LMFDB - Embedded newform 5544.2.a.bf.1.1

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 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-0.561553 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-0.561553 q^{5} -1.00000 q^{7} -1.00000 q^{11} +3.12311 q^{13} +2.00000 q^{17} -5.12311 q^{19} +1.43845 q^{23} -4.68466 q^{25} +2.00000 q^{29} -10.5616 q^{31} +0.561553 q^{35} +4.56155 q^{37} +10.0000 q^{41} +4.00000 q^{43} +6.24621 q^{47} +1.00000 q^{49} +4.24621 q^{53} +0.561553 q^{55} +5.43845 q^{59} -4.24621 q^{61} -1.75379 q^{65} -2.56155 q^{67} -3.68466 q^{71} -4.24621 q^{73} +1.00000 q^{77} +5.12311 q^{79} +8.00000 q^{83} -1.12311 q^{85} +12.5616 q^{89} -3.12311 q^{91} +2.87689 q^{95} +8.56155 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 3 * q^5 - 2 * q^7	$ 2 q + 3 q^{5} - 2 q^{7} - 2 q^{11} - 2 q^{13} + 4 q^{17} - 2 q^{19} + 7 q^{23} + 3 q^{25} + 4 q^{29} - 17 q^{31} - 3 q^{35} + 5 q^{37} + 20 q^{41} + 8 q^{43} - 4 q^{47} + 2 q^{49} - 8 q^{53} - 3 q^{55} + 15 q^{59} + 8 q^{61} - 20 q^{65} - q^{67} + 5 q^{71} + 8 q^{73} + 2 q^{77} + 2 q^{79} + 16 q^{83} + 6 q^{85} + 21 q^{89} + 2 q^{91} + 14 q^{95} + 13 q^{97}+O(q^{100}) $ 2 * q + 3 * q^5 - 2 * q^7 - 2 * q^11 - 2 * q^13 + 4 * q^17 - 2 * q^19 + 7 * q^23 + 3 * q^25 + 4 * q^29 - 17 * q^31 - 3 * q^35 + 5 * q^37 + 20 * q^41 + 8 * q^43 - 4 * q^47 + 2 * q^49 - 8 * q^53 - 3 * q^55 + 15 * q^59 + 8 * q^61 - 20 * q^65 - q^67 + 5 * q^71 + 8 * q^73 + 2 * q^77 + 2 * q^79 + 16 * q^83 + 6 * q^85 + 21 * q^89 + 2 * q^91 + 14 * q^95 + 13 * 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
$5$	−0.561553	−0.251134	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	−0.561553	−0.251134	−0.125567	+	0.992085i	$0.540075\pi$
$6$	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
$13$	3.12311	0.866194	0.433097	−	0.901347i	$-0.357421\pi$
$13$	3.12311	0.866194	0.433097	+	0.901347i	$0.357421\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.43845	0.299937	0.149968	−	0.988691i	$-0.452083\pi$
$23$	1.43845	0.299937	0.149968	+	0.988691i	$0.452083\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−10.5616	−1.89691	−0.948455	−	0.316911i	$-0.897355\pi$
$31$	−10.5616	−1.89691	−0.948455	+	0.316911i	$0.897355\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.561553	0.0949197
$36$	0	0
$37$	4.56155	0.749915	0.374957	−	0.927042i	$-0.377657\pi$
$37$	4.56155	0.749915	0.374957	+	0.927042i	$0.377657\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.24621	0.911104	0.455552	−	0.890209i	$-0.349442\pi$
$47$	6.24621	0.911104	0.455552	+	0.890209i	$0.349442\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.24621	0.583262	0.291631	−	0.956531i	$-0.405802\pi$
$53$	4.24621	0.583262	0.291631	+	0.956531i	$0.405802\pi$
$54$	0	0
$55$	0.561553	0.0757198
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.43845	0.708026	0.354013	−	0.935241i	$-0.384817\pi$
$59$	5.43845	0.708026	0.354013	+	0.935241i	$0.384817\pi$
$60$	0	0
$61$	−4.24621	−0.543672	−0.271836	−	0.962344i	$-0.587631\pi$
$61$	−4.24621	−0.543672	−0.271836	+	0.962344i	$0.587631\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.75379	−0.217531
$66$	0	0
$67$	−2.56155	−0.312943	−0.156472	−	0.987682i	$-0.550012\pi$
$67$	−2.56155	−0.312943	−0.156472	+	0.987682i	$0.550012\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.68466	−0.437289	−0.218644	−	0.975805i	$-0.570163\pi$
$71$	−3.68466	−0.437289	−0.218644	+	0.975805i	$0.570163\pi$
$72$	0	0
$73$	−4.24621	−0.496981	−0.248491	−	0.968634i	$-0.579935\pi$
$73$	−4.24621	−0.496981	−0.248491	+	0.968634i	$0.579935\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	5.12311	0.576394	0.288197	−	0.957571i	$-0.406944\pi$
$79$	5.12311	0.576394	0.288197	+	0.957571i	$0.406944\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	−1.12311	−0.121818
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.5616	1.33152	0.665761	−	0.746165i	$-0.268107\pi$
$89$	12.5616	1.33152	0.665761	+	0.746165i	$0.268107\pi$
$90$	0	0
$91$	−3.12311	−0.327390
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.87689	0.295163
$96$	0	0
