LMFDB - Embedded newform 2200.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,2,Mod(1,2200)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2200 = 2^{3} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2200.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$17.5670884447$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	2200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} +2.56155 q^{7} +3.56155 q^{9} +O(q^{10})$	$q+2.56155 q^{3} +2.56155 q^{7} +3.56155 q^{9} +1.00000 q^{11} -2.00000 q^{13} +0.561553 q^{17} +2.56155 q^{19} +6.56155 q^{21} -5.12311 q^{23} +1.43845 q^{27} +9.68466 q^{29} +6.56155 q^{31} +2.56155 q^{33} +5.68466 q^{37} -5.12311 q^{39} +2.00000 q^{41} -10.2462 q^{43} +13.1231 q^{47} -0.438447 q^{49} +1.43845 q^{51} -4.56155 q^{53} +6.56155 q^{57} +1.12311 q^{59} +2.31534 q^{61} +9.12311 q^{63} -6.24621 q^{67} -13.1231 q^{69} +3.68466 q^{71} +2.00000 q^{73} +2.56155 q^{77} -15.3693 q^{79} -7.00000 q^{81} -5.12311 q^{83} +24.8078 q^{87} -12.5616 q^{89} -5.12311 q^{91} +16.8078 q^{93} -7.12311 q^{97} +3.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + q^7 + 3 * q^9	$ 2 q + q^{3} + q^{7} + 3 q^{9} + 2 q^{11} - 4 q^{13} - 3 q^{17} + q^{19} + 9 q^{21} - 2 q^{23} + 7 q^{27} + 7 q^{29} + 9 q^{31} + q^{33} - q^{37} - 2 q^{39} + 4 q^{41} - 4 q^{43} + 18 q^{47} - 5 q^{49} + 7 q^{51} - 5 q^{53} + 9 q^{57} - 6 q^{59} + 17 q^{61} + 10 q^{63} + 4 q^{67} - 18 q^{69} - 5 q^{71} + 4 q^{73} + q^{77} - 6 q^{79} - 14 q^{81} - 2 q^{83} + 29 q^{87} - 21 q^{89} - 2 q^{91} + 13 q^{93} - 6 q^{97} + 3 q^{99}+O(q^{100}) $ 2 * q + q^3 + q^7 + 3 * q^9 + 2 * q^11 - 4 * q^13 - 3 * q^17 + q^19 + 9 * q^21 - 2 * q^23 + 7 * q^27 + 7 * q^29 + 9 * q^31 + q^33 - q^37 - 2 * q^39 + 4 * q^41 - 4 * q^43 + 18 * q^47 - 5 * q^49 + 7 * q^51 - 5 * q^53 + 9 * q^57 - 6 * q^59 + 17 * q^61 + 10 * q^63 + 4 * q^67 - 18 * q^69 - 5 * q^71 + 4 * q^73 + q^77 - 6 * q^79 - 14 * q^81 - 2 * q^83 + 29 * q^87 - 21 * q^89 - 2 * q^91 + 13 * q^93 - 6 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.56155	0.968176	0.484088	−	0.875019i	$-0.339151\pi$
$7$	2.56155	0.968176	0.484088	+	0.875019i	$0.339151\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.561553	0.136197	0.0680983	−	0.997679i	$-0.478307\pi$
$17$	0.561553	0.136197	0.0680983	+	0.997679i	$0.478307\pi$
$18$	0	0
$19$	2.56155	0.587661	0.293830	−	0.955858i	$-0.405070\pi$
$19$	2.56155	0.587661	0.293830	+	0.955858i	$0.405070\pi$
$20$	0	0
$21$	6.56155	1.43185
$22$	0	0
$23$	−5.12311	−1.06824	−0.534121	−	0.845408i	$-0.679357\pi$
$23$	−5.12311	−1.06824	−0.534121	+	0.845408i	$0.679357\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	9.68466	1.79840	0.899198	−	0.437542i	$-0.144151\pi$
$29$	9.68466	1.79840	0.899198	+	0.437542i	$0.144151\pi$
$30$	0	0
$31$	6.56155	1.17849	0.589245	−	0.807955i	$-0.299425\pi$
$31$	6.56155	1.17849	0.589245	+	0.807955i	$0.299425\pi$
$32$	0	0
$33$	2.56155	0.445909
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.68466	0.934552	0.467276	−	0.884111i	$-0.345235\pi$
$37$	5.68466	0.934552	0.467276	+	0.884111i	$0.345235\pi$
$38$	0	0
$39$	−5.12311	−0.820353
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−10.2462	−1.56253	−0.781266	−	0.624198i	$-0.785426\pi$
$43$	−10.2462	−1.56253	−0.781266	+	0.624198i	$0.785426\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	13.1231	1.91420	0.957101	−	0.289755i	$-0.0935738\pi$
$47$	13.1231	1.91420	0.957101	+	0.289755i	$0.0935738\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	1.43845	0.201423
$52$	0	0
$53$	−4.56155	−0.626577	−0.313289	−	0.949658i	$-0.601431\pi$
$53$	−4.56155	−0.626577	−0.313289	+	0.949658i	$0.601431\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.56155	0.869099
$58$	0	0
$59$	1.12311	0.146216	0.0731079	−	0.997324i	$-0.476708\pi$
$59$	1.12311	0.146216	0.0731079	+	0.997324i	$0.476708\pi$
$60$	0	0
$61$	2.31534	0.296449	0.148225	−	0.988954i	$-0.452644\pi$
$61$	2.31534	0.296449	0.148225	+	0.988954i	$0.452644\pi$
$62$	0	0
$63$	9.12311	1.14940
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.24621	−0.763096	−0.381548	−	0.924349i	$-0.624609\pi$
$67$	−6.24621	−0.763096	−0.381548	+	0.924349i	$0.624609\pi$
$68$	0	0
$69$	−13.1231	−1.57984
$70$	0	0
$71$	3.68466	0.437289	0.218644	−	0.975805i	$-0.429837\pi$
$71$	3.68466	0.437289	0.218644	+	0.975805i	$0.429837\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.56155	0.291916
$78$	0	0
