LMFDB - Embedded newform 6422.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 4x^{6} + 22x^{5} + 3x^{4} - 28x^{3} + 7x^{2} + 6x - 2 $ x^8 - 4x^7 - 4x^6 + 22x^5 + 3x^4 - 28x^3 + 7x^2 + 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.898616$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.898616 q^{3} +1.00000 q^{4} +0.684128 q^{5} +0.898616 q^{6} -3.14167 q^{7} -1.00000 q^{8} -2.19249 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.898616 q^{3} +1.00000 q^{4} +0.684128 q^{5} +0.898616 q^{6} -3.14167 q^{7} -1.00000 q^{8} -2.19249 q^{9} -0.684128 q^{10} +1.39846 q^{11} -0.898616 q^{12} +3.14167 q^{14} -0.614768 q^{15} +1.00000 q^{16} -1.95166 q^{17} +2.19249 q^{18} +1.00000 q^{19} +0.684128 q^{20} +2.82316 q^{21} -1.39846 q^{22} +2.21610 q^{23} +0.898616 q^{24} -4.53197 q^{25} +4.66605 q^{27} -3.14167 q^{28} +3.85332 q^{29} +0.614768 q^{30} +1.71795 q^{31} -1.00000 q^{32} -1.25668 q^{33} +1.95166 q^{34} -2.14931 q^{35} -2.19249 q^{36} +2.52893 q^{37} -1.00000 q^{38} -0.684128 q^{40} +6.44519 q^{41} -2.82316 q^{42} -2.64916 q^{43} +1.39846 q^{44} -1.49994 q^{45} -2.21610 q^{46} -1.10350 q^{47} -0.898616 q^{48} +2.87011 q^{49} +4.53197 q^{50} +1.75380 q^{51} -9.51153 q^{53} -4.66605 q^{54} +0.956729 q^{55} +3.14167 q^{56} -0.898616 q^{57} -3.85332 q^{58} +4.41644 q^{59} -0.614768 q^{60} +12.3006 q^{61} -1.71795 q^{62} +6.88809 q^{63} +1.00000 q^{64} +1.25668 q^{66} +7.06905 q^{67} -1.95166 q^{68} -1.99142 q^{69} +2.14931 q^{70} -8.27705 q^{71} +2.19249 q^{72} +11.1434 q^{73} -2.52893 q^{74} +4.07250 q^{75} +1.00000 q^{76} -4.39352 q^{77} -5.19848 q^{79} +0.684128 q^{80} +2.38448 q^{81} -6.44519 q^{82} -1.27105 q^{83} +2.82316 q^{84} -1.33519 q^{85} +2.64916 q^{86} -3.46266 q^{87} -1.39846 q^{88} -0.674698 q^{89} +1.49994 q^{90} +2.21610 q^{92} -1.54378 q^{93} +1.10350 q^{94} +0.684128 q^{95} +0.898616 q^{96} +5.68196 q^{97} -2.87011 q^{98} -3.06612 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^5 + 4 * q^6 + 2 * q^7 - 8 * q^8	$ 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} - 8 q^{8} - 2 q^{10} + 10 q^{11} - 4 q^{12} - 2 q^{14} + 8 q^{16} + 2 q^{17} + 8 q^{19} + 2 q^{20} - 12 q^{21} - 10 q^{22} - 8 q^{23} + 4 q^{24} - 14 q^{25} - 22 q^{27} + 2 q^{28} + 8 q^{29} + 12 q^{31} - 8 q^{32} - 4 q^{33} - 2 q^{34} - 10 q^{35} - 8 q^{38} - 2 q^{40} - 2 q^{41} + 12 q^{42} - 16 q^{43} + 10 q^{44} + 12 q^{45} + 8 q^{46} - 12 q^{47} - 4 q^{48} - 14 q^{49} + 14 q^{50} - 22 q^{51} - 24 q^{53} + 22 q^{54} - 10 q^{55} - 2 q^{56} - 4 q^{57} - 8 q^{58} + 4 q^{59} - 26 q^{61} - 12 q^{62} + 12 q^{63} + 8 q^{64} + 4 q^{66} - 10 q^{67} + 2 q^{68} + 6 q^{69} + 10 q^{70} + 36 q^{71} - 20 q^{73} + 6 q^{75} + 8 q^{76} - 12 q^{77} - 22 q^{79} + 2 q^{80} + 36 q^{81} + 2 q^{82} - 18 q^{83} - 12 q^{84} - 26 q^{85} + 16 q^{86} - 38 q^{87} - 10 q^{88} + 18 q^{89} - 12 q^{90} - 8 q^{92} - 12 q^{93} + 12 q^{94} + 2 q^{95} + 4 q^{96} + 8 q^{97} + 14 q^{98} + 8 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^5 + 4 * q^6 + 2 * q^7 - 8 * q^8 - 2 * q^10 + 10 * q^11 - 4 * q^12 - 2 * q^14 + 8 * q^16 + 2 * q^17 + 8 * q^19 + 2 * q^20 - 12 * q^21 - 10 * q^22 - 8 * q^23 + 4 * q^24 - 14 * q^25 - 22 * q^27 + 2 * q^28 + 8 * q^29 + 12 * q^31 - 8 * q^32 - 4 * q^33 - 2 * q^34 - 10 * q^35 - 8 * q^38 - 2 * q^40 - 2 * q^41 + 12 * q^42 - 16 * q^43 + 10 * q^44 + 12 * q^45 + 8 * q^46 - 12 * q^47 - 4 * q^48 - 14 * q^49 + 14 * q^50 - 22 * q^51 - 24 * q^53 + 22 * q^54 - 10 * q^55 - 2 * q^56 - 4 * q^57 - 8 * q^58 + 4 * q^59 - 26 * q^61 - 12 * q^62 + 12 * q^63 + 8 * q^64 + 4 * q^66 - 10 * q^67 + 2 * q^68 + 6 * q^69 + 10 * q^70 + 36 * q^71 - 20 * q^73 + 6 * q^75 + 8 * q^76 - 12 * q^77 - 22 * q^79 + 2 * q^80 + 36 * q^81 + 2 * q^82 - 18 * q^83 - 12 * q^84 - 26 * q^85 + 16 * q^86 - 38 * q^87 - 10 * q^88 + 18 * q^89 - 12 * q^90 - 8 * q^92 - 12 * q^93 + 12 * q^94 + 2 * q^95 + 4 * q^96 + 8 * q^97 + 14 * q^98 + 8 * 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
$3$	−0.898616	−0.518816	−0.259408	−	0.965768i	$-0.583527\pi$
$3$	−0.898616	−0.518816	−0.259408	+	0.965768i	$0.583527\pi$
$4$	1.00000	0.500000
$5$	0.684128	0.305951	0.152976	−	0.988230i	$-0.451114\pi$
$5$	0.684128	0.305951	0.152976	+	0.988230i	$0.451114\pi$
$6$	0.898616	0.366858
$7$	−3.14167	−1.18744	−0.593720	−	0.804671i	$-0.702341\pi$
$7$	−3.14167	−1.18744	−0.593720	+	0.804671i	$0.702341\pi$
$8$	−1.00000	−0.353553
$9$	−2.19249	−0.730830
$10$	−0.684128	−0.216340
$11$	1.39846	0.421653	0.210826	−	0.977524i	$-0.432385\pi$
$11$	1.39846	0.421653	0.210826	+	0.977524i	$0.432385\pi$
$12$	−0.898616	−0.259408
$13$	0	0
$13$	0	0
$14$	3.14167	0.839648
$15$	−0.614768	−0.158732
$16$	1.00000	0.250000
$17$	−1.95166	−0.473348	−0.236674	−	0.971589i	$-0.576057\pi$
$17$	−1.95166	−0.473348	−0.236674	+	0.971589i	$0.576057\pi$
$18$	2.19249	0.516775
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.684128	0.152976
$21$	2.82316	0.616063
$22$	−1.39846	−0.298154
$23$	2.21610	0.462088	0.231044	−	0.972943i	$-0.425786\pi$
$23$	2.21610	0.462088	0.231044	+	0.972943i	$0.425786\pi$
$24$	0.898616	0.183429
$25$	−4.53197	−0.906394
$26$	0	0
$27$	4.66605	0.897982
$28$	−3.14167	−0.593720
$29$	3.85332	0.715544	0.357772	−	0.933809i	$-0.383537\pi$
$29$	3.85332	0.715544	0.357772	+	0.933809i	$0.383537\pi$
$30$	0.614768	0.112241
$31$	1.71795	0.308553	0.154276	−	0.988028i	$-0.450695\pi$
$31$	1.71795	0.308553	0.154276	+	0.988028i	$0.450695\pi$
$32$	−1.00000	−0.176777
$33$	−1.25668	−0.218760
$34$	1.95166	0.334708
$35$	−2.14931	−0.363299
$36$	−2.19249	−0.365415
$37$	2.52893	0.415753	0.207877	−	0.978155i	$-0.433345\pi$
$37$	2.52893	0.415753	0.207877	+	0.978155i	$0.433345\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−0.684128	−0.108170
$41$	6.44519	1.00657	0.503285	−	0.864120i	$-0.332125\pi$
$41$	6.44519	1.00657	0.503285	+	0.864120i	$0.332125\pi$
$42$	−2.82316	−0.435622
$43$	−2.64916	−0.403992	−0.201996	−	0.979386i	$-0.564743\pi$
