LMFDB - Embedded newform 1680.4.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,4,Mod(1,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1680.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1680.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:	$99.1232088096$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+3.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -42.0000 q^{11} +20.0000 q^{13} +15.0000 q^{15} +66.0000 q^{17} -38.0000 q^{19} -21.0000 q^{21} -12.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -258.000 q^{29} -146.000 q^{31} -126.000 q^{33} -35.0000 q^{35} +434.000 q^{37} +60.0000 q^{39} -282.000 q^{41} -20.0000 q^{43} +45.0000 q^{45} +72.0000 q^{47} +49.0000 q^{49} +198.000 q^{51} +336.000 q^{53} -210.000 q^{55} -114.000 q^{57} +360.000 q^{59} -682.000 q^{61} -63.0000 q^{63} +100.000 q^{65} -812.000 q^{67} -36.0000 q^{69} -810.000 q^{71} -124.000 q^{73} +75.0000 q^{75} +294.000 q^{77} -1136.00 q^{79} +81.0000 q^{81} -156.000 q^{83} +330.000 q^{85} -774.000 q^{87} -1038.00 q^{89} -140.000 q^{91} -438.000 q^{93} -190.000 q^{95} +1208.00 q^{97} -378.000 q^{99} +O(q^{100})$

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
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	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	$-0.695236\pi$
$11$	−42.0000	−1.15123	−0.575613	+	0.817723i	$0.695236\pi$
$12$	0	0
$13$	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
$16$	0	0
$17$	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	$-0.573682\pi$
$19$	−38.0000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−12.0000	−0.108790	−0.0543951	−	0.998519i	$-0.517323\pi$
$23$	−12.0000	−0.108790	−0.0543951	+	0.998519i	$0.517323\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−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	$-0.639002\pi$
$31$	−146.000	−0.845883	−0.422942	+	0.906157i	$0.639002\pi$
$32$	0	0
$33$	−126.000	−0.664660
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	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	$-0.511291\pi$
$43$	−20.0000	−0.0709296	−0.0354648	+	0.999371i	$0.511291\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	72.0000	0.223453	0.111726	−	0.993739i	$-0.464362\pi$
$47$	72.0000	0.223453	0.111726	+	0.993739i	$0.464362\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	198.000	0.543638
$52$	0	0
$53$	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	$-0.369987\pi$
$59$	360.000	0.794373	0.397187	+	0.917738i	$0.369987\pi$
$60$	0	0
$61$	−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
$64$	0	0
$65$	100.000	0.190823
$66$	0	0
$67$	−812.000	−1.48062	−0.740310	−	0.672265i	$-0.765321\pi$
$67$	−812.000	−1.48062	−0.740310	+	0.672265i	$0.765321\pi$
$68$	0	0
$69$	−36.0000	−0.0628100
$70$	0	0
$71$	−810.000	−1.35393	−0.676967	−	0.736013i	$-0.736706\pi$
$71$	−810.000	−1.35393	−0.676967	+	0.736013i	$0.736706\pi$
$72$	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
$76$	0	0
$77$	294.000	0.435122
$78$	0	0
$79$	−1136.00	−1.61785	−0.808924	−	0.587913i	$-0.799950\pi$
$79$	−1136.00	−1.61785	−0.808924	+	0.587913i	$0.799950\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−156.000	−0.206304	−0.103152	−	0.994666i	$-0.532893\pi$
$83$	−156.000	−0.206304	−0.103152	+	0.994666i	$0.532893\pi$
$84$	0	0
