LMFDB - Embedded newform 1344.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +18.0000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} +18.0000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -44.0000 q^{11} -58.0000 q^{13} -54.0000 q^{15} -130.000 q^{17} -92.0000 q^{19} +21.0000 q^{21} +84.0000 q^{23} +199.000 q^{25} -27.0000 q^{27} +250.000 q^{29} -72.0000 q^{31} +132.000 q^{33} -126.000 q^{35} +354.000 q^{37} +174.000 q^{39} +334.000 q^{41} +416.000 q^{43} +162.000 q^{45} -464.000 q^{47} +49.0000 q^{49} +390.000 q^{51} +450.000 q^{53} -792.000 q^{55} +276.000 q^{57} +516.000 q^{59} -58.0000 q^{61} -63.0000 q^{63} -1044.00 q^{65} +656.000 q^{67} -252.000 q^{69} -940.000 q^{71} +178.000 q^{73} -597.000 q^{75} +308.000 q^{77} +1072.00 q^{79} +81.0000 q^{81} -660.000 q^{83} -2340.00 q^{85} -750.000 q^{87} +1254.00 q^{89} +406.000 q^{91} +216.000 q^{93} -1656.00 q^{95} +210.000 q^{97} -396.000 q^{99} +O(q^{100})$

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	18.0000	1.60997	0.804984	−	0.593296i	$-0.202174\pi$
$5$	18.0000	1.60997	0.804984	+	0.593296i	$0.202174\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−44.0000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−44.0000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−58.0000	−1.23741	−0.618704	−	0.785624i	$-0.712342\pi$
$13$	−58.0000	−1.23741	−0.618704	+	0.785624i	$0.712342\pi$
$14$	0	0
$15$	−54.0000	−0.929516
$16$	0	0
$17$	−130.000	−1.85468	−0.927342	−	0.374215i	$-0.877912\pi$
$17$	−130.000	−1.85468	−0.927342	+	0.374215i	$0.877912\pi$
$18$	0	0
$19$	−92.0000	−1.11086	−0.555428	−	0.831565i	$-0.687445\pi$
$19$	−92.0000	−1.11086	−0.555428	+	0.831565i	$0.687445\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	84.0000	0.761531	0.380765	−	0.924672i	$-0.375661\pi$
$23$	84.0000	0.761531	0.380765	+	0.924672i	$0.375661\pi$
$24$	0	0
$25$	199.000	1.59200
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	250.000	1.60082	0.800411	−	0.599452i	$-0.204615\pi$
$29$	250.000	1.60082	0.800411	+	0.599452i	$0.204615\pi$
$30$	0	0
$31$	−72.0000	−0.417148	−0.208574	−	0.978007i	$-0.566882\pi$
$31$	−72.0000	−0.417148	−0.208574	+	0.978007i	$0.566882\pi$
$32$	0	0
$33$	132.000	0.696311
$34$	0	0
$35$	−126.000	−0.608511
$36$	0	0
$37$	354.000	1.57290	0.786449	−	0.617655i	$-0.211917\pi$
$37$	354.000	1.57290	0.786449	+	0.617655i	$0.211917\pi$
$38$	0	0
$39$	174.000	0.714418
$40$	0	0
$41$	334.000	1.27224	0.636122	−	0.771588i	$-0.280537\pi$
$41$	334.000	1.27224	0.636122	+	0.771588i	$0.280537\pi$
$42$	0	0
$43$	416.000	1.47534	0.737668	−	0.675164i	$-0.235927\pi$
$43$	416.000	1.47534	0.737668	+	0.675164i	$0.235927\pi$
$44$	0	0
$45$	162.000	0.536656
$46$	0	0
$47$	−464.000	−1.44003	−0.720014	−	0.693959i	$-0.755865\pi$
$47$	−464.000	−1.44003	−0.720014	+	0.693959i	$0.755865\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	390.000	1.07080
$52$	0	0
$53$	450.000	1.16627	0.583134	−	0.812376i	$-0.301826\pi$
$53$	450.000	1.16627	0.583134	+	0.812376i	$0.301826\pi$
$54$	0	0
$55$	−792.000	−1.94170
$56$	0	0
$57$	276.000	0.641353
$58$	0	0
$59$	516.000	1.13860	0.569301	−	0.822129i	$-0.307214\pi$
$59$	516.000	1.13860	0.569301	+	0.822129i	$0.307214\pi$
$60$	0	0
$61$	−58.0000	−0.121740	−0.0608700	−	0.998146i	$-0.519388\pi$
$61$	−58.0000	−0.121740	−0.0608700	+	0.998146i	$0.519388\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−1044.00	−1.99219
$66$	0	0
$67$	656.000	1.19617	0.598083	−	0.801434i	$-0.295929\pi$
$67$	656.000	1.19617	0.598083	+	0.801434i	$0.295929\pi$
$68$	0	0
$69$	−252.000	−0.439670
$70$	0	0
$71$	−940.000	−1.57123	−0.785616	−	0.618714i	$-0.787654\pi$
$71$	−940.000	−1.57123	−0.785616	+	0.618714i	$0.787654\pi$
$72$	0	0
$73$	178.000	0.285388	0.142694	−	0.989767i	$-0.454424\pi$
$73$	178.000	0.285388	0.142694	+	0.989767i	$0.454424\pi$
$74$	0	0
$75$	−597.000	−0.919142
$76$	0	0
$77$	308.000	0.455842
$78$	0	0
$79$	1072.00	1.52670	0.763351	−	0.645984i	$-0.223553\pi$
$79$	1072.00	1.52670	0.763351	+	0.645984i	$0.223553\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−660.000	−0.872824	−0.436412	−	0.899747i	$-0.643751\pi$
$83$	−660.000	−0.872824	−0.436412	+	0.899747i	$0.643751\pi$
$84$	0	0
$85$	−2340.00	−2.98598
$86$	0	0
$87$	−750.000	−0.924235
$88$	0	0
$89$	1254.00	1.49353	0.746763	−	0.665091i	$-0.231607\pi$
