LMFDB - Embedded newform 1512.2.s.e.865.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1512,2,Mod(865,1512)]

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, 0, 4]))

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

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

chi := DirichletCharacter("1512.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.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:	$12.0733807856$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1512.865
Dual form			1512.2.s.e.1297.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{5} +(2.50000 + 0.866025i) q^{7} +O(q^{10})$	$q+(-1.00000 - 1.73205i) q^{5} +(2.50000 + 0.866025i) q^{7} +(-1.50000 + 2.59808i) q^{11} +6.00000 q^{13} +(-2.00000 + 3.46410i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(2.00000 + 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} +5.00000 q^{29} +(-3.50000 + 6.06218i) q^{31} +(-1.00000 - 5.19615i) q^{35} -2.00000 q^{41} +8.00000 q^{43} +(-1.00000 - 1.73205i) q^{47} +(5.50000 + 4.33013i) q^{49} +(5.00000 - 8.66025i) q^{53} +6.00000 q^{55} +(-4.50000 + 7.79423i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-6.00000 - 10.3923i) q^{65} +(3.00000 - 5.19615i) q^{67} +12.0000 q^{71} +(5.50000 - 9.52628i) q^{73} +(-6.00000 + 5.19615i) q^{77} +(-0.500000 - 0.866025i) q^{79} +15.0000 q^{83} +8.00000 q^{85} +(-5.00000 - 8.66025i) q^{89} +(15.0000 + 5.19615i) q^{91} +(-4.00000 + 6.92820i) q^{95} -5.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 5 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 5 * q^7	$ 2 q - 2 q^{5} + 5 q^{7} - 3 q^{11} + 12 q^{13} - 4 q^{17} - 4 q^{19} + 4 q^{23} + q^{25} + 10 q^{29} - 7 q^{31} - 2 q^{35} - 4 q^{41} + 16 q^{43} - 2 q^{47} + 11 q^{49} + 10 q^{53} + 12 q^{55} - 9 q^{59} + 8 q^{61} - 12 q^{65} + 6 q^{67} + 24 q^{71} + 11 q^{73} - 12 q^{77} - q^{79} + 30 q^{83} + 16 q^{85} - 10 q^{89} + 30 q^{91} - 8 q^{95} - 10 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 5 * q^7 - 3 * q^11 + 12 * q^13 - 4 * q^17 - 4 * q^19 + 4 * q^23 + q^25 + 10 * q^29 - 7 * q^31 - 2 * q^35 - 4 * q^41 + 16 * q^43 - 2 * q^47 + 11 * q^49 + 10 * q^53 + 12 * q^55 - 9 * q^59 + 8 * q^61 - 12 * q^65 + 6 * q^67 + 24 * q^71 + 11 * q^73 - 12 * q^77 - q^79 + 30 * q^83 + 16 * q^85 - 10 * q^89 + 30 * q^91 - 8 * q^95 - 10 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	0.550990	−	0.834512i	$-0.314250\pi$
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	−0.998203	+	0.0599153i	$0.980917\pi$
$6$		0			0
$7$	2.50000	+	0.866025i	0.944911	+	0.327327i
$7$	2.50000	+	0.866025i	0.944911	+	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	$-0.982718\pi$
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	0.546259	+	0.837616i	$0.316051\pi$
$12$		0			0
$13$	6.00000			1.66410			0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000			1.66410			0.832050	+	0.554700i	$0.187167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.00000	+	3.46410i	−0.485071	+	0.840168i	−0.999853	−	0.0171533i	$-0.994540\pi$
$17$	−2.00000	+	3.46410i	−0.485071	+	0.840168i	0.514782	+	0.857321i	$0.327873\pi$
$18$		0			0
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	0.540068	−	0.841621i	$-0.318398\pi$
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	−0.998899	+	0.0469020i	$0.985065\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.00000			0.928477			0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000			0.928477			0.464238	+	0.885710i	$0.346328\pi$
$30$		0			0
$31$	−3.50000	+	6.06218i	−0.628619	+	1.08880i	0.359211	+	0.933257i	$0.383046\pi$
$31$	−3.50000	+	6.06218i	−0.628619	+	1.08880i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.00000	−	5.19615i	−0.169031	−	0.878310i
$36$		0			0
$37$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$37$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.00000	−	1.73205i	−0.145865	−	0.252646i	0.783830	−	0.620975i	$-0.213263\pi$
$47$	−1.00000	−	1.73205i	−0.145865	−	0.252646i	−0.929695	+	0.368329i	$0.879930\pi$
$48$		0			0
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.00000	−	8.66025i	0.686803	−	1.18958i	−0.286064	−	0.958211i	$-0.592347\pi$
$53$	5.00000	−	8.66025i	0.686803	−	1.18958i	0.972867	−	0.231367i	$-0.0743197\pi$
$54$		0			0
$55$	6.00000			0.809040
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.50000	+	7.79423i	−0.585850	+	1.01472i	0.408919	+	0.912571i	$0.365906\pi$
$59$	−4.50000	+	7.79423i	−0.585850	+	1.01472i	−0.994769	+	0.102151i	$0.967427\pi$
$60$		0			0
$61$	4.00000	+	6.92820i	0.512148	+	0.887066i	0.999901	+	0.0140840i	$0.00448323\pi$
$61$	4.00000	+	6.92820i	0.512148	+	0.887066i	−0.487753	+	0.872982i	$0.662183\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000	−	10.3923i	−0.744208	−	1.28901i
$66$		0			0
$67$	3.00000	−	5.19615i	0.366508	−	0.634811i	−0.622509	−	0.782613i	$-0.713886\pi$
$67$	3.00000	−	5.19615i	0.366508	−	0.634811i	0.989017	+	0.147802i	$0.0472198\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	5.50000	−	9.52628i	0.643726	−	1.11497i	−0.340868	−	0.940111i	$-0.610721\pi$
$73$	5.50000	−	9.52628i	0.643726	−	1.11497i	0.984594	−	0.174855i	$-0.0559458\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.00000	+	5.19615i	−0.683763	+	0.592157i
$78$		0			0
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	0.836527	−	0.547926i	$-0.184582\pi$
