LMFDB - Embedded newform 8470.2.a.bw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} +4.46410 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -7.46410 q^{17} -2.00000 q^{18} +5.92820 q^{19} +1.00000 q^{20} -1.00000 q^{21} -4.46410 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.46410 q^{26} +5.00000 q^{27} +1.00000 q^{28} -6.92820 q^{29} -1.00000 q^{30} -0.535898 q^{31} +1.00000 q^{32} -7.46410 q^{34} +1.00000 q^{35} -2.00000 q^{36} -8.92820 q^{37} +5.92820 q^{38} -4.46410 q^{39} +1.00000 q^{40} -8.39230 q^{41} -1.00000 q^{42} -12.3923 q^{43} -2.00000 q^{45} -4.46410 q^{46} -0.535898 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +7.46410 q^{51} +4.46410 q^{52} -7.46410 q^{53} +5.00000 q^{54} +1.00000 q^{56} -5.92820 q^{57} -6.92820 q^{58} +9.92820 q^{59} -1.00000 q^{60} +4.92820 q^{61} -0.535898 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.46410 q^{65} +6.39230 q^{67} -7.46410 q^{68} +4.46410 q^{69} +1.00000 q^{70} +8.00000 q^{71} -2.00000 q^{72} -0.535898 q^{73} -8.92820 q^{74} -1.00000 q^{75} +5.92820 q^{76} -4.46410 q^{78} -7.53590 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.39230 q^{82} -1.00000 q^{83} -1.00000 q^{84} -7.46410 q^{85} -12.3923 q^{86} +6.92820 q^{87} -17.4641 q^{89} -2.00000 q^{90} +4.46410 q^{91} -4.46410 q^{92} +0.535898 q^{93} -0.535898 q^{94} +5.92820 q^{95} -1.00000 q^{96} -1.07180 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 - 4 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} - 4 q^{9} + 2 q^{10} - 2 q^{12} + 2 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} - 8 q^{17} - 4 q^{18} - 2 q^{19} + 2 q^{20} - 2 q^{21} - 2 q^{23} - 2 q^{24} + 2 q^{25} + 2 q^{26} + 10 q^{27} + 2 q^{28} - 2 q^{30} - 8 q^{31} + 2 q^{32} - 8 q^{34} + 2 q^{35} - 4 q^{36} - 4 q^{37} - 2 q^{38} - 2 q^{39} + 2 q^{40} + 4 q^{41} - 2 q^{42} - 4 q^{43} - 4 q^{45} - 2 q^{46} - 8 q^{47} - 2 q^{48} + 2 q^{49} + 2 q^{50} + 8 q^{51} + 2 q^{52} - 8 q^{53} + 10 q^{54} + 2 q^{56} + 2 q^{57} + 6 q^{59} - 2 q^{60} - 4 q^{61} - 8 q^{62} - 4 q^{63} + 2 q^{64} + 2 q^{65} - 8 q^{67} - 8 q^{68} + 2 q^{69} + 2 q^{70} + 16 q^{71} - 4 q^{72} - 8 q^{73} - 4 q^{74} - 2 q^{75} - 2 q^{76} - 2 q^{78} - 22 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} - 2 q^{83} - 2 q^{84} - 8 q^{85} - 4 q^{86} - 28 q^{89} - 4 q^{90} + 2 q^{91} - 2 q^{92} + 8 q^{93} - 8 q^{94} - 2 q^{95} - 2 q^{96} - 16 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 - 4 * q^9 + 2 * q^10 - 2 * q^12 + 2 * q^13 + 2 * q^14 - 2 * q^15 + 2 * q^16 - 8 * q^17 - 4 * q^18 - 2 * q^19 + 2 * q^20 - 2 * q^21 - 2 * q^23 - 2 * q^24 + 2 * q^25 + 2 * q^26 + 10 * q^27 + 2 * q^28 - 2 * q^30 - 8 * q^31 + 2 * q^32 - 8 * q^34 + 2 * q^35 - 4 * q^36 - 4 * q^37 - 2 * q^38 - 2 * q^39 + 2 * q^40 + 4 * q^41 - 2 * q^42 - 4 * q^43 - 4 * q^45 - 2 * q^46 - 8 * q^47 - 2 * q^48 + 2 * q^49 + 2 * q^50 + 8 * q^51 + 2 * q^52 - 8 * q^53 + 10 * q^54 + 2 * q^56 + 2 * q^57 + 6 * q^59 - 2 * q^60 - 4 * q^61 - 8 * q^62 - 4 * q^63 + 2 * q^64 + 2 * q^65 - 8 * q^67 - 8 * q^68 + 2 * q^69 + 2 * q^70 + 16 * q^71 - 4 * q^72 - 8 * q^73 - 4 * q^74 - 2 * q^75 - 2 * q^76 - 2 * q^78 - 22 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 - 2 * q^83 - 2 * q^84 - 8 * q^85 - 4 * q^86 - 28 * q^89 - 4 * q^90 + 2 * q^91 - 2 * q^92 + 8 * q^93 - 8 * q^94 - 2 * q^95 - 2 * q^96 - 16 * q^97 + 2 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−1.00000	−0.288675
$13$	4.46410	1.23812	0.619060	−	0.785344i	$-0.287514\pi$
$13$	4.46410	1.23812	0.619060	+	0.785344i	$0.287514\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−7.46410	−1.81031	−0.905155	−	0.425081i	$-0.860246\pi$
$17$	−7.46410	−1.81031	−0.905155	+	0.425081i	$0.860246\pi$
$18$	−2.00000	−0.471405
$19$	5.92820	1.36002	0.680012	−	0.733201i	$-0.261975\pi$
$19$	5.92820	1.36002	0.680012	+	0.733201i	$0.261975\pi$
$20$	1.00000	0.223607
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−4.46410	−0.930830	−0.465415	−	0.885093i	$-0.654095\pi$
$23$	−4.46410	−0.930830	−0.465415	+	0.885093i	$0.654095\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	4.46410	0.875482
$27$	5.00000	0.962250
$28$	1.00000	0.188982
$29$	−6.92820	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$30$	−1.00000	−0.182574
$31$	−0.535898	−0.0962502	−0.0481251	−	0.998841i	$-0.515325\pi$
$31$	−0.535898	−0.0962502	−0.0481251	+	0.998841i	$0.515325\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−7.46410	−1.28008
$35$	1.00000	0.169031
$36$	−2.00000	−0.333333
$37$	−8.92820	−1.46779	−0.733894	−	0.679264i	$-0.762299\pi$
$37$	−8.92820	−1.46779	−0.733894	+	0.679264i	$0.762299\pi$
$38$	5.92820	0.961682
$39$	−4.46410	−0.714828
$40$	1.00000	0.158114
$41$	−8.39230	−1.31066	−0.655329	−	0.755344i	$-0.727470\pi$
$41$	−8.39230	−1.31066	−0.655329	+	0.755344i	$0.727470\pi$
$42$	−1.00000	−0.154303
$43$	−12.3923	−1.88981	−0.944904	−	0.327346i	$-0.893846\pi$
$43$	−12.3923	−1.88981	−0.944904	+	0.327346i	$0.893846\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	−4.46410	−0.658196
$47$	−0.535898	−0.0781688	−0.0390844	−	0.999236i	$-0.512444\pi$
$47$	−0.535898	−0.0781688	−0.0390844	+	0.999236i	$0.512444\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	7.46410	1.04518
$52$	4.46410	0.619060
$53$	−7.46410	−1.02527	−0.512637	−	0.858606i	$-0.671331\pi$
$53$	−7.46410	−1.02527	−0.512637	+	0.858606i	$0.671331\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	1.00000	0.133631
$57$	−5.92820	−0.785210
$58$	−6.92820	−0.909718
$59$	9.92820	1.29254	0.646271	−	0.763108i	$-0.276328\pi$
