LMFDB - Embedded newform 168.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,4,Mod(1,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, 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("168.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	168.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:	$9.91232088096$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -10.0000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -10.0000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -52.0000 q^{11} -10.0000 q^{13} -30.0000 q^{15} -54.0000 q^{17} -52.0000 q^{19} -21.0000 q^{21} +48.0000 q^{23} -25.0000 q^{25} +27.0000 q^{27} -186.000 q^{29} +224.000 q^{31} -156.000 q^{33} +70.0000 q^{35} +94.0000 q^{37} -30.0000 q^{39} -478.000 q^{41} -316.000 q^{43} -90.0000 q^{45} +256.000 q^{47} +49.0000 q^{49} -162.000 q^{51} -66.0000 q^{53} +520.000 q^{55} -156.000 q^{57} +420.000 q^{59} +342.000 q^{61} -63.0000 q^{63} +100.000 q^{65} +668.000 q^{67} +144.000 q^{69} -272.000 q^{71} -86.0000 q^{73} -75.0000 q^{75} +364.000 q^{77} +1360.00 q^{79} +81.0000 q^{81} +188.000 q^{83} +540.000 q^{85} -558.000 q^{87} -366.000 q^{89} +70.0000 q^{91} +672.000 q^{93} +520.000 q^{95} +1554.00 q^{97} -468.000 q^{99} +O(q^{100})$

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
$5$	−10.0000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−10.0000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−52.0000	−1.42533	−0.712663	−	0.701506i	$-0.752511\pi$
$11$	−52.0000	−1.42533	−0.712663	+	0.701506i	$0.752511\pi$
$12$	0	0
$13$	−10.0000	−0.213346	−0.106673	−	0.994294i	$-0.534020\pi$
$13$	−10.0000	−0.213346	−0.106673	+	0.994294i	$0.534020\pi$
$14$	0	0
$15$	−30.0000	−0.516398
$16$	0	0
$17$	−54.0000	−0.770407	−0.385204	−	0.922832i	$-0.625869\pi$
$17$	−54.0000	−0.770407	−0.385204	+	0.922832i	$0.625869\pi$
$18$	0	0
$19$	−52.0000	−0.627875	−0.313937	−	0.949444i	$-0.601648\pi$
$19$	−52.0000	−0.627875	−0.313937	+	0.949444i	$0.601648\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	48.0000	0.435161	0.217580	−	0.976042i	$-0.430184\pi$
$23$	48.0000	0.435161	0.217580	+	0.976042i	$0.430184\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−186.000	−1.19101	−0.595506	−	0.803351i	$-0.703048\pi$
$29$	−186.000	−1.19101	−0.595506	+	0.803351i	$0.703048\pi$
$30$	0	0
$31$	224.000	1.29779	0.648897	−	0.760877i	$-0.275231\pi$
$31$	224.000	1.29779	0.648897	+	0.760877i	$0.275231\pi$
$32$	0	0
$33$	−156.000	−0.822913
$34$	0	0
$35$	70.0000	0.338062
$36$	0	0
$37$	94.0000	0.417662	0.208831	−	0.977952i	$-0.433034\pi$
$37$	94.0000	0.417662	0.208831	+	0.977952i	$0.433034\pi$
$38$	0	0
$39$	−30.0000	−0.123176
$40$	0	0
$41$	−478.000	−1.82076	−0.910379	−	0.413776i	$-0.864210\pi$
$41$	−478.000	−1.82076	−0.910379	+	0.413776i	$0.864210\pi$
$42$	0	0
$43$	−316.000	−1.12069	−0.560344	−	0.828260i	$-0.689331\pi$
$43$	−316.000	−1.12069	−0.560344	+	0.828260i	$0.689331\pi$
$44$	0	0
$45$	−90.0000	−0.298142
$46$	0	0
$47$	256.000	0.794499	0.397249	−	0.917711i	$-0.369965\pi$
$47$	256.000	0.794499	0.397249	+	0.917711i	$0.369965\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−162.000	−0.444795
$52$	0	0
$53$	−66.0000	−0.171053	−0.0855264	−	0.996336i	$-0.527257\pi$
$53$	−66.0000	−0.171053	−0.0855264	+	0.996336i	$0.527257\pi$
$54$	0	0
$55$	520.000	1.27485
$56$	0	0
$57$	−156.000	−0.362504
$58$	0	0
$59$	420.000	0.926769	0.463384	−	0.886157i	$-0.346635\pi$
$59$	420.000	0.926769	0.463384	+	0.886157i	$0.346635\pi$
$60$	0	0
$61$	342.000	0.717846	0.358923	−	0.933367i	$-0.383144\pi$
$61$	342.000	0.717846	0.358923	+	0.933367i	$0.383144\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	100.000	0.190823
$66$	0	0
$67$	668.000	1.21805	0.609024	−	0.793152i	$-0.291561\pi$
$67$	668.000	1.21805	0.609024	+	0.793152i	$0.291561\pi$
$68$	0	0
$69$	144.000	0.251240
$70$	0	0
$71$	−272.000	−0.454654	−0.227327	−	0.973818i	$-0.572999\pi$
$71$	−272.000	−0.454654	−0.227327	+	0.973818i	$0.572999\pi$
$72$	0	0
$73$	−86.0000	−0.137884	−0.0689420	−	0.997621i	$-0.521962\pi$
$73$	−86.0000	−0.137884	−0.0689420	+	0.997621i	$0.521962\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	364.000	0.538723
$78$	0	0
$79$	1360.00	1.93686	0.968430	−	0.249285i	$-0.0801957\pi$
$79$	1360.00	1.93686	0.968430	+	0.249285i	$0.0801957\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	188.000	0.248623	0.124311	−	0.992243i	$-0.460328\pi$
