LMFDB - Embedded newform 5580.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5580,2,Mod(1,5580)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5580, 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("5580.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5580 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5580.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:	$44.5565243279$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	5580.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -2.44949 q^{7} +0.449490 q^{13} -2.00000 q^{17} +4.89898 q^{19} -6.89898 q^{23} +1.00000 q^{25} +4.44949 q^{29} +1.00000 q^{31} -2.44949 q^{35} -8.44949 q^{37} +11.7980 q^{41} -12.8990 q^{43} +0.898979 q^{47} -1.00000 q^{49} +6.00000 q^{53} -7.34847 q^{59} +2.89898 q^{61} +0.449490 q^{65} -1.55051 q^{67} -6.44949 q^{71} -7.55051 q^{73} +6.89898 q^{79} +10.8990 q^{83} -2.00000 q^{85} -0.449490 q^{89} -1.10102 q^{91} +4.89898 q^{95} -14.8990 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 4 q^{13} - 4 q^{17} - 4 q^{23} + 2 q^{25} + 4 q^{29} + 2 q^{31} - 12 q^{37} + 4 q^{41} - 16 q^{43} - 8 q^{47} - 2 q^{49} + 12 q^{53} - 4 q^{61} - 4 q^{65} - 8 q^{67} - 8 q^{71} - 20 q^{73}+ \cdots - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^13 - 4 * q^17 - 4 * q^23 + 2 * q^25 + 4 * q^29 + 2 * q^31 - 12 * q^37 + 4 * q^41 - 16 * q^43 - 8 * q^47 - 2 * q^49 + 12 * q^53 - 4 * q^61 - 4 * q^65 - 8 * q^67 - 8 * q^71 - 20 * q^73 + 4 * q^79 + 12 * q^83 - 4 * q^85 + 4 * q^89 - 12 * q^91 - 20 * 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
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0.449490	0.124666	0.0623330	−	0.998055i	$-0.480146\pi$
$13$	0.449490	0.124666	0.0623330	+	0.998055i	$0.480146\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	4.89898	1.12390	0.561951	−	0.827170i	$-0.310051\pi$
$19$	4.89898	1.12390	0.561951	+	0.827170i	$0.310051\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.89898	−1.43854	−0.719268	−	0.694732i	$-0.755523\pi$
$23$	−6.89898	−1.43854	−0.719268	+	0.694732i	$0.755523\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.44949	0.826250	0.413125	−	0.910674i	$-0.364437\pi$
$29$	4.44949	0.826250	0.413125	+	0.910674i	$0.364437\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.44949	−0.414039
$36$	0	0
$37$	−8.44949	−1.38909	−0.694544	−	0.719450i	$-0.744394\pi$
$37$	−8.44949	−1.38909	−0.694544	+	0.719450i	$0.744394\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.7980	1.84253	0.921266	−	0.388934i	$-0.127156\pi$
$41$	11.7980	1.84253	0.921266	+	0.388934i	$0.127156\pi$
$42$	0	0
$43$	−12.8990	−1.96708	−0.983538	−	0.180702i	$-0.942163\pi$
$43$	−12.8990	−1.96708	−0.983538	+	0.180702i	$0.942163\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.898979	0.131130	0.0655648	−	0.997848i	$-0.479115\pi$
$47$	0.898979	0.131130	0.0655648	+	0.997848i	$0.479115\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.34847	−0.956689	−0.478345	−	0.878172i	$-0.658763\pi$
$59$	−7.34847	−0.956689	−0.478345	+	0.878172i	$0.658763\pi$
$60$	0	0
$61$	2.89898	0.371176	0.185588	−	0.982628i	$-0.440581\pi$
$61$	2.89898	0.371176	0.185588	+	0.982628i	$0.440581\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.449490	0.0557523
$66$	0	0
$67$	−1.55051	−0.189425	−0.0947125	−	0.995505i	$-0.530193\pi$
$67$	−1.55051	−0.189425	−0.0947125	+	0.995505i	$0.530193\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.44949	−0.765414	−0.382707	−	0.923870i	$-0.625008\pi$
$71$	−6.44949	−0.765414	−0.382707	+	0.923870i	$0.625008\pi$
$72$	0	0
$73$	−7.55051	−0.883720	−0.441860	−	0.897084i	$-0.645681\pi$
$73$	−7.55051	−0.883720	−0.441860	+	0.897084i	$0.645681\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.89898	0.776196	0.388098	−	0.921618i	$-0.373132\pi$
$79$	6.89898	0.776196	0.388098	+	0.921618i	$0.373132\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.8990	1.19632	0.598159	−	0.801377i	$-0.295899\pi$
$83$	10.8990	1.19632	0.598159	+	0.801377i	$0.295899\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.449490	−0.0476458	−0.0238229	−	0.999716i	$-0.507584\pi$
