LMFDB - Embedded newform 4015.2.a.i.1.13

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			1.13
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)

$f(q)$	$=$	$q-1.16964 q^{2} -3.05105 q^{3} -0.631953 q^{4} +1.00000 q^{5} +3.56861 q^{6} -2.34807 q^{7} +3.07843 q^{8} +6.30888 q^{9} +O(q^{10})$	\(q-1.16964 q^{2} -3.05105 q^{3} -0.631953 q^{4} +1.00000 q^{5} +3.56861 q^{6} -2.34807 q^{7} +3.07843 q^{8} +6.30888 q^{9} -1.16964 q^{10} -1.00000 q^{11} +1.92812 q^{12} -4.30309 q^{13} +2.74639 q^{14} -3.05105 q^{15} -2.33673 q^{16} -2.85096 q^{17} -7.37909 q^{18} +3.45483 q^{19} -0.631953 q^{20} +7.16407 q^{21} +1.16964 q^{22} +3.20269 q^{23} -9.39242 q^{24} +1.00000 q^{25} +5.03304 q^{26} -10.0955 q^{27} +1.48387 q^{28} +4.48490 q^{29} +3.56861 q^{30} +0.289434 q^{31} -3.42373 q^{32} +3.05105 q^{33} +3.33459 q^{34} -2.34807 q^{35} -3.98692 q^{36} +5.62886 q^{37} -4.04089 q^{38} +13.1289 q^{39} +3.07843 q^{40} -2.87724 q^{41} -8.37935 q^{42} -10.9241 q^{43} +0.631953 q^{44} +6.30888 q^{45} -3.74598 q^{46} +1.30319 q^{47} +7.12946 q^{48} -1.48657 q^{49} -1.16964 q^{50} +8.69842 q^{51} +2.71935 q^{52} -1.74702 q^{53} +11.8081 q^{54} -1.00000 q^{55} -7.22836 q^{56} -10.5408 q^{57} -5.24569 q^{58} +0.0486243 q^{59} +1.92812 q^{60} +11.4956 q^{61} -0.338532 q^{62} -14.8137 q^{63} +8.67797 q^{64} -4.30309 q^{65} -3.56861 q^{66} +0.776300 q^{67} +1.80168 q^{68} -9.77154 q^{69} +2.74639 q^{70} -7.35436 q^{71} +19.4214 q^{72} -1.00000 q^{73} -6.58371 q^{74} -3.05105 q^{75} -2.18329 q^{76} +2.34807 q^{77} -15.3560 q^{78} -6.39572 q^{79} -2.33673 q^{80} +11.8753 q^{81} +3.36532 q^{82} -3.15582 q^{83} -4.52736 q^{84} -2.85096 q^{85} +12.7772 q^{86} -13.6836 q^{87} -3.07843 q^{88} -6.17653 q^{89} -7.37909 q^{90} +10.1039 q^{91} -2.02395 q^{92} -0.883077 q^{93} -1.52426 q^{94} +3.45483 q^{95} +10.4460 q^{96} -4.07348 q^{97} +1.73874 q^{98} -6.30888 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.16964	−0.827057	−0.413528	−	0.910491i	$-0.635704\pi$
$2$	−1.16964	−0.827057	−0.413528	+	0.910491i	$0.635704\pi$
$3$	−3.05105	−1.76152	−0.880761	−	0.473561i	$-0.842968\pi$
$3$	−3.05105	−1.76152	−0.880761	+	0.473561i	$0.842968\pi$
$4$	−0.631953	−0.315977
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	3.56861	1.45688
$7$	−2.34807	−0.887487	−0.443744	−	0.896154i	$-0.646350\pi$
$7$	−2.34807	−0.887487	−0.443744	+	0.896154i	$0.646350\pi$
$8$	3.07843	1.08839
$9$	6.30888	2.10296
$10$	−1.16964	−0.369871
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.92812	0.556600
$13$	−4.30309	−1.19346	−0.596731	−	0.802442i	$-0.703534\pi$
$13$	−4.30309	−1.19346	−0.596731	+	0.802442i	$0.703534\pi$
$14$	2.74639	0.734002
$15$	−3.05105	−0.787777
$16$	−2.33673	−0.584182
$17$	−2.85096	−0.691460	−0.345730	−	0.938334i	$-0.612369\pi$
$17$	−2.85096	−0.691460	−0.345730	+	0.938334i	$0.612369\pi$
$18$	−7.37909	−1.73927
$19$	3.45483	0.792592	0.396296	−	0.918123i	$-0.370295\pi$
$19$	3.45483	0.792592	0.396296	+	0.918123i	$0.370295\pi$
$20$	−0.631953	−0.141309
$21$	7.16407	1.56333
$22$	1.16964	0.249367
$23$	3.20269	0.667806	0.333903	−	0.942607i	$-0.391634\pi$
$23$	3.20269	0.667806	0.333903	+	0.942607i	$0.391634\pi$
$24$	−9.39242	−1.91722
$25$	1.00000	0.200000
$26$	5.03304	0.987060
$27$	−10.0955	−1.94289
$28$	1.48387	0.280425
$29$	4.48490	0.832824	0.416412	−	0.909176i	$-0.363287\pi$
$29$	4.48490	0.832824	0.416412	+	0.909176i	$0.363287\pi$
$30$	3.56861	0.651536
$31$	0.289434	0.0519839	0.0259920	−	0.999662i	$-0.491726\pi$
$31$	0.289434	0.0519839	0.0259920	+	0.999662i	$0.491726\pi$
$32$	−3.42373	−0.605236
$33$	3.05105	0.531119
$34$	3.33459	0.571877
$35$	−2.34807	−0.396896
$36$	−3.98692	−0.664486
$37$	5.62886	0.925379	0.462690	−	0.886520i	$-0.346884\pi$
$37$	5.62886	0.925379	0.462690	+	0.886520i	$0.346884\pi$
$38$	−4.04089	−0.655519
$39$	13.1289	2.10231
$40$	3.07843	0.486742
$41$	−2.87724	−0.449349	−0.224674	−	0.974434i	$-0.572132\pi$
$41$	−2.87724	−0.449349	−0.224674	+	0.974434i	$0.572132\pi$
$42$	−8.37935	−1.29296
$43$	−10.9241	−1.66591	−0.832953	−	0.553344i	$-0.813352\pi$
$43$	−10.9241	−1.66591	−0.832953	+	0.553344i	$0.813352\pi$
$44$	0.631953	0.0952706
$45$	6.30888	0.940472
$46$	−3.74598	−0.552314
$47$	1.30319	0.190090	0.0950448	−	0.995473i	$-0.469701\pi$
$47$	1.30319	0.190090	0.0950448	+	0.995473i	$0.469701\pi$
$48$	7.12946	1.02905
$49$	−1.48657	−0.212366
$50$	−1.16964	−0.165411
$51$	8.69842	1.21802
$52$	2.71935	0.377106
$53$	−1.74702	−0.239971	−0.119986	−	0.992776i	$-0.538285\pi$
$53$	−1.74702	−0.239971	−0.119986	+	0.992776i	$0.538285\pi$
$54$	11.8081	1.60688
$55$	−1.00000	−0.134840
$56$	−7.22836	−0.965930
$57$	−10.5408	−1.39617
$58$	−5.24569	−0.688793
$59$	0.0486243	0.00633034	0.00316517	−	0.999995i	$-0.498992\pi$
$59$	0.0486243	0.00633034	0.00316517	+	0.999995i	$0.498992\pi$
$60$	1.92812	0.248919
$61$	11.4956	1.47186	0.735929	−	0.677059i	$-0.236746\pi$
$61$	11.4956	1.47186	0.735929	+	0.677059i	$0.236746\pi$
$62$	−0.338532	−0.0429937
$63$	−14.8137	−1.86635
$64$	8.67797	1.08475
$65$	−4.30309	−0.533732
$66$	−3.56861	−0.439266
$67$	0.776300	0.0948401	0.0474200	−	0.998875i	$-0.484900\pi$
$67$	0.776300	0.0948401	0.0474200	+	0.998875i	$0.484900\pi$
$68$	1.80168	0.218485
$69$	−9.77154	−1.17636
$70$	2.74639	0.328256
$71$	−7.35436	−0.872802	−0.436401	−	0.899752i	$-0.643747\pi$
$71$	−7.35436	−0.872802	−0.436401	+	0.899752i	$0.643747\pi$
$72$	19.4214	2.28883
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	−6.58371	−0.765341
$75$	−3.05105	−0.352304
$76$	−2.18329	−0.250441
$77$	2.34807	0.267587
