LMFDB - Embedded newform 6422.2.a.y.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1129.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 3 $ x^3 - 7*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			$2.83847$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.83847 q^{3} +1.00000 q^{4} +2.83847 q^{6} +3.62003 q^{7} +1.00000 q^{8} +5.05691 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.83847 q^{3} +1.00000 q^{4} +2.83847 q^{6} +3.62003 q^{7} +1.00000 q^{8} +5.05691 q^{9} +2.83847 q^{12} +3.62003 q^{14} +1.00000 q^{16} -1.16153 q^{17} +5.05691 q^{18} -1.00000 q^{19} +10.2753 q^{21} +7.73385 q^{23} +2.83847 q^{24} -5.00000 q^{25} +5.83847 q^{27} +3.62003 q^{28} +4.45850 q^{29} -0.436877 q^{31} +1.00000 q^{32} -1.16153 q^{34} +5.05691 q^{36} +10.6769 q^{37} -1.00000 q^{38} -11.7908 q^{41} +10.2753 q^{42} -1.67694 q^{43} +7.73385 q^{46} -10.3539 q^{47} +2.83847 q^{48} +6.10462 q^{49} -5.00000 q^{50} -3.29697 q^{51} -2.89538 q^{53} +5.83847 q^{54} +3.62003 q^{56} -2.83847 q^{57} +4.45850 q^{58} +6.78156 q^{59} -0.323061 q^{61} -0.436877 q^{62} +18.3062 q^{63} +1.00000 q^{64} -1.21844 q^{67} -1.16153 q^{68} +21.9523 q^{69} -14.1138 q^{71} +5.05691 q^{72} -14.1707 q^{73} +10.6769 q^{74} -14.1923 q^{75} -1.00000 q^{76} +10.1138 q^{79} +1.40159 q^{81} -11.7908 q^{82} -8.11382 q^{83} +10.2753 q^{84} -1.67694 q^{86} +12.6553 q^{87} -2.87375 q^{89} +7.73385 q^{92} -1.24006 q^{93} -10.3539 q^{94} +2.83847 q^{96} +1.67694 q^{97} +6.10462 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 4 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 4 * q^7 + 3 * q^8 + 5 * q^9	$ 3 q + 3 q^{2} + 3 q^{4} + 4 q^{7} + 3 q^{8} + 5 q^{9} + 4 q^{14} + 3 q^{16} - 12 q^{17} + 5 q^{18} - 3 q^{19} + 19 q^{21} - 4 q^{23} - 15 q^{25} + 9 q^{27} + 4 q^{28} - 2 q^{29} + 2 q^{31} + 3 q^{32} - 12 q^{34} + 5 q^{36} + 15 q^{37} - 3 q^{38} + 2 q^{41} + 19 q^{42} + 12 q^{43} - 4 q^{46} + 3 q^{47} + 37 q^{49} - 15 q^{50} + 14 q^{51} + 10 q^{53} + 9 q^{54} + 4 q^{56} - 2 q^{58} + 22 q^{59} - 18 q^{61} + 2 q^{62} - 8 q^{63} + 3 q^{64} - 2 q^{67} - 12 q^{68} + 37 q^{69} - 22 q^{71} + 5 q^{72} - 12 q^{73} + 15 q^{74} - 3 q^{76} + 10 q^{79} - q^{81} + 2 q^{82} - 4 q^{83} + 19 q^{84} + 12 q^{86} + 33 q^{87} - 2 q^{89} - 4 q^{92} + 10 q^{93} + 3 q^{94} - 12 q^{97} + 37 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 4 * q^7 + 3 * q^8 + 5 * q^9 + 4 * q^14 + 3 * q^16 - 12 * q^17 + 5 * q^18 - 3 * q^19 + 19 * q^21 - 4 * q^23 - 15 * q^25 + 9 * q^27 + 4 * q^28 - 2 * q^29 + 2 * q^31 + 3 * q^32 - 12 * q^34 + 5 * q^36 + 15 * q^37 - 3 * q^38 + 2 * q^41 + 19 * q^42 + 12 * q^43 - 4 * q^46 + 3 * q^47 + 37 * q^49 - 15 * q^50 + 14 * q^51 + 10 * q^53 + 9 * q^54 + 4 * q^56 - 2 * q^58 + 22 * q^59 - 18 * q^61 + 2 * q^62 - 8 * q^63 + 3 * q^64 - 2 * q^67 - 12 * q^68 + 37 * q^69 - 22 * q^71 + 5 * q^72 - 12 * q^73 + 15 * q^74 - 3 * q^76 + 10 * q^79 - q^81 + 2 * q^82 - 4 * q^83 + 19 * q^84 + 12 * q^86 + 33 * q^87 - 2 * q^89 - 4 * q^92 + 10 * q^93 + 3 * q^94 - 12 * q^97 + 37 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.83847	1.63879	0.819395	−	0.573229i	$-0.194309\pi$
$3$	2.83847	1.63879	0.819395	+	0.573229i	$0.194309\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	2.83847	1.15880
$7$	3.62003	1.36824	0.684122	−	0.729368i	$-0.260186\pi$
$7$	3.62003	1.36824	0.684122	+	0.729368i	$0.260186\pi$
$8$	1.00000	0.353553
$9$	5.05691	1.68564
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	2.83847	0.819395
$13$	0	0
$13$	0	0
$14$	3.62003	0.967494
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.16153	−0.281713	−0.140856	−	0.990030i	$-0.544986\pi$
$17$	−1.16153	−0.281713	−0.140856	+	0.990030i	$0.544986\pi$
$18$	5.05691	1.19192
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	10.2753	2.24226
$22$	0	0
$23$	7.73385	1.61262	0.806309	−	0.591494i	$-0.201462\pi$
$23$	7.73385	1.61262	0.806309	+	0.591494i	$0.201462\pi$
$24$	2.83847	0.579400
$25$	−5.00000	−1.00000
$26$	0	0
$27$	5.83847	1.12361
$28$	3.62003	0.684122
$29$	4.45850	0.827923	0.413961	−	0.910294i	$-0.364145\pi$
$29$	4.45850	0.827923	0.413961	+	0.910294i	$0.364145\pi$
$30$	0	0
$31$	−0.436877	−0.0784654	−0.0392327	−	0.999230i	$-0.512491\pi$
$31$	−0.436877	−0.0784654	−0.0392327	+	0.999230i	$0.512491\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.16153	−0.199201
$35$	0	0
$36$	5.05691	0.842818
$37$	10.6769	1.75528	0.877639	−	0.479322i	$-0.159118\pi$
$37$	10.6769	1.75528	0.877639	+	0.479322i	$0.159118\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	−11.7908	−1.84141	−0.920703	−	0.390264i	$-0.872384\pi$
$41$	−11.7908	−1.84141	−0.920703	+	0.390264i	$0.872384\pi$
$42$	10.2753	1.58552
$43$	−1.67694	−0.255731	−0.127865	−	0.991792i	$-0.540813\pi$
$43$	−1.67694	−0.255731	−0.127865	+	0.991792i	$0.540813\pi$
$44$	0	0
$45$	0	0
$46$	7.73385	1.14029
$47$	−10.3539	−1.51027	−0.755134	−	0.655570i	$-0.772428\pi$
$47$	−10.3539	−1.51027	−0.755134	+	0.655570i	$0.772428\pi$
$48$	2.83847	0.409698
$49$	6.10462	0.872089
$50$	−5.00000	−0.707107
$51$	−3.29697	−0.461668
$52$	0	0
$53$	−2.89538	−0.397711	−0.198855	−	0.980029i	$-0.563722\pi$
$53$	−2.89538	−0.397711	−0.198855	+	0.980029i	$0.563722\pi$
$54$	5.83847	0.794515
$55$	0	0
$56$	3.62003	0.483747
$57$	−2.83847	−0.375964
