LMFDB - Embedded newform 5586.2.a.bj.1.2

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			1.2
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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.58579 q^{11} -1.00000 q^{12} +2.82843 q^{13} +1.00000 q^{16} +1.17157 q^{17} +1.00000 q^{18} +1.00000 q^{19} -2.58579 q^{22} -4.24264 q^{23} -1.00000 q^{24} -5.00000 q^{25} +2.82843 q^{26} -1.00000 q^{27} -8.00000 q^{29} -3.65685 q^{31} +1.00000 q^{32} +2.58579 q^{33} +1.17157 q^{34} +1.00000 q^{36} -2.00000 q^{37} +1.00000 q^{38} -2.82843 q^{39} +2.58579 q^{41} -6.82843 q^{43} -2.58579 q^{44} -4.24264 q^{46} +7.65685 q^{47} -1.00000 q^{48} -5.00000 q^{50} -1.17157 q^{51} +2.82843 q^{52} -2.00000 q^{53} -1.00000 q^{54} -1.00000 q^{57} -8.00000 q^{58} -2.58579 q^{61} -3.65685 q^{62} +1.00000 q^{64} +2.58579 q^{66} -4.24264 q^{67} +1.17157 q^{68} +4.24264 q^{69} +14.1421 q^{71} +1.00000 q^{72} -9.41421 q^{73} -2.00000 q^{74} +5.00000 q^{75} +1.00000 q^{76} -2.82843 q^{78} +5.89949 q^{79} +1.00000 q^{81} +2.58579 q^{82} +9.65685 q^{83} -6.82843 q^{86} +8.00000 q^{87} -2.58579 q^{88} -2.58579 q^{89} -4.24264 q^{92} +3.65685 q^{93} +7.65685 q^{94} -1.00000 q^{96} -14.8284 q^{97} -2.58579 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	−2.58579	−0.779644	−0.389822	−	0.920890i	$-0.627463\pi$
$11$	−2.58579	−0.779644	−0.389822	+	0.920890i	$0.627463\pi$
$12$	−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
$17$	1.17157	0.284148	0.142074	−	0.989856i	$-0.454623\pi$
$17$	1.17157	0.284148	0.142074	+	0.989856i	$0.454623\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−2.58579	−0.551292
$23$	−4.24264	−0.884652	−0.442326	−	0.896854i	$-0.645847\pi$
$23$	−4.24264	−0.884652	−0.442326	+	0.896854i	$0.645847\pi$
$24$	−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
$31$	−3.65685	−0.656790	−0.328395	−	0.944540i	$-0.606508\pi$
$31$	−3.65685	−0.656790	−0.328395	+	0.944540i	$0.606508\pi$
$32$	1.00000	0.176777
$33$	2.58579	0.450128
$34$	1.17157	0.200923
$35$	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
$41$	2.58579	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$41$	2.58579	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$42$	0	0
$43$	−6.82843	−1.04133	−0.520663	−	0.853762i	$-0.674315\pi$
$43$	−6.82843	−1.04133	−0.520663	+	0.853762i	$0.674315\pi$
$44$	−2.58579	−0.389822
$45$	0	0
$46$	−4.24264	−0.625543
$47$	7.65685	1.11687	0.558433	−	0.829549i	$-0.311403\pi$
$47$	7.65685	1.11687	0.558433	+	0.829549i	$0.311403\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−5.00000	−0.707107
$51$	−1.17157	−0.164053
$52$	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
$61$	−2.58579	−0.331076	−0.165538	−	0.986203i	$-0.552936\pi$
$61$	−2.58579	−0.331076	−0.165538	+	0.986203i	$0.552936\pi$
$62$	−3.65685	−0.464421
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	2.58579	0.318288
$67$	−4.24264	−0.518321	−0.259161	−	0.965834i	$-0.583446\pi$
$67$	−4.24264	−0.518321	−0.259161	+	0.965834i	$0.583446\pi$
$68$	1.17157	0.142074
$69$	4.24264	0.510754
$70$	0	0
$71$	14.1421	1.67836	0.839181	−	0.543852i	$-0.183035\pi$
$71$	14.1421	1.67836	0.839181	+	0.543852i	$0.183035\pi$
$72$	1.00000	0.117851
$73$	−9.41421	−1.10185	−0.550925	−	0.834555i	$-0.685725\pi$
$73$	−9.41421	−1.10185	−0.550925	+	0.834555i	$0.685725\pi$
$74$	−2.00000	−0.232495
$75$	5.00000	0.577350
$76$	1.00000	0.114708
$77$	0	0
$78$	−2.82843	−0.320256
$79$	5.89949	0.663745	0.331873	−	0.943324i	$-0.392320\pi$
$79$	5.89949	0.663745	0.331873	+	0.943324i	$0.392320\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.58579	0.285552
$83$	9.65685	1.05998	0.529989	−	0.848005i	$-0.322196\pi$
$83$	9.65685	1.05998	0.529989	+	0.848005i	$0.322196\pi$
$84$	0	0
$85$	0	0
$86$	−6.82843	−0.736328
$87$	8.00000	0.857690
$88$	−2.58579	−0.275646
$89$	−2.58579	−0.274093	−0.137046	−	0.990565i	$-0.543761\pi$
$89$	−2.58579	−0.274093	−0.137046	+	0.990565i	$0.543761\pi$
$90$	0	0
$91$	0	0
$92$	−4.24264	−0.442326
$93$	3.65685	0.379198
$94$	7.65685	0.789744
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−14.8284	−1.50560	−0.752799	−	0.658250i	$-0.771297\pi$
$97$	−14.8284	−1.50560	−0.752799	+	0.658250i	$0.771297\pi$
$98$	0	0
$99$	−2.58579	−0.259881
$100$	−5.00000	−0.500000
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	−1.17157	−0.116003
$103$	7.65685	0.754452	0.377226	−	0.926121i	$-0.376878\pi$
$103$	7.65685	0.754452	0.377226	+	0.926121i	$0.376878\pi$
$104$	2.82843	0.277350
