LMFDB - Embedded newform 6006.2.a.cf.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6006,2,Mod(1,6006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6006 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6006.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:	$47.9581514540$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.72306708.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 11x^{4} - x^{3} + 10x^{2} - 2 $ x^6 - 11x^4 - x^3 + 10x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.717433$ of defining polynomial
Character	$\chi$	$=$	6006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.78772 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.78772 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.78772 q^{10} +1.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} -2.78772 q^{15} +1.00000 q^{16} +1.43487 q^{17} +1.00000 q^{18} +2.68789 q^{19} +2.78772 q^{20} -1.00000 q^{21} +1.00000 q^{22} -5.88710 q^{23} -1.00000 q^{24} +2.77137 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -3.89412 q^{29} -2.78772 q^{30} +7.96912 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.43487 q^{34} +2.78772 q^{35} +1.00000 q^{36} +3.35285 q^{37} +2.68789 q^{38} +1.00000 q^{39} +2.78772 q^{40} +10.5750 q^{41} -1.00000 q^{42} -4.85294 q^{43} +1.00000 q^{44} +2.78772 q^{45} -5.88710 q^{46} +6.97058 q^{47} -1.00000 q^{48} +1.00000 q^{49} +2.77137 q^{50} -1.43487 q^{51} -1.00000 q^{52} +1.26004 q^{53} -1.00000 q^{54} +2.78772 q^{55} +1.00000 q^{56} -2.68789 q^{57} -3.89412 q^{58} +11.2400 q^{59} -2.78772 q^{60} -6.91850 q^{61} +7.96912 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.78772 q^{65} -1.00000 q^{66} -1.98320 q^{67} +1.43487 q^{68} +5.88710 q^{69} +2.78772 q^{70} +5.66452 q^{71} +1.00000 q^{72} -11.3150 q^{73} +3.35285 q^{74} -2.77137 q^{75} +2.68789 q^{76} +1.00000 q^{77} +1.00000 q^{78} +16.2925 q^{79} +2.78772 q^{80} +1.00000 q^{81} +10.5750 q^{82} -12.4964 q^{83} -1.00000 q^{84} +4.00000 q^{85} -4.85294 q^{86} +3.89412 q^{87} +1.00000 q^{88} +2.46868 q^{89} +2.78772 q^{90} -1.00000 q^{91} -5.88710 q^{92} -7.96912 q^{93} +6.97058 q^{94} +7.49307 q^{95} -1.00000 q^{96} -12.1097 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{6} + 6 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^6 + 6 * q^7 + 6 * q^8 + 6 * q^9	$ 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{6} + 6 q^{7} + 6 q^{8} + 6 q^{9} + 6 q^{11} - 6 q^{12} - 6 q^{13} + 6 q^{14} + 6 q^{16} + 6 q^{18} + 2 q^{19} - 6 q^{21} + 6 q^{22} + 10 q^{23} - 6 q^{24} + 10 q^{25} - 6 q^{26} - 6 q^{27} + 6 q^{28} + 6 q^{29} + 2 q^{31} + 6 q^{32} - 6 q^{33} + 6 q^{36} + 12 q^{37} + 2 q^{38} + 6 q^{39} + 4 q^{41} - 6 q^{42} + 4 q^{43} + 6 q^{44} + 10 q^{46} + 4 q^{47} - 6 q^{48} + 6 q^{49} + 10 q^{50} - 6 q^{52} + 18 q^{53} - 6 q^{54} + 6 q^{56} - 2 q^{57} + 6 q^{58} + 14 q^{59} + 2 q^{62} + 6 q^{63} + 6 q^{64} - 6 q^{66} + 4 q^{67} - 10 q^{69} + 14 q^{71} + 6 q^{72} + 2 q^{73} + 12 q^{74} - 10 q^{75} + 2 q^{76} + 6 q^{77} + 6 q^{78} + 6 q^{79} + 6 q^{81} + 4 q^{82} + 24 q^{83} - 6 q^{84} + 24 q^{85} + 4 q^{86} - 6 q^{87} + 6 q^{88} - 14 q^{89} - 6 q^{91} + 10 q^{92} - 2 q^{93} + 4 q^{94} + 4 q^{95} - 6 q^{96} - 2 q^{97} + 6 q^{98} + 6 q^{99}+O(q^{100}) $ 6 * q + 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^6 + 6 * q^7 + 6 * q^8 + 6 * q^9 + 6 * q^11 - 6 * q^12 - 6 * q^13 + 6 * q^14 + 6 * q^16 + 6 * q^18 + 2 * q^19 - 6 * q^21 + 6 * q^22 + 10 * q^23 - 6 * q^24 + 10 * q^25 - 6 * q^26 - 6 * q^27 + 6 * q^28 + 6 * q^29 + 2 * q^31 + 6 * q^32 - 6 * q^33 + 6 * q^36 + 12 * q^37 + 2 * q^38 + 6 * q^39 + 4 * q^41 - 6 * q^42 + 4 * q^43 + 6 * q^44 + 10 * q^46 + 4 * q^47 - 6 * q^48 + 6 * q^49 + 10 * q^50 - 6 * q^52 + 18 * q^53 - 6 * q^54 + 6 * q^56 - 2 * q^57 + 6 * q^58 + 14 * q^59 + 2 * q^62 + 6 * q^63 + 6 * q^64 - 6 * q^66 + 4 * q^67 - 10 * q^69 + 14 * q^71 + 6 * q^72 + 2 * q^73 + 12 * q^74 - 10 * q^75 + 2 * q^76 + 6 * q^77 + 6 * q^78 + 6 * q^79 + 6 * q^81 + 4 * q^82 + 24 * q^83 - 6 * q^84 + 24 * q^85 + 4 * q^86 - 6 * q^87 + 6 * q^88 - 14 * q^89 - 6 * q^91 + 10 * q^92 - 2 * q^93 + 4 * q^94 + 4 * q^95 - 6 * q^96 - 2 * q^97 + 6 * q^98 + 6 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.78772	1.24670	0.623352	−	0.781941i	$-0.285770\pi$
$5$	2.78772	1.24670	0.623352	+	0.781941i	$0.285770\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.78772	0.881553
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.00000	0.267261
$15$	−2.78772	−0.719785
$16$	1.00000	0.250000
$17$	1.43487	0.348006	0.174003	−	0.984745i	$-0.444330\pi$
$17$	1.43487	0.348006	0.174003	+	0.984745i	$0.444330\pi$
$18$	1.00000	0.235702
$19$	2.68789	0.616644	0.308322	−	0.951282i	$-0.400233\pi$
$19$	2.68789	0.616644	0.308322	+	0.951282i	$0.400233\pi$
$20$	2.78772	0.623352
$21$	−1.00000	−0.218218
$22$	1.00000	0.213201
$23$	−5.88710	−1.22755	−0.613773	−	0.789483i	$-0.710349\pi$
$23$	−5.88710	−1.22755	−0.613773	+	0.789483i	$0.710349\pi$
$24$	−1.00000	−0.204124
$25$	2.77137	0.554273
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−3.89412	−0.723120	−0.361560	−	0.932349i	$-0.617756\pi$
$29$	−3.89412	−0.723120	−0.361560	+	0.932349i	$0.617756\pi$
$30$	−2.78772	−0.508965
$31$	7.96912	1.43130	0.715648	−	0.698461i	$-0.246132\pi$
$31$	7.96912	1.43130	0.715648	+	0.698461i	$0.246132\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	1.43487	0.246077
$35$	2.78772	0.471210
$36$	1.00000	0.166667
$37$	3.35285	0.551205	0.275603	−	0.961272i	$-0.411123\pi$
$37$	3.35285	0.551205	0.275603	+	0.961272i	$0.411123\pi$
$38$	2.68789	0.436033
$39$	1.00000	0.160128
$40$	2.78772	0.440777
$41$	10.5750	1.65154	0.825768	−	0.564010i	$-0.190742\pi$
$41$	10.5750	1.65154	0.825768	+	0.564010i	$0.190742\pi$
$42$	−1.00000	−0.154303
$43$	−4.85294	−0.740066	−0.370033	−	0.929019i	$-0.620654\pi$
$43$	−4.85294	−0.740066	−0.370033	+	0.929019i	$0.620654\pi$
$44$	1.00000	0.150756
$45$	2.78772	0.415568
$46$	−5.88710	−0.868006
$47$	6.97058	1.01676	0.508382	−	0.861132i	$-0.330244\pi$
