LMFDB - Embedded newform 5550.2.a.ca.1.2

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			1.2
Root			$3.37228$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.37228 q^{11} +1.00000 q^{12} -1.37228 q^{13} +3.37228 q^{14} +1.00000 q^{16} +1.37228 q^{17} +1.00000 q^{18} -1.37228 q^{19} +3.37228 q^{21} -1.37228 q^{22} -3.37228 q^{23} +1.00000 q^{24} -1.37228 q^{26} +1.00000 q^{27} +3.37228 q^{28} +6.00000 q^{29} +2.74456 q^{31} +1.00000 q^{32} -1.37228 q^{33} +1.37228 q^{34} +1.00000 q^{36} +1.00000 q^{37} -1.37228 q^{38} -1.37228 q^{39} +8.74456 q^{41} +3.37228 q^{42} +4.00000 q^{43} -1.37228 q^{44} -3.37228 q^{46} +4.74456 q^{47} +1.00000 q^{48} +4.37228 q^{49} +1.37228 q^{51} -1.37228 q^{52} -5.37228 q^{53} +1.00000 q^{54} +3.37228 q^{56} -1.37228 q^{57} +6.00000 q^{58} +14.7446 q^{59} -2.74456 q^{61} +2.74456 q^{62} +3.37228 q^{63} +1.00000 q^{64} -1.37228 q^{66} -2.74456 q^{67} +1.37228 q^{68} -3.37228 q^{69} +1.25544 q^{71} +1.00000 q^{72} +4.11684 q^{73} +1.00000 q^{74} -1.37228 q^{76} -4.62772 q^{77} -1.37228 q^{78} +4.00000 q^{79} +1.00000 q^{81} +8.74456 q^{82} -0.627719 q^{83} +3.37228 q^{84} +4.00000 q^{86} +6.00000 q^{87} -1.37228 q^{88} +13.3723 q^{89} -4.62772 q^{91} -3.37228 q^{92} +2.74456 q^{93} +4.74456 q^{94} +1.00000 q^{96} -13.4891 q^{97} +4.37228 q^{98} -1.37228 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	3.37228	1.27460	0.637301	−	0.770615i	$-0.280051\pi$
$7$	3.37228	1.27460	0.637301	+	0.770615i	$0.280051\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.37228	−0.413758	−0.206879	−	0.978366i	$-0.566331\pi$
$11$	−1.37228	−0.413758	−0.206879	+	0.978366i	$0.566331\pi$
$12$	1.00000	0.288675
$13$	−1.37228	−0.380602	−0.190301	−	0.981726i	$-0.560946\pi$
$13$	−1.37228	−0.380602	−0.190301	+	0.981726i	$0.560946\pi$
$14$	3.37228	0.901280
$15$	0	0
$16$	1.00000	0.250000
$17$	1.37228	0.332827	0.166414	−	0.986056i	$-0.446781\pi$
$17$	1.37228	0.332827	0.166414	+	0.986056i	$0.446781\pi$
$18$	1.00000	0.235702
$19$	−1.37228	−0.314823	−0.157411	−	0.987533i	$-0.550315\pi$
$19$	−1.37228	−0.314823	−0.157411	+	0.987533i	$0.550315\pi$
$20$	0	0
$21$	3.37228	0.735892
$22$	−1.37228	−0.292571
$23$	−3.37228	−0.703169	−0.351585	−	0.936156i	$-0.614357\pi$
$23$	−3.37228	−0.703169	−0.351585	+	0.936156i	$0.614357\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−1.37228	−0.269127
$27$	1.00000	0.192450
$28$	3.37228	0.637301
$29$	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
$31$	2.74456	0.492938	0.246469	−	0.969151i	$-0.420730\pi$
$31$	2.74456	0.492938	0.246469	+	0.969151i	$0.420730\pi$
$32$	1.00000	0.176777
$33$	−1.37228	−0.238884
$34$	1.37228	0.235344
$35$	0	0
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−1.37228	−0.222613
$39$	−1.37228	−0.219741
$40$	0	0
$41$	8.74456	1.36567	0.682836	−	0.730572i	$-0.260747\pi$
$41$	8.74456	1.36567	0.682836	+	0.730572i	$0.260747\pi$
$42$	3.37228	0.520354
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−1.37228	−0.206879
$45$	0	0
$46$	−3.37228	−0.497216
$47$	4.74456	0.692066	0.346033	−	0.938222i	$-0.387529\pi$
$47$	4.74456	0.692066	0.346033	+	0.938222i	$0.387529\pi$
$48$	1.00000	0.144338
$49$	4.37228	0.624612
$50$	0	0
$51$	1.37228	0.192158
$52$	−1.37228	−0.190301
$53$	−5.37228	−0.737940	−0.368970	−	0.929441i	$-0.620289\pi$
$53$	−5.37228	−0.737940	−0.368970	+	0.929441i	$0.620289\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	3.37228	0.450640
$57$	−1.37228	−0.181763
$58$	6.00000	0.787839
$59$	14.7446	1.91958	0.959789	−	0.280721i	$-0.0905737\pi$
$59$	14.7446	1.91958	0.959789	+	0.280721i	$0.0905737\pi$
$60$	0	0
$61$	−2.74456	−0.351405	−0.175703	−	0.984443i	$-0.556220\pi$
$61$	−2.74456	−0.351405	−0.175703	+	0.984443i	$0.556220\pi$
$62$	2.74456	0.348560
$63$	3.37228	0.424868
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.37228	−0.168916
$67$	−2.74456	−0.335302	−0.167651	−	0.985846i	$-0.553618\pi$
$67$	−2.74456	−0.335302	−0.167651	+	0.985846i	$0.553618\pi$
$68$	1.37228	0.166414
$69$	−3.37228	−0.405975
$70$	0	0
$71$	1.25544	0.148993	0.0744965	−	0.997221i	$-0.476265\pi$
$71$	1.25544	0.148993	0.0744965	+	0.997221i	$0.476265\pi$
$72$	1.00000	0.117851
$73$	4.11684	0.481840	0.240920	−	0.970545i	$-0.422551\pi$
$73$	4.11684	0.481840	0.240920	+	0.970545i	$0.422551\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−1.37228	−0.157411
$77$	−4.62772	−0.527377
$78$	−1.37228	−0.155380
$79$	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
$82$	8.74456	0.965675
$83$	−0.627719	−0.0689011	−0.0344505	−	0.999406i	$-0.510968\pi$
$83$	−0.627719	−0.0689011	−0.0344505	+	0.999406i	$0.510968\pi$
$84$	3.37228	0.367946
$85$	0	0
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	−1.37228	−0.146286
$89$	13.3723	1.41746	0.708729	−	0.705480i	$-0.249269\pi$
$89$	13.3723	1.41746	0.708729	+	0.705480i	$0.249269\pi$
$90$	0	0
$91$	−4.62772	−0.485117
$92$	−3.37228	−0.351585
