LMFDB - Embedded newform 468.4.l.d.217.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [468,4,Mod(217,468)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(468, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("468.217");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 468 = 2^{2} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	468.l (of order $3$, degree $2$, 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:	$27.6128938827$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 122x^{6} - 1188x^{5} + 15573x^{4} - 72468x^{3} + 268778x^{2} - 409266x + 474721 $ x^8 + 122x^6 - 1188x^5 + 15573x^4 - 72468x^3 + 268778x^2 - 409266x + 474721
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.2
Root			$-6.54584 - 11.3377i$ of defining polynomial
Character	$\chi$	$=$	468.217
Dual form			468.4.l.d.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.67342 q^{5} +(9.75498 + 16.8961i) q^{7} +(-2.78958 + 4.83170i) q^{11} +(-46.0772 - 8.59618i) q^{13} +(-52.7112 - 91.2985i) q^{17} +(67.7110 + 117.279i) q^{19} +(22.7699 - 39.4386i) q^{23} -92.8123 q^{25} +(113.179 - 196.031i) q^{29} +31.4566 q^{31} +(-55.3441 - 95.8588i) q^{35} +(-45.1476 + 78.1979i) q^{37} +(-36.3145 + 62.8986i) q^{41} +(-187.727 - 325.152i) q^{43} -637.171 q^{47} +(-18.8191 + 32.5957i) q^{49} -578.656 q^{53} +(15.8265 - 27.4123i) q^{55} +(-38.0433 - 65.8930i) q^{59} +(-378.260 - 655.165i) q^{61} +(261.415 + 48.7698i) q^{65} +(117.722 - 203.901i) q^{67} +(157.403 + 272.630i) q^{71} -407.548 q^{73} -108.849 q^{77} +185.914 q^{79} -694.924 q^{83} +(299.053 + 517.975i) q^{85} +(-6.86898 + 11.8974i) q^{89} +(-304.240 - 862.381i) q^{91} +(-384.153 - 665.373i) q^{95} +(-326.344 - 565.245i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 14 q^{5} - 11 q^{7} - 20 q^{11} + 48 q^{13} + 7 q^{17} - 18 q^{19} - 60 q^{23} + 134 q^{25} + 75 q^{29} + 174 q^{31} + 556 q^{35} + 51 q^{37} + 271 q^{41} + q^{43} + 64 q^{47} - 153 q^{49} - 1514 q^{53}+ \cdots - 2527 q^{97}+O(q^{100}) $ 8 * q - 14 * q^5 - 11 * q^7 - 20 * q^11 + 48 * q^13 + 7 * q^17 - 18 * q^19 - 60 * q^23 + 134 * q^25 + 75 * q^29 + 174 * q^31 + 556 * q^35 + 51 * q^37 + 271 * q^41 + q^43 + 64 * q^47 - 153 * q^49 - 1514 * q^53 - 626 * q^55 - 610 * q^59 - 1166 * q^61 + 649 * q^65 - 1191 * q^67 + 980 * q^71 + 964 * q^73 - 812 * q^77 + 4450 * q^79 + 3036 * q^83 + 2603 * q^85 + 1078 * q^89 - 2695 * q^91 + 154 * q^95 - 2527 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/468\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$235$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−5.67342			−0.507446			−0.253723	−	0.967277i	$-0.581655\pi$
$5$	−5.67342			−0.507446			−0.253723	+	0.967277i	$0.581655\pi$
$6$		0			0
$7$	9.75498	+	16.8961i	0.526719	+	0.912304i	0.999515	+	0.0311326i	$0.00991141\pi$
$7$	9.75498	+	16.8961i	0.526719	+	0.912304i	−0.472796	+	0.881172i	$0.656755\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.78958	+	4.83170i	−0.0764629	+	0.132438i	−0.901721	−	0.432317i	$-0.857696\pi$
$11$	−2.78958	+	4.83170i	−0.0764629	+	0.132438i	0.825259	+	0.564755i	$0.191029\pi$
$12$		0			0
$13$	−46.0772	−	8.59618i	−0.983039	−	0.183396i
$13$	−46.0772	−	8.59618i	−0.983039	−	0.183396i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−52.7112	−	91.2985i	−0.752021	−	1.30254i	−0.946842	−	0.321699i	$-0.895746\pi$
$17$	−52.7112	−	91.2985i	−0.752021	−	1.30254i	0.194821	−	0.980839i	$-0.437587\pi$
$18$		0			0
$19$	67.7110	+	117.279i	0.817578	+	1.41609i	0.907462	+	0.420134i	$0.138017\pi$
$19$	67.7110	+	117.279i	0.817578	+	1.41609i	−0.0898843	+	0.995952i	$0.528650\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	22.7699	−	39.4386i	0.206428	−	0.357544i	−0.744159	−	0.668003i	$-0.767149\pi$
$23$	22.7699	−	39.4386i	0.206428	−	0.357544i	0.950587	+	0.310459i	$0.100483\pi$
$24$		0			0
$25$	−92.8123			−0.742498
$26$		0			0
$27$		0			0
$28$		0			0
$29$	113.179	−	196.031i	0.724715	−	1.25524i	−0.234376	−	0.972146i	$-0.575305\pi$
$29$	113.179	−	196.031i	0.724715	−	1.25524i	0.959091	−	0.283098i	$-0.0913621\pi$
$30$		0			0
$31$	31.4566			0.182251			0.0911254	−	0.995839i	$-0.470954\pi$
$31$	31.4566			0.182251			0.0911254	+	0.995839i	$0.470954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−55.3441	−	95.8588i	−0.267282	−	0.462946i
$36$		0			0
$37$	−45.1476	+	78.1979i	−0.200600	+	0.347450i	−0.948722	−	0.316112i	$-0.897623\pi$
$37$	−45.1476	+	78.1979i	−0.200600	+	0.347450i	0.748122	+	0.663562i	$0.230956\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−36.3145	+	62.8986i	−0.138326	+	0.239588i	−0.926863	−	0.375399i	$-0.877506\pi$
$41$	−36.3145	+	62.8986i	−0.138326	+	0.239588i	0.788537	+	0.614987i	$0.210839\pi$
$42$		0			0
$43$	−187.727	−	325.152i	−0.665768	−	1.15314i	−0.979076	−	0.203493i	$-0.934771\pi$
$43$	−187.727	−	325.152i	−0.665768	−	1.15314i	0.313308	−	0.949652i	$-0.398563\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−637.171			−1.97747			−0.988734	−	0.149682i	$-0.952175\pi$
$47$	−637.171			−1.97747			−0.988734	+	0.149682i	$0.952175\pi$
$48$		0			0
$49$	−18.8191	+	32.5957i	−0.0548663	+	0.0950312i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−578.656			−1.49971			−0.749854	−	0.661603i	$-0.769877\pi$
$53$	−578.656			−1.49971			−0.749854	+	0.661603i	$0.769877\pi$
$54$		0			0
$55$	15.8265	−	27.4123i	0.0388008	−	0.0672050i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−38.0433	−	65.8930i	−0.0839461	−	0.145399i	0.820996	−	0.570935i	$-0.193419\pi$
$59$	−38.0433	−	65.8930i	−0.0839461	−	0.145399i	−0.904942	+	0.425536i	$0.860086\pi$
$60$		0			0
$61$	−378.260	−	655.165i	−0.793954	−	1.37517i	−0.923501	−	0.383596i	$-0.874686\pi$
$61$	−378.260	−	655.165i	−0.793954	−	1.37517i	0.129547	−	0.991573i	$-0.458648\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	261.415	+	48.7698i	0.498840	+	0.0930638i
$66$		0			0
$67$	117.722	−	203.901i	0.214657	−	0.371798i	−0.738509	−	0.674244i	$-0.764470\pi$
