LMFDB - Embedded newform 5586.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5586,2,Mod(1,5586)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5586, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5586.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5586.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:	$44.6044345691$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	5586.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -5.41421 q^{11} +1.00000 q^{12} +2.82843 q^{13} +1.00000 q^{16} -6.82843 q^{17} +1.00000 q^{18} -1.00000 q^{19} -5.41421 q^{22} +4.24264 q^{23} +1.00000 q^{24} -5.00000 q^{25} +2.82843 q^{26} +1.00000 q^{27} -8.00000 q^{29} -7.65685 q^{31} +1.00000 q^{32} -5.41421 q^{33} -6.82843 q^{34} +1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} +2.82843 q^{39} -5.41421 q^{41} -1.17157 q^{43} -5.41421 q^{44} +4.24264 q^{46} +3.65685 q^{47} +1.00000 q^{48} -5.00000 q^{50} -6.82843 q^{51} +2.82843 q^{52} -2.00000 q^{53} +1.00000 q^{54} -1.00000 q^{57} -8.00000 q^{58} +5.41421 q^{61} -7.65685 q^{62} +1.00000 q^{64} -5.41421 q^{66} +4.24264 q^{67} -6.82843 q^{68} +4.24264 q^{69} -14.1421 q^{71} +1.00000 q^{72} +6.58579 q^{73} -2.00000 q^{74} -5.00000 q^{75} -1.00000 q^{76} +2.82843 q^{78} -13.8995 q^{79} +1.00000 q^{81} -5.41421 q^{82} +1.65685 q^{83} -1.17157 q^{86} -8.00000 q^{87} -5.41421 q^{88} +5.41421 q^{89} +4.24264 q^{92} -7.65685 q^{93} +3.65685 q^{94} +1.00000 q^{96} +9.17157 q^{97} -5.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 8 q^{11} + 2 q^{12} + 2 q^{16} - 8 q^{17} + 2 q^{18} - 2 q^{19} - 8 q^{22} + 2 q^{24} - 10 q^{25} + 2 q^{27} - 16 q^{29} - 4 q^{31} + 2 q^{32} - 8 q^{33} - 8 q^{34} + 2 q^{36} - 4 q^{37} - 2 q^{38} - 8 q^{41} - 8 q^{43} - 8 q^{44} - 4 q^{47} + 2 q^{48} - 10 q^{50} - 8 q^{51} - 4 q^{53} + 2 q^{54} - 2 q^{57} - 16 q^{58} + 8 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{66} - 8 q^{68} + 2 q^{72} + 16 q^{73} - 4 q^{74} - 10 q^{75} - 2 q^{76} - 8 q^{79} + 2 q^{81} - 8 q^{82} - 8 q^{83} - 8 q^{86} - 16 q^{87} - 8 q^{88} + 8 q^{89} - 4 q^{93} - 4 q^{94} + 2 q^{96} + 24 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 8 * q^11 + 2 * q^12 + 2 * q^16 - 8 * q^17 + 2 * q^18 - 2 * q^19 - 8 * q^22 + 2 * q^24 - 10 * q^25 + 2 * q^27 - 16 * q^29 - 4 * q^31 + 2 * q^32 - 8 * q^33 - 8 * q^34 + 2 * q^36 - 4 * q^37 - 2 * q^38 - 8 * q^41 - 8 * q^43 - 8 * q^44 - 4 * q^47 + 2 * q^48 - 10 * q^50 - 8 * q^51 - 4 * q^53 + 2 * q^54 - 2 * q^57 - 16 * q^58 + 8 * q^61 - 4 * q^62 + 2 * q^64 - 8 * q^66 - 8 * q^68 + 2 * q^72 + 16 * q^73 - 4 * q^74 - 10 * q^75 - 2 * q^76 - 8 * q^79 + 2 * q^81 - 8 * q^82 - 8 * q^83 - 8 * q^86 - 16 * q^87 - 8 * q^88 + 8 * q^89 - 4 * q^93 - 4 * q^94 + 2 * q^96 + 24 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.41421	−1.63245	−0.816223	−	0.577736i	$-0.803936\pi$
$11$	−5.41421	−1.63245	−0.816223	+	0.577736i	$0.803936\pi$
$12$	1.00000	0.288675
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−5.41421	−1.15431
$23$	4.24264	0.884652	0.442326	−	0.896854i	$-0.354153\pi$
$23$	4.24264	0.884652	0.442326	+	0.896854i	$0.354153\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	2.82843	0.554700
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	−7.65685	−1.37521	−0.687606	−	0.726084i	$-0.741338\pi$
$31$	−7.65685	−1.37521	−0.687606	+	0.726084i	$0.741338\pi$
$32$	1.00000	0.176777
$33$	−5.41421	−0.942494
$34$	−6.82843	−1.17107
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−1.00000	−0.162221
$39$	2.82843	0.452911
$40$	0	0
$41$	−5.41421	−0.845558	−0.422779	−	0.906233i	$-0.638945\pi$
$41$	−5.41421	−0.845558	−0.422779	+	0.906233i	$0.638945\pi$
$42$	0	0
$43$	−1.17157	−0.178663	−0.0893316	−	0.996002i	$-0.528473\pi$
$43$	−1.17157	−0.178663	−0.0893316	+	0.996002i	$0.528473\pi$
$44$	−5.41421	−0.816223
$45$	0	0
$46$	4.24264	0.625543
$47$	3.65685	0.533407	0.266704	−	0.963779i	$-0.414066\pi$
$47$	3.65685	0.533407	0.266704	+	0.963779i	$0.414066\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−5.00000	−0.707107
$51$	−6.82843	−0.956171
$52$	2.82843	0.392232
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−8.00000	−1.05045
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	5.41421	0.693219	0.346610	−	0.938010i	$-0.387333\pi$
$61$	5.41421	0.693219	0.346610	+	0.938010i	$0.387333\pi$
$62$	−7.65685	−0.972421
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.41421	−0.666444
$67$	4.24264	0.518321	0.259161	−	0.965834i	$-0.416554\pi$
$67$	4.24264	0.518321	0.259161	+	0.965834i	$0.416554\pi$
$68$	−6.82843	−0.828068
$69$	4.24264	0.510754
$70$	0	0
$71$	−14.1421	−1.67836	−0.839181	−	0.543852i	$-0.816965\pi$
$71$	−14.1421	−1.67836	−0.839181	+	0.543852i	$0.816965\pi$
$72$	1.00000	0.117851
$73$	6.58579	0.770808	0.385404	−	0.922748i	$-0.374062\pi$
$73$	6.58579	0.770808	0.385404	+	0.922748i	$0.374062\pi$
$74$	−2.00000	−0.232495
$75$	−5.00000	−0.577350
$76$	−1.00000	−0.114708
$77$	0	0
$78$	2.82843	0.320256
