LMFDB - Embedded newform 1512.2.r.a.505.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1512,2,Mod(505,1512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1512, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1512.505");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.r (of order $3$, degree $2$, not 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:	$12.0733807856$
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 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			505.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1512.505
Dual form			1512.2.r.a.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{7} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(-3.00000 + 5.19615i) q^{11} +(-3.00000 - 5.19615i) q^{13} +2.00000 q^{17} +7.00000 q^{19} +(-0.500000 - 0.866025i) q^{23} +(2.00000 - 3.46410i) q^{25} +(1.00000 - 1.73205i) q^{29} +(-5.00000 - 8.66025i) q^{31} +1.00000 q^{35} -6.00000 q^{37} +(-4.00000 - 6.92820i) q^{41} +(5.00000 - 8.66025i) q^{43} +(4.00000 - 6.92820i) q^{47} +(-0.500000 - 0.866025i) q^{49} -2.00000 q^{53} +6.00000 q^{55} +(-3.50000 + 6.06218i) q^{61} +(-3.00000 + 5.19615i) q^{65} +(6.00000 + 10.3923i) q^{67} -15.0000 q^{71} -2.00000 q^{73} +(-3.00000 - 5.19615i) q^{77} +(-0.500000 + 0.866025i) q^{79} +(6.00000 - 10.3923i) q^{83} +(-1.00000 - 1.73205i) q^{85} -4.00000 q^{89} +6.00000 q^{91} +(-3.50000 - 6.06218i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} - q^{7}+O(q^{10}) $ 2 * q - q^5 - q^7	$ 2 q - q^{5} - q^{7} - 6 q^{11} - 6 q^{13} + 4 q^{17} + 14 q^{19} - q^{23} + 4 q^{25} + 2 q^{29} - 10 q^{31} + 2 q^{35} - 12 q^{37} - 8 q^{41} + 10 q^{43} + 8 q^{47} - q^{49} - 4 q^{53} + 12 q^{55} - 7 q^{61} - 6 q^{65} + 12 q^{67} - 30 q^{71} - 4 q^{73} - 6 q^{77} - q^{79} + 12 q^{83} - 2 q^{85} - 8 q^{89} + 12 q^{91} - 7 q^{95} + 2 q^{97}+O(q^{100}) $ 2 * q - q^5 - q^7 - 6 * q^11 - 6 * q^13 + 4 * q^17 + 14 * q^19 - q^23 + 4 * q^25 + 2 * q^29 - 10 * q^31 + 2 * q^35 - 12 * q^37 - 8 * q^41 + 10 * q^43 + 8 * q^47 - q^49 - 4 * q^53 + 12 * q^55 - 7 * q^61 - 6 * q^65 + 12 * q^67 - 30 * q^71 - 4 * q^73 - 6 * q^77 - q^79 + 12 * q^83 - 2 * q^85 - 8 * q^89 + 12 * q^91 - 7 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	0.732294	−	0.680989i	$-0.238450\pi$
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	−0.955901	+	0.293691i	$0.905116\pi$
$6$		0			0
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.00000	+	5.19615i	−0.904534	+	1.56670i	−0.0829925	+	0.996550i	$0.526448\pi$
$11$	−3.00000	+	5.19615i	−0.904534	+	1.56670i	−0.821541	+	0.570149i	$0.806886\pi$
$12$		0			0
$13$	−3.00000	−	5.19615i	−0.832050	−	1.44115i	−0.896410	−	0.443227i	$-0.853834\pi$
$13$	−3.00000	−	5.19615i	−0.832050	−	1.44115i	0.0643593	−	0.997927i	$-0.479500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$	7.00000			1.60591			0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000			1.60591			0.802955	+	0.596040i	$0.203260\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.500000	−	0.866025i	−0.104257	−	0.180579i	0.809177	−	0.587565i	$-0.199913\pi$
$23$	−0.500000	−	0.866025i	−0.104257	−	0.180579i	−0.913434	+	0.406986i	$0.866580\pi$
$24$		0			0
$25$	2.00000	−	3.46410i	0.400000	−	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	−0.758115	−	0.652121i	$-0.773880\pi$
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	0.943811	+	0.330487i	$0.107213\pi$
$30$		0			0
$31$	−5.00000	−	8.66025i	−0.898027	−	1.55543i	−0.830014	−	0.557743i	$-0.811667\pi$
$31$	−5.00000	−	8.66025i	−0.898027	−	1.55543i	−0.0680129	−	0.997684i	$-0.521666\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.00000	−	6.92820i	−0.624695	−	1.08200i	−0.988600	−	0.150567i	$-0.951890\pi$
$41$	−4.00000	−	6.92820i	−0.624695	−	1.08200i	0.363905	−	0.931436i	$-0.381443\pi$
$42$		0			0
$43$	5.00000	−	8.66025i	0.762493	−	1.32068i	−0.179069	−	0.983836i	$-0.557309\pi$
$43$	5.00000	−	8.66025i	0.762493	−	1.32068i	0.941562	−	0.336840i	$-0.109358\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	−0.411606	−	0.911362i	$-0.635032\pi$
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	0.995066	−	0.0992202i	$-0.0316348\pi$
$48$		0			0
$49$	−0.500000	−	0.866025i	−0.0714286	−	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$	6.00000			0.809040
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−3.50000	+	6.06218i	−0.448129	+	0.776182i	−0.998264	−	0.0588933i	$-0.981243\pi$
$61$	−3.50000	+	6.06218i	−0.448129	+	0.776182i	0.550135	+	0.835076i	$0.314576\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.00000	+	5.19615i	−0.372104	+	0.644503i
$66$		0			0
$67$	6.00000	+	10.3923i	0.733017	+	1.26962i	0.955588	+	0.294706i	$0.0952216\pi$
$67$	6.00000	+	10.3923i	0.733017	+	1.26962i	−0.222571	+	0.974916i	$0.571445\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−15.0000			−1.78017			−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000			−1.78017			−0.890086	+	0.455792i	$0.849356\pi$
$72$		0			0
$73$	−2.00000			−0.234082			−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000			−0.234082			−0.117041	+	0.993127i	$0.537341\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.00000	−	5.19615i	−0.341882	−	0.592157i
$78$		0			0
$79$	−0.500000	+	0.866025i	−0.0562544	+	0.0974355i	−0.892781	−	0.450490i	$-0.851249\pi$
$79$	−0.500000	+	0.866025i	−0.0562544	+	0.0974355i	0.836527	+	0.547926i	$0.184582\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.00000	−	10.3923i	0.658586	−	1.14070i	−0.322396	−	0.946605i	$-0.604488\pi$