$78$	−15.3560	−1.73873
$79$	−6.39572	−0.719575	−0.359788	−	0.933034i	$-0.617151\pi$
$79$	−6.39572	−0.719575	−0.359788	+	0.933034i	$0.617151\pi$
$80$	−2.33673	−0.261254
$81$	11.8753	1.31948
$82$	3.36532	0.371637
$83$	−3.15582	−0.346397	−0.173198	−	0.984887i	$-0.555410\pi$
$83$	−3.15582	−0.346397	−0.173198	+	0.984887i	$0.555410\pi$
$84$	−4.52736	−0.493975
$85$	−2.85096	−0.309230
$86$	12.7772	1.37780
$87$	−13.6836	−1.46704
$88$	−3.07843	−0.328161
$89$	−6.17653	−0.654711	−0.327356	−	0.944901i	$-0.606157\pi$
$89$	−6.17653	−0.654711	−0.327356	+	0.944901i	$0.606157\pi$
$90$	−7.37909	−0.777824
$91$	10.1039	1.05918
$92$	−2.02395	−0.211011
$93$	−0.883077	−0.0915708
$94$	−1.52426	−0.157215
$95$	3.45483	0.354458
$96$	10.4460	1.06614
$97$	−4.07348	−0.413599	−0.206800	−	0.978383i	$-0.566305\pi$
$97$	−4.07348	−0.413599	−0.206800	+	0.978383i	$0.566305\pi$
$98$	1.73874	0.175639
$99$	−6.30888	−0.634066
$100$	−0.631953	−0.0631953
$101$	−3.47377	−0.345653	−0.172827	−	0.984952i	$-0.555290\pi$
$101$	−3.47377	−0.345653	−0.172827	+	0.984952i	$0.555290\pi$
$102$	−10.1740	−1.00737
$103$	2.78647	0.274559	0.137280	−	0.990532i	$-0.456164\pi$
$103$	2.78647	0.274559	0.137280	+	0.990532i	$0.456164\pi$
$104$	−13.2467	−1.29895
$105$	7.16407	0.699142
$106$	2.04337	0.198470
$107$	−19.0483	−1.84147	−0.920733	−	0.390192i	$-0.872409\pi$
$107$	−19.0483	−1.84147	−0.920733	+	0.390192i	$0.872409\pi$
$108$	6.37991	0.613907
$109$	−6.82269	−0.653495	−0.326747	−	0.945112i	$-0.605953\pi$
$109$	−6.82269	−0.653495	−0.326747	+	0.945112i	$0.605953\pi$
$110$	1.16964	0.111520
$111$	−17.1739	−1.63008
$112$	5.48680	0.518454
$113$	2.21905	0.208750	0.104375	−	0.994538i	$-0.466716\pi$
$113$	2.21905	0.208750	0.104375	+	0.994538i	$0.466716\pi$
$114$	12.3289	1.15471
$115$	3.20269	0.298652
$116$	−2.83425	−0.263153
$117$	−27.1476	−2.50980
$118$	−0.0568726	−0.00523555
$119$	6.69426	0.613662
$120$	−9.39242	−0.857406
$121$	1.00000	0.0909091
$122$	−13.4456	−1.21731
$123$	8.77858	0.791538
$124$	−0.182909	−0.0164257
$125$	1.00000	0.0894427
$126$	17.3266	1.54358
$127$	−9.29170	−0.824505	−0.412252	−	0.911070i	$-0.635258\pi$
$127$	−9.29170	−0.824505	−0.412252	+	0.911070i	$0.635258\pi$
$128$	−3.30260	−0.291911
$129$	33.3298	2.93453
$130$	5.03304	0.441427
$131$	−14.7489	−1.28861	−0.644307	−	0.764767i	$-0.722854\pi$
$131$	−14.7489	−1.28861	−0.644307	+	0.764767i	$0.722854\pi$
$132$	−1.92812	−0.167821
$133$	−8.11219	−0.703416
$134$	−0.907987	−0.0784382
$135$	−10.0955	−0.868885
$136$	−8.77648	−0.752577
$137$	4.14169	0.353848	0.176924	−	0.984225i	$-0.443385\pi$
$137$	4.14169	0.353848	0.176924	+	0.984225i	$0.443385\pi$
$138$	11.4291	0.972913
$139$	−9.64480	−0.818061	−0.409031	−	0.912521i	$-0.634133\pi$
$139$	−9.64480	−0.818061	−0.409031	+	0.912521i	$0.634133\pi$
$140$	1.48387	0.125410
$141$	−3.97609	−0.334847
$142$	8.60192	0.721857
$143$	4.30309	0.359842
$144$	−14.7421	−1.22851
$145$	4.48490	0.372450
$146$	1.16964	0.0967997
$147$	4.53558	0.374088
$148$	−3.55718	−0.292398
$149$	−11.3292	−0.928123	−0.464062	−	0.885803i	$-0.653608\pi$
$149$	−11.3292	−0.928123	−0.464062	+	0.885803i	$0.653608\pi$
$150$	3.56861	0.291376
$151$	7.58864	0.617555	0.308778	−	0.951134i	$-0.400080\pi$
$151$	7.58864	0.617555	0.308778	+	0.951134i	$0.400080\pi$
$152$	10.6354	0.862648
$153$	−17.9864	−1.45411
$154$	−2.74639	−0.221310
$155$	0.289434	0.0232479
$156$	−8.29686	−0.664280
$157$	7.00980	0.559443	0.279722	−	0.960081i	$-0.409758\pi$
$157$	7.00980	0.559443	0.279722	+	0.960081i	$0.409758\pi$
$158$	7.48066	0.595130
$159$	5.33023	0.422715
$160$	−3.42373	−0.270670
$161$	−7.52014	−0.592670
$162$	−13.8898	−1.09128
$163$	−14.3303	−1.12244	−0.561219	−	0.827667i	$-0.689667\pi$
$163$	−14.3303	−1.12244	−0.561219	+	0.827667i	$0.689667\pi$
$164$	1.81828	0.141984
$165$	3.05105	0.237524
$166$	3.69116	0.286490
$167$	−25.2875	−1.95680	−0.978402	−	0.206709i	$-0.933725\pi$
$167$	−25.2875	−1.95680	−0.978402	+	0.206709i	$0.933725\pi$
$168$	22.0541	1.70151
$169$	5.51654	0.424349
$170$	3.33459	0.255751
$171$	21.7961	1.66679
$172$	6.90351	0.526387
$173$	16.4981	1.25433	0.627164	−	0.778887i	$-0.284216\pi$
$173$	16.4981	1.25433	0.627164	+	0.778887i	$0.284216\pi$
$174$	16.0048	1.21332
$175$	−2.34807	−0.177497
$176$	2.33673	0.176137
$177$	−0.148355	−0.0111510
$178$	7.22429	0.541483
$179$	−7.30034	−0.545653	−0.272827	−	0.962063i	$-0.587959\pi$
$179$	−7.30034	−0.545653	−0.272827	+	0.962063i	$0.587959\pi$
$180$	−3.98692	−0.297167
$181$	23.8023	1.76921	0.884606	−	0.466340i	$-0.154428\pi$
$181$	23.8023	1.76921	0.884606	+	0.466340i	$0.154428\pi$
$182$	−11.8179	−0.876003
$183$	−35.0735	−2.59271
$184$	9.85923	0.726832
$185$	5.62886	0.413842
$186$	1.03288	0.0757343
$187$	2.85096	0.208483
$188$	−0.823554	−0.0600639
$189$	23.7050	1.72429
$190$	−4.04089	−0.293157
$191$	7.35953	0.532517	0.266259	−	0.963902i	$-0.414212\pi$
$191$	7.35953	0.532517	0.266259	+	0.963902i	$0.414212\pi$
$192$	−26.4769	−1.91080
$193$	6.05148	0.435595	0.217798	−	0.975994i	$-0.430113\pi$
$193$	6.05148	0.435595	0.217798	+	0.975994i	$0.430113\pi$
$194$	4.76448	0.342070
$195$	13.1289	0.940181
$196$	0.939440	0.0671029
$197$	1.12547	0.0801867	0.0400933	−	0.999196i	$-0.487234\pi$
$197$	1.12547	0.0801867	0.0400933	+	0.999196i	$0.487234\pi$
$198$	7.37909	0.524409
$199$	−7.83864	−0.555666	−0.277833	−	0.960629i	$-0.589616\pi$
$199$	−7.83864	−0.555666	−0.277833	+	0.960629i	$0.589616\pi$
$200$	3.07843	0.217678
$201$	−2.36853	−0.167063
$202$	4.06305	0.285875
$203$	−10.5309	−0.739121
$204$	−5.49700	−0.384867
$205$	−2.87724	−0.200955
$206$	−3.25915	−0.227076