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−9.65685	−0.933563	−0.466782	−	0.884373i	$-0.654587\pi$
$107$	−9.65685	−0.933563	−0.466782	+	0.884373i	$0.654587\pi$
$108$	−1.00000	−0.0962250
$109$	−18.9706	−1.81705	−0.908525	−	0.417830i	$-0.862791\pi$
$109$	−18.9706	−1.81705	−0.908525	+	0.417830i	$0.862791\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	1.65685	0.155864	0.0779319	−	0.996959i	$-0.475168\pi$
$113$	1.65685	0.155864	0.0779319	+	0.996959i	$0.475168\pi$
$114$	−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
$121$	−4.31371	−0.392155
$122$	−2.58579	−0.234106
$123$	−2.58579	−0.233153
$124$	−3.65685	−0.328395
$125$	0	0
$126$	0	0
$127$	0.242641	0.0215309	0.0107654	−	0.999942i	$-0.496573\pi$
$127$	0.242641	0.0215309	0.0107654	+	0.999942i	$0.496573\pi$
$128$	1.00000	0.0883883
$129$	6.82843	0.601209
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	2.58579	0.225064
$133$	0	0
$134$	−4.24264	−0.366508
$135$	0	0
$136$	1.17157	0.100462
$137$	14.9706	1.27902	0.639511	−	0.768782i	$-0.279137\pi$
$137$	14.9706	1.27902	0.639511	+	0.768782i	$0.279137\pi$
$138$	4.24264	0.361158
$139$	−0.686292	−0.0582105	−0.0291052	−	0.999576i	$-0.509266\pi$
$139$	−0.686292	−0.0582105	−0.0291052	+	0.999576i	$0.509266\pi$
$140$	0	0
$141$	−7.65685	−0.644823
$142$	14.1421	1.18678
$143$	−7.31371	−0.611603
$144$	1.00000	0.0833333
$145$	0	0
$146$	−9.41421	−0.779126
$147$	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
$151$	−0.242641	−0.0197458	−0.00987291	−	0.999951i	$-0.503143\pi$
$151$	−0.242641	−0.0197458	−0.00987291	+	0.999951i	$0.503143\pi$
$152$	1.00000	0.0811107
$153$	1.17157	0.0947161
$154$	0	0
$155$	0	0
$156$	−2.82843	−0.226455
$157$	−16.2426	−1.29630	−0.648152	−	0.761511i	$-0.724458\pi$
$157$	−16.2426	−1.29630	−0.648152	+	0.761511i	$0.724458\pi$
$158$	5.89949	0.469339
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	4.48528	0.351314	0.175657	−	0.984451i	$-0.443795\pi$
$163$	4.48528	0.351314	0.175657	+	0.984451i	$0.443795\pi$
$164$	2.58579	0.201916
$165$	0	0
$166$	9.65685	0.749517
$167$	−8.97056	−0.694163	−0.347081	−	0.937835i	$-0.612827\pi$
$167$	−8.97056	−0.694163	−0.347081	+	0.937835i	$0.612827\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	1.00000	0.0764719
$172$	−6.82843	−0.520663
$173$	−12.2426	−0.930791	−0.465395	−	0.885103i	$-0.654088\pi$
$173$	−12.2426	−0.930791	−0.465395	+	0.885103i	$0.654088\pi$
$174$	8.00000	0.606478
$175$	0	0
$176$	−2.58579	−0.194911
$177$	0	0
$178$	−2.58579	−0.193813
$179$	0.686292	0.0512958	0.0256479	−	0.999671i	$-0.491835\pi$
$179$	0.686292	0.0512958	0.0256479	+	0.999671i	$0.491835\pi$
$180$	0	0
$181$	−3.31371	−0.246306	−0.123153	−	0.992388i	$-0.539301\pi$
$181$	−3.31371	−0.246306	−0.123153	+	0.992388i	$0.539301\pi$
$182$	0	0
$183$	2.58579	0.191147
$184$	−4.24264	−0.312772
$185$	0	0
$186$	3.65685	0.268134
$187$	−3.02944	−0.221534
$188$	7.65685	0.558433
$189$	0	0
$190$	0	0
$191$	−1.89949	−0.137443	−0.0687213	−	0.997636i	$-0.521892\pi$
$191$	−1.89949	−0.137443	−0.0687213	+	0.997636i	$0.521892\pi$
$192$	−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	$0.617192\pi$
$194$	−14.8284	−1.06462
$195$	0	0
$196$	0	0
$197$	−23.6569	−1.68548	−0.842741	−	0.538320i	$-0.819059\pi$
$197$	−23.6569	−1.68548	−0.842741	+	0.538320i	$0.819059\pi$
$198$	−2.58579	−0.183764
$199$	−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
$204$	−1.17157	−0.0820265
$205$	0	0
$206$	7.65685	0.533478
$207$	−4.24264	−0.294884
$208$	2.82843	0.196116
$209$	−2.58579	−0.178863
$210$	0	0
$211$	−6.10051	−0.419976	−0.209988	−	0.977704i	$-0.567343\pi$
$211$	−6.10051	−0.419976	−0.209988	+	0.977704i	$0.567343\pi$
$212$	−2.00000	−0.137361
$213$	−14.1421	−0.969003
$214$	−9.65685	−0.660129
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−18.9706	−1.28485
$219$	9.41421	0.636154
$220$	0	0
$221$	3.31371	0.222904
$222$	2.00000	0.134231
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	1.65685	0.110212
$227$	−17.6569	−1.17193	−0.585963	−	0.810338i	$-0.699284\pi$
$227$	−17.6569	−1.17193	−0.585963	+	0.810338i	$0.699284\pi$
$228$	−1.00000	−0.0662266
$229$	25.2132	1.66614	0.833068	−	0.553171i	$-0.186582\pi$
$229$	25.2132	1.66614	0.833068	+	0.553171i	$0.186582\pi$
$230$	0	0
$231$	0	0
$232$	−8.00000	−0.525226
$233$	13.3137	0.872210	0.436105	−	0.899896i	$-0.356358\pi$
$233$	13.3137	0.872210	0.436105	+	0.899896i	$0.356358\pi$
$234$	2.82843	0.184900
$235$	0	0
$236$	0	0
$237$	−5.89949	−0.383213
$238$	0	0
$239$	−21.2132	−1.37217	−0.686084	−	0.727522i	$-0.740672\pi$
$239$	−21.2132	−1.37217	−0.686084	+	0.727522i	$0.740672\pi$