$89$	−0.449490	−0.0476458	−0.0238229	+	0.999716i	$0.507584\pi$
$90$	0	0
$91$	−1.10102	−0.115418
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.89898	0.502625
$96$	0	0
$97$	−14.8990	−1.51276	−0.756381	−	0.654131i	$-0.773034\pi$
$97$	−14.8990	−1.51276	−0.756381	+	0.654131i	$0.773034\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−10.4495	−1.02962	−0.514809	−	0.857305i	$-0.672137\pi$
$103$	−10.4495	−1.02962	−0.514809	+	0.857305i	$0.672137\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.89898	−0.473602	−0.236801	−	0.971558i	$-0.576099\pi$
$107$	−4.89898	−0.473602	−0.236801	+	0.971558i	$0.576099\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.79796	−0.357282	−0.178641	−	0.983914i	$-0.557170\pi$
$113$	−3.79796	−0.357282	−0.178641	+	0.983914i	$0.557170\pi$
$114$	0	0
$115$	−6.89898	−0.643333
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.89898	0.449089
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.898979	0.0797715	0.0398858	−	0.999204i	$-0.487301\pi$
$127$	0.898979	0.0797715	0.0398858	+	0.999204i	$0.487301\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.2474	1.07006	0.535032	−	0.844832i	$-0.320299\pi$
$131$	12.2474	1.07006	0.535032	+	0.844832i	$0.320299\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.10102	0.435810	0.217905	−	0.975970i	$-0.430078\pi$
$137$	5.10102	0.435810	0.217905	+	0.975970i	$0.430078\pi$
$138$	0	0
$139$	−5.79796	−0.491776	−0.245888	−	0.969298i	$-0.579080\pi$
$139$	−5.79796	−0.491776	−0.245888	+	0.969298i	$0.579080\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.44949	0.369510
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.7980	−1.62191	−0.810956	−	0.585107i	$-0.801052\pi$
$149$	−19.7980	−1.62191	−0.810956	+	0.585107i	$0.801052\pi$
$150$	0	0
$151$	−1.10102	−0.0895998	−0.0447999	−	0.998996i	$-0.514265\pi$
$151$	−1.10102	−0.0895998	−0.0447999	+	0.998996i	$0.514265\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	2.89898	0.231364	0.115682	−	0.993286i	$-0.463095\pi$
$157$	2.89898	0.231364	0.115682	+	0.993286i	$0.463095\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	16.8990	1.33183
$162$	0	0
$163$	−21.1464	−1.65632	−0.828158	−	0.560495i	$-0.810611\pi$
$163$	−21.1464	−1.65632	−0.828158	+	0.560495i	$0.810611\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−12.7980	−0.984458
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.7980	1.50521	0.752605	−	0.658472i	$-0.228797\pi$
$173$	19.7980	1.50521	0.752605	+	0.658472i	$0.228797\pi$
$174$	0	0
$175$	−2.44949	−0.185164
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−23.5959	−1.76364	−0.881821	−	0.471585i	$-0.843682\pi$
$179$	−23.5959	−1.76364	−0.881821	+	0.471585i	$0.843682\pi$
$180$	0	0
$181$	6.89898	0.512797	0.256399	−	0.966571i	$-0.417464\pi$
$181$	6.89898	0.512797	0.256399	+	0.966571i	$0.417464\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.44949	−0.621219
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.44949	−0.466669	−0.233334	−	0.972397i	$-0.574964\pi$
$191$	−6.44949	−0.466669	−0.233334	+	0.972397i	$0.574964\pi$
$192$	0	0
$193$	20.6969	1.48980	0.744899	−	0.667177i	$-0.232498\pi$
$193$	20.6969	1.48980	0.744899	+	0.667177i	$0.232498\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.69694	−0.619631	−0.309816	−	0.950797i	$-0.600267\pi$
$197$	−8.69694	−0.619631	−0.309816	+	0.950797i	$0.600267\pi$
$198$	0	0
$199$	−4.69694	−0.332957	−0.166479	−	0.986045i	$-0.553240\pi$
$199$	−4.69694	−0.332957	−0.166479	+	0.986045i	$0.553240\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−10.8990	−0.764958
$204$	0	0
$205$	11.7980	0.824005
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−28.4949	−1.96167	−0.980835	−	0.194841i	$-0.937581\pi$
$211$	−28.4949	−1.96167	−0.980835	+	0.194841i	$0.937581\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−12.8990	−0.879703
$216$	0	0
$217$	−2.44949	−0.166282
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.898979	−0.0604719
$222$	0	0
$223$	9.79796	0.656120	0.328060	−	0.944657i	$-0.393605\pi$
