LMFDB - Embedded newform 441.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,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, 4, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$26.0198423125$
Analytic rank:	$0$
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)

$f(q)$

$=$

$q+3.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -21.0000 q^{8} +O(q^{10})$

$q+3.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -21.0000 q^{8} -9.00000 q^{10} +15.0000 q^{11} +64.0000 q^{13} -71.0000 q^{16} +84.0000 q^{17} +16.0000 q^{19} -3.00000 q^{20} +45.0000 q^{22} +84.0000 q^{23} -116.000 q^{25} +192.000 q^{26} +297.000 q^{29} +253.000 q^{31} -45.0000 q^{32} +252.000 q^{34} -316.000 q^{37} +48.0000 q^{38} +63.0000 q^{40} +360.000 q^{41} +26.0000 q^{43} +15.0000 q^{44} +252.000 q^{46} -30.0000 q^{47} -348.000 q^{50} +64.0000 q^{52} -363.000 q^{53} -45.0000 q^{55} +891.000 q^{58} -15.0000 q^{59} +118.000 q^{61} +759.000 q^{62} +433.000 q^{64} -192.000 q^{65} -370.000 q^{67} +84.0000 q^{68} +342.000 q^{71} -362.000 q^{73} -948.000 q^{74} +16.0000 q^{76} +467.000 q^{79} +213.000 q^{80} +1080.00 q^{82} +477.000 q^{83} -252.000 q^{85} +78.0000 q^{86} -315.000 q^{88} +906.000 q^{89} +84.0000 q^{92} -90.0000 q^{94} -48.0000 q^{95} -503.000 q^{97} +O(q^{100})$

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 $

$2$	3.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	3.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	−3.00000	−0.268328	−0.134164	−	0.990959i	$-0.542835\pi$
$5$	−3.00000	−0.268328	−0.134164	+	0.990959i	$0.542835\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−21.0000	−0.928078
$9$	0	0
$10$	−9.00000	−0.284605
$11$	15.0000	0.411152	0.205576	−	0.978641i	$-0.434093\pi$
$11$	15.0000	0.411152	0.205576	+	0.978641i	$0.434093\pi$
$12$	0	0
$13$	64.0000	1.36542	0.682708	−	0.730691i	$-0.260802\pi$
$13$	64.0000	1.36542	0.682708	+	0.730691i	$0.260802\pi$
$14$	0	0
$15$	0	0
$16$	−71.0000	−1.10938
$17$	84.0000	1.19841	0.599206	−	0.800595i	$-0.295483\pi$
$17$	84.0000	1.19841	0.599206	+	0.800595i	$0.295483\pi$
$18$	0	0
$19$	16.0000	0.193192	0.0965961	−	0.995324i	$-0.469204\pi$
$19$	16.0000	0.193192	0.0965961	+	0.995324i	$0.469204\pi$
$20$	−3.00000	−0.0335410
$21$	0	0
$22$	45.0000	0.436092
$23$	84.0000	0.761531	0.380765	−	0.924672i	$-0.375661\pi$
$23$	84.0000	0.761531	0.380765	+	0.924672i	$0.375661\pi$
$24$	0	0
$25$	−116.000	−0.928000
$26$	192.000	1.44824
$27$	0	0
$28$	0	0
$29$	297.000	1.90178	0.950888	−	0.309535i	$-0.100173\pi$
$29$	297.000	1.90178	0.950888	+	0.309535i	$0.100173\pi$
$30$	0	0
$31$	253.000	1.46581	0.732906	−	0.680330i	$-0.238164\pi$
$31$	253.000	1.46581	0.732906	+	0.680330i	$0.238164\pi$
$32$	−45.0000	−0.248592
$33$	0	0
$34$	252.000	1.27111
$35$	0	0
$36$	0	0
$37$	−316.000	−1.40406	−0.702028	−	0.712149i	$-0.747722\pi$
$37$	−316.000	−1.40406	−0.702028	+	0.712149i	$0.747722\pi$
$38$	48.0000	0.204911
$39$	0	0
$40$	63.0000	0.249029
$41$	360.000	1.37128	0.685641	−	0.727940i	$-0.259522\pi$
$41$	360.000	1.37128	0.685641	+	0.727940i	$0.259522\pi$
$42$	0	0
$43$	26.0000	0.0922084	0.0461042	−	0.998937i	$-0.485319\pi$
$43$	26.0000	0.0922084	0.0461042	+	0.998937i	$0.485319\pi$
$44$	15.0000	0.0513940
$45$	0	0
$46$	252.000	0.807725
$47$	−30.0000	−0.0931053	−0.0465527	−	0.998916i	$-0.514824\pi$
$47$	−30.0000	−0.0931053	−0.0465527	+	0.998916i	$0.514824\pi$
$48$	0	0
$49$	0	0
$50$	−348.000	−0.984293
$51$	0	0
$52$	64.0000	0.170677
$53$	−363.000	−0.940790	−0.470395	−	0.882456i	$-0.655889\pi$
$53$	−363.000	−0.940790	−0.470395	+	0.882456i	$0.655889\pi$
$54$	0	0
$55$	−45.0000	−0.110324
$56$	0	0
$57$	0	0
$58$	891.000	2.01714
$59$	−15.0000	−0.0330989	−0.0165494	−	0.999863i	$-0.505268\pi$
$59$	−15.0000	−0.0330989	−0.0165494	+	0.999863i	$0.505268\pi$
$60$	0	0
$61$	118.000	0.247678	0.123839	−	0.992302i	$-0.460479\pi$
$61$	118.000	0.247678	0.123839	+	0.992302i	$0.460479\pi$
$62$	759.000	1.55473
$63$	0	0
$64$	433.000	0.845703
$65$	−192.000	−0.366380
$66$	0	0
$67$	−370.000	−0.674667	−0.337334	−	0.941385i	$-0.609525\pi$
$67$	−370.000	−0.674667	−0.337334	+	0.941385i	$0.609525\pi$
$68$	84.0000	0.149801
$69$	0	0
$70$	0	0
$71$	342.000	0.571661	0.285831	−	0.958280i	$-0.407731\pi$
$71$	342.000	0.571661	0.285831	+	0.958280i	$0.407731\pi$
$72$	0	0
$73$	−362.000	−0.580396	−0.290198	−	0.956967i	$-0.593721\pi$
$73$	−362.000	−0.580396	−0.290198	+	0.956967i	$0.593721\pi$
$74$	−948.000	−1.48923
$75$	0	0
$76$	16.0000	0.0241490
$77$	0	0
$78$	0	0
$79$	467.000	0.665084	0.332542	−	0.943089i	$-0.392094\pi$
$79$	467.000	0.665084	0.332542	+	0.943089i	$0.392094\pi$
