LMFDB - Embedded newform 588.3.d.a.97.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,3,Mod(97,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("588.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	588.d (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:	$16.0218395444$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	588.97
Dual form			588.3.d.a.97.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -1.73205i q^{5} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -1.73205i q^{5} -3.00000 q^{9} -3.00000 q^{11} -6.92820i q^{13} +3.00000 q^{15} -17.3205i q^{17} -10.3923i q^{19} +36.0000 q^{23} +22.0000 q^{25} -5.19615i q^{27} -51.0000 q^{29} -12.1244i q^{31} -5.19615i q^{33} +22.0000 q^{37} +12.0000 q^{39} -24.2487i q^{41} +10.0000 q^{43} +5.19615i q^{45} -90.0666i q^{47} +30.0000 q^{51} +51.0000 q^{53} +5.19615i q^{55} +18.0000 q^{57} +74.4782i q^{59} -69.2820i q^{61} -12.0000 q^{65} +68.0000 q^{67} +62.3538i q^{69} -20.7846i q^{73} +38.1051i q^{75} +125.000 q^{79} +9.00000 q^{81} -154.153i q^{83} -30.0000 q^{85} -88.3346i q^{87} -72.7461i q^{89} +21.0000 q^{93} -18.0000 q^{95} +147.224i q^{97} +9.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{9}+O(q^{10}) $ 2 * q - 6 * q^9	$ 2 q - 6 q^{9} - 6 q^{11} + 6 q^{15} + 72 q^{23} + 44 q^{25} - 102 q^{29} + 44 q^{37} + 24 q^{39} + 20 q^{43} + 60 q^{51} + 102 q^{53} + 36 q^{57} - 24 q^{65} + 136 q^{67} + 250 q^{79} + 18 q^{81} - 60 q^{85} + 42 q^{93} - 36 q^{95} + 18 q^{99}+O(q^{100}) $ 2 * q - 6 * q^9 - 6 * q^11 + 6 * q^15 + 72 * q^23 + 44 * q^25 - 102 * q^29 + 44 * q^37 + 24 * q^39 + 20 * q^43 + 60 * q^51 + 102 * q^53 + 36 * q^57 - 24 * q^65 + 136 * q^67 + 250 * q^79 + 18 * q^81 - 60 * q^85 + 42 * q^93 - 36 * q^95 + 18 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	− 1.73205i	− 0.346410i	−0.984886	−	0.173205i	$-0.944588\pi$
$5$	− 1.73205i	− 0.346410i	0.984886	−	0.173205i	$-0.0554123\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−3.00000	−0.272727	−0.136364	−	0.990659i	$-0.543542\pi$
$11$	−3.00000	−0.272727	−0.136364	+	0.990659i	$0.543542\pi$
$12$	0	0
$13$	− 6.92820i	− 0.532939i	−0.963843	−	0.266469i	$-0.914143\pi$
$13$	− 6.92820i	− 0.532939i	0.963843	−	0.266469i	$-0.0858571\pi$
$14$	0	0
$15$	3.00000	0.200000
$16$	0	0
$17$	− 17.3205i	− 1.01885i	−0.860514	−	0.509427i	$-0.829857\pi$
$17$	− 17.3205i	− 1.01885i	0.860514	−	0.509427i	$-0.170143\pi$
$18$	0	0
$19$	− 10.3923i	− 0.546963i	−0.961877	−	0.273482i	$-0.911825\pi$
$19$	− 10.3923i	− 0.546963i	0.961877	−	0.273482i	$-0.0881753\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	36.0000	1.56522	0.782609	−	0.622514i	$-0.213889\pi$
$23$	36.0000	1.56522	0.782609	+	0.622514i	$0.213889\pi$
$24$	0	0
$25$	22.0000	0.880000
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−51.0000	−1.75862	−0.879310	−	0.476249i	$-0.841996\pi$
$29$	−51.0000	−1.75862	−0.879310	+	0.476249i	$0.841996\pi$
$30$	0	0
$31$	− 12.1244i	− 0.391108i	−0.980693	−	0.195554i	$-0.937349\pi$
$31$	− 12.1244i	− 0.391108i	0.980693	−	0.195554i	$-0.0626505\pi$
$32$	0	0
$33$	− 5.19615i	− 0.157459i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	22.0000	0.594595	0.297297	−	0.954785i	$-0.403915\pi$
$37$	22.0000	0.594595	0.297297	+	0.954785i	$0.403915\pi$
$38$	0	0
$39$	12.0000	0.307692
$40$	0	0
$41$	− 24.2487i	− 0.591432i	−0.955276	−	0.295716i	$-0.904442\pi$
$41$	− 24.2487i	− 0.591432i	0.955276	−	0.295716i	$-0.0955582\pi$
$42$	0	0
$43$	10.0000	0.232558	0.116279	−	0.993217i	$-0.462903\pi$
$43$	10.0000	0.232558	0.116279	+	0.993217i	$0.462903\pi$
$44$	0	0
$45$	5.19615i	0.115470i
$46$	0	0
$47$	− 90.0666i	− 1.91631i	−0.286247	−	0.958156i	$-0.592408\pi$
$47$	− 90.0666i	− 1.91631i	0.286247	−	0.958156i	$-0.407592\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	30.0000	0.588235
$52$	0	0
$53$	51.0000	0.962264	0.481132	−	0.876648i	$-0.340226\pi$
$53$	51.0000	0.962264	0.481132	+	0.876648i	$0.340226\pi$
$54$	0	0
$55$	5.19615i	0.0944755i
$56$	0	0
$57$	18.0000	0.315789
$58$	0	0
$59$	74.4782i	1.26234i	0.775644	+	0.631171i	$0.217425\pi$
$59$	74.4782i	1.26234i	−0.775644	+	0.631171i	$0.782575\pi$
$60$	0	0
$61$	− 69.2820i	− 1.13577i	−0.823108	−	0.567886i	$-0.807762\pi$
$61$	− 69.2820i	− 1.13577i	0.823108	−	0.567886i	$-0.192238\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.0000	−0.184615
$66$	0	0
$67$	68.0000	1.01493	0.507463	−	0.861674i	$-0.330584\pi$
$67$	68.0000	1.01493	0.507463	+	0.861674i	$0.330584\pi$
$68$	0	0
$69$	62.3538i	0.903679i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 20.7846i	− 0.284721i	−0.989815	−	0.142360i	$-0.954531\pi$
$73$	− 20.7846i	− 0.284721i	0.989815	−	0.142360i	$-0.0454692\pi$
$74$	0	0
$75$	38.1051i	0.508068i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	125.000	1.58228	0.791139	−	0.611636i	$-0.209488\pi$
