LMFDB - Embedded newform 847.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.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:	$6.76332905120$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7674048.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.10939$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.10939 q^{2} +1.69851 q^{3} +2.44952 q^{4} +0.492391 q^{5} -3.58282 q^{6} -1.00000 q^{7} -0.948212 q^{8} -0.115054 q^{9} +O(q^{10})$	\(q-2.10939 q^{2} +1.69851 q^{3} +2.44952 q^{4} +0.492391 q^{5} -3.58282 q^{6} -1.00000 q^{7} -0.948212 q^{8} -0.115054 q^{9} -1.03864 q^{10} +4.16054 q^{12} -5.30029 q^{13} +2.10939 q^{14} +0.836333 q^{15} -2.89889 q^{16} -3.03721 q^{17} +0.242693 q^{18} +4.66622 q^{19} +1.20612 q^{20} -1.69851 q^{21} -5.63835 q^{23} -1.61055 q^{24} -4.75755 q^{25} +11.1804 q^{26} -5.29096 q^{27} -2.44952 q^{28} -6.92295 q^{29} -1.76415 q^{30} -1.26565 q^{31} +8.01131 q^{32} +6.40665 q^{34} -0.492391 q^{35} -0.281826 q^{36} +10.8759 q^{37} -9.84288 q^{38} -9.00262 q^{39} -0.466891 q^{40} +1.44322 q^{41} +3.58282 q^{42} -2.88224 q^{43} -0.0566513 q^{45} +11.8935 q^{46} -8.75522 q^{47} -4.92381 q^{48} +1.00000 q^{49} +10.0355 q^{50} -5.15873 q^{51} -12.9832 q^{52} +6.63835 q^{53} +11.1607 q^{54} +0.948212 q^{56} +7.92564 q^{57} +14.6032 q^{58} -8.35733 q^{59} +2.04861 q^{60} +13.8953 q^{61} +2.66975 q^{62} +0.115054 q^{63} -11.1012 q^{64} -2.60982 q^{65} -9.70431 q^{67} -7.43970 q^{68} -9.57681 q^{69} +1.03864 q^{70} +5.94751 q^{71} +0.109095 q^{72} +3.77421 q^{73} -22.9414 q^{74} -8.08076 q^{75} +11.4300 q^{76} +18.9900 q^{78} +8.80383 q^{79} -1.42739 q^{80} -8.64160 q^{81} -3.04431 q^{82} -11.0898 q^{83} -4.16054 q^{84} -1.49549 q^{85} +6.07976 q^{86} -11.7587 q^{87} +3.10324 q^{89} +0.119500 q^{90} +5.30029 q^{91} -13.8113 q^{92} -2.14972 q^{93} +18.4682 q^{94} +2.29761 q^{95} +13.6073 q^{96} -6.31676 q^{97} -2.10939 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) $ 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9	\( 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9} + 8 q^{10} - 14 q^{12} - 4 q^{13} + 4 q^{14} + 2 q^{15} + 8 q^{16} - 22 q^{17} - 24 q^{18} - 6 q^{19} + 2 q^{20} + 2 q^{21} + 2 q^{23} + 20 q^{24} + 4 q^{25} + 6 q^{26} - 2 q^{27} - 4 q^{28} - 12 q^{29} - 20 q^{30} - 2 q^{31} - 8 q^{32} + 24 q^{34} + 4 q^{35} + 18 q^{36} + 14 q^{37} - 22 q^{38} - 20 q^{39} - 18 q^{40} - 26 q^{41} - 6 q^{42} + 4 q^{43} - 36 q^{45} - 12 q^{46} - 16 q^{47} - 24 q^{48} + 6 q^{49} + 4 q^{50} + 4 q^{51} - 12 q^{52} + 4 q^{53} + 32 q^{54} + 12 q^{56} - 20 q^{57} - 2 q^{58} - 4 q^{59} + 24 q^{60} + 8 q^{61} - 20 q^{62} - 8 q^{63} + 26 q^{64} - 24 q^{65} + 6 q^{67} - 12 q^{68} - 14 q^{69} - 8 q^{70} + 22 q^{71} - 16 q^{72} - 14 q^{73} - 44 q^{74} - 20 q^{75} + 30 q^{76} + 32 q^{78} + 28 q^{79} - 4 q^{80} - 6 q^{81} - 4 q^{82} - 22 q^{83} + 14 q^{84} + 24 q^{85} - 30 q^{86} - 22 q^{87} + 22 q^{90} + 4 q^{91} + 10 q^{92} - 50 q^{93} + 38 q^{94} + 24 q^{95} + 62 q^{96} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9 + 8 * q^10 - 14 * q^12 - 4 * q^13 + 4 * q^14 + 2 * q^15 + 8 * q^16 - 22 * q^17 - 24 * q^18 - 6 * q^19 + 2 * q^20 + 2 * q^21 + 2 * q^23 + 20 * q^24 + 4 * q^25 + 6 * q^26 - 2 * q^27 - 4 * q^28 - 12 * q^29 - 20 * q^30 - 2 * q^31 - 8 * q^32 + 24 * q^34 + 4 * q^35 + 18 * q^36 + 14 * q^37 - 22 * q^38 - 20 * q^39 - 18 * q^40 - 26 * q^41 - 6 * q^42 + 4 * q^43 - 36 * q^45 - 12 * q^46 - 16 * q^47 - 24 * q^48 + 6 * q^49 + 4 * q^50 + 4 * q^51 - 12 * q^52 + 4 * q^53 + 32 * q^54 + 12 * q^56 - 20 * q^57 - 2 * q^58 - 4 * q^59 + 24 * q^60 + 8 * q^61 - 20 * q^62 - 8 * q^63 + 26 * q^64 - 24 * q^65 + 6 * q^67 - 12 * q^68 - 14 * q^69 - 8 * q^70 + 22 * q^71 - 16 * q^72 - 14 * q^73 - 44 * q^74 - 20 * q^75 + 30 * q^76 + 32 * q^78 + 28 * q^79 - 4 * q^80 - 6 * q^81 - 4 * q^82 - 22 * q^83 + 14 * q^84 + 24 * q^85 - 30 * q^86 - 22 * q^87 + 22 * q^90 + 4 * q^91 + 10 * q^92 - 50 * q^93 + 38 * q^94 + 24 * q^95 + 62 * q^96 - 4 * q^97 - 4 * q^98

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 $

$2$	−2.10939	−1.49156	−0.745781	−	0.666191i	$-0.767924\pi$
$2$	−2.10939	−1.49156	−0.745781	+	0.666191i	$0.767924\pi$
$3$	1.69851	0.980637	0.490318	−	0.871543i	$-0.336880\pi$
$3$	1.69851	0.980637	0.490318	+	0.871543i	$0.336880\pi$
$4$	2.44952	1.22476
$5$	0.492391	0.220204	0.110102	−	0.993920i	$-0.464882\pi$
$5$	0.492391	0.220204	0.110102	+	0.993920i	$0.464882\pi$
$6$	−3.58282	−1.46268
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.948212	−0.335243
$9$	−0.115054	−0.0383512
$10$	−1.03864	−0.328448
$11$	0	0
$11$	0	0
$12$	4.16054	1.20104
$13$	−5.30029	−1.47004	−0.735018	−	0.678047i	$-0.762826\pi$
$13$	−5.30029	−1.47004	−0.735018	+	0.678047i	$0.762826\pi$
$14$	2.10939	0.563758
$15$	0.836333	0.215940
$16$	−2.89889	−0.724723
$17$	−3.03721	−0.736631	−0.368315	−	0.929701i	$-0.620065\pi$
$17$	−3.03721	−0.736631	−0.368315	+	0.929701i	$0.620065\pi$
$18$	0.242693	0.0572032
$19$	4.66622	1.07050	0.535252	−	0.844692i	$-0.320216\pi$
$19$	4.66622	1.07050	0.535252	+	0.844692i	$0.320216\pi$
$20$	1.20612	0.269697
$21$	−1.69851	−0.370646
$22$	0	0
$23$	−5.63835	−1.17568	−0.587839	−	0.808978i	$-0.700021\pi$
$23$	−5.63835	−1.17568	−0.587839	+	0.808978i	$0.700021\pi$
$24$	−1.61055	−0.328752
$25$	−4.75755	−0.951510
$26$	11.1804	2.19265
$27$	−5.29096	−1.01825
$28$	−2.44952	−0.462916
$29$	−6.92295	−1.28556	−0.642780	−	0.766051i	$-0.722219\pi$
$29$	−6.92295	−1.28556	−0.642780	+	0.766051i	$0.722219\pi$
$30$	−1.76415	−0.322088
$31$	−1.26565	−0.227317	−0.113659	−	0.993520i	$-0.536257\pi$
$31$	−1.26565	−0.227317	−0.113659	+	0.993520i	$0.536257\pi$
$32$	8.01131	1.41621
$33$	0	0
$34$	6.40665	1.09873
$35$	−0.492391	−0.0832293
$36$	−0.281826	−0.0469710
$37$	10.8759	1.78798	0.893990	−	0.448087i	$-0.147895\pi$
$37$	10.8759	1.78798	0.893990	+	0.448087i	$0.147895\pi$
$38$	−9.84288	−1.59673
$39$	−9.00262	−1.44157
$40$	−0.466891	−0.0738220
$41$	1.44322	0.225393	0.112696	−	0.993629i	$-0.464051\pi$
