LMFDB - Embedded newform 1664.2.b.e.833.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1664,2,Mod(833,1664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1664.833"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1664 = 2^{7} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1664.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-4,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.2871068963$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			833.1
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1664.833
Dual form			1664.2.b.e.833.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -3.82843i q^{5} -3.82843 q^{7} +2.00000 q^{9} -4.82843i q^{11} -1.00000i q^{13} -3.82843 q^{15} +6.65685 q^{17} -4.00000i q^{19} +3.82843i q^{21} +3.17157 q^{23} -9.65685 q^{25} -5.00000i q^{27} +3.17157i q^{29} -4.82843 q^{33} +14.6569i q^{35} -3.82843i q^{37} -1.00000 q^{39} -2.82843 q^{41} +3.00000i q^{43} -7.65685i q^{45} -11.4853 q^{47} +7.65685 q^{49} -6.65685i q^{51} -3.17157i q^{53} -18.4853 q^{55} -4.00000 q^{57} +5.17157i q^{59} +10.8284i q^{61} -7.65685 q^{63} -3.82843 q^{65} +3.65685i q^{67} -3.17157i q^{69} +10.1716 q^{71} -5.17157 q^{73} +9.65685i q^{75} +18.4853i q^{77} -7.65685 q^{79} +1.00000 q^{81} +6.00000i q^{83} -25.4853i q^{85} +3.17157 q^{87} +17.6569 q^{89} +3.82843i q^{91} -15.3137 q^{95} -14.4853 q^{97} -9.65685i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} + 8 q^{9} - 4 q^{15} + 4 q^{17} + 24 q^{23} - 16 q^{25} - 8 q^{33} - 4 q^{39} - 12 q^{47} + 8 q^{49} - 40 q^{55} - 16 q^{57} - 8 q^{63} - 4 q^{65} + 52 q^{71} - 32 q^{73} - 8 q^{79} + 4 q^{81}+ \cdots - 24 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 + 8 * q^9 - 4 * q^15 + 4 * q^17 + 24 * q^23 - 16 * q^25 - 8 * q^33 - 4 * q^39 - 12 * q^47 + 8 * q^49 - 40 * q^55 - 16 * q^57 - 8 * q^63 - 4 * q^65 + 52 * q^71 - 32 * q^73 - 8 * q^79 + 4 * q^81 + 24 * q^87 + 48 * q^89 - 16 * q^95 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1664\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	− 3.82843i	− 1.71212i	−0.516873	−	0.856062i	$-0.672904\pi$
$5$	− 3.82843i	− 1.71212i	0.516873	−	0.856062i	$-0.327096\pi$
$6$	0	0
$7$	−3.82843	−1.44701	−0.723505	−	0.690319i	$-0.757470\pi$
$7$	−3.82843	−1.44701	−0.723505	+	0.690319i	$0.757470\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	− 4.82843i	− 1.45583i	−0.685670	−	0.727913i	$-0.740491\pi$
$11$	− 4.82843i	− 1.45583i	0.685670	−	0.727913i	$-0.259509\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−3.82843	−0.988496
$16$	0	0
$17$	6.65685	1.61452	0.807262	−	0.590193i	$-0.200948\pi$
$17$	6.65685	1.61452	0.807262	+	0.590193i	$0.200948\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	3.82843i	0.835431i
$22$	0	0
$23$	3.17157	0.661319	0.330659	−	0.943750i	$-0.392729\pi$
$23$	3.17157	0.661319	0.330659	+	0.943750i	$0.392729\pi$
$24$	0	0
$25$	−9.65685	−1.93137
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	3.17157i	0.588946i	0.955660	+	0.294473i	$0.0951442\pi$
$29$	3.17157i	0.588946i	−0.955660	+	0.294473i	$0.904856\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−4.82843	−0.840521
$34$	0	0
$35$	14.6569i	2.47746i
$36$	0	0
$37$	− 3.82843i	− 0.629390i	−0.949193	−	0.314695i	$-0.898098\pi$
$37$	− 3.82843i	− 0.629390i	0.949193	−	0.314695i	$-0.101902\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.82843	−0.441726	−0.220863	−	0.975305i	$-0.570887\pi$
$41$	−2.82843	−0.441726	−0.220863	+	0.975305i	$0.570887\pi$
$42$	0	0
$43$	3.00000i	0.457496i	0.973486	+	0.228748i	$0.0734631\pi$
$43$	3.00000i	0.457496i	−0.973486	+	0.228748i	$0.926537\pi$
$44$	0	0
$45$	− 7.65685i	− 1.14142i
$46$	0	0
$47$	−11.4853	−1.67530	−0.837650	−	0.546207i	$-0.816071\pi$
$47$	−11.4853	−1.67530	−0.837650	+	0.546207i	$0.816071\pi$
$48$	0	0
$49$	7.65685	1.09384
$50$	0	0
$51$	− 6.65685i	− 0.932146i
$52$	0	0
$53$	− 3.17157i	− 0.435649i	−0.975988	−	0.217825i	$-0.930104\pi$
$53$	− 3.17157i	− 0.435649i	0.975988	−	0.217825i	$-0.0698960\pi$
$54$	0	0
$55$	−18.4853	−2.49255
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	5.17157i	0.673281i	0.941633	+	0.336641i	$0.109291\pi$
$59$	5.17157i	0.673281i	−0.941633	+	0.336641i	$0.890709\pi$
$60$	0	0
$61$	10.8284i	1.38644i	0.720727	+	0.693219i	$0.243808\pi$
$61$	10.8284i	1.38644i	−0.720727	+	0.693219i	$0.756192\pi$
$62$	0	0
$63$	−7.65685	−0.964673
$64$	0	0
$65$	−3.82843	−0.474858
$66$	0	0
$67$	3.65685i	0.446756i	0.974732	+	0.223378i	$0.0717084\pi$
$67$	3.65685i	0.446756i	−0.974732	+	0.223378i	$0.928292\pi$
$68$	0	0
$69$	− 3.17157i	− 0.381813i
$70$	0	0
$71$	10.1716	1.20714	0.603572	−	0.797309i	$-0.293744\pi$
$71$	10.1716	1.20714	0.603572	+	0.797309i	$0.293744\pi$
