LMFDB - Embedded newform 5070.2.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.24698 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.24698 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -0.801938 q^{11} -1.00000 q^{12} -4.24698 q^{14} -1.00000 q^{15} +1.00000 q^{16} +7.43296 q^{17} -1.00000 q^{18} +3.74094 q^{19} +1.00000 q^{20} -4.24698 q^{21} +0.801938 q^{22} -2.54288 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +4.24698 q^{28} +5.63102 q^{29} +1.00000 q^{30} -1.69202 q^{31} -1.00000 q^{32} +0.801938 q^{33} -7.43296 q^{34} +4.24698 q^{35} +1.00000 q^{36} -9.67456 q^{37} -3.74094 q^{38} -1.00000 q^{40} +8.85086 q^{41} +4.24698 q^{42} -6.29590 q^{43} -0.801938 q^{44} +1.00000 q^{45} +2.54288 q^{46} +7.53750 q^{47} -1.00000 q^{48} +11.0368 q^{49} -1.00000 q^{50} -7.43296 q^{51} +13.5646 q^{53} +1.00000 q^{54} -0.801938 q^{55} -4.24698 q^{56} -3.74094 q^{57} -5.63102 q^{58} +9.58211 q^{59} -1.00000 q^{60} -10.7899 q^{61} +1.69202 q^{62} +4.24698 q^{63} +1.00000 q^{64} -0.801938 q^{66} +7.82908 q^{67} +7.43296 q^{68} +2.54288 q^{69} -4.24698 q^{70} -2.04892 q^{71} -1.00000 q^{72} +2.17092 q^{73} +9.67456 q^{74} -1.00000 q^{75} +3.74094 q^{76} -3.40581 q^{77} -16.2349 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.85086 q^{82} -3.61356 q^{83} -4.24698 q^{84} +7.43296 q^{85} +6.29590 q^{86} -5.63102 q^{87} +0.801938 q^{88} -8.24459 q^{89} -1.00000 q^{90} -2.54288 q^{92} +1.69202 q^{93} -7.53750 q^{94} +3.74094 q^{95} +1.00000 q^{96} +6.07069 q^{97} -11.0368 q^{98} -0.801938 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 8 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 8 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 8 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 2 q^{11} - 3 q^{12} - 8 q^{14} - 3 q^{15} + 3 q^{16} + 3 q^{17} - 3 q^{18} - 3 q^{19} + 3 q^{20} - 8 q^{21} - 2 q^{22} + 11 q^{23} + 3 q^{24} + 3 q^{25} - 3 q^{27} + 8 q^{28} + 2 q^{29} + 3 q^{30} - 3 q^{32} - 2 q^{33} - 3 q^{34} + 8 q^{35} + 3 q^{36} - 8 q^{37} + 3 q^{38} - 3 q^{40} + 13 q^{41} + 8 q^{42} - 5 q^{43} + 2 q^{44} + 3 q^{45} - 11 q^{46} + 7 q^{47} - 3 q^{48} + 5 q^{49} - 3 q^{50} - 3 q^{51} + 19 q^{53} + 3 q^{54} + 2 q^{55} - 8 q^{56} + 3 q^{57} - 2 q^{58} + 23 q^{59} - 3 q^{60} - 9 q^{61} + 8 q^{63} + 3 q^{64} + 2 q^{66} + 13 q^{67} + 3 q^{68} - 11 q^{69} - 8 q^{70} + 3 q^{71} - 3 q^{72} + 17 q^{73} + 8 q^{74} - 3 q^{75} - 3 q^{76} + 3 q^{77} - 25 q^{79} + 3 q^{80} + 3 q^{81} - 13 q^{82} + 20 q^{83} - 8 q^{84} + 3 q^{85} + 5 q^{86} - 2 q^{87} - 2 q^{88} + 21 q^{89} - 3 q^{90} + 11 q^{92} - 7 q^{94} - 3 q^{95} + 3 q^{96} + 6 q^{97} - 5 q^{98} + 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 8 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 + 2 * q^11 - 3 * q^12 - 8 * q^14 - 3 * q^15 + 3 * q^16 + 3 * q^17 - 3 * q^18 - 3 * q^19 + 3 * q^20 - 8 * q^21 - 2 * q^22 + 11 * q^23 + 3 * q^24 + 3 * q^25 - 3 * q^27 + 8 * q^28 + 2 * q^29 + 3 * q^30 - 3 * q^32 - 2 * q^33 - 3 * q^34 + 8 * q^35 + 3 * q^36 - 8 * q^37 + 3 * q^38 - 3 * q^40 + 13 * q^41 + 8 * q^42 - 5 * q^43 + 2 * q^44 + 3 * q^45 - 11 * q^46 + 7 * q^47 - 3 * q^48 + 5 * q^49 - 3 * q^50 - 3 * q^51 + 19 * q^53 + 3 * q^54 + 2 * q^55 - 8 * q^56 + 3 * q^57 - 2 * q^58 + 23 * q^59 - 3 * q^60 - 9 * q^61 + 8 * q^63 + 3 * q^64 + 2 * q^66 + 13 * q^67 + 3 * q^68 - 11 * q^69 - 8 * q^70 + 3 * q^71 - 3 * q^72 + 17 * q^73 + 8 * q^74 - 3 * q^75 - 3 * q^76 + 3 * q^77 - 25 * q^79 + 3 * q^80 + 3 * q^81 - 13 * q^82 + 20 * q^83 - 8 * q^84 + 3 * q^85 + 5 * q^86 - 2 * q^87 - 2 * q^88 + 21 * q^89 - 3 * q^90 + 11 * q^92 - 7 * q^94 - 3 * q^95 + 3 * q^96 + 6 * q^97 - 5 * q^98 + 2 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	4.24698	1.60521	0.802604	−	0.596513i	$-0.203447\pi$
$7$	4.24698	1.60521	0.802604	+	0.596513i	$0.203447\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−0.801938	−0.241793	−0.120897	−	0.992665i	$-0.538577\pi$
$11$	−0.801938	−0.241793	−0.120897	+	0.992665i	$0.538577\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−4.24698	−1.13505
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	7.43296	1.80276	0.901379	−	0.433031i	$-0.142556\pi$
$17$	7.43296	1.80276	0.901379	+	0.433031i	$0.142556\pi$
$18$	−1.00000	−0.235702
$19$	3.74094	0.858230	0.429115	−	0.903250i	$-0.358825\pi$
$19$	3.74094	0.858230	0.429115	+	0.903250i	$0.358825\pi$
$20$	1.00000	0.223607
$21$	−4.24698	−0.926767
$22$	0.801938	0.170974
$23$	−2.54288	−0.530226	−0.265113	−	0.964217i	$-0.585409\pi$
$23$	−2.54288	−0.530226	−0.265113	+	0.964217i	$0.585409\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	4.24698	0.802604
$29$	5.63102	1.04565	0.522827	−	0.852439i	$-0.324877\pi$
$29$	5.63102	1.04565	0.522827	+	0.852439i	$0.324877\pi$
$30$	1.00000	0.182574
$31$	−1.69202	−0.303896	−0.151948	−	0.988388i	$-0.548555\pi$
$31$	−1.69202	−0.303896	−0.151948	+	0.988388i	$0.548555\pi$
$32$	−1.00000	−0.176777
$33$	0.801938	0.139599
$34$	−7.43296	−1.27474
$35$	4.24698	0.717871
$36$	1.00000	0.166667
$37$	−9.67456	−1.59049	−0.795244	−	0.606289i	$-0.792657\pi$
$37$	−9.67456	−1.59049	−0.795244	+	0.606289i	$0.792657\pi$
$38$	−3.74094	−0.606860
$39$	0	0
$40$	−1.00000	−0.158114
$41$	8.85086	1.38227	0.691136	−	0.722725i	$-0.257111\pi$
$41$	8.85086	1.38227	0.691136	+	0.722725i	$0.257111\pi$
$42$	4.24698	0.655323
$43$	−6.29590	−0.960115	−0.480058	−	0.877237i	$-0.659384\pi$
$43$	−6.29590	−0.960115	−0.480058	+	0.877237i	$0.659384\pi$
$44$	−0.801938	−0.120897
$45$	1.00000	0.149071
$46$	2.54288	0.374927
$47$	7.53750	1.09946	0.549729	−	0.835343i	$-0.314731\pi$
$47$	7.53750	1.09946	0.549729	+	0.835343i	$0.314731\pi$
$48$	−1.00000	−0.144338
$49$	11.0368	1.57669
$50$	−1.00000	−0.141421
$51$	−7.43296	−1.04082
$52$	0	0
$53$	13.5646	1.86325	0.931624	−	0.363424i	$-0.118392\pi$
$53$	13.5646	1.86325	0.931624	+	0.363424i	$0.118392\pi$
$54$	1.00000	0.136083