$41$	1.44322	0.225393	0.112696	+	0.993629i	$0.464051\pi$
$42$	3.58282	0.552842
$43$	−2.88224	−0.439537	−0.219769	−	0.975552i	$-0.570530\pi$
$43$	−2.88224	−0.439537	−0.219769	+	0.975552i	$0.570530\pi$
$44$	0	0
$45$	−0.0566513	−0.00844508
$46$	11.8935	1.75360
$47$	−8.75522	−1.27708	−0.638540	−	0.769589i	$-0.720461\pi$
$47$	−8.75522	−1.27708	−0.638540	+	0.769589i	$0.720461\pi$
$48$	−4.92381	−0.710690
$49$	1.00000	0.142857
$50$	10.0355	1.41924
$51$	−5.15873	−0.722367
$52$	−12.9832	−1.80044
$53$	6.63835	0.911848	0.455924	−	0.890019i	$-0.349309\pi$
$53$	6.63835	0.911848	0.455924	+	0.890019i	$0.349309\pi$
$54$	11.1607	1.51878
$55$	0	0
$56$	0.948212	0.126710
$57$	7.92564	1.04978
$58$	14.6032	1.91749
$59$	−8.35733	−1.08803	−0.544016	−	0.839075i	$-0.683097\pi$
$59$	−8.35733	−1.08803	−0.544016	+	0.839075i	$0.683097\pi$
$60$	2.04861	0.264475
$61$	13.8953	1.77911	0.889554	−	0.456829i	$-0.151015\pi$
$61$	13.8953	1.77911	0.889554	+	0.456829i	$0.151015\pi$
$62$	2.66975	0.339058
$63$	0.115054	0.0144954
$64$	−11.1012	−1.38765
$65$	−2.60982	−0.323708
$66$	0	0
$67$	−9.70431	−1.18557	−0.592785	−	0.805361i	$-0.701972\pi$
$67$	−9.70431	−1.18557	−0.592785	+	0.805361i	$0.701972\pi$
$68$	−7.43970	−0.902196
$69$	−9.57681	−1.15291
$70$	1.03864	0.124142
$71$	5.94751	0.705839	0.352920	−	0.935654i	$-0.385189\pi$
$71$	5.94751	0.705839	0.352920	+	0.935654i	$0.385189\pi$
$72$	0.109095	0.0128570
$73$	3.77421	0.441737	0.220869	−	0.975304i	$-0.429111\pi$
$73$	3.77421	0.441737	0.220869	+	0.975304i	$0.429111\pi$
$74$	−22.9414	−2.66688
$75$	−8.08076	−0.933086
$76$	11.4300	1.31111
$77$	0	0
$78$	18.9900	2.15020
$79$	8.80383	0.990508	0.495254	−	0.868748i	$-0.335075\pi$
$79$	8.80383	0.990508	0.495254	+	0.868748i	$0.335075\pi$
$80$	−1.42739	−0.159587
$81$	−8.64160	−0.960178
$82$	−3.04431	−0.336188
$83$	−11.0898	−1.21726	−0.608632	−	0.793453i	$-0.708281\pi$
$83$	−11.0898	−1.21726	−0.608632	+	0.793453i	$0.708281\pi$
$84$	−4.16054	−0.453952
$85$	−1.49549	−0.162209
$86$	6.07976	0.655597
$87$	−11.7587	−1.26067
$88$	0	0
$89$	3.10324	0.328943	0.164472	−	0.986382i	$-0.447408\pi$
$89$	3.10324	0.328943	0.164472	+	0.986382i	$0.447408\pi$
$90$	0.119500	0.0125964
$91$	5.30029	0.555622
$92$	−13.8113	−1.43992
$93$	−2.14972	−0.222916
$94$	18.4682	1.90484
$95$	2.29761	0.235730
$96$	13.6073	1.38879
$97$	−6.31676	−0.641370	−0.320685	−	0.947186i	$-0.603913\pi$
$97$	−6.31676	−0.641370	−0.320685	+	0.947186i	$0.603913\pi$
$98$	−2.10939	−0.213080
$99$	0	0
$100$	−11.6537	−1.16537
$101$	−11.7984	−1.17399	−0.586993	−	0.809592i	$-0.699688\pi$
$101$	−11.7984	−1.17399	−0.586993	+	0.809592i	$0.699688\pi$
$102$	10.8818	1.07746
$103$	7.00565	0.690287	0.345144	−	0.938550i	$-0.387830\pi$
$103$	7.00565	0.690287	0.345144	+	0.938550i	$0.387830\pi$
$104$	5.02580	0.492820
$105$	−0.836333	−0.0816177
$106$	−14.0029	−1.36008
$107$	−11.3547	−1.09770	−0.548850	−	0.835921i	$-0.684934\pi$
$107$	−11.3547	−1.09770	−0.548850	+	0.835921i	$0.684934\pi$
$108$	−12.9603	−1.24711
$109$	−18.9414	−1.81426	−0.907129	−	0.420853i	$-0.861731\pi$
$109$	−18.9414	−1.81426	−0.907129	+	0.420853i	$0.861731\pi$
$110$	0	0
$111$	18.4728	1.75336
$112$	2.89889	0.273920
$113$	−13.4961	−1.26961	−0.634804	−	0.772673i	$-0.718919\pi$
$113$	−13.4961	−1.26961	−0.634804	+	0.772673i	$0.718919\pi$
$114$	−16.7183	−1.56581
$115$	−2.77627	−0.258889
$116$	−16.9579	−1.57450
$117$	0.609817	0.0563776
$118$	17.6289	1.62287
$119$	3.03721	0.278420
$120$	−0.793021	−0.0723925
$121$	0	0
$122$	−29.3106	−2.65365
$123$	2.45133	0.221029
$124$	−3.10023	−0.278409
$125$	−4.80453	−0.429730
$126$	−0.242693	−0.0216208
$127$	4.66064	0.413565	0.206782	−	0.978387i	$-0.433701\pi$
$127$	4.66064	0.413565	0.206782	+	0.978387i	$0.433701\pi$
$128$	7.39409	0.653551
$129$	−4.89552	−0.431026
$130$	5.50512	0.482831
$131$	−9.03676	−0.789545	−0.394773	−	0.918779i	$-0.629177\pi$
$131$	−9.03676	−0.789545	−0.394773	+	0.918779i	$0.629177\pi$
$132$	0	0
$133$	−4.66622	−0.404613
$134$	20.4702	1.76835
$135$	−2.60522	−0.224222
$136$	2.87991	0.246951
$137$	−1.63772	−0.139920	−0.0699600	−	0.997550i	$-0.522287\pi$
$137$	−1.63772	−0.139920	−0.0699600	+	0.997550i	$0.522287\pi$
$138$	20.2012	1.71964
$139$	1.53472	0.130173	0.0650866	−	0.997880i	$-0.479268\pi$
$139$	1.53472	0.130173	0.0650866	+	0.997880i	$0.479268\pi$
$140$	−1.20612	−0.101936
$141$	−14.8709	−1.25235
$142$	−12.5456	−1.05280
$143$	0	0
$144$	0.333528	0.0277940
$145$	−3.40880	−0.283086
$146$	−7.96127	−0.658879
$147$	1.69851	0.140091
$148$	26.6406	2.18985
$149$	−13.4909	−1.10522	−0.552610	−	0.833440i	$-0.686368\pi$
$149$	−13.4909	−1.10522	−0.552610	+	0.833440i	$0.686368\pi$
$150$	17.0455	1.39176
$151$	12.2370	0.995835	0.497917	−	0.867225i	$-0.334098\pi$
$151$	12.2370	0.995835	0.497917	+	0.867225i	$0.334098\pi$
$152$	−4.42457	−0.358880
$153$	0.349441	0.0282507
$154$	0	0
$155$	−0.623194	−0.0500562
$156$	−22.0521	−1.76558
$157$	2.52042	0.201152	0.100576	−	0.994929i	$-0.467932\pi$
$157$	2.52042	0.201152	0.100576	+	0.994929i	$0.467932\pi$
$158$	−18.5707	−1.47741
$159$	11.2753	0.894191
$160$	3.94470	0.311856
$161$	5.63835	0.444364
$162$	18.2285	1.43217
$163$	7.87905	0.617135	0.308567	−	0.951203i	$-0.400151\pi$
$163$	7.87905	0.617135	0.308567	+	0.951203i	$0.400151\pi$
$164$	3.53519	0.276052
$165$	0	0
$166$	23.3927	1.81562
$167$	2.05485	0.159009	0.0795047	−	0.996834i	$-0.474666\pi$
$167$	2.05485	0.159009	0.0795047	+	0.996834i	$0.474666\pi$
$168$	1.61055	0.124257
$169$	15.0931	1.16101
$170$	3.15458	0.241945
$171$	−0.536865	−0.0410551
$172$	−7.06010	−0.538327
$173$	23.2707	1.76923	0.884617	−	0.466318i	$-0.154420\pi$
$173$	23.2707	1.76923	0.884617	+	0.466318i	$0.154420\pi$
$174$	24.8037	1.88037
$175$	4.75755	0.359637
$176$	0	0
$177$	−14.1950	−1.06696
$178$	−6.54595	−0.490640
$179$	17.6596	1.31994	0.659969	−	0.751293i	$-0.270569\pi$
$179$	17.6596	1.31994	0.659969	+	0.751293i	$0.270569\pi$
$180$	−0.138769	−0.0103432
$181$	15.4701	1.14988	0.574941	−	0.818195i	$-0.305025\pi$
$181$	15.4701	1.14988	0.574941	+	0.818195i	$0.305025\pi$
