LMFDB - Embedded newform 4015.2.a.i.1.12

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:	$0$
Dimension:	$38$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-1.49358 q^{2} +2.85677 q^{3} +0.230796 q^{4} +1.00000 q^{5} -4.26684 q^{6} +0.0973246 q^{7} +2.64246 q^{8} +5.16116 q^{9} +O(q^{10})$	\(q-1.49358 q^{2} +2.85677 q^{3} +0.230796 q^{4} +1.00000 q^{5} -4.26684 q^{6} +0.0973246 q^{7} +2.64246 q^{8} +5.16116 q^{9} -1.49358 q^{10} -1.00000 q^{11} +0.659332 q^{12} +7.01514 q^{13} -0.145363 q^{14} +2.85677 q^{15} -4.40832 q^{16} -1.04896 q^{17} -7.70864 q^{18} +2.98522 q^{19} +0.230796 q^{20} +0.278034 q^{21} +1.49358 q^{22} +0.0638906 q^{23} +7.54890 q^{24} +1.00000 q^{25} -10.4777 q^{26} +6.17396 q^{27} +0.0224621 q^{28} +3.54097 q^{29} -4.26684 q^{30} -0.932705 q^{31} +1.29929 q^{32} -2.85677 q^{33} +1.56671 q^{34} +0.0973246 q^{35} +1.19117 q^{36} +2.39409 q^{37} -4.45867 q^{38} +20.0407 q^{39} +2.64246 q^{40} -1.13570 q^{41} -0.415268 q^{42} -7.24581 q^{43} -0.230796 q^{44} +5.16116 q^{45} -0.0954261 q^{46} +10.8077 q^{47} -12.5936 q^{48} -6.99053 q^{49} -1.49358 q^{50} -2.99664 q^{51} +1.61906 q^{52} +12.0739 q^{53} -9.22133 q^{54} -1.00000 q^{55} +0.257176 q^{56} +8.52809 q^{57} -5.28874 q^{58} +2.44994 q^{59} +0.659332 q^{60} -9.47724 q^{61} +1.39307 q^{62} +0.502308 q^{63} +6.87604 q^{64} +7.01514 q^{65} +4.26684 q^{66} +1.84455 q^{67} -0.242095 q^{68} +0.182521 q^{69} -0.145363 q^{70} -1.56711 q^{71} +13.6381 q^{72} -1.00000 q^{73} -3.57578 q^{74} +2.85677 q^{75} +0.688975 q^{76} -0.0973246 q^{77} -29.9324 q^{78} +5.60956 q^{79} -4.40832 q^{80} +2.15412 q^{81} +1.69627 q^{82} -7.12354 q^{83} +0.0641692 q^{84} -1.04896 q^{85} +10.8222 q^{86} +10.1158 q^{87} -2.64246 q^{88} -17.4486 q^{89} -7.70864 q^{90} +0.682746 q^{91} +0.0147457 q^{92} -2.66453 q^{93} -16.1422 q^{94} +2.98522 q^{95} +3.71179 q^{96} +0.429873 q^{97} +10.4409 q^{98} -5.16116 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9}+O(q^{10}) $ 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9} + 4 q^{10} - 38 q^{11} + 12 q^{12} - q^{13} + 23 q^{14} + 5 q^{15} + 74 q^{16} + 26 q^{17} + 16 q^{18} - 10 q^{19} + 50 q^{20} + 21 q^{21} - 4 q^{22} + 10 q^{23} + 41 q^{24} + 38 q^{25} + 25 q^{26} + 5 q^{27} + 2 q^{28} + 28 q^{29} + 11 q^{30} + 24 q^{31} + 39 q^{32} - 5 q^{33} + 38 q^{34} + 111 q^{36} + 12 q^{37} + 19 q^{38} - 18 q^{39} + 15 q^{40} + 62 q^{41} - 17 q^{42} - 32 q^{43} - 50 q^{44} + 63 q^{45} - 9 q^{46} + 31 q^{47} + 53 q^{48} + 88 q^{49} + 4 q^{50} - 3 q^{51} - 21 q^{52} + 30 q^{53} + 49 q^{54} - 38 q^{55} + 32 q^{56} + 49 q^{57} + 12 q^{58} + 31 q^{59} + 12 q^{60} + 25 q^{61} + 12 q^{62} + 15 q^{63} + 137 q^{64} - q^{65} - 11 q^{66} + 20 q^{67} + 75 q^{68} + 92 q^{69} + 23 q^{70} + 32 q^{71} + 6 q^{72} - 38 q^{73} + 55 q^{74} + 5 q^{75} - 57 q^{76} - 17 q^{78} - 2 q^{79} + 74 q^{80} + 118 q^{81} + 14 q^{82} + 4 q^{83} + 22 q^{84} + 26 q^{85} + 5 q^{86} + 24 q^{87} - 15 q^{88} + 143 q^{89} + 16 q^{90} + 66 q^{91} + 29 q^{92} - 8 q^{93} - 7 q^{94} - 10 q^{95} + 59 q^{96} + 41 q^{97} - 10 q^{98} - 63 q^{99}+O(q^{100}) \) 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9 + 4 * q^10 - 38 * q^11 + 12 * q^12 - q^13 + 23 * q^14 + 5 * q^15 + 74 * q^16 + 26 * q^17 + 16 * q^18 - 10 * q^19 + 50 * q^20 + 21 * q^21 - 4 * q^22 + 10 * q^23 + 41 * q^24 + 38 * q^25 + 25 * q^26 + 5 * q^27 + 2 * q^28 + 28 * q^29 + 11 * q^30 + 24 * q^31 + 39 * q^32 - 5 * q^33 + 38 * q^34 + 111 * q^36 + 12 * q^37 + 19 * q^38 - 18 * q^39 + 15 * q^40 + 62 * q^41 - 17 * q^42 - 32 * q^43 - 50 * q^44 + 63 * q^45 - 9 * q^46 + 31 * q^47 + 53 * q^48 + 88 * q^49 + 4 * q^50 - 3 * q^51 - 21 * q^52 + 30 * q^53 + 49 * q^54 - 38 * q^55 + 32 * q^56 + 49 * q^57 + 12 * q^58 + 31 * q^59 + 12 * q^60 + 25 * q^61 + 12 * q^62 + 15 * q^63 + 137 * q^64 - q^65 - 11 * q^66 + 20 * q^67 + 75 * q^68 + 92 * q^69 + 23 * q^70 + 32 * q^71 + 6 * q^72 - 38 * q^73 + 55 * q^74 + 5 * q^75 - 57 * q^76 - 17 * q^78 - 2 * q^79 + 74 * q^80 + 118 * q^81 + 14 * q^82 + 4 * q^83 + 22 * q^84 + 26 * q^85 + 5 * q^86 + 24 * q^87 - 15 * q^88 + 143 * q^89 + 16 * q^90 + 66 * q^91 + 29 * q^92 - 8 * q^93 - 7 * q^94 - 10 * q^95 + 59 * q^96 + 41 * q^97 - 10 * q^98 - 63 * 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 $

$2$	−1.49358	−1.05612	−0.528062	−	0.849206i	$-0.677081\pi$
$2$	−1.49358	−1.05612	−0.528062	+	0.849206i	$0.677081\pi$
$3$	2.85677	1.64936	0.824680	−	0.565600i	$-0.191355\pi$
$3$	2.85677	1.64936	0.824680	+	0.565600i	$0.191355\pi$
$4$	0.230796	0.115398
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−4.26684	−1.74193
$7$	0.0973246	0.0367852	0.0183926	−	0.999831i	$-0.494145\pi$
$7$	0.0973246	0.0367852	0.0183926	+	0.999831i	$0.494145\pi$
$8$	2.64246	0.934250
$9$	5.16116	1.72039
$10$	−1.49358	−0.472313
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.659332	0.190333
$13$	7.01514	1.94565	0.972825	−	0.231542i	$-0.0743772\pi$
$13$	7.01514	1.94565	0.972825	+	0.231542i	$0.0743772\pi$
$14$	−0.145363	−0.0388498
$15$	2.85677	0.737616
$16$	−4.40832	−1.10208
$17$	−1.04896	−0.254410	−0.127205	−	0.991876i	$-0.540601\pi$
$17$	−1.04896	−0.254410	−0.127205	+	0.991876i	$0.540601\pi$
$18$	−7.70864	−1.81694
$19$	2.98522	0.684856	0.342428	−	0.939544i	$-0.388751\pi$
$19$	2.98522	0.684856	0.342428	+	0.939544i	$0.388751\pi$
$20$	0.230796	0.0516075
$21$	0.278034	0.0606721
$22$	1.49358	0.318433
$23$	0.0638906	0.0133221	0.00666106	−	0.999978i	$-0.497880\pi$
$23$	0.0638906	0.0133221	0.00666106	+	0.999978i	$0.497880\pi$
$24$	7.54890	1.54091
$25$	1.00000	0.200000
$26$	−10.4777	−2.05485
$27$	6.17396	1.18818
$28$	0.0224621	0.00424494
$29$	3.54097	0.657541	0.328771	−	0.944410i	$-0.393366\pi$
$29$	3.54097	0.657541	0.328771	+	0.944410i	$0.393366\pi$
$30$	−4.26684	−0.779014
$31$	−0.932705	−0.167519	−0.0837594	−	0.996486i	$-0.526693\pi$
$31$	−0.932705	−0.167519	−0.0837594	+	0.996486i	$0.526693\pi$
$32$	1.29929	0.229685
$33$	−2.85677	−0.497301
$34$	1.56671	0.268688
$35$	0.0973246	0.0164509
$36$	1.19117	0.198529
$37$	2.39409	0.393586	0.196793	−	0.980445i	$-0.436947\pi$
$37$	2.39409	0.393586	0.196793	+	0.980445i	$0.436947\pi$
$38$	−4.45867	−0.723292
$39$	20.0407	3.20908
