LMFDB - Embedded newform 6422.2.a.bl.1.8

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 12x^{7} + 14x^{6} + 35x^{5} - 35x^{4} - 28x^{3} + 10x^{2} + 4x - 1 $ x^9 - x^8 - 12x^7 + 14x^6 + 35x^5 - 35x^4 - 28x^3 + 10x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.490336$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.08053 q^{3} +1.00000 q^{4} -0.703813 q^{5} +2.08053 q^{6} -4.73726 q^{7} +1.00000 q^{8} +1.32860 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.08053 q^{3} +1.00000 q^{4} -0.703813 q^{5} +2.08053 q^{6} -4.73726 q^{7} +1.00000 q^{8} +1.32860 q^{9} -0.703813 q^{10} +2.29348 q^{11} +2.08053 q^{12} -4.73726 q^{14} -1.46430 q^{15} +1.00000 q^{16} -2.51438 q^{17} +1.32860 q^{18} -1.00000 q^{19} -0.703813 q^{20} -9.85600 q^{21} +2.29348 q^{22} +7.21013 q^{23} +2.08053 q^{24} -4.50465 q^{25} -3.47740 q^{27} -4.73726 q^{28} -5.26954 q^{29} -1.46430 q^{30} +2.87688 q^{31} +1.00000 q^{32} +4.77164 q^{33} -2.51438 q^{34} +3.33414 q^{35} +1.32860 q^{36} -7.70134 q^{37} -1.00000 q^{38} -0.703813 q^{40} -0.666842 q^{41} -9.85600 q^{42} +1.66233 q^{43} +2.29348 q^{44} -0.935083 q^{45} +7.21013 q^{46} -6.35151 q^{47} +2.08053 q^{48} +15.4416 q^{49} -4.50465 q^{50} -5.23124 q^{51} +3.11596 q^{53} -3.47740 q^{54} -1.61418 q^{55} -4.73726 q^{56} -2.08053 q^{57} -5.26954 q^{58} -13.7065 q^{59} -1.46430 q^{60} -0.958026 q^{61} +2.87688 q^{62} -6.29391 q^{63} +1.00000 q^{64} +4.77164 q^{66} -4.05864 q^{67} -2.51438 q^{68} +15.0009 q^{69} +3.33414 q^{70} +0.619808 q^{71} +1.32860 q^{72} -14.5204 q^{73} -7.70134 q^{74} -9.37205 q^{75} -1.00000 q^{76} -10.8648 q^{77} -4.34018 q^{79} -0.703813 q^{80} -11.2206 q^{81} -0.666842 q^{82} -5.33286 q^{83} -9.85600 q^{84} +1.76965 q^{85} +1.66233 q^{86} -10.9634 q^{87} +2.29348 q^{88} -8.16339 q^{89} -0.935083 q^{90} +7.21013 q^{92} +5.98544 q^{93} -6.35151 q^{94} +0.703813 q^{95} +2.08053 q^{96} +4.19470 q^{97} +15.4416 q^{98} +3.04711 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9	$ 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9} + q^{10} - 3 q^{11} + q^{12} - 13 q^{14} - 3 q^{15} + 9 q^{16} - 8 q^{17} - 2 q^{18} - 9 q^{19} + q^{20} - 24 q^{21} - 3 q^{22} - 10 q^{23} + q^{24} + 8 q^{25} + 10 q^{27} - 13 q^{28} - 20 q^{29} - 3 q^{30} - q^{31} + 9 q^{32} + 2 q^{33} - 8 q^{34} + 4 q^{35} - 2 q^{36} - 15 q^{37} - 9 q^{38} + q^{40} - 19 q^{41} - 24 q^{42} - 16 q^{43} - 3 q^{44} - 15 q^{45} - 10 q^{46} - 18 q^{47} + q^{48} + 18 q^{49} + 8 q^{50} - 11 q^{51} + 17 q^{53} + 10 q^{54} - 26 q^{55} - 13 q^{56} - q^{57} - 20 q^{58} - 24 q^{59} - 3 q^{60} + 6 q^{61} - q^{62} - q^{63} + 9 q^{64} + 2 q^{66} - 29 q^{67} - 8 q^{68} - 12 q^{69} + 4 q^{70} - 23 q^{71} - 2 q^{72} - 38 q^{73} - 15 q^{74} + 11 q^{75} - 9 q^{76} - 40 q^{77} - 20 q^{79} + q^{80} - 31 q^{81} - 19 q^{82} - 20 q^{83} - 24 q^{84} - 39 q^{85} - 16 q^{86} - 10 q^{87} - 3 q^{88} + 7 q^{89} - 15 q^{90} - 10 q^{92} - 11 q^{93} - 18 q^{94} - q^{95} + q^{96} - 28 q^{97} + 18 q^{98} - 25 q^{99}+O(q^{100}) $ 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9 + q^10 - 3 * q^11 + q^12 - 13 * q^14 - 3 * q^15 + 9 * q^16 - 8 * q^17 - 2 * q^18 - 9 * q^19 + q^20 - 24 * q^21 - 3 * q^22 - 10 * q^23 + q^24 + 8 * q^25 + 10 * q^27 - 13 * q^28 - 20 * q^29 - 3 * q^30 - q^31 + 9 * q^32 + 2 * q^33 - 8 * q^34 + 4 * q^35 - 2 * q^36 - 15 * q^37 - 9 * q^38 + q^40 - 19 * q^41 - 24 * q^42 - 16 * q^43 - 3 * q^44 - 15 * q^45 - 10 * q^46 - 18 * q^47 + q^48 + 18 * q^49 + 8 * q^50 - 11 * q^51 + 17 * q^53 + 10 * q^54 - 26 * q^55 - 13 * q^56 - q^57 - 20 * q^58 - 24 * q^59 - 3 * q^60 + 6 * q^61 - q^62 - q^63 + 9 * q^64 + 2 * q^66 - 29 * q^67 - 8 * q^68 - 12 * q^69 + 4 * q^70 - 23 * q^71 - 2 * q^72 - 38 * q^73 - 15 * q^74 + 11 * q^75 - 9 * q^76 - 40 * q^77 - 20 * q^79 + q^80 - 31 * q^81 - 19 * q^82 - 20 * q^83 - 24 * q^84 - 39 * q^85 - 16 * q^86 - 10 * q^87 - 3 * q^88 + 7 * q^89 - 15 * q^90 - 10 * q^92 - 11 * q^93 - 18 * q^94 - q^95 + q^96 - 28 * q^97 + 18 * q^98 - 25 * 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
$3$	2.08053	1.20119	0.600597	−	0.799552i	$-0.294930\pi$
$3$	2.08053	1.20119	0.600597	+	0.799552i	$0.294930\pi$
$4$	1.00000	0.500000
$5$	−0.703813	−0.314755	−0.157377	−	0.987539i	$-0.550304\pi$
$5$	−0.703813	−0.314755	−0.157377	+	0.987539i	$0.550304\pi$
$6$	2.08053	0.849372
$7$	−4.73726	−1.79052	−0.895258	−	0.445548i	$-0.853009\pi$
$7$	−4.73726	−1.79052	−0.895258	+	0.445548i	$0.853009\pi$
$8$	1.00000	0.353553
$9$	1.32860	0.442866
$10$	−0.703813	−0.222565
$11$	2.29348	0.691509	0.345755	−	0.938325i	$-0.387623\pi$
$11$	2.29348	0.691509	0.345755	+	0.938325i	$0.387623\pi$
$12$	2.08053	0.600597
$13$	0	0
$13$	0	0
$14$	−4.73726	−1.26609
$15$	−1.46430	−0.378081
$16$	1.00000	0.250000
$17$	−2.51438	−0.609827	−0.304914	−	0.952380i	$-0.598628\pi$
$17$	−2.51438	−0.609827	−0.304914	+	0.952380i	$0.598628\pi$
$18$	1.32860	0.313153
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.703813	−0.157377
$21$	−9.85600	−2.15076
$22$	2.29348	0.488971
$23$	7.21013	1.50342	0.751708	−	0.659496i	$-0.229230\pi$
$23$	7.21013	1.50342	0.751708	+	0.659496i	$0.229230\pi$
$24$	2.08053	0.424686
$25$	−4.50465	−0.900930
$26$	0	0
$27$	−3.47740	−0.669226
$28$	−4.73726	−0.895258
$29$	−5.26954	−0.978530	−0.489265	−	0.872135i	$-0.662735\pi$
$29$	−5.26954	−0.978530	−0.489265	+	0.872135i	$0.662735\pi$
$30$	−1.46430	−0.267344
$31$	2.87688	0.516704	0.258352	−	0.966051i	$-0.416821\pi$
$31$	2.87688	0.516704	0.258352	+	0.966051i	$0.416821\pi$
$32$	1.00000	0.176777
$33$	4.77164	0.830636
$34$	−2.51438	−0.431213
$35$	3.33414	0.563573
$36$	1.32860	0.221433
$37$	−7.70134	−1.26609	−0.633046	−	0.774114i	$-0.718196\pi$
$37$	−7.70134	−1.26609	−0.633046	+	0.774114i	$0.718196\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−0.703813	−0.111283
$41$	−0.666842	−0.104143	−0.0520716	−	0.998643i	$-0.516582\pi$
$41$	−0.666842	−0.104143	−0.0520716	+	0.998643i	$0.516582\pi$