$79$	−15.3693	−1.72918	−0.864592	−	0.502475i	$-0.832423\pi$
$79$	−15.3693	−1.72918	−0.864592	+	0.502475i	$0.832423\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−5.12311	−0.562334	−0.281167	−	0.959659i	$-0.590721\pi$
$83$	−5.12311	−0.562334	−0.281167	+	0.959659i	$0.590721\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	24.8078	2.65967
$88$	0	0
$89$	−12.5616	−1.33152	−0.665761	−	0.746165i	$-0.731893\pi$
$89$	−12.5616	−1.33152	−0.665761	+	0.746165i	$0.731893\pi$
$90$	0	0
$91$	−5.12311	−0.537047
$92$	0	0
$93$	16.8078	1.74288
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.12311	−0.723242	−0.361621	−	0.932325i	$-0.617777\pi$
$97$	−7.12311	−0.723242	−0.361621	+	0.932325i	$0.617777\pi$
$98$	0	0
$99$	3.56155	0.357950
$100$	0	0
$101$	−4.24621	−0.422514	−0.211257	−	0.977431i	$-0.567756\pi$
$101$	−4.24621	−0.422514	−0.211257	+	0.977431i	$0.567756\pi$
$102$	0	0
$103$	−10.2462	−1.00959	−0.504795	−	0.863239i	$-0.668432\pi$
$103$	−10.2462	−1.00959	−0.504795	+	0.863239i	$0.668432\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	18.4924	1.77125	0.885626	−	0.464398i	$-0.153729\pi$
$109$	18.4924	1.77125	0.885626	+	0.464398i	$0.153729\pi$
$110$	0	0
$111$	14.5616	1.38212
$112$	0	0
$113$	19.1231	1.79895	0.899475	−	0.436972i	$-0.143949\pi$
$113$	19.1231	1.79895	0.899475	+	0.436972i	$0.143949\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−7.12311	−0.658531
$118$	0	0
$119$	1.43845	0.131862
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.12311	0.461935
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.2462	−1.26415	−0.632073	−	0.774909i	$-0.717796\pi$
$127$	−14.2462	−1.26415	−0.632073	+	0.774909i	$0.717796\pi$
$128$	0	0
$129$	−26.2462	−2.31085
$130$	0	0
$131$	−7.68466	−0.671412	−0.335706	−	0.941967i	$-0.608975\pi$
$131$	−7.68466	−0.671412	−0.335706	+	0.941967i	$0.608975\pi$
$132$	0	0
$133$	6.56155	0.568959
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.6155	−1.67587	−0.837934	−	0.545772i	$-0.816237\pi$
$137$	−19.6155	−1.67587	−0.837934	+	0.545772i	$0.816237\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	33.6155	2.83094
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.12311	−0.0926322
$148$	0	0
$149$	2.31534	0.189680	0.0948401	−	0.995493i	$-0.469766\pi$
$149$	2.31534	0.189680	0.0948401	+	0.995493i	$0.469766\pi$
$150$	0	0
$151$	13.1231	1.06794	0.533972	−	0.845502i	$-0.320699\pi$
$151$	13.1231	1.06794	0.533972	+	0.845502i	$0.320699\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.8078	−1.18179	−0.590894	−	0.806749i	$-0.701225\pi$
$157$	−14.8078	−1.18179	−0.590894	+	0.806749i	$0.701225\pi$
$158$	0	0
$159$	−11.6847	−0.926654
$160$	0	0
$161$	−13.1231	−1.03425
$162$	0	0
$163$	−3.19224	−0.250035	−0.125018	−	0.992155i	$-0.539899\pi$
$163$	−3.19224	−0.250035	−0.125018	+	0.992155i	$0.539899\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.5616	0.817277	0.408639	−	0.912696i	$-0.366004\pi$
$167$	10.5616	0.817277	0.408639	+	0.912696i	$0.366004\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	9.12311	0.697661
$172$	0	0
$173$	−4.24621	−0.322833	−0.161417	−	0.986886i	$-0.551606\pi$
$173$	−4.24621	−0.322833	−0.161417	+	0.986886i	$0.551606\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.87689	0.216241
$178$	0	0
$179$	−6.24621	−0.466864	−0.233432	−	0.972373i	$-0.574996\pi$
$179$	−6.24621	−0.466864	−0.233432	+	0.972373i	$0.574996\pi$
$180$	0	0
$181$	−4.24621	−0.315618	−0.157809	−	0.987470i	$-0.550443\pi$
$181$	−4.24621	−0.315618	−0.157809	+	0.987470i	$0.550443\pi$
$182$	0	0
$183$	5.93087	0.438423
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.561553	0.0410648
$188$	0	0
$189$	3.68466	0.268019
$190$	0	0
$191$	10.2462	0.741390	0.370695	−	0.928755i	$-0.379120\pi$
$191$	10.2462	0.741390	0.370695	+	0.928755i	$0.379120\pi$
$192$	0	0
$193$	−5.19224	−0.373745	−0.186873	−	0.982384i	$-0.559835\pi$
$193$	−5.19224	−0.373745	−0.186873	+	0.982384i	$0.559835\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.36932	0.382548	0.191274	−	0.981537i	$-0.438738\pi$
$197$	5.36932	0.382548	0.191274	+	0.981537i	$0.438738\pi$
$198$	0	0
$199$	−0.807764	−0.0572609	−0.0286304	−	0.999590i	$-0.509115\pi$
$199$	−0.807764	−0.0572609	−0.0286304	+	0.999590i	$0.509115\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	0	0
$203$	24.8078	1.74116
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−18.2462	−1.26820
$208$	0	0
$209$	2.56155	0.177186
$210$	0	0
$211$	8.31534	0.572452	0.286226	−	0.958162i	$-0.407599\pi$