$47$	6.97058	1.01676	0.508382	+	0.861132i	$0.330244\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	2.77137	0.391930
$51$	−1.43487	−0.200921
$52$	−1.00000	−0.138675
$53$	1.26004	0.173079	0.0865397	−	0.996248i	$-0.472419\pi$
$53$	1.26004	0.173079	0.0865397	+	0.996248i	$0.472419\pi$
$54$	−1.00000	−0.136083
$55$	2.78772	0.375896
$56$	1.00000	0.133631
$57$	−2.68789	−0.356019
$58$	−3.89412	−0.511323
$59$	11.2400	1.46332	0.731659	−	0.681671i	$-0.238747\pi$
$59$	11.2400	1.46332	0.731659	+	0.681671i	$0.238747\pi$
$60$	−2.78772	−0.359893
$61$	−6.91850	−0.885824	−0.442912	−	0.896565i	$-0.646055\pi$
$61$	−6.91850	−0.885824	−0.442912	+	0.896565i	$0.646055\pi$
$62$	7.96912	1.01208
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	−2.78772	−0.345774
$66$	−1.00000	−0.123091
$67$	−1.98320	−0.242287	−0.121143	−	0.992635i	$-0.538656\pi$
$67$	−1.98320	−0.242287	−0.121143	+	0.992635i	$0.538656\pi$
$68$	1.43487	0.174003
$69$	5.88710	0.708724
$70$	2.78772	0.333196
$71$	5.66452	0.672255	0.336127	−	0.941817i	$-0.390883\pi$
$71$	5.66452	0.672255	0.336127	+	0.941817i	$0.390883\pi$
$72$	1.00000	0.117851
$73$	−11.3150	−1.32431	−0.662157	−	0.749365i	$-0.730359\pi$
$73$	−11.3150	−1.32431	−0.662157	+	0.749365i	$0.730359\pi$
$74$	3.35285	0.389761
$75$	−2.77137	−0.320010
$76$	2.68789	0.308322
$77$	1.00000	0.113961
$78$	1.00000	0.113228
$79$	16.2925	1.83305	0.916527	−	0.399972i	$-0.130980\pi$
$79$	16.2925	1.83305	0.916527	+	0.399972i	$0.130980\pi$
$80$	2.78772	0.311676
$81$	1.00000	0.111111
$82$	10.5750	1.16781
$83$	−12.4964	−1.37165	−0.685826	−	0.727765i	$-0.740559\pi$
$83$	−12.4964	−1.37165	−0.685826	+	0.727765i	$0.740559\pi$
$84$	−1.00000	−0.109109
$85$	4.00000	0.433861
$86$	−4.85294	−0.523306
$87$	3.89412	0.417493
$88$	1.00000	0.106600
$89$	2.46868	0.261680	0.130840	−	0.991404i	$-0.458233\pi$
$89$	2.46868	0.261680	0.130840	+	0.991404i	$0.458233\pi$
$90$	2.78772	0.293851
$91$	−1.00000	−0.104828
$92$	−5.88710	−0.613773
$93$	−7.96912	−0.826359
$94$	6.97058	0.718960
$95$	7.49307	0.768773
$96$	−1.00000	−0.102062
$97$	−12.1097	−1.22955	−0.614776	−	0.788702i	$-0.710754\pi$
$97$	−12.1097	−1.22955	−0.614776	+	0.788702i	$0.710754\pi$
$98$	1.00000	0.101015
$99$	1.00000	0.100504
$100$	2.77137	0.277137
$101$	17.8398	1.77512	0.887562	−	0.460689i	$-0.152397\pi$
$101$	17.8398	1.77512	0.887562	+	0.460689i	$0.152397\pi$
$102$	−1.43487	−0.142073
$103$	−6.97058	−0.686832	−0.343416	−	0.939183i	$-0.611584\pi$
$103$	−6.97058	−0.686832	−0.343416	+	0.939183i	$0.611584\pi$
$104$	−1.00000	−0.0980581
$105$	−2.78772	−0.272053
$106$	1.26004	0.122386
$107$	16.0999	1.55644	0.778218	−	0.627994i	$-0.216124\pi$
$107$	16.0999	1.55644	0.778218	+	0.627994i	$0.216124\pi$
$108$	−1.00000	−0.0962250
$109$	17.9382	1.71817	0.859086	−	0.511832i	$-0.171033\pi$
$109$	17.9382	1.71817	0.859086	+	0.511832i	$0.171033\pi$
$110$	2.78772	0.265798
$111$	−3.35285	−0.318239
$112$	1.00000	0.0944911
$113$	−13.9739	−1.31455	−0.657275	−	0.753651i	$-0.728291\pi$
$113$	−13.9739	−1.31455	−0.657275	+	0.753651i	$0.728291\pi$
$114$	−2.68789	−0.251744
$115$	−16.4116	−1.53039
$116$	−3.89412	−0.361560
$117$	−1.00000	−0.0924500
$118$	11.2400	1.03472
$119$	1.43487	0.131534
$120$	−2.78772	−0.254483
$121$	1.00000	0.0909091
$122$	−6.91850	−0.626372
$123$	−10.5750	−0.953515
$124$	7.96912	0.715648
$125$	−6.21280	−0.555690
$126$	1.00000	0.0890871
$127$	15.1744	1.34651	0.673254	−	0.739411i	$-0.264896\pi$
$127$	15.1744	1.34651	0.673254	+	0.739411i	$0.264896\pi$
$128$	1.00000	0.0883883
$129$	4.85294	0.427277
$130$	−2.78772	−0.244499
$131$	−7.45077	−0.650977	−0.325489	−	0.945546i	$-0.605529\pi$
$131$	−7.45077	−0.650977	−0.325489	+	0.945546i	$0.605529\pi$
$132$	−1.00000	−0.0870388
$133$	2.68789	0.233069
$134$	−1.98320	−0.171323
$135$	−2.78772	−0.239928
$136$	1.43487	0.123039
$137$	−12.7095	−1.08585	−0.542923	−	0.839782i	$-0.682683\pi$
$137$	−12.7095	−1.08585	−0.542923	+	0.839782i	$0.682683\pi$
$138$	5.88710	0.501143
$139$	−0.203804	−0.0172864	−0.00864321	−	0.999963i	$-0.502751\pi$
$139$	−0.203804	−0.0172864	−0.00864321	+	0.999963i	$0.502751\pi$
$140$	2.78772	0.235605
$141$	−6.97058	−0.587029
$142$	5.66452	0.475356
$143$	−1.00000	−0.0836242
$144$	1.00000	0.0833333
$145$	−10.8557	−0.901517
$146$	−11.3150	−0.936432
$147$	−1.00000	−0.0824786
$148$	3.35285	0.275603
$149$	−14.4977	−1.18770	−0.593848	−	0.804578i	$-0.702392\pi$
$149$	−14.4977	−1.18770	−0.593848	+	0.804578i	$0.702392\pi$
$150$	−2.77137	−0.226281
$151$	14.1342	1.15022	0.575111	−	0.818075i	$-0.304959\pi$
$151$	14.1342	1.15022	0.575111	+	0.818075i	$0.304959\pi$
$152$	2.68789	0.218016
$153$	1.43487	0.116002
$154$	1.00000	0.0805823
$155$	22.2156	1.78440
$156$	1.00000	0.0800641
$157$	−0.604854	−0.0482726	−0.0241363	−	0.999709i	$-0.507684\pi$
$157$	−0.604854	−0.0482726	−0.0241363	+	0.999709i	$0.507684\pi$
$158$	16.2925	1.29617
$159$	−1.26004	−0.0999275
$160$	2.78772	0.220388
$161$	−5.88710	−0.463969
$162$	1.00000	0.0785674
$163$	−5.05259	−0.395750	−0.197875	−	0.980227i	$-0.563404\pi$
$163$	−5.05259	−0.395750	−0.197875	+	0.980227i	$0.563404\pi$
$164$	10.5750	0.825768
$165$	−2.78772	−0.217023
$166$	−12.4964	−0.969905
$167$	−9.29714	−0.719434	−0.359717	−	0.933061i	$-0.617127\pi$
$167$	−9.29714	−0.719434	−0.359717	+	0.933061i	$0.617127\pi$
$168$	−1.00000	−0.0771517
$169$	1.00000	0.0769231
$170$	4.00000	0.306786
$171$	2.68789	0.205548
$172$	−4.85294	−0.370033
$173$	−21.3469	−1.62297	−0.811486	−	0.584371i	$-0.801341\pi$
$173$	−21.3469	−1.62297	−0.811486	+	0.584371i	$0.801341\pi$
$174$	3.89412	0.295212
$175$	2.77137	0.209496
$176$	1.00000	0.0753778
$177$	−11.2400	−0.844847
$178$	2.46868	0.185036
$179$	−14.2653	−1.06624	−0.533119	−	0.846040i	$-0.678980\pi$
$179$	−14.2653	−1.06624	−0.533119	+	0.846040i	$0.678980\pi$
$180$	2.78772	0.207784
$181$	4.99956	0.371614	0.185807	−	0.982586i	$-0.440510\pi$
$181$	4.99956	0.371614	0.185807	+	0.982586i	$0.440510\pi$
$182$	−1.00000	−0.0741249
$183$	6.91850	0.511431
$184$	−5.88710	−0.434003
$185$	9.34680	0.687190