$59$	9.92820	1.29254	0.646271	+	0.763108i	$0.276328\pi$
$60$	−1.00000	−0.129099
$61$	4.92820	0.630992	0.315496	−	0.948927i	$-0.397829\pi$
$61$	4.92820	0.630992	0.315496	+	0.948927i	$0.397829\pi$
$62$	−0.535898	−0.0680592
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	4.46410	0.553704
$66$	0	0
$67$	6.39230	0.780944	0.390472	−	0.920615i	$-0.372312\pi$
$67$	6.39230	0.780944	0.390472	+	0.920615i	$0.372312\pi$
$68$	−7.46410	−0.905155
$69$	4.46410	0.537415
$70$	1.00000	0.119523
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−2.00000	−0.235702
$73$	−0.535898	−0.0627222	−0.0313611	−	0.999508i	$-0.509984\pi$
$73$	−0.535898	−0.0627222	−0.0313611	+	0.999508i	$0.509984\pi$
$74$	−8.92820	−1.03788
$75$	−1.00000	−0.115470
$76$	5.92820	0.680012
$77$	0	0
$78$	−4.46410	−0.505460
$79$	−7.53590	−0.847855	−0.423927	−	0.905696i	$-0.639349\pi$
$79$	−7.53590	−0.847855	−0.423927	+	0.905696i	$0.639349\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−8.39230	−0.926775
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	−1.00000	−0.109109
$85$	−7.46410	−0.809595
$86$	−12.3923	−1.33630
$87$	6.92820	0.742781
$88$	0	0
$89$	−17.4641	−1.85119	−0.925596	−	0.378514i	$-0.876435\pi$
$89$	−17.4641	−1.85119	−0.925596	+	0.378514i	$0.876435\pi$
$90$	−2.00000	−0.210819
$91$	4.46410	0.467965
$92$	−4.46410	−0.465415
$93$	0.535898	0.0555701
$94$	−0.535898	−0.0552737
$95$	5.92820	0.608221
$96$	−1.00000	−0.102062
$97$	−1.07180	−0.108824	−0.0544122	−	0.998519i	$-0.517329\pi$
$97$	−1.07180	−0.108824	−0.0544122	+	0.998519i	$0.517329\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−11.5359	−1.14786	−0.573932	−	0.818903i	$-0.694583\pi$
$101$	−11.5359	−1.14786	−0.573932	+	0.818903i	$0.694583\pi$
$102$	7.46410	0.739056
$103$	−1.46410	−0.144262	−0.0721311	−	0.997395i	$-0.522980\pi$
$103$	−1.46410	−0.144262	−0.0721311	+	0.997395i	$0.522980\pi$
$104$	4.46410	0.437741
$105$	−1.00000	−0.0975900
$106$	−7.46410	−0.724978
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	5.00000	0.481125
$109$	18.3923	1.76166	0.880832	−	0.473430i	$-0.156984\pi$
$109$	18.3923	1.76166	0.880832	+	0.473430i	$0.156984\pi$
$110$	0	0
$111$	8.92820	0.847428
$112$	1.00000	0.0944911
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	−5.92820	−0.555227
$115$	−4.46410	−0.416280
$116$	−6.92820	−0.643268
$117$	−8.92820	−0.825413
$118$	9.92820	0.913965
$119$	−7.46410	−0.684233
$120$	−1.00000	−0.0912871
$121$	0	0
$122$	4.92820	0.446179
$123$	8.39230	0.756709
$124$	−0.535898	−0.0481251
$125$	1.00000	0.0894427
$126$	−2.00000	−0.178174
$127$	13.3923	1.18837	0.594187	−	0.804327i	$-0.297474\pi$
$127$	13.3923	1.18837	0.594187	+	0.804327i	$0.297474\pi$
$128$	1.00000	0.0883883
$129$	12.3923	1.09108
$130$	4.46410	0.391528
$131$	−10.8564	−0.948529	−0.474264	−	0.880383i	$-0.657286\pi$
$131$	−10.8564	−0.948529	−0.474264	+	0.880383i	$0.657286\pi$
$132$	0	0
$133$	5.92820	0.514040
$134$	6.39230	0.552211
$135$	5.00000	0.430331
$136$	−7.46410	−0.640041
$137$	7.92820	0.677352	0.338676	−	0.940903i	$-0.390021\pi$
$137$	7.92820	0.677352	0.338676	+	0.940903i	$0.390021\pi$
$138$	4.46410	0.380010
$139$	16.8564	1.42974	0.714871	−	0.699256i	$-0.246485\pi$
$139$	16.8564	1.42974	0.714871	+	0.699256i	$0.246485\pi$
$140$	1.00000	0.0845154
$141$	0.535898	0.0451308
$142$	8.00000	0.671345
$143$	0	0
$144$	−2.00000	−0.166667
$145$	−6.92820	−0.575356
$146$	−0.535898	−0.0443513
$147$	−1.00000	−0.0824786
$148$	−8.92820	−0.733894
$149$	−3.07180	−0.251651	−0.125826	−	0.992052i	$-0.540158\pi$
$149$	−3.07180	−0.251651	−0.125826	+	0.992052i	$0.540158\pi$
$150$	−1.00000	−0.0816497
$151$	−5.53590	−0.450505	−0.225253	−	0.974300i	$-0.572321\pi$
$151$	−5.53590	−0.450505	−0.225253	+	0.974300i	$0.572321\pi$
$152$	5.92820	0.480841
$153$	14.9282	1.20687
$154$	0	0
$155$	−0.535898	−0.0430444
$156$	−4.46410	−0.357414
$157$	−20.3205	−1.62175	−0.810877	−	0.585217i	$-0.801009\pi$
$157$	−20.3205	−1.62175	−0.810877	+	0.585217i	$0.801009\pi$
$158$	−7.53590	−0.599524
$159$	7.46410	0.591942
$160$	1.00000	0.0790569
$161$	−4.46410	−0.351820
$162$	1.00000	0.0785674
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	−8.39230	−0.655329
$165$	0	0
$166$	−1.00000	−0.0776151
$167$	−4.39230	−0.339887	−0.169943	−	0.985454i	$-0.554358\pi$
$167$	−4.39230	−0.339887	−0.169943	+	0.985454i	$0.554358\pi$
$168$	−1.00000	−0.0771517
$169$	6.92820	0.532939
$170$	−7.46410	−0.572470
$171$	−11.8564	−0.906682
$172$	−12.3923	−0.944904
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	6.92820	0.525226
$175$	1.00000	0.0755929
$176$	0	0
$177$	−9.92820	−0.746249
$178$	−17.4641	−1.30899
$179$	−8.39230	−0.627270	−0.313635	−	0.949544i	$-0.601547\pi$
$179$	−8.39230	−0.627270	−0.313635	+	0.949544i	$0.601547\pi$
$180$	−2.00000	−0.149071
$181$	10.4641	0.777791	0.388895	−	0.921282i	$-0.372857\pi$
$181$	10.4641	0.777791	0.388895	+	0.921282i	$0.372857\pi$
$182$	4.46410	0.330901
$183$	−4.92820	−0.364303
$184$	−4.46410	−0.329098
$185$	−8.92820	−0.656415
$186$	0.535898	0.0392940
$187$	0	0
$188$	−0.535898	−0.0390844
$189$	5.00000	0.363696
$190$	5.92820	0.430077
$191$	3.39230	0.245459	0.122729	−	0.992440i	$-0.460835\pi$
$191$	3.39230	0.245459	0.122729	+	0.992440i	$0.460835\pi$
$192$	−1.00000	−0.0721688
$193$	19.0000	1.36765	0.683825	−	0.729646i	$-0.260315\pi$
$193$	19.0000	1.36765	0.683825	+	0.729646i	$0.260315\pi$
$194$	−1.07180	−0.0769505
$195$	−4.46410	−0.319681
$196$	1.00000	0.0714286
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	0	0
$199$	−25.8564	−1.83291	−0.916456	−	0.400135i	$-0.868963\pi$
$199$	−25.8564	−1.83291	−0.916456	+	0.400135i	$0.868963\pi$
