LMFDB - Embedded newform 5586.2.a.bp.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:	$0$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 798)
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.82843 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.82843 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.82843 q^{10} +2.00000 q^{11} +1.00000 q^{12} +2.82843 q^{13} -2.82843 q^{15} +1.00000 q^{16} +1.00000 q^{18} -1.00000 q^{19} -2.82843 q^{20} +2.00000 q^{22} +4.82843 q^{23} +1.00000 q^{24} +3.00000 q^{25} +2.82843 q^{26} +1.00000 q^{27} -3.65685 q^{29} -2.82843 q^{30} -1.17157 q^{31} +1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{36} -6.48528 q^{37} -1.00000 q^{38} +2.82843 q^{39} -2.82843 q^{40} +7.65685 q^{41} +8.00000 q^{43} +2.00000 q^{44} -2.82843 q^{45} +4.82843 q^{46} -2.82843 q^{47} +1.00000 q^{48} +3.00000 q^{50} +2.82843 q^{52} +7.65685 q^{53} +1.00000 q^{54} -5.65685 q^{55} -1.00000 q^{57} -3.65685 q^{58} -1.65685 q^{59} -2.82843 q^{60} -9.31371 q^{61} -1.17157 q^{62} +1.00000 q^{64} -8.00000 q^{65} +2.00000 q^{66} +10.0000 q^{67} +4.82843 q^{69} +1.65685 q^{71} +1.00000 q^{72} +11.6569 q^{73} -6.48528 q^{74} +3.00000 q^{75} -1.00000 q^{76} +2.82843 q^{78} -8.82843 q^{79} -2.82843 q^{80} +1.00000 q^{81} +7.65685 q^{82} +11.3137 q^{83} +8.00000 q^{86} -3.65685 q^{87} +2.00000 q^{88} +10.0000 q^{89} -2.82843 q^{90} +4.82843 q^{92} -1.17157 q^{93} -2.82843 q^{94} +2.82843 q^{95} +1.00000 q^{96} -7.31371 q^{97} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{11} + 2 q^{12} + 2 q^{16} + 2 q^{18} - 2 q^{19} + 4 q^{22} + 4 q^{23} + 2 q^{24} + 6 q^{25} + 2 q^{27} + 4 q^{29} - 8 q^{31} + 2 q^{32} + 4 q^{33} + 2 q^{36} + 4 q^{37} - 2 q^{38} + 4 q^{41} + 16 q^{43} + 4 q^{44} + 4 q^{46} + 2 q^{48} + 6 q^{50} + 4 q^{53} + 2 q^{54} - 2 q^{57} + 4 q^{58} + 8 q^{59} + 4 q^{61} - 8 q^{62} + 2 q^{64} - 16 q^{65} + 4 q^{66} + 20 q^{67} + 4 q^{69} - 8 q^{71} + 2 q^{72} + 12 q^{73} + 4 q^{74} + 6 q^{75} - 2 q^{76} - 12 q^{79} + 2 q^{81} + 4 q^{82} + 16 q^{86} + 4 q^{87} + 4 q^{88} + 20 q^{89} + 4 q^{92} - 8 q^{93} + 2 q^{96} + 8 q^{97} + 4 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 + 4 * q^11 + 2 * q^12 + 2 * q^16 + 2 * q^18 - 2 * q^19 + 4 * q^22 + 4 * q^23 + 2 * q^24 + 6 * q^25 + 2 * q^27 + 4 * q^29 - 8 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^36 + 4 * q^37 - 2 * q^38 + 4 * q^41 + 16 * q^43 + 4 * q^44 + 4 * q^46 + 2 * q^48 + 6 * q^50 + 4 * q^53 + 2 * q^54 - 2 * q^57 + 4 * q^58 + 8 * q^59 + 4 * q^61 - 8 * q^62 + 2 * q^64 - 16 * q^65 + 4 * q^66 + 20 * q^67 + 4 * q^69 - 8 * q^71 + 2 * q^72 + 12 * q^73 + 4 * q^74 + 6 * q^75 - 2 * q^76 - 12 * q^79 + 2 * q^81 + 4 * q^82 + 16 * q^86 + 4 * q^87 + 4 * q^88 + 20 * q^89 + 4 * q^92 - 8 * q^93 + 2 * q^96 + 8 * q^97 + 4 * 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
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.82843	−0.894427
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	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
$15$	−2.82843	−0.730297
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.82843	−0.632456
$21$	0	0
$22$	2.00000	0.426401
$23$	4.82843	1.00680	0.503398	−	0.864054i	$-0.332083\pi$
$23$	4.82843	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$24$	1.00000	0.204124
$25$	3.00000	0.600000
$26$	2.82843	0.554700
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	−2.82843	−0.516398
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	1.00000	0.176777
$33$	2.00000	0.348155
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	−6.48528	−1.06617	−0.533087	−	0.846061i	$-0.678968\pi$
$37$	−6.48528	−1.06617	−0.533087	+	0.846061i	$0.678968\pi$
$38$	−1.00000	−0.162221
$39$	2.82843	0.452911
$40$	−2.82843	−0.447214
$41$	7.65685	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$41$	7.65685	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	2.00000	0.301511
$45$	−2.82843	−0.421637
$46$	4.82843	0.711913
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	3.00000	0.424264
$51$	0	0
$52$	2.82843	0.392232
$53$	7.65685	1.05175	0.525875	−	0.850562i	$-0.323738\pi$
$53$	7.65685	1.05175	0.525875	+	0.850562i	$0.323738\pi$
$54$	1.00000	0.136083
$55$	−5.65685	−0.762770
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−3.65685	−0.480168
$59$	−1.65685	−0.215704	−0.107852	−	0.994167i	$-0.534397\pi$
$59$	−1.65685	−0.215704	−0.107852	+	0.994167i	$0.534397\pi$
$60$	−2.82843	−0.365148
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	−1.17157	−0.148790
$63$	0	0
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	2.00000	0.246183
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	0	0
$69$	4.82843	0.581274
$70$	0	0
$71$	1.65685	0.196632	0.0983162	−	0.995155i	$-0.468654\pi$
$71$	1.65685	0.196632	0.0983162	+	0.995155i	$0.468654\pi$
$72$	1.00000	0.117851
$73$	11.6569	1.36433	0.682166	−	0.731198i	$-0.261038\pi$
$73$	11.6569	1.36433	0.682166	+	0.731198i	$0.261038\pi$
$74$	−6.48528	−0.753899
$75$	3.00000	0.346410
$76$	−1.00000	−0.114708
$77$	0	0
$78$	2.82843	0.320256
