LMFDB - Embedded newform 5148.2.a.t.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5148,2,Mod(1,5148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5148.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:	$41.1069869606$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.815952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 9x - 2 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 9x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.59369$ of defining polynomial
Character	$\chi$	$=$	5148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.95214 q^{5} -2.34634 q^{7} +O(q^{10})$	$q+3.95214 q^{5} -2.34634 q^{7} -1.00000 q^{11} -1.00000 q^{13} -7.66724 q^{17} -1.89650 q^{19} +6.65965 q^{23} +10.6194 q^{25} -2.03141 q^{29} +5.72450 q^{31} -9.27304 q^{35} -0.448456 q^{37} -7.75591 q^{41} +10.1619 q^{43} +7.14113 q^{47} -1.49470 q^{49} +4.44983 q^{53} -3.95214 q^{55} +14.6418 q^{59} +11.5401 q^{61} -3.95214 q^{65} -2.87245 q^{67} -2.35273 q^{71} +15.4622 q^{73} +2.34634 q^{77} +8.11708 q^{79} +13.2026 q^{83} -30.3020 q^{85} +17.6750 q^{89} +2.34634 q^{91} -7.49524 q^{95} -1.34812 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 5 * q + 2 * q^5 - 2 * q^7	$ 5 q + 2 q^{5} - 2 q^{7} - 5 q^{11} - 5 q^{13} - 2 q^{17} - 2 q^{19} + 12 q^{23} - q^{25} + 4 q^{29} - 2 q^{31} - 2 q^{35} + 6 q^{41} + 14 q^{43} + 14 q^{47} - 7 q^{49} + 20 q^{53} - 2 q^{55} + 20 q^{59} - 2 q^{65} + 10 q^{67} + 26 q^{71} + 16 q^{73} + 2 q^{77} + 2 q^{79} + 16 q^{83} - 18 q^{85} + 24 q^{89} + 2 q^{91} + 22 q^{95}+O(q^{100}) $ 5 * q + 2 * q^5 - 2 * q^7 - 5 * q^11 - 5 * q^13 - 2 * q^17 - 2 * q^19 + 12 * q^23 - q^25 + 4 * q^29 - 2 * q^31 - 2 * q^35 + 6 * q^41 + 14 * q^43 + 14 * q^47 - 7 * q^49 + 20 * q^53 - 2 * q^55 + 20 * q^59 - 2 * q^65 + 10 * q^67 + 26 * q^71 + 16 * q^73 + 2 * q^77 + 2 * q^79 + 16 * q^83 - 18 * q^85 + 24 * q^89 + 2 * q^91 + 22 * q^95

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.95214	1.76745	0.883725	−	0.468007i	$-0.155028\pi$
$5$	3.95214	1.76745	0.883725	+	0.468007i	$0.155028\pi$
$6$	0	0
$7$	−2.34634	−0.886832	−0.443416	−	0.896316i	$-0.646234\pi$
$7$	−2.34634	−0.886832	−0.443416	+	0.896316i	$0.646234\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.66724	−1.85958	−0.929790	−	0.368091i	$-0.880012\pi$
$17$	−7.66724	−1.85958	−0.929790	+	0.368091i	$0.880012\pi$
$18$	0	0
$19$	−1.89650	−0.435088	−0.217544	−	0.976051i	$-0.569804\pi$
$19$	−1.89650	−0.435088	−0.217544	+	0.976051i	$0.569804\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.65965	1.38863	0.694316	−	0.719670i	$-0.255707\pi$
$23$	6.65965	1.38863	0.694316	+	0.719670i	$0.255707\pi$
$24$	0	0
$25$	10.6194	2.12388
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.03141	−0.377224	−0.188612	−	0.982052i	$-0.560399\pi$
$29$	−2.03141	−0.377224	−0.188612	+	0.982052i	$0.560399\pi$
$30$	0	0
$31$	5.72450	1.02815	0.514075	−	0.857745i	$-0.328135\pi$
$31$	5.72450	1.02815	0.514075	+	0.857745i	$0.328135\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−9.27304	−1.56743
$36$	0	0
$37$	−0.448456	−0.0737257	−0.0368628	−	0.999320i	$-0.511736\pi$
$37$	−0.448456	−0.0737257	−0.0368628	+	0.999320i	$0.511736\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.75591	−1.21127	−0.605635	−	0.795743i	$-0.707081\pi$
$41$	−7.75591	−1.21127	−0.605635	+	0.795743i	$0.707081\pi$
$42$	0	0
$43$	10.1619	1.54968	0.774841	−	0.632156i	$-0.217830\pi$
$43$	10.1619	1.54968	0.774841	+	0.632156i	$0.217830\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.14113	1.04164	0.520820	−	0.853666i	$-0.325626\pi$
$47$	7.14113	1.04164	0.520820	+	0.853666i	$0.325626\pi$
$48$	0	0
$49$	−1.49470	−0.213529
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.44983	0.611232	0.305616	−	0.952155i	$-0.401138\pi$
$53$	4.44983	0.611232	0.305616	+	0.952155i	$0.401138\pi$
$54$	0	0
$55$	−3.95214	−0.532906
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.6418	1.90620	0.953101	−	0.302653i	$-0.0978722\pi$
$59$	14.6418	1.90620	0.953101	+	0.302653i	$0.0978722\pi$
$60$	0	0
$61$	11.5401	1.47756	0.738780	−	0.673947i	$-0.235402\pi$
$61$	11.5401	1.47756	0.738780	+	0.673947i	$0.235402\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.95214	−0.490202
$66$	0	0
$67$	−2.87245	−0.350926	−0.175463	−	0.984486i	$-0.556142\pi$
$67$	−2.87245	−0.350926	−0.175463	+	0.984486i	$0.556142\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.35273	−0.279217	−0.139609	−	0.990207i	$-0.544584\pi$
$71$	−2.35273	−0.279217	−0.139609	+	0.990207i	$0.544584\pi$
