LMFDB - Embedded newform 6552.2.a.bs.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6552,2,Mod(1,6552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6552, 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("6552.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6552.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:	$52.3179834043$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.138892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 2x + 12 $ x^4 - x^3 - 10x^2 + 2x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.52690$ of defining polynomial
Character	$\chi$	$=$	6552.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.152457 q^{5} -1.00000 q^{7} -0.385245 q^{11} +1.00000 q^{13} -7.43905 q^{17} -7.20627 q^{19} -2.90135 q^{23} -4.97676 q^{25} +5.20627 q^{29} +1.76721 q^{31} -0.152457 q^{35} +7.43905 q^{37} -7.05381 q^{41} +2.90135 q^{43} +3.59151 q^{47} +1.00000 q^{49} +10.9502 q^{53} -0.0587335 q^{55} +5.82430 q^{59} +12.5145 q^{61} +0.152457 q^{65} +9.80270 q^{67} -5.82430 q^{71} +3.09865 q^{73} +0.385245 q^{77} +12.6453 q^{79} -11.8964 q^{83} -1.13414 q^{85} +5.59151 q^{89} -1.00000 q^{91} -1.09865 q^{95} +18.9502 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} - 4 q^{7} + 3 q^{11} + 4 q^{13} - 3 q^{17} - 4 q^{19} + 8 q^{23} + 14 q^{25} - 4 q^{29} + 9 q^{31} + 2 q^{35} + 3 q^{37} - 6 q^{41} - 8 q^{43} - 15 q^{47} + 4 q^{49} - 13 q^{53} + 7 q^{55}+ \cdots + 19 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 - 4 * q^7 + 3 * q^11 + 4 * q^13 - 3 * q^17 - 4 * q^19 + 8 * q^23 + 14 * q^25 - 4 * q^29 + 9 * q^31 + 2 * q^35 + 3 * q^37 - 6 * q^41 - 8 * q^43 - 15 * q^47 + 4 * q^49 - 13 * q^53 + 7 * q^55 - 8 * q^59 + 9 * q^61 - 2 * q^65 + 8 * q^71 + 32 * q^73 - 3 * q^77 - q^79 - 13 * q^83 + 17 * q^85 - 7 * q^89 - 4 * q^91 - 24 * q^95 + 19 * q^97

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
$5$	0.152457	0.0681810	0.0340905	−	0.999419i	$-0.489147\pi$
$5$	0.152457	0.0681810	0.0340905	+	0.999419i	$0.489147\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.385245	−0.116156	−0.0580779	−	0.998312i	$-0.518497\pi$
$11$	−0.385245	−0.116156	−0.0580779	+	0.998312i	$0.518497\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.43905	−1.80424	−0.902118	−	0.431490i	$-0.857988\pi$
$17$	−7.43905	−1.80424	−0.902118	+	0.431490i	$0.857988\pi$
$18$	0	0
$19$	−7.20627	−1.65323	−0.826615	−	0.562767i	$-0.809737\pi$
$19$	−7.20627	−1.65323	−0.826615	+	0.562767i	$0.809737\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.90135	−0.604974	−0.302487	−	0.953154i	$-0.597817\pi$
$23$	−2.90135	−0.604974	−0.302487	+	0.953154i	$0.597817\pi$
$24$	0	0
$25$	−4.97676	−0.995351
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.20627	0.966779	0.483390	−	0.875405i	$-0.339405\pi$
$29$	5.20627	0.966779	0.483390	+	0.875405i	$0.339405\pi$
$30$	0	0
$31$	1.76721	0.317401	0.158700	−	0.987327i	$-0.449270\pi$
$31$	1.76721	0.317401	0.158700	+	0.987327i	$0.449270\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.152457	−0.0257700
$36$	0	0
$37$	7.43905	1.22297	0.611486	−	0.791255i	$-0.290572\pi$
$37$	7.43905	1.22297	0.611486	+	0.791255i	$0.290572\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.05381	−1.10162	−0.550810	−	0.834631i	$-0.685681\pi$
$41$	−7.05381	−1.10162	−0.550810	+	0.834631i	$0.685681\pi$
$42$	0	0
$43$	2.90135	0.442452	0.221226	−	0.975223i	$-0.428994\pi$
$43$	2.90135	0.442452	0.221226	+	0.975223i	$0.428994\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.59151	0.523876	0.261938	−	0.965085i	$-0.415638\pi$
$47$	3.59151	0.523876	0.261938	+	0.965085i	$0.415638\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.9502	1.50413	0.752065	−	0.659089i	$-0.229058\pi$
$53$	10.9502	1.50413	0.752065	+	0.659089i	$0.229058\pi$
$54$	0	0
$55$	−0.0587335	−0.00791963
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.82430	0.758259	0.379130	−	0.925344i	$-0.376223\pi$
$59$	5.82430	0.758259	0.379130	+	0.925344i	$0.376223\pi$
$60$	0	0
$61$	12.5145	1.60231	0.801156	−	0.598455i	$-0.204219\pi$
$61$	12.5145	1.60231	0.801156	+	0.598455i	$0.204219\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.152457	0.0189100
$66$	0	0
$67$	9.80270	1.19759	0.598795	−	0.800902i	$-0.295646\pi$
$67$	9.80270	1.19759	0.598795	+	0.800902i	$0.295646\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.82430	−0.691217	−0.345609	−	0.938379i	$-0.612328\pi$
$71$	−5.82430	−0.691217	−0.345609	+	0.938379i	$0.612328\pi$
$72$	0	0
$73$	3.09865	0.362669	0.181335	−	0.983421i	$-0.441958\pi$
