LMFDB - Embedded newform 6002.2.a.a.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6002,2,Mod(1,6002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6002 = 2 \cdot 3001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6002.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:	$47.9262112932$
Analytic rank:	$1$
Dimension:	$47$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.49159 q^{3} +1.00000 q^{4} +1.25928 q^{5} -1.49159 q^{6} -0.926965 q^{7} +1.00000 q^{8} -0.775168 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.49159 q^{3} +1.00000 q^{4} +1.25928 q^{5} -1.49159 q^{6} -0.926965 q^{7} +1.00000 q^{8} -0.775168 q^{9} +1.25928 q^{10} -3.21428 q^{11} -1.49159 q^{12} +0.860701 q^{13} -0.926965 q^{14} -1.87833 q^{15} +1.00000 q^{16} +5.37802 q^{17} -0.775168 q^{18} -0.0479057 q^{19} +1.25928 q^{20} +1.38265 q^{21} -3.21428 q^{22} -0.825262 q^{23} -1.49159 q^{24} -3.41421 q^{25} +0.860701 q^{26} +5.63099 q^{27} -0.926965 q^{28} -8.73682 q^{29} -1.87833 q^{30} +4.59166 q^{31} +1.00000 q^{32} +4.79438 q^{33} +5.37802 q^{34} -1.16731 q^{35} -0.775168 q^{36} +4.81096 q^{37} -0.0479057 q^{38} -1.28381 q^{39} +1.25928 q^{40} -0.558394 q^{41} +1.38265 q^{42} +3.23505 q^{43} -3.21428 q^{44} -0.976153 q^{45} -0.825262 q^{46} -8.74211 q^{47} -1.49159 q^{48} -6.14074 q^{49} -3.41421 q^{50} -8.02178 q^{51} +0.860701 q^{52} +5.71575 q^{53} +5.63099 q^{54} -4.04768 q^{55} -0.926965 q^{56} +0.0714556 q^{57} -8.73682 q^{58} +1.04345 q^{59} -1.87833 q^{60} -12.1728 q^{61} +4.59166 q^{62} +0.718553 q^{63} +1.00000 q^{64} +1.08386 q^{65} +4.79438 q^{66} -1.36388 q^{67} +5.37802 q^{68} +1.23095 q^{69} -1.16731 q^{70} -0.724563 q^{71} -0.775168 q^{72} -0.635731 q^{73} +4.81096 q^{74} +5.09260 q^{75} -0.0479057 q^{76} +2.97952 q^{77} -1.28381 q^{78} +13.6754 q^{79} +1.25928 q^{80} -6.07361 q^{81} -0.558394 q^{82} +13.8105 q^{83} +1.38265 q^{84} +6.77243 q^{85} +3.23505 q^{86} +13.0317 q^{87} -3.21428 q^{88} -9.45802 q^{89} -0.976153 q^{90} -0.797840 q^{91} -0.825262 q^{92} -6.84886 q^{93} -8.74211 q^{94} -0.0603267 q^{95} -1.49159 q^{96} -17.7590 q^{97} -6.14074 q^{98} +2.49160 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 47 q + 47 q^{2} - 13 q^{3} + 47 q^{4} - 14 q^{5} - 13 q^{6} - 17 q^{7} + 47 q^{8} + 12 q^{9}+O(q^{10}) $ 47 * q + 47 * q^2 - 13 * q^3 + 47 * q^4 - 14 * q^5 - 13 * q^6 - 17 * q^7 + 47 * q^8 + 12 * q^9	\( 47 q + 47 q^{2} - 13 q^{3} + 47 q^{4} - 14 q^{5} - 13 q^{6} - 17 q^{7} + 47 q^{8} + 12 q^{9} - 14 q^{10} - 30 q^{11} - 13 q^{12} - 39 q^{13} - 17 q^{14} - 18 q^{15} + 47 q^{16} - 26 q^{17} + 12 q^{18} - 23 q^{19} - 14 q^{20} - 39 q^{21} - 30 q^{22} - 25 q^{23} - 13 q^{24} - 19 q^{25} - 39 q^{26} - 46 q^{27} - 17 q^{28} - 53 q^{29} - 18 q^{30} - 23 q^{31} + 47 q^{32} - 26 q^{33} - 26 q^{34} - 31 q^{35} + 12 q^{36} - 83 q^{37} - 23 q^{38} - 9 q^{39} - 14 q^{40} - 48 q^{41} - 39 q^{42} - 78 q^{43} - 30 q^{44} - 27 q^{45} - 25 q^{46} - 15 q^{47} - 13 q^{48} - 12 q^{49} - 19 q^{50} - 47 q^{51} - 39 q^{52} - 76 q^{53} - 46 q^{54} - 39 q^{55} - 17 q^{56} - 44 q^{57} - 53 q^{58} - 33 q^{59} - 18 q^{60} - 33 q^{61} - 23 q^{62} - 7 q^{63} + 47 q^{64} - 67 q^{65} - 26 q^{66} - 85 q^{67} - 26 q^{68} - 33 q^{69} - 31 q^{70} - 17 q^{71} + 12 q^{72} - 59 q^{73} - 83 q^{74} - 21 q^{75} - 23 q^{76} - 59 q^{77} - 9 q^{78} - 49 q^{79} - 14 q^{80} - 41 q^{81} - 48 q^{82} - 30 q^{83} - 39 q^{84} - 84 q^{85} - 78 q^{86} + 9 q^{87} - 30 q^{88} - 50 q^{89} - 27 q^{90} - 42 q^{91} - 25 q^{92} - 43 q^{93} - 15 q^{94} + 8 q^{95} - 13 q^{96} - 49 q^{97} - 12 q^{98} - 49 q^{99}+O(q^{100}) \) 47 * q + 47 * q^2 - 13 * q^3 + 47 * q^4 - 14 * q^5 - 13 * q^6 - 17 * q^7 + 47 * q^8 + 12 * q^9 - 14 * q^10 - 30 * q^11 - 13 * q^12 - 39 * q^13 - 17 * q^14 - 18 * q^15 + 47 * q^16 - 26 * q^17 + 12 * q^18 - 23 * q^19 - 14 * q^20 - 39 * q^21 - 30 * q^22 - 25 * q^23 - 13 * q^24 - 19 * q^25 - 39 * q^26 - 46 * q^27 - 17 * q^28 - 53 * q^29 - 18 * q^30 - 23 * q^31 + 47 * q^32 - 26 * q^33 - 26 * q^34 - 31 * q^35 + 12 * q^36 - 83 * q^37 - 23 * q^38 - 9 * q^39 - 14 * q^40 - 48 * q^41 - 39 * q^42 - 78 * q^43 - 30 * q^44 - 27 * q^45 - 25 * q^46 - 15 * q^47 - 13 * q^48 - 12 * q^49 - 19 * q^50 - 47 * q^51 - 39 * q^52 - 76 * q^53 - 46 * q^54 - 39 * q^55 - 17 * q^56 - 44 * q^57 - 53 * q^58 - 33 * q^59 - 18 * q^60 - 33 * q^61 - 23 * q^62 - 7 * q^63 + 47 * q^64 - 67 * q^65 - 26 * q^66 - 85 * q^67 - 26 * q^68 - 33 * q^69 - 31 * q^70 - 17 * q^71 + 12 * q^72 - 59 * q^73 - 83 * q^74 - 21 * q^75 - 23 * q^76 - 59 * q^77 - 9 * q^78 - 49 * q^79 - 14 * q^80 - 41 * q^81 - 48 * q^82 - 30 * q^83 - 39 * q^84 - 84 * q^85 - 78 * q^86 + 9 * q^87 - 30 * q^88 - 50 * q^89 - 27 * q^90 - 42 * q^91 - 25 * q^92 - 43 * q^93 - 15 * q^94 + 8 * q^95 - 13 * q^96 - 49 * q^97 - 12 * q^98 - 49 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.49159	−0.861168	−0.430584	−	0.902550i	$-0.641692\pi$
$3$	−1.49159	−0.861168	−0.430584	+	0.902550i	$0.641692\pi$
$4$	1.00000	0.500000
$5$	1.25928	0.563167	0.281584	−	0.959537i	$-0.409140\pi$
$5$	1.25928	0.563167	0.281584	+	0.959537i	$0.409140\pi$
$6$	−1.49159	−0.608938
$7$	−0.926965	−0.350360	−0.175180	−	0.984536i	$-0.556051\pi$
$7$	−0.926965	−0.350360	−0.175180	+	0.984536i	$0.556051\pi$
$8$	1.00000	0.353553
$9$	−0.775168	−0.258389
$10$	1.25928	0.398219
$11$	−3.21428	−0.969141	−0.484571	−	0.874752i	$-0.661024\pi$
$11$	−3.21428	−0.969141	−0.484571	+	0.874752i	$0.661024\pi$
$12$	−1.49159	−0.430584
$13$	0.860701	0.238716	0.119358	−	0.992851i	$-0.461916\pi$
$13$	0.860701	0.238716	0.119358	+	0.992851i	$0.461916\pi$
$14$	−0.926965	−0.247742
$15$	−1.87833	−0.484982
$16$	1.00000	0.250000
$17$	5.37802	1.30436	0.652181	−	0.758064i	$-0.273854\pi$
$17$	5.37802	1.30436	0.652181	+	0.758064i	$0.273854\pi$
$18$	−0.775168	−0.182709
$19$	−0.0479057	−0.0109903	−0.00549517	−	0.999985i	$-0.501749\pi$
$19$	−0.0479057	−0.0109903	−0.00549517	+	0.999985i	$0.501749\pi$
$20$	1.25928	0.281584
$21$	1.38265	0.301719
$22$	−3.21428	−0.685286
$23$	−0.825262	−0.172079	−0.0860395	−	0.996292i	$-0.527421\pi$
$23$	−0.825262	−0.172079	−0.0860395	+	0.996292i	$0.527421\pi$
$24$	−1.49159	−0.304469
$25$	−3.41421	−0.682843
$26$	0.860701	0.168797
$27$	5.63099	1.08368
$28$	−0.926965	−0.175180
$29$	−8.73682	−1.62239	−0.811193	−	0.584778i	$-0.801181\pi$
