LMFDB - Embedded newform 1568.3.c.e.97.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1568,3,Mod(97,1568)] 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("1568.97"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-36,0,-48,0,0,0,0,0,0,0,0,0,0,0,128,0,-124, 0,0,0,0,0,0,0,0,0,0,0,-128,0,-160,0,0,0,464,0,0,0,0,0,0,0,400,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(53)] == traces)

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

Self dual:	no
Analytic conductor:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 64x^{10} + 1436x^{8} + 13392x^{6} + 45220x^{4} + 30880x^{2} + 1568 $ x^12 + 64x^10 + 1436x^8 + 13392x^6 + 45220x^4 + 30880*x^2 + 1568
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.12
Root			$3.86019i$ of defining polynomial
Character	$\chi$	$=$	1568.97
Dual form			1568.3.c.e.97.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.94258i q^{3} -0.228417i q^{5} -15.4291 q^{9} -18.3758 q^{11} -13.4283i q^{13} +1.12897 q^{15} -16.5172i q^{17} -12.8838i q^{19} -8.58814 q^{23} +24.9478 q^{25} -31.7765i q^{27} +56.5368 q^{29} +35.5478i q^{31} -90.8237i q^{33} -37.3980 q^{37} +66.3705 q^{39} +30.1061i q^{41} +59.4089 q^{43} +3.52427i q^{45} +69.9424i q^{47} +81.6374 q^{51} +68.6698 q^{53} +4.19733i q^{55} +63.6795 q^{57} -57.9361i q^{59} -88.9801i q^{61} -3.06724 q^{65} +0.939118 q^{67} -42.4476i q^{69} +51.3474 q^{71} -6.64003i q^{73} +123.307i q^{75} -88.4092 q^{79} +18.1960 q^{81} -59.3984i q^{83} -3.77279 q^{85} +279.438i q^{87} +59.5411i q^{89} -175.698 q^{93} -2.94288 q^{95} -49.6639i q^{97} +283.522 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 36 q^{9} - 48 q^{11} + 128 q^{23} - 124 q^{25} - 128 q^{37} - 160 q^{39} + 464 q^{43} + 400 q^{51} + 16 q^{53} + 224 q^{57} - 64 q^{65} - 64 q^{67} + 544 q^{71} + 288 q^{79} - 180 q^{81} - 272 q^{85}+ \cdots + 1328 q^{99}+O(q^{100}) $ 12 * q - 36 * q^9 - 48 * q^11 + 128 * q^23 - 124 * q^25 - 128 * q^37 - 160 * q^39 + 464 * q^43 + 400 * q^51 + 16 * q^53 + 224 * q^57 - 64 * q^65 - 64 * q^67 + 544 * q^71 + 288 * q^79 - 180 * q^81 - 272 * q^85 - 352 * q^93 - 576 * q^95 + 1328 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	4.94258i	1.64753i	0.566933	+	0.823764i	$0.308130\pi$
$3$	4.94258i	1.64753i	−0.566933	+	0.823764i	$0.691870\pi$
$4$	0	0
$5$	− 0.228417i	− 0.0456833i	−0.999739	−	0.0228417i	$-0.992729\pi$
$5$	− 0.228417i	− 0.0456833i	0.999739	−	0.0228417i	$-0.00727136\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−15.4291	−1.71435
$10$	0	0
$11$	−18.3758	−1.67052	−0.835262	−	0.549853i	$-0.814684\pi$
$11$	−18.3758	−1.67052	−0.835262	+	0.549853i	$0.814684\pi$
$12$	0	0
$13$	− 13.4283i	− 1.03295i	−0.856304	−	0.516473i	$-0.827245\pi$
$13$	− 13.4283i	− 1.03295i	0.856304	−	0.516473i	$-0.172755\pi$
$14$	0	0
$15$	1.12897	0.0752645
$16$	0	0
$17$	− 16.5172i	− 0.971597i	−0.874071	−	0.485799i	$-0.838529\pi$
$17$	− 16.5172i	− 0.971597i	0.874071	−	0.485799i	$-0.161471\pi$
$18$	0	0
$19$	− 12.8838i	− 0.678097i	−0.940769	−	0.339048i	$-0.889895\pi$
$19$	− 12.8838i	− 0.678097i	0.940769	−	0.339048i	$-0.110105\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.58814	−0.373397	−0.186699	−	0.982417i	$-0.559779\pi$
$23$	−8.58814	−0.373397	−0.186699	+	0.982417i	$0.559779\pi$
$24$	0	0
$25$	24.9478	0.997913
$26$	0	0
$27$	− 31.7765i	− 1.17691i
$28$	0	0
$29$	56.5368	1.94955	0.974773	−	0.223199i	$-0.0716500\pi$
$29$	56.5368	1.94955	0.974773	+	0.223199i	$0.0716500\pi$
$30$	0	0
$31$	35.5478i	1.14670i	0.819310	+	0.573351i	$0.194357\pi$
$31$	35.5478i	1.14670i	−0.819310	+	0.573351i	$0.805643\pi$
$32$	0	0
$33$	− 90.8237i	− 2.75223i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−37.3980	−1.01076	−0.505378	−	0.862898i	$-0.668647\pi$
$37$	−37.3980	−1.01076	−0.505378	+	0.862898i	$0.668647\pi$
$38$	0	0
$39$	66.3705	1.70181
$40$	0	0
$41$	30.1061i	0.734296i	0.930163	+	0.367148i	$0.119666\pi$
$41$	30.1061i	0.734296i	−0.930163	+	0.367148i	$0.880334\pi$
$42$	0	0
$43$	59.4089	1.38160	0.690801	−	0.723045i	$-0.257258\pi$
$43$	59.4089	1.38160	0.690801	+	0.723045i	$0.257258\pi$
$44$	0	0
$45$	3.52427i	0.0783171i
$46$	0	0
$47$	69.9424i	1.48814i	0.668103	+	0.744069i	$0.267106\pi$
$47$	69.9424i	1.48814i	−0.668103	+	0.744069i	$0.732894\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	81.6374	1.60073
$52$	0	0
$53$	68.6698	1.29566	0.647828	−	0.761786i	$-0.275677\pi$
$53$	68.6698	1.29566	0.647828	+	0.761786i	$0.275677\pi$
$54$	0	0
$55$	4.19733i	0.0763150i
$56$	0	0
$57$	63.6795	1.11718
$58$	0	0
$59$	− 57.9361i	− 0.981967i	−0.871169	−	0.490984i	$-0.836638\pi$
$59$	− 57.9361i	− 0.981967i	0.871169	−	0.490984i	$-0.163362\pi$
$60$	0	0
$61$	− 88.9801i	− 1.45869i	−0.684146	−	0.729345i	$-0.739825\pi$
$61$	− 88.9801i	− 1.45869i	0.684146	−	0.729345i	$-0.260175\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.06724	−0.0471884
