LMFDB - Embedded newform 4400.2.a.cc.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,2,0,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.52434$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.52434 q^{3} +3.46410 q^{7} +3.37228 q^{9} +1.00000 q^{11} +5.04868 q^{17} -4.00000 q^{19} +8.74456 q^{21} -2.52434 q^{23} +0.939764 q^{27} -2.74456 q^{29} +2.37228 q^{31} +2.52434 q^{33} +11.0371 q^{37} -2.74456 q^{41} +3.46410 q^{43} +6.63325 q^{47} +5.00000 q^{49} +12.7446 q^{51} +3.16915 q^{53} -10.0974 q^{57} +1.62772 q^{59} +10.7446 q^{61} +11.6819 q^{63} -0.644810 q^{67} -6.37228 q^{69} -7.11684 q^{71} +6.92820 q^{73} +3.46410 q^{77} -12.7446 q^{79} -7.74456 q^{81} -6.63325 q^{83} -6.92820 q^{87} -4.37228 q^{89} +5.98844 q^{93} +4.10891 q^{97} +3.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{9} + 4 q^{11} - 16 q^{19} + 12 q^{21} + 12 q^{29} - 2 q^{31} + 12 q^{41} + 20 q^{49} + 28 q^{51} + 18 q^{59} + 20 q^{61} - 14 q^{69} + 6 q^{71} - 28 q^{79} - 8 q^{81} - 6 q^{89} + 2 q^{99}+O(q^{100}) $ 4 * q + 2 * q^9 + 4 * q^11 - 16 * q^19 + 12 * q^21 + 12 * q^29 - 2 * q^31 + 12 * q^41 + 20 * q^49 + 28 * q^51 + 18 * q^59 + 20 * q^61 - 14 * q^69 + 6 * q^71 - 28 * q^79 - 8 * q^81 - 6 * q^89 + 2 * q^99

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
$3$	2.52434	1.45743	0.728714	−	0.684819i	$-0.240119\pi$
$3$	2.52434	1.45743	0.728714	+	0.684819i	$0.240119\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	0	0
$9$	3.37228	1.12409
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.04868	1.22448	0.612242	−	0.790671i	$-0.290268\pi$
$17$	5.04868	1.22448	0.612242	+	0.790671i	$0.290268\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	8.74456	1.90822
$22$	0	0
$23$	−2.52434	−0.526361	−0.263180	−	0.964747i	$-0.584771\pi$
$23$	−2.52434	−0.526361	−0.263180	+	0.964747i	$0.584771\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0.939764	0.180858
$28$	0	0
$29$	−2.74456	−0.509652	−0.254826	−	0.966987i	$-0.582018\pi$
$29$	−2.74456	−0.509652	−0.254826	+	0.966987i	$0.582018\pi$
$30$	0	0
$31$	2.37228	0.426074	0.213037	−	0.977044i	$-0.431664\pi$
$31$	2.37228	0.426074	0.213037	+	0.977044i	$0.431664\pi$
$32$	0	0
$33$	2.52434	0.439431
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.0371	1.81449	0.907245	−	0.420602i	$-0.138181\pi$
$37$	11.0371	1.81449	0.907245	+	0.420602i	$0.138181\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.74456	−0.428629	−0.214314	−	0.976765i	$-0.568752\pi$
$41$	−2.74456	−0.428629	−0.214314	+	0.976765i	$0.568752\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.63325	0.967559	0.483779	−	0.875190i	$-0.339264\pi$
$47$	6.63325	0.967559	0.483779	+	0.875190i	$0.339264\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	12.7446	1.78460
$52$	0	0
$53$	3.16915	0.435316	0.217658	−	0.976025i	$-0.430158\pi$
$53$	3.16915	0.435316	0.217658	+	0.976025i	$0.430158\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−10.0974	−1.33743
$58$	0	0
$59$	1.62772	0.211911	0.105955	−	0.994371i	$-0.466210\pi$
$59$	1.62772	0.211911	0.105955	+	0.994371i	$0.466210\pi$
$60$	0	0
$61$	10.7446	1.37570	0.687850	−	0.725853i	$-0.258555\pi$
$61$	10.7446	1.37570	0.687850	+	0.725853i	$0.258555\pi$
$62$	0	0
$63$	11.6819	1.47178
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.644810	−0.0787761	−0.0393880	−	0.999224i	$-0.512541\pi$
$67$	−0.644810	−0.0787761	−0.0393880	+	0.999224i	$0.512541\pi$
$68$	0	0
$69$	−6.37228	−0.767133
$70$	0	0
$71$	−7.11684	−0.844614	−0.422307	−	0.906453i	$-0.638780\pi$
$71$	−7.11684	−0.844614	−0.422307	+	0.906453i	$0.638780\pi$
$72$	0	0
$73$	6.92820	0.810885	0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820	0.810885	0.405442	+	0.914121i	$0.367117\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.46410	0.394771
$78$	0	0
$79$	−12.7446	−1.43388	−0.716938	−	0.697137i	$-0.754457\pi$
$79$	−12.7446	−1.43388	−0.716938	+	0.697137i	$0.754457\pi$
$80$	0	0
$81$	−7.74456	−0.860507
$82$	0	0
$83$	−6.63325	−0.728094	−0.364047	−	0.931381i	$-0.618605\pi$
$83$	−6.63325	−0.728094	−0.364047	+	0.931381i	$0.618605\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.92820	−0.742781
$88$	0	0
$89$	−4.37228	−0.463461	−0.231730	−	0.972780i	$-0.574439\pi$
$89$	−4.37228	−0.463461	−0.231730	+	0.972780i	$0.574439\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.98844	0.620972
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.10891	0.417197	0.208598	−	0.978001i	$-0.433110\pi$
$97$	4.10891	0.417197	0.208598	+	0.978001i	$0.433110\pi$
$98$	0	0