$240$	0	0
$241$	−17.6569	−1.13738	−0.568689	−	0.822553i	$-0.692549\pi$
$241$	−17.6569	−1.13738	−0.568689	+	0.822553i	$0.692549\pi$
$242$	−4.31371	−0.277296
$243$	−1.00000	−0.0641500
$244$	−2.58579	−0.165538
$245$	0	0
$246$	−2.58579	−0.164864
$247$	2.82843	0.179969
$248$	−3.65685	−0.232210
$249$	−9.65685	−0.611978
$250$	0	0
$251$	16.6274	1.04951	0.524757	−	0.851252i	$-0.324156\pi$
$251$	16.6274	1.04951	0.524757	+	0.851252i	$0.324156\pi$
$252$	0	0
$253$	10.9706	0.689713
$254$	0.242641	0.0152246
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.0416	1.74919	0.874594	−	0.484855i	$-0.161128\pi$
$257$	28.0416	1.74919	0.874594	+	0.484855i	$0.161128\pi$
$258$	6.82843	0.425119
$259$	0	0
$260$	0	0
$261$	−8.00000	−0.495188
$262$	10.0000	0.617802
$263$	3.75736	0.231689	0.115844	−	0.993267i	$-0.463043\pi$
$263$	3.75736	0.231689	0.115844	+	0.993267i	$0.463043\pi$
$264$	2.58579	0.159144
$265$	0	0
$266$	0	0
$267$	2.58579	0.158248
$268$	−4.24264	−0.259161
$269$	−1.89949	−0.115814	−0.0579071	−	0.998322i	$-0.518443\pi$
$269$	−1.89949	−0.115814	−0.0579071	+	0.998322i	$0.518443\pi$
$270$	0	0
$271$	13.6569	0.829595	0.414797	−	0.909914i	$-0.363852\pi$
$271$	13.6569	0.829595	0.414797	+	0.909914i	$0.363852\pi$
$272$	1.17157	0.0710370
$273$	0	0
$274$	14.9706	0.904405
$275$	12.9289	0.779644
$276$	4.24264	0.255377
$277$	13.3137	0.799943	0.399972	−	0.916528i	$-0.369020\pi$
$277$	13.3137	0.799943	0.399972	+	0.916528i	$0.369020\pi$
$278$	−0.686292	−0.0411610
$279$	−3.65685	−0.218930
$280$	0	0
$281$	−16.6274	−0.991909	−0.495954	−	0.868349i	$-0.665182\pi$
$281$	−16.6274	−0.991909	−0.495954	+	0.868349i	$0.665182\pi$
$282$	−7.65685	−0.455959
$283$	−4.97056	−0.295469	−0.147735	−	0.989027i	$-0.547198\pi$
$283$	−4.97056	−0.295469	−0.147735	+	0.989027i	$0.547198\pi$
$284$	14.1421	0.839181
$285$	0	0
$286$	−7.31371	−0.432469
$287$	0	0
$288$	1.00000	0.0589256
$289$	−15.6274	−0.919260
$290$	0	0
$291$	14.8284	0.869258
$292$	−9.41421	−0.550925
$293$	6.58579	0.384746	0.192373	−	0.981322i	$-0.438382\pi$
$293$	6.58579	0.384746	0.192373	+	0.981322i	$0.438382\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	2.58579	0.150043
$298$	−14.0000	−0.810998
$299$	−12.0000	−0.693978
$300$	5.00000	0.288675
$301$	0	0
$302$	−0.242641	−0.0139624
$303$	8.00000	0.459588
$304$	1.00000	0.0573539
$305$	0	0
$306$	1.17157	0.0669744
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	−7.65685	−0.435583
$310$	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	$0.590260\pi$
$314$	−16.2426	−0.916625
$315$	0	0
$316$	5.89949	0.331873
$317$	3.31371	0.186116	0.0930582	−	0.995661i	$-0.470336\pi$
$317$	3.31371	0.186116	0.0930582	+	0.995661i	$0.470336\pi$
$318$	2.00000	0.112154
$319$	20.6863	1.15821
$320$	0	0
$321$	9.65685	0.538993
$322$	0	0
$323$	1.17157	0.0651881
$324$	1.00000	0.0555556
$325$	−14.1421	−0.784465
$326$	4.48528	0.248417
$327$	18.9706	1.04907
$328$	2.58579	0.142776
$329$	0	0
$330$	0	0
$331$	1.41421	0.0777322	0.0388661	−	0.999244i	$-0.487625\pi$
$331$	1.41421	0.0777322	0.0388661	+	0.999244i	$0.487625\pi$
$332$	9.65685	0.529989
$333$	−2.00000	−0.109599
$334$	−8.97056	−0.490847
$335$	0	0
$336$	0	0
$337$	12.6274	0.687859	0.343930	−	0.938995i	$-0.388242\pi$
$337$	12.6274	0.687859	0.343930	+	0.938995i	$0.388242\pi$
$338$	−5.00000	−0.271964
$339$	−1.65685	−0.0899880
$340$	0	0
$341$	9.45584	0.512063
$342$	1.00000	0.0540738
$343$	0	0
$344$	−6.82843	−0.368164
$345$	0	0
$346$	−12.2426	−0.658168
$347$	2.58579	0.138812	0.0694061	−	0.997588i	$-0.477890\pi$
$347$	2.58579	0.138812	0.0694061	+	0.997588i	$0.477890\pi$
$348$	8.00000	0.428845
$349$	−7.75736	−0.415242	−0.207621	−	0.978209i	$-0.566572\pi$
$349$	−7.75736	−0.415242	−0.207621	+	0.978209i	$0.566572\pi$
$350$	0	0
$351$	−2.82843	−0.150970
$352$	−2.58579	−0.137823
$353$	−2.14214	−0.114014	−0.0570072	−	0.998374i	$-0.518156\pi$
$353$	−2.14214	−0.114014	−0.0570072	+	0.998374i	$0.518156\pi$
$354$	0	0
$355$	0	0
$356$	−2.58579	−0.137046
$357$	0	0
$358$	0.686292	0.0362716
$359$	1.41421	0.0746393	0.0373197	−	0.999303i	$-0.488118\pi$
$359$	1.41421	0.0746393	0.0373197	+	0.999303i	$0.488118\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−3.31371	−0.174165
$363$	4.31371	0.226411
$364$	0	0
$365$	0	0
$366$	2.58579	0.135161
$367$	−25.6569	−1.33928	−0.669638	−	0.742687i	$-0.733551\pi$
$367$	−25.6569	−1.33928	−0.669638	+	0.742687i	$0.733551\pi$
$368$	−4.24264	−0.221163
$369$	2.58579	0.134611
$370$	0	0
$371$	0	0
$372$	3.65685	0.189599
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−3.02944	−0.156648