$80$	213.000	0.297677
$81$	0	0
$82$	1080.00	1.45446
$83$	477.000	0.630814	0.315407	−	0.948957i	$-0.397859\pi$
$83$	477.000	0.630814	0.315407	+	0.948957i	$0.397859\pi$
$84$	0	0
$85$	−252.000	−0.321568
$86$	78.0000	0.0978018
$87$	0	0
$88$	−315.000	−0.381581
$89$	906.000	1.07905	0.539527	−	0.841968i	$-0.318603\pi$
$89$	906.000	1.07905	0.539527	+	0.841968i	$0.318603\pi$
$90$	0	0
$91$	0	0
$92$	84.0000	0.0951914
$93$	0	0
$94$	−90.0000	−0.0987531
$95$	−48.0000	−0.0518389
$96$	0	0
$97$	−503.000	−0.526515	−0.263257	−	0.964726i	$-0.584797\pi$
$97$	−503.000	−0.526515	−0.263257	+	0.964726i	$0.584797\pi$
$98$	0	0
$99$	0	0
$100$	−116.000	−0.116000
$101$	−1086.00	−1.06991	−0.534956	−	0.844880i	$-0.679672\pi$
$101$	−1086.00	−1.06991	−0.534956	+	0.844880i	$0.679672\pi$
$102$	0	0
$103$	−1736.00	−1.66071	−0.830355	−	0.557235i	$-0.811862\pi$
$103$	−1736.00	−1.66071	−0.830355	+	0.557235i	$0.811862\pi$
$104$	−1344.00	−1.26721
$105$	0	0
$106$	−1089.00	−0.997859
$107$	1353.00	1.22242	0.611212	−	0.791467i	$-0.290682\pi$
$107$	1353.00	1.22242	0.611212	+	0.791467i	$0.290682\pi$
$108$	0	0
$109$	−370.000	−0.325134	−0.162567	−	0.986698i	$-0.551977\pi$
$109$	−370.000	−0.325134	−0.162567	+	0.986698i	$0.551977\pi$
$110$	−135.000	−0.117016
$111$	0	0
$112$	0	0
$113$	648.000	0.539458	0.269729	−	0.962936i	$-0.413066\pi$
$113$	648.000	0.539458	0.269729	+	0.962936i	$0.413066\pi$
$114$	0	0
$115$	−252.000	−0.204340
$116$	297.000	0.237722
$117$	0	0
$118$	−45.0000	−0.0351067
$119$	0	0
$120$	0	0
$121$	−1106.00	−0.830954
$122$	354.000	0.262702
$123$	0	0
$124$	253.000	0.183226
$125$	723.000	0.517337
$126$	0	0
$127$	377.000	0.263412	0.131706	−	0.991289i	$-0.457954\pi$
$127$	377.000	0.263412	0.131706	+	0.991289i	$0.457954\pi$
$128$	1659.00	1.14560
$129$	0	0
$130$	−576.000	−0.388604
$131$	−651.000	−0.434184	−0.217092	−	0.976151i	$-0.569657\pi$
$131$	−651.000	−0.434184	−0.217092	+	0.976151i	$0.569657\pi$
$132$	0	0
$133$	0	0
$134$	−1110.00	−0.715593
$135$	0	0
$136$	−1764.00	−1.11222
$137$	1770.00	1.10381	0.551903	−	0.833909i	$-0.313902\pi$
$137$	1770.00	1.10381	0.551903	+	0.833909i	$0.313902\pi$
$138$	0	0
$139$	1558.00	0.950704	0.475352	−	0.879796i	$-0.342321\pi$
$139$	1558.00	0.950704	0.475352	+	0.879796i	$0.342321\pi$
$140$	0	0
$141$	0	0
$142$	1026.00	0.606338
$143$	960.000	0.561393
$144$	0	0
$145$	−891.000	−0.510300
$146$	−1086.00	−0.615603
$147$	0	0
$148$	−316.000	−0.175507
$149$	−2454.00	−1.34926	−0.674629	−	0.738157i	$-0.735696\pi$
$149$	−2454.00	−1.34926	−0.674629	+	0.738157i	$0.735696\pi$
$150$	0	0
$151$	1259.00	0.678516	0.339258	−	0.940693i	$-0.389824\pi$
$151$	1259.00	0.678516	0.339258	+	0.940693i	$0.389824\pi$
$152$	−336.000	−0.179297
$153$	0	0
$154$	0	0
$155$	−759.000	−0.393318
$156$	0	0
$157$	196.000	0.0996338	0.0498169	−	0.998758i	$-0.484136\pi$
$157$	196.000	0.0996338	0.0498169	+	0.998758i	$0.484136\pi$
$158$	1401.00	0.705428
$159$	0	0
$160$	135.000	0.0667043
$161$	0	0
$162$	0	0
$163$	−1252.00	−0.601621	−0.300810	−	0.953684i	$-0.597257\pi$
$163$	−1252.00	−0.601621	−0.300810	+	0.953684i	$0.597257\pi$
$164$	360.000	0.171410
$165$	0	0
$166$	1431.00	0.669079
$167$	−2646.00	−1.22607	−0.613035	−	0.790056i	$-0.710051\pi$
$167$	−2646.00	−1.22607	−0.613035	+	0.790056i	$0.710051\pi$
$168$	0	0
$169$	1899.00	0.864360
$170$	−756.000	−0.341074
$171$	0	0
$172$	26.0000	0.0115261
$173$	−786.000	−0.345425	−0.172712	−	0.984972i	$-0.555253\pi$
$173$	−786.000	−0.345425	−0.172712	+	0.984972i	$0.555253\pi$
$174$	0	0
$175$	0	0
$176$	−1065.00	−0.456122
$177$	0	0
$178$	2718.00	1.14451
$179$	−2892.00	−1.20759	−0.603794	−	0.797140i	$-0.706345\pi$
$179$	−2892.00	−1.20759	−0.603794	+	0.797140i	$0.706345\pi$
$180$	0	0
$181$	−1352.00	−0.555212	−0.277606	−	0.960695i	$-0.589541\pi$
$181$	−1352.00	−0.555212	−0.277606	+	0.960695i	$0.589541\pi$
$182$	0	0
$183$	0	0
$184$	−1764.00	−0.706760
$185$	948.000	0.376748
$186$	0	0
$187$	1260.00	0.492729
$188$	−30.0000	−0.0116382
$189$	0	0
$190$	−144.000	−0.0549835
$191$	−3912.00	−1.48200	−0.741001	−	0.671504i	$-0.765649\pi$
$191$	−3912.00	−1.48200	−0.741001	+	0.671504i	$0.765649\pi$
$192$	0	0
$193$	1493.00	0.556832	0.278416	−	0.960461i	$-0.410191\pi$
$193$	1493.00	0.556832	0.278416	+	0.960461i	$0.410191\pi$
$194$	−1509.00	−0.558453
$195$	0	0
$196$	0	0
$197$	4086.00	1.47774	0.738872	−	0.673846i	$-0.235359\pi$
$197$	4086.00	1.47774	0.738872	+	0.673846i	$0.235359\pi$
$198$	0	0
$199$	3556.00	1.26672	0.633362	−	0.773855i	$-0.281674\pi$
$199$	3556.00	1.26672	0.633362	+	0.773855i	$0.281674\pi$
$200$	2436.00	0.861256
$201$	0	0
$202$	−3258.00	−1.13481
$203$	0	0
$204$	0	0