$79$	−8.82843	−0.993276	−0.496638	−	0.867958i	$-0.665432\pi$
$79$	−8.82843	−0.993276	−0.496638	+	0.867958i	$0.665432\pi$
$80$	−2.82843	−0.316228
$81$	1.00000	0.111111
$82$	7.65685	0.845558
$83$	11.3137	1.24184	0.620920	−	0.783874i	$-0.286759\pi$
$83$	11.3137	1.24184	0.620920	+	0.783874i	$0.286759\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	−3.65685	−0.392056
$88$	2.00000	0.213201
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	−2.82843	−0.298142
$91$	0	0
$92$	4.82843	0.503398
$93$	−1.17157	−0.121486
$94$	−2.82843	−0.291730
$95$	2.82843	0.290191
$96$	1.00000	0.102062
$97$	−7.31371	−0.742595	−0.371297	−	0.928514i	$-0.621087\pi$
$97$	−7.31371	−0.742595	−0.371297	+	0.928514i	$0.621087\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	3.00000	0.300000
$101$	16.4853	1.64035	0.820173	−	0.572115i	$-0.193877\pi$
$101$	16.4853	1.64035	0.820173	+	0.572115i	$0.193877\pi$
$102$	0	0
$103$	12.4853	1.23021	0.615106	−	0.788445i	$-0.289113\pi$
$103$	12.4853	1.23021	0.615106	+	0.788445i	$0.289113\pi$
$104$	2.82843	0.277350
$105$	0	0
$106$	7.65685	0.743699
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	1.00000	0.0962250
$109$	2.48528	0.238047	0.119023	−	0.992891i	$-0.462024\pi$
$109$	2.48528	0.238047	0.119023	+	0.992891i	$0.462024\pi$
$110$	−5.65685	−0.539360
$111$	−6.48528	−0.615556
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−1.00000	−0.0936586
$115$	−13.6569	−1.27351
$116$	−3.65685	−0.339530
$117$	2.82843	0.261488
$118$	−1.65685	−0.152526
$119$	0	0
$120$	−2.82843	−0.258199
$121$	−7.00000	−0.636364
$122$	−9.31371	−0.843224
$123$	7.65685	0.690395
$124$	−1.17157	−0.105210
$125$	5.65685	0.505964
$126$	0	0
$127$	8.82843	0.783396	0.391698	−	0.920094i	$-0.371888\pi$
$127$	8.82843	0.783396	0.391698	+	0.920094i	$0.371888\pi$
$128$	1.00000	0.0883883
$129$	8.00000	0.704361
$130$	−8.00000	−0.701646
$131$	15.3137	1.33796	0.668982	−	0.743278i	$-0.266730\pi$
$131$	15.3137	1.33796	0.668982	+	0.743278i	$0.266730\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	10.0000	0.863868
$135$	−2.82843	−0.243432
$136$	0	0
$137$	0.343146	0.0293169	0.0146585	−	0.999893i	$-0.495334\pi$
$137$	0.343146	0.0293169	0.0146585	+	0.999893i	$0.495334\pi$
$138$	4.82843	0.411023
$139$	1.65685	0.140533	0.0702663	−	0.997528i	$-0.477615\pi$
$139$	1.65685	0.140533	0.0702663	+	0.997528i	$0.477615\pi$
$140$	0	0
$141$	−2.82843	−0.238197
$142$	1.65685	0.139040
$143$	5.65685	0.473050
$144$	1.00000	0.0833333
$145$	10.3431	0.858952
$146$	11.6569	0.964728
$147$	0	0
$148$	−6.48528	−0.533087
$149$	−18.4853	−1.51437	−0.757187	−	0.653199i	$-0.773427\pi$
$149$	−18.4853	−1.51437	−0.757187	+	0.653199i	$0.773427\pi$
$150$	3.00000	0.244949
$151$	−4.14214	−0.337082	−0.168541	−	0.985695i	$-0.553906\pi$
$151$	−4.14214	−0.337082	−0.168541	+	0.985695i	$0.553906\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	0	0
$155$	3.31371	0.266163
$156$	2.82843	0.226455
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−8.82843	−0.702352
$159$	7.65685	0.607228
$160$	−2.82843	−0.223607
$161$	0	0
$162$	1.00000	0.0785674
$163$	3.31371	0.259550	0.129775	−	0.991543i	$-0.458575\pi$
$163$	3.31371	0.259550	0.129775	+	0.991543i	$0.458575\pi$
$164$	7.65685	0.597900
$165$	−5.65685	−0.440386
$166$	11.3137	0.878114
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	8.00000	0.609994
$173$	19.6569	1.49448	0.747241	−	0.664553i	$-0.231378\pi$
$173$	19.6569	1.49448	0.747241	+	0.664553i	$0.231378\pi$
$174$	−3.65685	−0.277225
$175$	0	0
$176$	2.00000	0.150756
$177$	−1.65685	−0.124537
$178$	10.0000	0.749532
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−2.82843	−0.210819
$181$	−3.51472	−0.261247	−0.130623	−	0.991432i	$-0.541698\pi$
$181$	−3.51472	−0.261247	−0.130623	+	0.991432i	$0.541698\pi$
$182$	0	0
$183$	−9.31371	−0.688489
$184$	4.82843	0.355956
$185$	18.3431	1.34861
$186$	−1.17157	−0.0859039
$187$	0	0
$188$	−2.82843	−0.206284
$189$	0	0
$190$	2.82843	0.205196
$191$	16.1421	1.16800	0.584002	−	0.811752i	$-0.301486\pi$
$191$	16.1421	1.16800	0.584002	+	0.811752i	$0.301486\pi$
$192$	1.00000	0.0721688
$193$	26.9706	1.94138	0.970692	−	0.240328i	$-0.0772549\pi$
$193$	26.9706	1.94138	0.970692	+	0.240328i	$0.0772549\pi$
$194$	−7.31371	−0.525094
$195$	−8.00000	−0.572892
$196$	0	0
$197$	0.828427	0.0590230	0.0295115	−	0.999564i	$-0.490605\pi$
$197$	0.828427	0.0590230	0.0295115	+	0.999564i	$0.490605\pi$
$198$	2.00000	0.142134
$199$	16.9706	1.20301	0.601506	−	0.798869i	$-0.294568\pi$
$199$	16.9706	1.20301	0.601506	+	0.798869i	$0.294568\pi$
$200$	3.00000	0.212132
$201$	10.0000	0.705346
$202$	16.4853	1.15990
$203$	0	0
$204$	0	0
$205$	−21.6569	−1.51258
$206$	12.4853	0.869891
$207$	4.82843	0.335599
$208$	2.82843	0.196116
$209$	−2.00000	−0.138343
$210$	0	0
$211$	4.34315	0.298994	0.149497	−	0.988762i	$-0.452234\pi$
$211$	4.34315	0.298994	0.149497	+	0.988762i	$0.452234\pi$
$212$	7.65685	0.525875
$213$	1.65685	0.113526
$214$	−4.00000	−0.273434
$215$	−22.6274	−1.54318
$216$	1.00000	0.0680414
$217$	0	0
$218$	2.48528	0.168324
$219$	11.6569	0.787697
$220$	−5.65685	−0.381385
$221$	0	0
$222$	−6.48528	−0.435264
