LMFDB - Embedded newform 5544.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5544,2,Mod(1,5544)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5544, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5544.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5544 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5544.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:	$44.2690628806$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1848)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	5544.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+1.56155 q^{5} -1.00000 q^{7} +1.00000 q^{11} +3.56155 q^{13} -3.12311 q^{17} -0.438447 q^{19} +4.00000 q^{23} -2.56155 q^{25} -3.56155 q^{29} +2.00000 q^{31} -1.56155 q^{35} -0.438447 q^{37} +6.24621 q^{41} +6.24621 q^{43} +1.31534 q^{47} +1.00000 q^{49} +5.12311 q^{53} +1.56155 q^{55} +14.9309 q^{59} -2.00000 q^{61} +5.56155 q^{65} -1.56155 q^{67} +14.2462 q^{71} -8.68466 q^{73} -1.00000 q^{77} -8.00000 q^{79} -0.246211 q^{83} -4.87689 q^{85} -0.246211 q^{89} -3.56155 q^{91} -0.684658 q^{95} -1.12311 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - q^5 - 2 * q^7	$ 2 q - q^{5} - 2 q^{7} + 2 q^{11} + 3 q^{13} + 2 q^{17} - 5 q^{19} + 8 q^{23} - q^{25} - 3 q^{29} + 4 q^{31} + q^{35} - 5 q^{37} - 4 q^{41} - 4 q^{43} + 15 q^{47} + 2 q^{49} + 2 q^{53} - q^{55} + q^{59} - 4 q^{61} + 7 q^{65} + q^{67} + 12 q^{71} - 5 q^{73} - 2 q^{77} - 16 q^{79} + 16 q^{83} - 18 q^{85} + 16 q^{89} - 3 q^{91} + 11 q^{95} + 6 q^{97}+O(q^{100}) $ 2 * q - q^5 - 2 * q^7 + 2 * q^11 + 3 * q^13 + 2 * q^17 - 5 * q^19 + 8 * q^23 - q^25 - 3 * q^29 + 4 * q^31 + q^35 - 5 * q^37 - 4 * q^41 - 4 * q^43 + 15 * q^47 + 2 * q^49 + 2 * q^53 - q^55 + q^59 - 4 * q^61 + 7 * q^65 + q^67 + 12 * q^71 - 5 * q^73 - 2 * q^77 - 16 * q^79 + 16 * q^83 - 18 * q^85 + 16 * q^89 - 3 * q^91 + 11 * q^95 + 6 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.56155	0.698348	0.349174	−	0.937058i	$-0.386462\pi$
$5$	1.56155	0.698348	0.349174	+	0.937058i	$0.386462\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.56155	0.987797	0.493899	−	0.869520i	$-0.335571\pi$
$13$	3.56155	0.987797	0.493899	+	0.869520i	$0.335571\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.12311	−0.757464	−0.378732	−	0.925506i	$-0.623640\pi$
$17$	−3.12311	−0.757464	−0.378732	+	0.925506i	$0.623640\pi$
$18$	0	0
$19$	−0.438447	−0.100587	−0.0502933	−	0.998734i	$-0.516016\pi$
$19$	−0.438447	−0.100587	−0.0502933	+	0.998734i	$0.516016\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−2.56155	−0.512311
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.56155	−0.661364	−0.330682	−	0.943742i	$-0.607279\pi$
$29$	−3.56155	−0.661364	−0.330682	+	0.943742i	$0.607279\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.56155	−0.263951
$36$	0	0
$37$	−0.438447	−0.0720803	−0.0360401	−	0.999350i	$-0.511474\pi$
$37$	−0.438447	−0.0720803	−0.0360401	+	0.999350i	$0.511474\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.24621	0.975494	0.487747	−	0.872985i	$-0.337819\pi$
$41$	6.24621	0.975494	0.487747	+	0.872985i	$0.337819\pi$
$42$	0	0
$43$	6.24621	0.952538	0.476269	−	0.879300i	$-0.341989\pi$
$43$	6.24621	0.952538	0.476269	+	0.879300i	$0.341989\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.31534	0.191862	0.0959311	−	0.995388i	$-0.469417\pi$
$47$	1.31534	0.191862	0.0959311	+	0.995388i	$0.469417\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.12311	0.703713	0.351856	−	0.936054i	$-0.385551\pi$
$53$	5.12311	0.703713	0.351856	+	0.936054i	$0.385551\pi$
$54$	0	0
$55$	1.56155	0.210560
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.9309	1.94383	0.971917	−	0.235325i	$-0.0756153\pi$
$59$	14.9309	1.94383	0.971917	+	0.235325i	$0.0756153\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.56155	0.689826
$66$	0	0
$67$	−1.56155	−0.190774	−0.0953870	−	0.995440i	$-0.530409\pi$
$67$	−1.56155	−0.190774	−0.0953870	+	0.995440i	$0.530409\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.2462	1.69071	0.845357	−	0.534202i	$-0.179388\pi$
$71$	14.2462	1.69071	0.845357	+	0.534202i	$0.179388\pi$
$72$	0	0
$73$	−8.68466	−1.01646	−0.508231	−	0.861221i	$-0.669700\pi$