$89$	1254.00	1.49353	0.746763	+	0.665091i	$0.231607\pi$
$90$	0	0
$91$	406.000	0.467696
$92$	0	0
$93$	216.000	0.240840
$94$	0	0
$95$	−1656.00	−1.78844
$96$	0	0
$97$	210.000	0.219817	0.109909	−	0.993942i	$-0.464944\pi$
$97$	210.000	0.219817	0.109909	+	0.993942i	$0.464944\pi$
$98$	0	0
$99$	−396.000	−0.402015
$100$	0	0
$101$	186.000	0.183244	0.0916222	−	0.995794i	$-0.470795\pi$
$101$	186.000	0.183244	0.0916222	+	0.995794i	$0.470795\pi$
$102$	0	0
$103$	472.000	0.451530	0.225765	−	0.974182i	$-0.427512\pi$
$103$	472.000	0.451530	0.225765	+	0.974182i	$0.427512\pi$
$104$	0	0
$105$	378.000	0.351324
$106$	0	0
$107$	1212.00	1.09503	0.547516	−	0.836795i	$-0.315573\pi$
$107$	1212.00	1.09503	0.547516	+	0.836795i	$0.315573\pi$
$108$	0	0
$109$	1386.00	1.21793	0.608967	−	0.793196i	$-0.291584\pi$
$109$	1386.00	1.21793	0.608967	+	0.793196i	$0.291584\pi$
$110$	0	0
$111$	−1062.00	−0.908113
$112$	0	0
$113$	114.000	0.0949046	0.0474523	−	0.998874i	$-0.484890\pi$
$113$	114.000	0.0949046	0.0474523	+	0.998874i	$0.484890\pi$
$114$	0	0
$115$	1512.00	1.22604
$116$	0	0
$117$	−522.000	−0.412469
$118$	0	0
$119$	910.000	0.701005
$120$	0	0
$121$	605.000	0.454545
$122$	0	0
$123$	−1002.00	−0.734531
$124$	0	0
$125$	1332.00	0.953102
$126$	0	0
$127$	792.000	0.553375	0.276688	−	0.960960i	$-0.410763\pi$
$127$	792.000	0.553375	0.276688	+	0.960960i	$0.410763\pi$
$128$	0	0
$129$	−1248.00	−0.851785
$130$	0	0
$131$	428.000	0.285454	0.142727	−	0.989762i	$-0.454413\pi$
$131$	428.000	0.285454	0.142727	+	0.989762i	$0.454413\pi$
$132$	0	0
$133$	644.000	0.419864
$134$	0	0
$135$	−486.000	−0.309839
$136$	0	0
$137$	−2238.00	−1.39566	−0.697829	−	0.716264i	$-0.745851\pi$
$137$	−2238.00	−1.39566	−0.697829	+	0.716264i	$0.745851\pi$
$138$	0	0
$139$	−300.000	−0.183062	−0.0915312	−	0.995802i	$-0.529176\pi$
$139$	−300.000	−0.183062	−0.0915312	+	0.995802i	$0.529176\pi$
$140$	0	0
$141$	1392.00	0.831401
$142$	0	0
$143$	2552.00	1.49237
$144$	0	0
$145$	4500.00	2.57727
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	−1646.00	−0.905004	−0.452502	−	0.891763i	$-0.649468\pi$
$149$	−1646.00	−0.905004	−0.452502	+	0.891763i	$0.649468\pi$
$150$	0	0
$151$	−1184.00	−0.638096	−0.319048	−	0.947738i	$-0.603363\pi$
$151$	−1184.00	−0.638096	−0.319048	+	0.947738i	$0.603363\pi$
$152$	0	0
$153$	−1170.00	−0.618228
$154$	0	0
$155$	−1296.00	−0.671595
$156$	0	0
$157$	1150.00	0.584586	0.292293	−	0.956329i	$-0.405582\pi$
$157$	1150.00	0.584586	0.292293	+	0.956329i	$0.405582\pi$
$158$	0	0
$159$	−1350.00	−0.673346
$160$	0	0
$161$	−588.000	−0.287832
$162$	0	0
$163$	344.000	0.165302	0.0826508	−	0.996579i	$-0.473661\pi$
$163$	344.000	0.165302	0.0826508	+	0.996579i	$0.473661\pi$
$164$	0	0
$165$	2376.00	1.12104
$166$	0	0
$167$	−2304.00	−1.06760	−0.533799	−	0.845611i	$-0.679236\pi$
$167$	−2304.00	−1.06760	−0.533799	+	0.845611i	$0.679236\pi$
$168$	0	0
$169$	1167.00	0.531179
$170$	0	0
$171$	−828.000	−0.370285
$172$	0	0
$173$	−2742.00	−1.20503	−0.602516	−	0.798107i	$-0.705835\pi$
$173$	−2742.00	−1.20503	−0.602516	+	0.798107i	$0.705835\pi$
$174$	0	0
$175$	−1393.00	−0.601719
$176$	0	0
$177$	−1548.00	−0.657372
$178$	0	0
$179$	3940.00	1.64519	0.822596	−	0.568626i	$-0.192525\pi$
$179$	3940.00	1.64519	0.822596	+	0.568626i	$0.192525\pi$
$180$	0	0
$181$	−1970.00	−0.809000	−0.404500	−	0.914538i	$-0.632554\pi$
$181$	−1970.00	−0.809000	−0.404500	+	0.914538i	$0.632554\pi$
$182$	0	0
$183$	174.000	0.0702866
$184$	0	0
$185$	6372.00	2.53232
$186$	0	0
$187$	5720.00	2.23683
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−476.000	−0.180325	−0.0901627	−	0.995927i	$-0.528739\pi$
$191$	−476.000	−0.180325	−0.0901627	+	0.995927i	$0.528739\pi$
$192$	0	0
$193$	−782.000	−0.291656	−0.145828	−	0.989310i	$-0.546585\pi$
$193$	−782.000	−0.291656	−0.145828	+	0.989310i	$0.546585\pi$
$194$	0	0
$195$	3132.00	1.15019
$196$	0	0
$197$	2066.00	0.747190	0.373595	−	0.927592i	$-0.378125\pi$
$197$	2066.00	0.747190	0.373595	+	0.927592i	$0.378125\pi$
$198$	0	0
$199$	768.000	0.273578	0.136789	−	0.990600i	$-0.456322\pi$
$199$	768.000	0.273578	0.136789	+	0.990600i	$0.456322\pi$
$200$	0	0
$201$	−1968.00	−0.690607
$202$	0	0
$203$	−1750.00	−0.605054
$204$	0	0
$205$	6012.00	2.04827
$206$	0	0
$207$	756.000	0.253844
$208$	0	0
$209$	4048.00	1.33974
$210$	0	0
$211$	−4248.00	−1.38599	−0.692996	−	0.720941i	$-0.743710\pi$