$205$	−1080.00	−0.367954
$206$	−5208.00	−1.76145
$207$	0	0
$208$	−4544.00	−1.51476
$209$	240.000	0.0794313
$210$	0	0
$211$	1250.00	0.407837	0.203918	−	0.978988i	$-0.434632\pi$
$211$	1250.00	0.407837	0.203918	+	0.978988i	$0.434632\pi$
$212$	−363.000	−0.117599
$213$	0	0
$214$	4059.00	1.29658
$215$	−78.0000	−0.0247421
$216$	0	0
$217$	0	0
$218$	−1110.00	−0.344856
$219$	0	0
$220$	−45.0000	−0.0137905
$221$	5376.00	1.63633
$222$	0	0
$223$	−425.000	−0.127624	−0.0638119	−	0.997962i	$-0.520326\pi$
$223$	−425.000	−0.127624	−0.0638119	+	0.997962i	$0.520326\pi$
$224$	0	0
$225$	0	0
$226$	1944.00	0.572181
$227$	3855.00	1.12716	0.563580	−	0.826061i	$-0.309424\pi$
$227$	3855.00	1.12716	0.563580	+	0.826061i	$0.309424\pi$
$228$	0	0
$229$	2188.00	0.631385	0.315692	−	0.948862i	$-0.397763\pi$
$229$	2188.00	0.631385	0.315692	+	0.948862i	$0.397763\pi$
$230$	−756.000	−0.216735
$231$	0	0
$232$	−6237.00	−1.76500
$233$	−852.000	−0.239555	−0.119778	−	0.992801i	$-0.538218\pi$
$233$	−852.000	−0.239555	−0.119778	+	0.992801i	$0.538218\pi$
$234$	0	0
$235$	90.0000	0.0249828
$236$	−15.0000	−0.00413736
$237$	0	0
$238$	0	0
$239$	−5508.00	−1.49072	−0.745362	−	0.666660i	$-0.767723\pi$
$239$	−5508.00	−1.49072	−0.745362	+	0.666660i	$0.767723\pi$
$240$	0	0
$241$	−791.000	−0.211422	−0.105711	−	0.994397i	$-0.533712\pi$
$241$	−791.000	−0.211422	−0.105711	+	0.994397i	$0.533712\pi$
$242$	−3318.00	−0.881360
$243$	0	0
$244$	118.000	0.0309597
$245$	0	0
$246$	0	0
$247$	1024.00	0.263788
$248$	−5313.00	−1.36039
$249$	0	0
$250$	2169.00	0.548718
$251$	5265.00	1.32400	0.662000	−	0.749504i	$-0.269708\pi$
$251$	5265.00	1.32400	0.662000	+	0.749504i	$0.269708\pi$
$252$	0	0
$253$	1260.00	0.313105
$254$	1131.00	0.279391
$255$	0	0
$256$	1513.00	0.369385
$257$	−6870.00	−1.66747	−0.833733	−	0.552168i	$-0.813801\pi$
$257$	−6870.00	−1.66747	−0.833733	+	0.552168i	$0.813801\pi$
$258$	0	0
$259$	0	0
$260$	−192.000	−0.0457974
$261$	0	0
$262$	−1953.00	−0.460522
$263$	222.000	0.0520498	0.0260249	−	0.999661i	$-0.491715\pi$
$263$	222.000	0.0520498	0.0260249	+	0.999661i	$0.491715\pi$
$264$	0	0
$265$	1089.00	0.252441
$266$	0	0
$267$	0	0
$268$	−370.000	−0.0843334
$269$	7851.00	1.77949	0.889747	−	0.456454i	$-0.150881\pi$
$269$	7851.00	1.77949	0.889747	+	0.456454i	$0.150881\pi$
$270$	0	0
$271$	−5183.00	−1.16179	−0.580895	−	0.813979i	$-0.697297\pi$
$271$	−5183.00	−1.16179	−0.580895	+	0.813979i	$0.697297\pi$
$272$	−5964.00	−1.32949
$273$	0	0
$274$	5310.00	1.17076
$275$	−1740.00	−0.381549
$276$	0	0
$277$	−4960.00	−1.07588	−0.537938	−	0.842985i	$-0.680796\pi$
$277$	−4960.00	−1.07588	−0.537938	+	0.842985i	$0.680796\pi$
$278$	4674.00	1.00837
$279$	0	0
$280$	0	0
$281$	774.000	0.164317	0.0821583	−	0.996619i	$-0.473819\pi$
$281$	774.000	0.164317	0.0821583	+	0.996619i	$0.473819\pi$
$282$	0	0
$283$	−3698.00	−0.776761	−0.388380	−	0.921499i	$-0.626965\pi$
$283$	−3698.00	−0.776761	−0.388380	+	0.921499i	$0.626965\pi$
$284$	342.000	0.0714576
$285$	0	0
$286$	2880.00	0.595447
$287$	0	0
$288$	0	0
$289$	2143.00	0.436190
$290$	−2673.00	−0.541255
$291$	0	0
$292$	−362.000	−0.0725495
$293$	−6273.00	−1.25076	−0.625380	−	0.780321i	$-0.715056\pi$
$293$	−6273.00	−1.25076	−0.625380	+	0.780321i	$0.715056\pi$
$294$	0	0
$295$	45.0000	0.00888136
$296$	6636.00	1.30307
$297$	0	0
$298$	−7362.00	−1.43110
$299$	5376.00	1.03981
$300$	0	0
$301$	0	0
$302$	3777.00	0.719675
$303$	0	0
$304$	−1136.00	−0.214323
$305$	−354.000	−0.0664590
$306$	0	0
$307$	1684.00	0.313065	0.156533	−	0.987673i	$-0.449968\pi$
$307$	1684.00	0.313065	0.156533	+	0.987673i	$0.449968\pi$
$308$	0	0
$309$	0	0
$310$	−2277.00	−0.417177
$311$	−1320.00	−0.240676	−0.120338	−	0.992733i	$-0.538398\pi$
$311$	−1320.00	−0.240676	−0.120338	+	0.992733i	$0.538398\pi$
$312$	0	0
$313$	8503.00	1.53552	0.767760	−	0.640737i	$-0.221371\pi$
$313$	8503.00	1.53552	0.767760	+	0.640737i	$0.221371\pi$
$314$	588.000	0.105678
$315$	0	0
$316$	467.000	0.0831355
$317$	2577.00	0.456589	0.228295	−	0.973592i	$-0.426685\pi$
$317$	2577.00	0.456589	0.228295	+	0.973592i	$0.426685\pi$
$318$	0	0
$319$	4455.00	0.781919
$320$	−1299.00	−0.226926
$321$	0	0
$322$	0	0
$323$	1344.00	0.231524
$324$	0	0
$325$	−7424.00	−1.26711
$326$	−3756.00	−0.638115
$327$	0	0
$328$	−7560.00	−1.27266
$329$	0	0
$330$	0	0
$331$	−484.000	−0.0803717	−0.0401859	−	0.999192i	$-0.512795\pi$
$331$	−484.000	−0.0803717	−0.0401859	+	0.999192i	$0.512795\pi$
$332$	477.000	0.0788517
$333$	0	0
$334$	−7938.00	−1.30044
$335$	1110.00	0.181032
$336$	0	0
$337$	−8359.00	−1.35117	−0.675584	−	0.737283i	$-0.736109\pi$
$337$	−8359.00	−1.35117	−0.675584	+	0.737283i	$0.736109\pi$
