LMFDB - Embedded newform 2240.4.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,4,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2240.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:	$132.164278413$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} -26.0000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} -26.0000 q^{9} +7.00000 q^{11} +23.0000 q^{13} -5.00000 q^{15} -25.0000 q^{17} +62.0000 q^{19} +7.00000 q^{21} -86.0000 q^{23} +25.0000 q^{25} +53.0000 q^{27} +29.0000 q^{29} -12.0000 q^{31} -7.00000 q^{33} -35.0000 q^{35} +150.000 q^{37} -23.0000 q^{39} +204.000 q^{41} +178.000 q^{43} -130.000 q^{45} +33.0000 q^{47} +49.0000 q^{49} +25.0000 q^{51} -452.000 q^{53} +35.0000 q^{55} -62.0000 q^{57} -120.000 q^{59} -920.000 q^{61} +182.000 q^{63} +115.000 q^{65} +300.000 q^{67} +86.0000 q^{69} +520.000 q^{71} +370.000 q^{73} -25.0000 q^{75} -49.0000 q^{77} -1013.00 q^{79} +649.000 q^{81} +636.000 q^{83} -125.000 q^{85} -29.0000 q^{87} +292.000 q^{89} -161.000 q^{91} +12.0000 q^{93} +310.000 q^{95} -1381.00 q^{97} -182.000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.192450	−0.0962250	−	0.995360i	$-0.530677\pi$
$3$	−1.00000	−0.192450	−0.0962250	+	0.995360i	$0.530677\pi$
$4$	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
$9$	−26.0000	−0.962963
$10$	0	0
$11$	7.00000	0.191871	0.0959354	−	0.995388i	$-0.469416\pi$
$11$	7.00000	0.191871	0.0959354	+	0.995388i	$0.469416\pi$
$12$	0	0
$13$	23.0000	0.490696	0.245348	−	0.969435i	$-0.421098\pi$
$13$	23.0000	0.490696	0.245348	+	0.969435i	$0.421098\pi$
$14$	0	0
$15$	−5.00000	−0.0860663
$16$	0	0
$17$	−25.0000	−0.356670	−0.178335	−	0.983970i	$-0.557071\pi$
$17$	−25.0000	−0.356670	−0.178335	+	0.983970i	$0.557071\pi$
$18$	0	0
$19$	62.0000	0.748620	0.374310	−	0.927304i	$-0.377880\pi$
$19$	62.0000	0.748620	0.374310	+	0.927304i	$0.377880\pi$
$20$	0	0
$21$	7.00000	0.0727393
$22$	0	0
$23$	−86.0000	−0.779663	−0.389831	−	0.920886i	$-0.627467\pi$
$23$	−86.0000	−0.779663	−0.389831	+	0.920886i	$0.627467\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	53.0000	0.377772
$28$	0	0
$29$	29.0000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	29.0000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−12.0000	−0.0695246	−0.0347623	−	0.999396i	$-0.511067\pi$
$31$	−12.0000	−0.0695246	−0.0347623	+	0.999396i	$0.511067\pi$
$32$	0	0
$33$	−7.00000	−0.0369256
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	150.000	0.666482	0.333241	−	0.942842i	$-0.391858\pi$
$37$	150.000	0.666482	0.333241	+	0.942842i	$0.391858\pi$
$38$	0	0
$39$	−23.0000	−0.0944346
$40$	0	0
$41$	204.000	0.777060	0.388530	−	0.921436i	$-0.372983\pi$
$41$	204.000	0.777060	0.388530	+	0.921436i	$0.372983\pi$
$42$	0	0
$43$	178.000	0.631273	0.315637	−	0.948880i	$-0.397782\pi$
$43$	178.000	0.631273	0.315637	+	0.948880i	$0.397782\pi$
$44$	0	0
$45$	−130.000	−0.430650
$46$	0	0
$47$	33.0000	0.102416	0.0512079	−	0.998688i	$-0.483693\pi$
$47$	33.0000	0.102416	0.0512079	+	0.998688i	$0.483693\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	25.0000	0.0686412
$52$	0	0
$53$	−452.000	−1.17145	−0.585726	−	0.810509i	$-0.699191\pi$
$53$	−452.000	−1.17145	−0.585726	+	0.810509i	$0.699191\pi$
$54$	0	0
$55$	35.0000	0.0858073
$56$	0	0
$57$	−62.0000	−0.144072
$58$	0	0
$59$	−120.000	−0.264791	−0.132396	−	0.991197i	$-0.542267\pi$
$59$	−120.000	−0.264791	−0.132396	+	0.991197i	$0.542267\pi$
$60$	0	0
$61$	−920.000	−1.93105	−0.965524	−	0.260314i	$-0.916174\pi$
$61$	−920.000	−1.93105	−0.965524	+	0.260314i	$0.916174\pi$
$62$	0	0
$63$	182.000	0.363966
$64$	0	0
$65$	115.000	0.219446
$66$	0	0
$67$	300.000	0.547027	0.273514	−	0.961868i	$-0.411814\pi$
$67$	300.000	0.547027	0.273514	+	0.961868i	$0.411814\pi$
$68$	0	0
$69$	86.0000	0.150046
$70$	0	0
$71$	520.000	0.869192	0.434596	−	0.900625i	$-0.356891\pi$
$71$	520.000	0.869192	0.434596	+	0.900625i	$0.356891\pi$
$72$	0	0
$73$	370.000	0.593222	0.296611	−	0.954998i	$-0.404143\pi$
$73$	370.000	0.593222	0.296611	+	0.954998i	$0.404143\pi$
$74$	0	0
$75$	−25.0000	−0.0384900
$76$	0	0
$77$	−49.0000	−0.0725204
$78$	0	0
$79$	−1013.00	−1.44268	−0.721338	−	0.692583i	$-0.756473\pi$
$79$	−1013.00	−1.44268	−0.721338	+	0.692583i	$0.756473\pi$
$80$	0	0
$81$	649.000	0.890261
$82$	0	0
$83$	636.000	0.841085	0.420543	−	0.907273i	$-0.361840\pi$
$83$	636.000	0.841085	0.420543	+	0.907273i	$0.361840\pi$
$84$	0	0
$85$	−125.000	−0.159508
$86$	0	0
$87$	−29.0000	−0.0357371
