LMFDB - Embedded newform 4140.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.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:	$33.0580664368$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1380)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	4140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.56155 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +2.56155 q^{7} -5.12311 q^{11} +2.00000 q^{13} +0.561553 q^{17} -2.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} +5.68466 q^{29} -7.68466 q^{31} -2.56155 q^{35} +6.56155 q^{37} -9.68466 q^{41} -4.00000 q^{47} -0.438447 q^{49} -3.43845 q^{53} +5.12311 q^{55} +0.561553 q^{59} +0.876894 q^{61} -2.00000 q^{65} -3.68466 q^{67} -4.56155 q^{71} +12.2462 q^{73} -13.1231 q^{77} -11.1231 q^{79} -0.315342 q^{83} -0.561553 q^{85} -3.12311 q^{89} +5.12311 q^{91} +2.00000 q^{95} +2.87689 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + q^7	$ 2 q - 2 q^{5} + q^{7} - 2 q^{11} + 4 q^{13} - 3 q^{17} - 4 q^{19} - 2 q^{23} + 2 q^{25} - q^{29} - 3 q^{31} - q^{35} + 9 q^{37} - 7 q^{41} - 8 q^{47} - 5 q^{49} - 11 q^{53} + 2 q^{55} - 3 q^{59} + 10 q^{61} - 4 q^{65} + 5 q^{67} - 5 q^{71} + 8 q^{73} - 18 q^{77} - 14 q^{79} - 13 q^{83} + 3 q^{85} + 2 q^{89} + 2 q^{91} + 4 q^{95} + 14 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + q^7 - 2 * q^11 + 4 * q^13 - 3 * q^17 - 4 * q^19 - 2 * q^23 + 2 * q^25 - q^29 - 3 * q^31 - q^35 + 9 * q^37 - 7 * q^41 - 8 * q^47 - 5 * q^49 - 11 * q^53 + 2 * q^55 - 3 * q^59 + 10 * q^61 - 4 * q^65 + 5 * q^67 - 5 * q^71 + 8 * q^73 - 18 * q^77 - 14 * q^79 - 13 * q^83 + 3 * q^85 + 2 * q^89 + 2 * q^91 + 4 * q^95 + 14 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	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
$9$	0	0
$10$	0	0
$11$	−5.12311	−1.54467	−0.772337	−	0.635213i	$-0.780912\pi$
$11$	−5.12311	−1.54467	−0.772337	+	0.635213i	$0.780912\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	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
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	−7.68466	−1.38021	−0.690103	−	0.723711i	$-0.742435\pi$
$31$	−7.68466	−1.38021	−0.690103	+	0.723711i	$0.742435\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.56155	−0.432981
$36$	0	0
$37$	6.56155	1.07871	0.539356	−	0.842078i	$-0.318668\pi$
$37$	6.56155	1.07871	0.539356	+	0.842078i	$0.318668\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.68466	−1.51249	−0.756245	−	0.654289i	$-0.772968\pi$
$41$	−9.68466	−1.51249	−0.756245	+	0.654289i	$0.772968\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.43845	−0.472307	−0.236154	−	0.971716i	$-0.575887\pi$
$53$	−3.43845	−0.472307	−0.236154	+	0.971716i	$0.575887\pi$
$54$	0	0
$55$	5.12311	0.690799
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.561553	0.0731079	0.0365540	−	0.999332i	$-0.488362\pi$
$59$	0.561553	0.0731079	0.0365540	+	0.999332i	$0.488362\pi$
$60$	0	0
$61$	0.876894	0.112275	0.0561374	−	0.998423i	$-0.482122\pi$
$61$	0.876894	0.112275	0.0561374	+	0.998423i	$0.482122\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−3.68466	−0.450153	−0.225076	−	0.974341i	$-0.572263\pi$
$67$	−3.68466	−0.450153	−0.225076	+	0.974341i	$0.572263\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.56155	−0.541357	−0.270678	−	0.962670i	$-0.587248\pi$
$71$	−4.56155	−0.541357	−0.270678	+	0.962670i	$0.587248\pi$
$72$	0	0
$73$	12.2462	1.43331	0.716655	−	0.697428i	$-0.245672\pi$
$73$	12.2462	1.43331	0.716655	+	0.697428i	$0.245672\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13.1231	−1.49552
$78$	0	0
$79$	−11.1231	−1.25145	−0.625724	−	0.780045i	$-0.715196\pi$
$79$	−11.1231	−1.25145	−0.625724	+	0.780045i	$0.715196\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.315342	−0.0346132	−0.0173066	−	0.999850i	$-0.505509\pi$
$83$	−0.315342	−0.0346132	−0.0173066	+	0.999850i	$0.505509\pi$
$84$	0	0
$85$	−0.561553	−0.0609090
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.12311	−0.331049	−0.165524	−	0.986206i	$-0.552932\pi$
$89$	−3.12311	−0.331049	−0.165524	+	0.986206i	$0.552932\pi$
$90$	0	0
$91$	5.12311	0.537047
