LMFDB - Embedded newform 2214.2.a.o.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2214,2,Mod(1,2214)] 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("2214.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2214, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2214 = 2 \cdot 3^{3} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2214.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,-3,0,-3,3,0,-3,-5] 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:	$17.6788790075$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.86620$ of defining polynomial
Character	$\chi$	$=$	2214.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} -0.482696 q^{5} -1.51730 q^{7} +1.00000 q^{8} -0.482696 q^{10} -6.21509 q^{11} +0.831590 q^{13} -1.51730 q^{14} +1.00000 q^{16} +1.73240 q^{17} +5.08129 q^{19} -0.482696 q^{20} -6.21509 q^{22} -1.86620 q^{23} -4.76700 q^{25} +0.831590 q^{26} -1.51730 q^{28} -4.51730 q^{29} -1.13380 q^{31} +1.00000 q^{32} +1.73240 q^{34} +0.732397 q^{35} -4.04668 q^{37} +5.08129 q^{38} -0.482696 q^{40} -1.00000 q^{41} -7.73240 q^{43} -6.21509 q^{44} -1.86620 q^{46} -6.31429 q^{47} -4.69779 q^{49} -4.76700 q^{50} +0.831590 q^{52} -3.83159 q^{53} +3.00000 q^{55} -1.51730 q^{56} -4.51730 q^{58} -3.73240 q^{59} +2.38350 q^{61} -1.13380 q^{62} +1.00000 q^{64} -0.401405 q^{65} -4.00000 q^{67} +1.73240 q^{68} +0.732397 q^{70} -9.41811 q^{71} +6.03461 q^{73} -4.04668 q^{74} +5.08129 q^{76} +9.43018 q^{77} +1.55191 q^{79} -0.482696 q^{80} -1.00000 q^{82} -4.41811 q^{83} -0.836221 q^{85} -7.73240 q^{86} -6.21509 q^{88} -18.1626 q^{89} -1.26178 q^{91} -1.86620 q^{92} -6.31429 q^{94} -2.45272 q^{95} +9.54608 q^{97} -4.69779 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} + 3 q^{8} - 3 q^{10} - 5 q^{11} - 2 q^{13} - 3 q^{14} + 3 q^{16} - 10 q^{17} - 6 q^{19} - 3 q^{20} - 5 q^{22} + 2 q^{23} + 4 q^{25} - 2 q^{26} - 3 q^{28} - 12 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 + 3 * q^8 - 3 * q^10 - 5 * q^11 - 2 * q^13 - 3 * q^14 + 3 * q^16 - 10 * q^17 - 6 * q^19 - 3 * q^20 - 5 * q^22 + 2 * q^23 + 4 * q^25 - 2 * q^26 - 3 * q^28 - 12 * q^29 - 11 * q^31 + 3 * q^32 - 10 * q^34 - 13 * q^35 + 6 * q^37 - 6 * q^38 - 3 * q^40 - 3 * q^41 - 8 * q^43 - 5 * q^44 + 2 * q^46 - 16 * q^47 - 2 * q^49 + 4 * q^50 - 2 * q^52 - 7 * q^53 + 9 * q^55 - 3 * q^56 - 12 * q^58 + 4 * q^59 - 2 * q^61 - 11 * q^62 + 3 * q^64 - 24 * q^65 - 12 * q^67 - 10 * q^68 - 13 * q^70 - 16 * q^71 + 15 * q^73 + 6 * q^74 - 6 * q^76 + q^77 - 3 * q^80 - 3 * q^82 - q^83 + 22 * q^85 - 8 * q^86 - 5 * q^88 - 12 * q^89 + 28 * q^91 + 2 * q^92 - 16 * q^94 + 8 * q^95 - 23 * q^97 - 2 * 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
$5$	−0.482696	−0.215868	−0.107934	−	0.994158i	$-0.534424\pi$
$5$	−0.482696	−0.215868	−0.107934	+	0.994158i	$0.534424\pi$
$6$	0	0
$7$	−1.51730	−0.573487	−0.286744	−	0.958007i	$-0.592573\pi$
$7$	−1.51730	−0.573487	−0.286744	+	0.958007i	$0.592573\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.482696	−0.152642
$11$	−6.21509	−1.87392	−0.936960	−	0.349435i	$-0.886373\pi$
$11$	−6.21509	−1.87392	−0.936960	+	0.349435i	$0.886373\pi$
$12$	0	0
$13$	0.831590	0.230642	0.115321	−	0.993328i	$-0.463210\pi$
$13$	0.831590	0.230642	0.115321	+	0.993328i	$0.463210\pi$
$14$	−1.51730	−0.405517
$15$	0	0
$16$	1.00000	0.250000
$17$	1.73240	0.420168	0.210084	−	0.977683i	$-0.432626\pi$
$17$	1.73240	0.420168	0.210084	+	0.977683i	$0.432626\pi$
$18$	0	0
$19$	5.08129	1.16573	0.582864	−	0.812570i	$-0.301932\pi$
$19$	5.08129	1.16573	0.582864	+	0.812570i	$0.301932\pi$
$20$	−0.482696	−0.107934
$21$	0	0
$22$	−6.21509	−1.32506
$23$	−1.86620	−0.389129	−0.194565	−	0.980890i	$-0.562329\pi$
$23$	−1.86620	−0.389129	−0.194565	+	0.980890i	$0.562329\pi$
$24$	0	0
$25$	−4.76700	−0.953401
$26$	0.831590	0.163088
$27$	0	0
$28$	−1.51730	−0.286744
$29$	−4.51730	−0.838842	−0.419421	−	0.907792i	$-0.637767\pi$
$29$	−4.51730	−0.838842	−0.419421	+	0.907792i	$0.637767\pi$
$30$	0	0
$31$	−1.13380	−0.203637	−0.101818	−	0.994803i	$-0.532466\pi$
$31$	−1.13380	−0.203637	−0.101818	+	0.994803i	$0.532466\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.73240	0.297104
$35$	0.732397	0.123798
$36$	0	0
$37$	−4.04668	−0.665271	−0.332635	−	0.943056i	$-0.607938\pi$
$37$	−4.04668	−0.665271	−0.332635	+	0.943056i	$0.607938\pi$
$38$	5.08129	0.824294
$39$	0	0
$40$	−0.482696	−0.0763209
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−7.73240	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$43$	−7.73240	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$44$	−6.21509	−0.936960
$45$	0	0
$46$	−1.86620	−0.275156
$47$	−6.31429	−0.921033	−0.460517	−	0.887651i	$-0.652336\pi$
$47$	−6.31429	−0.921033	−0.460517	+	0.887651i	$0.652336\pi$
$48$	0	0
$49$	−4.69779	−0.671113
$50$	−4.76700	−0.674156
$51$	0	0
$52$	0.831590	0.115321
$53$	−3.83159	−0.526309	−0.263155	−	0.964754i	$-0.584763\pi$
$53$	−3.83159	−0.526309	−0.263155	+	0.964754i	$0.584763\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	−1.51730	−0.202758
$57$	0	0
$58$	−4.51730	−0.593151
$59$	−3.73240	−0.485917	−0.242958	−	0.970037i	$-0.578118\pi$