$200$	1.00000	0.0707107
$201$	−6.39230	−0.450878
$202$	−11.5359	−0.811663
$203$	−6.92820	−0.486265
$204$	7.46410	0.522592
$205$	−8.39230	−0.586144
$206$	−1.46410	−0.102009
$207$	8.92820	0.620553
$208$	4.46410	0.309530
$209$	0	0
$210$	−1.00000	−0.0690066
$211$	−24.0000	−1.65223	−0.826114	−	0.563503i	$-0.809453\pi$
$211$	−24.0000	−1.65223	−0.826114	+	0.563503i	$0.809453\pi$
$212$	−7.46410	−0.512637
$213$	−8.00000	−0.548151
$214$	−6.00000	−0.410152
$215$	−12.3923	−0.845148
$216$	5.00000	0.340207
$217$	−0.535898	−0.0363792
$218$	18.3923	1.24568
$219$	0.535898	0.0362127
$220$	0	0
$221$	−33.3205	−2.24138
$222$	8.92820	0.599222
$223$	−3.46410	−0.231973	−0.115987	−	0.993251i	$-0.537003\pi$
$223$	−3.46410	−0.231973	−0.115987	+	0.993251i	$0.537003\pi$
$224$	1.00000	0.0668153
$225$	−2.00000	−0.133333
$226$	1.00000	0.0665190
$227$	−16.0000	−1.06196	−0.530979	−	0.847385i	$-0.678176\pi$
$227$	−16.0000	−1.06196	−0.530979	+	0.847385i	$0.678176\pi$
$228$	−5.92820	−0.392605
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	−4.46410	−0.294354
$231$	0	0
$232$	−6.92820	−0.454859
$233$	−18.8564	−1.23532	−0.617662	−	0.786444i	$-0.711920\pi$
$233$	−18.8564	−1.23532	−0.617662	+	0.786444i	$0.711920\pi$
$234$	−8.92820	−0.583655
$235$	−0.535898	−0.0349582
$236$	9.92820	0.646271
$237$	7.53590	0.489509
$238$	−7.46410	−0.483826
$239$	−12.4641	−0.806236	−0.403118	−	0.915148i	$-0.632074\pi$
$239$	−12.4641	−0.806236	−0.403118	+	0.915148i	$0.632074\pi$
$240$	−1.00000	−0.0645497
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	4.92820	0.315496
$245$	1.00000	0.0638877
$246$	8.39230	0.535074
$247$	26.4641	1.68387
$248$	−0.535898	−0.0340296
$249$	1.00000	0.0633724
$250$	1.00000	0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	13.3923	0.840308
$255$	7.46410	0.467420
$256$	1.00000	0.0625000
$257$	2.39230	0.149228	0.0746139	−	0.997212i	$-0.476228\pi$
$257$	2.39230	0.149228	0.0746139	+	0.997212i	$0.476228\pi$
$258$	12.3923	0.771511
$259$	−8.92820	−0.554772
$260$	4.46410	0.276852
$261$	13.8564	0.857690
$262$	−10.8564	−0.670711
$263$	16.4641	1.01522	0.507610	−	0.861587i	$-0.330529\pi$
$263$	16.4641	1.01522	0.507610	+	0.861587i	$0.330529\pi$
$264$	0	0
$265$	−7.46410	−0.458516
$266$	5.92820	0.363481
$267$	17.4641	1.06879
$268$	6.39230	0.390472
$269$	−10.3205	−0.629252	−0.314626	−	0.949216i	$-0.601879\pi$
$269$	−10.3205	−0.629252	−0.314626	+	0.949216i	$0.601879\pi$
$270$	5.00000	0.304290
$271$	29.8564	1.81365	0.906824	−	0.421510i	$-0.138500\pi$
$271$	29.8564	1.81365	0.906824	+	0.421510i	$0.138500\pi$
$272$	−7.46410	−0.452578
$273$	−4.46410	−0.270180
$274$	7.92820	0.478960
$275$	0	0
$276$	4.46410	0.268707
$277$	27.3205	1.64153	0.820765	−	0.571266i	$-0.193547\pi$
$277$	27.3205	1.64153	0.820765	+	0.571266i	$0.193547\pi$
$278$	16.8564	1.01098
$279$	1.07180	0.0641668
$280$	1.00000	0.0597614
$281$	17.0000	1.01413	0.507067	−	0.861906i	$-0.330729\pi$
$281$	17.0000	1.01413	0.507067	+	0.861906i	$0.330729\pi$
$282$	0.535898	0.0319123
$283$	−13.7846	−0.819410	−0.409705	−	0.912218i	$-0.634368\pi$
$283$	−13.7846	−0.819410	−0.409705	+	0.912218i	$0.634368\pi$
$284$	8.00000	0.474713
$285$	−5.92820	−0.351156
$286$	0	0
$287$	−8.39230	−0.495382
$288$	−2.00000	−0.117851
$289$	38.7128	2.27722
$290$	−6.92820	−0.406838
$291$	1.07180	0.0628298
$292$	−0.535898	−0.0313611
$293$	−20.4641	−1.19553	−0.597763	−	0.801673i	$-0.703944\pi$
$293$	−20.4641	−1.19553	−0.597763	+	0.801673i	$0.703944\pi$
$294$	−1.00000	−0.0583212
$295$	9.92820	0.578042
$296$	−8.92820	−0.518941
$297$	0	0
$298$	−3.07180	−0.177944
$299$	−19.9282	−1.15248
$300$	−1.00000	−0.0577350
$301$	−12.3923	−0.714281
$302$	−5.53590	−0.318555
$303$	11.5359	0.662720
$304$	5.92820	0.340006
$305$	4.92820	0.282188
$306$	14.9282	0.853389
$307$	−14.9282	−0.851998	−0.425999	−	0.904724i	$-0.640077\pi$
$307$	−14.9282	−0.851998	−0.425999	+	0.904724i	$0.640077\pi$
$308$	0	0
$309$	1.46410	0.0832898
$310$	−0.535898	−0.0304370
$311$	23.3205	1.32238	0.661192	−	0.750216i	$-0.270051\pi$
$311$	23.3205	1.32238	0.661192	+	0.750216i	$0.270051\pi$
$312$	−4.46410	−0.252730
$313$	14.7846	0.835676	0.417838	−	0.908522i	$-0.362788\pi$
$313$	14.7846	0.835676	0.417838	+	0.908522i	$0.362788\pi$
$314$	−20.3205	−1.14675
$315$	−2.00000	−0.112687
$316$	−7.53590	−0.423927
$317$	−22.3923	−1.25768	−0.628839	−	0.777536i	$-0.716469\pi$
$317$	−22.3923	−1.25768	−0.628839	+	0.777536i	$0.716469\pi$
$318$	7.46410	0.418566
$319$	0	0
$320$	1.00000	0.0559017
$321$	6.00000	0.334887
$322$	−4.46410	−0.248775
$323$	−44.2487	−2.46206
$324$	1.00000	0.0555556
$325$	4.46410	0.247624
$326$	−22.0000	−1.21847
$327$	−18.3923	−1.01710
$328$	−8.39230	−0.463388
$329$	−0.535898	−0.0295450
$330$	0	0
$331$	1.46410	0.0804743	0.0402372	−	0.999190i	$-0.487189\pi$
$331$	1.46410	0.0804743	0.0402372	+	0.999190i	$0.487189\pi$
$332$	−1.00000	−0.0548821
$333$	17.8564	0.978525
$334$	−4.39230	−0.240336
$335$	6.39230	0.349249
$336$	−1.00000	−0.0545545
$337$	31.7846	1.73142	0.865709	−	0.500548i	$-0.166868\pi$
$337$	31.7846	1.73142	0.865709	+	0.500548i	$0.166868\pi$
$338$	6.92820	0.376845
$339$	−1.00000	−0.0543125
$340$	−7.46410	−0.404798
$341$	0	0
$342$	−11.8564	−0.641121
$343$	1.00000	0.0539949
$344$	−12.3923	−0.668148
$345$	4.46410	0.240339
$346$	18.0000	0.967686
$347$	0.928203	0.0498286	0.0249143	−	0.999690i	$-0.492069\pi$
$347$	0.928203	0.0498286	0.0249143	+	0.999690i	$0.492069\pi$
$348$	6.92820	0.371391
$349$	−1.53590	−0.0822148	−0.0411074	−	0.999155i	$-0.513089\pi$
$349$	−1.53590	−0.0822148	−0.0411074	+	0.999155i	$0.513089\pi$
