LMFDB - Embedded newform 5550.2.a.ca.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5550,2,Mod(1,5550)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5550.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5550, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5550.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,2,2,0,2,1,2,2,0,3,2,3,1,0,2,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$44.3169731218$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	5550.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.37228 q^{11} +1.00000 q^{12} +4.37228 q^{13} -2.37228 q^{14} +1.00000 q^{16} -4.37228 q^{17} +1.00000 q^{18} +4.37228 q^{19} -2.37228 q^{21} +4.37228 q^{22} +2.37228 q^{23} +1.00000 q^{24} +4.37228 q^{26} +1.00000 q^{27} -2.37228 q^{28} +6.00000 q^{29} -8.74456 q^{31} +1.00000 q^{32} +4.37228 q^{33} -4.37228 q^{34} +1.00000 q^{36} +1.00000 q^{37} +4.37228 q^{38} +4.37228 q^{39} -2.74456 q^{41} -2.37228 q^{42} +4.00000 q^{43} +4.37228 q^{44} +2.37228 q^{46} -6.74456 q^{47} +1.00000 q^{48} -1.37228 q^{49} -4.37228 q^{51} +4.37228 q^{52} +0.372281 q^{53} +1.00000 q^{54} -2.37228 q^{56} +4.37228 q^{57} +6.00000 q^{58} +3.25544 q^{59} +8.74456 q^{61} -8.74456 q^{62} -2.37228 q^{63} +1.00000 q^{64} +4.37228 q^{66} +8.74456 q^{67} -4.37228 q^{68} +2.37228 q^{69} +12.7446 q^{71} +1.00000 q^{72} -13.1168 q^{73} +1.00000 q^{74} +4.37228 q^{76} -10.3723 q^{77} +4.37228 q^{78} +4.00000 q^{79} +1.00000 q^{81} -2.74456 q^{82} -6.37228 q^{83} -2.37228 q^{84} +4.00000 q^{86} +6.00000 q^{87} +4.37228 q^{88} +7.62772 q^{89} -10.3723 q^{91} +2.37228 q^{92} -8.74456 q^{93} -6.74456 q^{94} +1.00000 q^{96} +9.48913 q^{97} -1.37228 q^{98} +4.37228 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{11} + 2 q^{12} + 3 q^{13} + q^{14} + 2 q^{16} - 3 q^{17} + 2 q^{18} + 3 q^{19} + q^{21} + 3 q^{22} - q^{23} + 2 q^{24}+ \cdots + 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + q^7 + 2 * q^8 + 2 * q^9 + 3 * q^11 + 2 * q^12 + 3 * q^13 + q^14 + 2 * q^16 - 3 * q^17 + 2 * q^18 + 3 * q^19 + q^21 + 3 * q^22 - q^23 + 2 * q^24 + 3 * q^26 + 2 * q^27 + q^28 + 12 * q^29 - 6 * q^31 + 2 * q^32 + 3 * q^33 - 3 * q^34 + 2 * q^36 + 2 * q^37 + 3 * q^38 + 3 * q^39 + 6 * q^41 + q^42 + 8 * q^43 + 3 * q^44 - q^46 - 2 * q^47 + 2 * q^48 + 3 * q^49 - 3 * q^51 + 3 * q^52 - 5 * q^53 + 2 * q^54 + q^56 + 3 * q^57 + 12 * q^58 + 18 * q^59 + 6 * q^61 - 6 * q^62 + q^63 + 2 * q^64 + 3 * q^66 + 6 * q^67 - 3 * q^68 - q^69 + 14 * q^71 + 2 * q^72 - 9 * q^73 + 2 * q^74 + 3 * q^76 - 15 * q^77 + 3 * q^78 + 8 * q^79 + 2 * q^81 + 6 * q^82 - 7 * q^83 + q^84 + 8 * q^86 + 12 * q^87 + 3 * q^88 + 21 * q^89 - 15 * q^91 - q^92 - 6 * q^93 - 2 * q^94 + 2 * q^96 - 4 * q^97 + 3 * q^98 + 3 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−2.37228	−0.896638	−0.448319	−	0.893874i	$-0.647977\pi$
$7$	−2.37228	−0.896638	−0.448319	+	0.893874i	$0.647977\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	4.37228	1.31829	0.659146	−	0.752015i	$-0.270918\pi$
$11$	4.37228	1.31829	0.659146	+	0.752015i	$0.270918\pi$
$12$	1.00000	0.288675
$13$	4.37228	1.21265	0.606326	−	0.795216i	$-0.292643\pi$
$13$	4.37228	1.21265	0.606326	+	0.795216i	$0.292643\pi$
$14$	−2.37228	−0.634019
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.37228	−1.06043	−0.530217	−	0.847862i	$-0.677890\pi$
$17$	−4.37228	−1.06043	−0.530217	+	0.847862i	$0.677890\pi$
$18$	1.00000	0.235702
$19$	4.37228	1.00307	0.501535	−	0.865137i	$-0.332769\pi$
$19$	4.37228	1.00307	0.501535	+	0.865137i	$0.332769\pi$
$20$	0	0
$21$	−2.37228	−0.517674
$22$	4.37228	0.932174
$23$	2.37228	0.494655	0.247327	−	0.968932i	$-0.420448\pi$
$23$	2.37228	0.494655	0.247327	+	0.968932i	$0.420448\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	4.37228	0.857475
$27$	1.00000	0.192450
$28$	−2.37228	−0.448319
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.74456	−1.57057	−0.785285	−	0.619135i	$-0.787484\pi$
$31$	−8.74456	−1.57057	−0.785285	+	0.619135i	$0.787484\pi$
$32$	1.00000	0.176777
$33$	4.37228	0.761116
$34$	−4.37228	−0.749840
$35$	0	0
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	4.37228	0.709278
$39$	4.37228	0.700125
$40$	0	0
$41$	−2.74456	−0.428629	−0.214314	−	0.976765i	$-0.568752\pi$
$41$	−2.74456	−0.428629	−0.214314	+	0.976765i	$0.568752\pi$
$42$	−2.37228	−0.366051
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	4.37228	0.659146
$45$	0	0
$46$	2.37228	0.349774
$47$	−6.74456	−0.983796	−0.491898	−	0.870653i	$-0.663697\pi$
$47$	−6.74456	−0.983796	−0.491898	+	0.870653i	$0.663697\pi$
$48$	1.00000	0.144338
$49$	−1.37228	−0.196040
$50$	0	0
$51$	−4.37228	−0.612242
$52$	4.37228	0.606326
$53$	0.372281	0.0511368	0.0255684	−	0.999673i	$-0.491860\pi$
$53$	0.372281	0.0511368	0.0255684	+	0.999673i	$0.491860\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.37228	−0.317009
$57$	4.37228	0.579123
$58$	6.00000	0.787839
$59$	3.25544	0.423822	0.211911	−	0.977289i	$-0.432031\pi$
$59$	3.25544	0.423822	0.211911	+	0.977289i	$0.432031\pi$
$60$	0	0
$61$	8.74456	1.11963	0.559813	−	0.828619i	$-0.310873\pi$
$61$	8.74456	1.11963	0.559813	+	0.828619i	$0.310873\pi$
$62$	−8.74456	−1.11056
$63$	−2.37228	−0.298879