$375$	0	0
$376$	7.65685	0.394872
$377$	−22.6274	−1.16537
$378$	0	0
$379$	25.8995	1.33037	0.665184	−	0.746680i	$-0.268353\pi$
$379$	25.8995	1.33037	0.665184	+	0.746680i	$0.268353\pi$
$380$	0	0
$381$	−0.242641	−0.0124309
$382$	−1.89949	−0.0971866
$383$	29.6569	1.51539	0.757697	−	0.652606i	$-0.226324\pi$
$383$	29.6569	1.51539	0.757697	+	0.652606i	$0.226324\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−10.0000	−0.508987
$387$	−6.82843	−0.347108
$388$	−14.8284	−0.752799
$389$	30.2843	1.53547	0.767737	−	0.640765i	$-0.221383\pi$
$389$	30.2843	1.53547	0.767737	+	0.640765i	$0.221383\pi$
$390$	0	0
$391$	−4.97056	−0.251372
$392$	0	0
$393$	−10.0000	−0.504433
$394$	−23.6569	−1.19182
$395$	0	0
$396$	−2.58579	−0.129941
$397$	36.0416	1.80888	0.904439	−	0.426603i	$-0.140290\pi$
$397$	36.0416	1.80888	0.904439	+	0.426603i	$0.140290\pi$
$398$	−11.3137	−0.567105
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−5.65685	−0.282490	−0.141245	−	0.989975i	$-0.545111\pi$
$401$	−5.65685	−0.282490	−0.141245	+	0.989975i	$0.545111\pi$
$402$	4.24264	0.211604
$403$	−10.3431	−0.515229
$404$	−8.00000	−0.398015
$405$	0	0
$406$	0	0
$407$	5.17157	0.256345
$408$	−1.17157	−0.0580015
$409$	26.1421	1.29265	0.646323	−	0.763064i	$-0.276306\pi$
$409$	26.1421	1.29265	0.646323	+	0.763064i	$0.276306\pi$
$410$	0	0
$411$	−14.9706	−0.738443
$412$	7.65685	0.377226
$413$	0	0
$414$	−4.24264	−0.208514
$415$	0	0
$416$	2.82843	0.138675
$417$	0.686292	0.0336078
$418$	−2.58579	−0.126475
$419$	21.3137	1.04124	0.520621	−	0.853788i	$-0.325700\pi$
$419$	21.3137	1.04124	0.520621	+	0.853788i	$0.325700\pi$
$420$	0	0
$421$	−8.34315	−0.406620	−0.203310	−	0.979114i	$-0.565170\pi$
$421$	−8.34315	−0.406620	−0.203310	+	0.979114i	$0.565170\pi$
$422$	−6.10051	−0.296968
$423$	7.65685	0.372289
$424$	−2.00000	−0.0971286
$425$	−5.85786	−0.284148
$426$	−14.1421	−0.685189
$427$	0	0
$428$	−9.65685	−0.466782
$429$	7.31371	0.353109
$430$	0	0
$431$	17.4558	0.840818	0.420409	−	0.907335i	$-0.361887\pi$
$431$	17.4558	0.840818	0.420409	+	0.907335i	$0.361887\pi$
$432$	−1.00000	−0.0481125
$433$	−9.17157	−0.440758	−0.220379	−	0.975414i	$-0.570729\pi$
$433$	−9.17157	−0.440758	−0.220379	+	0.975414i	$0.570729\pi$
$434$	0	0
$435$	0	0
$436$	−18.9706	−0.908525
$437$	−4.24264	−0.202953
$438$	9.41421	0.449829
$439$	10.9706	0.523596	0.261798	−	0.965123i	$-0.415685\pi$
$439$	10.9706	0.523596	0.261798	+	0.965123i	$0.415685\pi$
$440$	0	0
$441$	0	0
$442$	3.31371	0.157617
$443$	−27.0711	−1.28619	−0.643093	−	0.765788i	$-0.722349\pi$
$443$	−27.0711	−1.28619	−0.643093	+	0.765788i	$0.722349\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	−16.0000	−0.757622
$447$	14.0000	0.662177
$448$	0	0
$449$	−24.9706	−1.17843	−0.589217	−	0.807975i	$-0.700564\pi$
$449$	−24.9706	−1.17843	−0.589217	+	0.807975i	$0.700564\pi$
$450$	−5.00000	−0.235702
$451$	−6.68629	−0.314845
$452$	1.65685	0.0779319
$453$	0.242641	0.0114003
$454$	−17.6569	−0.828677
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−35.3137	−1.65191	−0.825953	−	0.563739i	$-0.809362\pi$
$457$	−35.3137	−1.65191	−0.825953	+	0.563739i	$0.809362\pi$
$458$	25.2132	1.17814
$459$	−1.17157	−0.0546843
$460$	0	0
$461$	−16.0000	−0.745194	−0.372597	−	0.927993i	$-0.621533\pi$
$461$	−16.0000	−0.745194	−0.372597	+	0.927993i	$0.621533\pi$
$462$	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
$466$	13.3137	0.616746
$467$	−3.02944	−0.140186	−0.0700928	−	0.997540i	$-0.522330\pi$
$467$	−3.02944	−0.140186	−0.0700928	+	0.997540i	$0.522330\pi$
$468$	2.82843	0.130744
$469$	0	0
$470$	0	0
$471$	16.2426	0.748421
$472$	0	0
$473$	17.6569	0.811863
$474$	−5.89949	−0.270973
$475$	−5.00000	−0.229416
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	−21.2132	−0.970269
$479$	8.97056	0.409875	0.204938	−	0.978775i	$-0.434301\pi$
$479$	8.97056	0.409875	0.204938	+	0.978775i	$0.434301\pi$
$480$	0	0
$481$	−5.65685	−0.257930
$482$	−17.6569	−0.804248
$483$	0	0
$484$	−4.31371	−0.196078
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−20.0416	−0.908173	−0.454086	−	0.890958i	$-0.650034\pi$
$487$	−20.0416	−0.908173	−0.454086	+	0.890958i	$0.650034\pi$
$488$	−2.58579	−0.117053
$489$	−4.48528	−0.202831
$490$	0	0
$491$	−4.92893	−0.222440	−0.111220	−	0.993796i	$-0.535476\pi$
$491$	−4.92893	−0.222440	−0.111220	+	0.993796i	$0.535476\pi$
$492$	−2.58579	−0.116576
$493$	−9.37258	−0.422120
$494$	2.82843	0.127257
$495$	0	0
$496$	−3.65685	−0.164198
$497$	0	0
$498$	−9.65685	−0.432734
$499$	34.1421	1.52841	0.764206	−	0.644972i	$-0.223131\pi$
