LMFDB - Embedded newform 4840.2.a.bg.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.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:	$38.6475945783$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 $ x^8 - x^7 - 21x^6 + 15x^5 + 140x^4 - 60x^3 - 295x^2 + 50x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$3.21195$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.21195 q^{3} +1.00000 q^{5} -4.90937 q^{7} +7.31662 q^{9} +O(q^{10})$	$q+3.21195 q^{3} +1.00000 q^{5} -4.90937 q^{7} +7.31662 q^{9} -0.804699 q^{13} +3.21195 q^{15} +1.84749 q^{17} +5.61737 q^{19} -15.7686 q^{21} -1.60313 q^{23} +1.00000 q^{25} +13.8648 q^{27} +7.86486 q^{29} +3.86973 q^{31} -4.90937 q^{35} -6.67764 q^{37} -2.58465 q^{39} -5.23795 q^{41} +1.05407 q^{43} +7.31662 q^{45} +8.65181 q^{47} +17.1019 q^{49} +5.93404 q^{51} +7.82577 q^{53} +18.0427 q^{57} +3.89313 q^{59} +6.62282 q^{61} -35.9200 q^{63} -0.804699 q^{65} -5.52774 q^{67} -5.14916 q^{69} -10.9694 q^{71} +2.79991 q^{73} +3.21195 q^{75} -2.91983 q^{79} +22.5830 q^{81} +1.93633 q^{83} +1.84749 q^{85} +25.2615 q^{87} +7.32421 q^{89} +3.95056 q^{91} +12.4294 q^{93} +5.61737 q^{95} +8.15740 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9}+O(q^{10}) $ 8 * q + q^3 + 8 * q^5 - 6 * q^7 + 19 * q^9	$ 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9} + 12 q^{13} + q^{15} + 2 q^{17} - 6 q^{19} + 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} + 8 q^{29} + 19 q^{31} - 6 q^{35} + 12 q^{37} - 21 q^{39} - 3 q^{41} - 8 q^{43} + 19 q^{45} + 10 q^{47} + 22 q^{49} - 7 q^{51} + 28 q^{53} + 25 q^{57} + 25 q^{59} + 10 q^{61} - 64 q^{63} + 12 q^{65} - 2 q^{67} + 18 q^{69} + 25 q^{71} + 38 q^{73} + q^{75} - 38 q^{79} + 32 q^{81} - 28 q^{83} + 2 q^{85} + 15 q^{87} + 12 q^{89} + 8 q^{91} - 15 q^{93} - 6 q^{95} + 21 q^{97}+O(q^{100}) $ 8 * q + q^3 + 8 * q^5 - 6 * q^7 + 19 * q^9 + 12 * q^13 + q^15 + 2 * q^17 - 6 * q^19 + 6 * q^21 + 10 * q^23 + 8 * q^25 + 13 * q^27 + 8 * q^29 + 19 * q^31 - 6 * q^35 + 12 * q^37 - 21 * q^39 - 3 * q^41 - 8 * q^43 + 19 * q^45 + 10 * q^47 + 22 * q^49 - 7 * q^51 + 28 * q^53 + 25 * q^57 + 25 * q^59 + 10 * q^61 - 64 * q^63 + 12 * q^65 - 2 * q^67 + 18 * q^69 + 25 * q^71 + 38 * q^73 + q^75 - 38 * q^79 + 32 * q^81 - 28 * q^83 + 2 * q^85 + 15 * q^87 + 12 * q^89 + 8 * q^91 - 15 * q^93 - 6 * q^95 + 21 * q^97

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
$3$	3.21195	1.85442	0.927210	−	0.374542i	$-0.122200\pi$
$3$	3.21195	1.85442	0.927210	+	0.374542i	$0.122200\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.90937	−1.85557	−0.927783	−	0.373120i	$-0.878288\pi$
$7$	−4.90937	−1.85557	−0.927783	+	0.373120i	$0.878288\pi$
$8$	0	0
$9$	7.31662	2.43887
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.804699	−0.223183	−0.111592	−	0.993754i	$-0.535595\pi$
$13$	−0.804699	−0.223183	−0.111592	+	0.993754i	$0.535595\pi$
$14$	0	0
$15$	3.21195	0.829322
$16$	0	0
$17$	1.84749	0.448082	0.224041	−	0.974580i	$-0.428075\pi$
$17$	1.84749	0.448082	0.224041	+	0.974580i	$0.428075\pi$
$18$	0	0
$19$	5.61737	1.28871	0.644357	−	0.764725i	$-0.277125\pi$
$19$	5.61737	1.28871	0.644357	+	0.764725i	$0.277125\pi$
$20$	0	0
$21$	−15.7686	−3.44100
$22$	0	0
$23$	−1.60313	−0.334275	−0.167138	−	0.985934i	$-0.553452\pi$
$23$	−1.60313	−0.334275	−0.167138	+	0.985934i	$0.553452\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	13.8648	2.66827
$28$	0	0
$29$	7.86486	1.46047	0.730234	−	0.683197i	$-0.239411\pi$
$29$	7.86486	1.46047	0.730234	+	0.683197i	$0.239411\pi$
$30$	0	0
$31$	3.86973	0.695024	0.347512	−	0.937676i	$-0.387027\pi$
$31$	3.86973	0.695024	0.347512	+	0.937676i	$0.387027\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.90937	−0.829834
$36$	0	0
$37$	−6.67764	−1.09780	−0.548899	−	0.835889i	$-0.684953\pi$
$37$	−6.67764	−1.09780	−0.548899	+	0.835889i	$0.684953\pi$
$38$	0	0
$39$	−2.58465	−0.413876
$40$	0	0
$41$	−5.23795	−0.818031	−0.409015	−	0.912527i	$-0.634128\pi$
$41$	−5.23795	−0.818031	−0.409015	+	0.912527i	$0.634128\pi$
$42$	0	0
$43$	1.05407	0.160744	0.0803718	−	0.996765i	$-0.474389\pi$
$43$	1.05407	0.160744	0.0803718	+	0.996765i	$0.474389\pi$
$44$	0	0
$45$	7.31662	1.09070
$46$	0	0
$47$	8.65181	1.26200	0.630998	−	0.775784i	$-0.282645\pi$
$47$	8.65181	1.26200	0.630998	+	0.775784i	$0.282645\pi$
$48$	0	0
$49$	17.1019	2.44313
$50$	0	0
$51$	5.93404	0.830932
$52$	0	0
$53$	7.82577	1.07495	0.537476	−	0.843279i	$-0.319378\pi$
$53$	7.82577	1.07495	0.537476	+	0.843279i	$0.319378\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	18.0427	2.38982
$58$	0	0
$59$	3.89313	0.506842	0.253421	−	0.967356i	$-0.418444\pi$
$59$	3.89313	0.506842	0.253421	+	0.967356i	$0.418444\pi$
$60$	0	0
$61$	6.62282	0.847965	0.423983	−	0.905670i	$-0.360632\pi$
