LMFDB - Embedded newform 864.2.r.b.145.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(145,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("864.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$864 = 2^{5} \cdot 3^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	864.r (of order $6$ , 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:	$6.89907473464$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + \cdots + 256$ x^16 - x^15 + x^14 + 2x^12 - 4x^11 - 8x^9 + 4x^8 - 16x^7 - 32x^5 + 32x^4 + 64x^2 - 128*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{8}\cdot 3^{4}$
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.1
Root			$1.41411 + 0.0174668i$ of defining polynomial
Character	$\chi$	$=$	864.145
Dual form			864.2.r.b.721.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.17262 - 1.83171i) q^{5} +(0.191926 + 0.332426i) q^{7} +(-1.73849 + 1.00372i) q^{11} +(-0.397799 - 0.229669i) q^{13} +4.08495 q^{17} +4.72398i q^{19} +(-2.97594 + 5.15447i) q^{23} +(4.21034 + 7.29252i) q^{25} +(2.03783 - 1.17654i) q^{29} +(-0.592083 + 1.02552i) q^{31} -1.40621i q^{35} +5.74432i q^{37} +(-4.75281 + 8.23212i) q^{41} +(-1.03633 + 0.598327i) q^{43} +(3.27688 + 5.67572i) q^{47} +(3.42633 - 5.93458i) q^{49} +7.63807i q^{53} +7.35407 q^{55} +(0.603703 + 0.348548i) q^{59} +(4.23774 - 2.44666i) q^{61} +(0.841376 + 1.45731i) q^{65} +(-8.87932 - 5.12648i) q^{67} -3.73792 q^{71} -2.68275 q^{73} +(-0.667322 - 0.385279i) q^{77} +(5.35979 + 9.28342i) q^{79} +(-5.49039 + 3.16988i) q^{83} +(-12.9600 - 7.48246i) q^{85} -7.56802 q^{89} -0.176318i q^{91} +(8.65297 - 14.9874i) q^{95} +(-2.98511 - 5.17036i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 6 q^{7} + 28 q^{17} - 10 q^{23} + 2 q^{25} + 10 q^{31} + 8 q^{41} + 6 q^{47} + 18 q^{49} + 4 q^{55} + 14 q^{65} + 72 q^{71} - 44 q^{73} + 30 q^{79} - 64 q^{89} + 44 q^{95}+O(q^{100})$ 16 * q - 6 * q^7 + 28 * q^17 - 10 * q^23 + 2 * q^25 + 10 * q^31 + 8 * q^41 + 6 * q^47 + 18 * q^49 + 4 * q^55 + 14 * q^65 + 72 * q^71 - 44 * q^73 + 30 * q^79 - 64 * q^89 + 44 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$ .

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

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
$5$	−3.17262	−	1.83171i	−1.41884	−	0.819167i	−0.422641	−	0.906297i	$-0.638897\pi$
$5$	−3.17262	−	1.83171i	−1.41884	−	0.819167i	−0.996197	+	0.0871306i	$0.972230\pi$
$6$		0			0
$7$	0.191926	+	0.332426i	0.0725413	+	0.125645i	0.900014	−	0.435860i	$-0.143556\pi$
$7$	0.191926	+	0.332426i	0.0725413	+	0.125645i	−0.827473	+	0.561505i	$0.810222\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.73849	+	1.00372i	−0.524173	+	0.302632i	−0.738640	−	0.674100i	$-0.764532\pi$
$11$	−1.73849	+	1.00372i	−0.524173	+	0.302632i	0.214467	+	0.976731i	$0.431199\pi$
$12$		0			0
$13$	−0.397799	−	0.229669i	−0.110330	−	0.0636988i	0.443820	−	0.896116i	$-0.353623\pi$
$13$	−0.397799	−	0.229669i	−0.110330	−	0.0636988i	−0.554149	+	0.832417i	$0.686956\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.08495			0.990747			0.495373	−	0.868680i	$-0.335031\pi$
$17$	4.08495			0.990747			0.495373	+	0.868680i	$0.335031\pi$
$18$		0			0
$19$			4.72398i			1.08376i	0.840457	+	0.541878i	$0.182286\pi$
$19$			4.72398i			1.08376i	−0.840457	+	0.541878i	$0.817714\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.97594	+	5.15447i	−0.620525	+	1.07478i	0.368863	+	0.929484i	$0.379747\pi$
$23$	−2.97594	+	5.15447i	−0.620525	+	1.07478i	−0.989388	+	0.145298i	$0.953586\pi$
$24$		0			0
$25$	4.21034	+	7.29252i	0.842068	+	1.45850i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.03783	−	1.17654i	0.378416	−	0.218479i	−0.298713	−	0.954343i	$-0.596557\pi$
$29$	2.03783	−	1.17654i	0.378416	−	0.218479i	0.677129	+	0.735864i	$0.263224\pi$
$30$		0			0
$31$	−0.592083	+	1.02552i	−0.106341	+	0.184188i	−0.914285	−	0.405071i	$-0.867247\pi$
$31$	−0.592083	+	1.02552i	−0.106341	+	0.184188i	0.807944	+	0.589259i	$0.200580\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	1.40621i		−	0.237693i
$36$		0			0
$37$			5.74432i			0.944360i	0.881502	+	0.472180i	$0.156533\pi$
$37$			5.74432i			0.944360i	−0.881502	+	0.472180i	$0.843467\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.75281	+	8.23212i	−0.742265	+	1.28564i	0.209197	+	0.977874i	$0.432915\pi$
$41$	−4.75281	+	8.23212i	−0.742265	+	1.28564i	−0.951462	+	0.307767i	$0.900418\pi$
$42$		0			0
$43$	−1.03633	+	0.598327i	−0.158039	+	0.0912440i	−0.576934	−	0.816791i	$-0.695751\pi$
$43$	−1.03633	+	0.598327i	−0.158039	+	0.0912440i	0.418895	+	0.908035i	$0.362418\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.27688	+	5.67572i	0.477982	+	0.827889i	0.999681	−	0.0252403i	$-0.00803510\pi$
$47$	3.27688	+	5.67572i	0.477982	+	0.827889i	−0.521699	+	0.853129i	$0.674702\pi$
$48$		0			0
$49$	3.42633	−	5.93458i	0.489476	−	0.847796i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			7.63807i			1.04917i	0.851358	+	0.524585i	$0.175779\pi$
$53$			7.63807i			1.04917i	−0.851358	+	0.524585i	$0.824221\pi$
$54$		0			0
$55$	7.35407			0.991623
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.603703	+	0.348548i	0.0785954	+	0.0453771i	0.538783	−	0.842445i	$-0.318884\pi$
$59$	0.603703	+	0.348548i	0.0785954	+	0.0453771i	−0.460187	+	0.887822i	$0.652218\pi$
$60$		0			0
$61$	4.23774	−	2.44666i	0.542587	−	0.313263i	−0.203540	−	0.979067i	$-0.565245\pi$
$61$	4.23774	−	2.44666i	0.542587	−	0.313263i	0.746127	+	0.665804i	$0.231911\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.841376	+	1.45731i	0.104360	+	0.180757i
$66$		0			0
$67$	−8.87932	−	5.12648i	−1.08478	−	0.626299i	−0.152599	−	0.988288i	$-0.548764\pi$
$67$	−8.87932	−	5.12648i	−1.08478	−	0.626299i	−0.932182	+	0.361989i	$0.882098\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.73792			−0.443610			−0.221805	−	0.975091i	$-0.571195\pi$