$93$	2.74456	0.284598
$94$	4.74456	0.489364
$95$	0	0
$96$	1.00000	0.102062
$97$	−13.4891	−1.36961	−0.684807	−	0.728725i	$-0.740113\pi$
$97$	−13.4891	−1.36961	−0.684807	+	0.728725i	$0.740113\pi$
$98$	4.37228	0.441667
$99$	−1.37228	−0.137919
$100$	0	0
$101$	10.7446	1.06912	0.534562	−	0.845129i	$-0.320477\pi$
$101$	10.7446	1.06912	0.534562	+	0.845129i	$0.320477\pi$
$102$	1.37228	0.135876
$103$	0.744563	0.0733639	0.0366820	−	0.999327i	$-0.488321\pi$
$103$	0.744563	0.0733639	0.0366820	+	0.999327i	$0.488321\pi$
$104$	−1.37228	−0.134563
$105$	0	0
$106$	−5.37228	−0.521802
$107$	−3.37228	−0.326011	−0.163005	−	0.986625i	$-0.552119\pi$
$107$	−3.37228	−0.326011	−0.163005	+	0.986625i	$0.552119\pi$
$108$	1.00000	0.0962250
$109$	4.62772	0.443255	0.221628	−	0.975131i	$-0.428863\pi$
$109$	4.62772	0.443255	0.221628	+	0.975131i	$0.428863\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	3.37228	0.318651
$113$	−19.4891	−1.83338	−0.916691	−	0.399596i	$-0.869150\pi$
$113$	−19.4891	−1.83338	−0.916691	+	0.399596i	$0.869150\pi$
$114$	−1.37228	−0.128526
$115$	0	0
$116$	6.00000	0.557086
$117$	−1.37228	−0.126867
$118$	14.7446	1.35735
$119$	4.62772	0.424222
$120$	0	0
$121$	−9.11684	−0.828804
$122$	−2.74456	−0.248481
$123$	8.74456	0.788471
$124$	2.74456	0.246469
$125$	0	0
$126$	3.37228	0.300427
$127$	−3.37228	−0.299242	−0.149621	−	0.988743i	$-0.547805\pi$
$127$	−3.37228	−0.299242	−0.149621	+	0.988743i	$0.547805\pi$
$128$	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	$0.5\pi$
$132$	−1.37228	−0.119442
$133$	−4.62772	−0.401274
$134$	−2.74456	−0.237094
$135$	0	0
$136$	1.37228	0.117672
$137$	−10.7446	−0.917970	−0.458985	−	0.888444i	$-0.651787\pi$
$137$	−10.7446	−0.917970	−0.458985	+	0.888444i	$0.651787\pi$
$138$	−3.37228	−0.287068
$139$	2.74456	0.232791	0.116395	−	0.993203i	$-0.462866\pi$
$139$	2.74456	0.232791	0.116395	+	0.993203i	$0.462866\pi$
$140$	0	0
$141$	4.74456	0.399564
$142$	1.25544	0.105354
$143$	1.88316	0.157477
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.11684	0.340712
$147$	4.37228	0.360620
$148$	1.00000	0.0821995
$149$	9.25544	0.758235	0.379117	−	0.925349i	$-0.376228\pi$
$149$	9.25544	0.758235	0.379117	+	0.925349i	$0.376228\pi$
$150$	0	0
$151$	−3.37228	−0.274432	−0.137216	−	0.990541i	$-0.543816\pi$
$151$	−3.37228	−0.274432	−0.137216	+	0.990541i	$0.543816\pi$
$152$	−1.37228	−0.111307
$153$	1.37228	0.110942
$154$	−4.62772	−0.372912
$155$	0	0
$156$	−1.37228	−0.109870
$157$	−3.25544	−0.259812	−0.129906	−	0.991526i	$-0.541468\pi$
$157$	−3.25544	−0.259812	−0.129906	+	0.991526i	$0.541468\pi$
$158$	4.00000	0.318223
$159$	−5.37228	−0.426050
$160$	0	0
$161$	−11.3723	−0.896261
$162$	1.00000	0.0785674
$163$	−4.86141	−0.380775	−0.190387	−	0.981709i	$-0.560974\pi$
$163$	−4.86141	−0.380775	−0.190387	+	0.981709i	$0.560974\pi$
$164$	8.74456	0.682836
$165$	0	0
$166$	−0.627719	−0.0487204
$167$	1.88316	0.145723	0.0728615	−	0.997342i	$-0.476787\pi$
$167$	1.88316	0.145723	0.0728615	+	0.997342i	$0.476787\pi$
$168$	3.37228	0.260177
$169$	−11.1168	−0.855142
$170$	0	0
$171$	−1.37228	−0.104941
$172$	4.00000	0.304997
$173$	6.86141	0.521663	0.260832	−	0.965384i	$-0.416003\pi$
$173$	6.86141	0.521663	0.260832	+	0.965384i	$0.416003\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−1.37228	−0.103440
$177$	14.7446	1.10827
$178$	13.3723	1.00229
$179$	5.48913	0.410276	0.205138	−	0.978733i	$-0.434236\pi$
$179$	5.48913	0.410276	0.205138	+	0.978733i	$0.434236\pi$
$180$	0	0
$181$	−20.9783	−1.55930	−0.779651	−	0.626215i	$-0.784603\pi$
$181$	−20.9783	−1.55930	−0.779651	+	0.626215i	$0.784603\pi$
$182$	−4.62772	−0.343029
$183$	−2.74456	−0.202884
$184$	−3.37228	−0.248608
$185$	0	0
$186$	2.74456	0.201241
$187$	−1.88316	−0.137710
$188$	4.74456	0.346033
$189$	3.37228	0.245297
$190$	0	0
$191$	−19.3723	−1.40173	−0.700865	−	0.713294i	$-0.747202\pi$
$191$	−19.3723	−1.40173	−0.700865	+	0.713294i	$0.747202\pi$
$192$	1.00000	0.0721688
$193$	1.25544	0.0903684	0.0451842	−	0.998979i	$-0.485613\pi$
$193$	1.25544	0.0903684	0.0451842	+	0.998979i	$0.485613\pi$
$194$	−13.4891	−0.968463
$195$	0	0
$196$	4.37228	0.312306
$197$	18.8614	1.34382	0.671910	−	0.740633i	$-0.265474\pi$
$197$	18.8614	1.34382	0.671910	+	0.740633i	$0.265474\pi$
$198$	−1.37228	−0.0975238
$199$	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
$201$	−2.74456	−0.193587
$202$	10.7446	0.755985
$203$	20.2337	1.42013
$204$	1.37228	0.0960789
$205$	0	0
$206$	0.744563	0.0518761
$207$	−3.37228	−0.234390
$208$	−1.37228	−0.0951506
$209$	1.88316	0.130261
$210$	0	0
$211$	−9.25544	−0.637171	−0.318585	−	0.947894i	$-0.603208\pi$
$211$	−9.25544	−0.637171	−0.318585	+	0.947894i	$0.603208\pi$
$212$	−5.37228	−0.368970
$213$	1.25544	0.0860212
$214$	−3.37228	−0.230524
$215$	0	0
$216$	1.00000	0.0680414
$217$	9.25544	0.628300
$218$	4.62772	0.313429
$219$	4.11684	0.278191
$220$	0	0
$221$	−1.88316	−0.126675
$222$	1.00000	0.0671156
