LMFDB - Embedded newform 528.6.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,6,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("528.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	528.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:	$84.6826568613$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 66)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	528.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} +50.0000 q^{5} -2.00000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} +50.0000 q^{5} -2.00000 q^{7} +81.0000 q^{9} +121.000 q^{11} +966.000 q^{13} +450.000 q^{15} +1964.00 q^{17} -1246.00 q^{19} -18.0000 q^{21} -136.000 q^{23} -625.000 q^{25} +729.000 q^{27} -7824.00 q^{29} +4752.00 q^{31} +1089.00 q^{33} -100.000 q^{35} +4650.00 q^{37} +8694.00 q^{39} +7536.00 q^{41} +14582.0 q^{43} +4050.00 q^{45} -3984.00 q^{47} -16803.0 q^{49} +17676.0 q^{51} +12350.0 q^{53} +6050.00 q^{55} -11214.0 q^{57} +22380.0 q^{59} -15662.0 q^{61} -162.000 q^{63} +48300.0 q^{65} +29564.0 q^{67} -1224.00 q^{69} -55536.0 q^{71} -63258.0 q^{73} -5625.00 q^{75} -242.000 q^{77} +40606.0 q^{79} +6561.00 q^{81} -81808.0 q^{83} +98200.0 q^{85} -70416.0 q^{87} +116434. q^{89} -1932.00 q^{91} +42768.0 q^{93} -62300.0 q^{95} +20734.0 q^{97} +9801.00 q^{99} +O(q^{100})$

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	50.0000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	50.0000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−2.00000	−0.0154271	−0.00771356	−	0.999970i	$-0.502455\pi$
$7$	−2.00000	−0.0154271	−0.00771356	+	0.999970i	$0.502455\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	966.000	1.58533	0.792663	−	0.609660i	$-0.208694\pi$
$13$	966.000	1.58533	0.792663	+	0.609660i	$0.208694\pi$
$14$	0	0
$15$	450.000	0.516398
$16$	0	0
$17$	1964.00	1.64824	0.824118	−	0.566419i	$-0.191671\pi$
$17$	1964.00	1.64824	0.824118	+	0.566419i	$0.191671\pi$
$18$	0	0
$19$	−1246.00	−0.791834	−0.395917	−	0.918286i	$-0.629573\pi$
$19$	−1246.00	−0.791834	−0.395917	+	0.918286i	$0.629573\pi$
$20$	0	0
$21$	−18.0000	−0.00890685
$22$	0	0
$23$	−136.000	−0.0536067	−0.0268034	−	0.999641i	$-0.508533\pi$
$23$	−136.000	−0.0536067	−0.0268034	+	0.999641i	$0.508533\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−7824.00	−1.72756	−0.863781	−	0.503867i	$-0.831910\pi$
$29$	−7824.00	−1.72756	−0.863781	+	0.503867i	$0.831910\pi$
$30$	0	0
$31$	4752.00	0.888121	0.444061	−	0.895997i	$-0.353538\pi$
$31$	4752.00	0.888121	0.444061	+	0.895997i	$0.353538\pi$
$32$	0	0
$33$	1089.00	0.174078
$34$	0	0
$35$	−100.000	−0.0137984
$36$	0	0
$37$	4650.00	0.558404	0.279202	−	0.960232i	$-0.409930\pi$
$37$	4650.00	0.558404	0.279202	+	0.960232i	$0.409930\pi$
$38$	0	0
$39$	8694.00	0.915289
$40$	0	0
$41$	7536.00	0.700134	0.350067	−	0.936725i	$-0.386159\pi$
$41$	7536.00	0.700134	0.350067	+	0.936725i	$0.386159\pi$
$42$	0	0
$43$	14582.0	1.20267	0.601334	−	0.798998i	$-0.294636\pi$
$43$	14582.0	1.20267	0.601334	+	0.798998i	$0.294636\pi$
$44$	0	0
$45$	4050.00	0.298142
$46$	0	0
$47$	−3984.00	−0.263072	−0.131536	−	0.991311i	$-0.541991\pi$
$47$	−3984.00	−0.263072	−0.131536	+	0.991311i	$0.541991\pi$
$48$	0	0
$49$	−16803.0	−0.999762
$50$	0	0
$51$	17676.0	0.951609
$52$	0	0
$53$	12350.0	0.603917	0.301959	−	0.953321i	$-0.402360\pi$
$53$	12350.0	0.603917	0.301959	+	0.953321i	$0.402360\pi$
$54$	0	0
$55$	6050.00	0.269680
$56$	0	0
$57$	−11214.0	−0.457165
$58$	0	0
$59$	22380.0	0.837009	0.418504	−	0.908215i	$-0.362554\pi$
$59$	22380.0	0.837009	0.418504	+	0.908215i	$0.362554\pi$
$60$	0	0
$61$	−15662.0	−0.538918	−0.269459	−	0.963012i	$-0.586845\pi$
$61$	−15662.0	−0.538918	−0.269459	+	0.963012i	$0.586845\pi$
$62$	0	0
$63$	−162.000	−0.00514237
$64$	0	0
$65$	48300.0	1.41796
$66$	0	0
$67$	29564.0	0.804593	0.402296	−	0.915509i	$-0.368212\pi$
$67$	29564.0	0.804593	0.402296	+	0.915509i	$0.368212\pi$
$68$	0	0
$69$	−1224.00	−0.0309499
$70$	0	0
$71$	−55536.0	−1.30746	−0.653730	−	0.756727i	$-0.726797\pi$
$71$	−55536.0	−1.30746	−0.653730	+	0.756727i	$0.726797\pi$
$72$	0	0
$73$	−63258.0	−1.38934	−0.694670	−	0.719329i	$-0.744449\pi$
$73$	−63258.0	−1.38934	−0.694670	+	0.719329i	$0.744449\pi$
$74$	0	0
$75$	−5625.00	−0.115470
$76$	0	0
$77$	−242.000	−0.00465145
$78$	0	0
$79$	40606.0	0.732019	0.366010	−	0.930611i	$-0.380724\pi$
$79$	40606.0	0.732019	0.366010	+	0.930611i	$0.380724\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−81808.0	−1.30347	−0.651734	−	0.758447i	$-0.725958\pi$