$71$	−3.73792			−0.443610			−0.221805	+	0.975091i	$0.571195\pi$
$72$		0			0
$73$	−2.68275			−0.313992			−0.156996	−	0.987599i	$-0.550181\pi$
$73$	−2.68275			−0.313992			−0.156996	+	0.987599i	$0.550181\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.667322	−	0.385279i	−0.0760484	−	0.0439066i
$78$		0			0
$79$	5.35979	+	9.28342i	0.603023	+	1.04447i	0.992361	+	0.123372i	$0.0393707\pi$
$79$	5.35979	+	9.28342i	0.603023	+	1.04447i	−0.389337	+	0.921095i	$0.627296\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.49039	+	3.16988i	−0.602648	+	0.347939i	−0.770083	−	0.637944i	$-0.779785\pi$
$83$	−5.49039	+	3.16988i	−0.602648	+	0.347939i	0.167434	+	0.985883i	$0.446452\pi$
$84$		0			0
$85$	−12.9600	−	7.48246i	−1.40571	−	0.811586i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.56802			−0.802208			−0.401104	−	0.916032i	$-0.631373\pi$
$89$	−7.56802			−0.802208			−0.401104	+	0.916032i	$0.631373\pi$
$90$		0			0
$91$		−	0.176318i		−	0.0184832i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.65297	−	14.9874i	0.887776	−	1.53767i
$96$		0			0
$97$	−2.98511	−	5.17036i	−0.303092	−	0.524971i	0.673743	−	0.738966i	$-0.264686\pi$
$97$	−2.98511	−	5.17036i	−0.303092	−	0.524971i	−0.976835	+	0.213995i	$0.931352\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.81265	+	2.77859i	−0.478877	+	0.276480i	−0.719948	−	0.694028i	$-0.755835\pi$
$101$	−4.81265	+	2.77859i	−0.478877	+	0.276480i	0.241071	+	0.970507i	$0.422501\pi$
$102$		0			0
$103$	6.14380	−	10.6414i	0.605366	−	1.04853i	−0.386627	−	0.922236i	$-0.626360\pi$
$103$	6.14380	−	10.6414i	0.605366	−	1.04853i	0.991993	−	0.126289i	$-0.0403067\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.61773i		−	0.639760i	−0.947458	−	0.319880i	$-0.896357\pi$
$107$		−	6.61773i		−	0.639760i	0.947458	−	0.319880i	$-0.103643\pi$
$108$		0			0
$109$			7.01563i			0.671975i	0.941866	+	0.335988i	$0.109070\pi$
$109$			7.01563i			0.671975i	−0.941866	+	0.335988i	$0.890930\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.09419	−	7.09135i	0.385149	−	0.667098i	−0.606641	−	0.794976i	$-0.707483\pi$
$113$	4.09419	−	7.09135i	0.385149	−	0.667098i	0.991790	+	0.127878i	$0.0408167\pi$
$114$		0			0
$115$	18.8830	−	10.9021i	1.76085	−	1.01663i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.784009	+	1.35794i	0.0718700	+	0.124483i
$120$		0			0
$121$	−3.48511	+	6.03639i	−0.316828	+	0.548762i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.5314i		−	1.12084i
$126$		0			0
$127$	−21.0113			−1.86445			−0.932224	−	0.361882i	$-0.882134\pi$
$127$	−21.0113			−1.86445			−0.932224	+	0.361882i	$0.882134\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.49039	+	3.16988i	0.479697	+	0.276953i	0.720290	−	0.693673i	$-0.244009\pi$
$131$	5.49039	+	3.16988i	0.479697	+	0.276953i	−0.240593	+	0.970626i	$0.577342\pi$
$132$		0			0
$133$	−1.57037	+	0.906655i	−0.136169	+	0.0786170i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.483695	−	0.837785i	−0.0413249	−	0.0715768i	0.844623	−	0.535361i	$-0.179825\pi$
$137$	−0.483695	−	0.837785i	−0.0413249	−	0.0715768i	−0.885948	+	0.463784i	$0.846491\pi$
$138$		0			0
$139$	5.18167	+	2.99164i	0.439503	+	0.253747i	0.703387	−	0.710807i	$-0.251670\pi$
$139$	5.18167	+	2.99164i	0.439503	+	0.253747i	−0.263884	+	0.964554i	$0.585004\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.922090			0.0771091
$144$		0			0
$145$	−8.62036			−0.715882
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.43764	+	3.71678i	0.527392	+	0.304490i	0.739954	−	0.672658i	$-0.234847\pi$
$149$	6.43764	+	3.71678i	0.527392	+	0.304490i	−0.212562	+	0.977148i	$0.568181\pi$
$150$		0			0
$151$	0.492870	+	0.853676i	0.0401092	+	0.0694711i	0.885383	−	0.464862i	$-0.153896\pi$
$151$	0.492870	+	0.853676i	0.0401092	+	0.0694711i	−0.845274	+	0.534333i	$0.820563\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.75691	−	2.16905i	0.301762	−	0.174222i
$156$		0			0
$157$	15.2336	+	8.79510i	1.21577	+	0.701925i	0.964010	−	0.265864i	$-0.0856573\pi$
$157$	15.2336	+	8.79510i	1.21577	+	0.701925i	0.251760	+	0.967790i	$0.418991\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.28464			−0.180055
$162$		0			0
$163$			17.8852i			1.40088i	0.713711	+	0.700440i	$0.247013\pi$
$163$			17.8852i			1.40088i	−0.713711	+	0.700440i	$0.752987\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.04037	+	1.80197i	−0.0805062	+	0.139441i	−0.903467	−	0.428657i	$-0.858987\pi$
$167$	−1.04037	+	1.80197i	−0.0805062	+	0.139441i	0.822961	+	0.568097i	$0.192320\pi$
$168$		0			0
$169$	−6.39450	−	11.0756i	−0.491885	−	0.851970i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	16.4718	−	9.51000i	1.25233	−	0.723032i	0.280757	−	0.959779i	$-0.409415\pi$
$173$	16.4718	−	9.51000i	1.25233	−	0.723032i	0.971571	+	0.236747i	$0.0760813\pi$
$174$		0			0
$175$	−1.61615	+	2.79925i	−0.122169	+	0.211603i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2.12111i		−	0.158539i	−0.996853	−	0.0792697i	$-0.974741\pi$
$179$		−	2.12111i		−	0.158539i	0.996853	−	0.0792697i	$-0.0252588\pi$
$180$		0			0
$181$		−	1.66297i		−	0.123608i	−0.998088	−	0.0618039i	$-0.980315\pi$
$181$		−	1.66297i		−	0.123608i	0.998088	−	0.0618039i	$-0.0196853\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.5219	−	18.2245i	0.773588	−	1.33989i
$186$		0			0
$187$	−7.10164	+	4.10013i	−0.519323	+	0.299831i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.69755	−	15.0646i	−0.629333	−	1.09004i	−0.987686	−	0.156450i	$-0.949995\pi$
$191$	−8.69755	−	15.0646i	−0.629333	−	1.09004i	0.358353	−	0.933586i	$-0.383338\pi$
$192$		0			0
$193$	−1.41709	+	2.45447i	−0.102004	+	0.176677i	−0.912510	−	0.409054i	$-0.865859\pi$
$193$	−1.41709	+	2.45447i	−0.102004	+	0.176677i	0.810506	+	0.585730i	$0.199192\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			12.5991i			0.897646i	0.893621	+	0.448823i	$0.148157\pi$
