LMFDB - Embedded newform 2106.2.a.r.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2106,2,Mod(1,2106)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2106, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2106.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2106 = 2 \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2106.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,2,0,-2,3,0,2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$16.8164946657$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 234)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.69963$ of defining polynomial
Character	$\chi$	$=$	2106.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.28799 q^{5} -2.81089 q^{7} +1.00000 q^{8} -3.28799 q^{10} -5.17673 q^{11} +1.00000 q^{13} -2.81089 q^{14} +1.00000 q^{16} +0.699628 q^{17} +7.17673 q^{19} -3.28799 q^{20} -5.17673 q^{22} +6.00000 q^{23} +5.81089 q^{25} +1.00000 q^{26} -2.81089 q^{28} +2.22253 q^{29} +2.77747 q^{31} +1.00000 q^{32} +0.699628 q^{34} +9.24219 q^{35} -0.712008 q^{37} +7.17673 q^{38} -3.28799 q^{40} +0.823272 q^{41} +11.4981 q^{43} -5.17673 q^{44} +6.00000 q^{46} -1.52290 q^{47} +0.901116 q^{49} +5.81089 q^{50} +1.00000 q^{52} +11.0210 q^{53} +17.0210 q^{55} -2.81089 q^{56} +2.22253 q^{58} -14.7985 q^{59} +9.39926 q^{61} +2.77747 q^{62} +1.00000 q^{64} -3.28799 q^{65} +0.445057 q^{67} +0.699628 q^{68} +9.24219 q^{70} +1.18911 q^{71} -0.222528 q^{73} -0.712008 q^{74} +7.17673 q^{76} +14.5512 q^{77} -15.7527 q^{79} -3.28799 q^{80} +0.823272 q^{82} +7.97524 q^{83} -2.30037 q^{85} +11.4981 q^{86} -5.17673 q^{88} -7.39926 q^{89} -2.81089 q^{91} +6.00000 q^{92} -1.52290 q^{94} -23.5970 q^{95} +4.95420 q^{97} +0.901116 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 2 q^{7} + 3 q^{8} + 2 q^{10} - 4 q^{11} + 3 q^{13} - 2 q^{14} + 3 q^{16} - 4 q^{17} + 10 q^{19} + 2 q^{20} - 4 q^{22} + 18 q^{23} + 11 q^{25} + 3 q^{26} - 2 q^{28} + 6 q^{29}+ \cdots + 21 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 2 * q^7 + 3 * q^8 + 2 * q^10 - 4 * q^11 + 3 * q^13 - 2 * q^14 + 3 * q^16 - 4 * q^17 + 10 * q^19 + 2 * q^20 - 4 * q^22 + 18 * q^23 + 11 * q^25 + 3 * q^26 - 2 * q^28 + 6 * q^29 + 9 * q^31 + 3 * q^32 - 4 * q^34 + 5 * q^35 - 14 * q^37 + 10 * q^38 + 2 * q^40 + 14 * q^41 + 4 * q^43 - 4 * q^44 + 18 * q^46 - 10 * q^47 + 21 * q^49 + 11 * q^50 + 3 * q^52 + 8 * q^53 + 26 * q^55 - 2 * q^56 + 6 * q^58 - 20 * q^59 + 16 * q^61 + 9 * q^62 + 3 * q^64 + 2 * q^65 - 4 * q^68 + 5 * q^70 + 10 * q^71 - 14 * q^74 + 10 * q^76 - 16 * q^77 - 12 * q^79 + 2 * q^80 + 14 * q^82 - 12 * q^83 - 13 * q^85 + 4 * q^86 - 4 * q^88 - 10 * q^89 - 2 * q^91 + 18 * q^92 - 10 * q^94 - 22 * q^95 + 4 * q^97 + 21 * q^98

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.28799	−1.47043	−0.735217	−	0.677831i	$-0.762920\pi$
$5$	−3.28799	−1.47043	−0.735217	+	0.677831i	$0.762920\pi$
$6$	0	0
$7$	−2.81089	−1.06242	−0.531209	−	0.847241i	$-0.678262\pi$
$7$	−2.81089	−1.06242	−0.531209	+	0.847241i	$0.678262\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.28799	−1.03975
$11$	−5.17673	−1.56084	−0.780421	−	0.625254i	$-0.784995\pi$
$11$	−5.17673	−1.56084	−0.780421	+	0.625254i	$0.784995\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.81089	−0.751243
$15$	0	0
$16$	1.00000	0.250000
$17$	0.699628	0.169685	0.0848424	−	0.996394i	$-0.472961\pi$
$17$	0.699628	0.169685	0.0848424	+	0.996394i	$0.472961\pi$
$18$	0	0
$19$	7.17673	1.64645	0.823227	−	0.567712i	$-0.192171\pi$
$19$	7.17673	1.64645	0.823227	+	0.567712i	$0.192171\pi$
$20$	−3.28799	−0.735217
$21$	0	0
$22$	−5.17673	−1.10368
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	5.81089	1.16218
$26$	1.00000	0.196116
$27$	0	0
$28$	−2.81089	−0.531209
$29$	2.22253	0.412713	0.206357	−	0.978477i	$-0.433839\pi$
$29$	2.22253	0.412713	0.206357	+	0.978477i	$0.433839\pi$
$30$	0	0
$31$	2.77747	0.498849	0.249424	−	0.968394i	$-0.419759\pi$
$31$	2.77747	0.498849	0.249424	+	0.968394i	$0.419759\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.699628	0.119985
$35$	9.24219	1.56222
$36$	0	0
$37$	−0.712008	−0.117053	−0.0585267	−	0.998286i	$-0.518640\pi$
$37$	−0.712008	−0.117053	−0.0585267	+	0.998286i	$0.518640\pi$
$38$	7.17673	1.16422
$39$	0	0
$40$	−3.28799	−0.519877
$41$	0.823272	0.128573	0.0642867	−	0.997931i	$-0.479523\pi$
$41$	0.823272	0.128573	0.0642867	+	0.997931i	$0.479523\pi$
$42$	0	0
$43$	11.4981	1.75345	0.876725	−	0.480992i	$-0.159723\pi$
$43$	11.4981	1.75345	0.876725	+	0.480992i	$0.159723\pi$
$44$	−5.17673	−0.780421
$45$	0	0
$46$	6.00000	0.884652
$47$	−1.52290	−0.222138	−0.111069	−	0.993813i	$-0.535427\pi$
$47$	−1.52290	−0.222138	−0.111069	+	0.993813i	$0.535427\pi$
$48$	0	0
$49$	0.901116	0.128731
$50$	5.81089	0.821784
$51$	0	0
$52$	1.00000	0.138675
$53$	11.0210	1.51386	0.756928	−	0.653498i	$-0.226699\pi$
$53$	11.0210	1.51386	0.756928	+	0.653498i	$0.226699\pi$
$54$	0	0
$55$	17.0210	2.29512
$56$	−2.81089	−0.375621
$57$	0	0
$58$	2.22253	0.291832
$59$	−14.7985	−1.92660	−0.963301	−	0.268423i	$-0.913497\pi$
$59$	−14.7985	−1.92660	−0.963301	+	0.268423i	$0.913497\pi$
$60$	0	0
