LMFDB - Embedded newform 504.2.s.e.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,2,Mod(289,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("504.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	504.s (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:	$4.02446026187$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	504.289
Dual form			504.2.s.e.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{5} +(2.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{11} +2.00000 q^{13} +(1.50000 + 2.59808i) q^{17} +(-2.50000 + 4.33013i) q^{19} +(-1.50000 + 2.59808i) q^{23} +(2.00000 + 3.46410i) q^{25} +6.00000 q^{29} +(0.500000 + 0.866025i) q^{31} +(-2.50000 + 0.866025i) q^{35} +(2.50000 - 4.33013i) q^{37} +10.0000 q^{41} -4.00000 q^{43} +(0.500000 - 0.866025i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-4.50000 - 7.79423i) q^{53} +1.00000 q^{55} +(1.50000 + 2.59808i) q^{59} +(-1.50000 + 2.59808i) q^{61} +(-1.00000 + 1.73205i) q^{65} +(-5.50000 - 9.52628i) q^{67} -16.0000 q^{71} +(-3.50000 - 6.06218i) q^{73} +(0.500000 - 2.59808i) q^{77} +(5.50000 - 9.52628i) q^{79} +4.00000 q^{83} -3.00000 q^{85} +(-4.50000 + 7.79423i) q^{89} +(4.00000 + 3.46410i) q^{91} +(-2.50000 - 4.33013i) q^{95} +6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} + 4 q^{7} - q^{11} + 4 q^{13} + 3 q^{17} - 5 q^{19} - 3 q^{23} + 4 q^{25} + 12 q^{29} + q^{31} - 5 q^{35} + 5 q^{37} + 20 q^{41} - 8 q^{43} + q^{47} + 2 q^{49} - 9 q^{53} + 2 q^{55} + 3 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) $ 2 * q - q^5 + 4 * q^7 - q^11 + 4 * q^13 + 3 * q^17 - 5 * q^19 - 3 * q^23 + 4 * q^25 + 12 * q^29 + q^31 - 5 * q^35 + 5 * q^37 + 20 * q^41 - 8 * q^43 + q^47 + 2 * q^49 - 9 * q^53 + 2 * q^55 + 3 * q^59 - 3 * q^61 - 2 * q^65 - 11 * q^67 - 32 * q^71 - 7 * q^73 + q^77 + 11 * q^79 + 8 * q^83 - 6 * q^85 - 9 * q^89 + 8 * q^91 - 5 * q^95 + 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i	−0.955901	−	0.293691i	$-0.905116\pi$
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i	0.732294	+	0.680989i	$0.238450\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.500000	−	0.866025i	−0.150756	−	0.261116i	0.780750	−	0.624844i	$-0.214837\pi$
$11$	−0.500000	−	0.866025i	−0.150756	−	0.261116i	−0.931505	+	0.363727i	$0.881504\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		0			0
$19$	−2.50000	+	4.33013i	−0.573539	+	0.993399i	0.422659	+	0.906289i	$0.361097\pi$
$19$	−2.50000	+	4.33013i	−0.573539	+	0.993399i	−0.996199	+	0.0871106i	$0.972237\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.50000	+	2.59808i	−0.312772	+	0.541736i	−0.978961	−	0.204046i	$-0.934591\pi$
$23$	−1.50000	+	2.59808i	−0.312772	+	0.541736i	0.666190	+	0.745782i	$0.267924\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	0.907428	−	0.420208i	$-0.138043\pi$
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	−0.817625	+	0.575751i	$0.804710\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.50000	+	0.866025i	−0.422577	+	0.146385i
$36$		0			0
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	−0.584002	−	0.811752i	$-0.698514\pi$
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	0.994999	+	0.0998840i	$0.0318472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.0000			1.56174			0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000			1.56174			0.780869	+	0.624695i	$0.214777\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.500000	−	0.866025i	0.0729325	−	0.126323i	−0.827253	−	0.561830i	$-0.810098\pi$
$47$	0.500000	−	0.866025i	0.0729325	−	0.126323i	0.900185	+	0.435507i	$0.143431\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.50000	−	7.79423i	−0.618123	−	1.07062i	−0.989828	−	0.142269i	$-0.954560\pi$
$53$	−4.50000	−	7.79423i	−0.618123	−	1.07062i	0.371706	−	0.928351i	$-0.378773\pi$
$54$		0			0
$55$	1.00000			0.134840
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	0.946993	−	0.321253i	$-0.104104\pi$
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	−0.751710	+	0.659494i	$0.770771\pi$
$60$		0			0
$61$	−1.50000	+	2.59808i	−0.192055	+	0.332650i	−0.945931	−	0.324367i	$-0.894849\pi$
$61$	−1.50000	+	2.59808i	−0.192055	+	0.332650i	0.753876	+	0.657017i	$0.228182\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.00000	+	1.73205i	−0.124035	+	0.214834i
$66$		0			0
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	0.305424	−	0.952217i	$-0.401202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−16.0000			−1.89885			−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000			−1.89885			−0.949425	+	0.313993i	$0.898333\pi$
$72$		0			0
$73$	−3.50000	−	6.06218i	−0.409644	−	0.709524i	0.585206	−	0.810885i	$-0.301014\pi$
$73$	−3.50000	−	6.06218i	−0.409644	−	0.709524i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.500000	−	2.59808i	0.0569803	−	0.296078i
$78$		0			0
$79$	5.50000	−	9.52628i	0.618798	−	1.07179i	−0.370907	−	0.928670i	$-0.620953\pi$
$79$	5.50000	−	9.52628i	0.618798	−	1.07179i	0.989705	−	0.143120i	$-0.0457135\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		0			0
$85$	−3.00000			−0.325396
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.50000	+	7.79423i	−0.476999	+	0.826187i	−0.999653	−	0.0263586i	$-0.991609\pi$
$89$	−4.50000	+	7.79423i	−0.476999	+	0.826187i	0.522654	+	0.852545i	$0.324942\pi$
$90$		0			0
$91$	4.00000	+	3.46410i	0.419314	+	0.363137i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.50000	−	4.33013i	−0.256495	−	0.444262i
