LMFDB - Embedded newform 882.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,4,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.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:	$52.0396846251$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} +9.00000 q^{5} -8.00000 q^{8} -18.0000 q^{10} +57.0000 q^{11} -70.0000 q^{13} +16.0000 q^{16} -51.0000 q^{17} +5.00000 q^{19} +36.0000 q^{20} -114.000 q^{22} -69.0000 q^{23} -44.0000 q^{25} +140.000 q^{26} -114.000 q^{29} +23.0000 q^{31} -32.0000 q^{32} +102.000 q^{34} -253.000 q^{37} -10.0000 q^{38} -72.0000 q^{40} +42.0000 q^{41} -124.000 q^{43} +228.000 q^{44} +138.000 q^{46} -201.000 q^{47} +88.0000 q^{50} -280.000 q^{52} +393.000 q^{53} +513.000 q^{55} +228.000 q^{58} -219.000 q^{59} -709.000 q^{61} -46.0000 q^{62} +64.0000 q^{64} -630.000 q^{65} +419.000 q^{67} -204.000 q^{68} +96.0000 q^{71} -313.000 q^{73} +506.000 q^{74} +20.0000 q^{76} +461.000 q^{79} +144.000 q^{80} -84.0000 q^{82} +588.000 q^{83} -459.000 q^{85} +248.000 q^{86} -456.000 q^{88} +1017.00 q^{89} -276.000 q^{92} +402.000 q^{94} +45.0000 q^{95} -1834.00 q^{97} +O(q^{100})$

Coefficient data

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

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

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	9.00000	0.804984	0.402492	−	0.915423i	$-0.368144\pi$
$5$	9.00000	0.804984	0.402492	+	0.915423i	$0.368144\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−18.0000	−0.569210
$11$	57.0000	1.56238	0.781188	−	0.624295i	$-0.214614\pi$
$11$	57.0000	1.56238	0.781188	+	0.624295i	$0.214614\pi$
$12$	0	0
$13$	−70.0000	−1.49342	−0.746712	−	0.665148i	$-0.768369\pi$
$13$	−70.0000	−1.49342	−0.746712	+	0.665148i	$0.768369\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−51.0000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−51.0000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	5.00000	0.0603726	0.0301863	−	0.999544i	$-0.490390\pi$
$19$	5.00000	0.0603726	0.0301863	+	0.999544i	$0.490390\pi$
$20$	36.0000	0.402492
$21$	0	0
$22$	−114.000	−1.10477
$23$	−69.0000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−69.0000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−44.0000	−0.352000
$26$	140.000	1.05601
$27$	0	0
$28$	0	0
$29$	−114.000	−0.729975	−0.364987	−	0.931012i	$-0.618927\pi$
$29$	−114.000	−0.729975	−0.364987	+	0.931012i	$0.618927\pi$
$30$	0	0
$31$	23.0000	0.133256	0.0666278	−	0.997778i	$-0.478776\pi$
$31$	23.0000	0.133256	0.0666278	+	0.997778i	$0.478776\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	102.000	0.514496
$35$	0	0
$36$	0	0
$37$	−253.000	−1.12413	−0.562067	−	0.827092i	$-0.689994\pi$
$37$	−253.000	−1.12413	−0.562067	+	0.827092i	$0.689994\pi$
$38$	−10.0000	−0.0426898
$39$	0	0
$40$	−72.0000	−0.284605
$41$	42.0000	0.159983	0.0799914	−	0.996796i	$-0.474511\pi$
$41$	42.0000	0.159983	0.0799914	+	0.996796i	$0.474511\pi$
$42$	0	0
$43$	−124.000	−0.439763	−0.219882	−	0.975527i	$-0.570567\pi$
$43$	−124.000	−0.439763	−0.219882	+	0.975527i	$0.570567\pi$
$44$	228.000	0.781188
$45$	0	0
$46$	138.000	0.442326
$47$	−201.000	−0.623806	−0.311903	−	0.950114i	$-0.600966\pi$
$47$	−201.000	−0.623806	−0.311903	+	0.950114i	$0.600966\pi$
$48$	0	0
$49$	0	0
$50$	88.0000	0.248902
$51$	0	0
$52$	−280.000	−0.746712
$53$	393.000	1.01854	0.509271	−	0.860606i	$-0.329915\pi$
$53$	393.000	1.01854	0.509271	+	0.860606i	$0.329915\pi$
$54$	0	0
$55$	513.000	1.25769
$56$	0	0
$57$	0	0
$58$	228.000	0.516170
$59$	−219.000	−0.483244	−0.241622	−	0.970371i	$-0.577679\pi$
$59$	−219.000	−0.483244	−0.241622	+	0.970371i	$0.577679\pi$
$60$	0	0
$61$	−709.000	−1.48817	−0.744083	−	0.668087i	$-0.767113\pi$
$61$	−709.000	−1.48817	−0.744083	+	0.668087i	$0.767113\pi$
$62$	−46.0000	−0.0942259
$63$	0	0
$64$	64.0000	0.125000
$65$	−630.000	−1.20218
$66$	0	0
$67$	419.000	0.764015	0.382007	−	0.924159i	$-0.375233\pi$
$67$	419.000	0.764015	0.382007	+	0.924159i	$0.375233\pi$
$68$	−204.000	−0.363803
$69$	0	0
$70$	0	0
$71$	96.0000	0.160466	0.0802331	−	0.996776i	$-0.474434\pi$
$71$	96.0000	0.160466	0.0802331	+	0.996776i	$0.474434\pi$
$72$	0	0
$73$	−313.000	−0.501834	−0.250917	−	0.968009i	$-0.580732\pi$
$73$	−313.000	−0.501834	−0.250917	+	0.968009i	$0.580732\pi$
$74$	506.000	0.794883
$75$	0	0
$76$	20.0000	0.0301863
$77$	0	0
$78$	0	0
$79$	461.000	0.656539	0.328269	−	0.944584i	$-0.393535\pi$
$79$	461.000	0.656539	0.328269	+	0.944584i	$0.393535\pi$
$80$	144.000	0.201246
$81$	0	0
$82$	−84.0000	−0.113125
$83$	588.000	0.777607	0.388804	−	0.921321i	$-0.372888\pi$
$83$	588.000	0.777607	0.388804	+	0.921321i	$0.372888\pi$
$84$	0	0
$85$	−459.000	−0.585712
$86$	248.000	0.310960
$87$	0	0