$99$	3.37228	0.338927
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−10.3923	−1.02398	−0.511992	−	0.858990i	$-0.671092\pi$
$103$	−10.3923	−1.02398	−0.511992	+	0.858990i	$0.671092\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.63325	0.641260	0.320630	−	0.947204i	$-0.396105\pi$
$107$	6.63325	0.641260	0.320630	+	0.947204i	$0.396105\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	27.8614	2.64449
$112$	0	0
$113$	−16.0858	−1.51322	−0.756612	−	0.653864i	$-0.773147\pi$
$113$	−16.0858	−1.51322	−0.756612	+	0.653864i	$0.773147\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	17.4891	1.60323
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.92820	−0.624695
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.6819	1.03660	0.518302	−	0.855198i	$-0.326564\pi$
$127$	11.6819	1.03660	0.518302	+	0.855198i	$0.326564\pi$
$128$	0	0
$129$	8.74456	0.769916
$130$	0	0
$131$	−8.74456	−0.764016	−0.382008	−	0.924159i	$-0.624767\pi$
$131$	−8.74456	−0.764016	−0.382008	+	0.924159i	$0.624767\pi$
$132$	0	0
$133$	−13.8564	−1.20150
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.22938	0.190469	0.0952346	−	0.995455i	$-0.469640\pi$
$137$	2.22938	0.190469	0.0952346	+	0.995455i	$0.469640\pi$
$138$	0	0
$139$	−18.2337	−1.54656	−0.773281	−	0.634064i	$-0.781386\pi$
$139$	−18.2337	−1.54656	−0.773281	+	0.634064i	$0.781386\pi$
$140$	0	0
$141$	16.7446	1.41015
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	12.6217	1.04102
$148$	0	0
$149$	11.4891	0.941226	0.470613	−	0.882340i	$-0.344033\pi$
$149$	11.4891	0.941226	0.470613	+	0.882340i	$0.344033\pi$
$150$	0	0
$151$	−22.2337	−1.80935	−0.904676	−	0.426100i	$-0.859887\pi$
$151$	−22.2337	−1.80935	−0.904676	+	0.426100i	$0.859887\pi$
$152$	0	0
$153$	17.0256	1.37643
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.39853	−0.430850	−0.215425	−	0.976520i	$-0.569114\pi$
$157$	−5.39853	−0.430850	−0.215425	+	0.976520i	$0.569114\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	0	0
$161$	−8.74456	−0.689168
$162$	0	0
$163$	−3.46410	−0.271329	−0.135665	−	0.990755i	$-0.543317\pi$
$163$	−3.46410	−0.271329	−0.135665	+	0.990755i	$0.543317\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.3692	−1.73098	−0.865490	−	0.500927i	$-0.832993\pi$
$167$	−22.3692	−1.73098	−0.865490	+	0.500927i	$0.832993\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−13.4891	−1.03154
$172$	0	0
$173$	1.87953	0.142898	0.0714489	−	0.997444i	$-0.477238\pi$
$173$	1.87953	0.142898	0.0714489	+	0.997444i	$0.477238\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.10891	0.308845
$178$	0	0
$179$	15.8614	1.18554	0.592769	−	0.805373i	$-0.298035\pi$
$179$	15.8614	1.18554	0.592769	+	0.805373i	$0.298035\pi$
$180$	0	0
$181$	6.88316	0.511621	0.255810	−	0.966727i	$-0.417658\pi$
$181$	6.88316	0.511621	0.255810	+	0.966727i	$0.417658\pi$
$182$	0	0
$183$	27.1229	2.00498
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.04868	0.369196
$188$	0	0
$189$	3.25544	0.236798
$190$	0	0
$191$	13.6277	0.986067	0.493034	−	0.870010i	$-0.335888\pi$
$191$	13.6277	0.986067	0.493034	+	0.870010i	$0.335888\pi$
$192$	0	0
$193$	−23.3639	−1.68177	−0.840883	−	0.541216i	$-0.817964\pi$
$193$	−23.3639	−1.68177	−0.840883	+	0.541216i	$0.817964\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.87953	−0.133911	−0.0669554	−	0.997756i	$-0.521329\pi$
$197$	−1.87953	−0.133911	−0.0669554	+	0.997756i	$0.521329\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−1.62772	−0.114810
$202$	0	0
$203$	−9.50744	−0.667292
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.51278	−0.591679
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	21.4891	1.47937	0.739686	−	0.672952i	$-0.234974\pi$
$211$	21.4891	1.47937	0.739686	+	0.672952i	$0.234974\pi$
$212$	0	0
$213$	−17.9653	−1.23096
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.21782	0.557862
$218$	0	0
$219$	17.4891	1.18181
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.57301	0.507126	0.253563	−	0.967319i	$-0.418397\pi$
$223$	7.57301	0.507126	0.253563	+	0.967319i	$0.418397\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.80240	−0.650608	−0.325304	−	0.945609i	$-0.605467\pi$
$227$	−9.80240	−0.650608	−0.325304	+	0.945609i	$0.605467\pi$
$228$	0	0
$229$	20.3723	1.34624	0.673119	−	0.739534i	$-0.264954\pi$
$229$	20.3723	1.34624	0.673119	+	0.739534i	$0.264954\pi$
$230$	0	0
$231$	8.74456	0.575350
$232$	0	0
$233$	17.0256	1.11538	0.557691	−	0.830049i	$-0.311688\pi$
$233$	17.0256	1.11538	0.557691	+	0.830049i	$0.311688\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−32.1716	−2.08977