$67$	117.722	−	203.901i	0.214657	−	0.371798i	0.953167	+	0.302446i	$0.0978032\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	157.403	+	272.630i	0.263103	+	0.455708i	0.967065	−	0.254530i	$-0.0819207\pi$
$71$	157.403	+	272.630i	0.263103	+	0.455708i	−0.703962	+	0.710238i	$0.748587\pi$
$72$		0			0
$73$	−407.548			−0.653423			−0.326711	−	0.945124i	$-0.605941\pi$
$73$	−407.548			−0.653423			−0.326711	+	0.945124i	$0.605941\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−108.849			−0.161098
$78$		0			0
$79$	185.914			0.264772			0.132386	−	0.991198i	$-0.457736\pi$
$79$	185.914			0.264772			0.132386	+	0.991198i	$0.457736\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−694.924			−0.919010			−0.459505	−	0.888175i	$-0.651973\pi$
$83$	−694.924			−0.919010			−0.459505	+	0.888175i	$0.651973\pi$
$84$		0			0
$85$	299.053	+	517.975i	0.381610	+	0.660968i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.86898	+	11.8974i	−0.00818102	+	0.0141699i	−0.870087	−	0.492898i	$-0.835937\pi$
$89$	−6.86898	+	11.8974i	−0.00818102	+	0.0141699i	0.861906	+	0.507068i	$0.169271\pi$
$90$		0			0
$91$	−304.240	−	862.381i	−0.350472	−	0.993429i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−384.153	−	665.373i	−0.414877	−	0.718588i
$96$		0			0
$97$	−326.344	−	565.245i	−0.341601	−	0.591669i	0.643130	−	0.765757i	$-0.277636\pi$
$97$	−326.344	−	565.245i	−0.341601	−	0.591669i	−0.984730	+	0.174088i	$0.944302\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	191.938	−	332.446i	0.189095	−	0.327521i	−0.755854	−	0.654740i	$-0.772778\pi$
$101$	191.938	−	332.446i	0.189095	−	0.327521i	0.944949	+	0.327219i	$0.106111\pi$
$102$		0			0
$103$	970.879			0.928772			0.464386	−	0.885633i	$-0.346275\pi$
$103$	970.879			0.928772			0.464386	+	0.885633i	$0.346275\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−451.219	+	781.534i	−0.407673	+	0.706110i	−0.994629	−	0.103509i	$-0.966993\pi$
$107$	−451.219	+	781.534i	−0.407673	+	0.706110i	0.586956	+	0.809619i	$0.300326\pi$
$108$		0			0
$109$	−630.574			−0.554111			−0.277055	−	0.960854i	$-0.589359\pi$
$109$	−630.574			−0.554111			−0.277055	+	0.960854i	$0.589359\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−258.209	−	447.231i	−0.214958	−	0.372318i	0.738302	−	0.674471i	$-0.235628\pi$
$113$	−258.209	−	447.231i	−0.214958	−	0.372318i	−0.953260	+	0.302153i	$0.902295\pi$
$114$		0			0
$115$	−129.183	+	223.752i	−0.104751	+	0.181434i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1028.39	−	1781.23i	0.792207	−	1.37214i
$120$		0			0
$121$	649.936	+	1125.72i	0.488307	+	0.845772i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1235.74			0.884224
$126$		0			0
$127$	82.5811	−	143.035i	0.0576999	−	0.0999392i	−0.835733	−	0.549137i	$-0.814957\pi$
$127$	82.5811	−	143.035i	0.0576999	−	0.0999392i	0.893433	+	0.449197i	$0.148290\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2676.77			−1.78527			−0.892635	−	0.450779i	$-0.851146\pi$
$131$	−2676.77			−1.78527			−0.892635	+	0.450779i	$0.851146\pi$
$132$		0			0
$133$	−1321.04	+	2288.11i	−0.861268	+	1.49176i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−876.421	−	1518.01i	−0.546552	−	0.946656i	−0.998507	−	0.0546158i	$-0.982607\pi$
$137$	−876.421	−	1518.01i	−0.546552	−	0.946656i	0.451955	−	0.892041i	$-0.350727\pi$
$138$		0			0
$139$	−590.167	−	1022.20i	−0.360124	−	0.623754i	0.627857	−	0.778329i	$-0.283932\pi$
$139$	−590.167	−	1022.20i	−0.360124	−	0.623754i	−0.987981	+	0.154575i	$0.950599\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	170.070	−	198.651i	0.0994545	−	0.116168i
$144$		0			0
$145$	−642.110	+	1112.17i	−0.367754	+	0.636969i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1695.76	+	2937.14i	0.932361	+	1.61490i	0.779273	+	0.626684i	$0.215588\pi$
$149$	1695.76	+	2937.14i	0.932361	+	1.61490i	0.153088	+	0.988213i	$0.451078\pi$
$150$		0			0
$151$	−51.1288			−0.0275550			−0.0137775	−	0.999905i	$-0.504386\pi$
$151$	−51.1288			−0.0275550			−0.0137775	+	0.999905i	$0.504386\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−178.467			−0.0924825
$156$		0			0
$157$	−107.766			−0.0547812			−0.0273906	−	0.999625i	$-0.508720\pi$
$157$	−107.766			−0.0547812			−0.0273906	+	0.999625i	$0.508720\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	888.479			0.434919
$162$		0			0
$163$	743.236	+	1287.32i	0.357146	+	0.618594i	0.987483	−	0.157727i	$-0.0504167\pi$
$163$	743.236	+	1287.32i	0.357146	+	0.618594i	−0.630337	+	0.776321i	$0.717083\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1278.05	−	2213.64i	0.592206	−	1.02573i	−0.401729	−	0.915759i	$-0.631591\pi$
$167$	1278.05	−	2213.64i	0.592206	−	1.02573i	0.993935	−	0.109972i	$-0.0350760\pi$
$168$		0			0
$169$	2049.21	+	792.176i	0.932732	+	0.360571i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1764.18	+	3055.64i	0.775306	+	1.34287i	0.934623	+	0.355641i	$0.115738\pi$
$173$	1764.18	+	3055.64i	0.775306	+	1.34287i	−0.159317	+	0.987227i	$0.550929\pi$
$174$		0			0
$175$	−905.381	−	1568.17i	−0.391088	−	0.677384i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1198.38	−	2075.66i	0.500398	−	0.866715i	−0.499602	−	0.866255i	$-0.666520\pi$
$179$	1198.38	−	2075.66i	0.500398	−	0.866715i	1.00000		0.000459998i	$-0.000146422\pi$
$180$		0			0
$181$	1646.29			0.676064			0.338032	−	0.941135i	$-0.390239\pi$
$181$	1646.29			0.676064			0.338032	+	0.941135i	$0.390239\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	256.141	−	443.650i	0.101794	−	0.176312i
$186$		0			0
$187$	588.170			0.230007
$188$		0			0
$189$		0			0
$190$		0			0
$191$	413.742	+	716.622i	0.156740	+	0.271481i	0.933691	−	0.358079i	$-0.116568\pi$
$191$	413.742	+	716.622i	0.156740	+	0.271481i	−0.776951	+	0.629561i	$0.783235\pi$
$192$		0			0
$193$	1699.20	−	2943.09i	0.633735	−	1.09766i	−0.353047	−	0.935606i	$-0.614855\pi$
$193$	1699.20	−	2943.09i	0.633735	−	1.09766i	0.986782	−	0.162055i	$-0.0518122\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2658.46	+	4604.58i	−0.961458	+	1.66529i	−0.242612	+	0.970123i	$0.578004\pi$