$88$	−456.000	−0.552384
$89$	1017.00	1.21126	0.605628	−	0.795748i	$-0.292922\pi$
$89$	1017.00	1.21126	0.605628	+	0.795748i	$0.292922\pi$
$90$	0	0
$91$	0	0
$92$	−276.000	−0.312772
$93$	0	0
$94$	402.000	0.441097
$95$	45.0000	0.0485990
$96$	0	0
$97$	−1834.00	−1.91974	−0.959868	−	0.280451i	$-0.909516\pi$
$97$	−1834.00	−1.91974	−0.959868	+	0.280451i	$0.909516\pi$
$98$	0	0
$99$	0	0
$100$	−176.000	−0.176000
$101$	285.000	0.280778	0.140389	−	0.990096i	$-0.455165\pi$
$101$	285.000	0.280778	0.140389	+	0.990096i	$0.455165\pi$
$102$	0	0
$103$	−499.000	−0.477359	−0.238679	−	0.971098i	$-0.576714\pi$
$103$	−499.000	−0.477359	−0.238679	+	0.971098i	$0.576714\pi$
$104$	560.000	0.528005
$105$	0	0
$106$	−786.000	−0.720218
$107$	1107.00	1.00017	0.500083	−	0.865978i	$-0.333303\pi$
$107$	1107.00	1.00017	0.500083	+	0.865978i	$0.333303\pi$
$108$	0	0
$109$	923.000	0.811077	0.405538	−	0.914078i	$-0.367084\pi$
$109$	923.000	0.811077	0.405538	+	0.914078i	$0.367084\pi$
$110$	−1026.00	−0.889321
$111$	0	0
$112$	0	0
$113$	−1542.00	−1.28371	−0.641855	−	0.766826i	$-0.721835\pi$
$113$	−1542.00	−1.28371	−0.641855	+	0.766826i	$0.721835\pi$
$114$	0	0
$115$	−621.000	−0.503553
$116$	−456.000	−0.364987
$117$	0	0
$118$	438.000	0.341705
$119$	0	0
$120$	0	0
$121$	1918.00	1.44102
$122$	1418.00	1.05229
$123$	0	0
$124$	92.0000	0.0666278
$125$	−1521.00	−1.08834
$126$	0	0
$127$	−2056.00	−1.43654	−0.718270	−	0.695765i	$-0.755066\pi$
$127$	−2056.00	−1.43654	−0.718270	+	0.695765i	$0.755066\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	1260.00	0.850072
$131$	−2049.00	−1.36658	−0.683290	−	0.730147i	$-0.739451\pi$
$131$	−2049.00	−1.36658	−0.683290	+	0.730147i	$0.739451\pi$
$132$	0	0
$133$	0	0
$134$	−838.000	−0.540240
$135$	0	0
$136$	408.000	0.257248
$137$	141.000	0.0879302	0.0439651	−	0.999033i	$-0.486001\pi$
$137$	141.000	0.0879302	0.0439651	+	0.999033i	$0.486001\pi$
$138$	0	0
$139$	1484.00	0.905548	0.452774	−	0.891625i	$-0.350434\pi$
$139$	1484.00	0.905548	0.452774	+	0.891625i	$0.350434\pi$
$140$	0	0
$141$	0	0
$142$	−192.000	−0.113467
$143$	−3990.00	−2.33329
$144$	0	0
$145$	−1026.00	−0.587618
$146$	626.000	0.354850
$147$	0	0
$148$	−1012.00	−0.562067
$149$	57.0000	0.0313397	0.0156699	−	0.999877i	$-0.495012\pi$
$149$	57.0000	0.0313397	0.0156699	+	0.999877i	$0.495012\pi$
$150$	0	0
$151$	839.000	0.452165	0.226082	−	0.974108i	$-0.427408\pi$
$151$	839.000	0.452165	0.226082	+	0.974108i	$0.427408\pi$
$152$	−40.0000	−0.0213449
$153$	0	0
$154$	0	0
$155$	207.000	0.107269
$156$	0	0
$157$	−2833.00	−1.44011	−0.720057	−	0.693915i	$-0.755885\pi$
$157$	−2833.00	−1.44011	−0.720057	+	0.693915i	$0.755885\pi$
$158$	−922.000	−0.464243
$159$	0	0
$160$	−288.000	−0.142302
$161$	0	0
$162$	0	0
$163$	−2311.00	−1.11050	−0.555250	−	0.831684i	$-0.687377\pi$
$163$	−2311.00	−1.11050	−0.555250	+	0.831684i	$0.687377\pi$
$164$	168.000	0.0799914
$165$	0	0
$166$	−1176.00	−0.549851
$167$	−1260.00	−0.583843	−0.291921	−	0.956442i	$-0.594295\pi$
$167$	−1260.00	−0.583843	−0.291921	+	0.956442i	$0.594295\pi$
$168$	0	0
$169$	2703.00	1.23031
$170$	918.000	0.414161
$171$	0	0
$172$	−496.000	−0.219882
$173$	−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
$175$	0	0
$176$	912.000	0.390594
$177$	0	0
$178$	−2034.00	−0.856487
$179$	−1287.00	−0.537402	−0.268701	−	0.963224i	$-0.586594\pi$
$179$	−1287.00	−0.537402	−0.268701	+	0.963224i	$0.586594\pi$
$180$	0	0
$181$	−2674.00	−1.09810	−0.549052	−	0.835788i	$-0.685011\pi$
$181$	−2674.00	−1.09810	−0.549052	+	0.835788i	$0.685011\pi$
$182$	0	0
$183$	0	0
$184$	552.000	0.221163
$185$	−2277.00	−0.904910
$186$	0	0
$187$	−2907.00	−1.13680
$188$	−804.000	−0.311903
$189$	0	0
$190$	−90.0000	−0.0343647
$191$	−4185.00	−1.58542	−0.792712	−	0.609596i	$-0.791332\pi$
$191$	−4185.00	−1.58542	−0.792712	+	0.609596i	$0.791332\pi$
$192$	0	0
$193$	−85.0000	−0.0317017	−0.0158509	−	0.999874i	$-0.505046\pi$
$193$	−85.0000	−0.0317017	−0.0158509	+	0.999874i	$0.505046\pi$
$194$	3668.00	1.35746
$195$	0	0
$196$	0	0
$197$	390.000	0.141047	0.0705237	−	0.997510i	$-0.477533\pi$
$197$	390.000	0.141047	0.0705237	+	0.997510i	$0.477533\pi$
$198$	0	0
$199$	−2833.00	−1.00918	−0.504588	−	0.863360i	$-0.668356\pi$
$199$	−2833.00	−1.00918	−0.504588	+	0.863360i	$0.668356\pi$
$200$	352.000	0.124451
$201$	0	0
$202$	−570.000	−0.198540
$203$	0	0
$204$	0	0
$205$	378.000	0.128784
$206$	998.000	0.337543
$207$	0	0
$208$	−1120.00	−0.373356
$209$	285.000	0.0943247
$210$	0	0
$211$	−124.000	−0.0404574	−0.0202287	−	0.999795i	$-0.506439\pi$
$211$	−124.000	−0.0404574	−0.0202287	+	0.999795i	$0.506439\pi$
$212$	1572.00	0.509271
$213$	0	0
$214$	−2214.00	−0.707224
$215$	−1116.00	−0.354003
$216$	0	0
$217$	0	0
$218$	−1846.00	−0.573518