$83$	−81808.0	−1.30347	−0.651734	+	0.758447i	$0.725958\pi$
$84$	0	0
$85$	98200.0	1.47423
$86$	0	0
$87$	−70416.0	−0.997409
$88$	0	0
$89$	116434.	1.55813	0.779067	−	0.626941i	$-0.215693\pi$
$89$	116434.	1.55813	0.779067	+	0.626941i	$0.215693\pi$
$90$	0	0
$91$	−1932.00	−0.0244570
$92$	0	0
$93$	42768.0	0.512757
$94$	0	0
$95$	−62300.0	−0.708238
$96$	0	0
$97$	20734.0	0.223745	0.111873	−	0.993723i	$-0.464315\pi$
$97$	20734.0	0.223745	0.111873	+	0.993723i	$0.464315\pi$
$98$	0	0
$99$	9801.00	0.100504
$100$	0	0
$101$	−47464.0	−0.462979	−0.231489	−	0.972837i	$-0.574360\pi$
$101$	−47464.0	−0.462979	−0.231489	+	0.972837i	$0.574360\pi$
$102$	0	0
$103$	106524.	0.989360	0.494680	−	0.869075i	$-0.335285\pi$
$103$	106524.	0.989360	0.494680	+	0.869075i	$0.335285\pi$
$104$	0	0
$105$	−900.000	−0.00796653
$106$	0	0
$107$	74192.0	0.626466	0.313233	−	0.949676i	$-0.398588\pi$
$107$	74192.0	0.626466	0.313233	+	0.949676i	$0.398588\pi$
$108$	0	0
$109$	192846.	1.55469	0.777346	−	0.629073i	$-0.216565\pi$
$109$	192846.	1.55469	0.777346	+	0.629073i	$0.216565\pi$
$110$	0	0
$111$	41850.0	0.322395
$112$	0	0
$113$	−174442.	−1.28515	−0.642577	−	0.766221i	$-0.722135\pi$
$113$	−174442.	−1.28515	−0.642577	+	0.766221i	$0.722135\pi$
$114$	0	0
$115$	−6800.00	−0.0479473
$116$	0	0
$117$	78246.0	0.528442
$118$	0	0
$119$	−3928.00	−0.0254275
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	67824.0	0.404223
$124$	0	0
$125$	−187500.	−1.07331
$126$	0	0
$127$	−34554.0	−0.190103	−0.0950515	−	0.995472i	$-0.530302\pi$
$127$	−34554.0	−0.190103	−0.0950515	+	0.995472i	$0.530302\pi$
$128$	0	0
$129$	131238.	0.694361
$130$	0	0
$131$	143612.	0.731160	0.365580	−	0.930780i	$-0.380871\pi$
$131$	143612.	0.731160	0.365580	+	0.930780i	$0.380871\pi$
$132$	0	0
$133$	2492.00	0.0122157
$134$	0	0
$135$	36450.0	0.172133
$136$	0	0
$137$	136142.	0.619713	0.309857	−	0.950783i	$-0.399719\pi$
$137$	136142.	0.619713	0.309857	+	0.950783i	$0.399719\pi$
$138$	0	0
$139$	292114.	1.28238	0.641188	−	0.767384i	$-0.278442\pi$
$139$	292114.	1.28238	0.641188	+	0.767384i	$0.278442\pi$
$140$	0	0
$141$	−35856.0	−0.151885
$142$	0	0
$143$	116886.	0.477994
$144$	0	0
$145$	−391200.	−1.54518
$146$	0	0
$147$	−151227.	−0.577213
$148$	0	0
$149$	386444.	1.42600	0.713002	−	0.701162i	$-0.247335\pi$
$149$	386444.	1.42600	0.713002	+	0.701162i	$0.247335\pi$
$150$	0	0
$151$	143030.	0.510487	0.255244	−	0.966877i	$-0.417844\pi$
$151$	143030.	0.510487	0.255244	+	0.966877i	$0.417844\pi$
$152$	0	0
$153$	159084.	0.549412
$154$	0	0
$155$	237600.	0.794360
$156$	0	0
$157$	−138778.	−0.449336	−0.224668	−	0.974435i	$-0.572130\pi$
$157$	−138778.	−0.449336	−0.224668	+	0.974435i	$0.572130\pi$
$158$	0	0
$159$	111150.	0.348672
$160$	0	0
$161$	272.000	0.000826998 0
$162$	0	0
$163$	−423376.	−1.24812	−0.624061	−	0.781375i	$-0.714518\pi$
$163$	−423376.	−1.24812	−0.624061	+	0.781375i	$0.714518\pi$
$164$	0	0
$165$	54450.0	0.155700
$166$	0	0
$167$	519948.	1.44268	0.721338	−	0.692583i	$-0.243528\pi$
$167$	519948.	1.44268	0.721338	+	0.692583i	$0.243528\pi$
$168$	0	0
$169$	561863.	1.51326
$170$	0	0
$171$	−100926.	−0.263945
$172$	0	0
$173$	62456.0	0.158657	0.0793284	−	0.996849i	$-0.474722\pi$
$173$	62456.0	0.158657	0.0793284	+	0.996849i	$0.474722\pi$
$174$	0	0
$175$	1250.00	0.00308542
$176$	0	0
$177$	201420.	0.483247
$178$	0	0
$179$	319876.	0.746189	0.373095	−	0.927793i	$-0.378297\pi$
$179$	319876.	0.746189	0.373095	+	0.927793i	$0.378297\pi$
$180$	0	0
$181$	1798.00	0.00407937	0.00203969	−	0.999998i	$-0.499351\pi$
$181$	1798.00	0.00407937	0.00203969	+	0.999998i	$0.499351\pi$
$182$	0	0
$183$	−140958.	−0.311144
$184$	0	0
$185$	232500.	0.499452
$186$	0	0
$187$	237644.	0.496962
$188$	0	0
$189$	−1458.00	−0.00296895
$190$	0	0
$191$	−407688.	−0.808620	−0.404310	−	0.914622i	$-0.632488\pi$
$191$	−407688.	−0.808620	−0.404310	+	0.914622i	$0.632488\pi$
$192$	0	0
$193$	−806006.	−1.55756	−0.778780	−	0.627297i	$-0.784161\pi$
$193$	−806006.	−1.55756	−0.778780	+	0.627297i	$0.784161\pi$
$194$	0	0
$195$	434700.	0.818659
$196$	0	0
$197$	−140384.	−0.257722	−0.128861	−	0.991663i	$-0.541132\pi$
$197$	−140384.	−0.257722	−0.128861	+	0.991663i	$0.541132\pi$
$198$	0	0
$199$	−800696.	−1.43329	−0.716646	−	0.697437i	$-0.754324\pi$
$199$	−800696.	−1.43329	−0.716646	+	0.697437i	$0.754324\pi$
$200$	0	0
$201$	266076.	0.464532
$202$	0	0
$203$	15648.0	0.0266513
$204$	0	0
$205$	376800.	0.626219
$206$	0	0
$207$	−11016.0	−0.0178689
$208$	0	0
$209$	−150766.	−0.238747
$210$	0	0