$350$	1.00000	0.0534522
$351$	22.3205	1.19138
$352$	0	0
$353$	−20.5359	−1.09302	−0.546508	−	0.837454i	$-0.684043\pi$
$353$	−20.5359	−1.09302	−0.546508	+	0.837454i	$0.684043\pi$
$354$	−9.92820	−0.527678
$355$	8.00000	0.424596
$356$	−17.4641	−0.925596
$357$	7.46410	0.395042
$358$	−8.39230	−0.443547
$359$	−19.7128	−1.04040	−0.520201	−	0.854044i	$-0.674143\pi$
$359$	−19.7128	−1.04040	−0.520201	+	0.854044i	$0.674143\pi$
$360$	−2.00000	−0.105409
$361$	16.1436	0.849663
$362$	10.4641	0.549981
$363$	0	0
$364$	4.46410	0.233983
$365$	−0.535898	−0.0280502
$366$	−4.92820	−0.257601
$367$	−21.4641	−1.12042	−0.560208	−	0.828352i	$-0.689279\pi$
$367$	−21.4641	−1.12042	−0.560208	+	0.828352i	$0.689279\pi$
$368$	−4.46410	−0.232707
$369$	16.7846	0.873772
$370$	−8.92820	−0.464155
$371$	−7.46410	−0.387517
$372$	0.535898	0.0277850
$373$	24.9282	1.29073	0.645367	−	0.763873i	$-0.276705\pi$
$373$	24.9282	1.29073	0.645367	+	0.763873i	$0.276705\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−0.535898	−0.0276368
$377$	−30.9282	−1.59288
$378$	5.00000	0.257172
$379$	23.8564	1.22542	0.612711	−	0.790307i	$-0.290079\pi$
$379$	23.8564	1.22542	0.612711	+	0.790307i	$0.290079\pi$
$380$	5.92820	0.304110
$381$	−13.3923	−0.686109
$382$	3.39230	0.173565
$383$	−21.4641	−1.09676	−0.548382	−	0.836228i	$-0.684756\pi$
$383$	−21.4641	−1.09676	−0.548382	+	0.836228i	$0.684756\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	19.0000	0.967075
$387$	24.7846	1.25987
$388$	−1.07180	−0.0544122
$389$	−13.3205	−0.675377	−0.337688	−	0.941258i	$-0.609645\pi$
$389$	−13.3205	−0.675377	−0.337688	+	0.941258i	$0.609645\pi$
$390$	−4.46410	−0.226049
$391$	33.3205	1.68509
$392$	1.00000	0.0505076
$393$	10.8564	0.547633
$394$	4.00000	0.201517
$395$	−7.53590	−0.379172
$396$	0	0
$397$	−11.0718	−0.555678	−0.277839	−	0.960628i	$-0.589618\pi$
$397$	−11.0718	−0.555678	−0.277839	+	0.960628i	$0.589618\pi$
$398$	−25.8564	−1.29606
$399$	−5.92820	−0.296781
$400$	1.00000	0.0500000
$401$	4.14359	0.206921	0.103461	−	0.994634i	$-0.467008\pi$
$401$	4.14359	0.206921	0.103461	+	0.994634i	$0.467008\pi$
$402$	−6.39230	−0.318819
$403$	−2.39230	−0.119169
$404$	−11.5359	−0.573932
$405$	1.00000	0.0496904
$406$	−6.92820	−0.343841
$407$	0	0
$408$	7.46410	0.369528
$409$	−31.7128	−1.56810	−0.784049	−	0.620699i	$-0.786849\pi$
$409$	−31.7128	−1.56810	−0.784049	+	0.620699i	$0.786849\pi$
$410$	−8.39230	−0.414466
$411$	−7.92820	−0.391069
$412$	−1.46410	−0.0721311
$413$	9.92820	0.488535
$414$	8.92820	0.438797
$415$	−1.00000	−0.0490881
$416$	4.46410	0.218871
$417$	−16.8564	−0.825462
$418$	0	0
$419$	38.7128	1.89124	0.945622	−	0.325267i	$-0.105454\pi$
$419$	38.7128	1.89124	0.945622	+	0.325267i	$0.105454\pi$
$420$	−1.00000	−0.0487950
$421$	5.85641	0.285424	0.142712	−	0.989764i	$-0.454418\pi$
$421$	5.85641	0.285424	0.142712	+	0.989764i	$0.454418\pi$
$422$	−24.0000	−1.16830
$423$	1.07180	0.0521125
$424$	−7.46410	−0.362489
$425$	−7.46410	−0.362062
$426$	−8.00000	−0.387601
$427$	4.92820	0.238492
$428$	−6.00000	−0.290021
$429$	0	0
$430$	−12.3923	−0.597610
$431$	−10.3205	−0.497121	−0.248561	−	0.968616i	$-0.579958\pi$
$431$	−10.3205	−0.497121	−0.248561	+	0.968616i	$0.579958\pi$
$432$	5.00000	0.240563
$433$	22.2487	1.06920	0.534602	−	0.845104i	$-0.320461\pi$
$433$	22.2487	1.06920	0.534602	+	0.845104i	$0.320461\pi$
$434$	−0.535898	−0.0257239
$435$	6.92820	0.332182
$436$	18.3923	0.880832
$437$	−26.4641	−1.26595
$438$	0.535898	0.0256062
$439$	−22.5359	−1.07558	−0.537790	−	0.843079i	$-0.680741\pi$
$439$	−22.5359	−1.07558	−0.537790	+	0.843079i	$0.680741\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	−33.3205	−1.58489
$443$	−15.7128	−0.746538	−0.373269	−	0.927723i	$-0.621763\pi$
$443$	−15.7128	−0.746538	−0.373269	+	0.927723i	$0.621763\pi$
$444$	8.92820	0.423714
$445$	−17.4641	−0.827878
$446$	−3.46410	−0.164030
$447$	3.07180	0.145291
$448$	1.00000	0.0472456
$449$	−8.07180	−0.380932	−0.190466	−	0.981694i	$-0.561000\pi$
$449$	−8.07180	−0.380932	−0.190466	+	0.981694i	$0.561000\pi$
$450$	−2.00000	−0.0942809
$451$	0	0
$452$	1.00000	0.0470360
$453$	5.53590	0.260099
$454$	−16.0000	−0.750917
$455$	4.46410	0.209280
$456$	−5.92820	−0.277614
$457$	−39.6410	−1.85433	−0.927164	−	0.374655i	$-0.877761\pi$
$457$	−39.6410	−1.85433	−0.927164	+	0.374655i	$0.877761\pi$
$458$	−18.0000	−0.841085
$459$	−37.3205	−1.74197
$460$	−4.46410	−0.208140
$461$	12.1436	0.565584	0.282792	−	0.959181i	$-0.408739\pi$
$461$	12.1436	0.565584	0.282792	+	0.959181i	$0.408739\pi$
$462$	0	0
$463$	−8.46410	−0.393360	−0.196680	−	0.980468i	$-0.563016\pi$
$463$	−8.46410	−0.393360	−0.196680	+	0.980468i	$0.563016\pi$
$464$	−6.92820	−0.321634
$465$	0.535898	0.0248517
$466$	−18.8564	−0.873506
$467$	−15.9282	−0.737069	−0.368535	−	0.929614i	$-0.620140\pi$
$467$	−15.9282	−0.737069	−0.368535	+	0.929614i	$0.620140\pi$
$468$	−8.92820	−0.412706
$469$	6.39230	0.295169
$470$	−0.535898	−0.0247191
$471$	20.3205	0.936320
$472$	9.92820	0.456983
$473$	0	0
$474$	7.53590	0.346135
$475$	5.92820	0.272005
$476$	−7.46410	−0.342117
$477$	14.9282	0.683515
$478$	−12.4641	−0.570095
$479$	−2.53590	−0.115868	−0.0579341	−	0.998320i	$-0.518451\pi$
$479$	−2.53590	−0.115868	−0.0579341	+	0.998320i	$0.518451\pi$
$480$	−1.00000	−0.0456435
$481$	−39.8564	−1.81730
$482$	−4.00000	−0.182195
$483$	4.46410	0.203124
$484$	0	0
$485$	−1.07180	−0.0486678
$486$	−16.0000	−0.725775
$487$	2.32051	0.105152	0.0525761	−	0.998617i	$-0.483257\pi$
$487$	2.32051	0.105152	0.0525761	+	0.998617i	$0.483257\pi$
$488$	4.92820	0.223089
$489$	22.0000	0.994874