$219$	0	0
$220$	2052.00	0.628845
$221$	3570.00	1.08663
$222$	0	0
$223$	56.0000	0.0168163	0.00840816	−	0.999965i	$-0.497324\pi$
$223$	56.0000	0.0168163	0.00840816	+	0.999965i	$0.497324\pi$
$224$	0	0
$225$	0	0
$226$	3084.00	0.907720
$227$	3057.00	0.893834	0.446917	−	0.894576i	$-0.352522\pi$
$227$	3057.00	0.893834	0.446917	+	0.894576i	$0.352522\pi$
$228$	0	0
$229$	−961.000	−0.277313	−0.138656	−	0.990341i	$-0.544278\pi$
$229$	−961.000	−0.277313	−0.138656	+	0.990341i	$0.544278\pi$
$230$	1242.00	0.356065
$231$	0	0
$232$	912.000	0.258085
$233$	2829.00	0.795425	0.397712	−	0.917510i	$-0.369804\pi$
$233$	2829.00	0.795425	0.397712	+	0.917510i	$0.369804\pi$
$234$	0	0
$235$	−1809.00	−0.502154
$236$	−876.000	−0.241622
$237$	0	0
$238$	0	0
$239$	3540.00	0.958090	0.479045	−	0.877790i	$-0.340983\pi$
$239$	3540.00	0.958090	0.479045	+	0.877790i	$0.340983\pi$
$240$	0	0
$241$	5231.00	1.39817	0.699084	−	0.715040i	$-0.253591\pi$
$241$	5231.00	1.39817	0.699084	+	0.715040i	$0.253591\pi$
$242$	−3836.00	−1.01896
$243$	0	0
$244$	−2836.00	−0.744083
$245$	0	0
$246$	0	0
$247$	−350.000	−0.0901618
$248$	−184.000	−0.0471130
$249$	0	0
$250$	3042.00	0.769572
$251$	−5040.00	−1.26742	−0.633709	−	0.773571i	$-0.718468\pi$
$251$	−5040.00	−1.26742	−0.633709	+	0.773571i	$0.718468\pi$
$252$	0	0
$253$	−3933.00	−0.977334
$254$	4112.00	1.01579
$255$	0	0
$256$	256.000	0.0625000
$257$	1437.00	0.348784	0.174392	−	0.984676i	$-0.444204\pi$
$257$	1437.00	0.348784	0.174392	+	0.984676i	$0.444204\pi$
$258$	0	0
$259$	0	0
$260$	−2520.00	−0.601091
$261$	0	0
$262$	4098.00	0.966318
$263$	2325.00	0.545117	0.272558	−	0.962139i	$-0.412130\pi$
$263$	2325.00	0.545117	0.272558	+	0.962139i	$0.412130\pi$
$264$	0	0
$265$	3537.00	0.819910
$266$	0	0
$267$	0	0
$268$	1676.00	0.382007
$269$	2385.00	0.540580	0.270290	−	0.962779i	$-0.412880\pi$
$269$	2385.00	0.540580	0.270290	+	0.962779i	$0.412880\pi$
$270$	0	0
$271$	−331.000	−0.0741949	−0.0370975	−	0.999312i	$-0.511811\pi$
$271$	−331.000	−0.0741949	−0.0370975	+	0.999312i	$0.511811\pi$
$272$	−816.000	−0.181902
$273$	0	0
$274$	−282.000	−0.0621761
$275$	−2508.00	−0.549957
$276$	0	0
$277$	4871.00	1.05657	0.528285	−	0.849067i	$-0.322835\pi$
$277$	4871.00	1.05657	0.528285	+	0.849067i	$0.322835\pi$
$278$	−2968.00	−0.640319
$279$	0	0
$280$	0	0
$281$	7026.00	1.49159	0.745794	−	0.666177i	$-0.232070\pi$
$281$	7026.00	1.49159	0.745794	+	0.666177i	$0.232070\pi$
$282$	0	0
$283$	−5353.00	−1.12439	−0.562196	−	0.827004i	$-0.690043\pi$
$283$	−5353.00	−1.12439	−0.562196	+	0.827004i	$0.690043\pi$
$284$	384.000	0.0802331
$285$	0	0
$286$	7980.00	1.64989
$287$	0	0
$288$	0	0
$289$	−2312.00	−0.470588
$290$	2052.00	0.415509
$291$	0	0
$292$	−1252.00	−0.250917
$293$	−4158.00	−0.829054	−0.414527	−	0.910037i	$-0.636053\pi$
$293$	−4158.00	−0.829054	−0.414527	+	0.910037i	$0.636053\pi$
$294$	0	0
$295$	−1971.00	−0.389004
$296$	2024.00	0.397441
$297$	0	0
$298$	−114.000	−0.0221605
$299$	4830.00	0.934201
$300$	0	0
$301$	0	0
$302$	−1678.00	−0.319729
$303$	0	0
$304$	80.0000	0.0150931
$305$	−6381.00	−1.19795
$306$	0	0
$307$	−9604.00	−1.78544	−0.892719	−	0.450615i	$-0.851205\pi$
$307$	−9604.00	−1.78544	−0.892719	+	0.450615i	$0.851205\pi$
$308$	0	0
$309$	0	0
$310$	−414.000	−0.0758504
$311$	−10131.0	−1.84719	−0.923595	−	0.383369i	$-0.874764\pi$
$311$	−10131.0	−1.84719	−0.923595	+	0.383369i	$0.874764\pi$
$312$	0	0
$313$	10799.0	1.95015	0.975073	−	0.221885i	$-0.0712210\pi$
$313$	10799.0	1.95015	0.975073	+	0.221885i	$0.0712210\pi$
$314$	5666.00	1.01831
$315$	0	0
$316$	1844.00	0.328269
$317$	−531.000	−0.0940818	−0.0470409	−	0.998893i	$-0.514979\pi$
$317$	−531.000	−0.0940818	−0.0470409	+	0.998893i	$0.514979\pi$
$318$	0	0
$319$	−6498.00	−1.14050
$320$	576.000	0.100623
$321$	0	0
$322$	0	0
$323$	−255.000	−0.0439275
$324$	0	0
$325$	3080.00	0.525685
$326$	4622.00	0.785242
$327$	0	0
$328$	−336.000	−0.0565625
$329$	0	0
$330$	0	0
$331$	−7015.00	−1.16489	−0.582446	−	0.812869i	$-0.697904\pi$
$331$	−7015.00	−1.16489	−0.582446	+	0.812869i	$0.697904\pi$
$332$	2352.00	0.388804
$333$	0	0
$334$	2520.00	0.412839
$335$	3771.00	0.615020
$336$	0	0
$337$	8990.00	1.45316	0.726582	−	0.687079i	$-0.241108\pi$
$337$	8990.00	1.45316	0.726582	+	0.687079i	$0.241108\pi$
$338$	−5406.00	−0.869963
$339$	0	0
$340$	−1836.00	−0.292856
$341$	1311.00	0.208195
$342$	0	0
$343$	0	0
$344$	992.000	0.155480
$345$	0	0
$346$	6534.00	1.01523
$347$	8709.00	1.34733	0.673665	−	0.739037i	$-0.264719\pi$
$347$	8709.00	1.34733	0.673665	+	0.739037i	$0.264719\pi$
$348$	0	0
$349$	6482.00	0.994193	0.497097	−	0.867695i	$-0.334399\pi$
$349$	6482.00	0.994193	0.497097	+	0.867695i	$0.334399\pi$
$350$	0	0
$351$	0	0
$352$	−1824.00	−0.276192