$79$	−13.8995	−1.56382	−0.781908	−	0.623394i	$-0.785753\pi$
$79$	−13.8995	−1.56382	−0.781908	+	0.623394i	$0.785753\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.41421	−0.597900
$83$	1.65685	0.181863	0.0909317	−	0.995857i	$-0.471016\pi$
$83$	1.65685	0.181863	0.0909317	+	0.995857i	$0.471016\pi$
$84$	0	0
$85$	0	0
$86$	−1.17157	−0.126334
$87$	−8.00000	−0.857690
$88$	−5.41421	−0.577157
$89$	5.41421	0.573905	0.286953	−	0.957945i	$-0.407358\pi$
$89$	5.41421	0.573905	0.286953	+	0.957945i	$0.407358\pi$
$90$	0	0
$91$	0	0
$92$	4.24264	0.442326
$93$	−7.65685	−0.793979
$94$	3.65685	0.377176
$95$	0	0
$96$	1.00000	0.102062
$97$	9.17157	0.931232	0.465616	−	0.884987i	$-0.345833\pi$
$97$	9.17157	0.931232	0.465616	+	0.884987i	$0.345833\pi$
$98$	0	0
$99$	−5.41421	−0.544149
$100$	−5.00000	−0.500000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	−6.82843	−0.676115
$103$	3.65685	0.360321	0.180160	−	0.983637i	$-0.442338\pi$
$103$	3.65685	0.360321	0.180160	+	0.983637i	$0.442338\pi$
$104$	2.82843	0.277350
$105$	0	0
$106$	−2.00000	−0.194257
$107$	1.65685	0.160174	0.0800871	−	0.996788i	$-0.474480\pi$
$107$	1.65685	0.160174	0.0800871	+	0.996788i	$0.474480\pi$
$108$	1.00000	0.0962250
$109$	14.9706	1.43392	0.716960	−	0.697114i	$-0.245533\pi$
$109$	14.9706	1.43392	0.716960	+	0.697114i	$0.245533\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−9.65685	−0.908440	−0.454220	−	0.890889i	$-0.650082\pi$
$113$	−9.65685	−0.908440	−0.454220	+	0.890889i	$0.650082\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−8.00000	−0.742781
$117$	2.82843	0.261488
$118$	0	0
$119$	0	0
$120$	0	0
$121$	18.3137	1.66488
$122$	5.41421	0.490180
$123$	−5.41421	−0.488183
$124$	−7.65685	−0.687606
$125$	0	0
$126$	0	0
$127$	−8.24264	−0.731416	−0.365708	−	0.930730i	$-0.619173\pi$
$127$	−8.24264	−0.731416	−0.365708	+	0.930730i	$0.619173\pi$
$128$	1.00000	0.0883883
$129$	−1.17157	−0.103151
$130$	0	0
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	−5.41421	−0.471247
$133$	0	0
$134$	4.24264	0.366508
$135$	0	0
$136$	−6.82843	−0.585533
$137$	−18.9706	−1.62076	−0.810382	−	0.585901i	$-0.800741\pi$
$137$	−18.9706	−1.62076	−0.810382	+	0.585901i	$0.800741\pi$
$138$	4.24264	0.361158
$139$	23.3137	1.97744	0.988721	−	0.149766i	$-0.0478520\pi$
$139$	23.3137	1.97744	0.988721	+	0.149766i	$0.0478520\pi$
$140$	0	0
$141$	3.65685	0.307963
$142$	−14.1421	−1.18678
$143$	−15.3137	−1.28060
$144$	1.00000	0.0833333
$145$	0	0
$146$	6.58579	0.545044
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	−5.00000	−0.408248
$151$	8.24264	0.670777	0.335388	−	0.942080i	$-0.391132\pi$
$151$	8.24264	0.670777	0.335388	+	0.942080i	$0.391132\pi$
$152$	−1.00000	−0.0811107
$153$	−6.82843	−0.552046
$154$	0	0
$155$	0	0
$156$	2.82843	0.226455
$157$	7.75736	0.619105	0.309552	−	0.950882i	$-0.399821\pi$
$157$	7.75736	0.619105	0.309552	+	0.950882i	$0.399821\pi$
$158$	−13.8995	−1.10578
$159$	−2.00000	−0.158610
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−12.4853	−0.977923	−0.488961	−	0.872305i	$-0.662624\pi$
$163$	−12.4853	−0.977923	−0.488961	+	0.872305i	$0.662624\pi$
$164$	−5.41421	−0.422779
$165$	0	0
$166$	1.65685	0.128597
$167$	−24.9706	−1.93228	−0.966140	−	0.258018i	$-0.916931\pi$
$167$	−24.9706	−1.93228	−0.966140	+	0.258018i	$0.916931\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−1.17157	−0.0893316
$173$	3.75736	0.285667	0.142833	−	0.989747i	$-0.454379\pi$
$173$	3.75736	0.285667	0.142833	+	0.989747i	$0.454379\pi$
$174$	−8.00000	−0.606478
$175$	0	0
$176$	−5.41421	−0.408112
$177$	0	0
$178$	5.41421	0.405812
$179$	23.3137	1.74255	0.871274	−	0.490797i	$-0.163294\pi$
$179$	23.3137	1.74255	0.871274	+	0.490797i	$0.163294\pi$
$180$	0	0
$181$	−19.3137	−1.43558	−0.717788	−	0.696261i	$-0.754845\pi$
$181$	−19.3137	−1.43558	−0.717788	+	0.696261i	$0.754845\pi$
$182$	0	0
$183$	5.41421	0.400230
$184$	4.24264	0.312772
$185$	0	0
$186$	−7.65685	−0.561428
$187$	36.9706	2.70356
$188$	3.65685	0.266704
$189$	0	0
$190$	0	0
$191$	17.8995	1.29516	0.647581	−	0.761997i	$-0.275781\pi$
$191$	17.8995	1.29516	0.647581	+	0.761997i	$0.275781\pi$
$192$	1.00000	0.0721688
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	9.17157	0.658481
$195$	0	0
$196$	0	0
$197$	−12.3431	−0.879413	−0.439706	−	0.898142i	$-0.644917\pi$
$197$	−12.3431	−0.879413	−0.439706	+	0.898142i	$0.644917\pi$
$198$	−5.41421	−0.384771
$199$	−11.3137	−0.802008	−0.401004	−	0.916076i	$-0.631339\pi$
$199$	−11.3137	−0.802008	−0.401004	+	0.916076i	$0.631339\pi$
$200$	−5.00000	−0.353553
$201$	4.24264	0.299253
$202$	8.00000	0.562878
$203$	0	0
$204$	−6.82843	−0.478086
$205$	0	0
$206$	3.65685	0.254785
$207$	4.24264	0.294884
$208$	2.82843	0.196116
$209$	5.41421	0.374509
$210$	0	0
$211$	−25.8995	−1.78299	−0.891497	−	0.453026i	$-0.850345\pi$
$211$	−25.8995	−1.78299	−0.891497	+	0.453026i	$0.850345\pi$
$212$	−2.00000	−0.137361
$213$	−14.1421	−0.969003
$214$	1.65685	0.113260
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	14.9706	1.01393
$219$	6.58579	0.445026
$220$	0	0