$85$	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
$92$	0	0
$93$	−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
$100$	0	0
$101$	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	$-0.419989\pi$
$103$	520.000	0.497448	0.248724	+	0.968574i	$0.419989\pi$
$104$	0	0
$105$	−105.000	−0.0975900
$106$	0	0
$107$	−1212.00	−1.09503	−0.547516	−	0.836795i	$-0.684427\pi$
$107$	−1212.00	−1.09503	−0.547516	+	0.836795i	$0.684427\pi$
$108$	0	0
$109$	−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
$112$	0	0
$113$	−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
$116$	0	0
$117$	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
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1312.00	0.916702	0.458351	−	0.888771i	$-0.348440\pi$
$127$	1312.00	0.916702	0.458351	+	0.888771i	$0.348440\pi$
$128$	0	0
$129$	−60.0000	−0.0409512
$130$	0	0
$131$	1356.00	0.904384	0.452192	−	0.891921i	$-0.350642\pi$
$131$	1356.00	0.904384	0.452192	+	0.891921i	$0.350642\pi$
$132$	0	0
$133$	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	$-0.461643\pi$
$139$	394.000	0.240422	0.120211	+	0.992748i	$0.461643\pi$
$140$	0	0
$141$	216.000	0.129011
$142$	0	0
$143$	−840.000	−0.491219
$144$	0	0
$145$	−1290.00	−0.738818
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−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	$-0.319235\pi$
$151$	1996.00	1.07571	0.537855	+	0.843037i	$0.319235\pi$
$152$	0	0
$153$	594.000	0.313870
$154$	0	0
$155$	−730.000	−0.378290
$156$	0	0
$157$	−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	$-0.662703\pi$
$163$	−2036.00	−0.978355	−0.489177	+	0.872184i	$0.662703\pi$
$164$	0	0
$165$	−630.000	−0.297245
$166$	0	0
$167$	3936.00	1.82381	0.911907	−	0.410398i	$-0.134610\pi$
$167$	3936.00	1.82381	0.911907	+	0.410398i	$0.134610\pi$
$168$	0	0
$169$	−1797.00	−0.817934
$170$	0	0
$171$	−342.000	−0.152944
$172$	0	0
$173$	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
$176$	0	0
$177$	1080.00	0.458631
$178$	0	0
$179$	222.000	0.0926987	0.0463493	−	0.998925i	$-0.485241\pi$
$179$	222.000	0.0926987	0.0463493	+	0.998925i	$0.485241\pi$
$180$	0	0
$181$	−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
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−2214.00	−0.838740	−0.419370	−	0.907815i	$-0.637749\pi$
$191$	−2214.00	−0.838740	−0.419370	+	0.907815i	$0.637749\pi$
$192$	0	0
$193$	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
$196$	0	0
$197$	−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	$-0.657495\pi$
$199$	−2666.00	−0.949687	−0.474844	+	0.880070i	$0.657495\pi$
$200$	0	0
$201$	−2436.00	−0.854837
$202$	0	0
$203$	1806.00	0.624416
$204$	0	0
$205$	−1410.00	−0.480384
$206$	0	0
$207$	−108.000	−0.0362634
$208$	0	0
$209$	1596.00	0.528218
$210$	0	0
$211$	1348.00	0.439811	0.219906	−	0.975521i	$-0.429425\pi$
$211$	1348.00	0.439811	0.219906	+	0.975521i	$0.429425\pi$
$212$	0	0
$213$	−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
$220$	0	0
$221$	1320.00	0.401777
$222$	0	0
$223$	−3188.00	−0.957329	−0.478664	−	0.877998i	$-0.658879\pi$
$223$	−3188.00	−0.957329	−0.478664	+	0.877998i	$0.658879\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	3396.00	0.992953	0.496477	−	0.868050i	$-0.334627\pi$
$227$	3396.00	0.992953	0.496477	+	0.868050i	$0.334627\pi$
$228$	0	0
$229$	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