$197$			12.5991i			0.897646i	−0.893621	+	0.448823i	$0.851843\pi$
$198$		0			0
$199$	−17.2733			−1.22447			−0.612237	−	0.790674i	$-0.709730\pi$
$199$	−17.2733			−1.22447			−0.612237	+	0.790674i	$0.709730\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.782227	+	0.451619i	0.0549016	+	0.0316975i
$204$		0			0
$205$	30.1577	−	17.4116i	2.10631	−	1.21608i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.74153	−	8.21258i	−0.327979	−	0.568076i
$210$		0			0
$211$	−15.2192	−	8.78678i	−1.04773	−	0.604907i	−0.125717	−	0.992066i	$-0.540123\pi$
$211$	−15.2192	−	8.78678i	−1.04773	−	0.604907i	−0.922013	+	0.387159i	$0.873456\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.38385			0.298976
$216$		0			0
$217$	−0.454545			−0.0308565
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.62499	−	0.938188i	−0.109309	−	0.0631094i
$222$		0			0
$223$	−12.3137	−	21.3280i	−0.824587	−	1.42823i	−0.902235	−	0.431245i	$-0.858074\pi$
$223$	−12.3137	−	21.3280i	−0.824587	−	1.42823i	0.0776484	−	0.996981i	$-0.475259\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.9918	−	9.81024i	1.12779	−	0.651128i	0.184410	−	0.982849i	$-0.440963\pi$
$227$	16.9918	−	9.81024i	1.12779	−	0.651128i	0.943378	+	0.331721i	$0.107629\pi$
$228$		0			0
$229$	−21.3431	−	12.3224i	−1.41039	−	0.814289i	−0.414965	−	0.909837i	$-0.636206\pi$
$229$	−21.3431	−	12.3224i	−1.41039	−	0.814289i	−0.995425	+	0.0955486i	$0.969539\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	20.9222			1.37066			0.685330	−	0.728233i	$-0.259658\pi$
$233$	20.9222			1.37066			0.685330	+	0.728233i	$0.259658\pi$
$234$		0			0
$235$		−	24.0092i		−	1.56619i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.14584	+	8.91286i	−0.332857	+	0.576525i	−0.983071	−	0.183226i	$-0.941346\pi$
$239$	−5.14584	+	8.91286i	−0.332857	+	0.576525i	0.650214	+	0.759751i	$0.274679\pi$
$240$		0			0
$241$	10.2379	+	17.7326i	0.659483	+	1.14226i	0.980750	+	0.195269i	$0.0625579\pi$
$241$	10.2379	+	17.7326i	0.659483	+	1.14226i	−0.321267	+	0.946989i	$0.604109\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−21.7409	+	12.5521i	−1.38897	+	0.801924i
$246$		0			0
$247$	1.08495	−	1.87919i	0.0690339	−	0.119570i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			28.0987i			1.77358i	0.462177	+	0.886788i	$0.347068\pi$
$251$			28.0987i			1.77358i	−0.462177	+	0.886788i	$0.652932\pi$
$252$		0			0
$253$		−	11.9480i		−	0.751163i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.53682	+	9.59006i	−0.345378	+	0.598211i	−0.985422	−	0.170126i	$-0.945582\pi$
$257$	−5.53682	+	9.59006i	−0.345378	+	0.598211i	0.640045	+	0.768338i	$0.278916\pi$
$258$		0			0
$259$	−1.90956	+	1.10248i	−0.118654	+	0.0685050i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.7620	−	22.1044i	−0.786938	−	1.36302i	−0.927834	−	0.372992i	$-0.878332\pi$
$263$	−12.7620	−	22.1044i	−0.786938	−	1.36302i	0.140897	−	0.990024i	$-0.455002\pi$
$264$		0			0
$265$	13.9907	−	24.2327i	0.859444	−	1.48860i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	18.3998i		−	1.12185i	−0.827865	−	0.560927i	$-0.810445\pi$
$269$		−	18.3998i		−	1.12185i	0.827865	−	0.560927i	$-0.189555\pi$
$270$		0			0
$271$	22.4135			1.36152			0.680760	−	0.732506i	$-0.261650\pi$
$271$	22.4135			1.36152			0.680760	+	0.732506i	$0.261650\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−14.6392	−	8.45196i	−0.882779	−	0.509673i
$276$		0			0
$277$	−20.2421	+	11.6868i	−1.21623	+	0.702190i	−0.964109	−	0.265506i	$-0.914461\pi$
$277$	−20.2421	+	11.6868i	−1.21623	+	0.702190i	−0.252119	+	0.967696i	$0.581128\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.29466	−	9.17062i	−0.315853	−	0.547073i	0.663765	−	0.747941i	$-0.268957\pi$
$281$	−5.29466	−	9.17062i	−0.315853	−	0.547073i	−0.979618	+	0.200867i	$0.935624\pi$
$282$		0			0
$283$	7.69029	+	4.43999i	0.457140	+	0.263930i	0.710841	−	0.703353i	$-0.248314\pi$
$283$	7.69029	+	4.43999i	0.457140	+	0.263930i	−0.253701	+	0.967283i	$0.581648\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.64876			−0.215379
$288$		0			0
$289$	−0.313160			−0.0184212
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.82117	+	5.67026i	0.573759	+	0.331260i	0.758649	−	0.651499i	$-0.225860\pi$
$293$	9.82117	+	5.67026i	0.573759	+	0.331260i	−0.184890	+	0.982759i	$0.559193\pi$
$294$		0			0
$295$	−1.27688	−	2.21162i	−0.0743428	−	0.128765i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.36765	−	1.36696i	0.136925	−	0.0790534i
$300$		0			0
$301$	−0.397799	−	0.229669i	−0.0229287	−	0.0132379i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−17.9263			−1.02646
$306$		0			0
$307$		−	0.628678i		−	0.0358805i	−0.999839	−	0.0179403i	$-0.994289\pi$
$307$		−	0.628678i		−	0.0358805i	0.999839	−	0.0179403i	$-0.00571087\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.64443	+	16.7046i	−0.546885	+	0.947233i	0.451600	+	0.892220i	$0.350853\pi$
$311$	−9.64443	+	16.7046i	−0.546885	+	0.947233i	−0.998486	+	0.0550127i	$0.982480\pi$
$312$		0			0
$313$	−2.86959	−	4.97028i	−0.162199	−	0.280937i	0.773458	−	0.633847i	$-0.218525\pi$
$313$	−2.86959	−	4.97028i	−0.162199	−	0.280937i	−0.935657	+	0.352911i	$0.885192\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.8187	−	6.24618i	0.607639	−	0.350821i	−0.164402	−	0.986393i	$-0.552569\pi$
$317$	10.8187	−	6.24618i	0.607639	−	0.350821i	0.772041	+	0.635573i	$0.219236\pi$
$318$		0			0
$319$	−2.36183	+	4.09081i	−0.132237	+	0.229042i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			19.2972i			1.07373i
$324$		0			0
$325$		−	3.86794i		−	0.214555i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.25784	+	2.17864i	−0.0693468	+	0.120112i
$330$		0			0
$331$	3.00014	−	1.73213i	0.164902	−	0.0952065i	−0.415278	−	0.909695i	$-0.636316\pi$
$331$	3.00014	−	1.73213i	0.164902	−	0.0952065i	0.580180	+	0.814488i	$0.302982\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	18.7805	+	32.5287i	1.02609	+	1.77723i