$64$	1.00000	0.125000
$65$	0	0
$66$	4.37228	0.538191
$67$	8.74456	1.06832	0.534159	−	0.845384i	$-0.320628\pi$
$67$	8.74456	1.06832	0.534159	+	0.845384i	$0.320628\pi$
$68$	−4.37228	−0.530217
$69$	2.37228	0.285589
$70$	0	0
$71$	12.7446	1.51250	0.756251	−	0.654282i	$-0.227029\pi$
$71$	12.7446	1.51250	0.756251	+	0.654282i	$0.227029\pi$
$72$	1.00000	0.117851
$73$	−13.1168	−1.53521	−0.767605	−	0.640923i	$-0.778552\pi$
$73$	−13.1168	−1.53521	−0.767605	+	0.640923i	$0.778552\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	4.37228	0.501535
$77$	−10.3723	−1.18203
$78$	4.37228	0.495063
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.74456	−0.303086
$83$	−6.37228	−0.699449	−0.349724	−	0.936853i	$-0.613725\pi$
$83$	−6.37228	−0.699449	−0.349724	+	0.936853i	$0.613725\pi$
$84$	−2.37228	−0.258837
$85$	0	0
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	4.37228	0.466087
$89$	7.62772	0.808537	0.404268	−	0.914640i	$-0.367526\pi$
$89$	7.62772	0.808537	0.404268	+	0.914640i	$0.367526\pi$
$90$	0	0
$91$	−10.3723	−1.08731
$92$	2.37228	0.247327
$93$	−8.74456	−0.906769
$94$	−6.74456	−0.695649
$95$	0	0
$96$	1.00000	0.102062
$97$	9.48913	0.963475	0.481737	−	0.876316i	$-0.340006\pi$
$97$	9.48913	0.963475	0.481737	+	0.876316i	$0.340006\pi$
$98$	−1.37228	−0.138621
$99$	4.37228	0.439431
$100$	0	0
$101$	−0.744563	−0.0740868	−0.0370434	−	0.999314i	$-0.511794\pi$
$101$	−0.744563	−0.0740868	−0.0370434	+	0.999314i	$0.511794\pi$
$102$	−4.37228	−0.432920
$103$	−10.7446	−1.05869	−0.529347	−	0.848406i	$-0.677563\pi$
$103$	−10.7446	−1.05869	−0.529347	+	0.848406i	$0.677563\pi$
$104$	4.37228	0.428737
$105$	0	0
$106$	0.372281	0.0361592
$107$	2.37228	0.229337	0.114669	−	0.993404i	$-0.463419\pi$
$107$	2.37228	0.229337	0.114669	+	0.993404i	$0.463419\pi$
$108$	1.00000	0.0962250
$109$	10.3723	0.993484	0.496742	−	0.867898i	$-0.334529\pi$
$109$	10.3723	0.993484	0.496742	+	0.867898i	$0.334529\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	−2.37228	−0.224160
$113$	3.48913	0.328229	0.164115	−	0.986441i	$-0.447523\pi$
$113$	3.48913	0.328229	0.164115	+	0.986441i	$0.447523\pi$
$114$	4.37228	0.409502
$115$	0	0
$116$	6.00000	0.557086
$117$	4.37228	0.404218
$118$	3.25544	0.299687
$119$	10.3723	0.950825
$120$	0	0
$121$	8.11684	0.737895
$122$	8.74456	0.791696
$123$	−2.74456	−0.247469
$124$	−8.74456	−0.785285
$125$	0	0
$126$	−2.37228	−0.211340
$127$	2.37228	0.210506	0.105253	−	0.994445i	$-0.466435\pi$
$127$	2.37228	0.210506	0.105253	+	0.994445i	$0.466435\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	4.37228	0.380558
$133$	−10.3723	−0.899391
$134$	8.74456	0.755415
$135$	0	0
$136$	−4.37228	−0.374920
$137$	0.744563	0.0636123	0.0318061	−	0.999494i	$-0.489874\pi$
$137$	0.744563	0.0636123	0.0318061	+	0.999494i	$0.489874\pi$
$138$	2.37228	0.201942
$139$	−8.74456	−0.741704	−0.370852	−	0.928692i	$-0.620934\pi$
$139$	−8.74456	−0.741704	−0.370852	+	0.928692i	$0.620934\pi$
$140$	0	0
$141$	−6.74456	−0.567995
$142$	12.7446	1.06950
$143$	19.1168	1.59863
$144$	1.00000	0.0833333
$145$	0	0
$146$	−13.1168	−1.08556
$147$	−1.37228	−0.113184
$148$	1.00000	0.0821995
$149$	20.7446	1.69946	0.849730	−	0.527217i	$-0.176765\pi$
$149$	20.7446	1.69946	0.849730	+	0.527217i	$0.176765\pi$
$150$	0	0
$151$	2.37228	0.193054	0.0965268	−	0.995330i	$-0.469227\pi$
$151$	2.37228	0.193054	0.0965268	+	0.995330i	$0.469227\pi$
$152$	4.37228	0.354639
$153$	−4.37228	−0.353478
$154$	−10.3723	−0.835822
$155$	0	0
$156$	4.37228	0.350063
$157$	−14.7446	−1.17674	−0.588372	−	0.808590i	$-0.700231\pi$
$157$	−14.7446	−1.17674	−0.588372	+	0.808590i	$0.700231\pi$
$158$	4.00000	0.318223
$159$	0.372281	0.0295238
$160$	0	0
$161$	−5.62772	−0.443526
$162$	1.00000	0.0785674
$163$	23.8614	1.86897	0.934485	−	0.356003i	$-0.115861\pi$
$163$	23.8614	1.86897	0.934485	+	0.356003i	$0.115861\pi$
$164$	−2.74456	−0.214314
$165$	0	0
$166$	−6.37228	−0.494585
$167$	19.1168	1.47931	0.739653	−	0.672989i	$-0.234990\pi$
$167$	19.1168	1.47931	0.739653	+	0.672989i	$0.234990\pi$
$168$	−2.37228	−0.183025
$169$	6.11684	0.470526
$170$	0	0
$171$	4.37228	0.334357
$172$	4.00000	0.304997
$173$	−21.8614	−1.66209	−0.831046	−	0.556204i	$-0.812257\pi$
$173$	−21.8614	−1.66209	−0.831046	+	0.556204i	$0.812257\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	4.37228	0.329573
$177$	3.25544	0.244694
$178$	7.62772	0.571722
$179$	−17.4891	−1.30720	−0.653599	−	0.756841i	$-0.726742\pi$
$179$	−17.4891	−1.30720	−0.653599	+	0.756841i	$0.726742\pi$
$180$	0	0
$181$	24.9783	1.85662	0.928309	−	0.371809i	$-0.121262\pi$
$181$	24.9783	1.85662	0.928309	+	0.371809i	$0.121262\pi$
$182$	−10.3723	−0.768845
$183$	8.74456	0.646417
$184$	2.37228	0.174887
$185$	0	0
$186$	−8.74456	−0.641182
$187$	−19.1168	−1.39796
$188$	−6.74456	−0.491898
$189$	−2.37228	−0.172558
$190$	0	0
$191$	−13.6277	−0.986067	−0.493034	−	0.870010i	$-0.664112\pi$
$191$	−13.6277	−0.986067	−0.493034	+	0.870010i	$0.664112\pi$
$192$	1.00000	0.0721688
$193$	12.7446	0.917374	0.458687	−	0.888598i	$-0.348320\pi$
$193$	12.7446	0.917374	0.458687	+	0.888598i	$0.348320\pi$
$194$	9.48913	0.681279
$195$	0	0
$196$	−1.37228	−0.0980201
$197$	−9.86141	−0.702596	−0.351298	−	0.936264i	$-0.614260\pi$