$236$	0	0
$237$	−3408.00	−0.934065
$238$	0	0
$239$	−4866.00	−1.31697	−0.658484	−	0.752595i	$-0.728802\pi$
$239$	−4866.00	−1.31697	−0.658484	+	0.752595i	$0.728802\pi$
$240$	0	0
$241$	−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
$244$	0	0
$245$	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	$-0.453731\pi$
$251$	1152.00	0.289696	0.144848	+	0.989454i	$0.453731\pi$
$252$	0	0
$253$	504.000	0.125242
$254$	0	0
$255$	990.000	0.243122
$256$	0	0
$257$	−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
$260$	0	0
$261$	−2322.00	−0.550683
$262$	0	0
$263$	−1968.00	−0.461415	−0.230707	−	0.973023i	$-0.574104\pi$
$263$	−1968.00	−0.461415	−0.230707	+	0.973023i	$0.574104\pi$
$264$	0	0
$265$	1680.00	0.389440
$266$	0	0
$267$	−3114.00	−0.713759
$268$	0	0
$269$	−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	$-0.792568\pi$
$271$	−7094.00	−1.59015	−0.795073	+	0.606513i	$0.792568\pi$
$272$	0	0
$273$	−420.000	−0.0931119
$274$	0	0
$275$	−1050.00	−0.230245
$276$	0	0
$277$	−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	$-0.317536\pi$
$283$	5164.00	1.08469	0.542346	+	0.840155i	$0.317536\pi$
$284$	0	0
$285$	−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
$292$	0	0
$293$	−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
$300$	0	0
$301$	140.000	0.0268089
$302$	0	0
$303$	1638.00	0.310563
$304$	0	0
$305$	−3410.00	−0.640184
$306$	0	0
$307$	448.000	0.0832857	0.0416429	−	0.999133i	$-0.486741\pi$
$307$	448.000	0.0832857	0.0416429	+	0.999133i	$0.486741\pi$
$308$	0	0
$309$	1560.00	0.287202
$310$	0	0
$311$	5832.00	1.06335	0.531676	−	0.846948i	$-0.321562\pi$
$311$	5832.00	1.06335	0.531676	+	0.846948i	$0.321562\pi$
$312$	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
$316$	0	0
$317$	−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
$320$	0	0
$321$	−3636.00	−0.632217
$322$	0	0
$323$	−2508.00	−0.432040
$324$	0	0
$325$	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	$-0.511949\pi$
$331$	−452.000	−0.0750579	−0.0375290	+	0.999296i	$0.511949\pi$
$332$	0	0
$333$	3906.00	0.642785
$334$	0	0
$335$	−4060.00	−0.662154
$336$	0	0
$337$	−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
$340$	0	0
$341$	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	$-0.539100\pi$
$347$	−1584.00	−0.245054	−0.122527	+	0.992465i	$0.539100\pi$
$348$	0	0
$349$	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
$356$	0	0
$357$	−1386.00	−0.205476
$358$	0	0
$359$	2154.00	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	2154.00	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−5415.00	−0.789474
$362$	0	0
$363$	1299.00	0.187823
$364$	0	0
$365$	−620.000	−0.0889104
$366$	0	0
$367$	−6644.00	−0.944997	−0.472499	−	0.881331i	$-0.656648\pi$
$367$	−6644.00	−0.944997	−0.472499	+	0.881331i	$0.656648\pi$
$368$	0	0
$369$	−2538.00	−0.358057
$370$	0	0
$371$	−2352.00	−0.329137
$372$	0	0
$373$	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	$-0.574892\pi$
$379$	−3440.00	−0.466229	−0.233115	+	0.972449i	$0.574892\pi$
$380$	0	0
$381$	3936.00	0.529258
$382$	0	0
$383$	−12936.0	−1.72585	−0.862923	−	0.505336i	$-0.831369\pi$
$383$	−12936.0	−1.72585	−0.862923	+	0.505336i	$0.831369\pi$
$384$	0	0
$385$	1470.00	0.194593
$386$	0	0
$387$	−180.000	−0.0236432
$388$	0	0
$389$	−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