$97$	8.56155	0.869294	0.434647	−	0.900601i	$-0.356873\pi$
$97$	8.56155	0.869294	0.434647	+	0.900601i	$0.356873\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.3693	−1.33030	−0.665148	−	0.746711i	$-0.731632\pi$
$101$	−13.3693	−1.33030	−0.665148	+	0.746711i	$0.731632\pi$
$102$	0	0
$103$	16.4924	1.62505	0.812523	−	0.582929i	$-0.198093\pi$
$103$	16.4924	1.62505	0.812523	+	0.582929i	$0.198093\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.1231	−1.65535	−0.827677	−	0.561205i	$-0.810338\pi$
$107$	−17.1231	−1.65535	−0.827677	+	0.561205i	$0.810338\pi$
$108$	0	0
$109$	−9.36932	−0.897418	−0.448709	−	0.893678i	$-0.648116\pi$
$109$	−9.36932	−0.897418	−0.448709	+	0.893678i	$0.648116\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.68466	−0.534768	−0.267384	−	0.963590i	$-0.586159\pi$
$113$	−5.68466	−0.534768	−0.267384	+	0.963590i	$0.586159\pi$
$114$	0	0
$115$	−0.807764	−0.0753244
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.43845	0.486430
$126$	0	0
$127$	−5.12311	−0.454602	−0.227301	−	0.973825i	$-0.572990\pi$
$127$	−5.12311	−0.454602	−0.227301	+	0.973825i	$0.572990\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.1231	−1.14657	−0.573286	−	0.819356i	$-0.694331\pi$
$131$	−13.1231	−1.14657	−0.573286	+	0.819356i	$0.694331\pi$
$132$	0	0
$133$	5.12311	0.444230
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.31534	0.197813	0.0989065	−	0.995097i	$-0.468466\pi$
$137$	2.31534	0.197813	0.0989065	+	0.995097i	$0.468466\pi$
$138$	0	0
$139$	−5.12311	−0.434536	−0.217268	−	0.976112i	$-0.569715\pi$
$139$	−5.12311	−0.434536	−0.217268	+	0.976112i	$0.569715\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.12311	−0.261167
$144$	0	0
$145$	−1.12311	−0.0932688
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	5.12311	0.416912	0.208456	−	0.978032i	$-0.433156\pi$
$151$	5.12311	0.416912	0.208456	+	0.978032i	$0.433156\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.93087	0.476379
$156$	0	0
$157$	16.5616	1.32176	0.660878	−	0.750493i	$-0.270184\pi$
$157$	16.5616	1.32176	0.660878	+	0.750493i	$0.270184\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.43845	−0.113366
$162$	0	0
$163$	16.4924	1.29179	0.645893	−	0.763428i	$-0.276485\pi$
$163$	16.4924	1.29179	0.645893	+	0.763428i	$0.276485\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.36932	0.570255	0.285127	−	0.958490i	$-0.407964\pi$
$167$	7.36932	0.570255	0.285127	+	0.958490i	$0.407964\pi$
$168$	0	0
$169$	−3.24621	−0.249709
$170$	0	0
$171$	0	0
$172$	0	0
$173$	23.1231	1.75802	0.879009	−	0.476806i	$-0.158206\pi$
$173$	23.1231	1.75802	0.879009	+	0.476806i	$0.158206\pi$
$174$	0	0
$175$	4.68466	0.354127
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.8078	1.55525	0.777623	−	0.628731i	$-0.216425\pi$
$179$	20.8078	1.55525	0.777623	+	0.628731i	$0.216425\pi$
$180$	0	0
$181$	7.93087	0.589497	0.294748	−	0.955575i	$-0.404764\pi$
$181$	7.93087	0.589497	0.294748	+	0.955575i	$0.404764\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.56155	−0.188329
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	27.0540	1.95756	0.978778	−	0.204921i	$-0.0656938\pi$
$191$	27.0540	1.95756	0.978778	+	0.204921i	$0.0656938\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.2462	1.44248	0.721241	−	0.692684i	$-0.243572\pi$
$197$	20.2462	1.44248	0.721241	+	0.692684i	$0.243572\pi$
$198$	0	0
$199$	−14.2462	−1.00989	−0.504944	−	0.863152i	$-0.668487\pi$
$199$	−14.2462	−1.00989	−0.504944	+	0.863152i	$0.668487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	−5.61553	−0.392205
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.12311	0.354373
$210$	0	0
$211$	25.1231	1.72955	0.864773	−	0.502163i	$-0.167462\pi$
$211$	25.1231	1.72955	0.864773	+	0.502163i	$0.167462\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.24621	−0.153190
$216$	0	0
$217$	10.5616	0.716965
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.24621	0.420166
$222$	0	0
$223$	−26.5616	−1.77869	−0.889347	−	0.457234i	$-0.848840\pi$
$223$	−26.5616	−1.77869	−0.889347	+	0.457234i	$0.848840\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.2462	0.680065	0.340032	−	0.940414i	$-0.389562\pi$
