LMFDB - Embedded newform 8752.2.a.s.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8752,2,Mod(1,8752)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8752, 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("8752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8752 = 2^{4} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8752.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:	$69.8850718490$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 $ x^18 - 4x^17 - 18x^16 + 84x^15 + 116x^14 - 708x^13 - 282x^12 + 3104x^11 - 137x^10 - 7703x^9 + 2068x^8 + 11068x^7 - 4274x^6 - 9021x^5 + 4048x^4 + 3834x^3 - 1851x^2 - 654*x + 328
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.735255$ of defining polynomial
Character	$\chi$	$=$	8752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.544167 q^{3} +0.962787 q^{5} +3.25298 q^{7} -2.70388 q^{9} +O(q^{10})$	$q-0.544167 q^{3} +0.962787 q^{5} +3.25298 q^{7} -2.70388 q^{9} +0.883096 q^{11} -4.48154 q^{13} -0.523917 q^{15} -3.18294 q^{17} -4.12466 q^{19} -1.77017 q^{21} -3.73970 q^{23} -4.07304 q^{25} +3.10387 q^{27} +6.33388 q^{29} +8.47939 q^{31} -0.480552 q^{33} +3.13193 q^{35} +11.0375 q^{37} +2.43871 q^{39} +1.41138 q^{41} +9.31097 q^{43} -2.60326 q^{45} +5.82191 q^{47} +3.58188 q^{49} +1.73205 q^{51} -7.06826 q^{53} +0.850233 q^{55} +2.24450 q^{57} -12.9274 q^{59} -3.04777 q^{61} -8.79567 q^{63} -4.31477 q^{65} +9.15218 q^{67} +2.03502 q^{69} +4.57601 q^{71} -14.4453 q^{73} +2.21642 q^{75} +2.87269 q^{77} -7.20620 q^{79} +6.42262 q^{81} -7.39056 q^{83} -3.06450 q^{85} -3.44669 q^{87} +0.838265 q^{89} -14.5784 q^{91} -4.61421 q^{93} -3.97117 q^{95} +0.475381 q^{97} -2.38779 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) $ 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9	$ 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9} - 2 q^{11} - 25 q^{13} - 9 q^{15} - 30 q^{17} - 4 q^{19} - 16 q^{21} + 26 q^{23} + 31 q^{25} + 37 q^{27} - 18 q^{29} + 5 q^{31} - 10 q^{33} + 9 q^{35} - 18 q^{37} - 7 q^{39} - 17 q^{41} - 8 q^{43} - 44 q^{45} + 52 q^{47} + 29 q^{49} - 19 q^{51} - 60 q^{53} - 11 q^{55} + 4 q^{57} + 8 q^{59} - 26 q^{61} + q^{63} - 6 q^{65} - 12 q^{67} - 38 q^{69} + q^{71} - 2 q^{73} + 17 q^{75} - 73 q^{77} - 18 q^{79} + 18 q^{81} + 43 q^{83} + 51 q^{85} - 3 q^{87} - 28 q^{89} + q^{91} - 60 q^{93} + 18 q^{95} - 34 q^{97} - 15 q^{99}+O(q^{100}) $ 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9 - 2 * q^11 - 25 * q^13 - 9 * q^15 - 30 * q^17 - 4 * q^19 - 16 * q^21 + 26 * q^23 + 31 * q^25 + 37 * q^27 - 18 * q^29 + 5 * q^31 - 10 * q^33 + 9 * q^35 - 18 * q^37 - 7 * q^39 - 17 * q^41 - 8 * q^43 - 44 * q^45 + 52 * q^47 + 29 * q^49 - 19 * q^51 - 60 * q^53 - 11 * q^55 + 4 * q^57 + 8 * q^59 - 26 * q^61 + q^63 - 6 * q^65 - 12 * q^67 - 38 * q^69 + q^71 - 2 * q^73 + 17 * q^75 - 73 * q^77 - 18 * q^79 + 18 * q^81 + 43 * q^83 + 51 * q^85 - 3 * q^87 - 28 * q^89 + q^91 - 60 * q^93 + 18 * q^95 - 34 * q^97 - 15 * 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 $

$2$	0	0
$2$	0	0
$3$	−0.544167	−0.314175	−0.157088	−	0.987585i	$-0.550210\pi$
$3$	−0.544167	−0.314175	−0.157088	+	0.987585i	$0.550210\pi$
$4$	0	0
$5$	0.962787	0.430571	0.215286	−	0.976551i	$-0.430932\pi$
$5$	0.962787	0.430571	0.215286	+	0.976551i	$0.430932\pi$
$6$	0	0
$7$	3.25298	1.22951	0.614755	−	0.788718i	$-0.289255\pi$
$7$	3.25298	1.22951	0.614755	+	0.788718i	$0.289255\pi$
$8$	0	0
$9$	−2.70388	−0.901294
$10$	0	0
$11$	0.883096	0.266263	0.133132	−	0.991098i	$-0.457497\pi$
$11$	0.883096	0.266263	0.133132	+	0.991098i	$0.457497\pi$
$12$	0	0
$13$	−4.48154	−1.24296	−0.621478	−	0.783431i	$-0.713468\pi$
$13$	−4.48154	−1.24296	−0.621478	+	0.783431i	$0.713468\pi$
$14$	0	0
$15$	−0.523917	−0.135275
$16$	0	0
$17$	−3.18294	−0.771977	−0.385988	−	0.922504i	$-0.626140\pi$
$17$	−3.18294	−0.771977	−0.385988	+	0.922504i	$0.626140\pi$
$18$	0	0
$19$	−4.12466	−0.946261	−0.473131	−	0.880992i	$-0.656876\pi$
$19$	−4.12466	−0.946261	−0.473131	+	0.880992i	$0.656876\pi$
$20$	0	0
$21$	−1.77017	−0.386282
$22$	0	0
$23$	−3.73970	−0.779781	−0.389890	−	0.920861i	$-0.627487\pi$
$23$	−3.73970	−0.779781	−0.389890	+	0.920861i	$0.627487\pi$
$24$	0	0
$25$	−4.07304	−0.814608
$26$	0	0
$27$	3.10387	0.597339
$28$	0	0
$29$	6.33388	1.17617	0.588086	−	0.808798i	$-0.299881\pi$
$29$	6.33388	1.17617	0.588086	+	0.808798i	$0.299881\pi$
$30$	0	0
$31$	8.47939	1.52294	0.761472	−	0.648198i	$-0.224477\pi$
$31$	8.47939	1.52294	0.761472	+	0.648198i	$0.224477\pi$
$32$	0	0
$33$	−0.480552	−0.0836533
$34$	0	0
$35$	3.13193	0.529392
$36$	0	0
$37$	11.0375	1.81455	0.907276	−	0.420535i	$-0.138158\pi$
$37$	11.0375	1.81455	0.907276	+	0.420535i	$0.138158\pi$
$38$	0	0
$39$	2.43871	0.390506
$40$	0	0
$41$	1.41138	0.220421	0.110210	−	0.993908i	$-0.464848\pi$
$41$	1.41138	0.220421	0.110210	+	0.993908i	$0.464848\pi$
$42$	0	0