$197$	−2658.46	+	4604.58i	−0.961458	+	1.66529i	−0.718845	+	0.695170i	$0.755329\pi$
$198$		0			0
$199$	1590.77	+	2755.30i	0.566669	+	0.981499i	0.996892	+	0.0787764i	$0.0251013\pi$
$199$	1590.77	+	2755.30i	0.566669	+	0.981499i	−0.430224	+	0.902722i	$0.641565\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4416.22			1.52689
$204$		0			0
$205$	206.028	−	356.850i	0.0701931	−	0.121578i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−755.543			−0.250057
$210$		0			0
$211$	−335.227	+	580.630i	−0.109374	+	0.189442i	−0.915517	−	0.402279i	$-0.868218\pi$
$211$	−335.227	+	580.630i	−0.109374	+	0.189442i	0.806143	+	0.591721i	$0.201551\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1065.05	+	1844.73i	0.337842	+	0.585159i
$216$		0			0
$217$	306.858	+	531.494i	0.0959950	+	0.166268i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1643.97	+	4659.89i	0.500385	+	1.41836i
$222$		0			0
$223$	2518.25	−	4361.73i	0.756207	−	1.30979i	−0.188565	−	0.982061i	$-0.560384\pi$
$223$	2518.25	−	4361.73i	0.756207	−	1.30979i	0.944772	−	0.327729i	$-0.106283\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2518.77	+	4362.64i	0.736461	+	1.27559i	0.954079	+	0.299554i	$0.0968377\pi$
$227$	2518.77	+	4362.64i	0.736461	+	1.27559i	−0.217619	+	0.976034i	$0.569829\pi$
$228$		0			0
$229$	−4661.22			−1.34508			−0.672538	−	0.740063i	$-0.734796\pi$
$229$	−4661.22			−1.34508			−0.672538	+	0.740063i	$0.734796\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−132.855			−0.0373545			−0.0186772	−	0.999826i	$-0.505945\pi$
$233$	−132.855			−0.0373545			−0.0186772	+	0.999826i	$0.505945\pi$
$234$		0			0
$235$	3614.94			1.00346
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1398.04			0.378376			0.189188	−	0.981941i	$-0.439414\pi$
$239$	1398.04			0.378376			0.189188	+	0.981941i	$0.439414\pi$
$240$		0			0
$241$	1277.74	+	2213.11i	0.341520	+	0.591531i	0.984715	−	0.174172i	$-0.0557248\pi$
$241$	1277.74	+	2213.11i	0.341520	+	0.591531i	−0.643195	+	0.765703i	$0.722391\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	106.769	−	184.929i	0.0278417	−	0.0482233i
$246$		0			0
$247$	−2111.78	−	5985.94i	−0.544006	−	1.54201i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1141.08	−	1976.42i	−0.286951	−	0.497013i	0.686130	−	0.727479i	$-0.259308\pi$
$251$	−1141.08	−	1976.42i	−0.286951	−	0.497013i	−0.973080	+	0.230466i	$0.925975\pi$
$252$		0			0
$253$	127.037	+	220.035i	0.0315682	+	0.0546777i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2557.31	+	4429.40i	−0.620703	+	1.07509i	0.368652	+	0.929568i	$0.379820\pi$
$257$	−2557.31	+	4429.40i	−0.620703	+	1.07509i	−0.989355	+	0.145522i	$0.953514\pi$
$258$		0			0
$259$	−1761.65			−0.422640
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2991.76	−	5181.89i	0.701445	−	1.21494i	−0.266514	−	0.963831i	$-0.585872\pi$
$263$	2991.76	−	5181.89i	0.701445	−	1.21494i	0.967959	−	0.251107i	$-0.0807947\pi$
$264$		0			0
$265$	3282.96			0.761022
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−826.760	−	1431.99i	−0.187392	−	0.324573i	0.756988	−	0.653429i	$-0.226670\pi$
$269$	−826.760	−	1431.99i	−0.187392	−	0.324573i	−0.944380	+	0.328856i	$0.893337\pi$
$270$		0			0
$271$	−3186.99	+	5520.02i	−0.714375	+	1.23733i	0.248825	+	0.968548i	$0.419956\pi$
$271$	−3186.99	+	5520.02i	−0.714375	+	1.23733i	−0.963200	+	0.268785i	$0.913378\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	258.908	−	448.441i	0.0567735	−	0.0983346i
$276$		0			0
$277$	−918.452	−	1590.81i	−0.199222	−	0.345062i	0.749054	−	0.662508i	$-0.230508\pi$
$277$	−918.452	−	1590.81i	−0.199222	−	0.345062i	−0.948276	+	0.317446i	$0.897175\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4214.79			−0.894781			−0.447390	−	0.894339i	$-0.647647\pi$
$281$	−4214.79			−0.894781			−0.447390	+	0.894339i	$0.647647\pi$
$282$		0			0
$283$	−2507.68	+	4343.42i	−0.526735	+	0.912331i	0.472780	+	0.881180i	$0.343251\pi$
$283$	−2507.68	+	4343.42i	−0.526735	+	0.912331i	−0.999515	+	0.0311507i	$0.990083\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1416.99			−0.291436
$288$		0			0
$289$	−3100.45	+	5370.13i	−0.631070	+	1.09304i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−477.473	−	827.007i	−0.0952023	−	0.164895i	0.814491	−	0.580177i	$-0.197016\pi$
$293$	−477.473	−	827.007i	−0.0952023	−	0.164895i	−0.909693	+	0.415281i	$0.863683\pi$
$294$		0			0
$295$	215.836	+	373.839i	0.0425982	+	0.0737822i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1388.19	+	1621.48i	−0.268499	+	0.313622i
$300$		0			0
$301$	3662.54	−	6343.70i	0.701346	−	1.21477i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2146.03	+	3717.03i	0.402889	+	0.697825i
$306$		0			0
$307$	−7207.06			−1.33983			−0.669916	−	0.742437i	$-0.733670\pi$
$307$	−7207.06			−1.33983			−0.669916	+	0.742437i	$0.733670\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1176.61			−0.214532			−0.107266	−	0.994230i	$-0.534210\pi$
$311$	−1176.61			−0.214532			−0.107266	+	0.994230i	$0.534210\pi$
$312$		0			0
$313$	9578.32			1.72971			0.864854	−	0.502023i	$-0.167411\pi$
$313$	9578.32			1.72971			0.864854	+	0.502023i	$0.167411\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7243.53			1.28340			0.641699	−	0.766957i	$-0.278230\pi$
$317$	7243.53			1.28340			0.641699	+	0.766957i	$0.278230\pi$
$318$		0			0
$319$	631.443	+	1093.69i	0.110828	+	0.191959i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7138.26	−	12363.8i	1.22967	−	2.12985i
$324$		0			0
$325$	4276.53	+	797.831i	0.729905	+	0.136171i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6215.59	−	10765.7i	−1.04157	−	1.80405i
$330$		0			0
$331$	−2567.45	−	4446.95i	−0.426343	−	0.738448i	0.570202	−	0.821505i	$-0.306865\pi$
$331$	−2567.45	−	4446.95i	−0.426343	−	0.738448i	−0.996545	+	0.0830566i	$0.973532\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−667.888	+	1156.82i	−0.108927	+	0.188667i
$336$		0			0
$337$	3989.68			0.644901			0.322451	−	0.946586i	$-0.395493\pi$
