LMFDB - Embedded newform 5265.2.a.be.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5265,2,Mod(1,5265)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5265, 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("5265.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5265 = 3^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5265.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:	$42.0412366642$
Analytic rank:	$1$
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} - 10x^{6} + 7x^{5} + 33x^{4} - 14x^{3} - 38x^{2} + 7x + 9 $ x^8 - x^7 - 10x^6 + 7x^5 + 33x^4 - 14x^3 - 38x^2 + 7x + 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.97477$ of defining polynomial
Character	$\chi$	$=$	5265.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.97477 q^{2} +1.89973 q^{4} -1.00000 q^{5} -0.248978 q^{7} -0.198009 q^{8} +O(q^{10})$	\(q+1.97477 q^{2} +1.89973 q^{4} -1.00000 q^{5} -0.248978 q^{7} -0.198009 q^{8} -1.97477 q^{10} -0.193531 q^{11} +1.00000 q^{13} -0.491676 q^{14} -4.19048 q^{16} +0.423209 q^{17} -3.02339 q^{19} -1.89973 q^{20} -0.382179 q^{22} +3.73748 q^{23} +1.00000 q^{25} +1.97477 q^{26} -0.472992 q^{28} -5.02168 q^{29} +4.87728 q^{31} -7.87924 q^{32} +0.835743 q^{34} +0.248978 q^{35} -6.39914 q^{37} -5.97050 q^{38} +0.198009 q^{40} +1.46705 q^{41} +0.971363 q^{43} -0.367656 q^{44} +7.38068 q^{46} -8.27914 q^{47} -6.93801 q^{49} +1.97477 q^{50} +1.89973 q^{52} -5.91550 q^{53} +0.193531 q^{55} +0.0493000 q^{56} -9.91668 q^{58} +10.6045 q^{59} -12.1404 q^{61} +9.63151 q^{62} -7.17875 q^{64} -1.00000 q^{65} -6.74152 q^{67} +0.803984 q^{68} +0.491676 q^{70} -2.47498 q^{71} -11.5025 q^{73} -12.6369 q^{74} -5.74362 q^{76} +0.0481849 q^{77} -4.07088 q^{79} +4.19048 q^{80} +2.89709 q^{82} -5.59844 q^{83} -0.423209 q^{85} +1.91822 q^{86} +0.0383208 q^{88} +4.19813 q^{89} -0.248978 q^{91} +7.10021 q^{92} -16.3494 q^{94} +3.02339 q^{95} -3.50013 q^{97} -13.7010 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 5 q^{4} - 8 q^{5} - 6 q^{7} + 6 q^{8}+O(q^{10}) $ 8 * q + q^2 + 5 * q^4 - 8 * q^5 - 6 * q^7 + 6 * q^8	$ 8 q + q^{2} + 5 q^{4} - 8 q^{5} - 6 q^{7} + 6 q^{8} - q^{10} + 9 q^{11} + 8 q^{13} - 3 q^{14} - 13 q^{16} - 6 q^{17} - 11 q^{19} - 5 q^{20} - 4 q^{22} + 3 q^{23} + 8 q^{25} + q^{26} - 13 q^{28} + 8 q^{29} - 18 q^{31} + 3 q^{32} - 9 q^{34} + 6 q^{35} - 18 q^{37} - 8 q^{38} - 6 q^{40} - 17 q^{41} - 17 q^{43} - 5 q^{44} + 3 q^{46} + 11 q^{47} - 16 q^{49} + q^{50} + 5 q^{52} - 10 q^{53} - 9 q^{55} - q^{56} - 10 q^{58} + 7 q^{59} - 21 q^{61} + 29 q^{62} - 10 q^{64} - 8 q^{65} - 13 q^{67} - 16 q^{68} + 3 q^{70} + 34 q^{71} - 16 q^{73} - 4 q^{74} - 2 q^{76} + 18 q^{77} - 37 q^{79} + 13 q^{80} + q^{82} + 3 q^{83} + 6 q^{85} - 2 q^{86} - 19 q^{88} - 14 q^{89} - 6 q^{91} - 14 q^{92} - 44 q^{94} + 11 q^{95} - 17 q^{97} - 45 q^{98}+O(q^{100}) $ 8 * q + q^2 + 5 * q^4 - 8 * q^5 - 6 * q^7 + 6 * q^8 - q^10 + 9 * q^11 + 8 * q^13 - 3 * q^14 - 13 * q^16 - 6 * q^17 - 11 * q^19 - 5 * q^20 - 4 * q^22 + 3 * q^23 + 8 * q^25 + q^26 - 13 * q^28 + 8 * q^29 - 18 * q^31 + 3 * q^32 - 9 * q^34 + 6 * q^35 - 18 * q^37 - 8 * q^38 - 6 * q^40 - 17 * q^41 - 17 * q^43 - 5 * q^44 + 3 * q^46 + 11 * q^47 - 16 * q^49 + q^50 + 5 * q^52 - 10 * q^53 - 9 * q^55 - q^56 - 10 * q^58 + 7 * q^59 - 21 * q^61 + 29 * q^62 - 10 * q^64 - 8 * q^65 - 13 * q^67 - 16 * q^68 + 3 * q^70 + 34 * q^71 - 16 * q^73 - 4 * q^74 - 2 * q^76 + 18 * q^77 - 37 * q^79 + 13 * q^80 + q^82 + 3 * q^83 + 6 * q^85 - 2 * q^86 - 19 * q^88 - 14 * q^89 - 6 * q^91 - 14 * q^92 - 44 * q^94 + 11 * q^95 - 17 * q^97 - 45 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.97477	1.39638	0.698188	−	0.715915i	$-0.253990\pi$
$2$	1.97477	1.39638	0.698188	+	0.715915i	$0.253990\pi$
$3$	0	0
$3$	0	0
$4$	1.89973	0.949865
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.248978	−0.0941050	−0.0470525	−	0.998892i	$-0.514983\pi$
$7$	−0.248978	−0.0941050	−0.0470525	+	0.998892i	$0.514983\pi$
$8$	−0.198009	−0.0700068
$9$	0	0
$10$	−1.97477	−0.624478
$11$	−0.193531	−0.0583516	−0.0291758	−	0.999574i	$-0.509288\pi$
$11$	−0.193531	−0.0583516	−0.0291758	+	0.999574i	$0.509288\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.491676	−0.131406
$15$	0	0
$16$	−4.19048	−1.04762
$17$	0.423209	0.102643	0.0513217	−	0.998682i	$-0.483657\pi$
$17$	0.423209	0.102643	0.0513217	+	0.998682i	$0.483657\pi$
$18$	0	0
$19$	−3.02339	−0.693612	−0.346806	−	0.937937i	$-0.612734\pi$
$19$	−3.02339	−0.693612	−0.346806	+	0.937937i	$0.612734\pi$
$20$	−1.89973	−0.424793
$21$	0	0
$22$	−0.382179	−0.0814808
$23$	3.73748	0.779319	0.389659	−	0.920959i	$-0.372593\pi$
$23$	3.73748	0.779319	0.389659	+	0.920959i	$0.372593\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.97477	0.387285
$27$	0	0
$28$	−0.472992	−0.0893871
$29$	−5.02168	−0.932502	−0.466251	−	0.884652i	$-0.654396\pi$
$29$	−5.02168	−0.932502	−0.466251	+	0.884652i	$0.654396\pi$
$30$	0	0
$31$	4.87728	0.875985	0.437992	−	0.898979i	$-0.355690\pi$
$31$	4.87728	0.875985	0.437992	+	0.898979i	$0.355690\pi$
$32$	−7.87924	−1.39287
$33$	0	0
$34$	0.835743	0.143329
$35$	0.248978	0.0420850
$36$	0	0
$37$	−6.39914	−1.05201	−0.526006	−	0.850481i	$-0.676311\pi$
$37$	−6.39914	−1.05201	−0.526006	+	0.850481i	$0.676311\pi$
$38$	−5.97050	−0.968543
$39$	0	0
$40$	0.198009	0.0313080
$41$	1.46705	0.229115	0.114557	−	0.993417i	$-0.463455\pi$
$41$	1.46705	0.229115	0.114557	+	0.993417i	$0.463455\pi$