$29$	−8.73682	−1.62239	−0.811193	+	0.584778i	$0.801181\pi$
$30$	−1.87833	−0.342934
$31$	4.59166	0.824686	0.412343	−	0.911029i	$-0.364711\pi$
$31$	4.59166	0.824686	0.412343	+	0.911029i	$0.364711\pi$
$32$	1.00000	0.176777
$33$	4.79438	0.834594
$34$	5.37802	0.922323
$35$	−1.16731	−0.197311
$36$	−0.775168	−0.129195
$37$	4.81096	0.790917	0.395459	−	0.918484i	$-0.370586\pi$
$37$	4.81096	0.790917	0.395459	+	0.918484i	$0.370586\pi$
$38$	−0.0479057	−0.00777134
$39$	−1.28381	−0.205574
$40$	1.25928	0.199110
$41$	−0.558394	−0.0872066	−0.0436033	−	0.999049i	$-0.513884\pi$
$41$	−0.558394	−0.0872066	−0.0436033	+	0.999049i	$0.513884\pi$
$42$	1.38265	0.213347
$43$	3.23505	0.493340	0.246670	−	0.969100i	$-0.420664\pi$
$43$	3.23505	0.493340	0.246670	+	0.969100i	$0.420664\pi$
$44$	−3.21428	−0.484571
$45$	−0.976153	−0.145516
$46$	−0.825262	−0.121678
$47$	−8.74211	−1.27517	−0.637584	−	0.770381i	$-0.720066\pi$
$47$	−8.74211	−1.27517	−0.637584	+	0.770381i	$0.720066\pi$
$48$	−1.49159	−0.215292
$49$	−6.14074	−0.877248
$50$	−3.41421	−0.482843
$51$	−8.02178	−1.12327
$52$	0.860701	0.119358
$53$	5.71575	0.785118	0.392559	−	0.919727i	$-0.371590\pi$
$53$	5.71575	0.785118	0.392559	+	0.919727i	$0.371590\pi$
$54$	5.63099	0.766281
$55$	−4.04768	−0.545789
$56$	−0.926965	−0.123871
$57$	0.0714556	0.00946452
$58$	−8.73682	−1.14720
$59$	1.04345	0.135846	0.0679229	−	0.997691i	$-0.478363\pi$
$59$	1.04345	0.135846	0.0679229	+	0.997691i	$0.478363\pi$
$60$	−1.87833	−0.242491
$61$	−12.1728	−1.55856	−0.779281	−	0.626675i	$-0.784415\pi$
$61$	−12.1728	−1.55856	−0.779281	+	0.626675i	$0.784415\pi$
$62$	4.59166	0.583141
$63$	0.718553	0.0905292
$64$	1.00000	0.125000
$65$	1.08386	0.134437
$66$	4.79438	0.590147
$67$	−1.36388	−0.166624	−0.0833121	−	0.996524i	$-0.526550\pi$
$67$	−1.36388	−0.166624	−0.0833121	+	0.996524i	$0.526550\pi$
$68$	5.37802	0.652181
$69$	1.23095	0.148189
$70$	−1.16731	−0.139520
$71$	−0.724563	−0.0859898	−0.0429949	−	0.999075i	$-0.513690\pi$
$71$	−0.724563	−0.0859898	−0.0429949	+	0.999075i	$0.513690\pi$
$72$	−0.775168	−0.0913544
$73$	−0.635731	−0.0744066	−0.0372033	−	0.999308i	$-0.511845\pi$
$73$	−0.635731	−0.0744066	−0.0372033	+	0.999308i	$0.511845\pi$
$74$	4.81096	0.559263
$75$	5.09260	0.588042
$76$	−0.0479057	−0.00549517
$77$	2.97952	0.339548
$78$	−1.28381	−0.145363
$79$	13.6754	1.53860	0.769299	−	0.638889i	$-0.220606\pi$
$79$	13.6754	1.53860	0.769299	+	0.638889i	$0.220606\pi$
$80$	1.25928	0.140792
$81$	−6.07361	−0.674846
$82$	−0.558394	−0.0616644
$83$	13.8105	1.51590	0.757952	−	0.652310i	$-0.226200\pi$
$83$	13.8105	1.51590	0.757952	+	0.652310i	$0.226200\pi$
$84$	1.38265	0.150859
$85$	6.77243	0.734573
$86$	3.23505	0.348844
$87$	13.0317	1.39715
$88$	−3.21428	−0.342643
$89$	−9.45802	−1.00255	−0.501274	−	0.865289i	$-0.667135\pi$
$89$	−9.45802	−1.00255	−0.501274	+	0.865289i	$0.667135\pi$
$90$	−0.976153	−0.102896
$91$	−0.797840	−0.0836363
$92$	−0.825262	−0.0860395
$93$	−6.84886	−0.710193
$94$	−8.74211	−0.901680
$95$	−0.0603267	−0.00618939
$96$	−1.49159	−0.152234
$97$	−17.7590	−1.80316	−0.901578	−	0.432617i	$-0.857590\pi$
$97$	−17.7590	−1.80316	−0.901578	+	0.432617i	$0.857590\pi$
$98$	−6.14074	−0.620308
$99$	2.49160	0.250416
$100$	−3.41421	−0.341421
$101$	−13.9475	−1.38783	−0.693913	−	0.720059i	$-0.744115\pi$
$101$	−13.9475	−1.38783	−0.693913	+	0.720059i	$0.744115\pi$
$102$	−8.02178	−0.794275
$103$	−14.6167	−1.44023	−0.720113	−	0.693857i	$-0.755910\pi$
$103$	−14.6167	−1.44023	−0.720113	+	0.693857i	$0.755910\pi$
$104$	0.860701	0.0843987
$105$	1.74114	0.169918
$106$	5.71575	0.555162
$107$	6.74079	0.651657	0.325828	−	0.945429i	$-0.394357\pi$
$107$	6.74079	0.651657	0.325828	+	0.945429i	$0.394357\pi$
$108$	5.63099	0.541842
$109$	−20.1352	−1.92860	−0.964299	−	0.264814i	$-0.914689\pi$
$109$	−20.1352	−1.92860	−0.964299	+	0.264814i	$0.914689\pi$
$110$	−4.04768	−0.385931
$111$	−7.17597	−0.681113
$112$	−0.926965	−0.0875900
$113$	−13.0658	−1.22913	−0.614563	−	0.788868i	$-0.710668\pi$
$113$	−13.0658	−1.22913	−0.614563	+	0.788868i	$0.710668\pi$
$114$	0.0714556	0.00669243
$115$	−1.03924	−0.0969093
$116$	−8.73682	−0.811193
$117$	−0.667188	−0.0616815
$118$	1.04345	0.0960574
$119$	−4.98524	−0.456996
$120$	−1.87833	−0.171467
$121$	−0.668418	−0.0607652
$122$	−12.1728	−1.10207
$123$	0.832894	0.0750995
$124$	4.59166	0.412343
$125$	−10.5959	−0.947722
$126$	0.718553	0.0640138
$127$	5.90615	0.524086	0.262043	−	0.965056i	$-0.415604\pi$
$127$	5.90615	0.524086	0.262043	+	0.965056i	$0.415604\pi$
$128$	1.00000	0.0883883
$129$	−4.82535	−0.424849
$130$	1.08386	0.0950611
$131$	−5.88232	−0.513941	−0.256970	−	0.966419i	$-0.582724\pi$
$131$	−5.88232	−0.513941	−0.256970	+	0.966419i	$0.582724\pi$
$132$	4.79438	0.417297
$133$	0.0444069	0.00385057
$134$	−1.36388	−0.117821
$135$	7.09100	0.610296
$136$	5.37802	0.461161
$137$	4.10714	0.350896	0.175448	−	0.984489i	$-0.443863\pi$
$137$	4.10714	0.350896	0.175448	+	0.984489i	$0.443863\pi$
$138$	1.23095	0.104785
$139$	−16.6028	−1.40823	−0.704117	−	0.710084i	$-0.748657\pi$
$139$	−16.6028	−1.40823	−0.704117	+	0.710084i	$0.748657\pi$
$140$	−1.16731	−0.0986556
$141$	13.0396	1.09813
$142$	−0.724563	−0.0608039
$143$	−2.76653	−0.231349
$144$	−0.775168	−0.0645973
$145$	−11.0021	−0.913675
$146$	−0.635731	−0.0526134
$147$	9.15944	0.755458
$148$	4.81096	0.395459
$149$	−5.25565	−0.430560	−0.215280	−	0.976552i	$-0.569066\pi$
$149$	−5.25565	−0.430560	−0.215280	+	0.976552i	$0.569066\pi$
$150$	5.09260	0.415809
$151$	2.87351	0.233843	0.116921	−	0.993141i	$-0.462697\pi$
$151$	2.87351	0.233843	0.116921	+	0.993141i	$0.462697\pi$
$152$	−0.0479057	−0.00388567
$153$	−4.16887	−0.337033
$154$	2.97952	0.240097
$155$	5.78218	0.464436
$156$	−1.28381	−0.102787
$157$	5.23391	0.417712	0.208856	−	0.977946i	$-0.433026\pi$
$157$	5.23391	0.417712	0.208856	+	0.977946i	$0.433026\pi$
$158$	13.6754	1.08795
$159$	−8.52554	−0.676119
$160$	1.25928	0.0995548
$161$	0.764989	0.0602896
$162$	−6.07361	−0.477188
$163$	4.22964	0.331291	0.165645	−	0.986185i	$-0.447029\pi$
$163$	4.22964	0.331291	0.165645	+	0.986185i	$0.447029\pi$
$164$	−0.558394	−0.0436033
$165$	6.03746	0.470016
$166$	13.8105	1.07191
$167$	9.91848	0.767515	0.383758	−	0.923434i	$-0.374630\pi$
$167$	9.91848	0.767515	0.383758	+	0.923434i	$0.374630\pi$
$168$	1.38265	0.106674
$169$	−12.2592	−0.943015
$170$	6.77243	0.519422
$171$	0.0371350	0.00283978