$211$	8.31534	0.572452	0.286226	+	0.958162i	$0.407599\pi$
$212$	0	0
$213$	9.43845	0.646712
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.8078	1.14099
$218$	0	0
$219$	5.12311	0.346187
$220$	0	0
$221$	−1.12311	−0.0755483
$222$	0	0
$223$	−13.1231	−0.878788	−0.439394	−	0.898294i	$-0.644807\pi$
$223$	−13.1231	−0.878788	−0.439394	+	0.898294i	$0.644807\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.87689	−0.190946	−0.0954731	−	0.995432i	$-0.530436\pi$
$227$	−2.87689	−0.190946	−0.0954731	+	0.995432i	$0.530436\pi$
$228$	0	0
$229$	−24.7386	−1.63477	−0.817387	−	0.576088i	$-0.804578\pi$
$229$	−24.7386	−1.63477	−0.817387	+	0.576088i	$0.804578\pi$
$230$	0	0
$231$	6.56155	0.431718
$232$	0	0
$233$	−19.9309	−1.30571	−0.652857	−	0.757481i	$-0.726430\pi$
$233$	−19.9309	−1.30571	−0.652857	+	0.757481i	$0.726430\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−39.3693	−2.55731
$238$	0	0
$239$	25.6155	1.65693	0.828465	−	0.560040i	$-0.189214\pi$
$239$	25.6155	1.65693	0.828465	+	0.560040i	$0.189214\pi$
$240$	0	0
$241$	−16.2462	−1.04651	−0.523255	−	0.852176i	$-0.675283\pi$
$241$	−16.2462	−1.04651	−0.523255	+	0.852176i	$0.675283\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.12311	−0.325975
$248$	0	0
$249$	−13.1231	−0.831643
$250$	0	0
$251$	−29.6155	−1.86932	−0.934658	−	0.355549i	$-0.884294\pi$
$251$	−29.6155	−1.86932	−0.934658	+	0.355549i	$0.884294\pi$
$252$	0	0
$253$	−5.12311	−0.322087
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.7386	1.79267	0.896333	−	0.443381i	$-0.146221\pi$
$257$	28.7386	1.79267	0.896333	+	0.443381i	$0.146221\pi$
$258$	0	0
$259$	14.5616	0.904811
$260$	0	0
$261$	34.4924	2.13503
$262$	0	0
$263$	−4.80776	−0.296459	−0.148230	−	0.988953i	$-0.547357\pi$
$263$	−4.80776	−0.296459	−0.148230	+	0.988953i	$0.547357\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−32.1771	−1.96921
$268$	0	0
$269$	−19.6155	−1.19598	−0.597990	−	0.801504i	$-0.704034\pi$
$269$	−19.6155	−1.19598	−0.597990	+	0.801504i	$0.704034\pi$
$270$	0	0
$271$	28.4924	1.73079	0.865396	−	0.501089i	$-0.167067\pi$
$271$	28.4924	1.73079	0.865396	+	0.501089i	$0.167067\pi$
$272$	0	0
$273$	−13.1231	−0.794246
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.12311	−0.427986	−0.213993	−	0.976835i	$-0.568647\pi$
$277$	−7.12311	−0.427986	−0.213993	+	0.976835i	$0.568647\pi$
$278$	0	0
$279$	23.3693	1.39908
$280$	0	0
$281$	−8.24621	−0.491928	−0.245964	−	0.969279i	$-0.579104\pi$
$281$	−8.24621	−0.491928	−0.245964	+	0.969279i	$0.579104\pi$
$282$	0	0
$283$	23.3693	1.38916	0.694581	−	0.719415i	$-0.255590\pi$
$283$	23.3693	1.38916	0.694581	+	0.719415i	$0.255590\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.12311	0.302407
$288$	0	0
$289$	−16.6847	−0.981450
$290$	0	0
$291$	−18.2462	−1.06961
$292$	0	0
$293$	−14.4924	−0.846656	−0.423328	−	0.905976i	$-0.639138\pi$
$293$	−14.4924	−0.846656	−0.423328	+	0.905976i	$0.639138\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.43845	0.0834672
$298$	0	0
$299$	10.2462	0.592554
$300$	0	0
$301$	−26.2462	−1.51281
$302$	0	0
$303$	−10.8769	−0.624861
$304$	0	0
$305$	0	0
$306$	0	0
$307$	25.6155	1.46196	0.730978	−	0.682401i	$-0.239064\pi$
$307$	25.6155	1.46196	0.730978	+	0.682401i	$0.239064\pi$
$308$	0	0
$309$	−26.2462	−1.49309
$310$	0	0
$311$	−25.4384	−1.44248	−0.721241	−	0.692684i	$-0.756428\pi$
$311$	−25.4384	−1.44248	−0.721241	+	0.692684i	$0.756428\pi$
$312$	0	0
$313$	−30.4924	−1.72353	−0.861767	−	0.507305i	$-0.830642\pi$
$313$	−30.4924	−1.72353	−0.861767	+	0.507305i	$0.830642\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.1922	−0.740950	−0.370475	−	0.928842i	$-0.620805\pi$
$317$	−13.1922	−0.740950	−0.370475	+	0.928842i	$0.620805\pi$
$318$	0	0
$319$	9.68466	0.542237
$320$	0	0
$321$	40.9848	2.28755
$322$	0	0
$323$	1.43845	0.0800373
$324$	0	0
$325$	0	0
$326$	0	0
$327$	47.3693	2.61953
$328$	0	0
$329$	33.6155	1.85328
$330$	0	0
$331$	−8.49242	−0.466786	−0.233393	−	0.972383i	$-0.574983\pi$
$331$	−8.49242	−0.466786	−0.233393	+	0.972383i	$0.574983\pi$
$332$	0	0
$333$	20.2462	1.10949
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.31534	−0.126125	−0.0630623	−	0.998010i	$-0.520087\pi$
$337$	−2.31534	−0.126125	−0.0630623	+	0.998010i	$0.520087\pi$
$338$	0	0
$339$	48.9848	2.66049
$340$	0	0
$341$	6.56155	0.355328
$342$	0	0
$343$	−19.0540	−1.02882
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.8769	1.01336	0.506682	−	0.862133i	$-0.330872\pi$