$211$	954442.	1.47585	0.737927	−	0.674881i	$-0.235805\pi$
$211$	954442.	1.47585	0.737927	+	0.674881i	$0.235805\pi$
$212$	0	0
$213$	−499824.	−0.754863
$214$	0	0
$215$	729100.	1.07570
$216$	0	0
$217$	−9504.00	−0.0137012
$218$	0	0
$219$	−569322.	−0.802135
$220$	0	0
$221$	1.89722e6	2.61299
$222$	0	0
$223$	1.09732e6	1.47764	0.738822	−	0.673901i	$-0.235383\pi$
$223$	1.09732e6	1.47764	0.738822	+	0.673901i	$0.235383\pi$
$224$	0	0
$225$	−50625.0	−0.0666667
$226$	0	0
$227$	−604488.	−0.778615	−0.389308	−	0.921108i	$-0.627286\pi$
$227$	−604488.	−0.778615	−0.389308	+	0.921108i	$0.627286\pi$
$228$	0	0
$229$	781298.	0.984528	0.492264	−	0.870446i	$-0.336169\pi$
$229$	781298.	0.984528	0.492264	+	0.870446i	$0.336169\pi$
$230$	0	0
$231$	−2178.00	−0.00268552
$232$	0	0
$233$	−421568.	−0.508719	−0.254359	−	0.967110i	$-0.581865\pi$
$233$	−421568.	−0.508719	−0.254359	+	0.967110i	$0.581865\pi$
$234$	0	0
$235$	−199200.	−0.235299
$236$	0	0
$237$	365454.	0.422631
$238$	0	0
$239$	−399548.	−0.452454	−0.226227	−	0.974075i	$-0.572639\pi$
$239$	−399548.	−0.452454	−0.226227	+	0.974075i	$0.572639\pi$
$240$	0	0
$241$	−696290.	−0.772232	−0.386116	−	0.922450i	$-0.626183\pi$
$241$	−696290.	−0.772232	−0.386116	+	0.922450i	$0.626183\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−840150.	−0.894214
$246$	0	0
$247$	−1.20364e6	−1.25532
$248$	0	0
$249$	−736272.	−0.752558
$250$	0	0
$251$	955300.	0.957096	0.478548	−	0.878061i	$-0.341163\pi$
$251$	955300.	0.957096	0.478548	+	0.878061i	$0.341163\pi$
$252$	0	0
$253$	−16456.0	−0.0161630
$254$	0	0
$255$	883800.	0.851145
$256$	0	0
$257$	1.00013e6	0.944547	0.472274	−	0.881452i	$-0.343433\pi$
$257$	1.00013e6	0.944547	0.472274	+	0.881452i	$0.343433\pi$
$258$	0	0
$259$	−9300.00	−0.00861457
$260$	0	0
$261$	−633744.	−0.575854
$262$	0	0
$263$	−225300.	−0.200850	−0.100425	−	0.994945i	$-0.532020\pi$
$263$	−225300.	−0.200850	−0.100425	+	0.994945i	$0.532020\pi$
$264$	0	0
$265$	617500.	0.540160
$266$	0	0
$267$	1.04791e6	0.899589
$268$	0	0
$269$	1.31709e6	1.10977	0.554886	−	0.831927i	$-0.312762\pi$
$269$	1.31709e6	1.10977	0.554886	+	0.831927i	$0.312762\pi$
$270$	0	0
$271$	341098.	0.282134	0.141067	−	0.990000i	$-0.454947\pi$
$271$	341098.	0.282134	0.141067	+	0.990000i	$0.454947\pi$
$272$	0	0
$273$	−17388.0	−0.0141203
$274$	0	0
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	−1.62251e6	−1.27054	−0.635268	−	0.772292i	$-0.719110\pi$
$277$	−1.62251e6	−1.27054	−0.635268	+	0.772292i	$0.719110\pi$
$278$	0	0
$279$	384912.	0.296040
$280$	0	0
$281$	−1.54370e6	−1.16626	−0.583131	−	0.812378i	$-0.698172\pi$
$281$	−1.54370e6	−1.16626	−0.583131	+	0.812378i	$0.698172\pi$
$282$	0	0
$283$	−2.25858e6	−1.67637	−0.838183	−	0.545389i	$-0.816382\pi$
$283$	−2.25858e6	−1.67637	−0.838183	+	0.545389i	$0.816382\pi$
$284$	0	0
$285$	−560700.	−0.408901
$286$	0	0
$287$	−15072.0	−0.0108011
$288$	0	0
$289$	2.43744e6	1.71668
$290$	0	0
$291$	186606.	0.129179
$292$	0	0
$293$	2.89788e6	1.97202	0.986011	−	0.166678i	$-0.0533039\pi$
$293$	2.89788e6	1.97202	0.986011	+	0.166678i	$0.0533039\pi$
$294$	0	0
$295$	1.11900e6	0.748643
$296$	0	0
$297$	88209.0	0.0580259
$298$	0	0
$299$	−131376.	−0.0849842
$300$	0	0
$301$	−29164.0	−0.0185537
$302$	0	0
$303$	−427176.	−0.267301
$304$	0	0
$305$	−783100.	−0.482023
$306$	0	0
$307$	−2.22295e6	−1.34612	−0.673060	−	0.739588i	$-0.735020\pi$
$307$	−2.22295e6	−1.34612	−0.673060	+	0.739588i	$0.735020\pi$
$308$	0	0
$309$	958716.	0.571207
$310$	0	0
$311$	2.08297e6	1.22119	0.610593	−	0.791945i	$-0.290931\pi$
$311$	2.08297e6	1.22119	0.610593	+	0.791945i	$0.290931\pi$
$312$	0	0
$313$	−190354.	−0.109825	−0.0549125	−	0.998491i	$-0.517488\pi$
$313$	−190354.	−0.109825	−0.0549125	+	0.998491i	$0.517488\pi$
$314$	0	0
$315$	−8100.00	−0.00459948
$316$	0	0
$317$	−282078.	−0.157660	−0.0788299	−	0.996888i	$-0.525118\pi$
$317$	−282078.	−0.157660	−0.0788299	+	0.996888i	$0.525118\pi$
$318$	0	0
$319$	−946704.	−0.520880
$320$	0	0
$321$	667728.	0.361690
$322$	0	0
$323$	−2.44714e6	−1.30513
$324$	0	0
$325$	−603750.	−0.317065
$326$	0	0
$327$	1.73561e6	0.897602
$328$	0	0
$329$	7968.00	0.00405844
$330$	0	0
$331$	−3.28062e6	−1.64583	−0.822916	−	0.568163i	$-0.807654\pi$
$331$	−3.28062e6	−1.64583	−0.822916	+	0.568163i	$0.807654\pi$
$332$	0	0
$333$	376650.	0.186135
$334$	0	0
$335$	1.47820e6	0.719650
$336$	0	0
$337$	−1.92962e6	−0.925543	−0.462771	−	0.886478i	$-0.653145\pi$
$337$	−1.92962e6	−0.925543	−0.462771	+	0.886478i	$0.653145\pi$
$338$	0	0
$339$	−1.56998e6	−0.741984