$227$	10.2462	0.680065	0.340032	+	0.940414i	$0.389562\pi$
$228$	0	0
$229$	−2.31534	−0.153002	−0.0765010	−	0.997070i	$-0.524375\pi$
$229$	−2.31534	−0.153002	−0.0765010	+	0.997070i	$0.524375\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.49242	0.163284	0.0816420	−	0.996662i	$-0.473984\pi$
$233$	2.49242	0.163284	0.0816420	+	0.996662i	$0.473984\pi$
$234$	0	0
$235$	−3.50758	−0.228809
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	5.36932	0.345868	0.172934	−	0.984933i	$-0.444675\pi$
$241$	5.36932	0.345868	0.172934	+	0.984933i	$0.444675\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.561553	−0.0358763
$246$	0	0
$247$	−16.0000	−1.01806
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.0540	−0.950198	−0.475099	−	0.879932i	$-0.657588\pi$
$251$	−15.0540	−0.950198	−0.475099	+	0.879932i	$0.657588\pi$
$252$	0	0
$253$	−1.43845	−0.0904344
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	−4.56155	−0.283441
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.61553	−0.592919	−0.296459	−	0.955045i	$-0.595806\pi$
$263$	−9.61553	−0.592919	−0.296459	+	0.955045i	$0.595806\pi$
$264$	0	0
$265$	−2.38447	−0.146477
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.75379	−0.472757	−0.236378	−	0.971661i	$-0.575960\pi$
$269$	−7.75379	−0.472757	−0.236378	+	0.971661i	$0.575960\pi$
$270$	0	0
$271$	18.2462	1.10838	0.554189	−	0.832391i	$-0.313028\pi$
$271$	18.2462	1.10838	0.554189	+	0.832391i	$0.313028\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.68466	0.282496
$276$	0	0
$277$	8.87689	0.533361	0.266680	−	0.963785i	$-0.414073\pi$
$277$	8.87689	0.533361	0.266680	+	0.963785i	$0.414073\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−29.8617	−1.78140	−0.890701	−	0.454590i	$-0.849786\pi$
$281$	−29.8617	−1.78140	−0.890701	+	0.454590i	$0.849786\pi$
$282$	0	0
$283$	28.4924	1.69370	0.846849	−	0.531833i	$-0.178497\pi$
$283$	28.4924	1.69370	0.846849	+	0.531833i	$0.178497\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.2462	0.715431	0.357716	−	0.933831i	$-0.383556\pi$
$293$	12.2462	0.715431	0.357716	+	0.933831i	$0.383556\pi$
$294$	0	0
$295$	−3.05398	−0.177809
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.49242	0.259804
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.38447	0.136534
$306$	0	0
$307$	20.4924	1.16956	0.584782	−	0.811190i	$-0.301180\pi$
$307$	20.4924	1.16956	0.584782	+	0.811190i	$0.301180\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.7386	−1.06257	−0.531285	−	0.847193i	$-0.678291\pi$
$311$	−18.7386	−1.06257	−0.531285	+	0.847193i	$0.678291\pi$
$312$	0	0
$313$	15.9309	0.900466	0.450233	−	0.892911i	$-0.351341\pi$
$313$	15.9309	0.900466	0.450233	+	0.892911i	$0.351341\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.0540	0.733184	0.366592	−	0.930382i	$-0.380524\pi$
$317$	13.0540	0.733184	0.366592	+	0.930382i	$0.380524\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−10.2462	−0.570114
$324$	0	0
$325$	−14.6307	−0.811564
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.24621	−0.344365
$330$	0	0
$331$	12.8078	0.703978	0.351989	−	0.936004i	$-0.385505\pi$
$331$	12.8078	0.703978	0.351989	+	0.936004i	$0.385505\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.43845	0.0785908
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.5616	0.571940
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.87689	−0.369171	−0.184586	−	0.982816i	$-0.559094\pi$
$347$	−6.87689	−0.369171	−0.184586	+	0.982816i	$0.559094\pi$
$348$	0	0
$349$	−12.2462	−0.655525	−0.327762	−	0.944760i	$-0.606295\pi$
$349$	−12.2462	−0.655525	−0.327762	+	0.944760i	$0.606295\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.5616	1.09438	0.547191	−	0.837008i	$-0.315697\pi$
$353$	20.5616	1.09438	0.547191	+	0.837008i	$0.315697\pi$
$354$	0	0
$355$	2.06913	0.109818
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.24621	−0.118550	−0.0592752	−	0.998242i	$-0.518879\pi$
$359$	−2.24621	−0.118550	−0.0592752	+	0.998242i	$0.518879\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.38447	0.124809
