LMFDB - Embedded newform 5148.2.a.m.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.08613$ 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)

$f(q)$	$=$	$q-3.08613 q^{5} +0.648061 q^{7} +O(q^{10})$	$q-3.08613 q^{5} +0.648061 q^{7} -1.00000 q^{11} -1.00000 q^{13} -1.73419 q^{17} +3.35194 q^{19} +7.69646 q^{23} +4.52420 q^{25} -9.14195 q^{29} +5.08613 q^{31} -2.00000 q^{35} -1.29612 q^{37} +5.69646 q^{41} +2.43807 q^{43} +4.17226 q^{47} -6.58002 q^{49} -5.46838 q^{53} +3.08613 q^{55} -2.17226 q^{59} +7.58002 q^{61} +3.08613 q^{65} -8.55451 q^{67} +7.64064 q^{71} -4.11644 q^{73} -0.648061 q^{77} +3.20257 q^{79} +4.87614 q^{83} +5.35194 q^{85} -16.1345 q^{89} -0.648061 q^{91} -10.3445 q^{95} -2.70388 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{5} + 4 q^{7}+O(q^{10}) $ 3 * q - 2 * q^5 + 4 * q^7	$ 3 q - 2 q^{5} + 4 q^{7} - 3 q^{11} - 3 q^{13} + 8 q^{19} - 8 q^{23} - 3 q^{25} - 14 q^{29} + 8 q^{31} - 6 q^{35} - 8 q^{37} - 14 q^{41} - 2 q^{43} - 2 q^{47} + 3 q^{49} - 6 q^{53} + 2 q^{55} + 8 q^{59} + 2 q^{65} - 8 q^{67} - 2 q^{71} - 4 q^{73} - 4 q^{77} - 6 q^{79} - 4 q^{83} + 14 q^{85} - 8 q^{89} - 4 q^{91} - 2 q^{95} - 4 q^{97}+O(q^{100}) $ 3 * q - 2 * q^5 + 4 * q^7 - 3 * q^11 - 3 * q^13 + 8 * q^19 - 8 * q^23 - 3 * q^25 - 14 * q^29 + 8 * q^31 - 6 * q^35 - 8 * q^37 - 14 * q^41 - 2 * q^43 - 2 * q^47 + 3 * q^49 - 6 * q^53 + 2 * q^55 + 8 * q^59 + 2 * q^65 - 8 * q^67 - 2 * q^71 - 4 * q^73 - 4 * q^77 - 6 * q^79 - 4 * q^83 + 14 * q^85 - 8 * q^89 - 4 * q^91 - 2 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.08613	−1.38016	−0.690080	−	0.723733i	$-0.742425\pi$
$5$	−3.08613	−1.38016	−0.690080	+	0.723733i	$0.742425\pi$
$6$	0	0
$7$	0.648061	0.244944	0.122472	−	0.992472i	$-0.460918\pi$
$7$	0.648061	0.244944	0.122472	+	0.992472i	$0.460918\pi$
$8$	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
$17$	−1.73419	−0.420603	−0.210302	−	0.977637i	$-0.567445\pi$
$17$	−1.73419	−0.420603	−0.210302	+	0.977637i	$0.567445\pi$
$18$	0	0
$19$	3.35194	0.768988	0.384494	−	0.923128i	$-0.374376\pi$
$19$	3.35194	0.768988	0.384494	+	0.923128i	$0.374376\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.69646	1.60482	0.802411	−	0.596771i	$-0.203550\pi$
$23$	7.69646	1.60482	0.802411	+	0.596771i	$0.203550\pi$
$24$	0	0
$25$	4.52420	0.904840
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.14195	−1.69762	−0.848809	−	0.528700i	$-0.822680\pi$
$29$	−9.14195	−1.69762	−0.848809	+	0.528700i	$0.822680\pi$
$30$	0	0
$31$	5.08613	0.913496	0.456748	−	0.889596i	$-0.349014\pi$
$31$	5.08613	0.913496	0.456748	+	0.889596i	$0.349014\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−1.29612	−0.213081	−0.106541	−	0.994308i	$-0.533977\pi$
$37$	−1.29612	−0.213081	−0.106541	+	0.994308i	$0.533977\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.69646	0.889638	0.444819	−	0.895621i	$-0.353268\pi$
$41$	5.69646	0.889638	0.444819	+	0.895621i	$0.353268\pi$
$42$	0	0
$43$	2.43807	0.371802	0.185901	−	0.982568i	$-0.440480\pi$
$43$	2.43807	0.371802	0.185901	+	0.982568i	$0.440480\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.17226	0.608587	0.304293	−	0.952578i	$-0.401580\pi$
$47$	4.17226	0.608587	0.304293	+	0.952578i	$0.401580\pi$
$48$	0	0
$49$	−6.58002	−0.940002
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.46838	−0.751140	−0.375570	−	0.926794i	$-0.622553\pi$
$53$	−5.46838	−0.751140	−0.375570	+	0.926794i	$0.622553\pi$
$54$	0	0
$55$	3.08613	0.416134
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.17226	−0.282804	−0.141402	−	0.989952i	$-0.545161\pi$
$59$	−2.17226	−0.282804	−0.141402	+	0.989952i	$0.545161\pi$
$60$	0	0
$61$	7.58002	0.970522	0.485261	−	0.874369i	$-0.338725\pi$
$61$	7.58002	0.970522	0.485261	+	0.874369i	$0.338725\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.08613	0.382787
$66$	0	0
$67$	−8.55451	−1.04510	−0.522550	−	0.852609i	$-0.675019\pi$
$67$	−8.55451	−1.04510	−0.522550	+	0.852609i	$0.675019\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.64064	0.906777	0.453389	−	0.891313i	$-0.350215\pi$
$71$	7.64064	0.906777	0.453389	+	0.891313i	$0.350215\pi$
$72$	0	0
$73$	−4.11644	−0.481793	−0.240897	−	0.970551i	$-0.577441\pi$
$73$	−4.11644	−0.481793	−0.240897	+	0.970551i	$0.577441\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.648061	−0.0738534
$78$	0	0
$79$	3.20257	0.360318	0.180159	−	0.983638i	$-0.442339\pi$
$79$	3.20257	0.360318	0.180159	+	0.983638i	$0.442339\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.87614	0.535226	0.267613	−	0.963527i	$-0.413765\pi$