$72$	0	0
$73$	15.4622	1.80971	0.904857	−	0.425715i	$-0.139977\pi$
$73$	15.4622	1.80971	0.904857	+	0.425715i	$0.139977\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.34634	0.267390
$78$	0	0
$79$	8.11708	0.913243	0.456621	−	0.889661i	$-0.349059\pi$
$79$	8.11708	0.913243	0.456621	+	0.889661i	$0.349059\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.2026	1.44917	0.724585	−	0.689185i	$-0.242031\pi$
$83$	13.2026	1.44917	0.724585	+	0.689185i	$0.242031\pi$
$84$	0	0
$85$	−30.3020	−3.28671
$86$	0	0
$87$	0	0
$88$	0	0
$89$	17.6750	1.87355	0.936774	−	0.349935i	$-0.113796\pi$
$89$	17.6750	1.87355	0.936774	+	0.349935i	$0.113796\pi$
$90$	0	0
$91$	2.34634	0.245963
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.49524	−0.768995
$96$	0	0
$97$	−1.34812	−0.136881	−0.0684407	−	0.997655i	$-0.521802\pi$
$97$	−1.34812	−0.136881	−0.0684407	+	0.997655i	$0.521802\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.237028	0.0235852	0.0117926	−	0.999930i	$-0.496246\pi$
$101$	0.237028	0.0235852	0.0117926	+	0.999930i	$0.496246\pi$
$102$	0	0
$103$	11.0916	1.09289	0.546446	−	0.837494i	$-0.315980\pi$
$103$	11.0916	1.09289	0.546446	+	0.837494i	$0.315980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.4893	1.01403	0.507017	−	0.861936i	$-0.330748\pi$
$107$	10.4893	1.01403	0.507017	+	0.861936i	$0.330748\pi$
$108$	0	0
$109$	−5.78013	−0.553636	−0.276818	−	0.960922i	$-0.589280\pi$
$109$	−5.78013	−0.553636	−0.276818	+	0.960922i	$0.589280\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.24641	0.211325	0.105662	−	0.994402i	$-0.466304\pi$
$113$	2.24641	0.211325	0.105662	+	0.994402i	$0.466304\pi$
$114$	0	0
$115$	26.3198	2.45434
$116$	0	0
$117$	0	0
$118$	0	0
$119$	17.9899	1.64914
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	22.2086	1.98639
$126$	0	0
$127$	−15.9563	−1.41590	−0.707948	−	0.706265i	$-0.750379\pi$
$127$	−15.9563	−1.41590	−0.707948	+	0.706265i	$0.750379\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.995396	0.0869682	0.0434841	−	0.999054i	$-0.486154\pi$
$131$	0.995396	0.0869682	0.0434841	+	0.999054i	$0.486154\pi$
$132$	0	0
$133$	4.44983	0.385850
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.8344	−1.01108	−0.505540	−	0.862803i	$-0.668707\pi$
$137$	−11.8344	−1.01108	−0.505540	+	0.862803i	$0.668707\pi$
$138$	0	0
$139$	4.52611	0.383900	0.191950	−	0.981405i	$-0.438519\pi$
$139$	4.52611	0.383900	0.191950	+	0.981405i	$0.438519\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−8.02842	−0.666724
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.486933	0.0398911	0.0199455	−	0.999801i	$-0.493651\pi$
$149$	0.486933	0.0398911	0.0199455	+	0.999801i	$0.493651\pi$
$150$	0	0
$151$	1.04803	0.0852878	0.0426439	−	0.999090i	$-0.486422\pi$
$151$	1.04803	0.0852878	0.0426439	+	0.999090i	$0.486422\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	22.6240	1.81720
$156$	0	0
$157$	−22.1563	−1.76826	−0.884131	−	0.467239i	$-0.845249\pi$
$157$	−22.1563	−1.76826	−0.884131	+	0.467239i	$0.845249\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−15.6258	−1.23148
$162$	0	0
$163$	1.41663	0.110959	0.0554797	−	0.998460i	$-0.482331\pi$
$163$	1.41663	0.110959	0.0554797	+	0.998460i	$0.482331\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.47508	−0.114145	−0.0570727	−	0.998370i	$-0.518177\pi$
$167$	−1.47508	−0.114145	−0.0570727	+	0.998370i	$0.518177\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.03003	−0.306398	−0.153199	−	0.988195i	$-0.548958\pi$
$173$	−4.03003	−0.306398	−0.153199	+	0.988195i	$0.548958\pi$
$174$	0	0
$175$	−24.9166	−1.88352
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.8149	−1.18206	−0.591032	−	0.806648i	$-0.701279\pi$
$179$	−15.8149	−1.18206	−0.591032	+	0.806648i	$0.701279\pi$
$180$	0	0
$181$	−7.87124	−0.585065	−0.292532	−	0.956256i	$-0.594498\pi$
$181$	−7.87124	−0.585065	−0.292532	+	0.956256i	$0.594498\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.77236	−0.130306
$186$	0	0
$187$	7.66724	0.560684
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.7243	0.848337	0.424169	−	0.905583i	$-0.360566\pi$
$191$	11.7243	0.848337	0.424169	+	0.905583i	$0.360566\pi$
$192$	0	0
$193$	−17.4173	−1.25373	−0.626864	−	0.779129i	$-0.715662\pi$
$193$	−17.4173	−1.25373	−0.626864	+	0.779129i	$0.715662\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.51115	0.321406	0.160703	−	0.987003i	$-0.448624\pi$
$197$	4.51115	0.321406	0.160703	+	0.987003i	$0.448624\pi$
$198$	0	0
$199$	−8.88823	−0.630070	−0.315035	−	0.949080i	$-0.602016\pi$