$223$	9.79796	0.656120	0.328060	+	0.944657i	$0.393605\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.20204	−0.146155	−0.0730773	−	0.997326i	$-0.523282\pi$
$227$	−2.20204	−0.146155	−0.0730773	+	0.997326i	$0.523282\pi$
$228$	0	0
$229$	0.202041	0.0133512	0.00667562	−	0.999978i	$-0.497875\pi$
$229$	0.202041	0.0133512	0.00667562	+	0.999978i	$0.497875\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.7980	1.42803	0.714016	−	0.700129i	$-0.246874\pi$
$233$	21.7980	1.42803	0.714016	+	0.700129i	$0.246874\pi$
$234$	0	0
$235$	0.898979	0.0586430
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.10102	−0.459327	−0.229663	−	0.973270i	$-0.573763\pi$
$239$	−7.10102	−0.459327	−0.229663	+	0.973270i	$0.573763\pi$
$240$	0	0
$241$	−15.7980	−1.01764	−0.508818	−	0.860874i	$-0.669917\pi$
$241$	−15.7980	−1.01764	−0.508818	+	0.860874i	$0.669917\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	2.20204	0.140113
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.79796	−0.618442	−0.309221	−	0.950990i	$-0.600068\pi$
$251$	−9.79796	−0.618442	−0.309221	+	0.950990i	$0.600068\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.0000	−0.998053	−0.499026	−	0.866587i	$-0.666309\pi$
$257$	−16.0000	−0.998053	−0.499026	+	0.866587i	$0.666309\pi$
$258$	0	0
$259$	20.6969	1.28605
$260$	0	0
$261$	0	0
$262$	0	0
$263$	29.3939	1.81250	0.906252	−	0.422738i	$-0.138931\pi$
$263$	29.3939	1.81250	0.906252	+	0.422738i	$0.138931\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	32.0454	1.95384	0.976921	−	0.213599i	$-0.0685185\pi$
$269$	32.0454	1.95384	0.976921	+	0.213599i	$0.0685185\pi$
$270$	0	0
$271$	1.79796	0.109218	0.0546091	−	0.998508i	$-0.482609\pi$
$271$	1.79796	0.109218	0.0546091	+	0.998508i	$0.482609\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−23.1464	−1.39073	−0.695367	−	0.718655i	$-0.744758\pi$
$277$	−23.1464	−1.39073	−0.695367	+	0.718655i	$0.744758\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.10102	−0.304301	−0.152151	−	0.988357i	$-0.548620\pi$
$281$	−5.10102	−0.304301	−0.152151	+	0.988357i	$0.548620\pi$
$282$	0	0
$283$	−14.4495	−0.858933	−0.429467	−	0.903083i	$-0.641298\pi$
$283$	−14.4495	−0.858933	−0.429467	+	0.903083i	$0.641298\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−28.8990	−1.70585
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.20204	0.128645	0.0643223	−	0.997929i	$-0.479511\pi$
$293$	2.20204	0.128645	0.0643223	+	0.997929i	$0.479511\pi$
$294$	0	0
$295$	−7.34847	−0.427844
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.10102	−0.179337
$300$	0	0
$301$	31.5959	1.82116
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.89898	0.165995
$306$	0	0
$307$	−17.5505	−1.00166	−0.500830	−	0.865546i	$-0.666972\pi$
$307$	−17.5505	−1.00166	−0.500830	+	0.865546i	$0.666972\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.3485	−1.09715	−0.548576	−	0.836101i	$-0.684830\pi$
$311$	−19.3485	−1.09715	−0.548576	+	0.836101i	$0.684830\pi$
$312$	0	0
$313$	1.34847	0.0762200	0.0381100	−	0.999274i	$-0.487866\pi$
$313$	1.34847	0.0762200	0.0381100	+	0.999274i	$0.487866\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.5959	−1.32528	−0.662639	−	0.748939i	$-0.730564\pi$
$317$	−23.5959	−1.32528	−0.662639	+	0.748939i	$0.730564\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−9.79796	−0.545173
$324$	0	0
$325$	0.449490	0.0249332
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.20204	−0.121402
$330$	0	0
$331$	−10.8990	−0.599062	−0.299531	−	0.954087i	$-0.596830\pi$
$331$	−10.8990	−0.599062	−0.299531	+	0.954087i	$0.596830\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.55051	−0.0847134
$336$	0	0
$337$	15.1464	0.825079	0.412539	−	0.910940i	$-0.364642\pi$
$337$	15.1464	0.825079	0.412539	+	0.910940i	$0.364642\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.5959	1.05808
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	33.7980	1.80916	0.904582	−	0.426300i	$-0.140183\pi$
$349$	33.7980	1.80916	0.904582	+	0.426300i	$0.140183\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.79796	−0.202145	−0.101072	−	0.994879i	$-0.532227\pi$
$353$	−3.79796	−0.202145	−0.101072	+	0.994879i	$0.532227\pi$
