LMFDB - Embedded newform 1456.4.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1456,4,Mod(1,1456)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1456.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1456, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1456.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,2,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$85.9067809684$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1456.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{3} -5.00000 q^{5} +7.00000 q^{7} -23.0000 q^{9} +36.0000 q^{11} -13.0000 q^{13} -10.0000 q^{15} +26.0000 q^{17} +47.0000 q^{19} +14.0000 q^{21} +99.0000 q^{23} -100.000 q^{25} -100.000 q^{27} -61.0000 q^{29} +23.0000 q^{31} +72.0000 q^{33} -35.0000 q^{35} -50.0000 q^{37} -26.0000 q^{39} +70.0000 q^{41} +19.0000 q^{43} +115.000 q^{45} -191.000 q^{47} +49.0000 q^{49} +52.0000 q^{51} +195.000 q^{53} -180.000 q^{55} +94.0000 q^{57} -264.000 q^{59} +310.000 q^{61} -161.000 q^{63} +65.0000 q^{65} +190.000 q^{67} +198.000 q^{69} +166.000 q^{71} +873.000 q^{73} -200.000 q^{75} +252.000 q^{77} +1191.00 q^{79} +421.000 q^{81} -259.000 q^{83} -130.000 q^{85} -122.000 q^{87} -635.000 q^{89} -91.0000 q^{91} +46.0000 q^{93} -235.000 q^{95} +133.000 q^{97} -828.000 q^{99} +O(q^{100})$

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
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	0	0
$5$	−5.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−5.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−23.0000	−0.851852
$10$	0	0
$11$	36.0000	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$11$	36.0000	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−10.0000	−0.172133
$16$	0	0
$17$	26.0000	0.370937	0.185468	−	0.982650i	$-0.440620\pi$
$17$	26.0000	0.370937	0.185468	+	0.982650i	$0.440620\pi$
$18$	0	0
$19$	47.0000	0.567502	0.283751	−	0.958898i	$-0.408421\pi$
$19$	47.0000	0.567502	0.283751	+	0.958898i	$0.408421\pi$
$20$	0	0
$21$	14.0000	0.145479
$22$	0	0
$23$	99.0000	0.897519	0.448759	−	0.893653i	$-0.351866\pi$
$23$	99.0000	0.897519	0.448759	+	0.893653i	$0.351866\pi$
$24$	0	0
$25$	−100.000	−0.800000
$26$	0	0
$27$	−100.000	−0.712778
$28$	0	0
$29$	−61.0000	−0.390601	−0.195300	−	0.980743i	$-0.562568\pi$
$29$	−61.0000	−0.390601	−0.195300	+	0.980743i	$0.562568\pi$
$30$	0	0
$31$	23.0000	0.133256	0.0666278	−	0.997778i	$-0.478776\pi$
$31$	23.0000	0.133256	0.0666278	+	0.997778i	$0.478776\pi$
$32$	0	0
$33$	72.0000	0.379806
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	−50.0000	−0.222161	−0.111080	−	0.993811i	$-0.535431\pi$
$37$	−50.0000	−0.222161	−0.111080	+	0.993811i	$0.535431\pi$
$38$	0	0
$39$	−26.0000	−0.106752
$40$	0	0
$41$	70.0000	0.266638	0.133319	−	0.991073i	$-0.457436\pi$
$41$	70.0000	0.266638	0.133319	+	0.991073i	$0.457436\pi$
$42$	0	0
$43$	19.0000	0.0673831	0.0336915	−	0.999432i	$-0.489274\pi$
$43$	19.0000	0.0673831	0.0336915	+	0.999432i	$0.489274\pi$
$44$	0	0
$45$	115.000	0.380960
$46$	0	0
$47$	−191.000	−0.592770	−0.296385	−	0.955068i	$-0.595781\pi$
$47$	−191.000	−0.592770	−0.296385	+	0.955068i	$0.595781\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	52.0000	0.142774
$52$	0	0
$53$	195.000	0.505383	0.252692	−	0.967547i	$-0.418684\pi$
$53$	195.000	0.505383	0.252692	+	0.967547i	$0.418684\pi$
$54$	0	0
$55$	−180.000	−0.441294
$56$	0	0
$57$	94.0000	0.218432
$58$	0	0
$59$	−264.000	−0.582540	−0.291270	−	0.956641i	$-0.594078\pi$
$59$	−264.000	−0.582540	−0.291270	+	0.956641i	$0.594078\pi$
$60$	0	0
$61$	310.000	0.650679	0.325340	−	0.945597i	$-0.394521\pi$
$61$	310.000	0.650679	0.325340	+	0.945597i	$0.394521\pi$
$62$	0	0
$63$	−161.000	−0.321970
$64$	0	0
$65$	65.0000	0.124035
$66$	0	0
$67$	190.000	0.346451	0.173225	−	0.984882i	$-0.444581\pi$
$67$	190.000	0.346451	0.173225	+	0.984882i	$0.444581\pi$
$68$	0	0
$69$	198.000	0.345455
$70$	0	0
$71$	166.000	0.277473	0.138736	−	0.990329i	$-0.455696\pi$
$71$	166.000	0.277473	0.138736	+	0.990329i	$0.455696\pi$
$72$	0	0
$73$	873.000	1.39968	0.699842	−	0.714298i	$-0.253254\pi$
$73$	873.000	1.39968	0.699842	+	0.714298i	$0.253254\pi$
$74$	0	0
$75$	−200.000	−0.307920
$76$	0	0
$77$	252.000	0.372962
$78$	0	0
$79$	1191.00	1.69618	0.848088	−	0.529855i	$-0.177754\pi$
$79$	1191.00	1.69618	0.848088	+	0.529855i	$0.177754\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	−259.000	−0.342517	−0.171259	−	0.985226i	$-0.554783\pi$