$223$	−18.9783	−1.27088	−0.635439	−	0.772151i	$-0.719181\pi$
$223$	−18.9783	−1.27088	−0.635439	+	0.772151i	$0.719181\pi$
$224$	3.37228	0.225320
$225$	0	0
$226$	−19.4891	−1.29640
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	−1.37228	−0.0908816
$229$	−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
$231$	−4.62772	−0.304482
$232$	6.00000	0.393919
$233$	22.7446	1.49005	0.745023	−	0.667039i	$-0.232439\pi$
$233$	22.7446	1.49005	0.745023	+	0.667039i	$0.232439\pi$
$234$	−1.37228	−0.0897088
$235$	0	0
$236$	14.7446	0.959789
$237$	4.00000	0.259828
$238$	4.62772	0.299970
$239$	−5.48913	−0.355062	−0.177531	−	0.984115i	$-0.556811\pi$
$239$	−5.48913	−0.355062	−0.177531	+	0.984115i	$0.556811\pi$
$240$	0	0
$241$	−0.510875	−0.0329083	−0.0164542	−	0.999865i	$-0.505238\pi$
$241$	−0.510875	−0.0329083	−0.0164542	+	0.999865i	$0.505238\pi$
$242$	−9.11684	−0.586053
$243$	1.00000	0.0641500
$244$	−2.74456	−0.175703
$245$	0	0
$246$	8.74456	0.557533
$247$	1.88316	0.119822
$248$	2.74456	0.174280
$249$	−0.627719	−0.0397801
$250$	0	0
$251$	−26.7446	−1.68810	−0.844051	−	0.536263i	$-0.819836\pi$
$251$	−26.7446	−1.68810	−0.844051	+	0.536263i	$0.819836\pi$
$252$	3.37228	0.212434
$253$	4.62772	0.290942
$254$	−3.37228	−0.211596
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.37228	0.335114	0.167557	−	0.985862i	$-0.446412\pi$
$257$	5.37228	0.335114	0.167557	+	0.985862i	$0.446412\pi$
$258$	4.00000	0.249029
$259$	3.37228	0.209543
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−10.2337	−0.631036	−0.315518	−	0.948920i	$-0.602178\pi$
$263$	−10.2337	−0.631036	−0.315518	+	0.948920i	$0.602178\pi$
$264$	−1.37228	−0.0844581
$265$	0	0
$266$	−4.62772	−0.283744
$267$	13.3723	0.818370
$268$	−2.74456	−0.167651
$269$	0.627719	0.0382727	0.0191363	−	0.999817i	$-0.493908\pi$
$269$	0.627719	0.0382727	0.0191363	+	0.999817i	$0.493908\pi$
$270$	0	0
$271$	13.4891	0.819406	0.409703	−	0.912219i	$-0.365632\pi$
$271$	13.4891	0.819406	0.409703	+	0.912219i	$0.365632\pi$
$272$	1.37228	0.0832068
$273$	−4.62772	−0.280082
$274$	−10.7446	−0.649103
$275$	0	0
$276$	−3.37228	−0.202987
$277$	−25.6060	−1.53851	−0.769257	−	0.638940i	$-0.779373\pi$
$277$	−25.6060	−1.53851	−0.769257	+	0.638940i	$0.779373\pi$
$278$	2.74456	0.164608
$279$	2.74456	0.164313
$280$	0	0
$281$	1.37228	0.0818634	0.0409317	−	0.999162i	$-0.486967\pi$
$281$	1.37228	0.0818634	0.0409317	+	0.999162i	$0.486967\pi$
$282$	4.74456	0.282535
$283$	11.6060	0.689903	0.344952	−	0.938620i	$-0.387895\pi$
$283$	11.6060	0.689903	0.344952	+	0.938620i	$0.387895\pi$
$284$	1.25544	0.0744965
$285$	0	0
$286$	1.88316	0.111353
$287$	29.4891	1.74069
$288$	1.00000	0.0589256
$289$	−15.1168	−0.889226
$290$	0	0
$291$	−13.4891	−0.790747
$292$	4.11684	0.240920
$293$	4.11684	0.240509	0.120254	−	0.992743i	$-0.461629\pi$
$293$	4.11684	0.240509	0.120254	+	0.992743i	$0.461629\pi$
$294$	4.37228	0.254997
$295$	0	0
$296$	1.00000	0.0581238
$297$	−1.37228	−0.0796278
$298$	9.25544	0.536153
$299$	4.62772	0.267628
$300$	0	0
$301$	13.4891	0.777500
$302$	−3.37228	−0.194053
$303$	10.7446	0.617259
$304$	−1.37228	−0.0787057
$305$	0	0
$306$	1.37228	0.0784481
$307$	10.7446	0.613225	0.306612	−	0.951834i	$-0.400805\pi$
$307$	10.7446	0.613225	0.306612	+	0.951834i	$0.400805\pi$
$308$	−4.62772	−0.263689
$309$	0.744563	0.0423567
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	−1.37228	−0.0776901
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	−3.25544	−0.183715
$315$	0	0
$316$	4.00000	0.225018
$317$	16.9783	0.953594	0.476797	−	0.879014i	$-0.341798\pi$
$317$	16.9783	0.953594	0.476797	+	0.879014i	$0.341798\pi$
$318$	−5.37228	−0.301263
$319$	−8.23369	−0.460998
$320$	0	0
$321$	−3.37228	−0.188222
$322$	−11.3723	−0.633752
$323$	−1.88316	−0.104782
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.86141	−0.269248
$327$	4.62772	0.255913
$328$	8.74456	0.482838
$329$	16.0000	0.882109
$330$	0	0
$331$	−8.74456	−0.480645	−0.240322	−	0.970693i	$-0.577253\pi$
$331$	−8.74456	−0.480645	−0.240322	+	0.970693i	$0.577253\pi$
$332$	−0.627719	−0.0344505
$333$	1.00000	0.0547997
$334$	1.88316	0.103042
$335$	0	0
$336$	3.37228	0.183973
$337$	4.11684	0.224259	0.112129	−	0.993694i	$-0.464233\pi$
$337$	4.11684	0.224259	0.112129	+	0.993694i	$0.464233\pi$
$338$	−11.1168	−0.604677
$339$	−19.4891	−1.05850
$340$	0	0
$341$	−3.76631	−0.203957
$342$	−1.37228	−0.0742045
$343$	−8.86141	−0.478471
$344$	4.00000	0.215666
$345$	0	0
$346$	6.86141	0.368872
$347$	−1.48913	−0.0799404	−0.0399702	−	0.999201i	$-0.512726\pi$
$347$	−1.48913	−0.0799404	−0.0399702	+	0.999201i	$0.512726\pi$
$348$	6.00000	0.321634
$349$	16.7446	0.896316	0.448158	−	0.893954i	$-0.352080\pi$
$349$	16.7446	0.896316	0.448158	+	0.893954i	$0.352080\pi$
$350$	0	0
$351$	−1.37228	−0.0732470
$352$	−1.37228	−0.0731428
$353$	−3.48913	−0.185707	−0.0928537	−	0.995680i	$-0.529599\pi$
$353$	−3.48913	−0.185707	−0.0928537	+	0.995680i	$0.529599\pi$