$43$	−2.64916	−0.403992	−0.201996	+	0.979386i	$0.564743\pi$
$44$	1.39846	0.210826
$45$	−1.49994	−0.223598
$46$	−2.21610	−0.326746
$47$	−1.10350	−0.160963	−0.0804813	−	0.996756i	$-0.525646\pi$
$47$	−1.10350	−0.160963	−0.0804813	+	0.996756i	$0.525646\pi$
$48$	−0.898616	−0.129704
$49$	2.87011	0.410016
$50$	4.53197	0.640917
$51$	1.75380	0.245581
$52$	0	0
$53$	−9.51153	−1.30651	−0.653254	−	0.757138i	$-0.726597\pi$
$53$	−9.51153	−1.30651	−0.653254	+	0.757138i	$0.726597\pi$
$54$	−4.66605	−0.634969
$55$	0.956729	0.129005
$56$	3.14167	0.419824
$57$	−0.898616	−0.119025
$58$	−3.85332	−0.505966
$59$	4.41644	0.574971	0.287486	−	0.957785i	$-0.407181\pi$
$59$	4.41644	0.574971	0.287486	+	0.957785i	$0.407181\pi$
$60$	−0.614768	−0.0793662
$61$	12.3006	1.57493	0.787465	−	0.616359i	$-0.211393\pi$
$61$	12.3006	1.57493	0.787465	+	0.616359i	$0.211393\pi$
$62$	−1.71795	−0.218180
$63$	6.88809	0.867817
$64$	1.00000	0.125000
$65$	0	0
$66$	1.25668	0.154687
$67$	7.06905	0.863622	0.431811	−	0.901964i	$-0.357875\pi$
$67$	7.06905	0.863622	0.431811	+	0.901964i	$0.357875\pi$
$68$	−1.95166	−0.236674
$69$	−1.99142	−0.239739
$70$	2.14931	0.256891
$71$	−8.27705	−0.982306	−0.491153	−	0.871073i	$-0.663424\pi$
$71$	−8.27705	−0.982306	−0.491153	+	0.871073i	$0.663424\pi$
$72$	2.19249	0.258387
$73$	11.1434	1.30423	0.652117	−	0.758118i	$-0.273881\pi$
$73$	11.1434	1.30423	0.652117	+	0.758118i	$0.273881\pi$
$74$	−2.52893	−0.293982
$75$	4.07250	0.470252
$76$	1.00000	0.114708
$77$	−4.39352	−0.500688
$78$	0	0
$79$	−5.19848	−0.584875	−0.292437	−	0.956285i	$-0.594466\pi$
$79$	−5.19848	−0.584875	−0.292437	+	0.956285i	$0.594466\pi$
$80$	0.684128	0.0764879
$81$	2.38448	0.264943
$82$	−6.44519	−0.711752
$83$	−1.27105	−0.139516	−0.0697582	−	0.997564i	$-0.522223\pi$
$83$	−1.27105	−0.139516	−0.0697582	+	0.997564i	$0.522223\pi$
$84$	2.82316	0.308032
$85$	−1.33519	−0.144822
$86$	2.64916	0.285666
$87$	−3.46266	−0.371236
$88$	−1.39846	−0.149077
$89$	−0.674698	−0.0715178	−0.0357589	−	0.999360i	$-0.511385\pi$
$89$	−0.674698	−0.0715178	−0.0357589	+	0.999360i	$0.511385\pi$
$90$	1.49994	0.158108
$91$	0	0
$92$	2.21610	0.231044
$93$	−1.54378	−0.160082
$94$	1.10350	0.113818
$95$	0.684128	0.0701901
$96$	0.898616	0.0917146
$97$	5.68196	0.576916	0.288458	−	0.957493i	$-0.406857\pi$
$97$	5.68196	0.576916	0.288458	+	0.957493i	$0.406857\pi$
$98$	−2.87011	−0.289925
$99$	−3.06612	−0.308156
$100$	−4.53197	−0.453197
$101$	−5.12297	−0.509754	−0.254877	−	0.966973i	$-0.582035\pi$
$101$	−5.12297	−0.509754	−0.254877	+	0.966973i	$0.582035\pi$
$102$	−1.75380	−0.173652
$103$	−3.71806	−0.366352	−0.183176	−	0.983080i	$-0.558638\pi$
$103$	−3.71806	−0.366352	−0.183176	+	0.983080i	$0.558638\pi$
$104$	0	0
$105$	1.93140	0.188485
$106$	9.51153	0.923841
$107$	12.1626	1.17580	0.587901	−	0.808933i	$-0.299954\pi$
$107$	12.1626	1.17580	0.587901	+	0.808933i	$0.299954\pi$
$108$	4.66605	0.448991
$109$	6.70323	0.642053	0.321027	−	0.947070i	$-0.395972\pi$
$109$	6.70323	0.642053	0.321027	+	0.947070i	$0.395972\pi$
$110$	−0.956729	−0.0912205
$111$	−2.27253	−0.215699
$112$	−3.14167	−0.296860
$113$	−11.9286	−1.12215	−0.561075	−	0.827765i	$-0.689612\pi$
$113$	−11.9286	−1.12215	−0.561075	+	0.827765i	$0.689612\pi$
$114$	0.898616	0.0841631
$115$	1.51609	0.141377
$116$	3.85332	0.357772
$117$	0	0
$118$	−4.41644	−0.406566
$119$	6.13149	0.562073
$120$	0.614768	0.0561204
$121$	−9.04430	−0.822209
$122$	−12.3006	−1.11364
$123$	−5.79175	−0.522225
$124$	1.71795	0.154276
$125$	−6.52109	−0.583264
$126$	−6.88809	−0.613640
$127$	6.59497	0.585209	0.292604	−	0.956234i	$-0.405478\pi$
$127$	6.59497	0.585209	0.292604	+	0.956234i	$0.405478\pi$
$128$	−1.00000	−0.0883883
$129$	2.38057	0.209598
$130$	0	0
$131$	−8.81605	−0.770262	−0.385131	−	0.922862i	$-0.625844\pi$
$131$	−8.81605	−0.770262	−0.385131	+	0.922862i	$0.625844\pi$
$132$	−1.25668	−0.109380
$133$	−3.14167	−0.272418
$134$	−7.06905	−0.610673
$135$	3.19218	0.274739
$136$	1.95166	0.167354
$137$	−18.4002	−1.57204	−0.786019	−	0.618203i	$-0.787861\pi$
$137$	−18.4002	−1.57204	−0.786019	+	0.618203i	$0.787861\pi$
$138$	1.99142	0.169521
$139$	−9.47178	−0.803386	−0.401693	−	0.915774i	$-0.631578\pi$
$139$	−9.47178	−0.803386	−0.401693	+	0.915774i	$0.631578\pi$
$140$	−2.14931	−0.181650
$141$	0.991626	0.0835100
$142$	8.27705	0.694595
$143$	0	0
$144$	−2.19249	−0.182708
$145$	2.63617	0.218922
$146$	−11.1434	−0.922233
$147$	−2.57913	−0.212723
$148$	2.52893	0.207877
$149$	21.1970	1.73652	0.868262	−	0.496106i	$-0.165237\pi$
$149$	21.1970	1.73652	0.868262	+	0.496106i	$0.165237\pi$
$150$	−4.07250	−0.332518
$151$	0.671643	0.0546575	0.0273288	−	0.999626i	$-0.491300\pi$
$151$	0.671643	0.0546575	0.0273288	+	0.999626i	$0.491300\pi$
$152$	−1.00000	−0.0811107
$153$	4.27901	0.345937
$154$	4.39352	0.354040
$155$	1.17530	0.0944021
$156$	0	0
$157$	−20.1166	−1.60548	−0.802738	−	0.596331i	$-0.796624\pi$
$157$	−20.1166	−1.60548	−0.802738	+	0.596331i	$0.796624\pi$
$158$	5.19848	0.413569
$159$	8.54721	0.677838
$160$	−0.684128	−0.0540851
$161$	−6.96225	−0.548702
$162$	−2.38448	−0.187343
$163$	−6.83144	−0.535079	−0.267540	−	0.963547i	$-0.586211\pi$
$163$	−6.83144	−0.535079	−0.267540	+	0.963547i	$0.586211\pi$
$164$	6.44519	0.503285
$165$	−0.859731	−0.0669300
$166$	1.27105	0.0986529
$167$	−10.9661	−0.848584	−0.424292	−	0.905526i	$-0.639477\pi$
$167$	−10.9661	−0.848584	−0.424292	+	0.905526i	$0.639477\pi$
$168$	−2.82316	−0.217811
$169$	0	0
$170$	1.33519	0.102404
$171$	−2.19249	−0.167664
$172$	−2.64916	−0.201996
$173$	−6.34922	−0.482722	−0.241361	−	0.970435i	$-0.577594\pi$
$173$	−6.34922	−0.482722	−0.241361	+	0.970435i	$0.577594\pi$
$174$	3.46266	0.262503
$175$	14.2380	1.07629
$176$	1.39846	0.105413
$177$	−3.96868	−0.298304
$178$	0.674698	0.0505708
$179$	4.47871	0.334754	0.167377	−	0.985893i	$-0.446470\pi$
$179$	4.47871	0.334754	0.167377	+	0.985893i	$0.446470\pi$
$180$	−1.49994	−0.111799
$181$	−3.00372	−0.223265	−0.111633	−	0.993750i	$-0.535608\pi$