$73$	−8.68466	−1.01646	−0.508231	+	0.861221i	$0.669700\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.246211	−0.0270252	−0.0135126	−	0.999909i	$-0.504301\pi$
$83$	−0.246211	−0.0270252	−0.0135126	+	0.999909i	$0.504301\pi$
$84$	0	0
$85$	−4.87689	−0.528973
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.246211	−0.0260983	−0.0130492	−	0.999915i	$-0.504154\pi$
$89$	−0.246211	−0.0260983	−0.0130492	+	0.999915i	$0.504154\pi$
$90$	0	0
$91$	−3.56155	−0.373352
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.684658	−0.0702445
$96$	0	0
$97$	−1.12311	−0.114034	−0.0570170	−	0.998373i	$-0.518159\pi$
$97$	−1.12311	−0.114034	−0.0570170	+	0.998373i	$0.518159\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.2462	−1.61656	−0.808279	−	0.588799i	$-0.799601\pi$
$101$	−16.2462	−1.61656	−0.808279	+	0.588799i	$0.799601\pi$
$102$	0	0
$103$	−8.24621	−0.812523	−0.406262	−	0.913757i	$-0.633168\pi$
$103$	−8.24621	−0.812523	−0.406262	+	0.913757i	$0.633168\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.8078	1.91489	0.957444	−	0.288618i	$-0.0931957\pi$
$107$	19.8078	1.91489	0.957444	+	0.288618i	$0.0931957\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	6.24621	0.582462
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.12311	0.286295
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.8078	−1.05612
$126$	0	0
$127$	3.12311	0.277131	0.138565	−	0.990353i	$-0.455751\pi$
$127$	3.12311	0.277131	0.138565	+	0.990353i	$0.455751\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.87689	0.600837	0.300419	−	0.953807i	$-0.402874\pi$
$131$	6.87689	0.600837	0.300419	+	0.953807i	$0.402874\pi$
$132$	0	0
$133$	0.438447	0.0380182
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	23.3693	1.98216	0.991080	−	0.133270i	$-0.0425477\pi$
$139$	23.3693	1.98216	0.991080	+	0.133270i	$0.0425477\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.56155	0.297832
$144$	0	0
$145$	−5.56155	−0.461862
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.43845	0.691305	0.345652	−	0.938363i	$-0.387658\pi$
$149$	8.43845	0.691305	0.345652	+	0.938363i	$0.387658\pi$
$150$	0	0
$151$	−17.3693	−1.41349	−0.706747	−	0.707466i	$-0.749838\pi$
$151$	−17.3693	−1.41349	−0.706747	+	0.707466i	$0.749838\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.12311	0.250854
$156$	0	0
$157$	22.2462	1.77544	0.887720	−	0.460383i	$-0.152288\pi$
$157$	22.2462	1.77544	0.887720	+	0.460383i	$0.152288\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	−22.0540	−1.72740	−0.863700	−	0.504006i	$-0.831859\pi$
$163$	−22.0540	−1.72740	−0.863700	+	0.504006i	$0.831859\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.2462	1.41193	0.705967	−	0.708245i	$-0.250513\pi$
$167$	18.2462	1.41193	0.705967	+	0.708245i	$0.250513\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.36932	−0.560279	−0.280139	−	0.959959i	$-0.590381\pi$
$173$	−7.36932	−0.560279	−0.280139	+	0.959959i	$0.590381\pi$
$174$	0	0
$175$	2.56155	0.193635
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.4924	1.23270	0.616351	−	0.787472i	$-0.288610\pi$
$179$	16.4924	1.23270	0.616351	+	0.787472i	$0.288610\pi$
$180$	0	0
$181$	−2.24621	−0.166960	−0.0834798	−	0.996509i	$-0.526603\pi$
$181$	−2.24621	−0.166960	−0.0834798	+	0.996509i	$0.526603\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.684658	−0.0503371
$186$	0	0
$187$	−3.12311	−0.228384
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.36932	−0.388510	−0.194255	−	0.980951i	$-0.562229\pi$
$191$	−5.36932	−0.388510	−0.194255	+	0.980951i	$0.562229\pi$
$192$	0	0
$193$	−17.1231	−1.23255	−0.616274	−	0.787532i	$-0.711359\pi$
$193$	−17.1231	−1.23255	−0.616274	+	0.787532i	$0.711359\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.2462	1.15749	0.578747	−	0.815507i	$-0.303542\pi$
$197$	16.2462	1.15749	0.578747	+	0.815507i	$0.303542\pi$
$198$	0	0
$199$	9.12311	0.646720	0.323360	−	0.946276i	$-0.395188\pi$
$199$	9.12311	0.646720	0.323360	+	0.946276i	$0.395188\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.56155	0.249972
$204$	0	0
$205$	9.75379	0.681234
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.438447	−0.0303280
$210$	0	0
$211$	11.1231	0.765746	0.382873	−	0.923801i	$-0.374935\pi$