$83$	188.000	0.248623	0.124311	+	0.992243i	$0.460328\pi$
$84$	0	0
$85$	540.000	0.689073
$86$	0	0
$87$	−558.000	−0.687631
$88$	0	0
$89$	−366.000	−0.435909	−0.217955	−	0.975959i	$-0.569938\pi$
$89$	−366.000	−0.435909	−0.217955	+	0.975959i	$0.569938\pi$
$90$	0	0
$91$	70.0000	0.0806373
$92$	0	0
$93$	672.000	0.749281
$94$	0	0
$95$	520.000	0.561588
$96$	0	0
$97$	1554.00	1.62665	0.813324	−	0.581811i	$-0.197656\pi$
$97$	1554.00	1.62665	0.813324	+	0.581811i	$0.197656\pi$
$98$	0	0
$99$	−468.000	−0.475109
$100$	0	0
$101$	−1690.00	−1.66496	−0.832482	−	0.554053i	$-0.813081\pi$
$101$	−1690.00	−1.66496	−0.832482	+	0.554053i	$0.813081\pi$
$102$	0	0
$103$	248.000	0.237244	0.118622	−	0.992939i	$-0.462152\pi$
$103$	248.000	0.237244	0.118622	+	0.992939i	$0.462152\pi$
$104$	0	0
$105$	210.000	0.195180
$106$	0	0
$107$	−1620.00	−1.46366	−0.731829	−	0.681489i	$-0.761333\pi$
$107$	−1620.00	−1.46366	−0.731829	+	0.681489i	$0.761333\pi$
$108$	0	0
$109$	−1066.00	−0.936737	−0.468368	−	0.883533i	$-0.655158\pi$
$109$	−1066.00	−0.936737	−0.468368	+	0.883533i	$0.655158\pi$
$110$	0	0
$111$	282.000	0.241137
$112$	0	0
$113$	−1198.00	−0.997331	−0.498665	−	0.866795i	$-0.666176\pi$
$113$	−1198.00	−0.997331	−0.498665	+	0.866795i	$0.666176\pi$
$114$	0	0
$115$	−480.000	−0.389219
$116$	0	0
$117$	−90.0000	−0.0711154
$118$	0	0
$119$	378.000	0.291187
$120$	0	0
$121$	1373.00	1.03156
$122$	0	0
$123$	−1434.00	−1.05121
$124$	0	0
$125$	1500.00	1.07331
$126$	0	0
$127$	−2528.00	−1.76633	−0.883164	−	0.469064i	$-0.844591\pi$
$127$	−2528.00	−1.76633	−0.883164	+	0.469064i	$0.844591\pi$
$128$	0	0
$129$	−948.000	−0.647029
$130$	0	0
$131$	2156.00	1.43794	0.718972	−	0.695039i	$-0.244613\pi$
$131$	2156.00	1.43794	0.718972	+	0.695039i	$0.244613\pi$
$132$	0	0
$133$	364.000	0.237314
$134$	0	0
$135$	−270.000	−0.172133
$136$	0	0
$137$	234.000	0.145927	0.0729634	−	0.997335i	$-0.476754\pi$
$137$	234.000	0.145927	0.0729634	+	0.997335i	$0.476754\pi$
$138$	0	0
$139$	−2396.00	−1.46206	−0.731029	−	0.682346i	$-0.760960\pi$
$139$	−2396.00	−1.46206	−0.731029	+	0.682346i	$0.760960\pi$
$140$	0	0
$141$	768.000	0.458704
$142$	0	0
$143$	520.000	0.304088
$144$	0	0
$145$	1860.00	1.06527
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−1602.00	−0.880812	−0.440406	−	0.897799i	$-0.645165\pi$
$149$	−1602.00	−0.880812	−0.440406	+	0.897799i	$0.645165\pi$
$150$	0	0
$151$	−1320.00	−0.711391	−0.355696	−	0.934602i	$-0.615756\pi$
$151$	−1320.00	−0.711391	−0.355696	+	0.934602i	$0.615756\pi$
$152$	0	0
$153$	−486.000	−0.256802
$154$	0	0
$155$	−2240.00	−1.16078
$156$	0	0
$157$	−1482.00	−0.753353	−0.376677	−	0.926345i	$-0.622933\pi$
$157$	−1482.00	−0.753353	−0.376677	+	0.926345i	$0.622933\pi$
$158$	0	0
$159$	−198.000	−0.0987574
$160$	0	0
$161$	−336.000	−0.164475
$162$	0	0
$163$	−2820.00	−1.35509	−0.677544	−	0.735482i	$-0.736956\pi$
$163$	−2820.00	−1.35509	−0.677544	+	0.735482i	$0.736956\pi$
$164$	0	0
$165$	1560.00	0.736035
$166$	0	0
$167$	1768.00	0.819233	0.409617	−	0.912258i	$-0.365662\pi$
$167$	1768.00	0.819233	0.409617	+	0.912258i	$0.365662\pi$
$168$	0	0
$169$	−2097.00	−0.954483
$170$	0	0
$171$	−468.000	−0.209292
$172$	0	0
$173$	894.000	0.392888	0.196444	−	0.980515i	$-0.437061\pi$
$173$	894.000	0.392888	0.196444	+	0.980515i	$0.437061\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	1260.00	0.535070
$178$	0	0
$179$	−1996.00	−0.833453	−0.416726	−	0.909032i	$-0.636823\pi$
$179$	−1996.00	−0.833453	−0.416726	+	0.909032i	$0.636823\pi$
$180$	0	0
$181$	1582.00	0.649664	0.324832	−	0.945772i	$-0.394692\pi$
$181$	1582.00	0.649664	0.324832	+	0.945772i	$0.394692\pi$
$182$	0	0
$183$	1026.00	0.414449
$184$	0	0
$185$	−940.000	−0.373569
$186$	0	0
$187$	2808.00	1.09808
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	2456.00	0.930418	0.465209	−	0.885201i	$-0.345979\pi$
$191$	2456.00	0.930418	0.465209	+	0.885201i	$0.345979\pi$
$192$	0	0
$193$	−2942.00	−1.09725	−0.548626	−	0.836068i	$-0.684849\pi$
$193$	−2942.00	−1.09725	−0.548626	+	0.836068i	$0.684849\pi$
$194$	0	0
$195$	300.000	0.110172
$196$	0	0
$197$	−3666.00	−1.32585	−0.662923	−	0.748688i	$-0.730684\pi$
$197$	−3666.00	−1.32585	−0.662923	+	0.748688i	$0.730684\pi$
$198$	0	0
$199$	1224.00	0.436015	0.218008	−	0.975947i	$-0.430044\pi$
$199$	1224.00	0.436015	0.218008	+	0.975947i	$0.430044\pi$
$200$	0	0
$201$	2004.00	0.703240
$202$	0	0
$203$	1302.00	0.450160
$204$	0	0
