LMFDB - Embedded newform 4650.2.a.bz.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4650,2,Mod(1,4650)] 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("4650.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,-2,2,0,2,1,-2,2,0,5,-2,-12] 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:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.53113$ of defining polynomial
Character	$\chi$	$=$	4650.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.53113 q^{7} -1.00000 q^{8} +1.00000 q^{9} +6.53113 q^{11} -1.00000 q^{12} -6.00000 q^{13} -4.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +4.53113 q^{19} -4.53113 q^{21} -6.53113 q^{22} -6.53113 q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} +4.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} -6.53113 q^{33} -4.00000 q^{34} +1.00000 q^{36} +7.06226 q^{37} -4.53113 q^{38} +6.00000 q^{39} +7.06226 q^{41} +4.53113 q^{42} +8.53113 q^{43} +6.53113 q^{44} +6.53113 q^{46} +5.06226 q^{47} -1.00000 q^{48} +13.5311 q^{49} -4.00000 q^{51} -6.00000 q^{52} +2.53113 q^{53} +1.00000 q^{54} -4.53113 q^{56} -4.53113 q^{57} -9.06226 q^{59} +5.06226 q^{61} -1.00000 q^{62} +4.53113 q^{63} +1.00000 q^{64} +6.53113 q^{66} -5.06226 q^{67} +4.00000 q^{68} +6.53113 q^{69} -12.5311 q^{71} -1.00000 q^{72} +8.53113 q^{73} -7.06226 q^{74} +4.53113 q^{76} +29.5934 q^{77} -6.00000 q^{78} -8.53113 q^{79} +1.00000 q^{81} -7.06226 q^{82} +8.00000 q^{83} -4.53113 q^{84} -8.53113 q^{86} -6.53113 q^{88} -6.53113 q^{89} -27.1868 q^{91} -6.53113 q^{92} -1.00000 q^{93} -5.06226 q^{94} +1.00000 q^{96} -16.1245 q^{97} -13.5311 q^{98} +6.53113 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9} + 5 q^{11} - 2 q^{12} - 12 q^{13} - q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{18} + q^{19} - q^{21} - 5 q^{22} - 5 q^{23} + 2 q^{24}+ \cdots + 5 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 + 5 * q^11 - 2 * q^12 - 12 * q^13 - q^14 + 2 * q^16 + 8 * q^17 - 2 * q^18 + q^19 - q^21 - 5 * q^22 - 5 * q^23 + 2 * q^24 + 12 * q^26 - 2 * q^27 + q^28 + 2 * q^31 - 2 * q^32 - 5 * q^33 - 8 * q^34 + 2 * q^36 - 2 * q^37 - q^38 + 12 * q^39 - 2 * q^41 + q^42 + 9 * q^43 + 5 * q^44 + 5 * q^46 - 6 * q^47 - 2 * q^48 + 19 * q^49 - 8 * q^51 - 12 * q^52 - 3 * q^53 + 2 * q^54 - q^56 - q^57 - 2 * q^59 - 6 * q^61 - 2 * q^62 + q^63 + 2 * q^64 + 5 * q^66 + 6 * q^67 + 8 * q^68 + 5 * q^69 - 17 * q^71 - 2 * q^72 + 9 * q^73 + 2 * q^74 + q^76 + 35 * q^77 - 12 * q^78 - 9 * q^79 + 2 * q^81 + 2 * q^82 + 16 * q^83 - q^84 - 9 * q^86 - 5 * q^88 - 5 * q^89 - 6 * q^91 - 5 * q^92 - 2 * q^93 + 6 * q^94 + 2 * q^96 - 19 * q^98 + 5 * 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
$7$	4.53113	1.71261	0.856303	−	0.516474i	$-0.172756\pi$
$7$	4.53113	1.71261	0.856303	+	0.516474i	$0.172756\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	6.53113	1.96921	0.984605	−	0.174796i	$-0.0559265\pi$
$11$	6.53113	1.96921	0.984605	+	0.174796i	$0.0559265\pi$
$12$	−1.00000	−0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−4.53113	−1.21100
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	4.53113	1.03951	0.519756	−	0.854315i	$-0.326023\pi$
$19$	4.53113	1.03951	0.519756	+	0.854315i	$0.326023\pi$
$20$	0	0
$21$	−4.53113	−0.988773
$22$	−6.53113	−1.39244
$23$	−6.53113	−1.36183	−0.680917	−	0.732360i	$-0.738419\pi$
$23$	−6.53113	−1.36183	−0.680917	+	0.732360i	$0.738419\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	−1.00000	−0.192450
$28$	4.53113	0.856303
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	−6.53113	−1.13692
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	7.06226	1.16103	0.580514	−	0.814250i	$-0.302852\pi$
$37$	7.06226	1.16103	0.580514	+	0.814250i	$0.302852\pi$
$38$	−4.53113	−0.735046
$39$	6.00000	0.960769
$40$	0	0
$41$	7.06226	1.10294	0.551470	−	0.834195i	$-0.314067\pi$
$41$	7.06226	1.10294	0.551470	+	0.834195i	$0.314067\pi$
$42$	4.53113	0.699168
$43$	8.53113	1.30098	0.650492	−	0.759513i	$-0.274563\pi$
$43$	8.53113	1.30098	0.650492	+	0.759513i	$0.274563\pi$
$44$	6.53113	0.984605
$45$	0	0
$46$	6.53113	0.962962
$47$	5.06226	0.738406	0.369203	−	0.929349i	$-0.379631\pi$
$47$	5.06226	0.738406	0.369203	+	0.929349i	$0.379631\pi$
$48$	−1.00000	−0.144338
$49$	13.5311	1.93302
$50$	0	0
$51$	−4.00000	−0.560112
$52$	−6.00000	−0.832050
$53$	2.53113	0.347677	0.173839	−	0.984774i	$-0.444383\pi$
$53$	2.53113	0.347677	0.173839	+	0.984774i	$0.444383\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−4.53113	−0.605498
$57$	−4.53113	−0.600163
$58$	0	0
$59$	−9.06226	−1.17981	−0.589903	−	0.807474i	$-0.700834\pi$
$59$	−9.06226	−1.17981	−0.589903	+	0.807474i	$0.700834\pi$
$60$	0	0
$61$	5.06226	0.648156	0.324078	−	0.946030i	$-0.394946\pi$
$61$	5.06226	0.648156	0.324078	+	0.946030i	$0.394946\pi$
$62$	−1.00000	−0.127000
$63$	4.53113	0.570869
$64$	1.00000	0.125000
$65$	0	0
$66$	6.53113	0.803926
$67$	−5.06226	−0.618453	−0.309227	−	0.950988i	$-0.600070\pi$