$40$	2.64246	0.417809
$41$	−1.13570	−0.177367	−0.0886836	−	0.996060i	$-0.528266\pi$
$41$	−1.13570	−0.177367	−0.0886836	+	0.996060i	$0.528266\pi$
$42$	−0.415268	−0.0640772
$43$	−7.24581	−1.10498	−0.552488	−	0.833521i	$-0.686321\pi$
$43$	−7.24581	−1.10498	−0.552488	+	0.833521i	$0.686321\pi$
$44$	−0.230796	−0.0347938
$45$	5.16116	0.769381
$46$	−0.0954261	−0.0140698
$47$	10.8077	1.57646	0.788232	−	0.615379i	$-0.210997\pi$
$47$	10.8077	1.57646	0.788232	+	0.615379i	$0.210997\pi$
$48$	−12.5936	−1.81773
$49$	−6.99053	−0.998647
$50$	−1.49358	−0.211225
$51$	−2.99664	−0.419613
$52$	1.61906	0.224524
$53$	12.0739	1.65848	0.829238	−	0.558896i	$-0.188775\pi$
$53$	12.0739	1.65848	0.829238	+	0.558896i	$0.188775\pi$
$54$	−9.22133	−1.25486
$55$	−1.00000	−0.134840
$56$	0.257176	0.0343666
$57$	8.52809	1.12957
$58$	−5.28874	−0.694445
$59$	2.44994	0.318955	0.159477	−	0.987202i	$-0.449019\pi$
$59$	2.44994	0.318955	0.159477	+	0.987202i	$0.449019\pi$
$60$	0.659332	0.0851194
$61$	−9.47724	−1.21344	−0.606718	−	0.794917i	$-0.707514\pi$
$61$	−9.47724	−1.21344	−0.606718	+	0.794917i	$0.707514\pi$
$62$	1.39307	0.176921
$63$	0.502308	0.0632849
$64$	6.87604	0.859505
$65$	7.01514	0.870121
$66$	4.26684	0.525211
$67$	1.84455	0.225347	0.112674	−	0.993632i	$-0.464059\pi$
$67$	1.84455	0.225347	0.112674	+	0.993632i	$0.464059\pi$
$68$	−0.242095	−0.0293583
$69$	0.182521	0.0219730
$70$	−0.145363	−0.0173741
$71$	−1.56711	−0.185981	−0.0929907	−	0.995667i	$-0.529643\pi$
$71$	−1.56711	−0.185981	−0.0929907	+	0.995667i	$0.529643\pi$
$72$	13.6381	1.60727
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	−3.57578	−0.415676
$75$	2.85677	0.329872
$76$	0.688975	0.0790309
$77$	−0.0973246	−0.0110912
$78$	−29.9324	−3.38918
$79$	5.60956	0.631125	0.315562	−	0.948905i	$-0.397807\pi$
$79$	5.60956	0.631125	0.315562	+	0.948905i	$0.397807\pi$
$80$	−4.40832	−0.492866
$81$	2.15412	0.239346
$82$	1.69627	0.187322
$83$	−7.12354	−0.781910	−0.390955	−	0.920410i	$-0.627855\pi$
$83$	−7.12354	−0.781910	−0.390955	+	0.920410i	$0.627855\pi$
$84$	0.0641692	0.00700143
$85$	−1.04896	−0.113775
$86$	10.8222	1.16699
$87$	10.1158	1.08452
$88$	−2.64246	−0.281687
$89$	−17.4486	−1.84954	−0.924771	−	0.380523i	$-0.875744\pi$
$89$	−17.4486	−1.84954	−0.924771	+	0.380523i	$0.875744\pi$
$90$	−7.70864	−0.812562
$91$	0.682746	0.0715712
$92$	0.0147457	0.00153734
$93$	−2.66453	−0.276299
$94$	−16.1422	−1.66494
$95$	2.98522	0.306277
$96$	3.71179	0.378833
$97$	0.429873	0.0436470	0.0218235	−	0.999762i	$-0.493053\pi$
$97$	0.429873	0.0436470	0.0218235	+	0.999762i	$0.493053\pi$
$98$	10.4409	1.05469
$99$	−5.16116	−0.518716
$100$	0.230796	0.0230796
$101$	−3.18660	−0.317079	−0.158539	−	0.987353i	$-0.550678\pi$
$101$	−3.18660	−0.317079	−0.158539	+	0.987353i	$0.550678\pi$
$102$	4.47573	0.443163
$103$	13.6864	1.34856	0.674281	−	0.738475i	$-0.264454\pi$
$103$	13.6864	1.34856	0.674281	+	0.738475i	$0.264454\pi$
$104$	18.5372	1.81772
$105$	0.278034	0.0271334
$106$	−18.0334	−1.75156
$107$	1.83708	0.177597	0.0887987	−	0.996050i	$-0.471697\pi$
$107$	1.83708	0.177597	0.0887987	+	0.996050i	$0.471697\pi$
$108$	1.42492	0.137113
$109$	−12.2248	−1.17092	−0.585462	−	0.810700i	$-0.699087\pi$
$109$	−12.2248	−1.17092	−0.585462	+	0.810700i	$0.699087\pi$
$110$	1.49358	0.142408
$111$	6.83938	0.649166
$112$	−0.429038	−0.0405403
$113$	16.2159	1.52546	0.762732	−	0.646714i	$-0.223857\pi$
$113$	16.2159	1.52546	0.762732	+	0.646714i	$0.223857\pi$
$114$	−12.7374	−1.19297
$115$	0.0638906	0.00595783
$116$	0.817241	0.0758789
$117$	36.2063	3.34727
$118$	−3.65919	−0.336856
$119$	−0.102089	−0.00935852
$120$	7.54890	0.689118
$121$	1.00000	0.0909091
$122$	14.1551	1.28154
$123$	−3.24445	−0.292542
$124$	−0.215264	−0.0193313
$125$	1.00000	0.0894427
$126$	−0.750240	−0.0668367
$127$	−9.99595	−0.886997	−0.443499	−	0.896275i	$-0.646263\pi$
$127$	−9.99595	−0.886997	−0.443499	+	0.896275i	$0.646263\pi$
$128$	−12.8685	−1.13743
$129$	−20.6997	−1.82250
$130$	−10.4777	−0.918956
$131$	−10.8942	−0.951828	−0.475914	−	0.879492i	$-0.657883\pi$
$131$	−10.8942	−0.951828	−0.475914	+	0.879492i	$0.657883\pi$
$132$	−0.659332	−0.0573875
$133$	0.290535	0.0251926
$134$	−2.75498	−0.237994
$135$	6.17396	0.531370
$136$	−2.77183	−0.237682
$137$	−0.997936	−0.0852595	−0.0426297	−	0.999091i	$-0.513574\pi$
$137$	−0.997936	−0.0852595	−0.0426297	+	0.999091i	$0.513574\pi$
$138$	−0.272611	−0.0232062
$139$	−10.5479	−0.894659	−0.447330	−	0.894369i	$-0.647625\pi$
$139$	−10.5479	−0.894659	−0.447330	+	0.894369i	$0.647625\pi$
$140$	0.0224621	0.00189839
$141$	30.8751	2.60016
$142$	2.34061	0.196419
$143$	−7.01514	−0.586635
$144$	−22.7521	−1.89601
$145$	3.54097	0.294061
$146$	1.49358	0.123610
$147$	−19.9704	−1.64713
$148$	0.552546	0.0454190
$149$	11.8051	0.967108	0.483554	−	0.875315i	$-0.339346\pi$
$149$	11.8051	0.967108	0.483554	+	0.875315i	$0.339346\pi$
$150$	−4.26684	−0.348386
$151$	−6.18927	−0.503676	−0.251838	−	0.967769i	$-0.581035\pi$
$151$	−6.18927	−0.503676	−0.251838	+	0.967769i	$0.581035\pi$
$152$	7.88830	0.639826
$153$	−5.41384	−0.437683
$154$	0.145363	0.0117136
$155$	−0.932705	−0.0749167
$156$	4.62530	0.370321
$157$	3.78852	0.302356	0.151178	−	0.988507i	$-0.451693\pi$
$157$	3.78852	0.302356	0.151178	+	0.988507i	$0.451693\pi$
$158$	−8.37835	−0.666546
$159$	34.4924	2.73542
$160$	1.29929	0.102718
$161$	0.00621813	0.000490057 0
$162$	−3.21735	−0.252779
$163$	9.92998	0.777776	0.388888	−	0.921285i	$-0.372859\pi$
$163$	9.92998	0.777776	0.388888	+	0.921285i	$0.372859\pi$
$164$	−0.262116	−0.0204678
$165$	−2.85677	−0.222400
$166$	10.6396	0.825794
$167$	13.1170	1.01502	0.507512	−	0.861645i	$-0.330565\pi$
$167$	13.1170	1.01502	0.507512	+	0.861645i	$0.330565\pi$
$168$	0.734694	0.0566829
$169$	36.2122	2.78555
$170$	1.56671	0.120161
$171$	15.4072	1.17822
$172$	−1.67230	−0.127512
$173$	1.25760	0.0956132	0.0478066	−	0.998857i	$-0.484777\pi$
$173$	1.25760	0.0956132	0.0478066	+	0.998857i	$0.484777\pi$
$174$	−15.1087	−1.14539
$175$	0.0973246	0.00735705
$176$	4.40832	0.332290
$177$	6.99892	0.526071
$178$	26.0609	1.95335
$179$	−4.51643	−0.337574	−0.168787	−	0.985653i	$-0.553985\pi$
$179$	−4.51643	−0.337574	−0.168787	+	0.985653i	$0.553985\pi$