$353$	2133.00	0.321609	0.160805	−	0.986986i	$-0.448591\pi$
$353$	2133.00	0.321609	0.160805	+	0.986986i	$0.448591\pi$
$354$	0	0
$355$	864.000	0.129173
$356$	4068.00	0.605628
$357$	0	0
$358$	2574.00	0.380000
$359$	−3849.00	−0.565856	−0.282928	−	0.959141i	$-0.591306\pi$
$359$	−3849.00	−0.565856	−0.282928	+	0.959141i	$0.591306\pi$
$360$	0	0
$361$	−6834.00	−0.996355
$362$	5348.00	0.776477
$363$	0	0
$364$	0	0
$365$	−2817.00	−0.403969
$366$	0	0
$367$	6491.00	0.923236	0.461618	−	0.887079i	$-0.347269\pi$
$367$	6491.00	0.923236	0.461618	+	0.887079i	$0.347269\pi$
$368$	−1104.00	−0.156386
$369$	0	0
$370$	4554.00	0.639868
$371$	0	0
$372$	0	0
$373$	923.000	0.128126	0.0640632	−	0.997946i	$-0.479594\pi$
$373$	923.000	0.128126	0.0640632	+	0.997946i	$0.479594\pi$
$374$	5814.00	0.803836
$375$	0	0
$376$	1608.00	0.220549
$377$	7980.00	1.09016
$378$	0	0
$379$	6344.00	0.859814	0.429907	−	0.902873i	$-0.358546\pi$
$379$	6344.00	0.859814	0.429907	+	0.902873i	$0.358546\pi$
$380$	180.000	0.0242995
$381$	0	0
$382$	8370.00	1.12106
$383$	5007.00	0.668005	0.334002	−	0.942572i	$-0.391601\pi$
$383$	5007.00	0.668005	0.334002	+	0.942572i	$0.391601\pi$
$384$	0	0
$385$	0	0
$386$	170.000	0.0224165
$387$	0	0
$388$	−7336.00	−0.959868
$389$	−12291.0	−1.60200	−0.801001	−	0.598664i	$-0.795699\pi$
$389$	−12291.0	−1.60200	−0.801001	+	0.598664i	$0.795699\pi$
$390$	0	0
$391$	3519.00	0.455150
$392$	0	0
$393$	0	0
$394$	−780.000	−0.0997356
$395$	4149.00	0.528503
$396$	0	0
$397$	887.000	0.112134	0.0560671	−	0.998427i	$-0.482144\pi$
$397$	887.000	0.112134	0.0560671	+	0.998427i	$0.482144\pi$
$398$	5666.00	0.713595
$399$	0	0
$400$	−704.000	−0.0880000
$401$	−11955.0	−1.48879	−0.744394	−	0.667740i	$-0.767262\pi$
$401$	−11955.0	−1.48879	−0.744394	+	0.667740i	$0.767262\pi$
$402$	0	0
$403$	−1610.00	−0.199007
$404$	1140.00	0.140389
$405$	0	0
$406$	0	0
$407$	−14421.0	−1.75632
$408$	0	0
$409$	−3421.00	−0.413588	−0.206794	−	0.978384i	$-0.566303\pi$
$409$	−3421.00	−0.413588	−0.206794	+	0.978384i	$0.566303\pi$
$410$	−756.000	−0.0910639
$411$	0	0
$412$	−1996.00	−0.238679
$413$	0	0
$414$	0	0
$415$	5292.00	0.625962
$416$	2240.00	0.264002
$417$	0	0
$418$	−570.000	−0.0666976
$419$	5460.00	0.636607	0.318304	−	0.947989i	$-0.396887\pi$
$419$	5460.00	0.636607	0.318304	+	0.947989i	$0.396887\pi$
$420$	0	0
$421$	7730.00	0.894863	0.447431	−	0.894318i	$-0.352339\pi$
$421$	7730.00	0.894863	0.447431	+	0.894318i	$0.352339\pi$
$422$	248.000	0.0286077
$423$	0	0
$424$	−3144.00	−0.360109
$425$	2244.00	0.256118
$426$	0	0
$427$	0	0
$428$	4428.00	0.500083
$429$	0	0
$430$	2232.00	0.250318
$431$	11313.0	1.26433	0.632167	−	0.774832i	$-0.282166\pi$
$431$	11313.0	1.26433	0.632167	+	0.774832i	$0.282166\pi$
$432$	0	0
$433$	4214.00	0.467695	0.233847	−	0.972273i	$-0.424868\pi$
$433$	4214.00	0.467695	0.233847	+	0.972273i	$0.424868\pi$
$434$	0	0
$435$	0	0
$436$	3692.00	0.405538
$437$	−345.000	−0.0377656
$438$	0	0
$439$	16553.0	1.79962	0.899808	−	0.436286i	$-0.143706\pi$
$439$	16553.0	1.79962	0.899808	+	0.436286i	$0.143706\pi$
$440$	−4104.00	−0.444660
$441$	0	0
$442$	−7140.00	−0.768360
$443$	16395.0	1.75835	0.879176	−	0.476497i	$-0.158094\pi$
$443$	16395.0	1.75835	0.879176	+	0.476497i	$0.158094\pi$
$444$	0	0
$445$	9153.00	0.975042
$446$	−112.000	−0.0118909
$447$	0	0
$448$	0	0
$449$	15090.0	1.58606	0.793030	−	0.609182i	$-0.208502\pi$
$449$	15090.0	1.58606	0.793030	+	0.609182i	$0.208502\pi$
$450$	0	0
$451$	2394.00	0.249954
$452$	−6168.00	−0.641855
$453$	0	0
$454$	−6114.00	−0.632036
$455$	0	0
$456$	0	0
$457$	−14785.0	−1.51338	−0.756688	−	0.653776i	$-0.773184\pi$
$457$	−14785.0	−1.51338	−0.756688	+	0.653776i	$0.773184\pi$
$458$	1922.00	0.196090
$459$	0	0
$460$	−2484.00	−0.251776
$461$	−2898.00	−0.292784	−0.146392	−	0.989227i	$-0.546766\pi$
$461$	−2898.00	−0.292784	−0.146392	+	0.989227i	$0.546766\pi$
$462$	0	0
$463$	464.000	0.0465743	0.0232872	−	0.999729i	$-0.492587\pi$
$463$	464.000	0.0465743	0.0232872	+	0.999729i	$0.492587\pi$
$464$	−1824.00	−0.182494
$465$	0	0
$466$	−5658.00	−0.562450
$467$	−4233.00	−0.419443	−0.209721	−	0.977761i	$-0.567256\pi$
$467$	−4233.00	−0.419443	−0.209721	+	0.977761i	$0.567256\pi$
$468$	0	0
$469$	0	0
$470$	3618.00	0.355076
$471$	0	0
$472$	1752.00	0.170852
$473$	−7068.00	−0.687076
$474$	0	0
$475$	−220.000	−0.0212511
$476$	0	0
$477$	0	0
$478$	−7080.00	−0.677472
$479$	−2739.00	−0.261270	−0.130635	−	0.991431i	$-0.541702\pi$
$479$	−2739.00	−0.261270	−0.130635	+	0.991431i	$0.541702\pi$
$480$	0	0
$481$	17710.0	1.67881
$482$	−10462.0	−0.988654
$483$	0	0
$484$	7672.00	0.720511
$485$	−16506.0	−1.54536
$486$	0	0
$487$	17051.0	1.58656	0.793280	−	0.608857i	$-0.208372\pi$
$487$	17051.0	1.58656	0.793280	+	0.608857i	$0.208372\pi$