$172$	3.23505	0.246670
$173$	3.23816	0.246193	0.123096	−	0.992395i	$-0.460718\pi$
$173$	3.23816	0.246193	0.123096	+	0.992395i	$0.460718\pi$
$174$	13.0317	0.987932
$175$	3.16486	0.239241
$176$	−3.21428	−0.242285
$177$	−1.55640	−0.116986
$178$	−9.45802	−0.708909
$179$	1.40794	0.105234	0.0526170	−	0.998615i	$-0.483244\pi$
$179$	1.40794	0.105234	0.0526170	+	0.998615i	$0.483244\pi$
$180$	−0.976153	−0.0727582
$181$	2.81633	0.209336	0.104668	−	0.994507i	$-0.466622\pi$
$181$	2.81633	0.209336	0.104668	+	0.994507i	$0.466622\pi$
$182$	−0.797840	−0.0591398
$183$	18.1567	1.34218
$184$	−0.825262	−0.0608391
$185$	6.05835	0.445419
$186$	−6.84886	−0.502183
$187$	−17.2864	−1.26411
$188$	−8.74211	−0.637584
$189$	−5.21973	−0.379680
$190$	−0.0603267	−0.00437656
$191$	12.2940	0.889560	0.444780	−	0.895640i	$-0.353282\pi$
$191$	12.2940	0.889560	0.444780	+	0.895640i	$0.353282\pi$
$192$	−1.49159	−0.107646
$193$	−1.12960	−0.0813105	−0.0406552	−	0.999173i	$-0.512945\pi$
$193$	−1.12960	−0.0813105	−0.0406552	+	0.999173i	$0.512945\pi$
$194$	−17.7590	−1.27502
$195$	−1.61668	−0.115773
$196$	−6.14074	−0.438624
$197$	−7.15238	−0.509586	−0.254793	−	0.966996i	$-0.582007\pi$
$197$	−7.15238	−0.509586	−0.254793	+	0.966996i	$0.582007\pi$
$198$	2.49160	0.177071
$199$	2.39483	0.169765	0.0848826	−	0.996391i	$-0.472948\pi$
$199$	2.39483	0.169765	0.0848826	+	0.996391i	$0.472948\pi$
$200$	−3.41421	−0.241421
$201$	2.03434	0.143492
$202$	−13.9475	−0.981341
$203$	8.09872	0.568419
$204$	−8.02178	−0.561637
$205$	−0.703175	−0.0491119
$206$	−14.6167	−1.01839
$207$	0.639716	0.0444634
$208$	0.860701	0.0596789
$209$	0.153982	0.0106512
$210$	1.74114	0.120150
$211$	−6.59309	−0.453887	−0.226944	−	0.973908i	$-0.572873\pi$
$211$	−6.59309	−0.453887	−0.226944	+	0.973908i	$0.572873\pi$
$212$	5.71575	0.392559
$213$	1.08075	0.0740517
$214$	6.74079	0.460791
$215$	4.07383	0.277833
$216$	5.63099	0.383140
$217$	−4.25631	−0.288937
$218$	−20.1352	−1.36373
$219$	0.948248	0.0640766
$220$	−4.04768	−0.272894
$221$	4.62887	0.311371
$222$	−7.17597	−0.481620
$223$	−15.5685	−1.04255	−0.521273	−	0.853390i	$-0.674543\pi$
$223$	−15.5685	−1.04255	−0.521273	+	0.853390i	$0.674543\pi$
$224$	−0.926965	−0.0619355
$225$	2.64659	0.176439
$226$	−13.0658	−0.869124
$227$	4.85950	0.322536	0.161268	−	0.986911i	$-0.448442\pi$
$227$	4.85950	0.322536	0.161268	+	0.986911i	$0.448442\pi$
$228$	0.0714556	0.00473226
$229$	−2.02418	−0.133762	−0.0668809	−	0.997761i	$-0.521305\pi$
$229$	−2.02418	−0.133762	−0.0668809	+	0.997761i	$0.521305\pi$
$230$	−1.03924	−0.0685252
$231$	−4.44422	−0.292408
$232$	−8.73682	−0.573600
$233$	10.6458	0.697427	0.348714	−	0.937229i	$-0.386619\pi$
$233$	10.6458	0.697427	0.348714	+	0.937229i	$0.386619\pi$
$234$	−0.667188	−0.0436154
$235$	−11.0088	−0.718133
$236$	1.04345	0.0679229
$237$	−20.3980	−1.32499
$238$	−4.98524	−0.323145
$239$	−10.8499	−0.701822	−0.350911	−	0.936409i	$-0.614128\pi$
$239$	−10.8499	−0.701822	−0.350911	+	0.936409i	$0.614128\pi$
$240$	−1.87833	−0.121245
$241$	11.7169	0.754755	0.377377	−	0.926060i	$-0.376826\pi$
$241$	11.7169	0.754755	0.377377	+	0.926060i	$0.376826\pi$
$242$	−0.668418	−0.0429675
$243$	−7.83365	−0.502529
$244$	−12.1728	−0.779281
$245$	−7.73291	−0.494037
$246$	0.832894	0.0531034
$247$	−0.0412325	−0.00262356
$248$	4.59166	0.291571
$249$	−20.5996	−1.30545
$250$	−10.5959	−0.670141
$251$	−26.8091	−1.69218	−0.846089	−	0.533042i	$-0.821049\pi$
$251$	−26.8091	−1.69218	−0.846089	+	0.533042i	$0.821049\pi$
$252$	0.718553	0.0452646
$253$	2.65262	0.166769
$254$	5.90615	0.370585
$255$	−10.1017	−0.632591
$256$	1.00000	0.0625000
$257$	24.3278	1.51753	0.758764	−	0.651365i	$-0.225803\pi$
$257$	24.3278	1.51753	0.758764	+	0.651365i	$0.225803\pi$
$258$	−4.82535	−0.300413
$259$	−4.45959	−0.277106
$260$	1.08386	0.0672184
$261$	6.77250	0.419207
$262$	−5.88232	−0.363411
$263$	0.518224	0.0319551	0.0159775	−	0.999872i	$-0.494914\pi$
$263$	0.518224	0.0319551	0.0159775	+	0.999872i	$0.494914\pi$
$264$	4.79438	0.295073
$265$	7.19773	0.442153
$266$	0.0444069	0.00272276
$267$	14.1075	0.863363
$268$	−1.36388	−0.0833121
$269$	2.16527	0.132019	0.0660095	−	0.997819i	$-0.478973\pi$
$269$	2.16527	0.132019	0.0660095	+	0.997819i	$0.478973\pi$
$270$	7.09100	0.431544
$271$	10.0616	0.611200	0.305600	−	0.952160i	$-0.401143\pi$
$271$	10.0616	0.611200	0.305600	+	0.952160i	$0.401143\pi$
$272$	5.37802	0.326090
$273$	1.19005	0.0720249
$274$	4.10714	0.248121
$275$	10.9742	0.661771
$276$	1.23095	0.0740945
$277$	18.7101	1.12418	0.562090	−	0.827076i	$-0.309998\pi$
$277$	18.7101	1.12418	0.562090	+	0.827076i	$0.309998\pi$
$278$	−16.6028	−0.995771
$279$	−3.55930	−0.213090
$280$	−1.16731	−0.0697600
$281$	−18.3310	−1.09353	−0.546767	−	0.837284i	$-0.684142\pi$
$281$	−18.3310	−1.09353	−0.546767	+	0.837284i	$0.684142\pi$
$282$	13.0396	0.776498
$283$	10.1018	0.600491	0.300245	−	0.953862i	$-0.402931\pi$
$283$	10.1018	0.600491	0.300245	+	0.953862i	$0.402931\pi$
$284$	−0.724563	−0.0429949
$285$	0.0899826	0.00533011
$286$	−2.76653	−0.163588
$287$	0.517612	0.0305537
$288$	−0.775168	−0.0456772
$289$	11.9231	0.701358
$290$	−11.0021	−0.646065
$291$	26.4891	1.55282
$292$	−0.635731	−0.0372033
$293$	−19.7374	−1.15307	−0.576534	−	0.817073i	$-0.695595\pi$
$293$	−19.7374	−1.15307	−0.576534	+	0.817073i	$0.695595\pi$
$294$	9.15944	0.534190
$295$	1.31400	0.0765038
$296$	4.81096	0.279632
$297$	−18.0996	−1.05024
$298$	−5.25565	−0.304452
$299$	−0.710304	−0.0410779
$300$	5.09260	0.294021
$301$	−2.99877	−0.172846
$302$	2.87351	0.165352
$303$	20.8039	1.19515
$304$	−0.0479057	−0.00274758
$305$	−15.3289	−0.877731
$306$	−4.16887	−0.238318
$307$	−11.6220	−0.663300	−0.331650	−	0.943403i	$-0.607605\pi$
$307$	−11.6220	−0.663300	−0.331650	+	0.943403i	$0.607605\pi$
$308$	2.97952	0.169774
$309$	21.8021	1.24028
$310$	5.78218	0.328406
$311$	−29.2398	−1.65804	−0.829020	−	0.559219i	$-0.811101\pi$
$311$	−29.2398	−1.65804	−0.829020	+	0.559219i	$0.811101\pi$
$312$	−1.28381	−0.0726815
$313$	−25.7945	−1.45799	−0.728996	−	0.684518i	$-0.760013\pi$
$313$	−25.7945	−1.45799	−0.728996	+	0.684518i	$0.760013\pi$
$314$	5.23391	0.295367
$315$	0.904860	0.0509831
$316$	13.6754	0.769299
$317$	6.10681	0.342993	0.171496	−	0.985185i	$-0.445140\pi$
$317$	6.10681	0.342993	0.171496	+	0.985185i	$0.445140\pi$
$318$	−8.52554	−0.478088
$319$	28.0826	1.57232