$88$	0	0
$89$	292.000	0.347775	0.173887	−	0.984766i	$-0.444367\pi$
$89$	292.000	0.347775	0.173887	+	0.984766i	$0.444367\pi$
$90$	0	0
$91$	−161.000	−0.185466
$92$	0	0
$93$	12.0000	0.0133800
$94$	0	0
$95$	310.000	0.334793
$96$	0	0
$97$	−1381.00	−1.44556	−0.722780	−	0.691078i	$-0.757136\pi$
$97$	−1381.00	−1.44556	−0.722780	+	0.691078i	$0.757136\pi$
$98$	0	0
$99$	−182.000	−0.184765
$100$	0	0
$101$	−1278.00	−1.25907	−0.629533	−	0.776973i	$-0.716754\pi$
$101$	−1278.00	−1.25907	−0.629533	+	0.776973i	$0.716754\pi$
$102$	0	0
$103$	−875.000	−0.837052	−0.418526	−	0.908205i	$-0.637453\pi$
$103$	−875.000	−0.837052	−0.418526	+	0.908205i	$0.637453\pi$
$104$	0	0
$105$	35.0000	0.0325300
$106$	0	0
$107$	1986.00	1.79434	0.897168	−	0.441690i	$-0.145621\pi$
$107$	1986.00	1.79434	0.897168	+	0.441690i	$0.145621\pi$
$108$	0	0
$109$	223.000	0.195959	0.0979795	−	0.995188i	$-0.468762\pi$
$109$	223.000	0.195959	0.0979795	+	0.995188i	$0.468762\pi$
$110$	0	0
$111$	−150.000	−0.128265
$112$	0	0
$113$	−930.000	−0.774222	−0.387111	−	0.922033i	$-0.626527\pi$
$113$	−930.000	−0.774222	−0.387111	+	0.922033i	$0.626527\pi$
$114$	0	0
$115$	−430.000	−0.348676
$116$	0	0
$117$	−598.000	−0.472522
$118$	0	0
$119$	175.000	0.134809
$120$	0	0
$121$	−1282.00	−0.963186
$122$	0	0
$123$	−204.000	−0.149545
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−2056.00	−1.43654	−0.718270	−	0.695765i	$-0.755066\pi$
$127$	−2056.00	−1.43654	−0.718270	+	0.695765i	$0.755066\pi$
$128$	0	0
$129$	−178.000	−0.121489
$130$	0	0
$131$	−1442.00	−0.961741	−0.480871	−	0.876792i	$-0.659679\pi$
$131$	−1442.00	−0.961741	−0.480871	+	0.876792i	$0.659679\pi$
$132$	0	0
$133$	−434.000	−0.282952
$134$	0	0
$135$	265.000	0.168945
$136$	0	0
$137$	−908.000	−0.566246	−0.283123	−	0.959084i	$-0.591370\pi$
$137$	−908.000	−0.566246	−0.283123	+	0.959084i	$0.591370\pi$
$138$	0	0
$139$	1690.00	1.03125	0.515626	−	0.856814i	$-0.327560\pi$
$139$	1690.00	1.03125	0.515626	+	0.856814i	$0.327560\pi$
$140$	0	0
$141$	−33.0000	−0.0197099
$142$	0	0
$143$	161.000	0.0941503
$144$	0	0
$145$	145.000	0.0830455
$146$	0	0
$147$	−49.0000	−0.0274929
$148$	0	0
$149$	2014.00	1.10734	0.553669	−	0.832737i	$-0.313227\pi$
$149$	2014.00	1.10734	0.553669	+	0.832737i	$0.313227\pi$
$150$	0	0
$151$	619.000	0.333599	0.166800	−	0.985991i	$-0.446657\pi$
$151$	619.000	0.333599	0.166800	+	0.985991i	$0.446657\pi$
$152$	0	0
$153$	650.000	0.343460
$154$	0	0
$155$	−60.0000	−0.0310924
$156$	0	0
$157$	−1446.00	−0.735053	−0.367527	−	0.930013i	$-0.619795\pi$
$157$	−1446.00	−0.735053	−0.367527	+	0.930013i	$0.619795\pi$
$158$	0	0
$159$	452.000	0.225446
$160$	0	0
$161$	602.000	0.294685
$162$	0	0
$163$	2638.00	1.26763	0.633816	−	0.773484i	$-0.281488\pi$
$163$	2638.00	1.26763	0.633816	+	0.773484i	$0.281488\pi$
$164$	0	0
$165$	−35.0000	−0.0165136
$166$	0	0
$167$	−11.0000	−0.00509704	−0.00254852	−	0.999997i	$-0.500811\pi$
$167$	−11.0000	−0.00509704	−0.00254852	+	0.999997i	$0.500811\pi$
$168$	0	0
$169$	−1668.00	−0.759217
$170$	0	0
$171$	−1612.00	−0.720893
$172$	0	0
$173$	−3267.00	−1.43575	−0.717877	−	0.696170i	$-0.754886\pi$
$173$	−3267.00	−1.43575	−0.717877	+	0.696170i	$0.754886\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	120.000	0.0509591
$178$	0	0
$179$	−3676.00	−1.53496	−0.767478	−	0.641075i	$-0.778489\pi$
$179$	−3676.00	−1.53496	−0.767478	+	0.641075i	$0.778489\pi$
$180$	0	0
$181$	−780.000	−0.320315	−0.160157	−	0.987092i	$-0.551200\pi$
$181$	−780.000	−0.320315	−0.160157	+	0.987092i	$0.551200\pi$
$182$	0	0
$183$	920.000	0.371630
$184$	0	0
$185$	750.000	0.298060
$186$	0	0
$187$	−175.000	−0.0684346
$188$	0	0
$189$	−371.000	−0.142785
$190$	0	0
$191$	−283.000	−0.107210	−0.0536051	−	0.998562i	$-0.517071\pi$
$191$	−283.000	−0.107210	−0.0536051	+	0.998562i	$0.517071\pi$
$192$	0	0
$193$	−484.000	−0.180513	−0.0902567	−	0.995919i	$-0.528769\pi$
$193$	−484.000	−0.180513	−0.0902567	+	0.995919i	$0.528769\pi$
$194$	0	0
$195$	−115.000	−0.0422324
$196$	0	0
$197$	360.000	0.130198	0.0650988	−	0.997879i	$-0.479264\pi$
$197$	360.000	0.130198	0.0650988	+	0.997879i	$0.479264\pi$
$198$	0	0
$199$	1032.00	0.367621	0.183810	−	0.982962i	$-0.441157\pi$
$199$	1032.00	0.367621	0.183810	+	0.982962i	$0.441157\pi$
$200$	0	0
$201$	−300.000	−0.105275
$202$	0	0
$203$	−203.000	−0.0701862
$204$	0	0
$205$	1020.00	0.347512
$206$	0	0
$207$	2236.00	0.750786
$208$	0	0
$209$	434.000	0.143638
$210$	0	0
$211$	513.000	0.167376	0.0836881	−	0.996492i	$-0.473330\pi$