$67$	−5.06226	−0.618453	−0.309227	+	0.950988i	$0.600070\pi$
$68$	4.00000	0.485071
$69$	6.53113	0.786256
$70$	0	0
$71$	−12.5311	−1.48717	−0.743586	−	0.668641i	$-0.766876\pi$
$71$	−12.5311	−1.48717	−0.743586	+	0.668641i	$0.766876\pi$
$72$	−1.00000	−0.117851
$73$	8.53113	0.998493	0.499247	−	0.866460i	$-0.333610\pi$
$73$	8.53113	0.998493	0.499247	+	0.866460i	$0.333610\pi$
$74$	−7.06226	−0.820971
$75$	0	0
$76$	4.53113	0.519756
$77$	29.5934	3.37248
$78$	−6.00000	−0.679366
$79$	−8.53113	−0.959827	−0.479913	−	0.877316i	$-0.659332\pi$
$79$	−8.53113	−0.959827	−0.479913	+	0.877316i	$0.659332\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−7.06226	−0.779896
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	−4.53113	−0.494387
$85$	0	0
$86$	−8.53113	−0.919935
$87$	0	0
$88$	−6.53113	−0.696221
$89$	−6.53113	−0.692298	−0.346149	−	0.938180i	$-0.612511\pi$
$89$	−6.53113	−0.692298	−0.346149	+	0.938180i	$0.612511\pi$
$90$	0	0
$91$	−27.1868	−2.84995
$92$	−6.53113	−0.680917
$93$	−1.00000	−0.103695
$94$	−5.06226	−0.522132
$95$	0	0
$96$	1.00000	0.102062
$97$	−16.1245	−1.63720	−0.818598	−	0.574367i	$-0.805248\pi$
$97$	−16.1245	−1.63720	−0.818598	+	0.574367i	$0.805248\pi$
$98$	−13.5311	−1.36685
$99$	6.53113	0.656403
$100$	0	0
$101$	−9.46887	−0.942188	−0.471094	−	0.882083i	$-0.656141\pi$
$101$	−9.46887	−0.942188	−0.471094	+	0.882083i	$0.656141\pi$
$102$	4.00000	0.396059
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−2.53113	−0.245845
$107$	−9.59339	−0.927428	−0.463714	−	0.885985i	$-0.653483\pi$
$107$	−9.59339	−0.927428	−0.463714	+	0.885985i	$0.653483\pi$
$108$	−1.00000	−0.0962250
$109$	15.0623	1.44270	0.721351	−	0.692569i	$-0.243521\pi$
$109$	15.0623	1.44270	0.721351	+	0.692569i	$0.243521\pi$
$110$	0	0
$111$	−7.06226	−0.670320
$112$	4.53113	0.428151
$113$	7.59339	0.714326	0.357163	−	0.934042i	$-0.383744\pi$
$113$	7.59339	0.714326	0.357163	+	0.934042i	$0.383744\pi$
$114$	4.53113	0.424379
$115$	0	0
$116$	0	0
$117$	−6.00000	−0.554700
$118$	9.06226	0.834248
$119$	18.1245	1.66147
$120$	0	0
$121$	31.6556	2.87779
$122$	−5.06226	−0.458315
$123$	−7.06226	−0.636782
$124$	1.00000	0.0898027
$125$	0	0
$126$	−4.53113	−0.403665
$127$	−7.06226	−0.626674	−0.313337	−	0.949642i	$-0.601447\pi$
$127$	−7.06226	−0.626674	−0.313337	+	0.949642i	$0.601447\pi$
$128$	−1.00000	−0.0883883
$129$	−8.53113	−0.751124
$130$	0	0
$131$	−9.06226	−0.791773	−0.395887	−	0.918299i	$-0.629563\pi$
$131$	−9.06226	−0.791773	−0.395887	+	0.918299i	$0.629563\pi$
$132$	−6.53113	−0.568462
$133$	20.5311	1.78027
$134$	5.06226	0.437312
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−17.0623	−1.45773	−0.728864	−	0.684659i	$-0.759951\pi$
$137$	−17.0623	−1.45773	−0.728864	+	0.684659i	$0.759951\pi$
$138$	−6.53113	−0.555967
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	0	0
$141$	−5.06226	−0.426319
$142$	12.5311	1.05159
$143$	−39.1868	−3.27696
$144$	1.00000	0.0833333
$145$	0	0
$146$	−8.53113	−0.706041
$147$	−13.5311	−1.11603
$148$	7.06226	0.580514
$149$	−18.5311	−1.51813	−0.759065	−	0.651015i	$-0.774343\pi$
$149$	−18.5311	−1.51813	−0.759065	+	0.651015i	$0.774343\pi$
$150$	0	0
$151$	14.1245	1.14944	0.574718	−	0.818351i	$-0.305112\pi$
$151$	14.1245	1.14944	0.574718	+	0.818351i	$0.305112\pi$
$152$	−4.53113	−0.367523
$153$	4.00000	0.323381
$154$	−29.5934	−2.38470
$155$	0	0
$156$	6.00000	0.480384
$157$	22.5311	1.79818	0.899090	−	0.437764i	$-0.144229\pi$
$157$	22.5311	1.79818	0.899090	+	0.437764i	$0.144229\pi$
$158$	8.53113	0.678700
$159$	−2.53113	−0.200732
$160$	0	0
$161$	−29.5934	−2.33229
$162$	−1.00000	−0.0785674
$163$	−5.06226	−0.396507	−0.198253	−	0.980151i	$-0.563527\pi$
$163$	−5.06226	−0.396507	−0.198253	+	0.980151i	$0.563527\pi$
$164$	7.06226	0.551470
$165$	0	0
$166$	−8.00000	−0.620920
$167$	23.5934	1.82571	0.912856	−	0.408283i	$-0.133872\pi$
$167$	23.5934	1.82571	0.912856	+	0.408283i	$0.133872\pi$
$168$	4.53113	0.349584
$169$	23.0000	1.76923
$170$	0	0
$171$	4.53113	0.346504
$172$	8.53113	0.650492
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	0	0
$176$	6.53113	0.492302
$177$	9.06226	0.681161
$178$	6.53113	0.489529
$179$	−3.06226	−0.228884	−0.114442	−	0.993430i	$-0.536508\pi$
$179$	−3.06226	−0.228884	−0.114442	+	0.993430i	$0.536508\pi$
$180$	0	0
$181$	2.40661	0.178882	0.0894411	−	0.995992i	$-0.471492\pi$
$181$	2.40661	0.178882	0.0894411	+	0.995992i	$0.471492\pi$
$182$	27.1868	2.01522
$183$	−5.06226	−0.374213
$184$	6.53113	0.481481
$185$	0	0
$186$	1.00000	0.0733236
$187$	26.1245	1.91041
$188$	5.06226	0.369203
$189$	−4.53113	−0.329591
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	16.1245	1.15767
$195$	0	0
$196$	13.5311	0.966509
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−6.53113	−0.464147
$199$	−8.53113	−0.604756	−0.302378	−	0.953188i	$-0.597780\pi$
$199$	−8.53113	−0.604756	−0.302378	+	0.953188i	$0.597780\pi$
$200$	0	0
$201$	5.06226	0.357064
$202$	9.46887	0.666227
$203$	0	0
$204$	−4.00000	−0.280056
$205$	0	0
$206$	0	0
