	0.987686i	$0.449995\pi$
$588$	−0.177625	−0.00732513
$589$	−3.48433	−0.143569
$590$	13.5852	0.559293
$591$	−1.89256	−0.0778496
$592$	14.0724	0.578373
$593$	−31.1119	−1.27761	−0.638807	−	0.769367i	$-0.720572\pi$
$593$	−31.1119	−1.27761	−0.638807	+	0.769367i	$0.720572\pi$
$594$	3.21473	0.131902
$595$	12.4040	0.508513
$596$	−3.78293	−0.154955
$597$	−6.20242	−0.253848
$598$	−5.20106	−0.212687
$599$	9.53708	0.389675	0.194837	−	0.980836i	$-0.437582\pi$
$599$	9.53708	0.389675	0.194837	+	0.980836i	$0.437582\pi$
$600$	−1.37999	−0.0563377
$601$	24.7425	1.00927	0.504634	−	0.863333i	$-0.331627\pi$
$601$	24.7425	1.00927	0.504634	+	0.863333i	$0.331627\pi$
$602$	4.02095	0.163882
$603$	13.3022	0.541708
$604$	9.41461	0.383075
$605$	1.00000	0.0406558
$606$	3.36189	0.136568
$607$	30.7647	1.24870	0.624350	−	0.781145i	$-0.285364\pi$
$607$	30.7647	1.24870	0.624350	+	0.781145i	$0.285364\pi$
$608$	2.67239	0.108380
$609$	11.5328	0.467334
$610$	6.38006	0.258321
$611$	2.66157	0.107675
$612$	−6.80586	−0.275110
$613$	36.9692	1.49317	0.746587	−	0.665288i	$-0.231691\pi$
$613$	36.9692	1.49317	0.746587	+	0.665288i	$0.231691\pi$
$614$	−5.50153	−0.222024
$615$	−1.84562	−0.0744228
$616$	7.61384	0.306770
$617$	−19.5089	−0.785399	−0.392699	−	0.919667i	$-0.628459\pi$
$617$	−19.5089	−0.785399	−0.392699	+	0.919667i	$0.628459\pi$
$618$	0.357588	0.0143843
$619$	1.13185	0.0454927	0.0227464	−	0.999741i	$-0.492759\pi$
$619$	1.13185	0.0454927	0.0227464	+	0.999741i	$0.492759\pi$
$620$	−1.71544	−0.0688938
$621$	19.3132	0.775012
$622$	−19.9543	−0.800094
$623$	2.34637	0.0940053
$624$	0.719413	0.0287996
$625$	1.00000	0.0400000
$626$	−41.4883	−1.65821
$627$	−0.447356	−0.0178657
$628$	−1.45490	−0.0580569
$629$	25.1845	1.00417
$630$	−8.55707	−0.340922
$631$	5.64221	0.224613	0.112306	−	0.993674i	$-0.464176\pi$
$631$	5.64221	0.224613	0.112306	+	0.993674i	$0.464176\pi$
$632$	20.5063	0.815696
$633$	4.00674	0.159254
$634$	−15.1587	−0.602027
$635$	−13.9899	−0.555172
$636$	1.45297	0.0576141
$637$	−0.461854	−0.0182993
$638$	12.6317	0.500095
$639$	31.5326	1.24741
$640$	−5.53277	−0.218702
$641$	−9.45854	−0.373590	−0.186795	−	0.982399i	$-0.559810\pi$
$641$	−9.45854	−0.373590	−0.186795	+	0.982399i	$0.559810\pi$
$642$	11.3403	0.447567
$643$	−31.0332	−1.22383	−0.611914	−	0.790924i	$-0.709600\pi$
$643$	−31.0332	−1.22383	−0.611914	+	0.790924i	$0.709600\pi$
$644$	8.97604	0.353705
$645$	−0.592740	−0.0233391
$646$	−6.07772	−0.239125
$647$	3.37459	0.132669	0.0663345	−	0.997797i	$-0.478870\pi$
$647$	3.37459	0.132669	0.0663345	+	0.997797i	$0.478870\pi$