$79$	125.000	1.58228	0.791139	+	0.611636i	$0.209488\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 154.153i	− 1.85726i	−0.371008	−	0.928630i	$-0.620988\pi$
$83$	− 154.153i	− 1.85726i	0.371008	−	0.928630i	$-0.379012\pi$
$84$	0	0
$85$	−30.0000	−0.352941
$86$	0	0
$87$	− 88.3346i	− 1.01534i
$88$	0	0
$89$	− 72.7461i	− 0.817372i	−0.912675	−	0.408686i	$-0.865987\pi$
$89$	− 72.7461i	− 0.817372i	0.912675	−	0.408686i	$-0.134013\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	21.0000	0.225806
$94$	0	0
$95$	−18.0000	−0.189474
$96$	0	0
$97$	147.224i	1.51778i	0.651221	+	0.758888i	$0.274257\pi$
$97$	147.224i	1.51778i	−0.651221	+	0.758888i	$0.725743\pi$
$98$	0	0
$99$	9.00000	0.0909091
$100$	0	0
$101$	20.7846i	0.205788i	0.994692	+	0.102894i	$0.0328103\pi$
$101$	20.7846i	0.205788i	−0.994692	+	0.102894i	$0.967190\pi$
$102$	0	0
$103$	152.420i	1.47981i	0.672711	+	0.739905i	$0.265130\pi$
$103$	152.420i	1.47981i	−0.672711	+	0.739905i	$0.734870\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−33.0000	−0.308411	−0.154206	−	0.988039i	$-0.549282\pi$
$107$	−33.0000	−0.308411	−0.154206	+	0.988039i	$0.549282\pi$
$108$	0	0
$109$	−32.0000	−0.293578	−0.146789	−	0.989168i	$-0.546894\pi$
$109$	−32.0000	−0.293578	−0.146789	+	0.989168i	$0.546894\pi$
$110$	0	0
$111$	38.1051i	0.343289i
$112$	0	0
$113$	−186.000	−1.64602	−0.823009	−	0.568029i	$-0.807706\pi$
$113$	−186.000	−1.64602	−0.823009	+	0.568029i	$0.807706\pi$
$114$	0	0
$115$	− 62.3538i	− 0.542207i
$116$	0	0
$117$	20.7846i	0.177646i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−112.000	−0.925620
$122$	0	0
$123$	42.0000	0.341463
$124$	0	0
$125$	− 81.4064i	− 0.651251i
$126$	0	0
$127$	131.000	1.03150	0.515748	−	0.856740i	$-0.327514\pi$
$127$	131.000	1.03150	0.515748	+	0.856740i	$0.327514\pi$
$128$	0	0
$129$	17.3205i	0.134268i
$130$	0	0
$131$	− 98.7269i	− 0.753640i	−0.926286	−	0.376820i	$-0.877017\pi$
$131$	− 98.7269i	− 0.753640i	0.926286	−	0.376820i	$-0.122983\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−9.00000	−0.0666667
$136$	0	0
$137$	−144.000	−1.05109	−0.525547	−	0.850764i	$-0.676139\pi$
$137$	−144.000	−1.05109	−0.525547	+	0.850764i	$0.676139\pi$
$138$	0	0
$139$	− 148.956i	− 1.07163i	−0.844336	−	0.535814i	$-0.820005\pi$
$139$	− 148.956i	− 1.07163i	0.844336	−	0.535814i	$-0.179995\pi$
$140$	0	0
$141$	156.000	1.10638
$142$	0	0
$143$	20.7846i	0.145347i
$144$	0	0
$145$	88.3346i	0.609204i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−150.000	−1.00671	−0.503356	−	0.864079i	$-0.667901\pi$
$149$	−150.000	−1.00671	−0.503356	+	0.864079i	$0.667901\pi$
$150$	0	0
$151$	−233.000	−1.54305	−0.771523	−	0.636201i	$-0.780505\pi$
$151$	−233.000	−1.54305	−0.771523	+	0.636201i	$0.780505\pi$
$152$	0	0
$153$	51.9615i	0.339618i
$154$	0	0
$155$	−21.0000	−0.135484
$156$	0	0
$157$	145.492i	0.926702i	0.886175	+	0.463351i	$0.153353\pi$
$157$	145.492i	0.926702i	−0.886175	+	0.463351i	$0.846647\pi$
$158$	0	0
$159$	88.3346i	0.555563i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−148.000	−0.907975	−0.453988	−	0.891008i	$-0.649999\pi$
$163$	−148.000	−0.907975	−0.453988	+	0.891008i	$0.649999\pi$
$164$	0	0
$165$	−9.00000	−0.0545455
$166$	0	0
$167$	− 27.7128i	− 0.165945i	−0.996552	−	0.0829725i	$-0.973559\pi$
$167$	− 27.7128i	− 0.165945i	0.996552	−	0.0829725i	$-0.0264414\pi$
$168$	0	0
$169$	121.000	0.715976
$170$	0	0
$171$	31.1769i	0.182321i
$172$	0	0
$173$	187.061i	1.08128i	0.841254	+	0.540640i	$0.181818\pi$
$173$	187.061i	1.08128i	−0.841254	+	0.540640i	$0.818182\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−129.000	−0.728814
$178$	0	0
$179$	−162.000	−0.905028	−0.452514	−	0.891757i	$-0.649473\pi$
$179$	−162.000	−0.905028	−0.452514	+	0.891757i	$0.649473\pi$
$180$	0	0
$181$	259.808i	1.43540i	0.696352	+	0.717701i	$0.254805\pi$
$181$	259.808i	1.43540i	−0.696352	+	0.717701i	$0.745195\pi$
$182$	0	0
$183$	120.000	0.655738
$184$	0	0
$185$	− 38.1051i	− 0.205974i
$186$	0	0
$187$	51.9615i	0.277869i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	96.0000	0.502618	0.251309	−	0.967907i	$-0.419139\pi$
$191$	96.0000	0.502618	0.251309	+	0.967907i	$0.419139\pi$
$192$	0	0
$193$	295.000	1.52850	0.764249	−	0.644922i	$-0.223110\pi$
$193$	295.000	1.52850	0.764249	+	0.644922i	$0.223110\pi$
$194$	0	0
$195$	− 20.7846i	− 0.106588i
$196$	0	0
$197$	−330.000	−1.67513	−0.837563	−	0.546340i	$-0.816021\pi$
$197$	−330.000	−1.67513	−0.837563	+	0.546340i	$0.816021\pi$
$198$	0	0
$199$	− 117.779i	− 0.591857i	−0.955210	−	0.295928i	$-0.904371\pi$
$199$	− 117.779i	− 0.591857i	0.955210	−	0.295928i	$-0.0956289\pi$
$200$	0	0
$201$	117.779i	0.585967i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−42.0000	−0.204878
$206$	0	0
$207$	−108.000	−0.521739
$208$	0	0
$209$	31.1769i	0.149172i
$210$	0	0
$211$	220.000	1.04265	0.521327	−	0.853357i	$-0.325437\pi$
$211$	220.000	1.04265	0.521327	+	0.853357i	$0.325437\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 17.3205i	− 0.0805605i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	36.0000	0.164384