$207$	−6.53113	−0.453945
$208$	−6.00000	−0.416025
$209$	29.5934	2.04702
$210$	0	0
$211$	−1.59339	−0.109693	−0.0548466	−	0.998495i	$-0.517467\pi$
$211$	−1.59339	−0.109693	−0.0548466	+	0.998495i	$0.517467\pi$
$212$	2.53113	0.173839
$213$	12.5311	0.858619
$214$	9.59339	0.655790
$215$	0	0
$216$	1.00000	0.0680414
$217$	4.53113	0.307593
$218$	−15.0623	−1.02014
$219$	−8.53113	−0.576480
$220$	0	0
$221$	−24.0000	−1.61441
$222$	7.06226	0.473988
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−4.53113	−0.302749
$225$	0	0
$226$	−7.59339	−0.505105
$227$	24.5311	1.62819	0.814094	−	0.580733i	$-0.197234\pi$
$227$	24.5311	1.62819	0.814094	+	0.580733i	$0.197234\pi$
$228$	−4.53113	−0.300081
$229$	−5.59339	−0.369621	−0.184811	−	0.982774i	$-0.559167\pi$
$229$	−5.59339	−0.369621	−0.184811	+	0.982774i	$0.559167\pi$
$230$	0	0
$231$	−29.5934	−1.94710
$232$	0	0
$233$	−1.46887	−0.0962289	−0.0481145	−	0.998842i	$-0.515321\pi$
$233$	−1.46887	−0.0962289	−0.0481145	+	0.998842i	$0.515321\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−9.06226	−0.589903
$237$	8.53113	0.554156
$238$	−18.1245	−1.17484
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−31.6556	−2.03490
$243$	−1.00000	−0.0641500
$244$	5.06226	0.324078
$245$	0	0
$246$	7.06226	0.450273
$247$	−27.1868	−1.72985
$248$	−1.00000	−0.0635001
$249$	−8.00000	−0.506979
$250$	0	0
$251$	3.06226	0.193288	0.0966440	−	0.995319i	$-0.469189\pi$
$251$	3.06226	0.193288	0.0966440	+	0.995319i	$0.469189\pi$
$252$	4.53113	0.285434
$253$	−42.6556	−2.68174
$254$	7.06226	0.443125
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.6556	−0.789437	−0.394719	−	0.918802i	$-0.629158\pi$
$257$	−12.6556	−0.789437	−0.394719	+	0.918802i	$0.629158\pi$
$258$	8.53113	0.531125
$259$	32.0000	1.98838
$260$	0	0
$261$	0	0
$262$	9.06226	0.559868
$263$	23.0623	1.42208	0.711040	−	0.703152i	$-0.248225\pi$
$263$	23.0623	1.42208	0.711040	+	0.703152i	$0.248225\pi$
$264$	6.53113	0.401963
$265$	0	0
$266$	−20.5311	−1.25884
$267$	6.53113	0.399699
$268$	−5.06226	−0.309227
$269$	−19.1868	−1.16984	−0.584919	−	0.811092i	$-0.698874\pi$
$269$	−19.1868	−1.16984	−0.584919	+	0.811092i	$0.698874\pi$
$270$	0	0
$271$	−11.4689	−0.696684	−0.348342	−	0.937367i	$-0.613255\pi$
$271$	−11.4689	−0.696684	−0.348342	+	0.937367i	$0.613255\pi$
$272$	4.00000	0.242536
$273$	27.1868	1.64542
$274$	17.0623	1.03077
$275$	0	0
$276$	6.53113	0.393128
$277$	15.0623	0.905003	0.452502	−	0.891764i	$-0.350532\pi$
$277$	15.0623	0.905003	0.452502	+	0.891764i	$0.350532\pi$
$278$	−6.00000	−0.359856
$279$	1.00000	0.0598684
$280$	0	0
$281$	15.0623	0.898539	0.449269	−	0.893396i	$-0.351684\pi$
$281$	15.0623	0.898539	0.449269	+	0.893396i	$0.351684\pi$
$282$	5.06226	0.301453
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−12.5311	−0.743586
$285$	0	0
$286$	39.1868	2.31716
$287$	32.0000	1.88890
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	16.1245	0.945236
$292$	8.53113	0.499247
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	13.5311	0.789151
$295$	0	0
$296$	−7.06226	−0.410485
$297$	−6.53113	−0.378975
$298$	18.5311	1.07348
$299$	39.1868	2.26623
$300$	0	0
$301$	38.6556	2.22807
$302$	−14.1245	−0.812775
$303$	9.46887	0.543972
$304$	4.53113	0.259878
$305$	0	0
$306$	−4.00000	−0.228665
$307$	30.1245	1.71930	0.859648	−	0.510886i	$-0.170683\pi$
$307$	30.1245	1.71930	0.859648	+	0.510886i	$0.170683\pi$
$308$	29.5934	1.68624
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	−6.00000	−0.339683
$313$	−10.9377	−0.618238	−0.309119	−	0.951023i	$-0.600034\pi$
$313$	−10.9377	−0.618238	−0.309119	+	0.951023i	$0.600034\pi$
$314$	−22.5311	−1.27151
$315$	0	0
$316$	−8.53113	−0.479913
$317$	−12.9377	−0.726656	−0.363328	−	0.931661i	$-0.618360\pi$
$317$	−12.9377	−0.726656	−0.363328	+	0.931661i	$0.618360\pi$
$318$	2.53113	0.141939
$319$	0	0
$320$	0	0
$321$	9.59339	0.535451
$322$	29.5934	1.64917
$323$	18.1245	1.00848
$324$	1.00000	0.0555556
$325$	0	0
$326$	5.06226	0.280373
$327$	−15.0623	−0.832945
$328$	−7.06226	−0.389948
$329$	22.9377	1.26460
$330$	0	0
$331$	12.9377	0.711123	0.355561	−	0.934653i	$-0.384290\pi$
$331$	12.9377	0.711123	0.355561	+	0.934653i	$0.384290\pi$
$332$	8.00000	0.439057
$333$	7.06226	0.387009
$334$	−23.5934	−1.29097
$335$	0	0
$336$	−4.53113	−0.247193
$337$	19.1868	1.04517	0.522585	−	0.852587i	$-0.324968\pi$
$337$	19.1868	1.04517	0.522585	+	0.852587i	$0.324968\pi$
$338$	−23.0000	−1.25104
$339$	−7.59339	−0.412416
$340$	0	0
$341$	6.53113	0.353680
$342$	−4.53113	−0.245015
$343$	29.5934	1.59789
$344$	−8.53113	−0.459968
$345$	0	0
$346$	14.0000	0.752645
$347$	−10.1245	−0.543512	−0.271756	−	0.962366i	$-0.587604\pi$
$347$	−10.1245	−0.543512	−0.271756	+	0.962366i	$0.587604\pi$
$348$	0	0
$349$	8.93774	0.478426	0.239213	−	0.970967i	$-0.423111\pi$
$349$	8.93774	0.478426	0.239213	+	0.970967i	$0.423111\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	−6.53113	−0.348110
$353$	6.93774	0.369259	0.184629	−	0.982808i	$-0.440892\pi$