$211$	−4248.00	−1.38599	−0.692996	+	0.720941i	$0.743710\pi$
$212$	0	0
$213$	2820.00	0.907151
$214$	0	0
$215$	7488.00	2.37524
$216$	0	0
$217$	504.000	0.157667
$218$	0	0
$219$	−534.000	−0.164769
$220$	0	0
$221$	7540.00	2.29500
$222$	0	0
$223$	−3496.00	−1.04982	−0.524909	−	0.851158i	$-0.675901\pi$
$223$	−3496.00	−1.04982	−0.524909	+	0.851158i	$0.675901\pi$
$224$	0	0
$225$	1791.00	0.530667
$226$	0	0
$227$	5620.00	1.64323	0.821613	−	0.570045i	$-0.193074\pi$
$227$	5620.00	1.64323	0.821613	+	0.570045i	$0.193074\pi$
$228$	0	0
$229$	1982.00	0.571940	0.285970	−	0.958239i	$-0.407684\pi$
$229$	1982.00	0.571940	0.285970	+	0.958239i	$0.407684\pi$
$230$	0	0
$231$	−924.000	−0.263181
$232$	0	0
$233$	−1342.00	−0.377328	−0.188664	−	0.982042i	$-0.560416\pi$
$233$	−1342.00	−0.377328	−0.188664	+	0.982042i	$0.560416\pi$
$234$	0	0
$235$	−8352.00	−2.31840
$236$	0	0
$237$	−3216.00	−0.881442
$238$	0	0
$239$	−2828.00	−0.765390	−0.382695	−	0.923875i	$-0.625004\pi$
$239$	−2828.00	−0.765390	−0.382695	+	0.923875i	$0.625004\pi$
$240$	0	0
$241$	2002.00	0.535104	0.267552	−	0.963543i	$-0.413785\pi$
$241$	2002.00	0.535104	0.267552	+	0.963543i	$0.413785\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	882.000	0.229996
$246$	0	0
$247$	5336.00	1.37458
$248$	0	0
$249$	1980.00	0.503925
$250$	0	0
$251$	−1188.00	−0.298749	−0.149374	−	0.988781i	$-0.547726\pi$
$251$	−1188.00	−0.298749	−0.149374	+	0.988781i	$0.547726\pi$
$252$	0	0
$253$	−3696.00	−0.918441
$254$	0	0
$255$	7020.00	1.72396
$256$	0	0
$257$	−5506.00	−1.33640	−0.668200	−	0.743982i	$-0.732935\pi$
$257$	−5506.00	−1.33640	−0.668200	+	0.743982i	$0.732935\pi$
$258$	0	0
$259$	−2478.00	−0.594500
$260$	0	0
$261$	2250.00	0.533607
$262$	0	0
$263$	4076.00	0.955654	0.477827	−	0.878454i	$-0.341425\pi$
$263$	4076.00	0.955654	0.477827	+	0.878454i	$0.341425\pi$
$264$	0	0
$265$	8100.00	1.87766
$266$	0	0
$267$	−3762.00	−0.862287
$268$	0	0
$269$	5938.00	1.34590	0.672948	−	0.739689i	$-0.265028\pi$
$269$	5938.00	1.34590	0.672948	+	0.739689i	$0.265028\pi$
$270$	0	0
$271$	592.000	0.132699	0.0663495	−	0.997796i	$-0.478865\pi$
$271$	592.000	0.132699	0.0663495	+	0.997796i	$0.478865\pi$
$272$	0	0
$273$	−1218.00	−0.270025
$274$	0	0
$275$	−8756.00	−1.92002
$276$	0	0
$277$	−5254.00	−1.13965	−0.569824	−	0.821767i	$-0.692988\pi$
$277$	−5254.00	−1.13965	−0.569824	+	0.821767i	$0.692988\pi$
$278$	0	0
$279$	−648.000	−0.139049
$280$	0	0
$281$	3410.00	0.723927	0.361964	−	0.932192i	$-0.382106\pi$
$281$	3410.00	0.723927	0.361964	+	0.932192i	$0.382106\pi$
$282$	0	0
$283$	2212.00	0.464628	0.232314	−	0.972641i	$-0.425370\pi$
$283$	2212.00	0.464628	0.232314	+	0.972641i	$0.425370\pi$
$284$	0	0
$285$	4968.00	1.03256
$286$	0	0
$287$	−2338.00	−0.480863
$288$	0	0
$289$	11987.0	2.43985
$290$	0	0
$291$	−630.000	−0.126912
$292$	0	0
$293$	2122.00	0.423101	0.211550	−	0.977367i	$-0.432149\pi$
$293$	2122.00	0.423101	0.211550	+	0.977367i	$0.432149\pi$
$294$	0	0
$295$	9288.00	1.83311
$296$	0	0
$297$	1188.00	0.232104
$298$	0	0
$299$	−4872.00	−0.942325
$300$	0	0
$301$	−2912.00	−0.557624
$302$	0	0
$303$	−558.000	−0.105796
$304$	0	0
$305$	−1044.00	−0.195998
$306$	0	0
$307$	−2588.00	−0.481124	−0.240562	−	0.970634i	$-0.577332\pi$
$307$	−2588.00	−0.481124	−0.240562	+	0.970634i	$0.577332\pi$
$308$	0	0
$309$	−1416.00	−0.260691
$310$	0	0
$311$	2728.00	0.497398	0.248699	−	0.968581i	$-0.419997\pi$
$311$	2728.00	0.497398	0.248699	+	0.968581i	$0.419997\pi$
$312$	0	0
$313$	−6446.00	−1.16406	−0.582028	−	0.813169i	$-0.697741\pi$
$313$	−6446.00	−1.16406	−0.582028	+	0.813169i	$0.697741\pi$
$314$	0	0
$315$	−1134.00	−0.202837
$316$	0	0
$317$	4234.00	0.750174	0.375087	−	0.926990i	$-0.377613\pi$
$317$	4234.00	0.750174	0.375087	+	0.926990i	$0.377613\pi$
$318$	0	0
$319$	−11000.0	−1.93066
$320$	0	0
$321$	−3636.00	−0.632217
$322$	0	0
$323$	11960.0	2.06029
$324$	0	0
$325$	−11542.0	−1.96995
$326$	0	0
$327$	−4158.00	−0.703174
$328$	0	0
$329$	3248.00	0.544280
$330$	0	0
$331$	4592.00	0.762535	0.381268	−	0.924465i	$-0.375488\pi$
$331$	4592.00	0.762535	0.381268	+	0.924465i	$0.375488\pi$
$332$	0	0
$333$	3186.00	0.524299
$334$	0	0
$335$	11808.0	1.92579
$336$	0	0
$337$	−1006.00	−0.162612	−0.0813061	−	0.996689i	$-0.525909\pi$
$337$	−1006.00	−0.162612	−0.0813061	+	0.996689i	$0.525909\pi$
$338$	0	0
$339$	−342.000	−0.0547932
$340$	0	0