$220$	0	0
$221$	−120.000	−0.542986
$222$	0	0
$223$	150.688i	0.675733i	0.941194	+	0.337866i	$0.109705\pi$
$223$	150.688i	0.675733i	−0.941194	+	0.337866i	$0.890295\pi$
$224$	0	0
$225$	−66.0000	−0.293333
$226$	0	0
$227$	109.119i	0.480701i	0.970686	+	0.240351i	$0.0772624\pi$
$227$	109.119i	0.480701i	−0.970686	+	0.240351i	$0.922738\pi$
$228$	0	0
$229$	377.587i	1.64885i	0.565970	+	0.824426i	$0.308502\pi$
$229$	377.587i	1.64885i	−0.565970	+	0.824426i	$0.691498\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−210.000	−0.901288	−0.450644	−	0.892704i	$-0.648806\pi$
$233$	−210.000	−0.901288	−0.450644	+	0.892704i	$0.648806\pi$
$234$	0	0
$235$	−156.000	−0.663830
$236$	0	0
$237$	216.506i	0.913529i
$238$	0	0
$239$	120.000	0.502092	0.251046	−	0.967975i	$-0.419225\pi$
$239$	120.000	0.502092	0.251046	+	0.967975i	$0.419225\pi$
$240$	0	0
$241$	− 12.1244i	− 0.0503085i	−0.999684	−	0.0251543i	$-0.991992\pi$
$241$	− 12.1244i	− 0.0503085i	0.999684	−	0.0251543i	$-0.00800770\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−72.0000	−0.291498
$248$	0	0
$249$	267.000	1.07229
$250$	0	0
$251$	− 136.832i	− 0.545147i	−0.962135	−	0.272574i	$-0.912125\pi$
$251$	− 136.832i	− 0.545147i	0.962135	−	0.272574i	$-0.0878749\pi$
$252$	0	0
$253$	−108.000	−0.426877
$254$	0	0
$255$	− 51.9615i	− 0.203771i
$256$	0	0
$257$	− 135.100i	− 0.525681i	−0.964839	−	0.262840i	$-0.915341\pi$
$257$	− 135.100i	− 0.525681i	0.964839	−	0.262840i	$-0.0846593\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	153.000	0.586207
$262$	0	0
$263$	162.000	0.615970	0.307985	−	0.951391i	$-0.400345\pi$
$263$	162.000	0.615970	0.307985	+	0.951391i	$0.400345\pi$
$264$	0	0
$265$	− 88.3346i	− 0.333338i
$266$	0	0
$267$	126.000	0.471910
$268$	0	0
$269$	386.247i	1.43586i	0.696113	+	0.717932i	$0.254911\pi$
$269$	386.247i	1.43586i	−0.696113	+	0.717932i	$0.745089\pi$
$270$	0	0
$271$	− 216.506i	− 0.798916i	−0.916751	−	0.399458i	$-0.869198\pi$
$271$	− 216.506i	− 0.798916i	0.916751	−	0.399458i	$-0.130802\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−66.0000	−0.240000
$276$	0	0
$277$	−100.000	−0.361011	−0.180505	−	0.983574i	$-0.557773\pi$
$277$	−100.000	−0.361011	−0.180505	+	0.983574i	$0.557773\pi$
$278$	0	0
$279$	36.3731i	0.130369i
$280$	0	0
$281$	228.000	0.811388	0.405694	−	0.914009i	$-0.367030\pi$
$281$	228.000	0.811388	0.405694	+	0.914009i	$0.367030\pi$
$282$	0	0
$283$	17.3205i	0.0612032i	0.999532	+	0.0306016i	$0.00974231\pi$
$283$	17.3205i	0.0612032i	−0.999532	+	0.0306016i	$0.990258\pi$
$284$	0	0
$285$	− 31.1769i	− 0.109393i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−11.0000	−0.0380623
$290$	0	0
$291$	−255.000	−0.876289
$292$	0	0
$293$	− 278.860i	− 0.951741i	−0.879515	−	0.475871i	$-0.842133\pi$
$293$	− 278.860i	− 0.951741i	0.879515	−	0.475871i	$-0.157867\pi$
$294$	0	0
$295$	129.000	0.437288
$296$	0	0
$297$	15.5885i	0.0524864i
$298$	0	0
$299$	− 249.415i	− 0.834165i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−36.0000	−0.118812
$304$	0	0
$305$	−120.000	−0.393443
$306$	0	0
$307$	− 27.7128i	− 0.0902697i	−0.998981	−	0.0451349i	$-0.985628\pi$
$307$	− 27.7128i	− 0.0902697i	0.998981	−	0.0451349i	$-0.0143718\pi$
$308$	0	0
$309$	−264.000	−0.854369
$310$	0	0
$311$	65.8179i	0.211633i	0.994386	+	0.105817i	$0.0337456\pi$
$311$	65.8179i	0.211633i	−0.994386	+	0.105817i	$0.966254\pi$
$312$	0	0
$313$	− 119.512i	− 0.381826i	−0.981607	−	0.190913i	$-0.938855\pi$
$313$	− 119.512i	− 0.381826i	0.981607	−	0.190913i	$-0.0611448\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−201.000	−0.634069	−0.317035	−	0.948414i	$-0.602687\pi$
$317$	−201.000	−0.634069	−0.317035	+	0.948414i	$0.602687\pi$
$318$	0	0
$319$	153.000	0.479624
$320$	0	0
$321$	− 57.1577i	− 0.178061i
$322$	0	0
$323$	−180.000	−0.557276
$324$	0	0
$325$	− 152.420i	− 0.468986i
$326$	0	0
$327$	− 55.4256i	− 0.169497i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	448.000	1.35347	0.676737	−	0.736225i	$-0.263393\pi$
$331$	448.000	1.35347	0.676737	+	0.736225i	$0.263393\pi$
$332$	0	0
$333$	−66.0000	−0.198198
$334$	0	0
$335$	− 117.779i	− 0.351580i
$336$	0	0
$337$	−293.000	−0.869436	−0.434718	−	0.900567i	$-0.643152\pi$
$337$	−293.000	−0.869436	−0.434718	+	0.900567i	$0.643152\pi$
$338$	0	0
$339$	− 322.161i	− 0.950329i
$340$	0	0
$341$	36.3731i	0.106666i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	108.000	0.313043
$346$	0	0
$347$	390.000	1.12392	0.561960	−	0.827165i	$-0.310048\pi$
$347$	390.000	1.12392	0.561960	+	0.827165i	$0.310048\pi$
$348$	0	0
$349$	138.564i	0.397032i	0.980098	+	0.198516i	$0.0636121\pi$
$349$	138.564i	0.397032i	−0.980098	+	0.198516i	$0.936388\pi$
$350$	0	0
$351$	−36.0000	−0.102564
$352$	0	0
$353$	− 505.759i	− 1.43274i	−0.697718	−	0.716372i	$-0.745801\pi$