$490$	1.00000	0.0451754
$491$	23.4641	1.05892	0.529460	−	0.848335i	$-0.322395\pi$
$491$	23.4641	1.05892	0.529460	+	0.848335i	$0.322395\pi$
$492$	8.39230	0.378354
$493$	51.7128	2.32903
$494$	26.4641	1.19068
$495$	0	0
$496$	−0.535898	−0.0240625
$497$	8.00000	0.358849
$498$	1.00000	0.0448111
$499$	−8.92820	−0.399681	−0.199841	−	0.979828i	$-0.564042\pi$
$499$	−8.92820	−0.399681	−0.199841	+	0.979828i	$0.564042\pi$
$500$	1.00000	0.0447214
$501$	4.39230	0.196234
$502$	0	0
$503$	30.2487	1.34872	0.674362	−	0.738401i	$-0.264419\pi$
$503$	30.2487	1.34872	0.674362	+	0.738401i	$0.264419\pi$
$504$	−2.00000	−0.0890871
$505$	−11.5359	−0.513341
$506$	0	0
$507$	−6.92820	−0.307692
$508$	13.3923	0.594187
$509$	−26.3205	−1.16664	−0.583318	−	0.812244i	$-0.698246\pi$
$509$	−26.3205	−1.16664	−0.583318	+	0.812244i	$0.698246\pi$
$510$	7.46410	0.330516
$511$	−0.535898	−0.0237067
$512$	1.00000	0.0441942
$513$	29.6410	1.30868
$514$	2.39230	0.105520
$515$	−1.46410	−0.0645160
$516$	12.3923	0.545541
$517$	0	0
$518$	−8.92820	−0.392283
$519$	−18.0000	−0.790112
$520$	4.46410	0.195764
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	13.8564	0.606478
$523$	−3.92820	−0.171768	−0.0858842	−	0.996305i	$-0.527371\pi$
$523$	−3.92820	−0.171768	−0.0858842	+	0.996305i	$0.527371\pi$
$524$	−10.8564	−0.474264
$525$	−1.00000	−0.0436436
$526$	16.4641	0.717869
$527$	4.00000	0.174243
$528$	0	0
$529$	−3.07180	−0.133556
$530$	−7.46410	−0.324220
$531$	−19.8564	−0.861695
$532$	5.92820	0.257020
$533$	−37.4641	−1.62275
$534$	17.4641	0.755746
$535$	−6.00000	−0.259403
$536$	6.39230	0.276106
$537$	8.39230	0.362155
$538$	−10.3205	−0.444949
$539$	0	0
$540$	5.00000	0.215166
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	29.8564	1.28244
$543$	−10.4641	−0.449058
$544$	−7.46410	−0.320021
$545$	18.3923	0.787840
$546$	−4.46410	−0.191046
$547$	−34.5359	−1.47665	−0.738324	−	0.674446i	$-0.764383\pi$
$547$	−34.5359	−1.47665	−0.738324	+	0.674446i	$0.764383\pi$
$548$	7.92820	0.338676
$549$	−9.85641	−0.420661
$550$	0	0
$551$	−41.0718	−1.74972
$552$	4.46410	0.190005
$553$	−7.53590	−0.320459
$554$	27.3205	1.16074
$555$	8.92820	0.378981
$556$	16.8564	0.714871
$557$	21.3205	0.903379	0.451689	−	0.892175i	$-0.350822\pi$
$557$	21.3205	0.903379	0.451689	+	0.892175i	$0.350822\pi$
$558$	1.07180	0.0453728
$559$	−55.3205	−2.33981
$560$	1.00000	0.0422577
$561$	0	0
$562$	17.0000	0.717102
$563$	42.7128	1.80013	0.900065	−	0.435755i	$-0.143519\pi$
$563$	42.7128	1.80013	0.900065	+	0.435755i	$0.143519\pi$
$564$	0.535898	0.0225654
$565$	1.00000	0.0420703
$566$	−13.7846	−0.579410
$567$	1.00000	0.0419961
$568$	8.00000	0.335673
$569$	−32.7128	−1.37139	−0.685696	−	0.727888i	$-0.740502\pi$
$569$	−32.7128	−1.37139	−0.685696	+	0.727888i	$0.740502\pi$
$570$	−5.92820	−0.248305
$571$	1.85641	0.0776882	0.0388441	−	0.999245i	$-0.487632\pi$
$571$	1.85641	0.0776882	0.0388441	+	0.999245i	$0.487632\pi$
$572$	0	0
$573$	−3.39230	−0.141716
$574$	−8.39230	−0.350288
$575$	−4.46410	−0.186166
$576$	−2.00000	−0.0833333
$577$	35.3205	1.47041	0.735206	−	0.677844i	$-0.237085\pi$
$577$	35.3205	1.47041	0.735206	+	0.677844i	$0.237085\pi$
$578$	38.7128	1.61024
$579$	−19.0000	−0.789613
$580$	−6.92820	−0.287678
$581$	−1.00000	−0.0414870
$582$	1.07180	0.0444274
$583$	0	0
$584$	−0.535898	−0.0221756
$585$	−8.92820	−0.369136
$586$	−20.4641	−0.845364
$587$	−28.8564	−1.19103	−0.595516	−	0.803344i	$-0.703052\pi$
$587$	−28.8564	−1.19103	−0.595516	+	0.803344i	$0.703052\pi$
$588$	−1.00000	−0.0412393
$589$	−3.17691	−0.130902
$590$	9.92820	0.408738
$591$	−4.00000	−0.164538
$592$	−8.92820	−0.366947
$593$	28.0000	1.14982	0.574911	−	0.818216i	$-0.305037\pi$
$593$	28.0000	1.14982	0.574911	+	0.818216i	$0.305037\pi$
$594$	0	0
$595$	−7.46410	−0.305998
$596$	−3.07180	−0.125826
$597$	25.8564	1.05823
$598$	−19.9282	−0.814925
$599$	−25.3923	−1.03750	−0.518751	−	0.854926i	$-0.673603\pi$
$599$	−25.3923	−1.03750	−0.518751	+	0.854926i	$0.673603\pi$
$600$	−1.00000	−0.0408248
$601$	2.67949	0.109299	0.0546494	−	0.998506i	$-0.482596\pi$
$601$	2.67949	0.109299	0.0546494	+	0.998506i	$0.482596\pi$
$602$	−12.3923	−0.505073
$603$	−12.7846	−0.520630
$604$	−5.53590	−0.225253
$605$	0	0
$606$	11.5359	0.468614
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	5.92820	0.240420
$609$	6.92820	0.280745
$610$	4.92820	0.199537
$611$	−2.39230	−0.0967823
$612$	14.9282	0.603437
$613$	26.5359	1.07177	0.535887	−	0.844289i	$-0.319977\pi$
$613$	26.5359	1.07177	0.535887	+	0.844289i	$0.319977\pi$
$614$	−14.9282	−0.602453
$615$	8.39230	0.338410
$616$	0	0
$617$	−1.21539	−0.0489298	−0.0244649	−	0.999701i	$-0.507788\pi$
$617$	−1.21539	−0.0489298	−0.0244649	+	0.999701i	$0.507788\pi$
$618$	1.46410	0.0588948
$619$	6.07180	0.244046	0.122023	−	0.992527i	$-0.461062\pi$
$619$	6.07180	0.244046	0.122023	+	0.992527i	$0.461062\pi$
$620$	−0.535898	−0.0215222
$621$	−22.3205	−0.895691
$622$	23.3205	0.935067
$623$	−17.4641	−0.699684
$624$	−4.46410	−0.178707
$625$	1.00000	0.0400000
$626$	14.7846	0.590912
$627$	0	0
$628$	−20.3205	−0.810877
$629$	66.6410	2.65715
$630$	−2.00000	−0.0796819
$631$	38.9282	1.54971	0.774854	−	0.632141i	$-0.217824\pi$
$631$	38.9282	1.54971	0.774854	+	0.632141i	$0.217824\pi$
$632$	−7.53590	−0.299762
$633$	24.0000	0.953914
$634$	−22.3923	−0.889312
$635$	13.3923	0.531457
$636$	7.46410	0.295971
$637$	4.46410	0.176874
$638$	0	0
$639$	−16.0000	−0.632950
$640$	1.00000	0.0395285
$641$	−34.8564	−1.37675	−0.688373	−	0.725357i	$-0.741675\pi$
$641$	−34.8564	−1.37675	−0.688373	+	0.725357i	$0.741675\pi$
$642$	6.00000	0.236801