$223$	−8.48528	−0.568216	−0.284108	−	0.958792i	$-0.591698\pi$
$223$	−8.48528	−0.568216	−0.284108	+	0.958792i	$0.591698\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	−6.00000	−0.399114
$227$	15.3137	1.01641	0.508203	−	0.861237i	$-0.330310\pi$
$227$	15.3137	1.01641	0.508203	+	0.861237i	$0.330310\pi$
$228$	−1.00000	−0.0662266
$229$	−0.343146	−0.0226757	−0.0113379	−	0.999936i	$-0.503609\pi$
$229$	−0.343146	−0.0226757	−0.0113379	+	0.999936i	$0.503609\pi$
$230$	−13.6569	−0.900506
$231$	0	0
$232$	−3.65685	−0.240084
$233$	20.6274	1.35135	0.675674	−	0.737201i	$-0.263853\pi$
$233$	20.6274	1.35135	0.675674	+	0.737201i	$0.263853\pi$
$234$	2.82843	0.184900
$235$	8.00000	0.521862
$236$	−1.65685	−0.107852
$237$	−8.82843	−0.573468
$238$	0	0
$239$	−8.82843	−0.571063	−0.285532	−	0.958369i	$-0.592170\pi$
$239$	−8.82843	−0.571063	−0.285532	+	0.958369i	$0.592170\pi$
$240$	−2.82843	−0.182574
$241$	−22.6274	−1.45756	−0.728780	−	0.684748i	$-0.759912\pi$
$241$	−22.6274	−1.45756	−0.728780	+	0.684748i	$0.759912\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	−9.31371	−0.596249
$245$	0	0
$246$	7.65685	0.488183
$247$	−2.82843	−0.179969
$248$	−1.17157	−0.0743950
$249$	11.3137	0.716977
$250$	5.65685	0.357771
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	9.65685	0.607121
$254$	8.82843	0.553945
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	8.00000	0.498058
$259$	0	0
$260$	−8.00000	−0.496139
$261$	−3.65685	−0.226354
$262$	15.3137	0.946084
$263$	−16.1421	−0.995367	−0.497683	−	0.867359i	$-0.665816\pi$
$263$	−16.1421	−0.995367	−0.497683	+	0.867359i	$0.665816\pi$
$264$	2.00000	0.123091
$265$	−21.6569	−1.33037
$266$	0	0
$267$	10.0000	0.611990
$268$	10.0000	0.610847
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	−2.82843	−0.172133
$271$	−19.3137	−1.17322	−0.586612	−	0.809868i	$-0.699539\pi$
$271$	−19.3137	−1.17322	−0.586612	+	0.809868i	$0.699539\pi$
$272$	0	0
$273$	0	0
$274$	0.343146	0.0207302
$275$	6.00000	0.361814
$276$	4.82843	0.290637
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	1.65685	0.0993715
$279$	−1.17157	−0.0701402
$280$	0	0
$281$	−2.68629	−0.160251	−0.0801254	−	0.996785i	$-0.525532\pi$
$281$	−2.68629	−0.160251	−0.0801254	+	0.996785i	$0.525532\pi$
$282$	−2.82843	−0.168430
$283$	−17.6569	−1.04959	−0.524796	−	0.851228i	$-0.675858\pi$
$283$	−17.6569	−1.04959	−0.524796	+	0.851228i	$0.675858\pi$
$284$	1.65685	0.0983162
$285$	2.82843	0.167542
$286$	5.65685	0.334497
$287$	0	0
$288$	1.00000	0.0589256
$289$	−17.0000	−1.00000
$290$	10.3431	0.607370
$291$	−7.31371	−0.428737
$292$	11.6569	0.682166
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	0	0
$295$	4.68629	0.272846
$296$	−6.48528	−0.376949
$297$	2.00000	0.116052
$298$	−18.4853	−1.07082
$299$	13.6569	0.789796
$300$	3.00000	0.173205
$301$	0	0
$302$	−4.14214	−0.238353
$303$	16.4853	0.947055
$304$	−1.00000	−0.0573539
$305$	26.3431	1.50840
$306$	0	0
$307$	5.65685	0.322854	0.161427	−	0.986885i	$-0.448390\pi$
$307$	5.65685	0.322854	0.161427	+	0.986885i	$0.448390\pi$
$308$	0	0
$309$	12.4853	0.710263
$310$	3.31371	0.188206
$311$	20.4853	1.16161	0.580807	−	0.814041i	$-0.302737\pi$
$311$	20.4853	1.16161	0.580807	+	0.814041i	$0.302737\pi$
$312$	2.82843	0.160128
$313$	27.6569	1.56326	0.781629	−	0.623744i	$-0.214389\pi$
$313$	27.6569	1.56326	0.781629	+	0.623744i	$0.214389\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−8.82843	−0.496638
$317$	30.9706	1.73948	0.869740	−	0.493510i	$-0.164286\pi$
$317$	30.9706	1.73948	0.869740	+	0.493510i	$0.164286\pi$
$318$	7.65685	0.429375
$319$	−7.31371	−0.409489
$320$	−2.82843	−0.158114
$321$	−4.00000	−0.223258
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	8.48528	0.470679
$326$	3.31371	0.183529
$327$	2.48528	0.137436
$328$	7.65685	0.422779
$329$	0	0
$330$	−5.65685	−0.311400
$331$	12.6274	0.694066	0.347033	−	0.937853i	$-0.387189\pi$
$331$	12.6274	0.694066	0.347033	+	0.937853i	$0.387189\pi$
$332$	11.3137	0.620920
$333$	−6.48528	−0.355391
$334$	0	0
$335$	−28.2843	−1.54533
$336$	0	0
$337$	−13.3137	−0.725244	−0.362622	−	0.931936i	$-0.618118\pi$
$337$	−13.3137	−0.725244	−0.362622	+	0.931936i	$0.618118\pi$
$338$	−5.00000	−0.271964
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−2.34315	−0.126888
$342$	−1.00000	−0.0540738
$343$	0	0
$344$	8.00000	0.431331
$345$	−13.6569	−0.735260
$346$	19.6569	1.05676
$347$	14.0000	0.751559	0.375780	−	0.926709i	$-0.377375\pi$
$347$	14.0000	0.751559	0.375780	+	0.926709i	$0.377375\pi$
$348$	−3.65685	−0.196028
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	2.82843	0.150970
$352$	2.00000	0.106600
$353$	18.3431	0.976307	0.488154	−	0.872758i	$-0.337671\pi$
$353$	18.3431	0.976307	0.488154	+	0.872758i	$0.337671\pi$
$354$	−1.65685	−0.0880608
$355$	−4.68629	−0.248723
$356$	10.0000	0.529999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−10.4853	−0.553392	−0.276696	−	0.960958i	$-0.589239\pi$
$359$	−10.4853	−0.553392	−0.276696	+	0.960958i	$0.589239\pi$
$360$	−2.82843	−0.149071
$361$	1.00000	0.0526316
$362$	−3.51472	−0.184730
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−32.9706	−1.72576
