LMFDB - Embedded newform 441.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -31.0000 q^{4} -34.0000 q^{5} +63.0000 q^{8} +34.0000 q^{10} +340.000 q^{11} -454.000 q^{13} +929.000 q^{16} -798.000 q^{17} -892.000 q^{19} +1054.00 q^{20} -340.000 q^{22} +3192.00 q^{23} -1969.00 q^{25} +454.000 q^{26} +8242.00 q^{29} +2496.00 q^{31} -2945.00 q^{32} +798.000 q^{34} +9798.00 q^{37} +892.000 q^{38} -2142.00 q^{40} +19834.0 q^{41} -17236.0 q^{43} -10540.0 q^{44} -3192.00 q^{46} +8928.00 q^{47} +1969.00 q^{50} +14074.0 q^{52} -150.000 q^{53} -11560.0 q^{55} -8242.00 q^{58} -42396.0 q^{59} -14758.0 q^{61} -2496.00 q^{62} -26783.0 q^{64} +15436.0 q^{65} -1676.00 q^{67} +24738.0 q^{68} -14568.0 q^{71} -78378.0 q^{73} -9798.00 q^{74} +27652.0 q^{76} -2272.00 q^{79} -31586.0 q^{80} -19834.0 q^{82} -37764.0 q^{83} +27132.0 q^{85} +17236.0 q^{86} +21420.0 q^{88} -117286. q^{89} -98952.0 q^{92} -8928.00 q^{94} +30328.0 q^{95} -10002.0 q^{97} +O(q^{100})$

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 $

$2$	−1.00000	−0.176777	−0.0883883	−	0.996086i	$-0.528172\pi$
$2$	−1.00000	−0.176777	−0.0883883	+	0.996086i	$0.528172\pi$
$3$	0	0
$3$	0	0
$4$	−31.0000	−0.968750
$5$	−34.0000	−0.608210	−0.304105	−	0.952638i	$-0.598357\pi$
$5$	−34.0000	−0.608210	−0.304105	+	0.952638i	$0.598357\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	63.0000	0.348029
$9$	0	0
$10$	34.0000	0.107517
$11$	340.000	0.847222	0.423611	−	0.905844i	$-0.360762\pi$
$11$	340.000	0.847222	0.423611	+	0.905844i	$0.360762\pi$
$12$	0	0
$13$	−454.000	−0.745071	−0.372535	−	0.928018i	$-0.621511\pi$
$13$	−454.000	−0.745071	−0.372535	+	0.928018i	$0.621511\pi$
$14$	0	0
$15$	0	0
$16$	929.000	0.907227
$17$	−798.000	−0.669700	−0.334850	−	0.942271i	$-0.608686\pi$
$17$	−798.000	−0.669700	−0.334850	+	0.942271i	$0.608686\pi$
$18$	0	0
$19$	−892.000	−0.566867	−0.283433	−	0.958992i	$-0.591473\pi$
$19$	−892.000	−0.566867	−0.283433	+	0.958992i	$0.591473\pi$
$20$	1054.00	0.589204
$21$	0	0
$22$	−340.000	−0.149769
$23$	3192.00	1.25818	0.629091	−	0.777332i	$-0.283427\pi$
$23$	3192.00	1.25818	0.629091	+	0.777332i	$0.283427\pi$
$24$	0	0
$25$	−1969.00	−0.630080
$26$	454.000	0.131711
$27$	0	0
$28$	0	0
$29$	8242.00	1.81986	0.909929	−	0.414764i	$-0.136136\pi$
$29$	8242.00	1.81986	0.909929	+	0.414764i	$0.136136\pi$
$30$	0	0
$31$	2496.00	0.466488	0.233244	−	0.972418i	$-0.425066\pi$
$31$	2496.00	0.466488	0.233244	+	0.972418i	$0.425066\pi$
$32$	−2945.00	−0.508406
$33$	0	0
$34$	798.000	0.118387
$35$	0	0
$36$	0	0
$37$	9798.00	1.17661	0.588306	−	0.808639i	$-0.299795\pi$
$37$	9798.00	1.17661	0.588306	+	0.808639i	$0.299795\pi$
$38$	892.000	0.100209
$39$	0	0
$40$	−2142.00	−0.211675
$41$	19834.0	1.84268	0.921342	−	0.388754i	$-0.127094\pi$
$41$	19834.0	1.84268	0.921342	+	0.388754i	$0.127094\pi$
$42$	0	0
$43$	−17236.0	−1.42156	−0.710780	−	0.703414i	$-0.751658\pi$
$43$	−17236.0	−1.42156	−0.710780	+	0.703414i	$0.751658\pi$
$44$	−10540.0	−0.820746
$45$	0	0
$46$	−3192.00	−0.222417
$47$	8928.00	0.589535	0.294767	−	0.955569i	$-0.404758\pi$
$47$	8928.00	0.589535	0.294767	+	0.955569i	$0.404758\pi$
$48$	0	0
$49$	0	0
$50$	1969.00	0.111383
$51$	0	0
$52$	14074.0	0.721787
$53$	−150.000	−0.00733502	−0.00366751	−	0.999993i	$-0.501167\pi$
$53$	−150.000	−0.00733502	−0.00366751	+	0.999993i	$0.501167\pi$
$54$	0	0
$55$	−11560.0	−0.515289
$56$	0	0
$57$	0	0
$58$	−8242.00	−0.321709
$59$	−42396.0	−1.58560	−0.792802	−	0.609479i	$-0.791379\pi$
$59$	−42396.0	−1.58560	−0.792802	+	0.609479i	$0.791379\pi$
$60$	0	0
$61$	−14758.0	−0.507812	−0.253906	−	0.967229i	$-0.581715\pi$
$61$	−14758.0	−0.507812	−0.253906	+	0.967229i	$0.581715\pi$
$62$	−2496.00	−0.0824642
$63$	0	0
$64$	−26783.0	−0.817352
$65$	15436.0	0.453160
$66$	0	0
$67$	−1676.00	−0.0456128	−0.0228064	−	0.999740i	$-0.507260\pi$
$67$	−1676.00	−0.0456128	−0.0228064	+	0.999740i	$0.507260\pi$
$68$	24738.0	0.648772
$69$	0	0
$70$	0	0
$71$	−14568.0	−0.342968	−0.171484	−	0.985187i	$-0.554856\pi$
$71$	−14568.0	−0.342968	−0.171484	+	0.985187i	$0.554856\pi$
$72$	0	0
$73$	−78378.0	−1.72142	−0.860710	−	0.509095i	$-0.829980\pi$
$73$	−78378.0	−1.72142	−0.860710	+	0.509095i	$0.829980\pi$
$74$	−9798.00	−0.207998
$75$	0	0
$76$	27652.0	0.549152
$77$	0	0
$78$	0	0
$79$	−2272.00	−0.0409582	−0.0204791	−	0.999790i	$-0.506519\pi$
$79$	−2272.00	−0.0409582	−0.0204791	+	0.999790i	$0.506519\pi$
$80$	−31586.0	−0.551785
$81$	0	0
$82$	−19834.0	−0.325743
$83$	−37764.0	−0.601704	−0.300852	−	0.953671i	$-0.597271\pi$
$83$	−37764.0	−0.601704	−0.300852	+	0.953671i	$0.597271\pi$
$84$	0	0
$85$	27132.0	0.407319
$86$	17236.0	0.251299
$87$	0	0
$88$	21420.0	0.294858