$83$	4.87614	0.535226	0.267613	+	0.963527i	$0.413765\pi$
$84$	0	0
$85$	5.35194	0.580499
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.1345	−1.71026	−0.855128	−	0.518416i	$-0.826522\pi$
$89$	−16.1345	−1.71026	−0.855128	+	0.518416i	$0.826522\pi$
$90$	0	0
$91$	−0.648061	−0.0679352
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−10.3445	−1.06133
$96$	0	0
$97$	−2.70388	−0.274537	−0.137269	−	0.990534i	$-0.543832\pi$
$97$	−2.70388	−0.274537	−0.137269	+	0.990534i	$0.543832\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.96969	−0.494502	−0.247251	−	0.968951i	$-0.579527\pi$
$101$	−4.96969	−0.494502	−0.247251	+	0.968951i	$0.579527\pi$
$102$	0	0
$103$	−13.7523	−1.35505	−0.677526	−	0.735499i	$-0.736948\pi$
$103$	−13.7523	−1.35505	−0.677526	+	0.735499i	$0.736948\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.58002	0.346093	0.173047	−	0.984914i	$-0.444639\pi$
$107$	3.58002	0.346093	0.173047	+	0.984914i	$0.444639\pi$
$108$	0	0
$109$	−8.57260	−0.821106	−0.410553	−	0.911837i	$-0.634664\pi$
$109$	−8.57260	−0.821106	−0.410553	+	0.911837i	$0.634664\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.58002	0.901212	0.450606	−	0.892723i	$-0.351208\pi$
$113$	9.58002	0.901212	0.450606	+	0.892723i	$0.351208\pi$
$114$	0	0
$115$	−23.7523	−2.21491
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.12386	−0.103024
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.46838	0.131336
$126$	0	0
$127$	2.61033	0.231629	0.115815	−	0.993271i	$-0.463052\pi$
$127$	2.61033	0.231629	0.115815	+	0.993271i	$0.463052\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.11164	−0.708717	−0.354358	−	0.935110i	$-0.615301\pi$
$131$	−8.11164	−0.708717	−0.354358	+	0.935110i	$0.615301\pi$
$132$	0	0
$133$	2.17226	0.188359
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.1345	−1.37847	−0.689233	−	0.724540i	$-0.742052\pi$
$137$	−16.1345	−1.37847	−0.689233	+	0.724540i	$0.742052\pi$
$138$	0	0
$139$	−4.43807	−0.376432	−0.188216	−	0.982128i	$-0.560271\pi$
$139$	−4.43807	−0.376432	−0.188216	+	0.982128i	$0.560271\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	28.2132	2.34298
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.7449	−1.69949	−0.849743	−	0.527198i	$-0.823243\pi$
$149$	−20.7449	−1.69949	−0.849743	+	0.527198i	$0.823243\pi$
$150$	0	0
$151$	−2.05582	−0.167300	−0.0836500	−	0.996495i	$-0.526658\pi$
$151$	−2.05582	−0.167300	−0.0836500	+	0.996495i	$0.526658\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−15.6965	−1.26077
$156$	0	0
$157$	−7.17968	−0.573001	−0.286500	−	0.958080i	$-0.592492\pi$
$157$	−7.17968	−0.573001	−0.286500	+	0.958080i	$0.592492\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.98777	0.393092
$162$	0	0
$163$	−0.149366	−0.0116993	−0.00584963	−	0.999983i	$-0.501862\pi$
$163$	−0.149366	−0.0116993	−0.00584963	+	0.999983i	$0.501862\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.23289	0.637080	0.318540	−	0.947909i	$-0.396808\pi$
$167$	8.23289	0.637080	0.318540	+	0.947909i	$0.396808\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−17.2026	−1.30789	−0.653944	−	0.756543i	$-0.726887\pi$
$173$	−17.2026	−1.30789	−0.653944	+	0.756543i	$0.726887\pi$
$174$	0	0
$175$	2.93196	0.221635
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.47580	0.484024	0.242012	−	0.970273i	$-0.422193\pi$
$179$	6.47580	0.484024	0.242012	+	0.970273i	$0.422193\pi$
$180$	0	0
$181$	−11.7571	−0.873897	−0.436949	−	0.899486i	$-0.643941\pi$
$181$	−11.7571	−0.873897	−0.436949	+	0.899486i	$0.643941\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	1.73419	0.126817
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.06063	0.149102	0.0745508	−	0.997217i	$-0.476248\pi$
$191$	2.06063	0.149102	0.0745508	+	0.997217i	$0.476248\pi$
$192$	0	0
$193$	−4.22808	−0.304344	−0.152172	−	0.988354i	$-0.548627\pi$
$193$	−4.22808	−0.304344	−0.152172	+	0.988354i	$0.548627\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.1164	−0.720767	−0.360383	−	0.932804i	$-0.617354\pi$
$197$	−10.1164	−0.720767	−0.360383	+	0.932804i	$0.617354\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.92454	−0.415821
$204$	0	0
$205$	−17.5800	−1.22784
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.35194	−0.231858
$210$	0	0
$211$	11.6587	0.802620	0.401310	−	0.915942i	$-0.368555\pi$
$211$	11.6587	0.802620	0.401310	+	0.915942i	$0.368555\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.52420	−0.513146