$55$	−0.801938	−0.108133
$56$	−4.24698	−0.567527
$57$	−3.74094	−0.495499
$58$	−5.63102	−0.739389
$59$	9.58211	1.24748	0.623742	−	0.781630i	$-0.285612\pi$
$59$	9.58211	1.24748	0.623742	+	0.781630i	$0.285612\pi$
$60$	−1.00000	−0.129099
$61$	−10.7899	−1.38150	−0.690750	−	0.723094i	$-0.742719\pi$
$61$	−10.7899	−1.38150	−0.690750	+	0.723094i	$0.742719\pi$
$62$	1.69202	0.214887
$63$	4.24698	0.535069
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.801938	−0.0987117
$67$	7.82908	0.956475	0.478237	−	0.878231i	$-0.341276\pi$
$67$	7.82908	0.956475	0.478237	+	0.878231i	$0.341276\pi$
$68$	7.43296	0.901379
$69$	2.54288	0.306126
$70$	−4.24698	−0.507611
$71$	−2.04892	−0.243162	−0.121581	−	0.992582i	$-0.538796\pi$
$71$	−2.04892	−0.243162	−0.121581	+	0.992582i	$0.538796\pi$
$72$	−1.00000	−0.117851
$73$	2.17092	0.254086	0.127043	−	0.991897i	$-0.459451\pi$
$73$	2.17092	0.254086	0.127043	+	0.991897i	$0.459451\pi$
$74$	9.67456	1.12465
$75$	−1.00000	−0.115470
$76$	3.74094	0.429115
$77$	−3.40581	−0.388128
$78$	0	0
$79$	−16.2349	−1.82657	−0.913284	−	0.407323i	$-0.866462\pi$
$79$	−16.2349	−1.82657	−0.913284	+	0.407323i	$0.866462\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−8.85086	−0.977413
$83$	−3.61356	−0.396640	−0.198320	−	0.980137i	$-0.563549\pi$
$83$	−3.61356	−0.396640	−0.198320	+	0.980137i	$0.563549\pi$
$84$	−4.24698	−0.463383
$85$	7.43296	0.806218
$86$	6.29590	0.678904
$87$	−5.63102	−0.603709
$88$	0.801938	0.0854868
$89$	−8.24459	−0.873924	−0.436962	−	0.899480i	$-0.643946\pi$
$89$	−8.24459	−0.873924	−0.436962	+	0.899480i	$0.643946\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−2.54288	−0.265113
$93$	1.69202	0.175454
$94$	−7.53750	−0.777434
$95$	3.74094	0.383812
$96$	1.00000	0.102062
$97$	6.07069	0.616385	0.308192	−	0.951324i	$-0.400276\pi$
$97$	6.07069	0.616385	0.308192	+	0.951324i	$0.400276\pi$
$98$	−11.0368	−1.11489
$99$	−0.801938	−0.0805978
$100$	1.00000	0.100000
$101$	−5.38404	−0.535732	−0.267866	−	0.963456i	$-0.586318\pi$
$101$	−5.38404	−0.535732	−0.267866	+	0.963456i	$0.586318\pi$
$102$	7.43296	0.735973
$103$	−8.92154	−0.879066	−0.439533	−	0.898227i	$-0.644856\pi$
$103$	−8.92154	−0.879066	−0.439533	+	0.898227i	$0.644856\pi$
$104$	0	0
$105$	−4.24698	−0.414463
$106$	−13.5646	−1.31751
$107$	13.9976	1.35320	0.676600	−	0.736351i	$-0.263453\pi$
$107$	13.9976	1.35320	0.676600	+	0.736351i	$0.263453\pi$
$108$	−1.00000	−0.0962250
$109$	13.8605	1.32760	0.663800	−	0.747910i	$-0.268943\pi$
$109$	13.8605	1.32760	0.663800	+	0.747910i	$0.268943\pi$
$110$	0.801938	0.0764618
$111$	9.67456	0.918269
$112$	4.24698	0.401302
$113$	2.19806	0.206776	0.103388	−	0.994641i	$-0.467032\pi$
$113$	2.19806	0.206776	0.103388	+	0.994641i	$0.467032\pi$
$114$	3.74094	0.350371
$115$	−2.54288	−0.237124
$116$	5.63102	0.522827
$117$	0	0
$118$	−9.58211	−0.882104
$119$	31.5676	2.89380
$120$	1.00000	0.0912871
$121$	−10.3569	−0.941536
$122$	10.7899	0.976868
$123$	−8.85086	−0.798055
$124$	−1.69202	−0.151948
$125$	1.00000	0.0894427
$126$	−4.24698	−0.378351
$127$	−4.19136	−0.371923	−0.185961	−	0.982557i	$-0.559540\pi$
$127$	−4.19136	−0.371923	−0.185961	+	0.982557i	$0.559540\pi$
$128$	−1.00000	−0.0883883
$129$	6.29590	0.554323
$130$	0	0
$131$	−10.8140	−0.944825	−0.472413	−	0.881378i	$-0.656617\pi$
$131$	−10.8140	−0.944825	−0.472413	+	0.881378i	$0.656617\pi$
$132$	0.801938	0.0697997
$133$	15.8877	1.37764
$134$	−7.82908	−0.676330
$135$	−1.00000	−0.0860663
$136$	−7.43296	−0.637371
$137$	12.8509	1.09792	0.548961	−	0.835848i	$-0.315023\pi$
$137$	12.8509	1.09792	0.548961	+	0.835848i	$0.315023\pi$
$138$	−2.54288	−0.216464
$139$	16.6233	1.40997	0.704983	−	0.709224i	$-0.250955\pi$
$139$	16.6233	1.40997	0.704983	+	0.709224i	$0.250955\pi$
$140$	4.24698	0.358935
$141$	−7.53750	−0.634772
$142$	2.04892	0.171941
$143$	0	0
$144$	1.00000	0.0833333
$145$	5.63102	0.467631
$146$	−2.17092	−0.179666
$147$	−11.0368	−0.910303
$148$	−9.67456	−0.795244
$149$	−1.74094	−0.142623	−0.0713116	−	0.997454i	$-0.522718\pi$
$149$	−1.74094	−0.142623	−0.0713116	+	0.997454i	$0.522718\pi$
$150$	1.00000	0.0816497
$151$	−1.75840	−0.143096	−0.0715482	−	0.997437i	$-0.522794\pi$
$151$	−1.75840	−0.143096	−0.0715482	+	0.997437i	$0.522794\pi$
$152$	−3.74094	−0.303430
$153$	7.43296	0.600919
$154$	3.40581	0.274448
$155$	−1.69202	−0.135906
$156$	0	0
$157$	−15.9825	−1.27555	−0.637773	−	0.770225i	$-0.720144\pi$
$157$	−15.9825	−1.27555	−0.637773	+	0.770225i	$0.720144\pi$
$158$	16.2349	1.29158
$159$	−13.5646	−1.07575
$160$	−1.00000	−0.0790569
$161$	−10.7995	−0.851123
$162$	−1.00000	−0.0785674
$163$	−5.18598	−0.406197	−0.203099	−	0.979158i	$-0.565101\pi$
$163$	−5.18598	−0.406197	−0.203099	+	0.979158i	$0.565101\pi$
$164$	8.85086	0.691136
$165$	0.801938	0.0624308
$166$	3.61356	0.280467
$167$	−5.29590	−0.409809	−0.204904	−	0.978782i	$-0.565688\pi$
$167$	−5.29590	−0.409809	−0.204904	+	0.978782i	$0.565688\pi$
$168$	4.24698	0.327662
$169$	0	0
$170$	−7.43296	−0.570082
$171$	3.74094	0.286077
$172$	−6.29590	−0.480058
$173$	−19.6558	−1.49440	−0.747201	−	0.664599i	$-0.768603\pi$
$173$	−19.6558	−1.49440	−0.747201	+	0.664599i	$0.768603\pi$
$174$	5.63102	0.426887
$175$	4.24698	0.321041
$176$	−0.801938	−0.0604483
$177$	−9.58211	−0.720235
$178$	8.24459	0.617958
$179$	−17.9095	−1.33862	−0.669308	−	0.742985i	$-0.733409\pi$
$179$	−17.9095	−1.33862	−0.669308	+	0.742985i	$0.733409\pi$
$180$	1.00000	0.0745356
$181$	−8.74094	−0.649709	−0.324854	−	0.945764i	$-0.605315\pi$
$181$	−8.74094	−0.649709	−0.324854	+	0.945764i	$0.605315\pi$
$182$	0	0
$183$	10.7899	0.797609
$184$	2.54288	0.187463
$185$	−9.67456	−0.711288
$186$	−1.69202	−0.124065
$187$	−5.96077	−0.435895
$188$	7.53750	0.549729
$189$	−4.24698	−0.308922
$190$	−3.74094	−0.271396
$191$	−3.76510	−0.272433	−0.136217	−	0.990679i	$-0.543494\pi$
$191$	−3.76510	−0.272433	−0.136217	+	0.990679i	$0.543494\pi$
$192$	−1.00000	−0.0721688