$205$	4780.00	1.62854
$206$	0	0
$207$	432.000	0.145054
$208$	0	0
$209$	2704.00	0.894926
$210$	0	0
$211$	−4932.00	−1.60916	−0.804580	−	0.593844i	$-0.797610\pi$
$211$	−4932.00	−1.60916	−0.804580	+	0.593844i	$0.797610\pi$
$212$	0	0
$213$	−816.000	−0.262495
$214$	0	0
$215$	3160.00	1.00237
$216$	0	0
$217$	−1568.00	−0.490520
$218$	0	0
$219$	−258.000	−0.0796074
$220$	0	0
$221$	540.000	0.164363
$222$	0	0
$223$	−816.000	−0.245038	−0.122519	−	0.992466i	$-0.539097\pi$
$223$	−816.000	−0.245038	−0.122519	+	0.992466i	$0.539097\pi$
$224$	0	0
$225$	−225.000	−0.0666667
$226$	0	0
$227$	−1316.00	−0.384784	−0.192392	−	0.981318i	$-0.561625\pi$
$227$	−1316.00	−0.384784	−0.192392	+	0.981318i	$0.561625\pi$
$228$	0	0
$229$	−2786.00	−0.803948	−0.401974	−	0.915651i	$-0.631676\pi$
$229$	−2786.00	−0.803948	−0.401974	+	0.915651i	$0.631676\pi$
$230$	0	0
$231$	1092.00	0.311032
$232$	0	0
$233$	5162.00	1.45139	0.725695	−	0.688017i	$-0.241518\pi$
$233$	5162.00	1.45139	0.725695	+	0.688017i	$0.241518\pi$
$234$	0	0
$235$	−2560.00	−0.710621
$236$	0	0
$237$	4080.00	1.11825
$238$	0	0
$239$	3320.00	0.898548	0.449274	−	0.893394i	$-0.351683\pi$
$239$	3320.00	0.898548	0.449274	+	0.893394i	$0.351683\pi$
$240$	0	0
$241$	4066.00	1.08678	0.543390	−	0.839480i	$-0.317140\pi$
$241$	4066.00	1.08678	0.543390	+	0.839480i	$0.317140\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−490.000	−0.127775
$246$	0	0
$247$	520.000	0.133955
$248$	0	0
$249$	564.000	0.143542
$250$	0	0
$251$	−1116.00	−0.280643	−0.140321	−	0.990106i	$-0.544814\pi$
$251$	−1116.00	−0.280643	−0.140321	+	0.990106i	$0.544814\pi$
$252$	0	0
$253$	−2496.00	−0.620246
$254$	0	0
$255$	1620.00	0.397837
$256$	0	0
$257$	650.000	0.157766	0.0788830	−	0.996884i	$-0.474865\pi$
$257$	650.000	0.157766	0.0788830	+	0.996884i	$0.474865\pi$
$258$	0	0
$259$	−658.000	−0.157862
$260$	0	0
$261$	−1674.00	−0.397004
$262$	0	0
$263$	3648.00	0.855305	0.427653	−	0.903943i	$-0.359341\pi$
$263$	3648.00	0.855305	0.427653	+	0.903943i	$0.359341\pi$
$264$	0	0
$265$	660.000	0.152994
$266$	0	0
$267$	−1098.00	−0.251672
$268$	0	0
$269$	−7058.00	−1.59975	−0.799877	−	0.600164i	$-0.795102\pi$
$269$	−7058.00	−1.59975	−0.799877	+	0.600164i	$0.795102\pi$
$270$	0	0
$271$	6960.00	1.56011	0.780055	−	0.625711i	$-0.215191\pi$
$271$	6960.00	1.56011	0.780055	+	0.625711i	$0.215191\pi$
$272$	0	0
$273$	210.000	0.0465560
$274$	0	0
$275$	1300.00	0.285065
$276$	0	0
$277$	−5330.00	−1.15613	−0.578066	−	0.815990i	$-0.696192\pi$
$277$	−5330.00	−1.15613	−0.578066	+	0.815990i	$0.696192\pi$
$278$	0	0
$279$	2016.00	0.432598
$280$	0	0
$281$	5754.00	1.22155	0.610774	−	0.791805i	$-0.290858\pi$
$281$	5754.00	1.22155	0.610774	+	0.791805i	$0.290858\pi$
$282$	0	0
$283$	3076.00	0.646110	0.323055	−	0.946380i	$-0.395290\pi$
$283$	3076.00	0.646110	0.323055	+	0.946380i	$0.395290\pi$
$284$	0	0
$285$	1560.00	0.324233
$286$	0	0
$287$	3346.00	0.688182
$288$	0	0
$289$	−1997.00	−0.406473
$290$	0	0
$291$	4662.00	0.939145
$292$	0	0
$293$	2982.00	0.594574	0.297287	−	0.954788i	$-0.403918\pi$
$293$	2982.00	0.594574	0.297287	+	0.954788i	$0.403918\pi$
$294$	0	0
$295$	−4200.00	−0.828927
$296$	0	0
$297$	−1404.00	−0.274304
$298$	0	0
$299$	−480.000	−0.0928399
$300$	0	0
$301$	2212.00	0.423580
$302$	0	0
$303$	−5070.00	−0.961267
$304$	0	0
$305$	−3420.00	−0.642061
$306$	0	0
$307$	−1460.00	−0.271422	−0.135711	−	0.990748i	$-0.543332\pi$
$307$	−1460.00	−0.271422	−0.135711	+	0.990748i	$0.543332\pi$
$308$	0	0
$309$	744.000	0.136973
$310$	0	0
$311$	1496.00	0.272766	0.136383	−	0.990656i	$-0.456452\pi$
$311$	1496.00	0.272766	0.136383	+	0.990656i	$0.456452\pi$
$312$	0	0
$313$	−9494.00	−1.71448	−0.857241	−	0.514916i	$-0.827823\pi$
$313$	−9494.00	−1.71448	−0.857241	+	0.514916i	$0.827823\pi$
$314$	0	0
$315$	630.000	0.112687
$316$	0	0
$317$	1974.00	0.349750	0.174875	−	0.984591i	$-0.444048\pi$
$317$	1974.00	0.349750	0.174875	+	0.984591i	$0.444048\pi$
$318$	0	0
$319$	9672.00	1.69758
$320$	0	0
$321$	−4860.00	−0.845043
$322$	0	0
$323$	2808.00	0.483719
$324$	0	0
$325$	250.000	0.0426692
$326$	0	0
$327$	−3198.00	−0.540825
$328$	0	0
$329$	−1792.00	−0.300292
$330$	0	0
$331$	−5884.00	−0.977081	−0.488541	−	0.872541i	$-0.662471\pi$
$331$	−5884.00	−0.977081	−0.488541	+	0.872541i	$0.662471\pi$
$332$	0	0
$333$	846.000	0.139221
$334$	0	0
$335$	−6680.00	−1.08945
$336$	0	0
$337$	−10030.0	−1.62127	−0.810636	−	0.585550i	$-0.800879\pi$