$366$	0	0
$367$	−0.315342	−0.0164607	−0.00823035	−	0.999966i	$-0.502620\pi$
$367$	−0.315342	−0.0164607	−0.00823035	+	0.999966i	$0.502620\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.24621	−0.220452
$372$	0	0
$373$	−11.6155	−0.601429	−0.300715	−	0.953714i	$-0.597225\pi$
$373$	−11.6155	−0.601429	−0.300715	+	0.953714i	$0.597225\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.24621	0.321696
$378$	0	0
$379$	−25.9309	−1.33198	−0.665990	−	0.745961i	$-0.731991\pi$
$379$	−25.9309	−1.33198	−0.665990	+	0.745961i	$0.731991\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.93087	−0.507444	−0.253722	−	0.967277i	$-0.581655\pi$
$383$	−9.93087	−0.507444	−0.253722	+	0.967277i	$0.581655\pi$
$384$	0	0
$385$	−0.561553	−0.0286194
$386$	0	0
$387$	0	0
$388$	0	0
$389$	29.0540	1.47310	0.736548	−	0.676386i	$-0.236455\pi$
$389$	29.0540	1.47310	0.736548	+	0.676386i	$0.236455\pi$
$390$	0	0
$391$	2.87689	0.145491
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.87689	−0.144752
$396$	0	0
$397$	−10.4924	−0.526600	−0.263300	−	0.964714i	$-0.584811\pi$
$397$	−10.4924	−0.526600	−0.263300	+	0.964714i	$0.584811\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	−32.9848	−1.64309
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.56155	−0.226108
$408$	0	0
$409$	34.4924	1.70554	0.852770	−	0.522286i	$-0.174921\pi$
$409$	34.4924	1.70554	0.852770	+	0.522286i	$0.174921\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.43845	−0.267608
$414$	0	0
$415$	−4.49242	−0.220524
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.7386	−0.915442	−0.457721	−	0.889096i	$-0.651334\pi$
$419$	−18.7386	−0.915442	−0.457721	+	0.889096i	$0.651334\pi$
$420$	0	0
$421$	30.9848	1.51011	0.755054	−	0.655662i	$-0.227610\pi$
$421$	30.9848	1.51011	0.755054	+	0.655662i	$0.227610\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−9.36932	−0.454479
$426$	0	0
$427$	4.24621	0.205489
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.7386	0.709935	0.354968	−	0.934879i	$-0.384492\pi$
$431$	14.7386	0.709935	0.354968	+	0.934879i	$0.384492\pi$
$432$	0	0
$433$	−1.68466	−0.0809595	−0.0404798	−	0.999180i	$-0.512889\pi$
$433$	−1.68466	−0.0809595	−0.0404798	+	0.999180i	$0.512889\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.36932	−0.352522
$438$	0	0
$439$	17.6155	0.840743	0.420372	−	0.907352i	$-0.361900\pi$
$439$	17.6155	0.840743	0.420372	+	0.907352i	$0.361900\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.1771	−0.578551	−0.289275	−	0.957246i	$-0.593414\pi$
$443$	−12.1771	−0.578551	−0.289275	+	0.957246i	$0.593414\pi$
$444$	0	0
$445$	−7.05398	−0.334390
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17.6847	0.834591	0.417295	−	0.908771i	$-0.362978\pi$
$449$	17.6847	0.834591	0.417295	+	0.908771i	$0.362978\pi$
$450$	0	0
$451$	−10.0000	−0.470882
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.75379	0.0822189
$456$	0	0
$457$	37.8617	1.77110	0.885549	−	0.464546i	$-0.153783\pi$
$457$	37.8617	1.77110	0.885549	+	0.464546i	$0.153783\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−39.5464	−1.83788	−0.918938	−	0.394401i	$-0.870952\pi$
$463$	−39.5464	−1.83788	−0.918938	+	0.394401i	$0.870952\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.4233	1.40782	0.703911	−	0.710288i	$-0.251435\pi$
$467$	30.4233	1.40782	0.703911	+	0.710288i	$0.251435\pi$
$468$	0	0
$469$	2.56155	0.118282
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.4924	−0.936323	−0.468161	−	0.883643i	$-0.655083\pi$
$479$	−20.4924	−0.936323	−0.468161	+	0.883643i	$0.655083\pi$
$480$	0	0
$481$	14.2462	0.649571
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.80776	−0.218309
$486$	0	0
$487$	−17.4384	−0.790211	−0.395106	−	0.918636i	$-0.629292\pi$
$487$	−17.4384	−0.790211	−0.395106	+	0.918636i	$0.629292\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.1231	0.772755	0.386377	−	0.922341i	$-0.373726\pi$
$491$	17.1231	0.772755	0.386377	+	0.922341i	$0.373726\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.68466	0.165280
$498$	0	0
$499$	−32.4924	−1.45456	−0.727280	−	0.686341i	$-0.759216\pi$