$396$	0	0
$397$	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
$400$	0	0
$401$	−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
$404$	0	0
$405$	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
$412$	0	0
$413$	−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	$-0.468999\pi$
$419$	1668.00	0.194480	0.0972400	+	0.995261i	$0.468999\pi$
$420$	0	0
$421$	−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
$428$	0	0
$429$	−2520.00	−0.283605
$430$	0	0
$431$	−15186.0	−1.69718	−0.848589	−	0.529052i	$-0.822548\pi$
$431$	−15186.0	−1.69718	−0.848589	+	0.529052i	$0.822548\pi$
$432$	0	0
$433$	−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
$436$	0	0
$437$	456.000	0.0499163
$438$	0	0
$439$	17206.0	1.87061	0.935305	−	0.353843i	$-0.115125\pi$
$439$	17206.0	1.87061	0.935305	+	0.353843i	$0.115125\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−3456.00	−0.370654	−0.185327	−	0.982677i	$-0.559334\pi$
$443$	−3456.00	−0.370654	−0.185327	+	0.982677i	$0.559334\pi$
$444$	0	0
$445$	−5190.00	−0.552875
$446$	0	0
$447$	−3042.00	−0.321883
$448$	0	0
$449$	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
$452$	0	0
$453$	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
$460$	0	0
$461$	−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	$-0.332199\pi$
$463$	10024.0	1.00617	0.503083	+	0.864238i	$0.332199\pi$
$464$	0	0
$465$	−2190.00	−0.218406
$466$	0	0
$467$	2460.00	0.243759	0.121879	−	0.992545i	$-0.461108\pi$
$467$	2460.00	0.243759	0.121879	+	0.992545i	$0.461108\pi$
$468$	0	0
$469$	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
$476$	0	0
$477$	3024.00	0.290271
$478$	0	0
$479$	−19320.0	−1.84291	−0.921454	−	0.388486i	$-0.872998\pi$
$479$	−19320.0	−1.84291	−0.921454	+	0.388486i	$0.872998\pi$
$480$	0	0
$481$	8680.00	0.822815
$482$	0	0
$483$	252.000	0.0237400
$484$	0	0
$485$	6040.00	0.565489
$486$	0	0
$487$	12544.0	1.16719	0.583596	−	0.812044i	$-0.301645\pi$
$487$	12544.0	1.16719	0.583596	+	0.812044i	$0.301645\pi$
$488$	0	0
$489$	−6108.00	−0.564853
$490$	0	0
$491$	15510.0	1.42557	0.712787	−	0.701381i	$-0.247433\pi$
$491$	15510.0	1.42557	0.712787	+	0.701381i	$0.247433\pi$
$492$	0	0
$493$	−17028.0	−1.55558
$494$	0	0
$495$	−1890.00	−0.171615
$496$	0	0
$497$	5670.00	0.511739
$498$	0	0
$499$	14344.0	1.28682	0.643412	−	0.765520i	$-0.277518\pi$
$499$	14344.0	1.28682	0.643412	+	0.765520i	$0.277518\pi$
$500$	0	0
$501$	11808.0	1.05298
$502$	0	0
$503$	21384.0	1.89556	0.947779	−	0.318929i	$-0.103323\pi$
$503$	21384.0	1.89556	0.947779	+	0.318929i	$0.103323\pi$
$504$	0	0
$505$	2730.00	0.240561
$506$	0	0
$507$	−5391.00	−0.472234
$508$	0	0
$509$	−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
$516$	0	0
$517$	−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	$-0.273168\pi$
$523$	15640.0	1.30763	0.653814	+	0.756655i	$0.273168\pi$
$524$	0	0
$525$	−525.000	−0.0436436
$526$	0	0
$527$	−9636.00	−0.796491
$528$	0	0
$529$	−12023.0	−0.988165
$530$	0	0
$531$	3240.00	0.264791
$532$	0	0
$533$	−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
$540$	0	0
$541$	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	$-0.444432\pi$
$547$	4444.00	0.347371	0.173685	+	0.984801i	$0.444432\pi$
$548$	0	0
$549$	−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
$556$	0	0