$207$	20.2054	1.40437
$208$	10.0551	0.697198
$209$	−3.45483	−0.238976
$210$	−8.37935	−0.578230
$211$	14.4188	0.992634	0.496317	−	0.868141i	$-0.334685\pi$
$211$	14.4188	0.992634	0.496317	+	0.868141i	$0.334685\pi$
$212$	1.10403	0.0758254
$213$	22.4385	1.53746
$214$	22.2795	1.52300
$215$	−10.9241	−0.745016
$216$	−31.0784	−2.11461
$217$	−0.679612	−0.0461351
$218$	7.98005	0.540477
$219$	3.05105	0.206171
$220$	0.631953	0.0426063
$221$	12.2679	0.825231
$222$	20.0872	1.34817
$223$	14.9795	1.00310	0.501552	−	0.865128i	$-0.332763\pi$
$223$	14.9795	1.00310	0.501552	+	0.865128i	$0.332763\pi$
$224$	8.03916	0.537139
$225$	6.30888	0.420592
$226$	−2.59548	−0.172648
$227$	9.51419	0.631479	0.315740	−	0.948846i	$-0.397747\pi$
$227$	9.51419	0.631479	0.315740	+	0.948846i	$0.397747\pi$
$228$	6.66132	0.441157
$229$	9.56834	0.632294	0.316147	−	0.948710i	$-0.397611\pi$
$229$	9.56834	0.632294	0.316147	+	0.948710i	$0.397611\pi$
$230$	−3.74598	−0.247002
$231$	−7.16407	−0.471361
$232$	13.8064	0.906436
$233$	−1.87739	−0.122992	−0.0614961	−	0.998107i	$-0.519587\pi$
$233$	−1.87739	−0.122992	−0.0614961	+	0.998107i	$0.519587\pi$
$234$	31.7528	2.07575
$235$	1.30319	0.0850106
$236$	−0.0307283	−0.00200024
$237$	19.5136	1.26755
$238$	−7.82984	−0.507533
$239$	29.5537	1.91167	0.955837	−	0.293899i	$-0.0949529\pi$
$239$	29.5537	1.91167	0.955837	+	0.293899i	$0.0949529\pi$
$240$	7.12946	0.460205
$241$	27.0501	1.74245	0.871226	−	0.490882i	$-0.163326\pi$
$241$	27.0501	1.74245	0.871226	+	0.490882i	$0.163326\pi$
$242$	−1.16964	−0.0751870
$243$	−5.94548	−0.381403
$244$	−7.26467	−0.465073
$245$	−1.48657	−0.0949732
$246$	−10.2677	−0.654647
$247$	−14.8664	−0.945928
$248$	0.891002	0.0565787
$249$	9.62856	0.610185
$250$	−1.16964	−0.0739742
$251$	8.68332	0.548086	0.274043	−	0.961717i	$-0.411639\pi$
$251$	8.68332	0.548086	0.274043	+	0.961717i	$0.411639\pi$
$252$	9.36156	0.589723
$253$	−3.20269	−0.201351
$254$	10.8679	0.681912
$255$	8.69842	0.544716
$256$	−13.4931	−0.843319
$257$	16.8488	1.05100	0.525498	−	0.850795i	$-0.323879\pi$
$257$	16.8488	1.05100	0.525498	+	0.850795i	$0.323879\pi$
$258$	−38.9838	−2.42702
$259$	−13.2170	−0.821262
$260$	2.71935	0.168647
$261$	28.2947	1.75140
$262$	17.2508	1.06576
$263$	−11.0782	−0.683109	−0.341555	−	0.939862i	$-0.610953\pi$
$263$	−11.0782	−0.683109	−0.341555	+	0.939862i	$0.610953\pi$
$264$	9.39242	0.578063
$265$	−1.74702	−0.107318
$266$	9.48830	0.581765
$267$	18.8449	1.15329
$268$	−0.490585	−0.0299673
$269$	−4.02418	−0.245359	−0.122679	−	0.992446i	$-0.539149\pi$
$269$	−4.02418	−0.245359	−0.122679	+	0.992446i	$0.539149\pi$
$270$	11.8081	0.718618
$271$	25.0889	1.52404	0.762022	−	0.647551i	$-0.224207\pi$
$271$	25.0889	1.52404	0.762022	+	0.647551i	$0.224207\pi$
$272$	6.66192	0.403939
$273$	−30.8276	−1.86577
$274$	−4.84426	−0.292653
$275$	−1.00000	−0.0603023
$276$	6.17516	0.371701
$277$	−16.6031	−0.997585	−0.498792	−	0.866721i	$-0.666223\pi$
$277$	−16.6031	−0.997585	−0.498792	+	0.866721i	$0.666223\pi$
$278$	11.2809	0.676583
$279$	1.82600	0.109320
$280$	−7.22836	−0.431977
$281$	5.47660	0.326707	0.163353	−	0.986568i	$-0.447769\pi$
$281$	5.47660	0.326707	0.163353	+	0.986568i	$0.447769\pi$
$282$	4.65057	0.276937
$283$	6.45321	0.383603	0.191802	−	0.981434i	$-0.438567\pi$
$283$	6.45321	0.383603	0.191802	+	0.981434i	$0.438567\pi$
$284$	4.64762	0.275785
$285$	−10.5408	−0.624386
$286$	−5.03304	−0.297610
$287$	6.75596	0.398791
$288$	−21.5999	−1.27279
$289$	−8.87201	−0.521883
$290$	−5.24569	−0.308038
$291$	12.4284	0.728564
$292$	0.631953	0.0369823
$293$	19.4342	1.13536	0.567678	−	0.823250i	$-0.307842\pi$
$293$	19.4342	1.13536	0.567678	+	0.823250i	$0.307842\pi$
$294$	−5.30497	−0.309392
$295$	0.0486243	0.00283101
$296$	17.3280	1.00717
$297$	10.0955	0.585802
$298$	13.2510	0.767611
$299$	−13.7814	−0.797001
$300$	1.92812	0.111320
$301$	25.6505	1.47847
$302$	−8.87594	−0.510753
$303$	10.5986	0.608876
$304$	−8.07300	−0.463018
$305$	11.4956	0.658235
$306$	21.0375	1.20263
$307$	−29.7058	−1.69540	−0.847701	−	0.530475i	$-0.822014\pi$
$307$	−29.7058	−1.69540	−0.847701	+	0.530475i	$0.822014\pi$
$308$	−1.48387	−0.0845514
$309$	−8.50165	−0.483642
$310$	−0.338532	−0.0192273
$311$	1.80175	0.102168	0.0510839	−	0.998694i	$-0.483732\pi$
$311$	1.80175	0.102168	0.0510839	+	0.998694i	$0.483732\pi$
$312$	40.4164	2.28813
$313$	14.6167	0.826183	0.413092	−	0.910689i	$-0.364449\pi$
$313$	14.6167	0.826183	0.413092	+	0.910689i	$0.364449\pi$
$314$	−8.19891	−0.462691
$315$	−14.8137	−0.834657
$316$	4.04180	0.227369
$317$	3.36561	0.189031	0.0945156	−	0.995523i	$-0.469870\pi$
$317$	3.36561	0.189031	0.0945156	+	0.995523i	$0.469870\pi$
$318$	−6.23443	−0.349609
$319$	−4.48490	−0.251106
$320$	8.67797	0.485113
$321$	58.1172	3.24378
$322$	8.79582	0.490172
$323$	−9.84959	−0.548046
$324$	−7.50464	−0.416924
$325$	−4.30309	−0.238692
$326$	16.7613	0.928320
$327$	20.8163	1.15115
$328$	−8.85736	−0.489066
$329$	−3.05998	−0.168702
$330$	−3.56861	−0.196446
$331$	−3.55214	−0.195243	−0.0976216	−	0.995224i	$-0.531123\pi$
$331$	−3.55214	−0.195243	−0.0976216	+	0.995224i	$0.531123\pi$
$332$	1.99433	0.109453
$333$	35.5118	1.94603
$334$	29.5771	1.61839
$335$	0.776300	0.0424138
$336$	−16.7405	−0.913268
$337$	−25.9584	−1.41405	−0.707023	−	0.707190i	$-0.749962\pi$
$337$	−25.9584	−1.41405	−0.707023	+	0.707190i	$0.749962\pi$
$338$	−6.45234	−0.350961
$339$	−6.77041	−0.367718
$340$	1.80168	0.0977096
$341$	−0.289434	−0.0156737
$342$	−25.4935	−1.37853
$343$	19.9271	1.07596
$344$	−33.6290	−1.81315
$345$	−9.77154	−0.526082
$346$	−19.2968	−1.03740
