LMFDB - Embedded newform 252.5.z.f.145.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [252,5,Mod(73,252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("252.73"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$N$	$=$	$252 = 2^{2} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	252.z (of order $6$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,27,0,66] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$26.0492306971$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.11337408.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} + 18x^{4} + 81x^{2} + 12$ x^6 + 18x^4 + 81x^2 + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{2}\cdot 3^{3}\cdot 7^{2}$
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.1
Root			$0.391571i$ of defining polynomial
Character	$\chi$	$=$	252.145
Dual form			252.5.z.f.73.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-38.3327 - 22.1314i) q^{5} +(42.3967 - 24.5667i) q^{7} +(11.7557 + 20.3615i) q^{11} -136.269i q^{13} +(227.333 - 131.251i) q^{17} +(-387.162 - 223.528i) q^{19} +(-374.585 + 648.800i) q^{23} +(667.098 + 1155.45i) q^{25} -406.524 q^{29} +(-584.199 + 337.287i) q^{31} +(-2168.87 + 3.40909i) q^{35} +(-372.666 + 645.476i) q^{37} +2476.20i q^{41} -2636.68 q^{43} +(579.099 + 334.343i) q^{47} +(1193.96 - 2083.09i) q^{49} +(-1014.61 - 1757.36i) q^{53} -1040.68i q^{55} +(1014.22 - 585.561i) q^{59} +(-2022.05 - 1167.43i) q^{61} +(-3015.83 + 5223.58i) q^{65} +(3779.79 + 6546.78i) q^{67} -7575.36 q^{71} +(3284.18 - 1896.12i) q^{73} +(998.615 + 574.460i) q^{77} +(-3897.33 + 6750.37i) q^{79} +2181.41i q^{83} -11619.0 q^{85} +(-8050.38 - 4647.89i) q^{89} +(-3347.69 - 5777.37i) q^{91} +(9893.98 + 17136.9i) q^{95} +9981.90i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 27 q^{5} + 66 q^{7} - 135 q^{11} + 1107 q^{17} - 747 q^{19} - 243 q^{23} + 1878 q^{25} + 540 q^{29} - 5355 q^{31} - 6021 q^{35} + 2355 q^{37} - 948 q^{43} + 9747 q^{47} + 8430 q^{49} - 6291 q^{53}+ \cdots + 20655 q^{95}+O(q^{100})$ 6 * q + 27 * q^5 + 66 * q^7 - 135 * q^11 + 1107 * q^17 - 747 * q^19 - 243 * q^23 + 1878 * q^25 + 540 * q^29 - 5355 * q^31 - 6021 * q^35 + 2355 * q^37 - 948 * q^43 + 9747 * q^47 + 8430 * q^49 - 6291 * q^53 + 2943 * q^59 + 4041 * q^61 - 7668 * q^65 + 1659 * q^67 - 2268 * q^71 - 8703 * q^73 + 10665 * q^77 - 7773 * q^79 - 702 * q^85 - 14985 * q^89 - 20952 * q^91 + 20655 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$ .

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$

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
$5$	−38.3327	−	22.1314i	−1.53331	−	0.885256i	−0.999206	−	0.0398360i	$-0.987316\pi$
$5$	−38.3327	−	22.1314i	−1.53331	−	0.885256i	−0.534102	−	0.845420i	$-0.679350\pi$
$6$		0			0
$7$	42.3967	−	24.5667i	0.865238	−	0.501361i
$7$	42.3967	−	24.5667i	0.865238	−	0.501361i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	11.7557	+	20.3615i	0.0971545	+	0.168276i	0.910506	−	0.413496i	$-0.135693\pi$
$11$	11.7557	+	20.3615i	0.0971545	+	0.168276i	−0.813351	+	0.581773i	$0.802359\pi$
$12$		0			0
$13$		−	136.269i		−	0.806328i	−0.915128	−	0.403164i	$-0.867910\pi$
$13$		−	136.269i		−	0.806328i	0.915128	−	0.403164i	$-0.132090\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	227.333	−	131.251i	0.786618	−	0.454154i	−0.0521523	−	0.998639i	$-0.516608\pi$
$17$	227.333	−	131.251i	0.786618	−	0.454154i	0.838771	+	0.544485i	$0.183275\pi$
$18$		0			0
$19$	−387.162	−	223.528i	−1.07247	−	0.619191i	−0.143615	−	0.989634i	$-0.545873\pi$
$19$	−387.162	−	223.528i	−1.07247	−	0.619191i	−0.928855	+	0.370442i	$0.879206\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−374.585	+	648.800i	−0.708100	+	1.22646i	0.257462	+	0.966288i	$0.417114\pi$
$23$	−374.585	+	648.800i	−0.708100	+	1.22646i	−0.965561	+	0.260176i	$0.916219\pi$
$24$		0			0
$25$	667.098	+	1155.45i	1.06736	+	1.84872i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−406.524			−0.483382			−0.241691	−	0.970353i	$-0.577702\pi$
$29$	−406.524			−0.483382			−0.241691	+	0.970353i	$0.577702\pi$
$30$		0			0
$31$	−584.199	+	337.287i	−0.607907	+	0.350975i	−0.772146	−	0.635445i	$-0.780817\pi$
$31$	−584.199	+	337.287i	−0.607907	+	0.350975i	0.164239	+	0.986421i	$0.447483\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2168.87	+	3.40909i	−1.77051	+	0.00278293i
$36$		0			0
$37$	−372.666	+	645.476i	−0.272218	+	0.471495i	−0.969429	−	0.245370i	$-0.921090\pi$
$37$	−372.666	+	645.476i	−0.272218	+	0.471495i	0.697212	+	0.716865i	$0.254424\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2476.20i			1.47305i	0.676410	+	0.736526i	$0.263535\pi$
$41$			2476.20i			1.47305i	−0.676410	+	0.736526i	$0.736465\pi$
$42$		0			0
$43$	−2636.68			−1.42601			−0.713003	−	0.701161i	$-0.752665\pi$
$43$	−2636.68			−1.42601			−0.713003	+	0.701161i	$0.752665\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	579.099	+	334.343i	0.262154	+	0.151355i	0.625317	−	0.780371i	$-0.284970\pi$
$47$	579.099	+	334.343i	0.262154	+	0.151355i	−0.363163	+	0.931726i	$0.618303\pi$
$48$		0			0
$49$	1193.96	−	2083.09i	0.497275	−	0.867593i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1014.61	−	1757.36i	−0.361200	−	0.625617i	0.626959	−	0.779053i	$-0.284300\pi$
$53$	−1014.61	−	1757.36i	−0.361200	−	0.625617i	−0.988159	+	0.153436i	$0.950966\pi$
$54$		0			0
$55$		−	1040.68i		−	0.344026i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1014.22	−	585.561i	0.291359	−	0.168216i	−0.347195	−	0.937793i	$-0.612866\pi$
$59$	1014.22	−	585.561i	0.291359	−	0.168216i	0.638555	+	0.769576i	$0.279533\pi$