$337$	−10030.0	−1.62127	−0.810636	+	0.585550i	$0.800879\pi$
$338$	0	0
$339$	−3594.00	−0.575809
$340$	0	0
$341$	−11648.0	−1.84978
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−1440.00	−0.224716
$346$	0	0
$347$	972.000	0.150374	0.0751869	−	0.997169i	$-0.476045\pi$
$347$	972.000	0.150374	0.0751869	+	0.997169i	$0.476045\pi$
$348$	0	0
$349$	−4362.00	−0.669033	−0.334516	−	0.942390i	$-0.608573\pi$
$349$	−4362.00	−0.669033	−0.334516	+	0.942390i	$0.608573\pi$
$350$	0	0
$351$	−270.000	−0.0410585
$352$	0	0
$353$	−3686.00	−0.555768	−0.277884	−	0.960615i	$-0.589633\pi$
$353$	−3686.00	−0.555768	−0.277884	+	0.960615i	$0.589633\pi$
$354$	0	0
$355$	2720.00	0.406655
$356$	0	0
$357$	1134.00	0.168117
$358$	0	0
$359$	−11280.0	−1.65832	−0.829158	−	0.559014i	$-0.811180\pi$
$359$	−11280.0	−1.65832	−0.829158	+	0.559014i	$0.811180\pi$
$360$	0	0
$361$	−4155.00	−0.605773
$362$	0	0
$363$	4119.00	0.595569
$364$	0	0
$365$	860.000	0.123327
$366$	0	0
$367$	6496.00	0.923947	0.461973	−	0.886894i	$-0.347142\pi$
$367$	6496.00	0.923947	0.461973	+	0.886894i	$0.347142\pi$
$368$	0	0
$369$	−4302.00	−0.606919
$370$	0	0
$371$	462.000	0.0646519
$372$	0	0
$373$	13582.0	1.88539	0.942693	−	0.333660i	$-0.108284\pi$
$373$	13582.0	1.88539	0.942693	+	0.333660i	$0.108284\pi$
$374$	0	0
$375$	4500.00	0.619677
$376$	0	0
$377$	1860.00	0.254098
$378$	0	0
$379$	6068.00	0.822407	0.411203	−	0.911544i	$-0.365109\pi$
$379$	6068.00	0.822407	0.411203	+	0.911544i	$0.365109\pi$
$380$	0	0
$381$	−7584.00	−1.01979
$382$	0	0
$383$	−8736.00	−1.16551	−0.582753	−	0.812649i	$-0.698024\pi$
$383$	−8736.00	−1.16551	−0.582753	+	0.812649i	$0.698024\pi$
$384$	0	0
$385$	−3640.00	−0.481848
$386$	0	0
$387$	−2844.00	−0.373562
$388$	0	0
$389$	2206.00	0.287529	0.143764	−	0.989612i	$-0.454079\pi$
$389$	2206.00	0.287529	0.143764	+	0.989612i	$0.454079\pi$
$390$	0	0
$391$	−2592.00	−0.335251
$392$	0	0
$393$	6468.00	0.830197
$394$	0	0
$395$	−13600.0	−1.73238
$396$	0	0
$397$	11974.0	1.51375	0.756874	−	0.653561i	$-0.226726\pi$
$397$	11974.0	1.51375	0.756874	+	0.653561i	$0.226726\pi$
$398$	0	0
$399$	1092.00	0.137013
$400$	0	0
$401$	12642.0	1.57434	0.787171	−	0.616734i	$-0.211545\pi$
$401$	12642.0	1.57434	0.787171	+	0.616734i	$0.211545\pi$
$402$	0	0
$403$	−2240.00	−0.276879
$404$	0	0
$405$	−810.000	−0.0993808
$406$	0	0
$407$	−4888.00	−0.595305
$408$	0	0
$409$	14170.0	1.71311	0.856554	−	0.516057i	$-0.172601\pi$
$409$	14170.0	1.71311	0.856554	+	0.516057i	$0.172601\pi$
$410$	0	0
$411$	702.000	0.0842509
$412$	0	0
$413$	−2940.00	−0.350286
$414$	0	0
$415$	−1880.00	−0.222375
$416$	0	0
$417$	−7188.00	−0.844120
$418$	0	0
$419$	4604.00	0.536802	0.268401	−	0.963307i	$-0.413505\pi$
$419$	4604.00	0.536802	0.268401	+	0.963307i	$0.413505\pi$
$420$	0	0
$421$	11998.0	1.38895	0.694474	−	0.719518i	$-0.255637\pi$
$421$	11998.0	1.38895	0.694474	+	0.719518i	$0.255637\pi$
$422$	0	0
$423$	2304.00	0.264833
$424$	0	0
$425$	1350.00	0.154081
$426$	0	0
$427$	−2394.00	−0.271320
$428$	0	0
$429$	1560.00	0.175565
$430$	0	0
$431$	712.000	0.0795727	0.0397863	−	0.999208i	$-0.487332\pi$
$431$	712.000	0.0795727	0.0397863	+	0.999208i	$0.487332\pi$
$432$	0	0
$433$	−10574.0	−1.17357	−0.586783	−	0.809744i	$-0.699606\pi$
$433$	−10574.0	−1.17357	−0.586783	+	0.809744i	$0.699606\pi$
$434$	0	0
$435$	5580.00	0.615036
$436$	0	0
$437$	−2496.00	−0.273226
$438$	0	0
$439$	−10536.0	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−10536.0	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−7956.00	−0.853275	−0.426638	−	0.904423i	$-0.640302\pi$
$443$	−7956.00	−0.853275	−0.426638	+	0.904423i	$0.640302\pi$
$444$	0	0
$445$	3660.00	0.389889
$446$	0	0
$447$	−4806.00	−0.508537
$448$	0	0
$449$	6178.00	0.649349	0.324675	−	0.945826i	$-0.394745\pi$
$449$	6178.00	0.649349	0.324675	+	0.945826i	$0.394745\pi$
$450$	0	0
$451$	24856.0	2.59517
$452$	0	0
$453$	−3960.00	−0.410722
$454$	0	0
$455$	−700.000	−0.0721242
$456$	0	0
$457$	10330.0	1.05737	0.528684	−	0.848819i	$-0.322686\pi$
$457$	10330.0	1.05737	0.528684	+	0.848819i	$0.322686\pi$
$458$	0	0
$459$	−1458.00	−0.148265
$460$	0	0
$461$	14142.0	1.42876	0.714380	−	0.699758i	$-0.246709\pi$
$461$	14142.0	1.42876	0.714380	+	0.699758i	$0.246709\pi$
$462$	0	0
$463$	−1936.00	−0.194327	−0.0971637	−	0.995268i	$-0.530977\pi$
$463$	−1936.00	−0.194327	−0.0971637	+	0.995268i	$0.530977\pi$
$464$	0	0
$465$	−6720.00	−0.670178
$466$	0	0