$338$	5697.00	0.916793
$339$	0	0
$340$	−252.000	−0.0401959
$341$	3795.00	0.602671
$342$	0	0
$343$	0	0
$344$	−546.000	−0.0855766
$345$	0	0
$346$	−2358.00	−0.366378
$347$	1860.00	0.287752	0.143876	−	0.989596i	$-0.454043\pi$
$347$	1860.00	0.287752	0.143876	+	0.989596i	$0.454043\pi$
$348$	0	0
$349$	1918.00	0.294178	0.147089	−	0.989123i	$-0.453010\pi$
$349$	1918.00	0.294178	0.147089	+	0.989123i	$0.453010\pi$
$350$	0	0
$351$	0	0
$352$	−675.000	−0.102209
$353$	−3048.00	−0.459571	−0.229786	−	0.973241i	$-0.573803\pi$
$353$	−3048.00	−0.459571	−0.229786	+	0.973241i	$0.573803\pi$
$354$	0	0
$355$	−1026.00	−0.153393
$356$	906.000	0.134882
$357$	0	0
$358$	−8676.00	−1.28084
$359$	30.0000	0.00441042	0.00220521	−	0.999998i	$-0.499298\pi$
$359$	30.0000	0.00441042	0.00220521	+	0.999998i	$0.499298\pi$
$360$	0	0
$361$	−6603.00	−0.962677
$362$	−4056.00	−0.588891
$363$	0	0
$364$	0	0
$365$	1086.00	0.155737
$366$	0	0
$367$	11311.0	1.60880	0.804400	−	0.594088i	$-0.202487\pi$
$367$	11311.0	1.60880	0.804400	+	0.594088i	$0.202487\pi$
$368$	−5964.00	−0.844823
$369$	0	0
$370$	2844.00	0.399601
$371$	0	0
$372$	0	0
$373$	1208.00	0.167689	0.0838443	−	0.996479i	$-0.473280\pi$
$373$	1208.00	0.167689	0.0838443	+	0.996479i	$0.473280\pi$
$374$	3780.00	0.522618
$375$	0	0
$376$	630.000	0.0864090
$377$	19008.0	2.59672
$378$	0	0
$379$	7640.00	1.03546	0.517731	−	0.855543i	$-0.326777\pi$
$379$	7640.00	1.03546	0.517731	+	0.855543i	$0.326777\pi$
$380$	−48.0000	−0.00647986
$381$	0	0
$382$	−11736.0	−1.57190
$383$	12750.0	1.70103	0.850515	−	0.525951i	$-0.176290\pi$
$383$	12750.0	1.70103	0.850515	+	0.525951i	$0.176290\pi$
$384$	0	0
$385$	0	0
$386$	4479.00	0.590609
$387$	0	0
$388$	−503.000	−0.0658143
$389$	−3126.00	−0.407441	−0.203720	−	0.979029i	$-0.565303\pi$
$389$	−3126.00	−0.407441	−0.203720	+	0.979029i	$0.565303\pi$
$390$	0	0
$391$	7056.00	0.912627
$392$	0	0
$393$	0	0
$394$	12258.0	1.56738
$395$	−1401.00	−0.178461
$396$	0	0
$397$	5932.00	0.749921	0.374960	−	0.927041i	$-0.377656\pi$
$397$	5932.00	0.749921	0.374960	+	0.927041i	$0.377656\pi$
$398$	10668.0	1.34356
$399$	0	0
$400$	8236.00	1.02950
$401$	−1608.00	−0.200249	−0.100124	−	0.994975i	$-0.531924\pi$
$401$	−1608.00	−0.200249	−0.100124	+	0.994975i	$0.531924\pi$
$402$	0	0
$403$	16192.0	2.00144
$404$	−1086.00	−0.133739
$405$	0	0
$406$	0	0
$407$	−4740.00	−0.577280
$408$	0	0
$409$	4465.00	0.539805	0.269902	−	0.962888i	$-0.413009\pi$
$409$	4465.00	0.539805	0.269902	+	0.962888i	$0.413009\pi$
$410$	−3240.00	−0.390274
$411$	0	0
$412$	−1736.00	−0.207589
$413$	0	0
$414$	0	0
$415$	−1431.00	−0.169265
$416$	−2880.00	−0.339432
$417$	0	0
$418$	720.000	0.0842496
$419$	−1584.00	−0.184686	−0.0923430	−	0.995727i	$-0.529436\pi$
$419$	−1584.00	−0.184686	−0.0923430	+	0.995727i	$0.529436\pi$
$420$	0	0
$421$	−1330.00	−0.153967	−0.0769837	−	0.997032i	$-0.524529\pi$
$421$	−1330.00	−0.153967	−0.0769837	+	0.997032i	$0.524529\pi$
$422$	3750.00	0.432576
$423$	0	0
$424$	7623.00	0.873126
$425$	−9744.00	−1.11213
$426$	0	0
$427$	0	0
$428$	1353.00	0.152803
$429$	0	0
$430$	−234.000	−0.0262430
$431$	−9588.00	−1.07155	−0.535775	−	0.844361i	$-0.679980\pi$
$431$	−9588.00	−1.07155	−0.535775	+	0.844361i	$0.679980\pi$
$432$	0	0
$433$	−494.000	−0.0548271	−0.0274135	−	0.999624i	$-0.508727\pi$
$433$	−494.000	−0.0548271	−0.0274135	+	0.999624i	$0.508727\pi$
$434$	0	0
$435$	0	0
$436$	−370.000	−0.0406417
$437$	1344.00	0.147122
$438$	0	0
$439$	16009.0	1.74047	0.870237	−	0.492634i	$-0.163966\pi$
$439$	16009.0	1.74047	0.870237	+	0.492634i	$0.163966\pi$
$440$	945.000	0.102389
$441$	0	0
$442$	16128.0	1.73559
$443$	−7773.00	−0.833649	−0.416824	−	0.908987i	$-0.636857\pi$
$443$	−7773.00	−0.833649	−0.416824	+	0.908987i	$0.636857\pi$
$444$	0	0
$445$	−2718.00	−0.289541
$446$	−1275.00	−0.135365
$447$	0	0
$448$	0	0
$449$	−864.000	−0.0908122	−0.0454061	−	0.998969i	$-0.514458\pi$
$449$	−864.000	−0.0908122	−0.0454061	+	0.998969i	$0.514458\pi$
$450$	0	0
$451$	5400.00	0.563805
$452$	648.000	0.0674322
$453$	0	0
$454$	11565.0	1.19553
$455$	0	0
$456$	0	0
$457$	2519.00	0.257842	0.128921	−	0.991655i	$-0.458849\pi$
$457$	2519.00	0.257842	0.128921	+	0.991655i	$0.458849\pi$
$458$	6564.00	0.669685
$459$	0	0
$460$	−252.000	−0.0255425
$461$	−342.000	−0.0345521	−0.0172761	−	0.999851i	$-0.505499\pi$
$461$	−342.000	−0.0345521	−0.0172761	+	0.999851i	$0.505499\pi$
$462$	0	0
$463$	−4336.00	−0.435229	−0.217614	−	0.976035i	$-0.569828\pi$
$463$	−4336.00	−0.435229	−0.217614	+	0.976035i	$0.569828\pi$
$464$	−21087.0	−2.10978
$465$	0	0
$466$	−2556.00	−0.254087
$467$	18636.0	1.84662	0.923310	−	0.384056i	$-0.125473\pi$