$73$	3.09865	0.362669	0.181335	+	0.983421i	$0.441958\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.385245	0.0439028
$78$	0	0
$79$	12.6453	1.42271	0.711355	−	0.702833i	$-0.248082\pi$
$79$	12.6453	1.42271	0.711355	+	0.702833i	$0.248082\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.8964	−1.30580	−0.652901	−	0.757443i	$-0.726448\pi$
$83$	−11.8964	−1.30580	−0.652901	+	0.757443i	$0.726448\pi$
$84$	0	0
$85$	−1.13414	−0.123015
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.59151	0.592699	0.296350	−	0.955080i	$-0.404231\pi$
$89$	5.59151	0.592699	0.296350	+	0.955080i	$0.404231\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.09865	−0.112719
$96$	0	0
$97$	18.9502	1.92410	0.962052	−	0.272865i	$-0.0879711\pi$
$97$	18.9502	1.92410	0.962052	+	0.272865i	$0.0879711\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.3803	−1.13238	−0.566192	−	0.824273i	$-0.691584\pi$
$101$	−11.3803	−1.13238	−0.566192	+	0.824273i	$0.691584\pi$
$102$	0	0
$103$	5.13414	0.505882	0.252941	−	0.967482i	$-0.418602\pi$
$103$	5.13414	0.505882	0.252941	+	0.967482i	$0.418602\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.8781	1.82502	0.912508	−	0.409059i	$-0.134143\pi$
$107$	18.8781	1.82502	0.912508	+	0.409059i	$0.134143\pi$
$108$	0	0
$109$	19.7440	1.89113	0.945565	−	0.325434i	$-0.105511\pi$
$109$	19.7440	1.89113	0.945565	+	0.325434i	$0.105511\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.842618	−0.0792668	−0.0396334	−	0.999214i	$-0.512619\pi$
$113$	−0.842618	−0.0792668	−0.0396334	+	0.999214i	$0.512619\pi$
$114$	0	0
$115$	−0.442333	−0.0412477
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.43905	0.681937
$120$	0	0
$121$	−10.8516	−0.986508
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.52103	−0.136045
$126$	0	0
$127$	−20.9857	−1.86218	−0.931091	−	0.364787i	$-0.881142\pi$
$127$	−20.9857	−1.86218	−0.931091	+	0.364787i	$0.881142\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.13414	−0.798053	−0.399027	−	0.916939i	$-0.630652\pi$
$131$	−9.13414	−0.798053	−0.399027	+	0.916939i	$0.630652\pi$
$132$	0	0
$133$	7.20627	0.624863
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−22.6005	−1.93089	−0.965445	−	0.260608i	$-0.916077\pi$
$137$	−22.6005	−1.93089	−0.965445	+	0.260608i	$0.916077\pi$
$138$	0	0
$139$	−19.7929	−1.67881	−0.839404	−	0.543508i	$-0.817096\pi$
$139$	−19.7929	−1.67881	−0.839404	+	0.543508i	$0.817096\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.385245	−0.0322158
$144$	0	0
$145$	0.793734	0.0659160
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.5948	1.03181	0.515903	−	0.856647i	$-0.327457\pi$
$149$	12.5948	1.03181	0.515903	+	0.856647i	$0.327457\pi$
$150$	0	0
$151$	21.1341	1.71987	0.859936	−	0.510402i	$-0.170503\pi$
$151$	21.1341	1.71987	0.859936	+	0.510402i	$0.170503\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.269425	0.0216407
$156$	0	0
$157$	15.4391	1.23217	0.616085	−	0.787679i	$-0.288718\pi$
$157$	15.4391	1.23217	0.616085	+	0.787679i	$0.288718\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.90135	0.228659
$162$	0	0
$163$	−3.22951	−0.252955	−0.126477	−	0.991969i	$-0.540367\pi$
$163$	−3.22951	−0.252955	−0.126477	+	0.991969i	$0.540367\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.3355	−1.34146	−0.670730	−	0.741702i	$-0.734019\pi$
$167$	−17.3355	−1.34146	−0.670730	+	0.741702i	$0.734019\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.57319	−0.651808	−0.325904	−	0.945403i	$-0.605669\pi$
$173$	−8.57319	−0.651808	−0.325904	+	0.945403i	$0.605669\pi$
$174$	0	0
$175$	4.97676	0.376207
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.15738	−0.0865068	−0.0432534	−	0.999064i	$-0.513772\pi$
$179$	−1.15738	−0.0865068	−0.0432534	+	0.999064i	$0.513772\pi$
$180$	0	0
$181$	6.60983	0.491305	0.245652	−	0.969358i	$-0.420998\pi$
$181$	6.60983	0.491305	0.245652	+	0.969358i	$0.420998\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.13414	0.0833836
$186$	0	0
$187$	2.86586	0.209573
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.0122	1.44804	0.724018	−	0.689781i	$-0.242293\pi$
$191$	20.0122	1.44804	0.724018	+	0.689781i	$0.242293\pi$
$192$	0	0
$193$	−0.878108	−0.0632076	−0.0316038	−	0.999500i	$-0.510061\pi$
$193$	−0.878108	−0.0632076	−0.0316038	+	0.999500i	$0.510061\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.336362	0.0239648	0.0119824	−	0.999928i	$-0.496186\pi$
$197$	0.336362	0.0239648	0.0119824	+	0.999928i	$0.496186\pi$
$198$	0	0