$72$	0	0
$73$	−5.17157	−0.605287	−0.302643	−	0.953104i	$-0.597869\pi$
$73$	−5.17157	−0.605287	−0.302643	+	0.953104i	$0.597869\pi$
$74$	0	0
$75$	9.65685i	1.11508i
$76$	0	0
$77$	18.4853i	2.10659i
$78$	0	0
$79$	−7.65685	−0.861463	−0.430732	−	0.902480i	$-0.641744\pi$
$79$	−7.65685	−0.861463	−0.430732	+	0.902480i	$0.641744\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	− 25.4853i	− 2.76427i
$86$	0	0
$87$	3.17157	0.340028
$88$	0	0
$89$	17.6569	1.87162	0.935811	−	0.352501i	$-0.114669\pi$
$89$	17.6569	1.87162	0.935811	+	0.352501i	$0.114669\pi$
$90$	0	0
$91$	3.82843i	0.401328i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−15.3137	−1.57115
$96$	0	0
$97$	−14.4853	−1.47076	−0.735379	−	0.677656i	$-0.762996\pi$
$97$	−14.4853	−1.47076	−0.735379	+	0.677656i	$0.762996\pi$
$98$	0	0
$99$	− 9.65685i	− 0.970550i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	14.6569	1.43036
$106$	0	0
$107$	15.3137i	1.48043i	0.672369	+	0.740216i	$0.265277\pi$
$107$	15.3137i	1.48043i	−0.672369	+	0.740216i	$0.734723\pi$
$108$	0	0
$109$	3.82843i	0.366697i	0.983048	+	0.183348i	$0.0586936\pi$
$109$	3.82843i	0.366697i	−0.983048	+	0.183348i	$0.941306\pi$
$110$	0	0
$111$	−3.82843	−0.363378
$112$	0	0
$113$	−3.65685	−0.344008	−0.172004	−	0.985096i	$-0.555024\pi$
$113$	−3.65685	−0.344008	−0.172004	+	0.985096i	$0.555024\pi$
$114$	0	0
$115$	− 12.1421i	− 1.13226i
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−25.4853	−2.33623
$120$	0	0
$121$	−12.3137	−1.11943
$122$	0	0
$123$	2.82843i	0.255031i
$124$	0	0
$125$	17.8284i	1.59462i
$126$	0	0
$127$	18.4853	1.64030	0.820152	−	0.572146i	$-0.193889\pi$
$127$	18.4853	1.64030	0.820152	+	0.572146i	$0.193889\pi$
$128$	0	0
$129$	3.00000	0.264135
$130$	0	0
$131$	2.65685i	0.232130i	0.993242	+	0.116065i	$0.0370282\pi$
$131$	2.65685i	0.232130i	−0.993242	+	0.116065i	$0.962972\pi$
$132$	0	0
$133$	15.3137i	1.32787i
$134$	0	0
$135$	−19.1421	−1.64749
$136$	0	0
$137$	20.1421	1.72086	0.860429	−	0.509570i	$-0.170195\pi$
$137$	20.1421	1.72086	0.860429	+	0.509570i	$0.170195\pi$
$138$	0	0
$139$	− 18.6569i	− 1.58245i	−0.611523	−	0.791227i	$-0.709443\pi$
$139$	− 18.6569i	− 1.58245i	0.611523	−	0.791227i	$-0.290557\pi$
$140$	0	0
$141$	11.4853i	0.967235i
$142$	0	0
$143$	−4.82843	−0.403773
$144$	0	0
$145$	12.1421	1.00835
$146$	0	0
$147$	− 7.65685i	− 0.631527i
$148$	0	0
$149$	1.31371i	0.107623i	0.998551	+	0.0538116i	$0.0171370\pi$
$149$	1.31371i	0.107623i	−0.998551	+	0.0538116i	$0.982863\pi$
$150$	0	0
$151$	−2.51472	−0.204645	−0.102322	−	0.994751i	$-0.532627\pi$
$151$	−2.51472	−0.204645	−0.102322	+	0.994751i	$0.532627\pi$
$152$	0	0
$153$	13.3137	1.07635
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.48528i	− 0.357964i	−0.983852	−	0.178982i	$-0.942720\pi$
$157$	− 4.48528i	− 0.357964i	0.983852	−	0.178982i	$-0.0572805\pi$
$158$	0	0
$159$	−3.17157	−0.251522
$160$	0	0
$161$	−12.1421	−0.956934
$162$	0	0
$163$	− 12.8284i	− 1.00480i	−0.864635	−	0.502400i	$-0.832451\pi$
$163$	− 12.8284i	− 1.00480i	0.864635	−	0.502400i	$-0.167549\pi$
$164$	0	0
$165$	18.4853i	1.43908i
$166$	0	0
$167$	−21.6569	−1.67586	−0.837929	−	0.545779i	$-0.816234\pi$
$167$	−21.6569	−1.67586	−0.837929	+	0.545779i	$0.816234\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 8.00000i	− 0.611775i
$172$	0	0
$173$	− 22.9706i	− 1.74642i	−0.487345	−	0.873210i	$-0.662034\pi$
$173$	− 22.9706i	− 1.74642i	0.487345	−	0.873210i	$-0.337966\pi$
$174$	0	0
$175$	36.9706	2.79471
$176$	0	0
$177$	5.17157	0.388719
$178$	0	0
$179$	8.31371i	0.621396i	0.950509	+	0.310698i	$0.100563\pi$
$179$	8.31371i	0.621396i	−0.950509	+	0.310698i	$0.899437\pi$
$180$	0	0
$181$	− 22.9706i	− 1.70739i	−0.520775	−	0.853694i	$-0.674357\pi$
$181$	− 22.9706i	− 1.70739i	0.520775	−	0.853694i	$-0.325643\pi$
$182$	0	0
$183$	10.8284	0.800460
$184$	0	0
$185$	−14.6569	−1.07759
$186$	0	0
$187$	− 32.1421i	− 2.35047i
$188$	0	0
$189$	19.1421i	1.39239i
$190$	0	0
$191$	3.17157	0.229487	0.114743	−	0.993395i	$-0.463395\pi$
$191$	3.17157	0.229487	0.114743	+	0.993395i	$0.463395\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	3.82843i	0.274159i
$196$	0	0
$197$	− 17.8284i	− 1.27022i	−0.772421	−	0.635111i	$-0.780954\pi$
$197$	− 17.8284i	− 1.27022i	0.772421	−	0.635111i	$-0.219046\pi$
$198$	0	0
$199$	22.9706	1.62834	0.814170	−	0.580627i	$-0.197192\pi$
$199$	22.9706	1.62834	0.814170	+	0.580627i	$0.197192\pi$
$200$	0	0
$201$	3.65685	0.257935
$202$	0	0
$203$	− 12.1421i	− 0.852211i
$204$	0	0
$205$	10.8284i	0.756290i
$206$	0	0
$207$	6.34315	0.440879
$208$	0	0
$209$	−19.3137	−1.33596
$210$	0	0
$211$	− 1.34315i	− 0.0924660i	−0.998931	−	0.0462330i	$-0.985278\pi$
$211$	− 1.34315i	− 0.0924660i	0.998931	−	0.0462330i	$-0.0147217\pi$