$221$	−19.3137	−1.29918
$222$	−2.00000	−0.134231
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	−9.65685	−0.642364
$227$	6.34315	0.421009	0.210505	−	0.977593i	$-0.432489\pi$
$227$	6.34315	0.421009	0.210505	+	0.977593i	$0.432489\pi$
$228$	−1.00000	−0.0662266
$229$	17.2132	1.13748	0.568740	−	0.822517i	$-0.307431\pi$
$229$	17.2132	1.13748	0.568740	+	0.822517i	$0.307431\pi$
$230$	0	0
$231$	0	0
$232$	−8.00000	−0.525226
$233$	−9.31371	−0.610161	−0.305081	−	0.952327i	$-0.598683\pi$
$233$	−9.31371	−0.610161	−0.305081	+	0.952327i	$0.598683\pi$
$234$	2.82843	0.184900
$235$	0	0
$236$	0	0
$237$	−13.8995	−0.902869
$238$	0	0
$239$	21.2132	1.37217	0.686084	−	0.727522i	$-0.259328\pi$
$239$	21.2132	1.37217	0.686084	+	0.727522i	$0.259328\pi$
$240$	0	0
$241$	6.34315	0.408598	0.204299	−	0.978909i	$-0.434509\pi$
$241$	6.34315	0.408598	0.204299	+	0.978909i	$0.434509\pi$
$242$	18.3137	1.17725
$243$	1.00000	0.0641500
$244$	5.41421	0.346610
$245$	0	0
$246$	−5.41421	−0.345198
$247$	−2.82843	−0.179969
$248$	−7.65685	−0.486211
$249$	1.65685	0.104999
$250$	0	0
$251$	28.6274	1.80695	0.903473	−	0.428644i	$-0.141009\pi$
$251$	28.6274	1.80695	0.903473	+	0.428644i	$0.141009\pi$
$252$	0	0
$253$	−22.9706	−1.44415
$254$	−8.24264	−0.517189
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.0416	1.25016	0.625081	−	0.780560i	$-0.285066\pi$
$257$	20.0416	1.25016	0.625081	+	0.780560i	$0.285066\pi$
$258$	−1.17157	−0.0729389
$259$	0	0
$260$	0	0
$261$	−8.00000	−0.495188
$262$	−10.0000	−0.617802
$263$	12.2426	0.754914	0.377457	−	0.926027i	$-0.376799\pi$
$263$	12.2426	0.754914	0.377457	+	0.926027i	$0.376799\pi$
$264$	−5.41421	−0.333222
$265$	0	0
$266$	0	0
$267$	5.41421	0.331344
$268$	4.24264	0.259161
$269$	−17.8995	−1.09135	−0.545676	−	0.837996i	$-0.683727\pi$
$269$	−17.8995	−1.09135	−0.545676	+	0.837996i	$0.683727\pi$
$270$	0	0
$271$	−2.34315	−0.142336	−0.0711680	−	0.997464i	$-0.522673\pi$
$271$	−2.34315	−0.142336	−0.0711680	+	0.997464i	$0.522673\pi$
$272$	−6.82843	−0.414034
$273$	0	0
$274$	−18.9706	−1.14605
$275$	27.0711	1.63245
$276$	4.24264	0.255377
$277$	−9.31371	−0.559607	−0.279803	−	0.960057i	$-0.590269\pi$
$277$	−9.31371	−0.559607	−0.279803	+	0.960057i	$0.590269\pi$
$278$	23.3137	1.39826
$279$	−7.65685	−0.458404
$280$	0	0
$281$	28.6274	1.70777	0.853884	−	0.520463i	$-0.174241\pi$
$281$	28.6274	1.70777	0.853884	+	0.520463i	$0.174241\pi$
$282$	3.65685	0.217763
$283$	−28.9706	−1.72212	−0.861061	−	0.508502i	$-0.830199\pi$
$283$	−28.9706	−1.72212	−0.861061	+	0.508502i	$0.830199\pi$
$284$	−14.1421	−0.839181
$285$	0	0
$286$	−15.3137	−0.905519
$287$	0	0
$288$	1.00000	0.0589256
$289$	29.6274	1.74279
$290$	0	0
$291$	9.17157	0.537647
$292$	6.58579	0.385404
$293$	−9.41421	−0.549984	−0.274992	−	0.961446i	$-0.588675\pi$
$293$	−9.41421	−0.549984	−0.274992	+	0.961446i	$0.588675\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	−5.41421	−0.314165
$298$	−14.0000	−0.810998
$299$	12.0000	0.693978
$300$	−5.00000	−0.288675
$301$	0	0
$302$	8.24264	0.474311
$303$	8.00000	0.459588
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−6.82843	−0.390355
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	0	0
$309$	3.65685	0.208031
$310$	0	0
$311$	−16.9706	−0.962312	−0.481156	−	0.876635i	$-0.659783\pi$
$311$	−16.9706	−0.962312	−0.481156	+	0.876635i	$0.659783\pi$
$312$	2.82843	0.160128
$313$	−9.89949	−0.559553	−0.279776	−	0.960065i	$-0.590260\pi$
$313$	−9.89949	−0.559553	−0.279776	+	0.960065i	$0.590260\pi$
$314$	7.75736	0.437773
$315$	0	0
$316$	−13.8995	−0.781908
$317$	−19.3137	−1.08477	−0.542383	−	0.840131i	$-0.682478\pi$
$317$	−19.3137	−1.08477	−0.542383	+	0.840131i	$0.682478\pi$
$318$	−2.00000	−0.112154
$319$	43.3137	2.42510
$320$	0	0
$321$	1.65685	0.0924766
$322$	0	0
$323$	6.82843	0.379944
$324$	1.00000	0.0555556
$325$	−14.1421	−0.784465
$326$	−12.4853	−0.691496
$327$	14.9706	0.827874
$328$	−5.41421	−0.298950
$329$	0	0
$330$	0	0
$331$	−1.41421	−0.0777322	−0.0388661	−	0.999244i	$-0.512375\pi$
$331$	−1.41421	−0.0777322	−0.0388661	+	0.999244i	$0.512375\pi$
$332$	1.65685	0.0909317
$333$	−2.00000	−0.109599
$334$	−24.9706	−1.36633
$335$	0	0
$336$	0	0
$337$	−32.6274	−1.77733	−0.888664	−	0.458558i	$-0.848366\pi$
$337$	−32.6274	−1.77733	−0.888664	+	0.458558i	$0.848366\pi$
$338$	−5.00000	−0.271964
$339$	−9.65685	−0.524488
$340$	0	0
$341$	41.4558	2.24496
$342$	−1.00000	−0.0540738
$343$	0	0
$344$	−1.17157	−0.0631670
$345$	0	0
$346$	3.75736	0.201997
$347$	5.41421	0.290650	0.145325	−	0.989384i	$-0.453577\pi$
$347$	5.41421	0.290650	0.145325	+	0.989384i	$0.453577\pi$
$348$	−8.00000	−0.428845
$349$	16.2426	0.869449	0.434724	−	0.900564i	$-0.356846\pi$
$349$	16.2426	0.869449	0.434724	+	0.900564i	$0.356846\pi$
$350$	0	0
$351$	2.82843	0.150970
$352$	−5.41421	−0.288579
$353$	−26.1421	−1.39141	−0.695703	−	0.718330i	$-0.744907\pi$
$353$	−26.1421	−1.39141	−0.695703	+	0.718330i	$0.744907\pi$
$354$	0	0
$355$	0	0
$356$	5.41421	0.286953
$357$	0	0
$358$	23.3137	1.23217
$359$	−1.41421	−0.0746393	−0.0373197	−	0.999303i	$-0.511882\pi$