$66$	0	0
$67$	0.939118	0.0140167	0.00700834	−	0.999975i	$-0.497769\pi$
$67$	0.939118	0.0140167	0.00700834	+	0.999975i	$0.497769\pi$
$68$	0	0
$69$	− 42.4476i	− 0.615183i
$70$	0	0
$71$	51.3474	0.723203	0.361602	−	0.932333i	$-0.382230\pi$
$71$	51.3474	0.723203	0.361602	+	0.932333i	$0.382230\pi$
$72$	0	0
$73$	− 6.64003i	− 0.0909594i	−0.998965	−	0.0454797i	$-0.985518\pi$
$73$	− 6.64003i	− 0.0909594i	0.998965	−	0.0454797i	$-0.0144816\pi$
$74$	0	0
$75$	123.307i	1.64409i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−88.4092	−1.11910	−0.559552	−	0.828795i	$-0.689027\pi$
$79$	−88.4092	−1.11910	−0.559552	+	0.828795i	$0.689027\pi$
$80$	0	0
$81$	18.1960	0.224642
$82$	0	0
$83$	− 59.3984i	− 0.715644i	−0.933790	−	0.357822i	$-0.883520\pi$
$83$	− 59.3984i	− 0.715644i	0.933790	−	0.357822i	$-0.116480\pi$
$84$	0	0
$85$	−3.77279	−0.0443858
$86$	0	0
$87$	279.438i	3.21193i
$88$	0	0
$89$	59.5411i	0.669001i	0.942396	+	0.334500i	$0.108568\pi$
$89$	59.5411i	0.669001i	−0.942396	+	0.334500i	$0.891432\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−175.698	−1.88923
$94$	0	0
$95$	−2.94288	−0.0309777
$96$	0	0
$97$	− 49.6639i	− 0.511999i	−0.966677	−	0.256000i	$-0.917595\pi$
$97$	− 49.6639i	− 0.511999i	0.966677	−	0.256000i	$-0.0824046\pi$
$98$	0	0
$99$	283.522	2.86386
$100$	0	0
$101$	− 167.262i	− 1.65606i	−0.560683	−	0.828031i	$-0.689461\pi$
$101$	− 167.262i	− 1.65606i	0.560683	−	0.828031i	$-0.310539\pi$
$102$	0	0
$103$	− 67.2970i	− 0.653369i	−0.945133	−	0.326684i	$-0.894069\pi$
$103$	− 67.2970i	− 0.653369i	0.945133	−	0.326684i	$-0.105931\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	154.078	1.43998	0.719991	−	0.693984i	$-0.244146\pi$
$107$	154.078	1.43998	0.719991	+	0.693984i	$0.244146\pi$
$108$	0	0
$109$	−181.141	−1.66185	−0.830923	−	0.556387i	$-0.812187\pi$
$109$	−181.141	−1.66185	−0.830923	+	0.556387i	$0.812187\pi$
$110$	0	0
$111$	− 184.843i	− 1.66525i
$112$	0	0
$113$	−20.2896	−0.179554	−0.0897768	−	0.995962i	$-0.528615\pi$
$113$	−20.2896	−0.179554	−0.0897768	+	0.995962i	$0.528615\pi$
$114$	0	0
$115$	1.96167i	0.0170580i
$116$	0	0
$117$	207.187i	1.77083i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	216.668	1.79065
$122$	0	0
$123$	−148.802	−1.20977
$124$	0	0
$125$	− 11.4089i	− 0.0912713i
$126$	0	0
$127$	−38.7432	−0.305064	−0.152532	−	0.988299i	$-0.548743\pi$
$127$	−38.7432	−0.305064	−0.152532	+	0.988299i	$0.548743\pi$
$128$	0	0
$129$	293.633i	2.27623i
$130$	0	0
$131$	− 108.725i	− 0.829961i	−0.909830	−	0.414980i	$-0.863788\pi$
$131$	− 108.725i	− 0.829961i	0.909830	−	0.414980i	$-0.136212\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−7.25829	−0.0537651
$136$	0	0
$137$	38.1937	0.278786	0.139393	−	0.990237i	$-0.455485\pi$
$137$	38.1937	0.278786	0.139393	+	0.990237i	$0.455485\pi$
$138$	0	0
$139$	− 252.096i	− 1.81364i	−0.421517	−	0.906820i	$-0.638502\pi$
$139$	− 252.096i	− 1.81364i	0.421517	−	0.906820i	$-0.361498\pi$
$140$	0	0
$141$	−345.696	−2.45175
$142$	0	0
$143$	246.755i	1.72556i
$144$	0	0
$145$	− 12.9139i	− 0.0890617i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	181.778	1.21998	0.609992	−	0.792408i	$-0.291173\pi$
$149$	181.778	1.21998	0.609992	+	0.792408i	$0.291173\pi$
$150$	0	0
$151$	156.791	1.03835	0.519176	−	0.854667i	$-0.326239\pi$
$151$	156.791	1.03835	0.519176	+	0.854667i	$0.326239\pi$
$152$	0	0
$153$	254.845i	1.66566i
$154$	0	0
$155$	8.11970	0.0523852
$156$	0	0
$157$	14.6817i	0.0935139i	0.998906	+	0.0467569i	$0.0148886\pi$
$157$	14.6817i	0.0935139i	−0.998906	+	0.0467569i	$0.985111\pi$
$158$	0	0
$159$	339.406i	2.13463i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	26.3021	0.161363	0.0806814	−	0.996740i	$-0.474290\pi$
$163$	26.3021	0.161363	0.0806814	+	0.996740i	$0.474290\pi$
$164$	0	0
$165$	−20.7456	−0.125731
$166$	0	0
$167$	12.0409i	0.0721013i	0.999350	+	0.0360507i	$0.0114778\pi$
$167$	12.0409i	0.0721013i	−0.999350	+	0.0360507i	$0.988522\pi$
$168$	0	0
$169$	−11.3190	−0.0669765
$170$	0	0
$171$	198.787i	1.16249i
$172$	0	0
$173$	284.151i	1.64249i	0.570575	+	0.821245i	$0.306720\pi$
$173$	284.151i	1.64249i	−0.570575	+	0.821245i	$0.693280\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	286.354	1.61782
$178$	0	0
$179$	113.645	0.634886	0.317443	−	0.948277i	$-0.397176\pi$
$179$	113.645	0.634886	0.317443	+	0.948277i	$0.397176\pi$
$180$	0	0
$181$	− 59.3265i	− 0.327771i	−0.986479	−	0.163885i	$-0.947597\pi$
$181$	− 59.3265i	− 0.327771i	0.986479	−	0.163885i	$-0.0524027\pi$
$182$	0	0
$183$	439.791	2.40323
$184$	0	0
$185$	8.54231i	0.0461746i
$186$	0	0
$187$	303.515i	1.62308i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	177.823	0.931011	0.465505	−	0.885045i	$-0.345873\pi$
$191$	177.823	0.931011	0.465505	+	0.885045i	$0.345873\pi$