$337$	3989.68			0.644901			0.322451	+	0.946586i	$0.395493\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−87.7509	+	151.989i	−0.0139354	+	0.0241368i
$342$		0			0
$343$	5957.59			0.937842
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2937.05	−	5087.12i	−0.454378	−	0.787005i	0.544275	−	0.838907i	$-0.316805\pi$
$347$	−2937.05	−	5087.12i	−0.454378	−	0.787005i	−0.998652	+	0.0519020i	$0.983472\pi$
$348$		0			0
$349$	−5046.74	+	8741.21i	−0.774057	+	1.34071i	0.161266	+	0.986911i	$0.448442\pi$
$349$	−5046.74	+	8741.21i	−0.774057	+	1.34071i	−0.935323	+	0.353795i	$0.884891\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3220.21	+	5577.57i	−0.485537	+	0.840974i	−0.999862	−	0.0166209i	$-0.994709\pi$
$353$	−3220.21	+	5577.57i	−0.485537	+	0.840974i	0.514325	+	0.857595i	$0.328042\pi$
$354$		0			0
$355$	−893.015	−	1546.75i	−0.133511	−	0.231247i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3177.72			0.467170			0.233585	−	0.972336i	$-0.424954\pi$
$359$	3177.72			0.467170			0.233585	+	0.972336i	$0.424954\pi$
$360$		0			0
$361$	−5740.07	+	9942.09i	−0.836867	+	1.44950i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2312.19			0.331577
$366$		0			0
$367$	−169.071	+	292.840i	−0.0240475	+	0.0416515i	−0.877799	−	0.479030i	$-0.840989\pi$
$367$	−169.071	+	292.840i	−0.0240475	+	0.0416515i	0.853751	+	0.520681i	$0.174322\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5644.78	−	9777.05i	−0.789926	−	1.36819i
$372$		0			0
$373$	−4893.29	−	8475.42i	−0.679262	−	1.17652i	−0.975203	−	0.221311i	$-0.928967\pi$
$373$	−4893.29	−	8475.42i	−0.679262	−	1.17652i	0.295941	−	0.955206i	$-0.404367\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6900.07	+	8059.66i	−0.942631	+	1.10104i
$378$		0			0
$379$	−4476.01	+	7752.68i	−0.606642	+	1.05073i	0.385148	+	0.922855i	$0.374150\pi$
$379$	−4476.01	+	7752.68i	−0.606642	+	1.05073i	−0.991790	+	0.127880i	$0.959183\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3616.66	−	6264.24i	−0.482514	−	0.835738i	0.517285	−	0.855813i	$-0.326943\pi$
$383$	−3616.66	−	6264.24i	−0.482514	−	0.835738i	−0.999798	+	0.0200753i	$0.993609\pi$
$384$		0			0
$385$	617.548			0.0817485
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10733.4			−1.39898			−0.699491	−	0.714641i	$-0.746590\pi$
$389$	−10733.4			−1.39898			−0.699491	+	0.714641i	$0.746590\pi$
$390$		0			0
$391$	−4800.91			−0.620953
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1054.77			−0.134357
$396$		0			0
$397$	−3590.24	−	6218.48i	−0.453877	−	0.786138i	0.544746	−	0.838601i	$-0.316626\pi$
$397$	−3590.24	−	6218.48i	−0.453877	−	0.786138i	−0.998623	+	0.0524631i	$0.983293\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	306.752	−	531.310i	0.0382006	−	0.0661654i	−0.846293	−	0.532718i	$-0.821171\pi$
$401$	306.752	−	531.310i	0.0382006	−	0.0661654i	0.884494	+	0.466552i	$0.154504\pi$
$402$		0			0
$403$	−1449.43	−	270.407i	−0.179160	−	0.0334241i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−251.886	−	436.279i	−0.0306769	−	0.0531340i
$408$		0			0
$409$	−6286.23	−	10888.1i	−0.759985	−	1.31633i	−0.942858	−	0.333196i	$-0.891873\pi$
$409$	−6286.23	−	10888.1i	−0.759985	−	1.31633i	0.182873	−	0.983137i	$-0.441460\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	742.224	−	1285.57i	0.0884321	−	0.153169i
$414$		0			0
$415$	3942.60			0.466348
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4841.03	+	8384.91i	−0.564439	+	0.977636i	0.432663	+	0.901556i	$0.357574\pi$
$419$	−4841.03	+	8384.91i	−0.564439	+	0.977636i	−0.997102	+	0.0760806i	$0.975759\pi$
$420$		0			0
$421$	−3930.59			−0.455025			−0.227512	−	0.973775i	$-0.573059\pi$
$421$	−3930.59			−0.455025			−0.227512	+	0.973775i	$0.573059\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4892.25	+	8473.62i	0.558374	+	0.967132i
$426$		0			0
$427$	7379.83	−	12782.2i	0.836382	−	1.44866i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−469.276	+	812.809i	−0.0524460	+	0.0908391i	−0.891057	−	0.453892i	$-0.850035\pi$
$431$	−469.276	+	812.809i	−0.0524460	+	0.0908391i	0.838611	+	0.544731i	$0.183368\pi$
$432$		0			0
$433$	3548.42	+	6146.04i	0.393825	+	0.682125i	0.992950	−	0.118530i	$-0.0378183\pi$
$433$	3548.42	+	6146.04i	0.393825	+	0.682125i	−0.599126	+	0.800655i	$0.704485\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6167.09			0.675084
$438$		0			0
$439$	7091.42	−	12282.7i	0.770968	−	1.33536i	−0.166065	−	0.986115i	$-0.553106\pi$
$439$	7091.42	−	12282.7i	0.770968	−	1.33536i	0.937033	−	0.349241i	$-0.113561\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12640.7			−1.35571			−0.677855	−	0.735195i	$-0.737090\pi$
$443$	−12640.7			−1.35571			−0.677855	+	0.735195i	$0.737090\pi$
$444$		0			0
$445$	38.9706	−	67.4991i	0.00415143	−	0.00719049i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−8741.53	−	15140.8i	−0.918793	−	1.59140i	−0.801251	−	0.598329i	$-0.795832\pi$
$449$	−8741.53	−	15140.8i	−0.918793	−	1.59140i	−0.117543	−	0.993068i	$-0.537502\pi$
$450$		0			0
$451$	−202.605	−	350.922i	−0.0211536	−	0.0366391i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1726.08	+	4892.65i	0.177846	+	0.504112i
$456$		0			0
$457$	−276.863	+	479.540i	−0.0283394	+	0.0490852i	−0.879847	−	0.475257i	$-0.842355\pi$
$457$	−276.863	+	479.540i	−0.0283394	+	0.0490852i	0.851508	+	0.524342i	$0.175689\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4619.71	+	8001.57i	0.466727	+	0.808396i	0.999278	−	0.0380028i	$-0.0120996\pi$
$461$	4619.71	+	8001.57i	0.466727	+	0.808396i	−0.532550	+	0.846398i	$0.678766\pi$
$462$		0			0
$463$	−5386.78			−0.540701			−0.270351	−	0.962762i	$-0.587140\pi$
$463$	−5386.78			−0.540701			−0.270351	+	0.962762i	$0.587140\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10755.4			−1.06574			−0.532871	−	0.846197i	$-0.678887\pi$
$467$	−10755.4			−1.06574			−0.532871	+	0.846197i	$0.678887\pi$
$468$		0			0
$469$	4593.51			0.452257
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2094.72			0.203626