$181$	−3.00372	−0.223265	−0.111633	+	0.993750i	$0.535608\pi$
$182$	0	0
$183$	−11.0535	−0.817099
$184$	−2.21610	−0.163373
$185$	1.73011	0.127200
$186$	1.54378	0.113195
$187$	−2.72933	−0.199589
$188$	−1.10350	−0.0804813
$189$	−14.6592	−1.06630
$190$	−0.684128	−0.0496319
$191$	−22.3399	−1.61646	−0.808228	−	0.588869i	$-0.799573\pi$
$191$	−22.3399	−1.61646	−0.808228	+	0.588869i	$0.799573\pi$
$192$	−0.898616	−0.0648520
$193$	9.41832	0.677946	0.338973	−	0.940796i	$-0.389920\pi$
$193$	9.41832	0.677946	0.338973	+	0.940796i	$0.389920\pi$
$194$	−5.68196	−0.407941
$195$	0	0
$196$	2.87011	0.205008
$197$	3.69904	0.263546	0.131773	−	0.991280i	$-0.457933\pi$
$197$	3.69904	0.263546	0.131773	+	0.991280i	$0.457933\pi$
$198$	3.06612	0.217900
$199$	−2.32306	−0.164678	−0.0823388	−	0.996604i	$-0.526239\pi$
$199$	−2.32306	−0.164678	−0.0823388	+	0.996604i	$0.526239\pi$
$200$	4.53197	0.320459
$201$	−6.35236	−0.448061
$202$	5.12297	0.360451
$203$	−12.1059	−0.849666
$204$	1.75380	0.122790
$205$	4.40934	0.307962
$206$	3.71806	0.259050
$207$	−4.85877	−0.337708
$208$	0	0
$209$	1.39846	0.0967338
$210$	−1.93140	−0.133279
$211$	17.3814	1.19659	0.598293	−	0.801277i	$-0.295846\pi$
$211$	17.3814	1.19659	0.598293	+	0.801277i	$0.295846\pi$
$212$	−9.51153	−0.653254
$213$	7.43789	0.509636
$214$	−12.1626	−0.831418
$215$	−1.81236	−0.123602
$216$	−4.66605	−0.317485
$217$	−5.39723	−0.366388
$218$	−6.70323	−0.454000
$219$	−10.0136	−0.676658
$220$	0.956729	0.0645026
$221$	0	0
$222$	2.27253	0.152522
$223$	−23.6297	−1.58236	−0.791181	−	0.611582i	$-0.790534\pi$
$223$	−23.6297	−1.58236	−0.791181	+	0.611582i	$0.790534\pi$
$224$	3.14167	0.209912
$225$	9.93630	0.662420
$226$	11.9286	0.793480
$227$	18.2136	1.20888	0.604439	−	0.796652i	$-0.293397\pi$
$227$	18.2136	1.20888	0.604439	+	0.796652i	$0.293397\pi$
$228$	−0.898616	−0.0595123
$229$	12.1571	0.803361	0.401680	−	0.915780i	$-0.368426\pi$
$229$	12.1571	0.803361	0.401680	+	0.915780i	$0.368426\pi$
$230$	−1.51609	−0.0999683
$231$	3.94808	0.259765
$232$	−3.85332	−0.252983
$233$	10.5555	0.691517	0.345758	−	0.938324i	$-0.387622\pi$
$233$	10.5555	0.691517	0.345758	+	0.938324i	$0.387622\pi$
$234$	0	0
$235$	−0.754939	−0.0492468
$236$	4.41644	0.287486
$237$	4.67143	0.303442
$238$	−6.13149	−0.397446
$239$	20.2308	1.30862	0.654310	−	0.756227i	$-0.272959\pi$
$239$	20.2308	1.30862	0.654310	+	0.756227i	$0.272959\pi$
$240$	−0.614768	−0.0396831
$241$	−22.9058	−1.47549	−0.737746	−	0.675078i	$-0.764110\pi$
$241$	−22.9058	−1.47549	−0.737746	+	0.675078i	$0.764110\pi$
$242$	9.04430	0.581390
$243$	−16.1409	−1.03544
$244$	12.3006	0.787465
$245$	1.96352	0.125445
$246$	5.79175	0.369269
$247$	0	0
$248$	−1.71795	−0.109090
$249$	1.14219	0.0723833
$250$	6.52109	0.412430
$251$	30.2819	1.91138	0.955688	−	0.294380i	$-0.0951131\pi$
$251$	30.2819	1.91138	0.955688	+	0.294380i	$0.0951131\pi$
$252$	6.88809	0.433909
$253$	3.09913	0.194841
$254$	−6.59497	−0.413805
$255$	1.19982	0.0751357
$256$	1.00000	0.0625000
$257$	−18.2494	−1.13837	−0.569184	−	0.822210i	$-0.692741\pi$
$257$	−18.2494	−1.13837	−0.569184	+	0.822210i	$0.692741\pi$
$258$	−2.38057	−0.148208
$259$	−7.94506	−0.493682
$260$	0	0
$261$	−8.44837	−0.522941
$262$	8.81605	0.544657
$263$	3.24165	0.199889	0.0999443	−	0.994993i	$-0.468134\pi$
$263$	3.24165	0.199889	0.0999443	+	0.994993i	$0.468134\pi$
$264$	1.25668	0.0773434
$265$	−6.50710	−0.399728
$266$	3.14167	0.192628
$267$	0.606294	0.0371046
$268$	7.06905	0.431811
$269$	−12.6051	−0.768546	−0.384273	−	0.923219i	$-0.625548\pi$
$269$	−12.6051	−0.768546	−0.384273	+	0.923219i	$0.625548\pi$
$270$	−3.19218	−0.194270
$271$	−24.2287	−1.47179	−0.735896	−	0.677095i	$-0.763239\pi$
$271$	−24.2287	−1.47179	−0.735896	+	0.677095i	$0.763239\pi$
$272$	−1.95166	−0.118337
$273$	0	0
$274$	18.4002	1.11160
$275$	−6.33779	−0.382183
$276$	−1.99142	−0.119869
$277$	−20.2080	−1.21418	−0.607090	−	0.794633i	$-0.707663\pi$
$277$	−20.2080	−1.21418	−0.607090	+	0.794633i	$0.707663\pi$
$278$	9.47178	0.568080
$279$	−3.76659	−0.225500
$280$	2.14931	0.128446
$281$	1.15015	0.0686119	0.0343060	−	0.999411i	$-0.489078\pi$
$281$	1.15015	0.0686119	0.0343060	+	0.999411i	$0.489078\pi$
$282$	−0.991626	−0.0590505
$283$	−1.19155	−0.0708306	−0.0354153	−	0.999373i	$-0.511275\pi$
$283$	−1.19155	−0.0708306	−0.0354153	+	0.999373i	$0.511275\pi$
$284$	−8.27705	−0.491153
$285$	−0.614768	−0.0364157
$286$	0	0
$287$	−20.2487	−1.19524
$288$	2.19249	0.129194
$289$	−13.1910	−0.775941
$290$	−2.63617	−0.154801
$291$	−5.10590	−0.299313
$292$	11.1434	0.652117
$293$	−29.4420	−1.72002	−0.860009	−	0.510278i	$-0.829542\pi$
$293$	−29.4420	−1.72002	−0.860009	+	0.510278i	$0.829542\pi$
$294$	2.57913	0.150418
$295$	3.02141	0.175913
$296$	−2.52893	−0.146991
$297$	6.52531	0.378637
$298$	−21.1970	−1.22791
$299$	0	0
$300$	4.07250	0.235126
$301$	8.32278	0.479717
$302$	−0.671643	−0.0386487
$303$	4.60358	0.264469
$304$	1.00000	0.0573539
$305$	8.41518	0.481852
$306$	−4.27901	−0.244614
$307$	−3.29710	−0.188175	−0.0940876	−	0.995564i	$-0.529993\pi$
$307$	−3.29710	−0.188175	−0.0940876	+	0.995564i	$0.529993\pi$
$308$	−4.39352	−0.250344
$309$	3.34111	0.190069
$310$	−1.17530	−0.0667524
$311$	−2.72596	−0.154575	−0.0772874	−	0.997009i	$-0.524626\pi$
$311$	−2.72596	−0.154575	−0.0772874	+	0.997009i	$0.524626\pi$
$312$	0	0
$313$	−12.7175	−0.718838	−0.359419	−	0.933176i	$-0.617025\pi$
$313$	−12.7175	−0.718838	−0.359419	+	0.933176i	$0.617025\pi$
$314$	20.1166	1.13524
$315$	4.71233	0.265510
$316$	−5.19848	−0.292437
$317$	23.6990	1.33107	0.665533	−	0.746368i	$-0.268204\pi$
$317$	23.6990	1.33107	0.665533	+	0.746368i	$0.268204\pi$
$318$	−8.54721	−0.479304
$319$	5.38873	0.301711
$320$	0.684128	0.0382439
$321$	−10.9295	−0.610025
$322$	6.96225	0.387991
$323$	−1.95166	−0.108594
$324$	2.38448	0.132471
$325$	0	0
$326$	6.83144	0.378358
$327$	−6.02363	−0.333108
$328$	−6.44519	−0.355876
$329$	3.46685	0.191134
$330$	0.859731	0.0473266
$331$	30.6993	1.68739	0.843694	−	0.536824i	$-0.180376\pi$
