LMFDB - Embedded newform 192.3.b.a.31.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,3,Mod(31,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("192.31");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	192.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$5.23162107572$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.3
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	192.31
Dual form			192.3.b.a.31.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} -3.46410i q^{5} -2.00000i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.73205 q^{3} -3.46410i q^{5} -2.00000i q^{7} +3.00000 q^{9} +13.8564 q^{11} -20.7846i q^{13} -6.00000i q^{15} -18.0000 q^{17} +20.7846 q^{19} -3.46410i q^{21} +36.0000i q^{23} +13.0000 q^{25} +5.19615 q^{27} -31.1769i q^{29} -22.0000i q^{31} +24.0000 q^{33} -6.92820 q^{35} +41.5692i q^{37} -36.0000i q^{39} -54.0000 q^{41} -20.7846 q^{43} -10.3923i q^{45} +36.0000i q^{47} +45.0000 q^{49} -31.1769 q^{51} +100.459i q^{53} -48.0000i q^{55} +36.0000 q^{57} -62.3538 q^{59} -6.00000i q^{63} -72.0000 q^{65} -62.3538 q^{67} +62.3538i q^{69} +108.000i q^{71} +10.0000 q^{73} +22.5167 q^{75} -27.7128i q^{77} +50.0000i q^{79} +9.00000 q^{81} -13.8564 q^{83} +62.3538i q^{85} -54.0000i q^{87} -18.0000 q^{89} -41.5692 q^{91} -38.1051i q^{93} -72.0000i q^{95} -34.0000 q^{97} +41.5692 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} - 72 q^{17} + 52 q^{25} + 96 q^{33} - 216 q^{41} + 180 q^{49} + 144 q^{57} - 288 q^{65} + 40 q^{73} + 36 q^{81} - 72 q^{89} - 136 q^{97}+O(q^{100}) $ 4 * q + 12 * q^9 - 72 * q^17 + 52 * q^25 + 96 * q^33 - 216 * q^41 + 180 * q^49 + 144 * q^57 - 288 * q^65 + 40 * q^73 + 36 * q^81 - 72 * q^89 - 136 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	1.73205	0.577350
$3$	1.73205	0.577350
$4$	0	0
$5$	− 3.46410i	− 0.692820i	−0.938083	−	0.346410i	$-0.887401\pi$
$5$	− 3.46410i	− 0.692820i	0.938083	−	0.346410i	$-0.112599\pi$
$6$	0	0
$7$	− 2.00000i	− 0.285714i	−0.989743	−	0.142857i	$-0.954371\pi$
$7$	− 2.00000i	− 0.285714i	0.989743	−	0.142857i	$-0.0456289\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	13.8564	1.25967	0.629837	−	0.776728i	$-0.283122\pi$
$11$	13.8564	1.25967	0.629837	+	0.776728i	$0.283122\pi$
$12$	0	0
$13$	− 20.7846i	− 1.59882i	−0.600788	−	0.799408i	$-0.705147\pi$
$13$	− 20.7846i	− 1.59882i	0.600788	−	0.799408i	$-0.294853\pi$
$14$	0	0
$15$	− 6.00000i	− 0.400000i
$16$	0	0
$17$	−18.0000	−1.05882	−0.529412	−	0.848365i	$-0.677587\pi$
$17$	−18.0000	−1.05882	−0.529412	+	0.848365i	$0.677587\pi$
$18$	0	0
$19$	20.7846	1.09393	0.546963	−	0.837157i	$-0.315784\pi$
$19$	20.7846	1.09393	0.546963	+	0.837157i	$0.315784\pi$
$20$	0	0
$21$	− 3.46410i	− 0.164957i
$22$	0	0
$23$	36.0000i	1.56522i	0.622514	+	0.782609i	$0.286111\pi$
$23$	36.0000i	1.56522i	−0.622514	+	0.782609i	$0.713889\pi$
$24$	0	0
$25$	13.0000	0.520000
$26$	0	0
$27$	5.19615	0.192450
$28$	0	0
$29$	− 31.1769i	− 1.07507i	−0.843243	−	0.537533i	$-0.819356\pi$
$29$	− 31.1769i	− 1.07507i	0.843243	−	0.537533i	$-0.180644\pi$
$30$	0	0
$31$	− 22.0000i	− 0.709677i	−0.934928	−	0.354839i	$-0.884536\pi$
$31$	− 22.0000i	− 0.709677i	0.934928	−	0.354839i	$-0.115464\pi$
$32$	0	0
$33$	24.0000	0.727273
$34$	0	0
$35$	−6.92820	−0.197949
$36$	0	0
$37$	41.5692i	1.12349i	0.827310	+	0.561746i	$0.189870\pi$
$37$	41.5692i	1.12349i	−0.827310	+	0.561746i	$0.810130\pi$
$38$	0	0
$39$	− 36.0000i	− 0.923077i
$40$	0	0
$41$	−54.0000	−1.31707	−0.658537	−	0.752549i	$-0.728824\pi$
$41$	−54.0000	−1.31707	−0.658537	+	0.752549i	$0.728824\pi$
$42$	0	0
$43$	−20.7846	−0.483363	−0.241682	−	0.970356i	$-0.577699\pi$
$43$	−20.7846	−0.483363	−0.241682	+	0.970356i	$0.577699\pi$
$44$	0	0
$45$	− 10.3923i	− 0.230940i
$46$	0	0
$47$	36.0000i	0.765957i	0.923757	+	0.382979i	$0.125102\pi$
$47$	36.0000i	0.765957i	−0.923757	+	0.382979i	$0.874898\pi$
$48$	0	0
$49$	45.0000	0.918367
$50$	0	0
$51$	−31.1769	−0.611312
$52$	0	0
$53$	100.459i	1.89545i	0.319086	+	0.947726i	$0.396624\pi$
$53$	100.459i	1.89545i	−0.319086	+	0.947726i	$0.603376\pi$
$54$	0	0
$55$	− 48.0000i	− 0.872727i
$56$	0	0
$57$	36.0000	0.631579
$58$	0	0
$59$	−62.3538	−1.05684	−0.528422	−	0.848982i	$-0.677216\pi$
$59$	−62.3538	−1.05684	−0.528422	+	0.848982i	$0.677216\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	− 6.00000i	− 0.0952381i
$64$	0	0
$65$	−72.0000	−1.10769
$66$	0	0
$67$	−62.3538	−0.930654	−0.465327	−	0.885139i	$-0.654063\pi$
$67$	−62.3538	−0.930654	−0.465327	+	0.885139i	$0.654063\pi$
$68$	0	0
$69$	62.3538i	0.903679i
$70$	0	0
$71$	108.000i	1.52113i	0.649264	+	0.760563i	$0.275077\pi$
$71$	108.000i	1.52113i	−0.649264	+	0.760563i	$0.724923\pi$
$72$	0	0
$73$	10.0000	0.136986	0.0684932	−	0.997652i	$-0.478181\pi$
$73$	10.0000	0.136986	0.0684932	+	0.997652i	$0.478181\pi$
$74$	0	0
$75$	22.5167	0.300222
$76$	0	0
$77$	− 27.7128i	− 0.359907i