$199$	−8.88823	−0.630070	−0.315035	+	0.949080i	$0.602016\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.76638	0.334534
$204$	0	0
$205$	−30.6524	−2.14086
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.89650	0.131184
$210$	0	0
$211$	−6.52474	−0.449181	−0.224591	−	0.974453i	$-0.572104\pi$
$211$	−6.52474	−0.449181	−0.224591	+	0.974453i	$0.572104\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	40.1614	2.73898
$216$	0	0
$217$	−13.4316	−0.911796
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.66724	0.515755
$222$	0	0
$223$	−28.9032	−1.93550	−0.967750	−	0.251911i	$-0.918941\pi$
$223$	−28.9032	−1.93550	−0.967750	+	0.251911i	$0.918941\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.62536	−0.572485	−0.286243	−	0.958157i	$-0.592406\pi$
$227$	−8.62536	−0.572485	−0.286243	+	0.958157i	$0.592406\pi$
$228$	0	0
$229$	−9.04540	−0.597737	−0.298869	−	0.954294i	$-0.596609\pi$
$229$	−9.04540	−0.597737	−0.298869	+	0.954294i	$0.596609\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.5701	1.15106	0.575529	−	0.817781i	$-0.304796\pi$
$233$	17.5701	1.15106	0.575529	+	0.817781i	$0.304796\pi$
$234$	0	0
$235$	28.2227	1.84105
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.9277	0.706857	0.353428	−	0.935462i	$-0.385016\pi$
$239$	10.9277	0.706857	0.353428	+	0.935462i	$0.385016\pi$
$240$	0	0
$241$	−5.54693	−0.357309	−0.178655	−	0.983912i	$-0.557174\pi$
$241$	−5.54693	−0.357309	−0.178655	+	0.983912i	$0.557174\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.90727	−0.377402
$246$	0	0
$247$	1.89650	0.120672
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.97336	0.629513	0.314757	−	0.949172i	$-0.398077\pi$
$251$	9.97336	0.629513	0.314757	+	0.949172i	$0.398077\pi$
$252$	0	0
$253$	−6.65965	−0.418688
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.0702	1.25194	0.625971	−	0.779846i	$-0.284703\pi$
$257$	20.0702	1.25194	0.625971	+	0.779846i	$0.284703\pi$
$258$	0	0
$259$	1.05223	0.0653823
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−20.6097	−1.27085	−0.635425	−	0.772162i	$-0.719175\pi$
$263$	−20.6097	−1.27085	−0.635425	+	0.772162i	$0.719175\pi$
$264$	0	0
$265$	17.5864	1.08032
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.36739	0.205314	0.102657	−	0.994717i	$-0.467266\pi$
$269$	3.36739	0.205314	0.102657	+	0.994717i	$0.467266\pi$
$270$	0	0
$271$	−4.20574	−0.255481	−0.127740	−	0.991808i	$-0.540772\pi$
$271$	−4.20574	−0.255481	−0.127740	+	0.991808i	$0.540772\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.6194	−0.640373
$276$	0	0
$277$	−29.2872	−1.75970	−0.879849	−	0.475254i	$-0.842356\pi$
$277$	−29.2872	−1.75970	−0.879849	+	0.475254i	$0.842356\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.4728	−0.982685	−0.491343	−	0.870966i	$-0.663494\pi$
$281$	−16.4728	−0.982685	−0.491343	+	0.870966i	$0.663494\pi$
$282$	0	0
$283$	1.67131	0.0993493	0.0496746	−	0.998765i	$-0.484182\pi$
$283$	1.67131	0.0993493	0.0496746	+	0.998765i	$0.484182\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	18.1980	1.07419
$288$	0	0
$289$	41.7866	2.45804
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−13.4677	−0.786789	−0.393394	−	0.919370i	$-0.628699\pi$
$293$	−13.4677	−0.786789	−0.393394	+	0.919370i	$0.628699\pi$
$294$	0	0
$295$	57.8664	3.36911
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.65965	−0.385137
$300$	0	0
$301$	−23.8434	−1.37431
$302$	0	0
$303$	0	0
$304$	0	0
$305$	45.6081	2.61151
$306$	0	0
$307$	27.1388	1.54890	0.774448	−	0.632638i	$-0.218028\pi$
$307$	27.1388	1.54890	0.774448	+	0.632638i	$0.218028\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.2824	−0.923289	−0.461645	−	0.887065i	$-0.652740\pi$
$311$	−16.2824	−0.923289	−0.461645	+	0.887065i	$0.652740\pi$
$312$	0	0
$313$	18.7134	1.05775	0.528873	−	0.848701i	$-0.322615\pi$
$313$	18.7134	1.05775	0.528873	+	0.848701i	$0.322615\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.2165	1.64096	0.820481	−	0.571674i	$-0.193706\pi$
$317$	29.2165	1.64096	0.820481	+	0.571674i	$0.193706\pi$
$318$	0	0
$319$	2.03141	0.113737
$320$	0	0
$321$	0	0
$322$	0	0
$323$	14.5409	0.809080
$324$	0	0
$325$	−10.6194	−0.589057
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−16.7555	−0.923760
$330$	0	0
$331$	9.52515	0.523549	0.261775	−	0.965129i	$-0.415692\pi$
$331$	9.52515	0.523549	0.261775	+	0.965129i	$0.415692\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.3523	−0.620243
$336$	0	0
$337$	−16.2790	−0.886775	−0.443388	−	0.896330i	$-0.646223\pi$