$499$	−32.4924	−1.45456	−0.727280	+	0.686341i	$0.759216\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14.7386	−0.657163	−0.328582	−	0.944476i	$-0.606571\pi$
$503$	−14.7386	−0.657163	−0.328582	+	0.944476i	$0.606571\pi$
$504$	0	0
$505$	7.50758	0.334083
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.8078	0.656343	0.328171	−	0.944618i	$-0.393568\pi$
$509$	14.8078	0.656343	0.328171	+	0.944618i	$0.393568\pi$
$510$	0	0
$511$	4.24621	0.187841
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.26137	−0.408105
$516$	0	0
$517$	−6.24621	−0.274708
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22.3153	−0.977653	−0.488826	−	0.872381i	$-0.662575\pi$
$521$	−22.3153	−0.977653	−0.488826	+	0.872381i	$0.662575\pi$
$522$	0	0
$523$	2.24621	0.0982200	0.0491100	−	0.998793i	$-0.484362\pi$
$523$	2.24621	0.0982200	0.0491100	+	0.998793i	$0.484362\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21.1231	−0.920137
$528$	0	0
$529$	−20.9309	−0.910038
$530$	0	0
$531$	0	0
$532$	0	0
$533$	31.2311	1.35277
$534$	0	0
$535$	9.61553	0.415716
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−15.1231	−0.650193	−0.325097	−	0.945681i	$-0.605397\pi$
$541$	−15.1231	−0.650193	−0.325097	+	0.945681i	$0.605397\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.26137	0.225372
$546$	0	0
$547$	34.7386	1.48532	0.742658	−	0.669670i	$-0.233565\pi$
$547$	34.7386	1.48532	0.742658	+	0.669670i	$0.233565\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10.2462	−0.436503
$552$	0	0
$553$	−5.12311	−0.217857
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−40.2462	−1.70529	−0.852643	−	0.522493i	$-0.825002\pi$
$557$	−40.2462	−1.70529	−0.852643	+	0.522493i	$0.825002\pi$
$558$	0	0
$559$	12.4924	0.528373
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.7386	1.29548	0.647739	−	0.761862i	$-0.275715\pi$
$563$	30.7386	1.29548	0.647739	+	0.761862i	$0.275715\pi$
$564$	0	0
$565$	3.19224	0.134298
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	−40.4924	−1.69456	−0.847278	−	0.531150i	$-0.821760\pi$
$571$	−40.4924	−1.69456	−0.847278	+	0.531150i	$0.821760\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.73863	−0.281020
$576$	0	0
$577$	6.31534	0.262911	0.131456	−	0.991322i	$-0.458035\pi$
$577$	6.31534	0.262911	0.131456	+	0.991322i	$0.458035\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	−4.24621	−0.175860
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	54.1080	2.22948
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.24621	−0.338631	−0.169316	−	0.985562i	$-0.554156\pi$
$593$	−8.24621	−0.338631	−0.169316	+	0.985562i	$0.554156\pi$
$594$	0	0
$595$	1.12311	0.0460428
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	43.1231	1.75903	0.879514	−	0.475873i	$-0.157868\pi$
$601$	43.1231	1.75903	0.879514	+	0.475873i	$0.157868\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.561553	−0.0228304
$606$	0	0
$607$	−48.3542	−1.96263	−0.981317	−	0.192396i	$-0.938374\pi$
$607$	−48.3542	−1.96263	−0.981317	+	0.192396i	$0.938374\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	19.5076	0.789192
$612$	0	0
$613$	23.6155	0.953822	0.476911	−	0.878952i	$-0.341756\pi$
$613$	23.6155	0.953822	0.476911	+	0.878952i	$0.341756\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.4924	1.06654	0.533272	−	0.845944i	$-0.320962\pi$
$617$	26.4924	1.06654	0.533272	+	0.845944i	$0.320962\pi$
$618$	0	0
$619$	27.1922	1.09295	0.546474	−	0.837476i	$-0.315970\pi$
$619$	27.1922	1.09295	0.546474	+	0.837476i	$0.315970\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.5616	−0.503268
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.12311	0.363762
$630$	0	0
$631$	−37.9309	−1.51000	−0.755002	−	0.655722i	$-0.772364\pi$
$631$	−37.9309	−1.51000	−0.755002	+	0.655722i	$0.772364\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.87689	0.114166
$636$	0	0
$637$	3.12311	0.123742
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−34.8078	−1.37482	−0.687412	−	0.726268i	$-0.741253\pi$
$641$	−34.8078	−1.37482	−0.687412	+	0.726268i	$0.741253\pi$
$642$	0	0
$643$	43.5464	1.71730	0.858651	−	0.512560i	$-0.171303\pi$