$83$	−259.000	−0.342517	−0.171259	+	0.985226i	$0.554783\pi$
$84$	0	0
$85$	−130.000	−0.165888
$86$	0	0
$87$	−122.000	−0.150342
$88$	0	0
$89$	−635.000	−0.756291	−0.378145	−	0.925746i	$-0.623438\pi$
$89$	−635.000	−0.756291	−0.378145	+	0.925746i	$0.623438\pi$
$90$	0	0
$91$	−91.0000	−0.104828
$92$	0	0
$93$	46.0000	0.0512901
$94$	0	0
$95$	−235.000	−0.253795
$96$	0	0
$97$	133.000	0.139218	0.0696088	−	0.997574i	$-0.477825\pi$
$97$	133.000	0.139218	0.0696088	+	0.997574i	$0.477825\pi$
$98$	0	0
$99$	−828.000	−0.840577
$100$	0	0
$101$	152.000	0.149748	0.0748741	−	0.997193i	$-0.476145\pi$
$101$	152.000	0.149748	0.0748741	+	0.997193i	$0.476145\pi$
$102$	0	0
$103$	1664.00	1.59183	0.795916	−	0.605406i	$-0.206989\pi$
$103$	1664.00	1.59183	0.795916	+	0.605406i	$0.206989\pi$
$104$	0	0
$105$	−70.0000	−0.0650600
$106$	0	0
$107$	36.0000	0.0325257	0.0162629	−	0.999868i	$-0.494823\pi$
$107$	36.0000	0.0325257	0.0162629	+	0.999868i	$0.494823\pi$
$108$	0	0
$109$	−232.000	−0.203868	−0.101934	−	0.994791i	$-0.532503\pi$
$109$	−232.000	−0.203868	−0.101934	+	0.994791i	$0.532503\pi$
$110$	0	0
$111$	−100.000	−0.0855097
$112$	0	0
$113$	1353.00	1.12637	0.563184	−	0.826332i	$-0.309576\pi$
$113$	1353.00	1.12637	0.563184	+	0.826332i	$0.309576\pi$
$114$	0	0
$115$	−495.000	−0.401383
$116$	0	0
$117$	299.000	0.236261
$118$	0	0
$119$	182.000	0.140201
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	140.000	0.102629
$124$	0	0
$125$	1125.00	0.804984
$126$	0	0
$127$	−576.000	−0.402455	−0.201227	−	0.979545i	$-0.564493\pi$
$127$	−576.000	−0.402455	−0.201227	+	0.979545i	$0.564493\pi$
$128$	0	0
$129$	38.0000	0.0259358
$130$	0	0
$131$	2056.00	1.37125	0.685624	−	0.727956i	$-0.259529\pi$
$131$	2056.00	1.37125	0.685624	+	0.727956i	$0.259529\pi$
$132$	0	0
$133$	329.000	0.214496
$134$	0	0
$135$	500.000	0.318764
$136$	0	0
$137$	−1842.00	−1.14871	−0.574353	−	0.818608i	$-0.694746\pi$
$137$	−1842.00	−1.14871	−0.574353	+	0.818608i	$0.694746\pi$
$138$	0	0
$139$	1288.00	0.785948	0.392974	−	0.919550i	$-0.371446\pi$
$139$	1288.00	0.785948	0.392974	+	0.919550i	$0.371446\pi$
$140$	0	0
$141$	−382.000	−0.228157
$142$	0	0
$143$	−468.000	−0.273679
$144$	0	0
$145$	305.000	0.174682
$146$	0	0
$147$	98.0000	0.0549857
$148$	0	0
$149$	1196.00	0.657585	0.328792	−	0.944402i	$-0.393358\pi$
$149$	1196.00	0.657585	0.328792	+	0.944402i	$0.393358\pi$
$150$	0	0
$151$	890.000	0.479650	0.239825	−	0.970816i	$-0.422910\pi$
$151$	890.000	0.479650	0.239825	+	0.970816i	$0.422910\pi$
$152$	0	0
$153$	−598.000	−0.315983
$154$	0	0
$155$	−115.000	−0.0595937
$156$	0	0
$157$	138.000	0.0701503	0.0350752	−	0.999385i	$-0.488833\pi$
$157$	138.000	0.0701503	0.0350752	+	0.999385i	$0.488833\pi$
$158$	0	0
$159$	390.000	0.194522
$160$	0	0
$161$	693.000	0.339230
$162$	0	0
$163$	3956.00	1.90097	0.950484	−	0.310773i	$-0.100588\pi$
$163$	3956.00	1.90097	0.950484	+	0.310773i	$0.100588\pi$
$164$	0	0
$165$	−360.000	−0.169854
$166$	0	0
$167$	3675.00	1.70287	0.851437	−	0.524457i	$-0.175731\pi$
$167$	3675.00	1.70287	0.851437	+	0.524457i	$0.175731\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1081.00	−0.483428
$172$	0	0
$173$	−2826.00	−1.24195	−0.620973	−	0.783832i	$-0.713263\pi$
$173$	−2826.00	−1.24195	−0.620973	+	0.783832i	$0.713263\pi$
$174$	0	0
$175$	−700.000	−0.302372
$176$	0	0
$177$	−528.000	−0.224220
$178$	0	0
$179$	3235.00	1.35081	0.675406	−	0.737446i	$-0.263969\pi$
$179$	3235.00	1.35081	0.675406	+	0.737446i	$0.263969\pi$
$180$	0	0
$181$	1260.00	0.517431	0.258716	−	0.965954i	$-0.416701\pi$
$181$	1260.00	0.517431	0.258716	+	0.965954i	$0.416701\pi$
$182$	0	0
$183$	620.000	0.250447
$184$	0	0
$185$	250.000	0.0993533
$186$	0	0
$187$	936.000	0.366027
$188$	0	0
$189$	−700.000	−0.269405
$190$	0	0
$191$	232.000	0.0878897	0.0439448	−	0.999034i	$-0.486007\pi$
$191$	232.000	0.0878897	0.0439448	+	0.999034i	$0.486007\pi$
$192$	0	0
$193$	−5342.00	−1.99236	−0.996180	−	0.0873208i	$-0.972169\pi$
$193$	−5342.00	−1.99236	−0.996180	+	0.0873208i	$0.972169\pi$
$194$	0	0
$195$	130.000	0.0477410
$196$	0	0
$197$	1542.00	0.557680	0.278840	−	0.960338i	$-0.410050\pi$
$197$	1542.00	0.557680	0.278840	+	0.960338i	$0.410050\pi$
$198$	0	0
$199$	−2182.00	−0.777276	−0.388638	−	0.921391i	$-0.627054\pi$
$199$	−2182.00	−0.777276	−0.388638	+	0.921391i	$0.627054\pi$
$200$	0	0
$201$	380.000	0.133349
$202$	0	0