$353$	− 505.759i	− 1.43274i	0.697718	−	0.716372i	$-0.254199\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	300.000	0.835655	0.417827	−	0.908526i	$-0.362792\pi$
$359$	300.000	0.835655	0.417827	+	0.908526i	$0.362792\pi$
$360$	0	0
$361$	253.000	0.700831
$362$	0	0
$363$	− 193.990i	− 0.534407i
$364$	0	0
$365$	−36.0000	−0.0986301
$366$	0	0
$367$	− 202.650i	− 0.552180i	−0.961132	−	0.276090i	$-0.910961\pi$
$367$	− 202.650i	− 0.552180i	0.961132	−	0.276090i	$-0.0890387\pi$
$368$	0	0
$369$	72.7461i	0.197144i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	410.000	1.09920	0.549598	−	0.835429i	$-0.314781\pi$
$373$	410.000	1.09920	0.549598	+	0.835429i	$0.314781\pi$
$374$	0	0
$375$	141.000	0.376000
$376$	0	0
$377$	353.338i	0.937237i
$378$	0	0
$379$	−374.000	−0.986807	−0.493404	−	0.869800i	$-0.664247\pi$
$379$	−374.000	−0.986807	−0.493404	+	0.869800i	$0.664247\pi$
$380$	0	0
$381$	226.899i	0.595535i
$382$	0	0
$383$	287.520i	0.750706i	0.926882	+	0.375353i	$0.122479\pi$
$383$	287.520i	0.750706i	−0.926882	+	0.375353i	$0.877521\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−30.0000	−0.0775194
$388$	0	0
$389$	−138.000	−0.354756	−0.177378	−	0.984143i	$-0.556761\pi$
$389$	−138.000	−0.354756	−0.177378	+	0.984143i	$0.556761\pi$
$390$	0	0
$391$	− 623.538i	− 1.59473i
$392$	0	0
$393$	171.000	0.435115
$394$	0	0
$395$	− 216.506i	− 0.548117i
$396$	0	0
$397$	− 616.610i	− 1.55317i	−0.630010	−	0.776587i	$-0.716949\pi$
$397$	− 616.610i	− 1.55317i	0.630010	−	0.776587i	$-0.283051\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	516.000	1.28678	0.643392	−	0.765537i	$-0.277527\pi$
$401$	516.000	1.28678	0.643392	+	0.765537i	$0.277527\pi$
$402$	0	0
$403$	−84.0000	−0.208437
$404$	0	0
$405$	− 15.5885i	− 0.0384900i
$406$	0	0
$407$	−66.0000	−0.162162
$408$	0	0
$409$	261.540i	0.639461i	0.947508	+	0.319731i	$0.103592\pi$
$409$	261.540i	0.639461i	−0.947508	+	0.319731i	$0.896408\pi$
$410$	0	0
$411$	− 249.415i	− 0.606850i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−267.000	−0.643373
$416$	0	0
$417$	258.000	0.618705
$418$	0	0
$419$	533.472i	1.27320i	0.771193	+	0.636601i	$0.219660\pi$
$419$	533.472i	1.27320i	−0.771193	+	0.636601i	$0.780340\pi$
$420$	0	0
$421$	−76.0000	−0.180523	−0.0902613	−	0.995918i	$-0.528770\pi$
$421$	−76.0000	−0.180523	−0.0902613	+	0.995918i	$0.528770\pi$
$422$	0	0
$423$	270.200i	0.638771i
$424$	0	0
$425$	− 381.051i	− 0.896591i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−36.0000	−0.0839161
$430$	0	0
$431$	606.000	1.40603	0.703016	−	0.711174i	$-0.251836\pi$
$431$	606.000	1.40603	0.703016	+	0.711174i	$0.251836\pi$
$432$	0	0
$433$	− 34.6410i	− 0.0800023i	−0.999200	−	0.0400012i	$-0.987264\pi$
$433$	− 34.6410i	− 0.0800023i	0.999200	−	0.0400012i	$-0.0127362\pi$
$434$	0	0
$435$	−153.000	−0.351724
$436$	0	0
$437$	− 374.123i	− 0.856117i
$438$	0	0
$439$	− 98.7269i	− 0.224890i	−0.993658	−	0.112445i	$-0.964132\pi$
$439$	− 98.7269i	− 0.224890i	0.993658	−	0.112445i	$-0.0358683\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	621.000	1.40181	0.700903	−	0.713257i	$-0.252781\pi$
$443$	621.000	1.40181	0.700903	+	0.713257i	$0.252781\pi$
$444$	0	0
$445$	−126.000	−0.283146
$446$	0	0
$447$	− 259.808i	− 0.581225i
$448$	0	0
$449$	504.000	1.12249	0.561247	−	0.827648i	$-0.310322\pi$
$449$	504.000	1.12249	0.561247	+	0.827648i	$0.310322\pi$
$450$	0	0
$451$	72.7461i	0.161300i
$452$	0	0
$453$	− 403.568i	− 0.890878i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−443.000	−0.969365	−0.484683	−	0.874690i	$-0.661065\pi$
$457$	−443.000	−0.969365	−0.484683	+	0.874690i	$0.661065\pi$
$458$	0	0
$459$	−90.0000	−0.196078
$460$	0	0
$461$	− 180.133i	− 0.390745i	−0.980729	−	0.195372i	$-0.937408\pi$
$461$	− 180.133i	− 0.390745i	0.980729	−	0.195372i	$-0.0625915\pi$
$462$	0	0
$463$	158.000	0.341253	0.170626	−	0.985336i	$-0.445421\pi$
$463$	158.000	0.341253	0.170626	+	0.985336i	$0.445421\pi$
$464$	0	0
$465$	− 36.3731i	− 0.0782216i
$466$	0	0
$467$	− 20.7846i	− 0.0445067i	−0.999752	−	0.0222533i	$-0.992916\pi$
$467$	− 20.7846i	− 0.0445067i	0.999752	−	0.0222533i	$-0.00708404\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−252.000	−0.535032
$472$	0	0
$473$	−30.0000	−0.0634249
$474$	0	0
$475$	− 228.631i	− 0.481328i
$476$	0	0
$477$	−153.000	−0.320755
$478$	0	0
$479$	412.228i	0.860601i	0.902686	+	0.430301i	$0.141592\pi$
$479$	412.228i	0.860601i	−0.902686	+	0.430301i	$0.858408\pi$
$480$	0	0
$481$	− 152.420i	− 0.316882i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	255.000	0.525773
$486$	0	0
$487$	115.000	0.236140	0.118070	−	0.993005i	$-0.462329\pi$
$487$	115.000	0.236140	0.118070	+	0.993005i	$0.462329\pi$
$488$	0	0
$489$	− 256.344i	− 0.524220i
$490$	0	0
$491$	273.000	0.556008	0.278004	−	0.960580i	$-0.410327\pi$
$491$	273.000	0.556008	0.278004	+	0.960580i	$0.410327\pi$
$492$	0	0
$493$	883.346i	1.79178i
$494$	0	0
$495$	− 15.5885i	− 0.0314918i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	466.000	0.933868	0.466934	−	0.884292i	$-0.345359\pi$