$648$	22.0590	0.866559
$649$	11.0492	0.433719
$650$	−0.704119	−0.0276178
$651$	3.94379	0.154569
$652$	−4.71715	−0.184738
$653$	−34.2479	−1.34023	−0.670113	−	0.742259i	$-0.733754\pi$
$653$	−34.2479	−1.34023	−0.670113	+	0.742259i	$0.733754\pi$
$654$	−0.591410	−0.0231260
$655$	5.31506	0.207676
$656$	−11.3955	−0.444918
$657$	−2.79654	−0.109103
$658$	14.2210	0.554393
$659$	0.974268	0.0379521	0.0189761	−	0.999820i	$-0.493959\pi$
$659$	0.974268	0.0379521	0.0189761	+	0.999820i	$0.493959\pi$
$660$	−0.220247	−0.00857310
$661$	−2.25429	−0.0876816	−0.0438408	−	0.999039i	$-0.513959\pi$
$661$	−2.25429	−0.0876816	−0.0438408	+	0.999039i	$0.513959\pi$
$662$	−2.10216	−0.0817026
$663$	1.28748	0.0500017
$664$	−2.69516	−0.104592
$665$	2.46821	0.0957131
$666$	−17.3739	−0.673225
$667$	75.8880	2.93840
$668$	8.54350	0.330558
$669$	6.82409	0.263835
$670$	5.84842	0.225944
$671$	5.18907	0.200322
$672$	−3.02478	−0.116684
$673$	44.9426	1.73241	0.866205	−	0.499689i	$-0.166552\pi$
$673$	44.9426	1.73241	0.866205	+	0.499689i	$0.166552\pi$
$674$	−8.44075	−0.325125
$675$	2.61462	0.100637
$676$	6.18752	0.237982
$677$	−0.00946534	−0.000363782 0	−0.000181891	−	1.00000i	$-0.500058\pi$
$677$	−0.00946534	−0.000363782 0	−0.000181891		1.00000i	$0.500058\pi$
$678$	−6.41623	−0.246414
$679$	10.0841	0.386993
$680$	−15.2485	−0.584752
$681$	−6.27212	−0.240348
$682$	4.31958	0.165405
$683$	−27.7655	−1.06242	−0.531209	−	0.847241i	$-0.678262\pi$
$683$	−27.7655	−1.06242	−0.531209	+	0.847241i	$0.678262\pi$
$684$	−1.35427	−0.0517818
$685$	5.56667	0.212692
$686$	−23.8869	−0.912005
$687$	11.4962	0.438606
$688$	−3.65976	−0.139527
$689$	3.77796	0.143929
$690$	4.09657	0.155954
$691$	42.3306	1.61033	0.805165	−	0.593050i	$-0.202077\pi$
$691$	42.3306	1.61033	0.805165	+	0.593050i	$0.202077\pi$
$692$	7.06788	0.268680
$693$	−6.95969	−0.264377
$694$	10.8420	0.411558
$695$	−6.18958	−0.234784
$696$	−14.1776	−0.537400
$697$	−20.3937	−0.772465
$698$	38.9171	1.47303
$699$	−11.4401	−0.432706
$700$	1.21518	0.0459293
$701$	32.0261	1.20961	0.604805	−	0.796374i	$-0.293251\pi$
$701$	32.0261	1.20961	0.604805	+	0.796374i	$0.293251\pi$
$702$	−1.84100	−0.0694842
$703$	5.01135	0.189007
$704$	−8.88304	−0.334792
$705$	−2.09636	−0.0789536
$706$	13.9946	0.526692
$707$	−15.0861	−0.567372
$708$	−2.43355	−0.0914584
$709$	−18.1271	−0.680779	−0.340390	−	0.940284i	$-0.610559\pi$
$709$	−18.1271	−0.680779	−0.340390	+	0.940284i	$0.610559\pi$
$710$	13.8635	0.520289
$711$	−18.7445	−0.702972
$712$	−2.88445	−0.108099
$713$	25.9508	0.971866
$714$	6.87916	0.257446
