LMFDB - Embedded newform 105.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [105,4,Mod(1,105)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(105, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("105.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 105 = 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	105.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:	$6.19520055060$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	105.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} -8.00000 q^{4} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

\(q-3.00000 q^{3} -8.00000 q^{4} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +42.0000 q^{11} +24.0000 q^{12} +20.0000 q^{13} -15.0000 q^{15} +64.0000 q^{16} +66.0000 q^{17} +38.0000 q^{19} -40.0000 q^{20} -21.0000 q^{21} +12.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -56.0000 q^{28} -258.000 q^{29} +146.000 q^{31} -126.000 q^{33} +35.0000 q^{35} -72.0000 q^{36} +434.000 q^{37} -60.0000 q^{39} -282.000 q^{41} +20.0000 q^{43} -336.000 q^{44} +45.0000 q^{45} -72.0000 q^{47} -192.000 q^{48} +49.0000 q^{49} -198.000 q^{51} -160.000 q^{52} +336.000 q^{53} +210.000 q^{55} -114.000 q^{57} -360.000 q^{59} +120.000 q^{60} -682.000 q^{61} +63.0000 q^{63} -512.000 q^{64} +100.000 q^{65} +812.000 q^{67} -528.000 q^{68} -36.0000 q^{69} +810.000 q^{71} -124.000 q^{73} -75.0000 q^{75} -304.000 q^{76} +294.000 q^{77} +1136.00 q^{79} +320.000 q^{80} +81.0000 q^{81} +156.000 q^{83} +168.000 q^{84} +330.000 q^{85} +774.000 q^{87} -1038.00 q^{89} +140.000 q^{91} -96.0000 q^{92} -438.000 q^{93} +190.000 q^{95} +1208.00 q^{97} +378.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−8.00000	−1.00000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	42.0000	1.15123	0.575613	−	0.817723i	$-0.304764\pi$
$11$	42.0000	1.15123	0.575613	+	0.817723i	$0.304764\pi$
$12$	24.0000	0.577350
$13$	20.0000	0.426692	0.213346	−	0.976977i	$-0.431564\pi$
$13$	20.0000	0.426692	0.213346	+	0.976977i	$0.431564\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	64.0000	1.00000
$17$	66.0000	0.941609	0.470804	−	0.882238i	$-0.343964\pi$
$17$	66.0000	0.941609	0.470804	+	0.882238i	$0.343964\pi$
$18$	0	0
$19$	38.0000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	38.0000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−40.0000	−0.447214
$21$	−21.0000	−0.218218
$22$	0	0
$23$	12.0000	0.108790	0.0543951	−	0.998519i	$-0.482677\pi$
$23$	12.0000	0.108790	0.0543951	+	0.998519i	$0.482677\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	−56.0000	−0.377964
$29$	−258.000	−1.65205	−0.826024	−	0.563635i	$-0.809403\pi$
$29$	−258.000	−1.65205	−0.826024	+	0.563635i	$0.809403\pi$
$30$	0	0
$31$	146.000	0.845883	0.422942	−	0.906157i	$-0.360998\pi$
$31$	146.000	0.845883	0.422942	+	0.906157i	$0.360998\pi$
$32$	0	0
$33$	−126.000	−0.664660
$34$	0	0
$35$	35.0000	0.169031
$36$	−72.0000	−0.333333
$37$	434.000	1.92836	0.964178	−	0.265257i	$-0.0854567\pi$
$37$	434.000	1.92836	0.964178	+	0.265257i	$0.0854567\pi$
$38$	0	0
$39$	−60.0000	−0.246351
$40$	0	0
$41$	−282.000	−1.07417	−0.537085	−	0.843528i	$-0.680475\pi$
$41$	−282.000	−1.07417	−0.537085	+	0.843528i	$0.680475\pi$
$42$	0	0
$43$	20.0000	0.0709296	0.0354648	−	0.999371i	$-0.488709\pi$
$43$	20.0000	0.0709296	0.0354648	+	0.999371i	$0.488709\pi$
$44$	−336.000	−1.15123
$45$	45.0000	0.149071
$46$	0	0
$47$	−72.0000	−0.223453	−0.111726	−	0.993739i	$-0.535638\pi$
$47$	−72.0000	−0.223453	−0.111726	+	0.993739i	$0.535638\pi$
$48$	−192.000	−0.577350
$49$	49.0000	0.142857
$50$	0	0
$51$	−198.000	−0.543638
$52$	−160.000	−0.426692
$53$	336.000	0.870814	0.435407	−	0.900234i	$-0.356604\pi$
$53$	336.000	0.870814	0.435407	+	0.900234i	$0.356604\pi$
$54$	0	0
$55$	210.000	0.514844
$56$	0	0
$57$	−114.000	−0.264906
$58$	0	0
$59$	−360.000	−0.794373	−0.397187	−	0.917738i	$-0.630013\pi$
$59$	−360.000	−0.794373	−0.397187	+	0.917738i	$0.630013\pi$
$60$	120.000	0.258199
$61$	−682.000	−1.43149	−0.715747	−	0.698360i	$-0.753914\pi$
$61$	−682.000	−1.43149	−0.715747	+	0.698360i	$0.753914\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	−512.000	−1.00000
$65$	100.000	0.190823
$66$	0	0
$67$	812.000	1.48062	0.740310	−	0.672265i	$-0.234679\pi$
$67$	812.000	1.48062	0.740310	+	0.672265i	$0.234679\pi$
$68$	−528.000	−0.941609
$69$	−36.0000	−0.0628100
$70$	0	0
$71$	810.000	1.35393	0.676967	−	0.736013i	$-0.263294\pi$
$71$	810.000	1.35393	0.676967	+	0.736013i	$0.263294\pi$
$72$	0	0
$73$	−124.000	−0.198810	−0.0994048	−	0.995047i	$-0.531694\pi$
$73$	−124.000	−0.198810	−0.0994048	+	0.995047i	$0.531694\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	−304.000	−0.458831
$77$	294.000	0.435122
$78$	0	0
$79$	1136.00	1.61785	0.808924	−	0.587913i	$-0.200050\pi$