$61$	6.62282	0.847965	0.423983	+	0.905670i	$0.360632\pi$
$62$	0	0
$63$	−35.9200	−4.52549
$64$	0	0
$65$	−0.804699	−0.0998107
$66$	0	0
$67$	−5.52774	−0.675321	−0.337661	−	0.941268i	$-0.609636\pi$
$67$	−5.52774	−0.675321	−0.337661	+	0.941268i	$0.609636\pi$
$68$	0	0
$69$	−5.14916	−0.619887
$70$	0	0
$71$	−10.9694	−1.30183	−0.650915	−	0.759150i	$-0.725615\pi$
$71$	−10.9694	−1.30183	−0.650915	+	0.759150i	$0.725615\pi$
$72$	0	0
$73$	2.79991	0.327705	0.163853	−	0.986485i	$-0.447608\pi$
$73$	2.79991	0.327705	0.163853	+	0.986485i	$0.447608\pi$
$74$	0	0
$75$	3.21195	0.370884
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.91983	−0.328506	−0.164253	−	0.986418i	$-0.552521\pi$
$79$	−2.91983	−0.328506	−0.164253	+	0.986418i	$0.552521\pi$
$80$	0	0
$81$	22.5830	2.50923
$82$	0	0
$83$	1.93633	0.212540	0.106270	−	0.994337i	$-0.466109\pi$
$83$	1.93633	0.212540	0.106270	+	0.994337i	$0.466109\pi$
$84$	0	0
$85$	1.84749	0.200388
$86$	0	0
$87$	25.2615	2.70832
$88$	0	0
$89$	7.32421	0.776365	0.388183	−	0.921583i	$-0.373103\pi$
$89$	7.32421	0.776365	0.388183	+	0.921583i	$0.373103\pi$
$90$	0	0
$91$	3.95056	0.414132
$92$	0	0
$93$	12.4294	1.28887
$94$	0	0
$95$	5.61737	0.576330
$96$	0	0
$97$	8.15740	0.828258	0.414129	−	0.910218i	$-0.364086\pi$
$97$	8.15740	0.828258	0.414129	+	0.910218i	$0.364086\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.38078	−0.634912	−0.317456	−	0.948273i	$-0.602829\pi$
$101$	−6.38078	−0.634912	−0.317456	+	0.948273i	$0.602829\pi$
$102$	0	0
$103$	−15.0414	−1.48208	−0.741038	−	0.671463i	$-0.765666\pi$
$103$	−15.0414	−1.48208	−0.741038	+	0.671463i	$0.765666\pi$
$104$	0	0
$105$	−15.7686	−1.53886
$106$	0	0
$107$	−5.61756	−0.543070	−0.271535	−	0.962429i	$-0.587531\pi$
$107$	−5.61756	−0.543070	−0.271535	+	0.962429i	$0.587531\pi$
$108$	0	0
$109$	14.2032	1.36042	0.680208	−	0.733019i	$-0.261890\pi$
$109$	14.2032	1.36042	0.680208	+	0.733019i	$0.261890\pi$
$110$	0	0
$111$	−21.4482	−2.03578
$112$	0	0
$113$	6.01592	0.565930	0.282965	−	0.959130i	$-0.408682\pi$
$113$	6.01592	0.565930	0.282965	+	0.959130i	$0.408682\pi$
$114$	0	0
$115$	−1.60313	−0.149492
$116$	0	0
$117$	−5.88768	−0.544316
$118$	0	0
$119$	−9.07001	−0.831446
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−16.8240	−1.51697
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.26040	−0.111842	−0.0559211	−	0.998435i	$-0.517810\pi$
$127$	−1.26040	−0.111842	−0.0559211	+	0.998435i	$0.517810\pi$
$128$	0	0
$129$	3.38561	0.298086
$130$	0	0
$131$	12.0760	1.05509	0.527543	−	0.849528i	$-0.323113\pi$
$131$	12.0760	1.05509	0.527543	+	0.849528i	$0.323113\pi$
$132$	0	0
$133$	−27.5777	−2.39129
$134$	0	0
$135$	13.8648	1.19329
$136$	0	0
$137$	−9.04007	−0.772345	−0.386173	−	0.922426i	$-0.626203\pi$
$137$	−9.04007	−0.772345	−0.386173	+	0.922426i	$0.626203\pi$
$138$	0	0
$139$	21.6951	1.84015	0.920076	−	0.391740i	$-0.128127\pi$
$139$	21.6951	1.84015	0.920076	+	0.391740i	$0.128127\pi$
$140$	0	0
$141$	27.7892	2.34027
$142$	0	0
$143$	0	0
$144$	0	0
$145$	7.86486	0.653141
$146$	0	0
$147$	54.9304	4.53058
$148$	0	0
$149$	−6.62139	−0.542445	−0.271223	−	0.962517i	$-0.587428\pi$
$149$	−6.62139	−0.542445	−0.271223	+	0.962517i	$0.587428\pi$
$150$	0	0
$151$	1.12497	0.0915489	0.0457745	−	0.998952i	$-0.485424\pi$
$151$	1.12497	0.0915489	0.0457745	+	0.998952i	$0.485424\pi$
$152$	0	0
$153$	13.5174	1.09282
$154$	0	0
$155$	3.86973	0.310824
$156$	0	0
$157$	−5.84097	−0.466160	−0.233080	−	0.972458i	$-0.574880\pi$
$157$	−5.84097	−0.466160	−0.233080	+	0.972458i	$0.574880\pi$
$158$	0	0
$159$	25.1360	1.99341
$160$	0	0
$161$	7.87034	0.620270
$162$	0	0
$163$	−14.6105	−1.14438	−0.572190	−	0.820121i	$-0.693906\pi$
$163$	−14.6105	−1.14438	−0.572190	+	0.820121i	$0.693906\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.73740	−0.521356	−0.260678	−	0.965426i	$-0.583946\pi$
$167$	−6.73740	−0.521356	−0.260678	+	0.965426i	$0.583946\pi$
$168$	0	0
$169$	−12.3525	−0.950189
$170$	0	0
$171$	41.1002	3.14301
$172$	0	0
$173$	2.48506	0.188936	0.0944678	−	0.995528i	$-0.469885\pi$
$173$	2.48506	0.188936	0.0944678	+	0.995528i	$0.469885\pi$
$174$	0	0
$175$	−4.90937	−0.371113
$176$	0	0
$177$	12.5045	0.939898
$178$	0	0
$179$	−25.8751	−1.93400	−0.966999	−	0.254782i	$-0.917996\pi$
$179$	−25.8751	−1.93400	−0.966999	+	0.254782i	$0.917996\pi$
$180$	0	0
$181$	14.2925	1.06235	0.531177	−	0.847261i	$-0.321750\pi$
$181$	14.2925	1.06235	0.531177	+	0.847261i	$0.321750\pi$
$182$	0	0
$183$	21.2722	1.57248
$184$	0	0
$185$	−6.67764	−0.490950
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−68.0671	−4.95116
$190$	0	0
$191$	3.92551	0.284040	0.142020	−	0.989864i	$-0.454640\pi$
$191$	3.92551	0.284040	0.142020	+	0.989864i	$0.454640\pi$
$192$	0	0
$193$	10.5814	0.761669	0.380835	−	0.924643i	$-0.375637\pi$