$216$	0	0
$217$	3.29612	0.223755
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.73419	0.116654
$222$	0	0
$223$	−24.3674	−1.63176	−0.815881	−	0.578219i	$-0.803748\pi$
$223$	−24.3674	−1.63176	−0.815881	+	0.578219i	$0.803748\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.24030	−0.0823219	−0.0411609	−	0.999153i	$-0.513106\pi$
$227$	−1.24030	−0.0823219	−0.0411609	+	0.999153i	$0.513106\pi$
$228$	0	0
$229$	9.10902	0.601941	0.300971	−	0.953633i	$-0.402689\pi$
$229$	9.10902	0.601941	0.300971	+	0.953633i	$0.402689\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.7826	1.75459	0.877293	−	0.479955i	$-0.159347\pi$
$233$	26.7826	1.75459	0.877293	+	0.479955i	$0.159347\pi$
$234$	0	0
$235$	−12.8761	−0.839947
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.3519	−0.863665	−0.431833	−	0.901954i	$-0.642133\pi$
$239$	−13.3519	−0.863665	−0.431833	+	0.901954i	$0.642133\pi$
$240$	0	0
$241$	5.23550	0.337248	0.168624	−	0.985680i	$-0.446068\pi$
$241$	5.23550	0.337248	0.168624	+	0.985680i	$0.446068\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.3068	1.29735
$246$	0	0
$247$	−3.35194	−0.213279
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.22066	−0.329525	−0.164763	−	0.986333i	$-0.552686\pi$
$251$	−5.22066	−0.329525	−0.164763	+	0.986333i	$0.552686\pi$
$252$	0	0
$253$	−7.69646	−0.483872
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.3929	1.33445	0.667227	−	0.744855i	$-0.267481\pi$
$257$	21.3929	1.33445	0.667227	+	0.744855i	$0.267481\pi$
$258$	0	0
$259$	−0.839966	−0.0521929
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.76450	−0.540442	−0.270221	−	0.962798i	$-0.587097\pi$
$263$	−8.76450	−0.540442	−0.270221	+	0.962798i	$0.587097\pi$
$264$	0	0
$265$	16.8761	1.03669
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.22066	0.196367	0.0981835	−	0.995168i	$-0.468697\pi$
$269$	3.22066	0.196367	0.0981835	+	0.995168i	$0.468697\pi$
$270$	0	0
$271$	−25.5242	−1.55048	−0.775242	−	0.631664i	$-0.782372\pi$
$271$	−25.5242	−1.55048	−0.775242	+	0.631664i	$0.782372\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.52420	−0.272820
$276$	0	0
$277$	−29.7374	−1.78675	−0.893375	−	0.449312i	$-0.851669\pi$
$277$	−29.7374	−1.78675	−0.893375	+	0.449312i	$0.851669\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.93196	0.532836	0.266418	−	0.963858i	$-0.414160\pi$
$281$	8.93196	0.532836	0.266418	+	0.963858i	$0.414160\pi$
$282$	0	0
$283$	25.2387	1.50029	0.750144	−	0.661275i	$-0.229984\pi$
$283$	25.2387	1.50029	0.750144	+	0.661275i	$0.229984\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.69165	0.217911
$288$	0	0
$289$	−13.9926	−0.823093
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−26.6842	−1.55891	−0.779455	−	0.626458i	$-0.784504\pi$
$293$	−26.6842	−1.55891	−0.779455	+	0.626458i	$0.784504\pi$
$294$	0	0
$295$	6.70388	0.390315
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.69646	−0.445098
$300$	0	0
$301$	1.58002	0.0910707
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−23.3929	−1.33947
$306$	0	0
$307$	−20.7597	−1.18482	−0.592409	−	0.805637i	$-0.701823\pi$
$307$	−20.7597	−1.18482	−0.592409	+	0.805637i	$0.701823\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.71130	0.550677	0.275339	−	0.961347i	$-0.411210\pi$
$311$	9.71130	0.550677	0.275339	+	0.961347i	$0.411210\pi$
$312$	0	0
$313$	−12.2887	−0.694599	−0.347299	−	0.937754i	$-0.612901\pi$
$313$	−12.2887	−0.694599	−0.347299	+	0.937754i	$0.612901\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.9500	−1.06434	−0.532170	−	0.846637i	$-0.678623\pi$
$317$	−18.9500	−1.06434	−0.532170	+	0.846637i	$0.678623\pi$
$318$	0	0
$319$	9.14195	0.511851
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.81290	−0.323439
$324$	0	0
$325$	−4.52420	−0.250957
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.70388	0.149070
$330$	0	0
$331$	−17.7900	−0.977827	−0.488914	−	0.872332i	$-0.662607\pi$
$331$	−17.7900	−0.977827	−0.488914	+	0.872332i	$0.662607\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	26.4003	1.44240
$336$	0	0
$337$	−2.07546	−0.113058	−0.0565288	−	0.998401i	$-0.518003\pi$
$337$	−2.07546	−0.113058	−0.0565288	+	0.998401i	$0.518003\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.08613	−0.275429
$342$	0	0
$343$	−8.80068	−0.475192
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.7039	0.681980	0.340990	−	0.940067i	$-0.389238\pi$
$347$	12.7039	0.681980	0.340990	+	0.940067i	$0.389238\pi$
$348$	0	0
$349$	−29.2058	−1.56335	−0.781676	−	0.623685i	$-0.785635\pi$
