LMFDB - Embedded newform 5292.2.i.e.2125.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5292,2,Mod(1549,5292)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5292, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5292.1549"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$5292 = 2^{2} \cdot 3^{3} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5292.i (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.309123.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$ x^6 - 3x^5 + 10x^4 - 15x^3 + 19x^2 - 12*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$3$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2125.3
Root			$0.500000 + 0.224437i$ of defining polynomial
Character	$\chi$	$=$	5292.2125
Dual form			5292.2.i.e.1549.3

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.849814 - 1.47192i) q^{5} +(1.23855 + 2.14523i) q^{11} +(0.388736 + 0.673310i) q^{13} +(-1.40545 + 2.43430i) q^{17} +(-2.49381 - 4.31941i) q^{19} +(0.356004 - 0.616617i) q^{23} +(1.05563 + 1.82841i) q^{25} +(2.25526 - 3.90623i) q^{29} -5.09888 q^{31} +(3.43818 + 5.95510i) q^{37} +(2.93818 + 5.08907i) q^{41} +(2.32691 - 4.03033i) q^{43} +12.9876 q^{47} +(0.944368 - 1.63569i) q^{53} +4.21015 q^{55} +14.2880 q^{59} -14.3090 q^{61} +1.32141 q^{65} +7.98762 q^{67} +10.2632 q^{71} +(2.49381 - 4.31941i) q^{73} -9.21015 q^{79} +(-4.40545 + 7.63046i) q^{83} +(2.38874 + 4.13741i) q^{85} +(-4.82691 - 8.36046i) q^{89} -8.47710 q^{95} +(4.32072 - 7.48371i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - q^{5} + 2 q^{11} + 3 q^{13} - 2 q^{17} + 3 q^{19} + 14 q^{23} + 6 q^{25} + q^{29} + 6 q^{31} + 3 q^{37} - 3 q^{43} + 42 q^{47} + 6 q^{53} - 12 q^{55} + 62 q^{59} - 12 q^{61} - 30 q^{65} + 12 q^{67}+ \cdots - 9 q^{97}+O(q^{100})$ 6 * q - q^5 + 2 * q^11 + 3 * q^13 - 2 * q^17 + 3 * q^19 + 14 * q^23 + 6 * q^25 + q^29 + 6 * q^31 + 3 * q^37 - 3 * q^43 + 42 * q^47 + 6 * q^53 - 12 * q^55 + 62 * q^59 - 12 * q^61 - 30 * q^65 + 12 * q^67 - 34 * q^71 - 3 * q^73 - 18 * q^79 - 20 * q^83 + 15 * q^85 - 12 * q^89 - 40 * q^95 - 9 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$ .

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	0.849814	−	1.47192i	0.380048	−	0.658263i	−0.611020	−	0.791615i	$-0.709241\pi$
$5$	0.849814	−	1.47192i	0.380048	−	0.658263i	0.991069	+	0.133352i	$0.0425740\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.23855	+	2.14523i	0.373437	+	0.646812i	0.990092	−	0.140422i	$-0.0448459\pi$
$11$	1.23855	+	2.14523i	0.373437	+	0.646812i	−0.616655	+	0.787234i	$0.711513\pi$
$12$		0			0
$13$	0.388736	+	0.673310i	0.107816	+	0.186743i	0.914885	−	0.403714i	$-0.132281\pi$
$13$	0.388736	+	0.673310i	0.107816	+	0.186743i	−0.807069	+	0.590457i	$0.798948\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.40545	+	2.43430i	−0.340871	+	0.590405i	−0.984595	−	0.174852i	$-0.944055\pi$
$17$	−1.40545	+	2.43430i	−0.340871	+	0.590405i	0.643724	+	0.765258i	$0.277389\pi$
$18$		0			0
$19$	−2.49381	−	4.31941i	−0.572119	−	0.990940i	−0.996348	−	0.0853846i	$-0.972788\pi$
$19$	−2.49381	−	4.31941i	−0.572119	−	0.990940i	0.424229	−	0.905555i	$-0.360545\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.356004	−	0.616617i	0.0742320	−	0.128574i	−0.826520	−	0.562907i	$-0.809683\pi$
$23$	0.356004	−	0.616617i	0.0742320	−	0.128574i	0.900752	+	0.434334i	$0.143016\pi$
$24$		0			0
$25$	1.05563	+	1.82841i	0.211126	+	0.365682i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.25526	−	3.90623i	0.418791	−	0.725368i	−0.577027	−	0.816725i	$-0.695787\pi$
$29$	2.25526	−	3.90623i	0.418791	−	0.725368i	0.995818	+	0.0913573i	$0.0291205\pi$
$30$		0			0
$31$	−5.09888			−0.915787			−0.457893	−	0.889007i	$-0.651396\pi$
$31$	−5.09888			−0.915787			−0.457893	+	0.889007i	$0.651396\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.43818	+	5.95510i	0.565233	+	0.979012i	0.997028	+	0.0770405i	$0.0245471\pi$
$37$	3.43818	+	5.95510i	0.565233	+	0.979012i	−0.431795	+	0.901972i	$0.642120\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.93818	+	5.08907i	0.458866	+	0.794780i	0.998901	−	0.0468628i	$-0.0149223\pi$
$41$	2.93818	+	5.08907i	0.458866	+	0.794780i	−0.540035	+	0.841643i	$0.681589\pi$
$42$		0			0
$43$	2.32691	−	4.03033i	0.354851	−	0.614620i	−0.632241	−	0.774771i	$-0.717865\pi$
$43$	2.32691	−	4.03033i	0.354851	−	0.614620i	0.987092	+	0.160151i	$0.0511982\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	12.9876			1.89444			0.947220	−	0.320586i	$-0.103880\pi$
$47$	12.9876			1.89444			0.947220	+	0.320586i	$0.103880\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.944368	−	1.63569i	0.129719	−	0.224680i	−0.793849	−	0.608115i	$-0.791926\pi$
$53$	0.944368	−	1.63569i	0.129719	−	0.224680i	0.923568	+	0.383436i	$0.125259\pi$
$54$		0			0
$55$	4.21015			0.567696
$56$		0			0
$57$		0			0
$58$		0			0
$59$	14.2880			1.86014			0.930069	−	0.367385i	$-0.119747\pi$
$59$	14.2880			1.86014			0.930069	+	0.367385i	$0.119747\pi$
$60$		0			0
$61$	−14.3090			−1.83208			−0.916042	−	0.401082i	$-0.868634\pi$
$61$	−14.3090			−1.83208			−0.916042	+	0.401082i	$0.868634\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.32141			0.163901
$66$		0			0
$67$	7.98762			0.975843			0.487922	−	0.872887i	$-0.337755\pi$
$67$	7.98762			0.975843			0.487922	+	0.872887i	$0.337755\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.2632			1.21802			0.609011	−	0.793162i	$-0.291567\pi$
$71$	10.2632			1.21802			0.609011	+	0.793162i	$0.291567\pi$
$72$		0			0
$73$	2.49381	−	4.31941i	0.291878	−	0.505548i	−0.682376	−	0.731002i	$-0.739053\pi$