$197$	−9.86141	−0.702596	−0.351298	+	0.936264i	$0.614260\pi$
$198$	4.37228	0.310725
$199$	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
$201$	8.74456	0.616794
$202$	−0.744563	−0.0523872
$203$	−14.2337	−0.999009
$204$	−4.37228	−0.306121
$205$	0	0
$206$	−10.7446	−0.748609
$207$	2.37228	0.164885
$208$	4.37228	0.303163
$209$	19.1168	1.32234
$210$	0	0
$211$	−20.7446	−1.42811	−0.714057	−	0.700087i	$-0.753144\pi$
$211$	−20.7446	−1.42811	−0.714057	+	0.700087i	$0.753144\pi$
$212$	0.372281	0.0255684
$213$	12.7446	0.873243
$214$	2.37228	0.162166
$215$	0	0
$216$	1.00000	0.0680414
$217$	20.7446	1.40823
$218$	10.3723	0.702500
$219$	−13.1168	−0.886354
$220$	0	0
$221$	−19.1168	−1.28594
$222$	1.00000	0.0671156
$223$	26.9783	1.80660	0.903299	−	0.429012i	$-0.141138\pi$
$223$	26.9783	1.80660	0.903299	+	0.429012i	$0.141138\pi$
$224$	−2.37228	−0.158505
$225$	0	0
$226$	3.48913	0.232093
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	4.37228	0.289561
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−10.3723	−0.682446
$232$	6.00000	0.393919
$233$	11.2554	0.737368	0.368684	−	0.929555i	$-0.379808\pi$
$233$	11.2554	0.737368	0.368684	+	0.929555i	$0.379808\pi$
$234$	4.37228	0.285825
$235$	0	0
$236$	3.25544	0.211911
$237$	4.00000	0.259828
$238$	10.3723	0.672335
$239$	17.4891	1.13128	0.565639	−	0.824653i	$-0.308630\pi$
$239$	17.4891	1.13128	0.565639	+	0.824653i	$0.308630\pi$
$240$	0	0
$241$	−23.4891	−1.51307	−0.756534	−	0.653955i	$-0.773109\pi$
$241$	−23.4891	−1.51307	−0.756534	+	0.653955i	$0.773109\pi$
$242$	8.11684	0.521770
$243$	1.00000	0.0641500
$244$	8.74456	0.559813
$245$	0	0
$246$	−2.74456	−0.174987
$247$	19.1168	1.21638
$248$	−8.74456	−0.555280
$249$	−6.37228	−0.403827
$250$	0	0
$251$	−15.2554	−0.962915	−0.481457	−	0.876470i	$-0.659892\pi$
$251$	−15.2554	−0.962915	−0.481457	+	0.876470i	$0.659892\pi$
$252$	−2.37228	−0.149440
$253$	10.3723	0.652100
$254$	2.37228	0.148850
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.372281	−0.0232223	−0.0116111	−	0.999933i	$-0.503696\pi$
$257$	−0.372281	−0.0232223	−0.0116111	+	0.999933i	$0.503696\pi$
$258$	4.00000	0.249029
$259$	−2.37228	−0.147406
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	24.2337	1.49431	0.747157	−	0.664648i	$-0.231418\pi$
$263$	24.2337	1.49431	0.747157	+	0.664648i	$0.231418\pi$
$264$	4.37228	0.269095
$265$	0	0
$266$	−10.3723	−0.635965
$267$	7.62772	0.466809
$268$	8.74456	0.534159
$269$	6.37228	0.388525	0.194262	−	0.980950i	$-0.437769\pi$
$269$	6.37228	0.388525	0.194262	+	0.980950i	$0.437769\pi$
$270$	0	0
$271$	−9.48913	−0.576423	−0.288212	−	0.957567i	$-0.593061\pi$
$271$	−9.48913	−0.576423	−0.288212	+	0.957567i	$0.593061\pi$
$272$	−4.37228	−0.265108
$273$	−10.3723	−0.627759
$274$	0.744563	0.0449807
$275$	0	0
$276$	2.37228	0.142795
$277$	14.6060	0.877588	0.438794	−	0.898588i	$-0.355406\pi$
$277$	14.6060	0.877588	0.438794	+	0.898588i	$0.355406\pi$
$278$	−8.74456	−0.524464
$279$	−8.74456	−0.523523
$280$	0	0
$281$	−4.37228	−0.260828	−0.130414	−	0.991460i	$-0.541631\pi$
$281$	−4.37228	−0.260828	−0.130414	+	0.991460i	$0.541631\pi$
$282$	−6.74456	−0.401633
$283$	−28.6060	−1.70045	−0.850224	−	0.526421i	$-0.823534\pi$
$283$	−28.6060	−1.70045	−0.850224	+	0.526421i	$0.823534\pi$
$284$	12.7446	0.756251
$285$	0	0
$286$	19.1168	1.13040
$287$	6.51087	0.384325
$288$	1.00000	0.0589256
$289$	2.11684	0.124520
$290$	0	0
$291$	9.48913	0.556262
$292$	−13.1168	−0.767605
$293$	−13.1168	−0.766294	−0.383147	−	0.923687i	$-0.625160\pi$
$293$	−13.1168	−0.766294	−0.383147	+	0.923687i	$0.625160\pi$
$294$	−1.37228	−0.0800331
$295$	0	0
$296$	1.00000	0.0581238
$297$	4.37228	0.253705
$298$	20.7446	1.20170
$299$	10.3723	0.599845
$300$	0	0
$301$	−9.48913	−0.546944
$302$	2.37228	0.136509
$303$	−0.744563	−0.0427740
$304$	4.37228	0.250768
$305$	0	0
$306$	−4.37228	−0.249947
$307$	−0.744563	−0.0424944	−0.0212472	−	0.999774i	$-0.506764\pi$
$307$	−0.744563	−0.0424944	−0.0212472	+	0.999774i	$0.506764\pi$
$308$	−10.3723	−0.591016
$309$	−10.7446	−0.611237
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	4.37228	0.247532
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	−14.7446	−0.832084
$315$	0	0
$316$	4.00000	0.225018
$317$	−28.9783	−1.62758	−0.813790	−	0.581159i	$-0.802600\pi$
$317$	−28.9783	−1.62758	−0.813790	+	0.581159i	$0.802600\pi$
$318$	0.372281	0.0208765
$319$	26.2337	1.46880
$320$	0	0
$321$	2.37228	0.132408
$322$	−5.62772	−0.313621
$323$	−19.1168	−1.06369
$324$	1.00000	0.0555556
$325$	0	0
$326$	23.8614	1.32156
$327$	10.3723	0.573588
$328$	−2.74456	−0.151543
$329$	16.0000	0.882109
$330$	0	0
$331$	2.74456	0.150855	0.0754274	−	0.997151i	$-0.475968\pi$
$331$	2.74456	0.150855	0.0754274	+	0.997151i	$0.475968\pi$
$332$	−6.37228	−0.349724
$333$	1.00000	0.0547997
$334$	19.1168	1.04603
$335$	0	0
$336$	−2.37228	−0.129419
$337$	−13.1168	−0.714520	−0.357260	−	0.934005i	$-0.616289\pi$
$337$	−13.1168	−0.714520	−0.357260	+	0.934005i	$0.616289\pi$
$338$	6.11684	0.332712
$339$	3.48913	0.189503
$340$	0	0
$341$	−38.2337	−2.07047
$342$	4.37228	0.236426
$343$	19.8614	1.07242
$344$	4.00000	0.215666
$345$	0	0
$346$	−21.8614	−1.17528
$347$	21.4891	1.15360	0.576798	−	0.816887i	$-0.304302\pi$