$83$	6.00000	−	10.3923i	0.658586	−	1.14070i	0.980982	−	0.194099i	$-0.0621783\pi$
$84$		0			0
$85$	−1.00000	−	1.73205i	−0.108465	−	0.187867i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.00000			−0.423999			−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000			−0.423999			−0.212000	+	0.977270i	$0.567998\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.50000	−	6.06218i	−0.359092	−	0.621966i
$96$		0			0
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	−0.810782	−	0.585348i	$-0.800958\pi$
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	0.912317	+	0.409484i	$0.134291\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	−0.781697	−	0.623658i	$-0.785646\pi$
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	0.930953	+	0.365140i	$0.118979\pi$
$102$		0			0
$103$	−3.00000	−	5.19615i	−0.295599	−	0.511992i	0.679525	−	0.733652i	$-0.262186\pi$
$103$	−3.00000	−	5.19615i	−0.295599	−	0.511992i	−0.975124	+	0.221660i	$0.928852\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.50000	−	7.79423i	−0.423324	−	0.733219i	0.572938	−	0.819599i	$-0.305804\pi$
$113$	−4.50000	−	7.79423i	−0.423324	−	0.733219i	−0.996262	+	0.0863794i	$0.972470\pi$
$114$		0			0
$115$	−0.500000	+	0.866025i	−0.0466252	+	0.0807573i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.00000	+	1.73205i	−0.0916698	+	0.158777i
$120$		0			0
$121$	−12.5000	−	21.6506i	−1.13636	−	1.96824i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	5.00000			0.443678			0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000			0.443678			0.221839	+	0.975083i	$0.428794\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.50000	+	14.7224i	0.742648	+	1.28630i	0.951285	+	0.308312i	$0.0997640\pi$
$131$	8.50000	+	14.7224i	0.742648	+	1.28630i	−0.208637	+	0.977993i	$0.566903\pi$
$132$		0			0
$133$	−3.50000	+	6.06218i	−0.303488	+	0.525657i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.00000	−	12.1244i	0.598050	−	1.03585i	−0.395058	−	0.918656i	$-0.629276\pi$
$137$	7.00000	−	12.1244i	0.598050	−	1.03585i	0.993109	−	0.117198i	$-0.0373911\pi$
$138$		0			0
$139$	−1.50000	−	2.59808i	−0.127228	−	0.220366i	0.795373	−	0.606120i	$-0.207275\pi$
$139$	−1.50000	−	2.59808i	−0.127228	−	0.220366i	−0.922602	+	0.385754i	$0.873941\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	36.0000			3.01047
$144$		0			0
$145$	−2.00000			−0.166091
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.00000	−	12.1244i	−0.573462	−	0.993266i	−0.996207	−	0.0870170i	$-0.972267\pi$
$149$	−7.00000	−	12.1244i	−0.573462	−	0.993266i	0.422744	−	0.906249i	$-0.361067\pi$
$150$		0			0
$151$	2.50000	−	4.33013i	0.203447	−	0.352381i	−0.746190	−	0.665733i	$-0.768119\pi$
$151$	2.50000	−	4.33013i	0.203447	−	0.352381i	0.949637	+	0.313353i	$0.101452\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.00000	+	8.66025i	−0.401610	+	0.695608i
$156$		0			0
$157$	7.50000	+	12.9904i	0.598565	+	1.03675i	0.993033	+	0.117836i	$0.0375956\pi$
$157$	7.50000	+	12.9904i	0.598565	+	1.03675i	−0.394468	+	0.918910i	$0.629071\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.00000			0.0788110
$162$		0			0
$163$	10.0000			0.783260			0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000			0.783260			0.391630	+	0.920123i	$0.371911\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	0.999169	−	0.0407502i	$-0.0129748\pi$
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	−0.534875	+	0.844931i	$0.679641\pi$
$168$		0			0
$169$	−11.5000	+	19.9186i	−0.884615	+	1.53220i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.00000	−	12.1244i	0.532200	−	0.921798i	−0.467093	−	0.884208i	$-0.654699\pi$
$173$	7.00000	−	12.1244i	0.532200	−	0.921798i	0.999293	−	0.0375896i	$-0.0119679\pi$
$174$		0			0
$175$	2.00000	+	3.46410i	0.151186	+	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.0000			0.747435			0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000			0.747435			0.373718	+	0.927543i	$0.378083\pi$
$180$		0			0
$181$	5.00000			0.371647			0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000			0.371647			0.185824	+	0.982583i	$0.440505\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.00000	+	5.19615i	0.220564	+	0.382029i
$186$		0			0
$187$	−6.00000	+	10.3923i	−0.438763	+	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.50000	+	4.33013i	−0.180894	+	0.313317i	−0.942185	−	0.335093i	$-0.891232\pi$
$191$	−2.50000	+	4.33013i	−0.180894	+	0.313317i	0.761291	+	0.648410i	$0.224566\pi$
$192$		0			0
$193$	−4.50000	−	7.79423i	−0.323917	−	0.561041i	0.657376	−	0.753563i	$-0.271667\pi$
$193$	−4.50000	−	7.79423i	−0.323917	−	0.561041i	−0.981293	+	0.192522i	$0.938333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	2.00000			0.141776			0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000			0.141776			0.0708881	+	0.997484i	$0.477417\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.00000	+	1.73205i	0.0701862	+	0.121566i
$204$		0			0
$205$	−4.00000	+	6.92820i	−0.279372	+	0.483887i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−21.0000	+	36.3731i	−1.45260	+	2.51598i
$210$		0			0
$211$	2.00000	+	3.46410i	0.137686	+	0.238479i	0.926620	−	0.375999i	$-0.122700\pi$
$211$	2.00000	+	3.46410i	0.137686	+	0.238479i	−0.788935	+	0.614477i	$0.789367\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10.0000			−0.681994