$78$	0	0
$79$	50.0000i	0.632911i	0.948607	+	0.316456i	$0.102493\pi$
$79$	50.0000i	0.632911i	−0.948607	+	0.316456i	$0.897507\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−13.8564	−0.166945	−0.0834723	−	0.996510i	$-0.526601\pi$
$83$	−13.8564	−0.166945	−0.0834723	+	0.996510i	$0.526601\pi$
$84$	0	0
$85$	62.3538i	0.733574i
$86$	0	0
$87$	− 54.0000i	− 0.620690i
$88$	0	0
$89$	−18.0000	−0.202247	−0.101124	−	0.994874i	$-0.532244\pi$
$89$	−18.0000	−0.202247	−0.101124	+	0.994874i	$0.532244\pi$
$90$	0	0
$91$	−41.5692	−0.456805
$92$	0	0
$93$	− 38.1051i	− 0.409732i
$94$	0	0
$95$	− 72.0000i	− 0.757895i
$96$	0	0
$97$	−34.0000	−0.350515	−0.175258	−	0.984523i	$-0.556076\pi$
$97$	−34.0000	−0.350515	−0.175258	+	0.984523i	$0.556076\pi$
$98$	0	0
$99$	41.5692	0.419891
$100$	0	0
$101$	17.3205i	0.171490i	0.996317	+	0.0857451i	$0.0273271\pi$
$101$	17.3205i	0.171490i	−0.996317	+	0.0857451i	$0.972673\pi$
$102$	0	0
$103$	− 118.000i	− 1.14563i	−0.819684	−	0.572816i	$-0.805851\pi$
$103$	− 118.000i	− 1.14563i	0.819684	−	0.572816i	$-0.194149\pi$
$104$	0	0
$105$	−12.0000	−0.114286
$106$	0	0
$107$	131.636	1.23024	0.615121	−	0.788433i	$-0.289107\pi$
$107$	131.636	1.23024	0.615121	+	0.788433i	$0.289107\pi$
$108$	0	0
$109$	− 145.492i	− 1.33479i	−0.744703	−	0.667396i	$-0.767409\pi$
$109$	− 145.492i	− 1.33479i	0.744703	−	0.667396i	$-0.232591\pi$
$110$	0	0
$111$	72.0000i	0.648649i
$112$	0	0
$113$	90.0000	0.796460	0.398230	−	0.917286i	$-0.369625\pi$
$113$	90.0000	0.796460	0.398230	+	0.917286i	$0.369625\pi$
$114$	0	0
$115$	124.708	1.08441
$116$	0	0
$117$	− 62.3538i	− 0.532939i
$118$	0	0
$119$	36.0000i	0.302521i
$120$	0	0
$121$	71.0000	0.586777
$122$	0	0
$123$	−93.5307	−0.760413
$124$	0	0
$125$	− 131.636i	− 1.05309i
$126$	0	0
$127$	118.000i	0.929134i	0.885538	+	0.464567i	$0.153790\pi$
$127$	118.000i	0.929134i	−0.885538	+	0.464567i	$0.846210\pi$
$128$	0	0
$129$	−36.0000	−0.279070
$130$	0	0
$131$	34.6410	0.264435	0.132218	−	0.991221i	$-0.457790\pi$
$131$	34.6410	0.264435	0.132218	+	0.991221i	$0.457790\pi$
$132$	0	0
$133$	− 41.5692i	− 0.312551i
$134$	0	0
$135$	− 18.0000i	− 0.133333i
$136$	0	0
$137$	90.0000	0.656934	0.328467	−	0.944515i	$-0.393468\pi$
$137$	90.0000	0.656934	0.328467	+	0.944515i	$0.393468\pi$
$138$	0	0
$139$	270.200	1.94388	0.971942	−	0.235220i	$-0.0755810\pi$
$139$	270.200	1.94388	0.971942	+	0.235220i	$0.0755810\pi$
$140$	0	0
$141$	62.3538i	0.442226i
$142$	0	0
$143$	− 288.000i	− 2.01399i
$144$	0	0
$145$	−108.000	−0.744828
$146$	0	0
$147$	77.9423	0.530220
$148$	0	0
$149$	− 183.597i	− 1.23220i	−0.787669	−	0.616099i	$-0.788712\pi$
$149$	− 183.597i	− 1.23220i	0.787669	−	0.616099i	$-0.211288\pi$
$150$	0	0
$151$	142.000i	0.940397i	0.882561	+	0.470199i	$0.155818\pi$
$151$	142.000i	0.940397i	−0.882561	+	0.470199i	$0.844182\pi$
$152$	0	0
$153$	−54.0000	−0.352941
$154$	0	0
$155$	−76.2102	−0.491679
$156$	0	0
$157$	41.5692i	0.264772i	0.991198	+	0.132386i	$0.0422639\pi$
$157$	41.5692i	0.264772i	−0.991198	+	0.132386i	$0.957736\pi$
$158$	0	0
$159$	174.000i	1.09434i
$160$	0	0
$161$	72.0000	0.447205
$162$	0	0
$163$	−103.923	−0.637565	−0.318782	−	0.947828i	$-0.603274\pi$
$163$	−103.923	−0.637565	−0.318782	+	0.947828i	$0.603274\pi$
$164$	0	0
$165$	− 83.1384i	− 0.503869i
$166$	0	0
$167$	72.0000i	0.431138i	0.976489	+	0.215569i	$0.0691606\pi$
$167$	72.0000i	0.431138i	−0.976489	+	0.215569i	$0.930839\pi$
$168$	0	0
$169$	−263.000	−1.55621
$170$	0	0
$171$	62.3538	0.364642
$172$	0	0
$173$	169.741i	0.981162i	0.871396	+	0.490581i	$0.163215\pi$
$173$	169.741i	0.981162i	−0.871396	+	0.490581i	$0.836785\pi$
$174$	0	0
$175$	− 26.0000i	− 0.148571i
$176$	0	0
$177$	−108.000	−0.610169
$178$	0	0
$179$	131.636	0.735396	0.367698	−	0.929945i	$-0.380146\pi$
$179$	131.636	0.735396	0.367698	+	0.929945i	$0.380146\pi$
$180$	0	0
$181$	− 20.7846i	− 0.114832i	−0.998350	−	0.0574160i	$-0.981714\pi$
$181$	− 20.7846i	− 0.114832i	0.998350	−	0.0574160i	$-0.0182862\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	144.000	0.778378
$186$	0	0
$187$	−249.415	−1.33377
$188$	0	0
$189$	− 10.3923i	− 0.0549857i
$190$	0	0
$191$	− 360.000i	− 1.88482i	−0.334464	−	0.942408i	$-0.608555\pi$
$191$	− 360.000i	− 1.88482i	0.334464	−	0.942408i	$-0.391445\pi$
$192$	0	0
$193$	194.000	1.00518	0.502591	−	0.864525i	$-0.332380\pi$
$193$	194.000	1.00518	0.502591	+	0.864525i	$0.332380\pi$
$194$	0	0
$195$	−124.708	−0.639526
$196$	0	0
$197$	86.6025i	0.439607i	0.975544	+	0.219803i	$0.0705416\pi$
$197$	86.6025i	0.439607i	−0.975544	+	0.219803i	$0.929458\pi$
$198$	0	0
$199$	26.0000i	0.130653i	0.997864	+	0.0653266i	$0.0208089\pi$
$199$	26.0000i	0.130653i	−0.997864	+	0.0653266i	$0.979191\pi$
$200$	0	0
$201$	−108.000	−0.537313
$202$	0	0
$203$	−62.3538	−0.307162
$204$	0	0
$205$	187.061i	0.912495i
$206$	0	0
$207$	108.000i	0.521739i
$208$	0	0
$209$	288.000	1.37799
$210$	0	0