$79$	1136.00	1.61785	0.808924	+	0.587913i	$0.200050\pi$
$80$	320.000	0.447214
$81$	81.0000	0.111111
$82$	0	0
$83$	156.000	0.206304	0.103152	−	0.994666i	$-0.467107\pi$
$83$	156.000	0.206304	0.103152	+	0.994666i	$0.467107\pi$
$84$	168.000	0.218218
$85$	330.000	0.421100
$86$	0	0
$87$	774.000	0.953810
$88$	0	0
$89$	−1038.00	−1.23627	−0.618134	−	0.786073i	$-0.712111\pi$
$89$	−1038.00	−1.23627	−0.618134	+	0.786073i	$0.712111\pi$
$90$	0	0
$91$	140.000	0.161275
$92$	−96.0000	−0.108790
$93$	−438.000	−0.488371
$94$	0	0
$95$	190.000	0.205196
$96$	0	0
$97$	1208.00	1.26447	0.632236	−	0.774776i	$-0.282137\pi$
$97$	1208.00	1.26447	0.632236	+	0.774776i	$0.282137\pi$
$98$	0	0
$99$	378.000	0.383742
$100$	−200.000	−0.200000
$101$	546.000	0.537911	0.268956	−	0.963153i	$-0.413322\pi$
$101$	546.000	0.537911	0.268956	+	0.963153i	$0.413322\pi$
$102$	0	0
$103$	−520.000	−0.497448	−0.248724	−	0.968574i	$-0.580011\pi$
$103$	−520.000	−0.497448	−0.248724	+	0.968574i	$0.580011\pi$
$104$	0	0
$105$	−105.000	−0.0975900
$106$	0	0
$107$	1212.00	1.09503	0.547516	−	0.836795i	$-0.315573\pi$
$107$	1212.00	1.09503	0.547516	+	0.836795i	$0.315573\pi$
$108$	216.000	0.192450
$109$	−1078.00	−0.947281	−0.473641	−	0.880718i	$-0.657060\pi$
$109$	−1078.00	−0.947281	−0.473641	+	0.880718i	$0.657060\pi$
$110$	0	0
$111$	−1302.00	−1.11334
$112$	448.000	0.377964
$113$	−1452.00	−1.20878	−0.604392	−	0.796687i	$-0.706584\pi$
$113$	−1452.00	−1.20878	−0.604392	+	0.796687i	$0.706584\pi$
$114$	0	0
$115$	60.0000	0.0486524
$116$	2064.00	1.65205
$117$	180.000	0.142231
$118$	0	0
$119$	462.000	0.355895
$120$	0	0
$121$	433.000	0.325319
$122$	0	0
$123$	846.000	0.620173
$124$	−1168.00	−0.845883
$125$	125.000	0.0894427
$126$	0	0
$127$	−1312.00	−0.916702	−0.458351	−	0.888771i	$-0.651560\pi$
$127$	−1312.00	−0.916702	−0.458351	+	0.888771i	$0.651560\pi$
$128$	0	0
$129$	−60.0000	−0.0409512
$130$	0	0
$131$	−1356.00	−0.904384	−0.452192	−	0.891921i	$-0.649358\pi$
$131$	−1356.00	−0.904384	−0.452192	+	0.891921i	$0.649358\pi$
$132$	1008.00	0.664660
$133$	266.000	0.173422
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	−984.000	−0.613641	−0.306820	−	0.951767i	$-0.599265\pi$
$137$	−984.000	−0.613641	−0.306820	+	0.951767i	$0.599265\pi$
$138$	0	0
$139$	−394.000	−0.240422	−0.120211	−	0.992748i	$-0.538357\pi$
$139$	−394.000	−0.240422	−0.120211	+	0.992748i	$0.538357\pi$
$140$	−280.000	−0.169031
$141$	216.000	0.129011
$142$	0	0
$143$	840.000	0.491219
$144$	576.000	0.333333
$145$	−1290.00	−0.738818
$146$	0	0
$147$	−147.000	−0.0824786
$148$	−3472.00	−1.92836
$149$	−1014.00	−0.557518	−0.278759	−	0.960361i	$-0.589923\pi$
$149$	−1014.00	−0.557518	−0.278759	+	0.960361i	$0.589923\pi$
$150$	0	0
$151$	−1996.00	−1.07571	−0.537855	−	0.843037i	$-0.680765\pi$
$151$	−1996.00	−1.07571	−0.537855	+	0.843037i	$0.680765\pi$
$152$	0	0
$153$	594.000	0.313870
$154$	0	0
$155$	730.000	0.378290
$156$	480.000	0.246351
$157$	−2392.00	−1.21594	−0.607969	−	0.793960i	$-0.708016\pi$
$157$	−2392.00	−1.21594	−0.607969	+	0.793960i	$0.708016\pi$
$158$	0	0
$159$	−1008.00	−0.502765
$160$	0	0
$161$	84.0000	0.0411188
$162$	0	0
$163$	2036.00	0.978355	0.489177	−	0.872184i	$-0.337297\pi$
$163$	2036.00	0.978355	0.489177	+	0.872184i	$0.337297\pi$
$164$	2256.00	1.07417
$165$	−630.000	−0.297245
$166$	0	0
$167$	−3936.00	−1.82381	−0.911907	−	0.410398i	$-0.865390\pi$
$167$	−3936.00	−1.82381	−0.911907	+	0.410398i	$0.865390\pi$
$168$	0	0
$169$	−1797.00	−0.817934
$170$	0	0
$171$	342.000	0.152944
$172$	−160.000	−0.0709296
$173$	378.000	0.166120	0.0830601	−	0.996545i	$-0.473531\pi$
$173$	378.000	0.166120	0.0830601	+	0.996545i	$0.473531\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	2688.00	1.15123
$177$	1080.00	0.458631
$178$	0	0
$179$	−222.000	−0.0926987	−0.0463493	−	0.998925i	$-0.514759\pi$
$179$	−222.000	−0.0926987	−0.0463493	+	0.998925i	$0.514759\pi$
$180$	−360.000	−0.149071
$181$	−2590.00	−1.06361	−0.531804	−	0.846867i	$-0.678486\pi$
$181$	−2590.00	−1.06361	−0.531804	+	0.846867i	$0.678486\pi$
$182$	0	0
$183$	2046.00	0.826474
$184$	0	0
$185$	2170.00	0.862387
$186$	0	0
$187$	2772.00	1.08400
$188$	576.000	0.223453
$189$	−189.000	−0.0727393
$190$	0	0
$191$	2214.00	0.838740	0.419370	−	0.907815i	$-0.362251\pi$
$191$	2214.00	0.838740	0.419370	+	0.907815i	$0.362251\pi$
$192$	1536.00	0.577350
$193$	4178.00	1.55823	0.779117	−	0.626879i	$-0.215668\pi$
$193$	4178.00	1.55823	0.779117	+	0.626879i	$0.215668\pi$
$194$	0	0
$195$	−300.000	−0.110172
$196$	−392.000	−0.142857
$197$	−3060.00	−1.10668	−0.553340	−	0.832955i	$-0.686647\pi$
$197$	−3060.00	−1.10668	−0.553340	+	0.832955i	$0.686647\pi$
$198$	0	0
$199$	2666.00	0.949687	0.474844	−	0.880070i	$-0.342505\pi$
$199$	2666.00	0.949687	0.474844	+	0.880070i	$0.342505\pi$