$211$	11.1231	0.765746	0.382873	+	0.923801i	$0.374935\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.75379	0.665203
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−11.1231	−0.748221
$222$	0	0
$223$	3.75379	0.251372	0.125686	−	0.992070i	$-0.459887\pi$
$223$	3.75379	0.251372	0.125686	+	0.992070i	$0.459887\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.246211	0.0163416	0.00817081	−	0.999967i	$-0.497399\pi$
$227$	0.246211	0.0163416	0.00817081	+	0.999967i	$0.497399\pi$
$228$	0	0
$229$	10.2462	0.677089	0.338544	−	0.940950i	$-0.390065\pi$
$229$	10.2462	0.677089	0.338544	+	0.940950i	$0.390065\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.75379	−0.507968	−0.253984	−	0.967208i	$-0.581741\pi$
$233$	−7.75379	−0.507968	−0.253984	+	0.967208i	$0.581741\pi$
$234$	0	0
$235$	2.05398	0.133987
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.0540	−0.650338	−0.325169	−	0.945656i	$-0.605421\pi$
$239$	−10.0540	−0.650338	−0.325169	+	0.945656i	$0.605421\pi$
$240$	0	0
$241$	14.9309	0.961782	0.480891	−	0.876780i	$-0.340313\pi$
$241$	14.9309	0.961782	0.480891	+	0.876780i	$0.340313\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.56155	0.0997639
$246$	0	0
$247$	−1.56155	−0.0993592
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.9309	−0.942428	−0.471214	−	0.882019i	$-0.656184\pi$
$251$	−14.9309	−0.942428	−0.471214	+	0.882019i	$0.656184\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.6847	0.916004	0.458002	−	0.888951i	$-0.348565\pi$
$257$	14.6847	0.916004	0.458002	+	0.888951i	$0.348565\pi$
$258$	0	0
$259$	0.438447	0.0272438
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.06913	0.0659254	0.0329627	−	0.999457i	$-0.489506\pi$
$263$	1.06913	0.0659254	0.0329627	+	0.999457i	$0.489506\pi$
$264$	0	0
$265$	8.00000	0.491436
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.3693	0.815142	0.407571	−	0.913174i	$-0.366376\pi$
$269$	13.3693	0.815142	0.407571	+	0.913174i	$0.366376\pi$
$270$	0	0
$271$	6.43845	0.391108	0.195554	−	0.980693i	$-0.437350\pi$
$271$	6.43845	0.391108	0.195554	+	0.980693i	$0.437350\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.56155	−0.154467
$276$	0	0
$277$	5.12311	0.307818	0.153909	−	0.988085i	$-0.450814\pi$
$277$	5.12311	0.307818	0.153909	+	0.988085i	$0.450814\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.6847	1.11463	0.557317	−	0.830300i	$-0.311831\pi$
$281$	18.6847	1.11463	0.557317	+	0.830300i	$0.311831\pi$
$282$	0	0
$283$	−15.1771	−0.902184	−0.451092	−	0.892478i	$-0.648965\pi$
$283$	−15.1771	−0.902184	−0.451092	+	0.892478i	$0.648965\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.24621	−0.368702
$288$	0	0
$289$	−7.24621	−0.426248
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−18.8769	−1.10280	−0.551400	−	0.834241i	$-0.685906\pi$
$293$	−18.8769	−1.10280	−0.551400	+	0.834241i	$0.685906\pi$
$294$	0	0
$295$	23.3153	1.35747
$296$	0	0
$297$	0	0
$298$	0	0
$299$	14.2462	0.823880
$300$	0	0
$301$	−6.24621	−0.360026
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.12311	−0.178829
$306$	0	0
$307$	4.63068	0.264287	0.132144	−	0.991231i	$-0.457814\pi$
$307$	4.63068	0.264287	0.132144	+	0.991231i	$0.457814\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.1231	−0.744143	−0.372072	−	0.928204i	$-0.621352\pi$
$311$	−13.1231	−0.744143	−0.372072	+	0.928204i	$0.621352\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.87689	0.386245	0.193122	−	0.981175i	$-0.438139\pi$
$317$	6.87689	0.386245	0.193122	+	0.981175i	$0.438139\pi$
$318$	0	0
$319$	−3.56155	−0.199409
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.36932	0.0761908
$324$	0	0
$325$	−9.12311	−0.506059
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.31534	−0.0725171
$330$	0	0
$331$	18.2462	1.00290	0.501451	−	0.865186i	$-0.332800\pi$
$331$	18.2462	1.00290	0.501451	+	0.865186i	$0.332800\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.43845	−0.133227
$336$	0	0
$337$	−6.87689	−0.374608	−0.187304	−	0.982302i	$-0.559975\pi$
$337$	−6.87689	−0.374608	−0.187304	+	0.982302i	$0.559975\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	0	0
$349$	3.06913	0.164287	0.0821434	−	0.996621i	$-0.473823\pi$
$349$	3.06913	0.164287	0.0821434	+	0.996621i	$0.473823\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.6847	−0.994484	−0.497242	−	0.867612i	$-0.665654\pi$