$353$	6.93774	0.369259	0.184629	+	0.982808i	$0.440892\pi$
$354$	−9.06226	−0.481654
$355$	0	0
$356$	−6.53113	−0.346149
$357$	−18.1245	−0.959251
$358$	3.06226	0.161845
$359$	3.46887	0.183080	0.0915400	−	0.995801i	$-0.470821\pi$
$359$	3.46887	0.183080	0.0915400	+	0.995801i	$0.470821\pi$
$360$	0	0
$361$	1.53113	0.0805857
$362$	−2.40661	−0.126489
$363$	−31.6556	−1.66149
$364$	−27.1868	−1.42497
$365$	0	0
$366$	5.06226	0.264608
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	−6.53113	−0.340459
$369$	7.06226	0.367646
$370$	0	0
$371$	11.4689	0.595434
$372$	−1.00000	−0.0518476
$373$	18.5311	0.959505	0.479753	−	0.877404i	$-0.340726\pi$
$373$	18.5311	0.959505	0.479753	+	0.877404i	$0.340726\pi$
$374$	−26.1245	−1.35087
$375$	0	0
$376$	−5.06226	−0.261066
$377$	0	0
$378$	4.53113	0.233056
$379$	21.5934	1.10918	0.554589	−	0.832124i	$-0.312876\pi$
$379$	21.5934	1.10918	0.554589	+	0.832124i	$0.312876\pi$
$380$	0	0
$381$	7.06226	0.361810
$382$	8.00000	0.409316
$383$	3.06226	0.156474	0.0782370	−	0.996935i	$-0.475071\pi$
$383$	3.06226	0.156474	0.0782370	+	0.996935i	$0.475071\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	8.53113	0.433662
$388$	−16.1245	−0.818598
$389$	10.9377	0.554566	0.277283	−	0.960788i	$-0.410566\pi$
$389$	10.9377	0.554566	0.277283	+	0.960788i	$0.410566\pi$
$390$	0	0
$391$	−26.1245	−1.32117
$392$	−13.5311	−0.683425
$393$	9.06226	0.457130
$394$	6.00000	0.302276
$395$	0	0
$396$	6.53113	0.328202
$397$	−37.7179	−1.89301	−0.946504	−	0.322693i	$-0.895412\pi$
$397$	−37.7179	−1.89301	−0.946504	+	0.322693i	$0.895412\pi$
$398$	8.53113	0.427627
$399$	−20.5311	−1.02784
$400$	0	0
$401$	21.7179	1.08454	0.542270	−	0.840204i	$-0.317565\pi$
$401$	21.7179	1.08454	0.542270	+	0.840204i	$0.317565\pi$
$402$	−5.06226	−0.252482
$403$	−6.00000	−0.298881
$404$	−9.46887	−0.471094
$405$	0	0
$406$	0	0
$407$	46.1245	2.28631
$408$	4.00000	0.198030
$409$	7.06226	0.349206	0.174603	−	0.984639i	$-0.444136\pi$
$409$	7.06226	0.349206	0.174603	+	0.984639i	$0.444136\pi$
$410$	0	0
$411$	17.0623	0.841619
$412$	0	0
$413$	−41.0623	−2.02054
$414$	6.53113	0.320987
$415$	0	0
$416$	6.00000	0.294174
$417$	−6.00000	−0.293821
$418$	−29.5934	−1.44746
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−26.2490	−1.27930	−0.639650	−	0.768667i	$-0.720921\pi$
$421$	−26.2490	−1.27930	−0.639650	+	0.768667i	$0.720921\pi$
$422$	1.59339	0.0775648
$423$	5.06226	0.246135
$424$	−2.53113	−0.122922
$425$	0	0
$426$	−12.5311	−0.607135
$427$	22.9377	1.11004
$428$	−9.59339	−0.463714
$429$	39.1868	1.89196
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−1.00000	−0.0481125
$433$	25.5934	1.22994	0.614970	−	0.788551i	$-0.289168\pi$
$433$	25.5934	1.22994	0.614970	+	0.788551i	$0.289168\pi$
$434$	−4.53113	−0.217501
$435$	0	0
$436$	15.0623	0.721351
$437$	−29.5934	−1.41564
$438$	8.53113	0.407633
$439$	−25.0623	−1.19616	−0.598078	−	0.801438i	$-0.704069\pi$
$439$	−25.0623	−1.19616	−0.598078	+	0.801438i	$0.704069\pi$
$440$	0	0
$441$	13.5311	0.644339
$442$	24.0000	1.14156
$443$	−14.4066	−0.684479	−0.342239	−	0.939613i	$-0.611185\pi$
$443$	−14.4066	−0.684479	−0.342239	+	0.939613i	$0.611185\pi$
$444$	−7.06226	−0.335160
$445$	0	0
$446$	2.00000	0.0947027
$447$	18.5311	0.876492
$448$	4.53113	0.214076
$449$	−4.12452	−0.194648	−0.0973240	−	0.995253i	$-0.531028\pi$
$449$	−4.12452	−0.194648	−0.0973240	+	0.995253i	$0.531028\pi$
$450$	0	0
$451$	46.1245	2.17192
$452$	7.59339	0.357163
$453$	−14.1245	−0.663628
$454$	−24.5311	−1.15130
$455$	0	0
$456$	4.53113	0.212190
$457$	−14.9377	−0.698758	−0.349379	−	0.936981i	$-0.613607\pi$
$457$	−14.9377	−0.698758	−0.349379	+	0.936981i	$0.613607\pi$
$458$	5.59339	0.261362
$459$	−4.00000	−0.186704
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	29.5934	1.37681
$463$	−13.1868	−0.612841	−0.306421	−	0.951896i	$-0.599131\pi$
$463$	−13.1868	−0.612841	−0.306421	+	0.951896i	$0.599131\pi$
$464$	0	0
$465$	0	0
$466$	1.46887	0.0680441
$467$	22.1245	1.02380	0.511900	−	0.859045i	$-0.328942\pi$
$467$	22.1245	1.02380	0.511900	+	0.859045i	$0.328942\pi$
$468$	−6.00000	−0.277350
$469$	−22.9377	−1.05917
$470$	0	0
$471$	−22.5311	−1.03818
$472$	9.06226	0.417124
$473$	55.7179	2.56191
$474$	−8.53113	−0.391848
$475$	0	0
$476$	18.1245	0.830736
$477$	2.53113	0.115892
$478$	8.00000	0.365911
$479$	1.34436	0.0614252	0.0307126	−	0.999528i	$-0.490222\pi$
$479$	1.34436	0.0614252	0.0307126	+	0.999528i	$0.490222\pi$
$480$	0	0
$481$	−42.3735	−1.93207
$482$	−2.00000	−0.0910975
$483$	29.5934	1.34655
$484$	31.6556	1.43889
$485$	0	0
$486$	1.00000	0.0453609
$487$	−23.0623	−1.04505	−0.522525	−	0.852624i	$-0.675010\pi$
$487$	−23.0623	−1.04505	−0.522525	+	0.852624i	$0.675010\pi$
$488$	−5.06226	−0.229158
$489$	5.06226	0.228923
$490$	0	0
$491$	−7.59339	−0.342685	−0.171342	−	0.985212i	$-0.554810\pi$
$491$	−7.59339	−0.342685	−0.171342	+	0.985212i	$0.554810\pi$
$492$	−7.06226	−0.318391
$493$	0	0
$494$	27.1868	1.22319
$495$	0	0
$496$	1.00000	0.0449013
$497$	−56.7802	−2.54694