$216$		0			0
$217$	10.0000			0.678844
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.00000	−	10.3923i	−0.403604	−	0.699062i
$222$		0			0
$223$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$223$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.500000	+	0.866025i	−0.0331862	+	0.0574801i	−0.882141	−	0.470985i	$-0.843899\pi$
$227$	−0.500000	+	0.866025i	−0.0331862	+	0.0574801i	0.848955	+	0.528465i	$0.177232\pi$
$228$		0			0
$229$	−2.50000	−	4.33013i	−0.165205	−	0.286143i	0.771523	−	0.636201i	$-0.219495\pi$
$229$	−2.50000	−	4.33013i	−0.165205	−	0.286143i	−0.936728	+	0.350058i	$0.886162\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.00000			0.458585			0.229293	−	0.973358i	$-0.426359\pi$
$233$	7.00000			0.458585			0.229293	+	0.973358i	$0.426359\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.50000	+	4.33013i	0.161712	+	0.280093i	0.935483	−	0.353373i	$-0.114965\pi$
$239$	2.50000	+	4.33013i	0.161712	+	0.280093i	−0.773771	+	0.633465i	$0.781632\pi$
$240$		0			0
$241$	−2.00000	+	3.46410i	−0.128831	+	0.223142i	−0.923224	−	0.384262i	$-0.874456\pi$
$241$	−2.00000	+	3.46410i	−0.128831	+	0.223142i	0.794393	+	0.607404i	$0.207789\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.500000	+	0.866025i	−0.0319438	+	0.0553283i
$246$		0			0
$247$	−21.0000	−	36.3731i	−1.33620	−	2.31436i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−13.0000			−0.820553			−0.410276	−	0.911961i	$-0.634568\pi$
$251$	−13.0000			−0.820553			−0.410276	+	0.911961i	$0.634568\pi$
$252$		0			0
$253$	6.00000			0.377217
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	0.997374	+	0.0724199i	$0.0230722\pi$
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	−0.435970	+	0.899961i	$0.643595\pi$
$258$		0			0
$259$	3.00000	−	5.19615i	0.186411	−	0.322873i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.50000	−	12.9904i	0.462470	−	0.801021i	−0.536614	−	0.843828i	$-0.680297\pi$
$263$	7.50000	−	12.9904i	0.462470	−	0.801021i	0.999083	+	0.0428069i	$0.0136300\pi$
$264$		0			0
$265$	1.00000	+	1.73205i	0.0614295	+	0.106399i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.00000			0.182913			0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000			0.182913			0.0914566	+	0.995809i	$0.470848\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.0000	+	20.7846i	0.723627	+	1.25336i
$276$		0			0
$277$	−2.00000	+	3.46410i	−0.120168	+	0.208138i	−0.919834	−	0.392308i	$-0.871677\pi$
$277$	−2.00000	+	3.46410i	−0.120168	+	0.208138i	0.799666	+	0.600446i	$0.205010\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.50000	+	2.59808i	−0.0894825	+	0.154988i	−0.907293	−	0.420500i	$-0.861855\pi$
$281$	−1.50000	+	2.59808i	−0.0894825	+	0.154988i	0.817810	+	0.575488i	$0.195188\pi$
$282$		0			0
$283$	6.50000	+	11.2583i	0.386385	+	0.669238i	0.991960	−	0.126550i	$-0.0403903\pi$
$283$	6.50000	+	11.2583i	0.386385	+	0.669238i	−0.605575	+	0.795788i	$0.707057\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.00000			0.472225
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.50000	−	7.79423i	−0.262893	−	0.455344i	0.704117	−	0.710084i	$-0.251343\pi$
$293$	−4.50000	−	7.79423i	−0.262893	−	0.455344i	−0.967009	+	0.254741i	$0.918010\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$	5.00000	+	8.66025i	0.288195	+	0.499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.00000			0.400819
$306$		0			0
$307$	−23.0000			−1.31268			−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000			−1.31268			−0.656340	+	0.754466i	$0.727896\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	0.974841	+	0.222900i	$0.0715523\pi$
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	−0.294384	+	0.955687i	$0.595114\pi$
$312$		0			0
$313$	−1.00000	+	1.73205i	−0.0565233	+	0.0979013i	−0.892903	−	0.450250i	$-0.851335\pi$
$313$	−1.00000	+	1.73205i	−0.0565233	+	0.0979013i	0.836379	+	0.548151i	$0.184668\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		0			0
$319$	6.00000	+	10.3923i	0.335936	+	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	14.0000			0.778981
$324$		0			0
$325$	−24.0000			−1.33128
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.00000	+	6.92820i	0.220527	+	0.381964i
$330$		0			0
$331$	−7.00000	+	12.1244i	−0.384755	+	0.666415i	−0.991735	−	0.128302i	$-0.959047\pi$
$331$	−7.00000	+	12.1244i	−0.384755	+	0.666415i	0.606980	+	0.794717i	$0.292381\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.00000	−	10.3923i	0.327815	−	0.567792i
$336$		0			0
$337$	−1.00000	−	1.73205i	−0.0544735	−	0.0943508i	0.837503	−	0.546433i	$-0.184015\pi$
$337$	−1.00000	−	1.73205i	−0.0544735	−	0.0943508i	−0.891976	+	0.452082i	$0.850681\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	60.0000			3.24918
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	0.984487	+	0.175457i	$0.0561403\pi$
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	−0.340293	+	0.940319i	$0.610526\pi$
$348$		0			0
$349$	5.00000	−	8.66025i	0.267644	−	0.463573i	−0.700609	−	0.713545i	$-0.747088\pi$
$349$	5.00000	−	8.66025i	0.267644	−	0.463573i	0.968253	+	0.249973i	$0.0804216\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.00000	−	5.19615i	0.159674	−	0.276563i	−0.775077	−	0.631867i	$-0.782289\pi$
$353$	3.00000	−	5.19615i	0.159674	−	0.276563i	0.934751	+	0.355303i	$0.115622\pi$
$354$		0			0
$355$	7.50000	+	12.9904i	0.398059	+	0.689458i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−33.0000			−1.74167			−0.870837	−	0.491572i	$-0.836422\pi$