$467$	−5204.00	−0.515658	−0.257829	−	0.966191i	$-0.583007\pi$
$467$	−5204.00	−0.515658	−0.257829	+	0.966191i	$0.583007\pi$
$468$	0	0
$469$	−4676.00	−0.460379
$470$	0	0
$471$	−4446.00	−0.434949
$472$	0	0
$473$	16432.0	1.59734
$474$	0	0
$475$	1300.00	0.125575
$476$	0	0
$477$	−594.000	−0.0570176
$478$	0	0
$479$	8016.00	0.764635	0.382318	−	0.924031i	$-0.375126\pi$
$479$	8016.00	0.764635	0.382318	+	0.924031i	$0.375126\pi$
$480$	0	0
$481$	−940.000	−0.0891067
$482$	0	0
$483$	−1008.00	−0.0949598
$484$	0	0
$485$	−15540.0	−1.45492
$486$	0	0
$487$	−9112.00	−0.847852	−0.423926	−	0.905697i	$-0.639348\pi$
$487$	−9112.00	−0.847852	−0.423926	+	0.905697i	$0.639348\pi$
$488$	0	0
$489$	−8460.00	−0.782361
$490$	0	0
$491$	17340.0	1.59377	0.796887	−	0.604128i	$-0.206478\pi$
$491$	17340.0	1.59377	0.796887	+	0.604128i	$0.206478\pi$
$492$	0	0
$493$	10044.0	0.917564
$494$	0	0
$495$	4680.00	0.424950
$496$	0	0
$497$	1904.00	0.171843
$498$	0	0
$499$	2108.00	0.189112	0.0945562	−	0.995520i	$-0.469857\pi$
$499$	2108.00	0.189112	0.0945562	+	0.995520i	$0.469857\pi$
$500$	0	0
$501$	5304.00	0.472985
$502$	0	0
$503$	−18872.0	−1.67288	−0.836442	−	0.548055i	$-0.815368\pi$
$503$	−18872.0	−1.67288	−0.836442	+	0.548055i	$0.815368\pi$
$504$	0	0
$505$	16900.0	1.48919
$506$	0	0
$507$	−6291.00	−0.551071
$508$	0	0
$509$	−1346.00	−0.117211	−0.0586055	−	0.998281i	$-0.518665\pi$
$509$	−1346.00	−0.117211	−0.0586055	+	0.998281i	$0.518665\pi$
$510$	0	0
$511$	602.000	0.0521153
$512$	0	0
$513$	−1404.00	−0.120835
$514$	0	0
$515$	−2480.00	−0.212198
$516$	0	0
$517$	−13312.0	−1.13242
$518$	0	0
$519$	2682.00	0.226834
$520$	0	0
$521$	22674.0	1.90665	0.953326	−	0.301942i	$-0.0976349\pi$
$521$	22674.0	1.90665	0.953326	+	0.301942i	$0.0976349\pi$
$522$	0	0
$523$	16228.0	1.35679	0.678395	−	0.734698i	$-0.262676\pi$
$523$	16228.0	1.35679	0.678395	+	0.734698i	$0.262676\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	−12096.0	−0.999829
$528$	0	0
$529$	−9863.00	−0.810635
$530$	0	0
$531$	3780.00	0.308923
$532$	0	0
$533$	4780.00	0.388452
$534$	0	0
$535$	16200.0	1.30913
$536$	0	0
$537$	−5988.00	−0.481194
$538$	0	0
$539$	−2548.00	−0.203618
$540$	0	0
$541$	11846.0	0.941404	0.470702	−	0.882292i	$-0.344001\pi$
$541$	11846.0	0.941404	0.470702	+	0.882292i	$0.344001\pi$
$542$	0	0
$543$	4746.00	0.375084
$544$	0	0
$545$	10660.0	0.837843
$546$	0	0
$547$	−10692.0	−0.835753	−0.417877	−	0.908504i	$-0.637226\pi$
$547$	−10692.0	−0.835753	−0.417877	+	0.908504i	$0.637226\pi$
$548$	0	0
$549$	3078.00	0.239282
$550$	0	0
$551$	9672.00	0.747806
$552$	0	0
$553$	−9520.00	−0.732064
$554$	0	0
$555$	−2820.00	−0.215680
$556$	0	0
$557$	−2458.00	−0.186982	−0.0934908	−	0.995620i	$-0.529803\pi$
$557$	−2458.00	−0.186982	−0.0934908	+	0.995620i	$0.529803\pi$
$558$	0	0
$559$	3160.00	0.239094
$560$	0	0
$561$	8424.00	0.633978
$562$	0	0
$563$	1196.00	0.0895300	0.0447650	−	0.998998i	$-0.485746\pi$
$563$	1196.00	0.0895300	0.0447650	+	0.998998i	$0.485746\pi$
$564$	0	0
$565$	11980.0	0.892040
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−19174.0	−1.41268	−0.706341	−	0.707872i	$-0.749655\pi$
$569$	−19174.0	−1.41268	−0.706341	+	0.707872i	$0.749655\pi$
$570$	0	0
$571$	9892.00	0.724987	0.362493	−	0.931986i	$-0.381926\pi$
$571$	9892.00	0.724987	0.362493	+	0.931986i	$0.381926\pi$
$572$	0	0
$573$	7368.00	0.537177
$574$	0	0
$575$	−1200.00	−0.0870321
$576$	0	0
$577$	−11870.0	−0.856420	−0.428210	−	0.903679i	$-0.640856\pi$
$577$	−11870.0	−0.856420	−0.428210	+	0.903679i	$0.640856\pi$
$578$	0	0
$579$	−8826.00	−0.633499
$580$	0	0
$581$	−1316.00	−0.0939705
$582$	0	0
$583$	3432.00	0.243806
$584$	0	0
$585$	900.000	0.0636076
$586$	0	0
$587$	−26892.0	−1.89089	−0.945444	−	0.325784i	$-0.894372\pi$
$587$	−26892.0	−1.89089	−0.945444	+	0.325784i	$0.894372\pi$
$588$	0	0
$589$	−11648.0	−0.814851
$590$	0	0
$591$	−10998.0	−0.765478
$592$	0	0
$593$	−16790.0	−1.16270	−0.581351	−	0.813653i	$-0.697476\pi$
$593$	−16790.0	−1.16270	−0.581351	+	0.813653i	$0.697476\pi$
$594$	0	0
$595$	−3780.00	−0.260445
$596$	0	0
$597$	3672.00	0.251734
$598$	0	0
$599$	3264.00	0.222643	0.111322	−	0.993784i	$-0.464492\pi$
$599$	3264.00	0.222643	0.111322	+	0.993784i	$0.464492\pi$
$600$	0	0
$601$	5162.00	0.350353	0.175177	−	0.984537i	$-0.443950\pi$
$601$	5162.00	0.350353	0.175177	+	0.984537i	$0.443950\pi$
$602$	0	0
$603$	6012.00	0.406016
$604$	0	0
$605$	−13730.0	−0.922651
$606$	0	0