$186$	−7.96912	−0.584324
$187$	1.43487	0.104928
$188$	6.97058	0.508382
$189$	−1.00000	−0.0727393
$190$	7.49307	0.543604
$191$	−9.67534	−0.700083	−0.350041	−	0.936734i	$-0.613833\pi$
$191$	−9.67534	−0.700083	−0.350041	+	0.936734i	$0.613833\pi$
$192$	−1.00000	−0.0721688
$193$	−0.508402	−0.0365956	−0.0182978	−	0.999833i	$-0.505825\pi$
$193$	−0.508402	−0.0365956	−0.0182978	+	0.999833i	$0.505825\pi$
$194$	−12.1097	−0.869425
$195$	2.78772	0.199633
$196$	1.00000	0.0714286
$197$	3.81810	0.272028	0.136014	−	0.990707i	$-0.456571\pi$
$197$	3.81810	0.272028	0.136014	+	0.990707i	$0.456571\pi$
$198$	1.00000	0.0710669
$199$	22.0033	1.55977	0.779885	−	0.625923i	$-0.215278\pi$
$199$	22.0033	1.55977	0.779885	+	0.625923i	$0.215278\pi$
$200$	2.77137	0.195965
$201$	1.98320	0.139884
$202$	17.8398	1.25520
$203$	−3.89412	−0.273314
$204$	−1.43487	−0.100461
$205$	29.4801	2.05898
$206$	−6.97058	−0.485663
$207$	−5.88710	−0.409182
$208$	−1.00000	−0.0693375
$209$	2.68789	0.185925
$210$	−2.78772	−0.192371
$211$	19.8041	1.36337	0.681687	−	0.731644i	$-0.261247\pi$
$211$	19.8041	1.36337	0.681687	+	0.731644i	$0.261247\pi$
$212$	1.26004	0.0865397
$213$	−5.66452	−0.388126
$214$	16.0999	1.10057
$215$	−13.5286	−0.922644
$216$	−1.00000	−0.0680414
$217$	7.96912	0.540979
$218$	17.9382	1.21493
$219$	11.3150	0.764594
$220$	2.78772	0.187948
$221$	−1.43487	−0.0965195
$222$	−3.35285	−0.225029
$223$	0.426387	0.0285530	0.0142765	−	0.999898i	$-0.495456\pi$
$223$	0.426387	0.0285530	0.0142765	+	0.999898i	$0.495456\pi$
$224$	1.00000	0.0668153
$225$	2.77137	0.184758
$226$	−13.9739	−0.929528
$227$	−20.1346	−1.33638	−0.668190	−	0.743991i	$-0.732931\pi$
$227$	−20.1346	−1.33638	−0.668190	+	0.743991i	$0.732931\pi$
$228$	−2.68789	−0.178010
$229$	−17.0544	−1.12699	−0.563494	−	0.826120i	$-0.690543\pi$
$229$	−17.0544	−1.12699	−0.563494	+	0.826120i	$0.690543\pi$
$230$	−16.4116	−1.08215
$231$	−1.00000	−0.0657952
$232$	−3.89412	−0.255661
$233$	25.8085	1.69077	0.845387	−	0.534155i	$-0.179370\pi$
$233$	25.8085	1.69077	0.845387	+	0.534155i	$0.179370\pi$
$234$	−1.00000	−0.0653720
$235$	19.4320	1.26760
$236$	11.2400	0.731659
$237$	−16.2925	−1.05831
$238$	1.43487	0.0930085
$239$	16.9381	1.09563	0.547816	−	0.836599i	$-0.315459\pi$
$239$	16.9381	1.09563	0.547816	+	0.836599i	$0.315459\pi$
$240$	−2.78772	−0.179946
$241$	−15.7406	−1.01394	−0.506969	−	0.861964i	$-0.669234\pi$
$241$	−15.7406	−1.01394	−0.506969	+	0.861964i	$0.669234\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	−6.91850	−0.442912
$245$	2.78772	0.178101
$246$	−10.5750	−0.674237
$247$	−2.68789	−0.171026
$248$	7.96912	0.506039
$249$	12.4964	0.791924
$250$	−6.21280	−0.392932
$251$	7.95034	0.501821	0.250910	−	0.968010i	$-0.419270\pi$
$251$	7.95034	0.501821	0.250910	+	0.968010i	$0.419270\pi$
$252$	1.00000	0.0629941
$253$	−5.88710	−0.370119
$254$	15.1744	0.952126
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	23.7873	1.48381	0.741907	−	0.670503i	$-0.233922\pi$
$257$	23.7873	1.48381	0.741907	+	0.670503i	$0.233922\pi$
$258$	4.85294	0.302131
$259$	3.35285	0.208336
$260$	−2.78772	−0.172887
$261$	−3.89412	−0.241040
$262$	−7.45077	−0.460310
$263$	6.02821	0.371715	0.185858	−	0.982577i	$-0.440494\pi$
$263$	6.02821	0.371715	0.185858	+	0.982577i	$0.440494\pi$
$264$	−1.00000	−0.0615457
$265$	3.51263	0.215779
$266$	2.68789	0.164805
$267$	−2.46868	−0.151081
$268$	−1.98320	−0.121143
$269$	1.55389	0.0947424	0.0473712	−	0.998877i	$-0.484916\pi$
$269$	1.55389	0.0947424	0.0473712	+	0.998877i	$0.484916\pi$
$270$	−2.78772	−0.169655
$271$	−12.4123	−0.753992	−0.376996	−	0.926215i	$-0.623043\pi$
$271$	−12.4123	−0.753992	−0.376996	+	0.926215i	$0.623043\pi$
$272$	1.43487	0.0870015
$273$	1.00000	0.0605228
$274$	−12.7095	−0.767809
$275$	2.77137	0.167120
$276$	5.88710	0.354362
$277$	−15.8285	−0.951045	−0.475523	−	0.879703i	$-0.657741\pi$
$277$	−15.8285	−0.951045	−0.475523	+	0.879703i	$0.657741\pi$
$278$	−0.203804	−0.0122233
$279$	7.96912	0.477099
$280$	2.78772	0.166598
$281$	30.3975	1.81337	0.906683	−	0.421814i	$-0.138606\pi$
$281$	30.3975	1.81337	0.906683	+	0.421814i	$0.138606\pi$
$282$	−6.97058	−0.415092
$283$	3.59654	0.213792	0.106896	−	0.994270i	$-0.465909\pi$
$283$	3.59654	0.213792	0.106896	+	0.994270i	$0.465909\pi$
$284$	5.66452	0.336127
$285$	−7.49307	−0.443851
$286$	−1.00000	−0.0591312
$287$	10.5750	0.624222
$288$	1.00000	0.0589256
$289$	−14.9412	−0.878892
$290$	−10.8557	−0.637469
$291$	12.1097	0.709882
$292$	−11.3150	−0.662157
$293$	−17.2323	−1.00672	−0.503362	−	0.864076i	$-0.667904\pi$
$293$	−17.2323	−1.00672	−0.503362	+	0.864076i	$0.667904\pi$
$294$	−1.00000	−0.0583212
$295$	31.3338	1.82432
$296$	3.35285	0.194880
$297$	−1.00000	−0.0580259
$298$	−14.4977	−0.839827
$299$	5.88710	0.340460
$300$	−2.77137	−0.160005
$301$	−4.85294	−0.279719
$302$	14.1342	0.813330
$303$	−17.8398	−1.02487
$304$	2.68789	0.154161
$305$	−19.2868	−1.10436
$306$	1.43487	0.0820258
$307$	26.5146	1.51327	0.756634	−	0.653839i	$-0.226843\pi$
$307$	26.5146	1.51327	0.756634	+	0.653839i	$0.226843\pi$
$308$	1.00000	0.0569803
$309$	6.97058	0.396542
$310$	22.2156	1.26176
$311$	−12.4799	−0.707667	−0.353834	−	0.935308i	$-0.615122\pi$
$311$	−12.4799	−0.707667	−0.353834	+	0.935308i	$0.615122\pi$
$312$	1.00000	0.0566139
$313$	−11.0336	−0.623654	−0.311827	−	0.950139i	$-0.600941\pi$
$313$	−11.0336	−0.623654	−0.311827	+	0.950139i	$0.600941\pi$
$314$	−0.604854	−0.0341339
$315$	2.78772	0.157070
$316$	16.2925	0.916527
$317$	23.6225	1.32677	0.663385	−	0.748278i	$-0.269119\pi$
$317$	23.6225	1.32677	0.663385	+	0.748278i	$0.269119\pi$
$318$	−1.26004	−0.0706594
$319$	−3.89412	−0.218029
$320$	2.78772	0.155838
$321$	−16.0999	−0.898609
$322$	−5.88710	−0.328075
$323$	3.85676	0.214596
$324$	1.00000	0.0555556
$325$	−2.77137	−0.153728
$326$	−5.05259	−0.279837
$327$	−17.9382	−0.991987
$328$	10.5750	0.583906
$329$	6.97058	0.384301
$330$	−2.78772	−0.153459
$331$	−24.7422	−1.35995	−0.679977	−	0.733233i	$-0.738010\pi$
$331$	−24.7422	−1.35995	−0.679977	+	0.733233i	$0.738010\pi$
$332$	−12.4964	−0.685826