$354$	14.7446	0.783665
$355$	0	0
$356$	13.3723	0.708729
$357$	4.62772	0.244925
$358$	5.48913	0.290109
$359$	32.4674	1.71356	0.856781	−	0.515680i	$-0.172461\pi$
$359$	32.4674	1.71356	0.856781	+	0.515680i	$0.172461\pi$
$360$	0	0
$361$	−17.1168	−0.900887
$362$	−20.9783	−1.10259
$363$	−9.11684	−0.478510
$364$	−4.62772	−0.242558
$365$	0	0
$366$	−2.74456	−0.143461
$367$	−20.6277	−1.07676	−0.538379	−	0.842703i	$-0.680963\pi$
$367$	−20.6277	−1.07676	−0.538379	+	0.842703i	$0.680963\pi$
$368$	−3.37228	−0.175792
$369$	8.74456	0.455224
$370$	0	0
$371$	−18.1168	−0.940580
$372$	2.74456	0.142299
$373$	−3.48913	−0.180660	−0.0903300	−	0.995912i	$-0.528792\pi$
$373$	−3.48913	−0.180660	−0.0903300	+	0.995912i	$0.528792\pi$
$374$	−1.88316	−0.0973757
$375$	0	0
$376$	4.74456	0.244682
$377$	−8.23369	−0.424057
$378$	3.37228	0.173451
$379$	24.2337	1.24480	0.622400	−	0.782699i	$-0.286158\pi$
$379$	24.2337	1.24480	0.622400	+	0.782699i	$0.286158\pi$
$380$	0	0
$381$	−3.37228	−0.172767
$382$	−19.3723	−0.991172
$383$	−27.6060	−1.41060	−0.705300	−	0.708909i	$-0.749188\pi$
$383$	−27.6060	−1.41060	−0.705300	+	0.708909i	$0.749188\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	1.25544	0.0639001
$387$	4.00000	0.203331
$388$	−13.4891	−0.684807
$389$	23.4891	1.19095	0.595473	−	0.803375i	$-0.296965\pi$
$389$	23.4891	1.19095	0.595473	+	0.803375i	$0.296965\pi$
$390$	0	0
$391$	−4.62772	−0.234034
$392$	4.37228	0.220834
$393$	0	0
$394$	18.8614	0.950224
$395$	0	0
$396$	−1.37228	−0.0689597
$397$	−20.7446	−1.04114	−0.520570	−	0.853819i	$-0.674280\pi$
$397$	−20.7446	−1.04114	−0.520570	+	0.853819i	$0.674280\pi$
$398$	8.00000	0.401004
$399$	−4.62772	−0.231676
$400$	0	0
$401$	−36.1168	−1.80359	−0.901795	−	0.432165i	$-0.857750\pi$
$401$	−36.1168	−1.80359	−0.901795	+	0.432165i	$0.857750\pi$
$402$	−2.74456	−0.136886
$403$	−3.76631	−0.187613
$404$	10.7446	0.534562
$405$	0	0
$406$	20.2337	1.00418
$407$	−1.37228	−0.0680215
$408$	1.37228	0.0679380
$409$	−8.74456	−0.432391	−0.216195	−	0.976350i	$-0.569365\pi$
$409$	−8.74456	−0.432391	−0.216195	+	0.976350i	$0.569365\pi$
$410$	0	0
$411$	−10.7446	−0.529990
$412$	0.744563	0.0366820
$413$	49.7228	2.44670
$414$	−3.37228	−0.165739
$415$	0	0
$416$	−1.37228	−0.0672816
$417$	2.74456	0.134402
$418$	1.88316	0.0921082
$419$	7.88316	0.385117	0.192559	−	0.981285i	$-0.438321\pi$
$419$	7.88316	0.385117	0.192559	+	0.981285i	$0.438321\pi$
$420$	0	0
$421$	24.2337	1.18108	0.590539	−	0.807009i	$-0.298915\pi$
$421$	24.2337	1.18108	0.590539	+	0.807009i	$0.298915\pi$
$422$	−9.25544	−0.450548
$423$	4.74456	0.230689
$424$	−5.37228	−0.260901
$425$	0	0
$426$	1.25544	0.0608261
$427$	−9.25544	−0.447902
$428$	−3.37228	−0.163005
$429$	1.88316	0.0909196
$430$	0	0
$431$	−31.6060	−1.52241	−0.761203	−	0.648514i	$-0.775391\pi$
$431$	−31.6060	−1.52241	−0.761203	+	0.648514i	$0.775391\pi$
$432$	1.00000	0.0481125
$433$	10.8614	0.521966	0.260983	−	0.965343i	$-0.415953\pi$
$433$	10.8614	0.521966	0.260983	+	0.965343i	$0.415953\pi$
$434$	9.25544	0.444275
$435$	0	0
$436$	4.62772	0.221628
$437$	4.62772	0.221374
$438$	4.11684	0.196710
$439$	−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
$441$	4.37228	0.208204
$442$	−1.88316	−0.0895726
$443$	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
$446$	−18.9783	−0.898646
$447$	9.25544	0.437767
$448$	3.37228	0.159325
$449$	36.9783	1.74511	0.872556	−	0.488515i	$-0.162461\pi$
$449$	36.9783	1.74511	0.872556	+	0.488515i	$0.162461\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	−19.4891	−0.916691
$453$	−3.37228	−0.158444
$454$	−4.00000	−0.187729
$455$	0	0
$456$	−1.37228	−0.0642630
$457$	40.4674	1.89298	0.946492	−	0.322727i	$-0.104600\pi$
$457$	40.4674	1.89298	0.946492	+	0.322727i	$0.104600\pi$
$458$	−6.00000	−0.280362
$459$	1.37228	0.0640526
$460$	0	0
$461$	7.48913	0.348803	0.174402	−	0.984675i	$-0.444201\pi$
$461$	7.48913	0.348803	0.174402	+	0.984675i	$0.444201\pi$
$462$	−4.62772	−0.215301
$463$	19.7228	0.916597	0.458298	−	0.888798i	$-0.348459\pi$
$463$	19.7228	0.916597	0.458298	+	0.888798i	$0.348459\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	22.7446	1.05362
$467$	−1.48913	−0.0689085	−0.0344543	−	0.999406i	$-0.510969\pi$
$467$	−1.48913	−0.0689085	−0.0344543	+	0.999406i	$0.510969\pi$
$468$	−1.37228	−0.0634337
$469$	−9.25544	−0.427376
$470$	0	0
$471$	−3.25544	−0.150003
$472$	14.7446	0.678674
$473$	−5.48913	−0.252390
$474$	4.00000	0.183726
$475$	0	0
$476$	4.62772	0.212111
$477$	−5.37228	−0.245980
$478$	−5.48913	−0.251067
$479$	−18.1168	−0.827780	−0.413890	−	0.910327i	$-0.635830\pi$
$479$	−18.1168	−0.827780	−0.413890	+	0.910327i	$0.635830\pi$
$480$	0	0
$481$	−1.37228	−0.0625706
$482$	−0.510875	−0.0232697
$483$	−11.3723	−0.517457
$484$	−9.11684	−0.414402
$485$	0	0
$486$	1.00000	0.0453609
$487$	30.4674	1.38061	0.690304	−	0.723519i	$-0.257477\pi$
$487$	30.4674	1.38061	0.690304	+	0.723519i	$0.257477\pi$
$488$	−2.74456	−0.124241
