LMFDB - Embedded newform 192.3.b.b.31.2

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			31.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	192.31
Dual form			192.3.b.b.31.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +3.46410i q^{5} -10.0000i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} +3.46410i q^{5} -10.0000i q^{7} +3.00000 q^{9} +13.8564 q^{11} +6.92820i q^{13} -6.00000i q^{15} +30.0000 q^{17} +6.92820 q^{19} +17.3205i q^{21} -12.0000i q^{23} +13.0000 q^{25} -5.19615 q^{27} -51.9615i q^{29} -14.0000i q^{31} -24.0000 q^{33} +34.6410 q^{35} +55.4256i q^{37} -12.0000i q^{39} -6.00000 q^{41} -62.3538 q^{43} +10.3923i q^{45} +84.0000i q^{47} -51.0000 q^{49} -51.9615 q^{51} -17.3205i q^{53} +48.0000i q^{55} -12.0000 q^{57} +62.3538 q^{59} -96.9948i q^{61} -30.0000i q^{63} -24.0000 q^{65} -48.4974 q^{67} +20.7846i q^{69} +60.0000i q^{71} -86.0000 q^{73} -22.5167 q^{75} -138.564i q^{77} -38.0000i q^{79} +9.00000 q^{81} -13.8564 q^{83} +103.923i q^{85} +90.0000i q^{87} +78.0000 q^{89} +69.2820 q^{91} +24.2487i q^{93} +24.0000i q^{95} +62.0000 q^{97} +41.5692 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} + 120 q^{17} + 52 q^{25} - 96 q^{33} - 24 q^{41} - 204 q^{49} - 48 q^{57} - 96 q^{65} - 344 q^{73} + 36 q^{81} + 312 q^{89} + 248 q^{97}+O(q^{100}) $ 4 * q + 12 * q^9 + 120 * q^17 + 52 * q^25 - 96 * q^33 - 24 * q^41 - 204 * q^49 - 48 * q^57 - 96 * q^65 - 344 * q^73 + 36 * q^81 + 312 * q^89 + 248 * 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	$0.112599\pi$
$5$	3.46410i	0.692820i	−0.938083	+	0.346410i	$0.887401\pi$
$6$	0	0
$7$	− 10.0000i	− 1.42857i	−0.699854	−	0.714286i	$-0.746752\pi$
$7$	− 10.0000i	− 1.42857i	0.699854	−	0.714286i	$-0.253248\pi$
$8$	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
$13$	6.92820i	0.532939i	0.963843	+	0.266469i	$0.0858571\pi$
$13$	6.92820i	0.532939i	−0.963843	+	0.266469i	$0.914143\pi$
$14$	0	0
$15$	− 6.00000i	− 0.400000i
$16$	0	0
$17$	30.0000	1.76471	0.882353	−	0.470588i	$-0.155958\pi$
$17$	30.0000	1.76471	0.882353	+	0.470588i	$0.155958\pi$
$18$	0	0
$19$	6.92820	0.364642	0.182321	−	0.983239i	$-0.441639\pi$
$19$	6.92820	0.364642	0.182321	+	0.983239i	$0.441639\pi$
$20$	0	0
$21$	17.3205i	0.824786i
$22$	0	0
$23$	− 12.0000i	− 0.521739i	−0.965374	−	0.260870i	$-0.915991\pi$
$23$	− 12.0000i	− 0.521739i	0.965374	−	0.260870i	$-0.0840093\pi$
$24$	0	0
$25$	13.0000	0.520000
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	− 51.9615i	− 1.79178i	−0.444279	−	0.895888i	$-0.646540\pi$
$29$	− 51.9615i	− 1.79178i	0.444279	−	0.895888i	$-0.353460\pi$
$30$	0	0
$31$	− 14.0000i	− 0.451613i	−0.974172	−	0.225806i	$-0.927498\pi$
$31$	− 14.0000i	− 0.451613i	0.974172	−	0.225806i	$-0.0725017\pi$
$32$	0	0
$33$	−24.0000	−0.727273
$34$	0	0
$35$	34.6410	0.989743
$36$	0	0
$37$	55.4256i	1.49799i	0.662576	+	0.748995i	$0.269463\pi$
$37$	55.4256i	1.49799i	−0.662576	+	0.748995i	$0.730537\pi$
$38$	0	0
$39$	− 12.0000i	− 0.307692i
$40$	0	0
$41$	−6.00000	−0.146341	−0.0731707	−	0.997319i	$-0.523312\pi$
$41$	−6.00000	−0.146341	−0.0731707	+	0.997319i	$0.523312\pi$
$42$	0	0
$43$	−62.3538	−1.45009	−0.725045	−	0.688702i	$-0.758181\pi$
$43$	−62.3538	−1.45009	−0.725045	+	0.688702i	$0.758181\pi$
$44$	0	0
$45$	10.3923i	0.230940i
$46$	0	0
$47$	84.0000i	1.78723i	0.448830	+	0.893617i	$0.351841\pi$
$47$	84.0000i	1.78723i	−0.448830	+	0.893617i	$0.648159\pi$
$48$	0	0
$49$	−51.0000	−1.04082
$50$	0	0
$51$	−51.9615	−1.01885
$52$	0	0
$53$	− 17.3205i	− 0.326802i	−0.986560	−	0.163401i	$-0.947754\pi$
$53$	− 17.3205i	− 0.326802i	0.986560	−	0.163401i	$-0.0522464\pi$
$54$	0	0
$55$	48.0000i	0.872727i
$56$	0	0
$57$	−12.0000	−0.210526
$58$	0	0
$59$	62.3538	1.05684	0.528422	−	0.848982i	$-0.322784\pi$
$59$	62.3538	1.05684	0.528422	+	0.848982i	$0.322784\pi$
$60$	0	0
$61$	− 96.9948i	− 1.59008i	−0.606557	−	0.795040i	$-0.707450\pi$
$61$	− 96.9948i	− 1.59008i	0.606557	−	0.795040i	$-0.292550\pi$
$62$	0	0
$63$	− 30.0000i	− 0.476190i
$64$	0	0
$65$	−24.0000	−0.369231
$66$	0	0
$67$	−48.4974	−0.723842	−0.361921	−	0.932209i	$-0.617879\pi$
$67$	−48.4974	−0.723842	−0.361921	+	0.932209i	$0.617879\pi$
$68$	0	0
$69$	20.7846i	0.301226i
$70$	0	0
$71$	60.0000i	0.845070i	0.906347	+	0.422535i	$0.138860\pi$
$71$	60.0000i	0.845070i	−0.906347	+	0.422535i	$0.861140\pi$
$72$	0	0
$73$	−86.0000	−1.17808	−0.589041	−	0.808103i	$-0.700494\pi$
$73$	−86.0000	−1.17808	−0.589041	+	0.808103i	$0.700494\pi$
$74$	0	0
$75$	−22.5167	−0.300222
$76$	0	0
$77$	− 138.564i	− 1.79953i
$78$	0	0
$79$	− 38.0000i	− 0.481013i	−0.970648	−	0.240506i	$-0.922687\pi$