$333$	3.35285	0.183735
$334$	−9.29714	−0.508717
$335$	−5.52861	−0.302060
$336$	−1.00000	−0.0545545
$337$	−4.01314	−0.218610	−0.109305	−	0.994008i	$-0.534862\pi$
$337$	−4.01314	−0.218610	−0.109305	+	0.994008i	$0.534862\pi$
$338$	1.00000	0.0543928
$339$	13.9739	0.758956
$340$	4.00000	0.216930
$341$	7.96912	0.431552
$342$	2.68789	0.145344
$343$	1.00000	0.0539949
$344$	−4.85294	−0.261653
$345$	16.4116	0.883569
$346$	−21.3469	−1.14762
$347$	4.76894	0.256010	0.128005	−	0.991774i	$-0.459143\pi$
$347$	4.76894	0.256010	0.128005	+	0.991774i	$0.459143\pi$
$348$	3.89412	0.208747
$349$	−21.1181	−1.13042	−0.565212	−	0.824945i	$-0.691206\pi$
$349$	−21.1181	−1.13042	−0.565212	+	0.824945i	$0.691206\pi$
$350$	2.77137	0.148136
$351$	1.00000	0.0533761
$352$	1.00000	0.0533002
$353$	3.45641	0.183966	0.0919830	−	0.995761i	$-0.470679\pi$
$353$	3.45641	0.183966	0.0919830	+	0.995761i	$0.470679\pi$
$354$	−11.2400	−0.597397
$355$	15.7911	0.838103
$356$	2.46868	0.130840
$357$	−1.43487	−0.0759412
$358$	−14.2653	−0.753945
$359$	−22.1230	−1.16761	−0.583803	−	0.811895i	$-0.698436\pi$
$359$	−22.1230	−1.16761	−0.583803	+	0.811895i	$0.698436\pi$
$360$	2.78772	0.146926
$361$	−11.7753	−0.619751
$362$	4.99956	0.262771
$363$	−1.00000	−0.0524864
$364$	−1.00000	−0.0524142
$365$	−31.5429	−1.65103
$366$	6.91850	0.361636
$367$	7.24968	0.378430	0.189215	−	0.981936i	$-0.439406\pi$
$367$	7.24968	0.378430	0.189215	+	0.981936i	$0.439406\pi$
$368$	−5.88710	−0.306886
$369$	10.5750	0.550512
$370$	9.34680	0.485917
$371$	1.26004	0.0654179
$372$	−7.96912	−0.413179
$373$	−0.759158	−0.0393077	−0.0196539	−	0.999807i	$-0.506256\pi$
$373$	−0.759158	−0.0393077	−0.0196539	+	0.999807i	$0.506256\pi$
$374$	1.43487	0.0741952
$375$	6.21280	0.320828
$376$	6.97058	0.359480
$377$	3.89412	0.200557
$378$	−1.00000	−0.0514344
$379$	−0.793056	−0.0407365	−0.0203683	−	0.999793i	$-0.506484\pi$
$379$	−0.793056	−0.0407365	−0.0203683	+	0.999793i	$0.506484\pi$
$380$	7.49307	0.384386
$381$	−15.1744	−0.777407
$382$	−9.67534	−0.495033
$383$	31.0749	1.58785	0.793926	−	0.608014i	$-0.208034\pi$
$383$	31.0749	1.58785	0.793926	+	0.608014i	$0.208034\pi$
$384$	−1.00000	−0.0510310
$385$	2.78772	0.142075
$386$	−0.508402	−0.0258770
$387$	−4.85294	−0.246689
$388$	−12.1097	−0.614776
$389$	−21.3549	−1.08274	−0.541369	−	0.840785i	$-0.682094\pi$
$389$	−21.3549	−1.08274	−0.541369	+	0.840785i	$0.682094\pi$
$390$	2.78772	0.141162
$391$	−8.44720	−0.427193
$392$	1.00000	0.0505076
$393$	7.45077	0.375842
$394$	3.81810	0.192353
$395$	45.4190	2.28528
$396$	1.00000	0.0502519
$397$	24.5636	1.23281	0.616405	−	0.787430i	$-0.288589\pi$
$397$	24.5636	1.23281	0.616405	+	0.787430i	$0.288589\pi$
$398$	22.0033	1.10292
$399$	−2.68789	−0.134563
$400$	2.77137	0.138568
$401$	33.3469	1.66526	0.832632	−	0.553827i	$-0.186833\pi$
$401$	33.3469	1.66526	0.832632	+	0.553827i	$0.186833\pi$
$402$	1.98320	0.0989132
$403$	−7.96912	−0.396970
$404$	17.8398	0.887562
$405$	2.78772	0.138523
$406$	−3.89412	−0.193262
$407$	3.35285	0.166195
$408$	−1.43487	−0.0710365
$409$	−19.3506	−0.956825	−0.478412	−	0.878135i	$-0.658788\pi$
$409$	−19.3506	−0.956825	−0.478412	+	0.878135i	$0.658788\pi$
$410$	29.4801	1.45592
$411$	12.7095	0.626914
$412$	−6.97058	−0.343416
$413$	11.2400	0.553082
$414$	−5.88710	−0.289335
$415$	−34.8363	−1.71005
$416$	−1.00000	−0.0490290
$417$	0.203804	0.00998032
$418$	2.68789	0.131469
$419$	−33.4190	−1.63263	−0.816313	−	0.577610i	$-0.803985\pi$
$419$	−33.4190	−1.63263	−0.816313	+	0.577610i	$0.803985\pi$
$420$	−2.78772	−0.136027
$421$	29.6707	1.44606	0.723030	−	0.690817i	$-0.242749\pi$
$421$	29.6707	1.44606	0.723030	+	0.690817i	$0.242749\pi$
$422$	19.8041	0.964051
$423$	6.97058	0.338921
$424$	1.26004	0.0611928
$425$	3.97654	0.192890
$426$	−5.66452	−0.274447
$427$	−6.91850	−0.334810
$428$	16.0999	0.778218
$429$	1.00000	0.0482805
$430$	−13.5286	−0.652408
$431$	−23.2355	−1.11921	−0.559606	−	0.828759i	$-0.689048\pi$
$431$	−23.2355	−1.11921	−0.559606	+	0.828759i	$0.689048\pi$
$432$	−1.00000	−0.0481125
$433$	4.97473	0.239070	0.119535	−	0.992830i	$-0.461860\pi$
$433$	4.97473	0.239070	0.119535	+	0.992830i	$0.461860\pi$
$434$	7.96912	0.382530
$435$	10.8557	0.520491
$436$	17.9382	0.859086
$437$	−15.8239	−0.756958
$438$	11.3150	0.540649
$439$	7.58477	0.362002	0.181001	−	0.983483i	$-0.442066\pi$
$439$	7.58477	0.362002	0.181001	+	0.983483i	$0.442066\pi$
$440$	2.78772	0.132899
$441$	1.00000	0.0476190
$442$	−1.43487	−0.0682496
$443$	29.7592	1.41390	0.706951	−	0.707263i	$-0.250070\pi$
$443$	29.7592	1.41390	0.706951	+	0.707263i	$0.250070\pi$
$444$	−3.35285	−0.159119
$445$	6.88199	0.326237
$446$	0.426387	0.0201900
$447$	14.4977	0.685716
$448$	1.00000	0.0472456
$449$	22.8725	1.07942	0.539710	−	0.841851i	$-0.318534\pi$
$449$	22.8725	1.07942	0.539710	+	0.841851i	$0.318534\pi$
$450$	2.77137	0.130643
$451$	10.5750	0.497957
$452$	−13.9739	−0.657275
$453$	−14.1342	−0.664081
$454$	−20.1346	−0.944963
$455$	−2.78772	−0.130690
$456$	−2.68789	−0.125872
$457$	19.5173	0.912979	0.456489	−	0.889729i	$-0.349107\pi$
$457$	19.5173	0.912979	0.456489	+	0.889729i	$0.349107\pi$
$458$	−17.0544	−0.796900
$459$	−1.43487	−0.0669738
$460$	−16.4116	−0.765194
$461$	29.0555	1.35325	0.676625	−	0.736328i	$-0.263442\pi$
$461$	29.0555	1.35325	0.676625	+	0.736328i	$0.263442\pi$
$462$	−1.00000	−0.0465242
$463$	−26.1300	−1.21436	−0.607182	−	0.794563i	$-0.707700\pi$
$463$	−26.1300	−1.21436	−0.607182	+	0.794563i	$0.707700\pi$
$464$	−3.89412	−0.180780
$465$	−22.2156	−1.03023
$466$	25.8085	1.19556
$467$	−38.2224	−1.76872	−0.884362	−	0.466802i	$-0.845406\pi$
$467$	−38.2224	−1.76872	−0.884362	+	0.466802i	$0.845406\pi$
$468$	−1.00000	−0.0462250
$469$	−1.98320	−0.0915758
$470$	19.4320	0.896331
$471$	0.604854	0.0278702
$472$	11.2400	0.517361
$473$	−4.85294	−0.223138
$474$	−16.2925	−0.748341
$475$	7.44912	0.341789
$476$	1.43487	0.0657670
$477$	1.26004	0.0576932
$478$	16.9381	0.774729
$479$	4.01403	0.183406	0.0917029	−	0.995786i	$-0.470769\pi$
$479$	4.01403	0.183406	0.0917029	+	0.995786i	$0.470769\pi$