$643$	43.5464	1.71730	0.858651	+	0.512560i	$0.171303\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.1771	1.42227	0.711134	−	0.703057i	$-0.248182\pi$
$647$	36.1771	1.42227	0.711134	+	0.703057i	$0.248182\pi$
$648$	0	0
$649$	−5.43845	−0.213478
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−43.3002	−1.69447	−0.847234	−	0.531220i	$-0.821734\pi$
$653$	−43.3002	−1.69447	−0.847234	+	0.531220i	$0.821734\pi$
$654$	0	0
$655$	7.36932	0.287943
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21.6155	0.842021	0.421011	−	0.907056i	$-0.361675\pi$
$659$	21.6155	0.842021	0.421011	+	0.907056i	$0.361675\pi$
$660$	0	0
$661$	−34.6695	−1.34849	−0.674244	−	0.738509i	$-0.735530\pi$
$661$	−34.6695	−1.34849	−0.674244	+	0.738509i	$0.735530\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.87689	−0.111561
$666$	0	0
$667$	2.87689	0.111394
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.24621	0.163923
$672$	0	0
$673$	33.3693	1.28629	0.643146	−	0.765743i	$-0.277629\pi$
$673$	33.3693	1.28629	0.643146	+	0.765743i	$0.277629\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.2462	1.08559	0.542795	−	0.839865i	$-0.317366\pi$
$677$	28.2462	1.08559	0.542795	+	0.839865i	$0.317366\pi$
$678$	0	0
$679$	−8.56155	−0.328562
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.4924	1.24329	0.621644	−	0.783300i	$-0.286465\pi$
$683$	32.4924	1.24329	0.621644	+	0.783300i	$0.286465\pi$
$684$	0	0
$685$	−1.30019	−0.0496776
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13.2614	0.505218
$690$	0	0
$691$	−28.1771	−1.07191	−0.535953	−	0.844248i	$-0.680048\pi$
$691$	−28.1771	−1.07191	−0.535953	+	0.844248i	$0.680048\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.87689	0.109127
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−36.1080	−1.36378	−0.681889	−	0.731455i	$-0.738841\pi$
$701$	−36.1080	−1.36378	−0.681889	+	0.731455i	$0.738841\pi$
$702$	0	0
$703$	−23.3693	−0.881390
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.3693	0.502805
$708$	0	0
$709$	3.30019	0.123941	0.0619706	−	0.998078i	$-0.480262\pi$
$709$	3.30019	0.123941	0.0619706	+	0.998078i	$0.480262\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−15.1922	−0.568954
$714$	0	0
$715$	1.75379	0.0655880
$716$	0	0
$717$	0	0
$718$	0	0
$719$	41.3002	1.54024	0.770119	−	0.637901i	$-0.220197\pi$
$719$	41.3002	1.54024	0.770119	+	0.637901i	$0.220197\pi$
$720$	0	0
$721$	−16.4924	−0.614210
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.36932	−0.347968
$726$	0	0
$727$	3.19224	0.118393	0.0591967	−	0.998246i	$-0.481146\pi$
$727$	3.19224	0.118393	0.0591967	+	0.998246i	$0.481146\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−23.1231	−0.854071	−0.427036	−	0.904235i	$-0.640442\pi$
$733$	−23.1231	−0.854071	−0.427036	+	0.904235i	$0.640442\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.56155	0.0943560
$738$	0	0
$739$	−20.6307	−0.758912	−0.379456	−	0.925210i	$-0.623889\pi$
$739$	−20.6307	−0.758912	−0.379456	+	0.925210i	$0.623889\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	−5.61553	−0.205737
$746$	0	0
$747$	0	0
$748$	0	0
$749$	17.1231	0.625665
$750$	0	0
$751$	−37.3002	−1.36110	−0.680552	−	0.732700i	$-0.738260\pi$
$751$	−37.3002	−1.36110	−0.680552	+	0.732700i	$0.738260\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.87689	−0.104701
$756$	0	0
$757$	−26.9848	−0.980781	−0.490390	−	0.871503i	$-0.663146\pi$
$757$	−26.9848	−0.980781	−0.490390	+	0.871503i	$0.663146\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.1080	1.45391	0.726956	−	0.686684i	$-0.240934\pi$
$761$	40.1080	1.45391	0.726956	+	0.686684i	$0.240934\pi$
$762$	0	0
$763$	9.36932	0.339192
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.9848	0.613287
$768$	0	0
$769$	−18.6307	−0.671840	−0.335920	−	0.941891i	$-0.609047\pi$
$769$	−18.6307	−0.671840	−0.335920	+	0.941891i	$0.609047\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13.5076	−0.485834	−0.242917	−	0.970047i	$-0.578104\pi$
$773$	−13.5076	−0.485834	−0.242917	+	0.970047i	$0.578104\pi$
$774$	0	0
$775$	49.4773	1.77728