$489$	−4.86141	−0.219840
$490$	0	0
$491$	2.62772	0.118587	0.0592936	−	0.998241i	$-0.481115\pi$
$491$	2.62772	0.118587	0.0592936	+	0.998241i	$0.481115\pi$
$492$	8.74456	0.394235
$493$	8.23369	0.370827
$494$	1.88316	0.0847272
$495$	0	0
$496$	2.74456	0.123235
$497$	4.23369	0.189907
$498$	−0.627719	−0.0281287
$499$	20.3505	0.911015	0.455507	−	0.890232i	$-0.349458\pi$
$499$	20.3505	0.911015	0.455507	+	0.890232i	$0.349458\pi$
$500$	0	0
$501$	1.88316	0.0841332
$502$	−26.7446	−1.19367
$503$	−5.48913	−0.244748	−0.122374	−	0.992484i	$-0.539051\pi$
$503$	−5.48913	−0.244748	−0.122374	+	0.992484i	$0.539051\pi$
$504$	3.37228	0.150213
$505$	0	0
$506$	4.62772	0.205727
$507$	−11.1168	−0.493716
$508$	−3.37228	−0.149621
$509$	18.3505	0.813373	0.406687	−	0.913568i	$-0.366684\pi$
$509$	18.3505	0.813373	0.406687	+	0.913568i	$0.366684\pi$
$510$	0	0
$511$	13.8832	0.614155
$512$	1.00000	0.0441942
$513$	−1.37228	−0.0605877
$514$	5.37228	0.236961
$515$	0	0
$516$	4.00000	0.176090
$517$	−6.51087	−0.286348
$518$	3.37228	0.148170
$519$	6.86141	0.301182
$520$	0	0
$521$	−5.76631	−0.252627	−0.126313	−	0.991990i	$-0.540314\pi$
$521$	−5.76631	−0.252627	−0.126313	+	0.991990i	$0.540314\pi$
$522$	6.00000	0.262613
$523$	6.97825	0.305138	0.152569	−	0.988293i	$-0.451245\pi$
$523$	6.97825	0.305138	0.152569	+	0.988293i	$0.451245\pi$
$524$	0	0
$525$	0	0
$526$	−10.2337	−0.446210
$527$	3.76631	0.164063
$528$	−1.37228	−0.0597209
$529$	−11.6277	−0.505553
$530$	0	0
$531$	14.7446	0.639860
$532$	−4.62772	−0.200637
$533$	−12.0000	−0.519778
$534$	13.3723	0.578675
$535$	0	0
$536$	−2.74456	−0.118547
$537$	5.48913	0.236873
$538$	0.627719	0.0270629
$539$	−6.00000	−0.258438
$540$	0	0
$541$	4.86141	0.209008	0.104504	−	0.994524i	$-0.466674\pi$
$541$	4.86141	0.209008	0.104504	+	0.994524i	$0.466674\pi$
$542$	13.4891	0.579408
$543$	−20.9783	−0.900263
$544$	1.37228	0.0588361
$545$	0	0
$546$	−4.62772	−0.198048
$547$	2.11684	0.0905097	0.0452549	−	0.998975i	$-0.485590\pi$
$547$	2.11684	0.0905097	0.0452549	+	0.998975i	$0.485590\pi$
$548$	−10.7446	−0.458985
$549$	−2.74456	−0.117135
$550$	0	0
$551$	−8.23369	−0.350767
$552$	−3.37228	−0.143534
$553$	13.4891	0.573616
$554$	−25.6060	−1.08789
$555$	0	0
$556$	2.74456	0.116395
$557$	−40.7446	−1.72640	−0.863201	−	0.504860i	$-0.831544\pi$
$557$	−40.7446	−1.72640	−0.863201	+	0.504860i	$0.831544\pi$
$558$	2.74456	0.116187
$559$	−5.48913	−0.232165
$560$	0	0
$561$	−1.88316	−0.0795069
$562$	1.37228	0.0578862
$563$	−13.2554	−0.558650	−0.279325	−	0.960197i	$-0.590111\pi$
$563$	−13.2554	−0.558650	−0.279325	+	0.960197i	$0.590111\pi$
$564$	4.74456	0.199782
$565$	0	0
$566$	11.6060	0.487835
$567$	3.37228	0.141623
$568$	1.25544	0.0526770
$569$	−32.3505	−1.35620	−0.678102	−	0.734967i	$-0.737197\pi$
$569$	−32.3505	−1.35620	−0.678102	+	0.734967i	$0.737197\pi$
$570$	0	0
$571$	−1.25544	−0.0525384	−0.0262692	−	0.999655i	$-0.508363\pi$
$571$	−1.25544	−0.0525384	−0.0262692	+	0.999655i	$0.508363\pi$
$572$	1.88316	0.0787387
$573$	−19.3723	−0.809289
$574$	29.4891	1.23085
$575$	0	0
$576$	1.00000	0.0416667
$577$	−26.7446	−1.11339	−0.556695	−	0.830717i	$-0.687931\pi$
$577$	−26.7446	−1.11339	−0.556695	+	0.830717i	$0.687931\pi$
$578$	−15.1168	−0.628778
$579$	1.25544	0.0521742
$580$	0	0
$581$	−2.11684	−0.0878215
$582$	−13.4891	−0.559142
$583$	7.37228	0.305329
$584$	4.11684	0.170356
$585$	0	0
$586$	4.11684	0.170065
$587$	−1.48913	−0.0614628	−0.0307314	−	0.999528i	$-0.509784\pi$
$587$	−1.48913	−0.0614628	−0.0307314	+	0.999528i	$0.509784\pi$
$588$	4.37228	0.180310
$589$	−3.76631	−0.155188
$590$	0	0
$591$	18.8614	0.775855
$592$	1.00000	0.0410997
$593$	22.9783	0.943604	0.471802	−	0.881705i	$-0.343604\pi$
$593$	22.9783	0.943604	0.471802	+	0.881705i	$0.343604\pi$
$594$	−1.37228	−0.0563054
$595$	0	0
$596$	9.25544	0.379117
$597$	8.00000	0.327418
$598$	4.62772	0.189241
$599$	−22.9783	−0.938866	−0.469433	−	0.882968i	$-0.655542\pi$
$599$	−22.9783	−0.938866	−0.469433	+	0.882968i	$0.655542\pi$
$600$	0	0
$601$	30.8614	1.25886	0.629432	−	0.777056i	$-0.283288\pi$
$601$	30.8614	1.25886	0.629432	+	0.777056i	$0.283288\pi$
$602$	13.4891	0.549776
$603$	−2.74456	−0.111767
$604$	−3.37228	−0.137216
$605$	0	0
$606$	10.7446	0.436468
$607$	−38.4674	−1.56134	−0.780671	−	0.624942i	$-0.785123\pi$
$607$	−38.4674	−1.56134	−0.780671	+	0.624942i	$0.785123\pi$
$608$	−1.37228	−0.0556534
$609$	20.2337	0.819910
$610$	0	0
$611$	−6.51087	−0.263402
$612$	1.37228	0.0554712
$613$	−30.4674	−1.23057	−0.615283	−	0.788306i	$-0.710958\pi$
$613$	−30.4674	−1.23057	−0.615283	+	0.788306i	$0.710958\pi$
$614$	10.7446	0.433615
$615$	0	0
$616$	−4.62772	−0.186456
$617$	−32.2337	−1.29768	−0.648840	−	0.760925i	$-0.724745\pi$
$617$	−32.2337	−1.29768	−0.648840	+	0.760925i	$0.724745\pi$
$618$	0.744563	0.0299507
$619$	−26.9783	−1.08435	−0.542174	−	0.840266i	$-0.682399\pi$
$619$	−26.9783	−1.08435	−0.542174	+	0.840266i	$0.682399\pi$
$620$	0	0