$366$	−9.31371	−0.486835
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	4.82843	0.251699
$369$	7.65685	0.398600
$370$	18.3431	0.953615
$371$	0	0
$372$	−1.17157	−0.0607432
$373$	−34.4853	−1.78558	−0.892790	−	0.450473i	$-0.851255\pi$
$373$	−34.4853	−1.78558	−0.892790	+	0.450473i	$0.851255\pi$
$374$	0	0
$375$	5.65685	0.292119
$376$	−2.82843	−0.145865
$377$	−10.3431	−0.532699
$378$	0	0
$379$	20.6274	1.05956	0.529780	−	0.848135i	$-0.322275\pi$
$379$	20.6274	1.05956	0.529780	+	0.848135i	$0.322275\pi$
$380$	2.82843	0.145095
$381$	8.82843	0.452294
$382$	16.1421	0.825904
$383$	30.6274	1.56499	0.782494	−	0.622658i	$-0.213947\pi$
$383$	30.6274	1.56499	0.782494	+	0.622658i	$0.213947\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	26.9706	1.37277
$387$	8.00000	0.406663
$388$	−7.31371	−0.371297
$389$	21.7990	1.10525	0.552626	−	0.833429i	$-0.313626\pi$
$389$	21.7990	1.10525	0.552626	+	0.833429i	$0.313626\pi$
$390$	−8.00000	−0.405096
$391$	0	0
$392$	0	0
$393$	15.3137	0.772474
$394$	0.828427	0.0417356
$395$	24.9706	1.25641
$396$	2.00000	0.100504
$397$	−38.2843	−1.92143	−0.960716	−	0.277533i	$-0.910483\pi$
$397$	−38.2843	−1.92143	−0.960716	+	0.277533i	$0.910483\pi$
$398$	16.9706	0.850657
$399$	0	0
$400$	3.00000	0.150000
$401$	−16.6274	−0.830334	−0.415167	−	0.909745i	$-0.636277\pi$
$401$	−16.6274	−0.830334	−0.415167	+	0.909745i	$0.636277\pi$
$402$	10.0000	0.498755
$403$	−3.31371	−0.165068
$404$	16.4853	0.820173
$405$	−2.82843	−0.140546
$406$	0	0
$407$	−12.9706	−0.642927
$408$	0	0
$409$	0.686292	0.0339349	0.0169675	−	0.999856i	$-0.494599\pi$
$409$	0.686292	0.0339349	0.0169675	+	0.999856i	$0.494599\pi$
$410$	−21.6569	−1.06956
$411$	0.343146	0.0169261
$412$	12.4853	0.615106
$413$	0	0
$414$	4.82843	0.237304
$415$	−32.0000	−1.57082
$416$	2.82843	0.138675
$417$	1.65685	0.0811365
$418$	−2.00000	−0.0978232
$419$	−20.9706	−1.02448	−0.512240	−	0.858843i	$-0.671184\pi$
$419$	−20.9706	−1.02448	−0.512240	+	0.858843i	$0.671184\pi$
$420$	0	0
$421$	8.82843	0.430271	0.215136	−	0.976584i	$-0.430981\pi$
$421$	8.82843	0.430271	0.215136	+	0.976584i	$0.430981\pi$
$422$	4.34315	0.211421
$423$	−2.82843	−0.137523
$424$	7.65685	0.371850
$425$	0	0
$426$	1.65685	0.0802749
$427$	0	0
$428$	−4.00000	−0.193347
$429$	5.65685	0.273115
$430$	−22.6274	−1.09119
$431$	6.34315	0.305539	0.152769	−	0.988262i	$-0.451181\pi$
$431$	6.34315	0.305539	0.152769	+	0.988262i	$0.451181\pi$
$432$	1.00000	0.0481125
$433$	−7.31371	−0.351474	−0.175737	−	0.984437i	$-0.556231\pi$
$433$	−7.31371	−0.351474	−0.175737	+	0.984437i	$0.556231\pi$
$434$	0	0
$435$	10.3431	0.495916
$436$	2.48528	0.119023
$437$	−4.82843	−0.230975
$438$	11.6569	0.556986
$439$	3.51472	0.167748	0.0838742	−	0.996476i	$-0.473271\pi$
$439$	3.51472	0.167748	0.0838742	+	0.996476i	$0.473271\pi$
$440$	−5.65685	−0.269680
$441$	0	0
$442$	0	0
$443$	−17.3137	−0.822599	−0.411300	−	0.911500i	$-0.634925\pi$
$443$	−17.3137	−0.822599	−0.411300	+	0.911500i	$0.634925\pi$
$444$	−6.48528	−0.307778
$445$	−28.2843	−1.34080
$446$	−8.48528	−0.401790
$447$	−18.4853	−0.874324
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	3.00000	0.141421
$451$	15.3137	0.721094
$452$	−6.00000	−0.282216
$453$	−4.14214	−0.194615
$454$	15.3137	0.718708
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−32.6274	−1.52625	−0.763123	−	0.646253i	$-0.776335\pi$
$457$	−32.6274	−1.52625	−0.763123	+	0.646253i	$0.776335\pi$
$458$	−0.343146	−0.0160341
$459$	0	0
$460$	−13.6569	−0.636754
$461$	−9.85786	−0.459127	−0.229563	−	0.973294i	$-0.573730\pi$
$461$	−9.85786	−0.459127	−0.229563	+	0.973294i	$0.573730\pi$
$462$	0	0
$463$	−4.68629	−0.217790	−0.108895	−	0.994053i	$-0.534731\pi$
$463$	−4.68629	−0.217790	−0.108895	+	0.994053i	$0.534731\pi$
$464$	−3.65685	−0.169765
$465$	3.31371	0.153670
$466$	20.6274	0.955547
$467$	−12.6863	−0.587052	−0.293526	−	0.955951i	$-0.594829\pi$
$467$	−12.6863	−0.587052	−0.293526	+	0.955951i	$0.594829\pi$
$468$	2.82843	0.130744
$469$	0	0
$470$	8.00000	0.369012
$471$	−10.0000	−0.460776
$472$	−1.65685	−0.0762629
$473$	16.0000	0.735681
$474$	−8.82843	−0.405503
$475$	−3.00000	−0.137649
$476$	0	0
$477$	7.65685	0.350583
$478$	−8.82843	−0.403803
$479$	−27.1127	−1.23881	−0.619405	−	0.785071i	$-0.712626\pi$
$479$	−27.1127	−1.23881	−0.619405	+	0.785071i	$0.712626\pi$
$480$	−2.82843	−0.129099
$481$	−18.3431	−0.836375
$482$	−22.6274	−1.03065
$483$	0	0
$484$	−7.00000	−0.318182
$485$	20.6863	0.939316
$486$	1.00000	0.0453609
$487$	−24.1421	−1.09398	−0.546992	−	0.837138i	$-0.684227\pi$
$487$	−24.1421	−1.09398	−0.546992	+	0.837138i	$0.684227\pi$
$488$	−9.31371	−0.421612
$489$	3.31371	0.149851
$490$	0	0
$491$	−4.34315	−0.196003	−0.0980017	−	0.995186i	$-0.531245\pi$
$491$	−4.34315	−0.196003	−0.0980017	+	0.995186i	$0.531245\pi$
$492$	7.65685	0.345198
$493$	0	0
$494$	−2.82843	−0.127257
$495$	−5.65685	−0.254257
$496$	−1.17157	−0.0526052
$497$	0	0
$498$	11.3137	0.506979
$499$	4.68629	0.209787	0.104894	−	0.994483i	$-0.466550\pi$
$499$	4.68629	0.209787	0.104894	+	0.994483i	$0.466550\pi$
$500$	5.65685	0.252982