$480$	−2.78772	−0.127241
$481$	−3.35285	−0.152877
$482$	−15.7406	−0.716963
$483$	5.88710	0.267872
$484$	1.00000	0.0454545
$485$	−33.7584	−1.53289
$486$	−1.00000	−0.0453609
$487$	−34.9482	−1.58365	−0.791826	−	0.610747i	$-0.790869\pi$
$487$	−34.9482	−1.58365	−0.791826	+	0.610747i	$0.790869\pi$
$488$	−6.91850	−0.313186
$489$	5.05259	0.228486
$490$	2.78772	0.125936
$491$	2.70131	0.121908	0.0609541	−	0.998141i	$-0.480586\pi$
$491$	2.70131	0.121908	0.0609541	+	0.998141i	$0.480586\pi$
$492$	−10.5750	−0.476757
$493$	−5.58754	−0.251650
$494$	−2.68789	−0.120934
$495$	2.78772	0.125299
$496$	7.96912	0.357824
$497$	5.66452	0.254088
$498$	12.4964	0.559975
$499$	30.2625	1.35473	0.677367	−	0.735646i	$-0.263121\pi$
$499$	30.2625	1.35473	0.677367	+	0.735646i	$0.263121\pi$
$500$	−6.21280	−0.277845
$501$	9.29714	0.415365
$502$	7.95034	0.354841
$503$	35.6788	1.59084	0.795421	−	0.606058i	$-0.207250\pi$
$503$	35.6788	1.59084	0.795421	+	0.606058i	$0.207250\pi$
$504$	1.00000	0.0445435
$505$	49.7322	2.21306
$506$	−5.88710	−0.261714
$507$	−1.00000	−0.0444116
$508$	15.1744	0.673254
$509$	−34.2002	−1.51590	−0.757948	−	0.652315i	$-0.773798\pi$
$509$	−34.2002	−1.51590	−0.757948	+	0.652315i	$0.773798\pi$
$510$	−4.00000	−0.177123
$511$	−11.3150	−0.500544
$512$	1.00000	0.0441942
$513$	−2.68789	−0.118673
$514$	23.7873	1.04921
$515$	−19.4320	−0.856276
$516$	4.85294	0.213639
$517$	6.97058	0.306566
$518$	3.35285	0.147316
$519$	21.3469	0.937024
$520$	−2.78772	−0.122249
$521$	−13.0937	−0.573648	−0.286824	−	0.957983i	$-0.592599\pi$
$521$	−13.0937	−0.573648	−0.286824	+	0.957983i	$0.592599\pi$
$522$	−3.89412	−0.170441
$523$	25.6192	1.12025	0.560124	−	0.828409i	$-0.310753\pi$
$523$	25.6192	1.12025	0.560124	+	0.828409i	$0.310753\pi$
$524$	−7.45077	−0.325489
$525$	−2.77137	−0.120952
$526$	6.02821	0.262842
$527$	11.4346	0.498100
$528$	−1.00000	−0.0435194
$529$	11.6580	0.506868
$530$	3.51263	0.152579
$531$	11.2400	0.487772
$532$	2.68789	0.116535
$533$	−10.5750	−0.458054
$534$	−2.46868	−0.106830
$535$	44.8820	1.94042
$536$	−1.98320	−0.0856614
$537$	14.2653	0.615593
$538$	1.55389	0.0669930
$539$	1.00000	0.0430730
$540$	−2.78772	−0.119964
$541$	−43.5157	−1.87089	−0.935443	−	0.353477i	$-0.884999\pi$
$541$	−43.5157	−1.87089	−0.935443	+	0.353477i	$0.884999\pi$
$542$	−12.4123	−0.533153
$543$	−4.99956	−0.214551
$544$	1.43487	0.0615194
$545$	50.0067	2.14205
$546$	1.00000	0.0427960
$547$	−33.1079	−1.41559	−0.707797	−	0.706416i	$-0.750311\pi$
$547$	−33.1079	−1.41559	−0.707797	+	0.706416i	$0.750311\pi$
$548$	−12.7095	−0.542923
$549$	−6.91850	−0.295275
$550$	2.77137	0.118171
$551$	−10.4670	−0.445907
$552$	5.88710	0.250572
$553$	16.2925	0.692830
$554$	−15.8285	−0.672491
$555$	−9.34680	−0.396750
$556$	−0.203804	−0.00864321
$557$	−17.6222	−0.746675	−0.373338	−	0.927696i	$-0.621787\pi$
$557$	−17.6222	−0.746675	−0.373338	+	0.927696i	$0.621787\pi$
$558$	7.96912	0.337360
$559$	4.85294	0.205257
$560$	2.78772	0.117803
$561$	−1.43487	−0.0605801
$562$	30.3975	1.28224
$563$	10.8492	0.457239	0.228619	−	0.973516i	$-0.426579\pi$
$563$	10.8492	0.457239	0.228619	+	0.973516i	$0.426579\pi$
$564$	−6.97058	−0.293514
$565$	−38.9552	−1.63886
$566$	3.59654	0.151174
$567$	1.00000	0.0419961
$568$	5.66452	0.237678
$569$	−41.0053	−1.71903	−0.859515	−	0.511111i	$-0.829234\pi$
$569$	−41.0053	−1.71903	−0.859515	+	0.511111i	$0.829234\pi$
$570$	−7.49307	−0.313850
$571$	30.5595	1.27888	0.639438	−	0.768843i	$-0.279167\pi$
$571$	30.5595	1.27888	0.639438	+	0.768843i	$0.279167\pi$
$572$	−1.00000	−0.0418121
$573$	9.67534	0.404193
$574$	10.5750	0.441392
$575$	−16.3153	−0.680395
$576$	1.00000	0.0416667
$577$	−6.08909	−0.253492	−0.126746	−	0.991935i	$-0.540453\pi$
$577$	−6.08909	−0.253492	−0.126746	+	0.991935i	$0.540453\pi$
$578$	−14.9412	−0.621470
$579$	0.508402	0.0211285
$580$	−10.8557	−0.450758
$581$	−12.4964	−0.518436
$582$	12.1097	0.501963
$583$	1.26004	0.0521854
$584$	−11.3150	−0.468216
$585$	−2.78772	−0.115258
$586$	−17.2323	−0.711862
$587$	5.20725	0.214926	0.107463	−	0.994209i	$-0.465727\pi$
$587$	5.20725	0.214926	0.107463	+	0.994209i	$0.465727\pi$
$588$	−1.00000	−0.0412393
$589$	21.4201	0.882599
$590$	31.3338	1.28999
$591$	−3.81810	−0.157056
$592$	3.35285	0.137801
$593$	−40.5284	−1.66430	−0.832151	−	0.554549i	$-0.812891\pi$
$593$	−40.5284	−1.66430	−0.832151	+	0.554549i	$0.812891\pi$
$594$	−1.00000	−0.0410305
$595$	4.00000	0.163984
$596$	−14.4977	−0.593848
$597$	−22.0033	−0.900534
$598$	5.88710	0.240741
$599$	36.6286	1.49660	0.748302	−	0.663359i	$-0.230870\pi$
$599$	36.6286	1.49660	0.748302	+	0.663359i	$0.230870\pi$
$600$	−2.77137	−0.113141
$601$	−25.5702	−1.04303	−0.521516	−	0.853242i	$-0.674633\pi$
$601$	−25.5702	−1.04303	−0.521516	+	0.853242i	$0.674633\pi$
$602$	−4.85294	−0.197791
$603$	−1.98320	−0.0807623
$604$	14.1342	0.575111
$605$	2.78772	0.113337
$606$	−17.8398	−0.724691
$607$	−18.1547	−0.736876	−0.368438	−	0.929652i	$-0.620107\pi$
$607$	−18.1547	−0.736876	−0.368438	+	0.929652i	$0.620107\pi$
$608$	2.68789	0.109008
$609$	3.89412	0.157798
$610$	−19.2868	−0.780901
$611$	−6.97058	−0.281999
$612$	1.43487	0.0580010
$613$	−14.5069	−0.585929	−0.292965	−	0.956123i	$-0.594642\pi$
$613$	−14.5069	−0.585929	−0.292965	+	0.956123i	$0.594642\pi$
$614$	26.5146	1.07004
$615$	−29.4801	−1.18875
$616$	1.00000	0.0402911
$617$	37.2502	1.49964	0.749818	−	0.661644i	$-0.230141\pi$
$617$	37.2502	1.49964	0.749818	+	0.661644i	$0.230141\pi$
$618$	6.97058	0.280398
$619$	−17.3523	−0.697446	−0.348723	−	0.937226i	$-0.613385\pi$
$619$	−17.3523	−0.697446	−0.348723	+	0.937226i	$0.613385\pi$
$620$	22.2156	0.892202
$621$	5.88710	0.236241
$622$	−12.4799	−0.500396
$623$	2.46868	0.0989057
$624$	1.00000	0.0400320
$625$	−31.1764	−1.24705
$626$	−11.0336	−0.440990
$627$	−2.68789	−0.107344
$628$	−0.604854	−0.0241363
$629$	4.81089	0.191823
$630$	2.78772	0.111065
$631$	−20.0549	−0.798373	−0.399186	−	0.916870i	$-0.630707\pi$
$631$	−20.0549	−0.798373	−0.399186	+	0.916870i	$0.630707\pi$
$632$	16.2925	0.648083