$347$	21.4891	1.15360	0.576798	+	0.816887i	$0.304302\pi$
$348$	6.00000	0.321634
$349$	5.25544	0.281317	0.140659	−	0.990058i	$-0.455078\pi$
$349$	5.25544	0.281317	0.140659	+	0.990058i	$0.455078\pi$
$350$	0	0
$351$	4.37228	0.233375
$352$	4.37228	0.233043
$353$	19.4891	1.03730	0.518651	−	0.854986i	$-0.326435\pi$
$353$	19.4891	1.03730	0.518651	+	0.854986i	$0.326435\pi$
$354$	3.25544	0.173025
$355$	0	0
$356$	7.62772	0.404268
$357$	10.3723	0.548959
$358$	−17.4891	−0.924329
$359$	−36.4674	−1.92467	−0.962337	−	0.271858i	$-0.912362\pi$
$359$	−36.4674	−1.92467	−0.962337	+	0.271858i	$0.912362\pi$
$360$	0	0
$361$	0.116844	0.00614968
$362$	24.9783	1.31283
$363$	8.11684	0.426024
$364$	−10.3723	−0.543655
$365$	0	0
$366$	8.74456	0.457086
$367$	−26.3723	−1.37662	−0.688311	−	0.725416i	$-0.741648\pi$
$367$	−26.3723	−1.37662	−0.688311	+	0.725416i	$0.741648\pi$
$368$	2.37228	0.123664
$369$	−2.74456	−0.142876
$370$	0	0
$371$	−0.883156	−0.0458512
$372$	−8.74456	−0.453384
$373$	19.4891	1.00911	0.504554	−	0.863380i	$-0.331657\pi$
$373$	19.4891	1.00911	0.504554	+	0.863380i	$0.331657\pi$
$374$	−19.1168	−0.988508
$375$	0	0
$376$	−6.74456	−0.347824
$377$	26.2337	1.35110
$378$	−2.37228	−0.122017
$379$	−10.2337	−0.525669	−0.262835	−	0.964841i	$-0.584657\pi$
$379$	−10.2337	−0.525669	−0.262835	+	0.964841i	$0.584657\pi$
$380$	0	0
$381$	2.37228	0.121536
$382$	−13.6277	−0.697255
$383$	12.6060	0.644135	0.322067	−	0.946717i	$-0.395622\pi$
$383$	12.6060	0.644135	0.322067	+	0.946717i	$0.395622\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	12.7446	0.648681
$387$	4.00000	0.203331
$388$	9.48913	0.481737
$389$	0.510875	0.0259024	0.0129512	−	0.999916i	$-0.495877\pi$
$389$	0.510875	0.0259024	0.0129512	+	0.999916i	$0.495877\pi$
$390$	0	0
$391$	−10.3723	−0.524549
$392$	−1.37228	−0.0693107
$393$	0	0
$394$	−9.86141	−0.496810
$395$	0	0
$396$	4.37228	0.219715
$397$	−9.25544	−0.464517	−0.232259	−	0.972654i	$-0.574612\pi$
$397$	−9.25544	−0.464517	−0.232259	+	0.972654i	$0.574612\pi$
$398$	8.00000	0.401004
$399$	−10.3723	−0.519264
$400$	0	0
$401$	−18.8832	−0.942980	−0.471490	−	0.881871i	$-0.656284\pi$
$401$	−18.8832	−0.942980	−0.471490	+	0.881871i	$0.656284\pi$
$402$	8.74456	0.436139
$403$	−38.2337	−1.90456
$404$	−0.744563	−0.0370434
$405$	0	0
$406$	−14.2337	−0.706406
$407$	4.37228	0.216726
$408$	−4.37228	−0.216460
$409$	2.74456	0.135710	0.0678549	−	0.997695i	$-0.478384\pi$
$409$	2.74456	0.135710	0.0678549	+	0.997695i	$0.478384\pi$
$410$	0	0
$411$	0.744563	0.0367266
$412$	−10.7446	−0.529347
$413$	−7.72281	−0.380015
$414$	2.37228	0.116591
$415$	0	0
$416$	4.37228	0.214369
$417$	−8.74456	−0.428223
$418$	19.1168	0.935035
$419$	25.1168	1.22704	0.613519	−	0.789680i	$-0.289753\pi$
$419$	25.1168	1.22704	0.613519	+	0.789680i	$0.289753\pi$
$420$	0	0
$421$	−10.2337	−0.498759	−0.249380	−	0.968406i	$-0.580227\pi$
$421$	−10.2337	−0.498759	−0.249380	+	0.968406i	$0.580227\pi$
$422$	−20.7446	−1.00983
$423$	−6.74456	−0.327932
$424$	0.372281	0.0180796
$425$	0	0
$426$	12.7446	0.617476
$427$	−20.7446	−1.00390
$428$	2.37228	0.114669
$429$	19.1168	0.922970
$430$	0	0
$431$	8.60597	0.414535	0.207267	−	0.978284i	$-0.433543\pi$
$431$	8.60597	0.414535	0.207267	+	0.978284i	$0.433543\pi$
$432$	1.00000	0.0481125
$433$	−17.8614	−0.858364	−0.429182	−	0.903218i	$-0.641198\pi$
$433$	−17.8614	−0.858364	−0.429182	+	0.903218i	$0.641198\pi$
$434$	20.7446	0.995771
$435$	0	0
$436$	10.3723	0.496742
$437$	10.3723	0.496174
$438$	−13.1168	−0.626747
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	−1.37228	−0.0653467
$442$	−19.1168	−0.909296
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	1.00000	0.0474579
$445$	0	0
$446$	26.9783	1.27746
$447$	20.7446	0.981184
$448$	−2.37228	−0.112080
$449$	−8.97825	−0.423710	−0.211855	−	0.977301i	$-0.567950\pi$
$449$	−8.97825	−0.423710	−0.211855	+	0.977301i	$0.567950\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	3.48913	0.164115
$453$	2.37228	0.111459
$454$	−4.00000	−0.187729
$455$	0	0
$456$	4.37228	0.204751
$457$	−28.4674	−1.33165	−0.665824	−	0.746109i	$-0.731920\pi$
$457$	−28.4674	−1.33165	−0.665824	+	0.746109i	$0.731920\pi$
$458$	−6.00000	−0.280362
$459$	−4.37228	−0.204081
$460$	0	0
$461$	−15.4891	−0.721400	−0.360700	−	0.932682i	$-0.617462\pi$
$461$	−15.4891	−0.721400	−0.360700	+	0.932682i	$0.617462\pi$
$462$	−10.3723	−0.482562
$463$	−37.7228	−1.75313	−0.876564	−	0.481285i	$-0.840170\pi$
$463$	−37.7228	−1.75313	−0.876564	+	0.481285i	$0.840170\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	11.2554	0.521398
$467$	21.4891	0.994398	0.497199	−	0.867636i	$-0.334362\pi$
$467$	21.4891	0.994398	0.497199	+	0.867636i	$0.334362\pi$
$468$	4.37228	0.202109
$469$	−20.7446	−0.957895
$470$	0	0
$471$	−14.7446	−0.679394
$472$	3.25544	0.149844
$473$	17.4891	0.804151
$474$	4.00000	0.183726
$475$	0	0
$476$	10.3723	0.475413
$477$	0.372281	0.0170456
$478$	17.4891	0.799934
$479$	−0.883156	−0.0403524	−0.0201762	−	0.999796i	$-0.506423\pi$
$479$	−0.883156	−0.0403524	−0.0201762	+	0.999796i	$0.506423\pi$
$480$	0	0
$481$	4.37228	0.199359
$482$	−23.4891	−1.06990
$483$	−5.62772	−0.256070
$484$	8.11684	0.368947
$485$	0	0
$486$	1.00000	0.0453609