$353$	−18.6847	−0.994484	−0.497242	+	0.867612i	$0.665654\pi$
$354$	0	0
$355$	22.2462	1.18071
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.2462	−0.751886	−0.375943	−	0.926643i	$-0.622681\pi$
$359$	−14.2462	−0.751886	−0.375943	+	0.926643i	$0.622681\pi$
$360$	0	0
$361$	−18.8078	−0.989882
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−13.5616	−0.709844
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.12311	−0.265978
$372$	0	0
$373$	24.7386	1.28092	0.640459	−	0.767992i	$-0.278744\pi$
$373$	24.7386	1.28092	0.640459	+	0.767992i	$0.278744\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.6847	−0.653293
$378$	0	0
$379$	30.0540	1.54377	0.771885	−	0.635763i	$-0.219314\pi$
$379$	30.0540	1.54377	0.771885	+	0.635763i	$0.219314\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	27.8617	1.42367	0.711834	−	0.702348i	$-0.247865\pi$
$383$	27.8617	1.42367	0.711834	+	0.702348i	$0.247865\pi$
$384$	0	0
$385$	−1.56155	−0.0795841
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.36932	0.373639	0.186820	−	0.982394i	$-0.440182\pi$
$389$	7.36932	0.373639	0.186820	+	0.982394i	$0.440182\pi$
$390$	0	0
$391$	−12.4924	−0.631769
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−12.4924	−0.628562
$396$	0	0
$397$	−1.36932	−0.0687240	−0.0343620	−	0.999409i	$-0.510940\pi$
$397$	−1.36932	−0.0687240	−0.0343620	+	0.999409i	$0.510940\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.61553	0.480177	0.240088	−	0.970751i	$-0.422824\pi$
$401$	9.61553	0.480177	0.240088	+	0.970751i	$0.422824\pi$
$402$	0	0
$403$	7.12311	0.354827
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.438447	−0.0217330
$408$	0	0
$409$	0.876894	0.0433596	0.0216798	−	0.999765i	$-0.493099\pi$
$409$	0.876894	0.0433596	0.0216798	+	0.999765i	$0.493099\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.9309	−0.734700
$414$	0	0
$415$	−0.384472	−0.0188730
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−8.19224	−0.400217	−0.200108	−	0.979774i	$-0.564129\pi$
$419$	−8.19224	−0.400217	−0.200108	+	0.979774i	$0.564129\pi$
$420$	0	0
$421$	−4.05398	−0.197579	−0.0987893	−	0.995108i	$-0.531497\pi$
$421$	−4.05398	−0.197579	−0.0987893	+	0.995108i	$0.531497\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.00000	0.388057
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.56155	0.267891	0.133945	−	0.990989i	$-0.457235\pi$
$431$	5.56155	0.267891	0.133945	+	0.990989i	$0.457235\pi$
$432$	0	0
$433$	−4.63068	−0.222536	−0.111268	−	0.993790i	$-0.535491\pi$
$433$	−4.63068	−0.222536	−0.111268	+	0.993790i	$0.535491\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.75379	−0.0838951
$438$	0	0
$439$	−6.05398	−0.288940	−0.144470	−	0.989509i	$-0.546148\pi$
$439$	−6.05398	−0.288940	−0.144470	+	0.989509i	$0.546148\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.4924	−0.783579	−0.391789	−	0.920055i	$-0.628144\pi$
$443$	−16.4924	−0.783579	−0.391789	+	0.920055i	$0.628144\pi$
$444$	0	0
$445$	−0.384472	−0.0182257
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.87689	0.324541	0.162270	−	0.986746i	$-0.448118\pi$
$449$	6.87689	0.324541	0.162270	+	0.986746i	$0.448118\pi$
$450$	0	0
$451$	6.24621	0.294123
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.56155	−0.260730
$456$	0	0
$457$	−10.8769	−0.508800	−0.254400	−	0.967099i	$-0.581878\pi$
$457$	−10.8769	−0.508800	−0.254400	+	0.967099i	$0.581878\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.8769	−0.692886	−0.346443	−	0.938071i	$-0.612611\pi$
$461$	−14.8769	−0.692886	−0.346443	+	0.938071i	$0.612611\pi$
$462$	0	0
$463$	32.6847	1.51898	0.759492	−	0.650516i	$-0.225447\pi$
$463$	32.6847	1.51898	0.759492	+	0.650516i	$0.225447\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.4384	0.668132	0.334066	−	0.942550i	$-0.391579\pi$
$467$	14.4384	0.668132	0.334066	+	0.942550i	$0.391579\pi$
$468$	0	0
$469$	1.56155	0.0721058
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.24621	0.287201
$474$	0	0
$475$	1.12311	0.0515316
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.12311	−0.142698	−0.0713492	−	0.997451i	$-0.522730\pi$
$479$	−3.12311	−0.142698	−0.0713492	+	0.997451i	$0.522730\pi$
$480$	0	0
$481$	−1.56155	−0.0712007