$349$	−29.2058	−1.56335	−0.781676	+	0.623685i	$0.785635\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−34.6661	−1.84509	−0.922546	−	0.385886i	$-0.873896\pi$
$353$	−34.6661	−1.84509	−0.922546	+	0.385886i	$0.873896\pi$
$354$	0	0
$355$	−23.5800	−1.25150
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.6965	1.14510	0.572548	−	0.819871i	$-0.305955\pi$
$359$	21.6965	1.14510	0.572548	+	0.819871i	$0.305955\pi$
$360$	0	0
$361$	−7.76450	−0.408658
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.7039	0.664951
$366$	0	0
$367$	−4.18710	−0.218565	−0.109282	−	0.994011i	$-0.534855\pi$
$367$	−4.18710	−0.218565	−0.109282	+	0.994011i	$0.534855\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.54384	−0.183987
$372$	0	0
$373$	15.1090	0.782316	0.391158	−	0.920324i	$-0.372075\pi$
$373$	15.1090	0.782316	0.391158	+	0.920324i	$0.372075\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.14195	0.470834
$378$	0	0
$379$	−29.2584	−1.50290	−0.751451	−	0.659789i	$-0.770646\pi$
$379$	−29.2584	−1.50290	−0.751451	+	0.659789i	$0.770646\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11.8129	0.603611	0.301806	−	0.953369i	$-0.402411\pi$
$383$	11.8129	0.603611	0.301806	+	0.953369i	$0.402411\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.23550	0.0626422	0.0313211	−	0.999509i	$-0.490029\pi$
$389$	1.23550	0.0626422	0.0313211	+	0.999509i	$0.490029\pi$
$390$	0	0
$391$	−13.3471	−0.674993
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−9.88356	−0.497296
$396$	0	0
$397$	8.23289	0.413197	0.206598	−	0.978426i	$-0.433761\pi$
$397$	8.23289	0.413197	0.206598	+	0.978426i	$0.433761\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.30679	−0.414821	−0.207411	−	0.978254i	$-0.566504\pi$
$401$	−8.30679	−0.414821	−0.207411	+	0.978254i	$0.566504\pi$
$402$	0	0
$403$	−5.08613	−0.253358
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.29612	0.0642464
$408$	0	0
$409$	3.88356	0.192030	0.0960148	−	0.995380i	$-0.469390\pi$
$409$	3.88356	0.192030	0.0960148	+	0.995380i	$0.469390\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.40776	−0.0692712
$414$	0	0
$415$	−15.0484	−0.738697
$416$	0	0
$417$	0	0
$418$	0	0
$419$	35.8539	1.75158	0.875788	−	0.482695i	$-0.160342\pi$
$419$	35.8539	1.75158	0.875788	+	0.482695i	$0.160342\pi$
$420$	0	0
$421$	−2.11164	−0.102915	−0.0514574	−	0.998675i	$-0.516387\pi$
$421$	−2.11164	−0.102915	−0.0514574	+	0.998675i	$0.516387\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.84583	−0.380578
$426$	0	0
$427$	4.91231	0.237723
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.0484	−0.917529	−0.458765	−	0.888558i	$-0.651708\pi$
$431$	−19.0484	−0.917529	−0.458765	+	0.888558i	$0.651708\pi$
$432$	0	0
$433$	36.7858	1.76781	0.883907	−	0.467662i	$-0.154904\pi$
$433$	36.7858	1.76781	0.883907	+	0.467662i	$0.154904\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	25.7981	1.23409
$438$	0	0
$439$	−40.3478	−1.92569	−0.962847	−	0.270047i	$-0.912961\pi$
$439$	−40.3478	−1.92569	−0.962847	+	0.270047i	$0.912961\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.75228	−0.273299	−0.136649	−	0.990619i	$-0.543633\pi$
$443$	−5.75228	−0.273299	−0.136649	+	0.990619i	$0.543633\pi$
$444$	0	0
$445$	49.7933	2.36043
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.3068	0.958337	0.479168	−	0.877723i	$-0.340938\pi$
$449$	20.3068	0.958337	0.479168	+	0.877723i	$0.340938\pi$
$450$	0	0
$451$	−5.69646	−0.268236
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	−33.7374	−1.57817	−0.789085	−	0.614283i	$-0.789445\pi$
$457$	−33.7374	−1.57817	−0.789085	+	0.614283i	$0.789445\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17.5848	−0.819007	−0.409503	−	0.912309i	$-0.634298\pi$
$461$	−17.5848	−0.819007	−0.409503	+	0.912309i	$0.634298\pi$
$462$	0	0
$463$	12.3068	0.571945	0.285973	−	0.958238i	$-0.407683\pi$
$463$	12.3068	0.571945	0.285973	+	0.958238i	$0.407683\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	40.7300	1.88476	0.942380	−	0.334543i	$-0.108582\pi$
$467$	40.7300	1.88476	0.942380	+	0.334543i	$0.108582\pi$
$468$	0	0
$469$	−5.54384	−0.255991
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.43807	−0.112103
$474$	0	0
$475$	15.1648	0.695811
$476$	0	0
$477$	0	0
$478$	0	0
$479$	23.1042	1.05566	0.527829	−	0.849350i	$-0.323006\pi$
$479$	23.1042	1.05566	0.527829	+	0.849350i	$0.323006\pi$
$480$	0	0
$481$	1.29612	0.0590981
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.34452	0.378905