$359$	−1.41421	−0.0746393	−0.0373197	+	0.999303i	$0.511882\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−19.3137	−1.01511
$363$	18.3137	0.961220
$364$	0	0
$365$	0	0
$366$	5.41421	0.283005
$367$	14.3431	0.748706	0.374353	−	0.927286i	$-0.377865\pi$
$367$	14.3431	0.748706	0.374353	+	0.927286i	$0.377865\pi$
$368$	4.24264	0.221163
$369$	−5.41421	−0.281853
$370$	0	0
$371$	0	0
$372$	−7.65685	−0.396989
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	36.9706	1.91170
$375$	0	0
$376$	3.65685	0.188588
$377$	−22.6274	−1.16537
$378$	0	0
$379$	6.10051	0.313362	0.156681	−	0.987649i	$-0.449921\pi$
$379$	6.10051	0.313362	0.156681	+	0.987649i	$0.449921\pi$
$380$	0	0
$381$	−8.24264	−0.422283
$382$	17.8995	0.915818
$383$	−18.3431	−0.937291	−0.468645	−	0.883386i	$-0.655258\pi$
$383$	−18.3431	−0.937291	−0.468645	+	0.883386i	$0.655258\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−10.0000	−0.508987
$387$	−1.17157	−0.0595544
$388$	9.17157	0.465616
$389$	−26.2843	−1.33267	−0.666333	−	0.745655i	$-0.732137\pi$
$389$	−26.2843	−1.33267	−0.666333	+	0.745655i	$0.732137\pi$
$390$	0	0
$391$	−28.9706	−1.46510
$392$	0	0
$393$	−10.0000	−0.504433
$394$	−12.3431	−0.621839
$395$	0	0
$396$	−5.41421	−0.272074
$397$	12.0416	0.604352	0.302176	−	0.953252i	$-0.402287\pi$
$397$	12.0416	0.604352	0.302176	+	0.953252i	$0.402287\pi$
$398$	−11.3137	−0.567105
$399$	0	0
$400$	−5.00000	−0.250000
$401$	5.65685	0.282490	0.141245	−	0.989975i	$-0.454889\pi$
$401$	5.65685	0.282490	0.141245	+	0.989975i	$0.454889\pi$
$402$	4.24264	0.211604
$403$	−21.6569	−1.07880
$404$	8.00000	0.398015
$405$	0	0
$406$	0	0
$407$	10.8284	0.536745
$408$	−6.82843	−0.338058
$409$	2.14214	0.105922	0.0529609	−	0.998597i	$-0.483134\pi$
$409$	2.14214	0.105922	0.0529609	+	0.998597i	$0.483134\pi$
$410$	0	0
$411$	−18.9706	−0.935749
$412$	3.65685	0.180160
$413$	0	0
$414$	4.24264	0.208514
$415$	0	0
$416$	2.82843	0.138675
$417$	23.3137	1.14168
$418$	5.41421	0.264818
$419$	1.31371	0.0641789	0.0320894	−	0.999485i	$-0.489784\pi$
$419$	1.31371	0.0641789	0.0320894	+	0.999485i	$0.489784\pi$
$420$	0	0
$421$	−19.6569	−0.958016	−0.479008	−	0.877810i	$-0.659004\pi$
$421$	−19.6569	−0.958016	−0.479008	+	0.877810i	$0.659004\pi$
$422$	−25.8995	−1.26077
$423$	3.65685	0.177802
$424$	−2.00000	−0.0971286
$425$	34.1421	1.65614
$426$	−14.1421	−0.685189
$427$	0	0
$428$	1.65685	0.0800871
$429$	−15.3137	−0.739353
$430$	0	0
$431$	−33.4558	−1.61151	−0.805756	−	0.592248i	$-0.798241\pi$
$431$	−33.4558	−1.61151	−0.805756	+	0.592248i	$0.798241\pi$
$432$	1.00000	0.0481125
$433$	14.8284	0.712609	0.356304	−	0.934370i	$-0.384037\pi$
$433$	14.8284	0.712609	0.356304	+	0.934370i	$0.384037\pi$
$434$	0	0
$435$	0	0
$436$	14.9706	0.716960
$437$	−4.24264	−0.202953
$438$	6.58579	0.314681
$439$	22.9706	1.09633	0.548163	−	0.836372i	$-0.315328\pi$
$439$	22.9706	1.09633	0.548163	+	0.836372i	$0.315328\pi$
$440$	0	0
$441$	0	0
$442$	−19.3137	−0.918659
$443$	−12.9289	−0.614272	−0.307136	−	0.951666i	$-0.599371\pi$
$443$	−12.9289	−0.614272	−0.307136	+	0.951666i	$0.599371\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	16.0000	0.757622
$447$	−14.0000	−0.662177
$448$	0	0
$449$	8.97056	0.423347	0.211674	−	0.977340i	$-0.432109\pi$
$449$	8.97056	0.423347	0.211674	+	0.977340i	$0.432109\pi$
$450$	−5.00000	−0.235702
$451$	29.3137	1.38033
$452$	−9.65685	−0.454220
$453$	8.24264	0.387273
$454$	6.34315	0.297699
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−12.6863	−0.593440	−0.296720	−	0.954965i	$-0.595893\pi$
$457$	−12.6863	−0.593440	−0.296720	+	0.954965i	$0.595893\pi$
$458$	17.2132	0.804320
$459$	−6.82843	−0.318724
$460$	0	0
$461$	16.0000	0.745194	0.372597	−	0.927993i	$-0.378467\pi$
$461$	16.0000	0.745194	0.372597	+	0.927993i	$0.378467\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−9.31371	−0.431449
$467$	36.9706	1.71079	0.855397	−	0.517973i	$-0.173313\pi$
$467$	36.9706	1.71079	0.855397	+	0.517973i	$0.173313\pi$
$468$	2.82843	0.130744
$469$	0	0
$470$	0	0
$471$	7.75736	0.357440
$472$	0	0
$473$	6.34315	0.291658
$474$	−13.8995	−0.638425
$475$	5.00000	0.229416
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	21.2132	0.970269
$479$	24.9706	1.14093	0.570467	−	0.821320i	$-0.306762\pi$
$479$	24.9706	1.14093	0.570467	+	0.821320i	$0.306762\pi$
$480$	0	0
$481$	−5.65685	−0.257930
$482$	6.34315	0.288922
$483$	0	0
$484$	18.3137	0.832441
$485$	0	0
$486$	1.00000	0.0453609
$487$	28.0416	1.27069	0.635344	−	0.772229i	$-0.280858\pi$
$487$	28.0416	1.27069	0.635344	+	0.772229i	$0.280858\pi$
$488$	5.41421	0.245090
$489$	−12.4853	−0.564604
$490$	0	0
$491$	−19.0711	−0.860665	−0.430333	−	0.902670i	$-0.641604\pi$
$491$	−19.0711	−0.860665	−0.430333	+	0.902670i	$0.641604\pi$
$492$	−5.41421	−0.244092
$493$	54.6274	2.46030
$494$	−2.82843	−0.127257
$495$	0	0
$496$	−7.65685	−0.343803
$497$	0	0
$498$	1.65685	0.0742454
$499$	5.85786	0.262234	0.131117	−	0.991367i	$-0.458144\pi$
$499$	5.85786	0.262234	0.131117	+	0.991367i	$0.458144\pi$