$192$	0	0
$193$	−222.636	−1.15356	−0.576778	−	0.816901i	$-0.695690\pi$
$193$	−222.636	−1.15356	−0.576778	+	0.816901i	$0.695690\pi$
$194$	0	0
$195$	− 15.1601i	− 0.0777442i
$196$	0	0
$197$	33.4090	0.169589	0.0847945	−	0.996398i	$-0.472977\pi$
$197$	33.4090	0.169589	0.0847945	+	0.996398i	$0.472977\pi$
$198$	0	0
$199$	− 280.456i	− 1.40933i	−0.709542	−	0.704664i	$-0.751098\pi$
$199$	− 280.456i	− 1.40933i	0.709542	−	0.704664i	$-0.248902\pi$
$200$	0	0
$201$	4.64167i	0.0230929i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	6.87674	0.0335451
$206$	0	0
$207$	132.508	0.640133
$208$	0	0
$209$	236.750i	1.13278i
$210$	0	0
$211$	243.027	1.15179	0.575894	−	0.817524i	$-0.304654\pi$
$211$	243.027	1.15179	0.575894	+	0.817524i	$0.304654\pi$
$212$	0	0
$213$	253.789i	1.19150i
$214$	0	0
$215$	− 13.5700i	− 0.0631161i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	32.8189	0.149858
$220$	0	0
$221$	−221.797	−1.00361
$222$	0	0
$223$	172.863i	0.775169i	0.921834	+	0.387585i	$0.126691\pi$
$223$	172.863i	0.775169i	−0.921834	+	0.387585i	$0.873309\pi$
$224$	0	0
$225$	−384.923	−1.71077
$226$	0	0
$227$	113.711i	0.500931i	0.968126	+	0.250465i	$0.0805837\pi$
$227$	113.711i	0.500931i	−0.968126	+	0.250465i	$0.919416\pi$
$228$	0	0
$229$	361.301i	1.57773i	0.614565	+	0.788866i	$0.289332\pi$
$229$	361.301i	1.57773i	−0.614565	+	0.788866i	$0.710668\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	191.895	0.823585	0.411792	−	0.911278i	$-0.364903\pi$
$233$	191.895	0.823585	0.411792	+	0.911278i	$0.364903\pi$
$234$	0	0
$235$	15.9760	0.0679830
$236$	0	0
$237$	− 436.970i	− 1.84375i
$238$	0	0
$239$	16.3527	0.0684212	0.0342106	−	0.999415i	$-0.489108\pi$
$239$	16.3527	0.0684212	0.0342106	+	0.999415i	$0.489108\pi$
$240$	0	0
$241$	− 308.408i	− 1.27970i	−0.768500	−	0.639850i	$-0.778996\pi$
$241$	− 308.408i	− 1.27970i	0.768500	−	0.639850i	$-0.221004\pi$
$242$	0	0
$243$	− 196.054i	− 0.806805i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−173.008	−0.700437
$248$	0	0
$249$	293.582	1.17904
$250$	0	0
$251$	159.093i	0.633838i	0.948453	+	0.316919i	$0.102648\pi$
$251$	159.093i	0.633838i	−0.948453	+	0.316919i	$0.897352\pi$
$252$	0	0
$253$	157.814	0.623769
$254$	0	0
$255$	− 18.6473i	− 0.0731268i
$256$	0	0
$257$	− 261.398i	− 1.01711i	−0.861029	−	0.508556i	$-0.830180\pi$
$257$	− 261.398i	− 1.01711i	0.861029	−	0.508556i	$-0.169820\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−872.314	−3.34220
$262$	0	0
$263$	−104.087	−0.395769	−0.197885	−	0.980225i	$-0.563407\pi$
$263$	−104.087	−0.395769	−0.197885	+	0.980225i	$0.563407\pi$
$264$	0	0
$265$	− 15.6853i	− 0.0591899i
$266$	0	0
$267$	−294.287	−1.10220
$268$	0	0
$269$	− 418.217i	− 1.55471i	−0.629062	−	0.777355i	$-0.716561\pi$
$269$	− 418.217i	− 1.55471i	0.629062	−	0.777355i	$-0.283439\pi$
$270$	0	0
$271$	125.684i	0.463779i	0.972742	+	0.231889i	$0.0744907\pi$
$271$	125.684i	0.463779i	−0.972742	+	0.231889i	$0.925509\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−458.435	−1.66704
$276$	0	0
$277$	58.3242	0.210557	0.105278	−	0.994443i	$-0.466427\pi$
$277$	58.3242	0.210557	0.105278	+	0.994443i	$0.466427\pi$
$278$	0	0
$279$	− 548.472i	− 1.96585i
$280$	0	0
$281$	191.806	0.682585	0.341292	−	0.939957i	$-0.389135\pi$
$281$	191.806	0.682585	0.341292	+	0.939957i	$0.389135\pi$
$282$	0	0
$283$	413.545i	1.46129i	0.682758	+	0.730644i	$0.260780\pi$
$283$	413.545i	1.46129i	−0.682758	+	0.730644i	$0.739220\pi$
$284$	0	0
$285$	− 14.5454i	− 0.0510366i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.1836	0.0559986
$290$	0	0
$291$	245.468	0.843533
$292$	0	0
$293$	− 411.566i	− 1.40466i	−0.711850	−	0.702331i	$-0.752142\pi$
$293$	− 411.566i	− 1.40466i	0.711850	−	0.702331i	$-0.247858\pi$
$294$	0	0
$295$	−13.2336	−0.0448595
$296$	0	0
$297$	583.918i	1.96605i
$298$	0	0
$299$	115.324i	0.385699i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	826.708	2.72841
$304$	0	0
$305$	−20.3245	−0.0666378
$306$	0	0
$307$	− 288.340i	− 0.939217i	−0.882875	−	0.469609i	$-0.844395\pi$
$307$	− 288.340i	− 0.939217i	0.882875	−	0.469609i	$-0.155605\pi$
$308$	0	0
$309$	332.621	1.07644
$310$	0	0
$311$	− 191.385i	− 0.615385i	−0.951486	−	0.307692i	$-0.900443\pi$
$311$	− 191.385i	− 0.615385i	0.951486	−	0.307692i	$-0.0995568\pi$
$312$	0	0
$313$	− 445.975i	− 1.42484i	−0.701753	−	0.712420i	$-0.747599\pi$
$313$	− 445.975i	− 1.42484i	0.701753	−	0.712420i	$-0.252401\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.9512	−0.0881741	−0.0440871	−	0.999028i	$-0.514038\pi$
$317$	−27.9512	−0.0881741	−0.0440871	+	0.999028i	$0.514038\pi$
$318$	0	0
$319$	−1038.91	−3.25676
$320$	0	0
$321$	761.544i	2.37241i
$322$	0	0
$323$	−212.804	−0.658837
$324$	0	0
$325$	− 335.007i	− 1.03079i
$326$	0	0
$327$	− 895.306i	− 2.73794i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	80.6909	0.243779	0.121890	−	0.992544i	$-0.461105\pi$