$60$		0			0
$61$	−2022.05	−	1167.43i	−0.543416	−	0.313741i	0.203047	−	0.979169i	$-0.434916\pi$
$61$	−2022.05	−	1167.43i	−0.543416	−	0.313741i	−0.746462	+	0.665428i	$0.768249\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3015.83	+	5223.58i	−0.713807	+	1.23635i
$66$		0			0
$67$	3779.79	+	6546.78i	0.842011	+	1.45841i	0.888192	+	0.459472i	$0.151961\pi$
$67$	3779.79	+	6546.78i	0.842011	+	1.45841i	−0.0461815	+	0.998933i	$0.514705\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7575.36			−1.50275			−0.751375	−	0.659876i	$-0.770609\pi$
$71$	−7575.36			−1.50275			−0.751375	+	0.659876i	$0.770609\pi$
$72$		0			0
$73$	3284.18	−	1896.12i	0.616284	−	0.355812i	−0.159137	−	0.987257i	$-0.550871\pi$
$73$	3284.18	−	1896.12i	0.616284	−	0.355812i	0.775421	+	0.631445i	$0.217538\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	998.615	+	574.460i	0.168429	+	0.0968898i
$78$		0			0
$79$	−3897.33	+	6750.37i	−0.624472	+	1.08162i	0.364171	+	0.931332i	$0.381353\pi$
$79$	−3897.33	+	6750.37i	−0.624472	+	1.08162i	−0.988643	+	0.150285i	$0.951981\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			2181.41i			0.316651i	0.987387	+	0.158326i	$0.0506096\pi$
$83$			2181.41i			0.316651i	−0.987387	+	0.158326i	$0.949390\pi$
$84$		0			0
$85$	−11619.0			−1.60817
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8050.38	−	4647.89i	−1.01633	−	0.586781i	−0.103294	−	0.994651i	$-0.532938\pi$
$89$	−8050.38	−	4647.89i	−1.01633	−	0.586781i	−0.913040	+	0.407870i	$0.866272\pi$
$90$		0			0
$91$	−3347.69	−	5777.37i	−0.404261	−	0.697666i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9893.98	+	17136.9i	1.09629	+	1.89882i
$96$		0			0
$97$			9981.90i			1.06089i	0.847720	+	0.530444i	$0.177975\pi$
$97$			9981.90i			1.06089i	−0.847720	+	0.530444i	$0.822025\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	675.552	−	390.030i	0.0662241	−	0.0382345i	−0.466522	−	0.884509i	$-0.654493\pi$
$101$	675.552	−	390.030i	0.0662241	−	0.0382345i	0.532746	+	0.846275i	$0.321160\pi$
$102$		0			0
$103$	2383.90	+	1376.35i	0.224706	+	0.129734i	0.608127	−	0.793840i	$-0.291921\pi$
$103$	2383.90	+	1376.35i	0.224706	+	0.129734i	−0.383422	+	0.923573i	$0.625254\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7429.46	−	12868.2i	0.648917	−	1.12396i	−0.334464	−	0.942408i	$-0.608555\pi$
$107$	7429.46	−	12868.2i	0.648917	−	1.12396i	0.983382	−	0.181550i	$-0.0581113\pi$
$108$		0			0
$109$	−1378.33	−	2387.33i	−0.116011	−	0.200937i	0.802172	−	0.597092i	$-0.203677\pi$
$109$	−1378.33	−	2387.33i	−0.116011	−	0.200937i	−0.918183	+	0.396155i	$0.870344\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6264.79			−0.490625			−0.245312	−	0.969444i	$-0.578891\pi$
$113$	−6264.79			−0.490625			−0.245312	+	0.969444i	$0.578891\pi$
$114$		0			0
$115$	28717.7	−	16580.2i	2.17147	−	1.25370i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6413.76	−	11149.4i	0.452917	−	0.787331i
$120$		0			0
$121$	7044.11	−	12200.8i	0.481122	−	0.833328i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	31391.0i		−	2.00902i
$126$		0			0
$127$	12855.0			0.797010			0.398505	−	0.917166i	$-0.369529\pi$
$127$	12855.0			0.797010			0.398505	+	0.917166i	$0.369529\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	21412.5	+	12362.5i	1.24774	+	0.720383i	0.970658	−	0.240464i	$-0.0772995\pi$
$131$	21412.5	+	12362.5i	1.24774	+	0.720383i	0.277082	+	0.960846i	$0.410633\pi$
$132$		0			0
$133$	−21905.7	+	34.4320i	−1.23838	+	0.00194652i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	661.078	+	1145.02i	0.0352218	+	0.0610059i	0.883099	−	0.469187i	$-0.155453\pi$
$137$	661.078	+	1145.02i	0.0352218	+	0.0610059i	−0.847877	+	0.530193i	$0.822120\pi$
$138$		0			0
$139$		−	5119.78i		−	0.264985i	−0.991184	−	0.132493i	$-0.957702\pi$
$139$		−	5119.78i		−	0.264985i	0.991184	−	0.132493i	$-0.0422981\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2774.64	−	1601.94i	0.135686	−	0.0783384i
$144$		0			0
$145$	15583.2	+	8996.95i	0.741173	+	0.427917i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5104.93	−	8842.00i	0.229942	−	0.398270i	−0.727849	−	0.685737i	$-0.759480\pi$
$149$	5104.93	−	8842.00i	0.229942	−	0.398270i	0.957791	+	0.287467i	$0.0928132\pi$
$150$		0			0
$151$	−14672.1	−	25412.8i	−0.643483	−	1.11455i	−0.984650	−	0.174543i	$-0.944155\pi$
$151$	−14672.1	−	25412.8i	−0.643483	−	1.11455i	0.341166	−	0.940003i	$-0.389178\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	29858.6			1.24281
$156$		0			0
$157$	−26033.1	+	15030.2i	−1.05615	+	0.609770i	−0.924366	−	0.381508i	$-0.875405\pi$
$157$	−26033.1	+	15030.2i	−1.05615	+	0.609770i	−0.131787	+	0.991278i	$0.542072\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	57.7005	+	36709.3i	0.00222602	+	1.41620i
$162$		0			0
$163$	−15439.7	+	26742.3i	−0.581115	+	1.00652i	0.414232	+	0.910171i	$0.364050\pi$
$163$	−15439.7	+	26742.3i	−0.581115	+	1.00652i	−0.995348	+	0.0963500i	$0.969283\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	12858.8i		−	0.461070i	−0.973064	−	0.230535i	$-0.925952\pi$
$167$		−	12858.8i		−	0.461070i	0.973064	−	0.230535i	$-0.0740477\pi$
$168$		0			0
$169$	9991.63			0.349835
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−13588.8	−	7845.51i	−0.454035	−	0.262137i	0.255498	−	0.966810i	$-0.417761\pi$
$173$	−13588.8	−	7845.51i	−0.454035	−	0.262137i	−0.709533	+	0.704672i	$0.751094\pi$
$174$		0			0
$175$	56668.2	+	32598.8i	1.85039	+	1.06445i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15993.2	−	27701.0i	−0.499148	−	0.864550i	0.500851	−	0.865533i	$-0.333020\pi$