$58$	4.45850	0.585430
$59$	6.78156	0.882884	0.441442	−	0.897290i	$-0.354467\pi$
$59$	6.78156	0.882884	0.441442	+	0.897290i	$0.354467\pi$
$60$	0	0
$61$	−0.323061	−0.0413638	−0.0206819	−	0.999786i	$-0.506584\pi$
$61$	−0.323061	−0.0413638	−0.0206819	+	0.999786i	$0.506584\pi$
$62$	−0.436877	−0.0554834
$63$	18.3062	2.30636
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−1.21844	−0.148856	−0.0744280	−	0.997226i	$-0.523713\pi$
$67$	−1.21844	−0.148856	−0.0744280	+	0.997226i	$0.523713\pi$
$68$	−1.16153	−0.140856
$69$	21.9523	2.64274
$70$	0	0
$71$	−14.1138	−1.67500	−0.837501	−	0.546436i	$-0.815984\pi$
$71$	−14.1138	−1.67500	−0.837501	+	0.546436i	$0.815984\pi$
$72$	5.05691	0.595962
$73$	−14.1707	−1.65856	−0.829279	−	0.558835i	$-0.811248\pi$
$73$	−14.1707	−1.65856	−0.829279	+	0.558835i	$0.811248\pi$
$74$	10.6769	1.24117
$75$	−14.1923	−1.63879
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	10.1138	1.13789	0.568947	−	0.822374i	$-0.307351\pi$
$79$	10.1138	1.13789	0.568947	+	0.822374i	$0.307351\pi$
$80$	0	0
$81$	1.40159	0.155732
$82$	−11.7908	−1.30207
$83$	−8.11382	−0.890607	−0.445303	−	0.895380i	$-0.646904\pi$
$83$	−8.11382	−0.890607	−0.445303	+	0.895380i	$0.646904\pi$
$84$	10.2753	1.12113
$85$	0	0
$86$	−1.67694	−0.180829
$87$	12.6553	1.35679
$88$	0	0
$89$	−2.87375	−0.304617	−0.152309	−	0.988333i	$-0.548671\pi$
$89$	−2.87375	−0.304617	−0.152309	+	0.988333i	$0.548671\pi$
$90$	0	0
$91$	0	0
$92$	7.73385	0.806309
$93$	−1.24006	−0.128588
$94$	−10.3539	−1.06792
$95$	0	0
$96$	2.83847	0.289700
$97$	1.67694	0.170267	0.0851337	−	0.996370i	$-0.472868\pi$
$97$	1.67694	0.170267	0.0851337	+	0.996370i	$0.472868\pi$
$98$	6.10462	0.616660
$99$	0	0
$100$	−5.00000	−0.500000
$101$	−1.56312	−0.155537	−0.0777683	−	0.996971i	$-0.524779\pi$
$101$	−1.56312	−0.155537	−0.0777683	+	0.996971i	$0.524779\pi$
$102$	−3.29697	−0.326449
$103$	14.5507	1.43372	0.716861	−	0.697216i	$-0.245578\pi$
$103$	14.5507	1.43372	0.716861	+	0.697216i	$0.245578\pi$
$104$	0	0
$105$	0	0
$106$	−2.89538	−0.281224
$107$	1.16153	0.112289	0.0561447	−	0.998423i	$-0.482119\pi$
$107$	1.16153	0.112289	0.0561447	+	0.998423i	$0.482119\pi$
$108$	5.83847	0.561807
$109$	8.45850	0.810177	0.405089	−	0.914277i	$-0.367241\pi$
$109$	8.45850	0.810177	0.405089	+	0.914277i	$0.367241\pi$
$110$	0	0
$111$	30.3062	2.87653
$112$	3.62003	0.342061
$113$	−2.87375	−0.270340	−0.135170	−	0.990822i	$-0.543158\pi$
$113$	−2.87375	−0.270340	−0.135170	+	0.990822i	$0.543158\pi$
$114$	−2.83847	−0.265847
$115$	0	0
$116$	4.45850	0.413961
$117$	0	0
$118$	6.78156	0.624293
$119$	−4.20478	−0.385451
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−0.323061	−0.0292486
$123$	−33.4677	−3.01768
$124$	−0.436877	−0.0392327
$125$	0	0
$126$	18.3062	1.63084
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	−4.75994	−0.419089
$130$	0	0
$131$	2.11382	0.184685	0.0923425	−	0.995727i	$-0.470565\pi$
$131$	2.11382	0.184685	0.0923425	+	0.995727i	$0.470565\pi$
$132$	0	0
$133$	−3.62003	−0.313896
$134$	−1.21844	−0.105257
$135$	0	0
$136$	−1.16153	−0.0996004
$137$	16.2493	1.38827	0.694134	−	0.719846i	$-0.255788\pi$
$137$	16.2493	1.38827	0.694134	+	0.719846i	$0.255788\pi$
$138$	21.9523	1.86870
$139$	10.9170	0.925968	0.462984	−	0.886367i	$-0.346779\pi$
$139$	10.9170	0.925968	0.462984	+	0.886367i	$0.346779\pi$
$140$	0	0
$141$	−29.3892	−2.47501
$142$	−14.1138	−1.18441
$143$	0	0
$144$	5.05691	0.421409
$145$	0	0
$146$	−14.1707	−1.17278
$147$	17.3278	1.42917
$148$	10.6769	0.877639
$149$	2.11382	0.173171	0.0865853	−	0.996244i	$-0.472405\pi$
$149$	2.11382	0.173171	0.0865853	+	0.996244i	$0.472405\pi$
$150$	−14.1923	−1.15880
$151$	6.32306	0.514563	0.257282	−	0.966336i	$-0.417173\pi$
$151$	6.32306	0.514563	0.257282	+	0.966336i	$0.417173\pi$
$152$	−1.00000	−0.0811107
$153$	−5.87375	−0.474865
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.5631	1.24207	0.621036	−	0.783782i	$-0.286712\pi$
$157$	15.5631	1.24207	0.621036	+	0.783782i	$0.286712\pi$
$158$	10.1138	0.804612
$159$	−8.21844	−0.651765
$160$	0	0
$161$	27.9968	2.20645
$162$	1.40159	0.110119
$163$	−4.91700	−0.385129	−0.192565	−	0.981284i	$-0.561680\pi$
$163$	−4.91700	−0.385129	−0.192565	+	0.981284i	$0.561680\pi$
$164$	−11.7908	−0.920703
$165$	0	0
$166$	−8.11382	−0.629754
$167$	−3.12625	−0.241916	−0.120958	−	0.992658i	$-0.538597\pi$
$167$	−3.12625	−0.241916	−0.120958	+	0.992658i	$0.538597\pi$
$168$	10.2753	0.792760
$169$	0	0
$170$	0	0
$171$	−5.05691	−0.386711
$172$	−1.67694	−0.127865
$173$	22.2276	1.68994	0.844968	−	0.534817i	$-0.179620\pi$
$173$	22.2276	1.68994	0.844968	+	0.534817i	$0.179620\pi$
$174$	12.6553	0.959397
$175$	−18.1002	−1.36824
$176$	0	0
$177$	19.2493	1.44686
$178$	−2.87375	−0.215397
$179$	2.19682	0.164198	0.0820988	−	0.996624i	$-0.473838\pi$
$179$	2.19682	0.164198	0.0820988	+	0.996624i	$0.473838\pi$
$180$	0	0
$181$	−5.35388	−0.397951	−0.198975	−	0.980005i	$-0.563761\pi$
$181$	−5.35388	−0.397951	−0.198975	+	0.980005i	$0.563761\pi$
$182$	0	0
$183$	−0.917000	−0.0677866
$184$	7.73385	0.570147
$185$	0	0
$186$	−1.24006	−0.0909257
$187$	0	0
$188$	−10.3539	−0.755134
$189$	21.1354	1.53738
$190$	0	0
$191$	−18.9523	−1.37134	−0.685670	−	0.727913i	$-0.740490\pi$
$191$	−18.9523	−1.37134	−0.685670	+	0.727913i	$0.740490\pi$
$192$	2.83847	0.204849