$487$	−38.4674	−1.74312	−0.871562	−	0.490286i	$-0.836892\pi$
$487$	−38.4674	−1.74312	−0.871562	+	0.490286i	$0.836892\pi$
$488$	8.74456	0.395848
$489$	23.8614	1.07905
$490$	0	0
$491$	8.37228	0.377836	0.188918	−	0.981993i	$-0.439502\pi$
$491$	8.37228	0.377836	0.188918	+	0.981993i	$0.439502\pi$
$492$	−2.74456	−0.123734
$493$	−26.2337	−1.18151
$494$	19.1168	0.860107
$495$	0	0
$496$	−8.74456	−0.392642
$497$	−30.2337	−1.35617
$498$	−6.37228	−0.285549
$499$	−31.3505	−1.40344	−0.701721	−	0.712452i	$-0.747585\pi$
$499$	−31.3505	−1.40344	−0.701721	+	0.712452i	$0.747585\pi$
$500$	0	0
$501$	19.1168	0.854078
$502$	−15.2554	−0.680883
$503$	17.4891	0.779802	0.389901	−	0.920857i	$-0.372509\pi$
$503$	17.4891	0.779802	0.389901	+	0.920857i	$0.372509\pi$
$504$	−2.37228	−0.105670
$505$	0	0
$506$	10.3723	0.461104
$507$	6.11684	0.271659
$508$	2.37228	0.105253
$509$	−33.3505	−1.47824	−0.739118	−	0.673576i	$-0.764758\pi$
$509$	−33.3505	−1.47824	−0.739118	+	0.673576i	$0.764758\pi$
$510$	0	0
$511$	31.1168	1.37653
$512$	1.00000	0.0441942
$513$	4.37228	0.193041
$514$	−0.372281	−0.0164206
$515$	0	0
$516$	4.00000	0.176090
$517$	−29.4891	−1.29693
$518$	−2.37228	−0.104232
$519$	−21.8614	−0.959609
$520$	0	0
$521$	−40.2337	−1.76267	−0.881335	−	0.472492i	$-0.843355\pi$
$521$	−40.2337	−1.76267	−0.881335	+	0.472492i	$0.843355\pi$
$522$	6.00000	0.262613
$523$	−38.9783	−1.70440	−0.852200	−	0.523216i	$-0.824732\pi$
$523$	−38.9783	−1.70440	−0.852200	+	0.523216i	$0.824732\pi$
$524$	0	0
$525$	0	0
$526$	24.2337	1.05664
$527$	38.2337	1.66549
$528$	4.37228	0.190279
$529$	−17.3723	−0.755317
$530$	0	0
$531$	3.25544	0.141274
$532$	−10.3723	−0.449695
$533$	−12.0000	−0.519778
$534$	7.62772	0.330084
$535$	0	0
$536$	8.74456	0.377708
$537$	−17.4891	−0.754711
$538$	6.37228	0.274729
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−23.8614	−1.02588	−0.512941	−	0.858424i	$-0.671444\pi$
$541$	−23.8614	−1.02588	−0.512941	+	0.858424i	$0.671444\pi$
$542$	−9.48913	−0.407593
$543$	24.9783	1.07192
$544$	−4.37228	−0.187460
$545$	0	0
$546$	−10.3723	−0.443893
$547$	−15.1168	−0.646350	−0.323175	−	0.946339i	$-0.604750\pi$
$547$	−15.1168	−0.646350	−0.323175	+	0.946339i	$0.604750\pi$
$548$	0.744563	0.0318061
$549$	8.74456	0.373209
$550$	0	0
$551$	26.2337	1.11759
$552$	2.37228	0.100971
$553$	−9.48913	−0.403519
$554$	14.6060	0.620548
$555$	0	0
$556$	−8.74456	−0.370852
$557$	−29.2554	−1.23959	−0.619796	−	0.784763i	$-0.712785\pi$
$557$	−29.2554	−1.23959	−0.619796	+	0.784763i	$0.712785\pi$
$558$	−8.74456	−0.370187
$559$	17.4891	0.739711
$560$	0	0
$561$	−19.1168	−0.807114
$562$	−4.37228	−0.184434
$563$	−24.7446	−1.04286	−0.521429	−	0.853294i	$-0.674601\pi$
$563$	−24.7446	−1.04286	−0.521429	+	0.853294i	$0.674601\pi$
$564$	−6.74456	−0.283997
$565$	0	0
$566$	−28.6060	−1.20240
$567$	−2.37228	−0.0996265
$568$	12.7446	0.534750
$569$	19.3505	0.811216	0.405608	−	0.914047i	$-0.367060\pi$
$569$	19.3505	0.811216	0.405608	+	0.914047i	$0.367060\pi$
$570$	0	0
$571$	−12.7446	−0.533343	−0.266672	−	0.963787i	$-0.585924\pi$
$571$	−12.7446	−0.533343	−0.266672	+	0.963787i	$0.585924\pi$
$572$	19.1168	0.799315
$573$	−13.6277	−0.569306
$574$	6.51087	0.271759
$575$	0	0
$576$	1.00000	0.0416667
$577$	−15.2554	−0.635092	−0.317546	−	0.948243i	$-0.602859\pi$
$577$	−15.2554	−0.635092	−0.317546	+	0.948243i	$0.602859\pi$
$578$	2.11684	0.0880491
$579$	12.7446	0.529646
$580$	0	0
$581$	15.1168	0.627152
$582$	9.48913	0.393337
$583$	1.62772	0.0674132
$584$	−13.1168	−0.542779
$585$	0	0
$586$	−13.1168	−0.541852
$587$	21.4891	0.886951	0.443476	−	0.896286i	$-0.353745\pi$
$587$	21.4891	0.886951	0.443476	+	0.896286i	$0.353745\pi$
$588$	−1.37228	−0.0565919
$589$	−38.2337	−1.57539
$590$	0	0
$591$	−9.86141	−0.405644
$592$	1.00000	0.0410997
$593$	−22.9783	−0.943604	−0.471802	−	0.881705i	$-0.656396\pi$
$593$	−22.9783	−0.943604	−0.471802	+	0.881705i	$0.656396\pi$
$594$	4.37228	0.179397
$595$	0	0
$596$	20.7446	0.849730
$597$	8.00000	0.327418
$598$	10.3723	0.424154
$599$	22.9783	0.938866	0.469433	−	0.882968i	$-0.344458\pi$
$599$	22.9783	0.938866	0.469433	+	0.882968i	$0.344458\pi$
$600$	0	0
$601$	2.13859	0.0872350	0.0436175	−	0.999048i	$-0.486112\pi$
$601$	2.13859	0.0872350	0.0436175	+	0.999048i	$0.486112\pi$
$602$	−9.48913	−0.386748
$603$	8.74456	0.356106
$604$	2.37228	0.0965268
$605$	0	0
$606$	−0.744563	−0.0302458
$607$	30.4674	1.23663	0.618316	−	0.785929i	$-0.287815\pi$
$607$	30.4674	1.23663	0.618316	+	0.785929i	$0.287815\pi$
$608$	4.37228	0.177319
$609$	−14.2337	−0.576778
$610$	0	0
$611$	−29.4891	−1.19300
$612$	−4.37228	−0.176739
$613$	38.4674	1.55368	0.776841	−	0.629696i	$-0.216821\pi$
$613$	38.4674	1.55368	0.776841	+	0.629696i	$0.216821\pi$
$614$	−0.744563	−0.0300481
$615$	0	0
$616$	−10.3723	−0.417911
$617$	2.23369	0.0899249	0.0449624	−	0.998989i	$-0.485683\pi$
$617$	2.23369	0.0899249	0.0449624	+	0.998989i	$0.485683\pi$
$618$	−10.7446	−0.432210
$619$	18.9783	0.762800	0.381400	−	0.924410i	$-0.375442\pi$
$619$	18.9783	0.762800	0.381400	+	0.924410i	$0.375442\pi$
$620$	0	0
$621$	2.37228	0.0951964
$622$	−8.00000	−0.320771
$623$	−18.0951	−0.724965
$624$	4.37228	0.175031
$625$	0	0
$626$	−8.00000	−0.319744