$193$	21.9584	1.58060	0.790299	−	0.612721i	$-0.209925\pi$
$193$	21.9584	1.58060	0.790299	+	0.612721i	$0.209925\pi$
$194$	−6.07069	−0.435850
$195$	0	0
$196$	11.0368	0.788345
$197$	−18.2446	−1.29987	−0.649936	−	0.759989i	$-0.725205\pi$
$197$	−18.2446	−1.29987	−0.649936	+	0.759989i	$0.725205\pi$
$198$	0.801938	0.0569912
$199$	−1.77479	−0.125812	−0.0629058	−	0.998019i	$-0.520037\pi$
$199$	−1.77479	−0.125812	−0.0629058	+	0.998019i	$0.520037\pi$
$200$	−1.00000	−0.0707107
$201$	−7.82908	−0.552221
$202$	5.38404	0.378820
$203$	23.9148	1.67849
$204$	−7.43296	−0.520411
$205$	8.85086	0.618171
$206$	8.92154	0.621593
$207$	−2.54288	−0.176742
$208$	0	0
$209$	−3.00000	−0.207514
$210$	4.24698	0.293069
$211$	0.0978347	0.00673522	0.00336761	−	0.999994i	$-0.498928\pi$
$211$	0.0978347	0.00673522	0.00336761	+	0.999994i	$0.498928\pi$
$212$	13.5646	0.931624
$213$	2.04892	0.140390
$214$	−13.9976	−0.956857
$215$	−6.29590	−0.429377
$216$	1.00000	0.0680414
$217$	−7.18598	−0.487816
$218$	−13.8605	−0.938754
$219$	−2.17092	−0.146697
$220$	−0.801938	−0.0540666
$221$	0	0
$222$	−9.67456	−0.649314
$223$	−19.5080	−1.30635	−0.653175	−	0.757207i	$-0.726563\pi$
$223$	−19.5080	−1.30635	−0.653175	+	0.757207i	$0.726563\pi$
$224$	−4.24698	−0.283763
$225$	1.00000	0.0666667
$226$	−2.19806	−0.146213
$227$	3.23059	0.214422	0.107211	−	0.994236i	$-0.465808\pi$
$227$	3.23059	0.214422	0.107211	+	0.994236i	$0.465808\pi$
$228$	−3.74094	−0.247750
$229$	−4.58748	−0.303149	−0.151575	−	0.988446i	$-0.548434\pi$
$229$	−4.58748	−0.303149	−0.151575	+	0.988446i	$0.548434\pi$
$230$	2.54288	0.167672
$231$	3.40581	0.224086
$232$	−5.63102	−0.369695
$233$	−6.18359	−0.405100	−0.202550	−	0.979272i	$-0.564923\pi$
$233$	−6.18359	−0.405100	−0.202550	+	0.979272i	$0.564923\pi$
$234$	0	0
$235$	7.53750	0.491692
$236$	9.58211	0.623742
$237$	16.2349	1.05457
$238$	−31.5676	−2.04623
$239$	8.42998	0.545290	0.272645	−	0.962115i	$-0.412102\pi$
$239$	8.42998	0.545290	0.272645	+	0.962115i	$0.412102\pi$
$240$	−1.00000	−0.0645497
$241$	20.8485	1.34297	0.671484	−	0.741019i	$-0.265657\pi$
$241$	20.8485	1.34297	0.671484	+	0.741019i	$0.265657\pi$
$242$	10.3569	0.665766
$243$	−1.00000	−0.0641500
$244$	−10.7899	−0.690750
$245$	11.0368	0.705118
$246$	8.85086	0.564310
$247$	0	0
$248$	1.69202	0.107443
$249$	3.61356	0.229000
$250$	−1.00000	−0.0632456
$251$	−6.39075	−0.403380	−0.201690	−	0.979449i	$-0.564643\pi$
$251$	−6.39075	−0.403380	−0.201690	+	0.979449i	$0.564643\pi$
$252$	4.24698	0.267535
$253$	2.03923	0.128205
$254$	4.19136	0.262989
$255$	−7.43296	−0.465470
$256$	1.00000	0.0625000
$257$	9.78687	0.610488	0.305244	−	0.952274i	$-0.401262\pi$
$257$	9.78687	0.610488	0.305244	+	0.952274i	$0.401262\pi$
$258$	−6.29590	−0.391965
$259$	−41.0877	−2.55306
$260$	0	0
$261$	5.63102	0.348552
$262$	10.8140	0.668092
$263$	14.0465	0.866146	0.433073	−	0.901359i	$-0.357429\pi$
$263$	14.0465	0.866146	0.433073	+	0.901359i	$0.357429\pi$
$264$	−0.801938	−0.0493559
$265$	13.5646	0.833270
$266$	−15.8877	−0.974137
$267$	8.24459	0.504561
$268$	7.82908	0.478237
$269$	−17.2597	−1.05234	−0.526170	−	0.850380i	$-0.676372\pi$
$269$	−17.2597	−1.05234	−0.526170	+	0.850380i	$0.676372\pi$
$270$	1.00000	0.0608581
$271$	−17.1782	−1.04350	−0.521751	−	0.853098i	$-0.674721\pi$
$271$	−17.1782	−1.04350	−0.521751	+	0.853098i	$0.674721\pi$
$272$	7.43296	0.450689
$273$	0	0
$274$	−12.8509	−0.776349
$275$	−0.801938	−0.0483587
$276$	2.54288	0.153063
$277$	−21.1444	−1.27044	−0.635221	−	0.772331i	$-0.719091\pi$
$277$	−21.1444	−1.27044	−0.635221	+	0.772331i	$0.719091\pi$
$278$	−16.6233	−0.996996
$279$	−1.69202	−0.101299
$280$	−4.24698	−0.253806
$281$	4.09246	0.244136	0.122068	−	0.992522i	$-0.461047\pi$
$281$	4.09246	0.244136	0.122068	+	0.992522i	$0.461047\pi$
$282$	7.53750	0.448852
$283$	12.1468	0.722049	0.361025	−	0.932556i	$-0.382427\pi$
$283$	12.1468	0.722049	0.361025	+	0.932556i	$0.382427\pi$
$284$	−2.04892	−0.121581
$285$	−3.74094	−0.221594
$286$	0	0
$287$	37.5894	2.21883
$288$	−1.00000	−0.0589256
$289$	38.2489	2.24994
$290$	−5.63102	−0.330665
$291$	−6.07069	−0.355870
$292$	2.17092	0.127043
$293$	20.6789	1.20807	0.604036	−	0.796957i	$-0.293558\pi$
$293$	20.6789	1.20807	0.604036	+	0.796957i	$0.293558\pi$
$294$	11.0368	0.643681
$295$	9.58211	0.557892
$296$	9.67456	0.562323
$297$	0.801938	0.0465331
$298$	1.74094	0.100850
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−26.7385	−1.54118
$302$	1.75840	0.101184
$303$	5.38404	0.309305
$304$	3.74094	0.214558
$305$	−10.7899	−0.617825
$306$	−7.43296	−0.424914
$307$	11.3123	0.645627	0.322813	−	0.946463i	$-0.395371\pi$
$307$	11.3123	0.645627	0.322813	+	0.946463i	$0.395371\pi$
$308$	−3.40581	−0.194064
$309$	8.92154	0.507529
$310$	1.69202	0.0961004
$311$	−26.2282	−1.48726	−0.743632	−	0.668589i	$-0.766899\pi$
$311$	−26.2282	−1.48726	−0.743632	+	0.668589i	$0.766899\pi$
$312$	0	0
$313$	29.5023	1.66757	0.833785	−	0.552089i	$-0.186169\pi$
$313$	29.5023	1.66757	0.833785	+	0.552089i	$0.186169\pi$
$314$	15.9825	0.901947
$315$	4.24698	0.239290
$316$	−16.2349	−0.913284
$317$	−7.06398	−0.396753	−0.198376	−	0.980126i	$-0.563567\pi$
$317$	−7.06398	−0.396753	−0.198376	+	0.980126i	$0.563567\pi$
$318$	13.5646	0.760668
$319$	−4.51573	−0.252832
$320$	1.00000	0.0559017
$321$	−13.9976	−0.781270
$322$	10.7995	0.601835
$323$	27.8062	1.54718
$324$	1.00000	0.0555556
$325$	0	0
$326$	5.18598	0.287225
$327$	−13.8605	−0.766490
$328$	−8.85086	−0.488707
$329$	32.0116	1.76486
$330$	−0.801938	−0.0441452
$331$	10.3134	0.566873	0.283437	−	0.958991i	$-0.408525\pi$
$331$	10.3134	0.566873	0.283437	+	0.958991i	$0.408525\pi$
$332$	−3.61356	−0.198320
$333$	−9.67456	−0.530163
$334$	5.29590	0.289779
$335$	7.82908	0.427749
$336$	−4.24698	−0.231692
$337$	30.3163	1.65144	0.825718	−	0.564083i	$-0.190770\pi$
$337$	30.3163	1.65144	0.825718	+	0.564083i	$0.190770\pi$
$338$	0	0
$339$	−2.19806	−0.119382
$340$	7.43296	0.403109