$347$	−11.2288	−0.602793	−0.301396	−	0.953499i	$-0.597453\pi$
$347$	−11.2288	−0.602793	−0.301396	+	0.953499i	$0.597453\pi$
$348$	8.64741	0.463550
$349$	−12.7538	−0.682696	−0.341348	−	0.939937i	$-0.610883\pi$
$349$	−12.7538	−0.682696	−0.341348	+	0.939937i	$0.610883\pi$
$350$	2.74639	0.146800
$351$	43.4419	2.31876
$352$	3.42373	0.182486
$353$	22.5006	1.19758	0.598792	−	0.800904i	$-0.295647\pi$
$353$	22.5006	1.19758	0.598792	+	0.800904i	$0.295647\pi$
$354$	0.173521	0.00922254
$355$	−7.35436	−0.390329
$356$	3.90328	0.206873
$357$	−20.4245	−1.08098
$358$	8.53874	0.451286
$359$	3.96180	0.209096	0.104548	−	0.994520i	$-0.466660\pi$
$359$	3.96180	0.209096	0.104548	+	0.994520i	$0.466660\pi$
$360$	19.4214	1.02360
$361$	−7.06415	−0.371797
$362$	−27.8400	−1.46324
$363$	−3.05105	−0.160138
$364$	−6.38522	−0.334677
$365$	−1.00000	−0.0523424
$366$	41.0232	2.14432
$367$	26.4212	1.37917	0.689587	−	0.724203i	$-0.257792\pi$
$367$	26.4212	1.37917	0.689587	+	0.724203i	$0.257792\pi$
$368$	−7.48381	−0.390121
$369$	−18.1521	−0.944962
$370$	−6.58371	−0.342271
$371$	4.10212	0.212972
$372$	0.558063	0.0289342
$373$	19.2583	0.997154	0.498577	−	0.866845i	$-0.333856\pi$
$373$	19.2583	0.997154	0.498577	+	0.866845i	$0.333856\pi$
$374$	−3.33459	−0.172427
$375$	−3.05105	−0.157555
$376$	4.01177	0.206891
$377$	−19.2989	−0.993944
$378$	−27.7262	−1.42608
$379$	−20.6265	−1.05951	−0.529756	−	0.848150i	$-0.677717\pi$
$379$	−20.6265	−1.05951	−0.529756	+	0.848150i	$0.677717\pi$
$380$	−2.18329	−0.112001
$381$	28.3494	1.45238
$382$	−8.60797	−0.440422
$383$	0.376655	0.0192462	0.00962308	−	0.999954i	$-0.496937\pi$
$383$	0.376655	0.0192462	0.00962308	+	0.999954i	$0.496937\pi$
$384$	10.0764	0.514208
$385$	2.34807	0.119669
$386$	−7.07802	−0.360262
$387$	−68.9187	−3.50333
$388$	2.57425	0.130688
$389$	20.3306	1.03080	0.515402	−	0.856948i	$-0.327643\pi$
$389$	20.3306	1.03080	0.515402	+	0.856948i	$0.327643\pi$
$390$	−15.3560	−0.777583
$391$	−9.13074	−0.461762
$392$	−4.57628	−0.231137
$393$	44.9994	2.26992
$394$	−1.31639	−0.0663190
$395$	−6.39572	−0.321804
$396$	3.98692	0.200350
$397$	1.58273	0.0794352	0.0397176	−	0.999211i	$-0.487354\pi$
$397$	1.58273	0.0794352	0.0397176	+	0.999211i	$0.487354\pi$
$398$	9.16835	0.459568
$399$	24.7506	1.23908
$400$	−2.33673	−0.116836
$401$	1.15158	0.0575072	0.0287536	−	0.999587i	$-0.490846\pi$
$401$	1.15158	0.0575072	0.0287536	+	0.999587i	$0.490846\pi$
$402$	2.77031	0.138171
$403$	−1.24546	−0.0620408
$404$	2.19526	0.109218
$405$	11.8753	0.590089
$406$	12.3173	0.611295
$407$	−5.62886	−0.279012
$408$	26.7774	1.32568
$409$	−30.5024	−1.50825	−0.754124	−	0.656732i	$-0.771938\pi$
$409$	−30.5024	−1.50825	−0.754124	+	0.656732i	$0.771938\pi$
$410$	3.36532	0.166201
$411$	−12.6365	−0.623311
$412$	−1.76092	−0.0867543
$413$	−0.114173	−0.00561809
$414$	−23.6329	−1.16149
$415$	−3.15582	−0.154913
$416$	14.7326	0.722326
$417$	29.4267	1.44103
$418$	4.04089	0.197646
$419$	−17.0975	−0.835267	−0.417633	−	0.908616i	$-0.637140\pi$
$419$	−17.0975	−0.835267	−0.417633	+	0.908616i	$0.637140\pi$
$420$	−4.52736	−0.220912
$421$	3.80850	0.185615	0.0928074	−	0.995684i	$-0.470416\pi$
$421$	3.80850	0.185615	0.0928074	+	0.995684i	$0.470416\pi$
$422$	−16.8648	−0.820965
$423$	8.22166	0.399751
$424$	−5.37807	−0.261182
$425$	−2.85096	−0.138292
$426$	−26.2449	−1.27157
$427$	−26.9924	−1.30626
$428$	12.0376	0.581861
$429$	−13.1289	−0.633870
$430$	12.7772	0.616170
$431$	−2.74726	−0.132331	−0.0661655	−	0.997809i	$-0.521077\pi$
$431$	−2.74726	−0.132331	−0.0661655	+	0.997809i	$0.521077\pi$
$432$	23.5905	1.13500
$433$	17.3562	0.834088	0.417044	−	0.908886i	$-0.363066\pi$
$433$	17.3562	0.834088	0.417044	+	0.908886i	$0.363066\pi$
$434$	0.794898	0.0381563
$435$	−13.6836	−0.656080
$436$	4.31162	0.206489
$437$	11.0647	0.529298
$438$	−3.56861	−0.170515
$439$	35.2657	1.68314	0.841570	−	0.540148i	$-0.181632\pi$
$439$	35.2657	1.68314	0.841570	+	0.540148i	$0.181632\pi$
$440$	−3.07843	−0.146758
$441$	−9.37856	−0.446598
$442$	−14.3490	−0.682513
$443$	17.7303	0.842391	0.421195	−	0.906970i	$-0.361611\pi$
$443$	17.7303	0.842391	0.421195	+	0.906970i	$0.361611\pi$
$444$	10.8531	0.515066
$445$	−6.17653	−0.292796
$446$	−17.5206	−0.829624
$447$	34.5659	1.63491
$448$	−20.3765	−0.962699
$449$	−33.9737	−1.60332	−0.801658	−	0.597783i	$-0.796049\pi$
$449$	−33.9737	−1.60332	−0.801658	+	0.597783i	$0.796049\pi$
$450$	−7.37909	−0.347853
$451$	2.87724	0.135484
$452$	−1.40233	−0.0659603
$453$	−23.1533	−1.08784
$454$	−11.1281	−0.522269
$455$	10.1039	0.473680
$456$	−32.4492	−1.51957
$457$	−20.9243	−0.978795	−0.489398	−	0.872061i	$-0.662783\pi$
$457$	−20.9243	−0.978795	−0.489398	+	0.872061i	$0.662783\pi$
$458$	−11.1915	−0.522943
$459$	28.7820	1.34343
$460$	−2.02395	−0.0943671
$461$	4.42571	0.206126	0.103063	−	0.994675i	$-0.467136\pi$
$461$	4.42571	0.206126	0.103063	+	0.994675i	$0.467136\pi$
$462$	8.37935	0.389843
$463$	−29.4136	−1.36697	−0.683484	−	0.729966i	$-0.739536\pi$
$463$	−29.4136	−1.36697	−0.683484	+	0.729966i	$0.739536\pi$
$464$	−10.4800	−0.486521
$465$	−0.883077	−0.0409517
$466$	2.19587	0.101722
$467$	−22.8233	−1.05613	−0.528067	−	0.849202i	$-0.677083\pi$
$467$	−22.8233	−1.05613	−0.528067	+	0.849202i	$0.677083\pi$
$468$	17.1560	0.793038
$469$	−1.82281	−0.0841694
$470$	−1.52426	−0.0703086
$471$	−21.3872	−0.985471
$472$	0.149686	0.00688986
$473$	10.9241	0.502290
$474$	−22.8238	−1.04833
$475$	3.45483	0.158518
$476$	−4.23046	−0.193903
$477$	−11.0217	−0.504650
$478$	−34.5671	−1.58106
$479$	6.19202	0.282921	0.141460	−	0.989944i	$-0.454820\pi$