$499$	34.1421	1.52841	0.764206	+	0.644972i	$0.223131\pi$
$500$	0	0
$501$	8.97056	0.400775
$502$	16.6274	0.742118
$503$	18.3431	0.817880	0.408940	−	0.912561i	$-0.365898\pi$
$503$	18.3431	0.817880	0.408940	+	0.912561i	$0.365898\pi$
$504$	0	0
$505$	0	0
$506$	10.9706	0.487701
$507$	5.00000	0.222058
$508$	0.242641	0.0107654
$509$	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
$514$	28.0416	1.23686
$515$	0	0
$516$	6.82843	0.300605
$517$	−19.7990	−0.870759
$518$	0	0
$519$	12.2426	0.537392
$520$	0	0
$521$	−5.89949	−0.258462	−0.129231	−	0.991615i	$-0.541251\pi$
$521$	−5.89949	−0.258462	−0.129231	+	0.991615i	$0.541251\pi$
$522$	−8.00000	−0.350150
$523$	−31.9411	−1.39669	−0.698344	−	0.715762i	$-0.746079\pi$
$523$	−31.9411	−1.39669	−0.698344	+	0.715762i	$0.746079\pi$
$524$	10.0000	0.436852
$525$	0	0
$526$	3.75736	0.163829
$527$	−4.28427	−0.186626
$528$	2.58579	0.112532
$529$	−5.00000	−0.217391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.31371	0.316792
$534$	2.58579	0.111898
$535$	0	0
$536$	−4.24264	−0.183254
$537$	−0.686292	−0.0296157
$538$	−1.89949	−0.0818930
$539$	0	0
$540$	0	0
$541$	−22.3431	−0.960607	−0.480303	−	0.877102i	$-0.659473\pi$
$541$	−22.3431	−0.960607	−0.480303	+	0.877102i	$0.659473\pi$
$542$	13.6569	0.586612
$543$	3.31371	0.142205
$544$	1.17157	0.0502308
$545$	0	0
$546$	0	0
$547$	0.443651	0.0189691	0.00948457	−	0.999955i	$-0.496981\pi$
$547$	0.443651	0.0189691	0.00948457	+	0.999955i	$0.496981\pi$
$548$	14.9706	0.639511
$549$	−2.58579	−0.110359
$550$	12.9289	0.551292
$551$	−8.00000	−0.340811
$552$	4.24264	0.180579
$553$	0	0
$554$	13.3137	0.565645
$555$	0	0
$556$	−0.686292	−0.0291052
$557$	−23.6569	−1.00237	−0.501187	−	0.865339i	$-0.667103\pi$
$557$	−23.6569	−1.00237	−0.501187	+	0.865339i	$0.667103\pi$
$558$	−3.65685	−0.154807
$559$	−19.3137	−0.816883
$560$	0	0
$561$	3.02944	0.127903
$562$	−16.6274	−0.701385
$563$	11.3137	0.476816	0.238408	−	0.971165i	$-0.423374\pi$
$563$	11.3137	0.476816	0.238408	+	0.971165i	$0.423374\pi$
$564$	−7.65685	−0.322412
$565$	0	0
$566$	−4.97056	−0.208928
$567$	0	0
$568$	14.1421	0.593391
$569$	−16.2843	−0.682672	−0.341336	−	0.939941i	$-0.610879\pi$
$569$	−16.2843	−0.682672	−0.341336	+	0.939941i	$0.610879\pi$
$570$	0	0
$571$	−28.9706	−1.21238	−0.606190	−	0.795320i	$-0.707303\pi$
$571$	−28.9706	−1.21238	−0.606190	+	0.795320i	$0.707303\pi$
$572$	−7.31371	−0.305802
$573$	1.89949	0.0793525
$574$	0	0
$575$	21.2132	0.884652
$576$	1.00000	0.0416667
$577$	−33.8995	−1.41125	−0.705627	−	0.708583i	$-0.749335\pi$
$577$	−33.8995	−1.41125	−0.705627	+	0.708583i	$0.749335\pi$
$578$	−15.6274	−0.650015
$579$	10.0000	0.415586
$580$	0	0
$581$	0	0
$582$	14.8284	0.614658
$583$	5.17157	0.214185
$584$	−9.41421	−0.389563
$585$	0	0
$586$	6.58579	0.272056
$587$	6.34315	0.261810	0.130905	−	0.991395i	$-0.458212\pi$
$587$	6.34315	0.261810	0.130905	+	0.991395i	$0.458212\pi$
$588$	0	0
$589$	−3.65685	−0.150678
$590$	0	0
$591$	23.6569	0.973113
$592$	−2.00000	−0.0821995
$593$	−2.62742	−0.107895	−0.0539475	−	0.998544i	$-0.517180\pi$
$593$	−2.62742	−0.107895	−0.0539475	+	0.998544i	$0.517180\pi$
$594$	2.58579	0.106096
$595$	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	$-0.229484\pi$
$599$	36.7696	1.50236	0.751182	+	0.660096i	$0.229484\pi$
$600$	5.00000	0.204124
$601$	26.1421	1.06636	0.533180	−	0.846002i	$-0.320997\pi$
$601$	26.1421	1.06636	0.533180	+	0.846002i	$0.320997\pi$
$602$	0	0
$603$	−4.24264	−0.172774
$604$	−0.242641	−0.00987291
$605$	0	0
$606$	8.00000	0.324978
$607$	−19.3137	−0.783919	−0.391960	−	0.919982i	$-0.628203\pi$
$607$	−19.3137	−0.783919	−0.391960	+	0.919982i	$0.628203\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	21.6569	0.876143
$612$	1.17157	0.0473580
$613$	18.3431	0.740873	0.370436	−	0.928858i	$-0.379208\pi$
$613$	18.3431	0.740873	0.370436	+	0.928858i	$0.379208\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	0	0
$617$	−18.9706	−0.763726	−0.381863	−	0.924219i	$-0.624717\pi$
$617$	−18.9706	−0.763726	−0.381863	+	0.924219i	$0.624717\pi$
$618$	−7.65685	−0.308004
$619$	21.6569	0.870462	0.435231	−	0.900319i	$-0.356667\pi$
$619$	21.6569	0.870462	0.435231	+	0.900319i	$0.356667\pi$
$620$	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
$627$	2.58579	0.103266
$628$	−16.2426	−0.648152
$629$	−2.34315	−0.0934273
$630$	0	0
$631$	39.1127	1.55705	0.778526	−	0.627612i	$-0.215968\pi$
$631$	39.1127	1.55705	0.778526	+	0.627612i	$0.215968\pi$
$632$	5.89949	0.234669
$633$	6.10051	0.242473
$634$	3.31371	0.131604
$635$	0	0
$636$	2.00000	0.0793052