$341$	1.35690	0.0734800
$342$	−3.74094	−0.202287
$343$	17.1444	0.925708
$344$	6.29590	0.339452
$345$	2.54288	0.136904
$346$	19.6558	1.05670
$347$	28.8592	1.54924	0.774622	−	0.632425i	$-0.217940\pi$
$347$	28.8592	1.54924	0.774622	+	0.632425i	$0.217940\pi$
$348$	−5.63102	−0.301854
$349$	−27.4566	−1.46972	−0.734860	−	0.678218i	$-0.762752\pi$
$349$	−27.4566	−1.46972	−0.734860	+	0.678218i	$0.762752\pi$
$350$	−4.24698	−0.227011
$351$	0	0
$352$	0.801938	0.0427434
$353$	0.885772	0.0471449	0.0235724	−	0.999722i	$-0.492496\pi$
$353$	0.885772	0.0471449	0.0235724	+	0.999722i	$0.492496\pi$
$354$	9.58211	0.509283
$355$	−2.04892	−0.108745
$356$	−8.24459	−0.436962
$357$	−31.5676	−1.67074
$358$	17.9095	0.946544
$359$	7.07069	0.373177	0.186588	−	0.982438i	$-0.440257\pi$
$359$	7.07069	0.373177	0.186588	+	0.982438i	$0.440257\pi$
$360$	−1.00000	−0.0527046
$361$	−5.00538	−0.263441
$362$	8.74094	0.459414
$363$	10.3569	0.543596
$364$	0	0
$365$	2.17092	0.113631
$366$	−10.7899	−0.563995
$367$	27.3588	1.42812	0.714059	−	0.700085i	$-0.246855\pi$
$367$	27.3588	1.42812	0.714059	+	0.700085i	$0.246855\pi$
$368$	−2.54288	−0.132557
$369$	8.85086	0.460757
$370$	9.67456	0.502957
$371$	57.6088	2.99090
$372$	1.69202	0.0877272
$373$	18.3370	0.949456	0.474728	−	0.880133i	$-0.342546\pi$
$373$	18.3370	0.949456	0.474728	+	0.880133i	$0.342546\pi$
$374$	5.96077	0.308224
$375$	−1.00000	−0.0516398
$376$	−7.53750	−0.388717
$377$	0	0
$378$	4.24698	0.218441
$379$	−14.3690	−0.738085	−0.369042	−	0.929413i	$-0.620314\pi$
$379$	−14.3690	−0.738085	−0.369042	+	0.929413i	$0.620314\pi$
$380$	3.74094	0.191906
$381$	4.19136	0.214730
$382$	3.76510	0.192639
$383$	34.6920	1.77268	0.886340	−	0.463035i	$-0.153240\pi$
$383$	34.6920	1.77268	0.886340	+	0.463035i	$0.153240\pi$
$384$	1.00000	0.0510310
$385$	−3.40581	−0.173576
$386$	−21.9584	−1.11765
$387$	−6.29590	−0.320038
$388$	6.07069	0.308192
$389$	−32.6872	−1.65731	−0.828654	−	0.559761i	$-0.810893\pi$
$389$	−32.6872	−1.65731	−0.828654	+	0.559761i	$0.810893\pi$
$390$	0	0
$391$	−18.9011	−0.955870
$392$	−11.0368	−0.557444
$393$	10.8140	0.545495
$394$	18.2446	0.919149
$395$	−16.2349	−0.816866
$396$	−0.801938	−0.0402989
$397$	12.3080	0.617720	0.308860	−	0.951108i	$-0.400053\pi$
$397$	12.3080	0.617720	0.308860	+	0.951108i	$0.400053\pi$
$398$	1.77479	0.0889622
$399$	−15.8877	−0.795379
$400$	1.00000	0.0500000
$401$	19.2524	0.961417	0.480708	−	0.876881i	$-0.340380\pi$
$401$	19.2524	0.961417	0.480708	+	0.876881i	$0.340380\pi$
$402$	7.82908	0.390479
$403$	0	0
$404$	−5.38404	−0.267866
$405$	1.00000	0.0496904
$406$	−23.9148	−1.18687
$407$	7.75840	0.384569
$408$	7.43296	0.367986
$409$	−22.8672	−1.13071	−0.565356	−	0.824847i	$-0.691261\pi$
$409$	−22.8672	−1.13071	−0.565356	+	0.824847i	$0.691261\pi$
$410$	−8.85086	−0.437113
$411$	−12.8509	−0.633886
$412$	−8.92154	−0.439533
$413$	40.6950	2.00247
$414$	2.54288	0.124976
$415$	−3.61356	−0.177383
$416$	0	0
$417$	−16.6233	−0.814044
$418$	3.00000	0.146735
$419$	−17.1806	−0.839327	−0.419664	−	0.907680i	$-0.637852\pi$
$419$	−17.1806	−0.839327	−0.419664	+	0.907680i	$0.637852\pi$
$420$	−4.24698	−0.207231
$421$	36.9922	1.80289	0.901445	−	0.432893i	$-0.142507\pi$
$421$	36.9922	1.80289	0.901445	+	0.432893i	$0.142507\pi$
$422$	−0.0978347	−0.00476252
$423$	7.53750	0.366486
$424$	−13.5646	−0.658757
$425$	7.43296	0.360552
$426$	−2.04892	−0.0992704
$427$	−45.8243	−2.21759
$428$	13.9976	0.676600
$429$	0	0
$430$	6.29590	0.303615
$431$	−19.0834	−0.919213	−0.459607	−	0.888123i	$-0.652010\pi$
$431$	−19.0834	−0.919213	−0.459607	+	0.888123i	$0.652010\pi$
$432$	−1.00000	−0.0481125
$433$	11.5084	0.553060	0.276530	−	0.961005i	$-0.410815\pi$
$433$	11.5084	0.553060	0.276530	+	0.961005i	$0.410815\pi$
$434$	7.18598	0.344938
$435$	−5.63102	−0.269987
$436$	13.8605	0.663800
$437$	−9.51275	−0.455056
$438$	2.17092	0.103730
$439$	8.87502	0.423582	0.211791	−	0.977315i	$-0.432070\pi$
$439$	8.87502	0.423582	0.211791	+	0.977315i	$0.432070\pi$
$440$	0.801938	0.0382309
$441$	11.0368	0.525564
$442$	0	0
$443$	24.4838	1.16326	0.581630	−	0.813453i	$-0.302415\pi$
$443$	24.4838	1.16326	0.581630	+	0.813453i	$0.302415\pi$
$444$	9.67456	0.459134
$445$	−8.24459	−0.390831
$446$	19.5080	0.923729
$447$	1.74094	0.0823436
$448$	4.24698	0.200651
$449$	10.2567	0.484042	0.242021	−	0.970271i	$-0.422190\pi$
$449$	10.2567	0.484042	0.242021	+	0.970271i	$0.422190\pi$
$450$	−1.00000	−0.0471405
$451$	−7.09783	−0.334224
$452$	2.19806	0.103388
$453$	1.75840	0.0826167
$454$	−3.23059	−0.151619
$455$	0	0
$456$	3.74094	0.175186
$457$	38.6359	1.80731	0.903656	−	0.428259i	$-0.140873\pi$
$457$	38.6359	1.80731	0.903656	+	0.428259i	$0.140873\pi$
$458$	4.58748	0.214359
$459$	−7.43296	−0.346941
$460$	−2.54288	−0.118562
$461$	24.8073	1.15539	0.577696	−	0.816252i	$-0.303952\pi$
$461$	24.8073	1.15539	0.577696	+	0.816252i	$0.303952\pi$
$462$	−3.40581	−0.158453
$463$	−18.0713	−0.839844	−0.419922	−	0.907560i	$-0.637942\pi$
$463$	−18.0713	−0.839844	−0.419922	+	0.907560i	$0.637942\pi$
$464$	5.63102	0.261414
$465$	1.69202	0.0784656
$466$	6.18359	0.286449
$467$	−16.8834	−0.781270	−0.390635	−	0.920546i	$-0.627744\pi$
$467$	−16.8834	−0.781270	−0.390635	+	0.920546i	$0.627744\pi$
$468$	0	0
$469$	33.2500	1.53534
$470$	−7.53750	−0.347679
$471$	15.9825	0.736437
$472$	−9.58211	−0.441052
$473$	5.04892	0.232149
$474$	−16.2349	−0.745694
$475$	3.74094	0.171646
$476$	31.5676	1.44690
$477$	13.5646	0.621083
$478$	−8.42998	−0.385578
$479$	13.9879	0.639124	0.319562	−	0.947565i	$-0.396464\pi$
$479$	13.9879	0.639124	0.319562	+	0.947565i	$0.396464\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−20.8485	−0.949621
$483$	10.7995	0.491396
$484$	−10.3569	−0.470768
$485$	6.07069	0.275656
$486$	1.00000	0.0453609
$487$	23.2760	1.05474	0.527369	−	0.849636i	$-0.323178\pi$
$487$	23.2760	1.05474	0.527369	+	0.849636i	$0.323178\pi$
$488$	10.7899	0.488434