$89$	−117286.	−1.56954	−0.784768	−	0.619790i	$-0.787218\pi$
$89$	−117286.	−1.56954	−0.784768	+	0.619790i	$0.787218\pi$
$90$	0	0
$91$	0	0
$92$	−98952.0	−1.21886
$93$	0	0
$94$	−8928.00	−0.104216
$95$	30328.0	0.344774
$96$	0	0
$97$	−10002.0	−0.107934	−0.0539669	−	0.998543i	$-0.517187\pi$
$97$	−10002.0	−0.107934	−0.0539669	+	0.998543i	$0.517187\pi$
$98$	0	0
$99$	0	0
$100$	61039.0	0.610390
$101$	−108770.	−1.06098	−0.530488	−	0.847692i	$-0.677991\pi$
$101$	−108770.	−1.06098	−0.530488	+	0.847692i	$0.677991\pi$
$102$	0	0
$103$	199192.	1.85003	0.925015	−	0.379930i	$-0.124052\pi$
$103$	199192.	1.85003	0.925015	+	0.379930i	$0.124052\pi$
$104$	−28602.0	−0.259306
$105$	0	0
$106$	150.000	0.00129666
$107$	79972.0	0.675272	0.337636	−	0.941277i	$-0.390373\pi$
$107$	79972.0	0.675272	0.337636	+	0.941277i	$0.390373\pi$
$108$	0	0
$109$	−46098.0	−0.371634	−0.185817	−	0.982584i	$-0.559493\pi$
$109$	−46098.0	−0.371634	−0.185817	+	0.982584i	$0.559493\pi$
$110$	11560.0	0.0910911
$111$	0	0
$112$	0	0
$113$	−262706.	−1.93541	−0.967707	−	0.252078i	$-0.918886\pi$
$113$	−262706.	−1.93541	−0.967707	+	0.252078i	$0.918886\pi$
$114$	0	0
$115$	−108528.	−0.765239
$116$	−255502.	−1.76299
$117$	0	0
$118$	42396.0	0.280298
$119$	0	0
$120$	0	0
$121$	−45451.0	−0.282215
$122$	14758.0	0.0897693
$123$	0	0
$124$	−77376.0	−0.451910
$125$	173196.	0.991432
$126$	0	0
$127$	196608.	1.08166	0.540831	−	0.841131i	$-0.318110\pi$
$127$	196608.	1.08166	0.540831	+	0.841131i	$0.318110\pi$
$128$	121023.	0.652894
$129$	0	0
$130$	−15436.0	−0.0801081
$131$	−77140.0	−0.392737	−0.196368	−	0.980530i	$-0.562915\pi$
$131$	−77140.0	−0.392737	−0.196368	+	0.980530i	$0.562915\pi$
$132$	0	0
$133$	0	0
$134$	1676.00	0.00806329
$135$	0	0
$136$	−50274.0	−0.233075
$137$	−208170.	−0.947582	−0.473791	−	0.880637i	$-0.657115\pi$
$137$	−208170.	−0.947582	−0.473791	+	0.880637i	$0.657115\pi$
$138$	0	0
$139$	275580.	1.20979	0.604896	−	0.796304i	$-0.293215\pi$
$139$	275580.	1.20979	0.604896	+	0.796304i	$0.293215\pi$
$140$	0	0
$141$	0	0
$142$	14568.0	0.0606288
$143$	−154360.	−0.631240
$144$	0	0
$145$	−280228.	−1.10686
$146$	78378.0	0.304307
$147$	0	0
$148$	−303738.	−1.13984
$149$	296106.	1.09265	0.546326	−	0.837573i	$-0.316026\pi$
$149$	296106.	1.09265	0.546326	+	0.837573i	$0.316026\pi$
$150$	0	0
$151$	−426472.	−1.52212	−0.761059	−	0.648683i	$-0.775320\pi$
$151$	−426472.	−1.52212	−0.761059	+	0.648683i	$0.775320\pi$
$152$	−56196.0	−0.197286
$153$	0	0
$154$	0	0
$155$	−84864.0	−0.283723
$156$	0	0
$157$	−178486.	−0.577903	−0.288952	−	0.957344i	$-0.593307\pi$
$157$	−178486.	−0.577903	−0.288952	+	0.957344i	$0.593307\pi$
$158$	2272.00	0.00724045
$159$	0	0
$160$	100130.	0.309218
$161$	0	0
$162$	0	0
$163$	252772.	0.745178	0.372589	−	0.927996i	$-0.378470\pi$
$163$	252772.	0.745178	0.372589	+	0.927996i	$0.378470\pi$
$164$	−614854.	−1.78510
$165$	0	0
$166$	37764.0	0.106367
$167$	508088.	1.40977	0.704884	−	0.709322i	$-0.250999\pi$
$167$	508088.	1.40977	0.704884	+	0.709322i	$0.250999\pi$
$168$	0	0
$169$	−165177.	−0.444870
$170$	−27132.0	−0.0720045
$171$	0	0
$172$	534316.	1.37714
$173$	−221834.	−0.563525	−0.281762	−	0.959484i	$-0.590919\pi$
$173$	−221834.	−0.563525	−0.281762	+	0.959484i	$0.590919\pi$
$174$	0	0
$175$	0	0
$176$	315860.	0.768622
$177$	0	0
$178$	117286.	0.277457
$179$	113564.	0.264916	0.132458	−	0.991189i	$-0.457713\pi$
$179$	113564.	0.264916	0.132458	+	0.991189i	$0.457713\pi$
$180$	0	0
$181$	−663118.	−1.50451	−0.752254	−	0.658873i	$-0.771033\pi$
$181$	−663118.	−1.50451	−0.752254	+	0.658873i	$0.771033\pi$
$182$	0	0
$183$	0	0
$184$	201096.	0.437884
$185$	−333132.	−0.715628
$186$	0	0
$187$	−271320.	−0.567385
$188$	−276768.	−0.571112
$189$	0	0
$190$	−30328.0	−0.0609480
$191$	−505664.	−1.00295	−0.501474	−	0.865173i	$-0.667209\pi$
$191$	−505664.	−1.00295	−0.501474	+	0.865173i	$0.667209\pi$
$192$	0	0
$193$	−432382.	−0.835554	−0.417777	−	0.908550i	$-0.637191\pi$
$193$	−432382.	−0.835554	−0.417777	+	0.908550i	$0.637191\pi$
$194$	10002.0	0.0190802
$195$	0	0
$196$	0	0
$197$	131962.	0.242261	0.121130	−	0.992637i	$-0.461348\pi$
$197$	131962.	0.242261	0.121130	+	0.992637i	$0.461348\pi$
$198$	0	0
$199$	−298536.	−0.534397	−0.267199	−	0.963642i	$-0.586098\pi$
$199$	−298536.	−0.534397	−0.267199	+	0.963642i	$0.586098\pi$
$200$	−124047.	−0.219286
$201$	0	0
$202$	108770.	0.187556
$203$	0	0
$204$	0	0
$205$	−674356.	−1.12074
$206$	−199192.	−0.327042
$207$	0	0
$208$	−421766.	−0.675948
$209$	−303280.	−0.480262
$210$	0	0
$211$	−1.17062e6	−1.81013	−0.905065	−	0.425273i	$-0.860178\pi$
$211$	−1.17062e6	−1.81013	−0.905065	+	0.425273i	$0.860178\pi$
$212$	4650.00	0.00710581
$213$	0	0
$214$	−79972.0	−0.119372
$215$	586024.	0.864608