$474$		0			0
$475$	−6284.41	−	10884.9i	−0.607050	−	1.05144i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5844.53	−	10123.0i	0.557502	−	0.965621i	−0.440203	−	0.897898i	$-0.645093\pi$
$479$	5844.53	−	10123.0i	0.557502	−	0.965621i	0.997704	−	0.0677226i	$-0.0215733\pi$
$480$		0			0
$481$	2752.47	−	3215.04i	0.260919	−	0.304768i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1851.49	+	3206.87i	0.173344	+	0.300241i
$486$		0			0
$487$	−268.835	−	465.636i	−0.0250145	−	0.0433264i	0.853247	−	0.521507i	$-0.174630\pi$
$487$	−268.835	−	465.636i	−0.0250145	−	0.0433264i	−0.878262	+	0.478180i	$0.841297\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7085.19	+	12271.9i	−0.651222	+	1.12795i	0.331604	+	0.943419i	$0.392410\pi$
$491$	−7085.19	+	12271.9i	−0.651222	+	1.12795i	−0.982827	+	0.184532i	$0.940923\pi$
$492$		0			0
$493$	−23863.1			−2.18000
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3070.93	+	5319.01i	−0.277163	+	0.480060i
$498$		0			0
$499$	11824.1			1.06076			0.530382	−	0.847759i	$-0.322049\pi$
$499$	11824.1			1.06076			0.530382	+	0.847759i	$0.322049\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3885.60	−	6730.06i	−0.344434	−	0.596578i	0.640817	−	0.767694i	$-0.278596\pi$
$503$	−3885.60	−	6730.06i	−0.344434	−	0.596578i	−0.985251	+	0.171116i	$0.945263\pi$
$504$		0			0
$505$	−1088.95	+	1886.11i	−0.0959554	+	0.166200i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5602.87	−	9704.46i	0.487903	−	0.845074i	−0.512000	−	0.858986i	$-0.671095\pi$
$509$	5602.87	−	9704.46i	0.487903	−	0.845074i	0.999903	+	0.0139121i	$0.00442849\pi$
$510$		0			0
$511$	−3975.62	−	6885.98i	−0.344170	−	0.596121i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5508.21			−0.471302
$516$		0			0
$517$	1777.44	−	3078.62i	0.151203	−	0.261891i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10763.7			0.905118			0.452559	−	0.891734i	$-0.350511\pi$
$521$	10763.7			0.905118			0.452559	+	0.891734i	$0.350511\pi$
$522$		0			0
$523$	−10603.0	+	18365.0i	−0.886496	+	1.53546i	−0.0425069	+	0.999096i	$0.513534\pi$
$523$	−10603.0	+	18365.0i	−0.886496	+	1.53546i	−0.843989	+	0.536360i	$0.819799\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1658.12	−	2871.94i	−0.137056	−	0.237388i
$528$		0			0
$529$	5046.57	+	8740.91i	0.414775	+	0.718411i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2213.96	−	2586.02i	0.179920	−	0.210156i
$534$		0			0
$535$	2559.96	−	4433.98i	0.206872	−	0.358313i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−104.995	−	181.857i	−0.00839047	−	0.0145327i
$540$		0			0
$541$	8378.58			0.665847			0.332924	−	0.942954i	$-0.391965\pi$
$541$	8378.58			0.665847			0.332924	+	0.942954i	$0.391965\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3577.52			0.281182
$546$		0			0
$547$	17579.7			1.37414			0.687069	−	0.726592i	$-0.258897\pi$
$547$	17579.7			1.37414			0.687069	+	0.726592i	$0.258897\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	30653.8			2.37004
$552$		0			0
$553$	1813.59	+	3141.23i	0.139460	+	0.241552i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1484.68	+	2571.54i	−0.112941	+	0.195619i	−0.916955	−	0.398991i	$-0.869360\pi$
$557$	−1484.68	+	2571.54i	−0.112941	+	0.195619i	0.804014	+	0.594610i	$0.202694\pi$
$558$		0			0
$559$	5854.84	+	16595.8i	0.442994	+	1.25569i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4562.19	+	7901.94i	0.341516	+	0.591522i	0.984714	−	0.174177i	$-0.0557265\pi$
$563$	4562.19	+	7901.94i	0.341516	+	0.591522i	−0.643199	+	0.765699i	$0.722393\pi$
$564$		0			0
$565$	1464.93	+	2537.33i	0.109080	+	0.188931i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8172.71	−	14155.5i	0.602140	−	1.04294i	−0.390356	−	0.920664i	$-0.627648\pi$
$569$	8172.71	−	14155.5i	0.602140	−	1.04294i	0.992496	−	0.122273i	$-0.0390185\pi$
$570$		0			0
$571$	9870.90			0.723440			0.361720	−	0.932287i	$-0.382190\pi$
$571$	9870.90			0.723440			0.361720	+	0.932287i	$0.382190\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2113.32	+	3660.38i	−0.153273	+	0.265476i
$576$		0			0
$577$	−11297.7			−0.815126			−0.407563	−	0.913177i	$-0.633621\pi$
$577$	−11297.7			−0.815126			−0.407563	+	0.913177i	$0.633621\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6778.97	−	11741.5i	−0.484060	−	0.838417i
$582$		0			0
$583$	1614.21	−	2795.90i	0.114672	−	0.198618i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10612.8	+	18381.9i	−0.746230	+	1.29251i	0.203387	+	0.979098i	$0.434805\pi$
$587$	−10612.8	+	18381.9i	−0.746230	+	1.29251i	−0.949618	+	0.313411i	$0.898528\pi$
$588$		0			0
$589$	2129.96	+	3689.20i	0.149004	+	0.258083i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−21708.9			−1.50333			−0.751666	−	0.659544i	$-0.770750\pi$
$593$	−21708.9			−1.50333			−0.751666	+	0.659544i	$0.770750\pi$
$594$		0			0
$595$	−5834.51	+	10105.7i	−0.402003	+	0.696289i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1041.35			−0.0710324			−0.0355162	−	0.999369i	$-0.511308\pi$
$599$	−1041.35			−0.0710324			−0.0355162	+	0.999369i	$0.511308\pi$
$600$		0			0
$601$	−6469.69	+	11205.8i	−0.439108	+	0.760558i	−0.997621	−	0.0689381i	$-0.978039\pi$
$601$	−6469.69	+	11205.8i	−0.439108	+	0.760558i	0.558513	+	0.829496i	$0.311372\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3687.36	−	6386.70i	−0.247790	−	0.429184i
$606$		0			0
$607$	−9943.59	−	17222.8i	−0.664906	−	1.15165i	−0.979311	−	0.202362i	$-0.935138\pi$
$607$	−9943.59	−	17222.8i	−0.664906	−	1.15165i	0.314405	−	0.949289i	$-0.398195\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	29359.1	+	5477.24i	1.94393	+	0.362660i
$612$		0			0
$613$	−10189.8	+	17649.3i	−0.671390	+	1.16288i	0.306120	+	0.951993i	$0.400969\pi$
$613$	−10189.8	+	17649.3i	−0.671390	+	1.16288i	−0.977510	+	0.210889i	$0.932364\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2533.78	+	4388.64i	0.165326	+	0.286353i	0.936771	−	0.349943i	$-0.113799\pi$
$617$	2533.78	+	4388.64i	0.165326	+	0.286353i	−0.771445	+	0.636296i	$0.780466\pi$
