LMFDB - Embedded newform 2760.2.a.w.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(1,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2760.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:	$22.0387109579$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.20087896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 21x^{3} + 5x^{2} + 84x - 4 $ x^5 - x^4 - 21x^3 + 5x^2 + 84*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4.50582$ of defining polynomial
Character	$\chi$	$=$	2760.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +2.89831 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +2.89831 q^{7} +1.00000 q^{9} +3.43123 q^{11} -4.72973 q^{13} -1.00000 q^{15} +2.61639 q^{17} +2.28192 q^{19} -2.89831 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.53292 q^{29} +6.19681 q^{31} -3.43123 q^{33} +2.89831 q^{35} +4.89831 q^{37} +4.72973 q^{39} +10.4787 q^{41} -9.01165 q^{43} +1.00000 q^{45} -2.28192 q^{47} +1.40019 q^{49} -2.61639 q^{51} +0.682109 q^{53} +3.43123 q^{55} -2.28192 q^{57} -14.2753 q^{59} +4.98342 q^{61} +2.89831 q^{63} -4.72973 q^{65} +1.74900 q^{67} +1.00000 q^{69} -6.32954 q^{71} -1.51470 q^{73} -1.00000 q^{75} +9.94476 q^{77} -3.86234 q^{79} +1.00000 q^{81} -4.61639 q^{83} +2.61639 q^{85} -2.53292 q^{87} -6.31015 q^{89} -13.7082 q^{91} -6.19681 q^{93} +2.28192 q^{95} +17.8741 q^{97} +3.43123 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 5 q^{5} - 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 5 * q^5 - 4 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + 5 q^{5} - 4 q^{7} + 5 q^{9} - 4 q^{11} + 4 q^{13} - 5 q^{15} + 10 q^{17} - 4 q^{19} + 4 q^{21} - 5 q^{23} + 5 q^{25} - 5 q^{27} + 10 q^{29} + 6 q^{31} + 4 q^{33} - 4 q^{35} + 6 q^{37} - 4 q^{39} + 12 q^{41} - 2 q^{43} + 5 q^{45} + 4 q^{47} + 19 q^{49} - 10 q^{51} - 4 q^{55} + 4 q^{57} + 6 q^{59} + 16 q^{61} - 4 q^{63} + 4 q^{65} - 4 q^{67} + 5 q^{69} + 8 q^{71} + 14 q^{73} - 5 q^{75} + 16 q^{77} + 18 q^{79} + 5 q^{81} - 20 q^{83} + 10 q^{85} - 10 q^{87} + 18 q^{89} + 20 q^{91} - 6 q^{93} - 4 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 5 * q^5 - 4 * q^7 + 5 * q^9 - 4 * q^11 + 4 * q^13 - 5 * q^15 + 10 * q^17 - 4 * q^19 + 4 * q^21 - 5 * q^23 + 5 * q^25 - 5 * q^27 + 10 * q^29 + 6 * q^31 + 4 * q^33 - 4 * q^35 + 6 * q^37 - 4 * q^39 + 12 * q^41 - 2 * q^43 + 5 * q^45 + 4 * q^47 + 19 * q^49 - 10 * q^51 - 4 * q^55 + 4 * q^57 + 6 * q^59 + 16 * q^61 - 4 * q^63 + 4 * q^65 - 4 * q^67 + 5 * q^69 + 8 * q^71 + 14 * q^73 - 5 * q^75 + 16 * q^77 + 18 * q^79 + 5 * q^81 - 20 * q^83 + 10 * q^85 - 10 * q^87 + 18 * q^89 + 20 * q^91 - 6 * q^93 - 4 * q^95 + 4 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.89831	1.09546	0.547729	−	0.836656i	$-0.315493\pi$
$7$	2.89831	1.09546	0.547729	+	0.836656i	$0.315493\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.43123	1.03455	0.517277	−	0.855818i	$-0.326946\pi$
$11$	3.43123	1.03455	0.517277	+	0.855818i	$0.326946\pi$
$12$	0	0
$13$	−4.72973	−1.31179	−0.655895	−	0.754852i	$-0.727709\pi$
$13$	−4.72973	−1.31179	−0.655895	+	0.754852i	$0.727709\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	2.61639	0.634568	0.317284	−	0.948331i	$-0.397229\pi$
$17$	2.61639	0.634568	0.317284	+	0.948331i	$0.397229\pi$
$18$	0	0
$19$	2.28192	0.523508	0.261754	−	0.965135i	$-0.415699\pi$
$19$	2.28192	0.523508	0.261754	+	0.965135i	$0.415699\pi$
$20$	0	0
$21$	−2.89831	−0.632463
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.53292	0.470351	0.235176	−	0.971953i	$-0.424433\pi$
$29$	2.53292	0.470351	0.235176	+	0.971953i	$0.424433\pi$
$30$	0	0
$31$	6.19681	1.11298	0.556490	−	0.830854i	$-0.312148\pi$
$31$	6.19681	1.11298	0.556490	+	0.830854i	$0.312148\pi$
$32$	0	0
$33$	−3.43123	−0.597300
$34$	0	0
$35$	2.89831	0.489904
$36$	0	0
$37$	4.89831	0.805277	0.402638	−	0.915359i	$-0.368093\pi$
$37$	4.89831	0.805277	0.402638	+	0.915359i	$0.368093\pi$
$38$	0	0
$39$	4.72973	0.757363
$40$	0	0
$41$	10.4787	1.63650	0.818251	−	0.574861i	$-0.194944\pi$
$41$	10.4787	1.63650	0.818251	+	0.574861i	$0.194944\pi$
$42$	0	0
$43$	−9.01165	−1.37426	−0.687132	−	0.726533i	$-0.741130\pi$
$43$	−9.01165	−1.37426	−0.687132	+	0.726533i	$0.741130\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−2.28192	−0.332852	−0.166426	−	0.986054i	$-0.553223\pi$
$47$	−2.28192	−0.332852	−0.166426	+	0.986054i	$0.553223\pi$
$48$	0	0
$49$	1.40019	0.200027
$50$	0	0
$51$	−2.61639	−0.366368
$52$	0	0
$53$	0.682109	0.0936949	0.0468475	−	0.998902i	$-0.485083\pi$
$53$	0.682109	0.0936949	0.0468475	+	0.998902i	$0.485083\pi$
$54$	0	0
$55$	3.43123	0.462667
$56$	0	0
$57$	−2.28192	−0.302247
$58$	0	0
$59$	−14.2753	−1.85849	−0.929246	−	0.369462i	$-0.879542\pi$
$59$	−14.2753	−1.85849	−0.929246	+	0.369462i	$0.879542\pi$
$60$	0	0
