LMFDB - Embedded newform 5220.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5220,2,Mod(1,5220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5220, 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("5220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5220.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:	$41.6819098551$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 580)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	5220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -0.630898 q^{7} +0.630898 q^{11} +4.34017 q^{13} -4.63090 q^{17} -1.70928 q^{19} -1.70928 q^{23} +1.00000 q^{25} -1.00000 q^{29} +1.70928 q^{31} +0.630898 q^{35} +3.36910 q^{37} -6.49693 q^{41} +6.38962 q^{43} -9.12783 q^{47} -6.60197 q^{49} -5.23513 q^{53} -0.630898 q^{55} +4.49693 q^{59} +12.4391 q^{61} -4.34017 q^{65} +1.70928 q^{67} -0.183417 q^{71} -1.70928 q^{73} -0.398032 q^{77} -13.8927 q^{79} +13.3112 q^{83} +4.63090 q^{85} -1.10504 q^{89} -2.73820 q^{91} +1.70928 q^{95} -7.28458 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 2 q^{7} - 2 q^{11} + 2 q^{13} - 10 q^{17} + 2 q^{19} + 2 q^{23} + 3 q^{25} - 3 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} - 2 q^{41} - 10 q^{43} - 6 q^{47} - q^{49} - 6 q^{53} + 2 q^{55}+ \cdots + 6 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 2 * q^7 - 2 * q^11 + 2 * q^13 - 10 * q^17 + 2 * q^19 + 2 * q^23 + 3 * q^25 - 3 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 - 2 * q^41 - 10 * q^43 - 6 * q^47 - q^49 - 6 * q^53 + 2 * q^55 - 4 * q^59 - 10 * q^61 - 2 * q^65 - 2 * q^67 + 4 * q^71 + 2 * q^73 - 20 * q^77 - 30 * q^79 + 14 * q^83 + 10 * q^85 - 2 * q^89 - 16 * q^91 - 2 * q^95 + 6 * q^97

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.630898	−0.238457	−0.119228	−	0.992867i	$-0.538042\pi$
$7$	−0.630898	−0.238457	−0.119228	+	0.992867i	$0.538042\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.630898	0.190223	0.0951114	−	0.995467i	$-0.469679\pi$
$11$	0.630898	0.190223	0.0951114	+	0.995467i	$0.469679\pi$
$12$	0	0
$13$	4.34017	1.20375	0.601874	−	0.798591i	$-0.294421\pi$
$13$	4.34017	1.20375	0.601874	+	0.798591i	$0.294421\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.63090	−1.12316	−0.561579	−	0.827423i	$-0.689806\pi$
$17$	−4.63090	−1.12316	−0.561579	+	0.827423i	$0.689806\pi$
$18$	0	0
$19$	−1.70928	−0.392135	−0.196067	−	0.980590i	$-0.562817\pi$
$19$	−1.70928	−0.392135	−0.196067	+	0.980590i	$0.562817\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.70928	−0.356409	−0.178204	−	0.983994i	$-0.557029\pi$
$23$	−1.70928	−0.356409	−0.178204	+	0.983994i	$0.557029\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.70928	0.306995	0.153497	−	0.988149i	$-0.450946\pi$
$31$	1.70928	0.306995	0.153497	+	0.988149i	$0.450946\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.630898	0.106641
$36$	0	0
$37$	3.36910	0.553877	0.276939	−	0.960888i	$-0.410680\pi$
$37$	3.36910	0.553877	0.276939	+	0.960888i	$0.410680\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.49693	−1.01465	−0.507325	−	0.861755i	$-0.669366\pi$
$41$	−6.49693	−1.01465	−0.507325	+	0.861755i	$0.669366\pi$
$42$	0	0
$43$	6.38962	0.974408	0.487204	−	0.873288i	$-0.338017\pi$
$43$	6.38962	0.974408	0.487204	+	0.873288i	$0.338017\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.12783	−1.33143	−0.665715	−	0.746206i	$-0.731873\pi$
$47$	−9.12783	−1.33143	−0.665715	+	0.746206i	$0.731873\pi$
$48$	0	0
$49$	−6.60197	−0.943138
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.23513	−0.719101	−0.359550	−	0.933126i	$-0.617070\pi$
$53$	−5.23513	−0.719101	−0.359550	+	0.933126i	$0.617070\pi$
$54$	0	0
$55$	−0.630898	−0.0850702
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.49693	0.585450	0.292725	−	0.956197i	$-0.405438\pi$
$59$	4.49693	0.585450	0.292725	+	0.956197i	$0.405438\pi$
$60$	0	0
$61$	12.4391	1.59266	0.796330	−	0.604863i	$-0.206772\pi$
$61$	12.4391	1.59266	0.796330	+	0.604863i	$0.206772\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.34017	−0.538332
$66$	0	0
$67$	1.70928	0.208821	0.104411	−	0.994534i	$-0.466704\pi$
$67$	1.70928	0.208821	0.104411	+	0.994534i	$0.466704\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.183417	−0.0217677	−0.0108838	−	0.999941i	$-0.503464\pi$
$71$	−0.183417	−0.0217677	−0.0108838	+	0.999941i	$0.503464\pi$
$72$	0	0
$73$	−1.70928	−0.200056	−0.100028	−	0.994985i	$-0.531893\pi$
$73$	−1.70928	−0.200056	−0.100028	+	0.994985i	$0.531893\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.398032	−0.0453599
$78$	0	0