$59$	−3.73240	−0.485917	−0.242958	+	0.970037i	$0.578118\pi$
$60$	0	0
$61$	2.38350	0.305176	0.152588	−	0.988290i	$-0.451239\pi$
$61$	2.38350	0.305176	0.152588	+	0.988290i	$0.451239\pi$
$62$	−1.13380	−0.143993
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.401405	−0.0497882
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	1.73240	0.210084
$69$	0	0
$70$	0.732397	0.0875381
$71$	−9.41811	−1.11772	−0.558862	−	0.829261i	$-0.688762\pi$
$71$	−9.41811	−1.11772	−0.558862	+	0.829261i	$0.688762\pi$
$72$	0	0
$73$	6.03461	0.706297	0.353149	−	0.935567i	$-0.385111\pi$
$73$	6.03461	0.706297	0.353149	+	0.935567i	$0.385111\pi$
$74$	−4.04668	−0.470417
$75$	0	0
$76$	5.08129	0.582864
$77$	9.43018	1.07467
$78$	0	0
$79$	1.55191	0.174604	0.0873019	−	0.996182i	$-0.472176\pi$
$79$	1.55191	0.174604	0.0873019	+	0.996182i	$0.472176\pi$
$80$	−0.482696	−0.0539670
$81$	0	0
$82$	−1.00000	−0.110432
$83$	−4.41811	−0.484951	−0.242475	−	0.970158i	$-0.577959\pi$
$83$	−4.41811	−0.484951	−0.242475	+	0.970158i	$0.577959\pi$
$84$	0	0
$85$	−0.836221	−0.0907009
$86$	−7.73240	−0.833806
$87$	0	0
$88$	−6.21509	−0.662531
$89$	−18.1626	−1.92523	−0.962615	−	0.270874i	$-0.912687\pi$
$89$	−18.1626	−1.92523	−0.962615	+	0.270874i	$0.912687\pi$
$90$	0	0
$91$	−1.26178	−0.132270
$92$	−1.86620	−0.194565
$93$	0	0
$94$	−6.31429	−0.651269
$95$	−2.45272	−0.251644
$96$	0	0
$97$	9.54608	0.969258	0.484629	−	0.874720i	$-0.338955\pi$
$97$	9.54608	0.969258	0.484629	+	0.874720i	$0.338955\pi$
$98$	−4.69779	−0.474548
$99$	0	0
$100$	−4.76700	−0.476700
$101$	11.5461	1.14888	0.574439	−	0.818547i	$-0.305220\pi$
$101$	11.5461	1.14888	0.574439	+	0.818547i	$0.305220\pi$
$102$	0	0
$103$	−5.66318	−0.558010	−0.279005	−	0.960290i	$-0.590005\pi$
$103$	−5.66318	−0.558010	−0.279005	+	0.960290i	$0.590005\pi$
$104$	0.831590	0.0815441
$105$	0	0
$106$	−3.83159	−0.372157
$107$	−3.56399	−0.344544	−0.172272	−	0.985049i	$-0.555111\pi$
$107$	−3.56399	−0.344544	−0.172272	+	0.985049i	$0.555111\pi$
$108$	0	0
$109$	16.3656	1.56754	0.783770	−	0.621051i	$-0.213294\pi$
$109$	16.3656	1.56754	0.783770	+	0.621051i	$0.213294\pi$
$110$	3.00000	0.286039
$111$	0	0
$112$	−1.51730	−0.143372
$113$	9.03461	0.849904	0.424952	−	0.905216i	$-0.360291\pi$
$113$	9.03461	0.849904	0.424952	+	0.905216i	$0.360291\pi$
$114$	0	0
$115$	0.900806	0.0840006
$116$	−4.51730	−0.419421
$117$	0	0
$118$	−3.73240	−0.343595
$119$	−2.62857	−0.240961
$120$	0	0
$121$	27.6274	2.51158
$122$	2.38350	0.215792
$123$	0	0
$124$	−1.13380	−0.101818
$125$	4.71449	0.421677
$126$	0	0
$127$	10.7791	0.956489	0.478244	−	0.878227i	$-0.341273\pi$
$127$	10.7791	0.956489	0.478244	+	0.878227i	$0.341273\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−0.401405	−0.0352056
$131$	14.9596	1.30702	0.653512	−	0.756917i	$-0.273295\pi$
$131$	14.9596	1.30702	0.653512	+	0.756917i	$0.273295\pi$
$132$	0	0
$133$	−7.70986	−0.668530
$134$	−4.00000	−0.345547
$135$	0	0
$136$	1.73240	0.148552
$137$	−1.79698	−0.153527	−0.0767633	−	0.997049i	$-0.524459\pi$
$137$	−1.79698	−0.153527	−0.0767633	+	0.997049i	$0.524459\pi$
$138$	0	0
$139$	0.0645856	0.00547808	0.00273904	−	0.999996i	$-0.499128\pi$
$139$	0.0645856	0.00547808	0.00273904	+	0.999996i	$0.499128\pi$
$140$	0.732397	0.0618988
$141$	0	0
$142$	−9.41811	−0.790350
$143$	−5.16841	−0.432204
$144$	0	0
$145$	2.18048	0.181079
$146$	6.03461	0.499428
$147$	0	0
$148$	−4.04668	−0.332635
$149$	−7.54608	−0.618199	−0.309100	−	0.951030i	$-0.600028\pi$
$149$	−7.54608	−0.618199	−0.309100	+	0.951030i	$0.600028\pi$
$150$	0	0
$151$	−9.06922	−0.738042	−0.369021	−	0.929421i	$-0.620307\pi$
$151$	−9.06922	−0.738042	−0.369021	+	0.929421i	$0.620307\pi$
$152$	5.08129	0.412147
$153$	0	0
$154$	9.43018	0.759906
$155$	0.547282	0.0439587
$156$	0	0
$157$	1.56982	0.125285	0.0626424	−	0.998036i	$-0.480047\pi$
$157$	1.56982	0.125285	0.0626424	+	0.998036i	$0.480047\pi$
$158$	1.55191	0.123464
$159$	0	0
$160$	−0.482696	−0.0381605
$161$	2.83159	0.223161
$162$	0	0
$163$	−1.93541	−0.151593	−0.0757967	−	0.997123i	$-0.524150\pi$
$163$	−1.93541	−0.151593	−0.0757967	+	0.997123i	$0.524150\pi$
$164$	−1.00000	−0.0780869
$165$	0	0
$166$	−4.41811	−0.342912
$167$	5.19719	0.402171	0.201085	−	0.979574i	$-0.435553\pi$
$167$	5.19719	0.402171	0.201085	+	0.979574i	$0.435553\pi$
$168$	0	0
$169$	−12.3085	−0.946804
$170$	−0.836221	−0.0641352
$171$	0	0
$172$	−7.73240	−0.589590
$173$	7.36097	0.559644	0.279822	−	0.960052i	$-0.409725\pi$
$173$	7.36097	0.559644	0.279822	+	0.960052i	$0.409725\pi$
$174$	0	0
$175$	7.23300	0.546763
$176$	−6.21509	−0.468480
$177$	0	0
$178$	−18.1626	−1.36134
$179$	−14.1459	−1.05731	−0.528656	−	0.848836i	$-0.677304\pi$
$179$	−14.1459	−1.05731	−0.528656	+	0.848836i	$0.677304\pi$
$180$	0	0
$181$	6.40141	0.475813	0.237906	−	0.971288i	$-0.423539\pi$
$181$	6.40141	0.475813	0.237906	+	0.971288i	$0.423539\pi$
$182$	−1.26178	−0.0935290
$183$	0	0
$184$	−1.86620	−0.137578
$185$	1.95332	0.143611
$186$	0	0
$187$	−10.7670	−0.787361
$188$	−6.31429	−0.460517
$189$	0	0
$190$	−2.45272	−0.177939
$191$	−14.4994	−1.04914	−0.524570	−	0.851367i	$-0.675774\pi$