$203$	−427.000	−0.147633
$204$	0	0
$205$	−350.000	−0.119244
$206$	0	0
$207$	−2277.00	−0.764553
$208$	0	0
$209$	1692.00	0.559991
$210$	0	0
$211$	523.000	0.170639	0.0853194	−	0.996354i	$-0.472809\pi$
$211$	523.000	0.170639	0.0853194	+	0.996354i	$0.472809\pi$
$212$	0	0
$213$	332.000	0.106799
$214$	0	0
$215$	−95.0000	−0.0301346
$216$	0	0
$217$	161.000	0.0503659
$218$	0	0
$219$	1746.00	0.538739
$220$	0	0
$221$	−338.000	−0.102879
$222$	0	0
$223$	1981.00	0.594877	0.297439	−	0.954741i	$-0.403868\pi$
$223$	1981.00	0.594877	0.297439	+	0.954741i	$0.403868\pi$
$224$	0	0
$225$	2300.00	0.681481
$226$	0	0
$227$	−4352.00	−1.27248	−0.636239	−	0.771492i	$-0.719511\pi$
$227$	−4352.00	−1.27248	−0.636239	+	0.771492i	$0.719511\pi$
$228$	0	0
$229$	2130.00	0.614648	0.307324	−	0.951605i	$-0.400566\pi$
$229$	2130.00	0.614648	0.307324	+	0.951605i	$0.400566\pi$
$230$	0	0
$231$	504.000	0.143553
$232$	0	0
$233$	2687.00	0.755499	0.377749	−	0.925908i	$-0.376698\pi$
$233$	2687.00	0.755499	0.377749	+	0.925908i	$0.376698\pi$
$234$	0	0
$235$	955.000	0.265095
$236$	0	0
$237$	2382.00	0.652859
$238$	0	0
$239$	−3852.00	−1.04253	−0.521266	−	0.853394i	$-0.674540\pi$
$239$	−3852.00	−1.04253	−0.521266	+	0.853394i	$0.674540\pi$
$240$	0	0
$241$	1069.00	0.285728	0.142864	−	0.989742i	$-0.454369\pi$
$241$	1069.00	0.285728	0.142864	+	0.989742i	$0.454369\pi$
$242$	0	0
$243$	3542.00	0.935059
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	−611.000	−0.157397
$248$	0	0
$249$	−518.000	−0.131835
$250$	0	0
$251$	−5460.00	−1.37304	−0.686518	−	0.727113i	$-0.740862\pi$
$251$	−5460.00	−1.37304	−0.686518	+	0.727113i	$0.740862\pi$
$252$	0	0
$253$	3564.00	0.885639
$254$	0	0
$255$	−260.000	−0.0638503
$256$	0	0
$257$	2172.00	0.527181	0.263591	−	0.964635i	$-0.415093\pi$
$257$	2172.00	0.527181	0.263591	+	0.964635i	$0.415093\pi$
$258$	0	0
$259$	−350.000	−0.0839689
$260$	0	0
$261$	1403.00	0.332734
$262$	0	0
$263$	3417.00	0.801145	0.400573	−	0.916265i	$-0.368811\pi$
$263$	3417.00	0.801145	0.400573	+	0.916265i	$0.368811\pi$
$264$	0	0
$265$	−975.000	−0.226014
$266$	0	0
$267$	−1270.00	−0.291096
$268$	0	0
$269$	3792.00	0.859488	0.429744	−	0.902951i	$-0.358604\pi$
$269$	3792.00	0.859488	0.429744	+	0.902951i	$0.358604\pi$
$270$	0	0
$271$	4408.00	0.988070	0.494035	−	0.869442i	$-0.335521\pi$
$271$	4408.00	0.988070	0.494035	+	0.869442i	$0.335521\pi$
$272$	0	0
$273$	−182.000	−0.0403485
$274$	0	0
$275$	−3600.00	−0.789412
$276$	0	0
$277$	−1023.00	−0.221899	−0.110950	−	0.993826i	$-0.535389\pi$
$277$	−1023.00	−0.221899	−0.110950	+	0.993826i	$0.535389\pi$
$278$	0	0
$279$	−529.000	−0.113514
$280$	0	0
$281$	7912.00	1.67968	0.839840	−	0.542833i	$-0.182648\pi$
$281$	7912.00	1.67968	0.839840	+	0.542833i	$0.182648\pi$
$282$	0	0
$283$	5336.00	1.12082	0.560410	−	0.828215i	$-0.310643\pi$
$283$	5336.00	1.12082	0.560410	+	0.828215i	$0.310643\pi$
$284$	0	0
$285$	−470.000	−0.0976856
$286$	0	0
$287$	490.000	0.100780
$288$	0	0
$289$	−4237.00	−0.862406
$290$	0	0
$291$	266.000	0.0535849
$292$	0	0
$293$	6615.00	1.31895	0.659475	−	0.751726i	$-0.270779\pi$
$293$	6615.00	1.31895	0.659475	+	0.751726i	$0.270779\pi$
$294$	0	0
$295$	1320.00	0.260520
$296$	0	0
$297$	−3600.00	−0.703344
$298$	0	0
$299$	−1287.00	−0.248927
$300$	0	0
$301$	133.000	0.0254684
$302$	0	0
$303$	304.000	0.0576381
$304$	0	0
$305$	−1550.00	−0.290993
$306$	0	0
$307$	−3843.00	−0.714435	−0.357218	−	0.934021i	$-0.616275\pi$
$307$	−3843.00	−0.714435	−0.357218	+	0.934021i	$0.616275\pi$
$308$	0	0
$309$	3328.00	0.612697
$310$	0	0
$311$	−1748.00	−0.318714	−0.159357	−	0.987221i	$-0.550942\pi$
$311$	−1748.00	−0.318714	−0.159357	+	0.987221i	$0.550942\pi$
$312$	0	0
$313$	−9392.00	−1.69606	−0.848031	−	0.529947i	$-0.822212\pi$
$313$	−9392.00	−1.69606	−0.848031	+	0.529947i	$0.822212\pi$
$314$	0	0
$315$	805.000	0.143989
$316$	0	0
$317$	10380.0	1.83911	0.919557	−	0.392958i	$-0.128548\pi$
$317$	10380.0	1.83911	0.919557	+	0.392958i	$0.128548\pi$
$318$	0	0
$319$	−2196.00	−0.385431
$320$	0	0
$321$	72.0000	0.0125192
$322$	0	0
$323$	1222.00	0.210507
$324$	0	0
$325$	1300.00	0.221880
$326$	0	0
$327$	−464.000	−0.0784687
$328$	0	0
$329$	−1337.00	−0.224046
$330$	0	0
$331$	−6250.00	−1.03786	−0.518929	−	0.854817i	$-0.673669\pi$
$331$	−6250.00	−1.03786	−0.518929	+	0.854817i	$0.673669\pi$
$332$	0	0
$333$	1150.00	0.189248
$334$	0	0
$335$	−950.000	−0.154937
$336$	0	0