$79$	− 38.0000i	− 0.481013i	0.970648	−	0.240506i	$-0.0773135\pi$
$80$	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
$85$	103.923i	1.22262i
$86$	0	0
$87$	90.0000i	1.03448i
$88$	0	0
$89$	78.0000	0.876404	0.438202	−	0.898876i	$-0.355615\pi$
$89$	78.0000	0.876404	0.438202	+	0.898876i	$0.355615\pi$
$90$	0	0
$91$	69.2820	0.761341
$92$	0	0
$93$	24.2487i	0.260739i
$94$	0	0
$95$	24.0000i	0.252632i
$96$	0	0
$97$	62.0000	0.639175	0.319588	−	0.947557i	$-0.396456\pi$
$97$	62.0000	0.639175	0.319588	+	0.947557i	$0.396456\pi$
$98$	0	0
$99$	41.5692	0.419891
$100$	0	0
$101$	65.8179i	0.651663i	0.945428	+	0.325831i	$0.105644\pi$
$101$	65.8179i	0.651663i	−0.945428	+	0.325831i	$0.894356\pi$
$102$	0	0
$103$	− 14.0000i	− 0.135922i	−0.997688	−	0.0679612i	$-0.978351\pi$
$103$	− 14.0000i	− 0.135922i	0.997688	−	0.0679612i	$-0.0216494\pi$
$104$	0	0
$105$	−60.0000	−0.571429
$106$	0	0
$107$	−76.2102	−0.712245	−0.356123	−	0.934439i	$-0.615901\pi$
$107$	−76.2102	−0.712245	−0.356123	+	0.934439i	$0.615901\pi$
$108$	0	0
$109$	20.7846i	0.190684i	0.995445	+	0.0953422i	$0.0303945\pi$
$109$	20.7846i	0.190684i	−0.995445	+	0.0953422i	$0.969605\pi$
$110$	0	0
$111$	− 96.0000i	− 0.864865i
$112$	0	0
$113$	−6.00000	−0.0530973	−0.0265487	−	0.999648i	$-0.508452\pi$
$113$	−6.00000	−0.0530973	−0.0265487	+	0.999648i	$0.508452\pi$
$114$	0	0
$115$	41.5692	0.361471
$116$	0	0
$117$	20.7846i	0.177646i
$118$	0	0
$119$	− 300.000i	− 2.52101i
$120$	0	0
$121$	71.0000	0.586777
$122$	0	0
$123$	10.3923	0.0844903
$124$	0	0
$125$	131.636i	1.05309i
$126$	0	0
$127$	− 82.0000i	− 0.645669i	−0.946455	−	0.322835i	$-0.895364\pi$
$127$	− 82.0000i	− 0.645669i	0.946455	−	0.322835i	$-0.104636\pi$
$128$	0	0
$129$	108.000	0.837209
$130$	0	0
$131$	76.2102	0.581758	0.290879	−	0.956760i	$-0.406052\pi$
$131$	76.2102	0.581758	0.290879	+	0.956760i	$0.406052\pi$
$132$	0	0
$133$	− 69.2820i	− 0.520918i
$134$	0	0
$135$	− 18.0000i	− 0.133333i
$136$	0	0
$137$	42.0000	0.306569	0.153285	−	0.988182i	$-0.451015\pi$
$137$	42.0000	0.306569	0.153285	+	0.988182i	$0.451015\pi$
$138$	0	0
$139$	−48.4974	−0.348902	−0.174451	−	0.984666i	$-0.555815\pi$
$139$	−48.4974	−0.348902	−0.174451	+	0.984666i	$0.555815\pi$
$140$	0	0
$141$	− 145.492i	− 1.03186i
$142$	0	0
$143$	96.0000i	0.671329i
$144$	0	0
$145$	180.000	1.24138
$146$	0	0
$147$	88.3346	0.600916
$148$	0	0
$149$	183.597i	1.23220i	0.787669	+	0.616099i	$0.211288\pi$
$149$	183.597i	1.23220i	−0.787669	+	0.616099i	$0.788712\pi$
$150$	0	0
$151$	134.000i	0.887417i	0.896171	+	0.443709i	$0.146337\pi$
$151$	134.000i	0.887417i	−0.896171	+	0.443709i	$0.853663\pi$
$152$	0	0
$153$	90.0000	0.588235
$154$	0	0
$155$	48.4974	0.312887
$156$	0	0
$157$	83.1384i	0.529544i	0.964311	+	0.264772i	$0.0852967\pi$
$157$	83.1384i	0.529544i	−0.964311	+	0.264772i	$0.914703\pi$
$158$	0	0
$159$	30.0000i	0.188679i
$160$	0	0
$161$	−120.000	−0.745342
$162$	0	0
$163$	76.2102	0.467547	0.233774	−	0.972291i	$-0.424893\pi$
$163$	76.2102	0.467547	0.233774	+	0.972291i	$0.424893\pi$
$164$	0	0
$165$	− 83.1384i	− 0.503869i
$166$	0	0
$167$	− 216.000i	− 1.29341i	−0.762739	−	0.646707i	$-0.776146\pi$
$167$	− 216.000i	− 1.29341i	0.762739	−	0.646707i	$-0.223854\pi$
$168$	0	0
$169$	121.000	0.715976
$170$	0	0
$171$	20.7846	0.121547
$172$	0	0
$173$	162.813i	0.941114i	0.882370	+	0.470557i	$0.155947\pi$
$173$	162.813i	0.941114i	−0.882370	+	0.470557i	$0.844053\pi$
$174$	0	0
$175$	− 130.000i	− 0.742857i
$176$	0	0
$177$	−108.000	−0.610169
$178$	0	0
$179$	−242.487	−1.35468	−0.677338	−	0.735672i	$-0.736867\pi$
$179$	−242.487	−1.35468	−0.677338	+	0.735672i	$0.736867\pi$
$180$	0	0
$181$	− 297.913i	− 1.64593i	−0.568094	−	0.822963i	$-0.692319\pi$
$181$	− 297.913i	− 1.64593i	0.568094	−	0.822963i	$-0.307681\pi$
$182$	0	0
$183$	168.000i	0.918033i
$184$	0	0
$185$	−192.000	−1.03784
$186$	0	0
$187$	415.692	2.22295
$188$	0	0
$189$	51.9615i	0.274929i
$190$	0	0
$191$	216.000i	1.13089i	0.824786	+	0.565445i	$0.191296\pi$
$191$	216.000i	1.13089i	−0.824786	+	0.565445i	$0.808704\pi$
$192$	0	0
$193$	−286.000	−1.48187	−0.740933	−	0.671579i	$-0.765616\pi$
$193$	−286.000	−1.48187	−0.740933	+	0.671579i	$0.765616\pi$
$194$	0	0
$195$	41.5692	0.213175
$196$	0	0
$197$	− 169.741i	− 0.861629i	−0.902440	−	0.430815i	$-0.858226\pi$
$197$	− 169.741i	− 0.861629i	0.902440	−	0.430815i	$-0.141774\pi$
$198$	0	0
$199$	130.000i	0.653266i	0.945151	+	0.326633i	$0.105914\pi$
$199$	130.000i	0.653266i	−0.945151	+	0.326633i	$0.894086\pi$
$200$	0	0
$201$	84.0000	0.417910
$202$	0	0
$203$	−519.615	−2.55968
$204$	0	0
$205$	− 20.7846i	− 0.101388i
$206$	0	0