$354$	0	0
$355$	−6.44949	−0.342303
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.4495	−0.762615	−0.381307	−	0.924448i	$-0.624526\pi$
$359$	−14.4495	−0.762615	−0.381307	+	0.924448i	$0.624526\pi$
$360$	0	0
$361$	5.00000	0.263158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.55051	−0.395212
$366$	0	0
$367$	27.5959	1.44050	0.720248	−	0.693717i	$-0.244028\pi$
$367$	27.5959	1.44050	0.720248	+	0.693717i	$0.244028\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14.6969	−0.763027
$372$	0	0
$373$	−32.6969	−1.69298	−0.846492	−	0.532402i	$-0.821289\pi$
$373$	−32.6969	−1.69298	−0.846492	+	0.532402i	$0.821289\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	7.10102	0.364755	0.182377	−	0.983229i	$-0.441621\pi$
$379$	7.10102	0.364755	0.182377	+	0.983229i	$0.441621\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.10102	−0.0562595	−0.0281298	−	0.999604i	$-0.508955\pi$
$383$	−1.10102	−0.0562595	−0.0281298	+	0.999604i	$0.508955\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	21.3485	1.08241	0.541205	−	0.840891i	$-0.317968\pi$
$389$	21.3485	1.08241	0.541205	+	0.840891i	$0.317968\pi$
$390$	0	0
$391$	13.7980	0.697793
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.89898	0.347125
$396$	0	0
$397$	−7.79796	−0.391368	−0.195684	−	0.980667i	$-0.562693\pi$
$397$	−7.79796	−0.391368	−0.195684	+	0.980667i	$0.562693\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	33.8434	1.69006	0.845029	−	0.534721i	$-0.179583\pi$
$401$	33.8434	1.69006	0.845029	+	0.534721i	$0.179583\pi$
$402$	0	0
$403$	0.449490	0.0223907
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	28.6969	1.41897	0.709486	−	0.704719i	$-0.248927\pi$
$409$	28.6969	1.41897	0.709486	+	0.704719i	$0.248927\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	18.0000	0.885722
$414$	0	0
$415$	10.8990	0.535010
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−21.5505	−1.05281	−0.526406	−	0.850234i	$-0.676461\pi$
$419$	−21.5505	−1.05281	−0.526406	+	0.850234i	$0.676461\pi$
$420$	0	0
$421$	−13.7980	−0.672471	−0.336236	−	0.941778i	$-0.609154\pi$
$421$	−13.7980	−0.672471	−0.336236	+	0.941778i	$0.609154\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−7.10102	−0.343642
$428$	0	0
$429$	0	0
$430$	0	0
$431$	22.4495	1.08135	0.540677	−	0.841230i	$-0.318168\pi$
$431$	22.4495	1.08135	0.540677	+	0.841230i	$0.318168\pi$
$432$	0	0
$433$	−3.14643	−0.151208	−0.0756038	−	0.997138i	$-0.524088\pi$
$433$	−3.14643	−0.151208	−0.0756038	+	0.997138i	$0.524088\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−33.7980	−1.61678
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−0.449490	−0.0213079
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−31.1464	−1.46989	−0.734945	−	0.678126i	$-0.762792\pi$
$449$	−31.1464	−1.46989	−0.734945	+	0.678126i	$0.762792\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.10102	−0.0516166
$456$	0	0
$457$	−9.34847	−0.437303	−0.218651	−	0.975803i	$-0.570166\pi$
$457$	−9.34847	−0.437303	−0.218651	+	0.975803i	$0.570166\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8.44949	−0.393532	−0.196766	−	0.980450i	$-0.563044\pi$
$461$	−8.44949	−0.393532	−0.196766	+	0.980450i	$0.563044\pi$
$462$	0	0
$463$	30.6969	1.42661	0.713304	−	0.700855i	$-0.247198\pi$
$463$	30.6969	1.42661	0.713304	+	0.700855i	$0.247198\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.8990	−1.33729	−0.668643	−	0.743584i	$-0.733124\pi$
$467$	−28.8990	−1.33729	−0.668643	+	0.743584i	$0.733124\pi$
$468$	0	0
$469$	3.79796	0.175373
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.89898	0.224781
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.65153	−0.212534	−0.106267	−	0.994338i	$-0.533890\pi$
$479$	−4.65153	−0.212534	−0.106267	+	0.994338i	$0.533890\pi$
$480$	0	0
$481$	−3.79796	−0.173172
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.8990	−0.676528
$486$	0	0
$487$	−23.1010	−1.04681	−0.523404	−	0.852085i	$-0.675338\pi$
$487$	−23.1010	−1.04681	−0.523404	+	0.852085i	$0.675338\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	20.8990	0.943158	0.471579	−	0.881824i	$-0.343684\pi$