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	2.87689	0.292104	0.146052	−	0.989277i	$-0.453343\pi$
$97$	2.87689	0.292104	0.146052	+	0.989277i	$0.453343\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.8078	−1.87144	−0.935721	−	0.352740i	$-0.885250\pi$
$101$	−18.8078	−1.87144	−0.935721	+	0.352740i	$0.885250\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.315342	−0.0304852	−0.0152426	−	0.999884i	$-0.504852\pi$
$107$	−0.315342	−0.0304852	−0.0152426	+	0.999884i	$0.504852\pi$
$108$	0	0
$109$	−16.2462	−1.55610	−0.778052	−	0.628199i	$-0.783792\pi$
$109$	−16.2462	−1.55610	−0.778052	+	0.628199i	$0.783792\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.43845	−0.323462	−0.161731	−	0.986835i	$-0.551708\pi$
$113$	−3.43845	−0.323462	−0.161731	+	0.986835i	$0.551708\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.43845	0.131862
$120$	0	0
$121$	15.2462	1.38602
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.2462	−0.909204	−0.454602	−	0.890695i	$-0.650219\pi$
$127$	−10.2462	−0.909204	−0.454602	+	0.890695i	$0.650219\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.3693	1.16808	0.584041	−	0.811724i	$-0.301471\pi$
$131$	13.3693	1.16808	0.584041	+	0.811724i	$0.301471\pi$
$132$	0	0
$133$	−5.12311	−0.444230
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.4924	0.896428	0.448214	−	0.893926i	$-0.352060\pi$
$137$	10.4924	0.896428	0.448214	+	0.893926i	$0.352060\pi$
$138$	0	0
$139$	9.93087	0.842325	0.421163	−	0.906985i	$-0.361622\pi$
$139$	9.93087	0.842325	0.421163	+	0.906985i	$0.361622\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−10.2462	−0.856831
$144$	0	0
$145$	−5.68466	−0.472085
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.3693	−1.42295	−0.711475	−	0.702711i	$-0.751972\pi$
$149$	−17.3693	−1.42295	−0.711475	+	0.702711i	$0.751972\pi$
$150$	0	0
$151$	−10.2462	−0.833825	−0.416912	−	0.908947i	$-0.636888\pi$
$151$	−10.2462	−0.833825	−0.416912	+	0.908947i	$0.636888\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.68466	0.617247
$156$	0	0
$157$	17.9309	1.43104	0.715520	−	0.698593i	$-0.246190\pi$
$157$	17.9309	1.43104	0.715520	+	0.698593i	$0.246190\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.56155	−0.201879
$162$	0	0
$163$	3.36932	0.263905	0.131953	−	0.991256i	$-0.457875\pi$
$163$	3.36932	0.263905	0.131953	+	0.991256i	$0.457875\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.24621	0.170776	0.0853881	−	0.996348i	$-0.472787\pi$
$173$	2.24621	0.170776	0.0853881	+	0.996348i	$0.472787\pi$
$174$	0	0
$175$	2.56155	0.193635
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.12311	−0.233432	−0.116716	−	0.993165i	$-0.537237\pi$
$179$	−3.12311	−0.233432	−0.116716	+	0.993165i	$0.537237\pi$
$180$	0	0
$181$	7.12311	0.529456	0.264728	−	0.964323i	$-0.414718\pi$
$181$	7.12311	0.529456	0.264728	+	0.964323i	$0.414718\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.56155	−0.482415
$186$	0	0
$187$	−2.87689	−0.210379
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.4924	1.48278	0.741390	−	0.671075i	$-0.234167\pi$
$191$	20.4924	1.48278	0.741390	+	0.671075i	$0.234167\pi$
$192$	0	0
$193$	−4.87689	−0.351047	−0.175523	−	0.984475i	$-0.556162\pi$
$193$	−4.87689	−0.351047	−0.175523	+	0.984475i	$0.556162\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.3693	−1.66499	−0.832497	−	0.554029i	$-0.813090\pi$
$197$	−23.3693	−1.66499	−0.832497	+	0.554029i	$0.813090\pi$
$198$	0	0
$199$	−3.75379	−0.266099	−0.133050	−	0.991109i	$-0.542477\pi$
$199$	−3.75379	−0.266099	−0.133050	+	0.991109i	$0.542477\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	14.5616	1.02202
$204$	0	0
$205$	9.68466	0.676406
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.2462	0.708745
$210$	0	0
$211$	−21.9309	−1.50978	−0.754892	−	0.655850i	$-0.772311\pi$
$211$	−21.9309	−1.50978	−0.754892	+	0.655850i	$0.772311\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−19.6847	−1.33628
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.12311	0.0755483
$222$	0	0
$223$	−21.1231	−1.41451	−0.707254	−	0.706960i	$-0.750066\pi$
$223$	−21.1231	−1.41451	−0.707254	+	0.706960i	$0.750066\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−25.3693	−1.67645	−0.838226	−	0.545323i	$-0.816407\pi$