$467$	18636.0	1.84662	0.923310	+	0.384056i	$0.125473\pi$
$468$	0	0
$469$	0	0
$470$	270.000	0.0264982
$471$	0	0
$472$	315.000	0.0307183
$473$	390.000	0.0379117
$474$	0	0
$475$	−1856.00	−0.179282
$476$	0	0
$477$	0	0
$478$	−16524.0	−1.58115
$479$	15078.0	1.43827	0.719135	−	0.694870i	$-0.244538\pi$
$479$	15078.0	1.43827	0.719135	+	0.694870i	$0.244538\pi$
$480$	0	0
$481$	−20224.0	−1.91712
$482$	−2373.00	−0.224247
$483$	0	0
$484$	−1106.00	−0.103869
$485$	1509.00	0.141279
$486$	0	0
$487$	6221.00	0.578851	0.289425	−	0.957201i	$-0.406536\pi$
$487$	6221.00	0.578851	0.289425	+	0.957201i	$0.406536\pi$
$488$	−2478.00	−0.229864
$489$	0	0
$490$	0	0
$491$	7371.00	0.677492	0.338746	−	0.940878i	$-0.389997\pi$
$491$	7371.00	0.677492	0.338746	+	0.940878i	$0.389997\pi$
$492$	0	0
$493$	24948.0	2.27911
$494$	3072.00	0.279789
$495$	0	0
$496$	−17963.0	−1.62613
$497$	0	0
$498$	0	0
$499$	4274.00	0.383428	0.191714	−	0.981451i	$-0.438595\pi$
$499$	4274.00	0.383428	0.191714	+	0.981451i	$0.438595\pi$
$500$	723.000	0.0646671
$501$	0	0
$502$	15795.0	1.40431
$503$	−2520.00	−0.223382	−0.111691	−	0.993743i	$-0.535627\pi$
$503$	−2520.00	−0.223382	−0.111691	+	0.993743i	$0.535627\pi$
$504$	0	0
$505$	3258.00	0.287087
$506$	3780.00	0.332098
$507$	0	0
$508$	377.000	0.0329265
$509$	−14277.0	−1.24326	−0.621628	−	0.783313i	$-0.713528\pi$
$509$	−14277.0	−1.24326	−0.621628	+	0.783313i	$0.713528\pi$
$510$	0	0
$511$	0	0
$512$	−8733.00	−0.753804
$513$	0	0
$514$	−20610.0	−1.76862
$515$	5208.00	0.445615
$516$	0	0
$517$	−450.000	−0.0382804
$518$	0	0
$519$	0	0
$520$	4032.00	0.340029
$521$	−6306.00	−0.530270	−0.265135	−	0.964211i	$-0.585417\pi$
$521$	−6306.00	−0.530270	−0.265135	+	0.964211i	$0.585417\pi$
$522$	0	0
$523$	−8072.00	−0.674883	−0.337442	−	0.941346i	$-0.609562\pi$
$523$	−8072.00	−0.674883	−0.337442	+	0.941346i	$0.609562\pi$
$524$	−651.000	−0.0542730
$525$	0	0
$526$	666.000	0.0552072
$527$	21252.0	1.75664
$528$	0	0
$529$	−5111.00	−0.420071
$530$	3267.00	0.267754
$531$	0	0
$532$	0	0
$533$	23040.0	1.87237
$534$	0	0
$535$	−4059.00	−0.328011
$536$	7770.00	0.626143
$537$	0	0
$538$	23553.0	1.88744
$539$	0	0
$540$	0	0
$541$	−22858.0	−1.81653	−0.908264	−	0.418396i	$-0.862592\pi$
$541$	−22858.0	−1.81653	−0.908264	+	0.418396i	$0.862592\pi$
$542$	−15549.0	−1.23226
$543$	0	0
$544$	−3780.00	−0.297916
$545$	1110.00	0.0872425
$546$	0	0
$547$	−24724.0	−1.93258	−0.966291	−	0.257454i	$-0.917116\pi$
$547$	−24724.0	−1.93258	−0.966291	+	0.257454i	$0.917116\pi$
$548$	1770.00	0.137976
$549$	0	0
$550$	−5220.00	−0.404694
$551$	4752.00	0.367408
$552$	0	0
$553$	0	0
$554$	−14880.0	−1.14114
$555$	0	0
$556$	1558.00	0.118838
$557$	9843.00	0.748764	0.374382	−	0.927275i	$-0.377855\pi$
$557$	9843.00	0.748764	0.374382	+	0.927275i	$0.377855\pi$
$558$	0	0
$559$	1664.00	0.125903
$560$	0	0
$561$	0	0
$562$	2322.00	0.174284
$563$	−13371.0	−1.00092	−0.500462	−	0.865758i	$-0.666837\pi$
$563$	−13371.0	−1.00092	−0.500462	+	0.865758i	$0.666837\pi$
$564$	0	0
$565$	−1944.00	−0.144752
$566$	−11094.0	−0.823879
$567$	0	0
$568$	−7182.00	−0.530546
$569$	5232.00	0.385478	0.192739	−	0.981250i	$-0.438263\pi$
$569$	5232.00	0.385478	0.192739	+	0.981250i	$0.438263\pi$
$570$	0	0
$571$	−14398.0	−1.05523	−0.527616	−	0.849483i	$-0.676914\pi$
$571$	−14398.0	−1.05523	−0.527616	+	0.849483i	$0.676914\pi$
$572$	960.000	0.0701742
$573$	0	0
$574$	0	0
$575$	−9744.00	−0.706701
$576$	0	0
$577$	−19871.0	−1.43369	−0.716846	−	0.697231i	$-0.754415\pi$
$577$	−19871.0	−1.43369	−0.716846	+	0.697231i	$0.754415\pi$
$578$	6429.00	0.462649
$579$	0	0
$580$	−891.000	−0.0637875
$581$	0	0
$582$	0	0
$583$	−5445.00	−0.386808
$584$	7602.00	0.538652
$585$	0	0
$586$	−18819.0	−1.32663
$587$	−16137.0	−1.13466	−0.567330	−	0.823491i	$-0.692024\pi$
$587$	−16137.0	−1.13466	−0.567330	+	0.823491i	$0.692024\pi$
$588$	0	0
$589$	4048.00	0.283183
$590$	135.000	0.00942011
$591$	0	0
$592$	22436.0	1.55762
$593$	−21324.0	−1.47668	−0.738340	−	0.674428i	$-0.764390\pi$
$593$	−21324.0	−1.47668	−0.738340	+	0.674428i	$0.764390\pi$
$594$	0	0
$595$	0	0
$596$	−2454.00	−0.168657
$597$	0	0
$598$	16128.0	1.10288
$599$	8646.00	0.589760	0.294880	−	0.955534i	$-0.404720\pi$
$599$	8646.00	0.589760	0.294880	+	0.955534i	$0.404720\pi$
$600$	0	0
$601$	−11195.0	−0.759823	−0.379911	−	0.925023i	$-0.624046\pi$
$601$	−11195.0	−0.759823	−0.379911	+	0.925023i	$0.624046\pi$
$602$	0	0
$603$	0	0
$604$	1259.00	0.0848145
$605$	3318.00	0.222968
$606$	0	0
$607$	8971.00	0.599871	0.299935	−	0.953959i	$-0.403035\pi$
$607$	8971.00	0.599871	0.299935	+	0.953959i	$0.403035\pi$
$608$	−720.000	−0.0480261
$609$	0	0
$610$	−1062.00	−0.0704904
$611$	−1920.00	−0.127127