$491$	20.8990	0.943158	0.471579	+	0.881824i	$0.343684\pi$
$492$	0	0
$493$	−8.89898	−0.400790
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.7980	0.708635
$498$	0	0
$499$	−5.79796	−0.259552	−0.129776	−	0.991543i	$-0.541426\pi$
$499$	−5.79796	−0.259552	−0.129776	+	0.991543i	$0.541426\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−36.8990	−1.64524	−0.822622	−	0.568589i	$-0.807490\pi$
$503$	−36.8990	−1.64524	−0.822622	+	0.568589i	$0.807490\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18.2474	0.808804	0.404402	−	0.914581i	$-0.367480\pi$
$509$	18.2474	0.808804	0.404402	+	0.914581i	$0.367480\pi$
$510$	0	0
$511$	18.4949	0.818166
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.4495	−0.460460
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−20.8990	−0.913849	−0.456924	−	0.889506i	$-0.651049\pi$
$523$	−20.8990	−0.913849	−0.456924	+	0.889506i	$0.651049\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.00000	−0.0871214
$528$	0	0
$529$	24.5959	1.06939
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.30306	0.229701
$534$	0	0
$535$	−4.89898	−0.211801
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−39.5959	−1.70236	−0.851181	−	0.524873i	$-0.824113\pi$
$541$	−39.5959	−1.70236	−0.851181	+	0.524873i	$0.824113\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	−22.9444	−0.981031	−0.490516	−	0.871432i	$-0.663192\pi$
$547$	−22.9444	−0.981031	−0.490516	+	0.871432i	$0.663192\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	21.7980	0.928624
$552$	0	0
$553$	−16.8990	−0.718618
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.5959	0.745563	0.372781	−	0.927919i	$-0.378404\pi$
$557$	17.5959	0.745563	0.372781	+	0.927919i	$0.378404\pi$
$558$	0	0
$559$	−5.79796	−0.245228
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.89898	0.375047	0.187524	−	0.982260i	$-0.439954\pi$
$563$	8.89898	0.375047	0.187524	+	0.982260i	$0.439954\pi$
$564$	0	0
$565$	−3.79796	−0.159781
$566$	0	0
$567$	0	0
$568$	0	0
$569$	35.5505	1.49035	0.745177	−	0.666866i	$-0.232365\pi$
$569$	35.5505	1.49035	0.745177	+	0.666866i	$0.232365\pi$
$570$	0	0
$571$	−29.7980	−1.24701	−0.623503	−	0.781821i	$-0.714291\pi$
$571$	−29.7980	−1.24701	−0.623503	+	0.781821i	$0.714291\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.89898	−0.287707
$576$	0	0
$577$	41.5959	1.73166	0.865830	−	0.500338i	$-0.166791\pi$
$577$	41.5959	1.73166	0.865830	+	0.500338i	$0.166791\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−26.6969	−1.10758
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.5959	0.973908	0.486954	−	0.873428i	$-0.338108\pi$
$587$	23.5959	0.973908	0.486954	+	0.873428i	$0.338108\pi$
$588$	0	0
$589$	4.89898	0.201859
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5.59592	−0.229797	−0.114898	−	0.993377i	$-0.536654\pi$
$593$	−5.59592	−0.229797	−0.114898	+	0.993377i	$0.536654\pi$
$594$	0	0
$595$	4.89898	0.200839
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−38.9444	−1.59122	−0.795612	−	0.605806i	$-0.792851\pi$
$599$	−38.9444	−1.59122	−0.795612	+	0.605806i	$0.792851\pi$
$600$	0	0
$601$	32.6969	1.33374	0.666868	−	0.745176i	$-0.267634\pi$
$601$	32.6969	1.33374	0.666868	+	0.745176i	$0.267634\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11.0000	−0.447214
$606$	0	0
$607$	10.9444	0.444219	0.222109	−	0.975022i	$-0.428706\pi$
$607$	10.9444	0.444219	0.222109	+	0.975022i	$0.428706\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.404082	0.0163474
$612$	0	0
$613$	−8.04541	−0.324951	−0.162475	−	0.986713i	$-0.551948\pi$
$613$	−8.04541	−0.324951	−0.162475	+	0.986713i	$0.551948\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−2.49490	−0.100278	−0.0501392	−	0.998742i	$-0.515966\pi$
$619$	−2.49490	−0.100278	−0.0501392	+	0.998742i	$0.515966\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.10102	0.0441115
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.8990	0.673806
$630$	0	0
$631$	−35.5959	−1.41705	−0.708526	−	0.705685i	$-0.750639\pi$
$631$	−35.5959	−1.41705	−0.708526	+	0.705685i	$0.750639\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.898979	0.0356749