$182$	−11.1804	−0.828745
$183$	23.6013	1.74466
$184$	5.34635	0.394138
$185$	5.35518	0.393720
$186$	4.53460	0.332493
$187$	0	0
$188$	−21.4461	−1.56412
$189$	5.29096	0.384861
$190$	−4.84655	−0.351605
$191$	15.9385	1.15327	0.576635	−	0.817002i	$-0.304366\pi$
$191$	15.9385	1.15327	0.576635	+	0.817002i	$0.304366\pi$
$192$	−18.8555	−1.36078
$193$	−8.45386	−0.608522	−0.304261	−	0.952589i	$-0.598410\pi$
$193$	−8.45386	−0.608522	−0.304261	+	0.952589i	$0.598410\pi$
$194$	13.3245	0.956643
$195$	−4.43281	−0.317440
$196$	2.44952	0.174966
$197$	14.3384	1.02157	0.510785	−	0.859708i	$-0.329355\pi$
$197$	14.3384	1.02157	0.510785	+	0.859708i	$0.329355\pi$
$198$	0	0
$199$	−22.1343	−1.56906	−0.784528	−	0.620094i	$-0.787095\pi$
$199$	−22.1343	−1.56906	−0.784528	+	0.620094i	$0.787095\pi$
$200$	4.51117	0.318988
$201$	−16.4829	−1.16261
$202$	24.8874	1.75107
$203$	6.92295	0.485896
$204$	−12.6364	−0.884727
$205$	0.710628	0.0496324
$206$	−14.7776	−1.02961
$207$	0.648712	0.0450886
$208$	15.3650	1.06537
$209$	0	0
$210$	1.76415	0.121738
$211$	−2.18302	−0.150286	−0.0751428	−	0.997173i	$-0.523941\pi$
$211$	−2.18302	−0.150286	−0.0751428	+	0.997173i	$0.523941\pi$
$212$	16.2608	1.11679
$213$	10.1019	0.692172
$214$	23.9515	1.63729
$215$	−1.41919	−0.0967878
$216$	5.01695	0.341360
$217$	1.26565	0.0859178
$218$	39.9548	2.70608
$219$	6.41054	0.433184
$220$	0	0
$221$	16.0981	1.08287
$222$	−38.9663	−2.61525
$223$	27.2603	1.82549	0.912744	−	0.408533i	$-0.133960\pi$
$223$	27.2603	1.82549	0.912744	+	0.408533i	$0.133960\pi$
$224$	−8.01131	−0.535278
$225$	0.547373	0.0364915
$226$	28.4685	1.89370
$227$	13.5892	0.901945	0.450973	−	0.892538i	$-0.351077\pi$
$227$	13.5892	0.901945	0.450973	+	0.892538i	$0.351077\pi$
$228$	19.4140	1.28572
$229$	−8.30141	−0.548573	−0.274286	−	0.961648i	$-0.588442\pi$
$229$	−8.30141	−0.548573	−0.274286	+	0.961648i	$0.588442\pi$
$230$	5.85624	0.386149
$231$	0	0
$232$	6.56443	0.430976
$233$	−12.9476	−0.848229	−0.424114	−	0.905609i	$-0.639415\pi$
$233$	−12.9476	−0.848229	−0.424114	+	0.905609i	$0.639415\pi$
$234$	−1.28634	−0.0840908
$235$	−4.31099	−0.281218
$236$	−20.4715	−1.33258
$237$	14.9534	0.971329
$238$	−6.40665	−0.415281
$239$	−1.89342	−0.122475	−0.0612375	−	0.998123i	$-0.519505\pi$
$239$	−1.89342	−0.122475	−0.0612375	+	0.998123i	$0.519505\pi$
$240$	−2.42444	−0.156497
$241$	11.6983	0.753557	0.376778	−	0.926303i	$-0.377032\pi$
$241$	11.6983	0.753557	0.376778	+	0.926303i	$0.377032\pi$
$242$	0	0
$243$	1.19500	0.0766595
$244$	34.0368	2.17898
$245$	0.492391	0.0314577
$246$	−5.17080	−0.329678
$247$	−24.7323	−1.57368
$248$	1.20010	0.0762066
$249$	−18.8362	−1.19369
$250$	10.1346	0.640970
$251$	−13.1860	−0.832291	−0.416146	−	0.909298i	$-0.636619\pi$
$251$	−13.1860	−0.832291	−0.416146	+	0.909298i	$0.636619\pi$
$252$	0.281826	0.0177534
$253$	0	0
$254$	−9.83110	−0.616858
$255$	−2.54012	−0.159068
$256$	6.60537	0.412836
$257$	−15.3445	−0.957165	−0.478582	−	0.878043i	$-0.658849\pi$
$257$	−15.3445	−0.957165	−0.478582	+	0.878043i	$0.658849\pi$
$258$	10.3265	0.642903
$259$	−10.8759	−0.675793
$260$	−6.39280	−0.396465
$261$	0.796510	0.0493027
$262$	19.0620	1.17766
$263$	−10.4197	−0.642507	−0.321254	−	0.946993i	$-0.604104\pi$
$263$	−10.4197	−0.642507	−0.321254	+	0.946993i	$0.604104\pi$
$264$	0	0
$265$	3.26867	0.200792
$266$	9.84288	0.603506
$267$	5.27090	0.322574
$268$	−23.7709	−1.45204
$269$	16.2712	0.992074	0.496037	−	0.868301i	$-0.334788\pi$
$269$	16.2712	0.992074	0.496037	+	0.868301i	$0.334788\pi$
$270$	5.49542	0.334441
$271$	5.83922	0.354707	0.177354	−	0.984147i	$-0.443246\pi$
$271$	5.83922	0.354707	0.177354	+	0.984147i	$0.443246\pi$
$272$	8.80454	0.533853
$273$	9.00262	0.544863
$274$	3.45459	0.208700
$275$	0	0
$276$	−23.4586	−1.41204
$277$	−15.9255	−0.956868	−0.478434	−	0.878123i	$-0.658795\pi$
$277$	−15.9255	−0.956868	−0.478434	+	0.878123i	$0.658795\pi$
$278$	−3.23732	−0.194162
$279$	0.145617	0.00871788
$280$	0.466891	0.0279021
$281$	−10.2004	−0.608504	−0.304252	−	0.952592i	$-0.598406\pi$
$281$	−10.2004	−0.608504	−0.304252	+	0.952592i	$0.598406\pi$
$282$	31.3684	1.86796
$283$	16.1634	0.960812	0.480406	−	0.877046i	$-0.340489\pi$
$283$	16.1634	0.960812	0.480406	+	0.877046i	$0.340489\pi$
$284$	14.5685	0.864483
$285$	3.90252	0.231165
$286$	0	0
$287$	−1.44322	−0.0851905
$288$	−0.921730	−0.0543135
$289$	−7.77538	−0.457375
$290$	7.19049	0.422240
$291$	−10.7291	−0.628951
$292$	9.24499	0.541022
$293$	−27.0517	−1.58038	−0.790188	−	0.612864i	$-0.790017\pi$
$293$	−27.0517	−1.58038	−0.790188	+	0.612864i	$0.790017\pi$
$294$	−3.58282	−0.208955
$295$	−4.11508	−0.239589
$296$	−10.3126	−0.599408
$297$	0	0
$298$	28.4576	1.64851
$299$	29.8849	1.72829
$300$	−19.7940	−1.14281
$301$	2.88224	0.166129
$302$	−25.8126	−1.48535
$303$	−20.0398	−1.15125
$304$	−13.5269	−0.775820
$305$	6.84192	0.391767
$306$	−0.737107	−0.0421376
$307$	−29.7251	−1.69650	−0.848250	−	0.529596i	$-0.822343\pi$
$307$	−29.7251	−1.69650	−0.848250	+	0.529596i	$0.822343\pi$
$308$	0	0
$309$	11.8992	0.676921
$310$	1.31456	0.0746619
$311$	22.4029	1.27035	0.635176	−	0.772367i	$-0.280928\pi$
$311$	22.4029	1.27035	0.635176	+	0.772367i	$0.280928\pi$
$312$	8.53639	0.483278
$313$	9.94833	0.562313	0.281157	−	0.959662i	$-0.409282\pi$
$313$	9.94833	0.562313	0.281157	+	0.959662i	$0.409282\pi$
$314$	−5.31655	−0.300030
$315$	0.0566513	0.00319194
$316$	21.5652	1.21313
$317$	−11.1420	−0.625796	−0.312898	−	0.949787i	$-0.601300\pi$
$317$	−11.1420	−0.625796	−0.312898	+	0.949787i	$0.601300\pi$
$318$	−23.7840	−1.33374
$319$	0	0
$320$	−5.46613	−0.305566
$321$	−19.2861	−1.07645
$322$	−11.8935	−0.662797
$323$	−14.1723	−0.788567
$324$	−21.1678	−1.17599
$325$	25.2164	1.39875
$326$	−16.6200	−0.920495
$327$	−32.1722	−1.77913
$328$	−1.36848	−0.0755615
$329$	8.75522	0.482691
$330$	0	0
$331$	14.5950	0.802214	0.401107	−	0.916031i	$-0.368626\pi$
$331$	14.5950	0.802214	0.401107	+	0.916031i	$0.368626\pi$
$332$	−27.1647	−1.49086
$333$	−1.25131	−0.0685711
$334$	−4.33449	−0.237172