$340$	0	0
$341$	574992.	0.267779
$342$	0	0
$343$	67220.0	0.0308506
$344$	0	0
$345$	−61200.0	−0.0276824
$346$	0	0
$347$	−3.80586e6	−1.69679	−0.848396	−	0.529362i	$-0.822431\pi$
$347$	−3.80586e6	−1.69679	−0.848396	+	0.529362i	$0.822431\pi$
$348$	0	0
$349$	−3.86237e6	−1.69743	−0.848713	−	0.528854i	$-0.822622\pi$
$349$	−3.86237e6	−1.69743	−0.848713	+	0.528854i	$0.822622\pi$
$350$	0	0
$351$	704214.	0.305096
$352$	0	0
$353$	1.12703e6	0.481394	0.240697	−	0.970600i	$-0.422624\pi$
$353$	1.12703e6	0.481394	0.240697	+	0.970600i	$0.422624\pi$
$354$	0	0
$355$	−2.77680e6	−1.16943
$356$	0	0
$357$	−35352.0	−0.0146806
$358$	0	0
$359$	3.74695e6	1.53441	0.767206	−	0.641401i	$-0.221646\pi$
$359$	3.74695e6	1.53441	0.767206	+	0.641401i	$0.221646\pi$
$360$	0	0
$361$	−923583.	−0.372999
$362$	0	0
$363$	131769.	0.0524864
$364$	0	0
$365$	−3.16290e6	−1.24266
$366$	0	0
$367$	1.09493e6	0.424348	0.212174	−	0.977232i	$-0.431946\pi$
$367$	1.09493e6	0.424348	0.212174	+	0.977232i	$0.431946\pi$
$368$	0	0
$369$	610416.	0.233378
$370$	0	0
$371$	−24700.0	−0.00931670
$372$	0	0
$373$	−3.92543e6	−1.46088	−0.730441	−	0.682976i	$-0.760685\pi$
$373$	−3.92543e6	−1.46088	−0.730441	+	0.682976i	$0.760685\pi$
$374$	0	0
$375$	−1.68750e6	−0.619677
$376$	0	0
$377$	−7.55798e6	−2.73875
$378$	0	0
$379$	−3.83192e6	−1.37031	−0.685154	−	0.728398i	$-0.740265\pi$
$379$	−3.83192e6	−1.37031	−0.685154	+	0.728398i	$0.740265\pi$
$380$	0	0
$381$	−310986.	−0.109756
$382$	0	0
$383$	1.90266e6	0.662774	0.331387	−	0.943495i	$-0.392484\pi$
$383$	1.90266e6	0.662774	0.331387	+	0.943495i	$0.392484\pi$
$384$	0	0
$385$	−12100.0	−0.00416039
$386$	0	0
$387$	1.18114e6	0.400890
$388$	0	0
$389$	93570.0	0.0313518	0.0156759	−	0.999877i	$-0.495010\pi$
$389$	93570.0	0.0313518	0.0156759	+	0.999877i	$0.495010\pi$
$390$	0	0
$391$	−267104.	−0.0883565
$392$	0	0
$393$	1.29251e6	0.422135
$394$	0	0
$395$	2.03030e6	0.654738
$396$	0	0
$397$	−4.89097e6	−1.55747	−0.778734	−	0.627354i	$-0.784138\pi$
$397$	−4.89097e6	−1.55747	−0.778734	+	0.627354i	$0.784138\pi$
$398$	0	0
$399$	22428.0	0.00705275
$400$	0	0
$401$	−5.20637e6	−1.61687	−0.808433	−	0.588588i	$-0.799684\pi$
$401$	−5.20637e6	−1.61687	−0.808433	+	0.588588i	$0.799684\pi$
$402$	0	0
$403$	4.59043e6	1.40796
$404$	0	0
$405$	328050.	0.0993808
$406$	0	0
$407$	562650.	0.168365
$408$	0	0
$409$	−2.35975e6	−0.697523	−0.348761	−	0.937212i	$-0.613398\pi$
$409$	−2.35975e6	−0.697523	−0.348761	+	0.937212i	$0.613398\pi$
$410$	0	0
$411$	1.22528e6	0.357792
$412$	0	0
$413$	−44760.0	−0.0129126
$414$	0	0
$415$	−4.09040e6	−1.16586
$416$	0	0
$417$	2.62903e6	0.740380
$418$	0	0
$419$	1.89052e6	0.526072	0.263036	−	0.964786i	$-0.415276\pi$
$419$	1.89052e6	0.526072	0.263036	+	0.964786i	$0.415276\pi$
$420$	0	0
$421$	2.15591e6	0.592823	0.296412	−	0.955060i	$-0.404210\pi$
$421$	2.15591e6	0.592823	0.296412	+	0.955060i	$0.404210\pi$
$422$	0	0
$423$	−322704.	−0.0876907
$424$	0	0
$425$	−1.22750e6	−0.329647
$426$	0	0
$427$	31324.0	0.00831395
$428$	0	0
$429$	1.05197e6	0.275970
$430$	0	0
$431$	−6.70626e6	−1.73895	−0.869475	−	0.493977i	$-0.835543\pi$
$431$	−6.70626e6	−1.73895	−0.869475	+	0.493977i	$0.835543\pi$
$432$	0	0
$433$	7.70613e6	1.97522	0.987612	−	0.156916i	$-0.0501550\pi$
$433$	7.70613e6	1.97522	0.987612	+	0.156916i	$0.0501550\pi$
$434$	0	0
$435$	−3.52080e6	−0.892110
$436$	0	0
$437$	169456.	0.0424476
$438$	0	0
$439$	−3.83731e6	−0.950311	−0.475155	−	0.879902i	$-0.657608\pi$
$439$	−3.83731e6	−0.950311	−0.475155	+	0.879902i	$0.657608\pi$
$440$	0	0
$441$	−1.36104e6	−0.333254
$442$	0	0
$443$	−2.66516e6	−0.645228	−0.322614	−	0.946531i	$-0.604562\pi$
$443$	−2.66516e6	−0.645228	−0.322614	+	0.946531i	$0.604562\pi$
$444$	0	0
$445$	5.82170e6	1.39364
$446$	0	0
$447$	3.47800e6	0.823304
$448$	0	0
$449$	6.35499e6	1.48764	0.743822	−	0.668378i	$-0.233011\pi$
$449$	6.35499e6	1.48764	0.743822	+	0.668378i	$0.233011\pi$
$450$	0	0
$451$	911856.	0.211098
$452$	0	0
$453$	1.28727e6	0.294730
$454$	0	0
$455$	−96600.0	−0.0218750
$456$	0	0
$457$	−4.11425e6	−0.921510	−0.460755	−	0.887527i	$-0.652421\pi$
$457$	−4.11425e6	−0.921510	−0.460755	+	0.887527i	$0.652421\pi$
$458$	0	0
$459$	1.43176e6	0.317203
$460$	0	0
$461$	−4.12875e6	−0.904828	−0.452414	−	0.891808i	$-0.649437\pi$
$461$	−4.12875e6	−0.904828	−0.452414	+	0.891808i	$0.649437\pi$
$462$	0	0
$463$	4.24316e6	0.919893	0.459947	−	0.887947i	$-0.347869\pi$
$463$	4.24316e6	0.919893	0.459947	+	0.887947i	$0.347869\pi$
$464$	0	0
$465$	2.13840e6	0.458624
$466$	0	0