$43$	9.31097	1.41991	0.709955	−	0.704247i	$-0.248715\pi$
$43$	9.31097	1.41991	0.709955	+	0.704247i	$0.248715\pi$
$44$	0	0
$45$	−2.60326	−0.388071
$46$	0	0
$47$	5.82191	0.849213	0.424606	−	0.905378i	$-0.360413\pi$
$47$	5.82191	0.849213	0.424606	+	0.905378i	$0.360413\pi$
$48$	0	0
$49$	3.58188	0.511697
$50$	0	0
$51$	1.73205	0.242536
$52$	0	0
$53$	−7.06826	−0.970900	−0.485450	−	0.874264i	$-0.661344\pi$
$53$	−7.06826	−0.970900	−0.485450	+	0.874264i	$0.661344\pi$
$54$	0	0
$55$	0.850233	0.114645
$56$	0	0
$57$	2.24450	0.297292
$58$	0	0
$59$	−12.9274	−1.68301	−0.841503	−	0.540252i	$-0.818329\pi$
$59$	−12.9274	−1.68301	−0.841503	+	0.540252i	$0.818329\pi$
$60$	0	0
$61$	−3.04777	−0.390227	−0.195114	−	0.980781i	$-0.562508\pi$
$61$	−3.04777	−0.390227	−0.195114	+	0.980781i	$0.562508\pi$
$62$	0	0
$63$	−8.79567	−1.10815
$64$	0	0
$65$	−4.31477	−0.535182
$66$	0	0
$67$	9.15218	1.11812	0.559059	−	0.829128i	$-0.311163\pi$
$67$	9.15218	1.11812	0.559059	+	0.829128i	$0.311163\pi$
$68$	0	0
$69$	2.03502	0.244988
$70$	0	0
$71$	4.57601	0.543072	0.271536	−	0.962428i	$-0.412468\pi$
$71$	4.57601	0.543072	0.271536	+	0.962428i	$0.412468\pi$
$72$	0	0
$73$	−14.4453	−1.69070	−0.845349	−	0.534215i	$-0.820607\pi$
$73$	−14.4453	−1.69070	−0.845349	+	0.534215i	$0.820607\pi$
$74$	0	0
$75$	2.21642	0.255930
$76$	0	0
$77$	2.87269	0.327374
$78$	0	0
$79$	−7.20620	−0.810761	−0.405381	−	0.914148i	$-0.632861\pi$
$79$	−7.20620	−0.810761	−0.405381	+	0.914148i	$0.632861\pi$
$80$	0	0
$81$	6.42262	0.713625
$82$	0	0
$83$	−7.39056	−0.811219	−0.405610	−	0.914046i	$-0.632941\pi$
$83$	−7.39056	−0.811219	−0.405610	+	0.914046i	$0.632941\pi$
$84$	0	0
$85$	−3.06450	−0.332391
$86$	0	0
$87$	−3.44669	−0.369524
$88$	0	0
$89$	0.838265	0.0888559	0.0444280	−	0.999013i	$-0.485853\pi$
$89$	0.838265	0.0888559	0.0444280	+	0.999013i	$0.485853\pi$
$90$	0	0
$91$	−14.5784	−1.52823
$92$	0	0
$93$	−4.61421	−0.478471
$94$	0	0
$95$	−3.97117	−0.407433
$96$	0	0
$97$	0.475381	0.0482676	0.0241338	−	0.999709i	$-0.492317\pi$
$97$	0.475381	0.0482676	0.0241338	+	0.999709i	$0.492317\pi$
$98$	0	0
$99$	−2.38779	−0.239982
$100$	0	0
$101$	−3.07939	−0.306411	−0.153205	−	0.988194i	$-0.548960\pi$
$101$	−3.07939	−0.306411	−0.153205	+	0.988194i	$0.548960\pi$
$102$	0	0
$103$	3.71378	0.365929	0.182965	−	0.983119i	$-0.441431\pi$
$103$	3.71378	0.365929	0.182965	+	0.983119i	$0.441431\pi$
$104$	0	0
$105$	−1.70429	−0.166322
$106$	0	0
$107$	−16.1036	−1.55680	−0.778398	−	0.627771i	$-0.783968\pi$
$107$	−16.1036	−1.55680	−0.778398	+	0.627771i	$0.783968\pi$
$108$	0	0
$109$	2.61098	0.250087	0.125043	−	0.992151i	$-0.460093\pi$
$109$	2.61098	0.250087	0.125043	+	0.992151i	$0.460093\pi$
$110$	0	0
$111$	−6.00624	−0.570087
$112$	0	0
$113$	−12.2204	−1.14960	−0.574798	−	0.818295i	$-0.694919\pi$
$113$	−12.2204	−1.14960	−0.574798	+	0.818295i	$0.694919\pi$
$114$	0	0
$115$	−3.60053	−0.335751
$116$	0	0
$117$	12.1176	1.12027
$118$	0	0
$119$	−10.3540	−0.949154
$120$	0	0
$121$	−10.2201	−0.929104
$122$	0	0
$123$	−0.768028	−0.0692508
$124$	0	0
$125$	−8.73541	−0.781318
$126$	0	0
$127$	−5.90845	−0.524290	−0.262145	−	0.965028i	$-0.584430\pi$
$127$	−5.90845	−0.524290	−0.262145	+	0.965028i	$0.584430\pi$
$128$	0	0
$129$	−5.06673	−0.446100
$130$	0	0
$131$	−4.91027	−0.429012	−0.214506	−	0.976723i	$-0.568814\pi$
$131$	−4.91027	−0.429012	−0.214506	+	0.976723i	$0.568814\pi$
$132$	0	0
$133$	−13.4174	−1.16344
$134$	0	0
$135$	2.98836	0.257197
$136$	0	0
$137$	−8.29905	−0.709036	−0.354518	−	0.935049i	$-0.615355\pi$
$137$	−8.29905	−0.709036	−0.354518	+	0.935049i	$0.615355\pi$
$138$	0	0
$139$	−21.0776	−1.78778	−0.893889	−	0.448288i	$-0.852034\pi$
$139$	−21.0776	−1.78778	−0.893889	+	0.448288i	$0.852034\pi$
$140$	0	0
$141$	−3.16809	−0.266802
$142$	0	0
$143$	−3.95763	−0.330954
$144$	0	0
$145$	6.09818	0.506426
$146$	0	0
$147$	−1.94914	−0.160762
$148$	0	0
$149$	−20.9236	−1.71413	−0.857063	−	0.515212i	$-0.827713\pi$
$149$	−20.9236	−1.71413	−0.857063	+	0.515212i	$0.827713\pi$
$150$	0	0
$151$	12.2666	0.998241	0.499121	−	0.866533i	$-0.333656\pi$
$151$	12.2666	0.998241	0.499121	+	0.866533i	$0.333656\pi$
$152$	0	0
$153$	8.60630	0.695778
$154$	0	0
$155$	8.16385	0.655736
$156$	0	0
$157$	9.63793	0.769191	0.384595	−	0.923085i	$-0.374341\pi$
$157$	9.63793	0.769191	0.384595	+	0.923085i	$0.374341\pi$
$158$	0	0
$159$	3.84632	0.305033
$160$	0	0
$161$	−12.1652	−0.958749
$162$	0	0
$163$	−11.8440	−0.927693	−0.463847	−	0.885915i	$-0.653531\pi$
$163$	−11.8440	−0.927693	−0.463847	+	0.885915i	$0.653531\pi$
$164$	0	0
$165$	−0.462669	−0.0360187
$166$	0	0
$167$	18.8572	1.45922	0.729609	−	0.683865i	$-0.239702\pi$
$167$	18.8572	1.45922	0.729609	+	0.683865i	$0.239702\pi$
$168$	0	0
$169$	7.08424	0.544941
$170$	0	0
$171$	11.1526	0.852859
$172$	0	0
$173$	−15.1822	−1.15428	−0.577140	−	0.816645i	$-0.695831\pi$
$173$	−15.1822	−1.15428	−0.577140	+	0.816645i	$0.695831\pi$
$174$	0	0
$175$	−13.2495	−1.00157