$200$	0	0
$201$	−2436.00	−0.854837
$202$	0	0
$203$	−1806.00	−0.624416
$204$	1584.00	0.543638
$205$	−1410.00	−0.480384
$206$	0	0
$207$	108.000	0.0362634
$208$	1280.00	0.426692
$209$	1596.00	0.528218
$210$	0	0
$211$	−1348.00	−0.439811	−0.219906	−	0.975521i	$-0.570575\pi$
$211$	−1348.00	−0.439811	−0.219906	+	0.975521i	$0.570575\pi$
$212$	−2688.00	−0.870814
$213$	−2430.00	−0.781694
$214$	0	0
$215$	100.000	0.0317207
$216$	0	0
$217$	1022.00	0.319714
$218$	0	0
$219$	372.000	0.114783
$220$	−1680.00	−0.514844
$221$	1320.00	0.401777
$222$	0	0
$223$	3188.00	0.957329	0.478664	−	0.877998i	$-0.341121\pi$
$223$	3188.00	0.957329	0.478664	+	0.877998i	$0.341121\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−3396.00	−0.992953	−0.496477	−	0.868050i	$-0.665373\pi$
$227$	−3396.00	−0.992953	−0.496477	+	0.868050i	$0.665373\pi$
$228$	912.000	0.264906
$229$	5294.00	1.52767	0.763837	−	0.645409i	$-0.223313\pi$
$229$	5294.00	1.52767	0.763837	+	0.645409i	$0.223313\pi$
$230$	0	0
$231$	−882.000	−0.251218
$232$	0	0
$233$	852.000	0.239555	0.119778	−	0.992801i	$-0.461782\pi$
$233$	852.000	0.239555	0.119778	+	0.992801i	$0.461782\pi$
$234$	0	0
$235$	−360.000	−0.0999311
$236$	2880.00	0.794373
$237$	−3408.00	−0.934065
$238$	0	0
$239$	4866.00	1.31697	0.658484	−	0.752595i	$-0.271198\pi$
$239$	4866.00	1.31697	0.658484	+	0.752595i	$0.271198\pi$
$240$	−960.000	−0.258199
$241$	−2050.00	−0.547934	−0.273967	−	0.961739i	$-0.588336\pi$
$241$	−2050.00	−0.547934	−0.273967	+	0.961739i	$0.588336\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	5456.00	1.43149
$245$	245.000	0.0638877
$246$	0	0
$247$	760.000	0.195780
$248$	0	0
$249$	−468.000	−0.119110
$250$	0	0
$251$	−1152.00	−0.289696	−0.144848	−	0.989454i	$-0.546269\pi$
$251$	−1152.00	−0.289696	−0.144848	+	0.989454i	$0.546269\pi$
$252$	−504.000	−0.125988
$253$	504.000	0.125242
$254$	0	0
$255$	−990.000	−0.243122
$256$	4096.00	1.00000
$257$	−6450.00	−1.56553	−0.782763	−	0.622321i	$-0.786190\pi$
$257$	−6450.00	−1.56553	−0.782763	+	0.622321i	$0.786190\pi$
$258$	0	0
$259$	3038.00	0.728850
$260$	−800.000	−0.190823
$261$	−2322.00	−0.550683
$262$	0	0
$263$	1968.00	0.461415	0.230707	−	0.973023i	$-0.425896\pi$
$263$	1968.00	0.461415	0.230707	+	0.973023i	$0.425896\pi$
$264$	0	0
$265$	1680.00	0.389440
$266$	0	0
$267$	3114.00	0.713759
$268$	−6496.00	−1.48062
$269$	−3894.00	−0.882607	−0.441304	−	0.897358i	$-0.645484\pi$
$269$	−3894.00	−0.882607	−0.441304	+	0.897358i	$0.645484\pi$
$270$	0	0
$271$	7094.00	1.59015	0.795073	−	0.606513i	$-0.207432\pi$
$271$	7094.00	1.59015	0.795073	+	0.606513i	$0.207432\pi$
$272$	4224.00	0.941609
$273$	−420.000	−0.0931119
$274$	0	0
$275$	1050.00	0.230245
$276$	288.000	0.0628100
$277$	−3310.00	−0.717973	−0.358987	−	0.933343i	$-0.616878\pi$
$277$	−3310.00	−0.717973	−0.358987	+	0.933343i	$0.616878\pi$
$278$	0	0
$279$	1314.00	0.281961
$280$	0	0
$281$	7158.00	1.51961	0.759805	−	0.650151i	$-0.225294\pi$
$281$	7158.00	1.51961	0.759805	+	0.650151i	$0.225294\pi$
$282$	0	0
$283$	−5164.00	−1.08469	−0.542346	−	0.840155i	$-0.682464\pi$
$283$	−5164.00	−1.08469	−0.542346	+	0.840155i	$0.682464\pi$
$284$	−6480.00	−1.35393
$285$	−570.000	−0.118470
$286$	0	0
$287$	−1974.00	−0.405998
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	−3624.00	−0.730043
$292$	992.000	0.198810
$293$	−8598.00	−1.71434	−0.857168	−	0.515037i	$-0.827778\pi$
$293$	−8598.00	−1.71434	−0.857168	+	0.515037i	$0.827778\pi$
$294$	0	0
$295$	−1800.00	−0.355254
$296$	0	0
$297$	−1134.00	−0.221553
$298$	0	0
$299$	240.000	0.0464199
$300$	600.000	0.115470
$301$	140.000	0.0268089
$302$	0	0
$303$	−1638.00	−0.310563
$304$	2432.00	0.458831
$305$	−3410.00	−0.640184
$306$	0	0
$307$	−448.000	−0.0832857	−0.0416429	−	0.999133i	$-0.513259\pi$
$307$	−448.000	−0.0832857	−0.0416429	+	0.999133i	$0.513259\pi$
$308$	−2352.00	−0.435122
$309$	1560.00	0.287202
$310$	0	0
$311$	−5832.00	−1.06335	−0.531676	−	0.846948i	$-0.678438\pi$
$311$	−5832.00	−1.06335	−0.531676	+	0.846948i	$0.678438\pi$
$312$	0	0
$313$	9848.00	1.77841	0.889204	−	0.457510i	$-0.151259\pi$
$313$	9848.00	1.77841	0.889204	+	0.457510i	$0.151259\pi$
$314$	0	0
$315$	315.000	0.0563436
$316$	−9088.00	−1.61785
$317$	−5616.00	−0.995035	−0.497517	−	0.867454i	$-0.665755\pi$
$317$	−5616.00	−0.995035	−0.497517	+	0.867454i	$0.665755\pi$
$318$	0	0
$319$	−10836.0	−1.90188
$320$	−2560.00	−0.447214
$321$	−3636.00	−0.632217
$322$	0	0
$323$	2508.00	0.432040
$324$	−648.000	−0.111111
$325$	500.000	0.0853385
$326$	0	0
$327$	3234.00	0.546913
$328$	0	0
$329$	−504.000	−0.0844572
$330$	0	0
$331$	452.000	0.0750579	0.0375290	−	0.999296i	$-0.488051\pi$
$331$	452.000	0.0750579	0.0375290	+	0.999296i	$0.488051\pi$
$332$	−1248.00	−0.206304
$333$	3906.00	0.642785
$334$	0	0