$337$	−2315.00	−0.374202	−0.187101	−	0.982341i	$-0.559909\pi$
$337$	−2315.00	−0.374202	−0.187101	+	0.982341i	$0.559909\pi$
$338$	0	0
$339$	2706.00	0.433539
$340$	0	0
$341$	828.000	0.131492
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−990.000	−0.154492
$346$	0	0
$347$	−3296.00	−0.509909	−0.254955	−	0.966953i	$-0.582061\pi$
$347$	−3296.00	−0.509909	−0.254955	+	0.966953i	$0.582061\pi$
$348$	0	0
$349$	−5607.00	−0.859988	−0.429994	−	0.902832i	$-0.641484\pi$
$349$	−5607.00	−0.859988	−0.429994	+	0.902832i	$0.641484\pi$
$350$	0	0
$351$	1300.00	0.197689
$352$	0	0
$353$	3722.00	0.561196	0.280598	−	0.959825i	$-0.409467\pi$
$353$	3722.00	0.561196	0.280598	+	0.959825i	$0.409467\pi$
$354$	0	0
$355$	−830.000	−0.124090
$356$	0	0
$357$	364.000	0.0539634
$358$	0	0
$359$	2430.00	0.357244	0.178622	−	0.983918i	$-0.442836\pi$
$359$	2430.00	0.357244	0.178622	+	0.983918i	$0.442836\pi$
$360$	0	0
$361$	−4650.00	−0.677941
$362$	0	0
$363$	−70.0000	−0.0101213
$364$	0	0
$365$	−4365.00	−0.625958
$366$	0	0
$367$	8584.00	1.22093	0.610465	−	0.792043i	$-0.290983\pi$
$367$	8584.00	1.22093	0.610465	+	0.792043i	$0.290983\pi$
$368$	0	0
$369$	−1610.00	−0.227136
$370$	0	0
$371$	1365.00	0.191017
$372$	0	0
$373$	−7274.00	−1.00974	−0.504871	−	0.863195i	$-0.668460\pi$
$373$	−7274.00	−1.00974	−0.504871	+	0.863195i	$0.668460\pi$
$374$	0	0
$375$	2250.00	0.309839
$376$	0	0
$377$	793.000	0.108333
$378$	0	0
$379$	1160.00	0.157217	0.0786084	−	0.996906i	$-0.474952\pi$
$379$	1160.00	0.157217	0.0786084	+	0.996906i	$0.474952\pi$
$380$	0	0
$381$	−1152.00	−0.154905
$382$	0	0
$383$	5528.00	0.737513	0.368757	−	0.929526i	$-0.379784\pi$
$383$	5528.00	0.737513	0.368757	+	0.929526i	$0.379784\pi$
$384$	0	0
$385$	−1260.00	−0.166794
$386$	0	0
$387$	−437.000	−0.0574004
$388$	0	0
$389$	9666.00	1.25986	0.629930	−	0.776652i	$-0.283084\pi$
$389$	9666.00	1.25986	0.629930	+	0.776652i	$0.283084\pi$
$390$	0	0
$391$	2574.00	0.332923
$392$	0	0
$393$	4112.00	0.527794
$394$	0	0
$395$	−5955.00	−0.758553
$396$	0	0
$397$	4125.00	0.521481	0.260740	−	0.965409i	$-0.416033\pi$
$397$	4125.00	0.521481	0.260740	+	0.965409i	$0.416033\pi$
$398$	0	0
$399$	658.000	0.0825594
$400$	0	0
$401$	−11796.0	−1.46899	−0.734494	−	0.678615i	$-0.762580\pi$
$401$	−11796.0	−1.46899	−0.734494	+	0.678615i	$0.762580\pi$
$402$	0	0
$403$	−299.000	−0.0369584
$404$	0	0
$405$	−2105.00	−0.258267
$406$	0	0
$407$	−1800.00	−0.219220
$408$	0	0
$409$	8181.00	0.989057	0.494529	−	0.869161i	$-0.335341\pi$
$409$	8181.00	0.989057	0.494529	+	0.869161i	$0.335341\pi$
$410$	0	0
$411$	−3684.00	−0.442137
$412$	0	0
$413$	−1848.00	−0.220180
$414$	0	0
$415$	1295.00	0.153178
$416$	0	0
$417$	2576.00	0.302511
$418$	0	0
$419$	9296.00	1.08386	0.541932	−	0.840422i	$-0.317693\pi$
$419$	9296.00	1.08386	0.541932	+	0.840422i	$0.317693\pi$
$420$	0	0
$421$	9074.00	1.05045	0.525225	−	0.850963i	$-0.323981\pi$
$421$	9074.00	1.05045	0.525225	+	0.850963i	$0.323981\pi$
$422$	0	0
$423$	4393.00	0.504953
$424$	0	0
$425$	−2600.00	−0.296749
$426$	0	0
$427$	2170.00	0.245934
$428$	0	0
$429$	−936.000	−0.105339
$430$	0	0
$431$	−9358.00	−1.04584	−0.522922	−	0.852380i	$-0.675158\pi$
$431$	−9358.00	−1.04584	−0.522922	+	0.852380i	$0.675158\pi$
$432$	0	0
$433$	−14392.0	−1.59731	−0.798655	−	0.601789i	$-0.794455\pi$
$433$	−14392.0	−1.59731	−0.798655	+	0.601789i	$0.794455\pi$
$434$	0	0
$435$	610.000	0.0672351
$436$	0	0
$437$	4653.00	0.509344
$438$	0	0
$439$	6074.00	0.660356	0.330178	−	0.943919i	$-0.392891\pi$
$439$	6074.00	0.660356	0.330178	+	0.943919i	$0.392891\pi$
$440$	0	0
$441$	−1127.00	−0.121693
$442$	0	0
$443$	6483.00	0.695297	0.347649	−	0.937625i	$-0.386980\pi$
$443$	6483.00	0.695297	0.347649	+	0.937625i	$0.386980\pi$
$444$	0	0
$445$	3175.00	0.338223
$446$	0	0
$447$	2392.00	0.253105
$448$	0	0
$449$	15388.0	1.61738	0.808691	−	0.588234i	$-0.200176\pi$
$449$	15388.0	1.61738	0.808691	+	0.588234i	$0.200176\pi$
$450$	0	0
$451$	2520.00	0.263109
$452$	0	0
$453$	1780.00	0.184617
$454$	0	0
$455$	455.000	0.0468807
$456$	0	0
$457$	−4642.00	−0.475150	−0.237575	−	0.971369i	$-0.576353\pi$
$457$	−4642.00	−0.475150	−0.237575	+	0.971369i	$0.576353\pi$
$458$	0	0
$459$	−2600.00	−0.264396
$460$	0	0
$461$	−14266.0	−1.44129	−0.720644	−	0.693305i	$-0.756154\pi$
$461$	−14266.0	−1.44129	−0.720644	+	0.693305i	$0.756154\pi$
$462$	0	0
$463$	−1472.00	−0.147753	−0.0738765	−	0.997267i	$-0.523537\pi$
$463$	−1472.00	−0.147753	−0.0738765	+	0.997267i	$0.523537\pi$