$42$	0	0
$43$	0.971363	0.148131	0.0740657	−	0.997253i	$-0.476403\pi$
$43$	0.971363	0.148131	0.0740657	+	0.997253i	$0.476403\pi$
$44$	−0.367656	−0.0554262
$45$	0	0
$46$	7.38068	1.08822
$47$	−8.27914	−1.20764	−0.603818	−	0.797122i	$-0.706355\pi$
$47$	−8.27914	−1.20764	−0.603818	+	0.797122i	$0.706355\pi$
$48$	0	0
$49$	−6.93801	−0.991144
$50$	1.97477	0.279275
$51$	0	0
$52$	1.89973	0.263445
$53$	−5.91550	−0.812557	−0.406278	−	0.913749i	$-0.633174\pi$
$53$	−5.91550	−0.812557	−0.406278	+	0.913749i	$0.633174\pi$
$54$	0	0
$55$	0.193531	0.0260956
$56$	0.0493000	0.00658799
$57$	0	0
$58$	−9.91668	−1.30212
$59$	10.6045	1.38058	0.690292	−	0.723531i	$-0.257482\pi$
$59$	10.6045	1.38058	0.690292	+	0.723531i	$0.257482\pi$
$60$	0	0
$61$	−12.1404	−1.55442	−0.777208	−	0.629244i	$-0.783365\pi$
$61$	−12.1404	−1.55442	−0.777208	+	0.629244i	$0.783365\pi$
$62$	9.63151	1.22320
$63$	0	0
$64$	−7.17875	−0.897343
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−6.74152	−0.823608	−0.411804	−	0.911272i	$-0.635101\pi$
$67$	−6.74152	−0.823608	−0.411804	+	0.911272i	$0.635101\pi$
$68$	0.803984	0.0974974
$69$	0	0
$70$	0.491676	0.0587665
$71$	−2.47498	−0.293725	−0.146863	−	0.989157i	$-0.546918\pi$
$71$	−2.47498	−0.293725	−0.146863	+	0.989157i	$0.546918\pi$
$72$	0	0
$73$	−11.5025	−1.34626	−0.673131	−	0.739523i	$-0.735051\pi$
$73$	−11.5025	−1.34626	−0.673131	+	0.739523i	$0.735051\pi$
$74$	−12.6369	−1.46900
$75$	0	0
$76$	−5.74362	−0.658838
$77$	0.0481849	0.00549118
$78$	0	0
$79$	−4.07088	−0.458010	−0.229005	−	0.973425i	$-0.573547\pi$
$79$	−4.07088	−0.458010	−0.229005	+	0.973425i	$0.573547\pi$
$80$	4.19048	0.468510
$81$	0	0
$82$	2.89709	0.319931
$83$	−5.59844	−0.614509	−0.307254	−	0.951627i	$-0.599410\pi$
$83$	−5.59844	−0.614509	−0.307254	+	0.951627i	$0.599410\pi$
$84$	0	0
$85$	−0.423209	−0.0459035
$86$	1.91822	0.206847
$87$	0	0
$88$	0.0383208	0.00408501
$89$	4.19813	0.445001	0.222501	−	0.974933i	$-0.428578\pi$
$89$	4.19813	0.445001	0.222501	+	0.974933i	$0.428578\pi$
$90$	0	0
$91$	−0.248978	−0.0261000
$92$	7.10021	0.740248
$93$	0	0
$94$	−16.3494	−1.68631
$95$	3.02339	0.310193
$96$	0	0
$97$	−3.50013	−0.355384	−0.177692	−	0.984086i	$-0.556863\pi$
$97$	−3.50013	−0.355384	−0.177692	+	0.984086i	$0.556863\pi$
$98$	−13.7010	−1.38401
$99$	0	0
$100$	1.89973	0.189973
$101$	−17.7896	−1.77013	−0.885067	−	0.465463i	$-0.845888\pi$
$101$	−17.7896	−1.77013	−0.885067	+	0.465463i	$0.845888\pi$
$102$	0	0
$103$	10.4538	1.03004	0.515022	−	0.857177i	$-0.327784\pi$
$103$	10.4538	1.03004	0.515022	+	0.857177i	$0.327784\pi$
$104$	−0.198009	−0.0194164
$105$	0	0
$106$	−11.6818	−1.13463
$107$	5.89359	0.569754	0.284877	−	0.958564i	$-0.408047\pi$
$107$	5.89359	0.569754	0.284877	+	0.958564i	$0.408047\pi$
$108$	0	0
$109$	−0.452700	−0.0433608	−0.0216804	−	0.999765i	$-0.506902\pi$
$109$	−0.452700	−0.0433608	−0.0216804	+	0.999765i	$0.506902\pi$
$110$	0.382179	0.0364393
$111$	0	0
$112$	1.04334	0.0985864
$113$	−4.31274	−0.405708	−0.202854	−	0.979209i	$-0.565022\pi$
$113$	−4.31274	−0.405708	−0.202854	+	0.979209i	$0.565022\pi$
$114$	0	0
$115$	−3.73748	−0.348522
$116$	−9.53984	−0.885752
$117$	0	0
$118$	20.9414	1.92781
$119$	−0.105370	−0.00965925
$120$	0	0
$121$	−10.9625	−0.996595
$122$	−23.9745	−2.17055
$123$	0	0
$124$	9.26551	0.832067
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.56358	−0.138745	−0.0693727	−	0.997591i	$-0.522100\pi$
$127$	−1.56358	−0.138745	−0.0693727	+	0.997591i	$0.522100\pi$
$128$	1.58208	0.139838
$129$	0	0
$130$	−1.97477	−0.173199
$131$	7.34973	0.642149	0.321074	−	0.947054i	$-0.395956\pi$
$131$	7.34973	0.642149	0.321074	+	0.947054i	$0.395956\pi$
$132$	0	0
$133$	0.752758	0.0652724
$134$	−13.3130	−1.15007
$135$	0	0
$136$	−0.0837993	−0.00718573
$137$	9.40143	0.803219	0.401609	−	0.915811i	$-0.368451\pi$
$137$	9.40143	0.803219	0.401609	+	0.915811i	$0.368451\pi$
$138$	0	0
$139$	0.982710	0.0833524	0.0416762	−	0.999131i	$-0.486730\pi$
$139$	0.982710	0.0833524	0.0416762	+	0.999131i	$0.486730\pi$
$140$	0.472992	0.0399751
$141$	0	0
$142$	−4.88752	−0.410151
$143$	−0.193531	−0.0161838
$144$	0	0
$145$	5.02168	0.417028
$146$	−22.7148	−1.87989
$147$	0	0
$148$	−12.1566	−0.999270
$149$	−3.16327	−0.259145	−0.129572	−	0.991570i	$-0.541360\pi$
$149$	−3.16327	−0.259145	−0.129572	+	0.991570i	$0.541360\pi$
$150$	0	0
$151$	−20.5979	−1.67623	−0.838115	−	0.545494i	$-0.816342\pi$
$151$	−20.5979	−1.67623	−0.838115	+	0.545494i	$0.816342\pi$
$152$	0.598658	0.0485576
$153$	0	0
$154$	0.0951543	0.00766775
$155$	−4.87728	−0.391752
$156$	0	0
$157$	−5.41330	−0.432028	−0.216014	−	0.976390i	$-0.569306\pi$
$157$	−5.41330	−0.432028	−0.216014	+	0.976390i	$0.569306\pi$
$158$	−8.03907	−0.639554
$159$	0	0
$160$	7.87924	0.622909
$161$	−0.930552	−0.0733378
$162$	0	0
$163$	6.27893	0.491804	0.245902	−	0.969295i	$-0.420916\pi$
$163$	6.27893	0.491804	0.245902	+	0.969295i	$0.420916\pi$
$164$	2.78700	0.217628
$165$	0	0
$166$	−11.0557	−0.858085
$167$	−15.6217	−1.20884	−0.604421	−	0.796665i	$-0.706596\pi$
$167$	−15.6217	−1.20884	−0.604421	+	0.796665i	$0.706596\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−0.835743	−0.0640985
$171$	0	0
$172$	1.84533	0.140705
$173$	14.9941	1.13998	0.569991	−	0.821651i	$-0.306947\pi$
$173$	14.9941	1.13998	0.569991	+	0.821651i	$0.306947\pi$
$174$	0	0
$175$	−0.248978	−0.0188210
$176$	0.810987	0.0611304
$177$	0	0
$178$	8.29036	0.621389