$176$	0	0
$177$	7.03468	0.528759
$178$	0	0
$179$	2.57897	0.192761	0.0963806	−	0.995345i	$-0.469273\pi$
$179$	2.57897	0.192761	0.0963806	+	0.995345i	$0.469273\pi$
$180$	0	0
$181$	−9.50776	−0.706706	−0.353353	−	0.935490i	$-0.614959\pi$
$181$	−9.50776	−0.706706	−0.353353	+	0.935490i	$0.614959\pi$
$182$	0	0
$183$	1.65850	0.122600
$184$	0	0
$185$	10.6268	0.781295
$186$	0	0
$187$	−2.81084	−0.205549
$188$	0	0
$189$	10.0968	0.734435
$190$	0	0
$191$	−6.20885	−0.449257	−0.224628	−	0.974444i	$-0.572117\pi$
$191$	−6.20885	−0.449257	−0.224628	+	0.974444i	$0.572117\pi$
$192$	0	0
$193$	−19.6307	−1.41305	−0.706526	−	0.707687i	$-0.749739\pi$
$193$	−19.6307	−1.41305	−0.706526	+	0.707687i	$0.749739\pi$
$194$	0	0
$195$	2.34796	0.168141
$196$	0	0
$197$	26.3638	1.87834	0.939171	−	0.343449i	$-0.111595\pi$
$197$	26.3638	1.87834	0.939171	+	0.343449i	$0.111595\pi$
$198$	0	0
$199$	5.17965	0.367176	0.183588	−	0.983003i	$-0.441229\pi$
$199$	5.17965	0.367176	0.183588	+	0.983003i	$0.441229\pi$
$200$	0	0
$201$	−4.98032	−0.351285
$202$	0	0
$203$	20.6040	1.44612
$204$	0	0
$205$	1.35886	0.0949069
$206$	0	0
$207$	10.1117	0.702812
$208$	0	0
$209$	−3.64247	−0.251955
$210$	0	0
$211$	−11.3624	−0.782217	−0.391109	−	0.920344i	$-0.627908\pi$
$211$	−11.3624	−0.782217	−0.391109	+	0.920344i	$0.627908\pi$
$212$	0	0
$213$	−2.49011	−0.170620
$214$	0	0
$215$	8.96448	0.611373
$216$	0	0
$217$	27.5833	1.87248
$218$	0	0
$219$	7.86068	0.531175
$220$	0	0
$221$	14.2645	0.959534
$222$	0	0
$223$	−18.9951	−1.27201	−0.636003	−	0.771686i	$-0.719413\pi$
$223$	−18.9951	−1.27201	−0.636003	+	0.771686i	$0.719413\pi$
$224$	0	0
$225$	11.0130	0.734201
$226$	0	0
$227$	−11.7708	−0.781254	−0.390627	−	0.920549i	$-0.627742\pi$
$227$	−11.7708	−0.781254	−0.390627	+	0.920549i	$0.627742\pi$
$228$	0	0
$229$	6.31464	0.417283	0.208641	−	0.977992i	$-0.433096\pi$
$229$	6.31464	0.417283	0.208641	+	0.977992i	$0.433096\pi$
$230$	0	0
$231$	−1.56323	−0.102853
$232$	0	0
$233$	20.7839	1.36160	0.680799	−	0.732470i	$-0.261633\pi$
$233$	20.7839	1.36160	0.680799	+	0.732470i	$0.261633\pi$
$234$	0	0
$235$	5.60526	0.365647
$236$	0	0
$237$	3.92138	0.254721
$238$	0	0
$239$	23.8405	1.54212	0.771058	−	0.636765i	$-0.219728\pi$
$239$	23.8405	1.54212	0.771058	+	0.636765i	$0.219728\pi$
$240$	0	0
$241$	1.62031	0.104374	0.0521868	−	0.998637i	$-0.483381\pi$
$241$	1.62031	0.104374	0.0521868	+	0.998637i	$0.483381\pi$
$242$	0	0
$243$	−12.8066	−0.821543
$244$	0	0
$245$	3.44859	0.220322
$246$	0	0
$247$	18.4848	1.17616
$248$	0	0
$249$	4.02170	0.254865
$250$	0	0
$251$	10.7109	0.676064	0.338032	−	0.941135i	$-0.390239\pi$
$251$	10.7109	0.676064	0.338032	+	0.941135i	$0.390239\pi$
$252$	0	0
$253$	−3.30251	−0.207627
$254$	0	0
$255$	1.66760	0.104429
$256$	0	0
$257$	−28.7954	−1.79621	−0.898105	−	0.439781i	$-0.855056\pi$
$257$	−28.7954	−1.79621	−0.898105	+	0.439781i	$0.855056\pi$
$258$	0	0
$259$	35.9047	2.23101
$260$	0	0
$261$	−17.1261	−1.06008
$262$	0	0
$263$	27.1652	1.67508	0.837539	−	0.546378i	$-0.183994\pi$
$263$	27.1652	1.67508	0.837539	+	0.546378i	$0.183994\pi$
$264$	0	0
$265$	−6.80523	−0.418042
$266$	0	0
$267$	−0.456156	−0.0279163
$268$	0	0
$269$	−19.2182	−1.17176	−0.585878	−	0.810399i	$-0.699250\pi$
$269$	−19.2182	−1.17176	−0.585878	+	0.810399i	$0.699250\pi$
$270$	0	0
$271$	9.73786	0.591533	0.295766	−	0.955260i	$-0.404425\pi$
$271$	9.73786	0.591533	0.295766	+	0.955260i	$0.404425\pi$
$272$	0	0
$273$	7.93308	0.480132
$274$	0	0
$275$	−3.59689	−0.216900
$276$	0	0
$277$	−26.8931	−1.61585	−0.807925	−	0.589285i	$-0.799409\pi$
$277$	−26.8931	−1.61585	−0.807925	+	0.589285i	$0.799409\pi$
$278$	0	0
$279$	−22.9273	−1.37262
$280$	0	0
$281$	−4.21282	−0.251316	−0.125658	−	0.992074i	$-0.540104\pi$
$281$	−4.21282	−0.251316	−0.125658	+	0.992074i	$0.540104\pi$
$282$	0	0
$283$	−31.3587	−1.86408	−0.932039	−	0.362357i	$-0.881972\pi$
$283$	−31.3587	−1.86408	−0.932039	+	0.362357i	$0.881972\pi$
$284$	0	0
$285$	2.16098	0.128005
$286$	0	0
$287$	4.59120	0.271010
$288$	0	0
$289$	−6.86888	−0.404052
$290$	0	0
$291$	−0.258687	−0.0151645
$292$	0	0
$293$	−15.8890	−0.928244	−0.464122	−	0.885771i	$-0.653630\pi$
$293$	−15.8890	−0.928244	−0.464122	+	0.885771i	$0.653630\pi$
$294$	0	0
$295$	−12.4463	−0.724654
$296$	0	0
$297$	2.74101	0.159050
$298$	0	0
$299$	16.7596	0.969234
$300$	0	0
$301$	30.2884	1.74579
$302$	0	0
$303$	1.67570	0.0962667
$304$	0	0
$305$	−2.93436	−0.168021
$306$	0	0
$307$	13.7030	0.782070	0.391035	−	0.920376i	$-0.372117\pi$
$307$	13.7030	0.782070	0.391035	+	0.920376i	$0.372117\pi$
$308$	0	0
$309$	−2.02092	−0.114966
$310$	0	0
$311$	−6.49422	−0.368253	−0.184127	−	0.982903i	$-0.558946\pi$
$311$	−6.49422	−0.368253	−0.184127	+	0.982903i	$0.558946\pi$
$312$	0	0
$313$	8.26403	0.467111	0.233555	−	0.972344i	$-0.424964\pi$
$313$	8.26403	0.467111	0.233555	+	0.972344i	$0.424964\pi$
$314$	0	0
$315$	−8.46836	−0.477138
$316$	0	0