$180$	1.19117	0.0887849
$181$	8.08910	0.601258	0.300629	−	0.953741i	$-0.402803\pi$
$181$	8.08910	0.601258	0.300629	+	0.953741i	$0.402803\pi$
$182$	−1.01974	−0.0755880
$183$	−27.0743	−2.00139
$184$	0.168828	0.0124462
$185$	2.39409	0.176017
$186$	3.97970	0.291806
$187$	1.04896	0.0767074
$188$	2.49437	0.181921
$189$	0.600878	0.0437074
$190$	−4.45867	−0.323466
$191$	7.29503	0.527850	0.263925	−	0.964543i	$-0.414983\pi$
$191$	7.29503	0.527850	0.263925	+	0.964543i	$0.414983\pi$
$192$	19.6433	1.41763
$193$	−13.6081	−0.979532	−0.489766	−	0.871854i	$-0.662918\pi$
$193$	−13.6081	−0.979532	−0.489766	+	0.871854i	$0.662918\pi$
$194$	−0.642052	−0.0460967
$195$	20.0407	1.43514
$196$	−1.61338	−0.115242
$197$	−13.2183	−0.941762	−0.470881	−	0.882197i	$-0.656064\pi$
$197$	−13.2183	−0.941762	−0.470881	+	0.882197i	$0.656064\pi$
$198$	7.70864	0.547829
$199$	13.4003	0.949922	0.474961	−	0.880007i	$-0.342462\pi$
$199$	13.4003	0.949922	0.474961	+	0.880007i	$0.342462\pi$
$200$	2.64246	0.186850
$201$	5.26945	0.371678
$202$	4.75946	0.334874
$203$	0.344623	0.0241878
$204$	−0.691611	−0.0484225
$205$	−1.13570	−0.0793210
$206$	−20.4418	−1.42425
$207$	0.329750	0.0229192
$208$	−30.9250	−2.14426
$209$	−2.98522	−0.206492
$210$	−0.415268	−0.0286562
$211$	5.75913	0.396475	0.198237	−	0.980154i	$-0.436478\pi$
$211$	5.75913	0.396475	0.198237	+	0.980154i	$0.436478\pi$
$212$	2.78660	0.191385
$213$	−4.47687	−0.306750
$214$	−2.74384	−0.187565
$215$	−7.24581	−0.494160
$216$	16.3144	1.11006
$217$	−0.0907751	−0.00616222
$218$	18.2588	1.23664
$219$	−2.85677	−0.193043
$220$	−0.230796	−0.0155602
$221$	−7.35859	−0.494992
$222$	−10.2152	−0.685599
$223$	−0.621870	−0.0416435	−0.0208217	−	0.999783i	$-0.506628\pi$
$223$	−0.621870	−0.0416435	−0.0208217	+	0.999783i	$0.506628\pi$
$224$	0.126453	0.00844901
$225$	5.16116	0.344078
$226$	−24.2198	−1.61108
$227$	6.96693	0.462411	0.231206	−	0.972905i	$-0.425733\pi$
$227$	6.96693	0.462411	0.231206	+	0.972905i	$0.425733\pi$
$228$	1.96825	0.130350
$229$	−9.75008	−0.644304	−0.322152	−	0.946688i	$-0.604406\pi$
$229$	−9.75008	−0.644304	−0.322152	+	0.946688i	$0.604406\pi$
$230$	−0.0954261	−0.00629221
$231$	−0.278034	−0.0182933
$232$	9.35686	0.614308
$233$	−9.01389	−0.590519	−0.295260	−	0.955417i	$-0.595406\pi$
$233$	−9.01389	−0.590519	−0.295260	+	0.955417i	$0.595406\pi$
$234$	−54.0772	−3.53513
$235$	10.8077	0.705016
$236$	0.565435	0.0368067
$237$	16.0252	1.04095
$238$	0.152479	0.00988376
$239$	−5.75438	−0.372220	−0.186110	−	0.982529i	$-0.559588\pi$
$239$	−5.75438	−0.372220	−0.186110	+	0.982529i	$0.559588\pi$
$240$	−12.5936	−0.812913
$241$	15.8043	1.01804	0.509022	−	0.860754i	$-0.330007\pi$
$241$	15.8043	1.01804	0.509022	+	0.860754i	$0.330007\pi$
$242$	−1.49358	−0.0960113
$243$	−12.3680	−0.793411
$244$	−2.18731	−0.140028
$245$	−6.99053	−0.446608
$246$	4.84586	0.308961
$247$	20.9417	1.33249
$248$	−2.46463	−0.156504
$249$	−20.3504	−1.28965
$250$	−1.49358	−0.0944626
$251$	3.75305	0.236891	0.118445	−	0.992961i	$-0.462209\pi$
$251$	3.75305	0.236891	0.118445	+	0.992961i	$0.462209\pi$
$252$	0.115931	0.00730294
$253$	−0.0638906	−0.00401677
$254$	14.9298	0.936779
$255$	−2.99664	−0.187657
$256$	5.46817	0.341761
$257$	24.8065	1.54739	0.773695	−	0.633558i	$-0.218406\pi$
$257$	24.8065	1.54739	0.773695	+	0.633558i	$0.218406\pi$
$258$	30.9167	1.92479
$259$	0.233004	0.0144782
$260$	1.61906	0.100410
$261$	18.2755	1.13123
$262$	16.2714	1.00525
$263$	20.6659	1.27432	0.637158	−	0.770733i	$-0.280110\pi$
$263$	20.6659	1.27432	0.637158	+	0.770733i	$0.280110\pi$
$264$	−7.54890	−0.464603
$265$	12.0739	0.741693
$266$	−0.433939	−0.0266065
$267$	−49.8466	−3.05056
$268$	0.425713	0.0260046
$269$	11.2183	0.683993	0.341997	−	0.939701i	$-0.388897\pi$
$269$	11.2183	0.683993	0.341997	+	0.939701i	$0.388897\pi$
$270$	−9.22133	−0.561192
$271$	−30.4904	−1.85216	−0.926080	−	0.377327i	$-0.876843\pi$
$271$	−30.4904	−1.85216	−0.926080	+	0.377327i	$0.876843\pi$
$272$	4.62415	0.280380
$273$	1.95045	0.118047
$274$	1.49050	0.0900446
$275$	−1.00000	−0.0603023
$276$	0.0421251	0.00253563
$277$	14.1622	0.850922	0.425461	−	0.904977i	$-0.360112\pi$
$277$	14.1622	0.850922	0.425461	+	0.904977i	$0.360112\pi$
$278$	15.7542	0.944871
$279$	−4.81384	−0.288197
$280$	0.257176	0.0153692
$281$	16.8770	1.00680	0.503400	−	0.864053i	$-0.332082\pi$
$281$	16.8770	1.00680	0.503400	+	0.864053i	$0.332082\pi$
$282$	−46.1146	−2.74609
$283$	−26.3387	−1.56567	−0.782836	−	0.622228i	$-0.786228\pi$
$283$	−26.3387	−1.56567	−0.782836	+	0.622228i	$0.786228\pi$
$284$	−0.361682	−0.0214619
$285$	8.52809	0.505161
$286$	10.4777	0.619560
$287$	−0.110532	−0.00652449
$288$	6.70587	0.395147
$289$	−15.8997	−0.935276
$290$	−5.28874	−0.310565
$291$	1.22805	0.0719896
$292$	−0.230796	−0.0135063
$293$	7.17074	0.418919	0.209460	−	0.977817i	$-0.432830\pi$
$293$	7.17074	0.418919	0.209460	+	0.977817i	$0.432830\pi$
$294$	29.8274	1.73957
$295$	2.44994	0.142641
$296$	6.32629	0.367708
$297$	−6.17396	−0.358249
$298$	−17.6319	−1.02139
$299$	0.448202	0.0259202
$300$	0.659332	0.0380665
$301$	−0.705196	−0.0406468
$302$	9.24421	0.531944
$303$	−9.10340	−0.522977
$304$	−13.1598	−0.754766
$305$	−9.47724	−0.542665
$306$	8.08603	0.462248
$307$	10.6487	0.607752	0.303876	−	0.952712i	$-0.401719\pi$
$307$	10.6487	0.607752	0.303876	+	0.952712i	$0.401719\pi$
$308$	−0.0224621	−0.00127990
$309$	39.0990	2.22426
$310$	1.39307	0.0791213
$311$	−26.4728	−1.50114	−0.750568	−	0.660793i	$-0.770220\pi$
$311$	−26.4728	−1.50114	−0.750568	+	0.660793i	$0.770220\pi$
$312$	52.9566	2.99808
$313$	7.01186	0.396334	0.198167	−	0.980168i	$-0.436501\pi$
$313$	7.01186	0.396334	0.198167	+	0.980168i	$0.436501\pi$
$314$	−5.65847	−0.319326
$315$	0.502308	0.0283019
$316$	1.29466	0.0728305
$317$	−33.8345	−1.90034	−0.950168	−	0.311739i	$-0.899088\pi$
$317$	−33.8345	−1.90034	−0.950168	+	0.311739i	$0.899088\pi$
$318$	−51.5173	−2.88895
$319$	−3.54097	−0.198256
$320$	6.87604	0.384383
$321$	5.24813	0.292922
$322$	−0.00928730	−0.000517561 0
$323$	−3.13137	−0.174234
$324$	0.497161	0.0276201
$325$	7.01514	0.389130
$326$	−14.8313	−0.821428
$327$	−34.9235	−1.93127
$328$	−3.00105	−0.165705
$329$	1.05185	0.0579906
$330$	4.26684	0.234882