$179$	19.9121	1.48830	0.744150	−	0.668013i	$-0.232855\pi$
$179$	19.9121	1.48830	0.744150	+	0.668013i	$0.232855\pi$
$180$	0	0
$181$	15.4291	1.14683	0.573416	−	0.819264i	$-0.305618\pi$
$181$	15.4291	1.14683	0.573416	+	0.819264i	$0.305618\pi$
$182$	−0.491676	−0.0364454
$183$	0	0
$184$	−0.740055	−0.0545576
$185$	6.39914	0.470474
$186$	0	0
$187$	−0.0819039	−0.00598941
$188$	−15.7281	−1.14709
$189$	0	0
$190$	5.97050	0.433146
$191$	−4.44112	−0.321348	−0.160674	−	0.987007i	$-0.551367\pi$
$191$	−4.44112	−0.321348	−0.160674	+	0.987007i	$0.551367\pi$
$192$	0	0
$193$	22.8917	1.64778	0.823889	−	0.566752i	$-0.191800\pi$
$193$	22.8917	1.64778	0.823889	+	0.566752i	$0.191800\pi$
$194$	−6.91196	−0.496250
$195$	0	0
$196$	−13.1804	−0.941454
$197$	−5.87506	−0.418581	−0.209290	−	0.977854i	$-0.567115\pi$
$197$	−5.87506	−0.418581	−0.209290	+	0.977854i	$0.567115\pi$
$198$	0	0
$199$	−15.0439	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$199$	−15.0439	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$200$	−0.198009	−0.0140014
$201$	0	0
$202$	−35.1305	−2.47177
$203$	1.25029	0.0877531
$204$	0	0
$205$	−1.46705	−0.102463
$206$	20.6439	1.43833
$207$	0	0
$208$	−4.19048	−0.290558
$209$	0.585117	0.0404734
$210$	0	0
$211$	−19.8897	−1.36927	−0.684633	−	0.728888i	$-0.740037\pi$
$211$	−19.8897	−1.36927	−0.684633	+	0.728888i	$0.740037\pi$
$212$	−11.2379	−0.771819
$213$	0	0
$214$	11.6385	0.795591
$215$	−0.971363	−0.0662464
$216$	0	0
$217$	−1.21434	−0.0824345
$218$	−0.893980	−0.0605480
$219$	0	0
$220$	0.367656	0.0247874
$221$	0.423209	0.0284681
$222$	0	0
$223$	6.93534	0.464424	0.232212	−	0.972665i	$-0.425404\pi$
$223$	6.93534	0.464424	0.232212	+	0.972665i	$0.425404\pi$
$224$	1.96176	0.131076
$225$	0	0
$226$	−8.51668	−0.566521
$227$	22.4237	1.48831	0.744155	−	0.668007i	$-0.232852\pi$
$227$	22.4237	1.48831	0.744155	+	0.668007i	$0.232852\pi$
$228$	0	0
$229$	27.1729	1.79564	0.897819	−	0.440364i	$-0.145151\pi$
$229$	27.1729	1.79564	0.897819	+	0.440364i	$0.145151\pi$
$230$	−7.38068	−0.486668
$231$	0	0
$232$	0.994338	0.0652815
$233$	−0.882364	−0.0578056	−0.0289028	−	0.999582i	$-0.509201\pi$
$233$	−0.882364	−0.0578056	−0.0289028	+	0.999582i	$0.509201\pi$
$234$	0	0
$235$	8.27914	0.540071
$236$	20.1456	1.31137
$237$	0	0
$238$	−0.208082	−0.0134879
$239$	6.41292	0.414817	0.207409	−	0.978254i	$-0.433497\pi$
$239$	6.41292	0.414817	0.207409	+	0.978254i	$0.433497\pi$
$240$	0	0
$241$	4.75278	0.306153	0.153077	−	0.988214i	$-0.451082\pi$
$241$	4.75278	0.306153	0.153077	+	0.988214i	$0.451082\pi$
$242$	−21.6485	−1.39162
$243$	0	0
$244$	−23.0634	−1.47649
$245$	6.93801	0.443253
$246$	0	0
$247$	−3.02339	−0.192373
$248$	−0.965745	−0.0613249
$249$	0	0
$250$	−1.97477	−0.124896
$251$	−22.6706	−1.43096	−0.715478	−	0.698636i	$-0.753791\pi$
$251$	−22.6706	−1.43096	−0.715478	+	0.698636i	$0.753791\pi$
$252$	0	0
$253$	−0.723317	−0.0454745
$254$	−3.08772	−0.193741
$255$	0	0
$256$	17.4817	1.09261
$257$	−17.5838	−1.09685	−0.548423	−	0.836201i	$-0.684772\pi$
$257$	−17.5838	−1.09685	−0.548423	+	0.836201i	$0.684772\pi$
$258$	0	0
$259$	1.59325	0.0989996
$260$	−1.89973	−0.117816
$261$	0	0
$262$	14.5140	0.896681
$263$	−8.27123	−0.510026	−0.255013	−	0.966938i	$-0.582080\pi$
$263$	−8.27123	−0.510026	−0.255013	+	0.966938i	$0.582080\pi$
$264$	0	0
$265$	5.91550	0.363386
$266$	1.48653	0.0911448
$267$	0	0
$268$	−12.8071	−0.782317
$269$	29.6933	1.81043	0.905217	−	0.424950i	$-0.139708\pi$
$269$	29.6933	1.81043	0.905217	+	0.424950i	$0.139708\pi$
$270$	0	0
$271$	−21.3158	−1.29484	−0.647420	−	0.762133i	$-0.724152\pi$
$271$	−21.3158	−1.29484	−0.647420	+	0.762133i	$0.724152\pi$
$272$	−1.77345	−0.107531
$273$	0	0
$274$	18.5657	1.12160
$275$	−0.193531	−0.0116703
$276$	0	0
$277$	12.5569	0.754468	0.377234	−	0.926118i	$-0.376875\pi$
$277$	12.5569	0.754468	0.377234	+	0.926118i	$0.376875\pi$
$278$	1.94063	0.116391
$279$	0	0
$280$	−0.0493000	−0.00294624
$281$	8.51685	0.508072	0.254036	−	0.967195i	$-0.418242\pi$
$281$	8.51685	0.508072	0.254036	+	0.967195i	$0.418242\pi$
$282$	0	0
$283$	13.9790	0.830965	0.415483	−	0.909601i	$-0.363613\pi$
$283$	13.9790	0.830965	0.415483	+	0.909601i	$0.363613\pi$
$284$	−4.70179	−0.279000
$285$	0	0
$286$	−0.382179	−0.0225987
$287$	−0.365264	−0.0215609
$288$	0	0
$289$	−16.8209	−0.989464
$290$	9.91668	0.582327
$291$	0	0
$292$	−21.8516	−1.27877
$293$	−21.6411	−1.26429	−0.632144	−	0.774851i	$-0.717825\pi$
$293$	−21.6411	−1.26429	−0.632144	+	0.774851i	$0.717825\pi$
$294$	0	0
$295$	−10.6045	−0.617416
$296$	1.26709	0.0736480
$297$	0	0
$298$	−6.24673	−0.361864
$299$	3.73748	0.216144
$300$	0	0
$301$	−0.241848	−0.0139399
$302$	−40.6761	−2.34065
$303$	0	0
$304$	12.6695	0.726643
$305$	12.1404	0.695156
$306$	0	0
$307$	−13.9106	−0.793922	−0.396961	−	0.917835i	$-0.629935\pi$
$307$	−13.9106	−0.793922	−0.396961	+	0.917835i	$0.629935\pi$
$308$	0.0915384	0.00521588
$309$	0	0
$310$	−9.63151	−0.547033
$311$	15.3473	0.870263	0.435132	−	0.900367i	$-0.356702\pi$
$311$	15.3473	0.870263	0.435132	+	0.900367i	$0.356702\pi$
$312$	0	0
$313$	−25.3354	−1.43204	−0.716021	−	0.698079i	$-0.754038\pi$
$313$	−25.3354	−1.43204	−0.716021	+	0.698079i	$0.754038\pi$
$314$	−10.6900	−0.603274
$315$	0	0
$316$	−7.73358	−0.435048
$317$	−2.75353	−0.154654	−0.0773269	−	0.997006i	$-0.524638\pi$
$317$	−2.75353	−0.154654	−0.0773269	+	0.997006i	$0.524638\pi$
$318$	0	0
$319$	0.971848	0.0544130
$320$	7.17875	0.401304