$337$	−16.2790	−0.886775	−0.443388	+	0.896330i	$0.646223\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.72450	−0.309999
$342$	0	0
$343$	19.9314	1.07620
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.9848	0.804426	0.402213	−	0.915546i	$-0.368241\pi$
$347$	14.9848	0.804426	0.402213	+	0.915546i	$0.368241\pi$
$348$	0	0
$349$	1.50288	0.0804474	0.0402237	−	0.999191i	$-0.487193\pi$
$349$	1.50288	0.0804474	0.0402237	+	0.999191i	$0.487193\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.0796	−1.22840	−0.614202	−	0.789149i	$-0.710522\pi$
$353$	−23.0796	−1.22840	−0.614202	+	0.789149i	$0.710522\pi$
$354$	0	0
$355$	−9.29830	−0.493503
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.9754	0.684814	0.342407	−	0.939552i	$-0.388758\pi$
$359$	12.9754	0.684814	0.342407	+	0.939552i	$0.388758\pi$
$360$	0	0
$361$	−15.4033	−0.810699
$362$	0	0
$363$	0	0
$364$	0	0
$365$	61.1088	3.19858
$366$	0	0
$367$	−1.59779	−0.0834040	−0.0417020	−	0.999130i	$-0.513278\pi$
$367$	−1.59779	−0.0834040	−0.0417020	+	0.999130i	$0.513278\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.4408	−0.542060
$372$	0	0
$373$	26.9268	1.39422	0.697109	−	0.716965i	$-0.254469\pi$
$373$	26.9268	1.39422	0.697109	+	0.716965i	$0.254469\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.03141	0.104623
$378$	0	0
$379$	−32.2157	−1.65481	−0.827403	−	0.561608i	$-0.810183\pi$
$379$	−32.2157	−1.65481	−0.827403	+	0.561608i	$0.810183\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.7411	0.855431	0.427716	−	0.903913i	$-0.359318\pi$
$383$	16.7411	0.855431	0.427716	+	0.903913i	$0.359318\pi$
$384$	0	0
$385$	9.27304	0.472598
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−25.7753	−1.30686	−0.653430	−	0.756987i	$-0.726671\pi$
$389$	−25.7753	−1.30686	−0.653430	+	0.756987i	$0.726671\pi$
$390$	0	0
$391$	−51.0611	−2.58227
$392$	0	0
$393$	0	0
$394$	0	0
$395$	32.0798	1.61411
$396$	0	0
$397$	−19.5287	−0.980116	−0.490058	−	0.871690i	$-0.663024\pi$
$397$	−19.5287	−0.980116	−0.490058	+	0.871690i	$0.663024\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.5491	−1.22592	−0.612961	−	0.790113i	$-0.710022\pi$
$401$	−24.5491	−1.22592	−0.612961	+	0.790113i	$0.710022\pi$
$402$	0	0
$403$	−5.72450	−0.285157
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.448456	0.0222291
$408$	0	0
$409$	9.09584	0.449761	0.224880	−	0.974386i	$-0.427801\pi$
$409$	9.09584	0.449761	0.224880	+	0.974386i	$0.427801\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−34.3546	−1.69048
$414$	0	0
$415$	52.1784	2.56134
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−19.9250	−0.973402	−0.486701	−	0.873569i	$-0.661800\pi$
$419$	−19.9250	−0.973402	−0.486701	+	0.873569i	$0.661800\pi$
$420$	0	0
$421$	−32.7350	−1.59541	−0.797704	−	0.603049i	$-0.793952\pi$
$421$	−32.7350	−1.59541	−0.797704	+	0.603049i	$0.793952\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−81.4214	−3.94952
$426$	0	0
$427$	−27.0770	−1.31035
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.04844	0.195007	0.0975033	−	0.995235i	$-0.468914\pi$
$431$	4.04844	0.195007	0.0975033	+	0.995235i	$0.468914\pi$
$432$	0	0
$433$	−25.1243	−1.20739	−0.603697	−	0.797214i	$-0.706306\pi$
$433$	−25.1243	−1.20739	−0.603697	+	0.797214i	$0.706306\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.6300	−0.604176
$438$	0	0
$439$	28.5742	1.36377	0.681887	−	0.731457i	$-0.261159\pi$
$439$	28.5742	1.36377	0.681887	+	0.731457i	$0.261159\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.6492	1.40867	0.704337	−	0.709865i	$-0.251244\pi$
$443$	29.6492	1.40867	0.704337	+	0.709865i	$0.251244\pi$
$444$	0	0
$445$	69.8541	3.31140
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.97121	−0.234606	−0.117303	−	0.993096i	$-0.537425\pi$
$449$	−4.97121	−0.234606	−0.117303	+	0.993096i	$0.537425\pi$
$450$	0	0
$451$	7.75591	0.365211
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.27304	0.434727
$456$	0	0
$457$	40.5287	1.89585	0.947927	−	0.318488i	$-0.103175\pi$
$457$	40.5287	1.89585	0.947927	+	0.318488i	$0.103175\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.70828	0.265861	0.132931	−	0.991125i	$-0.457561\pi$
$461$	5.70828	0.265861	0.132931	+	0.991125i	$0.457561\pi$
$462$	0	0
$463$	8.53574	0.396689	0.198345	−	0.980132i	$-0.436443\pi$
$463$	8.53574	0.396689	0.198345	+	0.980132i	$0.436443\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−41.0474	−1.89945	−0.949724	−	0.313087i	$-0.898637\pi$
$467$	−41.0474	−1.89945	−0.949724	+	0.313087i	$0.898637\pi$
$468$	0	0
$469$	6.73974	0.311212
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10.1619	−0.467247