$212$	0	0
$213$	− 10.1716i	− 0.696945i
$214$	0	0
$215$	11.4853	0.783290
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5.17157i	0.349463i
$220$	0	0
$221$	− 6.65685i	− 0.447788i
$222$	0	0
$223$	−10.1716	−0.681139	−0.340569	−	0.940219i	$-0.610620\pi$
$223$	−10.1716	−0.681139	−0.340569	+	0.940219i	$0.610620\pi$
$224$	0	0
$225$	−19.3137	−1.28758
$226$	0	0
$227$	9.17157i	0.608739i	0.952554	+	0.304369i	$0.0984457\pi$
$227$	9.17157i	0.608739i	−0.952554	+	0.304369i	$0.901554\pi$
$228$	0	0
$229$	10.1716i	0.672156i	0.941834	+	0.336078i	$0.109101\pi$
$229$	10.1716i	0.672156i	−0.941834	+	0.336078i	$0.890899\pi$
$230$	0	0
$231$	18.4853	1.21624
$232$	0	0
$233$	−20.3137	−1.33080	−0.665398	−	0.746489i	$-0.731738\pi$
$233$	−20.3137	−1.33080	−0.665398	+	0.746489i	$0.731738\pi$
$234$	0	0
$235$	43.9706i	2.86832i
$236$	0	0
$237$	7.65685i	0.497366i
$238$	0	0
$239$	19.1421	1.23820	0.619101	−	0.785311i	$-0.287497\pi$
$239$	19.1421	1.23820	0.619101	+	0.785311i	$0.287497\pi$
$240$	0	0
$241$	3.31371	0.213455	0.106727	−	0.994288i	$-0.465963\pi$
$241$	3.31371	0.213455	0.106727	+	0.994288i	$0.465963\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	− 29.3137i	− 1.87278i
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	− 26.6274i	− 1.68071i	−0.542038	−	0.840354i	$-0.682347\pi$
$251$	− 26.6274i	− 1.68071i	0.542038	−	0.840354i	$-0.317653\pi$
$252$	0	0
$253$	− 15.3137i	− 0.962765i
$254$	0	0
$255$	−25.4853	−1.59595
$256$	0	0
$257$	−1.34315	−0.0837831	−0.0418916	−	0.999122i	$-0.513338\pi$
$257$	−1.34315	−0.0837831	−0.0418916	+	0.999122i	$0.513338\pi$
$258$	0	0
$259$	14.6569i	0.910733i
$260$	0	0
$261$	6.34315i	0.392631i
$262$	0	0
$263$	−26.1421	−1.61199	−0.805997	−	0.591920i	$-0.798370\pi$
$263$	−26.1421	−1.61199	−0.805997	+	0.591920i	$0.798370\pi$
$264$	0	0
$265$	−12.1421	−0.745885
$266$	0	0
$267$	− 17.6569i	− 1.08058i
$268$	0	0
$269$	− 19.7990i	− 1.20717i	−0.797300	−	0.603583i	$-0.793739\pi$
$269$	− 19.7990i	− 1.20717i	0.797300	−	0.603583i	$-0.206261\pi$
$270$	0	0
$271$	−2.51472	−0.152758	−0.0763791	−	0.997079i	$-0.524336\pi$
$271$	−2.51472	−0.152758	−0.0763791	+	0.997079i	$0.524336\pi$
$272$	0	0
$273$	3.82843	0.231707
$274$	0	0
$275$	46.6274i	2.81174i
$276$	0	0
$277$	4.48528i	0.269494i	0.990880	+	0.134747i	$0.0430222\pi$
$277$	4.48528i	0.269494i	−0.990880	+	0.134747i	$0.956978\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.97056	0.296519	0.148259	−	0.988948i	$-0.452633\pi$
$281$	4.97056	0.296519	0.148259	+	0.988948i	$0.452633\pi$
$282$	0	0
$283$	14.6274i	0.869510i	0.900549	+	0.434755i	$0.143165\pi$
$283$	14.6274i	0.869510i	−0.900549	+	0.434755i	$0.856835\pi$
$284$	0	0
$285$	15.3137i	0.907106i
$286$	0	0
$287$	10.8284	0.639182
$288$	0	0
$289$	27.3137	1.60669
$290$	0	0
$291$	14.4853i	0.849142i
$292$	0	0
$293$	− 12.7990i	− 0.747725i	−0.927484	−	0.373862i	$-0.878033\pi$
$293$	− 12.7990i	− 0.747725i	0.927484	−	0.373862i	$-0.121967\pi$
$294$	0	0
$295$	19.7990	1.15274
$296$	0	0
$297$	−24.1421	−1.40087
$298$	0	0
$299$	− 3.17157i	− 0.183417i
$300$	0	0
$301$	− 11.4853i	− 0.662001i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	41.4558	2.37375
$306$	0	0
$307$	13.1716i	0.751741i	0.926672	+	0.375871i	$0.122656\pi$
$307$	13.1716i	0.751741i	−0.926672	+	0.375871i	$0.877344\pi$
$308$	0	0
$309$	− 14.0000i	− 0.796432i
$310$	0	0
$311$	24.2843	1.37703	0.688517	−	0.725220i	$-0.258262\pi$
$311$	24.2843	1.37703	0.688517	+	0.725220i	$0.258262\pi$
$312$	0	0
$313$	−19.9706	−1.12880	−0.564401	−	0.825500i	$-0.690893\pi$
$313$	−19.9706	−1.12880	−0.564401	+	0.825500i	$0.690893\pi$
$314$	0	0
$315$	29.3137i	1.65164i
$316$	0	0
$317$	29.3137i	1.64642i	0.567736	+	0.823211i	$0.307820\pi$
$317$	29.3137i	1.64642i	−0.567736	+	0.823211i	$0.692180\pi$
$318$	0	0
$319$	15.3137	0.857403
$320$	0	0
$321$	15.3137	0.854728
$322$	0	0
$323$	− 26.6274i	− 1.48159i
$324$	0	0
$325$	9.65685i	0.535666i
$326$	0	0
$327$	3.82843	0.211713
$328$	0	0
$329$	43.9706	2.42418
$330$	0	0
$331$	− 10.9706i	− 0.602997i	−0.953467	−	0.301498i	$-0.902513\pi$
$331$	− 10.9706i	− 0.602997i	0.953467	−	0.301498i	$-0.0974868\pi$
$332$	0	0
$333$	− 7.65685i	− 0.419593i
$334$	0	0
$335$	14.0000	0.764902
$336$	0	0
$337$	−1.97056	−0.107343	−0.0536717	−	0.998559i	$-0.517092\pi$
$337$	−1.97056	−0.107343	−0.0536717	+	0.998559i	$0.517092\pi$
$338$	0	0
$339$	3.65685i	0.198613i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−2.51472	−0.135782
$344$	0	0
$345$	−12.1421	−0.653711
$346$	0	0
$347$	15.0000i	0.805242i	0.915367	+	0.402621i	$0.131901\pi$
$347$	15.0000i	0.805242i	−0.915367	+	0.402621i	$0.868099\pi$
$348$	0	0
$349$	25.4853i	1.36420i	0.731261	+	0.682098i	$0.238932\pi$