$238$	0	0
$239$	3.25544	0.210577	0.105288	−	0.994442i	$-0.466423\pi$
$239$	3.25544	0.210577	0.105288	+	0.994442i	$0.466423\pi$
$240$	0	0
$241$	5.25544	0.338532	0.169266	−	0.985570i	$-0.445860\pi$
$241$	5.25544	0.338532	0.169266	+	0.985570i	$0.445860\pi$
$242$	0	0
$243$	−22.3692	−1.43498
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−16.7446	−1.06114
$250$	0	0
$251$	4.88316	0.308222	0.154111	−	0.988054i	$-0.450749\pi$
$251$	4.88316	0.308222	0.154111	+	0.988054i	$0.450749\pi$
$252$	0	0
$253$	−2.52434	−0.158704
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.9538	−1.49419	−0.747097	−	0.664715i	$-0.768553\pi$
$257$	−23.9538	−1.49419	−0.747097	+	0.664715i	$0.768553\pi$
$258$	0	0
$259$	38.2337	2.37573
$260$	0	0
$261$	−9.25544	−0.572897
$262$	0	0
$263$	14.1514	0.872610	0.436305	−	0.899799i	$-0.356287\pi$
$263$	14.1514	0.872610	0.436305	+	0.899799i	$0.356287\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−11.0371	−0.675460
$268$	0	0
$269$	11.4891	0.700504	0.350252	−	0.936655i	$-0.386096\pi$
$269$	11.4891	0.700504	0.350252	+	0.936655i	$0.386096\pi$
$270$	0	0
$271$	9.48913	0.576423	0.288212	−	0.957567i	$-0.406939\pi$
$271$	9.48913	0.576423	0.288212	+	0.957567i	$0.406939\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.21782	−0.493761	−0.246881	−	0.969046i	$-0.579406\pi$
$277$	−8.21782	−0.493761	−0.246881	+	0.969046i	$0.579406\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	−23.4891	−1.40124	−0.700622	−	0.713533i	$-0.747094\pi$
$281$	−23.4891	−1.40124	−0.700622	+	0.713533i	$0.747094\pi$
$282$	0	0
$283$	−4.75372	−0.282579	−0.141290	−	0.989968i	$-0.545125\pi$
$283$	−4.75372	−0.282579	−0.141290	+	0.989968i	$0.545125\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.50744	−0.561207
$288$	0	0
$289$	8.48913	0.499360
$290$	0	0
$291$	10.3723	0.608034
$292$	0	0
$293$	10.0974	0.589894	0.294947	−	0.955514i	$-0.404698\pi$
$293$	10.0974	0.589894	0.294947	+	0.955514i	$0.404698\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.939764	0.0545306
$298$	0	0
$299$	0	0
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	15.1460	0.870117
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−28.1176	−1.60475	−0.802377	−	0.596817i	$-0.796432\pi$
$307$	−28.1176	−1.60475	−0.802377	+	0.596817i	$0.796432\pi$
$308$	0	0
$309$	−26.2337	−1.49238
$310$	0	0
$311$	17.4891	0.991717	0.495859	−	0.868403i	$-0.334853\pi$
$311$	17.4891	0.991717	0.495859	+	0.868403i	$0.334853\pi$
$312$	0	0
$313$	31.8217	1.79867	0.899335	−	0.437260i	$-0.144051\pi$
$313$	31.8217	1.79867	0.899335	+	0.437260i	$0.144051\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.51900	0.197647	0.0988235	−	0.995105i	$-0.468492\pi$
$317$	3.51900	0.197647	0.0988235	+	0.995105i	$0.468492\pi$
$318$	0	0
$319$	−2.74456	−0.153666
$320$	0	0
$321$	16.7446	0.934590
$322$	0	0
$323$	−20.1947	−1.12366
$324$	0	0
$325$	0	0
$326$	0	0
$327$	25.2434	1.39596
$328$	0	0
$329$	22.9783	1.26683
$330$	0	0
$331$	−3.11684	−0.171317	−0.0856586	−	0.996325i	$-0.527299\pi$
$331$	−3.11684	−0.171317	−0.0856586	+	0.996325i	$0.527299\pi$
$332$	0	0
$333$	37.2203	2.03966
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.5668	0.684556	0.342278	−	0.939599i	$-0.388801\pi$
$337$	12.5668	0.684556	0.342278	+	0.939599i	$0.388801\pi$
$338$	0	0
$339$	−40.6060	−2.20541
$340$	0	0
$341$	2.37228	0.128466
$342$	0	0
$343$	−6.92820	−0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.2974	−1.57277	−0.786383	−	0.617739i	$-0.788049\pi$
$347$	−29.2974	−1.57277	−0.786383	+	0.617739i	$0.788049\pi$
$348$	0	0
$349$	−7.48913	−0.400884	−0.200442	−	0.979706i	$-0.564238\pi$
$349$	−7.48913	−0.400884	−0.200442	+	0.979706i	$0.564238\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.7244	−1.15627	−0.578136	−	0.815941i	$-0.696220\pi$
$353$	−21.7244	−1.15627	−0.578136	+	0.815941i	$0.696220\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	44.1485	2.33658
$358$	0	0
$359$	−6.51087	−0.343631	−0.171815	−	0.985129i	$-0.554963\pi$
$359$	−6.51087	−0.343631	−0.171815	+	0.985129i	$0.554963\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	2.52434	0.132493
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−24.0087	−1.25324	−0.626621	−	0.779324i	$-0.715563\pi$
$367$	−24.0087	−1.25324	−0.626621	+	0.779324i	$0.715563\pi$
$368$	0	0
$369$	−9.25544	−0.481819
$370$	0	0
$371$	10.9783	0.569962
$372$	0	0
$373$	−8.21782	−0.425503	−0.212751	−	0.977106i	$-0.568242\pi$