$621$	−3.37228	−0.135325
$622$	−8.00000	−0.320771
$623$	45.0951	1.80670
$624$	−1.37228	−0.0549352
$625$	0	0
$626$	−8.00000	−0.319744
$627$	1.88316	0.0752060
$628$	−3.25544	−0.129906
$629$	1.37228	0.0547164
$630$	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
$633$	−9.25544	−0.367871
$634$	16.9783	0.674292
$635$	0	0
$636$	−5.37228	−0.213025
$637$	−6.00000	−0.237729
$638$	−8.23369	−0.325975
$639$	1.25544	0.0496643
$640$	0	0
$641$	43.7228	1.72695	0.863474	−	0.504394i	$-0.168284\pi$
$641$	43.7228	1.72695	0.863474	+	0.504394i	$0.168284\pi$
$642$	−3.37228	−0.133093
$643$	−23.6060	−0.930929	−0.465464	−	0.885067i	$-0.654113\pi$
$643$	−23.6060	−0.930929	−0.465464	+	0.885067i	$0.654113\pi$
$644$	−11.3723	−0.448131
$645$	0	0
$646$	−1.88316	−0.0740918
$647$	39.6060	1.55707	0.778536	−	0.627600i	$-0.215963\pi$
$647$	39.6060	1.55707	0.778536	+	0.627600i	$0.215963\pi$
$648$	1.00000	0.0392837
$649$	−20.2337	−0.794242
$650$	0	0
$651$	9.25544	0.362749
$652$	−4.86141	−0.190387
$653$	−19.7228	−0.771813	−0.385907	−	0.922538i	$-0.626111\pi$
$653$	−19.7228	−0.771813	−0.385907	+	0.922538i	$0.626111\pi$
$654$	4.62772	0.180958
$655$	0	0
$656$	8.74456	0.341418
$657$	4.11684	0.160613
$658$	16.0000	0.623745
$659$	−2.23369	−0.0870121	−0.0435061	−	0.999053i	$-0.513853\pi$
$659$	−2.23369	−0.0870121	−0.0435061	+	0.999053i	$0.513853\pi$
$660$	0	0
$661$	−26.1168	−1.01583	−0.507914	−	0.861408i	$-0.669583\pi$
$661$	−26.1168	−1.01583	−0.507914	+	0.861408i	$0.669583\pi$
$662$	−8.74456	−0.339867
$663$	−1.88316	−0.0731357
$664$	−0.627719	−0.0243602
$665$	0	0
$666$	1.00000	0.0387492
$667$	−20.2337	−0.783452
$668$	1.88316	0.0728615
$669$	−18.9783	−0.733742
$670$	0	0
$671$	3.76631	0.145397
$672$	3.37228	0.130089
$673$	−2.86141	−0.110299	−0.0551496	−	0.998478i	$-0.517564\pi$
$673$	−2.86141	−0.110299	−0.0551496	+	0.998478i	$0.517564\pi$
$674$	4.11684	0.158575
$675$	0	0
$676$	−11.1168	−0.427571
$677$	2.86141	0.109973	0.0549864	−	0.998487i	$-0.482488\pi$
$677$	2.86141	0.109973	0.0549864	+	0.998487i	$0.482488\pi$
$678$	−19.4891	−0.748475
$679$	−45.4891	−1.74571
$680$	0	0
$681$	−4.00000	−0.153280
$682$	−3.76631	−0.144220
$683$	−30.9783	−1.18535	−0.592675	−	0.805442i	$-0.701928\pi$
$683$	−30.9783	−1.18535	−0.592675	+	0.805442i	$0.701928\pi$
$684$	−1.37228	−0.0524705
$685$	0	0
$686$	−8.86141	−0.338330
$687$	−6.00000	−0.228914
$688$	4.00000	0.152499
$689$	7.37228	0.280862
$690$	0	0
$691$	22.7446	0.865244	0.432622	−	0.901575i	$-0.357588\pi$
$691$	22.7446	0.865244	0.432622	+	0.901575i	$0.357588\pi$
$692$	6.86141	0.260832
$693$	−4.62772	−0.175792
$694$	−1.48913	−0.0565264
$695$	0	0
$696$	6.00000	0.227429
$697$	12.0000	0.454532
$698$	16.7446	0.633791
$699$	22.7446	0.860278
$700$	0	0
$701$	−47.7228	−1.80247	−0.901233	−	0.433335i	$-0.857337\pi$
$701$	−47.7228	−1.80247	−0.901233	+	0.433335i	$0.857337\pi$
$702$	−1.37228	−0.0517934
$703$	−1.37228	−0.0517566
$704$	−1.37228	−0.0517198
$705$	0	0
$706$	−3.48913	−0.131315
$707$	36.2337	1.36271
$708$	14.7446	0.554135
$709$	−20.6277	−0.774690	−0.387345	−	0.921935i	$-0.626608\pi$
$709$	−20.6277	−0.774690	−0.387345	+	0.921935i	$0.626608\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	13.3723	0.501147
$713$	−9.25544	−0.346619
$714$	4.62772	0.173188
$715$	0	0
$716$	5.48913	0.205138
$717$	−5.48913	−0.204995
$718$	32.4674	1.21167
$719$	−13.7228	−0.511775	−0.255887	−	0.966707i	$-0.582368\pi$
$719$	−13.7228	−0.511775	−0.255887	+	0.966707i	$0.582368\pi$
$720$	0	0
$721$	2.51087	0.0935099
$722$	−17.1168	−0.637023
$723$	−0.510875	−0.0189996
$724$	−20.9783	−0.779651
$725$	0	0
$726$	−9.11684	−0.338358
$727$	−8.97825	−0.332985	−0.166492	−	0.986043i	$-0.553244\pi$
$727$	−8.97825	−0.332985	−0.166492	+	0.986043i	$0.553244\pi$
$728$	−4.62772	−0.171515
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.48913	0.203023
$732$	−2.74456	−0.101442
$733$	−3.48913	−0.128874	−0.0644369	−	0.997922i	$-0.520525\pi$
$733$	−3.48913	−0.128874	−0.0644369	+	0.997922i	$0.520525\pi$
$734$	−20.6277	−0.761383
$735$	0	0
$736$	−3.37228	−0.124304
$737$	3.76631	0.138734
$738$	8.74456	0.321892
$739$	25.7228	0.946229	0.473114	−	0.881001i	$-0.343130\pi$
$739$	25.7228	0.946229	0.473114	+	0.881001i	$0.343130\pi$
$740$	0	0
$741$	1.88316	0.0691795
$742$	−18.1168	−0.665090
$743$	−22.4674	−0.824248	−0.412124	−	0.911128i	$-0.635213\pi$
$743$	−22.4674	−0.824248	−0.412124	+	0.911128i	$0.635213\pi$
$744$	2.74456	0.100621
$745$	0	0
$746$	−3.48913	−0.127746
$747$	−0.627719	−0.0229670
$748$	−1.88316	−0.0688550
$749$	−11.3723	−0.415534
$750$	0	0
$751$	−10.5109	−0.383547	−0.191774	−	0.981439i	$-0.561424\pi$
$751$	−10.5109	−0.383547	−0.191774	+	0.981439i	$0.561424\pi$
$752$	4.74456	0.173016
$753$	−26.7446	−0.974626
$754$	−8.23369	−0.299853
$755$	0	0
$756$	3.37228	0.122649
$757$	−44.1168	−1.60345	−0.801727	−	0.597690i	$-0.796085\pi$
$757$	−44.1168	−1.60345	−0.801727	+	0.597690i	$0.796085\pi$