$211$	−228.631	−1.08356	−0.541779	−	0.840521i	$-0.682249\pi$
$211$	−228.631	−1.08356	−0.541779	+	0.840521i	$0.682249\pi$
$212$	0	0
$213$	187.061i	0.878223i
$214$	0	0
$215$	72.0000i	0.334884i
$216$	0	0
$217$	−44.0000	−0.202765
$218$	0	0
$219$	17.3205	0.0790891
$220$	0	0
$221$	374.123i	1.69286i
$222$	0	0
$223$	310.000i	1.39013i	0.718945	+	0.695067i	$0.244625\pi$
$223$	310.000i	1.39013i	−0.718945	+	0.695067i	$0.755375\pi$
$224$	0	0
$225$	39.0000	0.173333
$226$	0	0
$227$	−249.415	−1.09875	−0.549373	−	0.835577i	$-0.685133\pi$
$227$	−249.415	−1.09875	−0.549373	+	0.835577i	$0.685133\pi$
$228$	0	0
$229$	− 228.631i	− 0.998387i	−0.866490	−	0.499194i	$-0.833630\pi$
$229$	− 228.631i	− 0.998387i	0.866490	−	0.499194i	$-0.166370\pi$
$230$	0	0
$231$	− 48.0000i	− 0.207792i
$232$	0	0
$233$	198.000	0.849785	0.424893	−	0.905244i	$-0.360312\pi$
$233$	198.000	0.849785	0.424893	+	0.905244i	$0.360312\pi$
$234$	0	0
$235$	124.708	0.530671
$236$	0	0
$237$	86.6025i	0.365412i
$238$	0	0
$239$	− 288.000i	− 1.20502i	−0.798111	−	0.602510i	$-0.794167\pi$
$239$	− 288.000i	− 1.20502i	0.798111	−	0.602510i	$-0.205833\pi$
$240$	0	0
$241$	−374.000	−1.55187	−0.775934	−	0.630815i	$-0.782721\pi$
$241$	−374.000	−1.55187	−0.775934	+	0.630815i	$0.782721\pi$
$242$	0	0
$243$	15.5885	0.0641500
$244$	0	0
$245$	− 155.885i	− 0.636264i
$246$	0	0
$247$	− 432.000i	− 1.74899i
$248$	0	0
$249$	−24.0000	−0.0963855
$250$	0	0
$251$	−374.123	−1.49053	−0.745265	−	0.666769i	$-0.767677\pi$
$251$	−374.123	−1.49053	−0.745265	+	0.666769i	$0.767677\pi$
$252$	0	0
$253$	498.831i	1.97166i
$254$	0	0
$255$	108.000i	0.423529i
$256$	0	0
$257$	−270.000	−1.05058	−0.525292	−	0.850922i	$-0.676044\pi$
$257$	−270.000	−1.05058	−0.525292	+	0.850922i	$0.676044\pi$
$258$	0	0
$259$	83.1384	0.320998
$260$	0	0
$261$	− 93.5307i	− 0.358355i
$262$	0	0
$263$	144.000i	0.547529i	0.961797	+	0.273764i	$0.0882688\pi$
$263$	144.000i	0.547529i	−0.961797	+	0.273764i	$0.911731\pi$
$264$	0	0
$265$	348.000	1.31321
$266$	0	0
$267$	−31.1769	−0.116767
$268$	0	0
$269$	− 31.1769i	− 0.115899i	−0.998320	−	0.0579497i	$-0.981544\pi$
$269$	− 31.1769i	− 0.115899i	0.998320	−	0.0579497i	$-0.0184563\pi$
$270$	0	0
$271$	262.000i	0.966790i	0.875402	+	0.483395i	$0.160596\pi$
$271$	262.000i	0.966790i	−0.875402	+	0.483395i	$0.839404\pi$
$272$	0	0
$273$	−72.0000	−0.263736
$274$	0	0
$275$	180.133	0.655030
$276$	0	0
$277$	− 103.923i	− 0.375173i	−0.982248	−	0.187587i	$-0.939933\pi$
$277$	− 103.923i	− 0.375173i	0.982248	−	0.187587i	$-0.0600666\pi$
$278$	0	0
$279$	− 66.0000i	− 0.236559i
$280$	0	0
$281$	−234.000	−0.832740	−0.416370	−	0.909195i	$-0.636698\pi$
$281$	−234.000	−0.832740	−0.416370	+	0.909195i	$0.636698\pi$
$282$	0	0
$283$	−353.338	−1.24855	−0.624273	−	0.781206i	$-0.714605\pi$
$283$	−353.338	−1.24855	−0.624273	+	0.781206i	$0.714605\pi$
$284$	0	0
$285$	− 124.708i	− 0.437571i
$286$	0	0
$287$	108.000i	0.376307i
$288$	0	0
$289$	35.0000	0.121107
$290$	0	0
$291$	−58.8897	−0.202370
$292$	0	0
$293$	− 93.5307i	− 0.319218i	−0.987180	−	0.159609i	$-0.948977\pi$
$293$	− 93.5307i	− 0.319218i	0.987180	−	0.159609i	$-0.0510233\pi$
$294$	0	0
$295$	216.000i	0.732203i
$296$	0	0
$297$	72.0000	0.242424
$298$	0	0
$299$	748.246	2.50249
$300$	0	0
$301$	41.5692i	0.138104i
$302$	0	0
$303$	30.0000i	0.0990099i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	436.477	1.42175	0.710874	−	0.703319i	$-0.248299\pi$
$307$	436.477	1.42175	0.710874	+	0.703319i	$0.248299\pi$
$308$	0	0
$309$	− 204.382i	− 0.661430i
$310$	0	0
$311$	216.000i	0.694534i	0.937766	+	0.347267i	$0.112890\pi$
$311$	216.000i	0.694534i	−0.937766	+	0.347267i	$0.887110\pi$
$312$	0	0
$313$	−290.000	−0.926518	−0.463259	−	0.886223i	$-0.653320\pi$
$313$	−290.000	−0.926518	−0.463259	+	0.886223i	$0.653320\pi$
$314$	0	0
$315$	−20.7846	−0.0659829
$316$	0	0
$317$	− 142.028i	− 0.448038i	−0.974585	−	0.224019i	$-0.928082\pi$
$317$	− 142.028i	− 0.448038i	0.974585	−	0.224019i	$-0.0719178\pi$
$318$	0	0
$319$	− 432.000i	− 1.35423i
$320$	0	0
$321$	228.000	0.710280
$322$	0	0
$323$	−374.123	−1.15828
$324$	0	0
$325$	− 270.200i	− 0.831384i
$326$	0	0
$327$	− 252.000i	− 0.770642i
$328$	0	0
$329$	72.0000	0.218845
$330$	0	0
$331$	353.338	1.06749	0.533744	−	0.845646i	$-0.320785\pi$
$331$	353.338	1.06749	0.533744	+	0.845646i	$0.320785\pi$
$332$	0	0
$333$	124.708i	0.374497i
$334$	0	0
$335$	216.000i	0.644776i
$336$	0	0
$337$	566.000	1.67953	0.839763	−	0.542954i	$-0.182694\pi$
$337$	566.000	1.67953	0.839763	+	0.542954i	$0.182694\pi$
$338$	0	0
$339$	155.885	0.459836
$340$	0	0
$341$	− 304.841i	− 0.893962i
$342$	0	0
$343$	− 188.000i	− 0.548105i
$344$	0	0
$345$	216.000	0.626087
$346$	0	0
$347$	27.7128	0.0798640	0.0399320	−	0.999202i	$-0.487286\pi$
$347$	27.7128	0.0798640	0.0399320	+	0.999202i	$0.487286\pi$
$348$	0	0
$349$	− 581.969i	− 1.66753i	−0.552117	−	0.833767i	$-0.686180\pi$