$73$	2.49381	−	4.31941i	0.291878	−	0.505548i	0.974254	+	0.225454i	$0.0723864\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−9.21015			−1.03622			−0.518111	−	0.855313i	$-0.673365\pi$
$79$	−9.21015			−1.03622			−0.518111	+	0.855313i	$0.673365\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.40545	+	7.63046i	−0.483561	+	0.837551i	−0.999822	−	0.0188798i	$-0.993990\pi$
$83$	−4.40545	+	7.63046i	−0.483561	+	0.837551i	0.516261	+	0.856431i	$0.327323\pi$
$84$		0			0
$85$	2.38874	+	4.13741i	0.259095	+	0.448765i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.82691	−	8.36046i	−0.511652	−	0.886207i	−0.999909	−	0.0135071i	$-0.995700\pi$
$89$	−4.82691	−	8.36046i	−0.511652	−	0.886207i	0.488257	−	0.872700i	$-0.337633\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.47710			−0.869732
$96$		0			0
$97$	4.32072	−	7.48371i	0.438703	−	0.759856i	−0.558887	−	0.829244i	$-0.688771\pi$
$97$	4.32072	−	7.48371i	0.438703	−	0.759856i	0.997590	+	0.0693880i	$0.0221047\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.20582	−	2.08854i	−0.119983	−	0.207817i	0.799777	−	0.600297i	$-0.204951\pi$
$101$	−1.20582	−	2.08854i	−0.119983	−	0.207817i	−0.919761	+	0.392479i	$0.871617\pi$
$102$		0			0
$103$	−2.16690	+	3.75317i	−0.213511	+	0.369811i	−0.952811	−	0.303565i	$-0.901823\pi$
$103$	−2.16690	+	3.75317i	−0.213511	+	0.369811i	0.739300	+	0.673376i	$0.235156\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.59888	+	16.6258i	0.927959	+	1.60727i	0.786732	+	0.617295i	$0.211772\pi$
$107$	9.59888	+	16.6258i	0.927959	+	1.60727i	0.141228	+	0.989977i	$0.454895\pi$
$108$		0			0
$109$	−9.48143	+	16.4223i	−0.908156	+	1.57297i	−0.0915329	+	0.995802i	$0.529177\pi$
$109$	−9.48143	+	16.4223i	−0.908156	+	1.57297i	−0.816623	+	0.577171i	$0.804157\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.46472	+	11.1972i	0.608150	+	1.05335i	0.991545	+	0.129762i	$0.0414213\pi$
$113$	6.46472	+	11.1972i	0.608150	+	1.05335i	−0.383395	+	0.923584i	$0.625245\pi$
$114$		0			0
$115$	−0.605074	−	1.04802i	−0.0564235	−	0.0977283i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.43199	−	4.21233i	0.221090	−	0.382939i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0865			1.08105
$126$		0			0
$127$	17.6291			1.56433			0.782163	−	0.623073i	$-0.214116\pi$
$127$	17.6291			1.56433			0.782163	+	0.623073i	$0.214116\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.84362	−	4.92530i	0.248449	−	0.430326i	−0.714647	−	0.699485i	$-0.753413\pi$
$131$	2.84362	−	4.92530i	0.248449	−	0.430326i	0.963096	+	0.269160i	$0.0867460\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.72617	−	16.8462i	−0.830963	−	1.43927i	−0.897276	−	0.441471i	$-0.854457\pi$
$137$	−9.72617	−	16.8462i	−0.830963	−	1.43927i	0.0663128	−	0.997799i	$-0.478876\pi$
$138$		0			0
$139$	−1.49381	−	2.58736i	−0.126703	−	0.219457i	0.795694	−	0.605699i	$-0.207106\pi$
$139$	−1.49381	−	2.58736i	−0.126703	−	0.219457i	−0.922397	+	0.386242i	$0.873773\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.962937	+	1.66786i	−0.0805249	+	0.139473i
$144$		0			0
$145$	−3.83310	−	6.63913i	−0.318322	−	0.551350i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.04944	−	7.01384i	0.331743	−	0.574596i	−0.651111	−	0.758983i	$-0.725697\pi$
$149$	4.04944	−	7.01384i	0.331743	−	0.574596i	0.982854	+	0.184387i	$0.0590299\pi$
$150$		0			0
$151$	4.43199	+	7.67643i	0.360670	+	0.624699i	0.988071	−	0.153997i	$-0.0492147\pi$
$151$	4.43199	+	7.67643i	0.360670	+	0.624699i	−0.627401	+	0.778696i	$0.715881\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.33310	+	7.50516i	−0.348043	+	0.602829i
$156$		0			0
$157$	8.76509			0.699530			0.349765	−	0.936837i	$-0.386261\pi$
$157$	8.76509			0.699530			0.349765	+	0.936837i	$0.386261\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.993810	+	1.72133i	0.0778412	+	0.134825i	0.902318	−	0.431070i	$-0.141864\pi$
$163$	0.993810	+	1.72133i	0.0778412	+	0.134825i	−0.824477	+	0.565895i	$0.808531\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.31089	−	2.27053i	−0.101440	−	0.175699i	0.810838	−	0.585270i	$-0.199012\pi$
$167$	−1.31089	−	2.27053i	−0.101440	−	0.175699i	−0.912278	+	0.409571i	$0.865678\pi$
$168$		0			0
$169$	6.19777	−	10.7349i	0.476751	−	0.825758i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.22981			−0.397615			−0.198808	−	0.980039i	$-0.563707\pi$
$173$	−5.22981			−0.397615			−0.198808	+	0.980039i	$0.563707\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.38255	+	4.12669i	−0.178080	+	0.308443i	−0.941223	−	0.337786i	$-0.890322\pi$
$179$	−2.38255	+	4.12669i	−0.178080	+	0.308443i	0.763143	+	0.646230i	$0.223655\pi$
$180$		0			0
$181$	10.4313			0.775352			0.387676	−	0.921796i	$-0.373278\pi$
$181$	10.4313			0.775352			0.387676	+	0.921796i	$0.373278\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	11.6872			0.859264
$186$		0			0
$187$	−6.96286			−0.509175
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.3214			−0.963904			−0.481952	−	0.876198i	$-0.660072\pi$
$191$	−13.3214			−0.963904			−0.481952	+	0.876198i	$0.660072\pi$
$192$		0			0
$193$	−14.6414			−1.05391			−0.526957	−	0.849892i	$-0.676667\pi$
$193$	−14.6414			−1.05391			−0.526957	+	0.849892i	$0.676667\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.4858			1.31706			0.658528	−	0.752556i	$-0.271179\pi$
$197$	18.4858			1.31706			0.658528	+	0.752556i	$0.271179\pi$
$198$		0			0
$199$	11.8083	−	20.4527i	0.837071	−	1.44985i	−0.0552614	−	0.998472i	$-0.517599\pi$