$216$	0	0
$217$	0	0
$218$	46098.0	0.0656963
$219$	0	0
$220$	358360.	0.499186
$221$	362292.	0.498974
$222$	0	0
$223$	−399376.	−0.537799	−0.268899	−	0.963168i	$-0.586660\pi$
$223$	−399376.	−0.537799	−0.268899	+	0.963168i	$0.586660\pi$
$224$	0	0
$225$	0	0
$226$	262706.	0.342136
$227$	707916.	0.911837	0.455918	−	0.890022i	$-0.349311\pi$
$227$	707916.	0.911837	0.455918	+	0.890022i	$0.349311\pi$
$228$	0	0
$229$	735778.	0.927167	0.463584	−	0.886053i	$-0.346563\pi$
$229$	735778.	0.927167	0.463584	+	0.886053i	$0.346563\pi$
$230$	108528.	0.135276
$231$	0	0
$232$	519246.	0.633364
$233$	208758.	0.251915	0.125957	−	0.992036i	$-0.459800\pi$
$233$	208758.	0.251915	0.125957	+	0.992036i	$0.459800\pi$
$234$	0	0
$235$	−303552.	−0.358561
$236$	1.31428e6	1.53605
$237$	0	0
$238$	0	0
$239$	−713376.	−0.807837	−0.403919	−	0.914795i	$-0.632352\pi$
$239$	−713376.	−0.807837	−0.403919	+	0.914795i	$0.632352\pi$
$240$	0	0
$241$	505246.	0.560351	0.280176	−	0.959949i	$-0.409607\pi$
$241$	505246.	0.560351	0.280176	+	0.959949i	$0.409607\pi$
$242$	45451.0	0.0498890
$243$	0	0
$244$	457498.	0.491943
$245$	0	0
$246$	0	0
$247$	404968.	0.422356
$248$	157248.	0.162351
$249$	0	0
$250$	−173196.	−0.175262
$251$	317108.	0.317704	0.158852	−	0.987302i	$-0.449221\pi$
$251$	317108.	0.317704	0.158852	+	0.987302i	$0.449221\pi$
$252$	0	0
$253$	1.08528e6	1.06596
$254$	−196608.	−0.191213
$255$	0	0
$256$	736033.	0.701936
$257$	−1.44285e6	−1.36266	−0.681329	−	0.731977i	$-0.738598\pi$
$257$	−1.44285e6	−1.36266	−0.681329	+	0.731977i	$0.738598\pi$
$258$	0	0
$259$	0	0
$260$	−478516.	−0.438999
$261$	0	0
$262$	77140.0	0.0694267
$263$	−271496.	−0.242033	−0.121016	−	0.992651i	$-0.538615\pi$
$263$	−271496.	−0.242033	−0.121016	+	0.992651i	$0.538615\pi$
$264$	0	0
$265$	5100.00	0.00446124
$266$	0	0
$267$	0	0
$268$	51956.0	0.0441874
$269$	850614.	0.716724	0.358362	−	0.933583i	$-0.383335\pi$
$269$	850614.	0.716724	0.358362	+	0.933583i	$0.383335\pi$
$270$	0	0
$271$	540128.	0.446759	0.223380	−	0.974732i	$-0.428291\pi$
$271$	540128.	0.446759	0.223380	+	0.974732i	$0.428291\pi$
$272$	−741342.	−0.607570
$273$	0	0
$274$	208170.	0.167510
$275$	−669460.	−0.533818
$276$	0	0
$277$	513574.	0.402164	0.201082	−	0.979574i	$-0.435554\pi$
$277$	513574.	0.402164	0.201082	+	0.979574i	$0.435554\pi$
$278$	−275580.	−0.213863
$279$	0	0
$280$	0	0
$281$	1.35642e6	1.02478	0.512388	−	0.858754i	$-0.328761\pi$
$281$	1.35642e6	1.02478	0.512388	+	0.858754i	$0.328761\pi$
$282$	0	0
$283$	−286756.	−0.212837	−0.106418	−	0.994321i	$-0.533938\pi$
$283$	−286756.	−0.212837	−0.106418	+	0.994321i	$0.533938\pi$
$284$	451608.	0.332251
$285$	0	0
$286$	154360.	0.111589
$287$	0	0
$288$	0	0
$289$	−783053.	−0.551501
$290$	280228.	0.195667
$291$	0	0
$292$	2.42972e6	1.66763
$293$	−1.70727e6	−1.16180	−0.580901	−	0.813974i	$-0.697300\pi$
$293$	−1.70727e6	−1.16180	−0.580901	+	0.813974i	$0.697300\pi$
$294$	0	0
$295$	1.44146e6	0.964381
$296$	617274.	0.409495
$297$	0	0
$298$	−296106.	−0.193155
$299$	−1.44917e6	−0.937434
$300$	0	0
$301$	0	0
$302$	426472.	0.269075
$303$	0	0
$304$	−828668.	−0.514276
$305$	501772.	0.308857
$306$	0	0
$307$	546788.	0.331111	0.165555	−	0.986201i	$-0.447058\pi$
$307$	546788.	0.331111	0.165555	+	0.986201i	$0.447058\pi$
$308$	0	0
$309$	0	0
$310$	84864.0	0.0501556
$311$	3.23426e6	1.89616	0.948079	−	0.318035i	$-0.103023\pi$
$311$	3.23426e6	1.89616	0.948079	+	0.318035i	$0.103023\pi$
$312$	0	0
$313$	−1.81313e6	−1.04609	−0.523044	−	0.852306i	$-0.675204\pi$
$313$	−1.81313e6	−1.04609	−0.523044	+	0.852306i	$0.675204\pi$
$314$	178486.	0.102160
$315$	0	0
$316$	70432.0	0.0396782
$317$	1.27658e6	0.713509	0.356754	−	0.934198i	$-0.383883\pi$
$317$	1.27658e6	0.713509	0.356754	+	0.934198i	$0.383883\pi$
$318$	0	0
$319$	2.80228e6	1.54182
$320$	910622.	0.497122
$321$	0	0
$322$	0	0
$323$	711816.	0.379631
$324$	0	0
$325$	893926.	0.469454
$326$	−252772.	−0.131730
$327$	0	0
$328$	1.24954e6	0.641307
$329$	0	0
$330$	0	0
$331$	−1.73621e6	−0.871029	−0.435515	−	0.900182i	$-0.643434\pi$
$331$	−1.73621e6	−0.871029	−0.435515	+	0.900182i	$0.643434\pi$
$332$	1.17068e6	0.582901
$333$	0	0
$334$	−508088.	−0.249214
$335$	56984.0	0.0277422
$336$	0	0
$337$	2.07215e6	0.993907	0.496953	−	0.867777i	$-0.334452\pi$
$337$	2.07215e6	0.993907	0.496953	+	0.867777i	$0.334452\pi$
$338$	165177.	0.0786426
$339$	0	0
$340$	−841092.	−0.394590
$341$	848640.	0.395219
$342$	0	0
$343$	0	0
$344$	−1.08587e6	−0.494744
$345$	0	0
$346$	221834.	0.0996180
$347$	1.65146e6	0.736282	0.368141	−	0.929770i	$-0.379994\pi$
$347$	1.65146e6	0.736282	0.368141	+	0.929770i	$0.379994\pi$
$348$	0	0
$349$	−1.26645e6	−0.556578	−0.278289	−	0.960497i	$-0.589767\pi$