$61$	9.39926	1.20345	0.601726	−	0.798703i	$-0.294480\pi$
$61$	9.39926	1.20345	0.601726	+	0.798703i	$0.294480\pi$
$62$	2.77747	0.352739
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.28799	−0.407825
$66$	0	0
$67$	0.445057	0.0543723	0.0271862	−	0.999630i	$-0.491345\pi$
$67$	0.445057	0.0543723	0.0271862	+	0.999630i	$0.491345\pi$
$68$	0.699628	0.0848424
$69$	0	0
$70$	9.24219	1.10465
$71$	1.18911	0.141121	0.0705606	−	0.997507i	$-0.477521\pi$
$71$	1.18911	0.141121	0.0705606	+	0.997507i	$0.477521\pi$
$72$	0	0
$73$	−0.222528	−0.0260450	−0.0130225	−	0.999915i	$-0.504145\pi$
$73$	−0.222528	−0.0260450	−0.0130225	+	0.999915i	$0.504145\pi$
$74$	−0.712008	−0.0827692
$75$	0	0
$76$	7.17673	0.823227
$77$	14.5512	1.65827
$78$	0	0
$79$	−15.7527	−1.77232	−0.886159	−	0.463381i	$-0.846636\pi$
$79$	−15.7527	−1.77232	−0.886159	+	0.463381i	$0.846636\pi$
$80$	−3.28799	−0.367609
$81$	0	0
$82$	0.823272	0.0909152
$83$	7.97524	0.875396	0.437698	−	0.899122i	$-0.355794\pi$
$83$	7.97524	0.875396	0.437698	+	0.899122i	$0.355794\pi$
$84$	0	0
$85$	−2.30037	−0.249510
$86$	11.4981	1.23988
$87$	0	0
$88$	−5.17673	−0.551841
$89$	−7.39926	−0.784320	−0.392160	−	0.919897i	$-0.628272\pi$
$89$	−7.39926	−0.784320	−0.392160	+	0.919897i	$0.628272\pi$
$90$	0	0
$91$	−2.81089	−0.294662
$92$	6.00000	0.625543
$93$	0	0
$94$	−1.52290	−0.157075
$95$	−23.5970	−2.42100
$96$	0	0
$97$	4.95420	0.503023	0.251511	−	0.967854i	$-0.419072\pi$
$97$	4.95420	0.503023	0.251511	+	0.967854i	$0.419072\pi$
$98$	0.901116	0.0910264
$99$	0	0
$100$	5.81089	0.581089
$101$	2.95420	0.293954	0.146977	−	0.989140i	$-0.453046\pi$
$101$	2.95420	0.293954	0.146977	+	0.989140i	$0.453046\pi$
$102$	0	0
$103$	9.39926	0.926136	0.463068	−	0.886323i	$-0.346749\pi$
$103$	9.39926	0.926136	0.463068	+	0.886323i	$0.346749\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	11.0210	1.07046
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−6.67487	−0.639336	−0.319668	−	0.947530i	$-0.603571\pi$
$109$	−6.67487	−0.639336	−0.319668	+	0.947530i	$0.603571\pi$
$110$	17.0210	1.62289
$111$	0	0
$112$	−2.81089	−0.265604
$113$	3.00000	0.282216	0.141108	−	0.989994i	$-0.454933\pi$
$113$	3.00000	0.282216	0.141108	+	0.989994i	$0.454933\pi$
$114$	0	0
$115$	−19.7280	−1.83964
$116$	2.22253	0.206357
$117$	0	0
$118$	−14.7985	−1.36231
$119$	−1.96658	−0.180276
$120$	0	0
$121$	15.7985	1.43623
$122$	9.39926	0.850969
$123$	0	0
$124$	2.77747	0.249424
$125$	−2.66621	−0.238473
$126$	0	0
$127$	−3.42402	−0.303832	−0.151916	−	0.988393i	$-0.548544\pi$
$127$	−3.42402	−0.303832	−0.151916	+	0.988393i	$0.548544\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.28799	−0.288376
$131$	−8.25457	−0.721205	−0.360603	−	0.932720i	$-0.617429\pi$
$131$	−8.25457	−0.721205	−0.360603	+	0.932720i	$0.617429\pi$
$132$	0	0
$133$	−20.1730	−1.74922
$134$	0.445057	0.0384470
$135$	0	0
$136$	0.699628	0.0599926
$137$	4.60074	0.393068	0.196534	−	0.980497i	$-0.437031\pi$
$137$	4.60074	0.393068	0.196534	+	0.980497i	$0.437031\pi$
$138$	0	0
$139$	10.5884	0.898093	0.449047	−	0.893508i	$-0.351764\pi$
$139$	10.5884	0.898093	0.449047	+	0.893508i	$0.351764\pi$
$140$	9.24219	0.781108
$141$	0	0
$142$	1.18911	0.0997877
$143$	−5.17673	−0.432900
$144$	0	0
$145$	−7.30766	−0.606868
$146$	−0.222528	−0.0184166
$147$	0	0
$148$	−0.712008	−0.0585267
$149$	16.9752	1.39067	0.695333	−	0.718688i	$-0.255257\pi$
$149$	16.9752	1.39067	0.695333	+	0.718688i	$0.255257\pi$
$150$	0	0
$151$	22.2174	1.80803	0.904015	−	0.427502i	$-0.140606\pi$
$151$	22.2174	1.80803	0.904015	+	0.427502i	$0.140606\pi$
$152$	7.17673	0.582110
$153$	0	0
$154$	14.5512	1.17257
$155$	−9.13231	−0.733524
$156$	0	0
$157$	−4.66758	−0.372514	−0.186257	−	0.982501i	$-0.559636\pi$
$157$	−4.66758	−0.372514	−0.186257	+	0.982501i	$0.559636\pi$
$158$	−15.7527	−1.25322
$159$	0	0
$160$	−3.28799	−0.259939
$161$	−16.8654	−1.32918
$162$	0	0
$163$	−11.9752	−0.937973	−0.468987	−	0.883205i	$-0.655381\pi$
$163$	−11.9752	−0.937973	−0.468987	+	0.883205i	$0.655381\pi$
$164$	0.823272	0.0642867
$165$	0	0
$166$	7.97524	0.618999
$167$	6.77747	0.524457	0.262228	−	0.965006i	$-0.415543\pi$
$167$	6.77747	0.524457	0.262228	+	0.965006i	$0.415543\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−2.30037	−0.176430
$171$	0	0
$172$	11.4981	0.876725
$173$	−19.1520	−1.45610	−0.728049	−	0.685526i	$-0.759573\pi$
$173$	−19.1520	−1.45610	−0.728049	+	0.685526i	$0.759573\pi$
$174$	0	0
$175$	−16.3338	−1.23472
$176$	−5.17673	−0.390211
$177$	0	0
$178$	−7.39926	−0.554598
$179$	0.210149	0.0157072	0.00785362	−	0.999969i	$-0.497500\pi$
$179$	0.210149	0.0157072	0.00785362	+	0.999969i	$0.497500\pi$
$180$	0	0
$181$	6.44506	0.479057	0.239529	−	0.970889i	$-0.423007\pi$
$181$	6.44506	0.479057	0.239529	+	0.970889i	$0.423007\pi$
$182$	−2.81089	−0.208357
$183$	0	0
$184$	6.00000	0.442326
$185$	2.34108	0.172119
$186$	0	0
$187$	−3.62178	−0.264851
$188$	−1.52290	−0.111069
$189$	0	0
$190$	−23.5970	−1.71191
$191$	23.3497	1.68953	0.844764	−	0.535139i	$-0.179741\pi$
$191$	23.3497	1.68953	0.844764	+	0.535139i	$0.179741\pi$
$192$	0	0
$193$	24.9294	1.79446	0.897230	−	0.441563i	$-0.145576\pi$
$193$	24.9294	1.79446	0.897230	+	0.441563i	$0.145576\pi$