$488$	5672.00	0.526146
$489$	0	0
$490$	0	0
$491$	4296.00	0.394859	0.197429	−	0.980317i	$-0.436741\pi$
$491$	4296.00	0.394859	0.197429	+	0.980317i	$0.436741\pi$
$492$	0	0
$493$	5814.00	0.531135
$494$	700.000	0.0637540
$495$	0	0
$496$	368.000	0.0333139
$497$	0	0
$498$	0	0
$499$	3401.00	0.305110	0.152555	−	0.988295i	$-0.451250\pi$
$499$	3401.00	0.305110	0.152555	+	0.988295i	$0.451250\pi$
$500$	−6084.00	−0.544170
$501$	0	0
$502$	10080.0	0.896200
$503$	−16800.0	−1.48921	−0.744607	−	0.667503i	$-0.767363\pi$
$503$	−16800.0	−1.48921	−0.744607	+	0.667503i	$0.767363\pi$
$504$	0	0
$505$	2565.00	0.226022
$506$	7866.00	0.691080
$507$	0	0
$508$	−8224.00	−0.718270
$509$	−1839.00	−0.160142	−0.0800710	−	0.996789i	$-0.525515\pi$
$509$	−1839.00	−0.160142	−0.0800710	+	0.996789i	$0.525515\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−2874.00	−0.246628
$515$	−4491.00	−0.384266
$516$	0	0
$517$	−11457.0	−0.974620
$518$	0	0
$519$	0	0
$520$	5040.00	0.425036
$521$	−303.000	−0.0254792	−0.0127396	−	0.999919i	$-0.504055\pi$
$521$	−303.000	−0.0254792	−0.0127396	+	0.999919i	$0.504055\pi$
$522$	0	0
$523$	−21667.0	−1.81153	−0.905767	−	0.423777i	$-0.860704\pi$
$523$	−21667.0	−1.81153	−0.905767	+	0.423777i	$0.860704\pi$
$524$	−8196.00	−0.683290
$525$	0	0
$526$	−4650.00	−0.385456
$527$	−1173.00	−0.0969577
$528$	0	0
$529$	−7406.00	−0.608696
$530$	−7074.00	−0.579764
$531$	0	0
$532$	0	0
$533$	−2940.00	−0.238922
$534$	0	0
$535$	9963.00	0.805118
$536$	−3352.00	−0.270120
$537$	0	0
$538$	−4770.00	−0.382248
$539$	0	0
$540$	0	0
$541$	5039.00	0.400450	0.200225	−	0.979750i	$-0.435833\pi$
$541$	5039.00	0.400450	0.200225	+	0.979750i	$0.435833\pi$
$542$	662.000	0.0524637
$543$	0	0
$544$	1632.00	0.128624
$545$	8307.00	0.652904
$546$	0	0
$547$	−2392.00	−0.186974	−0.0934868	−	0.995621i	$-0.529801\pi$
$547$	−2392.00	−0.186974	−0.0934868	+	0.995621i	$0.529801\pi$
$548$	564.000	0.0439651
$549$	0	0
$550$	5016.00	0.388878
$551$	−570.000	−0.0440704
$552$	0	0
$553$	0	0
$554$	−9742.00	−0.747108
$555$	0	0
$556$	5936.00	0.452774
$557$	22149.0	1.68489	0.842445	−	0.538783i	$-0.181116\pi$
$557$	22149.0	1.68489	0.842445	+	0.538783i	$0.181116\pi$
$558$	0	0
$559$	8680.00	0.656753
$560$	0	0
$561$	0	0
$562$	−14052.0	−1.05471
$563$	8349.00	0.624988	0.312494	−	0.949920i	$-0.398836\pi$
$563$	8349.00	0.624988	0.312494	+	0.949920i	$0.398836\pi$
$564$	0	0
$565$	−13878.0	−1.03337
$566$	10706.0	0.795065
$567$	0	0
$568$	−768.000	−0.0567334
$569$	15345.0	1.13057	0.565286	−	0.824895i	$-0.308766\pi$
$569$	15345.0	1.13057	0.565286	+	0.824895i	$0.308766\pi$
$570$	0	0
$571$	−11593.0	−0.849653	−0.424827	−	0.905275i	$-0.639665\pi$
$571$	−11593.0	−0.849653	−0.424827	+	0.905275i	$0.639665\pi$
$572$	−15960.0	−1.16665
$573$	0	0
$574$	0	0
$575$	3036.00	0.220191
$576$	0	0
$577$	−14593.0	−1.05288	−0.526442	−	0.850211i	$-0.676474\pi$
$577$	−14593.0	−1.05288	−0.526442	+	0.850211i	$0.676474\pi$
$578$	4624.00	0.332756
$579$	0	0
$580$	−4104.00	−0.293809
$581$	0	0
$582$	0	0
$583$	22401.0	1.59135
$584$	2504.00	0.177425
$585$	0	0
$586$	8316.00	0.586230
$587$	15372.0	1.08087	0.540435	−	0.841386i	$-0.318260\pi$
$587$	15372.0	1.08087	0.540435	+	0.841386i	$0.318260\pi$
$588$	0	0
$589$	115.000	0.00804498
$590$	3942.00	0.275067
$591$	0	0
$592$	−4048.00	−0.281033
$593$	14373.0	0.995326	0.497663	−	0.867370i	$-0.334192\pi$
$593$	14373.0	0.995326	0.497663	+	0.867370i	$0.334192\pi$
$594$	0	0
$595$	0	0
$596$	228.000	0.0156699
$597$	0	0
$598$	−9660.00	−0.660580
$599$	−2547.00	−0.173736	−0.0868678	−	0.996220i	$-0.527686\pi$
$599$	−2547.00	−0.173736	−0.0868678	+	0.996220i	$0.527686\pi$
$600$	0	0
$601$	−7042.00	−0.477952	−0.238976	−	0.971025i	$-0.576812\pi$
$601$	−7042.00	−0.477952	−0.238976	+	0.971025i	$0.576812\pi$
$602$	0	0
$603$	0	0
$604$	3356.00	0.226082
$605$	17262.0	1.16000
$606$	0	0
$607$	−22591.0	−1.51061	−0.755305	−	0.655373i	$-0.772511\pi$
$607$	−22591.0	−1.51061	−0.755305	+	0.655373i	$0.772511\pi$
$608$	−160.000	−0.0106725
$609$	0	0
$610$	12762.0	0.847079
$611$	14070.0	0.931606
$612$	0	0
$613$	−8485.00	−0.559063	−0.279532	−	0.960136i	$-0.590179\pi$
$613$	−8485.00	−0.559063	−0.279532	+	0.960136i	$0.590179\pi$
$614$	19208.0	1.26249
$615$	0	0
$616$	0	0
$617$	18282.0	1.19288	0.596439	−	0.802658i	$-0.296582\pi$
$617$	18282.0	1.19288	0.596439	+	0.802658i	$0.296582\pi$
$618$	0	0
$619$	2291.00	0.148761	0.0743805	−	0.997230i	$-0.476302\pi$
$619$	2291.00	0.148761	0.0743805	+	0.997230i	$0.476302\pi$
$620$	828.000	0.0536343
$621$	0	0
$622$	20262.0	1.30616
$623$	0	0
$624$	0	0
$625$	−8189.00	−0.524096
$626$	−21598.0	−1.37896
$627$	0	0
$628$	−11332.0	−0.720057
$629$	12903.0	0.817927
$630$	0	0