$331$	80.6909	0.243779	0.121890	+	0.992544i	$0.461105\pi$
$332$	0	0
$333$	577.018	1.73279
$334$	0	0
$335$	− 0.214510i	0 0.000640328i
$336$	0	0
$337$	486.290	1.44300	0.721498	−	0.692416i	$-0.243454\pi$
$337$	486.290	1.44300	0.721498	+	0.692416i	$0.243454\pi$
$338$	0	0
$339$	− 100.283i	− 0.295820i
$340$	0	0
$341$	− 653.217i	− 1.91559i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−9.69573	−0.0281036
$346$	0	0
$347$	241.169	0.695011	0.347505	−	0.937678i	$-0.387029\pi$
$347$	241.169	0.695011	0.347505	+	0.937678i	$0.387029\pi$
$348$	0	0
$349$	251.212i	0.719805i	0.932990	+	0.359903i	$0.117190\pi$
$349$	251.212i	0.719805i	−0.932990	+	0.359903i	$0.882810\pi$
$350$	0	0
$351$	−426.705	−1.21568
$352$	0	0
$353$	− 25.9644i	− 0.0735536i	−0.999324	−	0.0367768i	$-0.988291\pi$
$353$	− 25.9644i	− 0.0735536i	0.999324	−	0.0367768i	$-0.0117091\pi$
$354$	0	0
$355$	− 11.7286i	− 0.0330383i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−340.320	−0.947965	−0.473983	−	0.880534i	$-0.657184\pi$
$359$	−340.320	−0.947965	−0.473983	+	0.880534i	$0.657184\pi$
$360$	0	0
$361$	195.007	0.540184
$362$	0	0
$363$	1070.90i	2.95014i
$364$	0	0
$365$	−1.51669	−0.00415533
$366$	0	0
$367$	− 287.856i	− 0.784349i	−0.919891	−	0.392174i	$-0.871723\pi$
$367$	− 287.856i	− 0.784349i	0.919891	−	0.392174i	$-0.128277\pi$
$368$	0	0
$369$	− 464.512i	− 1.25884i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	86.6033	0.232180	0.116090	−	0.993239i	$-0.462964\pi$
$373$	86.6033	0.232180	0.116090	+	0.993239i	$0.462964\pi$
$374$	0	0
$375$	56.3895	0.150372
$376$	0	0
$377$	− 759.193i	− 2.01377i
$378$	0	0
$379$	−635.808	−1.67759	−0.838796	−	0.544446i	$-0.816740\pi$
$379$	−635.808	−1.67759	−0.838796	+	0.544446i	$0.816740\pi$
$380$	0	0
$381$	− 191.491i	− 0.502602i
$382$	0	0
$383$	− 30.5799i	− 0.0798431i	−0.999203	−	0.0399215i	$-0.987289\pi$
$383$	− 30.5799i	− 0.0798431i	0.999203	−	0.0399215i	$-0.0127108\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−916.627	−2.36855
$388$	0	0
$389$	−588.587	−1.51308	−0.756538	−	0.653949i	$-0.773111\pi$
$389$	−588.587	−1.51308	−0.756538	+	0.653949i	$0.773111\pi$
$390$	0	0
$391$	141.852i	0.362792i
$392$	0	0
$393$	537.382	1.36738
$394$	0	0
$395$	20.1941i	0.0511243i
$396$	0	0
$397$	− 317.394i	− 0.799482i	−0.916628	−	0.399741i	$-0.869100\pi$
$397$	− 317.394i	− 0.799482i	0.916628	−	0.399741i	$-0.130900\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	75.8181	0.189072	0.0945362	−	0.995521i	$-0.469863\pi$
$401$	75.8181	0.189072	0.0945362	+	0.995521i	$0.469863\pi$
$402$	0	0
$403$	477.346	1.18448
$404$	0	0
$405$	− 4.15627i	− 0.0102624i
$406$	0	0
$407$	687.216	1.68849
$408$	0	0
$409$	69.4749i	0.169865i	0.996387	+	0.0849326i	$0.0270675\pi$
$409$	69.4749i	0.169865i	−0.996387	+	0.0849326i	$0.972932\pi$
$410$	0	0
$411$	188.775i	0.459308i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−13.5676	−0.0326930
$416$	0	0
$417$	1246.01	2.98802
$418$	0	0
$419$	29.8461i	0.0712317i	0.999366	+	0.0356158i	$0.0113393\pi$
$419$	29.8461i	0.0712317i	−0.999366	+	0.0356158i	$0.988661\pi$
$420$	0	0
$421$	−28.7671	−0.0683303	−0.0341651	−	0.999416i	$-0.510877\pi$
$421$	−28.7671	−0.0683303	−0.0341651	+	0.999416i	$0.510877\pi$
$422$	0	0
$423$	− 1079.15i	− 2.55119i
$424$	0	0
$425$	− 412.067i	− 0.969570i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1219.61	−2.84291
$430$	0	0
$431$	180.698	0.419252	0.209626	−	0.977782i	$-0.432775\pi$
$431$	180.698	0.419252	0.209626	+	0.977782i	$0.432775\pi$
$432$	0	0
$433$	124.250i	0.286952i	0.989654	+	0.143476i	$0.0458279\pi$
$433$	124.250i	0.286952i	−0.989654	+	0.143476i	$0.954172\pi$
$434$	0	0
$435$	63.8283	0.146732
$436$	0	0
$437$	110.648i	0.253200i
$438$	0	0
$439$	472.310i	1.07588i	0.842984	+	0.537938i	$0.180797\pi$
$439$	472.310i	1.07588i	−0.842984	+	0.537938i	$0.819203\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	741.905	1.67473	0.837364	−	0.546646i	$-0.184095\pi$
$443$	741.905	1.67473	0.837364	+	0.546646i	$0.184095\pi$
$444$	0	0
$445$	13.6002	0.0305622
$446$	0	0
$447$	898.451i	2.00996i
$448$	0	0
$449$	341.817	0.761286	0.380643	−	0.924722i	$-0.375703\pi$
$449$	341.817	0.761286	0.380643	+	0.924722i	$0.375703\pi$
$450$	0	0
$451$	− 553.223i	− 1.22666i
$452$	0	0
$453$	774.953i	1.71071i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	377.286	0.825571	0.412785	−	0.910828i	$-0.364556\pi$
$457$	377.286	0.825571	0.412785	+	0.910828i	$0.364556\pi$
$458$	0	0
$459$	−524.858	−1.14348
$460$	0	0
$461$	233.771i	0.507095i	0.967323	+	0.253547i	$0.0815974\pi$
$461$	233.771i	0.507095i	−0.967323	+	0.253547i	$0.918403\pi$
$462$	0	0
$463$	−528.842	−1.14221	−0.571104	−	0.820878i	$-0.693485\pi$
$463$	−528.842	−1.14221	−0.571104	+	0.820878i	$0.693485\pi$
$464$	0	0