$331$	−18.3229	−1.00712	−0.503559	−	0.863961i	$-0.667976\pi$
$331$	−18.3229	−1.00712	−0.503559	+	0.863961i	$0.667976\pi$
$332$	−1.64408	−0.0902308
$333$	12.3563	0.677121
$334$	−19.5914	−1.07199
$335$	1.84455	0.100778
$336$	−1.22567	−0.0668656
$337$	−0.770207	−0.0419558	−0.0209779	−	0.999780i	$-0.506678\pi$
$337$	−0.770207	−0.0419558	−0.0209779	+	0.999780i	$0.506678\pi$
$338$	−54.0860	−2.94189
$339$	46.3252	2.51604
$340$	−0.242095	−0.0131294
$341$	0.932705	0.0505088
$342$	−23.0119	−1.24434
$343$	−1.36162	−0.0735207
$344$	−19.1467	−1.03232
$345$	0.182521	0.00982661
$346$	−1.87833	−0.100979
$347$	10.6136	0.569765	0.284883	−	0.958562i	$-0.408045\pi$
$347$	10.6136	0.569765	0.284883	+	0.958562i	$0.408045\pi$
$348$	2.33467	0.125152
$349$	23.3019	1.24732	0.623662	−	0.781694i	$-0.285644\pi$
$349$	23.3019	1.24732	0.623662	+	0.781694i	$0.285644\pi$
$350$	−0.145363	−0.00776995
$351$	43.3112	2.31178
$352$	−1.29929	−0.0692526
$353$	2.70025	0.143720	0.0718598	−	0.997415i	$-0.477107\pi$
$353$	2.70025	0.143720	0.0718598	+	0.997415i	$0.477107\pi$
$354$	−10.4535	−0.555596
$355$	−1.56711	−0.0831734
$356$	−4.02705	−0.213433
$357$	−0.291646	−0.0154356
$358$	6.74568	0.356520
$359$	3.31736	0.175083	0.0875417	−	0.996161i	$-0.472099\pi$
$359$	3.31736	0.175083	0.0875417	+	0.996161i	$0.472099\pi$
$360$	13.6381	0.718794
$361$	−10.0885	−0.530973
$362$	−12.0818	−0.635003
$363$	2.85677	0.149942
$364$	0.157575	0.00825916
$365$	−1.00000	−0.0523424
$366$	40.4378	2.11372
$367$	−27.3935	−1.42993	−0.714964	−	0.699162i	$-0.753557\pi$
$367$	−27.3935	−1.42993	−0.714964	+	0.699162i	$0.753557\pi$
$368$	−0.281651	−0.0146821
$369$	−5.86155	−0.305140
$370$	−3.57578	−0.185896
$371$	1.17509	0.0610074
$372$	−0.614962	−0.0318843
$373$	−30.2160	−1.56452	−0.782262	−	0.622950i	$-0.785934\pi$
$373$	−30.2160	−1.56452	−0.782262	+	0.622950i	$0.785934\pi$
$374$	−1.56671	−0.0810125
$375$	2.85677	0.147523
$376$	28.5588	1.47281
$377$	24.8404	1.27935
$378$	−0.897462	−0.0461605
$379$	−27.4537	−1.41020	−0.705102	−	0.709106i	$-0.749099\pi$
$379$	−27.4537	−1.41020	−0.705102	+	0.709106i	$0.749099\pi$
$380$	0.688975	0.0353437
$381$	−28.5562	−1.46298
$382$	−10.8958	−0.557475
$383$	−7.25387	−0.370655	−0.185328	−	0.982677i	$-0.559335\pi$
$383$	−7.25387	−0.370655	−0.185328	+	0.982677i	$0.559335\pi$
$384$	−36.7625	−1.87603
$385$	−0.0973246	−0.00496012
$386$	20.3248	1.03451
$387$	−37.3968	−1.90099
$388$	0.0992130	0.00503678
$389$	−21.0891	−1.06926	−0.534631	−	0.845086i	$-0.679549\pi$
$389$	−21.0891	−1.06926	−0.534631	+	0.845086i	$0.679549\pi$
$390$	−29.9324	−1.51569
$391$	−0.0670186	−0.00338927
$392$	−18.4722	−0.932985
$393$	−31.1222	−1.56991
$394$	19.7426	0.994617
$395$	5.60956	0.282248
$396$	−1.19117	−0.0598588
$397$	29.5409	1.48261	0.741307	−	0.671166i	$-0.234206\pi$
$397$	29.5409	1.48261	0.741307	+	0.671166i	$0.234206\pi$
$398$	−20.0145	−1.00324
$399$	0.829993	0.0415516
$400$	−4.40832	−0.220416
$401$	0.0117831	0.000588421 0	0.000294210	−	1.00000i	$-0.499906\pi$
$401$	0.0117831	0.000588421 0	0.000294210		1.00000i	$0.499906\pi$
$402$	−7.87037	−0.392538
$403$	−6.54306	−0.325933
$404$	−0.735454	−0.0365902
$405$	2.15412	0.107039
$406$	−0.514724	−0.0255453
$407$	−2.39409	−0.118671
$408$	−7.91848	−0.392023
$409$	5.31317	0.262719	0.131360	−	0.991335i	$-0.458066\pi$
$409$	5.31317	0.262719	0.131360	+	0.991335i	$0.458066\pi$
$410$	1.69627	0.0837728
$411$	−2.85088	−0.140624
$412$	3.15877	0.155621
$413$	0.238439	0.0117328
$414$	−0.492509	−0.0242055
$415$	−7.12354	−0.349681
$416$	9.11473	0.446886
$417$	−30.1329	−1.47562
$418$	4.45867	0.218081
$419$	14.2547	0.696385	0.348193	−	0.937423i	$-0.386795\pi$
$419$	14.2547	0.696385	0.348193	+	0.937423i	$0.386795\pi$
$420$	0.0641692	0.00313114
$421$	17.9432	0.874498	0.437249	−	0.899341i	$-0.355953\pi$
$421$	17.9432	0.874498	0.437249	+	0.899341i	$0.355953\pi$
$422$	−8.60175	−0.418727
$423$	55.7802	2.71213
$424$	31.9047	1.54943
$425$	−1.04896	−0.0508819
$426$	6.68659	0.323966
$427$	−0.922369	−0.0446365
$428$	0.423991	0.0204944
$429$	−20.0407	−0.967573
$430$	10.8222	0.521894
$431$	−3.04320	−0.146586	−0.0732929	−	0.997310i	$-0.523351\pi$
$431$	−3.04320	−0.146586	−0.0732929	+	0.997310i	$0.523351\pi$
$432$	−27.2168	−1.30947
$433$	38.9707	1.87281	0.936407	−	0.350915i	$-0.114129\pi$
$433$	38.9707	1.87281	0.936407	+	0.350915i	$0.114129\pi$
$434$	0.135580	0.00650807
$435$	10.1158	0.485013
$436$	−2.82143	−0.135122
$437$	0.190727	0.00912372
$438$	4.26684	0.203877
$439$	23.0163	1.09851	0.549254	−	0.835656i	$-0.314912\pi$
$439$	23.0163	1.09851	0.549254	+	0.835656i	$0.314912\pi$
$440$	−2.64246	−0.125974
$441$	−36.0793	−1.71806
$442$	10.9907	0.522773
$443$	39.5749	1.88026	0.940130	−	0.340817i	$-0.110704\pi$
$443$	39.5749	1.88026	0.940130	+	0.340817i	$0.110704\pi$
$444$	1.57850	0.0749123
$445$	−17.4486	−0.827141
$446$	0.928816	0.0439807
$447$	33.7244	1.59511
$448$	0.669208	0.0316171
$449$	7.84123	0.370051	0.185025	−	0.982734i	$-0.440763\pi$
$449$	7.84123	0.370051	0.185025	+	0.982734i	$0.440763\pi$
$450$	−7.70864	−0.363389
$451$	1.13570	0.0534782
$452$	3.74256	0.176035
$453$	−17.6814	−0.830743
$454$	−10.4057	−0.488364
$455$	0.682746	0.0320076
$456$	22.5351	1.05530
$457$	−3.76953	−0.176331	−0.0881656	−	0.996106i	$-0.528100\pi$
$457$	−3.76953	−0.176331	−0.0881656	+	0.996106i	$0.528100\pi$
$458$	14.5626	0.680464
$459$	−6.47622	−0.302284
$460$	0.0147457	0.000687521 0
$461$	9.62143	0.448115	0.224057	−	0.974576i	$-0.428070\pi$
$461$	9.62143	0.448115	0.224057	+	0.974576i	$0.428070\pi$
$462$	0.415268	0.0193200
$463$	−18.0799	−0.840245	−0.420123	−	0.907467i	$-0.638013\pi$
$463$	−18.0799	−0.840245	−0.420123	+	0.907467i	$0.638013\pi$
$464$	−15.6097	−0.724664
$465$	−2.66453	−0.123565
$466$	13.4630	0.623662
$467$	−26.0978	−1.20766	−0.603832	−	0.797112i	$-0.706360\pi$
$467$	−26.0978	−1.20766	−0.603832	+	0.797112i	$0.706360\pi$
$468$	8.35626	0.386268
$469$	0.179520	0.00828944
$470$	−16.1422	−0.744584
$471$	10.8229	0.498695
$472$	6.47386	0.297983
$473$	7.24581	0.333163
$474$	−23.9351	−1.09937
$475$	2.98522	0.136971
$476$	−0.0235618	−0.00107995
$477$	62.3153	2.85322
$478$	8.59466	0.393110
$479$	−3.45019	−0.157643	−0.0788216	−	0.996889i	$-0.525116\pi$