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.75379	−0.0796354
$486$	0	0
$487$	22.7386	1.03039	0.515193	−	0.857074i	$-0.327720\pi$
$487$	22.7386	1.03039	0.515193	+	0.857074i	$0.327720\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.9309	−1.21537	−0.607687	−	0.794177i	$-0.707902\pi$
$491$	−26.9309	−1.21537	−0.607687	+	0.794177i	$0.707902\pi$
$492$	0	0
$493$	11.1231	0.500959
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.2462	−0.639030
$498$	0	0
$499$	−39.4233	−1.76483	−0.882414	−	0.470473i	$-0.844083\pi$
$499$	−39.4233	−1.76483	−0.882414	+	0.470473i	$0.844083\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	−25.3693	−1.12892
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.36932	0.237991	0.118995	−	0.992895i	$-0.462033\pi$
$509$	5.36932	0.237991	0.118995	+	0.992895i	$0.462033\pi$
$510$	0	0
$511$	8.68466	0.384187
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.8769	−0.567424
$516$	0	0
$517$	1.31534	0.0578487
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5.80776	−0.254443	−0.127221	−	0.991874i	$-0.540606\pi$
$521$	−5.80776	−0.254443	−0.127221	+	0.991874i	$0.540606\pi$
$522$	0	0
$523$	−31.5616	−1.38009	−0.690045	−	0.723766i	$-0.742409\pi$
$523$	−31.5616	−1.38009	−0.690045	+	0.723766i	$0.742409\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.24621	−0.272089
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	22.2462	0.963590
$534$	0	0
$535$	30.9309	1.33726
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	15.3693	0.660779	0.330389	−	0.943845i	$-0.392820\pi$
$541$	15.3693	0.660779	0.330389	+	0.943845i	$0.392820\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.12311	0.133779
$546$	0	0
$547$	18.2462	0.780152	0.390076	−	0.920783i	$-0.372449\pi$
$547$	18.2462	0.780152	0.390076	+	0.920783i	$0.372449\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.56155	0.0665244
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.80776	−0.0765974	−0.0382987	−	0.999266i	$-0.512194\pi$
$557$	−1.80776	−0.0765974	−0.0382987	+	0.999266i	$0.512194\pi$
$558$	0	0
$559$	22.2462	0.940914
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.36932	0.142000	0.0709999	−	0.997476i	$-0.477381\pi$
$563$	3.36932	0.142000	0.0709999	+	0.997476i	$0.477381\pi$
$564$	0	0
$565$	3.12311	0.131390
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−38.9848	−1.63433	−0.817165	−	0.576404i	$-0.804455\pi$
$569$	−38.9848	−1.63433	−0.817165	+	0.576404i	$0.804455\pi$
$570$	0	0
$571$	−14.7386	−0.616793	−0.308396	−	0.951258i	$-0.599792\pi$
$571$	−14.7386	−0.616793	−0.308396	+	0.951258i	$0.599792\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−10.2462	−0.427297
$576$	0	0
$577$	−32.2462	−1.34243	−0.671214	−	0.741264i	$-0.734227\pi$
$577$	−32.2462	−1.34243	−0.671214	+	0.741264i	$0.734227\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.246211	0.0102146
$582$	0	0
$583$	5.12311	0.212177
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.82292	0.116514	0.0582572	−	0.998302i	$-0.481446\pi$
$587$	2.82292	0.116514	0.0582572	+	0.998302i	$0.481446\pi$
$588$	0	0
$589$	−0.876894	−0.0361318
$590$	0	0
$591$	0	0
$592$	0	0
$593$	40.0000	1.64260	0.821302	−	0.570494i	$-0.193248\pi$
$593$	40.0000	1.64260	0.821302	+	0.570494i	$0.193248\pi$
$594$	0	0
$595$	4.87689	0.199933
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	15.8078	0.644812	0.322406	−	0.946601i	$-0.395508\pi$
$601$	15.8078	0.644812	0.322406	+	0.946601i	$0.395508\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.56155	0.0634861
$606$	0	0
$607$	−27.8078	−1.12868	−0.564341	−	0.825542i	$-0.690870\pi$
$607$	−27.8078	−1.12868	−0.564341	+	0.825542i	$0.690870\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.68466	0.189521
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	7.12311	0.286302	0.143151	−	0.989701i	$-0.454277\pi$
$619$	7.12311	0.286302	0.143151	+	0.989701i	$0.454277\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.246211	0.00986425
$624$	0	0
$625$	−5.63068	−0.225227
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.36932	0.0545982
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.87689	0.193534
$636$	0	0
$637$	3.56155	0.141114