$499$	466.000	0.933868	0.466934	+	0.884292i	$0.345359\pi$
$500$	0	0
$501$	48.0000	0.0958084
$502$	0	0
$503$	723.997i	1.43936i	0.694307	+	0.719679i	$0.255711\pi$
$503$	723.997i	1.43936i	−0.694307	+	0.719679i	$0.744289\pi$
$504$	0	0
$505$	36.0000	0.0712871
$506$	0	0
$507$	209.578i	0.413369i
$508$	0	0
$509$	355.070i	0.697584i	0.937200	+	0.348792i	$0.113408\pi$
$509$	355.070i	0.697584i	−0.937200	+	0.348792i	$0.886592\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−54.0000	−0.105263
$514$	0	0
$515$	264.000	0.512621
$516$	0	0
$517$	270.200i	0.522630i
$518$	0	0
$519$	−324.000	−0.624277
$520$	0	0
$521$	− 917.987i	− 1.76197i	−0.473143	−	0.880986i	$-0.656881\pi$
$521$	− 917.987i	− 1.76197i	0.473143	−	0.880986i	$-0.343119\pi$
$522$	0	0
$523$	− 484.974i	− 0.927293i	−0.886020	−	0.463646i	$-0.846541\pi$
$523$	− 484.974i	− 0.927293i	0.886020	−	0.463646i	$-0.153459\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−210.000	−0.398482
$528$	0	0
$529$	767.000	1.44991
$530$	0	0
$531$	− 223.435i	− 0.420781i
$532$	0	0
$533$	−168.000	−0.315197
$534$	0	0
$535$	57.1577i	0.106837i
$536$	0	0
$537$	− 280.592i	− 0.522518i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	62.0000	0.114603	0.0573013	−	0.998357i	$-0.481750\pi$
$541$	62.0000	0.114603	0.0573013	+	0.998357i	$0.481750\pi$
$542$	0	0
$543$	−450.000	−0.828729
$544$	0	0
$545$	55.4256i	0.101698i
$546$	0	0
$547$	−802.000	−1.46618	−0.733090	−	0.680132i	$-0.761922\pi$
$547$	−802.000	−1.46618	−0.733090	+	0.680132i	$0.761922\pi$
$548$	0	0
$549$	207.846i	0.378590i
$550$	0	0
$551$	530.008i	0.961901i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	66.0000	0.118919
$556$	0	0
$557$	−111.000	−0.199282	−0.0996409	−	0.995023i	$-0.531769\pi$
$557$	−111.000	−0.199282	−0.0996409	+	0.995023i	$0.531769\pi$
$558$	0	0
$559$	− 69.2820i	− 0.123939i
$560$	0	0
$561$	−90.0000	−0.160428
$562$	0	0
$563$	192.258i	0.341488i	0.985315	+	0.170744i	$0.0546171\pi$
$563$	192.258i	0.341488i	−0.985315	+	0.170744i	$0.945383\pi$
$564$	0	0
$565$	322.161i	0.570197i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−426.000	−0.748682	−0.374341	−	0.927291i	$-0.622131\pi$
$569$	−426.000	−0.748682	−0.374341	+	0.927291i	$0.622131\pi$
$570$	0	0
$571$	−580.000	−1.01576	−0.507881	−	0.861427i	$-0.669571\pi$
$571$	−580.000	−1.01576	−0.507881	+	0.861427i	$0.669571\pi$
$572$	0	0
$573$	166.277i	0.290187i
$574$	0	0
$575$	792.000	1.37739
$576$	0	0
$577$	− 964.752i	− 1.67201i	−0.548718	−	0.836007i	$-0.684884\pi$
$577$	− 964.752i	− 1.67201i	0.548718	−	0.836007i	$-0.315116\pi$
$578$	0	0
$579$	510.955i	0.882478i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−153.000	−0.262436
$584$	0	0
$585$	36.0000	0.0615385
$586$	0	0
$587$	1002.86i	1.70845i	0.519907	+	0.854223i	$0.325966\pi$
$587$	1002.86i	1.70845i	−0.519907	+	0.854223i	$0.674034\pi$
$588$	0	0
$589$	−126.000	−0.213922
$590$	0	0
$591$	− 571.577i	− 0.967135i
$592$	0	0
$593$	879.882i	1.48378i	0.670521	+	0.741890i	$0.266070\pi$
$593$	879.882i	1.48378i	−0.670521	+	0.741890i	$0.733930\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	204.000	0.341709
$598$	0	0
$599$	−156.000	−0.260434	−0.130217	−	0.991486i	$-0.541567\pi$
$599$	−156.000	−0.260434	−0.130217	+	0.991486i	$0.541567\pi$
$600$	0	0
$601$	− 1124.10i	− 1.87038i	−0.354141	−	0.935192i	$-0.615227\pi$
$601$	− 1124.10i	− 1.87038i	0.354141	−	0.935192i	$-0.384773\pi$
$602$	0	0
$603$	−204.000	−0.338308
$604$	0	0
$605$	193.990i	0.320644i
$606$	0	0
$607$	400.104i	0.659149i	0.944129	+	0.329575i	$0.106905\pi$
$607$	400.104i	0.659149i	−0.944129	+	0.329575i	$0.893095\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−624.000	−1.02128
$612$	0	0
$613$	−856.000	−1.39641	−0.698206	−	0.715897i	$-0.746018\pi$
$613$	−856.000	−1.39641	−0.698206	+	0.715897i	$0.746018\pi$
$614$	0	0
$615$	− 72.7461i	− 0.118286i
$616$	0	0
$617$	444.000	0.719611	0.359806	−	0.933027i	$-0.382843\pi$
$617$	444.000	0.719611	0.359806	+	0.933027i	$0.382843\pi$
$618$	0	0
$619$	− 256.344i	− 0.414125i	−0.978328	−	0.207063i	$-0.933610\pi$
$619$	− 256.344i	− 0.414125i	0.978328	−	0.207063i	$-0.0663904\pi$
$620$	0	0
$621$	− 187.061i	− 0.301226i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	409.000	0.654400
$626$	0	0
$627$	−54.0000	−0.0861244
$628$	0	0
$629$	− 381.051i	− 0.605805i
$630$	0	0
$631$	−85.0000	−0.134707	−0.0673534	−	0.997729i	$-0.521455\pi$
$631$	−85.0000	−0.134707	−0.0673534	+	0.997729i	$0.521455\pi$
$632$	0	0
$633$	381.051i	0.601977i
$634$	0	0
$635$	− 226.899i	− 0.357321i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−132.000	−0.205928	−0.102964	−	0.994685i	$-0.532833\pi$
$641$	−132.000	−0.205928	−0.102964	+	0.994685i	$0.532833\pi$
$642$	0	0
$643$	713.605i	1.10981i	0.831915	+	0.554903i	$0.187245\pi$
$643$	713.605i	1.10981i	−0.831915	+	0.554903i	$0.812755\pi$
$644$	0	0