$349$	−1.26645e6	−0.556578	−0.278289	+	0.960497i	$0.589767\pi$
$350$	0	0
$351$	0	0
$352$	−1.00130e6	−0.430732
$353$	573218.	0.244840	0.122420	−	0.992478i	$-0.460934\pi$
$353$	573218.	0.244840	0.122420	+	0.992478i	$0.460934\pi$
$354$	0	0
$355$	495312.	0.208597
$356$	3.63587e6	1.52049
$357$	0	0
$358$	−113564.	−0.0468310
$359$	−4.46322e6	−1.82773	−0.913866	−	0.406016i	$-0.866918\pi$
$359$	−4.46322e6	−1.82773	−0.913866	+	0.406016i	$0.866918\pi$
$360$	0	0
$361$	−1.68044e6	−0.678662
$362$	663118.	0.265962
$363$	0	0
$364$	0	0
$365$	2.66485e6	1.04699
$366$	0	0
$367$	4.50797e6	1.74709	0.873546	−	0.486742i	$-0.161815\pi$
$367$	4.50797e6	1.74709	0.873546	+	0.486742i	$0.161815\pi$
$368$	2.96537e6	1.14146
$369$	0	0
$370$	333132.	0.126506
$371$	0	0
$372$	0	0
$373$	1.66535e6	0.619774	0.309887	−	0.950773i	$-0.399709\pi$
$373$	1.66535e6	0.619774	0.309887	+	0.950773i	$0.399709\pi$
$374$	271320.	0.100300
$375$	0	0
$376$	562464.	0.205175
$377$	−3.74187e6	−1.35592
$378$	0	0
$379$	−2.53232e6	−0.905568	−0.452784	−	0.891620i	$-0.649569\pi$
$379$	−2.53232e6	−0.905568	−0.452784	+	0.891620i	$0.649569\pi$
$380$	−940168.	−0.334000
$381$	0	0
$382$	505664.	0.177298
$383$	796368.	0.277407	0.138703	−	0.990334i	$-0.455707\pi$
$383$	796368.	0.277407	0.138703	+	0.990334i	$0.455707\pi$
$384$	0	0
$385$	0	0
$386$	432382.	0.147706
$387$	0	0
$388$	310062.	0.104561
$389$	−1.94799e6	−0.652699	−0.326349	−	0.945249i	$-0.605819\pi$
$389$	−1.94799e6	−0.652699	−0.326349	+	0.945249i	$0.605819\pi$
$390$	0	0
$391$	−2.54722e6	−0.842605
$392$	0	0
$393$	0	0
$394$	−131962.	−0.0428261
$395$	77248.0	0.0249112
$396$	0	0
$397$	−1.08116e6	−0.344281	−0.172140	−	0.985072i	$-0.555068\pi$
$397$	−1.08116e6	−0.344281	−0.172140	+	0.985072i	$0.555068\pi$
$398$	298536.	0.0944689
$399$	0	0
$400$	−1.82920e6	−0.571625
$401$	−2.76770e6	−0.859524	−0.429762	−	0.902942i	$-0.641402\pi$
$401$	−2.76770e6	−0.859524	−0.429762	+	0.902942i	$0.641402\pi$
$402$	0	0
$403$	−1.13318e6	−0.347566
$404$	3.37187e6	1.02782
$405$	0	0
$406$	0	0
$407$	3.33132e6	0.996851
$408$	0	0
$409$	−2.36350e6	−0.698630	−0.349315	−	0.937005i	$-0.613586\pi$
$409$	−2.36350e6	−0.698630	−0.349315	+	0.937005i	$0.613586\pi$
$410$	674356.	0.198121
$411$	0	0
$412$	−6.17495e6	−1.79222
$413$	0	0
$414$	0	0
$415$	1.28398e6	0.365963
$416$	1.33703e6	0.378798
$417$	0	0
$418$	303280.	0.0848991
$419$	−2.98669e6	−0.831104	−0.415552	−	0.909569i	$-0.636412\pi$
$419$	−2.98669e6	−0.831104	−0.415552	+	0.909569i	$0.636412\pi$
$420$	0	0
$421$	−3.46331e6	−0.952326	−0.476163	−	0.879357i	$-0.657973\pi$
$421$	−3.46331e6	−0.952326	−0.476163	+	0.879357i	$0.657973\pi$
$422$	1.17062e6	0.319989
$423$	0	0
$424$	−9450.00	−0.00255280
$425$	1.57126e6	0.421965
$426$	0	0
$427$	0	0
$428$	−2.47913e6	−0.654169
$429$	0	0
$430$	−586024.	−0.152843
$431$	−2.33693e6	−0.605971	−0.302986	−	0.952995i	$-0.597983\pi$
$431$	−2.33693e6	−0.605971	−0.302986	+	0.952995i	$0.597983\pi$
$432$	0	0
$433$	3.50838e6	0.899264	0.449632	−	0.893214i	$-0.351555\pi$
$433$	3.50838e6	0.899264	0.449632	+	0.893214i	$0.351555\pi$
$434$	0	0
$435$	0	0
$436$	1.42904e6	0.360021
$437$	−2.84726e6	−0.713221
$438$	0	0
$439$	−3.54833e6	−0.878744	−0.439372	−	0.898305i	$-0.644799\pi$
$439$	−3.54833e6	−0.878744	−0.439372	+	0.898305i	$0.644799\pi$
$440$	−728280.	−0.179336
$441$	0	0
$442$	−362292.	−0.0882070
$443$	−1.76833e6	−0.428109	−0.214055	−	0.976822i	$-0.568667\pi$
$443$	−1.76833e6	−0.428109	−0.214055	+	0.976822i	$0.568667\pi$
$444$	0	0
$445$	3.98772e6	0.954608
$446$	399376.	0.0950703
$447$	0	0
$448$	0	0
$449$	5.52579e6	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$449$	5.52579e6	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$450$	0	0
$451$	6.74356e6	1.56116
$452$	8.14389e6	1.87493
$453$	0	0
$454$	−707916.	−0.161191
$455$	0	0
$456$	0	0
$457$	−2.96226e6	−0.663488	−0.331744	−	0.943369i	$-0.607637\pi$
$457$	−2.96226e6	−0.663488	−0.331744	+	0.943369i	$0.607637\pi$
$458$	−735778.	−0.163902
$459$	0	0
$460$	3.36437e6	0.741325
$461$	2.11884e6	0.464350	0.232175	−	0.972674i	$-0.425416\pi$
$461$	2.11884e6	0.464350	0.232175	+	0.972674i	$0.425416\pi$
$462$	0	0
$463$	3.19226e6	0.692062	0.346031	−	0.938223i	$-0.387529\pi$
$463$	3.19226e6	0.692062	0.346031	+	0.938223i	$0.387529\pi$
$464$	7.65682e6	1.65102
$465$	0	0
$466$	−208758.	−0.0445326
$467$	−7.42621e6	−1.57571	−0.787853	−	0.615863i	$-0.788807\pi$
$467$	−7.42621e6	−1.57571	−0.787853	+	0.615863i	$0.788807\pi$
$468$	0	0
$469$	0	0
$470$	303552.	0.0633853
$471$	0	0
$472$	−2.67095e6	−0.551837
$473$	−5.86024e6	−1.20438
$474$	0	0
$475$	1.75635e6	0.357171
$476$	0	0
$477$	0	0
$478$	713376.	0.142807
$479$	−3.39685e6	−0.676453	−0.338226	−	0.941065i	$-0.609827\pi$