$633$	−19.8041	−0.787144
$634$	23.6225	0.938168
$635$	42.3019	1.67870
$636$	−1.26004	−0.0499637
$637$	−1.00000	−0.0396214
$638$	−3.89412	−0.154170
$639$	5.66452	0.224085
$640$	2.78772	0.110194
$641$	−36.9512	−1.45948	−0.729742	−	0.683722i	$-0.760360\pi$
$641$	−36.9512	−1.45948	−0.729742	+	0.683722i	$0.760360\pi$
$642$	−16.0999	−0.635413
$643$	−5.07112	−0.199986	−0.0999928	−	0.994988i	$-0.531882\pi$
$643$	−5.07112	−0.199986	−0.0999928	+	0.994988i	$0.531882\pi$
$644$	−5.88710	−0.231984
$645$	13.5286	0.532689
$646$	3.85676	0.151742
$647$	21.5248	0.846228	0.423114	−	0.906076i	$-0.360937\pi$
$647$	21.5248	0.846228	0.423114	+	0.906076i	$0.360937\pi$
$648$	1.00000	0.0392837
$649$	11.2400	0.441207
$650$	−2.77137	−0.108702
$651$	−7.96912	−0.312334
$652$	−5.05259	−0.197875
$653$	−3.14666	−0.123138	−0.0615692	−	0.998103i	$-0.519610\pi$
$653$	−3.14666	−0.123138	−0.0615692	+	0.998103i	$0.519610\pi$
$654$	−17.9382	−0.701440
$655$	−20.7706	−0.811576
$656$	10.5750	0.412884
$657$	−11.3150	−0.441438
$658$	6.97058	0.271741
$659$	25.4017	0.989508	0.494754	−	0.869033i	$-0.335258\pi$
$659$	25.4017	0.989508	0.494754	+	0.869033i	$0.335258\pi$
$660$	−2.78772	−0.108512
$661$	−33.2345	−1.29267	−0.646336	−	0.763053i	$-0.723700\pi$
$661$	−33.2345	−1.29267	−0.646336	+	0.763053i	$0.723700\pi$
$662$	−24.7422	−0.961633
$663$	1.43487	0.0557256
$664$	−12.4964	−0.484952
$665$	7.49307	0.290569
$666$	3.35285	0.129920
$667$	22.9251	0.887662
$668$	−9.29714	−0.359717
$669$	−0.426387	−0.0164851
$670$	−5.52861	−0.213589
$671$	−6.91850	−0.267086
$672$	−1.00000	−0.0385758
$673$	−46.6685	−1.79894	−0.899469	−	0.436984i	$-0.856047\pi$
$673$	−46.6685	−1.79894	−0.899469	+	0.436984i	$0.856047\pi$
$674$	−4.01314	−0.154581
$675$	−2.77137	−0.106670
$676$	1.00000	0.0384615
$677$	33.5924	1.29106	0.645530	−	0.763735i	$-0.276637\pi$
$677$	33.5924	1.29106	0.645530	+	0.763735i	$0.276637\pi$
$678$	13.9739	0.536663
$679$	−12.1097	−0.464727
$680$	4.00000	0.153393
$681$	20.1346	0.771559
$682$	7.96912	0.305153
$683$	3.95237	0.151233	0.0756167	−	0.997137i	$-0.475907\pi$
$683$	3.95237	0.151233	0.0756167	+	0.997137i	$0.475907\pi$
$684$	2.68789	0.102774
$685$	−35.4305	−1.35373
$686$	1.00000	0.0381802
$687$	17.0544	0.650667
$688$	−4.85294	−0.185016
$689$	−1.26004	−0.0480036
$690$	16.4116	0.624778
$691$	−16.4796	−0.626914	−0.313457	−	0.949602i	$-0.601487\pi$
$691$	−16.4796	−0.626914	−0.313457	+	0.949602i	$0.601487\pi$
$692$	−21.3469	−0.811486
$693$	1.00000	0.0379869
$694$	4.76894	0.181026
$695$	−0.568148	−0.0215511
$696$	3.89412	0.147606
$697$	15.1737	0.574745
$698$	−21.1181	−0.799331
$699$	−25.8085	−0.976169
$700$	2.77137	0.104748
$701$	35.6301	1.34573	0.672865	−	0.739765i	$-0.265064\pi$
$701$	35.6301	1.34573	0.672865	+	0.739765i	$0.265064\pi$
$702$	1.00000	0.0377426
$703$	9.01208	0.339897
$704$	1.00000	0.0376889
$705$	−19.4320	−0.731852
$706$	3.45641	0.130084
$707$	17.8398	0.670934
$708$	−11.2400	−0.422423
$709$	−8.43539	−0.316798	−0.158399	−	0.987375i	$-0.550633\pi$
$709$	−8.43539	−0.316798	−0.158399	+	0.987375i	$0.550633\pi$
$710$	15.7911	0.592628
$711$	16.2925	0.611018
$712$	2.46868	0.0925178
$713$	−46.9150	−1.75698
$714$	−1.43487	−0.0536985
$715$	−2.78772	−0.104255
$716$	−14.2653	−0.533119
$717$	−16.9381	−0.632564
$718$	−22.1230	−0.825622
$719$	−9.56972	−0.356890	−0.178445	−	0.983950i	$-0.557107\pi$
$719$	−9.56972	−0.356890	−0.178445	+	0.983950i	$0.557107\pi$
$720$	2.78772	0.103892
$721$	−6.97058	−0.259598
$722$	−11.7753	−0.438230
$723$	15.7406	0.585398
$724$	4.99956	0.185807
$725$	−10.7920	−0.400806
$726$	−1.00000	−0.0371135
$727$	−3.72302	−0.138079	−0.0690395	−	0.997614i	$-0.521993\pi$
$727$	−3.72302	−0.138079	−0.0690395	+	0.997614i	$0.521993\pi$
$728$	−1.00000	−0.0370625
$729$	1.00000	0.0370370
$730$	−31.5429	−1.16745
$731$	−6.96331	−0.257547
$732$	6.91850	0.255715
$733$	−24.4509	−0.903116	−0.451558	−	0.892242i	$-0.649132\pi$
$733$	−24.4509	−0.903116	−0.451558	+	0.892242i	$0.649132\pi$
$734$	7.24968	0.267591
$735$	−2.78772	−0.102826
$736$	−5.88710	−0.217001
$737$	−1.98320	−0.0730522
$738$	10.5750	0.389271
$739$	−18.1008	−0.665848	−0.332924	−	0.942954i	$-0.608035\pi$
$739$	−18.1008	−0.665848	−0.332924	+	0.942954i	$0.608035\pi$
$740$	9.34680	0.343595
$741$	2.68789	0.0987420
$742$	1.26004	0.0462574
$743$	−24.1380	−0.885539	−0.442769	−	0.896635i	$-0.646004\pi$
$743$	−24.1380	−0.885539	−0.442769	+	0.896635i	$0.646004\pi$
$744$	−7.96912	−0.292162
$745$	−40.4154	−1.48071
$746$	−0.759158	−0.0277948
$747$	−12.4964	−0.457218
$748$	1.43487	0.0524639
$749$	16.0999	0.588278
$750$	6.21280	0.226859
$751$	26.5814	0.969968	0.484984	−	0.874523i	$-0.338825\pi$
$751$	26.5814	0.969968	0.484984	+	0.874523i	$0.338825\pi$
$752$	6.97058	0.254191
$753$	−7.95034	−0.289726
$754$	3.89412	0.141815
$755$	39.4021	1.43399
$756$	−1.00000	−0.0363696
$757$	38.3694	1.39456	0.697279	−	0.716800i	$-0.254394\pi$
$757$	38.3694	1.39456	0.697279	+	0.716800i	$0.254394\pi$
$758$	−0.793056	−0.0288051
$759$	5.88710	0.213688
$760$	7.49307	0.271802
$761$	−13.8731	−0.502900	−0.251450	−	0.967870i	$-0.580907\pi$
$761$	−13.8731	−0.502900	−0.251450	+	0.967870i	$0.580907\pi$
$762$	−15.1744	−0.549710
$763$	17.9382	0.649408
$764$	−9.67534	−0.350041
$765$	4.00000	0.144620
$766$	31.0749	1.12278
$767$	−11.2400	−0.405851
$768$	−1.00000	−0.0360844
$769$	−52.5730	−1.89583	−0.947916	−	0.318520i	$-0.896814\pi$
$769$	−52.5730	−1.89583	−0.947916	+	0.318520i	$0.896814\pi$
$770$	2.78772	0.100462
$771$	−23.7873	−0.856680
$772$	−0.508402	−0.0182978
$773$	−20.0820	−0.722300	−0.361150	−	0.932508i	$-0.617616\pi$
$773$	−20.0820	−0.722300	−0.361150	+	0.932508i	$0.617616\pi$
$774$	−4.85294	−0.174435
$775$	22.0853	0.793329
$776$	−12.1097	−0.434712
$777$	−3.35285	−0.120283
$778$	−21.3549	−0.765611
$779$	28.4244	1.01841
$780$	2.78772	0.0998163
$781$	5.66452	0.202692
$782$	−8.44720	−0.302071
$783$	3.89412	0.139164
$784$	1.00000	0.0357143
$785$	−1.68616	−0.0601817
$786$	7.45077	0.265760
$787$	27.1163	0.966593	0.483296	−	0.875457i	$-0.339439\pi$