$474$	0	0
$475$	−20.1397	−0.924072
$476$	0	0
$477$	0	0
$478$	0	0
$479$	18.8535	0.861441	0.430720	−	0.902485i	$-0.358260\pi$
$479$	18.8535	0.861441	0.430720	+	0.902485i	$0.358260\pi$
$480$	0	0
$481$	0.448456	0.0204478
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.32797	−0.241931
$486$	0	0
$487$	−27.6957	−1.25501	−0.627505	−	0.778612i	$-0.715924\pi$
$487$	−27.6957	−1.25501	−0.627505	+	0.778612i	$0.715924\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.5447	0.566135	0.283067	−	0.959100i	$-0.408648\pi$
$491$	12.5447	0.566135	0.283067	+	0.959100i	$0.408648\pi$
$492$	0	0
$493$	15.5753	0.701478
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.52029	0.247619
$498$	0	0
$499$	−28.1417	−1.25980	−0.629898	−	0.776678i	$-0.716903\pi$
$499$	−28.1417	−1.25980	−0.629898	+	0.776678i	$0.716903\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	36.3214	1.61949	0.809745	−	0.586782i	$-0.199605\pi$
$503$	36.3214	1.61949	0.809745	+	0.586782i	$0.199605\pi$
$504$	0	0
$505$	0.936767	0.0416856
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6.57193	−0.291295	−0.145648	−	0.989337i	$-0.546527\pi$
$509$	−6.57193	−0.291295	−0.145648	+	0.989337i	$0.546527\pi$
$510$	0	0
$511$	−36.2796	−1.60491
$512$	0	0
$513$	0	0
$514$	0	0
$515$	43.8357	1.93163
$516$	0	0
$517$	−7.14113	−0.314067
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.26383	−0.186802	−0.0934008	−	0.995629i	$-0.529774\pi$
$521$	−4.26383	−0.186802	−0.0934008	+	0.995629i	$0.529774\pi$
$522$	0	0
$523$	−0.527152	−0.0230507	−0.0115254	−	0.999934i	$-0.503669\pi$
$523$	−0.527152	−0.0230507	−0.0115254	+	0.999934i	$0.503669\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−43.8911	−1.91193
$528$	0	0
$529$	21.3509	0.928299
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.75591	0.335946
$534$	0	0
$535$	41.4550	1.79225
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.49470	0.0643814
$540$	0	0
$541$	41.6146	1.78915	0.894575	−	0.446917i	$-0.147478\pi$
$541$	41.6146	1.78915	0.894575	+	0.446917i	$0.147478\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−22.8439	−0.978523
$546$	0	0
$547$	−23.6063	−1.00933	−0.504667	−	0.863314i	$-0.668385\pi$
$547$	−23.6063	−1.00933	−0.504667	+	0.863314i	$0.668385\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.85258	0.164125
$552$	0	0
$553$	−19.0454	−0.809893
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.6653	0.579017	0.289508	−	0.957175i	$-0.406508\pi$
$557$	13.6653	0.579017	0.289508	+	0.957175i	$0.406508\pi$
$558$	0	0
$559$	−10.1619	−0.429805
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.4316	1.28254	0.641269	−	0.767316i	$-0.278408\pi$
$563$	30.4316	1.28254	0.641269	+	0.767316i	$0.278408\pi$
$564$	0	0
$565$	8.87814	0.373506
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.0621	0.715282	0.357641	−	0.933859i	$-0.383581\pi$
$569$	17.0621	0.715282	0.357641	+	0.933859i	$0.383581\pi$
$570$	0	0
$571$	−7.61335	−0.318609	−0.159304	−	0.987230i	$-0.550925\pi$
$571$	−7.61335	−0.318609	−0.159304	+	0.987230i	$0.550925\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	70.7213	2.94928
$576$	0	0
$577$	41.6772	1.73504	0.867521	−	0.497400i	$-0.165712\pi$
$577$	41.6772	1.73504	0.867521	+	0.497400i	$0.165712\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−30.9777	−1.28517
$582$	0	0
$583$	−4.44983	−0.184293
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.62386	0.355945	0.177972	−	0.984035i	$-0.443046\pi$
$587$	8.62386	0.355945	0.177972	+	0.984035i	$0.443046\pi$
$588$	0	0
$589$	−10.8565	−0.447335
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−18.1163	−0.743946	−0.371973	−	0.928243i	$-0.621319\pi$
$593$	−18.1163	−0.743946	−0.371973	+	0.928243i	$0.621319\pi$
$594$	0	0
$595$	71.0987	2.91476
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.03653	−0.328364	−0.164182	−	0.986430i	$-0.552498\pi$
$599$	−8.03653	−0.328364	−0.164182	+	0.986430i	$0.552498\pi$
$600$	0	0
$601$	35.1218	1.43265	0.716325	−	0.697767i	$-0.245823\pi$
$601$	35.1218	1.43265	0.716325	+	0.697767i	$0.245823\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.95214	0.160677
$606$	0	0
$607$	9.05700	0.367612	0.183806	−	0.982963i	$-0.441158\pi$
$607$	9.05700	0.367612	0.183806	+	0.982963i	$0.441158\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.14113	−0.288899
$612$	0	0
$613$	28.9457	1.16911	0.584553	−	0.811356i	$-0.301270\pi$
$613$	28.9457	1.16911	0.584553	+	0.811356i	$0.301270\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.6023	1.55407	0.777036	−	0.629456i	$-0.216722\pi$