$618$		0			0
$619$	−6642.99			−0.431348			−0.215674	−	0.976465i	$-0.569195\pi$
$619$	−6642.99			−0.431348			−0.215674	+	0.976465i	$0.569195\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−268.027			−0.0172364
$624$		0			0
$625$	4590.65			0.293801
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9519.13			0.603422
$630$		0			0
$631$	13818.4	+	23934.1i	0.871793	+	1.50999i	0.860141	+	0.510057i	$0.170376\pi$
$631$	13818.4	+	23934.1i	0.871793	+	1.50999i	0.0116521	+	0.999932i	$0.496291\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−468.518	+	811.496i	−0.0292796	+	0.0507138i
$636$		0			0
$637$	1147.33	−	1340.15i	0.0713641	−	0.0833571i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5288.26	−	9159.54i	−0.325856	−	0.564399i	0.655829	−	0.754909i	$-0.272319\pi$
$641$	−5288.26	−	9159.54i	−0.325856	−	0.564399i	−0.981685	+	0.190510i	$0.938986\pi$
$642$		0			0
$643$	8026.14	+	13901.7i	0.492255	+	0.852611i	0.999960	−	0.00892009i	$-0.00283939\pi$
$643$	8026.14	+	13901.7i	0.492255	+	0.852611i	−0.507705	+	0.861531i	$0.669506\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6438.69	+	11152.1i	−0.391238	+	0.677644i	−0.992613	−	0.121323i	$-0.961286\pi$
$647$	−6438.69	+	11152.1i	−0.391238	+	0.677644i	0.601375	+	0.798967i	$0.294620\pi$
$648$		0			0
$649$	424.500			0.0256750
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3837.44	+	6646.65i	−0.229971	+	0.398321i	−0.957799	−	0.287438i	$-0.907196\pi$
$653$	−3837.44	+	6646.65i	−0.229971	+	0.398321i	0.727829	+	0.685759i	$0.240530\pi$
$654$		0			0
$655$	15186.5			0.905929
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9308.28	−	16122.4i	−0.550227	−	0.953020i	−0.998258	−	0.0590025i	$-0.981208\pi$
$659$	−9308.28	−	16122.4i	−0.550227	−	0.953020i	0.448031	−	0.894018i	$-0.352125\pi$
$660$		0			0
$661$	10650.3	−	18446.8i	0.626699	−	1.08547i	−0.361510	−	0.932368i	$-0.617739\pi$
$661$	10650.3	−	18446.8i	0.626699	−	1.08547i	0.988210	−	0.153107i	$-0.0489279\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7494.82	−	12981.4i	0.437047	−	0.756988i
$666$		0			0
$667$	−5154.13	−	8927.21i	−0.299203	−	0.518235i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4220.75			0.242832
$672$		0			0
$673$	6115.77	−	10592.8i	0.350291	−	0.606721i	−0.636010	−	0.771681i	$-0.719416\pi$
$673$	6115.77	−	10592.8i	0.350291	−	0.606721i	0.986300	+	0.164960i	$0.0527495\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	33485.1			1.90094			0.950471	−	0.310813i	$-0.100601\pi$
$677$	33485.1			1.90094			0.950471	+	0.310813i	$0.100601\pi$
$678$		0			0
$679$	6366.96	−	11027.9i	0.359855	−	0.623287i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10433.6	+	18071.4i	0.584522	+	1.01242i	0.994935	+	0.100522i	$0.0320514\pi$
$683$	10433.6	+	18071.4i	0.584522	+	1.01242i	−0.410412	+	0.911900i	$0.634615\pi$
$684$		0			0
$685$	4972.31	+	8612.29i	0.277346	+	0.480377i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	26662.8	+	4974.24i	1.47427	+	0.275041i
$690$		0			0
$691$	2841.78	−	4922.10i	0.156449	−	0.270978i	−0.777137	−	0.629332i	$-0.783329\pi$
$691$	2841.78	−	4922.10i	0.156449	−	0.270978i	0.933586	+	0.358354i	$0.116662\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3348.27	+	5799.37i	0.182744	+	0.316522i
$696$		0			0
$697$	7656.73			0.416096
$698$		0			0
$699$		0			0
$700$		0			0
$701$	14124.9			0.761039			0.380519	−	0.924773i	$-0.375745\pi$
$701$	14124.9			0.761039			0.380519	+	0.924773i	$0.375745\pi$
$702$		0			0
$703$	−12228.0			−0.656025
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7489.41			0.398399
$708$		0			0
$709$	−9815.96	−	17001.7i	−0.519952	−	0.900584i	−0.999731	−	0.0231942i	$-0.992616\pi$
$709$	−9815.96	−	17001.7i	−0.519952	−	0.900584i	0.479779	−	0.877390i	$-0.340717\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	716.263	−	1240.60i	0.0376217	−	0.0651627i
$714$		0			0
$715$	−964.881	+	1127.03i	−0.0504679	+	0.0589492i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5311.54	+	9199.85i	0.275503	+	0.477186i	0.970262	−	0.242057i	$-0.0778221\pi$
$719$	5311.54	+	9199.85i	0.275503	+	0.477186i	−0.694759	+	0.719243i	$0.744489\pi$
$720$		0			0
$721$	9470.90	+	16404.1i	0.489202	+	0.847323i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−10504.4	+	18194.1i	−0.538100	+	0.932016i
$726$		0			0
$727$	−3361.06			−0.171464			−0.0857322	−	0.996318i	$-0.527323\pi$
$727$	−3361.06			−0.171464			−0.0857322	+	0.996318i	$0.527323\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−19790.6	+	34278.3i	−1.00134	+	1.73438i
$732$		0			0
$733$	−8833.43			−0.445116			−0.222558	−	0.974919i	$-0.571441\pi$
$733$	−8833.43			−0.445116			−0.222558	+	0.974919i	$0.571441\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	656.792	+	1137.60i	0.0328266	+	0.0568574i
$738$		0			0
$739$	−14428.7	+	24991.2i	−0.718223	+	1.24400i	0.243481	+	0.969906i	$0.421711\pi$
$739$	−14428.7	+	24991.2i	−0.718223	+	1.24400i	−0.961703	+	0.274092i	$0.911623\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6771.93	−	11729.3i	0.334371	−	0.579148i	−0.648992	−	0.760795i	$-0.724809\pi$
$743$	6771.93	−	11729.3i	0.334371	−	0.579148i	0.983364	+	0.181647i	$0.0581427\pi$
$744$		0			0
$745$	−9620.75	−	16663.6i	−0.473123	−	0.819474i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−17606.5			−0.858916
$750$		0			0
$751$	3697.48	−	6404.22i	0.179658	−	0.311176i	−0.762106	−	0.647453i	$-0.775834\pi$
$751$	3697.48	−	6404.22i	0.179658	−	0.311176i	0.941763	+	0.336276i	$0.109168\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	290.075			0.0139827
$756$		0			0
$757$	11892.5	−	20598.5i	0.570992	−	0.988988i	−0.425472	−	0.904972i	$-0.639892\pi$
$757$	11892.5	−	20598.5i	0.570992	−	0.988988i	0.996464	−	0.0840162i	$-0.0267748\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	493.433	+	854.652i	0.0235045	+	0.0407110i	0.877538	−	0.479506i	$-0.159184\pi$
$761$	493.433	+	854.652i	0.0235045	+	0.0407110i	−0.854034	+	0.520217i	$0.825851\pi$
$762$		0			0