$636$	0	0
$637$	−0.449490	−0.0178094
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−0.0454077	−0.00179350	−0.000896748	−	1.00000i	$-0.500285\pi$
$641$	−0.0454077	−0.00179350	−0.000896748		1.00000i	$0.500285\pi$
$642$	0	0
$643$	28.4949	1.12373	0.561865	−	0.827229i	$-0.310084\pi$
$643$	28.4949	1.12373	0.561865	+	0.827229i	$0.310084\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.5959	0.770395	0.385198	−	0.922834i	$-0.374133\pi$
$647$	19.5959	0.770395	0.385198	+	0.922834i	$0.374133\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.00000	−0.156532	−0.0782660	−	0.996933i	$-0.524938\pi$
$653$	−4.00000	−0.156532	−0.0782660	+	0.996933i	$0.524938\pi$
$654$	0	0
$655$	12.2474	0.478547
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.3485	−0.597891	−0.298946	−	0.954270i	$-0.596635\pi$
$659$	−15.3485	−0.597891	−0.298946	+	0.954270i	$0.596635\pi$
$660$	0	0
$661$	−19.3939	−0.754334	−0.377167	−	0.926145i	$-0.623102\pi$
$661$	−19.3939	−0.754334	−0.377167	+	0.926145i	$0.623102\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	−30.6969	−1.18859
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−24.9444	−0.961535	−0.480768	−	0.876848i	$-0.659642\pi$
$673$	−24.9444	−0.961535	−0.480768	+	0.876848i	$0.659642\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27.7980	1.06836	0.534181	−	0.845370i	$-0.320620\pi$
$677$	27.7980	1.06836	0.534181	+	0.845370i	$0.320620\pi$
$678$	0	0
$679$	36.4949	1.40055
$680$	0	0
$681$	0	0
$682$	0	0
$683$	35.5959	1.36204	0.681020	−	0.732265i	$-0.261537\pi$
$683$	35.5959	1.36204	0.681020	+	0.732265i	$0.261537\pi$
$684$	0	0
$685$	5.10102	0.194900
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.69694	0.102745
$690$	0	0
$691$	13.3939	0.509527	0.254764	−	0.967003i	$-0.418002\pi$
$691$	13.3939	0.509527	0.254764	+	0.967003i	$0.418002\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.79796	−0.219929
$696$	0	0
$697$	−23.5959	−0.893759
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−15.3031	−0.577989	−0.288994	−	0.957331i	$-0.593321\pi$
$701$	−15.3031	−0.577989	−0.288994	+	0.957331i	$0.593321\pi$
$702$	0	0
$703$	−41.3939	−1.56120
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.89898	0.184245
$708$	0	0
$709$	11.7980	0.443082	0.221541	−	0.975151i	$-0.428891\pi$
$709$	11.7980	0.443082	0.221541	+	0.975151i	$0.428891\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.89898	−0.258369
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.79796	0.0670526	0.0335263	−	0.999438i	$-0.489326\pi$
$719$	1.79796	0.0670526	0.0335263	+	0.999438i	$0.489326\pi$
$720$	0	0
$721$	25.5959	0.953242
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.44949	0.165250
$726$	0	0
$727$	−17.1464	−0.635926	−0.317963	−	0.948103i	$-0.602999\pi$
$727$	−17.1464	−0.635926	−0.317963	+	0.948103i	$0.602999\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	25.7980	0.954172
$732$	0	0
$733$	49.5959	1.83187	0.915934	−	0.401330i	$-0.131452\pi$
$733$	49.5959	1.83187	0.915934	+	0.401330i	$0.131452\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−11.3031	−0.415790	−0.207895	−	0.978151i	$-0.566661\pi$
$739$	−11.3031	−0.415790	−0.207895	+	0.978151i	$0.566661\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.7980	−0.652944	−0.326472	−	0.945207i	$-0.605860\pi$
$743$	−17.7980	−0.652944	−0.326472	+	0.945207i	$0.605860\pi$
$744$	0	0
$745$	−19.7980	−0.725341
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	13.7980	0.503495	0.251747	−	0.967793i	$-0.418995\pi$
$751$	13.7980	0.503495	0.251747	+	0.967793i	$0.418995\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.10102	−0.0400702
$756$	0	0
$757$	−7.55051	−0.274428	−0.137214	−	0.990541i	$-0.543815\pi$
$757$	−7.55051	−0.274428	−0.137214	+	0.990541i	$0.543815\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−25.8434	−0.936821	−0.468411	−	0.883511i	$-0.655173\pi$
$761$	−25.8434	−0.936821	−0.468411	+	0.883511i	$0.655173\pi$
$762$	0	0
$763$	9.79796	0.354710
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.30306	−0.119267
$768$	0	0