$96$		0			0
$97$	6.00000			0.609208			0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000			0.609208			0.304604	+	0.952479i	$0.401476\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.50000	−	11.2583i	−0.646774	−	1.12025i	−0.983889	−	0.178782i	$-0.942784\pi$
$101$	−6.50000	−	11.2583i	−0.646774	−	1.12025i	0.337115	−	0.941464i	$-0.390549\pi$
$102$		0			0
$103$	−2.50000	+	4.33013i	−0.246332	+	0.426660i	−0.962505	−	0.271263i	$-0.912559\pi$
$103$	−2.50000	+	4.33013i	−0.246332	+	0.426660i	0.716173	+	0.697923i	$0.245892\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.50000	+	2.59808i	−0.145010	+	0.251166i	−0.929377	−	0.369132i	$-0.879655\pi$
$107$	−1.50000	+	2.59808i	−0.145010	+	0.251166i	0.784366	+	0.620298i	$0.212988\pi$
$108$		0			0
$109$	−5.50000	−	9.52628i	−0.526804	−	0.912452i	−0.999512	−	0.0312328i	$-0.990057\pi$
$109$	−5.50000	−	9.52628i	−0.526804	−	0.912452i	0.472708	−	0.881219i	$-0.343277\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.0000			0.940721			0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000			0.940721			0.470360	+	0.882474i	$0.344124\pi$
$114$		0			0
$115$	−1.50000	−	2.59808i	−0.139876	−	0.242272i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.50000	+	7.79423i	−0.137505	+	0.714496i
$120$		0			0
$121$	5.00000	−	8.66025i	0.454545	−	0.787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.50000	−	14.7224i	0.742648	−	1.28630i	−0.208637	−	0.977993i	$-0.566903\pi$
$131$	8.50000	−	14.7224i	0.742648	−	1.28630i	0.951285	−	0.308312i	$-0.0997640\pi$
$132$		0			0
$133$	−12.5000	+	4.33013i	−1.08389	+	0.375470i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.50000	+	2.59808i	0.128154	+	0.221969i	0.922961	−	0.384893i	$-0.125762\pi$
$137$	1.50000	+	2.59808i	0.128154	+	0.221969i	−0.794808	+	0.606861i	$0.792428\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.00000	−	1.73205i	−0.0836242	−	0.144841i
$144$		0			0
$145$	−3.00000	+	5.19615i	−0.249136	+	0.431517i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.50000	−	12.9904i	0.614424	−	1.06421i	−0.376061	−	0.926595i	$-0.622722\pi$
$149$	7.50000	−	12.9904i	0.614424	−	1.06421i	0.990485	−	0.137619i	$-0.0439449\pi$
$150$		0			0
$151$	−7.50000	−	12.9904i	−0.610341	−	1.05714i	−0.991183	−	0.132502i	$-0.957699\pi$
$151$	−7.50000	−	12.9904i	−0.610341	−	1.05714i	0.380841	−	0.924640i	$-0.375634\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.00000			−0.0803219
$156$		0			0
$157$	−7.50000	−	12.9904i	−0.598565	−	1.03675i	−0.993033	−	0.117836i	$-0.962404\pi$
$157$	−7.50000	−	12.9904i	−0.598565	−	1.03675i	0.394468	−	0.918910i	$-0.370929\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7.50000	+	2.59808i	−0.591083	+	0.204757i
$162$		0			0
$163$	−4.50000	+	7.79423i	−0.352467	+	0.610491i	−0.986681	−	0.162667i	$-0.947991\pi$
$163$	−4.50000	+	7.79423i	−0.352467	+	0.610491i	0.634214	+	0.773158i	$0.281324\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	20.0000			1.54765			0.773823	−	0.633402i	$-0.218342\pi$
$167$	20.0000			1.54765			0.773823	+	0.633402i	$0.218342\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.5000	+	18.1865i	−0.798300	+	1.38270i	0.122422	+	0.992478i	$0.460934\pi$
$173$	−10.5000	+	18.1865i	−0.798300	+	1.38270i	−0.920722	+	0.390218i	$0.872399\pi$
$174$		0			0
$175$	−2.00000	+	10.3923i	−0.151186	+	0.785584i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.500000	−	0.866025i	−0.0373718	−	0.0647298i	0.846735	−	0.532016i	$-0.178565\pi$
$179$	−0.500000	−	0.866025i	−0.0373718	−	0.0647298i	−0.884106	+	0.467286i	$0.845232\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.50000	+	4.33013i	0.183804	+	0.318357i
$186$		0			0
$187$	1.50000	−	2.59808i	0.109691	−	0.189990i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.50000	−	14.7224i	0.615038	−	1.06528i	−0.375339	−	0.926887i	$-0.622474\pi$
$191$	8.50000	−	14.7224i	0.615038	−	1.06528i	0.990378	−	0.138390i	$-0.0441928\pi$
$192$		0			0
$193$	2.50000	+	4.33013i	0.179954	+	0.311689i	0.941865	−	0.335993i	$-0.109072\pi$
$193$	2.50000	+	4.33013i	0.179954	+	0.311689i	−0.761911	+	0.647682i	$0.775738\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$	4.50000	+	7.79423i	0.318997	+	0.552518i	0.980279	−	0.197619i	$-0.0633208\pi$
$199$	4.50000	+	7.79423i	0.318997	+	0.552518i	−0.661282	+	0.750137i	$0.729987\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.0000	+	10.3923i	0.842235	+	0.729397i
$204$		0			0
$205$	−5.00000	+	8.66025i	−0.349215	+	0.604858i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.00000			0.345857
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.00000	−	3.46410i	0.136399	−	0.236250i
$216$		0			0
$217$	−0.500000	+	2.59808i	−0.0339422	+	0.176369i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.00000	+	5.19615i	0.201802	+	0.349531i
$222$		0			0
$223$	24.0000			1.60716			0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000			1.60716			0.803579	+	0.595198i	$0.202926\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.50000	+	6.06218i	0.232303	+	0.402361i	0.958485	−	0.285141i	$-0.0920405\pi$
$227$	3.50000	+	6.06218i	0.232303	+	0.402361i	−0.726182	+	0.687502i	$0.758707\pi$
$228$		0			0
$229$	−3.50000	+	6.06218i	−0.231287	+	0.400600i	−0.958187	−	0.286143i	$-0.907627\pi$