$335$	4060.00	0.662154
$336$	−1344.00	−0.218218
$337$	−2302.00	−0.372101	−0.186050	−	0.982540i	$-0.559569\pi$
$337$	−2302.00	−0.372101	−0.186050	+	0.982540i	$0.559569\pi$
$338$	0	0
$339$	4356.00	0.697892
$340$	−2640.00	−0.421100
$341$	6132.00	0.973802
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−180.000	−0.0280895
$346$	0	0
$347$	1584.00	0.245054	0.122527	−	0.992465i	$-0.460900\pi$
$347$	1584.00	0.245054	0.122527	+	0.992465i	$0.460900\pi$
$348$	−6192.00	−0.953810
$349$	8174.00	1.25371	0.626854	−	0.779137i	$-0.284342\pi$
$349$	8174.00	1.25371	0.626854	+	0.779137i	$0.284342\pi$
$350$	0	0
$351$	−540.000	−0.0821170
$352$	0	0
$353$	8610.00	1.29820	0.649099	−	0.760704i	$-0.275146\pi$
$353$	8610.00	1.29820	0.649099	+	0.760704i	$0.275146\pi$
$354$	0	0
$355$	4050.00	0.605498
$356$	8304.00	1.23627
$357$	−1386.00	−0.205476
$358$	0	0
$359$	−2154.00	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−2154.00	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	−5415.00	−0.789474
$362$	0	0
$363$	−1299.00	−0.187823
$364$	−1120.00	−0.161275
$365$	−620.000	−0.0889104
$366$	0	0
$367$	6644.00	0.944997	0.472499	−	0.881331i	$-0.343352\pi$
$367$	6644.00	0.944997	0.472499	+	0.881331i	$0.343352\pi$
$368$	768.000	0.108790
$369$	−2538.00	−0.358057
$370$	0	0
$371$	2352.00	0.329137
$372$	3504.00	0.488371
$373$	7958.00	1.10469	0.552345	−	0.833615i	$-0.313733\pi$
$373$	7958.00	1.10469	0.552345	+	0.833615i	$0.313733\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−5160.00	−0.704917
$378$	0	0
$379$	3440.00	0.466229	0.233115	−	0.972449i	$-0.425108\pi$
$379$	3440.00	0.466229	0.233115	+	0.972449i	$0.425108\pi$
$380$	−1520.00	−0.205196
$381$	3936.00	0.529258
$382$	0	0
$383$	12936.0	1.72585	0.862923	−	0.505336i	$-0.168631\pi$
$383$	12936.0	1.72585	0.862923	+	0.505336i	$0.168631\pi$
$384$	0	0
$385$	1470.00	0.194593
$386$	0	0
$387$	180.000	0.0236432
$388$	−9664.00	−1.26447
$389$	−14862.0	−1.93710	−0.968552	−	0.248812i	$-0.919960\pi$
$389$	−14862.0	−1.93710	−0.968552	+	0.248812i	$0.919960\pi$
$390$	0	0
$391$	792.000	0.102438
$392$	0	0
$393$	4068.00	0.522146
$394$	0	0
$395$	5680.00	0.723524
$396$	−3024.00	−0.383742
$397$	10460.0	1.32235	0.661174	−	0.750232i	$-0.270058\pi$
$397$	10460.0	1.32235	0.661174	+	0.750232i	$0.270058\pi$
$398$	0	0
$399$	−798.000	−0.100125
$400$	1600.00	0.200000
$401$	−9150.00	−1.13947	−0.569737	−	0.821827i	$-0.692955\pi$
$401$	−9150.00	−1.13947	−0.569737	+	0.821827i	$0.692955\pi$
$402$	0	0
$403$	2920.00	0.360932
$404$	−4368.00	−0.537911
$405$	405.000	0.0496904
$406$	0	0
$407$	18228.0	2.21997
$408$	0	0
$409$	−4894.00	−0.591669	−0.295835	−	0.955239i	$-0.595598\pi$
$409$	−4894.00	−0.591669	−0.295835	+	0.955239i	$0.595598\pi$
$410$	0	0
$411$	2952.00	0.354286
$412$	4160.00	0.497448
$413$	−2520.00	−0.300245
$414$	0	0
$415$	780.000	0.0922619
$416$	0	0
$417$	1182.00	0.138808
$418$	0	0
$419$	−1668.00	−0.194480	−0.0972400	−	0.995261i	$-0.531001\pi$
$419$	−1668.00	−0.194480	−0.0972400	+	0.995261i	$0.531001\pi$
$420$	840.000	0.0975900
$421$	−12418.0	−1.43757	−0.718784	−	0.695233i	$-0.755301\pi$
$421$	−12418.0	−1.43757	−0.718784	+	0.695233i	$0.755301\pi$
$422$	0	0
$423$	−648.000	−0.0744843
$424$	0	0
$425$	1650.00	0.188322
$426$	0	0
$427$	−4774.00	−0.541054
$428$	−9696.00	−1.09503
$429$	−2520.00	−0.283605
$430$	0	0
$431$	15186.0	1.69718	0.848589	−	0.529052i	$-0.177452\pi$
$431$	15186.0	1.69718	0.848589	+	0.529052i	$0.177452\pi$
$432$	−1728.00	−0.192450
$433$	−5704.00	−0.633064	−0.316532	−	0.948582i	$-0.602518\pi$
$433$	−5704.00	−0.633064	−0.316532	+	0.948582i	$0.602518\pi$
$434$	0	0
$435$	3870.00	0.426557
$436$	8624.00	0.947281
$437$	456.000	0.0499163
$438$	0	0
$439$	−17206.0	−1.87061	−0.935305	−	0.353843i	$-0.884875\pi$
$439$	−17206.0	−1.87061	−0.935305	+	0.353843i	$0.884875\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	3456.00	0.370654	0.185327	−	0.982677i	$-0.440666\pi$
$443$	3456.00	0.370654	0.185327	+	0.982677i	$0.440666\pi$
$444$	10416.0	1.11334
$445$	−5190.00	−0.552875
$446$	0	0
$447$	3042.00	0.321883
$448$	−3584.00	−0.377964
$449$	16074.0	1.68949	0.844743	−	0.535173i	$-0.179753\pi$
$449$	16074.0	1.68949	0.844743	+	0.535173i	$0.179753\pi$
$450$	0	0
$451$	−11844.0	−1.23661
$452$	11616.0	1.20878
$453$	5988.00	0.621061
$454$	0	0
$455$	700.000	0.0721242
$456$	0	0
$457$	7526.00	0.770353	0.385177	−	0.922843i	$-0.374141\pi$
$457$	7526.00	0.770353	0.385177	+	0.922843i	$0.374141\pi$
$458$	0	0
$459$	−1782.00	−0.181213
$460$	−480.000	−0.0486524
$461$	−2274.00	−0.229741	−0.114871	−	0.993380i	$-0.536645\pi$
$461$	−2274.00	−0.229741	−0.114871	+	0.993380i	$0.536645\pi$
$462$	0	0
$463$	−10024.0	−1.00617	−0.503083	−	0.864238i	$-0.667801\pi$
$463$	−10024.0	−1.00617	−0.503083	+	0.864238i	$0.667801\pi$