$193$	−1.24006	−0.0892616	−0.0446308	−	0.999004i	$-0.514211\pi$
$193$	−1.24006	−0.0892616	−0.0446308	+	0.999004i	$0.514211\pi$
$194$	1.67694	0.120397
$195$	0	0
$196$	6.10462	0.436045
$197$	−23.0308	−1.64088	−0.820439	−	0.571734i	$-0.806271\pi$
$197$	−23.0308	−1.64088	−0.820439	+	0.571734i	$0.806271\pi$
$198$	0	0
$199$	−22.9523	−1.62704	−0.813522	−	0.581534i	$-0.802453\pi$
$199$	−22.9523	−1.62704	−0.813522	+	0.581534i	$0.802453\pi$
$200$	−5.00000	−0.353553
$201$	−3.45850	−0.243944
$202$	−1.56312	−0.109981
$203$	16.1399	1.13280
$204$	−3.29697	−0.230834
$205$	0	0
$206$	14.5507	1.01379
$207$	39.1093	2.71829
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1.90781	−0.131339	−0.0656695	−	0.997841i	$-0.520918\pi$
$211$	−1.90781	−0.131339	−0.0656695	+	0.997841i	$0.520918\pi$
$212$	−2.89538	−0.198855
$213$	−40.0616	−2.74498
$214$	1.16153	0.0794006
$215$	0	0
$216$	5.83847	0.397258
$217$	−1.58151	−0.107360
$218$	8.45850	0.572882
$219$	−40.2232	−2.71803
$220$	0	0
$221$	0	0
$222$	30.3062	2.03402
$223$	−25.7908	−1.72708	−0.863538	−	0.504283i	$-0.831757\pi$
$223$	−25.7908	−1.72708	−0.863538	+	0.504283i	$0.831757\pi$
$224$	3.62003	0.241873
$225$	−25.2845	−1.68564
$226$	−2.87375	−0.191159
$227$	−3.48459	−0.231281	−0.115640	−	0.993291i	$-0.536892\pi$
$227$	−3.48459	−0.231281	−0.115640	+	0.993291i	$0.536892\pi$
$228$	−2.83847	−0.187982
$229$	−9.90457	−0.654512	−0.327256	−	0.944936i	$-0.606124\pi$
$229$	−9.90457	−0.654512	−0.327256	+	0.944936i	$0.606124\pi$
$230$	0	0
$231$	0	0
$232$	4.45850	0.292715
$233$	−1.22763	−0.0804248	−0.0402124	−	0.999191i	$-0.512803\pi$
$233$	−1.22763	−0.0804248	−0.0402124	+	0.999191i	$0.512803\pi$
$234$	0	0
$235$	0	0
$236$	6.78156	0.441442
$237$	28.7078	1.86477
$238$	−4.20478	−0.272555
$239$	−12.3800	−0.800794	−0.400397	−	0.916342i	$-0.631128\pi$
$239$	−12.3800	−0.800794	−0.400397	+	0.916342i	$0.631128\pi$
$240$	0	0
$241$	−19.0308	−1.22588	−0.612941	−	0.790128i	$-0.710014\pi$
$241$	−19.0308	−1.22588	−0.612941	+	0.790128i	$0.710014\pi$
$242$	−11.0000	−0.707107
$243$	−13.5370	−0.868401
$244$	−0.323061	−0.0206819
$245$	0	0
$246$	−33.4677	−2.13382
$247$	0	0
$248$	−0.436877	−0.0277417
$249$	−23.0308	−1.45952
$250$	0	0
$251$	0.759938	0.0479669	0.0239834	−	0.999712i	$-0.492365\pi$
$251$	0.759938	0.0479669	0.0239834	+	0.999712i	$0.492365\pi$
$252$	18.3062	1.15318
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−26.1571	−1.63163	−0.815816	−	0.578311i	$-0.803712\pi$
$257$	−26.1571	−1.63163	−0.815816	+	0.578311i	$0.803712\pi$
$258$	−4.75994	−0.296341
$259$	38.6508	2.40165
$260$	0	0
$261$	22.5462	1.39558
$262$	2.11382	0.130592
$263$	25.2276	1.55560	0.777801	−	0.628510i	$-0.216335\pi$
$263$	25.2276	1.55560	0.777801	+	0.628510i	$0.216335\pi$
$264$	0	0
$265$	0	0
$266$	−3.62003	−0.221958
$267$	−8.15706	−0.499204
$268$	−1.21844	−0.0744280
$269$	28.9046	1.76234	0.881171	−	0.472797i	$-0.156756\pi$
$269$	28.9046	1.76234	0.881171	+	0.472797i	$0.156756\pi$
$270$	0	0
$271$	−5.27535	−0.320454	−0.160227	−	0.987080i	$-0.551223\pi$
$271$	−5.27535	−0.320454	−0.160227	+	0.987080i	$0.551223\pi$
$272$	−1.16153	−0.0704281
$273$	0	0
$274$	16.2493	0.981653
$275$	0	0
$276$	21.9523	1.32137
$277$	28.6645	1.72228	0.861142	−	0.508365i	$-0.169750\pi$
$277$	28.6645	1.72228	0.861142	+	0.508365i	$0.169750\pi$
$278$	10.9170	0.654758
$279$	−2.20925	−0.132264
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	−29.3892	−1.75010
$283$	−22.3414	−1.32806	−0.664031	−	0.747705i	$-0.731156\pi$
$283$	−22.3414	−1.32806	−0.664031	+	0.747705i	$0.731156\pi$
$284$	−14.1138	−0.837501
$285$	0	0
$286$	0	0
$287$	−42.6829	−2.51949
$288$	5.05691	0.297981
$289$	−15.6508	−0.920638
$290$	0	0
$291$	4.75994	0.279033
$292$	−14.1707	−0.829279
$293$	8.74628	0.510963	0.255481	−	0.966814i	$-0.417766\pi$
$293$	8.74628	0.510963	0.255481	+	0.966814i	$0.417766\pi$
$294$	17.3278	1.01058
$295$	0	0
$296$	10.6769	0.620584
$297$	0	0
$298$	2.11382	0.122450
$299$	0	0
$300$	−14.1923	−0.819395
$301$	−6.07057	−0.349902
$302$	6.32306	0.363851
$303$	−4.43688	−0.254892
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−5.87375	−0.335780
$307$	−20.2584	−1.15621	−0.578105	−	0.815962i	$-0.696208\pi$
$307$	−20.2584	−1.15621	−0.578105	+	0.815962i	$0.696208\pi$
$308$	0	0
$309$	41.3017	2.34957
$310$	0	0
$311$	−9.65532	−0.547503	−0.273751	−	0.961800i	$-0.588265\pi$
$311$	−9.65532	−0.547503	−0.273751	+	0.961800i	$0.588265\pi$
$312$	0	0
$313$	23.8693	1.34917	0.674586	−	0.738196i	$-0.264322\pi$
$313$	23.8693	1.34917	0.674586	+	0.738196i	$0.264322\pi$
$314$	15.5631	0.878278
$315$	0	0
$316$	10.1138	0.568947
$317$	−2.72465	−0.153032	−0.0765159	−	0.997068i	$-0.524380\pi$
$317$	−2.72465	−0.153032	−0.0765159	+	0.997068i	$0.524380\pi$
$318$	−8.21844	−0.460867
$319$	0	0
$320$	0	0
$321$	3.29697	0.184019
$322$	27.9968	1.56020
$323$	1.16153	0.0646293
$324$	1.40159	0.0778662
$325$	0	0
$326$	−4.91700	−0.272327
$327$	24.0092	1.32771
$328$	−11.7908	−0.651035
$329$	−37.4814	−2.06641
$330$	0	0
$331$	27.9399	1.53571	0.767857	−	0.640622i	$-0.221323\pi$
$331$	27.9399	1.53571	0.767857	+	0.640622i	$0.221323\pi$
$332$	−8.11382	−0.445303
$333$	53.9923	2.95876
$334$	−3.12625	−0.171061
$335$	0	0
$336$	10.2753	0.560566
$337$	−10.5507	−0.574733	−0.287366	−	0.957821i	$-0.592780\pi$
$337$	−10.5507	−0.574733	−0.287366	+	0.957821i	$0.592780\pi$