$341$	3168.00	0.503099
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−4536.00	−0.707855
$346$	0	0
$347$	−4644.00	−0.718452	−0.359226	−	0.933251i	$-0.616959\pi$
$347$	−4644.00	−0.718452	−0.359226	+	0.933251i	$0.616959\pi$
$348$	0	0
$349$	−4786.00	−0.734065	−0.367033	−	0.930208i	$-0.619626\pi$
$349$	−4786.00	−0.734065	−0.367033	+	0.930208i	$0.619626\pi$
$350$	0	0
$351$	1566.00	0.238139
$352$	0	0
$353$	1302.00	0.196313	0.0981565	−	0.995171i	$-0.468705\pi$
$353$	1302.00	0.196313	0.0981565	+	0.995171i	$0.468705\pi$
$354$	0	0
$355$	−16920.0	−2.52963
$356$	0	0
$357$	−2730.00	−0.404725
$358$	0	0
$359$	11260.0	1.65538	0.827688	−	0.561188i	$-0.189656\pi$
$359$	11260.0	1.65538	0.827688	+	0.561188i	$0.189656\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	0	0
$363$	−1815.00	−0.262432
$364$	0	0
$365$	3204.00	0.459466
$366$	0	0
$367$	2792.00	0.397115	0.198558	−	0.980089i	$-0.436374\pi$
$367$	2792.00	0.397115	0.198558	+	0.980089i	$0.436374\pi$
$368$	0	0
$369$	3006.00	0.424082
$370$	0	0
$371$	−3150.00	−0.440808
$372$	0	0
$373$	−4118.00	−0.571641	−0.285820	−	0.958283i	$-0.592266\pi$
$373$	−4118.00	−0.571641	−0.285820	+	0.958283i	$0.592266\pi$
$374$	0	0
$375$	−3996.00	−0.550273
$376$	0	0
$377$	−14500.0	−1.98087
$378$	0	0
$379$	−8624.00	−1.16883	−0.584413	−	0.811456i	$-0.698675\pi$
$379$	−8624.00	−1.16883	−0.584413	+	0.811456i	$0.698675\pi$
$380$	0	0
$381$	−2376.00	−0.319491
$382$	0	0
$383$	6488.00	0.865591	0.432795	−	0.901492i	$-0.357527\pi$
$383$	6488.00	0.865591	0.432795	+	0.901492i	$0.357527\pi$
$384$	0	0
$385$	5544.00	0.733892
$386$	0	0
$387$	3744.00	0.491778
$388$	0	0
$389$	−1406.00	−0.183257	−0.0916286	−	0.995793i	$-0.529207\pi$
$389$	−1406.00	−0.183257	−0.0916286	+	0.995793i	$0.529207\pi$
$390$	0	0
$391$	−10920.0	−1.41240
$392$	0	0
$393$	−1284.00	−0.164807
$394$	0	0
$395$	19296.0	2.45794
$396$	0	0
$397$	−9378.00	−1.18556	−0.592781	−	0.805363i	$-0.701970\pi$
$397$	−9378.00	−1.18556	−0.592781	+	0.805363i	$0.701970\pi$
$398$	0	0
$399$	−1932.00	−0.242408
$400$	0	0
$401$	2890.00	0.359900	0.179950	−	0.983676i	$-0.442406\pi$
$401$	2890.00	0.359900	0.179950	+	0.983676i	$0.442406\pi$
$402$	0	0
$403$	4176.00	0.516182
$404$	0	0
$405$	1458.00	0.178885
$406$	0	0
$407$	−15576.0	−1.89699
$408$	0	0
$409$	−10582.0	−1.27933	−0.639665	−	0.768654i	$-0.720927\pi$
$409$	−10582.0	−1.27933	−0.639665	+	0.768654i	$0.720927\pi$
$410$	0	0
$411$	6714.00	0.805784
$412$	0	0
$413$	−3612.00	−0.430351
$414$	0	0
$415$	−11880.0	−1.40522
$416$	0	0
$417$	900.000	0.105691
$418$	0	0
$419$	9500.00	1.10765	0.553825	−	0.832633i	$-0.313168\pi$
$419$	9500.00	1.10765	0.553825	+	0.832633i	$0.313168\pi$
$420$	0	0
$421$	−598.000	−0.0692274	−0.0346137	−	0.999401i	$-0.511020\pi$
$421$	−598.000	−0.0692274	−0.0346137	+	0.999401i	$0.511020\pi$
$422$	0	0
$423$	−4176.00	−0.480010
$424$	0	0
$425$	−25870.0	−2.95266
$426$	0	0
$427$	406.000	0.0460134
$428$	0	0
$429$	−7656.00	−0.861620
$430$	0	0
$431$	3708.00	0.414404	0.207202	−	0.978298i	$-0.433564\pi$
$431$	3708.00	0.414404	0.207202	+	0.978298i	$0.433564\pi$
$432$	0	0
$433$	13706.0	1.52117	0.760587	−	0.649236i	$-0.224911\pi$
$433$	13706.0	1.52117	0.760587	+	0.649236i	$0.224911\pi$
$434$	0	0
$435$	−13500.0	−1.48799
$436$	0	0
$437$	−7728.00	−0.845951
$438$	0	0
$439$	−8232.00	−0.894970	−0.447485	−	0.894291i	$-0.647680\pi$
$439$	−8232.00	−0.894970	−0.447485	+	0.894291i	$0.647680\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−2524.00	−0.270697	−0.135349	−	0.990798i	$-0.543215\pi$
$443$	−2524.00	−0.270697	−0.135349	+	0.990798i	$0.543215\pi$
$444$	0	0
$445$	22572.0	2.40453
$446$	0	0
$447$	4938.00	0.522504
$448$	0	0
$449$	−3630.00	−0.381537	−0.190769	−	0.981635i	$-0.561098\pi$
$449$	−3630.00	−0.381537	−0.190769	+	0.981635i	$0.561098\pi$
$450$	0	0
$451$	−14696.0	−1.53438
$452$	0	0
$453$	3552.00	0.368405
$454$	0	0
$455$	7308.00	0.752977
$456$	0	0
$457$	5386.00	0.551305	0.275653	−	0.961257i	$-0.411106\pi$
$457$	5386.00	0.551305	0.275653	+	0.961257i	$0.411106\pi$
$458$	0	0
$459$	3510.00	0.356934
$460$	0	0
$461$	−11766.0	−1.18871	−0.594357	−	0.804201i	$-0.702593\pi$
$461$	−11766.0	−1.18871	−0.594357	+	0.804201i	$0.702593\pi$
$462$	0	0
$463$	−10240.0	−1.02785	−0.513923	−	0.857836i	$-0.671808\pi$
$463$	−10240.0	−1.02785	−0.513923	+	0.857836i	$0.671808\pi$
$464$	0	0
$465$	3888.00	0.387746