$229$	−3.50000	+	6.06218i	−0.231287	+	0.400600i	0.726900	+	0.686743i	$0.240960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.50000	+	11.2583i	−0.425829	+	0.737558i	−0.996497	−	0.0836229i	$-0.973351\pi$
$233$	−6.50000	+	11.2583i	−0.425829	+	0.737558i	0.570668	+	0.821181i	$0.306684\pi$
$234$		0			0
$235$	0.500000	+	0.866025i	0.0326164	+	0.0564933i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.00000			0.258738			0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000			0.258738			0.129369	+	0.991596i	$0.458705\pi$
$240$		0			0
$241$	8.50000	+	14.7224i	0.547533	+	0.948355i	0.998443	+	0.0557856i	$0.0177663\pi$
$241$	8.50000	+	14.7224i	0.547533	+	0.948355i	−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.50000	−	2.59808i	−0.415270	−	0.165985i
$246$		0			0
$247$	−5.00000	+	8.66025i	−0.318142	+	0.551039i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$	3.00000			0.188608
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.50000	+	11.2583i	−0.405459	+	0.702275i	−0.994375	−	0.105919i	$-0.966222\pi$
$257$	−6.50000	+	11.2583i	−0.405459	+	0.702275i	0.588916	+	0.808194i	$0.299555\pi$
$258$		0			0
$259$	12.5000	−	4.33013i	0.776712	−	0.269061i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.50000	+	2.59808i	0.0924940	+	0.160204i	0.908560	−	0.417755i	$-0.137183\pi$
$263$	1.50000	+	2.59808i	0.0924940	+	0.160204i	−0.816066	+	0.577959i	$0.803849\pi$
$264$		0			0
$265$	9.00000			0.552866
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.50000	−	14.7224i	−0.518254	−	0.897643i	−0.999775	−	0.0212079i	$-0.993249\pi$
$269$	−8.50000	−	14.7224i	−0.518254	−	0.897643i	0.481521	−	0.876435i	$-0.340085\pi$
$270$		0			0
$271$	1.50000	−	2.59808i	0.0911185	−	0.157822i	−0.816864	−	0.576831i	$-0.804289\pi$
$271$	1.50000	−	2.59808i	0.0911185	−	0.157822i	0.907982	+	0.419009i	$0.137622\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.00000	−	3.46410i	0.120605	−	0.208893i
$276$		0			0
$277$	−3.50000	−	6.06218i	−0.210295	−	0.364241i	0.741512	−	0.670940i	$-0.234109\pi$
$277$	−3.50000	−	6.06218i	−0.210295	−	0.364241i	−0.951807	+	0.306699i	$0.900776\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$	8.50000	+	14.7224i	0.505273	+	0.875158i	0.999981	+	0.00609896i	$0.00194137\pi$
$283$	8.50000	+	14.7224i	0.505273	+	0.875158i	−0.494709	+	0.869059i	$0.664725\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	20.0000	+	17.3205i	1.18056	+	1.02240i
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$	−3.00000			−0.174667
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$	−8.00000	−	6.92820i	−0.461112	−	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.50000	−	2.59808i	−0.0858898	−	0.148765i
$306$		0			0
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.50000	+	9.52628i	0.311876	+	0.540186i	0.978769	−	0.204968i	$-0.0657092\pi$
$311$	5.50000	+	9.52628i	0.311876	+	0.540186i	−0.666892	+	0.745154i	$0.732376\pi$
$312$		0			0
$313$	−15.5000	+	26.8468i	−0.876112	+	1.51747i	−0.0205381	+	0.999789i	$0.506538\pi$
$313$	−15.5000	+	26.8468i	−0.876112	+	1.51747i	−0.855574	+	0.517681i	$0.826795\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.5000	−	23.3827i	0.758236	−	1.31330i	−0.185514	−	0.982642i	$-0.559395\pi$
$317$	13.5000	−	23.3827i	0.758236	−	1.31330i	0.943750	−	0.330661i	$-0.107272\pi$
$318$		0			0
$319$	−3.00000	−	5.19615i	−0.167968	−	0.290929i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−15.0000			−0.834622
$324$		0			0
$325$	4.00000	+	6.92820i	0.221880	+	0.384308i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.50000	−	0.866025i	0.137829	−	0.0477455i
$330$		0			0
$331$	3.50000	−	6.06218i	0.192377	−	0.333207i	−0.753660	−	0.657264i	$-0.771714\pi$
$331$	3.50000	−	6.06218i	0.192377	−	0.333207i	0.946038	+	0.324057i	$0.105047\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11.0000			0.600994
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.500000	−	0.866025i	0.0270765	−	0.0468979i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.50000	+	2.59808i	0.0805242	+	0.139472i	0.903475	−	0.428640i	$-0.141007\pi$
$347$	1.50000	+	2.59808i	0.0805242	+	0.139472i	−0.822951	+	0.568112i	$0.807674\pi$
$348$		0			0
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.50000	−	4.33013i	−0.133062	−	0.230469i	0.791794	−	0.610789i	$-0.209147\pi$
$353$	−2.50000	−	4.33013i	−0.133062	−	0.230469i	−0.924855	+	0.380319i	$0.875814\pi$
$354$		0			0
$355$	8.00000	−	13.8564i	0.424596	−	0.735422i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.50000	+	12.9904i	−0.395835	+	0.685606i	−0.993207	−	0.116358i	$-0.962878\pi$
$359$	−7.50000	+	12.9904i	−0.395835	+	0.685606i	0.597372	+	0.801964i	$0.296211\pi$
$360$		0			0
$361$	−3.00000	−	5.19615i	−0.157895	−	0.273482i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.00000			0.366397
$366$		0			0
$367$	−9.50000	−	16.4545i	−0.495896	−	0.858917i	0.504093	−	0.863649i	$-0.331827\pi$
$367$	−9.50000	−	16.4545i	−0.495896	−	0.858917i	−0.999989	+	0.00473247i	$0.998494\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.50000	−	23.3827i	0.233628	−	1.21397i
$372$		0			0
$373$	−9.50000	+	16.4545i	−0.491891	+	0.851981i	−0.999956	−	0.00933789i	$-0.997028\pi$
$373$	−9.50000	+	16.4545i	−0.491891	+	0.851981i	0.508065	+	0.861319i	$0.330361\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.50000	−	7.79423i	0.229939	−	0.398266i	−0.727851	−	0.685736i	$-0.759481\pi$
$383$	4.50000	−	7.79423i	0.229939	−	0.398266i	0.957790	+	0.287469i	$0.0928139\pi$
$384$		0			0