$61$	4.98342	0.638061	0.319031	−	0.947744i	$-0.396643\pi$
$61$	4.98342	0.638061	0.319031	+	0.947744i	$0.396643\pi$
$62$	0	0
$63$	2.89831	0.365153
$64$	0	0
$65$	−4.72973	−0.586651
$66$	0	0
$67$	1.74900	0.213674	0.106837	−	0.994277i	$-0.465928\pi$
$67$	1.74900	0.213674	0.106837	+	0.994277i	$0.465928\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−6.32954	−0.751178	−0.375589	−	0.926786i	$-0.622560\pi$
$71$	−6.32954	−0.751178	−0.375589	+	0.926786i	$0.622560\pi$
$72$	0	0
$73$	−1.51470	−0.177282	−0.0886410	−	0.996064i	$-0.528252\pi$
$73$	−1.51470	−0.177282	−0.0886410	+	0.996064i	$0.528252\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	9.94476	1.13331
$78$	0	0
$79$	−3.86234	−0.434547	−0.217273	−	0.976111i	$-0.569716\pi$
$79$	−3.86234	−0.434547	−0.217273	+	0.976111i	$0.569716\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.61639	−0.506715	−0.253357	−	0.967373i	$-0.581535\pi$
$83$	−4.61639	−0.506715	−0.253357	+	0.967373i	$0.581535\pi$
$84$	0	0
$85$	2.61639	0.283787
$86$	0	0
$87$	−2.53292	−0.271557
$88$	0	0
$89$	−6.31015	−0.668874	−0.334437	−	0.942418i	$-0.608546\pi$
$89$	−6.31015	−0.668874	−0.334437	+	0.942418i	$0.608546\pi$
$90$	0	0
$91$	−13.7082	−1.43701
$92$	0	0
$93$	−6.19681	−0.642579
$94$	0	0
$95$	2.28192	0.234120
$96$	0	0
$97$	17.8741	1.81484	0.907420	−	0.420225i	$-0.138049\pi$
$97$	17.8741	1.81484	0.907420	+	0.420225i	$0.138049\pi$
$98$	0	0
$99$	3.43123	0.344851
$100$	0	0
$101$	3.76570	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$101$	3.76570	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$102$	0	0
$103$	9.09512	0.896168	0.448084	−	0.893991i	$-0.352106\pi$
$103$	9.09512	0.896168	0.448084	+	0.893991i	$0.352106\pi$
$104$	0	0
$105$	−2.89831	−0.282846
$106$	0	0
$107$	2.31777	0.224067	0.112034	−	0.993704i	$-0.464264\pi$
$107$	2.31777	0.224067	0.112034	+	0.993704i	$0.464264\pi$
$108$	0	0
$109$	12.6463	1.21129	0.605646	−	0.795734i	$-0.292915\pi$
$109$	12.6463	1.21129	0.605646	+	0.795734i	$0.292915\pi$
$110$	0	0
$111$	−4.89831	−0.464927
$112$	0	0
$113$	6.11439	0.575193	0.287597	−	0.957752i	$-0.407144\pi$
$113$	6.11439	0.575193	0.287597	+	0.957752i	$0.407144\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−4.72973	−0.437263
$118$	0	0
$119$	7.58311	0.695142
$120$	0	0
$121$	0.773325	0.0703023
$122$	0	0
$123$	−10.4787	−0.944835
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.13273	0.366720	0.183360	−	0.983046i	$-0.441303\pi$
$127$	4.13273	0.366720	0.183360	+	0.983046i	$0.441303\pi$
$128$	0	0
$129$	9.01165	0.793431
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	6.61370	0.573481
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	4.49812	0.384300	0.192150	−	0.981366i	$-0.438454\pi$
$137$	4.49812	0.384300	0.192150	+	0.981366i	$0.438454\pi$
$138$	0	0
$139$	14.7606	1.25198	0.625991	−	0.779830i	$-0.284695\pi$
$139$	14.7606	1.25198	0.625991	+	0.779830i	$0.284695\pi$
$140$	0	0
$141$	2.28192	0.192172
$142$	0	0
$143$	−16.2288	−1.35712
$144$	0	0
$145$	2.53292	0.210348
$146$	0	0
$147$	−1.40019	−0.115486
$148$	0	0
$149$	9.34776	0.765798	0.382899	−	0.923790i	$-0.374926\pi$
$149$	9.34776	0.765798	0.382899	+	0.923790i	$0.374926\pi$
$150$	0	0
$151$	12.2267	0.994993	0.497496	−	0.867466i	$-0.334253\pi$
$151$	12.2267	0.994993	0.497496	+	0.867466i	$0.334253\pi$
$152$	0	0
$153$	2.61639	0.211523
$154$	0	0
$155$	6.19681	0.497740
$156$	0	0
$157$	0.334473	0.0266938	0.0133469	−	0.999911i	$-0.495751\pi$
$157$	0.334473	0.0266938	0.0133469	+	0.999911i	$0.495751\pi$
$158$	0	0
$159$	−0.682109	−0.0540948
$160$	0	0
$161$	−2.89831	−0.228419
$162$	0	0
$163$	−12.0952	−0.947372	−0.473686	−	0.880694i	$-0.657077\pi$
$163$	−12.0952	−0.947372	−0.473686	+	0.880694i	$0.657077\pi$
$164$	0	0
$165$	−3.43123	−0.267121
$166$	0	0
$167$	−19.3346	−1.49616	−0.748078	−	0.663610i	$-0.769023\pi$
$167$	−19.3346	−1.49616	−0.748078	+	0.663610i	$0.769023\pi$
$168$	0	0
$169$	9.37032	0.720794
$170$	0	0
$171$	2.28192	0.174503
$172$	0	0
$173$	−0.203383	−0.0154629	−0.00773147	−	0.999970i	$-0.502461\pi$
$173$	−0.203383	−0.0154629	−0.00773147	+	0.999970i	$0.502461\pi$
$174$	0	0
$175$	2.89831	0.219092
$176$	0	0
$177$	14.2753	1.07300
$178$	0	0
$179$	−5.62968	−0.420782	−0.210391	−	0.977617i	$-0.567474\pi$
$179$	−5.62968	−0.420782	−0.210391	+	0.977617i	$0.567474\pi$
$180$	0	0
$181$	14.9282	1.10960	0.554801	−	0.831983i	$-0.312794\pi$
$181$	14.9282	1.10960	0.554801	+	0.831983i	$0.312794\pi$
$182$	0	0
$183$	−4.98342	−0.368385
$184$	0	0
$185$	4.89831	0.360131
$186$	0	0
$187$	8.97743	0.656495
$188$	0	0
$189$	−2.89831	−0.210821
$190$	0	0
$191$	15.7414	1.13901	0.569503	−	0.821989i	$-0.307136\pi$
$191$	15.7414	1.13901	0.569503	+	0.821989i	$0.307136\pi$
$192$	0	0
$193$	20.3605	1.46558	0.732789	−	0.680456i	$-0.238218\pi$