$347$	18.8769	1.01336	0.506682	+	0.862133i	$0.330872\pi$
$348$	0	0
$349$	10.4924	0.561646	0.280823	−	0.959760i	$-0.409393\pi$
$349$	10.4924	0.561646	0.280823	+	0.959760i	$0.409393\pi$
$350$	0	0
$351$	−2.87689	−0.153557
$352$	0	0
$353$	34.4924	1.83585	0.917923	−	0.396758i	$-0.129865\pi$
$353$	34.4924	1.83585	0.917923	+	0.396758i	$0.129865\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.68466	0.195013
$358$	0	0
$359$	26.2462	1.38522	0.692611	−	0.721311i	$-0.256460\pi$
$359$	26.2462	1.38522	0.692611	+	0.721311i	$0.256460\pi$
$360$	0	0
$361$	−12.4384	−0.654655
$362$	0	0
$363$	2.56155	0.134447
$364$	0	0
$365$	0	0
$366$	0	0
$367$	14.7386	0.769350	0.384675	−	0.923052i	$-0.374313\pi$
$367$	14.7386	0.769350	0.384675	+	0.923052i	$0.374313\pi$
$368$	0	0
$369$	7.12311	0.370814
$370$	0	0
$371$	−11.6847	−0.606637
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.3693	−0.997571
$378$	0	0
$379$	−25.1231	−1.29049	−0.645244	−	0.763977i	$-0.723244\pi$
$379$	−25.1231	−1.29049	−0.645244	+	0.763977i	$0.723244\pi$
$380$	0	0
$381$	−36.4924	−1.86956
$382$	0	0
$383$	31.3693	1.60290	0.801449	−	0.598064i	$-0.204063\pi$
$383$	31.3693	1.60290	0.801449	+	0.598064i	$0.204063\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−36.4924	−1.85501
$388$	0	0
$389$	32.2462	1.63495	0.817474	−	0.575966i	$-0.195374\pi$
$389$	32.2462	1.63495	0.817474	+	0.575966i	$0.195374\pi$
$390$	0	0
$391$	−2.87689	−0.145491
$392$	0	0
$393$	−19.6847	−0.992960
$394$	0	0
$395$	0	0
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	0	0
$399$	16.8078	0.841441
$400$	0	0
$401$	−7.43845	−0.371458	−0.185729	−	0.982601i	$-0.559465\pi$
$401$	−7.43845	−0.371458	−0.185729	+	0.982601i	$0.559465\pi$
$402$	0	0
$403$	−13.1231	−0.653708
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.68466	0.281778
$408$	0	0
$409$	7.12311	0.352215	0.176107	−	0.984371i	$-0.443649\pi$
$409$	7.12311	0.352215	0.176107	+	0.984371i	$0.443649\pi$
$410$	0	0
$411$	−50.2462	−2.47846
$412$	0	0
$413$	2.87689	0.141563
$414$	0	0
$415$	0	0
$416$	0	0
$417$	30.7386	1.50528
$418$	0	0
$419$	−26.7386	−1.30627	−0.653134	−	0.757242i	$-0.726546\pi$
$419$	−26.7386	−1.30627	−0.653134	+	0.757242i	$0.726546\pi$
$420$	0	0
$421$	7.61553	0.371158	0.185579	−	0.982629i	$-0.440584\pi$
$421$	7.61553	0.371158	0.185579	+	0.982629i	$0.440584\pi$
$422$	0	0
$423$	46.7386	2.27251
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.93087	0.287015
$428$	0	0
$429$	−5.12311	−0.247346
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	−33.3693	−1.60363	−0.801814	−	0.597574i	$-0.796131\pi$
$433$	−33.3693	−1.60363	−0.801814	+	0.597574i	$0.796131\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13.1231	−0.627763
$438$	0	0
$439$	−34.2462	−1.63448	−0.817241	−	0.576296i	$-0.804498\pi$
$439$	−34.2462	−1.63448	−0.817241	+	0.576296i	$0.804498\pi$
$440$	0	0
$441$	−1.56155	−0.0743597
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.93087	0.280521
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	0	0
$453$	33.6155	1.57940
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.5616	−1.33605	−0.668027	−	0.744137i	$-0.732861\pi$
$457$	−28.5616	−1.33605	−0.668027	+	0.744137i	$0.732861\pi$
$458$	0	0
$459$	0.807764	0.0377032
$460$	0	0
$461$	20.5616	0.957647	0.478823	−	0.877911i	$-0.341063\pi$
$461$	20.5616	0.957647	0.478823	+	0.877911i	$0.341063\pi$
$462$	0	0
$463$	7.36932	0.342481	0.171241	−	0.985229i	$-0.445222\pi$
$463$	7.36932	0.342481	0.171241	+	0.985229i	$0.445222\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.93087	−0.459546	−0.229773	−	0.973244i	$-0.573798\pi$
$467$	−9.93087	−0.459546	−0.229773	+	0.973244i	$0.573798\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−37.9309	−1.74776
$472$	0	0
$473$	−10.2462	−0.471121
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−16.2462	−0.743863
$478$	0	0
$479$	−32.9848	−1.50712	−0.753558	−	0.657381i	$-0.771664\pi$
$479$	−32.9848	−1.50712	−0.753558	+	0.657381i	$0.771664\pi$
$480$	0	0
$481$	−11.3693	−0.518396
$482$	0	0
$483$	−33.6155	−1.52956
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.24621	0.101786	0.0508928	−	0.998704i	$-0.483793\pi$
$487$	2.24621	0.101786	0.0508928	+	0.998704i	$0.483793\pi$
$488$	0	0
$489$	−8.17708	−0.369780
$490$	0	0
$491$	7.05398	0.318341	0.159171	−	0.987251i	$-0.449118\pi$
$491$	7.05398	0.318341	0.159171	+	0.987251i	$0.449118\pi$
$492$	0	0
$493$	5.43845	0.244935
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.43845	0.423372