$229$	−25.3693	−1.67645	−0.838226	+	0.545323i	$0.816407\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	0	0
$238$	0	0
$239$	18.8078	1.21657	0.608287	−	0.793717i	$-0.291857\pi$
$239$	18.8078	1.21657	0.608287	+	0.793717i	$0.291857\pi$
$240$	0	0
$241$	−14.4924	−0.933539	−0.466769	−	0.884379i	$-0.654582\pi$
$241$	−14.4924	−0.933539	−0.466769	+	0.884379i	$0.654582\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.438447	0.0280114
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−11.3693	−0.717625	−0.358812	−	0.933410i	$-0.616818\pi$
$251$	−11.3693	−0.717625	−0.358812	+	0.933410i	$0.616818\pi$
$252$	0	0
$253$	5.12311	0.322087
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.4924	−1.02877	−0.514385	−	0.857560i	$-0.671980\pi$
$257$	−16.4924	−1.02877	−0.514385	+	0.857560i	$0.671980\pi$
$258$	0	0
$259$	16.8078	1.04438
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.93087	−0.365713	−0.182857	−	0.983140i	$-0.558534\pi$
$263$	−5.93087	−0.365713	−0.182857	+	0.983140i	$0.558534\pi$
$264$	0	0
$265$	3.43845	0.211222
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.5616	1.25366	0.626830	−	0.779156i	$-0.284352\pi$
$269$	20.5616	1.25366	0.626830	+	0.779156i	$0.284352\pi$
$270$	0	0
$271$	5.93087	0.360275	0.180137	−	0.983641i	$-0.442346\pi$
$271$	5.93087	0.360275	0.180137	+	0.983641i	$0.442346\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.12311	−0.308935
$276$	0	0
$277$	8.24621	0.495467	0.247733	−	0.968828i	$-0.420314\pi$
$277$	8.24621	0.495467	0.247733	+	0.968828i	$0.420314\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.2462	0.969168	0.484584	−	0.874745i	$-0.338971\pi$
$281$	16.2462	0.969168	0.484584	+	0.874745i	$0.338971\pi$
$282$	0	0
$283$	−18.5616	−1.10337	−0.551685	−	0.834053i	$-0.686015\pi$
$283$	−18.5616	−1.10337	−0.551685	+	0.834053i	$0.686015\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.8078	−1.46436
$288$	0	0
$289$	−16.6847	−0.981450
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.9309	1.16437	0.582187	−	0.813055i	$-0.302197\pi$
$293$	19.9309	1.16437	0.582187	+	0.813055i	$0.302197\pi$
$294$	0	0
$295$	−0.561553	−0.0326949
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.876894	−0.0502108
$306$	0	0
$307$	9.12311	0.520683	0.260342	−	0.965517i	$-0.416165\pi$
$307$	9.12311	0.520683	0.260342	+	0.965517i	$0.416165\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.3693	−0.984924	−0.492462	−	0.870334i	$-0.663903\pi$
$311$	−17.3693	−0.984924	−0.492462	+	0.870334i	$0.663903\pi$
$312$	0	0
$313$	5.43845	0.307399	0.153700	−	0.988118i	$-0.450881\pi$
$313$	5.43845	0.307399	0.153700	+	0.988118i	$0.450881\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.00000	0.224662	0.112331	−	0.993671i	$-0.464168\pi$
$317$	4.00000	0.224662	0.112331	+	0.993671i	$0.464168\pi$
$318$	0	0
$319$	−29.1231	−1.63058
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.12311	−0.0624913
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−10.2462	−0.564892
$330$	0	0
$331$	2.56155	0.140796	0.0703978	−	0.997519i	$-0.477573\pi$
$331$	2.56155	0.140796	0.0703978	+	0.997519i	$0.477573\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.68466	0.201314
$336$	0	0
$337$	−21.1231	−1.15065	−0.575324	−	0.817925i	$-0.695124\pi$
$337$	−21.1231	−1.15065	−0.575324	+	0.817925i	$0.695124\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	39.3693	2.13197
$342$	0	0
$343$	−19.0540	−1.02882
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	0	0
$349$	−1.68466	−0.0901777	−0.0450888	−	0.998983i	$-0.514357\pi$
$349$	−1.68466	−0.0901777	−0.0450888	+	0.998983i	$0.514357\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−34.7386	−1.84895	−0.924475	−	0.381242i	$-0.875497\pi$
$353$	−34.7386	−1.84895	−0.924475	+	0.381242i	$0.875497\pi$
$354$	0	0
$355$	4.56155	0.242102
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.24621	−0.329662	−0.164831	−	0.986322i	$-0.552708\pi$
$359$	−6.24621	−0.329662	−0.164831	+	0.986322i	$0.552708\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−12.2462	−0.640996
$366$	0	0