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−51.2311	−1.83554
$780$	0	0
$781$	3.68466	0.131847
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−9.30019	−0.331938
$786$	0	0
$787$	−10.8769	−0.387719	−0.193860	−	0.981029i	$-0.562101\pi$
$787$	−10.8769	−0.387719	−0.193860	+	0.981029i	$0.562101\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.68466	0.202123
$792$	0	0
$793$	−13.2614	−0.470925
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.0691	−0.427511	−0.213755	−	0.976887i	$-0.568570\pi$
$797$	−12.0691	−0.427511	−0.213755	+	0.976887i	$0.568570\pi$
$798$	0	0
$799$	12.4924	0.441950
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.24621	0.149846
$804$	0	0
$805$	0.807764	0.0284699
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20.7386	0.729132	0.364566	−	0.931178i	$-0.381217\pi$
$809$	20.7386	0.729132	0.364566	+	0.931178i	$0.381217\pi$
$810$	0	0
$811$	−5.12311	−0.179897	−0.0899483	−	0.995946i	$-0.528670\pi$
$811$	−5.12311	−0.179897	−0.0899483	+	0.995946i	$0.528670\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.26137	−0.324412
$816$	0	0
$817$	−20.4924	−0.716939
$818$	0	0
$819$	0	0
$820$	0	0
$821$	36.2462	1.26500	0.632501	−	0.774560i	$-0.282029\pi$
$821$	36.2462	1.26500	0.632501	+	0.774560i	$0.282029\pi$
$822$	0	0
$823$	−52.6695	−1.83594	−0.917972	−	0.396646i	$-0.870174\pi$
$823$	−52.6695	−1.83594	−0.917972	+	0.396646i	$0.870174\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−47.2311	−1.64238	−0.821192	−	0.570651i	$-0.806691\pi$
$827$	−47.2311	−1.64238	−0.821192	+	0.570651i	$0.806691\pi$
$828$	0	0
$829$	−6.17708	−0.214539	−0.107269	−	0.994230i	$-0.534211\pi$
$829$	−6.17708	−0.214539	−0.107269	+	0.994230i	$0.534211\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	−4.13826	−0.143210
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.68466	−0.265304	−0.132652	−	0.991163i	$-0.542349\pi$
$839$	−7.68466	−0.265304	−0.132652	+	0.991163i	$0.542349\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.82292	0.0627103
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.56155	0.224927
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−32.8769	−1.12305	−0.561527	−	0.827459i	$-0.689786\pi$
$857$	−32.8769	−1.12305	−0.561527	+	0.827459i	$0.689786\pi$
$858$	0	0
$859$	36.8078	1.25586	0.627932	−	0.778268i	$-0.283901\pi$
$859$	36.8078	1.25586	0.627932	+	0.778268i	$0.283901\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−51.2311	−1.74393	−0.871963	−	0.489572i	$-0.837153\pi$
$863$	−51.2311	−1.74393	−0.871963	+	0.489572i	$0.837153\pi$
$864$	0	0
$865$	−12.9848	−0.441498
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5.12311	−0.173789
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.43845	−0.183853
$876$	0	0
$877$	−44.2462	−1.49409	−0.747044	−	0.664774i	$-0.768528\pi$
$877$	−44.2462	−1.49409	−0.747044	+	0.664774i	$0.768528\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	19.9309	0.671488	0.335744	−	0.941953i	$-0.391012\pi$
$881$	19.9309	0.671488	0.335744	+	0.941953i	$0.391012\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.3693	1.05328	0.526639	−	0.850089i	$-0.323452\pi$
$887$	31.3693	1.05328	0.526639	+	0.850089i	$0.323452\pi$
$888$	0	0
$889$	5.12311	0.171823
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−32.0000	−1.07084
$894$	0	0
$895$	−11.6847	−0.390575
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−21.1231	−0.704495
$900$	0	0
$901$	8.49242	0.282924
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.45360	−0.148043
$906$	0	0
$907$	1.75379	0.0582336	0.0291168	−	0.999576i	$-0.490731\pi$
$907$	1.75379	0.0582336	0.0291168	+	0.999576i	$0.490731\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.1231	0.433363
$918$	0	0
$919$	−38.7386	−1.27787	−0.638935	−	0.769261i	$-0.720625\pi$
$919$	−38.7386	−1.27787	−0.638935	+	0.769261i	$0.720625\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−11.5076	−0.378777
$924$	0	0
$925$	−21.3693	−0.702619
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	−5.12311	−0.167903
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.12311	0.0367295
$936$	0	0