$385$	2.00000	+	1.73205i	0.101929	+	0.0882735i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.50000	+	16.4545i	0.481669	+	0.834275i	0.999779	−	0.0210389i	$-0.00669738\pi$
$389$	9.50000	+	16.4545i	0.481669	+	0.834275i	−0.518110	+	0.855314i	$0.673364\pi$
$390$		0			0
$391$	−9.00000			−0.455150
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.50000	+	9.52628i	0.276735	+	0.479319i
$396$		0			0
$397$	8.50000	−	14.7224i	0.426603	−	0.738898i	−0.569966	−	0.821668i	$-0.693044\pi$
$397$	8.50000	−	14.7224i	0.426603	−	0.738898i	0.996569	+	0.0827707i	$0.0263769\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.50000	−	2.59808i	0.0749064	−	0.129742i	−0.826139	−	0.563466i	$-0.809468\pi$
$401$	1.50000	−	2.59808i	0.0749064	−	0.129742i	0.901046	+	0.433724i	$0.142801\pi$
$402$		0			0
$403$	1.00000	+	1.73205i	0.0498135	+	0.0862796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.00000			−0.247841
$408$		0			0
$409$	−9.50000	−	16.4545i	−0.469745	−	0.813622i	0.529657	−	0.848212i	$-0.322321\pi$
$409$	−9.50000	−	16.4545i	−0.469745	−	0.813622i	−0.999402	+	0.0345902i	$0.988987\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.50000	+	7.79423i	−0.0738102	+	0.383529i
$414$		0			0
$415$	−2.00000	+	3.46410i	−0.0981761	+	0.170046i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.00000	+	10.3923i	−0.291043	+	0.504101i
$426$		0			0
$427$	−7.50000	+	2.59808i	−0.362950	+	0.125730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−20.5000	−	35.5070i	−0.987450	−	1.71031i	−0.630497	−	0.776192i	$-0.717149\pi$
$431$	−20.5000	−	35.5070i	−0.987450	−	1.71031i	−0.356953	−	0.934122i	$-0.616185\pi$
$432$		0			0
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.50000	−	12.9904i	−0.358774	−	0.621414i
$438$		0			0
$439$	7.50000	−	12.9904i	0.357955	−	0.619997i	−0.629664	−	0.776868i	$-0.716807\pi$
$439$	7.50000	−	12.9904i	0.357955	−	0.619997i	0.987619	+	0.156871i	$0.0501406\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.5000	+	23.3827i	−0.641404	+	1.11094i	0.343715	+	0.939074i	$0.388315\pi$
$443$	−13.5000	+	23.3827i	−0.641404	+	1.11094i	−0.985119	+	0.171871i	$0.945019\pi$
$444$		0			0
$445$	−4.50000	−	7.79423i	−0.213320	−	0.369482i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$		0			0
$451$	−5.00000	−	8.66025i	−0.235441	−	0.407795i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.00000	+	1.73205i	−0.234404	+	0.0811998i
$456$		0			0
$457$	8.50000	−	14.7224i	0.397613	−	0.688686i	−0.595818	−	0.803120i	$-0.703172\pi$
$457$	8.50000	−	14.7224i	0.397613	−	0.688686i	0.993431	+	0.114433i	$0.0365053\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.00000			0.279448			0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000			0.279448			0.139724	+	0.990190i	$0.455378\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.5000	−	21.6506i	0.578431	−	1.00187i	−0.417229	−	0.908802i	$-0.636999\pi$
$467$	12.5000	−	21.6506i	0.578431	−	1.00187i	0.995660	−	0.0930703i	$-0.0296681\pi$
$468$		0			0
$469$	5.50000	−	28.5788i	0.253966	−	1.31965i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.00000	+	3.46410i	0.0919601	+	0.159280i
$474$		0			0
$475$	−20.0000			−0.917663
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.5000	−	18.1865i	−0.479757	−	0.830964i	0.519973	−	0.854183i	$-0.325942\pi$
$479$	−10.5000	−	18.1865i	−0.479757	−	0.830964i	−0.999730	+	0.0232187i	$0.992609\pi$
$480$		0			0
$481$	5.00000	−	8.66025i	0.227980	−	0.394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.00000	+	5.19615i	−0.136223	+	0.235945i
$486$		0			0
$487$	6.50000	+	11.2583i	0.294543	+	0.510164i	0.974879	−	0.222737i	$-0.0714992\pi$
$487$	6.50000	+	11.2583i	0.294543	+	0.510164i	−0.680335	+	0.732901i	$0.738166\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	9.00000	+	15.5885i	0.405340	+	0.702069i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−32.0000	−	27.7128i	−1.43540	−	1.24309i
$498$		0			0
$499$	3.50000	−	6.06218i	0.156682	−	0.271380i	−0.776989	−	0.629515i	$-0.783254\pi$
$499$	3.50000	−	6.06218i	0.156682	−	0.271380i	0.933670	+	0.358134i	$0.116587\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16.0000			−0.713405			−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000			−0.713405			−0.356702	+	0.934218i	$0.616099\pi$
$504$		0			0
$505$	13.0000			0.578492
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.50000	−	6.06218i	0.155135	−	0.268701i	−0.777973	−	0.628297i	$-0.783752\pi$
$509$	3.50000	−	6.06218i	0.155135	−	0.268701i	0.933108	+	0.359596i	$0.117085\pi$
$510$		0			0
$511$	3.50000	−	18.1865i	0.154831	−	0.804525i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.50000	−	4.33013i	−0.110163	−	0.190808i
$516$		0			0
$517$	−1.00000			−0.0439799
$518$		0			0
$519$		0			0
$520$		0			0
$521$	7.50000	+	12.9904i	0.328581	+	0.569119i	0.982231	−	0.187678i	$-0.0600963\pi$
$521$	7.50000	+	12.9904i	0.328581	+	0.569119i	−0.653650	+	0.756797i	$0.726763\pi$
$522$		0			0
$523$	−6.50000	+	11.2583i	−0.284225	+	0.492292i	−0.972421	−	0.233233i	$-0.925070\pi$
$523$	−6.50000	+	11.2583i	−0.284225	+	0.492292i	0.688196	+	0.725525i	$0.258403\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.50000	+	2.59808i	−0.0653410	+	0.113174i
$528$		0			0
$529$	7.00000	+	12.1244i	0.304348	+	0.527146i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	20.0000			0.866296
$534$		0			0