$367$	−6.56155	−0.342510	−0.171255	−	0.985227i	$-0.554782\pi$
$367$	−6.56155	−0.342510	−0.171255	+	0.985227i	$0.554782\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.80776	−0.457276
$372$	0	0
$373$	19.3693	1.00291	0.501453	−	0.865185i	$-0.332799\pi$
$373$	19.3693	1.00291	0.501453	+	0.865185i	$0.332799\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.3693	0.585550
$378$	0	0
$379$	6.63068	0.340595	0.170298	−	0.985393i	$-0.445527\pi$
$379$	6.63068	0.340595	0.170298	+	0.985393i	$0.445527\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.5616	0.744061	0.372030	−	0.928221i	$-0.378662\pi$
$383$	14.5616	0.744061	0.372030	+	0.928221i	$0.378662\pi$
$384$	0	0
$385$	13.1231	0.668815
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−0.561553	−0.0283989
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.1231	0.559664
$396$	0	0
$397$	16.8769	0.847027	0.423514	−	0.905890i	$-0.360797\pi$
$397$	16.8769	0.847027	0.423514	+	0.905890i	$0.360797\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	−15.3693	−0.765600
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−33.6155	−1.66626
$408$	0	0
$409$	−8.56155	−0.423342	−0.211671	−	0.977341i	$-0.567890\pi$
$409$	−8.56155	−0.423342	−0.211671	+	0.977341i	$0.567890\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.43845	0.0707814
$414$	0	0
$415$	0.315342	0.0154795
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13.6155	0.665162	0.332581	−	0.943075i	$-0.392081\pi$
$419$	13.6155	0.665162	0.332581	+	0.943075i	$0.392081\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.561553	0.0272393
$426$	0	0
$427$	2.24621	0.108702
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.24621	−0.108196	−0.0540981	−	0.998536i	$-0.517228\pi$
$431$	−2.24621	−0.108196	−0.0540981	+	0.998536i	$0.517228\pi$
$432$	0	0
$433$	−21.9309	−1.05393	−0.526965	−	0.849887i	$-0.676670\pi$
$433$	−21.9309	−1.05393	−0.526965	+	0.849887i	$0.676670\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.4924	0.593533	0.296766	−	0.954950i	$-0.404092\pi$
$443$	12.4924	0.593533	0.296766	+	0.954950i	$0.404092\pi$
$444$	0	0
$445$	3.12311	0.148049
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−24.5616	−1.15913	−0.579566	−	0.814926i	$-0.696778\pi$
$449$	−24.5616	−1.15913	−0.579566	+	0.814926i	$0.696778\pi$
$450$	0	0
$451$	49.6155	2.33630
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.12311	−0.240175
$456$	0	0
$457$	−19.0540	−0.891307	−0.445654	−	0.895205i	$-0.647029\pi$
$457$	−19.0540	−0.891307	−0.445654	+	0.895205i	$0.647029\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.7386	1.33849	0.669246	−	0.743041i	$-0.266617\pi$
$461$	28.7386	1.33849	0.669246	+	0.743041i	$0.266617\pi$
$462$	0	0
$463$	−39.3693	−1.82965	−0.914824	−	0.403854i	$-0.867671\pi$
$463$	−39.3693	−1.82965	−0.914824	+	0.403854i	$0.867671\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16.3153	−0.754984	−0.377492	−	0.926013i	$-0.623213\pi$
$467$	−16.3153	−0.754984	−0.377492	+	0.926013i	$0.623213\pi$
$468$	0	0
$469$	−9.43845	−0.435827
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32.4924	1.48462	0.742308	−	0.670058i	$-0.233731\pi$
$479$	32.4924	1.48462	0.742308	+	0.670058i	$0.233731\pi$
$480$	0	0
$481$	13.1231	0.598362
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.87689	−0.130633
$486$	0	0
$487$	−26.2462	−1.18933	−0.594665	−	0.803974i	$-0.702715\pi$
$487$	−26.2462	−1.18933	−0.594665	+	0.803974i	$0.702715\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	20.4233	0.921690	0.460845	−	0.887481i	$-0.347546\pi$
$491$	20.4233	0.921690	0.460845	+	0.887481i	$0.347546\pi$
$492$	0	0
$493$	3.19224	0.143771
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−11.6847	−0.524129
$498$	0	0
$499$	4.17708	0.186992	0.0934959	−	0.995620i	$-0.470196\pi$
$499$	4.17708	0.186992	0.0934959	+	0.995620i	$0.470196\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.31534	0.192412	0.0962058	−	0.995361i	$-0.469329\pi$
$503$	4.31534	0.192412	0.0962058	+	0.995361i	$0.469329\pi$
$504$	0	0
$505$	18.8078	0.836935
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−28.7386	−1.27382	−0.636909	−	0.770939i	$-0.719787\pi$