$758$	24.2337	0.880207
$759$	4.62772	0.167976
$760$	0	0
$761$	−12.5109	−0.453519	−0.226759	−	0.973951i	$-0.572813\pi$
$761$	−12.5109	−0.453519	−0.226759	+	0.973951i	$0.572813\pi$
$762$	−3.37228	−0.122165
$763$	15.6060	0.564974
$764$	−19.3723	−0.700865
$765$	0	0
$766$	−27.6060	−0.997444
$767$	−20.2337	−0.730596
$768$	1.00000	0.0360844
$769$	−51.4891	−1.85675	−0.928373	−	0.371651i	$-0.878792\pi$
$769$	−51.4891	−1.85675	−0.928373	+	0.371651i	$0.878792\pi$
$770$	0	0
$771$	5.37228	0.193478
$772$	1.25544	0.0451842
$773$	−20.1168	−0.723553	−0.361776	−	0.932265i	$-0.617830\pi$
$773$	−20.1168	−0.723553	−0.361776	+	0.932265i	$0.617830\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−13.4891	−0.484231
$777$	3.37228	0.120980
$778$	23.4891	0.842126
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−1.72281	−0.0616471
$782$	−4.62772	−0.165487
$783$	6.00000	0.214423
$784$	4.37228	0.156153
$785$	0	0
$786$	0	0
$787$	−10.7446	−0.383002	−0.191501	−	0.981492i	$-0.561336\pi$
$787$	−10.7446	−0.383002	−0.191501	+	0.981492i	$0.561336\pi$
$788$	18.8614	0.671910
$789$	−10.2337	−0.364329
$790$	0	0
$791$	−65.7228	−2.33683
$792$	−1.37228	−0.0487619
$793$	3.76631	0.133746
$794$	−20.7446	−0.736197
$795$	0	0
$796$	8.00000	0.283552
$797$	−4.51087	−0.159783	−0.0798917	−	0.996804i	$-0.525457\pi$
$797$	−4.51087	−0.159783	−0.0798917	+	0.996804i	$0.525457\pi$
$798$	−4.62772	−0.163819
$799$	6.51087	0.230338
$800$	0	0
$801$	13.3723	0.472486
$802$	−36.1168	−1.27533
$803$	−5.64947	−0.199365
$804$	−2.74456	−0.0967933
$805$	0	0
$806$	−3.76631	−0.132663
$807$	0.627719	0.0220967
$808$	10.7446	0.377992
$809$	−24.1168	−0.847903	−0.423952	−	0.905685i	$-0.639357\pi$
$809$	−24.1168	−0.847903	−0.423952	+	0.905685i	$0.639357\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	20.2337	0.710063
$813$	13.4891	0.473084
$814$	−1.37228	−0.0480984
$815$	0	0
$816$	1.37228	0.0480395
$817$	−5.48913	−0.192040
$818$	−8.74456	−0.305746
$819$	−4.62772	−0.161706
$820$	0	0
$821$	26.3505	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$821$	26.3505	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$822$	−10.7446	−0.374760
$823$	−25.8832	−0.902230	−0.451115	−	0.892466i	$-0.648974\pi$
$823$	−25.8832	−0.902230	−0.451115	+	0.892466i	$0.648974\pi$
$824$	0.744563	0.0259381
$825$	0	0
$826$	49.7228	1.73008
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	−3.37228	−0.117195
$829$	−35.8397	−1.24476	−0.622381	−	0.782714i	$-0.713835\pi$
$829$	−35.8397	−1.24476	−0.622381	+	0.782714i	$0.713835\pi$
$830$	0	0
$831$	−25.6060	−0.888261
$832$	−1.37228	−0.0475753
$833$	6.00000	0.207888
$834$	2.74456	0.0950364
$835$	0	0
$836$	1.88316	0.0651303
$837$	2.74456	0.0948660
$838$	7.88316	0.272319
$839$	−9.25544	−0.319533	−0.159767	−	0.987155i	$-0.551074\pi$
$839$	−9.25544	−0.319533	−0.159767	+	0.987155i	$0.551074\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	24.2337	0.835148
$843$	1.37228	0.0472639
$844$	−9.25544	−0.318585
$845$	0	0
$846$	4.74456	0.163121
$847$	−30.7446	−1.05640
$848$	−5.37228	−0.184485
$849$	11.6060	0.398316
$850$	0	0
$851$	−3.37228	−0.115600
$852$	1.25544	0.0430106
$853$	−40.3505	−1.38158	−0.690788	−	0.723057i	$-0.742736\pi$
$853$	−40.3505	−1.38158	−0.690788	+	0.723057i	$0.742736\pi$
$854$	−9.25544	−0.316715
$855$	0	0
$856$	−3.37228	−0.115262
$857$	−17.8397	−0.609391	−0.304696	−	0.952450i	$-0.598555\pi$
$857$	−17.8397	−0.609391	−0.304696	+	0.952450i	$0.598555\pi$
$858$	1.88316	0.0642899
$859$	17.8397	0.608681	0.304341	−	0.952563i	$-0.401564\pi$
$859$	17.8397	0.608681	0.304341	+	0.952563i	$0.401564\pi$
$860$	0	0
$861$	29.4891	1.00499
$862$	−31.6060	−1.07650
$863$	−3.25544	−0.110816	−0.0554082	−	0.998464i	$-0.517646\pi$
$863$	−3.25544	−0.110816	−0.0554082	+	0.998464i	$0.517646\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	10.8614	0.369086
$867$	−15.1168	−0.513395
$868$	9.25544	0.314150
$869$	−5.48913	−0.186206
$870$	0	0
$871$	3.76631	0.127617
$872$	4.62772	0.156714
$873$	−13.4891	−0.456538
$874$	4.62772	0.156535
$875$	0	0
$876$	4.11684	0.139095
$877$	54.2337	1.83134	0.915671	−	0.401929i	$-0.131660\pi$
$877$	54.2337	1.83134	0.915671	+	0.401929i	$0.131660\pi$
$878$	−32.0000	−1.07995
$879$	4.11684	0.138858
$880$	0	0
$881$	−14.2337	−0.479545	−0.239773	−	0.970829i	$-0.577073\pi$
$881$	−14.2337	−0.479545	−0.239773	+	0.970829i	$0.577073\pi$
$882$	4.37228	0.147222
$883$	50.1168	1.68657	0.843283	−	0.537470i	$-0.180620\pi$
$883$	50.1168	1.68657	0.843283	+	0.537470i	$0.180620\pi$
$884$	−1.88316	−0.0633374
$885$	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
$889$	−11.3723	−0.381414
$890$	0	0
$891$	−1.37228	−0.0459732
$892$	−18.9783	−0.635439
$893$	−6.51087	−0.217878
$894$	9.25544	0.309548
$895$	0	0
$896$	3.37228	0.112660
$897$	4.62772	0.154515
$898$	36.9783	1.23398
$899$	16.4674	0.549218
$900$	0	0
$901$	−7.37228	−0.245606
$902$	−12.0000	−0.399556
$903$	13.4891	0.448890