$349$	25.4853i	1.36420i	−0.731261	+	0.682098i	$0.761068\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−17.3137	−0.921516	−0.460758	−	0.887526i	$-0.652422\pi$
$353$	−17.3137	−0.921516	−0.460758	+	0.887526i	$0.652422\pi$
$354$	0	0
$355$	− 38.9411i	− 2.06678i
$356$	0	0
$357$	25.4853i	1.34882i
$358$	0	0
$359$	21.6569	1.14301	0.571503	−	0.820600i	$-0.306361\pi$
$359$	21.6569	1.14301	0.571503	+	0.820600i	$0.306361\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	12.3137i	0.646302i
$364$	0	0
$365$	19.7990i	1.03633i
$366$	0	0
$367$	31.9411	1.66731	0.833657	−	0.552283i	$-0.186243\pi$
$367$	31.9411	1.66731	0.833657	+	0.552283i	$0.186243\pi$
$368$	0	0
$369$	−5.65685	−0.294484
$370$	0	0
$371$	12.1421i	0.630388i
$372$	0	0
$373$	8.97056i	0.464478i	0.972659	+	0.232239i	$0.0746052\pi$
$373$	8.97056i	0.464478i	−0.972659	+	0.232239i	$0.925395\pi$
$374$	0	0
$375$	17.8284	0.920656
$376$	0	0
$377$	3.17157	0.163344
$378$	0	0
$379$	23.4558i	1.20485i	0.798177	+	0.602423i	$0.205798\pi$
$379$	23.4558i	1.20485i	−0.798177	+	0.602423i	$0.794202\pi$
$380$	0	0
$381$	− 18.4853i	− 0.947030i
$382$	0	0
$383$	17.8284	0.910990	0.455495	−	0.890238i	$-0.349462\pi$
$383$	17.8284	0.910990	0.455495	+	0.890238i	$0.349462\pi$
$384$	0	0
$385$	70.7696	3.60675
$386$	0	0
$387$	6.00000i	0.304997i
$388$	0	0
$389$	− 26.1421i	− 1.32546i	−0.748859	−	0.662729i	$-0.769398\pi$
$389$	− 26.1421i	− 1.32546i	0.748859	−	0.662729i	$-0.230602\pi$
$390$	0	0
$391$	21.1127	1.06772
$392$	0	0
$393$	2.65685	0.134021
$394$	0	0
$395$	29.3137i	1.47493i
$396$	0	0
$397$	1.31371i	0.0659331i	0.999456	+	0.0329666i	$0.0104955\pi$
$397$	1.31371i	0.0659331i	−0.999456	+	0.0329666i	$0.989505\pi$
$398$	0	0
$399$	15.3137	0.766644
$400$	0	0
$401$	−11.1716	−0.557882	−0.278941	−	0.960308i	$-0.589983\pi$
$401$	−11.1716	−0.557882	−0.278941	+	0.960308i	$0.589983\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 3.82843i	− 0.190236i
$406$	0	0
$407$	−18.4853	−0.916281
$408$	0	0
$409$	1.51472	0.0748980	0.0374490	−	0.999299i	$-0.488077\pi$
$409$	1.51472	0.0748980	0.0374490	+	0.999299i	$0.488077\pi$
$410$	0	0
$411$	− 20.1421i	− 0.993538i
$412$	0	0
$413$	− 19.7990i	− 0.974245i
$414$	0	0
$415$	22.9706	1.12758
$416$	0	0
$417$	−18.6569	−0.913630
$418$	0	0
$419$	− 19.3431i	− 0.944975i	−0.881338	−	0.472487i	$-0.843356\pi$
$419$	− 19.3431i	− 0.944975i	0.881338	−	0.472487i	$-0.156644\pi$
$420$	0	0
$421$	8.85786i	0.431706i	0.976426	+	0.215853i	$0.0692532\pi$
$421$	8.85786i	0.431706i	−0.976426	+	0.215853i	$0.930747\pi$
$422$	0	0
$423$	−22.9706	−1.11687
$424$	0	0
$425$	−64.2843	−3.11825
$426$	0	0
$427$	− 41.4558i	− 2.00619i
$428$	0	0
$429$	4.82843i	0.233119i
$430$	0	0
$431$	−17.8284	−0.858765	−0.429383	−	0.903123i	$-0.641269\pi$
$431$	−17.8284	−0.858765	−0.429383	+	0.903123i	$0.641269\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	0	0
$435$	− 12.1421i	− 0.582171i
$436$	0	0
$437$	− 12.6863i	− 0.606868i
$438$	0	0
$439$	4.48528	0.214071	0.107035	−	0.994255i	$-0.465864\pi$
$439$	4.48528	0.214071	0.107035	+	0.994255i	$0.465864\pi$
$440$	0	0
$441$	15.3137	0.729224
$442$	0	0
$443$	0.313708i	0.0149047i	0.999972	+	0.00745237i	$0.00237219\pi$
$443$	0.313708i	0.0149047i	−0.999972	+	0.00745237i	$0.997628\pi$
$444$	0	0
$445$	− 67.5980i	− 3.20445i
$446$	0	0
$447$	1.31371	0.0621363
$448$	0	0
$449$	12.1421	0.573023	0.286511	−	0.958077i	$-0.407504\pi$
$449$	12.1421	0.573023	0.286511	+	0.958077i	$0.407504\pi$
$450$	0	0
$451$	13.6569i	0.643076i
$452$	0	0
$453$	2.51472i	0.118152i
$454$	0	0
$455$	14.6569	0.687124
$456$	0	0
$457$	20.9706	0.980962	0.490481	−	0.871452i	$-0.336821\pi$
$457$	20.9706	0.980962	0.490481	+	0.871452i	$0.336821\pi$
$458$	0	0
$459$	− 33.2843i	− 1.55358i
$460$	0	0
$461$	− 11.4853i	− 0.534923i	−0.963569	−	0.267461i	$-0.913815\pi$
$461$	− 11.4853i	− 0.534923i	0.963569	−	0.267461i	$-0.0861848\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 30.3431i	− 1.40411i	−0.712121	−	0.702057i	$-0.752265\pi$
$467$	− 30.3431i	− 1.40411i	0.712121	−	0.702057i	$-0.247735\pi$
$468$	0	0
$469$	− 14.0000i	− 0.646460i
$470$	0	0
$471$	−4.48528	−0.206671
$472$	0	0
$473$	14.4853	0.666034
$474$	0	0
$475$	38.6274i	1.77235i
$476$	0	0
$477$	− 6.34315i	− 0.290433i
$478$	0	0
$479$	19.1421	0.874627	0.437313	−	0.899309i	$-0.355930\pi$
$479$	19.1421	0.874627	0.437313	+	0.899309i	$0.355930\pi$
$480$	0	0
$481$	−3.82843	−0.174561
$482$	0	0
$483$	12.1421i	0.552486i
$484$	0	0
$485$	55.4558i	2.51812i
$486$	0	0
$487$	8.97056	0.406495	0.203247	−	0.979127i	$-0.434850\pi$
$487$	8.97056	0.406495	0.203247	+	0.979127i	$0.434850\pi$
$488$	0	0
$489$	−12.8284	−0.580122
$490$	0	0
$491$	− 17.6863i	− 0.798171i	−0.916914	−	0.399086i	$-0.869328\pi$