$479$	−3.39685e6	−0.676453	−0.338226	+	0.941065i	$0.609827\pi$
$480$	0	0
$481$	−4.44829e6	−0.876659
$482$	−505246.	−0.0990570
$483$	0	0
$484$	1.40898e6	0.273396
$485$	340068.	0.0656465
$486$	0	0
$487$	−3.71382e6	−0.709574	−0.354787	−	0.934947i	$-0.615447\pi$
$487$	−3.71382e6	−0.709574	−0.354787	+	0.934947i	$0.615447\pi$
$488$	−929754.	−0.176733
$489$	0	0
$490$	0	0
$491$	−5.57494e6	−1.04361	−0.521803	−	0.853066i	$-0.674740\pi$
$491$	−5.57494e6	−1.04361	−0.521803	+	0.853066i	$0.674740\pi$
$492$	0	0
$493$	−6.57712e6	−1.21876
$494$	−404968.	−0.0746626
$495$	0	0
$496$	2.31878e6	0.423210
$497$	0	0
$498$	0	0
$499$	3.92698e6	0.706004	0.353002	−	0.935623i	$-0.385161\pi$
$499$	3.92698e6	0.706004	0.353002	+	0.935623i	$0.385161\pi$
$500$	−5.36908e6	−0.960450
$501$	0	0
$502$	−317108.	−0.0561627
$503$	6.42079e6	1.13154	0.565768	−	0.824564i	$-0.308580\pi$
$503$	6.42079e6	1.13154	0.565768	+	0.824564i	$0.308580\pi$
$504$	0	0
$505$	3.69818e6	0.645297
$506$	−1.08528e6	−0.188437
$507$	0	0
$508$	−6.09485e6	−1.04786
$509$	146278.	0.0250256	0.0125128	−	0.999922i	$-0.496017\pi$
$509$	146278.	0.0250256	0.0125128	+	0.999922i	$0.496017\pi$
$510$	0	0
$511$	0	0
$512$	−4.60877e6	−0.776980
$513$	0	0
$514$	1.44285e6	0.240886
$515$	−6.77253e6	−1.12521
$516$	0	0
$517$	3.03552e6	0.499467
$518$	0	0
$519$	0	0
$520$	972468.	0.157713
$521$	7.70937e6	1.24430	0.622149	−	0.782899i	$-0.286260\pi$
$521$	7.70937e6	1.24430	0.622149	+	0.782899i	$0.286260\pi$
$522$	0	0
$523$	569420.	0.0910287	0.0455144	−	0.998964i	$-0.485507\pi$
$523$	569420.	0.0910287	0.0455144	+	0.998964i	$0.485507\pi$
$524$	2.39134e6	0.380464
$525$	0	0
$526$	271496.	0.0427857
$527$	−1.99181e6	−0.312407
$528$	0	0
$529$	3.75252e6	0.583021
$530$	−5100.00	−0.000788643 0
$531$	0	0
$532$	0	0
$533$	−9.00464e6	−1.37293
$534$	0	0
$535$	−2.71905e6	−0.410707
$536$	−105588.	−0.0158746
$537$	0	0
$538$	−850614.	−0.126700
$539$	0	0
$540$	0	0
$541$	−9.44802e6	−1.38787	−0.693933	−	0.720040i	$-0.744124\pi$
$541$	−9.44802e6	−1.38787	−0.693933	+	0.720040i	$0.744124\pi$
$542$	−540128.	−0.0789766
$543$	0	0
$544$	2.35011e6	0.340479
$545$	1.56733e6	0.226032
$546$	0	0
$547$	−1.35321e6	−0.193374	−0.0966869	−	0.995315i	$-0.530825\pi$
$547$	−1.35321e6	−0.193374	−0.0966869	+	0.995315i	$0.530825\pi$
$548$	6.45327e6	0.917970
$549$	0	0
$550$	669460.	0.0943665
$551$	−7.35186e6	−1.03162
$552$	0	0
$553$	0	0
$554$	−513574.	−0.0710933
$555$	0	0
$556$	−8.54298e6	−1.17199
$557$	−8.19390e6	−1.11906	−0.559529	−	0.828811i	$-0.689018\pi$
$557$	−8.19390e6	−1.11906	−0.559529	+	0.828811i	$0.689018\pi$
$558$	0	0
$559$	7.82514e6	1.05916
$560$	0	0
$561$	0	0
$562$	−1.35642e6	−0.181157
$563$	−1.05796e7	−1.40669	−0.703347	−	0.710847i	$-0.748312\pi$
$563$	−1.05796e7	−1.40669	−0.703347	+	0.710847i	$0.748312\pi$
$564$	0	0
$565$	8.93200e6	1.17714
$566$	286756.	0.0376246
$567$	0	0
$568$	−917784.	−0.119363
$569$	1.20205e7	1.55648	0.778238	−	0.627969i	$-0.216114\pi$
$569$	1.20205e7	1.55648	0.778238	+	0.627969i	$0.216114\pi$
$570$	0	0
$571$	−2.48948e6	−0.319534	−0.159767	−	0.987155i	$-0.551074\pi$
$571$	−2.48948e6	−0.319534	−0.159767	+	0.987155i	$0.551074\pi$
$572$	4.78516e6	0.611514
$573$	0	0
$574$	0	0
$575$	−6.28505e6	−0.792755
$576$	0	0
$577$	−8.21322e6	−1.02701	−0.513504	−	0.858087i	$-0.671653\pi$
$577$	−8.21322e6	−1.02701	−0.513504	+	0.858087i	$0.671653\pi$
$578$	783053.	0.0974926
$579$	0	0
$580$	8.68707e6	1.07227
$581$	0	0
$582$	0	0
$583$	−51000.0	−0.00621439
$584$	−4.93781e6	−0.599105
$585$	0	0
$586$	1.70727e6	0.205380
$587$	−1.21827e6	−0.145931	−0.0729655	−	0.997334i	$-0.523246\pi$
$587$	−1.21827e6	−0.145931	−0.0729655	+	0.997334i	$0.523246\pi$
$588$	0	0
$589$	−2.22643e6	−0.264436
$590$	−1.44146e6	−0.170480
$591$	0	0
$592$	9.10234e6	1.06745
$593$	−8.42379e6	−0.983718	−0.491859	−	0.870675i	$-0.663683\pi$
$593$	−8.42379e6	−0.983718	−0.491859	+	0.870675i	$0.663683\pi$
$594$	0	0
$595$	0	0
$596$	−9.17929e6	−1.05851
$597$	0	0
$598$	1.44917e6	0.165717
$599$	−8.21254e6	−0.935212	−0.467606	−	0.883937i	$-0.654883\pi$
$599$	−8.21254e6	−0.935212	−0.467606	+	0.883937i	$0.654883\pi$
$600$	0	0
$601$	−3.25478e6	−0.367566	−0.183783	−	0.982967i	$-0.558834\pi$
$601$	−3.25478e6	−0.367566	−0.183783	+	0.982967i	$0.558834\pi$
$602$	0	0
$603$	0	0
$604$	1.32206e7	1.47455
$605$	1.54533e6	0.171646
$606$	0	0
$607$	−7.82101e6	−0.861571	−0.430785	−	0.902454i	$-0.641763\pi$
$607$	−7.82101e6	−0.861571	−0.430785	+	0.902454i	$0.641763\pi$
$608$	2.62694e6	0.288198
$609$	0	0
$610$	−501772.	−0.0545986
$611$	−4.05331e6	−0.439245
$612$	0	0
$613$	−9.51670e6	−1.02290	−0.511452	−	0.859312i	$-0.670892\pi$
$613$	−9.51670e6	−1.02290	−0.511452	+	0.859312i	$0.670892\pi$