$500$	0	0
$501$	−24.9706	−1.11560
$502$	28.6274	1.27770
$503$	−29.6569	−1.32233	−0.661167	−	0.750239i	$-0.729938\pi$
$503$	−29.6569	−1.32233	−0.661167	+	0.750239i	$0.729938\pi$
$504$	0	0
$505$	0	0
$506$	−22.9706	−1.02117
$507$	−5.00000	−0.222058
$508$	−8.24264	−0.365708
$509$	7.07107	0.313420	0.156710	−	0.987645i	$-0.449911\pi$
$509$	7.07107	0.313420	0.156710	+	0.987645i	$0.449911\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	20.0416	0.883998
$515$	0	0
$516$	−1.17157	−0.0515756
$517$	−19.7990	−0.870759
$518$	0	0
$519$	3.75736	0.164930
$520$	0	0
$521$	−13.8995	−0.608948	−0.304474	−	0.952521i	$-0.598481\pi$
$521$	−13.8995	−0.608948	−0.304474	+	0.952521i	$0.598481\pi$
$522$	−8.00000	−0.350150
$523$	−35.9411	−1.57160	−0.785798	−	0.618483i	$-0.787747\pi$
$523$	−35.9411	−1.57160	−0.785798	+	0.618483i	$0.787747\pi$
$524$	−10.0000	−0.436852
$525$	0	0
$526$	12.2426	0.533805
$527$	52.2843	2.27754
$528$	−5.41421	−0.235623
$529$	−5.00000	−0.217391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−15.3137	−0.663310
$534$	5.41421	0.234296
$535$	0	0
$536$	4.24264	0.183254
$537$	23.3137	1.00606
$538$	−17.8995	−0.771702
$539$	0	0
$540$	0	0
$541$	−33.6569	−1.44702	−0.723511	−	0.690313i	$-0.757473\pi$
$541$	−33.6569	−1.44702	−0.723511	+	0.690313i	$0.757473\pi$
$542$	−2.34315	−0.100647
$543$	−19.3137	−0.828831
$544$	−6.82843	−0.292766
$545$	0	0
$546$	0	0
$547$	31.5563	1.34925	0.674626	−	0.738160i	$-0.264305\pi$
$547$	31.5563	1.34925	0.674626	+	0.738160i	$0.264305\pi$
$548$	−18.9706	−0.810382
$549$	5.41421	0.231073
$550$	27.0711	1.15431
$551$	8.00000	0.340811
$552$	4.24264	0.180579
$553$	0	0
$554$	−9.31371	−0.395702
$555$	0	0
$556$	23.3137	0.988721
$557$	−12.3431	−0.522996	−0.261498	−	0.965204i	$-0.584216\pi$
$557$	−12.3431	−0.522996	−0.261498	+	0.965204i	$0.584216\pi$
$558$	−7.65685	−0.324140
$559$	−3.31371	−0.140155
$560$	0	0
$561$	36.9706	1.56090
$562$	28.6274	1.20757
$563$	11.3137	0.476816	0.238408	−	0.971165i	$-0.423374\pi$
$563$	11.3137	0.476816	0.238408	+	0.971165i	$0.423374\pi$
$564$	3.65685	0.153981
$565$	0	0
$566$	−28.9706	−1.21772
$567$	0	0
$568$	−14.1421	−0.593391
$569$	40.2843	1.68880	0.844402	−	0.535710i	$-0.179956\pi$
$569$	40.2843	1.68880	0.844402	+	0.535710i	$0.179956\pi$
$570$	0	0
$571$	4.97056	0.208012	0.104006	−	0.994577i	$-0.466834\pi$
$571$	4.97056	0.208012	0.104006	+	0.994577i	$0.466834\pi$
$572$	−15.3137	−0.640298
$573$	17.8995	0.747762
$574$	0	0
$575$	−21.2132	−0.884652
$576$	1.00000	0.0416667
$577$	14.1005	0.587012	0.293506	−	0.955957i	$-0.405178\pi$
$577$	14.1005	0.587012	0.293506	+	0.955957i	$0.405178\pi$
$578$	29.6274	1.23234
$579$	−10.0000	−0.415586
$580$	0	0
$581$	0	0
$582$	9.17157	0.380174
$583$	10.8284	0.448468
$584$	6.58579	0.272522
$585$	0	0
$586$	−9.41421	−0.388898
$587$	−17.6569	−0.728776	−0.364388	−	0.931247i	$-0.618722\pi$
$587$	−17.6569	−0.728776	−0.364388	+	0.931247i	$0.618722\pi$
$588$	0	0
$589$	7.65685	0.315495
$590$	0	0
$591$	−12.3431	−0.507729
$592$	−2.00000	−0.0821995
$593$	−42.6274	−1.75050	−0.875249	−	0.483672i	$-0.839303\pi$
$593$	−42.6274	−1.75050	−0.875249	+	0.483672i	$0.839303\pi$
$594$	−5.41421	−0.222148
$595$	0	0
$596$	−14.0000	−0.573462
$597$	−11.3137	−0.463039
$598$	12.0000	0.490716
$599$	−36.7696	−1.50236	−0.751182	−	0.660096i	$-0.770516\pi$
$599$	−36.7696	−1.50236	−0.751182	+	0.660096i	$0.770516\pi$
$600$	−5.00000	−0.204124
$601$	2.14214	0.0873795	0.0436898	−	0.999045i	$-0.486089\pi$
$601$	2.14214	0.0873795	0.0436898	+	0.999045i	$0.486089\pi$
$602$	0	0
$603$	4.24264	0.172774
$604$	8.24264	0.335388
$605$	0	0
$606$	8.00000	0.324978
$607$	−3.31371	−0.134499	−0.0672496	−	0.997736i	$-0.521422\pi$
$607$	−3.31371	−0.134499	−0.0672496	+	0.997736i	$0.521422\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	10.3431	0.418439
$612$	−6.82843	−0.276023
$613$	29.6569	1.19783	0.598915	−	0.800813i	$-0.295599\pi$
$613$	29.6569	1.19783	0.598915	+	0.800813i	$0.295599\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	0	0
$617$	14.9706	0.602692	0.301346	−	0.953515i	$-0.402564\pi$
$617$	14.9706	0.602692	0.301346	+	0.953515i	$0.402564\pi$
$618$	3.65685	0.147100
$619$	−10.3431	−0.415726	−0.207863	−	0.978158i	$-0.566651\pi$
$619$	−10.3431	−0.415726	−0.207863	+	0.978158i	$0.566651\pi$
$620$	0	0
$621$	4.24264	0.170251
$622$	−16.9706	−0.680458
$623$	0	0
$624$	2.82843	0.113228
$625$	25.0000	1.00000
$626$	−9.89949	−0.395663
$627$	5.41421	0.216223
$628$	7.75736	0.309552
$629$	13.6569	0.544534
$630$	0	0
$631$	−23.1127	−0.920102	−0.460051	−	0.887892i	$-0.652169\pi$
$631$	−23.1127	−0.920102	−0.460051	+	0.887892i	$0.652169\pi$
$632$	−13.8995	−0.552892
$633$	−25.8995	−1.02941
$634$	−19.3137	−0.767045
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	0	0
$638$	43.3137	1.71481
$639$	−14.1421	−0.559454
$640$	0	0
$641$	10.6863	0.422083	0.211042	−	0.977477i	$-0.432314\pi$
$641$	10.6863	0.422083	0.211042	+	0.977477i	$0.432314\pi$
$642$	1.65685	0.0653908
$643$	40.9706	1.61572	0.807861	−	0.589374i	$-0.200625\pi$