$612$	0	0
$613$	−12772.0	−0.841527	−0.420764	−	0.907170i	$-0.638238\pi$
$613$	−12772.0	−0.841527	−0.420764	+	0.907170i	$0.638238\pi$
$614$	5052.00	0.332056
$615$	0	0
$616$	0	0
$617$	−12762.0	−0.832705	−0.416352	−	0.909203i	$-0.636692\pi$
$617$	−12762.0	−0.832705	−0.416352	+	0.909203i	$0.636692\pi$
$618$	0	0
$619$	−12842.0	−0.833867	−0.416933	−	0.908937i	$-0.636895\pi$
$619$	−12842.0	−0.833867	−0.416933	+	0.908937i	$0.636895\pi$
$620$	−759.000	−0.0491648
$621$	0	0
$622$	−3960.00	−0.255276
$623$	0	0
$624$	0	0
$625$	12331.0	0.789184
$626$	25509.0	1.62867
$627$	0	0
$628$	196.000	0.0124542
$629$	−26544.0	−1.68264
$630$	0	0
$631$	21365.0	1.34790	0.673952	−	0.738775i	$-0.264596\pi$
$631$	21365.0	1.34790	0.673952	+	0.738775i	$0.264596\pi$
$632$	−9807.00	−0.617249
$633$	0	0
$634$	7731.00	0.484286
$635$	−1131.00	−0.0706809
$636$	0	0
$637$	0	0
$638$	13365.0	0.829350
$639$	0	0
$640$	−4977.00	−0.307396
$641$	−8274.00	−0.509834	−0.254917	−	0.966963i	$-0.582048\pi$
$641$	−8274.00	−0.509834	−0.254917	+	0.966963i	$0.582048\pi$
$642$	0	0
$643$	−27998.0	−1.71716	−0.858580	−	0.512680i	$-0.828653\pi$
$643$	−27998.0	−1.71716	−0.858580	+	0.512680i	$0.828653\pi$
$644$	0	0
$645$	0	0
$646$	4032.00	0.245568
$647$	−17466.0	−1.06130	−0.530649	−	0.847592i	$-0.678052\pi$
$647$	−17466.0	−1.06130	−0.530649	+	0.847592i	$0.678052\pi$
$648$	0	0
$649$	−225.000	−0.0136087
$650$	−22272.0	−1.34397
$651$	0	0
$652$	−1252.00	−0.0752026
$653$	−2157.00	−0.129265	−0.0646324	−	0.997909i	$-0.520587\pi$
$653$	−2157.00	−0.129265	−0.0646324	+	0.997909i	$0.520587\pi$
$654$	0	0
$655$	1953.00	0.116504
$656$	−25560.0	−1.52127
$657$	0	0
$658$	0	0
$659$	−19944.0	−1.17892	−0.589460	−	0.807798i	$-0.700659\pi$
$659$	−19944.0	−1.17892	−0.589460	+	0.807798i	$0.700659\pi$
$660$	0	0
$661$	−27506.0	−1.61855	−0.809273	−	0.587432i	$-0.800139\pi$
$661$	−27506.0	−1.61855	−0.809273	+	0.587432i	$0.800139\pi$
$662$	−1452.00	−0.0852471
$663$	0	0
$664$	−10017.0	−0.585444
$665$	0	0
$666$	0	0
$667$	24948.0	1.44826
$668$	−2646.00	−0.153259
$669$	0	0
$670$	3330.00	0.192014
$671$	1770.00	0.101833
$672$	0	0
$673$	−19123.0	−1.09530	−0.547650	−	0.836707i	$-0.684478\pi$
$673$	−19123.0	−1.09530	−0.547650	+	0.836707i	$0.684478\pi$
$674$	−25077.0	−1.43313
$675$	0	0
$676$	1899.00	0.108045
$677$	13857.0	0.786658	0.393329	−	0.919398i	$-0.371323\pi$
$677$	13857.0	0.786658	0.393329	+	0.919398i	$0.371323\pi$
$678$	0	0
$679$	0	0
$680$	5292.00	0.298440
$681$	0	0
$682$	11385.0	0.639229
$683$	22245.0	1.24624	0.623120	−	0.782127i	$-0.285865\pi$
$683$	22245.0	1.24624	0.623120	+	0.782127i	$0.285865\pi$
$684$	0	0
$685$	−5310.00	−0.296182
$686$	0	0
$687$	0	0
$688$	−1846.00	−0.102294
$689$	−23232.0	−1.28457
$690$	0	0
$691$	640.000	0.0352341	0.0176170	−	0.999845i	$-0.494392\pi$
$691$	640.000	0.0352341	0.0176170	+	0.999845i	$0.494392\pi$
$692$	−786.000	−0.0431781
$693$	0	0
$694$	5580.00	0.305207
$695$	−4674.00	−0.255101
$696$	0	0
$697$	30240.0	1.64336
$698$	5754.00	0.312023
$699$	0	0
$700$	0	0
$701$	15561.0	0.838418	0.419209	−	0.907890i	$-0.362307\pi$
$701$	15561.0	0.838418	0.419209	+	0.907890i	$0.362307\pi$
$702$	0	0
$703$	−5056.00	−0.271253
$704$	6495.00	0.347712
$705$	0	0
$706$	−9144.00	−0.487449
$707$	0	0
$708$	0	0
$709$	5534.00	0.293136	0.146568	−	0.989201i	$-0.453177\pi$
$709$	5534.00	0.293136	0.146568	+	0.989201i	$0.453177\pi$
$710$	−3078.00	−0.162698
$711$	0	0
$712$	−19026.0	−1.00145
$713$	21252.0	1.11626
$714$	0	0
$715$	−2880.00	−0.150638
$716$	−2892.00	−0.150948
$717$	0	0
$718$	90.0000	0.00467795
$719$	−21846.0	−1.13313	−0.566564	−	0.824018i	$-0.691727\pi$
$719$	−21846.0	−1.13313	−0.566564	+	0.824018i	$0.691727\pi$
$720$	0	0
$721$	0	0
$722$	−19809.0	−1.02107
$723$	0	0
$724$	−1352.00	−0.0694015
$725$	−34452.0	−1.76485
$726$	0	0
$727$	11089.0	0.565706	0.282853	−	0.959163i	$-0.408719\pi$
$727$	11089.0	0.565706	0.282853	+	0.959163i	$0.408719\pi$
$728$	0	0
$729$	0	0
$730$	3258.00	0.165184
$731$	2184.00	0.110504
$732$	0	0
$733$	−11762.0	−0.592687	−0.296343	−	0.955081i	$-0.595767\pi$
$733$	−11762.0	−0.592687	−0.296343	+	0.955081i	$0.595767\pi$
$734$	33933.0	1.70639
$735$	0	0
$736$	−3780.00	−0.189311
$737$	−5550.00	−0.277391
$738$	0	0
$739$	−22726.0	−1.13124	−0.565622	−	0.824665i	$-0.691364\pi$
$739$	−22726.0	−1.13124	−0.565622	+	0.824665i	$0.691364\pi$
$740$	948.000	0.0470935
$741$	0	0
$742$	0	0
$743$	−6678.00	−0.329734	−0.164867	−	0.986316i	$-0.552719\pi$
$743$	−6678.00	−0.329734	−0.164867	+	0.986316i	$0.552719\pi$
$744$	0	0
$745$	7362.00	0.362044
$746$	3624.00	0.177861
$747$	0	0
$748$	1260.00	0.0615911
$749$	0	0
$750$	0	0