$211$	513.000	0.167376	0.0836881	+	0.996492i	$0.473330\pi$
$212$	0	0
$213$	−520.000	−0.167276
$214$	0	0
$215$	890.000	0.282314
$216$	0	0
$217$	84.0000	0.0262778
$218$	0	0
$219$	−370.000	−0.114166
$220$	0	0
$221$	−575.000	−0.175017
$222$	0	0
$223$	−907.000	−0.272364	−0.136182	−	0.990684i	$-0.543483\pi$
$223$	−907.000	−0.272364	−0.136182	+	0.990684i	$0.543483\pi$
$224$	0	0
$225$	−650.000	−0.192593
$226$	0	0
$227$	−5701.00	−1.66691	−0.833455	−	0.552587i	$-0.813641\pi$
$227$	−5701.00	−1.66691	−0.833455	+	0.552587i	$0.813641\pi$
$228$	0	0
$229$	2044.00	0.589831	0.294916	−	0.955523i	$-0.404708\pi$
$229$	2044.00	0.589831	0.294916	+	0.955523i	$0.404708\pi$
$230$	0	0
$231$	49.0000	0.0139566
$232$	0	0
$233$	−2728.00	−0.767027	−0.383513	−	0.923535i	$-0.625286\pi$
$233$	−2728.00	−0.767027	−0.383513	+	0.923535i	$0.625286\pi$
$234$	0	0
$235$	165.000	0.0458018
$236$	0	0
$237$	1013.00	0.277643
$238$	0	0
$239$	7095.00	1.92024	0.960120	−	0.279588i	$-0.0901979\pi$
$239$	7095.00	1.92024	0.960120	+	0.279588i	$0.0901979\pi$
$240$	0	0
$241$	−2618.00	−0.699752	−0.349876	−	0.936796i	$-0.613776\pi$
$241$	−2618.00	−0.699752	−0.349876	+	0.936796i	$0.613776\pi$
$242$	0	0
$243$	−2080.00	−0.549103
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	1426.00	0.367345
$248$	0	0
$249$	−636.000	−0.161867
$250$	0	0
$251$	3318.00	0.834384	0.417192	−	0.908818i	$-0.363014\pi$
$251$	3318.00	0.834384	0.417192	+	0.908818i	$0.363014\pi$
$252$	0	0
$253$	−602.000	−0.149595
$254$	0	0
$255$	125.000	0.0306973
$256$	0	0
$257$	1678.00	0.407279	0.203640	−	0.979046i	$-0.434723\pi$
$257$	1678.00	0.407279	0.203640	+	0.979046i	$0.434723\pi$
$258$	0	0
$259$	−1050.00	−0.251907
$260$	0	0
$261$	−754.000	−0.178818
$262$	0	0
$263$	−898.000	−0.210544	−0.105272	−	0.994443i	$-0.533571\pi$
$263$	−898.000	−0.210544	−0.105272	+	0.994443i	$0.533571\pi$
$264$	0	0
$265$	−2260.00	−0.523889
$266$	0	0
$267$	−292.000	−0.0669293
$268$	0	0
$269$	4902.00	1.11108	0.555539	−	0.831490i	$-0.312512\pi$
$269$	4902.00	1.11108	0.555539	+	0.831490i	$0.312512\pi$
$270$	0	0
$271$	−7040.00	−1.57804	−0.789021	−	0.614366i	$-0.789412\pi$
$271$	−7040.00	−1.57804	−0.789021	+	0.614366i	$0.789412\pi$
$272$	0	0
$273$	161.000	0.0356929
$274$	0	0
$275$	175.000	0.0383742
$276$	0	0
$277$	2402.00	0.521019	0.260509	−	0.965471i	$-0.416110\pi$
$277$	2402.00	0.521019	0.260509	+	0.965471i	$0.416110\pi$
$278$	0	0
$279$	312.000	0.0669496
$280$	0	0
$281$	−6525.00	−1.38523	−0.692614	−	0.721309i	$-0.743541\pi$
$281$	−6525.00	−1.38523	−0.692614	+	0.721309i	$0.743541\pi$
$282$	0	0
$283$	4109.00	0.863091	0.431545	−	0.902091i	$-0.357968\pi$
$283$	4109.00	0.863091	0.431545	+	0.902091i	$0.357968\pi$
$284$	0	0
$285$	−310.000	−0.0644309
$286$	0	0
$287$	−1428.00	−0.293701
$288$	0	0
$289$	−4288.00	−0.872786
$290$	0	0
$291$	1381.00	0.278198
$292$	0	0
$293$	−3495.00	−0.696860	−0.348430	−	0.937335i	$-0.613285\pi$
$293$	−3495.00	−0.696860	−0.348430	+	0.937335i	$0.613285\pi$
$294$	0	0
$295$	−600.000	−0.118418
$296$	0	0
$297$	371.000	0.0724835
$298$	0	0
$299$	−1978.00	−0.382578
$300$	0	0
$301$	−1246.00	−0.238599
$302$	0	0
$303$	1278.00	0.242308
$304$	0	0
$305$	−4600.00	−0.863591
$306$	0	0
$307$	−803.000	−0.149282	−0.0746411	−	0.997210i	$-0.523781\pi$
$307$	−803.000	−0.149282	−0.0746411	+	0.997210i	$0.523781\pi$
$308$	0	0
$309$	875.000	0.161091
$310$	0	0
$311$	−5438.00	−0.991513	−0.495757	−	0.868461i	$-0.665109\pi$
$311$	−5438.00	−0.991513	−0.495757	+	0.868461i	$0.665109\pi$
$312$	0	0
$313$	681.000	0.122979	0.0614895	−	0.998108i	$-0.480415\pi$
$313$	681.000	0.122979	0.0614895	+	0.998108i	$0.480415\pi$
$314$	0	0
$315$	910.000	0.162770
$316$	0	0
$317$	−9214.00	−1.63252	−0.816262	−	0.577683i	$-0.803957\pi$
$317$	−9214.00	−1.63252	−0.816262	+	0.577683i	$0.803957\pi$
$318$	0	0
$319$	203.000	0.0356295
$320$	0	0
$321$	−1986.00	−0.345320
$322$	0	0
$323$	−1550.00	−0.267010
$324$	0	0
$325$	575.000	0.0981393
$326$	0	0
$327$	−223.000	−0.0377123
$328$	0	0
$329$	−231.000	−0.0387096
$330$	0	0
$331$	2700.00	0.448355	0.224177	−	0.974548i	$-0.428031\pi$
$331$	2700.00	0.448355	0.224177	+	0.974548i	$0.428031\pi$
$332$	0	0
$333$	−3900.00	−0.641798
$334$	0	0
$335$	1500.00	0.244638
$336$	0	0
$337$	−3834.00	−0.619737	−0.309868	−	0.950779i	$-0.600285\pi$
$337$	−3834.00	−0.619737	−0.309868	+	0.950779i	$0.600285\pi$
$338$	0	0
$339$	930.000	0.148999
$340$	0	0
$341$	−84.0000	−0.0133398
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	430.000	0.0671027