$336$		0			0
$337$	−9.30453	+	16.1159i	−0.506850	+	0.877890i	0.493119	+	0.869962i	$0.335857\pi$
$337$	−9.30453	+	16.1159i	−0.506850	+	0.877890i	−0.999969	+	0.00792778i	$0.997476\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	2.37713i		−	0.128729i
$342$		0			0
$343$	5.31737			0.287111
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.0408752	−	0.0235993i	−0.00219430	−	0.00126688i	0.498902	−	0.866658i	$-0.333737\pi$
$347$	−0.0408752	−	0.0235993i	−0.00219430	−	0.00126688i	−0.501097	+	0.865391i	$0.667070\pi$
$348$		0			0
$349$	−12.8884	+	7.44113i	−0.689901	+	0.398314i	−0.803575	−	0.595204i	$-0.797071\pi$
$349$	−12.8884	+	7.44113i	−0.689901	+	0.398314i	0.113674	+	0.993518i	$0.463738\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.25423	+	16.0288i	0.492553	+	0.853127i	0.999963	−	0.00857792i	$-0.00273047\pi$
$353$	9.25423	+	16.0288i	0.492553	+	0.853127i	−0.507410	+	0.861705i	$0.669397\pi$
$354$		0			0
$355$	11.8590	+	6.84680i	0.629411	+	0.363390i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	31.3426			1.65420			0.827101	−	0.562054i	$-0.189989\pi$
$359$	31.3426			1.65420			0.827101	+	0.562054i	$0.189989\pi$
$360$		0			0
$361$	−3.31599			−0.174526
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.51135	+	4.91403i	0.445504	+	0.257212i
$366$		0			0
$367$	10.2308	+	17.7203i	0.534043	+	0.924990i	0.999209	+	0.0397663i	$0.0126613\pi$
$367$	10.2308	+	17.7203i	0.534043	+	0.924990i	−0.465166	+	0.885224i	$0.654005\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.53909	+	1.46594i	−0.131823	+	0.0761081i
$372$		0			0
$373$	20.6021	+	11.8946i	1.06674	+	0.615880i	0.927288	−	0.374349i	$-0.122134\pi$
$373$	20.6021	+	11.8946i	1.06674	+	0.615880i	0.139448	+	0.990229i	$0.455467\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.08086			−0.0556673
$378$		0			0
$379$			24.0988i			1.23787i	0.785441	+	0.618937i	$0.212436\pi$
$379$			24.0988i			1.23787i	−0.785441	+	0.618937i	$0.787564\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.39161	−	16.2667i	0.479889	−	0.831192i	−0.519845	−	0.854261i	$-0.674010\pi$
$383$	9.39161	−	16.2667i	0.479889	−	0.831192i	0.999734	+	0.0230686i	$0.00734361\pi$
$384$		0			0
$385$	1.41144	+	2.44468i	0.0719336	+	0.124593i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.0467	+	8.68720i	−0.762897	+	0.440459i	−0.830335	−	0.557265i	$-0.811851\pi$
$389$	−15.0467	+	8.68720i	−0.762897	+	0.440459i	0.0674382	+	0.997723i	$0.478517\pi$
$390$		0			0
$391$	−12.1566	+	21.0558i	−0.614783	+	1.06484i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	39.2703i		−	1.97591i
$396$		0			0
$397$			26.2401i			1.31696i	0.752600	+	0.658478i	$0.228799\pi$
$397$			26.2401i			1.31696i	−0.752600	+	0.658478i	$0.771201\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.8194	+	18.7398i	−0.540296	+	0.935820i	0.458591	+	0.888648i	$0.348354\pi$
$401$	−10.8194	+	18.7398i	−0.540296	+	0.935820i	−0.998887	+	0.0471725i	$0.984979\pi$
$402$		0			0
$403$	0.471060	−	0.271967i	0.0234652	−	0.0135476i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.76566	−	9.98642i	−0.285793	−	0.495008i
$408$		0			0
$409$	−6.00563	+	10.4021i	−0.296959	+	0.514348i	−0.975439	−	0.220271i	$-0.929306\pi$
$409$	−6.00563	+	10.4021i	−0.296959	+	0.514348i	0.678480	+	0.734619i	$0.262639\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			0.267582i			0.0131668i
$414$		0			0
$415$	23.2252			1.14008
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.38092	−	0.797277i	−0.0674625	−	0.0389495i	0.465889	−	0.884843i	$-0.345734\pi$
$419$	−1.38092	−	0.797277i	−0.0674625	−	0.0389495i	−0.533352	+	0.845893i	$0.679068\pi$
$420$		0			0
$421$	−15.6612	+	9.04197i	−0.763278	+	0.440679i	−0.830471	−	0.557061i	$-0.811929\pi$
$421$	−15.6612	+	9.04197i	−0.763278	+	0.440679i	0.0671934	+	0.997740i	$0.478596\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	17.1990	+	29.7896i	0.834276	+	1.44501i
$426$		0			0
$427$	1.62667	+	0.939156i	0.0787199	+	0.0454489i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−34.7451			−1.67361			−0.836806	−	0.547499i	$-0.815580\pi$
$431$	−34.7451			−1.67361			−0.836806	+	0.547499i	$0.815580\pi$
$432$		0			0
$433$	12.7197			0.611270			0.305635	−	0.952149i	$-0.401131\pi$
$433$	12.7197			0.611270			0.305635	+	0.952149i	$0.401131\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−24.3496	−	14.0583i	−1.16480	−	0.672498i
$438$		0			0
$439$	7.45900	+	12.9194i	0.355999	+	0.616608i	0.987288	−	0.158939i	$-0.0508074\pi$
$439$	7.45900	+	12.9194i	0.355999	+	0.616608i	−0.631290	+	0.775547i	$0.717474\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.48405	+	2.01152i	−0.165532	+	0.0955702i	−0.580478	−	0.814276i	$-0.697134\pi$
$443$	−3.48405	+	2.01152i	−0.165532	+	0.0955702i	0.414945	+	0.909846i	$0.363801\pi$
$444$		0			0
$445$	24.0104	+	13.8624i	1.13820	+	0.657142i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−10.9179			−0.515246			−0.257623	−	0.966245i	$-0.582939\pi$
$449$	−10.9179			−0.515246			−0.257623	+	0.966245i	$0.582939\pi$
$450$		0			0
$451$		−	19.0819i		−	0.898531i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.322964	+	0.559390i	−0.0151408	+	0.0262246i
$456$		0			0
$457$	−0.815204	−	1.41197i	−0.0381336	−	0.0660494i	0.846329	−	0.532661i	$-0.178808\pi$
$457$	−0.815204	−	1.41197i	−0.0381336	−	0.0660494i	−0.884462	+	0.466612i	$0.845475\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.1812	+	5.87811i	−0.474185	+	0.273771i	−0.717990	−	0.696054i	$-0.754938\pi$
$461$	−10.1812	+	5.87811i	−0.474185	+	0.273771i	0.243805	+	0.969824i	$0.421604\pi$
$462$		0			0
$463$	10.6473	−	18.4416i	0.494820	−	0.857053i	−0.505162	−	0.863024i	$-0.668567\pi$
$463$	10.6473	−	18.4416i	0.494820	−	0.857053i	0.999982	+	0.00597113i	$0.00190068\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	18.9230i		−	0.875650i	−0.899060	−	0.437825i	$-0.855749\pi$