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	−0.892781	+	0.450490i	$0.851249\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	15.0000			1.64646			0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000			1.64646			0.823232	+	0.567705i	$0.192169\pi$
$84$		0			0
$85$	8.00000			0.867722
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.00000	−	8.66025i	−0.529999	−	0.917985i	−0.999388	−	0.0349934i	$-0.988859\pi$
$89$	−5.00000	−	8.66025i	−0.529999	−	0.917985i	0.469389	−	0.882992i	$-0.344474\pi$
$90$		0			0
$91$	15.0000	+	5.19615i	1.57243	+	0.544705i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.00000	+	6.92820i	−0.410391	+	0.710819i
$96$		0			0
$97$	−5.00000			−0.507673			−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000			−0.507673			−0.253837	+	0.967247i	$0.581693\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.50000	−	12.9904i	0.746278	−	1.29259i	−0.203317	−	0.979113i	$-0.565172\pi$
$101$	7.50000	−	12.9904i	0.746278	−	1.29259i	0.949595	−	0.313478i	$-0.101494\pi$
$102$		0			0
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	0.995474	+	0.0950377i	$0.0302972\pi$
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	−0.415432	+	0.909624i	$0.636370\pi$
$108$		0			0
$109$	−8.00000	+	13.8564i	−0.766261	+	1.32720i	0.173316	+	0.984866i	$0.444552\pi$
$109$	−8.00000	+	13.8564i	−0.766261	+	1.32720i	−0.939577	+	0.342337i	$0.888782\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$		0			0
$115$	4.00000	−	6.92820i	0.373002	−	0.646058i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.00000	+	6.92820i	−0.733359	+	0.635107i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.0000			−1.07331
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.50000	+	11.2583i	0.567908	+	0.983645i	0.996773	+	0.0802763i	$0.0255803\pi$
$131$	6.50000	+	11.2583i	0.567908	+	0.983645i	−0.428865	+	0.903369i	$0.641086\pi$
$132$		0			0
$133$	−2.00000	−	10.3923i	−0.173422	−	0.901127i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.00000	−	13.8564i	0.683486	−	1.18383i	−0.290424	−	0.956898i	$-0.593796\pi$
$137$	8.00000	−	13.8564i	0.683486	−	1.18383i	0.973910	−	0.226935i	$-0.0728704\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−9.00000	+	15.5885i	−0.752618	+	1.30357i
$144$		0			0
$145$	−5.00000	−	8.66025i	−0.415227	−	0.719195i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.50000	+	4.33013i	0.204808	+	0.354738i	0.950072	−	0.312032i	$-0.101010\pi$
$149$	2.50000	+	4.33013i	0.204808	+	0.354738i	−0.745264	+	0.666770i	$0.767676\pi$
$150$		0			0
$151$	−9.50000	+	16.4545i	−0.773099	+	1.33905i	0.162758	+	0.986666i	$0.447961\pi$
$151$	−9.50000	+	16.4545i	−0.773099	+	1.33905i	−0.935857	+	0.352381i	$0.885372\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	14.0000			1.12451
$156$		0			0
$157$	9.00000	−	15.5885i	0.718278	−	1.24409i	−0.243403	−	0.969925i	$-0.578264\pi$
$157$	9.00000	−	15.5885i	0.718278	−	1.24409i	0.961681	−	0.274169i	$-0.0884028\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.00000	+	10.3923i	0.157622	+	0.819028i
$162$		0			0
$163$	7.00000	+	12.1244i	0.548282	+	0.949653i	0.998392	+	0.0566798i	$0.0180514\pi$
$163$	7.00000	+	12.1244i	0.548282	+	0.949653i	−0.450110	+	0.892973i	$0.648615\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−18.0000			−1.39288			−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000			−1.39288			−0.696441	+	0.717614i	$0.745234\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.50000	−	9.52628i	−0.418157	−	0.724270i	0.577597	−	0.816322i	$-0.303991\pi$
$173$	−5.50000	−	9.52628i	−0.418157	−	0.724270i	−0.995754	+	0.0920525i	$0.970657\pi$
$174$		0			0
$175$	2.00000	−	1.73205i	0.151186	−	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.500000	−	0.866025i	0.0373718	−	0.0647298i	−0.846735	−	0.532016i	$-0.821435\pi$
$179$	0.500000	−	0.866025i	0.0373718	−	0.0647298i	0.884106	+	0.467286i	$0.154768\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−6.00000	−	10.3923i	−0.438763	−	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.00000	−	8.66025i	−0.361787	−	0.626634i	0.626468	−	0.779447i	$-0.284500\pi$
$191$	−5.00000	−	8.66025i	−0.361787	−	0.626634i	−0.988255	+	0.152813i	$0.951167\pi$
$192$		0			0
$193$	−10.5000	+	18.1865i	−0.755807	+	1.30910i	0.189166	+	0.981945i	$0.439422\pi$
$193$	−10.5000	+	18.1865i	−0.755807	+	1.30910i	−0.944972	+	0.327150i	$0.893912\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.00000			−0.213741			−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000			−0.213741			−0.106871	+	0.994273i	$0.534083\pi$
$198$		0			0
$199$	3.50000	−	6.06218i	0.248108	−	0.429736i	−0.714893	−	0.699234i	$-0.753524\pi$
$199$	3.50000	−	6.06218i	0.248108	−	0.429736i	0.963001	+	0.269498i	$0.0868577\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.5000	+	4.33013i	0.877328	+	0.303915i
$204$		0			0
$205$	2.00000	+	3.46410i	0.139686	+	0.241943i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	−22.0000			−1.51454			−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000			−1.51454			−0.757271	+	0.653101i	$0.773468\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.00000	−	13.8564i	−0.545595	−	0.944999i
$216$		0			0
$217$	−14.0000	+	12.1244i	−0.950382	+	0.823055i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.0000	+	20.7846i	−0.807207	+	1.39812i
$222$		0			0