$199$	11.8083	−	20.4527i	0.837071	−	1.44985i	0.892333	−	0.451378i	$-0.149067\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	9.98762			0.697566
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.17742	−	10.6996i	0.427301	−	0.740107i
$210$		0			0
$211$	7.27747	+	12.6050i	0.501002	+	0.867761i	0.999999	+	0.00115718i	$0.000368342\pi$
$211$	7.27747	+	12.6050i	0.501002	+	0.867761i	−0.498998	+	0.866603i	$0.666298\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.95489	−	6.85007i	−0.269721	−	0.467171i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.18539			−0.147005
$222$		0			0
$223$	4.72253	−	8.17966i	0.316244	−	0.547750i	−0.663457	−	0.748214i	$-0.730912\pi$
$223$	4.72253	−	8.17966i	0.316244	−	0.547750i	0.979701	+	0.200464i	$0.0642449\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.55563	−	16.5508i	−0.634230	−	1.09852i	−0.986678	−	0.162687i	$-0.947984\pi$
$227$	−9.55563	−	16.5508i	−0.634230	−	1.09852i	0.352448	−	0.935831i	$-0.385349\pi$
$228$		0			0
$229$	5.72253	−	9.91171i	0.378155	−	0.654984i	−0.612639	−	0.790363i	$-0.709892\pi$
$229$	5.72253	−	9.91171i	0.378155	−	0.654984i	0.990794	+	0.135379i	$0.0432252\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.595243	−	1.03099i	−0.0389956	−	0.0675424i	0.845869	−	0.533391i	$-0.179083\pi$
$233$	−0.595243	−	1.03099i	−0.0389956	−	0.0675424i	−0.884865	+	0.465848i	$0.845749\pi$
$234$		0			0
$235$	11.0371	−	19.1168i	0.719979	−	1.24704i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.1414	+	21.0296i	0.785365	+	1.36029i	0.928781	+	0.370630i	$0.120858\pi$
$239$	12.1414	+	21.0296i	0.785365	+	1.36029i	−0.143416	+	0.989663i	$0.545809\pi$
$240$		0			0
$241$	−10.7095	−	18.5493i	−0.689857	−	1.19487i	−0.971884	−	0.235461i	$-0.924340\pi$
$241$	−10.7095	−	18.5493i	−0.689857	−	1.19487i	0.282027	−	0.959406i	$-0.408993\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	1.93887	−	3.35822i	0.123367	−	0.213678i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.67996			−0.169158			−0.0845789	−	0.996417i	$-0.526955\pi$
$251$	−2.67996			−0.169158			−0.0845789	+	0.996417i	$0.526955\pi$
$252$		0			0
$253$	1.76371			0.110884
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.54256	−	9.60000i	0.345736	−	0.598832i	−0.639752	−	0.768582i	$-0.720963\pi$
$257$	5.54256	−	9.60000i	0.345736	−	0.598832i	0.985487	+	0.169750i	$0.0542961\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.70396	+	11.6116i	0.413384	+	0.716002i	0.995257	−	0.0972776i	$-0.0310135\pi$
$263$	6.70396	+	11.6116i	0.413384	+	0.716002i	−0.581873	+	0.813279i	$0.697680\pi$
$264$		0			0
$265$	−1.60507	−	2.78007i	−0.0985989	−	0.170778i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.04511	+	3.54224i	−0.124693	+	0.215974i	−0.921613	−	0.388111i	$-0.873128\pi$
$269$	−2.04511	+	3.54224i	−0.124693	+	0.215974i	0.796920	+	0.604085i	$0.206461\pi$
$270$		0			0
$271$	3.06182	+	5.30323i	0.185992	+	0.322148i	0.943910	−	0.330201i	$-0.107117\pi$
$271$	3.06182	+	5.30323i	0.185992	+	0.322148i	−0.757918	+	0.652350i	$0.773783\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.61491	+	4.52915i	−0.157685	+	0.273118i
$276$		0			0
$277$	−7.88255	−	13.6530i	−0.473616	−	0.820327i	0.525928	−	0.850529i	$-0.323718\pi$
$277$	−7.88255	−	13.6530i	−0.473616	−	0.820327i	−0.999544	+	0.0302019i	$0.990385\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−10.5946	+	18.3503i	−0.632018	+	1.09469i	0.355120	+	0.934821i	$0.384440\pi$
$281$	−10.5946	+	18.3503i	−0.632018	+	1.09469i	−0.987139	+	0.159867i	$0.948893\pi$
$282$		0			0
$283$	−6.87636			−0.408757			−0.204378	−	0.978892i	$-0.565517\pi$
$283$	−6.87636			−0.408757			−0.204378	+	0.978892i	$0.565517\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4.54944	+	7.87987i	0.267614	+	0.463521i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.7534	+	23.8216i	0.803482	+	1.39167i	0.917311	+	0.398172i	$0.130355\pi$
$293$	13.7534	+	23.8216i	0.803482	+	1.39167i	−0.113829	+	0.993500i	$0.536311\pi$
$294$		0			0
$295$	12.1421	−	21.0308i	0.706943	−	1.22446i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.553566			0.0320135
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.1600	+	21.0618i	−0.696281	+	1.20599i
$306$		0			0
$307$	21.5178			1.22809			0.614043	−	0.789273i	$-0.289542\pi$
$307$	21.5178			1.22809			0.614043	+	0.789273i	$0.289542\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.3855			1.04255			0.521273	−	0.853390i	$-0.325457\pi$
$311$	18.3855			1.04255			0.521273	+	0.853390i	$0.325457\pi$
$312$		0			0
$313$	0.00137742			7.78563e−5			3.89281e−5	−	1.00000i	$-0.499988\pi$
$313$	0.00137742			7.78563e−5			3.89281e−5		1.00000i	$0.499988\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.0989			−0.791872			−0.395936	−	0.918278i	$-0.629580\pi$
$317$	−14.0989			−0.791872			−0.395936	+	0.918278i	$0.629580\pi$
$318$		0			0
$319$	11.1730			0.625568
$320$		0			0
$321$		0			0
$322$		0			0
$323$	14.0197			0.780075
$324$		0			0
$325$	−0.820724	+	1.42154i	−0.0455256	+	0.0788526i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	13.9629			0.767468			0.383734	−	0.923444i	$-0.374638\pi$
$331$	13.9629			0.767468			0.383734	+	0.923444i	$0.374638\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.78799	−	11.7571i	0.370868	−	0.642362i
$336$		0			0
$337$	−12.0982	−	20.9547i	−0.659031	−	1.14147i	−0.980867	−	0.194679i	$-0.937633\pi$
$337$	−12.0982	−	20.9547i	−0.659031	−	1.14147i	0.321836	−	0.946795i	$-0.395700\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.31522	−	10.9383i	−0.341988	−	0.592341i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	32.7156			1.75626			0.878132	−	0.478418i	$-0.158790\pi$