$617$	38.6023	1.55407	0.777036	+	0.629456i	$0.216722\pi$
$618$	0	0
$619$	−29.5996	−1.18971	−0.594854	−	0.803834i	$-0.702790\pi$
$619$	−29.5996	−1.18971	−0.594854	+	0.803834i	$0.702790\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−41.4715	−1.66152
$624$	0	0
$625$	34.6743	1.38697
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.43842	0.137099
$630$	0	0
$631$	17.5964	0.700500	0.350250	−	0.936656i	$-0.386097\pi$
$631$	17.5964	0.700500	0.350250	+	0.936656i	$0.386097\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−63.0616	−2.50252
$636$	0	0
$637$	1.49470	0.0592223
$638$	0	0
$639$	0	0
$640$	0	0
$641$	27.5485	1.08810	0.544051	−	0.839052i	$-0.316890\pi$
$641$	27.5485	1.08810	0.544051	+	0.839052i	$0.316890\pi$
$642$	0	0
$643$	−6.59664	−0.260146	−0.130073	−	0.991504i	$-0.541521\pi$
$643$	−6.59664	−0.260146	−0.130073	+	0.991504i	$0.541521\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.86946	0.348694	0.174347	−	0.984684i	$-0.444219\pi$
$647$	8.86946	0.348694	0.174347	+	0.984684i	$0.444219\pi$
$648$	0	0
$649$	−14.6418	−0.574741
$650$	0	0
$651$	0	0
$652$	0	0
$653$	45.9347	1.79756	0.898782	−	0.438395i	$-0.144453\pi$
$653$	45.9347	1.79756	0.898782	+	0.438395i	$0.144453\pi$
$654$	0	0
$655$	3.93394	0.153712
$656$	0	0
$657$	0	0
$658$	0	0
$659$	17.6361	0.687004	0.343502	−	0.939152i	$-0.388387\pi$
$659$	17.6361	0.687004	0.343502	+	0.939152i	$0.388387\pi$
$660$	0	0
$661$	36.4207	1.41660	0.708300	−	0.705911i	$-0.249462\pi$
$661$	36.4207	1.41660	0.708300	+	0.705911i	$0.249462\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	17.5864	0.681969
$666$	0	0
$667$	−13.5285	−0.523825
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.5401	−0.445501
$672$	0	0
$673$	31.0163	1.19559	0.597794	−	0.801649i	$-0.296044\pi$
$673$	31.0163	1.19559	0.597794	+	0.801649i	$0.296044\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−23.8492	−0.916598	−0.458299	−	0.888798i	$-0.651541\pi$
$677$	−23.8492	−0.916598	−0.458299	+	0.888798i	$0.651541\pi$
$678$	0	0
$679$	3.16315	0.121391
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−39.3665	−1.50632	−0.753159	−	0.657838i	$-0.771471\pi$
$683$	−39.3665	−1.50632	−0.753159	+	0.657838i	$0.771471\pi$
$684$	0	0
$685$	−46.7711	−1.78703
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.44983	−0.169525
$690$	0	0
$691$	−40.6325	−1.54573	−0.772867	−	0.634568i	$-0.781178\pi$
$691$	−40.6325	−1.54573	−0.772867	+	0.634568i	$0.781178\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.8878	0.678524
$696$	0	0
$697$	59.4664	2.25245
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−7.41896	−0.280210	−0.140105	−	0.990137i	$-0.544744\pi$
$701$	−7.41896	−0.280210	−0.140105	+	0.990137i	$0.544744\pi$
$702$	0	0
$703$	0.850498	0.0320771
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.556148	−0.0209161
$708$	0	0
$709$	1.75221	0.0658055	0.0329027	−	0.999459i	$-0.489525\pi$
$709$	1.75221	0.0658055	0.0329027	+	0.999459i	$0.489525\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	38.1231	1.42772
$714$	0	0
$715$	3.95214	0.147802
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.4792	−0.912921	−0.456460	−	0.889744i	$-0.650883\pi$
$719$	−24.4792	−0.912921	−0.456460	+	0.889744i	$0.650883\pi$
$720$	0	0
$721$	−26.0247	−0.969212
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−21.5723	−0.801176
$726$	0	0
$727$	2.75443	0.102156	0.0510781	−	0.998695i	$-0.483734\pi$
$727$	2.75443	0.102156	0.0510781	+	0.998695i	$0.483734\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−77.9141	−2.88176
$732$	0	0
$733$	−30.9977	−1.14493	−0.572463	−	0.819930i	$-0.694012\pi$
$733$	−30.9977	−1.14493	−0.572463	+	0.819930i	$0.694012\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.87245	0.105808
$738$	0	0
$739$	−27.8397	−1.02410	−0.512050	−	0.858955i	$-0.671114\pi$
$739$	−27.8397	−1.02410	−0.512050	+	0.858955i	$0.671114\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.8292	1.09433	0.547163	−	0.837026i	$-0.315708\pi$
$743$	29.8292	1.09433	0.547163	+	0.837026i	$0.315708\pi$
$744$	0	0
$745$	1.92442	0.0705055
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24.6113	−0.899278
$750$	0	0
$751$	2.86588	0.104577	0.0522887	−	0.998632i	$-0.483348\pi$
$751$	2.86588	0.104577	0.0522887	+	0.998632i	$0.483348\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.14197	0.150742
$756$	0	0
$757$	−26.5247	−0.964056	−0.482028	−	0.876156i	$-0.660100\pi$
$757$	−26.5247	−0.964056	−0.482028	+	0.876156i	$0.660100\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.4450	1.21238	0.606191	−	0.795319i	$-0.292697\pi$