$498$	0	0
$499$	16.4924	0.738302	0.369151	−	0.929369i	$-0.379648\pi$
$499$	16.4924	0.738302	0.369151	+	0.929369i	$0.379648\pi$
$500$	0	0
$501$	27.0540	1.20868
$502$	0	0
$503$	14.2462	0.635207	0.317604	−	0.948224i	$-0.397122\pi$
$503$	14.2462	0.635207	0.317604	+	0.948224i	$0.397122\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−23.0540	−1.02386
$508$	0	0
$509$	21.3693	0.947178	0.473589	−	0.880746i	$-0.342958\pi$
$509$	21.3693	0.947178	0.473589	+	0.880746i	$0.342958\pi$
$510$	0	0
$511$	5.12311	0.226633
$512$	0	0
$513$	3.68466	0.162682
$514$	0	0
$515$	0	0
$516$	0	0
$517$	13.1231	0.577154
$518$	0	0
$519$	−10.8769	−0.477443
$520$	0	0
$521$	8.73863	0.382846	0.191423	−	0.981508i	$-0.438690\pi$
$521$	8.73863	0.382846	0.191423	+	0.981508i	$0.438690\pi$
$522$	0	0
$523$	3.50758	0.153376	0.0766878	−	0.997055i	$-0.475566\pi$
$523$	3.50758	0.153376	0.0766878	+	0.997055i	$0.475566\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.68466	0.160506
$528$	0	0
$529$	3.24621	0.141140
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−4.00000	−0.173259
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−16.0000	−0.690451
$538$	0	0
$539$	−0.438447	−0.0188853
$540$	0	0
$541$	−33.5464	−1.44227	−0.721136	−	0.692793i	$-0.756380\pi$
$541$	−33.5464	−1.44227	−0.721136	+	0.692793i	$0.756380\pi$
$542$	0	0
$543$	−10.8769	−0.466772
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	0	0
$549$	8.24621	0.351940
$550$	0	0
$551$	24.8078	1.05685
$552$	0	0
$553$	−39.3693	−1.67415
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−46.4924	−1.96995	−0.984974	−	0.172705i	$-0.944749\pi$
$557$	−46.4924	−1.96995	−0.984974	+	0.172705i	$0.944749\pi$
$558$	0	0
$559$	20.4924	0.866737
$560$	0	0
$561$	1.43845	0.0607313
$562$	0	0
$563$	2.87689	0.121247	0.0606233	−	0.998161i	$-0.480691\pi$
$563$	2.87689	0.121247	0.0606233	+	0.998161i	$0.480691\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−17.9309	−0.753026
$568$	0	0
$569$	−28.1080	−1.17835	−0.589173	−	0.808007i	$-0.700546\pi$
$569$	−28.1080	−1.17835	−0.589173	+	0.808007i	$0.700546\pi$
$570$	0	0
$571$	30.4233	1.27318	0.636588	−	0.771204i	$-0.280345\pi$
$571$	30.4233	1.27318	0.636588	+	0.771204i	$0.280345\pi$
$572$	0	0
$573$	26.2462	1.09645
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−24.7386	−1.02988	−0.514941	−	0.857225i	$-0.672186\pi$
$577$	−24.7386	−1.02988	−0.514941	+	0.857225i	$0.672186\pi$
$578$	0	0
$579$	−13.3002	−0.552737
$580$	0	0
$581$	−13.1231	−0.544438
$582$	0	0
$583$	−4.56155	−0.188920
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−32.3153	−1.33380	−0.666898	−	0.745149i	$-0.732379\pi$
$587$	−32.3153	−1.33380	−0.666898	+	0.745149i	$0.732379\pi$
$588$	0	0
$589$	16.8078	0.692552
$590$	0	0
$591$	13.7538	0.565755
$592$	0	0
$593$	−1.50758	−0.0619088	−0.0309544	−	0.999521i	$-0.509855\pi$
$593$	−1.50758	−0.0619088	−0.0309544	+	0.999521i	$0.509855\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.06913	−0.0846839
$598$	0	0
$599$	−17.4384	−0.712516	−0.356258	−	0.934388i	$-0.615948\pi$
$599$	−17.4384	−0.712516	−0.356258	+	0.934388i	$0.615948\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	−22.2462	−0.905936
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−20.8078	−0.844561	−0.422281	−	0.906465i	$-0.638770\pi$
$607$	−20.8078	−0.844561	−0.422281	+	0.906465i	$0.638770\pi$
$608$	0	0
$609$	63.5464	2.57503
$610$	0	0
$611$	−26.2462	−1.06181
$612$	0	0
$613$	−16.7386	−0.676067	−0.338034	−	0.941134i	$-0.609762\pi$
$613$	−16.7386	−0.676067	−0.338034	+	0.941134i	$0.609762\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.63068	0.266941	0.133471	−	0.991053i	$-0.457388\pi$
$617$	6.63068	0.266941	0.133471	+	0.991053i	$0.457388\pi$
$618$	0	0
$619$	−8.49242	−0.341339	−0.170670	−	0.985328i	$-0.554593\pi$
$619$	−8.49242	−0.341339	−0.170670	+	0.985328i	$0.554593\pi$
$620$	0	0
$621$	−7.36932	−0.295720
$622$	0	0
$623$	−32.1771	−1.28915
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.56155	0.262043
$628$	0	0
$629$	3.19224	0.127283
$630$	0	0
$631$	−16.8078	−0.669107	−0.334553	−	0.942377i	$-0.608585\pi$
$631$	−16.8078	−0.669107	−0.334553	+	0.942377i	$0.608585\pi$
$632$	0	0
$633$	21.3002	0.846606
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.876894	0.0347438
$638$	0	0
$639$	13.1231	0.519142
$640$	0	0
$641$	−38.1771	−1.50790	−0.753952	−	0.656929i	$-0.771855\pi$
$641$	−38.1771	−1.50790	−0.753952	+	0.656929i	$0.771855\pi$