$320$	1.25928	0.0703959
$321$	−10.0545	−0.561186
$322$	0.764989	0.0426312
$323$	−0.257638	−0.0143354
$324$	−6.07361	−0.337423
$325$	−2.93862	−0.163005
$326$	4.22964	0.234258
$327$	30.0334	1.66085
$328$	−0.558394	−0.0308322
$329$	8.10363	0.446768
$330$	6.03746	0.332351
$331$	18.2000	1.00037	0.500183	−	0.865920i	$-0.333266\pi$
$331$	18.2000	1.00037	0.500183	+	0.865920i	$0.333266\pi$
$332$	13.8105	0.757952
$333$	−3.72930	−0.204365
$334$	9.91848	0.542715
$335$	−1.71751	−0.0938373
$336$	1.38265	0.0754297
$337$	−1.46347	−0.0797206	−0.0398603	−	0.999205i	$-0.512691\pi$
$337$	−1.46347	−0.0797206	−0.0398603	+	0.999205i	$0.512691\pi$
$338$	−12.2592	−0.666812
$339$	19.4888	1.05848
$340$	6.77243	0.367287
$341$	−14.7589	−0.799237
$342$	0.0371350	0.00200803
$343$	12.1810	0.657712
$344$	3.23505	0.174422
$345$	1.55011	0.0834552
$346$	3.23816	0.174085
$347$	−22.9011	−1.22940	−0.614698	−	0.788763i	$-0.710722\pi$
$347$	−22.9011	−1.22940	−0.614698	+	0.788763i	$0.710722\pi$
$348$	13.0317	0.698574
$349$	−6.34915	−0.339862	−0.169931	−	0.985456i	$-0.554354\pi$
$349$	−6.34915	−0.339862	−0.169931	+	0.985456i	$0.554354\pi$
$350$	3.16486	0.169169
$351$	4.84660	0.258692
$352$	−3.21428	−0.171322
$353$	2.15016	0.114442	0.0572208	−	0.998362i	$-0.481776\pi$
$353$	2.15016	0.114442	0.0572208	+	0.998362i	$0.481776\pi$
$354$	−1.55640	−0.0827216
$355$	−0.912427	−0.0484266
$356$	−9.45802	−0.501274
$357$	7.43591	0.393550
$358$	1.40794	0.0744117
$359$	−22.6151	−1.19358	−0.596790	−	0.802397i	$-0.703558\pi$
$359$	−22.6151	−1.19358	−0.596790	+	0.802397i	$0.703558\pi$
$360$	−0.976153	−0.0514478
$361$	−18.9977	−0.999879
$362$	2.81633	0.148023
$363$	0.997003	0.0523291
$364$	−0.797840	−0.0418182
$365$	−0.800563	−0.0419034
$366$	18.1567	0.949067
$367$	−10.2679	−0.535980	−0.267990	−	0.963422i	$-0.586359\pi$
$367$	−10.2679	−0.535980	−0.267990	+	0.963422i	$0.586359\pi$
$368$	−0.825262	−0.0430198
$369$	0.432849	0.0225332
$370$	6.05835	0.314959
$371$	−5.29830	−0.275074
$372$	−6.84886	−0.355097
$373$	−6.28462	−0.325405	−0.162703	−	0.986675i	$-0.552021\pi$
$373$	−6.28462	−0.325405	−0.162703	+	0.986675i	$0.552021\pi$
$374$	−17.2864	−0.893861
$375$	15.8046	0.816148
$376$	−8.74211	−0.450840
$377$	−7.51979	−0.387289
$378$	−5.21973	−0.268474
$379$	0.136134	0.00699276	0.00349638	−	0.999994i	$-0.498887\pi$
$379$	0.136134	0.00699276	0.00349638	+	0.999994i	$0.498887\pi$
$380$	−0.0603267	−0.00309470
$381$	−8.80953	−0.451326
$382$	12.2940	0.629014
$383$	11.5785	0.591632	0.295816	−	0.955245i	$-0.404409\pi$
$383$	11.5785	0.591632	0.295816	+	0.955245i	$0.404409\pi$
$384$	−1.49159	−0.0761172
$385$	3.75205	0.191222
$386$	−1.12960	−0.0574952
$387$	−2.50770	−0.127474
$388$	−17.7590	−0.901578
$389$	−6.50278	−0.329704	−0.164852	−	0.986318i	$-0.552715\pi$
$389$	−6.50278	−0.329704	−0.164852	+	0.986318i	$0.552715\pi$
$390$	−1.61668	−0.0818636
$391$	−4.43827	−0.224453
$392$	−6.14074	−0.310154
$393$	8.77399	0.442589
$394$	−7.15238	−0.360331
$395$	17.2211	0.866488
$396$	2.49160	0.125208
$397$	18.5934	0.933176	0.466588	−	0.884475i	$-0.345483\pi$
$397$	18.5934	0.933176	0.466588	+	0.884475i	$0.345483\pi$
$398$	2.39483	0.120042
$399$	−0.0662368	−0.00331599
$400$	−3.41421	−0.170711
$401$	14.4724	0.722716	0.361358	−	0.932427i	$-0.382313\pi$
$401$	14.4724	0.722716	0.361358	+	0.932427i	$0.382313\pi$
$402$	2.03434	0.101464
$403$	3.95204	0.196865
$404$	−13.9475	−0.693913
$405$	−7.64838	−0.380051
$406$	8.09872	0.401933
$407$	−15.4638	−0.766511
$408$	−8.02178	−0.397137
$409$	−32.8688	−1.62526	−0.812629	−	0.582781i	$-0.801965\pi$
$409$	−32.8688	−1.62526	−0.812629	+	0.582781i	$0.801965\pi$
$410$	−0.703175	−0.0347273
$411$	−6.12615	−0.302181
$412$	−14.6167	−0.720113
$413$	−0.967242	−0.0475949
$414$	0.639716	0.0314403
$415$	17.3913	0.853707
$416$	0.860701	0.0421993
$417$	24.7646	1.21273
$418$	0.153982	0.00753152
$419$	−6.76631	−0.330556	−0.165278	−	0.986247i	$-0.552852\pi$
$419$	−6.76631	−0.330556	−0.165278	+	0.986247i	$0.552852\pi$
$420$	1.74114	0.0849591
$421$	−6.62325	−0.322797	−0.161399	−	0.986889i	$-0.551600\pi$
$421$	−6.62325	−0.322797	−0.161399	+	0.986889i	$0.551600\pi$
$422$	−6.59309	−0.320947
$423$	6.77661	0.329490
$424$	5.71575	0.277581
$425$	−18.3617	−0.890674
$426$	1.08075	0.0523624
$427$	11.2837	0.546057
$428$	6.74079	0.325828
$429$	4.12652	0.199230
$430$	4.07383	0.196457
$431$	−27.6307	−1.33093	−0.665463	−	0.746431i	$-0.731766\pi$
$431$	−27.6307	−1.33093	−0.665463	+	0.746431i	$0.731766\pi$
$432$	5.63099	0.270921
$433$	30.0686	1.44501	0.722503	−	0.691368i	$-0.242991\pi$
$433$	30.0686	1.44501	0.722503	+	0.691368i	$0.242991\pi$
$434$	−4.25631	−0.204309
$435$	16.4106	0.786828
$436$	−20.1352	−0.964299
$437$	0.0395348	0.00189121
$438$	0.948248	0.0453090
$439$	36.9779	1.76486	0.882430	−	0.470444i	$-0.155906\pi$
$439$	36.9779	1.76486	0.882430	+	0.470444i	$0.155906\pi$
$440$	−4.04768	−0.192965
$441$	4.76010	0.226671
$442$	4.62887	0.220173
$443$	−1.86279	−0.0885037	−0.0442518	−	0.999020i	$-0.514090\pi$
$443$	−1.86279	−0.0885037	−0.0442518	+	0.999020i	$0.514090\pi$
$444$	−7.17597	−0.340556
$445$	−11.9103	−0.564602
$446$	−15.5685	−0.737191
$447$	7.83927	0.370785
$448$	−0.926965	−0.0437950
$449$	38.1097	1.79851	0.899255	−	0.437425i	$-0.144110\pi$
$449$	38.1097	1.79851	0.899255	+	0.437425i	$0.144110\pi$
$450$	2.64659	0.124761
$451$	1.79484	0.0845155
$452$	−13.0658	−0.614563
$453$	−4.28609	−0.201378
$454$	4.85950	0.228067
$455$	−1.00470	−0.0471012
$456$	0.0714556	0.00334621
$457$	−17.9985	−0.841934	−0.420967	−	0.907076i	$-0.638309\pi$
$457$	−17.9985	−0.841934	−0.420967	+	0.907076i	$0.638309\pi$
$458$	−2.02418	−0.0945839
$459$	30.2836	1.41352
$460$	−1.03924	−0.0484546
$461$	3.13484	0.146004	0.0730021	−	0.997332i	$-0.476742\pi$
$461$	3.13484	0.146004	0.0730021	+	0.997332i	$0.476742\pi$
$462$	−4.44422	−0.206764
$463$	29.2577	1.35972	0.679861	−	0.733341i	$-0.262040\pi$
$463$	29.2577	1.35972	0.679861	+	0.733341i	$0.262040\pi$
$464$	−8.73682	−0.405597
$465$	−8.62463	−0.399958
$466$	10.6458	0.493156
$467$	−18.6225	−0.861746	−0.430873	−	0.902413i	$-0.641794\pi$
$467$	−18.6225	−0.861746	−0.430873	+	0.902413i	$0.641794\pi$
$468$	−0.667188	−0.0308408
$469$	1.26427	0.0583785
$470$	−11.0088	−0.507797
$471$	−7.80684	−0.359720
$472$	1.04345	0.0480287
$473$	−10.3983	−0.478116
$474$	−20.3980	−0.936911
$475$	0.163560	0.00750467