$223$	−21.0000			−1.40626			−0.703132	−	0.711059i	$-0.748216\pi$
$223$	−21.0000			−1.40626			−0.703132	+	0.711059i	$0.748216\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.50000	−	11.2583i	0.431420	−	0.747242i	−0.565576	−	0.824696i	$-0.691346\pi$
$227$	6.50000	−	11.2583i	0.431420	−	0.747242i	0.996996	+	0.0774548i	$0.0246793\pi$
$228$		0			0
$229$	−10.0000	−	17.3205i	−0.660819	−	1.14457i	−0.980401	−	0.197013i	$-0.936876\pi$
$229$	−10.0000	−	17.3205i	−0.660819	−	1.14457i	0.319582	−	0.947559i	$-0.396457\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.00000	+	8.66025i	0.327561	+	0.567352i	0.982027	−	0.188739i	$-0.0604400\pi$
$233$	5.00000	+	8.66025i	0.327561	+	0.567352i	−0.654466	+	0.756091i	$0.727107\pi$
$234$		0			0
$235$	−2.00000	+	3.46410i	−0.130466	+	0.225973i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	−0.956456	−	0.291877i	$-0.905720\pi$
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	0.731001	+	0.682376i	$0.239053\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.00000	−	13.8564i	0.127775	−	0.885253i
$246$		0			0
$247$	−12.0000	−	20.7846i	−0.763542	−	1.32249i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	13.0000			0.820553			0.410276	−	0.911961i	$-0.365432\pi$
$251$	13.0000			0.820553			0.410276	+	0.911961i	$0.365432\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	−0.619586	−	0.784929i	$-0.712699\pi$
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	0.989561	+	0.144112i	$0.0460326\pi$
$264$		0			0
$265$	−20.0000			−1.22859
$266$		0			0
$267$		0			0
$268$		0			0
$269$	13.5000	−	23.3827i	0.823110	−	1.42567i	−0.0802460	−	0.996775i	$-0.525571\pi$
$269$	13.5000	−	23.3827i	0.823110	−	1.42567i	0.903356	−	0.428892i	$-0.141096\pi$
$270$		0			0
$271$	−4.00000	−	6.92820i	−0.242983	−	0.420858i	0.718580	−	0.695444i	$-0.244792\pi$
$271$	−4.00000	−	6.92820i	−0.242983	−	0.420858i	−0.961563	+	0.274586i	$0.911459\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.50000	+	2.59808i	0.0904534	+	0.156670i
$276$		0			0
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	−0.976231	−	0.216731i	$-0.930460\pi$
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	0.675810	+	0.737075i	$0.263794\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	−4.00000	+	6.92820i	−0.237775	+	0.411839i	−0.960076	−	0.279741i	$-0.909752\pi$
$283$	−4.00000	+	6.92820i	−0.237775	+	0.411839i	0.722300	+	0.691580i	$0.243085\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.00000	−	1.73205i	−0.295141	−	0.102240i
$288$		0			0
$289$	0.500000	+	0.866025i	0.0294118	+	0.0509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.00000			0.525786			0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000			0.525786			0.262893	+	0.964825i	$0.415323\pi$
$294$		0			0
$295$	18.0000			1.04800
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000	+	20.7846i	0.693978	+	1.20201i
$300$		0			0
$301$	20.0000	+	6.92820i	1.15278	+	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.00000	−	13.8564i	0.458079	−	0.793416i
$306$		0			0
$307$	−2.00000			−0.114146			−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000			−0.114146			−0.0570730	+	0.998370i	$0.518177\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.00000	+	15.5885i	−0.510343	+	0.883940i	0.489585	+	0.871956i	$0.337148\pi$
$311$	−9.00000	+	15.5885i	−0.510343	+	0.883940i	−0.999928	+	0.0119847i	$0.996185\pi$
$312$		0			0
$313$	−13.0000	−	22.5167i	−0.734803	−	1.27272i	−0.954810	−	0.297218i	$-0.903941\pi$
$313$	−13.0000	−	22.5167i	−0.734803	−	1.27272i	0.220006	−	0.975499i	$-0.429392\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.5000	+	23.3827i	0.758236	+	1.31330i	0.943750	+	0.330661i	$0.107272\pi$
$317$	13.5000	+	23.3827i	0.758236	+	1.31330i	−0.185514	+	0.982642i	$0.559395\pi$
$318$		0			0
$319$	−7.50000	+	12.9904i	−0.419919	+	0.727322i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.0000			0.890264
$324$		0			0
$325$	3.00000	−	5.19615i	0.166410	−	0.288231i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.00000	−	5.19615i	−0.0551318	−	0.286473i
$330$		0			0
$331$	−16.0000	−	27.7128i	−0.879440	−	1.52323i	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−16.0000	−	27.7128i	−0.879440	−	1.52323i	−0.0274825	−	0.999622i	$-0.508749\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$	29.0000			1.57973			0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000			1.57973			0.789865	+	0.613280i	$0.210150\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.5000	−	18.1865i	−0.568607	−	0.984856i
$342$		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	$-0.858993\pi$
$347$	−1.50000	+	2.59808i	−0.0805242	+	0.139472i	0.822951	+	0.568112i	$0.192326\pi$
$348$		0			0
$349$	−4.00000			−0.214115			−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000			−0.214115			−0.107058	+	0.994253i	$0.534143\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	−0.934751	−	0.355303i	$-0.884378\pi$
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	0.775077	+	0.631867i	$0.217711\pi$
$354$		0			0
$355$	−12.0000	−	20.7846i	−0.636894	−	1.10313i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.0000	−	20.7846i	−0.633336	−	1.09697i	−0.986865	−	0.161546i	$-0.948352\pi$
$359$	−12.0000	−	20.7846i	−0.633336	−	1.09697i	0.353529	−	0.935423i	$-0.384981\pi$
$360$		0			0
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−22.0000			−1.15153
$366$		0			0