$466$	0	0
$467$	−6076.00	−0.602064	−0.301032	−	0.953614i	$-0.597331\pi$
$467$	−6076.00	−0.602064	−0.301032	+	0.953614i	$0.597331\pi$
$468$	0	0
$469$	−4592.00	−0.452108
$470$	0	0
$471$	−3450.00	−0.337511
$472$	0	0
$473$	−18304.0	−1.77932
$474$	0	0
$475$	−18308.0	−1.76848
$476$	0	0
$477$	4050.00	0.388756
$478$	0	0
$479$	6480.00	0.618118	0.309059	−	0.951043i	$-0.399986\pi$
$479$	6480.00	0.618118	0.309059	+	0.951043i	$0.399986\pi$
$480$	0	0
$481$	−20532.0	−1.94632
$482$	0	0
$483$	1764.00	0.166180
$484$	0	0
$485$	3780.00	0.353899
$486$	0	0
$487$	4240.00	0.394523	0.197262	−	0.980351i	$-0.436795\pi$
$487$	4240.00	0.394523	0.197262	+	0.980351i	$0.436795\pi$
$488$	0	0
$489$	−1032.00	−0.0954369
$490$	0	0
$491$	17892.0	1.64451	0.822255	−	0.569119i	$-0.192716\pi$
$491$	17892.0	1.64451	0.822255	+	0.569119i	$0.192716\pi$
$492$	0	0
$493$	−32500.0	−2.96902
$494$	0	0
$495$	−7128.00	−0.647232
$496$	0	0
$497$	6580.00	0.593870
$498$	0	0
$499$	4616.00	0.414109	0.207055	−	0.978329i	$-0.433612\pi$
$499$	4616.00	0.414109	0.207055	+	0.978329i	$0.433612\pi$
$500$	0	0
$501$	6912.00	0.616378
$502$	0	0
$503$	3696.00	0.327627	0.163814	−	0.986491i	$-0.447620\pi$
$503$	3696.00	0.327627	0.163814	+	0.986491i	$0.447620\pi$
$504$	0	0
$505$	3348.00	0.295018
$506$	0	0
$507$	−3501.00	−0.306676
$508$	0	0
$509$	16738.0	1.45756	0.728781	−	0.684747i	$-0.240087\pi$
$509$	16738.0	1.45756	0.728781	+	0.684747i	$0.240087\pi$
$510$	0	0
$511$	−1246.00	−0.107867
$512$	0	0
$513$	2484.00	0.213784
$514$	0	0
$515$	8496.00	0.726949
$516$	0	0
$517$	20416.0	1.73674
$518$	0	0
$519$	8226.00	0.695725
$520$	0	0
$521$	19062.0	1.60292	0.801460	−	0.598048i	$-0.204057\pi$
$521$	19062.0	1.60292	0.801460	+	0.598048i	$0.204057\pi$
$522$	0	0
$523$	12268.0	1.02570	0.512851	−	0.858478i	$-0.328589\pi$
$523$	12268.0	1.02570	0.512851	+	0.858478i	$0.328589\pi$
$524$	0	0
$525$	4179.00	0.347403
$526$	0	0
$527$	9360.00	0.773677
$528$	0	0
$529$	−5111.00	−0.420071
$530$	0	0
$531$	4644.00	0.379534
$532$	0	0
$533$	−19372.0	−1.57429
$534$	0	0
$535$	21816.0	1.76297
$536$	0	0
$537$	−11820.0	−0.949852
$538$	0	0
$539$	−2156.00	−0.172292
$540$	0	0
$541$	17042.0	1.35433	0.677165	−	0.735831i	$-0.263208\pi$
$541$	17042.0	1.35433	0.677165	+	0.735831i	$0.263208\pi$
$542$	0	0
$543$	5910.00	0.467076
$544$	0	0
$545$	24948.0	1.96083
$546$	0	0
$547$	3656.00	0.285776	0.142888	−	0.989739i	$-0.454361\pi$
$547$	3656.00	0.285776	0.142888	+	0.989739i	$0.454361\pi$
$548$	0	0
$549$	−522.000	−0.0405800
$550$	0	0
$551$	−23000.0	−1.77828
$552$	0	0
$553$	−7504.00	−0.577039
$554$	0	0
$555$	−19116.0	−1.46203
$556$	0	0
$557$	−14038.0	−1.06788	−0.533940	−	0.845522i	$-0.679289\pi$
$557$	−14038.0	−1.06788	−0.533940	+	0.845522i	$0.679289\pi$
$558$	0	0
$559$	−24128.0	−1.82559
$560$	0	0
$561$	−17160.0	−1.29144
$562$	0	0
$563$	18332.0	1.37229	0.686147	−	0.727463i	$-0.259301\pi$
$563$	18332.0	1.37229	0.686147	+	0.727463i	$0.259301\pi$
$564$	0	0
$565$	2052.00	0.152793
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−10046.0	−0.740159	−0.370079	−	0.929000i	$-0.620669\pi$
$569$	−10046.0	−0.740159	−0.370079	+	0.929000i	$0.620669\pi$
$570$	0	0
$571$	5704.00	0.418047	0.209024	−	0.977911i	$-0.432971\pi$
$571$	5704.00	0.418047	0.209024	+	0.977911i	$0.432971\pi$
$572$	0	0
$573$	1428.00	0.104111
$574$	0	0
$575$	16716.0	1.21236
$576$	0	0
$577$	24610.0	1.77561	0.887806	−	0.460219i	$-0.152229\pi$
$577$	24610.0	1.77561	0.887806	+	0.460219i	$0.152229\pi$
$578$	0	0
$579$	2346.00	0.168388
$580$	0	0
$581$	4620.00	0.329897
$582$	0	0
$583$	−19800.0	−1.40657
$584$	0	0
$585$	−9396.00	−0.664063
$586$	0	0
$587$	−18516.0	−1.30194	−0.650969	−	0.759105i	$-0.725637\pi$
$587$	−18516.0	−1.30194	−0.650969	+	0.759105i	$0.725637\pi$
$588$	0	0
$589$	6624.00	0.463391
$590$	0	0
$591$	−6198.00	−0.431390
$592$	0	0
$593$	20038.0	1.38763	0.693813	−	0.720155i	$-0.255929\pi$
$593$	20038.0	1.38763	0.693813	+	0.720155i	$0.255929\pi$
$594$	0	0
$595$	16380.0	1.12860
$596$	0	0
$597$	−2304.00	−0.157950
$598$	0	0
$599$	2596.00	0.177078	0.0885390	−	0.996073i	$-0.471780\pi$
$599$	2596.00	0.177078	0.0885390	+	0.996073i	$0.471780\pi$
$600$	0	0
$601$	−5190.00	−0.352254	−0.176127	−	0.984367i	$-0.556357\pi$
$601$	−5190.00	−0.352254	−0.176127	+	0.984367i	$0.556357\pi$
$602$	0	0
$603$	5904.00	0.398722
$604$	0	0
$605$	10890.0	0.731804