$479$	6.19202	0.282921	0.141460	+	0.989944i	$0.454820\pi$
$480$	10.4460	0.476791
$481$	−24.2215	−1.10440
$482$	−31.6388	−1.44111
$483$	22.9443	1.04400
$484$	−0.631953	−0.0287252
$485$	−4.07348	−0.184967
$486$	6.95404	0.315442
$487$	18.6224	0.843862	0.421931	−	0.906628i	$-0.361352\pi$
$487$	18.6224	0.843862	0.421931	+	0.906628i	$0.361352\pi$
$488$	35.3883	1.60195
$489$	43.7225	1.97720
$490$	1.73874	0.0785482
$491$	21.5327	0.971755	0.485878	−	0.874027i	$-0.338500\pi$
$491$	21.5327	0.971755	0.485878	+	0.874027i	$0.338500\pi$
$492$	−5.54765	−0.250108
$493$	−12.7863	−0.575865
$494$	17.3883	0.782337
$495$	−6.30888	−0.283563
$496$	−0.676329	−0.0303681
$497$	17.2686	0.774601
$498$	−11.2619	−0.504658
$499$	−4.38270	−0.196197	−0.0980983	−	0.995177i	$-0.531276\pi$
$499$	−4.38270	−0.196197	−0.0980983	+	0.995177i	$0.531276\pi$
$500$	−0.631953	−0.0282618
$501$	77.1533	3.44695
$502$	−10.1563	−0.453298
$503$	−9.01710	−0.402053	−0.201026	−	0.979586i	$-0.564428\pi$
$503$	−9.01710	−0.402053	−0.201026	+	0.979586i	$0.564428\pi$
$504$	−45.6028	−2.03131
$505$	−3.47377	−0.154581
$506$	3.74598	0.166529
$507$	−16.8312	−0.747501
$508$	5.87192	0.260524
$509$	7.29333	0.323271	0.161635	−	0.986851i	$-0.448323\pi$
$509$	7.29333	0.323271	0.161635	+	0.986851i	$0.448323\pi$
$510$	−10.1740	−0.450511
$511$	2.34807	0.103873
$512$	22.3872	0.989384
$513$	−34.8784	−1.53992
$514$	−19.7069	−0.869234
$515$	2.78647	0.122787
$516$	−21.0629	−0.927243
$517$	−1.30319	−0.0573142
$518$	15.4590	0.679231
$519$	−50.3365	−2.20953
$520$	−13.2467	−0.580907
$521$	13.2454	0.580290	0.290145	−	0.956983i	$-0.406296\pi$
$521$	13.2454	0.580290	0.290145	+	0.956983i	$0.406296\pi$
$522$	−33.0944	−1.44850
$523$	−25.4068	−1.11096	−0.555480	−	0.831530i	$-0.687466\pi$
$523$	−25.4068	−1.11096	−0.555480	+	0.831530i	$0.687466\pi$
$524$	9.32059	0.407172
$525$	7.16407	0.312666
$526$	12.9574	0.564970
$527$	−0.825166	−0.0359448
$528$	−7.12946	−0.310270
$529$	−12.7428	−0.554035
$530$	2.04337	0.0887585
$531$	0.306764	0.0133124
$532$	5.12652	0.222263
$533$	12.3810	0.536280
$534$	−22.0416	−0.953835
$535$	−19.0483	−0.823529
$536$	2.38978	0.103223
$537$	22.2737	0.961180
$538$	4.70682	0.202926
$539$	1.48657	0.0640309
$540$	6.37991	0.274548
$541$	38.8665	1.67100	0.835502	−	0.549488i	$-0.185177\pi$
$541$	38.8665	1.67100	0.835502	+	0.549488i	$0.185177\pi$
$542$	−29.3449	−1.26047
$543$	−72.6219	−3.11650
$544$	9.76093	0.418497
$545$	−6.82269	−0.292252
$546$	36.0570	1.54310
$547$	25.6326	1.09597	0.547986	−	0.836487i	$-0.315395\pi$
$547$	25.6326	1.09597	0.547986	+	0.836487i	$0.315395\pi$
$548$	−2.61735	−0.111808
$549$	72.5242	3.09526
$550$	1.16964	0.0498734
$551$	15.4946	0.660090
$552$	−30.0810	−1.28033
$553$	15.0176	0.638614
$554$	19.4196	0.825060
$555$	−17.1739	−0.728992
$556$	6.09506	0.258488
$557$	1.22439	0.0518791	0.0259395	−	0.999664i	$-0.491742\pi$
$557$	1.22439	0.0518791	0.0259395	+	0.999664i	$0.491742\pi$
$558$	−2.13576	−0.0904139
$559$	47.0072	1.98819
$560$	5.48680	0.231860
$561$	−8.69842	−0.367247
$562$	−6.40562	−0.270205
$563$	32.3555	1.36362	0.681810	−	0.731529i	$-0.261193\pi$
$563$	32.3555	1.36362	0.681810	+	0.731529i	$0.261193\pi$
$564$	2.51270	0.105804
$565$	2.21905	0.0933560
$566$	−7.54790	−0.317262
$567$	−27.8840	−1.17102
$568$	−22.6399	−0.949948
$569$	11.0891	0.464881	0.232440	−	0.972611i	$-0.425329\pi$
$569$	11.0891	0.464881	0.232440	+	0.972611i	$0.425329\pi$
$570$	12.3289	0.516403
$571$	−13.3302	−0.557854	−0.278927	−	0.960312i	$-0.589979\pi$
$571$	−13.3302	−0.557854	−0.278927	+	0.960312i	$0.589979\pi$
$572$	−2.71935	−0.113702
$573$	−22.4543	−0.938041
$574$	−7.90200	−0.329823
$575$	3.20269	0.133561
$576$	54.7483	2.28118
$577$	4.87509	0.202953	0.101476	−	0.994838i	$-0.467643\pi$
$577$	4.87509	0.202953	0.101476	+	0.994838i	$0.467643\pi$
$578$	10.3770	0.431627
$579$	−18.4633	−0.767310
$580$	−2.83425	−0.117686
$581$	7.41009	0.307423
$582$	−14.5367	−0.602564
$583$	1.74702	0.0723541
$584$	−3.07843	−0.127386
$585$	−27.1476	−1.12242
$586$	−22.7309	−0.939005
$587$	−0.396308	−0.0163574	−0.00817868	−	0.999967i	$-0.502603\pi$
$587$	−0.396308	−0.0163574	−0.00817868	+	0.999967i	$0.502603\pi$
$588$	−2.86627	−0.118203
$589$	0.999946	0.0412021
$590$	−0.0568726	−0.00234141
$591$	−3.43387	−0.141251
$592$	−13.1531	−0.540590
$593$	38.3388	1.57438	0.787192	−	0.616708i	$-0.211534\pi$
$593$	38.3388	1.57438	0.787192	+	0.616708i	$0.211534\pi$
$594$	−11.8081	−0.484492
$595$	6.69426	0.274438
$596$	7.15952	0.293265
$597$	23.9160	0.978818
$598$	16.1193	0.659165
$599$	32.8162	1.34083	0.670417	−	0.741984i	$-0.266115\pi$
$599$	32.8162	1.34083	0.670417	+	0.741984i	$0.266115\pi$
$600$	−9.39242	−0.383444
$601$	21.6404	0.882729	0.441364	−	0.897328i	$-0.354495\pi$
$601$	21.6404	0.882729	0.441364	+	0.897328i	$0.354495\pi$
$602$	−30.0017	−1.22278
$603$	4.89758	0.199445
$604$	−4.79567	−0.195133
$605$	1.00000	0.0406558
$606$	−12.3965	−0.503575
$607$	−26.9943	−1.09567	−0.547833	−	0.836588i	$-0.684547\pi$
$607$	−26.9943	−1.09567	−0.547833	+	0.836588i	$0.684547\pi$
$608$	−11.8284	−0.479705
$609$	32.1301	1.30198
$610$	−13.4456	−0.544398
$611$	−5.60773	−0.226865
$612$	11.3666	0.459466
$613$	0.986909	0.0398609	0.0199304	−	0.999801i	$-0.493656\pi$
$613$	0.986909	0.0398609	0.0199304	+	0.999801i	$0.493656\pi$
$614$	34.7450	1.40219
$615$	8.77858	0.353987
$616$	7.22836	0.291239
$617$	25.3010	1.01858	0.509290	−	0.860595i	$-0.329908\pi$
$617$	25.3010	1.01858	0.509290	+	0.860595i	$0.329908\pi$
$618$	9.94383	0.399999
$619$	−7.83744	−0.315013	−0.157507	−	0.987518i	$-0.550346\pi$