$751$	−19987.0	−0.971153	−0.485577	−	0.874194i	$-0.661390\pi$
$751$	−19987.0	−0.971153	−0.485577	+	0.874194i	$0.661390\pi$
$752$	2130.00	0.103289
$753$	0	0
$754$	57024.0	2.75423
$755$	−3777.00	−0.182065
$756$	0	0
$757$	314.000	0.0150760	0.00753799	−	0.999972i	$-0.497601\pi$
$757$	314.000	0.0150760	0.00753799	+	0.999972i	$0.497601\pi$
$758$	22920.0	1.09827
$759$	0	0
$760$	1008.00	0.0481105
$761$	−11496.0	−0.547608	−0.273804	−	0.961786i	$-0.588282\pi$
$761$	−11496.0	−0.547608	−0.273804	+	0.961786i	$0.588282\pi$
$762$	0	0
$763$	0	0
$764$	−3912.00	−0.185250
$765$	0	0
$766$	38250.0	1.80421
$767$	−960.000	−0.0451937
$768$	0	0
$769$	−2765.00	−0.129660	−0.0648299	−	0.997896i	$-0.520650\pi$
$769$	−2765.00	−0.129660	−0.0648299	+	0.997896i	$0.520650\pi$
$770$	0	0
$771$	0	0
$772$	1493.00	0.0696039
$773$	−14046.0	−0.653557	−0.326778	−	0.945101i	$-0.605963\pi$
$773$	−14046.0	−0.653557	−0.326778	+	0.945101i	$0.605963\pi$
$774$	0	0
$775$	−29348.0	−1.36027
$776$	10563.0	0.488646
$777$	0	0
$778$	−9378.00	−0.432156
$779$	5760.00	0.264921
$780$	0	0
$781$	5130.00	0.235039
$782$	21168.0	0.967987
$783$	0	0
$784$	0	0
$785$	−588.000	−0.0267345
$786$	0	0
$787$	18514.0	0.838568	0.419284	−	0.907855i	$-0.362281\pi$
$787$	18514.0	0.838568	0.419284	+	0.907855i	$0.362281\pi$
$788$	4086.00	0.184718
$789$	0	0
$790$	−4203.00	−0.189286
$791$	0	0
$792$	0	0
$793$	7552.00	0.338183
$794$	17796.0	0.795411
$795$	0	0
$796$	3556.00	0.158341
$797$	−27495.0	−1.22199	−0.610993	−	0.791636i	$-0.709230\pi$
$797$	−27495.0	−1.22199	−0.610993	+	0.791636i	$0.709230\pi$
$798$	0	0
$799$	−2520.00	−0.111578
$800$	5220.00	0.230694
$801$	0	0
$802$	−4824.00	−0.212396
$803$	−5430.00	−0.238631
$804$	0	0
$805$	0	0
$806$	48576.0	2.12285
$807$	0	0
$808$	22806.0	0.992961
$809$	7944.00	0.345236	0.172618	−	0.984989i	$-0.444777\pi$
$809$	7944.00	0.345236	0.172618	+	0.984989i	$0.444777\pi$
$810$	0	0
$811$	28942.0	1.25313	0.626567	−	0.779368i	$-0.284460\pi$
$811$	28942.0	1.25313	0.626567	+	0.779368i	$0.284460\pi$
$812$	0	0
$813$	0	0
$814$	−14220.0	−0.612298
$815$	3756.00	0.161432
$816$	0	0
$817$	416.000	0.0178140
$818$	13395.0	0.572549
$819$	0	0
$820$	−1080.00	−0.0459942
$821$	−8187.00	−0.348025	−0.174012	−	0.984743i	$-0.555673\pi$
$821$	−8187.00	−0.348025	−0.174012	+	0.984743i	$0.555673\pi$
$822$	0	0
$823$	−280.000	−0.0118593	−0.00592964	−	0.999982i	$-0.501887\pi$
$823$	−280.000	−0.0118593	−0.00592964	+	0.999982i	$0.501887\pi$
$824$	36456.0	1.54127
$825$	0	0
$826$	0	0
$827$	−25317.0	−1.06452	−0.532260	−	0.846581i	$-0.678657\pi$
$827$	−25317.0	−1.06452	−0.532260	+	0.846581i	$0.678657\pi$
$828$	0	0
$829$	−15320.0	−0.641840	−0.320920	−	0.947106i	$-0.603992\pi$
$829$	−15320.0	−0.641840	−0.320920	+	0.947106i	$0.603992\pi$
$830$	−4293.00	−0.179533
$831$	0	0
$832$	27712.0	1.15474
$833$	0	0
$834$	0	0
$835$	7938.00	0.328989
$836$	240.000	0.00992892
$837$	0	0
$838$	−4752.00	−0.195889
$839$	34092.0	1.40284	0.701422	−	0.712746i	$-0.252549\pi$
$839$	34092.0	1.40284	0.701422	+	0.712746i	$0.252549\pi$
$840$	0	0
$841$	63820.0	2.61675
$842$	−3990.00	−0.163307
$843$	0	0
$844$	1250.00	0.0509796
$845$	−5697.00	−0.231932
$846$	0	0
$847$	0	0
$848$	25773.0	1.04369
$849$	0	0
$850$	−29232.0	−1.17959
$851$	−26544.0	−1.06923
$852$	0	0
$853$	7378.00	0.296152	0.148076	−	0.988976i	$-0.452692\pi$
$853$	7378.00	0.296152	0.148076	+	0.988976i	$0.452692\pi$
$854$	0	0
$855$	0	0
$856$	−28413.0	−1.13451
$857$	−15594.0	−0.621565	−0.310782	−	0.950481i	$-0.600591\pi$
$857$	−15594.0	−0.621565	−0.310782	+	0.950481i	$0.600591\pi$
$858$	0	0
$859$	30538.0	1.21297	0.606486	−	0.795094i	$-0.292579\pi$
$859$	30538.0	1.21297	0.606486	+	0.795094i	$0.292579\pi$
$860$	−78.0000	−0.00309277
$861$	0	0
$862$	−28764.0	−1.13655
$863$	822.000	0.0324232	0.0162116	−	0.999869i	$-0.494839\pi$
$863$	822.000	0.0324232	0.0162116	+	0.999869i	$0.494839\pi$
$864$	0	0
$865$	2358.00	0.0926872
$866$	−1482.00	−0.0581529
$867$	0	0
$868$	0	0
$869$	7005.00	0.273450
$870$	0	0
$871$	−23680.0	−0.921201
$872$	7770.00	0.301749
$873$	0	0
$874$	4032.00	0.156046
$875$	0	0
$876$	0	0
$877$	−41824.0	−1.61037	−0.805186	−	0.593022i	$-0.797935\pi$
$877$	−41824.0	−1.61037	−0.805186	+	0.593022i	$0.797935\pi$
$878$	48027.0	1.84605
$879$	0	0
$880$	3195.00	0.122390
$881$	−46098.0	−1.76286	−0.881431	−	0.472313i	$-0.843419\pi$
$881$	−46098.0	−1.76286	−0.881431	+	0.472313i	$0.843419\pi$
$882$	0	0
$883$	21008.0	0.800652	0.400326	−	0.916373i	$-0.368897\pi$
$883$	21008.0	0.800652	0.400326	+	0.916373i	$0.368897\pi$
$884$	5376.00	0.204541
$885$	0	0
$886$	−23319.0	−0.884218
$887$	24036.0	0.909865	0.454932	−	0.890526i	$-0.349663\pi$