$607$	−27088.0	−1.81131	−0.905657	−	0.424010i	$-0.860622\pi$
$607$	−27088.0	−1.81131	−0.905657	+	0.424010i	$0.860622\pi$
$608$	0	0
$609$	3906.00	0.259900
$610$	0	0
$611$	−2560.00	−0.169503
$612$	0	0
$613$	−26178.0	−1.72483	−0.862414	−	0.506204i	$-0.831048\pi$
$613$	−26178.0	−1.72483	−0.862414	+	0.506204i	$0.831048\pi$
$614$	0	0
$615$	14340.0	0.940235
$616$	0	0
$617$	28330.0	1.84850	0.924249	−	0.381791i	$-0.124693\pi$
$617$	28330.0	1.84850	0.924249	+	0.381791i	$0.124693\pi$
$618$	0	0
$619$	17956.0	1.16593	0.582967	−	0.812496i	$-0.301892\pi$
$619$	17956.0	1.16593	0.582967	+	0.812496i	$0.301892\pi$
$620$	0	0
$621$	1296.00	0.0837467
$622$	0	0
$623$	2562.00	0.164758
$624$	0	0
$625$	−11875.0	−0.760000
$626$	0	0
$627$	8112.00	0.516686
$628$	0	0
$629$	−5076.00	−0.321770
$630$	0	0
$631$	6792.00	0.428503	0.214251	−	0.976779i	$-0.431269\pi$
$631$	6792.00	0.428503	0.214251	+	0.976779i	$0.431269\pi$
$632$	0	0
$633$	−14796.0	−0.929049
$634$	0	0
$635$	25280.0	1.57985
$636$	0	0
$637$	−490.000	−0.0304780
$638$	0	0
$639$	−2448.00	−0.151551
$640$	0	0
$641$	10098.0	0.622226	0.311113	−	0.950373i	$-0.399298\pi$
$641$	10098.0	0.622226	0.311113	+	0.950373i	$0.399298\pi$
$642$	0	0
$643$	−16292.0	−0.999213	−0.499606	−	0.866253i	$-0.666522\pi$
$643$	−16292.0	−0.999213	−0.499606	+	0.866253i	$0.666522\pi$
$644$	0	0
$645$	9480.00	0.578720
$646$	0	0
$647$	−31544.0	−1.91673	−0.958364	−	0.285550i	$-0.907824\pi$
$647$	−31544.0	−1.91673	−0.958364	+	0.285550i	$0.907824\pi$
$648$	0	0
$649$	−21840.0	−1.32095
$650$	0	0
$651$	−4704.00	−0.283202
$652$	0	0
$653$	−170.000	−0.0101878	−0.00509389	−	0.999987i	$-0.501621\pi$
$653$	−170.000	−0.0101878	−0.00509389	+	0.999987i	$0.501621\pi$
$654$	0	0
$655$	−21560.0	−1.28614
$656$	0	0
$657$	−774.000	−0.0459614
$658$	0	0
$659$	29220.0	1.72724	0.863619	−	0.504145i	$-0.168192\pi$
$659$	29220.0	1.72724	0.863619	+	0.504145i	$0.168192\pi$
$660$	0	0
$661$	−10402.0	−0.612089	−0.306045	−	0.952017i	$-0.599006\pi$
$661$	−10402.0	−0.612089	−0.306045	+	0.952017i	$0.599006\pi$
$662$	0	0
$663$	1620.00	0.0948953
$664$	0	0
$665$	−3640.00	−0.212260
$666$	0	0
$667$	−8928.00	−0.518281
$668$	0	0
$669$	−2448.00	−0.141473
$670$	0	0
$671$	−17784.0	−1.02316
$672$	0	0
$673$	2338.00	0.133913	0.0669564	−	0.997756i	$-0.478671\pi$
$673$	2338.00	0.133913	0.0669564	+	0.997756i	$0.478671\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−18330.0	−1.04059	−0.520295	−	0.853987i	$-0.674178\pi$
$677$	−18330.0	−1.04059	−0.520295	+	0.853987i	$0.674178\pi$
$678$	0	0
$679$	−10878.0	−0.614815
$680$	0	0
$681$	−3948.00	−0.222155
$682$	0	0
$683$	4956.00	0.277652	0.138826	−	0.990317i	$-0.455667\pi$
$683$	4956.00	0.277652	0.138826	+	0.990317i	$0.455667\pi$
$684$	0	0
$685$	−2340.00	−0.130521
$686$	0	0
$687$	−8358.00	−0.464160
$688$	0	0
$689$	660.000	0.0364935
$690$	0	0
$691$	−6692.00	−0.368416	−0.184208	−	0.982887i	$-0.558972\pi$
$691$	−6692.00	−0.368416	−0.184208	+	0.982887i	$0.558972\pi$
$692$	0	0
$693$	3276.00	0.179574
$694$	0	0
$695$	23960.0	1.30770
$696$	0	0
$697$	25812.0	1.40272
$698$	0	0
$699$	15486.0	0.837960
$700$	0	0
$701$	−33306.0	−1.79451	−0.897254	−	0.441515i	$-0.854441\pi$
$701$	−33306.0	−1.79451	−0.897254	+	0.441515i	$0.854441\pi$
$702$	0	0
$703$	−4888.00	−0.262240
$704$	0	0
$705$	−7680.00	−0.410277
$706$	0	0
$707$	11830.0	0.629297
$708$	0	0
$709$	−13218.0	−0.700159	−0.350079	−	0.936720i	$-0.613845\pi$
$709$	−13218.0	−0.700159	−0.350079	+	0.936720i	$0.613845\pi$
$710$	0	0
$711$	12240.0	0.645620
$712$	0	0
$713$	10752.0	0.564748
$714$	0	0
$715$	−5200.00	−0.271985
$716$	0	0
$717$	9960.00	0.518777
$718$	0	0
$719$	−6640.00	−0.344409	−0.172205	−	0.985061i	$-0.555089\pi$
$719$	−6640.00	−0.344409	−0.172205	+	0.985061i	$0.555089\pi$
$720$	0	0
$721$	−1736.00	−0.0896699
$722$	0	0
$723$	12198.0	0.627453
$724$	0	0
$725$	4650.00	0.238202
$726$	0	0
$727$	−1304.00	−0.0665236	−0.0332618	−	0.999447i	$-0.510590\pi$
$727$	−1304.00	−0.0665236	−0.0332618	+	0.999447i	$0.510590\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	17064.0	0.863386
$732$	0	0
$733$	8294.00	0.417934	0.208967	−	0.977923i	$-0.432990\pi$
$733$	8294.00	0.417934	0.208967	+	0.977923i	$0.432990\pi$
$734$	0	0
$735$	−1470.00	−0.0737711
$736$	0	0
$737$	−34736.0	−1.73612
$738$	0	0
$739$	−8420.00	−0.419127	−0.209563	−	0.977795i	$-0.567204\pi$
$739$	−8420.00	−0.419127	−0.209563	+	0.977795i	$0.567204\pi$
$740$	0	0
$741$	1560.00	0.0773388