$467$		−	18.9230i		−	0.875650i	0.899060	−	0.437825i	$-0.144251\pi$
$468$		0			0
$469$		−	3.93562i		−	0.181730i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.20110	−	2.08037i	0.0552267	−	0.0956554i
$474$		0			0
$475$	−34.4497	+	19.8896i	−1.58066	+	0.912595i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	14.9759	+	25.9391i	0.684268	+	1.18519i	0.973666	+	0.227978i	$0.0732115\pi$
$479$	14.9759	+	25.9391i	0.684268	+	1.18519i	−0.289398	+	0.957209i	$0.593455\pi$
$480$		0			0
$481$	1.31929	−	2.28508i	0.0601546	−	0.104191i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			21.8714i			0.993131i
$486$		0			0
$487$	17.3370			0.785614			0.392807	−	0.919621i	$-0.371504\pi$
$487$	17.3370			0.785614			0.392807	+	0.919621i	$0.371504\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−31.1204	−	17.9674i	−1.40444	−	0.810856i	−0.409599	−	0.912265i	$-0.634332\pi$
$491$	−31.1204	−	17.9674i	−1.40444	−	0.810856i	−0.994845	+	0.101409i	$0.967665\pi$
$492$		0			0
$493$	8.32446	−	4.80613i	0.374915	−	0.216457i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.717405	−	1.24258i	−0.0321800	−	0.0557374i
$498$		0			0
$499$	10.4956	+	6.05962i	0.469846	+	0.271266i	0.716175	−	0.697920i	$-0.245891\pi$
$499$	10.4956	+	6.05962i	0.469846	+	0.271266i	−0.246329	+	0.969186i	$0.579224\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	32.3442			1.44215			0.721077	−	0.692855i	$-0.243647\pi$
$503$	32.3442			1.44215			0.721077	+	0.692855i	$0.243647\pi$
$504$		0			0
$505$	20.3583			0.905932
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.90310	+	1.67610i	0.128677	+	0.0742919i	0.562957	−	0.826486i	$-0.309664\pi$
$509$	2.90310	+	1.67610i	0.128677	+	0.0742919i	−0.434280	+	0.900778i	$0.642997\pi$
$510$		0			0
$511$	−0.514890	−	0.891816i	−0.0227774	−	0.0394516i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−38.9838	+	22.5073i	−1.71783	+	0.991792i
$516$		0			0
$517$	−11.3936	−	6.57811i	−0.501091	−	0.289305i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21.5651			−0.944786			−0.472393	−	0.881388i	$-0.656610\pi$
$521$	−21.5651			−0.944786			−0.472393	+	0.881388i	$0.656610\pi$
$522$		0			0
$523$		−	24.9549i		−	1.09120i	−0.838046	−	0.545600i	$-0.816302\pi$
$523$		−	24.9549i		−	1.09120i	0.838046	−	0.545600i	$-0.183698\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.41863	+	4.18919i	−0.105357	+	0.182484i
$528$		0			0
$529$	−6.21238	−	10.7602i	−0.270104	−	0.467833i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.78133	−	2.18315i	0.163787	−	0.0945627i
$534$		0			0
$535$	−12.1218	+	20.9955i	−0.524070	+	0.907716i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			13.7562i			0.592523i
$540$		0			0
$541$		−	41.9065i		−	1.80170i	−0.434131	−	0.900850i	$-0.642945\pi$
$541$		−	41.9065i		−	1.80170i	0.434131	−	0.900850i	$-0.357055\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12.8506	−	22.2579i	0.550460	−	0.953424i
$546$		0			0
$547$	−25.7251	+	14.8524i	−1.09993	+	0.635043i	−0.936202	−	0.351462i	$-0.885684\pi$
$547$	−25.7251	+	14.8524i	−1.09993	+	0.635043i	−0.163726	+	0.986506i	$0.552351\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.55797	+	9.62669i	0.236778	+	0.410111i
$552$		0			0
$553$	−2.05737	+	3.56346i	−0.0874881	+	0.151534i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	21.9632i		−	0.930609i	−0.885151	−	0.465305i	$-0.845945\pi$
$557$		−	21.9632i		−	0.930609i	0.885151	−	0.465305i	$-0.154055\pi$
$558$		0			0
$559$	0.549669			0.0232485
$560$		0			0
$561$		0			0
$562$		0			0
$563$	23.8394	+	13.7637i	1.00471	+	0.580069i	0.909638	−	0.415401i	$-0.136359\pi$
$563$	23.8394	+	13.7637i	1.00471	+	0.580069i	0.0950710	+	0.995470i	$0.469692\pi$
$564$		0			0
$565$	−25.9786	+	14.9988i	−1.09293	+	0.631002i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.83572	−	15.3039i	−0.370413	−	0.641574i	0.619216	−	0.785221i	$-0.287450\pi$
$569$	−8.83572	−	15.3039i	−0.370413	−	0.641574i	−0.989629	+	0.143647i	$0.954117\pi$
$570$		0			0
$571$	3.92630	+	2.26685i	0.164311	+	0.0948648i	0.579900	−	0.814687i	$-0.303091\pi$
$571$	3.92630	+	2.26685i	0.164311	+	0.0948648i	−0.415590	+	0.909552i	$0.636425\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−50.1188			−2.09010
$576$		0			0
$577$	23.3909			0.973776			0.486888	−	0.873464i	$-0.338132\pi$
$577$	23.3909			0.973776			0.486888	+	0.873464i	$0.338132\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.10750	−	1.21676i	−0.0874337	−	0.0504799i
$582$		0			0
$583$	−7.66645	−	13.2787i	−0.317512	−	0.549947i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.18259	+	4.14687i	−0.296457	+	0.171160i	−0.640850	−	0.767666i	$-0.721418\pi$
$587$	−7.18259	+	4.14687i	−0.296457	+	0.171160i	0.344393	+	0.938826i	$0.388085\pi$
$588$		0			0
$589$	−4.84453	−	2.79699i	−0.199615	−	0.115248i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	43.5169			1.78702			0.893512	−	0.449039i	$-0.148234\pi$
$593$	43.5169			1.78702			0.893512	+	0.449039i	$0.148234\pi$
$594$		0			0
$595$		−	5.74432i		−	0.235494i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	0.527649	−	0.913915i	0.0215592	−	0.0373416i	−0.855045	−	0.518555i	$-0.826470\pi$
$599$	0.527649	−	0.913915i	0.0215592	−	0.0373416i	0.876604	+	0.481213i	$0.159804\pi$
$600$		0			0
$601$	17.9922	+	31.1633i	0.733915	+	1.27118i	0.955198	+	0.295969i	$0.0956424\pi$
$601$	17.9922	+	31.1633i	0.733915	+	1.27118i	−0.221282	+	0.975210i	$0.571024\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	22.1138	−	12.7674i	0.899056	−	0.519070i
$606$		0			0
$607$	9.81512	−	17.0003i	0.398383	−	0.690020i	−0.595143	−	0.803620i	$-0.702905\pi$
$607$	9.81512	−	17.0003i	0.398383	−	0.690020i	0.993527	+	0.113599i	$0.0362381\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	3.01039i		−	0.121787i
$612$		0			0
$613$		−	0.0630655i		−	0.00254719i	−0.999999	−	0.00127360i	$-0.999595\pi$