$347$	32.7156			1.75626			0.878132	+	0.478418i	$0.158790\pi$
$348$		0			0
$349$	−11.8887	+	20.5919i	−0.636389	+	1.10226i	0.349830	+	0.936813i	$0.386240\pi$
$349$	−11.8887	+	20.5919i	−0.636389	+	1.10226i	−0.986219	+	0.165445i	$0.947094\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10.0309	−	17.3740i	−0.533889	−	0.924724i	−0.999216	−	0.0395847i	$-0.987396\pi$
$353$	−10.0309	−	17.3740i	−0.533889	−	0.924724i	0.465327	−	0.885139i	$-0.345937\pi$
$354$		0			0
$355$	8.72184	−	15.1067i	0.462907	−	0.801779i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.15087	+	7.18953i	0.219075	+	0.379449i	0.954525	−	0.298130i	$-0.0963628\pi$
$359$	4.15087	+	7.18953i	0.219075	+	0.379449i	−0.735451	+	0.677578i	$0.763029\pi$
$360$		0			0
$361$	−2.93818	+	5.08907i	−0.154641	+	0.267846i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.23855	−	7.34138i	−0.221856	−	0.384266i
$366$		0			0
$367$	5.77197	+	9.99735i	0.301294	+	0.521857i	0.976429	−	0.215837i	$-0.0692480\pi$
$367$	5.77197	+	9.99735i	0.301294	+	0.521857i	−0.675135	+	0.737694i	$0.735915\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.42580	+	2.46956i	−0.0738250	+	0.127869i	−0.900575	−	0.434701i	$-0.856854\pi$
$373$	−1.42580	+	2.46956i	−0.0738250	+	0.127869i	0.826750	+	0.562570i	$0.190187\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.50680			0.180609
$378$		0			0
$379$	35.9519			1.84672			0.923361	−	0.383932i	$-0.125430\pi$
$379$	35.9519			1.84672			0.923361	+	0.383932i	$0.125430\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.915278	−	1.58531i	0.0467685	−	0.0810054i	−0.841694	−	0.539956i	$-0.818441\pi$
$383$	0.915278	−	1.58531i	0.0467685	−	0.0810054i	0.888462	+	0.458950i	$0.151774\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.69530	−	9.86454i	−0.288763	−	0.500152i	0.684752	−	0.728776i	$-0.259911\pi$
$389$	−5.69530	−	9.86454i	−0.288763	−	0.500152i	−0.973515	+	0.228624i	$0.926577\pi$
$390$		0			0
$391$	1.00069	+	1.73324i	0.0506070	+	0.0876539i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.82691	+	13.5566i	−0.393815	+	0.682107i
$396$		0			0
$397$	−5.21565	−	9.03377i	−0.261766	−	0.453392i	0.704945	−	0.709262i	$-0.250972\pi$
$397$	−5.21565	−	9.03377i	−0.261766	−	0.453392i	−0.966711	+	0.255870i	$0.917638\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−17.0371	+	29.5091i	−0.850790	+	1.47361i	0.0297058	+	0.999559i	$0.490543\pi$
$401$	−17.0371	+	29.5091i	−0.850790	+	1.47361i	−0.880496	+	0.474053i	$0.842790\pi$
$402$		0			0
$403$	−1.98212	−	3.43313i	−0.0987364	−	0.171016i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.51671	+	14.7514i	−0.422158	+	0.731199i
$408$		0			0
$409$	−3.97524			−0.196563			−0.0982815	−	0.995159i	$-0.531335\pi$
$409$	−3.97524			−0.196563			−0.0982815	+	0.995159i	$0.531335\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	7.48762	+	12.9689i	0.367553	+	0.636620i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.72184	+	8.17847i	0.230677	+	0.399544i	0.958008	−	0.286743i	$-0.0925726\pi$
$419$	4.72184	+	8.17847i	0.230677	+	0.399544i	−0.727331	+	0.686287i	$0.759239\pi$
$420$		0			0
$421$	3.16002	−	5.47331i	0.154010	−	0.266753i	−0.778688	−	0.627411i	$-0.784115\pi$
$421$	3.16002	−	5.47331i	0.154010	−	0.266753i	0.932698	+	0.360658i	$0.117448\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.93454			−0.287867
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−13.8770	+	24.0357i	−0.668434	+	1.15776i	0.309908	+	0.950766i	$0.399702\pi$
$431$	−13.8770	+	24.0357i	−0.668434	+	1.15776i	−0.978342	+	0.206995i	$0.933632\pi$
$432$		0			0
$433$	−11.2473			−0.540510			−0.270255	−	0.962789i	$-0.587108\pi$
$433$	−11.2473			−0.540510			−0.270255	+	0.962789i	$0.587108\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.55122			−0.169878
$438$		0			0
$439$	15.0865			0.720040			0.360020	−	0.932945i	$-0.382770\pi$
$439$	15.0865			0.720040			0.360020	+	0.932945i	$0.382770\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.93316			−0.376916			−0.188458	−	0.982081i	$-0.560349\pi$
$443$	−7.93316			−0.376916			−0.188458	+	0.982081i	$0.560349\pi$
$444$		0			0
$445$	−16.4079			−0.777810
$446$		0			0
$447$		0			0
$448$		0			0
$449$	32.5636			1.53677			0.768386	−	0.639987i	$-0.221060\pi$
$449$	32.5636			1.53677			0.768386	+	0.639987i	$0.221060\pi$
$450$		0			0
$451$	−7.27816	+	12.6061i	−0.342715	+	0.593600i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.4079			0.533640			0.266820	−	0.963746i	$-0.414027\pi$
$457$	11.4079			0.533640			0.266820	+	0.963746i	$0.414027\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.45853	+	4.25830i	−0.114505	+	0.198329i	−0.917582	−	0.397547i	$-0.869862\pi$
$461$	−2.45853	+	4.25830i	−0.114505	+	0.198329i	0.803077	+	0.595876i	$0.203195\pi$
$462$		0			0
$463$	−7.59957	−	13.1628i	−0.353182	−	0.611729i	0.633623	−	0.773642i	$-0.281567\pi$
$463$	−7.59957	−	13.1628i	−0.353182	−	0.611729i	−0.986805	+	0.161913i	$0.948234\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11.8905	−	20.5950i	−0.550228	−	0.953022i	−0.998258	−	0.0590037i	$-0.981208\pi$
$467$	−11.8905	−	20.5950i	−0.550228	−	0.953022i	0.448030	−	0.894018i	$-0.352126\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	11.5280			0.530058
$474$		0			0
$475$	5.26509	−	9.11941i	0.241579	−	0.418427i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.02909	−	5.24654i	−0.138403	−	0.239720i	0.788489	−	0.615048i	$-0.210863\pi$
$479$	−3.02909	−	5.24654i	−0.138403	−	0.239720i	−0.926892	+	0.375328i	$0.877530\pi$