$476$	−4.98524	−0.228498
$477$	−4.43066	−0.202866
$478$	−10.8499	−0.496263
$479$	−3.31455	−0.151446	−0.0757228	−	0.997129i	$-0.524126\pi$
$479$	−3.31455	−0.151446	−0.0757228	+	0.997129i	$0.524126\pi$
$480$	−1.87833	−0.0857335
$481$	4.14080	0.188804
$482$	11.7169	0.533692
$483$	−1.14105	−0.0519195
$484$	−0.668418	−0.0303826
$485$	−22.3636	−1.01548
$486$	−7.83365	−0.355342
$487$	−1.84793	−0.0837376	−0.0418688	−	0.999123i	$-0.513331\pi$
$487$	−1.84793	−0.0837376	−0.0418688	+	0.999123i	$0.513331\pi$
$488$	−12.1728	−0.551035
$489$	−6.30887	−0.285297
$490$	−7.73291	−0.349337
$491$	0.962265	0.0434264	0.0217132	−	0.999764i	$-0.493088\pi$
$491$	0.962265	0.0434264	0.0217132	+	0.999764i	$0.493088\pi$
$492$	0.832894	0.0375498
$493$	−46.9868	−2.11618
$494$	−0.0412325	−0.00185514
$495$	3.13763	0.141026
$496$	4.59166	0.206171
$497$	0.671644	0.0301274
$498$	−20.5996	−0.923091
$499$	−35.0608	−1.56954	−0.784769	−	0.619788i	$-0.787218\pi$
$499$	−35.0608	−1.56954	−0.784769	+	0.619788i	$0.787218\pi$
$500$	−10.5959	−0.473861
$501$	−14.7943	−0.660960
$502$	−26.8091	−1.19655
$503$	19.0292	0.848469	0.424235	−	0.905552i	$-0.360543\pi$
$503$	19.0292	0.848469	0.424235	+	0.905552i	$0.360543\pi$
$504$	0.718553	0.0320069
$505$	−17.5638	−0.781578
$506$	2.65262	0.117923
$507$	18.2857	0.812094
$508$	5.90615	0.262043
$509$	−5.57341	−0.247037	−0.123519	−	0.992342i	$-0.539418\pi$
$509$	−5.57341	−0.247037	−0.123519	+	0.992342i	$0.539418\pi$
$510$	−10.1017	−0.447310
$511$	0.589300	0.0260691
$512$	1.00000	0.0441942
$513$	−0.269757	−0.0119101
$514$	24.3278	1.07305
$515$	−18.4065	−0.811088
$516$	−4.82535	−0.212424
$517$	28.0996	1.23582
$518$	−4.45959	−0.195943
$519$	−4.83000	−0.212014
$520$	1.08386	0.0475306
$521$	−16.0959	−0.705173	−0.352586	−	0.935779i	$-0.614698\pi$
$521$	−16.0959	−0.705173	−0.352586	+	0.935779i	$0.614698\pi$
$522$	6.77250	0.296424
$523$	−13.4045	−0.586137	−0.293069	−	0.956091i	$-0.594676\pi$
$523$	−13.4045	−0.586137	−0.293069	+	0.956091i	$0.594676\pi$
$524$	−5.88232	−0.256970
$525$	−4.72066	−0.206026
$526$	0.518224	0.0225957
$527$	24.6940	1.07569
$528$	4.79438	0.208648
$529$	−22.3189	−0.970389
$530$	7.19773	0.312649
$531$	−0.808849	−0.0351011
$532$	0.0444069	0.00192529
$533$	−0.480611	−0.0208176
$534$	14.1075	0.610490
$535$	8.48854	0.366992
$536$	−1.36388	−0.0589106
$537$	−2.10006	−0.0906242
$538$	2.16527	0.0933515
$539$	19.7380	0.850177
$540$	7.09100	0.305148
$541$	−42.8941	−1.84416	−0.922080	−	0.386998i	$-0.873512\pi$
$541$	−42.8941	−1.84416	−0.922080	+	0.386998i	$0.873512\pi$
$542$	10.0616	0.432183
$543$	−4.20080	−0.180274
$544$	5.37802	0.230581
$545$	−25.3558	−1.08612
$546$	1.19005	0.0509293
$547$	−21.1887	−0.905962	−0.452981	−	0.891520i	$-0.649639\pi$
$547$	−21.1887	−0.905962	−0.452981	+	0.891520i	$0.649639\pi$
$548$	4.10714	0.175448
$549$	9.43593	0.402716
$550$	10.9742	0.467943
$551$	0.418544	0.0178306
$552$	1.23095	0.0523927
$553$	−12.6766	−0.539063
$554$	18.7101	0.794915
$555$	−9.03656	−0.383580
$556$	−16.6028	−0.704117
$557$	−5.60774	−0.237608	−0.118804	−	0.992918i	$-0.537906\pi$
$557$	−5.60774	−0.237608	−0.118804	+	0.992918i	$0.537906\pi$
$558$	−3.55930	−0.150677
$559$	2.78441	0.117768
$560$	−1.16731	−0.0493278
$561$	25.7842	1.08861
$562$	−18.3310	−0.773246
$563$	35.0362	1.47660	0.738299	−	0.674473i	$-0.235629\pi$
$563$	35.0362	1.47660	0.738299	+	0.674473i	$0.235629\pi$
$564$	13.0396	0.549067
$565$	−16.4535	−0.692204
$566$	10.1018	0.424611
$567$	5.63003	0.236439
$568$	−0.724563	−0.0304020
$569$	37.1041	1.55549	0.777743	−	0.628583i	$-0.216365\pi$
$569$	37.1041	1.55549	0.777743	+	0.628583i	$0.216365\pi$
$570$	0.0899826	0.00376896
$571$	11.8049	0.494018	0.247009	−	0.969013i	$-0.420552\pi$
$571$	11.8049	0.494018	0.247009	+	0.969013i	$0.420552\pi$
$572$	−2.76653	−0.115675
$573$	−18.3375	−0.766061
$574$	0.517612	0.0216047
$575$	2.81762	0.117503
$576$	−0.775168	−0.0322987
$577$	17.5151	0.729163	0.364582	−	0.931171i	$-0.381212\pi$
$577$	17.5151	0.729163	0.364582	+	0.931171i	$0.381212\pi$
$578$	11.9231	0.495935
$579$	1.68490	0.0700220
$580$	−11.0021	−0.456837
$581$	−12.8019	−0.531112
$582$	26.4891	1.09801
$583$	−18.3720	−0.760891
$584$	−0.635731	−0.0263067
$585$	−0.840176	−0.0347370
$586$	−19.7374	−0.815342
$587$	−2.89901	−0.119655	−0.0598275	−	0.998209i	$-0.519055\pi$
$587$	−2.89901	−0.119655	−0.0598275	+	0.998209i	$0.519055\pi$
$588$	9.15944	0.377729
$589$	−0.219967	−0.00906357
$590$	1.31400	0.0540964
$591$	10.6684	0.438839
$592$	4.81096	0.197729
$593$	−29.9756	−1.23095	−0.615474	−	0.788157i	$-0.711036\pi$
$593$	−29.9756	−1.23095	−0.615474	+	0.788157i	$0.711036\pi$
$594$	−18.0996	−0.742634
$595$	−6.27781	−0.257365
$596$	−5.25565	−0.215280
$597$	−3.57210	−0.146196
$598$	−0.710304	−0.0290465
$599$	−9.92498	−0.405524	−0.202762	−	0.979228i	$-0.564992\pi$
$599$	−9.92498	−0.405524	−0.202762	+	0.979228i	$0.564992\pi$
$600$	5.09260	0.207904
$601$	−27.8508	−1.13606	−0.568029	−	0.823009i	$-0.692294\pi$
$601$	−27.8508	−1.13606	−0.568029	+	0.823009i	$0.692294\pi$
$602$	−2.99877	−0.122221
$603$	1.05723	0.0430539
$604$	2.87351	0.116921
$605$	−0.841725	−0.0342210
$606$	20.8039	0.845100
$607$	21.5897	0.876299	0.438150	−	0.898902i	$-0.355634\pi$
$607$	21.5897	0.876299	0.438150	+	0.898902i	$0.355634\pi$
$608$	−0.0479057	−0.00194283
$609$	−12.0800	−0.489504
$610$	−15.3289	−0.620649
$611$	−7.52435	−0.304402
$612$	−4.16887	−0.168516
$613$	−35.4912	−1.43347	−0.716737	−	0.697343i	$-0.754365\pi$
$613$	−35.4912	−1.43347	−0.716737	+	0.697343i	$0.754365\pi$
$614$	−11.6220	−0.469024
$615$	1.04885	0.0422936
$616$	2.97952	0.120048
$617$	42.1877	1.69841	0.849206	−	0.528062i	$-0.177081\pi$
$617$	42.1877	1.69841	0.849206	+	0.528062i	$0.177081\pi$
$618$	21.8021	0.877008
$619$	0.700779	0.0281667	0.0140833	−	0.999901i	$-0.495517\pi$
$619$	0.700779	0.0281667	0.0140833	+	0.999901i	$0.495517\pi$
$620$	5.78218	0.232218
$621$	−4.64704	−0.186479
$622$	−29.2398	−1.17241
$623$	8.76725	0.351253
$624$	−1.28381	−0.0513936
$625$	3.72792	0.149117
$626$	−25.7945	−1.03096
$627$	−0.229678	−0.00917246
$628$	5.23391	0.208856
$629$	25.8734	1.03164
$630$	0.904860	0.0360505
$631$	26.6451	1.06072	0.530362	−	0.847771i	$-0.322056\pi$
$631$	26.6451	1.06072	0.530362	+	0.847771i	$0.322056\pi$
$632$	13.6754	0.543977
$633$	9.83417	0.390873
$634$	6.10681	0.242533
$635$	7.43749	0.295148