$557$	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
$560$	0	0
$561$	−8316.00	−0.625850
$562$	0	0
$563$	16452.0	1.23156	0.615781	−	0.787918i	$-0.288841\pi$
$563$	16452.0	1.23156	0.615781	+	0.787918i	$0.288841\pi$
$564$	0	0
$565$	−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	$-0.530093\pi$
$571$	−2576.00	−0.188796	−0.0943978	+	0.995535i	$0.530093\pi$
$572$	0	0
$573$	−6642.00	−0.484247
$574$	0	0
$575$	−300.000	−0.0217580
$576$	0	0
$577$	−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
$580$	0	0
$581$	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	$-0.516256\pi$
$587$	−1452.00	−0.102096	−0.0510481	+	0.998696i	$0.516256\pi$
$588$	0	0
$589$	5548.00	0.388118
$590$	0	0
$591$	−9180.00	−0.638942
$592$	0	0
$593$	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
$596$	0	0
$597$	−7998.00	−0.548302
$598$	0	0
$599$	8730.00	0.595489	0.297745	−	0.954646i	$-0.403766\pi$
$599$	8730.00	0.595489	0.297745	+	0.954646i	$0.403766\pi$
$600$	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
$604$	0	0
$605$	2165.00	0.145487
$606$	0	0
$607$	5596.00	0.374192	0.187096	−	0.982342i	$-0.440092\pi$
$607$	5596.00	0.374192	0.187096	+	0.982342i	$0.440092\pi$
$608$	0	0
$609$	5418.00	0.360506
$610$	0	0
$611$	1440.00	0.0953456
$612$	0	0
$613$	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	$-0.570158\pi$
$619$	−6734.00	−0.437257	−0.218629	+	0.975808i	$0.570158\pi$
$620$	0	0
$621$	−324.000	−0.0209367
$622$	0	0
$623$	7266.00	0.467265
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	4788.00	0.304967
$628$	0	0
$629$	28644.0	1.81576
$630$	0	0
$631$	−7184.00	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$631$	−7184.00	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$632$	0	0
$633$	4044.00	0.253925
$634$	0	0
$635$	6560.00	0.409962
$636$	0	0
$637$	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	$-0.280433\pi$
$643$	20752.0	1.27275	0.636376	+	0.771379i	$0.280433\pi$
$644$	0	0
$645$	−300.000	−0.0183139
$646$	0	0
$647$	−21072.0	−1.28041	−0.640205	−	0.768204i	$-0.721151\pi$
$647$	−21072.0	−1.28041	−0.640205	+	0.768204i	$0.721151\pi$
$648$	0	0
$649$	−15120.0	−0.914502
$650$	0	0
$651$	3066.00	0.184587
$652$	0	0
$653$	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
$656$	0	0
$657$	−1116.00	−0.0662699
$658$	0	0
$659$	−750.000	−0.0443336	−0.0221668	−	0.999754i	$-0.507056\pi$
$659$	−750.000	−0.0443336	−0.0221668	+	0.999754i	$0.507056\pi$
$660$	0	0
$661$	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
$668$	0	0
$669$	−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
$676$	0	0
$677$	−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	$-0.571649\pi$
$683$	−7968.00	−0.446394	−0.223197	+	0.974773i	$0.571649\pi$
$684$	0	0
$685$	−4920.00	−0.274429
$686$	0	0
$687$	15882.0	0.882003
$688$	0	0
$689$	6720.00	0.371570
$690$	0	0
$691$	14398.0	0.792657	0.396328	−	0.918109i	$-0.370284\pi$
$691$	14398.0	0.792657	0.396328	+	0.918109i	$0.370284\pi$
$692$	0	0
$693$	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
$700$	0	0
$701$	−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
$704$	0	0
$705$	1080.00	0.0576953
$706$	0	0
$707$	−3822.00	−0.203311
$708$	0	0
$709$	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
$716$	0	0
$717$	−14598.0	−0.760352
$718$	0	0
$719$	−27060.0	−1.40357	−0.701786	−	0.712388i	$-0.747614\pi$