$79$	−13.8927	−1.56305	−0.781525	−	0.623874i	$-0.785558\pi$
$79$	−13.8927	−1.56305	−0.781525	+	0.623874i	$0.785558\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.3112	1.46110	0.730549	−	0.682860i	$-0.239264\pi$
$83$	13.3112	1.46110	0.730549	+	0.682860i	$0.239264\pi$
$84$	0	0
$85$	4.63090	0.502291
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.10504	−0.117134	−0.0585670	−	0.998283i	$-0.518653\pi$
$89$	−1.10504	−0.117134	−0.0585670	+	0.998283i	$0.518653\pi$
$90$	0	0
$91$	−2.73820	−0.287042
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.70928	0.175368
$96$	0	0
$97$	−7.28458	−0.739637	−0.369819	−	0.929104i	$-0.620580\pi$
$97$	−7.28458	−0.739637	−0.369819	+	0.929104i	$0.620580\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.26180	−0.324561	−0.162280	−	0.986745i	$-0.551885\pi$
$101$	−3.26180	−0.324561	−0.162280	+	0.986745i	$0.551885\pi$
$102$	0	0
$103$	−15.2846	−1.50603	−0.753017	−	0.658001i	$-0.771402\pi$
$103$	−15.2846	−1.50603	−0.753017	+	0.658001i	$0.771402\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.10731	−0.203721	−0.101861	−	0.994799i	$-0.532480\pi$
$107$	−2.10731	−0.203721	−0.101861	+	0.994799i	$0.532480\pi$
$108$	0	0
$109$	−9.23513	−0.884565	−0.442283	−	0.896876i	$-0.645831\pi$
$109$	−9.23513	−0.884565	−0.442283	+	0.896876i	$0.645831\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.86603	0.363686	0.181843	−	0.983328i	$-0.441794\pi$
$113$	3.86603	0.363686	0.181843	+	0.983328i	$0.441794\pi$
$114$	0	0
$115$	1.70928	0.159391
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.92162	0.267825
$120$	0	0
$121$	−10.6020	−0.963815
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.54638	0.758368	0.379184	−	0.925321i	$-0.376205\pi$
$127$	8.54638	0.758368	0.379184	+	0.925321i	$0.376205\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.133969	0.0117049	0.00585247	−	0.999983i	$-0.498137\pi$
$131$	0.133969	0.0117049	0.00585247	+	0.999983i	$0.498137\pi$
$132$	0	0
$133$	1.07838	0.0935072
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.63090	−0.395644	−0.197822	−	0.980238i	$-0.563387\pi$
$137$	−4.63090	−0.395644	−0.197822	+	0.980238i	$0.563387\pi$
$138$	0	0
$139$	10.2557	0.869873	0.434937	−	0.900461i	$-0.356771\pi$
$139$	10.2557	0.869873	0.434937	+	0.900461i	$0.356771\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.73820	0.228980
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−1.65983	−0.135075	−0.0675374	−	0.997717i	$-0.521514\pi$
$151$	−1.65983	−0.135075	−0.0675374	+	0.997717i	$0.521514\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.70928	−0.137292
$156$	0	0
$157$	−11.6514	−0.929884	−0.464942	−	0.885341i	$-0.653925\pi$
$157$	−11.6514	−0.929884	−0.464942	+	0.885341i	$0.653925\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.07838	0.0849881
$162$	0	0
$163$	8.23287	0.644848	0.322424	−	0.946595i	$-0.395502\pi$
$163$	8.23287	0.644848	0.322424	+	0.946595i	$0.395502\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.133969	−0.0103668	−0.00518342	−	0.999987i	$-0.501650\pi$
$167$	−0.133969	−0.0103668	−0.00518342	+	0.999987i	$0.501650\pi$
$168$	0	0
$169$	5.83710	0.449008
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.18342	−0.470117	−0.235058	−	0.971981i	$-0.575528\pi$
$173$	−6.18342	−0.470117	−0.235058	+	0.971981i	$0.575528\pi$
$174$	0	0
$175$	−0.630898	−0.0476914
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−22.0410	−1.64742	−0.823712	−	0.567008i	$-0.808101\pi$
$179$	−22.0410	−1.64742	−0.823712	+	0.567008i	$0.808101\pi$
$180$	0	0
$181$	10.5958	0.787581	0.393791	−	0.919200i	$-0.371163\pi$
$181$	10.5958	0.787581	0.393791	+	0.919200i	$0.371163\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.36910	−0.247701
$186$	0	0
$187$	−2.92162	−0.213650
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.26406	−0.308537	−0.154268	−	0.988029i	$-0.549302\pi$
$191$	−4.26406	−0.308537	−0.154268	+	0.988029i	$0.549302\pi$
$192$	0	0
$193$	−1.31124	−0.0943854	−0.0471927	−	0.998886i	$-0.515027\pi$
$193$	−1.31124	−0.0943854	−0.0471927	+	0.998886i	$0.515027\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.6225	−1.75428	−0.877140	−	0.480235i	$-0.840551\pi$
$197$	−24.6225	−1.75428	−0.877140	+	0.480235i	$0.840551\pi$
$198$	0	0
$199$	−9.97334	−0.706991	−0.353496	−	0.935436i	$-0.615007\pi$
$199$	−9.97334	−0.706991	−0.353496	+	0.935436i	$0.615007\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.630898	0.0442803
$204$	0	0
$205$	6.49693	0.453765
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.07838	−0.0745929
$210$	0	0