$338$	0	0
$339$	−8.15706	−0.443031
$340$	0	0
$341$	0	0
$342$	−5.05691	−0.273446
$343$	−3.24129	−0.175013
$344$	−1.67694	−0.0904145
$345$	0	0
$346$	22.2276	1.19496
$347$	21.4244	1.15012	0.575062	−	0.818110i	$-0.304978\pi$
$347$	21.4244	1.15012	0.575062	+	0.818110i	$0.304978\pi$
$348$	12.6553	0.678396
$349$	−28.1138	−1.50490	−0.752449	−	0.658650i	$-0.771128\pi$
$349$	−28.1138	−1.50490	−0.752449	+	0.658650i	$0.771128\pi$
$350$	−18.1002	−0.967494
$351$	0	0
$352$	0	0
$353$	−1.94309	−0.103420	−0.0517102	−	0.998662i	$-0.516467\pi$
$353$	−1.94309	−0.103420	−0.0517102	+	0.998662i	$0.516467\pi$
$354$	19.2493	1.02309
$355$	0	0
$356$	−2.87375	−0.152309
$357$	−11.9351	−0.631674
$358$	2.19682	0.116105
$359$	25.9262	1.36833	0.684166	−	0.729326i	$-0.260166\pi$
$359$	25.9262	1.36833	0.684166	+	0.729326i	$0.260166\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.35388	−0.281394
$363$	−31.2232	−1.63879
$364$	0	0
$365$	0	0
$366$	−0.917000	−0.0479324
$367$	−23.2276	−1.21247	−0.606236	−	0.795285i	$-0.707321\pi$
$367$	−23.2276	−1.21247	−0.606236	+	0.795285i	$0.707321\pi$
$368$	7.73385	0.403155
$369$	−59.6248	−3.10394
$370$	0	0
$371$	−10.4814	−0.544165
$372$	−1.24006	−0.0642942
$373$	3.37078	0.174532	0.0872661	−	0.996185i	$-0.472187\pi$
$373$	3.37078	0.174532	0.0872661	+	0.996185i	$0.472187\pi$
$374$	0	0
$375$	0	0
$376$	−10.3539	−0.533960
$377$	0	0
$378$	21.1354	1.08709
$379$	25.8909	1.32993	0.664963	−	0.746876i	$-0.268447\pi$
$379$	25.8909	1.32993	0.664963	+	0.746876i	$0.268447\pi$
$380$	0	0
$381$	22.7078	1.16335
$382$	−18.9523	−0.969683
$383$	15.6769	0.801054	0.400527	−	0.916285i	$-0.368827\pi$
$383$	15.6769	0.801054	0.400527	+	0.916285i	$0.368827\pi$
$384$	2.83847	0.144850
$385$	0	0
$386$	−1.24006	−0.0631175
$387$	−8.48012	−0.431069
$388$	1.67694	0.0851337
$389$	−11.0308	−0.559285	−0.279642	−	0.960104i	$-0.590216\pi$
$389$	−11.0308	−0.559285	−0.279642	+	0.960104i	$0.590216\pi$
$390$	0	0
$391$	−8.98310	−0.454295
$392$	6.10462	0.308330
$393$	6.00000	0.302660
$394$	−23.0308	−1.16028
$395$	0	0
$396$	0	0
$397$	−20.8032	−1.04408	−0.522041	−	0.852920i	$-0.674829\pi$
$397$	−20.8032	−1.04408	−0.522041	+	0.852920i	$0.674829\pi$
$398$	−22.9523	−1.15049
$399$	−10.2753	−0.514411
$400$	−5.00000	−0.250000
$401$	−13.5631	−0.677310	−0.338655	−	0.940911i	$-0.609972\pi$
$401$	−13.5631	−0.677310	−0.338655	+	0.940911i	$0.609972\pi$
$402$	−3.45850	−0.172494
$403$	0	0
$404$	−1.56312	−0.0777683
$405$	0	0
$406$	16.1399	0.801010
$407$	0	0
$408$	−3.29697	−0.163224
$409$	−24.2276	−1.19798	−0.598990	−	0.800757i	$-0.704431\pi$
$409$	−24.2276	−1.19798	−0.598990	+	0.800757i	$0.704431\pi$
$410$	0	0
$411$	46.1230	2.27508
$412$	14.5507	0.716861
$413$	24.5495	1.20800
$414$	39.1093	1.92212
$415$	0	0
$416$	0	0
$417$	30.9876	1.51747
$418$	0	0
$419$	11.0308	0.538891	0.269445	−	0.963016i	$-0.413160\pi$
$419$	11.0308	0.538891	0.269445	+	0.963016i	$0.413160\pi$
$420$	0	0
$421$	14.6076	0.711931	0.355965	−	0.934499i	$-0.384152\pi$
$421$	14.6076	0.711931	0.355965	+	0.934499i	$0.384152\pi$
$422$	−1.90781	−0.0928706
$423$	−52.3586	−2.54576
$424$	−2.89538	−0.140612
$425$	5.80765	0.281713
$426$	−40.0616	−1.94099
$427$	−1.16949	−0.0565957
$428$	1.16153	0.0561447
$429$	0	0
$430$	0	0
$431$	39.2584	1.89101	0.945506	−	0.325603i	$-0.105567\pi$
$431$	39.2584	1.89101	0.945506	+	0.325603i	$0.105567\pi$
$432$	5.83847	0.280903
$433$	9.14463	0.439463	0.219731	−	0.975560i	$-0.429482\pi$
$433$	9.14463	0.439463	0.219731	+	0.975560i	$0.429482\pi$
$434$	−1.58151	−0.0759148
$435$	0	0
$436$	8.45850	0.405089
$437$	−7.73385	−0.369960
$438$	−40.2232	−1.92194
$439$	13.0308	0.621927	0.310963	−	0.950422i	$-0.399348\pi$
$439$	13.0308	0.621927	0.310963	+	0.950422i	$0.399348\pi$
$440$	0	0
$441$	30.8705	1.47002
$442$	0	0
$443$	−21.7202	−1.03196	−0.515979	−	0.856601i	$-0.672572\pi$
$443$	−21.7202	−1.03196	−0.515979	+	0.856601i	$0.672572\pi$
$444$	30.3062	1.43827
$445$	0	0
$446$	−25.7908	−1.22123
$447$	6.00000	0.283790
$448$	3.62003	0.171030
$449$	14.1138	0.666072	0.333036	−	0.942914i	$-0.391927\pi$
$449$	14.1138	0.666072	0.333036	+	0.942914i	$0.391927\pi$
$450$	−25.2845	−1.19192
$451$	0	0
$452$	−2.87375	−0.135170
$453$	17.9478	0.843262
$454$	−3.48459	−0.163540
$455$	0	0
$456$	−2.83847	−0.132924
$457$	26.4200	1.23587	0.617937	−	0.786227i	$-0.287969\pi$
$457$	26.4200	1.23587	0.617937	+	0.786227i	$0.287969\pi$
$458$	−9.90457	−0.462810
$459$	−6.78156	−0.316536
$460$	0	0
$461$	17.8340	0.830612	0.415306	−	0.909682i	$-0.363674\pi$
$461$	17.8340	0.830612	0.415306	+	0.909682i	$0.363674\pi$
$462$	0	0
$463$	−12.6461	−0.587715	−0.293858	−	0.955849i	$-0.594939\pi$
$463$	−12.6461	−0.587715	−0.293858	+	0.955849i	$0.594939\pi$
$464$	4.45850	0.206981
$465$	0	0
$466$	−1.22763	−0.0568689
$467$	16.0184	0.741242	0.370621	−	0.928784i	$-0.379145\pi$
$467$	16.0184	0.741242	0.370621	+	0.928784i	$0.379145\pi$
$468$	0	0
$469$	−4.41078	−0.203671
$470$	0	0
$471$	44.1754	2.03550
$472$	6.78156	0.312147
$473$	0	0
$474$	28.7078	1.31859
$475$	5.00000	0.229416
$476$	−4.20478	−0.192726
$477$	−14.6417	−0.670395
$478$	−12.3800	−0.566247
$479$	−7.35388	−0.336007	−0.168004	−	0.985786i	$-0.553732\pi$
$479$	−7.35388	−0.336007	−0.168004	+	0.985786i	$0.553732\pi$
$480$	0	0
$481$	0	0
$482$	−19.0308	−0.866830