$479$	−3.45019	−0.157643	−0.0788216	+	0.996889i	$0.525116\pi$
$480$	3.71179	0.169419
$481$	16.7949	0.765781
$482$	−23.6050	−1.07518
$483$	0.0177638	0.000808281 0
$484$	0.230796	0.0104907
$485$	0.429873	0.0195195
$486$	18.4727	0.837940
$487$	9.49287	0.430163	0.215081	−	0.976596i	$-0.430998\pi$
$487$	9.49287	0.430163	0.215081	+	0.976596i	$0.430998\pi$
$488$	−25.0432	−1.13365
$489$	28.3677	1.28283
$490$	10.4409	0.471674
$491$	−25.6488	−1.15751	−0.578757	−	0.815500i	$-0.696462\pi$
$491$	−25.6488	−1.15751	−0.578757	+	0.815500i	$0.696462\pi$
$492$	−0.748805	−0.0337588
$493$	−3.71433	−0.167285
$494$	−31.2782	−1.40727
$495$	−5.16116	−0.231977
$496$	4.11167	0.184619
$497$	−0.152518	−0.00684137
$498$	30.3950	1.36203
$499$	30.8253	1.37993	0.689966	−	0.723842i	$-0.257626\pi$
$499$	30.8253	1.37993	0.689966	+	0.723842i	$0.257626\pi$
$500$	0.230796	0.0103215
$501$	37.4723	1.67414
$502$	−5.60551	−0.250186
$503$	−5.97588	−0.266451	−0.133226	−	0.991086i	$-0.542534\pi$
$503$	−5.97588	−0.266451	−0.133226	+	0.991086i	$0.542534\pi$
$504$	1.32733	0.0591239
$505$	−3.18660	−0.141802
$506$	0.0954261	0.00424221
$507$	103.450	4.59438
$508$	−2.30702	−0.102358
$509$	35.3283	1.56590	0.782949	−	0.622085i	$-0.213714\pi$
$509$	35.3283	1.56590	0.782949	+	0.622085i	$0.213714\pi$
$510$	4.47573	0.198189
$511$	−0.0973246	−0.00430539
$512$	17.5699	0.776487
$513$	18.4306	0.813731
$514$	−37.0507	−1.63424
$515$	13.6864	0.603095
$516$	−4.77739	−0.210313
$517$	−10.8077	−0.475322
$518$	−0.348011	−0.0152907
$519$	3.59267	0.157701
$520$	18.5372	0.812910
$521$	21.3400	0.934921	0.467460	−	0.884014i	$-0.345169\pi$
$521$	21.3400	0.934921	0.467460	+	0.884014i	$0.345169\pi$
$522$	−27.2960	−1.19472
$523$	−5.34366	−0.233662	−0.116831	−	0.993152i	$-0.537274\pi$
$523$	−5.34366	−0.233662	−0.116831	+	0.993152i	$0.537274\pi$
$524$	−2.51433	−0.109839
$525$	0.278034	0.0121344
$526$	−30.8663	−1.34584
$527$	0.978368	0.0426184
$528$	12.5936	0.548066
$529$	−22.9959	−0.999823
$530$	−18.0334	−0.783320
$531$	12.6445	0.548726
$532$	0.0670542	0.00290717
$533$	−7.96712	−0.345094
$534$	74.4501	3.22177
$535$	1.83708	0.0794240
$536$	4.87413	0.210530
$537$	−12.9024	−0.556781
$538$	−16.7555	−0.722382
$539$	6.99053	0.301103
$540$	1.42492	0.0613189
$541$	−11.2167	−0.482244	−0.241122	−	0.970495i	$-0.577515\pi$
$541$	−11.2167	−0.482244	−0.241122	+	0.970495i	$0.577515\pi$
$542$	45.5400	1.95611
$543$	23.1087	0.991691
$544$	−1.36290	−0.0584341
$545$	−12.2248	−0.523653
$546$	−2.91316	−0.124672
$547$	−29.3676	−1.25567	−0.627834	−	0.778347i	$-0.716058\pi$
$547$	−29.3676	−1.25567	−0.627834	+	0.778347i	$0.716058\pi$
$548$	−0.230320	−0.00983876
$549$	−48.9136	−2.08758
$550$	1.49358	0.0636867
$551$	10.5706	0.450321
$552$	0.482304	0.0205282
$553$	0.545948	0.0232161
$554$	−21.1524	−0.898679
$555$	6.83938	0.290316
$556$	−2.43441	−0.103242
$557$	19.5284	0.827445	0.413723	−	0.910403i	$-0.364228\pi$
$557$	19.5284	0.827445	0.413723	+	0.910403i	$0.364228\pi$
$558$	7.18988	0.304372
$559$	−50.8304	−2.14990
$560$	−0.429038	−0.0181302
$561$	2.99664	0.126518
$562$	−25.2073	−1.06331
$563$	−13.6999	−0.577382	−0.288691	−	0.957422i	$-0.593220\pi$
$563$	−13.6999	−0.577382	−0.288691	+	0.957422i	$0.593220\pi$
$564$	7.12585	0.300052
$565$	16.2159	0.682209
$566$	39.3391	1.65354
$567$	0.209648	0.00880441
$568$	−4.14101	−0.173753
$569$	−13.1947	−0.553150	−0.276575	−	0.960992i	$-0.589199\pi$
$569$	−13.1947	−0.553150	−0.276575	+	0.960992i	$0.589199\pi$
$570$	−12.7374	−0.533512
$571$	33.1635	1.38785	0.693925	−	0.720047i	$-0.255880\pi$
$571$	33.1635	1.38785	0.693925	+	0.720047i	$0.255880\pi$
$572$	−1.61906	−0.0676965
$573$	20.8403	0.870615
$574$	0.165089	0.00689067
$575$	0.0638906	0.00266442
$576$	35.4884	1.47868
$577$	−25.5502	−1.06367	−0.531834	−	0.846849i	$-0.678497\pi$
$577$	−25.5502	−1.06367	−0.531834	+	0.846849i	$0.678497\pi$
$578$	23.7475	0.987767
$579$	−38.8753	−1.61560
$580$	0.817241	0.0339341
$581$	−0.693296	−0.0287628
$582$	−1.83420	−0.0760300
$583$	−12.0739	−0.500049
$584$	−2.64246	−0.109346
$585$	36.2063	1.49695
$586$	−10.7101	−0.442431
$587$	15.8217	0.653030	0.326515	−	0.945192i	$-0.394126\pi$
$587$	15.8217	0.653030	0.326515	+	0.945192i	$0.394126\pi$
$588$	−4.60908	−0.190075
$589$	−2.78433	−0.114726
$590$	−3.65919	−0.150646
$591$	−37.7616	−1.55330
$592$	−10.5539	−0.433764
$593$	6.38566	0.262227	0.131114	−	0.991367i	$-0.458145\pi$
$593$	6.38566	0.262227	0.131114	+	0.991367i	$0.458145\pi$
$594$	9.22133	0.378356
$595$	−0.102089	−0.00418526
$596$	2.72456	0.111602
$597$	38.2816	1.56676
$598$	−0.669427	−0.0273749
$599$	34.1340	1.39468	0.697339	−	0.716741i	$-0.254367\pi$
$599$	34.1340	1.39468	0.697339	+	0.716741i	$0.254367\pi$
$600$	7.54890	0.308183
$601$	25.1815	1.02718	0.513588	−	0.858037i	$-0.328316\pi$
$601$	25.1815	1.02718	0.513588	+	0.858037i	$0.328316\pi$
$602$	1.05327	0.0429281
$603$	9.52000	0.387684
$604$	−1.42846	−0.0581231
$605$	1.00000	0.0406558
$606$	13.5967	0.552328
$607$	−30.7986	−1.25008	−0.625039	−	0.780593i	$-0.714917\pi$
$607$	−30.7986	−1.25008	−0.625039	+	0.780593i	$0.714917\pi$
$608$	3.87867	0.157301
$609$	0.984511	0.0398944
$610$	14.1551	0.573122
$611$	75.8174	3.06725
$612$	−1.24949	−0.0505077
$613$	−21.4159	−0.864979	−0.432489	−	0.901639i	$-0.642365\pi$
$613$	−21.4159	−0.864979	−0.432489	+	0.901639i	$0.642365\pi$
$614$	−15.9047	−0.641861
$615$	−3.24445	−0.130829
$616$	−0.257176	−0.0103619
$617$	20.7288	0.834508	0.417254	−	0.908790i	$-0.362993\pi$
$617$	20.7288	0.834508	0.417254	+	0.908790i	$0.362993\pi$
$618$	−58.3977	−2.34910
$619$	−25.7989	−1.03695	−0.518473	−	0.855094i	$-0.673499\pi$
$619$	−25.7989	−1.03695	−0.518473	+	0.855094i	$0.673499\pi$
$620$	−0.215264	−0.00864523
$621$	0.394458	0.0158291
$622$	39.5394	1.58539
$623$	−1.69817	−0.0680359
$624$	−88.3458	−3.53666
$625$	1.00000	0.0400000
$626$	−10.4728	−0.418578
$627$	−8.52809	−0.340579
$628$	0.874373	0.0348913
$629$	−2.51130	−0.100132
$630$	−0.750240	−0.0298903
$631$	−34.9301	−1.39054	−0.695272	−	0.718747i	$-0.744716\pi$
$631$	−34.9301	−1.39054	−0.695272	+	0.718747i	$0.744716\pi$
$632$	14.8230	0.589628
$633$	16.4525	0.653930
$634$	50.5347	2.00699
$635$	−9.99595	−0.396677
$636$	7.96070	0.315662