$211$	−13.4101	−0.923192	−0.461596	−	0.887090i	$-0.652723\pi$
$211$	−13.4101	−0.923192	−0.461596	+	0.887090i	$0.652723\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.38962	−0.435769
$216$	0	0
$217$	−1.07838	−0.0732051
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−20.0989	−1.35200
$222$	0	0
$223$	6.07611	0.406886	0.203443	−	0.979087i	$-0.434787\pi$
$223$	6.07611	0.406886	0.203443	+	0.979087i	$0.434787\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.2329	−1.34290	−0.671451	−	0.741049i	$-0.734329\pi$
$227$	−20.2329	−1.34290	−0.671451	+	0.741049i	$0.734329\pi$
$228$	0	0
$229$	−21.8843	−1.44615	−0.723077	−	0.690767i	$-0.757273\pi$
$229$	−21.8843	−1.44615	−0.723077	+	0.690767i	$0.757273\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.1978	0.668080	0.334040	−	0.942559i	$-0.391588\pi$
$233$	10.1978	0.668080	0.334040	+	0.942559i	$0.391588\pi$
$234$	0	0
$235$	9.12783	0.595434
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.17727	−0.334890	−0.167445	−	0.985881i	$-0.553552\pi$
$239$	−5.17727	−0.334890	−0.167445	+	0.985881i	$0.553552\pi$
$240$	0	0
$241$	−25.5174	−1.64372	−0.821862	−	0.569687i	$-0.807064\pi$
$241$	−25.5174	−1.64372	−0.821862	+	0.569687i	$0.807064\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.60197	0.421784
$246$	0	0
$247$	−7.41855	−0.472031
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.0494	−1.51799	−0.758994	−	0.651098i	$-0.774309\pi$
$251$	−24.0494	−1.51799	−0.758994	+	0.651098i	$0.774309\pi$
$252$	0	0
$253$	−1.07838	−0.0677970
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.0144	1.37322	0.686610	−	0.727026i	$-0.259098\pi$
$257$	22.0144	1.37322	0.686610	+	0.727026i	$0.259098\pi$
$258$	0	0
$259$	−2.12556	−0.132076
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.12783	−0.0695447	−0.0347724	−	0.999395i	$-0.511071\pi$
$263$	−1.12783	−0.0695447	−0.0347724	+	0.999395i	$0.511071\pi$
$264$	0	0
$265$	5.23513	0.321592
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.85762	0.479088	0.239544	−	0.970886i	$-0.423002\pi$
$269$	7.85762	0.479088	0.239544	+	0.970886i	$0.423002\pi$
$270$	0	0
$271$	−6.47414	−0.393276	−0.196638	−	0.980476i	$-0.563002\pi$
$271$	−6.47414	−0.393276	−0.196638	+	0.980476i	$0.563002\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.630898	0.0380446
$276$	0	0
$277$	−18.5958	−1.11731	−0.558657	−	0.829399i	$-0.688683\pi$
$277$	−18.5958	−1.11731	−0.558657	+	0.829399i	$0.688683\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8.24128	−0.491633	−0.245817	−	0.969316i	$-0.579056\pi$
$281$	−8.24128	−0.491633	−0.245817	+	0.969316i	$0.579056\pi$
$282$	0	0
$283$	−21.5936	−1.28360	−0.641802	−	0.766870i	$-0.721813\pi$
$283$	−21.5936	−1.28360	−0.641802	+	0.766870i	$0.721813\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.09890	0.241950
$288$	0	0
$289$	4.44521	0.261483
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−31.0166	−1.81201	−0.906006	−	0.423265i	$-0.860884\pi$
$293$	−31.0166	−1.81201	−0.906006	+	0.423265i	$0.860884\pi$
$294$	0	0
$295$	−4.49693	−0.261821
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.41855	−0.429026
$300$	0	0
$301$	−4.03120	−0.232354
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12.4391	−0.712259
$306$	0	0
$307$	13.3958	0.764537	0.382268	−	0.924051i	$-0.375143\pi$
$307$	13.3958	0.764537	0.382268	+	0.924051i	$0.375143\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.2784	1.37670	0.688352	−	0.725377i	$-0.258334\pi$
$311$	24.2784	1.37670	0.688352	+	0.725377i	$0.258334\pi$
$312$	0	0
$313$	−0.0266620	−0.00150702	−0.000753512	−	1.00000i	$-0.500240\pi$
$313$	−0.0266620	−0.00150702	−0.000753512		1.00000i	$0.500240\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.9854	1.29099	0.645496	−	0.763764i	$-0.276651\pi$
$317$	22.9854	1.29099	0.645496	+	0.763764i	$0.276651\pi$
$318$	0	0
$319$	−0.630898	−0.0353235
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.91548	0.440429
$324$	0	0
$325$	4.34017	0.240749
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.75872	0.317489
$330$	0	0
$331$	25.9071	1.42398	0.711991	−	0.702189i	$-0.247794\pi$
$331$	25.9071	1.42398	0.711991	+	0.702189i	$0.247794\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.70928	−0.0933877
$336$	0	0
$337$	12.8599	0.700523	0.350261	−	0.936652i	$-0.386093\pi$
$337$	12.8599	0.700523	0.350261	+	0.936652i	$0.386093\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.07838	0.0583974
$342$	0	0
$343$	8.58145	0.463355
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.2579	−0.711722	−0.355861	−	0.934539i	$-0.615812\pi$