$367$	−10.0000	+	17.3205i	−0.521996	+	0.904123i	0.477677	+	0.878536i	$0.341479\pi$
$367$	−10.0000	+	17.3205i	−0.521996	+	0.904123i	−0.999673	+	0.0255875i	$0.991854\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	20.0000	−	17.3205i	1.03835	−	0.899236i
$372$		0			0
$373$	2.00000	+	3.46410i	0.103556	+	0.179364i	0.913147	−	0.407630i	$-0.133645\pi$
$373$	2.00000	+	3.46410i	0.103556	+	0.179364i	−0.809591	+	0.586994i	$0.800311\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	30.0000			1.54508
$378$		0			0
$379$	−6.00000			−0.308199			−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000			−0.308199			−0.154100	+	0.988055i	$0.549248\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	18.0000	+	31.1769i	0.919757	+	1.59307i	0.799783	+	0.600289i	$0.204948\pi$
$383$	18.0000	+	31.1769i	0.919757	+	1.59307i	0.119974	+	0.992777i	$0.461719\pi$
$384$		0			0
$385$	15.0000	+	5.19615i	0.764471	+	0.264820i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.50000	+	16.4545i	−0.481669	+	0.834275i	−0.999779	−	0.0210389i	$-0.993303\pi$
$389$	−9.50000	+	16.4545i	−0.481669	+	0.834275i	0.518110	+	0.855314i	$0.326636\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.00000	+	1.73205i	−0.0503155	+	0.0871489i
$396$		0			0
$397$	−11.0000	−	19.0526i	−0.552074	−	0.956221i	−0.998125	−	0.0612128i	$-0.980503\pi$
$397$	−11.0000	−	19.0526i	−0.552074	−	0.956221i	0.446051	−	0.895008i	$-0.352830\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	0.548911	−	0.835881i	$-0.315043\pi$
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	−0.998350	+	0.0574304i	$0.981709\pi$
$402$		0			0
$403$	−21.0000	+	36.3731i	−1.04608	+	1.81187i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−3.00000	+	5.19615i	−0.148340	+	0.256933i	−0.930614	−	0.366002i	$-0.880726\pi$
$409$	−3.00000	+	5.19615i	−0.148340	+	0.256933i	0.782274	+	0.622935i	$0.214060\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−18.0000	+	15.5885i	−0.885722	+	0.767058i
$414$		0			0
$415$	−15.0000	−	25.9808i	−0.736321	−	1.27535i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−28.0000			−1.36789			−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000			−1.36789			−0.683945	+	0.729534i	$0.739737\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.00000	+	3.46410i	0.0970143	+	0.168034i
$426$		0			0
$427$	4.00000	+	20.7846i	0.193574	+	1.00584i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.00000	+	15.5885i	−0.433515	+	0.750870i	−0.997173	−	0.0751385i	$-0.976060\pi$
$431$	−9.00000	+	15.5885i	−0.433515	+	0.750870i	0.563658	+	0.826008i	$0.309393\pi$
$432$		0			0
$433$	−23.0000			−1.10531			−0.552655	−	0.833410i	$-0.686385\pi$
$433$	−23.0000			−1.10531			−0.552655	+	0.833410i	$0.686385\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.00000	−	13.8564i	0.382692	−	0.662842i
$438$		0			0
$439$	−7.50000	−	12.9904i	−0.357955	−	0.619997i	0.629664	−	0.776868i	$-0.283193\pi$
$439$	−7.50000	−	12.9904i	−0.357955	−	0.619997i	−0.987619	+	0.156871i	$0.949859\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−14.5000	−	25.1147i	−0.688916	−	1.19324i	−0.972189	−	0.234198i	$-0.924754\pi$
$443$	−14.5000	−	25.1147i	−0.688916	−	1.19324i	0.283273	−	0.959039i	$-0.408580\pi$
$444$		0			0
$445$	−10.0000	+	17.3205i	−0.474045	+	0.821071i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$	3.00000	−	5.19615i	0.141264	−	0.244677i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.00000	−	31.1769i	−0.281284	−	1.46160i
$456$		0			0
$457$	−11.0000	−	19.0526i	−0.514558	−	0.891241i	−0.999857	−	0.0168929i	$-0.994623\pi$
$457$	−11.0000	−	19.0526i	−0.514558	−	0.891241i	0.485299	−	0.874348i	$-0.338711\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−21.0000			−0.978068			−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000			−0.978068			−0.489034	+	0.872265i	$0.662651\pi$
$462$		0			0
$463$	−1.00000			−0.0464739			−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000			−0.0464739			−0.0232370	+	0.999730i	$0.507397\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.50000	+	12.9904i	0.347059	+	0.601123i	0.985726	−	0.168360i	$-0.0538472\pi$
$467$	7.50000	+	12.9904i	0.347059	+	0.601123i	−0.638667	+	0.769483i	$0.720514\pi$
$468$		0			0
$469$	12.0000	−	10.3923i	0.554109	−	0.479872i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−12.0000	+	20.7846i	−0.551761	+	0.955677i
$474$		0			0
$475$	−4.00000			−0.183533
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.00000	−	1.73205i	0.0456912	−	0.0791394i	−0.842275	−	0.539048i	$-0.818784\pi$
$479$	1.00000	−	1.73205i	0.0456912	−	0.0791394i	0.887967	+	0.459908i	$0.152118\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.00000	+	8.66025i	0.227038	+	0.393242i
$486$		0			0
$487$	−3.50000	+	6.06218i	−0.158600	+	0.274703i	−0.934364	−	0.356320i	$-0.884031\pi$
$487$	−3.50000	+	6.06218i	−0.158600	+	0.274703i	0.775764	+	0.631023i	$0.217365\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	36.0000			1.62466			0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000			1.62466			0.812329	+	0.583200i	$0.198200\pi$
$492$		0			0
$493$	−10.0000	+	17.3205i	−0.450377	+	0.780076i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	30.0000	+	10.3923i	1.34568	+	0.466159i
$498$		0			0
$499$	4.00000	+	6.92820i	0.179065	+	0.310149i	0.941560	−	0.336844i	$-0.109360\pi$