$193$	10.5814	0.761669	0.380835	+	0.924643i	$0.375637\pi$
$194$	0	0
$195$	−2.58465	−0.185091
$196$	0	0
$197$	−1.89648	−0.135119	−0.0675594	−	0.997715i	$-0.521521\pi$
$197$	−1.89648	−0.135119	−0.0675594	+	0.997715i	$0.521521\pi$
$198$	0	0
$199$	13.9966	0.992189	0.496095	−	0.868268i	$-0.334767\pi$
$199$	13.9966	0.992189	0.496095	+	0.868268i	$0.334767\pi$
$200$	0	0
$201$	−17.7548	−1.25233
$202$	0	0
$203$	−38.6115	−2.70999
$204$	0	0
$205$	−5.23795	−0.365834
$206$	0	0
$207$	−11.7295	−0.815255
$208$	0	0
$209$	0	0
$210$	0	0
$211$	27.2000	1.87252	0.936262	−	0.351302i	$-0.114261\pi$
$211$	27.2000	1.87252	0.936262	+	0.351302i	$0.114261\pi$
$212$	0	0
$213$	−35.2332	−2.41414
$214$	0	0
$215$	1.05407	0.0718867
$216$	0	0
$217$	−18.9979	−1.28966
$218$	0	0
$219$	8.99318	0.607703
$220$	0	0
$221$	−1.48667	−0.100005
$222$	0	0
$223$	12.1911	0.816378	0.408189	−	0.912897i	$-0.366160\pi$
$223$	12.1911	0.816378	0.408189	+	0.912897i	$0.366160\pi$
$224$	0	0
$225$	7.31662	0.487774
$226$	0	0
$227$	−17.3556	−1.15193	−0.575966	−	0.817474i	$-0.695374\pi$
$227$	−17.3556	−1.15193	−0.575966	+	0.817474i	$0.695374\pi$
$228$	0	0
$229$	0.676698	0.0447175	0.0223587	−	0.999750i	$-0.492882\pi$
$229$	0.676698	0.0447175	0.0223587	+	0.999750i	$0.492882\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.38766	0.156421	0.0782105	−	0.996937i	$-0.475079\pi$
$233$	2.38766	0.156421	0.0782105	+	0.996937i	$0.475079\pi$
$234$	0	0
$235$	8.65181	0.564382
$236$	0	0
$237$	−9.37834	−0.609189
$238$	0	0
$239$	−8.18939	−0.529727	−0.264864	−	0.964286i	$-0.585327\pi$
$239$	−8.18939	−0.529727	−0.264864	+	0.964286i	$0.585327\pi$
$240$	0	0
$241$	9.43609	0.607832	0.303916	−	0.952699i	$-0.401706\pi$
$241$	9.43609	0.607832	0.303916	+	0.952699i	$0.401706\pi$
$242$	0	0
$243$	30.9413	1.98488
$244$	0	0
$245$	17.1019	1.09260
$246$	0	0
$247$	−4.52030	−0.287619
$248$	0	0
$249$	6.21940	0.394138
$250$	0	0
$251$	18.5177	1.16882	0.584412	−	0.811457i	$-0.301325\pi$
$251$	18.5177	1.16882	0.584412	+	0.811457i	$0.301325\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	5.93404	0.371604
$256$	0	0
$257$	27.0138	1.68507	0.842536	−	0.538640i	$-0.181062\pi$
$257$	27.0138	1.68507	0.842536	+	0.538640i	$0.181062\pi$
$258$	0	0
$259$	32.7830	2.03704
$260$	0	0
$261$	57.5442	3.56189
$262$	0	0
$263$	−17.4169	−1.07397	−0.536985	−	0.843592i	$-0.680437\pi$
$263$	−17.4169	−1.07397	−0.536985	+	0.843592i	$0.680437\pi$
$264$	0	0
$265$	7.82577	0.480734
$266$	0	0
$267$	23.5250	1.43971
$268$	0	0
$269$	−10.4185	−0.635226	−0.317613	−	0.948220i	$-0.602881\pi$
$269$	−10.4185	−0.635226	−0.317613	+	0.948220i	$0.602881\pi$
$270$	0	0
$271$	−30.2995	−1.84056	−0.920282	−	0.391255i	$-0.872041\pi$
$271$	−30.2995	−1.84056	−0.920282	+	0.391255i	$0.872041\pi$
$272$	0	0
$273$	12.6890	0.767974
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.13844	−0.188571	−0.0942854	−	0.995545i	$-0.530057\pi$
$277$	−3.13844	−0.188571	−0.0942854	+	0.995545i	$0.530057\pi$
$278$	0	0
$279$	28.3133	1.69507
$280$	0	0
$281$	11.6302	0.693801	0.346901	−	0.937902i	$-0.387234\pi$
$281$	11.6302	0.693801	0.346901	+	0.937902i	$0.387234\pi$
$282$	0	0
$283$	−2.14782	−0.127675	−0.0638374	−	0.997960i	$-0.520334\pi$
$283$	−2.14782	−0.127675	−0.0638374	+	0.997960i	$0.520334\pi$
$284$	0	0
$285$	18.0427	1.06876
$286$	0	0
$287$	25.7150	1.51791
$288$	0	0
$289$	−13.5868	−0.799222
$290$	0	0
$291$	26.2011	1.53594
$292$	0	0
$293$	−26.1399	−1.52711	−0.763553	−	0.645745i	$-0.776547\pi$
$293$	−26.1399	−1.52711	−0.763553	+	0.645745i	$0.776547\pi$
$294$	0	0
$295$	3.89313	0.226667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.29004	0.0746047
$300$	0	0
$301$	−5.17480	−0.298270
$302$	0	0
$303$	−20.4947	−1.17739
$304$	0	0
$305$	6.62282	0.379222
$306$	0	0
$307$	−5.59232	−0.319171	−0.159585	−	0.987184i	$-0.551016\pi$
$307$	−5.59232	−0.319171	−0.159585	+	0.987184i	$0.551016\pi$
$308$	0	0
$309$	−48.3123	−2.74839
$310$	0	0
$311$	1.71451	0.0972211	0.0486106	−	0.998818i	$-0.484521\pi$
$311$	1.71451	0.0972211	0.0486106	+	0.998818i	$0.484521\pi$
$312$	0	0
$313$	−5.29655	−0.299379	−0.149689	−	0.988733i	$-0.547827\pi$
$313$	−5.29655	−0.299379	−0.149689	+	0.988733i	$0.547827\pi$
$314$	0	0
$315$	−35.9200	−2.02386
$316$	0	0
$317$	7.73781	0.434599	0.217299	−	0.976105i	$-0.430275\pi$
$317$	7.73781	0.434599	0.217299	+	0.976105i	$0.430275\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−18.0433	−1.00708
$322$	0	0
$323$	10.3780	0.577450
$324$	0	0
$325$	−0.804699	−0.0446367
$326$	0	0
$327$	45.6198	2.52278
$328$	0	0
$329$	−42.4749	−2.34172
$330$	0	0
$331$	−18.7346	−1.02975	−0.514874	−	0.857266i	$-0.672161\pi$
$331$	−18.7346	−1.02975	−0.514874	+	0.857266i	$0.672161\pi$
$332$	0	0
$333$	−48.8577	−2.67739
$334$	0	0
$335$	−5.52774	−0.302013
$336$	0	0