$467$	348972.	0.0740455	0.0370227	−	0.999314i	$-0.488213\pi$
$467$	348972.	0.0740455	0.0370227	+	0.999314i	$0.488213\pi$
$468$	0	0
$469$	−59128.0	−0.0124126
$470$	0	0
$471$	−1.24900e6	−0.259424
$472$	0	0
$473$	1.76442e6	0.362618
$474$	0	0
$475$	778750.	0.158367
$476$	0	0
$477$	1.00035e6	0.201306
$478$	0	0
$479$	−9.88060e6	−1.96763	−0.983817	−	0.179176i	$-0.942657\pi$
$479$	−9.88060e6	−1.96763	−0.983817	+	0.179176i	$0.942657\pi$
$480$	0	0
$481$	4.49190e6	0.885253
$482$	0	0
$483$	2448.00	0.000477467 0
$484$	0	0
$485$	1.03670e6	0.200124
$486$	0	0
$487$	−4.64960e6	−0.888369	−0.444184	−	0.895935i	$-0.646506\pi$
$487$	−4.64960e6	−0.888369	−0.444184	+	0.895935i	$0.646506\pi$
$488$	0	0
$489$	−3.81038e6	−0.720604
$490$	0	0
$491$	−7.78491e6	−1.45730	−0.728652	−	0.684884i	$-0.759853\pi$
$491$	−7.78491e6	−1.45730	−0.728652	+	0.684884i	$0.759853\pi$
$492$	0	0
$493$	−1.53663e7	−2.84743
$494$	0	0
$495$	490050.	0.0898933
$496$	0	0
$497$	111072.	0.0201704
$498$	0	0
$499$	−8.12784e6	−1.46125	−0.730623	−	0.682781i	$-0.760770\pi$
$499$	−8.12784e6	−1.46125	−0.730623	+	0.682781i	$0.760770\pi$
$500$	0	0
$501$	4.67953e6	0.832929
$502$	0	0
$503$	5.14635e6	0.906941	0.453471	−	0.891271i	$-0.350186\pi$
$503$	5.14635e6	0.906941	0.453471	+	0.891271i	$0.350186\pi$
$504$	0	0
$505$	−2.37320e6	−0.414101
$506$	0	0
$507$	5.05677e6	0.873681
$508$	0	0
$509$	9.08329e6	1.55399	0.776996	−	0.629505i	$-0.216742\pi$
$509$	9.08329e6	1.55399	0.776996	+	0.629505i	$0.216742\pi$
$510$	0	0
$511$	126516.	0.0214335
$512$	0	0
$513$	−908334.	−0.152388
$514$	0	0
$515$	5.32620e6	0.884911
$516$	0	0
$517$	−482064.	−0.0793192
$518$	0	0
$519$	562104.	0.0916006
$520$	0	0
$521$	7.23778e6	1.16818	0.584092	−	0.811688i	$-0.301451\pi$
$521$	7.23778e6	1.16818	0.584092	+	0.811688i	$0.301451\pi$
$522$	0	0
$523$	−3.71095e6	−0.593241	−0.296621	−	0.954995i	$-0.595860\pi$
$523$	−3.71095e6	−0.593241	−0.296621	+	0.954995i	$0.595860\pi$
$524$	0	0
$525$	11250.0	0.00178137
$526$	0	0
$527$	9.33293e6	1.46383
$528$	0	0
$529$	−6.41785e6	−0.997126
$530$	0	0
$531$	1.81278e6	0.279003
$532$	0	0
$533$	7.27978e6	1.10994
$534$	0	0
$535$	3.70960e6	0.560328
$536$	0	0
$537$	2.87888e6	0.430813
$538$	0	0
$539$	−2.03316e6	−0.301440
$540$	0	0
$541$	3.39271e6	0.498372	0.249186	−	0.968456i	$-0.419837\pi$
$541$	3.39271e6	0.498372	0.249186	+	0.968456i	$0.419837\pi$
$542$	0	0
$543$	16182.0	0.00235523
$544$	0	0
$545$	9.64230e6	1.39056
$546$	0	0
$547$	1.37004e6	0.195778	0.0978891	−	0.995197i	$-0.468791\pi$
$547$	1.37004e6	0.195778	0.0978891	+	0.995197i	$0.468791\pi$
$548$	0	0
$549$	−1.26862e6	−0.179639
$550$	0	0
$551$	9.74870e6	1.36794
$552$	0	0
$553$	−81212.0	−0.0112929
$554$	0	0
$555$	2.09250e6	0.288359
$556$	0	0
$557$	−3.25741e6	−0.444872	−0.222436	−	0.974947i	$-0.571401\pi$
$557$	−3.25741e6	−0.444872	−0.222436	+	0.974947i	$0.571401\pi$
$558$	0	0
$559$	1.40862e7	1.90662
$560$	0	0
$561$	2.13880e6	0.286921
$562$	0	0
$563$	−1.74615e6	−0.232172	−0.116086	−	0.993239i	$-0.537035\pi$
$563$	−1.74615e6	−0.232172	−0.116086	+	0.993239i	$0.537035\pi$
$564$	0	0
$565$	−8.72210e6	−1.14948
$566$	0	0
$567$	−13122.0	−0.00171412
$568$	0	0
$569$	6.70083e6	0.867656	0.433828	−	0.900996i	$-0.357162\pi$
$569$	6.70083e6	0.867656	0.433828	+	0.900996i	$0.357162\pi$
$570$	0	0
$571$	−6.62816e6	−0.850752	−0.425376	−	0.905017i	$-0.639858\pi$
$571$	−6.62816e6	−0.850752	−0.425376	+	0.905017i	$0.639858\pi$
$572$	0	0
$573$	−3.66919e6	−0.466857
$574$	0	0
$575$	85000.0	0.0107213
$576$	0	0
$577$	−7.70361e6	−0.963285	−0.481643	−	0.876368i	$-0.659960\pi$
$577$	−7.70361e6	−0.963285	−0.481643	+	0.876368i	$0.659960\pi$
$578$	0	0
$579$	−7.25405e6	−0.899258
$580$	0	0
$581$	163616.	0.0201088
$582$	0	0
$583$	1.49435e6	0.182088
$584$	0	0
$585$	3.91230e6	0.472653
$586$	0	0
$587$	3.71704e6	0.445248	0.222624	−	0.974904i	$-0.428538\pi$
$587$	3.71704e6	0.445248	0.222624	+	0.974904i	$0.428538\pi$
$588$	0	0
$589$	−5.92099e6	−0.703244
$590$	0	0
$591$	−1.26346e6	−0.148796
$592$	0	0
$593$	−6.54492e6	−0.764307	−0.382153	−	0.924099i	$-0.624817\pi$
$593$	−6.54492e6	−0.764307	−0.382153	+	0.924099i	$0.624817\pi$
$594$	0	0
$595$	−196400.	−0.0227431
$596$	0	0
$597$	−7.20626e6	−0.827512
$598$	0	0
$599$	6.46776e6	0.736524	0.368262	−	0.929722i	$-0.379953\pi$
$599$	6.46776e6	0.736524	0.368262	+	0.929722i	$0.379953\pi$
$600$	0	0
$601$	−1.30683e7	−1.47582	−0.737909	−	0.674900i	$-0.764187\pi$
$601$	−1.30683e7	−1.47582	−0.737909	+	0.674900i	$0.764187\pi$
$602$	0	0
$603$	2.39468e6	0.268198