$480$		0			0
$481$	−2.67309	+	4.62992i	−0.121882	+	0.211106i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.34362	−	12.7195i	−0.333457	−	0.577564i
$486$		0			0
$487$	−0.568012	+	0.983825i	−0.0257391	+	0.0445814i	−0.878608	−	0.477544i	$-0.841527\pi$
$487$	−0.568012	+	0.983825i	−0.0257391	+	0.0445814i	0.852869	+	0.522125i	$0.174861\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−16.4382	−	28.4718i	−0.741845	−	1.28491i	−0.951655	−	0.307170i	$-0.900618\pi$
$491$	−16.4382	−	28.4718i	−0.741845	−	1.28491i	0.209810	−	0.977742i	$-0.432715\pi$
$492$		0			0
$493$	6.33929	+	10.9800i	0.285507	+	0.494513i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−13.0989	+	22.6879i	−0.586387	+	1.01565i	0.408314	+	0.912841i	$0.366117\pi$
$499$	−13.0989	+	22.6879i	−0.586387	+	1.01565i	−0.994701	+	0.102810i	$0.967217\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−25.8516			−1.15267			−0.576333	−	0.817215i	$-0.695517\pi$
$503$	−25.8516			−1.15267			−0.576333	+	0.817215i	$0.695517\pi$
$504$		0			0
$505$	−4.09888			−0.182398
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−17.5858	+	30.4595i	−0.779478	+	1.35009i	0.152766	+	0.988262i	$0.451182\pi$
$509$	−17.5858	+	30.4595i	−0.779478	+	1.35009i	−0.932243	+	0.361832i	$0.882151\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.68292	+	6.37900i	0.162289	+	0.281092i
$516$		0			0
$517$	16.0858	+	27.8615i	0.707453	+	1.22535i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8.93130	−	15.4695i	0.391287	−	0.677730i	−0.601332	−	0.798999i	$-0.705363\pi$
$521$	8.93130	−	15.4695i	0.391287	−	0.677730i	0.992620	+	0.121270i	$0.0386965\pi$
$522$		0			0
$523$	−11.4320	−	19.8008i	−0.499886	−	0.865828i	0.500114	−	0.865960i	$-0.333291\pi$
$523$	−11.4320	−	19.8008i	−0.499886	−	0.865828i	−1.00000		0.000131698i	$0.999958\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.16621	−	12.4122i	0.312165	−	0.540685i
$528$		0			0
$529$	11.2465	+	19.4795i	0.488979	+	0.846937i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.28435	+	3.95661i	−0.0989462	+	0.171380i
$534$		0			0
$535$	32.6291			1.41068
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−11.1538	−	19.3190i	−0.479541	−	0.830589i	0.520184	−	0.854054i	$-0.325863\pi$
$541$	−11.1538	−	19.3190i	−0.479541	−	0.830589i	−0.999725	+	0.0234656i	$0.992530\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	16.1149	+	27.9118i	0.690287	+	1.19561i
$546$		0			0
$547$	−10.8083	+	18.7206i	−0.462131	+	0.800435i	−0.999067	−	0.0431882i	$-0.986249\pi$
$547$	−10.8083	+	18.7206i	−0.462131	+	0.800435i	0.536936	+	0.843623i	$0.319582\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−22.4968			−0.958394
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.58768	−	2.74993i	0.0672720	−	0.116518i	−0.830428	−	0.557127i	$-0.811904\pi$
$557$	1.58768	−	2.74993i	0.0672720	−	0.116518i	0.897700	+	0.440608i	$0.145237\pi$
$558$		0			0
$559$	3.61822			0.153034
$560$		0			0
$561$		0			0
$562$		0			0
$563$	43.7628			1.84438			0.922190	−	0.386737i	$-0.126398\pi$
$563$	43.7628			1.84438			0.922190	+	0.386737i	$0.126398\pi$
$564$		0			0
$565$	21.9752			0.924505
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.8626			1.00037			0.500186	−	0.865918i	$-0.333265\pi$
$569$	23.8626			1.00037			0.500186	+	0.865918i	$0.333265\pi$
$570$		0			0
$571$	10.2212			0.427742			0.213871	−	0.976862i	$-0.431393\pi$
$571$	10.2212			0.427742			0.213871	+	0.976862i	$0.431393\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.50324			0.0626893
$576$		0			0
$577$	−18.0185	+	31.2089i	−0.750120	+	1.29925i	0.197645	+	0.980274i	$0.436671\pi$
$577$	−18.0185	+	31.2089i	−0.750120	+	1.29925i	−0.947764	+	0.318972i	$0.896663\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	4.67859			0.193767
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.5142	−	18.2111i	0.433966	−	0.751651i	−0.563245	−	0.826290i	$-0.690447\pi$
$587$	10.5142	−	18.2111i	0.433966	−	0.751651i	0.997211	+	0.0746391i	$0.0237805\pi$
$588$		0			0
$589$	12.7156	+	22.0242i	0.523939	+	0.907489i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	12.5803	+	21.7897i	0.516612	+	0.894798i	0.999814	+	0.0192889i	$0.00614021\pi$
$593$	12.5803	+	21.7897i	0.516612	+	0.894798i	−0.483202	+	0.875509i	$0.660526\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.22253			0.0908100			0.0454050	−	0.998969i	$-0.485542\pi$
$599$	2.22253			0.0908100			0.0454050	+	0.998969i	$0.485542\pi$
$600$		0			0
$601$	14.0494	−	24.3343i	0.573089	−	0.992619i	−0.423158	−	0.906056i	$-0.639078\pi$
$601$	14.0494	−	24.3343i	0.573089	−	0.992619i	0.996246	−	0.0865627i	$-0.0275883\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.13348	−	7.15939i	−0.168050	−	0.291071i
$606$		0			0
$607$	−3.26509	+	5.65531i	−0.132526	+	0.229542i	−0.924650	−	0.380819i	$-0.875642\pi$
$607$	−3.26509	+	5.65531i	−0.132526	+	0.229542i	0.792124	+	0.610361i	$0.208975\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.04875	+	8.74470i	0.204251	+	0.353773i
$612$		0			0
$613$	−5.36398	+	9.29068i	−0.216649	+	0.375247i	−0.953781	−	0.300501i	$-0.902846\pi$
$613$	−5.36398	+	9.29068i	−0.216649	+	0.375247i	0.737132	+	0.675748i	$0.236179\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.5265	−	26.8928i	−0.625075	−	1.08266i	−0.988526	−	0.151049i	$-0.951735\pi$
$617$	−15.5265	−	26.8928i	−0.625075	−	1.08266i	0.363451	−	0.931613i	$-0.381598\pi$
$618$		0			0
$619$	−0.723217	−	1.25265i	−0.0290685	−	0.0503482i	0.851125	−	0.524963i	$-0.175921\pi$