$349$	− 581.969i	− 1.66753i	0.552117	−	0.833767i	$-0.313820\pi$
$350$	0	0
$351$	− 108.000i	− 0.307692i
$352$	0	0
$353$	−342.000	−0.968839	−0.484419	−	0.874836i	$-0.660969\pi$
$353$	−342.000	−0.968839	−0.484419	+	0.874836i	$0.660969\pi$
$354$	0	0
$355$	374.123	1.05387
$356$	0	0
$357$	62.3538i	0.174661i
$358$	0	0
$359$	108.000i	0.300836i	0.988623	+	0.150418i	$0.0480619\pi$
$359$	108.000i	0.300836i	−0.988623	+	0.150418i	$0.951938\pi$
$360$	0	0
$361$	71.0000	0.196676
$362$	0	0
$363$	122.976	0.338776
$364$	0	0
$365$	− 34.6410i	− 0.0949069i
$366$	0	0
$367$	74.0000i	0.201635i	0.994905	+	0.100817i	$0.0321458\pi$
$367$	74.0000i	0.201635i	−0.994905	+	0.100817i	$0.967854\pi$
$368$	0	0
$369$	−162.000	−0.439024
$370$	0	0
$371$	200.918	0.541558
$372$	0	0
$373$	83.1384i	0.222891i	0.993771	+	0.111446i	$0.0355481\pi$
$373$	83.1384i	0.222891i	−0.993771	+	0.111446i	$0.964452\pi$
$374$	0	0
$375$	− 228.000i	− 0.608000i
$376$	0	0
$377$	−648.000	−1.71883
$378$	0	0
$379$	−228.631	−0.603247	−0.301624	−	0.953427i	$-0.597529\pi$
$379$	−228.631	−0.603247	−0.301624	+	0.953427i	$0.597529\pi$
$380$	0	0
$381$	204.382i	0.536436i
$382$	0	0
$383$	216.000i	0.563969i	0.959419	+	0.281984i	$0.0909926\pi$
$383$	216.000i	0.563969i	−0.959419	+	0.281984i	$0.909007\pi$
$384$	0	0
$385$	−96.0000	−0.249351
$386$	0	0
$387$	−62.3538	−0.161121
$388$	0	0
$389$	− 142.028i	− 0.365111i	−0.983196	−	0.182555i	$-0.941563\pi$
$389$	− 142.028i	− 0.365111i	0.983196	−	0.182555i	$-0.0584369\pi$
$390$	0	0
$391$	− 648.000i	− 1.65729i
$392$	0	0
$393$	60.0000	0.152672
$394$	0	0
$395$	173.205	0.438494
$396$	0	0
$397$	− 332.554i	− 0.837667i	−0.908063	−	0.418833i	$-0.862439\pi$
$397$	− 332.554i	− 0.837667i	0.908063	−	0.418833i	$-0.137561\pi$
$398$	0	0
$399$	− 72.0000i	− 0.180451i
$400$	0	0
$401$	126.000	0.314214	0.157107	−	0.987582i	$-0.449783\pi$
$401$	126.000	0.314214	0.157107	+	0.987582i	$0.449783\pi$
$402$	0	0
$403$	−457.261	−1.13464
$404$	0	0
$405$	− 31.1769i	− 0.0769800i
$406$	0	0
$407$	576.000i	1.41523i
$408$	0	0
$409$	562.000	1.37408	0.687042	−	0.726618i	$-0.258909\pi$
$409$	562.000	1.37408	0.687042	+	0.726618i	$0.258909\pi$
$410$	0	0
$411$	155.885	0.379281
$412$	0	0
$413$	124.708i	0.301956i
$414$	0	0
$415$	48.0000i	0.115663i
$416$	0	0
$417$	468.000	1.12230
$418$	0	0
$419$	498.831	1.19053	0.595263	−	0.803531i	$-0.297048\pi$
$419$	498.831	1.19053	0.595263	+	0.803531i	$0.297048\pi$
$420$	0	0
$421$	644.323i	1.53046i	0.643758	+	0.765229i	$0.277374\pi$
$421$	644.323i	1.53046i	−0.643758	+	0.765229i	$0.722626\pi$
$422$	0	0
$423$	108.000i	0.255319i
$424$	0	0
$425$	−234.000	−0.550588
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 498.831i	− 1.16278i
$430$	0	0
$431$	252.000i	0.584687i	0.956313	+	0.292343i	$0.0944350\pi$
$431$	252.000i	0.584687i	−0.956313	+	0.292343i	$0.905565\pi$
$432$	0	0
$433$	−386.000	−0.891455	−0.445727	−	0.895169i	$-0.647055\pi$
$433$	−386.000	−0.891455	−0.445727	+	0.895169i	$0.647055\pi$
$434$	0	0
$435$	−187.061	−0.430026
$436$	0	0
$437$	748.246i	1.71223i
$438$	0	0
$439$	866.000i	1.97267i	0.164767	+	0.986333i	$0.447313\pi$
$439$	866.000i	1.97267i	−0.164767	+	0.986333i	$0.552687\pi$
$440$	0	0
$441$	135.000	0.306122
$442$	0	0
$443$	−748.246	−1.68904	−0.844521	−	0.535522i	$-0.820115\pi$
$443$	−748.246	−1.68904	−0.844521	+	0.535522i	$0.820115\pi$
$444$	0	0
$445$	62.3538i	0.140121i
$446$	0	0
$447$	− 318.000i	− 0.711409i
$448$	0	0
$449$	−450.000	−1.00223	−0.501114	−	0.865382i	$-0.667076\pi$
$449$	−450.000	−1.00223	−0.501114	+	0.865382i	$0.667076\pi$
$450$	0	0
$451$	−748.246	−1.65908
$452$	0	0
$453$	245.951i	0.542939i
$454$	0	0
$455$	144.000i	0.316484i
$456$	0	0
$457$	−254.000	−0.555799	−0.277899	−	0.960610i	$-0.589638\pi$
$457$	−254.000	−0.555799	−0.277899	+	0.960610i	$0.589638\pi$
$458$	0	0
$459$	−93.5307	−0.203771
$460$	0	0
$461$	17.3205i	0.0375716i	0.999824	+	0.0187858i	$0.00598006\pi$
$461$	17.3205i	0.0375716i	−0.999824	+	0.0187858i	$0.994020\pi$
$462$	0	0
$463$	− 626.000i	− 1.35205i	−0.736878	−	0.676026i	$-0.763701\pi$
$463$	− 626.000i	− 1.35205i	0.736878	−	0.676026i	$-0.236299\pi$
$464$	0	0
$465$	−132.000	−0.283871
$466$	0	0
$467$	−304.841	−0.652764	−0.326382	−	0.945238i	$-0.605830\pi$
$467$	−304.841	−0.652764	−0.326382	+	0.945238i	$0.605830\pi$
$468$	0	0
$469$	124.708i	0.265901i
$470$	0	0
$471$	72.0000i	0.152866i
$472$	0	0
$473$	−288.000	−0.608879
$474$	0	0
$475$	270.200	0.568842
$476$	0	0
$477$	301.377i	0.631817i
$478$	0	0
$479$	− 252.000i	− 0.526096i	−0.964783	−	0.263048i	$-0.915272\pi$
$479$	− 252.000i	− 0.526096i	0.964783	−	0.263048i	$-0.0847278\pi$
$480$	0	0
$481$	864.000	1.79626
$482$	0	0
$483$	124.708	0.258194
$484$	0	0
$485$	117.779i	0.242844i
$486$	0	0
$487$	− 218.000i	− 0.447639i	−0.974631	−	0.223819i	$-0.928147\pi$
$487$	− 218.000i	− 0.447639i	0.974631	−	0.223819i	$-0.0718525\pi$
$488$	0	0
$489$	−180.000	−0.368098