$359$	−33.0000			−1.74167			−0.870837	+	0.491572i	$0.836422\pi$
$360$		0			0
$361$	30.0000			1.57895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.00000	+	1.73205i	0.0523424	+	0.0906597i
$366$		0			0
$367$	−16.0000	+	27.7128i	−0.835193	+	1.44660i	0.0586798	+	0.998277i	$0.481311\pi$
$367$	−16.0000	+	27.7128i	−0.835193	+	1.44660i	−0.893873	+	0.448320i	$0.852022\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.00000	−	1.73205i	0.0519174	−	0.0899236i
$372$		0			0
$373$	−1.00000	−	1.73205i	−0.0517780	−	0.0896822i	0.838975	−	0.544170i	$-0.183156\pi$
$373$	−1.00000	−	1.73205i	−0.0517780	−	0.0896822i	−0.890753	+	0.454488i	$0.849822\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.0000			−0.618031
$378$		0			0
$379$	6.00000			0.308199			0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000			0.308199			0.154100	+	0.988055i	$0.450752\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	0.779143	−	0.626846i	$-0.215654\pi$
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	−0.932436	+	0.361335i	$0.882321\pi$
$384$		0			0
$385$	−3.00000	+	5.19615i	−0.152894	+	0.264820i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.0000	+	17.3205i	−0.507020	+	0.878185i	0.492947	+	0.870059i	$0.335920\pi$
$389$	−10.0000	+	17.3205i	−0.507020	+	0.878185i	−0.999967	+	0.00812520i	$0.997414\pi$
$390$		0			0
$391$	−1.00000	−	1.73205i	−0.0505722	−	0.0875936i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.00000			0.0503155
$396$		0			0
$397$	22.0000			1.10415			0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000			1.10415			0.552074	+	0.833795i	$0.313837\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.5000	−	18.1865i	−0.524345	−	0.908192i	−0.999598	−	0.0283431i	$-0.990977\pi$
$401$	−10.5000	−	18.1865i	−0.524345	−	0.908192i	0.475253	−	0.879849i	$-0.342356\pi$
$402$		0			0
$403$	−30.0000	+	51.9615i	−1.49441	+	2.58839i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.0000	−	31.1769i	0.892227	−	1.54538i
$408$		0			0
$409$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$409$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−12.0000			−0.589057
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.50000	−	6.06218i	−0.170986	−	0.296157i	0.767779	−	0.640715i	$-0.221362\pi$
$419$	−3.50000	−	6.06218i	−0.170986	−	0.296157i	−0.938765	+	0.344558i	$0.888029\pi$
$420$		0			0
$421$	−5.00000	+	8.66025i	−0.243685	+	0.422075i	−0.961761	−	0.273890i	$-0.911690\pi$
$421$	−5.00000	+	8.66025i	−0.243685	+	0.422075i	0.718076	+	0.695965i	$0.245023\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.00000	−	6.92820i	0.194029	−	0.336067i
$426$		0			0
$427$	−3.50000	−	6.06218i	−0.169377	−	0.293369i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−36.0000			−1.73406			−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000			−1.73406			−0.867029	+	0.498257i	$0.833974\pi$
$432$		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
$437$	−3.50000	−	6.06218i	−0.167428	−	0.289993i
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.0000	+	29.4449i	−0.807694	+	1.39897i	0.106763	+	0.994285i	$0.465952\pi$
$443$	−17.0000	+	29.4449i	−0.807694	+	1.39897i	−0.914457	+	0.404683i	$0.867382\pi$
$444$		0			0
$445$	2.00000	+	3.46410i	0.0948091	+	0.164214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.00000			−0.0471929			−0.0235965	−	0.999722i	$-0.507512\pi$
$449$	−1.00000			−0.0471929			−0.0235965	+	0.999722i	$0.507512\pi$
$450$		0			0
$451$	48.0000			2.26023
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.00000	−	5.19615i	−0.140642	−	0.243599i
$456$		0			0
$457$	5.50000	−	9.52628i	0.257279	−	0.445621i	−0.708233	−	0.705979i	$-0.750507\pi$
$457$	5.50000	−	9.52628i	0.257279	−	0.445621i	0.965512	+	0.260358i	$0.0838407\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.50000	+	2.59808i	−0.0698620	+	0.121004i	−0.898840	−	0.438276i	$-0.855589\pi$
$461$	−1.50000	+	2.59808i	−0.0698620	+	0.121004i	0.828978	+	0.559281i	$0.188923\pi$
$462$		0			0
$463$	−11.5000	−	19.9186i	−0.534450	−	0.925695i	−0.999190	−	0.0402476i	$-0.987185\pi$
$463$	−11.5000	−	19.9186i	−0.534450	−	0.925695i	0.464739	−	0.885448i	$-0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	−12.0000			−0.554109
$470$		0			0
$471$		0			0
$472$		0			0
$473$	30.0000	+	51.9615i	1.37940	+	2.38919i
$474$		0			0
$475$	14.0000	−	24.2487i	0.642364	−	1.11261i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.0000	−	29.4449i	0.776750	−	1.34537i	−0.157056	−	0.987590i	$-0.550200\pi$
$479$	17.0000	−	29.4449i	0.776750	−	1.34537i	0.933806	−	0.357780i	$-0.116466\pi$
$480$		0			0
$481$	18.0000	+	31.1769i	0.820729	+	1.42154i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.00000			−0.0908153
$486$		0			0
$487$	1.00000			0.0453143			0.0226572	−	0.999743i	$-0.492787\pi$
$487$	1.00000			0.0453143			0.0226572	+	0.999743i	$0.492787\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.00000	−	5.19615i	−0.135388	−	0.234499i	0.790358	−	0.612646i	$-0.209895\pi$
$491$	−3.00000	−	5.19615i	−0.135388	−	0.234499i	−0.925746	+	0.378147i	$0.876561\pi$
$492$		0			0
$493$	2.00000	−	3.46410i	0.0900755	−	0.156015i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.50000	−	12.9904i	0.336421	−	0.582698i
$498$		0			0
$499$	−8.00000	−	13.8564i	−0.358129	−	0.620298i	0.629519	−	0.776985i	$-0.283252\pi$