$346$	0	0
$347$	2626.00	0.406257	0.203128	−	0.979152i	$-0.434889\pi$
$347$	2626.00	0.406257	0.203128	+	0.979152i	$0.434889\pi$
$348$	0	0
$349$	−8354.00	−1.28132	−0.640658	−	0.767826i	$-0.721338\pi$
$349$	−8354.00	−1.28132	−0.640658	+	0.767826i	$0.721338\pi$
$350$	0	0
$351$	1219.00	0.185372
$352$	0	0
$353$	−8213.00	−1.23834	−0.619170	−	0.785257i	$-0.712531\pi$
$353$	−8213.00	−1.23834	−0.619170	+	0.785257i	$0.712531\pi$
$354$	0	0
$355$	2600.00	0.388715
$356$	0	0
$357$	−175.000	−0.0259439
$358$	0	0
$359$	6432.00	0.945593	0.472797	−	0.881172i	$-0.343245\pi$
$359$	6432.00	0.945593	0.472797	+	0.881172i	$0.343245\pi$
$360$	0	0
$361$	−3015.00	−0.439568
$362$	0	0
$363$	1282.00	0.185365
$364$	0	0
$365$	1850.00	0.265297
$366$	0	0
$367$	11617.0	1.65232	0.826161	−	0.563434i	$-0.190520\pi$
$367$	11617.0	1.65232	0.826161	+	0.563434i	$0.190520\pi$
$368$	0	0
$369$	−5304.00	−0.748280
$370$	0	0
$371$	3164.00	0.442767
$372$	0	0
$373$	−11084.0	−1.53863	−0.769313	−	0.638872i	$-0.779401\pi$
$373$	−11084.0	−1.53863	−0.769313	+	0.638872i	$0.779401\pi$
$374$	0	0
$375$	−125.000	−0.0172133
$376$	0	0
$377$	667.000	0.0911200
$378$	0	0
$379$	−12292.0	−1.66596	−0.832978	−	0.553306i	$-0.813366\pi$
$379$	−12292.0	−1.66596	−0.832978	+	0.553306i	$0.813366\pi$
$380$	0	0
$381$	2056.00	0.276462
$382$	0	0
$383$	−10820.0	−1.44354	−0.721770	−	0.692133i	$-0.756671\pi$
$383$	−10820.0	−1.44354	−0.721770	+	0.692133i	$0.756671\pi$
$384$	0	0
$385$	−245.000	−0.0324321
$386$	0	0
$387$	−4628.00	−0.607893
$388$	0	0
$389$	−7501.00	−0.977676	−0.488838	−	0.872375i	$-0.662579\pi$
$389$	−7501.00	−0.977676	−0.488838	+	0.872375i	$0.662579\pi$
$390$	0	0
$391$	2150.00	0.278082
$392$	0	0
$393$	1442.00	0.185087
$394$	0	0
$395$	−5065.00	−0.645184
$396$	0	0
$397$	−7899.00	−0.998588	−0.499294	−	0.866433i	$-0.666407\pi$
$397$	−7899.00	−0.998588	−0.499294	+	0.866433i	$0.666407\pi$
$398$	0	0
$399$	434.000	0.0544541
$400$	0	0
$401$	1753.00	0.218306	0.109153	−	0.994025i	$-0.465186\pi$
$401$	1753.00	0.218306	0.109153	+	0.994025i	$0.465186\pi$
$402$	0	0
$403$	−276.000	−0.0341155
$404$	0	0
$405$	3245.00	0.398137
$406$	0	0
$407$	1050.00	0.127879
$408$	0	0
$409$	−11834.0	−1.43069	−0.715347	−	0.698770i	$-0.753731\pi$
$409$	−11834.0	−1.43069	−0.715347	+	0.698770i	$0.753731\pi$
$410$	0	0
$411$	908.000	0.108974
$412$	0	0
$413$	840.000	0.100082
$414$	0	0
$415$	3180.00	0.376145
$416$	0	0
$417$	−1690.00	−0.198464
$418$	0	0
$419$	−6908.00	−0.805436	−0.402718	−	0.915324i	$-0.631935\pi$
$419$	−6908.00	−0.805436	−0.402718	+	0.915324i	$0.631935\pi$
$420$	0	0
$421$	−8377.00	−0.969762	−0.484881	−	0.874580i	$-0.661137\pi$
$421$	−8377.00	−0.969762	−0.484881	+	0.874580i	$0.661137\pi$
$422$	0	0
$423$	−858.000	−0.0986227
$424$	0	0
$425$	−625.000	−0.0713340
$426$	0	0
$427$	6440.00	0.729868
$428$	0	0
$429$	−161.000	−0.0181192
$430$	0	0
$431$	−4287.00	−0.479113	−0.239556	−	0.970882i	$-0.577002\pi$
$431$	−4287.00	−0.479113	−0.239556	+	0.970882i	$0.577002\pi$
$432$	0	0
$433$	15874.0	1.76179	0.880896	−	0.473310i	$-0.156941\pi$
$433$	15874.0	1.76179	0.880896	+	0.473310i	$0.156941\pi$
$434$	0	0
$435$	−145.000	−0.0159821
$436$	0	0
$437$	−5332.00	−0.583671
$438$	0	0
$439$	−7646.00	−0.831261	−0.415631	−	0.909534i	$-0.636439\pi$
$439$	−7646.00	−0.831261	−0.415631	+	0.909534i	$0.636439\pi$
$440$	0	0
$441$	−1274.00	−0.137566
$442$	0	0
$443$	1626.00	0.174387	0.0871937	−	0.996191i	$-0.472210\pi$
$443$	1626.00	0.174387	0.0871937	+	0.996191i	$0.472210\pi$
$444$	0	0
$445$	1460.00	0.155530
$446$	0	0
$447$	−2014.00	−0.213107
$448$	0	0
$449$	−841.000	−0.0883948	−0.0441974	−	0.999023i	$-0.514073\pi$
$449$	−841.000	−0.0883948	−0.0441974	+	0.999023i	$0.514073\pi$
$450$	0	0
$451$	1428.00	0.149095
$452$	0	0
$453$	−619.000	−0.0642012
$454$	0	0
$455$	−805.000	−0.0829428
$456$	0	0
$457$	15424.0	1.57878	0.789392	−	0.613889i	$-0.210396\pi$
$457$	15424.0	1.57878	0.789392	+	0.613889i	$0.210396\pi$
$458$	0	0
$459$	−1325.00	−0.134740
$460$	0	0
$461$	−10568.0	−1.06768	−0.533840	−	0.845585i	$-0.679252\pi$
$461$	−10568.0	−1.06768	−0.533840	+	0.845585i	$0.679252\pi$
$462$	0	0
$463$	4368.00	0.438441	0.219220	−	0.975675i	$-0.429649\pi$
$463$	4368.00	0.438441	0.219220	+	0.975675i	$0.429649\pi$
$464$	0	0
$465$	60.0000	0.00598373
$466$	0	0
$467$	4329.00	0.428956	0.214478	−	0.976729i	$-0.431195\pi$
$467$	4329.00	0.428956	0.214478	+	0.976729i	$0.431195\pi$
$468$	0	0
$469$	−2100.00	−0.206757
$470$	0	0
$471$	1446.00	0.141461