$887$	24036.0	0.909865	0.454932	+	0.890526i	$0.349663\pi$
$888$	0	0
$889$	0	0
$890$	−8154.00	−0.307104
$891$	0	0
$892$	−425.000	−0.0159530
$893$	−480.000	−0.0179872
$894$	0	0
$895$	8676.00	0.324030
$896$	0	0
$897$	0	0
$898$	−2592.00	−0.0963209
$899$	75141.0	2.78764
$900$	0	0
$901$	−30492.0	−1.12745
$902$	16200.0	0.598006
$903$	0	0
$904$	−13608.0	−0.500659
$905$	4056.00	0.148979
$906$	0	0
$907$	13292.0	0.486608	0.243304	−	0.969950i	$-0.421769\pi$
$907$	13292.0	0.486608	0.243304	+	0.969950i	$0.421769\pi$
$908$	3855.00	0.140895
$909$	0	0
$910$	0	0
$911$	9306.00	0.338443	0.169221	−	0.985578i	$-0.445875\pi$
$911$	9306.00	0.338443	0.169221	+	0.985578i	$0.445875\pi$
$912$	0	0
$913$	7155.00	0.259360
$914$	7557.00	0.273483
$915$	0	0
$916$	2188.00	0.0789231
$917$	0	0
$918$	0	0
$919$	16496.0	0.592114	0.296057	−	0.955170i	$-0.404328\pi$
$919$	16496.0	0.592114	0.296057	+	0.955170i	$0.404328\pi$
$920$	5292.00	0.189644
$921$	0	0
$922$	−1026.00	−0.0366481
$923$	21888.0	0.780555
$924$	0	0
$925$	36656.0	1.30296
$926$	−13008.0	−0.461630
$927$	0	0
$928$	−13365.0	−0.472767
$929$	14154.0	0.499868	0.249934	−	0.968263i	$-0.419591\pi$
$929$	14154.0	0.499868	0.249934	+	0.968263i	$0.419591\pi$
$930$	0	0
$931$	0	0
$932$	−852.000	−0.0299444
$933$	0	0
$934$	55908.0	1.95864
$935$	−3780.00	−0.132213
$936$	0	0
$937$	3781.00	0.131825	0.0659124	−	0.997825i	$-0.479004\pi$
$937$	3781.00	0.131825	0.0659124	+	0.997825i	$0.479004\pi$
$938$	0	0
$939$	0	0
$940$	90.0000	0.00312285
$941$	−25863.0	−0.895972	−0.447986	−	0.894041i	$-0.647859\pi$
$941$	−25863.0	−0.895972	−0.447986	+	0.894041i	$0.647859\pi$
$942$	0	0
$943$	30240.0	1.04427
$944$	1065.00	0.0367191
$945$	0	0
$946$	1170.00	0.0402114
$947$	42384.0	1.45438	0.727188	−	0.686438i	$-0.240827\pi$
$947$	42384.0	1.45438	0.727188	+	0.686438i	$0.240827\pi$
$948$	0	0
$949$	−23168.0	−0.792482
$950$	−5568.00	−0.190158
$951$	0	0
$952$	0	0
$953$	−10530.0	−0.357923	−0.178961	−	0.983856i	$-0.557274\pi$
$953$	−10530.0	−0.357923	−0.178961	+	0.983856i	$0.557274\pi$
$954$	0	0
$955$	11736.0	0.397663
$956$	−5508.00	−0.186340
$957$	0	0
$958$	45234.0	1.52552
$959$	0	0
$960$	0	0
$961$	34218.0	1.14860
$962$	−60672.0	−2.03341
$963$	0	0
$964$	−791.000	−0.0264278
$965$	−4479.00	−0.149414
$966$	0	0
$967$	−38341.0	−1.27504	−0.637520	−	0.770434i	$-0.720040\pi$
$967$	−38341.0	−1.27504	−0.637520	+	0.770434i	$0.720040\pi$
$968$	23226.0	0.771190
$969$	0	0
$970$	4527.00	0.149849
$971$	1923.00	0.0635551	0.0317776	−	0.999495i	$-0.489883\pi$
$971$	1923.00	0.0635551	0.0317776	+	0.999495i	$0.489883\pi$
$972$	0	0
$973$	0	0
$974$	18663.0	0.613964
$975$	0	0
$976$	−8378.00	−0.274768
$977$	−57090.0	−1.86947	−0.934734	−	0.355347i	$-0.884363\pi$
$977$	−57090.0	−1.86947	−0.934734	+	0.355347i	$0.884363\pi$
$978$	0	0
$979$	13590.0	0.443655
$980$	0	0
$981$	0	0
$982$	22113.0	0.718589
$983$	5484.00	0.177937	0.0889687	−	0.996034i	$-0.471643\pi$
$983$	5484.00	0.177937	0.0889687	+	0.996034i	$0.471643\pi$
$984$	0	0
$985$	−12258.0	−0.396520
$986$	74844.0	2.41736
$987$	0	0
$988$	1024.00	0.0329735
$989$	2184.00	0.0702196
$990$	0	0
$991$	−22465.0	−0.720105	−0.360053	−	0.932932i	$-0.617241\pi$
$991$	−22465.0	−0.720105	−0.360053	+	0.932932i	$0.617241\pi$
$992$	−11385.0	−0.364389
$993$	0	0
$994$	0	0
$995$	−10668.0	−0.339898
$996$	0	0
$997$	−29366.0	−0.932829	−0.466415	−	0.884566i	$-0.654454\pi$
$997$	−29366.0	−0.932829	−0.466415	+	0.884566i	$0.654454\pi$
$998$	12822.0	0.406687
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.k.1.1		1
3.2	odd	2		147.4.a.a.1.1		1
7.2	even	3		441.4.e.c.361.1		2
7.3	odd	6		63.4.e.a.37.1		2
7.4	even	3		441.4.e.c.226.1		2
7.5	odd	6		63.4.e.a.46.1		2
7.6	odd	2		441.4.a.l.1.1		1
12.11	even	2		2352.4.a.bd.1.1		1
21.2	odd	6		147.4.e.h.67.1		2
21.5	even	6		21.4.e.a.4.1	✓	2
21.11	odd	6		147.4.e.h.79.1		2
21.17	even	6		21.4.e.a.16.1	yes	2
21.20	even	2		147.4.a.b.1.1		1
84.47	odd	6		336.4.q.e.193.1		2
84.59	odd	6		336.4.q.e.289.1		2
84.83	odd	2		2352.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.e.a.4.1	✓	2	21.5	even	6
21.4.e.a.16.1	yes	2	21.17	even	6
63.4.e.a.37.1		2	7.3	odd	6
63.4.e.a.46.1		2	7.5	odd	6
147.4.a.a.1.1		1	3.2	odd	2
147.4.a.b.1.1		1	21.20	even	2
147.4.e.h.67.1		2	21.2	odd	6
147.4.e.h.79.1		2	21.11	odd	6
336.4.q.e.193.1		2	84.47	odd	6
336.4.q.e.289.1		2	84.59	odd	6
441.4.a.k.1.1		1	1.1	even	1	trivial
441.4.a.l.1.1		1	7.6	odd	2
441.4.e.c.226.1		2	7.4	even	3
441.4.e.c.361.1		2	7.2	even	3
2352.4.a.i.1.1		1	84.83	odd	2
2352.4.a.bd.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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