$207$	− 36.0000i	− 0.173913i
$208$	0	0
$209$	96.0000	0.459330
$210$	0	0
$211$	−325.626	−1.54325	−0.771625	−	0.636078i	$-0.780556\pi$
$211$	−325.626	−1.54325	−0.771625	+	0.636078i	$0.780556\pi$
$212$	0	0
$213$	− 103.923i	− 0.487902i
$214$	0	0
$215$	− 216.000i	− 1.00465i
$216$	0	0
$217$	−140.000	−0.645161
$218$	0	0
$219$	148.956	0.680166
$220$	0	0
$221$	207.846i	0.940480i
$222$	0	0
$223$	302.000i	1.35426i	0.735863	+	0.677130i	$0.236777\pi$
$223$	302.000i	1.35426i	−0.735863	+	0.677130i	$0.763223\pi$
$224$	0	0
$225$	39.0000	0.173333
$226$	0	0
$227$	−166.277	−0.732497	−0.366249	−	0.930517i	$-0.619358\pi$
$227$	−166.277	−0.732497	−0.366249	+	0.930517i	$0.619358\pi$
$228$	0	0
$229$	20.7846i	0.0907625i	0.998970	+	0.0453812i	$0.0144503\pi$
$229$	20.7846i	0.0907625i	−0.998970	+	0.0453812i	$0.985550\pi$
$230$	0	0
$231$	240.000i	1.03896i
$232$	0	0
$233$	−282.000	−1.21030	−0.605150	−	0.796111i	$-0.706887\pi$
$233$	−282.000	−1.21030	−0.605150	+	0.796111i	$0.706887\pi$
$234$	0	0
$235$	−290.985	−1.23823
$236$	0	0
$237$	65.8179i	0.277713i
$238$	0	0
$239$	192.000i	0.803347i	0.915783	+	0.401674i	$0.131571\pi$
$239$	192.000i	0.803347i	−0.915783	+	0.401674i	$0.868429\pi$
$240$	0	0
$241$	−182.000	−0.755187	−0.377593	−	0.925972i	$-0.623248\pi$
$241$	−182.000	−0.755187	−0.377593	+	0.925972i	$0.623248\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	− 176.669i	− 0.721099i
$246$	0	0
$247$	48.0000i	0.194332i
$248$	0	0
$249$	24.0000	0.0963855
$250$	0	0
$251$	290.985	1.15930	0.579650	−	0.814865i	$-0.303189\pi$
$251$	290.985	1.15930	0.579650	+	0.814865i	$0.303189\pi$
$252$	0	0
$253$	− 166.277i	− 0.657221i
$254$	0	0
$255$	− 180.000i	− 0.705882i
$256$	0	0
$257$	−78.0000	−0.303502	−0.151751	−	0.988419i	$-0.548491\pi$
$257$	−78.0000	−0.303502	−0.151751	+	0.988419i	$0.548491\pi$
$258$	0	0
$259$	554.256	2.13999
$260$	0	0
$261$	− 155.885i	− 0.597259i
$262$	0	0
$263$	48.0000i	0.182510i	0.995828	+	0.0912548i	$0.0290878\pi$
$263$	48.0000i	0.182510i	−0.995828	+	0.0912548i	$0.970912\pi$
$264$	0	0
$265$	60.0000	0.226415
$266$	0	0
$267$	−135.100	−0.505992
$268$	0	0
$269$	280.592i	1.04309i	0.853223	+	0.521547i	$0.174645\pi$
$269$	280.592i	1.04309i	−0.853223	+	0.521547i	$0.825355\pi$
$270$	0	0
$271$	− 130.000i	− 0.479705i	−0.970809	−	0.239852i	$-0.922901\pi$
$271$	− 130.000i	− 0.479705i	0.970809	−	0.239852i	$-0.0770990\pi$
$272$	0	0
$273$	−120.000	−0.439560
$274$	0	0
$275$	180.133	0.655030
$276$	0	0
$277$	− 270.200i	− 0.975451i	−0.872997	−	0.487725i	$-0.837827\pi$
$277$	− 270.200i	− 0.975451i	0.872997	−	0.487725i	$-0.162173\pi$
$278$	0	0
$279$	− 42.0000i	− 0.150538i
$280$	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
$283$	−311.769	−1.10166	−0.550829	−	0.834618i	$-0.685688\pi$
$283$	−311.769	−1.10166	−0.550829	+	0.834618i	$0.685688\pi$
$284$	0	0
$285$	− 41.5692i	− 0.145857i
$286$	0	0
$287$	60.0000i	0.209059i
$288$	0	0
$289$	611.000	2.11419
$290$	0	0
$291$	−107.387	−0.369028
$292$	0	0
$293$	− 155.885i	− 0.532029i	−0.963969	−	0.266015i	$-0.914293\pi$
$293$	− 155.885i	− 0.532029i	0.963969	−	0.266015i	$-0.0857069\pi$
$294$	0	0
$295$	216.000i	0.732203i
$296$	0	0
$297$	−72.0000	−0.242424
$298$	0	0
$299$	83.1384	0.278055
$300$	0	0
$301$	623.538i	2.07156i
$302$	0	0
$303$	− 114.000i	− 0.376238i
$304$	0	0
$305$	336.000	1.10164
$306$	0	0
$307$	−103.923	−0.338512	−0.169256	−	0.985572i	$-0.554136\pi$
$307$	−103.923	−0.338512	−0.169256	+	0.985572i	$0.554136\pi$
$308$	0	0
$309$	24.2487i	0.0784748i
$310$	0	0
$311$	24.0000i	0.0771704i	0.999255	+	0.0385852i	$0.0122851\pi$
$311$	24.0000i	0.0771704i	−0.999255	+	0.0385852i	$0.987715\pi$
$312$	0	0
$313$	190.000	0.607029	0.303514	−	0.952827i	$-0.401840\pi$
$313$	190.000	0.607029	0.303514	+	0.952827i	$0.401840\pi$
$314$	0	0
$315$	103.923	0.329914
$316$	0	0
$317$	− 107.387i	− 0.338761i	−0.985551	−	0.169380i	$-0.945823\pi$
$317$	− 107.387i	− 0.338761i	0.985551	−	0.169380i	$-0.0541766\pi$
$318$	0	0
$319$	− 720.000i	− 2.25705i
$320$	0	0
$321$	132.000	0.411215
$322$	0	0
$323$	207.846	0.643486
$324$	0	0
$325$	90.0666i	0.277128i
$326$	0	0
$327$	− 36.0000i	− 0.110092i
$328$	0	0
$329$	840.000	2.55319
$330$	0	0
$331$	34.6410	0.104656	0.0523278	−	0.998630i	$-0.483336\pi$
$331$	34.6410	0.104656	0.0523278	+	0.998630i	$0.483336\pi$
$332$	0	0
$333$	166.277i	0.499330i
$334$	0	0
$335$	− 168.000i	− 0.501493i
$336$	0	0
$337$	−490.000	−1.45401	−0.727003	−	0.686634i	$-0.759087\pi$
$337$	−490.000	−1.45401	−0.727003	+	0.686634i	$0.759087\pi$
$338$	0	0
$339$	10.3923	0.0306558
$340$	0	0
$341$	− 193.990i	− 0.568885i
$342$	0	0
$343$	20.0000i	0.0583090i