$193$	20.3605	1.46558	0.732789	+	0.680456i	$0.238218\pi$
$194$	0	0
$195$	4.72973	0.338703
$196$	0	0
$197$	−7.16108	−0.510206	−0.255103	−	0.966914i	$-0.582109\pi$
$197$	−7.16108	−0.510206	−0.255103	+	0.966914i	$0.582109\pi$
$198$	0	0
$199$	21.0951	1.49539	0.747697	−	0.664041i	$-0.231160\pi$
$199$	21.0951	1.49539	0.747697	+	0.664041i	$0.231160\pi$
$200$	0	0
$201$	−1.74900	−0.123365
$202$	0	0
$203$	7.34118	0.515250
$204$	0	0
$205$	10.4787	0.731866
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	7.82978	0.541597
$210$	0	0
$211$	−6.76064	−0.465422	−0.232711	−	0.972546i	$-0.574760\pi$
$211$	−6.76064	−0.465422	−0.232711	+	0.972546i	$0.574760\pi$
$212$	0	0
$213$	6.32954	0.433693
$214$	0	0
$215$	−9.01165	−0.614589
$216$	0	0
$217$	17.9603	1.21922
$218$	0	0
$219$	1.51470	0.102354
$220$	0	0
$221$	−12.3748	−0.832420
$222$	0	0
$223$	−1.58535	−0.106163	−0.0530816	−	0.998590i	$-0.516904\pi$
$223$	−1.58535	−0.106163	−0.0530816	+	0.998590i	$0.516904\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	14.3279	0.950976	0.475488	−	0.879722i	$-0.342272\pi$
$227$	14.3279	0.950976	0.475488	+	0.879722i	$0.342272\pi$
$228$	0	0
$229$	−16.8200	−1.11150	−0.555749	−	0.831350i	$-0.687569\pi$
$229$	−16.8200	−1.11150	−0.555749	+	0.831350i	$0.687569\pi$
$230$	0	0
$231$	−9.94476	−0.654317
$232$	0	0
$233$	−14.2267	−0.932020	−0.466010	−	0.884779i	$-0.654309\pi$
$233$	−14.2267	−0.932020	−0.466010	+	0.884779i	$0.654309\pi$
$234$	0	0
$235$	−2.28192	−0.148856
$236$	0	0
$237$	3.86234	0.250886
$238$	0	0
$239$	18.6279	1.20494	0.602470	−	0.798142i	$-0.294183\pi$
$239$	18.6279	1.20494	0.602470	+	0.798142i	$0.294183\pi$
$240$	0	0
$241$	13.3478	0.859805	0.429902	−	0.902875i	$-0.358548\pi$
$241$	13.3478	0.859805	0.429902	+	0.902875i	$0.358548\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.40019	0.0894550
$246$	0	0
$247$	−10.7929	−0.686733
$248$	0	0
$249$	4.61639	0.292552
$250$	0	0
$251$	12.7425	0.804302	0.402151	−	0.915573i	$-0.368263\pi$
$251$	12.7425	0.804302	0.402151	+	0.915573i	$0.368263\pi$
$252$	0	0
$253$	−3.43123	−0.215719
$254$	0	0
$255$	−2.61639	−0.163845
$256$	0	0
$257$	−17.1079	−1.06716	−0.533582	−	0.845748i	$-0.679154\pi$
$257$	−17.1079	−1.06716	−0.533582	+	0.845748i	$0.679154\pi$
$258$	0	0
$259$	14.1968	0.882147
$260$	0	0
$261$	2.53292	0.156784
$262$	0	0
$263$	0.141565	0.00872927	0.00436463	−	0.999990i	$-0.498611\pi$
$263$	0.141565	0.00872927	0.00436463	+	0.999990i	$0.498611\pi$
$264$	0	0
$265$	0.682109	0.0419016
$266$	0	0
$267$	6.31015	0.386175
$268$	0	0
$269$	−9.69376	−0.591039	−0.295519	−	0.955337i	$-0.595493\pi$
$269$	−9.69376	−0.591039	−0.295519	+	0.955337i	$0.595493\pi$
$270$	0	0
$271$	7.59981	0.461655	0.230828	−	0.972995i	$-0.425857\pi$
$271$	7.59981	0.461655	0.230828	+	0.972995i	$0.425857\pi$
$272$	0	0
$273$	13.7082	0.829659
$274$	0	0
$275$	3.43123	0.206911
$276$	0	0
$277$	−19.2383	−1.15592	−0.577959	−	0.816066i	$-0.696151\pi$
$277$	−19.2383	−1.15592	−0.577959	+	0.816066i	$0.696151\pi$
$278$	0	0
$279$	6.19681	0.370993
$280$	0	0
$281$	10.3887	0.619737	0.309868	−	0.950779i	$-0.399715\pi$
$281$	10.3887	0.619737	0.309868	+	0.950779i	$0.399715\pi$
$282$	0	0
$283$	−20.1095	−1.19538	−0.597691	−	0.801726i	$-0.703915\pi$
$283$	−20.1095	−1.19538	−0.597691	+	0.801726i	$0.703915\pi$
$284$	0	0
$285$	−2.28192	−0.135169
$286$	0	0
$287$	30.3706	1.79272
$288$	0	0
$289$	−10.1545	−0.597324
$290$	0	0
$291$	−17.8741	−1.04780
$292$	0	0
$293$	5.27935	0.308423	0.154211	−	0.988038i	$-0.450716\pi$
$293$	5.27935	0.308423	0.154211	+	0.988038i	$0.450716\pi$
$294$	0	0
$295$	−14.2753	−0.831143
$296$	0	0
$297$	−3.43123	−0.199100
$298$	0	0
$299$	4.72973	0.273527
$300$	0	0
$301$	−26.1185	−1.50545
$302$	0	0
$303$	−3.76570	−0.216334
$304$	0	0
$305$	4.98342	0.285350
$306$	0	0
$307$	−11.7414	−0.670116	−0.335058	−	0.942198i	$-0.608756\pi$
$307$	−11.7414	−0.670116	−0.335058	+	0.942198i	$0.608756\pi$
$308$	0	0
$309$	−9.09512	−0.517403
$310$	0	0
$311$	−23.7039	−1.34412	−0.672062	−	0.740495i	$-0.734591\pi$
$311$	−23.7039	−1.34412	−0.672062	+	0.740495i	$0.734591\pi$
$312$	0	0
$313$	12.6446	0.714716	0.357358	−	0.933967i	$-0.383678\pi$
$313$	12.6446	0.714716	0.357358	+	0.933967i	$0.383678\pi$
$314$	0	0
$315$	2.89831	0.163301
$316$	0	0
$317$	−21.1079	−1.18554	−0.592770	−	0.805372i	$-0.701966\pi$
$317$	−21.1079	−1.18554	−0.592770	+	0.805372i	$0.701966\pi$
$318$	0	0
$319$	8.69102	0.486604
$320$	0	0
$321$	−2.31777	−0.129365
$322$	0	0
$323$	5.97039	0.332201
$324$	0	0
$325$	−4.72973	−0.262358
$326$	0	0
$327$	−12.6463	−0.699340
$328$	0	0
$329$	−6.61370	−0.364625
$330$	0	0
$331$	−10.0266	−0.551111	−0.275556	−	0.961285i	$-0.588862\pi$
$331$	−10.0266	−0.551111	−0.275556	+	0.961285i	$0.588862\pi$
$332$	0	0
$333$	4.89831	0.268426
$334$	0	0
$335$	1.74900	0.0955580
$336$	0	0