$643$	−41.5692	−1.63933	−0.819665	−	0.572843i	$-0.805840\pi$
$643$	−41.5692	−1.63933	−0.819665	+	0.572843i	$0.805840\pi$
$644$	−4.46410	−0.175910
$645$	12.3923	0.487947
$646$	−44.2487	−1.74094
$647$	−11.3205	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$647$	−11.3205	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	4.46410	0.175096
$651$	0.535898	0.0210035
$652$	−22.0000	−0.861586
$653$	−25.4641	−0.996487	−0.498244	−	0.867037i	$-0.666021\pi$
$653$	−25.4641	−0.996487	−0.498244	+	0.867037i	$0.666021\pi$
$654$	−18.3923	−0.719196
$655$	−10.8564	−0.424195
$656$	−8.39230	−0.327664
$657$	1.07180	0.0418148
$658$	−0.535898	−0.0208915
$659$	−29.3205	−1.14216	−0.571082	−	0.820893i	$-0.693476\pi$
$659$	−29.3205	−1.14216	−0.571082	+	0.820893i	$0.693476\pi$
$660$	0	0
$661$	−0.607695	−0.0236366	−0.0118183	−	0.999930i	$-0.503762\pi$
$661$	−0.607695	−0.0236366	−0.0118183	+	0.999930i	$0.503762\pi$
$662$	1.46410	0.0569039
$663$	33.3205	1.29406
$664$	−1.00000	−0.0388075
$665$	5.92820	0.229886
$666$	17.8564	0.691922
$667$	30.9282	1.19754
$668$	−4.39230	−0.169943
$669$	3.46410	0.133930
$670$	6.39230	0.246956
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	38.8564	1.49780	0.748902	−	0.662681i	$-0.230581\pi$
$673$	38.8564	1.49780	0.748902	+	0.662681i	$0.230581\pi$
$674$	31.7846	1.22430
$675$	5.00000	0.192450
$676$	6.92820	0.266469
$677$	−33.3923	−1.28337	−0.641685	−	0.766968i	$-0.721764\pi$
$677$	−33.3923	−1.28337	−0.641685	+	0.766968i	$0.721764\pi$
$678$	−1.00000	−0.0384048
$679$	−1.07180	−0.0411318
$680$	−7.46410	−0.286235
$681$	16.0000	0.613121
$682$	0	0
$683$	−32.7846	−1.25447	−0.627234	−	0.778831i	$-0.715813\pi$
$683$	−32.7846	−1.25447	−0.627234	+	0.778831i	$0.715813\pi$
$684$	−11.8564	−0.453341
$685$	7.92820	0.302921
$686$	1.00000	0.0381802
$687$	18.0000	0.686743
$688$	−12.3923	−0.472452
$689$	−33.3205	−1.26941
$690$	4.46410	0.169945
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	0.928203	0.0352341
$695$	16.8564	0.639400
$696$	6.92820	0.262613
$697$	62.6410	2.37270
$698$	−1.53590	−0.0581346
$699$	18.8564	0.713215
$700$	1.00000	0.0377964
$701$	30.3923	1.14790	0.573951	−	0.818890i	$-0.305410\pi$
$701$	30.3923	1.14790	0.573951	+	0.818890i	$0.305410\pi$
$702$	22.3205	0.842433
$703$	−52.9282	−1.99622
$704$	0	0
$705$	0.535898	0.0201831
$706$	−20.5359	−0.772879
$707$	−11.5359	−0.433852
$708$	−9.92820	−0.373125
$709$	19.8564	0.745723	0.372861	−	0.927887i	$-0.378377\pi$
$709$	19.8564	0.745723	0.372861	+	0.927887i	$0.378377\pi$
$710$	8.00000	0.300235
$711$	15.0718	0.565237
$712$	−17.4641	−0.654495
$713$	2.39230	0.0895925
$714$	7.46410	0.279337
$715$	0	0
$716$	−8.39230	−0.313635
$717$	12.4641	0.465480
$718$	−19.7128	−0.735676
$719$	39.7128	1.48104	0.740519	−	0.672035i	$-0.234580\pi$
$719$	39.7128	1.48104	0.740519	+	0.672035i	$0.234580\pi$
$720$	−2.00000	−0.0745356
$721$	−1.46410	−0.0545260
$722$	16.1436	0.600802
$723$	4.00000	0.148762
$724$	10.4641	0.388895
$725$	−6.92820	−0.257307
$726$	0	0
$727$	−20.7846	−0.770859	−0.385429	−	0.922737i	$-0.625947\pi$
$727$	−20.7846	−0.770859	−0.385429	+	0.922737i	$0.625947\pi$
$728$	4.46410	0.165451
$729$	13.0000	0.481481
$730$	−0.535898	−0.0198345
$731$	92.4974	3.42114
$732$	−4.92820	−0.182152
$733$	−48.1769	−1.77945	−0.889727	−	0.456492i	$-0.849106\pi$
$733$	−48.1769	−1.77945	−0.889727	+	0.456492i	$0.849106\pi$
$734$	−21.4641	−0.792254
$735$	−1.00000	−0.0368856
$736$	−4.46410	−0.164549
$737$	0	0
$738$	16.7846	0.617850
$739$	−20.2487	−0.744861	−0.372430	−	0.928060i	$-0.621475\pi$
$739$	−20.2487	−0.744861	−0.372430	+	0.928060i	$0.621475\pi$
$740$	−8.92820	−0.328207
$741$	−26.4641	−0.972183
$742$	−7.46410	−0.274016
$743$	33.0718	1.21329	0.606643	−	0.794974i	$-0.292516\pi$
$743$	33.0718	1.21329	0.606643	+	0.794974i	$0.292516\pi$
$744$	0.535898	0.0196470
$745$	−3.07180	−0.112542
$746$	24.9282	0.912686
$747$	2.00000	0.0731762
$748$	0	0
$749$	−6.00000	−0.219235
$750$	−1.00000	−0.0365148
$751$	−26.1769	−0.955209	−0.477605	−	0.878575i	$-0.658495\pi$
$751$	−26.1769	−0.955209	−0.477605	+	0.878575i	$0.658495\pi$
$752$	−0.535898	−0.0195422
$753$	0	0
$754$	−30.9282	−1.12634
$755$	−5.53590	−0.201472
$756$	5.00000	0.181848
$757$	−27.3205	−0.992981	−0.496490	−	0.868042i	$-0.665378\pi$
$757$	−27.3205	−0.992981	−0.496490	+	0.868042i	$0.665378\pi$
$758$	23.8564	0.866504
$759$	0	0
$760$	5.92820	0.215039
$761$	48.2487	1.74901	0.874507	−	0.485013i	$-0.161185\pi$
$761$	48.2487	1.74901	0.874507	+	0.485013i	$0.161185\pi$
$762$	−13.3923	−0.485152
$763$	18.3923	0.665846
$764$	3.39230	0.122729
$765$	14.9282	0.539730
$766$	−21.4641	−0.775530
$767$	44.3205	1.60032
$768$	−1.00000	−0.0360844
$769$	−22.7846	−0.821634	−0.410817	−	0.911718i	$-0.634756\pi$
$769$	−22.7846	−0.821634	−0.410817	+	0.911718i	$0.634756\pi$
$770$	0	0
$771$	−2.39230	−0.0861568
$772$	19.0000	0.683825
$773$	2.46410	0.0886276	0.0443138	−	0.999018i	$-0.485890\pi$
$773$	2.46410	0.0886276	0.0443138	+	0.999018i	$0.485890\pi$
$774$	24.7846	0.890864
$775$	−0.535898	−0.0192500
$776$	−1.07180	−0.0384753
$777$	8.92820	0.320298
$778$	−13.3205	−0.477563
$779$	−49.7513	−1.78252
$780$	−4.46410	−0.159840
$781$	0	0
$782$	33.3205	1.19154
$783$	−34.6410	−1.23797
$784$	1.00000	0.0357143
$785$	−20.3205	−0.725270
$786$	10.8564	0.387235
$787$	48.4974	1.72875	0.864373	−	0.502851i	$-0.167715\pi$
$787$	48.4974	1.72875	0.864373	+	0.502851i	$0.167715\pi$
$788$	4.00000	0.142494
$789$	−16.4641	−0.586138
$790$	−7.53590	−0.268115
$791$	1.00000	0.0355559
$792$	0	0
$793$	22.0000	0.781243
$794$	−11.0718	−0.392923
$795$	7.46410	0.264724