$465$	40.1323i	0.0863060i
$466$	0	0
$467$	− 686.650i	− 1.47034i	−0.677881	−	0.735171i	$-0.737102\pi$
$467$	− 686.650i	− 1.47034i	0.677881	−	0.735171i	$-0.262898\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−72.5654	−0.154067
$472$	0	0
$473$	−1091.68	−2.30800
$474$	0	0
$475$	− 321.424i	− 0.676682i
$476$	0	0
$477$	−1059.52	−2.22121
$478$	0	0
$479$	− 286.317i	− 0.597738i	−0.954294	−	0.298869i	$-0.903391\pi$
$479$	− 286.317i	− 0.597738i	0.954294	−	0.298869i	$-0.0966094\pi$
$480$	0	0
$481$	502.191i	1.04406i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11.3441	−0.0233898
$486$	0	0
$487$	307.583	0.631588	0.315794	−	0.948828i	$-0.397729\pi$
$487$	307.583	0.631588	0.315794	+	0.948828i	$0.397729\pi$
$488$	0	0
$489$	130.001i	0.265850i
$490$	0	0
$491$	−778.757	−1.58606	−0.793031	−	0.609181i	$-0.791498\pi$
$491$	−778.757	−1.58606	−0.793031	+	0.609181i	$0.791498\pi$
$492$	0	0
$493$	− 933.828i	− 1.89417i
$494$	0	0
$495$	− 64.7611i	− 0.130831i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−748.074	−1.49915	−0.749573	−	0.661921i	$-0.769741\pi$
$499$	−748.074	−1.49915	−0.749573	+	0.661921i	$0.769741\pi$
$500$	0	0
$501$	−59.5132	−0.118789
$502$	0	0
$503$	574.649i	1.14244i	0.820796	+	0.571221i	$0.193530\pi$
$503$	574.649i	1.14244i	−0.820796	+	0.571221i	$0.806470\pi$
$504$	0	0
$505$	−38.2055	−0.0756544
$506$	0	0
$507$	− 55.9453i	− 0.110346i
$508$	0	0
$509$	435.284i	0.855174i	0.903974	+	0.427587i	$0.140636\pi$
$509$	435.284i	0.855174i	−0.903974	+	0.427587i	$0.859364\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−409.404	−0.798059
$514$	0	0
$515$	−15.3717	−0.0298480
$516$	0	0
$517$	− 1285.25i	− 2.48597i
$518$	0	0
$519$	−1404.44	−2.70605
$520$	0	0
$521$	256.067i	0.491491i	0.969334	+	0.245746i	$0.0790328\pi$
$521$	256.067i	0.491491i	−0.969334	+	0.245746i	$0.920967\pi$
$522$	0	0
$523$	− 96.3409i	− 0.184208i	−0.995749	−	0.0921041i	$-0.970641\pi$
$523$	− 96.3409i	− 0.184208i	0.995749	−	0.0921041i	$-0.0293593\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	587.148	1.11413
$528$	0	0
$529$	−455.244	−0.860574
$530$	0	0
$531$	893.903i	1.68343i
$532$	0	0
$533$	404.274	0.758488
$534$	0	0
$535$	− 35.1940i	− 0.0657831i
$536$	0	0
$537$	561.698i	1.04599i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	697.203	1.28873	0.644365	−	0.764718i	$-0.277122\pi$
$541$	697.203	1.28873	0.644365	+	0.764718i	$0.277122\pi$
$542$	0	0
$543$	293.226	0.540011
$544$	0	0
$545$	41.3756i	0.0759186i
$546$	0	0
$547$	−376.839	−0.688920	−0.344460	−	0.938801i	$-0.611938\pi$
$547$	−376.839	−0.688920	−0.344460	+	0.938801i	$0.611938\pi$
$548$	0	0
$549$	1372.89i	2.50070i
$550$	0	0
$551$	− 728.412i	− 1.32198i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−42.2211	−0.0760740
$556$	0	0
$557$	254.338	0.456621	0.228310	−	0.973588i	$-0.426680\pi$
$557$	254.338	0.456621	0.228310	+	0.973588i	$0.426680\pi$
$558$	0	0
$559$	− 797.760i	− 1.42712i
$560$	0	0
$561$	−1500.15	−2.67406
$562$	0	0
$563$	− 2.32968i	− 0.00413797i	−0.999998	−	0.00206899i	$-0.999341\pi$
$563$	− 2.32968i	− 0.00413797i	0.999998	−	0.00206899i	$-0.000658579\pi$
$564$	0	0
$565$	4.63447i	0.00820260i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	629.717	1.10671	0.553354	−	0.832946i	$-0.313348\pi$
$569$	629.717	1.10671	0.553354	+	0.832946i	$0.313348\pi$
$570$	0	0
$571$	−716.412	−1.25466	−0.627331	−	0.778753i	$-0.715853\pi$
$571$	−716.412	−1.25466	−0.627331	+	0.778753i	$0.715853\pi$
$572$	0	0
$573$	878.906i	1.53387i
$574$	0	0
$575$	−214.255	−0.372618
$576$	0	0
$577$	− 789.618i	− 1.36849i	−0.729253	−	0.684245i	$-0.760132\pi$
$577$	− 789.618i	− 1.36849i	0.729253	−	0.684245i	$-0.239868\pi$
$578$	0	0
$579$	− 1100.40i	− 1.90051i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1261.86	−2.16442
$584$	0	0
$585$	47.3249	0.0808973
$586$	0	0
$587$	− 494.461i	− 0.842352i	−0.906979	−	0.421176i	$-0.861617\pi$
$587$	− 494.461i	− 0.842352i	0.906979	−	0.421176i	$-0.138383\pi$
$588$	0	0
$589$	457.992	0.777576
$590$	0	0
$591$	165.127i	0.279403i
$592$	0	0
$593$	− 70.3602i	− 0.118651i	−0.998239	−	0.0593256i	$-0.981105\pi$
$593$	− 70.3602i	− 0.118651i	0.998239	−	0.0593256i	$-0.0188950\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1386.18	2.32191
$598$	0	0
$599$	−492.972	−0.822992	−0.411496	−	0.911412i	$-0.634994\pi$
$599$	−492.972	−0.822992	−0.411496	+	0.911412i	$0.634994\pi$
$600$	0	0
$601$	596.738i	0.992909i	0.868063	+	0.496454i	$0.165365\pi$
$601$	596.738i	0.992909i	−0.868063	+	0.496454i	$0.834635\pi$
$602$	0	0
$603$	−14.4898	−0.0240295
$604$	0	0
$605$	− 49.4906i	− 0.0818027i
$606$	0	0
$607$	313.101i	0.515817i	0.966169	+	0.257909i	$0.0830334\pi$
$607$	313.101i	0.515817i	−0.966169	+	0.257909i	$0.916967\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	939.208	1.53716
$612$	0	0