$337$	−10.8034	−0.588499	−0.294250	−	0.955729i	$-0.595070\pi$
$337$	−10.8034	−0.588499	−0.294250	+	0.955729i	$0.595070\pi$
$338$	0	0
$339$	19.3228	1.04947
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−49.5938	−2.67782
$344$	0	0
$345$	−5.14916	−0.277222
$346$	0	0
$347$	−15.4530	−0.829560	−0.414780	−	0.909922i	$-0.636141\pi$
$347$	−15.4530	−0.829560	−0.414780	+	0.909922i	$0.636141\pi$
$348$	0	0
$349$	−28.4964	−1.52538	−0.762690	−	0.646765i	$-0.776122\pi$
$349$	−28.4964	−1.52538	−0.762690	+	0.646765i	$0.776122\pi$
$350$	0	0
$351$	−11.1570	−0.595514
$352$	0	0
$353$	−16.7619	−0.892148	−0.446074	−	0.894996i	$-0.647178\pi$
$353$	−16.7619	−0.892148	−0.446074	+	0.894996i	$0.647178\pi$
$354$	0	0
$355$	−10.9694	−0.582196
$356$	0	0
$357$	−29.1324	−1.54185
$358$	0	0
$359$	−15.0160	−0.792514	−0.396257	−	0.918140i	$-0.629691\pi$
$359$	−15.0160	−0.792514	−0.396257	+	0.918140i	$0.629691\pi$
$360$	0	0
$361$	12.5549	0.660783
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.79991	0.146554
$366$	0	0
$367$	−20.4306	−1.06647	−0.533236	−	0.845967i	$-0.679024\pi$
$367$	−20.4306	−1.06647	−0.533236	+	0.845967i	$0.679024\pi$
$368$	0	0
$369$	−38.3241	−1.99507
$370$	0	0
$371$	−38.4196	−1.99465
$372$	0	0
$373$	16.8624	0.873101	0.436550	−	0.899680i	$-0.356200\pi$
$373$	16.8624	0.873101	0.436550	+	0.899680i	$0.356200\pi$
$374$	0	0
$375$	3.21195	0.165864
$376$	0	0
$377$	−6.32885	−0.325952
$378$	0	0
$379$	−24.3238	−1.24943	−0.624715	−	0.780853i	$-0.714785\pi$
$379$	−24.3238	−1.24943	−0.624715	+	0.780853i	$0.714785\pi$
$380$	0	0
$381$	−4.04833	−0.207402
$382$	0	0
$383$	12.5978	0.643716	0.321858	−	0.946788i	$-0.395693\pi$
$383$	12.5978	0.643716	0.321858	+	0.946788i	$0.395693\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.71220	0.392033
$388$	0	0
$389$	−20.7471	−1.05192	−0.525961	−	0.850509i	$-0.676294\pi$
$389$	−20.7471	−1.05192	−0.525961	+	0.850509i	$0.676294\pi$
$390$	0	0
$391$	−2.96176	−0.149783
$392$	0	0
$393$	38.7876	1.95657
$394$	0	0
$395$	−2.91983	−0.146913
$396$	0	0
$397$	−3.76055	−0.188737	−0.0943683	−	0.995537i	$-0.530083\pi$
$397$	−3.76055	−0.188737	−0.0943683	+	0.995537i	$0.530083\pi$
$398$	0	0
$399$	−88.5783	−4.43446
$400$	0	0
$401$	−16.8516	−0.841530	−0.420765	−	0.907170i	$-0.638238\pi$
$401$	−16.8516	−0.841530	−0.420765	+	0.907170i	$0.638238\pi$
$402$	0	0
$403$	−3.11397	−0.155118
$404$	0	0
$405$	22.5830	1.12216
$406$	0	0
$407$	0	0
$408$	0	0
$409$	21.7119	1.07358	0.536792	−	0.843715i	$-0.319636\pi$
$409$	21.7119	1.07358	0.536792	+	0.843715i	$0.319636\pi$
$410$	0	0
$411$	−29.0362	−1.43225
$412$	0	0
$413$	−19.1128	−0.940479
$414$	0	0
$415$	1.93633	0.0950507
$416$	0	0
$417$	69.6835	3.41241
$418$	0	0
$419$	−39.1675	−1.91346	−0.956729	−	0.290981i	$-0.906018\pi$
$419$	−39.1675	−1.91346	−0.956729	+	0.290981i	$0.906018\pi$
$420$	0	0
$421$	−8.44175	−0.411426	−0.205713	−	0.978612i	$-0.565951\pi$
$421$	−8.44175	−0.411426	−0.205713	+	0.978612i	$0.565951\pi$
$422$	0	0
$423$	63.3020	3.07785
$424$	0	0
$425$	1.84749	0.0896164
$426$	0	0
$427$	−32.5138	−1.57346
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.69401	0.0815977	0.0407988	−	0.999167i	$-0.487010\pi$
$431$	1.69401	0.0815977	0.0407988	+	0.999167i	$0.487010\pi$
$432$	0	0
$433$	39.8944	1.91720	0.958601	−	0.284754i	$-0.0919118\pi$
$433$	39.8944	1.91720	0.958601	+	0.284754i	$0.0919118\pi$
$434$	0	0
$435$	25.2615	1.21120
$436$	0	0
$437$	−9.00536	−0.430785
$438$	0	0
$439$	−0.598520	−0.0285658	−0.0142829	−	0.999898i	$-0.504547\pi$
$439$	−0.598520	−0.0285658	−0.0142829	+	0.999898i	$0.504547\pi$
$440$	0	0
$441$	125.128	5.95847
$442$	0	0
$443$	−36.5608	−1.73706	−0.868529	−	0.495639i	$-0.834934\pi$
$443$	−36.5608	−1.73706	−0.868529	+	0.495639i	$0.834934\pi$
$444$	0	0
$445$	7.32421	0.347201
$446$	0	0
$447$	−21.2676	−1.00592
$448$	0	0
$449$	−36.0092	−1.69938	−0.849689	−	0.527284i	$-0.823210\pi$
$449$	−36.0092	−1.69938	−0.849689	+	0.527284i	$0.823210\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.61335	0.169770
$454$	0	0
$455$	3.95056	0.185205
$456$	0	0
$457$	−10.8344	−0.506812	−0.253406	−	0.967360i	$-0.581551\pi$
$457$	−10.8344	−0.506812	−0.253406	+	0.967360i	$0.581551\pi$
$458$	0	0
$459$	25.6150	1.19561
$460$	0	0
$461$	5.70020	0.265485	0.132742	−	0.991151i	$-0.457622\pi$
$461$	5.70020	0.265485	0.132742	+	0.991151i	$0.457622\pi$
$462$	0	0
$463$	−6.28911	−0.292280	−0.146140	−	0.989264i	$-0.546685\pi$
$463$	−6.28911	−0.292280	−0.146140	+	0.989264i	$0.546685\pi$
$464$	0	0
$465$	12.4294	0.576398
$466$	0	0
$467$	0.785087	0.0363295	0.0181647	−	0.999835i	$-0.494218\pi$
$467$	0.785087	0.0363295	0.0181647	+	0.999835i	$0.494218\pi$
$468$	0	0
$469$	27.1377	1.25310
$470$	0	0
$471$	−18.7609	−0.864456
$472$	0	0
$473$	0	0
$474$	0	0
$475$	5.61737	0.257743
$476$	0	0
$477$	57.2582	2.62167