$715$	−0.572678	−0.0214170
$716$	3.16019	0.118102
$717$	8.61042	0.321562
$718$	−3.44766	−0.128666
$719$	−7.77496	−0.289957	−0.144979	−	0.989435i	$-0.546311\pi$
$719$	−7.77496	−0.289957	−0.144979	+	0.989435i	$0.546311\pi$
$720$	7.78841	0.290257
$721$	−1.60463	−0.0597597
$722$	22.1515	0.824393
$723$	7.11642	0.264663
$724$	10.4395	0.387981
$725$	10.2737	0.381556
$726$	0.554594	0.0205829
$727$	45.6647	1.69361	0.846805	−	0.531904i	$-0.178523\pi$
$727$	45.6647	1.69361	0.846805	+	0.531904i	$0.178523\pi$
$728$	−4.36028	−0.161603
$729$	−16.6257	−0.615765
$730$	−1.22952	−0.0455066
$731$	−6.54961	−0.242246
$732$	−1.14288	−0.0422420
$733$	16.3609	0.604305	0.302153	−	0.953260i	$-0.402295\pi$
$733$	16.3609	0.604305	0.302153	+	0.953260i	$0.402295\pi$
$734$	22.4424	0.828366
$735$	0.363775	0.0134181
$736$	−19.9036	−0.733656
$737$	4.75667	0.175214
$738$	14.0689	0.517883
$739$	32.8863	1.20974	0.604871	−	0.796324i	$-0.293225\pi$
$739$	32.8863	1.20974	0.604871	+	0.796324i	$0.293225\pi$
$740$	2.46724	0.0906976
$741$	0.256191	0.00941142
$742$	20.1860	0.741053
$743$	36.9905	1.35705	0.678525	−	0.734577i	$-0.262619\pi$
$743$	36.9905	1.35705	0.678525	+	0.734577i	$0.262619\pi$
$744$	−4.84820	−0.177744
$745$	7.74743	0.283844
$746$	−24.5924	−0.900392
$747$	2.46360	0.0901384
$748$	−2.43367	−0.0889838
$749$	−50.8885	−1.85942
$750$	0.554594	0.0202509
$751$	16.8110	0.613442	0.306721	−	0.951799i	$-0.400768\pi$
$751$	16.8110	0.613442	0.306721	+	0.951799i	$0.400768\pi$
$752$	−12.9436	−0.472004
$753$	6.26521	0.228317
$754$	−7.23392	−0.263444
$755$	−19.2811	−0.701711
$756$	3.17722	0.115554
$757$	0.282691	0.0102746	0.00513730	−	0.999987i	$-0.498365\pi$
$757$	0.282691	0.0102746	0.00513730	+	0.999987i	$0.498365\pi$
$758$	10.4508	0.379589
$759$	3.33185	0.120938
$760$	−3.03423	−0.110063
$761$	−38.6453	−1.40089	−0.700445	−	0.713706i	$-0.747015\pi$
$761$	−38.6453	−1.40089	−0.700445	+	0.713706i	$0.747015\pi$
$762$	−7.75871	−0.281068
$763$	2.65389	0.0960771
$764$	−11.8569	−0.428968
$765$	13.9384	0.503943
$766$	20.4962	0.740559
$767$	−6.32763	−0.228477
$768$	4.94522	0.178445
$769$	−8.84886	−0.319098	−0.159549	−	0.987190i	$-0.551004\pi$
$769$	−8.84886	−0.319098	−0.159549	+	0.987190i	$0.551004\pi$
$770$	−3.05988	−0.110270
$771$	−8.35779	−0.300999
$772$	−1.51428	−0.0545000
$773$	10.3577	0.372542	0.186271	−	0.982498i	$-0.440360\pi$
$773$	10.3577	0.372542	0.186271	+	0.982498i	$0.440360\pi$
$774$	4.51836	0.162409
$775$	3.51322	0.126199
$776$	−12.3967	−0.445014
$777$	−5.67218	−0.203488
$778$	38.6998	1.38746