$321$	0	0
$322$	−1.83763	−0.102407
$323$	−1.27953	−0.0711947
$324$	0	0
$325$	1.00000	0.0554700
$326$	12.3995	0.686743
$327$	0	0
$328$	−0.290490	−0.0160396
$329$	2.06133	0.113645
$330$	0	0
$331$	−20.5533	−1.12971	−0.564855	−	0.825190i	$-0.691068\pi$
$331$	−20.5533	−1.12971	−0.564855	+	0.825190i	$0.691068\pi$
$332$	−10.6355	−0.583700
$333$	0	0
$334$	−30.8493	−1.68800
$335$	6.74152	0.368329
$336$	0	0
$337$	−0.0793257	−0.00432115	−0.00216057	−	0.999998i	$-0.500688\pi$
$337$	−0.0793257	−0.00432115	−0.00216057	+	0.999998i	$0.500688\pi$
$338$	1.97477	0.107414
$339$	0	0
$340$	−0.803984	−0.0436022
$341$	−0.943902	−0.0511151
$342$	0	0
$343$	3.47026	0.187377
$344$	−0.192339	−0.0103702
$345$	0	0
$346$	29.6100	1.59184
$347$	33.4548	1.79595	0.897973	−	0.440050i	$-0.145039\pi$
$347$	33.4548	1.79595	0.897973	+	0.440050i	$0.145039\pi$
$348$	0	0
$349$	18.1521	0.971661	0.485831	−	0.874053i	$-0.338517\pi$
$349$	18.1521	0.971661	0.485831	+	0.874053i	$0.338517\pi$
$350$	−0.491676	−0.0262812
$351$	0	0
$352$	1.52487	0.0812760
$353$	−12.6540	−0.673507	−0.336753	−	0.941593i	$-0.609329\pi$
$353$	−12.6540	−0.673507	−0.336753	+	0.941593i	$0.609329\pi$
$354$	0	0
$355$	2.47498	0.131358
$356$	7.97532	0.422691
$357$	0	0
$358$	39.3219	2.07823
$359$	2.22297	0.117324	0.0586620	−	0.998278i	$-0.481317\pi$
$359$	2.22297	0.117324	0.0586620	+	0.998278i	$0.481317\pi$
$360$	0	0
$361$	−9.85914	−0.518902
$362$	30.4689	1.60141
$363$	0	0
$364$	−0.472992	−0.0247915
$365$	11.5025	0.602067
$366$	0	0
$367$	−28.1324	−1.46850	−0.734249	−	0.678880i	$-0.762466\pi$
$367$	−28.1324	−1.46850	−0.734249	+	0.678880i	$0.762466\pi$
$368$	−15.6619	−0.816431
$369$	0	0
$370$	12.6369	0.656959
$371$	1.47283	0.0764656
$372$	0	0
$373$	28.9954	1.50132	0.750661	−	0.660687i	$-0.229735\pi$
$373$	28.9954	1.50132	0.750661	+	0.660687i	$0.229735\pi$
$374$	−0.161742	−0.00836347
$375$	0	0
$376$	1.63934	0.0845428
$377$	−5.02168	−0.258630
$378$	0	0
$379$	19.3233	0.992572	0.496286	−	0.868159i	$-0.334697\pi$
$379$	19.3233	0.992572	0.496286	+	0.868159i	$0.334697\pi$
$380$	5.74362	0.294641
$381$	0	0
$382$	−8.77021	−0.448723
$383$	−9.31316	−0.475880	−0.237940	−	0.971280i	$-0.576472\pi$
$383$	−9.31316	−0.475880	−0.237940	+	0.971280i	$0.576472\pi$
$384$	0	0
$385$	−0.0481849	−0.00245573
$386$	45.2058	2.30092
$387$	0	0
$388$	−6.64930	−0.337567
$389$	−15.9922	−0.810836	−0.405418	−	0.914131i	$-0.632874\pi$
$389$	−15.9922	−0.810836	−0.405418	+	0.914131i	$0.632874\pi$
$390$	0	0
$391$	1.58174	0.0799919
$392$	1.37379	0.0693868
$393$	0	0
$394$	−11.6019	−0.584496
$395$	4.07088	0.204828
$396$	0	0
$397$	0.554561	0.0278326	0.0139163	−	0.999903i	$-0.495570\pi$
$397$	0.554561	0.0278326	0.0139163	+	0.999903i	$0.495570\pi$
$398$	−29.7084	−1.48915
$399$	0	0
$400$	−4.19048	−0.209524
$401$	33.6137	1.67859	0.839295	−	0.543676i	$-0.182968\pi$
$401$	33.6137	1.67859	0.839295	+	0.543676i	$0.182968\pi$
$402$	0	0
$403$	4.87728	0.242954
$404$	−33.7955	−1.68139
$405$	0	0
$406$	2.46904	0.122536
$407$	1.23843	0.0613866
$408$	0	0
$409$	−0.0244952	−0.00121121	−0.000605604	−	1.00000i	$-0.500193\pi$
$409$	−0.0244952	−0.00121121	−0.000605604		1.00000i	$0.500193\pi$
$410$	−2.89709	−0.143077
$411$	0	0
$412$	19.8594	0.978404
$413$	−2.64028	−0.129920
$414$	0	0
$415$	5.59844	0.274817
$416$	−7.87924	−0.386312
$417$	0	0
$418$	1.15547	0.0565161
$419$	27.7265	1.35453	0.677263	−	0.735741i	$-0.263166\pi$
$419$	27.7265	1.35453	0.677263	+	0.735741i	$0.263166\pi$
$420$	0	0
$421$	2.59574	0.126509	0.0632543	−	0.997997i	$-0.479852\pi$
$421$	2.59574	0.126509	0.0632543	+	0.997997i	$0.479852\pi$
$422$	−39.2777	−1.91201
$423$	0	0
$424$	1.17132	0.0568845
$425$	0.423209	0.0205287
$426$	0	0
$427$	3.02269	0.146278
$428$	11.1962	0.541190
$429$	0	0
$430$	−1.91822	−0.0925049
$431$	−24.4572	−1.17806	−0.589031	−	0.808111i	$-0.700490\pi$
$431$	−24.4572	−1.17806	−0.589031	+	0.808111i	$0.700490\pi$
$432$	0	0
$433$	−25.7985	−1.23980	−0.619899	−	0.784681i	$-0.712827\pi$
$433$	−25.7985	−1.23980	−0.619899	+	0.784681i	$0.712827\pi$
$434$	−2.39804	−0.115110
$435$	0	0
$436$	−0.860008	−0.0411869
$437$	−11.2998	−0.540545
$438$	0	0
$439$	−25.9276	−1.23746	−0.618728	−	0.785606i	$-0.712352\pi$
$439$	−25.9276	−1.23746	−0.618728	+	0.785606i	$0.712352\pi$
$440$	−0.0383208	−0.00182687
$441$	0	0
$442$	0.835743	0.0397522
$443$	10.6589	0.506422	0.253211	−	0.967411i	$-0.418513\pi$
$443$	10.6589	0.506422	0.253211	+	0.967411i	$0.418513\pi$
$444$	0	0
$445$	−4.19813	−0.199011
$446$	13.6957	0.648511
$447$	0	0
$448$	1.78735	0.0844445
$449$	−6.37651	−0.300926	−0.150463	−	0.988616i	$-0.548076\pi$
$449$	−6.37651	−0.300926	−0.150463	+	0.988616i	$0.548076\pi$
$450$	0	0
$451$	−0.283919	−0.0133692
$452$	−8.19304	−0.385368
$453$	0	0
$454$	44.2817	2.07824
$455$	0.248978	0.0116723
$456$	0	0
$457$	24.0046	1.12289	0.561445	−	0.827514i	$-0.310246\pi$
$457$	24.0046	1.12289	0.561445	+	0.827514i	$0.310246\pi$
$458$	53.6604	2.50739
$459$	0	0
$460$	−7.10021	−0.331049
$461$	39.1329	1.82260	0.911300	−	0.411744i	$-0.135080\pi$
$461$	39.1329	1.82260	0.911300	+	0.411744i	$0.135080\pi$
$462$	0	0
$463$	29.4152	1.36704	0.683520	−	0.729931i	$-0.260448\pi$
$463$	29.4152	1.36704	0.683520	+	0.729931i	$0.260448\pi$
$464$	21.0433	0.976909
$465$	0	0
$466$	−1.74247	−0.0807184
$467$	−16.7547	−0.775314	−0.387657	−	0.921804i	$-0.626715\pi$
$467$	−16.7547	−0.775314	−0.387657	+	0.921804i	$0.626715\pi$
$468$	0	0