$464$	−16512.0	−1.65205
$465$	−2190.00	−0.218406
$466$	0	0
$467$	−2460.00	−0.243759	−0.121879	−	0.992545i	$-0.538892\pi$
$467$	−2460.00	−0.243759	−0.121879	+	0.992545i	$0.538892\pi$
$468$	−1440.00	−0.142231
$469$	5684.00	0.559622
$470$	0	0
$471$	7176.00	0.702023
$472$	0	0
$473$	840.000	0.0816559
$474$	0	0
$475$	950.000	0.0917663
$476$	−3696.00	−0.355895
$477$	3024.00	0.290271
$478$	0	0
$479$	19320.0	1.84291	0.921454	−	0.388486i	$-0.127002\pi$
$479$	19320.0	1.84291	0.921454	+	0.388486i	$0.127002\pi$
$480$	0	0
$481$	8680.00	0.822815
$482$	0	0
$483$	−252.000	−0.0237400
$484$	−3464.00	−0.325319
$485$	6040.00	0.565489
$486$	0	0
$487$	−12544.0	−1.16719	−0.583596	−	0.812044i	$-0.698355\pi$
$487$	−12544.0	−1.16719	−0.583596	+	0.812044i	$0.698355\pi$
$488$	0	0
$489$	−6108.00	−0.564853
$490$	0	0
$491$	−15510.0	−1.42557	−0.712787	−	0.701381i	$-0.752567\pi$
$491$	−15510.0	−1.42557	−0.712787	+	0.701381i	$0.752567\pi$
$492$	−6768.00	−0.620173
$493$	−17028.0	−1.55558
$494$	0	0
$495$	1890.00	0.171615
$496$	9344.00	0.845883
$497$	5670.00	0.511739
$498$	0	0
$499$	−14344.0	−1.28682	−0.643412	−	0.765520i	$-0.722482\pi$
$499$	−14344.0	−1.28682	−0.643412	+	0.765520i	$0.722482\pi$
$500$	−1000.00	−0.0894427
$501$	11808.0	1.05298
$502$	0	0
$503$	−21384.0	−1.89556	−0.947779	−	0.318929i	$-0.896677\pi$
$503$	−21384.0	−1.89556	−0.947779	+	0.318929i	$0.896677\pi$
$504$	0	0
$505$	2730.00	0.240561
$506$	0	0
$507$	5391.00	0.472234
$508$	10496.0	0.916702
$509$	−7134.00	−0.621236	−0.310618	−	0.950535i	$-0.600536\pi$
$509$	−7134.00	−0.621236	−0.310618	+	0.950535i	$0.600536\pi$
$510$	0	0
$511$	−868.000	−0.0751430
$512$	0	0
$513$	−1026.00	−0.0883022
$514$	0	0
$515$	−2600.00	−0.222465
$516$	480.000	0.0409512
$517$	−3024.00	−0.257244
$518$	0	0
$519$	−1134.00	−0.0959096
$520$	0	0
$521$	−19122.0	−1.60797	−0.803983	−	0.594653i	$-0.797290\pi$
$521$	−19122.0	−1.60797	−0.803983	+	0.594653i	$0.797290\pi$
$522$	0	0
$523$	−15640.0	−1.30763	−0.653814	−	0.756655i	$-0.726832\pi$
$523$	−15640.0	−1.30763	−0.653814	+	0.756655i	$0.726832\pi$
$524$	10848.0	0.904384
$525$	−525.000	−0.0436436
$526$	0	0
$527$	9636.00	0.796491
$528$	−8064.00	−0.664660
$529$	−12023.0	−0.988165
$530$	0	0
$531$	−3240.00	−0.264791
$532$	−2128.00	−0.173422
$533$	−5640.00	−0.458341
$534$	0	0
$535$	6060.00	0.489713
$536$	0	0
$537$	666.000	0.0535196
$538$	0	0
$539$	2058.00	0.164461
$540$	1080.00	0.0860663
$541$	2846.00	0.226172	0.113086	−	0.993585i	$-0.463926\pi$
$541$	2846.00	0.226172	0.113086	+	0.993585i	$0.463926\pi$
$542$	0	0
$543$	7770.00	0.614075
$544$	0	0
$545$	−5390.00	−0.423637
$546$	0	0
$547$	−4444.00	−0.347371	−0.173685	−	0.984801i	$-0.555568\pi$
$547$	−4444.00	−0.347371	−0.173685	+	0.984801i	$0.555568\pi$
$548$	7872.00	0.613641
$549$	−6138.00	−0.477165
$550$	0	0
$551$	−9804.00	−0.758012
$552$	0	0
$553$	7952.00	0.611489
$554$	0	0
$555$	−6510.00	−0.497899
$556$	3152.00	0.240422
$557$	18552.0	1.41126	0.705631	−	0.708579i	$-0.250663\pi$
$557$	18552.0	1.41126	0.705631	+	0.708579i	$0.250663\pi$
$558$	0	0
$559$	400.000	0.0302651
$560$	2240.00	0.169031
$561$	−8316.00	−0.625850
$562$	0	0
$563$	−16452.0	−1.23156	−0.615781	−	0.787918i	$-0.711159\pi$
$563$	−16452.0	−1.23156	−0.615781	+	0.787918i	$0.711159\pi$
$564$	−1728.00	−0.129011
$565$	−7260.00	−0.540585
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	7722.00	0.568933	0.284467	−	0.958686i	$-0.408183\pi$
$569$	7722.00	0.568933	0.284467	+	0.958686i	$0.408183\pi$
$570$	0	0
$571$	2576.00	0.188796	0.0943978	−	0.995535i	$-0.469907\pi$
$571$	2576.00	0.188796	0.0943978	+	0.995535i	$0.469907\pi$
$572$	−6720.00	−0.491219
$573$	−6642.00	−0.484247
$574$	0	0
$575$	300.000	0.0217580
$576$	−4608.00	−0.333333
$577$	−2464.00	−0.177778	−0.0888888	−	0.996042i	$-0.528332\pi$
$577$	−2464.00	−0.177778	−0.0888888	+	0.996042i	$0.528332\pi$
$578$	0	0
$579$	−12534.0	−0.899646
$580$	10320.0	0.738818
$581$	1092.00	0.0779755
$582$	0	0
$583$	14112.0	1.00250
$584$	0	0
$585$	900.000	0.0636076
$586$	0	0
$587$	1452.00	0.102096	0.0510481	−	0.998696i	$-0.483744\pi$
$587$	1452.00	0.102096	0.0510481	+	0.998696i	$0.483744\pi$
$588$	1176.00	0.0824786
$589$	5548.00	0.388118
$590$	0	0
$591$	9180.00	0.638942
$592$	27776.0	1.92836
$593$	10698.0	0.740833	0.370417	−	0.928866i	$-0.379215\pi$
$593$	10698.0	0.740833	0.370417	+	0.928866i	$0.379215\pi$
$594$	0	0
$595$	2310.00	0.159161
$596$	8112.00	0.557518
$597$	−7998.00	−0.548302
$598$	0	0
$599$	−8730.00	−0.595489	−0.297745	−	0.954646i	$-0.596234\pi$
$599$	−8730.00	−0.595489	−0.297745	+	0.954646i	$0.596234\pi$
$600$	0	0
$601$	1910.00	0.129635	0.0648174	−	0.997897i	$-0.479354\pi$
$601$	1910.00	0.129635	0.0648174	+	0.997897i	$0.479354\pi$
$602$	0	0
$603$	7308.00	0.493540
$604$	15968.0	1.07571