$373$	−8.21782	−0.425503	−0.212751	+	0.977106i	$0.568242\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	6.37228	0.327322	0.163661	−	0.986517i	$-0.447670\pi$
$379$	6.37228	0.327322	0.163661	+	0.986517i	$0.447670\pi$
$380$	0	0
$381$	29.4891	1.51077
$382$	0	0
$383$	5.69349	0.290924	0.145462	−	0.989364i	$-0.453533\pi$
$383$	5.69349	0.290924	0.145462	+	0.989364i	$0.453533\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.6819	0.593826
$388$	0	0
$389$	−9.86141	−0.499993	−0.249997	−	0.968247i	$-0.580429\pi$
$389$	−9.86141	−0.499993	−0.249997	+	0.968247i	$0.580429\pi$
$390$	0	0
$391$	−12.7446	−0.644520
$392$	0	0
$393$	−22.0742	−1.11350
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−23.3639	−1.17260	−0.586299	−	0.810095i	$-0.699416\pi$
$397$	−23.3639	−1.17260	−0.586299	+	0.810095i	$0.699416\pi$
$398$	0	0
$399$	−34.9783	−1.75110
$400$	0	0
$401$	11.4891	0.573740	0.286870	−	0.957970i	$-0.407385\pi$
$401$	11.4891	0.573740	0.286870	+	0.957970i	$0.407385\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.0371	0.547089
$408$	0	0
$409$	4.51087	0.223048	0.111524	−	0.993762i	$-0.464427\pi$
$409$	4.51087	0.223048	0.111524	+	0.993762i	$0.464427\pi$
$410$	0	0
$411$	5.62772	0.277595
$412$	0	0
$413$	5.63858	0.277457
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−46.0280	−2.25400
$418$	0	0
$419$	−22.9783	−1.12256	−0.561280	−	0.827626i	$-0.689691\pi$
$419$	−22.9783	−1.12256	−0.561280	+	0.827626i	$0.689691\pi$
$420$	0	0
$421$	31.4891	1.53469	0.767343	−	0.641237i	$-0.221578\pi$
$421$	31.4891	1.53469	0.767343	+	0.641237i	$0.221578\pi$
$422$	0	0
$423$	22.3692	1.08763
$424$	0	0
$425$	0	0
$426$	0	0
$427$	37.2203	1.80121
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.7228	−1.52803	−0.764017	−	0.645196i	$-0.776776\pi$
$431$	−31.7228	−1.52803	−0.764017	+	0.645196i	$0.776776\pi$
$432$	0	0
$433$	20.5446	0.987308	0.493654	−	0.869658i	$-0.335661\pi$
$433$	20.5446	0.987308	0.493654	+	0.869658i	$0.335661\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.0974	0.483022
$438$	0	0
$439$	1.48913	0.0710721	0.0355360	−	0.999368i	$-0.488686\pi$
$439$	1.48913	0.0710721	0.0355360	+	0.999368i	$0.488686\pi$
$440$	0	0
$441$	16.8614	0.802924
$442$	0	0
$443$	−15.0911	−0.717001	−0.358500	−	0.933530i	$-0.616712\pi$
$443$	−15.0911	−0.717001	−0.358500	+	0.933530i	$0.616712\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	29.0024	1.37177
$448$	0	0
$449$	−21.8614	−1.03170	−0.515852	−	0.856678i	$-0.672524\pi$
$449$	−21.8614	−1.03170	−0.515852	+	0.856678i	$0.672524\pi$
$450$	0	0
$451$	−2.74456	−0.129236
$452$	0	0
$453$	−56.1253	−2.63700
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.7846	0.972263	0.486132	−	0.873886i	$-0.338408\pi$
$457$	20.7846	0.972263	0.486132	+	0.873886i	$0.338408\pi$
$458$	0	0
$459$	4.74456	0.221457
$460$	0	0
$461$	−32.2337	−1.50127	−0.750636	−	0.660716i	$-0.770253\pi$
$461$	−32.2337	−1.50127	−0.750636	+	0.660716i	$0.770253\pi$
$462$	0	0
$463$	−20.1398	−0.935976	−0.467988	−	0.883735i	$-0.655021\pi$
$463$	−20.1398	−0.935976	−0.467988	+	0.883735i	$0.655021\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.40387	−0.203787	−0.101893	−	0.994795i	$-0.532490\pi$
$467$	−4.40387	−0.203787	−0.101893	+	0.994795i	$0.532490\pi$
$468$	0	0
$469$	−2.23369	−0.103142
$470$	0	0
$471$	−13.6277	−0.627932
$472$	0	0
$473$	3.46410	0.159280
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.6873	0.489336
$478$	0	0
$479$	−17.4891	−0.799099	−0.399549	−	0.916712i	$-0.630833\pi$
$479$	−17.4891	−0.799099	−0.399549	+	0.916712i	$0.630833\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−22.0742	−1.00441
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.7190	1.02950	0.514749	−	0.857341i	$-0.327885\pi$
$487$	22.7190	1.02950	0.514749	+	0.857341i	$0.327885\pi$
$488$	0	0
$489$	−8.74456	−0.395443
$490$	0	0
$491$	−29.4891	−1.33083	−0.665413	−	0.746476i	$-0.731744\pi$
$491$	−29.4891	−1.33083	−0.665413	+	0.746476i	$0.731744\pi$
$492$	0	0
$493$	−13.8564	−0.624061
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.6535	−1.10586
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−56.4674	−2.52278
$502$	0	0
$503$	0.294954	0.0131513	0.00657567	−	0.999978i	$-0.497907\pi$
$503$	0.294954	0.0131513	0.00657567	+	0.999978i	$0.497907\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−32.8164	−1.45743
$508$	0	0
$509$	28.3723	1.25758	0.628790	−	0.777575i	$-0.283551\pi$
$509$	28.3723	1.25758	0.628790	+	0.777575i	$0.283551\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	0	0
$513$	−3.75906	−0.165966