$499$	−8.00000	−	13.8564i	−0.358129	−	0.620298i	−0.987648	+	0.156687i	$0.949919\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	32.0000			1.42681			0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000			1.42681			0.713405	+	0.700752i	$0.247152\pi$
$504$		0			0
$505$	−3.00000			−0.133498
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−15.0000	−	25.9808i	−0.664863	−	1.15158i	−0.979322	−	0.202306i	$-0.935156\pi$
$509$	−15.0000	−	25.9808i	−0.664863	−	1.15158i	0.314459	−	0.949271i	$-0.398177\pi$
$510$		0			0
$511$	1.00000	−	1.73205i	0.0442374	−	0.0766214i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.00000	+	5.19615i	−0.132196	+	0.228970i
$516$		0			0
$517$	24.0000	+	41.5692i	1.05552	+	1.82821i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.0000			0.525730			0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000			0.525730			0.262865	+	0.964833i	$0.415333\pi$
$522$		0			0
$523$	29.0000			1.26808			0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000			1.26808			0.634041	+	0.773300i	$0.281395\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.0000	−	17.3205i	−0.435607	−	0.754493i
$528$		0			0
$529$	11.0000	−	19.0526i	0.478261	−	0.828372i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−24.0000	+	41.5692i	−1.03956	+	1.80056i
$534$		0			0
$535$	6.00000	+	10.3923i	0.259403	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.00000			0.258438
$540$		0			0
$541$	−14.0000			−0.601907			−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000			−0.601907			−0.300954	+	0.953639i	$0.597305\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.00000	−	8.66025i	−0.214176	−	0.370965i
$546$		0			0
$547$	4.00000	−	6.92820i	0.171028	−	0.296229i	−0.767752	−	0.640747i	$-0.778625\pi$
$547$	4.00000	−	6.92820i	0.171028	−	0.296229i	0.938779	+	0.344519i	$0.111958\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7.00000	−	12.1244i	0.298210	−	0.516515i
$552$		0			0
$553$	−0.500000	−	0.866025i	−0.0212622	−	0.0368271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	44.0000			1.86434			0.932170	−	0.362021i	$-0.117913\pi$
$557$	44.0000			1.86434			0.932170	+	0.362021i	$0.117913\pi$
$558$		0			0
$559$	−60.0000			−2.53773
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−11.5000	−	19.9186i	−0.484667	−	0.839468i	0.515178	−	0.857083i	$-0.327726\pi$
$563$	−11.5000	−	19.9186i	−0.484667	−	0.839468i	−0.999845	+	0.0176152i	$0.994393\pi$
$564$		0			0
$565$	−4.50000	+	7.79423i	−0.189316	+	0.327906i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.0000	−	25.9808i	0.628833	−	1.08917i	−0.358954	−	0.933355i	$-0.616866\pi$
$569$	15.0000	−	25.9808i	0.628833	−	1.08917i	0.987786	−	0.155815i	$-0.0498003\pi$
$570$		0			0
$571$	20.0000	+	34.6410i	0.836974	+	1.44968i	0.892413	+	0.451219i	$0.149011\pi$
$571$	20.0000	+	34.6410i	0.836974	+	1.44968i	−0.0554391	+	0.998462i	$0.517656\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.00000			−0.166812
$576$		0			0
$577$	36.0000			1.49870			0.749350	−	0.662174i	$-0.230366\pi$
$577$	36.0000			1.49870			0.749350	+	0.662174i	$0.230366\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	6.00000	+	10.3923i	0.248922	+	0.431145i
$582$		0			0
$583$	6.00000	−	10.3923i	0.248495	−	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.50000	+	4.33013i	−0.103186	+	0.178723i	−0.912996	−	0.407969i	$-0.866237\pi$
$587$	−2.50000	+	4.33013i	−0.103186	+	0.178723i	0.809810	+	0.586693i	$0.199570\pi$
$588$		0			0
$589$	−35.0000	−	60.6218i	−1.44215	−	2.49788i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	12.0000			0.492781			0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000			0.492781			0.246390	+	0.969171i	$0.420755\pi$
$594$		0			0
$595$	2.00000			0.0819920
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−16.0000	−	27.7128i	−0.653742	−	1.13231i	−0.982208	−	0.187799i	$-0.939865\pi$
$599$	−16.0000	−	27.7128i	−0.653742	−	1.13231i	0.328465	−	0.944516i	$-0.393469\pi$
$600$		0			0
$601$	7.00000	−	12.1244i	0.285536	−	0.494563i	−0.687203	−	0.726465i	$-0.741162\pi$
$601$	7.00000	−	12.1244i	0.285536	−	0.494563i	0.972739	+	0.231903i	$0.0744951\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.5000	+	21.6506i	−0.508197	+	0.880223i
$606$		0			0
$607$	1.00000	+	1.73205i	0.0405887	+	0.0703018i	0.885606	−	0.464437i	$-0.153743\pi$
$607$	1.00000	+	1.73205i	0.0405887	+	0.0703018i	−0.845017	+	0.534739i	$0.820410\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−48.0000			−1.94187
$612$		0			0
$613$	−20.0000			−0.807792			−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000			−0.807792			−0.403896	+	0.914805i	$0.632344\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.0000	−	25.9808i	−0.603877	−	1.04595i	−0.992228	−	0.124434i	$-0.960288\pi$
$617$	−15.0000	−	25.9808i	−0.603877	−	1.04595i	0.388351	−	0.921512i	$-0.373045\pi$
$618$		0			0
$619$	16.5000	−	28.5788i	0.663191	−	1.14868i	−0.316581	−	0.948565i	$-0.602535\pi$
$619$	16.5000	−	28.5788i	0.663191	−	1.14868i	0.979772	−	0.200115i	$-0.0641316\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.00000	−	3.46410i	0.0801283	−	0.138786i
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	5.00000			0.199047			0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000			0.199047			0.0995234	+	0.995035i	$0.468268\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.50000	−	4.33013i	−0.0992095	−	0.171836i
$636$		0			0