$331$	30.6993	1.68739	0.843694	+	0.536824i	$0.180376\pi$
$332$	−1.27105	−0.0697582
$333$	−5.54465	−0.303845
$334$	10.9661	0.600039
$335$	4.83614	0.264226
$336$	2.82316	0.154016
$337$	−20.0651	−1.09302	−0.546508	−	0.837454i	$-0.684043\pi$
$337$	−20.0651	−1.09302	−0.546508	+	0.837454i	$0.684043\pi$
$338$	0	0
$339$	10.7192	0.582190
$340$	−1.33519	−0.0724108
$341$	2.40249	0.130102
$342$	2.19249	0.118556
$343$	12.9748	0.700571
$344$	2.64916	0.142833
$345$	−1.36239	−0.0733484
$346$	6.34922	0.341336
$347$	−16.2689	−0.873359	−0.436679	−	0.899617i	$-0.643846\pi$
$347$	−16.2689	−0.873359	−0.436679	+	0.899617i	$0.643846\pi$
$348$	−3.46266	−0.185618
$349$	−0.0170360	−0.000911917 0	−0.000455959	−	1.00000i	$-0.500145\pi$
$349$	−0.0170360	−0.000911917 0	−0.000455959		1.00000i	$0.500145\pi$
$350$	−14.2380	−0.761051
$351$	0	0
$352$	−1.39846	−0.0745384
$353$	−21.5851	−1.14886	−0.574429	−	0.818554i	$-0.694776\pi$
$353$	−21.5851	−1.14886	−0.574429	+	0.818554i	$0.694776\pi$
$354$	3.96868	0.210933
$355$	−5.66257	−0.300538
$356$	−0.674698	−0.0357589
$357$	−5.50986	−0.291612
$358$	−4.47871	−0.236707
$359$	18.9767	1.00155	0.500776	−	0.865577i	$-0.333048\pi$
$359$	18.9767	1.00155	0.500776	+	0.865577i	$0.333048\pi$
$360$	1.49994	0.0790540
$361$	1.00000	0.0526316
$362$	3.00372	0.157872
$363$	8.12735	0.426575
$364$	0	0
$365$	7.62350	0.399032
$366$	11.0535	0.577776
$367$	−25.1722	−1.31398	−0.656990	−	0.753899i	$-0.728171\pi$
$367$	−25.1722	−1.31398	−0.656990	+	0.753899i	$0.728171\pi$
$368$	2.21610	0.115522
$369$	−14.1310	−0.735632
$370$	−1.73011	−0.0899441
$371$	29.8821	1.55140
$372$	−1.54378	−0.0800410
$373$	9.01060	0.466551	0.233276	−	0.972411i	$-0.425056\pi$
$373$	9.01060	0.466551	0.233276	+	0.972411i	$0.425056\pi$
$374$	2.72933	0.141130
$375$	5.85995	0.302607
$376$	1.10350	0.0569089
$377$	0	0
$378$	14.6592	0.753989
$379$	−5.00587	−0.257134	−0.128567	−	0.991701i	$-0.541038\pi$
$379$	−5.00587	−0.257134	−0.128567	+	0.991701i	$0.541038\pi$
$380$	0.684128	0.0350950
$381$	−5.92634	−0.303616
$382$	22.3399	1.14301
$383$	16.9802	0.867646	0.433823	−	0.900998i	$-0.357164\pi$
$383$	16.9802	0.867646	0.433823	+	0.900998i	$0.357164\pi$
$384$	0.898616	0.0458573
$385$	−3.00573	−0.153186
$386$	−9.41832	−0.479380
$387$	5.80825	0.295250
$388$	5.68196	0.288458
$389$	18.4784	0.936891	0.468446	−	0.883492i	$-0.344814\pi$
$389$	18.4784	0.936891	0.468446	+	0.883492i	$0.344814\pi$
$390$	0	0
$391$	−4.32508	−0.218729
$392$	−2.87011	−0.144962
$393$	7.92224	0.399624
$394$	−3.69904	−0.186355
$395$	−3.55643	−0.178943
$396$	−3.06612	−0.154078
$397$	9.77064	0.490374	0.245187	−	0.969476i	$-0.421151\pi$
$397$	9.77064	0.490374	0.245187	+	0.969476i	$0.421151\pi$
$398$	2.32306	0.116445
$399$	2.82316	0.141335
$400$	−4.53197	−0.226598
$401$	−35.4743	−1.77150	−0.885752	−	0.464159i	$-0.846357\pi$
$401$	−35.4743	−1.77150	−0.885752	+	0.464159i	$0.846357\pi$
$402$	6.35236	0.316827
$403$	0	0
$404$	−5.12297	−0.254877
$405$	1.63129	0.0810596
$406$	12.1059	0.600805
$407$	3.53661	0.175303
$408$	−1.75380	−0.0868259
$409$	14.8392	0.733749	0.366875	−	0.930270i	$-0.380428\pi$
$409$	14.8392	0.733749	0.366875	+	0.930270i	$0.380428\pi$
$410$	−4.40934	−0.217762
$411$	16.5347	0.815598
$412$	−3.71806	−0.183176
$413$	−13.8750	−0.682744
$414$	4.85877	0.238796
$415$	−0.869564	−0.0426852
$416$	0	0
$417$	8.51149	0.416810
$418$	−1.39846	−0.0684011
$419$	−1.37185	−0.0670191	−0.0335096	−	0.999438i	$-0.510668\pi$
$419$	−1.37185	−0.0670191	−0.0335096	+	0.999438i	$0.510668\pi$
$420$	1.93140	0.0942427
$421$	−5.24590	−0.255670	−0.127835	−	0.991795i	$-0.540803\pi$
$421$	−5.24590	−0.255670	−0.127835	+	0.991795i	$0.540803\pi$
$422$	−17.3814	−0.846114
$423$	2.41942	0.117636
$424$	9.51153	0.461921
$425$	8.84488	0.429040
$426$	−7.43789	−0.360367
$427$	−38.6445	−1.87014
$428$	12.1626	0.587901
$429$	0	0
$430$	1.81236	0.0873999
$431$	17.8400	0.859324	0.429662	−	0.902990i	$-0.358633\pi$
$431$	17.8400	0.859324	0.429662	+	0.902990i	$0.358633\pi$
$432$	4.66605	0.224496
$433$	−22.2206	−1.06785	−0.533926	−	0.845531i	$-0.679284\pi$
$433$	−22.2206	−1.06785	−0.533926	+	0.845531i	$0.679284\pi$
$434$	5.39723	0.259076
$435$	−2.36890	−0.113580
$436$	6.70323	0.321027
$437$	2.21610	0.106010
$438$	10.0136	0.478469
$439$	−20.7084	−0.988359	−0.494179	−	0.869360i	$-0.664531\pi$
$439$	−20.7084	−0.988359	−0.494179	+	0.869360i	$0.664531\pi$
$440$	−0.956729	−0.0456102
$441$	−6.29269	−0.299652
$442$	0	0
$443$	26.6113	1.26434	0.632169	−	0.774830i	$-0.282165\pi$
$443$	26.6113	1.26434	0.632169	+	0.774830i	$0.282165\pi$
$444$	−2.27253	−0.107850
$445$	−0.461580	−0.0218810
$446$	23.6297	1.11890
$447$	−19.0479	−0.900936
$448$	−3.14167	−0.148430
$449$	−14.7616	−0.696642	−0.348321	−	0.937375i	$-0.613248\pi$
$449$	−14.7616	−0.696642	−0.348321	+	0.937375i	$0.613248\pi$
$450$	−9.93630	−0.468402
$451$	9.01337	0.424423
$452$	−11.9286	−0.561075
$453$	−0.603549	−0.0283572
$454$	−18.2136	−0.854805
$455$	0	0
$456$	0.898616	0.0420815
$457$	−32.5445	−1.52237	−0.761184	−	0.648536i	$-0.775382\pi$
$457$	−32.5445	−1.52237	−0.761184	+	0.648536i	$0.775382\pi$
$458$	−12.1571	−0.568062
$459$	−9.10657	−0.425058
$460$	1.51609	0.0706883
$461$	−40.1818	−1.87145	−0.935727	−	0.352724i	$-0.885255\pi$
$461$	−40.1818	−1.87145	−0.935727	+	0.352724i	$0.885255\pi$
$462$	−3.94808	−0.183681
$463$	23.9788	1.11439	0.557194	−	0.830382i	$-0.311878\pi$
$463$	23.9788	1.11439	0.557194	+	0.830382i	$0.311878\pi$
$464$	3.85332	0.178886
$465$	−1.05614	−0.0489773
$466$	−10.5555	−0.488976
$467$	5.03262	0.232882	0.116441	−	0.993198i	$-0.462851\pi$
$467$	5.03262	0.232882	0.116441	+	0.993198i	$0.462851\pi$
$468$	0	0
$469$	−22.2086	−1.02550
$470$	0.754939	0.0348227
$471$	18.0771	0.832947
$472$	−4.41644	−0.203283
$473$	−3.70475	−0.170345
$474$	−4.67143	−0.214566
$475$	−4.53197	−0.207941
$476$	6.13149	0.281037
$477$	20.8539	0.954836
$478$	−20.2308	−0.925333
$479$	5.07610	0.231933	0.115967	−	0.993253i	$-0.463003\pi$