$619$	−0.723217	−	1.25265i	−0.0290685	−	0.0503482i	−0.880194	+	0.474615i	$0.842587\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.99312	−	8.64834i	0.199725	−	0.345934i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−19.3287			−0.770686
$630$		0			0
$631$	0.0741250			0.00295087			0.00147544	−	0.999999i	$-0.499530\pi$
$631$	0.0741250			0.00295087			0.00147544	+	0.999999i	$0.499530\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.9814	−	25.9486i	0.594520	−	1.02974i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−23.5204	−	40.7384i	−0.928998	−	1.60907i	−0.785002	−	0.619494i	$-0.787338\pi$
$641$	−23.5204	−	40.7384i	−0.928998	−	1.60907i	−0.143996	−	0.989578i	$-0.545995\pi$
$642$		0			0
$643$	−16.8647	−	29.2105i	−0.665077	−	1.15195i	−0.979264	−	0.202587i	$-0.935065\pi$
$643$	−16.8647	−	29.2105i	−0.665077	−	1.15195i	0.314187	−	0.949361i	$-0.398268\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−22.4814	+	38.9390i	−0.883836	+	1.53085i	−0.0367945	+	0.999323i	$0.511715\pi$
$647$	−22.4814	+	38.9390i	−0.883836	+	1.53085i	−0.847042	+	0.531526i	$0.821619\pi$
$648$		0			0
$649$	17.6964	+	30.6510i	0.694644	+	1.20316i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	20.8578	−	36.1267i	0.816228	−	1.41375i	−0.0922143	−	0.995739i	$-0.529394\pi$
$653$	20.8578	−	36.1267i	0.816228	−	1.41375i	0.908443	−	0.418010i	$-0.137272\pi$
$654$		0			0
$655$	−4.83310	−	8.37118i	−0.188845	−	0.327089i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10.5259	+	18.2313i	−0.410029	+	0.710191i	−0.994892	−	0.100941i	$-0.967815\pi$
$659$	−10.5259	+	18.2313i	−0.410029	+	0.710191i	0.584863	+	0.811132i	$0.301148\pi$
$660$		0			0
$661$	−22.4437			−0.872958			−0.436479	−	0.899714i	$-0.643775\pi$
$661$	−22.4437			−0.872958			−0.436479	+	0.899714i	$0.643775\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.60576	−	2.78126i	−0.0621754	−	0.107691i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−17.7225	−	30.6962i	−0.684168	−	1.18501i
$672$		0			0
$673$	5.83929	−	10.1140i	0.225088	−	0.389864i	−0.731258	−	0.682101i	$-0.761066\pi$
$673$	5.83929	−	10.1140i	0.225088	−	0.389864i	0.956346	+	0.292237i	$0.0943996\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10.4684			0.402335			0.201167	−	0.979557i	$-0.435526\pi$
$677$	10.4684			0.402335			0.201167	+	0.979557i	$0.435526\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	16.4079	−	28.4193i	0.627832	−	1.08744i	−0.360154	−	0.932893i	$-0.617276\pi$
$683$	16.4079	−	28.4193i	0.627832	−	1.08744i	0.987986	−	0.154543i	$-0.0493906\pi$
$684$		0			0
$685$	−33.0617			−1.26322
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.46844			0.0559431
$690$		0			0
$691$	−5.90112			−0.224489			−0.112245	−	0.993681i	$-0.535804\pi$
$691$	−5.90112			−0.224489			−0.112245	+	0.993681i	$0.535804\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5.07784			−0.192614
$696$		0			0
$697$	−16.5178			−0.625656
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.3782			0.467519			0.233759	−	0.972294i	$-0.424897\pi$
$701$	12.3782			0.467519			0.233759	+	0.972294i	$0.424897\pi$
$702$		0			0
$703$	17.1483	−	29.7018i	0.646761	−	1.12022i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−13.2829			−0.498850			−0.249425	−	0.968394i	$-0.580242\pi$
$709$	−13.2829			−0.498850			−0.249425	+	0.968394i	$0.580242\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.81522	+	3.14406i	−0.0679806	+	0.117746i
$714$		0			0
$715$	1.63664	+	2.83474i	0.0612067	+	0.106013i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12.1847	+	21.1045i	0.454413	+	0.787066i	0.998654	−	0.0518628i	$-0.0165158\pi$
$719$	12.1847	+	21.1045i	0.454413	+	0.787066i	−0.544242	+	0.838929i	$0.683183\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.52290			0.353672
$726$		0			0
$727$	−7.99450	+	13.8469i	−0.296500	+	0.513552i	−0.975333	−	0.220740i	$-0.929153\pi$
$727$	−7.99450	+	13.8469i	−0.296500	+	0.513552i	0.678833	+	0.734293i	$0.262486\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.54070	+	11.3288i	0.241917	+	0.419012i
$732$		0			0
$733$	−21.1414	+	36.6181i	−0.780877	+	1.35252i	0.150554	+	0.988602i	$0.451894\pi$
$733$	−21.1414	+	36.6181i	−0.780877	+	1.35252i	−0.931431	+	0.363917i	$0.881439\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.89307	+	17.1353i	0.364416	+	0.631187i
$738$		0			0
$739$	1.54325	−	2.67299i	0.0567695	−	0.0983276i	−0.836244	−	0.548357i	$-0.815253\pi$
$739$	1.54325	−	2.67299i	0.0567695	−	0.0983276i	0.893014	+	0.450030i	$0.148587\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.31522	+	5.74213i	0.121624	+	0.210658i	0.920408	−	0.390959i	$-0.127857\pi$
$743$	3.31522	+	5.74213i	0.121624	+	0.210658i	−0.798784	+	0.601617i	$0.794523\pi$
$744$		0			0
$745$	−6.88255	−	11.9209i	−0.252157	−	0.436749i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	21.3702	−	37.0142i	0.779808	−	1.35067i	−0.152243	−	0.988343i	$-0.548650\pi$
$751$	21.3702	−	37.0142i	0.779808	−	1.35067i	0.932052	−	0.362325i	$-0.118017\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.0655			0.548288
$756$		0			0
$757$	−31.0232			−1.12756			−0.563779	−	0.825926i	$-0.690653\pi$
$757$	−31.0232			−1.12756			−0.563779	+	0.825926i	$0.690653\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.8182	+	20.4697i	−0.428409	+	0.742025i	−0.996732	−	0.0807799i	$-0.974259\pi$
$761$	−11.8182	+	20.4697i	−0.428409	+	0.742025i	0.568323	+	0.822805i	$0.307592\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.55425	+	9.62025i	0.200553	+	0.347367i
$768$		0			0
$769$	−1.73422	−	3.00376i	−0.0625375	−	0.108318i	0.833061	−	0.553180i	$-0.186586\pi$