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−11.5147	−0.513416	−0.256708	−	0.966489i	$-0.582638\pi$
$503$	−11.5147	−0.513416	−0.256708	+	0.966489i	$0.582638\pi$
$504$	0	0
$505$	−46.6274	−2.07489
$506$	9.65685	0.429300
$507$	−5.00000	−0.222058
$508$	8.82843	0.391698
$509$	−29.3137	−1.29931	−0.649654	−	0.760230i	$-0.725086\pi$
$509$	−29.3137	−1.29931	−0.649654	+	0.760230i	$0.725086\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−18.0000	−0.793946
$515$	−35.3137	−1.55611
$516$	8.00000	0.352180
$517$	−5.65685	−0.248788
$518$	0	0
$519$	19.6569	0.862840
$520$	−8.00000	−0.350823
$521$	−23.9411	−1.04888	−0.524440	−	0.851447i	$-0.675725\pi$
$521$	−23.9411	−1.04888	−0.524440	+	0.851447i	$0.675725\pi$
$522$	−3.65685	−0.160056
$523$	−13.6569	−0.597173	−0.298586	−	0.954383i	$-0.596515\pi$
$523$	−13.6569	−0.597173	−0.298586	+	0.954383i	$0.596515\pi$
$524$	15.3137	0.668982
$525$	0	0
$526$	−16.1421	−0.703831
$527$	0	0
$528$	2.00000	0.0870388
$529$	0.313708	0.0136395
$530$	−21.6569	−0.940714
$531$	−1.65685	−0.0719014
$532$	0	0
$533$	21.6569	0.938062
$534$	10.0000	0.432742
$535$	11.3137	0.489134
$536$	10.0000	0.431934
$537$	−4.00000	−0.172613
$538$	14.0000	0.603583
$539$	0	0
$540$	−2.82843	−0.121716
$541$	−41.3137	−1.77622	−0.888108	−	0.459636i	$-0.847980\pi$
$541$	−41.3137	−1.77622	−0.888108	+	0.459636i	$0.847980\pi$
$542$	−19.3137	−0.829595
$543$	−3.51472	−0.150831
$544$	0	0
$545$	−7.02944	−0.301108
$546$	0	0
$547$	43.9411	1.87879	0.939393	−	0.342841i	$-0.111389\pi$
$547$	43.9411	1.87879	0.939393	+	0.342841i	$0.111389\pi$
$548$	0.343146	0.0146585
$549$	−9.31371	−0.397499
$550$	6.00000	0.255841
$551$	3.65685	0.155787
$552$	4.82843	0.205512
$553$	0	0
$554$	6.00000	0.254916
$555$	18.3431	0.778623
$556$	1.65685	0.0702663
$557$	16.8284	0.713043	0.356522	−	0.934287i	$-0.383963\pi$
$557$	16.8284	0.713043	0.356522	+	0.934287i	$0.383963\pi$
$558$	−1.17157	−0.0495966
$559$	22.6274	0.957038
$560$	0	0
$561$	0	0
$562$	−2.68629	−0.113314
$563$	−35.5980	−1.50028	−0.750138	−	0.661281i	$-0.770013\pi$
$563$	−35.5980	−1.50028	−0.750138	+	0.661281i	$0.770013\pi$
$564$	−2.82843	−0.119098
$565$	16.9706	0.713957
$566$	−17.6569	−0.742173
$567$	0	0
$568$	1.65685	0.0695201
$569$	−5.31371	−0.222762	−0.111381	−	0.993778i	$-0.535527\pi$
$569$	−5.31371	−0.222762	−0.111381	+	0.993778i	$0.535527\pi$
$570$	2.82843	0.118470
$571$	42.6274	1.78390	0.891951	−	0.452132i	$-0.149336\pi$
$571$	42.6274	1.78390	0.891951	+	0.452132i	$0.149336\pi$
$572$	5.65685	0.236525
$573$	16.1421	0.674347
$574$	0	0
$575$	14.4853	0.604078
$576$	1.00000	0.0416667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	−17.0000	−0.707107
$579$	26.9706	1.12086
$580$	10.3431	0.429476
$581$	0	0
$582$	−7.31371	−0.303163
$583$	15.3137	0.634229
$584$	11.6569	0.482364
$585$	−8.00000	−0.330759
$586$	−10.0000	−0.413096
$587$	−10.3431	−0.426907	−0.213454	−	0.976953i	$-0.568471\pi$
$587$	−10.3431	−0.426907	−0.213454	+	0.976953i	$0.568471\pi$
$588$	0	0
$589$	1.17157	0.0482738
$590$	4.68629	0.192932
$591$	0.828427	0.0340769
$592$	−6.48528	−0.266543
$593$	−33.6569	−1.38212	−0.691061	−	0.722797i	$-0.742856\pi$
$593$	−33.6569	−1.38212	−0.691061	+	0.722797i	$0.742856\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	−18.4853	−0.757187
$597$	16.9706	0.694559
$598$	13.6569	0.558470
$599$	20.9706	0.856834	0.428417	−	0.903581i	$-0.359071\pi$
$599$	20.9706	0.856834	0.428417	+	0.903581i	$0.359071\pi$
$600$	3.00000	0.122474
$601$	33.6569	1.37289	0.686446	−	0.727181i	$-0.259170\pi$
$601$	33.6569	1.37289	0.686446	+	0.727181i	$0.259170\pi$
$602$	0	0
$603$	10.0000	0.407231
$604$	−4.14214	−0.168541
$605$	19.7990	0.804943
$606$	16.4853	0.669669
$607$	−26.8284	−1.08893	−0.544466	−	0.838783i	$-0.683268\pi$
$607$	−26.8284	−1.08893	−0.544466	+	0.838783i	$0.683268\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	26.3431	1.06660
$611$	−8.00000	−0.323645
$612$	0	0
$613$	−30.2843	−1.22317	−0.611585	−	0.791179i	$-0.709468\pi$
$613$	−30.2843	−1.22317	−0.611585	+	0.791179i	$0.709468\pi$
$614$	5.65685	0.228292
$615$	−21.6569	−0.873289
$616$	0	0
$617$	10.6863	0.430214	0.215107	−	0.976590i	$-0.430990\pi$
$617$	10.6863	0.430214	0.215107	+	0.976590i	$0.430990\pi$
$618$	12.4853	0.502232
$619$	−7.31371	−0.293963	−0.146981	−	0.989139i	$-0.546956\pi$
$619$	−7.31371	−0.293963	−0.146981	+	0.989139i	$0.546956\pi$
$620$	3.31371	0.133082
$621$	4.82843	0.193758
$622$	20.4853	0.821385
$623$	0	0
$624$	2.82843	0.113228
$625$	−31.0000	−1.24000
$626$	27.6569	1.10539
$627$	−2.00000	−0.0798723
$628$	−10.0000	−0.399043
$629$	0	0
$630$	0	0
$631$	−41.6569	−1.65833	−0.829167	−	0.559002i	$-0.811185\pi$
$631$	−41.6569	−1.65833	−0.829167	+	0.559002i	$0.811185\pi$
$632$	−8.82843	−0.351176
$633$	4.34315	0.172625
$634$	30.9706	1.23000
$635$	−24.9706	−0.990927
$636$	7.65685	0.303614
$637$	0	0
$638$	−7.31371	−0.289552
$639$	1.65685	0.0655441
$640$	−2.82843	−0.111803
$641$	−26.6863	−1.05405	−0.527023	−	0.849851i	$-0.676692\pi$
$641$	−26.6863	−1.05405	−0.527023	+	0.849851i	$0.676692\pi$
$642$	−4.00000	−0.157867
$643$	44.9706	1.77347	0.886733	−	0.462282i	$-0.152969\pi$