$763$	−6151.24	−	10654.3i	−0.291861	−	0.505518i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1186.50	+	3363.19i	0.0558567	+	0.158328i
$768$		0			0
$769$	6429.73	−	11136.6i	0.301511	−	0.522232i	−0.674967	−	0.737847i	$-0.735842\pi$
$769$	6429.73	−	11136.6i	0.301511	−	0.522232i	0.976478	+	0.215615i	$0.0691757\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16669.5	+	28872.5i	0.775630	+	1.34343i	0.934440	+	0.356121i	$0.115901\pi$
$773$	16669.5	+	28872.5i	0.775630	+	1.34343i	−0.158810	+	0.987309i	$0.550766\pi$
$774$		0			0
$775$	−2919.56			−0.135321
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9835.57			−0.452369
$780$		0			0
$781$	−1756.36			−0.0804705
$782$		0			0
$783$		0			0
$784$		0			0
$785$	611.401			0.0277985
$786$		0			0
$787$	−20290.6	−	35144.4i	−0.919037	−	1.59182i	−0.800880	−	0.598824i	$-0.795635\pi$
$787$	−20290.6	−	35144.4i	−0.919037	−	1.59182i	−0.118157	−	0.992995i	$-0.537699\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5037.64	−	8725.45i	0.226445	−	0.392214i
$792$		0			0
$793$	11797.2	+	33439.8i	0.528287	+	1.49745i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11082.4	−	19195.3i	−0.492547	−	0.853116i	0.507416	−	0.861701i	$-0.330601\pi$
$797$	−11082.4	−	19195.3i	−0.492547	−	0.853116i	−0.999963	+	0.00858488i	$0.997267\pi$
$798$		0			0
$799$	33586.1	+	58172.8i	1.48710	+	2.57573i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1136.89	−	1969.15i	0.0499626	−	0.0865377i
$804$		0			0
$805$	−5040.72			−0.220698
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−20370.1	+	35282.0i	−0.885258	+	1.53331i	−0.0398397	+	0.999206i	$0.512685\pi$
$809$	−20370.1	+	35282.0i	−0.885258	+	1.53331i	−0.845418	+	0.534105i	$0.820649\pi$
$810$		0			0
$811$	32854.7			1.42255			0.711273	−	0.702916i	$-0.248119\pi$
$811$	32854.7			1.42255			0.711273	+	0.702916i	$0.248119\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4216.69	−	7303.53i	−0.181232	−	0.313903i
$816$		0			0
$817$	25422.3	−	44032.8i	1.08863	−	1.88557i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6627.82	−	11479.7i	0.281745	−	0.487996i	−0.690070	−	0.723743i	$-0.742420\pi$
$821$	6627.82	−	11479.7i	0.281745	−	0.487996i	0.971815	+	0.235747i	$0.0757536\pi$
$822$		0			0
$823$	−14345.9	−	24847.9i	−0.607615	−	1.05242i	−0.991632	−	0.129094i	$-0.958793\pi$
$823$	−14345.9	−	24847.9i	−0.607615	−	1.05242i	0.384017	−	0.923326i	$-0.374540\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−41617.9			−1.74994			−0.874968	−	0.484181i	$-0.839117\pi$
$827$	−41617.9			−1.74994			−0.874968	+	0.484181i	$0.839117\pi$
$828$		0			0
$829$	−2790.25	+	4832.86i	−0.116899	+	0.202475i	−0.918537	−	0.395334i	$-0.870629\pi$
$829$	−2790.25	+	4832.86i	−0.116899	+	0.202475i	0.801638	+	0.597810i	$0.203962\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3967.92			0.165042
$834$		0			0
$835$	−7250.91	+	12558.9i	−0.300513	+	0.520503i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3383.47	+	5860.34i	0.139226	+	0.241146i	0.927204	−	0.374557i	$-0.122205\pi$
$839$	3383.47	+	5860.34i	0.139226	+	0.241146i	−0.787978	+	0.615703i	$0.788872\pi$
$840$		0			0
$841$	−13424.3	−	23251.6i	−0.550425	−	0.953363i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−11626.0	−	4494.35i	−0.473311	−	0.182971i
$846$		0			0
$847$	−12680.2	+	21962.8i	−0.514401	+	0.890969i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2056.01	+	3561.11i	0.0828191	+	0.143447i
$852$		0			0
$853$	23869.0			0.958102			0.479051	−	0.877787i	$-0.340981\pi$
$853$	23869.0			0.958102			0.479051	+	0.877787i	$0.340981\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	11507.3			0.458670			0.229335	−	0.973348i	$-0.426345\pi$
$857$	11507.3			0.458670			0.229335	+	0.973348i	$0.426345\pi$
$858$		0			0
$859$	−40591.0			−1.61228			−0.806139	−	0.591726i	$-0.798447\pi$
$859$	−40591.0			−1.61228			−0.806139	+	0.591726i	$0.798447\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24143.3			0.952315			0.476158	−	0.879360i	$-0.342029\pi$
$863$	24143.3			0.952315			0.476158	+	0.879360i	$0.342029\pi$
$864$		0			0
$865$	−10008.9	−	17336.0i	−0.393426	−	0.681434i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−518.623	+	898.282i	−0.0202452	+	0.0350657i
$870$		0			0
$871$	−7177.07	+	8383.21i	−0.279203	+	0.326124i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	12054.6	+	20879.2i	0.465738	+	0.806682i
$876$		0			0
$877$	−8879.93	−	15380.5i	−0.341909	−	0.592203i	0.642879	−	0.765968i	$-0.277740\pi$
$877$	−8879.93	−	15380.5i	−0.341909	−	0.592203i	−0.984787	+	0.173765i	$0.944407\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	13763.7	−	23839.3i	0.526344	−	0.911655i	−0.473185	−	0.880963i	$-0.656896\pi$
$881$	13763.7	−	23839.3i	0.526344	−	0.911655i	0.999529	−	0.0306917i	$-0.00977099\pi$
$882$		0			0
$883$	14882.4			0.567193			0.283596	−	0.958944i	$-0.408472\pi$
$883$	14882.4			0.567193			0.283596	+	0.958944i	$0.408472\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−22176.8	+	38411.4i	−0.839487	+	1.45403i	0.0508370	+	0.998707i	$0.483811\pi$
$887$	−22176.8	+	38411.4i	−0.839487	+	1.45403i	−0.890324	+	0.455327i	$0.849522\pi$
$888$		0			0
$889$	3222.31			0.121567
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−43143.5	−	74726.8i	−1.61673	−	2.80027i
$894$		0			0
$895$	−6798.93	+	11776.1i	−0.253925	+	0.439812i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3560.22	−	6166.47i	0.132080	−	0.228769i
$900$		0			0
$901$	30501.7	+	52830.5i	1.12781	+	1.95343i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9340.08			−0.343066
$906$		0			0
$907$	−23145.6	+	40089.4i	−0.847340	+	1.46764i	0.0362327	+	0.999343i	$0.488464\pi$
$907$	−23145.6	+	40089.4i	−0.847340	+	1.46764i	−0.883573	+	0.468293i	$0.844869\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	48843.4			1.77635			0.888175	−	0.459506i	$-0.151973\pi$
$911$	48843.4			1.77635			0.888175	+	0.459506i	$0.151973\pi$
$912$		0			0