$742$	0	0
$743$	22352.0	1.10365	0.551827	−	0.833958i	$-0.313931\pi$
$743$	22352.0	1.10365	0.551827	+	0.833958i	$0.313931\pi$
$744$	0	0
$745$	16020.0	0.787822
$746$	0	0
$747$	1692.00	0.0828742
$748$	0	0
$749$	11340.0	0.553210
$750$	0	0
$751$	24160.0	1.17392	0.586958	−	0.809617i	$-0.300325\pi$
$751$	24160.0	1.17392	0.586958	+	0.809617i	$0.300325\pi$
$752$	0	0
$753$	−3348.00	−0.162029
$754$	0	0
$755$	13200.0	0.636288
$756$	0	0
$757$	−35858.0	−1.72164	−0.860820	−	0.508910i	$-0.830049\pi$
$757$	−35858.0	−1.72164	−0.860820	+	0.508910i	$0.830049\pi$
$758$	0	0
$759$	−7488.00	−0.358099
$760$	0	0
$761$	10770.0	0.513025	0.256513	−	0.966541i	$-0.417426\pi$
$761$	10770.0	0.513025	0.256513	+	0.966541i	$0.417426\pi$
$762$	0	0
$763$	7462.00	0.354053
$764$	0	0
$765$	4860.00	0.229691
$766$	0	0
$767$	−4200.00	−0.197723
$768$	0	0
$769$	−36190.0	−1.69707	−0.848534	−	0.529141i	$-0.822514\pi$
$769$	−36190.0	−1.69707	−0.848534	+	0.529141i	$0.822514\pi$
$770$	0	0
$771$	1950.00	0.0910863
$772$	0	0
$773$	−8026.00	−0.373448	−0.186724	−	0.982412i	$-0.559787\pi$
$773$	−8026.00	−0.373448	−0.186724	+	0.982412i	$0.559787\pi$
$774$	0	0
$775$	−5600.00	−0.259559
$776$	0	0
$777$	−1974.00	−0.0911414
$778$	0	0
$779$	24856.0	1.14321
$780$	0	0
$781$	14144.0	0.648031
$782$	0	0
$783$	−5022.00	−0.229210
$784$	0	0
$785$	14820.0	0.673820
$786$	0	0
$787$	9308.00	0.421594	0.210797	−	0.977530i	$-0.432394\pi$
$787$	9308.00	0.421594	0.210797	+	0.977530i	$0.432394\pi$
$788$	0	0
$789$	10944.0	0.493811
$790$	0	0
$791$	8386.00	0.376956
$792$	0	0
$793$	−3420.00	−0.153150
$794$	0	0
$795$	1980.00	0.0883313
$796$	0	0
$797$	22734.0	1.01039	0.505194	−	0.863006i	$-0.331421\pi$
$797$	22734.0	1.01039	0.505194	+	0.863006i	$0.331421\pi$
$798$	0	0
$799$	−13824.0	−0.612088
$800$	0	0
$801$	−3294.00	−0.145303
$802$	0	0
$803$	4472.00	0.196530
$804$	0	0
$805$	3360.00	0.147111
$806$	0	0
$807$	−21174.0	−0.923618
$808$	0	0
$809$	21818.0	0.948183	0.474091	−	0.880476i	$-0.342777\pi$
$809$	21818.0	0.948183	0.474091	+	0.880476i	$0.342777\pi$
$810$	0	0
$811$	25252.0	1.09336	0.546682	−	0.837341i	$-0.315891\pi$
$811$	25252.0	1.09336	0.546682	+	0.837341i	$0.315891\pi$
$812$	0	0
$813$	20880.0	0.900730
$814$	0	0
$815$	28200.0	1.21203
$816$	0	0
$817$	16432.0	0.703651
$818$	0	0
$819$	630.000	0.0268791
$820$	0	0
$821$	−4642.00	−0.197329	−0.0986644	−	0.995121i	$-0.531457\pi$
$821$	−4642.00	−0.197329	−0.0986644	+	0.995121i	$0.531457\pi$
$822$	0	0
$823$	−19512.0	−0.826422	−0.413211	−	0.910635i	$-0.635593\pi$
$823$	−19512.0	−0.826422	−0.413211	+	0.910635i	$0.635593\pi$
$824$	0	0
$825$	3900.00	0.164583
$826$	0	0
$827$	−12484.0	−0.524923	−0.262461	−	0.964942i	$-0.584534\pi$
$827$	−12484.0	−0.524923	−0.262461	+	0.964942i	$0.584534\pi$
$828$	0	0
$829$	24902.0	1.04328	0.521642	−	0.853165i	$-0.325320\pi$
$829$	24902.0	1.04328	0.521642	+	0.853165i	$0.325320\pi$
$830$	0	0
$831$	−15990.0	−0.667493
$832$	0	0
$833$	−2646.00	−0.110058
$834$	0	0
$835$	−17680.0	−0.732744
$836$	0	0
$837$	6048.00	0.249760
$838$	0	0
$839$	14376.0	0.591555	0.295777	−	0.955257i	$-0.404421\pi$
$839$	14376.0	0.591555	0.295777	+	0.955257i	$0.404421\pi$
$840$	0	0
$841$	10207.0	0.418508
$842$	0	0
$843$	17262.0	0.705261
$844$	0	0
$845$	20970.0	0.853716
$846$	0	0
$847$	−9611.00	−0.389891
$848$	0	0
$849$	9228.00	0.373032
$850$	0	0
$851$	4512.00	0.181750
$852$	0	0
$853$	6878.00	0.276082	0.138041	−	0.990426i	$-0.455919\pi$
$853$	6878.00	0.276082	0.138041	+	0.990426i	$0.455919\pi$
$854$	0	0
$855$	4680.00	0.187196
$856$	0	0
$857$	−9870.00	−0.393410	−0.196705	−	0.980463i	$-0.563024\pi$
$857$	−9870.00	−0.393410	−0.196705	+	0.980463i	$0.563024\pi$
$858$	0	0
$859$	26660.0	1.05894	0.529469	−	0.848329i	$-0.322391\pi$
$859$	26660.0	1.05894	0.529469	+	0.848329i	$0.322391\pi$
$860$	0	0
$861$	10038.0	0.397322
$862$	0	0
$863$	33544.0	1.32312	0.661559	−	0.749893i	$-0.269895\pi$
$863$	33544.0	1.32312	0.661559	+	0.749893i	$0.269895\pi$
$864$	0	0
$865$	−8940.00	−0.351409
$866$	0	0
$867$	−5991.00	−0.234677
$868$	0	0
$869$	−70720.0	−2.76066
$870$	0	0
$871$	−6680.00	−0.259866
$872$	0	0
$873$	13986.0	0.542216
$874$	0	0
$875$	−10500.0	−0.405674
$876$	0	0
$877$	49494.0	1.90569	0.952847	−	0.303451i	$-0.0981389\pi$
$877$	49494.0	1.90569	0.952847	+	0.303451i	$0.0981389\pi$
$878$	0	0
$879$	8946.00	0.343278
$880$	0	0
$881$	−3798.00	−0.145242	−0.0726208	−	0.997360i	$-0.523136\pi$