$613$		−	0.0630655i		−	0.00254719i	0.999999	−	0.00127360i	$-0.000405399\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.0926	+	26.1412i	−0.607606	+	1.05240i	0.384028	+	0.923322i	$0.374537\pi$
$617$	−15.0926	+	26.1412i	−0.607606	+	1.05240i	−0.991634	+	0.129083i	$0.958797\pi$
$618$		0			0
$619$	38.2663	−	22.0930i	1.53805	−	0.887994i	0.539099	−	0.842243i	$-0.318765\pi$
$619$	38.2663	−	22.0930i	1.53805	−	0.887994i	0.998953	−	0.0457517i	$-0.0145683\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.45250	−	2.51580i	−0.0581932	−	0.100794i
$624$		0			0
$625$	−1.90220	+	3.29471i	−0.0760881	+	0.131788i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			23.4653i			0.935621i
$630$		0			0
$631$	−19.6743			−0.783221			−0.391610	−	0.920131i	$-0.628082\pi$
$631$	−19.6743			−0.783221			−0.391610	+	0.920131i	$0.628082\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	66.6607	+	38.4866i	2.64535	+	1.52729i
$636$		0			0
$637$	−2.72598	+	1.57384i	−0.108007	+	0.0623580i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.994371	+	1.72230i	0.0392753	+	0.0680268i	0.884995	−	0.465601i	$-0.154162\pi$
$641$	0.994371	+	1.72230i	0.0392753	+	0.0680268i	−0.845720	+	0.533628i	$0.820828\pi$
$642$		0			0
$643$	12.9370	+	7.46917i	0.510185	+	0.294555i	0.732910	−	0.680326i	$-0.238162\pi$
$643$	12.9370	+	7.46917i	0.510185	+	0.294555i	−0.222725	+	0.974881i	$0.571495\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.81283			−0.346468			−0.173234	−	0.984881i	$-0.555422\pi$
$647$	−8.81283			−0.346468			−0.173234	+	0.984881i	$0.555422\pi$
$648$		0			0
$649$	−1.39937			−0.0549302
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−18.1802	−	10.4964i	−0.711447	−	0.410754i	0.100150	−	0.994972i	$-0.468068\pi$
$653$	−18.1802	−	10.4964i	−0.711447	−	0.410754i	−0.811597	+	0.584218i	$0.801401\pi$
$654$		0			0
$655$	−11.6126	−	20.1136i	−0.453742	−	0.785904i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	40.6209	−	23.4525i	1.58237	−	0.913580i	0.587853	−	0.808968i	$-0.299973\pi$
$659$	40.6209	−	23.4525i	1.58237	−	0.913580i	0.994513	−	0.104612i	$-0.0333601\pi$
$660$		0			0
$661$	−0.736985	−	0.425498i	−0.0286654	−	0.0165500i	0.485599	−	0.874182i	$-0.338602\pi$
$661$	−0.736985	−	0.425498i	−0.0286654	−	0.0165500i	−0.514264	+	0.857632i	$0.671935\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.64293			0.257602
$666$		0			0
$667$			14.0053i			0.542287i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.91150	+	8.50697i	−0.189606	+	0.328408i
$672$		0			0
$673$	−22.4873	−	38.9491i	−0.866820	−	1.50138i	−0.865229	−	0.501378i	$-0.832827\pi$
$673$	−22.4873	−	38.9491i	−0.866820	−	1.50138i	−0.00159139	−	0.999999i	$-0.500507\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.29713	−	4.21300i	0.280451	−	0.161919i	−0.353176	−	0.935557i	$-0.614898\pi$
$677$	7.29713	−	4.21300i	0.280451	−	0.161919i	0.633628	+	0.773638i	$0.281565\pi$
$678$		0			0
$679$	1.14584	−	1.98466i	0.0439733	−	0.0761641i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			23.3602i			0.893853i	0.894571	+	0.446926i	$0.147481\pi$
$683$			23.3602i			0.893853i	−0.894571	+	0.446926i	$0.852519\pi$
$684$		0			0
$685$			3.54396i			0.135408i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.75423	−	3.03841i	0.0668308	−	0.115754i
$690$		0			0
$691$	−25.9304	+	14.9709i	−0.986439	+	0.569521i	−0.904208	−	0.427092i	$-0.859538\pi$
$691$	−25.9304	+	14.9709i	−0.986439	+	0.569521i	−0.0822311	+	0.996613i	$0.526205\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.9596	−	18.9826i	−0.415722	−	0.720052i
$696$		0			0
$697$	−19.4150	+	33.6278i	−0.735396	+	1.27374i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			0.718775i			0.0271478i	0.999908	+	0.0135739i	$0.00432083\pi$
$701$			0.718775i			0.0271478i	−0.999908	+	0.0135739i	$0.995679\pi$
$702$		0			0
$703$	−27.1360			−1.02346
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.84735	−	1.06657i	−0.0694767	−	0.0401124i
$708$		0			0
$709$	10.9918	−	6.34613i	0.412807	−	0.238334i	−0.279188	−	0.960236i	$-0.590065\pi$
$709$	10.9918	−	6.34613i	0.412807	−	0.238334i	0.691995	+	0.721902i	$0.256732\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.52400	−	6.10375i	−0.131975	−	0.228587i
$714$		0			0
$715$	−2.92544	−	1.68900i	−0.109405	−	0.0631652i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7.74226			0.288738			0.144369	−	0.989524i	$-0.453885\pi$
$719$	7.74226			0.288738			0.144369	+	0.989524i	$0.453885\pi$
$720$		0			0
$721$	4.71662			0.175656
$722$		0			0
$723$		0			0
$724$		0			0
$725$	17.1599	+	9.90730i	0.637304	+	0.367948i
$726$		0			0
$727$	−9.91355	−	17.1708i	−0.367673	−	0.636828i	0.621528	−	0.783392i	$-0.286512\pi$
$727$	−9.91355	−	17.1708i	−0.367673	−	0.636828i	−0.989201	+	0.146563i	$0.953179\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.23337	+	2.44414i	−0.156577	+	0.0903997i
$732$		0			0
$733$	32.7535	+	18.9102i	1.20978	+	0.698466i	0.962711	−	0.270532i	$-0.0871997\pi$
$733$	32.7535	+	18.9102i	1.20978	+	0.698466i	0.247068	+	0.968998i	$0.420533\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	20.5821			0.758152
$738$		0			0
$739$		−	6.61431i		−	0.243311i	−0.992572	−	0.121656i	$-0.961180\pi$
$739$		−	6.61431i		−	0.243311i	0.992572	−	0.121656i	$-0.0388204\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.30245	+	5.72000i	−0.121155	+	0.209847i	−0.920223	−	0.391394i	$-0.871993\pi$
$743$	−3.30245	+	5.72000i	−0.121155	+	0.209847i	0.799068	+	0.601240i	$0.205326\pi$
$744$		0			0
$745$	−13.6161	−	23.5838i	−0.498856	−	0.864044i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.19990	−	1.27012i	0.0803828	−	0.0464090i
$750$		0			0
$751$	−12.8675	+	22.2871i	−0.469540	+	0.813268i	−0.999394	−	0.0348217i	$-0.988914\pi$
$751$	−12.8675	+	22.2871i	−0.469540	+	0.813268i	0.529853	+	0.848089i	$0.322247\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	3.61118i		−	0.131424i