$335$	−4.77832	−0.261067
$336$	4.92381	0.268616
$337$	−12.6059	−0.686688	−0.343344	−	0.939210i	$-0.611560\pi$
$337$	−12.6059	−0.686688	−0.343344	+	0.939210i	$0.611560\pi$
$338$	−31.8372	−1.73172
$339$	−22.9233	−1.24502
$340$	−3.66324	−0.198667
$341$	0	0
$342$	1.13246	0.0612363
$343$	−1.00000	−0.0539949
$344$	2.73297	0.147352
$345$	−4.71554	−0.253876
$346$	−49.0868	−2.63893
$347$	−0.410734	−0.0220494	−0.0110247	−	0.999939i	$-0.503509\pi$
$347$	−0.410734	−0.0220494	−0.0110247	+	0.999939i	$0.503509\pi$
$348$	−28.8032	−1.54402
$349$	−13.4025	−0.717422	−0.358711	−	0.933449i	$-0.616784\pi$
$349$	−13.4025	−0.717422	−0.358711	+	0.933449i	$0.616784\pi$
$350$	−10.0355	−0.536421
$351$	28.0436	1.49686
$352$	0	0
$353$	12.3419	0.656892	0.328446	−	0.944523i	$-0.393475\pi$
$353$	12.3419	0.656892	0.328446	+	0.944523i	$0.393475\pi$
$354$	29.9429	1.59144
$355$	2.92850	0.155429
$356$	7.60146	0.402877
$357$	5.15873	0.273029
$358$	−37.2509	−1.96877
$359$	25.0097	1.31996	0.659981	−	0.751283i	$-0.270565\pi$
$359$	25.0097	1.31996	0.659981	+	0.751283i	$0.270565\pi$
$360$	0.0537175	0.00283116
$361$	2.77364	0.145981
$362$	−32.6324	−1.71512
$363$	0	0
$364$	12.9832	0.680503
$365$	1.85839	0.0972724
$366$	−49.7844	−2.60227
$367$	−2.06915	−0.108009	−0.0540043	−	0.998541i	$-0.517198\pi$
$367$	−2.06915	−0.108009	−0.0540043	+	0.998541i	$0.517198\pi$
$368$	16.3450	0.852041
$369$	−0.166047	−0.00864408
$370$	−11.2961	−0.587259
$371$	−6.63835	−0.344646
$372$	−5.26578	−0.273018
$373$	14.7623	0.764365	0.382183	−	0.924087i	$-0.375172\pi$
$373$	14.7623	0.764365	0.382183	+	0.924087i	$0.375172\pi$
$374$	0	0
$375$	−8.16056	−0.421410
$376$	8.30180	0.428133
$377$	36.6937	1.88982
$378$	−11.1607	−0.574044
$379$	27.7508	1.42546	0.712730	−	0.701438i	$-0.247458\pi$
$379$	27.7508	1.42546	0.712730	+	0.701438i	$0.247458\pi$
$380$	5.62803	0.288712
$381$	7.91615	0.405557
$382$	−33.6205	−1.72017
$383$	−18.0334	−0.921464	−0.460732	−	0.887539i	$-0.652413\pi$
$383$	−18.0334	−0.921464	−0.460732	+	0.887539i	$0.652413\pi$
$384$	12.5590	0.640897
$385$	0	0
$386$	17.8325	0.907649
$387$	0.331612	0.0168568
$388$	−15.4730	−0.785524
$389$	13.5412	0.686564	0.343282	−	0.939232i	$-0.388461\pi$
$389$	13.5412	0.686564	0.343282	+	0.939232i	$0.388461\pi$
$390$	9.35052	0.473482
$391$	17.1248	0.866040
$392$	−0.948212	−0.0478919
$393$	−15.3491	−0.774257
$394$	−30.2453	−1.52374
$395$	4.33493	0.218114
$396$	0	0
$397$	−24.6525	−1.23727	−0.618637	−	0.785677i	$-0.712315\pi$
$397$	−24.6525	−1.23727	−0.618637	+	0.785677i	$0.712315\pi$
$398$	46.6897	2.34035
$399$	−7.92564	−0.396778
$400$	13.7916	0.689581
$401$	18.7389	0.935778	0.467889	−	0.883787i	$-0.345015\pi$
$401$	18.7389	0.935778	0.467889	+	0.883787i	$0.345015\pi$
$402$	34.7688	1.73411
$403$	6.70831	0.334165
$404$	−28.9004	−1.43785
$405$	−4.25505	−0.211435
$406$	−14.6032	−0.724745
$407$	0	0
$408$	4.89157	0.242169
$409$	−15.7098	−0.776801	−0.388401	−	0.921491i	$-0.626972\pi$
$409$	−15.7098	−0.776801	−0.388401	+	0.921491i	$0.626972\pi$
$410$	−1.49899	−0.0740299
$411$	−2.78169	−0.137211
$412$	17.1605	0.845436
$413$	8.35733	0.411238
$414$	−1.36839	−0.0672525
$415$	−5.46052	−0.268046
$416$	−42.4623	−2.08189
$417$	2.60674	0.127653
$418$	0	0
$419$	22.6034	1.10425	0.552125	−	0.833761i	$-0.313817\pi$
$419$	22.6034	1.10425	0.552125	+	0.833761i	$0.313817\pi$
$420$	−2.04861	−0.0999621
$421$	−23.3311	−1.13709	−0.568544	−	0.822653i	$-0.692493\pi$
$421$	−23.3311	−1.13709	−0.568544	+	0.822653i	$0.692493\pi$
$422$	4.60485	0.224160
$423$	1.00732	0.0489775
$424$	−6.29456	−0.305691
$425$	14.4497	0.700912
$426$	−21.3089	−1.03242
$427$	−13.8953	−0.672440
$428$	−27.8136	−1.34442
$429$	0	0
$430$	2.99362	0.144365
$431$	9.53898	0.459476	0.229738	−	0.973252i	$-0.426213\pi$
$431$	9.53898	0.459476	0.229738	+	0.973252i	$0.426213\pi$
$432$	15.3379	0.737946
$433$	−23.0105	−1.10581	−0.552907	−	0.833243i	$-0.686482\pi$
$433$	−23.0105	−1.10581	−0.552907	+	0.833243i	$0.686482\pi$
$434$	−2.66975	−0.128152
$435$	−5.78989	−0.277604
$436$	−46.3973	−2.22203
$437$	−26.3098	−1.25857
$438$	−13.5223	−0.646121
$439$	27.6434	1.31935	0.659673	−	0.751553i	$-0.270695\pi$
$439$	27.6434	1.31935	0.659673	+	0.751553i	$0.270695\pi$
$440$	0	0
$441$	−0.115054	−0.00547874
$442$	−33.9571	−1.61518
$443$	14.7713	0.701807	0.350904	−	0.936412i	$-0.385875\pi$
$443$	14.7713	0.701807	0.350904	+	0.936412i	$0.385875\pi$
$444$	45.2494	2.14744
$445$	1.52801	0.0724346
$446$	−57.5026	−2.72283
$447$	−22.9145	−1.08382
$448$	11.1012	0.524482
$449$	30.2669	1.42838	0.714192	−	0.699950i	$-0.246795\pi$
$449$	30.2669	1.42838	0.714192	+	0.699950i	$0.246795\pi$
$450$	−1.15462	−0.0544294
$451$	0	0
$452$	−33.0590	−1.55496
$453$	20.7847	0.976552
$454$	−28.6648	−1.34531
$455$	2.60982	0.122350
$456$	−7.51518	−0.351931
$457$	20.2565	0.947558	0.473779	−	0.880644i	$-0.342889\pi$
$457$	20.2565	0.947558	0.473779	+	0.880644i	$0.342889\pi$
$458$	17.5109	0.818231
$459$	16.0697	0.750071
$460$	−6.80054	−0.317077
$461$	−8.51184	−0.396436	−0.198218	−	0.980158i	$-0.563515\pi$
$461$	−8.51184	−0.396436	−0.198218	+	0.980158i	$0.563515\pi$
$462$	0	0
$463$	−0.591469	−0.0274879	−0.0137440	−	0.999906i	$-0.504375\pi$
$463$	−0.591469	−0.0274879	−0.0137440	+	0.999906i	$0.504375\pi$
$464$	20.0689	0.931675
$465$	−1.05850	−0.0490869
$466$	27.3116	1.26519
$467$	−41.0347	−1.89886	−0.949430	−	0.313979i	$-0.898338\pi$
$467$	−41.0347	−1.89886	−0.949430	+	0.313979i	$0.898338\pi$
$468$	1.49376	0.0690491
$469$	9.70431	0.448103
$470$	9.09356	0.419454
$471$	4.28097	0.197257
$472$	7.92452	0.364756
$473$	0	0
$474$	−31.5426	−1.44880
$475$	−22.1998	−1.01860
$476$	7.43970	0.340998
$477$	−0.763766	−0.0349704
$478$	3.99395	0.182679
$479$	−20.3437	−0.929527	−0.464763	−	0.885435i	$-0.653861\pi$
$479$	−20.3437	−0.929527	−0.464763	+	0.885435i	$0.653861\pi$
$480$	6.70012	0.305817
$481$	−57.6452	−2.62840
$482$	−24.6764	−1.12398
$483$	9.57681	0.435760
$484$	0	0
$485$	−3.11032	−0.141232
$486$	−2.52073	−0.114342
$487$	−28.8165	−1.30580	−0.652900	−	0.757444i	$-0.726448\pi$