$179$	−15993.2	−	27701.0i	−0.499148	−	0.864550i	−1.00000		0.000983488i	$0.999687\pi$
$180$		0			0
$181$		−	47967.6i		−	1.46417i	−0.681215	−	0.732083i	$-0.738548\pi$
$181$		−	47967.6i		−	1.46417i	0.681215	−	0.732083i	$-0.261452\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	28570.6	−	16495.2i	0.834787	−	0.481965i
$186$		0			0
$187$	5344.91	+	3085.88i	0.152847	+	0.0882462i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2227.93	−	3858.88i	0.0610709	−	0.105778i	−0.833873	−	0.551956i	$-0.813882\pi$
$191$	2227.93	−	3858.88i	0.0610709	−	0.105778i	0.894944	+	0.446178i	$0.147215\pi$
$192$		0			0
$193$	14699.9	+	25460.9i	0.394638	+	0.683533i	0.993055	−	0.117652i	$-0.0375366\pi$
$193$	14699.9	+	25460.9i	0.394638	+	0.683533i	−0.598417	+	0.801185i	$0.704203\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−23816.0			−0.613671			−0.306836	−	0.951763i	$-0.599270\pi$
$197$	−23816.0			−0.613671			−0.306836	+	0.951763i	$0.599270\pi$
$198$		0			0
$199$	−42280.9	+	24410.9i	−1.06767	+	0.616421i	−0.927545	−	0.373713i	$-0.878085\pi$
$199$	−42280.9	+	24410.9i	−1.06767	+	0.616421i	−0.140128	+	0.990133i	$0.544751\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−17235.3	+	9986.94i	−0.418241	+	0.242349i
$204$		0			0
$205$	54801.8	−	94919.4i	1.30403	−	2.25864i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	10510.9i		−	0.240629i
$210$		0			0
$211$	−9582.07			−0.215226			−0.107613	−	0.994193i	$-0.534321\pi$
$211$	−9582.07			−0.215226			−0.107613	+	0.994193i	$0.534321\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	101071.	+	58353.5i	2.18651	+	1.26238i
$216$		0			0
$217$	−16482.1	+	28651.7i	−0.350019	+	0.608458i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−17885.5	−	30978.5i	−0.366197	−	0.634273i
$222$		0			0
$223$		−	54978.9i		−	1.10557i	−0.833324	−	0.552785i	$-0.813565\pi$
$223$		−	54978.9i		−	1.10557i	0.833324	−	0.552785i	$-0.186435\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7518.24	−	4340.66i	0.145903	−	0.0842372i	−0.425271	−	0.905066i	$-0.639821\pi$
$227$	7518.24	−	4340.66i	0.145903	−	0.0842372i	0.571174	+	0.820829i	$0.306488\pi$
$228$		0			0
$229$	−6262.86	−	3615.86i	−0.119427	−	0.0689511i	0.439097	−	0.898440i	$-0.355299\pi$
$229$	−6262.86	−	3615.86i	−0.119427	−	0.0689511i	−0.558523	+	0.829489i	$0.688632\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4422.96	−	7660.79i	0.0814707	−	0.141111i	−0.822411	−	0.568894i	$-0.807372\pi$
$233$	4422.96	−	7660.79i	0.0814707	−	0.141111i	0.903882	+	0.427782i	$0.140705\pi$
$234$		0			0
$235$	−14799.0	−	25632.5i	−0.267976	−	0.464147i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−88491.7			−1.54920			−0.774599	−	0.632453i	$-0.782048\pi$
$239$	−88491.7			−1.54920			−0.774599	+	0.632453i	$0.782048\pi$
$240$		0			0
$241$	−53365.8	+	30810.7i	−0.918816	+	0.530479i	−0.883257	−	0.468889i	$-0.844654\pi$
$241$	−53365.8	+	30810.7i	−0.918816	+	0.530479i	−0.0355589	+	0.999368i	$0.511321\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−91869.3	+	53426.6i	−1.53052	+	0.890072i
$246$		0			0
$247$	−30460.0	+	52758.4i	−0.499271	+	0.864763i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			57796.3i			0.917386i	0.888595	+	0.458693i	$0.151682\pi$
$251$			57796.3i			0.917386i	−0.888595	+	0.458693i	$0.848318\pi$
$252$		0			0
$253$	−17614.0			−0.275180
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−60734.3	−	35065.0i	−0.919534	−	0.530893i	−0.0360479	−	0.999350i	$-0.511477\pi$
$257$	−60734.3	−	35065.0i	−0.919534	−	0.530893i	−0.883486	+	0.468457i	$0.844810\pi$
$258$		0			0
$259$	57.4050	+	36521.2i	0.000855756	+	0.544435i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	11120.9	+	19262.0i	0.160779	+	0.278477i	0.935148	−	0.354257i	$-0.115266\pi$
$263$	11120.9	+	19262.0i	0.160779	+	0.278477i	−0.774369	+	0.632734i	$0.781933\pi$
$264$		0			0
$265$			89819.0i			1.27902i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−40332.3	+	23285.9i	−0.557376	+	0.321801i	−0.752092	−	0.659058i	$-0.770955\pi$
$269$	−40332.3	+	23285.9i	−0.557376	+	0.321801i	0.194715	+	0.980860i	$0.437622\pi$
$270$		0			0
$271$	71958.3	+	41545.1i	0.979811	+	0.565694i	0.902213	−	0.431291i	$-0.141942\pi$
$271$	71958.3	+	41545.1i	0.979811	+	0.565694i	0.0775980	+	0.996985i	$0.475275\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−15684.4	+	27166.2i	−0.207397	+	0.359222i
$276$		0			0
$277$	4514.06	+	7818.59i	0.0588313	+	0.101899i	0.893941	−	0.448185i	$-0.147929\pi$
$277$	4514.06	+	7818.59i	0.0588313	+	0.101899i	−0.835110	+	0.550083i	$0.814596\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−64188.4			−0.812912			−0.406456	−	0.913670i	$-0.633236\pi$
$281$	−64188.4			−0.812912			−0.406456	+	0.913670i	$0.633236\pi$
$282$		0			0
$283$	19563.7	−	11295.1i	0.244274	−	0.141032i	−0.372865	−	0.927885i	$-0.621625\pi$
$283$	19563.7	−	11295.1i	0.244274	−	0.141032i	0.617140	+	0.786854i	$0.288291\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	60832.0	+	104983.i	0.738530	+	1.27454i
$288$		0			0
$289$	−7307.06	+	12656.2i	−0.0874877	+	0.151533i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	130473.i		−	1.51980i	−0.650040	−	0.759900i	$-0.725248\pi$
$293$		−	130473.i		−	1.51980i	0.650040	−	0.759900i	$-0.274752\pi$
$294$		0			0
$295$	−51837.2			−0.595658
$296$		0			0
$297$		0			0
$298$		0			0
$299$	88411.6	+	51044.5i	0.988933	+	0.570961i
$300$		0			0
$301$	−111787.	+	64774.5i	−1.23383	+	0.714943i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	51673.7	+	89501.6i	0.555482	+	0.962124i