$637$	−49.0395	−1.94302
$638$	5.28874	0.209383
$639$	−8.08810	−0.319960
$640$	−12.8685	−0.508674
$641$	41.3642	1.63379	0.816894	−	0.576788i	$-0.195694\pi$
$641$	41.3642	1.63379	0.816894	+	0.576788i	$0.195694\pi$
$642$	−7.83853	−0.309362
$643$	26.8773	1.05994	0.529968	−	0.848018i	$-0.322204\pi$
$643$	26.8773	1.05994	0.529968	+	0.848018i	$0.322204\pi$
$644$	0.00143512	5.65516e−5 0
$645$	−20.6997	−0.815048
$646$	4.67696	0.184013
$647$	−48.6976	−1.91450	−0.957249	−	0.289264i	$-0.906589\pi$
$647$	−48.6976	−1.91450	−0.957249	+	0.289264i	$0.906589\pi$
$648$	5.69216	0.223609
$649$	−2.44994	−0.0961685
$650$	−10.4777	−0.410969
$651$	−0.259324	−0.0101637
$652$	2.29180	0.0897538
$653$	18.6881	0.731324	0.365662	−	0.930748i	$-0.380843\pi$
$653$	18.6881	0.731324	0.365662	+	0.930748i	$0.380843\pi$
$654$	52.1612	2.03967
$655$	−10.8942	−0.425670
$656$	5.00655	0.195473
$657$	−5.16116	−0.201356
$658$	−1.57103	−0.0612452
$659$	−4.98588	−0.194222	−0.0971111	−	0.995274i	$-0.530960\pi$
$659$	−4.98588	−0.194222	−0.0971111	+	0.995274i	$0.530960\pi$
$660$	−0.659332	−0.0256645
$661$	18.1768	0.706997	0.353499	−	0.935435i	$-0.384992\pi$
$661$	18.1768	0.706997	0.353499	+	0.935435i	$0.384992\pi$
$662$	27.3668	1.06364
$663$	−21.0218	−0.816420
$664$	−18.8237	−0.730499
$665$	0.290535	0.0112665
$666$	−18.4552	−0.715124
$667$	0.226235	0.00875984
$668$	3.02735	0.117132
$669$	−1.77654	−0.0686851
$670$	−2.75498	−0.106434
$671$	9.47724	0.365865
$672$	0.361249	0.0139355
$673$	−13.6629	−0.526667	−0.263333	−	0.964705i	$-0.584822\pi$
$673$	−13.6629	−0.526667	−0.263333	+	0.964705i	$0.584822\pi$
$674$	1.15037	0.0443106
$675$	6.17396	0.237636
$676$	8.35762	0.321447
$677$	−39.1147	−1.50330	−0.751650	−	0.659562i	$-0.770742\pi$
$677$	−39.1147	−1.50330	−0.751650	+	0.659562i	$0.770742\pi$
$678$	−69.1906	−2.65725
$679$	0.0418372	0.00160557
$680$	−2.77183	−0.106295
$681$	19.9029	0.762683
$682$	−1.39307	−0.0533436
$683$	4.66319	0.178432	0.0892161	−	0.996012i	$-0.471564\pi$
$683$	4.66319	0.178432	0.0892161	+	0.996012i	$0.471564\pi$
$684$	3.55591	0.135964
$685$	−0.997936	−0.0381292
$686$	2.03370	0.0776470
$687$	−27.8538	−1.06269
$688$	31.9419	1.21777
$689$	84.7000	3.22681
$690$	−0.272611	−0.0103781
$691$	−32.9508	−1.25351	−0.626754	−	0.779217i	$-0.715617\pi$
$691$	−32.9508	−1.25351	−0.626754	+	0.779217i	$0.715617\pi$
$692$	0.290248	0.0110336
$693$	−0.502308	−0.0190811
$694$	−15.8522	−0.601743
$695$	−10.5479	−0.400104
$696$	26.7304	1.01321
$697$	1.19131	0.0451239
$698$	−34.8034	−1.31733
$699$	−25.7507	−0.973979
$700$	0.0224621	0.000848988 0
$701$	52.2212	1.97237	0.986183	−	0.165660i	$-0.0529755\pi$
$701$	52.2212	1.97237	0.986183	+	0.165660i	$0.0529755\pi$
$702$	−64.6889	−2.44153
$703$	7.14688	0.269550
$704$	−6.87604	−0.259151
$705$	30.8751	1.16282
$706$	−4.03305	−0.151786
$707$	−0.310135	−0.0116638
$708$	1.61532	0.0607075
$709$	43.6868	1.64069	0.820346	−	0.571868i	$-0.193781\pi$
$709$	43.6868	1.64069	0.820346	+	0.571868i	$0.193781\pi$
$710$	2.34061	0.0878415
$711$	28.9519	1.08578
$712$	−46.1070	−1.72793
$713$	−0.0595911	−0.00223170
$714$	0.435599	0.0163019
$715$	−7.01514	−0.262351
$716$	−1.04237	−0.0389553
$717$	−16.4390	−0.613925
$718$	−4.95476	−0.184910
$719$	−12.7198	−0.474370	−0.237185	−	0.971464i	$-0.576225\pi$
$719$	−12.7198	−0.474370	−0.237185	+	0.971464i	$0.576225\pi$
$720$	−22.7521	−0.847920
$721$	1.33202	0.0496072
$722$	15.0680	0.560773
$723$	45.1493	1.67912
$724$	1.86693	0.0693839
$725$	3.54097	0.131508
$726$	−4.26684	−0.158357
$727$	−6.24691	−0.231685	−0.115842	−	0.993268i	$-0.536957\pi$
$727$	−6.24691	−0.231685	−0.115842	+	0.993268i	$0.536957\pi$
$728$	1.80413	0.0668653
$729$	−41.7951	−1.54797
$730$	1.49358	0.0552801
$731$	7.60055	0.281117
$732$	−6.24865	−0.230957
$733$	−29.2930	−1.08196	−0.540981	−	0.841034i	$-0.681947\pi$
$733$	−29.2930	−1.08196	−0.540981	+	0.841034i	$0.681947\pi$
$734$	40.9145	1.51018
$735$	−19.9704	−0.736618
$736$	0.0830127	0.00305989
$737$	−1.84455	−0.0679447
$738$	8.75473	0.322266
$739$	−33.1734	−1.22030	−0.610151	−	0.792285i	$-0.708891\pi$
$739$	−33.1734	−1.22030	−0.610151	+	0.792285i	$0.708891\pi$
$740$	0.552546	0.0203120
$741$	59.8258	2.19775
$742$	−1.75509	−0.0644314
$743$	−21.9925	−0.806827	−0.403414	−	0.915018i	$-0.632176\pi$
$743$	−21.9925	−0.806827	−0.403414	+	0.915018i	$0.632176\pi$
$744$	−7.04090	−0.258132
$745$	11.8051	0.432504
$746$	45.1301	1.65233
$747$	−36.7658	−1.34519
$748$	0.242095	0.00885187
$749$	0.178793	0.00653296
$750$	−4.26684	−0.155803
$751$	−35.7050	−1.30289	−0.651447	−	0.758694i	$-0.725838\pi$
$751$	−35.7050	−1.30289	−0.651447	+	0.758694i	$0.725838\pi$
$752$	−47.6438	−1.73739
$753$	10.7216	0.390718
$754$	−37.1012	−1.35115
$755$	−6.18927	−0.225251
$756$	0.138680	0.00504375
$757$	−44.8186	−1.62896	−0.814479	−	0.580193i	$-0.802977\pi$
$757$	−44.8186	−1.62896	−0.814479	+	0.580193i	$0.802977\pi$
$758$	41.0045	1.48935
$759$	−0.182521	−0.00662510
$760$	7.88830	0.286139
$761$	27.5744	0.999570	0.499785	−	0.866149i	$-0.333412\pi$
$761$	27.5744	0.999570	0.499785	+	0.866149i	$0.333412\pi$
$762$	42.6511	1.54509
$763$	−1.18977	−0.0430727
$764$	1.68366	0.0609128
$765$	−5.41384	−0.195738
$766$	10.8343	0.391458
$767$	17.1867	0.620574
$768$	15.6213	0.563687
$769$	−29.9741	−1.08089	−0.540447	−	0.841378i	$-0.681745\pi$
$769$	−29.9741	−1.08089	−0.540447	+	0.841378i	$0.681745\pi$
$770$	0.145363	0.00523850
$771$	70.8667	2.55220
$772$	−3.14069	−0.113036
$773$	−16.0713	−0.578045	−0.289022	−	0.957322i	$-0.593330\pi$
$773$	−16.0713	−0.578045	−0.289022	+	0.957322i	$0.593330\pi$
$774$	55.8553	2.00768
$775$	−0.932705	−0.0335038
$776$	1.13592	0.0407772
$777$	0.665640	0.0238797
$778$	31.4984	1.12927
$779$	−3.39032	−0.121471
$780$	4.62530	0.165612
$781$	1.56711	0.0560755
$782$	0.100098	0.00357949
$783$	21.8618	0.781277
$784$	30.8165	1.10059
$785$	3.78852	0.135218
$786$	46.4836	1.65802
$787$	−15.7464	−0.561299	−0.280649	−	0.959810i	$-0.590550\pi$
$787$	−15.7464	−0.561299	−0.280649	+	0.959810i	$0.590550\pi$
$788$	−3.05072	−0.108677
$789$	59.0379	2.10180
$790$	−8.37835	−0.298088
$791$	1.57821	0.0561146
$792$	−13.6381	−0.484611
$793$	−66.4842	−2.36092
$794$	−44.1218	−1.56582