$483$	79.4679	3.61592
$484$	−11.0000	−0.500000
$485$	0	0
$486$	−13.5370	−0.614052
$487$	2.68937	0.121867	0.0609335	−	0.998142i	$-0.480592\pi$
$487$	2.68937	0.121867	0.0609335	+	0.998142i	$0.480592\pi$
$488$	−0.323061	−0.0146243
$489$	−13.9568	−0.631146
$490$	0	0
$491$	−38.1571	−1.72200	−0.861002	−	0.508601i	$-0.830163\pi$
$491$	−38.1571	−1.72200	−0.861002	+	0.508601i	$0.830163\pi$
$492$	−33.4677	−1.50884
$493$	−5.17868	−0.233236
$494$	0	0
$495$	0	0
$496$	−0.436877	−0.0196164
$497$	−51.0924	−2.29181
$498$	−23.0308	−1.03204
$499$	36.1138	1.61668	0.808338	−	0.588718i	$-0.200367\pi$
$499$	36.1138	1.61668	0.808338	+	0.588718i	$0.200367\pi$
$500$	0	0
$501$	−8.87375	−0.396450
$502$	0.759938	0.0339177
$503$	−23.2617	−1.03719	−0.518594	−	0.855021i	$-0.673544\pi$
$503$	−23.2617	−1.03719	−0.518594	+	0.855021i	$0.673544\pi$
$504$	18.3062	0.815421
$505$	0	0
$506$	0	0
$507$	0	0
$508$	8.00000	0.354943
$509$	−2.87375	−0.127377	−0.0636884	−	0.997970i	$-0.520286\pi$
$509$	−2.87375	−0.127377	−0.0636884	+	0.997970i	$0.520286\pi$
$510$	0	0
$511$	−51.2985	−2.26931
$512$	1.00000	0.0441942
$513$	−5.83847	−0.257775
$514$	−26.1571	−1.15374
$515$	0	0
$516$	−4.75994	−0.209545
$517$	0	0
$518$	38.6508	1.69822
$519$	63.0924	2.76945
$520$	0	0
$521$	13.1014	0.573982	0.286991	−	0.957933i	$-0.407345\pi$
$521$	13.1014	0.573982	0.286991	+	0.957933i	$0.407345\pi$
$522$	22.5462	0.986821
$523$	15.1832	0.663913	0.331956	−	0.943295i	$-0.392291\pi$
$523$	15.1832	0.663913	0.331956	+	0.943295i	$0.392291\pi$
$524$	2.11382	0.0923425
$525$	−51.3767	−2.24226
$526$	25.2276	1.09998
$527$	0.507446	0.0221047
$528$	0	0
$529$	36.8124	1.60054
$530$	0	0
$531$	34.2937	1.48822
$532$	−3.62003	−0.156948
$533$	0	0
$534$	−8.15706	−0.352991
$535$	0	0
$536$	−1.21844	−0.0526285
$537$	6.23559	0.269086
$538$	28.9046	1.24616
$539$	0	0
$540$	0	0
$541$	23.6953	1.01874	0.509371	−	0.860547i	$-0.329878\pi$
$541$	23.6953	1.01874	0.509371	+	0.860547i	$0.329878\pi$
$542$	−5.27535	−0.226596
$543$	−15.1968	−0.652158
$544$	−1.16153	−0.0498002
$545$	0	0
$546$	0	0
$547$	−9.74751	−0.416773	−0.208387	−	0.978047i	$-0.566821\pi$
$547$	−9.74751	−0.416773	−0.208387	+	0.978047i	$0.566821\pi$
$548$	16.2493	0.694134
$549$	−1.63369	−0.0697243
$550$	0	0
$551$	−4.45850	−0.189938
$552$	21.9523	0.934351
$553$	36.6123	1.55691
$554$	28.6645	1.21784
$555$	0	0
$556$	10.9170	0.462984
$557$	−42.1754	−1.78703	−0.893516	−	0.449032i	$-0.851769\pi$
$557$	−42.1754	−1.78703	−0.893516	+	0.449032i	$0.851769\pi$
$558$	−2.20925	−0.0935248
$559$	0	0
$560$	0	0
$561$	0	0
$562$	12.0000	0.506189
$563$	−29.8340	−1.25735	−0.628677	−	0.777667i	$-0.716403\pi$
$563$	−29.8340	−1.25735	−0.628677	+	0.777667i	$0.716403\pi$
$564$	−29.3892	−1.23751
$565$	0	0
$566$	−22.3414	−0.939081
$567$	5.07381	0.213080
$568$	−14.1138	−0.592203
$569$	−10.4369	−0.437537	−0.218768	−	0.975777i	$-0.570204\pi$
$569$	−10.4369	−0.437537	−0.218768	+	0.975777i	$0.570204\pi$
$570$	0	0
$571$	29.1262	1.21890	0.609448	−	0.792826i	$-0.291391\pi$
$571$	29.1262	1.21890	0.609448	+	0.792826i	$0.291391\pi$
$572$	0	0
$573$	−53.7955	−2.24734
$574$	−42.6829	−1.78155
$575$	−38.6692	−1.61262
$576$	5.05691	0.210704
$577$	39.6031	1.64870	0.824350	−	0.566081i	$-0.191541\pi$
$577$	39.6031	1.64870	0.824350	+	0.566081i	$0.191541\pi$
$578$	−15.6508	−0.650989
$579$	−3.51988	−0.146281
$580$	0	0
$581$	−29.3723	−1.21857
$582$	4.75994	0.197306
$583$	0	0
$584$	−14.1707	−0.586389
$585$	0	0
$586$	8.74628	0.361305
$587$	−3.12625	−0.129034	−0.0645170	−	0.997917i	$-0.520551\pi$
$587$	−3.12625	−0.129034	−0.0645170	+	0.997917i	$0.520551\pi$
$588$	17.3278	0.714586
$589$	0.436877	0.0180012
$590$	0	0
$591$	−65.3723	−2.68906
$592$	10.6769	0.438819
$593$	10.6064	0.435551	0.217776	−	0.975999i	$-0.430120\pi$
$593$	10.6064	0.435551	0.217776	+	0.975999i	$0.430120\pi$
$594$	0	0
$595$	0	0
$596$	2.11382	0.0865853
$597$	−65.1494	−2.66639
$598$	0	0
$599$	−28.2276	−1.15335	−0.576675	−	0.816974i	$-0.695650\pi$
$599$	−28.2276	−1.15335	−0.576675	+	0.816974i	$0.695650\pi$
$600$	−14.1923	−0.579400
$601$	15.5631	0.634833	0.317417	−	0.948286i	$-0.397185\pi$
$601$	15.5631	0.634833	0.317417	+	0.948286i	$0.397185\pi$
$602$	−6.07057	−0.247418
$603$	−6.16153	−0.250917
$604$	6.32306	0.257282
$605$	0	0
$606$	−4.43688	−0.180236
$607$	−36.1138	−1.46581	−0.732907	−	0.680329i	$-0.761837\pi$
$607$	−36.1138	−1.46581	−0.732907	+	0.680329i	$0.761837\pi$
$608$	−1.00000	−0.0405554
$609$	45.8126	1.85642
$610$	0	0
$611$	0	0
$612$	−5.87375	−0.237432
$613$	43.0492	1.73874	0.869370	−	0.494161i	$-0.164525\pi$
$613$	43.0492	1.73874	0.869370	+	0.494161i	$0.164525\pi$
$614$	−20.2584	−0.817564
$615$	0	0
$616$	0	0
$617$	−12.1262	−0.488184	−0.244092	−	0.969752i	$-0.578490\pi$
$617$	−12.1262	−0.488184	−0.244092	+	0.969752i	$0.578490\pi$
$618$	41.3017	1.66140
$619$	−27.9046	−1.12158	−0.560790	−	0.827958i	$-0.689502\pi$
$619$	−27.9046	−1.12158	−0.560790	+	0.827958i	$0.689502\pi$
$620$	0	0
$621$	45.1538	1.81196
$622$	−9.65532	−0.387143
$623$	−10.4031	−0.416790
$624$	0	0
$625$	25.0000	1.00000
$626$	23.8693	0.954008
$627$	0	0
$628$	15.5631	0.621036
$629$	−12.4016	−0.494484
$630$	0	0
$631$	16.1262	0.641976	0.320988	−	0.947083i	$-0.395985\pi$
$631$	16.1262	0.641976	0.320988	+	0.947083i	$0.395985\pi$
$632$	10.1138	0.402306