$604$	0	0
$605$	732050.	0.0813116
$606$	0	0
$607$	7.81053e6	0.860416	0.430208	−	0.902730i	$-0.358440\pi$
$607$	7.81053e6	0.860416	0.430208	+	0.902730i	$0.358440\pi$
$608$	0	0
$609$	140832.	0.0153871
$610$	0	0
$611$	−3.84854e6	−0.417055
$612$	0	0
$613$	1.09116e7	1.17284	0.586418	−	0.810009i	$-0.300538\pi$
$613$	1.09116e7	1.17284	0.586418	+	0.810009i	$0.300538\pi$
$614$	0	0
$615$	3.39120e6	0.361548
$616$	0	0
$617$	5.09399e6	0.538698	0.269349	−	0.963043i	$-0.413192\pi$
$617$	5.09399e6	0.538698	0.269349	+	0.963043i	$0.413192\pi$
$618$	0	0
$619$	−1.63836e7	−1.71863	−0.859315	−	0.511447i	$-0.829110\pi$
$619$	−1.63836e7	−1.71863	−0.859315	+	0.511447i	$0.829110\pi$
$620$	0	0
$621$	−99144.0	−0.0103166
$622$	0	0
$623$	−232868.	−0.0240375
$624$	0	0
$625$	−7.42188e6	−0.760000
$626$	0	0
$627$	−1.35689e6	−0.137841
$628$	0	0
$629$	9.13260e6	0.920381
$630$	0	0
$631$	−1.29287e7	−1.29266	−0.646328	−	0.763060i	$-0.723696\pi$
$631$	−1.29287e7	−1.29266	−0.646328	+	0.763060i	$0.723696\pi$
$632$	0	0
$633$	8.58998e6	0.852085
$634$	0	0
$635$	−1.72770e6	−0.170033
$636$	0	0
$637$	−1.62317e7	−1.58495
$638$	0	0
$639$	−4.49842e6	−0.435820
$640$	0	0
$641$	−4.58477e6	−0.440730	−0.220365	−	0.975417i	$-0.570725\pi$
$641$	−4.58477e6	−0.440730	−0.220365	+	0.975417i	$0.570725\pi$
$642$	0	0
$643$	9.46470e6	0.902775	0.451388	−	0.892328i	$-0.350929\pi$
$643$	9.46470e6	0.902775	0.451388	+	0.892328i	$0.350929\pi$
$644$	0	0
$645$	6.56190e6	0.621055
$646$	0	0
$647$	2.44554e6	0.229676	0.114838	−	0.993384i	$-0.463365\pi$
$647$	2.44554e6	0.229676	0.114838	+	0.993384i	$0.463365\pi$
$648$	0	0
$649$	2.70798e6	0.252368
$650$	0	0
$651$	−85536.0	−0.00791036
$652$	0	0
$653$	−1.01815e7	−0.934388	−0.467194	−	0.884155i	$-0.654735\pi$
$653$	−1.01815e7	−0.934388	−0.467194	+	0.884155i	$0.654735\pi$
$654$	0	0
$655$	7.18060e6	0.653969
$656$	0	0
$657$	−5.12390e6	−0.463113
$658$	0	0
$659$	3.21476e6	0.288360	0.144180	−	0.989551i	$-0.453946\pi$
$659$	3.21476e6	0.288360	0.144180	+	0.989551i	$0.453946\pi$
$660$	0	0
$661$	1.30189e7	1.15896	0.579482	−	0.814985i	$-0.303255\pi$
$661$	1.30189e7	1.15896	0.579482	+	0.814985i	$0.303255\pi$
$662$	0	0
$663$	1.70750e7	1.50861
$664$	0	0
$665$	124600.	0.0109261
$666$	0	0
$667$	1.06406e6	0.0926090
$668$	0	0
$669$	9.87584e6	0.853118
$670$	0	0
$671$	−1.89510e6	−0.162490
$672$	0	0
$673$	1.17603e7	1.00088	0.500439	−	0.865772i	$-0.333172\pi$
$673$	1.17603e7	1.00088	0.500439	+	0.865772i	$0.333172\pi$
$674$	0	0
$675$	−455625.	−0.0384900
$676$	0	0
$677$	−4.58024e6	−0.384075	−0.192038	−	0.981388i	$-0.561510\pi$
$677$	−4.58024e6	−0.384075	−0.192038	+	0.981388i	$0.561510\pi$
$678$	0	0
$679$	−41468.0	−0.00345174
$680$	0	0
$681$	−5.44039e6	−0.449534
$682$	0	0
$683$	−1.98214e7	−1.62586	−0.812929	−	0.582363i	$-0.802128\pi$
$683$	−1.98214e7	−1.62586	−0.812929	+	0.582363i	$0.802128\pi$
$684$	0	0
$685$	6.80710e6	0.554288
$686$	0	0
$687$	7.03168e6	0.568417
$688$	0	0
$689$	1.19301e7	0.957406
$690$	0	0
$691$	−3.67404e6	−0.292718	−0.146359	−	0.989232i	$-0.546755\pi$
$691$	−3.67404e6	−0.292718	−0.146359	+	0.989232i	$0.546755\pi$
$692$	0	0
$693$	−19602.0	−0.00155048
$694$	0	0
$695$	1.46057e7	1.14699
$696$	0	0
$697$	1.48007e7	1.15399
$698$	0	0
$699$	−3.79411e6	−0.293709
$700$	0	0
$701$	−1.06350e7	−0.817417	−0.408709	−	0.912665i	$-0.634021\pi$
$701$	−1.06350e7	−0.817417	−0.408709	+	0.912665i	$0.634021\pi$
$702$	0	0
$703$	−5.79390e6	−0.442163
$704$	0	0
$705$	−1.79280e6	−0.135850
$706$	0	0
$707$	94928.0	0.00714243
$708$	0	0
$709$	1.05850e7	0.790818	0.395409	−	0.918505i	$-0.370603\pi$
$709$	1.05850e7	0.790818	0.395409	+	0.918505i	$0.370603\pi$
$710$	0	0
$711$	3.28909e6	0.244006
$712$	0	0
$713$	−646272.	−0.0476093
$714$	0	0
$715$	5.84430e6	0.427531
$716$	0	0
$717$	−3.59593e6	−0.261224
$718$	0	0
$719$	8.17800e6	0.589963	0.294982	−	0.955503i	$-0.404686\pi$
$719$	8.17800e6	0.589963	0.294982	+	0.955503i	$0.404686\pi$
$720$	0	0
$721$	−213048.	−0.0152630
$722$	0	0
$723$	−6.26661e6	−0.445848
$724$	0	0
$725$	4.89000e6	0.345513
$726$	0	0
$727$	2.15741e6	0.151390	0.0756948	−	0.997131i	$-0.475883\pi$
$727$	2.15741e6	0.151390	0.0756948	+	0.997131i	$0.475883\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.86390e7	1.98228
$732$	0	0
$733$	−1.15251e6	−0.0792294	−0.0396147	−	0.999215i	$-0.512613\pi$
$733$	−1.15251e6	−0.0792294	−0.0396147	+	0.999215i	$0.512613\pi$
$734$	0	0
$735$	−7.56135e6	−0.516275
$736$	0	0
$737$	3.57724e6	0.242594
$738$	0	0