$337$	9.08359	0.494815	0.247407	−	0.968912i	$-0.420421\pi$
$337$	9.08359	0.494815	0.247407	+	0.968912i	$0.420421\pi$
$338$	0	0
$339$	−6.11439	−0.332088
$340$	0	0
$341$	21.2627	1.15144
$342$	0	0
$343$	−16.2300	−0.876336
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	−18.7541	−1.00677	−0.503386	−	0.864062i	$-0.667912\pi$
$347$	−18.7541	−1.00677	−0.503386	+	0.864062i	$0.667912\pi$
$348$	0	0
$349$	−12.9276	−0.691998	−0.345999	−	0.938235i	$-0.612460\pi$
$349$	−12.9276	−0.691998	−0.345999	+	0.938235i	$0.612460\pi$
$350$	0	0
$351$	4.72973	0.252454
$352$	0	0
$353$	13.5712	0.722320	0.361160	−	0.932504i	$-0.382381\pi$
$353$	13.5712	0.722320	0.361160	+	0.932504i	$0.382381\pi$
$354$	0	0
$355$	−6.32954	−0.335937
$356$	0	0
$357$	−7.58311	−0.401341
$358$	0	0
$359$	−27.3346	−1.44267	−0.721333	−	0.692589i	$-0.756470\pi$
$359$	−27.3346	−1.44267	−0.721333	+	0.692589i	$0.756470\pi$
$360$	0	0
$361$	−13.7929	−0.725940
$362$	0	0
$363$	−0.773325	−0.0405891
$364$	0	0
$365$	−1.51470	−0.0792830
$366$	0	0
$367$	14.0591	0.733881	0.366941	−	0.930244i	$-0.380405\pi$
$367$	14.0591	0.733881	0.366941	+	0.930244i	$0.380405\pi$
$368$	0	0
$369$	10.4787	0.545501
$370$	0	0
$371$	1.97696	0.102639
$372$	0	0
$373$	15.6590	0.810790	0.405395	−	0.914142i	$-0.367134\pi$
$373$	15.6590	0.810790	0.405395	+	0.914142i	$0.367134\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−11.9800	−0.617002
$378$	0	0
$379$	38.5121	1.97824	0.989118	−	0.147124i	$-0.0470018\pi$
$379$	38.5121	1.97824	0.989118	+	0.147124i	$0.0470018\pi$
$380$	0	0
$381$	−4.13273	−0.211726
$382$	0	0
$383$	1.61627	0.0825875	0.0412938	−	0.999147i	$-0.486852\pi$
$383$	1.61627	0.0825875	0.0412938	+	0.999147i	$0.486852\pi$
$384$	0	0
$385$	9.94476	0.506832
$386$	0	0
$387$	−9.01165	−0.458088
$388$	0	0
$389$	25.1844	1.27690	0.638449	−	0.769664i	$-0.279576\pi$
$389$	25.1844	1.27690	0.638449	+	0.769664i	$0.279576\pi$
$390$	0	0
$391$	−2.61639	−0.132317
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.86234	−0.194335
$396$	0	0
$397$	−22.3065	−1.11953	−0.559766	−	0.828651i	$-0.689109\pi$
$397$	−22.3065	−1.11953	−0.559766	+	0.828651i	$0.689109\pi$
$398$	0	0
$399$	−6.61370	−0.331099
$400$	0	0
$401$	−37.4014	−1.86774	−0.933868	−	0.357618i	$-0.883589\pi$
$401$	−37.4014	−1.86774	−0.933868	+	0.357618i	$0.883589\pi$
$402$	0	0
$403$	−29.3092	−1.46000
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	16.8072	0.833103
$408$	0	0
$409$	32.1869	1.59154	0.795771	−	0.605598i	$-0.207066\pi$
$409$	32.1869	1.59154	0.795771	+	0.605598i	$0.207066\pi$
$410$	0	0
$411$	−4.49812	−0.221876
$412$	0	0
$413$	−41.3743	−2.03590
$414$	0	0
$415$	−4.61639	−0.226610
$416$	0	0
$417$	−14.7606	−0.722832
$418$	0	0
$419$	−13.0012	−0.635149	−0.317574	−	0.948233i	$-0.602868\pi$
$419$	−13.0012	−0.635149	−0.317574	+	0.948233i	$0.602868\pi$
$420$	0	0
$421$	16.4064	0.799601	0.399800	−	0.916602i	$-0.369079\pi$
$421$	16.4064	0.799601	0.399800	+	0.916602i	$0.369079\pi$
$422$	0	0
$423$	−2.28192	−0.110951
$424$	0	0
$425$	2.61639	0.126914
$426$	0	0
$427$	14.4435	0.698969
$428$	0	0
$429$	16.2288	0.783533
$430$	0	0
$431$	−15.7581	−0.759040	−0.379520	−	0.925184i	$-0.623911\pi$
$431$	−15.7581	−0.759040	−0.379520	+	0.925184i	$0.623911\pi$
$432$	0	0
$433$	20.9101	1.00487	0.502437	−	0.864614i	$-0.332437\pi$
$433$	20.9101	1.00487	0.502437	+	0.864614i	$0.332437\pi$
$434$	0	0
$435$	−2.53292	−0.121444
$436$	0	0
$437$	−2.28192	−0.109159
$438$	0	0
$439$	−9.85307	−0.470261	−0.235131	−	0.971964i	$-0.575552\pi$
$439$	−9.85307	−0.470261	−0.235131	+	0.971964i	$0.575552\pi$
$440$	0	0
$441$	1.40019	0.0666758
$442$	0	0
$443$	−22.2069	−1.05508	−0.527542	−	0.849529i	$-0.676886\pi$
$443$	−22.2069	−1.05508	−0.527542	+	0.849529i	$0.676886\pi$
$444$	0	0
$445$	−6.31015	−0.299130
$446$	0	0
$447$	−9.34776	−0.442134
$448$	0	0
$449$	−37.6783	−1.77815	−0.889075	−	0.457761i	$-0.848652\pi$
$449$	−37.6783	−1.77815	−0.889075	+	0.457761i	$0.848652\pi$
$450$	0	0
$451$	35.9549	1.69305
$452$	0	0
$453$	−12.2267	−0.574459
$454$	0	0
$455$	−13.7082	−0.642651
$456$	0	0
$457$	12.8116	0.599299	0.299650	−	0.954049i	$-0.403130\pi$
$457$	12.8116	0.599299	0.299650	+	0.954049i	$0.403130\pi$
$458$	0	0
$459$	−2.61639	−0.122123
$460$	0	0
$461$	25.8610	1.20446	0.602232	−	0.798321i	$-0.294278\pi$
$461$	25.8610	1.20446	0.602232	+	0.798321i	$0.294278\pi$
$462$	0	0
$463$	−10.8999	−0.506564	−0.253282	−	0.967393i	$-0.581510\pi$
$463$	−10.8999	−0.506564	−0.253282	+	0.967393i	$0.581510\pi$
$464$	0	0
$465$	−6.19681	−0.287370
$466$	0	0
$467$	15.8757	0.734642	0.367321	−	0.930094i	$-0.380275\pi$
$467$	15.8757	0.734642	0.367321	+	0.930094i	$0.380275\pi$
$468$	0	0
$469$	5.06914	0.234071
$470$	0	0
$471$	−0.334473	−0.0154117
$472$	0	0
$473$	−30.9210	−1.42175
$474$	0	0
$475$	2.28192	0.104702
$476$	0	0