$619$	−7.83744	−0.315013	−0.157507	+	0.987518i	$0.550346\pi$
$620$	−0.182909	−0.00734580
$621$	−32.3328	−1.29747
$622$	−2.10739	−0.0844986
$623$	14.5029	0.581048
$624$	−30.6787	−1.22813
$625$	1.00000	0.0400000
$626$	−17.0962	−0.683301
$627$	10.5408	0.420961
$628$	−4.42987	−0.176771
$629$	−16.0477	−0.639863
$630$	17.3266	0.690309
$631$	12.7556	0.507793	0.253896	−	0.967231i	$-0.418288\pi$
$631$	12.7556	0.507793	0.253896	+	0.967231i	$0.418288\pi$
$632$	−19.6888	−0.783177
$633$	−43.9925	−1.74855
$634$	−3.93653	−0.156340
$635$	−9.29170	−0.368730
$636$	−3.36846	−0.133568
$637$	6.39682	0.253451
$638$	5.24569	0.207679
$639$	−46.3978	−1.83547
$640$	−3.30260	−0.130547
$641$	−17.6704	−0.697939	−0.348969	−	0.937134i	$-0.613468\pi$
$641$	−17.6704	−0.697939	−0.348969	+	0.937134i	$0.613468\pi$
$642$	−67.9759	−2.68279
$643$	19.8401	0.782415	0.391208	−	0.920302i	$-0.372057\pi$
$643$	19.8401	0.782415	0.391208	+	0.920302i	$0.372057\pi$
$644$	4.75238	0.187270
$645$	33.3298	1.31236
$646$	11.5204	0.453265
$647$	22.5102	0.884966	0.442483	−	0.896777i	$-0.354098\pi$
$647$	22.5102	0.884966	0.442483	+	0.896777i	$0.354098\pi$
$648$	36.5572	1.43610
$649$	−0.0486243	−0.00190867
$650$	5.03304	0.197412
$651$	2.07353	0.0812679
$652$	9.05610	0.354664
$653$	−27.6422	−1.08172	−0.540862	−	0.841111i	$-0.681902\pi$
$653$	−27.6422	−1.08172	−0.540862	+	0.841111i	$0.681902\pi$
$654$	−24.3475	−0.952063
$655$	−14.7489	−0.576285
$656$	6.72332	0.262502
$657$	−6.30888	−0.246133
$658$	3.57906	0.139526
$659$	36.3392	1.41558	0.707788	−	0.706425i	$-0.249693\pi$
$659$	36.3392	1.41558	0.707788	+	0.706425i	$0.249693\pi$
$660$	−1.92812	−0.0750519
$661$	−45.1675	−1.75681	−0.878407	−	0.477914i	$-0.841393\pi$
$661$	−45.1675	−1.75681	−0.878407	+	0.477914i	$0.841393\pi$
$662$	4.15471	0.161477
$663$	−37.4300	−1.45366
$664$	−9.71497	−0.377014
$665$	−8.11219	−0.314577
$666$	−41.5358	−1.60948
$667$	14.3637	0.556166
$668$	15.9805	0.618305
$669$	−45.7032	−1.76699
$670$	−0.907987	−0.0350786
$671$	−11.4956	−0.443782
$672$	−24.5279	−0.946182
$673$	4.67596	0.180245	0.0901224	−	0.995931i	$-0.471274\pi$
$673$	4.67596	0.180245	0.0901224	+	0.995931i	$0.471274\pi$
$674$	30.3619	1.16950
$675$	−10.0955	−0.388577
$676$	−3.48620	−0.134085
$677$	4.08472	0.156989	0.0784944	−	0.996915i	$-0.474989\pi$
$677$	4.08472	0.156989	0.0784944	+	0.996915i	$0.474989\pi$
$678$	7.91891	0.304124
$679$	9.56482	0.367064
$680$	−8.77648	−0.336563
$681$	−29.0282	−1.11236
$682$	0.338532	0.0129631
$683$	17.1788	0.657329	0.328664	−	0.944447i	$-0.393402\pi$
$683$	17.1788	0.657329	0.328664	+	0.944447i	$0.393402\pi$
$684$	−13.7741	−0.526667
$685$	4.14169	0.158246
$686$	−23.3074	−0.889880
$687$	−29.1934	−1.11380
$688$	25.5266	0.973192
$689$	7.51757	0.286397
$690$	11.4291	0.435100
$691$	−0.197774	−0.00752369	−0.00376185	−	0.999993i	$-0.501197\pi$
$691$	−0.197774	−0.00752369	−0.00376185	+	0.999993i	$0.501197\pi$
$692$	−10.4260	−0.396338
$693$	14.8137	0.562725
$694$	13.1336	0.498544
$695$	−9.64480	−0.365848
$696$	−42.1240	−1.59671
$697$	8.20290	0.310707
$698$	14.9173	0.564628
$699$	5.72801	0.216653
$700$	1.48387	0.0560851
$701$	3.47807	0.131365	0.0656825	−	0.997841i	$-0.479078\pi$
$701$	3.47807	0.131365	0.0656825	+	0.997841i	$0.479078\pi$
$702$	−50.8112	−1.91775
$703$	19.4468	0.733449
$704$	−8.67797	−0.327063
$705$	−3.97609	−0.149748
$706$	−26.3175	−0.990471
$707$	8.15666	0.306763
$708$	0.0937533	0.00352347
$709$	−2.26025	−0.0848854	−0.0424427	−	0.999099i	$-0.513514\pi$
$709$	−2.26025	−0.0848854	−0.0424427	+	0.999099i	$0.513514\pi$
$710$	8.60192	0.322824
$711$	−40.3498	−1.51324
$712$	−19.0140	−0.712580
$713$	0.926967	0.0347152
$714$	23.8892	0.894031
$715$	4.30309	0.160926
$716$	4.61348	0.172414
$717$	−90.1698	−3.36745
$718$	−4.63386	−0.172934
$719$	−45.2318	−1.68686	−0.843431	−	0.537238i	$-0.819468\pi$
$719$	−45.2318	−1.68686	−0.843431	+	0.537238i	$0.819468\pi$
$720$	−14.7421	−0.549407
$721$	−6.54283	−0.243668
$722$	8.26247	0.307497
$723$	−82.5312	−3.06937
$724$	−15.0419	−0.559030
$725$	4.48490	0.166565
$726$	3.56861	0.132444
$727$	15.8677	0.588500	0.294250	−	0.955728i	$-0.404930\pi$
$727$	15.8677	0.588500	0.294250	+	0.955728i	$0.404930\pi$
$728$	31.1042	1.15280
$729$	−17.4860	−0.647629
$730$	1.16964	0.0432901
$731$	31.1441	1.15191
$732$	22.1648	0.819236
$733$	−27.8147	−1.02736	−0.513680	−	0.857982i	$-0.671718\pi$
$733$	−27.8147	−1.02736	−0.513680	+	0.857982i	$0.671718\pi$
$734$	−30.9031	−1.14066
$735$	4.53558	0.167297
$736$	−10.9651	−0.404181
$737$	−0.776300	−0.0285954
$738$	21.2314	0.781538
$739$	−48.2236	−1.77393	−0.886966	−	0.461834i	$-0.847191\pi$
$739$	−48.2236	−1.77393	−0.886966	+	0.461834i	$0.847191\pi$
$740$	−3.55718	−0.130764
$741$	45.3582	1.66627
$742$	−4.79799	−0.176140
$743$	−18.9169	−0.693993	−0.346996	−	0.937866i	$-0.612798\pi$
$743$	−18.9169	−0.693993	−0.346996	+	0.937866i	$0.612798\pi$
$744$	−2.71849	−0.0996645
$745$	−11.3292	−0.415069
$746$	−22.5251	−0.824704
$747$	−19.9097	−0.728458
$748$	−1.80168	−0.0658758
$749$	44.7267	1.63428
$750$	3.56861	0.130307
$751$	1.33664	0.0487748	0.0243874	−	0.999703i	$-0.492236\pi$
$751$	1.33664	0.0487748	0.0243874	+	0.999703i	$0.492236\pi$
$752$	−3.04520	−0.111047
$753$	−26.4932	−0.965466
$754$	22.5727	0.822048
$755$	7.58864	0.276179
$756$	−14.9805	−0.544835
$757$	0.0859254	0.00312301	0.00156151	−	0.999999i	$-0.499503\pi$
$757$	0.0859254	0.00312301	0.00156151	+	0.999999i	$0.499503\pi$
$758$	24.1255	0.876277
$759$	9.77154	0.354685
$760$	10.6354	0.385788
$761$	11.3153	0.410177	0.205089	−	0.978743i	$-0.434252\pi$
$761$	11.3153	0.410177	0.205089	+	0.978743i	$0.434252\pi$