$769$	14.2020	0.512139	0.256069	−	0.966658i	$-0.417572\pi$
$769$	14.2020	0.512139	0.256069	+	0.966658i	$0.417572\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	0	0
$778$	0	0
$779$	57.7980	2.07083
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.89898	0.103469
$786$	0	0
$787$	31.1918	1.11187	0.555934	−	0.831226i	$-0.312361\pi$
$787$	31.1918	1.11187	0.555934	+	0.831226i	$0.312361\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.30306	0.330779
$792$	0	0
$793$	1.30306	0.0462731
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.4949	0.796810	0.398405	−	0.917210i	$-0.369564\pi$
$797$	22.4949	0.796810	0.398405	+	0.917210i	$0.369564\pi$
$798$	0	0
$799$	−1.79796	−0.0636072
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	16.8990	0.595611
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.65153	0.0932229	0.0466114	−	0.998913i	$-0.485158\pi$
$809$	2.65153	0.0932229	0.0466114	+	0.998913i	$0.485158\pi$
$810$	0	0
$811$	−22.2929	−0.782808	−0.391404	−	0.920219i	$-0.628011\pi$
$811$	−22.2929	−0.782808	−0.391404	+	0.920219i	$0.628011\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−21.1464	−0.740727
$816$	0	0
$817$	−63.1918	−2.21080
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.6413	−0.964689	−0.482344	−	0.875982i	$-0.660215\pi$
$821$	−27.6413	−0.964689	−0.482344	+	0.875982i	$0.660215\pi$
$822$	0	0
$823$	−24.4949	−0.853838	−0.426919	−	0.904290i	$-0.640401\pi$
$823$	−24.4949	−0.853838	−0.426919	+	0.904290i	$0.640401\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−46.0908	−1.60273	−0.801367	−	0.598173i	$-0.795894\pi$
$827$	−46.0908	−1.60273	−0.801367	+	0.598173i	$0.795894\pi$
$828$	0	0
$829$	−6.89898	−0.239611	−0.119806	−	0.992797i	$-0.538227\pi$
$829$	−6.89898	−0.239611	−0.119806	+	0.992797i	$0.538227\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.55051	0.329720	0.164860	−	0.986317i	$-0.447283\pi$
$839$	9.55051	0.329720	0.164860	+	0.986317i	$0.447283\pi$
$840$	0	0
$841$	−9.20204	−0.317312
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.7980	−0.440263
$846$	0	0
$847$	26.9444	0.925820
$848$	0	0
$849$	0	0
$850$	0	0
$851$	58.2929	1.99825
$852$	0	0
$853$	39.3939	1.34882	0.674410	−	0.738357i	$-0.264398\pi$
$853$	39.3939	1.34882	0.674410	+	0.738357i	$0.264398\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.1918	1.54372	0.771862	−	0.635790i	$-0.219326\pi$
$857$	45.1918	1.54372	0.771862	+	0.635790i	$0.219326\pi$
$858$	0	0
$859$	−8.40408	−0.286744	−0.143372	−	0.989669i	$-0.545794\pi$
$859$	−8.40408	−0.286744	−0.143372	+	0.989669i	$0.545794\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	56.0000	1.90626	0.953131	−	0.302558i	$-0.0978405\pi$
$863$	56.0000	1.90626	0.953131	+	0.302558i	$0.0978405\pi$
$864$	0	0
$865$	19.7980	0.673151
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−0.696938	−0.0236149
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.44949	−0.0828079
$876$	0	0
$877$	−35.7980	−1.20881	−0.604406	−	0.796677i	$-0.706589\pi$
$877$	−35.7980	−1.20881	−0.604406	+	0.796677i	$0.706589\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−26.2474	−0.884299	−0.442150	−	0.896941i	$-0.645784\pi$
$881$	−26.2474	−0.884299	−0.442150	+	0.896941i	$0.645784\pi$
$882$	0	0
$883$	37.3939	1.25840	0.629202	−	0.777242i	$-0.283382\pi$
$883$	37.3939	1.25840	0.629202	+	0.777242i	$0.283382\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	0	0
$889$	−2.20204	−0.0738541
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.40408	0.147377
$894$	0	0
$895$	−23.5959	−0.788725
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.44949	0.148399
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.89898	0.229330
$906$	0	0
$907$	2.85357	0.0947513	0.0473756	−	0.998877i	$-0.484914\pi$
$907$	2.85357	0.0947513	0.0473756	+	0.998877i	$0.484914\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	25.7980	0.854725	0.427362	−	0.904080i	$-0.359443\pi$
$911$	25.7980	0.854725	0.427362	+	0.904080i	$0.359443\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−30.0000	−0.990687
$918$	0	0
$919$	54.6969	1.80429	0.902143	−	0.431438i	$-0.141994\pi$