$489$	5.18598	0.234518
$490$	−11.0368	−0.498593
$491$	−32.5918	−1.47085	−0.735424	−	0.677608i	$-0.763017\pi$
$491$	−32.5918	−1.47085	−0.735424	+	0.677608i	$0.763017\pi$
$492$	−8.85086	−0.399027
$493$	41.8552	1.88506
$494$	0	0
$495$	−0.801938	−0.0360444
$496$	−1.69202	−0.0759740
$497$	−8.70171	−0.390325
$498$	−3.61356	−0.161928
$499$	−1.48666	−0.0665522	−0.0332761	−	0.999446i	$-0.510594\pi$
$499$	−1.48666	−0.0665522	−0.0332761	+	0.999446i	$0.510594\pi$
$500$	1.00000	0.0447214
$501$	5.29590	0.236603
$502$	6.39075	0.285233
$503$	38.3629	1.71052	0.855258	−	0.518203i	$-0.173399\pi$
$503$	38.3629	1.71052	0.855258	+	0.518203i	$0.173399\pi$
$504$	−4.24698	−0.189176
$505$	−5.38404	−0.239587
$506$	−2.03923	−0.0906548
$507$	0	0
$508$	−4.19136	−0.185961
$509$	0.770479	0.0341509	0.0170754	−	0.999854i	$-0.494564\pi$
$509$	0.770479	0.0341509	0.0170754	+	0.999854i	$0.494564\pi$
$510$	7.43296	0.329137
$511$	9.21983	0.407861
$512$	−1.00000	−0.0441942
$513$	−3.74094	−0.165166
$514$	−9.78687	−0.431680
$515$	−8.92154	−0.393130
$516$	6.29590	0.277161
$517$	−6.04461	−0.265841
$518$	41.0877	1.80529
$519$	19.6558	0.862793
$520$	0	0
$521$	−13.7657	−0.603086	−0.301543	−	0.953453i	$-0.597502\pi$
$521$	−13.7657	−0.603086	−0.301543	+	0.953453i	$0.597502\pi$
$522$	−5.63102	−0.246463
$523$	−17.5991	−0.769554	−0.384777	−	0.923009i	$-0.625722\pi$
$523$	−17.5991	−0.769554	−0.384777	+	0.923009i	$0.625722\pi$
$524$	−10.8140	−0.472413
$525$	−4.24698	−0.185353
$526$	−14.0465	−0.612458
$527$	−12.5767	−0.547851
$528$	0.801938	0.0348999
$529$	−16.5338	−0.718860
$530$	−13.5646	−0.589211
$531$	9.58211	0.415828
$532$	15.8877	0.688819
$533$	0	0
$534$	−8.24459	−0.356778
$535$	13.9976	0.605169
$536$	−7.82908	−0.338165
$537$	17.9095	0.772850
$538$	17.2597	0.744116
$539$	−8.85086	−0.381233
$540$	−1.00000	−0.0430331
$541$	−20.9259	−0.899673	−0.449836	−	0.893111i	$-0.648518\pi$
$541$	−20.9259	−0.899673	−0.449836	+	0.893111i	$0.648518\pi$
$542$	17.1782	0.737867
$543$	8.74094	0.375110
$544$	−7.43296	−0.318686
$545$	13.8605	0.593720
$546$	0	0
$547$	42.5599	1.81973	0.909864	−	0.414906i	$-0.136186\pi$
$547$	42.5599	1.81973	0.909864	+	0.414906i	$0.136186\pi$
$548$	12.8509	0.548961
$549$	−10.7899	−0.460500
$550$	0.801938	0.0341947
$551$	21.0653	0.897412
$552$	−2.54288	−0.108232
$553$	−68.9493	−2.93202
$554$	21.1444	0.898338
$555$	9.67456	0.410662
$556$	16.6233	0.704983
$557$	33.1637	1.40519	0.702596	−	0.711589i	$-0.252024\pi$
$557$	33.1637	1.40519	0.702596	+	0.711589i	$0.252024\pi$
$558$	1.69202	0.0716290
$559$	0	0
$560$	4.24698	0.179468
$561$	5.96077	0.251664
$562$	−4.09246	−0.172630
$563$	−7.27173	−0.306467	−0.153234	−	0.988190i	$-0.548969\pi$
$563$	−7.27173	−0.306467	−0.153234	+	0.988190i	$0.548969\pi$
$564$	−7.53750	−0.317386
$565$	2.19806	0.0924732
$566$	−12.1468	−0.510566
$567$	4.24698	0.178356
$568$	2.04892	0.0859707
$569$	38.9269	1.63190	0.815951	−	0.578122i	$-0.196214\pi$
$569$	38.9269	1.63190	0.815951	+	0.578122i	$0.196214\pi$
$570$	3.74094	0.156691
$571$	38.6045	1.61555	0.807773	−	0.589493i	$-0.200673\pi$
$571$	38.6045	1.61555	0.807773	+	0.589493i	$0.200673\pi$
$572$	0	0
$573$	3.76510	0.157289
$574$	−37.5894	−1.56895
$575$	−2.54288	−0.106045
$576$	1.00000	0.0416667
$577$	27.8646	1.16002	0.580009	−	0.814610i	$-0.303049\pi$
$577$	27.8646	1.16002	0.580009	+	0.814610i	$0.303049\pi$
$578$	−38.2489	−1.59094
$579$	−21.9584	−0.912559
$580$	5.63102	0.233815
$581$	−15.3467	−0.636690
$582$	6.07069	0.251638
$583$	−10.8780	−0.450521
$584$	−2.17092	−0.0898331
$585$	0	0
$586$	−20.6789	−0.854236
$587$	−26.1648	−1.07994	−0.539968	−	0.841685i	$-0.681564\pi$
$587$	−26.1648	−1.07994	−0.539968	+	0.841685i	$0.681564\pi$
$588$	−11.0368	−0.455151
$589$	−6.32975	−0.260813
$590$	−9.58211	−0.394489
$591$	18.2446	0.750482
$592$	−9.67456	−0.397622
$593$	−8.61117	−0.353618	−0.176809	−	0.984245i	$-0.556578\pi$
$593$	−8.61117	−0.353618	−0.176809	+	0.984245i	$0.556578\pi$
$594$	−0.801938	−0.0329039
$595$	31.5676	1.29415
$596$	−1.74094	−0.0713116
$597$	1.77479	0.0726373
$598$	0	0
$599$	11.1709	0.456431	0.228216	−	0.973611i	$-0.426711\pi$
$599$	11.1709	0.456431	0.228216	+	0.973611i	$0.426711\pi$
$600$	1.00000	0.0408248
$601$	8.91856	0.363796	0.181898	−	0.983317i	$-0.441776\pi$
$601$	8.91856	0.363796	0.181898	+	0.983317i	$0.441776\pi$
$602$	26.7385	1.08978
$603$	7.82908	0.318825
$604$	−1.75840	−0.0715482
$605$	−10.3569	−0.421068
$606$	−5.38404	−0.218712
$607$	−28.0277	−1.13761	−0.568805	−	0.822472i	$-0.692594\pi$
$607$	−28.0277	−1.13761	−0.568805	+	0.822472i	$0.692594\pi$
$608$	−3.74094	−0.151715
$609$	−23.9148	−0.969078
$610$	10.7899	0.436869
$611$	0	0
$612$	7.43296	0.300460
$613$	1.10885	0.0447861	0.0223930	−	0.999749i	$-0.492871\pi$
$613$	1.10885	0.0447861	0.0223930	+	0.999749i	$0.492871\pi$
$614$	−11.3123	−0.456527
$615$	−8.85086	−0.356901
$616$	3.40581	0.137224
$617$	−16.2905	−0.655832	−0.327916	−	0.944707i	$-0.606346\pi$
$617$	−16.2905	−0.655832	−0.327916	+	0.944707i	$0.606346\pi$
$618$	−8.92154	−0.358877
$619$	−23.2252	−0.933500	−0.466750	−	0.884389i	$-0.654575\pi$
$619$	−23.2252	−0.933500	−0.466750	+	0.884389i	$0.654575\pi$
$620$	−1.69202	−0.0679532
$621$	2.54288	0.102042
$622$	26.2282	1.05165
$623$	−35.0146	−1.40283
$624$	0	0
$625$	1.00000	0.0400000
$626$	−29.5023	−1.17915
$627$	3.00000	0.119808
$628$	−15.9825	−0.637773
$629$	−71.9106	−2.86727
$630$	−4.24698	−0.169204
$631$	0.357961	0.0142502	0.00712510	−	0.999975i	$-0.497732\pi$
$631$	0.357961	0.0142502	0.00712510	+	0.999975i	$0.497732\pi$
$632$	16.2349	0.645790
$633$	−0.0978347	−0.00388858
$634$	7.06398	0.280547
$635$	−4.19136	−0.166329
$636$	−13.5646	−0.537873
$637$	0	0
$638$	4.51573	0.178779
$639$	−2.04892	−0.0810539
$640$	−1.00000	−0.0395285
$641$	33.2814	1.31454	0.657269	−	0.753656i	$-0.271712\pi$
$641$	33.2814	1.31454	0.657269	+	0.753656i	$0.271712\pi$
$642$	13.9976	0.552441