$762$	−33.1584	−1.20120
$763$	16.0201	0.579968
$764$	−4.65088	−0.168263
$765$	−17.9864	−0.650299
$766$	−0.440549	−0.0159177
$767$	−0.209234	−0.00755501
$768$	41.1681	1.48553
$769$	39.5724	1.42702	0.713509	−	0.700646i	$-0.247105\pi$
$769$	39.5724	1.42702	0.713509	+	0.700646i	$0.247105\pi$
$770$	−2.74639	−0.0989729
$771$	−51.4063	−1.85135
$772$	−3.82425	−0.137638
$773$	17.8803	0.643109	0.321554	−	0.946891i	$-0.395795\pi$
$773$	17.8803	0.643109	0.321554	+	0.946891i	$0.395795\pi$
$774$	80.6097	2.89746
$775$	0.289434	0.0103968
$776$	−12.5399	−0.450156
$777$	40.3256	1.44667
$778$	−23.7794	−0.852534
$779$	−9.94037	−0.356151
$780$	−8.29686	−0.297075
$781$	7.35436	0.263160
$782$	10.6796	0.381903
$783$	−45.2774	−1.61808
$784$	3.47370	0.124061
$785$	7.00980	0.250191
$786$	−52.6329	−1.87735
$787$	20.6683	0.736744	0.368372	−	0.929679i	$-0.379915\pi$
$787$	20.6683	0.736744	0.368372	+	0.929679i	$0.379915\pi$
$788$	−0.711247	−0.0253371
$789$	33.8000	1.20331
$790$	7.48066	0.266150
$791$	−5.21048	−0.185263
$792$	−19.4214	−0.690110
$793$	−49.4664	−1.75661
$794$	−1.85122	−0.0656974
$795$	5.33023	0.189044
$796$	4.95365	0.175578
$797$	−22.2893	−0.789526	−0.394763	−	0.918783i	$-0.629173\pi$
$797$	−22.2893	−0.789526	−0.394763	+	0.918783i	$0.629173\pi$
$798$	−28.9492	−1.02479
$799$	−3.71534	−0.131439
$800$	−3.42373	−0.121047
$801$	−38.9670	−1.37683
$802$	−1.34693	−0.0475618
$803$	1.00000	0.0352892
$804$	1.49680	0.0527880
$805$	−7.52014	−0.265050
$806$	1.45673	0.0513113
$807$	12.2780	0.432204
$808$	−10.6938	−0.376205
$809$	39.6770	1.39497	0.697484	−	0.716601i	$-0.254303\pi$
$809$	39.6770	1.39497	0.697484	+	0.716601i	$0.254303\pi$
$810$	−13.8898	−0.488037
$811$	−43.3396	−1.52186	−0.760930	−	0.648834i	$-0.775257\pi$
$811$	−43.3396	−1.52186	−0.760930	+	0.648834i	$0.775257\pi$
$812$	6.65501	0.233545
$813$	−76.5475	−2.68464
$814$	6.58371	0.230759
$815$	−14.3303	−0.501970
$816$	−20.3258	−0.711547
$817$	−37.7408	−1.32038
$818$	35.6767	1.24741
$819$	63.7446	2.22742
$820$	1.81828	0.0634971
$821$	−7.25758	−0.253291	−0.126646	−	0.991948i	$-0.540421\pi$
$821$	−7.25758	−0.253291	−0.126646	+	0.991948i	$0.540421\pi$
$822$	14.7801	0.515514
$823$	1.42485	0.0496673	0.0248336	−	0.999692i	$-0.492094\pi$
$823$	1.42485	0.0496673	0.0248336	+	0.999692i	$0.492094\pi$
$824$	8.57794	0.298827
$825$	3.05105	0.106224
$826$	0.133541	0.00464648
$827$	−13.6159	−0.473470	−0.236735	−	0.971574i	$-0.576077\pi$
$827$	−13.6159	−0.473470	−0.236735	+	0.971574i	$0.576077\pi$
$828$	−12.7688	−0.443748
$829$	−1.44135	−0.0500601	−0.0250300	−	0.999687i	$-0.507968\pi$
$829$	−1.44135	−0.0500601	−0.0250300	+	0.999687i	$0.507968\pi$
$830$	3.69116	0.128122
$831$	50.6569	1.75727
$832$	−37.3421	−1.29460
$833$	4.23814	0.146843
$834$	−34.4185	−1.19182
$835$	−25.2875	−0.875110
$836$	2.18329	0.0755107
$837$	−2.92199	−0.100999
$838$	19.9978	0.690813
$839$	23.2092	0.801270	0.400635	−	0.916238i	$-0.368789\pi$
$839$	23.2092	0.801270	0.400635	+	0.916238i	$0.368789\pi$
$840$	22.0541	0.760937
$841$	−8.88570	−0.306403
$842$	−4.45455	−0.153514
$843$	−16.7094	−0.575501
$844$	−9.11204	−0.313649
$845$	5.51654	0.189775
$846$	−9.61634	−0.330617
$847$	−2.34807	−0.0806807
$848$	4.08231	0.140187
$849$	−19.6890	−0.675726
$850$	3.33459	0.114375
$851$	18.0275	0.617974
$852$	−14.1801	−0.485802
$853$	51.6136	1.76722	0.883608	−	0.468228i	$-0.155108\pi$
$853$	51.6136	1.76722	0.883608	+	0.468228i	$0.155108\pi$
$854$	31.5713	1.08035
$855$	21.7961	0.745411
$856$	−58.6387	−2.00423
$857$	35.8298	1.22392	0.611962	−	0.790887i	$-0.290381\pi$
$857$	35.8298	1.22392	0.611962	+	0.790887i	$0.290381\pi$
$858$	15.3560	0.524246
$859$	4.92760	0.168128	0.0840639	−	0.996460i	$-0.473210\pi$
$859$	4.92760	0.168128	0.0840639	+	0.996460i	$0.473210\pi$
$860$	6.90351	0.235408
$861$	−20.6127	−0.702480
$862$	3.21330	0.109445
$863$	12.9625	0.441249	0.220625	−	0.975359i	$-0.429190\pi$
$863$	12.9625	0.441249	0.220625	+	0.975359i	$0.429190\pi$
$864$	34.5644	1.17590
$865$	16.4981	0.560953
$866$	−20.3005	−0.689838
$867$	27.0689	0.919308
$868$	0.429483	0.0145776
$869$	6.39572	0.216960
$870$	16.0048	0.542615
$871$	−3.34048	−0.113188
$872$	−21.0031	−0.711256
$873$	−25.6991	−0.869782
$874$	−12.9417	−0.437760
$875$	−2.34807	−0.0793793
$876$	−1.92812	−0.0651451
$877$	15.0028	0.506608	0.253304	−	0.967387i	$-0.418483\pi$
$877$	15.0028	0.506608	0.253304	+	0.967387i	$0.418483\pi$
$878$	−41.2480	−1.39205
$879$	−59.2946	−1.99996
$880$	2.33673	0.0787711
$881$	2.22320	0.0749014	0.0374507	−	0.999298i	$-0.488076\pi$
$881$	2.22320	0.0749014	0.0374507	+	0.999298i	$0.488076\pi$
$882$	10.9695	0.369362
$883$	8.34458	0.280817	0.140409	−	0.990094i	$-0.455158\pi$
$883$	8.34458	0.280817	0.140409	+	0.990094i	$0.455158\pi$
$884$	−7.75277	−0.260754
$885$	−0.148355	−0.00498689
$886$	−20.7380	−0.696705
$887$	49.5005	1.66206	0.831032	−	0.556224i	$-0.187750\pi$
$887$	49.5005	1.66206	0.831032	+	0.556224i	$0.187750\pi$
$888$	−52.8686	−1.77415
$889$	21.8176	0.731737
$890$	7.22429	0.242159
$891$	−11.8753	−0.397838
$892$	−9.46637	−0.316957
$893$	4.50230	0.150664
$894$	−40.4295	−1.35216
$895$	−7.30034	−0.244024
$896$	7.75474	0.259068
$897$	42.0478	1.40393
$898$	39.7368	1.32603
$899$	1.29808	0.0432935
$900$	−3.98692	−0.132897
$901$	4.98068	0.165931
$902$	−3.36532	−0.112053
$903$	−78.2608	−2.60436
$904$	6.83117	0.227201
$905$	23.8023	0.791215
$906$	27.0809	0.899703
$907$	−55.9667	−1.85834	−0.929172	−	0.369647i	$-0.879479\pi$
$907$	−55.9667	−1.85834	−0.929172	+	0.369647i	$0.879479\pi$
$908$	−6.01253	−0.199533