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.7386	−0.503146	−0.251573	−	0.967838i	$-0.580948\pi$
$641$	−12.7386	−0.503146	−0.251573	+	0.967838i	$0.580948\pi$
$642$	0	0
$643$	−17.7538	−0.700141	−0.350071	−	0.936723i	$-0.613842\pi$
$643$	−17.7538	−0.700141	−0.350071	+	0.936723i	$0.613842\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.06913	−0.120660	−0.0603300	−	0.998178i	$-0.519215\pi$
$647$	−3.06913	−0.120660	−0.0603300	+	0.998178i	$0.519215\pi$
$648$	0	0
$649$	14.9309	0.586088
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.630683	−0.0246805	−0.0123403	−	0.999924i	$-0.503928\pi$
$653$	−0.630683	−0.0246805	−0.0123403	+	0.999924i	$0.503928\pi$
$654$	0	0
$655$	10.7386	0.419593
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.05398	0.0800115	0.0400058	−	0.999199i	$-0.487262\pi$
$659$	2.05398	0.0800115	0.0400058	+	0.999199i	$0.487262\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.684658	0.0265499
$666$	0	0
$667$	−14.2462	−0.551616
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	21.6155	0.833217	0.416609	−	0.909086i	$-0.363219\pi$
$673$	21.6155	0.833217	0.416609	+	0.909086i	$0.363219\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	46.4924	1.78685	0.893424	−	0.449213i	$-0.148296\pi$
$677$	46.4924	1.78685	0.893424	+	0.449213i	$0.148296\pi$
$678$	0	0
$679$	1.12311	0.0431008
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−23.1231	−0.884781	−0.442391	−	0.896823i	$-0.645869\pi$
$683$	−23.1231	−0.884781	−0.442391	+	0.896823i	$0.645869\pi$
$684$	0	0
$685$	15.6155	0.596639
$686$	0	0
$687$	0	0
$688$	0	0
$689$	18.2462	0.695125
$690$	0	0
$691$	39.6155	1.50705	0.753523	−	0.657422i	$-0.228353\pi$
$691$	39.6155	1.50705	0.753523	+	0.657422i	$0.228353\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	36.4924	1.38424
$696$	0	0
$697$	−19.5076	−0.738902
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	0.192236	0.00725032
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.2462	0.611002
$708$	0	0
$709$	−24.0540	−0.903366	−0.451683	−	0.892178i	$-0.649176\pi$
$709$	−24.0540	−0.903366	−0.451683	+	0.892178i	$0.649176\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	5.56155	0.207990
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.68466	0.249296	0.124648	−	0.992201i	$-0.460220\pi$
$719$	6.68466	0.249296	0.124648	+	0.992201i	$0.460220\pi$
$720$	0	0
$721$	8.24621	0.307105
$722$	0	0
$723$	0	0
$724$	0	0
$725$	9.12311	0.338824
$726$	0	0
$727$	24.6307	0.913501	0.456751	−	0.889595i	$-0.349013\pi$
$727$	24.6307	0.913501	0.456751	+	0.889595i	$0.349013\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−19.5076	−0.721514
$732$	0	0
$733$	−47.4773	−1.75361	−0.876806	−	0.480843i	$-0.840331\pi$
$733$	−47.4773	−1.75361	−0.876806	+	0.480843i	$0.840331\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.56155	−0.0575205
$738$	0	0
$739$	2.24621	0.0826282	0.0413141	−	0.999146i	$-0.486846\pi$
$739$	2.24621	0.0826282	0.0413141	+	0.999146i	$0.486846\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.05398	0.0753530	0.0376765	−	0.999290i	$-0.488004\pi$
$743$	2.05398	0.0753530	0.0376765	+	0.999290i	$0.488004\pi$
$744$	0	0
$745$	13.1771	0.482771
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−19.8078	−0.723760
$750$	0	0
$751$	1.94602	0.0710114	0.0355057	−	0.999369i	$-0.488696\pi$
$751$	1.94602	0.0710114	0.0355057	+	0.999369i	$0.488696\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−27.1231	−0.987111
$756$	0	0
$757$	−11.9460	−0.434186	−0.217093	−	0.976151i	$-0.569657\pi$
$757$	−11.9460	−0.434186	−0.217093	+	0.976151i	$0.569657\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.24621	0.0814251	0.0407125	−	0.999171i	$-0.487037\pi$
$761$	2.24621	0.0814251	0.0407125	+	0.999171i	$0.487037\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	53.1771	1.92011
$768$	0	0
$769$	−11.3153	−0.408042	−0.204021	−	0.978967i	$-0.565401\pi$
$769$	−11.3153	−0.408042	−0.204021	+	0.978967i	$0.565401\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−51.4233	−1.84957	−0.924784	−	0.380493i	$-0.875754\pi$
$773$	−51.4233	−1.84957	−0.924784	+	0.380493i	$0.875754\pi$
$774$	0	0
$775$	−5.12311	−0.184027
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.73863	−0.0981217
$780$	0	0
$781$	14.2462	0.509770
$782$	0	0