$42$	−9.85600	−1.52081
$43$	1.66233	0.253503	0.126751	−	0.991935i	$-0.459545\pi$
$43$	1.66233	0.253503	0.126751	+	0.991935i	$0.459545\pi$
$44$	2.29348	0.345755
$45$	−0.935083	−0.139394
$46$	7.21013	1.06308
$47$	−6.35151	−0.926463	−0.463232	−	0.886237i	$-0.653310\pi$
$47$	−6.35151	−0.926463	−0.463232	+	0.886237i	$0.653310\pi$
$48$	2.08053	0.300298
$49$	15.4416	2.20595
$50$	−4.50465	−0.637053
$51$	−5.23124	−0.732521
$52$	0	0
$53$	3.11596	0.428010	0.214005	−	0.976833i	$-0.431349\pi$
$53$	3.11596	0.428010	0.214005	+	0.976833i	$0.431349\pi$
$54$	−3.47740	−0.473214
$55$	−1.61418	−0.217656
$56$	−4.73726	−0.633043
$57$	−2.08053	−0.275573
$58$	−5.26954	−0.691925
$59$	−13.7065	−1.78444	−0.892218	−	0.451605i	$-0.850852\pi$
$59$	−13.7065	−1.78444	−0.892218	+	0.451605i	$0.850852\pi$
$60$	−1.46430	−0.189041
$61$	−0.958026	−0.122663	−0.0613313	−	0.998117i	$-0.519535\pi$
$61$	−0.958026	−0.122663	−0.0613313	+	0.998117i	$0.519535\pi$
$62$	2.87688	0.365365
$63$	−6.29391	−0.792958
$64$	1.00000	0.125000
$65$	0	0
$66$	4.77164	0.587348
$67$	−4.05864	−0.495841	−0.247921	−	0.968780i	$-0.579747\pi$
$67$	−4.05864	−0.495841	−0.247921	+	0.968780i	$0.579747\pi$
$68$	−2.51438	−0.304914
$69$	15.0009	1.80589
$70$	3.33414	0.398506
$71$	0.619808	0.0735577	0.0367788	−	0.999323i	$-0.488290\pi$
$71$	0.619808	0.0735577	0.0367788	+	0.999323i	$0.488290\pi$
$72$	1.32860	0.156577
$73$	−14.5204	−1.69949	−0.849745	−	0.527194i	$-0.823244\pi$
$73$	−14.5204	−1.69949	−0.849745	+	0.527194i	$0.823244\pi$
$74$	−7.70134	−0.895263
$75$	−9.37205	−1.08219
$76$	−1.00000	−0.114708
$77$	−10.8648	−1.23816
$78$	0	0
$79$	−4.34018	−0.488308	−0.244154	−	0.969736i	$-0.578510\pi$
$79$	−4.34018	−0.488308	−0.244154	+	0.969736i	$0.578510\pi$
$80$	−0.703813	−0.0786886
$81$	−11.2206	−1.24674
$82$	−0.666842	−0.0736404
$83$	−5.33286	−0.585358	−0.292679	−	0.956211i	$-0.594547\pi$
$83$	−5.33286	−0.585358	−0.292679	+	0.956211i	$0.594547\pi$
$84$	−9.85600	−1.07538
$85$	1.76965	0.191946
$86$	1.66233	0.179254
$87$	−10.9634	−1.17540
$88$	2.29348	0.244485
$89$	−8.16339	−0.865318	−0.432659	−	0.901558i	$-0.642425\pi$
$89$	−8.16339	−0.865318	−0.432659	+	0.901558i	$0.642425\pi$
$90$	−0.935083	−0.0985664
$91$	0	0
$92$	7.21013	0.751708
$93$	5.98544	0.620661
$94$	−6.35151	−0.655108
$95$	0.703813	0.0722097
$96$	2.08053	0.212343
$97$	4.19470	0.425908	0.212954	−	0.977062i	$-0.431692\pi$
$97$	4.19470	0.425908	0.212954	+	0.977062i	$0.431692\pi$
$98$	15.4416	1.55984
$99$	3.04711	0.306246
$100$	−4.50465	−0.450465
$101$	1.27101	0.126471	0.0632353	−	0.997999i	$-0.479858\pi$
$101$	1.27101	0.126471	0.0632353	+	0.997999i	$0.479858\pi$
$102$	−5.23124	−0.517970
$103$	10.5670	1.04120	0.520599	−	0.853801i	$-0.325709\pi$
$103$	10.5670	1.04120	0.520599	+	0.853801i	$0.325709\pi$
$104$	0	0
$105$	6.93678	0.676960
$106$	3.11596	0.302648
$107$	4.52269	0.437225	0.218612	−	0.975812i	$-0.429847\pi$
$107$	4.52269	0.437225	0.218612	+	0.975812i	$0.429847\pi$
$108$	−3.47740	−0.334613
$109$	−20.1405	−1.92911	−0.964553	−	0.263887i	$-0.914995\pi$
$109$	−20.1405	−1.92911	−0.964553	+	0.263887i	$0.914995\pi$
$110$	−1.61418	−0.153906
$111$	−16.0229	−1.52082
$112$	−4.73726	−0.447629
$113$	−9.48823	−0.892577	−0.446289	−	0.894889i	$-0.647255\pi$
$113$	−9.48823	−0.892577	−0.446289	+	0.894889i	$0.647255\pi$
$114$	−2.08053	−0.194859
$115$	−5.07458	−0.473207
$116$	−5.26954	−0.489265
$117$	0	0
$118$	−13.7065	−1.26179
$119$	11.9113	1.09191
$120$	−1.46430	−0.133672
$121$	−5.73997	−0.521815
$122$	−0.958026	−0.0867356
$123$	−1.38738	−0.125096
$124$	2.87688	0.258352
$125$	6.68949	0.598326
$126$	−6.29391	−0.560706
$127$	19.8190	1.75865	0.879324	−	0.476224i	$-0.157995\pi$
$127$	19.8190	1.75865	0.879324	+	0.476224i	$0.157995\pi$
$128$	1.00000	0.0883883
$129$	3.45852	0.304506
$130$	0	0
$131$	10.6907	0.934050	0.467025	−	0.884244i	$-0.345326\pi$
$131$	10.6907	0.934050	0.467025	+	0.884244i	$0.345326\pi$
$132$	4.77164	0.415318
$133$	4.73726	0.410773
$134$	−4.05864	−0.350613
$135$	2.44744	0.210642
$136$	−2.51438	−0.215607
$137$	−1.19696	−0.102264	−0.0511318	−	0.998692i	$-0.516283\pi$
$137$	−1.19696	−0.102264	−0.0511318	+	0.998692i	$0.516283\pi$
$138$	15.0009	1.27696
$139$	13.7767	1.16853	0.584264	−	0.811564i	$-0.301383\pi$
$139$	13.7767	1.16853	0.584264	+	0.811564i	$0.301383\pi$
$140$	3.33414	0.281787
$141$	−13.2145	−1.11286
$142$	0.619808	0.0520131
$143$	0	0
$144$	1.32860	0.110716
$145$	3.70877	0.307997
$146$	−14.5204	−1.20172
$147$	32.1268	2.64977
$148$	−7.70134	−0.633046
$149$	11.1757	0.915548	0.457774	−	0.889069i	$-0.348647\pi$
$149$	11.1757	0.915548	0.457774	+	0.889069i	$0.348647\pi$
$150$	−9.37205	−0.765224
$151$	−22.7574	−1.85197	−0.925985	−	0.377561i	$-0.876763\pi$
$151$	−22.7574	−1.85197	−0.925985	+	0.377561i	$0.876763\pi$
$152$	−1.00000	−0.0811107
$153$	−3.34060	−0.270072
$154$	−10.8648	−0.875510
$155$	−2.02479	−0.162635
$156$	0	0
$157$	−23.2766	−1.85768	−0.928838	−	0.370486i	$-0.879191\pi$
$157$	−23.2766	−1.85768	−0.928838	+	0.370486i	$0.879191\pi$
$158$	−4.34018	−0.345286
$159$	6.48284	0.514122
$160$	−0.703813	−0.0556413
$161$	−34.1563	−2.69189
$162$	−11.2206	−0.881575
$163$	6.94439	0.543926	0.271963	−	0.962308i	$-0.412327\pi$
$163$	6.94439	0.543926	0.271963	+	0.962308i	$0.412327\pi$
$164$	−0.666842	−0.0520716
$165$	−3.35834	−0.261447
$166$	−5.33286	−0.413910
$167$	5.01311	0.387926	0.193963	−	0.981009i	$-0.437866\pi$
$167$	5.01311	0.387926	0.193963	+	0.981009i	$0.437866\pi$
$168$	−9.85600	−0.760407
$169$	0	0
$170$	1.76965	0.135726
$171$	−1.32860	−0.101600
$172$	1.66233	0.126751
$173$	−7.97095	−0.606020	−0.303010	−	0.952987i	$-0.597992\pi$
$173$	−7.97095	−0.606020	−0.303010	+	0.952987i	$0.597992\pi$
$174$	−10.9634	−0.831136
$175$	21.3397	1.61313
$176$	2.29348	0.172877
$177$	−28.5168	−2.14345
$178$	−8.16339	−0.611872
$179$	−2.29200	−0.171312	−0.0856560	−	0.996325i	$-0.527299\pi$
$179$	−2.29200	−0.171312	−0.0856560	+	0.996325i	$0.527299\pi$