$347$	−13.2579	−0.711722	−0.355861	+	0.934539i	$0.615812\pi$
$348$	0	0
$349$	4.21008	0.225360	0.112680	−	0.993631i	$-0.464056\pi$
$349$	4.21008	0.225360	0.112680	+	0.993631i	$0.464056\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.24128	−0.438639	−0.219319	−	0.975653i	$-0.570384\pi$
$353$	−8.24128	−0.438639	−0.219319	+	0.975653i	$0.570384\pi$
$354$	0	0
$355$	0.183417	0.00973479
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.2329	−0.645626	−0.322813	−	0.946463i	$-0.604628\pi$
$359$	−12.2329	−0.645626	−0.322813	+	0.946463i	$0.604628\pi$
$360$	0	0
$361$	−16.0784	−0.846230
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.70928	0.0894676
$366$	0	0
$367$	−3.55252	−0.185440	−0.0927200	−	0.995692i	$-0.529556\pi$
$367$	−3.55252	−0.185440	−0.0927200	+	0.995692i	$0.529556\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.30283	0.171475
$372$	0	0
$373$	7.94214	0.411228	0.205614	−	0.978633i	$-0.434081\pi$
$373$	7.94214	0.411228	0.205614	+	0.978633i	$0.434081\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.34017	−0.223530
$378$	0	0
$379$	21.5259	1.10571	0.552855	−	0.833278i	$-0.313539\pi$
$379$	21.5259	1.10571	0.552855	+	0.833278i	$0.313539\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	26.2062	1.33907	0.669537	−	0.742778i	$-0.266492\pi$
$383$	26.2062	1.33907	0.669537	+	0.742778i	$0.266492\pi$
$384$	0	0
$385$	0.398032	0.0202856
$386$	0	0
$387$	0	0
$388$	0	0
$389$	22.6947	1.15067	0.575334	−	0.817919i	$-0.304872\pi$
$389$	22.6947	1.15067	0.575334	+	0.817919i	$0.304872\pi$
$390$	0	0
$391$	7.91548	0.400303
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13.8927	0.699017
$396$	0	0
$397$	−14.4124	−0.723338	−0.361669	−	0.932307i	$-0.617793\pi$
$397$	−14.4124	−0.723338	−0.361669	+	0.932307i	$0.617793\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−33.1194	−1.65390	−0.826952	−	0.562272i	$-0.809927\pi$
$401$	−33.1194	−1.65390	−0.826952	+	0.562272i	$0.809927\pi$
$402$	0	0
$403$	7.41855	0.369544
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.12556	0.105360
$408$	0	0
$409$	17.2351	0.852222	0.426111	−	0.904671i	$-0.359883\pi$
$409$	17.2351	0.852222	0.426111	+	0.904671i	$0.359883\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.83710	−0.139605
$414$	0	0
$415$	−13.3112	−0.653423
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.96880	0.193889	0.0969444	−	0.995290i	$-0.469093\pi$
$419$	3.96880	0.193889	0.0969444	+	0.995290i	$0.469093\pi$
$420$	0	0
$421$	−24.9216	−1.21461	−0.607303	−	0.794470i	$-0.707749\pi$
$421$	−24.9216	−1.21461	−0.607303	+	0.794470i	$0.707749\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.63090	−0.224632
$426$	0	0
$427$	−7.84778	−0.379781
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.67420	−0.273317	−0.136658	−	0.990618i	$-0.543636\pi$
$431$	−5.67420	−0.273317	−0.136658	+	0.990618i	$0.543636\pi$
$432$	0	0
$433$	12.3174	0.591936	0.295968	−	0.955198i	$-0.404358\pi$
$433$	12.3174	0.591936	0.295968	+	0.955198i	$0.404358\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.92162	0.139760
$438$	0	0
$439$	−26.3857	−1.25932	−0.629661	−	0.776870i	$-0.716806\pi$
$439$	−26.3857	−1.25932	−0.629661	+	0.776870i	$0.716806\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−38.1750	−1.81375	−0.906875	−	0.421400i	$-0.861539\pi$
$443$	−38.1750	−1.81375	−0.906875	+	0.421400i	$0.861539\pi$
$444$	0	0
$445$	1.10504	0.0523839
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.96880	0.281685	0.140843	−	0.990032i	$-0.455019\pi$
$449$	5.96880	0.281685	0.140843	+	0.990032i	$0.455019\pi$
$450$	0	0
$451$	−4.09890	−0.193010
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.73820	0.128369
$456$	0	0
$457$	−14.7649	−0.690671	−0.345336	−	0.938479i	$-0.612235\pi$
$457$	−14.7649	−0.690671	−0.345336	+	0.938479i	$0.612235\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13.6163	−0.634176	−0.317088	−	0.948396i	$-0.602705\pi$
$461$	−13.6163	−0.634176	−0.317088	+	0.948396i	$0.602705\pi$
$462$	0	0
$463$	27.3835	1.27262	0.636309	−	0.771434i	$-0.280460\pi$
$463$	27.3835	1.27262	0.636309	+	0.771434i	$0.280460\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.4824	1.27173	0.635866	−	0.771799i	$-0.280643\pi$
$467$	27.4824	1.27173	0.635866	+	0.771799i	$0.280643\pi$
$468$	0	0
$469$	−1.07838	−0.0497949
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.03120	0.185355
$474$	0	0
$475$	−1.70928	−0.0784269
$476$	0	0
$477$	0	0
$478$	0	0
$479$	23.1012	1.05552	0.527760	−	0.849394i	$-0.323032\pi$
$479$	23.1012	1.05552	0.527760	+	0.849394i	$0.323032\pi$