$631$	−6928.00	−0.437083	−0.218541	−	0.975828i	$-0.570130\pi$
$631$	−6928.00	−0.437083	−0.218541	+	0.975828i	$0.570130\pi$
$632$	−3688.00	−0.232121
$633$	0	0
$634$	1062.00	0.0665259
$635$	−18504.0	−1.15639
$636$	0	0
$637$	0	0
$638$	12996.0	0.806452
$639$	0	0
$640$	−1152.00	−0.0711512
$641$	−24975.0	−1.53893	−0.769464	−	0.638690i	$-0.779477\pi$
$641$	−24975.0	−1.53893	−0.769464	+	0.638690i	$0.779477\pi$
$642$	0	0
$643$	9548.00	0.585593	0.292797	−	0.956175i	$-0.405414\pi$
$643$	9548.00	0.585593	0.292797	+	0.956175i	$0.405414\pi$
$644$	0	0
$645$	0	0
$646$	510.000	0.0310614
$647$	−10131.0	−0.615596	−0.307798	−	0.951452i	$-0.599592\pi$
$647$	−10131.0	−0.615596	−0.307798	+	0.951452i	$0.599592\pi$
$648$	0	0
$649$	−12483.0	−0.755009
$650$	−6160.00	−0.371716
$651$	0	0
$652$	−9244.00	−0.555250
$653$	−16659.0	−0.998342	−0.499171	−	0.866504i	$-0.666362\pi$
$653$	−16659.0	−0.998342	−0.499171	+	0.866504i	$0.666362\pi$
$654$	0	0
$655$	−18441.0	−1.10008
$656$	672.000	0.0399957
$657$	0	0
$658$	0	0
$659$	−29556.0	−1.74710	−0.873550	−	0.486735i	$-0.838188\pi$
$659$	−29556.0	−1.74710	−0.873550	+	0.486735i	$0.838188\pi$
$660$	0	0
$661$	191.000	0.0112391	0.00561955	−	0.999984i	$-0.498211\pi$
$661$	191.000	0.0112391	0.00561955	+	0.999984i	$0.498211\pi$
$662$	14030.0	0.823703
$663$	0	0
$664$	−4704.00	−0.274926
$665$	0	0
$666$	0	0
$667$	7866.00	0.456631
$668$	−5040.00	−0.291921
$669$	0	0
$670$	−7542.00	−0.434885
$671$	−40413.0	−2.32508
$672$	0	0
$673$	2606.00	0.149263	0.0746314	−	0.997211i	$-0.476222\pi$
$673$	2606.00	0.149263	0.0746314	+	0.997211i	$0.476222\pi$
$674$	−17980.0	−1.02754
$675$	0	0
$676$	10812.0	0.615157
$677$	4209.00	0.238944	0.119472	−	0.992838i	$-0.461880\pi$
$677$	4209.00	0.238944	0.119472	+	0.992838i	$0.461880\pi$
$678$	0	0
$679$	0	0
$680$	3672.00	0.207081
$681$	0	0
$682$	−2622.00	−0.147216
$683$	−24303.0	−1.36154	−0.680768	−	0.732500i	$-0.738354\pi$
$683$	−24303.0	−1.36154	−0.680768	+	0.732500i	$0.738354\pi$
$684$	0	0
$685$	1269.00	0.0707825
$686$	0	0
$687$	0	0
$688$	−1984.00	−0.109941
$689$	−27510.0	−1.52111
$690$	0	0
$691$	15041.0	0.828056	0.414028	−	0.910264i	$-0.364122\pi$
$691$	15041.0	0.828056	0.414028	+	0.910264i	$0.364122\pi$
$692$	−13068.0	−0.717877
$693$	0	0
$694$	−17418.0	−0.952706
$695$	13356.0	0.728952
$696$	0	0
$697$	−2142.00	−0.116405
$698$	−12964.0	−0.703001
$699$	0	0
$700$	0	0
$701$	−24726.0	−1.33222	−0.666111	−	0.745852i	$-0.732042\pi$
$701$	−24726.0	−1.33222	−0.666111	+	0.745852i	$0.732042\pi$
$702$	0	0
$703$	−1265.00	−0.0678668
$704$	3648.00	0.195297
$705$	0	0
$706$	−4266.00	−0.227412
$707$	0	0
$708$	0	0
$709$	−4957.00	−0.262573	−0.131286	−	0.991344i	$-0.541911\pi$
$709$	−4957.00	−0.262573	−0.131286	+	0.991344i	$0.541911\pi$
$710$	−1728.00	−0.0913390
$711$	0	0
$712$	−8136.00	−0.428244
$713$	−1587.00	−0.0833571
$714$	0	0
$715$	−35910.0	−1.87826
$716$	−5148.00	−0.268701
$717$	0	0
$718$	7698.00	0.400121
$719$	−27669.0	−1.43516	−0.717580	−	0.696476i	$-0.754750\pi$
$719$	−27669.0	−1.43516	−0.717580	+	0.696476i	$0.754750\pi$
$720$	0	0
$721$	0	0
$722$	13668.0	0.704529
$723$	0	0
$724$	−10696.0	−0.549052
$725$	5016.00	0.256951
$726$	0	0
$727$	−13888.0	−0.708497	−0.354249	−	0.935151i	$-0.615263\pi$
$727$	−13888.0	−0.708497	−0.354249	+	0.935151i	$0.615263\pi$
$728$	0	0
$729$	0	0
$730$	5634.00	0.285649
$731$	6324.00	0.319975
$732$	0	0
$733$	14243.0	0.717704	0.358852	−	0.933394i	$-0.383168\pi$
$733$	14243.0	0.717704	0.358852	+	0.933394i	$0.383168\pi$
$734$	−12982.0	−0.652826
$735$	0	0
$736$	2208.00	0.110581
$737$	23883.0	1.19368
$738$	0	0
$739$	36959.0	1.83973	0.919864	−	0.392238i	$-0.128299\pi$
$739$	36959.0	1.83973	0.919864	+	0.392238i	$0.128299\pi$
$740$	−9108.00	−0.452455
$741$	0	0
$742$	0	0
$743$	12528.0	0.618584	0.309292	−	0.950967i	$-0.399908\pi$
$743$	12528.0	0.618584	0.309292	+	0.950967i	$0.399908\pi$
$744$	0	0
$745$	513.000	0.0252280
$746$	−1846.00	−0.0905990
$747$	0	0
$748$	−11628.0	−0.568398
$749$	0	0
$750$	0	0
$751$	−17767.0	−0.863285	−0.431643	−	0.902045i	$-0.642066\pi$
$751$	−17767.0	−0.863285	−0.431643	+	0.902045i	$0.642066\pi$
$752$	−3216.00	−0.155951
$753$	0	0
$754$	−15960.0	−0.770861
$755$	7551.00	0.363985
$756$	0	0
$757$	−28726.0	−1.37921	−0.689606	−	0.724184i	$-0.742216\pi$
$757$	−28726.0	−1.37921	−0.689606	+	0.724184i	$0.742216\pi$
$758$	−12688.0	−0.607980
$759$	0	0
$760$	−360.000	−0.0171823
$761$	26469.0	1.26084	0.630421	−	0.776254i	$-0.282882\pi$
$761$	26469.0	1.26084	0.630421	+	0.776254i	$0.282882\pi$
$762$	0	0
$763$	0	0
$764$	−16740.0	−0.792712
$765$	0	0
$766$	−10014.0	−0.472351
$767$	15330.0	0.721687
$768$	0	0
$769$	5054.00	0.236999	0.118499	−	0.992954i	$-0.462192\pi$
$769$	5054.00	0.236999	0.118499	+	0.992954i	$0.462192\pi$