$486$	0	0
$487$	−2.81707	−0.127654	−0.0638268	−	0.997961i	$-0.520331\pi$
$487$	−2.81707	−0.127654	−0.0638268	+	0.997961i	$0.520331\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.0632	0.679794	0.339897	−	0.940463i	$-0.389608\pi$
$491$	15.0632	0.679794	0.339897	+	0.940463i	$0.389608\pi$
$492$	0	0
$493$	15.8539	0.714023
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.95160	0.222110
$498$	0	0
$499$	29.9984	1.34291	0.671457	−	0.741043i	$-0.265669\pi$
$499$	29.9984	1.34291	0.671457	+	0.741043i	$0.265669\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−35.9245	−1.60180	−0.800898	−	0.598801i	$-0.795644\pi$
$503$	−35.9245	−1.60180	−0.800898	+	0.598801i	$0.795644\pi$
$504$	0	0
$505$	15.3371	0.682492
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−28.8990	−1.28093	−0.640464	−	0.767989i	$-0.721258\pi$
$509$	−28.8990	−1.28093	−0.640464	+	0.767989i	$0.721258\pi$
$510$	0	0
$511$	−2.66771	−0.118012
$512$	0	0
$513$	0	0
$514$	0	0
$515$	42.4413	1.87019
$516$	0	0
$517$	−4.17226	−0.183496
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.9368	1.44299	0.721493	−	0.692422i	$-0.243456\pi$
$521$	32.9368	1.44299	0.721493	+	0.692422i	$0.243456\pi$
$522$	0	0
$523$	−23.1271	−1.01128	−0.505639	−	0.862745i	$-0.668743\pi$
$523$	−23.1271	−1.01128	−0.505639	+	0.862745i	$0.668743\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.82032	−0.384219
$528$	0	0
$529$	36.2355	1.57546
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.69646	−0.246741
$534$	0	0
$535$	−11.0484	−0.477664
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.58002	0.283421
$540$	0	0
$541$	31.4078	1.35032	0.675162	−	0.737669i	$-0.264074\pi$
$541$	31.4078	1.35032	0.675162	+	0.737669i	$0.264074\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	26.4562	1.13326
$546$	0	0
$547$	14.1903	0.606735	0.303368	−	0.952874i	$-0.401889\pi$
$547$	14.1903	0.606735	0.303368	+	0.952874i	$0.401889\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−30.6433	−1.30545
$552$	0	0
$553$	2.07546	0.0882576
$554$	0	0
$555$	0	0
$556$	0	0
$557$	32.5120	1.37758	0.688788	−	0.724963i	$-0.258143\pi$
$557$	32.5120	1.37758	0.688788	+	0.724963i	$0.258143\pi$
$558$	0	0
$559$	−2.43807	−0.103119
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.51939	0.401195	0.200597	−	0.979674i	$-0.435712\pi$
$563$	9.51939	0.401195	0.200597	+	0.979674i	$0.435712\pi$
$564$	0	0
$565$	−29.5652	−1.24382
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.54709	−0.232546	−0.116273	−	0.993217i	$-0.537095\pi$
$569$	−5.54709	−0.232546	−0.116273	+	0.993217i	$0.537095\pi$
$570$	0	0
$571$	−7.73419	−0.323666	−0.161833	−	0.986818i	$-0.551741\pi$
$571$	−7.73419	−0.323666	−0.161833	+	0.986818i	$0.551741\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	34.8203	1.45211
$576$	0	0
$577$	4.82513	0.200873	0.100436	−	0.994943i	$-0.467976\pi$
$577$	4.82513	0.200873	0.100436	+	0.994943i	$0.467976\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.16003	0.131100
$582$	0	0
$583$	5.46838	0.226477
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.23550	0.133543	0.0667716	−	0.997768i	$-0.478730\pi$
$587$	3.23550	0.133543	0.0667716	+	0.997768i	$0.478730\pi$
$588$	0	0
$589$	17.0484	0.702467
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.40295	−0.304003	−0.152001	−	0.988380i	$-0.548572\pi$
$593$	−7.40295	−0.304003	−0.152001	+	0.988380i	$0.548572\pi$
$594$	0	0
$595$	3.46838	0.142190
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.7252	−1.50055	−0.750276	−	0.661125i	$-0.770079\pi$
$599$	−36.7252	−1.50055	−0.750276	+	0.661125i	$0.770079\pi$
$600$	0	0
$601$	−32.4413	−1.32331	−0.661655	−	0.749809i	$-0.730145\pi$
$601$	−32.4413	−1.32331	−0.661655	+	0.749809i	$0.730145\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.08613	−0.125469
$606$	0	0
$607$	37.2026	1.51001	0.755003	−	0.655721i	$-0.227635\pi$
$607$	37.2026	1.51001	0.755003	+	0.655721i	$0.227635\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.17226	−0.168792
$612$	0	0
$613$	30.8203	1.24482	0.622411	−	0.782691i	$-0.286153\pi$
$613$	30.8203	1.24482	0.622411	+	0.782691i	$0.286153\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21.7900	−0.877233	−0.438616	−	0.898674i	$-0.644531\pi$
$617$	−21.7900	−0.877233	−0.438616	+	0.898674i	$0.644531\pi$
$618$	0	0
$619$	−10.5545	−0.424222	−0.212111	−	0.977246i	$-0.568034\pi$
$619$	−10.5545	−0.424222	−0.212111	+	0.977246i	$0.568034\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.4562	−0.418917