$613$	−463.328	−0.755837	−0.377918	−	0.925839i	$-0.623360\pi$
$613$	−463.328	−0.755837	−0.377918	+	0.925839i	$0.623360\pi$
$614$	0	0
$615$	33.9889i	0.0552664i
$616$	0	0
$617$	−828.523	−1.34283	−0.671413	−	0.741084i	$-0.734312\pi$
$617$	−828.523	−1.34283	−0.671413	+	0.741084i	$0.734312\pi$
$618$	0	0
$619$	− 1010.47i	− 1.63242i	−0.577756	−	0.816210i	$-0.696071\pi$
$619$	− 1010.47i	− 1.63242i	0.577756	−	0.816210i	$-0.303929\pi$
$620$	0	0
$621$	272.901i	0.439455i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	621.090	0.993743
$626$	0	0
$627$	−1170.16	−1.86628
$628$	0	0
$629$	617.708i	0.982047i
$630$	0	0
$631$	815.831	1.29292	0.646458	−	0.762949i	$-0.276249\pi$
$631$	815.831	1.29292	0.646458	+	0.762949i	$0.276249\pi$
$632$	0	0
$633$	1201.18i	1.89760i
$634$	0	0
$635$	8.84958i	0.0139363i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−792.246	−1.23982
$640$	0	0
$641$	−462.732	−0.721891	−0.360946	−	0.932587i	$-0.617546\pi$
$641$	−462.732	−0.721891	−0.360946	+	0.932587i	$0.617546\pi$
$642$	0	0
$643$	− 293.891i	− 0.457062i	−0.973537	−	0.228531i	$-0.926608\pi$
$643$	− 293.891i	− 0.457062i	0.973537	−	0.228531i	$-0.0733922\pi$
$644$	0	0
$645$	67.0707	0.103986
$646$	0	0
$647$	944.250i	1.45943i	0.683753	+	0.729714i	$0.260347\pi$
$647$	944.250i	1.45943i	−0.683753	+	0.729714i	$0.739653\pi$
$648$	0	0
$649$	1064.62i	1.64040i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	386.170	0.591379	0.295689	−	0.955284i	$-0.404451\pi$
$653$	386.170	0.591379	0.295689	+	0.955284i	$0.404451\pi$
$654$	0	0
$655$	−24.8346	−0.0379153
$656$	0	0
$657$	102.450i	0.155936i
$658$	0	0
$659$	480.966	0.729843	0.364921	−	0.931038i	$-0.381096\pi$
$659$	480.966	0.729843	0.364921	+	0.931038i	$0.381096\pi$
$660$	0	0
$661$	922.865i	1.39617i	0.716017	+	0.698083i	$0.245963\pi$
$661$	922.865i	1.39617i	−0.716017	+	0.698083i	$0.754037\pi$
$662$	0	0
$663$	− 1096.25i	− 1.65347i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−485.546	−0.727955
$668$	0	0
$669$	−854.389	−1.27711
$670$	0	0
$671$	1635.08i	2.43677i
$672$	0	0
$673$	−939.147	−1.39546	−0.697732	−	0.716359i	$-0.745807\pi$
$673$	−939.147	−1.39546	−0.697732	+	0.716359i	$0.745807\pi$
$674$	0	0
$675$	− 792.756i	− 1.17445i
$676$	0	0
$677$	724.693i	1.07045i	0.844710	+	0.535224i	$0.179773\pi$
$677$	724.693i	1.07045i	−0.844710	+	0.535224i	$0.820227\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−562.028	−0.825298
$682$	0	0
$683$	385.228	0.564023	0.282011	−	0.959411i	$-0.408998\pi$
$683$	385.228	0.564023	0.282011	+	0.959411i	$0.408998\pi$
$684$	0	0
$685$	− 8.72407i	− 0.0127359i
$686$	0	0
$687$	−1785.76	−2.59936
$688$	0	0
$689$	− 922.118i	− 1.33834i
$690$	0	0
$691$	643.566i	0.931354i	0.884955	+	0.465677i	$0.154189\pi$
$691$	643.566i	0.931354i	−0.884955	+	0.465677i	$0.845811\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−57.5829	−0.0828531
$696$	0	0
$697$	497.268	0.713440
$698$	0	0
$699$	948.459i	1.35688i
$700$	0	0
$701$	−314.892	−0.449204	−0.224602	−	0.974451i	$-0.572108\pi$
$701$	−314.892	−0.449204	−0.224602	+	0.974451i	$0.572108\pi$
$702$	0	0
$703$	481.829i	0.685390i
$704$	0	0
$705$	78.9628i	0.112004i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−226.316	−0.319204	−0.159602	−	0.987181i	$-0.551021\pi$
$709$	−226.316	−0.319204	−0.159602	+	0.987181i	$0.551021\pi$
$710$	0	0
$711$	1364.08	1.91853
$712$	0	0
$713$	− 305.289i	− 0.428176i
$714$	0	0
$715$	56.3629	0.0788293
$716$	0	0
$717$	80.8245i	0.112726i
$718$	0	0
$719$	571.504i	0.794860i	0.917633	+	0.397430i	$0.130098\pi$
$719$	571.504i	0.794860i	−0.917633	+	0.397430i	$0.869902\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1524.33	2.10834
$724$	0	0
$725$	1410.47	1.94548
$726$	0	0
$727$	270.254i	0.371739i	0.982574	+	0.185869i	$0.0595101\pi$
$727$	270.254i	0.371739i	−0.982574	+	0.185869i	$0.940490\pi$
$728$	0	0
$729$	1132.78	1.55388
$730$	0	0
$731$	− 981.265i	− 1.34236i
$732$	0	0
$733$	− 767.053i	− 1.04646i	−0.852192	−	0.523229i	$-0.824727\pi$
$733$	− 767.053i	− 1.04646i	0.852192	−	0.523229i	$-0.175273\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.2570	−0.0234152
$738$	0	0
$739$	982.844	1.32996	0.664982	−	0.746859i	$-0.268439\pi$
$739$	982.844	1.32996	0.664982	+	0.746859i	$0.268439\pi$
$740$	0	0
$741$	− 855.107i	− 1.15399i
$742$	0	0
$743$	−440.189	−0.592449	−0.296224	−	0.955118i	$-0.595728\pi$
$743$	−440.189	−0.592449	−0.296224	+	0.955118i	$0.595728\pi$
$744$	0	0
$745$	− 41.5210i	− 0.0557329i
$746$	0	0
$747$	916.466i	1.22686i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1091.22	−1.45302	−0.726509	−	0.687157i	$-0.758858\pi$
$751$	−1091.22	−1.45302	−0.726509	+	0.687157i	$0.758858\pi$
$752$	0	0
$753$	−786.332	−1.04427
$754$	0	0
$755$	− 35.8137i	− 0.0474353i
$756$	0	0
$757$	−482.625	−0.637550	−0.318775	−	0.947830i	$-0.603271\pi$