$770$	0	0
$771$	0	0
$772$	−340.000	−0.0158509
$773$	35565.0	1.65483	0.827415	−	0.561590i	$-0.189810\pi$
$773$	35565.0	1.65483	0.827415	+	0.561590i	$0.189810\pi$
$774$	0	0
$775$	−1012.00	−0.0469060
$776$	14672.0	0.678730
$777$	0	0
$778$	24582.0	1.13279
$779$	210.000	0.00965858
$780$	0	0
$781$	5472.00	0.250709
$782$	−7038.00	−0.321839
$783$	0	0
$784$	0	0
$785$	−25497.0	−1.15927
$786$	0	0
$787$	−8629.00	−0.390839	−0.195420	−	0.980720i	$-0.562607\pi$
$787$	−8629.00	−0.390839	−0.195420	+	0.980720i	$0.562607\pi$
$788$	1560.00	0.0705237
$789$	0	0
$790$	−8298.00	−0.373708
$791$	0	0
$792$	0	0
$793$	49630.0	2.22246
$794$	−1774.00	−0.0792908
$795$	0	0
$796$	−11332.0	−0.504588
$797$	−20706.0	−0.920256	−0.460128	−	0.887853i	$-0.652197\pi$
$797$	−20706.0	−0.920256	−0.460128	+	0.887853i	$0.652197\pi$
$798$	0	0
$799$	10251.0	0.453885
$800$	1408.00	0.0622254
$801$	0	0
$802$	23910.0	1.05273
$803$	−17841.0	−0.784054
$804$	0	0
$805$	0	0
$806$	3220.00	0.140719
$807$	0	0
$808$	−2280.00	−0.0992700
$809$	16185.0	0.703380	0.351690	−	0.936117i	$-0.385607\pi$
$809$	16185.0	0.703380	0.351690	+	0.936117i	$0.385607\pi$
$810$	0	0
$811$	−11788.0	−0.510398	−0.255199	−	0.966889i	$-0.582141\pi$
$811$	−11788.0	−0.510398	−0.255199	+	0.966889i	$0.582141\pi$
$812$	0	0
$813$	0	0
$814$	28842.0	1.24191
$815$	−20799.0	−0.893935
$816$	0	0
$817$	−620.000	−0.0265496
$818$	6842.00	0.292451
$819$	0	0
$820$	1512.00	0.0643919
$821$	29793.0	1.26648	0.633242	−	0.773954i	$-0.281724\pi$
$821$	29793.0	1.26648	0.633242	+	0.773954i	$0.281724\pi$
$822$	0	0
$823$	30323.0	1.28432	0.642159	−	0.766572i	$-0.278039\pi$
$823$	30323.0	1.28432	0.642159	+	0.766572i	$0.278039\pi$
$824$	3992.00	0.168772
$825$	0	0
$826$	0	0
$827$	−21156.0	−0.889560	−0.444780	−	0.895640i	$-0.646718\pi$
$827$	−21156.0	−0.889560	−0.444780	+	0.895640i	$0.646718\pi$
$828$	0	0
$829$	−5269.00	−0.220748	−0.110374	−	0.993890i	$-0.535205\pi$
$829$	−5269.00	−0.220748	−0.110374	+	0.993890i	$0.535205\pi$
$830$	−10584.0	−0.442622
$831$	0	0
$832$	−4480.00	−0.186678
$833$	0	0
$834$	0	0
$835$	−11340.0	−0.469984
$836$	1140.00	0.0471623
$837$	0	0
$838$	−10920.0	−0.450149
$839$	39816.0	1.63838	0.819190	−	0.573522i	$-0.194423\pi$
$839$	39816.0	1.63838	0.819190	+	0.573522i	$0.194423\pi$
$840$	0	0
$841$	−11393.0	−0.467137
$842$	−15460.0	−0.632763
$843$	0	0
$844$	−496.000	−0.0202287
$845$	24327.0	0.990384
$846$	0	0
$847$	0	0
$848$	6288.00	0.254635
$849$	0	0
$850$	−4488.00	−0.181103
$851$	17457.0	0.703194
$852$	0	0
$853$	14546.0	0.583875	0.291938	−	0.956437i	$-0.405700\pi$
$853$	14546.0	0.583875	0.291938	+	0.956437i	$0.405700\pi$
$854$	0	0
$855$	0	0
$856$	−8856.00	−0.353612
$857$	31449.0	1.25353	0.626766	−	0.779207i	$-0.284378\pi$
$857$	31449.0	1.25353	0.626766	+	0.779207i	$0.284378\pi$
$858$	0	0
$859$	−24523.0	−0.974056	−0.487028	−	0.873386i	$-0.661919\pi$
$859$	−24523.0	−0.974056	−0.487028	+	0.873386i	$0.661919\pi$
$860$	−4464.00	−0.177001
$861$	0	0
$862$	−22626.0	−0.894019
$863$	8163.00	0.321983	0.160992	−	0.986956i	$-0.448531\pi$
$863$	8163.00	0.321983	0.160992	+	0.986956i	$0.448531\pi$
$864$	0	0
$865$	−29403.0	−1.15576
$866$	−8428.00	−0.330710
$867$	0	0
$868$	0	0
$869$	26277.0	1.02576
$870$	0	0
$871$	−29330.0	−1.14100
$872$	−7384.00	−0.286759
$873$	0	0
$874$	690.000	0.0267043
$875$	0	0
$876$	0	0
$877$	4367.00	0.168145	0.0840725	−	0.996460i	$-0.473207\pi$
$877$	4367.00	0.168145	0.0840725	+	0.996460i	$0.473207\pi$
$878$	−33106.0	−1.27252
$879$	0	0
$880$	8208.00	0.314422
$881$	50190.0	1.91935	0.959673	−	0.281118i	$-0.0907053\pi$
$881$	50190.0	1.91935	0.959673	+	0.281118i	$0.0907053\pi$
$882$	0	0
$883$	12308.0	0.469079	0.234540	−	0.972107i	$-0.424642\pi$
$883$	12308.0	0.469079	0.234540	+	0.972107i	$0.424642\pi$
$884$	14280.0	0.543313
$885$	0	0
$886$	−32790.0	−1.24334
$887$	−31617.0	−1.19684	−0.598419	−	0.801183i	$-0.704204\pi$
$887$	−31617.0	−1.19684	−0.598419	+	0.801183i	$0.704204\pi$
$888$	0	0
$889$	0	0
$890$	−18306.0	−0.689459
$891$	0	0
$892$	224.000	0.00840816
$893$	−1005.00	−0.0376607
$894$	0	0
$895$	−11583.0	−0.432600
$896$	0	0
$897$	0	0
$898$	−30180.0	−1.12151
$899$	−2622.00	−0.0972732
$900$	0	0
$901$	−20043.0	−0.741098
$902$	−4788.00	−0.176744
$903$	0	0
$904$	12336.0	0.453860
$905$	−24066.0	−0.883957
$906$	0	0
$907$	−13525.0	−0.495138	−0.247569	−	0.968870i	$-0.579632\pi$
$907$	−13525.0	−0.495138	−0.247569	+	0.968870i	$0.579632\pi$
$908$	12228.0	0.446917
$909$	0	0
$910$	0	0
$911$	19248.0	0.700016	0.350008	−	0.936747i	$-0.386179\pi$
$911$	19248.0	0.700016	0.350008	+	0.936747i	$0.386179\pi$
$912$	0	0
$913$	33516.0	1.21492
$914$	29570.0	1.07012
$915$	0	0
$916$	−3844.00	−0.138656
$917$	0	0