$499$	4.00000	+	6.92820i	0.179065	+	0.310149i	−0.762496	+	0.646993i	$0.776026\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.0000			−0.624229			−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000			−0.624229			−0.312115	+	0.950044i	$0.601037\pi$
$504$		0			0
$505$	−30.0000			−1.33498
$506$		0			0
$507$		0			0
$508$		0			0
$509$	19.5000	+	33.7750i	0.864322	+	1.49705i	0.867719	+	0.497056i	$0.165586\pi$
$509$	19.5000	+	33.7750i	0.864322	+	1.49705i	−0.00339621	+	0.999994i	$0.501081\pi$
$510$		0			0
$511$	22.0000	−	19.0526i	0.973223	−	0.842836i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	6.00000			0.263880
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12.0000	+	20.7846i	−0.525730	+	0.910590i	0.473821	+	0.880621i	$0.342874\pi$
$521$	−12.0000	+	20.7846i	−0.525730	+	0.910590i	−0.999551	+	0.0299693i	$0.990459\pi$
$522$		0			0
$523$	−1.00000	−	1.73205i	−0.0437269	−	0.0757373i	0.843334	−	0.537390i	$-0.180590\pi$
$523$	−1.00000	−	1.73205i	−0.0437269	−	0.0757373i	−0.887061	+	0.461653i	$0.847256\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14.0000	−	24.2487i	−0.609850	−	1.05629i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.0000			−0.519778
$534$		0			0
$535$	12.0000	−	20.7846i	0.518805	−	0.898597i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−19.5000	+	7.79423i	−0.839924	+	0.335721i
$540$		0			0
$541$	−5.00000	−	8.66025i	−0.214967	−	0.372333i	0.738296	−	0.674477i	$-0.235631\pi$
$541$	−5.00000	−	8.66025i	−0.214967	−	0.372333i	−0.953262	+	0.302144i	$0.902298\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	32.0000			1.37073
$546$		0			0
$547$	22.0000			0.940652			0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000			0.940652			0.470326	+	0.882493i	$0.344136\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.0000	−	17.3205i	−0.426014	−	0.737878i
$552$		0			0
$553$	−0.500000	−	2.59808i	−0.0212622	−	0.110481i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.50000	+	6.06218i	−0.148300	+	0.256863i	−0.930599	−	0.366040i	$-0.880713\pi$
$557$	−3.50000	+	6.06218i	−0.148300	+	0.256863i	0.782299	+	0.622903i	$0.214047\pi$
$558$		0			0
$559$	48.0000			2.03018
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.0000	−	17.3205i	0.421450	−	0.729972i	−0.574632	−	0.818412i	$-0.694855\pi$
$563$	10.0000	−	17.3205i	0.421450	−	0.729972i	0.996082	+	0.0884397i	$0.0281881\pi$
$564$		0			0
$565$	12.0000	+	20.7846i	0.504844	+	0.874415i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.0000	+	31.1769i	0.754599	+	1.30700i	0.945573	+	0.325409i	$0.105502\pi$
$569$	18.0000	+	31.1769i	0.754599	+	1.30700i	−0.190974	+	0.981595i	$0.561165\pi$
$570$		0			0
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	0.308443	+	0.951243i	$0.400192\pi$
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	−0.978022	+	0.208502i	$0.933141\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.00000			0.166812
$576$		0			0
$577$	−1.50000	+	2.59808i	−0.0624458	+	0.108159i	−0.895558	−	0.444945i	$-0.853223\pi$
$577$	−1.50000	+	2.59808i	−0.0624458	+	0.108159i	0.833112	+	0.553104i	$0.186557\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	37.5000	+	12.9904i	1.55576	+	0.538932i
$582$		0			0
$583$	15.0000	+	25.9808i	0.621237	+	1.07601i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.0000			−0.825488			−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000			−0.825488			−0.412744	+	0.910847i	$0.635430\pi$
$588$		0			0
$589$	28.0000			1.15372
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.00000	+	10.3923i	0.246390	+	0.426761i	0.962522	−	0.271205i	$-0.0874221\pi$
$593$	6.00000	+	10.3923i	0.246390	+	0.426761i	−0.716131	+	0.697966i	$0.754089\pi$
$594$		0			0
$595$	20.0000	+	6.92820i	0.819920	+	0.284029i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.00000	−	12.1244i	0.286012	−	0.495388i	−0.686842	−	0.726807i	$-0.741004\pi$
$599$	7.00000	−	12.1244i	0.286012	−	0.495388i	0.972854	+	0.231419i	$0.0743369\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.00000	−	3.46410i	0.0813116	−	0.140836i
$606$		0			0
$607$	−9.50000	−	16.4545i	−0.385593	−	0.667867i	0.606258	−	0.795268i	$-0.292670\pi$
$607$	−9.50000	−	16.4545i	−0.385593	−	0.667867i	−0.991851	+	0.127401i	$0.959336\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.00000	−	10.3923i	−0.242734	−	0.420428i
$612$		0			0
$613$	−17.0000	+	29.4449i	−0.686624	+	1.18927i	0.286300	+	0.958140i	$0.407575\pi$
$613$	−17.0000	+	29.4449i	−0.686624	+	1.18927i	−0.972924	+	0.231127i	$0.925759\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−42.0000			−1.69086			−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000			−1.69086			−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$	18.0000	−	31.1769i	0.723481	−	1.25311i	−0.236115	−	0.971725i	$-0.575874\pi$
$619$	18.0000	−	31.1769i	0.723481	−	1.25311i	0.959596	−	0.281381i	$-0.0907924\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.00000	−	25.9808i	−0.200321	−	1.04090i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	41.0000			1.63218			0.816092	−	0.577922i	$-0.196136\pi$
$631$	41.0000			1.63218			0.816092	+	0.577922i	$0.196136\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.00000	−	13.8564i	−0.317470	−	0.549875i
$636$		0			0
$637$	33.0000	+	25.9808i	1.30751	+	1.02940i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.0000	−	32.9090i	0.750455	−	1.29983i	−0.197148	−	0.980374i	$-0.563168\pi$