$636$	−8.52554	−0.338059
$637$	−5.28534	−0.209413
$638$	28.0826	1.11180
$639$	0.561658	0.0222188
$640$	1.25928	0.0497774
$641$	42.0284	1.66002	0.830011	−	0.557747i	$-0.188334\pi$
$641$	42.0284	1.66002	0.830011	+	0.557747i	$0.188334\pi$
$642$	−10.0545	−0.396818
$643$	36.9077	1.45550	0.727749	−	0.685844i	$-0.240567\pi$
$643$	36.9077	1.45550	0.727749	+	0.685844i	$0.240567\pi$
$644$	0.764989	0.0301448
$645$	−6.07647	−0.239261
$646$	−0.257638	−0.0101366
$647$	3.37419	0.132653	0.0663266	−	0.997798i	$-0.478872\pi$
$647$	3.37419	0.132653	0.0663266	+	0.997798i	$0.478872\pi$
$648$	−6.07361	−0.238594
$649$	−3.35394	−0.131654
$650$	−2.93862	−0.115262
$651$	6.34865	0.248823
$652$	4.22964	0.165645
$653$	5.58280	0.218472	0.109236	−	0.994016i	$-0.465160\pi$
$653$	5.58280	0.218472	0.109236	+	0.994016i	$0.465160\pi$
$654$	30.0334	1.17440
$655$	−7.40749	−0.289435
$656$	−0.558394	−0.0218016
$657$	0.492798	0.0192259
$658$	8.10363	0.315913
$659$	−32.9848	−1.28490	−0.642452	−	0.766326i	$-0.722083\pi$
$659$	−32.9848	−1.28490	−0.642452	+	0.766326i	$0.722083\pi$
$660$	6.03746	0.235008
$661$	43.1599	1.67873	0.839364	−	0.543570i	$-0.182928\pi$
$661$	43.1599	1.67873	0.839364	+	0.543570i	$0.182928\pi$
$662$	18.2000	0.707365
$663$	−6.90436	−0.268143
$664$	13.8105	0.535953
$665$	0.0559208	0.00216851
$666$	−3.72930	−0.144508
$667$	7.21016	0.279179
$668$	9.91848	0.383758
$669$	23.2218	0.897807
$670$	−1.71751	−0.0663530
$671$	39.1266	1.51047
$672$	1.38265	0.0533368
$673$	−16.9463	−0.653233	−0.326617	−	0.945157i	$-0.605909\pi$
$673$	−16.9463	−0.653233	−0.326617	+	0.945157i	$0.605909\pi$
$674$	−1.46347	−0.0563709
$675$	−19.2254	−0.739986
$676$	−12.2592	−0.471507
$677$	−16.1379	−0.620231	−0.310115	−	0.950699i	$-0.600368\pi$
$677$	−16.1379	−0.620231	−0.310115	+	0.950699i	$0.600368\pi$
$678$	19.4888	0.748462
$679$	16.4620	0.631753
$680$	6.77243	0.259711
$681$	−7.24836	−0.277758
$682$	−14.7589	−0.565146
$683$	0.297181	0.0113713	0.00568566	−	0.999984i	$-0.498190\pi$
$683$	0.297181	0.0113713	0.00568566	+	0.999984i	$0.498190\pi$
$684$	0.0371350	0.00141989
$685$	5.17204	0.197613
$686$	12.1810	0.465073
$687$	3.01925	0.115191
$688$	3.23505	0.123335
$689$	4.91955	0.187420
$690$	1.55011	0.0590117
$691$	42.0107	1.59816	0.799081	−	0.601223i	$-0.205320\pi$
$691$	42.0107	1.59816	0.799081	+	0.601223i	$0.205320\pi$
$692$	3.23816	0.123096
$693$	−2.30963	−0.0877356
$694$	−22.9011	−0.869314
$695$	−20.9076	−0.793071
$696$	13.0317	0.493966
$697$	−3.00306	−0.113749
$698$	−6.34915	−0.240319
$699$	−15.8791	−0.600602
$700$	3.16486	0.119620
$701$	4.35242	0.164388	0.0821942	−	0.996616i	$-0.473807\pi$
$701$	4.35242	0.164388	0.0821942	+	0.996616i	$0.473807\pi$
$702$	4.84660	0.182923
$703$	−0.230473	−0.00869244
$704$	−3.21428	−0.121143
$705$	16.4205	0.618433
$706$	2.15016	0.0809224
$707$	12.9288	0.486239
$708$	−1.55640	−0.0584930
$709$	5.29092	0.198705	0.0993523	−	0.995052i	$-0.468323\pi$
$709$	5.29092	0.198705	0.0993523	+	0.995052i	$0.468323\pi$
$710$	−0.912427	−0.0342428
$711$	−10.6007	−0.397557
$712$	−9.45802	−0.354454
$713$	−3.78932	−0.141911
$714$	7.43591	0.278282
$715$	−3.48384	−0.130288
$716$	1.40794	0.0526170
$717$	16.1836	0.604387
$718$	−22.6151	−0.843989
$719$	−38.2633	−1.42698	−0.713489	−	0.700666i	$-0.752886\pi$
$719$	−38.2633	−1.42698	−0.713489	+	0.700666i	$0.752886\pi$
$720$	−0.976153	−0.0363791
$721$	13.5492	0.504597
$722$	−18.9977	−0.707021
$723$	−17.4768	−0.649971
$724$	2.81633	0.104668
$725$	29.8294	1.10783
$726$	0.997003	0.0370023
$727$	−18.8092	−0.697594	−0.348797	−	0.937198i	$-0.613410\pi$
$727$	−18.8092	−0.697594	−0.348797	+	0.937198i	$0.613410\pi$
$728$	−0.797840	−0.0295699
$729$	29.9054	1.10761
$730$	−0.800563	−0.0296302
$731$	17.3981	0.643493
$732$	18.1567	0.671092
$733$	−8.93348	−0.329966	−0.164983	−	0.986296i	$-0.552757\pi$
$733$	−8.93348	−0.329966	−0.164983	+	0.986296i	$0.552757\pi$
$734$	−10.2679	−0.378995
$735$	11.5343	0.425449
$736$	−0.825262	−0.0304196
$737$	4.38388	0.161482
$738$	0.432849	0.0159334
$739$	−20.9150	−0.769369	−0.384684	−	0.923048i	$-0.625690\pi$
$739$	−20.9150	−0.769369	−0.384684	+	0.923048i	$0.625690\pi$
$740$	6.05835	0.222709
$741$	0.0615019	0.00225933
$742$	−5.29830	−0.194507
$743$	−0.522929	−0.0191844	−0.00959219	−	0.999954i	$-0.503053\pi$
$743$	−0.522929	−0.0191844	−0.00959219	+	0.999954i	$0.503053\pi$
$744$	−6.84886	−0.251091
$745$	−6.61834	−0.242477
$746$	−6.28462	−0.230096
$747$	−10.7055	−0.391693
$748$	−17.2864	−0.632055
$749$	−6.24847	−0.228314
$750$	15.8046	0.577104
$751$	−4.86279	−0.177446	−0.0887230	−	0.996056i	$-0.528279\pi$
$751$	−4.86279	−0.177446	−0.0887230	+	0.996056i	$0.528279\pi$
$752$	−8.74211	−0.318792
$753$	39.9882	1.45725
$754$	−7.51979	−0.273854
$755$	3.61855	0.131693
$756$	−5.21973	−0.189840
$757$	50.5020	1.83553	0.917763	−	0.397129i	$-0.129994\pi$
$757$	50.5020	1.83553	0.917763	+	0.397129i	$0.129994\pi$
$758$	0.136134	0.00494463
$759$	−3.95662	−0.143616
$760$	−0.0603267	−0.00218828
$761$	43.0793	1.56162	0.780812	−	0.624766i	$-0.214806\pi$
$761$	43.0793	1.56162	0.780812	+	0.624766i	$0.214806\pi$
$762$	−8.80953	−0.319136
$763$	18.6646	0.675704
$764$	12.2940	0.444780
$765$	−5.24977	−0.189806
$766$	11.5785	0.418347
$767$	0.898099	0.0324285
$768$	−1.49159	−0.0538230
$769$	−11.6918	−0.421617	−0.210809	−	0.977527i	$-0.567610\pi$
$769$	−11.6918	−0.421617	−0.210809	+	0.977527i	$0.567610\pi$
$770$	3.75205	0.135215
$771$	−36.2871	−1.30685
$772$	−1.12960	−0.0406552
$773$	−2.48337	−0.0893206	−0.0446603	−	0.999002i	$-0.514221\pi$
$773$	−2.48337	−0.0893206	−0.0446603	+	0.999002i	$0.514221\pi$
$774$	−2.50770	−0.0901375
$775$	−15.6769	−0.563131
$776$	−17.7590	−0.637512
$777$	6.65187	0.238635
$778$	−6.50278	−0.233136
$779$	0.0267503	0.000958429 0
$780$	−1.61668	−0.0578863
$781$	2.32895	0.0833362
$782$	−4.43827	−0.158712
$783$	−49.1969	−1.75816
$784$	−6.14074	−0.219312
$785$	6.59096	0.235241
$786$	8.77399	0.312958
$787$	−4.48889	−0.160011	−0.0800057	−	0.996794i	$-0.525494\pi$
$787$	−4.48889	−0.160011	−0.0800057	+	0.996794i	$0.525494\pi$
$788$	−7.15238	−0.254793
$789$	−0.772977	−0.0275187
$790$	17.2211	0.612700
$791$	12.1115	0.430637
$792$	2.49160	0.0885353
$793$	−10.4771	−0.372053
$794$	18.5934	0.659855
$795$	−10.7360	−0.380768
$796$	2.39483	0.0848826
$797$	34.2860	1.21447	0.607236	−	0.794521i	$-0.292278\pi$