$191$	−14.4994	−1.04914	−0.524570	+	0.851367i	$0.675774\pi$
$192$	0	0
$193$	18.3310	1.31949	0.659747	−	0.751488i	$-0.270664\pi$
$193$	18.3310	1.31949	0.659747	+	0.751488i	$0.270664\pi$
$194$	9.54608	0.685369
$195$	0	0
$196$	−4.69779	−0.335556
$197$	−18.7145	−1.33335	−0.666676	−	0.745347i	$-0.732284\pi$
$197$	−18.7145	−1.33335	−0.666676	+	0.745347i	$0.732284\pi$
$198$	0	0
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	−4.76700	−0.337078
$201$	0	0
$202$	11.5461	0.812380
$203$	6.85412	0.481065
$204$	0	0
$205$	0.482696	0.0337129
$206$	−5.66318	−0.394572
$207$	0	0
$208$	0.831590	0.0576604
$209$	−31.5807	−2.18448
$210$	0	0
$211$	−18.5461	−1.27677	−0.638383	−	0.769719i	$-0.720396\pi$
$211$	−18.5461	−1.27677	−0.638383	+	0.769719i	$0.720396\pi$
$212$	−3.83159	−0.263155
$213$	0	0
$214$	−3.56399	−0.243629
$215$	3.73240	0.254547
$216$	0	0
$217$	1.72032	0.116783
$218$	16.3656	1.10842
$219$	0	0
$220$	3.00000	0.202260
$221$	1.44064	0.0969082
$222$	0	0
$223$	3.32636	0.222750	0.111375	−	0.993778i	$-0.464475\pi$
$223$	3.32636	0.222750	0.111375	+	0.993778i	$0.464475\pi$
$224$	−1.51730	−0.101379
$225$	0	0
$226$	9.03461	0.600973
$227$	22.2738	1.47837	0.739184	−	0.673504i	$-0.235212\pi$
$227$	22.2738	1.47837	0.739184	+	0.673504i	$0.235212\pi$
$228$	0	0
$229$	20.7670	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$229$	20.7670	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$230$	0.900806	0.0593974
$231$	0	0
$232$	−4.51730	−0.296576
$233$	−15.1280	−0.991066	−0.495533	−	0.868589i	$-0.665027\pi$
$233$	−15.1280	−0.991066	−0.495533	+	0.868589i	$0.665027\pi$
$234$	0	0
$235$	3.04788	0.198822
$236$	−3.73240	−0.242958
$237$	0	0
$238$	−2.62857	−0.170385
$239$	−12.2318	−0.791209	−0.395605	−	0.918421i	$-0.629465\pi$
$239$	−12.2318	−0.791209	−0.395605	+	0.918421i	$0.629465\pi$
$240$	0	0
$241$	21.2151	1.36658	0.683292	−	0.730145i	$-0.260548\pi$
$241$	21.2151	1.36658	0.683292	+	0.730145i	$0.260548\pi$
$242$	27.6274	1.77595
$243$	0	0
$244$	2.38350	0.152588
$245$	2.26760	0.144872
$246$	0	0
$247$	4.22555	0.268865
$248$	−1.13380	−0.0719965
$249$	0	0
$250$	4.71449	0.298171
$251$	1.71449	0.108218	0.0541089	−	0.998535i	$-0.482768\pi$
$251$	1.71449	0.108218	0.0541089	+	0.998535i	$0.482768\pi$
$252$	0	0
$253$	11.5986	0.729197
$254$	10.7791	0.676340
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.2272	−1.13698	−0.568490	−	0.822690i	$-0.692472\pi$
$257$	−18.2272	−1.13698	−0.568490	+	0.822690i	$0.692472\pi$
$258$	0	0
$259$	6.14005	0.381524
$260$	−0.401405	−0.0248941
$261$	0	0
$262$	14.9596	0.924205
$263$	1.44064	0.0888339	0.0444170	−	0.999013i	$-0.485857\pi$
$263$	1.44064	0.0888339	0.0444170	+	0.999013i	$0.485857\pi$
$264$	0	0
$265$	1.84949	0.113613
$266$	−7.70986	−0.472722
$267$	0	0
$268$	−4.00000	−0.244339
$269$	23.7386	1.44737	0.723685	−	0.690130i	$-0.242447\pi$
$269$	23.7386	1.44737	0.723685	+	0.690130i	$0.242447\pi$
$270$	0	0
$271$	11.3835	0.691499	0.345749	−	0.938327i	$-0.387625\pi$
$271$	11.3835	0.691499	0.345749	+	0.938327i	$0.387625\pi$
$272$	1.73240	0.105042
$273$	0	0
$274$	−1.79698	−0.108560
$275$	29.6274	1.78660
$276$	0	0
$277$	22.1400	1.33027	0.665133	−	0.746725i	$-0.268375\pi$
$277$	22.1400	1.33027	0.665133	+	0.746725i	$0.268375\pi$
$278$	0.0645856	0.00387358
$279$	0	0
$280$	0.732397	0.0437691
$281$	2.49477	0.148826	0.0744128	−	0.997228i	$-0.476292\pi$
$281$	2.49477	0.148826	0.0744128	+	0.997228i	$0.476292\pi$
$282$	0	0
$283$	−16.4948	−0.980512	−0.490256	−	0.871578i	$-0.663097\pi$
$283$	−16.4948	−0.980512	−0.490256	+	0.871578i	$0.663097\pi$
$284$	−9.41811	−0.558862
$285$	0	0
$286$	−5.16841	−0.305614
$287$	1.51730	0.0895636
$288$	0	0
$289$	−13.9988	−0.823459
$290$	2.18048	0.128042
$291$	0	0
$292$	6.03461	0.353149
$293$	1.50060	0.0876659	0.0438330	−	0.999039i	$-0.486043\pi$
$293$	1.50060	0.0876659	0.0438330	+	0.999039i	$0.486043\pi$
$294$	0	0
$295$	1.80161	0.104894
$296$	−4.04668	−0.235209
$297$	0	0
$298$	−7.54608	−0.437133
$299$	−1.55191	−0.0897494
$300$	0	0
$301$	11.7324	0.676244
$302$	−9.06922	−0.521875
$303$	0	0
$304$	5.08129	0.291432
$305$	−1.15051	−0.0658778
$306$	0	0
$307$	1.93541	0.110460	0.0552300	−	0.998474i	$-0.482411\pi$
$307$	1.93541	0.110460	0.0552300	+	0.998474i	$0.482411\pi$
$308$	9.43018	0.537335
$309$	0	0
$310$	0.547282	0.0310835
$311$	−11.5565	−0.655311	−0.327656	−	0.944797i	$-0.606259\pi$
$311$	−11.5565	−0.655311	−0.327656	+	0.944797i	$0.606259\pi$
$312$	0	0
$313$	−28.8662	−1.63161	−0.815807	−	0.578324i	$-0.803707\pi$
$313$	−28.8662	−1.63161	−0.815807	+	0.578324i	$0.803707\pi$
$314$	1.56982	0.0885898
$315$	0	0
$316$	1.55191	0.0873019
$317$	−32.9238	−1.84918	−0.924591	−	0.380961i	$-0.875593\pi$
$317$	−32.9238	−1.84918	−0.924591	+	0.380961i	$0.875593\pi$
$318$	0	0
$319$	28.0755	1.57192
$320$	−0.482696	−0.0269835
$321$	0	0
$322$	2.83159	0.157798
$323$	8.80281	0.489801
$324$	0	0
$325$	−3.96419	−0.219894
$326$	−1.93541	−0.107193
$327$	0	0
$328$	−1.00000	−0.0552158
$329$	9.58069	0.528201
$330$	0	0
$331$	−7.37143	−0.405170	−0.202585	−	0.979265i	$-0.564934\pi$
$331$	−7.37143	−0.405170	−0.202585	+	0.979265i	$0.564934\pi$
$332$	−4.41811	−0.242475