$606$	0	0
$607$	6536.00	0.437048	0.218524	−	0.975832i	$-0.429876\pi$
$607$	6536.00	0.437048	0.218524	+	0.975832i	$0.429876\pi$
$608$	0	0
$609$	5250.00	0.349328
$610$	0	0
$611$	26912.0	1.78190
$612$	0	0
$613$	−4702.00	−0.309807	−0.154904	−	0.987930i	$-0.549507\pi$
$613$	−4702.00	−0.309807	−0.154904	+	0.987930i	$0.549507\pi$
$614$	0	0
$615$	−18036.0	−1.18257
$616$	0	0
$617$	−8638.00	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−8638.00	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	0	0
$619$	19676.0	1.27762	0.638809	−	0.769366i	$-0.279428\pi$
$619$	19676.0	1.27762	0.638809	+	0.769366i	$0.279428\pi$
$620$	0	0
$621$	−2268.00	−0.146557
$622$	0	0
$623$	−8778.00	−0.564499
$624$	0	0
$625$	−899.000	−0.0575360
$626$	0	0
$627$	−12144.0	−0.773500
$628$	0	0
$629$	−46020.0	−2.91723
$630$	0	0
$631$	26720.0	1.68575	0.842874	−	0.538112i	$-0.180862\pi$
$631$	26720.0	1.68575	0.842874	+	0.538112i	$0.180862\pi$
$632$	0	0
$633$	12744.0	0.800203
$634$	0	0
$635$	14256.0	0.890917
$636$	0	0
$637$	−2842.00	−0.176773
$638$	0	0
$639$	−8460.00	−0.523744
$640$	0	0
$641$	−1990.00	−0.122621	−0.0613107	−	0.998119i	$-0.519528\pi$
$641$	−1990.00	−0.122621	−0.0613107	+	0.998119i	$0.519528\pi$
$642$	0	0
$643$	6956.00	0.426622	0.213311	−	0.976984i	$-0.431575\pi$
$643$	6956.00	0.426622	0.213311	+	0.976984i	$0.431575\pi$
$644$	0	0
$645$	−22464.0	−1.37135
$646$	0	0
$647$	−15984.0	−0.971246	−0.485623	−	0.874168i	$-0.661407\pi$
$647$	−15984.0	−0.971246	−0.485623	+	0.874168i	$0.661407\pi$
$648$	0	0
$649$	−22704.0	−1.37320
$650$	0	0
$651$	−1512.00	−0.0910291
$652$	0	0
$653$	−6614.00	−0.396364	−0.198182	−	0.980165i	$-0.563504\pi$
$653$	−6614.00	−0.396364	−0.198182	+	0.980165i	$0.563504\pi$
$654$	0	0
$655$	7704.00	0.459573
$656$	0	0
$657$	1602.00	0.0951293
$658$	0	0
$659$	29364.0	1.73575	0.867875	−	0.496783i	$-0.165485\pi$
$659$	29364.0	1.73575	0.867875	+	0.496783i	$0.165485\pi$
$660$	0	0
$661$	3150.00	0.185357	0.0926784	−	0.995696i	$-0.470457\pi$
$661$	3150.00	0.185357	0.0926784	+	0.995696i	$0.470457\pi$
$662$	0	0
$663$	−22620.0	−1.32502
$664$	0	0
$665$	11592.0	0.675968
$666$	0	0
$667$	21000.0	1.21908
$668$	0	0
$669$	10488.0	0.606113
$670$	0	0
$671$	2552.00	0.146824
$672$	0	0
$673$	8402.00	0.481238	0.240619	−	0.970620i	$-0.422650\pi$
$673$	8402.00	0.481238	0.240619	+	0.970620i	$0.422650\pi$
$674$	0	0
$675$	−5373.00	−0.306381
$676$	0	0
$677$	−7854.00	−0.445870	−0.222935	−	0.974833i	$-0.571564\pi$
$677$	−7854.00	−0.445870	−0.222935	+	0.974833i	$0.571564\pi$
$678$	0	0
$679$	−1470.00	−0.0830831
$680$	0	0
$681$	−16860.0	−0.948717
$682$	0	0
$683$	14244.0	0.797996	0.398998	−	0.916952i	$-0.369358\pi$
$683$	14244.0	0.797996	0.398998	+	0.916952i	$0.369358\pi$
$684$	0	0
$685$	−40284.0	−2.24697
$686$	0	0
$687$	−5946.00	−0.330210
$688$	0	0
$689$	−26100.0	−1.44315
$690$	0	0
$691$	−22420.0	−1.23429	−0.617147	−	0.786848i	$-0.711712\pi$
$691$	−22420.0	−1.23429	−0.617147	+	0.786848i	$0.711712\pi$
$692$	0	0
$693$	2772.00	0.151947
$694$	0	0
$695$	−5400.00	−0.294725
$696$	0	0
$697$	−43420.0	−2.35961
$698$	0	0
$699$	4026.00	0.217850
$700$	0	0
$701$	−19814.0	−1.06757	−0.533783	−	0.845621i	$-0.679230\pi$
$701$	−19814.0	−1.06757	−0.533783	+	0.845621i	$0.679230\pi$
$702$	0	0
$703$	−32568.0	−1.74726
$704$	0	0
$705$	25056.0	1.33853
$706$	0	0
$707$	−1302.00	−0.0692599
$708$	0	0
$709$	15986.0	0.846780	0.423390	−	0.905948i	$-0.360840\pi$
$709$	15986.0	0.846780	0.423390	+	0.905948i	$0.360840\pi$
$710$	0	0
$711$	9648.00	0.508901
$712$	0	0
$713$	−6048.00	−0.317671
$714$	0	0
$715$	45936.0	2.40267
$716$	0	0
$717$	8484.00	0.441898
$718$	0	0
$719$	22440.0	1.16394	0.581969	−	0.813211i	$-0.302283\pi$
$719$	22440.0	1.16394	0.581969	+	0.813211i	$0.302283\pi$
$720$	0	0
$721$	−3304.00	−0.170662
$722$	0	0
$723$	−6006.00	−0.308943
$724$	0	0
$725$	49750.0	2.54851
$726$	0	0
$727$	10264.0	0.523619	0.261809	−	0.965120i	$-0.415681\pi$
$727$	10264.0	0.523619	0.261809	+	0.965120i	$0.415681\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−54080.0	−2.73628
$732$	0	0
$733$	−9282.00	−0.467720	−0.233860	−	0.972270i	$-0.575136\pi$
$733$	−9282.00	−0.467720	−0.233860	+	0.972270i	$0.575136\pi$
$734$	0	0
$735$	−2646.00	−0.132788
$736$	0	0
$737$	−28864.0	−1.44263
$738$	0	0
$739$	−12792.0	−0.636754	−0.318377	−	0.947964i	$-0.603138\pi$