$904$	−19.4891	−0.648199
$905$	0	0
$906$	−3.37228	−0.112037
$907$	3.60597	0.119734	0.0598671	−	0.998206i	$-0.480932\pi$
$907$	3.60597	0.119734	0.0598671	+	0.998206i	$0.480932\pi$
$908$	−4.00000	−0.132745
$909$	10.7446	0.356375
$910$	0	0
$911$	34.9783	1.15888	0.579441	−	0.815014i	$-0.303271\pi$
$911$	34.9783	1.15888	0.579441	+	0.815014i	$0.303271\pi$
$912$	−1.37228	−0.0454408
$913$	0.861407	0.0285084
$914$	40.4674	1.33854
$915$	0	0
$916$	−6.00000	−0.198246
$917$	0	0
$918$	1.37228	0.0452920
$919$	−34.5109	−1.13841	−0.569204	−	0.822196i	$-0.692749\pi$
$919$	−34.5109	−1.13841	−0.569204	+	0.822196i	$0.692749\pi$
$920$	0	0
$921$	10.7446	0.354045
$922$	7.48913	0.246641
$923$	−1.72281	−0.0567071
$924$	−4.62772	−0.152241
$925$	0	0
$926$	19.7228	0.648132
$927$	0.744563	0.0244546
$928$	6.00000	0.196960
$929$	−12.5109	−0.410468	−0.205234	−	0.978713i	$-0.565796\pi$
$929$	−12.5109	−0.410468	−0.205234	+	0.978713i	$0.565796\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	22.7446	0.745023
$933$	−8.00000	−0.261908
$934$	−1.48913	−0.0487257
$935$	0	0
$936$	−1.37228	−0.0448544
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	−9.25544	−0.302201
$939$	−8.00000	−0.261070
$940$	0	0
$941$	15.7663	0.513967	0.256984	−	0.966416i	$-0.417271\pi$
$941$	15.7663	0.513967	0.256984	+	0.966416i	$0.417271\pi$
$942$	−3.25544	−0.106068
$943$	−29.4891	−0.960298
$944$	14.7446	0.479895
$945$	0	0
$946$	−5.48913	−0.178467
$947$	28.4674	0.925065	0.462533	−	0.886602i	$-0.346941\pi$
$947$	28.4674	0.925065	0.462533	+	0.886602i	$0.346941\pi$
$948$	4.00000	0.129914
$949$	−5.64947	−0.183389
$950$	0	0
$951$	16.9783	0.550557
$952$	4.62772	0.149985
$953$	50.7446	1.64378	0.821889	−	0.569648i	$-0.192920\pi$
$953$	50.7446	1.64378	0.821889	+	0.569648i	$0.192920\pi$
$954$	−5.37228	−0.173934
$955$	0	0
$956$	−5.48913	−0.177531
$957$	−8.23369	−0.266157
$958$	−18.1168	−0.585329
$959$	−36.2337	−1.17005
$960$	0	0
$961$	−23.4674	−0.757012
$962$	−1.37228	−0.0442441
$963$	−3.37228	−0.108670
$964$	−0.510875	−0.0164542
$965$	0	0
$966$	−11.3723	−0.365897
$967$	16.9783	0.545984	0.272992	−	0.962016i	$-0.411987\pi$
$967$	16.9783	0.545984	0.272992	+	0.962016i	$0.411987\pi$
$968$	−9.11684	−0.293026
$969$	−1.88316	−0.0604957
$970$	0	0
$971$	50.2337	1.61208	0.806038	−	0.591864i	$-0.201608\pi$
$971$	50.2337	1.61208	0.806038	+	0.591864i	$0.201608\pi$
$972$	1.00000	0.0320750
$973$	9.25544	0.296716
$974$	30.4674	0.976238
$975$	0	0
$976$	−2.74456	−0.0878513
$977$	−25.1386	−0.804255	−0.402127	−	0.915584i	$-0.631729\pi$
$977$	−25.1386	−0.804255	−0.402127	+	0.915584i	$0.631729\pi$
$978$	−4.86141	−0.155451
$979$	−18.3505	−0.586486
$980$	0	0
$981$	4.62772	0.147752
$982$	2.62772	0.0838539
$983$	12.5109	0.399035	0.199517	−	0.979894i	$-0.436063\pi$
$983$	12.5109	0.399035	0.199517	+	0.979894i	$0.436063\pi$
$984$	8.74456	0.278766
$985$	0	0
$986$	8.23369	0.262214
$987$	16.0000	0.509286
$988$	1.88316	0.0599112
$989$	−13.4891	−0.428929
$990$	0	0
$991$	22.9783	0.729928	0.364964	−	0.931022i	$-0.381081\pi$
$991$	22.9783	0.729928	0.364964	+	0.931022i	$0.381081\pi$
$992$	2.74456	0.0871400
$993$	−8.74456	−0.277500
$994$	4.23369	0.134284
$995$	0	0
$996$	−0.627719	−0.0198900
$997$	−16.1168	−0.510426	−0.255213	−	0.966885i	$-0.582146\pi$
$997$	−16.1168	−0.510426	−0.255213	+	0.966885i	$0.582146\pi$
$998$	20.3505	0.644185
$999$	1.00000	0.0316386

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.37228 q^{11} +1.00000 q^{12} -1.37228 q^{13} +3.37228 q^{14} +1.00000 q^{16} +1.37228 q^{17} +1.00000 q^{18} -1.37228 q^{19} +3.37228 q^{21} -1.37228 q^{22} -3.37228 q^{23} +1.00000 q^{24} -1.37228 q^{26} +1.00000 q^{27} +3.37228 q^{28} +6.00000 q^{29} +2.74456 q^{31} +1.00000 q^{32} -1.37228 q^{33} +1.37228 q^{34} +1.00000 q^{36} +1.00000 q^{37} -1.37228 q^{38} -1.37228 q^{39} +8.74456 q^{41} +3.37228 q^{42} +4.00000 q^{43} -1.37228 q^{44} -3.37228 q^{46} +4.74456 q^{47} +1.00000 q^{48} +4.37228 q^{49} +1.37228 q^{51} -1.37228 q^{52} -5.37228 q^{53} +1.00000 q^{54} +3.37228 q^{56} -1.37228 q^{57} +6.00000 q^{58} +14.7446 q^{59} -2.74456 q^{61} +2.74456 q^{62} +3.37228 q^{63} +1.00000 q^{64} -1.37228 q^{66} -2.74456 q^{67} +1.37228 q^{68} -3.37228 q^{69} +1.25544 q^{71} +1.00000 q^{72} +4.11684 q^{73} +1.00000 q^{74} -1.37228 q^{76} -4.62772 q^{77} -1.37228 q^{78} +4.00000 q^{79} +1.00000 q^{81} +8.74456 q^{82} -0.627719 q^{83} +3.37228 q^{84} +4.00000 q^{86} +6.00000 q^{87} -1.37228 q^{88} +13.3723 q^{89} -4.62772 q^{91} -3.37228 q^{92} +2.74456 q^{93} +4.74456 q^{94} +1.00000 q^{96} -13.4891 q^{97} +4.37228 q^{98} -1.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{11} + 2 q^{12} + 3 q^{13} + q^{14} + 2 q^{16} - 3 q^{17} + 2 q^{18} + 3 q^{19} + q^{21} + 3 q^{22} - q^{23} + 2 q^{24}+ \cdots + 3 q^{99}+O(q^{100}) \) 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

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