$645$	30.0000	0.0465116
$646$	0	0
$647$	997.661i	1.54198i	0.636847	+	0.770990i	$0.280238\pi$
$647$	997.661i	1.54198i	−0.636847	+	0.770990i	$0.719762\pi$
$648$	0	0
$649$	− 223.435i	− 0.344275i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−375.000	−0.574273	−0.287136	−	0.957890i	$-0.592703\pi$
$653$	−375.000	−0.574273	−0.287136	+	0.957890i	$0.592703\pi$
$654$	0	0
$655$	−171.000	−0.261069
$656$	0	0
$657$	62.3538i	0.0949069i
$658$	0	0
$659$	1038.00	1.57511	0.787557	−	0.616242i	$-0.211346\pi$
$659$	1038.00	1.57511	0.787557	+	0.616242i	$0.211346\pi$
$660$	0	0
$661$	446.869i	0.676050i	0.941137	+	0.338025i	$0.109759\pi$
$661$	446.869i	0.676050i	−0.941137	+	0.338025i	$0.890241\pi$
$662$	0	0
$663$	− 207.846i	− 0.313493i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1836.00	−2.75262
$668$	0	0
$669$	−261.000	−0.390135
$670$	0	0
$671$	207.846i	0.309756i
$672$	0	0
$673$	1093.00	1.62407	0.812036	−	0.583608i	$-0.198359\pi$
$673$	1093.00	1.62407	0.812036	+	0.583608i	$0.198359\pi$
$674$	0	0
$675$	− 114.315i	− 0.169356i
$676$	0	0
$677$	1210.70i	1.78834i	0.447732	+	0.894168i	$0.352232\pi$
$677$	1210.70i	1.78834i	−0.447732	+	0.894168i	$0.647768\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−189.000	−0.277533
$682$	0	0
$683$	−3.00000	−0.00439239	−0.00219619	−	0.999998i	$-0.500699\pi$
$683$	−3.00000	−0.00439239	−0.00219619	+	0.999998i	$0.500699\pi$
$684$	0	0
$685$	249.415i	0.364110i
$686$	0	0
$687$	−654.000	−0.951965
$688$	0	0
$689$	− 353.338i	− 0.512828i
$690$	0	0
$691$	1139.69i	1.64933i	0.565619	+	0.824667i	$0.308637\pi$
$691$	1139.69i	1.64933i	−0.565619	+	0.824667i	$0.691363\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−258.000	−0.371223
$696$	0	0
$697$	−420.000	−0.602582
$698$	0	0
$699$	− 363.731i	− 0.520359i
$700$	0	0
$701$	447.000	0.637660	0.318830	−	0.947812i	$-0.396710\pi$
$701$	447.000	0.637660	0.318830	+	0.947812i	$0.396710\pi$
$702$	0	0
$703$	− 228.631i	− 0.325221i
$704$	0	0
$705$	− 270.200i	− 0.383262i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−422.000	−0.595205	−0.297602	−	0.954690i	$-0.596187\pi$
$709$	−422.000	−0.595205	−0.297602	+	0.954690i	$0.596187\pi$
$710$	0	0
$711$	−375.000	−0.527426
$712$	0	0
$713$	− 436.477i	− 0.612169i
$714$	0	0
$715$	36.0000	0.0503497
$716$	0	0
$717$	207.846i	0.289883i
$718$	0	0
$719$	− 169.741i	− 0.236079i	−0.993009	−	0.118040i	$-0.962339\pi$
$719$	− 169.741i	− 0.236079i	0.993009	−	0.118040i	$-0.0376610\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	21.0000	0.0290456
$724$	0	0
$725$	−1122.00	−1.54759
$726$	0	0
$727$	− 50.2295i	− 0.0690914i	−0.999403	−	0.0345457i	$-0.989002\pi$
$727$	− 50.2295i	− 0.0690914i	0.999403	−	0.0345457i	$-0.0109984\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 173.205i	− 0.236943i
$732$	0	0
$733$	− 294.449i	− 0.401703i	−0.979622	−	0.200852i	$-0.935629\pi$
$733$	− 294.449i	− 0.401703i	0.979622	−	0.200852i	$-0.0643709\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−204.000	−0.276798
$738$	0	0
$739$	98.0000	0.132612	0.0663058	−	0.997799i	$-0.478879\pi$
$739$	98.0000	0.132612	0.0663058	+	0.997799i	$0.478879\pi$
$740$	0	0
$741$	− 124.708i	− 0.168296i
$742$	0	0
$743$	−180.000	−0.242261	−0.121131	−	0.992637i	$-0.538652\pi$
$743$	−180.000	−0.242261	−0.121131	+	0.992637i	$0.538652\pi$
$744$	0	0
$745$	259.808i	0.348735i
$746$	0	0
$747$	462.458i	0.619086i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	721.000	0.960053	0.480027	−	0.877254i	$-0.340627\pi$
$751$	721.000	0.960053	0.480027	+	0.877254i	$0.340627\pi$
$752$	0	0
$753$	237.000	0.314741
$754$	0	0
$755$	403.568i	0.534527i
$756$	0	0
$757$	−644.000	−0.850727	−0.425363	−	0.905023i	$-0.639854\pi$
$757$	−644.000	−0.850727	−0.425363	+	0.905023i	$0.639854\pi$
$758$	0	0
$759$	− 187.061i	− 0.246458i
$760$	0	0
$761$	568.113i	0.746534i	0.927724	+	0.373267i	$0.121763\pi$
$761$	568.113i	0.746534i	−0.927724	+	0.373267i	$0.878237\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	90.0000	0.117647
$766$	0	0
$767$	516.000	0.672751
$768$	0	0
$769$	275.396i	0.358122i	0.983838	+	0.179061i	$0.0573060\pi$
$769$	275.396i	0.358122i	−0.983838	+	0.179061i	$0.942694\pi$
$770$	0	0
$771$	234.000	0.303502
$772$	0	0
$773$	− 1004.59i	− 1.29960i	−0.760106	−	0.649799i	$-0.774853\pi$
$773$	− 1004.59i	− 1.29960i	0.760106	−	0.649799i	$-0.225147\pi$
$774$	0	0
$775$	− 266.736i	− 0.344175i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−252.000	−0.323492
$780$	0	0
$781$	0	0
$782$	0	0
$783$	265.004i	0.338447i
$784$	0	0
$785$	252.000	0.321019
$786$	0	0
$787$	796.743i	1.01238i	0.862422	+	0.506190i	$0.168947\pi$
$787$	796.743i	1.01238i	−0.862422	+	0.506190i	$0.831053\pi$
$788$	0	0
$789$	280.592i	0.355630i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−480.000	−0.605296
$794$	0	0
$795$	153.000	0.192453
$796$	0	0
$797$	545.596i	0.684562i	0.939598	+	0.342281i	$0.111199\pi$