$783$	0	0
$784$	0	0
$785$	34.7386	1.23988
$786$	0	0
$787$	−37.3153	−1.33015	−0.665074	−	0.746777i	$-0.731600\pi$
$787$	−37.3153	−1.33015	−0.665074	+	0.746777i	$0.731600\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	−7.12311	−0.252949
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.0540	−0.356130	−0.178065	−	0.984019i	$-0.556984\pi$
$797$	−10.0540	−0.356130	−0.178065	+	0.984019i	$0.556984\pi$
$798$	0	0
$799$	−4.10795	−0.145329
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8.68466	−0.306475
$804$	0	0
$805$	−6.24621	−0.220150
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.80776	−0.204190	−0.102095	−	0.994775i	$-0.532555\pi$
$809$	−5.80776	−0.204190	−0.102095	+	0.994775i	$0.532555\pi$
$810$	0	0
$811$	−48.5464	−1.70469	−0.852347	−	0.522976i	$-0.824822\pi$
$811$	−48.5464	−1.70469	−0.852347	+	0.522976i	$0.824822\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−34.4384	−1.20633
$816$	0	0
$817$	−2.73863	−0.0958127
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−50.6847	−1.76891	−0.884453	−	0.466629i	$-0.845468\pi$
$821$	−50.6847	−1.76891	−0.884453	+	0.466629i	$0.845468\pi$
$822$	0	0
$823$	−1.56155	−0.0544323	−0.0272162	−	0.999630i	$-0.508664\pi$
$823$	−1.56155	−0.0544323	−0.0272162	+	0.999630i	$0.508664\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.93087	−0.101916	−0.0509582	−	0.998701i	$-0.516228\pi$
$827$	−2.93087	−0.101916	−0.0509582	+	0.998701i	$0.516228\pi$
$828$	0	0
$829$	13.7538	0.477689	0.238844	−	0.971058i	$-0.423231\pi$
$829$	13.7538	0.477689	0.238844	+	0.971058i	$0.423231\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.12311	−0.108209
$834$	0	0
$835$	28.4924	0.986021
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.42329	−0.187233	−0.0936164	−	0.995608i	$-0.529843\pi$
$839$	−5.42329	−0.187233	−0.0936164	+	0.995608i	$0.529843\pi$
$840$	0	0
$841$	−16.3153	−0.562598
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.492423	−0.0169398
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.75379	−0.0601191
$852$	0	0
$853$	3.26137	0.111667	0.0558335	−	0.998440i	$-0.482218\pi$
$853$	3.26137	0.111667	0.0558335	+	0.998440i	$0.482218\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.3693	1.54979	0.774893	−	0.632092i	$-0.217804\pi$
$857$	45.3693	1.54979	0.774893	+	0.632092i	$0.217804\pi$
$858$	0	0
$859$	28.4924	0.972149	0.486074	−	0.873917i	$-0.338428\pi$
$859$	28.4924	0.972149	0.486074	+	0.873917i	$0.338428\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46.7386	−1.59100	−0.795501	−	0.605953i	$-0.792792\pi$
$863$	−46.7386	−1.59100	−0.795501	+	0.605953i	$0.792792\pi$
$864$	0	0
$865$	−11.5076	−0.391269
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−5.56155	−0.188446
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.8078	0.399175
$876$	0	0
$877$	−31.3693	−1.05927	−0.529633	−	0.848227i	$-0.677670\pi$
$877$	−31.3693	−1.05927	−0.529633	+	0.848227i	$0.677670\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	49.9157	1.68170	0.840852	−	0.541266i	$-0.182055\pi$
$881$	49.9157	1.68170	0.840852	+	0.541266i	$0.182055\pi$
$882$	0	0
$883$	−2.93087	−0.0986316	−0.0493158	−	0.998783i	$-0.515704\pi$
$883$	−2.93087	−0.0986316	−0.0493158	+	0.998783i	$0.515704\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.1231	0.642091	0.321046	−	0.947064i	$-0.395966\pi$
$887$	19.1231	0.642091	0.321046	+	0.947064i	$0.395966\pi$
$888$	0	0
$889$	−3.12311	−0.104746
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.576708	−0.0192988
$894$	0	0
$895$	25.7538	0.860854
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.12311	−0.237569
$900$	0	0
$901$	−16.0000	−0.533037
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.50758	−0.116596
$906$	0	0
$907$	−52.9848	−1.75933	−0.879666	−	0.475591i	$-0.842234\pi$
$907$	−52.9848	−1.75933	−0.879666	+	0.475591i	$0.842234\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−15.1231	−0.501051	−0.250525	−	0.968110i	$-0.580603\pi$
$911$	−15.1231	−0.501051	−0.250525	+	0.968110i	$0.580603\pi$
$912$	0	0
$913$	−0.246211	−0.00814840
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.87689	−0.227095
$918$	0	0
$919$	−30.2462	−0.997730	−0.498865	−	0.866680i	$-0.666250\pi$