$756$		0			0
$757$		−	48.0424i		−	1.74613i	−0.487602	−	0.873066i	$-0.662128\pi$
$757$		−	48.0424i		−	1.74613i	0.487602	−	0.873066i	$-0.337872\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.20688	−	2.09038i	0.0437493	−	0.0757761i	−0.843322	−	0.537409i	$-0.819403\pi$
$761$	1.20688	−	2.09038i	0.0437493	−	0.0757761i	0.887071	+	0.461633i	$0.152736\pi$
$762$		0			0
$763$	−2.33218	+	1.34648i	−0.0844305	+	0.0487459i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.160101	−	0.277304i	−0.00578093	−	0.0100129i
$768$		0			0
$769$	16.8464	−	29.1788i	0.607496	−	1.05221i	−0.384156	−	0.923268i	$-0.625507\pi$
$769$	16.8464	−	29.1788i	0.607496	−	1.05221i	0.991652	−	0.128945i	$-0.0411592\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	18.8545i		−	0.678149i	−0.940759	−	0.339075i	$-0.889886\pi$
$773$		−	18.8545i		−	0.678149i	0.940759	−	0.339075i	$-0.110114\pi$
$774$		0			0
$775$	−9.97148			−0.358186
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−38.8884	−	22.4522i	−1.39332	−	0.804434i
$780$		0			0
$781$	6.49833	−	3.75181i	0.232529	−	0.134250i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−32.2202	−	55.8070i	−1.14999	−	1.99184i
$786$		0			0
$787$	14.8124	+	8.55193i	0.528004	+	0.304843i	0.740203	−	0.672383i	$-0.234729\pi$
$787$	14.8124	+	8.55193i	0.528004	+	0.304843i	−0.212199	+	0.977226i	$0.568063\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.14313			0.111757
$792$		0			0
$793$	−2.24769			−0.0798178
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.11800	−	1.80018i	−0.110445	−	0.0637656i	0.443760	−	0.896146i	$-0.353644\pi$
$797$	−3.11800	−	1.80018i	−0.110445	−	0.0637656i	−0.554205	+	0.832380i	$0.686978\pi$
$798$		0			0
$799$	13.3859	+	23.1851i	0.473559	+	0.820228i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.66393	−	2.69272i	0.164586	−	0.0950240i
$804$		0			0
$805$	7.24829	+	4.18480i	0.255469	+	0.147495i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13.5918			−0.477864			−0.238932	−	0.971036i	$-0.576797\pi$
$809$	−13.5918			−0.477864			−0.238932	+	0.971036i	$0.576797\pi$
$810$		0			0
$811$		−	10.0627i		−	0.353349i	−0.984269	−	0.176674i	$-0.943466\pi$
$811$		−	10.0627i		−	0.353349i	0.984269	−	0.176674i	$-0.0565339\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	32.7606	−	56.7430i	1.14755	−	1.98762i
$816$		0			0
$817$	−2.82649	−	4.89562i	−0.0988862	−	0.171276i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	33.4098	−	19.2891i	1.16601	−	0.673196i	0.213273	−	0.976993i	$-0.431588\pi$
$821$	33.4098	−	19.2891i	1.16601	−	0.673196i	0.952737	+	0.303797i	$0.0982544\pi$
$822$		0			0
$823$	16.1120	−	27.9068i	0.561628	−	0.972768i	−0.435727	−	0.900079i	$-0.643509\pi$
$823$	16.1120	−	27.9068i	0.561628	−	0.972768i	0.997355	−	0.0726892i	$-0.0231581\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	27.7190i		−	0.963886i	−0.876203	−	0.481943i	$-0.839931\pi$
$827$		−	27.7190i		−	0.963886i	0.876203	−	0.481943i	$-0.160069\pi$
$828$		0			0
$829$			10.3137i			0.358208i	0.983830	+	0.179104i	$0.0573199\pi$
$829$			10.3137i			0.358208i	−0.983830	+	0.179104i	$0.942680\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	13.9964	−	24.2425i	0.484946	−	0.839951i
$834$		0			0
$835$	6.60139	−	3.81131i	0.228451	−	0.131896i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−19.1961	−	33.2487i	−0.662724	−	1.14787i	−0.979897	−	0.199504i	$-0.936067\pi$
$839$	−19.1961	−	33.2487i	−0.662724	−	1.14787i	0.317173	−	0.948368i	$-0.397267\pi$
$840$		0			0
$841$	−11.7315	+	20.3195i	−0.404534	+	0.700673i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			46.8516i			1.61174i
$846$		0			0
$847$	−2.67553			−0.0919325
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−29.6089	−	17.0947i	−1.01498	−	0.585999i
$852$		0			0
$853$	24.4212	−	14.0996i	0.836166	−	0.482761i	−0.0197931	−	0.999804i	$-0.506301\pi$
$853$	24.4212	−	14.0996i	0.836166	−	0.482761i	0.855959	+	0.517043i	$0.172967\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−15.7011	−	27.1951i	−0.536339	−	0.928966i	−0.999097	−	0.0424814i	$-0.986474\pi$
$857$	−15.7011	−	27.1951i	−0.536339	−	0.928966i	0.462759	−	0.886484i	$-0.346860\pi$
$858$		0			0
$859$	−41.6368	−	24.0390i	−1.42063	−	0.820201i	−0.424278	−	0.905532i	$-0.639472\pi$
$859$	−41.6368	−	24.0390i	−1.42063	−	0.820201i	−0.996353	+	0.0853309i	$0.972805\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	37.6968			1.28321			0.641607	−	0.767034i	$-0.278268\pi$
$863$	37.6968			1.28321			0.641607	+	0.767034i	$0.278268\pi$
$864$		0			0
$865$	−69.6783			−2.36913
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−18.6358	−	10.7594i	−0.632177	−	0.364988i
$870$		0			0
$871$	2.35479	+	4.07861i	0.0797890	+	0.138199i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.16576	−	2.40510i	0.140828	−	0.0813072i
$876$		0			0
$877$	−21.6823	−	12.5183i	−0.732158	−	0.422711i	0.0870533	−	0.996204i	$-0.472255\pi$
$877$	−21.6823	−	12.5183i	−0.732158	−	0.422711i	−0.819211	+	0.573492i	$0.805588\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	21.0382			0.708795			0.354398	−	0.935095i	$-0.384686\pi$
$881$	21.0382			0.708795			0.354398	+	0.935095i	$0.384686\pi$
$882$		0			0
$883$		−	55.5904i		−	1.87077i	−0.353636	−	0.935383i	$-0.615055\pi$
$883$		−	55.5904i		−	1.87077i	0.353636	−	0.935383i	$-0.384945\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.695385	−	1.20444i	0.0233487	−	0.0404412i	−0.854115	−	0.520084i	$-0.825901\pi$
$887$	0.695385	−	1.20444i	0.0233487	−	0.0404412i	0.877464	+	0.479643i	$0.159234\pi$
$888$		0			0
$889$	−4.03261	−	6.98468i	−0.135249	−	0.234259i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−26.8120	+	15.4799i	−0.897229	+	0.518015i
$894$		0			0
$895$	−3.88527	+	6.72948i	−0.129870	+	0.224942i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			2.78645i			0.0929332i