$344$	0	0
$345$	−72.0000	−0.208696
$346$	0	0
$347$	443.405	1.27782	0.638912	−	0.769280i	$-0.279385\pi$
$347$	443.405	1.27782	0.638912	+	0.769280i	$0.279385\pi$
$348$	0	0
$349$	96.9948i	0.277922i	0.990298	+	0.138961i	$0.0443763\pi$
$349$	96.9948i	0.277922i	−0.990298	+	0.138961i	$0.955624\pi$
$350$	0	0
$351$	− 36.0000i	− 0.102564i
$352$	0	0
$353$	−246.000	−0.696884	−0.348442	−	0.937330i	$-0.613289\pi$
$353$	−246.000	−0.696884	−0.348442	+	0.937330i	$0.613289\pi$
$354$	0	0
$355$	−207.846	−0.585482
$356$	0	0
$357$	519.615i	1.45550i
$358$	0	0
$359$	− 708.000i	− 1.97214i	−0.166316	−	0.986072i	$-0.553187\pi$
$359$	− 708.000i	− 1.97214i	0.166316	−	0.986072i	$-0.446813\pi$
$360$	0	0
$361$	−313.000	−0.867036
$362$	0	0
$363$	−122.976	−0.338776
$364$	0	0
$365$	− 297.913i	− 0.816199i
$366$	0	0
$367$	82.0000i	0.223433i	0.993740	+	0.111717i	$0.0356349\pi$
$367$	82.0000i	0.223433i	−0.993740	+	0.111717i	$0.964365\pi$
$368$	0	0
$369$	−18.0000	−0.0487805
$370$	0	0
$371$	−173.205	−0.466860
$372$	0	0
$373$	512.687i	1.37450i	0.726423	+	0.687248i	$0.241181\pi$
$373$	512.687i	1.37450i	−0.726423	+	0.687248i	$0.758819\pi$
$374$	0	0
$375$	− 228.000i	− 0.608000i
$376$	0	0
$377$	360.000	0.954907
$378$	0	0
$379$	533.472	1.40758	0.703788	−	0.710410i	$-0.251490\pi$
$379$	533.472	1.40758	0.703788	+	0.710410i	$0.251490\pi$
$380$	0	0
$381$	142.028i	0.372777i
$382$	0	0
$383$	408.000i	1.06527i	0.846344	+	0.532637i	$0.178799\pi$
$383$	408.000i	1.06527i	−0.846344	+	0.532637i	$0.821201\pi$
$384$	0	0
$385$	480.000	1.24675
$386$	0	0
$387$	−187.061	−0.483363
$388$	0	0
$389$	142.028i	0.365111i	0.983196	+	0.182555i	$0.0584369\pi$
$389$	142.028i	0.365111i	−0.983196	+	0.182555i	$0.941563\pi$
$390$	0	0
$391$	− 360.000i	− 0.920716i
$392$	0	0
$393$	−132.000	−0.335878
$394$	0	0
$395$	131.636	0.333255
$396$	0	0
$397$	− 41.5692i	− 0.104708i	−0.998629	−	0.0523542i	$-0.983328\pi$
$397$	− 41.5692i	− 0.104708i	0.998629	−	0.0523542i	$-0.0166725\pi$
$398$	0	0
$399$	120.000i	0.300752i
$400$	0	0
$401$	270.000	0.673317	0.336658	−	0.941627i	$-0.390703\pi$
$401$	270.000	0.673317	0.336658	+	0.941627i	$0.390703\pi$
$402$	0	0
$403$	96.9948	0.240682
$404$	0	0
$405$	31.1769i	0.0769800i
$406$	0	0
$407$	768.000i	1.88698i
$408$	0	0
$409$	−110.000	−0.268949	−0.134474	−	0.990917i	$-0.542935\pi$
$409$	−110.000	−0.268949	−0.134474	+	0.990917i	$0.542935\pi$
$410$	0	0
$411$	−72.7461	−0.176998
$412$	0	0
$413$	− 623.538i	− 1.50978i
$414$	0	0
$415$	− 48.0000i	− 0.115663i
$416$	0	0
$417$	84.0000	0.201439
$418$	0	0
$419$	415.692	0.992105	0.496053	−	0.868292i	$-0.334782\pi$
$419$	415.692	0.992105	0.496053	+	0.868292i	$0.334782\pi$
$420$	0	0
$421$	− 575.041i	− 1.36589i	−0.730468	−	0.682946i	$-0.760698\pi$
$421$	− 575.041i	− 1.36589i	0.730468	−	0.682946i	$-0.239302\pi$
$422$	0	0
$423$	252.000i	0.595745i
$424$	0	0
$425$	390.000	0.917647
$426$	0	0
$427$	−969.948	−2.27154
$428$	0	0
$429$	− 166.277i	− 0.387592i
$430$	0	0
$431$	396.000i	0.918794i	0.888231	+	0.459397i	$0.151934\pi$
$431$	396.000i	0.918794i	−0.888231	+	0.459397i	$0.848066\pi$
$432$	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
$435$	−311.769	−0.716711
$436$	0	0
$437$	− 83.1384i	− 0.190248i
$438$	0	0
$439$	− 470.000i	− 1.07062i	−0.844657	−	0.535308i	$-0.820196\pi$
$439$	− 470.000i	− 1.07062i	0.844657	−	0.535308i	$-0.179804\pi$
$440$	0	0
$441$	−153.000	−0.346939
$442$	0	0
$443$	332.554	0.750686	0.375343	−	0.926886i	$-0.377525\pi$
$443$	332.554	0.750686	0.375343	+	0.926886i	$0.377525\pi$
$444$	0	0
$445$	270.200i	0.607191i
$446$	0	0
$447$	− 318.000i	− 0.711409i
$448$	0	0
$449$	−786.000	−1.75056	−0.875278	−	0.483619i	$-0.839322\pi$
$449$	−786.000	−1.75056	−0.875278	+	0.483619i	$0.839322\pi$
$450$	0	0
$451$	−83.1384	−0.184342
$452$	0	0
$453$	− 232.095i	− 0.512351i
$454$	0	0
$455$	240.000i	0.527473i
$456$	0	0
$457$	34.0000	0.0743982	0.0371991	−	0.999308i	$-0.488156\pi$
$457$	34.0000	0.0743982	0.0371991	+	0.999308i	$0.488156\pi$
$458$	0	0
$459$	−155.885	−0.339618
$460$	0	0
$461$	− 349.874i	− 0.758946i	−0.925203	−	0.379473i	$-0.876105\pi$
$461$	− 349.874i	− 0.758946i	0.925203	−	0.379473i	$-0.123895\pi$
$462$	0	0
$463$	614.000i	1.32613i	0.748560	+	0.663067i	$0.230746\pi$
$463$	614.000i	1.32613i	−0.748560	+	0.663067i	$0.769254\pi$
$464$	0	0
$465$	−84.0000	−0.180645
$466$	0	0
$467$	443.405	0.949475	0.474738	−	0.880127i	$-0.342543\pi$
$467$	443.405	0.949475	0.474738	+	0.880127i	$0.342543\pi$
$468$	0	0
$469$	484.974i	1.03406i
$470$	0	0
$471$	− 144.000i	− 0.305732i
$472$	0	0
$473$	−864.000	−1.82664
$474$	0	0
$475$	90.0666	0.189614
$476$	0	0