$477$	0.682109	0.0312316
$478$	0	0
$479$	13.0129	0.594576	0.297288	−	0.954788i	$-0.403918\pi$
$479$	13.0129	0.594576	0.297288	+	0.954788i	$0.403918\pi$
$480$	0	0
$481$	−23.1677	−1.05635
$482$	0	0
$483$	2.89831	0.131878
$484$	0	0
$485$	17.8741	0.811621
$486$	0	0
$487$	−1.56865	−0.0710824	−0.0355412	−	0.999368i	$-0.511315\pi$
$487$	−1.56865	−0.0710824	−0.0355412	+	0.999368i	$0.511315\pi$
$488$	0	0
$489$	12.0952	0.546966
$490$	0	0
$491$	13.9382	0.629021	0.314511	−	0.949254i	$-0.398160\pi$
$491$	13.9382	0.629021	0.314511	+	0.949254i	$0.398160\pi$
$492$	0	0
$493$	6.62711	0.298470
$494$	0	0
$495$	3.43123	0.154222
$496$	0	0
$497$	−18.3449	−0.822883
$498$	0	0
$499$	7.56103	0.338478	0.169239	−	0.985575i	$-0.445869\pi$
$499$	7.56103	0.338478	0.169239	+	0.985575i	$0.445869\pi$
$500$	0	0
$501$	19.3346	0.863807
$502$	0	0
$503$	7.34118	0.327327	0.163664	−	0.986516i	$-0.447669\pi$
$503$	7.34118	0.327327	0.163664	+	0.986516i	$0.447669\pi$
$504$	0	0
$505$	3.76570	0.167571
$506$	0	0
$507$	−9.37032	−0.416151
$508$	0	0
$509$	2.74619	0.121723	0.0608614	−	0.998146i	$-0.480615\pi$
$509$	2.74619	0.121723	0.0608614	+	0.998146i	$0.480615\pi$
$510$	0	0
$511$	−4.39006	−0.194205
$512$	0	0
$513$	−2.28192	−0.100749
$514$	0	0
$515$	9.09512	0.400779
$516$	0	0
$517$	−7.82978	−0.344353
$518$	0	0
$519$	0.203383	0.00892753
$520$	0	0
$521$	−14.4971	−0.635128	−0.317564	−	0.948237i	$-0.602865\pi$
$521$	−14.4971	−0.635128	−0.317564	+	0.948237i	$0.602865\pi$
$522$	0	0
$523$	−39.4286	−1.72409	−0.862045	−	0.506832i	$-0.830817\pi$
$523$	−39.4286	−1.72409	−0.862045	+	0.506832i	$0.830817\pi$
$524$	0	0
$525$	−2.89831	−0.126493
$526$	0	0
$527$	16.2133	0.706261
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−14.2753	−0.619497
$532$	0	0
$533$	−49.5615	−2.14675
$534$	0	0
$535$	2.31777	0.100206
$536$	0	0
$537$	5.62968	0.242938
$538$	0	0
$539$	4.80438	0.206939
$540$	0	0
$541$	−6.74393	−0.289944	−0.144972	−	0.989436i	$-0.546309\pi$
$541$	−6.74393	−0.289944	−0.144972	+	0.989436i	$0.546309\pi$
$542$	0	0
$543$	−14.9282	−0.640629
$544$	0	0
$545$	12.6463	0.541706
$546$	0	0
$547$	−25.7249	−1.09992	−0.549959	−	0.835192i	$-0.685357\pi$
$547$	−25.7249	−1.09992	−0.549959	+	0.835192i	$0.685357\pi$
$548$	0	0
$549$	4.98342	0.212687
$550$	0	0
$551$	5.77991	0.246233
$552$	0	0
$553$	−11.1942	−0.476027
$554$	0	0
$555$	−4.89831	−0.207922
$556$	0	0
$557$	−38.3374	−1.62441	−0.812204	−	0.583373i	$-0.801733\pi$
$557$	−38.3374	−1.62441	−0.812204	+	0.583373i	$0.801733\pi$
$558$	0	0
$559$	42.6226	1.80275
$560$	0	0
$561$	−8.97743	−0.379027
$562$	0	0
$563$	−31.8757	−1.34340	−0.671701	−	0.740822i	$-0.734436\pi$
$563$	−31.8757	−1.34340	−0.671701	+	0.740822i	$0.734436\pi$
$564$	0	0
$565$	6.11439	0.257234
$566$	0	0
$567$	2.89831	0.121718
$568$	0	0
$569$	41.4249	1.73662	0.868311	−	0.496020i	$-0.165206\pi$
$569$	41.4249	1.73662	0.868311	+	0.496020i	$0.165206\pi$
$570$	0	0
$571$	−44.5674	−1.86509	−0.932544	−	0.361057i	$-0.882416\pi$
$571$	−44.5674	−1.86509	−0.932544	+	0.361057i	$0.882416\pi$
$572$	0	0
$573$	−15.7414	−0.657605
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	30.1418	1.25482	0.627410	−	0.778689i	$-0.284115\pi$
$577$	30.1418	1.25482	0.627410	+	0.778689i	$0.284115\pi$
$578$	0	0
$579$	−20.3605	−0.846152
$580$	0	0
$581$	−13.3797	−0.555084
$582$	0	0
$583$	2.34047	0.0969325
$584$	0	0
$585$	−4.72973	−0.195550
$586$	0	0
$587$	−15.5215	−0.640642	−0.320321	−	0.947309i	$-0.603791\pi$
$587$	−15.5215	−0.640642	−0.320321	+	0.947309i	$0.603791\pi$
$588$	0	0
$589$	14.1406	0.582654
$590$	0	0
$591$	7.16108	0.294567
$592$	0	0
$593$	−25.4042	−1.04323	−0.521613	−	0.853182i	$-0.674669\pi$
$593$	−25.4042	−1.04323	−0.521613	+	0.853182i	$0.674669\pi$
$594$	0	0
$595$	7.58311	0.310877
$596$	0	0
$597$	−21.0951	−0.863366
$598$	0	0
$599$	−19.3721	−0.791522	−0.395761	−	0.918353i	$-0.629519\pi$
$599$	−19.3721	−0.791522	−0.395761	+	0.918353i	$0.629519\pi$
$600$	0	0
$601$	6.62687	0.270316	0.135158	−	0.990824i	$-0.456846\pi$
$601$	6.62687	0.270316	0.135158	+	0.990824i	$0.456846\pi$
$602$	0	0
$603$	1.74900	0.0712247
$604$	0	0
$605$	0.773325	0.0314401
$606$	0	0
$607$	5.99895	0.243490	0.121745	−	0.992561i	$-0.461151\pi$
$607$	5.99895	0.243490	0.121745	+	0.992561i	$0.461151\pi$
$608$	0	0
$609$	−7.34118	−0.297480
$610$	0	0
$611$	10.7929	0.436632
$612$	0	0
$613$	13.5640	0.547843	0.273922	−	0.961752i	$-0.411679\pi$
$613$	13.5640	0.547843	0.273922	+	0.961752i	$0.411679\pi$
$614$	0	0
$615$	−10.4787	−0.422543
$616$	0	0
$617$	−27.9089	−1.12357	−0.561785	−	0.827283i	$-0.689885\pi$
$617$	−27.9089	−1.12357	−0.561785	+	0.827283i	$0.689885\pi$
$618$	0	0
$619$	23.7482	0.954521	0.477260	−	0.878762i	$-0.341630\pi$
$619$	23.7482	0.954521	0.477260	+	0.878762i	$0.341630\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−18.2887	−0.732723