$796$	−25.8564	−0.916456
$797$	−21.3923	−0.757754	−0.378877	−	0.925447i	$-0.623690\pi$
$797$	−21.3923	−0.757754	−0.378877	+	0.925447i	$0.623690\pi$
$798$	−5.92820	−0.209856
$799$	4.00000	0.141510
$800$	1.00000	0.0353553
$801$	34.9282	1.23413
$802$	4.14359	0.146315
$803$	0	0
$804$	−6.39230	−0.225439
$805$	−4.46410	−0.157339
$806$	−2.39230	−0.0842653
$807$	10.3205	0.363299
$808$	−11.5359	−0.405831
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	1.00000	0.0351364
$811$	35.7128	1.25405	0.627023	−	0.779001i	$-0.284273\pi$
$811$	35.7128	1.25405	0.627023	+	0.779001i	$0.284273\pi$
$812$	−6.92820	−0.243132
$813$	−29.8564	−1.04711
$814$	0	0
$815$	−22.0000	−0.770626
$816$	7.46410	0.261296
$817$	−73.4641	−2.57018
$818$	−31.7128	−1.10881
$819$	−8.92820	−0.311977
$820$	−8.39230	−0.293072
$821$	36.9282	1.28880	0.644402	−	0.764687i	$-0.277107\pi$
$821$	36.9282	1.28880	0.644402	+	0.764687i	$0.277107\pi$
$822$	−7.92820	−0.276528
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	−1.46410	−0.0510044
$825$	0	0
$826$	9.92820	0.345446
$827$	−18.3923	−0.639563	−0.319782	−	0.947491i	$-0.603610\pi$
$827$	−18.3923	−0.639563	−0.319782	+	0.947491i	$0.603610\pi$
$828$	8.92820	0.310277
$829$	−21.3923	−0.742985	−0.371493	−	0.928436i	$-0.621154\pi$
$829$	−21.3923	−0.742985	−0.371493	+	0.928436i	$0.621154\pi$
$830$	−1.00000	−0.0347105
$831$	−27.3205	−0.947738
$832$	4.46410	0.154765
$833$	−7.46410	−0.258616
$834$	−16.8564	−0.583690
$835$	−4.39230	−0.152002
$836$	0	0
$837$	−2.67949	−0.0926168
$838$	38.7128	1.33731
$839$	6.24871	0.215729	0.107865	−	0.994166i	$-0.465599\pi$
$839$	6.24871	0.215729	0.107865	+	0.994166i	$0.465599\pi$
$840$	−1.00000	−0.0345033
$841$	19.0000	0.655172
$842$	5.85641	0.201825
$843$	−17.0000	−0.585511
$844$	−24.0000	−0.826114
$845$	6.92820	0.238337
$846$	1.07180	0.0368491
$847$	0	0
$848$	−7.46410	−0.256318
$849$	13.7846	0.473087
$850$	−7.46410	−0.256017
$851$	39.8564	1.36626
$852$	−8.00000	−0.274075
$853$	9.53590	0.326503	0.163251	−	0.986584i	$-0.447802\pi$
$853$	9.53590	0.326503	0.163251	+	0.986584i	$0.447802\pi$
$854$	4.92820	0.168640
$855$	−11.8564	−0.405481
$856$	−6.00000	−0.205076
$857$	6.92820	0.236663	0.118331	−	0.992974i	$-0.462245\pi$
$857$	6.92820	0.236663	0.118331	+	0.992974i	$0.462245\pi$
$858$	0	0
$859$	−26.6410	−0.908980	−0.454490	−	0.890752i	$-0.650178\pi$
$859$	−26.6410	−0.908980	−0.454490	+	0.890752i	$0.650178\pi$
$860$	−12.3923	−0.422574
$861$	8.39230	0.286009
$862$	−10.3205	−0.351518
$863$	0.784610	0.0267084	0.0133542	−	0.999911i	$-0.495749\pi$
$863$	0.784610	0.0267084	0.0133542	+	0.999911i	$0.495749\pi$
$864$	5.00000	0.170103
$865$	18.0000	0.612018
$866$	22.2487	0.756042
$867$	−38.7128	−1.31476
$868$	−0.535898	−0.0181896
$869$	0	0
$870$	6.92820	0.234888
$871$	28.5359	0.966902
$872$	18.3923	0.622842
$873$	2.14359	0.0725496
$874$	−26.4641	−0.895162
$875$	1.00000	0.0338062
$876$	0.535898	0.0181063
$877$	−8.39230	−0.283388	−0.141694	−	0.989911i	$-0.545255\pi$
$877$	−8.39230	−0.283388	−0.141694	+	0.989911i	$0.545255\pi$
$878$	−22.5359	−0.760550
$879$	20.4641	0.690237
$880$	0	0
$881$	−5.71281	−0.192470	−0.0962348	−	0.995359i	$-0.530680\pi$
$881$	−5.71281	−0.192470	−0.0962348	+	0.995359i	$0.530680\pi$
$882$	−2.00000	−0.0673435
$883$	−34.7846	−1.17060	−0.585298	−	0.810819i	$-0.699022\pi$
$883$	−34.7846	−1.17060	−0.585298	+	0.810819i	$0.699022\pi$
$884$	−33.3205	−1.12069
$885$	−9.92820	−0.333733
$886$	−15.7128	−0.527882
$887$	−26.5359	−0.890988	−0.445494	−	0.895285i	$-0.646972\pi$
$887$	−26.5359	−0.890988	−0.445494	+	0.895285i	$0.646972\pi$
$888$	8.92820	0.299611
$889$	13.3923	0.449163
$890$	−17.4641	−0.585398
$891$	0	0
$892$	−3.46410	−0.115987
$893$	−3.17691	−0.106311
$894$	3.07180	0.102736
$895$	−8.39230	−0.280524
$896$	1.00000	0.0334077
$897$	19.9282	0.665383
$898$	−8.07180	−0.269359
$899$	3.71281	0.123829
$900$	−2.00000	−0.0666667
$901$	55.7128	1.85606
$902$	0	0
$903$	12.3923	0.412390
$904$	1.00000	0.0332595
$905$	10.4641	0.347839
$906$	5.53590	0.183918
$907$	−42.6410	−1.41587	−0.707936	−	0.706277i	$-0.750373\pi$
$907$	−42.6410	−1.41587	−0.707936	+	0.706277i	$0.750373\pi$
$908$	−16.0000	−0.530979
$909$	23.0718	0.765243
$910$	4.46410	0.147984
$911$	9.39230	0.311181	0.155590	−	0.987822i	$-0.450272\pi$
$911$	9.39230	0.311181	0.155590	+	0.987822i	$0.450272\pi$
$912$	−5.92820	−0.196302
$913$	0	0
$914$	−39.6410	−1.31121
$915$	−4.92820	−0.162921
$916$	−18.0000	−0.594737
$917$	−10.8564	−0.358510
$918$	−37.3205	−1.23176
$919$	3.71281	0.122474	0.0612372	−	0.998123i	$-0.480495\pi$
$919$	3.71281	0.122474	0.0612372	+	0.998123i	$0.480495\pi$
$920$	−4.46410	−0.147177
$921$	14.9282	0.491901
$922$	12.1436	0.399928
$923$	35.7128	1.17550
$924$	0	0
$925$	−8.92820	−0.293558
$926$	−8.46410	−0.278148
$927$	2.92820	0.0961748
$928$	−6.92820	−0.227429
$929$	−25.8564	−0.848321	−0.424161	−	0.905587i	$-0.639431\pi$
$929$	−25.8564	−0.848321	−0.424161	+	0.905587i	$0.639431\pi$
$930$	0.535898	0.0175728
$931$	5.92820	0.194289
$932$	−18.8564	−0.617662
$933$	−23.3205	−0.763479
$934$	−15.9282	−0.521187
$935$	0	0
$936$	−8.92820	−0.291827
$937$	6.14359	0.200702	0.100351	−	0.994952i	$-0.468003\pi$
$937$	6.14359	0.200702	0.100351	+	0.994952i	$0.468003\pi$
$938$	6.39230	0.208716
$939$	−14.7846	−0.482478
$940$	−0.535898	−0.0174791
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	20.3205	0.662078
$943$	37.4641	1.22000
$944$	9.92820	0.323135
$945$	5.00000	0.162650
$946$	0	0
$947$	29.6077	0.962121	0.481060	−	0.876687i	$-0.340252\pi$
$947$	29.6077	0.962121	0.481060	+	0.876687i	$0.340252\pi$