$761$	33.4450	1.21238	0.606191	+	0.795319i	$0.292697\pi$
$762$	0	0
$763$	13.5621	0.490982
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.6418	−0.528685
$768$	0	0
$769$	33.4634	1.20672	0.603360	−	0.797469i	$-0.293828\pi$
$769$	33.4634	1.20672	0.603360	+	0.797469i	$0.293828\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.5839	0.524547	0.262273	−	0.964994i	$-0.415528\pi$
$773$	14.5839	0.524547	0.262273	+	0.964994i	$0.415528\pi$
$774$	0	0
$775$	60.7906	2.18366
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.7091	0.527008
$780$	0	0
$781$	2.35273	0.0841872
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−87.5646	−3.12531
$786$	0	0
$787$	−15.2901	−0.545032	−0.272516	−	0.962151i	$-0.587856\pi$
$787$	−15.2901	−0.545032	−0.272516	+	0.962151i	$0.587856\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.27085	−0.187410
$792$	0	0
$793$	−11.5401	−0.409801
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.8445	−0.632084	−0.316042	−	0.948745i	$-0.602354\pi$
$797$	−17.8445	−0.632084	−0.316042	+	0.948745i	$0.602354\pi$
$798$	0	0
$799$	−54.7528	−1.93701
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−15.4622	−0.545649
$804$	0	0
$805$	−61.7552	−2.17658
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−28.3298	−0.996022	−0.498011	−	0.867171i	$-0.665936\pi$
$809$	−28.3298	−0.996022	−0.498011	+	0.867171i	$0.665936\pi$
$810$	0	0
$811$	−53.5484	−1.88034	−0.940170	−	0.340707i	$-0.889334\pi$
$811$	−53.5484	−1.88034	−0.940170	+	0.340707i	$0.889334\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.59873	0.196115
$816$	0	0
$817$	−19.2722	−0.674248
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.39396	−0.153350	−0.0766752	−	0.997056i	$-0.524430\pi$
$821$	−4.39396	−0.153350	−0.0766752	+	0.997056i	$0.524430\pi$
$822$	0	0
$823$	13.2713	0.462609	0.231305	−	0.972881i	$-0.425701\pi$
$823$	13.2713	0.462609	0.231305	+	0.972881i	$0.425701\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.2740	1.29614	0.648072	−	0.761579i	$-0.275576\pi$
$827$	37.2740	1.29614	0.648072	+	0.761579i	$0.275576\pi$
$828$	0	0
$829$	1.05189	0.0365336	0.0182668	−	0.999833i	$-0.494185\pi$
$829$	1.05189	0.0365336	0.0182668	+	0.999833i	$0.494185\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	11.4602	0.397074
$834$	0	0
$835$	−5.82973	−0.201746
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32.1004	−1.10823	−0.554114	−	0.832441i	$-0.686943\pi$
$839$	−32.1004	−1.10823	−0.554114	+	0.832441i	$0.686943\pi$
$840$	0	0
$841$	−24.8734	−0.857702
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.95214	0.135958
$846$	0	0
$847$	−2.34634	−0.0806211
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.98656	−0.102378
$852$	0	0
$853$	29.6319	1.01458	0.507289	−	0.861776i	$-0.330648\pi$
$853$	29.6319	1.01458	0.507289	+	0.861776i	$0.330648\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.7570	−0.367452	−0.183726	−	0.982978i	$-0.558816\pi$
$857$	−10.7570	−0.367452	−0.183726	+	0.982978i	$0.558816\pi$
$858$	0	0
$859$	−19.1458	−0.653246	−0.326623	−	0.945155i	$-0.605911\pi$
$859$	−19.1458	−0.653246	−0.326623	+	0.945155i	$0.605911\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37.1142	−1.26338	−0.631691	−	0.775221i	$-0.717639\pi$
$863$	−37.1142	−1.26338	−0.631691	+	0.775221i	$0.717639\pi$
$864$	0	0
$865$	−15.9272	−0.541542
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.11708	−0.275353
$870$	0	0
$871$	2.87245	0.0973293
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−52.1088	−1.76160
$876$	0	0
$877$	−40.8893	−1.38073	−0.690367	−	0.723460i	$-0.742551\pi$
$877$	−40.8893	−1.38073	−0.690367	+	0.723460i	$0.742551\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	53.9205	1.81663	0.908314	−	0.418288i	$-0.137370\pi$
$881$	53.9205	1.81663	0.908314	+	0.418288i	$0.137370\pi$
$882$	0	0
$883$	24.5229	0.825262	0.412631	−	0.910898i	$-0.364610\pi$
$883$	24.5229	0.825262	0.412631	+	0.910898i	$0.364610\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.9384	1.27385	0.636924	−	0.770926i	$-0.280206\pi$
$887$	37.9384	1.27385	0.636924	+	0.770926i	$0.280206\pi$
$888$	0	0
$889$	37.4389	1.25566
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−13.5432	−0.453205
$894$	0	0
$895$	−62.5028	−2.08924
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11.6288	−0.387842
$900$	0	0
$901$	−34.1180	−1.13663
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31.1082	−1.03407
$906$	0	0
$907$	32.1587	1.06781	0.533907	−	0.845543i	$-0.320723\pi$