$535$	−1.50000	−	2.59808i	−0.0648507	−	0.112325i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.50000	−	4.33013i	0.236902	−	0.186512i
$540$		0			0
$541$	12.5000	−	21.6506i	0.537417	−	0.930834i	−0.461625	−	0.887075i	$-0.652733\pi$
$541$	12.5000	−	21.6506i	0.537417	−	0.930834i	0.999042	−	0.0437584i	$-0.0139332\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	11.0000			0.471188
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−15.0000	+	25.9808i	−0.639021	+	1.10682i
$552$		0			0
$553$	27.5000	−	9.52628i	1.16942	−	0.405099i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.50000	+	9.52628i	0.233042	+	0.403641i	0.958702	−	0.284413i	$-0.0917985\pi$
$557$	5.50000	+	9.52628i	0.233042	+	0.403641i	−0.725660	+	0.688054i	$0.758465\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.50000	+	9.52628i	0.231797	+	0.401485i	0.958337	−	0.285640i	$-0.0922060\pi$
$563$	5.50000	+	9.52628i	0.231797	+	0.401485i	−0.726540	+	0.687124i	$0.758873\pi$
$564$		0			0
$565$	−5.00000	+	8.66025i	−0.210352	+	0.364340i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.500000	+	0.866025i	−0.0209611	+	0.0363057i	−0.876316	−	0.481737i	$-0.840006\pi$
$569$	−0.500000	+	0.866025i	−0.0209611	+	0.0363057i	0.855355	+	0.518043i	$0.173339\pi$
$570$		0			0
$571$	8.50000	+	14.7224i	0.355714	+	0.616115i	0.987240	−	0.159240i	$-0.0509044\pi$
$571$	8.50000	+	14.7224i	0.355714	+	0.616115i	−0.631526	+	0.775355i	$0.717571\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12.0000			−0.500435
$576$		0			0
$577$	−15.5000	−	26.8468i	−0.645273	−	1.11765i	−0.984238	−	0.176847i	$-0.943410\pi$
$577$	−15.5000	−	26.8468i	−0.645273	−	1.11765i	0.338965	−	0.940799i	$-0.389923\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.00000	+	6.92820i	0.331896	+	0.287430i
$582$		0			0
$583$	−4.50000	+	7.79423i	−0.186371	+	0.322804i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$		0			0
$589$	−5.00000			−0.206021
$590$		0			0
$591$		0			0
$592$		0			0
$593$	21.5000	−	37.2391i	0.882899	−	1.52923i	0.0347964	−	0.999394i	$-0.488922\pi$
$593$	21.5000	−	37.2391i	0.882899	−	1.52923i	0.848103	−	0.529832i	$-0.177745\pi$
$594$		0			0
$595$	−6.00000	−	5.19615i	−0.245976	−	0.213021i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.5000	−	18.1865i	−0.429018	−	0.743082i	0.567768	−	0.823189i	$-0.307807\pi$
$599$	−10.5000	−	18.1865i	−0.429018	−	0.743082i	−0.996786	+	0.0801071i	$0.974474\pi$
$600$		0			0
$601$	−34.0000			−1.38689			−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000			−1.38689			−0.693444	+	0.720510i	$0.743908\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.00000	+	8.66025i	0.203279	+	0.352089i
$606$		0			0
$607$	3.50000	−	6.06218i	0.142061	−	0.246056i	−0.786212	−	0.617957i	$-0.787961\pi$
$607$	3.50000	−	6.06218i	0.142061	−	0.246056i	0.928272	+	0.371901i	$0.121294\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.00000	−	1.73205i	0.0404557	−	0.0700713i
$612$		0			0
$613$	10.5000	+	18.1865i	0.424091	+	0.734547i	0.996335	−	0.0855362i	$-0.0272603\pi$
$613$	10.5000	+	18.1865i	0.424091	+	0.734547i	−0.572244	+	0.820083i	$0.693927\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.0000			−0.885687			−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000			−0.885687			−0.442843	+	0.896599i	$0.646030\pi$
$618$		0			0
$619$	2.50000	+	4.33013i	0.100483	+	0.174042i	0.911884	−	0.410448i	$-0.134628\pi$
$619$	2.50000	+	4.33013i	0.100483	+	0.174042i	−0.811400	+	0.584491i	$0.801294\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−22.5000	+	7.79423i	−0.901443	+	0.312269i
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	15.0000			0.598089
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.00000	−	6.92820i	0.158735	−	0.274937i
$636$		0			0
$637$	2.00000	+	13.8564i	0.0792429	+	0.549011i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.50000	+	12.9904i	0.296232	+	0.513089i	0.975271	−	0.221013i	$-0.0709364\pi$
$641$	7.50000	+	12.9904i	0.296232	+	0.513089i	−0.679039	+	0.734103i	$0.737603\pi$
$642$		0			0
$643$	−44.0000			−1.73519			−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000			−1.73519			−0.867595	+	0.497271i	$0.834335\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	21.5000	+	37.2391i	0.845252	+	1.46402i	0.885402	+	0.464826i	$0.153883\pi$
$647$	21.5000	+	37.2391i	0.845252	+	1.46402i	−0.0401498	+	0.999194i	$0.512784\pi$
$648$		0			0
$649$	1.50000	−	2.59808i	0.0588802	−	0.101983i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.50000	+	4.33013i	−0.0978326	+	0.169451i	−0.910787	−	0.412876i	$-0.864524\pi$
$653$	−2.50000	+	4.33013i	−0.0978326	+	0.169451i	0.812955	+	0.582327i	$0.197858\pi$
$654$		0			0
$655$	8.50000	+	14.7224i	0.332122	+	0.575253i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	36.0000			1.40236			0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000			1.40236			0.701180	+	0.712984i	$0.252657\pi$
$660$		0			0
$661$	0.500000	+	0.866025i	0.0194477	+	0.0336845i	0.875585	−	0.483063i	$-0.160476\pi$
$661$	0.500000	+	0.866025i	0.0194477	+	0.0336845i	−0.856138	+	0.516748i	$0.827143\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.50000	−	12.9904i	0.0969458	−	0.503745i
$666$		0			0
$667$	−9.00000	+	15.5885i	−0.348481	+	0.603587i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3.00000			0.115814
$672$		0			0
$673$	−2.00000			−0.0770943			−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000			−0.0770943			−0.0385472	+	0.999257i	$0.512273\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.50000	+	7.79423i	−0.172949	+	0.299557i	−0.939450	−	0.342687i	$-0.888663\pi$