$491$	− 17.6863i	− 0.798171i	0.916914	−	0.399086i	$-0.130672\pi$
$492$	0	0
$493$	21.1127i	0.950868i
$494$	0	0
$495$	−36.9706	−1.66170
$496$	0	0
$497$	−38.9411	−1.74675
$498$	0	0
$499$	− 18.1421i	− 0.812154i	−0.913839	−	0.406077i	$-0.866897\pi$
$499$	− 18.1421i	− 0.812154i	0.913839	−	0.406077i	$-0.133103\pi$
$500$	0	0
$501$	21.6569i	0.967557i
$502$	0	0
$503$	−15.3137	−0.682805	−0.341402	−	0.939917i	$-0.610902\pi$
$503$	−15.3137	−0.682805	−0.341402	+	0.939917i	$0.610902\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	0	0
$509$	38.2843i	1.69692i	0.529259	+	0.848460i	$0.322470\pi$
$509$	38.2843i	1.69692i	−0.529259	+	0.848460i	$0.677530\pi$
$510$	0	0
$511$	19.7990	0.875856
$512$	0	0
$513$	−20.0000	−0.883022
$514$	0	0
$515$	− 53.5980i	− 2.36181i
$516$	0	0
$517$	55.4558i	2.43895i
$518$	0	0
$519$	−22.9706	−1.00830
$520$	0	0
$521$	−14.3137	−0.627095	−0.313547	−	0.949573i	$-0.601517\pi$
$521$	−14.3137	−0.627095	−0.313547	+	0.949573i	$0.601517\pi$
$522$	0	0
$523$	− 9.65685i	− 0.422265i	−0.977457	−	0.211132i	$-0.932285\pi$
$523$	− 9.65685i	− 0.422265i	0.977457	−	0.211132i	$-0.0677151\pi$
$524$	0	0
$525$	− 36.9706i	− 1.61353i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12.9411	−0.562658
$530$	0	0
$531$	10.3431i	0.448854i
$532$	0	0
$533$	2.82843i	0.122513i
$534$	0	0
$535$	58.6274	2.53468
$536$	0	0
$537$	8.31371	0.358763
$538$	0	0
$539$	− 36.9706i	− 1.59243i
$540$	0	0
$541$	− 16.5147i	− 0.710023i	−0.934862	−	0.355012i	$-0.884477\pi$
$541$	− 16.5147i	− 0.710023i	0.934862	−	0.355012i	$-0.115523\pi$
$542$	0	0
$543$	−22.9706	−0.985761
$544$	0	0
$545$	14.6569	0.627831
$546$	0	0
$547$	5.62742i	0.240611i	0.992737	+	0.120305i	$0.0383874\pi$
$547$	5.62742i	0.240611i	−0.992737	+	0.120305i	$0.961613\pi$
$548$	0	0
$549$	21.6569i	0.924292i
$550$	0	0
$551$	12.6863	0.540454
$552$	0	0
$553$	29.3137	1.24655
$554$	0	0
$555$	14.6569i	0.622149i
$556$	0	0
$557$	5.14214i	0.217879i	0.994048	+	0.108940i	$0.0347455\pi$
$557$	5.14214i	0.217879i	−0.994048	+	0.108940i	$0.965254\pi$
$558$	0	0
$559$	3.00000	0.126886
$560$	0	0
$561$	−32.1421	−1.35704
$562$	0	0
$563$	− 38.5980i	− 1.62671i	−0.581767	−	0.813355i	$-0.697638\pi$
$563$	− 38.5980i	− 1.62671i	0.581767	−	0.813355i	$-0.302362\pi$
$564$	0	0
$565$	14.0000i	0.588984i
$566$	0	0
$567$	−3.82843	−0.160779
$568$	0	0
$569$	1.97056	0.0826103	0.0413051	−	0.999147i	$-0.486848\pi$
$569$	1.97056	0.0826103	0.0413051	+	0.999147i	$0.486848\pi$
$570$	0	0
$571$	22.6569i	0.948160i	0.880482	+	0.474080i	$0.157219\pi$
$571$	22.6569i	0.948160i	−0.880482	+	0.474080i	$0.842781\pi$
$572$	0	0
$573$	− 3.17157i	− 0.132494i
$574$	0	0
$575$	−30.6274	−1.27725
$576$	0	0
$577$	33.4558	1.39279	0.696393	−	0.717661i	$-0.254787\pi$
$577$	33.4558	1.39279	0.696393	+	0.717661i	$0.254787\pi$
$578$	0	0
$579$	6.00000i	0.249351i
$580$	0	0
$581$	− 22.9706i	− 0.952980i
$582$	0	0
$583$	−15.3137	−0.634229
$584$	0	0
$585$	−7.65685	−0.316572
$586$	0	0
$587$	25.1127i	1.03651i	0.855226	+	0.518256i	$0.173419\pi$
$587$	25.1127i	1.03651i	−0.855226	+	0.518256i	$0.826581\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−17.8284	−0.733363
$592$	0	0
$593$	−8.14214	−0.334357	−0.167179	−	0.985927i	$-0.553466\pi$
$593$	−8.14214	−0.334357	−0.167179	+	0.985927i	$0.553466\pi$
$594$	0	0
$595$	97.5685i	3.99992i
$596$	0	0
$597$	− 22.9706i	− 0.940123i
$598$	0	0
$599$	1.85786	0.0759103	0.0379551	−	0.999279i	$-0.487916\pi$
$599$	1.85786	0.0759103	0.0379551	+	0.999279i	$0.487916\pi$
$600$	0	0
$601$	−1.68629	−0.0687853	−0.0343926	−	0.999408i	$-0.510950\pi$
$601$	−1.68629	−0.0687853	−0.0343926	+	0.999408i	$0.510950\pi$
$602$	0	0
$603$	7.31371i	0.297837i
$604$	0	0
$605$	47.1421i	1.91660i
$606$	0	0
$607$	−10.8284	−0.439512	−0.219756	−	0.975555i	$-0.570526\pi$
$607$	−10.8284	−0.439512	−0.219756	+	0.975555i	$0.570526\pi$
$608$	0	0
$609$	−12.1421	−0.492024
$610$	0	0
$611$	11.4853i	0.464645i
$612$	0	0
$613$	42.0000i	1.69636i	0.529705	+	0.848182i	$0.322303\pi$
$613$	42.0000i	1.69636i	−0.529705	+	0.848182i	$0.677697\pi$
$614$	0	0
$615$	10.8284	0.436644
$616$	0	0
$617$	0.686292	0.0276291	0.0138145	−	0.999905i	$-0.495603\pi$
$617$	0.686292	0.0276291	0.0138145	+	0.999905i	$0.495603\pi$
$618$	0	0
$619$	− 18.1421i	− 0.729194i	−0.931165	−	0.364597i	$-0.881207\pi$
$619$	− 18.1421i	− 0.729194i	0.931165	−	0.364597i	$-0.118793\pi$
$620$	0	0
$621$	− 15.8579i	− 0.636354i
$622$	0	0
$623$	−67.5980	−2.70826
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	19.3137i	0.771315i
$628$	0	0
$629$	− 25.4853i	− 1.01616i
$630$	0	0
$631$	12.7990	0.509520	0.254760	−	0.967004i	$-0.418004\pi$
$631$	12.7990	0.509520	0.254760	+	0.967004i	$0.418004\pi$
$632$	0	0
$633$	−1.34315	−0.0533853
$634$	0	0