$919$	−30.2462	−0.997730	−0.498865	+	0.866680i	$0.666250\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	50.7386	1.67008
$924$	0	0
$925$	1.12311	0.0369275
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−27.6695	−0.907807	−0.453903	−	0.891051i	$-0.649969\pi$
$929$	−27.6695	−0.907807	−0.453903	+	0.891051i	$0.649969\pi$
$930$	0	0
$931$	−0.438447	−0.0143695
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.87689	−0.159492
$936$	0	0
$937$	−47.6155	−1.55553	−0.777766	−	0.628554i	$-0.783647\pi$
$937$	−47.6155	−1.55553	−0.777766	+	0.628554i	$0.783647\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5.61553	−0.183061	−0.0915305	−	0.995802i	$-0.529176\pi$
$941$	−5.61553	−0.183061	−0.0915305	+	0.995802i	$0.529176\pi$
$942$	0	0
$943$	24.9848	0.813618
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.1231	−1.27133	−0.635665	−	0.771965i	$-0.719274\pi$
$947$	−39.1231	−1.27133	−0.635665	+	0.771965i	$0.719274\pi$
$948$	0	0
$949$	−30.9309	−1.00406
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−37.8078	−1.22471	−0.612357	−	0.790582i	$-0.709778\pi$
$953$	−37.8078	−1.22471	−0.612357	+	0.790582i	$0.709778\pi$
$954$	0	0
$955$	−8.38447	−0.271315
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.0000	−0.322917
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−26.7386	−0.860747
$966$	0	0
$967$	6.63068	0.213228	0.106614	−	0.994300i	$-0.465999\pi$
$967$	6.63068	0.213228	0.106614	+	0.994300i	$0.465999\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	49.5616	1.59051	0.795253	−	0.606278i	$-0.207338\pi$
$971$	49.5616	1.59051	0.795253	+	0.606278i	$0.207338\pi$
$972$	0	0
$973$	−23.3693	−0.749186
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.38447	0.204257	0.102129	−	0.994771i	$-0.467435\pi$
$977$	6.38447	0.204257	0.102129	+	0.994771i	$0.467435\pi$
$978$	0	0
$979$	−0.246211	−0.00786895
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−39.3693	−1.25569	−0.627843	−	0.778340i	$-0.716062\pi$
$983$	−39.3693	−1.25569	−0.627843	+	0.778340i	$0.716062\pi$
$984$	0	0
$985$	25.3693	0.808334
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24.9848	0.794472
$990$	0	0
$991$	32.6847	1.03826	0.519131	−	0.854695i	$-0.326256\pi$
$991$	32.6847	1.03826	0.519131	+	0.854695i	$0.326256\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14.2462	0.451635
$996$	0	0
$997$	−54.0000	−1.71020	−0.855099	−	0.518465i	$-0.826503\pi$
$997$	−54.0000	−1.71020	−0.855099	+	0.518465i	$0.826503\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5544.2.a.bd.1.2		2
3.2	odd	2		1848.2.a.p.1.1	✓	2
12.11	even	2		3696.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1848.2.a.p.1.1	✓	2	3.2	odd	2
3696.2.a.bi.1.1		2	12.11	even	2
5544.2.a.bd.1.2		2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5544 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5544.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+1.56155 q^{5} -1.00000 q^{7} +1.00000 q^{11} +3.56155 q^{13} -3.12311 q^{17} -0.438447 q^{19} +4.00000 q^{23} -2.56155 q^{25} -3.56155 q^{29} +2.00000 q^{31} -1.56155 q^{35} -0.438447 q^{37} +6.24621 q^{41} +6.24621 q^{43} +1.31534 q^{47} +1.00000 q^{49} +5.12311 q^{53} +1.56155 q^{55} +14.9309 q^{59} -2.00000 q^{61} +5.56155 q^{65} -1.56155 q^{67} +14.2462 q^{71} -8.68466 q^{73} -1.00000 q^{77} -8.00000 q^{79} -0.246211 q^{83} -4.87689 q^{85} -0.246211 q^{89} -3.56155 q^{91} -0.684658 q^{95} -1.12311 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - q^5 - 2 * q^7	\( 2 q - q^{5} - 2 q^{7} + 2 q^{11} + 3 q^{13} + 2 q^{17} - 5 q^{19} + 8 q^{23} - q^{25} - 3 q^{29} + 4 q^{31} + q^{35} - 5 q^{37} - 4 q^{41} - 4 q^{43} + 15 q^{47} + 2 q^{49} + 2 q^{53} - q^{55} + q^{59} - 4 q^{61} + 7 q^{65} + q^{67} + 12 q^{71} - 5 q^{73} - 2 q^{77} - 16 q^{79} + 16 q^{83} - 18 q^{85} + 16 q^{89} - 3 q^{91} + 11 q^{95} + 6 q^{97}+O(q^{100}) \) 2 * q - q^5 - 2 * q^7 + 2 * q^11 + 3 * q^13 + 2 * q^17 - 5 * q^19 + 8 * q^23 - q^25 - 3 * q^29 + 4 * q^31 + q^35 - 5 * q^37 - 4 * q^41 - 4 * q^43 + 15 * q^47 + 2 * q^49 + 2 * q^53 - q^55 + q^59 - 4 * q^61 + 7 * q^65 + q^67 + 12 * q^71 - 5 * q^73 - 2 * q^77 - 16 * q^79 + 16 * q^83 - 18 * q^85 + 16 * q^89 - 3 * q^91 + 11 * q^95 + 6 * q^97

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