$677$	−4.50000	+	7.79423i	−0.172949	+	0.299557i	0.766501	+	0.642244i	$0.221996\pi$
$678$		0			0
$679$	12.0000	+	10.3923i	0.460518	+	0.398820i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	19.5000	+	33.7750i	0.746147	+	1.29236i	0.949657	+	0.313291i	$0.101432\pi$
$683$	19.5000	+	33.7750i	0.746147	+	1.29236i	−0.203510	+	0.979073i	$0.565235\pi$
$684$		0			0
$685$	−3.00000			−0.114624
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.00000	−	15.5885i	−0.342873	−	0.593873i
$690$		0			0
$691$	23.5000	−	40.7032i	0.893982	−	1.54842i	0.0589228	−	0.998263i	$-0.481233\pi$
$691$	23.5000	−	40.7032i	0.893982	−	1.54842i	0.835059	−	0.550160i	$-0.185433\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.00000	+	3.46410i	−0.0758643	+	0.131401i
$696$		0			0
$697$	15.0000	+	25.9808i	0.568166	+	0.984092i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	12.5000	+	21.6506i	0.471446	+	0.816569i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.50000	−	33.7750i	0.244458	−	1.27024i
$708$		0			0
$709$	−13.5000	+	23.3827i	−0.507003	+	0.878155i	0.492964	+	0.870050i	$0.335913\pi$
$709$	−13.5000	+	23.3827i	−0.507003	+	0.878155i	−0.999967	+	0.00810550i	$0.997420\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.00000			−0.112351
$714$		0			0
$715$	2.00000			0.0747958
$716$		0			0
$717$		0			0
$718$		0			0
$719$	14.5000	−	25.1147i	0.540759	−	0.936622i	−0.458102	−	0.888900i	$-0.651471\pi$
$719$	14.5000	−	25.1147i	0.540759	−	0.936622i	0.998861	−	0.0477220i	$-0.0151961\pi$
$720$		0			0
$721$	−12.5000	+	4.33013i	−0.465524	+	0.161262i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12.0000	+	20.7846i	0.445669	+	0.771921i
$726$		0			0
$727$	16.0000			0.593407			0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000			0.593407			0.296704	+	0.954970i	$0.404113\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−6.00000	−	10.3923i	−0.221918	−	0.384373i
$732$		0			0
$733$	−5.50000	+	9.52628i	−0.203147	+	0.351861i	−0.949541	−	0.313644i	$-0.898450\pi$
$733$	−5.50000	+	9.52628i	−0.203147	+	0.351861i	0.746394	+	0.665505i	$0.231784\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.50000	+	9.52628i	−0.202595	+	0.350905i
$738$		0			0
$739$	20.5000	+	35.5070i	0.754105	+	1.30615i	0.945818	+	0.324697i	$0.105262\pi$
$739$	20.5000	+	35.5070i	0.754105	+	1.30615i	−0.191714	+	0.981451i	$0.561404\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−32.0000			−1.17397			−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000			−1.17397			−0.586983	+	0.809599i	$0.699684\pi$
$744$		0			0
$745$	7.50000	+	12.9904i	0.274779	+	0.475931i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.50000	+	2.59808i	−0.274044	+	0.0949316i
$750$		0			0
$751$	23.5000	−	40.7032i	0.857527	−	1.48528i	−0.0167534	−	0.999860i	$-0.505333\pi$
$751$	23.5000	−	40.7032i	0.857527	−	1.48528i	0.874281	−	0.485421i	$-0.161334\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.0000			0.545906
$756$		0			0
$757$	34.0000			1.23575			0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000			1.23575			0.617876	+	0.786276i	$0.287994\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	13.5000	−	23.3827i	0.489375	−	0.847622i	−0.510551	−	0.859848i	$-0.670558\pi$
$761$	13.5000	−	23.3827i	0.489375	−	0.847622i	0.999925	+	0.0122260i	$0.00389175\pi$
$762$		0			0
$763$	5.50000	−	28.5788i	0.199113	−	1.03462i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.00000	+	5.19615i	0.108324	+	0.187622i
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	17.5000	+	30.3109i	0.629431	+	1.09021i	0.987666	+	0.156575i	$0.0500454\pi$
$773$	17.5000	+	30.3109i	0.629431	+	1.09021i	−0.358235	+	0.933632i	$0.616621\pi$
$774$		0			0
$775$	−2.00000	+	3.46410i	−0.0718421	+	0.124434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−25.0000	+	43.3013i	−0.895718	+	1.55143i
$780$		0			0
$781$	8.00000	+	13.8564i	0.286263	+	0.495821i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	15.0000			0.535373
$786$		0			0
$787$	6.50000	+	11.2583i	0.231700	+	0.401316i	0.958308	−	0.285736i	$-0.0922379\pi$
$787$	6.50000	+	11.2583i	0.231700	+	0.401316i	−0.726609	+	0.687052i	$0.758905\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	20.0000	+	17.3205i	0.711118	+	0.615846i
$792$		0			0
$793$	−3.00000	+	5.19615i	−0.106533	+	0.184521i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−42.0000			−1.48772			−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000			−1.48772			−0.743858	+	0.668338i	$0.767006\pi$
$798$		0			0
$799$	3.00000			0.106132
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.50000	+	6.06218i	−0.123512	+	0.213930i
$804$		0			0
$805$	1.50000	−	7.79423i	0.0528681	−	0.274710i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−12.5000	−	21.6506i	−0.439477	−	0.761196i	0.558173	−	0.829725i	$-0.311503\pi$
$809$	−12.5000	−	21.6506i	−0.439477	−	0.761196i	−0.997649	+	0.0685291i	$0.978169\pi$
$810$		0			0
$811$	−20.0000			−0.702295			−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000			−0.702295			−0.351147	+	0.936320i	$0.614208\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.50000	−	7.79423i	−0.157628	−	0.273020i
$816$		0			0
$817$	10.0000	−	17.3205i	0.349856	−	0.605968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12.5000	+	21.6506i	−0.436253	+	0.755612i	−0.997397	−	0.0721058i	$-0.977028\pi$