$478$	0	0
$479$	38.5558	1.76166	0.880829	−	0.473435i	$-0.156986\pi$
$479$	38.5558	1.76166	0.880829	+	0.473435i	$0.156986\pi$
$480$	0	0
$481$	5.37349	0.245010
$482$	0	0
$483$	25.2791	1.15024
$484$	0	0
$485$	8.15740	0.370408
$486$	0	0
$487$	15.9940	0.724759	0.362379	−	0.932031i	$-0.381965\pi$
$487$	15.9940	0.724759	0.362379	+	0.932031i	$0.381965\pi$
$488$	0	0
$489$	−46.9280	−2.12216
$490$	0	0
$491$	−25.2541	−1.13970	−0.569850	−	0.821749i	$-0.692999\pi$
$491$	−25.2541	−1.13970	−0.569850	+	0.821749i	$0.692999\pi$
$492$	0	0
$493$	14.5302	0.654409
$494$	0	0
$495$	0	0
$496$	0	0
$497$	53.8529	2.41563
$498$	0	0
$499$	−4.96527	−0.222276	−0.111138	−	0.993805i	$-0.535450\pi$
$499$	−4.96527	−0.222276	−0.111138	+	0.993805i	$0.535450\pi$
$500$	0	0
$501$	−21.6402	−0.966812
$502$	0	0
$503$	−20.1890	−0.900185	−0.450093	−	0.892982i	$-0.648609\pi$
$503$	−20.1890	−0.900185	−0.450093	+	0.892982i	$0.648609\pi$
$504$	0	0
$505$	−6.38078	−0.283941
$506$	0	0
$507$	−39.6755	−1.76205
$508$	0	0
$509$	36.2638	1.60736	0.803682	−	0.595059i	$-0.202871\pi$
$509$	36.2638	1.60736	0.803682	+	0.595059i	$0.202871\pi$
$510$	0	0
$511$	−13.7458	−0.608079
$512$	0	0
$513$	77.8835	3.43864
$514$	0	0
$515$	−15.0414	−0.662805
$516$	0	0
$517$	0	0
$518$	0	0
$519$	7.98188	0.350366
$520$	0	0
$521$	−11.0780	−0.485337	−0.242668	−	0.970109i	$-0.578023\pi$
$521$	−11.0780	−0.485337	−0.242668	+	0.970109i	$0.578023\pi$
$522$	0	0
$523$	−23.5405	−1.02935	−0.514676	−	0.857385i	$-0.672088\pi$
$523$	−23.5405	−1.02935	−0.514676	+	0.857385i	$0.672088\pi$
$524$	0	0
$525$	−15.7686	−0.688200
$526$	0	0
$527$	7.14928	0.311428
$528$	0	0
$529$	−20.4300	−0.888260
$530$	0	0
$531$	28.4845	1.23612
$532$	0	0
$533$	4.21498	0.182571
$534$	0	0
$535$	−5.61756	−0.242868
$536$	0	0
$537$	−83.1096	−3.58644
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−25.6020	−1.10072	−0.550358	−	0.834929i	$-0.685509\pi$
$541$	−25.6020	−1.10072	−0.550358	+	0.834929i	$0.685509\pi$
$542$	0	0
$543$	45.9069	1.97005
$544$	0	0
$545$	14.2032	0.608396
$546$	0	0
$547$	−27.7459	−1.18633	−0.593164	−	0.805081i	$-0.702122\pi$
$547$	−27.7459	−1.18633	−0.593164	+	0.805081i	$0.702122\pi$
$548$	0	0
$549$	48.4566	2.06808
$550$	0	0
$551$	44.1798	1.88212
$552$	0	0
$553$	14.3345	0.609565
$554$	0	0
$555$	−21.4482	−0.910427
$556$	0	0
$557$	7.80957	0.330902	0.165451	−	0.986218i	$-0.447092\pi$
$557$	7.80957	0.330902	0.165451	+	0.986218i	$0.447092\pi$
$558$	0	0
$559$	−0.848206	−0.0358753
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−30.3975	−1.28110	−0.640551	−	0.767916i	$-0.721294\pi$
$563$	−30.3975	−1.28110	−0.640551	+	0.767916i	$0.721294\pi$
$564$	0	0
$565$	6.01592	0.253092
$566$	0	0
$567$	−110.868	−4.65603
$568$	0	0
$569$	−1.98020	−0.0830141	−0.0415071	−	0.999138i	$-0.513216\pi$
$569$	−1.98020	−0.0830141	−0.0415071	+	0.999138i	$0.513216\pi$
$570$	0	0
$571$	−33.2773	−1.39261	−0.696307	−	0.717744i	$-0.745175\pi$
$571$	−33.2773	−1.39261	−0.696307	+	0.717744i	$0.745175\pi$
$572$	0	0
$573$	12.6085	0.526729
$574$	0	0
$575$	−1.60313	−0.0668550
$576$	0	0
$577$	35.9736	1.49760	0.748800	−	0.662796i	$-0.230630\pi$
$577$	35.9736	1.49760	0.748800	+	0.662796i	$0.230630\pi$
$578$	0	0
$579$	33.9871	1.41245
$580$	0	0
$581$	−9.50616	−0.394382
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−5.88768	−0.243425
$586$	0	0
$587$	21.4760	0.886411	0.443205	−	0.896420i	$-0.353841\pi$
$587$	21.4760	0.886411	0.443205	+	0.896420i	$0.353841\pi$
$588$	0	0
$589$	21.7377	0.895686
$590$	0	0
$591$	−6.09140	−0.250567
$592$	0	0
$593$	14.2405	0.584787	0.292393	−	0.956298i	$-0.405548\pi$
$593$	14.2405	0.584787	0.292393	+	0.956298i	$0.405548\pi$
$594$	0	0
$595$	−9.07001	−0.371834
$596$	0	0
$597$	44.9562	1.83994
$598$	0	0
$599$	42.8230	1.74970	0.874851	−	0.484393i	$-0.160959\pi$
$599$	42.8230	1.74970	0.874851	+	0.484393i	$0.160959\pi$
$600$	0	0
$601$	15.8267	0.645584	0.322792	−	0.946470i	$-0.395379\pi$
$601$	15.8267	0.645584	0.322792	+	0.946470i	$0.395379\pi$
$602$	0	0
$603$	−40.4444	−1.64702
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.2675	1.47205	0.736025	−	0.676954i	$-0.236700\pi$
$607$	36.2675	1.47205	0.736025	+	0.676954i	$0.236700\pi$
$608$	0	0
$609$	−124.018	−5.02547
$610$	0	0
$611$	−6.96211	−0.281657
$612$	0	0
$613$	38.9422	1.57286	0.786430	−	0.617679i	$-0.211927\pi$
$613$	38.9422	1.57286	0.786430	+	0.617679i	$0.211927\pi$
$614$	0	0
$615$	−16.8240	−0.678411
$616$	0	0
$617$	−30.2869	−1.21931	−0.609653	−	0.792669i	$-0.708691\pi$
$617$	−30.2869	−1.21931	−0.609653	+	0.792669i	$0.708691\pi$
$618$	0	0
$619$	20.0538	0.806032	0.403016	−	0.915193i	$-0.367962\pi$
$619$	20.0538	0.806032	0.403016	+	0.915193i	$0.367962\pi$
$620$	0	0
$621$	−22.2270	−0.891937
$622$	0	0
$623$	−35.9572	−1.44060