$464$	0	0
$465$	−230.000	−0.0229376
$466$	0	0
$467$	−1332.00	−0.131986	−0.0659932	−	0.997820i	$-0.521022\pi$
$467$	−1332.00	−0.131986	−0.0659932	+	0.997820i	$0.521022\pi$
$468$	0	0
$469$	1330.00	0.130946
$470$	0	0
$471$	276.000	0.0270009
$472$	0	0
$473$	684.000	0.0664912
$474$	0	0
$475$	−4700.00	−0.454002
$476$	0	0
$477$	−4485.00	−0.430512
$478$	0	0
$479$	−1629.00	−0.155388	−0.0776941	−	0.996977i	$-0.524756\pi$
$479$	−1629.00	−0.155388	−0.0776941	+	0.996977i	$0.524756\pi$
$480$	0	0
$481$	650.000	0.0616163
$482$	0	0
$483$	1386.00	0.130570
$484$	0	0
$485$	−665.000	−0.0622600
$486$	0	0
$487$	−13754.0	−1.27978	−0.639890	−	0.768466i	$-0.721020\pi$
$487$	−13754.0	−1.27978	−0.639890	+	0.768466i	$0.721020\pi$
$488$	0	0
$489$	7912.00	0.731683
$490$	0	0
$491$	10904.0	1.00222	0.501111	−	0.865383i	$-0.332925\pi$
$491$	10904.0	1.00222	0.501111	+	0.865383i	$0.332925\pi$
$492$	0	0
$493$	−1586.00	−0.144888
$494$	0	0
$495$	4140.00	0.375917
$496$	0	0
$497$	1162.00	0.104875
$498$	0	0
$499$	−10394.0	−0.932464	−0.466232	−	0.884663i	$-0.654389\pi$
$499$	−10394.0	−0.932464	−0.466232	+	0.884663i	$0.654389\pi$
$500$	0	0
$501$	7350.00	0.655437
$502$	0	0
$503$	5754.00	0.510056	0.255028	−	0.966934i	$-0.417915\pi$
$503$	5754.00	0.510056	0.255028	+	0.966934i	$0.417915\pi$
$504$	0	0
$505$	−760.000	−0.0669694
$506$	0	0
$507$	338.000	0.0296077
$508$	0	0
$509$	13547.0	1.17969	0.589843	−	0.807518i	$-0.299190\pi$
$509$	13547.0	1.17969	0.589843	+	0.807518i	$0.299190\pi$
$510$	0	0
$511$	6111.00	0.529031
$512$	0	0
$513$	−4700.00	−0.404503
$514$	0	0
$515$	−8320.00	−0.711889
$516$	0	0
$517$	−6876.00	−0.584925
$518$	0	0
$519$	−5652.00	−0.478026
$520$	0	0
$521$	−7044.00	−0.592329	−0.296164	−	0.955137i	$-0.595708\pi$
$521$	−7044.00	−0.592329	−0.296164	+	0.955137i	$0.595708\pi$
$522$	0	0
$523$	−9858.00	−0.824207	−0.412103	−	0.911137i	$-0.635206\pi$
$523$	−9858.00	−0.824207	−0.412103	+	0.911137i	$0.635206\pi$
$524$	0	0
$525$	−1400.00	−0.116383
$526$	0	0
$527$	598.000	0.0494294
$528$	0	0
$529$	−2366.00	−0.194460
$530$	0	0
$531$	6072.00	0.496238
$532$	0	0
$533$	−910.000	−0.0739521
$534$	0	0
$535$	−180.000	−0.0145459
$536$	0	0
$537$	6470.00	0.519928
$538$	0	0
$539$	1764.00	0.140966
$540$	0	0
$541$	−5650.00	−0.449006	−0.224503	−	0.974473i	$-0.572076\pi$
$541$	−5650.00	−0.449006	−0.224503	+	0.974473i	$0.572076\pi$
$542$	0	0
$543$	2520.00	0.199159
$544$	0	0
$545$	1160.00	0.0911724
$546$	0	0
$547$	14285.0	1.11660	0.558302	−	0.829638i	$-0.311453\pi$
$547$	14285.0	1.11660	0.558302	+	0.829638i	$0.311453\pi$
$548$	0	0
$549$	−7130.00	−0.554282
$550$	0	0
$551$	−2867.00	−0.221667
$552$	0	0
$553$	8337.00	0.641095
$554$	0	0
$555$	500.000	0.0382411
$556$	0	0
$557$	−2472.00	−0.188047	−0.0940233	−	0.995570i	$-0.529973\pi$
$557$	−2472.00	−0.188047	−0.0940233	+	0.995570i	$0.529973\pi$
$558$	0	0
$559$	−247.000	−0.0186887
$560$	0	0
$561$	1872.00	0.140884
$562$	0	0
$563$	3936.00	0.294641	0.147320	−	0.989089i	$-0.452935\pi$
$563$	3936.00	0.294641	0.147320	+	0.989089i	$0.452935\pi$
$564$	0	0
$565$	−6765.00	−0.503727
$566$	0	0
$567$	2947.00	0.218276
$568$	0	0
$569$	2995.00	0.220662	0.110331	−	0.993895i	$-0.464809\pi$
$569$	2995.00	0.220662	0.110331	+	0.993895i	$0.464809\pi$
$570$	0	0
$571$	−19221.0	−1.40871	−0.704355	−	0.709848i	$-0.748764\pi$
$571$	−19221.0	−1.40871	−0.704355	+	0.709848i	$0.748764\pi$
$572$	0	0
$573$	464.000	0.0338288
$574$	0	0
$575$	−9900.00	−0.718015
$576$	0	0
$577$	−3166.00	−0.228427	−0.114213	−	0.993456i	$-0.536435\pi$
$577$	−3166.00	−0.228427	−0.114213	+	0.993456i	$0.536435\pi$
$578$	0	0
$579$	−10684.0	−0.766860
$580$	0	0
$581$	−1813.00	−0.129459
$582$	0	0
$583$	7020.00	0.498694
$584$	0	0
$585$	−1495.00	−0.105659
$586$	0	0
$587$	−13951.0	−0.980953	−0.490476	−	0.871454i	$-0.663177\pi$
$587$	−13951.0	−0.980953	−0.490476	+	0.871454i	$0.663177\pi$
$588$	0	0
$589$	1081.00	0.0756228
$590$	0	0
$591$	3084.00	0.214651
$592$	0	0
$593$	−10645.0	−0.737163	−0.368582	−	0.929595i	$-0.620156\pi$
$593$	−10645.0	−0.737163	−0.368582	+	0.929595i	$0.620156\pi$
$594$	0	0
$595$	−910.000	−0.0626998
$596$	0	0
$597$	−4364.00	−0.299174
$598$	0	0
$599$	−6313.00	−0.430621	−0.215311	−	0.976546i	$-0.569076\pi$
$599$	−6313.00	−0.430621	−0.215311	+	0.976546i	$0.569076\pi$
$600$	0	0
$601$	−22162.0	−1.50417	−0.752086	−	0.659065i	$-0.770952\pi$
$601$	−22162.0	−1.50417	−0.752086	+	0.659065i	$0.770952\pi$