$739$	−12792.0	−0.636754	−0.318377	+	0.947964i	$0.603138\pi$
$740$	0	0
$741$	−16008.0	−0.793615
$742$	0	0
$743$	25644.0	1.26620	0.633100	−	0.774070i	$-0.281782\pi$
$743$	25644.0	1.26620	0.633100	+	0.774070i	$0.281782\pi$
$744$	0	0
$745$	−29628.0	−1.45703
$746$	0	0
$747$	−5940.00	−0.290941
$748$	0	0
$749$	−8484.00	−0.413883
$750$	0	0
$751$	−4528.00	−0.220012	−0.110006	−	0.993931i	$-0.535087\pi$
$751$	−4528.00	−0.220012	−0.110006	+	0.993931i	$0.535087\pi$
$752$	0	0
$753$	3564.00	0.172483
$754$	0	0
$755$	−21312.0	−1.02732
$756$	0	0
$757$	−31310.0	−1.50328	−0.751639	−	0.659575i	$-0.770736\pi$
$757$	−31310.0	−1.50328	−0.751639	+	0.659575i	$0.770736\pi$
$758$	0	0
$759$	11088.0	0.530262
$760$	0	0
$761$	16622.0	0.791783	0.395892	−	0.918297i	$-0.370436\pi$
$761$	16622.0	0.791783	0.395892	+	0.918297i	$0.370436\pi$
$762$	0	0
$763$	−9702.00	−0.460335
$764$	0	0
$765$	−21060.0	−0.995328
$766$	0	0
$767$	−29928.0	−1.40891
$768$	0	0
$769$	−9814.00	−0.460211	−0.230105	−	0.973166i	$-0.573907\pi$
$769$	−9814.00	−0.460211	−0.230105	+	0.973166i	$0.573907\pi$
$770$	0	0
$771$	16518.0	0.771571
$772$	0	0
$773$	−7686.00	−0.357628	−0.178814	−	0.983883i	$-0.557226\pi$
$773$	−7686.00	−0.357628	−0.178814	+	0.983883i	$0.557226\pi$
$774$	0	0
$775$	−14328.0	−0.664099
$776$	0	0
$777$	7434.00	0.343235
$778$	0	0
$779$	−30728.0	−1.41328
$780$	0	0
$781$	41360.0	1.89498
$782$	0	0
$783$	−6750.00	−0.308078
$784$	0	0
$785$	20700.0	0.941165
$786$	0	0
$787$	−5860.00	−0.265421	−0.132711	−	0.991155i	$-0.542368\pi$
$787$	−5860.00	−0.265421	−0.132711	+	0.991155i	$0.542368\pi$
$788$	0	0
$789$	−12228.0	−0.551747
$790$	0	0
$791$	−798.000	−0.0358706
$792$	0	0
$793$	3364.00	0.150642
$794$	0	0
$795$	−24300.0	−1.08407
$796$	0	0
$797$	15450.0	0.686659	0.343329	−	0.939215i	$-0.388445\pi$
$797$	15450.0	0.686659	0.343329	+	0.939215i	$0.388445\pi$
$798$	0	0
$799$	60320.0	2.67080
$800$	0	0
$801$	11286.0	0.497842
$802$	0	0
$803$	−7832.00	−0.344191
$804$	0	0
$805$	−10584.0	−0.463400
$806$	0	0
$807$	−17814.0	−0.777054
$808$	0	0
$809$	−26726.0	−1.16148	−0.580739	−	0.814090i	$-0.697236\pi$
$809$	−26726.0	−1.16148	−0.580739	+	0.814090i	$0.697236\pi$
$810$	0	0
$811$	3052.00	0.132146	0.0660729	−	0.997815i	$-0.478953\pi$
$811$	3052.00	0.132146	0.0660729	+	0.997815i	$0.478953\pi$
$812$	0	0
$813$	−1776.00	−0.0766138
$814$	0	0
$815$	6192.00	0.266130
$816$	0	0
$817$	−38272.0	−1.63888
$818$	0	0
$819$	3654.00	0.155899
$820$	0	0
$821$	−23838.0	−1.01334	−0.506670	−	0.862140i	$-0.669124\pi$
$821$	−23838.0	−1.01334	−0.506670	+	0.862140i	$0.669124\pi$
$822$	0	0
$823$	19136.0	0.810497	0.405248	−	0.914207i	$-0.367185\pi$
$823$	19136.0	0.810497	0.405248	+	0.914207i	$0.367185\pi$
$824$	0	0
$825$	26268.0	1.10853
$826$	0	0
$827$	32556.0	1.36890	0.684452	−	0.729058i	$-0.260042\pi$
$827$	32556.0	1.36890	0.684452	+	0.729058i	$0.260042\pi$
$828$	0	0
$829$	33086.0	1.38616	0.693079	−	0.720862i	$-0.256254\pi$
$829$	33086.0	1.38616	0.693079	+	0.720862i	$0.256254\pi$
$830$	0	0
$831$	15762.0	0.657976
$832$	0	0
$833$	−6370.00	−0.264955
$834$	0	0
$835$	−41472.0	−1.71880
$836$	0	0
$837$	1944.00	0.0802801
$838$	0	0
$839$	35248.0	1.45041	0.725206	−	0.688532i	$-0.241744\pi$
$839$	35248.0	1.45041	0.725206	+	0.688532i	$0.241744\pi$
$840$	0	0
$841$	38111.0	1.56263
$842$	0	0
$843$	−10230.0	−0.417960
$844$	0	0
$845$	21006.0	0.855182
$846$	0	0
$847$	−4235.00	−0.171802
$848$	0	0
$849$	−6636.00	−0.268253
$850$	0	0
$851$	29736.0	1.19781
$852$	0	0
$853$	−8922.00	−0.358128	−0.179064	−	0.983837i	$-0.557307\pi$
$853$	−8922.00	−0.358128	−0.179064	+	0.983837i	$0.557307\pi$
$854$	0	0
$855$	−14904.0	−0.596147
$856$	0	0
$857$	28126.0	1.12108	0.560540	−	0.828127i	$-0.310594\pi$
$857$	28126.0	1.12108	0.560540	+	0.828127i	$0.310594\pi$
$858$	0	0
$859$	−28916.0	−1.14855	−0.574273	−	0.818664i	$-0.694715\pi$
$859$	−28916.0	−1.14855	−0.574273	+	0.818664i	$0.694715\pi$
$860$	0	0
$861$	7014.00	0.277627
$862$	0	0
$863$	−22308.0	−0.879923	−0.439961	−	0.898017i	$-0.645008\pi$
$863$	−22308.0	−0.879923	−0.439961	+	0.898017i	$0.645008\pi$
$864$	0	0
$865$	−49356.0	−1.94006
$866$	0	0
$867$	−35961.0	−1.40865
$868$	0	0
$869$	−47168.0	−1.84127
$870$	0	0
$871$	−38048.0	−1.48015
$872$	0	0
$873$	1890.00	0.0732724
$874$	0	0
$875$	−9324.00	−0.360239
$876$	0	0
$877$	34970.0	1.34647	0.673234	−	0.739429i	$-0.264905\pi$