$605$	2165.00	0.145487
$606$	0	0
$607$	−5596.00	−0.374192	−0.187096	−	0.982342i	$-0.559908\pi$
$607$	−5596.00	−0.374192	−0.187096	+	0.982342i	$0.559908\pi$
$608$	0	0
$609$	5418.00	0.360506
$610$	0	0
$611$	−1440.00	−0.0953456
$612$	−4752.00	−0.313870
$613$	28586.0	1.88349	0.941744	−	0.336332i	$-0.109186\pi$
$613$	28586.0	1.88349	0.941744	+	0.336332i	$0.109186\pi$
$614$	0	0
$615$	4230.00	0.277350
$616$	0	0
$617$	−19236.0	−1.25513	−0.627563	−	0.778566i	$-0.715947\pi$
$617$	−19236.0	−1.25513	−0.627563	+	0.778566i	$0.715947\pi$
$618$	0	0
$619$	6734.00	0.437257	0.218629	−	0.975808i	$-0.429842\pi$
$619$	6734.00	0.437257	0.218629	+	0.975808i	$0.429842\pi$
$620$	−5840.00	−0.378290
$621$	−324.000	−0.0209367
$622$	0	0
$623$	−7266.00	−0.467265
$624$	−3840.00	−0.246351
$625$	625.000	0.0400000
$626$	0	0
$627$	−4788.00	−0.304967
$628$	19136.0	1.21594
$629$	28644.0	1.81576
$630$	0	0
$631$	7184.00	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$631$	7184.00	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$632$	0	0
$633$	4044.00	0.253925
$634$	0	0
$635$	−6560.00	−0.409962
$636$	8064.00	0.502765
$637$	980.000	0.0609561
$638$	0	0
$639$	7290.00	0.451311
$640$	0	0
$641$	510.000	0.0314256	0.0157128	−	0.999877i	$-0.494998\pi$
$641$	510.000	0.0314256	0.0157128	+	0.999877i	$0.494998\pi$
$642$	0	0
$643$	−20752.0	−1.27275	−0.636376	−	0.771379i	$-0.719567\pi$
$643$	−20752.0	−1.27275	−0.636376	+	0.771379i	$0.719567\pi$
$644$	−672.000	−0.0411188
$645$	−300.000	−0.0183139
$646$	0	0
$647$	21072.0	1.28041	0.640205	−	0.768204i	$-0.278849\pi$
$647$	21072.0	1.28041	0.640205	+	0.768204i	$0.278849\pi$
$648$	0	0
$649$	−15120.0	−0.914502
$650$	0	0
$651$	−3066.00	−0.184587
$652$	−16288.0	−0.978355
$653$	2892.00	0.173312	0.0866560	−	0.996238i	$-0.472382\pi$
$653$	2892.00	0.173312	0.0866560	+	0.996238i	$0.472382\pi$
$654$	0	0
$655$	−6780.00	−0.404453
$656$	−18048.0	−1.07417
$657$	−1116.00	−0.0662699
$658$	0	0
$659$	750.000	0.0443336	0.0221668	−	0.999754i	$-0.492944\pi$
$659$	750.000	0.0443336	0.0221668	+	0.999754i	$0.492944\pi$
$660$	5040.00	0.297245
$661$	30062.0	1.76895	0.884475	−	0.466587i	$-0.154517\pi$
$661$	30062.0	1.76895	0.884475	+	0.466587i	$0.154517\pi$
$662$	0	0
$663$	−3960.00	−0.231966
$664$	0	0
$665$	1330.00	0.0775567
$666$	0	0
$667$	−3096.00	−0.179727
$668$	31488.0	1.82381
$669$	−9564.00	−0.552714
$670$	0	0
$671$	−28644.0	−1.64797
$672$	0	0
$673$	15446.0	0.884695	0.442347	−	0.896844i	$-0.354146\pi$
$673$	15446.0	0.884695	0.442347	+	0.896844i	$0.354146\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	14376.0	0.817934
$677$	−25110.0	−1.42549	−0.712744	−	0.701424i	$-0.752548\pi$
$677$	−25110.0	−1.42549	−0.712744	+	0.701424i	$0.752548\pi$
$678$	0	0
$679$	8456.00	0.477926
$680$	0	0
$681$	10188.0	0.573282
$682$	0	0
$683$	7968.00	0.446394	0.223197	−	0.974773i	$-0.428351\pi$
$683$	7968.00	0.446394	0.223197	+	0.974773i	$0.428351\pi$
$684$	−2736.00	−0.152944
$685$	−4920.00	−0.274429
$686$	0	0
$687$	−15882.0	−0.882003
$688$	1280.00	0.0709296
$689$	6720.00	0.371570
$690$	0	0
$691$	−14398.0	−0.792657	−0.396328	−	0.918109i	$-0.629716\pi$
$691$	−14398.0	−0.792657	−0.396328	+	0.918109i	$0.629716\pi$
$692$	−3024.00	−0.166120
$693$	2646.00	0.145041
$694$	0	0
$695$	−1970.00	−0.107520
$696$	0	0
$697$	−18612.0	−1.01145
$698$	0	0
$699$	−2556.00	−0.138307
$700$	−1400.00	−0.0755929
$701$	−9234.00	−0.497523	−0.248761	−	0.968565i	$-0.580023\pi$
$701$	−9234.00	−0.497523	−0.248761	+	0.968565i	$0.580023\pi$
$702$	0	0
$703$	16492.0	0.884790
$704$	−21504.0	−1.15123
$705$	1080.00	0.0576953
$706$	0	0
$707$	3822.00	0.203311
$708$	−8640.00	−0.458631
$709$	8030.00	0.425350	0.212675	−	0.977123i	$-0.431782\pi$
$709$	8030.00	0.425350	0.212675	+	0.977123i	$0.431782\pi$
$710$	0	0
$711$	10224.0	0.539283
$712$	0	0
$713$	1752.00	0.0920237
$714$	0	0
$715$	4200.00	0.219680
$716$	1776.00	0.0926987
$717$	−14598.0	−0.760352
$718$	0	0
$719$	27060.0	1.40357	0.701786	−	0.712388i	$-0.252386\pi$
$719$	27060.0	1.40357	0.701786	+	0.712388i	$0.252386\pi$
$720$	2880.00	0.149071
$721$	−3640.00	−0.188018
$722$	0	0
$723$	6150.00	0.316350
$724$	20720.0	1.06361
$725$	−6450.00	−0.330410
$726$	0	0
$727$	−3724.00	−0.189980	−0.0949900	−	0.995478i	$-0.530282\pi$
$727$	−3724.00	−0.189980	−0.0949900	+	0.995478i	$0.530282\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	1320.00	0.0667879
$732$	−16368.0	−0.826474
$733$	−5668.00	−0.285610	−0.142805	−	0.989751i	$-0.545612\pi$
$733$	−5668.00	−0.285610	−0.142805	+	0.989751i	$0.545612\pi$
$734$	0	0
$735$	−735.000	−0.0368856
$736$	0	0
$737$	34104.0	1.70453
$738$	0	0
$739$	−16072.0	−0.800024	−0.400012	−	0.916510i	$-0.630994\pi$
$739$	−16072.0	−0.800024	−0.400012	+	0.916510i	$0.630994\pi$
$740$	−17360.0	−0.862387