$602$	0	0
$603$	−4370.00	−0.295125
$604$	0	0
$605$	175.000	0.0117599
$606$	0	0
$607$	−28716.0	−1.92018	−0.960088	−	0.279699i	$-0.909765\pi$
$607$	−28716.0	−1.92018	−0.960088	+	0.279699i	$0.909765\pi$
$608$	0	0
$609$	−854.000	−0.0568240
$610$	0	0
$611$	2483.00	0.164405
$612$	0	0
$613$	−5860.00	−0.386106	−0.193053	−	0.981188i	$-0.561839\pi$
$613$	−5860.00	−0.386106	−0.193053	+	0.981188i	$0.561839\pi$
$614$	0	0
$615$	−700.000	−0.0458971
$616$	0	0
$617$	−16154.0	−1.05403	−0.527014	−	0.849856i	$-0.676689\pi$
$617$	−16154.0	−1.05403	−0.527014	+	0.849856i	$0.676689\pi$
$618$	0	0
$619$	−9644.00	−0.626212	−0.313106	−	0.949718i	$-0.601369\pi$
$619$	−9644.00	−0.626212	−0.313106	+	0.949718i	$0.601369\pi$
$620$	0	0
$621$	−9900.00	−0.639732
$622$	0	0
$623$	−4445.00	−0.285851
$624$	0	0
$625$	6875.00	0.440000
$626$	0	0
$627$	3384.00	0.215541
$628$	0	0
$629$	−1300.00	−0.0824076
$630$	0	0
$631$	−8682.00	−0.547742	−0.273871	−	0.961766i	$-0.588304\pi$
$631$	−8682.00	−0.547742	−0.273871	+	0.961766i	$0.588304\pi$
$632$	0	0
$633$	1046.00	0.0656789
$634$	0	0
$635$	2880.00	0.179983
$636$	0	0
$637$	−637.000	−0.0396214
$638$	0	0
$639$	−3818.00	−0.236366
$640$	0	0
$641$	−13791.0	−0.849784	−0.424892	−	0.905244i	$-0.639688\pi$
$641$	−13791.0	−0.849784	−0.424892	+	0.905244i	$0.639688\pi$
$642$	0	0
$643$	15316.0	0.939353	0.469677	−	0.882839i	$-0.344371\pi$
$643$	15316.0	0.939353	0.469677	+	0.882839i	$0.344371\pi$
$644$	0	0
$645$	−190.000	−0.0115988
$646$	0	0
$647$	−15244.0	−0.926280	−0.463140	−	0.886285i	$-0.653277\pi$
$647$	−15244.0	−0.926280	−0.463140	+	0.886285i	$0.653277\pi$
$648$	0	0
$649$	−9504.00	−0.574830
$650$	0	0
$651$	322.000	0.0193858
$652$	0	0
$653$	90.0000	0.00539353	0.00269676	−	0.999996i	$-0.499142\pi$
$653$	90.0000	0.00539353	0.00269676	+	0.999996i	$0.499142\pi$
$654$	0	0
$655$	−10280.0	−0.613241
$656$	0	0
$657$	−20079.0	−1.19232
$658$	0	0
$659$	−10887.0	−0.643547	−0.321773	−	0.946817i	$-0.604279\pi$
$659$	−10887.0	−0.643547	−0.321773	+	0.946817i	$0.604279\pi$
$660$	0	0
$661$	−4475.00	−0.263324	−0.131662	−	0.991295i	$-0.542031\pi$
$661$	−4475.00	−0.263324	−0.131662	+	0.991295i	$0.542031\pi$
$662$	0	0
$663$	−676.000	−0.0395983
$664$	0	0
$665$	−1645.00	−0.0959254
$666$	0	0
$667$	−6039.00	−0.350571
$668$	0	0
$669$	3962.00	0.228968
$670$	0	0
$671$	11160.0	0.642067
$672$	0	0
$673$	−33451.0	−1.91596	−0.957980	−	0.286834i	$-0.907397\pi$
$673$	−33451.0	−1.91596	−0.957980	+	0.286834i	$0.907397\pi$
$674$	0	0
$675$	10000.0	0.570222
$676$	0	0
$677$	−5556.00	−0.315413	−0.157706	−	0.987486i	$-0.550410\pi$
$677$	−5556.00	−0.315413	−0.157706	+	0.987486i	$0.550410\pi$
$678$	0	0
$679$	931.000	0.0526193
$680$	0	0
$681$	−8704.00	−0.489777
$682$	0	0
$683$	6504.00	0.364376	0.182188	−	0.983264i	$-0.441682\pi$
$683$	6504.00	0.364376	0.182188	+	0.983264i	$0.441682\pi$
$684$	0	0
$685$	9210.00	0.513717
$686$	0	0
$687$	4260.00	0.236578
$688$	0	0
$689$	−2535.00	−0.140168
$690$	0	0
$691$	−16963.0	−0.933868	−0.466934	−	0.884292i	$-0.654641\pi$
$691$	−16963.0	−0.933868	−0.466934	+	0.884292i	$0.654641\pi$
$692$	0	0
$693$	−5796.00	−0.317708
$694$	0	0
$695$	−6440.00	−0.351487
$696$	0	0
$697$	1820.00	0.0989059
$698$	0	0
$699$	5374.00	0.290792
$700$	0	0
$701$	12805.0	0.689926	0.344963	−	0.938616i	$-0.387891\pi$
$701$	12805.0	0.689926	0.344963	+	0.938616i	$0.387891\pi$
$702$	0	0
$703$	−2350.00	−0.126077
$704$	0	0
$705$	1910.00	0.102035
$706$	0	0
$707$	1064.00	0.0565995
$708$	0	0
$709$	10772.0	0.570594	0.285297	−	0.958439i	$-0.407908\pi$
$709$	10772.0	0.570594	0.285297	+	0.958439i	$0.407908\pi$
$710$	0	0
$711$	−27393.0	−1.44489
$712$	0	0
$713$	2277.00	0.119599
$714$	0	0
$715$	2340.00	0.122393
$716$	0	0
$717$	−7704.00	−0.401271
$718$	0	0
$719$	22524.0	1.16829	0.584147	−	0.811648i	$-0.301429\pi$
$719$	22524.0	1.16829	0.584147	+	0.811648i	$0.301429\pi$
$720$	0	0
$721$	11648.0	0.601656
$722$	0	0
$723$	2138.00	0.109977
$724$	0	0
$725$	6100.00	0.312480
$726$	0	0
$727$	5754.00	0.293541	0.146770	−	0.989171i	$-0.453112\pi$
$727$	5754.00	0.293541	0.146770	+	0.989171i	$0.453112\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	0	0
$731$	494.000	0.0249949
$732$	0	0
$733$	31817.0	1.60326	0.801629	−	0.597822i	$-0.203967\pi$
$733$	31817.0	1.60326	0.801629	+	0.597822i	$0.203967\pi$
$734$	0	0
$735$	−490.000	−0.0245904
$736$	0	0
$737$	6840.00	0.341865
$738$	0	0