$779$	−4.05805	−0.145395
$780$	0.126131	0.00451620
$781$	11.2756	0.403472
$782$	45.2660	1.61871
$783$	26.8619	0.959965
$784$	2.24606	0.0802165
$785$	2.97964	0.106348
$786$	2.94770	0.105141
$787$	−29.3651	−1.04675	−0.523376	−	0.852102i	$-0.675328\pi$
$787$	−29.3651	−1.04675	−0.523376	+	0.852102i	$0.675328\pi$
$788$	−2.04871	−0.0729823
$789$	11.7932	0.419850
$790$	−8.24114	−0.293207
$791$	28.7921	1.02373
$792$	8.55571	0.304014
$793$	−2.97167	−0.105527
$794$	37.9726	1.34760
$795$	−2.97568	−0.105537
$796$	−6.71417	−0.237977
$797$	−25.8753	−0.916549	−0.458274	−	0.888811i	$-0.651532\pi$
$797$	−25.8753	−0.916549	−0.458274	+	0.888811i	$0.651532\pi$
$798$	1.36886	0.0484570
$799$	−23.1642	−0.819492
$800$	−2.69455	−0.0952667
$801$	2.63663	0.0931606
$802$	33.2305	1.17341
$803$	−1.00000	−0.0352892
$804$	−1.04764	−0.0369475
$805$	−18.3829	−0.647912
$806$	−2.47373	−0.0871333
$807$	6.89702	0.242787
$808$	18.5457	0.652436
$809$	−16.2802	−0.572380	−0.286190	−	0.958173i	$-0.592389\pi$
$809$	−16.2802	−0.572380	−0.286190	+	0.958173i	$0.592389\pi$
$810$	−8.86515	−0.311490
$811$	9.40600	0.330289	0.165145	−	0.986269i	$-0.447191\pi$
$811$	9.40600	0.330289	0.165145	+	0.986269i	$0.447191\pi$
$812$	12.4844	0.438116
$813$	2.28882	0.0802725
$814$	−6.21265	−0.217753
$815$	9.66071	0.338400
$816$	−6.26122	−0.219187
$817$	−1.30328	−0.0455960
$818$	25.9478	0.907245
$819$	3.98566	0.139270
$820$	−1.99790	−0.0697697
$821$	−33.6668	−1.17498	−0.587491	−	0.809231i	$-0.699884\pi$
$821$	−33.6668	−1.17498	−0.587491	+	0.809231i	$0.699884\pi$
$822$	3.08724	0.107680
$823$	−44.2729	−1.54325	−0.771627	−	0.636075i	$-0.780557\pi$
$823$	−44.2729	−1.54325	−0.771627	+	0.636075i	$0.780557\pi$
$824$	1.97261	0.0687193
$825$	0.451066	0.0157041
$826$	−33.8092	−1.17637
$827$	23.3343	0.811412	0.405706	−	0.914004i	$-0.367026\pi$
$827$	23.3343	0.811412	0.405706	+	0.914004i	$0.367026\pi$
$828$	10.0864	0.350527
$829$	−53.6198	−1.86229	−0.931145	−	0.364648i	$-0.881189\pi$
$829$	−53.6198	−1.86229	−0.931145	+	0.364648i	$0.881189\pi$
$830$	1.08314	0.0375963
$831$	−8.50042	−0.294876
$832$	5.08712	0.176364
$833$	4.01962	0.139272
$834$	−3.43270	−0.118865
$835$	−17.4971	−0.605512
$836$	−0.484266	−0.0167487
$837$	9.18574	0.317506
$838$	−17.1851	−0.593651
$839$	−33.1614	−1.14486	−0.572429	−	0.819954i	$-0.693999\pi$
$839$	−33.1614	−1.14486	−0.572429	+	0.819954i	$0.693999\pi$
$840$	3.43434	0.118496
$841$	76.5493	2.63963
$842$	31.3962	1.08198
$843$	13.6487	0.470088
$844$	4.33732	0.149297
$845$	−12.6720	−0.435931
$846$	15.9802	0.549412
$847$	−2.48868	−0.0855120