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.82978	−0.312691
$628$	0	0
$629$	12.8159	0.511003
$630$	0	0
$631$	−13.5108	−0.537857	−0.268928	−	0.963160i	$-0.586670\pi$
$631$	−13.5108	−0.537857	−0.268928	+	0.963160i	$0.586670\pi$
$632$	0	0
$633$	6.76064	0.268711
$634$	0	0
$635$	4.13273	0.164002
$636$	0	0
$637$	−6.62252	−0.262394
$638$	0	0
$639$	−6.32954	−0.250393
$640$	0	0
$641$	−38.5556	−1.52286	−0.761428	−	0.648250i	$-0.775501\pi$
$641$	−38.5556	−1.52286	−0.761428	+	0.648250i	$0.775501\pi$
$642$	0	0
$643$	−20.3848	−0.803897	−0.401949	−	0.915662i	$-0.631667\pi$
$643$	−20.3848	−0.803897	−0.401949	+	0.915662i	$0.631667\pi$
$644$	0	0
$645$	9.01165	0.354833
$646$	0	0
$647$	−13.3710	−0.525670	−0.262835	−	0.964841i	$-0.584658\pi$
$647$	−13.3710	−0.525670	−0.262835	+	0.964841i	$0.584658\pi$
$648$	0	0
$649$	−48.9820	−1.92271
$650$	0	0
$651$	−17.9603	−0.703918
$652$	0	0
$653$	−6.63699	−0.259726	−0.129863	−	0.991532i	$-0.541454\pi$
$653$	−6.63699	−0.259726	−0.129863	+	0.991532i	$0.541454\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.51470	−0.0590940
$658$	0	0
$659$	−25.2897	−0.985146	−0.492573	−	0.870271i	$-0.663944\pi$
$659$	−25.2897	−0.985146	−0.492573	+	0.870271i	$0.663944\pi$
$660$	0	0
$661$	11.8292	0.460101	0.230051	−	0.973179i	$-0.426111\pi$
$661$	11.8292	0.460101	0.230051	+	0.973179i	$0.426111\pi$
$662$	0	0
$663$	12.3748	0.480598
$664$	0	0
$665$	6.61370	0.256468
$666$	0	0
$667$	−2.53292	−0.0980750
$668$	0	0
$669$	1.58535	0.0612933
$670$	0	0
$671$	17.0992	0.660109
$672$	0	0
$673$	12.2819	0.473433	0.236717	−	0.971579i	$-0.423929\pi$
$673$	12.2819	0.473433	0.236717	+	0.971579i	$0.423929\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	17.8065	0.684360	0.342180	−	0.939635i	$-0.388835\pi$
$677$	17.8065	0.684360	0.342180	+	0.939635i	$0.388835\pi$
$678$	0	0
$679$	51.8047	1.98808
$680$	0	0
$681$	−14.3279	−0.549046
$682$	0	0
$683$	−41.3678	−1.58289	−0.791447	−	0.611238i	$-0.790672\pi$
$683$	−41.3678	−1.58289	−0.791447	+	0.611238i	$0.790672\pi$
$684$	0	0
$685$	4.49812	0.171864
$686$	0	0
$687$	16.8200	0.641724
$688$	0	0
$689$	−3.22619	−0.122908
$690$	0	0
$691$	−43.6530	−1.66064	−0.830319	−	0.557289i	$-0.811842\pi$
$691$	−43.6530	−1.66064	−0.830319	+	0.557289i	$0.811842\pi$
$692$	0	0
$693$	9.94476	0.377770
$694$	0	0
$695$	14.7606	0.559903
$696$	0	0
$697$	27.4164	1.03847
$698$	0	0
$699$	14.2267	0.538102
$700$	0	0
$701$	−43.2396	−1.63314	−0.816569	−	0.577248i	$-0.804127\pi$
$701$	−43.2396	−1.63314	−0.816569	+	0.577248i	$0.804127\pi$
$702$	0	0
$703$	11.1775	0.421569
$704$	0	0
$705$	2.28192	0.0859420
$706$	0	0
$707$	10.9142	0.410469
$708$	0	0
$709$	50.2595	1.88754	0.943768	−	0.330609i	$-0.107254\pi$
$709$	50.2595	1.88754	0.943768	+	0.330609i	$0.107254\pi$
$710$	0	0
$711$	−3.86234	−0.144849
$712$	0	0
$713$	−6.19681	−0.232072
$714$	0	0
$715$	−16.2288	−0.606922
$716$	0	0
$717$	−18.6279	−0.695672
$718$	0	0
$719$	37.8254	1.41065	0.705325	−	0.708884i	$-0.250801\pi$
$719$	37.8254	1.41065	0.705325	+	0.708884i	$0.250801\pi$
$720$	0	0
$721$	26.3605	0.981715
$722$	0	0
$723$	−13.3478	−0.496408
$724$	0	0
$725$	2.53292	0.0940703
$726$	0	0
$727$	−30.9548	−1.14805	−0.574024	−	0.818838i	$-0.694619\pi$
$727$	−30.9548	−1.14805	−0.574024	+	0.818838i	$0.694619\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−23.5780	−0.872063
$732$	0	0
$733$	42.9781	1.58743	0.793715	−	0.608289i	$-0.208144\pi$
$733$	42.9781	1.58743	0.793715	+	0.608289i	$0.208144\pi$
$734$	0	0
$735$	−1.40019	−0.0516468
$736$	0	0
$737$	6.00121	0.221057
$738$	0	0
$739$	12.0322	0.442612	0.221306	−	0.975204i	$-0.428968\pi$
$739$	12.0322	0.442612	0.221306	+	0.975204i	$0.428968\pi$
$740$	0	0
$741$	10.7929	0.396485
$742$	0	0
$743$	11.8686	0.435415	0.217708	−	0.976014i	$-0.430142\pi$
$743$	11.8686	0.435415	0.217708	+	0.976014i	$0.430142\pi$
$744$	0	0
$745$	9.34776	0.342475
$746$	0	0
$747$	−4.61639	−0.168905
$748$	0	0
$749$	6.71761	0.245456
$750$	0	0
$751$	43.7022	1.59472	0.797358	−	0.603506i	$-0.206230\pi$
$751$	43.7022	1.59472	0.797358	+	0.603506i	$0.206230\pi$
$752$	0	0
$753$	−12.7425	−0.464364
$754$	0	0
$755$	12.2267	0.444974
$756$	0	0
$757$	−37.3155	−1.35625	−0.678127	−	0.734945i	$-0.737208\pi$
$757$	−37.3155	−1.35625	−0.678127	+	0.734945i	$0.737208\pi$
$758$	0	0
$759$	3.43123	0.124546
$760$	0	0
$761$	−9.45167	−0.342623	−0.171311	−	0.985217i	$-0.554800\pi$
$761$	−9.45167	−0.342623	−0.171311	+	0.985217i	$0.554800\pi$
$762$	0	0
$763$	36.6528	1.32692
$764$	0	0
$765$	2.61639	0.0945958
$766$	0	0
$767$	67.5185	2.43795
$768$	0	0
$769$	−4.74724	−0.171190	−0.0855949	−	0.996330i	$-0.527279\pi$
$769$	−4.74724	−0.171190	−0.0855949	+	0.996330i	$0.527279\pi$
$770$	0	0
$771$	17.1079	0.616127
$772$	0	0
$773$	11.1841	0.402265	0.201133	−	0.979564i	$-0.435538\pi$