$306$		0			0
$307$		−	56173.9i		−	0.596015i	−0.954563	−	0.298008i	$-0.903678\pi$
$307$		−	56173.9i		−	0.596015i	0.954563	−	0.298008i	$-0.0963221\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−40087.2	+	23144.4i	−0.414462	+	0.239290i	−0.692705	−	0.721221i	$-0.743581\pi$
$311$	−40087.2	+	23144.4i	−0.414462	+	0.239290i	0.278243	+	0.960511i	$0.410248\pi$
$312$		0			0
$313$	−119814.	−	69174.4i	−1.22298	−	0.706085i	−0.257424	−	0.966299i	$-0.582874\pi$
$313$	−119814.	−	69174.4i	−1.22298	−	0.706085i	−0.965551	+	0.260213i	$0.916207\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−29152.7	+	50493.9i	−0.290108	+	0.502482i	−0.973835	−	0.227256i	$-0.927025\pi$
$317$	−29152.7	+	50493.9i	−0.290108	+	0.502482i	0.683727	+	0.729738i	$0.260358\pi$
$318$		0			0
$319$	−4778.97	−	8277.42i	−0.0469627	−	0.0813418i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−117353.			−1.12483
$324$		0			0
$325$	157452.	−	90905.1i	1.49067	−	0.860640i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	32765.6	−	51.5018i	0.302709	−	0.000475806i
$330$		0			0
$331$	−494.602	+	856.675i	−0.00451440	+	0.00781916i	−0.868274	−	0.496085i	$-0.834770\pi$
$331$	−494.602	+	856.675i	−0.00451440	+	0.00781916i	0.863759	+	0.503904i	$0.168104\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	334608.i		−	2.98158i
$336$		0			0
$337$	−55958.7			−0.492728			−0.246364	−	0.969177i	$-0.579236\pi$
$337$	−55958.7			−0.492728			−0.246364	+	0.969177i	$0.579236\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−13735.3	−	7930.09i	−0.118122	−	0.0681976i
$342$		0			0
$343$	−554.769	−	117648.i	−0.00471546	−	0.999989i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	65322.2	+	113141.i	0.542502	+	0.939641i	0.998760	+	0.0497936i	$0.0158563\pi$
$347$	65322.2	+	113141.i	0.542502	+	0.939641i	−0.456257	+	0.889848i	$0.650810\pi$
$348$		0			0
$349$		−	110948.i		−	0.910899i	−0.890262	−	0.455450i	$-0.849478\pi$
$349$		−	110948.i		−	0.910899i	0.890262	−	0.455450i	$-0.150522\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14727.9	+	8503.18i	−0.118193	+	0.0682389i	−0.557931	−	0.829887i	$-0.688405\pi$
$353$	−14727.9	+	8503.18i	−0.118193	+	0.0682389i	0.439738	+	0.898126i	$0.355071\pi$
$354$		0			0
$355$	290384.	+	167653.i	2.30418	+	1.33032i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	35732.4	−	61890.3i	0.277251	−	0.480212i	−0.693450	−	0.720505i	$-0.743910\pi$
$359$	35732.4	−	61890.3i	0.277251	−	0.480212i	0.970701	+	0.240293i	$0.0772434\pi$
$360$		0			0
$361$	34769.0	+	60221.8i	0.266795	+	0.462103i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−167855.			−1.25994
$366$		0			0
$367$	147543.	−	85183.8i	1.09543	−	0.632448i	0.160415	−	0.987050i	$-0.448717\pi$
$367$	147543.	−	85183.8i	1.09543	−	0.632448i	0.935017	+	0.354602i	$0.115383\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−86188.6	−	49580.5i	−0.626184	−	0.360216i
$372$		0			0
$373$	−75346.0	+	130503.i	−0.541555	+	0.938001i	0.457260	+	0.889333i	$0.348831\pi$
$373$	−75346.0	+	130503.i	−0.541555	+	0.938001i	−0.998815	+	0.0486680i	$0.984502\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			55396.8i			0.389764i
$378$		0			0
$379$	−211477.			−1.47226			−0.736129	−	0.676841i	$-0.763348\pi$
$379$	−211477.			−1.47226			−0.736129	+	0.676841i	$0.763348\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	102769.	+	59333.6i	0.700590	+	0.404486i	0.807567	−	0.589776i	$-0.200784\pi$
$383$	102769.	+	59333.6i	0.700590	+	0.404486i	−0.106977	+	0.994261i	$0.534117\pi$
$384$		0			0
$385$	−25566.0	−	44121.4i	−0.172481	−	0.297665i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	107858.	+	186816.i	0.712778	+	1.23457i	0.963810	+	0.266588i	$0.0858964\pi$
$389$	107858.	+	186816.i	0.712778	+	1.23457i	−0.251033	+	0.967979i	$0.580770\pi$
$390$		0			0
$391$			196658.i			1.28635i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	298790.	−	172507.i	1.91502	−	1.10564i
$396$		0			0
$397$	−134011.	−	77371.1i	−0.850273	−	0.490905i	0.0104699	−	0.999945i	$-0.496667\pi$
$397$	−134011.	−	77371.1i	−0.850273	−	0.490905i	−0.860743	+	0.509040i	$0.830001\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	100029.	−	173255.i	0.622067	−	1.07745i	−0.367033	−	0.930208i	$-0.619626\pi$
$401$	100029.	−	173255.i	0.622067	−	1.07745i	0.989100	−	0.147244i	$-0.0470403\pi$
$402$		0			0
$403$	45962.0	+	79608.5i	0.283001	+	0.490173i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−17523.8			−0.105789
$408$		0			0
$409$	190382.	−	109917.i	1.13810	−	0.657080i	0.192137	−	0.981368i	$-0.438458\pi$
$409$	190382.	−	109917.i	1.13810	−	0.657080i	0.945958	+	0.324288i	$0.105125\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	28614.4	−	49741.9i	0.167758	−	0.291623i
$414$		0			0
$415$	48277.7	−	83619.4i	0.280318	−	0.485524i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			31359.8i			0.178626i	0.996004	+	0.0893132i	$0.0284672\pi$
$419$			31359.8i			0.178626i	−0.996004	+	0.0893132i	$0.971533\pi$
$420$		0			0
$421$	−73690.7			−0.415766			−0.207883	−	0.978154i	$-0.566657\pi$
$421$	−73690.7			−0.415766			−0.207883	+	0.978154i	$0.566657\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	303306.	+	175114.i	1.67920	+	0.969489i
$426$		0			0
$427$	−114408.	+	179.830i	−0.627481	+	0.000986291i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−132586.	−	229646.i	−0.713746	−	1.23624i	−0.963441	−	0.267920i	$-0.913664\pi$
$431$	−132586.	−	229646.i	−0.713746	−	1.23624i	0.249695	−	0.968325i	$-0.419670\pi$