$194$	4.95420	0.355691
$195$	0	0
$196$	0.901116	0.0643654
$197$	10.2349	0.729207	0.364604	−	0.931163i	$-0.381205\pi$
$197$	10.2349	0.729207	0.364604	+	0.931163i	$0.381205\pi$
$198$	0	0
$199$	26.9047	1.90722	0.953611	−	0.301041i	$-0.0973342\pi$
$199$	26.9047	1.90722	0.953611	+	0.301041i	$0.0973342\pi$
$200$	5.81089	0.410892
$201$	0	0
$202$	2.95420	0.207857
$203$	−6.24729	−0.438474
$204$	0	0
$205$	−2.70691	−0.189059
$206$	9.39926	0.654877
$207$	0	0
$208$	1.00000	0.0693375
$209$	−37.1520	−2.56986
$210$	0	0
$211$	4.71201	0.324388	0.162194	−	0.986759i	$-0.448143\pi$
$211$	4.71201	0.324388	0.162194	+	0.986759i	$0.448143\pi$
$212$	11.0210	0.756928
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−37.8058	−2.57833
$216$	0	0
$217$	−7.80717	−0.529985
$218$	−6.67487	−0.452079
$219$	0	0
$220$	17.0210	1.14756
$221$	0.699628	0.0470621
$222$	0	0
$223$	7.62907	0.510880	0.255440	−	0.966825i	$-0.417780\pi$
$223$	7.62907	0.510880	0.255440	+	0.966825i	$0.417780\pi$
$224$	−2.81089	−0.187811
$225$	0	0
$226$	3.00000	0.199557
$227$	−22.0421	−1.46298	−0.731492	−	0.681850i	$-0.761176\pi$
$227$	−22.0421	−1.46298	−0.731492	+	0.681850i	$0.761176\pi$
$228$	0	0
$229$	−4.48948	−0.296673	−0.148337	−	0.988937i	$-0.547392\pi$
$229$	−4.48948	−0.296673	−0.148337	+	0.988937i	$0.547392\pi$
$230$	−19.7280	−1.30082
$231$	0	0
$232$	2.22253	0.145916
$233$	−5.54256	−0.363105	−0.181553	−	0.983381i	$-0.558112\pi$
$233$	−5.54256	−0.363105	−0.181553	+	0.983381i	$0.558112\pi$
$234$	0	0
$235$	5.00728	0.326639
$236$	−14.7985	−0.963301
$237$	0	0
$238$	−1.96658	−0.127474
$239$	−7.43130	−0.480691	−0.240345	−	0.970687i	$-0.577261\pi$
$239$	−7.43130	−0.480691	−0.240345	+	0.970687i	$0.577261\pi$
$240$	0	0
$241$	9.39926	0.605459	0.302730	−	0.953076i	$-0.402102\pi$
$241$	9.39926	0.605459	0.302730	+	0.953076i	$0.402102\pi$
$242$	15.7985	1.01557
$243$	0	0
$244$	9.39926	0.601726
$245$	−2.96286	−0.189290
$246$	0	0
$247$	7.17673	0.456644
$248$	2.77747	0.176370
$249$	0	0
$250$	−2.66621	−0.168626
$251$	−15.9963	−1.00968	−0.504838	−	0.863214i	$-0.668448\pi$
$251$	−15.9963	−1.00968	−0.504838	+	0.863214i	$0.668448\pi$
$252$	0	0
$253$	−31.0604	−1.95275
$254$	−3.42402	−0.214842
$255$	0	0
$256$	1.00000	0.0625000
$257$	0.0865047	0.00539602	0.00269801	−	0.999996i	$-0.499141\pi$
$257$	0.0865047	0.00539602	0.00269801	+	0.999996i	$0.499141\pi$
$258$	0	0
$259$	2.00138	0.124360
$260$	−3.28799	−0.203913
$261$	0	0
$262$	−8.25457	−0.509969
$263$	−0.978959	−0.0603652	−0.0301826	−	0.999544i	$-0.509609\pi$
$263$	−0.978959	−0.0603652	−0.0301826	+	0.999544i	$0.509609\pi$
$264$	0	0
$265$	−36.2371	−2.22603
$266$	−20.1730	−1.23689
$267$	0	0
$268$	0.445057	0.0271862
$269$	25.5723	1.55917	0.779584	−	0.626297i	$-0.215430\pi$
$269$	25.5723	1.55917	0.779584	+	0.626297i	$0.215430\pi$
$270$	0	0
$271$	16.9766	1.03126	0.515628	−	0.856813i	$-0.327559\pi$
$271$	16.9766	1.03126	0.515628	+	0.856813i	$0.327559\pi$
$272$	0.699628	0.0424212
$273$	0	0
$274$	4.60074	0.277941
$275$	−30.0814	−1.81398
$276$	0	0
$277$	3.39926	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$277$	3.39926	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$278$	10.5884	0.635048
$279$	0	0
$280$	9.24219	0.552327
$281$	28.6822	1.71103	0.855517	−	0.517775i	$-0.173240\pi$
$281$	28.6822	1.71103	0.855517	+	0.517775i	$0.173240\pi$
$282$	0	0
$283$	−11.1978	−0.665638	−0.332819	−	0.942991i	$-0.608000\pi$
$283$	−11.1978	−0.665638	−0.332819	+	0.942991i	$0.608000\pi$
$284$	1.18911	0.0705606
$285$	0	0
$286$	−5.17673	−0.306106
$287$	−2.31413	−0.136599
$288$	0	0
$289$	−16.5105	−0.971207
$290$	−7.30766	−0.429120
$291$	0	0
$292$	−0.222528	−0.0130225
$293$	−6.94182	−0.405545	−0.202773	−	0.979226i	$-0.564995\pi$
$293$	−6.94182	−0.405545	−0.202773	+	0.979226i	$0.564995\pi$
$294$	0	0
$295$	48.6574	2.83294
$296$	−0.712008	−0.0413846
$297$	0	0
$298$	16.9752	0.983349
$299$	6.00000	0.346989
$300$	0	0
$301$	−32.3200	−1.86290
$302$	22.2174	1.27847
$303$	0	0
$304$	7.17673	0.411614
$305$	−30.9047	−1.76960
$306$	0	0
$307$	−15.0210	−0.857296	−0.428648	−	0.903472i	$-0.641010\pi$
$307$	−15.0210	−0.857296	−0.428648	+	0.903472i	$0.641010\pi$
$308$	14.5512	0.829133
$309$	0	0
$310$	−9.13231	−0.518680
$311$	7.97524	0.452234	0.226117	−	0.974100i	$-0.427397\pi$
$311$	7.97524	0.452234	0.226117	+	0.974100i	$0.427397\pi$
$312$	0	0
$313$	13.9949	0.791039	0.395519	−	0.918458i	$-0.370565\pi$
$313$	13.9949	0.791039	0.395519	+	0.918458i	$0.370565\pi$
$314$	−4.66758	−0.263407
$315$	0	0
$316$	−15.7527	−0.886159
$317$	9.42030	0.529097	0.264548	−	0.964372i	$-0.414777\pi$
$317$	9.42030	0.529097	0.264548	+	0.964372i	$0.414777\pi$
$318$	0	0
$319$	−11.5054	−0.644180
$320$	−3.28799	−0.183804
$321$	0	0
$322$	−16.8654	−0.939869
$323$	5.02104	0.279378
$324$	0	0
$325$	5.81089	0.322330
$326$	−11.9752	−0.663247
$327$	0	0
$328$	0.823272	0.0454576
$329$	4.28071	0.236003
$330$	0	0
$331$	−11.5549	−0.635117	−0.317559	−	0.948239i	$-0.602863\pi$
$331$	−11.5549	−0.635117	−0.317559	+	0.948239i	$0.602863\pi$
$332$	7.97524	0.437698
$333$	0	0
$334$	6.77747	0.370847
$335$	−1.46334	−0.0799510
$336$	0	0
$337$	0.810892	0.0441721	0.0220861	−	0.999756i	$-0.492969\pi$