$637$	−3.00000	+	5.19615i	−0.118864	+	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	21.5000	−	37.2391i	0.849199	−	1.47086i	−0.0327252	−	0.999464i	$-0.510419\pi$
$641$	21.5000	−	37.2391i	0.849199	−	1.47086i	0.881924	−	0.471391i	$-0.156248\pi$
$642$		0			0
$643$	−2.00000	−	3.46410i	−0.0788723	−	0.136611i	0.823891	−	0.566748i	$-0.191799\pi$
$643$	−2.00000	−	3.46410i	−0.0788723	−	0.136611i	−0.902764	+	0.430137i	$0.858465\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−12.0000			−0.471769			−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	22.0000	+	38.1051i	0.860927	+	1.49117i	0.871036	+	0.491220i	$0.163449\pi$
$653$	22.0000	+	38.1051i	0.860927	+	1.49117i	−0.0101092	+	0.999949i	$0.503218\pi$
$654$		0			0
$655$	8.50000	−	14.7224i	0.332122	−	0.575253i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−17.0000	+	29.4449i	−0.662226	+	1.14701i	0.317803	+	0.948157i	$0.397055\pi$
$659$	−17.0000	+	29.4449i	−0.662226	+	1.14701i	−0.980029	+	0.198852i	$0.936279\pi$
$660$		0			0
$661$	12.5000	+	21.6506i	0.486194	+	0.842112i	0.999874	−	0.0158695i	$-0.00505163\pi$
$661$	12.5000	+	21.6506i	0.486194	+	0.842112i	−0.513680	+	0.857982i	$0.671718\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7.00000			0.271448
$666$		0			0
$667$	−2.00000			−0.0774403
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−21.0000	−	36.3731i	−0.810696	−	1.40417i
$672$		0			0
$673$	−9.50000	+	16.4545i	−0.366198	+	0.634274i	−0.988968	−	0.148132i	$-0.952674\pi$
$673$	−9.50000	+	16.4545i	−0.366198	+	0.634274i	0.622770	+	0.782405i	$0.286007\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−21.0000	+	36.3731i	−0.807096	+	1.39793i	0.107772	+	0.994176i	$0.465628\pi$
$677$	−21.0000	+	36.3731i	−0.807096	+	1.39793i	−0.914867	+	0.403755i	$0.867705\pi$
$678$		0			0
$679$	1.00000	+	1.73205i	0.0383765	+	0.0664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18.0000			−0.688751			−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000			−0.688751			−0.344375	+	0.938832i	$0.611909\pi$
$684$		0			0
$685$	−14.0000			−0.534913
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.00000	+	10.3923i	0.228582	+	0.395915i
$690$		0			0
$691$	−20.5000	+	35.5070i	−0.779857	+	1.35075i	0.152167	+	0.988355i	$0.451375\pi$
$691$	−20.5000	+	35.5070i	−0.779857	+	1.35075i	−0.932024	+	0.362397i	$0.881959\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.50000	+	2.59808i	−0.0568982	+	0.0985506i
$696$		0			0
$697$	−8.00000	−	13.8564i	−0.303022	−	0.524849i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26.0000			0.982006			0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000			0.982006			0.491003	+	0.871158i	$0.336630\pi$
$702$		0			0
$703$	−42.0000			−1.58406
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.50000	+	2.59808i	0.0564133	+	0.0977107i
$708$		0			0
$709$	22.0000	−	38.1051i	0.826227	−	1.43107i	−0.0747503	−	0.997202i	$-0.523816\pi$
$709$	22.0000	−	38.1051i	0.826227	−	1.43107i	0.900978	−	0.433865i	$-0.142851\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.00000	+	8.66025i	−0.187251	+	0.324329i
$714$		0			0
$715$	−18.0000	−	31.1769i	−0.673162	−	1.16595i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12.0000			0.447524			0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000			0.447524			0.223762	+	0.974644i	$0.428166\pi$
$720$		0			0
$721$	6.00000			0.223452
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.00000	−	6.92820i	−0.148556	−	0.257307i
$726$		0			0
$727$	14.0000	−	24.2487i	0.519231	−	0.899335i	−0.480519	−	0.876984i	$-0.659552\pi$
$727$	14.0000	−	24.2487i	0.519231	−	0.899335i	0.999750	−	0.0223506i	$-0.00711500\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.0000	−	17.3205i	0.369863	−	0.640622i
$732$		0			0
$733$	−15.5000	−	26.8468i	−0.572506	−	0.991609i	−0.996308	−	0.0858539i	$-0.972638\pi$
$733$	−15.5000	−	26.8468i	−0.572506	−	0.991609i	0.423802	−	0.905755i	$-0.360695\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−72.0000			−2.65215
$738$		0			0
$739$	−38.0000			−1.39785			−0.698926	−	0.715194i	$-0.746338\pi$
$739$	−38.0000			−1.39785			−0.698926	+	0.715194i	$0.746338\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	22.0000	+	38.1051i	0.807102	+	1.39794i	0.914863	+	0.403764i	$0.132298\pi$
$743$	22.0000	+	38.1051i	0.807102	+	1.39794i	−0.107761	+	0.994177i	$0.534368\pi$
$744$		0			0
$745$	−7.00000	+	12.1244i	−0.256460	+	0.444202i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.00000	−	10.3923i	0.219235	−	0.379727i
$750$		0			0
$751$	−2.50000	−	4.33013i	−0.0912263	−	0.158009i	0.816801	−	0.576919i	$-0.195745\pi$
$751$	−2.50000	−	4.33013i	−0.0912263	−	0.158009i	−0.908027	+	0.418911i	$0.862412\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−5.00000			−0.181969
$756$		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
$761$	3.00000	+	5.19615i	0.108750	+	0.188360i	0.915264	−	0.402854i	$-0.131982\pi$
$761$	3.00000	+	5.19615i	0.108750	+	0.188360i	−0.806514	+	0.591215i	$0.798649\pi$
$762$		0			0
$763$	−5.00000	+	8.66025i	−0.181012	+	0.313522i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−5.00000	−	8.66025i	−0.180305	−	0.312297i	0.761680	−	0.647954i	$-0.224375\pi$
$769$	−5.00000	−	8.66025i	−0.180305	−	0.312297i	−0.941984	+	0.335657i	$0.891042\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−31.0000			−1.11499			−0.557496	−	0.830179i	$-0.688238\pi$
$773$	−31.0000			−1.11499			−0.557496	+	0.830179i	$0.688238\pi$