$795$	34.4924	1.22332
$796$	3.09273	0.109619
$797$	27.0518	0.958223	0.479112	−	0.877754i	$-0.340959\pi$
$797$	27.0518	0.958223	0.479112	+	0.877754i	$0.340959\pi$
$798$	−1.23966	−0.0438837
$799$	−11.3368	−0.401067
$800$	1.29929	0.0459370
$801$	−90.0548	−3.18193
$802$	−0.0175991	−0.000621445 0
$803$	1.00000	0.0352892
$804$	1.21617	0.0428909
$805$	0.00621813	0.000219160 0
$806$	9.77261	0.344226
$807$	32.0482	1.12815
$808$	−8.42046	−0.296231
$809$	38.5427	1.35509	0.677545	−	0.735481i	$-0.263044\pi$
$809$	38.5427	1.35509	0.677545	+	0.735481i	$0.263044\pi$
$810$	−3.21735	−0.113046
$811$	4.90745	0.172324	0.0861619	−	0.996281i	$-0.472540\pi$
$811$	4.90745	0.172324	0.0861619	+	0.996281i	$0.472540\pi$
$812$	0.0795376	0.00279122
$813$	−87.1042	−3.05488
$814$	3.57578	0.125331
$815$	9.92998	0.347832
$816$	13.2101	0.462448
$817$	−21.6303	−0.756749
$818$	−7.93567	−0.277464
$819$	3.52376	0.123130
$820$	−0.262116	−0.00915348
$821$	−0.863221	−0.0301266	−0.0150633	−	0.999887i	$-0.504795\pi$
$821$	−0.863221	−0.0301266	−0.0150633	+	0.999887i	$0.504795\pi$
$822$	4.25803	0.148516
$823$	31.5052	1.09820	0.549100	−	0.835756i	$-0.314971\pi$
$823$	31.5052	1.09820	0.549100	+	0.835756i	$0.314971\pi$
$824$	36.1657	1.25989
$825$	−2.85677	−0.0994601
$826$	−0.356129	−0.0123913
$827$	−0.950453	−0.0330505	−0.0165252	−	0.999863i	$-0.505260\pi$
$827$	−0.950453	−0.0330505	−0.0165252	+	0.999863i	$0.505260\pi$
$828$	0.0761049	0.00264483
$829$	−51.5358	−1.78991	−0.894955	−	0.446156i	$-0.852793\pi$
$829$	−51.5358	−1.78991	−0.894955	+	0.446156i	$0.852793\pi$
$830$	10.6396	0.369306
$831$	40.4581	1.40348
$832$	48.2364	1.67230
$833$	7.33277	0.254065
$834$	45.0061	1.55843
$835$	13.1170	0.453933
$836$	−0.688975	−0.0238287
$837$	−5.75848	−0.199042
$838$	−21.2905	−0.735469
$839$	6.87672	0.237411	0.118705	−	0.992930i	$-0.462126\pi$
$839$	6.87672	0.237411	0.118705	+	0.992930i	$0.462126\pi$
$840$	0.734694	0.0253493
$841$	−16.4615	−0.567639
$842$	−26.7997	−0.923578
$843$	48.2139	1.66058
$844$	1.32918	0.0457524
$845$	36.2122	1.24574
$846$	−83.3125	−2.86434
$847$	0.0973246	0.00334411
$848$	−53.2256	−1.82778
$849$	−75.2437	−2.58236
$850$	1.56671	0.0537376
$851$	0.152960	0.00524340
$852$	−1.03324	−0.0353983
$853$	15.8523	0.542774	0.271387	−	0.962470i	$-0.412518\pi$
$853$	15.8523	0.542774	0.271387	+	0.962470i	$0.412518\pi$
$854$	1.37764	0.0471417
$855$	15.4072	0.526915
$856$	4.85441	0.165920
$857$	−47.4226	−1.61992	−0.809962	−	0.586482i	$-0.800512\pi$
$857$	−47.4226	−1.61992	−0.809962	+	0.586482i	$0.800512\pi$
$858$	29.9324	1.02188
$859$	−13.8643	−0.473043	−0.236521	−	0.971626i	$-0.576007\pi$
$859$	−13.8643	−0.473043	−0.236521	+	0.971626i	$0.576007\pi$
$860$	−1.67230	−0.0570251
$861$	−0.315765	−0.0107612
$862$	4.54528	0.154813
$863$	−17.3499	−0.590599	−0.295300	−	0.955405i	$-0.595419\pi$
$863$	−17.3499	−0.590599	−0.295300	+	0.955405i	$0.595419\pi$
$864$	8.02179	0.272907
$865$	1.25760	0.0427595
$866$	−58.2061	−1.97792
$867$	−45.4218	−1.54261
$868$	−0.0209505	−0.000711107 0
$869$	−5.60956	−0.190291
$870$	−15.1087	−0.512234
$871$	12.9397	0.438446
$872$	−32.3035	−1.09393
$873$	2.21865	0.0750898
$874$	−0.284867	−0.00963578
$875$	0.0973246	0.00329017
$876$	−0.659332	−0.0222768
$877$	18.7510	0.633177	0.316588	−	0.948563i	$-0.397463\pi$
$877$	18.7510	0.633177	0.316588	+	0.948563i	$0.397463\pi$
$878$	−34.3768	−1.16016
$879$	20.4852	0.690949
$880$	4.40832	0.148605
$881$	−19.8798	−0.669767	−0.334884	−	0.942260i	$-0.608697\pi$
$881$	−19.8798	−0.669767	−0.334884	+	0.942260i	$0.608697\pi$
$882$	53.8874	1.81448
$883$	−41.7062	−1.40353	−0.701763	−	0.712411i	$-0.747603\pi$
$883$	−41.7062	−1.40353	−0.701763	+	0.712411i	$0.747603\pi$
$884$	−1.69833	−0.0571210
$885$	6.99892	0.235266
$886$	−59.1084	−1.98579
$887$	14.9263	0.501175	0.250588	−	0.968094i	$-0.419376\pi$
$887$	14.9263	0.501175	0.250588	+	0.968094i	$0.419376\pi$
$888$	18.0728	0.606483
$889$	−0.972852	−0.0326284
$890$	26.0609	0.873563
$891$	−2.15412	−0.0721656
$892$	−0.143525	−0.00480557
$893$	32.2633	1.07965
$894$	−50.3702	−1.68463
$895$	−4.51643	−0.150968
$896$	−1.25243	−0.0418406
$897$	1.28041	0.0427517
$898$	−11.7115	−0.390819
$899$	−3.30268	−0.110151
$900$	1.19117	0.0397058
$901$	−12.6650	−0.421932
$902$	−1.69627	−0.0564796
$903$	−2.01459	−0.0670412
$904$	42.8498	1.42516
$905$	8.08910	0.268891
$906$	26.4086	0.877367
$907$	−9.02465	−0.299659	−0.149829	−	0.988712i	$-0.547872\pi$
$907$	−9.02465	−0.299659	−0.149829	+	0.988712i	$0.547872\pi$
$908$	1.60794	0.0533613
$909$	−16.4466	−0.545498
$910$	−1.01974	−0.0338040
$911$	−23.6927	−0.784975	−0.392487	−	0.919757i	$-0.628385\pi$
$911$	−23.6927	−0.784975	−0.392487	+	0.919757i	$0.628385\pi$
$912$	−37.5946	−1.24488
$913$	7.12354	0.235755
$914$	5.63012	0.186228
$915$	−27.0743	−0.895050
$916$	−2.25028	−0.0743513
$917$	−1.06027	−0.0350132
$918$	9.67278	0.319249
$919$	10.2958	0.339628	0.169814	−	0.985476i	$-0.445683\pi$
$919$	10.2958	0.339628	0.169814	+	0.985476i	$0.445683\pi$
$920$	0.168828	0.00556610
$921$	30.4209	1.00240
$922$	−14.3704	−0.473265
$923$	−10.9935	−0.361855
$924$	−0.0641692	−0.00211101
$925$	2.39409	0.0787173
$926$	27.0039	0.887403
$927$	70.6378	2.32005
$928$	4.60076	0.151027
$929$	−41.3087	−1.35529	−0.677647	−	0.735387i	$-0.737000\pi$
$929$	−41.3087	−1.35529	−0.677647	+	0.735387i	$0.737000\pi$
$930$	3.97970	0.130499
$931$	−20.8682	−0.683929
$932$	−2.08037	−0.0681447
$933$	−75.6269	−2.47591
$934$	38.9793	1.27544
$935$	1.04896	0.0343046
$936$	95.6735	3.12719
$937$	24.8031	0.810281	0.405140	−	0.914255i	$-0.367223\pi$
$937$	24.8031	0.810281	0.405140	+	0.914255i	$0.367223\pi$
$938$	−0.268128	−0.00875468
$939$	20.0313	0.653697
$940$	2.49437	0.0813574
$941$	6.01452	0.196068	0.0980339	−	0.995183i	$-0.468745\pi$
$941$	6.01452	0.196068	0.0980339	+	0.995183i	$0.468745\pi$
$942$	−16.1650	−0.526683
$943$	−0.0725608	−0.00236291
$944$	−10.8001	−0.351514
$945$	0.600878	0.0195466
$946$	−10.8222	−0.351861
$947$	42.0469	1.36634	0.683170	−	0.730260i	$-0.260601\pi$
$947$	42.0469	1.36634	0.683170	+	0.730260i	$0.260601\pi$
$948$	3.69856	0.120124
$949$	−7.01514	−0.227721
$950$	−4.45867	−0.144658
$951$	−96.6576	−3.13434
$952$	−0.269767	−0.00874319