$337$	0.810892	0.0441721	0.0220861	+	0.999756i	$0.492969\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	−2.30037	−0.124755
$341$	−14.3782	−0.778624
$342$	0	0
$343$	17.1433	0.925652
$344$	11.4981	0.619938
$345$	0	0
$346$	−19.1520	−1.02962
$347$	−14.3090	−0.768149	−0.384075	−	0.923302i	$-0.625479\pi$
$347$	−14.3090	−0.768149	−0.384075	+	0.923302i	$0.625479\pi$
$348$	0	0
$349$	−15.5105	−0.830259	−0.415130	−	0.909762i	$-0.636264\pi$
$349$	−15.5105	−0.830259	−0.415130	+	0.909762i	$0.636264\pi$
$350$	−16.3338	−0.873078
$351$	0	0
$352$	−5.17673	−0.275921
$353$	34.7738	1.85082	0.925410	−	0.378967i	$-0.123721\pi$
$353$	34.7738	1.85082	0.925410	+	0.378967i	$0.123721\pi$
$354$	0	0
$355$	−3.90978	−0.207509
$356$	−7.39926	−0.392160
$357$	0	0
$358$	0.210149	0.0111067
$359$	24.2829	1.28160	0.640801	−	0.767707i	$-0.278602\pi$
$359$	24.2829	1.28160	0.640801	+	0.767707i	$0.278602\pi$
$360$	0	0
$361$	32.5054	1.71081
$362$	6.44506	0.338745
$363$	0	0
$364$	−2.81089	−0.147331
$365$	0.731671	0.0382974
$366$	0	0
$367$	−15.9084	−0.830412	−0.415206	−	0.909727i	$-0.636290\pi$
$367$	−15.9084	−0.830412	−0.415206	+	0.909727i	$0.636290\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	2.34108	0.121707
$371$	−30.9790	−1.60835
$372$	0	0
$373$	11.4661	0.593692	0.296846	−	0.954925i	$-0.404065\pi$
$373$	11.4661	0.593692	0.296846	+	0.954925i	$0.404065\pi$
$374$	−3.62178	−0.187278
$375$	0	0
$376$	−1.52290	−0.0785376
$377$	2.22253	0.114466
$378$	0	0
$379$	3.71063	0.190602	0.0953011	−	0.995448i	$-0.469619\pi$
$379$	3.71063	0.190602	0.0953011	+	0.995448i	$0.469619\pi$
$380$	−23.5970	−1.21050
$381$	0	0
$382$	23.3497	1.19468
$383$	−3.56732	−0.182282	−0.0911408	−	0.995838i	$-0.529051\pi$
$383$	−3.56732	−0.182282	−0.0911408	+	0.995838i	$0.529051\pi$
$384$	0	0
$385$	−47.8443	−2.43837
$386$	24.9294	1.26888
$387$	0	0
$388$	4.95420	0.251511
$389$	5.42402	0.275009	0.137504	−	0.990501i	$-0.456092\pi$
$389$	5.42402	0.275009	0.137504	+	0.990501i	$0.456092\pi$
$390$	0	0
$391$	4.19777	0.212290
$392$	0.901116	0.0455132
$393$	0	0
$394$	10.2349	0.515627
$395$	51.7948	2.60608
$396$	0	0
$397$	−28.1272	−1.41166	−0.705832	−	0.708379i	$-0.749427\pi$
$397$	−28.1272	−1.41166	−0.705832	+	0.708379i	$0.749427\pi$
$398$	26.9047	1.34861
$399$	0	0
$400$	5.81089	0.290545
$401$	14.8626	0.742203	0.371101	−	0.928592i	$-0.378980\pi$
$401$	14.8626	0.742203	0.371101	+	0.928592i	$0.378980\pi$
$402$	0	0
$403$	2.77747	0.138356
$404$	2.95420	0.146977
$405$	0	0
$406$	−6.24729	−0.310048
$407$	3.68587	0.182702
$408$	0	0
$409$	−17.9752	−0.888818	−0.444409	−	0.895824i	$-0.646586\pi$
$409$	−17.9752	−0.888818	−0.444409	+	0.895824i	$0.646586\pi$
$410$	−2.70691	−0.133685
$411$	0	0
$412$	9.39926	0.463068
$413$	41.5970	2.04686
$414$	0	0
$415$	−26.2225	−1.28721
$416$	1.00000	0.0490290
$417$	0	0
$418$	−37.1520	−1.81716
$419$	27.4610	1.34156	0.670779	−	0.741657i	$-0.265960\pi$
$419$	27.4610	1.34156	0.670779	+	0.741657i	$0.265960\pi$
$420$	0	0
$421$	2.94182	0.143376	0.0716878	−	0.997427i	$-0.477161\pi$
$421$	2.94182	0.143376	0.0716878	+	0.997427i	$0.477161\pi$
$422$	4.71201	0.229377
$423$	0	0
$424$	11.0210	0.535229
$425$	4.06546	0.197204
$426$	0	0
$427$	−26.4203	−1.27857
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−37.8058	−1.82316
$431$	−3.37959	−0.162789	−0.0813946	−	0.996682i	$-0.525937\pi$
$431$	−3.37959	−0.162789	−0.0813946	+	0.996682i	$0.525937\pi$
$432$	0	0
$433$	8.36584	0.402036	0.201018	−	0.979588i	$-0.435575\pi$
$433$	8.36584	0.402036	0.201018	+	0.979588i	$0.435575\pi$
$434$	−7.80717	−0.374756
$435$	0	0
$436$	−6.67487	−0.319668
$437$	43.0604	2.05986
$438$	0	0
$439$	−28.5760	−1.36386	−0.681929	−	0.731419i	$-0.738859\pi$
$439$	−28.5760	−1.36386	−0.681929	+	0.731419i	$0.738859\pi$
$440$	17.0210	0.811446
$441$	0	0
$442$	0.699628	0.0332779
$443$	18.5388	0.880807	0.440404	−	0.897800i	$-0.354835\pi$
$443$	18.5388	0.880807	0.440404	+	0.897800i	$0.354835\pi$
$444$	0	0
$445$	24.3287	1.15329
$446$	7.62907	0.361247
$447$	0	0
$448$	−2.81089	−0.132802
$449$	14.8626	0.701409	0.350705	−	0.936486i	$-0.385942\pi$
$449$	14.8626	0.701409	0.350705	+	0.936486i	$0.385942\pi$
$450$	0	0
$451$	−4.26186	−0.200683
$452$	3.00000	0.141108
$453$	0	0
$454$	−22.0421	−1.03449
$455$	9.24219	0.433281
$456$	0	0
$457$	22.3039	1.04333	0.521667	−	0.853149i	$-0.325310\pi$
$457$	22.3039	1.04333	0.521667	+	0.853149i	$0.325310\pi$
$458$	−4.48948	−0.209780
$459$	0	0
$460$	−19.7280	−0.919821
$461$	−9.74033	−0.453653	−0.226826	−	0.973935i	$-0.572835\pi$
$461$	−9.74033	−0.453653	−0.226826	+	0.973935i	$0.572835\pi$
$462$	0	0
$463$	−3.62550	−0.168491	−0.0842457	−	0.996445i	$-0.526848\pi$
$463$	−3.62550	−0.168491	−0.0842457	+	0.996445i	$0.526848\pi$
$464$	2.22253	0.103178
$465$	0	0
$466$	−5.54256	−0.256754
$467$	0.0458003	0.00211939	0.00105969	−	0.999999i	$-0.499663\pi$
$467$	0.0458003	0.00211939	0.00105969	+	0.999999i	$0.499663\pi$
$468$	0	0
$469$	−1.25101	−0.0577661
$470$	5.00728	0.230969
$471$	0	0
$472$	−14.7985	−0.681157
$473$	−59.5227	−2.73686
$474$	0	0
$475$	41.7032	1.91347
$476$	−1.96658	−0.0901380
$477$	0	0
$478$	−7.43130	−0.339900