$637$	0	0
$638$	20.6863	0.818978
$639$	14.1421	0.559454
$640$	0	0
$641$	33.3137	1.31581	0.657906	−	0.753100i	$-0.271442\pi$
$641$	33.3137	1.31581	0.657906	+	0.753100i	$0.271442\pi$
$642$	9.65685	0.381126
$643$	−7.02944	−0.277214	−0.138607	−	0.990347i	$-0.544262\pi$
$643$	−7.02944	−0.277214	−0.138607	+	0.990347i	$0.544262\pi$
$644$	0	0
$645$	0	0
$646$	1.17157	0.0460949
$647$	26.3431	1.03566	0.517828	−	0.855485i	$-0.326741\pi$
$647$	26.3431	1.03566	0.517828	+	0.855485i	$0.326741\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−14.1421	−0.554700
$651$	0	0
$652$	4.48528	0.175657
$653$	−30.9706	−1.21197	−0.605986	−	0.795475i	$-0.707221\pi$
$653$	−30.9706	−1.21197	−0.605986	+	0.795475i	$0.707221\pi$
$654$	18.9706	0.741808
$655$	0	0
$656$	2.58579	0.100958
$657$	−9.41421	−0.367283
$658$	0	0
$659$	−16.2843	−0.634345	−0.317173	−	0.948368i	$-0.602733\pi$
$659$	−16.2843	−0.634345	−0.317173	+	0.948368i	$0.602733\pi$
$660$	0	0
$661$	−20.2843	−0.788967	−0.394483	−	0.918903i	$-0.629076\pi$
$661$	−20.2843	−0.788967	−0.394483	+	0.918903i	$0.629076\pi$
$662$	1.41421	0.0549650
$663$	−3.31371	−0.128694
$664$	9.65685	0.374759
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	33.9411	1.31421
$668$	−8.97056	−0.347081
$669$	16.0000	0.618596
$670$	0	0
$671$	6.68629	0.258121
$672$	0	0
$673$	13.3137	0.513206	0.256603	−	0.966517i	$-0.417397\pi$
$673$	13.3137	0.513206	0.256603	+	0.966517i	$0.417397\pi$
$674$	12.6274	0.486390
$675$	5.00000	0.192450
$676$	−5.00000	−0.192308
$677$	20.7279	0.796639	0.398319	−	0.917247i	$-0.369594\pi$
$677$	20.7279	0.796639	0.398319	+	0.917247i	$0.369594\pi$
$678$	−1.65685	−0.0636311
$679$	0	0
$680$	0	0
$681$	17.6569	0.676612
$682$	9.45584	0.362083
$683$	12.4853	0.477736	0.238868	−	0.971052i	$-0.423224\pi$
$683$	12.4853	0.477736	0.238868	+	0.971052i	$0.423224\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	0	0
$687$	−25.2132	−0.961944
$688$	−6.82843	−0.260331
$689$	−5.65685	−0.215509
$690$	0	0
$691$	−19.3137	−0.734728	−0.367364	−	0.930077i	$-0.619740\pi$
$691$	−19.3137	−0.734728	−0.367364	+	0.930077i	$0.619740\pi$
$692$	−12.2426	−0.465395
$693$	0	0
$694$	2.58579	0.0981551
$695$	0	0
$696$	8.00000	0.303239
$697$	3.02944	0.114748
$698$	−7.75736	−0.293620
$699$	−13.3137	−0.503571
$700$	0	0
$701$	6.97056	0.263275	0.131637	−	0.991298i	$-0.457977\pi$
$701$	6.97056	0.263275	0.131637	+	0.991298i	$0.457977\pi$
$702$	−2.82843	−0.106752
$703$	−2.00000	−0.0754314
$704$	−2.58579	−0.0974555
$705$	0	0
$706$	−2.14214	−0.0806203
$707$	0	0
$708$	0	0
$709$	49.3137	1.85202	0.926008	−	0.377505i	$-0.123218\pi$
$709$	49.3137	1.85202	0.926008	+	0.377505i	$0.123218\pi$
$710$	0	0
$711$	5.89949	0.221248
$712$	−2.58579	−0.0969064
$713$	15.5147	0.581031
$714$	0	0
$715$	0	0
$716$	0.686292	0.0256479
$717$	21.2132	0.792222
$718$	1.41421	0.0527780
$719$	−12.2843	−0.458126	−0.229063	−	0.973412i	$-0.573566\pi$
$719$	−12.2843	−0.458126	−0.229063	+	0.973412i	$0.573566\pi$
$720$	0	0
$721$	0	0
$722$	1.00000	0.0372161
$723$	17.6569	0.656665
$724$	−3.31371	−0.123153
$725$	40.0000	1.48556
$726$	4.31371	0.160097
$727$	−18.3431	−0.680310	−0.340155	−	0.940369i	$-0.610479\pi$
$727$	−18.3431	−0.680310	−0.340155	+	0.940369i	$0.610479\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	2.58579	0.0955734
$733$	25.2132	0.931271	0.465635	−	0.884977i	$-0.345826\pi$
$733$	25.2132	0.931271	0.465635	+	0.884977i	$0.345826\pi$
$734$	−25.6569	−0.947012
$735$	0	0
$736$	−4.24264	−0.156386
$737$	10.9706	0.404106
$738$	2.58579	0.0951841
$739$	−29.9411	−1.10140	−0.550701	−	0.834703i	$-0.685640\pi$
$739$	−29.9411	−1.10140	−0.550701	+	0.834703i	$0.685640\pi$
$740$	0	0
$741$	−2.82843	−0.103905
$742$	0	0
$743$	26.3431	0.966436	0.483218	−	0.875500i	$-0.339468\pi$
$743$	26.3431	0.966436	0.483218	+	0.875500i	$0.339468\pi$
$744$	3.65685	0.134067
$745$	0	0
$746$	10.0000	0.366126
$747$	9.65685	0.353326
$748$	−3.02944	−0.110767
$749$	0	0
$750$	0	0
$751$	−41.2132	−1.50389	−0.751946	−	0.659225i	$-0.770884\pi$
$751$	−41.2132	−1.50389	−0.751946	+	0.659225i	$0.770884\pi$
$752$	7.65685	0.279217
$753$	−16.6274	−0.605937
$754$	−22.6274	−0.824042
$755$	0	0
$756$	0	0
$757$	28.2843	1.02801	0.514005	−	0.857787i	$-0.328161\pi$
$757$	28.2843	1.02801	0.514005	+	0.857787i	$0.328161\pi$
$758$	25.8995	0.940712
$759$	−10.9706	−0.398206
$760$	0	0
$761$	5.37258	0.194756	0.0973780	−	0.995247i	$-0.468954\pi$
$761$	5.37258	0.194756	0.0973780	+	0.995247i	$0.468954\pi$
$762$	−0.242641	−0.00878994
$763$	0	0
$764$	−1.89949	−0.0687213
$765$	0	0