$317$	−28.1441	−1.58073	−0.790364	−	0.612638i	$-0.790108\pi$
$317$	−28.1441	−1.58073	−0.790364	+	0.612638i	$0.790108\pi$
$318$	0	0
$319$	5.59343	0.313172
$320$	0	0
$321$	8.76307	0.489107
$322$	0	0
$323$	13.1285	0.730492
$324$	0	0
$325$	18.2535	1.01252
$326$	0	0
$327$	−1.42081	−0.0785710
$328$	0	0
$329$	18.9386	1.04412
$330$	0	0
$331$	1.78319	0.0980130	0.0490065	−	0.998798i	$-0.484394\pi$
$331$	1.78319	0.0980130	0.0490065	+	0.998798i	$0.484394\pi$
$332$	0	0
$333$	−29.8441	−1.63545
$334$	0	0
$335$	8.81160	0.481429
$336$	0	0
$337$	17.5528	0.956160	0.478080	−	0.878316i	$-0.341333\pi$
$337$	17.5528	0.956160	0.478080	+	0.878316i	$0.341333\pi$
$338$	0	0
$339$	6.64993	0.361175
$340$	0	0
$341$	7.48811	0.405504
$342$	0	0
$343$	−11.1191	−0.600374
$344$	0	0
$345$	1.95929	0.105485
$346$	0	0
$347$	−14.8308	−0.796160	−0.398080	−	0.917351i	$-0.630323\pi$
$347$	−14.8308	−0.796160	−0.398080	+	0.917351i	$0.630323\pi$
$348$	0	0
$349$	−34.3697	−1.83977	−0.919884	−	0.392190i	$-0.871718\pi$
$349$	−34.3697	−1.83977	−0.919884	+	0.392190i	$0.871718\pi$
$350$	0	0
$351$	−13.9101	−0.742467
$352$	0	0
$353$	−16.1758	−0.860949	−0.430474	−	0.902603i	$-0.641654\pi$
$353$	−16.1758	−0.860949	−0.430474	+	0.902603i	$0.641654\pi$
$354$	0	0
$355$	4.40572	0.233831
$356$	0	0
$357$	5.63433	0.298201
$358$	0	0
$359$	−17.7167	−0.935053	−0.467526	−	0.883979i	$-0.654855\pi$
$359$	−17.7167	−0.935053	−0.467526	+	0.883979i	$0.654855\pi$
$360$	0	0
$361$	−1.98721	−0.104590
$362$	0	0
$363$	5.56147	0.291901
$364$	0	0
$365$	−13.9078	−0.727966
$366$	0	0
$367$	−7.29148	−0.380612	−0.190306	−	0.981725i	$-0.560948\pi$
$367$	−7.29148	−0.380612	−0.190306	+	0.981725i	$0.560948\pi$
$368$	0	0
$369$	−3.81621	−0.198664
$370$	0	0
$371$	−22.9929	−1.19373
$372$	0	0
$373$	19.4135	1.00520	0.502598	−	0.864520i	$-0.332378\pi$
$373$	19.4135	1.00520	0.502598	+	0.864520i	$0.332378\pi$
$374$	0	0
$375$	4.75352	0.245471
$376$	0	0
$377$	−28.3856	−1.46193
$378$	0	0
$379$	−21.5543	−1.10717	−0.553586	−	0.832792i	$-0.686741\pi$
$379$	−21.5543	−1.10717	−0.553586	+	0.832792i	$0.686741\pi$
$380$	0	0
$381$	3.21519	0.164719
$382$	0	0
$383$	8.69462	0.444274	0.222137	−	0.975015i	$-0.428697\pi$
$383$	8.69462	0.444274	0.222137	+	0.975015i	$0.428697\pi$
$384$	0	0
$385$	2.76579	0.140958
$386$	0	0
$387$	−25.1758	−1.27976
$388$	0	0
$389$	20.0986	1.01904	0.509520	−	0.860459i	$-0.329823\pi$
$389$	20.0986	1.01904	0.509520	+	0.860459i	$0.329823\pi$
$390$	0	0
$391$	11.9032	0.601973
$392$	0	0
$393$	2.67201	0.134785
$394$	0	0
$395$	−6.93804	−0.349091
$396$	0	0
$397$	−22.3318	−1.12080	−0.560402	−	0.828221i	$-0.689353\pi$
$397$	−22.3318	−1.12080	−0.560402	+	0.828221i	$0.689353\pi$
$398$	0	0
$399$	7.30132	0.365523
$400$	0	0
$401$	−1.95708	−0.0977321	−0.0488661	−	0.998805i	$-0.515561\pi$
$401$	−1.95708	−0.0977321	−0.0488661	+	0.998805i	$0.515561\pi$
$402$	0	0
$403$	−38.0008	−1.89295
$404$	0	0
$405$	6.18362	0.307266
$406$	0	0
$407$	9.74716	0.483149
$408$	0	0
$409$	14.4716	0.715576	0.357788	−	0.933803i	$-0.383531\pi$
$409$	14.4716	0.715576	0.357788	+	0.933803i	$0.383531\pi$
$410$	0	0
$411$	4.51607	0.222761
$412$	0	0
$413$	−42.0526	−2.06927
$414$	0	0
$415$	−7.11554	−0.349288
$416$	0	0
$417$	11.4697	0.561676
$418$	0	0
$419$	4.92102	0.240408	0.120204	−	0.992749i	$-0.461645\pi$
$419$	4.92102	0.240408	0.120204	+	0.992749i	$0.461645\pi$
$420$	0	0
$421$	16.7810	0.817857	0.408928	−	0.912566i	$-0.365903\pi$
$421$	16.7810	0.817857	0.408928	+	0.912566i	$0.365903\pi$
$422$	0	0
$423$	−15.7418	−0.765390
$424$	0	0
$425$	12.9643	0.628859
$426$	0	0
$427$	−9.91435	−0.479789
$428$	0	0
$429$	2.15361	0.103977
$430$	0	0
$431$	−10.5214	−0.506800	−0.253400	−	0.967362i	$-0.581549\pi$
$431$	−10.5214	−0.506800	−0.253400	+	0.967362i	$0.581549\pi$
$432$	0	0
$433$	−11.8135	−0.567721	−0.283860	−	0.958866i	$-0.591615\pi$
$433$	−11.8135	−0.567721	−0.283860	+	0.958866i	$0.591615\pi$
$434$	0	0
$435$	−3.31843	−0.159107
$436$	0	0
$437$	15.4250	0.737876
$438$	0	0
$439$	−8.18885	−0.390833	−0.195416	−	0.980720i	$-0.562606\pi$
$439$	−8.18885	−0.390833	−0.195416	+	0.980720i	$0.562606\pi$
$440$	0	0
$441$	−9.68497	−0.461189
$442$	0	0
$443$	15.7119	0.746496	0.373248	−	0.927732i	$-0.378244\pi$
$443$	15.7119	0.746496	0.373248	+	0.927732i	$0.378244\pi$
$444$	0	0
$445$	0.807071	0.0382588
$446$	0	0
$447$	11.3859	0.538536
$448$	0	0
$449$	9.89465	0.466957	0.233479	−	0.972362i	$-0.424989\pi$
$449$	9.89465	0.466957	0.233479	+	0.972362i	$0.424989\pi$
$450$	0	0
$451$	1.24639	0.0586900
$452$	0	0
$453$	−6.67508	−0.313623
$454$	0	0
$455$	−14.0359	−0.658012
$456$	0	0
$457$	19.4319	0.908985	0.454492	−	0.890751i	$-0.349821\pi$
$457$	19.4319	0.908985	0.454492	+	0.890751i	$0.349821\pi$
$458$	0	0
$459$	−9.87943	−0.461132
$460$	0	0
$461$	28.4760	1.32626	0.663130	−	0.748504i	$-0.269228\pi$
$461$	28.4760	1.32626	0.663130	+	0.748504i	$0.269228\pi$
$462$	0	0
$463$	10.8095	0.502361	0.251181	−	0.967940i	$-0.419181\pi$