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.3369	−0.491903
$630$	0	0
$631$	−12.1601	−0.484085	−0.242043	−	0.970266i	$-0.577817\pi$
$631$	−12.1601	−0.484085	−0.242043	+	0.970266i	$0.577817\pi$
$632$	0	0
$633$	87.3650	3.47245
$634$	0	0
$635$	−1.26040	−0.0500174
$636$	0	0
$637$	−13.7619	−0.545265
$638$	0	0
$639$	−80.2590	−3.17500
$640$	0	0
$641$	17.1883	0.678897	0.339449	−	0.940625i	$-0.389760\pi$
$641$	17.1883	0.678897	0.339449	+	0.940625i	$0.389760\pi$
$642$	0	0
$643$	−11.3045	−0.445807	−0.222904	−	0.974840i	$-0.571554\pi$
$643$	−11.3045	−0.445807	−0.222904	+	0.974840i	$0.571554\pi$
$644$	0	0
$645$	3.38561	0.133308
$646$	0	0
$647$	21.1833	0.832801	0.416400	−	0.909181i	$-0.363291\pi$
$647$	21.1833	0.832801	0.416400	+	0.909181i	$0.363291\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−61.0203	−2.39158
$652$	0	0
$653$	12.4832	0.488507	0.244253	−	0.969711i	$-0.421457\pi$
$653$	12.4832	0.488507	0.244253	+	0.969711i	$0.421457\pi$
$654$	0	0
$655$	12.0760	0.471849
$656$	0	0
$657$	20.4859	0.799231
$658$	0	0
$659$	−10.2784	−0.400391	−0.200195	−	0.979756i	$-0.564158\pi$
$659$	−10.2784	−0.400391	−0.200195	+	0.979756i	$0.564158\pi$
$660$	0	0
$661$	−39.9264	−1.55296	−0.776478	−	0.630145i	$-0.782996\pi$
$661$	−39.9264	−1.55296	−0.776478	+	0.630145i	$0.782996\pi$
$662$	0	0
$663$	−4.77512	−0.185450
$664$	0	0
$665$	−27.5777	−1.06942
$666$	0	0
$667$	−12.6084	−0.488198
$668$	0	0
$669$	39.1573	1.51391
$670$	0	0
$671$	0	0
$672$	0	0
$673$	20.9293	0.806765	0.403382	−	0.915031i	$-0.367834\pi$
$673$	20.9293	0.806765	0.403382	+	0.915031i	$0.367834\pi$
$674$	0	0
$675$	13.8648	0.533655
$676$	0	0
$677$	18.8878	0.725917	0.362958	−	0.931805i	$-0.381767\pi$
$677$	18.8878	0.725917	0.362958	+	0.931805i	$0.381767\pi$
$678$	0	0
$679$	−40.0477	−1.53689
$680$	0	0
$681$	−55.7453	−2.13616
$682$	0	0
$683$	11.5145	0.440590	0.220295	−	0.975433i	$-0.429298\pi$
$683$	11.5145	0.440590	0.220295	+	0.975433i	$0.429298\pi$
$684$	0	0
$685$	−9.04007	−0.345403
$686$	0	0
$687$	2.17352	0.0829249
$688$	0	0
$689$	−6.29740	−0.239912
$690$	0	0
$691$	22.7689	0.866170	0.433085	−	0.901353i	$-0.357425\pi$
$691$	22.7689	0.866170	0.433085	+	0.901353i	$0.357425\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	21.6951	0.822941
$696$	0	0
$697$	−9.67707	−0.366545
$698$	0	0
$699$	7.66905	0.290070
$700$	0	0
$701$	−6.63323	−0.250534	−0.125267	−	0.992123i	$-0.539979\pi$
$701$	−6.63323	−0.250534	−0.125267	+	0.992123i	$0.539979\pi$
$702$	0	0
$703$	−37.5108	−1.41475
$704$	0	0
$705$	27.7892	1.04660
$706$	0	0
$707$	31.3256	1.17812
$708$	0	0
$709$	−37.3101	−1.40121	−0.700604	−	0.713550i	$-0.747086\pi$
$709$	−37.3101	−1.40121	−0.700604	+	0.713550i	$0.747086\pi$
$710$	0	0
$711$	−21.3633	−0.801185
$712$	0	0
$713$	−6.20367	−0.232329
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−26.3039	−0.982337
$718$	0	0
$719$	24.6334	0.918670	0.459335	−	0.888263i	$-0.348088\pi$
$719$	24.6334	0.918670	0.459335	+	0.888263i	$0.348088\pi$
$720$	0	0
$721$	73.8439	2.75009
$722$	0	0
$723$	30.3082	1.12717
$724$	0	0
$725$	7.86486	0.292094
$726$	0	0
$727$	−31.1865	−1.15664	−0.578321	−	0.815809i	$-0.696292\pi$
$727$	−31.1865	−1.15664	−0.578321	+	0.815809i	$0.696292\pi$
$728$	0	0
$729$	31.6327	1.17158
$730$	0	0
$731$	1.94738	0.0720263
$732$	0	0
$733$	−12.2957	−0.454152	−0.227076	−	0.973877i	$-0.572917\pi$
$733$	−12.2957	−0.454152	−0.227076	+	0.973877i	$0.572917\pi$
$734$	0	0
$735$	54.9304	2.02614
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1.49260	0.0549063	0.0274532	−	0.999623i	$-0.491260\pi$
$739$	1.49260	0.0549063	0.0274532	+	0.999623i	$0.491260\pi$
$740$	0	0
$741$	−14.5190	−0.533367
$742$	0	0
$743$	9.96694	0.365652	0.182826	−	0.983145i	$-0.441476\pi$
$743$	9.96694	0.365652	0.182826	+	0.983145i	$0.441476\pi$
$744$	0	0
$745$	−6.62139	−0.242589
$746$	0	0
$747$	14.1674	0.518358
$748$	0	0
$749$	27.5786	1.00770
$750$	0	0
$751$	11.1885	0.408275	0.204137	−	0.978942i	$-0.434561\pi$
$751$	11.1885	0.408275	0.204137	+	0.978942i	$0.434561\pi$
$752$	0	0
$753$	59.4778	2.16749
$754$	0	0
$755$	1.12497	0.0409419
$756$	0	0
$757$	43.7830	1.59132	0.795661	−	0.605742i	$-0.207124\pi$
$757$	43.7830	1.59132	0.795661	+	0.605742i	$0.207124\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.41151	−0.0511670	−0.0255835	−	0.999673i	$-0.508144\pi$
$761$	−1.41151	−0.0511670	−0.0255835	+	0.999673i	$0.508144\pi$
$762$	0	0
$763$	−69.7285	−2.52434
$764$	0	0
$765$	13.5174	0.488722
$766$	0	0
$767$	−3.13280	−0.113119
$768$	0	0
$769$	42.9335	1.54822	0.774111	−	0.633049i	$-0.218197\pi$
$769$	42.9335	1.54822	0.774111	+	0.633049i	$0.218197\pi$
$770$	0	0
$771$	86.7668	3.12483
$772$	0	0
$773$	7.09741	0.255276	0.127638	−	0.991821i	$-0.459260\pi$
$773$	7.09741	0.255276	0.127638	+	0.991821i	$0.459260\pi$
$774$	0	0