$487$	−28.8165	−1.30580	−0.652900	+	0.757444i	$0.726448\pi$
$488$	−13.1757	−0.596435
$489$	13.3827	0.605185
$490$	−1.03864	−0.0469212
$491$	−2.68045	−0.120967	−0.0604834	−	0.998169i	$-0.519264\pi$
$491$	−2.68045	−0.120967	−0.0604834	+	0.998169i	$0.519264\pi$
$492$	6.00457	0.270707
$493$	21.0264	0.946983
$494$	52.1701	2.34725
$495$	0	0
$496$	3.66898	0.164742
$497$	−5.94751	−0.266782
$498$	39.7328	1.78047
$499$	22.3425	1.00019	0.500095	−	0.865971i	$-0.333298\pi$
$499$	22.3425	1.00019	0.500095	+	0.865971i	$0.333298\pi$
$500$	−11.7688	−0.526317
$501$	3.49020	0.155930
$502$	27.8143	1.24141
$503$	4.47599	0.199575	0.0997873	−	0.995009i	$-0.468184\pi$
$503$	4.47599	0.199575	0.0997873	+	0.995009i	$0.468184\pi$
$504$	−0.109095	−0.00485948
$505$	−5.80943	−0.258516
$506$	0	0
$507$	25.6358	1.13853
$508$	11.4163	0.506518
$509$	−17.1547	−0.760368	−0.380184	−	0.924911i	$-0.624139\pi$
$509$	−17.1547	−0.760368	−0.380184	+	0.924911i	$0.624139\pi$
$510$	5.35809	0.237260
$511$	−3.77421	−0.166961
$512$	−28.7215	−1.26932
$513$	−24.6888	−1.09004
$514$	32.3676	1.42767
$515$	3.44952	0.152004
$516$	−11.9917	−0.527904
$517$	0	0
$518$	22.9414	1.00799
$519$	39.5255	1.73498
$520$	2.47466	0.108521
$521$	1.00957	0.0442300	0.0221150	−	0.999755i	$-0.492960\pi$
$521$	1.00957	0.0442300	0.0221150	+	0.999755i	$0.492960\pi$
$522$	−1.68015	−0.0735382
$523$	13.6433	0.596578	0.298289	−	0.954476i	$-0.403584\pi$
$523$	13.6433	0.596578	0.298289	+	0.954476i	$0.403584\pi$
$524$	−22.1357	−0.967004
$525$	8.08076	0.352673
$526$	21.9792	0.958340
$527$	3.84404	0.167449
$528$	0	0
$529$	8.79099	0.382217
$530$	−6.89488	−0.299495
$531$	0.961541	0.0417273
$532$	−11.4300	−0.495554
$533$	−7.64948	−0.331336
$534$	−11.1184	−0.481139
$535$	−5.59095	−0.241718
$536$	9.20174	0.397455
$537$	29.9950	1.29438
$538$	−34.3223	−1.47974
$539$	0	0
$540$	−6.38154	−0.274618
$541$	2.76335	0.118806	0.0594028	−	0.998234i	$-0.481080\pi$
$541$	2.76335	0.118806	0.0594028	+	0.998234i	$0.481080\pi$
$542$	−12.3172	−0.529068
$543$	26.2761	1.12762
$544$	−24.3320	−1.04323
$545$	−9.32658	−0.399507
$546$	−18.9900	−0.812698
$547$	−15.2417	−0.651687	−0.325843	−	0.945424i	$-0.605648\pi$
$547$	−15.2417	−0.651687	−0.325843	+	0.945424i	$0.605648\pi$
$548$	−4.01163	−0.171368
$549$	−1.59870	−0.0682309
$550$	0	0
$551$	−32.3041	−1.37620
$552$	9.08084	0.386506
$553$	−8.80383	−0.374377
$554$	33.5930	1.42723
$555$	9.09584	0.386097
$556$	3.75933	0.159431
$557$	−3.56730	−0.151151	−0.0755757	−	0.997140i	$-0.524079\pi$
$557$	−3.56730	−0.151151	−0.0755757	+	0.997140i	$0.524079\pi$
$558$	−0.307164	−0.0130033
$559$	15.2767	0.646136
$560$	1.42739	0.0603182
$561$	0	0
$562$	21.5166	0.907621
$563$	−36.7500	−1.54883	−0.774414	−	0.632679i	$-0.781955\pi$
$563$	−36.7500	−1.54883	−0.774414	+	0.632679i	$0.781955\pi$
$564$	−36.4264	−1.53383
$565$	−6.64537	−0.279573
$566$	−34.0948	−1.43311
$567$	8.64160	0.362913
$568$	−5.63949	−0.236628
$569$	34.0802	1.42872	0.714359	−	0.699779i	$-0.246718\pi$
$569$	34.0802	1.42872	0.714359	+	0.699779i	$0.246718\pi$
$570$	−8.23192	−0.344797
$571$	5.79312	0.242434	0.121217	−	0.992626i	$-0.461320\pi$
$571$	5.79312	0.242434	0.121217	+	0.992626i	$0.461320\pi$
$572$	0	0
$573$	27.0718	1.13094
$574$	3.04431	0.127067
$575$	26.8247	1.11867
$576$	1.27723	0.0532179
$577$	−31.1231	−1.29567	−0.647837	−	0.761779i	$-0.724326\pi$
$577$	−31.1231	−1.29567	−0.647837	+	0.761779i	$0.724326\pi$
$578$	16.4013	0.682204
$579$	−14.3590	−0.596739
$580$	−8.34993	−0.346712
$581$	11.0898	0.460082
$582$	22.6318	0.938120
$583$	0	0
$584$	−3.57875	−0.148090
$585$	0.300269	0.0124146
$586$	57.0625	2.35723
$587$	−47.6946	−1.96857	−0.984283	−	0.176601i	$-0.943490\pi$
$587$	−47.6946	−1.96857	−0.984283	+	0.176601i	$0.943490\pi$
$588$	4.16054	0.171578
$589$	−5.90580	−0.243344
$590$	8.68030	0.357362
$591$	24.3540	1.00179
$592$	−31.5279	−1.29579
$593$	−44.5859	−1.83092	−0.915462	−	0.402404i	$-0.868175\pi$
$593$	−44.5859	−1.83092	−0.915462	+	0.402404i	$0.868175\pi$
$594$	0	0
$595$	1.49549	0.0613093
$596$	−33.0463	−1.35363
$597$	−37.5953	−1.53867
$598$	−63.0389	−2.57785
$599$	−8.06937	−0.329706	−0.164853	−	0.986318i	$-0.552715\pi$
$599$	−8.06937	−0.329706	−0.164853	+	0.986318i	$0.552715\pi$
$600$	7.66227	0.312811
$601$	28.9086	1.17921	0.589603	−	0.807693i	$-0.299284\pi$
$601$	28.9086	1.17921	0.589603	+	0.807693i	$0.299284\pi$
$602$	−6.07976	−0.247792
$603$	1.11651	0.0454680
$604$	29.9748	1.21966
$605$	0	0
$606$	42.2716	1.71717
$607$	9.85310	0.399925	0.199962	−	0.979804i	$-0.435918\pi$
$607$	9.85310	0.399925	0.199962	+	0.979804i	$0.435918\pi$
$608$	37.3826	1.51606
$609$	11.7587	0.476488
$610$	−14.4323	−0.584345
$611$	46.4052	1.87735
$612$	0.855963	0.0346003
$613$	35.8329	1.44728	0.723638	−	0.690180i	$-0.242469\pi$
$613$	35.8329	1.44728	0.723638	+	0.690180i	$0.242469\pi$
$614$	62.7017	2.53044
$615$	1.20701	0.0486714
$616$	0	0
$617$	−38.4398	−1.54753	−0.773764	−	0.633474i	$-0.781628\pi$
$617$	−38.4398	−1.54753	−0.773764	+	0.633474i	$0.781628\pi$
$618$	−25.1000	−1.00967
$619$	4.19552	0.168632	0.0843162	−	0.996439i	$-0.473129\pi$
$619$	4.19552	0.168632	0.0843162	+	0.996439i	$0.473129\pi$
$620$	−1.52653	−0.0613068
$621$	29.8323	1.19713
$622$	−47.2564	−1.89481
$623$	−3.10324	−0.124329
$624$	26.0976	1.04474
$625$	21.4220	0.856882
$626$	−20.9849	−0.838725
$627$	0	0
$628$	6.17382	0.246362
$629$	−33.0322	−1.31708
$630$	−0.119500	−0.00476098
$631$	−3.51798	−0.140049	−0.0700243	−	0.997545i	$-0.522308\pi$
$631$	−3.51798	−0.140049	−0.0700243	+	0.997545i	$0.522308\pi$
$632$	−8.34790	−0.332061
$633$	−3.70789	−0.147376
$634$	23.5028	0.933414
$635$	2.29486	0.0910686
$636$	27.6191	1.09517
$637$	−5.30029	−0.210005
$638$	0	0
$639$	−0.684281	−0.0270698
$640$	3.64078	0.143915
$641$	16.0952	0.635723	0.317862	−	0.948137i	$-0.397035\pi$
$641$	16.0952	0.635723	0.317862	+	0.948137i	$0.397035\pi$
$642$	40.6819	1.60559
$643$	−34.8261	−1.37341	−0.686704	−	0.726938i	$-0.740943\pi$
$643$	−34.8261	−1.37341	−0.686704	+	0.726938i	$0.740943\pi$