$641$	19.0000	−	32.9090i	0.750455	−	1.29983i	0.947602	−	0.319452i	$-0.103499\pi$
$642$		0			0
$643$	−26.0000			−1.02534			−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000			−1.02534			−0.512670	+	0.858586i	$0.671344\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.00000	+	15.5885i	−0.353827	+	0.612845i	−0.986916	−	0.161233i	$-0.948453\pi$
$647$	−9.00000	+	15.5885i	−0.353827	+	0.612845i	0.633090	+	0.774078i	$0.281786\pi$
$648$		0			0
$649$	−13.5000	−	23.3827i	−0.529921	−	0.917851i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.00000	−	12.1244i	−0.273931	−	0.474463i	0.695934	−	0.718106i	$-0.254991\pi$
$653$	−7.00000	−	12.1244i	−0.273931	−	0.474463i	−0.969865	+	0.243643i	$0.921657\pi$
$654$		0			0
$655$	13.0000	−	22.5167i	0.507952	−	0.879799i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	23.0000			0.895953			0.447976	−	0.894045i	$-0.352145\pi$
$659$	23.0000			0.895953			0.447976	+	0.894045i	$0.352145\pi$
$660$		0			0
$661$	23.0000	−	39.8372i	0.894596	−	1.54949i	0.0602929	−	0.998181i	$-0.480797\pi$
$661$	23.0000	−	39.8372i	0.894596	−	1.54949i	0.834303	−	0.551306i	$-0.185870\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−16.0000	+	13.8564i	−0.620453	+	0.537328i
$666$		0			0
$667$	10.0000	+	17.3205i	0.387202	+	0.670653i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−24.0000			−0.926510
$672$		0			0
$673$	34.0000			1.31060			0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000			1.31060			0.655302	+	0.755367i	$0.272541\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.50000	−	2.59808i	−0.0576497	−	0.0998522i	0.835760	−	0.549095i	$-0.185027\pi$
$677$	−1.50000	−	2.59808i	−0.0576497	−	0.0998522i	−0.893410	+	0.449242i	$0.851694\pi$
$678$		0			0
$679$	−12.5000	−	4.33013i	−0.479706	−	0.166175i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	22.5000	−	38.9711i	0.860939	−	1.49119i	−0.0100856	−	0.999949i	$-0.503210\pi$
$683$	22.5000	−	38.9711i	0.860939	−	1.49119i	0.871024	−	0.491240i	$-0.163456\pi$
$684$		0			0
$685$	−32.0000			−1.22266
$686$		0			0
$687$		0			0
$688$		0			0
$689$	30.0000	−	51.9615i	1.14291	−	1.97958i
$690$		0			0
$691$	8.00000	+	13.8564i	0.304334	+	0.527123i	0.977113	−	0.212721i	$-0.0682327\pi$
$691$	8.00000	+	13.8564i	0.304334	+	0.527123i	−0.672779	+	0.739844i	$0.734899\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	4.00000	−	6.92820i	0.151511	−	0.262424i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	46.0000			1.73740			0.868698	−	0.495342i	$-0.164957\pi$
$701$	46.0000			1.73740			0.868698	+	0.495342i	$0.164957\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	30.0000	−	25.9808i	1.12827	−	0.977107i
$708$		0			0
$709$	−20.0000	−	34.6410i	−0.751116	−	1.30097i	−0.947282	−	0.320400i	$-0.896183\pi$
$709$	−20.0000	−	34.6410i	−0.751116	−	1.30097i	0.196167	−	0.980571i	$-0.437151\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−28.0000			−1.04861
$714$		0			0
$715$	36.0000			1.34632
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.0000	−	36.3731i	−0.783168	−	1.35649i	−0.930087	−	0.367338i	$-0.880269\pi$
$719$	−21.0000	−	36.3731i	−0.783168	−	1.35649i	0.146920	−	0.989148i	$-0.453064\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.50000	−	4.33013i	0.0928477	−	0.160817i
$726$		0			0
$727$	−40.0000			−1.48352			−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000			−1.48352			−0.741759	+	0.670667i	$0.766008\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−16.0000	+	27.7128i	−0.591781	+	1.02500i
$732$		0			0
$733$	10.0000	+	17.3205i	0.369358	+	0.639748i	0.989465	−	0.144770i	$-0.0462441\pi$
$733$	10.0000	+	17.3205i	0.369358	+	0.639748i	−0.620107	+	0.784517i	$0.712911\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.00000	+	15.5885i	0.331519	+	0.574208i
$738$		0			0
$739$	7.00000	−	12.1244i	0.257499	−	0.446002i	−0.708072	−	0.706140i	$-0.750435\pi$
$739$	7.00000	−	12.1244i	0.257499	−	0.446002i	0.965571	+	0.260138i	$0.0837682\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−34.0000			−1.24734			−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000			−1.24734			−0.623670	+	0.781688i	$0.714359\pi$
$744$		0			0
$745$	5.00000	−	8.66025i	0.183186	−	0.317287i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.00000	+	31.1769i	0.219235	+	1.13918i
$750$		0			0
$751$	14.0000	+	24.2487i	0.510867	+	0.884848i	0.999921	+	0.0125942i	$0.00400897\pi$
$751$	14.0000	+	24.2487i	0.510867	+	0.884848i	−0.489053	+	0.872254i	$0.662658\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	38.0000			1.38296
$756$		0			0
$757$	46.0000			1.67190			0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000			1.67190			0.835949	+	0.548807i	$0.184918\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	24.0000	+	41.5692i	0.869999	+	1.50688i	0.861996	+	0.506915i	$0.169214\pi$
$761$	24.0000	+	41.5692i	0.869999	+	1.50688i	0.00800331	+	0.999968i	$0.497452\pi$
$762$		0			0
$763$	−32.0000	+	27.7128i	−1.15848	+	1.00327i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−27.0000	+	46.7654i	−0.974913	+	1.68860i
$768$		0			0
$769$	25.0000			0.901523			0.450762	−	0.892644i	$-0.351152\pi$
$769$	25.0000			0.901523			0.450762	+	0.892644i	$0.351152\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−11.0000	+	19.0526i	−0.395643	+	0.685273i	−0.993183	−	0.116566i	$-0.962811\pi$