$643$	44.9706	1.77347	0.886733	+	0.462282i	$0.152969\pi$
$644$	0	0
$645$	−22.6274	−0.890954
$646$	0	0
$647$	−5.85786	−0.230296	−0.115148	−	0.993348i	$-0.536734\pi$
$647$	−5.85786	−0.230296	−0.115148	+	0.993348i	$0.536734\pi$
$648$	1.00000	0.0392837
$649$	−3.31371	−0.130074
$650$	8.48528	0.332820
$651$	0	0
$652$	3.31371	0.129775
$653$	25.5147	0.998468	0.499234	−	0.866467i	$-0.333615\pi$
$653$	25.5147	0.998468	0.499234	+	0.866467i	$0.333615\pi$
$654$	2.48528	0.0971822
$655$	−43.3137	−1.69241
$656$	7.65685	0.298950
$657$	11.6569	0.454777
$658$	0	0
$659$	−29.9411	−1.16634	−0.583170	−	0.812350i	$-0.698188\pi$
$659$	−29.9411	−1.16634	−0.583170	+	0.812350i	$0.698188\pi$
$660$	−5.65685	−0.220193
$661$	−15.7990	−0.614509	−0.307255	−	0.951627i	$-0.599410\pi$
$661$	−15.7990	−0.614509	−0.307255	+	0.951627i	$0.599410\pi$
$662$	12.6274	0.490778
$663$	0	0
$664$	11.3137	0.439057
$665$	0	0
$666$	−6.48528	−0.251300
$667$	−17.6569	−0.683676
$668$	0	0
$669$	−8.48528	−0.328060
$670$	−28.2843	−1.09272
$671$	−18.6274	−0.719103
$672$	0	0
$673$	−34.9706	−1.34802	−0.674008	−	0.738724i	$-0.735429\pi$
$673$	−34.9706	−1.34802	−0.674008	+	0.738724i	$0.735429\pi$
$674$	−13.3137	−0.512825
$675$	3.00000	0.115470
$676$	−5.00000	−0.192308
$677$	14.6863	0.564440	0.282220	−	0.959350i	$-0.408929\pi$
$677$	14.6863	0.564440	0.282220	+	0.959350i	$0.408929\pi$
$678$	−6.00000	−0.230429
$679$	0	0
$680$	0	0
$681$	15.3137	0.586823
$682$	−2.34315	−0.0897237
$683$	33.9411	1.29872	0.649361	−	0.760481i	$-0.275037\pi$
$683$	33.9411	1.29872	0.649361	+	0.760481i	$0.275037\pi$
$684$	−1.00000	−0.0382360
$685$	−0.970563	−0.0370833
$686$	0	0
$687$	−0.343146	−0.0130918
$688$	8.00000	0.304997
$689$	21.6569	0.825060
$690$	−13.6569	−0.519908
$691$	0.686292	0.0261078	0.0130539	−	0.999915i	$-0.495845\pi$
$691$	0.686292	0.0261078	0.0130539	+	0.999915i	$0.495845\pi$
$692$	19.6569	0.747241
$693$	0	0
$694$	14.0000	0.531433
$695$	−4.68629	−0.177761
$696$	−3.65685	−0.138613
$697$	0	0
$698$	−18.0000	−0.681310
$699$	20.6274	0.780201
$700$	0	0
$701$	23.1716	0.875178	0.437589	−	0.899175i	$-0.355832\pi$
$701$	23.1716	0.875178	0.437589	+	0.899175i	$0.355832\pi$
$702$	2.82843	0.106752
$703$	6.48528	0.244597
$704$	2.00000	0.0753778
$705$	8.00000	0.301297
$706$	18.3431	0.690353
$707$	0	0
$708$	−1.65685	−0.0622684
$709$	−18.2843	−0.686680	−0.343340	−	0.939211i	$-0.611558\pi$
$709$	−18.2843	−0.686680	−0.343340	+	0.939211i	$0.611558\pi$
$710$	−4.68629	−0.175873
$711$	−8.82843	−0.331092
$712$	10.0000	0.374766
$713$	−5.65685	−0.211851
$714$	0	0
$715$	−16.0000	−0.598366
$716$	−4.00000	−0.149487
$717$	−8.82843	−0.329704
$718$	−10.4853	−0.391307
$719$	−2.14214	−0.0798882	−0.0399441	−	0.999202i	$-0.512718\pi$
$719$	−2.14214	−0.0798882	−0.0399441	+	0.999202i	$0.512718\pi$
$720$	−2.82843	−0.105409
$721$	0	0
$722$	1.00000	0.0372161
$723$	−22.6274	−0.841523
$724$	−3.51472	−0.130623
$725$	−10.9706	−0.407436
$726$	−7.00000	−0.259794
$727$	0.970563	0.0359962	0.0179981	−	0.999838i	$-0.494271\pi$
$727$	0.970563	0.0359962	0.0179981	+	0.999838i	$0.494271\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−32.9706	−1.22030
$731$	0	0
$732$	−9.31371	−0.344245
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	4.82843	0.177978
$737$	20.0000	0.736709
$738$	7.65685	0.281853
$739$	10.6274	0.390936	0.195468	−	0.980710i	$-0.437377\pi$
$739$	10.6274	0.390936	0.195468	+	0.980710i	$0.437377\pi$
$740$	18.3431	0.674307
$741$	−2.82843	−0.103905
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	−1.17157	−0.0429519
$745$	52.2843	1.91555
$746$	−34.4853	−1.26260
$747$	11.3137	0.413947
$748$	0	0
$749$	0	0
$750$	5.65685	0.206559
$751$	13.1127	0.478489	0.239245	−	0.970959i	$-0.423100\pi$
$751$	13.1127	0.478489	0.239245	+	0.970959i	$0.423100\pi$
$752$	−2.82843	−0.103142
$753$	−12.0000	−0.437304
$754$	−10.3431	−0.376675
$755$	11.7157	0.426379
$756$	0	0
$757$	48.9117	1.77773	0.888863	−	0.458174i	$-0.151496\pi$
$757$	48.9117	1.77773	0.888863	+	0.458174i	$0.151496\pi$
$758$	20.6274	0.749222
$759$	9.65685	0.350522
$760$	2.82843	0.102598
$761$	42.6274	1.54524	0.772621	−	0.634867i	$-0.218945\pi$
$761$	42.6274	1.54524	0.772621	+	0.634867i	$0.218945\pi$
$762$	8.82843	0.319820
$763$	0	0
$764$	16.1421	0.584002
$765$	0	0
$766$	30.6274	1.10661
$767$	−4.68629	−0.169212
$768$	1.00000	0.0360844
$769$	−33.5980	−1.21157	−0.605787	−	0.795627i	$-0.707142\pi$
$769$	−33.5980	−1.21157	−0.605787	+	0.795627i	$0.707142\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	26.9706	0.970692
$773$	−40.6274	−1.46127	−0.730633	−	0.682770i	$-0.760775\pi$
$773$	−40.6274	−1.46127	−0.730633	+	0.682770i	$0.760775\pi$
$774$	8.00000	0.287554
$775$	−3.51472	−0.126252
$776$	−7.31371	−0.262547
$777$	0	0
$778$	21.7990	0.781532
$779$	−7.65685	−0.274335
$780$	−8.00000	−0.286446
$781$	3.31371	0.118574
$782$	0	0
$783$	−3.65685	−0.130685
$784$	0	0
$785$	28.2843	1.00951
$786$	15.3137	0.546222
$787$	−35.3137	−1.25880	−0.629399	−	0.777082i	$-0.716699\pi$
$787$	−35.3137	−1.25880	−0.629399	+	0.777082i	$0.716699\pi$
$788$	0.828427	0.0295115
$789$	−16.1421	−0.574675
$790$	24.9706	0.888413