$180$	−0.935083	−0.0696970
$181$	20.3731	1.51432	0.757162	−	0.653228i	$-0.226586\pi$
$181$	20.3731	1.51432	0.757162	+	0.653228i	$0.226586\pi$
$182$	0	0
$183$	−1.99320	−0.147342
$184$	7.21013	0.531538
$185$	5.42030	0.398509
$186$	5.98544	0.438874
$187$	−5.76668	−0.421701
$188$	−6.35151	−0.463232
$189$	16.4734	1.19826
$190$	0.703813	0.0510599
$191$	−6.46055	−0.467469	−0.233734	−	0.972300i	$-0.575095\pi$
$191$	−6.46055	−0.467469	−0.233734	+	0.972300i	$0.575095\pi$
$192$	2.08053	0.150149
$193$	−4.90598	−0.353141	−0.176570	−	0.984288i	$-0.556500\pi$
$193$	−4.90598	−0.353141	−0.176570	+	0.984288i	$0.556500\pi$
$194$	4.19470	0.301162
$195$	0	0
$196$	15.4416	1.10297
$197$	2.53763	0.180798	0.0903991	−	0.995906i	$-0.471186\pi$
$197$	2.53763	0.180798	0.0903991	+	0.995906i	$0.471186\pi$
$198$	3.04711	0.216548
$199$	3.83100	0.271572	0.135786	−	0.990738i	$-0.456644\pi$
$199$	3.83100	0.271572	0.135786	+	0.990738i	$0.456644\pi$
$200$	−4.50465	−0.318527
$201$	−8.44410	−0.595601
$202$	1.27101	0.0894282
$203$	24.9632	1.75207
$204$	−5.23124	−0.366260
$205$	0.469332	0.0327796
$206$	10.5670	0.736238
$207$	9.57936	0.665811
$208$	0	0
$209$	−2.29348	−0.158643
$210$	6.93678	0.478683
$211$	4.96934	0.342103	0.171052	−	0.985262i	$-0.445284\pi$
$211$	4.96934	0.342103	0.171052	+	0.985262i	$0.445284\pi$
$212$	3.11596	0.214005
$213$	1.28953	0.0883570
$214$	4.52269	0.309165
$215$	−1.16997	−0.0797912
$216$	−3.47740	−0.236607
$217$	−13.6286	−0.925167
$218$	−20.1405	−1.36408
$219$	−30.2102	−2.04142
$220$	−1.61418	−0.108828
$221$	0	0
$222$	−16.0229	−1.07538
$223$	5.56696	0.372791	0.186396	−	0.982475i	$-0.440319\pi$
$223$	5.56696	0.372791	0.186396	+	0.982475i	$0.440319\pi$
$224$	−4.73726	−0.316522
$225$	−5.98486	−0.398991
$226$	−9.48823	−0.631147
$227$	−21.7588	−1.44418	−0.722091	−	0.691798i	$-0.756819\pi$
$227$	−21.7588	−1.44418	−0.722091	+	0.691798i	$0.756819\pi$
$228$	−2.08053	−0.137786
$229$	23.5240	1.55451	0.777255	−	0.629185i	$-0.216611\pi$
$229$	23.5240	1.55451	0.777255	+	0.629185i	$0.216611\pi$
$230$	−5.07458	−0.334608
$231$	−22.6045	−1.48727
$232$	−5.26954	−0.345962
$233$	−24.1390	−1.58140	−0.790698	−	0.612206i	$-0.790282\pi$
$233$	−24.1390	−1.58140	−0.790698	+	0.612206i	$0.790282\pi$
$234$	0	0
$235$	4.47027	0.291609
$236$	−13.7065	−0.892218
$237$	−9.02987	−0.586553
$238$	11.9113	0.772094
$239$	14.9738	0.968573	0.484286	−	0.874910i	$-0.339079\pi$
$239$	14.9738	0.968573	0.484286	+	0.874910i	$0.339079\pi$
$240$	−1.46430	−0.0945203
$241$	23.0621	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$241$	23.0621	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$242$	−5.73997	−0.368979
$243$	−12.9126	−0.828344
$244$	−0.958026	−0.0613313
$245$	−10.8680	−0.694333
$246$	−1.38738	−0.0884564
$247$	0	0
$248$	2.87688	0.182682
$249$	−11.0952	−0.703128
$250$	6.68949	0.423081
$251$	−30.2351	−1.90842	−0.954211	−	0.299135i	$-0.903302\pi$
$251$	−30.2351	−1.90842	−0.954211	+	0.299135i	$0.903302\pi$
$252$	−6.29391	−0.396479
$253$	16.5363	1.03963
$254$	19.8190	1.24355
$255$	3.68182	0.230564
$256$	1.00000	0.0625000
$257$	−5.20501	−0.324680	−0.162340	−	0.986735i	$-0.551904\pi$
$257$	−5.20501	−0.324680	−0.162340	+	0.986735i	$0.551904\pi$
$258$	3.45852	0.215318
$259$	36.4833	2.26696
$260$	0	0
$261$	−7.00110	−0.433357
$262$	10.6907	0.660473
$263$	−18.5363	−1.14300	−0.571498	−	0.820603i	$-0.693638\pi$
$263$	−18.5363	−1.14300	−0.571498	+	0.820603i	$0.693638\pi$
$264$	4.77164	0.293674
$265$	−2.19305	−0.134718
$266$	4.73726	0.290460
$267$	−16.9842	−1.03941
$268$	−4.05864	−0.247921
$269$	9.09490	0.554526	0.277263	−	0.960794i	$-0.410573\pi$
$269$	9.09490	0.554526	0.277263	+	0.960794i	$0.410573\pi$
$270$	2.44744	0.148946
$271$	0.949705	0.0576904	0.0288452	−	0.999584i	$-0.490817\pi$
$271$	0.949705	0.0576904	0.0288452	+	0.999584i	$0.490817\pi$
$272$	−2.51438	−0.152457
$273$	0	0
$274$	−1.19696	−0.0723113
$275$	−10.3313	−0.623001
$276$	15.0009	0.902947
$277$	−7.35711	−0.442046	−0.221023	−	0.975269i	$-0.570940\pi$
$277$	−7.35711	−0.442046	−0.221023	+	0.975269i	$0.570940\pi$
$278$	13.7767	0.826274
$279$	3.82222	0.228830
$280$	3.33414	0.199253
$281$	4.65033	0.277415	0.138708	−	0.990333i	$-0.455705\pi$
$281$	4.65033	0.277415	0.138708	+	0.990333i	$0.455705\pi$
$282$	−13.2145	−0.786912
$283$	24.5320	1.45827	0.729137	−	0.684368i	$-0.239922\pi$
$283$	24.5320	1.45827	0.729137	+	0.684368i	$0.239922\pi$
$284$	0.619808	0.0367788
$285$	1.46430	0.0867378
$286$	0	0
$287$	3.15901	0.186470
$288$	1.32860	0.0782883
$289$	−10.6779	−0.628110
$290$	3.70877	0.217787
$291$	8.72720	0.511597
$292$	−14.5204	−0.849745
$293$	33.0376	1.93008	0.965038	−	0.262109i	$-0.0844180\pi$
$293$	33.0376	1.93008	0.965038	+	0.262109i	$0.0844180\pi$
$294$	32.1268	1.87367
$295$	9.64682	0.561659
$296$	−7.70134	−0.447631
$297$	−7.97534	−0.462776
$298$	11.1757	0.647390
$299$	0	0
$300$	−9.37205	−0.541095
$301$	−7.87489	−0.453901
$302$	−22.7574	−1.30954
$303$	2.64438	0.151916
$304$	−1.00000	−0.0573539
$305$	0.674271	0.0386086
$306$	−3.34060	−0.190969
$307$	−14.3368	−0.818245	−0.409123	−	0.912479i	$-0.634165\pi$
$307$	−14.3368	−0.818245	−0.409123	+	0.912479i	$0.634165\pi$
$308$	−10.8648	−0.619079
$309$	21.9850	1.25068
$310$	−2.02479	−0.115000
$311$	−15.0764	−0.854906	−0.427453	−	0.904038i	$-0.640589\pi$
$311$	−15.0764	−0.854906	−0.427453	+	0.904038i	$0.640589\pi$
$312$	0	0
$313$	32.3362	1.82775	0.913875	−	0.405996i	$-0.133075\pi$
$313$	32.3362	1.82775	0.913875	+	0.405996i	$0.133075\pi$
$314$	−23.2766	−1.31358
$315$	4.42973	0.249587
$316$	−4.34018	−0.244154
$317$	33.1238	1.86042	0.930210	−	0.367028i	$-0.119625\pi$
$317$	33.1238	1.86042	0.930210	+	0.367028i	$0.119625\pi$
$318$	6.48284	0.363539
$319$	−12.0856	−0.676662
$320$	−0.703813	−0.0393443
$321$	9.40958	0.525192
$322$	−34.1563	−1.90345
$323$	2.51438	0.139904
$324$	−11.2206	−0.623368
$325$	0	0
$326$	6.94439	0.384614
$327$	−41.9028	−2.31723
$328$	−0.666842	−0.0368202