$490$	0	0
$491$	−713.605	−1.45337	−0.726685	−	0.686971i	$-0.758940\pi$
$491$	−713.605	−1.45337	−0.726685	+	0.686971i	$0.758940\pi$
$492$	0	0
$493$	561.184i	1.13831i
$494$	0	0
$495$	− 144.000i	− 0.290909i
$496$	0	0
$497$	216.000	0.434608
$498$	0	0
$499$	561.184	1.12462	0.562309	−	0.826927i	$-0.309913\pi$
$499$	561.184	1.12462	0.562309	+	0.826927i	$0.309913\pi$
$500$	0	0
$501$	124.708i	0.248917i
$502$	0	0
$503$	− 324.000i	− 0.644135i	−0.946717	−	0.322068i	$-0.895622\pi$
$503$	− 324.000i	− 0.644135i	0.946717	−	0.322068i	$-0.104378\pi$
$504$	0	0
$505$	60.0000	0.118812
$506$	0	0
$507$	−455.529	−0.898480
$508$	0	0
$509$	− 751.710i	− 1.47684i	−0.674343	−	0.738419i	$-0.735573\pi$
$509$	− 751.710i	− 1.47684i	0.674343	−	0.738419i	$-0.264427\pi$
$510$	0	0
$511$	− 20.0000i	− 0.0391389i
$512$	0	0
$513$	108.000	0.210526
$514$	0	0
$515$	−408.764	−0.793716
$516$	0	0
$517$	498.831i	0.964856i
$518$	0	0
$519$	294.000i	0.566474i
$520$	0	0
$521$	738.000	1.41651	0.708253	−	0.705958i	$-0.249483\pi$
$521$	738.000	1.41651	0.708253	+	0.705958i	$0.249483\pi$
$522$	0	0
$523$	−62.3538	−0.119223	−0.0596117	−	0.998222i	$-0.518986\pi$
$523$	−62.3538	−0.119223	−0.0596117	+	0.998222i	$0.518986\pi$
$524$	0	0
$525$	− 45.0333i	− 0.0857778i
$526$	0	0
$527$	396.000i	0.751423i
$528$	0	0
$529$	−767.000	−1.44991
$530$	0	0
$531$	−187.061	−0.352282
$532$	0	0
$533$	1122.37i	2.10576i
$534$	0	0
$535$	− 456.000i	− 0.852336i
$536$	0	0
$537$	228.000	0.424581
$538$	0	0
$539$	623.538	1.15684
$540$	0	0
$541$	− 228.631i	− 0.422608i	−0.977420	−	0.211304i	$-0.932229\pi$
$541$	− 228.631i	− 0.422608i	0.977420	−	0.211304i	$-0.0677709\pi$
$542$	0	0
$543$	− 36.0000i	− 0.0662983i
$544$	0	0
$545$	−504.000	−0.924771
$546$	0	0
$547$	−436.477	−0.797947	−0.398973	−	0.916963i	$-0.630633\pi$
$547$	−436.477	−0.797947	−0.398973	+	0.916963i	$0.630633\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 648.000i	− 1.17604i
$552$	0	0
$553$	100.000	0.180832
$554$	0	0
$555$	249.415	0.449397
$556$	0	0
$557$	− 273.664i	− 0.491318i	−0.969356	−	0.245659i	$-0.920996\pi$
$557$	− 273.664i	− 0.491318i	0.969356	−	0.245659i	$-0.0790043\pi$
$558$	0	0
$559$	432.000i	0.772809i
$560$	0	0
$561$	−432.000	−0.770053
$562$	0	0
$563$	−27.7128	−0.0492235	−0.0246117	−	0.999697i	$-0.507835\pi$
$563$	−27.7128	−0.0492235	−0.0246117	+	0.999697i	$0.507835\pi$
$564$	0	0
$565$	− 311.769i	− 0.551804i
$566$	0	0
$567$	− 18.0000i	− 0.0317460i
$568$	0	0
$569$	666.000	1.17047	0.585237	−	0.810862i	$-0.301001\pi$
$569$	666.000	1.17047	0.585237	+	0.810862i	$0.301001\pi$
$570$	0	0
$571$	−270.200	−0.473205	−0.236602	−	0.971607i	$-0.576034\pi$
$571$	−270.200	−0.473205	−0.236602	+	0.971607i	$0.576034\pi$
$572$	0	0
$573$	− 623.538i	− 1.08820i
$574$	0	0
$575$	468.000i	0.813913i
$576$	0	0
$577$	142.000	0.246101	0.123050	−	0.992400i	$-0.460732\pi$
$577$	142.000	0.246101	0.123050	+	0.992400i	$0.460732\pi$
$578$	0	0
$579$	336.018	0.580342
$580$	0	0
$581$	27.7128i	0.0476985i
$582$	0	0
$583$	1392.00i	2.38765i
$584$	0	0
$585$	−216.000	−0.369231
$586$	0	0
$587$	422.620	0.719967	0.359983	−	0.932959i	$-0.382782\pi$
$587$	422.620	0.719967	0.359983	+	0.932959i	$0.382782\pi$
$588$	0	0
$589$	− 457.261i	− 0.776335i
$590$	0	0
$591$	150.000i	0.253807i
$592$	0	0
$593$	954.000	1.60877	0.804384	−	0.594109i	$-0.202495\pi$
$593$	954.000	1.60877	0.804384	+	0.594109i	$0.202495\pi$
$594$	0	0
$595$	124.708	0.209593
$596$	0	0
$597$	45.0333i	0.0754327i
$598$	0	0
$599$	− 1044.00i	− 1.74290i	−0.490480	−	0.871452i	$-0.663179\pi$
$599$	− 1044.00i	− 1.74290i	0.490480	−	0.871452i	$-0.336821\pi$
$600$	0	0
$601$	−430.000	−0.715474	−0.357737	−	0.933822i	$-0.616452\pi$
$601$	−430.000	−0.715474	−0.357737	+	0.933822i	$0.616452\pi$
$602$	0	0
$603$	−187.061	−0.310218
$604$	0	0
$605$	− 245.951i	− 0.406531i
$606$	0	0
$607$	− 458.000i	− 0.754530i	−0.926105	−	0.377265i	$-0.876865\pi$
$607$	− 458.000i	− 0.754530i	0.926105	−	0.377265i	$-0.123135\pi$
$608$	0	0
$609$	−108.000	−0.177340
$610$	0	0
$611$	748.246	1.22463
$612$	0	0
$613$	249.415i	0.406877i	0.979088	+	0.203438i	$0.0652116\pi$
$613$	249.415i	0.406877i	−0.979088	+	0.203438i	$0.934788\pi$
$614$	0	0
$615$	324.000i	0.526829i
$616$	0	0
$617$	−234.000	−0.379254	−0.189627	−	0.981856i	$-0.560728\pi$
$617$	−234.000	−0.379254	−0.189627	+	0.981856i	$0.560728\pi$
$618$	0	0
$619$	311.769	0.503666	0.251833	−	0.967771i	$-0.418967\pi$
$619$	311.769	0.503666	0.251833	+	0.967771i	$0.418967\pi$
$620$	0	0
$621$	187.061i	0.301226i
$622$	0	0
$623$	36.0000i	0.0577849i
$624$	0	0
$625$	−131.000	−0.209600
$626$	0	0
$627$	498.831	0.795583
$628$	0	0
$629$	− 748.246i	− 1.18958i
$630$	0	0
$631$	− 26.0000i	− 0.0412044i	−0.999788	−	0.0206022i	$-0.993442\pi$
$631$	− 26.0000i	− 0.0412044i	0.999788	−	0.0206022i	$-0.00655835\pi$
$632$	0	0
$633$	−396.000	−0.625592
$634$	0	0
$635$	408.764	0.643723
$636$	0	0