$472$	0	0
$473$	1246.00	0.121123
$474$	0	0
$475$	1550.00	0.149724
$476$	0	0
$477$	11752.0	1.12807
$478$	0	0
$479$	−11062.0	−1.05519	−0.527595	−	0.849496i	$-0.676906\pi$
$479$	−11062.0	−1.05519	−0.527595	+	0.849496i	$0.676906\pi$
$480$	0	0
$481$	3450.00	0.327040
$482$	0	0
$483$	−602.000	−0.0567121
$484$	0	0
$485$	−6905.00	−0.646474
$486$	0	0
$487$	−18698.0	−1.73981	−0.869905	−	0.493220i	$-0.835820\pi$
$487$	−18698.0	−1.73981	−0.869905	+	0.493220i	$0.835820\pi$
$488$	0	0
$489$	−2638.00	−0.243956
$490$	0	0
$491$	−3003.00	−0.276015	−0.138008	−	0.990431i	$-0.544070\pi$
$491$	−3003.00	−0.276015	−0.138008	+	0.990431i	$0.544070\pi$
$492$	0	0
$493$	−725.000	−0.0662320
$494$	0	0
$495$	−910.000	−0.0826292
$496$	0	0
$497$	−3640.00	−0.328524
$498$	0	0
$499$	3291.00	0.295241	0.147621	−	0.989044i	$-0.452839\pi$
$499$	3291.00	0.295241	0.147621	+	0.989044i	$0.452839\pi$
$500$	0	0
$501$	11.0000	0.000980926 0
$502$	0	0
$503$	−12533.0	−1.11097	−0.555486	−	0.831526i	$-0.687468\pi$
$503$	−12533.0	−1.11097	−0.555486	+	0.831526i	$0.687468\pi$
$504$	0	0
$505$	−6390.00	−0.563072
$506$	0	0
$507$	1668.00	0.146111
$508$	0	0
$509$	2406.00	0.209517	0.104758	−	0.994498i	$-0.466593\pi$
$509$	2406.00	0.209517	0.104758	+	0.994498i	$0.466593\pi$
$510$	0	0
$511$	−2590.00	−0.224217
$512$	0	0
$513$	3286.00	0.282808
$514$	0	0
$515$	−4375.00	−0.374341
$516$	0	0
$517$	231.000	0.0196506
$518$	0	0
$519$	3267.00	0.276311
$520$	0	0
$521$	−10098.0	−0.849139	−0.424569	−	0.905395i	$-0.639575\pi$
$521$	−10098.0	−0.849139	−0.424569	+	0.905395i	$0.639575\pi$
$522$	0	0
$523$	16100.0	1.34609	0.673044	−	0.739603i	$-0.264987\pi$
$523$	16100.0	1.34609	0.673044	+	0.739603i	$0.264987\pi$
$524$	0	0
$525$	175.000	0.0145479
$526$	0	0
$527$	300.000	0.0247974
$528$	0	0
$529$	−4771.00	−0.392126
$530$	0	0
$531$	3120.00	0.254984
$532$	0	0
$533$	4692.00	0.381300
$534$	0	0
$535$	9930.00	0.802451
$536$	0	0
$537$	3676.00	0.295402
$538$	0	0
$539$	343.000	0.0274101
$540$	0	0
$541$	−16835.0	−1.33788	−0.668940	−	0.743316i	$-0.733252\pi$
$541$	−16835.0	−1.33788	−0.668940	+	0.743316i	$0.733252\pi$
$542$	0	0
$543$	780.000	0.0616446
$544$	0	0
$545$	1115.00	0.0876355
$546$	0	0
$547$	4104.00	0.320794	0.160397	−	0.987053i	$-0.448723\pi$
$547$	4104.00	0.320794	0.160397	+	0.987053i	$0.448723\pi$
$548$	0	0
$549$	23920.0	1.85953
$550$	0	0
$551$	1798.00	0.139015
$552$	0	0
$553$	7091.00	0.545280
$554$	0	0
$555$	−750.000	−0.0573617
$556$	0	0
$557$	20280.0	1.54271	0.771357	−	0.636403i	$-0.219579\pi$
$557$	20280.0	1.54271	0.771357	+	0.636403i	$0.219579\pi$
$558$	0	0
$559$	4094.00	0.309763
$560$	0	0
$561$	175.000	0.0131702
$562$	0	0
$563$	17092.0	1.27947	0.639735	−	0.768595i	$-0.279044\pi$
$563$	17092.0	1.27947	0.639735	+	0.768595i	$0.279044\pi$
$564$	0	0
$565$	−4650.00	−0.346242
$566$	0	0
$567$	−4543.00	−0.336487
$568$	0	0
$569$	242.000	0.0178298	0.00891491	−	0.999960i	$-0.497162\pi$
$569$	242.000	0.0178298	0.00891491	+	0.999960i	$0.497162\pi$
$570$	0	0
$571$	−1180.00	−0.0864824	−0.0432412	−	0.999065i	$-0.513768\pi$
$571$	−1180.00	−0.0864824	−0.0432412	+	0.999065i	$0.513768\pi$
$572$	0	0
$573$	283.000	0.0206326
$574$	0	0
$575$	−2150.00	−0.155933
$576$	0	0
$577$	18421.0	1.32907	0.664537	−	0.747255i	$-0.268629\pi$
$577$	18421.0	1.32907	0.664537	+	0.747255i	$0.268629\pi$
$578$	0	0
$579$	484.000	0.0347398
$580$	0	0
$581$	−4452.00	−0.317900
$582$	0	0
$583$	−3164.00	−0.224768
$584$	0	0
$585$	−2990.00	−0.211318
$586$	0	0
$587$	19776.0	1.39053	0.695267	−	0.718752i	$-0.255286\pi$
$587$	19776.0	1.39053	0.695267	+	0.718752i	$0.255286\pi$
$588$	0	0
$589$	−744.000	−0.0520475
$590$	0	0
$591$	−360.000	−0.0250566
$592$	0	0
$593$	−2451.00	−0.169731	−0.0848655	−	0.996392i	$-0.527046\pi$
$593$	−2451.00	−0.169731	−0.0848655	+	0.996392i	$0.527046\pi$
$594$	0	0
$595$	875.000	0.0602882
$596$	0	0
$597$	−1032.00	−0.0707487
$598$	0	0
$599$	18081.0	1.23334	0.616669	−	0.787222i	$-0.288482\pi$
$599$	18081.0	1.23334	0.616669	+	0.787222i	$0.288482\pi$
$600$	0	0
$601$	18238.0	1.23784	0.618921	−	0.785453i	$-0.287570\pi$
$601$	18238.0	1.23784	0.618921	+	0.785453i	$0.287570\pi$
$602$	0	0
$603$	−7800.00	−0.526767
$604$	0	0
$605$	−6410.00	−0.430750
$606$	0	0
$607$	−10925.0	−0.730531	−0.365265	−	0.930903i	$-0.619022\pi$
$607$	−10925.0	−0.730531	−0.365265	+	0.930903i	$0.619022\pi$
$608$	0	0
$609$	203.000	0.0135073
$610$	0	0
$611$	759.000	0.0502551
$612$	0	0
$613$	−1506.00	−0.0992280	−0.0496140	−	0.998768i	$-0.515799\pi$