$199$	5.90463	0.418568	0.209284	−	0.977855i	$-0.432887\pi$
$199$	5.90463	0.418568	0.209284	+	0.977855i	$0.432887\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.20627	−0.365408
$204$	0	0
$205$	−1.07541	−0.0751096
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.77618	0.192032
$210$	0	0
$211$	3.67184	0.252780	0.126390	−	0.991981i	$-0.459661\pi$
$211$	3.67184	0.252780	0.126390	+	0.991981i	$0.459661\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.442333	0.0301668
$216$	0	0
$217$	−1.76721	−0.119966
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.43905	−0.500405
$222$	0	0
$223$	−12.6453	−0.846793	−0.423397	−	0.905944i	$-0.639162\pi$
$223$	−12.6453	−0.846793	−0.423397	+	0.905944i	$0.639162\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.59479	−0.437712	−0.218856	−	0.975757i	$-0.570232\pi$
$227$	−6.59479	−0.437712	−0.218856	+	0.975757i	$0.570232\pi$
$228$	0	0
$229$	−2.60983	−0.172462	−0.0862312	−	0.996275i	$-0.527482\pi$
$229$	−2.60983	−0.172462	−0.0862312	+	0.996275i	$0.527482\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.35468	0.0887480	0.0443740	−	0.999015i	$-0.485871\pi$
$233$	1.35468	0.0887480	0.0443740	+	0.999015i	$0.485871\pi$
$234$	0	0
$235$	0.547553	0.0357184
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.4341	−0.933666	−0.466833	−	0.884345i	$-0.654605\pi$
$239$	−14.4341	−0.933666	−0.466833	+	0.884345i	$0.654605\pi$
$240$	0	0
$241$	17.7306	1.14213	0.571063	−	0.820906i	$-0.306531\pi$
$241$	17.7306	1.14213	0.571063	+	0.820906i	$0.306531\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.152457	0.00974015
$246$	0	0
$247$	−7.20627	−0.458524
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.05873	−0.256185	−0.128092	−	0.991762i	$-0.540885\pi$
$251$	−4.05873	−0.256185	−0.128092	+	0.991762i	$0.540885\pi$
$252$	0	0
$253$	1.11773	0.0702712
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.2584	1.88747	0.943734	−	0.330704i	$-0.107286\pi$
$257$	30.2584	1.88747	0.943734	+	0.330704i	$0.107286\pi$
$258$	0	0
$259$	−7.43905	−0.462240
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.9502	−0.798546	−0.399273	−	0.916832i	$-0.630737\pi$
$263$	−12.9502	−0.798546	−0.399273	+	0.916832i	$0.630737\pi$
$264$	0	0
$265$	1.66944	0.102553
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−17.7929	−1.08485	−0.542425	−	0.840104i	$-0.682494\pi$
$269$	−17.7929	−1.08485	−0.542425	+	0.840104i	$0.682494\pi$
$270$	0	0
$271$	17.9470	1.09020	0.545100	−	0.838371i	$-0.316492\pi$
$271$	17.9470	1.09020	0.545100	+	0.838371i	$0.316492\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.91727	0.115616
$276$	0	0
$277$	−5.87483	−0.352984	−0.176492	−	0.984302i	$-0.556475\pi$
$277$	−5.87483	−0.352984	−0.176492	+	0.984302i	$0.556475\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.5417	1.58335	0.791674	−	0.610944i	$-0.209210\pi$
$281$	26.5417	1.58335	0.791674	+	0.610944i	$0.209210\pi$
$282$	0	0
$283$	17.7562	1.05550	0.527749	−	0.849401i	$-0.323036\pi$
$283$	17.7562	1.05550	0.527749	+	0.849401i	$0.323036\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.05381	0.416373
$288$	0	0
$289$	38.3395	2.25527
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.0139	0.818700	0.409350	−	0.912377i	$-0.365755\pi$
$293$	14.0139	0.818700	0.409350	+	0.912377i	$0.365755\pi$
$294$	0	0
$295$	0.887958	0.0516989
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.90135	−0.167789
$300$	0	0
$301$	−2.90135	−0.167231
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.90792	0.109247
$306$	0	0
$307$	22.4912	1.28364	0.641821	−	0.766855i	$-0.278179\pi$
$307$	22.4912	1.28364	0.641821	+	0.766855i	$0.278179\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	33.1300	1.87863	0.939314	−	0.343058i	$-0.111463\pi$
$311$	33.1300	1.87863	0.939314	+	0.343058i	$0.111463\pi$
$312$	0	0
$313$	19.3371	1.09300	0.546500	−	0.837459i	$-0.315960\pi$
$313$	19.3371	1.09300	0.546500	+	0.837459i	$0.315960\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.0860	0.903481	0.451740	−	0.892149i	$-0.350803\pi$
$317$	16.0860	0.903481	0.451740	+	0.892149i	$0.350803\pi$
$318$	0	0
$319$	−2.00569	−0.112297
$320$	0	0
$321$	0	0
$322$	0	0
$323$	53.6078	2.98282
$324$	0	0
$325$	−4.97676	−0.276061
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.59151	−0.198006
$330$	0	0
$331$	−27.9535	−1.53646	−0.768232	−	0.640172i	$-0.778863\pi$
$331$	−27.9535	−1.53646	−0.768232	+	0.640172i	$0.778863\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.49449	0.0816530
$336$	0	0
$337$	7.31388	0.398413	0.199206	−	0.979958i	$-0.436164\pi$