$627$	19.1168	0.763453
$628$	−14.7446	−0.588372
$629$	−4.37228	−0.174334
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	4.00000	0.159111
$633$	−20.7446	−0.824522
$634$	−28.9783	−1.15087
$635$	0	0
$636$	0.372281	0.0147619
$637$	−6.00000	−0.237729
$638$	26.2337	1.03860
$639$	12.7446	0.504167
$640$	0	0
$641$	−13.7228	−0.542019	−0.271009	−	0.962577i	$-0.587357\pi$
$641$	−13.7228	−0.542019	−0.271009	+	0.962577i	$0.587357\pi$
$642$	2.37228	0.0936265
$643$	16.6060	0.654876	0.327438	−	0.944873i	$-0.393815\pi$
$643$	16.6060	0.654876	0.327438	+	0.944873i	$0.393815\pi$
$644$	−5.62772	−0.221763
$645$	0	0
$646$	−19.1168	−0.752142
$647$	−0.605969	−0.0238231	−0.0119116	−	0.999929i	$-0.503792\pi$
$647$	−0.605969	−0.0238231	−0.0119116	+	0.999929i	$0.503792\pi$
$648$	1.00000	0.0392837
$649$	14.2337	0.558721
$650$	0	0
$651$	20.7446	0.813044
$652$	23.8614	0.934485
$653$	37.7228	1.47621	0.738104	−	0.674687i	$-0.235721\pi$
$653$	37.7228	1.47621	0.738104	+	0.674687i	$0.235721\pi$
$654$	10.3723	0.405588
$655$	0	0
$656$	−2.74456	−0.107157
$657$	−13.1168	−0.511737
$658$	16.0000	0.623745
$659$	32.2337	1.25565	0.627823	−	0.778356i	$-0.283946\pi$
$659$	32.2337	1.25565	0.627823	+	0.778356i	$0.283946\pi$
$660$	0	0
$661$	−8.88316	−0.345515	−0.172757	−	0.984964i	$-0.555268\pi$
$661$	−8.88316	−0.345515	−0.172757	+	0.984964i	$0.555268\pi$
$662$	2.74456	0.106670
$663$	−19.1168	−0.742437
$664$	−6.37228	−0.247292
$665$	0	0
$666$	1.00000	0.0387492
$667$	14.2337	0.551131
$668$	19.1168	0.739653
$669$	26.9783	1.04304
$670$	0	0
$671$	38.2337	1.47600
$672$	−2.37228	−0.0915127
$673$	25.8614	0.996884	0.498442	−	0.866923i	$-0.333906\pi$
$673$	25.8614	0.996884	0.498442	+	0.866923i	$0.333906\pi$
$674$	−13.1168	−0.505242
$675$	0	0
$676$	6.11684	0.235263
$677$	−25.8614	−0.993935	−0.496967	−	0.867769i	$-0.665553\pi$
$677$	−25.8614	−0.993935	−0.496967	+	0.867769i	$0.665553\pi$
$678$	3.48913	0.133999
$679$	−22.5109	−0.863888
$680$	0	0
$681$	−4.00000	−0.153280
$682$	−38.2337	−1.46404
$683$	14.9783	0.573127	0.286563	−	0.958061i	$-0.407487\pi$
$683$	14.9783	0.573127	0.286563	+	0.958061i	$0.407487\pi$
$684$	4.37228	0.167178
$685$	0	0
$686$	19.8614	0.758312
$687$	−6.00000	−0.228914
$688$	4.00000	0.152499
$689$	1.62772	0.0620111
$690$	0	0
$691$	11.2554	0.428177	0.214089	−	0.976814i	$-0.431322\pi$
$691$	11.2554	0.428177	0.214089	+	0.976814i	$0.431322\pi$
$692$	−21.8614	−0.831046
$693$	−10.3723	−0.394010
$694$	21.4891	0.815716
$695$	0	0
$696$	6.00000	0.227429
$697$	12.0000	0.454532
$698$	5.25544	0.198921
$699$	11.2554	0.425720
$700$	0	0
$701$	9.72281	0.367226	0.183613	−	0.982999i	$-0.441221\pi$
$701$	9.72281	0.367226	0.183613	+	0.982999i	$0.441221\pi$
$702$	4.37228	0.165021
$703$	4.37228	0.164904
$704$	4.37228	0.164787
$705$	0	0
$706$	19.4891	0.733483
$707$	1.76631	0.0664290
$708$	3.25544	0.122347
$709$	−26.3723	−0.990432	−0.495216	−	0.868770i	$-0.664911\pi$
$709$	−26.3723	−0.990432	−0.495216	+	0.868770i	$0.664911\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	7.62772	0.285861
$713$	−20.7446	−0.776890
$714$	10.3723	0.388173
$715$	0	0
$716$	−17.4891	−0.653599
$717$	17.4891	0.653143
$718$	−36.4674	−1.36095
$719$	43.7228	1.63059	0.815293	−	0.579049i	$-0.196576\pi$
$719$	43.7228	1.63059	0.815293	+	0.579049i	$0.196576\pi$
$720$	0	0
$721$	25.4891	0.949265
$722$	0.116844	0.00434848
$723$	−23.4891	−0.873570
$724$	24.9783	0.928309
$725$	0	0
$726$	8.11684	0.301244
$727$	36.9783	1.37145	0.685724	−	0.727862i	$-0.259486\pi$
$727$	36.9783	1.37145	0.685724	+	0.727862i	$0.259486\pi$
$728$	−10.3723	−0.384422
$729$	1.00000	0.0370370
$730$	0	0
$731$	−17.4891	−0.646859
$732$	8.74456	0.323208
$733$	19.4891	0.719847	0.359924	−	0.932982i	$-0.382803\pi$
$733$	19.4891	0.719847	0.359924	+	0.932982i	$0.382803\pi$
$734$	−26.3723	−0.973419
$735$	0	0
$736$	2.37228	0.0874434
$737$	38.2337	1.40836
$738$	−2.74456	−0.101029
$739$	−31.7228	−1.16694	−0.583471	−	0.812134i	$-0.698306\pi$
$739$	−31.7228	−1.16694	−0.583471	+	0.812134i	$0.698306\pi$
$740$	0	0
$741$	19.1168	0.702275
$742$	−0.883156	−0.0324217
$743$	46.4674	1.70472	0.852361	−	0.522953i	$-0.175170\pi$
$743$	46.4674	1.70472	0.852361	+	0.522953i	$0.175170\pi$
$744$	−8.74456	−0.320591
$745$	0	0
$746$	19.4891	0.713548
$747$	−6.37228	−0.233150
$748$	−19.1168	−0.698981
$749$	−5.62772	−0.205632
$750$	0	0
$751$	−33.4891	−1.22204	−0.611018	−	0.791617i	$-0.709240\pi$
$751$	−33.4891	−1.22204	−0.611018	+	0.791617i	$0.709240\pi$
$752$	−6.74456	−0.245949
$753$	−15.2554	−0.555939
$754$	26.2337	0.955375
$755$	0	0
$756$	−2.37228	−0.0862790
$757$	−26.8832	−0.977085	−0.488542	−	0.872540i	$-0.662471\pi$
$757$	−26.8832	−0.977085	−0.488542	+	0.872540i	$0.662471\pi$
$758$	−10.2337	−0.371704
$759$	10.3723	0.376490
$760$	0	0
$761$	−35.4891	−1.28648	−0.643240	−	0.765665i	$-0.722410\pi$
$761$	−35.4891	−1.28648	−0.643240	+	0.765665i	$0.722410\pi$
$762$	2.37228	0.0859387
$763$	−24.6060	−0.890796
$764$	−13.6277	−0.493034
$765$	0	0
$766$	12.6060	0.455472
$767$	14.2337	0.513949
$768$	1.00000	0.0360844
$769$	−28.5109	−1.02813	−0.514064	−	0.857752i	$-0.671861\pi$
$769$	−28.5109	−1.02813	−0.514064	+	0.857752i	$0.671861\pi$
$770$	0	0
$771$	−0.372281	−0.0134074