$479$	−17.6835	−0.807981	−0.403991	−	0.914763i	$-0.632377\pi$
$479$	−17.6835	−0.807981	−0.403991	+	0.914763i	$0.632377\pi$
$480$	0	0
$481$	−0.712008	−0.0324648
$482$	9.39926	0.428124
$483$	0	0
$484$	15.7985	0.718114
$485$	−16.2894	−0.739662
$486$	0	0
$487$	−23.2436	−1.05327	−0.526633	−	0.850093i	$-0.676546\pi$
$487$	−23.2436	−1.05327	−0.526633	+	0.850093i	$0.676546\pi$
$488$	9.39926	0.425484
$489$	0	0
$490$	−2.96286	−0.133848
$491$	18.5439	0.836876	0.418438	−	0.908245i	$-0.362578\pi$
$491$	18.5439	0.836876	0.418438	+	0.908245i	$0.362578\pi$
$492$	0	0
$493$	1.55494	0.0700311
$494$	7.17673	0.322896
$495$	0	0
$496$	2.77747	0.124712
$497$	−3.34245	−0.149930
$498$	0	0
$499$	−31.5475	−1.41226	−0.706130	−	0.708082i	$-0.749561\pi$
$499$	−31.5475	−1.41226	−0.706130	+	0.708082i	$0.749561\pi$
$500$	−2.66621	−0.119236
$501$	0	0
$502$	−15.9963	−0.713949
$503$	−21.2188	−0.946100	−0.473050	−	0.881036i	$-0.656847\pi$
$503$	−21.2188	−0.946100	−0.473050	+	0.881036i	$0.656847\pi$
$504$	0	0
$505$	−9.71339	−0.432240
$506$	−31.0604	−1.38080
$507$	0	0
$508$	−3.42402	−0.151916
$509$	−3.42030	−0.151602	−0.0758010	−	0.997123i	$-0.524151\pi$
$509$	−3.42030	−0.151602	−0.0758010	+	0.997123i	$0.524151\pi$
$510$	0	0
$511$	0.625503	0.0276706
$512$	1.00000	0.0441942
$513$	0	0
$514$	0.0865047	0.00381556
$515$	−30.9047	−1.36182
$516$	0	0
$517$	7.88364	0.346722
$518$	2.00138	0.0879355
$519$	0	0
$520$	−3.28799	−0.144188
$521$	−25.5498	−1.11936	−0.559680	−	0.828709i	$-0.689076\pi$
$521$	−25.5498	−1.11936	−0.559680	+	0.828709i	$0.689076\pi$
$522$	0	0
$523$	−16.0458	−0.701634	−0.350817	−	0.936444i	$-0.614096\pi$
$523$	−16.0458	−0.701634	−0.350817	+	0.936444i	$0.614096\pi$
$524$	−8.25457	−0.360603
$525$	0	0
$526$	−0.978959	−0.0426846
$527$	1.94320	0.0846470
$528$	0	0
$529$	13.0000	0.565217
$530$	−36.2371	−1.57404
$531$	0	0
$532$	−20.1730	−0.874611
$533$	0.823272	0.0356599
$534$	0	0
$535$	39.4559	1.70583
$536$	0.445057	0.0192235
$537$	0	0
$538$	25.5723	1.10250
$539$	−4.66483	−0.200928
$540$	0	0
$541$	−0.899738	−0.0386828	−0.0193414	−	0.999813i	$-0.506157\pi$
$541$	−0.899738	−0.0386828	−0.0193414	+	0.999813i	$0.506157\pi$
$542$	16.9766	0.729208
$543$	0	0
$544$	0.699628	0.0299963
$545$	21.9469	0.940103
$546$	0	0
$547$	−33.4189	−1.42889	−0.714445	−	0.699692i	$-0.753321\pi$
$547$	−33.4189	−1.42889	−0.714445	+	0.699692i	$0.753321\pi$
$548$	4.60074	0.196534
$549$	0	0
$550$	−30.0814	−1.28268
$551$	15.9505	0.679513
$552$	0	0
$553$	44.2792	1.88294
$554$	3.39926	0.144421
$555$	0	0
$556$	10.5884	0.449047
$557$	0.123644	0.00523896	0.00261948	−	0.999997i	$-0.499166\pi$
$557$	0.123644	0.00523896	0.00261948	+	0.999997i	$0.499166\pi$
$558$	0	0
$559$	11.4981	0.486320
$560$	9.24219	0.390554
$561$	0	0
$562$	28.6822	1.20988
$563$	−2.58836	−0.109087	−0.0545433	−	0.998511i	$-0.517370\pi$
$563$	−2.58836	−0.109087	−0.0545433	+	0.998511i	$0.517370\pi$
$564$	0	0
$565$	−9.86398	−0.414981
$566$	−11.1978	−0.470677
$567$	0	0
$568$	1.18911	0.0498939
$569$	7.46844	0.313093	0.156547	−	0.987671i	$-0.449964\pi$
$569$	7.46844	0.313093	0.156547	+	0.987671i	$0.449964\pi$
$570$	0	0
$571$	−32.6501	−1.36636	−0.683182	−	0.730248i	$-0.739405\pi$
$571$	−32.6501	−1.36636	−0.683182	+	0.730248i	$0.739405\pi$
$572$	−5.17673	−0.216450
$573$	0	0
$574$	−2.31413	−0.0965899
$575$	34.8654	1.45399
$576$	0	0
$577$	−18.9542	−0.789074	−0.394537	−	0.918880i	$-0.629095\pi$
$577$	−18.9542	−0.789074	−0.394537	+	0.918880i	$0.629095\pi$
$578$	−16.5105	−0.686747
$579$	0	0
$580$	−7.30766	−0.303434
$581$	−22.4175	−0.930036
$582$	0	0
$583$	−57.0529	−2.36289
$584$	−0.222528	−0.00920829
$585$	0	0
$586$	−6.94182	−0.286764
$587$	31.1520	1.28578	0.642890	−	0.765959i	$-0.277735\pi$
$587$	31.1520	1.28578	0.642890	+	0.765959i	$0.277735\pi$
$588$	0	0
$589$	19.9332	0.821331
$590$	48.6574	2.00319
$591$	0	0
$592$	−0.712008	−0.0292633
$593$	−39.0631	−1.60413	−0.802065	−	0.597237i	$-0.796265\pi$
$593$	−39.0631	−1.60413	−0.802065	+	0.597237i	$0.796265\pi$
$594$	0	0
$595$	6.46610	0.265084
$596$	16.9752	0.695333
$597$	0	0
$598$	6.00000	0.245358
$599$	−19.7280	−0.806062	−0.403031	−	0.915186i	$-0.632043\pi$
$599$	−19.7280	−0.806062	−0.403031	+	0.915186i	$0.632043\pi$
$600$	0	0
$601$	−1.40654	−0.0573740	−0.0286870	−	0.999588i	$-0.509133\pi$
$601$	−1.40654	−0.0573740	−0.0286870	+	0.999588i	$0.509133\pi$
$602$	−32.3200	−1.31727
$603$	0	0
$604$	22.2174	0.904015
$605$	−51.9454	−2.11188
$606$	0	0
$607$	−10.2473	−0.415925	−0.207962	−	0.978137i	$-0.566683\pi$
$607$	−10.2473	−0.415925	−0.207962	+	0.978137i	$0.566683\pi$
$608$	7.17673	0.291055
$609$	0	0
$610$	−30.9047	−1.25129
$611$	−1.52290	−0.0616099
$612$	0	0
$613$	−43.3287	−1.75003	−0.875015	−	0.484096i	$-0.839148\pi$
$613$	−43.3287	−1.75003	−0.875015	+	0.484096i	$0.839148\pi$
$614$	−15.0210	−0.606200
$615$	0	0
$616$	14.5512	0.586286
$617$	−2.89011	−0.116352	−0.0581758	−	0.998306i	$-0.518528\pi$
$617$	−2.89011	−0.116352	−0.0581758	+	0.998306i	$0.518528\pi$
$618$	0	0
$619$	0.692344	0.0278277	0.0139138	−	0.999903i	$-0.495571\pi$
$619$	0.692344	0.0278277	0.0139138	+	0.999903i	$0.495571\pi$
$620$	−9.13231	−0.366762
$621$	0	0
$622$	7.97524	0.319778