$469$	1.67849	0.0775056
$470$	16.3494	0.754143
$471$	0	0
$472$	−2.09978	−0.0966502
$473$	−0.187988	−0.00864371
$474$	0	0
$475$	−3.02339	−0.138722
$476$	−0.200175	−0.00917499
$477$	0	0
$478$	12.6641	0.579241
$479$	33.2846	1.52081	0.760407	−	0.649447i	$-0.225000\pi$
$479$	33.2846	1.52081	0.760407	+	0.649447i	$0.225000\pi$
$480$	0	0
$481$	−6.39914	−0.291776
$482$	9.38566	0.427505
$483$	0	0
$484$	−20.8259	−0.946631
$485$	3.50013	0.158933
$486$	0	0
$487$	−1.95921	−0.0887805	−0.0443902	−	0.999014i	$-0.514134\pi$
$487$	−1.95921	−0.0887805	−0.0443902	+	0.999014i	$0.514134\pi$
$488$	2.40390	0.108820
$489$	0	0
$490$	13.7010	0.618948
$491$	34.2483	1.54560	0.772802	−	0.634648i	$-0.218855\pi$
$491$	34.2483	1.54560	0.772802	+	0.634648i	$0.218855\pi$
$492$	0	0
$493$	−2.12522	−0.0957152
$494$	−5.97050	−0.268626
$495$	0	0
$496$	−20.4381	−0.917700
$497$	0.616215	0.0276410
$498$	0	0
$499$	−37.1932	−1.66500	−0.832499	−	0.554026i	$-0.813091\pi$
$499$	−37.1932	−1.66500	−0.832499	+	0.554026i	$0.813091\pi$
$500$	−1.89973	−0.0849585
$501$	0	0
$502$	−44.7693	−1.99815
$503$	−3.90366	−0.174056	−0.0870278	−	0.996206i	$-0.527737\pi$
$503$	−3.90366	−0.174056	−0.0870278	+	0.996206i	$0.527737\pi$
$504$	0	0
$505$	17.7896	0.791628
$506$	−1.42839	−0.0634995
$507$	0	0
$508$	−2.97038	−0.131789
$509$	16.3953	0.726709	0.363355	−	0.931651i	$-0.381631\pi$
$509$	16.3953	0.726709	0.363355	+	0.931651i	$0.381631\pi$
$510$	0	0
$511$	2.86387	0.126690
$512$	31.3583	1.38586
$513$	0	0
$514$	−34.7240	−1.53161
$515$	−10.4538	−0.460650
$516$	0	0
$517$	1.60227	0.0704676
$518$	3.14630	0.138241
$519$	0	0
$520$	0.198009	0.00868327
$521$	−2.13168	−0.0933905	−0.0466952	−	0.998909i	$-0.514869\pi$
$521$	−2.13168	−0.0933905	−0.0466952	+	0.998909i	$0.514869\pi$
$522$	0	0
$523$	−2.48494	−0.108659	−0.0543294	−	0.998523i	$-0.517302\pi$
$523$	−2.48494	−0.108659	−0.0543294	+	0.998523i	$0.517302\pi$
$524$	13.9625	0.609955
$525$	0	0
$526$	−16.3338	−0.712187
$527$	2.06411	0.0899140
$528$	0	0
$529$	−9.03123	−0.392662
$530$	11.6818	0.507424
$531$	0	0
$532$	1.43004	0.0620000
$533$	1.46705	0.0635451
$534$	0	0
$535$	−5.89359	−0.254802
$536$	1.33488	0.0576582
$537$	0	0
$538$	58.6376	2.52805
$539$	1.34272	0.0578349
$540$	0	0
$541$	−39.5888	−1.70206	−0.851028	−	0.525121i	$-0.824020\pi$
$541$	−39.5888	−1.70206	−0.851028	+	0.525121i	$0.824020\pi$
$542$	−42.0938	−1.80808
$543$	0	0
$544$	−3.33457	−0.142968
$545$	0.452700	0.0193915
$546$	0	0
$547$	−20.3819	−0.871467	−0.435733	−	0.900076i	$-0.643511\pi$
$547$	−20.3819	−0.871467	−0.435733	+	0.900076i	$0.643511\pi$
$548$	17.8602	0.762950
$549$	0	0
$550$	−0.382179	−0.0162962
$551$	15.1825	0.646795
$552$	0	0
$553$	1.01356	0.0431010
$554$	24.7969	1.05352
$555$	0	0
$556$	1.86689	0.0791736
$557$	15.9725	0.676778	0.338389	−	0.941006i	$-0.390118\pi$
$557$	15.9725	0.676778	0.338389	+	0.941006i	$0.390118\pi$
$558$	0	0
$559$	0.971363	0.0410843
$560$	−1.04334	−0.0440892
$561$	0	0
$562$	16.8188	0.709460
$563$	21.1008	0.889291	0.444646	−	0.895707i	$-0.353330\pi$
$563$	21.1008	0.889291	0.444646	+	0.895707i	$0.353330\pi$
$564$	0	0
$565$	4.31274	0.181438
$566$	27.6054	1.16034
$567$	0	0
$568$	0.490068	0.0205628
$569$	27.5521	1.15504	0.577521	−	0.816376i	$-0.304020\pi$
$569$	27.5521	1.15504	0.577521	+	0.816376i	$0.304020\pi$
$570$	0	0
$571$	−7.23680	−0.302851	−0.151425	−	0.988469i	$-0.548386\pi$
$571$	−7.23680	−0.302851	−0.151425	+	0.988469i	$0.548386\pi$
$572$	−0.367656	−0.0153725
$573$	0	0
$574$	−0.721314	−0.0301071
$575$	3.73748	0.155864
$576$	0	0
$577$	−4.28406	−0.178348	−0.0891738	−	0.996016i	$-0.528423\pi$
$577$	−4.28406	−0.178348	−0.0891738	+	0.996016i	$0.528423\pi$
$578$	−33.2175	−1.38166
$579$	0	0
$580$	9.53984	0.396120
$581$	1.39389	0.0578283
$582$	0	0
$583$	1.14483	0.0474140
$584$	2.27759	0.0942475
$585$	0	0
$586$	−42.7363	−1.76542
$587$	−3.25264	−0.134251	−0.0671253	−	0.997745i	$-0.521383\pi$
$587$	−3.25264	−0.134251	−0.0671253	+	0.997745i	$0.521383\pi$
$588$	0	0
$589$	−14.7459	−0.607594
$590$	−20.9414	−0.862144
$591$	0	0
$592$	26.8155	1.10211
$593$	10.9308	0.448875	0.224437	−	0.974489i	$-0.427946\pi$
$593$	10.9308	0.448875	0.224437	+	0.974489i	$0.427946\pi$
$594$	0	0
$595$	0.105370	0.00431975
$596$	−6.00935	−0.246153
$597$	0	0
$598$	7.38068	0.301818
$599$	13.2183	0.540086	0.270043	−	0.962848i	$-0.412962\pi$
$599$	13.2183	0.540086	0.270043	+	0.962848i	$0.412962\pi$
$600$	0	0
$601$	31.5575	1.28726	0.643629	−	0.765338i	$-0.277428\pi$
$601$	31.5575	1.28726	0.643629	+	0.765338i	$0.277428\pi$
$602$	−0.477596	−0.0194653
$603$	0	0
$604$	−39.1304	−1.59219
$605$	10.9625	0.445691
$606$	0	0
$607$	−32.7134	−1.32780	−0.663899	−	0.747822i	$-0.731100\pi$
$607$	−32.7134	−1.32780	−0.663899	+	0.747822i	$0.731100\pi$
$608$	23.8220	0.966109
$609$	0	0
$610$	23.9745	0.970699
$611$	−8.27914	−0.334938
$612$	0	0
$613$	13.2514	0.535219	0.267609	−	0.963527i	$-0.413766\pi$
$613$	13.2514	0.535219	0.267609	+	0.963527i	$0.413766\pi$
$614$	−27.4704	−1.10861
$615$	0	0
$616$	−0.00954105	−0.000384420 0
$617$	−14.0668	−0.566308	−0.283154	−	0.959074i	$-0.591381\pi$
$617$	−14.0668	−0.566308	−0.283154	+	0.959074i	$0.591381\pi$
$618$	0	0
$619$	−22.8438	−0.918170	−0.459085	−	0.888392i	$-0.651823\pi$
$619$	−22.8438	−0.918170	−0.459085	+	0.888392i	$0.651823\pi$
$620$	−9.26551	−0.372112
$621$	0	0
$622$	30.3074	1.21521
$623$	−1.04524	−0.0418768
$624$	0	0
$625$	1.00000	0.0400000