$337$	7.31388	0.398413	0.199206	+	0.979958i	$0.436164\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.680810	−0.0368679
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.1076	1.40153	0.700765	−	0.713392i	$-0.252842\pi$
$347$	26.1076	1.40153	0.700765	+	0.713392i	$0.252842\pi$
$348$	0	0
$349$	−3.76721	−0.201654	−0.100827	−	0.994904i	$-0.532149\pi$
$349$	−3.76721	−0.201654	−0.100827	+	0.994904i	$0.532149\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.8243	0.629344	0.314672	−	0.949200i	$-0.398105\pi$
$353$	11.8243	0.629344	0.314672	+	0.949200i	$0.398105\pi$
$354$	0	0
$355$	−0.887958	−0.0471279
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.4341	−0.761804	−0.380902	−	0.924615i	$-0.624387\pi$
$359$	−14.4341	−0.761804	−0.380902	+	0.924615i	$0.624387\pi$
$360$	0	0
$361$	32.9303	1.73317
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.472412	0.0247272
$366$	0	0
$367$	−13.9634	−0.728882	−0.364441	−	0.931227i	$-0.618740\pi$
$367$	−13.9634	−0.728882	−0.364441	+	0.931227i	$0.618740\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.9502	−0.568508
$372$	0	0
$373$	−31.0289	−1.60662	−0.803308	−	0.595563i	$-0.796929\pi$
$373$	−31.0289	−1.60662	−0.803308	+	0.595563i	$0.796929\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.20627	0.268136
$378$	0	0
$379$	−25.9634	−1.33365	−0.666824	−	0.745215i	$-0.732347\pi$
$379$	−25.9634	−1.33365	−0.666824	+	0.745215i	$0.732347\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−13.4175	−0.685600	−0.342800	−	0.939408i	$-0.611375\pi$
$383$	−13.4175	−0.685600	−0.342800	+	0.939408i	$0.611375\pi$
$384$	0	0
$385$	0.0587335	0.00299334
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	21.5833	1.09151
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.92787	0.0970018
$396$	0	0
$397$	−25.5234	−1.28098	−0.640492	−	0.767965i	$-0.721270\pi$
$397$	−25.5234	−1.28098	−0.640492	+	0.767965i	$0.721270\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.00732	−0.150178	−0.0750892	−	0.997177i	$-0.523924\pi$
$401$	−3.00732	−0.150178	−0.0750892	+	0.997177i	$0.523924\pi$
$402$	0	0
$403$	1.76721	0.0880311
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.86586	−0.142055
$408$	0	0
$409$	−22.8483	−1.12978	−0.564888	−	0.825168i	$-0.691081\pi$
$409$	−22.8483	−1.12978	−0.564888	+	0.825168i	$0.691081\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.82430	−0.286595
$414$	0	0
$415$	−1.81370	−0.0890310
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−16.1565	−0.789297	−0.394648	−	0.918832i	$-0.629134\pi$
$419$	−16.1565	−0.789297	−0.394648	+	0.918832i	$0.629134\pi$
$420$	0	0
$421$	21.7929	1.06212	0.531059	−	0.847335i	$-0.321794\pi$
$421$	21.7929	1.06212	0.531059	+	0.847335i	$0.321794\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	37.0224	1.79585
$426$	0	0
$427$	−12.5145	−0.605617
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.80194	0.327638	0.163819	−	0.986490i	$-0.447619\pi$
$431$	6.80194	0.327638	0.163819	+	0.986490i	$0.447619\pi$
$432$	0	0
$433$	6.26172	0.300919	0.150460	−	0.988616i	$-0.451925\pi$
$433$	6.26172	0.300919	0.150460	+	0.988616i	$0.451925\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.9079	1.00016
$438$	0	0
$439$	−21.0877	−1.00646	−0.503229	−	0.864153i	$-0.667855\pi$
$439$	−21.0877	−1.00646	−0.503229	+	0.864153i	$0.667855\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.39345	0.303762	0.151881	−	0.988399i	$-0.451467\pi$
$443$	6.39345	0.303762	0.151881	+	0.988399i	$0.451467\pi$
$444$	0	0
$445$	0.852467	0.0404108
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.49286	−0.212031	−0.106016	−	0.994364i	$-0.533809\pi$
$449$	−4.49286	−0.212031	−0.106016	+	0.994364i	$0.533809\pi$
$450$	0	0
$451$	2.71745	0.127960
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.152457	−0.00714731
$456$	0	0
$457$	−31.7398	−1.48473	−0.742363	−	0.669998i	$-0.766295\pi$
$457$	−31.7398	−1.48473	−0.742363	+	0.669998i	$0.766295\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.76886	−0.175533	−0.0877666	−	0.996141i	$-0.527973\pi$
$461$	−3.76886	−0.175533	−0.0877666	+	0.996141i	$0.527973\pi$
$462$	0	0
$463$	18.1664	0.844262	0.422131	−	0.906535i	$-0.361282\pi$
$463$	18.1664	0.844262	0.422131	+	0.906535i	$0.361282\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.0057	0.648106	0.324053	−	0.946039i	$-0.394954\pi$
$467$	14.0057	0.648106	0.324053	+	0.946039i	$0.394954\pi$
$468$	0	0
$469$	−9.80270	−0.452647
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.11773	−0.0513934