$644$	13.8113	0.544239
$645$	−2.41051	−0.0949137
$646$	29.8948	1.17620
$647$	16.9051	0.664607	0.332304	−	0.943172i	$-0.392174\pi$
$647$	16.9051	0.664607	0.332304	+	0.943172i	$0.392174\pi$
$648$	8.19407	0.321893
$649$	0	0
$650$	−53.1912	−2.08633
$651$	2.14972	0.0842542
$652$	19.2999	0.755842
$653$	3.52799	0.138061	0.0690304	−	0.997615i	$-0.478009\pi$
$653$	3.52799	0.138061	0.0690304	+	0.997615i	$0.478009\pi$
$654$	67.8637	2.65368
$655$	−4.44962	−0.173861
$656$	−4.18374	−0.163347
$657$	−0.434236	−0.0169411
$658$	−18.4682	−0.719964
$659$	−29.4409	−1.14686	−0.573428	−	0.819256i	$-0.694387\pi$
$659$	−29.4409	−1.14686	−0.573428	+	0.819256i	$0.694387\pi$
$660$	0	0
$661$	15.2989	0.595059	0.297529	−	0.954713i	$-0.403837\pi$
$661$	15.2989	0.595059	0.297529	+	0.954713i	$0.403837\pi$
$662$	−30.7865	−1.19655
$663$	27.3428	1.06191
$664$	10.5155	0.408080
$665$	−2.29761	−0.0890974
$666$	2.63949	0.102278
$667$	39.0340	1.51140
$668$	5.03341	0.194748
$669$	46.3020	1.79014
$670$	10.0793	0.389398
$671$	0	0
$672$	−13.6073	−0.524914
$673$	−12.1652	−0.468936	−0.234468	−	0.972124i	$-0.575335\pi$
$673$	−12.1652	−0.468936	−0.234468	+	0.972124i	$0.575335\pi$
$674$	26.5908	1.02424
$675$	25.1720	0.968871
$676$	36.9709	1.42196
$677$	0.313619	0.0120533	0.00602667	−	0.999982i	$-0.498082\pi$
$677$	0.313619	0.0120533	0.00602667	+	0.999982i	$0.498082\pi$
$678$	48.3542	1.85703
$679$	6.31676	0.242415
$680$	1.41804	0.0543795
$681$	23.0814	0.884481
$682$	0	0
$683$	38.2419	1.46328	0.731642	−	0.681689i	$-0.238754\pi$
$683$	38.2419	1.46328	0.731642	+	0.681689i	$0.238754\pi$
$684$	−1.31506	−0.0502827
$685$	−0.806400	−0.0308110
$686$	2.10939	0.0805368
$687$	−14.1001	−0.537951
$688$	8.35530	0.318543
$689$	−35.1852	−1.34045
$690$	9.94690	0.378672
$691$	−10.2754	−0.390895	−0.195448	−	0.980714i	$-0.562616\pi$
$691$	−10.2754	−0.390895	−0.195448	+	0.980714i	$0.562616\pi$
$692$	57.0019	2.16689
$693$	0	0
$694$	0.866398	0.0328880
$695$	0.755683	0.0286647
$696$	11.1498	0.422631
$697$	−4.38335	−0.166031
$698$	28.2712	1.07008
$699$	−21.9917	−0.831804
$700$	11.6537	0.440469
$701$	−18.9188	−0.714552	−0.357276	−	0.933999i	$-0.616294\pi$
$701$	−18.9188	−0.714552	−0.357276	+	0.933999i	$0.616294\pi$
$702$	−59.1549	−2.23266
$703$	50.7492	1.91404
$704$	0	0
$705$	−7.32228	−0.275773
$706$	−26.0338	−0.979795
$707$	11.7984	0.443725
$708$	−34.7710	−1.30678
$709$	−14.1498	−0.531409	−0.265704	−	0.964055i	$-0.585605\pi$
$709$	−14.1498	−0.531409	−0.265704	+	0.964055i	$0.585605\pi$
$710$	−6.17734	−0.231832
$711$	−1.01291	−0.0379872
$712$	−2.94253	−0.110276
$713$	7.13617	0.267252
$714$	−10.8818	−0.407240
$715$	0	0
$716$	43.2575	1.61661
$717$	−3.21599	−0.120104
$718$	−52.7552	−1.96880
$719$	−15.9330	−0.594201	−0.297101	−	0.954846i	$-0.596020\pi$
$719$	−15.9330	−0.594201	−0.297101	+	0.954846i	$0.596020\pi$
$720$	0.164226	0.00612035
$721$	−7.00565	−0.260904
$722$	−5.85068	−0.217740
$723$	19.8698	0.738965
$724$	37.8943	1.40833
$725$	32.9363	1.22322
$726$	0	0
$727$	−4.20455	−0.155938	−0.0779691	−	0.996956i	$-0.524844\pi$
$727$	−4.20455	−0.155938	−0.0779691	+	0.996956i	$0.524844\pi$
$728$	−5.02580	−0.186269
$729$	27.9545	1.03535
$730$	−3.92006	−0.145088
$731$	8.75395	0.323777
$732$	57.8119	2.13679
$733$	−34.0777	−1.25869	−0.629345	−	0.777126i	$-0.716677\pi$
$733$	−34.0777	−1.25869	−0.629345	+	0.777126i	$0.716677\pi$
$734$	4.36464	0.161102
$735$	0.836333	0.0308486
$736$	−45.1706	−1.66501
$737$	0	0
$738$	0.350258	0.0128932
$739$	−26.1306	−0.961230	−0.480615	−	0.876932i	$-0.659587\pi$
$739$	−26.1306	−0.961230	−0.480615	+	0.876932i	$0.659587\pi$
$740$	13.1176	0.482213
$741$	−42.0082	−1.54321
$742$	14.0029	0.514061
$743$	23.6263	0.866765	0.433383	−	0.901210i	$-0.357320\pi$
$743$	23.6263	0.866765	0.433383	+	0.901210i	$0.357320\pi$
$744$	2.03839	0.0747310
$745$	−6.64282	−0.243374
$746$	−31.1395	−1.14010
$747$	1.27592	0.0466835
$748$	0	0
$749$	11.3547	0.414892
$750$	17.2138	0.628559
$751$	−2.32359	−0.0847889	−0.0423945	−	0.999101i	$-0.513499\pi$
$751$	−2.32359	−0.0847889	−0.0423945	+	0.999101i	$0.513499\pi$
$752$	25.3804	0.925529
$753$	−22.3965	−0.816175
$754$	−77.4012	−2.81879
$755$	6.02540	0.219287
$756$	12.9603	0.471362
$757$	−14.7651	−0.536647	−0.268324	−	0.963329i	$-0.586470\pi$
$757$	−14.7651	−0.536647	−0.268324	+	0.963329i	$0.586470\pi$
$758$	−58.5371	−2.12616
$759$	0	0
$760$	−2.17862	−0.0790268
$761$	47.1876	1.71055	0.855274	−	0.518175i	$-0.173389\pi$
$761$	47.1876	1.71055	0.855274	+	0.518175i	$0.173389\pi$
$762$	−16.6982	−0.604914
$763$	18.9414	0.685725
$764$	39.0417	1.41248
$765$	0.172062	0.00622091
$766$	38.0395	1.37442
$767$	44.2963	1.59945
$768$	11.2193	0.404842
$769$	12.4418	0.448662	0.224331	−	0.974513i	$-0.427980\pi$
$769$	12.4418	0.448662	0.224331	+	0.974513i	$0.427980\pi$
$770$	0	0
$771$	−26.0629	−0.938631
$772$	−20.7079	−0.745294
$773$	−35.2698	−1.26857	−0.634284	−	0.773101i	$-0.718705\pi$
$773$	−35.2698	−1.26857	−0.634284	+	0.773101i	$0.718705\pi$
$774$	−0.699497	−0.0251429
$775$	6.02139	0.216295
$776$	5.98962	0.215015
$777$	−18.4728	−0.662707
$778$	−28.5636	−1.02405
$779$	6.73438	0.241284
$780$	−10.8583	−0.388788
$781$	0	0
$782$	−36.1229	−1.29175
$783$	36.6291	1.30902
$784$	−2.89889	−0.103532
$785$	1.24103	0.0442944
$786$	32.3771	1.15485
$787$	16.3383	0.582397	0.291198	−	0.956663i	$-0.405946\pi$
$787$	16.3383	0.582397	0.291198	+	0.956663i	$0.405946\pi$
$788$	35.1223	1.25118
$789$	−17.6980	−0.630067
$790$	−9.14405	−0.325331
$791$	13.4961	0.479867
$792$	0	0
$793$	−73.6491	−2.61536
$794$	52.0017	1.84547
$795$	5.55187	0.196905
$796$	−54.2183	−1.92172
$797$	−0.292902	−0.0103751	−0.00518755	−	0.999987i	$-0.501651\pi$
$797$	−0.292902	−0.0103751	−0.00518755	+	0.999987i	$0.501651\pi$
$798$	16.7183	0.591820
$799$	26.5914	0.940736
$800$	−38.1142	−1.34754
$801$	−0.357039	−0.0126154
$802$	−39.5277	−1.39577
$803$	0	0
$804$	−40.3752	−1.42392
$805$	2.77627	0.0978508
$806$	−14.1504	−0.498428
$807$	27.6369	0.972865
$808$	11.1874	0.393571