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.63325	0.291730
$518$	0	0
$519$	4.74456	0.208263
$520$	0	0
$521$	18.6060	0.815142	0.407571	−	0.913173i	$-0.366376\pi$
$521$	18.6060	0.815142	0.407571	+	0.913173i	$0.366376\pi$
$522$	0	0
$523$	9.10268	0.398033	0.199016	−	0.979996i	$-0.436225\pi$
$523$	9.10268	0.398033	0.199016	+	0.979996i	$0.436225\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.9769	0.521721
$528$	0	0
$529$	−16.6277	−0.722944
$530$	0	0
$531$	5.48913	0.238208
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	40.0395	1.72783
$538$	0	0
$539$	5.00000	0.215365
$540$	0	0
$541$	−0.233688	−0.0100470	−0.00502351	−	0.999987i	$-0.501599\pi$
$541$	−0.233688	−0.0100470	−0.00502351	+	0.999987i	$0.501599\pi$
$542$	0	0
$543$	17.3754	0.745650
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−9.10268	−0.389203	−0.194601	−	0.980882i	$-0.562341\pi$
$547$	−9.10268	−0.389203	−0.194601	+	0.980882i	$0.562341\pi$
$548$	0	0
$549$	36.2337	1.54642
$550$	0	0
$551$	10.9783	0.467689
$552$	0	0
$553$	−44.1485	−1.87738
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−32.1716	−1.36315	−0.681577	−	0.731747i	$-0.738705\pi$
$557$	−32.1716	−1.36315	−0.681577	+	0.731747i	$0.738705\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	12.7446	0.538076
$562$	0	0
$563$	−12.2718	−0.517196	−0.258598	−	0.965985i	$-0.583261\pi$
$563$	−12.2718	−0.517196	−0.258598	+	0.965985i	$0.583261\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−26.8280	−1.12667
$568$	0	0
$569$	−38.7446	−1.62426	−0.812128	−	0.583479i	$-0.801691\pi$
$569$	−38.7446	−1.62426	−0.812128	+	0.583479i	$0.801691\pi$
$570$	0	0
$571$	21.4891	0.899292	0.449646	−	0.893207i	$-0.351550\pi$
$571$	21.4891	0.899292	0.449646	+	0.893207i	$0.351550\pi$
$572$	0	0
$573$	34.4010	1.43712
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−31.8217	−1.32476	−0.662378	−	0.749170i	$-0.730453\pi$
$577$	−31.8217	−1.32476	−0.662378	+	0.749170i	$0.730453\pi$
$578$	0	0
$579$	−58.9783	−2.45105
$580$	0	0
$581$	−22.9783	−0.953298
$582$	0	0
$583$	3.16915	0.131253
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.0078	−1.15600	−0.578002	−	0.816035i	$-0.696167\pi$
$587$	−28.0078	−1.15600	−0.578002	+	0.816035i	$0.696167\pi$
$588$	0	0
$589$	−9.48913	−0.390993
$590$	0	0
$591$	−4.74456	−0.195165
$592$	0	0
$593$	−43.5586	−1.78874	−0.894368	−	0.447333i	$-0.852374\pi$
$593$	−43.5586	−1.78874	−0.894368	+	0.447333i	$0.852374\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.1947	0.826514
$598$	0	0
$599$	34.9783	1.42917	0.714586	−	0.699547i	$-0.246615\pi$
$599$	34.9783	1.42917	0.714586	+	0.699547i	$0.246615\pi$
$600$	0	0
$601$	30.4674	1.24279	0.621395	−	0.783497i	$-0.286566\pi$
$601$	30.4674	1.24279	0.621395	+	0.783497i	$0.286566\pi$
$602$	0	0
$603$	−2.17448	−0.0885517
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.46410	0.140604	0.0703018	−	0.997526i	$-0.477604\pi$
$607$	3.46410	0.140604	0.0703018	+	0.997526i	$0.477604\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	0	0
$612$	0	0
$613$	44.1485	1.78314	0.891570	−	0.452883i	$-0.149605\pi$
$613$	44.1485	1.78314	0.891570	+	0.452883i	$0.149605\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.75906	0.151334	0.0756669	−	0.997133i	$-0.475891\pi$
$617$	3.75906	0.151334	0.0756669	+	0.997133i	$0.475891\pi$
$618$	0	0
$619$	3.11684	0.125277	0.0626383	−	0.998036i	$-0.480049\pi$
$619$	3.11684	0.125277	0.0626383	+	0.998036i	$0.480049\pi$
$620$	0	0
$621$	−2.37228	−0.0951964
$622$	0	0
$623$	−15.1460	−0.606813
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−10.0974	−0.403249
$628$	0	0
$629$	55.7228	2.22181
$630$	0	0
$631$	16.6060	0.661073	0.330537	−	0.943793i	$-0.392770\pi$
$631$	16.6060	0.661073	0.330537	+	0.943793i	$0.392770\pi$
$632$	0	0
$633$	54.2458	2.15608
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−24.0000	−0.949425
$640$	0	0
$641$	−19.6277	−0.775248	−0.387624	−	0.921818i	$-0.626704\pi$
$641$	−19.6277	−0.775248	−0.387624	+	0.921818i	$0.626704\pi$
$642$	0	0
$643$	39.1547	1.54411	0.772055	−	0.635556i	$-0.219229\pi$
$643$	39.1547	1.54411	0.772055	+	0.635556i	$0.219229\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.0342	1.61322	0.806611	−	0.591083i	$-0.201299\pi$
$647$	41.0342	1.61322	0.806611	+	0.591083i	$0.201299\pi$
$648$	0	0
$649$	1.62772	0.0638935
$650$	0	0
$651$	20.7446	0.813044
$652$	0	0
$653$	−25.5932	−1.00154	−0.500770	−	0.865580i	$-0.666950\pi$
$653$	−25.5932	−1.00154	−0.500770	+	0.865580i	$0.666950\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	23.3639	0.911511
$658$	0	0