$463$	10.8095	0.502361	0.251181	+	0.967940i	$0.419181\pi$
$464$	0	0
$465$	−4.44250	−0.206016
$466$	0	0
$467$	−26.3662	−1.22008	−0.610041	−	0.792369i	$-0.708847\pi$
$467$	−26.3662	−1.22008	−0.610041	+	0.792369i	$0.708847\pi$
$468$	0	0
$469$	29.7719	1.37474
$470$	0	0
$471$	−5.24465	−0.241661
$472$	0	0
$473$	8.22248	0.378070
$474$	0	0
$475$	16.7999	0.770832
$476$	0	0
$477$	19.1117	0.875067
$478$	0	0
$479$	0.194030	0.00886547	0.00443274	−	0.999990i	$-0.498589\pi$
$479$	0.194030	0.00886547	0.00443274	+	0.999990i	$0.498589\pi$
$480$	0	0
$481$	−49.4650	−2.25541
$482$	0	0
$483$	6.61988	0.301215
$484$	0	0
$485$	0.457691	0.0207827
$486$	0	0
$487$	2.65682	0.120392	0.0601959	−	0.998187i	$-0.480827\pi$
$487$	2.65682	0.120392	0.0601959	+	0.998187i	$0.480827\pi$
$488$	0	0
$489$	6.44512	0.291458
$490$	0	0
$491$	−11.0398	−0.498219	−0.249109	−	0.968475i	$-0.580138\pi$
$491$	−11.0398	−0.498219	−0.249109	+	0.968475i	$0.580138\pi$
$492$	0	0
$493$	−20.1604	−0.907978
$494$	0	0
$495$	−2.29893	−0.103329
$496$	0	0
$497$	14.8857	0.667713
$498$	0	0
$499$	−30.9966	−1.38760	−0.693799	−	0.720168i	$-0.744065\pi$
$499$	−30.9966	−1.38760	−0.693799	+	0.720168i	$0.744065\pi$
$500$	0	0
$501$	−10.2615	−0.458450
$502$	0	0
$503$	37.0982	1.65413	0.827063	−	0.562109i	$-0.190010\pi$
$503$	37.0982	1.65413	0.827063	+	0.562109i	$0.190010\pi$
$504$	0	0
$505$	−2.96480	−0.131932
$506$	0	0
$507$	−3.85501	−0.171207
$508$	0	0
$509$	−8.10275	−0.359148	−0.179574	−	0.983744i	$-0.557472\pi$
$509$	−8.10275	−0.359148	−0.179574	+	0.983744i	$0.557472\pi$
$510$	0	0
$511$	−46.9904	−2.07873
$512$	0	0
$513$	−12.8024	−0.565239
$514$	0	0
$515$	3.57558	0.157559
$516$	0	0
$517$	5.14130	0.226114
$518$	0	0
$519$	8.26165	0.362646
$520$	0	0
$521$	9.96721	0.436671	0.218336	−	0.975874i	$-0.429937\pi$
$521$	9.96721	0.436671	0.218336	+	0.975874i	$0.429937\pi$
$522$	0	0
$523$	7.12073	0.311368	0.155684	−	0.987807i	$-0.450242\pi$
$523$	7.12073	0.311368	0.155684	+	0.987807i	$0.450242\pi$
$524$	0	0
$525$	7.20996	0.314668
$526$	0	0
$527$	−26.9894	−1.17568
$528$	0	0
$529$	−9.01467	−0.391942
$530$	0	0
$531$	34.9542	1.51688
$532$	0	0
$533$	−6.32517	−0.273974
$534$	0	0
$535$	−15.5044	−0.670312
$536$	0	0
$537$	−1.40339	−0.0605608
$538$	0	0
$539$	3.16314	0.136246
$540$	0	0
$541$	39.1006	1.68107	0.840533	−	0.541760i	$-0.182242\pi$
$541$	39.1006	1.68107	0.840533	+	0.541760i	$0.182242\pi$
$542$	0	0
$543$	5.17381	0.222030
$544$	0	0
$545$	2.51382	0.107680
$546$	0	0
$547$	1.00000	0.0427569
$547$	1.00000	0.0427569
$548$	0	0
$549$	8.24082	0.351710
$550$	0	0
$551$	−26.1251	−1.11297
$552$	0	0
$553$	−23.4416	−0.996840
$554$	0	0
$555$	−5.78273	−0.245463
$556$	0	0
$557$	39.1227	1.65768	0.828841	−	0.559485i	$-0.189001\pi$
$557$	39.1227	1.65768	0.828841	+	0.559485i	$0.189001\pi$
$558$	0	0
$559$	−41.7275	−1.76489
$560$	0	0
$561$	1.52957	0.0645784
$562$	0	0
$563$	−32.3381	−1.36289	−0.681445	−	0.731869i	$-0.738648\pi$
$563$	−32.3381	−1.36289	−0.681445	+	0.731869i	$0.738648\pi$
$564$	0	0
$565$	−11.7656	−0.494983
$566$	0	0
$567$	20.8927	0.877409
$568$	0	0
$569$	0.316254	0.0132581	0.00662903	−	0.999978i	$-0.497890\pi$
$569$	0.316254	0.0132581	0.00662903	+	0.999978i	$0.497890\pi$
$570$	0	0
$571$	−4.67642	−0.195702	−0.0978511	−	0.995201i	$-0.531197\pi$
$571$	−4.67642	−0.195702	−0.0978511	+	0.995201i	$0.531197\pi$
$572$	0	0
$573$	3.37866	0.141145
$574$	0	0
$575$	15.2319	0.635216
$576$	0	0
$577$	38.4107	1.59906	0.799529	−	0.600627i	$-0.205082\pi$
$577$	38.4107	1.59906	0.799529	+	0.600627i	$0.205082\pi$
$578$	0	0
$579$	10.6824	0.443946
$580$	0	0
$581$	−24.0413	−0.997403
$582$	0	0
$583$	−6.24195	−0.258515
$584$	0	0
$585$	11.6666	0.482356
$586$	0	0
$587$	−6.56995	−0.271171	−0.135586	−	0.990766i	$-0.543292\pi$
$587$	−6.56995	−0.271171	−0.135586	+	0.990766i	$0.543292\pi$
$588$	0	0
$589$	−34.9746	−1.44110
$590$	0	0
$591$	−14.3463	−0.590129
$592$	0	0
$593$	−0.570902	−0.0234441	−0.0117221	−	0.999931i	$-0.503731\pi$
$593$	−0.570902	−0.0234441	−0.0117221	+	0.999931i	$0.503731\pi$
$594$	0	0
$595$	−9.96874	−0.408679
$596$	0	0
$597$	−2.81860	−0.115357
$598$	0	0
$599$	−8.95352	−0.365831	−0.182915	−	0.983129i	$-0.558553\pi$
$599$	−8.95352	−0.365831	−0.182915	+	0.983129i	$0.558553\pi$
$600$	0	0
$601$	−12.7284	−0.519202	−0.259601	−	0.965716i	$-0.583591\pi$
$601$	−12.7284	−0.519202	−0.259601	+	0.965716i	$0.583591\pi$
$602$	0	0
$603$	−24.7464	−1.00775
$604$	0	0
$605$	−9.83982	−0.400046
$606$	0	0
$607$	−26.7859	−1.08721	−0.543603	−	0.839342i	$-0.682940\pi$
$607$	−26.7859	−1.08721	−0.543603	+	0.839342i	$0.682940\pi$
$608$	0	0
$609$	−11.2120	−0.454334
$610$	0	0
$611$	−26.0911	−1.05553
$612$	0	0
$613$	31.3273	1.26530	0.632649	−	0.774439i	$-0.281968\pi$
$613$	31.3273	1.26530	0.632649	+	0.774439i	$0.281968\pi$
$614$	0	0
$615$	−0.739448	−0.0298174
$616$	0	0
$617$	−31.6103	−1.27258	−0.636292	−	0.771448i	$-0.719533\pi$