$477$	− 51.9615i	− 0.108934i
$478$	0	0
$479$	− 300.000i	− 0.626305i	−0.949703	−	0.313152i	$-0.898615\pi$
$479$	− 300.000i	− 0.626305i	0.949703	−	0.313152i	$-0.101385\pi$
$480$	0	0
$481$	−384.000	−0.798337
$482$	0	0
$483$	207.846	0.430323
$484$	0	0
$485$	214.774i	0.442834i
$486$	0	0
$487$	638.000i	1.31006i	0.755602	+	0.655031i	$0.227344\pi$
$487$	638.000i	1.31006i	−0.755602	+	0.655031i	$0.772656\pi$
$488$	0	0
$489$	−132.000	−0.269939
$490$	0	0
$491$	−672.036	−1.36871	−0.684354	−	0.729150i	$-0.739916\pi$
$491$	−672.036	−1.36871	−0.684354	+	0.729150i	$0.739916\pi$
$492$	0	0
$493$	− 1558.85i	− 3.16196i
$494$	0	0
$495$	144.000i	0.290909i
$496$	0	0
$497$	600.000	1.20724
$498$	0	0
$499$	−561.184	−1.12462	−0.562309	−	0.826927i	$-0.690087\pi$
$499$	−561.184	−1.12462	−0.562309	+	0.826927i	$0.690087\pi$
$500$	0	0
$501$	374.123i	0.746752i
$502$	0	0
$503$	588.000i	1.16899i	0.811399	+	0.584493i	$0.198707\pi$
$503$	588.000i	1.16899i	−0.811399	+	0.584493i	$0.801293\pi$
$504$	0	0
$505$	−228.000	−0.451485
$506$	0	0
$507$	−209.578	−0.413369
$508$	0	0
$509$	169.741i	0.333479i	0.986001	+	0.166740i	$0.0533239\pi$
$509$	169.741i	0.333479i	−0.986001	+	0.166740i	$0.946676\pi$
$510$	0	0
$511$	860.000i	1.68297i
$512$	0	0
$513$	−36.0000	−0.0701754
$514$	0	0
$515$	48.4974	0.0941698
$516$	0	0
$517$	1163.94i	2.25133i
$518$	0	0
$519$	− 282.000i	− 0.543353i
$520$	0	0
$521$	−366.000	−0.702495	−0.351248	−	0.936283i	$-0.614242\pi$
$521$	−366.000	−0.702495	−0.351248	+	0.936283i	$0.614242\pi$
$522$	0	0
$523$	367.195	0.702093	0.351047	−	0.936358i	$-0.385826\pi$
$523$	367.195	0.702093	0.351047	+	0.936358i	$0.385826\pi$
$524$	0	0
$525$	225.167i	0.428889i
$526$	0	0
$527$	− 420.000i	− 0.796964i
$528$	0	0
$529$	385.000	0.727788
$530$	0	0
$531$	187.061	0.352282
$532$	0	0
$533$	− 41.5692i	− 0.0779910i
$534$	0	0
$535$	− 264.000i	− 0.493458i
$536$	0	0
$537$	420.000	0.782123
$538$	0	0
$539$	−706.677	−1.31109
$540$	0	0
$541$	48.4974i	0.0896440i	0.998995	+	0.0448220i	$0.0142721\pi$
$541$	48.4974i	0.0896440i	−0.998995	+	0.0448220i	$0.985728\pi$
$542$	0	0
$543$	516.000i	0.950276i
$544$	0	0
$545$	−72.0000	−0.132110
$546$	0	0
$547$	−422.620	−0.772615	−0.386307	−	0.922370i	$-0.626250\pi$
$547$	−422.620	−0.772615	−0.386307	+	0.922370i	$0.626250\pi$
$548$	0	0
$549$	− 290.985i	− 0.530026i
$550$	0	0
$551$	− 360.000i	− 0.653358i
$552$	0	0
$553$	−380.000	−0.687161
$554$	0	0
$555$	332.554	0.599196
$556$	0	0
$557$	− 557.720i	− 1.00129i	−0.865652	−	0.500647i	$-0.833096\pi$
$557$	− 557.720i	− 1.00129i	0.865652	−	0.500647i	$-0.166904\pi$
$558$	0	0
$559$	− 432.000i	− 0.772809i
$560$	0	0
$561$	−720.000	−1.28342
$562$	0	0
$563$	−775.959	−1.37826	−0.689129	−	0.724639i	$-0.742006\pi$
$563$	−775.959	−1.37826	−0.689129	+	0.724639i	$0.742006\pi$
$564$	0	0
$565$	− 20.7846i	− 0.0367869i
$566$	0	0
$567$	− 90.0000i	− 0.158730i
$568$	0	0
$569$	−54.0000	−0.0949033	−0.0474517	−	0.998874i	$-0.515110\pi$
$569$	−54.0000	−0.0949033	−0.0474517	+	0.998874i	$0.515110\pi$
$570$	0	0
$571$	214.774	0.376137	0.188069	−	0.982156i	$-0.439777\pi$
$571$	214.774	0.376137	0.188069	+	0.982156i	$0.439777\pi$
$572$	0	0
$573$	− 374.123i	− 0.652920i
$574$	0	0
$575$	− 156.000i	− 0.271304i
$576$	0	0
$577$	46.0000	0.0797227	0.0398614	−	0.999205i	$-0.487308\pi$
$577$	46.0000	0.0797227	0.0398614	+	0.999205i	$0.487308\pi$
$578$	0	0
$579$	495.367	0.855555
$580$	0	0
$581$	138.564i	0.238492i
$582$	0	0
$583$	− 240.000i	− 0.411664i
$584$	0	0
$585$	−72.0000	−0.123077
$586$	0	0
$587$	630.466	1.07405	0.537024	−	0.843567i	$-0.319548\pi$
$587$	630.466	1.07405	0.537024	+	0.843567i	$0.319548\pi$
$588$	0	0
$589$	− 96.9948i	− 0.164677i
$590$	0	0
$591$	294.000i	0.497462i
$592$	0	0
$593$	858.000	1.44688	0.723440	−	0.690387i	$-0.242560\pi$
$593$	858.000	1.44688	0.723440	+	0.690387i	$0.242560\pi$
$594$	0	0
$595$	1039.23	1.74661
$596$	0	0
$597$	− 225.167i	− 0.377163i
$598$	0	0
$599$	− 228.000i	− 0.380634i	−0.981723	−	0.190317i	$-0.939048\pi$
$599$	− 228.000i	− 0.380634i	0.981723	−	0.190317i	$-0.0609516\pi$
$600$	0	0
$601$	1010.00	1.68053	0.840266	−	0.542174i	$-0.182399\pi$
$601$	1010.00	1.68053	0.840266	+	0.542174i	$0.182399\pi$
$602$	0	0
$603$	−145.492	−0.241281
$604$	0	0
$605$	245.951i	0.406531i
$606$	0	0
$607$	− 850.000i	− 1.40033i	−0.713981	−	0.700165i	$-0.753110\pi$
$607$	− 850.000i	− 1.40033i	0.713981	−	0.700165i	$-0.246890\pi$
$608$	0	0
$609$	900.000	1.47783
$610$	0	0
$611$	−581.969	−0.952486
$612$	0	0
$613$	− 429.549i	− 0.700732i	−0.936613	−	0.350366i	$-0.886057\pi$
$613$	− 429.549i	− 0.700732i	0.936613	−	0.350366i	$-0.113943\pi$