$773$	−11.0000	+	19.0526i	−0.395643	+	0.685273i	0.597540	+	0.801839i	$0.296145\pi$
$774$		0			0
$775$	3.50000	+	6.06218i	0.125724	+	0.217760i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.00000	+	6.92820i	0.143315	+	0.248229i
$780$		0			0
$781$	−18.0000	+	31.1769i	−0.644091	+	1.11560i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−36.0000			−1.28490
$786$		0			0
$787$	7.00000	−	12.1244i	0.249523	−	0.432187i	−0.713871	−	0.700278i	$-0.753059\pi$
$787$	7.00000	−	12.1244i	0.249523	−	0.432187i	0.963394	+	0.268091i	$0.0863928\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−30.0000	−	10.3923i	−1.06668	−	0.369508i
$792$		0			0
$793$	24.0000	+	41.5692i	0.852265	+	1.47617i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.00000			0.106265			0.0531327	−	0.998587i	$-0.483079\pi$
$797$	3.00000			0.106265			0.0531327	+	0.998587i	$0.483079\pi$
$798$		0			0
$799$	8.00000			0.283020
$800$		0			0
$801$		0			0
$802$		0			0
$803$	16.5000	+	28.5788i	0.582272	+	1.00853i
$804$		0			0
$805$	16.0000	−	13.8564i	0.563926	−	0.488374i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.00000	+	5.19615i	−0.105474	+	0.182687i	−0.913932	−	0.405868i	$-0.866969\pi$
$809$	−3.00000	+	5.19615i	−0.105474	+	0.182687i	0.808458	+	0.588555i	$0.200303\pi$
$810$		0			0
$811$	6.00000			0.210688			0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000			0.210688			0.105344	+	0.994436i	$0.466406\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.0000	−	24.2487i	0.490399	−	0.849395i
$816$		0			0
$817$	−16.0000	−	27.7128i	−0.559769	−	0.969549i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13.5000	+	23.3827i	0.471153	+	0.816061i	0.999456	−	0.0329950i	$-0.0105045\pi$
$821$	13.5000	+	23.3827i	0.471153	+	0.816061i	−0.528302	+	0.849056i	$0.677171\pi$
$822$		0			0
$823$	−8.50000	+	14.7224i	−0.296291	+	0.513192i	−0.975284	−	0.220953i	$-0.929083\pi$
$823$	−8.50000	+	14.7224i	−0.296291	+	0.513192i	0.678993	+	0.734145i	$0.262417\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	45.0000			1.56480			0.782402	−	0.622774i	$-0.213994\pi$
$827$	45.0000			1.56480			0.782402	+	0.622774i	$0.213994\pi$
$828$		0			0
$829$	17.0000	−	29.4449i	0.590434	−	1.02266i	−0.403739	−	0.914874i	$-0.632290\pi$
$829$	17.0000	−	29.4449i	0.590434	−	1.02266i	0.994174	−	0.107788i	$-0.0343769\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−26.0000	+	10.3923i	−0.900847	+	0.360072i
$834$		0			0
$835$	18.0000	+	31.1769i	0.622916	+	1.07892i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−23.0000	−	39.8372i	−0.791224	−	1.37044i
$846$		0			0
$847$	1.00000	+	5.19615i	0.0343604	+	0.178542i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	10.0000			0.342393			0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000			0.342393			0.171197	+	0.985237i	$0.445237\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.00000	+	15.5885i	−0.307434	+	0.532492i	−0.977800	−	0.209539i	$-0.932804\pi$
$857$	−9.00000	+	15.5885i	−0.307434	+	0.532492i	0.670366	+	0.742030i	$0.266137\pi$
$858$		0			0
$859$	−8.00000	−	13.8564i	−0.272956	−	0.472774i	0.696661	−	0.717400i	$-0.254668\pi$
$859$	−8.00000	−	13.8564i	−0.272956	−	0.472774i	−0.969618	+	0.244626i	$0.921335\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.00000	−	12.1244i	−0.238283	−	0.412718i	0.721939	−	0.691957i	$-0.243251\pi$
$863$	−7.00000	−	12.1244i	−0.238283	−	0.412718i	−0.960222	+	0.279239i	$0.909918\pi$
$864$		0			0
$865$	−11.0000	+	19.0526i	−0.374011	+	0.647806i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.00000			0.101768
$870$		0			0
$871$	18.0000	−	31.1769i	0.609907	−	1.05639i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−30.0000	−	10.3923i	−1.01419	−	0.351324i
$876$		0			0
$877$	−1.00000	−	1.73205i	−0.0337676	−	0.0584872i	0.848648	−	0.528958i	$-0.177417\pi$
$877$	−1.00000	−	1.73205i	−0.0337676	−	0.0584872i	−0.882415	+	0.470471i	$0.844084\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.00000			−0.0673817			−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000			−0.0673817			−0.0336909	+	0.999432i	$0.510726\pi$
$882$		0			0
$883$	10.0000			0.336527			0.168263	−	0.985742i	$-0.446184\pi$
$883$	10.0000			0.336527			0.168263	+	0.985742i	$0.446184\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−9.00000	−	15.5885i	−0.302190	−	0.523409i	0.674441	−	0.738328i	$-0.264385\pi$
$887$	−9.00000	−	15.5885i	−0.302190	−	0.523409i	−0.976632	+	0.214919i	$0.931051\pi$
$888$		0			0
$889$	20.0000	+	6.92820i	0.670778	+	0.232364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.00000	+	6.92820i	−0.133855	+	0.231843i
$894$		0			0
$895$	−2.00000			−0.0668526
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−17.5000	+	30.3109i	−0.583658	+	1.01092i
$900$		0			0
$901$	20.0000	+	34.6410i	0.666297	+	1.15406i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	16.0000	+	27.7128i	0.531858	+	0.921205i
$906$		0			0
$907$	5.00000	−	8.66025i	0.166022	−	0.287559i	−0.770996	−	0.636841i	$-0.780241\pi$
$907$	5.00000	−	8.66025i	0.166022	−	0.287559i	0.937018	+	0.349281i	$0.113574\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−44.0000			−1.45779			−0.728893	−	0.684628i	$-0.759965\pi$
$911$	−44.0000			−1.45779			−0.728893	+	0.684628i	$0.759965\pi$
$912$		0			0
$913$	−22.5000	+	38.9711i	−0.744641	+	1.28976i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	6.50000	+	33.7750i	0.214649	+	1.11535i
$918$		0			0