$937$	−12.8769	−0.420670	−0.210335	−	0.977629i	$-0.567455\pi$
$937$	−12.8769	−0.420670	−0.210335	+	0.977629i	$0.567455\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4.73863	−0.154475	−0.0772375	−	0.997013i	$-0.524610\pi$
$941$	−4.73863	−0.154475	−0.0772375	+	0.997013i	$0.524610\pi$
$942$	0	0
$943$	14.3845	0.468423
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.9309	−0.582675	−0.291337	−	0.956620i	$-0.594100\pi$
$947$	−17.9309	−0.582675	−0.291337	+	0.956620i	$0.594100\pi$
$948$	0	0
$949$	−13.2614	−0.430482
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54.3542	1.76070	0.880352	−	0.474321i	$-0.157306\pi$
$953$	54.3542	1.76070	0.880352	+	0.474321i	$0.157306\pi$
$954$	0	0
$955$	−15.1922	−0.491609
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.31534	−0.0747663
$960$	0	0
$961$	80.5464	2.59827
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.12311	−0.0361540
$966$	0	0
$967$	−37.1231	−1.19380	−0.596899	−	0.802316i	$-0.703601\pi$
$967$	−37.1231	−1.19380	−0.596899	+	0.802316i	$0.703601\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	47.0540	1.51003	0.755017	−	0.655705i	$-0.227629\pi$
$971$	47.0540	1.51003	0.755017	+	0.655705i	$0.227629\pi$
$972$	0	0
$973$	5.12311	0.164239
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.42329	−0.141514	−0.0707568	−	0.997494i	$-0.522541\pi$
$977$	−4.42329	−0.141514	−0.0707568	+	0.997494i	$0.522541\pi$
$978$	0	0
$979$	−12.5616	−0.401469
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−47.6847	−1.52090	−0.760452	−	0.649394i	$-0.775023\pi$
$983$	−47.6847	−1.52090	−0.760452	+	0.649394i	$0.775023\pi$
$984$	0	0
$985$	−11.3693	−0.362257
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.75379	0.182960
$990$	0	0
$991$	−3.50758	−0.111422	−0.0557109	−	0.998447i	$-0.517743\pi$
$991$	−3.50758	−0.111422	−0.0557109	+	0.998447i	$0.517743\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	13.3693	0.423411	0.211705	−	0.977334i	$-0.432098\pi$
$997$	13.3693	0.423411	0.211705	+	0.977334i	$0.432098\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5544.2.a.bf.1.1		2
3.2	odd	2		616.2.a.f.1.1	✓	2
12.11	even	2		1232.2.a.o.1.2		2
21.20	even	2		4312.2.a.t.1.2		2
24.5	odd	2		4928.2.a.bs.1.2		2
24.11	even	2		4928.2.a.bo.1.1		2
33.32	even	2		6776.2.a.l.1.1		2
84.83	odd	2		8624.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.a.f.1.1	✓	2	3.2	odd	2
1232.2.a.o.1.2		2	12.11	even	2
4312.2.a.t.1.2		2	21.20	even	2
4928.2.a.bo.1.1		2	24.11	even	2
4928.2.a.bs.1.2		2	24.5	odd	2
5544.2.a.bf.1.1		2	1.1	even	1	trivial
6776.2.a.l.1.1		2	33.32	even	2
8624.2.a.bi.1.1		2	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.561553 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-0.561553 q^{5} -1.00000 q^{7} -1.00000 q^{11} +3.12311 q^{13} +2.00000 q^{17} -5.12311 q^{19} +1.43845 q^{23} -4.68466 q^{25} +2.00000 q^{29} -10.5616 q^{31} +0.561553 q^{35} +4.56155 q^{37} +10.0000 q^{41} +4.00000 q^{43} +6.24621 q^{47} +1.00000 q^{49} +4.24621 q^{53} +0.561553 q^{55} +5.43845 q^{59} -4.24621 q^{61} -1.75379 q^{65} -2.56155 q^{67} -3.68466 q^{71} -4.24621 q^{73} +1.00000 q^{77} +5.12311 q^{79} +8.00000 q^{83} -1.12311 q^{85} +12.5616 q^{89} -3.12311 q^{91} +2.87689 q^{95} +8.56155 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 3 * q^5 - 2 * q^7	\( 2 q + 3 q^{5} - 2 q^{7} - 2 q^{11} - 2 q^{13} + 4 q^{17} - 2 q^{19} + 7 q^{23} + 3 q^{25} + 4 q^{29} - 17 q^{31} - 3 q^{35} + 5 q^{37} + 20 q^{41} + 8 q^{43} - 4 q^{47} + 2 q^{49} - 8 q^{53} - 3 q^{55} + 15 q^{59} + 8 q^{61} - 20 q^{65} - q^{67} + 5 q^{71} + 8 q^{73} + 2 q^{77} + 2 q^{79} + 16 q^{83} + 6 q^{85} + 21 q^{89} + 2 q^{91} + 14 q^{95} + 13 q^{97}+O(q^{100}) \) 2 * q + 3 * q^5 - 2 * q^7 - 2 * q^11 - 2 * q^13 + 4 * q^17 - 2 * q^19 + 7 * q^23 + 3 * q^25 + 4 * q^29 - 17 * q^31 - 3 * q^35 + 5 * q^37 + 20 * q^41 + 8 * q^43 - 4 * q^47 + 2 * q^49 - 8 * q^53 - 3 * q^55 + 15 * q^59 + 8 * q^61 - 20 * q^65 - q^67 + 5 * q^71 + 8 * q^73 + 2 * q^77 + 2 * q^79 + 16 * q^83 + 6 * q^85 + 21 * q^89 + 2 * q^91 + 14 * q^95 + 13 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more