$474$	0	0
$475$	35.8638	1.64555
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−36.1704	−1.65267	−0.826334	−	0.563181i	$-0.809577\pi$
$479$	−36.1704	−1.65267	−0.826334	+	0.563181i	$0.809577\pi$
$480$	0	0
$481$	7.43905	0.339192
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.88910	0.131187
$486$	0	0
$487$	5.03221	0.228031	0.114016	−	0.993479i	$-0.463629\pi$
$487$	5.03221	0.228031	0.114016	+	0.993479i	$0.463629\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10.8781	−0.490922	−0.245461	−	0.969406i	$-0.578939\pi$
$491$	−10.8781	−0.490922	−0.245461	+	0.969406i	$0.578939\pi$
$492$	0	0
$493$	−38.7297	−1.74430
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.82430	0.261256
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13.8459	−0.617358	−0.308679	−	0.951166i	$-0.599887\pi$
$503$	−13.8459	−0.617358	−0.308679	+	0.951166i	$0.599887\pi$
$504$	0	0
$505$	−1.73501	−0.0772071
$506$	0	0
$507$	0	0
$508$	0	0
$509$	37.1814	1.64804	0.824018	−	0.566564i	$-0.191728\pi$
$509$	37.1814	1.64804	0.824018	+	0.566564i	$0.191728\pi$
$510$	0	0
$511$	−3.09865	−0.137076
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.782738	0.0344915
$516$	0	0
$517$	−1.38361	−0.0608512
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−43.4936	−1.90549	−0.952745	−	0.303771i	$-0.901754\pi$
$521$	−43.4936	−1.90549	−0.952745	+	0.303771i	$0.901754\pi$
$522$	0	0
$523$	−25.9634	−1.13530	−0.567649	−	0.823270i	$-0.692147\pi$
$523$	−25.9634	−1.13530	−0.567649	+	0.823270i	$0.692147\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13.1464	−0.572666
$528$	0	0
$529$	−14.5822	−0.634007
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.05381	−0.305534
$534$	0	0
$535$	2.87811	0.124431
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.385245	−0.0165937
$540$	0	0
$541$	−14.1565	−0.608636	−0.304318	−	0.952571i	$-0.598428\pi$
$541$	−14.1565	−0.608636	−0.304318	+	0.952571i	$0.598428\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.01011	0.128939
$546$	0	0
$547$	−14.3338	−0.612871	−0.306436	−	0.951891i	$-0.599136\pi$
$547$	−14.3338	−0.612871	−0.306436	+	0.951891i	$0.599136\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−37.5177	−1.59831
$552$	0	0
$553$	−12.6453	−0.537734
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.4953	1.63110	0.815548	−	0.578689i	$-0.196436\pi$
$557$	38.4953	1.63110	0.815548	+	0.578689i	$0.196436\pi$
$558$	0	0
$559$	2.90135	0.122714
$560$	0	0
$561$	0	0
$562$	0	0
$563$	39.9494	1.68366	0.841832	−	0.539739i	$-0.181477\pi$
$563$	39.9494	1.68366	0.841832	+	0.539739i	$0.181477\pi$
$564$	0	0
$565$	−0.128463	−0.00540449
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.83606	−0.118894	−0.0594469	−	0.998231i	$-0.518934\pi$
$569$	−2.83606	−0.118894	−0.0594469	+	0.998231i	$0.518934\pi$
$570$	0	0
$571$	−15.9890	−0.669119	−0.334559	−	0.942375i	$-0.608588\pi$
$571$	−15.9890	−0.669119	−0.334559	+	0.942375i	$0.608588\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	14.4393	0.602161
$576$	0	0
$577$	15.2826	0.636221	0.318111	−	0.948054i	$-0.396952\pi$
$577$	15.2826	0.636221	0.318111	+	0.948054i	$0.396952\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11.8964	0.493547
$582$	0	0
$583$	−4.21853	−0.174714
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44.5340	1.83812	0.919058	−	0.394122i	$-0.128951\pi$
$587$	44.5340	1.83812	0.919058	+	0.394122i	$0.128951\pi$
$588$	0	0
$589$	−12.7350	−0.524737
$590$	0	0
$591$	0	0
$592$	0	0
$593$	23.9063	0.981713	0.490857	−	0.871240i	$-0.336684\pi$
$593$	23.9063	0.981713	0.490857	+	0.871240i	$0.336684\pi$
$594$	0	0
$595$	1.13414	0.0464952
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23.9303	−0.977764	−0.488882	−	0.872350i	$-0.662595\pi$
$599$	−23.9303	−0.977764	−0.488882	+	0.872350i	$0.662595\pi$
$600$	0	0
$601$	6.51206	0.265633	0.132816	−	0.991141i	$-0.457598\pi$
$601$	6.51206	0.265633	0.132816	+	0.991141i	$0.457598\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.65440	−0.0672611
$606$	0	0
$607$	40.5756	1.64691	0.823456	−	0.567380i	$-0.192043\pi$
$607$	40.5756	1.64691	0.823456	+	0.567380i	$0.192043\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.59151	0.145297
$612$	0	0
$613$	39.1422	1.58094	0.790470	−	0.612501i	$-0.209836\pi$
$613$	39.1422	1.58094	0.790470	+	0.612501i	$0.209836\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.7191	1.27696	0.638481	−	0.769638i	$-0.279563\pi$
$617$	31.7191	1.27696	0.638481	+	0.769638i	$0.279563\pi$
$618$	0	0