$623$	20.7985	0.833275
$624$	0	0
$625$	−20.2880	−0.811520
$626$	13.9949	0.559349
$627$	0	0
$628$	−4.66758	−0.186257
$629$	−0.498141	−0.0198622
$630$	0	0
$631$	30.4771	1.21327	0.606637	−	0.794979i	$-0.292518\pi$
$631$	30.4771	1.21327	0.606637	+	0.794979i	$0.292518\pi$
$632$	−15.7527	−0.626609
$633$	0	0
$634$	9.42030	0.374128
$635$	11.2581	0.446766
$636$	0	0
$637$	0.901116	0.0357035
$638$	−11.5054	−0.455504
$639$	0	0
$640$	−3.28799	−0.129969
$641$	38.1941	1.50857	0.754287	−	0.656545i	$-0.227983\pi$
$641$	38.1941	1.50857	0.754287	+	0.656545i	$0.227983\pi$
$642$	0	0
$643$	12.6181	0.497608	0.248804	−	0.968554i	$-0.419963\pi$
$643$	12.6181	0.497608	0.248804	+	0.968554i	$0.419963\pi$
$644$	−16.8654	−0.664588
$645$	0	0
$646$	5.02104	0.197550
$647$	−35.9257	−1.41239	−0.706193	−	0.708019i	$-0.749589\pi$
$647$	−35.9257	−1.41239	−0.706193	+	0.708019i	$0.749589\pi$
$648$	0	0
$649$	76.6079	3.00712
$650$	5.81089	0.227922
$651$	0	0
$652$	−11.9752	−0.468987
$653$	21.4386	0.838957	0.419478	−	0.907765i	$-0.362213\pi$
$653$	21.4386	0.838957	0.419478	+	0.907765i	$0.362213\pi$
$654$	0	0
$655$	27.1410	1.06049
$656$	0.823272	0.0321434
$657$	0	0
$658$	4.28071	0.166879
$659$	−32.7527	−1.27586	−0.637932	−	0.770092i	$-0.720210\pi$
$659$	−32.7527	−1.27586	−0.637932	+	0.770092i	$0.720210\pi$
$660$	0	0
$661$	0.536657	0.0208735	0.0104368	−	0.999946i	$-0.496678\pi$
$661$	0.536657	0.0208735	0.0104368	+	0.999946i	$0.496678\pi$
$662$	−11.5549	−0.449096
$663$	0	0
$664$	7.97524	0.309499
$665$	66.3287	2.57212
$666$	0	0
$667$	13.3352	0.516340
$668$	6.77747	0.262228
$669$	0	0
$670$	−1.46334	−0.0565339
$671$	−48.6574	−1.87840
$672$	0	0
$673$	33.9061	1.30698	0.653491	−	0.756934i	$-0.273304\pi$
$673$	33.9061	1.30698	0.653491	+	0.756934i	$0.273304\pi$
$674$	0.810892	0.0312344
$675$	0	0
$676$	1.00000	0.0384615
$677$	26.7985	1.02995	0.514975	−	0.857205i	$-0.327801\pi$
$677$	26.7985	1.02995	0.514975	+	0.857205i	$0.327801\pi$
$678$	0	0
$679$	−13.9257	−0.534420
$680$	−2.30037	−0.0882152
$681$	0	0
$682$	−14.3782	−0.550570
$683$	−0.978959	−0.0374588	−0.0187294	−	0.999825i	$-0.505962\pi$
$683$	−0.978959	−0.0374588	−0.0187294	+	0.999825i	$0.505962\pi$
$684$	0	0
$685$	−15.1272	−0.577981
$686$	17.1433	0.654535
$687$	0	0
$688$	11.4981	0.438363
$689$	11.0210	0.419868
$690$	0	0
$691$	−50.5265	−1.92212	−0.961059	−	0.276344i	$-0.910877\pi$
$691$	−50.5265	−1.92212	−0.961059	+	0.276344i	$0.910877\pi$
$692$	−19.1520	−0.728049
$693$	0	0
$694$	−14.3090	−0.543163
$695$	−34.8145	−1.32059
$696$	0	0
$697$	0.575984	0.0218170
$698$	−15.5105	−0.587082
$699$	0	0
$700$	−16.3338	−0.617359
$701$	9.37450	0.354070	0.177035	−	0.984205i	$-0.443349\pi$
$701$	9.37450	0.354070	0.177035	+	0.984205i	$0.443349\pi$
$702$	0	0
$703$	−5.10989	−0.192723
$704$	−5.17673	−0.195105
$705$	0	0
$706$	34.7738	1.30873
$707$	−8.30394	−0.312302
$708$	0	0
$709$	−14.5723	−0.547273	−0.273636	−	0.961833i	$-0.588227\pi$
$709$	−14.5723	−0.547273	−0.273636	+	0.961833i	$0.588227\pi$
$710$	−3.90978	−0.146731
$711$	0	0
$712$	−7.39926	−0.277299
$713$	16.6648	0.624103
$714$	0	0
$715$	17.0210	0.636551
$716$	0.210149	0.00785362
$717$	0	0
$718$	24.2829	0.906230
$719$	33.3104	1.24227	0.621134	−	0.783704i	$-0.286672\pi$
$719$	33.3104	1.24227	0.621134	+	0.783704i	$0.286672\pi$
$720$	0	0
$721$	−26.4203	−0.983943
$722$	32.5054	1.20973
$723$	0	0
$724$	6.44506	0.239529
$725$	12.9149	0.479646
$726$	0	0
$727$	−33.2857	−1.23450	−0.617248	−	0.786768i	$-0.711753\pi$
$727$	−33.2857	−1.23450	−0.617248	+	0.786768i	$0.711753\pi$
$728$	−2.81089	−0.104179
$729$	0	0
$730$	0.731671	0.0270804
$731$	8.04442	0.297534
$732$	0	0
$733$	46.9171	1.73292	0.866461	−	0.499245i	$-0.166389\pi$
$733$	46.9171	1.73292	0.866461	+	0.499245i	$0.166389\pi$
$734$	−15.9084	−0.587190
$735$	0	0
$736$	6.00000	0.221163
$737$	−2.30394	−0.0848666
$738$	0	0
$739$	−43.6116	−1.60428	−0.802139	−	0.597137i	$-0.796305\pi$
$739$	−43.6116	−1.60428	−0.802139	+	0.597137i	$0.796305\pi$
$740$	2.34108	0.0860597
$741$	0	0
$742$	−30.9790	−1.13727
$743$	40.0741	1.47018	0.735089	−	0.677971i	$-0.237141\pi$
$743$	40.0741	1.47018	0.735089	+	0.677971i	$0.237141\pi$
$744$	0	0
$745$	−55.8145	−2.04488
$746$	11.4661	0.419804
$747$	0	0
$748$	−3.62178	−0.132426
$749$	33.7307	1.23249
$750$	0	0
$751$	45.3077	1.65330	0.826650	−	0.562717i	$-0.190244\pi$
$751$	45.3077	1.65330	0.826650	+	0.562717i	$0.190244\pi$
$752$	−1.52290	−0.0555345
$753$	0	0
$754$	2.22253	0.0809397
$755$	−73.0507	−2.65859
$756$	0	0
$757$	38.3287	1.39308	0.696540	−	0.717518i	$-0.254722\pi$
$757$	38.3287	1.39308	0.696540	+	0.717518i	$0.254722\pi$
$758$	3.71063	0.134776
$759$	0	0
$760$	−23.5970	−0.855954
$761$	10.6822	0.387228	0.193614	−	0.981078i	$-0.437979\pi$
$761$	10.6822	0.387228	0.193614	+	0.981078i	$0.437979\pi$
$762$	0	0
$763$	18.7623	0.679242
$764$	23.3497	0.844764
$765$	0	0
$766$	−3.56732	−0.128893
$767$	−14.7985	−0.534343
$768$	0	0
$769$	−48.5685	−1.75143	−0.875713	−	0.482831i	$-0.839608\pi$
$769$	−48.5685	−1.75143	−0.875713	+	0.482831i	$0.839608\pi$
$770$	−47.8443	−1.72419
$771$	0	0
$772$	24.9294	0.897230
$773$	−1.45881	−0.0524699	−0.0262349	−	0.999656i	$-0.508352\pi$