$329$	30.0888	1.65885
$330$	−3.35834	−0.184871
$331$	8.26787	0.454443	0.227222	−	0.973843i	$-0.427036\pi$
$331$	8.26787	0.454443	0.227222	+	0.973843i	$0.427036\pi$
$332$	−5.33286	−0.292679
$333$	−10.2320	−0.560709
$334$	5.01311	0.274305
$335$	2.85652	0.156068
$336$	−9.85600	−0.537689
$337$	−15.0286	−0.818658	−0.409329	−	0.912387i	$-0.634237\pi$
$337$	−15.0286	−0.818658	−0.409329	+	0.912387i	$0.634237\pi$
$338$	0	0
$339$	−19.7405	−1.07216
$340$	1.76965	0.0959730
$341$	6.59807	0.357305
$342$	−1.32860	−0.0718423
$343$	−39.9903	−2.15927
$344$	1.66233	0.0896268
$345$	−10.5578	−0.568413
$346$	−7.97095	−0.428521
$347$	−30.4687	−1.63564	−0.817822	−	0.575471i	$-0.804819\pi$
$347$	−30.4687	−1.63564	−0.817822	+	0.575471i	$0.804819\pi$
$348$	−10.9634	−0.587702
$349$	−16.5345	−0.885073	−0.442536	−	0.896751i	$-0.645921\pi$
$349$	−16.5345	−0.885073	−0.442536	+	0.896751i	$0.645921\pi$
$350$	21.3397	1.14065
$351$	0	0
$352$	2.29348	0.122243
$353$	−20.4143	−1.08654	−0.543271	−	0.839557i	$-0.682814\pi$
$353$	−20.4143	−1.08654	−0.543271	+	0.839557i	$0.682814\pi$
$354$	−28.5168	−1.51565
$355$	−0.436229	−0.0231526
$356$	−8.16339	−0.432659
$357$	24.7818	1.31159
$358$	−2.29200	−0.121136
$359$	33.5566	1.77105	0.885525	−	0.464592i	$-0.153799\pi$
$359$	33.5566	1.77105	0.885525	+	0.464592i	$0.153799\pi$
$360$	−0.935083	−0.0492832
$361$	1.00000	0.0526316
$362$	20.3731	1.07079
$363$	−11.9422	−0.626801
$364$	0	0
$365$	10.2197	0.534922
$366$	−1.99320	−0.104186
$367$	−1.68216	−0.0878082	−0.0439041	−	0.999036i	$-0.513980\pi$
$367$	−1.68216	−0.0878082	−0.0439041	+	0.999036i	$0.513980\pi$
$368$	7.21013	0.375854
$369$	−0.885964	−0.0461215
$370$	5.42030	0.281788
$371$	−14.7611	−0.766358
$372$	5.98544	0.310331
$373$	−16.2710	−0.842483	−0.421241	−	0.906949i	$-0.638405\pi$
$373$	−16.2710	−0.842483	−0.421241	+	0.906949i	$0.638405\pi$
$374$	−5.76668	−0.298188
$375$	13.9177	0.718706
$376$	−6.35151	−0.327554
$377$	0	0
$378$	16.4734	0.847298
$379$	2.88749	0.148321	0.0741603	−	0.997246i	$-0.476372\pi$
$379$	2.88749	0.148321	0.0741603	+	0.997246i	$0.476372\pi$
$380$	0.703813	0.0361048
$381$	41.2339	2.11248
$382$	−6.46055	−0.330550
$383$	30.7828	1.57293	0.786463	−	0.617637i	$-0.211910\pi$
$383$	30.7828	1.57293	0.786463	+	0.617637i	$0.211910\pi$
$384$	2.08053	0.106172
$385$	7.64678	0.389716
$386$	−4.90598	−0.249708
$387$	2.20857	0.112268
$388$	4.19470	0.212954
$389$	−23.0738	−1.16989	−0.584943	−	0.811074i	$-0.698883\pi$
$389$	−23.0738	−1.16989	−0.584943	+	0.811074i	$0.698883\pi$
$390$	0	0
$391$	−18.1290	−0.916824
$392$	15.4416	0.779921
$393$	22.2423	1.12198
$394$	2.53763	0.127844
$395$	3.05467	0.153697
$396$	3.04711	0.153123
$397$	3.27143	0.164188	0.0820942	−	0.996625i	$-0.473839\pi$
$397$	3.27143	0.164188	0.0820942	+	0.996625i	$0.473839\pi$
$398$	3.83100	0.192030
$399$	9.85600	0.493417
$400$	−4.50465	−0.225232
$401$	−0.0300522	−0.00150073	−0.000750367	−	1.00000i	$-0.500239\pi$
$401$	−0.0300522	−0.00150073	−0.000750367		1.00000i	$0.500239\pi$
$402$	−8.44410	−0.421154
$403$	0	0
$404$	1.27101	0.0632353
$405$	7.89722	0.392416
$406$	24.9632	1.23890
$407$	−17.6628	−0.875515
$408$	−5.23124	−0.258985
$409$	26.0490	1.28804	0.644019	−	0.765009i	$-0.277266\pi$
$409$	26.0490	1.28804	0.644019	+	0.765009i	$0.277266\pi$
$410$	0.469332	0.0231787
$411$	−2.49032	−0.122838
$412$	10.5670	0.520599
$413$	64.9313	3.19506
$414$	9.57936	0.470800
$415$	3.75334	0.184244
$416$	0	0
$417$	28.6629	1.40363
$418$	−2.29348	−0.112178
$419$	2.94148	0.143701	0.0718504	−	0.997415i	$-0.477110\pi$
$419$	2.94148	0.143701	0.0718504	+	0.997415i	$0.477110\pi$
$420$	6.93678	0.338480
$421$	11.6767	0.569089	0.284544	−	0.958663i	$-0.408158\pi$
$421$	11.6767	0.569089	0.284544	+	0.958663i	$0.408158\pi$
$422$	4.96934	0.241903
$423$	−8.43860	−0.410299
$424$	3.11596	0.151324
$425$	11.3264	0.549412
$426$	1.28953	0.0624778
$427$	4.53842	0.219629
$428$	4.52269	0.218612
$429$	0	0
$430$	−1.16997	−0.0564209
$431$	−19.6368	−0.945873	−0.472936	−	0.881097i	$-0.656806\pi$
$431$	−19.6368	−0.945873	−0.472936	+	0.881097i	$0.656806\pi$
$432$	−3.47740	−0.167307
$433$	−21.3203	−1.02459	−0.512293	−	0.858811i	$-0.671204\pi$
$433$	−21.3203	−1.02459	−0.512293	+	0.858811i	$0.671204\pi$
$434$	−13.6286	−0.654192
$435$	7.71620	0.369964
$436$	−20.1405	−0.964553
$437$	−7.21013	−0.344907
$438$	−30.2102	−1.44350
$439$	24.5778	1.17304	0.586518	−	0.809936i	$-0.300498\pi$
$439$	24.5778	1.17304	0.586518	+	0.809936i	$0.300498\pi$
$440$	−1.61418	−0.0769529
$441$	20.5157	0.976939
$442$	0	0
$443$	19.3916	0.921323	0.460662	−	0.887576i	$-0.347612\pi$
$443$	19.3916	0.921323	0.460662	+	0.887576i	$0.347612\pi$
$444$	−16.0229	−0.760411
$445$	5.74550	0.272363
$446$	5.56696	0.263603
$447$	23.2513	1.09975
$448$	−4.73726	−0.223815
$449$	−33.6024	−1.58580	−0.792899	−	0.609354i	$-0.791429\pi$
$449$	−33.6024	−1.58580	−0.792899	+	0.609354i	$0.791429\pi$
$450$	−5.98486	−0.282129
$451$	−1.52939	−0.0720160
$452$	−9.48823	−0.446289
$453$	−47.3474	−2.22457
$454$	−21.7588	−1.02119
$455$	0	0
$456$	−2.08053	−0.0974297
$457$	30.5305	1.42816	0.714078	−	0.700066i	$-0.246846\pi$
$457$	30.5305	1.42816	0.714078	+	0.700066i	$0.246846\pi$
$458$	23.5240	1.09921
$459$	8.74352	0.408113
$460$	−5.07458	−0.236604
$461$	21.0261	0.979283	0.489642	−	0.871924i	$-0.337128\pi$
$461$	21.0261	0.979283	0.489642	+	0.871924i	$0.337128\pi$
$462$	−22.6045	−1.05166
$463$	1.34855	0.0626723	0.0313361	−	0.999509i	$-0.490024\pi$
$463$	1.34855	0.0626723	0.0313361	+	0.999509i	$0.490024\pi$
$464$	−5.26954	−0.244632
$465$	−4.21263	−0.195356
$466$	−24.1390	−1.11822
$467$	12.4188	0.574674	0.287337	−	0.957830i	$-0.407230\pi$
$467$	12.4188	0.574674	0.287337	+	0.957830i	$0.407230\pi$
$468$	0	0
$469$	19.2268	0.887812
$470$	4.47027	0.206198
$471$	−48.4276	−2.23143
$472$	−13.7065	−0.630893
$473$	3.81251	0.175300
$474$	−9.02987	−0.414755
$475$	4.50465	0.206687
$476$	11.9113	0.545953
$477$	4.13985	0.189551