$769$	−1.73422	−	3.00376i	−0.0625375	−	0.108318i	−0.895599	+	0.444862i	$0.853253\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−17.2985	+	29.9619i	−0.622184	+	1.07765i	0.366894	+	0.930263i	$0.380421\pi$
$773$	−17.2985	+	29.9619i	−0.622184	+	1.07765i	−0.989078	+	0.147392i	$0.952912\pi$
$774$		0			0
$775$	−5.38255	−	9.32284i	−0.193347	−	0.334886i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14.6545	−	25.3824i	0.525053	−	0.909418i
$780$		0			0
$781$	12.7115	+	22.0170i	0.454854	+	0.787831i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.44870	−	12.9015i	0.265855	−	0.460475i
$786$		0			0
$787$	12.1593			0.433431			0.216715	−	0.976235i	$-0.430466\pi$
$787$	12.1593			0.433431			0.216715	+	0.976235i	$0.430466\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−5.56243	−	9.63442i	−0.197528	−	0.342128i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.89493	−	5.01416i	−0.102544	−	0.177611i	0.810188	−	0.586170i	$-0.199365\pi$
$797$	−2.89493	−	5.01416i	−0.102544	−	0.177611i	−0.912732	+	0.408559i	$0.866031\pi$
$798$		0			0
$799$	−18.2534	+	31.6158i	−0.645759	+	1.11849i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.3548			0.435993
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−24.5908	+	42.5926i	−0.864568	+	1.49748i	0.00290803	+	0.999996i	$0.499074\pi$
$809$	−24.5908	+	42.5926i	−0.864568	+	1.49748i	−0.867476	+	0.497479i	$0.834259\pi$
$810$		0			0
$811$	−40.7266			−1.43010			−0.715052	−	0.699072i	$-0.753597\pi$
$811$	−40.7266			−1.43010			−0.715052	+	0.699072i	$0.753597\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.37822			0.118334
$816$		0			0
$817$	−23.2115			−0.812069
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.0938			−0.526777			−0.263388	−	0.964690i	$-0.584840\pi$
$821$	−15.0938			−0.526777			−0.263388	+	0.964690i	$0.584840\pi$
$822$		0			0
$823$	16.0000			0.557725			0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000			0.557725			0.278862	+	0.960331i	$0.410043\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.2348			−1.22523			−0.612616	−	0.790381i	$-0.709883\pi$
$827$	−35.2348			−1.22523			−0.612616	+	0.790381i	$0.709883\pi$
$828$		0			0
$829$	−1.61745	+	2.80151i	−0.0561765	+	0.0973006i	−0.892746	−	0.450560i	$-0.851224\pi$
$829$	−1.61745	+	2.80151i	−0.0561765	+	0.0973006i	0.836570	+	0.547861i	$0.184558\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−4.45606			−0.154208
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15.5197	−	26.8808i	0.535798	−	0.928030i	−0.463326	−	0.886188i	$-0.653344\pi$
$839$	15.5197	−	26.8808i	0.535798	−	0.928030i	0.999124	−	0.0418419i	$-0.0133226\pi$
$840$		0			0
$841$	4.32760	+	7.49563i	0.149228	+	0.258470i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−10.5339	−	18.2453i	−0.362377	−	0.627656i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.89602			0.167833
$852$		0			0
$853$	−8.03637	+	13.9194i	−0.275160	+	0.476591i	−0.970176	−	0.242403i	$-0.922064\pi$
$853$	−8.03637	+	13.9194i	−0.275160	+	0.476591i	0.695015	+	0.718995i	$0.255398\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.61058	−	16.6460i	−0.328291	−	0.568617i	0.653882	−	0.756597i	$-0.273139\pi$
$857$	−9.61058	−	16.6460i	−0.328291	−	0.568617i	−0.982173	+	0.187980i	$0.939806\pi$
$858$		0			0
$859$	−7.40112	+	12.8191i	−0.252523	+	0.437382i	−0.964220	−	0.265104i	$-0.914594\pi$
$859$	−7.40112	+	12.8191i	−0.252523	+	0.437382i	0.711697	+	0.702487i	$0.247927\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	7.38441	+	12.7902i	0.251368	+	0.435382i	0.963903	−	0.266255i	$-0.0857862\pi$
$863$	7.38441	+	12.7902i	0.251368	+	0.435382i	−0.712535	+	0.701637i	$0.752453\pi$
$864$		0			0
$865$	−4.44437	+	7.69787i	−0.151113	+	0.261735i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−11.4072	−	19.7579i	−0.386964	−	0.670241i
$870$		0			0
$871$	3.10507	+	5.37815i	0.105211	+	0.182232i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−26.1916	+	45.3651i	−0.884427	+	1.53187i	−0.0380575	+	0.999276i	$0.512117\pi$
$877$	−26.1916	+	45.3651i	−0.884427	+	1.53187i	−0.846369	+	0.532597i	$0.821216\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−31.3214			−1.05525			−0.527623	−	0.849479i	$-0.676916\pi$
$881$	−31.3214			−1.05525			−0.527623	+	0.849479i	$0.676916\pi$
$882$		0			0
$883$	−43.0494			−1.44873			−0.724363	−	0.689419i	$-0.757866\pi$
$883$	−43.0494			−1.44873			−0.724363	+	0.689419i	$0.757866\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7.48831	−	12.9701i	0.251433	−	0.435494i	−0.712488	−	0.701685i	$-0.752432\pi$
$887$	7.48831	−	12.9701i	0.251433	−	0.435494i	0.963921	+	0.266190i	$0.0857649\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−32.3887	−	56.0988i	−1.08385	−	1.87727i
$894$		0			0
$895$	4.04944	+	7.01384i	0.135358	+	0.234447i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.4993	+	19.9174i	−0.383524	+	0.664282i
$900$		0			0
$901$	2.65452	+	4.59776i	0.0884348	+	0.153174i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8.86467	−	15.3541i	0.294671	−	0.510386i
$906$		0			0
$907$	−15.2280	−	26.3756i	−0.505636	−	0.875787i	−0.999979	−	0.00652002i	$-0.997925\pi$
$907$	−15.2280	−	26.3756i	−0.505636	−	0.875787i	0.494343	−	0.869267i	$-0.335409\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−9.97593	+	17.2788i	−0.330517	+	0.572473i	−0.982613	−	0.185664i	$-0.940557\pi$
$911$	−9.97593	+	17.2788i	−0.330517	+	0.572473i	0.652096	+	0.758136i	$0.273890\pi$
$912$		0			0
$913$	−21.8255			−0.722317