$848$	−18.3728	−0.630924
$849$	−7.32146	−0.251272
$850$	6.12811	0.210193
$851$	−37.3239	−1.27945
$852$	−2.48341	−0.0850802
$853$	44.9427	1.53881	0.769405	−	0.638761i	$-0.220553\pi$
$853$	44.9427	1.53881	0.769405	+	0.638761i	$0.220553\pi$
$854$	−15.8779	−0.543332
$855$	2.77354	0.0948531
$856$	62.5584	2.13820
$857$	37.8231	1.29201	0.646006	−	0.763333i	$-0.276438\pi$
$857$	37.8231	1.29201	0.646006	+	0.763333i	$0.276438\pi$
$858$	−0.317604	−0.0108428
$859$	11.4686	0.391302	0.195651	−	0.980674i	$-0.437318\pi$
$859$	11.4686	0.391302	0.195651	+	0.980674i	$0.437318\pi$
$860$	−0.641645	−0.0218799
$861$	4.59317	0.156535
$862$	−28.8032	−0.981039
$863$	9.50509	0.323557	0.161778	−	0.986827i	$-0.448277\pi$
$863$	9.50509	0.323557	0.161778	+	0.986827i	$0.448277\pi$
$864$	−7.04522	−0.239683
$865$	−14.4750	−0.492165
$866$	25.6193	0.870578
$867$	−3.53716	−0.120128
$868$	4.26918	0.144906
$869$	−6.70273	−0.227375
$870$	5.69774	0.193172
$871$	−2.72404	−0.0923006
$872$	−3.26248	−0.110482
$873$	11.3316	0.383516
$874$	9.00730	0.304676
$875$	−2.48868	−0.0841327
$876$	0.220247	0.00744146
$877$	−2.01062	−0.0678937	−0.0339468	−	0.999424i	$-0.510808\pi$
$877$	−2.01062	−0.0678937	−0.0339468	+	0.999424i	$0.510808\pi$
$878$	−25.1094	−0.847400
$879$	12.0899	0.407783
$880$	2.78502	0.0938829
$881$	−48.8329	−1.64522	−0.822611	−	0.568605i	$-0.807483\pi$
$881$	−48.8329	−1.64522	−0.822611	+	0.568605i	$0.807483\pi$
$882$	−2.77300	−0.0933718
$883$	2.83453	0.0953896	0.0476948	−	0.998862i	$-0.484813\pi$
$883$	2.83453	0.0953896	0.0476948	+	0.998862i	$0.484813\pi$
$884$	1.39371	0.0468756
$885$	4.98391	0.167532
$886$	17.7673	0.596903
$887$	−54.0379	−1.81442	−0.907208	−	0.420682i	$-0.861791\pi$
$887$	−54.0379	−1.81442	−0.907208	+	0.420682i	$0.861791\pi$
$888$	6.97294	0.233997
$889$	34.8163	1.16770
$890$	1.15921	0.0388569
$891$	−7.21025	−0.241553
$892$	7.38713	0.247339
$893$	−4.60936	−0.154246
$894$	4.29668	0.143702
$895$	−6.47207	−0.216337
$896$	13.7693	0.459999
$897$	−1.90808	−0.0637088
$898$	−15.3962	−0.513780
$899$	36.0939	1.20380
$900$	1.36550	0.0455166
$901$	−32.8805	−1.09541
$902$	5.03082	0.167508
$903$	1.47514	0.0490895
$904$	−35.3948	−1.17722
$905$	−21.3801	−0.710697
$906$	−10.6932	−0.355257
$907$	8.17696	0.271511	0.135756	−	0.990742i	$-0.456654\pi$
$907$	8.17696	0.271511	0.135756	+	0.990742i	$0.456654\pi$
$908$	−6.78961	−0.225321
$909$	−16.9523	−0.562274
$910$	1.75233	0.0580890
$911$	−37.9194	−1.25632	−0.628162	−	0.778082i	$-0.716193\pi$
$911$	−37.9194	−1.25632	−0.628162	+	0.778082i	$0.716193\pi$