$719$	−27060.0	−1.40357	−0.701786	+	0.712388i	$0.747614\pi$
$720$	0	0
$721$	−3640.00	−0.188018
$722$	0	0
$723$	−6150.00	−0.316350
$724$	0	0
$725$	−6450.00	−0.330410
$726$	0	0
$727$	3724.00	0.189980	0.0949900	−	0.995478i	$-0.469718\pi$
$727$	3724.00	0.189980	0.0949900	+	0.995478i	$0.469718\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1320.00	−0.0667879
$732$	0	0
$733$	−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	$-0.369006\pi$
$739$	16072.0	0.800024	0.400012	+	0.916510i	$0.369006\pi$
$740$	0	0
$741$	−2280.00	−0.113034
$742$	0	0
$743$	8256.00	0.407649	0.203825	−	0.979007i	$-0.434663\pi$
$743$	8256.00	0.407649	0.203825	+	0.979007i	$0.434663\pi$
$744$	0	0
$745$	−5070.00	−0.249329
$746$	0	0
$747$	−1404.00	−0.0687680
$748$	0	0
$749$	8484.00	0.413883
$750$	0	0
$751$	6352.00	0.308639	0.154319	−	0.988021i	$-0.450682\pi$
$751$	6352.00	0.308639	0.154319	+	0.988021i	$0.450682\pi$
$752$	0	0
$753$	3456.00	0.167256
$754$	0	0
$755$	9980.00	0.481072
$756$	0	0
$757$	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
$764$	0	0
$765$	2970.00	0.140367
$766$	0	0
$767$	7200.00	0.338953
$768$	0	0
$769$	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
$772$	0	0
$773$	−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
$780$	0	0
$781$	34020.0	1.55868
$782$	0	0
$783$	−6966.00	−0.317937
$784$	0	0
$785$	−11960.0	−0.543784
$786$	0	0
$787$	17080.0	0.773617	0.386808	−	0.922160i	$-0.373578\pi$
$787$	17080.0	0.773617	0.386808	+	0.922160i	$0.373578\pi$
$788$	0	0
$789$	−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
$796$	0	0
$797$	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
$804$	0	0
$805$	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	$-0.296688\pi$
$811$	27538.0	1.19234	0.596171	+	0.802857i	$0.296688\pi$
$812$	0	0
$813$	−21282.0	−0.918072
$814$	0	0
$815$	−10180.0	−0.437534
$816$	0	0
$817$	760.000	0.0325447
$818$	0	0
$819$	−1260.00	−0.0537582
$820$	0	0
$821$	−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	$-0.419939\pi$
$823$	11752.0	0.497751	0.248875	+	0.968536i	$0.419939\pi$
$824$	0	0
$825$	−3150.00	−0.132932
$826$	0	0
$827$	28692.0	1.20643	0.603216	−	0.797578i	$-0.293886\pi$
$827$	28692.0	1.20643	0.603216	+	0.797578i	$0.293886\pi$
$828$	0	0
$829$	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
$832$	0	0
$833$	3234.00	0.134516
$834$	0	0
$835$	19680.0	0.815634
$836$	0	0
$837$	−3942.00	−0.162790
$838$	0	0
$839$	−20172.0	−0.830053	−0.415027	−	0.909809i	$-0.636228\pi$
$839$	−20172.0	−0.830053	−0.415027	+	0.909809i	$0.636228\pi$
$840$	0	0
$841$	42175.0	1.72926
$842$	0	0
$843$	21474.0	0.877347
$844$	0	0
$845$	−8985.00	−0.365791
$846$	0	0
$847$	−3031.00	−0.122959
$848$	0	0
$849$	15492.0	0.626247
$850$	0	0
$851$	−5208.00	−0.209786
$852$	0	0
$853$	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	$-0.283998\pi$
$859$	31606.0	1.25539	0.627697	+	0.778458i	$0.283998\pi$
$860$	0	0
$861$	5922.00	0.234403
$862$	0	0
$863$	−23172.0	−0.914002	−0.457001	−	0.889466i	$-0.651076\pi$
$863$	−23172.0	−0.914002	−0.457001	+	0.889466i	$0.651076\pi$
$864$	0	0
$865$	1890.00	0.0742912
$866$	0	0
$867$	−1671.00	−0.0654558
$868$	0	0
$869$	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
$876$	0	0
$877$	−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
$880$	0	0