$659$	−32.7446	−1.27555	−0.637774	−	0.770224i	$-0.720144\pi$
$659$	−32.7446	−1.27555	−0.637774	+	0.770224i	$0.720144\pi$
$660$	0	0
$661$	35.3505	1.37498	0.687488	−	0.726196i	$-0.258713\pi$
$661$	35.3505	1.37498	0.687488	+	0.726196i	$0.258713\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.92820	0.268261
$668$	0	0
$669$	19.1168	0.739100
$670$	0	0
$671$	10.7446	0.414789
$672$	0	0
$673$	1.28962	0.0497112	0.0248556	−	0.999691i	$-0.492087\pi$
$673$	1.28962	0.0497112	0.0248556	+	0.999691i	$0.492087\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−43.4487	−1.66987	−0.834935	−	0.550348i	$-0.814495\pi$
$677$	−43.4487	−1.66987	−0.834935	+	0.550348i	$0.814495\pi$
$678$	0	0
$679$	14.2337	0.546239
$680$	0	0
$681$	−24.7446	−0.948214
$682$	0	0
$683$	44.4434	1.70058	0.850290	−	0.526314i	$-0.176427\pi$
$683$	44.4434	1.70058	0.850290	+	0.526314i	$0.176427\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	51.4265	1.96204
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−16.1386	−0.613941	−0.306971	−	0.951719i	$-0.599315\pi$
$691$	−16.1386	−0.613941	−0.306971	+	0.951719i	$0.599315\pi$
$692$	0	0
$693$	11.6819	0.443760
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−13.8564	−0.524849
$698$	0	0
$699$	42.9783	1.62559
$700$	0	0
$701$	−35.4891	−1.34041	−0.670203	−	0.742178i	$-0.733793\pi$
$701$	−35.4891	−1.34041	−0.670203	+	0.742178i	$0.733793\pi$
$702$	0	0
$703$	−44.1485	−1.66509
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.7846	0.781686
$708$	0	0
$709$	41.1168	1.54418	0.772088	−	0.635516i	$-0.219213\pi$
$709$	41.1168	1.54418	0.772088	+	0.635516i	$0.219213\pi$
$710$	0	0
$711$	−42.9783	−1.61181
$712$	0	0
$713$	−5.98844	−0.224269
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.21782	0.306900
$718$	0	0
$719$	−21.3505	−0.796240	−0.398120	−	0.917333i	$-0.630337\pi$
$719$	−21.3505	−0.796240	−0.398120	+	0.917333i	$0.630337\pi$
$720$	0	0
$721$	−36.0000	−1.34071
$722$	0	0
$723$	13.2665	0.493386
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−15.7908	−0.585650	−0.292825	−	0.956166i	$-0.594595\pi$
$727$	−15.7908	−0.585650	−0.292825	+	0.956166i	$0.594595\pi$
$728$	0	0
$729$	−33.2337	−1.23088
$730$	0	0
$731$	17.4891	0.646859
$732$	0	0
$733$	−30.2921	−1.11886	−0.559431	−	0.828877i	$-0.688980\pi$
$733$	−30.2921	−1.11886	−0.559431	+	0.828877i	$0.688980\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.644810	−0.0237519
$738$	0	0
$739$	−0.744563	−0.0273892	−0.0136946	−	0.999906i	$-0.504359\pi$
$739$	−0.744563	−0.0273892	−0.0136946	+	0.999906i	$0.504359\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.7793	−0.799004	−0.399502	−	0.916732i	$-0.630817\pi$
$743$	−21.7793	−0.799004	−0.399502	+	0.916732i	$0.630817\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−22.3692	−0.818446
$748$	0	0
$749$	22.9783	0.839607
$750$	0	0
$751$	−21.6277	−0.789207	−0.394603	−	0.918852i	$-0.629118\pi$
$751$	−21.6277	−0.789207	−0.394603	+	0.918852i	$0.629118\pi$
$752$	0	0
$753$	12.3267	0.449211
$754$	0	0
$755$	0	0
$756$	0	0
$757$	39.7995	1.44654	0.723269	−	0.690567i	$-0.242639\pi$
$757$	39.7995	1.44654	0.723269	+	0.690567i	$0.242639\pi$
$758$	0	0
$759$	−6.37228	−0.231299
$760$	0	0
$761$	21.2554	0.770509	0.385255	−	0.922810i	$-0.374114\pi$
$761$	21.2554	0.770509	0.385255	+	0.922810i	$0.374114\pi$
$762$	0	0
$763$	34.6410	1.25409
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−51.2119	−1.84675	−0.923375	−	0.383900i	$-0.874581\pi$
$769$	−51.2119	−1.84675	−0.923375	+	0.383900i	$0.874581\pi$
$770$	0	0
$771$	−60.4674	−2.17768
$772$	0	0
$773$	30.8820	1.11075	0.555373	−	0.831601i	$-0.312575\pi$
$773$	30.8820	1.11075	0.555373	+	0.831601i	$0.312575\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	96.5147	3.46245
$778$	0	0
$779$	10.9783	0.393337
$780$	0	0
$781$	−7.11684	−0.254661
$782$	0	0
$783$	−2.57924	−0.0921745
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.75372	0.169452	0.0847259	−	0.996404i	$-0.472999\pi$
$787$	4.75372	0.169452	0.0847259	+	0.996404i	$0.472999\pi$
$788$	0	0
$789$	35.7228	1.27177
$790$	0	0
$791$	−55.7228	−1.98128
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.2318	1.10629	0.553144	−	0.833086i	$-0.313428\pi$
$797$	31.2318	1.10629	0.553144	+	0.833086i	$0.313428\pi$
$798$	0	0
$799$	33.4891	1.18476
$800$	0	0
$801$	−14.7446	−0.520974
$802$	0	0
$803$	6.92820	0.244491
$804$	0	0
$805$	0	0
$806$	0	0
$807$	29.0024	1.02093
$808$	0	0
$809$	−21.2554	−0.747301	−0.373651	−	0.927569i	$-0.621894\pi$
$809$	−21.2554	−0.747301	−0.373651	+	0.927569i	$0.621894\pi$