$797$	34.2860	1.21447	0.607236	+	0.794521i	$0.292278\pi$
$798$	−0.0662368	−0.00234476
$799$	−47.0153	−1.66328
$800$	−3.41421	−0.120711
$801$	7.33155	0.259048
$802$	14.4724	0.511038
$803$	2.04341	0.0721105
$804$	2.03434	0.0717458
$805$	0.963335	0.0339531
$806$	3.95204	0.139205
$807$	−3.22969	−0.113691
$808$	−13.9475	−0.490671
$809$	−1.38904	−0.0488360	−0.0244180	−	0.999702i	$-0.507773\pi$
$809$	−1.38904	−0.0488360	−0.0244180	+	0.999702i	$0.507773\pi$
$810$	−7.64838	−0.268737
$811$	16.8982	0.593377	0.296688	−	0.954974i	$-0.404118\pi$
$811$	16.8982	0.593377	0.296688	+	0.954974i	$0.404118\pi$
$812$	8.09872	0.284209
$813$	−15.0078	−0.526346
$814$	−15.4638	−0.542005
$815$	5.32630	0.186572
$816$	−8.02178	−0.280819
$817$	−0.154977	−0.00542197
$818$	−32.8688	−1.14923
$819$	0.618460	0.0216107
$820$	−0.703175	−0.0245559
$821$	45.1628	1.57619	0.788096	−	0.615552i	$-0.211067\pi$
$821$	45.1628	1.57619	0.788096	+	0.615552i	$0.211067\pi$
$822$	−6.12615	−0.213674
$823$	2.53373	0.0883201	0.0441601	−	0.999024i	$-0.485939\pi$
$823$	2.53373	0.0883201	0.0441601	+	0.999024i	$0.485939\pi$
$824$	−14.6167	−0.509196
$825$	−16.3690	−0.569896
$826$	−0.967242	−0.0336547
$827$	18.6967	0.650149	0.325075	−	0.945688i	$-0.394611\pi$
$827$	18.6967	0.650149	0.325075	+	0.945688i	$0.394611\pi$
$828$	0.639716	0.0222317
$829$	25.2550	0.877141	0.438571	−	0.898697i	$-0.355485\pi$
$829$	25.2550	0.877141	0.438571	+	0.898697i	$0.355485\pi$
$830$	17.3913	0.603662
$831$	−27.9077	−0.968108
$832$	0.860701	0.0298394
$833$	−33.0250	−1.14425
$834$	24.7646	0.857527
$835$	12.4902	0.432239
$836$	0.153982	0.00532559
$837$	25.8556	0.893700
$838$	−6.76631	−0.233738
$839$	9.72504	0.335746	0.167873	−	0.985809i	$-0.446310\pi$
$839$	9.72504	0.335746	0.167873	+	0.985809i	$0.446310\pi$
$840$	1.74114	0.0600751
$841$	47.3320	1.63214
$842$	−6.62325	−0.228252
$843$	27.3423	0.941718
$844$	−6.59309	−0.226944
$845$	−15.4378	−0.531075
$846$	6.77661	0.232984
$847$	0.619600	0.0212897
$848$	5.71575	0.196280
$849$	−15.0677	−0.517123
$850$	−18.3617	−0.629801
$851$	−3.97030	−0.136100
$852$	1.08075	0.0370258
$853$	19.7500	0.676229	0.338114	−	0.941105i	$-0.390211\pi$
$853$	19.7500	0.676229	0.338114	+	0.941105i	$0.390211\pi$
$854$	11.2837	0.386121
$855$	0.0467634	0.00159927
$856$	6.74079	0.230395
$857$	−17.2447	−0.589068	−0.294534	−	0.955641i	$-0.595164\pi$
$857$	−17.2447	−0.589068	−0.294534	+	0.955641i	$0.595164\pi$
$858$	4.12652	0.140877
$859$	34.6996	1.18394	0.591968	−	0.805962i	$-0.298351\pi$
$859$	34.6996	1.18394	0.591968	+	0.805962i	$0.298351\pi$
$860$	4.07383	0.138916
$861$	−0.772064	−0.0263119
$862$	−27.6307	−0.941107
$863$	47.2597	1.60874	0.804369	−	0.594130i	$-0.202504\pi$
$863$	47.2597	1.60874	0.804369	+	0.594130i	$0.202504\pi$
$864$	5.63099	0.191570
$865$	4.07775	0.138648
$866$	30.0686	1.02177
$867$	−17.7843	−0.603987
$868$	−4.25631	−0.144468
$869$	−43.9564	−1.49112
$870$	16.4106	0.556371
$871$	−1.17389	−0.0397758
$872$	−20.1352	−0.681863
$873$	13.7662	0.465916
$874$	0.0395348	0.00133728
$875$	9.82198	0.332044
$876$	0.948248	0.0320383
$877$	−53.8289	−1.81767	−0.908837	−	0.417152i	$-0.863028\pi$
$877$	−53.8289	−1.81767	−0.908837	+	0.417152i	$0.863028\pi$
$878$	36.9779	1.24794
$879$	29.4400	0.992986
$880$	−4.04768	−0.136447
$881$	45.6715	1.53871	0.769355	−	0.638821i	$-0.220578\pi$
$881$	45.6715	1.53871	0.769355	+	0.638821i	$0.220578\pi$
$882$	4.76010	0.160281
$883$	−34.2398	−1.15226	−0.576130	−	0.817358i	$-0.695438\pi$
$883$	−34.2398	−1.15226	−0.576130	+	0.817358i	$0.695438\pi$
$884$	4.62887	0.155686
$885$	−1.95994	−0.0658827
$886$	−1.86279	−0.0625815
$887$	−3.65022	−0.122562	−0.0612812	−	0.998121i	$-0.519519\pi$
$887$	−3.65022	−0.122562	−0.0612812	+	0.998121i	$0.519519\pi$
$888$	−7.17597	−0.240810
$889$	−5.47479	−0.183619
$890$	−11.9103	−0.399234
$891$	19.5223	0.654021
$892$	−15.5685	−0.521273
$893$	0.418797	0.0140145
$894$	7.83927	0.262184
$895$	1.77299	0.0592644
$896$	−0.926965	−0.0309677
$897$	1.05948	0.0353750
$898$	38.1097	1.27174
$899$	−40.1165	−1.33796
$900$	2.64659	0.0882196
$901$	30.7394	1.02408
$902$	1.79484	0.0597615
$903$	4.47293	0.148850
$904$	−13.0658	−0.434562
$905$	3.54655	0.117891
$906$	−4.28609	−0.142396
$907$	−22.9014	−0.760428	−0.380214	−	0.924899i	$-0.624150\pi$
$907$	−22.9014	−0.760428	−0.380214	+	0.924899i	$0.624150\pi$
$908$	4.85950	0.161268
$909$	10.8116	0.358599
$910$	−1.00470	−0.0333056
$911$	10.9607	0.363143	0.181572	−	0.983378i	$-0.441882\pi$
$911$	10.9607	0.363143	0.181572	+	0.983378i	$0.441882\pi$
$912$	0.0714556	0.00236613
$913$	−44.3909	−1.46913
$914$	−17.9985	−0.595338
$915$	22.8644	0.755874
$916$	−2.02418	−0.0668809
$917$	5.45270	0.180064
$918$	30.2836	0.999507
$919$	−20.5092	−0.676536	−0.338268	−	0.941050i	$-0.609841\pi$
$919$	−20.5092	−0.676536	−0.338268	+	0.941050i	$0.609841\pi$
$920$	−1.03924	−0.0342626
$921$	17.3352	0.571213
$922$	3.13484	0.103241
$923$	−0.623632	−0.0205271
$924$	−4.44422	−0.146204
$925$	−16.4257	−0.540072
$926$	29.2577	0.961469
$927$	11.3304	0.372139
$928$	−8.73682	−0.286800
$929$	−27.1856	−0.891930	−0.445965	−	0.895050i	$-0.647139\pi$
$929$	−27.1856	−0.891930	−0.445965	+	0.895050i	$0.647139\pi$
$930$	−8.62463	−0.282813
$931$	0.294177	0.00964125
$932$	10.6458	0.348714
$933$	43.6138	1.42785
$934$	−18.6225	−0.609346
$935$	−21.7685	−0.711905
$936$	−0.667188	−0.0218077
$937$	30.3798	0.992463	0.496232	−	0.868190i	$-0.334717\pi$
$937$	30.3798	0.992463	0.496232	+	0.868190i	$0.334717\pi$
$938$	1.26427	0.0412798
$939$	38.4747	1.25558
$940$	−11.0088	−0.359067
$941$	−2.81186	−0.0916640	−0.0458320	−	0.998949i	$-0.514594\pi$
$941$	−2.81186	−0.0916640	−0.0458320	+	0.998949i	$0.514594\pi$
$942$	−7.80684	−0.254360
$943$	0.460822	0.0150064
$944$	1.04345	0.0339614
$945$	−6.57310	−0.213823
$946$	−10.3983	−0.338079
$947$	33.1259	1.07645	0.538223	−	0.842802i	$-0.319096\pi$
$947$	33.1259	1.07645	0.538223	+	0.842802i	$0.319096\pi$
$948$	−20.3980	−0.662496
$949$	−0.547174	−0.0177620
$950$	0.163560	0.00530660
$951$	−9.10885	−0.295375
$952$	−4.98524	−0.161572
$953$	42.9998	1.39290	0.696450	−	0.717605i	$-0.254762\pi$
$953$	42.9998	1.39290	0.696450	+	0.717605i	$0.254762\pi$
$954$	−4.43066	−0.143448
$955$	15.4816	0.500971
$956$	−10.8499	−0.350911
$957$	−41.8876	−1.35403
$958$	−3.31455	−0.107088
$959$	−3.80717	−0.122940
$960$	−1.87833	−0.0606227