$509$	−28.7386	−1.27382	−0.636909	+	0.770939i	$0.719787\pi$
$510$	0	0
$511$	31.3693	1.38770
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	20.4924	0.901256
$518$	0	0
$519$	0	0
$520$	0	0
$521$	42.4924	1.86163	0.930813	−	0.365495i	$-0.119100\pi$
$521$	42.4924	1.86163	0.930813	+	0.365495i	$0.119100\pi$
$522$	0	0
$523$	44.9848	1.96705	0.983525	−	0.180772i	$-0.0578597\pi$
$523$	44.9848	1.96705	0.983525	+	0.180772i	$0.0578597\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.31534	−0.187979
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−19.3693	−0.838978
$534$	0	0
$535$	0.315342	0.0136334
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.24621	0.0967512
$540$	0	0
$541$	44.2462	1.90229	0.951147	−	0.308740i	$-0.0999072\pi$
$541$	44.2462	1.90229	0.951147	+	0.308740i	$0.0999072\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16.2462	0.695911
$546$	0	0
$547$	−6.24621	−0.267069	−0.133534	−	0.991044i	$-0.542633\pi$
$547$	−6.24621	−0.267069	−0.133534	+	0.991044i	$0.542633\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.3693	−0.484349
$552$	0	0
$553$	−28.4924	−1.21162
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.0540	1.06157	0.530786	−	0.847506i	$-0.321897\pi$
$557$	25.0540	1.06157	0.530786	+	0.847506i	$0.321897\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15.6847	−0.661030	−0.330515	−	0.943801i	$-0.607222\pi$
$563$	−15.6847	−0.661030	−0.330515	+	0.943801i	$0.607222\pi$
$564$	0	0
$565$	3.43845	0.144657
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.50758	0.230890	0.115445	−	0.993314i	$-0.463171\pi$
$569$	5.50758	0.230890	0.115445	+	0.993314i	$0.463171\pi$
$570$	0	0
$571$	13.3693	0.559488	0.279744	−	0.960075i	$-0.409750\pi$
$571$	13.3693	0.559488	0.279744	+	0.960075i	$0.409750\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	11.1231	0.463061	0.231530	−	0.972828i	$-0.425627\pi$
$577$	11.1231	0.463061	0.231530	+	0.972828i	$0.425627\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.807764	−0.0335117
$582$	0	0
$583$	17.6155	0.729561
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.6155	−1.22236	−0.611182	−	0.791490i	$-0.709306\pi$
$587$	−29.6155	−1.22236	−0.611182	+	0.791490i	$0.709306\pi$
$588$	0	0
$589$	15.3693	0.633282
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.2462	0.913542	0.456771	−	0.889584i	$-0.349006\pi$
$593$	22.2462	0.913542	0.456771	+	0.889584i	$0.349006\pi$
$594$	0	0
$595$	−1.43845	−0.0589706
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.12311	−0.127607	−0.0638033	−	0.997962i	$-0.520323\pi$
$599$	−3.12311	−0.127607	−0.0638033	+	0.997962i	$0.520323\pi$
$600$	0	0
$601$	47.7926	1.94950	0.974751	−	0.223296i	$-0.0716818\pi$
$601$	47.7926	1.94950	0.974751	+	0.223296i	$0.0716818\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−15.2462	−0.619847
$606$	0	0
$607$	−46.7386	−1.89706	−0.948531	−	0.316683i	$-0.897431\pi$
$607$	−46.7386	−1.89706	−0.948531	+	0.316683i	$0.897431\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	31.8617	1.28688	0.643442	−	0.765495i	$-0.277506\pi$
$613$	31.8617	1.28688	0.643442	+	0.765495i	$0.277506\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.4233	1.30531	0.652656	−	0.757654i	$-0.273655\pi$
$617$	32.4233	1.30531	0.652656	+	0.757654i	$0.273655\pi$
$618$	0	0
$619$	14.6307	0.588057	0.294028	−	0.955797i	$-0.405004\pi$
$619$	14.6307	0.588057	0.294028	+	0.955797i	$0.405004\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.00000	−0.320513
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.68466	0.146917
$630$	0	0
$631$	33.3693	1.32841	0.664206	−	0.747550i	$-0.268770\pi$
$631$	33.3693	1.32841	0.664206	+	0.747550i	$0.268770\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.2462	0.406608
$636$	0	0
$637$	−0.876894	−0.0347438
$638$	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
$643$	−37.9309	−1.49585	−0.747924	−	0.663785i	$-0.768949\pi$
$643$	−37.9309	−1.49585	−0.747924	+	0.663785i	$0.768949\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	39.8617	1.56713	0.783563	−	0.621312i	$-0.213400\pi$
$647$	39.8617	1.56713	0.783563	+	0.621312i	$0.213400\pi$
$648$	0	0