$881$	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	$-0.315226\pi$
$883$	28780.0	1.09686	0.548428	+	0.836198i	$0.315226\pi$
$884$	0	0
$885$	5400.00	0.205106
$886$	0	0
$887$	−22872.0	−0.865802	−0.432901	−	0.901441i	$-0.642510\pi$
$887$	−22872.0	−0.865802	−0.432901	+	0.901441i	$0.642510\pi$
$888$	0	0
$889$	−9184.00	−0.346481
$890$	0	0
$891$	−3402.00	−0.127914
$892$	0	0
$893$	−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
$900$	0	0
$901$	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	$-0.437203\pi$
$907$	10708.0	0.392010	0.196005	+	0.980603i	$0.437203\pi$
$908$	0	0
$909$	4914.00	0.179304
$910$	0	0
$911$	−1326.00	−0.0482243	−0.0241122	−	0.999709i	$-0.507676\pi$
$911$	−1326.00	−0.0482243	−0.0241122	+	0.999709i	$0.507676\pi$
$912$	0	0
$913$	6552.00	0.237502
$914$	0	0
$915$	−10230.0	−0.369610
$916$	0	0
$917$	−9492.00	−0.341825
$918$	0	0
$919$	13696.0	0.491610	0.245805	−	0.969319i	$-0.420948\pi$
$919$	13696.0	0.491610	0.245805	+	0.969319i	$0.420948\pi$
$920$	0	0
$921$	1344.00	0.0480850
$922$	0	0
$923$	−16200.0	−0.577713
$924$	0	0
$925$	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
$932$	0	0
$933$	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
$940$	0	0
$941$	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
$944$	0	0
$945$	−945.000	−0.0325300
$946$	0	0
$947$	−49320.0	−1.69238	−0.846190	−	0.532881i	$-0.821109\pi$
$947$	−49320.0	−1.69238	−0.846190	+	0.532881i	$0.821109\pi$
$948$	0	0
$949$	−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
$956$	0	0
$957$	32508.0	1.09805
$958$	0	0
$959$	6888.00	0.231934
$960$	0	0
$961$	−8475.00	−0.284482
$962$	0	0
$963$	−10908.0	−0.365011
$964$	0	0
$965$	20890.0	0.696863
$966$	0	0
$967$	−47216.0	−1.57018	−0.785090	−	0.619382i	$-0.787383\pi$
$967$	−47216.0	−1.57018	−0.785090	+	0.619382i	$0.787383\pi$
$968$	0	0
$969$	−7524.00	−0.249438
$970$	0	0
$971$	−12552.0	−0.414843	−0.207422	−	0.978252i	$-0.566507\pi$
$971$	−12552.0	−0.414843	−0.207422	+	0.978252i	$0.566507\pi$
$972$	0	0
$973$	−2758.00	−0.0908709
$974$	0	0
$975$	1500.00	0.0492702
$976$	0	0
$977$	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
$980$	0	0
$981$	−9702.00	−0.315760
$982$	0	0
$983$	46128.0	1.49670	0.748349	−	0.663305i	$-0.230847\pi$
$983$	46128.0	1.49670	0.748349	+	0.663305i	$0.230847\pi$
$984$	0	0
$985$	−15300.0	−0.494922
$986$	0	0
$987$	−1512.00	−0.0487614
$988$	0	0
$989$	240.000	0.00771644
$990$	0	0
$991$	12184.0	0.390552	0.195276	−	0.980748i	$-0.437440\pi$
$991$	12184.0	0.390552	0.195276	+	0.980748i	$0.437440\pi$
$992$	0	0
$993$	−1356.00	−0.0433347
$994$	0	0
$995$	−13330.0	−0.424713
$996$	0	0
$997$	−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	1680.4.a.s.1.1		1
4.3	odd	2		105.4.a.a.1.1	✓	1
12.11	even	2		315.4.a.d.1.1		1
20.3	even	4		525.4.d.f.274.1		2
20.7	even	4		525.4.d.f.274.2		2
20.19	odd	2		525.4.a.e.1.1		1
28.27	even	2		735.4.a.c.1.1		1
60.59	even	2		1575.4.a.f.1.1		1
84.83	odd	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	4.3	odd	2
315.4.a.d.1.1		1	12.11	even	2
525.4.a.e.1.1		1	20.19	odd	2
525.4.d.f.274.1		2	20.3	even	4
525.4.d.f.274.2		2	20.7	even	4
735.4.a.c.1.1		1	28.27	even	2
1575.4.a.f.1.1		1	60.59	even	2
1680.4.a.s.1.1		1	1.1	even	1	trivial
2205.4.a.o.1.1		1	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1680.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more