$773$	−1.45881	−0.0524699	−0.0262349	+	0.999656i	$0.508352\pi$
$774$	0	0
$775$	16.1396	0.579751
$776$	4.95420	0.177845
$777$	0	0
$778$	5.42402	0.194460
$779$	5.90840	0.211690
$780$	0	0
$781$	−6.15569	−0.220268
$782$	4.19777	0.150112
$783$	0	0
$784$	0.901116	0.0321827
$785$	15.3470	0.547757
$786$	0	0
$787$	−14.1062	−0.502831	−0.251415	−	0.967879i	$-0.580896\pi$
$787$	−14.1062	−0.502831	−0.251415	+	0.967879i	$0.580896\pi$
$788$	10.2349	0.364604
$789$	0	0
$790$	51.7948	1.84278
$791$	−8.43268	−0.299831
$792$	0	0
$793$	9.39926	0.333777
$794$	−28.1272	−0.998197
$795$	0	0
$796$	26.9047	0.953611
$797$	−29.1941	−1.03411	−0.517053	−	0.855953i	$-0.672971\pi$
$797$	−29.1941	−1.03411	−0.517053	+	0.855953i	$0.672971\pi$
$798$	0	0
$799$	−1.06546	−0.0376934
$800$	5.81089	0.205446
$801$	0	0
$802$	14.8626	0.524817
$803$	1.15197	0.0406521
$804$	0	0
$805$	55.4532	1.95447
$806$	2.77747	0.0978323
$807$	0	0
$808$	2.95420	0.103928
$809$	24.4523	0.859699	0.429849	−	0.902901i	$-0.358567\pi$
$809$	24.4523	0.859699	0.429849	+	0.902901i	$0.358567\pi$
$810$	0	0
$811$	−21.5970	−0.758374	−0.379187	−	0.925320i	$-0.623796\pi$
$811$	−21.5970	−0.758374	−0.379187	+	0.925320i	$0.623796\pi$
$812$	−6.24729	−0.219237
$813$	0	0
$814$	3.68587	0.129190
$815$	39.3745	1.37923
$816$	0	0
$817$	82.5190	2.88698
$818$	−17.9752	−0.628490
$819$	0	0
$820$	−2.70691	−0.0945295
$821$	4.32141	0.150818	0.0754092	−	0.997153i	$-0.475974\pi$
$821$	4.32141	0.150818	0.0754092	+	0.997153i	$0.475974\pi$
$822$	0	0
$823$	−27.7527	−0.967399	−0.483699	−	0.875234i	$-0.660707\pi$
$823$	−27.7527	−0.967399	−0.483699	+	0.875234i	$0.660707\pi$
$824$	9.39926	0.327439
$825$	0	0
$826$	41.5970	1.44735
$827$	−8.55122	−0.297355	−0.148678	−	0.988886i	$-0.547502\pi$
$827$	−8.55122	−0.297355	−0.148678	+	0.988886i	$0.547502\pi$
$828$	0	0
$829$	38.2371	1.32803	0.664015	−	0.747720i	$-0.268851\pi$
$829$	38.2371	1.32803	0.664015	+	0.747720i	$0.268851\pi$
$830$	−26.2225	−0.910197
$831$	0	0
$832$	1.00000	0.0346688
$833$	0.630446	0.0218437
$834$	0	0
$835$	−22.2843	−0.771179
$836$	−37.1520	−1.28493
$837$	0	0
$838$	27.4610	0.948625
$839$	45.0814	1.55638	0.778192	−	0.628027i	$-0.216137\pi$
$839$	45.0814	1.55638	0.778192	+	0.628027i	$0.216137\pi$
$840$	0	0
$841$	−24.0604	−0.829668
$842$	2.94182	0.101382
$843$	0	0
$844$	4.71201	0.162194
$845$	−3.28799	−0.113110
$846$	0	0
$847$	−44.4079	−1.52587
$848$	11.0210	0.378464
$849$	0	0
$850$	4.06546	0.139444
$851$	−4.27205	−0.146444
$852$	0	0
$853$	25.9680	0.889126	0.444563	−	0.895748i	$-0.353359\pi$
$853$	25.9680	0.889126	0.444563	+	0.895748i	$0.353359\pi$
$854$	−26.4203	−0.904084
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−18.1099	−0.618622	−0.309311	−	0.950961i	$-0.600098\pi$
$857$	−18.1099	−0.618622	−0.309311	+	0.950961i	$0.600098\pi$
$858$	0	0
$859$	−32.2161	−1.09920	−0.549599	−	0.835429i	$-0.685219\pi$
$859$	−32.2161	−1.09920	−0.549599	+	0.835429i	$0.685219\pi$
$860$	−37.8058	−1.28917
$861$	0	0
$862$	−3.37959	−0.115109
$863$	5.01594	0.170745	0.0853724	−	0.996349i	$-0.472792\pi$
$863$	5.01594	0.170745	0.0853724	+	0.996349i	$0.472792\pi$
$864$	0	0
$865$	62.9715	2.14110
$866$	8.36584	0.284283
$867$	0	0
$868$	−7.80717	−0.264993
$869$	81.5475	2.76631
$870$	0	0
$871$	0.445057	0.0150802
$872$	−6.67487	−0.226040
$873$	0	0
$874$	43.0604	1.45654
$875$	7.49442	0.253358
$876$	0	0
$877$	19.5105	0.658823	0.329412	−	0.944186i	$-0.393150\pi$
$877$	19.5105	0.658823	0.329412	+	0.944186i	$0.393150\pi$
$878$	−28.5760	−0.964393
$879$	0	0
$880$	17.0210	0.573779
$881$	30.4895	1.02722	0.513608	−	0.858025i	$-0.328308\pi$
$881$	30.4895	1.02722	0.513608	+	0.858025i	$0.328308\pi$
$882$	0	0
$883$	−0.217432	−0.00731718	−0.00365859	−	0.999993i	$-0.501165\pi$
$883$	−0.217432	−0.00731718	−0.00365859	+	0.999993i	$0.501165\pi$
$884$	0.699628	0.0235310
$885$	0	0
$886$	18.5388	0.622825
$887$	4.44506	0.149250	0.0746252	−	0.997212i	$-0.476224\pi$
$887$	4.44506	0.149250	0.0746252	+	0.997212i	$0.476224\pi$
$888$	0	0
$889$	9.62454	0.322797
$890$	24.3287	0.815500
$891$	0	0
$892$	7.62907	0.255440
$893$	−10.9294	−0.365740
$894$	0	0
$895$	−0.690967	−0.0230965
$896$	−2.81089	−0.0939053
$897$	0	0
$898$	14.8626	0.495971
$899$	6.17301	0.205881
$900$	0	0
$901$	7.71063	0.256878
$902$	−4.26186	−0.141904
$903$	0	0
$904$	3.00000	0.0997785
$905$	−21.1913	−0.704422
$906$	0	0
$907$	−18.5141	−0.614750	−0.307375	−	0.951588i	$-0.599451\pi$
$907$	−18.5141	−0.614750	−0.307375	+	0.951588i	$0.599451\pi$
$908$	−22.0421	−0.731492
$909$	0	0
$910$	9.24219	0.306376
$911$	4.28937	0.142113	0.0710566	−	0.997472i	$-0.477363\pi$
$911$	4.28937	0.142113	0.0710566	+	0.997472i	$0.477363\pi$
$912$	0	0
$913$	−41.2857	−1.36636
$914$	22.3039	0.737749
$915$	0	0
$916$	−4.48948	−0.148337
$917$	23.2027	0.766221
$918$	0	0
$919$	53.5302	1.76580	0.882899	−	0.469563i	$-0.155589\pi$
$919$	53.5302	1.76580	0.882899	+	0.469563i	$0.155589\pi$
$920$	−19.7280	−0.650411
$921$	0	0
$922$	−9.74033	−0.320781
$923$	1.18911	0.0391400
$924$	0	0
$925$	−4.13740	−0.136037
$926$	−3.62550	−0.119141
$927$	0	0
$928$	2.22253	0.0729581
$929$	45.7948	1.50248	0.751239	−	0.660030i	$-0.229456\pi$