$637$	− 935.307i	− 1.46830i
$638$	0	0
$639$	324.000i	0.507042i
$640$	0	0
$641$	−378.000	−0.589704	−0.294852	−	0.955543i	$-0.595270\pi$
$641$	−378.000	−0.589704	−0.294852	+	0.955543i	$0.595270\pi$
$642$	0	0
$643$	62.3538	0.0969733	0.0484866	−	0.998824i	$-0.484560\pi$
$643$	62.3538	0.0969733	0.0484866	+	0.998824i	$0.484560\pi$
$644$	0	0
$645$	124.708i	0.193345i
$646$	0	0
$647$	− 684.000i	− 1.05719i	−0.848875	−	0.528594i	$-0.822720\pi$
$647$	− 684.000i	− 1.05719i	0.848875	−	0.528594i	$-0.177280\pi$
$648$	0	0
$649$	−864.000	−1.33128
$650$	0	0
$651$	−76.2102	−0.117066
$652$	0	0
$653$	− 495.367i	− 0.758601i	−0.925274	−	0.379301i	$-0.876165\pi$
$653$	− 495.367i	− 0.758601i	0.925274	−	0.379301i	$-0.123835\pi$
$654$	0	0
$655$	− 120.000i	− 0.183206i
$656$	0	0
$657$	30.0000	0.0456621
$658$	0	0
$659$	−325.626	−0.494121	−0.247060	−	0.969000i	$-0.579465\pi$
$659$	−325.626	−0.494121	−0.247060	+	0.969000i	$0.579465\pi$
$660$	0	0
$661$	− 374.123i	− 0.565995i	−0.959121	−	0.282998i	$-0.908671\pi$
$661$	− 374.123i	− 0.565995i	0.959121	−	0.282998i	$-0.0913289\pi$
$662$	0	0
$663$	648.000i	0.977376i
$664$	0	0
$665$	−144.000	−0.216541
$666$	0	0
$667$	1122.37	1.68271
$668$	0	0
$669$	536.936i	0.802595i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−146.000	−0.216939	−0.108470	−	0.994100i	$-0.534595\pi$
$673$	−146.000	−0.216939	−0.108470	+	0.994100i	$0.534595\pi$
$674$	0	0
$675$	67.5500	0.100074
$676$	0	0
$677$	− 530.008i	− 0.782877i	−0.920204	−	0.391438i	$-0.871978\pi$
$677$	− 530.008i	− 0.782877i	0.920204	−	0.391438i	$-0.128022\pi$
$678$	0	0
$679$	68.0000i	0.100147i
$680$	0	0
$681$	−432.000	−0.634361
$682$	0	0
$683$	−124.708	−0.182588	−0.0912940	−	0.995824i	$-0.529100\pi$
$683$	−124.708	−0.182588	−0.0912940	+	0.995824i	$0.529100\pi$
$684$	0	0
$685$	− 311.769i	− 0.455137i
$686$	0	0
$687$	− 396.000i	− 0.576419i
$688$	0	0
$689$	2088.00	3.03048
$690$	0	0
$691$	478.046	0.691818	0.345909	−	0.938268i	$-0.387571\pi$
$691$	478.046	0.691818	0.345909	+	0.938268i	$0.387571\pi$
$692$	0	0
$693$	− 83.1384i	− 0.119969i
$694$	0	0
$695$	− 936.000i	− 1.34676i
$696$	0	0
$697$	972.000	1.39455
$698$	0	0
$699$	342.946	0.490624
$700$	0	0
$701$	1285.18i	1.83335i	0.399628	+	0.916677i	$0.369139\pi$
$701$	1285.18i	1.83335i	−0.399628	+	0.916677i	$0.630861\pi$
$702$	0	0
$703$	864.000i	1.22902i
$704$	0	0
$705$	216.000	0.306383
$706$	0	0
$707$	34.6410	0.0489972
$708$	0	0
$709$	− 394.908i	− 0.556992i	−0.960437	−	0.278496i	$-0.910164\pi$
$709$	− 394.908i	− 0.556992i	0.960437	−	0.278496i	$-0.0898360\pi$
$710$	0	0
$711$	150.000i	0.210970i
$712$	0	0
$713$	792.000	1.11080
$714$	0	0
$715$	−997.661	−1.39533
$716$	0	0
$717$	− 498.831i	− 0.695719i
$718$	0	0
$719$	− 612.000i	− 0.851182i	−0.904916	−	0.425591i	$-0.860066\pi$
$719$	− 612.000i	− 0.851182i	0.904916	−	0.425591i	$-0.139934\pi$
$720$	0	0
$721$	−236.000	−0.327323
$722$	0	0
$723$	−647.787	−0.895971
$724$	0	0
$725$	− 405.300i	− 0.559034i
$726$	0	0
$727$	502.000i	0.690509i	0.938509	+	0.345254i	$0.112207\pi$
$727$	502.000i	0.690509i	−0.938509	+	0.345254i	$0.887793\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	374.123	0.511796
$732$	0	0
$733$	− 20.7846i	− 0.0283555i	−0.999899	−	0.0141778i	$-0.995487\pi$
$733$	− 20.7846i	− 0.0283555i	0.999899	−	0.0141778i	$-0.00451308\pi$
$734$	0	0
$735$	− 270.000i	− 0.367347i
$736$	0	0
$737$	−864.000	−1.17232
$738$	0	0
$739$	644.323	0.871885	0.435942	−	0.899975i	$-0.356415\pi$
$739$	644.323	0.871885	0.435942	+	0.899975i	$0.356415\pi$
$740$	0	0
$741$	− 748.246i	− 1.00978i
$742$	0	0
$743$	− 720.000i	− 0.969044i	−0.874779	−	0.484522i	$-0.838993\pi$
$743$	− 720.000i	− 0.969044i	0.874779	−	0.484522i	$-0.161007\pi$
$744$	0	0
$745$	−636.000	−0.853691
$746$	0	0
$747$	−41.5692	−0.0556482
$748$	0	0
$749$	− 263.272i	− 0.351498i
$750$	0	0
$751$	− 310.000i	− 0.412783i	−0.978469	−	0.206391i	$-0.933828\pi$
$751$	− 310.000i	− 0.412783i	0.978469	−	0.206391i	$-0.0661720\pi$
$752$	0	0
$753$	−648.000	−0.860558
$754$	0	0
$755$	491.902	0.651526
$756$	0	0
$757$	1184.72i	1.56502i	0.622636	+	0.782512i	$0.286062\pi$
$757$	1184.72i	1.56502i	−0.622636	+	0.782512i	$0.713938\pi$
$758$	0	0
$759$	864.000i	1.13834i
$760$	0	0
$761$	450.000	0.591327	0.295664	−	0.955292i	$-0.404459\pi$
$761$	450.000	0.591327	0.295664	+	0.955292i	$0.404459\pi$
$762$	0	0
$763$	−290.985	−0.381369
$764$	0	0
$765$	187.061i	0.244525i
$766$	0	0
$767$	1296.00i	1.68970i
$768$	0	0
$769$	50.0000	0.0650195	0.0325098	−	0.999471i	$-0.489650\pi$
$769$	50.0000	0.0650195	0.0325098	+	0.999471i	$0.489650\pi$
$770$	0	0
$771$	−467.654	−0.606555
$772$	0	0
$773$	779.423i	1.00831i	0.863613	+	0.504155i	$0.168196\pi$
$773$	779.423i	1.00831i	−0.863613	+	0.504155i	$0.831804\pi$
$774$	0	0
$775$	− 286.000i	− 0.369032i
$776$	0	0
$777$	144.000	0.185328
$778$	0	0
$779$	−1122.37	−1.44078
$780$	0	0
$781$	1496.49i	1.91612i
$782$	0	0