$635$	− 70.7696i	− 2.80840i
$636$	0	0
$637$	− 7.65685i	− 0.303376i
$638$	0	0
$639$	20.3431	0.804762
$640$	0	0
$641$	−22.9706	−0.907283	−0.453641	−	0.891184i	$-0.649875\pi$
$641$	−22.9706	−0.907283	−0.453641	+	0.891184i	$0.649875\pi$
$642$	0	0
$643$	− 48.6274i	− 1.91768i	−0.283950	−	0.958839i	$-0.591645\pi$
$643$	− 48.6274i	− 1.91768i	0.283950	−	0.958839i	$-0.408355\pi$
$644$	0	0
$645$	− 11.4853i	− 0.452233i
$646$	0	0
$647$	−17.1716	−0.675084	−0.337542	−	0.941310i	$-0.609596\pi$
$647$	−17.1716	−0.675084	−0.337542	+	0.941310i	$0.609596\pi$
$648$	0	0
$649$	24.9706	0.980180
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15.3137i	0.599272i	0.954054	+	0.299636i	$0.0968652\pi$
$653$	15.3137i	0.599272i	−0.954054	+	0.299636i	$0.903135\pi$
$654$	0	0
$655$	10.1716	0.397436
$656$	0	0
$657$	−10.3431	−0.403525
$658$	0	0
$659$	− 4.97056i	− 0.193626i	−0.995303	−	0.0968128i	$-0.969135\pi$
$659$	− 4.97056i	− 0.193626i	0.995303	−	0.0968128i	$-0.0308648\pi$
$660$	0	0
$661$	− 1.31371i	− 0.0510973i	−0.999674	−	0.0255487i	$-0.991867\pi$
$661$	− 1.31371i	− 0.0510973i	0.999674	−	0.0255487i	$-0.00813328\pi$
$662$	0	0
$663$	−6.65685	−0.258531
$664$	0	0
$665$	58.6274	2.27347
$666$	0	0
$667$	10.0589i	0.389481i
$668$	0	0
$669$	10.1716i	0.393256i
$670$	0	0
$671$	52.2843	2.01841
$672$	0	0
$673$	12.3137	0.474659	0.237329	−	0.971429i	$-0.423728\pi$
$673$	12.3137	0.474659	0.237329	+	0.971429i	$0.423728\pi$
$674$	0	0
$675$	48.2843i	1.85846i
$676$	0	0
$677$	29.3137i	1.12662i	0.826247	+	0.563309i	$0.190472\pi$
$677$	29.3137i	1.12662i	−0.826247	+	0.563309i	$0.809528\pi$
$678$	0	0
$679$	55.4558	2.12820
$680$	0	0
$681$	9.17157	0.351455
$682$	0	0
$683$	39.6569i	1.51743i	0.651424	+	0.758714i	$0.274172\pi$
$683$	39.6569i	1.51743i	−0.651424	+	0.758714i	$0.725828\pi$
$684$	0	0
$685$	− 77.1127i	− 2.94632i
$686$	0	0
$687$	10.1716	0.388070
$688$	0	0
$689$	−3.17157	−0.120827
$690$	0	0
$691$	− 7.02944i	− 0.267412i	−0.991021	−	0.133706i	$-0.957312\pi$
$691$	− 7.02944i	− 0.267412i	0.991021	−	0.133706i	$-0.0426878\pi$
$692$	0	0
$693$	36.9706i	1.40440i
$694$	0	0
$695$	−71.4264	−2.70936
$696$	0	0
$697$	−18.8284	−0.713178
$698$	0	0
$699$	20.3137i	0.768335i
$700$	0	0
$701$	− 12.1421i	− 0.458602i	−0.973356	−	0.229301i	$-0.926356\pi$
$701$	− 12.1421i	− 0.458602i	0.973356	−	0.229301i	$-0.0736440\pi$
$702$	0	0
$703$	−15.3137	−0.577567
$704$	0	0
$705$	43.9706	1.65603
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 7.65685i	− 0.287559i	−0.989610	−	0.143780i	$-0.954074\pi$
$709$	− 7.65685i	− 0.287559i	0.989610	−	0.143780i	$-0.0459256\pi$
$710$	0	0
$711$	−15.3137	−0.574309
$712$	0	0
$713$	0	0
$714$	0	0
$715$	18.4853i	0.691310i
$716$	0	0
$717$	− 19.1421i	− 0.714876i
$718$	0	0
$719$	−22.9706	−0.856657	−0.428329	−	0.903623i	$-0.640897\pi$
$719$	−22.9706	−0.856657	−0.428329	+	0.903623i	$0.640897\pi$
$720$	0	0
$721$	−53.5980	−1.99609
$722$	0	0
$723$	− 3.31371i	− 0.123238i
$724$	0	0
$725$	− 30.6274i	− 1.13747i
$726$	0	0
$727$	18.4853	0.685581	0.342791	−	0.939412i	$-0.388628\pi$
$727$	18.4853	0.685581	0.342791	+	0.939412i	$0.388628\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	19.9706i	0.738638i
$732$	0	0
$733$	35.7696i	1.32118i	0.750747	+	0.660589i	$0.229694\pi$
$733$	35.7696i	1.32118i	−0.750747	+	0.660589i	$0.770306\pi$
$734$	0	0
$735$	−29.3137	−1.08125
$736$	0	0
$737$	17.6569	0.650399
$738$	0	0
$739$	31.5147i	1.15929i	0.814870	+	0.579644i	$0.196808\pi$
$739$	31.5147i	1.15929i	−0.814870	+	0.579644i	$0.803192\pi$
$740$	0	0
$741$	4.00000i	0.146944i
$742$	0	0
$743$	−40.7990	−1.49677	−0.748385	−	0.663265i	$-0.769170\pi$
$743$	−40.7990	−1.49677	−0.748385	+	0.663265i	$0.769170\pi$
$744$	0	0
$745$	5.02944	0.184264
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	− 58.6274i	− 2.14220i
$750$	0	0
$751$	−23.5147	−0.858064	−0.429032	−	0.903289i	$-0.641145\pi$
$751$	−23.5147	−0.858064	−0.429032	+	0.903289i	$0.641145\pi$
$752$	0	0
$753$	−26.6274	−0.970357
$754$	0	0
$755$	9.62742i	0.350378i
$756$	0	0
$757$	− 31.1716i	− 1.13295i	−0.824079	−	0.566475i	$-0.808307\pi$
$757$	− 31.1716i	− 1.13295i	0.824079	−	0.566475i	$-0.191693\pi$
$758$	0	0
$759$	−15.3137	−0.555852
$760$	0	0
$761$	33.1716	1.20247	0.601234	−	0.799073i	$-0.294676\pi$
$761$	33.1716	1.20247	0.601234	+	0.799073i	$0.294676\pi$
$762$	0	0
$763$	− 14.6569i	− 0.530614i
$764$	0	0
$765$	− 50.9706i	− 1.84284i
$766$	0	0
$767$	5.17157	0.186735
$768$	0	0
$769$	−13.6569	−0.492479	−0.246239	−	0.969209i	$-0.579195\pi$
$769$	−13.6569	−0.492479	−0.246239	+	0.969209i	$0.579195\pi$
$770$	0	0
$771$	1.34315i	0.0483722i
$772$	0	0
$773$	26.7990i	0.963893i	0.876201	+	0.481946i	$0.160070\pi$