$643$	40.9706	1.61572	0.807861	+	0.589374i	$0.200625\pi$
$644$	0	0
$645$	0	0
$646$	6.82843	0.268661
$647$	−37.6569	−1.48044	−0.740222	−	0.672363i	$-0.765280\pi$
$647$	−37.6569	−1.48044	−0.740222	+	0.672363i	$0.765280\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−14.1421	−0.554700
$651$	0	0
$652$	−12.4853	−0.488961
$653$	2.97056	0.116247	0.0581236	−	0.998309i	$-0.481488\pi$
$653$	2.97056	0.116247	0.0581236	+	0.998309i	$0.481488\pi$
$654$	14.9706	0.585395
$655$	0	0
$656$	−5.41421	−0.211390
$657$	6.58579	0.256936
$658$	0	0
$659$	40.2843	1.56925	0.784626	−	0.619969i	$-0.212855\pi$
$659$	40.2843	1.56925	0.784626	+	0.619969i	$0.212855\pi$
$660$	0	0
$661$	−36.2843	−1.41129	−0.705647	−	0.708563i	$-0.749344\pi$
$661$	−36.2843	−1.41129	−0.705647	+	0.708563i	$0.749344\pi$
$662$	−1.41421	−0.0549650
$663$	−19.3137	−0.750082
$664$	1.65685	0.0642984
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−33.9411	−1.31421
$668$	−24.9706	−0.966140
$669$	16.0000	0.618596
$670$	0	0
$671$	−29.3137	−1.13164
$672$	0	0
$673$	−9.31371	−0.359017	−0.179509	−	0.983756i	$-0.557451\pi$
$673$	−9.31371	−0.359017	−0.179509	+	0.983756i	$0.557451\pi$
$674$	−32.6274	−1.25676
$675$	−5.00000	−0.192450
$676$	−5.00000	−0.192308
$677$	4.72792	0.181709	0.0908544	−	0.995864i	$-0.471040\pi$
$677$	4.72792	0.181709	0.0908544	+	0.995864i	$0.471040\pi$
$678$	−9.65685	−0.370869
$679$	0	0
$680$	0	0
$681$	6.34315	0.243070
$682$	41.4558	1.58743
$683$	−4.48528	−0.171625	−0.0858123	−	0.996311i	$-0.527349\pi$
$683$	−4.48528	−0.171625	−0.0858123	+	0.996311i	$0.527349\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	0	0
$687$	17.2132	0.656725
$688$	−1.17157	−0.0446658
$689$	−5.65685	−0.215509
$690$	0	0
$691$	−3.31371	−0.126059	−0.0630297	−	0.998012i	$-0.520076\pi$
$691$	−3.31371	−0.126059	−0.0630297	+	0.998012i	$0.520076\pi$
$692$	3.75736	0.142833
$693$	0	0
$694$	5.41421	0.205521
$695$	0	0
$696$	−8.00000	−0.303239
$697$	36.9706	1.40036
$698$	16.2426	0.614793
$699$	−9.31371	−0.352277
$700$	0	0
$701$	−26.9706	−1.01866	−0.509332	−	0.860570i	$-0.670107\pi$
$701$	−26.9706	−1.01866	−0.509332	+	0.860570i	$0.670107\pi$
$702$	2.82843	0.106752
$703$	2.00000	0.0754314
$704$	−5.41421	−0.204056
$705$	0	0
$706$	−26.1421	−0.983872
$707$	0	0
$708$	0	0
$709$	26.6863	1.00222	0.501112	−	0.865382i	$-0.332924\pi$
$709$	26.6863	1.00222	0.501112	+	0.865382i	$0.332924\pi$
$710$	0	0
$711$	−13.8995	−0.521272
$712$	5.41421	0.202906
$713$	−32.4853	−1.21658
$714$	0	0
$715$	0	0
$716$	23.3137	0.871274
$717$	21.2132	0.792222
$718$	−1.41421	−0.0527780
$719$	−44.2843	−1.65152	−0.825762	−	0.564018i	$-0.809255\pi$
$719$	−44.2843	−1.65152	−0.825762	+	0.564018i	$0.809255\pi$
$720$	0	0
$721$	0	0
$722$	1.00000	0.0372161
$723$	6.34315	0.235904
$724$	−19.3137	−0.717788
$725$	40.0000	1.48556
$726$	18.3137	0.679685
$727$	29.6569	1.09991	0.549956	−	0.835194i	$-0.314645\pi$
$727$	29.6569	1.09991	0.549956	+	0.835194i	$0.314645\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	5.41421	0.200115
$733$	17.2132	0.635784	0.317892	−	0.948127i	$-0.397025\pi$
$733$	17.2132	0.635784	0.317892	+	0.948127i	$0.397025\pi$
$734$	14.3431	0.529415
$735$	0	0
$736$	4.24264	0.156386
$737$	−22.9706	−0.846132
$738$	−5.41421	−0.199300
$739$	37.9411	1.39569	0.697843	−	0.716250i	$-0.254143\pi$
$739$	37.9411	1.39569	0.697843	+	0.716250i	$0.254143\pi$
$740$	0	0
$741$	−2.82843	−0.103905
$742$	0	0
$743$	37.6569	1.38150	0.690748	−	0.723096i	$-0.257281\pi$
$743$	37.6569	1.38150	0.690748	+	0.723096i	$0.257281\pi$
$744$	−7.65685	−0.280714
$745$	0	0
$746$	10.0000	0.366126
$747$	1.65685	0.0606211
$748$	36.9706	1.35178
$749$	0	0
$750$	0	0
$751$	1.21320	0.0442704	0.0221352	−	0.999755i	$-0.492954\pi$
$751$	1.21320	0.0442704	0.0221352	+	0.999755i	$0.492954\pi$
$752$	3.65685	0.133352
$753$	28.6274	1.04324
$754$	−22.6274	−0.824042
$755$	0	0
$756$	0	0
$757$	−28.2843	−1.02801	−0.514005	−	0.857787i	$-0.671839\pi$
$757$	−28.2843	−1.02801	−0.514005	+	0.857787i	$0.671839\pi$
$758$	6.10051	0.221580
$759$	−22.9706	−0.833779
$760$	0	0
$761$	−50.6274	−1.83524	−0.917621	−	0.397456i	$-0.869893\pi$
$761$	−50.6274	−1.83524	−0.917621	+	0.397456i	$0.869893\pi$
$762$	−8.24264	−0.298599
$763$	0	0
$764$	17.8995	0.647581
$765$	0	0
$766$	−18.3431	−0.662765
$767$	0	0
$768$	1.00000	0.0360844
$769$	32.5269	1.17295	0.586475	−	0.809967i	$-0.300515\pi$
$769$	32.5269	1.17295	0.586475	+	0.809967i	$0.300515\pi$
$770$	0	0
$771$	20.0416	0.721782
$772$	−10.0000	−0.359908
$773$	−28.7279	−1.03327	−0.516636	−	0.856205i	$-0.672816\pi$
$773$	−28.7279	−1.03327	−0.516636	+	0.856205i	$0.672816\pi$
$774$	−1.17157	−0.0421113
$775$	38.2843	1.37521
$776$	9.17157	0.329240
$777$	0	0
$778$	−26.2843	−0.942337
$779$	5.41421	0.193984
$780$	0	0
$781$	76.5685	2.73984
$782$	−28.9706	−1.03599
$783$	−8.00000	−0.285897
$784$	0	0
$785$	0	0
$786$	−10.0000	−0.356688
$787$	36.6274	1.30563	0.652813	−	0.757519i	$-0.273589\pi$
$787$	36.6274	1.30563	0.652813	+	0.757519i	$0.273589\pi$