$633$	−5.41525	−0.215237
$634$	−2.72465	−0.108210
$635$	0	0
$636$	−8.21844	−0.325882
$637$	0	0
$638$	0	0
$639$	−71.3723	−2.82344
$640$	0	0
$641$	5.19682	0.205262	0.102631	−	0.994720i	$-0.467274\pi$
$641$	5.19682	0.205262	0.102631	+	0.994720i	$0.467274\pi$
$642$	3.29697	0.130121
$643$	−28.7078	−1.13212	−0.566062	−	0.824363i	$-0.691534\pi$
$643$	−28.7078	−1.13212	−0.566062	+	0.824363i	$0.691534\pi$
$644$	27.9968	1.10323
$645$	0	0
$646$	1.16153	0.0456998
$647$	26.3447	1.03572	0.517858	−	0.855466i	$-0.326729\pi$
$647$	26.3447	1.03572	0.517858	+	0.855466i	$0.326729\pi$
$648$	1.40159	0.0550597
$649$	0	0
$650$	0	0
$651$	−4.48906	−0.175940
$652$	−4.91700	−0.192565
$653$	−1.56312	−0.0611697	−0.0305849	−	0.999532i	$-0.509737\pi$
$653$	−1.56312	−0.0611697	−0.0305849	+	0.999532i	$0.509737\pi$
$654$	24.0092	0.938834
$655$	0	0
$656$	−11.7908	−0.460352
$657$	−71.6600	−2.79572
$658$	−37.4814	−1.46118
$659$	−38.8785	−1.51449	−0.757245	−	0.653131i	$-0.773455\pi$
$659$	−38.8785	−1.51449	−0.757245	+	0.653131i	$0.773455\pi$
$660$	0	0
$661$	−25.9615	−1.00978	−0.504892	−	0.863182i	$-0.668468\pi$
$661$	−25.9615	−1.00978	−0.504892	+	0.863182i	$0.668468\pi$
$662$	27.9399	1.08591
$663$	0	0
$664$	−8.11382	−0.314877
$665$	0	0
$666$	53.9923	2.09216
$667$	34.4814	1.33512
$668$	−3.12625	−0.120958
$669$	−73.2063	−2.83032
$670$	0	0
$671$	0	0
$672$	10.2753	0.396380
$673$	32.3414	1.24667	0.623336	−	0.781954i	$-0.285777\pi$
$673$	32.3414	1.24667	0.623336	+	0.781954i	$0.285777\pi$
$674$	−10.5507	−0.406397
$675$	−29.1923	−1.12361
$676$	0	0
$677$	−10.7952	−0.414894	−0.207447	−	0.978246i	$-0.566515\pi$
$677$	−10.7952	−0.414894	−0.207447	+	0.978246i	$0.566515\pi$
$678$	−8.15706	−0.313270
$679$	6.07057	0.232967
$680$	0	0
$681$	−9.89091	−0.379020
$682$	0	0
$683$	5.15706	0.197329	0.0986647	−	0.995121i	$-0.468543\pi$
$683$	5.15706	0.197329	0.0986647	+	0.995121i	$0.468543\pi$
$684$	−5.05691	−0.193356
$685$	0	0
$686$	−3.24129	−0.123753
$687$	−28.1138	−1.07261
$688$	−1.67694	−0.0639327
$689$	0	0
$690$	0	0
$691$	−19.5383	−0.743270	−0.371635	−	0.928379i	$-0.621203\pi$
$691$	−19.5383	−0.743270	−0.371635	+	0.928379i	$0.621203\pi$
$692$	22.2276	0.844968
$693$	0	0
$694$	21.4244	0.813261
$695$	0	0
$696$	12.6553	0.479698
$697$	13.6953	0.518747
$698$	−28.1138	−1.06412
$699$	−3.48459	−0.131799
$700$	−18.1002	−0.684122
$701$	44.7510	1.69022	0.845111	−	0.534591i	$-0.179534\pi$
$701$	44.7510	1.69022	0.845111	+	0.534591i	$0.179534\pi$
$702$	0	0
$703$	−10.6769	−0.402688
$704$	0	0
$705$	0	0
$706$	−1.94309	−0.0731292
$707$	−5.65855	−0.212812
$708$	19.2493	0.723431
$709$	9.44931	0.354876	0.177438	−	0.984132i	$-0.443219\pi$
$709$	9.44931	0.354876	0.177438	+	0.984132i	$0.443219\pi$
$710$	0	0
$711$	51.1446	1.91807
$712$	−2.87375	−0.107698
$713$	−3.37874	−0.126535
$714$	−11.9351	−0.446661
$715$	0	0
$716$	2.19682	0.0820988
$717$	−35.1402	−1.31233
$718$	25.9262	0.967557
$719$	−22.1002	−0.824197	−0.412098	−	0.911139i	$-0.635204\pi$
$719$	−22.1002	−0.824197	−0.412098	+	0.911139i	$0.635204\pi$
$720$	0	0
$721$	52.6740	1.96168
$722$	1.00000	0.0372161
$723$	−54.0184	−2.00897
$724$	−5.35388	−0.198975
$725$	−22.2925	−0.827923
$726$	−31.2232	−1.15880
$727$	36.6941	1.36091	0.680454	−	0.732791i	$-0.261782\pi$
$727$	36.6941	1.36091	0.680454	+	0.732791i	$0.261782\pi$
$728$	0	0
$729$	−42.6292	−1.57886
$730$	0	0
$731$	1.94782	0.0720426
$732$	−0.917000	−0.0338933
$733$	35.1879	1.29969	0.649847	−	0.760065i	$-0.274833\pi$
$733$	35.1879	1.29969	0.649847	+	0.760065i	$0.274833\pi$
$734$	−23.2276	−0.857347
$735$	0	0
$736$	7.73385	0.285073
$737$	0	0
$738$	−59.6248	−2.19482
$739$	−15.1014	−0.555513	−0.277757	−	0.960651i	$-0.589591\pi$
$739$	−15.1014	−0.555513	−0.277757	+	0.960651i	$0.589591\pi$
$740$	0	0
$741$	0	0
$742$	−10.4814	−0.384783
$743$	−42.7261	−1.56747	−0.783735	−	0.621095i	$-0.786688\pi$
$743$	−42.7261	−1.56747	−0.783735	+	0.621095i	$0.786688\pi$
$744$	−1.24006	−0.0454629
$745$	0	0
$746$	3.37078	0.123413
$747$	−41.0308	−1.50124
$748$	0	0
$749$	4.20478	0.153639
$750$	0	0
$751$	−25.6769	−0.936965	−0.468482	−	0.883473i	$-0.655199\pi$
$751$	−25.6769	−0.936965	−0.468482	+	0.883473i	$0.655199\pi$
$752$	−10.3539	−0.377567
$753$	2.15706	0.0786077
$754$	0	0
$755$	0	0
$756$	21.1354	0.768688
$757$	−29.9046	−1.08690	−0.543450	−	0.839442i	$-0.682882\pi$
$757$	−29.9046	−1.08690	−0.543450	+	0.839442i	$0.682882\pi$
$758$	25.8909	0.940400
$759$	0	0
$760$	0	0
$761$	6.90904	0.250452	0.125226	−	0.992128i	$-0.460034\pi$
$761$	6.90904	0.250452	0.125226	+	0.992128i	$0.460034\pi$
$762$	22.7078	0.822615
$763$	30.6200	1.10852
$764$	−18.9523	−0.685670
$765$	0	0
$766$	15.6769	0.566431
$767$	0	0
$768$	2.83847	0.102424
$769$	51.1414	1.84421	0.922103	−	0.386945i	$-0.126470\pi$
$769$	51.1414	1.84421	0.922103	+	0.386945i	$0.126470\pi$
$770$	0	0
$771$	−74.2460	−2.67390
$772$	−1.24006	−0.0446308
$773$	20.6861	0.744028	0.372014	−	0.928227i	$-0.378667\pi$
$773$	20.6861	0.744028	0.372014	+	0.928227i	$0.378667\pi$
$774$	−8.48012	−0.304812
$775$	2.18438	0.0784654
$776$	1.67694	0.0601986
$777$	109.709	3.93580
$778$	−11.0308	−0.395474
$779$	11.7908	0.422448
$780$	0	0
$781$	0	0
$782$	−8.98310	−0.321235
$783$	26.0308	0.930265
$784$	6.10462	0.218022
$785$	0	0
$786$	6.00000	0.214013
$787$	33.2448	1.18505	0.592524	−	0.805553i	$-0.298131\pi$