$773$	26.7990i	0.963893i	−0.876201	+	0.481946i	$0.839930\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	14.6569	0.525812
$778$	0	0
$779$	11.3137i	0.405356i
$780$	0	0
$781$	− 49.1127i	− 1.75739i
$782$	0	0
$783$	15.8579	0.566714
$784$	0	0
$785$	−17.1716	−0.612880
$786$	0	0
$787$	− 11.5147i	− 0.410455i	−0.978714	−	0.205228i	$-0.934207\pi$
$787$	− 11.5147i	− 0.410455i	0.978714	−	0.205228i	$-0.0657935\pi$
$788$	0	0
$789$	26.1421i	0.930685i
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	10.8284	0.384529
$794$	0	0
$795$	12.1421i	0.430637i
$796$	0	0
$797$	34.3431i	1.21650i	0.793747	+	0.608248i	$0.208128\pi$
$797$	34.3431i	1.21650i	−0.793747	+	0.608248i	$0.791872\pi$
$798$	0	0
$799$	−76.4558	−2.70481
$800$	0	0
$801$	35.3137	1.24775
$802$	0	0
$803$	24.9706i	0.881192i
$804$	0	0
$805$	46.4853i	1.63839i
$806$	0	0
$807$	−19.7990	−0.696957
$808$	0	0
$809$	−27.0000	−0.949269	−0.474635	−	0.880183i	$-0.657420\pi$
$809$	−27.0000	−0.949269	−0.474635	+	0.880183i	$0.657420\pi$
$810$	0	0
$811$	− 28.9706i	− 1.01729i	−0.860975	−	0.508647i	$-0.830146\pi$
$811$	− 28.9706i	− 1.01729i	0.860975	−	0.508647i	$-0.169854\pi$
$812$	0	0
$813$	2.51472i	0.0881950i
$814$	0	0
$815$	−49.1127	−1.72034
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	7.65685i	0.267552i
$820$	0	0
$821$	11.4853i	0.400839i	0.979710	+	0.200420i	$0.0642305\pi$
$821$	11.4853i	0.400839i	−0.979710	+	0.200420i	$0.935769\pi$
$822$	0	0
$823$	−28.7696	−1.00284	−0.501422	−	0.865203i	$-0.667189\pi$
$823$	−28.7696	−1.00284	−0.501422	+	0.865203i	$0.667189\pi$
$824$	0	0
$825$	46.6274	1.62336
$826$	0	0
$827$	− 30.6274i	− 1.06502i	−0.846424	−	0.532510i	$-0.821249\pi$
$827$	− 30.6274i	− 1.06502i	0.846424	−	0.532510i	$-0.178751\pi$
$828$	0	0
$829$	− 22.9706i	− 0.797801i	−0.916994	−	0.398900i	$-0.869392\pi$
$829$	− 22.9706i	− 0.797801i	0.916994	−	0.398900i	$-0.130608\pi$
$830$	0	0
$831$	4.48528	0.155593
$832$	0	0
$833$	50.9706	1.76603
$834$	0	0
$835$	82.9117i	2.86928i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12.6863	0.437979	0.218990	−	0.975727i	$-0.429724\pi$
$839$	12.6863	0.437979	0.218990	+	0.975727i	$0.429724\pi$
$840$	0	0
$841$	18.9411	0.653142
$842$	0	0
$843$	− 4.97056i	− 0.171195i
$844$	0	0
$845$	3.82843i	0.131702i
$846$	0	0
$847$	47.1421	1.61982
$848$	0	0
$849$	14.6274	0.502012
$850$	0	0
$851$	− 12.1421i	− 0.416227i
$852$	0	0
$853$	− 35.7696i	− 1.22473i	−0.790577	−	0.612363i	$-0.790219\pi$
$853$	− 35.7696i	− 1.22473i	0.790577	−	0.612363i	$-0.209781\pi$
$854$	0	0
$855$	−30.6274	−1.04744
$856$	0	0
$857$	−41.3137	−1.41125	−0.705625	−	0.708586i	$-0.749334\pi$
$857$	−41.3137	−1.41125	−0.705625	+	0.708586i	$0.749334\pi$
$858$	0	0
$859$	4.97056i	0.169593i	0.996398	+	0.0847967i	$0.0270241\pi$
$859$	4.97056i	0.169593i	−0.996398	+	0.0847967i	$0.972976\pi$
$860$	0	0
$861$	− 10.8284i	− 0.369032i
$862$	0	0
$863$	−14.1127	−0.480402	−0.240201	−	0.970723i	$-0.577213\pi$
$863$	−14.1127	−0.480402	−0.240201	+	0.970723i	$0.577213\pi$
$864$	0	0
$865$	−87.9411	−2.99009
$866$	0	0
$867$	− 27.3137i	− 0.927622i
$868$	0	0
$869$	36.9706i	1.25414i
$870$	0	0
$871$	3.65685	0.123908
$872$	0	0
$873$	−28.9706	−0.980505
$874$	0	0
$875$	− 68.2548i	− 2.30743i
$876$	0	0
$877$	− 16.5147i	− 0.557662i	−0.960340	−	0.278831i	$-0.910053\pi$
$877$	− 16.5147i	− 0.557662i	0.960340	−	0.278831i	$-0.0899470\pi$
$878$	0	0
$879$	−12.7990	−0.431699
$880$	0	0
$881$	40.3137	1.35820	0.679102	−	0.734044i	$-0.262370\pi$
$881$	40.3137	1.35820	0.679102	+	0.734044i	$0.262370\pi$
$882$	0	0
$883$	− 1.97056i	− 0.0663147i	−0.999450	−	0.0331574i	$-0.989444\pi$
$883$	− 1.97056i	− 0.0663147i	0.999450	−	0.0331574i	$-0.0105562\pi$
$884$	0	0
$885$	− 19.7990i	− 0.665536i
$886$	0	0
$887$	26.1421	0.877767	0.438884	−	0.898544i	$-0.355374\pi$
$887$	26.1421	0.877767	0.438884	+	0.898544i	$0.355374\pi$
$888$	0	0
$889$	−70.7696	−2.37353
$890$	0	0
$891$	− 4.82843i	− 0.161758i
$892$	0	0
$893$	45.9411i	1.53736i
$894$	0	0
$895$	31.8284	1.06391
$896$	0	0
$897$	−3.17157	−0.105896
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 21.1127i	− 0.703366i
$902$	0	0
$903$	−11.4853	−0.382206
$904$	0	0
$905$	−87.9411	−2.92326
$906$	0	0
$907$	24.9411i	0.828156i	0.910241	+	0.414078i	$0.135896\pi$
$907$	24.9411i	0.828156i	−0.910241	+	0.414078i	$0.864104\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	55.4558	1.83733	0.918667	−	0.395033i	$-0.129267\pi$
$911$	55.4558	1.83733	0.918667	+	0.395033i	$0.129267\pi$
$912$	0	0
$913$	28.9706	0.958786
$914$	0	0
$915$	− 41.4558i	− 1.37049i
$916$	0	0
$917$	− 10.1716i	− 0.335895i
$918$	0	0
$919$	−15.3137	−0.505153	−0.252576	−	0.967577i	$-0.581278\pi$
$919$	−15.3137	−0.505153	−0.252576	+	0.967577i	$0.581278\pi$
$920$	0	0
$921$	13.1716	0.434018
$922$	0	0
$923$	− 10.1716i	− 0.334801i