$432$		0			0
$433$		−	124873.i		−	0.666027i	−0.942922	−	0.333014i	$-0.891934\pi$
$433$		−	124873.i		−	0.666027i	0.942922	−	0.333014i	$-0.108066\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	290050.	−	167460.i	1.51883	−	0.876898i
$438$		0			0
$439$	−120989.	−	69852.8i	−0.627792	−	0.362456i	0.152105	−	0.988364i	$-0.451395\pi$
$439$	−120989.	−	69852.8i	−0.627792	−	0.362456i	−0.779896	+	0.625909i	$0.784728\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−110595.	+	191556.i	−0.563544	+	0.976087i	0.433639	+	0.901087i	$0.357229\pi$
$443$	−110595.	+	191556.i	−0.563544	+	0.976087i	−0.997183	+	0.0750005i	$0.976104\pi$
$444$		0			0
$445$	205729.	+	356332.i	1.03890	+	1.79943i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−255878.			−1.26923			−0.634615	−	0.772828i	$-0.718841\pi$
$449$	−255878.			−1.26923			−0.634615	+	0.772828i	$0.718841\pi$
$450$		0			0
$451$	−50419.0	+	29109.4i	−0.247880	+	0.143114i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	464.555	+	295551.i	0.00224396	+	1.42761i
$456$		0			0
$457$	159964.	−	277065.i	0.765929	−	1.32663i	−0.173825	−	0.984777i	$-0.555613\pi$
$457$	159964.	−	277065.i	0.765929	−	1.32663i	0.939754	−	0.341852i	$-0.111054\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	369557.i		−	1.73892i	−0.494002	−	0.869461i	$-0.664466\pi$
$461$		−	369557.i		−	1.73892i	0.494002	−	0.869461i	$-0.335534\pi$
$462$		0			0
$463$	32715.0			0.152611			0.0763053	−	0.997084i	$-0.475688\pi$
$463$	32715.0			0.152611			0.0763053	+	0.997084i	$0.475688\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−265325.	−	153186.i	−1.21659	−	0.702400i	−0.252405	−	0.967622i	$-0.581221\pi$
$467$	−265325.	−	153186.i	−1.21659	−	0.702400i	−0.964187	+	0.265222i	$0.914555\pi$
$468$		0			0
$469$	321083.	+	184705.i	1.45973	+	0.839717i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−30996.0	−	53686.7i	−0.138543	−	0.239963i
$474$		0			0
$475$		−	596460.i		−	2.64359i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	226801.	−	130944.i	0.988493	−	0.570707i	0.0836697	−	0.996494i	$-0.473336\pi$
$479$	226801.	−	130944.i	0.988493	−	0.570707i	0.904824	+	0.425787i	$0.140003\pi$
$480$		0			0
$481$	87958.7	+	50783.0i	0.380180	+	0.219497i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	220913.	−	382633.i	0.939158	−	1.62667i
$486$		0			0
$487$	−41127.7	−	71235.3i	−0.173411	−	0.300357i	0.766199	−	0.642603i	$-0.222146\pi$
$487$	−41127.7	−	71235.3i	−0.173411	−	0.300357i	−0.939610	+	0.342246i	$0.888812\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	102923.			0.426924			0.213462	−	0.976951i	$-0.431526\pi$
$491$	102923.			0.426924			0.213462	+	0.976951i	$0.431526\pi$
$492$		0			0
$493$	−92416.2	+	53356.5i	−0.380237	+	0.219530i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−321170.	+	186101.i	−1.30024	+	0.753420i
$498$		0			0
$499$	−75028.2	+	129953.i	−0.301317	+	0.521896i	−0.976434	−	0.215814i	$-0.930759\pi$
$499$	−75028.2	+	129953.i	−0.301317	+	0.521896i	0.675118	+	0.737710i	$0.264093\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			91806.6i			0.362859i	0.983404	+	0.181430i	$0.0580725\pi$
$503$			91806.6i			0.362859i	−0.983404	+	0.181430i	$0.941928\pi$
$504$		0			0
$505$	−34527.6			−0.135389
$506$		0			0
$507$		0			0
$508$		0			0
$509$	412600.	+	238215.i	1.59255	+	0.919460i	0.992867	+	0.119224i	$0.0380405\pi$
$509$	412600.	+	238215.i	1.59255	+	0.919460i	0.599684	+	0.800237i	$0.295293\pi$
$510$		0			0
$511$	92656.9	−	161071.i	0.354843	−	0.616843i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−60921.0	−	105518.i	−0.229696	−	0.397844i
$516$		0			0
$517$			15721.7i			0.0588192i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−147768.	+	85314.0i	−0.544384	+	0.314300i	−0.746854	−	0.664988i	$-0.768437\pi$
$521$	−147768.	+	85314.0i	−0.544384	+	0.314300i	0.202470	+	0.979289i	$0.435103\pi$
$522$		0			0
$523$	85040.5	+	49098.2i	0.310901	+	0.179499i	0.647330	−	0.762210i	$-0.275886\pi$
$523$	85040.5	+	49098.2i	0.310901	+	0.179499i	−0.336428	+	0.941709i	$0.609219\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−88538.3	+	153353.i	−0.318794	+	0.552167i
$528$		0			0
$529$	−140707.	−	243711.i	−0.502810	−	0.870892i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	337430.			1.18776
$534$		0			0
$535$	−569582.	+	328849.i	−1.98998	+	1.14892i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	56450.5	−	177.461i	0.194308	−	0.000610838i
$540$		0			0
$541$	52833.7	−	91510.6i	0.180516	−	0.312663i	−0.761540	−	0.648118i	$-0.775557\pi$
$541$	52833.7	−	91510.6i	0.180516	−	0.312663i	0.942056	+	0.335454i	$0.108890\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			122017.i			0.410798i
$546$		0			0
$547$	−127121.			−0.424855			−0.212428	−	0.977177i	$-0.568137\pi$
$547$	−127121.			−0.424855			−0.212428	+	0.977177i	$0.568137\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	157391.	+	90869.5i	0.518413	+	0.299306i
$552$		0			0
$553$	600.340	+	381938.i	0.00196312	+	1.24894i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15246.5	+	26407.6i	0.0491427	+	0.0851176i	0.889550	−	0.456837i	$-0.151018\pi$
$557$	15246.5	+	26407.6i	0.0491427	+	0.0851176i	−0.840408	+	0.541955i	$0.817684\pi$
$558$		0			0
$559$			359300.i			1.14983i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−459404.	+	265237.i	−1.44936	+	0.836791i	−0.998444	−	0.0557725i	$-0.982238\pi$
$563$	−459404.	+	265237.i	−1.44936	+	0.836791i	−0.450921	+	0.892564i	$0.648905\pi$
$564$		0			0