$642$	0	0
$643$	−23.6847	−0.934032	−0.467016	−	0.884249i	$-0.654671\pi$
$643$	−23.6847	−0.934032	−0.467016	+	0.884249i	$0.654671\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.1231	0.515923	0.257961	−	0.966155i	$-0.416949\pi$
$647$	13.1231	0.515923	0.257961	+	0.966155i	$0.416949\pi$
$648$	0	0
$649$	1.12311	0.0440858
$650$	0	0
$651$	43.0540	1.68742
$652$	0	0
$653$	10.8078	0.422940	0.211470	−	0.977384i	$-0.432175\pi$
$653$	10.8078	0.422940	0.211470	+	0.977384i	$0.432175\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.12311	0.277899
$658$	0	0
$659$	19.1922	0.747623	0.373812	−	0.927505i	$-0.378051\pi$
$659$	19.1922	0.747623	0.373812	+	0.927505i	$0.378051\pi$
$660$	0	0
$661$	16.2462	0.631904	0.315952	−	0.948775i	$-0.397676\pi$
$661$	16.2462	0.631904	0.315952	+	0.948775i	$0.397676\pi$
$662$	0	0
$663$	−2.87689	−0.111729
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−49.6155	−1.92112
$668$	0	0
$669$	−33.6155	−1.29965
$670$	0	0
$671$	2.31534	0.0893828
$672$	0	0
$673$	35.7926	1.37970	0.689852	−	0.723951i	$-0.257676\pi$
$673$	35.7926	1.37970	0.689852	+	0.723951i	$0.257676\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.6155	−1.06135	−0.530675	−	0.847575i	$-0.678062\pi$
$677$	−27.6155	−1.06135	−0.530675	+	0.847575i	$0.678062\pi$
$678$	0	0
$679$	−18.2462	−0.700225
$680$	0	0
$681$	−7.36932	−0.282393
$682$	0	0
$683$	25.9309	0.992217	0.496109	−	0.868260i	$-0.334762\pi$
$683$	25.9309	0.992217	0.496109	+	0.868260i	$0.334762\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−63.3693	−2.41769
$688$	0	0
$689$	9.12311	0.347563
$690$	0	0
$691$	3.36932	0.128175	0.0640874	−	0.997944i	$-0.479586\pi$
$691$	3.36932	0.128175	0.0640874	+	0.997944i	$0.479586\pi$
$692$	0	0
$693$	9.12311	0.346558
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.12311	0.0425407
$698$	0	0
$699$	−51.0540	−1.93104
$700$	0	0
$701$	34.3153	1.29607	0.648036	−	0.761609i	$-0.275591\pi$
$701$	34.3153	1.29607	0.648036	+	0.761609i	$0.275591\pi$
$702$	0	0
$703$	14.5616	0.549199
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.8769	−0.409068
$708$	0	0
$709$	1.50758	0.0566183	0.0283091	−	0.999599i	$-0.490988\pi$
$709$	1.50758	0.0566183	0.0283091	+	0.999599i	$0.490988\pi$
$710$	0	0
$711$	−54.7386	−2.05286
$712$	0	0
$713$	−33.6155	−1.25891
$714$	0	0
$715$	0	0
$716$	0	0
$717$	65.6155	2.45046
$718$	0	0
$719$	7.82292	0.291746	0.145873	−	0.989303i	$-0.453401\pi$
$719$	7.82292	0.291746	0.145873	+	0.989303i	$0.453401\pi$
$720$	0	0
$721$	−26.2462	−0.977460
$722$	0	0
$723$	−41.6155	−1.54770
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−17.6155	−0.653324	−0.326662	−	0.945141i	$-0.605924\pi$
$727$	−17.6155	−0.653324	−0.326662	+	0.945141i	$0.605924\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−5.75379	−0.212812
$732$	0	0
$733$	19.1231	0.706328	0.353164	−	0.935561i	$-0.385106\pi$
$733$	19.1231	0.706328	0.353164	+	0.935561i	$0.385106\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.24621	−0.230082
$738$	0	0
$739$	−42.7386	−1.57217	−0.786083	−	0.618121i	$-0.787894\pi$
$739$	−42.7386	−1.57217	−0.786083	+	0.618121i	$0.787894\pi$
$740$	0	0
$741$	−13.1231	−0.482089
$742$	0	0
$743$	2.56155	0.0939743	0.0469871	−	0.998895i	$-0.485038\pi$
$743$	2.56155	0.0939743	0.0469871	+	0.998895i	$0.485038\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−18.2462	−0.667594
$748$	0	0
$749$	40.9848	1.49755
$750$	0	0
$751$	−16.1771	−0.590310	−0.295155	−	0.955449i	$-0.595371\pi$
$751$	−16.1771	−0.590310	−0.295155	+	0.955449i	$0.595371\pi$
$752$	0	0
$753$	−75.8617	−2.76456
$754$	0	0
$755$	0	0
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	0	0
$759$	−13.1231	−0.476339
$760$	0	0
$761$	−36.7386	−1.33177	−0.665887	−	0.746052i	$-0.731947\pi$
$761$	−36.7386	−1.33177	−0.665887	+	0.746052i	$0.731947\pi$
$762$	0	0
$763$	47.3693	1.71488
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.24621	−0.0811060
$768$	0	0
$769$	24.7386	0.892098	0.446049	−	0.895009i	$-0.352831\pi$
$769$	24.7386	0.892098	0.446049	+	0.895009i	$0.352831\pi$
$770$	0	0
$771$	73.6155	2.65120
$772$	0	0
$773$	37.6847	1.35542	0.677711	−	0.735328i	$-0.262972\pi$
$773$	37.6847	1.35542	0.677711	+	0.735328i	$0.262972\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	37.3002	1.33814
$778$	0	0
$779$	5.12311	0.183554
$780$	0	0
$781$	3.68466	0.131847
$782$	0	0
$783$	13.9309	0.497849
$784$	0	0
$785$	0	0
$786$	0	0
$787$	54.7386	1.95122	0.975611	−	0.219508i	$-0.0704451\pi$