$912$	−1.24589	−0.0412557
$913$	0.880945	0.0291550
$914$	8.59711	0.284367
$915$	2.34061	0.0773782
$916$	12.4447	0.411184
$917$	−13.2275	−0.436809
$918$	16.0227	0.528828
$919$	33.2636	1.09727	0.548633	−	0.836063i	$-0.315148\pi$
$919$	33.2636	1.09727	0.548633	+	0.836063i	$0.315148\pi$
$920$	22.5985	0.745052
$921$	−2.01831	−0.0665056
$922$	11.6722	0.384405
$923$	−6.45727	−0.212544
$924$	0.548124	0.0180320
$925$	−5.05291	−0.166139
$926$	−48.7238	−1.60116
$927$	−1.80314	−0.0592227
$928$	−27.6830	−0.908740
$929$	−8.84585	−0.290223	−0.145111	−	0.989415i	$-0.546354\pi$
$929$	−8.84585	−0.290223	−0.145111	+	0.989415i	$0.546354\pi$
$930$	1.94841	0.0638910
$931$	0.799848	0.0262140
$932$	−12.3840	−0.405652
$933$	−7.32050	−0.239662
$934$	−7.92409	−0.259284
$935$	4.98415	0.162999
$936$	−4.89967	−0.160151
$937$	58.9818	1.92685	0.963427	−	0.267973i	$-0.0863537\pi$
$937$	58.9818	1.92685	0.963427	+	0.267973i	$0.0863537\pi$
$938$	−14.5548	−0.475232
$939$	−15.2205	−0.496704
$940$	−2.26933	−0.0740173
$941$	16.5424	0.539267	0.269633	−	0.962963i	$-0.413098\pi$
$941$	16.5424	0.539267	0.269633	+	0.962963i	$0.413098\pi$
$942$	1.65249	0.0538410
$943$	30.2238	0.984222
$944$	30.7722	1.00155
$945$	−6.50695	−0.211671
$946$	1.61570	0.0525308
$947$	−26.8983	−0.874077	−0.437038	−	0.899443i	$-0.643973\pi$
$947$	−26.8983	−0.874077	−0.437038	+	0.899443i	$0.643973\pi$
$948$	1.47626	0.0479466
$949$	0.572678	0.0185899
$950$	1.21941	0.0395628
$951$	−5.56116	−0.180333
$952$	37.9486	1.22992
$953$	−30.3312	−0.982524	−0.491262	−	0.871012i	$-0.663464\pi$
$953$	−30.3312	−0.982524	−0.491262	+	0.871012i	$0.663464\pi$
$954$	22.6832	0.734394
$955$	24.2829	0.785777
$956$	9.32083	0.301457
$957$	4.63412	0.149800
$958$	27.0243	0.873115
$959$	−13.8537	−0.447358
$960$	−4.00683	−0.129320
$961$	−18.6573	−0.601847
$962$	3.55785	0.114710
$963$	−57.1836	−1.84272
$964$	7.70358	0.248115
$965$	3.10124	0.0998324
$966$	−10.1951	−0.328020
$967$	−21.5407	−0.692702	−0.346351	−	0.938105i	$-0.612579\pi$
$967$	−21.5407	−0.692702	−0.346351	+	0.938105i	$0.612579\pi$
$968$	3.05939	0.0983325
$969$	−2.22969	−0.0716280
$970$	4.98202	0.159963
$971$	4.80185	0.154099	0.0770494	−	0.997027i	$-0.475450\pi$
$971$	4.80185	0.154099	0.0770494	+	0.997027i	$0.475450\pi$
$972$	5.41805	0.173784
$973$	15.4039	0.493825
$974$	30.3646	0.972944
$975$	−0.258315	−0.00827272
$976$	14.4517	0.462586
$977$	2.71341	0.0868098	0.0434049	−	0.999058i	$-0.486179\pi$
$977$	2.71341	0.0868098	0.0434049	+	0.999058i	$0.486179\pi$
$978$	5.35777	0.171323
$979$	0.942817	0.0301326