$626$	−50.0317	−1.99967
$627$	0	0
$628$	−10.2838	−0.410369
$629$	−2.70818	−0.107982
$630$	0	0
$631$	20.0750	0.799172	0.399586	−	0.916696i	$-0.369154\pi$
$631$	20.0750	0.799172	0.399586	+	0.916696i	$0.369154\pi$
$632$	0.806072	0.0320638
$633$	0	0
$634$	−5.43760	−0.215955
$635$	1.56358	0.0620488
$636$	0	0
$637$	−6.93801	−0.274894
$638$	1.91918	0.0759811
$639$	0	0
$640$	−1.58208	−0.0625373
$641$	9.03808	0.356983	0.178491	−	0.983941i	$-0.442878\pi$
$641$	9.03808	0.356983	0.178491	+	0.983941i	$0.442878\pi$
$642$	0	0
$643$	−35.7744	−1.41081	−0.705403	−	0.708806i	$-0.749234\pi$
$643$	−35.7744	−1.41081	−0.705403	+	0.708806i	$0.749234\pi$
$644$	−1.76780	−0.0696610
$645$	0	0
$646$	−2.52677	−0.0994146
$647$	12.3837	0.486855	0.243428	−	0.969919i	$-0.421728\pi$
$647$	12.3837	0.486855	0.243428	+	0.969919i	$0.421728\pi$
$648$	0	0
$649$	−2.05229	−0.0805593
$650$	1.97477	0.0774570
$651$	0	0
$652$	11.9283	0.467147
$653$	−22.9605	−0.898512	−0.449256	−	0.893403i	$-0.648311\pi$
$653$	−22.9605	−0.898512	−0.449256	+	0.893403i	$0.648311\pi$
$654$	0	0
$655$	−7.34973	−0.287178
$656$	−6.14766	−0.240026
$657$	0	0
$658$	4.07065	0.158691
$659$	−10.8975	−0.424507	−0.212254	−	0.977215i	$-0.568080\pi$
$659$	−10.8975	−0.424507	−0.212254	+	0.977215i	$0.568080\pi$
$660$	0	0
$661$	−36.3252	−1.41289	−0.706443	−	0.707770i	$-0.749701\pi$
$661$	−36.3252	−1.41289	−0.706443	+	0.707770i	$0.749701\pi$
$662$	−40.5880	−1.57750
$663$	0	0
$664$	1.10854	0.0430198
$665$	−0.752758	−0.0291907
$666$	0	0
$667$	−18.7684	−0.726716
$668$	−29.6770	−1.14824
$669$	0	0
$670$	13.3130	0.514325
$671$	2.34953	0.0907027
$672$	0	0
$673$	6.22822	0.240080	0.120040	−	0.992769i	$-0.461698\pi$
$673$	6.22822	0.240080	0.120040	+	0.992769i	$0.461698\pi$
$674$	−0.156650	−0.00603395
$675$	0	0
$676$	1.89973	0.0730666
$677$	42.7012	1.64114	0.820570	−	0.571546i	$-0.193656\pi$
$677$	42.7012	1.64114	0.820570	+	0.571546i	$0.193656\pi$
$678$	0	0
$679$	0.871456	0.0334434
$680$	0.0837993	0.00321356
$681$	0	0
$682$	−1.86399	−0.0713759
$683$	−20.9400	−0.801246	−0.400623	−	0.916243i	$-0.631206\pi$
$683$	−20.9400	−0.801246	−0.400623	+	0.916243i	$0.631206\pi$
$684$	0	0
$685$	−9.40143	−0.359210
$686$	6.85298	0.261648
$687$	0	0
$688$	−4.07048	−0.155186
$689$	−5.91550	−0.225363
$690$	0	0
$691$	24.6453	0.937551	0.468775	−	0.883317i	$-0.344695\pi$
$691$	24.6453	0.937551	0.468775	+	0.883317i	$0.344695\pi$
$692$	28.4848	1.08283
$693$	0	0
$694$	66.0656	2.50782
$695$	−0.982710	−0.0372763
$696$	0	0
$697$	0.620870	0.0235171
$698$	35.8463	1.35680
$699$	0	0
$700$	−0.472992	−0.0178774
$701$	−10.9522	−0.413660	−0.206830	−	0.978377i	$-0.566315\pi$
$701$	−10.9522	−0.413660	−0.206830	+	0.978377i	$0.566315\pi$
$702$	0	0
$703$	19.3471	0.729689
$704$	1.38931	0.0523615
$705$	0	0
$706$	−24.9889	−0.940468
$707$	4.42924	0.166579
$708$	0	0
$709$	20.9949	0.788481	0.394241	−	0.919007i	$-0.371008\pi$
$709$	20.9949	0.788481	0.394241	+	0.919007i	$0.371008\pi$
$710$	4.88752	0.183425
$711$	0	0
$712$	−0.831268	−0.0311531
$713$	18.2287	0.682671
$714$	0	0
$715$	0.193531	0.00723763
$716$	37.8276	1.41368
$717$	0	0
$718$	4.38987	0.163828
$719$	−20.2852	−0.756508	−0.378254	−	0.925702i	$-0.623475\pi$
$719$	−20.2852	−0.756508	−0.378254	+	0.925702i	$0.623475\pi$
$720$	0	0
$721$	−2.60277	−0.0969324
$722$	−19.4696	−0.724582
$723$	0	0
$724$	29.3110	1.08934
$725$	−5.02168	−0.186500
$726$	0	0
$727$	−39.2681	−1.45637	−0.728187	−	0.685379i	$-0.759637\pi$
$727$	−39.2681	−1.45637	−0.728187	+	0.685379i	$0.759637\pi$
$728$	0.0493000	0.00182718
$729$	0	0
$730$	22.7148	0.840712
$731$	0.411090	0.0152047
$732$	0	0
$733$	50.6628	1.87127	0.935637	−	0.352964i	$-0.114826\pi$
$733$	50.6628	1.87127	0.935637	+	0.352964i	$0.114826\pi$
$734$	−55.5551	−2.05058
$735$	0	0
$736$	−29.4485	−1.08549
$737$	1.30469	0.0480589
$738$	0	0
$739$	29.7662	1.09497	0.547484	−	0.836816i	$-0.315586\pi$
$739$	29.7662	1.09497	0.547484	+	0.836816i	$0.315586\pi$
$740$	12.1566	0.446887
$741$	0	0
$742$	2.90851	0.106775
$743$	42.9021	1.57393	0.786963	−	0.617000i	$-0.211652\pi$
$743$	42.9021	1.57393	0.786963	+	0.617000i	$0.211652\pi$
$744$	0	0
$745$	3.16327	0.115893
$746$	57.2593	2.09641
$747$	0	0
$748$	−0.155595	−0.00568913
$749$	−1.46738	−0.0536167
$750$	0	0
$751$	−41.9570	−1.53103	−0.765516	−	0.643417i	$-0.777516\pi$
$751$	−41.9570	−1.53103	−0.765516	+	0.643417i	$0.777516\pi$
$752$	34.6936	1.26515
$753$	0	0
$754$	−9.91668	−0.361144
$755$	20.5979	0.749633
$756$	0	0
$757$	−27.9453	−1.01569	−0.507845	−	0.861449i	$-0.669558\pi$
$757$	−27.9453	−1.01569	−0.507845	+	0.861449i	$0.669558\pi$
$758$	38.1592	1.38600
$759$	0	0
$760$	−0.598658	−0.0217156
$761$	−25.4372	−0.922099	−0.461049	−	0.887374i	$-0.652527\pi$
$761$	−25.4372	−0.922099	−0.461049	+	0.887374i	$0.652527\pi$
$762$	0	0
$763$	0.112713	0.00408047
$764$	−8.43694	−0.305238
$765$	0	0
$766$	−18.3914	−0.664508
$767$	10.6045	0.382905
$768$	0	0
$769$	−40.1288	−1.44708	−0.723541	−	0.690282i	$-0.757487\pi$
$769$	−40.1288	−1.44708	−0.723541	+	0.690282i	$0.757487\pi$
$770$	−0.0951543	−0.00342912
$771$	0	0
$772$	43.4880	1.56517
$773$	41.8266	1.50440	0.752199	−	0.658936i	$-0.228993\pi$
$773$	41.8266	1.50440	0.752199	+	0.658936i	$0.228993\pi$
$774$	0	0
$775$	4.87728	0.175197
$776$	0.693057	0.0248793
$777$	0	0
$778$	−31.5809	−1.13223
$779$	−4.43546	−0.158917
$780$	0	0
$781$	0.478983	0.0171394
$782$	3.12357	0.111699
$783$	0	0
$784$	29.0736	1.03834