$797$	545.596i	0.684562i	−0.939598	+	0.342281i	$0.888801\pi$
$798$	0	0
$799$	−1560.00	−1.95244
$800$	0	0
$801$	218.238i	0.272457i
$802$	0	0
$803$	62.3538i	0.0776511i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−669.000	−0.828996
$808$	0	0
$809$	48.0000	0.0593325	0.0296663	−	0.999560i	$-0.490556\pi$
$809$	48.0000	0.0593325	0.0296663	+	0.999560i	$0.490556\pi$
$810$	0	0
$811$	− 436.477i	− 0.538196i	−0.963113	−	0.269098i	$-0.913274\pi$
$811$	− 436.477i	− 0.538196i	0.963113	−	0.269098i	$-0.0867255\pi$
$812$	0	0
$813$	375.000	0.461255
$814$	0	0
$815$	256.344i	0.314532i
$816$	0	0
$817$	− 103.923i	− 0.127201i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1191.00	1.45067	0.725335	−	0.688396i	$-0.241685\pi$
$821$	1191.00	1.45067	0.725335	+	0.688396i	$0.241685\pi$
$822$	0	0
$823$	−250.000	−0.303767	−0.151883	−	0.988398i	$-0.548534\pi$
$823$	−250.000	−0.303767	−0.151883	+	0.988398i	$0.548534\pi$
$824$	0	0
$825$	− 114.315i	− 0.138564i
$826$	0	0
$827$	−1137.00	−1.37485	−0.687424	−	0.726256i	$-0.741259\pi$
$827$	−1137.00	−1.37485	−0.687424	+	0.726256i	$0.741259\pi$
$828$	0	0
$829$	72.7461i	0.0877517i	0.999037	+	0.0438758i	$0.0139706\pi$
$829$	72.7461i	0.0877517i	−0.999037	+	0.0438758i	$0.986029\pi$
$830$	0	0
$831$	− 173.205i	− 0.208430i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−48.0000	−0.0574850
$836$	0	0
$837$	−63.0000	−0.0752688
$838$	0	0
$839$	1416.82i	1.68870i	0.535794	+	0.844349i	$0.320012\pi$
$839$	1416.82i	1.68870i	−0.535794	+	0.844349i	$0.679988\pi$
$840$	0	0
$841$	1760.00	2.09275
$842$	0	0
$843$	394.908i	0.468455i
$844$	0	0
$845$	− 209.578i	− 0.248021i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−30.0000	−0.0353357
$850$	0	0
$851$	792.000	0.930670
$852$	0	0
$853$	− 872.954i	− 1.02339i	−0.859166	−	0.511696i	$-0.829017\pi$
$853$	− 872.954i	− 1.02339i	0.859166	−	0.511696i	$-0.170983\pi$
$854$	0	0
$855$	54.0000	0.0631579
$856$	0	0
$857$	− 692.820i	− 0.808425i	−0.914665	−	0.404213i	$-0.867546\pi$
$857$	− 692.820i	− 0.808425i	0.914665	−	0.404213i	$-0.132454\pi$
$858$	0	0
$859$	890.274i	1.03641i	0.855257	+	0.518204i	$0.173399\pi$
$859$	890.274i	1.03641i	−0.855257	+	0.518204i	$0.826601\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	630.000	0.730012	0.365006	−	0.931005i	$-0.381067\pi$
$863$	630.000	0.730012	0.365006	+	0.931005i	$0.381067\pi$
$864$	0	0
$865$	324.000	0.374566
$866$	0	0
$867$	− 19.0526i	− 0.0219753i
$868$	0	0
$869$	−375.000	−0.431530
$870$	0	0
$871$	− 471.118i	− 0.540893i
$872$	0	0
$873$	− 441.673i	− 0.505925i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1640.00	1.87001	0.935006	−	0.354633i	$-0.115394\pi$
$877$	1640.00	1.87001	0.935006	+	0.354633i	$0.115394\pi$
$878$	0	0
$879$	483.000	0.549488
$880$	0	0
$881$	− 1267.86i	− 1.43912i	−0.694432	−	0.719558i	$-0.744344\pi$
$881$	− 1267.86i	− 1.43912i	0.694432	−	0.719558i	$-0.255656\pi$
$882$	0	0
$883$	226.000	0.255946	0.127973	−	0.991778i	$-0.459153\pi$
$883$	226.000	0.255946	0.127973	+	0.991778i	$0.459153\pi$
$884$	0	0
$885$	223.435i	0.252468i
$886$	0	0
$887$	− 6.92820i	− 0.00781083i	−0.999992	−	0.00390541i	$-0.998757\pi$
$887$	− 6.92820i	− 0.00781083i	0.999992	−	0.00390541i	$-0.00124313\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−27.0000	−0.0303030
$892$	0	0
$893$	−936.000	−1.04815
$894$	0	0
$895$	280.592i	0.313511i
$896$	0	0
$897$	432.000	0.481605
$898$	0	0
$899$	618.342i	0.687811i
$900$	0	0
$901$	− 883.346i	− 0.980406i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	450.000	0.497238
$906$	0	0
$907$	1144.00	1.26130	0.630650	−	0.776067i	$-0.282788\pi$
$907$	1144.00	1.26130	0.630650	+	0.776067i	$0.282788\pi$
$908$	0	0
$909$	− 62.3538i	− 0.0685961i
$910$	0	0
$911$	−480.000	−0.526894	−0.263447	−	0.964674i	$-0.584859\pi$
$911$	−480.000	−0.526894	−0.263447	+	0.964674i	$0.584859\pi$
$912$	0	0
$913$	462.458i	0.506525i
$914$	0	0
$915$	− 207.846i	− 0.227154i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1742.00	−1.89554	−0.947769	−	0.318957i	$-0.896668\pi$
$919$	−1742.00	−1.89554	−0.947769	+	0.318957i	$0.896668\pi$
$920$	0	0
$921$	48.0000	0.0521173
$922$	0	0
$923$	0	0
$924$	0	0
$925$	484.000	0.523243
$926$	0	0
$927$	− 457.261i	− 0.493270i
$928$	0	0
$929$	− 1160.47i	− 1.24916i	−0.780959	−	0.624582i	$-0.785269\pi$
$929$	− 1160.47i	− 1.24916i	0.780959	−	0.624582i	$-0.214731\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−114.000	−0.122186
$934$	0	0
$935$	90.0000	0.0962567
$936$	0	0
$937$	1331.95i	1.42150i	0.703444	+	0.710751i	$0.251645\pi$
$937$	1331.95i	1.42150i	−0.703444	+	0.710751i	$0.748355\pi$
$938$	0	0
$939$	207.000	0.220447
$940$	0	0
$941$	812.332i	0.863264i	0.902050	+	0.431632i	$0.142062\pi$
$941$	812.332i	0.863264i	−0.902050	+	0.431632i	$0.857938\pi$
$942$	0	0
$943$	− 872.954i	− 0.925720i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1290.00	−1.36220	−0.681098	−	0.732192i	$-0.738497\pi$