$614$	−546788.	−0.0585326
$615$	0	0
$616$	0	0
$617$	7.04895e6	0.745438	0.372719	−	0.927944i	$-0.378426\pi$
$617$	7.04895e6	0.745438	0.372719	+	0.927944i	$0.378426\pi$
$618$	0	0
$619$	6.32174e6	0.663147	0.331574	−	0.943429i	$-0.392420\pi$
$619$	6.32174e6	0.663147	0.331574	+	0.943429i	$0.392420\pi$
$620$	2.63078e6	0.274856
$621$	0	0
$622$	−3.23426e6	−0.335197
$623$	0	0
$624$	0	0
$625$	264461.	0.0270808
$626$	1.81313e6	0.184924
$627$	0	0
$628$	5.53307e6	0.559844
$629$	−7.81880e6	−0.787977
$630$	0	0
$631$	8.61236e6	0.861090	0.430545	−	0.902569i	$-0.358321\pi$
$631$	8.61236e6	0.861090	0.430545	+	0.902569i	$0.358321\pi$
$632$	−143136.	−0.0142546
$633$	0	0
$634$	−1.27658e6	−0.126132
$635$	−6.68467e6	−0.657879
$636$	0	0
$637$	0	0
$638$	−2.80228e6	−0.272559
$639$	0	0
$640$	−4.11478e6	−0.397097
$641$	5.22829e6	0.502590	0.251295	−	0.967910i	$-0.419143\pi$
$641$	5.22829e6	0.502590	0.251295	+	0.967910i	$0.419143\pi$
$642$	0	0
$643$	−1.61373e7	−1.53923	−0.769615	−	0.638508i	$-0.779552\pi$
$643$	−1.61373e7	−1.53923	−0.769615	+	0.638508i	$0.779552\pi$
$644$	0	0
$645$	0	0
$646$	−711816.	−0.0671099
$647$	−1.58749e7	−1.49090	−0.745451	−	0.666560i	$-0.767766\pi$
$647$	−1.58749e7	−1.49090	−0.745451	+	0.666560i	$0.767766\pi$
$648$	0	0
$649$	−1.44146e7	−1.34336
$650$	−893926.	−0.0829886
$651$	0	0
$652$	−7.83593e6	−0.721891
$653$	5.94112e6	0.545237	0.272619	−	0.962122i	$-0.412110\pi$
$653$	5.94112e6	0.545237	0.272619	+	0.962122i	$0.412110\pi$
$654$	0	0
$655$	2.62276e6	0.238867
$656$	1.84258e7	1.67173
$657$	0	0
$658$	0	0
$659$	7.64430e6	0.685684	0.342842	−	0.939393i	$-0.388610\pi$
$659$	7.64430e6	0.685684	0.342842	+	0.939393i	$0.388610\pi$
$660$	0	0
$661$	7.58688e6	0.675398	0.337699	−	0.941254i	$-0.390351\pi$
$661$	7.58688e6	0.675398	0.337699	+	0.941254i	$0.390351\pi$
$662$	1.73621e6	0.153978
$663$	0	0
$664$	−2.37913e6	−0.209410
$665$	0	0
$666$	0	0
$667$	2.63085e7	2.28971
$668$	−1.57507e7	−1.36571
$669$	0	0
$670$	−56984.0	−0.00490417
$671$	−5.01772e6	−0.430229
$672$	0	0
$673$	−2.06681e7	−1.75899	−0.879494	−	0.475910i	$-0.842119\pi$
$673$	−2.06681e7	−1.75899	−0.879494	+	0.475910i	$0.842119\pi$
$674$	−2.07215e6	−0.175700
$675$	0	0
$676$	5.12049e6	0.430968
$677$	7.89541e6	0.662068	0.331034	−	0.943619i	$-0.392602\pi$
$677$	7.89541e6	0.662068	0.331034	+	0.943619i	$0.392602\pi$
$678$	0	0
$679$	0	0
$680$	1.70932e6	0.141759
$681$	0	0
$682$	−848640.	−0.0698655
$683$	1.96015e7	1.60782	0.803911	−	0.594750i	$-0.202749\pi$
$683$	1.96015e7	1.60782	0.803911	+	0.594750i	$0.202749\pi$
$684$	0	0
$685$	7.07778e6	0.576329
$686$	0	0
$687$	0	0
$688$	−1.60122e7	−1.28968
$689$	68100.0	0.00546511
$690$	0	0
$691$	1.72710e7	1.37601	0.688005	−	0.725706i	$-0.258487\pi$
$691$	1.72710e7	1.37601	0.688005	+	0.725706i	$0.258487\pi$
$692$	6.87685e6	0.545914
$693$	0	0
$694$	−1.65146e6	−0.130158
$695$	−9.36972e6	−0.735808
$696$	0	0
$697$	−1.58275e7	−1.23405
$698$	1.26645e6	0.0983900
$699$	0	0
$700$	0	0
$701$	5.36344e6	0.412238	0.206119	−	0.978527i	$-0.433917\pi$
$701$	5.36344e6	0.412238	0.206119	+	0.978527i	$0.433917\pi$
$702$	0	0
$703$	−8.73982e6	−0.666982
$704$	−9.10622e6	−0.692479
$705$	0	0
$706$	−573218.	−0.0432821
$707$	0	0
$708$	0	0
$709$	−1.73733e7	−1.29798	−0.648988	−	0.760798i	$-0.724808\pi$
$709$	−1.73733e7	−1.29798	−0.648988	+	0.760798i	$0.724808\pi$
$710$	−495312.	−0.0368751
$711$	0	0
$712$	−7.38902e6	−0.546244
$713$	7.96723e6	0.586926
$714$	0	0
$715$	5.24824e6	0.383927
$716$	−3.52048e6	−0.256637
$717$	0	0
$718$	4.46322e6	0.323100
$719$	424608.	0.0306313	0.0153157	−	0.999883i	$-0.495125\pi$
$719$	424608.	0.0306313	0.0153157	+	0.999883i	$0.495125\pi$
$720$	0	0
$721$	0	0
$722$	1.68044e6	0.119972
$723$	0	0
$724$	2.05567e7	1.45749
$725$	−1.62285e7	−1.14666
$726$	0	0
$727$	−2.18290e7	−1.53179	−0.765893	−	0.642968i	$-0.777703\pi$
$727$	−2.18290e7	−1.53179	−0.765893	+	0.642968i	$0.777703\pi$
$728$	0	0
$729$	0	0
$730$	−2.66485e6	−0.185083
$731$	1.37543e7	0.952020
$732$	0	0
$733$	−2.17675e7	−1.49640	−0.748202	−	0.663470i	$-0.769083\pi$
$733$	−2.17675e7	−1.49640	−0.748202	+	0.663470i	$0.769083\pi$
$734$	−4.50797e6	−0.308845
$735$	0	0
$736$	−9.40044e6	−0.639667
$737$	−569840.	−0.0386442
$738$	0	0
$739$	6.21786e6	0.418822	0.209411	−	0.977828i	$-0.432845\pi$
$739$	6.21786e6	0.418822	0.209411	+	0.977828i	$0.432845\pi$
$740$	1.03271e7	0.693264
$741$	0	0
$742$	0	0
$743$	−3.77647e6	−0.250966	−0.125483	−	0.992096i	$-0.540048\pi$
$743$	−3.77647e6	−0.250966	−0.125483	+	0.992096i	$0.540048\pi$
$744$	0	0
$745$	−1.00676e7	−0.664562
$746$	−1.66535e6	−0.109562
$747$	0	0
$748$	8.41092e6	0.549654
$749$	0	0
$750$	0	0