$980$	0.393789	0.0125791
$981$	2.98218	0.0952138
$982$	29.5200	0.942021
$983$	34.5733	1.10272	0.551359	−	0.834268i	$-0.314110\pi$
$983$	34.5733	1.10272	0.551359	+	0.834268i	$0.314110\pi$
$984$	−5.64649	−0.180003
$985$	4.19576	0.133688
$986$	62.9585	2.00501
$987$	5.21717	0.166064
$988$	0.277329	0.00882300
$989$	9.70666	0.308654
$990$	−3.43840	−0.109280
$991$	49.4481	1.57077	0.785386	−	0.619007i	$-0.212465\pi$
$991$	49.4481	1.57077	0.785386	+	0.619007i	$0.212465\pi$
$992$	−9.46655	−0.300563
$993$	−0.771204	−0.0244734
$994$	−34.5019	−1.09433
$995$	13.7506	0.435923
$996$	−0.194026	−0.00614794
$997$	−46.5683	−1.47483	−0.737417	−	0.675438i	$-0.763955\pi$
$997$	−46.5683	−1.47483	−0.737417	+	0.675438i	$0.763955\pi$
$998$	−14.7826	−0.467937
$999$	−13.2114	−0.417991

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.c.1.7	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.c.1.7	✓	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)

\(f(q)\)	\(=\)	\(q-1.22952 q^{2} -0.451066 q^{3} -0.488282 q^{4} +1.00000 q^{5} +0.554594 q^{6} -2.48868 q^{7} +3.05939 q^{8} -2.79654 q^{9} +O(q^{10})\)	\(q-1.22952 q^{2} -0.451066 q^{3} -0.488282 q^{4} +1.00000 q^{5} +0.554594 q^{6} -2.48868 q^{7} +3.05939 q^{8} -2.79654 q^{9} -1.22952 q^{10} -1.00000 q^{11} +0.220247 q^{12} +0.572678 q^{13} +3.05988 q^{14} -0.451066 q^{15} -2.78502 q^{16} -4.98415 q^{17} +3.43840 q^{18} -0.991776 q^{19} -0.488282 q^{20} +1.12256 q^{21} +1.22952 q^{22} +7.38662 q^{23} -1.37999 q^{24} +1.00000 q^{25} -0.704119 q^{26} +2.61462 q^{27} +1.21518 q^{28} +10.2737 q^{29} +0.554594 q^{30} +3.51322 q^{31} -2.69455 q^{32} +0.451066 q^{33} +6.12811 q^{34} -2.48868 q^{35} +1.36550 q^{36} -5.05291 q^{37} +1.21941 q^{38} -0.258315 q^{39} +3.05939 q^{40} +4.09170 q^{41} -1.38021 q^{42} +1.31409 q^{43} +0.488282 q^{44} -2.79654 q^{45} -9.08199 q^{46} +4.64758 q^{47} +1.25623 q^{48} -0.806480 q^{49} -1.22952 q^{50} +2.24818 q^{51} -0.279628 q^{52} +6.59701 q^{53} -3.21473 q^{54} -1.00000 q^{55} -7.61384 q^{56} +0.447356 q^{57} -12.6317 q^{58} -11.0492 q^{59} +0.220247 q^{60} -5.18907 q^{61} -4.31958 q^{62} +6.95969 q^{63} +8.88304 q^{64} +0.572678 q^{65} -0.554594 q^{66} -4.75667 q^{67} +2.43367 q^{68} -3.33185 q^{69} +3.05988 q^{70} -11.2756 q^{71} -8.55571 q^{72} +1.00000 q^{73} +6.21265 q^{74} -0.451066 q^{75} +0.484266 q^{76} +2.48868 q^{77} +0.317604 q^{78} +6.70273 q^{79} -2.78502 q^{80} +7.21025 q^{81} -5.03082 q^{82} -0.880945 q^{83} -0.548124 q^{84} -4.98415 q^{85} -1.61570 q^{86} -4.63412 q^{87} -3.05939 q^{88} -0.942817 