$565$	240146.	+	138649.i	0.752279	+	0.434329i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−169366.	+	293351.i	−0.523121	+	0.906072i	0.476517	+	0.879165i	$0.341899\pi$
$569$	−169366.	+	293351.i	−0.523121	+	0.906072i	−0.999638	+	0.0269067i	$0.991434\pi$
$570$		0			0
$571$	53455.6	+	92587.8i	0.163954	+	0.283976i	0.936283	−	0.351246i	$-0.114242\pi$
$571$	53455.6	+	92587.8i	0.163954	+	0.283976i	−0.772330	+	0.635222i	$0.780909\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−999538.			−3.02318
$576$		0			0
$577$	−104337.	+	60238.8i	−0.313390	+	0.180936i	−0.648443	−	0.761264i	$-0.724579\pi$
$577$	−104337.	+	60238.8i	−0.313390	+	0.180936i	0.335052	+	0.942200i	$0.391246\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	53590.0	+	92484.6i	0.158757	+	0.273979i
$582$		0			0
$583$	23854.9	−	41317.9i	0.0701844	−	0.121563i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	525818.i		−	1.52602i	−0.646389	−	0.763008i	$-0.723722\pi$
$587$		−	525818.i		−	1.52602i	0.646389	−	0.763008i	$-0.276278\pi$
$588$		0			0
$589$	301573.			0.869283
$590$		0			0
$591$		0			0
$592$		0			0
$593$	64283.0	+	37113.8i	0.182804	+	0.105542i	0.588610	−	0.808417i	$-0.299676\pi$
$593$	64283.0	+	37113.8i	0.182804	+	0.105542i	−0.405805	+	0.913960i	$0.633009\pi$
$594$		0			0
$595$	−492609.	+	285441.i	−1.39145	+	0.806274i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10990.7	+	19036.5i	0.0306318	+	0.0530558i	0.880935	−	0.473237i	$-0.156915\pi$
$599$	10990.7	+	19036.5i	0.0306318	+	0.0530558i	−0.850303	+	0.526293i	$0.823581\pi$
$600$		0			0
$601$		−	77418.2i		−	0.214336i	−0.994241	−	0.107168i	$-0.965822\pi$
$601$		−	77418.2i		−	0.214336i	0.994241	−	0.107168i	$-0.0341782\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−540039.	+	311792.i	−1.47542	+	0.851832i
$606$		0			0
$607$	155765.	+	89931.2i	0.422760	+	0.244081i	0.696257	−	0.717792i	$-0.254847\pi$
$607$	155765.	+	89931.2i	0.422760	+	0.244081i	−0.273498	+	0.961873i	$0.588181\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	45560.7	−	78913.5i	0.122042	−	0.211382i
$612$		0			0
$613$	−314637.	−	544968.i	−0.837316	−	1.45027i	−0.892131	−	0.451777i	$-0.850790\pi$
$613$	−314637.	−	544968.i	−0.837316	−	1.45027i	0.0548147	−	0.998497i	$-0.482543\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	52877.1			0.138899			0.0694493	−	0.997585i	$-0.477876\pi$
$617$	52877.1			0.138899			0.0694493	+	0.997585i	$0.477876\pi$
$618$		0			0
$619$	447040.	−	258099.i	1.16672	−	0.673603i	0.213811	−	0.976875i	$-0.431412\pi$
$619$	447040.	−	258099.i	1.16672	−	0.673603i	0.952904	+	0.303271i	$0.0980789\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−455493.	+	715.955i	−1.17356	+	0.00184463i
$624$		0			0
$625$	−277790.	+	481147.i	−0.711143	+	1.23174i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			195651.i			0.494515i
$630$		0			0
$631$	193439.			0.485831			0.242915	−	0.970047i	$-0.421896\pi$
$631$	193439.			0.485831			0.242915	+	0.970047i	$0.421896\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−492766.	−	284499.i	−1.22206	−	0.705558i
$636$		0			0
$637$	−283862.	−	162700.i	−0.699565	−	0.400967i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	376563.	+	652226.i	0.916477	+	1.58738i	0.804725	+	0.593648i	$0.202313\pi$
$641$	376563.	+	652226.i	0.916477	+	1.58738i	0.111751	+	0.993736i	$0.464354\pi$
$642$		0			0
$643$			7110.23i			0.0171974i	0.999963	+	0.00859868i	$0.00273708\pi$
$643$			7110.23i			0.0171974i	−0.999963	+	0.00859868i	$0.997263\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	341808.	−	197343.i	0.816534	−	0.471426i	−0.0326859	−	0.999466i	$-0.510406\pi$
$647$	341808.	−	197343.i	0.816534	−	0.471426i	0.849220	+	0.528040i	$0.177073\pi$
$648$		0			0
$649$	23845.8	+	13767.4i	0.0566137	+	0.0326860i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	115578.	−	200186.i	0.271049	−	0.469470i	−0.698082	−	0.716018i	$-0.745963\pi$
$653$	115578.	−	200186.i	0.271049	−	0.469470i	0.969131	+	0.246548i	$0.0792963\pi$
$654$		0			0
$655$	−547198.	−	947775.i	−1.27545	−	2.20914i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	772498.			1.77880			0.889399	−	0.457131i	$-0.151123\pi$
$659$	772498.			1.77880			0.889399	+	0.457131i	$0.151123\pi$
$660$		0			0
$661$	−195287.	+	112749.i	−0.446962	+	0.258053i	−0.706546	−	0.707667i	$-0.749748\pi$
$661$	−195287.	+	112749.i	−0.446962	+	0.258053i	0.259584	+	0.965720i	$0.416414\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	840468.	+	483484.i	1.90054	+	1.09330i
$666$		0			0
$667$	152278.	−	263753.i	0.342282	−	0.592851i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	54895.8i		−	0.121925i
$672$		0			0
$673$	59564.6			0.131510			0.0657549	−	0.997836i	$-0.479054\pi$
$673$	59564.6			0.131510			0.0657549	+	0.997836i	$0.479054\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−687131.	−	396715.i	−1.49921	−	0.865569i	−0.499210	−	0.866481i	$-0.666376\pi$
$677$	−687131.	−	396715.i	−1.49921	−	0.865569i	−1.00000		0.000912440i	$0.999710\pi$
$678$		0			0
$679$	245222.	+	423199.i	0.531888	+	0.917921i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−46304.3	−	80201.3i	−0.0992612	−	0.171925i	0.812118	−	0.583493i	$-0.198315\pi$
$683$	−46304.3	−	80201.3i	−0.0992612	−	0.171925i	−0.911379	+	0.411568i	$0.864981\pi$
$684$		0			0
$685$		−	58522.3i		−	0.124721i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−239474.	+	138261.i	−0.504453	+	0.291246i
$690$		0			0
$691$	−162273.	−	93688.2i	−0.339852	−	0.196213i	0.320355	−	0.947298i	$-0.396198\pi$