$751$	−2.88795e6	−0.186849	−0.0934244	−	0.995626i	$-0.529781\pi$
$751$	−2.88795e6	−0.186849	−0.0934244	+	0.995626i	$0.529781\pi$
$752$	8.29411e6	0.534842
$753$	0	0
$754$	3.74187e6	0.239696
$755$	1.45000e7	0.925768
$756$	0	0
$757$	1.25519e6	0.0796104	0.0398052	−	0.999207i	$-0.487326\pi$
$757$	1.25519e6	0.0796104	0.0398052	+	0.999207i	$0.487326\pi$
$758$	2.53232e6	0.160083
$759$	0	0
$760$	1.91066e6	0.119991
$761$	−1.42623e7	−0.892746	−0.446373	−	0.894847i	$-0.647284\pi$
$761$	−1.42623e7	−0.892746	−0.446373	+	0.894847i	$0.647284\pi$
$762$	0	0
$763$	0	0
$764$	1.56756e7	0.971606
$765$	0	0
$766$	−796368.	−0.0490390
$767$	1.92478e7	1.18139
$768$	0	0
$769$	2.02261e7	1.23338	0.616689	−	0.787207i	$-0.288474\pi$
$769$	2.02261e7	1.23338	0.616689	+	0.787207i	$0.288474\pi$
$770$	0	0
$771$	0	0
$772$	1.34038e7	0.809443
$773$	2.62288e7	1.57881	0.789406	−	0.613872i	$-0.210389\pi$
$773$	2.62288e7	1.57881	0.789406	+	0.613872i	$0.210389\pi$
$774$	0	0
$775$	−4.91462e6	−0.293925
$776$	−630126.	−0.0375641
$777$	0	0
$778$	1.94799e6	0.115382
$779$	−1.76919e7	−1.04456
$780$	0	0
$781$	−4.95312e6	−0.290570
$782$	2.54722e6	0.148953
$783$	0	0
$784$	0	0
$785$	6.06852e6	0.351487
$786$	0	0
$787$	9.92829e6	0.571397	0.285698	−	0.958320i	$-0.407774\pi$
$787$	9.92829e6	0.571397	0.285698	+	0.958320i	$0.407774\pi$
$788$	−4.09082e6	−0.234690
$789$	0	0
$790$	−77248.0	−0.00440372
$791$	0	0
$792$	0	0
$793$	6.70013e6	0.378356
$794$	1.08116e6	0.0608608
$795$	0	0
$796$	9.25462e6	0.517697
$797$	1.09033e7	0.608014	0.304007	−	0.952670i	$-0.401675\pi$
$797$	1.09033e7	0.608014	0.304007	+	0.952670i	$0.401675\pi$
$798$	0	0
$799$	−7.12454e6	−0.394812
$800$	5.79871e6	0.320336
$801$	0	0
$802$	2.76770e6	0.151944
$803$	−2.66485e7	−1.45843
$804$	0	0
$805$	0	0
$806$	1.13318e6	0.0614416
$807$	0	0
$808$	−6.85251e6	−0.369251
$809$	−6.06398e6	−0.325751	−0.162876	−	0.986647i	$-0.552077\pi$
$809$	−6.06398e6	−0.325751	−0.162876	+	0.986647i	$0.552077\pi$
$810$	0	0
$811$	8.59438e6	0.458841	0.229421	−	0.973327i	$-0.426317\pi$
$811$	8.59438e6	0.458841	0.229421	+	0.973327i	$0.426317\pi$
$812$	0	0
$813$	0	0
$814$	−3.33132e6	−0.176220
$815$	−8.59425e6	−0.453225
$816$	0	0
$817$	1.53745e7	0.805835
$818$	2.36350e6	0.123501
$819$	0	0
$820$	2.09050e7	1.08572
$821$	2.01396e6	0.104278	0.0521391	−	0.998640i	$-0.483396\pi$
$821$	2.01396e6	0.104278	0.0521391	+	0.998640i	$0.483396\pi$
$822$	0	0
$823$	−2.64679e7	−1.36213	−0.681067	−	0.732221i	$-0.738484\pi$
$823$	−2.64679e7	−1.36213	−0.681067	+	0.732221i	$0.738484\pi$
$824$	1.25491e7	0.643864
$825$	0	0
$826$	0	0
$827$	3.90229e6	0.198407	0.0992033	−	0.995067i	$-0.468371\pi$
$827$	3.90229e6	0.198407	0.0992033	+	0.995067i	$0.468371\pi$
$828$	0	0
$829$	1.95595e7	0.988487	0.494244	−	0.869323i	$-0.335445\pi$
$829$	1.95595e7	0.988487	0.494244	+	0.869323i	$0.335445\pi$
$830$	−1.28398e6	−0.0646937
$831$	0	0
$832$	1.21595e7	0.608985
$833$	0	0
$834$	0	0
$835$	−1.72750e7	−0.857436
$836$	9.40168e6	0.465254
$837$	0	0
$838$	2.98669e6	0.146920
$839$	−2.45448e7	−1.20380	−0.601901	−	0.798570i	$-0.705590\pi$
$839$	−2.45448e7	−1.20380	−0.601901	+	0.798570i	$0.705590\pi$
$840$	0	0
$841$	4.74194e7	2.31188
$842$	3.46331e6	0.168349
$843$	0	0
$844$	3.62892e7	1.75356
$845$	5.61602e6	0.270574
$846$	0	0
$847$	0	0
$848$	−139350.	−0.00665453
$849$	0	0
$850$	−1.57126e6	−0.0745936
$851$	3.12752e7	1.48039
$852$	0	0
$853$	−3.38305e7	−1.59197	−0.795987	−	0.605314i	$-0.793048\pi$
$853$	−3.38305e7	−1.59197	−0.795987	+	0.605314i	$0.793048\pi$
$854$	0	0
$855$	0	0
$856$	5.03824e6	0.235014
$857$	3.18009e7	1.47907	0.739534	−	0.673120i	$-0.235046\pi$
$857$	3.18009e7	1.47907	0.739534	+	0.673120i	$0.235046\pi$
$858$	0	0
$859$	−638420.	−0.0295205	−0.0147602	−	0.999891i	$-0.504699\pi$
$859$	−638420.	−0.0295205	−0.0147602	+	0.999891i	$0.504699\pi$
$860$	−1.81667e7	−0.837589
$861$	0	0
$862$	2.33693e6	0.107122
$863$	4.22256e6	0.192996	0.0964981	−	0.995333i	$-0.469236\pi$
$863$	4.22256e6	0.192996	0.0964981	+	0.995333i	$0.469236\pi$
$864$	0	0
$865$	7.54236e6	0.342742
$866$	−3.50838e6	−0.158969
$867$	0	0
$868$	0	0
$869$	−772480.	−0.0347007
$870$	0	0
$871$	760904.	0.0339848
$872$	−2.90417e6	−0.129340
$873$	0	0
$874$	2.84726e6	0.126081
$875$	0	0
$876$	0	0
$877$	−2.45043e7	−1.07583	−0.537915	−	0.842999i	$-0.680788\pi$
$877$	−2.45043e7	−1.07583	−0.537915	+	0.842999i	$0.680788\pi$
$878$	3.54833e6	0.155341
$879$	0	0
$880$	−1.07392e7	−0.467484
$881$	−2.77630e7	−1.20511	−0.602555	−	0.798078i	$-0.705850\pi$
$881$	−2.77630e7	−1.20511	−0.602555	+	0.798078i	$0.705850\pi$
$882$	0	0
$883$	3.30170e7	1.42507	0.712534	−	0.701638i	$-0.247548\pi$
$883$	3.30170e7	1.42507	0.712534	+	0.701638i	$0.247548\pi$