$785$	5.41330	0.193209
$786$	0	0
$787$	−8.42735	−0.300403	−0.150201	−	0.988655i	$-0.547992\pi$
$787$	−8.42735	−0.300403	−0.150201	+	0.988655i	$0.547992\pi$
$788$	−11.1610	−0.397595
$789$	0	0
$790$	8.03907	0.286017
$791$	1.07378	0.0381792
$792$	0	0
$793$	−12.1404	−0.431117
$794$	1.09513	0.0388648
$795$	0	0
$796$	−28.5794	−1.01297
$797$	−25.0711	−0.888064	−0.444032	−	0.896011i	$-0.646452\pi$
$797$	−25.0711	−0.888064	−0.444032	+	0.896011i	$0.646452\pi$
$798$	0	0
$799$	−3.50381	−0.123956
$800$	−7.87924	−0.278573
$801$	0	0
$802$	66.3795	2.34394
$803$	2.22608	0.0785567
$804$	0	0
$805$	0.930552	0.0327977
$806$	9.63151	0.339256
$807$	0	0
$808$	3.52251	0.123921
$809$	15.4188	0.542098	0.271049	−	0.962566i	$-0.412629\pi$
$809$	15.4188	0.542098	0.271049	+	0.962566i	$0.412629\pi$
$810$	0	0
$811$	19.5623	0.686924	0.343462	−	0.939167i	$-0.388400\pi$
$811$	19.5623	0.686924	0.343462	+	0.939167i	$0.388400\pi$
$812$	2.37521	0.0833536
$813$	0	0
$814$	2.44562	0.0857188
$815$	−6.27893	−0.219941
$816$	0	0
$817$	−2.93680	−0.102746
$818$	−0.0483724	−0.00169130
$819$	0	0
$820$	−2.78700	−0.0973264
$821$	51.8508	1.80960	0.904802	−	0.425832i	$-0.140018\pi$
$821$	51.8508	1.80960	0.904802	+	0.425832i	$0.140018\pi$
$822$	0	0
$823$	−2.26237	−0.0788613	−0.0394307	−	0.999222i	$-0.512554\pi$
$823$	−2.26237	−0.0788613	−0.0394307	+	0.999222i	$0.512554\pi$
$824$	−2.06995	−0.0721101
$825$	0	0
$826$	−5.21396	−0.181417
$827$	−13.8977	−0.483271	−0.241636	−	0.970367i	$-0.577684\pi$
$827$	−13.8977	−0.483271	−0.241636	+	0.970367i	$0.577684\pi$
$828$	0	0
$829$	53.7072	1.86533	0.932665	−	0.360744i	$-0.117477\pi$
$829$	53.7072	1.86533	0.932665	+	0.360744i	$0.117477\pi$
$830$	11.0557	0.383747
$831$	0	0
$832$	−7.17875	−0.248878
$833$	−2.93623	−0.101734
$834$	0	0
$835$	15.6217	0.540611
$836$	1.11157	0.0384443
$837$	0	0
$838$	54.7535	1.89143
$839$	−29.8273	−1.02975	−0.514877	−	0.857264i	$-0.672162\pi$
$839$	−29.8273	−1.02975	−0.514877	+	0.857264i	$0.672162\pi$
$840$	0	0
$841$	−3.78275	−0.130440
$842$	5.12599	0.176653
$843$	0	0
$844$	−37.7851	−1.30062
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	2.72944	0.0937846
$848$	24.7888	0.851252
$849$	0	0
$850$	0.835743	0.0286657
$851$	−23.9167	−0.819853
$852$	0	0
$853$	11.5983	0.397118	0.198559	−	0.980089i	$-0.436374\pi$
$853$	11.5983	0.397118	0.198559	+	0.980089i	$0.436374\pi$
$854$	5.96913	0.204259
$855$	0	0
$856$	−1.16698	−0.0398867
$857$	−45.9946	−1.57115	−0.785574	−	0.618768i	$-0.787632\pi$
$857$	−45.9946	−1.57115	−0.785574	+	0.618768i	$0.787632\pi$
$858$	0	0
$859$	53.4572	1.82394	0.911968	−	0.410261i	$-0.134562\pi$
$859$	53.4572	1.82394	0.911968	+	0.410261i	$0.134562\pi$
$860$	−1.84533	−0.0629252
$861$	0	0
$862$	−48.2974	−1.64502
$863$	−30.2950	−1.03125	−0.515627	−	0.856813i	$-0.672441\pi$
$863$	−30.2950	−1.03125	−0.515627	+	0.856813i	$0.672441\pi$
$864$	0	0
$865$	−14.9941	−0.509816
$866$	−50.9463	−1.73123
$867$	0	0
$868$	−2.30691	−0.0783017
$869$	0.787840	0.0267256
$870$	0	0
$871$	−6.74152	−0.228428
$872$	0.0896388	0.00303555
$873$	0	0
$874$	−22.3146	−0.754804
$875$	0.248978	0.00841701
$876$	0	0
$877$	−28.2893	−0.955262	−0.477631	−	0.878561i	$-0.658504\pi$
$877$	−28.2893	−0.955262	−0.477631	+	0.878561i	$0.658504\pi$
$878$	−51.2011	−1.72795
$879$	0	0
$880$	−0.810987	−0.0273384
$881$	0.478075	0.0161067	0.00805337	−	0.999968i	$-0.497437\pi$
$881$	0.478075	0.0161067	0.00805337	+	0.999968i	$0.497437\pi$
$882$	0	0
$883$	33.6966	1.13398	0.566991	−	0.823724i	$-0.308108\pi$
$883$	33.6966	1.13398	0.566991	+	0.823724i	$0.308108\pi$
$884$	0.803984	0.0270409
$885$	0	0
$886$	21.0490	0.707155
$887$	−27.6118	−0.927115	−0.463557	−	0.886067i	$-0.653427\pi$
$887$	−27.6118	−0.927115	−0.463557	+	0.886067i	$0.653427\pi$
$888$	0	0
$889$	0.389298	0.0130566
$890$	−8.29036	−0.277894
$891$	0	0
$892$	13.1753	0.441141
$893$	25.0310	0.837631
$894$	0	0
$895$	−19.9121	−0.665588
$896$	−0.393904	−0.0131594
$897$	0	0
$898$	−12.5922	−0.420206
$899$	−24.4921	−0.816858
$900$	0	0
$901$	−2.50350	−0.0834035
$902$	−0.560676	−0.0186685
$903$	0	0
$904$	0.853962	0.0284023
$905$	−15.4291	−0.512879
$906$	0	0
$907$	−37.1164	−1.23243	−0.616214	−	0.787579i	$-0.711334\pi$
$907$	−37.1164	−1.23243	−0.616214	+	0.787579i	$0.711334\pi$
$908$	42.5989	1.41369
$909$	0	0
$910$	0.491676	0.0162989
$911$	57.0497	1.89014	0.945071	−	0.326866i	$-0.105992\pi$
$911$	57.0497	1.89014	0.945071	+	0.326866i	$0.105992\pi$
$912$	0	0
$913$	1.08347	0.0358576
$914$	47.4037	1.56798
$915$	0	0
$916$	51.6213	1.70562
$917$	−1.82992	−0.0604294
$918$	0	0
$919$	5.88089	0.193993	0.0969963	−	0.995285i	$-0.469077\pi$
$919$	5.88089	0.193993	0.0969963	+	0.995285i	$0.469077\pi$
$920$	0.740055	0.0243989
$921$	0	0
$922$	77.2785	2.54503
$923$	−2.47498	−0.0814648
$924$	0	0
$925$	−6.39914	−0.210402
$926$	58.0884	1.90890
$927$	0	0
$928$	39.5670	1.29885
$929$	−21.9449	−0.719988	−0.359994	−	0.932955i	$-0.617221\pi$
$929$	−21.9449	−0.719988	−0.359994	+	0.932955i	$0.617221\pi$
$930$	0	0
$931$	20.9763	0.687470
$932$	−1.67625	−0.0549075
$933$	0	0
$934$	−33.0867	−1.08263
$935$	0.0819039	0.00267855
$936$	0	0
$937$	−38.2930	−1.25098	−0.625489	−	0.780233i	$-0.715101\pi$
$937$	−38.2930	−1.25098	−0.625489	+	0.780233i	$0.715101\pi$
$938$	3.31464	0.108227
$939$	0	0
$940$	15.7281	0.512995
$941$	−36.9255	−1.20374	−0.601868	−	0.798595i	$-0.705577\pi$
$941$	−36.9255	−1.20374	−0.601868	+	0.798595i	$0.705577\pi$