$691$	−162273.	−	93688.2i	−0.339852	−	0.196213i	−0.660206	+	0.751084i	$0.729531\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−113308.	+	196255.i	−0.234580	+	0.406304i
$696$		0			0
$697$	325003.	+	562921.i	0.668993	+	1.15873i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−326056.			−0.663522			−0.331761	−	0.943363i	$-0.607643\pi$
$701$	−326056.			−0.663522			−0.331761	+	0.943363i	$0.607643\pi$
$702$		0			0
$703$	288564.	−	166603.i	0.583891	−	0.337110i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	19059.4	−	33132.0i	0.0381303	−	0.0662841i
$708$		0			0
$709$	−32372.6	+	56070.9i	−0.0643998	+	0.111544i	−0.896428	−	0.443190i	$-0.853847\pi$
$709$	−32372.6	+	56070.9i	−0.0643998	+	0.111544i	0.832028	+	0.554734i	$0.187180\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	505371.i		−	0.994102i
$714$		0			0
$715$	−141813.			−0.277398
$716$		0			0
$717$		0			0
$718$		0			0
$719$	616355.	+	355853.i	1.19227	+	0.688355i	0.958820	−	0.284014i	$-0.0916664\pi$
$719$	616355.	+	355853.i	1.19227	+	0.688355i	0.233446	+	0.972370i	$0.425000\pi$
$720$		0			0
$721$	134882.	−	212.011i	0.259468	−	0.000407838i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−271191.	−	469717.i	−0.515941	−	0.893635i
$726$		0			0
$727$			763134.i			1.44388i	0.691954	+	0.721941i	$0.256750\pi$
$727$			763134.i			1.44388i	−0.691954	+	0.721941i	$0.743250\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−599404.	+	346066.i	−1.12172	+	0.647626i
$732$		0			0
$733$	−17848.0	−	10304.6i	−0.0332187	−	0.0191788i	0.483299	−	0.875456i	$-0.339439\pi$
$733$	−17848.0	−	10304.6i	−0.0332187	−	0.0191788i	−0.516517	+	0.856277i	$0.672772\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−88868.0	+	153924.i	−0.163610	+	0.283381i
$738$		0			0
$739$	393194.	+	681032.i	0.719976	+	1.24703i	0.961009	+	0.276518i	$0.0891805\pi$
$739$	393194.	+	681032.i	0.719976	+	1.24703i	−0.241033	+	0.970517i	$0.577486\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−740479.			−1.34133			−0.670664	−	0.741761i	$-0.733991\pi$
$743$	−740479.			−1.34133			−0.670664	+	0.741761i	$0.733991\pi$
$744$		0			0
$745$	−391372.	+	225959.i	−0.705143	+	0.407114i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1144.42	−	728086.i	−0.00203997	−	1.29783i
$750$		0			0
$751$	247292.	−	428322.i	0.438460	−	0.759436i	−0.559111	−	0.829093i	$-0.688857\pi$
$751$	247292.	−	428322.i	0.438460	−	0.759436i	0.997571	+	0.0696574i	$0.0221906\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1.29885e6i			2.27859i
$756$		0			0
$757$	−139399.			−0.243258			−0.121629	−	0.992576i	$-0.538812\pi$
$757$	−139399.			−0.243258			−0.121629	+	0.992576i	$0.538812\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	207340.	+	119708.i	0.358025	+	0.206706i	0.668214	−	0.743969i	$-0.267059\pi$
$761$	207340.	+	119708.i	0.358025	+	0.206706i	−0.310189	+	0.950675i	$0.600392\pi$
$762$		0			0
$763$	−117085.	−	67354.1i	−0.201119	−	0.115695i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−79794.1	−	138208.i	−0.135638	−	0.234931i
$768$		0			0
$769$			98701.5i			0.166906i	0.996512	+	0.0834528i	$0.0265948\pi$
$769$			98701.5i			0.166906i	−0.996512	+	0.0834528i	$0.973405\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	507921.	−	293248.i	0.850036	−	0.490769i	−0.0106268	−	0.999944i	$-0.503383\pi$
$773$	507921.	−	293248.i	0.850036	−	0.490769i	0.860663	+	0.509175i	$0.170049\pi$
$774$		0			0
$775$	−779435.	−	450007.i	−1.29771	−	0.749232i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	553500.	−	958690.i	0.912100	−	1.57980i
$780$		0			0
$781$	−89053.6	−	154245.i	−0.145999	−	0.252877i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.33056e6			2.15921
$786$		0			0
$787$	−129536.	+	74787.4i	−0.209141	+	0.120748i	−0.600912	−	0.799315i	$-0.705196\pi$
$787$	−129536.	+	74787.4i	−0.209141	+	0.120748i	0.391771	+	0.920063i	$0.371863\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−265606.	+	153905.i	−0.424507	+	0.245980i
$792$		0			0
$793$	−159085.	+	275544.i	−0.252978	+	0.438171i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	241092.i		−	0.379547i	−0.981828	−	0.189774i	$-0.939225\pi$
$797$		−	241092.i		−	0.379547i	0.981828	−	0.189774i	$-0.0607754\pi$
$798$		0			0
$799$	175531.			0.274954
$800$		0			0
$801$		0			0
$802$		0			0
$803$	77215.6	+	44580.4i	0.119750	+	0.0691374i
$804$		0			0
$805$	810215.	−	1.40844e6i	1.25028	−	2.17344i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	100933.	+	174820.i	0.154218	+	0.267113i	0.932774	−	0.360462i	$-0.117381\pi$
$809$	100933.	+	174820.i	0.154218	+	0.267113i	−0.778556	+	0.627575i	$0.784048\pi$
$810$		0			0
$811$		−	1.16747e6i		−	1.77503i	−0.460779	−	0.887515i	$-0.652430\pi$
$811$		−	1.16747e6i		−	1.77503i	0.460779	−	0.887515i	$-0.347570\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.18369e6	−	683402.i	1.78206	−	1.02887i
$816$		0			0
$817$	1.02082e6	+	589373.i	1.52935	+	0.882970i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−324585.	+	562198.i	−0.481551	+	0.834071i	−0.999776	−	0.0211734i	$-0.993260\pi$
$821$	−324585.	+	562198.i	−0.481551	+	0.834071i	0.518225	+	0.855245i	$0.326593\pi$
$822$		0			0
$823$	245486.	+	425193.i	0.362432	+	0.627750i	0.988360	−	0.152130i	$-0.0486133\pi$
$823$	245486.	+	425193.i	0.362432	+	0.627750i	−0.625929	+	0.779880i	$0.715280\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	709017.			1.03668			0.518341	−	0.855174i	$-0.326550\pi$
$827$	709017.			1.03668			0.518341	+	0.855174i	$0.326550\pi$
$828$		0			0
$829$	534645.	−	308677.i	0.777958	−	0.449154i	−0.0577480	−	0.998331i	$-0.518392\pi$