$613$	−1506.00	−0.0992280	−0.0496140	+	0.998768i	$0.515799\pi$
$614$	0	0
$615$	−1020.00	−0.0668787
$616$	0	0
$617$	−24142.0	−1.57524	−0.787618	−	0.616164i	$-0.788686\pi$
$617$	−24142.0	−1.57524	−0.787618	+	0.616164i	$0.788686\pi$
$618$	0	0
$619$	16558.0	1.07516	0.537579	−	0.843214i	$-0.319339\pi$
$619$	16558.0	1.07516	0.537579	+	0.843214i	$0.319339\pi$
$620$	0	0
$621$	−4558.00	−0.294535
$622$	0	0
$623$	−2044.00	−0.131446
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−434.000	−0.0276432
$628$	0	0
$629$	−3750.00	−0.237714
$630$	0	0
$631$	25025.0	1.57881	0.789405	−	0.613872i	$-0.210389\pi$
$631$	25025.0	1.57881	0.789405	+	0.613872i	$0.210389\pi$
$632$	0	0
$633$	−513.000	−0.0322116
$634$	0	0
$635$	−10280.0	−0.642440
$636$	0	0
$637$	1127.00	0.0700995
$638$	0	0
$639$	−13520.0	−0.837000
$640$	0	0
$641$	9154.00	0.564058	0.282029	−	0.959406i	$-0.408993\pi$
$641$	9154.00	0.564058	0.282029	+	0.959406i	$0.408993\pi$
$642$	0	0
$643$	−8225.00	−0.504452	−0.252226	−	0.967668i	$-0.581163\pi$
$643$	−8225.00	−0.504452	−0.252226	+	0.967668i	$0.581163\pi$
$644$	0	0
$645$	−890.000	−0.0543313
$646$	0	0
$647$	23688.0	1.43937	0.719684	−	0.694302i	$-0.244287\pi$
$647$	23688.0	1.43937	0.719684	+	0.694302i	$0.244287\pi$
$648$	0	0
$649$	−840.000	−0.0508057
$650$	0	0
$651$	−84.0000	−0.00505717
$652$	0	0
$653$	14518.0	0.870036	0.435018	−	0.900422i	$-0.356742\pi$
$653$	14518.0	0.870036	0.435018	+	0.900422i	$0.356742\pi$
$654$	0	0
$655$	−7210.00	−0.430104
$656$	0	0
$657$	−9620.00	−0.571251
$658$	0	0
$659$	−30381.0	−1.79587	−0.897933	−	0.440132i	$-0.854932\pi$
$659$	−30381.0	−1.79587	−0.897933	+	0.440132i	$0.854932\pi$
$660$	0	0
$661$	−25568.0	−1.50451	−0.752254	−	0.658873i	$-0.771033\pi$
$661$	−25568.0	−1.50451	−0.752254	+	0.658873i	$0.771033\pi$
$662$	0	0
$663$	575.000	0.0336820
$664$	0	0
$665$	−2170.00	−0.126540
$666$	0	0
$667$	−2494.00	−0.144780
$668$	0	0
$669$	907.000	0.0524165
$670$	0	0
$671$	−6440.00	−0.370512
$672$	0	0
$673$	19428.0	1.11277	0.556385	−	0.830925i	$-0.312188\pi$
$673$	19428.0	1.11277	0.556385	+	0.830925i	$0.312188\pi$
$674$	0	0
$675$	1325.00	0.0755545
$676$	0	0
$677$	−937.000	−0.0531933	−0.0265966	−	0.999646i	$-0.508467\pi$
$677$	−937.000	−0.0531933	−0.0265966	+	0.999646i	$0.508467\pi$
$678$	0	0
$679$	9667.00	0.546370
$680$	0	0
$681$	5701.00	0.320797
$682$	0	0
$683$	−15960.0	−0.894132	−0.447066	−	0.894501i	$-0.647531\pi$
$683$	−15960.0	−0.894132	−0.447066	+	0.894501i	$0.647531\pi$
$684$	0	0
$685$	−4540.00	−0.253233
$686$	0	0
$687$	−2044.00	−0.113513
$688$	0	0
$689$	−10396.0	−0.574827
$690$	0	0
$691$	−10260.0	−0.564846	−0.282423	−	0.959290i	$-0.591138\pi$
$691$	−10260.0	−0.564846	−0.282423	+	0.959290i	$0.591138\pi$
$692$	0	0
$693$	1274.00	0.0698344
$694$	0	0
$695$	8450.00	0.461190
$696$	0	0
$697$	−5100.00	−0.277154
$698$	0	0
$699$	2728.00	0.147614
$700$	0	0
$701$	15465.0	0.833245	0.416623	−	0.909080i	$-0.363214\pi$
$701$	15465.0	0.833245	0.416623	+	0.909080i	$0.363214\pi$
$702$	0	0
$703$	9300.00	0.498942
$704$	0	0
$705$	−165.000	−0.00881455
$706$	0	0
$707$	8946.00	0.475883
$708$	0	0
$709$	21005.0	1.11264	0.556318	−	0.830969i	$-0.312214\pi$
$709$	21005.0	1.11264	0.556318	+	0.830969i	$0.312214\pi$
$710$	0	0
$711$	26338.0	1.38924
$712$	0	0
$713$	1032.00	0.0542058
$714$	0	0
$715$	805.000	0.0421053
$716$	0	0
$717$	−7095.00	−0.369550
$718$	0	0
$719$	−22.0000	−0.00114111	−0.000570557	−	1.00000i	$-0.500182\pi$
$719$	−22.0000	−0.00114111	−0.000570557		1.00000i	$0.500182\pi$
$720$	0	0
$721$	6125.00	0.316376
$722$	0	0
$723$	2618.00	0.134667
$724$	0	0
$725$	725.000	0.0371391
$726$	0	0
$727$	−1816.00	−0.0926433	−0.0463217	−	0.998927i	$-0.514750\pi$
$727$	−1816.00	−0.0926433	−0.0463217	+	0.998927i	$0.514750\pi$
$728$	0	0
$729$	−15443.0	−0.784586
$730$	0	0
$731$	−4450.00	−0.225156
$732$	0	0
$733$	−24397.0	−1.22936	−0.614682	−	0.788775i	$-0.710716\pi$
$733$	−24397.0	−1.22936	−0.614682	+	0.788775i	$0.710716\pi$
$734$	0	0
$735$	−245.000	−0.0122952
$736$	0	0
$737$	2100.00	0.104959
$738$	0	0
$739$	−6065.00	−0.301901	−0.150950	−	0.988541i	$-0.548233\pi$
$739$	−6065.00	−0.301901	−0.150950	+	0.988541i	$0.548233\pi$
$740$	0	0
$741$	−1426.00	−0.0706956
$742$	0	0
$743$	3660.00	0.180717	0.0903583	−	0.995909i	$-0.471199\pi$
$743$	3660.00	0.180717	0.0903583	+	0.995909i	$0.471199\pi$
$744$	0	0
$745$	10070.0	0.495216
$746$	0	0
$747$	−16536.0	−0.809934
$748$	0	0
$749$	−13902.0	−0.678195
$750$	0	0