$643$	−48.4172	−1.90939	−0.954693	−	0.297592i	$-0.903817\pi$
$643$	−48.4172	−1.90939	−0.954693	+	0.297592i	$0.903817\pi$
$644$	−10.7995	−0.425562
$645$	6.29590	0.247901
$646$	−27.8062	−1.09402
$647$	−47.4814	−1.86669	−0.933343	−	0.358985i	$-0.883123\pi$
$647$	−47.4814	−1.86669	−0.933343	+	0.358985i	$0.883123\pi$
$648$	−1.00000	−0.0392837
$649$	−7.68425	−0.301633
$650$	0	0
$651$	7.18598	0.281641
$652$	−5.18598	−0.203099
$653$	−25.4336	−0.995292	−0.497646	−	0.867380i	$-0.665802\pi$
$653$	−25.4336	−0.995292	−0.497646	+	0.867380i	$0.665802\pi$
$654$	13.8605	0.541990
$655$	−10.8140	−0.422539
$656$	8.85086	0.345568
$657$	2.17092	0.0846955
$658$	−32.0116	−1.24794
$659$	−2.55496	−0.0995270	−0.0497635	−	0.998761i	$-0.515847\pi$
$659$	−2.55496	−0.0995270	−0.0497635	+	0.998761i	$0.515847\pi$
$660$	0.801938	0.0312154
$661$	26.3394	1.02449	0.512243	−	0.858841i	$-0.328815\pi$
$661$	26.3394	1.02449	0.512243	+	0.858841i	$0.328815\pi$
$662$	−10.3134	−0.400840
$663$	0	0
$664$	3.61356	0.140233
$665$	15.8877	0.616098
$666$	9.67456	0.374882
$667$	−14.3190	−0.554434
$668$	−5.29590	−0.204904
$669$	19.5080	0.754221
$670$	−7.82908	−0.302464
$671$	8.65279	0.334037
$672$	4.24698	0.163831
$673$	−23.7767	−0.916525	−0.458262	−	0.888817i	$-0.651528\pi$
$673$	−23.7767	−0.916525	−0.458262	+	0.888817i	$0.651528\pi$
$674$	−30.3163	−1.16774
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−40.3250	−1.54981	−0.774907	−	0.632075i	$-0.782203\pi$
$677$	−40.3250	−1.54981	−0.774907	+	0.632075i	$0.782203\pi$
$678$	2.19806	0.0844161
$679$	25.7821	0.989426
$680$	−7.43296	−0.285041
$681$	−3.23059	−0.123796
$682$	−1.35690	−0.0519582
$683$	−4.51573	−0.172790	−0.0863948	−	0.996261i	$-0.527535\pi$
$683$	−4.51573	−0.172790	−0.0863948	+	0.996261i	$0.527535\pi$
$684$	3.74094	0.143038
$685$	12.8509	0.491006
$686$	−17.1444	−0.654575
$687$	4.58748	0.175023
$688$	−6.29590	−0.240029
$689$	0	0
$690$	−2.54288	−0.0968057
$691$	5.44398	0.207099	0.103549	−	0.994624i	$-0.466980\pi$
$691$	5.44398	0.207099	0.103549	+	0.994624i	$0.466980\pi$
$692$	−19.6558	−0.747201
$693$	−3.40581	−0.129376
$694$	−28.8592	−1.09548
$695$	16.6233	0.630556
$696$	5.63102	0.213443
$697$	65.7881	2.49190
$698$	27.4566	1.03925
$699$	6.18359	0.233885
$700$	4.24698	0.160521
$701$	−19.8670	−0.750366	−0.375183	−	0.926951i	$-0.622420\pi$
$701$	−19.8670	−0.750366	−0.375183	+	0.926951i	$0.622420\pi$
$702$	0	0
$703$	−36.1919	−1.36501
$704$	−0.801938	−0.0302242
$705$	−7.53750	−0.283879
$706$	−0.885772	−0.0333365
$707$	−22.8659	−0.859961
$708$	−9.58211	−0.360118
$709$	5.64204	0.211891	0.105946	−	0.994372i	$-0.466213\pi$
$709$	5.64204	0.211891	0.105946	+	0.994372i	$0.466213\pi$
$710$	2.04892	0.0768945
$711$	−16.2349	−0.608856
$712$	8.24459	0.308979
$713$	4.30260	0.161134
$714$	31.5676	1.18139
$715$	0	0
$716$	−17.9095	−0.669308
$717$	−8.42998	−0.314823
$718$	−7.07069	−0.263876
$719$	9.49289	0.354025	0.177013	−	0.984209i	$-0.443357\pi$
$719$	9.49289	0.354025	0.177013	+	0.984209i	$0.443357\pi$
$720$	1.00000	0.0372678
$721$	−37.8896	−1.41108
$722$	5.00538	0.186281
$723$	−20.8485	−0.775363
$724$	−8.74094	−0.324854
$725$	5.63102	0.209131
$726$	−10.3569	−0.384380
$727$	−18.2338	−0.676255	−0.338128	−	0.941100i	$-0.609794\pi$
$727$	−18.2338	−0.676255	−0.338128	+	0.941100i	$0.609794\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−2.17092	−0.0803492
$731$	−46.7972	−1.73086
$732$	10.7899	0.398805
$733$	36.4172	1.34510	0.672549	−	0.740052i	$-0.265199\pi$
$733$	36.4172	1.34510	0.672549	+	0.740052i	$0.265199\pi$
$734$	−27.3588	−1.00983
$735$	−11.0368	−0.407100
$736$	2.54288	0.0937317
$737$	−6.27844	−0.231269
$738$	−8.85086	−0.325804
$739$	−48.2312	−1.77421	−0.887106	−	0.461565i	$-0.847288\pi$
$739$	−48.2312	−1.77421	−0.887106	+	0.461565i	$0.847288\pi$
$740$	−9.67456	−0.355644
$741$	0	0
$742$	−57.6088	−2.11488
$743$	41.6859	1.52931	0.764654	−	0.644441i	$-0.222910\pi$
$743$	41.6859	1.52931	0.764654	+	0.644441i	$0.222910\pi$
$744$	−1.69202	−0.0620325
$745$	−1.74094	−0.0637831
$746$	−18.3370	−0.671367
$747$	−3.61356	−0.132213
$748$	−5.96077	−0.217947
$749$	59.4476	2.17217
$750$	1.00000	0.0365148
$751$	−0.486663	−0.0177586	−0.00887930	−	0.999961i	$-0.502826\pi$
$751$	−0.486663	−0.0177586	−0.00887930	+	0.999961i	$0.502826\pi$
$752$	7.53750	0.274864
$753$	6.39075	0.232892
$754$	0	0
$755$	−1.75840	−0.0639946
$756$	−4.24698	−0.154461
$757$	−46.2784	−1.68202	−0.841009	−	0.541021i	$-0.818038\pi$
$757$	−46.2784	−1.68202	−0.841009	+	0.541021i	$0.818038\pi$
$758$	14.3690	0.521905
$759$	−2.03923	−0.0740193
$760$	−3.74094	−0.135698
$761$	9.76915	0.354131	0.177066	−	0.984199i	$-0.443339\pi$
$761$	9.76915	0.354131	0.177066	+	0.984199i	$0.443339\pi$
$762$	−4.19136	−0.151837
$763$	58.8654	2.13107
$764$	−3.76510	−0.136217
$765$	7.43296	0.268739
$766$	−34.6920	−1.25347
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−18.5888	−0.670330	−0.335165	−	0.942160i	$-0.608792\pi$
$769$	−18.5888	−0.670330	−0.335165	+	0.942160i	$0.608792\pi$
$770$	3.40581	0.122737
$771$	−9.78687	−0.352466
$772$	21.9584	0.790299
$773$	23.4480	0.843367	0.421683	−	0.906743i	$-0.361439\pi$
$773$	23.4480	0.843367	0.421683	+	0.906743i	$0.361439\pi$
$774$	6.29590	0.226301
$775$	−1.69202	−0.0607792
$776$	−6.07069	−0.217925
$777$	41.0877	1.47401
$778$	32.6872	1.17189
$779$	33.1105	1.18631
$780$	0	0
$781$	1.64310	0.0587949
$782$	18.9011	0.675902
$783$	−5.63102	−0.201236
$784$	11.0368	0.394173
$785$	−15.9825	−0.570441
$786$	−10.8140	−0.385723
$787$	28.0030	0.998199	0.499099	−	0.866545i	$-0.333664\pi$
$787$	28.0030	0.998199	0.499099	+	0.866545i	$0.333664\pi$
$788$	−18.2446	−0.649936
$789$	−14.0465	−0.500070
$790$	16.2349	0.577612
$791$	9.33513	0.331919
$792$	0.801938	0.0284956
$793$	0	0
$794$	−12.3080	−0.436794
$795$	−13.5646	−0.481088
$796$	−1.77479	−0.0629058
$797$	42.9452	1.52120	0.760599	−	0.649222i	$-0.224905\pi$