$791$	0	0
$792$	2.00000	0.0710669
$793$	−26.3431	−0.935473
$794$	−38.2843	−1.35866
$795$	−21.6569	−0.768089
$796$	16.9706	0.601506
$797$	−10.9706	−0.388597	−0.194299	−	0.980942i	$-0.562243\pi$
$797$	−10.9706	−0.388597	−0.194299	+	0.980942i	$0.562243\pi$
$798$	0	0
$799$	0	0
$800$	3.00000	0.106066
$801$	10.0000	0.353333
$802$	−16.6274	−0.587135
$803$	23.3137	0.822723
$804$	10.0000	0.352673
$805$	0	0
$806$	−3.31371	−0.116720
$807$	14.0000	0.492823
$808$	16.4853	0.579950
$809$	−34.2843	−1.20537	−0.602685	−	0.797979i	$-0.705903\pi$
$809$	−34.2843	−1.20537	−0.602685	+	0.797979i	$0.705903\pi$
$810$	−2.82843	−0.0993808
$811$	−5.65685	−0.198639	−0.0993195	−	0.995056i	$-0.531667\pi$
$811$	−5.65685	−0.198639	−0.0993195	+	0.995056i	$0.531667\pi$
$812$	0	0
$813$	−19.3137	−0.677361
$814$	−12.9706	−0.454618
$815$	−9.37258	−0.328307
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0.686292	0.0239956
$819$	0	0
$820$	−21.6569	−0.756290
$821$	−35.4558	−1.23742	−0.618709	−	0.785620i	$-0.712344\pi$
$821$	−35.4558	−1.23742	−0.618709	+	0.785620i	$0.712344\pi$
$822$	0.343146	0.0119686
$823$	30.6274	1.06760	0.533802	−	0.845609i	$-0.320763\pi$
$823$	30.6274	1.06760	0.533802	+	0.845609i	$0.320763\pi$
$824$	12.4853	0.434945
$825$	6.00000	0.208893
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	4.82843	0.167799
$829$	−17.4558	−0.606267	−0.303133	−	0.952948i	$-0.598033\pi$
$829$	−17.4558	−0.606267	−0.303133	+	0.952948i	$0.598033\pi$
$830$	−32.0000	−1.11074
$831$	6.00000	0.208138
$832$	2.82843	0.0980581
$833$	0	0
$834$	1.65685	0.0573722
$835$	0	0
$836$	−2.00000	−0.0691714
$837$	−1.17157	−0.0404955
$838$	−20.9706	−0.724416
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	8.82843	0.304248
$843$	−2.68629	−0.0925208
$844$	4.34315	0.149497
$845$	14.1421	0.486504
$846$	−2.82843	−0.0972433
$847$	0	0
$848$	7.65685	0.262937
$849$	−17.6569	−0.605982
$850$	0	0
$851$	−31.3137	−1.07342
$852$	1.65685	0.0567629
$853$	21.3137	0.729767	0.364884	−	0.931053i	$-0.381109\pi$
$853$	21.3137	0.729767	0.364884	+	0.931053i	$0.381109\pi$
$854$	0	0
$855$	2.82843	0.0967302
$856$	−4.00000	−0.136717
$857$	−4.62742	−0.158070	−0.0790348	−	0.996872i	$-0.525184\pi$
$857$	−4.62742	−0.158070	−0.0790348	+	0.996872i	$0.525184\pi$
$858$	5.65685	0.193122
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	−22.6274	−0.771589
$861$	0	0
$862$	6.34315	0.216048
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	1.00000	0.0340207
$865$	−55.5980	−1.89039
$866$	−7.31371	−0.248530
$867$	−17.0000	−0.577350
$868$	0	0
$869$	−17.6569	−0.598968
$870$	10.3431	0.350665
$871$	28.2843	0.958376
$872$	2.48528	0.0841622
$873$	−7.31371	−0.247532
$874$	−4.82843	−0.163324
$875$	0	0
$876$	11.6569	0.393849
$877$	35.4558	1.19726	0.598629	−	0.801026i	$-0.295712\pi$
$877$	35.4558	1.19726	0.598629	+	0.801026i	$0.295712\pi$
$878$	3.51472	0.118616
$879$	−10.0000	−0.337292
$880$	−5.65685	−0.190693
$881$	−32.9706	−1.11081	−0.555403	−	0.831581i	$-0.687436\pi$
$881$	−32.9706	−1.11081	−0.555403	+	0.831581i	$0.687436\pi$
$882$	0	0
$883$	−28.6863	−0.965371	−0.482685	−	0.875794i	$-0.660338\pi$
$883$	−28.6863	−0.965371	−0.482685	+	0.875794i	$0.660338\pi$
$884$	0	0
$885$	4.68629	0.157528
$886$	−17.3137	−0.581665
$887$	−34.3431	−1.15313	−0.576565	−	0.817051i	$-0.695607\pi$
$887$	−34.3431	−1.15313	−0.576565	+	0.817051i	$0.695607\pi$
$888$	−6.48528	−0.217632
$889$	0	0
$890$	−28.2843	−0.948091
$891$	2.00000	0.0670025
$892$	−8.48528	−0.284108
$893$	2.82843	0.0946497
$894$	−18.4853	−0.618240
$895$	11.3137	0.378176
$896$	0	0
$897$	13.6569	0.455989
$898$	−14.0000	−0.467186
$899$	4.28427	0.142888
$900$	3.00000	0.100000
$901$	0	0
$902$	15.3137	0.509891
$903$	0	0
$904$	−6.00000	−0.199557
$905$	9.94113	0.330454
$906$	−4.14214	−0.137613
$907$	12.6274	0.419286	0.209643	−	0.977778i	$-0.432770\pi$
$907$	12.6274	0.419286	0.209643	+	0.977778i	$0.432770\pi$
$908$	15.3137	0.508203
$909$	16.4853	0.546782
$910$	0	0
$911$	−49.6569	−1.64520	−0.822602	−	0.568617i	$-0.807479\pi$
$911$	−49.6569	−1.64520	−0.822602	+	0.568617i	$0.807479\pi$
$912$	−1.00000	−0.0331133
$913$	22.6274	0.748858
$914$	−32.6274	−1.07922
$915$	26.3431	0.870878
$916$	−0.343146	−0.0113379
$917$	0	0
$918$	0	0
$919$	43.3137	1.42879	0.714394	−	0.699744i	$-0.246703\pi$
$919$	43.3137	1.42879	0.714394	+	0.699744i	$0.246703\pi$
$920$	−13.6569	−0.450253
$921$	5.65685	0.186400
$922$	−9.85786	−0.324651
$923$	4.68629	0.154251
$924$	0	0
$925$	−19.4558	−0.639704
$926$	−4.68629	−0.154001
$927$	12.4853	0.410070
$928$	−3.65685	−0.120042
$929$	27.3137	0.896134	0.448067	−	0.894000i	$-0.352113\pi$
$929$	27.3137	0.896134	0.448067	+	0.894000i	$0.352113\pi$
$930$	3.31371	0.108661
$931$	0	0
$932$	20.6274	0.675674
$933$	20.4853	0.670658
$934$	−12.6863	−0.415108
$935$	0	0
$936$	2.82843	0.0924500
$937$	22.2843	0.727995	0.363998	−	0.931400i	$-0.381412\pi$
$937$	22.2843	0.727995	0.363998	+	0.931400i	$0.381412\pi$
$938$	0	0
$939$	27.6569	0.902547
$940$	8.00000	0.260931
$941$	−5.31371	−0.173222	−0.0866110	−	0.996242i	$-0.527604\pi$
$941$	−5.31371	−0.173222	−0.0866110	+	0.996242i	$0.527604\pi$