$884$	−1.12311e7	−0.483381
$885$	0	0
$886$	1.76833e6	0.0756797
$887$	4.34462e6	0.185414	0.0927070	−	0.995693i	$-0.470448\pi$
$887$	4.34462e6	0.185414	0.0927070	+	0.995693i	$0.470448\pi$
$888$	0	0
$889$	0	0
$890$	−3.98772e6	−0.168752
$891$	0	0
$892$	1.23807e7	0.520993
$893$	−7.96378e6	−0.334188
$894$	0	0
$895$	−3.86118e6	−0.161125
$896$	0	0
$897$	0	0
$898$	−5.52579e6	−0.228667
$899$	2.05720e7	0.848942
$900$	0	0
$901$	119700.	0.00491227
$902$	−6.74356e6	−0.275977
$903$	0	0
$904$	−1.65505e7	−0.673580
$905$	2.25460e7	0.915057
$906$	0	0
$907$	1.96499e7	0.793128	0.396564	−	0.918007i	$-0.370203\pi$
$907$	1.96499e7	0.793128	0.396564	+	0.918007i	$0.370203\pi$
$908$	−2.19454e7	−0.883342
$909$	0	0
$910$	0	0
$911$	7.26518e6	0.290035	0.145018	−	0.989429i	$-0.453676\pi$
$911$	7.26518e6	0.290035	0.145018	+	0.989429i	$0.453676\pi$
$912$	0	0
$913$	−1.28398e7	−0.509777
$914$	2.96226e6	0.117289
$915$	0	0
$916$	−2.28091e7	−0.898193
$917$	0	0
$918$	0	0
$919$	9.82532e6	0.383758	0.191879	−	0.981419i	$-0.438542\pi$
$919$	9.82532e6	0.383758	0.191879	+	0.981419i	$0.438542\pi$
$920$	−6.83726e6	−0.266326
$921$	0	0
$922$	−2.11884e6	−0.0820863
$923$	6.61387e6	0.255536
$924$	0	0
$925$	−1.92923e7	−0.741359
$926$	−3.19226e6	−0.122340
$927$	0	0
$928$	−2.42727e7	−0.925226
$929$	2.71152e7	1.03080	0.515399	−	0.856951i	$-0.327644\pi$
$929$	2.71152e7	1.03080	0.515399	+	0.856951i	$0.327644\pi$
$930$	0	0
$931$	0	0
$932$	−6.47150e6	−0.244042
$933$	0	0
$934$	7.42621e6	0.278548
$935$	9.22488e6	0.345089
$936$	0	0
$937$	4.53522e7	1.68752	0.843761	−	0.536720i	$-0.180337\pi$
$937$	4.53522e7	1.68752	0.843761	+	0.536720i	$0.180337\pi$
$938$	0	0
$939$	0	0
$940$	9.41011e6	0.347356
$941$	4.65780e7	1.71477	0.857387	−	0.514672i	$-0.172086\pi$
$941$	4.65780e7	1.71477	0.857387	+	0.514672i	$0.172086\pi$
$942$	0	0
$943$	6.33101e7	2.31843
$944$	−3.93859e7	−1.43850
$945$	0	0
$946$	5.86024e6	0.212906
$947$	−2.53799e7	−0.919632	−0.459816	−	0.888014i	$-0.652085\pi$
$947$	−2.53799e7	−0.919632	−0.459816	+	0.888014i	$0.652085\pi$
$948$	0	0
$949$	3.55836e7	1.28258
$950$	−1.75635e6	−0.0631396
$951$	0	0
$952$	0	0
$953$	−1.52948e7	−0.545520	−0.272760	−	0.962082i	$-0.587937\pi$
$953$	−1.52948e7	−0.545520	−0.272760	+	0.962082i	$0.587937\pi$
$954$	0	0
$955$	1.71926e7	0.610004
$956$	2.21147e7	0.782592
$957$	0	0
$958$	3.39685e6	0.119581
$959$	0	0
$960$	0	0
$961$	−2.23991e7	−0.782389
$962$	4.44829e6	0.154973
$963$	0	0
$964$	−1.56626e7	−0.542840
$965$	1.47010e7	0.508192
$966$	0	0
$967$	−5.71465e6	−0.196527	−0.0982637	−	0.995160i	$-0.531329\pi$
$967$	−5.71465e6	−0.196527	−0.0982637	+	0.995160i	$0.531329\pi$
$968$	−2.86341e6	−0.0982190
$969$	0	0
$970$	−340068.	−0.0116048
$971$	1.30250e7	0.443332	0.221666	−	0.975123i	$-0.428851\pi$
$971$	1.30250e7	0.443332	0.221666	+	0.975123i	$0.428851\pi$
$972$	0	0
$973$	0	0
$974$	3.71382e6	0.125436
$975$	0	0
$976$	−1.37102e7	−0.460700
$977$	1.70360e7	0.570992	0.285496	−	0.958380i	$-0.407842\pi$
$977$	1.70360e7	0.570992	0.285496	+	0.958380i	$0.407842\pi$
$978$	0	0
$979$	−3.98772e7	−1.32974
$980$	0	0
$981$	0	0
$982$	5.57494e6	0.184485
$983$	−1.36985e7	−0.452156	−0.226078	−	0.974109i	$-0.572590\pi$
$983$	−1.36985e7	−0.452156	−0.226078	+	0.974109i	$0.572590\pi$
$984$	0	0
$985$	−4.48671e6	−0.147346
$986$	6.57712e6	0.215448
$987$	0	0
$988$	−1.25540e7	−0.409157
$989$	−5.50173e7	−1.78858
$990$	0	0
$991$	−3.49088e7	−1.12915	−0.564574	−	0.825383i	$-0.690959\pi$
$991$	−3.49088e7	−1.12915	−0.564574	+	0.825383i	$0.690959\pi$
$992$	−7.35072e6	−0.237165
$993$	0	0
$994$	0	0
$995$	1.01502e7	0.325026
$996$	0	0
$997$	−875662.	−0.0278996	−0.0139498	−	0.999903i	$-0.504441\pi$
$997$	−875662.	−0.0278996	−0.0139498	+	0.999903i	$0.504441\pi$
$998$	−3.92698e6	−0.124805
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.d.1.1		1
3.2	odd	2		147.6.a.e.1.1		1
7.6	odd	2		63.6.a.c.1.1		1
21.2	odd	6		147.6.e.e.67.1		2
21.5	even	6		147.6.e.f.67.1		2
21.11	odd	6		147.6.e.e.79.1		2
21.17	even	6		147.6.e.f.79.1		2
21.20	even	2		21.6.a.b.1.1	✓	1
28.27	even	2		1008.6.a.t.1.1		1
84.83	odd	2		336.6.a.l.1.1		1
105.62	odd	4		525.6.d.d.274.2		2
105.83	odd	4		525.6.d.d.274.1		2
105.104	even	2		525.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.b.1.1	✓	1	21.20	even	2
63.6.a.c.1.1		1	7.6	odd	2
147.6.a.e.1.1		1	3.2	odd	2
147.6.e.e.67.1		2	21.2	odd	6
147.6.e.e.79.1		2	21.11	odd	6
147.6.e.f.67.1		2	21.5	even	6
147.6.e.f.79.1		2	21.17	even	6
336.6.a.l.1.1		1	84.83	odd	2
441.6.a.d.1.1		1	1.1	even	1	trivial
525.6.a.c.1.1		1	105.104	even	2
525.6.d.d.274.1		2	105.83	odd	4
525.6.d.d.274.2		2	105.62	odd	4
1008.6.a.t.1.1		1	28.27	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more