$788$	−12.3431	−0.439706
$789$	12.2426	0.435850
$790$	0	0
$791$	0	0
$792$	−5.41421	−0.192386
$793$	15.3137	0.543806
$794$	12.0416	0.427341
$795$	0	0
$796$	−11.3137	−0.401004
$797$	−4.24264	−0.150282	−0.0751410	−	0.997173i	$-0.523941\pi$
$797$	−4.24264	−0.150282	−0.0751410	+	0.997173i	$0.523941\pi$
$798$	0	0
$799$	−24.9706	−0.883395
$800$	−5.00000	−0.176777
$801$	5.41421	0.191302
$802$	5.65685	0.199750
$803$	−35.6569	−1.25830
$804$	4.24264	0.149626
$805$	0	0
$806$	−21.6569	−0.762830
$807$	−17.8995	−0.630092
$808$	8.00000	0.281439
$809$	−22.2843	−0.783473	−0.391737	−	0.920077i	$-0.628126\pi$
$809$	−22.2843	−0.783473	−0.391737	+	0.920077i	$0.628126\pi$
$810$	0	0
$811$	14.0000	0.491606	0.245803	−	0.969320i	$-0.420948\pi$
$811$	14.0000	0.491606	0.245803	+	0.969320i	$0.420948\pi$
$812$	0	0
$813$	−2.34315	−0.0821777
$814$	10.8284	0.379536
$815$	0	0
$816$	−6.82843	−0.239043
$817$	1.17157	0.0409881
$818$	2.14214	0.0748980
$819$	0	0
$820$	0	0
$821$	50.2843	1.75493	0.877467	−	0.479638i	$-0.159232\pi$
$821$	50.2843	1.75493	0.877467	+	0.479638i	$0.159232\pi$
$822$	−18.9706	−0.661674
$823$	24.9706	0.870419	0.435210	−	0.900329i	$-0.356674\pi$
$823$	24.9706	0.870419	0.435210	+	0.900329i	$0.356674\pi$
$824$	3.65685	0.127393
$825$	27.0711	0.942494
$826$	0	0
$827$	7.31371	0.254323	0.127161	−	0.991882i	$-0.459413\pi$
$827$	7.31371	0.254323	0.127161	+	0.991882i	$0.459413\pi$
$828$	4.24264	0.147442
$829$	−25.4558	−0.884118	−0.442059	−	0.896986i	$-0.645752\pi$
$829$	−25.4558	−0.884118	−0.442059	+	0.896986i	$0.645752\pi$
$830$	0	0
$831$	−9.31371	−0.323089
$832$	2.82843	0.0980581
$833$	0	0
$834$	23.3137	0.807288
$835$	0	0
$836$	5.41421	0.187254
$837$	−7.65685	−0.264660
$838$	1.31371	0.0453813
$839$	24.9706	0.862080	0.431040	−	0.902333i	$-0.358147\pi$
$839$	24.9706	0.862080	0.431040	+	0.902333i	$0.358147\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−19.6569	−0.677420
$843$	28.6274	0.985981
$844$	−25.8995	−0.891497
$845$	0	0
$846$	3.65685	0.125725
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	−28.9706	−0.994267
$850$	34.1421	1.17107
$851$	−8.48528	−0.290872
$852$	−14.1421	−0.484502
$853$	−50.1838	−1.71826	−0.859130	−	0.511758i	$-0.828994\pi$
$853$	−50.1838	−1.71826	−0.859130	+	0.511758i	$0.828994\pi$
$854$	0	0
$855$	0	0
$856$	1.65685	0.0566301
$857$	9.21320	0.314717	0.157359	−	0.987542i	$-0.449702\pi$
$857$	9.21320	0.314717	0.157359	+	0.987542i	$0.449702\pi$
$858$	−15.3137	−0.522801
$859$	−23.3137	−0.795453	−0.397727	−	0.917504i	$-0.630201\pi$
$859$	−23.3137	−0.795453	−0.397727	+	0.917504i	$0.630201\pi$
$860$	0	0
$861$	0	0
$862$	−33.4558	−1.13951
$863$	53.6569	1.82650	0.913250	−	0.407399i	$-0.133564\pi$
$863$	53.6569	1.82650	0.913250	+	0.407399i	$0.133564\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	14.8284	0.503890
$867$	29.6274	1.00620
$868$	0	0
$869$	75.2548	2.55285
$870$	0	0
$871$	12.0000	0.406604
$872$	14.9706	0.506967
$873$	9.17157	0.310411
$874$	−4.24264	−0.143509
$875$	0	0
$876$	6.58579	0.222513
$877$	−29.3137	−0.989854	−0.494927	−	0.868935i	$-0.664805\pi$
$877$	−29.3137	−0.989854	−0.494927	+	0.868935i	$0.664805\pi$
$878$	22.9706	0.775219
$879$	−9.41421	−0.317534
$880$	0	0
$881$	19.1127	0.643923	0.321962	−	0.946753i	$-0.395658\pi$
$881$	19.1127	0.643923	0.321962	+	0.946753i	$0.395658\pi$
$882$	0	0
$883$	−12.4853	−0.420163	−0.210082	−	0.977684i	$-0.567373\pi$
$883$	−12.4853	−0.420163	−0.210082	+	0.977684i	$0.567373\pi$
$884$	−19.3137	−0.649590
$885$	0	0
$886$	−12.9289	−0.434356
$887$	9.37258	0.314701	0.157350	−	0.987543i	$-0.449705\pi$
$887$	9.37258	0.314701	0.157350	+	0.987543i	$0.449705\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	0	0
$891$	−5.41421	−0.181383
$892$	16.0000	0.535720
$893$	−3.65685	−0.122372
$894$	−14.0000	−0.468230
$895$	0	0
$896$	0	0
$897$	12.0000	0.400668
$898$	8.97056	0.299352
$899$	61.2548	2.04296
$900$	−5.00000	−0.166667
$901$	13.6569	0.454976
$902$	29.3137	0.976040
$903$	0	0
$904$	−9.65685	−0.321182
$905$	0	0
$906$	8.24264	0.273843
$907$	−35.3553	−1.17395	−0.586977	−	0.809603i	$-0.699682\pi$
$907$	−35.3553	−1.17395	−0.586977	+	0.809603i	$0.699682\pi$
$908$	6.34315	0.210505
$909$	8.00000	0.265343
$910$	0	0
$911$	8.48528	0.281130	0.140565	−	0.990071i	$-0.455108\pi$
$911$	8.48528	0.281130	0.140565	+	0.990071i	$0.455108\pi$
$912$	−1.00000	−0.0331133
$913$	−8.97056	−0.296882
$914$	−12.6863	−0.419625
$915$	0	0
$916$	17.2132	0.568740
$917$	0	0
$918$	−6.82843	−0.225372
$919$	18.3431	0.605085	0.302542	−	0.953136i	$-0.402165\pi$
$919$	18.3431	0.605085	0.302542	+	0.953136i	$0.402165\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	16.0000	0.526932
$923$	−40.0000	−1.31662
$924$	0	0
$925$	10.0000	0.328798
$926$	8.00000	0.262896
$927$	3.65685	0.120107
$928$	−8.00000	−0.262613
$929$	28.4853	0.934572	0.467286	−	0.884106i	$-0.345232\pi$
$929$	28.4853	0.934572	0.467286	+	0.884106i	$0.345232\pi$
$930$	0	0
$931$	0	0
$932$	−9.31371	−0.305081
$933$	−16.9706	−0.555591
$934$	36.9706	1.20971
$935$	0	0