$739$	−30820.0	−1.53414	−0.767072	−	0.641561i	$-0.778287\pi$
$739$	−30820.0	−1.53414	−0.767072	+	0.641561i	$0.778287\pi$
$740$	0	0
$741$	−1222.00	−0.0605820
$742$	0	0
$743$	5724.00	0.282629	0.141314	−	0.989965i	$-0.454867\pi$
$743$	5724.00	0.282629	0.141314	+	0.989965i	$0.454867\pi$
$744$	0	0
$745$	−5980.00	−0.294081
$746$	0	0
$747$	5957.00	0.291774
$748$	0	0
$749$	252.000	0.0122936
$750$	0	0
$751$	−20397.0	−0.991075	−0.495537	−	0.868587i	$-0.665029\pi$
$751$	−20397.0	−0.991075	−0.495537	+	0.868587i	$0.665029\pi$
$752$	0	0
$753$	−10920.0	−0.528482
$754$	0	0
$755$	−4450.00	−0.214506
$756$	0	0
$757$	−21103.0	−1.01321	−0.506606	−	0.862178i	$-0.669100\pi$
$757$	−21103.0	−1.01321	−0.506606	+	0.862178i	$0.669100\pi$
$758$	0	0
$759$	7128.00	0.340883
$760$	0	0
$761$	−22209.0	−1.05792	−0.528959	−	0.848647i	$-0.677417\pi$
$761$	−22209.0	−1.05792	−0.528959	+	0.848647i	$0.677417\pi$
$762$	0	0
$763$	−1624.00	−0.0770547
$764$	0	0
$765$	2990.00	0.141312
$766$	0	0
$767$	3432.00	0.161568
$768$	0	0
$769$	27895.0	1.30809	0.654044	−	0.756457i	$-0.273071\pi$
$769$	27895.0	1.30809	0.654044	+	0.756457i	$0.273071\pi$
$770$	0	0
$771$	4344.00	0.202912
$772$	0	0
$773$	24722.0	1.15031	0.575154	−	0.818045i	$-0.304942\pi$
$773$	24722.0	1.15031	0.575154	+	0.818045i	$0.304942\pi$
$774$	0	0
$775$	−2300.00	−0.106604
$776$	0	0
$777$	−700.000	−0.0323196
$778$	0	0
$779$	3290.00	0.151318
$780$	0	0
$781$	5976.00	0.273800
$782$	0	0
$783$	6100.00	0.278412
$784$	0	0
$785$	−690.000	−0.0313722
$786$	0	0
$787$	40091.0	1.81587	0.907935	−	0.419111i	$-0.137658\pi$
$787$	40091.0	1.81587	0.907935	+	0.419111i	$0.137658\pi$
$788$	0	0
$789$	6834.00	0.308361
$790$	0	0
$791$	9471.00	0.425727
$792$	0	0
$793$	−4030.00	−0.180466
$794$	0	0
$795$	−1950.00	−0.0869929
$796$	0	0
$797$	602.000	0.0267552	0.0133776	−	0.999911i	$-0.495742\pi$
$797$	602.000	0.0267552	0.0133776	+	0.999911i	$0.495742\pi$
$798$	0	0
$799$	−4966.00	−0.219880
$800$	0	0
$801$	14605.0	0.644248
$802$	0	0
$803$	31428.0	1.38116
$804$	0	0
$805$	−3465.00	−0.151708
$806$	0	0
$807$	7584.00	0.330817
$808$	0	0
$809$	−38963.0	−1.69328	−0.846642	−	0.532163i	$-0.821379\pi$
$809$	−38963.0	−1.69328	−0.846642	+	0.532163i	$0.821379\pi$
$810$	0	0
$811$	−18116.0	−0.784388	−0.392194	−	0.919882i	$-0.628284\pi$
$811$	−18116.0	−0.784388	−0.392194	+	0.919882i	$0.628284\pi$
$812$	0	0
$813$	8816.00	0.380308
$814$	0	0
$815$	−19780.0	−0.850139
$816$	0	0
$817$	893.000	0.0382400
$818$	0	0
$819$	2093.00	0.0892983
$820$	0	0
$821$	8070.00	0.343051	0.171526	−	0.985180i	$-0.445130\pi$
$821$	8070.00	0.343051	0.171526	+	0.985180i	$0.445130\pi$
$822$	0	0
$823$	−14664.0	−0.621087	−0.310544	−	0.950559i	$-0.600511\pi$
$823$	−14664.0	−0.621087	−0.310544	+	0.950559i	$0.600511\pi$
$824$	0	0
$825$	−7200.00	−0.303845
$826$	0	0
$827$	12836.0	0.539724	0.269862	−	0.962899i	$-0.413022\pi$
$827$	12836.0	0.539724	0.269862	+	0.962899i	$0.413022\pi$
$828$	0	0
$829$	22664.0	0.949521	0.474761	−	0.880115i	$-0.342535\pi$
$829$	22664.0	0.949521	0.474761	+	0.880115i	$0.342535\pi$
$830$	0	0
$831$	−2046.00	−0.0854091
$832$	0	0
$833$	1274.00	0.0529910
$834$	0	0
$835$	−18375.0	−0.761549
$836$	0	0
$837$	−2300.00	−0.0949816
$838$	0	0
$839$	420.000	0.0172825	0.00864125	−	0.999963i	$-0.497249\pi$
$839$	420.000	0.0172825	0.00864125	+	0.999963i	$0.497249\pi$
$840$	0	0
$841$	−20668.0	−0.847431
$842$	0	0
$843$	15824.0	0.646510
$844$	0	0
$845$	−845.000	−0.0344010
$846$	0	0
$847$	−245.000	−0.00993896
$848$	0	0
$849$	10672.0	0.431404
$850$	0	0
$851$	−4950.00	−0.199393
$852$	0	0
$853$	−5425.00	−0.217759	−0.108880	−	0.994055i	$-0.534726\pi$
$853$	−5425.00	−0.217759	−0.108880	+	0.994055i	$0.534726\pi$
$854$	0	0
$855$	5405.00	0.216195
$856$	0	0
$857$	−33294.0	−1.32707	−0.663536	−	0.748144i	$-0.730945\pi$
$857$	−33294.0	−1.32707	−0.663536	+	0.748144i	$0.730945\pi$
$858$	0	0
$859$	−27386.0	−1.08777	−0.543887	−	0.839158i	$-0.683048\pi$
$859$	−27386.0	−1.08777	−0.543887	+	0.839158i	$0.683048\pi$
$860$	0	0
$861$	980.000	0.0387901
$862$	0	0
$863$	−12004.0	−0.473489	−0.236744	−	0.971572i	$-0.576080\pi$
$863$	−12004.0	−0.473489	−0.236744	+	0.971572i	$0.576080\pi$
$864$	0	0
$865$	14130.0	0.555416
$866$	0	0
$867$	−8474.00	−0.331940
$868$	0	0
$869$	42876.0	1.67373
$870$	0	0
$871$	−2470.00	−0.0960881
$872$	0	0
$873$	−3059.00	−0.118593
$874$	0	0
$875$	7875.00	0.304256
$876$	0	0