$478$	14.9738	0.684884
$479$	−28.0760	−1.28283	−0.641413	−	0.767195i	$-0.721652\pi$
$479$	−28.0760	−1.28283	−0.641413	+	0.767195i	$0.721652\pi$
$480$	−1.46430	−0.0668359
$481$	0	0
$482$	23.0621	1.05045
$483$	−71.0631	−3.23348
$484$	−5.73997	−0.260908
$485$	−2.95228	−0.134056
$486$	−12.9126	−0.585728
$487$	21.7475	0.985472	0.492736	−	0.870179i	$-0.335997\pi$
$487$	21.7475	0.985472	0.492736	+	0.870179i	$0.335997\pi$
$488$	−0.958026	−0.0433678
$489$	14.4480	0.653361
$490$	−10.8680	−0.490967
$491$	−20.4269	−0.921855	−0.460927	−	0.887438i	$-0.652483\pi$
$491$	−20.4269	−0.921855	−0.460927	+	0.887438i	$0.652483\pi$
$492$	−1.38738	−0.0625481
$493$	13.2497	0.596734
$494$	0	0
$495$	−2.14459	−0.0963922
$496$	2.87688	0.129176
$497$	−2.93619	−0.131706
$498$	−11.0952	−0.497186
$499$	7.41060	0.331744	0.165872	−	0.986147i	$-0.446956\pi$
$499$	7.41060	0.331744	0.165872	+	0.986147i	$0.446956\pi$
$500$	6.68949	0.299163
$501$	10.4299	0.465974
$502$	−30.2351	−1.34946
$503$	−16.7114	−0.745126	−0.372563	−	0.928007i	$-0.621521\pi$
$503$	−16.7114	−0.745126	−0.372563	+	0.928007i	$0.621521\pi$
$504$	−6.29391	−0.280353
$505$	−0.894555	−0.0398072
$506$	16.5363	0.735127
$507$	0	0
$508$	19.8190	0.879324
$509$	23.3081	1.03311	0.516556	−	0.856254i	$-0.327214\pi$
$509$	23.3081	1.03311	0.516556	+	0.856254i	$0.327214\pi$
$510$	3.68182	0.163034
$511$	68.7871	3.04296
$512$	1.00000	0.0441942
$513$	3.47740	0.153531
$514$	−5.20501	−0.229583
$515$	−7.43719	−0.327722
$516$	3.45852	0.152253
$517$	−14.5670	−0.640658
$518$	36.4833	1.60298
$519$	−16.5838	−0.727947
$520$	0	0
$521$	−4.73908	−0.207623	−0.103811	−	0.994597i	$-0.533104\pi$
$521$	−4.73908	−0.207623	−0.103811	+	0.994597i	$0.533104\pi$
$522$	−7.00110	−0.306430
$523$	−13.9573	−0.610309	−0.305154	−	0.952303i	$-0.598708\pi$
$523$	−13.9573	−0.610309	−0.305154	+	0.952303i	$0.598708\pi$
$524$	10.6907	0.467025
$525$	44.3978	1.93768
$526$	−18.5363	−0.808221
$527$	−7.23359	−0.315100
$528$	4.77164	0.207659
$529$	28.9860	1.26026
$530$	−2.19305	−0.0952600
$531$	−18.2104	−0.790265
$532$	4.73726	0.205386
$533$	0	0
$534$	−16.9842	−0.734977
$535$	−3.18313	−0.137619
$536$	−4.05864	−0.175306
$537$	−4.76857	−0.205779
$538$	9.09490	0.392109
$539$	35.4150	1.52543
$540$	2.44744	0.105321
$541$	−37.8426	−1.62698	−0.813491	−	0.581577i	$-0.802436\pi$
$541$	−37.8426	−1.62698	−0.813491	+	0.581577i	$0.802436\pi$
$542$	0.949705	0.0407933
$543$	42.3869	1.81899
$544$	−2.51438	−0.107803
$545$	14.1751	0.607195
$546$	0	0
$547$	18.8788	0.807200	0.403600	−	0.914936i	$-0.367759\pi$
$547$	18.8788	0.807200	0.403600	+	0.914936i	$0.367759\pi$
$548$	−1.19696	−0.0511318
$549$	−1.27283	−0.0543231
$550$	−10.3313	−0.440528
$551$	5.26954	0.224490
$552$	15.0009	0.638480
$553$	20.5606	0.874324
$554$	−7.35711	−0.312574
$555$	11.2771	0.478686
$556$	13.7767	0.584264
$557$	36.6434	1.55263	0.776316	−	0.630344i	$-0.217086\pi$
$557$	36.6434	1.55263	0.776316	+	0.630344i	$0.217086\pi$
$558$	3.82222	0.161807
$559$	0	0
$560$	3.33414	0.140893
$561$	−11.9977	−0.506545
$562$	4.65033	0.196162
$563$	25.6874	1.08259	0.541297	−	0.840831i	$-0.317933\pi$
$563$	25.6874	1.08259	0.541297	+	0.840831i	$0.317933\pi$
$564$	−13.2145	−0.556431
$565$	6.67793	0.280943
$566$	24.5320	1.03116
$567$	53.1550	2.23230
$568$	0.619808	0.0260066
$569$	−22.5313	−0.944563	−0.472281	−	0.881448i	$-0.656569\pi$
$569$	−22.5313	−0.944563	−0.472281	+	0.881448i	$0.656569\pi$
$570$	1.46430	0.0613329
$571$	4.25412	0.178029	0.0890147	−	0.996030i	$-0.471628\pi$
$571$	4.25412	0.178029	0.0890147	+	0.996030i	$0.471628\pi$
$572$	0	0
$573$	−13.4413	−0.561520
$574$	3.15901	0.131854
$575$	−32.4791	−1.35447
$576$	1.32860	0.0553582
$577$	28.5023	1.18656	0.593282	−	0.804994i	$-0.297832\pi$
$577$	28.5023	1.18656	0.593282	+	0.804994i	$0.297832\pi$
$578$	−10.6779	−0.444141
$579$	−10.2070	−0.424190
$580$	3.70877	0.153998
$581$	25.2632	1.04809
$582$	8.72720	0.361754
$583$	7.14637	0.295973
$584$	−14.5204	−0.600860
$585$	0	0
$586$	33.0376	1.36477
$587$	5.30356	0.218901	0.109451	−	0.993992i	$-0.465091\pi$
$587$	5.30356	0.218901	0.109451	+	0.993992i	$0.465091\pi$
$588$	32.1268	1.32489
$589$	−2.87688	−0.118540
$590$	9.64682	0.397153
$591$	5.27960	0.217174
$592$	−7.70134	−0.316523
$593$	−15.5892	−0.640172	−0.320086	−	0.947389i	$-0.603712\pi$
$593$	−15.5892	−0.640172	−0.320086	+	0.947389i	$0.603712\pi$
$594$	−7.97534	−0.327232
$595$	−8.38332	−0.343682
$596$	11.1757	0.457774
$597$	7.97050	0.326211
$598$	0	0
$599$	22.9220	0.936566	0.468283	−	0.883578i	$-0.344873\pi$
$599$	22.9220	0.936566	0.468283	+	0.883578i	$0.344873\pi$
$600$	−9.37205	−0.382612
$601$	9.58163	0.390843	0.195421	−	0.980719i	$-0.437393\pi$
$601$	9.58163	0.390843	0.195421	+	0.980719i	$0.437393\pi$
$602$	−7.87489	−0.320957
$603$	−5.39229	−0.219591
$604$	−22.7574	−0.925985
$605$	4.03986	0.164244
$606$	2.64438	0.107421
$607$	−2.13022	−0.0864631	−0.0432315	−	0.999065i	$-0.513765\pi$
$607$	−2.13022	−0.0864631	−0.0432315	+	0.999065i	$0.513765\pi$
$608$	−1.00000	−0.0405554
$609$	51.9366	2.10458
$610$	0.674271	0.0273004
$611$	0	0
$612$	−3.34060	−0.135036
$613$	4.37172	0.176572	0.0882860	−	0.996095i	$-0.471861\pi$
$613$	4.37172	0.176572	0.0882860	+	0.996095i	$0.471861\pi$
$614$	−14.3368	−0.578587
$615$	0.976458	0.0393746
$616$	−10.8648	−0.437755
$617$	0.587927	0.0236690	0.0118345	−	0.999930i	$-0.496233\pi$
$617$	0.587927	0.0236690	0.0118345	+	0.999930i	$0.496233\pi$
$618$	21.9850	0.884365
$619$	28.8138	1.15812	0.579062	−	0.815284i	$-0.303419\pi$
$619$	28.8138	1.15812	0.579062	+	0.815284i	$0.303419\pi$
$620$	−2.02479	−0.0813174
$621$	−25.0725	−1.00613
$622$	−15.0764	−0.604510
$623$	38.6721	1.54937
$624$	0	0
$625$	17.8151	0.712604
$626$	32.3362	1.29241
$627$	−4.77164	−0.190561
$628$	−23.2766	−0.928838
$629$	19.3641	0.772098
$630$	4.42973	0.176485
$631$	−17.8631	−0.711117	−0.355559	−	0.934654i	$-0.615709\pi$
$631$	−17.8631	−0.711117	−0.355559	+	0.934654i	$0.615709\pi$
$632$	−4.34018	−0.172643