$772$	12.7446	0.458687
$773$	−2.88316	−0.103700	−0.0518500	−	0.998655i	$-0.516512\pi$
$773$	−2.88316	−0.103700	−0.0518500	+	0.998655i	$0.516512\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	9.48913	0.340640
$777$	−2.37228	−0.0851051
$778$	0.510875	0.0183157
$779$	−12.0000	−0.429945
$780$	0	0
$781$	55.7228	1.99392
$782$	−10.3723	−0.370912
$783$	6.00000	0.214423
$784$	−1.37228	−0.0490100
$785$	0	0
$786$	0	0
$787$	0.744563	0.0265408	0.0132704	−	0.999912i	$-0.495776\pi$
$787$	0.744563	0.0265408	0.0132704	+	0.999912i	$0.495776\pi$
$788$	−9.86141	−0.351298
$789$	24.2337	0.862742
$790$	0	0
$791$	−8.27719	−0.294303
$792$	4.37228	0.155362
$793$	38.2337	1.35772
$794$	−9.25544	−0.328463
$795$	0	0
$796$	8.00000	0.283552
$797$	−27.4891	−0.973715	−0.486857	−	0.873481i	$-0.661857\pi$
$797$	−27.4891	−0.973715	−0.486857	+	0.873481i	$0.661857\pi$
$798$	−10.3723	−0.367175
$799$	29.4891	1.04325
$800$	0	0
$801$	7.62772	0.269512
$802$	−18.8832	−0.666787
$803$	−57.3505	−2.02386
$804$	8.74456	0.308397
$805$	0	0
$806$	−38.2337	−1.34672
$807$	6.37228	0.224315
$808$	−0.744563	−0.0261936
$809$	−6.88316	−0.241999	−0.120999	−	0.992653i	$-0.538610\pi$
$809$	−6.88316	−0.241999	−0.120999	+	0.992653i	$0.538610\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−14.2337	−0.499505
$813$	−9.48913	−0.332798
$814$	4.37228	0.153248
$815$	0	0
$816$	−4.37228	−0.153060
$817$	17.4891	0.611867
$818$	2.74456	0.0959614
$819$	−10.3723	−0.362437
$820$	0	0
$821$	−25.3505	−0.884740	−0.442370	−	0.896833i	$-0.645862\pi$
$821$	−25.3505	−0.884740	−0.442370	+	0.896833i	$0.645862\pi$
$822$	0.744563	0.0259696
$823$	−43.1168	−1.50296	−0.751479	−	0.659757i	$-0.770659\pi$
$823$	−43.1168	−1.50296	−0.751479	+	0.659757i	$0.770659\pi$
$824$	−10.7446	−0.374305
$825$	0	0
$826$	−7.72281	−0.268711
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	2.37228	0.0824425
$829$	38.8397	1.34896	0.674479	−	0.738294i	$-0.264369\pi$
$829$	38.8397	1.34896	0.674479	+	0.738294i	$0.264369\pi$
$830$	0	0
$831$	14.6060	0.506675
$832$	4.37228	0.151582
$833$	6.00000	0.207888
$834$	−8.74456	−0.302799
$835$	0	0
$836$	19.1168	0.661170
$837$	−8.74456	−0.302256
$838$	25.1168	0.867647
$839$	−20.7446	−0.716182	−0.358091	−	0.933687i	$-0.616572\pi$
$839$	−20.7446	−0.716182	−0.358091	+	0.933687i	$0.616572\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−10.2337	−0.352676
$843$	−4.37228	−0.150589
$844$	−20.7446	−0.714057
$845$	0	0
$846$	−6.74456	−0.231883
$847$	−19.2554	−0.661625
$848$	0.372281	0.0127842
$849$	−28.6060	−0.981754
$850$	0	0
$851$	2.37228	0.0813208
$852$	12.7446	0.436622
$853$	11.3505	0.388635	0.194317	−	0.980939i	$-0.437751\pi$
$853$	11.3505	0.388635	0.194317	+	0.980939i	$0.437751\pi$
$854$	−20.7446	−0.709864
$855$	0	0
$856$	2.37228	0.0810829
$857$	56.8397	1.94161	0.970803	−	0.239879i	$-0.0771077\pi$
$857$	56.8397	1.94161	0.970803	+	0.239879i	$0.0771077\pi$
$858$	19.1168	0.652638
$859$	−56.8397	−1.93934	−0.969672	−	0.244410i	$-0.921406\pi$
$859$	−56.8397	−1.93934	−0.969672	+	0.244410i	$0.921406\pi$
$860$	0	0
$861$	6.51087	0.221890
$862$	8.60597	0.293120
$863$	−14.7446	−0.501911	−0.250955	−	0.967999i	$-0.580745\pi$
$863$	−14.7446	−0.501911	−0.250955	+	0.967999i	$0.580745\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−17.8614	−0.606955
$867$	2.11684	0.0718918
$868$	20.7446	0.704116
$869$	17.4891	0.593278
$870$	0	0
$871$	38.2337	1.29550
$872$	10.3723	0.351250
$873$	9.48913	0.321158
$874$	10.3723	0.350848
$875$	0	0
$876$	−13.1168	−0.443177
$877$	19.7663	0.667461	0.333730	−	0.942669i	$-0.391692\pi$
$877$	19.7663	0.667461	0.333730	+	0.942669i	$0.391692\pi$
$878$	−32.0000	−1.07995
$879$	−13.1168	−0.442420
$880$	0	0
$881$	20.2337	0.681690	0.340845	−	0.940119i	$-0.389287\pi$
$881$	20.2337	0.681690	0.340845	+	0.940119i	$0.389287\pi$
$882$	−1.37228	−0.0462071
$883$	32.8832	1.10661	0.553303	−	0.832980i	$-0.313367\pi$
$883$	32.8832	1.10661	0.553303	+	0.832980i	$0.313367\pi$
$884$	−19.1168	−0.642969
$885$	0	0
$886$	4.00000	0.134383
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	1.00000	0.0335578
$889$	−5.62772	−0.188748
$890$	0	0
$891$	4.37228	0.146477
$892$	26.9783	0.903299
$893$	−29.4891	−0.986816
$894$	20.7446	0.693802
$895$	0	0
$896$	−2.37228	−0.0792524
$897$	10.3723	0.346320
$898$	−8.97825	−0.299608
$899$	−52.4674	−1.74988
$900$	0	0
$901$	−1.62772	−0.0542272
$902$	−12.0000	−0.399556
$903$	−9.48913	−0.315778
$904$	3.48913	0.116047
$905$	0	0
$906$	2.37228	0.0788138
$907$	−36.6060	−1.21548	−0.607741	−	0.794136i	$-0.707924\pi$
$907$	−36.6060	−1.21548	−0.607741	+	0.794136i	$0.707924\pi$
$908$	−4.00000	−0.132745
$909$	−0.744563	−0.0246956
$910$	0	0
$911$	−10.9783	−0.363726	−0.181863	−	0.983324i	$-0.558213\pi$
$911$	−10.9783	−0.363726	−0.181863	+	0.983324i	$0.558213\pi$
$912$	4.37228	0.144781
$913$	−27.8614	−0.922078
$914$	−28.4674	−0.941617
$915$	0	0
$916$	−6.00000	−0.198246
$917$	0	0
$918$	−4.37228	−0.144307
$919$	−57.4891	−1.89639	−0.948196	−	0.317687i	$-0.897094\pi$
$919$	−57.4891	−1.89639	−0.948196	+	0.317687i	$0.897094\pi$
$920$	0	0
$921$	−0.744563	−0.0245342
$922$	−15.4891	−0.510107
$923$	55.7228	1.83414
$924$	−10.3723	−0.341223
$925$	0	0
$926$	−37.7228	−1.23965