$929$	45.7948	1.50248	0.751239	+	0.660030i	$0.229456\pi$
$930$	0	0
$931$	6.46706	0.211949
$932$	−5.54256	−0.181553
$933$	0	0
$934$	0.0458003	0.00149863
$935$	11.9084	0.389446
$936$	0	0
$937$	−32.0604	−1.04737	−0.523683	−	0.851913i	$-0.675442\pi$
$937$	−32.0604	−1.04737	−0.523683	+	0.851913i	$0.675442\pi$
$938$	−1.25101	−0.0408468
$939$	0	0
$940$	5.00728	0.163320
$941$	14.3686	0.468403	0.234201	−	0.972188i	$-0.424753\pi$
$941$	14.3686	0.468403	0.234201	+	0.972188i	$0.424753\pi$
$942$	0	0
$943$	4.93963	0.160857
$944$	−14.7985	−0.481651
$945$	0	0
$946$	−59.5227	−1.93525
$947$	2.15844	0.0701399	0.0350700	−	0.999385i	$-0.488835\pi$
$947$	2.15844	0.0701399	0.0350700	+	0.999385i	$0.488835\pi$
$948$	0	0
$949$	−0.222528	−0.00722357
$950$	41.7032	1.35303
$951$	0	0
$952$	−1.96658	−0.0637372
$953$	53.0480	1.71839	0.859196	−	0.511646i	$-0.170964\pi$
$953$	53.0480	1.71839	0.859196	+	0.511646i	$0.170964\pi$
$954$	0	0
$955$	−76.7738	−2.48434
$956$	−7.43130	−0.240345
$957$	0	0
$958$	−17.6835	−0.571329
$959$	−12.9322	−0.417602
$960$	0	0
$961$	−23.2857	−0.751150
$962$	−0.712008	−0.0229561
$963$	0	0
$964$	9.39926	0.302730
$965$	−81.9678	−2.63864
$966$	0	0
$967$	27.0558	0.870057	0.435029	−	0.900417i	$-0.356738\pi$
$967$	27.0558	0.870057	0.435029	+	0.900417i	$0.356738\pi$
$968$	15.7985	0.507783
$969$	0	0
$970$	−16.2894	−0.523020
$971$	−30.3979	−0.975514	−0.487757	−	0.872979i	$-0.662185\pi$
$971$	−30.3979	−0.975514	−0.487757	+	0.872979i	$0.662185\pi$
$972$	0	0
$973$	−29.7628	−0.954150
$974$	−23.2436	−0.744772
$975$	0	0
$976$	9.39926	0.300863
$977$	17.1126	0.547482	0.273741	−	0.961803i	$-0.411739\pi$
$977$	17.1126	0.547482	0.273741	+	0.961803i	$0.411739\pi$
$978$	0	0
$979$	38.3039	1.22420
$980$	−2.96286	−0.0946451
$981$	0	0
$982$	18.5439	0.591761
$983$	−41.3722	−1.31957	−0.659783	−	0.751456i	$-0.729352\pi$
$983$	−41.3722	−1.31957	−0.659783	+	0.751456i	$0.729352\pi$
$984$	0	0
$985$	−33.6523	−1.07225
$986$	1.55494	0.0495195
$987$	0	0
$988$	7.17673	0.228322
$989$	68.9888	2.19372
$990$	0	0
$991$	−38.8552	−1.23427	−0.617137	−	0.786855i	$-0.711708\pi$
$991$	−38.8552	−1.23427	−0.617137	+	0.786855i	$0.711708\pi$
$992$	2.77747	0.0881848
$993$	0	0
$994$	−3.34245	−0.106016
$995$	−88.4624	−2.80445
$996$	0	0
$997$	−5.46334	−0.173026	−0.0865129	−	0.996251i	$-0.527572\pi$
$997$	−5.46334	−0.173026	−0.0865129	+	0.996251i	$0.527572\pi$
$998$	−31.5475	−0.998619
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2106.2.a.r.1.1		3
3.2	odd	2		2106.2.a.q.1.3		3
9.2	odd	6		234.2.e.d.157.3	yes	6
9.4	even	3		702.2.e.d.235.3		6
9.5	odd	6		234.2.e.d.79.3	✓	6
9.7	even	3		702.2.e.d.469.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.e.d.79.3	✓	6	9.5	odd	6
234.2.e.d.157.3	yes	6	9.2	odd	6
702.2.e.d.235.3		6	9.4	even	3
702.2.e.d.469.3		6	9.7	even	3
2106.2.a.q.1.3		3	3.2	odd	2
2106.2.a.r.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2106 = 2 \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2106.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.28799 q^{5} -2.81089 q^{7} +1.00000 q^{8} -3.28799 q^{10} -5.17673 q^{11} +1.00000 q^{13} -2.81089 q^{14} +1.00000 q^{16} +0.699628 q^{17} +7.17673 q^{19} -3.28799 q^{20} -5.17673 q^{22} +6.00000 q^{23} +5.81089 q^{25} +1.00000 q^{26} -2.81089 q^{28} +2.22253 q^{29} +2.77747 q^{31} +1.00000 q^{32} +0.699628 q^{34} +9.24219 q^{35} -0.712008 q^{37} +7.17673 q^{38} -3.28799 q^{40} +0.823272 q^{41} +11.4981 q^{43} -5.17673 q^{44} +6.00000 q^{46} -1.52290 q^{47} +0.901116 q^{49} +5.81089 q^{50} +1.00000 q^{52} +11.0210 q^{53} +17.0210 q^{55} -2.81089 q^{56} +2.22253 q^{58} -14.7985 q^{59} +9.39926 q^{61} +2.77747 q^{62} +1.00000 q^{64} -3.28799 q^{65} +0.445057 q^{67} +0.699628 q^{68} +9.24219 q^{70} +1.18911 q^{71} -0.222528 q^{73} -0.712008 q^{74} +7.17673 q^{76} +14.5512 q^{77} -15.7527 q^{79} -3.28799 q^{80} +0.823272 q^{82} +7.97524 q^{83} -2.30037 q^{85} +11.4981 q^{86} -5.17673 q^{88} -7.39926 q^{89} -2.81089 q^{91} +6.00000 q^{92} -1.52290 q^{94} -23.5970 q^{95} +4.95420 q^{97} +0.901116 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 2 q^{7} + 3 q^{8} + 2 q^{10} - 4 q^{11} + 3 q^{13} - 2 q^{14} + 3 q^{16} - 4 q^{17} + 10 q^{19} + 2 q^{20} - 4 q^{22} + 18 q^{23} + 11 q^{25} + 3 q^{26} - 2 q^{28} + 6 q^{29}+ \cdots + 21 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 2 * q^7 + 3 * q^8 + 2 * q^10 - 4 * q^11 + 3 * q^13 - 2 * q^14 + 3 * q^16 - 4 * q^17 + 10 * q^19 + 2 * q^20 - 4 * q^22 + 18 * q^23 + 11 * q^25 + 3 * q^26 - 2 * q^28 + 6 * q^29 + 9 * q^31 + 3 * q^32 - 4 * q^34 + 5 * q^35 - 14 * q^37 + 10 * q^38 + 2 * q^40 + 14 * q^41 + 4 * q^43 - 4 * q^44 + 18 * q^46 - 10 * q^47 + 21 * q^49 + 11 * q^50 + 3 * q^52 + 8 * q^53 + 26 * q^55 - 2 * q^56 + 6 * q^58 - 20 * q^59 + 16 * q^61 + 9 * q^62 + 3 * q^64 + 2 * q^65 - 4 * q^68 + 5 * q^70 + 10 * q^71 - 14 * q^74 + 10 * q^76 - 16 * q^77 - 12 * q^79 + 2 * q^80 + 14 * q^82 - 12 * q^83 - 13 * q^85 + 4 * q^86 - 4 * q^88 - 10 * q^89 - 2 * q^91 + 18 * q^92 - 10 * q^94 - 22 * q^95 + 4 * q^97 + 21 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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