$614$	0	0
$615$	36.0000i	0.0585366i
$616$	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
$619$	1129.30	1.82439	0.912195	−	0.409757i	$-0.134386\pi$
$619$	1129.30	1.82439	0.912195	+	0.409757i	$0.134386\pi$
$620$	0	0
$621$	62.3538i	0.100409i
$622$	0	0
$623$	− 780.000i	− 1.25201i
$624$	0	0
$625$	−131.000	−0.209600
$626$	0	0
$627$	−166.277	−0.265194
$628$	0	0
$629$	1662.77i	2.64351i
$630$	0	0
$631$	446.000i	0.706815i	0.935470	+	0.353407i	$0.114977\pi$
$631$	446.000i	0.706815i	−0.935470	+	0.353407i	$0.885023\pi$
$632$	0	0
$633$	564.000	0.890995
$634$	0	0
$635$	284.056	0.447333
$636$	0	0
$637$	− 353.338i	− 0.554691i
$638$	0	0
$639$	180.000i	0.281690i
$640$	0	0
$641$	−522.000	−0.814353	−0.407176	−	0.913350i	$-0.633487\pi$
$641$	−522.000	−0.814353	−0.407176	+	0.913350i	$0.633487\pi$
$642$	0	0
$643$	−145.492	−0.226271	−0.113136	−	0.993580i	$-0.536089\pi$
$643$	−145.492	−0.226271	−0.113136	+	0.993580i	$0.536089\pi$
$644$	0	0
$645$	374.123i	0.580036i
$646$	0	0
$647$	996.000i	1.53941i	0.638398	+	0.769706i	$0.279597\pi$
$647$	996.000i	1.53941i	−0.638398	+	0.769706i	$0.720403\pi$
$648$	0	0
$649$	864.000	1.33128
$650$	0	0
$651$	242.487	0.372484
$652$	0	0
$653$	994.197i	1.52251i	0.648454	+	0.761254i	$0.275416\pi$
$653$	994.197i	1.52251i	−0.648454	+	0.761254i	$0.724584\pi$
$654$	0	0
$655$	264.000i	0.403053i
$656$	0	0
$657$	−258.000	−0.392694
$658$	0	0
$659$	48.4974	0.0735924	0.0367962	−	0.999323i	$-0.488285\pi$
$659$	48.4974	0.0735924	0.0367962	+	0.999323i	$0.488285\pi$
$660$	0	0
$661$	− 138.564i	− 0.209628i	−0.994492	−	0.104814i	$-0.966575\pi$
$661$	− 138.564i	− 0.209628i	0.994492	−	0.104814i	$-0.0334247\pi$
$662$	0	0
$663$	− 360.000i	− 0.542986i
$664$	0	0
$665$	240.000	0.360902
$666$	0	0
$667$	−623.538	−0.934840
$668$	0	0
$669$	− 523.079i	− 0.781882i
$670$	0	0
$671$	− 1344.00i	− 2.00298i
$672$	0	0
$673$	46.0000	0.0683507	0.0341753	−	0.999416i	$-0.489120\pi$
$673$	46.0000	0.0683507	0.0341753	+	0.999416i	$0.489120\pi$
$674$	0	0
$675$	−67.5500	−0.100074
$676$	0	0
$677$	696.284i	1.02849i	0.857645	+	0.514243i	$0.171927\pi$
$677$	696.284i	1.02849i	−0.857645	+	0.514243i	$0.828073\pi$
$678$	0	0
$679$	− 620.000i	− 0.913108i
$680$	0	0
$681$	288.000	0.422907
$682$	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
$685$	145.492i	0.212397i
$686$	0	0
$687$	− 36.0000i	− 0.0524017i
$688$	0	0
$689$	120.000	0.174165
$690$	0	0
$691$	602.754	0.872292	0.436146	−	0.899876i	$-0.356343\pi$
$691$	602.754	0.872292	0.436146	+	0.899876i	$0.356343\pi$
$692$	0	0
$693$	− 415.692i	− 0.599844i
$694$	0	0
$695$	− 168.000i	− 0.241727i
$696$	0	0
$697$	−180.000	−0.258250
$698$	0	0
$699$	488.438	0.698767
$700$	0	0
$701$	128.172i	0.182841i	0.995812	+	0.0914207i	$0.0291408\pi$
$701$	128.172i	0.182841i	−0.995812	+	0.0914207i	$0.970859\pi$
$702$	0	0
$703$	384.000i	0.546230i
$704$	0	0
$705$	504.000	0.714894
$706$	0	0
$707$	658.179	0.930947
$708$	0	0
$709$	− 533.472i	− 0.752428i	−0.926533	−	0.376214i	$-0.877226\pi$
$709$	− 533.472i	− 0.752428i	0.926533	−	0.376214i	$-0.122774\pi$
$710$	0	0
$711$	− 114.000i	− 0.160338i
$712$	0	0
$713$	−168.000	−0.235624
$714$	0	0
$715$	−332.554	−0.465110
$716$	0	0
$717$	− 332.554i	− 0.463813i
$718$	0	0
$719$	− 1332.00i	− 1.85257i	−0.376820	−	0.926287i	$-0.622982\pi$
$719$	− 1332.00i	− 1.85257i	0.376820	−	0.926287i	$-0.377018\pi$
$720$	0	0
$721$	−140.000	−0.194175
$722$	0	0
$723$	315.233	0.436007
$724$	0	0
$725$	− 675.500i	− 0.931724i
$726$	0	0
$727$	− 754.000i	− 1.03714i	−0.855036	−	0.518569i	$-0.826465\pi$
$727$	− 754.000i	− 1.03714i	0.855036	−	0.518569i	$-0.173535\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	−1870.61	−2.55898
$732$	0	0
$733$	727.461i	0.992444i	0.868196	+	0.496222i	$0.165280\pi$
$733$	727.461i	0.992444i	−0.868196	+	0.496222i	$0.834720\pi$
$734$	0	0
$735$	306.000i	0.416327i
$736$	0	0
$737$	−672.000	−0.911805
$738$	0	0
$739$	−145.492	−0.196877	−0.0984386	−	0.995143i	$-0.531385\pi$
$739$	−145.492	−0.196877	−0.0984386	+	0.995143i	$0.531385\pi$
$740$	0	0
$741$	− 83.1384i	− 0.112198i
$742$	0	0
$743$	816.000i	1.09825i	0.835740	+	0.549125i	$0.185039\pi$
$743$	816.000i	1.09825i	−0.835740	+	0.549125i	$0.814961\pi$
$744$	0	0
$745$	−636.000	−0.853691
$746$	0	0
$747$	−41.5692	−0.0556482
$748$	0	0
$749$	762.102i	1.01749i
$750$	0	0
$751$	466.000i	0.620506i	0.950654	+	0.310253i	$0.100414\pi$
$751$	466.000i	0.620506i	−0.950654	+	0.310253i	$0.899586\pi$
$752$	0	0
$753$	−504.000	−0.669323
$754$	0	0
$755$	−464.190	−0.614821
$756$	0	0
$757$	− 450.333i	− 0.594892i	−0.954739	−	0.297446i	$-0.903865\pi$