$909$	−21.9156	−0.726895
$910$	−11.8179	−0.391761
$911$	13.5284	0.448217	0.224109	−	0.974564i	$-0.428053\pi$
$911$	13.5284	0.448217	0.224109	+	0.974564i	$0.428053\pi$
$912$	24.6311	0.815617
$913$	3.15582	0.104442
$914$	24.4737	0.809519
$915$	−35.0735	−1.15950
$916$	−6.04674	−0.199790
$917$	34.6313	1.14363
$918$	−33.6644	−1.11109
$919$	−43.7232	−1.44230	−0.721148	−	0.692781i	$-0.756385\pi$
$919$	−43.7232	−1.44230	−0.721148	+	0.692781i	$0.756385\pi$
$920$	9.85923	0.325049
$921$	90.6338	2.98649
$922$	−5.17647	−0.170478
$923$	31.6465	1.04166
$924$	4.52736	0.148939
$925$	5.62886	0.185076
$926$	34.4032	1.13056
$927$	17.5795	0.577387
$928$	−15.3551	−0.504055
$929$	14.1090	0.462900	0.231450	−	0.972847i	$-0.425653\pi$
$929$	14.1090	0.462900	0.231450	+	0.972847i	$0.425653\pi$
$930$	1.03288	0.0338694
$931$	−5.13583	−0.168320
$932$	1.18643	0.0388627
$933$	−5.49722	−0.179971
$934$	26.6949	0.873484
$935$	2.85096	0.0932365
$936$	−83.5720	−2.73164
$937$	50.3739	1.64564	0.822822	−	0.568299i	$-0.192398\pi$
$937$	50.3739	1.64564	0.822822	+	0.568299i	$0.192398\pi$
$938$	2.13202	0.0696129
$939$	−44.5961	−1.45534
$940$	−0.823554	−0.0268614
$941$	−2.34700	−0.0765100	−0.0382550	−	0.999268i	$-0.512180\pi$
$941$	−2.34700	−0.0765100	−0.0382550	+	0.999268i	$0.512180\pi$
$942$	25.0152	0.815041
$943$	−9.21489	−0.300078
$944$	−0.113622	−0.00369807
$945$	23.7050	0.771125
$946$	−12.7772	−0.415422
$947$	9.10010	0.295714	0.147857	−	0.989009i	$-0.452763\pi$
$947$	9.10010	0.295714	0.147857	+	0.989009i	$0.452763\pi$
$948$	−12.3317	−0.400515
$949$	4.30309	0.139684
$950$	−4.04089	−0.131104
$951$	−10.2686	−0.332983
$952$	20.6078	0.667902
$953$	9.32550	0.302083	0.151041	−	0.988527i	$-0.451737\pi$
$953$	9.32550	0.302083	0.151041	+	0.988527i	$0.451737\pi$
$954$	12.8914	0.417374
$955$	7.35953	0.238149
$956$	−18.6766	−0.604044
$957$	13.6836	0.442329
$958$	−7.24240	−0.233991
$959$	−9.72497	−0.314036
$960$	−26.4769	−0.854538
$961$	−30.9162	−0.997298
$962$	28.3303	0.913405
$963$	−120.173	−3.87253
$964$	−17.0944	−0.550574
$965$	6.05148	0.194804
$966$	−26.8364	−0.863448
$967$	−18.3048	−0.588642	−0.294321	−	0.955707i	$-0.595093\pi$
$967$	−18.3048	−0.588642	−0.294321	+	0.955707i	$0.595093\pi$
$968$	3.07843	0.0989443
$969$	30.0516	0.965395
$970$	4.76448	0.152978
$971$	−48.9550	−1.57104	−0.785520	−	0.618836i	$-0.787604\pi$
$971$	−48.9550	−1.57104	−0.785520	+	0.618836i	$0.787604\pi$
$972$	3.75727	0.120514
$973$	22.6467	0.726019
$974$	−21.7814	−0.697922
$975$	13.1289	0.420462
$976$	−26.8620	−0.859833
$977$	14.0971	0.451007	0.225503	−	0.974242i	$-0.427597\pi$
$977$	14.0971	0.451007	0.225503	+	0.974242i	$0.427597\pi$
$978$	−51.1394	−1.63526
$979$	6.17653	0.197403
$980$	0.939440	0.0300093
$981$	−43.0435	−1.37427
$982$	−25.1854	−0.803697
$983$	53.1154	1.69412	0.847060	−	0.531498i	$-0.178371\pi$
$983$	53.1154	1.69412	0.847060	+	0.531498i	$0.178371\pi$
$984$	27.0242	0.861500
$985$	1.12547	0.0358606
$986$	14.9553	0.476273
$987$	9.33613	0.297172
$988$	9.39489	0.298891
$989$	−34.9864	−1.11250
$990$	7.37909	0.234523
$991$	59.8398	1.90087	0.950437	−	0.310916i	$-0.100636\pi$
$991$	59.8398	1.90087	0.950437	+	0.310916i	$0.100636\pi$
$992$	−0.990945	−0.0314625
$993$	10.8377	0.343925
$994$	−20.1979	−0.640639
$995$	−7.83864	−0.248502
$996$	−6.08480	−0.192804
$997$	−6.41848	−0.203275	−0.101638	−	0.994821i	$-0.532408\pi$
$997$	−6.41848	−0.203275	−0.101638	+	0.994821i	$0.532408\pi$
$998$	5.12616	0.162266
$999$	−56.8264	−1.79791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.i.1.13	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.i.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-1.16964 q^{2} -3.05105 q^{3} -0.631953 q^{4} +1.00000 q^{5} +3.56861 q^{6} -2.34807 q^{7} +3.07843 q^{8} +6.30888 q^{9} +O(q^{10})\)	\(q-1.16964 q^{2} -3.05105 q^{3} -0.631953 q^{4} +1.00000 q^{5} +3.56861 q^{6} -2.34807 q^{7} +3.07843 q^{8} +6.30888 q^{9} -1.16964 q^{10} -1.00000 q^{11} +1.92812 q^{12} -4.30309 q^{13} +2.74639 q^{14} -3.05105 q^{15} -2.33673 q^{16} -2.85096 q^{17} -7.37909 q^{18} +3.45483 q^{19} -0.631953 q^{20} +7.16407 q^{21} +1.16964 q^{22} +3.20269 q^{23} -9.39242 q^{24} +1.00000 q^{25} +5.03304 q^{26} -10.0955 q^{27} +1.48387 q^{28} +4.48490 q^{29} +3.56861 q^{30} +0.289434 q^{31} -3.42373 q^{32} +3.05105 q^{33} +3.33459 q^{34} -2.34807 q^{35} -3.98692 q^{36} +5.62886 q^{37} -4.04089 q^{38} +13.1289 q^{39} +3.07843 q^{40} -2.87724 q^{41} -8.37935 q^{42} -10.9241 q^{43} +0.631953 q^{44} +6.30888 q^{45} -3.74598 q^{46} +1.30319 q^{47} +7.12946 q^{48} -1.48657 q^{49} -1.16964 q^{50} +8.69842 q^{51} +2.71935 q^{52} -1.74702 q^{53} +11.8081 q^{54} -1.00000 q^{55} -7.22836 q^{56} -10.5408 q^{57} -5.24569 q^{58} +0.0486243 q^{59} +1.92812 q^{60} +11.4956 q^{61} -0.338532 q^{62} -14.8137 q^{63} +8.67797 q^{64} -4.30309 q^{65} -3.56861 q^{66} +0.776300 q^{67} +1.80168 q^{68} -9.77154 q^{69} +2.74639 q^{70} -7.35436 q^{71} +19.4214 q^{72} -1.00000 q^{73} -6.58371 q^{74} -3.05105 q^{75} -2.18329 q^{76} +2.34807 q^{77} -15.3560 q^{78} -6.39572 q^{79} -2.33673 q^{80} +11.8753 q^{81} +3.36532 q^{82} -3.15582 q^{83} -4.52736 q^{84} -2.85096 q^{85} +12.7772 q^{86} -13.6836 q^{87} -3.07843 q^{88} -6.17653 q^{89} -7.37909 q^{90} +10.1039 q^{91} -2.02395 q^{92} -0.883077 q^{93} -1.52426 q^{94} +3.45483 q^{95} +10.4460 q^{96} -4.07348 q^{97} +1.73874 q^{98} -6.30888 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9}+O(q^{10}) \) 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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