$633$	10.3388	0.410932
$634$	33.1238	1.31552
$635$	−13.9488	−0.553542
$636$	6.48284	0.257061
$637$	0	0
$638$	−12.0856	−0.478472
$639$	0.823475	0.0325762
$640$	−0.703813	−0.0278206
$641$	4.89400	0.193301	0.0966506	−	0.995318i	$-0.469187\pi$
$641$	4.89400	0.193301	0.0966506	+	0.995318i	$0.469187\pi$
$642$	9.40958	0.371367
$643$	−28.8503	−1.13775	−0.568873	−	0.822426i	$-0.692620\pi$
$643$	−28.8503	−1.13775	−0.568873	+	0.822426i	$0.692620\pi$
$644$	−34.1563	−1.34595
$645$	−2.43415	−0.0958447
$646$	2.51438	0.0989271
$647$	20.2493	0.796081	0.398040	−	0.917368i	$-0.369690\pi$
$647$	20.2493	0.796081	0.398040	+	0.917368i	$0.369690\pi$
$648$	−11.2206	−0.440788
$649$	−31.4356	−1.23395
$650$	0	0
$651$	−28.3546	−1.11130
$652$	6.94439	0.271963
$653$	−13.6602	−0.534563	−0.267282	−	0.963618i	$-0.586125\pi$
$653$	−13.6602	−0.534563	−0.267282	+	0.963618i	$0.586125\pi$
$654$	−41.9028	−1.63853
$655$	−7.52425	−0.293997
$656$	−0.666842	−0.0260358
$657$	−19.2918	−0.752645
$658$	30.0888	1.17298
$659$	5.57577	0.217201	0.108601	−	0.994085i	$-0.465363\pi$
$659$	5.57577	0.217201	0.108601	+	0.994085i	$0.465363\pi$
$660$	−3.35834	−0.130723
$661$	25.9913	1.01094	0.505472	−	0.862843i	$-0.331318\pi$
$661$	25.9913	1.01094	0.505472	+	0.862843i	$0.331318\pi$
$662$	8.26787	0.321340
$663$	0	0
$664$	−5.33286	−0.206955
$665$	−3.33414	−0.129293
$666$	−10.2320	−0.396481
$667$	−37.9941	−1.47114
$668$	5.01311	0.193963
$669$	11.5822	0.447794
$670$	2.85652	0.110357
$671$	−2.19721	−0.0848223
$672$	−9.85600	−0.380204
$673$	−42.4963	−1.63811	−0.819057	−	0.573712i	$-0.805503\pi$
$673$	−42.4963	−1.63811	−0.819057	+	0.573712i	$0.805503\pi$
$674$	−15.0286	−0.578879
$675$	15.6645	0.602926
$676$	0	0
$677$	49.3763	1.89769	0.948843	−	0.315750i	$-0.102256\pi$
$677$	49.3763	1.89769	0.948843	+	0.315750i	$0.102256\pi$
$678$	−19.7405	−0.758130
$679$	−19.8714	−0.762594
$680$	1.76965	0.0678632
$681$	−45.2698	−1.73474
$682$	6.59807	0.252653
$683$	30.6153	1.17146	0.585732	−	0.810505i	$-0.300807\pi$
$683$	30.6153	1.17146	0.585732	+	0.810505i	$0.300807\pi$
$684$	−1.32860	−0.0508002
$685$	0.842439	0.0321879
$686$	−39.9903	−1.52684
$687$	48.9424	1.86727
$688$	1.66233	0.0633757
$689$	0	0
$690$	−10.5578	−0.401929
$691$	−34.9322	−1.32888	−0.664441	−	0.747341i	$-0.731330\pi$
$691$	−34.9322	−1.32888	−0.664441	+	0.747341i	$0.731330\pi$
$692$	−7.97095	−0.303010
$693$	−14.4349	−0.548338
$694$	−30.4687	−1.15657
$695$	−9.69625	−0.367800
$696$	−10.9634	−0.415568
$697$	1.67670	0.0635094
$698$	−16.5345	−0.625841
$699$	−50.2218	−1.89956
$700$	21.3397	0.806565
$701$	−33.2795	−1.25695	−0.628474	−	0.777830i	$-0.716320\pi$
$701$	−33.2795	−1.25695	−0.628474	+	0.777830i	$0.716320\pi$
$702$	0	0
$703$	7.70134	0.290462
$704$	2.29348	0.0864386
$705$	9.30053	0.350278
$706$	−20.4143	−0.768301
$707$	−6.02112	−0.226448
$708$	−28.5168	−1.07173
$709$	17.7358	0.666083	0.333041	−	0.942912i	$-0.391925\pi$
$709$	17.7358	0.666083	0.333041	+	0.942912i	$0.391925\pi$
$710$	−0.436229	−0.0163714
$711$	−5.76635	−0.216255
$712$	−8.16339	−0.305936
$713$	20.7427	0.776821
$714$	24.7818	0.927434
$715$	0	0
$716$	−2.29200	−0.0856560
$717$	31.1533	1.16344
$718$	33.5566	1.25232
$719$	−3.04487	−0.113555	−0.0567773	−	0.998387i	$-0.518082\pi$
$719$	−3.04487	−0.113555	−0.0567773	+	0.998387i	$0.518082\pi$
$720$	−0.935083	−0.0348485
$721$	−50.0587	−1.86428
$722$	1.00000	0.0372161
$723$	47.9814	1.78445
$724$	20.3731	0.757162
$725$	23.7374	0.881586
$726$	−11.9422	−0.443215
$727$	−32.9233	−1.22106	−0.610529	−	0.791994i	$-0.709043\pi$
$727$	−32.9233	−1.22106	−0.610529	+	0.791994i	$0.709043\pi$
$728$	0	0
$729$	6.79681	0.251734
$730$	10.2197	0.378247
$731$	−4.17973	−0.154593
$732$	−1.99320	−0.0736708
$733$	18.0102	0.665221	0.332610	−	0.943064i	$-0.392071\pi$
$733$	18.0102	0.665221	0.332610	+	0.943064i	$0.392071\pi$
$734$	−1.68216	−0.0620897
$735$	−22.6112	−0.834028
$736$	7.21013	0.265769
$737$	−9.30838	−0.342879
$738$	−0.885964	−0.0326128
$739$	−1.63728	−0.0602283	−0.0301142	−	0.999546i	$-0.509587\pi$
$739$	−1.63728	−0.0602283	−0.0301142	+	0.999546i	$0.509587\pi$
$740$	5.42030	0.199254
$741$	0	0
$742$	−14.7611	−0.541897
$743$	28.8306	1.05769	0.528846	−	0.848718i	$-0.322625\pi$
$743$	28.8306	1.05769	0.528846	+	0.848718i	$0.322625\pi$
$744$	5.98544	0.219437
$745$	−7.86559	−0.288173
$746$	−16.2710	−0.595725
$747$	−7.08522	−0.259235
$748$	−5.76668	−0.210851
$749$	−21.4252	−0.782858
$750$	13.9177	0.508202
$751$	6.15875	0.224736	0.112368	−	0.993667i	$-0.464156\pi$
$751$	6.15875	0.224736	0.112368	+	0.993667i	$0.464156\pi$
$752$	−6.35151	−0.231616
$753$	−62.9049	−2.29238
$754$	0	0
$755$	16.0169	0.582916
$756$	16.4734	0.599130
$757$	13.5593	0.492821	0.246411	−	0.969165i	$-0.420749\pi$
$757$	13.5593	0.492821	0.246411	+	0.969165i	$0.420749\pi$
$758$	2.88749	0.104879
$759$	34.4042	1.24879
$760$	0.703813	0.0255300
$761$	34.8024	1.26158	0.630792	−	0.775952i	$-0.282730\pi$
$761$	34.8024	1.26158	0.630792	+	0.775952i	$0.282730\pi$
$762$	41.2339	1.49375
$763$	95.4107	3.45410
$764$	−6.46055	−0.233734
$765$	2.35116	0.0850063
$766$	30.7828	1.11223
$767$	0	0
$768$	2.08053	0.0750746
$769$	−23.5064	−0.847661	−0.423831	−	0.905741i	$-0.639315\pi$
$769$	−23.5064	−0.847661	−0.423831	+	0.905741i	$0.639315\pi$
$770$	7.64678	0.275571
$771$	−10.8292	−0.390003
$772$	−4.90598	−0.176570
$773$	−27.4800	−0.988389	−0.494194	−	0.869351i	$-0.664537\pi$
$773$	−27.4800	−0.988389	−0.494194	+	0.869351i	$0.664537\pi$
$774$	2.20857	0.0793853
$775$	−12.9594	−0.465514
$776$	4.19470	0.150581
$777$	75.9045	2.72306
$778$	−23.0738	−0.827234
$779$	0.666842	0.0238921
$780$	0	0
$781$	1.42152	0.0508658
$782$	−18.1290	−0.648293
$783$	18.3243	0.654858
$784$	15.4416	0.551487
$785$	16.3824	0.584712
$786$	22.2423	0.793356
$787$	−36.5676	−1.30349	−0.651747	−	0.758437i	$-0.725964\pi$
$787$	−36.5676	−1.30349	−0.651747	+	0.758437i	$0.725964\pi$
$788$	2.53763	0.0903991
$789$	−38.5653	−1.37296