$773$	11.1841	0.402265	0.201133	+	0.979564i	$0.435538\pi$
$774$	0	0
$775$	6.19681	0.222596
$776$	0	0
$777$	−14.1968	−0.509308
$778$	0	0
$779$	23.9116	0.856722
$780$	0	0
$781$	−21.7181	−0.777134
$782$	0	0
$783$	−2.53292	−0.0905192
$784$	0	0
$785$	0.334473	0.0119378
$786$	0	0
$787$	−31.9780	−1.13989	−0.569947	−	0.821682i	$-0.693036\pi$
$787$	−31.9780	−1.13989	−0.569947	+	0.821682i	$0.693036\pi$
$788$	0	0
$789$	−0.141565	−0.00503985
$790$	0	0
$791$	17.7214	0.630100
$792$	0	0
$793$	−23.5702	−0.837003
$794$	0	0
$795$	−0.682109	−0.0241919
$796$	0	0
$797$	40.5354	1.43584	0.717919	−	0.696126i	$-0.245095\pi$
$797$	40.5354	1.43584	0.717919	+	0.696126i	$0.245095\pi$
$798$	0	0
$799$	−5.97039	−0.211217
$800$	0	0
$801$	−6.31015	−0.222958
$802$	0	0
$803$	−5.19728	−0.183408
$804$	0	0
$805$	−2.89831	−0.102152
$806$	0	0
$807$	9.69376	0.341236
$808$	0	0
$809$	−17.1178	−0.601829	−0.300915	−	0.953651i	$-0.597292\pi$
$809$	−17.1178	−0.601829	−0.300915	+	0.953651i	$0.597292\pi$
$810$	0	0
$811$	29.6198	1.04009	0.520046	−	0.854138i	$-0.325915\pi$
$811$	29.6198	1.04009	0.520046	+	0.854138i	$0.325915\pi$
$812$	0	0
$813$	−7.59981	−0.266537
$814$	0	0
$815$	−12.0952	−0.423678
$816$	0	0
$817$	−20.5638	−0.719438
$818$	0	0
$819$	−13.7082	−0.479004
$820$	0	0
$821$	−28.1958	−0.984039	−0.492020	−	0.870584i	$-0.663741\pi$
$821$	−28.1958	−0.984039	−0.492020	+	0.870584i	$0.663741\pi$
$822$	0	0
$823$	41.1535	1.43452	0.717260	−	0.696806i	$-0.245396\pi$
$823$	41.1535	1.43452	0.717260	+	0.696806i	$0.245396\pi$
$824$	0	0
$825$	−3.43123	−0.119460
$826$	0	0
$827$	14.8433	0.516152	0.258076	−	0.966125i	$-0.416911\pi$
$827$	14.8433	0.516152	0.258076	+	0.966125i	$0.416911\pi$
$828$	0	0
$829$	37.1778	1.29124	0.645619	−	0.763660i	$-0.276599\pi$
$829$	37.1778	1.29124	0.645619	+	0.763660i	$0.276599\pi$
$830$	0	0
$831$	19.2383	0.667370
$832$	0	0
$833$	3.66345	0.126931
$834$	0	0
$835$	−19.3346	−0.669102
$836$	0	0
$837$	−6.19681	−0.214193
$838$	0	0
$839$	−35.4131	−1.22260	−0.611299	−	0.791400i	$-0.709353\pi$
$839$	−35.4131	−1.22260	−0.611299	+	0.791400i	$0.709353\pi$
$840$	0	0
$841$	−22.5843	−0.778770
$842$	0	0
$843$	−10.3887	−0.357805
$844$	0	0
$845$	9.37032	0.322349
$846$	0	0
$847$	2.24134	0.0770132
$848$	0	0
$849$	20.1095	0.690155
$850$	0	0
$851$	−4.89831	−0.167912
$852$	0	0
$853$	7.58863	0.259830	0.129915	−	0.991525i	$-0.458530\pi$
$853$	7.58863	0.259830	0.129915	+	0.991525i	$0.458530\pi$
$854$	0	0
$855$	2.28192	0.0780399
$856$	0	0
$857$	−14.3219	−0.489227	−0.244614	−	0.969621i	$-0.578661\pi$
$857$	−14.3219	−0.489227	−0.244614	+	0.969621i	$0.578661\pi$
$858$	0	0
$859$	−50.9977	−1.74002	−0.870010	−	0.493035i	$-0.835887\pi$
$859$	−50.9977	−1.74002	−0.870010	+	0.493035i	$0.835887\pi$
$860$	0	0
$861$	−30.3706	−1.03503
$862$	0	0
$863$	−45.4463	−1.54701	−0.773505	−	0.633790i	$-0.781498\pi$
$863$	−45.4463	−1.54701	−0.773505	+	0.633790i	$0.781498\pi$
$864$	0	0
$865$	−0.203383	−0.00691524
$866$	0	0
$867$	10.1545	0.344865
$868$	0	0
$869$	−13.2526	−0.449562
$870$	0	0
$871$	−8.27229	−0.280296
$872$	0	0
$873$	17.8741	0.604947
$874$	0	0
$875$	2.89831	0.0979807
$876$	0	0
$877$	−12.8832	−0.435036	−0.217518	−	0.976056i	$-0.569796\pi$
$877$	−12.8832	−0.435036	−0.217518	+	0.976056i	$0.569796\pi$
$878$	0	0
$879$	−5.27935	−0.178068
$880$	0	0
$881$	−33.0480	−1.11342	−0.556708	−	0.830709i	$-0.687936\pi$
$881$	−33.0480	−1.11342	−0.556708	+	0.830709i	$0.687936\pi$
$882$	0	0
$883$	−7.94710	−0.267441	−0.133721	−	0.991019i	$-0.542692\pi$
$883$	−7.94710	−0.267441	−0.133721	+	0.991019i	$0.542692\pi$
$884$	0	0
$885$	14.2753	0.479860
$886$	0	0
$887$	35.5314	1.19303	0.596514	−	0.802603i	$-0.296552\pi$
$887$	35.5314	1.19303	0.596514	+	0.802603i	$0.296552\pi$
$888$	0	0
$889$	11.9779	0.401727
$890$	0	0
$891$	3.43123	0.114950
$892$	0	0
$893$	−5.20715	−0.174251
$894$	0	0
$895$	−5.62968	−0.188179
$896$	0	0
$897$	−4.72973	−0.157921
$898$	0	0
$899$	15.6960	0.523491
$900$	0	0
$901$	1.78466	0.0594558
$902$	0	0
$903$	26.1185	0.869170
$904$	0	0
$905$	14.9282	0.496229
$906$	0	0
$907$	−55.2938	−1.83600	−0.918001	−	0.396579i	$-0.870197\pi$
$907$	−55.2938	−1.83600	−0.918001	+	0.396579i	$0.870197\pi$
$908$	0	0
$909$	3.76570	0.124900
$910$	0	0
$911$	40.6203	1.34581	0.672905	−	0.739729i	$-0.265046\pi$
$911$	40.6203	1.34581	0.672905	+	0.739729i	$0.265046\pi$
$912$	0	0
$913$	−15.8399	−0.524224
$914$	0	0
$915$	−4.98342	−0.164747
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−40.0994	−1.32276	−0.661378	−	0.750052i	$-0.730028\pi$
$919$	−40.0994	−1.32276	−0.661378	+	0.750052i	$0.730028\pi$
$920$	0	0
$921$	11.7414	0.386891
$922$	0	0
$923$	29.9370	0.985388
$924$	0	0
$925$	4.89831	0.161055
$926$	0	0
$927$	9.09512	0.298723
$928$	0	0
$929$	21.3976	0.702034	0.351017	−	0.936369i	$-0.385836\pi$