$741$	−2280.00	−0.113034
$742$	0	0
$743$	−8256.00	−0.407649	−0.203825	−	0.979007i	$-0.565337\pi$
$743$	−8256.00	−0.407649	−0.203825	+	0.979007i	$0.565337\pi$
$744$	0	0
$745$	−5070.00	−0.249329
$746$	0	0
$747$	1404.00	0.0687680
$748$	−22176.0	−1.08400
$749$	8484.00	0.413883
$750$	0	0
$751$	−6352.00	−0.308639	−0.154319	−	0.988021i	$-0.549318\pi$
$751$	−6352.00	−0.308639	−0.154319	+	0.988021i	$0.549318\pi$
$752$	−4608.00	−0.223453
$753$	3456.00	0.167256
$754$	0	0
$755$	−9980.00	−0.481072
$756$	1512.00	0.0727393
$757$	11558.0	0.554931	0.277465	−	0.960736i	$-0.410506\pi$
$757$	11558.0	0.554931	0.277465	+	0.960736i	$0.410506\pi$
$758$	0	0
$759$	−1512.00	−0.0723085
$760$	0	0
$761$	7770.00	0.370121	0.185061	−	0.982727i	$-0.440752\pi$
$761$	7770.00	0.370121	0.185061	+	0.982727i	$0.440752\pi$
$762$	0	0
$763$	−7546.00	−0.358039
$764$	−17712.0	−0.838740
$765$	2970.00	0.140367
$766$	0	0
$767$	−7200.00	−0.338953
$768$	−12288.0	−0.577350
$769$	22646.0	1.06194	0.530972	−	0.847389i	$-0.321827\pi$
$769$	22646.0	1.06194	0.530972	+	0.847389i	$0.321827\pi$
$770$	0	0
$771$	19350.0	0.903856
$772$	−33424.0	−1.55823
$773$	−35502.0	−1.65190	−0.825950	−	0.563744i	$-0.809361\pi$
$773$	−35502.0	−1.65190	−0.825950	+	0.563744i	$0.809361\pi$
$774$	0	0
$775$	3650.00	0.169177
$776$	0	0
$777$	−9114.00	−0.420802
$778$	0	0
$779$	−10716.0	−0.492863
$780$	2400.00	0.110172
$781$	34020.0	1.55868
$782$	0	0
$783$	6966.00	0.317937
$784$	3136.00	0.142857
$785$	−11960.0	−0.543784
$786$	0	0
$787$	−17080.0	−0.773617	−0.386808	−	0.922160i	$-0.626422\pi$
$787$	−17080.0	−0.773617	−0.386808	+	0.922160i	$0.626422\pi$
$788$	24480.0	1.10668
$789$	−5904.00	−0.266398
$790$	0	0
$791$	−10164.0	−0.456878
$792$	0	0
$793$	−13640.0	−0.610808
$794$	0	0
$795$	−5040.00	−0.224843
$796$	−21328.0	−0.949687
$797$	5730.00	0.254664	0.127332	−	0.991860i	$-0.459359\pi$
$797$	5730.00	0.254664	0.127332	+	0.991860i	$0.459359\pi$
$798$	0	0
$799$	−4752.00	−0.210405
$800$	0	0
$801$	−9342.00	−0.412089
$802$	0	0
$803$	−5208.00	−0.228875
$804$	19488.0	0.854837
$805$	420.000	0.0183889
$806$	0	0
$807$	11682.0	0.509574
$808$	0	0
$809$	2550.00	0.110820	0.0554099	−	0.998464i	$-0.482353\pi$
$809$	2550.00	0.110820	0.0554099	+	0.998464i	$0.482353\pi$
$810$	0	0
$811$	−27538.0	−1.19234	−0.596171	−	0.802857i	$-0.703312\pi$
$811$	−27538.0	−1.19234	−0.596171	+	0.802857i	$0.703312\pi$
$812$	14448.0	0.624416
$813$	−21282.0	−0.918072
$814$	0	0
$815$	10180.0	0.437534
$816$	−12672.0	−0.543638
$817$	760.000	0.0325447
$818$	0	0
$819$	1260.00	0.0537582
$820$	11280.0	0.480384
$821$	−19242.0	−0.817966	−0.408983	−	0.912542i	$-0.634117\pi$
$821$	−19242.0	−0.817966	−0.408983	+	0.912542i	$0.634117\pi$
$822$	0	0
$823$	−11752.0	−0.497751	−0.248875	−	0.968536i	$-0.580061\pi$
$823$	−11752.0	−0.497751	−0.248875	+	0.968536i	$0.580061\pi$
$824$	0	0
$825$	−3150.00	−0.132932
$826$	0	0
$827$	−28692.0	−1.20643	−0.603216	−	0.797578i	$-0.706114\pi$
$827$	−28692.0	−1.20643	−0.603216	+	0.797578i	$0.706114\pi$
$828$	−864.000	−0.0362634
$829$	28442.0	1.19159	0.595797	−	0.803135i	$-0.296836\pi$
$829$	28442.0	1.19159	0.595797	+	0.803135i	$0.296836\pi$
$830$	0	0
$831$	9930.00	0.414522
$832$	−10240.0	−0.426692
$833$	3234.00	0.134516
$834$	0	0
$835$	−19680.0	−0.815634
$836$	−12768.0	−0.528218
$837$	−3942.00	−0.162790
$838$	0	0
$839$	20172.0	0.830053	0.415027	−	0.909809i	$-0.363772\pi$
$839$	20172.0	0.830053	0.415027	+	0.909809i	$0.363772\pi$
$840$	0	0
$841$	42175.0	1.72926
$842$	0	0
$843$	−21474.0	−0.877347
$844$	10784.0	0.439811
$845$	−8985.00	−0.365791
$846$	0	0
$847$	3031.00	0.122959
$848$	21504.0	0.870814
$849$	15492.0	0.626247
$850$	0	0
$851$	5208.00	0.209786
$852$	19440.0	0.781694
$853$	19820.0	0.795573	0.397787	−	0.917478i	$-0.369778\pi$
$853$	19820.0	0.795573	0.397787	+	0.917478i	$0.369778\pi$
$854$	0	0
$855$	1710.00	0.0683986
$856$	0	0
$857$	−10290.0	−0.410151	−0.205076	−	0.978746i	$-0.565744\pi$
$857$	−10290.0	−0.410151	−0.205076	+	0.978746i	$0.565744\pi$
$858$	0	0
$859$	−31606.0	−1.25539	−0.627697	−	0.778458i	$-0.716002\pi$
$859$	−31606.0	−1.25539	−0.627697	+	0.778458i	$0.716002\pi$
$860$	−800.000	−0.0317207
$861$	5922.00	0.234403
$862$	0	0
$863$	23172.0	0.914002	0.457001	−	0.889466i	$-0.348924\pi$
$863$	23172.0	0.914002	0.457001	+	0.889466i	$0.348924\pi$
$864$	0	0
$865$	1890.00	0.0742912
$866$	0	0
$867$	1671.00	0.0654558
$868$	−8176.00	−0.319714
$869$	47712.0	1.86251
$870$	0	0
$871$	16240.0	0.631770
$872$	0	0
$873$	10872.0	0.421491
$874$	0	0
$875$	875.000	0.0338062
$876$	−2976.00	−0.114783
$877$	−15550.0	−0.598730	−0.299365	−	0.954139i	$-0.596775\pi$
$877$	−15550.0	−0.598730	−0.299365	+	0.954139i	$0.596775\pi$
$878$	0	0
$879$	25794.0	0.989772
$880$	13440.0	0.514844