$797$	42.9452	1.52120	0.760599	+	0.649222i	$0.224905\pi$
$798$	15.8877	0.562418
$799$	56.0259	1.98206
$800$	−1.00000	−0.0353553
$801$	−8.24459	−0.291308
$802$	−19.2524	−0.679824
$803$	−1.74094	−0.0614364
$804$	−7.82908	−0.276111
$805$	−10.7995	−0.380634
$806$	0	0
$807$	17.2597	0.607569
$808$	5.38404	0.189410
$809$	36.8622	1.29601	0.648003	−	0.761638i	$-0.275604\pi$
$809$	36.8622	1.29601	0.648003	+	0.761638i	$0.275604\pi$
$810$	−1.00000	−0.0351364
$811$	27.9952	0.983045	0.491522	−	0.870865i	$-0.336441\pi$
$811$	27.9952	0.983045	0.491522	+	0.870865i	$0.336441\pi$
$812$	23.9148	0.839246
$813$	17.1782	0.602466
$814$	−7.75840	−0.271932
$815$	−5.18598	−0.181657
$816$	−7.43296	−0.260206
$817$	−23.5526	−0.824000
$818$	22.8672	0.799535
$819$	0	0
$820$	8.85086	0.309085
$821$	18.5593	0.647723	0.323861	−	0.946105i	$-0.395019\pi$
$821$	18.5593	0.647723	0.323861	+	0.946105i	$0.395019\pi$
$822$	12.8509	0.448225
$823$	−43.2234	−1.50667	−0.753337	−	0.657635i	$-0.771557\pi$
$823$	−43.2234	−1.50667	−0.753337	+	0.657635i	$0.771557\pi$
$824$	8.92154	0.310797
$825$	0.801938	0.0279199
$826$	−40.6950	−1.41596
$827$	−16.7162	−0.581279	−0.290639	−	0.956833i	$-0.593868\pi$
$827$	−16.7162	−0.581279	−0.290639	+	0.956833i	$0.593868\pi$
$828$	−2.54288	−0.0883711
$829$	−10.6735	−0.370706	−0.185353	−	0.982672i	$-0.559343\pi$
$829$	−10.6735	−0.370706	−0.185353	+	0.982672i	$0.559343\pi$
$830$	3.61356	0.125429
$831$	21.1444	0.733490
$832$	0	0
$833$	82.0364	2.84239
$834$	16.6233	0.575616
$835$	−5.29590	−0.183272
$836$	−3.00000	−0.103757
$837$	1.69202	0.0584848
$838$	17.1806	0.593494
$839$	−44.2465	−1.52756	−0.763780	−	0.645477i	$-0.776659\pi$
$839$	−44.2465	−1.52756	−0.763780	+	0.645477i	$0.776659\pi$
$840$	4.24698	0.146535
$841$	2.70841	0.0933936
$842$	−36.9922	−1.27484
$843$	−4.09246	−0.140952
$844$	0.0978347	0.00336761
$845$	0	0
$846$	−7.53750	−0.259145
$847$	−43.9855	−1.51136
$848$	13.5646	0.465812
$849$	−12.1468	−0.416875
$850$	−7.43296	−0.254948
$851$	24.6012	0.843319
$852$	2.04892	0.0701948
$853$	−42.5437	−1.45667	−0.728335	−	0.685221i	$-0.759705\pi$
$853$	−42.5437	−1.45667	−0.728335	+	0.685221i	$0.759705\pi$
$854$	45.8243	1.56808
$855$	3.74094	0.127937
$856$	−13.9976	−0.478428
$857$	8.95061	0.305747	0.152873	−	0.988246i	$-0.451147\pi$
$857$	8.95061	0.305747	0.152873	+	0.988246i	$0.451147\pi$
$858$	0	0
$859$	−24.8810	−0.848928	−0.424464	−	0.905445i	$-0.639538\pi$
$859$	−24.8810	−0.848928	−0.424464	+	0.905445i	$0.639538\pi$
$860$	−6.29590	−0.214688
$861$	−37.5894	−1.28104
$862$	19.0834	0.649982
$863$	53.7531	1.82978	0.914889	−	0.403705i	$-0.132278\pi$
$863$	53.7531	1.82978	0.914889	+	0.403705i	$0.132278\pi$
$864$	1.00000	0.0340207
$865$	−19.6558	−0.668316
$866$	−11.5084	−0.391072
$867$	−38.2489	−1.29900
$868$	−7.18598	−0.243908
$869$	13.0194	0.441652
$870$	5.63102	0.190910
$871$	0	0
$872$	−13.8605	−0.469377
$873$	6.07069	0.205462
$874$	9.51275	0.321773
$875$	4.24698	0.143574
$876$	−2.17092	−0.0733484
$877$	−36.1782	−1.22165	−0.610826	−	0.791765i	$-0.709162\pi$
$877$	−36.1782	−1.22165	−0.610826	+	0.791765i	$0.709162\pi$
$878$	−8.87502	−0.299517
$879$	−20.6789	−0.697481
$880$	−0.801938	−0.0270333
$881$	−24.3075	−0.818941	−0.409470	−	0.912323i	$-0.634286\pi$
$881$	−24.3075	−0.818941	−0.409470	+	0.912323i	$0.634286\pi$
$882$	−11.0368	−0.371630
$883$	33.9764	1.14340	0.571699	−	0.820463i	$-0.306285\pi$
$883$	33.9764	1.14340	0.571699	+	0.820463i	$0.306285\pi$
$884$	0	0
$885$	−9.58211	−0.322099
$886$	−24.4838	−0.822549
$887$	−4.97929	−0.167188	−0.0835942	−	0.996500i	$-0.526640\pi$
$887$	−4.97929	−0.167188	−0.0835942	+	0.996500i	$0.526640\pi$
$888$	−9.67456	−0.324657
$889$	−17.8006	−0.597013
$890$	8.24459	0.276359
$891$	−0.801938	−0.0268659
$892$	−19.5080	−0.653175
$893$	28.1973	0.943588
$894$	−1.74094	−0.0582257
$895$	−17.9095	−0.598647
$896$	−4.24698	−0.141882
$897$	0	0
$898$	−10.2567	−0.342269
$899$	−9.52781	−0.317770
$900$	1.00000	0.0333333
$901$	100.825	3.35898
$902$	7.09783	0.236332
$903$	26.7385	0.889803
$904$	−2.19806	−0.0731065
$905$	−8.74094	−0.290559
$906$	−1.75840	−0.0584188
$907$	−24.1919	−0.803280	−0.401640	−	0.915798i	$-0.631560\pi$
$907$	−24.1919	−0.803280	−0.401640	+	0.915798i	$0.631560\pi$
$908$	3.23059	0.107211
$909$	−5.38404	−0.178577
$910$	0	0
$911$	−25.3284	−0.839168	−0.419584	−	0.907717i	$-0.637824\pi$
$911$	−25.3284	−0.839168	−0.419584	+	0.907717i	$0.637824\pi$
$912$	−3.74094	−0.123875
$913$	2.89785	0.0959050
$914$	−38.6359	−1.27796
$915$	10.7899	0.356702
$916$	−4.58748	−0.151575
$917$	−45.9269	−1.51664
$918$	7.43296	0.245324
$919$	24.4316	0.805925	0.402963	−	0.915216i	$-0.367980\pi$
$919$	24.4316	0.805925	0.402963	+	0.915216i	$0.367980\pi$
$920$	2.54288	0.0838362
$921$	−11.3123	−0.372753
$922$	−24.8073	−0.816985
$923$	0	0
$924$	3.40581	0.112043
$925$	−9.67456	−0.318098
$926$	18.0713	0.593859
$927$	−8.92154	−0.293022
$928$	−5.63102	−0.184847
$929$	1.78448	0.0585469	0.0292734	−	0.999571i	$-0.490681\pi$
$929$	1.78448	0.0585469	0.0292734	+	0.999571i	$0.490681\pi$
$930$	−1.69202	−0.0554836
$931$	41.2881	1.35316
$932$	−6.18359	−0.202550
$933$	26.2282	0.858672
$934$	16.8834	0.552441
$935$	−5.96077	−0.194938
$936$	0	0
$937$	46.9463	1.53367	0.766834	−	0.641845i	$-0.221831\pi$
$937$	46.9463	1.53367	0.766834	+	0.641845i	$0.221831\pi$
$938$	−33.2500	−1.08565
$939$	−29.5023	−0.962772
$940$	7.53750	0.245846
$941$	27.4047	0.893369	0.446685	−	0.894691i	$-0.352605\pi$
$941$	27.4047	0.893369	0.446685	+	0.894691i	$0.352605\pi$
$942$	−15.9825	−0.520739
$943$	−22.5066	−0.732917
$944$	9.58211	0.311871
$945$	−4.24698	−0.138154
$946$	−5.04892	−0.164154
$947$	−34.6055	−1.12453	−0.562264	−	0.826958i	$-0.690070\pi$
$947$	−34.6055	−1.12453	−0.562264	+	0.826958i	$0.690070\pi$
$948$	16.2349	0.527285
$949$	0	0
$950$	−3.74094	−0.121372
$951$	7.06398	0.229065
$952$	−31.5676	−1.02311
$953$	42.3435	1.37164	0.685820	−	0.727771i	$-0.259444\pi$