q^{89} +3.43840 q^{90} -1.42521 q^{91} -3.60675 q^{92} -1.58469 q^{93} -5.71429 q^{94} -0.991776 q^{95} +1.21542 q^{96} -4.05200 q^{97} +0.991583 q^{98} +2.79654 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) 23 * q - 3 * q^2 - 5 * q^3 + 15 * q^4 + 23 * q^5 - 5 * q^6 - 6 * q^8 + 8 * q^9	\( 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 23 q^{11} - 18 q^{12} - 5 q^{13} - 17 q^{14} - 5 q^{15} - q^{16} - 36 q^{17} + q^{18} + 6 q^{19} + 15 q^{20} - 18 q^{21} + 3 q^{22} - 14 q^{23} - 7 q^{24} + 23 q^{25} - 21 q^{26} - 29 q^{27} + 28 q^{28} - 36 q^{29} - 5 q^{30} - 16 q^{31} - 5 q^{32} + 5 q^{33} - 28 q^{34} - 14 q^{36} - 24 q^{37} + q^{38} - 10 q^{39} - 6 q^{40} - 36 q^{41} - 5 q^{42} + 17 q^{43} - 15 q^{44} + 8 q^{45} - 25 q^{46} - 21 q^{47} - 17 q^{48} - 27 q^{49} - 3 q^{50} + 19 q^{51} - 21 q^{52} - 28 q^{53} - 15 q^{54} - 23 q^{55} - 46 q^{56} - 23 q^{57} - 16 q^{58} - 61 q^{59} - 18 q^{60} - 17 q^{61} - 22 q^{62} - 9 q^{63} - 18 q^{64} - 5 q^{65} + 5 q^{66} + 2 q^{67} - 39 q^{68} - 36 q^{69} - 17 q^{70} - 50 q^{71} + 15 q^{72} + 23 q^{73} + 17 q^{74} - 5 q^{75} - 21 q^{76} + 49 q^{78} - 18 q^{79} - q^{80} - 57 q^{81} + 14 q^{82} - 20 q^{83} - 38 q^{84} - 36 q^{85} - 45 q^{86} + 37 q^{87} + 6 q^{88} - 93 q^{89} + q^{90} - 42 q^{91} - 39 q^{92} - 18 q^{93} - 6 q^{94} + 6 q^{95} - 9 q^{96} - 31 q^{97} - 31 q^{98} - 8 q^{99}+O(q^{100}) \) 23 * q - 3 * q^2 - 5 * q^3 + 15 * q^4 + 23 * q^5 - 5 * q^6 - 6 * q^8 + 8 * q^9 - 3 * q^10 - 23 * q^11 - 18 * q^12 - 5 * q^13 - 17 * q^14 - 5 * q^15 - q^16 - 36 * q^17 + q^18 + 6 * q^19 + 15 * q^20 - 18 * q^21 + 3 * q^22 - 14 * q^23 - 7 * q^24 + 23 * q^25 - 21 * q^26 - 29 * q^27 + 28 * q^28 - 36 * q^29 - 5 * q^30 - 16 * q^31 - 5 * q^32 + 5 * q^33 - 28 * q^34 - 14 * q^36 - 24 * q^37 + q^38 - 10 * q^39 - 6 * q^40 - 36 * q^41 - 5 * q^42 + 17 * q^43 - 15 * q^44 + 8 * q^45 - 25 * q^46 - 21 * q^47 - 17 * q^48 - 27 * q^49 - 3 * q^50 + 19 * q^51 - 21 * q^52 - 28 * q^53 - 15 * q^54 - 23 * q^55 - 46 * q^56 - 23 * q^57 - 16 * q^58 - 61 * q^59 - 18 * q^60 - 17 * q^61 - 22 * q^62 - 9 * q^63 - 18 * q^64 - 5 * q^65 + 5 * q^66 + 2 * q^67 - 39 * q^68 - 36 * q^69 - 17 * q^70 - 50 * q^71 + 15 * q^72 + 23 * q^73 + 17 * q^74 - 5 * q^75 - 21 * q^76 + 49 * q^78 - 18 * q^79 - q^80 - 57 * q^81 + 14 * q^82 - 20 * q^83 - 38 * q^84 - 36 * q^85 - 45 * q^86 + 37 * q^87 + 6 * q^88 - 93 * q^89 + q^90 - 42 * q^91 - 39 * q^92 - 18 * q^93 - 6 * q^94 + 6 * q^95 - 9 * q^96 - 31 * q^97 - 31 * q^98 - 8 * q^99

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