$757$	−482.625	−0.637550	−0.318775	+	0.947830i	$0.603271\pi$
$758$	0	0
$759$	780.007i	1.02768i
$760$	0	0
$761$	405.089i	0.532311i	0.963930	+	0.266156i	$0.0857535\pi$
$761$	405.089i	0.532311i	−0.963930	+	0.266156i	$0.914246\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	58.2109	0.0760927
$766$	0	0
$767$	−777.982	−1.01432
$768$	0	0
$769$	− 35.3891i	− 0.0460196i	−0.999735	−	0.0230098i	$-0.992675\pi$
$769$	− 35.3891i	− 0.0460196i	0.999735	−	0.0230098i	$-0.00732490\pi$
$770$	0	0
$771$	1291.98	1.67572
$772$	0	0
$773$	− 929.071i	− 1.20190i	−0.799286	−	0.600951i	$-0.794789\pi$
$773$	− 929.071i	− 1.20190i	0.799286	−	0.600951i	$-0.205211\pi$
$774$	0	0
$775$	886.840i	1.14431i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	387.883	0.497924
$780$	0	0
$781$	−943.548	−1.20813
$782$	0	0
$783$	− 1796.55i	− 2.29444i
$784$	0	0
$785$	3.35354	0.00427202
$786$	0	0
$787$	− 1232.01i	− 1.56545i	−0.622369	−	0.782724i	$-0.713830\pi$
$787$	− 1232.01i	− 1.56545i	0.622369	−	0.782724i	$-0.286170\pi$
$788$	0	0
$789$	− 514.460i	− 0.652041i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1194.85	−1.50675
$794$	0	0
$795$	77.5260	0.0975170
$796$	0	0
$797$	155.667i	0.195317i	0.995220	+	0.0976583i	$0.0311352\pi$
$797$	155.667i	0.195317i	−0.995220	+	0.0976583i	$0.968865\pi$
$798$	0	0
$799$	1155.25	1.44587
$800$	0	0
$801$	− 918.667i	− 1.14690i
$802$	0	0
$803$	122.016i	0.151950i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2067.07	2.56143
$808$	0	0
$809$	−162.214	−0.200512	−0.100256	−	0.994962i	$-0.531966\pi$
$809$	−162.214	−0.200512	−0.100256	+	0.994962i	$0.531966\pi$
$810$	0	0
$811$	− 1257.67i	− 1.55076i	−0.631495	−	0.775380i	$-0.717558\pi$
$811$	− 1257.67i	− 1.55076i	0.631495	−	0.775380i	$-0.282442\pi$
$812$	0	0
$813$	−621.204	−0.764088
$814$	0	0
$815$	− 6.00784i	− 0.00737159i
$816$	0	0
$817$	− 765.414i	− 0.936860i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	110.853	0.135022	0.0675109	−	0.997719i	$-0.478494\pi$
$821$	110.853	0.135022	0.0675109	+	0.997719i	$0.478494\pi$
$822$	0	0
$823$	33.0230	0.0401251	0.0200626	−	0.999799i	$-0.493613\pi$
$823$	33.0230	0.0401251	0.0200626	+	0.999799i	$0.493613\pi$
$824$	0	0
$825$	− 2265.85i	− 2.74649i
$826$	0	0
$827$	588.109	0.711136	0.355568	−	0.934650i	$-0.384288\pi$
$827$	588.109	0.711136	0.355568	+	0.934650i	$0.384288\pi$
$828$	0	0
$829$	566.067i	0.682831i	0.939912	+	0.341415i	$0.110906\pi$
$829$	566.067i	0.682831i	−0.939912	+	0.341415i	$0.889094\pi$
$830$	0	0
$831$	288.272i	0.346898i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.75034	0.00329383
$836$	0	0
$837$	1129.59	1.34957
$838$	0	0
$839$	411.575i	0.490554i	0.969453	+	0.245277i	$0.0788790\pi$
$839$	411.575i	0.490554i	−0.969453	+	0.245277i	$0.921121\pi$
$840$	0	0
$841$	2355.41	2.80073
$842$	0	0
$843$	948.019i	1.12458i
$844$	0	0
$845$	2.58545i	0.00305971i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2043.98	−2.40751
$850$	0	0
$851$	321.179	0.377413
$852$	0	0
$853$	1489.13i	1.74576i	0.487934	+	0.872881i	$0.337751\pi$
$853$	1489.13i	1.74576i	−0.487934	+	0.872881i	$0.662249\pi$
$854$	0	0
$855$	45.4061	0.0531066
$856$	0	0
$857$	− 435.577i	− 0.508258i	−0.967170	−	0.254129i	$-0.918211\pi$
$857$	− 435.577i	− 0.508258i	0.967170	−	0.254129i	$-0.0817888\pi$
$858$	0	0
$859$	− 896.123i	− 1.04322i	−0.853185	−	0.521608i	$-0.825332\pi$
$859$	− 896.123i	− 1.04322i	0.853185	−	0.521608i	$-0.174668\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1633.84	−1.89321	−0.946605	−	0.322395i	$-0.895512\pi$
$863$	−1633.84	−1.89321	−0.946605	+	0.322395i	$0.895512\pi$
$864$	0	0
$865$	64.9048	0.0750344
$866$	0	0
$867$	79.9888i	0.0922592i
$868$	0	0
$869$	1624.59	1.86949
$870$	0	0
$871$	− 12.6107i	− 0.0144785i
$872$	0	0
$873$	766.272i	0.877745i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−65.1657	−0.0743053	−0.0371526	−	0.999310i	$-0.511829\pi$
$877$	−65.1657	−0.0743053	−0.0371526	+	0.999310i	$0.511829\pi$
$878$	0	0
$879$	2034.20	2.31422
$880$	0	0
$881$	− 456.877i	− 0.518590i	−0.965798	−	0.259295i	$-0.916510\pi$
$881$	− 456.877i	− 0.518590i	0.965798	−	0.259295i	$-0.0834901\pi$
$882$	0	0
$883$	−1292.73	−1.46402	−0.732012	−	0.681291i	$-0.761419\pi$
$883$	−1292.73	−1.46402	−0.732012	+	0.681291i	$0.761419\pi$
$884$	0	0
$885$	− 65.4079i	− 0.0739073i
$886$	0	0
$887$	− 223.900i	− 0.252424i	−0.992003	−	0.126212i	$-0.959718\pi$
$887$	− 223.900i	− 0.252424i	0.992003	−	0.126212i	$-0.0402820\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−334.366	−0.375270
$892$	0	0
$893$	901.127	1.00910
$894$	0	0
$895$	− 25.9583i	− 0.0290037i
$896$	0	0
$897$	−569.999	−0.635450
$898$	0	0
$899$	2009.76i	2.23555i
$900$	0	0
$901$	− 1134.23i	− 1.25886i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.5512	−0.0149737
$906$	0	0
$907$	−1631.06	−1.79830	−0.899150	−	0.437641i	$-0.855814\pi$