$774$		0			0
$775$	−40.0000			−1.43684
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−28.0000	−	48.4974i	−1.00320	−	1.73760i
$780$		0			0
$781$	45.0000	−	77.9423i	1.61023	−	2.78899i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.50000	−	12.9904i	0.267686	−	0.463647i
$786$		0			0
$787$	10.0000	+	17.3205i	0.356462	+	0.617409i	0.987367	−	0.158450i	$-0.0506498\pi$
$787$	10.0000	+	17.3205i	0.356462	+	0.617409i	−0.630905	+	0.775860i	$0.717316\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.00000			0.320003
$792$		0			0
$793$	42.0000			1.49146
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.50000	+	7.79423i	0.159398	+	0.276086i	0.934652	−	0.355564i	$-0.115711\pi$
$797$	4.50000	+	7.79423i	0.159398	+	0.276086i	−0.775254	+	0.631650i	$0.782378\pi$
$798$		0			0
$799$	8.00000	−	13.8564i	0.283020	−	0.490204i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6.00000	−	10.3923i	0.211735	−	0.366736i
$804$		0			0
$805$	−0.500000	−	0.866025i	−0.0176227	−	0.0305234i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.00000			0.210949			0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000			0.210949			0.105474	+	0.994422i	$0.466364\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.00000	−	8.66025i	−0.175142	−	0.303355i
$816$		0			0
$817$	35.0000	−	60.6218i	1.22449	−	2.12089i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$		0			0
$823$	8.00000	+	13.8564i	0.278862	+	0.483004i	0.971102	−	0.238664i	$-0.0767093\pi$
$823$	8.00000	+	13.8564i	0.278862	+	0.483004i	−0.692240	+	0.721668i	$0.743376\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	50.0000			1.73657			0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000			1.73657			0.868286	+	0.496064i	$0.165222\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.00000	−	1.73205i	−0.0346479	−	0.0600120i
$834$		0			0
$835$	6.00000	−	10.3923i	0.207639	−	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−6.00000	+	10.3923i	−0.207143	+	0.358782i	−0.950813	−	0.309764i	$-0.899750\pi$
$839$	−6.00000	+	10.3923i	−0.207143	+	0.358782i	0.743670	+	0.668546i	$0.233083\pi$
$840$		0			0
$841$	12.5000	+	21.6506i	0.431034	+	0.746574i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	23.0000			0.791224
$846$		0			0
$847$	25.0000			0.859010
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.00000	+	5.19615i	0.102839	+	0.178122i
$852$		0			0
$853$	11.5000	−	19.9186i	0.393753	−	0.681999i	−0.599189	−	0.800608i	$-0.704510\pi$
$853$	11.5000	−	19.9186i	0.393753	−	0.681999i	0.992941	+	0.118609i	$0.0378434\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.00000	+	5.19615i	−0.102478	+	0.177497i	−0.912705	−	0.408619i	$-0.866010\pi$
$857$	−3.00000	+	5.19615i	−0.102478	+	0.177497i	0.810227	+	0.586116i	$0.199344\pi$
$858$		0			0
$859$	10.0000	+	17.3205i	0.341196	+	0.590968i	0.984655	−	0.174512i	$-0.0558348\pi$
$859$	10.0000	+	17.3205i	0.341196	+	0.590968i	−0.643459	+	0.765480i	$0.722501\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	43.0000			1.46374			0.731869	−	0.681446i	$-0.238649\pi$
$863$	43.0000			1.46374			0.731869	+	0.681446i	$0.238649\pi$
$864$		0			0
$865$	−14.0000			−0.476014
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.00000	−	5.19615i	−0.101768	−	0.176267i
$870$		0			0
$871$	36.0000	−	62.3538i	1.21981	−	2.11278i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.50000	−	7.79423i	0.152128	−	0.263493i
$876$		0			0
$877$	−7.00000	−	12.1244i	−0.236373	−	0.409410i	0.723298	−	0.690536i	$-0.242625\pi$
$877$	−7.00000	−	12.1244i	−0.236373	−	0.409410i	−0.959671	+	0.281126i	$0.909292\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−40.0000			−1.34763			−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000			−1.34763			−0.673817	+	0.738898i	$0.735346\pi$
$882$		0			0
$883$	−26.0000			−0.874970			−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000			−0.874970			−0.437485	+	0.899226i	$0.644131\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	−2.50000	+	4.33013i	−0.0838473	+	0.145228i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	28.0000	−	48.4974i	0.936984	−	1.62290i
$894$		0			0
$895$	−5.00000	−	8.66025i	−0.167132	−	0.289480i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.0000			−0.667037
$900$		0			0
$901$	−4.00000			−0.133259
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.50000	−	4.33013i	−0.0831028	−	0.143938i
$906$		0			0
$907$	14.0000	−	24.2487i	0.464862	−	0.805165i	−0.534333	−	0.845274i	$-0.679437\pi$
$907$	14.0000	−	24.2487i	0.464862	−	0.805165i	0.999195	+	0.0401089i	$0.0127705\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.50000	−	16.4545i	0.314749	−	0.545161i	−0.664635	−	0.747168i	$-0.731413\pi$
$911$	9.50000	−	16.4545i	0.314749	−	0.545161i	0.979384	+	0.202007i	$0.0647463\pi$
$912$		0			0
$913$	36.0000	+	62.3538i	1.19143	+	2.06361i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−17.0000			−0.561389
$918$		0			0
$919$	−17.0000			−0.560778			−0.280389	−	0.959886i	$-0.590464\pi$
$919$	−17.0000			−0.560778			−0.280389	+	0.959886i	$0.590464\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	45.0000	+	77.9423i	1.48119	+	2.56550i
$924$		0			0
$925$	−12.0000	+	20.7846i	−0.394558	+	0.683394i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.00000	−	3.46410i	0.0656179	−	0.113653i	−0.831350	−	0.555749i	$-0.812431\pi$
$929$	2.00000	−	3.46410i	0.0656179	−	0.113653i	0.896968	+	0.442096i	$0.145765\pi$