$617$	−31.6103	−1.27258	−0.636292	+	0.771448i	$0.719533\pi$
$618$	0	0
$619$	−7.99712	−0.321431	−0.160716	−	0.987001i	$-0.551380\pi$
$619$	−7.99712	−0.321431	−0.160716	+	0.987001i	$0.551380\pi$
$620$	0	0
$621$	−11.6075	−0.465794
$622$	0	0
$623$	2.72686	0.109249
$624$	0	0
$625$	11.9549	0.478195
$626$	0	0
$627$	1.98211	0.0791579
$628$	0	0
$629$	−35.1317	−1.40079
$630$	0	0
$631$	44.2164	1.76023	0.880113	−	0.474764i	$-0.157466\pi$
$631$	44.2164	1.76023	0.880113	+	0.474764i	$0.157466\pi$
$632$	0	0
$633$	6.18303	0.245753
$634$	0	0
$635$	−5.68858	−0.225744
$636$	0	0
$637$	−16.0523	−0.636017
$638$	0	0
$639$	−12.3730	−0.489468
$640$	0	0
$641$	−16.8320	−0.664824	−0.332412	−	0.943134i	$-0.607862\pi$
$641$	−16.8320	−0.664824	−0.332412	+	0.943134i	$0.607862\pi$
$642$	0	0
$643$	12.4597	0.491362	0.245681	−	0.969351i	$-0.420988\pi$
$643$	12.4597	0.491362	0.245681	+	0.969351i	$0.420988\pi$
$644$	0	0
$645$	−4.87818	−0.192078
$646$	0	0
$647$	48.1533	1.89310	0.946551	−	0.322555i	$-0.104542\pi$
$647$	48.1533	1.89310	0.946551	+	0.322555i	$0.104542\pi$
$648$	0	0
$649$	−11.4161	−0.448123
$650$	0	0
$651$	−15.0099	−0.588285
$652$	0	0
$653$	−12.7505	−0.498967	−0.249484	−	0.968379i	$-0.580261\pi$
$653$	−12.7505	−0.498967	−0.249484	+	0.968379i	$0.580261\pi$
$654$	0	0
$655$	−4.72754	−0.184720
$656$	0	0
$657$	39.0585	1.52382
$658$	0	0
$659$	29.8939	1.16450	0.582250	−	0.813010i	$-0.302172\pi$
$659$	29.8939	1.16450	0.582250	+	0.813010i	$0.302172\pi$
$660$	0	0
$661$	22.0815	0.858872	0.429436	−	0.903097i	$-0.358712\pi$
$661$	22.0815	0.858872	0.429436	+	0.903097i	$0.358712\pi$
$662$	0	0
$663$	−7.76227	−0.301462
$664$	0	0
$665$	−12.9181	−0.500943
$666$	0	0
$667$	−23.6868	−0.917157
$668$	0	0
$669$	10.3365	0.399633
$670$	0	0
$671$	−2.69148	−0.103903
$672$	0	0
$673$	24.6133	0.948772	0.474386	−	0.880317i	$-0.342670\pi$
$673$	24.6133	0.948772	0.474386	+	0.880317i	$0.342670\pi$
$674$	0	0
$675$	−12.6422	−0.486598
$676$	0	0
$677$	15.1930	0.583913	0.291956	−	0.956432i	$-0.405694\pi$
$677$	15.1930	0.583913	0.291956	+	0.956432i	$0.405694\pi$
$678$	0	0
$679$	1.54640	0.0593456
$680$	0	0
$681$	6.40527	0.245451
$682$	0	0
$683$	23.5519	0.901189	0.450594	−	0.892729i	$-0.351212\pi$
$683$	23.5519	0.901189	0.450594	+	0.892729i	$0.351212\pi$
$684$	0	0
$685$	−7.99022	−0.305291
$686$	0	0
$687$	−3.43622	−0.131100
$688$	0	0
$689$	31.6767	1.20679
$690$	0	0
$691$	37.0300	1.40869	0.704343	−	0.709860i	$-0.251242\pi$
$691$	37.0300	1.40869	0.704343	+	0.709860i	$0.251242\pi$
$692$	0	0
$693$	−7.76742	−0.295060
$694$	0	0
$695$	−20.2932	−0.769766
$696$	0	0
$697$	−4.49235	−0.170160
$698$	0	0
$699$	−11.3099	−0.427781
$700$	0	0
$701$	35.5829	1.34395	0.671973	−	0.740576i	$-0.265447\pi$
$701$	35.5829	1.34395	0.671973	+	0.740576i	$0.265447\pi$
$702$	0	0
$703$	−45.5259	−1.71704
$704$	0	0
$705$	−3.05020	−0.114877
$706$	0	0
$707$	−10.0172	−0.376736
$708$	0	0
$709$	40.5125	1.52148	0.760739	−	0.649058i	$-0.224837\pi$
$709$	40.5125	1.52148	0.760739	+	0.649058i	$0.224837\pi$
$710$	0	0
$711$	19.4847	0.730734
$712$	0	0
$713$	−31.7103	−1.18756
$714$	0	0
$715$	−3.81036	−0.142499
$716$	0	0
$717$	−12.9732	−0.484494
$718$	0	0
$719$	−6.36743	−0.237465	−0.118733	−	0.992926i	$-0.537883\pi$
$719$	−6.36743	−0.237465	−0.118733	+	0.992926i	$0.537883\pi$
$720$	0	0
$721$	12.0808	0.449914
$722$	0	0
$723$	−0.881722	−0.0327916
$724$	0	0
$725$	−25.7982	−0.958120
$726$	0	0
$727$	35.1164	1.30240	0.651198	−	0.758908i	$-0.274267\pi$
$727$	35.1164	1.30240	0.651198	+	0.758908i	$0.274267\pi$
$728$	0	0
$729$	−12.2989	−0.455516
$730$	0	0
$731$	−29.6363	−1.09614
$732$	0	0
$733$	26.1891	0.967315	0.483658	−	0.875257i	$-0.339308\pi$
$733$	26.1891	0.967315	0.483658	+	0.875257i	$0.339308\pi$
$734$	0	0
$735$	−1.87661	−0.0692197
$736$	0	0
$737$	8.08225	0.297714
$738$	0	0
$739$	−15.8483	−0.582988	−0.291494	−	0.956573i	$-0.594152\pi$
$739$	−15.8483	−0.582988	−0.291494	+	0.956573i	$0.594152\pi$
$740$	0	0
$741$	−10.0588	−0.369521
$742$	0	0
$743$	−13.1980	−0.484189	−0.242095	−	0.970253i	$-0.577834\pi$
$743$	−13.1980	−0.484189	−0.242095	+	0.970253i	$0.577834\pi$
$744$	0	0
$745$	−20.1449	−0.738054
$746$	0	0
$747$	19.9832	0.731147
$748$	0	0
$749$	−52.3848	−1.91410
$750$	0	0
$751$	17.3729	0.633945	0.316973	−	0.948435i	$-0.397334\pi$
$751$	17.3729	0.633945	0.316973	+	0.948435i	$0.397334\pi$
$752$	0	0
$753$	−5.82851	−0.212403
$754$	0	0
$755$	11.8101	0.429814
$756$	0	0
$757$	−38.7514	−1.40844	−0.704222	−	0.709980i	$-0.748704\pi$
$757$	−38.7514	−1.40844	−0.704222	+	0.709980i	$0.748704\pi$
$758$	0	0
$759$	1.79712	0.0652313
$760$	0	0
$761$	−29.0280	−1.05226	−0.526132	−	0.850403i	$-0.676358\pi$
$761$	−29.0280	−1.05226	−0.526132	+	0.850403i	$0.676358\pi$
$762$	0	0
$763$	8.49347	0.307484
$764$	0	0
$765$	8.28603	0.299582
$766$	0	0
$767$	57.9348	2.09190
$768$	0	0
$769$	−8.49451	−0.306320	−0.153160	−	0.988201i	$-0.548945\pi$