$624$	0	0
$625$	−27.1526	−1.08610
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.24772	0.0896226
$630$	0	0
$631$	−10.4791	−0.417164	−0.208582	−	0.978005i	$-0.566885\pi$
$631$	−10.4791	−0.417164	−0.208582	+	0.978005i	$0.566885\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.05582	−0.319685
$636$	0	0
$637$	6.58002	0.260710
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−0.582628	−0.0230124	−0.0115062	−	0.999934i	$-0.503663\pi$
$641$	−0.582628	−0.0230124	−0.0115062	+	0.999934i	$0.503663\pi$
$642$	0	0
$643$	−32.3578	−1.27607	−0.638034	−	0.770009i	$-0.720252\pi$
$643$	−32.3578	−1.27607	−0.638034	+	0.770009i	$0.720252\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.37158	0.0539225	0.0269613	−	0.999636i	$-0.491417\pi$
$647$	1.37158	0.0539225	0.0269613	+	0.999636i	$0.491417\pi$
$648$	0	0
$649$	2.17226	0.0852687
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.0484	1.13675	0.568376	−	0.822769i	$-0.307572\pi$
$653$	29.0484	1.13675	0.568376	+	0.822769i	$0.307572\pi$
$654$	0	0
$655$	25.0336	0.978142
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.4832	−0.603141	−0.301570	−	0.953444i	$-0.597511\pi$
$659$	−15.4832	−0.603141	−0.301570	+	0.953444i	$0.597511\pi$
$660$	0	0
$661$	38.5019	1.49755	0.748776	−	0.662823i	$-0.230642\pi$
$661$	38.5019	1.49755	0.748776	+	0.662823i	$0.230642\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.70388	−0.259965
$666$	0	0
$667$	−70.3606	−2.72437
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.58002	−0.292623
$672$	0	0
$673$	16.6071	0.640156	0.320078	−	0.947391i	$-0.396291\pi$
$673$	16.6071	0.640156	0.320078	+	0.947391i	$0.396291\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.32643	0.243145	0.121572	−	0.992583i	$-0.461206\pi$
$677$	6.32643	0.243145	0.121572	+	0.992583i	$0.461206\pi$
$678$	0	0
$679$	−1.75228	−0.0672462
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−29.5142	−1.12933	−0.564664	−	0.825321i	$-0.690994\pi$
$683$	−29.5142	−1.12933	−0.564664	+	0.825321i	$0.690994\pi$
$684$	0	0
$685$	49.7933	1.90250
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.46838	0.208329
$690$	0	0
$691$	13.7145	0.521726	0.260863	−	0.965376i	$-0.415993\pi$
$691$	13.7145	0.521726	0.260863	+	0.965376i	$0.415993\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.6965	0.519536
$696$	0	0
$697$	−9.87875	−0.374184
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24.9400	−0.941971	−0.470986	−	0.882141i	$-0.656102\pi$
$701$	−24.9400	−0.941971	−0.470986	+	0.882141i	$0.656102\pi$
$702$	0	0
$703$	−4.34452	−0.163857
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.22066	−0.121125
$708$	0	0
$709$	−32.7252	−1.22902	−0.614511	−	0.788909i	$-0.710646\pi$
$709$	−32.7252	−1.22902	−0.614511	+	0.788909i	$0.710646\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	39.1452	1.46600
$714$	0	0
$715$	−3.08613	−0.115415
$716$	0	0
$717$	0	0
$718$	0	0
$719$	32.9123	1.22742	0.613711	−	0.789531i	$-0.289676\pi$
$719$	32.9123	1.22742	0.613711	+	0.789531i	$0.289676\pi$
$720$	0	0
$721$	−8.91231	−0.331912
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−41.3600	−1.53607
$726$	0	0
$727$	30.6284	1.13595	0.567973	−	0.823048i	$-0.307728\pi$
$727$	30.6284	1.13595	0.567973	+	0.823048i	$0.307728\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.22808	−0.156381
$732$	0	0
$733$	−15.9655	−0.589700	−0.294850	−	0.955544i	$-0.595270\pi$
$733$	−15.9655	−0.589700	−0.294850	+	0.955544i	$0.595270\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.55451	0.315110
$738$	0	0
$739$	40.7300	1.49828	0.749139	−	0.662413i	$-0.230467\pi$
$739$	40.7300	1.49828	0.749139	+	0.662413i	$0.230467\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−31.2355	−1.14592	−0.572960	−	0.819584i	$-0.694205\pi$
$743$	−31.2355	−1.14592	−0.572960	+	0.819584i	$0.694205\pi$
$744$	0	0
$745$	64.0213	2.34556
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.32007	0.0847735
$750$	0	0
$751$	12.8761	0.469857	0.234928	−	0.972013i	$-0.424514\pi$
$751$	12.8761	0.469857	0.234928	+	0.972013i	$0.424514\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.34452	0.230901
$756$	0	0
$757$	−20.7087	−0.752670	−0.376335	−	0.926484i	$-0.622816\pi$
$757$	−20.7087	−0.752670	−0.376335	+	0.926484i	$0.622816\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.1164	0.656721	0.328360	−	0.944553i	$-0.393504\pi$
$761$	18.1164	0.656721	0.328360	+	0.944553i	$0.393504\pi$
$762$	0	0
$763$	−5.55557	−0.201125
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.17226	0.0784358