$961$	−9.91669	−0.319893
$962$	4.14080	0.133505
$963$	−5.22524	−0.168381
$964$	11.7169	0.377377
$965$	−1.42248	−0.0457914
$966$	−1.14105	−0.0367126
$967$	36.8531	1.18512	0.592558	−	0.805528i	$-0.298118\pi$
$967$	36.8531	1.18512	0.592558	+	0.805528i	$0.298118\pi$
$968$	−0.668418	−0.0214838
$969$	0.384290	0.0123452
$970$	−22.3636	−0.718051
$971$	−1.00162	−0.0321434	−0.0160717	−	0.999871i	$-0.505116\pi$
$971$	−1.00162	−0.0321434	−0.0160717	+	0.999871i	$0.505116\pi$
$972$	−7.83365	−0.251265
$973$	15.3902	0.493388
$974$	−1.84793	−0.0592114
$975$	4.38320	0.140375
$976$	−12.1728	−0.389640
$977$	15.6513	0.500729	0.250364	−	0.968152i	$-0.419450\pi$
$977$	15.6513	0.500729	0.250364	+	0.968152i	$0.419450\pi$
$978$	−6.30887	−0.201736
$979$	30.4007	0.971611
$980$	−7.73291	−0.247019
$981$	15.6081	0.498329
$982$	0.962265	0.0307071
$983$	−40.1627	−1.28099	−0.640496	−	0.767962i	$-0.721271\pi$
$983$	−40.1627	−1.28099	−0.640496	+	0.767962i	$0.721271\pi$
$984$	0.832894	0.0265517
$985$	−9.00684	−0.286982
$986$	−46.9868	−1.49636
$987$	−12.0873	−0.384742
$988$	−0.0412325	−0.00131178
$989$	−2.66976	−0.0848934
$990$	3.13763	0.0997204
$991$	12.4660	0.395995	0.197997	−	0.980203i	$-0.436556\pi$
$991$	12.4660	0.395995	0.197997	+	0.980203i	$0.436556\pi$
$992$	4.59166	0.145785
$993$	−27.1470	−0.861483
$994$	0.671644	0.0213033
$995$	3.01576	0.0956062
$996$	−20.5996	−0.652724
$997$	14.6473	0.463883	0.231942	−	0.972730i	$-0.425492\pi$
$997$	14.6473	0.463883	0.231942	+	0.972730i	$0.425492\pi$
$998$	−35.0608	−1.10983
$999$	27.0905	0.857105

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6002.2.a.a.1.15	✓	47

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6002.2.a.a.1.15	✓	47	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:	\( N \)	\(=\)	\( 6002 = 2 \cdot 3001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.49159 q^{3} +1.00000 q^{4} +1.25928 q^{5} -1.49159 q^{6} -0.926965 q^{7} +1.00000 q^{8} -0.775168 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.49159 q^{3} +1.00000 q^{4} +1.25928 q^{5} -1.49159 q^{6} -0.926965 q^{7} +1.00000 q^{8} -0.775168 q^{9} +1.25928 q^{10} -3.21428 q^{11} -1.49159 q^{12} +0.860701 q^{13} -0.926965 q^{14} -1.87833 q^{15} +1.00000 q^{16} +5.37802 q^{17} -0.775168 q^{18} -0.0479057 q^{19} +1.25928 q^{20} +1.38265 q^{21} -3.21428 q^{22} -0.825262 q^{23} -1.49159 q^{24} -3.41421 q^{25} +0.860701 q^{26} +5.63099 q^{27} -0.926965 q^{28} -8.73682 q^{29} -1.87833 q^{30} +4.59166 q^{31} +1.00000 q^{32} +4.79438 q^{33} +5.37802 q^{34} -1.16731 q^{35} -0.775168 q^{36} +4.81096 q^{37} -0.0479057 q^{38} -1.28381 q^{39} +1.25928 q^{40} -0.558394 q^{41} +1.38265 q^{42} +3.23505 q^{43} -3.21428 q^{44} -0.976153 q^{45} -0.825262 q^{46} -8.74211 q^{47} -1.49159 q^{48} -6.14074 q^{49} -3.41421 q^{50} -8.02178 q^{51} +0.860701 q^{52} +5.71575 q^{53} +5.63099 q^{54} -4.04768 q^{55} -0.926965 q^{56} +0.0714556 q^{57} -8.73682 q^{58} +1.04345 q^{59} -1.87833 q^{60} -12.1728 q^{61} +4.59166 q^{62} +0.718553 q^{63} +1.00000 q^{64} +1.08386 q^{65} +4.79438 q^{66} -1.36388 q^{67} +5.37802 q^{68} +1.23095 q^{69} -1.16731 q^{70} -0.724563 q^{71} -0.775168 q^{72} -0.635731 q^{73} +4.81096 q^{74} +5.09260 q^{75} -0.0479057 q^{76} +2.97952 q^{77} -1.28381 q^{78} +13.6754 q^{79} +1.25928 q^{80} -6.07361 q^{81} -0.558394 q^{82} +13.8105 q^{83} +1.38265 q^{84} +6.77243 q^{85} +3.23505 q^{86} +13.0317 q^{87} -3.21428 q^{88} -9.45802 q^{89} -0.976153 q^{90} -0.797840 q^{91} -0.825262 q^{92} -6.84886 q^{93} -8.74211 q^{94} -0.0603267 q^{95} -1.49159 q^{96} -17.7590 q^{97} -6.14074 q^{98} +2.49160 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 47 q + 47 q^{2} - 13 q^{3} + 47 q^{4} - 14 q^{5} - 13 q^{6} - 17 q^{7} + 47 q^{8} + 12 q^{9}+O(q^{10}) \) 47 * q + 47 * q^2 - 13 * q^3 + 47 * q^4 - 14 * q^5 - 13 * q^6 - 17 * q^7 + 47 * q^8 + 12 * q^9	\( 47 q + 47 q^{2} - 13 q^{3} + 47 q^{4} - 14 q^{5} - 13 q^{6} - 17 q^{7} + 47 q^{8} + 12 q^{9} - 14 q^{10} - 30 q^{11} - 13 q^{12} - 39 q^{13} - 17 q^{14} - 18 q^{15} + 47 q^{16} - 26 q^{17} + 12 q^{18} - 23 q^{19} - 14 q^{20} - 39 q^{21} - 30 q^{22} - 25 q^{23} - 13 q^{24} - 19 q^{25} - 39 q^{26} - 46 q^{27} - 17 q^{28} - 53 q^{29} - 18 q^{30} - 23 q^{31} + 47 q^{32} - 26 q^{33} - 26 q^{34} - 31 q^{35} + 12 q^{36} - 83 q^{37} - 23 q^{38} - 9 q^{39} - 14 q^{40} - 48 q^{41} - 39 q^{42} - 78 q^{43} - 30 q^{44} - 27 q^{45} - 25 q^{46} - 15 q^{47} - 13 q^{48} - 12 q^{49} - 19 q^{50} - 47 q^{51} - 39 q^{52} - 76 q^{53} - 46 q^{54} - 39 q^{55} - 17 q^{56} - 44 q^{57} - 53 q^{58} - 33 q^{59} - 18 q^{60} - 33 q^{61} - 23 q^{62} - 7 q^{63} + 47 q^{64} - 67 q^{65} - 26 q^{66} - 85 q^{67} - 26 q^{68} - 33 q^{69} - 31 q^{70} - 17 q^{71} + 12 q^{72} - 59 q^{73} - 83 q^{74} - 21 q^{75} - 23 q^{76} - 59 q^{77} - 9 q^{78} - 49 q^{79} - 14 q^{80} - 41 q^{81} - 48 q^{82} - 30 q^{83} - 39 q^{84} - 84 q^{85} - 78 q^{86} + 9 q^{87} - 30 q^{88} - 50 q^{89} - 27 q^{90} - 42 q^{91} - 25 q^{92} - 43 q^{93} - 15 q^{94} + 8 q^{95} - 13 q^{96} - 49 q^{97} - 12 q^{98} - 49 q^{99}+O(q^{100}) \) 47 * q + 47 * q^2 - 13 * q^3 + 47 * q^4 - 14 * q^5 - 13 * q^6 - 17 * q^7 + 47 * q^8 + 12 * q^9 - 14 * q^10 - 30 * q^11 - 13 * q^12 - 39 * q^13 - 17 * q^14 - 18 * q^15 + 47 * q^16 - 26 * q^17 + 12 * q^18 - 23 * q^19 - 14 * q^20 - 39 * q^21 - 30 * q^22 - 25 * q^23 - 13 * q^24 - 19 * q^25 - 39 * q^26 - 46 * q^27 - 17 * q^28 - 53 * q^29 - 18 * q^30 - 23 * q^31 + 47 * q^32 - 26 * q^33 - 26 * q^34 - 31 * q^35 + 12 * q^36 - 83 * q^37 - 23 * q^38 - 9 * q^39 - 14 * q^40 - 48 * q^41 - 39 * q^42 - 78 * q^43 - 30 * q^44 - 27 * q^45 - 25 * q^46 - 15 * q^47 - 13 * q^48 - 12 * q^49 - 19 * q^50 - 47 * q^51 - 39 * q^52 - 76 * q^53 - 46 * q^54 - 39 * q^55 - 17 * q^56 - 44 * q^57 - 53 * q^58 - 33 * q^59 - 18 * q^60 - 33 * q^61 - 23 * q^62 - 7 * q^63 + 47 * q^64 - 67 * q^65 - 26 * q^66 - 85 * q^67 - 26 * q^68 - 33 * q^69 - 31 * q^70 - 17 * q^71 + 12 * q^72 - 59 * q^73 - 83 * q^74 - 21 * q^75 - 23 * q^76 - 59 * q^77 - 9 * q^78 - 49 * q^79 - 14 * q^80 - 41 * q^81 - 48 * q^82 - 30 * q^83 - 39 * q^84 - 84 * q^85 - 78 * q^86 + 9 * q^87 - 30 * q^88 - 50 * q^89 - 27 * q^90 - 42 * q^91 - 25 * q^92 - 43 * q^93 - 15 * q^94 + 8 * q^95 - 13 * q^96 - 49 * q^97 - 12 * q^98 - 49 * q^99

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