$809$	11.0373	0.388050	0.194025	−	0.980997i	$-0.437846\pi$
$809$	11.0373	0.388050	0.194025	+	0.980997i	$0.437846\pi$
$810$	8.97555	0.315369
$811$	5.21763	0.183216	0.0916078	−	0.995795i	$-0.470799\pi$
$811$	5.21763	0.183216	0.0916078	+	0.995795i	$0.470799\pi$
$812$	16.9579	0.595106
$813$	9.91799	0.347839
$814$	0	0
$815$	3.87957	0.135896
$816$	14.9546	0.523516
$817$	−13.4492	−0.470527
$818$	33.1381	1.15865
$819$	−0.609817	−0.0213087
$820$	1.74070	0.0607878
$821$	−26.4468	−0.923000	−0.461500	−	0.887140i	$-0.652689\pi$
$821$	−26.4468	−0.923000	−0.461500	+	0.887140i	$0.652689\pi$
$822$	5.86767	0.204658
$823$	−49.1895	−1.71464	−0.857319	−	0.514786i	$-0.827871\pi$
$823$	−49.1895	−1.71464	−0.857319	+	0.514786i	$0.827871\pi$
$824$	−6.64284	−0.231414
$825$	0	0
$826$	−17.6289	−0.613387
$827$	41.8006	1.45355	0.726774	−	0.686876i	$-0.241019\pi$
$827$	41.8006	1.45355	0.726774	+	0.686876i	$0.241019\pi$
$828$	1.58903	0.0552227
$829$	−24.8988	−0.864771	−0.432385	−	0.901689i	$-0.642328\pi$
$829$	−24.8988	−0.864771	−0.432385	+	0.901689i	$0.642328\pi$
$830$	11.5184	0.399808
$831$	−27.0496	−0.938340
$832$	58.8395	2.03989
$833$	−3.03721	−0.105233
$834$	−5.49863	−0.190402
$835$	1.01179	0.0350145
$836$	0	0
$837$	6.69650	0.231465
$838$	−47.6794	−1.64706
$839$	−53.2912	−1.83982	−0.919908	−	0.392135i	$-0.871737\pi$
$839$	−53.2912	−1.83982	−0.919908	+	0.392135i	$0.871737\pi$
$840$	0.793021	0.0273618
$841$	18.9273	0.652666
$842$	49.2143	1.69604
$843$	−17.3255	−0.596721
$844$	−5.34736	−0.184064
$845$	7.43171	0.255659
$846$	−2.12483	−0.0730530
$847$	0	0
$848$	−19.2439	−0.660837
$849$	27.4537	0.942207
$850$	−30.4800	−1.04545
$851$	−61.3219	−2.10209
$852$	24.7448	0.847744
$853$	−13.3706	−0.457800	−0.228900	−	0.973450i	$-0.573513\pi$
$853$	−13.3706	−0.457800	−0.228900	+	0.973450i	$0.573513\pi$
$854$	29.3106	1.00299
$855$	−0.264348	−0.00904050
$856$	10.7667	0.367997
$857$	−43.4419	−1.48395	−0.741974	−	0.670429i	$-0.766110\pi$
$857$	−43.4419	−1.48395	−0.741974	+	0.670429i	$0.766110\pi$
$858$	0	0
$859$	−7.96898	−0.271898	−0.135949	−	0.990716i	$-0.543408\pi$
$859$	−7.96898	−0.271898	−0.135949	+	0.990716i	$0.543408\pi$
$860$	−3.47633	−0.118542
$861$	−2.45133	−0.0835410
$862$	−20.1214	−0.685338
$863$	50.3471	1.71384	0.856918	−	0.515452i	$-0.172376\pi$
$863$	50.3471	1.71384	0.856918	+	0.515452i	$0.172376\pi$
$864$	−42.3875	−1.44205
$865$	11.4583	0.389593
$866$	48.5381	1.64939
$867$	−13.2066	−0.448519
$868$	3.10023	0.105229
$869$	0	0
$870$	12.2131	0.414064
$871$	51.4357	1.74283
$872$	17.9605	0.608218
$873$	0.726765	0.0245973
$874$	55.4976	1.87723
$875$	4.80453	0.162423
$876$	15.7027	0.530546
$877$	55.1801	1.86330	0.931649	−	0.363359i	$-0.118370\pi$
$877$	55.1801	1.86330	0.931649	+	0.363359i	$0.118370\pi$
$878$	−58.3106	−1.96789
$879$	−45.9476	−1.54978
$880$	0	0
$881$	−22.6475	−0.763014	−0.381507	−	0.924366i	$-0.624595\pi$
$881$	−22.6475	−0.763014	−0.381507	+	0.924366i	$0.624595\pi$
$882$	0.242693	0.00817188
$883$	−12.7050	−0.427557	−0.213779	−	0.976882i	$-0.568577\pi$
$883$	−12.7050	−0.427557	−0.213779	+	0.976882i	$0.568577\pi$
$884$	39.4326	1.32626
$885$	−6.98951	−0.234950
$886$	−31.1585	−1.04679
$887$	−6.70004	−0.224965	−0.112483	−	0.993654i	$-0.535880\pi$
$887$	−6.70004	−0.224965	−0.112483	+	0.993654i	$0.535880\pi$
$888$	−17.5161	−0.587802
$889$	−4.66064	−0.156313
$890$	−3.22317	−0.108041
$891$	0	0
$892$	66.7747	2.23578
$893$	−40.8538	−1.36712
$894$	48.3357	1.61659
$895$	8.69542	0.290656
$896$	−7.39409	−0.247019
$897$	50.7599	1.69482
$898$	−63.8446	−2.13052
$899$	8.76203	0.292230
$900$	1.34080	0.0446934
$901$	−20.1620	−0.671695
$902$	0	0
$903$	4.89552	0.162913
$904$	12.7972	0.425628
$905$	7.61733	0.253209
$906$	−43.8431	−1.45659
$907$	−57.7184	−1.91651	−0.958254	−	0.285918i	$-0.907701\pi$
$907$	−57.7184	−1.91651	−0.958254	+	0.285918i	$0.907701\pi$
$908$	33.2869	1.10467
$909$	1.35745	0.0450237
$910$	−5.50512	−0.182493
$911$	−2.84362	−0.0942134	−0.0471067	−	0.998890i	$-0.515000\pi$
$911$	−2.84362	−0.0942134	−0.0471067	+	0.998890i	$0.515000\pi$
$912$	−22.9756	−0.760798
$913$	0	0
$914$	−42.7288	−1.41334
$915$	11.6211	0.384181
$916$	−20.3345	−0.671870
$917$	9.03676	0.298420
$918$	−33.8973	−1.11878
$919$	9.36824	0.309030	0.154515	−	0.987990i	$-0.450619\pi$
$919$	9.36824	0.309030	0.154515	+	0.987990i	$0.450619\pi$
$920$	2.63250	0.0867908
$921$	−50.4884	−1.66365
$922$	17.9548	0.591309
$923$	−31.5235	−1.03761
$924$	0	0
$925$	−51.7424	−1.70128
$926$	1.24764	0.0410000
$927$	−0.806025	−0.0264733
$928$	−55.4620	−1.82063
$929$	−43.8437	−1.43847	−0.719233	−	0.694769i	$-0.755506\pi$
$929$	−43.8437	−1.43847	−0.719233	+	0.694769i	$0.755506\pi$
$930$	2.23280	0.0732162
$931$	4.66622	0.152929
$932$	−31.7155	−1.03888
$933$	38.0516	1.24575
$934$	86.5581	2.83227
$935$	0	0
$936$	−0.578236	−0.0189002
$937$	−3.20925	−0.104842	−0.0524208	−	0.998625i	$-0.516694\pi$
$937$	−3.20925	−0.104842	−0.0524208	+	0.998625i	$0.516694\pi$
$938$	−20.4702	−0.668374
$939$	16.8974	0.551425
$940$	−10.5599	−0.344425
$941$	−11.1320	−0.362893	−0.181446	−	0.983401i	$-0.558078\pi$
$941$	−11.1320	−0.362893	−0.181446	+	0.983401i	$0.558078\pi$
$942$	−9.03022	−0.294221
$943$	−8.13737	−0.264989
$944$	24.2270	0.788522
$945$	2.60522	0.0847479
$946$	0	0
$947$	51.6934	1.67981	0.839905	−	0.542733i	$-0.182610\pi$
$947$	51.6934	1.67981	0.839905	+	0.542733i	$0.182610\pi$
$948$	36.6287	1.18964
$949$	−20.0044	−0.649370
$950$	46.8280	1.51930
$951$	−18.9248	−0.613679
$952$	−2.87991	−0.0933386
$953$	−28.0305	−0.907996	−0.453998	−	0.891003i	$-0.650003\pi$
$953$	−28.0305	−0.907996	−0.453998	+	0.891003i	$0.650003\pi$
$954$	1.61108	0.0521606
$955$	7.84798	0.253955
$956$	−4.63796	−0.150002
$957$	0	0
$958$	42.9127	1.38645
$959$	1.63772	0.0528848
$960$	−9.28429	−0.299649
$961$	−29.3981	−0.948327
$962$	121.596	3.92042
$963$	1.30640	0.0420981
$964$	28.6553	0.922926
$965$	−4.16261	−0.133999
$966$	−20.2012	−0.649963
$967$	25.5912	0.822957	0.411479	−	0.911419i	$-0.365012\pi$
$967$	25.5912	0.822957	0.411479	+	0.911419i	$0.365012\pi$