$907$	32.1587	1.06781	0.533907	+	0.845543i	$0.320723\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	35.5713	1.17853	0.589265	−	0.807939i	$-0.299417\pi$
$911$	35.5713	1.17853	0.589265	+	0.807939i	$0.299417\pi$
$912$	0	0
$913$	−13.2026	−0.436941
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.33554	−0.0771262
$918$	0	0
$919$	−21.5960	−0.712385	−0.356192	−	0.934413i	$-0.615925\pi$
$919$	−21.5960	−0.712385	−0.356192	+	0.934413i	$0.615925\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.35273	0.0774410
$924$	0	0
$925$	−4.76232	−0.156584
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16.7452	0.549392	0.274696	−	0.961531i	$-0.411423\pi$
$929$	16.7452	0.549392	0.274696	+	0.961531i	$0.411423\pi$
$930$	0	0
$931$	2.83471	0.0929038
$932$	0	0
$933$	0	0
$934$	0	0
$935$	30.3020	0.990981
$936$	0	0
$937$	−27.2318	−0.889623	−0.444811	−	0.895624i	$-0.646729\pi$
$937$	−27.2318	−0.889623	−0.444811	+	0.895624i	$0.646729\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.9395	−1.39979	−0.699894	−	0.714247i	$-0.746769\pi$
$941$	−42.9395	−1.39979	−0.699894	+	0.714247i	$0.746769\pi$
$942$	0	0
$943$	−51.6516	−1.68201
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43.0962	1.40044	0.700220	−	0.713927i	$-0.253085\pi$
$947$	43.0962	1.40044	0.700220	+	0.713927i	$0.253085\pi$
$948$	0	0
$949$	−15.4622	−0.501925
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−32.1255	−1.04065	−0.520323	−	0.853970i	$-0.674188\pi$
$953$	−32.1255	−1.04065	−0.520323	+	0.853970i	$0.674188\pi$
$954$	0	0
$955$	46.3359	1.49939
$956$	0	0
$957$	0	0
$958$	0	0
$959$	27.7675	0.896658
$960$	0	0
$961$	1.76984	0.0570917
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−68.8357	−2.21590
$966$	0	0
$967$	−30.3808	−0.976982	−0.488491	−	0.872569i	$-0.662452\pi$
$967$	−30.3808	−0.976982	−0.488491	+	0.872569i	$0.662452\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−60.5947	−1.94458	−0.972288	−	0.233788i	$-0.924888\pi$
$971$	−60.5947	−1.94458	−0.972288	+	0.233788i	$0.924888\pi$
$972$	0	0
$973$	−10.6198	−0.340455
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.1017	0.995030	0.497515	−	0.867455i	$-0.334246\pi$
$977$	31.1017	0.995030	0.497515	+	0.867455i	$0.334246\pi$
$978$	0	0
$979$	−17.6750	−0.564896
$980$	0	0
$981$	0	0
$982$	0	0
$983$	50.8162	1.62078	0.810392	−	0.585888i	$-0.199254\pi$
$983$	50.8162	1.62078	0.810392	+	0.585888i	$0.199254\pi$
$984$	0	0
$985$	17.8287	0.568070
$986$	0	0
$987$	0	0
$988$	0	0
$989$	67.6750	2.15194
$990$	0	0
$991$	9.96112	0.316426	0.158213	−	0.987405i	$-0.449427\pi$
$991$	9.96112	0.316426	0.158213	+	0.987405i	$0.449427\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−35.1275	−1.11362
$996$	0	0
$997$	34.6753	1.09818	0.549089	−	0.835764i	$-0.314975\pi$
$997$	34.6753	1.09818	0.549089	+	0.835764i	$0.314975\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.t.1.5	yes	5
3.2	odd	2		5148.2.a.s.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5148.2.a.s.1.1	✓	5	3.2	odd	2
5148.2.a.t.1.5	yes	5	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:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q+3.95214 q^{5} -2.34634 q^{7} +O(q^{10})\)	\(q+3.95214 q^{5} -2.34634 q^{7} -1.00000 q^{11} -1.00000 q^{13} -7.66724 q^{17} -1.89650 q^{19} +6.65965 q^{23} +10.6194 q^{25} -2.03141 q^{29} +5.72450 q^{31} -9.27304 q^{35} -0.448456 q^{37} -7.75591 q^{41} +10.1619 q^{43} +7.14113 q^{47} -1.49470 q^{49} +4.44983 q^{53} -3.95214 q^{55} +14.6418 q^{59} +11.5401 q^{61} -3.95214 q^{65} -2.87245 q^{67} -2.35273 q^{71} +15.4622 q^{73} +2.34634 q^{77} +8.11708 q^{79} +13.2026 q^{83} -30.3020 q^{85} +17.6750 q^{89} +2.34634 q^{91} -7.49524 q^{95} -1.34812 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 5 * q + 2 * q^5 - 2 * q^7	\( 5 q + 2 q^{5} - 2 q^{7} - 5 q^{11} - 5 q^{13} - 2 q^{17} - 2 q^{19} + 12 q^{23} - q^{25} + 4 q^{29} - 2 q^{31} - 2 q^{35} + 6 q^{41} + 14 q^{43} + 14 q^{47} - 7 q^{49} + 20 q^{53} - 2 q^{55} + 20 q^{59} - 2 q^{65} + 10 q^{67} + 26 q^{71} + 16 q^{73} + 2 q^{77} + 2 q^{79} + 16 q^{83} - 18 q^{85} + 24 q^{89} + 2 q^{91} + 22 q^{95}+O(q^{100}) \) 5 * q + 2 * q^5 - 2 * q^7 - 5 * q^11 - 5 * q^13 - 2 * q^17 - 2 * q^19 + 12 * q^23 - q^25 + 4 * q^29 - 2 * q^31 - 2 * q^35 + 6 * q^41 + 14 * q^43 + 14 * q^47 - 7 * q^49 + 20 * q^53 - 2 * q^55 + 20 * q^59 - 2 * q^65 + 10 * q^67 + 26 * q^71 + 16 * q^73 + 2 * q^77 + 2 * q^79 + 16 * q^83 - 18 * q^85 + 24 * q^89 + 2 * q^91 + 22 * q^95

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