$498$	8.00000	0.358489
$499$	−7.06226	−0.316150	−0.158075	−	0.987427i	$-0.550529\pi$
$499$	−7.06226	−0.316150	−0.158075	+	0.987427i	$0.550529\pi$
$500$	0	0
$501$	−23.5934	−1.05407
$502$	−3.06226	−0.136675
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	−4.53113	−0.201833
$505$	0	0
$506$	42.6556	1.89627
$507$	−23.0000	−1.02147
$508$	−7.06226	−0.313337
$509$	38.1245	1.68984	0.844920	−	0.534893i	$-0.179648\pi$
$509$	38.1245	1.68984	0.844920	+	0.534893i	$0.179648\pi$
$510$	0	0
$511$	38.6556	1.71003
$512$	−1.00000	−0.0441942
$513$	−4.53113	−0.200054
$514$	12.6556	0.558217
$515$	0	0
$516$	−8.53113	−0.375562
$517$	33.0623	1.45408
$518$	−32.0000	−1.40600
$519$	14.0000	0.614532
$520$	0	0
$521$	−12.1245	−0.531185	−0.265592	−	0.964085i	$-0.585568\pi$
$521$	−12.1245	−0.531185	−0.265592	+	0.964085i	$0.585568\pi$
$522$	0	0
$523$	−9.59339	−0.419490	−0.209745	−	0.977756i	$-0.567263\pi$
$523$	−9.59339	−0.419490	−0.209745	+	0.977756i	$0.567263\pi$
$524$	−9.06226	−0.395887
$525$	0	0
$526$	−23.0623	−1.00556
$527$	4.00000	0.174243
$528$	−6.53113	−0.284231
$529$	19.6556	0.854593
$530$	0	0
$531$	−9.06226	−0.393268
$532$	20.5311	0.890137
$533$	−42.3735	−1.83540
$534$	−6.53113	−0.282630
$535$	0	0
$536$	5.06226	0.218656
$537$	3.06226	0.132146
$538$	19.1868	0.827201
$539$	88.3735	3.80652
$540$	0	0
$541$	−4.12452	−0.177327	−0.0886634	−	0.996062i	$-0.528260\pi$
$541$	−4.12452	−0.177327	−0.0886634	+	0.996062i	$0.528260\pi$
$542$	11.4689	0.492630
$543$	−2.40661	−0.103278
$544$	−4.00000	−0.171499
$545$	0	0
$546$	−27.1868	−1.16349
$547$	7.18677	0.307284	0.153642	−	0.988127i	$-0.450900\pi$
$547$	7.18677	0.307284	0.153642	+	0.988127i	$0.450900\pi$
$548$	−17.0623	−0.728864
$549$	5.06226	0.216052
$550$	0	0
$551$	0	0
$552$	−6.53113	−0.277983
$553$	−38.6556	−1.64381
$554$	−15.0623	−0.639934
$555$	0	0
$556$	6.00000	0.254457
$557$	26.7802	1.13471	0.567356	−	0.823473i	$-0.307966\pi$
$557$	26.7802	1.13471	0.567356	+	0.823473i	$0.307966\pi$
$558$	−1.00000	−0.0423334
$559$	−51.1868	−2.16497
$560$	0	0
$561$	−26.1245	−1.10298
$562$	−15.0623	−0.635363
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	−5.06226	−0.213160
$565$	0	0
$566$	−12.0000	−0.504398
$567$	4.53113	0.190290
$568$	12.5311	0.525794
$569$	9.46887	0.396956	0.198478	−	0.980105i	$-0.436400\pi$
$569$	9.46887	0.396956	0.198478	+	0.980105i	$0.436400\pi$
$570$	0	0
$571$	36.1245	1.51176	0.755882	−	0.654708i	$-0.227208\pi$
$571$	36.1245	1.51176	0.755882	+	0.654708i	$0.227208\pi$
$572$	−39.1868	−1.63848
$573$	8.00000	0.334205
$574$	−32.0000	−1.33565
$575$	0	0
$576$	1.00000	0.0416667
$577$	−37.1868	−1.54811	−0.774053	−	0.633121i	$-0.781774\pi$
$577$	−37.1868	−1.54811	−0.774053	+	0.633121i	$0.781774\pi$
$578$	1.00000	0.0415945
$579$	14.0000	0.581820
$580$	0	0
$581$	36.2490	1.50386
$582$	−16.1245	−0.668383
$583$	16.5311	0.684649
$584$	−8.53113	−0.353021
$585$	0	0
$586$	−14.0000	−0.578335
$587$	44.2490	1.82635	0.913176	−	0.407564i	$-0.133622\pi$
$587$	44.2490	1.82635	0.913176	+	0.407564i	$0.133622\pi$
$588$	−13.5311	−0.558014
$589$	4.53113	0.186702
$590$	0	0
$591$	6.00000	0.246807
$592$	7.06226	0.290257
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	6.53113	0.267975
$595$	0	0
$596$	−18.5311	−0.759065
$597$	8.53113	0.349156
$598$	−39.1868	−1.60247
$599$	−13.5934	−0.555411	−0.277705	−	0.960666i	$-0.589574\pi$
$599$	−13.5934	−0.555411	−0.277705	+	0.960666i	$0.589574\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	−38.6556	−1.57549
$603$	−5.06226	−0.206151
$604$	14.1245	0.574718
$605$	0	0
$606$	−9.46887	−0.384647
$607$	14.4066	0.584746	0.292373	−	0.956304i	$-0.405555\pi$
$607$	14.4066	0.584746	0.292373	+	0.956304i	$0.405555\pi$
$608$	−4.53113	−0.183762
$609$	0	0
$610$	0	0
$611$	−30.3735	−1.22878
$612$	4.00000	0.161690
$613$	30.0000	1.21169	0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000	1.21169	0.605844	+	0.795583i	$0.292835\pi$
$614$	−30.1245	−1.21573
$615$	0	0
$616$	−29.5934	−1.19235
$617$	3.59339	0.144664	0.0723321	−	0.997381i	$-0.476956\pi$
$617$	3.59339	0.144664	0.0723321	+	0.997381i	$0.476956\pi$
$618$	0	0
$619$	−34.0000	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$619$	−34.0000	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$620$	0	0
$621$	6.53113	0.262085
$622$	8.00000	0.320771
$623$	−29.5934	−1.18563
$624$	6.00000	0.240192
$625$	0	0
$626$	10.9377	0.437160
$627$	−29.5934	−1.18185
$628$	22.5311	0.899090
$629$	28.2490	1.12636
$630$	0	0
$631$	30.6556	1.22038	0.610191	−	0.792254i	$-0.291093\pi$
$631$	30.6556	1.22038	0.610191	+	0.792254i	$0.291093\pi$
$632$	8.53113	0.339350
$633$	1.59339	0.0633314
$634$	12.9377	0.513823
$635$	0	0
$636$	−2.53113	−0.100366
$637$	−81.1868	−3.21674
$638$	0	0
$639$	−12.5311	−0.495724
$640$	0	0
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	−9.59339	−0.378621
$643$	17.5934	0.693815	0.346908	−	0.937899i	$-0.387232\pi$
$643$	17.5934	0.693815	0.346908	+	0.937899i	$0.387232\pi$
$644$	−29.5934	−1.16614