$739$	−2.02937e7	−1.36694	−0.683470	−	0.729979i	$-0.739530\pi$
$739$	−2.02937e7	−1.36694	−0.683470	+	0.729979i	$0.739530\pi$
$740$	0	0
$741$	−1.08327e7	−0.724757
$742$	0	0
$743$	1.80798e7	1.20149	0.600746	−	0.799440i	$-0.294870\pi$
$743$	1.80798e7	1.20149	0.600746	+	0.799440i	$0.294870\pi$
$744$	0	0
$745$	1.93222e7	1.27546
$746$	0	0
$747$	−6.62645e6	−0.434490
$748$	0	0
$749$	−148384.	−0.00966457
$750$	0	0
$751$	−1.01462e7	−0.656456	−0.328228	−	0.944599i	$-0.606451\pi$
$751$	−1.01462e7	−0.656456	−0.328228	+	0.944599i	$0.606451\pi$
$752$	0	0
$753$	8.59770e6	0.552580
$754$	0	0
$755$	7.15150e6	0.456594
$756$	0	0
$757$	5.96121e6	0.378089	0.189045	−	0.981968i	$-0.439461\pi$
$757$	5.96121e6	0.378089	0.189045	+	0.981968i	$0.439461\pi$
$758$	0	0
$759$	−148104.	−0.00933173
$760$	0	0
$761$	−1.51013e7	−0.945263	−0.472631	−	0.881260i	$-0.656696\pi$
$761$	−1.51013e7	−0.945263	−0.472631	+	0.881260i	$0.656696\pi$
$762$	0	0
$763$	−385692.	−0.0239844
$764$	0	0
$765$	7.95420e6	0.491409
$766$	0	0
$767$	2.16191e7	1.32693
$768$	0	0
$769$	1.82633e7	1.11369	0.556843	−	0.830618i	$-0.312013\pi$
$769$	1.82633e7	1.11369	0.556843	+	0.830618i	$0.312013\pi$
$770$	0	0
$771$	9.00117e6	0.545335
$772$	0	0
$773$	2.53652e7	1.52682	0.763412	−	0.645912i	$-0.223523\pi$
$773$	2.53652e7	1.52682	0.763412	+	0.645912i	$0.223523\pi$
$774$	0	0
$775$	−2.97000e6	−0.177624
$776$	0	0
$777$	−83700.0	−0.00497362
$778$	0	0
$779$	−9.38986e6	−0.554390
$780$	0	0
$781$	−6.71986e6	−0.394214
$782$	0	0
$783$	−5.70370e6	−0.332470
$784$	0	0
$785$	−6.93890e6	−0.401899
$786$	0	0
$787$	4.17652e6	0.240368	0.120184	−	0.992752i	$-0.461651\pi$
$787$	4.17652e6	0.240368	0.120184	+	0.992752i	$0.461651\pi$
$788$	0	0
$789$	−2.02770e6	−0.115961
$790$	0	0
$791$	348884.	0.0198262
$792$	0	0
$793$	−1.51295e7	−0.854361
$794$	0	0
$795$	5.55750e6	0.311861
$796$	0	0
$797$	1.16289e6	0.0648474	0.0324237	−	0.999474i	$-0.489677\pi$
$797$	1.16289e6	0.0648474	0.0324237	+	0.999474i	$0.489677\pi$
$798$	0	0
$799$	−7.82458e6	−0.433605
$800$	0	0
$801$	9.43115e6	0.519378
$802$	0	0
$803$	−7.65422e6	−0.418902
$804$	0	0
$805$	13600.0	0.000739689 0
$806$	0	0
$807$	1.18538e7	0.640727
$808$	0	0
$809$	−7.61001e6	−0.408803	−0.204401	−	0.978887i	$-0.565525\pi$
$809$	−7.61001e6	−0.408803	−0.204401	+	0.978887i	$0.565525\pi$
$810$	0	0
$811$	3.71132e6	0.198142	0.0990710	−	0.995080i	$-0.468413\pi$
$811$	3.71132e6	0.198142	0.0990710	+	0.995080i	$0.468413\pi$
$812$	0	0
$813$	3.06988e6	0.162890
$814$	0	0
$815$	−2.11688e7	−1.11635
$816$	0	0
$817$	−1.81692e7	−0.952314
$818$	0	0
$819$	−156492.	−0.00815234
$820$	0	0
$821$	−7.53697e6	−0.390246	−0.195123	−	0.980779i	$-0.562511\pi$
$821$	−7.53697e6	−0.390246	−0.195123	+	0.980779i	$0.562511\pi$
$822$	0	0
$823$	3.64495e7	1.87582	0.937911	−	0.346877i	$-0.112758\pi$
$823$	3.64495e7	1.87582	0.937911	+	0.346877i	$0.112758\pi$
$824$	0	0
$825$	−680625.	−0.0348155
$826$	0	0
$827$	1.14515e7	0.582234	0.291117	−	0.956687i	$-0.405973\pi$
$827$	1.14515e7	0.582234	0.291117	+	0.956687i	$0.405973\pi$
$828$	0	0
$829$	1.85780e7	0.938883	0.469442	−	0.882964i	$-0.344455\pi$
$829$	1.85780e7	0.938883	0.469442	+	0.882964i	$0.344455\pi$
$830$	0	0
$831$	−1.46026e7	−0.733544
$832$	0	0
$833$	−3.30011e7	−1.64784
$834$	0	0
$835$	2.59974e7	1.29037
$836$	0	0
$837$	3.46421e6	0.170919
$838$	0	0
$839$	1.14159e7	0.559894	0.279947	−	0.960015i	$-0.409683\pi$
$839$	1.14159e7	0.559894	0.279947	+	0.960015i	$0.409683\pi$
$840$	0	0
$841$	4.07038e7	1.98447
$842$	0	0
$843$	−1.38933e7	−0.673341
$844$	0	0
$845$	2.80931e7	1.35350
$846$	0	0
$847$	−29282.0	−0.00140247
$848$	0	0
$849$	−2.03272e7	−0.967851
$850$	0	0
$851$	−632400.	−0.0299342
$852$	0	0
$853$	−3.30590e7	−1.55567	−0.777835	−	0.628469i	$-0.783682\pi$
$853$	−3.30590e7	−1.55567	−0.777835	+	0.628469i	$0.783682\pi$
$854$	0	0
$855$	−5.04630e6	−0.236079
$856$	0	0
$857$	2.64603e7	1.23067	0.615336	−	0.788265i	$-0.289020\pi$
$857$	2.64603e7	1.23067	0.615336	+	0.788265i	$0.289020\pi$
$858$	0	0
$859$	−3.65684e7	−1.69092	−0.845459	−	0.534040i	$-0.820673\pi$
$859$	−3.65684e7	−1.69092	−0.845459	+	0.534040i	$0.820673\pi$
$860$	0	0
$861$	−135648.	−0.00623599
$862$	0	0
$863$	1.94034e7	0.886850	0.443425	−	0.896311i	$-0.353763\pi$
$863$	1.94034e7	0.886850	0.443425	+	0.896311i	$0.353763\pi$
$864$	0	0
$865$	3.12280e6	0.141907
$866$	0	0
$867$	2.19370e7	0.991125
$868$	0	0
$869$	4.91333e6	0.220712
$870$	0	0
$871$	2.85588e7	1.27554
$872$	0	0
$873$	1.67945e6	0.0745817
$874$	0	0
$875$	375000.	0.0165581
$876$	0	0