$953$	42.3435	1.37164	0.685820	+	0.727771i	$0.259444\pi$
$954$	−13.5646	−0.439172
$955$	−3.76510	−0.121836
$956$	8.42998	0.272645
$957$	4.51573	0.145973
$958$	−13.9879	−0.451929
$959$	54.5773	1.76239
$960$	−1.00000	−0.0322749
$961$	−28.1371	−0.907647
$962$	0	0
$963$	13.9976	0.451067
$964$	20.8485	0.671484
$965$	21.9584	0.706865
$966$	−10.7995	−0.347470
$967$	−31.6437	−1.01759	−0.508796	−	0.860887i	$-0.669909\pi$
$967$	−31.6437	−1.01759	−0.508796	+	0.860887i	$0.669909\pi$
$968$	10.3569	0.332883
$969$	−27.8062	−0.893265
$970$	−6.07069	−0.194918
$971$	8.41849	0.270162	0.135081	−	0.990835i	$-0.456871\pi$
$971$	8.41849	0.270162	0.135081	+	0.990835i	$0.456871\pi$
$972$	−1.00000	−0.0320750
$973$	70.5986	2.26329
$974$	−23.2760	−0.745813
$975$	0	0
$976$	−10.7899	−0.345375
$977$	−22.0814	−0.706448	−0.353224	−	0.935539i	$-0.614915\pi$
$977$	−22.0814	−0.706448	−0.353224	+	0.935539i	$0.614915\pi$
$978$	−5.18598	−0.165829
$979$	6.61165	0.211309
$980$	11.0368	0.352559
$981$	13.8605	0.442533
$982$	32.5918	1.04005
$983$	−32.7217	−1.04366	−0.521830	−	0.853050i	$-0.674750\pi$
$983$	−32.7217	−1.04366	−0.521830	+	0.853050i	$0.674750\pi$
$984$	8.85086	0.282155
$985$	−18.2446	−0.581321
$986$	−41.8552	−1.33294
$987$	−32.0116	−1.01894
$988$	0	0
$989$	16.0097	0.509078
$990$	0.801938	0.0254873
$991$	−52.9329	−1.68147	−0.840734	−	0.541448i	$-0.817876\pi$
$991$	−52.9329	−1.68147	−0.840734	+	0.541448i	$0.817876\pi$
$992$	1.69202	0.0537217
$993$	−10.3134	−0.327284
$994$	8.70171	0.276001
$995$	−1.77479	−0.0562646
$996$	3.61356	0.114500
$997$	−10.8823	−0.344646	−0.172323	−	0.985040i	$-0.555127\pi$
$997$	−10.8823	−0.344646	−0.172323	+	0.985040i	$0.555127\pi$
$998$	1.48666	0.0470595
$999$	9.67456	0.306090

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bm.1.3	✓	3
13.5	odd	4		5070.2.b.u.1351.6		6
13.8	odd	4		5070.2.b.u.1351.1		6
13.12	even	2		5070.2.a.br.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bm.1.3	✓	3	1.1	even	1	trivial
5070.2.a.br.1.1	yes	3	13.12	even	2
5070.2.b.u.1351.1		6	13.8	odd	4
5070.2.b.u.1351.6		6	13.5	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.24698 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.24698 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -0.801938 q^{11} -1.00000 q^{12} -4.24698 q^{14} -1.00000 q^{15} +1.00000 q^{16} +7.43296 q^{17} -1.00000 q^{18} +3.74094 q^{19} +1.00000 q^{20} -4.24698 q^{21} +0.801938 q^{22} -2.54288 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +4.24698 q^{28} +5.63102 q^{29} +1.00000 q^{30} -1.69202 q^{31} -1.00000 q^{32} +0.801938 q^{33} -7.43296 q^{34} +4.24698 q^{35} +1.00000 q^{36} -9.67456 q^{37} -3.74094 q^{38} -1.00000 q^{40} +8.85086 q^{41} +4.24698 q^{42} -6.29590 q^{43} -0.801938 q^{44} +1.00000 q^{45} +2.54288 q^{46} +7.53750 q^{47} -1.00000 q^{48} +11.0368 q^{49} -1.00000 q^{50} -7.43296 q^{51} +13.5646 q^{53} +1.00000 q^{54} -0.801938 q^{55} -4.24698 q^{56} -3.74094 q^{57} -5.63102 q^{58} +9.58211 q^{59} -1.00000 q^{60} -10.7899 q^{61} +1.69202 q^{62} +4.24698 q^{63} +1.00000 q^{64} -0.801938 q^{66} +7.82908 q^{67} +7.43296 q^{68} +2.54288 q^{69} -4.24698 q^{70} -2.04892 q^{71} -1.00000 q^{72} +2.17092 q^{73} +9.67456 q^{74} -1.00000 q^{75} +3.74094 q^{76} -3.40581 q^{77} -16.2349 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.85086 q^{82} -3.61356 q^{83} -4.24698 q^{84} +7.43296 q^{85} +6.29590 q^{86} -5.63102 q^{87} +0.801938 q^{88} -8.24459 q^{89} -1.00000 q^{90} -2.54288 q^{92} +1.69202 q^{93} -7.53750 q^{94} +3.74094 q^{95} +1.00000 q^{96} +6.07069 q^{97} -11.0368 q^{98} -0.801938 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 8 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 8 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 8 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 2 q^{11} - 3 q^{12} - 8 q^{14} - 3 q^{15} + 3 q^{16} + 3 q^{17} - 3 q^{18} - 3 q^{19} + 3 q^{20} - 8 q^{21} - 2 q^{22} + 11 q^{23} + 3 q^{24} + 3 q^{25} - 3 q^{27} + 8 q^{28} + 2 q^{29} + 3 q^{30} - 3 q^{32} - 2 q^{33} - 3 q^{34} + 8 q^{35} + 3 q^{36} - 8 q^{37} + 3 q^{38} - 3 q^{40} + 13 q^{41} + 8 q^{42} - 5 q^{43} + 2 q^{44} + 3 q^{45} - 11 q^{46} + 7 q^{47} - 3 q^{48} + 5 q^{49} - 3 q^{50} - 3 q^{51} + 19 q^{53} + 3 q^{54} + 2 q^{55} - 8 q^{56} + 3 q^{57} - 2 q^{58} + 23 q^{59} - 3 q^{60} - 9 q^{61} + 8 q^{63} + 3 q^{64} + 2 q^{66} + 13 q^{67} + 3 q^{68} - 11 q^{69} - 8 q^{70} + 3 q^{71} - 3 q^{72} + 17 q^{73} + 8 q^{74} - 3 q^{75} - 3 q^{76} + 3 q^{77} - 25 q^{79} + 3 q^{80} + 3 q^{81} - 13 q^{82} + 20 q^{83} - 8 q^{84} + 3 q^{85} + 5 q^{86} - 2 q^{87} - 2 q^{88} + 21 q^{89} - 3 q^{90} + 11 q^{92} - 7 q^{94} - 3 q^{95} + 3 q^{96} + 6 q^{97} - 5 q^{98} + 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 8 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 + 2 * q^11 - 3 * q^12 - 8 * q^14 - 3 * q^15 + 3 * q^16 + 3 * q^17 - 3 * q^18 - 3 * q^19 + 3 * q^20 - 8 * q^21 - 2 * q^22 + 11 * q^23 + 3 * q^24 + 3 * q^25 - 3 * q^27 + 8 * q^28 + 2 * q^29 + 3 * q^30 - 3 * q^32 - 2 * q^33 - 3 * q^34 + 8 * q^35 + 3 * q^36 - 8 * q^37 + 3 * q^38 - 3 * q^40 + 13 * q^41 + 8 * q^42 - 5 * q^43 + 2 * q^44 + 3 * q^45 - 11 * q^46 + 7 * q^47 - 3 * q^48 + 5 * q^49 - 3 * q^50 - 3 * q^51 + 19 * q^53 + 3 * q^54 + 2 * q^55 - 8 * q^56 + 3 * q^57 - 2 * q^58 + 23 * q^59 - 3 * q^60 - 9 * q^61 + 8 * q^63 + 3 * q^64 + 2 * q^66 + 13 * q^67 + 3 * q^68 - 11 * q^69 - 8 * q^70 + 3 * q^71 - 3 * q^72 + 17 * q^73 + 8 * q^74 - 3 * q^75 - 3 * q^76 + 3 * q^77 - 25 * q^79 + 3 * q^80 + 3 * q^81 - 13 * q^82 + 20 * q^83 - 8 * q^84 + 3 * q^85 + 5 * q^86 - 2 * q^87 - 2 * q^88 + 21 * q^89 - 3 * q^90 + 11 * q^92 - 7 * q^94 - 3 * q^95 + 3 * q^96 + 6 * q^97 - 5 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

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Database

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