$757$	− 450.333i	− 0.594892i	0.954739	−	0.297446i	$-0.0961348\pi$
$758$	0	0
$759$	288.000i	0.379447i
$760$	0	0
$761$	18.0000	0.0236531	0.0118265	−	0.999930i	$-0.496235\pi$
$761$	18.0000	0.0236531	0.0118265	+	0.999930i	$0.496235\pi$
$762$	0	0
$763$	207.846	0.272406
$764$	0	0
$765$	311.769i	0.407541i
$766$	0	0
$767$	432.000i	0.563233i
$768$	0	0
$769$	1298.00	1.68791	0.843953	−	0.536417i	$-0.180222\pi$
$769$	1298.00	1.68791	0.843953	+	0.536417i	$0.180222\pi$
$770$	0	0
$771$	135.100	0.175227
$772$	0	0
$773$	− 197.454i	− 0.255438i	−0.991810	−	0.127719i	$-0.959234\pi$
$773$	− 197.454i	− 0.255438i	0.991810	−	0.127719i	$-0.0407656\pi$
$774$	0	0
$775$	− 182.000i	− 0.234839i
$776$	0	0
$777$	−960.000	−1.23552
$778$	0	0
$779$	−41.5692	−0.0533623
$780$	0	0
$781$	831.384i	1.06451i
$782$	0	0
$783$	270.000i	0.344828i
$784$	0	0
$785$	−288.000	−0.366879
$786$	0	0
$787$	256.344	0.325722	0.162861	−	0.986649i	$-0.447928\pi$
$787$	256.344	0.325722	0.162861	+	0.986649i	$0.447928\pi$
$788$	0	0
$789$	− 83.1384i	− 0.105372i
$790$	0	0
$791$	60.0000i	0.0758534i
$792$	0	0
$793$	672.000	0.847415
$794$	0	0
$795$	−103.923	−0.130721
$796$	0	0
$797$	− 959.556i	− 1.20396i	−0.798511	−	0.601980i	$-0.794379\pi$
$797$	− 959.556i	− 1.20396i	0.798511	−	0.601980i	$-0.205621\pi$
$798$	0	0
$799$	2520.00i	3.15394i
$800$	0	0
$801$	234.000	0.292135
$802$	0	0
$803$	−1191.65	−1.48400
$804$	0	0
$805$	− 415.692i	− 0.516388i
$806$	0	0
$807$	− 486.000i	− 0.602230i
$808$	0	0
$809$	378.000	0.467244	0.233622	−	0.972328i	$-0.424942\pi$
$809$	378.000	0.467244	0.233622	+	0.972328i	$0.424942\pi$
$810$	0	0
$811$	228.631	0.281912	0.140956	−	0.990016i	$-0.454982\pi$
$811$	228.631	0.281912	0.140956	+	0.990016i	$0.454982\pi$
$812$	0	0
$813$	225.167i	0.276958i
$814$	0	0
$815$	264.000i	0.323926i
$816$	0	0
$817$	−432.000	−0.528764
$818$	0	0
$819$	207.846	0.253780
$820$	0	0
$821$	− 1236.68i	− 1.50631i	−0.657840	−	0.753157i	$-0.728530\pi$
$821$	− 1236.68i	− 1.50631i	0.657840	−	0.753157i	$-0.271470\pi$
$822$	0	0
$823$	758.000i	0.921021i	0.887654	+	0.460510i	$0.152334\pi$
$823$	758.000i	0.921021i	−0.887654	+	0.460510i	$0.847666\pi$
$824$	0	0
$825$	−312.000	−0.378182
$826$	0	0
$827$	644.323	0.779109	0.389554	−	0.921003i	$-0.372629\pi$
$827$	644.323	0.779109	0.389554	+	0.921003i	$0.372629\pi$
$828$	0	0
$829$	− 921.451i	− 1.11152i	−0.831342	−	0.555761i	$-0.812427\pi$
$829$	− 921.451i	− 1.11152i	0.831342	−	0.555761i	$-0.187573\pi$
$830$	0	0
$831$	468.000i	0.563177i
$832$	0	0
$833$	−1530.00	−1.83673
$834$	0	0
$835$	748.246	0.896103
$836$	0	0
$837$	72.7461i	0.0869129i
$838$	0	0
$839$	612.000i	0.729440i	0.931117	+	0.364720i	$0.118835\pi$
$839$	612.000i	0.729440i	−0.931117	+	0.364720i	$0.881165\pi$
$840$	0	0
$841$	−1859.00	−2.21046
$842$	0	0
$843$	405.300	0.480783
$844$	0	0
$845$	419.156i	0.496043i
$846$	0	0
$847$	− 710.000i	− 0.838253i
$848$	0	0
$849$	540.000	0.636042
$850$	0	0
$851$	665.108	0.781560
$852$	0	0
$853$	55.4256i	0.0649773i	0.999472	+	0.0324886i	$0.0103433\pi$
$853$	55.4256i	0.0649773i	−0.999472	+	0.0324886i	$0.989657\pi$
$854$	0	0
$855$	72.0000i	0.0842105i
$856$	0	0
$857$	−582.000	−0.679113	−0.339557	−	0.940586i	$-0.610277\pi$
$857$	−582.000	−0.679113	−0.339557	+	0.940586i	$0.610277\pi$
$858$	0	0
$859$	−1364.86	−1.58889	−0.794445	−	0.607336i	$-0.792238\pi$
$859$	−1364.86	−1.58889	−0.794445	+	0.607336i	$0.792238\pi$
$860$	0	0
$861$	− 103.923i	− 0.120700i
$862$	0	0
$863$	− 456.000i	− 0.528389i	−0.964469	−	0.264195i	$-0.914894\pi$
$863$	− 456.000i	− 0.528389i	0.964469	−	0.264195i	$-0.0851061\pi$
$864$	0	0
$865$	−564.000	−0.652023
$866$	0	0
$867$	−1058.28	−1.22063
$868$	0	0
$869$	− 526.543i	− 0.605919i
$870$	0	0
$871$	− 336.000i	− 0.385763i
$872$	0	0
$873$	186.000	0.213058
$874$	0	0
$875$	1316.36	1.50441
$876$	0	0
$877$	13.8564i	0.0157998i	0.999969	+	0.00789989i	$0.00251464\pi$
$877$	13.8564i	0.0157998i	−0.999969	+	0.00789989i	$0.997485\pi$
$878$	0	0
$879$	270.000i	0.307167i
$880$	0	0
$881$	1362.00	1.54597	0.772985	−	0.634424i	$-0.218763\pi$
$881$	1362.00	1.54597	0.772985	+	0.634424i	$0.218763\pi$
$882$	0	0
$883$	−1586.56	−1.79678	−0.898391	−	0.439197i	$-0.855263\pi$
$883$	−1586.56	−1.79678	−0.898391	+	0.439197i	$0.855263\pi$
$884$	0	0
$885$	− 374.123i	− 0.422738i
$886$	0	0
$887$	240.000i	0.270575i	0.990806	+	0.135287i	$0.0431958\pi$
$887$	240.000i	0.270575i	−0.990806	+	0.135287i	$0.956804\pi$
$888$	0	0
$889$	−820.000	−0.922385
$890$	0	0
$891$	124.708	0.139964
$892$	0	0
$893$	581.969i	0.651701i
$894$	0	0
$895$	− 840.000i	− 0.938547i
$896$	0	0
$897$	−144.000	−0.160535
$898$	0	0
$899$	−727.461	−0.809189