$810$	0	0
$811$	−34.2337	−1.20211	−0.601054	−	0.799209i	$-0.705252\pi$
$811$	−34.2337	−1.20211	−0.601054	+	0.799209i	$0.705252\pi$
$812$	0	0
$813$	23.9538	0.840095
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−13.8564	−0.484774
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	−33.5161	−1.16830	−0.584149	−	0.811646i	$-0.698572\pi$
$823$	−33.5161	−1.16830	−0.584149	+	0.811646i	$0.698572\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.0202	−0.626624	−0.313312	−	0.949650i	$-0.601439\pi$
$827$	−18.0202	−0.626624	−0.313312	+	0.949650i	$0.601439\pi$
$828$	0	0
$829$	31.3505	1.08885	0.544424	−	0.838810i	$-0.316748\pi$
$829$	31.3505	1.08885	0.544424	+	0.838810i	$0.316748\pi$
$830$	0	0
$831$	−20.7446	−0.719621
$832$	0	0
$833$	25.2434	0.874631
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.22938	0.0770588
$838$	0	0
$839$	−7.11684	−0.245701	−0.122850	−	0.992425i	$-0.539204\pi$
$839$	−7.11684	−0.245701	−0.122850	+	0.992425i	$0.539204\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	0	0
$843$	−59.2945	−2.04221
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.46410	0.119028
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−27.8614	−0.955077
$852$	0	0
$853$	24.6535	0.844119	0.422059	−	0.906568i	$-0.361307\pi$
$853$	24.6535	0.844119	0.422059	+	0.906568i	$0.361307\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.6873	0.365070	0.182535	−	0.983199i	$-0.441570\pi$
$857$	10.6873	0.365070	0.182535	+	0.983199i	$0.441570\pi$
$858$	0	0
$859$	−11.1168	−0.379302	−0.189651	−	0.981852i	$-0.560736\pi$
$859$	−11.1168	−0.379302	−0.189651	+	0.981852i	$0.560736\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	0	0
$863$	23.6588	0.805355	0.402678	−	0.915342i	$-0.368080\pi$
$863$	23.6588	0.805355	0.402678	+	0.915342i	$0.368080\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	21.4294	0.727781
$868$	0	0
$869$	−12.7446	−0.432330
$870$	0	0
$871$	0	0
$872$	0	0
$873$	13.8564	0.468968
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	25.4891	0.859727
$880$	0	0
$881$	21.8614	0.736530	0.368265	−	0.929721i	$-0.379952\pi$
$881$	21.8614	0.736530	0.368265	+	0.929721i	$0.379952\pi$
$882$	0	0
$883$	24.2487	0.816034	0.408017	−	0.912974i	$-0.366220\pi$
$883$	24.2487	0.816034	0.408017	+	0.912974i	$0.366220\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.1514	−0.475156	−0.237578	−	0.971368i	$-0.576354\pi$
$887$	−14.1514	−0.475156	−0.237578	+	0.971368i	$0.576354\pi$
$888$	0	0
$889$	40.4674	1.35723
$890$	0	0
$891$	−7.74456	−0.259453
$892$	0	0
$893$	−26.5330	−0.887893
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.51087	−0.217150
$900$	0	0
$901$	16.0000	0.533037
$902$	0	0
$903$	30.2921	1.00806
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19.8997	0.660760	0.330380	−	0.943848i	$-0.392823\pi$
$907$	19.8997	0.660760	0.330380	+	0.943848i	$0.392823\pi$
$908$	0	0
$909$	20.2337	0.671109
$910$	0	0
$911$	30.5109	1.01087	0.505435	−	0.862865i	$-0.331332\pi$
$911$	30.5109	1.01087	0.505435	+	0.862865i	$0.331332\pi$
$912$	0	0
$913$	−6.63325	−0.219529
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−30.2921	−1.00033
$918$	0	0
$919$	−6.23369	−0.205630	−0.102815	−	0.994700i	$-0.532785\pi$
$919$	−6.23369	−0.205630	−0.102815	+	0.994700i	$0.532785\pi$
$920$	0	0
$921$	−70.9783	−2.33881
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−35.0458	−1.15105
$928$	0	0
$929$	−52.9783	−1.73816	−0.869080	−	0.494672i	$-0.835288\pi$
$929$	−52.9783	−1.73816	−0.869080	+	0.494672i	$0.835288\pi$
$930$	0	0
$931$	−20.0000	−0.655474
$932$	0	0
$933$	44.1485	1.44536
$934$	0	0
$935$	0	0
$936$	0	0
$937$	25.9431	0.847524	0.423762	−	0.905774i	$-0.360709\pi$
$937$	25.9431	0.847524	0.423762	+	0.905774i	$0.360709\pi$
$938$	0	0
$939$	80.3288	2.62143
$940$	0	0
$941$	10.4674	0.341227	0.170613	−	0.985338i	$-0.445425\pi$
$941$	10.4674	0.341227	0.170613	+	0.985338i	$0.445425\pi$
$942$	0	0
$943$	6.92820	0.225613
$944$	0	0
$945$	0	0
$946$	0	0
$947$	56.1802	1.82561	0.912806	−	0.408393i	$-0.133911\pi$
$947$	56.1802	1.82561	0.912806	+	0.408393i	$0.133911\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	8.88316	0.288056
$952$	0	0
$953$	41.6790	1.35012	0.675058	−	0.737765i	$-0.264119\pi$
$953$	41.6790	1.35012	0.675058	+	0.737765i	$0.264119\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−6.92820	−0.223957
$958$	0	0
$959$	7.72281	0.249383
$960$	0	0
$961$	−25.3723	−0.818461
$962$	0	0
$963$	22.3692	0.720837
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−46.3229	−1.48965	−0.744823	−	0.667262i	$-0.767466\pi$