$751$	−27077.0	−1.31565	−0.657825	−	0.753170i	$-0.728524\pi$
$751$	−27077.0	−1.31565	−0.657825	+	0.753170i	$0.728524\pi$
$752$	0	0
$753$	−3318.00	−0.160577
$754$	0	0
$755$	3095.00	0.149190
$756$	0	0
$757$	31424.0	1.50875	0.754376	−	0.656443i	$-0.227940\pi$
$757$	31424.0	1.50875	0.754376	+	0.656443i	$0.227940\pi$
$758$	0	0
$759$	602.000	0.0287895
$760$	0	0
$761$	−16898.0	−0.804930	−0.402465	−	0.915435i	$-0.631847\pi$
$761$	−16898.0	−0.804930	−0.402465	+	0.915435i	$0.631847\pi$
$762$	0	0
$763$	−1561.00	−0.0740655
$764$	0	0
$765$	3250.00	0.153600
$766$	0	0
$767$	−2760.00	−0.129932
$768$	0	0
$769$	16294.0	0.764079	0.382039	−	0.924146i	$-0.375222\pi$
$769$	16294.0	0.764079	0.382039	+	0.924146i	$0.375222\pi$
$770$	0	0
$771$	−1678.00	−0.0783809
$772$	0	0
$773$	35279.0	1.64152	0.820762	−	0.571271i	$-0.193549\pi$
$773$	35279.0	1.64152	0.820762	+	0.571271i	$0.193549\pi$
$774$	0	0
$775$	−300.000	−0.0139049
$776$	0	0
$777$	1050.00	0.0484795
$778$	0	0
$779$	12648.0	0.581722
$780$	0	0
$781$	3640.00	0.166773
$782$	0	0
$783$	1537.00	0.0701506
$784$	0	0
$785$	−7230.00	−0.328726
$786$	0	0
$787$	−37829.0	−1.71342	−0.856708	−	0.515802i	$-0.827494\pi$
$787$	−37829.0	−1.71342	−0.856708	+	0.515802i	$0.827494\pi$
$788$	0	0
$789$	898.000	0.0405192
$790$	0	0
$791$	6510.00	0.292628
$792$	0	0
$793$	−21160.0	−0.947558
$794$	0	0
$795$	2260.00	0.100823
$796$	0	0
$797$	27741.0	1.23292	0.616460	−	0.787387i	$-0.288566\pi$
$797$	27741.0	1.23292	0.616460	+	0.787387i	$0.288566\pi$
$798$	0	0
$799$	−825.000	−0.0365287
$800$	0	0
$801$	−7592.00	−0.334894
$802$	0	0
$803$	2590.00	0.113822
$804$	0	0
$805$	3010.00	0.131787
$806$	0	0
$807$	−4902.00	−0.213827
$808$	0	0
$809$	27521.0	1.19603	0.598014	−	0.801486i	$-0.295957\pi$
$809$	27521.0	1.19603	0.598014	+	0.801486i	$0.295957\pi$
$810$	0	0
$811$	10694.0	0.463030	0.231515	−	0.972831i	$-0.425632\pi$
$811$	10694.0	0.463030	0.231515	+	0.972831i	$0.425632\pi$
$812$	0	0
$813$	7040.00	0.303694
$814$	0	0
$815$	13190.0	0.566903
$816$	0	0
$817$	11036.0	0.472584
$818$	0	0
$819$	4186.00	0.178597
$820$	0	0
$821$	13835.0	0.588118	0.294059	−	0.955787i	$-0.404994\pi$
$821$	13835.0	0.588118	0.294059	+	0.955787i	$0.404994\pi$
$822$	0	0
$823$	−5196.00	−0.220074	−0.110037	−	0.993927i	$-0.535097\pi$
$823$	−5196.00	−0.220074	−0.110037	+	0.993927i	$0.535097\pi$
$824$	0	0
$825$	−175.000	−0.00738511
$826$	0	0
$827$	22346.0	0.939597	0.469798	−	0.882774i	$-0.344327\pi$
$827$	22346.0	0.939597	0.469798	+	0.882774i	$0.344327\pi$
$828$	0	0
$829$	22044.0	0.923546	0.461773	−	0.886998i	$-0.347213\pi$
$829$	22044.0	0.923546	0.461773	+	0.886998i	$0.347213\pi$
$830$	0	0
$831$	−2402.00	−0.100270
$832$	0	0
$833$	−1225.00	−0.0509529
$834$	0	0
$835$	−55.0000	−0.00227947
$836$	0	0
$837$	−636.000	−0.0262645
$838$	0	0
$839$	21566.0	0.887415	0.443707	−	0.896172i	$-0.353663\pi$
$839$	21566.0	0.887415	0.443707	+	0.896172i	$0.353663\pi$
$840$	0	0
$841$	−23548.0	−0.965517
$842$	0	0
$843$	6525.00	0.266587
$844$	0	0
$845$	−8340.00	−0.339532
$846$	0	0
$847$	8974.00	0.364050
$848$	0	0
$849$	−4109.00	−0.166102
$850$	0	0
$851$	−12900.0	−0.519631
$852$	0	0
$853$	6098.00	0.244773	0.122387	−	0.992483i	$-0.460945\pi$
$853$	6098.00	0.244773	0.122387	+	0.992483i	$0.460945\pi$
$854$	0	0
$855$	−8060.00	−0.322393
$856$	0	0
$857$	40514.0	1.61486	0.807428	−	0.589966i	$-0.200859\pi$
$857$	40514.0	1.61486	0.807428	+	0.589966i	$0.200859\pi$
$858$	0	0
$859$	24884.0	0.988395	0.494197	−	0.869350i	$-0.335462\pi$
$859$	24884.0	0.988395	0.494197	+	0.869350i	$0.335462\pi$
$860$	0	0
$861$	1428.00	0.0565228
$862$	0	0
$863$	−35968.0	−1.41873	−0.709366	−	0.704841i	$-0.751018\pi$
$863$	−35968.0	−1.41873	−0.709366	+	0.704841i	$0.751018\pi$
$864$	0	0
$865$	−16335.0	−0.642089
$866$	0	0
$867$	4288.00	0.167968
$868$	0	0
$869$	−7091.00	−0.276807
$870$	0	0
$871$	6900.00	0.268424
$872$	0	0
$873$	35906.0	1.39202
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	9790.00	0.376950	0.188475	−	0.982078i	$-0.439646\pi$
$877$	9790.00	0.376950	0.188475	+	0.982078i	$0.439646\pi$
$878$	0	0
$879$	3495.00	0.134111
$880$	0	0
$881$	−47008.0	−1.79766	−0.898831	−	0.438296i	$-0.855582\pi$
$881$	−47008.0	−1.79766	−0.898831	+	0.438296i	$0.855582\pi$
$882$	0	0
$883$	−23828.0	−0.908127	−0.454063	−	0.890969i	$-0.650026\pi$
$883$	−23828.0	−0.908127	−0.454063	+	0.890969i	$0.650026\pi$
$884$	0	0
$885$	600.000	0.0227896
$886$	0	0
$887$	22616.0	0.856112	0.428056	−	0.903752i	$-0.359199\pi$