$924$	0	0
$925$	36.9706i	1.21558i
$926$	0	0
$927$	28.0000	0.919641
$928$	0	0
$929$	−28.6863	−0.941167	−0.470583	−	0.882356i	$-0.655956\pi$
$929$	−28.6863	−0.941167	−0.470583	+	0.882356i	$0.655956\pi$
$930$	0	0
$931$	− 30.6274i	− 1.00377i
$932$	0	0
$933$	− 24.2843i	− 0.795031i
$934$	0	0
$935$	−123.054	−4.02429
$936$	0	0
$937$	15.6569	0.511487	0.255744	−	0.966745i	$-0.417680\pi$
$937$	15.6569	0.511487	0.255744	+	0.966745i	$0.417680\pi$
$938$	0	0
$939$	19.9706i	0.651715i
$940$	0	0
$941$	− 40.7990i	− 1.33001i	−0.746839	−	0.665005i	$-0.768430\pi$
$941$	− 40.7990i	− 1.33001i	0.746839	−	0.665005i	$-0.231570\pi$
$942$	0	0
$943$	−8.97056	−0.292122
$944$	0	0
$945$	73.2843	2.38394
$946$	0	0
$947$	− 37.1127i	− 1.20600i	−0.797741	−	0.603000i	$-0.793972\pi$
$947$	− 37.1127i	− 1.20600i	0.797741	−	0.603000i	$-0.206028\pi$
$948$	0	0
$949$	5.17157i	0.167876i
$950$	0	0
$951$	29.3137	0.950562
$952$	0	0
$953$	5.68629	0.184197	0.0920985	−	0.995750i	$-0.470643\pi$
$953$	5.68629	0.184197	0.0920985	+	0.995750i	$0.470643\pi$
$954$	0	0
$955$	− 12.1421i	− 0.392910i
$956$	0	0
$957$	− 15.3137i	− 0.495022i
$958$	0	0
$959$	−77.1127	−2.49010
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	30.6274i	0.986955i
$964$	0	0
$965$	22.9706i	0.739449i
$966$	0	0
$967$	−15.2010	−0.488832	−0.244416	−	0.969671i	$-0.578596\pi$
$967$	−15.2010	−0.488832	−0.244416	+	0.969671i	$0.578596\pi$
$968$	0	0
$969$	−26.6274	−0.855396
$970$	0	0
$971$	36.5980i	1.17449i	0.809411	+	0.587243i	$0.199787\pi$
$971$	36.5980i	1.17449i	−0.809411	+	0.587243i	$0.800213\pi$
$972$	0	0
$973$	71.4264i	2.28983i
$974$	0	0
$975$	9.65685	0.309267
$976$	0	0
$977$	2.28427	0.0730803	0.0365402	−	0.999332i	$-0.488366\pi$
$977$	2.28427	0.0730803	0.0365402	+	0.999332i	$0.488366\pi$
$978$	0	0
$979$	− 85.2548i	− 2.72476i
$980$	0	0
$981$	7.65685i	0.244465i
$982$	0	0
$983$	−21.7696	−0.694341	−0.347170	−	0.937802i	$-0.612857\pi$
$983$	−21.7696	−0.694341	−0.347170	+	0.937802i	$0.612857\pi$
$984$	0	0
$985$	−68.2548	−2.17478
$986$	0	0
$987$	− 43.9706i	− 1.39960i
$988$	0	0
$989$	9.51472i	0.302550i
$990$	0	0
$991$	19.0294	0.604490	0.302245	−	0.953230i	$-0.402264\pi$
$991$	19.0294	0.604490	0.302245	+	0.953230i	$0.402264\pi$
$992$	0	0
$993$	−10.9706	−0.348140
$994$	0	0
$995$	− 87.9411i	− 2.78792i
$996$	0	0
$997$	− 27.4558i	− 0.869535i	−0.900543	−	0.434768i	$-0.856830\pi$
$997$	− 27.4558i	− 0.869535i	0.900543	−	0.434768i	$-0.143170\pi$
$998$	0	0
$999$	−19.1421	−0.605630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1664.2.b.e.833.1	✓	4
4.3	odd	2		1664.2.b.i.833.3	yes	4
8.3	odd	2		1664.2.b.i.833.2	yes	4
8.5	even	2	inner	1664.2.b.e.833.4	yes	4
16.3	odd	4		3328.2.a.o.1.1		2
16.5	even	4		3328.2.a.r.1.2		2
16.11	odd	4		3328.2.a.ba.1.2		2
16.13	even	4		3328.2.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.e.833.1	✓	4	1.1	even	1	trivial
1664.2.b.e.833.4	yes	4	8.5	even	2	inner
1664.2.b.i.833.2	yes	4	8.3	odd	2
1664.2.b.i.833.3	yes	4	4.3	odd	2
3328.2.a.o.1.1		2	16.3	odd	4
3328.2.a.r.1.2		2	16.5	even	4
3328.2.a.z.1.1		2	16.13	even	4
3328.2.a.ba.1.2		2	16.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1664 = 2^{7} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1664.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -3.82843i q^{5} -3.82843 q^{7} +2.00000 q^{9} -4.82843i q^{11} -1.00000i q^{13} -3.82843 q^{15} +6.65685 q^{17} -4.00000i q^{19} +3.82843i q^{21} +3.17157 q^{23} -9.65685 q^{25} -5.00000i q^{27} +3.17157i q^{29} -4.82843 q^{33} +14.6569i q^{35} -3.82843i q^{37} -1.00000 q^{39} -2.82843 q^{41} +3.00000i q^{43} -7.65685i q^{45} -11.4853 q^{47} +7.65685 q^{49} -6.65685i q^{51} -3.17157i q^{53} -18.4853 q^{55} -4.00000 q^{57} +5.17157i q^{59} +10.8284i q^{61} -7.65685 q^{63} -3.82843 q^{65} +3.65685i q^{67} -3.17157i q^{69} +10.1716 q^{71} -5.17157 q^{73} +9.65685i q^{75} +18.4853i q^{77} -7.65685 q^{79} +1.00000 q^{81} +6.00000i q^{83} -25.4853i q^{85} +3.17157 q^{87} +17.6569 q^{89} +3.82843i q^{91} -15.3137 q^{95} -14.4853 q^{97} -9.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} + 8 q^{9} - 4 q^{15} + 4 q^{17} + 24 q^{23} - 16 q^{25} - 8 q^{33} - 4 q^{39} - 12 q^{47} + 8 q^{49} - 40 q^{55} - 16 q^{57} - 8 q^{63} - 4 q^{65} + 52 q^{71} - 32 q^{73} - 8 q^{79} + 4 q^{81}+ \cdots - 24 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 + 8 * q^9 - 4 * q^15 + 4 * q^17 + 24 * q^23 - 16 * q^25 - 8 * q^33 - 4 * q^39 - 12 * q^47 + 8 * q^49 - 40 * q^55 - 16 * q^57 - 8 * q^63 - 4 * q^65 + 52 * q^71 - 32 * q^73 - 8 * q^79 + 4 * q^81 + 24 * q^87 + 48 * q^89 - 16 * q^95 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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