$783$	− 162.000i	− 0.206897i
$784$	0	0
$785$	144.000	0.183439
$786$	0	0
$787$	−1060.02	−1.34691	−0.673453	−	0.739230i	$-0.735190\pi$
$787$	−1060.02	−1.34691	−0.673453	+	0.739230i	$0.735190\pi$
$788$	0	0
$789$	249.415i	0.316116i
$790$	0	0
$791$	− 180.000i	− 0.227560i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	602.754	0.758181
$796$	0	0
$797$	128.172i	0.160818i	0.996762	+	0.0804089i	$0.0256226\pi$
$797$	128.172i	0.160818i	−0.996762	+	0.0804089i	$0.974377\pi$
$798$	0	0
$799$	− 648.000i	− 0.811014i
$800$	0	0
$801$	−54.0000	−0.0674157
$802$	0	0
$803$	138.564	0.172558
$804$	0	0
$805$	− 249.415i	− 0.309833i
$806$	0	0
$807$	− 54.0000i	− 0.0669145i
$808$	0	0
$809$	234.000	0.289246	0.144623	−	0.989487i	$-0.453803\pi$
$809$	234.000	0.289246	0.144623	+	0.989487i	$0.453803\pi$
$810$	0	0
$811$	−644.323	−0.794480	−0.397240	−	0.917715i	$-0.630032\pi$
$811$	−644.323	−0.794480	−0.397240	+	0.917715i	$0.630032\pi$
$812$	0	0
$813$	453.797i	0.558176i
$814$	0	0
$815$	360.000i	0.441718i
$816$	0	0
$817$	−432.000	−0.528764
$818$	0	0
$819$	−124.708	−0.152268
$820$	0	0
$821$	− 841.777i	− 1.02531i	−0.858596	−	0.512653i	$-0.828663\pi$
$821$	− 841.777i	− 1.02531i	0.858596	−	0.512653i	$-0.171337\pi$
$822$	0	0
$823$	382.000i	0.464156i	0.972697	+	0.232078i	$0.0745524\pi$
$823$	382.000i	0.464156i	−0.972697	+	0.232078i	$0.925448\pi$
$824$	0	0
$825$	312.000	0.378182
$826$	0	0
$827$	−810.600	−0.980169	−0.490085	−	0.871675i	$-0.663034\pi$
$827$	−810.600	−0.980169	−0.490085	+	0.871675i	$0.663034\pi$
$828$	0	0
$829$	1018.45i	1.22852i	0.789102	+	0.614262i	$0.210546\pi$
$829$	1018.45i	1.22852i	−0.789102	+	0.614262i	$0.789454\pi$
$830$	0	0
$831$	− 180.000i	− 0.216606i
$832$	0	0
$833$	−810.000	−0.972389
$834$	0	0
$835$	249.415	0.298701
$836$	0	0
$837$	− 114.315i	− 0.136577i
$838$	0	0
$839$	− 396.000i	− 0.471990i	−0.971754	−	0.235995i	$-0.924165\pi$
$839$	− 396.000i	− 0.471990i	0.971754	−	0.235995i	$-0.0758350\pi$
$840$	0	0
$841$	−131.000	−0.155767
$842$	0	0
$843$	−405.300	−0.480783
$844$	0	0
$845$	911.059i	1.07818i
$846$	0	0
$847$	− 142.000i	− 0.167651i
$848$	0	0
$849$	−612.000	−0.720848
$850$	0	0
$851$	−1496.49	−1.75851
$852$	0	0
$853$	− 623.538i	− 0.730994i	−0.930812	−	0.365497i	$-0.880899\pi$
$853$	− 623.538i	− 0.730994i	0.930812	−	0.365497i	$-0.119101\pi$
$854$	0	0
$855$	− 216.000i	− 0.252632i
$856$	0	0
$857$	−1206.00	−1.40723	−0.703617	−	0.710579i	$-0.748433\pi$
$857$	−1206.00	−1.40723	−0.703617	+	0.710579i	$0.748433\pi$
$858$	0	0
$859$	1226.29	1.42758	0.713790	−	0.700359i	$-0.246977\pi$
$859$	1226.29	1.42758	0.713790	+	0.700359i	$0.246977\pi$
$860$	0	0
$861$	187.061i	0.217261i
$862$	0	0
$863$	− 72.0000i	− 0.0834299i	−0.999130	−	0.0417149i	$-0.986718\pi$
$863$	− 72.0000i	− 0.0834299i	0.999130	−	0.0417149i	$-0.0132821\pi$
$864$	0	0
$865$	588.000	0.679769
$866$	0	0
$867$	60.6218	0.0699213
$868$	0	0
$869$	692.820i	0.797262i
$870$	0	0
$871$	1296.00i	1.48794i
$872$	0	0
$873$	−102.000	−0.116838
$874$	0	0
$875$	−263.272	−0.300882
$876$	0	0
$877$	665.108i	0.758389i	0.925317	+	0.379195i	$0.123799\pi$
$877$	665.108i	0.758389i	−0.925317	+	0.379195i	$0.876201\pi$
$878$	0	0
$879$	− 162.000i	− 0.184300i
$880$	0	0
$881$	594.000	0.674234	0.337117	−	0.941463i	$-0.390548\pi$
$881$	594.000	0.674234	0.337117	+	0.941463i	$0.390548\pi$
$882$	0	0
$883$	−769.031	−0.870929	−0.435465	−	0.900206i	$-0.643416\pi$
$883$	−769.031	−0.870929	−0.435465	+	0.900206i	$0.643416\pi$
$884$	0	0
$885$	374.123i	0.422738i
$886$	0	0
$887$	720.000i	0.811725i	0.913934	+	0.405862i	$0.133029\pi$
$887$	720.000i	0.811725i	−0.913934	+	0.405862i	$0.866971\pi$
$888$	0	0
$889$	236.000	0.265467
$890$	0	0
$891$	124.708	0.139964
$892$	0	0
$893$	748.246i	0.837901i
$894$	0	0
$895$	− 456.000i	− 0.509497i
$896$	0	0
$897$	1296.00	1.44482
$898$	0	0
$899$	−685.892	−0.762950
$900$	0	0
$901$	− 1808.26i	− 2.00695i
$902$	0	0
$903$	72.0000i	0.0797342i
$904$	0	0
$905$	−72.0000	−0.0795580
$906$	0	0
$907$	−1517.28	−1.67285	−0.836426	−	0.548080i	$-0.815359\pi$
$907$	−1517.28	−1.67285	−0.836426	+	0.548080i	$0.815359\pi$
$908$	0	0
$909$	51.9615i	0.0571634i
$910$	0	0
$911$	720.000i	0.790340i	0.918608	+	0.395170i	$0.129314\pi$
$911$	720.000i	0.790340i	−0.918608	+	0.395170i	$0.870686\pi$
$912$	0	0
$913$	−192.000	−0.210296
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 69.2820i	− 0.0755529i
$918$	0	0
$919$	46.0000i	0.0500544i	0.999687	+	0.0250272i	$0.00796724\pi$
$919$	46.0000i	0.0500544i	−0.999687	+	0.0250272i	$0.992033\pi$
$920$	0	0
$921$	756.000	0.820847
$922$	0	0
$923$	2244.74	2.43200
$924$	0	0
$925$	540.400i	0.584216i
$926$	0	0
$927$	− 354.000i	− 0.381877i
$928$	0	0
$929$	−18.0000	−0.0193757	−0.00968784	−	0.999953i	$-0.503084\pi$
$929$	−18.0000	−0.0193757	−0.00968784	+	0.999953i	$0.503084\pi$
$930$	0	0
$931$	935.307	1.00463
$932$	0	0
$933$	374.123i	0.400989i
$934$	0	0