$881$	28530.0	1.09103	0.545517	−	0.838100i	$-0.316334\pi$
$881$	28530.0	1.09103	0.545517	+	0.838100i	$0.316334\pi$
$882$	0	0
$883$	−28780.0	−1.09686	−0.548428	−	0.836198i	$-0.684774\pi$
$883$	−28780.0	−1.09686	−0.548428	+	0.836198i	$0.684774\pi$
$884$	−10560.0	−0.401777
$885$	5400.00	0.205106
$886$	0	0
$887$	22872.0	0.865802	0.432901	−	0.901441i	$-0.357490\pi$
$887$	22872.0	0.865802	0.432901	+	0.901441i	$0.357490\pi$
$888$	0	0
$889$	−9184.00	−0.346481
$890$	0	0
$891$	3402.00	0.127914
$892$	−25504.0	−0.957329
$893$	−2736.00	−0.102527
$894$	0	0
$895$	−1110.00	−0.0414561
$896$	0	0
$897$	−720.000	−0.0268006
$898$	0	0
$899$	−37668.0	−1.39744
$900$	−1800.00	−0.0666667
$901$	22176.0	0.819966
$902$	0	0
$903$	−420.000	−0.0154781
$904$	0	0
$905$	−12950.0	−0.475660
$906$	0	0
$907$	−10708.0	−0.392010	−0.196005	−	0.980603i	$-0.562797\pi$
$907$	−10708.0	−0.392010	−0.196005	+	0.980603i	$0.562797\pi$
$908$	27168.0	0.992953
$909$	4914.00	0.179304
$910$	0	0
$911$	1326.00	0.0482243	0.0241122	−	0.999709i	$-0.492324\pi$
$911$	1326.00	0.0482243	0.0241122	+	0.999709i	$0.492324\pi$
$912$	−7296.00	−0.264906
$913$	6552.00	0.237502
$914$	0	0
$915$	10230.0	0.369610
$916$	−42352.0	−1.52767
$917$	−9492.00	−0.341825
$918$	0	0
$919$	−13696.0	−0.491610	−0.245805	−	0.969319i	$-0.579052\pi$
$919$	−13696.0	−0.491610	−0.245805	+	0.969319i	$0.579052\pi$
$920$	0	0
$921$	1344.00	0.0480850
$922$	0	0
$923$	16200.0	0.577713
$924$	7056.00	0.251218
$925$	10850.0	0.385671
$926$	0	0
$927$	−4680.00	−0.165816
$928$	0	0
$929$	42354.0	1.49579	0.747895	−	0.663817i	$-0.231064\pi$
$929$	42354.0	1.49579	0.747895	+	0.663817i	$0.231064\pi$
$930$	0	0
$931$	1862.00	0.0655474
$932$	−6816.00	−0.239555
$933$	17496.0	0.613926
$934$	0	0
$935$	13860.0	0.484781
$936$	0	0
$937$	6644.00	0.231644	0.115822	−	0.993270i	$-0.463050\pi$
$937$	6644.00	0.231644	0.115822	+	0.993270i	$0.463050\pi$
$938$	0	0
$939$	−29544.0	−1.02676
$940$	2880.00	0.0999311
$941$	1350.00	0.0467681	0.0233840	−	0.999727i	$-0.492556\pi$
$941$	1350.00	0.0467681	0.0233840	+	0.999727i	$0.492556\pi$
$942$	0	0
$943$	−3384.00	−0.116859
$944$	−23040.0	−0.794373
$945$	−945.000	−0.0325300
$946$	0	0
$947$	49320.0	1.69238	0.846190	−	0.532881i	$-0.178891\pi$
$947$	49320.0	1.69238	0.846190	+	0.532881i	$0.178891\pi$
$948$	27264.0	0.934065
$949$	−2480.00	−0.0848306
$950$	0	0
$951$	16848.0	0.574484
$952$	0	0
$953$	5940.00	0.201905	0.100953	−	0.994891i	$-0.467811\pi$
$953$	5940.00	0.201905	0.100953	+	0.994891i	$0.467811\pi$
$954$	0	0
$955$	11070.0	0.375096
$956$	−38928.0	−1.31697
$957$	32508.0	1.09805
$958$	0	0
$959$	−6888.00	−0.231934
$960$	7680.00	0.258199
$961$	−8475.00	−0.284482
$962$	0	0
$963$	10908.0	0.365011
$964$	16400.0	0.547934
$965$	20890.0	0.696863
$966$	0	0
$967$	47216.0	1.57018	0.785090	−	0.619382i	$-0.212617\pi$
$967$	47216.0	1.57018	0.785090	+	0.619382i	$0.212617\pi$
$968$	0	0
$969$	−7524.00	−0.249438
$970$	0	0
$971$	12552.0	0.414843	0.207422	−	0.978252i	$-0.433493\pi$
$971$	12552.0	0.414843	0.207422	+	0.978252i	$0.433493\pi$
$972$	1944.00	0.0641500
$973$	−2758.00	−0.0908709
$974$	0	0
$975$	−1500.00	−0.0492702
$976$	−43648.0	−1.43149
$977$	46908.0	1.53605	0.768025	−	0.640420i	$-0.221240\pi$
$977$	46908.0	1.53605	0.768025	+	0.640420i	$0.221240\pi$
$978$	0	0
$979$	−43596.0	−1.42322
$980$	−1960.00	−0.0638877
$981$	−9702.00	−0.315760
$982$	0	0
$983$	−46128.0	−1.49670	−0.748349	−	0.663305i	$-0.769153\pi$
$983$	−46128.0	−1.49670	−0.748349	+	0.663305i	$0.769153\pi$
$984$	0	0
$985$	−15300.0	−0.494922
$986$	0	0
$987$	1512.00	0.0487614
$988$	−6080.00	−0.195780
$989$	240.000	0.00771644
$990$	0	0
$991$	−12184.0	−0.390552	−0.195276	−	0.980748i	$-0.562560\pi$
$991$	−12184.0	−0.390552	−0.195276	+	0.980748i	$0.562560\pi$
$992$	0	0
$993$	−1356.00	−0.0433347
$994$	0	0
$995$	13330.0	0.424713
$996$	3744.00	0.119110
$997$	−5164.00	−0.164038	−0.0820188	−	0.996631i	$-0.526137\pi$
$997$	−5164.00	−0.164038	−0.0820188	+	0.996631i	$0.526137\pi$
$998$	0	0
$999$	−11718.0	−0.371112

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.a.a.1.1	✓	1
3.2	odd	2		315.4.a.d.1.1		1
4.3	odd	2		1680.4.a.s.1.1		1
5.2	odd	4		525.4.d.f.274.2		2
5.3	odd	4		525.4.d.f.274.1		2
5.4	even	2		525.4.a.e.1.1		1
7.6	odd	2		735.4.a.c.1.1		1
15.14	odd	2		1575.4.a.f.1.1		1
21.20	even	2		2205.4.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.a.1.1	✓	1	1.1	even	1	trivial
315.4.a.d.1.1		1	3.2	odd	2
525.4.a.e.1.1		1	5.4	even	2
525.4.d.f.274.1		2	5.3	odd	4
525.4.d.f.274.2		2	5.2	odd	4
735.4.a.c.1.1		1	7.6	odd	2
1575.4.a.f.1.1		1	15.14	odd	2
1680.4.a.s.1.1		1	4.3	odd	2
2205.4.a.o.1.1		1	21.20	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:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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