$936$	2.82843	0.0924500
$937$	−1.89949	−0.0620538	−0.0310269	−	0.999519i	$-0.509878\pi$
$937$	−1.89949	−0.0620538	−0.0310269	+	0.999519i	$0.509878\pi$
$938$	0	0
$939$	−9.89949	−0.323058
$940$	0	0
$941$	−44.2426	−1.44227	−0.721134	−	0.692795i	$-0.756379\pi$
$941$	−44.2426	−1.44227	−0.721134	+	0.692795i	$0.756379\pi$
$942$	7.75736	0.252748
$943$	−22.9706	−0.748024
$944$	0	0
$945$	0	0
$946$	6.34315	0.206233
$947$	−46.3848	−1.50730	−0.753651	−	0.657274i	$-0.771709\pi$
$947$	−46.3848	−1.50730	−0.753651	+	0.657274i	$0.771709\pi$
$948$	−13.8995	−0.451435
$949$	18.6274	0.604672
$950$	5.00000	0.162221
$951$	−19.3137	−0.626290
$952$	0	0
$953$	53.6569	1.73812	0.869058	−	0.494710i	$-0.164726\pi$
$953$	53.6569	1.73812	0.869058	+	0.494710i	$0.164726\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	21.2132	0.686084
$957$	43.3137	1.40013
$958$	24.9706	0.806762
$959$	0	0
$960$	0	0
$961$	27.6274	0.891207
$962$	−5.65685	−0.182384
$963$	1.65685	0.0533914
$964$	6.34315	0.204299
$965$	0	0
$966$	0	0
$967$	−1.45584	−0.0468168	−0.0234084	−	0.999726i	$-0.507452\pi$
$967$	−1.45584	−0.0468168	−0.0234084	+	0.999726i	$0.507452\pi$
$968$	18.3137	0.588625
$969$	6.82843	0.219361
$970$	0	0
$971$	−50.6274	−1.62471	−0.812356	−	0.583162i	$-0.801815\pi$
$971$	−50.6274	−1.62471	−0.812356	+	0.583162i	$0.801815\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	28.0416	0.898512
$975$	−14.1421	−0.452911
$976$	5.41421	0.173305
$977$	−5.65685	−0.180979	−0.0904894	−	0.995897i	$-0.528843\pi$
$977$	−5.65685	−0.180979	−0.0904894	+	0.995897i	$0.528843\pi$
$978$	−12.4853	−0.399235
$979$	−29.3137	−0.936870
$980$	0	0
$981$	14.9706	0.477973
$982$	−19.0711	−0.608582
$983$	12.9706	0.413697	0.206848	−	0.978373i	$-0.433679\pi$
$983$	12.9706	0.413697	0.206848	+	0.978373i	$0.433679\pi$
$984$	−5.41421	−0.172599
$985$	0	0
$986$	54.6274	1.73969
$987$	0	0
$988$	−2.82843	−0.0899843
$989$	−4.97056	−0.158055
$990$	0	0
$991$	6.38478	0.202819	0.101410	−	0.994845i	$-0.467665\pi$
$991$	6.38478	0.202819	0.101410	+	0.994845i	$0.467665\pi$
$992$	−7.65685	−0.243105
$993$	−1.41421	−0.0448787
$994$	0	0
$995$	0	0
$996$	1.65685	0.0524994
$997$	−43.0711	−1.36407	−0.682037	−	0.731317i	$-0.738906\pi$
$997$	−43.0711	−1.36407	−0.682037	+	0.731317i	$0.738906\pi$
$998$	5.85786	0.185427
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5586.2.a.bo.1.1	yes	2
7.6	odd	2		5586.2.a.bj.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5586.2.a.bj.1.1	✓	2	7.6	odd	2
5586.2.a.bo.1.1	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5586.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -5.41421 q^{11} +1.00000 q^{12} +2.82843 q^{13} +1.00000 q^{16} -6.82843 q^{17} +1.00000 q^{18} -1.00000 q^{19} -5.41421 q^{22} +4.24264 q^{23} +1.00000 q^{24} -5.00000 q^{25} +2.82843 q^{26} +1.00000 q^{27} -8.00000 q^{29} -7.65685 q^{31} +1.00000 q^{32} -5.41421 q^{33} -6.82843 q^{34} +1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} +2.82843 q^{39} -5.41421 q^{41} -1.17157 q^{43} -5.41421 q^{44} +4.24264 q^{46} +3.65685 q^{47} +1.00000 q^{48} -5.00000 q^{50} -6.82843 q^{51} +2.82843 q^{52} -2.00000 q^{53} +1.00000 q^{54} -1.00000 q^{57} -8.00000 q^{58} +5.41421 q^{61} -7.65685 q^{62} +1.00000 q^{64} -5.41421 q^{66} +4.24264 q^{67} -6.82843 q^{68} +4.24264 q^{69} -14.1421 q^{71} +1.00000 q^{72} +6.58579 q^{73} -2.00000 q^{74} -5.00000 q^{75} -1.00000 q^{76} +2.82843 q^{78} -13.8995 q^{79} +1.00000 q^{81} -5.41421 q^{82} +1.65685 q^{83} -1.17157 q^{86} -8.00000 q^{87} -5.41421 q^{88} +5.41421 q^{89} +4.24264 q^{92} -7.65685 q^{93} +3.65685 q^{94} +1.00000 q^{96} +9.17157 q^{97} -5.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 8 q^{11} + 2 q^{12} + 2 q^{16} - 8 q^{17} + 2 q^{18} - 2 q^{19} - 8 q^{22} + 2 q^{24} - 10 q^{25} + 2 q^{27} - 16 q^{29} - 4 q^{31} + 2 q^{32} - 8 q^{33} - 8 q^{34} + 2 q^{36} - 4 q^{37} - 2 q^{38} - 8 q^{41} - 8 q^{43} - 8 q^{44} - 4 q^{47} + 2 q^{48} - 10 q^{50} - 8 q^{51} - 4 q^{53} + 2 q^{54} - 2 q^{57} - 16 q^{58} + 8 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{66} - 8 q^{68} + 2 q^{72} + 16 q^{73} - 4 q^{74} - 10 q^{75} - 2 q^{76} - 8 q^{79} + 2 q^{81} - 8 q^{82} - 8 q^{83} - 8 q^{86} - 16 q^{87} - 8 q^{88} + 8 q^{89} - 4 q^{93} - 4 q^{94} + 2 q^{96} + 24 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 8 * q^11 + 2 * q^12 + 2 * q^16 - 8 * q^17 + 2 * q^18 - 2 * q^19 - 8 * q^22 + 2 * q^24 - 10 * q^25 + 2 * q^27 - 16 * q^29 - 4 * q^31 + 2 * q^32 - 8 * q^33 - 8 * q^34 + 2 * q^36 - 4 * q^37 - 2 * q^38 - 8 * q^41 - 8 * q^43 - 8 * q^44 - 4 * q^47 + 2 * q^48 - 10 * q^50 - 8 * q^51 - 4 * q^53 + 2 * q^54 - 2 * q^57 - 16 * q^58 + 8 * q^61 - 4 * q^62 + 2 * q^64 - 8 * q^66 - 8 * q^68 + 2 * q^72 + 16 * q^73 - 4 * q^74 - 10 * q^75 - 2 * q^76 - 8 * q^79 + 2 * q^81 - 8 * q^82 - 8 * q^83 - 8 * q^86 - 16 * q^87 - 8 * q^88 + 8 * q^89 - 4 * q^93 - 4 * q^94 + 2 * q^96 + 24 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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