$918$	0	0
$919$	−8695.00	−0.312102	−0.156051	−	0.987749i	$-0.549876\pi$
$919$	−8695.00	−0.312102	−0.156051	+	0.987749i	$0.549876\pi$
$920$	4968.00	0.178033
$921$	0	0
$922$	5796.00	0.207029
$923$	−6720.00	−0.239644
$924$	0	0
$925$	11132.0	0.395695
$926$	−928.000	−0.0329330
$927$	0	0
$928$	3648.00	0.129043
$929$	−19479.0	−0.687928	−0.343964	−	0.938983i	$-0.611770\pi$
$929$	−19479.0	−0.687928	−0.343964	+	0.938983i	$0.611770\pi$
$930$	0	0
$931$	0	0
$932$	11316.0	0.397712
$933$	0	0
$934$	8466.00	0.296591
$935$	−26163.0	−0.915103
$936$	0	0
$937$	−12502.0	−0.435883	−0.217942	−	0.975962i	$-0.569934\pi$
$937$	−12502.0	−0.435883	−0.217942	+	0.975962i	$0.569934\pi$
$938$	0	0
$939$	0	0
$940$	−7236.00	−0.251077
$941$	15993.0	0.554046	0.277023	−	0.960863i	$-0.410652\pi$
$941$	15993.0	0.554046	0.277023	+	0.960863i	$0.410652\pi$
$942$	0	0
$943$	−2898.00	−0.100076
$944$	−3504.00	−0.120811
$945$	0	0
$946$	14136.0	0.485836
$947$	−44001.0	−1.50986	−0.754932	−	0.655804i	$-0.772330\pi$
$947$	−44001.0	−1.50986	−0.754932	+	0.655804i	$0.772330\pi$
$948$	0	0
$949$	21910.0	0.749451
$950$	440.000	0.0150268
$951$	0	0
$952$	0	0
$953$	4002.00	0.136031	0.0680155	−	0.997684i	$-0.478333\pi$
$953$	4002.00	0.136031	0.0680155	+	0.997684i	$0.478333\pi$
$954$	0	0
$955$	−37665.0	−1.27624
$956$	14160.0	0.479045
$957$	0	0
$958$	5478.00	0.184745
$959$	0	0
$960$	0	0
$961$	−29262.0	−0.982243
$962$	−35420.0	−1.18710
$963$	0	0
$964$	20924.0	0.699084
$965$	−765.000	−0.0255194
$966$	0	0
$967$	10544.0	0.350643	0.175322	−	0.984511i	$-0.443903\pi$
$967$	10544.0	0.350643	0.175322	+	0.984511i	$0.443903\pi$
$968$	−15344.0	−0.509478
$969$	0	0
$970$	33012.0	1.09273
$971$	6183.00	0.204348	0.102174	−	0.994767i	$-0.467420\pi$
$971$	6183.00	0.204348	0.102174	+	0.994767i	$0.467420\pi$
$972$	0	0
$973$	0	0
$974$	−34102.0	−1.12187
$975$	0	0
$976$	−11344.0	−0.372042
$977$	−3723.00	−0.121913	−0.0609567	−	0.998140i	$-0.519415\pi$
$977$	−3723.00	−0.121913	−0.0609567	+	0.998140i	$0.519415\pi$
$978$	0	0
$979$	57969.0	1.89244
$980$	0	0
$981$	0	0
$982$	−8592.00	−0.279207
$983$	45897.0	1.48920	0.744602	−	0.667509i	$-0.232639\pi$
$983$	45897.0	1.48920	0.744602	+	0.667509i	$0.232639\pi$
$984$	0	0
$985$	3510.00	0.113541
$986$	−11628.0	−0.375569
$987$	0	0
$988$	−1400.00	−0.0450809
$989$	8556.00	0.275091
$990$	0	0
$991$	6467.00	0.207297	0.103648	−	0.994614i	$-0.466948\pi$
$991$	6467.00	0.207297	0.103648	+	0.994614i	$0.466948\pi$
$992$	−736.000	−0.0235565
$993$	0	0
$994$	0	0
$995$	−25497.0	−0.812371
$996$	0	0
$997$	23039.0	0.731848	0.365924	−	0.930645i	$-0.380753\pi$
$997$	23039.0	0.731848	0.365924	+	0.930645i	$0.380753\pi$
$998$	−6802.00	−0.215745
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.f.1.1		1
3.2	odd	2		98.4.a.d.1.1		1
7.2	even	3		126.4.g.d.109.1		2
7.3	odd	6		882.4.g.u.667.1		2
7.4	even	3		126.4.g.d.37.1		2
7.5	odd	6		882.4.g.u.361.1		2
7.6	odd	2		882.4.a.c.1.1		1
12.11	even	2		784.4.a.p.1.1		1
15.14	odd	2		2450.4.a.q.1.1		1
21.2	odd	6		14.4.c.a.11.1	yes	2
21.5	even	6		98.4.c.a.67.1		2
21.11	odd	6		14.4.c.a.9.1	✓	2
21.17	even	6		98.4.c.a.79.1		2
21.20	even	2		98.4.a.f.1.1		1
84.11	even	6		112.4.i.a.65.1		2
84.23	even	6		112.4.i.a.81.1		2
84.83	odd	2		784.4.a.c.1.1		1
105.2	even	12		350.4.j.b.249.2		4
105.23	even	12		350.4.j.b.249.1		4
105.32	even	12		350.4.j.b.149.1		4
105.44	odd	6		350.4.e.e.151.1		2
105.53	even	12		350.4.j.b.149.2		4
105.74	odd	6		350.4.e.e.51.1		2
105.104	even	2		2450.4.a.d.1.1		1
168.11	even	6		448.4.i.e.65.1		2
168.53	odd	6		448.4.i.b.65.1		2
168.107	even	6		448.4.i.e.193.1		2
168.149	odd	6		448.4.i.b.193.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.4.c.a.9.1	✓	2	21.11	odd	6
14.4.c.a.11.1	yes	2	21.2	odd	6
98.4.a.d.1.1		1	3.2	odd	2
98.4.a.f.1.1		1	21.20	even	2
98.4.c.a.67.1		2	21.5	even	6
98.4.c.a.79.1		2	21.17	even	6
112.4.i.a.65.1		2	84.11	even	6
112.4.i.a.81.1		2	84.23	even	6
126.4.g.d.37.1		2	7.4	even	3
126.4.g.d.109.1		2	7.2	even	3
350.4.e.e.51.1		2	105.74	odd	6
350.4.e.e.151.1		2	105.44	odd	6
350.4.j.b.149.1		4	105.32	even	12
350.4.j.b.149.2		4	105.53	even	12
350.4.j.b.249.1		4	105.23	even	12
350.4.j.b.249.2		4	105.2	even	12
448.4.i.b.65.1		2	168.53	odd	6
448.4.i.b.193.1		2	168.149	odd	6
448.4.i.e.65.1		2	168.11	even	6
448.4.i.e.193.1		2	168.107	even	6
784.4.a.c.1.1		1	84.83	odd	2
784.4.a.p.1.1		1	12.11	even	2
882.4.a.c.1.1		1	7.6	odd	2
882.4.a.f.1.1		1	1.1	even	1	trivial
882.4.g.u.361.1		2	7.5	odd	6
882.4.g.u.667.1		2	7.3	odd	6
2450.4.a.d.1.1		1	105.104	even	2
2450.4.a.q.1.1		1	15.14	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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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