$948$	7.53590	0.244755
$949$	−2.39230	−0.0776575
$950$	5.92820	0.192336
$951$	22.3923	0.726120
$952$	−7.46410	−0.241913
$953$	32.5692	1.05502	0.527510	−	0.849549i	$-0.323126\pi$
$953$	32.5692	1.05502	0.527510	+	0.849549i	$0.323126\pi$
$954$	14.9282	0.483318
$955$	3.39230	0.109772
$956$	−12.4641	−0.403118
$957$	0	0
$958$	−2.53590	−0.0819312
$959$	7.92820	0.256015
$960$	−1.00000	−0.0322749
$961$	−30.7128	−0.990736
$962$	−39.8564	−1.28502
$963$	12.0000	0.386695
$964$	−4.00000	−0.128831
$965$	19.0000	0.611632
$966$	4.46410	0.143630
$967$	11.7128	0.376659	0.188329	−	0.982106i	$-0.439693\pi$
$967$	11.7128	0.376659	0.188329	+	0.982106i	$0.439693\pi$
$968$	0	0
$969$	44.2487	1.42147
$970$	−1.07180	−0.0344133
$971$	12.7128	0.407974	0.203987	−	0.978974i	$-0.434610\pi$
$971$	12.7128	0.407974	0.203987	+	0.978974i	$0.434610\pi$
$972$	−16.0000	−0.513200
$973$	16.8564	0.540392
$974$	2.32051	0.0743539
$975$	−4.46410	−0.142966
$976$	4.92820	0.157748
$977$	18.8564	0.603270	0.301635	−	0.953423i	$-0.402468\pi$
$977$	18.8564	0.603270	0.301635	+	0.953423i	$0.402468\pi$
$978$	22.0000	0.703482
$979$	0	0
$980$	1.00000	0.0319438
$981$	−36.7846	−1.17444
$982$	23.4641	0.748770
$983$	14.7846	0.471556	0.235778	−	0.971807i	$-0.424236\pi$
$983$	14.7846	0.471556	0.235778	+	0.971807i	$0.424236\pi$
$984$	8.39230	0.267537
$985$	4.00000	0.127451
$986$	51.7128	1.64687
$987$	0.535898	0.0170578
$988$	26.4641	0.841935
$989$	55.3205	1.75909
$990$	0	0
$991$	43.1051	1.36928	0.684640	−	0.728882i	$-0.259960\pi$
$991$	43.1051	1.36928	0.684640	+	0.728882i	$0.259960\pi$
$992$	−0.535898	−0.0170148
$993$	−1.46410	−0.0464619
$994$	8.00000	0.253745
$995$	−25.8564	−0.819703
$996$	1.00000	0.0316862
$997$	−30.3205	−0.960260	−0.480130	−	0.877197i	$-0.659411\pi$
$997$	−30.3205	−0.960260	−0.480130	+	0.877197i	$0.659411\pi$
$998$	−8.92820	−0.282617
$999$	−44.6410	−1.41238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.bw.1.2	yes	2
11.10	odd	2		8470.2.a.bi.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.bi.1.1	✓	2	11.10	odd	2
8470.2.a.bw.1.2	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} +4.46410 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -7.46410 q^{17} -2.00000 q^{18} +5.92820 q^{19} +1.00000 q^{20} -1.00000 q^{21} -4.46410 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.46410 q^{26} +5.00000 q^{27} +1.00000 q^{28} -6.92820 q^{29} -1.00000 q^{30} -0.535898 q^{31} +1.00000 q^{32} -7.46410 q^{34} +1.00000 q^{35} -2.00000 q^{36} -8.92820 q^{37} +5.92820 q^{38} -4.46410 q^{39} +1.00000 q^{40} -8.39230 q^{41} -1.00000 q^{42} -12.3923 q^{43} -2.00000 q^{45} -4.46410 q^{46} -0.535898 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +7.46410 q^{51} +4.46410 q^{52} -7.46410 q^{53} +5.00000 q^{54} +1.00000 q^{56} -5.92820 q^{57} -6.92820 q^{58} +9.92820 q^{59} -1.00000 q^{60} +4.92820 q^{61} -0.535898 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.46410 q^{65} +6.39230 q^{67} -7.46410 q^{68} +4.46410 q^{69} +1.00000 q^{70} +8.00000 q^{71} -2.00000 q^{72} -0.535898 q^{73} -8.92820 q^{74} -1.00000 q^{75} +5.92820 q^{76} -4.46410 q^{78} -7.53590 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.39230 q^{82} -1.00000 q^{83} -1.00000 q^{84} -7.46410 q^{85} -12.3923 q^{86} +6.92820 q^{87} -17.4641 q^{89} -2.00000 q^{90} +4.46410 q^{91} -4.46410 q^{92} +0.535898 q^{93} -0.535898 q^{94} +5.92820 q^{95} -1.00000 q^{96} -1.07180 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 - 4 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} - 4 q^{9} + 2 q^{10} - 2 q^{12} + 2 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} - 8 q^{17} - 4 q^{18} - 2 q^{19} + 2 q^{20} - 2 q^{21} - 2 q^{23} - 2 q^{24} + 2 q^{25} + 2 q^{26} + 10 q^{27} + 2 q^{28} - 2 q^{30} - 8 q^{31} + 2 q^{32} - 8 q^{34} + 2 q^{35} - 4 q^{36} - 4 q^{37} - 2 q^{38} - 2 q^{39} + 2 q^{40} + 4 q^{41} - 2 q^{42} - 4 q^{43} - 4 q^{45} - 2 q^{46} - 8 q^{47} - 2 q^{48} + 2 q^{49} + 2 q^{50} + 8 q^{51} + 2 q^{52} - 8 q^{53} + 10 q^{54} + 2 q^{56} + 2 q^{57} + 6 q^{59} - 2 q^{60} - 4 q^{61} - 8 q^{62} - 4 q^{63} + 2 q^{64} + 2 q^{65} - 8 q^{67} - 8 q^{68} + 2 q^{69} + 2 q^{70} + 16 q^{71} - 4 q^{72} - 8 q^{73} - 4 q^{74} - 2 q^{75} - 2 q^{76} - 2 q^{78} - 22 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} - 2 q^{83} - 2 q^{84} - 8 q^{85} - 4 q^{86} - 28 q^{89} - 4 q^{90} + 2 q^{91} - 2 q^{92} + 8 q^{93} - 8 q^{94} - 2 q^{95} - 2 q^{96} - 16 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 - 4 * q^9 + 2 * q^10 - 2 * q^12 + 2 * q^13 + 2 * q^14 - 2 * q^15 + 2 * q^16 - 8 * q^17 - 4 * q^18 - 2 * q^19 + 2 * q^20 - 2 * q^21 - 2 * q^23 - 2 * q^24 + 2 * q^25 + 2 * q^26 + 10 * q^27 + 2 * q^28 - 2 * q^30 - 8 * q^31 + 2 * q^32 - 8 * q^34 + 2 * q^35 - 4 * q^36 - 4 * q^37 - 2 * q^38 - 2 * q^39 + 2 * q^40 + 4 * q^41 - 2 * q^42 - 4 * q^43 - 4 * q^45 - 2 * q^46 - 8 * q^47 - 2 * q^48 + 2 * q^49 + 2 * q^50 + 8 * q^51 + 2 * q^52 - 8 * q^53 + 10 * q^54 + 2 * q^56 + 2 * q^57 + 6 * q^59 - 2 * q^60 - 4 * q^61 - 8 * q^62 - 4 * q^63 + 2 * q^64 + 2 * q^65 - 8 * q^67 - 8 * q^68 + 2 * q^69 + 2 * q^70 + 16 * q^71 - 4 * q^72 - 8 * q^73 - 4 * q^74 - 2 * q^75 - 2 * q^76 - 2 * q^78 - 22 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 - 2 * q^83 - 2 * q^84 - 8 * q^85 - 4 * q^86 - 28 * q^89 - 4 * q^90 + 2 * q^91 - 2 * q^92 + 8 * q^93 - 8 * q^94 - 2 * q^95 - 2 * q^96 - 16 * q^97 + 2 * q^98

Introduction

L-functions

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Database

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