$953$	−42.8614	−1.38842	−0.694208	−	0.719775i	$-0.744245\pi$
$953$	−42.8614	−1.38842	−0.694208	+	0.719775i	$0.744245\pi$
$954$	−93.0732	−3.01336
$955$	7.29503	0.236062
$956$	−1.32809	−0.0429534
$957$	−10.1158	−0.326996
$958$	5.15315	0.166491
$959$	−0.0971237	−0.00313629
$960$	19.6433	0.633985
$961$	−30.1301	−0.971937
$962$	−25.0846	−0.808760
$963$	9.48148	0.305536
$964$	3.64756	0.117480
$965$	−13.6081	−0.438060
$966$	−0.0265317	−0.000853644 0
$967$	−29.9891	−0.964383	−0.482191	−	0.876066i	$-0.660159\pi$
$967$	−29.9891	−0.964383	−0.482191	+	0.876066i	$0.660159\pi$
$968$	2.64246	0.0849318
$969$	−8.94561	−0.287374
$970$	−0.642052	−0.0206151
$971$	13.1958	0.423472	0.211736	−	0.977327i	$-0.432088\pi$
$971$	13.1958	0.423472	0.211736	+	0.977327i	$0.432088\pi$
$972$	−2.85449	−0.0915579
$973$	−1.02657	−0.0329103
$974$	−14.1784	−0.454305
$975$	20.0407	0.641815
$976$	41.7788	1.33731
$977$	−38.0367	−1.21690	−0.608451	−	0.793592i	$-0.708209\pi$
$977$	−38.0367	−1.21690	−0.608451	+	0.793592i	$0.708209\pi$
$978$	−42.3696	−1.35483
$979$	17.4486	0.557658
$980$	−1.61338	−0.0515377
$981$	−63.0942	−2.01444
$982$	38.3087	1.22248
$983$	−25.4115	−0.810502	−0.405251	−	0.914206i	$-0.632816\pi$
$983$	−25.4115	−0.810502	−0.405251	+	0.914206i	$0.632816\pi$
$984$	−8.57332	−0.273307
$985$	−13.2183	−0.421169
$986$	5.54766	0.176674
$987$	3.00491	0.0956473
$988$	4.83326	0.153766
$989$	−0.462939	−0.0147206
$990$	7.70864	0.244997
$991$	14.2068	0.451295	0.225648	−	0.974209i	$-0.427550\pi$
$991$	14.2068	0.451295	0.225648	+	0.974209i	$0.427550\pi$
$992$	−1.21186	−0.0384765
$993$	−52.3444	−1.66110
$994$	0.227799	0.00722534
$995$	13.4003	0.424818
$996$	−4.69678	−0.148823
$997$	−5.64105	−0.178654	−0.0893270	−	0.996002i	$-0.528472\pi$
$997$	−5.64105	−0.178654	−0.0893270	+	0.996002i	$0.528472\pi$
$998$	−46.0403	−1.45738
$999$	14.7810	0.467651

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.i.1.12	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.i.1.12	✓	38	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)

\(f(q)\)	\(=\)	\(q-1.49358 q^{2} +2.85677 q^{3} +0.230796 q^{4} +1.00000 q^{5} -4.26684 q^{6} +0.0973246 q^{7} +2.64246 q^{8} +5.16116 q^{9} +O(q^{10})\)	\(q-1.49358 q^{2} +2.85677 q^{3} +0.230796 q^{4} +1.00000 q^{5} -4.26684 q^{6} +0.0973246 q^{7} +2.64246 q^{8} +5.16116 q^{9} -1.49358 q^{10} -1.00000 q^{11} +0.659332 q^{12} +7.01514 q^{13} -0.145363 q^{14} +2.85677 q^{15} -4.40832 q^{16} -1.04896 q^{17} -7.70864 q^{18} +2.98522 q^{19} +0.230796 q^{20} +0.278034 q^{21} +1.49358 q^{22} +0.0638906 q^{23} +7.54890 q^{24} +1.00000 q^{25} -10.4777 q^{26} +6.17396 q^{27} +0.0224621 q^{28} +3.54097 q^{29} -4.26684 q^{30} -0.932705 q^{31} +1.29929 q^{32} -2.85677 q^{33} +1.56671 q^{34} +0.0973246 q^{35} +1.19117 q^{36} +2.39409 q^{37} -4.45867 q^{38} +20.0407 q^{39} +2.64246 q^{40} -1.13570 q^{41} -0.415268 q^{42} -7.24581 q^{43} -0.230796 q^{44} +5.16116 q^{45} -0.0954261 q^{46} +10.8077 q^{47} -12.5936 q^{48} -6.99053 q^{49} -1.49358 q^{50} -2.99664 q^{51} +1.61906 q^{52} +12.0739 q^{53} -9.22133 q^{54} -1.00000 q^{55} +0.257176 q^{56} +8.52809 q^{57} -5.28874 q^{58} +2.44994 q^{59} +0.659332 q^{60} -9.47724 q^{61} +1.39307 q^{62} +0.502308 q^{63} +6.87604 q^{64} +7.01514 q^{65} +4.26684 q^{66} +1.84455 q^{67} -0.242095 q^{68} +0.182521 q^{69} -0.145363 q^{70} -1.56711 q^{71} +13.6381 q^{72} -1.00000 q^{73} -3.57578 q^{74} +2.85677 q^{75} +0.688975 q^{76} -0.0973246 q^{77} -29.9324 q^{78} +5.60956 q^{79} -4.40832 q^{80} +2.15412 q^{81} +1.69627 q^{82} -7.12354 q^{83} +0.0641692 q^{84} -1.04896 q^{85} +10.8222 q^{86} +10.1158 q^{87} -2.64246 q^{88} -17.4486 q^{89} -7.70864 q^{90} +0.682746 q^{91} +0.0147457 q^{92} -2.66453 q^{93} -16.1422 q^{94} +2.98522 q^{95} +3.71179 q^{96} +0.429873 q^{97} +10.4409 q^{98} -5.16116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9}+O(q^{10}) \) 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9} + 4 q^{10} - 38 q^{11} + 12 q^{12} - q^{13} + 23 q^{14} + 5 q^{15} + 74 q^{16} + 26 q^{17} + 16 q^{18} - 10 q^{19} + 50 q^{20} + 21 q^{21} - 4 q^{22} + 10 q^{23} + 41 q^{24} + 38 q^{25} + 25 q^{26} + 5 q^{27} + 2 q^{28} + 28 q^{29} + 11 q^{30} + 24 q^{31} + 39 q^{32} - 5 q^{33} + 38 q^{34} + 111 q^{36} + 12 q^{37} + 19 q^{38} - 18 q^{39} + 15 q^{40} + 62 q^{41} - 17 q^{42} - 32 q^{43} - 50 q^{44} + 63 q^{45} - 9 q^{46} + 31 q^{47} + 53 q^{48} + 88 q^{49} + 4 q^{50} - 3 q^{51} - 21 q^{52} + 30 q^{53} + 49 q^{54} - 38 q^{55} + 32 q^{56} + 49 q^{57} + 12 q^{58} + 31 q^{59} + 12 q^{60} + 25 q^{61} + 12 q^{62} + 15 q^{63} + 137 q^{64} - q^{65} - 11 q^{66} + 20 q^{67} + 75 q^{68} + 92 q^{69} + 23 q^{70} + 32 q^{71} + 6 q^{72} - 38 q^{73} + 55 q^{74} + 5 q^{75} - 57 q^{76} - 17 q^{78} - 2 q^{79} + 74 q^{80} + 118 q^{81} + 14 q^{82} + 4 q^{83} + 22 q^{84} + 26 q^{85} + 5 q^{86} + 24 q^{87} - 15 q^{88} + 143 q^{89} + 16 q^{90} + 66 q^{91} + 29 q^{92} - 8 q^{93} - 7 q^{94} - 10 q^{95} + 59 q^{96} + 41 q^{97} - 10 q^{98} - 63 q^{99}+O(q^{100}) \) 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9 + 4 * q^10 - 38 * q^11 + 12 * q^12 - q^13 + 23 * q^14 + 5 * q^15 + 74 * q^16 + 26 * q^17 + 16 * q^18 - 10 * q^19 + 50 * q^20 + 21 * q^21 - 4 * q^22 + 10 * q^23 + 41 * q^24 + 38 * q^25 + 25 * q^26 + 5 * q^27 + 2 * q^28 + 28 * q^29 + 11 * q^30 + 24 * q^31 + 39 * q^32 - 5 * q^33 + 38 * q^34 + 111 * q^36 + 12 * q^37 + 19 * q^38 - 18 * q^39 + 15 * q^40 + 62 * q^41 - 17 * q^42 - 32 * q^43 - 50 * q^44 + 63 * q^45 - 9 * q^46 + 31 * q^47 + 53 * q^48 + 88 * q^49 + 4 * q^50 - 3 * q^51 - 21 * q^52 + 30 * q^53 + 49 * q^54 - 38 * q^55 + 32 * q^56 + 49 * q^57 + 12 * q^58 + 31 * q^59 + 12 * q^60 + 25 * q^61 + 12 * q^62 + 15 * q^63 + 137 * q^64 - q^65 - 11 * q^66 + 20 * q^67 + 75 * q^68 + 92 * q^69 + 23 * q^70 + 32 * q^71 + 6 * q^72 - 38 * q^73 + 55 * q^74 + 5 * q^75 - 57 * q^76 - 17 * q^78 - 2 * q^79 + 74 * q^80 + 118 * q^81 + 14 * q^82 + 4 * q^83 + 22 * q^84 + 26 * q^85 + 5 * q^86 + 24 * q^87 - 15 * q^88 + 143 * q^89 + 16 * q^90 + 66 * q^91 + 29 * q^92 - 8 * q^93 - 7 * q^94 - 10 * q^95 + 59 * q^96 + 41 * q^97 - 10 * q^98 - 63 * q^99

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