$333$	0	0
$334$	5.19719	0.284378
$335$	1.93078	0.105490
$336$	0	0
$337$	18.1113	0.986584	0.493292	−	0.869864i	$-0.335793\pi$
$337$	18.1113	0.986584	0.493292	+	0.869864i	$0.335793\pi$
$338$	−12.3085	−0.669492
$339$	0	0
$340$	−0.836221	−0.0453504
$341$	7.04668	0.381599
$342$	0	0
$343$	17.7491	0.958361
$344$	−7.73240	−0.416903
$345$	0	0
$346$	7.36097	0.395728
$347$	22.8258	1.22535	0.612676	−	0.790335i	$-0.290093\pi$
$347$	22.8258	1.22535	0.612676	+	0.790335i	$0.290093\pi$
$348$	0	0
$349$	23.9175	1.28028	0.640138	−	0.768260i	$-0.278877\pi$
$349$	23.9175	1.28028	0.640138	+	0.768260i	$0.278877\pi$
$350$	7.23300	0.386620
$351$	0	0
$352$	−6.21509	−0.331266
$353$	17.7791	0.946285	0.473142	−	0.880986i	$-0.343120\pi$
$353$	17.7791	0.946285	0.473142	+	0.880986i	$0.343120\pi$
$354$	0	0
$355$	4.54608	0.241281
$356$	−18.1626	−0.962615
$357$	0	0
$358$	−14.1459	−0.747633
$359$	26.1914	1.38233	0.691164	−	0.722698i	$-0.257098\pi$
$359$	26.1914	1.38233	0.691164	+	0.722698i	$0.257098\pi$
$360$	0	0
$361$	6.81952	0.358922
$362$	6.40141	0.336450
$363$	0	0
$364$	−1.26178	−0.0661350
$365$	−2.91288	−0.152467
$366$	0	0
$367$	−19.1159	−0.997842	−0.498921	−	0.866648i	$-0.666270\pi$
$367$	−19.1159	−0.997842	−0.498921	+	0.866648i	$0.666270\pi$
$368$	−1.86620	−0.0972823
$369$	0	0
$370$	1.95332	0.101548
$371$	5.81369	0.301832
$372$	0	0
$373$	−33.0696	−1.71228	−0.856140	−	0.516743i	$-0.827144\pi$
$373$	−33.0696	−1.71228	−0.856140	+	0.516743i	$0.827144\pi$
$374$	−10.7670	−0.556749
$375$	0	0
$376$	−6.31429	−0.325634
$377$	−3.75655	−0.193472
$378$	0	0
$379$	−6.76700	−0.347598	−0.173799	−	0.984781i	$-0.555604\pi$
$379$	−6.76700	−0.347598	−0.173799	+	0.984781i	$0.555604\pi$
$380$	−2.45272	−0.125822
$381$	0	0
$382$	−14.4994	−0.741854
$383$	−14.2559	−0.728445	−0.364222	−	0.931312i	$-0.618665\pi$
$383$	−14.2559	−0.728445	−0.364222	+	0.931312i	$0.618665\pi$
$384$	0	0
$385$	−4.55191	−0.231987
$386$	18.3310	0.933023
$387$	0	0
$388$	9.54608	0.484629
$389$	−24.4648	−1.24041	−0.620207	−	0.784438i	$-0.712951\pi$
$389$	−24.4648	−1.24041	−0.620207	+	0.784438i	$0.712951\pi$
$390$	0	0
$391$	−3.23300	−0.163500
$392$	−4.69779	−0.237274
$393$	0	0
$394$	−18.7145	−0.942823
$395$	−0.749102	−0.0376914
$396$	0	0
$397$	−8.49477	−0.426340	−0.213170	−	0.977015i	$-0.568379\pi$
$397$	−8.49477	−0.426340	−0.213170	+	0.977015i	$0.568379\pi$
$398$	5.00000	0.250627
$399$	0	0
$400$	−4.76700	−0.238350
$401$	38.4636	1.92078	0.960390	−	0.278659i	$-0.0898899\pi$
$401$	38.4636	1.92078	0.960390	+	0.278659i	$0.0898899\pi$
$402$	0	0
$403$	−0.942858	−0.0469671
$404$	11.5461	0.574439
$405$	0	0
$406$	6.85412	0.340164
$407$	25.1505	1.24666
$408$	0	0
$409$	−1.51730	−0.0750259	−0.0375129	−	0.999296i	$-0.511944\pi$
$409$	−1.51730	−0.0750259	−0.0375129	+	0.999296i	$0.511944\pi$
$410$	0.482696	0.0238387
$411$	0	0
$412$	−5.66318	−0.279005
$413$	5.66318	0.278667
$414$	0	0
$415$	2.13260	0.104685
$416$	0.831590	0.0407721
$417$	0	0
$418$	−31.5807	−1.54466
$419$	1.07666	0.0525983	0.0262991	−	0.999654i	$-0.491628\pi$
$419$	1.07666	0.0525983	0.0262991	+	0.999654i	$0.491628\pi$
$420$	0	0
$421$	−18.4948	−0.901380	−0.450690	−	0.892681i	$-0.648822\pi$
$421$	−18.4948	−0.901380	−0.450690	+	0.892681i	$0.648822\pi$
$422$	−18.5461	−0.902809
$423$	0	0
$424$	−3.83159	−0.186078
$425$	−8.25834	−0.400588
$426$	0	0
$427$	−3.61650	−0.175015
$428$	−3.56399	−0.172272
$429$	0	0
$430$	3.73240	0.179992
$431$	22.6332	1.09020	0.545102	−	0.838370i	$-0.316491\pi$
$431$	22.6332	1.09020	0.545102	+	0.838370i	$0.316491\pi$
$432$	0	0
$433$	10.5340	0.506232	0.253116	−	0.967436i	$-0.418545\pi$
$433$	10.5340	0.506232	0.253116	+	0.967436i	$0.418545\pi$
$434$	1.72032	0.0825781
$435$	0	0
$436$	16.3656	0.783770
$437$	−9.48270	−0.453619
$438$	0	0
$439$	−30.0230	−1.43292	−0.716459	−	0.697630i	$-0.754238\pi$
$439$	−30.0230	−1.43292	−0.716459	+	0.697630i	$0.754238\pi$
$440$	3.00000	0.143019
$441$	0	0
$442$	1.44064	0.0685244
$443$	37.7658	1.79431	0.897154	−	0.441718i	$-0.145631\pi$
$443$	37.7658	1.79431	0.897154	+	0.441718i	$0.145631\pi$
$444$	0	0
$445$	8.76700	0.415596
$446$	3.32636	0.157508
$447$	0	0
$448$	−1.51730	−0.0716859
$449$	33.9191	1.60074	0.800371	−	0.599505i	$-0.204636\pi$
$449$	33.9191	1.60074	0.800371	+	0.599505i	$0.204636\pi$
$450$	0	0
$451$	6.21509	0.292657
$452$	9.03461	0.424952
$453$	0	0
$454$	22.2738	1.04536
$455$	0.609054	0.0285529
$456$	0	0
$457$	−34.8304	−1.62930	−0.814649	−	0.579955i	$-0.803070\pi$
$457$	−34.8304	−1.62930	−0.814649	+	0.579955i	$0.803070\pi$
$458$	20.7670	0.970378
$459$	0	0
$460$	0.900806	0.0420003
$461$	11.9988	0.558840	0.279420	−	0.960169i	$-0.409858\pi$
$461$	11.9988	0.558840	0.279420	+	0.960169i	$0.409858\pi$
$462$	0	0
$463$	−29.2318	−1.35852	−0.679258	−	0.733899i	$-0.737698\pi$
$463$	−29.2318	−1.35852	−0.679258	+	0.733899i	$0.737698\pi$
$464$	−4.51730	−0.209711
$465$	0	0
$466$	−15.1280	−0.700790
$467$	−17.4077	−0.805530	−0.402765	−	0.915303i	$-0.631951\pi$
$467$	−17.4077	−0.805530	−0.402765	+	0.915303i	$0.631951\pi$
$468$	0	0
$469$	6.06922	0.280250
$470$	3.04788	0.140588
$471$	0	0
$472$	−3.73240	−0.171797
$473$	48.0576	2.20969