$887$	22616.0	0.856112	0.428056	+	0.903752i	$0.359199\pi$
$888$	0	0
$889$	14392.0	0.542961
$890$	0	0
$891$	4543.00	0.170815
$892$	0	0
$893$	2046.00	0.0766705
$894$	0	0
$895$	−18380.0	−0.686453
$896$	0	0
$897$	1978.00	0.0736271
$898$	0	0
$899$	−348.000	−0.0129104
$900$	0	0
$901$	11300.0	0.417822
$902$	0	0
$903$	1246.00	0.0459184
$904$	0	0
$905$	−3900.00	−0.143249
$906$	0	0
$907$	40406.0	1.47923	0.739614	−	0.673032i	$-0.235008\pi$
$907$	40406.0	1.47923	0.739614	+	0.673032i	$0.235008\pi$
$908$	0	0
$909$	33228.0	1.21243
$910$	0	0
$911$	6368.00	0.231593	0.115797	−	0.993273i	$-0.463058\pi$
$911$	6368.00	0.231593	0.115797	+	0.993273i	$0.463058\pi$
$912$	0	0
$913$	4452.00	0.161380
$914$	0	0
$915$	4600.00	0.166198
$916$	0	0
$917$	10094.0	0.363504
$918$	0	0
$919$	−42305.0	−1.51851	−0.759256	−	0.650792i	$-0.774437\pi$
$919$	−42305.0	−1.51851	−0.759256	+	0.650792i	$0.774437\pi$
$920$	0	0
$921$	803.000	0.0287294
$922$	0	0
$923$	11960.0	0.426509
$924$	0	0
$925$	3750.00	0.133296
$926$	0	0
$927$	22750.0	0.806050
$928$	0	0
$929$	38668.0	1.36561	0.682807	−	0.730599i	$-0.260759\pi$
$929$	38668.0	1.36561	0.682807	+	0.730599i	$0.260759\pi$
$930$	0	0
$931$	3038.00	0.106946
$932$	0	0
$933$	5438.00	0.190817
$934$	0	0
$935$	−875.000	−0.0306049
$936$	0	0
$937$	16171.0	0.563803	0.281902	−	0.959443i	$-0.409035\pi$
$937$	16171.0	0.563803	0.281902	+	0.959443i	$0.409035\pi$
$938$	0	0
$939$	−681.000	−0.0236673
$940$	0	0
$941$	18240.0	0.631888	0.315944	−	0.948778i	$-0.397679\pi$
$941$	18240.0	0.631888	0.315944	+	0.948778i	$0.397679\pi$
$942$	0	0
$943$	−17544.0	−0.605844
$944$	0	0
$945$	−1855.00	−0.0638552
$946$	0	0
$947$	−45928.0	−1.57599	−0.787993	−	0.615684i	$-0.788880\pi$
$947$	−45928.0	−1.57599	−0.787993	+	0.615684i	$0.788880\pi$
$948$	0	0
$949$	8510.00	0.291092
$950$	0	0
$951$	9214.00	0.314179
$952$	0	0
$953$	−14104.0	−0.479405	−0.239703	−	0.970846i	$-0.577050\pi$
$953$	−14104.0	−0.479405	−0.239703	+	0.970846i	$0.577050\pi$
$954$	0	0
$955$	−1415.00	−0.0479459
$956$	0	0
$957$	−203.000	−0.00685690
$958$	0	0
$959$	6356.00	0.214021
$960$	0	0
$961$	−29647.0	−0.995166
$962$	0	0
$963$	−51636.0	−1.72788
$964$	0	0
$965$	−2420.00	−0.0807280
$966$	0	0
$967$	26914.0	0.895032	0.447516	−	0.894276i	$-0.352309\pi$
$967$	26914.0	0.895032	0.447516	+	0.894276i	$0.352309\pi$
$968$	0	0
$969$	1550.00	0.0513861
$970$	0	0
$971$	−10408.0	−0.343984	−0.171992	−	0.985098i	$-0.555020\pi$
$971$	−10408.0	−0.343984	−0.171992	+	0.985098i	$0.555020\pi$
$972$	0	0
$973$	−11830.0	−0.389776
$974$	0	0
$975$	−575.000	−0.0188869
$976$	0	0
$977$	39510.0	1.29379	0.646897	−	0.762577i	$-0.276066\pi$
$977$	39510.0	1.29379	0.646897	+	0.762577i	$0.276066\pi$
$978$	0	0
$979$	2044.00	0.0667278
$980$	0	0
$981$	−5798.00	−0.188701
$982$	0	0
$983$	52131.0	1.69148	0.845738	−	0.533599i	$-0.179161\pi$
$983$	52131.0	1.69148	0.845738	+	0.533599i	$0.179161\pi$
$984$	0	0
$985$	1800.00	0.0582262
$986$	0	0
$987$	231.000	0.00744966
$988$	0	0
$989$	−15308.0	−0.492180
$990$	0	0
$991$	−48512.0	−1.55503	−0.777515	−	0.628865i	$-0.783520\pi$
$991$	−48512.0	−1.55503	−0.777515	+	0.628865i	$0.783520\pi$
$992$	0	0
$993$	−2700.00	−0.0862859
$994$	0	0
$995$	5160.00	0.164405
$996$	0	0
$997$	−40951.0	−1.30083	−0.650417	−	0.759577i	$-0.725406\pi$
$997$	−40951.0	−1.30083	−0.650417	+	0.759577i	$0.725406\pi$
$998$	0	0
$999$	7950.00	0.251779

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.s.1.1		1
4.3	odd	2		2240.4.a.u.1.1		1
8.3	odd	2		560.4.a.h.1.1		1
8.5	even	2		140.4.a.d.1.1	✓	1
24.5	odd	2		1260.4.a.i.1.1		1
40.13	odd	4		700.4.e.i.449.2		2
40.29	even	2		700.4.a.g.1.1		1
40.37	odd	4		700.4.e.i.449.1		2
56.5	odd	6		980.4.i.k.361.1		2
56.13	odd	2		980.4.a.g.1.1		1
56.37	even	6		980.4.i.i.361.1		2
56.45	odd	6		980.4.i.k.961.1		2
56.53	even	6		980.4.i.i.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.4.a.d.1.1	✓	1	8.5	even	2
560.4.a.h.1.1		1	8.3	odd	2
700.4.a.g.1.1		1	40.29	even	2
700.4.e.i.449.1		2	40.37	odd	4
700.4.e.i.449.2		2	40.13	odd	4
980.4.a.g.1.1		1	56.13	odd	2
980.4.i.i.361.1		2	56.37	even	6
980.4.i.i.961.1		2	56.53	even	6
980.4.i.k.361.1		2	56.5	odd	6
980.4.i.k.961.1		2	56.45	odd	6
1260.4.a.i.1.1		1	24.5	odd	2
2240.4.a.s.1.1		1	1.1	even	1	trivial
2240.4.a.u.1.1		1	4.3	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

Introduction

L-functions

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Representations

Groups

Database

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