$768$	0	0
$769$	30.6890	1.10668	0.553338	−	0.832957i	$-0.313354\pi$
$769$	30.6890	1.10668	0.553338	+	0.832957i	$0.313354\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	23.7145	0.852953	0.426476	−	0.904499i	$-0.359755\pi$
$773$	23.7145	0.852953	0.426476	+	0.904499i	$0.359755\pi$
$774$	0	0
$775$	23.0107	0.826568
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.0942	0.684120
$780$	0	0
$781$	−7.64064	−0.273404
$782$	0	0
$783$	0	0
$784$	0	0
$785$	22.1574	0.790832
$786$	0	0
$787$	−33.3371	−1.18834	−0.594170	−	0.804340i	$-0.702519\pi$
$787$	−33.3371	−1.18834	−0.594170	+	0.804340i	$0.702519\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.20843	0.220746
$792$	0	0
$793$	−7.58002	−0.269174
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−45.5046	−1.61185	−0.805927	−	0.592015i	$-0.798333\pi$
$797$	−45.5046	−1.61185	−0.805927	+	0.592015i	$0.798333\pi$
$798$	0	0
$799$	−7.23550	−0.255973
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.11644	0.145266
$804$	0	0
$805$	−15.3929	−0.542529
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32.6710	1.14865	0.574325	−	0.818628i	$-0.305265\pi$
$809$	32.6710	1.14865	0.574325	+	0.818628i	$0.305265\pi$
$810$	0	0
$811$	17.7113	0.621928	0.310964	−	0.950422i	$-0.399348\pi$
$811$	17.7113	0.621928	0.310964	+	0.950422i	$0.399348\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.460964	0.0161468
$816$	0	0
$817$	8.17226	0.285911
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38.6938	−1.35042	−0.675212	−	0.737623i	$-0.735948\pi$
$821$	−38.6938	−1.35042	−0.675212	+	0.737623i	$0.735948\pi$
$822$	0	0
$823$	−23.3175	−0.812795	−0.406398	−	0.913696i	$-0.633215\pi$
$823$	−23.3175	−0.812795	−0.406398	+	0.913696i	$0.633215\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.5848	−0.611484	−0.305742	−	0.952114i	$-0.598905\pi$
$827$	−17.5848	−0.611484	−0.305742	+	0.952114i	$0.598905\pi$
$828$	0	0
$829$	39.7226	1.37962	0.689812	−	0.723989i	$-0.257693\pi$
$829$	39.7226	1.37962	0.689812	+	0.723989i	$0.257693\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	11.4110	0.395368
$834$	0	0
$835$	−25.4078	−0.879272
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−46.6742	−1.61137	−0.805686	−	0.592343i	$-0.798203\pi$
$839$	−46.6742	−1.61137	−0.805686	+	0.592343i	$0.798203\pi$
$840$	0	0
$841$	54.5752	1.88190
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.08613	−0.106166
$846$	0	0
$847$	0.648061	0.0222676
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9.97555	−0.341957
$852$	0	0
$853$	16.9271	0.579575	0.289787	−	0.957091i	$-0.406415\pi$
$853$	16.9271	0.579575	0.289787	+	0.957091i	$0.406415\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.2994	1.27412	0.637061	−	0.770813i	$-0.280150\pi$
$857$	37.2994	1.27412	0.637061	+	0.770813i	$0.280150\pi$
$858$	0	0
$859$	−0.531618	−0.0181386	−0.00906929	−	0.999959i	$-0.502887\pi$
$859$	−0.531618	−0.0181386	−0.00906929	+	0.999959i	$0.502887\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47.8129	−1.62757	−0.813785	−	0.581166i	$-0.802597\pi$
$863$	−47.8129	−1.62757	−0.813785	+	0.581166i	$0.802597\pi$
$864$	0	0
$865$	53.0894	1.80509
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.20257	−0.108640
$870$	0	0
$871$	8.55451	0.289859
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.951601	0.0321700
$876$	0	0
$877$	35.6768	1.20472	0.602360	−	0.798224i	$-0.294227\pi$
$877$	35.6768	1.20472	0.602360	+	0.798224i	$0.294227\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−37.8491	−1.27517	−0.637584	−	0.770381i	$-0.720066\pi$
$881$	−37.8491	−1.27517	−0.637584	+	0.770381i	$0.720066\pi$
$882$	0	0
$883$	−5.40776	−0.181986	−0.0909928	−	0.995852i	$-0.529004\pi$
$883$	−5.40776	−0.181986	−0.0909928	+	0.995852i	$0.529004\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.05101	0.270326	0.135163	−	0.990823i	$-0.456844\pi$
$887$	8.05101	0.270326	0.135163	+	0.990823i	$0.456844\pi$
$888$	0	0
$889$	1.69165	0.0567362
$890$	0	0
$891$	0	0
$892$	0	0
$893$	13.9852	0.467996
$894$	0	0
$895$	−19.9852	−0.668030
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−46.4971	−1.55077
$900$	0	0
$901$	9.48322	0.315932
$902$	0	0
$903$	0	0
$904$	0	0
$905$	36.2839	1.20612
$906$	0	0
$907$	−20.9974	−0.697207	−0.348603	−	0.937270i	$-0.613344\pi$
$907$	−20.9974	−0.697207	−0.348603	+	0.937270i	$0.613344\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−45.2207	−1.49823	−0.749114	−	0.662442i	$-0.769520\pi$