$821$	−12.5000	+	21.6506i	−0.436253	+	0.755612i	0.561144	+	0.827718i	$0.310361\pi$
$822$		0			0
$823$	10.5000	+	18.1865i	0.366007	+	0.633943i	0.988937	−	0.148335i	$-0.0473913\pi$
$823$	10.5000	+	18.1865i	0.366007	+	0.633943i	−0.622930	+	0.782277i	$0.714058\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.00000			0.139094			0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000			0.139094			0.0695468	+	0.997579i	$0.477845\pi$
$828$		0			0
$829$	18.5000	+	32.0429i	0.642532	+	1.11290i	0.984866	+	0.173319i	$0.0554492\pi$
$829$	18.5000	+	32.0429i	0.642532	+	1.11290i	−0.342334	+	0.939578i	$0.611217\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16.5000	+	12.9904i	−0.571691	+	0.450090i
$834$		0			0
$835$	−10.0000	+	17.3205i	−0.346064	+	0.599401i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.50000	−	7.79423i	0.154805	−	0.268130i
$846$		0			0
$847$	25.0000	−	8.66025i	0.859010	−	0.297570i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.50000	+	12.9904i	0.257097	+	0.445305i
$852$		0			0
$853$	−22.0000			−0.753266			−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000			−0.753266			−0.376633	+	0.926363i	$0.622918\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−28.5000	−	49.3634i	−0.973541	−	1.68622i	−0.684667	−	0.728856i	$-0.740052\pi$
$857$	−28.5000	−	49.3634i	−0.973541	−	1.68622i	−0.288875	−	0.957367i	$-0.593281\pi$
$858$		0			0
$859$	−2.50000	+	4.33013i	−0.0852989	+	0.147742i	−0.905519	−	0.424307i	$-0.860518\pi$
$859$	−2.50000	+	4.33013i	−0.0852989	+	0.147742i	0.820220	+	0.572049i	$0.193851\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	18.5000	−	32.0429i	0.629747	−	1.09075i	−0.357855	−	0.933777i	$-0.616492\pi$
$863$	18.5000	−	32.0429i	0.629747	−	1.09075i	0.987602	−	0.156977i	$-0.0501749\pi$
$864$		0			0
$865$	−10.5000	−	18.1865i	−0.357011	−	0.618361i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−11.0000			−0.373149
$870$		0			0
$871$	−11.0000	−	19.0526i	−0.372721	−	0.645571i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−18.0000	−	15.5885i	−0.608511	−	0.526986i
$876$		0			0
$877$	−19.5000	+	33.7750i	−0.658468	+	1.14050i	0.322544	+	0.946554i	$0.395462\pi$
$877$	−19.5000	+	33.7750i	−0.658468	+	1.14050i	−0.981012	+	0.193946i	$0.937871\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−27.5000	+	47.6314i	−0.923360	+	1.59931i	−0.129181	+	0.991621i	$0.541235\pi$
$887$	−27.5000	+	47.6314i	−0.923360	+	1.59931i	−0.794178	+	0.607685i	$0.792098\pi$
$888$		0			0
$889$	−16.0000	−	13.8564i	−0.536623	−	0.464729i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.50000	+	4.33013i	0.0836593	+	0.144902i
$894$		0			0
$895$	1.00000			0.0334263
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.00000	+	5.19615i	0.100056	+	0.173301i
$900$		0			0
$901$	13.5000	−	23.3827i	0.449750	−	0.778990i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−11.0000	+	19.0526i	−0.365652	+	0.633328i
$906$		0			0
$907$	6.50000	+	11.2583i	0.215829	+	0.373827i	0.953529	−	0.301302i	$-0.0974213\pi$
$907$	6.50000	+	11.2583i	0.215829	+	0.373827i	−0.737700	+	0.675129i	$0.764088\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$	−2.00000	−	3.46410i	−0.0661903	−	0.114645i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	42.5000	−	14.7224i	1.40347	−	0.486178i
$918$		0			0
$919$	−8.50000	+	14.7224i	−0.280389	+	0.485648i	−0.971481	−	0.237119i	$-0.923797\pi$
$919$	−8.50000	+	14.7224i	−0.280389	+	0.485648i	0.691091	+	0.722767i	$0.257130\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−32.0000			−1.05329
$924$		0			0
$925$	20.0000			0.657596
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−20.5000	+	35.5070i	−0.672583	+	1.16495i	0.304586	+	0.952485i	$0.401482\pi$
$929$	−20.5000	+	35.5070i	−0.672583	+	1.16495i	−0.977169	+	0.212463i	$0.931851\pi$
$930$		0			0
$931$	−32.5000	−	12.9904i	−1.06514	−	0.425743i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.50000	+	2.59808i	0.0490552	+	0.0849662i
$936$		0			0
$937$	−38.0000			−1.24141			−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000			−1.24141			−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.5000	+	33.7750i	0.635682	+	1.10103i	0.986370	+	0.164541i	$0.0526143\pi$
$941$	19.5000	+	33.7750i	0.635682	+	1.10103i	−0.350688	+	0.936492i	$0.614052\pi$
$942$		0			0
$943$	−15.0000	+	25.9808i	−0.488467	+	0.846050i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24.5000	−	42.4352i	0.796143	−	1.37896i	−0.125968	−	0.992034i	$-0.540204\pi$
$947$	24.5000	−	42.4352i	0.796143	−	1.37896i	0.922111	−	0.386926i	$-0.126463\pi$
$948$		0			0
$949$	−7.00000	−	12.1244i	−0.227230	−	0.393573i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−30.0000			−0.971795			−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000			−0.971795			−0.485898	+	0.874016i	$0.661507\pi$
$954$		0			0
$955$	8.50000	+	14.7224i	0.275054	+	0.476407i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.50000	+	7.79423i	−0.0484375	+	0.251689i
$960$		0			0
$961$	15.0000	−	25.9808i	0.483871	−	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.00000			−0.160956
$966$		0			0
$967$	32.0000			1.02905			0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000			1.02905			0.514525	+	0.857475i	$0.327968\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	28.5000	−	49.3634i	0.914609	−	1.58415i	0.107135	−	0.994244i	$-0.465832\pi$