$649$	−2.87689	−0.112928
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41.6155	1.62854	0.814271	−	0.580485i	$-0.197137\pi$
$653$	41.6155	1.62854	0.814271	+	0.580485i	$0.197137\pi$
$654$	0	0
$655$	−13.3693	−0.522382
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−23.3693	−0.910339	−0.455170	−	0.890405i	$-0.650421\pi$
$659$	−23.3693	−0.910339	−0.455170	+	0.890405i	$0.650421\pi$
$660$	0	0
$661$	−18.9848	−0.738425	−0.369212	−	0.929345i	$-0.620373\pi$
$661$	−18.9848	−0.738425	−0.369212	+	0.929345i	$0.620373\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.12311	0.198666
$666$	0	0
$667$	−5.68466	−0.220111
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.49242	−0.173428
$672$	0	0
$673$	47.1231	1.81646	0.908231	−	0.418469i	$-0.137433\pi$
$673$	47.1231	1.81646	0.908231	+	0.418469i	$0.137433\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.4384	1.36201	0.681005	−	0.732279i	$-0.261543\pi$
$677$	35.4384	1.36201	0.681005	+	0.732279i	$0.261543\pi$
$678$	0	0
$679$	7.36932	0.282808
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30.2462	1.15734	0.578670	−	0.815562i	$-0.303572\pi$
$683$	30.2462	1.15734	0.578670	+	0.815562i	$0.303572\pi$
$684$	0	0
$685$	−10.4924	−0.400895
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.87689	−0.261989
$690$	0	0
$691$	1.26137	0.0479846	0.0239923	−	0.999712i	$-0.492362\pi$
$691$	1.26137	0.0479846	0.0239923	+	0.999712i	$0.492362\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.93087	−0.376699
$696$	0	0
$697$	−5.43845	−0.205996
$698$	0	0
$699$	0	0
$700$	0	0
$701$	39.6155	1.49626	0.748129	−	0.663553i	$-0.230952\pi$
$701$	39.6155	1.49626	0.748129	+	0.663553i	$0.230952\pi$
$702$	0	0
$703$	−13.1231	−0.494947
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−48.1771	−1.81189
$708$	0	0
$709$	13.3693	0.502095	0.251048	−	0.967975i	$-0.419225\pi$
$709$	13.3693	0.502095	0.251048	+	0.967975i	$0.419225\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.68466	0.287793
$714$	0	0
$715$	10.2462	0.383187
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−15.9309	−0.594121	−0.297061	−	0.954859i	$-0.596006\pi$
$719$	−15.9309	−0.594121	−0.297061	+	0.954859i	$0.596006\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.68466	0.211123
$726$	0	0
$727$	44.1771	1.63844	0.819219	−	0.573481i	$-0.194407\pi$
$727$	44.1771	1.63844	0.819219	+	0.573481i	$0.194407\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−12.1771	−0.449771	−0.224885	−	0.974385i	$-0.572201\pi$
$733$	−12.1771	−0.449771	−0.224885	+	0.974385i	$0.572201\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.8769	0.695339
$738$	0	0
$739$	−40.6695	−1.49605	−0.748026	−	0.663669i	$-0.768998\pi$
$739$	−40.6695	−1.49605	−0.748026	+	0.663669i	$0.768998\pi$
$740$	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
$745$	17.3693	0.636363
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−0.807764	−0.0295151
$750$	0	0
$751$	−32.2462	−1.17668	−0.588340	−	0.808613i	$-0.700219\pi$
$751$	−32.2462	−1.17668	−0.588340	+	0.808613i	$0.700219\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10.2462	0.372898
$756$	0	0
$757$	1.43845	0.0522813	0.0261406	−	0.999658i	$-0.491678\pi$
$757$	1.43845	0.0522813	0.0261406	+	0.999658i	$0.491678\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.3002	0.409631	0.204816	−	0.978801i	$-0.434340\pi$
$761$	11.3002	0.409631	0.204816	+	0.978801i	$0.434340\pi$
$762$	0	0
$763$	−41.6155	−1.50658
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.12311	0.0405530
$768$	0	0
$769$	13.3693	0.482110	0.241055	−	0.970511i	$-0.422507\pi$
$769$	13.3693	0.482110	0.241055	+	0.970511i	$0.422507\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−11.7538	−0.422754	−0.211377	−	0.977405i	$-0.567795\pi$
$773$	−11.7538	−0.422754	−0.211377	+	0.977405i	$0.567795\pi$
$774$	0	0
$775$	−7.68466	−0.276041
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.3693	0.693978
$780$	0	0
$781$	23.3693	0.836220
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−17.9309	−0.639980
$786$	0	0
$787$	−26.4233	−0.941889	−0.470944	−	0.882163i	$-0.656087\pi$