$900$		0			0
$901$			31.2011i			1.03946i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.04608	+	5.27597i	−0.101255	+	0.175379i
$906$		0			0
$907$	37.3197	−	21.5465i	1.23918	−	0.715440i	0.270252	−	0.962789i	$-0.412893\pi$
$907$	37.3197	−	21.5465i	1.23918	−	0.715440i	0.968926	+	0.247349i	$0.0795595\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.96463	−	6.86694i	−0.131354	−	0.227512i	0.792845	−	0.609424i	$-0.208599\pi$
$911$	−3.96463	−	6.86694i	−0.131354	−	0.227512i	−0.924199	+	0.381912i	$0.875266\pi$
$912$		0			0
$913$	6.36331	−	11.0216i	0.210595	−	0.364761i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			2.43353i			0.0803622i
$918$		0			0
$919$	−1.66862			−0.0550426			−0.0275213	−	0.999621i	$-0.508761\pi$
$919$	−1.66862			−0.0550426			−0.0275213	+	0.999621i	$0.508761\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.48694	+	0.858486i	0.0489433	+	0.0282574i
$924$		0			0
$925$	−41.8905	+	24.1855i	−1.37735	+	0.795215i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23.3838	+	40.5020i	0.767199	+	1.32883i	0.939076	+	0.343709i	$0.111683\pi$
$929$	23.3838	+	40.5020i	0.767199	+	1.32883i	−0.171878	+	0.985118i	$0.554983\pi$
$930$		0			0
$931$	28.0348	+	16.1859i	0.918804	+	0.530472i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	30.0410			0.982447
$936$		0			0
$937$	−9.30185			−0.303878			−0.151939	−	0.988390i	$-0.548552\pi$
$937$	−9.30185			−0.303878			−0.151939	+	0.988390i	$0.548552\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.42009	+	3.12929i	0.176690	+	0.102012i	0.585736	−	0.810502i	$-0.300805\pi$
$941$	5.42009	+	3.12929i	0.176690	+	0.102012i	−0.409047	+	0.912513i	$0.634139\pi$
$942$		0			0
$943$	−28.2881	−	48.9965i	−0.921188	−	1.59555i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−38.4243	+	22.1843i	−1.24862	+	0.720893i	−0.970835	−	0.239749i	$-0.922935\pi$
$947$	−38.4243	+	22.1843i	−1.24862	+	0.720893i	−0.277789	+	0.960642i	$0.589601\pi$
$948$		0			0
$949$	1.06720	+	0.616146i	0.0346426	+	0.0200009i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−6.11599			−0.198116			−0.0990582	−	0.995082i	$-0.531583\pi$
$953$	−6.11599			−0.198116			−0.0990582	+	0.995082i	$0.531583\pi$
$954$		0			0
$955$			63.7257i			2.06211i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.185668	−	0.321586i	0.00599552	−	0.0103845i
$960$		0			0
$961$	14.7989	+	25.6324i	0.477383	+	0.826852i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.99178	−	5.19141i	0.289456	−	0.167117i
$966$		0			0
$967$	−16.9985	+	29.4423i	−0.546635	+	0.946799i	0.451867	+	0.892085i	$0.350758\pi$
$967$	−16.9985	+	29.4423i	−0.546635	+	0.946799i	−0.998502	+	0.0547141i	$0.982575\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	42.4798i		−	1.36324i	−0.731706	−	0.681621i	$-0.761275\pi$
$971$		−	42.4798i		−	1.36324i	0.731706	−	0.681621i	$-0.238725\pi$
$972$		0			0
$973$			2.29669i			0.0736286i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−7.66569	+	13.2774i	−0.245247	+	0.424781i	−0.962201	−	0.272340i	$-0.912202\pi$
$977$	−7.66569	+	13.2774i	−0.245247	+	0.424781i	0.716954	+	0.697121i	$0.245536\pi$
$978$		0			0
$979$	13.1569	−	7.59614i	0.420496	−	0.242774i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−14.8653	−	25.7474i	−0.474129	−	0.821216i	0.525432	−	0.850836i	$-0.323904\pi$
$983$	−14.8653	−	25.7474i	−0.474129	−	0.821216i	−0.999561	+	0.0296198i	$0.990570\pi$
$984$		0			0
$985$	23.0779	−	39.9720i	0.735322	−	1.27361i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	7.12233i		−	0.226477i
$990$		0			0
$991$	49.5495			1.57399			0.786996	−	0.616958i	$-0.211635\pi$
$991$	49.5495			1.57399			0.786996	+	0.616958i	$0.211635\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	54.8017	+	31.6398i	1.73733	+	1.00305i
$996$		0			0
$997$	34.4541	−	19.8921i	1.09117	−	0.629988i	0.157284	−	0.987553i	$-0.449726\pi$
$997$	34.4541	−	19.8921i	1.09117	−	0.629988i	0.933888	+	0.357565i	$0.116393\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.r.b.145.1		16
3.2	odd	2		288.2.r.b.49.4		16
4.3	odd	2		216.2.n.b.37.6		16
8.3	odd	2		216.2.n.b.37.1		16
8.5	even	2	inner	864.2.r.b.145.8		16
9.2	odd	6		288.2.r.b.241.5		16
9.4	even	3		2592.2.d.k.1297.8		8
9.5	odd	6		2592.2.d.j.1297.1		8
9.7	even	3	inner	864.2.r.b.721.8		16
12.11	even	2		72.2.n.b.13.3	✓	16
24.5	odd	2		288.2.r.b.49.5		16
24.11	even	2		72.2.n.b.13.8	yes	16
36.7	odd	6		216.2.n.b.181.1		16
36.11	even	6		72.2.n.b.61.8	yes	16
36.23	even	6		648.2.d.j.325.3		8
36.31	odd	6		648.2.d.k.325.6		8
72.5	odd	6		2592.2.d.j.1297.8		8
72.11	even	6		72.2.n.b.61.3	yes	16
72.13	even	6		2592.2.d.k.1297.1		8
72.29	odd	6		288.2.r.b.241.4		16
72.43	odd	6		216.2.n.b.181.6		16
72.59	even	6		648.2.d.j.325.4		8
72.61	even	6	inner	864.2.r.b.721.1		16
72.67	odd	6		648.2.d.k.325.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.n.b.13.3	✓	16	12.11	even	2
72.2.n.b.13.8	yes	16	24.11	even	2
72.2.n.b.61.3	yes	16	72.11	even	6
72.2.n.b.61.8	yes	16	36.11	even	6
216.2.n.b.37.1		16	8.3	odd	2
216.2.n.b.37.6		16	4.3	odd	2
216.2.n.b.181.1		16	36.7	odd	6
216.2.n.b.181.6		16	72.43	odd	6
288.2.r.b.49.4		16	3.2	odd	2
288.2.r.b.49.5		16	24.5	odd	2
288.2.r.b.241.4		16	72.29	odd	6
288.2.r.b.241.5		16	9.2	odd	6
648.2.d.j.325.3		8	36.23	even	6
648.2.d.j.325.4		8	72.59	even	6
648.2.d.k.325.5		8	72.67	odd	6
648.2.d.k.325.6		8	36.31	odd	6
864.2.r.b.145.1		16	1.1	even	1	trivial
864.2.r.b.145.8		16	8.5	even	2	inner
864.2.r.b.721.1		16	72.61	even	6	inner
864.2.r.b.721.8		16	9.7	even	3	inner
2592.2.d.j.1297.1		8	9.5	odd	6
2592.2.d.j.1297.8		8	72.5	odd	6
2592.2.d.k.1297.1		8	72.13	even	6
2592.2.d.k.1297.8		8	9.4	even	3

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	864.2.r.b.145.1

Level	$864$
Weight	$2$
Character	864.145
Analytic conductor	$6.899$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$