$927$	−10.7446	−0.352898
$928$	6.00000	0.196960
$929$	−35.4891	−1.16436	−0.582180	−	0.813060i	$-0.697800\pi$
$929$	−35.4891	−1.16436	−0.582180	+	0.813060i	$0.697800\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	11.2554	0.368684
$933$	−8.00000	−0.261908
$934$	21.4891	0.703146
$935$	0	0
$936$	4.37228	0.142912
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	−20.7446	−0.677334
$939$	−8.00000	−0.261070
$940$	0	0
$941$	50.2337	1.63757	0.818786	−	0.574099i	$-0.194648\pi$
$941$	50.2337	1.63757	0.818786	+	0.574099i	$0.194648\pi$
$942$	−14.7446	−0.480404
$943$	−6.51087	−0.212023
$944$	3.25544	0.105955
$945$	0	0
$946$	17.4891	0.568621
$947$	−40.4674	−1.31501	−0.657507	−	0.753449i	$-0.728389\pi$
$947$	−40.4674	−1.31501	−0.657507	+	0.753449i	$0.728389\pi$
$948$	4.00000	0.129914
$949$	−57.3505	−1.86168
$950$	0	0
$951$	−28.9783	−0.939684
$952$	10.3723	0.336168
$953$	39.2554	1.27161	0.635804	−	0.771850i	$-0.280669\pi$
$953$	39.2554	1.27161	0.635804	+	0.771850i	$0.280669\pi$
$954$	0.372281	0.0120531
$955$	0	0
$956$	17.4891	0.565639
$957$	26.2337	0.848015
$958$	−0.883156	−0.0285335
$959$	−1.76631	−0.0570372
$960$	0	0
$961$	45.4674	1.46669
$962$	4.37228	0.140968
$963$	2.37228	0.0764457
$964$	−23.4891	−0.756534
$965$	0	0
$966$	−5.62772	−0.181069
$967$	−28.9783	−0.931878	−0.465939	−	0.884817i	$-0.654283\pi$
$967$	−28.9783	−0.931878	−0.465939	+	0.884817i	$0.654283\pi$
$968$	8.11684	0.260885
$969$	−19.1168	−0.614122
$970$	0	0
$971$	15.7663	0.505965	0.252983	−	0.967471i	$-0.418588\pi$
$971$	15.7663	0.505965	0.252983	+	0.967471i	$0.418588\pi$
$972$	1.00000	0.0320750
$973$	20.7446	0.665040
$974$	−38.4674	−1.23257
$975$	0	0
$976$	8.74456	0.279907
$977$	−53.8614	−1.72318	−0.861590	−	0.507606i	$-0.830531\pi$
$977$	−53.8614	−1.72318	−0.861590	+	0.507606i	$0.830531\pi$
$978$	23.8614	0.763004
$979$	33.3505	1.06589
$980$	0	0
$981$	10.3723	0.331161
$982$	8.37228	0.267170
$983$	35.4891	1.13193	0.565964	−	0.824430i	$-0.308504\pi$
$983$	35.4891	1.13193	0.565964	+	0.824430i	$0.308504\pi$
$984$	−2.74456	−0.0874935
$985$	0	0
$986$	−26.2337	−0.835451
$987$	16.0000	0.509286
$988$	19.1168	0.608188
$989$	9.48913	0.301737
$990$	0	0
$991$	−22.9783	−0.729928	−0.364964	−	0.931022i	$-0.618919\pi$
$991$	−22.9783	−0.729928	−0.364964	+	0.931022i	$0.618919\pi$
$992$	−8.74456	−0.277640
$993$	2.74456	0.0870961
$994$	−30.2337	−0.958954
$995$	0	0
$996$	−6.37228	−0.201913
$997$	1.11684	0.0353708	0.0176854	−	0.999844i	$-0.494370\pi$
$997$	1.11684	0.0353708	0.0176854	+	0.999844i	$0.494370\pi$
$998$	−31.3505	−0.992384
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5550.2.a.ca.1.1		2
5.4	even	2		1110.2.a.p.1.2	✓	2
15.14	odd	2		3330.2.a.be.1.2		2
20.19	odd	2		8880.2.a.br.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.p.1.2	✓	2	5.4	even	2
3330.2.a.be.1.2		2	15.14	odd	2
5550.2.a.ca.1.1		2	1.1	even	1	trivial
8880.2.a.br.1.1		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.37228 q^{11} +1.00000 q^{12} +4.37228 q^{13} -2.37228 q^{14} +1.00000 q^{16} -4.37228 q^{17} +1.00000 q^{18} +4.37228 q^{19} -2.37228 q^{21} +4.37228 q^{22} +2.37228 q^{23} +1.00000 q^{24} +4.37228 q^{26} +1.00000 q^{27} -2.37228 q^{28} +6.00000 q^{29} -8.74456 q^{31} +1.00000 q^{32} +4.37228 q^{33} -4.37228 q^{34} +1.00000 q^{36} +1.00000 q^{37} +4.37228 q^{38} +4.37228 q^{39} -2.74456 q^{41} -2.37228 q^{42} +4.00000 q^{43} +4.37228 q^{44} +2.37228 q^{46} -6.74456 q^{47} +1.00000 q^{48} -1.37228 q^{49} -4.37228 q^{51} +4.37228 q^{52} +0.372281 q^{53} +1.00000 q^{54} -2.37228 q^{56} +4.37228 q^{57} +6.00000 q^{58} +3.25544 q^{59} +8.74456 q^{61} -8.74456 q^{62} -2.37228 q^{63} +1.00000 q^{64} +4.37228 q^{66} +8.74456 q^{67} -4.37228 q^{68} +2.37228 q^{69} +12.7446 q^{71} +1.00000 q^{72} -13.1168 q^{73} +1.00000 q^{74} +4.37228 q^{76} -10.3723 q^{77} +4.37228 q^{78} +4.00000 q^{79} +1.00000 q^{81} -2.74456 q^{82} -6.37228 q^{83} -2.37228 q^{84} +4.00000 q^{86} +6.00000 q^{87} +4.37228 q^{88} +7.62772 q^{89} -10.3723 q^{91} +2.37228 q^{92} -8.74456 q^{93} -6.74456 q^{94} +1.00000 q^{96} +9.48913 q^{97} -1.37228 q^{98} +4.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{11} + 2 q^{12} + 3 q^{13} + q^{14} + 2 q^{16} - 3 q^{17} + 2 q^{18} + 3 q^{19} + q^{21} + 3 q^{22} - q^{23} + 2 q^{24}+ \cdots + 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + q^7 + 2 * q^8 + 2 * q^9 + 3 * q^11 + 2 * q^12 + 3 * q^13 + q^14 + 2 * q^16 - 3 * q^17 + 2 * q^18 + 3 * q^19 + q^21 + 3 * q^22 - q^23 + 2 * q^24 + 3 * q^26 + 2 * q^27 + q^28 + 12 * q^29 - 6 * q^31 + 2 * q^32 + 3 * q^33 - 3 * q^34 + 2 * q^36 + 2 * q^37 + 3 * q^38 + 3 * q^39 + 6 * q^41 + q^42 + 8 * q^43 + 3 * q^44 - q^46 - 2 * q^47 + 2 * q^48 + 3 * q^49 - 3 * q^51 + 3 * q^52 - 5 * q^53 + 2 * q^54 + q^56 + 3 * q^57 + 12 * q^58 + 18 * q^59 + 6 * q^61 - 6 * q^62 + q^63 + 2 * q^64 + 3 * q^66 + 6 * q^67 - 3 * q^68 - q^69 + 14 * q^71 + 2 * q^72 - 9 * q^73 + 2 * q^74 + 3 * q^76 - 15 * q^77 + 3 * q^78 + 8 * q^79 + 2 * q^81 + 6 * q^82 - 7 * q^83 + q^84 + 8 * q^86 + 12 * q^87 + 3 * q^88 + 21 * q^89 - 15 * q^91 - q^92 - 6 * q^93 - 2 * q^94 + 2 * q^96 - 4 * q^97 + 3 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more