$919$	54.6969	1.80429	0.902143	+	0.431438i	$0.141994\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.89898	−0.0954211
$924$	0	0
$925$	−8.44949	−0.277818
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−12.9444	−0.424692	−0.212346	−	0.977195i	$-0.568110\pi$
$929$	−12.9444	−0.424692	−0.212346	+	0.977195i	$0.568110\pi$
$930$	0	0
$931$	−4.89898	−0.160558
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.4949	0.865551	0.432775	−	0.901502i	$-0.357534\pi$
$937$	26.4949	0.865551	0.432775	+	0.901502i	$0.357534\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−22.2474	−0.725246	−0.362623	−	0.931936i	$-0.618119\pi$
$941$	−22.2474	−0.725246	−0.362623	+	0.931936i	$0.618119\pi$
$942$	0	0
$943$	−81.3939	−2.65055
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42.4949	−1.38090	−0.690449	−	0.723381i	$-0.742587\pi$
$947$	−42.4949	−1.38090	−0.690449	+	0.723381i	$0.742587\pi$
$948$	0	0
$949$	−3.39388	−0.110170
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−11.3031	−0.366142	−0.183071	−	0.983100i	$-0.558604\pi$
$953$	−11.3031	−0.366142	−0.183071	+	0.983100i	$0.558604\pi$
$954$	0	0
$955$	−6.44949	−0.208701
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.4949	−0.403481
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	20.6969	0.666258
$966$	0	0
$967$	−29.3939	−0.945243	−0.472622	−	0.881265i	$-0.656692\pi$
$967$	−29.3939	−0.945243	−0.472622	+	0.881265i	$0.656692\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	26.4495	0.848805	0.424402	−	0.905474i	$-0.360484\pi$
$971$	26.4495	0.848805	0.424402	+	0.905474i	$0.360484\pi$
$972$	0	0
$973$	14.2020	0.455297
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−53.1918	−1.70176	−0.850879	−	0.525362i	$-0.823930\pi$
$977$	−53.1918	−1.70176	−0.850879	+	0.525362i	$0.823930\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.7980	−0.822827	−0.411414	−	0.911449i	$-0.634965\pi$
$983$	−25.7980	−0.822827	−0.411414	+	0.911449i	$0.634965\pi$
$984$	0	0
$985$	−8.69694	−0.277108
$986$	0	0
$987$	0	0
$988$	0	0
$989$	88.9898	2.82971
$990$	0	0
$991$	−21.3939	−0.679599	−0.339799	−	0.940498i	$-0.610359\pi$
$991$	−21.3939	−0.679599	−0.339799	+	0.940498i	$0.610359\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.69694	−0.148903
$996$	0	0
$997$	40.2929	1.27609	0.638044	−	0.770000i	$-0.279744\pi$
$997$	40.2929	1.27609	0.638044	+	0.770000i	$0.279744\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5580.2.a.g.1.1		2
3.2	odd	2		1860.2.a.d.1.1	✓	2
12.11	even	2		7440.2.a.bj.1.2		2
15.2	even	4		9300.2.g.l.3349.3		4
15.8	even	4		9300.2.g.l.3349.2		4
15.14	odd	2		9300.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.d.1.1	✓	2	3.2	odd	2
5580.2.a.g.1.1		2	1.1	even	1	trivial
7440.2.a.bj.1.2		2	12.11	even	2
9300.2.a.p.1.2		2	15.14	odd	2
9300.2.g.l.3349.2		4	15.8	even	4
9300.2.g.l.3349.3		4	15.2	even	4

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:	\( N \)	\(=\)	\( 5580 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5580.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -2.44949 q^{7} +0.449490 q^{13} -2.00000 q^{17} +4.89898 q^{19} -6.89898 q^{23} +1.00000 q^{25} +4.44949 q^{29} +1.00000 q^{31} -2.44949 q^{35} -8.44949 q^{37} +11.7980 q^{41} -12.8990 q^{43} +0.898979 q^{47} -1.00000 q^{49} +6.00000 q^{53} -7.34847 q^{59} +2.89898 q^{61} +0.449490 q^{65} -1.55051 q^{67} -6.44949 q^{71} -7.55051 q^{73} +6.89898 q^{79} +10.8990 q^{83} -2.00000 q^{85} -0.449490 q^{89} -1.10102 q^{91} +4.89898 q^{95} -14.8990 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{13} - 4 q^{17} - 4 q^{23} + 2 q^{25} + 4 q^{29} + 2 q^{31} - 12 q^{37} + 4 q^{41} - 16 q^{43} - 8 q^{47} - 2 q^{49} + 12 q^{53} - 4 q^{61} - 4 q^{65} - 8 q^{67} - 8 q^{71} - 20 q^{73}+ \cdots - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^13 - 4 * q^17 - 4 * q^23 + 2 * q^25 + 4 * q^29 + 2 * q^31 - 12 * q^37 + 4 * q^41 - 16 * q^43 - 8 * q^47 - 2 * q^49 + 12 * q^53 - 4 * q^61 - 4 * q^65 - 8 * q^67 - 8 * q^71 - 20 * q^73 + 4 * q^79 + 12 * q^83 - 4 * q^85 + 4 * q^89 - 12 * q^91 - 20 * q^97

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