$645$	0	0
$646$	−18.1245	−0.713100
$647$	6.53113	0.256765	0.128383	−	0.991725i	$-0.459021\pi$
$647$	6.53113	0.256765	0.128383	+	0.991725i	$0.459021\pi$
$648$	−1.00000	−0.0392837
$649$	−59.1868	−2.32328
$650$	0	0
$651$	−4.53113	−0.177589
$652$	−5.06226	−0.198253
$653$	−38.2490	−1.49680	−0.748400	−	0.663248i	$-0.769178\pi$
$653$	−38.2490	−1.49680	−0.748400	+	0.663248i	$0.769178\pi$
$654$	15.0623	0.588981
$655$	0	0
$656$	7.06226	0.275735
$657$	8.53113	0.332831
$658$	−22.9377	−0.894206
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−12.9377	−0.502840
$663$	24.0000	0.932083
$664$	−8.00000	−0.310460
$665$	0	0
$666$	−7.06226	−0.273657
$667$	0	0
$668$	23.5934	0.912856
$669$	2.00000	0.0773245
$670$	0	0
$671$	33.0623	1.27635
$672$	4.53113	0.174792
$673$	9.06226	0.349324	0.174662	−	0.984628i	$-0.444117\pi$
$673$	9.06226	0.349324	0.174662	+	0.984628i	$0.444117\pi$
$674$	−19.1868	−0.739047
$675$	0	0
$676$	23.0000	0.884615
$677$	−27.5934	−1.06050	−0.530250	−	0.847841i	$-0.677902\pi$
$677$	−27.5934	−1.06050	−0.530250	+	0.847841i	$0.677902\pi$
$678$	7.59339	0.291622
$679$	−73.0623	−2.80387
$680$	0	0
$681$	−24.5311	−0.940035
$682$	−6.53113	−0.250090
$683$	−7.46887	−0.285788	−0.142894	−	0.989738i	$-0.545641\pi$
$683$	−7.46887	−0.285788	−0.142894	+	0.989738i	$0.545641\pi$
$684$	4.53113	0.173252
$685$	0	0
$686$	−29.5934	−1.12988
$687$	5.59339	0.213401
$688$	8.53113	0.325246
$689$	−15.1868	−0.578570
$690$	0	0
$691$	0.531129	0.0202051	0.0101025	−	0.999949i	$-0.496784\pi$
$691$	0.531129	0.0202051	0.0101025	+	0.999949i	$0.496784\pi$
$692$	−14.0000	−0.532200
$693$	29.5934	1.12416
$694$	10.1245	0.384321
$695$	0	0
$696$	0	0
$697$	28.2490	1.07001
$698$	−8.93774	−0.338299
$699$	1.46887	0.0555578
$700$	0	0
$701$	−13.4689	−0.508712	−0.254356	−	0.967111i	$-0.581864\pi$
$701$	−13.4689	−0.508712	−0.254356	+	0.967111i	$0.581864\pi$
$702$	−6.00000	−0.226455
$703$	32.0000	1.20690
$704$	6.53113	0.246151
$705$	0	0
$706$	−6.93774	−0.261105
$707$	−42.9047	−1.61360
$708$	9.06226	0.340581
$709$	49.5934	1.86252	0.931259	−	0.364357i	$-0.118711\pi$
$709$	49.5934	1.86252	0.931259	+	0.364357i	$0.118711\pi$
$710$	0	0
$711$	−8.53113	−0.319942
$712$	6.53113	0.244764
$713$	−6.53113	−0.244593
$714$	18.1245	0.678293
$715$	0	0
$716$	−3.06226	−0.114442
$717$	8.00000	0.298765
$718$	−3.46887	−0.129457
$719$	15.1868	0.566371	0.283186	−	0.959065i	$-0.408609\pi$
$719$	15.1868	0.566371	0.283186	+	0.959065i	$0.408609\pi$
$720$	0	0
$721$	0	0
$722$	−1.53113	−0.0569827
$723$	−2.00000	−0.0743808
$724$	2.40661	0.0894411
$725$	0	0
$726$	31.6556	1.17485
$727$	−37.8424	−1.40350	−0.701749	−	0.712424i	$-0.747597\pi$
$727$	−37.8424	−1.40350	−0.701749	+	0.712424i	$0.747597\pi$
$728$	27.1868	1.00761
$729$	1.00000	0.0370370
$730$	0	0
$731$	34.1245	1.26214
$732$	−5.06226	−0.187106
$733$	28.1245	1.03880	0.519401	−	0.854530i	$-0.326155\pi$
$733$	28.1245	1.03880	0.519401	+	0.854530i	$0.326155\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	6.53113	0.240741
$737$	−33.0623	−1.21786
$738$	−7.06226	−0.259965
$739$	−21.1868	−0.779368	−0.389684	−	0.920949i	$-0.627416\pi$
$739$	−21.1868	−0.779368	−0.389684	+	0.920949i	$0.627416\pi$
$740$	0	0
$741$	27.1868	0.998731
$742$	−11.4689	−0.421036
$743$	28.6556	1.05127	0.525637	−	0.850709i	$-0.323827\pi$
$743$	28.6556	1.05127	0.525637	+	0.850709i	$0.323827\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	−18.5311	−0.678473
$747$	8.00000	0.292705
$748$	26.1245	0.955207
$749$	−43.4689	−1.58832
$750$	0	0
$751$	30.9377	1.12893	0.564467	−	0.825456i	$-0.309082\pi$
$751$	30.9377	1.12893	0.564467	+	0.825456i	$0.309082\pi$
$752$	5.06226	0.184602
$753$	−3.06226	−0.111595
$754$	0	0
$755$	0	0
$756$	−4.53113	−0.164796
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	−21.5934	−0.784307
$759$	42.6556	1.54830
$760$	0	0
$761$	11.5934	0.420260	0.210130	−	0.977673i	$-0.432611\pi$
$761$	11.5934	0.420260	0.210130	+	0.977673i	$0.432611\pi$
$762$	−7.06226	−0.255839
$763$	68.2490	2.47078
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−3.06226	−0.110644
$767$	54.3735	1.96331
$768$	−1.00000	−0.0360844
$769$	−4.40661	−0.158907	−0.0794533	−	0.996839i	$-0.525317\pi$
$769$	−4.40661	−0.158907	−0.0794533	+	0.996839i	$0.525317\pi$
$770$	0	0
$771$	12.6556	0.455782
$772$	−14.0000	−0.503871
$773$	−22.5311	−0.810388	−0.405194	−	0.914231i	$-0.632796\pi$
$773$	−22.5311	−0.810388	−0.405194	+	0.914231i	$0.632796\pi$
$774$	−8.53113	−0.306645
$775$	0	0
$776$	16.1245	0.578836
$777$	−32.0000	−1.14799
$778$	−10.9377	−0.392137
$779$	32.0000	1.14652
$780$	0	0
$781$	−81.8424	−2.92855
$782$	26.1245	0.934211
$783$	0	0
$784$	13.5311	0.483255
$785$	0	0
$786$	−9.06226	−0.323240
$787$	36.5311	1.30219	0.651097	−	0.758994i	$-0.274309\pi$
$787$	36.5311	1.30219	0.651097	+	0.758994i	$0.274309\pi$
$788$	−6.00000	−0.213741
$789$	−23.0623	−0.821038
$790$	0	0
$791$	34.4066	1.22336
$792$	−6.53113	−0.232074
$793$	−30.3735	−1.07860
$794$	37.7179	1.33856
$795$	0	0