$480$	0	0
$481$	14.6225	0.666728
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.28458	0.330776
$486$	0	0
$487$	19.5669	0.886661	0.443330	−	0.896358i	$-0.353797\pi$
$487$	19.5669	0.886661	0.443330	+	0.896358i	$0.353797\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−19.3835	−0.874764	−0.437382	−	0.899276i	$-0.644094\pi$
$491$	−19.3835	−0.874764	−0.437382	+	0.899276i	$0.644094\pi$
$492$	0	0
$493$	4.63090	0.208565
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.115718	0.00519065
$498$	0	0
$499$	−14.3545	−0.642598	−0.321299	−	0.946978i	$-0.604119\pi$
$499$	−14.3545	−0.642598	−0.321299	+	0.946978i	$0.604119\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	33.5936	1.49786	0.748931	−	0.662648i	$-0.230567\pi$
$503$	33.5936	1.49786	0.748931	+	0.662648i	$0.230567\pi$
$504$	0	0
$505$	3.26180	0.145148
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.7998	1.49815	0.749074	−	0.662486i	$-0.230499\pi$
$509$	33.7998	1.49815	0.749074	+	0.662486i	$0.230499\pi$
$510$	0	0
$511$	1.07838	0.0477046
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15.2846	0.673519
$516$	0	0
$517$	−5.75872	−0.253268
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.6491	1.16752	0.583760	−	0.811926i	$-0.301581\pi$
$521$	26.6491	1.16752	0.583760	+	0.811926i	$0.301581\pi$
$522$	0	0
$523$	10.2907	0.449982	0.224991	−	0.974361i	$-0.427765\pi$
$523$	10.2907	0.449982	0.224991	+	0.974361i	$0.427765\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.91548	−0.344804
$528$	0	0
$529$	−20.0784	−0.872973
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−28.1978	−1.22138
$534$	0	0
$535$	2.10731	0.0911068
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.16517	−0.179406
$540$	0	0
$541$	−33.3028	−1.43180	−0.715900	−	0.698203i	$-0.753984\pi$
$541$	−33.3028	−1.43180	−0.715900	+	0.698203i	$0.753984\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9.23513	0.395590
$546$	0	0
$547$	−38.2062	−1.63358	−0.816790	−	0.576936i	$-0.804248\pi$
$547$	−38.2062	−1.63358	−0.816790	+	0.576936i	$0.804248\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.70928	0.0728176
$552$	0	0
$553$	8.76487	0.372720
$554$	0	0
$555$	0	0
$556$	0	0
$557$	41.5174	1.75915	0.879575	−	0.475760i	$-0.157827\pi$
$557$	41.5174	1.75915	0.879575	+	0.475760i	$0.157827\pi$
$558$	0	0
$559$	27.7321	1.17294
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.6042	0.784075	0.392038	−	0.919949i	$-0.371770\pi$
$563$	18.6042	0.784075	0.392038	+	0.919949i	$0.371770\pi$
$564$	0	0
$565$	−3.86603	−0.162645
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19.4452	−0.815186	−0.407593	−	0.913164i	$-0.633632\pi$
$569$	−19.4452	−0.815186	−0.407593	+	0.913164i	$0.633632\pi$
$570$	0	0
$571$	32.5113	1.36056	0.680278	−	0.732954i	$-0.261859\pi$
$571$	32.5113	1.36056	0.680278	+	0.732954i	$0.261859\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.70928	−0.0712817
$576$	0	0
$577$	−16.9444	−0.705405	−0.352702	−	0.935736i	$-0.614737\pi$
$577$	−16.9444	−0.705405	−0.352702	+	0.935736i	$0.614737\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.39803	−0.348409
$582$	0	0
$583$	−3.30283	−0.136789
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.59809	0.313607	0.156803	−	0.987630i	$-0.449881\pi$
$587$	7.59809	0.313607	0.156803	+	0.987630i	$0.449881\pi$
$588$	0	0
$589$	−2.92162	−0.120383
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−13.7998	−0.566688	−0.283344	−	0.959018i	$-0.591444\pi$
$593$	−13.7998	−0.566688	−0.283344	+	0.959018i	$0.591444\pi$
$594$	0	0
$595$	−2.92162	−0.119775
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.47414	−0.264526	−0.132263	−	0.991215i	$-0.542224\pi$
$599$	−6.47414	−0.264526	−0.132263	+	0.991215i	$0.542224\pi$
$600$	0	0
$601$	−38.3812	−1.56560	−0.782801	−	0.622272i	$-0.786210\pi$
$601$	−38.3812	−1.56560	−0.782801	+	0.622272i	$0.786210\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.6020	0.431031
$606$	0	0
$607$	−3.81271	−0.154753	−0.0773765	−	0.997002i	$-0.524654\pi$
$607$	−3.81271	−0.154753	−0.0773765	+	0.997002i	$0.524654\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−39.6163	−1.60271
$612$	0	0
$613$	13.6332	0.550638	0.275319	−	0.961353i	$-0.411216\pi$
$613$	13.6332	0.550638	0.275319	+	0.961353i	$0.411216\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.70928	−0.390881	−0.195440	−	0.980716i	$-0.562614\pi$
$617$	−9.70928	−0.390881	−0.195440	+	0.980716i	$0.562614\pi$
$618$	0	0
$619$	13.9916	0.562369	0.281185	−	0.959654i	$-0.409273\pi$