$769$	−8.49451	−0.306320	−0.153160	+	0.988201i	$0.548945\pi$
$770$	0	0
$771$	15.6695	0.564325
$772$	0	0
$773$	−30.2676	−1.08865	−0.544325	−	0.838875i	$-0.683214\pi$
$773$	−30.2676	−1.08865	−0.544325	+	0.838875i	$0.683214\pi$
$774$	0	0
$775$	−34.5369	−1.24060
$776$	0	0
$777$	−19.5382	−0.700929
$778$	0	0
$779$	−5.82147	−0.208576
$780$	0	0
$781$	4.04105	0.144600
$782$	0	0
$783$	19.6595	0.702574
$784$	0	0
$785$	9.27927	0.331191
$786$	0	0
$787$	25.5339	0.910185	0.455093	−	0.890444i	$-0.349606\pi$
$787$	25.5339	0.910185	0.455093	+	0.890444i	$0.349606\pi$
$788$	0	0
$789$	−14.7824	−0.526268
$790$	0	0
$791$	−39.7526	−1.41344
$792$	0	0
$793$	13.6587	0.485036
$794$	0	0
$795$	3.70318	0.131338
$796$	0	0
$797$	−48.5429	−1.71948	−0.859738	−	0.510735i	$-0.829373\pi$
$797$	−48.5429	−1.71948	−0.859738	+	0.510735i	$0.829373\pi$
$798$	0	0
$799$	−18.5308	−0.655573
$800$	0	0
$801$	−2.26657	−0.0800853
$802$	0	0
$803$	−12.7566	−0.450171
$804$	0	0
$805$	−11.7125	−0.412810
$806$	0	0
$807$	10.4579	0.368137
$808$	0	0
$809$	30.1082	1.05855	0.529274	−	0.848451i	$-0.322464\pi$
$809$	30.1082	1.05855	0.529274	+	0.848451i	$0.322464\pi$
$810$	0	0
$811$	26.6166	0.934634	0.467317	−	0.884090i	$-0.345221\pi$
$811$	26.6166	0.934634	0.467317	+	0.884090i	$0.345221\pi$
$812$	0	0
$813$	−5.29902	−0.185845
$814$	0	0
$815$	−11.4032	−0.399438
$816$	0	0
$817$	−38.4046	−1.34361
$818$	0	0
$819$	39.4182	1.37738
$820$	0	0
$821$	−52.7820	−1.84210	−0.921052	−	0.389440i	$-0.872668\pi$
$821$	−52.7820	−1.84210	−0.921052	+	0.389440i	$0.872668\pi$
$822$	0	0
$823$	20.7992	0.725014	0.362507	−	0.931981i	$-0.381921\pi$
$823$	20.7992	0.725014	0.362507	+	0.931981i	$0.381921\pi$
$824$	0	0
$825$	1.95731	0.0681447
$826$	0	0
$827$	−12.6722	−0.440654	−0.220327	−	0.975426i	$-0.570713\pi$
$827$	−12.6722	−0.440654	−0.220327	+	0.975426i	$0.570713\pi$
$828$	0	0
$829$	24.4262	0.848357	0.424179	−	0.905579i	$-0.360563\pi$
$829$	24.4262	0.848357	0.424179	+	0.905579i	$0.360563\pi$
$830$	0	0
$831$	14.6343	0.507660
$832$	0	0
$833$	−11.4009	−0.395018
$834$	0	0
$835$	18.1555	0.628297
$836$	0	0
$837$	26.3189	0.909714
$838$	0	0
$839$	−41.6208	−1.43691	−0.718455	−	0.695574i	$-0.755150\pi$
$839$	−41.6208	−1.43691	−0.718455	+	0.695574i	$0.755150\pi$
$840$	0	0
$841$	11.1181	0.383383
$842$	0	0
$843$	2.29248	0.0789572
$844$	0	0
$845$	6.82061	0.234636
$846$	0	0
$847$	−33.2459	−1.14234
$848$	0	0
$849$	17.0644	0.585647
$850$	0	0
$851$	−41.2769	−1.41495
$852$	0	0
$853$	11.9436	0.408942	0.204471	−	0.978873i	$-0.434453\pi$
$853$	11.9436	0.408942	0.204471	+	0.978873i	$0.434453\pi$
$854$	0	0
$855$	10.7376	0.367217
$856$	0	0
$857$	−17.0803	−0.583452	−0.291726	−	0.956502i	$-0.594229\pi$
$857$	−17.0803	−0.583452	−0.291726	+	0.956502i	$0.594229\pi$
$858$	0	0
$859$	45.1499	1.54049	0.770247	−	0.637745i	$-0.220133\pi$
$859$	45.1499	1.54049	0.770247	+	0.637745i	$0.220133\pi$
$860$	0	0
$861$	−2.49838	−0.0851446
$862$	0	0
$863$	42.9192	1.46099	0.730494	−	0.682919i	$-0.239290\pi$
$863$	42.9192	1.46099	0.730494	+	0.682919i	$0.239290\pi$
$864$	0	0
$865$	−14.6172	−0.497000
$866$	0	0
$867$	3.73782	0.126943
$868$	0	0
$869$	−6.36377	−0.215876
$870$	0	0
$871$	−41.0159	−1.38977
$872$	0	0
$873$	−1.28537	−0.0435033
$874$	0	0
$875$	−28.4161	−0.960640
$876$	0	0
$877$	−22.1529	−0.748050	−0.374025	−	0.927419i	$-0.622023\pi$
$877$	−22.1529	−0.748050	−0.374025	+	0.927419i	$0.622023\pi$
$878$	0	0
$879$	8.64627	0.291631
$880$	0	0
$881$	−8.84872	−0.298121	−0.149060	−	0.988828i	$-0.547625\pi$
$881$	−8.84872	−0.298121	−0.149060	+	0.988828i	$0.547625\pi$
$882$	0	0
$883$	−14.6632	−0.493455	−0.246728	−	0.969085i	$-0.579355\pi$
$883$	−14.6632	−0.493455	−0.246728	+	0.969085i	$0.579355\pi$
$884$	0	0
$885$	6.77290	0.227668
$886$	0	0
$887$	−17.3973	−0.584143	−0.292072	−	0.956396i	$-0.594345\pi$
$887$	−17.3973	−0.584143	−0.292072	+	0.956396i	$0.594345\pi$
$888$	0	0
$889$	−19.2201	−0.644621
$890$	0	0
$891$	5.67179	0.190012
$892$	0	0
$893$	−24.0134	−0.803577
$894$	0	0
$895$	2.48300	0.0829975
$896$	0	0
$897$	−9.12004	−0.304509
$898$	0	0
$899$	53.7075	1.79124
$900$	0	0
$901$	22.4979	0.749512
$902$	0	0
$903$	−16.4820	−0.548485
$904$	0	0
$905$	−9.15395	−0.304288
$906$	0	0
$907$	33.7438	1.12045	0.560223	−	0.828342i	$-0.310715\pi$
$907$	33.7438	1.12045	0.560223	+	0.828342i	$0.310715\pi$
$908$	0	0
$909$	8.32631	0.276166
$910$	0	0
$911$	−12.1001	−0.400895	−0.200447	−	0.979704i	$-0.564240\pi$
$911$	−12.1001	−0.400895	−0.200447	+	0.979704i	$0.564240\pi$
$912$	0	0
$913$	−6.52657	−0.215998
$914$	0	0
$915$	1.59678	0.0527880
$916$	0	0
$917$	−15.9730	−0.527475
$918$	0	0
$919$	31.4919	1.03882	0.519412	−	0.854524i	$-0.326151\pi$
$919$	31.4919	1.03882	0.519412	+	0.854524i	$0.326151\pi$
$920$	0	0
$921$	−7.45671	−0.245707
$922$	0	0
$923$	−20.5076	−0.675015
$924$	0	0
$925$	−44.9562	−1.47815
$926$	0	0
$927$	−10.0416	−0.329810
$928$	0	0