$911$	−45.2207	−1.49823	−0.749114	+	0.662442i	$0.769520\pi$
$912$	0	0
$913$	−4.87614	−0.161377
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.25683	−0.173596
$918$	0	0
$919$	50.6103	1.66948	0.834740	−	0.550644i	$-0.185618\pi$
$919$	50.6103	1.66948	0.834740	+	0.550644i	$0.185618\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7.64064	−0.251495
$924$	0	0
$925$	−5.86391	−0.192804
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−35.6784	−1.17057	−0.585285	−	0.810828i	$-0.699017\pi$
$929$	−35.6784	−1.17057	−0.585285	+	0.810828i	$0.699017\pi$
$930$	0	0
$931$	−22.0558	−0.722850
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.35194	−0.175027
$936$	0	0
$937$	−35.8491	−1.17114	−0.585569	−	0.810622i	$-0.699129\pi$
$937$	−35.8491	−1.17114	−0.585569	+	0.810622i	$0.699129\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	29.5455	0.963157	0.481578	−	0.876403i	$-0.340064\pi$
$941$	29.5455	0.963157	0.481578	+	0.876403i	$0.340064\pi$
$942$	0	0
$943$	43.8426	1.42771
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.2839	−1.04909	−0.524543	−	0.851384i	$-0.675764\pi$
$947$	−32.2839	−1.04909	−0.524543	+	0.851384i	$0.675764\pi$
$948$	0	0
$949$	4.11644	0.133625
$950$	0	0
$951$	0	0
$952$	0	0
$953$	40.4987	1.31188	0.655941	−	0.754812i	$-0.272272\pi$
$953$	40.4987	1.31188	0.655941	+	0.754812i	$0.272272\pi$
$954$	0	0
$955$	−6.35936	−0.205784
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.4562	−0.337647
$960$	0	0
$961$	−5.13128	−0.165525
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.0484	0.420043
$966$	0	0
$967$	21.1797	0.681093	0.340546	−	0.940228i	$-0.389388\pi$
$967$	21.1797	0.681093	0.340546	+	0.940228i	$0.389388\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	56.8417	1.82414	0.912068	−	0.410039i	$-0.134485\pi$
$971$	56.8417	1.82414	0.912068	+	0.410039i	$0.134485\pi$
$972$	0	0
$973$	−2.87614	−0.0922048
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.8990	−0.668619	−0.334310	−	0.942463i	$-0.608503\pi$
$977$	−20.8990	−0.668619	−0.334310	+	0.942463i	$0.608503\pi$
$978$	0	0
$979$	16.1345	0.515662
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−53.1452	−1.69507	−0.847534	−	0.530741i	$-0.821914\pi$
$983$	−53.1452	−1.69507	−0.847534	+	0.530741i	$0.821914\pi$
$984$	0	0
$985$	31.2207	0.994773
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.7645	0.596677
$990$	0	0
$991$	−11.7013	−0.371703	−0.185852	−	0.982578i	$-0.559504\pi$
$991$	−11.7013	−0.371703	−0.185852	+	0.982578i	$0.559504\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	12.3445	0.391348
$996$	0	0
$997$	−16.5168	−0.523092	−0.261546	−	0.965191i	$-0.584232\pi$
$997$	−16.5168	−0.523092	−0.261546	+	0.965191i	$0.584232\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.m.1.1		3
3.2	odd	2		1716.2.a.g.1.3	✓	3
12.11	even	2		6864.2.a.br.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.g.1.3	✓	3	3.2	odd	2
5148.2.a.m.1.1		3	1.1	even	1	trivial
6864.2.a.br.1.3		3	12.11	even	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:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q-3.08613 q^{5} +0.648061 q^{7} +O(q^{10})\)	\(q-3.08613 q^{5} +0.648061 q^{7} -1.00000 q^{11} -1.00000 q^{13} -1.73419 q^{17} +3.35194 q^{19} +7.69646 q^{23} +4.52420 q^{25} -9.14195 q^{29} +5.08613 q^{31} -2.00000 q^{35} -1.29612 q^{37} +5.69646 q^{41} +2.43807 q^{43} +4.17226 q^{47} -6.58002 q^{49} -5.46838 q^{53} +3.08613 q^{55} -2.17226 q^{59} +7.58002 q^{61} +3.08613 q^{65} -8.55451 q^{67} +7.64064 q^{71} -4.11644 q^{73} -0.648061 q^{77} +3.20257 q^{79} +4.87614 q^{83} +5.35194 q^{85} -16.1345 q^{89} -0.648061 q^{91} -10.3445 q^{95} -2.70388 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) 3 * q - 2 * q^5 + 4 * q^7	\( 3 q - 2 q^{5} + 4 q^{7} - 3 q^{11} - 3 q^{13} + 8 q^{19} - 8 q^{23} - 3 q^{25} - 14 q^{29} + 8 q^{31} - 6 q^{35} - 8 q^{37} - 14 q^{41} - 2 q^{43} - 2 q^{47} + 3 q^{49} - 6 q^{53} + 2 q^{55} + 8 q^{59} + 2 q^{65} - 8 q^{67} - 2 q^{71} - 4 q^{73} - 4 q^{77} - 6 q^{79} - 4 q^{83} + 14 q^{85} - 8 q^{89} - 4 q^{91} - 2 q^{95} - 4 q^{97}+O(q^{100}) \) 3 * q - 2 * q^5 + 4 * q^7 - 3 * q^11 - 3 * q^13 + 8 * q^19 - 8 * q^23 - 3 * q^25 - 14 * q^29 + 8 * q^31 - 6 * q^35 - 8 * q^37 - 14 * q^41 - 2 * q^43 - 2 * q^47 + 3 * q^49 - 6 * q^53 + 2 * q^55 + 8 * q^59 + 2 * q^65 - 8 * q^67 - 2 * q^71 - 4 * q^73 - 4 * q^77 - 6 * q^79 - 4 * q^83 + 14 * q^85 - 8 * q^89 - 4 * q^91 - 2 * q^95 - 4 * q^97

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