$479$	5.07610	0.231933	0.115967	+	0.993253i	$0.463003\pi$
$480$	0.614768	0.0280602
$481$	0	0
$482$	22.9058	1.04333
$483$	6.25639	0.284676
$484$	−9.04430	−0.411104
$485$	3.88719	0.176508
$486$	16.1409	0.732166
$487$	−23.4707	−1.06356	−0.531780	−	0.846883i	$-0.678477\pi$
$487$	−23.4707	−1.06356	−0.531780	+	0.846883i	$0.678477\pi$
$488$	−12.3006	−0.556822
$489$	6.13884	0.277608
$490$	−1.96352	−0.0887030
$491$	0.146156	0.00659594	0.00329797	−	0.999995i	$-0.498950\pi$
$491$	0.146156	0.00659594	0.00329797	+	0.999995i	$0.498950\pi$
$492$	−5.79175	−0.261112
$493$	−7.52039	−0.338702
$494$	0	0
$495$	−2.09762	−0.0942809
$496$	1.71795	0.0771382
$497$	26.0038	1.16643
$498$	−1.14219	−0.0511827
$499$	22.7423	1.01808	0.509042	−	0.860742i	$-0.330000\pi$
$499$	22.7423	1.01808	0.509042	+	0.860742i	$0.330000\pi$
$500$	−6.52109	−0.291632
$501$	9.85432	0.440259
$502$	−30.2819	−1.35155
$503$	−0.380694	−0.0169743	−0.00848714	−	0.999964i	$-0.502702\pi$
$503$	−0.380694	−0.0169743	−0.00848714	+	0.999964i	$0.502702\pi$
$504$	−6.88809	−0.306820
$505$	−3.50477	−0.155960
$506$	−3.09913	−0.137773
$507$	0	0
$508$	6.59497	0.292604
$509$	25.6402	1.13648	0.568240	−	0.822863i	$-0.307624\pi$
$509$	25.6402	1.13648	0.568240	+	0.822863i	$0.307624\pi$
$510$	−1.19982	−0.0531290
$511$	−35.0089	−1.54870
$512$	−1.00000	−0.0441942
$513$	4.66605	0.206011
$514$	18.2494	0.804948
$515$	−2.54363	−0.112086
$516$	2.38057	0.104799
$517$	−1.54321	−0.0678704
$518$	7.94506	0.349086
$519$	5.70551	0.250444
$520$	0	0
$521$	−17.3228	−0.758925	−0.379462	−	0.925207i	$-0.623891\pi$
$521$	−17.3228	−0.758925	−0.379462	+	0.925207i	$0.623891\pi$
$522$	8.44837	0.369775
$523$	31.2955	1.36846	0.684228	−	0.729268i	$-0.260139\pi$
$523$	31.2955	1.36846	0.684228	+	0.729268i	$0.260139\pi$
$524$	−8.81605	−0.385131
$525$	−12.7945	−0.558396
$526$	−3.24165	−0.141343
$527$	−3.35286	−0.146053
$528$	−1.25668	−0.0546900
$529$	−18.0889	−0.786475
$530$	6.50710	0.282651
$531$	−9.68300	−0.420206
$532$	−3.14167	−0.136209
$533$	0	0
$534$	−0.606294	−0.0262369
$535$	8.32078	0.359739
$536$	−7.06905	−0.305336
$537$	−4.02463	−0.173676
$538$	12.6051	0.543444
$539$	4.01375	0.172884
$540$	3.19218	0.137369
$541$	−17.2961	−0.743617	−0.371809	−	0.928309i	$-0.621262\pi$
$541$	−17.2961	−0.743617	−0.371809	+	0.928309i	$0.621262\pi$
$542$	24.2287	1.04071
$543$	2.69919	0.115833
$544$	1.95166	0.0836769
$545$	4.58587	0.196437
$546$	0	0
$547$	−38.7902	−1.65855	−0.829275	−	0.558841i	$-0.811246\pi$
$547$	−38.7902	−1.65855	−0.829275	+	0.558841i	$0.811246\pi$
$548$	−18.4002	−0.786019
$549$	−26.9689	−1.15101
$550$	6.33779	0.270244
$551$	3.85332	0.164157
$552$	1.99142	0.0847604
$553$	16.3319	0.694504
$554$	20.2080	0.858554
$555$	−1.55470	−0.0659935
$556$	−9.47178	−0.401693
$557$	−17.9889	−0.762213	−0.381107	−	0.924531i	$-0.624457\pi$
$557$	−17.9889	−0.762213	−0.381107	+	0.924531i	$0.624457\pi$
$558$	3.76659	0.159452
$559$	0	0
$560$	−2.14931	−0.0908248
$561$	2.45262	0.103550
$562$	−1.15015	−0.0485160
$563$	0.923313	0.0389130	0.0194565	−	0.999811i	$-0.493806\pi$
$563$	0.923313	0.0389130	0.0194565	+	0.999811i	$0.493806\pi$
$564$	0.991626	0.0417550
$565$	−8.16071	−0.343324
$566$	1.19155	0.0500848
$567$	−7.49127	−0.314604
$568$	8.27705	0.347297
$569$	−15.4409	−0.647315	−0.323657	−	0.946174i	$-0.604913\pi$
$569$	−15.4409	−0.647315	−0.323657	+	0.946174i	$0.604913\pi$
$570$	0.614768	0.0257498
$571$	−25.7365	−1.07704	−0.538520	−	0.842612i	$-0.681017\pi$
$571$	−25.7365	−1.07704	−0.538520	+	0.842612i	$0.681017\pi$
$572$	0	0
$573$	20.0750	0.838643
$574$	20.2487	0.845164
$575$	−10.0433	−0.418834
$576$	−2.19249	−0.0913538
$577$	−10.4212	−0.433838	−0.216919	−	0.976190i	$-0.569601\pi$
$577$	−10.4212	−0.433838	−0.216919	+	0.976190i	$0.569601\pi$
$578$	13.1910	0.548673
$579$	−8.46345	−0.351729
$580$	2.63617	0.109461
$581$	3.99324	0.165667
$582$	5.10590	0.211646
$583$	−13.3015	−0.550893
$584$	−11.1434	−0.461116
$585$	0	0
$586$	29.4420	1.21624
$587$	−25.8107	−1.06532	−0.532661	−	0.846329i	$-0.678808\pi$
$587$	−25.8107	−1.06532	−0.532661	+	0.846329i	$0.678808\pi$
$588$	−2.57913	−0.106361
$589$	1.71795	0.0707869
$590$	−3.02141	−0.124389
$591$	−3.32401	−0.136732
$592$	2.52893	0.103938
$593$	−5.63446	−0.231380	−0.115690	−	0.993285i	$-0.536908\pi$
$593$	−5.63446	−0.231380	−0.115690	+	0.993285i	$0.536908\pi$
$594$	−6.52531	−0.267737
$595$	4.19473	0.171967
$596$	21.1970	0.868262
$597$	2.08754	0.0854373
$598$	0	0
$599$	8.30231	0.339223	0.169612	−	0.985511i	$-0.445749\pi$
$599$	8.30231	0.339223	0.169612	+	0.985511i	$0.445749\pi$
$600$	−4.07250	−0.166259
$601$	−31.5889	−1.28854	−0.644269	−	0.764799i	$-0.722838\pi$
$601$	−31.5889	−1.28854	−0.644269	+	0.764799i	$0.722838\pi$
$602$	−8.32278	−0.339211
$603$	−15.4988	−0.631161
$604$	0.671643	0.0273288
$605$	−6.18746	−0.251556
$606$	−4.60358	−0.187008
$607$	−15.5028	−0.629241	−0.314620	−	0.949218i	$-0.601877\pi$
$607$	−15.5028	−0.629241	−0.314620	+	0.949218i	$0.601877\pi$
$608$	−1.00000	−0.0405554
$609$	10.8785	0.440820
$610$	−8.41518	−0.340721
$611$	0	0
$612$	4.27901	0.172969
$613$	−13.4741	−0.544216	−0.272108	−	0.962267i	$-0.587721\pi$
$613$	−13.4741	−0.544216	−0.272108	+	0.962267i	$0.587721\pi$
$614$	3.29710	0.133060
$615$	−3.96230	−0.159775
$616$	4.39352	0.177020
$617$	−25.4074	−1.02286	−0.511431	−	0.859324i	$-0.670884\pi$
$617$	−25.4074	−1.02286	−0.511431	+	0.859324i	$0.670884\pi$
$618$	−3.34111	−0.134399
$619$	−6.22939	−0.250380	−0.125190	−	0.992133i	$-0.539954\pi$
$619$	−6.22939	−0.250380	−0.125190	+	0.992133i	$0.539954\pi$
$620$	1.17530	0.0472011
$621$	10.3404	0.414947
$622$	2.72596	0.109301
$623$	2.11968	0.0849232
$624$	0	0
$625$	18.1986	0.727943
$626$	12.7175	0.508295
$627$	−1.25668	−0.0501870
$628$	−20.1166	−0.802738
$629$	−4.93562	−0.196796
$630$	−4.71233	−0.187744
$631$	18.1627	0.723048	0.361524	−	0.932363i	$-0.382257\pi$
$631$	18.1627	0.723048	0.361524	+	0.932363i	$0.382257\pi$
$632$	5.19848	0.206784
$633$	−15.6192	−0.620808