$474$	0	0
$475$	−24.2225	−1.11141
$476$	−2.62857	−0.120480
$477$	0	0
$478$	−12.2318	−0.559469
$479$	−16.1159	−0.736354	−0.368177	−	0.929756i	$-0.620018\pi$
$479$	−16.1159	−0.736354	−0.368177	+	0.929756i	$0.620018\pi$
$480$	0	0
$481$	−3.36518	−0.153439
$482$	21.2151	0.966321
$483$	0	0
$484$	27.6274	1.25579
$485$	−4.60786	−0.209232
$486$	0	0
$487$	33.8529	1.53402	0.767011	−	0.641634i	$-0.221743\pi$
$487$	33.8529	1.53402	0.767011	+	0.641634i	$0.221743\pi$
$488$	2.38350	0.107896
$489$	0	0
$490$	2.26760	0.102440
$491$	6.12636	0.276479	0.138239	−	0.990399i	$-0.455856\pi$
$491$	6.12636	0.276479	0.138239	+	0.990399i	$0.455856\pi$
$492$	0	0
$493$	−7.82576	−0.352455
$494$	4.22555	0.190117
$495$	0	0
$496$	−1.13380	−0.0509092
$497$	14.2901	0.641000
$498$	0	0
$499$	−13.7549	−0.615755	−0.307878	−	0.951426i	$-0.599619\pi$
$499$	−13.7549	−0.615755	−0.307878	+	0.951426i	$0.599619\pi$
$500$	4.71449	0.210839
$501$	0	0
$502$	1.71449	0.0765216
$503$	−29.1521	−1.29983	−0.649914	−	0.760007i	$-0.725195\pi$
$503$	−29.1521	−1.29983	−0.649914	+	0.760007i	$0.725195\pi$
$504$	0	0
$505$	−5.57325	−0.248006
$506$	11.5986	0.515620
$507$	0	0
$508$	10.7791	0.478244
$509$	25.2030	1.11710	0.558552	−	0.829469i	$-0.311357\pi$
$509$	25.2030	1.11710	0.558552	+	0.829469i	$0.311357\pi$
$510$	0	0
$511$	−9.15634	−0.405052
$512$	1.00000	0.0441942
$513$	0	0
$514$	−18.2272	−0.803966
$515$	2.73359	0.120457
$516$	0	0
$517$	39.2439	1.72594
$518$	6.14005	0.269778
$519$	0	0
$520$	−0.401405	−0.0176028
$521$	−31.7900	−1.39274	−0.696372	−	0.717681i	$-0.745203\pi$
$521$	−31.7900	−1.39274	−0.696372	+	0.717681i	$0.745203\pi$
$522$	0	0
$523$	−10.4753	−0.458051	−0.229025	−	0.973420i	$-0.573554\pi$
$523$	−10.4753	−0.458051	−0.229025	+	0.973420i	$0.573554\pi$
$524$	14.9596	0.653512
$525$	0	0
$526$	1.44064	0.0628151
$527$	−1.96419	−0.0855616
$528$	0	0
$529$	−19.5173	−0.848578
$530$	1.84949	0.0803368
$531$	0	0
$532$	−7.70986	−0.334265
$533$	−0.831590	−0.0360202
$534$	0	0
$535$	1.72032	0.0743760
$536$	−4.00000	−0.172774
$537$	0	0
$538$	23.7386	1.02345
$539$	29.1972	1.25761
$540$	0	0
$541$	30.4877	1.31077	0.655385	−	0.755295i	$-0.272506\pi$
$541$	30.4877	1.31077	0.655385	+	0.755295i	$0.272506\pi$
$542$	11.3835	0.488963
$543$	0	0
$544$	1.73240	0.0742759
$545$	−7.89961	−0.338382
$546$	0	0
$547$	13.5940	0.581236	0.290618	−	0.956839i	$-0.406139\pi$
$547$	13.5940	0.581236	0.290618	+	0.956839i	$0.406139\pi$
$548$	−1.79698	−0.0767633
$549$	0	0
$550$	29.6274	1.26332
$551$	−22.9537	−0.977862
$552$	0	0
$553$	−2.35472	−0.100133
$554$	22.1400	0.940641
$555$	0	0
$556$	0.0645856	0.00273904
$557$	20.3823	0.863626	0.431813	−	0.901963i	$-0.357874\pi$
$557$	20.3823	0.863626	0.431813	+	0.901963i	$0.357874\pi$
$558$	0	0
$559$	−6.43018	−0.271968
$560$	0.732397	0.0309494
$561$	0	0
$562$	2.49477	0.105236
$563$	−0.906635	−0.0382101	−0.0191050	−	0.999817i	$-0.506082\pi$
$563$	−0.906635	−0.0382101	−0.0191050	+	0.999817i	$0.506082\pi$
$564$	0	0
$565$	−4.36097	−0.183467
$566$	−16.4948	−0.693327
$567$	0	0
$568$	−9.41811	−0.395175
$569$	−19.3730	−0.812160	−0.406080	−	0.913837i	$-0.633105\pi$
$569$	−19.3730	−0.812160	−0.406080	+	0.913837i	$0.633105\pi$
$570$	0	0
$571$	8.25433	0.345433	0.172717	−	0.984972i	$-0.444746\pi$
$571$	8.25433	0.345433	0.172717	+	0.984972i	$0.444746\pi$
$572$	−5.16841	−0.216102
$573$	0	0
$574$	1.51730	0.0633310
$575$	8.89618	0.370996
$576$	0	0
$577$	−9.19719	−0.382884	−0.191442	−	0.981504i	$-0.561316\pi$
$577$	−9.19719	−0.382884	−0.191442	+	0.981504i	$0.561316\pi$
$578$	−13.9988	−0.582273
$579$	0	0
$580$	2.18048	0.0905397
$581$	6.70362	0.278113
$582$	0	0
$583$	23.8137	0.986262
$584$	6.03461	0.249714
$585$	0	0
$586$	1.50060	0.0619892
$587$	4.07666	0.168262	0.0841309	−	0.996455i	$-0.473189\pi$
$587$	4.07666	0.168262	0.0841309	+	0.996455i	$0.473189\pi$
$588$	0	0
$589$	−5.76118	−0.237385
$590$	1.80161	0.0741712
$591$	0	0
$592$	−4.04668	−0.166318
$593$	0.812072	0.0333478	0.0166739	−	0.999861i	$-0.494692\pi$
$593$	0.812072	0.0333478	0.0166739	+	0.999861i	$0.494692\pi$
$594$	0	0
$595$	1.26880	0.0520158
$596$	−7.54608	−0.309100
$597$	0	0
$598$	−1.55191	−0.0634624
$599$	7.23300	0.295532	0.147766	−	0.989022i	$-0.452792\pi$
$599$	7.23300	0.295532	0.147766	+	0.989022i	$0.452792\pi$
$600$	0	0
$601$	−10.1505	−0.414048	−0.207024	−	0.978336i	$-0.566378\pi$
$601$	−10.1505	−0.414048	−0.207024	+	0.978336i	$0.566378\pi$
$602$	11.7324	0.478177
$603$	0	0
$604$	−9.06922	−0.369021
$605$	−13.3356	−0.542170
$606$	0	0
$607$	−40.6861	−1.65140	−0.825700	−	0.564110i	$-0.809219\pi$
$607$	−40.6861	−1.65140	−0.825700	+	0.564110i	$0.809219\pi$
$608$	5.08129	0.206074
$609$	0	0
$610$	−1.15051	−0.0465827
$611$	−5.25090	−0.212429
$612$	0	0
$613$	9.30221	0.375713	0.187856	−	0.982197i	$-0.439846\pi$
$613$	9.30221	0.375713	0.187856	+	0.982197i	$0.439846\pi$
$614$	1.93541	0.0781070
$615$	0	0
$616$	9.43018	0.379953
$617$	−10.9054	−0.439036	−0.219518	−	0.975608i	$-0.570449\pi$
$617$	−10.9054	−0.439036	−0.219518	+	0.975608i	$0.570449\pi$
$618$	0	0
$619$	−0.0645856	−0.00259591	−0.00129796	−	0.999999i	$-0.500413\pi$