$619$	13.9916	0.562369	0.281185	+	0.959654i	$0.409273\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.697167	0.0279314
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−15.6020	−0.622091
$630$	0	0
$631$	−29.8264	−1.18737	−0.593686	−	0.804697i	$-0.702328\pi$
$631$	−29.8264	−1.18737	−0.593686	+	0.804697i	$0.702328\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.54638	−0.339153
$636$	0	0
$637$	−28.6537	−1.13530
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−41.8453	−1.65279	−0.826396	−	0.563090i	$-0.809612\pi$
$641$	−41.8453	−1.65279	−0.826396	+	0.563090i	$0.809612\pi$
$642$	0	0
$643$	29.7548	1.17342	0.586708	−	0.809798i	$-0.300424\pi$
$643$	29.7548	1.17342	0.586708	+	0.809798i	$0.300424\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.27635	0.0894925	0.0447463	−	0.998998i	$-0.485752\pi$
$647$	2.27635	0.0894925	0.0447463	+	0.998998i	$0.485752\pi$
$648$	0	0
$649$	2.83710	0.111366
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.2678	1.06707	0.533535	−	0.845778i	$-0.320863\pi$
$653$	27.2678	1.06707	0.533535	+	0.845778i	$0.320863\pi$
$654$	0	0
$655$	−0.133969	−0.00523461
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.1750	−0.707998	−0.353999	−	0.935246i	$-0.615178\pi$
$659$	−18.1750	−0.707998	−0.353999	+	0.935246i	$0.615178\pi$
$660$	0	0
$661$	−49.2639	−1.91614	−0.958072	−	0.286529i	$-0.907498\pi$
$661$	−49.2639	−1.91614	−0.958072	+	0.286529i	$0.907498\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.07838	−0.0418177
$666$	0	0
$667$	1.70928	0.0661834
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.84778	0.302960
$672$	0	0
$673$	−9.78539	−0.377199	−0.188599	−	0.982054i	$-0.560395\pi$
$673$	−9.78539	−0.377199	−0.188599	+	0.982054i	$0.560395\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26.4352	1.01599	0.507994	−	0.861361i	$-0.330387\pi$
$677$	26.4352	1.01599	0.507994	+	0.861361i	$0.330387\pi$
$678$	0	0
$679$	4.59583	0.176372
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.7031	0.868711	0.434356	−	0.900741i	$-0.356976\pi$
$683$	22.7031	0.868711	0.434356	+	0.900741i	$0.356976\pi$
$684$	0	0
$685$	4.63090	0.176938
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−22.7214	−0.865616
$690$	0	0
$691$	−5.81205	−0.221101	−0.110550	−	0.993871i	$-0.535261\pi$
$691$	−5.81205	−0.221101	−0.110550	+	0.993871i	$0.535261\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.2557	−0.389019
$696$	0	0
$697$	30.0866	1.13961
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.3111	1.44699	0.723494	−	0.690330i	$-0.242535\pi$
$701$	38.3111	1.44699	0.723494	+	0.690330i	$0.242535\pi$
$702$	0	0
$703$	−5.75872	−0.217194
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.05786	0.0773937
$708$	0	0
$709$	−3.75872	−0.141162	−0.0705809	−	0.997506i	$-0.522485\pi$
$709$	−3.75872	−0.141162	−0.0705809	+	0.997506i	$0.522485\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.92162	−0.109416
$714$	0	0
$715$	−2.73820	−0.102403
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21.8576	−0.815152	−0.407576	−	0.913171i	$-0.633626\pi$
$719$	−21.8576	−0.815152	−0.407576	+	0.913171i	$0.633626\pi$
$720$	0	0
$721$	9.64301	0.359124
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−18.2739	−0.677742	−0.338871	−	0.940833i	$-0.610045\pi$
$727$	−18.2739	−0.677742	−0.338871	+	0.940833i	$0.610045\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−29.5897	−1.09441
$732$	0	0
$733$	49.9383	1.84451	0.922256	−	0.386580i	$-0.126344\pi$
$733$	49.9383	1.84451	0.922256	+	0.386580i	$0.126344\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.07838	0.0397226
$738$	0	0
$739$	32.6453	1.20088	0.600438	−	0.799671i	$-0.294993\pi$
$739$	32.6453	1.20088	0.600438	+	0.799671i	$0.294993\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	38.3896	1.40838	0.704189	−	0.710012i	$-0.251311\pi$
$743$	38.3896	1.40838	0.704189	+	0.710012i	$0.251311\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.32950	0.0485787
$750$	0	0
$751$	44.6453	1.62913	0.814565	−	0.580073i	$-0.196976\pi$
$751$	44.6453	1.62913	0.814565	+	0.580073i	$0.196976\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.65983	0.0604073
$756$	0	0
$757$	26.2907	0.955553	0.477776	−	0.878482i	$-0.341443\pi$
$757$	26.2907	0.955553	0.477776	+	0.878482i	$0.341443\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.15061	0.331709	0.165855	−	0.986150i	$-0.446962\pi$
$761$	9.15061	0.331709	0.165855	+	0.986150i	$0.446962\pi$
$762$	0	0
$763$	5.82642	0.210931
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19.5174	0.704734
$768$	0	0
$769$	−9.90110	−0.357043	−0.178521	−	0.983936i	$-0.557131\pi$