$634$	−23.6990	−0.941206
$635$	4.51180	0.179045
$636$	8.54721	0.338919
$637$	0	0
$638$	−5.38873	−0.213342
$639$	18.1474	0.717898
$640$	−0.684128	−0.0270425
$641$	−33.1023	−1.30746	−0.653732	−	0.756726i	$-0.726797\pi$
$641$	−33.1023	−1.30746	−0.653732	+	0.756726i	$0.726797\pi$
$642$	10.9295	0.431353
$643$	34.1813	1.34798	0.673989	−	0.738741i	$-0.264579\pi$
$643$	34.1813	1.34798	0.673989	+	0.738741i	$0.264579\pi$
$644$	−6.96225	−0.274351
$645$	1.62862	0.0641267
$646$	1.95166	0.0767872
$647$	20.4733	0.804887	0.402444	−	0.915445i	$-0.368161\pi$
$647$	20.4733	0.804887	0.402444	+	0.915445i	$0.368161\pi$
$648$	−2.38448	−0.0936713
$649$	6.17623	0.242438
$650$	0	0
$651$	4.85004	0.190088
$652$	−6.83144	−0.267540
$653$	34.5765	1.35308	0.676542	−	0.736404i	$-0.263478\pi$
$653$	34.5765	1.35308	0.676542	+	0.736404i	$0.263478\pi$
$654$	6.02363	0.235543
$655$	−6.03131	−0.235663
$656$	6.44519	0.251642
$657$	−24.4318	−0.953174
$658$	−3.46685	−0.135152
$659$	−2.73591	−0.106576	−0.0532879	−	0.998579i	$-0.516970\pi$
$659$	−2.73591	−0.106576	−0.0532879	+	0.998579i	$0.516970\pi$
$660$	−0.859731	−0.0334650
$661$	−30.4446	−1.18416	−0.592078	−	0.805880i	$-0.701692\pi$
$661$	−30.4446	−1.18416	−0.592078	+	0.805880i	$0.701692\pi$
$662$	−30.6993	−1.19316
$663$	0	0
$664$	1.27105	0.0493265
$665$	−2.14931	−0.0833466
$666$	5.54465	0.214851
$667$	8.53934	0.330644
$668$	−10.9661	−0.424292
$669$	21.2340	0.820955
$670$	−4.83614	−0.186836
$671$	17.2019	0.664074
$672$	−2.82316	−0.108906
$673$	−41.7506	−1.60937	−0.804684	−	0.593704i	$-0.797665\pi$
$673$	−41.7506	−1.60937	−0.804684	+	0.593704i	$0.797665\pi$
$674$	20.0651	0.772879
$675$	−21.1464	−0.813925
$676$	0	0
$677$	−31.4238	−1.20772	−0.603858	−	0.797092i	$-0.706370\pi$
$677$	−31.4238	−1.20772	−0.603858	+	0.797092i	$0.706370\pi$
$678$	−10.7192	−0.411670
$679$	−17.8509	−0.685053
$680$	1.33519	0.0512022
$681$	−16.3670	−0.627185
$682$	−2.40249	−0.0919961
$683$	−43.0979	−1.64909	−0.824547	−	0.565793i	$-0.808570\pi$
$683$	−43.0979	−1.64909	−0.824547	+	0.565793i	$0.808570\pi$
$684$	−2.19249	−0.0838320
$685$	−12.5881	−0.480967
$686$	−12.9748	−0.495379
$687$	−10.9245	−0.416796
$688$	−2.64916	−0.100998
$689$	0	0
$690$	1.36239	0.0518651
$691$	29.8427	1.13527	0.567635	−	0.823280i	$-0.307859\pi$
$691$	29.8427	1.13527	0.567635	+	0.823280i	$0.307859\pi$
$692$	−6.34922	−0.241361
$693$	9.63274	0.365918
$694$	16.2689	0.617558
$695$	−6.47991	−0.245797
$696$	3.46266	0.131252
$697$	−12.5789	−0.476458
$698$	0.0170360	0.000644823 0
$699$	−9.48537	−0.358770
$700$	14.2380	0.538144
$701$	30.7734	1.16229	0.581147	−	0.813798i	$-0.302604\pi$
$701$	30.7734	1.16229	0.581147	+	0.813798i	$0.302604\pi$
$702$	0	0
$703$	2.52893	0.0953803
$704$	1.39846	0.0527066
$705$	0.678400	0.0255500
$706$	21.5851	0.812366
$707$	16.0947	0.605303
$708$	−3.96868	−0.149152
$709$	−2.26396	−0.0850249	−0.0425125	−	0.999096i	$-0.513536\pi$
$709$	−2.26396	−0.0850249	−0.0425125	+	0.999096i	$0.513536\pi$
$710$	5.66257	0.212512
$711$	11.3976	0.427444
$712$	0.674698	0.0252854
$713$	3.80714	0.142579
$714$	5.50986	0.206201
$715$	0	0
$716$	4.47871	0.167377
$717$	−18.1797	−0.678932
$718$	−18.9767	−0.708204
$719$	34.8659	1.30028	0.650140	−	0.759814i	$-0.274710\pi$
$719$	34.8659	1.30028	0.650140	+	0.759814i	$0.274710\pi$
$720$	−1.49994	−0.0558996
$721$	11.6809	0.435021
$722$	−1.00000	−0.0372161
$723$	20.5835	0.765509
$724$	−3.00372	−0.111633
$725$	−17.4631	−0.648565
$726$	−8.12735	−0.301634
$727$	−50.2528	−1.86377	−0.931886	−	0.362751i	$-0.881837\pi$
$727$	−50.2528	−1.86377	−0.931886	+	0.362751i	$0.881837\pi$
$728$	0	0
$729$	7.35101	0.272259
$730$	−7.62350	−0.282158
$731$	5.17026	0.191229
$732$	−11.0535	−0.408549
$733$	−8.74945	−0.323168	−0.161584	−	0.986859i	$-0.551660\pi$
$733$	−8.74945	−0.323168	−0.161584	+	0.986859i	$0.551660\pi$
$734$	25.1722	0.929124
$735$	−1.76445	−0.0650828
$736$	−2.21610	−0.0816864
$737$	9.88581	0.364149
$738$	14.1310	0.520170
$739$	9.80778	0.360785	0.180393	−	0.983595i	$-0.442263\pi$
$739$	9.80778	0.360785	0.180393	+	0.983595i	$0.442263\pi$
$740$	1.73011	0.0636001
$741$	0	0
$742$	−29.8821	−1.09701
$743$	−29.4703	−1.08116	−0.540580	−	0.841292i	$-0.681795\pi$
$743$	−29.4703	−1.08116	−0.540580	+	0.841292i	$0.681795\pi$
$744$	1.54378	0.0565976
$745$	14.5014	0.531292
$746$	−9.01060	−0.329902
$747$	2.78677	0.101963
$748$	−2.72933	−0.0997943
$749$	−38.2109	−1.39620
$750$	−5.85995	−0.213975
$751$	11.1191	0.405742	0.202871	−	0.979205i	$-0.434973\pi$
$751$	11.1191	0.405742	0.202871	+	0.979205i	$0.434973\pi$
$752$	−1.10350	−0.0402407
$753$	−27.2118	−0.991653
$754$	0	0
$755$	0.459490	0.0167226
$756$	−14.6592	−0.533150
$757$	28.7330	1.04432	0.522160	−	0.852847i	$-0.325126\pi$
$757$	28.7330	1.04432	0.522160	+	0.852847i	$0.325126\pi$
$758$	5.00587	0.181821
$759$	−2.78493	−0.101086
$760$	−0.684128	−0.0248159
$761$	−48.0235	−1.74085	−0.870425	−	0.492301i	$-0.836156\pi$
$761$	−48.0235	−1.74085	−0.870425	+	0.492301i	$0.836156\pi$
$762$	5.92634	0.214689
$763$	−21.0594	−0.762401
$764$	−22.3399	−0.808228
$765$	2.92739	0.105840
$766$	−16.9802	−0.613518
$767$	0	0
$768$	−0.898616	−0.0324260
$769$	−23.4559	−0.845842	−0.422921	−	0.906167i	$-0.638995\pi$
$769$	−23.4559	−0.845842	−0.422921	+	0.906167i	$0.638995\pi$
$770$	3.00573	0.108319
$771$	16.3992	0.590604
$772$	9.41832	0.338973
$773$	−17.1742	−0.617713	−0.308857	−	0.951109i	$-0.599946\pi$
$773$	−17.1742	−0.617713	−0.308857	+	0.951109i	$0.599946\pi$
$774$	−5.80825	−0.208773
$775$	−7.78569	−0.279670
$776$	−5.68196	−0.203971
$777$	7.13956	0.256130
$778$	−18.4784	−0.662482
$779$	6.44519	0.230923
$780$	0	0
$781$	−11.5752	−0.414192
$782$	4.32508	0.154664
$783$	17.9798	0.642546
$784$	2.87011	0.102504
$785$	−13.7623	−0.491198
$786$	−7.92224	−0.282577
$787$	−20.7625	−0.740103	−0.370052	−	0.929011i	$-0.620660\pi$
$787$	−20.7625	−0.740103	−0.370052	+	0.929011i	$0.620660\pi$
$788$	3.69904	0.131773
$789$	−2.91300	−0.103705
$790$	3.55643	0.126532