$929$	28.6545	0.940123	0.470061	−	0.882634i	$-0.344232\pi$
$929$	28.6545	0.940123	0.470061	+	0.882634i	$0.344232\pi$
$930$	0	0
$931$	−14.7740	−0.484199
$932$	0	0
$933$	3.53394	0.115696
$934$	0	0
$935$	−2.70624	−0.0885036
$936$	0	0
$937$	6.14354	0.200701	0.100350	−	0.994952i	$-0.468004\pi$
$937$	6.14354	0.200701	0.100350	+	0.994952i	$0.468004\pi$
$938$	0	0
$939$	−4.49702	−0.146755
$940$	0	0
$941$	−25.9528	−0.846038	−0.423019	−	0.906121i	$-0.639030\pi$
$941$	−25.9528	−0.846038	−0.423019	+	0.906121i	$0.639030\pi$
$942$	0	0
$943$	−5.27814	−0.171880
$944$	0	0
$945$	9.72108	0.316227
$946$	0	0
$947$	−45.1337	−1.46665	−0.733324	−	0.679880i	$-0.762032\pi$
$947$	−45.1337	−1.46665	−0.733324	+	0.679880i	$0.762032\pi$
$948$	0	0
$949$	64.7374	2.10146
$950$	0	0
$951$	15.3151	0.496625
$952$	0	0
$953$	−6.09060	−0.197294	−0.0986469	−	0.995122i	$-0.531451\pi$
$953$	−6.09060	−0.197294	−0.0986469	+	0.995122i	$0.531451\pi$
$954$	0	0
$955$	−5.97781	−0.193437
$956$	0	0
$957$	−3.04376	−0.0983908
$958$	0	0
$959$	−26.9966	−0.871767
$960$	0	0
$961$	40.9000	1.31936
$962$	0	0
$963$	43.5423	1.40313
$964$	0	0
$965$	−18.9002	−0.608420
$966$	0	0
$967$	32.8474	1.05630	0.528151	−	0.849150i	$-0.322885\pi$
$967$	32.8474	1.05630	0.528151	+	0.849150i	$0.322885\pi$
$968$	0	0
$969$	−7.14412	−0.229502
$970$	0	0
$971$	−45.3642	−1.45581	−0.727903	−	0.685680i	$-0.759505\pi$
$971$	−45.3642	−1.45581	−0.727903	+	0.685680i	$0.759505\pi$
$972$	0	0
$973$	−68.5650	−2.19809
$974$	0	0
$975$	−9.93297	−0.318110
$976$	0	0
$977$	9.96515	0.318813	0.159407	−	0.987213i	$-0.449042\pi$
$977$	9.96515	0.318813	0.159407	+	0.987213i	$0.449042\pi$
$978$	0	0
$979$	0.740268	0.0236591
$980$	0	0
$981$	−7.05979	−0.225402
$982$	0	0
$983$	8.37150	0.267009	0.133505	−	0.991048i	$-0.457377\pi$
$983$	8.37150	0.267009	0.133505	+	0.991048i	$0.457377\pi$
$984$	0	0
$985$	25.3827	0.808761
$986$	0	0
$987$	−10.3057	−0.328035
$988$	0	0
$989$	−34.8202	−1.10722
$990$	0	0
$991$	−28.8111	−0.915215	−0.457608	−	0.889154i	$-0.651294\pi$
$991$	−28.8111	−0.915215	−0.457608	+	0.889154i	$0.651294\pi$
$992$	0	0
$993$	−0.970354	−0.0307933
$994$	0	0
$995$	4.98690	0.158095
$996$	0	0
$997$	12.3708	0.391788	0.195894	−	0.980625i	$-0.437239\pi$
$997$	12.3708	0.391788	0.195894	+	0.980625i	$0.437239\pi$
$998$	0	0
$999$	34.2589	1.08390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.s.1.6		18
4.3	odd	2		547.2.a.b.1.11	✓	18
12.11	even	2		4923.2.a.l.1.8		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.11	✓	18	4.3	odd	2
4923.2.a.l.1.8		18	12.11	even	2
8752.2.a.s.1.6		18	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:	\( N \)	\(=\)	\( 8752 = 2^{4} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8752.a (trivial)

\(f(q)\)	\(=\)	\(q-0.544167 q^{3} +0.962787 q^{5} +3.25298 q^{7} -2.70388 q^{9} +O(q^{10})\)	\(q-0.544167 q^{3} +0.962787 q^{5} +3.25298 q^{7} -2.70388 q^{9} +0.883096 q^{11} -4.48154 q^{13} -0.523917 q^{15} -3.18294 q^{17} -4.12466 q^{19} -1.77017 q^{21} -3.73970 q^{23} -4.07304 q^{25} +3.10387 q^{27} +6.33388 q^{29} +8.47939 q^{31} -0.480552 q^{33} +3.13193 q^{35} +11.0375 q^{37} +2.43871 q^{39} +1.41138 q^{41} +9.31097 q^{43} -2.60326 q^{45} +5.82191 q^{47} +3.58188 q^{49} +1.73205 q^{51} -7.06826 q^{53} +0.850233 q^{55} +2.24450 q^{57} -12.9274 q^{59} -3.04777 q^{61} -8.79567 q^{63} -4.31477 q^{65} +9.15218 q^{67} +2.03502 q^{69} +4.57601 q^{71} -14.4453 q^{73} +2.21642 q^{75} +2.87269 q^{77} -7.20620 q^{79} +6.42262 q^{81} -7.39056 q^{83} -3.06450 q^{85} -3.44669 q^{87} +0.838265 q^{89} -14.5784 q^{91} -4.61421 q^{93} -3.97117 q^{95} +0.475381 q^{97} -2.38779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) \) 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9} - 2 q^{11} - 25 q^{13} - 9 q^{15} - 30 q^{17} - 4 q^{19} - 16 q^{21} + 26 q^{23} + 31 q^{25} + 37 q^{27} - 18 q^{29} + 5 q^{31} - 10 q^{33} + 9 q^{35} - 18 q^{37} - 7 q^{39} - 17 q^{41} - 8 q^{43} - 44 q^{45} + 52 q^{47} + 29 q^{49} - 19 q^{51} - 60 q^{53} - 11 q^{55} + 4 q^{57} + 8 q^{59} - 26 q^{61} + q^{63} - 6 q^{65} - 12 q^{67} - 38 q^{69} + q^{71} - 2 q^{73} + 17 q^{75} - 73 q^{77} - 18 q^{79} + 18 q^{81} + 43 q^{83} + 51 q^{85} - 3 q^{87} - 28 q^{89} + q^{91} - 60 q^{93} + 18 q^{95} - 34 q^{97} - 15 q^{99}+O(q^{100}) \) 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9 - 2 * q^11 - 25 * q^13 - 9 * q^15 - 30 * q^17 - 4 * q^19 - 16 * q^21 + 26 * q^23 + 31 * q^25 + 37 * q^27 - 18 * q^29 + 5 * q^31 - 10 * q^33 + 9 * q^35 - 18 * q^37 - 7 * q^39 - 17 * q^41 - 8 * q^43 - 44 * q^45 + 52 * q^47 + 29 * q^49 - 19 * q^51 - 60 * q^53 - 11 * q^55 + 4 * q^57 + 8 * q^59 - 26 * q^61 + q^63 - 6 * q^65 - 12 * q^67 - 38 * q^69 + q^71 - 2 * q^73 + 17 * q^75 - 73 * q^77 - 18 * q^79 + 18 * q^81 + 43 * q^83 + 51 * q^85 - 3 * q^87 - 28 * q^89 + q^91 - 60 * q^93 + 18 * q^95 - 34 * q^97 - 15 * q^99

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