$769$	−9.90110	−0.357043	−0.178521	+	0.983936i	$0.557131\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−26.5197	−0.953848	−0.476924	−	0.878945i	$-0.658248\pi$
$773$	−26.5197	−0.953848	−0.476924	+	0.878945i	$0.658248\pi$
$774$	0	0
$775$	1.70928	0.0613990
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.1050	0.397879
$780$	0	0
$781$	−0.115718	−0.00414070
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11.6514	0.415857
$786$	0	0
$787$	−52.6141	−1.87549	−0.937745	−	0.347325i	$-0.887090\pi$
$787$	−52.6141	−1.87549	−0.937745	+	0.347325i	$0.887090\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.43907	−0.0867233
$792$	0	0
$793$	53.9877	1.91716
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.4885	−0.654897	−0.327448	−	0.944869i	$-0.606189\pi$
$797$	−18.4885	−0.654897	−0.327448	+	0.944869i	$0.606189\pi$
$798$	0	0
$799$	42.2700	1.49541
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.07838	−0.0380551
$804$	0	0
$805$	−1.07838	−0.0380078
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−54.9770	−1.93289	−0.966445	−	0.256874i	$-0.917307\pi$
$809$	−54.9770	−1.93289	−0.966445	+	0.256874i	$0.917307\pi$
$810$	0	0
$811$	51.5006	1.80843	0.904216	−	0.427076i	$-0.140456\pi$
$811$	51.5006	1.80843	0.904216	+	0.427076i	$0.140456\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.23287	−0.288385
$816$	0	0
$817$	−10.9216	−0.382099
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.7961	−0.376785	−0.188393	−	0.982094i	$-0.560328\pi$
$821$	−10.7961	−0.376785	−0.188393	+	0.982094i	$0.560328\pi$
$822$	0	0
$823$	13.2969	0.463500	0.231750	−	0.972775i	$-0.425555\pi$
$823$	13.2969	0.463500	0.231750	+	0.972775i	$0.425555\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.3896	1.05675	0.528375	−	0.849011i	$-0.322801\pi$
$827$	30.3896	1.05675	0.528375	+	0.849011i	$0.322801\pi$
$828$	0	0
$829$	−9.64754	−0.335073	−0.167536	−	0.985866i	$-0.553581\pi$
$829$	−9.64754	−0.335073	−0.167536	+	0.985866i	$0.553581\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	30.5730	1.05929
$834$	0	0
$835$	0.133969	0.00463619
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.3584	1.04809	0.524045	−	0.851691i	$-0.324423\pi$
$839$	30.3584	1.04809	0.524045	+	0.851691i	$0.324423\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−5.83710	−0.200802
$846$	0	0
$847$	6.68876	0.229828
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.75872	−0.197407
$852$	0	0
$853$	−16.2472	−0.556295	−0.278147	−	0.960538i	$-0.589720\pi$
$853$	−16.2472	−0.556295	−0.278147	+	0.960538i	$0.589720\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.2183	−0.861441	−0.430721	−	0.902485i	$-0.641741\pi$
$857$	−25.2183	−0.861441	−0.430721	+	0.902485i	$0.641741\pi$
$858$	0	0
$859$	−10.4040	−0.354980	−0.177490	−	0.984123i	$-0.556798\pi$
$859$	−10.4040	−0.354980	−0.177490	+	0.984123i	$0.556798\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.515180	−0.0175369	−0.00876846	−	0.999962i	$-0.502791\pi$
$863$	−0.515180	−0.0175369	−0.00876846	+	0.999962i	$0.502791\pi$
$864$	0	0
$865$	6.18342	0.210243
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.76487	−0.297328
$870$	0	0
$871$	7.41855	0.251368
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.630898	0.0213282
$876$	0	0
$877$	−44.3234	−1.49669	−0.748347	−	0.663308i	$-0.769152\pi$
$877$	−44.3234	−1.49669	−0.748347	+	0.663308i	$0.769152\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	48.4222	1.63139	0.815694	−	0.578484i	$-0.196356\pi$
$881$	48.4222	1.63139	0.815694	+	0.578484i	$0.196356\pi$
$882$	0	0
$883$	2.85780	0.0961726	0.0480863	−	0.998843i	$-0.484688\pi$
$883$	2.85780	0.0961726	0.0480863	+	0.998843i	$0.484688\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.6453	0.424587	0.212293	−	0.977206i	$-0.431907\pi$
$887$	12.6453	0.424587	0.212293	+	0.977206i	$0.431907\pi$
$888$	0	0
$889$	−5.39189	−0.180838
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15.6020	0.522100
$894$	0	0
$895$	22.0410	0.736751
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.70928	−0.0570075
$900$	0	0
$901$	24.2434	0.807664
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.5958	−0.352217
$906$	0	0
$907$	4.39416	0.145906	0.0729528	−	0.997335i	$-0.476758\pi$
$907$	4.39416	0.145906	0.0729528	+	0.997335i	$0.476758\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	25.2267	0.835799	0.417899	−	0.908493i	$-0.362766\pi$
$911$	25.2267	0.835799	0.417899	+	0.908493i	$0.362766\pi$
$912$	0	0
$913$	8.39803	0.277934
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.0845208	−0.00279112