$877$	6.28226e6	0.275814	0.137907	−	0.990445i	$-0.455962\pi$
$877$	6.28226e6	0.275814	0.137907	+	0.990445i	$0.455962\pi$
$878$	0	0
$879$	2.60810e7	1.13855
$880$	0	0
$881$	−1.58365e7	−0.687414	−0.343707	−	0.939077i	$-0.611683\pi$
$881$	−1.58365e7	−0.687414	−0.343707	+	0.939077i	$0.611683\pi$
$882$	0	0
$883$	−3.78701e7	−1.63454	−0.817269	−	0.576256i	$-0.804513\pi$
$883$	−3.78701e7	−1.63454	−0.817269	+	0.576256i	$0.804513\pi$
$884$	0	0
$885$	1.00710e7	0.432230
$886$	0	0
$887$	−3.89442e7	−1.66201	−0.831006	−	0.556264i	$-0.812234\pi$
$887$	−3.89442e7	−1.66201	−0.831006	+	0.556264i	$0.812234\pi$
$888$	0	0
$889$	69108.0	0.00293274
$890$	0	0
$891$	793881.	0.0335013
$892$	0	0
$893$	4.96406e6	0.208309
$894$	0	0
$895$	1.59938e7	0.667412
$896$	0	0
$897$	−1.18238e6	−0.0490656
$898$	0	0
$899$	−3.71796e7	−1.53429
$900$	0	0
$901$	2.42554e7	0.995397
$902$	0	0
$903$	−262476.	−0.0107120
$904$	0	0
$905$	89900.0	0.00364870
$906$	0	0
$907$	−4.23522e7	−1.70946	−0.854728	−	0.519075i	$-0.826276\pi$
$907$	−4.23522e7	−1.70946	−0.854728	+	0.519075i	$0.826276\pi$
$908$	0	0
$909$	−3.84458e6	−0.154326
$910$	0	0
$911$	−2.81715e7	−1.12464	−0.562320	−	0.826919i	$-0.690091\pi$
$911$	−2.81715e7	−1.12464	−0.562320	+	0.826919i	$0.690091\pi$
$912$	0	0
$913$	−9.89877e6	−0.393011
$914$	0	0
$915$	−7.04790e6	−0.278296
$916$	0	0
$917$	−287224.	−0.0112797
$918$	0	0
$919$	3.35481e7	1.31033	0.655163	−	0.755488i	$-0.272600\pi$
$919$	3.35481e7	1.31033	0.655163	+	0.755488i	$0.272600\pi$
$920$	0	0
$921$	−2.00066e7	−0.777183
$922$	0	0
$923$	−5.36478e7	−2.07275
$924$	0	0
$925$	−2.90625e6	−0.111681
$926$	0	0
$927$	8.62844e6	0.329787
$928$	0	0
$929$	1.07901e7	0.410190	0.205095	−	0.978742i	$-0.434250\pi$
$929$	1.07901e7	0.410190	0.205095	+	0.978742i	$0.434250\pi$
$930$	0	0
$931$	2.09365e7	0.791645
$932$	0	0
$933$	1.87467e7	0.705052
$934$	0	0
$935$	1.18822e7	0.444496
$936$	0	0
$937$	−1.75228e7	−0.652011	−0.326006	−	0.945368i	$-0.605703\pi$
$937$	−1.75228e7	−0.652011	−0.326006	+	0.945368i	$0.605703\pi$
$938$	0	0
$939$	−1.71319e6	−0.0634075
$940$	0	0
$941$	−6.39269e6	−0.235347	−0.117674	−	0.993052i	$-0.537544\pi$
$941$	−6.39269e6	−0.235347	−0.117674	+	0.993052i	$0.537544\pi$
$942$	0	0
$943$	−1.02490e6	−0.0375319
$944$	0	0
$945$	−72900.0	−0.00265551
$946$	0	0
$947$	−2.33957e7	−0.847737	−0.423868	−	0.905724i	$-0.639328\pi$
$947$	−2.33957e7	−0.847737	−0.423868	+	0.905724i	$0.639328\pi$
$948$	0	0
$949$	−6.11072e7	−2.20256
$950$	0	0
$951$	−2.53870e6	−0.0910249
$952$	0	0
$953$	−2.08499e7	−0.743655	−0.371828	−	0.928302i	$-0.621269\pi$
$953$	−2.08499e7	−0.743655	−0.371828	+	0.928302i	$0.621269\pi$
$954$	0	0
$955$	−2.03844e7	−0.723252
$956$	0	0
$957$	−8.52034e6	−0.300730
$958$	0	0
$959$	−272284.	−0.00956039
$960$	0	0
$961$	−6.04765e6	−0.211241
$962$	0	0
$963$	6.00955e6	0.208822
$964$	0	0
$965$	−4.03003e7	−1.39312
$966$	0	0
$967$	−4.36248e7	−1.50026	−0.750132	−	0.661288i	$-0.770010\pi$
$967$	−4.36248e7	−1.50026	−0.750132	+	0.661288i	$0.770010\pi$
$968$	0	0
$969$	−2.20243e7	−0.753516
$970$	0	0
$971$	4.66175e7	1.58672	0.793362	−	0.608750i	$-0.208329\pi$
$971$	4.66175e7	1.58672	0.793362	+	0.608750i	$0.208329\pi$
$972$	0	0
$973$	−584228.	−0.0197834
$974$	0	0
$975$	−5.43375e6	−0.183058
$976$	0	0
$977$	1.72165e7	0.577045	0.288522	−	0.957473i	$-0.406836\pi$
$977$	1.72165e7	0.577045	0.288522	+	0.957473i	$0.406836\pi$
$978$	0	0
$979$	1.40885e7	0.469795
$980$	0	0
$981$	1.56205e7	0.518231
$982$	0	0
$983$	1.11175e7	0.366965	0.183482	−	0.983023i	$-0.441263\pi$
$983$	1.11175e7	0.366965	0.183482	+	0.983023i	$0.441263\pi$
$984$	0	0
$985$	−7.01920e6	−0.230514
$986$	0	0
$987$	71712.0	0.00234314
$988$	0	0
$989$	−1.98315e6	−0.0644711
$990$	0	0
$991$	4.11903e7	1.33233	0.666164	−	0.745806i	$-0.267935\pi$
$991$	4.11903e7	1.33233	0.666164	+	0.745806i	$0.267935\pi$
$992$	0	0
$993$	−2.95255e7	−0.950221
$994$	0	0
$995$	−4.00348e7	−1.28198
$996$	0	0
$997$	−1.07046e7	−0.341060	−0.170530	−	0.985352i	$-0.554548\pi$
$997$	−1.07046e7	−0.341060	−0.170530	+	0.985352i	$0.554548\pi$
$998$	0	0
$999$	3.38985e6	0.107465

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.a.j.1.1		1
4.3	odd	2		66.6.a.d.1.1	✓	1
12.11	even	2		198.6.a.a.1.1		1
44.43	even	2		726.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.6.a.d.1.1	✓	1	4.3	odd	2
198.6.a.a.1.1		1	12.11	even	2
528.6.a.j.1.1		1	1.1	even	1	trivial
726.6.a.c.1.1		1	44.43	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:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more