$787$	27.1163	0.966593	0.483296	+	0.875457i	$0.339439\pi$
$788$	3.81810	0.136014
$789$	−6.02821	−0.214610
$790$	45.4190	1.61594
$791$	−13.9739	−0.496853
$792$	1.00000	0.0355335
$793$	6.91850	0.245683
$794$	24.5636	0.871728
$795$	−3.51263	−0.124580
$796$	22.0033	0.779885
$797$	−5.95128	−0.210805	−0.105403	−	0.994430i	$-0.533613\pi$
$797$	−5.95128	−0.210805	−0.105403	+	0.994430i	$0.533613\pi$
$798$	−2.68789	−0.0951502
$799$	10.0018	0.353840
$800$	2.77137	0.0979826
$801$	2.46868	0.0872266
$802$	33.3469	1.17752
$803$	−11.3150	−0.399296
$804$	1.98320	0.0699422
$805$	−16.4116	−0.578432
$806$	−7.96912	−0.280700
$807$	−1.55389	−0.0546996
$808$	17.8398	0.627601
$809$	4.84419	0.170313	0.0851563	−	0.996368i	$-0.472861\pi$
$809$	4.84419	0.170313	0.0851563	+	0.996368i	$0.472861\pi$
$810$	2.78772	0.0979504
$811$	21.8258	0.766409	0.383205	−	0.923663i	$-0.374820\pi$
$811$	21.8258	0.766409	0.383205	+	0.923663i	$0.374820\pi$
$812$	−3.89412	−0.136657
$813$	12.4123	0.435317
$814$	3.35285	0.117517
$815$	−14.0852	−0.493383
$816$	−1.43487	−0.0502304
$817$	−13.0441	−0.456357
$818$	−19.3506	−0.676577
$819$	−1.00000	−0.0349428
$820$	29.4801	1.02949
$821$	55.2391	1.92786	0.963928	−	0.266162i	$-0.0857557\pi$
$821$	55.2391	1.92786	0.963928	+	0.266162i	$0.0857557\pi$
$822$	12.7095	0.443295
$823$	−41.7873	−1.45662	−0.728308	−	0.685250i	$-0.759693\pi$
$823$	−41.7873	−1.45662	−0.728308	+	0.685250i	$0.759693\pi$
$824$	−6.97058	−0.242832
$825$	−2.77137	−0.0964865
$826$	11.2400	0.391088
$827$	9.72274	0.338093	0.169046	−	0.985608i	$-0.445931\pi$
$827$	9.72274	0.338093	0.169046	+	0.985608i	$0.445931\pi$
$828$	−5.88710	−0.204591
$829$	40.1122	1.39315	0.696577	−	0.717482i	$-0.254706\pi$
$829$	40.1122	1.39315	0.696577	+	0.717482i	$0.254706\pi$
$830$	−34.8363	−1.20919
$831$	15.8285	0.549086
$832$	−1.00000	−0.0346688
$833$	1.43487	0.0497152
$834$	0.203804	0.00705715
$835$	−25.9178	−0.896922
$836$	2.68789	0.0929625
$837$	−7.96912	−0.275453
$838$	−33.4190	−1.15444
$839$	21.7061	0.749376	0.374688	−	0.927151i	$-0.377750\pi$
$839$	21.7061	0.749376	0.374688	+	0.927151i	$0.377750\pi$
$840$	−2.78772	−0.0961854
$841$	−13.8358	−0.477098
$842$	29.6707	1.02252
$843$	−30.3975	−1.04695
$844$	19.8041	0.681687
$845$	2.78772	0.0959004
$846$	6.97058	0.239653
$847$	1.00000	0.0343604
$848$	1.26004	0.0432699
$849$	−3.59654	−0.123433
$850$	3.97654	0.136394
$851$	−19.7386	−0.676630
$852$	−5.66452	−0.194063
$853$	−3.55653	−0.121773	−0.0608866	−	0.998145i	$-0.519393\pi$
$853$	−3.55653	−0.121773	−0.0608866	+	0.998145i	$0.519393\pi$
$854$	−6.91850	−0.236746
$855$	7.49307	0.256258
$856$	16.0999	0.550283
$857$	13.0806	0.446823	0.223412	−	0.974724i	$-0.428281\pi$
$857$	13.0806	0.446823	0.223412	+	0.974724i	$0.428281\pi$
$858$	1.00000	0.0341394
$859$	−37.5305	−1.28053	−0.640263	−	0.768156i	$-0.721175\pi$
$859$	−37.5305	−1.28053	−0.640263	+	0.768156i	$0.721175\pi$
$860$	−13.5286	−0.461322
$861$	−10.5750	−0.360395
$862$	−23.2355	−0.791403
$863$	−0.645829	−0.0219843	−0.0109921	−	0.999940i	$-0.503499\pi$
$863$	−0.645829	−0.0219843	−0.0109921	+	0.999940i	$0.503499\pi$
$864$	−1.00000	−0.0340207
$865$	−59.5090	−2.02337
$866$	4.97473	0.169048
$867$	14.9412	0.507428
$868$	7.96912	0.270489
$869$	16.2925	0.552687
$870$	10.8557	0.368043
$871$	1.98320	0.0671983
$872$	17.9382	0.607465
$873$	−12.1097	−0.409851
$874$	−15.8239	−0.535250
$875$	−6.21280	−0.210031
$876$	11.3150	0.382297
$877$	−26.1151	−0.881843	−0.440922	−	0.897546i	$-0.645348\pi$
$877$	−26.1151	−0.881843	−0.440922	+	0.897546i	$0.645348\pi$
$878$	7.58477	0.255974
$879$	17.2323	0.581233
$880$	2.78772	0.0939739
$881$	45.9600	1.54843	0.774216	−	0.632922i	$-0.218145\pi$
$881$	45.9600	1.54843	0.774216	+	0.632922i	$0.218145\pi$
$882$	1.00000	0.0336718
$883$	15.1998	0.511515	0.255757	−	0.966741i	$-0.417675\pi$
$883$	15.1998	0.511515	0.255757	+	0.966741i	$0.417675\pi$
$884$	−1.43487	−0.0482598
$885$	−31.3338	−1.05327
$886$	29.7592	0.999779
$887$	−27.2943	−0.916452	−0.458226	−	0.888836i	$-0.651515\pi$
$887$	−27.2943	−0.916452	−0.458226	+	0.888836i	$0.651515\pi$
$888$	−3.35285	−0.112514
$889$	15.1744	0.508932
$890$	6.88199	0.230685
$891$	1.00000	0.0335013
$892$	0.426387	0.0142765
$893$	18.7361	0.626981
$894$	14.4977	0.484874
$895$	−39.7676	−1.32929
$896$	1.00000	0.0334077
$897$	−5.88710	−0.196565
$898$	22.8725	0.763265
$899$	−31.0327	−1.03500
$900$	2.77137	0.0923788
$901$	1.80799	0.0602327
$902$	10.5750	0.352109
$903$	4.85294	0.161496
$904$	−13.9739	−0.464764
$905$	13.9373	0.463293
$906$	−14.1342	−0.469576
$907$	−29.4808	−0.978893	−0.489447	−	0.872033i	$-0.662801\pi$
$907$	−29.4808	−0.978893	−0.489447	+	0.872033i	$0.662801\pi$
$908$	−20.1346	−0.668190
$909$	17.8398	0.591708
$910$	−2.78772	−0.0924119
$911$	3.66481	0.121420	0.0607102	−	0.998155i	$-0.480663\pi$
$911$	3.66481	0.121420	0.0607102	+	0.998155i	$0.480663\pi$
$912$	−2.68789	−0.0890048
$913$	−12.4964	−0.413569
$914$	19.5173	0.645574
$915$	19.2868	0.637603
$916$	−17.0544	−0.563494
$917$	−7.45077	−0.246046
$918$	−1.43487	−0.0473576
$919$	56.6882	1.86997	0.934986	−	0.354686i	$-0.115412\pi$
$919$	56.6882	1.86997	0.934986	+	0.354686i	$0.115412\pi$
$920$	−16.4116	−0.541074
$921$	−26.5146	−0.873685
$922$	29.0555	0.956892
$923$	−5.66452	−0.186450
$924$	−1.00000	−0.0328976
$925$	9.29197	0.305518
$926$	−26.1300	−0.858685
$927$	−6.97058	−0.228944
$928$	−3.89412	−0.127831
$929$	38.5979	1.26636	0.633179	−	0.774006i	$-0.281750\pi$
$929$	38.5979	1.26636	0.633179	+	0.774006i	$0.281750\pi$
$930$	−22.2156	−0.728480
$931$	2.68789	0.0880919
$932$	25.8085	0.845387
$933$	12.4799	0.408572
$934$	−38.2224	−1.25068
$935$	4.00000	0.130814
$936$	−1.00000	−0.0326860
$937$	−29.9977	−0.979983	−0.489992	−	0.871727i	$-0.663000\pi$
$937$	−29.9977	−0.979983	−0.489992	+	0.871727i	$0.663000\pi$
$938$	−1.98320	−0.0647539
$939$	11.0336	0.360067
$940$	19.4320	0.633802
$941$	−11.6111	−0.378510	−0.189255	−	0.981928i	$-0.560607\pi$
$941$	−11.6111	−0.378510	−0.189255	+	0.981928i	$0.560607\pi$
$942$	0.604854	0.0197072
$943$	−62.2560	−2.02734