$766$	29.6569	1.07155
$767$	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
$771$	−28.0416	−1.00989
$772$	−10.0000	−0.359908
$773$	3.27208	0.117688	0.0588442	−	0.998267i	$-0.481258\pi$
$773$	3.27208	0.117688	0.0588442	+	0.998267i	$0.481258\pi$
$774$	−6.82843	−0.245443
$775$	18.2843	0.656790
$776$	−14.8284	−0.532310
$777$	0	0
$778$	30.2843	1.08574
$779$	2.58579	0.0926454
$780$	0	0
$781$	−36.5685	−1.30853
$782$	−4.97056	−0.177747
$783$	8.00000	0.285897
$784$	0	0
$785$	0	0
$786$	−10.0000	−0.356688
$787$	8.62742	0.307534	0.153767	−	0.988107i	$-0.450859\pi$
$787$	8.62742	0.307534	0.153767	+	0.988107i	$0.450859\pi$
$788$	−23.6569	−0.842741
$789$	−3.75736	−0.133766
$790$	0	0
$791$	0	0
$792$	−2.58579	−0.0918819
$793$	−7.31371	−0.259717
$794$	36.0416	1.27907
$795$	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
$799$	8.97056	0.317356
$800$	−5.00000	−0.176777
$801$	−2.58579	−0.0913643
$802$	−5.65685	−0.199750
$803$	24.3431	0.859051
$804$	4.24264	0.149626
$805$	0	0
$806$	−10.3431	−0.364322
$807$	1.89949	0.0668654
$808$	−8.00000	−0.281439
$809$	34.2843	1.20537	0.602685	−	0.797979i	$-0.294097\pi$
$809$	34.2843	1.20537	0.602685	+	0.797979i	$0.294097\pi$
$810$	0	0
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	0	0
$813$	−13.6569	−0.478967
$814$	5.17157	0.181264
$815$	0	0
$816$	−1.17157	−0.0410133
$817$	−6.82843	−0.238896
$818$	26.1421	0.914038
$819$	0	0
$820$	0	0
$821$	−6.28427	−0.219323	−0.109661	−	0.993969i	$-0.534977\pi$
$821$	−6.28427	−0.219323	−0.109661	+	0.993969i	$0.534977\pi$
$822$	−14.9706	−0.522158
$823$	−8.97056	−0.312694	−0.156347	−	0.987702i	$-0.549972\pi$
$823$	−8.97056	−0.312694	−0.156347	+	0.987702i	$0.549972\pi$
$824$	7.65685	0.266739
$825$	−12.9289	−0.450128
$826$	0	0
$827$	−15.3137	−0.532510	−0.266255	−	0.963903i	$-0.585786\pi$
$827$	−15.3137	−0.532510	−0.266255	+	0.963903i	$0.585786\pi$
$828$	−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
$831$	−13.3137	−0.461847
$832$	2.82843	0.0980581
$833$	0	0
$834$	0.686292	0.0237643
$835$	0	0
$836$	−2.58579	−0.0894313
$837$	3.65685	0.126399
$838$	21.3137	0.736270
$839$	8.97056	0.309698	0.154849	−	0.987938i	$-0.450511\pi$
$839$	8.97056	0.309698	0.154849	+	0.987938i	$0.450511\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−8.34315	−0.287524
$843$	16.6274	0.572679
$844$	−6.10051	−0.209988
$845$	0	0
$846$	7.65685	0.263248
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	4.97056	0.170589
$850$	−5.85786	−0.200923
$851$	8.48528	0.290872
$852$	−14.1421	−0.484502
$853$	−26.1838	−0.896515	−0.448258	−	0.893904i	$-0.647955\pi$
$853$	−26.1838	−0.896515	−0.448258	+	0.893904i	$0.647955\pi$
$854$	0	0
$855$	0	0
$856$	−9.65685	−0.330064
$857$	33.2132	1.13454	0.567271	−	0.823531i	$-0.307999\pi$
$857$	33.2132	1.13454	0.567271	+	0.823531i	$0.307999\pi$
$858$	7.31371	0.249686
$859$	0.686292	0.0234160	0.0117080	−	0.999931i	$-0.496273\pi$
$859$	0.686292	0.0234160	0.0117080	+	0.999931i	$0.496273\pi$
$860$	0	0
$861$	0	0
$862$	17.4558	0.594548
$863$	42.3431	1.44138	0.720689	−	0.693259i	$-0.243826\pi$
$863$	42.3431	1.44138	0.720689	+	0.693259i	$0.243826\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−9.17157	−0.311663
$867$	15.6274	0.530735
$868$	0	0
$869$	−15.2548	−0.517485
$870$	0	0
$871$	−12.0000	−0.406604
$872$	−18.9706	−0.642424
$873$	−14.8284	−0.501866
$874$	−4.24264	−0.143509
$875$	0	0
$876$	9.41421	0.318077
$877$	−6.68629	−0.225780	−0.112890	−	0.993607i	$-0.536011\pi$
$877$	−6.68629	−0.225780	−0.112890	+	0.993607i	$0.536011\pi$
$878$	10.9706	0.370239
$879$	−6.58579	−0.222133
$880$	0	0
$881$	43.1127	1.45250	0.726252	−	0.687429i	$-0.241261\pi$
$881$	43.1127	1.45250	0.726252	+	0.687429i	$0.241261\pi$
$882$	0	0
$883$	4.48528	0.150942	0.0754709	−	0.997148i	$-0.475954\pi$
$883$	4.48528	0.150942	0.0754709	+	0.997148i	$0.475954\pi$
$884$	3.31371	0.111452
$885$	0	0
$886$	−27.0711	−0.909470
$887$	−54.6274	−1.83421	−0.917105	−	0.398647i	$-0.869480\pi$
$887$	−54.6274	−1.83421	−0.917105	+	0.398647i	$0.869480\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	0	0
$891$	−2.58579	−0.0866271
$892$	−16.0000	−0.535720
$893$	7.65685	0.256227
$894$	14.0000	0.468230
$895$	0	0
$896$	0	0
$897$	12.0000	0.400668
$898$	−24.9706	−0.833278
$899$	29.2548	0.975703
$900$	−5.00000	−0.166667
$901$	−2.34315	−0.0780615
$902$	−6.68629	−0.222629
$903$	0	0
$904$	1.65685	0.0551062
$905$	0	0
$906$	0.242641	0.00806120
$907$	35.3553	1.17395	0.586977	−	0.809603i	$-0.300318\pi$
$907$	35.3553	1.17395	0.586977	+	0.809603i	$0.300318\pi$