$791$	37.4758	1.33249
$792$	3.06612	0.108950
$793$	0	0
$794$	−9.77064	−0.346747
$795$	5.84738	0.207385
$796$	−2.32306	−0.0823388
$797$	23.0244	0.815565	0.407782	−	0.913079i	$-0.366302\pi$
$797$	23.0244	0.815565	0.407782	+	0.913079i	$0.366302\pi$
$798$	−2.82316	−0.0999387
$799$	2.15367	0.0761914
$800$	4.53197	0.160229
$801$	1.47927	0.0522674
$802$	35.4743	1.25264
$803$	15.5836	0.549934
$804$	−6.35236	−0.224030
$805$	−4.76307	−0.167876
$806$	0	0
$807$	11.3271	0.398734
$808$	5.12297	0.180225
$809$	−2.64397	−0.0929572	−0.0464786	−	0.998919i	$-0.514800\pi$
$809$	−2.64397	−0.0929572	−0.0464786	+	0.998919i	$0.514800\pi$
$810$	−1.63129	−0.0573178
$811$	−56.6735	−1.99007	−0.995037	−	0.0995050i	$-0.968274\pi$
$811$	−56.6735	−1.99007	−0.995037	+	0.0995050i	$0.968274\pi$
$812$	−12.1059	−0.424833
$813$	21.7723	0.763589
$814$	−3.53661	−0.123958
$815$	−4.67358	−0.163708
$816$	1.75380	0.0613952
$817$	−2.64916	−0.0926822
$818$	−14.8392	−0.518839
$819$	0	0
$820$	4.40934	0.153981
$821$	19.0253	0.663986	0.331993	−	0.943282i	$-0.392279\pi$
$821$	19.0253	0.663986	0.331993	+	0.943282i	$0.392279\pi$
$822$	−16.5347	−0.576715
$823$	54.3564	1.89474	0.947372	−	0.320135i	$-0.103728\pi$
$823$	54.3564	1.89474	0.947372	+	0.320135i	$0.103728\pi$
$824$	3.71806	0.129525
$825$	5.69524	0.198283
$826$	13.8750	0.482773
$827$	−42.4997	−1.47786	−0.738930	−	0.673782i	$-0.764669\pi$
$827$	−42.4997	−1.47786	−0.738930	+	0.673782i	$0.764669\pi$
$828$	−4.85877	−0.168854
$829$	−16.1103	−0.559533	−0.279766	−	0.960068i	$-0.590257\pi$
$829$	−16.1103	−0.559533	−0.279766	+	0.960068i	$0.590257\pi$
$830$	0.869564	0.0301830
$831$	18.1592	0.629935
$832$	0	0
$833$	−5.60150	−0.194080
$834$	−8.51149	−0.294729
$835$	−7.50223	−0.259625
$836$	1.39846	0.0483669
$837$	8.01604	0.277075
$838$	1.37185	0.0473897
$839$	3.16470	0.109258	0.0546289	−	0.998507i	$-0.482602\pi$
$839$	3.16470	0.109258	0.0546289	+	0.998507i	$0.482602\pi$
$840$	−1.93140	−0.0666397
$841$	−14.1519	−0.487997
$842$	5.24590	0.180786
$843$	−1.03354	−0.0355970
$844$	17.3814	0.598293
$845$	0	0
$846$	−2.41942	−0.0831815
$847$	28.4142	0.976325
$848$	−9.51153	−0.326627
$849$	1.07075	0.0367480
$850$	−8.84488	−0.303377
$851$	5.60435	0.192115
$852$	7.43789	0.254818
$853$	−3.12402	−0.106964	−0.0534822	−	0.998569i	$-0.517032\pi$
$853$	−3.12402	−0.106964	−0.0534822	+	0.998569i	$0.517032\pi$
$854$	38.6445	1.32239
$855$	−1.49994	−0.0512970
$856$	−12.1626	−0.415709
$857$	11.8072	0.403325	0.201663	−	0.979455i	$-0.435366\pi$
$857$	11.8072	0.403325	0.201663	+	0.979455i	$0.435366\pi$
$858$	0	0
$859$	1.32269	0.0451297	0.0225648	−	0.999745i	$-0.492817\pi$
$859$	1.32269	0.0451297	0.0225648	+	0.999745i	$0.492817\pi$
$860$	−1.81236	−0.0618010
$861$	18.1958	0.620111
$862$	−17.8400	−0.607634
$863$	−41.9019	−1.42636	−0.713179	−	0.700982i	$-0.752745\pi$
$863$	−41.9019	−1.42636	−0.713179	+	0.700982i	$0.752745\pi$
$864$	−4.66605	−0.158742
$865$	−4.34368	−0.147690
$866$	22.2206	0.755086
$867$	11.8536	0.402571
$868$	−5.39723	−0.183194
$869$	−7.26989	−0.246614
$870$	2.36890	0.0803132
$871$	0	0
$872$	−6.70323	−0.227000
$873$	−12.4576	−0.421627
$874$	−2.21610	−0.0749606
$875$	20.4871	0.692591
$876$	−10.0136	−0.338329
$877$	−41.6016	−1.40478	−0.702392	−	0.711790i	$-0.747885\pi$
$877$	−41.6016	−1.40478	−0.702392	+	0.711790i	$0.747885\pi$
$878$	20.7084	0.698875
$879$	26.4570	0.892373
$880$	0.956729	0.0322513
$881$	−10.2890	−0.346646	−0.173323	−	0.984865i	$-0.555450\pi$
$881$	−10.2890	−0.346646	−0.173323	+	0.984865i	$0.555450\pi$
$882$	6.29269	0.211886
$883$	46.1408	1.55276	0.776381	−	0.630264i	$-0.217053\pi$
$883$	46.1408	1.55276	0.776381	+	0.630264i	$0.217053\pi$
$884$	0	0
$885$	−2.71509	−0.0912666
$886$	−26.6113	−0.894022
$887$	−26.4432	−0.887876	−0.443938	−	0.896058i	$-0.646419\pi$
$887$	−26.4432	−0.887876	−0.443938	+	0.896058i	$0.646419\pi$
$888$	2.27253	0.0762612
$889$	−20.7192	−0.694901
$890$	0.461580	0.0154722
$891$	3.33461	0.111714
$892$	−23.6297	−0.791181
$893$	−1.10350	−0.0369274
$894$	19.0479	0.637058
$895$	3.06401	0.102419
$896$	3.14167	0.104956
$897$	0	0
$898$	14.7616	0.492601
$899$	6.61981	0.220783
$900$	9.93630	0.331210
$901$	18.5633	0.618434
$902$	−9.01337	−0.300112
$903$	−7.47898	−0.248885
$904$	11.9286	0.396740
$905$	−2.05493	−0.0683083
$906$	0.603549	0.0200516
$907$	52.0491	1.72826	0.864131	−	0.503267i	$-0.167869\pi$
$907$	52.0491	1.72826	0.864131	+	0.503267i	$0.167869\pi$
$908$	18.2136	0.604439
$909$	11.2321	0.372544
$910$	0	0
$911$	−37.7744	−1.25152	−0.625761	−	0.780015i	$-0.715212\pi$
$911$	−37.7744	−1.25152	−0.625761	+	0.780015i	$0.715212\pi$
$912$	−0.898616	−0.0297561
$913$	−1.77752	−0.0588274
$914$	32.5445	1.07648
$915$	−7.56202	−0.249993
$916$	12.1571	0.401680
$917$	27.6972	0.914641
$918$	9.10657	0.300562
$919$	−29.6199	−0.977072	−0.488536	−	0.872544i	$-0.662469\pi$
$919$	−29.6199	−0.977072	−0.488536	+	0.872544i	$0.662469\pi$
$920$	−1.51609	−0.0499841
$921$	2.96282	0.0976283
$922$	40.1818	1.32332
$923$	0	0
$924$	3.94808	0.129882
$925$	−11.4610	−0.376836
$926$	−23.9788	−0.787991
$927$	8.15182	0.267741
$928$	−3.85332	−0.126492
$929$	−24.1744	−0.793137	−0.396569	−	0.918005i	$-0.629799\pi$
$929$	−24.1744	−0.793137	−0.396569	+	0.918005i	$0.629799\pi$
$930$	1.05614	0.0346322
$931$	2.87011	0.0940641
$932$	10.5555	0.345758
$933$	2.44959	0.0801959
$934$	−5.03262	−0.164672
$935$	−1.86721	−0.0610644
$936$	0	0
$937$	37.1513	1.21368	0.606840	−	0.794824i	$-0.292437\pi$
$937$	37.1513	1.21368	0.606840	+	0.794824i	$0.292437\pi$
$938$	22.2086	0.725138
$939$	11.4282	0.372945
$940$	−0.754939	−0.0246234
$941$	−30.0724	−0.980332	−0.490166	−	0.871629i	$-0.663064\pi$
$941$	−30.0724	−0.980332	−0.490166	+	0.871629i	$0.663064\pi$
$942$	−18.0771	−0.588982
$943$	14.2832	0.465124
$944$	4.41644	0.143743
$945$	−10.0288	−0.326236
$946$	3.70475	0.120452
$947$	−19.9228	−0.647403	−0.323701	−	0.946159i	$-0.604927\pi$
$947$	−19.9228	−0.647403	−0.323701	+	0.946159i	$0.604927\pi$