$944$	11.2400	0.365829
$945$	−2.78772	−0.0906844
$946$	−4.85294	−0.157783
$947$	−23.1330	−0.751722	−0.375861	−	0.926676i	$-0.622653\pi$
$947$	−23.1330	−0.751722	−0.375861	+	0.926676i	$0.622653\pi$
$948$	−16.2925	−0.529157
$949$	11.3150	0.367299
$950$	7.44912	0.241681
$951$	−23.6225	−0.766011
$952$	1.43487	0.0465043
$953$	−39.6153	−1.28327	−0.641633	−	0.767012i	$-0.721743\pi$
$953$	−39.6153	−1.28327	−0.641633	+	0.767012i	$0.721743\pi$
$954$	1.26004	0.0407952
$955$	−26.9721	−0.872797
$956$	16.9381	0.547816
$957$	3.89412	0.125879
$958$	4.01403	0.129688
$959$	−12.7095	−0.410411
$960$	−2.78772	−0.0899732
$961$	32.5068	1.04861
$962$	−3.35285	−0.108100
$963$	16.0999	0.518812
$964$	−15.7406	−0.506969
$965$	−1.41728	−0.0456239
$966$	5.88710	0.189414
$967$	24.8566	0.799334	0.399667	−	0.916660i	$-0.369126\pi$
$967$	24.8566	0.799334	0.399667	+	0.916660i	$0.369126\pi$
$968$	1.00000	0.0321412
$969$	−3.85676	−0.123897
$970$	−33.7584	−1.08392
$971$	−9.99900	−0.320883	−0.160442	−	0.987045i	$-0.551292\pi$
$971$	−9.99900	−0.320883	−0.160442	+	0.987045i	$0.551292\pi$
$972$	−1.00000	−0.0320750
$973$	−0.203804	−0.00653365
$974$	−34.9482	−1.11981
$975$	2.77137	0.0887547
$976$	−6.91850	−0.221456
$977$	−3.75607	−0.120167	−0.0600837	−	0.998193i	$-0.519137\pi$
$977$	−3.75607	−0.120167	−0.0600837	+	0.998193i	$0.519137\pi$
$978$	5.05259	0.161564
$979$	2.46868	0.0788994
$980$	2.78772	0.0890503
$981$	17.9382	0.572724
$982$	2.70131	0.0862021
$983$	−36.8863	−1.17649	−0.588246	−	0.808682i	$-0.700181\pi$
$983$	−36.8863	−1.17649	−0.588246	+	0.808682i	$0.700181\pi$
$984$	−10.5750	−0.337118
$985$	10.6438	0.339139
$986$	−5.58754	−0.177943
$987$	−6.97058	−0.221876
$988$	−2.68789	−0.0855131
$989$	28.5697	0.908464
$990$	2.78772	0.0885995
$991$	−41.3767	−1.31438	−0.657188	−	0.753727i	$-0.728254\pi$
$991$	−41.3767	−1.31438	−0.657188	+	0.753727i	$0.728254\pi$
$992$	7.96912	0.253020
$993$	24.7422	0.785170
$994$	5.66452	0.179668
$995$	61.3389	1.94457
$996$	12.4964	0.395962
$997$	−59.9187	−1.89764	−0.948822	−	0.315813i	$-0.897723\pi$
$997$	−59.9187	−1.89764	−0.948822	+	0.315813i	$0.897723\pi$
$998$	30.2625	0.957941
$999$	−3.35285	−0.106080

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6006.2.a.cf.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6006.2.a.cf.1.5	✓	6	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:	\( N \)	\(=\)	\( 6006 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.78772 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.78772 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.78772 q^{10} +1.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} -2.78772 q^{15} +1.00000 q^{16} +1.43487 q^{17} +1.00000 q^{18} +2.68789 q^{19} +2.78772 q^{20} -1.00000 q^{21} +1.00000 q^{22} -5.88710 q^{23} -1.00000 q^{24} +2.77137 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -3.89412 q^{29} -2.78772 q^{30} +7.96912 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.43487 q^{34} +2.78772 q^{35} +1.00000 q^{36} +3.35285 q^{37} +2.68789 q^{38} +1.00000 q^{39} +2.78772 q^{40} +10.5750 q^{41} -1.00000 q^{42} -4.85294 q^{43} +1.00000 q^{44} +2.78772 q^{45} -5.88710 q^{46} +6.97058 q^{47} -1.00000 q^{48} +1.00000 q^{49} +2.77137 q^{50} -1.43487 q^{51} -1.00000 q^{52} +1.26004 q^{53} -1.00000 q^{54} +2.78772 q^{55} +1.00000 q^{56} -2.68789 q^{57} -3.89412 q^{58} +11.2400 q^{59} -2.78772 q^{60} -6.91850 q^{61} +7.96912 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.78772 q^{65} -1.00000 q^{66} -1.98320 q^{67} +1.43487 q^{68} +5.88710 q^{69} +2.78772 q^{70} +5.66452 q^{71} +1.00000 q^{72} -11.3150 q^{73} +3.35285 q^{74} -2.77137 q^{75} +2.68789 q^{76} +1.00000 q^{77} +1.00000 q^{78} +16.2925 q^{79} +2.78772 q^{80} +1.00000 q^{81} +10.5750 q^{82} -12.4964 q^{83} -1.00000 q^{84} +4.00000 q^{85} -4.85294 q^{86} +3.89412 q^{87} +1.00000 q^{88} +2.46868 q^{89} +2.78772 q^{90} -1.00000 q^{91} -5.88710 q^{92} -7.96912 q^{93} +6.97058 q^{94} +7.49307 q^{95} -1.00000 q^{96} -12.1097 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{6} + 6 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^6 + 6 * q^7 + 6 * q^8 + 6 * q^9	\( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{6} + 6 q^{7} + 6 q^{8} + 6 q^{9} + 6 q^{11} - 6 q^{12} - 6 q^{13} + 6 q^{14} + 6 q^{16} + 6 q^{18} + 2 q^{19} - 6 q^{21} + 6 q^{22} + 10 q^{23} - 6 q^{24} + 10 q^{25} - 6 q^{26} - 6 q^{27} + 6 q^{28} + 6 q^{29} + 2 q^{31} + 6 q^{32} - 6 q^{33} + 6 q^{36} + 12 q^{37} + 2 q^{38} + 6 q^{39} + 4 q^{41} - 6 q^{42} + 4 q^{43} + 6 q^{44} + 10 q^{46} + 4 q^{47} - 6 q^{48} + 6 q^{49} + 10 q^{50} - 6 q^{52} + 18 q^{53} - 6 q^{54} + 6 q^{56} - 2 q^{57} + 6 q^{58} + 14 q^{59} + 2 q^{62} + 6 q^{63} + 6 q^{64} - 6 q^{66} + 4 q^{67} - 10 q^{69} + 14 q^{71} + 6 q^{72} + 2 q^{73} + 12 q^{74} - 10 q^{75} + 2 q^{76} + 6 q^{77} + 6 q^{78} + 6 q^{79} + 6 q^{81} + 4 q^{82} + 24 q^{83} - 6 q^{84} + 24 q^{85} + 4 q^{86} - 6 q^{87} + 6 q^{88} - 14 q^{89} - 6 q^{91} + 10 q^{92} - 2 q^{93} + 4 q^{94} + 4 q^{95} - 6 q^{96} - 2 q^{97} + 6 q^{98} + 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^6 + 6 * q^7 + 6 * q^8 + 6 * q^9 + 6 * q^11 - 6 * q^12 - 6 * q^13 + 6 * q^14 + 6 * q^16 + 6 * q^18 + 2 * q^19 - 6 * q^21 + 6 * q^22 + 10 * q^23 - 6 * q^24 + 10 * q^25 - 6 * q^26 - 6 * q^27 + 6 * q^28 + 6 * q^29 + 2 * q^31 + 6 * q^32 - 6 * q^33 + 6 * q^36 + 12 * q^37 + 2 * q^38 + 6 * q^39 + 4 * q^41 - 6 * q^42 + 4 * q^43 + 6 * q^44 + 10 * q^46 + 4 * q^47 - 6 * q^48 + 6 * q^49 + 10 * q^50 - 6 * q^52 + 18 * q^53 - 6 * q^54 + 6 * q^56 - 2 * q^57 + 6 * q^58 + 14 * q^59 + 2 * q^62 + 6 * q^63 + 6 * q^64 - 6 * q^66 + 4 * q^67 - 10 * q^69 + 14 * q^71 + 6 * q^72 + 2 * q^73 + 12 * q^74 - 10 * q^75 + 2 * q^76 + 6 * q^77 + 6 * q^78 + 6 * q^79 + 6 * q^81 + 4 * q^82 + 24 * q^83 - 6 * q^84 + 24 * q^85 + 4 * q^86 - 6 * q^87 + 6 * q^88 - 14 * q^89 - 6 * q^91 + 10 * q^92 - 2 * q^93 + 4 * q^94 + 4 * q^95 - 6 * q^96 - 2 * q^97 + 6 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more