$619$	0.246181	0.00989486	0.00494743	−	0.999988i	$-0.498425\pi$
$619$	0.246181	0.00989486	0.00494743	+	0.999988i	$0.498425\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.59151	−0.224019
$624$	0	0
$625$	24.6519	0.986076
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−55.3395	−2.20653
$630$	0	0
$631$	24.0587	0.957763	0.478882	−	0.877880i	$-0.341042\pi$
$631$	24.0587	0.957763	0.478882	+	0.877880i	$0.341042\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.19943	−0.126965
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.1962	0.718705	0.359352	−	0.933202i	$-0.382998\pi$
$641$	18.1962	0.718705	0.359352	+	0.933202i	$0.382998\pi$
$642$	0	0
$643$	−21.3926	−0.843641	−0.421820	−	0.906679i	$-0.638609\pi$
$643$	−21.3926	−0.843641	−0.421820	+	0.906679i	$0.638609\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.8557	−0.780610	−0.390305	−	0.920686i	$-0.627630\pi$
$647$	−19.8557	−0.780610	−0.390305	+	0.920686i	$0.627630\pi$
$648$	0	0
$649$	−2.24378	−0.0880762
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.4182	1.34689	0.673445	−	0.739238i	$-0.264814\pi$
$653$	34.4182	1.34689	0.673445	+	0.739238i	$0.264814\pi$
$654$	0	0
$655$	−1.39257	−0.0544121
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−35.7851	−1.39399	−0.696996	−	0.717075i	$-0.745480\pi$
$659$	−35.7851	−1.39399	−0.696996	+	0.717075i	$0.745480\pi$
$660$	0	0
$661$	18.5988	0.723411	0.361705	−	0.932292i	$-0.382195\pi$
$661$	18.5988	0.723411	0.361705	+	0.932292i	$0.382195\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.09865	0.0426038
$666$	0	0
$667$	−15.1052	−0.584876
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.82114	−0.186118
$672$	0	0
$673$	−42.6576	−1.64433	−0.822164	−	0.569250i	$-0.807233\pi$
$673$	−42.6576	−1.64433	−0.822164	+	0.569250i	$0.807233\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−32.3660	−1.24393	−0.621964	−	0.783046i	$-0.713665\pi$
$677$	−32.3660	−1.24393	−0.621964	+	0.783046i	$0.713665\pi$
$678$	0	0
$679$	−18.9502	−0.727243
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7.40761	−0.283444	−0.141722	−	0.989906i	$-0.545264\pi$
$683$	−7.40761	−0.283444	−0.141722	+	0.989906i	$0.545264\pi$
$684$	0	0
$685$	−3.44561	−0.131650
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10.9502	0.417171
$690$	0	0
$691$	39.0147	1.48419	0.742094	−	0.670296i	$-0.233833\pi$
$691$	39.0147	1.48419	0.742094	+	0.670296i	$0.233833\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.01757	−0.114463
$696$	0	0
$697$	52.4737	1.98758
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−4.27598	−0.161502	−0.0807508	−	0.996734i	$-0.525732\pi$
$701$	−4.27598	−0.161502	−0.0807508	+	0.996734i	$0.525732\pi$
$702$	0	0
$703$	−53.6078	−2.02186
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.3803	0.428001
$708$	0	0
$709$	45.5425	1.71038	0.855192	−	0.518311i	$-0.173439\pi$
$709$	45.5425	1.71038	0.855192	+	0.518311i	$0.173439\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.12730	−0.192019
$714$	0	0
$715$	−0.0587335	−0.00219651
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−23.3005	−0.868962	−0.434481	−	0.900681i	$-0.643068\pi$
$719$	−23.3005	−0.868962	−0.434481	+	0.900681i	$0.643068\pi$
$720$	0	0
$721$	−5.13414	−0.191205
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−25.9103	−0.962285
$726$	0	0
$727$	1.55652	0.0577282	0.0288641	−	0.999583i	$-0.490811\pi$
$727$	1.55652	0.0577282	0.0288641	+	0.999583i	$0.490811\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−21.5833	−0.798287
$732$	0	0
$733$	32.6087	1.20443	0.602215	−	0.798334i	$-0.294285\pi$
$733$	32.6087	1.20443	0.602215	+	0.798334i	$0.294285\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.77645	−0.139107
$738$	0	0
$739$	28.5633	1.05072	0.525360	−	0.850880i	$-0.323931\pi$
$739$	28.5633	1.05072	0.525360	+	0.850880i	$0.323931\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20.4440	0.750017	0.375008	−	0.927021i	$-0.377640\pi$
$743$	20.4440	0.750017	0.375008	+	0.927021i	$0.377640\pi$
$744$	0	0
$745$	1.92017	0.0703496
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−18.8781	−0.689791
$750$	0	0
$751$	11.8748	0.433319	0.216659	−	0.976247i	$-0.430484\pi$
$751$	11.8748	0.433319	0.216659	+	0.976247i	$0.430484\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.22206	0.117263
$756$	0	0
$757$	−37.1720	−1.35104	−0.675520	−	0.737342i	$-0.736081\pi$
$757$	−37.1720	−1.35104	−0.675520	+	0.737342i	$0.736081\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	44.8920	1.62733	0.813667	−	0.581331i	$-0.197468\pi$