$796$	−8.53113	−0.302378
$797$	−12.1245	−0.429472	−0.214736	−	0.976672i	$-0.568889\pi$
$797$	−12.1245	−0.429472	−0.214736	+	0.976672i	$0.568889\pi$
$798$	20.5311	0.726794
$799$	20.2490	0.716359
$800$	0	0
$801$	−6.53113	−0.230766
$802$	−21.7179	−0.766886
$803$	55.7179	1.96624
$804$	5.06226	0.178532
$805$	0	0
$806$	6.00000	0.211341
$807$	19.1868	0.675406
$808$	9.46887	0.333114
$809$	15.5934	0.548234	0.274117	−	0.961696i	$-0.411614\pi$
$809$	15.5934	0.548234	0.274117	+	0.961696i	$0.411614\pi$
$810$	0	0
$811$	32.7802	1.15107	0.575534	−	0.817778i	$-0.304794\pi$
$811$	32.7802	1.15107	0.575534	+	0.817778i	$0.304794\pi$
$812$	0	0
$813$	11.4689	0.402231
$814$	−46.1245	−1.61666
$815$	0	0
$816$	−4.00000	−0.140028
$817$	38.6556	1.35239
$818$	−7.06226	−0.246926
$819$	−27.1868	−0.949983
$820$	0	0
$821$	42.1245	1.47016	0.735078	−	0.677983i	$-0.237146\pi$
$821$	42.1245	1.47016	0.735078	+	0.677983i	$0.237146\pi$
$822$	−17.0623	−0.595115
$823$	−40.1245	−1.39865	−0.699326	−	0.714803i	$-0.746517\pi$
$823$	−40.1245	−1.39865	−0.699326	+	0.714803i	$0.746517\pi$
$824$	0	0
$825$	0	0
$826$	41.0623	1.42874
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	−6.53113	−0.226972
$829$	30.6556	1.06471	0.532357	−	0.846520i	$-0.321306\pi$
$829$	30.6556	1.06471	0.532357	+	0.846520i	$0.321306\pi$
$830$	0	0
$831$	−15.0623	−0.522504
$832$	−6.00000	−0.208013
$833$	54.1245	1.87530
$834$	6.00000	0.207763
$835$	0	0
$836$	29.5934	1.02351
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	33.8424	1.16837	0.584185	−	0.811621i	$-0.301414\pi$
$839$	33.8424	1.16837	0.584185	+	0.811621i	$0.301414\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	26.2490	0.904601
$843$	−15.0623	−0.518772
$844$	−1.59339	−0.0548466
$845$	0	0
$846$	−5.06226	−0.174044
$847$	143.436	4.92851
$848$	2.53113	0.0869193
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−46.1245	−1.58113
$852$	12.5311	0.429309
$853$	−57.7179	−1.97622	−0.988112	−	0.153738i	$-0.950869\pi$
$853$	−57.7179	−1.97622	−0.988112	+	0.153738i	$0.950869\pi$
$854$	−22.9377	−0.784913
$855$	0	0
$856$	9.59339	0.327895
$857$	−14.0000	−0.478231	−0.239115	−	0.970991i	$-0.576857\pi$
$857$	−14.0000	−0.478231	−0.239115	+	0.970991i	$0.576857\pi$
$858$	−39.1868	−1.33781
$859$	−23.0623	−0.786874	−0.393437	−	0.919352i	$-0.628714\pi$
$859$	−23.0623	−0.786874	−0.393437	+	0.919352i	$0.628714\pi$
$860$	0	0
$861$	−32.0000	−1.09056
$862$	24.0000	0.817443
$863$	−33.4689	−1.13929	−0.569647	−	0.821890i	$-0.692920\pi$
$863$	−33.4689	−1.13929	−0.569647	+	0.821890i	$0.692920\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−25.5934	−0.869699
$867$	1.00000	0.0339618
$868$	4.53113	0.153797
$869$	−55.7179	−1.89010
$870$	0	0
$871$	30.3735	1.02917
$872$	−15.0623	−0.510072
$873$	−16.1245	−0.545732
$874$	29.5934	1.00101
$875$	0	0
$876$	−8.53113	−0.288240
$877$	−11.8755	−0.401007	−0.200503	−	0.979693i	$-0.564258\pi$
$877$	−11.8755	−0.401007	−0.200503	+	0.979693i	$0.564258\pi$
$878$	25.0623	0.845810
$879$	−14.0000	−0.472208
$880$	0	0
$881$	3.87548	0.130568	0.0652842	−	0.997867i	$-0.479205\pi$
$881$	3.87548	0.130568	0.0652842	+	0.997867i	$0.479205\pi$
$882$	−13.5311	−0.455617
$883$	13.3444	0.449073	0.224537	−	0.974466i	$-0.427913\pi$
$883$	13.3444	0.449073	0.224537	+	0.974466i	$0.427913\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	14.4066	0.484000
$887$	−23.1868	−0.778536	−0.389268	−	0.921125i	$-0.627272\pi$
$887$	−23.1868	−0.778536	−0.389268	+	0.921125i	$0.627272\pi$
$888$	7.06226	0.236994
$889$	−32.0000	−1.07325
$890$	0	0
$891$	6.53113	0.218801
$892$	−2.00000	−0.0669650
$893$	22.9377	0.767582
$894$	−18.5311	−0.619774
$895$	0	0
$896$	−4.53113	−0.151374
$897$	−39.1868	−1.30841
$898$	4.12452	0.137637
$899$	0	0
$900$	0	0
$901$	10.1245	0.337297
$902$	−46.1245	−1.53578
$903$	−38.6556	−1.28638
$904$	−7.59339	−0.252552
$905$	0	0
$906$	14.1245	0.469256
$907$	−32.2490	−1.07081	−0.535406	−	0.844595i	$-0.679841\pi$
$907$	−32.2490	−1.07081	−0.535406	+	0.844595i	$0.679841\pi$
$908$	24.5311	0.814094
$909$	−9.46887	−0.314063
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	−4.53113	−0.150041
$913$	52.2490	1.72919
$914$	14.9377	0.494097
$915$	0	0
$916$	−5.59339	−0.184811
$917$	−41.0623	−1.35600
$918$	4.00000	0.132020
$919$	−25.0623	−0.826728	−0.413364	−	0.910566i	$-0.635646\pi$
$919$	−25.0623	−0.826728	−0.413364	+	0.910566i	$0.635646\pi$
$920$	0	0
$921$	−30.1245	−0.992637
$922$	−24.0000	−0.790398
$923$	75.1868	2.47480
$924$	−29.5934	−0.973551
$925$	0	0
$926$	13.1868	0.433344
$927$	0	0
$928$	0	0
$929$	−51.8424	−1.70089	−0.850447	−	0.526060i	$-0.823669\pi$
$929$	−51.8424	−1.70089	−0.850447	+	0.526060i	$0.823669\pi$
$930$	0	0
$931$	61.3113	2.00940
$932$	−1.46887	−0.0481145
$933$	8.00000	0.261908
$934$	−22.1245	−0.723936
$935$	0	0
$936$	6.00000	0.196116
$937$	8.93774	0.291983	0.145992	−	0.989286i	$-0.453363\pi$
$937$	8.93774	0.291983	0.145992	+	0.989286i	$0.453363\pi$
$938$	22.9377	0.748944
$939$	10.9377	0.356940