$971$	28.5000	−	49.3634i	0.914609	−	1.58415i	0.807473	−	0.589904i	$-0.200834\pi$
$972$		0			0
$973$	8.00000	+	6.92820i	0.256468	+	0.222108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.5000	−	32.0429i	−0.591867	−	1.02514i	−0.993981	−	0.109555i	$-0.965058\pi$
$977$	−18.5000	−	32.0429i	−0.591867	−	1.02514i	0.402113	−	0.915590i	$-0.368276\pi$
$978$		0			0
$979$	9.00000			0.287641
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16.5000	−	28.5788i	−0.526268	−	0.911523i	−0.999532	−	0.0306024i	$-0.990257\pi$
$983$	−16.5000	−	28.5788i	−0.526268	−	0.911523i	0.473263	−	0.880921i	$-0.343076\pi$
$984$		0			0
$985$	9.00000	−	15.5885i	0.286764	−	0.496690i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.00000	−	10.3923i	0.190789	−	0.330456i
$990$		0			0
$991$	−29.5000	−	51.0955i	−0.937098	−	1.62310i	−0.770849	−	0.637018i	$-0.780168\pi$
$991$	−29.5000	−	51.0955i	−0.937098	−	1.62310i	−0.166250	−	0.986084i	$-0.553166\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9.00000			−0.285319
$996$		0			0
$997$	−15.5000	−	26.8468i	−0.490890	−	0.850246i	0.509055	−	0.860734i	$-0.329995\pi$
$997$	−15.5000	−	26.8468i	−0.490890	−	0.850246i	−0.999945	+	0.0104877i	$0.996662\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.s.e.289.1		2
3.2	odd	2		56.2.i.a.9.1	✓	2
4.3	odd	2		1008.2.s.e.289.1		2
7.2	even	3		3528.2.a.r.1.1		1
7.3	odd	6		3528.2.s.o.361.1		2
7.4	even	3	inner	504.2.s.e.361.1		2
7.5	odd	6		3528.2.a.k.1.1		1
7.6	odd	2		3528.2.s.o.3313.1		2
12.11	even	2		112.2.i.c.65.1		2
15.2	even	4		1400.2.bh.f.849.1		4
15.8	even	4		1400.2.bh.f.849.2		4
15.14	odd	2		1400.2.q.g.401.1		2
21.2	odd	6		392.2.a.f.1.1		1
21.5	even	6		392.2.a.a.1.1		1
21.11	odd	6		56.2.i.a.25.1	yes	2
21.17	even	6		392.2.i.f.361.1		2
21.20	even	2		392.2.i.f.177.1		2
24.5	odd	2		448.2.i.f.65.1		2
24.11	even	2		448.2.i.a.65.1		2
28.11	odd	6		1008.2.s.e.865.1		2
28.19	even	6		7056.2.a.s.1.1		1
28.23	odd	6		7056.2.a.bi.1.1		1
84.11	even	6		112.2.i.c.81.1		2
84.23	even	6		784.2.a.a.1.1		1
84.47	odd	6		784.2.a.j.1.1		1
84.59	odd	6		784.2.i.a.753.1		2
84.83	odd	2		784.2.i.a.177.1		2
105.32	even	12		1400.2.bh.f.249.2		4
105.44	odd	6		9800.2.a.b.1.1		1
105.53	even	12		1400.2.bh.f.249.1		4
105.74	odd	6		1400.2.q.g.1201.1		2
105.89	even	6		9800.2.a.bp.1.1		1
168.5	even	6		3136.2.a.bb.1.1		1
168.11	even	6		448.2.i.a.193.1		2
168.53	odd	6		448.2.i.f.193.1		2
168.107	even	6		3136.2.a.bc.1.1		1
168.131	odd	6		3136.2.a.a.1.1		1
168.149	odd	6		3136.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.i.a.9.1	✓	2	3.2	odd	2
56.2.i.a.25.1	yes	2	21.11	odd	6
112.2.i.c.65.1		2	12.11	even	2
112.2.i.c.81.1		2	84.11	even	6
392.2.a.a.1.1		1	21.5	even	6
392.2.a.f.1.1		1	21.2	odd	6
392.2.i.f.177.1		2	21.20	even	2
392.2.i.f.361.1		2	21.17	even	6
448.2.i.a.65.1		2	24.11	even	2
448.2.i.a.193.1		2	168.11	even	6
448.2.i.f.65.1		2	24.5	odd	2
448.2.i.f.193.1		2	168.53	odd	6
504.2.s.e.289.1		2	1.1	even	1	trivial
504.2.s.e.361.1		2	7.4	even	3	inner
784.2.a.a.1.1		1	84.23	even	6
784.2.a.j.1.1		1	84.47	odd	6
784.2.i.a.177.1		2	84.83	odd	2
784.2.i.a.753.1		2	84.59	odd	6
1008.2.s.e.289.1		2	4.3	odd	2
1008.2.s.e.865.1		2	28.11	odd	6
1400.2.q.g.401.1		2	15.14	odd	2
1400.2.q.g.1201.1		2	105.74	odd	6
1400.2.bh.f.249.1		4	105.53	even	12
1400.2.bh.f.249.2		4	105.32	even	12
1400.2.bh.f.849.1		4	15.2	even	4
1400.2.bh.f.849.2		4	15.8	even	4
3136.2.a.a.1.1		1	168.131	odd	6
3136.2.a.b.1.1		1	168.149	odd	6
3136.2.a.bb.1.1		1	168.5	even	6
3136.2.a.bc.1.1		1	168.107	even	6
3528.2.a.k.1.1		1	7.5	odd	6
3528.2.a.r.1.1		1	7.2	even	3
3528.2.s.o.361.1		2	7.3	odd	6
3528.2.s.o.3313.1		2	7.6	odd	2
7056.2.a.s.1.1		1	28.19	even	6
7056.2.a.bi.1.1		1	28.23	odd	6
9800.2.a.b.1.1		1	105.44	odd	6
9800.2.a.bp.1.1		1	105.89	even	6

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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{5} +(2.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{11} +2.00000 q^{13} +(1.50000 + 2.59808i) q^{17} +(-2.50000 + 4.33013i) q^{19} +(-1.50000 + 2.59808i) q^{23} +(2.00000 + 3.46410i) q^{25} +6.00000 q^{29} +(0.500000 + 0.866025i) q^{31} +(-2.50000 + 0.866025i) q^{35} +(2.50000 - 4.33013i) q^{37} +10.0000 q^{41} -4.00000 q^{43} +(0.500000 - 0.866025i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-4.50000 - 7.79423i) q^{53} +1.00000 q^{55} +(1.50000 + 2.59808i) q^{59} +(-1.50000 + 2.59808i) q^{61} +(-1.00000 + 1.73205i) q^{65} +(-5.50000 - 9.52628i) q^{67} -16.0000 q^{71} +(-3.50000 - 6.06218i) q^{73} +(0.500000 - 2.59808i) q^{77} +(5.50000 - 9.52628i) q^{79} +4.00000 q^{83} -3.00000 q^{85} +(-4.50000 + 7.79423i) q^{89} +(4.00000 + 3.46410i) q^{91} +(-2.50000 - 4.33013i) q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} + 4 q^{7} - q^{11} + 4 q^{13} + 3 q^{17} - 5 q^{19} - 3 q^{23} + 4 q^{25} + 12 q^{29} + q^{31} - 5 q^{35} + 5 q^{37} + 20 q^{41} - 8 q^{43} + q^{47} + 2 q^{49} - 9 q^{53} + 2 q^{55} + 3 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) \) 2 * q - q^5 + 4 * q^7 - q^11 + 4 * q^13 + 3 * q^17 - 5 * q^19 - 3 * q^23 + 4 * q^25 + 12 * q^29 + q^31 - 5 * q^35 + 5 * q^37 + 20 * q^41 - 8 * q^43 + q^47 + 2 * q^49 - 9 * q^53 + 2 * q^55 + 3 * q^59 - 3 * q^61 - 2 * q^65 - 11 * q^67 - 32 * q^71 - 7 * q^73 + q^77 + 11 * q^79 + 8 * q^83 - 6 * q^85 - 9 * q^89 + 8 * q^91 - 5 * q^95 + 12 * q^97

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