$907$	−1631.06	−1.79830	−0.899150	+	0.437641i	$0.855814\pi$
$908$	0	0
$909$	2580.71i	2.83907i
$910$	0	0
$911$	367.755	0.403682	0.201841	−	0.979418i	$-0.435307\pi$
$911$	367.755	0.403682	0.201841	+	0.979418i	$0.435307\pi$
$912$	0	0
$913$	1091.49i	1.19550i
$914$	0	0
$915$	− 100.456i	− 0.109788i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−437.228	−0.475765	−0.237882	−	0.971294i	$-0.576453\pi$
$919$	−437.228	−0.475765	−0.237882	+	0.971294i	$0.576453\pi$
$920$	0	0
$921$	1425.14	1.54739
$922$	0	0
$923$	− 689.508i	− 0.747030i
$924$	0	0
$925$	−932.998	−1.00865
$926$	0	0
$927$	1038.33i	1.12010i
$928$	0	0
$929$	− 1398.75i	− 1.50565i	−0.658222	−	0.752824i	$-0.728691\pi$
$929$	− 1398.75i	− 1.50565i	0.658222	−	0.752824i	$-0.271309\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	945.935	1.01386
$934$	0	0
$935$	69.3279	0.0741475
$936$	0	0
$937$	− 1377.17i	− 1.46976i	−0.678196	−	0.734881i	$-0.737238\pi$
$937$	− 1377.17i	− 1.46976i	0.678196	−	0.734881i	$-0.262762\pi$
$938$	0	0
$939$	2204.27	2.34746
$940$	0	0
$941$	− 102.598i	− 0.109030i	−0.998513	−	0.0545152i	$-0.982639\pi$
$941$	− 102.598i	− 0.109030i	0.998513	−	0.0545152i	$-0.0173613\pi$
$942$	0	0
$943$	− 258.556i	− 0.274184i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	629.056	0.664262	0.332131	−	0.943233i	$-0.392232\pi$
$947$	629.056	0.664262	0.332131	+	0.943233i	$0.392232\pi$
$948$	0	0
$949$	−89.1643	−0.0939561
$950$	0	0
$951$	− 138.151i	− 0.145269i
$952$	0	0
$953$	1284.44	1.34779	0.673895	−	0.738827i	$-0.264620\pi$
$953$	1284.44	1.34779	0.673895	+	0.738827i	$0.264620\pi$
$954$	0	0
$955$	− 40.6177i	− 0.0425317i
$956$	0	0
$957$	− 5134.88i	− 5.36561i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−302.645	−0.314927
$962$	0	0
$963$	−2377.29	−2.46863
$964$	0	0
$965$	50.8538i	0.0526982i
$966$	0	0
$967$	522.809	0.540650	0.270325	−	0.962769i	$-0.412869\pi$
$967$	522.809	0.540650	0.270325	+	0.962769i	$0.412869\pi$
$968$	0	0
$969$	− 1051.80i	− 1.08545i
$970$	0	0
$971$	− 345.493i	− 0.355812i	−0.984048	−	0.177906i	$-0.943068\pi$
$971$	− 345.493i	− 0.355812i	0.984048	−	0.177906i	$-0.0569323\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1655.80	1.69826
$976$	0	0
$977$	876.774	0.897415	0.448707	−	0.893679i	$-0.351885\pi$
$977$	876.774	0.897415	0.448707	+	0.893679i	$0.351885\pi$
$978$	0	0
$979$	− 1094.11i	− 1.11758i
$980$	0	0
$981$	2794.85	2.84898
$982$	0	0
$983$	632.697i	0.643639i	0.946801	+	0.321819i	$0.104294\pi$
$983$	632.697i	0.643639i	−0.946801	+	0.321819i	$0.895706\pi$
$984$	0	0
$985$	− 7.63118i	− 0.00774739i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−510.212	−0.515886
$990$	0	0
$991$	−1467.06	−1.48038	−0.740190	−	0.672398i	$-0.765265\pi$
$991$	−1467.06	−1.48038	−0.740190	+	0.672398i	$0.765265\pi$
$992$	0	0
$993$	398.821i	0.401633i
$994$	0	0
$995$	−64.0608	−0.0643827
$996$	0	0
$997$	13.1748i	0.0132144i	0.999978	+	0.00660720i	$0.00210315\pi$
$997$	13.1748i	0.0132144i	−0.999978	+	0.00660720i	$0.997897\pi$
$998$	0	0
$999$	1188.38i	1.18957i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.c.e.97.12	yes	12
4.3	odd	2		1568.3.c.f.97.1	yes	12
7.6	odd	2	inner	1568.3.c.e.97.1	✓	12
28.27	even	2		1568.3.c.f.97.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1568.3.c.e.97.1	✓	12	7.6	odd	2	inner
1568.3.c.e.97.12	yes	12	1.1	even	1	trivial
1568.3.c.f.97.1	yes	12	4.3	odd	2
1568.3.c.f.97.12	yes	12	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.94258i q^{3} -0.228417i q^{5} -15.4291 q^{9} -18.3758 q^{11} -13.4283i q^{13} +1.12897 q^{15} -16.5172i q^{17} -12.8838i q^{19} -8.58814 q^{23} +24.9478 q^{25} -31.7765i q^{27} +56.5368 q^{29} +35.5478i q^{31} -90.8237i q^{33} -37.3980 q^{37} +66.3705 q^{39} +30.1061i q^{41} +59.4089 q^{43} +3.52427i q^{45} +69.9424i q^{47} +81.6374 q^{51} +68.6698 q^{53} +4.19733i q^{55} +63.6795 q^{57} -57.9361i q^{59} -88.9801i q^{61} -3.06724 q^{65} +0.939118 q^{67} -42.4476i q^{69} +51.3474 q^{71} -6.64003i q^{73} +123.307i q^{75} -88.4092 q^{79} +18.1960 q^{81} -59.3984i q^{83} -3.77279 q^{85} +279.438i q^{87} +59.5411i q^{89} -175.698 q^{93} -2.94288 q^{95} -49.6639i q^{97} +283.522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 36 q^{9} - 48 q^{11} + 128 q^{23} - 124 q^{25} - 128 q^{37} - 160 q^{39} + 464 q^{43} + 400 q^{51} + 16 q^{53} + 224 q^{57} - 64 q^{65} - 64 q^{67} + 544 q^{71} + 288 q^{79} - 180 q^{81} - 272 q^{85}+ \cdots + 1328 q^{99}+O(q^{100}) \) 12 * q - 36 * q^9 - 48 * q^11 + 128 * q^23 - 124 * q^25 - 128 * q^37 - 160 * q^39 + 464 * q^43 + 400 * q^51 + 16 * q^53 + 224 * q^57 - 64 * q^65 - 64 * q^67 + 544 * q^71 + 288 * q^79 - 180 * q^81 - 272 * q^85 - 352 * q^93 - 576 * q^95 + 1328 * q^99

Introduction

L-functions

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