$877$	39346.0	1.51496	0.757480	−	0.652858i	$-0.226430\pi$
$877$	39346.0	1.51496	0.757480	+	0.652858i	$0.226430\pi$
$878$	0	0
$879$	13230.0	0.507664
$880$	0	0
$881$	50806.0	1.94290	0.971452	−	0.237238i	$-0.0762421\pi$
$881$	50806.0	1.94290	0.971452	+	0.237238i	$0.0762421\pi$
$882$	0	0
$883$	34592.0	1.31836	0.659181	−	0.751984i	$-0.270903\pi$
$883$	34592.0	1.31836	0.659181	+	0.751984i	$0.270903\pi$
$884$	0	0
$885$	2640.00	0.100274
$886$	0	0
$887$	−34624.0	−1.31067	−0.655333	−	0.755340i	$-0.727472\pi$
$887$	−34624.0	−1.31067	−0.655333	+	0.755340i	$0.727472\pi$
$888$	0	0
$889$	−4032.00	−0.152114
$890$	0	0
$891$	15156.0	0.569860
$892$	0	0
$893$	−8977.00	−0.336398
$894$	0	0
$895$	−16175.0	−0.604101
$896$	0	0
$897$	−2574.00	−0.0958120
$898$	0	0
$899$	−1403.00	−0.0520497
$900$	0	0
$901$	5070.00	0.187465
$902$	0	0
$903$	266.000	0.00980280
$904$	0	0
$905$	−6300.00	−0.231402
$906$	0	0
$907$	−10849.0	−0.397172	−0.198586	−	0.980083i	$-0.563635\pi$
$907$	−10849.0	−0.397172	−0.198586	+	0.980083i	$0.563635\pi$
$908$	0	0
$909$	−3496.00	−0.127563
$910$	0	0
$911$	−39671.0	−1.44276	−0.721382	−	0.692537i	$-0.756493\pi$
$911$	−39671.0	−1.44276	−0.721382	+	0.692537i	$0.756493\pi$
$912$	0	0
$913$	−9324.00	−0.337984
$914$	0	0
$915$	−3100.00	−0.112003
$916$	0	0
$917$	14392.0	0.518283
$918$	0	0
$919$	−27208.0	−0.976615	−0.488307	−	0.872672i	$-0.662386\pi$
$919$	−27208.0	−0.976615	−0.488307	+	0.872672i	$0.662386\pi$
$920$	0	0
$921$	−7686.00	−0.274986
$922$	0	0
$923$	−2158.00	−0.0769571
$924$	0	0
$925$	5000.00	0.177729
$926$	0	0
$927$	−38272.0	−1.35601
$928$	0	0
$929$	−27193.0	−0.960359	−0.480179	−	0.877170i	$-0.659428\pi$
$929$	−27193.0	−0.960359	−0.480179	+	0.877170i	$0.659428\pi$
$930$	0	0
$931$	2303.00	0.0810717
$932$	0	0
$933$	−3496.00	−0.122673
$934$	0	0
$935$	−4680.00	−0.163692
$936$	0	0
$937$	24262.0	0.845896	0.422948	−	0.906154i	$-0.360995\pi$
$937$	24262.0	0.845896	0.422948	+	0.906154i	$0.360995\pi$
$938$	0	0
$939$	−18784.0	−0.652814
$940$	0	0
$941$	−303.000	−0.0104968	−0.00524842	−	0.999986i	$-0.501671\pi$
$941$	−303.000	−0.0104968	−0.00524842	+	0.999986i	$0.501671\pi$
$942$	0	0
$943$	6930.00	0.239313
$944$	0	0
$945$	3500.00	0.120481
$946$	0	0
$947$	20098.0	0.689649	0.344824	−	0.938667i	$-0.387938\pi$
$947$	20098.0	0.689649	0.344824	+	0.938667i	$0.387938\pi$
$948$	0	0
$949$	−11349.0	−0.388202
$950$	0	0
$951$	20760.0	0.707875
$952$	0	0
$953$	13477.0	0.458093	0.229047	−	0.973415i	$-0.426439\pi$
$953$	13477.0	0.458093	0.229047	+	0.973415i	$0.426439\pi$
$954$	0	0
$955$	−1160.00	−0.0393055
$956$	0	0
$957$	−4392.00	−0.148352
$958$	0	0
$959$	−12894.0	−0.434170
$960$	0	0
$961$	−29262.0	−0.982243
$962$	0	0
$963$	−828.000	−0.0277071
$964$	0	0
$965$	26710.0	0.891011
$966$	0	0
$967$	14488.0	0.481802	0.240901	−	0.970550i	$-0.422557\pi$
$967$	14488.0	0.481802	0.240901	+	0.970550i	$0.422557\pi$
$968$	0	0
$969$	2444.00	0.0810243
$970$	0	0
$971$	−2830.00	−0.0935314	−0.0467657	−	0.998906i	$-0.514891\pi$
$971$	−2830.00	−0.0935314	−0.0467657	+	0.998906i	$0.514891\pi$
$972$	0	0
$973$	9016.00	0.297060
$974$	0	0
$975$	2600.00	0.0854017
$976$	0	0
$977$	−36016.0	−1.17938	−0.589690	−	0.807630i	$-0.700750\pi$
$977$	−36016.0	−1.17938	−0.589690	+	0.807630i	$0.700750\pi$
$978$	0	0
$979$	−22860.0	−0.746281
$980$	0	0
$981$	5336.00	0.173665
$982$	0	0
$983$	16907.0	0.548575	0.274288	−	0.961648i	$-0.411558\pi$
$983$	16907.0	0.548575	0.274288	+	0.961648i	$0.411558\pi$
$984$	0	0
$985$	−7710.00	−0.249402
$986$	0	0
$987$	−2674.00	−0.0862354
$988$	0	0
$989$	1881.00	0.0604776
$990$	0	0
$991$	51528.0	1.65171	0.825853	−	0.563885i	$-0.190694\pi$
$991$	51528.0	1.65171	0.825853	+	0.563885i	$0.190694\pi$
$992$	0	0
$993$	−12500.0	−0.399472
$994$	0	0
$995$	10910.0	0.347608
$996$	0	0
$997$	46492.0	1.47685	0.738423	−	0.674337i	$-0.235571\pi$
$997$	46492.0	1.47685	0.738423	+	0.674337i	$0.235571\pi$
$998$	0	0
$999$	5000.00	0.158351

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.4.a.f.1.1		1
4.3	odd	2		182.4.a.c.1.1	✓	1
12.11	even	2		1638.4.a.e.1.1		1
28.27	even	2		1274.4.a.g.1.1		1
52.51	odd	2		2366.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.4.a.c.1.1	✓	1	4.3	odd	2
1274.4.a.g.1.1		1	28.27	even	2
1456.4.a.f.1.1		1	1.1	even	1	trivial
1638.4.a.e.1.1		1	12.11	even	2
2366.4.a.b.1.1		1	52.51	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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