$935$	864.000i	0.924064i
$936$	0	0
$937$	1198.00	1.27855	0.639274	−	0.768979i	$-0.279235\pi$
$937$	1198.00	1.27855	0.639274	+	0.768979i	$0.279235\pi$
$938$	0	0
$939$	−502.295	−0.534925
$940$	0	0
$941$	682.428i	0.725216i	0.931942	+	0.362608i	$0.118114\pi$
$941$	682.428i	0.725216i	−0.931942	+	0.362608i	$0.881886\pi$
$942$	0	0
$943$	− 1944.00i	− 2.06151i
$944$	0	0
$945$	−36.0000	−0.0380952
$946$	0	0
$947$	1046.16	1.10471	0.552354	−	0.833610i	$-0.313730\pi$
$947$	1046.16	1.10471	0.552354	+	0.833610i	$0.313730\pi$
$948$	0	0
$949$	− 207.846i	− 0.219016i
$950$	0	0
$951$	− 246.000i	− 0.258675i
$952$	0	0
$953$	−414.000	−0.434418	−0.217209	−	0.976125i	$-0.569695\pi$
$953$	−414.000	−0.434418	−0.217209	+	0.976125i	$0.569695\pi$
$954$	0	0
$955$	−1247.08	−1.30584
$956$	0	0
$957$	− 748.246i	− 0.781866i
$958$	0	0
$959$	− 180.000i	− 0.187696i
$960$	0	0
$961$	477.000	0.496358
$962$	0	0
$963$	394.908	0.410081
$964$	0	0
$965$	− 672.036i	− 0.696410i
$966$	0	0
$967$	− 1174.00i	− 1.21406i	−0.794677	−	0.607032i	$-0.792360\pi$
$967$	− 1174.00i	− 1.21406i	0.794677	−	0.607032i	$-0.207640\pi$
$968$	0	0
$969$	−648.000	−0.668731
$970$	0	0
$971$	180.133	0.185513	0.0927566	−	0.995689i	$-0.470432\pi$
$971$	180.133	0.185513	0.0927566	+	0.995689i	$0.470432\pi$
$972$	0	0
$973$	− 540.400i	− 0.555396i
$974$	0	0
$975$	− 468.000i	− 0.480000i
$976$	0	0
$977$	−1746.00	−1.78710	−0.893552	−	0.448960i	$-0.851794\pi$
$977$	−1746.00	−1.78710	−0.893552	+	0.448960i	$0.851794\pi$
$978$	0	0
$979$	−249.415	−0.254765
$980$	0	0
$981$	− 436.477i	− 0.444930i
$982$	0	0
$983$	− 1008.00i	− 1.02543i	−0.858558	−	0.512716i	$-0.828639\pi$
$983$	− 1008.00i	− 1.02543i	0.858558	−	0.512716i	$-0.171361\pi$
$984$	0	0
$985$	300.000	0.304569
$986$	0	0
$987$	124.708	0.126350
$988$	0	0
$989$	− 748.246i	− 0.756568i
$990$	0	0
$991$	314.000i	0.316852i	0.987371	+	0.158426i	$0.0506419\pi$
$991$	314.000i	0.316852i	−0.987371	+	0.158426i	$0.949358\pi$
$992$	0	0
$993$	612.000	0.616314
$994$	0	0
$995$	90.0666	0.0905192
$996$	0	0
$997$	249.415i	0.250166i	0.992146	+	0.125083i	$0.0399197\pi$
$997$	249.415i	0.250166i	−0.992146	+	0.125083i	$0.960080\pi$
$998$	0	0
$999$	216.000i	0.216216i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.b.a.31.3	yes	4
3.2	odd	2		576.3.b.e.415.3		4
4.3	odd	2	inner	192.3.b.a.31.1	✓	4
8.3	odd	2	inner	192.3.b.a.31.4	yes	4
8.5	even	2	inner	192.3.b.a.31.2	yes	4
12.11	even	2		576.3.b.e.415.4		4
16.3	odd	4		768.3.g.d.511.1		4
16.5	even	4		768.3.g.d.511.2		4
16.11	odd	4		768.3.g.d.511.4		4
16.13	even	4		768.3.g.d.511.3		4
24.5	odd	2		576.3.b.e.415.1		4
24.11	even	2		576.3.b.e.415.2		4
48.5	odd	4		2304.3.g.v.1279.2		4
48.11	even	4		2304.3.g.v.1279.1		4
48.29	odd	4		2304.3.g.v.1279.4		4
48.35	even	4		2304.3.g.v.1279.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.b.a.31.1	✓	4	4.3	odd	2	inner
192.3.b.a.31.2	yes	4	8.5	even	2	inner
192.3.b.a.31.3	yes	4	1.1	even	1	trivial
192.3.b.a.31.4	yes	4	8.3	odd	2	inner
576.3.b.e.415.1		4	24.5	odd	2
576.3.b.e.415.2		4	24.11	even	2
576.3.b.e.415.3		4	3.2	odd	2
576.3.b.e.415.4		4	12.11	even	2
768.3.g.d.511.1		4	16.3	odd	4
768.3.g.d.511.2		4	16.5	even	4
768.3.g.d.511.3		4	16.13	even	4
768.3.g.d.511.4		4	16.11	odd	4
2304.3.g.v.1279.1		4	48.11	even	4
2304.3.g.v.1279.2		4	48.5	odd	4
2304.3.g.v.1279.3		4	48.35	even	4
2304.3.g.v.1279.4		4	48.29	odd	4

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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	192.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} -3.46410i q^{5} -2.00000i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{3} -3.46410i q^{5} -2.00000i q^{7} +3.00000 q^{9} +13.8564 q^{11} -20.7846i q^{13} -6.00000i q^{15} -18.0000 q^{17} +20.7846 q^{19} -3.46410i q^{21} +36.0000i q^{23} +13.0000 q^{25} +5.19615 q^{27} -31.1769i q^{29} -22.0000i q^{31} +24.0000 q^{33} -6.92820 q^{35} +41.5692i q^{37} -36.0000i q^{39} -54.0000 q^{41} -20.7846 q^{43} -10.3923i q^{45} +36.0000i q^{47} +45.0000 q^{49} -31.1769 q^{51} +100.459i q^{53} -48.0000i q^{55} +36.0000 q^{57} -62.3538 q^{59} -6.00000i q^{63} -72.0000 q^{65} -62.3538 q^{67} +62.3538i q^{69} +108.000i q^{71} +10.0000 q^{73} +22.5167 q^{75} -27.7128i q^{77} +50.0000i q^{79} +9.00000 q^{81} -13.8564 q^{83} +62.3538i q^{85} -54.0000i q^{87} -18.0000 q^{89} -41.5692 q^{91} -38.1051i q^{93} -72.0000i q^{95} -34.0000 q^{97} +41.5692 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} - 72 q^{17} + 52 q^{25} + 96 q^{33} - 216 q^{41} + 180 q^{49} + 144 q^{57} - 288 q^{65} + 40 q^{73} + 36 q^{81} - 72 q^{89} - 136 q^{97}+O(q^{100}) \) 4 * q + 12 * q^9 - 72 * q^17 + 52 * q^25 + 96 * q^33 - 216 * q^41 + 180 * q^49 + 144 * q^57 - 288 * q^65 + 40 * q^73 + 36 * q^81 - 72 * q^89 - 136 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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