$775$	3.86973	0.139005
$776$	0	0
$777$	105.297	3.77752
$778$	0	0
$779$	−29.4235	−1.05421
$780$	0	0
$781$	0	0
$782$	0	0
$783$	109.044	3.89693
$784$	0	0
$785$	−5.84097	−0.208473
$786$	0	0
$787$	−43.2598	−1.54205	−0.771023	−	0.636807i	$-0.780255\pi$
$787$	−43.2598	−1.54205	−0.771023	+	0.636807i	$0.780255\pi$
$788$	0	0
$789$	−55.9421	−1.99159
$790$	0	0
$791$	−29.5343	−1.05012
$792$	0	0
$793$	−5.32938	−0.189252
$794$	0	0
$795$	25.1360	0.891482
$796$	0	0
$797$	−35.3172	−1.25100	−0.625500	−	0.780224i	$-0.715105\pi$
$797$	−35.3172	−1.25100	−0.625500	+	0.780224i	$0.715105\pi$
$798$	0	0
$799$	15.9841	0.565478
$800$	0	0
$801$	53.5885	1.89346
$802$	0	0
$803$	0	0
$804$	0	0
$805$	7.87034	0.277393
$806$	0	0
$807$	−33.4636	−1.17798
$808$	0	0
$809$	4.08007	0.143448	0.0717238	−	0.997425i	$-0.477150\pi$
$809$	4.08007	0.143448	0.0717238	+	0.997425i	$0.477150\pi$
$810$	0	0
$811$	−36.1960	−1.27102	−0.635508	−	0.772095i	$-0.719209\pi$
$811$	−36.1960	−1.27102	−0.635508	+	0.772095i	$0.719209\pi$
$812$	0	0
$813$	−97.3205	−3.41318
$814$	0	0
$815$	−14.6105	−0.511782
$816$	0	0
$817$	5.92108	0.207152
$818$	0	0
$819$	28.9048	1.01001
$820$	0	0
$821$	−16.0761	−0.561060	−0.280530	−	0.959845i	$-0.590510\pi$
$821$	−16.0761	−0.561060	−0.280530	+	0.959845i	$0.590510\pi$
$822$	0	0
$823$	−30.0538	−1.04761	−0.523805	−	0.851838i	$-0.675488\pi$
$823$	−30.0538	−1.04761	−0.523805	+	0.851838i	$0.675488\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−0.628773	−0.0218646	−0.0109323	−	0.999940i	$-0.503480\pi$
$827$	−0.628773	−0.0218646	−0.0109323	+	0.999940i	$0.503480\pi$
$828$	0	0
$829$	4.92727	0.171131	0.0855655	−	0.996333i	$-0.472730\pi$
$829$	4.92727	0.171131	0.0855655	+	0.996333i	$0.472730\pi$
$830$	0	0
$831$	−10.0805	−0.349689
$832$	0	0
$833$	31.5956	1.09472
$834$	0	0
$835$	−6.73740	−0.233157
$836$	0	0
$837$	53.6528	1.85451
$838$	0	0
$839$	6.09432	0.210399	0.105200	−	0.994451i	$-0.466452\pi$
$839$	6.09432	0.210399	0.105200	+	0.994451i	$0.466452\pi$
$840$	0	0
$841$	32.8560	1.13297
$842$	0	0
$843$	37.3557	1.28660
$844$	0	0
$845$	−12.3525	−0.424938
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−6.89870	−0.236763
$850$	0	0
$851$	10.7051	0.366966
$852$	0	0
$853$	16.2383	0.555990	0.277995	−	0.960582i	$-0.410330\pi$
$853$	16.2383	0.555990	0.277995	+	0.960582i	$0.410330\pi$
$854$	0	0
$855$	41.1002	1.40560
$856$	0	0
$857$	42.8315	1.46310	0.731548	−	0.681790i	$-0.238798\pi$
$857$	42.8315	1.46310	0.731548	+	0.681790i	$0.238798\pi$
$858$	0	0
$859$	25.2561	0.861725	0.430863	−	0.902417i	$-0.358209\pi$
$859$	25.2561	0.861725	0.430863	+	0.902417i	$0.358209\pi$
$860$	0	0
$861$	82.5954	2.81484
$862$	0	0
$863$	22.1766	0.754901	0.377451	−	0.926030i	$-0.376801\pi$
$863$	22.1766	0.754901	0.377451	+	0.926030i	$0.376801\pi$
$864$	0	0
$865$	2.48506	0.0844946
$866$	0	0
$867$	−43.6400	−1.48209
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.44817	0.150720
$872$	0	0
$873$	59.6846	2.02002
$874$	0	0
$875$	−4.90937	−0.165967
$876$	0	0
$877$	19.3555	0.653589	0.326794	−	0.945095i	$-0.394032\pi$
$877$	19.3555	0.653589	0.326794	+	0.945095i	$0.394032\pi$
$878$	0	0
$879$	−83.9599	−2.83190
$880$	0	0
$881$	−34.6226	−1.16646	−0.583232	−	0.812306i	$-0.698212\pi$
$881$	−34.6226	−1.16646	−0.583232	+	0.812306i	$0.698212\pi$
$882$	0	0
$883$	−3.68596	−0.124042	−0.0620211	−	0.998075i	$-0.519755\pi$
$883$	−3.68596	−0.124042	−0.0620211	+	0.998075i	$0.519755\pi$
$884$	0	0
$885$	12.5045	0.420335
$886$	0	0
$887$	9.06129	0.304248	0.152124	−	0.988361i	$-0.451389\pi$
$887$	9.06129	0.304248	0.152124	+	0.988361i	$0.451389\pi$
$888$	0	0
$889$	6.18775	0.207531
$890$	0	0
$891$	0	0
$892$	0	0
$893$	48.6005	1.62635
$894$	0	0
$895$	−25.8751	−0.864910
$896$	0	0
$897$	4.14353	0.138348
$898$	0	0
$899$	30.4349	1.01506
$900$	0	0
$901$	14.4580	0.481667
$902$	0	0
$903$	−16.6212	−0.553118
$904$	0	0
$905$	14.2925	0.475100
$906$	0	0
$907$	32.4654	1.07799	0.538997	−	0.842308i	$-0.318803\pi$
$907$	32.4654	1.07799	0.538997	+	0.842308i	$0.318803\pi$
$908$	0	0
$909$	−46.6857	−1.54847
$910$	0	0
$911$	51.2941	1.69945	0.849724	−	0.527227i	$-0.176768\pi$
$911$	51.2941	1.69945	0.849724	+	0.527227i	$0.176768\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	21.2722	0.703236
$916$	0	0
$917$	−59.2856	−1.95778
$918$	0	0
$919$	12.3040	0.405870	0.202935	−	0.979192i	$-0.434952\pi$
$919$	12.3040	0.405870	0.202935	+	0.979192i	$0.434952\pi$
$920$	0	0
$921$	−17.9623	−0.591877
$922$	0	0
$923$	8.82708	0.290547
$924$	0	0
$925$	−6.67764	−0.219559
$926$	0	0
$927$	−110.052	−3.61460
$928$	0	0
$929$	18.3854	0.603207	0.301603	−	0.953434i	$-0.402478\pi$
$929$	18.3854	0.603207	0.301603	+	0.953434i	$0.402478\pi$
$930$	0	0
$931$	96.0676	3.14849
$932$	0	0
$933$	5.50693	0.180289