$787$	33.2448	1.18505	0.592524	+	0.805553i	$0.298131\pi$
$788$	−23.0308	−0.820439
$789$	71.6079	2.54931
$790$	0	0
$791$	−10.4031	−0.369891
$792$	0	0
$793$	0	0
$794$	−20.8032	−0.738277
$795$	0	0
$796$	−22.9523	−0.813522
$797$	−2.15234	−0.0762397	−0.0381199	−	0.999273i	$-0.512137\pi$
$797$	−2.15234	−0.0762397	−0.0381199	+	0.999273i	$0.512137\pi$
$798$	−10.2753	−0.363743
$799$	12.0263	0.425462
$800$	−5.00000	−0.176777
$801$	−14.5323	−0.513474
$802$	−13.5631	−0.478931
$803$	0	0
$804$	−3.45850	−0.121972
$805$	0	0
$806$	0	0
$807$	82.0447	2.88811
$808$	−1.56312	−0.0549905
$809$	−13.5462	−0.476260	−0.238130	−	0.971233i	$-0.576534\pi$
$809$	−13.5462	−0.476260	−0.238130	+	0.971233i	$0.576534\pi$
$810$	0	0
$811$	23.2937	0.817954	0.408977	−	0.912545i	$-0.365886\pi$
$811$	23.2937	0.817954	0.408977	+	0.912545i	$0.365886\pi$
$812$	16.1399	0.566400
$813$	−14.9739	−0.525158
$814$	0	0
$815$	0	0
$816$	−3.29697	−0.115417
$817$	1.67694	0.0586687
$818$	−24.2276	−0.847099
$819$	0	0
$820$	0	0
$821$	−9.08300	−0.316999	−0.158499	−	0.987359i	$-0.550666\pi$
$821$	−9.08300	−0.316999	−0.158499	+	0.987359i	$0.550666\pi$
$822$	46.1230	1.60872
$823$	−26.4632	−0.922450	−0.461225	−	0.887283i	$-0.652590\pi$
$823$	−26.4632	−0.922450	−0.461225	+	0.887283i	$0.652590\pi$
$824$	14.5507	0.506897
$825$	0	0
$826$	24.5495	0.854185
$827$	−50.8785	−1.76922	−0.884609	−	0.466333i	$-0.845575\pi$
$827$	−50.8785	−1.76922	−0.884609	+	0.466333i	$0.845575\pi$
$828$	39.1093	1.35914
$829$	−19.5952	−0.680568	−0.340284	−	0.940323i	$-0.610523\pi$
$829$	−19.5952	−0.680568	−0.340284	+	0.940323i	$0.610523\pi$
$830$	0	0
$831$	81.3633	2.82246
$832$	0	0
$833$	−7.09071	−0.245678
$834$	30.9876	1.07301
$835$	0	0
$836$	0	0
$837$	−2.55069	−0.0881648
$838$	11.0308	0.381053
$839$	−52.6123	−1.81638	−0.908190	−	0.418559i	$-0.862535\pi$
$839$	−52.6123	−1.81638	−0.908190	+	0.418559i	$0.862535\pi$
$840$	0	0
$841$	−9.12178	−0.314544
$842$	14.6076	0.503411
$843$	34.0616	1.17314
$844$	−1.90781	−0.0656695
$845$	0	0
$846$	−52.3586	−1.80013
$847$	−39.8203	−1.36824
$848$	−2.89538	−0.0994277
$849$	−63.4155	−2.17641
$850$	5.80765	0.199201
$851$	82.5738	2.83059
$852$	−40.0616	−1.37249
$853$	−27.9478	−0.956915	−0.478457	−	0.878111i	$-0.658804\pi$
$853$	−27.9478	−0.956915	−0.478457	+	0.878111i	$0.658804\pi$
$854$	−1.16949	−0.0400192
$855$	0	0
$856$	1.16153	0.0397003
$857$	25.3106	0.864595	0.432297	−	0.901731i	$-0.357703\pi$
$857$	25.3106	0.864595	0.432297	+	0.901731i	$0.357703\pi$
$858$	0	0
$859$	−2.68937	−0.0917601	−0.0458800	−	0.998947i	$-0.514609\pi$
$859$	−2.68937	−0.0917601	−0.0458800	+	0.998947i	$0.514609\pi$
$860$	0	0
$861$	−121.154	−4.12892
$862$	39.2584	1.33715
$863$	13.3106	0.453099	0.226550	−	0.974000i	$-0.427255\pi$
$863$	13.3106	0.453099	0.226550	+	0.974000i	$0.427255\pi$
$864$	5.83847	0.198629
$865$	0	0
$866$	9.14463	0.310747
$867$	−44.4244	−1.50873
$868$	−1.58151	−0.0536799
$869$	0	0
$870$	0	0
$871$	0	0
$872$	8.45850	0.286441
$873$	8.48012	0.287009
$874$	−7.73385	−0.261601
$875$	0	0
$876$	−40.2232	−1.35901
$877$	−16.7294	−0.564911	−0.282455	−	0.959280i	$-0.591149\pi$
$877$	−16.7294	−0.564911	−0.282455	+	0.959280i	$0.591149\pi$
$878$	13.0308	0.439769
$879$	24.8260	0.837361
$880$	0	0
$881$	−43.2276	−1.45638	−0.728188	−	0.685378i	$-0.759637\pi$
$881$	−43.2276	−1.45638	−0.728188	+	0.685378i	$0.759637\pi$
$882$	30.8705	1.03946
$883$	−22.0432	−0.741814	−0.370907	−	0.928670i	$-0.620953\pi$
$883$	−22.0432	−0.741814	−0.370907	+	0.928670i	$0.620953\pi$
$884$	0	0
$885$	0	0
$886$	−21.7202	−0.729704
$887$	21.4677	0.720815	0.360407	−	0.932795i	$-0.382638\pi$
$887$	21.4677	0.720815	0.360407	+	0.932795i	$0.382638\pi$
$888$	30.3062	1.01701
$889$	28.9602	0.971295
$890$	0	0
$891$	0	0
$892$	−25.7908	−0.863538
$893$	10.3539	0.346479
$894$	6.00000	0.200670
$895$	0	0
$896$	3.62003	0.120937
$897$	0	0
$898$	14.1138	0.470984
$899$	−1.94782	−0.0649633
$900$	−25.2845	−0.842818
$901$	3.36307	0.112040
$902$	0	0
$903$	−17.2311	−0.573416
$904$	−2.87375	−0.0955796
$905$	0	0
$906$	17.9478	0.596276
$907$	2.36307	0.0784644	0.0392322	−	0.999230i	$-0.487509\pi$
$907$	2.36307	0.0784644	0.0392322	+	0.999230i	$0.487509\pi$
$908$	−3.48459	−0.115640
$909$	−7.90457	−0.262178
$910$	0	0
$911$	5.53826	0.183491	0.0917454	−	0.995782i	$-0.470755\pi$
$911$	5.53826	0.183491	0.0917454	+	0.995782i	$0.470755\pi$
$912$	−2.83847	−0.0939911
$913$	0	0
$914$	26.4200	0.873895
$915$	0	0
$916$	−9.90457	−0.327256
$917$	7.65208	0.252694
$918$	−6.78156	−0.223825
$919$	44.8079	1.47808	0.739038	−	0.673663i	$-0.235280\pi$
$919$	44.8079	1.47808	0.739038	+	0.673663i	$0.235280\pi$
$920$	0	0
$921$	−57.5030	−1.89479
$922$	17.8340	0.587332
$923$	0	0
$924$	0	0
$925$	−53.3847	−1.75528
$926$	−12.6461	−0.415577
$927$	73.5815	2.41673
$928$	4.45850	0.146357
$929$	−54.9401	−1.80253	−0.901263	−	0.433272i	$-0.857359\pi$
$929$	−54.9401	−1.80253	−0.901263	+	0.433272i	$0.857359\pi$
$930$	0	0
$931$	−6.10462	−0.200071
$932$	−1.22763	−0.0402124
$933$	−27.4063	−0.897242
$934$	16.0184	0.524138
$935$	0	0
$936$	0	0
$937$	19.9182	0.650700	0.325350	−	0.945594i	$-0.394518\pi$
$937$	19.9182	0.650700	0.325350	+	0.945594i	$0.394518\pi$
$938$	−4.41078	−0.144017
$939$	67.7522	2.21101
$940$	0	0
$941$	−53.2015	−1.73432	−0.867160	−	0.498029i	$-0.834057\pi$