$787$	−26.4233	−0.941889	−0.470944	+	0.882163i	$0.656087\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.80776	−0.313168
$792$	0	0
$793$	1.75379	0.0622789
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.3002	−1.10871	−0.554355	−	0.832280i	$-0.687035\pi$
$797$	−31.3002	−1.10871	−0.554355	+	0.832280i	$0.687035\pi$
$798$	0	0
$799$	−2.24621	−0.0794652
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−62.7386	−2.21400
$804$	0	0
$805$	2.56155	0.0902829
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−20.5616	−0.722906	−0.361453	−	0.932390i	$-0.617719\pi$
$809$	−20.5616	−0.722906	−0.361453	+	0.932390i	$0.617719\pi$
$810$	0	0
$811$	16.3153	0.572909	0.286455	−	0.958094i	$-0.407523\pi$
$811$	16.3153	0.572909	0.286455	+	0.958094i	$0.407523\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.36932	−0.118022
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.7538	1.10821	0.554107	−	0.832445i	$-0.313060\pi$
$821$	31.7538	1.10821	0.554107	+	0.832445i	$0.313060\pi$
$822$	0	0
$823$	−46.7386	−1.62921	−0.814603	−	0.580019i	$-0.803045\pi$
$823$	−46.7386	−1.62921	−0.814603	+	0.580019i	$0.803045\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21.4384	−0.745488	−0.372744	−	0.927934i	$-0.621583\pi$
$827$	−21.4384	−0.745488	−0.372744	+	0.927934i	$0.621583\pi$
$828$	0	0
$829$	−45.5464	−1.58189	−0.790946	−	0.611886i	$-0.790411\pi$
$829$	−45.5464	−1.58189	−0.790946	+	0.611886i	$0.790411\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.246211	−0.00853071
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37.1231	−1.28163	−0.640816	−	0.767695i	$-0.721404\pi$
$839$	−37.1231	−1.28163	−0.640816	+	0.767695i	$0.721404\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	39.0540	1.34191
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.56155	−0.224927
$852$	0	0
$853$	−0.738634	−0.0252903	−0.0126452	−	0.999920i	$-0.504025\pi$
$853$	−0.738634	−0.0252903	−0.0126452	+	0.999920i	$0.504025\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39.3693	−1.34483	−0.672415	−	0.740174i	$-0.734743\pi$
$857$	−39.3693	−1.34483	−0.672415	+	0.740174i	$0.734743\pi$
$858$	0	0
$859$	40.8078	1.39234	0.696171	−	0.717876i	$-0.254885\pi$
$859$	40.8078	1.39234	0.696171	+	0.717876i	$0.254885\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.1231	0.719039	0.359519	−	0.933138i	$-0.382941\pi$
$863$	21.1231	0.719039	0.359519	+	0.933138i	$0.382941\pi$
$864$	0	0
$865$	−2.24621	−0.0763735
$866$	0	0
$867$	0	0
$868$	0	0
$869$	56.9848	1.93308
$870$	0	0
$871$	−7.36932	−0.249700
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.56155	−0.0865963
$876$	0	0
$877$	−7.26137	−0.245199	−0.122599	−	0.992456i	$-0.539123\pi$
$877$	−7.26137	−0.245199	−0.122599	+	0.992456i	$0.539123\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−19.6155	−0.660864	−0.330432	−	0.943830i	$-0.607194\pi$
$881$	−19.6155	−0.660864	−0.330432	+	0.943830i	$0.607194\pi$
$882$	0	0
$883$	47.2311	1.58945	0.794726	−	0.606969i	$-0.207615\pi$
$883$	47.2311	1.58945	0.794726	+	0.606969i	$0.207615\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41.6155	1.39731	0.698656	−	0.715457i	$-0.253782\pi$
$887$	41.6155	1.39731	0.698656	+	0.715457i	$0.253782\pi$
$888$	0	0
$889$	−26.2462	−0.880270
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	3.12311	0.104394
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−43.6847	−1.45696
$900$	0	0
$901$	−1.93087	−0.0643266
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.12311	−0.236780
$906$	0	0
$907$	19.5464	0.649027	0.324514	−	0.945881i	$-0.394799\pi$
$907$	19.5464	0.649027	0.324514	+	0.945881i	$0.394799\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−46.2462	−1.53221	−0.766103	−	0.642718i	$-0.777807\pi$
$911$	−46.2462	−1.53221	−0.766103	+	0.642718i	$0.777807\pi$
$912$	0	0
$913$	1.61553	0.0534662
$914$	0	0
$915$	0	0
$916$	0	0
$917$	34.2462	1.13091
$918$	0	0
$919$	−31.1231	−1.02666	−0.513328	−	0.858192i	$-0.671588\pi$
$919$	−31.1231	−1.02666	−0.513328	+	0.858192i	$0.671588\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.12311	−0.300291
$924$	0	0
$925$	6.56155	0.215743
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19.9309	0.653911	0.326955	−	0.945040i	$-0.393977\pi$