$900$	0	0
$901$	− 519.615i	− 0.576709i
$902$	0	0
$903$	− 1080.00i	− 1.19601i
$904$	0	0
$905$	1032.00	1.14033
$906$	0	0
$907$	159.349	0.175688	0.0878438	−	0.996134i	$-0.472002\pi$
$907$	159.349	0.175688	0.0878438	+	0.996134i	$0.472002\pi$
$908$	0	0
$909$	197.454i	0.217221i
$910$	0	0
$911$	1392.00i	1.52799i	0.645221	+	0.763996i	$0.276765\pi$
$911$	1392.00i	1.52799i	−0.645221	+	0.763996i	$0.723235\pi$
$912$	0	0
$913$	−192.000	−0.210296
$914$	0	0
$915$	−581.969	−0.636032
$916$	0	0
$917$	− 762.102i	− 0.831082i
$918$	0	0
$919$	422.000i	0.459195i	0.973286	+	0.229597i	$0.0737409\pi$
$919$	422.000i	0.459195i	−0.973286	+	0.229597i	$0.926259\pi$
$920$	0	0
$921$	180.000	0.195440
$922$	0	0
$923$	−415.692	−0.450371
$924$	0	0
$925$	720.533i	0.778955i
$926$	0	0
$927$	− 42.0000i	− 0.0453074i
$928$	0	0
$929$	−1026.00	−1.10441	−0.552207	−	0.833707i	$-0.686214\pi$
$929$	−1026.00	−1.10441	−0.552207	+	0.833707i	$0.686214\pi$
$930$	0	0
$931$	−353.338	−0.379526
$932$	0	0
$933$	− 41.5692i	− 0.0445544i
$934$	0	0
$935$	1440.00i	1.54011i
$936$	0	0
$937$	814.000	0.868730	0.434365	−	0.900737i	$-0.356973\pi$
$937$	814.000	0.868730	0.434365	+	0.900737i	$0.356973\pi$
$938$	0	0
$939$	−329.090	−0.350468
$940$	0	0
$941$	980.341i	1.04181i	0.853615	+	0.520904i	$0.174405\pi$
$941$	980.341i	1.04181i	−0.853615	+	0.520904i	$0.825595\pi$
$942$	0	0
$943$	72.0000i	0.0763521i
$944$	0	0
$945$	−180.000	−0.190476
$946$	0	0
$947$	1503.42	1.58756	0.793780	−	0.608204i	$-0.208110\pi$
$947$	1503.42	1.58756	0.793780	+	0.608204i	$0.208110\pi$
$948$	0	0
$949$	− 595.825i	− 0.627846i
$950$	0	0
$951$	186.000i	0.195584i
$952$	0	0
$953$	−366.000	−0.384050	−0.192025	−	0.981390i	$-0.561506\pi$
$953$	−366.000	−0.384050	−0.192025	+	0.981390i	$0.561506\pi$
$954$	0	0
$955$	−748.246	−0.783504
$956$	0	0
$957$	1247.08i	1.30311i
$958$	0	0
$959$	− 420.000i	− 0.437956i
$960$	0	0
$961$	765.000	0.796046
$962$	0	0
$963$	−228.631	−0.237415
$964$	0	0
$965$	− 990.733i	− 1.02667i
$966$	0	0
$967$	− 686.000i	− 0.709411i	−0.934978	−	0.354705i	$-0.884581\pi$
$967$	− 686.000i	− 0.709411i	0.934978	−	0.354705i	$-0.115419\pi$
$968$	0	0
$969$	−360.000	−0.371517
$970$	0	0
$971$	1344.07	1.38421	0.692107	−	0.721795i	$-0.256683\pi$
$971$	1344.07	1.38421	0.692107	+	0.721795i	$0.256683\pi$
$972$	0	0
$973$	484.974i	0.498432i
$974$	0	0
$975$	− 156.000i	− 0.160000i
$976$	0	0
$977$	1470.00	1.50461	0.752303	−	0.658817i	$-0.228943\pi$
$977$	1470.00	1.50461	0.752303	+	0.658817i	$0.228943\pi$
$978$	0	0
$979$	1080.80	1.10398
$980$	0	0
$981$	62.3538i	0.0635615i
$982$	0	0
$983$	− 720.000i	− 0.732452i	−0.930526	−	0.366226i	$-0.880650\pi$
$983$	− 720.000i	− 0.732452i	0.930526	−	0.366226i	$-0.119350\pi$
$984$	0	0
$985$	588.000	0.596954
$986$	0	0
$987$	−1454.92	−1.47409
$988$	0	0
$989$	748.246i	0.756568i
$990$	0	0
$991$	− 1598.00i	− 1.61251i	−0.591566	−	0.806256i	$-0.701490\pi$
$991$	− 1598.00i	− 1.61251i	0.591566	−	0.806256i	$-0.298510\pi$
$992$	0	0
$993$	−60.0000	−0.0604230
$994$	0	0
$995$	−450.333	−0.452596
$996$	0	0
$997$	1454.92i	1.45930i	0.683820	+	0.729650i	$0.260317\pi$
$997$	1454.92i	1.45930i	−0.683820	+	0.729650i	$0.739683\pi$
$998$	0	0
$999$	− 288.000i	− 0.288288i

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +3.46410i q^{5} -10.0000i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} +3.46410i q^{5} -10.0000i q^{7} +3.00000 q^{9} +13.8564 q^{11} +6.92820i q^{13} -6.00000i q^{15} +30.0000 q^{17} +6.92820 q^{19} +17.3205i q^{21} -12.0000i q^{23} +13.0000 q^{25} -5.19615 q^{27} -51.9615i q^{29} -14.0000i q^{31} -24.0000 q^{33} +34.6410 q^{35} +55.4256i q^{37} -12.0000i q^{39} -6.00000 q^{41} -62.3538 q^{43} +10.3923i q^{45} +84.0000i q^{47} -51.0000 q^{49} -51.9615 q^{51} -17.3205i q^{53} +48.0000i q^{55} -12.0000 q^{57} +62.3538 q^{59} -96.9948i q^{61} -30.0000i q^{63} -24.0000 q^{65} -48.4974 q^{67} +20.7846i q^{69} +60.0000i q^{71} -86.0000 q^{73} -22.5167 q^{75} -138.564i q^{77} -38.0000i q^{79} +9.00000 q^{81} -13.8564 q^{83} +103.923i q^{85} +90.0000i q^{87} +78.0000 q^{89} +69.2820 q^{91} +24.2487i q^{93} +24.0000i q^{95} +62.0000 q^{97} +41.5692 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} + 120 q^{17} + 52 q^{25} - 96 q^{33} - 24 q^{41} - 204 q^{49} - 48 q^{57} - 96 q^{65} - 344 q^{73} + 36 q^{81} + 312 q^{89} + 248 q^{97}+O(q^{100}) \) 4 * q + 12 * q^9 + 120 * q^17 + 52 * q^25 - 96 * q^33 - 24 * q^41 - 204 * q^49 - 48 * q^57 - 96 * q^65 - 344 * q^73 + 36 * q^81 + 312 * q^89 + 248 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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