$919$	−0.500000	−	0.866025i	−0.0164935	−	0.0285675i	0.857661	−	0.514216i	$-0.171917\pi$
$919$	−0.500000	−	0.866025i	−0.0164935	−	0.0285675i	−0.874154	+	0.485648i	$0.838584\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	72.0000			2.36991
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.00000	−	3.46410i	−0.0656179	−	0.113653i	0.831350	−	0.555749i	$-0.187569\pi$
$929$	−2.00000	−	3.46410i	−0.0656179	−	0.113653i	−0.896968	+	0.442096i	$0.854235\pi$
$930$		0			0
$931$	4.00000	−	27.7128i	0.131095	−	0.908251i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.0000	+	20.7846i	−0.392442	+	0.679729i
$936$		0			0
$937$	30.0000			0.980057			0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000			0.980057			0.490029	+	0.871706i	$0.336986\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	16.5000	−	28.5788i	0.537885	−	0.931644i	−0.461133	−	0.887331i	$-0.652557\pi$
$941$	16.5000	−	28.5788i	0.537885	−	0.931644i	0.999018	−	0.0443125i	$-0.0141097\pi$
$942$		0			0
$943$	−4.00000	−	6.92820i	−0.130258	−	0.225613i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.50000	−	6.06218i	−0.113735	−	0.196994i	0.803539	−	0.595253i	$-0.202948\pi$
$947$	−3.50000	−	6.06218i	−0.113735	−	0.196994i	−0.917273	+	0.398258i	$0.869615\pi$
$948$		0			0
$949$	33.0000	−	57.1577i	1.07123	−	1.85542i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−26.0000			−0.842223			−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000			−0.842223			−0.421111	+	0.907009i	$0.638360\pi$
$954$		0			0
$955$	−10.0000	+	17.3205i	−0.323592	+	0.560478i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	32.0000	−	27.7128i	1.03333	−	0.894893i
$960$		0			0
$961$	−9.00000	−	15.5885i	−0.290323	−	0.502853i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	42.0000			1.35203
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−26.0000	−	45.0333i	−0.834380	−	1.44519i	−0.894534	−	0.446999i	$-0.852493\pi$
$971$	−26.0000	−	45.0333i	−0.834380	−	1.44519i	0.0601548	−	0.998189i	$-0.480841\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	27.0000	−	46.7654i	0.863807	−	1.49616i	−0.00442082	−	0.999990i	$-0.501407\pi$
$977$	27.0000	−	46.7654i	0.863807	−	1.49616i	0.868227	−	0.496167i	$-0.165259\pi$
$978$		0			0
$979$	30.0000			0.958804
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	0.422027	+	0.906583i	$0.361319\pi$
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	−0.996138	+	0.0878058i	$0.972015\pi$
$984$		0			0
$985$	3.00000	+	5.19615i	0.0955879	+	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16.0000	+	27.7128i	0.508770	+	0.881216i
$990$		0			0
$991$	−24.0000	+	41.5692i	−0.762385	+	1.32049i	0.179233	+	0.983807i	$0.442638\pi$
$991$	−24.0000	+	41.5692i	−0.762385	+	1.32049i	−0.941618	+	0.336683i	$0.890695\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−14.0000			−0.443830
$996$		0			0
$997$	−23.0000	+	39.8372i	−0.728417	+	1.26166i	0.229135	+	0.973395i	$0.426410\pi$
$997$	−23.0000	+	39.8372i	−0.728417	+	1.26166i	−0.957552	+	0.288261i	$0.906923\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.s.e.865.1	✓	2
3.2	odd	2		1512.2.s.i.865.1	yes	2
7.2	even	3	inner	1512.2.s.e.1297.1	yes	2
21.2	odd	6		1512.2.s.i.1297.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.s.e.865.1	✓	2	1.1	even	1	trivial
1512.2.s.e.1297.1	yes	2	7.2	even	3	inner
1512.2.s.i.865.1	yes	2	3.2	odd	2
1512.2.s.i.1297.1	yes	2	21.2	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.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{5} +(2.50000 + 0.866025i) q^{7} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{5} +(2.50000 + 0.866025i) q^{7} +(-1.50000 + 2.59808i) q^{11} +6.00000 q^{13} +(-2.00000 + 3.46410i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(2.00000 + 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} +5.00000 q^{29} +(-3.50000 + 6.06218i) q^{31} +(-1.00000 - 5.19615i) q^{35} -2.00000 q^{41} +8.00000 q^{43} +(-1.00000 - 1.73205i) q^{47} +(5.50000 + 4.33013i) q^{49} +(5.00000 - 8.66025i) q^{53} +6.00000 q^{55} +(-4.50000 + 7.79423i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-6.00000 - 10.3923i) q^{65} +(3.00000 - 5.19615i) q^{67} +12.0000 q^{71} +(5.50000 - 9.52628i) q^{73} +(-6.00000 + 5.19615i) q^{77} +(-0.500000 - 0.866025i) q^{79} +15.0000 q^{83} +8.00000 q^{85} +(-5.00000 - 8.66025i) q^{89} +(15.0000 + 5.19615i) q^{91} +(-4.00000 + 6.92820i) q^{95} -5.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 5 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 5 * q^7	\( 2 q - 2 q^{5} + 5 q^{7} - 3 q^{11} + 12 q^{13} - 4 q^{17} - 4 q^{19} + 4 q^{23} + q^{25} + 10 q^{29} - 7 q^{31} - 2 q^{35} - 4 q^{41} + 16 q^{43} - 2 q^{47} + 11 q^{49} + 10 q^{53} + 12 q^{55} - 9 q^{59} + 8 q^{61} - 12 q^{65} + 6 q^{67} + 24 q^{71} + 11 q^{73} - 12 q^{77} - q^{79} + 30 q^{83} + 16 q^{85} - 10 q^{89} + 30 q^{91} - 8 q^{95} - 10 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 5 * q^7 - 3 * q^11 + 12 * q^13 - 4 * q^17 - 4 * q^19 + 4 * q^23 + q^25 + 10 * q^29 - 7 * q^31 - 2 * q^35 - 4 * q^41 + 16 * q^43 - 2 * q^47 + 11 * q^49 + 10 * q^53 + 12 * q^55 - 9 * q^59 + 8 * q^61 - 12 * q^65 + 6 * q^67 + 24 * q^71 + 11 * q^73 - 12 * q^77 - q^79 + 30 * q^83 + 16 * q^85 - 10 * q^89 + 30 * q^91 - 8 * q^95 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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