$967$	−46.3229	−1.48965	−0.744823	+	0.667262i	$0.767466\pi$
$968$	0	0
$969$	−50.9783	−1.63766
$970$	0	0
$971$	−54.0951	−1.73599	−0.867997	−	0.496569i	$-0.834593\pi$
$971$	−54.0951	−1.73599	−0.867997	+	0.496569i	$0.834593\pi$
$972$	0	0
$973$	−63.1633	−2.02492
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27.3630	0.875419	0.437709	−	0.899117i	$-0.355790\pi$
$977$	27.3630	0.875419	0.437709	+	0.899117i	$0.355790\pi$
$978$	0	0
$979$	−4.37228	−0.139739
$980$	0	0
$981$	33.7228	1.07669
$982$	0	0
$983$	−8.16292	−0.260357	−0.130178	−	0.991491i	$-0.541555\pi$
$983$	−8.16292	−0.260357	−0.130178	+	0.991491i	$0.541555\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	58.0049	1.84632
$988$	0	0
$989$	−8.74456	−0.278061
$990$	0	0
$991$	26.9783	0.856992	0.428496	−	0.903544i	$-0.359044\pi$
$991$	26.9783	0.856992	0.428496	+	0.903544i	$0.359044\pi$
$992$	0	0
$993$	−7.86797	−0.249682
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−22.0742	−0.699098	−0.349549	−	0.936918i	$-0.613665\pi$
$997$	−22.0742	−0.699098	−0.349549	+	0.936918i	$0.613665\pi$
$998$	0	0
$999$	10.3723	0.328164

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.cc.1.4		4
4.3	odd	2		275.2.a.h.1.2		4
5.2	odd	4		880.2.b.h.529.1		4
5.3	odd	4		880.2.b.h.529.4		4
5.4	even	2	inner	4400.2.a.cc.1.1		4
12.11	even	2		2475.2.a.bi.1.3		4
20.3	even	4		55.2.b.a.34.3	yes	4
20.7	even	4		55.2.b.a.34.2	✓	4
20.19	odd	2		275.2.a.h.1.3		4
44.43	even	2		3025.2.a.ba.1.3		4
60.23	odd	4		495.2.c.a.199.2		4
60.47	odd	4		495.2.c.a.199.3		4
60.59	even	2		2475.2.a.bi.1.2		4
220.3	even	20		605.2.j.i.9.3		16
220.7	odd	20		605.2.j.j.269.2		16
220.27	even	20		605.2.j.i.124.3		16
220.43	odd	4		605.2.b.c.364.2		4
220.47	even	20		605.2.j.i.9.2		16
220.63	odd	20		605.2.j.j.9.2		16
220.83	odd	20		605.2.j.j.124.3		16
220.87	odd	4		605.2.b.c.364.3		4
220.103	even	20		605.2.j.i.269.2		16
220.107	odd	20		605.2.j.j.9.3		16
220.123	odd	20		605.2.j.j.444.2		16
220.127	odd	20		605.2.j.j.124.2		16
220.147	even	20		605.2.j.i.269.3		16
220.163	even	20		605.2.j.i.444.3		16
220.167	odd	20		605.2.j.j.444.3		16
220.183	odd	20		605.2.j.j.269.3		16
220.203	even	20		605.2.j.i.124.2		16
220.207	even	20		605.2.j.i.444.2		16
220.219	even	2		3025.2.a.ba.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.b.a.34.2	✓	4	20.7	even	4
55.2.b.a.34.3	yes	4	20.3	even	4
275.2.a.h.1.2		4	4.3	odd	2
275.2.a.h.1.3		4	20.19	odd	2
495.2.c.a.199.2		4	60.23	odd	4
495.2.c.a.199.3		4	60.47	odd	4
605.2.b.c.364.2		4	220.43	odd	4
605.2.b.c.364.3		4	220.87	odd	4
605.2.j.i.9.2		16	220.47	even	20
605.2.j.i.9.3		16	220.3	even	20
605.2.j.i.124.2		16	220.203	even	20
605.2.j.i.124.3		16	220.27	even	20
605.2.j.i.269.2		16	220.103	even	20
605.2.j.i.269.3		16	220.147	even	20
605.2.j.i.444.2		16	220.207	even	20
605.2.j.i.444.3		16	220.163	even	20
605.2.j.j.9.2		16	220.63	odd	20
605.2.j.j.9.3		16	220.107	odd	20
605.2.j.j.124.2		16	220.127	odd	20
605.2.j.j.124.3		16	220.83	odd	20
605.2.j.j.269.2		16	220.7	odd	20
605.2.j.j.269.3		16	220.183	odd	20
605.2.j.j.444.2		16	220.123	odd	20
605.2.j.j.444.3		16	220.167	odd	20
880.2.b.h.529.1		4	5.2	odd	4
880.2.b.h.529.4		4	5.3	odd	4
2475.2.a.bi.1.2		4	60.59	even	2
2475.2.a.bi.1.3		4	12.11	even	2
3025.2.a.ba.1.2		4	220.219	even	2
3025.2.a.ba.1.3		4	44.43	even	2
4400.2.a.cc.1.1		4	5.4	even	2	inner
4400.2.a.cc.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q+2.52434 q^{3} +3.46410 q^{7} +3.37228 q^{9} +1.00000 q^{11} +5.04868 q^{17} -4.00000 q^{19} +8.74456 q^{21} -2.52434 q^{23} +0.939764 q^{27} -2.74456 q^{29} +2.37228 q^{31} +2.52434 q^{33} +11.0371 q^{37} -2.74456 q^{41} +3.46410 q^{43} +6.63325 q^{47} +5.00000 q^{49} +12.7446 q^{51} +3.16915 q^{53} -10.0974 q^{57} +1.62772 q^{59} +10.7446 q^{61} +11.6819 q^{63} -0.644810 q^{67} -6.37228 q^{69} -7.11684 q^{71} +6.92820 q^{73} +3.46410 q^{77} -12.7446 q^{79} -7.74456 q^{81} -6.63325 q^{83} -6.92820 q^{87} -4.37228 q^{89} +5.98844 q^{93} +4.10891 q^{97} +3.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{9} + 4 q^{11} - 16 q^{19} + 12 q^{21} + 12 q^{29} - 2 q^{31} + 12 q^{41} + 20 q^{49} + 28 q^{51} + 18 q^{59} + 20 q^{61} - 14 q^{69} + 6 q^{71} - 28 q^{79} - 8 q^{81} - 6 q^{89} + 2 q^{99}+O(q^{100}) \) 4 * q + 2 * q^9 + 4 * q^11 - 16 * q^19 + 12 * q^21 + 12 * q^29 - 2 * q^31 + 12 * q^41 + 20 * q^49 + 28 * q^51 + 18 * q^59 + 20 * q^61 - 14 * q^69 + 6 * q^71 - 28 * q^79 - 8 * q^81 - 6 * q^89 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more