$787$	54.7386	1.95122	0.975611	+	0.219508i	$0.0704451\pi$
$788$	0	0
$789$	−12.3153	−0.438438
$790$	0	0
$791$	48.9848	1.74170
$792$	0	0
$793$	−4.63068	−0.164440
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.7538	0.841402	0.420701	−	0.907199i	$-0.361784\pi$
$797$	23.7538	0.841402	0.420701	+	0.907199i	$0.361784\pi$
$798$	0	0
$799$	7.36932	0.260708
$800$	0	0
$801$	−44.7386	−1.58076
$802$	0	0
$803$	2.00000	0.0705785
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−50.2462	−1.76875
$808$	0	0
$809$	6.49242	0.228261	0.114131	−	0.993466i	$-0.463592\pi$
$809$	6.49242	0.228261	0.114131	+	0.993466i	$0.463592\pi$
$810$	0	0
$811$	27.1922	0.954849	0.477424	−	0.878673i	$-0.341570\pi$
$811$	27.1922	0.954849	0.477424	+	0.878673i	$0.341570\pi$
$812$	0	0
$813$	72.9848	2.55969
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−26.2462	−0.918239
$818$	0	0
$819$	−18.2462	−0.637574
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	52.4924	1.82977	0.914885	−	0.403714i	$-0.132281\pi$
$823$	52.4924	1.82977	0.914885	+	0.403714i	$0.132281\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.49242	0.156217	0.0781084	−	0.996945i	$-0.475112\pi$
$827$	4.49242	0.156217	0.0781084	+	0.996945i	$0.475112\pi$
$828$	0	0
$829$	5.36932	0.186484	0.0932420	−	0.995643i	$-0.470277\pi$
$829$	5.36932	0.186484	0.0932420	+	0.995643i	$0.470277\pi$
$830$	0	0
$831$	−18.2462	−0.632954
$832$	0	0
$833$	−0.246211	−0.00853071
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.43845	0.326240
$838$	0	0
$839$	11.5076	0.397286	0.198643	−	0.980072i	$-0.436347\pi$
$839$	11.5076	0.397286	0.198643	+	0.980072i	$0.436347\pi$
$840$	0	0
$841$	64.7926	2.23423
$842$	0	0
$843$	−21.1231	−0.727518
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.56155	0.0880160
$848$	0	0
$849$	59.8617	2.05445
$850$	0	0
$851$	−29.1231	−0.998327
$852$	0	0
$853$	3.75379	0.128527	0.0642636	−	0.997933i	$-0.479530\pi$
$853$	3.75379	0.128527	0.0642636	+	0.997933i	$0.479530\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−32.4233	−1.10756	−0.553779	−	0.832664i	$-0.686815\pi$
$857$	−32.4233	−1.10756	−0.553779	+	0.832664i	$0.686815\pi$
$858$	0	0
$859$	−38.8769	−1.32646	−0.663231	−	0.748415i	$-0.730815\pi$
$859$	−38.8769	−1.32646	−0.663231	+	0.748415i	$0.730815\pi$
$860$	0	0
$861$	13.1231	0.447234
$862$	0	0
$863$	−9.61553	−0.327316	−0.163658	−	0.986517i	$-0.552329\pi$
$863$	−9.61553	−0.327316	−0.163658	+	0.986517i	$0.552329\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−42.7386	−1.45148
$868$	0	0
$869$	−15.3693	−0.521368
$870$	0	0
$871$	12.4924	0.423290
$872$	0	0
$873$	−25.3693	−0.858621
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.2462	1.08888	0.544439	−	0.838801i	$-0.316743\pi$
$877$	32.2462	1.08888	0.544439	+	0.838801i	$0.316743\pi$
$878$	0	0
$879$	−37.1231	−1.25213
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	−6.06913	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$883$	−6.06913	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.2462	1.01557	0.507784	−	0.861484i	$-0.330465\pi$
$887$	30.2462	1.01557	0.507784	+	0.861484i	$0.330465\pi$
$888$	0	0
$889$	−36.4924	−1.22392
$890$	0	0
$891$	−7.00000	−0.234509
$892$	0	0
$893$	33.6155	1.12490
$894$	0	0
$895$	0	0
$896$	0	0
$897$	26.2462	0.876335
$898$	0	0
$899$	63.5464	2.11939
$900$	0	0
$901$	−2.56155	−0.0853377
$902$	0	0
$903$	−67.2311	−2.23731
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−8.94602	−0.297048	−0.148524	−	0.988909i	$-0.547452\pi$
$907$	−8.94602	−0.297048	−0.148524	+	0.988909i	$0.547452\pi$
$908$	0	0
$909$	−15.1231	−0.501602
$910$	0	0
$911$	−2.06913	−0.0685533	−0.0342767	−	0.999412i	$-0.510913\pi$
$911$	−2.06913	−0.0685533	−0.0342767	+	0.999412i	$0.510913\pi$
$912$	0	0
$913$	−5.12311	−0.169550
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19.6847	−0.650045
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	65.6155	2.16211
$922$	0	0
$923$	−7.36932	−0.242564
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−36.4924	−1.19857
$928$	0	0
$929$	45.0540	1.47817	0.739086	−	0.673611i	$-0.235257\pi$
$929$	45.0540	1.47817	0.739086	+	0.673611i	$0.235257\pi$
$930$	0	0
$931$	−1.12311	−0.0368083
$932$	0	0
$933$	−65.1619	−2.13331
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	−78.1080	−2.54896
$940$	0	0
$941$	−4.06913	−0.132650	−0.0663249	−	0.997798i	$-0.521127\pi$
$941$	−4.06913	−0.132650	−0.0663249	+	0.997798i	$0.521127\pi$