$942$	0	0
$943$	5.48308	0.178554
$944$	−44.4378	−1.44633
$945$	0	0
$946$	−0.371234	−0.0120699
$947$	−2.37168	−0.0770693	−0.0385346	−	0.999257i	$-0.512269\pi$
$947$	−2.37168	−0.0770693	−0.0385346	+	0.999257i	$0.512269\pi$
$948$	0	0
$949$	−11.5025	−0.373386
$950$	−5.97050	−0.193709
$951$	0	0
$952$	0.0208642	0.000676213 0
$953$	−2.90432	−0.0940802	−0.0470401	−	0.998893i	$-0.514979\pi$
$953$	−2.90432	−0.0940802	−0.0470401	+	0.998893i	$0.514979\pi$
$954$	0	0
$955$	4.44112	0.143711
$956$	12.1828	0.394020
$957$	0	0
$958$	65.7296	2.12363
$959$	−2.34075	−0.0755869
$960$	0	0
$961$	−7.21219	−0.232651
$962$	−12.6369	−0.407428
$963$	0	0
$964$	9.02900	0.290805
$965$	−22.8917	−0.736908
$966$	0	0
$967$	−36.5693	−1.17599	−0.587995	−	0.808865i	$-0.700082\pi$
$967$	−36.5693	−1.17599	−0.587995	+	0.808865i	$0.700082\pi$
$968$	2.17068	0.0697684
$969$	0	0
$970$	6.91196	0.221930
$971$	45.6524	1.46506	0.732528	−	0.680737i	$-0.238340\pi$
$971$	45.6524	1.46506	0.732528	+	0.680737i	$0.238340\pi$
$972$	0	0
$973$	−0.244674	−0.00784388
$974$	−3.86900	−0.123971
$975$	0	0
$976$	50.8740	1.62844
$977$	46.2824	1.48070	0.740352	−	0.672219i	$-0.234659\pi$
$977$	46.2824	1.48070	0.740352	+	0.672219i	$0.234659\pi$
$978$	0	0
$979$	−0.812467	−0.0259665
$980$	13.1804	0.421031
$981$	0	0
$982$	67.6326	2.15824
$983$	−55.7898	−1.77942	−0.889709	−	0.456528i	$-0.849093\pi$
$983$	−55.7898	−1.77942	−0.889709	+	0.456528i	$0.849093\pi$
$984$	0	0
$985$	5.87506	0.187195
$986$	−4.19683	−0.133654
$987$	0	0
$988$	−5.74362	−0.182729
$989$	3.63045	0.115442
$990$	0	0
$991$	9.68400	0.307622	0.153811	−	0.988100i	$-0.450845\pi$
$991$	9.68400	0.307622	0.153811	+	0.988100i	$0.450845\pi$
$992$	−38.4292	−1.22013
$993$	0	0
$994$	1.21689	0.0385973
$995$	15.0439	0.476925
$996$	0	0
$997$	−8.62012	−0.273002	−0.136501	−	0.990640i	$-0.543586\pi$
$997$	−8.62012	−0.273002	−0.136501	+	0.990640i	$0.543586\pi$
$998$	−73.4482	−2.32496
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.be.1.7		8
3.2	odd	2		5265.2.a.bb.1.2		8
9.2	odd	6		585.2.i.f.391.7	yes	16
9.4	even	3		1755.2.i.e.586.2		16
9.5	odd	6		585.2.i.f.196.7	✓	16
9.7	even	3		1755.2.i.e.1171.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.f.196.7	✓	16	9.5	odd	6
585.2.i.f.391.7	yes	16	9.2	odd	6
1755.2.i.e.586.2		16	9.4	even	3
1755.2.i.e.1171.2		16	9.7	even	3
5265.2.a.bb.1.2		8	3.2	odd	2
5265.2.a.be.1.7		8	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:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q+1.97477 q^{2} +1.89973 q^{4} -1.00000 q^{5} -0.248978 q^{7} -0.198009 q^{8} +O(q^{10})\)	\(q+1.97477 q^{2} +1.89973 q^{4} -1.00000 q^{5} -0.248978 q^{7} -0.198009 q^{8} -1.97477 q^{10} -0.193531 q^{11} +1.00000 q^{13} -0.491676 q^{14} -4.19048 q^{16} +0.423209 q^{17} -3.02339 q^{19} -1.89973 q^{20} -0.382179 q^{22} +3.73748 q^{23} +1.00000 q^{25} +1.97477 q^{26} -0.472992 q^{28} -5.02168 q^{29} +4.87728 q^{31} -7.87924 q^{32} +0.835743 q^{34} +0.248978 q^{35} -6.39914 q^{37} -5.97050 q^{38} +0.198009 q^{40} +1.46705 q^{41} +0.971363 q^{43} -0.367656 q^{44} +7.38068 q^{46} -8.27914 q^{47} -6.93801 q^{49} +1.97477 q^{50} +1.89973 q^{52} -5.91550 q^{53} +0.193531 q^{55} +0.0493000 q^{56} -9.91668 q^{58} +10.6045 q^{59} -12.1404 q^{61} +9.63151 q^{62} -7.17875 q^{64} -1.00000 q^{65} -6.74152 q^{67} +0.803984 q^{68} +0.491676 q^{70} -2.47498 q^{71} -11.5025 q^{73} -12.6369 q^{74} -5.74362 q^{76} +0.0481849 q^{77} -4.07088 q^{79} +4.19048 q^{80} +2.89709 q^{82} -5.59844 q^{83} -0.423209 q^{85} +1.91822 q^{86} +0.0383208 q^{88} +4.19813 q^{89} -0.248978 q^{91} +7.10021 q^{92} -16.3494 q^{94} +3.02339 q^{95} -3.50013 q^{97} -13.7010 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 5 q^{4} - 8 q^{5} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) 8 * q + q^2 + 5 * q^4 - 8 * q^5 - 6 * q^7 + 6 * q^8	\( 8 q + q^{2} + 5 q^{4} - 8 q^{5} - 6 q^{7} + 6 q^{8} - q^{10} + 9 q^{11} + 8 q^{13} - 3 q^{14} - 13 q^{16} - 6 q^{17} - 11 q^{19} - 5 q^{20} - 4 q^{22} + 3 q^{23} + 8 q^{25} + q^{26} - 13 q^{28} + 8 q^{29} - 18 q^{31} + 3 q^{32} - 9 q^{34} + 6 q^{35} - 18 q^{37} - 8 q^{38} - 6 q^{40} - 17 q^{41} - 17 q^{43} - 5 q^{44} + 3 q^{46} + 11 q^{47} - 16 q^{49} + q^{50} + 5 q^{52} - 10 q^{53} - 9 q^{55} - q^{56} - 10 q^{58} + 7 q^{59} - 21 q^{61} + 29 q^{62} - 10 q^{64} - 8 q^{65} - 13 q^{67} - 16 q^{68} + 3 q^{70} + 34 q^{71} - 16 q^{73} - 4 q^{74} - 2 q^{76} + 18 q^{77} - 37 q^{79} + 13 q^{80} + q^{82} + 3 q^{83} + 6 q^{85} - 2 q^{86} - 19 q^{88} - 14 q^{89} - 6 q^{91} - 14 q^{92} - 44 q^{94} + 11 q^{95} - 17 q^{97} - 45 q^{98}+O(q^{100}) \) 8 * q + q^2 + 5 * q^4 - 8 * q^5 - 6 * q^7 + 6 * q^8 - q^10 + 9 * q^11 + 8 * q^13 - 3 * q^14 - 13 * q^16 - 6 * q^17 - 11 * q^19 - 5 * q^20 - 4 * q^22 + 3 * q^23 + 8 * q^25 + q^26 - 13 * q^28 + 8 * q^29 - 18 * q^31 + 3 * q^32 - 9 * q^34 + 6 * q^35 - 18 * q^37 - 8 * q^38 - 6 * q^40 - 17 * q^41 - 17 * q^43 - 5 * q^44 + 3 * q^46 + 11 * q^47 - 16 * q^49 + q^50 + 5 * q^52 - 10 * q^53 - 9 * q^55 - q^56 - 10 * q^58 + 7 * q^59 - 21 * q^61 + 29 * q^62 - 10 * q^64 - 8 * q^65 - 13 * q^67 - 16 * q^68 + 3 * q^70 + 34 * q^71 - 16 * q^73 - 4 * q^74 - 2 * q^76 + 18 * q^77 - 37 * q^79 + 13 * q^80 + q^82 + 3 * q^83 + 6 * q^85 - 2 * q^86 - 19 * q^88 - 14 * q^89 - 6 * q^91 - 14 * q^92 - 44 * q^94 + 11 * q^95 - 17 * q^97 - 45 * q^98

Introduction

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