$948$	4.67143	0.151721
$949$	0	0
$950$	4.53197	0.147036
$951$	−21.2963	−0.690578
$952$	−6.13149	−0.198723
$953$	15.3552	0.497403	0.248701	−	0.968580i	$-0.419996\pi$
$953$	15.3552	0.497403	0.248701	+	0.968580i	$0.419996\pi$
$954$	−20.8539	−0.675171
$955$	−15.2833	−0.494557
$956$	20.2308	0.654310
$957$	−4.84240	−0.156533
$958$	−5.07610	−0.164001
$959$	57.8075	1.86670
$960$	−0.614768	−0.0198416
$961$	−28.0487	−0.904795
$962$	0	0
$963$	−26.6664	−0.859312
$964$	−22.9058	−0.737746
$965$	6.44334	0.207418
$966$	−6.25639	−0.201296
$967$	33.9418	1.09150	0.545748	−	0.837949i	$-0.316246\pi$
$967$	33.9418	1.09150	0.545748	+	0.837949i	$0.316246\pi$
$968$	9.04430	0.290695
$969$	1.75380	0.0563401
$970$	−3.88719	−0.124810
$971$	50.1155	1.60828	0.804141	−	0.594439i	$-0.202626\pi$
$971$	50.1155	1.60828	0.804141	+	0.594439i	$0.202626\pi$
$972$	−16.1409	−0.517719
$973$	29.7572	0.953974
$974$	23.4707	0.752050
$975$	0	0
$976$	12.3006	0.393732
$977$	19.1035	0.611174	0.305587	−	0.952164i	$-0.401147\pi$
$977$	19.1035	0.611174	0.305587	+	0.952164i	$0.401147\pi$
$978$	−6.13884	−0.196298
$979$	−0.943541	−0.0301557
$980$	1.96352	0.0627225
$981$	−14.6968	−0.469232
$982$	−0.146156	−0.00466403
$983$	27.4032	0.874028	0.437014	−	0.899455i	$-0.356036\pi$
$983$	27.4032	0.874028	0.437014	+	0.899455i	$0.356036\pi$
$984$	5.79175	0.184634
$985$	2.53062	0.0806321
$986$	7.52039	0.239498
$987$	−3.11537	−0.0991632
$988$	0	0
$989$	−5.87079	−0.186680
$990$	2.09762	0.0666667
$991$	−17.0129	−0.540432	−0.270216	−	0.962800i	$-0.587095\pi$
$991$	−17.0129	−0.540432	−0.270216	+	0.962800i	$0.587095\pi$
$992$	−1.71795	−0.0545449
$993$	−27.5869	−0.875444
$994$	−26.0038	−0.824790
$995$	−1.58927	−0.0503833
$996$	1.14219	0.0361916
$997$	24.4874	0.775524	0.387762	−	0.921760i	$-0.373248\pi$
$997$	24.4874	0.775524	0.387762	+	0.921760i	$0.373248\pi$
$998$	−22.7423	−0.719894
$999$	11.8001	0.373339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bg.1.3		8
13.6	odd	12		494.2.m.a.153.7	✓	16
13.11	odd	12		494.2.m.a.381.7	yes	16
13.12	even	2		6422.2.a.bh.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.m.a.153.7	✓	16	13.6	odd	12
494.2.m.a.381.7	yes	16	13.11	odd	12
6422.2.a.bg.1.3		8	1.1	even	1	trivial
6422.2.a.bh.1.3		8	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.898616 q^{3} +1.00000 q^{4} +0.684128 q^{5} +0.898616 q^{6} -3.14167 q^{7} -1.00000 q^{8} -2.19249 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.898616 q^{3} +1.00000 q^{4} +0.684128 q^{5} +0.898616 q^{6} -3.14167 q^{7} -1.00000 q^{8} -2.19249 q^{9} -0.684128 q^{10} +1.39846 q^{11} -0.898616 q^{12} +3.14167 q^{14} -0.614768 q^{15} +1.00000 q^{16} -1.95166 q^{17} +2.19249 q^{18} +1.00000 q^{19} +0.684128 q^{20} +2.82316 q^{21} -1.39846 q^{22} +2.21610 q^{23} +0.898616 q^{24} -4.53197 q^{25} +4.66605 q^{27} -3.14167 q^{28} +3.85332 q^{29} +0.614768 q^{30} +1.71795 q^{31} -1.00000 q^{32} -1.25668 q^{33} +1.95166 q^{34} -2.14931 q^{35} -2.19249 q^{36} +2.52893 q^{37} -1.00000 q^{38} -0.684128 q^{40} +6.44519 q^{41} -2.82316 q^{42} -2.64916 q^{43} +1.39846 q^{44} -1.49994 q^{45} -2.21610 q^{46} -1.10350 q^{47} -0.898616 q^{48} +2.87011 q^{49} +4.53197 q^{50} +1.75380 q^{51} -9.51153 q^{53} -4.66605 q^{54} +0.956729 q^{55} +3.14167 q^{56} -0.898616 q^{57} -3.85332 q^{58} +4.41644 q^{59} -0.614768 q^{60} +12.3006 q^{61} -1.71795 q^{62} +6.88809 q^{63} +1.00000 q^{64} +1.25668 q^{66} +7.06905 q^{67} -1.95166 q^{68} -1.99142 q^{69} +2.14931 q^{70} -8.27705 q^{71} +2.19249 q^{72} +11.1434 q^{73} -2.52893 q^{74} +4.07250 q^{75} +1.00000 q^{76} -4.39352 q^{77} -5.19848 q^{79} +0.684128 q^{80} +2.38448 q^{81} -6.44519 q^{82} -1.27105 q^{83} +2.82316 q^{84} -1.33519 q^{85} +2.64916 q^{86} -3.46266 q^{87} -1.39846 q^{88} -0.674698 q^{89} +1.49994 q^{90} +2.21610 q^{92} -1.54378 q^{93} +1.10350 q^{94} +0.684128 q^{95} +0.898616 q^{96} +5.68196 q^{97} -2.87011 q^{98} -3.06612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^5 + 4 * q^6 + 2 * q^7 - 8 * q^8	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} - 8 q^{8} - 2 q^{10} + 10 q^{11} - 4 q^{12} - 2 q^{14} + 8 q^{16} + 2 q^{17} + 8 q^{19} + 2 q^{20} - 12 q^{21} - 10 q^{22} - 8 q^{23} + 4 q^{24} - 14 q^{25} - 22 q^{27} + 2 q^{28} + 8 q^{29} + 12 q^{31} - 8 q^{32} - 4 q^{33} - 2 q^{34} - 10 q^{35} - 8 q^{38} - 2 q^{40} - 2 q^{41} + 12 q^{42} - 16 q^{43} + 10 q^{44} + 12 q^{45} + 8 q^{46} - 12 q^{47} - 4 q^{48} - 14 q^{49} + 14 q^{50} - 22 q^{51} - 24 q^{53} + 22 q^{54} - 10 q^{55} - 2 q^{56} - 4 q^{57} - 8 q^{58} + 4 q^{59} - 26 q^{61} - 12 q^{62} + 12 q^{63} + 8 q^{64} + 4 q^{66} - 10 q^{67} + 2 q^{68} + 6 q^{69} + 10 q^{70} + 36 q^{71} - 20 q^{73} + 6 q^{75} + 8 q^{76} - 12 q^{77} - 22 q^{79} + 2 q^{80} + 36 q^{81} + 2 q^{82} - 18 q^{83} - 12 q^{84} - 26 q^{85} + 16 q^{86} - 38 q^{87} - 10 q^{88} + 18 q^{89} - 12 q^{90} - 8 q^{92} - 12 q^{93} + 12 q^{94} + 2 q^{95} + 4 q^{96} + 8 q^{97} + 14 q^{98} + 8 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^5 + 4 * q^6 + 2 * q^7 - 8 * q^8 - 2 * q^10 + 10 * q^11 - 4 * q^12 - 2 * q^14 + 8 * q^16 + 2 * q^17 + 8 * q^19 + 2 * q^20 - 12 * q^21 - 10 * q^22 - 8 * q^23 + 4 * q^24 - 14 * q^25 - 22 * q^27 + 2 * q^28 + 8 * q^29 + 12 * q^31 - 8 * q^32 - 4 * q^33 - 2 * q^34 - 10 * q^35 - 8 * q^38 - 2 * q^40 - 2 * q^41 + 12 * q^42 - 16 * q^43 + 10 * q^44 + 12 * q^45 + 8 * q^46 - 12 * q^47 - 4 * q^48 - 14 * q^49 + 14 * q^50 - 22 * q^51 - 24 * q^53 + 22 * q^54 - 10 * q^55 - 2 * q^56 - 4 * q^57 - 8 * q^58 + 4 * q^59 - 26 * q^61 - 12 * q^62 + 12 * q^63 + 8 * q^64 + 4 * q^66 - 10 * q^67 + 2 * q^68 + 6 * q^69 + 10 * q^70 + 36 * q^71 - 20 * q^73 + 6 * q^75 + 8 * q^76 - 12 * q^77 - 22 * q^79 + 2 * q^80 + 36 * q^81 + 2 * q^82 - 18 * q^83 - 12 * q^84 - 26 * q^85 + 16 * q^86 - 38 * q^87 - 10 * q^88 + 18 * q^89 - 12 * q^90 - 8 * q^92 - 12 * q^93 + 12 * q^94 + 2 * q^95 + 4 * q^96 + 8 * q^97 + 14 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more