$929$	19.9309	0.653911	0.326955	+	0.945040i	$0.393977\pi$
$930$	0	0
$931$	0.876894	0.0287391
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.87689	0.0940845
$936$	0	0
$937$	42.1080	1.37561	0.687803	−	0.725897i	$-0.258575\pi$
$937$	42.1080	1.37561	0.687803	+	0.725897i	$0.258575\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.61553	−0.117863	−0.0589314	−	0.998262i	$-0.518769\pi$
$941$	−3.61553	−0.117863	−0.0589314	+	0.998262i	$0.518769\pi$
$942$	0	0
$943$	9.68466	0.315376
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.4924	1.44581	0.722905	−	0.690948i	$-0.242807\pi$
$947$	44.4924	1.44581	0.722905	+	0.690948i	$0.242807\pi$
$948$	0	0
$949$	24.4924	0.795058
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−30.9848	−1.00370	−0.501849	−	0.864955i	$-0.667347\pi$
$953$	−30.9848	−1.00370	−0.501849	+	0.864955i	$0.667347\pi$
$954$	0	0
$955$	−20.4924	−0.663119
$956$	0	0
$957$	0	0
$958$	0	0
$959$	26.8769	0.867900
$960$	0	0
$961$	28.0540	0.904967
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.87689	0.156993
$966$	0	0
$967$	20.4924	0.658992	0.329496	−	0.944157i	$-0.393121\pi$
$967$	20.4924	0.658992	0.329496	+	0.944157i	$0.393121\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14.2462	−0.457183	−0.228591	−	0.973522i	$-0.573412\pi$
$971$	−14.2462	−0.457183	−0.228591	+	0.973522i	$0.573412\pi$
$972$	0	0
$973$	25.4384	0.815519
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.8078	−0.345771	−0.172886	−	0.984942i	$-0.555309\pi$
$977$	−10.8078	−0.345771	−0.172886	+	0.984942i	$0.555309\pi$
$978$	0	0
$979$	16.0000	0.511362
$980$	0	0
$981$	0	0
$982$	0	0
$983$	52.6695	1.67990	0.839948	−	0.542667i	$-0.182585\pi$
$983$	52.6695	1.67990	0.839948	+	0.542667i	$0.182585\pi$
$984$	0	0
$985$	23.3693	0.744608
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−38.4233	−1.22056	−0.610278	−	0.792187i	$-0.708942\pi$
$991$	−38.4233	−1.22056	−0.610278	+	0.792187i	$0.708942\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.75379	0.119003
$996$	0	0
$997$	−43.1231	−1.36572	−0.682861	−	0.730548i	$-0.739265\pi$
$997$	−43.1231	−1.36572	−0.682861	+	0.730548i	$0.739265\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.a.n.1.2		2
3.2	odd	2		1380.2.a.g.1.2	✓	2
12.11	even	2		5520.2.a.br.1.1		2
15.2	even	4		6900.2.f.p.6349.4		4
15.8	even	4		6900.2.f.p.6349.1		4
15.14	odd	2		6900.2.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.g.1.2	✓	2	3.2	odd	2
4140.2.a.n.1.2		2	1.1	even	1	trivial
5520.2.a.br.1.1		2	12.11	even	2
6900.2.a.u.1.1		2	15.14	odd	2
6900.2.f.p.6349.1		4	15.8	even	4
6900.2.f.p.6349.4		4	15.2	even	4

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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.56155 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +2.56155 q^{7} -5.12311 q^{11} +2.00000 q^{13} +0.561553 q^{17} -2.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} +5.68466 q^{29} -7.68466 q^{31} -2.56155 q^{35} +6.56155 q^{37} -9.68466 q^{41} -4.00000 q^{47} -0.438447 q^{49} -3.43845 q^{53} +5.12311 q^{55} +0.561553 q^{59} +0.876894 q^{61} -2.00000 q^{65} -3.68466 q^{67} -4.56155 q^{71} +12.2462 q^{73} -13.1231 q^{77} -11.1231 q^{79} -0.315342 q^{83} -0.561553 q^{85} -3.12311 q^{89} +5.12311 q^{91} +2.00000 q^{95} +2.87689 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + q^7	\( 2 q - 2 q^{5} + q^{7} - 2 q^{11} + 4 q^{13} - 3 q^{17} - 4 q^{19} - 2 q^{23} + 2 q^{25} - q^{29} - 3 q^{31} - q^{35} + 9 q^{37} - 7 q^{41} - 8 q^{47} - 5 q^{49} - 11 q^{53} + 2 q^{55} - 3 q^{59} + 10 q^{61} - 4 q^{65} + 5 q^{67} - 5 q^{71} + 8 q^{73} - 18 q^{77} - 14 q^{79} - 13 q^{83} + 3 q^{85} + 2 q^{89} + 2 q^{91} + 4 q^{95} + 14 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + q^7 - 2 * q^11 + 4 * q^13 - 3 * q^17 - 4 * q^19 - 2 * q^23 + 2 * q^25 - q^29 - 3 * q^31 - q^35 + 9 * q^37 - 7 * q^41 - 8 * q^47 - 5 * q^49 - 11 * q^53 + 2 * q^55 - 3 * q^59 + 10 * q^61 - 4 * q^65 + 5 * q^67 - 5 * q^71 + 8 * q^73 - 18 * q^77 - 14 * q^79 - 13 * q^83 + 3 * q^85 + 2 * q^89 + 2 * q^91 + 4 * q^95 + 14 * q^97

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