$930$		0			0
$931$	−3.50000	−	6.06218i	−0.114708	−	0.198680i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.0000			0.392442
$936$		0			0
$937$	−42.0000			−1.37208			−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000			−1.37208			−0.686040	+	0.727564i	$0.740653\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−16.5000	−	28.5788i	−0.537885	−	0.931644i	−0.999018	−	0.0443125i	$-0.985890\pi$
$941$	−16.5000	−	28.5788i	−0.537885	−	0.931644i	0.461133	−	0.887331i	$-0.347443\pi$
$942$		0			0
$943$	−4.00000	+	6.92820i	−0.130258	+	0.225613i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.0000	−	24.2487i	0.454939	−	0.787977i	−0.543746	−	0.839250i	$-0.682994\pi$
$947$	14.0000	−	24.2487i	0.454939	−	0.787977i	0.998685	+	0.0512727i	$0.0163278\pi$
$948$		0			0
$949$	6.00000	+	10.3923i	0.194768	+	0.337348i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	14.0000			0.453504			0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000			0.453504			0.226752	+	0.973952i	$0.427189\pi$
$954$		0			0
$955$	5.00000			0.161796
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.00000	+	12.1244i	0.226042	+	0.391516i
$960$		0			0
$961$	−34.5000	+	59.7558i	−1.11290	+	1.92760i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.50000	+	7.79423i	−0.144860	+	0.250905i
$966$		0			0
$967$	1.50000	+	2.59808i	0.0482367	+	0.0835485i	0.889136	−	0.457644i	$-0.151306\pi$
$967$	1.50000	+	2.59808i	0.0482367	+	0.0835485i	−0.840899	+	0.541192i	$0.817973\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−55.0000			−1.76503			−0.882517	−	0.470281i	$-0.844153\pi$
$971$	−55.0000			−1.76503			−0.882517	+	0.470281i	$0.844153\pi$
$972$		0			0
$973$	3.00000			0.0961756
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.00000	+	5.19615i	0.0959785	+	0.166240i	0.910017	−	0.414572i	$-0.136069\pi$
$977$	3.00000	+	5.19615i	0.0959785	+	0.166240i	−0.814038	+	0.580812i	$0.802735\pi$
$978$		0			0
$979$	12.0000	−	20.7846i	0.383522	−	0.664279i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9.00000	−	15.5885i	0.287055	−	0.497195i	−0.686050	−	0.727554i	$-0.740657\pi$
$983$	9.00000	−	15.5885i	0.287055	−	0.497195i	0.973106	+	0.230360i	$0.0739903\pi$
$984$		0			0
$985$	3.00000	+	5.19615i	0.0955879	+	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−10.0000			−0.317982
$990$		0			0
$991$	−48.0000			−1.52477			−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000			−1.52477			−0.762385	+	0.647124i	$0.775972\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.00000	−	1.73205i	−0.0317021	−	0.0549097i
$996$		0			0
$997$	−9.50000	+	16.4545i	−0.300868	+	0.521119i	−0.976333	−	0.216274i	$-0.930610\pi$
$997$	−9.50000	+	16.4545i	−0.300868	+	0.521119i	0.675465	+	0.737392i	$0.263943\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.r.a.505.1		2
3.2	odd	2		504.2.r.a.169.1	✓	2
4.3	odd	2		3024.2.r.b.2017.1		2
9.2	odd	6		4536.2.a.d.1.1		1
9.4	even	3	inner	1512.2.r.a.1009.1		2
9.5	odd	6		504.2.r.a.337.1	yes	2
9.7	even	3		4536.2.a.g.1.1		1
12.11	even	2		1008.2.r.c.673.1		2
36.7	odd	6		9072.2.a.n.1.1		1
36.11	even	6		9072.2.a.i.1.1		1
36.23	even	6		1008.2.r.c.337.1		2
36.31	odd	6		3024.2.r.b.1009.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.a.169.1	✓	2	3.2	odd	2
504.2.r.a.337.1	yes	2	9.5	odd	6
1008.2.r.c.337.1		2	36.23	even	6
1008.2.r.c.673.1		2	12.11	even	2
1512.2.r.a.505.1		2	1.1	even	1	trivial
1512.2.r.a.1009.1		2	9.4	even	3	inner
3024.2.r.b.1009.1		2	36.31	odd	6
3024.2.r.b.2017.1		2	4.3	odd	2
4536.2.a.d.1.1		1	9.2	odd	6
4536.2.a.g.1.1		1	9.7	even	3
9072.2.a.i.1.1		1	36.11	even	6
9072.2.a.n.1.1		1	36.7	odd	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:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.r (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{7} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(-3.00000 + 5.19615i) q^{11} +(-3.00000 - 5.19615i) q^{13} +2.00000 q^{17} +7.00000 q^{19} +(-0.500000 - 0.866025i) q^{23} +(2.00000 - 3.46410i) q^{25} +(1.00000 - 1.73205i) q^{29} +(-5.00000 - 8.66025i) q^{31} +1.00000 q^{35} -6.00000 q^{37} +(-4.00000 - 6.92820i) q^{41} +(5.00000 - 8.66025i) q^{43} +(4.00000 - 6.92820i) q^{47} +(-0.500000 - 0.866025i) q^{49} -2.00000 q^{53} +6.00000 q^{55} +(-3.50000 + 6.06218i) q^{61} +(-3.00000 + 5.19615i) q^{65} +(6.00000 + 10.3923i) q^{67} -15.0000 q^{71} -2.00000 q^{73} +(-3.00000 - 5.19615i) q^{77} +(-0.500000 + 0.866025i) q^{79} +(6.00000 - 10.3923i) q^{83} +(-1.00000 - 1.73205i) q^{85} -4.00000 q^{89} +6.00000 q^{91} +(-3.50000 - 6.06218i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} - q^{7}+O(q^{10}) \) 2 * q - q^5 - q^7	\( 2 q - q^{5} - q^{7} - 6 q^{11} - 6 q^{13} + 4 q^{17} + 14 q^{19} - q^{23} + 4 q^{25} + 2 q^{29} - 10 q^{31} + 2 q^{35} - 12 q^{37} - 8 q^{41} + 10 q^{43} + 8 q^{47} - q^{49} - 4 q^{53} + 12 q^{55} - 7 q^{61} - 6 q^{65} + 12 q^{67} - 30 q^{71} - 4 q^{73} - 6 q^{77} - q^{79} + 12 q^{83} - 2 q^{85} - 8 q^{89} + 12 q^{91} - 7 q^{95} + 2 q^{97}+O(q^{100}) \) 2 * q - q^5 - q^7 - 6 * q^11 - 6 * q^13 + 4 * q^17 + 14 * q^19 - q^23 + 4 * q^25 + 2 * q^29 - 10 * q^31 + 2 * q^35 - 12 * q^37 - 8 * q^41 + 10 * q^43 + 8 * q^47 - q^49 - 4 * q^53 + 12 * q^55 - 7 * q^61 - 6 * q^65 + 12 * q^67 - 30 * q^71 - 4 * q^73 - 6 * q^77 - q^79 + 12 * q^83 - 2 * q^85 - 8 * q^89 + 12 * q^91 - 7 * q^95 + 2 * q^97

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