$619$	−0.0645856	−0.00259591	−0.00129796	+	0.999999i	$0.500413\pi$
$620$	0.547282	0.0219794
$621$	0	0
$622$	−11.5565	−0.463375
$623$	27.5582	1.10409
$624$	0	0
$625$	21.5594	0.862374
$626$	−28.8662	−1.15373
$627$	0	0
$628$	1.56982	0.0626424
$629$	−7.01046	−0.279525
$630$	0	0
$631$	−29.3835	−1.16974	−0.584869	−	0.811127i	$-0.698854\pi$
$631$	−29.3835	−1.16974	−0.584869	+	0.811127i	$0.698854\pi$
$632$	1.55191	0.0617318
$633$	0	0
$634$	−32.9238	−1.30757
$635$	−5.20302	−0.206475
$636$	0	0
$637$	−3.90663	−0.154787
$638$	28.0755	1.11152
$639$	0	0
$640$	−0.482696	−0.0190802
$641$	8.09337	0.319669	0.159834	−	0.987144i	$-0.448904\pi$
$641$	8.09337	0.319669	0.159834	+	0.987144i	$0.448904\pi$
$642$	0	0
$643$	36.6036	1.44351	0.721753	−	0.692150i	$-0.243336\pi$
$643$	36.6036	1.44351	0.721753	+	0.692150i	$0.243336\pi$
$644$	2.83159	0.111580
$645$	0	0
$646$	8.80281	0.346342
$647$	−22.3656	−0.879282	−0.439641	−	0.898173i	$-0.644894\pi$
$647$	−22.3656	−0.879282	−0.439641	+	0.898173i	$0.644894\pi$
$648$	0	0
$649$	23.1972	0.910569
$650$	−3.96419	−0.155488
$651$	0	0
$652$	−1.93541	−0.0757967
$653$	15.2451	0.596586	0.298293	−	0.954474i	$-0.403583\pi$
$653$	15.2451	0.596586	0.298293	+	0.954474i	$0.403583\pi$
$654$	0	0
$655$	−7.22092	−0.282145
$656$	−1.00000	−0.0390434
$657$	0	0
$658$	9.58069	0.373494
$659$	−13.2855	−0.517530	−0.258765	−	0.965940i	$-0.583315\pi$
$659$	−13.2855	−0.517530	−0.258765	+	0.965940i	$0.583315\pi$
$660$	0	0
$661$	−25.4873	−0.991342	−0.495671	−	0.868510i	$-0.665078\pi$
$661$	−25.4873	−0.991342	−0.495671	+	0.868510i	$0.665078\pi$
$662$	−7.37143	−0.286499
$663$	0	0
$664$	−4.41811	−0.171456
$665$	3.72152	0.144314
$666$	0	0
$667$	8.43018	0.326418
$668$	5.19719	0.201085
$669$	0	0
$670$	1.93078	0.0745927
$671$	−14.8137	−0.571876
$672$	0	0
$673$	23.2364	0.895698	0.447849	−	0.894109i	$-0.352190\pi$
$673$	23.2364	0.895698	0.447849	+	0.894109i	$0.352190\pi$
$674$	18.1113	0.697620
$675$	0	0
$676$	−12.3085	−0.473402
$677$	−2.29477	−0.0881951	−0.0440975	−	0.999027i	$-0.514041\pi$
$677$	−2.29477	−0.0881951	−0.0440975	+	0.999027i	$0.514041\pi$
$678$	0	0
$679$	−14.4843	−0.555857
$680$	−0.836221	−0.0320676
$681$	0	0
$682$	7.04668	0.269831
$683$	−18.9475	−0.725006	−0.362503	−	0.931983i	$-0.618078\pi$
$683$	−18.9475	−0.725006	−0.362503	+	0.931983i	$0.618078\pi$
$684$	0	0
$685$	0.867396	0.0331415
$686$	17.7491	0.677664
$687$	0	0
$688$	−7.73240	−0.294795
$689$	−3.18631	−0.121389
$690$	0	0
$691$	−38.9071	−1.48009	−0.740047	−	0.672555i	$-0.765197\pi$
$691$	−38.9071	−1.48009	−0.740047	+	0.672555i	$0.765197\pi$
$692$	7.36097	0.279822
$693$	0	0
$694$	22.8258	0.866454
$695$	−0.0311752	−0.00118254
$696$	0	0
$697$	−1.73240	−0.0656192
$698$	23.9175	0.905291
$699$	0	0
$700$	7.23300	0.273382
$701$	23.6348	0.892675	0.446337	−	0.894865i	$-0.352728\pi$
$701$	23.6348	0.892675	0.446337	+	0.894865i	$0.352728\pi$
$702$	0	0
$703$	−20.5624	−0.775525
$704$	−6.21509	−0.234240
$705$	0	0
$706$	17.7791	0.669124
$707$	−17.5189	−0.658867
$708$	0	0
$709$	37.2859	1.40030	0.700151	−	0.713995i	$-0.253116\pi$
$709$	37.2859	1.40030	0.700151	+	0.713995i	$0.253116\pi$
$710$	4.54608	0.170611
$711$	0	0
$712$	−18.1626	−0.680672
$713$	2.11590	0.0792410
$714$	0	0
$715$	2.49477	0.0932991
$716$	−14.1459	−0.528656
$717$	0	0
$718$	26.1914	0.977453
$719$	19.3022	0.719851	0.359926	−	0.932981i	$-0.382802\pi$
$719$	19.3022	0.719851	0.359926	+	0.932981i	$0.382802\pi$
$720$	0	0
$721$	8.59277	0.320011
$722$	6.81952	0.253796
$723$	0	0
$724$	6.40141	0.237906
$725$	21.5340	0.799753
$726$	0	0
$727$	1.69779	0.0629675	0.0314837	−	0.999504i	$-0.489977\pi$
$727$	1.69779	0.0629675	0.0314837	+	0.999504i	$0.489977\pi$
$728$	−1.26178	−0.0467645
$729$	0	0
$730$	−2.91288	−0.107811
$731$	−13.3956	−0.495453
$732$	0	0
$733$	−6.07083	−0.224231	−0.112116	−	0.993695i	$-0.535763\pi$
$733$	−6.07083	−0.224231	−0.112116	+	0.993695i	$0.535763\pi$
$734$	−19.1159	−0.705581
$735$	0	0
$736$	−1.86620	−0.0687890
$737$	24.8604	0.915743
$738$	0	0
$739$	−3.93541	−0.144767	−0.0723833	−	0.997377i	$-0.523060\pi$
$739$	−3.93541	−0.144767	−0.0723833	+	0.997377i	$0.523060\pi$
$740$	1.95332	0.0718054
$741$	0	0
$742$	5.81369	0.213427
$743$	−10.1431	−0.372113	−0.186056	−	0.982539i	$-0.559571\pi$
$743$	−10.1431	−0.372113	−0.186056	+	0.982539i	$0.559571\pi$
$744$	0	0
$745$	3.64246	0.133450
$746$	−33.0696	−1.21077
$747$	0	0
$748$	−10.7670	−0.393681
$749$	5.40765	0.197591
$750$	0	0
$751$	−54.2906	−1.98109	−0.990545	−	0.137186i	$-0.956194\pi$
$751$	−54.2906	−1.98109	−0.990545	+	0.137186i	$0.956194\pi$
$752$	−6.31429	−0.230258
$753$	0	0
$754$	−3.75655	−0.136805
$755$	4.37767	0.159320
$756$	0	0
$757$	−33.8950	−1.23193	−0.615967	−	0.787772i	$-0.711235\pi$
$757$	−33.8950	−1.23193	−0.615967	+	0.787772i	$0.711235\pi$
$758$	−6.76700	−0.245789
$759$	0	0
$760$	−2.45272	−0.0889695
$761$	−17.6950	−0.641442	−0.320721	−	0.947174i	$-0.603925\pi$
$761$	−17.6950	−0.641442	−0.320721	+	0.947174i	$0.603925\pi$
$762$	0	0
$763$	−24.8316	−0.898964
$764$	−14.4994	−0.524570
$765$	0	0
$766$	−14.2559	−0.515088
$767$	−3.10382	−0.112073
$768$	0	0
$769$	−32.0743	−1.15663	−0.578314	−	0.815814i	$-0.696289\pi$