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.8251	0.418979	0.209489	−	0.977811i	$-0.432820\pi$
$937$	12.8251	0.418979	0.209489	+	0.977811i	$0.432820\pi$
$938$	0	0
$939$	−17.0123	−0.555174
$940$	0	0
$941$	−8.62568	−0.281189	−0.140595	−	0.990067i	$-0.544901\pi$
$941$	−8.62568	−0.281189	−0.140595	+	0.990067i	$0.544901\pi$
$942$	0	0
$943$	8.39711	0.273447
$944$	0	0
$945$	−68.0671	−2.21422
$946$	0	0
$947$	22.1259	0.718996	0.359498	−	0.933146i	$-0.382948\pi$
$947$	22.1259	0.718996	0.359498	+	0.933146i	$0.382948\pi$
$948$	0	0
$949$	−2.25309	−0.0731384
$950$	0	0
$951$	24.8535	0.805929
$952$	0	0
$953$	−30.4842	−0.987481	−0.493741	−	0.869609i	$-0.664371\pi$
$953$	−30.4842	−0.987481	−0.493741	+	0.869609i	$0.664371\pi$
$954$	0	0
$955$	3.92551	0.127027
$956$	0	0
$957$	0	0
$958$	0	0
$959$	44.3810	1.43314
$960$	0	0
$961$	−16.0252	−0.516942
$962$	0	0
$963$	−41.1015	−1.32448
$964$	0	0
$965$	10.5814	0.340629
$966$	0	0
$967$	−45.7942	−1.47264	−0.736321	−	0.676632i	$-0.763439\pi$
$967$	−45.7942	−1.47264	−0.736321	+	0.676632i	$0.763439\pi$
$968$	0	0
$969$	33.3337	1.07083
$970$	0	0
$971$	58.7275	1.88465	0.942327	−	0.334694i	$-0.108633\pi$
$971$	58.7275	1.88465	0.942327	+	0.334694i	$0.108633\pi$
$972$	0	0
$973$	−106.509	−3.41452
$974$	0	0
$975$	−2.58465	−0.0827751
$976$	0	0
$977$	−22.7484	−0.727786	−0.363893	−	0.931441i	$-0.618553\pi$
$977$	−22.7484	−0.727786	−0.363893	+	0.931441i	$0.618553\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	103.919	3.31788
$982$	0	0
$983$	−33.1105	−1.05606	−0.528030	−	0.849226i	$-0.677069\pi$
$983$	−33.1105	−1.05606	−0.528030	+	0.849226i	$0.677069\pi$
$984$	0	0
$985$	−1.89648	−0.0604270
$986$	0	0
$987$	−136.427	−4.34253
$988$	0	0
$989$	−1.68980	−0.0537326
$990$	0	0
$991$	12.6263	0.401088	0.200544	−	0.979685i	$-0.435729\pi$
$991$	12.6263	0.401088	0.200544	+	0.979685i	$0.435729\pi$
$992$	0	0
$993$	−60.1747	−1.90959
$994$	0	0
$995$	13.9966	0.443721
$996$	0	0
$997$	−19.0865	−0.604477	−0.302238	−	0.953232i	$-0.597734\pi$
$997$	−19.0865	−0.604477	−0.302238	+	0.953232i	$0.597734\pi$
$998$	0	0
$999$	−92.5838	−2.92922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.bg.1.8		8
4.3	odd	2		9680.2.a.df.1.1		8
11.3	even	5		440.2.y.d.361.1	✓	16
11.4	even	5		440.2.y.d.401.1	yes	16
11.10	odd	2		4840.2.a.bh.1.8		8
44.3	odd	10		880.2.bo.k.801.4		16
44.15	odd	10		880.2.bo.k.401.4		16
44.43	even	2		9680.2.a.de.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.d.361.1	✓	16	11.3	even	5
440.2.y.d.401.1	yes	16	11.4	even	5
880.2.bo.k.401.4		16	44.15	odd	10
880.2.bo.k.801.4		16	44.3	odd	10
4840.2.a.bg.1.8		8	1.1	even	1	trivial
4840.2.a.bh.1.8		8	11.10	odd	2
9680.2.a.de.1.1		8	44.43	even	2
9680.2.a.df.1.1		8	4.3	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q+3.21195 q^{3} +1.00000 q^{5} -4.90937 q^{7} +7.31662 q^{9} +O(q^{10})\)	\(q+3.21195 q^{3} +1.00000 q^{5} -4.90937 q^{7} +7.31662 q^{9} -0.804699 q^{13} +3.21195 q^{15} +1.84749 q^{17} +5.61737 q^{19} -15.7686 q^{21} -1.60313 q^{23} +1.00000 q^{25} +13.8648 q^{27} +7.86486 q^{29} +3.86973 q^{31} -4.90937 q^{35} -6.67764 q^{37} -2.58465 q^{39} -5.23795 q^{41} +1.05407 q^{43} +7.31662 q^{45} +8.65181 q^{47} +17.1019 q^{49} +5.93404 q^{51} +7.82577 q^{53} +18.0427 q^{57} +3.89313 q^{59} +6.62282 q^{61} -35.9200 q^{63} -0.804699 q^{65} -5.52774 q^{67} -5.14916 q^{69} -10.9694 q^{71} +2.79991 q^{73} +3.21195 q^{75} -2.91983 q^{79} +22.5830 q^{81} +1.93633 q^{83} +1.84749 q^{85} +25.2615 q^{87} +7.32421 q^{89} +3.95056 q^{91} +12.4294 q^{93} +5.61737 q^{95} +8.15740 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9}+O(q^{10}) \) 8 * q + q^3 + 8 * q^5 - 6 * q^7 + 19 * q^9	\( 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9} + 12 q^{13} + q^{15} + 2 q^{17} - 6 q^{19} + 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} + 8 q^{29} + 19 q^{31} - 6 q^{35} + 12 q^{37} - 21 q^{39} - 3 q^{41} - 8 q^{43} + 19 q^{45} + 10 q^{47} + 22 q^{49} - 7 q^{51} + 28 q^{53} + 25 q^{57} + 25 q^{59} + 10 q^{61} - 64 q^{63} + 12 q^{65} - 2 q^{67} + 18 q^{69} + 25 q^{71} + 38 q^{73} + q^{75} - 38 q^{79} + 32 q^{81} - 28 q^{83} + 2 q^{85} + 15 q^{87} + 12 q^{89} + 8 q^{91} - 15 q^{93} - 6 q^{95} + 21 q^{97}+O(q^{100}) \) 8 * q + q^3 + 8 * q^5 - 6 * q^7 + 19 * q^9 + 12 * q^13 + q^15 + 2 * q^17 - 6 * q^19 + 6 * q^21 + 10 * q^23 + 8 * q^25 + 13 * q^27 + 8 * q^29 + 19 * q^31 - 6 * q^35 + 12 * q^37 - 21 * q^39 - 3 * q^41 - 8 * q^43 + 19 * q^45 + 10 * q^47 + 22 * q^49 - 7 * q^51 + 28 * q^53 + 25 * q^57 + 25 * q^59 + 10 * q^61 - 64 * q^63 + 12 * q^65 - 2 * q^67 + 18 * q^69 + 25 * q^71 + 38 * q^73 + q^75 - 38 * q^79 + 32 * q^81 - 28 * q^83 + 2 * q^85 + 15 * q^87 + 12 * q^89 + 8 * q^91 - 15 * q^93 - 6 * q^95 + 21 * q^97

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