$881$	−3798.00	−0.145242	−0.0726208	+	0.997360i	$0.523136\pi$
$882$	0	0
$883$	−31268.0	−1.19168	−0.595839	−	0.803104i	$-0.703180\pi$
$883$	−31268.0	−1.19168	−0.595839	+	0.803104i	$0.703180\pi$
$884$	0	0
$885$	−12600.0	−0.478581
$886$	0	0
$887$	−6184.00	−0.234091	−0.117045	−	0.993127i	$-0.537342\pi$
$887$	−6184.00	−0.234091	−0.117045	+	0.993127i	$0.537342\pi$
$888$	0	0
$889$	17696.0	0.667609
$890$	0	0
$891$	−4212.00	−0.158370
$892$	0	0
$893$	−13312.0	−0.498846
$894$	0	0
$895$	19960.0	0.745463
$896$	0	0
$897$	−1440.00	−0.0536011
$898$	0	0
$899$	−41664.0	−1.54569
$900$	0	0
$901$	3564.00	0.131780
$902$	0	0
$903$	6636.00	0.244554
$904$	0	0
$905$	−15820.0	−0.581077
$906$	0	0
$907$	−23452.0	−0.858557	−0.429278	−	0.903172i	$-0.641232\pi$
$907$	−23452.0	−0.858557	−0.429278	+	0.903172i	$0.641232\pi$
$908$	0	0
$909$	−15210.0	−0.554988
$910$	0	0
$911$	−16632.0	−0.604877	−0.302438	−	0.953169i	$-0.597801\pi$
$911$	−16632.0	−0.604877	−0.302438	+	0.953169i	$0.597801\pi$
$912$	0	0
$913$	−9776.00	−0.354368
$914$	0	0
$915$	−10260.0	−0.370694
$916$	0	0
$917$	−15092.0	−0.543492
$918$	0	0
$919$	34232.0	1.22874	0.614369	−	0.789019i	$-0.289411\pi$
$919$	34232.0	1.22874	0.614369	+	0.789019i	$0.289411\pi$
$920$	0	0
$921$	−4380.00	−0.156706
$922$	0	0
$923$	2720.00	0.0969988
$924$	0	0
$925$	−2350.00	−0.0835325
$926$	0	0
$927$	2232.00	0.0790814
$928$	0	0
$929$	−31958.0	−1.12864	−0.564321	−	0.825556i	$-0.690862\pi$
$929$	−31958.0	−1.12864	−0.564321	+	0.825556i	$0.690862\pi$
$930$	0	0
$931$	−2548.00	−0.0896964
$932$	0	0
$933$	4488.00	0.157482
$934$	0	0
$935$	−28080.0	−0.982154
$936$	0	0
$937$	−40262.0	−1.40374	−0.701869	−	0.712306i	$-0.747651\pi$
$937$	−40262.0	−1.40374	−0.701869	+	0.712306i	$0.747651\pi$
$938$	0	0
$939$	−28482.0	−0.989856
$940$	0	0
$941$	22830.0	0.790900	0.395450	−	0.918488i	$-0.370589\pi$
$941$	22830.0	0.790900	0.395450	+	0.918488i	$0.370589\pi$
$942$	0	0
$943$	−22944.0	−0.792322
$944$	0	0
$945$	1890.00	0.0650600
$946$	0	0
$947$	46852.0	1.60769	0.803847	−	0.594837i	$-0.202783\pi$
$947$	46852.0	1.60769	0.803847	+	0.594837i	$0.202783\pi$
$948$	0	0
$949$	860.000	0.0294171
$950$	0	0
$951$	5922.00	0.201929
$952$	0	0
$953$	−2934.00	−0.0997288	−0.0498644	−	0.998756i	$-0.515879\pi$
$953$	−2934.00	−0.0997288	−0.0498644	+	0.998756i	$0.515879\pi$
$954$	0	0
$955$	−24560.0	−0.832192
$956$	0	0
$957$	29016.0	0.980098
$958$	0	0
$959$	−1638.00	−0.0551551
$960$	0	0
$961$	20385.0	0.684267
$962$	0	0
$963$	−14580.0	−0.487886
$964$	0	0
$965$	29420.0	0.981413
$966$	0	0
$967$	−6376.00	−0.212036	−0.106018	−	0.994364i	$-0.533810\pi$
$967$	−6376.00	−0.212036	−0.106018	+	0.994364i	$0.533810\pi$
$968$	0	0
$969$	8424.00	0.279275
$970$	0	0
$971$	−28476.0	−0.941131	−0.470566	−	0.882365i	$-0.655950\pi$
$971$	−28476.0	−0.941131	−0.470566	+	0.882365i	$0.655950\pi$
$972$	0	0
$973$	16772.0	0.552606
$974$	0	0
$975$	750.000	0.0246351
$976$	0	0
$977$	19682.0	0.644507	0.322253	−	0.946653i	$-0.395560\pi$
$977$	19682.0	0.644507	0.322253	+	0.946653i	$0.395560\pi$
$978$	0	0
$979$	19032.0	0.621313
$980$	0	0
$981$	−9594.00	−0.312246
$982$	0	0
$983$	−15960.0	−0.517848	−0.258924	−	0.965898i	$-0.583368\pi$
$983$	−15960.0	−0.517848	−0.258924	+	0.965898i	$0.583368\pi$
$984$	0	0
$985$	36660.0	1.18587
$986$	0	0
$987$	−5376.00	−0.173374
$988$	0	0
$989$	−15168.0	−0.487679
$990$	0	0
$991$	52592.0	1.68581	0.842906	−	0.538061i	$-0.180843\pi$
$991$	52592.0	1.68581	0.842906	+	0.538061i	$0.180843\pi$
$992$	0	0
$993$	−17652.0	−0.564118
$994$	0	0
$995$	−12240.0	−0.389984
$996$	0	0
$997$	41806.0	1.32799	0.663997	−	0.747736i	$-0.268859\pi$
$997$	41806.0	1.32799	0.663997	+	0.747736i	$0.268859\pi$
$998$	0	0
$999$	2538.00	0.0803791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.4.a.e.1.1	✓	1
3.2	odd	2		504.4.a.e.1.1		1
4.3	odd	2		336.4.a.b.1.1		1
7.6	odd	2		1176.4.a.g.1.1		1
8.3	odd	2		1344.4.a.x.1.1		1
8.5	even	2		1344.4.a.k.1.1		1
12.11	even	2		1008.4.a.q.1.1		1
28.27	even	2		2352.4.a.bh.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.a.e.1.1	✓	1	1.1	even	1	trivial
336.4.a.b.1.1		1	4.3	odd	2
504.4.a.e.1.1		1	3.2	odd	2
1008.4.a.q.1.1		1	12.11	even	2
1176.4.a.g.1.1		1	7.6	odd	2
1344.4.a.k.1.1		1	8.5	even	2
1344.4.a.x.1.1		1	8.3	odd	2
2352.4.a.bh.1.1		1	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	168.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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