$968$	0	0
$969$	−24.0718	−0.773298
$970$	6.56086	0.210657
$971$	12.5737	0.403508	0.201754	−	0.979436i	$-0.435336\pi$
$971$	12.5737	0.403508	0.201754	+	0.979436i	$0.435336\pi$
$972$	2.92718	0.0938895
$973$	−1.53472	−0.0492009
$974$	60.7852	1.94768
$975$	42.8304	1.37167
$976$	−40.2809	−1.28936
$977$	−14.5164	−0.464420	−0.232210	−	0.972666i	$-0.574596\pi$
$977$	−14.5164	−0.464420	−0.232210	+	0.972666i	$0.574596\pi$
$978$	−28.2292	−0.902671
$979$	0	0
$980$	1.20612	0.0385281
$981$	2.17928	0.0695789
$982$	5.65410	0.180430
$983$	−12.9883	−0.414264	−0.207132	−	0.978313i	$-0.566413\pi$
$983$	−12.9883	−0.414264	−0.207132	+	0.978313i	$0.566413\pi$
$984$	−2.32438	−0.0740984
$985$	7.06012	0.224954
$986$	−44.3529	−1.41249
$987$	14.8709	0.473344
$988$	−60.5824	−1.92738
$989$	16.2511	0.516754
$990$	0	0
$991$	7.01006	0.222682	0.111341	−	0.993782i	$-0.464485\pi$
$991$	7.01006	0.222682	0.111341	+	0.993782i	$0.464485\pi$
$992$	−10.1395	−0.321930
$993$	24.7898	0.786681
$994$	12.5456	0.397922
$995$	−10.8987	−0.345512
$996$	−46.1396	−1.46199
$997$	31.8789	1.00961	0.504807	−	0.863232i	$-0.331564\pi$
$997$	31.8789	1.00961	0.504807	+	0.863232i	$0.331564\pi$
$998$	−47.1291	−1.49185
$999$	−57.5437	−1.82060

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.m.1.2	✓	6
3.2	odd	2		7623.2.a.cs.1.5		6
7.6	odd	2		5929.2.a.bj.1.2		6
11.2	odd	10		847.2.f.y.323.2		24
11.3	even	5		847.2.f.z.372.2		24
11.4	even	5		847.2.f.z.148.2		24
11.5	even	5		847.2.f.z.729.5		24
11.6	odd	10		847.2.f.y.729.2		24
11.7	odd	10		847.2.f.y.148.5		24
11.8	odd	10		847.2.f.y.372.5		24
11.9	even	5		847.2.f.z.323.5		24
11.10	odd	2		847.2.a.n.1.5	yes	6
33.32	even	2		7623.2.a.cp.1.2		6
77.76	even	2		5929.2.a.bm.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.m.1.2	✓	6	1.1	even	1	trivial
847.2.a.n.1.5	yes	6	11.10	odd	2
847.2.f.y.148.5		24	11.7	odd	10
847.2.f.y.323.2		24	11.2	odd	10
847.2.f.y.372.5		24	11.8	odd	10
847.2.f.y.729.2		24	11.6	odd	10
847.2.f.z.148.2		24	11.4	even	5
847.2.f.z.323.5		24	11.9	even	5
847.2.f.z.372.2		24	11.3	even	5
847.2.f.z.729.5		24	11.5	even	5
5929.2.a.bj.1.2		6	7.6	odd	2
5929.2.a.bm.1.5		6	77.76	even	2
7623.2.a.cp.1.2		6	33.32	even	2
7623.2.a.cs.1.5		6	3.2	odd	2

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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q-2.10939 q^{2} +1.69851 q^{3} +2.44952 q^{4} +0.492391 q^{5} -3.58282 q^{6} -1.00000 q^{7} -0.948212 q^{8} -0.115054 q^{9} +O(q^{10})\)	\(q-2.10939 q^{2} +1.69851 q^{3} +2.44952 q^{4} +0.492391 q^{5} -3.58282 q^{6} -1.00000 q^{7} -0.948212 q^{8} -0.115054 q^{9} -1.03864 q^{10} +4.16054 q^{12} -5.30029 q^{13} +2.10939 q^{14} +0.836333 q^{15} -2.89889 q^{16} -3.03721 q^{17} +0.242693 q^{18} +4.66622 q^{19} +1.20612 q^{20} -1.69851 q^{21} -5.63835 q^{23} -1.61055 q^{24} -4.75755 q^{25} +11.1804 q^{26} -5.29096 q^{27} -2.44952 q^{28} -6.92295 q^{29} -1.76415 q^{30} -1.26565 q^{31} +8.01131 q^{32} +6.40665 q^{34} -0.492391 q^{35} -0.281826 q^{36} +10.8759 q^{37} -9.84288 q^{38} -9.00262 q^{39} -0.466891 q^{40} +1.44322 q^{41} +3.58282 q^{42} -2.88224 q^{43} -0.0566513 q^{45} +11.8935 q^{46} -8.75522 q^{47} -4.92381 q^{48} +1.00000 q^{49} +10.0355 q^{50} -5.15873 q^{51} -12.9832 q^{52} +6.63835 q^{53} +11.1607 q^{54} +0.948212 q^{56} +7.92564 q^{57} +14.6032 q^{58} -8.35733 q^{59} +2.04861 q^{60} +13.8953 q^{61} +2.66975 q^{62} +0.115054 q^{63} -11.1012 q^{64} -2.60982 q^{65} -9.70431 q^{67} -7.43970 q^{68} -9.57681 q^{69} +1.03864 q^{70} +5.94751 q^{71} +0.109095 q^{72} +3.77421 q^{73} -22.9414 q^{74} -8.08076 q^{75} +11.4300 q^{76} +18.9900 q^{78} +8.80383 q^{79} -1.42739 q^{80} -8.64160 q^{81} -3.04431 q^{82} -11.0898 q^{83} -4.16054 q^{84} -1.49549 q^{85} +6.07976 q^{86} -11.7587 q^{87} +3.10324 q^{89} +0.119500 q^{90} +5.30029 q^{91} -13.8113 q^{92} -2.14972 q^{93} +18.4682 q^{94} +2.29761 q^{95} +13.6073 q^{96} -6.31676 q^{97} -2.10939 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9	\( 6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9} + 8 q^{10} - 14 q^{12} - 4 q^{13} + 4 q^{14} + 2 q^{15} + 8 q^{16} - 22 q^{17} - 24 q^{18} - 6 q^{19} + 2 q^{20} + 2 q^{21} + 2 q^{23} + 20 q^{24} + 4 q^{25} + 6 q^{26} - 2 q^{27} - 4 q^{28} - 12 q^{29} - 20 q^{30} - 2 q^{31} - 8 q^{32} + 24 q^{34} + 4 q^{35} + 18 q^{36} + 14 q^{37} - 22 q^{38} - 20 q^{39} - 18 q^{40} - 26 q^{41} - 6 q^{42} + 4 q^{43} - 36 q^{45} - 12 q^{46} - 16 q^{47} - 24 q^{48} + 6 q^{49} + 4 q^{50} + 4 q^{51} - 12 q^{52} + 4 q^{53} + 32 q^{54} + 12 q^{56} - 20 q^{57} - 2 q^{58} - 4 q^{59} + 24 q^{60} + 8 q^{61} - 20 q^{62} - 8 q^{63} + 26 q^{64} - 24 q^{65} + 6 q^{67} - 12 q^{68} - 14 q^{69} - 8 q^{70} + 22 q^{71} - 16 q^{72} - 14 q^{73} - 44 q^{74} - 20 q^{75} + 30 q^{76} + 32 q^{78} + 28 q^{79} - 4 q^{80} - 6 q^{81} - 4 q^{82} - 22 q^{83} + 14 q^{84} + 24 q^{85} - 30 q^{86} - 22 q^{87} + 22 q^{90} + 4 q^{91} + 10 q^{92} - 50 q^{93} + 38 q^{94} + 24 q^{95} + 62 q^{96} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9 + 8 * q^10 - 14 * q^12 - 4 * q^13 + 4 * q^14 + 2 * q^15 + 8 * q^16 - 22 * q^17 - 24 * q^18 - 6 * q^19 + 2 * q^20 + 2 * q^21 + 2 * q^23 + 20 * q^24 + 4 * q^25 + 6 * q^26 - 2 * q^27 - 4 * q^28 - 12 * q^29 - 20 * q^30 - 2 * q^31 - 8 * q^32 + 24 * q^34 + 4 * q^35 + 18 * q^36 + 14 * q^37 - 22 * q^38 - 20 * q^39 - 18 * q^40 - 26 * q^41 - 6 * q^42 + 4 * q^43 - 36 * q^45 - 12 * q^46 - 16 * q^47 - 24 * q^48 + 6 * q^49 + 4 * q^50 + 4 * q^51 - 12 * q^52 + 4 * q^53 + 32 * q^54 + 12 * q^56 - 20 * q^57 - 2 * q^58 - 4 * q^59 + 24 * q^60 + 8 * q^61 - 20 * q^62 - 8 * q^63 + 26 * q^64 - 24 * q^65 + 6 * q^67 - 12 * q^68 - 14 * q^69 - 8 * q^70 + 22 * q^71 - 16 * q^72 - 14 * q^73 - 44 * q^74 - 20 * q^75 + 30 * q^76 + 32 * q^78 + 28 * q^79 - 4 * q^80 - 6 * q^81 - 4 * q^82 - 22 * q^83 + 14 * q^84 + 24 * q^85 - 30 * q^86 - 22 * q^87 + 22 * q^90 + 4 * q^91 + 10 * q^92 - 50 * q^93 + 38 * q^94 + 24 * q^95 + 62 * q^96 - 4 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more