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−22.8145	−	39.5159i	−0.752582	−	1.30351i	−0.946567	−	0.322506i	$-0.895475\pi$
$919$	−22.8145	−	39.5159i	−0.752582	−	1.30351i	0.193985	−	0.981004i	$-0.437859\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3.98969	+	6.91034i	0.131322	+	0.227457i
$924$		0			0
$925$	−7.25890	+	12.5728i	−0.238671	+	0.413391i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	56.3722			1.84951			0.924755	−	0.380562i	$-0.124270\pi$
$929$	56.3722			1.84951			0.924755	+	0.380562i	$0.124270\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.91714	+	10.2488i	−0.193511	+	0.335171i
$936$		0			0
$937$	36.8530			1.20393			0.601967	−	0.798521i	$-0.294384\pi$
$937$	36.8530			1.20393			0.601967	+	0.798521i	$0.294384\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.76000			−0.285568			−0.142784	−	0.989754i	$-0.545605\pi$
$941$	−8.76000			−0.285568			−0.142784	+	0.989754i	$0.545605\pi$
$942$		0			0
$943$	4.18401			0.136250
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−26.6452			−0.865852			−0.432926	−	0.901430i	$-0.642519\pi$
$947$	−26.6452			−0.865852			−0.432926	+	0.901430i	$0.642519\pi$
$948$		0			0
$949$	3.87773			0.125877
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−24.3039			−0.787282			−0.393641	−	0.919264i	$-0.628785\pi$
$953$	−24.3039			−0.787282			−0.393641	+	0.919264i	$0.628785\pi$
$954$		0			0
$955$	−11.3207	+	19.6081i	−0.366330	+	0.634502i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−5.00138			−0.161335
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12.4425	+	21.5511i	−0.400539	+	0.693753i
$966$		0			0
$967$	5.22872	+	9.05641i	0.168144	+	0.291234i	0.937767	−	0.347264i	$-0.112889\pi$
$967$	5.22872	+	9.05641i	0.168144	+	0.291234i	−0.769623	+	0.638498i	$0.779556\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20.8578	+	36.1267i	0.669358	+	1.15936i	0.978084	+	0.208211i	$0.0667642\pi$
$971$	20.8578	+	36.1267i	0.669358	+	1.15936i	−0.308726	+	0.951151i	$0.599902\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.89011			−0.188441			−0.0942207	−	0.995551i	$-0.530036\pi$
$977$	−5.89011			−0.188441			−0.0942207	+	0.995551i	$0.530036\pi$
$978$		0			0
$979$	11.9567	−	20.7097i	0.382139	−	0.661885i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−20.9196	−	36.2338i	−0.667232	−	1.15568i	−0.978675	−	0.205415i	$-0.934146\pi$
$983$	−20.9196	−	36.2338i	−0.667232	−	1.15568i	0.311443	−	0.950265i	$-0.399188\pi$
$984$		0			0
$985$	15.7095	−	27.2096i	0.500545	−	0.866969i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.65678	−	2.86963i	−0.0526826	−	0.0912489i
$990$		0			0
$991$	27.3578	−	47.3851i	0.869049	−	1.50524i	0.00607865	−	0.999982i	$-0.498065\pi$
$991$	27.3578	−	47.3851i	0.869049	−	1.50524i	0.862970	−	0.505255i	$-0.168602\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.0698	−	34.7619i	−0.636255	−	1.10203i
$996$		0			0
$997$	−9.02476	−	15.6313i	−0.285817	−	0.495050i	0.686990	−	0.726667i	$-0.258932\pi$
$997$	−9.02476	−	15.6313i	−0.285817	−	0.495050i	−0.972807	+	0.231617i	$0.925598\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.i.e.2125.3		6
3.2	odd	2		1764.2.i.d.1537.3		6
7.2	even	3		5292.2.l.f.3313.1		6
7.3	odd	6		756.2.j.b.505.1		6
7.4	even	3		5292.2.j.d.3529.3		6
7.5	odd	6		5292.2.l.e.3313.3		6
7.6	odd	2		5292.2.i.f.2125.1		6
9.4	even	3		5292.2.l.f.361.1		6
9.5	odd	6		1764.2.l.f.949.3		6
21.2	odd	6		1764.2.l.f.961.3		6
21.5	even	6		1764.2.l.e.961.1		6
21.11	odd	6		1764.2.j.e.1177.1		6
21.17	even	6		252.2.j.a.169.3	yes	6
21.20	even	2		1764.2.i.g.1537.1		6
28.3	even	6		3024.2.r.j.2017.1		6
63.4	even	3		5292.2.j.d.1765.3		6
63.5	even	6		1764.2.i.g.373.1		6
63.13	odd	6		5292.2.l.e.361.3		6
63.23	odd	6		1764.2.i.d.373.3		6
63.31	odd	6		756.2.j.b.253.1		6
63.32	odd	6		1764.2.j.e.589.1		6
63.38	even	6		2268.2.a.i.1.1		3
63.40	odd	6		5292.2.i.f.1549.1		6
63.41	even	6		1764.2.l.e.949.1		6
63.52	odd	6		2268.2.a.h.1.3		3
63.58	even	3	inner	5292.2.i.e.1549.3		6
63.59	even	6		252.2.j.a.85.3	✓	6
84.59	odd	6		1008.2.r.j.673.1		6
252.31	even	6		3024.2.r.j.1009.1		6
252.59	odd	6		1008.2.r.j.337.1		6
252.115	even	6		9072.2.a.bv.1.3		3
252.227	odd	6		9072.2.a.by.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.a.85.3	✓	6	63.59	even	6
252.2.j.a.169.3	yes	6	21.17	even	6
756.2.j.b.253.1		6	63.31	odd	6
756.2.j.b.505.1		6	7.3	odd	6
1008.2.r.j.337.1		6	252.59	odd	6
1008.2.r.j.673.1		6	84.59	odd	6
1764.2.i.d.373.3		6	63.23	odd	6
1764.2.i.d.1537.3		6	3.2	odd	2
1764.2.i.g.373.1		6	63.5	even	6
1764.2.i.g.1537.1		6	21.20	even	2
1764.2.j.e.589.1		6	63.32	odd	6
1764.2.j.e.1177.1		6	21.11	odd	6
1764.2.l.e.949.1		6	63.41	even	6
1764.2.l.e.961.1		6	21.5	even	6
1764.2.l.f.949.3		6	9.5	odd	6
1764.2.l.f.961.3		6	21.2	odd	6
2268.2.a.h.1.3		3	63.52	odd	6
2268.2.a.i.1.1		3	63.38	even	6
3024.2.r.j.1009.1		6	252.31	even	6
3024.2.r.j.2017.1		6	28.3	even	6
5292.2.i.e.1549.3		6	63.58	even	3	inner
5292.2.i.e.2125.3		6	1.1	even	1	trivial
5292.2.i.f.1549.1		6	63.40	odd	6
5292.2.i.f.2125.1		6	7.6	odd	2
5292.2.j.d.1765.3		6	63.4	even	3
5292.2.j.d.3529.3		6	7.4	even	3
5292.2.l.e.361.3		6	63.13	odd	6
5292.2.l.e.3313.3		6	7.5	odd	6
5292.2.l.f.361.1		6	9.4	even	3
5292.2.l.f.3313.1		6	7.2	even	3
9072.2.a.bv.1.3		3	252.115	even	6
9072.2.a.by.1.1		3	252.227	odd	6

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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	5292.2.i.e.2125.3

Level	$5292$
Weight	$2$
Character	5292.2125
Analytic conductor	$42.257$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$