$942$	−10.0000	−0.325818
$943$	36.9706	1.20393
$944$	−1.65685	−0.0539260
$945$	0	0
$946$	16.0000	0.520205
$947$	−54.2843	−1.76400	−0.882001	−	0.471248i	$-0.843804\pi$
$947$	−54.2843	−1.76400	−0.882001	+	0.471248i	$0.843804\pi$
$948$	−8.82843	−0.286734
$949$	32.9706	1.07027
$950$	−3.00000	−0.0973329
$951$	30.9706	1.00429
$952$	0	0
$953$	−4.62742	−0.149897	−0.0749484	−	0.997187i	$-0.523879\pi$
$953$	−4.62742	−0.149897	−0.0749484	+	0.997187i	$0.523879\pi$
$954$	7.65685	0.247900
$955$	−45.6569	−1.47742
$956$	−8.82843	−0.285532
$957$	−7.31371	−0.236419
$958$	−27.1127	−0.875972
$959$	0	0
$960$	−2.82843	−0.0912871
$961$	−29.6274	−0.955723
$962$	−18.3431	−0.591407
$963$	−4.00000	−0.128898
$964$	−22.6274	−0.728780
$965$	−76.2843	−2.45568
$966$	0	0
$967$	30.3431	0.975770	0.487885	−	0.872908i	$-0.337769\pi$
$967$	30.3431	0.975770	0.487885	+	0.872908i	$0.337769\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	20.6863	0.664197
$971$	55.3137	1.77510	0.887551	−	0.460710i	$-0.152405\pi$
$971$	55.3137	1.77510	0.887551	+	0.460710i	$0.152405\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−24.1421	−0.773564
$975$	8.48528	0.271746
$976$	−9.31371	−0.298125
$977$	22.0000	0.703842	0.351921	−	0.936030i	$-0.385529\pi$
$977$	22.0000	0.703842	0.351921	+	0.936030i	$0.385529\pi$
$978$	3.31371	0.105961
$979$	20.0000	0.639203
$980$	0	0
$981$	2.48528	0.0793489
$982$	−4.34315	−0.138595
$983$	42.9117	1.36867	0.684335	−	0.729168i	$-0.260093\pi$
$983$	42.9117	1.36867	0.684335	+	0.729168i	$0.260093\pi$
$984$	7.65685	0.244092
$985$	−2.34315	−0.0746588
$986$	0	0
$987$	0	0
$988$	−2.82843	−0.0899843
$989$	38.6274	1.22828
$990$	−5.65685	−0.179787
$991$	6.20101	0.196982	0.0984908	−	0.995138i	$-0.468598\pi$
$991$	6.20101	0.196982	0.0984908	+	0.995138i	$0.468598\pi$
$992$	−1.17157	−0.0371975
$993$	12.6274	0.400719
$994$	0	0
$995$	−48.0000	−1.52170
$996$	11.3137	0.358489
$997$	4.62742	0.146552	0.0732759	−	0.997312i	$-0.476655\pi$
$997$	4.62742	0.146552	0.0732759	+	0.997312i	$0.476655\pi$
$998$	4.68629	0.148342
$999$	−6.48528	−0.205185

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5586.2.a.bp.1.1		2
7.6	odd	2		798.2.a.l.1.2	✓	2
21.20	even	2		2394.2.a.r.1.1		2
28.27	even	2		6384.2.a.bq.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.a.l.1.2	✓	2	7.6	odd	2
2394.2.a.r.1.1		2	21.20	even	2
5586.2.a.bp.1.1		2	1.1	even	1	trivial
6384.2.a.bq.1.2		2	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.82843 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.82843 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.82843 q^{10} +2.00000 q^{11} +1.00000 q^{12} +2.82843 q^{13} -2.82843 q^{15} +1.00000 q^{16} +1.00000 q^{18} -1.00000 q^{19} -2.82843 q^{20} +2.00000 q^{22} +4.82843 q^{23} +1.00000 q^{24} +3.00000 q^{25} +2.82843 q^{26} +1.00000 q^{27} -3.65685 q^{29} -2.82843 q^{30} -1.17157 q^{31} +1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{36} -6.48528 q^{37} -1.00000 q^{38} +2.82843 q^{39} -2.82843 q^{40} +7.65685 q^{41} +8.00000 q^{43} +2.00000 q^{44} -2.82843 q^{45} +4.82843 q^{46} -2.82843 q^{47} +1.00000 q^{48} +3.00000 q^{50} +2.82843 q^{52} +7.65685 q^{53} +1.00000 q^{54} -5.65685 q^{55} -1.00000 q^{57} -3.65685 q^{58} -1.65685 q^{59} -2.82843 q^{60} -9.31371 q^{61} -1.17157 q^{62} +1.00000 q^{64} -8.00000 q^{65} +2.00000 q^{66} +10.0000 q^{67} +4.82843 q^{69} +1.65685 q^{71} +1.00000 q^{72} +11.6569 q^{73} -6.48528 q^{74} +3.00000 q^{75} -1.00000 q^{76} +2.82843 q^{78} -8.82843 q^{79} -2.82843 q^{80} +1.00000 q^{81} +7.65685 q^{82} +11.3137 q^{83} +8.00000 q^{86} -3.65685 q^{87} +2.00000 q^{88} +10.0000 q^{89} -2.82843 q^{90} +4.82843 q^{92} -1.17157 q^{93} -2.82843 q^{94} +2.82843 q^{95} +1.00000 q^{96} -7.31371 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} + 2 q^{16} + 2 q^{18} - 2 q^{19} + 4 q^{22} + 4 q^{23} + 2 q^{24} + 6 q^{25} + 2 q^{27} + 4 q^{29} - 8 q^{31} + 2 q^{32} + 4 q^{33} + 2 q^{36} + 4 q^{37} - 2 q^{38} + 4 q^{41} + 16 q^{43} + 4 q^{44} + 4 q^{46} + 2 q^{48} + 6 q^{50} + 4 q^{53} + 2 q^{54} - 2 q^{57} + 4 q^{58} + 8 q^{59} + 4 q^{61} - 8 q^{62} + 2 q^{64} - 16 q^{65} + 4 q^{66} + 20 q^{67} + 4 q^{69} - 8 q^{71} + 2 q^{72} + 12 q^{73} + 4 q^{74} + 6 q^{75} - 2 q^{76} - 12 q^{79} + 2 q^{81} + 4 q^{82} + 16 q^{86} + 4 q^{87} + 4 q^{88} + 20 q^{89} + 4 q^{92} - 8 q^{93} + 2 q^{96} + 8 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 + 2 * q^16 + 2 * q^18 - 2 * q^19 + 4 * q^22 + 4 * q^23 + 2 * q^24 + 6 * q^25 + 2 * q^27 + 4 * q^29 - 8 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^36 + 4 * q^37 - 2 * q^38 + 4 * q^41 + 16 * q^43 + 4 * q^44 + 4 * q^46 + 2 * q^48 + 6 * q^50 + 4 * q^53 + 2 * q^54 - 2 * q^57 + 4 * q^58 + 8 * q^59 + 4 * q^61 - 8 * q^62 + 2 * q^64 - 16 * q^65 + 4 * q^66 + 20 * q^67 + 4 * q^69 - 8 * q^71 + 2 * q^72 + 12 * q^73 + 4 * q^74 + 6 * q^75 - 2 * q^76 - 12 * q^79 + 2 * q^81 + 4 * q^82 + 16 * q^86 + 4 * q^87 + 4 * q^88 + 20 * q^89 + 4 * q^92 - 8 * q^93 + 2 * q^96 + 8 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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