$941$	−53.2015	−1.73432	−0.867160	+	0.498029i	$0.834057\pi$
$942$	44.1754	1.43931
$943$	−91.1879	−2.96949
$944$	6.78156	0.220721
$945$	0	0
$946$	0	0
$947$	−24.0890	−0.782786	−0.391393	−	0.920224i	$-0.628007\pi$
$947$	−24.0890	−0.782786	−0.391393	+	0.920224i	$0.628007\pi$
$948$	28.7078	0.932385
$949$	0	0
$950$	5.00000	0.162221
$951$	−7.73385	−0.250787
$952$	−4.20478	−0.136278
$953$	−35.3723	−1.14582	−0.572910	−	0.819618i	$-0.694186\pi$
$953$	−35.3723	−1.14582	−0.572910	+	0.819618i	$0.694186\pi$
$954$	−14.6417	−0.474041
$955$	0	0
$956$	−12.3800	−0.400397
$957$	0	0
$958$	−7.35388	−0.237593
$959$	58.8228	1.89949
$960$	0	0
$961$	−30.8091	−0.993843
$962$	0	0
$963$	5.87375	0.189279
$964$	−19.0308	−0.612941
$965$	0	0
$966$	79.4679	2.55684
$967$	21.9602	0.706194	0.353097	−	0.935587i	$-0.385129\pi$
$967$	21.9602	0.706194	0.353097	+	0.935587i	$0.385129\pi$
$968$	−11.0000	−0.353553
$969$	3.29697	0.105914
$970$	0	0
$971$	−46.9046	−1.50524	−0.752620	−	0.658456i	$-0.771210\pi$
$971$	−46.9046	−1.50524	−0.752620	+	0.658456i	$0.771210\pi$
$972$	−13.5370	−0.434200
$973$	39.5199	1.26695
$974$	2.68937	0.0861730
$975$	0	0
$976$	−0.323061	−0.0103409
$977$	24.3847	0.780135	0.390068	−	0.920786i	$-0.372452\pi$
$977$	24.3847	0.780135	0.390068	+	0.920786i	$0.372452\pi$
$978$	−13.9568	−0.446288
$979$	0	0
$980$	0	0
$981$	42.7739	1.36566
$982$	−38.1571	−1.21764
$983$	16.6461	0.530929	0.265464	−	0.964121i	$-0.414475\pi$
$983$	16.6461	0.530929	0.265464	+	0.964121i	$0.414475\pi$
$984$	−33.4677	−1.06691
$985$	0	0
$986$	−5.17868	−0.164923
$987$	−106.390	−3.38642
$988$	0	0
$989$	−12.9692	−0.412396
$990$	0	0
$991$	−55.0492	−1.74870	−0.874348	−	0.485300i	$-0.838710\pi$
$991$	−55.0492	−1.74870	−0.874348	+	0.485300i	$0.838710\pi$
$992$	−0.436877	−0.0138709
$993$	79.3064	2.51671
$994$	−51.0924	−1.62055
$995$	0	0
$996$	−23.0308	−0.729759
$997$	47.8091	1.51413	0.757065	−	0.653339i	$-0.226632\pi$
$997$	47.8091	1.51413	0.757065	+	0.653339i	$0.226632\pi$
$998$	36.1138	1.14316
$999$	62.3370	1.97225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.y.1.3		3
13.4	even	6		494.2.g.c.419.1	yes	6
13.10	even	6		494.2.g.c.191.1	✓	6
13.12	even	2		6422.2.a.o.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.c.191.1	✓	6	13.10	even	6
494.2.g.c.419.1	yes	6	13.4	even	6
6422.2.a.o.1.3		3	13.12	even	2
6422.2.a.y.1.3		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.83847 q^{3} +1.00000 q^{4} +2.83847 q^{6} +3.62003 q^{7} +1.00000 q^{8} +5.05691 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.83847 q^{3} +1.00000 q^{4} +2.83847 q^{6} +3.62003 q^{7} +1.00000 q^{8} +5.05691 q^{9} +2.83847 q^{12} +3.62003 q^{14} +1.00000 q^{16} -1.16153 q^{17} +5.05691 q^{18} -1.00000 q^{19} +10.2753 q^{21} +7.73385 q^{23} +2.83847 q^{24} -5.00000 q^{25} +5.83847 q^{27} +3.62003 q^{28} +4.45850 q^{29} -0.436877 q^{31} +1.00000 q^{32} -1.16153 q^{34} +5.05691 q^{36} +10.6769 q^{37} -1.00000 q^{38} -11.7908 q^{41} +10.2753 q^{42} -1.67694 q^{43} +7.73385 q^{46} -10.3539 q^{47} +2.83847 q^{48} +6.10462 q^{49} -5.00000 q^{50} -3.29697 q^{51} -2.89538 q^{53} +5.83847 q^{54} +3.62003 q^{56} -2.83847 q^{57} +4.45850 q^{58} +6.78156 q^{59} -0.323061 q^{61} -0.436877 q^{62} +18.3062 q^{63} +1.00000 q^{64} -1.21844 q^{67} -1.16153 q^{68} +21.9523 q^{69} -14.1138 q^{71} +5.05691 q^{72} -14.1707 q^{73} +10.6769 q^{74} -14.1923 q^{75} -1.00000 q^{76} +10.1138 q^{79} +1.40159 q^{81} -11.7908 q^{82} -8.11382 q^{83} +10.2753 q^{84} -1.67694 q^{86} +12.6553 q^{87} -2.87375 q^{89} +7.73385 q^{92} -1.24006 q^{93} -10.3539 q^{94} +2.83847 q^{96} +1.67694 q^{97} +6.10462 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 4 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 4 * q^7 + 3 * q^8 + 5 * q^9	\( 3 q + 3 q^{2} + 3 q^{4} + 4 q^{7} + 3 q^{8} + 5 q^{9} + 4 q^{14} + 3 q^{16} - 12 q^{17} + 5 q^{18} - 3 q^{19} + 19 q^{21} - 4 q^{23} - 15 q^{25} + 9 q^{27} + 4 q^{28} - 2 q^{29} + 2 q^{31} + 3 q^{32} - 12 q^{34} + 5 q^{36} + 15 q^{37} - 3 q^{38} + 2 q^{41} + 19 q^{42} + 12 q^{43} - 4 q^{46} + 3 q^{47} + 37 q^{49} - 15 q^{50} + 14 q^{51} + 10 q^{53} + 9 q^{54} + 4 q^{56} - 2 q^{58} + 22 q^{59} - 18 q^{61} + 2 q^{62} - 8 q^{63} + 3 q^{64} - 2 q^{67} - 12 q^{68} + 37 q^{69} - 22 q^{71} + 5 q^{72} - 12 q^{73} + 15 q^{74} - 3 q^{76} + 10 q^{79} - q^{81} + 2 q^{82} - 4 q^{83} + 19 q^{84} + 12 q^{86} + 33 q^{87} - 2 q^{89} - 4 q^{92} + 10 q^{93} + 3 q^{94} - 12 q^{97} + 37 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 4 * q^7 + 3 * q^8 + 5 * q^9 + 4 * q^14 + 3 * q^16 - 12 * q^17 + 5 * q^18 - 3 * q^19 + 19 * q^21 - 4 * q^23 - 15 * q^25 + 9 * q^27 + 4 * q^28 - 2 * q^29 + 2 * q^31 + 3 * q^32 - 12 * q^34 + 5 * q^36 + 15 * q^37 - 3 * q^38 + 2 * q^41 + 19 * q^42 + 12 * q^43 - 4 * q^46 + 3 * q^47 + 37 * q^49 - 15 * q^50 + 14 * q^51 + 10 * q^53 + 9 * q^54 + 4 * q^56 - 2 * q^58 + 22 * q^59 - 18 * q^61 + 2 * q^62 - 8 * q^63 + 3 * q^64 - 2 * q^67 - 12 * q^68 + 37 * q^69 - 22 * q^71 + 5 * q^72 - 12 * q^73 + 15 * q^74 - 3 * q^76 + 10 * q^79 - q^81 + 2 * q^82 - 4 * q^83 + 19 * q^84 + 12 * q^86 + 33 * q^87 - 2 * q^89 - 4 * q^92 + 10 * q^93 + 3 * q^94 - 12 * q^97 + 37 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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