$908$	−17.6569	−0.585963
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−8.48528	−0.281130	−0.140565	−	0.990071i	$-0.544892\pi$
$911$	−8.48528	−0.281130	−0.140565	+	0.990071i	$0.544892\pi$
$912$	−1.00000	−0.0331133
$913$	−24.9706	−0.826405
$914$	−35.3137	−1.16807
$915$	0	0
$916$	25.2132	0.833068
$917$	0	0
$918$	−1.17157	−0.0386677
$919$	29.6569	0.978289	0.489145	−	0.872203i	$-0.337309\pi$
$919$	29.6569	0.978289	0.489145	+	0.872203i	$0.337309\pi$
$920$	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
$927$	7.65685	0.251484
$928$	−8.00000	−0.262613
$929$	−11.5147	−0.377786	−0.188893	−	0.981998i	$-0.560490\pi$
$929$	−11.5147	−0.377786	−0.188893	+	0.981998i	$0.560490\pi$
$930$	0	0
$931$	0	0
$932$	13.3137	0.436105
$933$	16.9706	0.555591
$934$	−3.02944	−0.0991262
$935$	0	0
$936$	2.82843	0.0924500
$937$	−17.8995	−0.584751	−0.292376	−	0.956304i	$-0.594446\pi$
$937$	−17.8995	−0.584751	−0.292376	+	0.956304i	$0.594446\pi$
$938$	0	0
$939$	9.89949	0.323058
$940$	0	0
$941$	35.7574	1.16566	0.582828	−	0.812595i	$-0.301946\pi$
$941$	35.7574	1.16566	0.582828	+	0.812595i	$0.301946\pi$
$942$	16.2426	0.529214
$943$	−10.9706	−0.357251
$944$	0	0
$945$	0	0
$946$	17.6569	0.574074
$947$	−9.61522	−0.312453	−0.156226	−	0.987721i	$-0.549933\pi$
$947$	−9.61522	−0.312453	−0.156226	+	0.987721i	$0.549933\pi$
$948$	−5.89949	−0.191607
$949$	−26.6274	−0.864363
$950$	−5.00000	−0.162221
$951$	−3.31371	−0.107454
$952$	0	0
$953$	42.3431	1.37163	0.685815	−	0.727776i	$-0.259446\pi$
$953$	42.3431	1.37163	0.685815	+	0.727776i	$0.259446\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−21.2132	−0.686084
$957$	−20.6863	−0.668693
$958$	8.97056	0.289826
$959$	0	0
$960$	0	0
$961$	−17.6274	−0.568626
$962$	−5.65685	−0.182384
$963$	−9.65685	−0.311188
$964$	−17.6569	−0.568689
$965$	0	0
$966$	0	0
$967$	49.4558	1.59039	0.795196	−	0.606352i	$-0.207368\pi$
$967$	49.4558	1.59039	0.795196	+	0.606352i	$0.207368\pi$
$968$	−4.31371	−0.138648
$969$	−1.17157	−0.0376363
$970$	0	0
$971$	5.37258	0.172414	0.0862072	−	0.996277i	$-0.472525\pi$
$971$	5.37258	0.172414	0.0862072	+	0.996277i	$0.472525\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−20.0416	−0.642175
$975$	14.1421	0.452911
$976$	−2.58579	−0.0827690
$977$	5.65685	0.180979	0.0904894	−	0.995897i	$-0.471157\pi$
$977$	5.65685	0.180979	0.0904894	+	0.995897i	$0.471157\pi$
$978$	−4.48528	−0.143423
$979$	6.68629	0.213695
$980$	0	0
$981$	−18.9706	−0.605683
$982$	−4.92893	−0.157289
$983$	20.9706	0.668857	0.334429	−	0.942421i	$-0.391457\pi$
$983$	20.9706	0.668857	0.334429	+	0.942421i	$0.391457\pi$
$984$	−2.58579	−0.0824319
$985$	0	0
$986$	−9.37258	−0.298484
$987$	0	0
$988$	2.82843	0.0899843
$989$	28.9706	0.921210
$990$	0	0
$991$	−30.3848	−0.965204	−0.482602	−	0.875840i	$-0.660308\pi$
$991$	−30.3848	−0.965204	−0.482602	+	0.875840i	$0.660308\pi$
$992$	−3.65685	−0.116105
$993$	−1.41421	−0.0448787
$994$	0	0
$995$	0	0
$996$	−9.65685	−0.305989
$997$	28.9289	0.916188	0.458094	−	0.888904i	$-0.348532\pi$
$997$	28.9289	0.916188	0.458094	+	0.888904i	$0.348532\pi$
$998$	34.1421	1.08075
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5586.2.a.bj.1.2	✓	2
7.6	odd	2		5586.2.a.bo.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5586.2.a.bj.1.2	✓	2	1.1	even	1	trivial
5586.2.a.bo.1.2	yes	2	7.6	odd	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.58579 q^{11} -1.00000 q^{12} +2.82843 q^{13} +1.00000 q^{16} +1.17157 q^{17} +1.00000 q^{18} +1.00000 q^{19} -2.58579 q^{22} -4.24264 q^{23} -1.00000 q^{24} -5.00000 q^{25} +2.82843 q^{26} -1.00000 q^{27} -8.00000 q^{29} -3.65685 q^{31} +1.00000 q^{32} +2.58579 q^{33} +1.17157 q^{34} +1.00000 q^{36} -2.00000 q^{37} +1.00000 q^{38} -2.82843 q^{39} +2.58579 q^{41} -6.82843 q^{43} -2.58579 q^{44} -4.24264 q^{46} +7.65685 q^{47} -1.00000 q^{48} -5.00000 q^{50} -1.17157 q^{51} +2.82843 q^{52} -2.00000 q^{53} -1.00000 q^{54} -1.00000 q^{57} -8.00000 q^{58} -2.58579 q^{61} -3.65685 q^{62} +1.00000 q^{64} +2.58579 q^{66} -4.24264 q^{67} +1.17157 q^{68} +4.24264 q^{69} +14.1421 q^{71} +1.00000 q^{72} -9.41421 q^{73} -2.00000 q^{74} +5.00000 q^{75} +1.00000 q^{76} -2.82843 q^{78} +5.89949 q^{79} +1.00000 q^{81} +2.58579 q^{82} +9.65685 q^{83} -6.82843 q^{86} +8.00000 q^{87} -2.58579 q^{88} -2.58579 q^{89} -4.24264 q^{92} +3.65685 q^{93} +7.65685 q^{94} -1.00000 q^{96} -14.8284 q^{97} -2.58579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 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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