$940$	0	0
$941$	−54.1245	−1.76441	−0.882204	−	0.470867i	$-0.843941\pi$
$941$	−54.1245	−1.76441	−0.882204	+	0.470867i	$0.843941\pi$
$942$	22.5311	0.734104
$943$	−46.1245	−1.50202
$944$	−9.06226	−0.294951
$945$	0	0
$946$	−55.7179	−1.81155
$947$	−18.9377	−0.615394	−0.307697	−	0.951484i	$-0.599558\pi$
$947$	−18.9377	−0.615394	−0.307697	+	0.951484i	$0.599558\pi$
$948$	8.53113	0.277078
$949$	−51.1868	−1.66159
$950$	0	0
$951$	12.9377	0.419535
$952$	−18.1245	−0.587419
$953$	−21.8755	−0.708616	−0.354308	−	0.935129i	$-0.615284\pi$
$953$	−21.8755	−0.708616	−0.354308	+	0.935129i	$0.615284\pi$
$954$	−2.53113	−0.0819483
$955$	0	0
$956$	−8.00000	−0.258738
$957$	0	0
$958$	−1.34436	−0.0434342
$959$	−77.3113	−2.49651
$960$	0	0
$961$	1.00000	0.0322581
$962$	42.3735	1.36618
$963$	−9.59339	−0.309143
$964$	2.00000	0.0644157
$965$	0	0
$966$	−29.5934	−0.952152
$967$	0.937742	0.0301558	0.0150779	−	0.999886i	$-0.495200\pi$
$967$	0.937742	0.0301558	0.0150779	+	0.999886i	$0.495200\pi$
$968$	−31.6556	−1.01745
$969$	−18.1245	−0.582243
$970$	0	0
$971$	15.1868	0.487367	0.243683	−	0.969855i	$-0.421644\pi$
$971$	15.1868	0.487367	0.243683	+	0.969855i	$0.421644\pi$
$972$	−1.00000	−0.0320750
$973$	27.1868	0.871568
$974$	23.0623	0.738962
$975$	0	0
$976$	5.06226	0.162039
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	−5.06226	−0.161873
$979$	−42.6556	−1.36328
$980$	0	0
$981$	15.0623	0.480901
$982$	7.59339	0.242315
$983$	−0.937742	−0.0299093	−0.0149547	−	0.999888i	$-0.504760\pi$
$983$	−0.937742	−0.0299093	−0.0149547	+	0.999888i	$0.504760\pi$
$984$	7.06226	0.225137
$985$	0	0
$986$	0	0
$987$	−22.9377	−0.730116
$988$	−27.1868	−0.864926
$989$	−55.7179	−1.77173
$990$	0	0
$991$	−32.5311	−1.03339	−0.516693	−	0.856171i	$-0.672837\pi$
$991$	−32.5311	−1.03339	−0.516693	+	0.856171i	$0.672837\pi$
$992$	−1.00000	−0.0317500
$993$	−12.9377	−0.410567
$994$	56.7802	1.80096
$995$	0	0
$996$	−8.00000	−0.253490
$997$	2.24903	0.0712275	0.0356138	−	0.999366i	$-0.488661\pi$
$997$	2.24903	0.0712275	0.0356138	+	0.999366i	$0.488661\pi$
$998$	7.06226	0.223552
$999$	−7.06226	−0.223440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.bz.1.2		2
5.2	odd	4		4650.2.d.bg.3349.2		4
5.3	odd	4		4650.2.d.bg.3349.3		4
5.4	even	2		930.2.a.q.1.1	✓	2
15.14	odd	2		2790.2.a.bf.1.1		2
20.19	odd	2		7440.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.q.1.1	✓	2	5.4	even	2
2790.2.a.bf.1.1		2	15.14	odd	2
4650.2.a.bz.1.2		2	1.1	even	1	trivial
4650.2.d.bg.3349.2		4	5.2	odd	4
4650.2.d.bg.3349.3		4	5.3	odd	4
7440.2.a.bd.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:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.53113 q^{7} -1.00000 q^{8} +1.00000 q^{9} +6.53113 q^{11} -1.00000 q^{12} -6.00000 q^{13} -4.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +4.53113 q^{19} -4.53113 q^{21} -6.53113 q^{22} -6.53113 q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} +4.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} -6.53113 q^{33} -4.00000 q^{34} +1.00000 q^{36} +7.06226 q^{37} -4.53113 q^{38} +6.00000 q^{39} +7.06226 q^{41} +4.53113 q^{42} +8.53113 q^{43} +6.53113 q^{44} +6.53113 q^{46} +5.06226 q^{47} -1.00000 q^{48} +13.5311 q^{49} -4.00000 q^{51} -6.00000 q^{52} +2.53113 q^{53} +1.00000 q^{54} -4.53113 q^{56} -4.53113 q^{57} -9.06226 q^{59} +5.06226 q^{61} -1.00000 q^{62} +4.53113 q^{63} +1.00000 q^{64} +6.53113 q^{66} -5.06226 q^{67} +4.00000 q^{68} +6.53113 q^{69} -12.5311 q^{71} -1.00000 q^{72} +8.53113 q^{73} -7.06226 q^{74} +4.53113 q^{76} +29.5934 q^{77} -6.00000 q^{78} -8.53113 q^{79} +1.00000 q^{81} -7.06226 q^{82} +8.00000 q^{83} -4.53113 q^{84} -8.53113 q^{86} -6.53113 q^{88} -6.53113 q^{89} -27.1868 q^{91} -6.53113 q^{92} -1.00000 q^{93} -5.06226 q^{94} +1.00000 q^{96} -16.1245 q^{97} -13.5311 q^{98} +6.53113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9} + 5 q^{11} - 2 q^{12} - 12 q^{13} - q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{18} + q^{19} - q^{21} - 5 q^{22} - 5 q^{23} + 2 q^{24}+ \cdots + 5 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9 + 5 * q^11 - 2 * q^12 - 12 * q^13 - q^14 + 2 * q^16 + 8 * q^17 - 2 * q^18 + q^19 - q^21 - 5 * q^22 - 5 * q^23 + 2 * q^24 + 12 * q^26 - 2 * q^27 + q^28 + 2 * q^31 - 2 * q^32 - 5 * q^33 - 8 * q^34 + 2 * q^36 - 2 * q^37 - q^38 + 12 * q^39 - 2 * q^41 + q^42 + 9 * q^43 + 5 * q^44 + 5 * q^46 - 6 * q^47 - 2 * q^48 + 19 * q^49 - 8 * q^51 - 12 * q^52 - 3 * q^53 + 2 * q^54 - q^56 - q^57 - 2 * q^59 - 6 * q^61 - 2 * q^62 + q^63 + 2 * q^64 + 5 * q^66 + 6 * q^67 + 8 * q^68 + 5 * q^69 - 17 * q^71 - 2 * q^72 + 9 * q^73 + 2 * q^74 + q^76 + 35 * q^77 - 12 * q^78 - 9 * q^79 + 2 * q^81 + 2 * q^82 + 16 * q^83 - q^84 - 9 * q^86 - 5 * q^88 - 5 * q^89 - 6 * q^91 - 5 * q^92 - 2 * q^93 + 6 * q^94 + 2 * q^96 - 19 * q^98 + 5 * q^99

Introduction

L-functions

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Database

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