$929$	21.3976	0.702034	0.351017	+	0.936369i	$0.385836\pi$
$930$	0	0
$931$	3.19512	0.104716
$932$	0	0
$933$	23.7039	0.776030
$934$	0	0
$935$	8.97743	0.293593
$936$	0	0
$937$	−48.0879	−1.57096	−0.785481	−	0.618886i	$-0.787584\pi$
$937$	−48.0879	−1.57096	−0.785481	+	0.618886i	$0.787584\pi$
$938$	0	0
$939$	−12.6446	−0.412642
$940$	0	0
$941$	35.9704	1.17260	0.586301	−	0.810094i	$-0.300584\pi$
$941$	35.9704	1.17260	0.586301	+	0.810094i	$0.300584\pi$
$942$	0	0
$943$	−10.4787	−0.341234
$944$	0	0
$945$	−2.89831	−0.0942820
$946$	0	0
$947$	38.1568	1.23993	0.619965	−	0.784629i	$-0.287147\pi$
$947$	38.1568	1.23993	0.619965	+	0.784629i	$0.287147\pi$
$948$	0	0
$949$	7.16411	0.232557
$950$	0	0
$951$	21.1079	0.684472
$952$	0	0
$953$	19.7187	0.638751	0.319375	−	0.947628i	$-0.396527\pi$
$953$	19.7187	0.638751	0.319375	+	0.947628i	$0.396527\pi$
$954$	0	0
$955$	15.7414	0.509379
$956$	0	0
$957$	−8.69102	−0.280941
$958$	0	0
$959$	13.0369	0.420984
$960$	0	0
$961$	7.40043	0.238724
$962$	0	0
$963$	2.31777	0.0746891
$964$	0	0
$965$	20.3605	0.655426
$966$	0	0
$967$	−49.9496	−1.60627	−0.803135	−	0.595797i	$-0.796836\pi$
$967$	−49.9496	−1.60627	−0.803135	+	0.595797i	$0.796836\pi$
$968$	0	0
$969$	−5.97039	−0.191797
$970$	0	0
$971$	−41.4016	−1.32864	−0.664321	−	0.747448i	$-0.731279\pi$
$971$	−41.4016	−1.32864	−0.664321	+	0.747448i	$0.731279\pi$
$972$	0	0
$973$	42.7809	1.37149
$974$	0	0
$975$	4.72973	0.151473
$976$	0	0
$977$	34.7018	1.11021	0.555104	−	0.831781i	$-0.312679\pi$
$977$	34.7018	1.11021	0.555104	+	0.831781i	$0.312679\pi$
$978$	0	0
$979$	−21.6515	−0.691986
$980$	0	0
$981$	12.6463	0.403764
$982$	0	0
$983$	−30.8239	−0.983131	−0.491565	−	0.870841i	$-0.663575\pi$
$983$	−30.8239	−0.983131	−0.491565	+	0.870841i	$0.663575\pi$
$984$	0	0
$985$	−7.16108	−0.228171
$986$	0	0
$987$	6.61370	0.210516
$988$	0	0
$989$	9.01165	0.286554
$990$	0	0
$991$	1.53774	0.0488478	0.0244239	−	0.999702i	$-0.492225\pi$
$991$	1.53774	0.0488478	0.0244239	+	0.999702i	$0.492225\pi$
$992$	0	0
$993$	10.0266	0.318184
$994$	0	0
$995$	21.0951	0.668760
$996$	0	0
$997$	−53.8478	−1.70538	−0.852688	−	0.522420i	$-0.825029\pi$
$997$	−53.8478	−1.70538	−0.852688	+	0.522420i	$0.825029\pi$
$998$	0	0
$999$	−4.89831	−0.154976

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2760.2.a.w.1.5	✓	5
3.2	odd	2		8280.2.a.br.1.5		5
4.3	odd	2		5520.2.a.cc.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.w.1.5	✓	5	1.1	even	1	trivial
5520.2.a.cc.1.1		5	4.3	odd	2
8280.2.a.br.1.5		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2760.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +2.89831 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +2.89831 q^{7} +1.00000 q^{9} +3.43123 q^{11} -4.72973 q^{13} -1.00000 q^{15} +2.61639 q^{17} +2.28192 q^{19} -2.89831 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.53292 q^{29} +6.19681 q^{31} -3.43123 q^{33} +2.89831 q^{35} +4.89831 q^{37} +4.72973 q^{39} +10.4787 q^{41} -9.01165 q^{43} +1.00000 q^{45} -2.28192 q^{47} +1.40019 q^{49} -2.61639 q^{51} +0.682109 q^{53} +3.43123 q^{55} -2.28192 q^{57} -14.2753 q^{59} +4.98342 q^{61} +2.89831 q^{63} -4.72973 q^{65} +1.74900 q^{67} +1.00000 q^{69} -6.32954 q^{71} -1.51470 q^{73} -1.00000 q^{75} +9.94476 q^{77} -3.86234 q^{79} +1.00000 q^{81} -4.61639 q^{83} +2.61639 q^{85} -2.53292 q^{87} -6.31015 q^{89} -13.7082 q^{91} -6.19681 q^{93} +2.28192 q^{95} +17.8741 q^{97} +3.43123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 5 q^{5} - 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 5 * q^5 - 4 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + 5 q^{5} - 4 q^{7} + 5 q^{9} - 4 q^{11} + 4 q^{13} - 5 q^{15} + 10 q^{17} - 4 q^{19} + 4 q^{21} - 5 q^{23} + 5 q^{25} - 5 q^{27} + 10 q^{29} + 6 q^{31} + 4 q^{33} - 4 q^{35} + 6 q^{37} - 4 q^{39} + 12 q^{41} - 2 q^{43} + 5 q^{45} + 4 q^{47} + 19 q^{49} - 10 q^{51} - 4 q^{55} + 4 q^{57} + 6 q^{59} + 16 q^{61} - 4 q^{63} + 4 q^{65} - 4 q^{67} + 5 q^{69} + 8 q^{71} + 14 q^{73} - 5 q^{75} + 16 q^{77} + 18 q^{79} + 5 q^{81} - 20 q^{83} + 10 q^{85} - 10 q^{87} + 18 q^{89} + 20 q^{91} - 6 q^{93} - 4 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 5 * q^5 - 4 * q^7 + 5 * q^9 - 4 * q^11 + 4 * q^13 - 5 * q^15 + 10 * q^17 - 4 * q^19 + 4 * q^21 - 5 * q^23 + 5 * q^25 - 5 * q^27 + 10 * q^29 + 6 * q^31 + 4 * q^33 - 4 * q^35 + 6 * q^37 - 4 * q^39 + 12 * q^41 - 2 * q^43 + 5 * q^45 + 4 * q^47 + 19 * q^49 - 10 * q^51 - 4 * q^55 + 4 * q^57 + 6 * q^59 + 16 * q^61 - 4 * q^63 + 4 * q^65 - 4 * q^67 + 5 * q^69 + 8 * q^71 + 14 * q^73 - 5 * q^75 + 16 * q^77 + 18 * q^79 + 5 * q^81 - 20 * q^83 + 10 * q^85 - 10 * q^87 + 18 * q^89 + 20 * q^91 - 6 * q^93 - 4 * q^95 + 4 * q^97 - 4 * q^99

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