$761$	44.8920	1.62733	0.813667	+	0.581331i	$0.197468\pi$
$762$	0	0
$763$	−19.7440	−0.714780
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.82430	0.210303
$768$	0	0
$769$	46.7008	1.68407	0.842036	−	0.539421i	$-0.181357\pi$
$769$	46.7008	1.68407	0.842036	+	0.539421i	$0.181357\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.56827	−0.344147	−0.172073	−	0.985084i	$-0.555047\pi$
$773$	−9.56827	−0.344147	−0.172073	+	0.985084i	$0.555047\pi$
$774$	0	0
$775$	−8.79498	−0.315925
$776$	0	0
$777$	0	0
$778$	0	0
$779$	50.8316	1.82123
$780$	0	0
$781$	2.24378	0.0802889
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.35380	0.0840107
$786$	0	0
$787$	−40.2898	−1.43617	−0.718087	−	0.695953i	$-0.754982\pi$
$787$	−40.2898	−1.43617	−0.718087	+	0.695953i	$0.754982\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.842618	0.0299600
$792$	0	0
$793$	12.5145	0.444401
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.6584	0.838025	0.419013	−	0.907980i	$-0.362376\pi$
$797$	23.6584	0.838025	0.419013	+	0.907980i	$0.362376\pi$
$798$	0	0
$799$	−26.7174	−0.945195
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.19374	−0.0421262
$804$	0	0
$805$	0.442333	0.0155902
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38.7431	1.36213	0.681067	−	0.732221i	$-0.261516\pi$
$809$	38.7431	1.36213	0.681067	+	0.732221i	$0.261516\pi$
$810$	0	0
$811$	−1.86144	−0.0653639	−0.0326819	−	0.999466i	$-0.510405\pi$
$811$	−1.86144	−0.0653639	−0.0326819	+	0.999466i	$0.510405\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.492363	−0.0172467
$816$	0	0
$817$	−20.9079	−0.731475
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.7024	−0.373517	−0.186758	−	0.982406i	$-0.559798\pi$
$821$	−10.7024	−0.373517	−0.186758	+	0.982406i	$0.559798\pi$
$822$	0	0
$823$	38.9759	1.35861	0.679307	−	0.733854i	$-0.262281\pi$
$823$	38.9759	1.35861	0.679307	+	0.733854i	$0.262281\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.1349	−0.839253	−0.419626	−	0.907697i	$-0.637839\pi$
$827$	−24.1349	−0.839253	−0.419626	+	0.907697i	$0.637839\pi$
$828$	0	0
$829$	10.3636	0.359944	0.179972	−	0.983672i	$-0.442399\pi$
$829$	10.3636	0.359944	0.179972	+	0.983672i	$0.442399\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.43905	−0.257748
$834$	0	0
$835$	−2.64292	−0.0914621
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−22.7855	−0.786644	−0.393322	−	0.919401i	$-0.628674\pi$
$839$	−22.7855	−0.786644	−0.393322	+	0.919401i	$0.628674\pi$
$840$	0	0
$841$	−1.89479	−0.0653377
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.152457	0.00524470
$846$	0	0
$847$	10.8516	0.372865
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−21.5833	−0.739866
$852$	0	0
$853$	−24.0289	−0.822735	−0.411367	−	0.911470i	$-0.634949\pi$
$853$	−24.0289	−0.822735	−0.411367	+	0.911470i	$0.634949\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26.8683	−0.917802	−0.458901	−	0.888487i	$-0.651757\pi$
$857$	−26.8683	−0.917802	−0.458901	+	0.888487i	$0.651757\pi$
$858$	0	0
$859$	18.6164	0.635183	0.317591	−	0.948228i	$-0.397126\pi$
$859$	18.6164	0.635183	0.317591	+	0.948228i	$0.397126\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.8341	−0.675162	−0.337581	−	0.941296i	$-0.609609\pi$
$863$	−19.8341	−0.675162	−0.337581	+	0.941296i	$0.609609\pi$
$864$	0	0
$865$	−1.30705	−0.0444409
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.87155	−0.165256
$870$	0	0
$871$	9.80270	0.332152
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.52103	0.0514202
$876$	0	0
$877$	−7.07541	−0.238919	−0.119460	−	0.992839i	$-0.538116\pi$
$877$	−7.07541	−0.238919	−0.119460	+	0.992839i	$0.538116\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	23.0412	0.776277	0.388138	−	0.921601i	$-0.373118\pi$
$881$	23.0412	0.776277	0.388138	+	0.921601i	$0.373118\pi$
$882$	0	0
$883$	−21.6462	−0.728453	−0.364226	−	0.931310i	$-0.618667\pi$
$883$	−21.6462	−0.728453	−0.364226	+	0.931310i	$0.618667\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	52.1223	1.75009	0.875047	−	0.484038i	$-0.160830\pi$
$887$	52.1223	1.75009	0.875047	+	0.484038i	$0.160830\pi$
$888$	0	0
$889$	20.9857	0.703839
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−25.8814	−0.866088
$894$	0	0
$895$	−0.176452	−0.00589812
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.20058	0.306856
$900$	0	0
$901$	−81.4594	−2.71381
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.00772	0.0334977
$906$	0	0
$907$	38.2841	1.27120	0.635601	−	0.772018i	$-0.280752\pi$