$769$	−32.0743	−1.15663	−0.578314	+	0.815814i	$0.696289\pi$
$770$	−4.55191	−0.164040
$771$	0	0
$772$	18.3310	0.659747
$773$	28.6044	1.02883	0.514415	−	0.857541i	$-0.328009\pi$
$773$	28.6044	1.02883	0.514415	+	0.857541i	$0.328009\pi$
$774$	0	0
$775$	5.40484	0.194148
$776$	9.54608	0.342684
$777$	0	0
$778$	−24.4648	−0.877105
$779$	−5.08129	−0.182056
$780$	0	0
$781$	58.5344	2.09453
$782$	−3.23300	−0.115612
$783$	0	0
$784$	−4.69779	−0.167778
$785$	−0.757743	−0.0270450
$786$	0	0
$787$	−22.4660	−0.800826	−0.400413	−	0.916335i	$-0.631133\pi$
$787$	−22.4660	−0.800826	−0.400413	+	0.916335i	$0.631133\pi$
$788$	−18.7145	−0.666676
$789$	0	0
$790$	−0.749102	−0.0266518
$791$	−13.7082	−0.487409
$792$	0	0
$793$	1.98210	0.0703863
$794$	−8.49477	−0.301468
$795$	0	0
$796$	5.00000	0.177220
$797$	29.8845	1.05856	0.529282	−	0.848446i	$-0.322461\pi$
$797$	29.8845	1.05856	0.529282	+	0.848446i	$0.322461\pi$
$798$	0	0
$799$	−10.9388	−0.386989
$800$	−4.76700	−0.168539
$801$	0	0
$802$	38.4636	1.35820
$803$	−37.5056	−1.32355
$804$	0	0
$805$	−1.36680	−0.0481733
$806$	−0.942858	−0.0332108
$807$	0	0
$808$	11.5461	0.406190
$809$	−0.401405	−0.0141127	−0.00705633	−	0.999975i	$-0.502246\pi$
$809$	−0.401405	−0.0141127	−0.00705633	+	0.999975i	$0.502246\pi$
$810$	0	0
$811$	−14.6978	−0.516109	−0.258055	−	0.966130i	$-0.583081\pi$
$811$	−14.6978	−0.516109	−0.258055	+	0.966130i	$0.583081\pi$
$812$	6.85412	0.240533
$813$	0	0
$814$	25.1505	0.881525
$815$	0.934217	0.0327242
$816$	0	0
$817$	−39.2906	−1.37460
$818$	−1.51730	−0.0530513
$819$	0	0
$820$	0.482696	0.0168565
$821$	−23.3956	−0.816511	−0.408256	−	0.912868i	$-0.633863\pi$
$821$	−23.3956	−0.816511	−0.408256	+	0.912868i	$0.633863\pi$
$822$	0	0
$823$	25.6378	0.893679	0.446839	−	0.894614i	$-0.352550\pi$
$823$	25.6378	0.893679	0.446839	+	0.894614i	$0.352550\pi$
$824$	−5.66318	−0.197286
$825$	0	0
$826$	5.66318	0.197047
$827$	−10.1354	−0.352443	−0.176221	−	0.984351i	$-0.556387\pi$
$827$	−10.1354	−0.352443	−0.176221	+	0.984351i	$0.556387\pi$
$828$	0	0
$829$	51.1054	1.77497	0.887483	−	0.460841i	$-0.152452\pi$
$829$	51.1054	1.77497	0.887483	+	0.460841i	$0.152452\pi$
$830$	2.13260	0.0740238
$831$	0	0
$832$	0.831590	0.0288302
$833$	−8.13843	−0.281980
$834$	0	0
$835$	−2.50866	−0.0868158
$836$	−31.5807	−1.09224
$837$	0	0
$838$	1.07666	0.0371926
$839$	9.86157	0.340459	0.170230	−	0.985404i	$-0.445549\pi$
$839$	9.86157	0.340459	0.170230	+	0.985404i	$0.445549\pi$
$840$	0	0
$841$	−8.59396	−0.296344
$842$	−18.4948	−0.637372
$843$	0	0
$844$	−18.5461	−0.638383
$845$	5.94124	0.204385
$846$	0	0
$847$	−41.9191	−1.44036
$848$	−3.83159	−0.131577
$849$	0	0
$850$	−8.25834	−0.283259
$851$	7.55191	0.258876
$852$	0	0
$853$	27.2797	0.934038	0.467019	−	0.884247i	$-0.345328\pi$
$853$	27.2797	0.934038	0.467019	+	0.884247i	$0.345328\pi$
$854$	−3.61650	−0.123754
$855$	0	0
$856$	−3.56399	−0.121815
$857$	−25.6857	−0.877407	−0.438704	−	0.898632i	$-0.644562\pi$
$857$	−25.6857	−0.877407	−0.438704	+	0.898632i	$0.644562\pi$
$858$	0	0
$859$	−5.75655	−0.196411	−0.0982054	−	0.995166i	$-0.531310\pi$
$859$	−5.75655	−0.196411	−0.0982054	+	0.995166i	$0.531310\pi$
$860$	3.73240	0.127274
$861$	0	0
$862$	22.6332	0.770890
$863$	−10.5352	−0.358623	−0.179311	−	0.983792i	$-0.557387\pi$
$863$	−10.5352	−0.358623	−0.179311	+	0.983792i	$0.557387\pi$
$864$	0	0
$865$	−3.55311	−0.120809
$866$	10.5340	0.357960
$867$	0	0
$868$	1.72032	0.0583915
$869$	−9.64528	−0.327194
$870$	0	0
$871$	−3.32636	−0.112709
$872$	16.3656	0.554209
$873$	0	0
$874$	−9.48270	−0.320757
$875$	−7.15332	−0.241826
$876$	0	0
$877$	−13.1972	−0.445637	−0.222819	−	0.974860i	$-0.571526\pi$
$877$	−13.1972	−0.445637	−0.222819	+	0.974860i	$0.571526\pi$
$878$	−30.0230	−1.01323
$879$	0	0
$880$	3.00000	0.101130
$881$	23.9400	0.806561	0.403280	−	0.915076i	$-0.367870\pi$
$881$	23.9400	0.806561	0.403280	+	0.915076i	$0.367870\pi$
$882$	0	0
$883$	−23.7682	−0.799864	−0.399932	−	0.916545i	$-0.630966\pi$
$883$	−23.7682	−0.799864	−0.399932	+	0.916545i	$0.630966\pi$
$884$	1.44064	0.0484541
$885$	0	0
$886$	37.7658	1.26877
$887$	4.06922	0.136631	0.0683155	−	0.997664i	$-0.478238\pi$
$887$	4.06922	0.136631	0.0683155	+	0.997664i	$0.478238\pi$
$888$	0	0
$889$	−16.3551	−0.548534
$890$	8.76700	0.293871
$891$	0	0
$892$	3.32636	0.111375
$893$	−32.0847	−1.07367
$894$	0	0
$895$	6.82816	0.228240
$896$	−1.51730	−0.0506896
$897$	0	0
$898$	33.9191	1.13190
$899$	5.12173	0.170819
$900$	0	0
$901$	−6.63783	−0.221138
$902$	6.21509	0.206940
$903$	0	0
$904$	9.03461	0.300487
$905$	−3.08993	−0.102713
$906$	0	0
$907$	−27.2167	−0.903716	−0.451858	−	0.892090i	$-0.649239\pi$
$907$	−27.2167	−0.903716	−0.451858	+	0.892090i	$0.649239\pi$
$908$	22.2738	0.739184
$909$	0	0
$910$	0.609054	0.0201899
$911$	39.0922	1.29518	0.647591	−	0.761988i	$-0.275777\pi$
$911$	39.0922	1.29518	0.647591	+	0.761988i	$0.275777\pi$
$912$	0	0
$913$	27.4590	0.908759
$914$	−34.8304	−1.15209
$915$	0	0
$916$	20.7670	0.686161
$917$	−22.6982	−0.749561
$918$	0	0
$919$	−51.8067	−1.70894	−0.854472	−	0.519497i	$-0.826119\pi$
$919$	−51.8067	−1.70894	−0.854472	+	0.519497i	$0.826119\pi$