$790$	3.05467	0.108680
$791$	44.9482	1.59817
$792$	3.04711	0.108274
$793$	0	0
$794$	3.27143	0.116099
$795$	−4.56270	−0.161822
$796$	3.83100	0.135786
$797$	−34.6569	−1.22761	−0.613804	−	0.789458i	$-0.710362\pi$
$797$	−34.6569	−1.22761	−0.613804	+	0.789458i	$0.710362\pi$
$798$	9.85600	0.348899
$799$	15.9701	0.564983
$800$	−4.50465	−0.159263
$801$	−10.8459	−0.383220
$802$	−0.0300522	−0.00106118
$803$	−33.3023	−1.17521
$804$	−8.44410	−0.297801
$805$	24.0396	0.847285
$806$	0	0
$807$	18.9222	0.666093
$808$	1.27101	0.0447141
$809$	35.1440	1.23560	0.617798	−	0.786337i	$-0.288025\pi$
$809$	35.1440	1.23560	0.617798	+	0.786337i	$0.288025\pi$
$810$	7.89722	0.277480
$811$	−17.9359	−0.629816	−0.314908	−	0.949122i	$-0.601974\pi$
$811$	−17.9359	−0.629816	−0.314908	+	0.949122i	$0.601974\pi$
$812$	24.9632	0.876037
$813$	1.97589	0.0692974
$814$	−17.6628	−0.619082
$815$	−4.88755	−0.171203
$816$	−5.23124	−0.183130
$817$	−1.66233	−0.0581576
$818$	26.0490	0.910781
$819$	0	0
$820$	0.469332	0.0163898
$821$	−47.7492	−1.66646	−0.833230	−	0.552927i	$-0.813511\pi$
$821$	−47.7492	−1.66646	−0.833230	+	0.552927i	$0.813511\pi$
$822$	−2.49032	−0.0868598
$823$	15.2333	0.531001	0.265501	−	0.964111i	$-0.414463\pi$
$823$	15.2333	0.531001	0.265501	+	0.964111i	$0.414463\pi$
$824$	10.5670	0.368119
$825$	−21.4946	−0.748345
$826$	64.9313	2.25925
$827$	−38.0549	−1.32330	−0.661650	−	0.749813i	$-0.730143\pi$
$827$	−38.0549	−1.32330	−0.661650	+	0.749813i	$0.730143\pi$
$828$	9.57936	0.332906
$829$	28.1637	0.978165	0.489082	−	0.872238i	$-0.337332\pi$
$829$	28.1637	0.978165	0.489082	+	0.872238i	$0.337332\pi$
$830$	3.75334	0.130280
$831$	−15.3067	−0.530983
$832$	0	0
$833$	−38.8262	−1.34525
$834$	28.6629	0.992515
$835$	−3.52829	−0.122101
$836$	−2.29348	−0.0793215
$837$	−10.0041	−0.345792
$838$	2.94148	0.101612
$839$	−17.8222	−0.615291	−0.307646	−	0.951501i	$-0.599541\pi$
$839$	−17.8222	−0.615291	−0.307646	+	0.951501i	$0.599541\pi$
$840$	6.93678	0.239342
$841$	−1.23191	−0.0424796
$842$	11.6767	0.402406
$843$	9.67514	0.333230
$844$	4.96934	0.171052
$845$	0	0
$846$	−8.43860	−0.290125
$847$	27.1917	0.934319
$848$	3.11596	0.107002
$849$	51.0394	1.75167
$850$	11.3264	0.388493
$851$	−55.5277	−1.90346
$852$	1.28953	0.0441785
$853$	−14.9434	−0.511652	−0.255826	−	0.966723i	$-0.582347\pi$
$853$	−14.9434	−0.511652	−0.255826	+	0.966723i	$0.582347\pi$
$854$	4.53842	0.155301
$855$	0.935083	0.0319792
$856$	4.52269	0.154582
$857$	−12.9926	−0.443820	−0.221910	−	0.975067i	$-0.571229\pi$
$857$	−12.9926	−0.443820	−0.221910	+	0.975067i	$0.571229\pi$
$858$	0	0
$859$	−29.1357	−0.994097	−0.497048	−	0.867723i	$-0.665583\pi$
$859$	−29.1357	−0.994097	−0.497048	+	0.867723i	$0.665583\pi$
$860$	−1.16997	−0.0398956
$861$	6.57240	0.223987
$862$	−19.6368	−0.668833
$863$	−51.3760	−1.74886	−0.874430	−	0.485152i	$-0.838764\pi$
$863$	−51.3760	−1.74886	−0.874430	+	0.485152i	$0.838764\pi$
$864$	−3.47740	−0.118304
$865$	5.61006	0.190748
$866$	−21.3203	−0.724492
$867$	−22.2156	−0.754482
$868$	−13.6286	−0.462583
$869$	−9.95410	−0.337670
$870$	7.71620	0.261604
$871$	0	0
$872$	−20.1405	−0.682042
$873$	5.57307	0.188620
$874$	−7.21013	−0.243886
$875$	−31.6899	−1.07131
$876$	−30.2102	−1.02071
$877$	−45.5294	−1.53742	−0.768710	−	0.639598i	$-0.779101\pi$
$877$	−45.5294	−1.53742	−0.768710	+	0.639598i	$0.779101\pi$
$878$	24.5778	0.829462
$879$	68.7356	2.31840
$880$	−1.61418	−0.0544139
$881$	1.34646	0.0453635	0.0226817	−	0.999743i	$-0.492780\pi$
$881$	1.34646	0.0453635	0.0226817	+	0.999743i	$0.492780\pi$
$882$	20.5157	0.690800
$883$	50.6337	1.70396	0.851980	−	0.523574i	$-0.175402\pi$
$883$	50.6337	1.70396	0.851980	+	0.523574i	$0.175402\pi$
$884$	0	0
$885$	20.0705	0.674662
$886$	19.3916	0.651474
$887$	−15.8716	−0.532916	−0.266458	−	0.963846i	$-0.585853\pi$
$887$	−15.8716	−0.532916	−0.266458	+	0.963846i	$0.585853\pi$
$888$	−16.0229	−0.537692
$889$	−93.8876	−3.14889
$890$	5.74550	0.192590
$891$	−25.7342	−0.862129
$892$	5.56696	0.186396
$893$	6.35151	0.212545
$894$	23.2513	0.777641
$895$	1.61314	0.0539212
$896$	−4.73726	−0.158261
$897$	0	0
$898$	−33.6024	−1.12133
$899$	−15.1599	−0.505610
$900$	−5.98486	−0.199495
$901$	−7.83471	−0.261012
$902$	−1.52939	−0.0509230
$903$	−16.3839	−0.545223
$904$	−9.48823	−0.315574
$905$	−14.3389	−0.476640
$906$	−47.3474	−1.57301
$907$	−6.76224	−0.224537	−0.112268	−	0.993678i	$-0.535812\pi$
$907$	−6.76224	−0.224537	−0.112268	+	0.993678i	$0.535812\pi$
$908$	−21.7588	−0.722091
$909$	1.68866	0.0560095
$910$	0	0
$911$	16.8476	0.558186	0.279093	−	0.960264i	$-0.409966\pi$
$911$	16.8476	0.558186	0.279093	+	0.960264i	$0.409966\pi$
$912$	−2.08053	−0.0688932
$913$	−12.2308	−0.404780
$914$	30.5305	1.00986
$915$	1.40284	0.0463764
$916$	23.5240	0.777255
$917$	−50.6446	−1.67243
$918$	8.74352	0.288579
$919$	52.3981	1.72845	0.864227	−	0.503101i	$-0.167808\pi$
$919$	52.3981	1.72845	0.864227	+	0.503101i	$0.167808\pi$
$920$	−5.07458	−0.167304
$921$	−29.8281	−0.982871
$922$	21.0261	0.692458
$923$	0	0
$924$	−22.6045	−0.743634
$925$	34.6918	1.14066
$926$	1.34855	0.0443160
$927$	14.0393	0.461111
$928$	−5.26954	−0.172981
$929$	0.591044	0.0193915	0.00969577	−	0.999953i	$-0.496914\pi$
$929$	0.591044	0.0193915	0.00969577	+	0.999953i	$0.496914\pi$
$930$	−4.21263	−0.138138
$931$	−15.4416	−0.506079
$932$	−24.1390	−0.790698
$933$	−31.3669	−1.02691
$934$	12.4188	0.406356
$935$	4.05866	0.132732
$936$	0	0
$937$	−43.4420	−1.41919	−0.709595	−	0.704610i	$-0.751122\pi$
$937$	−43.4420	−1.41919	−0.709595	+	0.704610i	$0.751122\pi$
$938$	19.2268	0.627778
$939$	67.2763	2.19548
$940$	4.47027	0.145804
$941$	−8.32836	−0.271497	−0.135748	−	0.990743i	$-0.543344\pi$
$941$	−8.32836	−0.271497	−0.135748	+	0.990743i	$0.543344\pi$
$942$	−48.4276	−1.57786
$943$	−4.80802	−0.156571
$944$	−13.7065	−0.446109
$945$	−11.5942	−0.377158
$946$	3.81251	0.123956
$947$	−43.9503	−1.42819	−0.714096	−	0.700048i	$-0.753162\pi$
$947$	−43.9503	−1.42819	−0.714096	+	0.700048i	$0.753162\pi$