$947$	−1290.00	−1.36220	−0.681098	+	0.732192i	$0.738497\pi$
$948$	0	0
$949$	−144.000	−0.151739
$950$	0	0
$951$	− 348.142i	− 0.366080i
$952$	0	0
$953$	174.000	0.182581	0.0912907	−	0.995824i	$-0.470901\pi$
$953$	174.000	0.182581	0.0912907	+	0.995824i	$0.470901\pi$
$954$	0	0
$955$	− 166.277i	− 0.174112i
$956$	0	0
$957$	265.004i	0.276911i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	814.000	0.847034
$962$	0	0
$963$	99.0000	0.102804
$964$	0	0
$965$	− 510.955i	− 0.529487i
$966$	0	0
$967$	41.0000	0.0423992	0.0211996	−	0.999775i	$-0.493251\pi$
$967$	41.0000	0.0423992	0.0211996	+	0.999775i	$0.493251\pi$
$968$	0	0
$969$	− 311.769i	− 0.321743i
$970$	0	0
$971$	− 985.537i	− 1.01497i	−0.861660	−	0.507486i	$-0.830575\pi$
$971$	− 985.537i	− 1.01497i	0.861660	−	0.507486i	$-0.169425\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	264.000	0.270769
$976$	0	0
$977$	1278.00	1.30809	0.654043	−	0.756457i	$-0.273072\pi$
$977$	1278.00	1.30809	0.654043	+	0.756457i	$0.273072\pi$
$978$	0	0
$979$	218.238i	0.222920i
$980$	0	0
$981$	96.0000	0.0978593
$982$	0	0
$983$	− 232.095i	− 0.236109i	−0.993007	−	0.118054i	$-0.962334\pi$
$983$	− 232.095i	− 0.236109i	0.993007	−	0.118054i	$-0.0376657\pi$
$984$	0	0
$985$	571.577i	0.580281i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	360.000	0.364004
$990$	0	0
$991$	−833.000	−0.840565	−0.420283	−	0.907393i	$-0.638069\pi$
$991$	−833.000	−0.840565	−0.420283	+	0.907393i	$0.638069\pi$
$992$	0	0
$993$	775.959i	0.781429i
$994$	0	0
$995$	−204.000	−0.205025
$996$	0	0
$997$	1170.87i	1.17439i	0.809446	+	0.587195i	$0.199768\pi$
$997$	1170.87i	1.17439i	−0.809446	+	0.587195i	$0.800232\pi$
$998$	0	0
$999$	− 114.315i	− 0.114430i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.3.d.a.97.2		2
3.2	odd	2		1764.3.d.c.685.2		2
4.3	odd	2		2352.3.f.c.97.1		2
7.2	even	3		588.3.m.a.325.1		2
7.3	odd	6		588.3.m.a.313.1		2
7.4	even	3		84.3.m.a.61.1	✓	2
7.5	odd	6		84.3.m.a.73.1	yes	2
7.6	odd	2	inner	588.3.d.a.97.1		2
21.2	odd	6		1764.3.z.e.325.1		2
21.5	even	6		252.3.z.b.73.1		2
21.11	odd	6		252.3.z.b.145.1		2
21.17	even	6		1764.3.z.e.901.1		2
21.20	even	2		1764.3.d.c.685.1		2
28.11	odd	6		336.3.bh.b.145.1		2
28.19	even	6		336.3.bh.b.241.1		2
28.27	even	2		2352.3.f.c.97.2		2
35.4	even	6		2100.3.bd.b.901.1		2
35.12	even	12		2100.3.be.c.1249.2		4
35.18	odd	12		2100.3.be.c.649.2		4
35.19	odd	6		2100.3.bd.b.1501.1		2
35.32	odd	12		2100.3.be.c.649.1		4
35.33	even	12		2100.3.be.c.1249.1		4
84.11	even	6		1008.3.cg.b.145.1		2
84.47	odd	6		1008.3.cg.b.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.m.a.61.1	✓	2	7.4	even	3
84.3.m.a.73.1	yes	2	7.5	odd	6
252.3.z.b.73.1		2	21.5	even	6
252.3.z.b.145.1		2	21.11	odd	6
336.3.bh.b.145.1		2	28.11	odd	6
336.3.bh.b.241.1		2	28.19	even	6
588.3.d.a.97.1		2	7.6	odd	2	inner
588.3.d.a.97.2		2	1.1	even	1	trivial
588.3.m.a.313.1		2	7.3	odd	6
588.3.m.a.325.1		2	7.2	even	3
1008.3.cg.b.145.1		2	84.11	even	6
1008.3.cg.b.577.1		2	84.47	odd	6
1764.3.d.c.685.1		2	21.20	even	2
1764.3.d.c.685.2		2	3.2	odd	2
1764.3.z.e.325.1		2	21.2	odd	6
1764.3.z.e.901.1		2	21.17	even	6
2100.3.bd.b.901.1		2	35.4	even	6
2100.3.bd.b.1501.1		2	35.19	odd	6
2100.3.be.c.649.1		4	35.32	odd	12
2100.3.be.c.649.2		4	35.18	odd	12
2100.3.be.c.1249.1		4	35.33	even	12
2100.3.be.c.1249.2		4	35.12	even	12
2352.3.f.c.97.1		2	4.3	odd	2
2352.3.f.c.97.2		2	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	588.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -1.73205i q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -1.73205i q^{5} -3.00000 q^{9} -3.00000 q^{11} -6.92820i q^{13} +3.00000 q^{15} -17.3205i q^{17} -10.3923i q^{19} +36.0000 q^{23} +22.0000 q^{25} -5.19615i q^{27} -51.0000 q^{29} -12.1244i q^{31} -5.19615i q^{33} +22.0000 q^{37} +12.0000 q^{39} -24.2487i q^{41} +10.0000 q^{43} +5.19615i q^{45} -90.0666i q^{47} +30.0000 q^{51} +51.0000 q^{53} +5.19615i q^{55} +18.0000 q^{57} +74.4782i q^{59} -69.2820i q^{61} -12.0000 q^{65} +68.0000 q^{67} +62.3538i q^{69} -20.7846i q^{73} +38.1051i q^{75} +125.000 q^{79} +9.00000 q^{81} -154.153i q^{83} -30.0000 q^{85} -88.3346i q^{87} -72.7461i q^{89} +21.0000 q^{93} -18.0000 q^{95} +147.224i q^{97} +9.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9}+O(q^{10}) \) 2 * q - 6 * q^9	\( 2 q - 6 q^{9} - 6 q^{11} + 6 q^{15} + 72 q^{23} + 44 q^{25} - 102 q^{29} + 44 q^{37} + 24 q^{39} + 20 q^{43} + 60 q^{51} + 102 q^{53} + 36 q^{57} - 24 q^{65} + 136 q^{67} + 250 q^{79} + 18 q^{81} - 60 q^{85} + 42 q^{93} - 36 q^{95} + 18 q^{99}+O(q^{100}) \) 2 * q - 6 * q^9 - 6 * q^11 + 6 * q^15 + 72 * q^23 + 44 * q^25 - 102 * q^29 + 44 * q^37 + 24 * q^39 + 20 * q^43 + 60 * q^51 + 102 * q^53 + 36 * q^57 - 24 * q^65 + 136 * q^67 + 250 * q^79 + 18 * q^81 - 60 * q^85 + 42 * q^93 - 36 * q^95 + 18 * q^99

Introduction

L-functions

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