$920$	0.900806	0.0296987
$921$	0	0
$922$	11.9988	0.395160
$923$	−7.83201	−0.257794
$924$	0	0
$925$	19.2906	0.634270
$926$	−29.2318	−0.960616
$927$	0	0
$928$	−4.51730	−0.148288
$929$	−49.0634	−1.60972	−0.804859	−	0.593466i	$-0.797759\pi$
$929$	−49.0634	−1.60972	−0.804859	+	0.593466i	$0.797759\pi$
$930$	0	0
$931$	−23.8708	−0.782335
$932$	−15.1280	−0.495533
$933$	0	0
$934$	−17.4077	−0.569596
$935$	5.19719	0.169966
$936$	0	0
$937$	−31.4232	−1.02655	−0.513275	−	0.858224i	$-0.671568\pi$
$937$	−31.4232	−1.02655	−0.513275	+	0.858224i	$0.671568\pi$
$938$	6.06922	0.198167
$939$	0	0
$940$	3.04788	0.0994109
$941$	−45.1205	−1.47089	−0.735444	−	0.677586i	$-0.763026\pi$
$941$	−45.1205	−1.47089	−0.735444	+	0.677586i	$0.763026\pi$
$942$	0	0
$943$	1.86620	0.0607718
$944$	−3.73240	−0.121479
$945$	0	0
$946$	48.0576	1.56249
$947$	−3.49779	−0.113663	−0.0568314	−	0.998384i	$-0.518100\pi$
$947$	−3.49779	−0.113663	−0.0568314	+	0.998384i	$0.518100\pi$
$948$	0	0
$949$	5.01832	0.162902
$950$	−24.2225	−0.785883
$951$	0	0
$952$	−2.62857	−0.0851925
$953$	45.6157	1.47764	0.738819	−	0.673904i	$-0.235384\pi$
$953$	45.6157	1.47764	0.738819	+	0.673904i	$0.235384\pi$
$954$	0	0
$955$	6.99880	0.226476
$956$	−12.2318	−0.395605
$957$	0	0
$958$	−16.1159	−0.520681
$959$	2.72657	0.0880455
$960$	0	0
$961$	−29.7145	−0.958532
$962$	−3.36518	−0.108498
$963$	0	0
$964$	21.2151	0.683292
$965$	−8.84830	−0.284837
$966$	0	0
$967$	43.9179	1.41231	0.706153	−	0.708060i	$-0.250429\pi$
$967$	43.9179	1.41231	0.706153	+	0.708060i	$0.250429\pi$
$968$	27.6274	0.887977
$969$	0	0
$970$	−4.60786	−0.147949
$971$	−15.2676	−0.489961	−0.244980	−	0.969528i	$-0.578782\pi$
$971$	−15.2676	−0.489961	−0.244980	+	0.969528i	$0.578782\pi$
$972$	0	0
$973$	−0.0979959	−0.00314161
$974$	33.8529	1.08472
$975$	0	0
$976$	2.38350	0.0762940
$977$	26.3539	0.843137	0.421569	−	0.906796i	$-0.361480\pi$
$977$	26.3539	0.843137	0.421569	+	0.906796i	$0.361480\pi$
$978$	0	0
$979$	112.882	3.60773
$980$	2.26760	0.0724359
$981$	0	0
$982$	6.12636	0.195500
$983$	18.9700	0.605050	0.302525	−	0.953141i	$-0.402170\pi$
$983$	18.9700	0.605050	0.302525	+	0.953141i	$0.402170\pi$
$984$	0	0
$985$	9.03341	0.287828
$986$	−7.82576	−0.249223
$987$	0	0
$988$	4.22555	0.134433
$989$	14.4302	0.458853
$990$	0	0
$991$	−37.1688	−1.18071	−0.590353	−	0.807145i	$-0.701012\pi$
$991$	−37.1688	−1.18071	−0.590353	+	0.807145i	$0.701012\pi$
$992$	−1.13380	−0.0359982
$993$	0	0
$994$	14.2901	0.453256
$995$	−2.41348	−0.0765125
$996$	0	0
$997$	50.5523	1.60101	0.800504	−	0.599327i	$-0.204565\pi$
$997$	50.5523	1.60101	0.800504	+	0.599327i	$0.204565\pi$
$998$	−13.7549	−0.435405
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2214.2.a.o.1.2	yes	3
3.2	odd	2		2214.2.a.n.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2214.2.a.n.1.2	✓	3	3.2	odd	2
2214.2.a.o.1.2	yes	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:	\( N \)	\(=\)	\( 2214 = 2 \cdot 3^{3} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2214.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.482696 q^{5} -1.51730 q^{7} +1.00000 q^{8} -0.482696 q^{10} -6.21509 q^{11} +0.831590 q^{13} -1.51730 q^{14} +1.00000 q^{16} +1.73240 q^{17} +5.08129 q^{19} -0.482696 q^{20} -6.21509 q^{22} -1.86620 q^{23} -4.76700 q^{25} +0.831590 q^{26} -1.51730 q^{28} -4.51730 q^{29} -1.13380 q^{31} +1.00000 q^{32} +1.73240 q^{34} +0.732397 q^{35} -4.04668 q^{37} +5.08129 q^{38} -0.482696 q^{40} -1.00000 q^{41} -7.73240 q^{43} -6.21509 q^{44} -1.86620 q^{46} -6.31429 q^{47} -4.69779 q^{49} -4.76700 q^{50} +0.831590 q^{52} -3.83159 q^{53} +3.00000 q^{55} -1.51730 q^{56} -4.51730 q^{58} -3.73240 q^{59} +2.38350 q^{61} -1.13380 q^{62} +1.00000 q^{64} -0.401405 q^{65} -4.00000 q^{67} +1.73240 q^{68} +0.732397 q^{70} -9.41811 q^{71} +6.03461 q^{73} -4.04668 q^{74} +5.08129 q^{76} +9.43018 q^{77} +1.55191 q^{79} -0.482696 q^{80} -1.00000 q^{82} -4.41811 q^{83} -0.836221 q^{85} -7.73240 q^{86} -6.21509 q^{88} -18.1626 q^{89} -1.26178 q^{91} -1.86620 q^{92} -6.31429 q^{94} -2.45272 q^{95} +9.54608 q^{97} -4.69779 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} + 3 q^{8} - 3 q^{10} - 5 q^{11} - 2 q^{13} - 3 q^{14} + 3 q^{16} - 10 q^{17} - 6 q^{19} - 3 q^{20} - 5 q^{22} + 2 q^{23} + 4 q^{25} - 2 q^{26} - 3 q^{28} - 12 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 + 3 * q^8 - 3 * q^10 - 5 * q^11 - 2 * q^13 - 3 * q^14 + 3 * q^16 - 10 * q^17 - 6 * q^19 - 3 * q^20 - 5 * q^22 + 2 * q^23 + 4 * q^25 - 2 * q^26 - 3 * q^28 - 12 * q^29 - 11 * q^31 + 3 * q^32 - 10 * q^34 - 13 * q^35 + 6 * q^37 - 6 * q^38 - 3 * q^40 - 3 * q^41 - 8 * q^43 - 5 * q^44 + 2 * q^46 - 16 * q^47 - 2 * q^49 + 4 * q^50 - 2 * q^52 - 7 * q^53 + 9 * q^55 - 3 * q^56 - 12 * q^58 + 4 * q^59 - 2 * q^61 - 11 * q^62 + 3 * q^64 - 24 * q^65 - 12 * q^67 - 10 * q^68 - 13 * q^70 - 16 * q^71 + 15 * q^73 + 6 * q^74 - 6 * q^76 + q^77 - 3 * q^80 - 3 * q^82 - q^83 + 22 * q^85 - 8 * q^86 - 5 * q^88 - 12 * q^89 + 28 * q^91 + 2 * q^92 - 16 * q^94 + 8 * q^95 - 23 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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