$948$	−9.02987	−0.293276
$949$	0	0
$950$	4.50465	0.146150
$951$	68.9151	2.23472
$952$	11.9113	0.386047
$953$	−50.3080	−1.62964	−0.814818	−	0.579717i	$-0.803163\pi$
$953$	−50.3080	−1.62964	−0.814818	+	0.579717i	$0.803163\pi$
$954$	4.13985	0.134033
$955$	4.54701	0.147138
$956$	14.9738	0.484286
$957$	−25.1444	−0.812802
$958$	−28.0760	−0.907096
$959$	5.67033	0.183105
$960$	−1.46430	−0.0472601
$961$	−22.7235	−0.733017
$962$	0	0
$963$	6.00883	0.193632
$964$	23.0621	0.742781
$965$	3.45289	0.111153
$966$	−71.0631	−2.28642
$967$	8.88150	0.285610	0.142805	−	0.989751i	$-0.454388\pi$
$967$	8.88150	0.285610	0.142805	+	0.989751i	$0.454388\pi$
$968$	−5.73997	−0.184490
$969$	5.23124	0.168052
$970$	−2.95228	−0.0947922
$971$	50.0642	1.60664	0.803318	−	0.595550i	$-0.203066\pi$
$971$	50.0642	1.60664	0.803318	+	0.595550i	$0.203066\pi$
$972$	−12.9126	−0.414172
$973$	−65.2640	−2.09227
$974$	21.7475	0.696834
$975$	0	0
$976$	−0.958026	−0.0306657
$977$	−28.8544	−0.923134	−0.461567	−	0.887105i	$-0.652713\pi$
$977$	−28.8544	−0.923134	−0.461567	+	0.887105i	$0.652713\pi$
$978$	14.4480	0.461996
$979$	−18.7225	−0.598375
$980$	−10.8680	−0.347166
$981$	−26.7586	−0.854335
$982$	−20.4269	−0.651850
$983$	−1.56411	−0.0498873	−0.0249436	−	0.999689i	$-0.507941\pi$
$983$	−1.56411	−0.0498873	−0.0249436	+	0.999689i	$0.507941\pi$
$984$	−1.38738	−0.0442282
$985$	−1.78601	−0.0569071
$986$	13.2497	0.421955
$987$	62.6005	1.99260
$988$	0	0
$989$	11.9856	0.381120
$990$	−2.14459	−0.0681596
$991$	10.0726	0.319967	0.159983	−	0.987120i	$-0.448856\pi$
$991$	10.0726	0.319967	0.159983	+	0.987120i	$0.448856\pi$
$992$	2.87688	0.0913412
$993$	17.2015	0.545874
$994$	−2.93619	−0.0931304
$995$	−2.69630	−0.0854786
$996$	−11.0952	−0.351564
$997$	−42.2836	−1.33913	−0.669567	−	0.742752i	$-0.733520\pi$
$997$	−42.2836	−1.33913	−0.669567	+	0.742752i	$0.733520\pi$
$998$	7.41060	0.234578
$999$	26.7807	0.847302

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bl.1.8	yes	9
13.12	even	2		6422.2.a.bj.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bj.1.8	✓	9	13.12	even	2
6422.2.a.bl.1.8	yes	9	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.08053 q^{3} +1.00000 q^{4} -0.703813 q^{5} +2.08053 q^{6} -4.73726 q^{7} +1.00000 q^{8} +1.32860 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.08053 q^{3} +1.00000 q^{4} -0.703813 q^{5} +2.08053 q^{6} -4.73726 q^{7} +1.00000 q^{8} +1.32860 q^{9} -0.703813 q^{10} +2.29348 q^{11} +2.08053 q^{12} -4.73726 q^{14} -1.46430 q^{15} +1.00000 q^{16} -2.51438 q^{17} +1.32860 q^{18} -1.00000 q^{19} -0.703813 q^{20} -9.85600 q^{21} +2.29348 q^{22} +7.21013 q^{23} +2.08053 q^{24} -4.50465 q^{25} -3.47740 q^{27} -4.73726 q^{28} -5.26954 q^{29} -1.46430 q^{30} +2.87688 q^{31} +1.00000 q^{32} +4.77164 q^{33} -2.51438 q^{34} +3.33414 q^{35} +1.32860 q^{36} -7.70134 q^{37} -1.00000 q^{38} -0.703813 q^{40} -0.666842 q^{41} -9.85600 q^{42} +1.66233 q^{43} +2.29348 q^{44} -0.935083 q^{45} +7.21013 q^{46} -6.35151 q^{47} +2.08053 q^{48} +15.4416 q^{49} -4.50465 q^{50} -5.23124 q^{51} +3.11596 q^{53} -3.47740 q^{54} -1.61418 q^{55} -4.73726 q^{56} -2.08053 q^{57} -5.26954 q^{58} -13.7065 q^{59} -1.46430 q^{60} -0.958026 q^{61} +2.87688 q^{62} -6.29391 q^{63} +1.00000 q^{64} +4.77164 q^{66} -4.05864 q^{67} -2.51438 q^{68} +15.0009 q^{69} +3.33414 q^{70} +0.619808 q^{71} +1.32860 q^{72} -14.5204 q^{73} -7.70134 q^{74} -9.37205 q^{75} -1.00000 q^{76} -10.8648 q^{77} -4.34018 q^{79} -0.703813 q^{80} -11.2206 q^{81} -0.666842 q^{82} -5.33286 q^{83} -9.85600 q^{84} +1.76965 q^{85} +1.66233 q^{86} -10.9634 q^{87} +2.29348 q^{88} -8.16339 q^{89} -0.935083 q^{90} +7.21013 q^{92} +5.98544 q^{93} -6.35151 q^{94} +0.703813 q^{95} +2.08053 q^{96} +4.19470 q^{97} +15.4416 q^{98} +3.04711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9	\( 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9} + q^{10} - 3 q^{11} + q^{12} - 13 q^{14} - 3 q^{15} + 9 q^{16} - 8 q^{17} - 2 q^{18} - 9 q^{19} + q^{20} - 24 q^{21} - 3 q^{22} - 10 q^{23} + q^{24} + 8 q^{25} + 10 q^{27} - 13 q^{28} - 20 q^{29} - 3 q^{30} - q^{31} + 9 q^{32} + 2 q^{33} - 8 q^{34} + 4 q^{35} - 2 q^{36} - 15 q^{37} - 9 q^{38} + q^{40} - 19 q^{41} - 24 q^{42} - 16 q^{43} - 3 q^{44} - 15 q^{45} - 10 q^{46} - 18 q^{47} + q^{48} + 18 q^{49} + 8 q^{50} - 11 q^{51} + 17 q^{53} + 10 q^{54} - 26 q^{55} - 13 q^{56} - q^{57} - 20 q^{58} - 24 q^{59} - 3 q^{60} + 6 q^{61} - q^{62} - q^{63} + 9 q^{64} + 2 q^{66} - 29 q^{67} - 8 q^{68} - 12 q^{69} + 4 q^{70} - 23 q^{71} - 2 q^{72} - 38 q^{73} - 15 q^{74} + 11 q^{75} - 9 q^{76} - 40 q^{77} - 20 q^{79} + q^{80} - 31 q^{81} - 19 q^{82} - 20 q^{83} - 24 q^{84} - 39 q^{85} - 16 q^{86} - 10 q^{87} - 3 q^{88} + 7 q^{89} - 15 q^{90} - 10 q^{92} - 11 q^{93} - 18 q^{94} - q^{95} + q^{96} - 28 q^{97} + 18 q^{98} - 25 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9 + q^10 - 3 * q^11 + q^12 - 13 * q^14 - 3 * q^15 + 9 * q^16 - 8 * q^17 - 2 * q^18 - 9 * q^19 + q^20 - 24 * q^21 - 3 * q^22 - 10 * q^23 + q^24 + 8 * q^25 + 10 * q^27 - 13 * q^28 - 20 * q^29 - 3 * q^30 - q^31 + 9 * q^32 + 2 * q^33 - 8 * q^34 + 4 * q^35 - 2 * q^36 - 15 * q^37 - 9 * q^38 + q^40 - 19 * q^41 - 24 * q^42 - 16 * q^43 - 3 * q^44 - 15 * q^45 - 10 * q^46 - 18 * q^47 + q^48 + 18 * q^49 + 8 * q^50 - 11 * q^51 + 17 * q^53 + 10 * q^54 - 26 * q^55 - 13 * q^56 - q^57 - 20 * q^58 - 24 * q^59 - 3 * q^60 + 6 * q^61 - q^62 - q^63 + 9 * q^64 + 2 * q^66 - 29 * q^67 - 8 * q^68 - 12 * q^69 + 4 * q^70 - 23 * q^71 - 2 * q^72 - 38 * q^73 - 15 * q^74 + 11 * q^75 - 9 * q^76 - 40 * q^77 - 20 * q^79 + q^80 - 31 * q^81 - 19 * q^82 - 20 * q^83 - 24 * q^84 - 39 * q^85 - 16 * q^86 - 10 * q^87 - 3 * q^88 + 7 * q^89 - 15 * q^90 - 10 * q^92 - 11 * q^93 - 18 * q^94 - q^95 + q^96 - 28 * q^97 + 18 * q^98 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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