$913$	1938.55	−	3357.66i	0.0702701	−	0.121711i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−26111.8	−	45227.0i	−0.940337	−	1.62871i
$918$		0			0
$919$	−6384.86	−	11058.9i	−0.229181	−	0.396952i	0.728385	−	0.685168i	$-0.240271\pi$
$919$	−6384.86	−	11058.9i	−0.229181	−	0.396952i	−0.957566	+	0.288216i	$0.906938\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4909.11	−	13915.1i	−0.175066	−	0.496231i
$924$		0			0
$925$	4190.25	−	7257.72i	0.148945	−	0.257981i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−18094.9	−	31341.3i	−0.639046	−	1.10686i	−0.985642	−	0.168846i	$-0.945996\pi$
$929$	−18094.9	−	31341.3i	−0.639046	−	1.10686i	0.346596	−	0.938014i	$-0.387337\pi$
$930$		0			0
$931$	−5097.05			−0.179430
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3336.94			−0.116716
$936$		0			0
$937$	−51638.7			−1.80039			−0.900193	−	0.435491i	$-0.856575\pi$
$937$	−51638.7			−1.80039			−0.900193	+	0.435491i	$0.856575\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	17015.4			0.589466			0.294733	−	0.955580i	$-0.404769\pi$
$941$	17015.4			0.589466			0.294733	+	0.955580i	$0.404769\pi$
$942$		0			0
$943$	1653.75	+	2864.38i	0.0571088	+	0.0989154i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13898.6	−	24073.1i	0.476921	−	0.826052i	−0.522729	−	0.852499i	$-0.675086\pi$
$947$	13898.6	−	24073.1i	0.476921	−	0.826052i	0.999650	+	0.0264471i	$0.00841935\pi$
$948$		0			0
$949$	18778.7	+	3503.36i	0.642340	+	0.119835i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−16261.3	−	28165.3i	−0.552732	−	0.957361i	−0.998076	−	0.0620009i	$-0.980252\pi$
$953$	−16261.3	−	28165.3i	−0.552732	−	0.957361i	0.445344	−	0.895360i	$-0.353082\pi$
$954$		0			0
$955$	−2347.33	−	4065.70i	−0.0795371	−	0.137762i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17098.9	−	29616.2i	0.575759	−	0.997244i
$960$		0			0
$961$	−28801.5			−0.966785
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9640.26	+	16697.4i	−0.321586	+	0.557004i
$966$		0			0
$967$	15192.7			0.505237			0.252619	−	0.967566i	$-0.418708\pi$
$967$	15192.7			0.505237			0.252619	+	0.967566i	$0.418708\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2032.99	+	3521.24i	0.0671903	+	0.116377i	0.897663	−	0.440682i	$-0.145263\pi$
$971$	2032.99	+	3521.24i	0.0671903	+	0.116377i	−0.830473	+	0.557059i	$0.811930\pi$
$972$		0			0
$973$	11514.1	−	19943.0i	0.379369	−	0.657086i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−7734.47	+	13396.5i	−0.253273	+	0.438681i	−0.964425	−	0.264357i	$-0.914840\pi$
$977$	−7734.47	+	13396.5i	−0.253273	+	0.438681i	0.711152	+	0.703038i	$0.248174\pi$
$978$		0			0
$979$	−38.3232	−	66.3777i	−0.00125109	−	0.00216695i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−15997.7			−0.519073			−0.259537	−	0.965733i	$-0.583570\pi$
$983$	−15997.7			−0.519073			−0.259537	+	0.965733i	$0.583570\pi$
$984$		0			0
$985$	15082.5	−	26123.7i	0.487888	−	0.845047i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−17098.1			−0.549733
$990$		0			0
$991$	3013.98	−	5220.36i	0.0966117	−	0.167336i	−0.813668	−	0.581329i	$-0.802533\pi$
$991$	3013.98	−	5220.36i	0.0966117	−	0.167336i	0.910280	+	0.413993i	$0.135866\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9025.14	−	15632.0i	−0.287554	−	0.498058i
$996$		0			0
$997$	10769.8	+	18653.9i	0.342110	+	0.592551i	0.984824	−	0.173554i	$-0.0555253\pi$
$997$	10769.8	+	18653.9i	0.342110	+	0.592551i	−0.642715	+	0.766106i	$0.722192\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	468.4.l.d.217.2		8
3.2	odd	2		156.4.i.a.61.3	✓	8
12.11	even	2		624.4.q.l.529.3		8
13.3	even	3	inner	468.4.l.d.289.2		8
39.17	odd	6		2028.4.a.m.1.2		4
39.20	even	12		2028.4.b.j.337.6		8
39.29	odd	6		156.4.i.a.133.3	yes	8
39.32	even	12		2028.4.b.j.337.3		8
39.35	odd	6		2028.4.a.n.1.3		4
156.107	even	6		624.4.q.l.289.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.4.i.a.61.3	✓	8	3.2	odd	2
156.4.i.a.133.3	yes	8	39.29	odd	6
468.4.l.d.217.2		8	1.1	even	1	trivial
468.4.l.d.289.2		8	13.3	even	3	inner
624.4.q.l.289.3		8	156.107	even	6
624.4.q.l.529.3		8	12.11	even	2
2028.4.a.m.1.2		4	39.17	odd	6
2028.4.a.n.1.3		4	39.35	odd	6
2028.4.b.j.337.3		8	39.32	even	12
2028.4.b.j.337.6		8	39.20	even	12

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:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	468.l (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-5.67342 q^{5} +(9.75498 + 16.8961i) q^{7} +(-2.78958 + 4.83170i) q^{11} +(-46.0772 - 8.59618i) q^{13} +(-52.7112 - 91.2985i) q^{17} +(67.7110 + 117.279i) q^{19} +(22.7699 - 39.4386i) q^{23} -92.8123 q^{25} +(113.179 - 196.031i) q^{29} +31.4566 q^{31} +(-55.3441 - 95.8588i) q^{35} +(-45.1476 + 78.1979i) q^{37} +(-36.3145 + 62.8986i) q^{41} +(-187.727 - 325.152i) q^{43} -637.171 q^{47} +(-18.8191 + 32.5957i) q^{49} -578.656 q^{53} +(15.8265 - 27.4123i) q^{55} +(-38.0433 - 65.8930i) q^{59} +(-378.260 - 655.165i) q^{61} +(261.415 + 48.7698i) q^{65} +(117.722 - 203.901i) q^{67} +(157.403 + 272.630i) q^{71} -407.548 q^{73} -108.849 q^{77} +185.914 q^{79} -694.924 q^{83} +(299.053 + 517.975i) q^{85} +(-6.86898 + 11.8974i) q^{89} +(-304.240 - 862.381i) q^{91} +(-384.153 - 665.373i) q^{95} +(-326.344 - 565.245i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 14 q^{5} - 11 q^{7} - 20 q^{11} + 48 q^{13} + 7 q^{17} - 18 q^{19} - 60 q^{23} + 134 q^{25} + 75 q^{29} + 174 q^{31} + 556 q^{35} + 51 q^{37} + 271 q^{41} + q^{43} + 64 q^{47} - 153 q^{49} - 1514 q^{53}+ \cdots - 2527 q^{97}+O(q^{100}) \) 8 * q - 14 * q^5 - 11 * q^7 - 20 * q^11 + 48 * q^13 + 7 * q^17 - 18 * q^19 - 60 * q^23 + 134 * q^25 + 75 * q^29 + 174 * q^31 + 556 * q^35 + 51 * q^37 + 271 * q^41 + q^43 + 64 * q^47 - 153 * q^49 - 1514 * q^53 - 626 * q^55 - 610 * q^59 - 1166 * q^61 + 649 * q^65 - 1191 * q^67 + 980 * q^71 + 964 * q^73 - 812 * q^77 + 4450 * q^79 + 3036 * q^83 + 2603 * q^85 + 1078 * q^89 - 2695 * q^91 + 154 * q^95 - 2527 * q^97

Introduction

L-functions

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