$918$	0	0
$919$	29.9733	0.988729	0.494365	−	0.869255i	$-0.335401\pi$
$919$	29.9733	0.988729	0.494365	+	0.869255i	$0.335401\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.796064	−0.0262028
$924$	0	0
$925$	3.36910	0.110775
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−52.8659	−1.73447	−0.867236	−	0.497897i	$-0.834106\pi$
$929$	−52.8659	−1.73447	−0.867236	+	0.497897i	$0.834106\pi$
$930$	0	0
$931$	11.2846	0.369837
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.92162	0.0955473
$936$	0	0
$937$	43.2762	1.41377	0.706885	−	0.707328i	$-0.250100\pi$
$937$	43.2762	1.41377	0.706885	+	0.707328i	$0.250100\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	53.1506	1.73266	0.866330	−	0.499472i	$-0.166473\pi$
$941$	53.1506	1.73266	0.866330	+	0.499472i	$0.166473\pi$
$942$	0	0
$943$	11.1050	0.361630
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24.8599	−0.807838	−0.403919	−	0.914795i	$-0.632352\pi$
$947$	−24.8599	−0.807838	−0.403919	+	0.914795i	$0.632352\pi$
$948$	0	0
$949$	−7.41855	−0.240816
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−8.47027	−0.274379	−0.137189	−	0.990545i	$-0.543807\pi$
$953$	−8.47027	−0.274379	−0.137189	+	0.990545i	$0.543807\pi$
$954$	0	0
$955$	4.26406	0.137982
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.92162	0.0943441
$960$	0	0
$961$	−28.0784	−0.905754
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.31124	0.0422104
$966$	0	0
$967$	−15.9116	−0.511683	−0.255841	−	0.966719i	$-0.582352\pi$
$967$	−15.9116	−0.511683	−0.255841	+	0.966719i	$0.582352\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.9071	1.08813	0.544065	−	0.839043i	$-0.316885\pi$
$971$	33.9071	1.08813	0.544065	+	0.839043i	$0.316885\pi$
$972$	0	0
$973$	−6.47027	−0.207427
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.1362	−0.420265	−0.210133	−	0.977673i	$-0.567390\pi$
$977$	−13.1362	−0.420265	−0.210133	+	0.977673i	$0.567390\pi$
$978$	0	0
$979$	−0.697167	−0.0222816
$980$	0	0
$981$	0	0
$982$	0	0
$983$	43.0700	1.37372	0.686859	−	0.726790i	$-0.258989\pi$
$983$	43.0700	1.37372	0.686859	+	0.726790i	$0.258989\pi$
$984$	0	0
$985$	24.6225	0.784538
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10.9216	−0.347287
$990$	0	0
$991$	−41.3217	−1.31263	−0.656314	−	0.754488i	$-0.727885\pi$
$991$	−41.3217	−1.31263	−0.656314	+	0.754488i	$0.727885\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.97334	0.316176
$996$	0	0
$997$	44.3605	1.40491	0.702456	−	0.711727i	$-0.252087\pi$
$997$	44.3605	1.40491	0.702456	+	0.711727i	$0.252087\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5220.2.a.w.1.2		3
3.2	odd	2		580.2.a.d.1.1	✓	3
12.11	even	2		2320.2.a.o.1.3		3
15.2	even	4		2900.2.c.g.349.5		6
15.8	even	4		2900.2.c.g.349.2		6
15.14	odd	2		2900.2.a.f.1.3		3
24.5	odd	2		9280.2.a.bh.1.3		3
24.11	even	2		9280.2.a.bt.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.a.d.1.1	✓	3	3.2	odd	2
2320.2.a.o.1.3		3	12.11	even	2
2900.2.a.f.1.3		3	15.14	odd	2
2900.2.c.g.349.2		6	15.8	even	4
2900.2.c.g.349.5		6	15.2	even	4
5220.2.a.w.1.2		3	1.1	even	1	trivial
9280.2.a.bh.1.3		3	24.5	odd	2
9280.2.a.bt.1.1		3	24.11	even	2

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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:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -0.630898 q^{7} +0.630898 q^{11} +4.34017 q^{13} -4.63090 q^{17} -1.70928 q^{19} -1.70928 q^{23} +1.00000 q^{25} -1.00000 q^{29} +1.70928 q^{31} +0.630898 q^{35} +3.36910 q^{37} -6.49693 q^{41} +6.38962 q^{43} -9.12783 q^{47} -6.60197 q^{49} -5.23513 q^{53} -0.630898 q^{55} +4.49693 q^{59} +12.4391 q^{61} -4.34017 q^{65} +1.70928 q^{67} -0.183417 q^{71} -1.70928 q^{73} -0.398032 q^{77} -13.8927 q^{79} +13.3112 q^{83} +4.63090 q^{85} -1.10504 q^{89} -2.73820 q^{91} +1.70928 q^{95} -7.28458 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 2 q^{7} - 2 q^{11} + 2 q^{13} - 10 q^{17} + 2 q^{19} + 2 q^{23} + 3 q^{25} - 3 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} - 2 q^{41} - 10 q^{43} - 6 q^{47} - q^{49} - 6 q^{53} + 2 q^{55}+ \cdots + 6 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 2 * q^7 - 2 * q^11 + 2 * q^13 - 10 * q^17 + 2 * q^19 + 2 * q^23 + 3 * q^25 - 3 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 - 2 * q^41 - 10 * q^43 - 6 * q^47 - q^49 - 6 * q^53 + 2 * q^55 - 4 * q^59 - 10 * q^61 - 2 * q^65 - 2 * q^67 + 4 * q^71 + 2 * q^73 - 20 * q^77 - 30 * q^79 + 14 * q^83 + 10 * q^85 - 2 * q^89 - 16 * q^91 - 2 * q^95 + 6 * q^97

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