$907$	38.2841	1.27120	0.635601	+	0.772018i	$0.280752\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−15.8161	−0.524011	−0.262005	−	0.965066i	$-0.584384\pi$
$911$	−15.8161	−0.524011	−0.262005	+	0.965066i	$0.584384\pi$
$912$	0	0
$913$	4.58304	0.151677
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.13414	0.301636
$918$	0	0
$919$	12.0546	0.397644	0.198822	−	0.980036i	$-0.436288\pi$
$919$	12.0546	0.397644	0.198822	+	0.980036i	$0.436288\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.82430	−0.191709
$924$	0	0
$925$	−37.0224	−1.21729
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.98940	0.0652701	0.0326350	−	0.999467i	$-0.489610\pi$
$929$	1.98940	0.0652701	0.0326350	+	0.999467i	$0.489610\pi$
$930$	0	0
$931$	−7.20627	−0.236176
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.436922	0.0142889
$936$	0	0
$937$	1.31919	0.0430961	0.0215480	−	0.999768i	$-0.493141\pi$
$937$	1.31919	0.0430961	0.0215480	+	0.999768i	$0.493141\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12.9107	−0.420877	−0.210438	−	0.977607i	$-0.567489\pi$
$941$	−12.9107	−0.420877	−0.210438	+	0.977607i	$0.567489\pi$
$942$	0	0
$943$	20.4656	0.666451
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−46.7081	−1.51781	−0.758905	−	0.651202i	$-0.774265\pi$
$947$	−46.7081	−1.51781	−0.758905	+	0.651202i	$0.774265\pi$
$948$	0	0
$949$	3.09865	0.100586
$950$	0	0
$951$	0	0
$952$	0	0
$953$	58.0355	1.87995	0.939977	−	0.341238i	$-0.110846\pi$
$953$	58.0355	1.87995	0.939977	+	0.341238i	$0.110846\pi$
$954$	0	0
$955$	3.05102	0.0987286
$956$	0	0
$957$	0	0
$958$	0	0
$959$	22.6005	0.729808
$960$	0	0
$961$	−27.8770	−0.899257
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.133874	−0.00430956
$966$	0	0
$967$	50.0057	1.60807	0.804037	−	0.594579i	$-0.202681\pi$
$967$	50.0057	1.60807	0.804037	+	0.594579i	$0.202681\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−52.3595	−1.68030	−0.840148	−	0.542357i	$-0.817532\pi$
$971$	−52.3595	−1.68030	−0.840148	+	0.542357i	$0.817532\pi$
$972$	0	0
$973$	19.7929	0.634530
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.3098	0.553791	0.276895	−	0.960900i	$-0.410694\pi$
$977$	17.3098	0.553791	0.276895	+	0.960900i	$0.410694\pi$
$978$	0	0
$979$	−2.15410	−0.0688455
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−47.5409	−1.51632	−0.758159	−	0.652070i	$-0.773901\pi$
$983$	−47.5409	−1.51632	−0.758159	+	0.652070i	$0.773901\pi$
$984$	0	0
$985$	0.0512808	0.00163394
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.41784	−0.267672
$990$	0	0
$991$	11.5809	0.367880	0.183940	−	0.982937i	$-0.441115\pi$
$991$	11.5809	0.367880	0.183940	+	0.982937i	$0.441115\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.900205	0.0285384
$996$	0	0
$997$	16.5103	0.522886	0.261443	−	0.965219i	$-0.415802\pi$
$997$	16.5103	0.522886	0.261443	+	0.965219i	$0.415802\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6552.2.a.bs.1.3		4
3.2	odd	2		2184.2.a.v.1.2	✓	4
12.11	even	2		4368.2.a.bs.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.a.v.1.2	✓	4	3.2	odd	2
4368.2.a.bs.1.2		4	12.11	even	2
6552.2.a.bs.1.3		4	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:	\( N \)	\(=\)	\( 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6552.a (trivial)

\(f(q)\)	\(=\)	\(q+0.152457 q^{5} -1.00000 q^{7} -0.385245 q^{11} +1.00000 q^{13} -7.43905 q^{17} -7.20627 q^{19} -2.90135 q^{23} -4.97676 q^{25} +5.20627 q^{29} +1.76721 q^{31} -0.152457 q^{35} +7.43905 q^{37} -7.05381 q^{41} +2.90135 q^{43} +3.59151 q^{47} +1.00000 q^{49} +10.9502 q^{53} -0.0587335 q^{55} +5.82430 q^{59} +12.5145 q^{61} +0.152457 q^{65} +9.80270 q^{67} -5.82430 q^{71} +3.09865 q^{73} +0.385245 q^{77} +12.6453 q^{79} -11.8964 q^{83} -1.13414 q^{85} +5.59151 q^{89} -1.00000 q^{91} -1.09865 q^{95} +18.9502 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} - 4 q^{7} + 3 q^{11} + 4 q^{13} - 3 q^{17} - 4 q^{19} + 8 q^{23} + 14 q^{25} - 4 q^{29} + 9 q^{31} + 2 q^{35} + 3 q^{37} - 6 q^{41} - 8 q^{43} - 15 q^{47} + 4 q^{49} - 13 q^{53} + 7 q^{55}+ \cdots + 19 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 - 4 * q^7 + 3 * q^11 + 4 * q^13 - 3 * q^17 - 4 * q^19 + 8 * q^23 + 14 * q^25 - 4 * q^29 + 9 * q^31 + 2 * q^35 + 3 * q^37 - 6 * q^41 - 8 * q^43 - 15 * q^47 + 4 * q^49 - 13 * q^53 + 7 * q^55 - 8 * q^59 + 9 * q^61 - 2 * q^65 + 8 * q^71 + 32 * q^73 - 3 * q^77 - q^79 - 13 * q^83 + 17 * q^85 - 7 * q^89 - 4 * q^91 - 24 * q^95 + 19 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more