$942$	0	0
$943$	−10.2462	−0.333663
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.6847	1.28958	0.644789	−	0.764361i	$-0.276945\pi$
$947$	39.6847	1.28958	0.644789	+	0.764361i	$0.276945\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−33.7926	−1.09580
$952$	0	0
$953$	34.1771	1.10710	0.553552	−	0.832815i	$-0.313272\pi$
$953$	34.1771	1.10710	0.553552	+	0.832815i	$0.313272\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	24.8078	0.801921
$958$	0	0
$959$	−50.2462	−1.62253
$960$	0	0
$961$	12.0540	0.388838
$962$	0	0
$963$	56.9848	1.83631
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−11.5464	−0.371307	−0.185654	−	0.982615i	$-0.559440\pi$
$967$	−11.5464	−0.371307	−0.185654	+	0.982615i	$0.559440\pi$
$968$	0	0
$969$	3.68466	0.118368
$970$	0	0
$971$	6.24621	0.200450	0.100225	−	0.994965i	$-0.468044\pi$
$971$	6.24621	0.200450	0.100225	+	0.994965i	$0.468044\pi$
$972$	0	0
$973$	30.7386	0.985435
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.8769	0.795882	0.397941	−	0.917411i	$-0.369725\pi$
$977$	24.8769	0.795882	0.397941	+	0.917411i	$0.369725\pi$
$978$	0	0
$979$	−12.5616	−0.401469
$980$	0	0
$981$	65.8617	2.10280
$982$	0	0
$983$	−18.8769	−0.602079	−0.301040	−	0.953612i	$-0.597334\pi$
$983$	−18.8769	−0.602079	−0.301040	+	0.953612i	$0.597334\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	86.1080	2.74085
$988$	0	0
$989$	52.4924	1.66916
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	−21.7538	−0.690336
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−13.8617	−0.439006	−0.219503	−	0.975612i	$-0.570444\pi$
$997$	−13.8617	−0.439006	−0.219503	+	0.975612i	$0.570444\pi$
$998$	0	0
$999$	8.17708	0.258711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2200.2.a.s.1.2		2
4.3	odd	2		4400.2.a.bj.1.1		2
5.2	odd	4		2200.2.b.i.1849.1		4
5.3	odd	4		2200.2.b.i.1849.4		4
5.4	even	2		440.2.a.e.1.1	✓	2
15.14	odd	2		3960.2.a.w.1.1		2
20.3	even	4		4400.2.b.t.4049.1		4
20.7	even	4		4400.2.b.t.4049.4		4
20.19	odd	2		880.2.a.o.1.2		2
40.19	odd	2		3520.2.a.bk.1.1		2
40.29	even	2		3520.2.a.bp.1.2		2
55.54	odd	2		4840.2.a.j.1.1		2
60.59	even	2		7920.2.a.bu.1.2		2
220.219	even	2		9680.2.a.bs.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.e.1.1	✓	2	5.4	even	2
880.2.a.o.1.2		2	20.19	odd	2
2200.2.a.s.1.2		2	1.1	even	1	trivial
2200.2.b.i.1849.1		4	5.2	odd	4
2200.2.b.i.1849.4		4	5.3	odd	4
3520.2.a.bk.1.1		2	40.19	odd	2
3520.2.a.bp.1.2		2	40.29	even	2
3960.2.a.w.1.1		2	15.14	odd	2
4400.2.a.bj.1.1		2	4.3	odd	2
4400.2.b.t.4049.1		4	20.3	even	4
4400.2.b.t.4049.4		4	20.7	even	4
4840.2.a.j.1.1		2	55.54	odd	2
7920.2.a.bu.1.2		2	60.59	even	2
9680.2.a.bs.1.2		2	220.219	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2200.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} +2.56155 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q+2.56155 q^{3} +2.56155 q^{7} +3.56155 q^{9} +1.00000 q^{11} -2.00000 q^{13} +0.561553 q^{17} +2.56155 q^{19} +6.56155 q^{21} -5.12311 q^{23} +1.43845 q^{27} +9.68466 q^{29} +6.56155 q^{31} +2.56155 q^{33} +5.68466 q^{37} -5.12311 q^{39} +2.00000 q^{41} -10.2462 q^{43} +13.1231 q^{47} -0.438447 q^{49} +1.43845 q^{51} -4.56155 q^{53} +6.56155 q^{57} +1.12311 q^{59} +2.31534 q^{61} +9.12311 q^{63} -6.24621 q^{67} -13.1231 q^{69} +3.68466 q^{71} +2.00000 q^{73} +2.56155 q^{77} -15.3693 q^{79} -7.00000 q^{81} -5.12311 q^{83} +24.8078 q^{87} -12.5616 q^{89} -5.12311 q^{91} +16.8078 q^{93} -7.12311 q^{97} +3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + q^7 + 3 * q^9	\( 2 q + q^{3} + q^{7} + 3 q^{9} + 2 q^{11} - 4 q^{13} - 3 q^{17} + q^{19} + 9 q^{21} - 2 q^{23} + 7 q^{27} + 7 q^{29} + 9 q^{31} + q^{33} - q^{37} - 2 q^{39} + 4 q^{41} - 4 q^{43} + 18 q^{47} - 5 q^{49} + 7 q^{51} - 5 q^{53} + 9 q^{57} - 6 q^{59} + 17 q^{61} + 10 q^{63} + 4 q^{67} - 18 q^{69} - 5 q^{71} + 4 q^{73} + q^{77} - 6 q^{79} - 14 q^{81} - 2 q^{83} + 29 q^{87} - 21 q^{89} - 2 q^{91} + 13 q^{93} - 6 q^{97} + 3 q^{99}+O(q^{100}) \) 2 * q + q^3 + q^7 + 3 * q^9 + 2 * q^11 - 4 * q^13 - 3 * q^17 + q^19 + 9 * q^21 - 2 * q^23 + 7 * q^27 + 7 * q^29 + 9 * q^31 + q^33 - q^37 - 2 * q^39 + 4 * q^41 - 4 * q^43 + 18 * q^47 - 5 * q^49 + 7 * q^51 - 5 * q^53 + 9 * q^57 - 6 * q^59 + 17 * q^61 + 10 * q^63 + 4 * q^67 - 18 * q^69 - 5 * q^71 + 4 * q^73 + q^77 - 6 * q^79 - 14 * q^81 - 2 * q^83 + 29 * q^87 - 21 * q^89 - 2 * q^91 + 13 * q^93 - 6 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more