$877$	34970.0	1.34647	0.673234	+	0.739429i	$0.264905\pi$
$878$	0	0
$879$	−6366.00	−0.244277
$880$	0	0
$881$	−858.000	−0.0328113	−0.0164056	−	0.999865i	$-0.505222\pi$
$881$	−858.000	−0.0328113	−0.0164056	+	0.999865i	$0.505222\pi$
$882$	0	0
$883$	−24088.0	−0.918036	−0.459018	−	0.888427i	$-0.651799\pi$
$883$	−24088.0	−0.918036	−0.459018	+	0.888427i	$0.651799\pi$
$884$	0	0
$885$	−27864.0	−1.05835
$886$	0	0
$887$	−30960.0	−1.17197	−0.585984	−	0.810323i	$-0.699292\pi$
$887$	−30960.0	−1.17197	−0.585984	+	0.810323i	$0.699292\pi$
$888$	0	0
$889$	−5544.00	−0.209156
$890$	0	0
$891$	−3564.00	−0.134005
$892$	0	0
$893$	42688.0	1.59966
$894$	0	0
$895$	70920.0	2.64871
$896$	0	0
$897$	14616.0	0.544051
$898$	0	0
$899$	−18000.0	−0.667779
$900$	0	0
$901$	−58500.0	−2.16306
$902$	0	0
$903$	8736.00	0.321944
$904$	0	0
$905$	−35460.0	−1.30246
$906$	0	0
$907$	37048.0	1.35629	0.678147	−	0.734926i	$-0.262783\pi$
$907$	37048.0	1.35629	0.678147	+	0.734926i	$0.262783\pi$
$908$	0	0
$909$	1674.00	0.0610815
$910$	0	0
$911$	25228.0	0.917498	0.458749	−	0.888566i	$-0.348298\pi$
$911$	25228.0	0.917498	0.458749	+	0.888566i	$0.348298\pi$
$912$	0	0
$913$	29040.0	1.05267
$914$	0	0
$915$	3132.00	0.113159
$916$	0	0
$917$	−2996.00	−0.107892
$918$	0	0
$919$	−19336.0	−0.694054	−0.347027	−	0.937855i	$-0.612809\pi$
$919$	−19336.0	−0.694054	−0.347027	+	0.937855i	$0.612809\pi$
$920$	0	0
$921$	7764.00	0.277777
$922$	0	0
$923$	54520.0	1.94426
$924$	0	0
$925$	70446.0	2.50405
$926$	0	0
$927$	4248.00	0.150510
$928$	0	0
$929$	11926.0	0.421183	0.210592	−	0.977574i	$-0.432461\pi$
$929$	11926.0	0.421183	0.210592	+	0.977574i	$0.432461\pi$
$930$	0	0
$931$	−4508.00	−0.158694
$932$	0	0
$933$	−8184.00	−0.287173
$934$	0	0
$935$	102960.	3.60123
$936$	0	0
$937$	4698.00	0.163796	0.0818981	−	0.996641i	$-0.473902\pi$
$937$	4698.00	0.163796	0.0818981	+	0.996641i	$0.473902\pi$
$938$	0	0
$939$	19338.0	0.672068
$940$	0	0
$941$	12986.0	0.449874	0.224937	−	0.974373i	$-0.427782\pi$
$941$	12986.0	0.449874	0.224937	+	0.974373i	$0.427782\pi$
$942$	0	0
$943$	28056.0	0.968854
$944$	0	0
$945$	3402.00	0.117108
$946$	0	0
$947$	17972.0	0.616696	0.308348	−	0.951274i	$-0.400224\pi$
$947$	17972.0	0.616696	0.308348	+	0.951274i	$0.400224\pi$
$948$	0	0
$949$	−10324.0	−0.353141
$950$	0	0
$951$	−12702.0	−0.433113
$952$	0	0
$953$	−5414.00	−0.184026	−0.0920129	−	0.995758i	$-0.529330\pi$
$953$	−5414.00	−0.184026	−0.0920129	+	0.995758i	$0.529330\pi$
$954$	0	0
$955$	−8568.00	−0.290318
$956$	0	0
$957$	33000.0	1.11467
$958$	0	0
$959$	15666.0	0.527509
$960$	0	0
$961$	−24607.0	−0.825988
$962$	0	0
$963$	10908.0	0.365011
$964$	0	0
$965$	−14076.0	−0.469557
$966$	0	0
$967$	−57496.0	−1.91204	−0.956022	−	0.293295i	$-0.905248\pi$
$967$	−57496.0	−1.91204	−0.956022	+	0.293295i	$0.905248\pi$
$968$	0	0
$969$	−35880.0	−1.18951
$970$	0	0
$971$	36812.0	1.21664	0.608318	−	0.793693i	$-0.291845\pi$
$971$	36812.0	1.21664	0.608318	+	0.793693i	$0.291845\pi$
$972$	0	0
$973$	2100.00	0.0691911
$974$	0	0
$975$	34626.0	1.13735
$976$	0	0
$977$	26442.0	0.865870	0.432935	−	0.901425i	$-0.357478\pi$
$977$	26442.0	0.865870	0.432935	+	0.901425i	$0.357478\pi$
$978$	0	0
$979$	−55176.0	−1.80126
$980$	0	0
$981$	12474.0	0.405978
$982$	0	0
$983$	35240.0	1.14342	0.571710	−	0.820456i	$-0.306280\pi$
$983$	35240.0	1.14342	0.571710	+	0.820456i	$0.306280\pi$
$984$	0	0
$985$	37188.0	1.20295
$986$	0	0
$987$	−9744.00	−0.314240
$988$	0	0
$989$	34944.0	1.12351
$990$	0	0
$991$	−36472.0	−1.16909	−0.584547	−	0.811360i	$-0.698728\pi$
$991$	−36472.0	−1.16909	−0.584547	+	0.811360i	$0.698728\pi$
$992$	0	0
$993$	−13776.0	−0.440250
$994$	0	0
$995$	13824.0	0.440453
$996$	0	0
$997$	−25090.0	−0.796999	−0.398500	−	0.917168i	$-0.630469\pi$
$997$	−25090.0	−0.796999	−0.398500	+	0.917168i	$0.630469\pi$
$998$	0	0
$999$	−9558.00	−0.302704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.m.1.1		1
4.3	odd	2		1344.4.a.bb.1.1		1
8.3	odd	2		672.4.a.a.1.1	✓	1
8.5	even	2		672.4.a.c.1.1	yes	1
24.5	odd	2		2016.4.a.e.1.1		1
24.11	even	2		2016.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.a.1.1	✓	1	8.3	odd	2
672.4.a.c.1.1	yes	1	8.5	even	2
1344.4.a.m.1.1		1	1.1	even	1	trivial
1344.4.a.bb.1.1		1	4.3	odd	2
2016.4.a.e.1.1		1	24.5	odd	2
2016.4.a.f.1.1		1	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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