$829$	534645.	−	308677.i	0.777958	−	0.449154i	0.835706	+	0.549177i	$0.185059\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1981.33	−	630262.i	−0.00285540	−	0.908304i
$834$		0			0
$835$	−284583.	+	492912.i	−0.408165	+	0.706963i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			493961.i			0.701728i	0.936426	+	0.350864i	$0.114112\pi$
$839$			493961.i			0.701728i	−0.936426	+	0.350864i	$0.885888\pi$
$840$		0			0
$841$	−542019.			−0.766342
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−383006.	−	221129.i	−0.536404	−	0.309693i
$846$		0			0
$847$	−1085.07	−	690322.i	−0.00151248	−	0.962243i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−279190.	−	483571.i	−0.385514	−	0.667730i
$852$		0			0
$853$		−	505142.i		−	0.694249i	−0.937819	−	0.347124i	$-0.887158\pi$
$853$		−	505142.i		−	0.694249i	0.937819	−	0.347124i	$-0.112842\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	969980.	−	560018.i	1.32069	−	0.762501i	0.336852	−	0.941557i	$-0.390638\pi$
$857$	969980.	−	560018.i	1.32069	−	0.762501i	0.983839	+	0.179056i	$0.0573043\pi$
$858$		0			0
$859$	−605141.	−	349379.i	−0.820107	−	0.473489i	0.0303463	−	0.999539i	$-0.490339\pi$
$859$	−605141.	−	349379.i	−0.820107	−	0.473489i	−0.850453	+	0.526050i	$0.823672\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	479533.	−	830576.i	0.643868	−	1.11521i	−0.340693	−	0.940175i	$-0.610662\pi$
$863$	479533.	−	830576.i	0.643868	−	1.11521i	0.984562	−	0.175038i	$-0.0560049\pi$
$864$		0			0
$865$	347264.	+	601479.i	0.464117	+	0.803875i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−183263.			−0.242681
$870$		0			0
$871$	892127.	−	515069.i	1.17595	−	0.678937i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−771172.	−	1.33087e6i	−1.00724	−	1.73828i
$876$		0			0
$877$	498442.	−	863326.i	0.648060	−	1.12247i	−0.335526	−	0.942031i	$-0.608914\pi$
$877$	498442.	−	863326.i	0.648060	−	1.12247i	0.983586	−	0.180442i	$-0.0577527\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			494568.i			0.637198i	0.947890	+	0.318599i	$0.103212\pi$
$881$			494568.i			0.637198i	−0.947890	+	0.318599i	$0.896788\pi$
$882$		0			0
$883$	887278.			1.13799			0.568995	−	0.822341i	$-0.307333\pi$
$883$	887278.			1.13799			0.568995	+	0.822341i	$0.307333\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−542062.	−	312960.i	−0.688973	−	0.397779i	0.114254	−	0.993452i	$-0.463552\pi$
$887$	−542062.	−	312960.i	−0.688973	−	0.397779i	−0.803227	+	0.595673i	$0.796885\pi$
$888$		0			0
$889$	545008.	−	315804.i	0.689604	−	0.399589i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−149470.	−	258890.i	−0.187435	−	0.324647i
$894$		0			0
$895$			1.41581e6i			1.76750i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	237491.	−	137115.i	0.293851	−	0.169655i
$900$		0			0
$901$	−461309.	−	266337.i	−0.568253	−	0.328081i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.06159e6	+	1.83873e6i	−1.29616	+	2.24502i
$906$		0			0
$907$	634131.	+	1.09835e6i	0.770840	+	1.33513i	0.937103	+	0.349052i	$0.113496\pi$
$907$	634131.	+	1.09835e6i	0.770840	+	1.33513i	−0.166263	+	0.986081i	$0.553170\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.09985e6			1.32525			0.662625	−	0.748951i	$-0.269442\pi$
$911$	1.09985e6			1.32525			0.662625	+	0.748951i	$0.269442\pi$
$912$		0			0
$913$	−44416.7	+	25644.0i	−0.0532850	+	0.0307641i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.21152e6	−	1904.30i	1.44076	−	0.00226463i
$918$		0			0
$919$	−430660.	+	745925.i	−0.509922	+	0.883211i	0.490012	+	0.871716i	$0.336992\pi$
$919$	−430660.	+	745925.i	−0.509922	+	0.883211i	−0.999934	+	0.0114950i	$0.996341\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1.03229e6i			1.21171i
$924$		0			0
$925$	−994418.			−1.16221
$926$		0			0
$927$		0			0
$928$		0			0
$929$	326247.	+	188359.i	0.378021	+	0.218250i	0.676957	−	0.736023i	$-0.263298\pi$
$929$	326247.	+	188359.i	0.378021	+	0.218250i	−0.298936	+	0.954273i	$0.596632\pi$
$930$		0			0
$931$	−927884.	+	539610.i	−1.07052	+	0.622560i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−136590.	−	236580.i	−0.156241	−	0.270617i
$936$		0			0
$937$			417813.i			0.475885i	0.971279	+	0.237943i	$0.0764731\pi$
$937$			417813.i			0.475885i	−0.971279	+	0.237943i	$0.923527\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−213031.	+	122994.i	−0.240582	+	0.138900i	−0.615444	−	0.788180i	$-0.711023\pi$
$941$	−213031.	+	122994.i	−0.240582	+	0.138900i	0.374862	+	0.927081i	$0.377690\pi$
$942$		0			0
$943$	−1.60656e6	−	927546.i	−1.80665	−	1.04307i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−722964.	+	1.25221e6i	−0.806152	+	1.39630i	0.109359	+	0.994002i	$0.465120\pi$
$947$	−722964.	+	1.25221e6i	−0.806152	+	1.39630i	−0.915511	+	0.402294i	$0.868213\pi$
$948$		0			0
$949$	−258384.	−	447533.i	−0.286901	−	0.496928i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−907377.			−0.999084			−0.499542	−	0.866290i	$-0.666498\pi$
$953$	−907377.			−0.999084			−0.499542	+	0.866290i	$0.666498\pi$
$954$		0			0
$955$	−170805.	+	98614.2i	−0.187281	+	0.108127i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	56156.9	+	32304.6i	0.0610612	+	0.0351259i
$960$		0			0
$961$	−234235.	+	405707.i	−0.253633	+	0.439305i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.30132e6i		−	1.39742i
$966$		0			0
$967$	−50286.2			−0.0537770			−0.0268885	−	0.999638i	$-0.508560\pi$
$967$	−50286.2			−0.0537770			−0.0268885	+	0.999638i	$0.508560\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−677500.	−	391155.i	−0.718573	−	0.414868i	0.0956544	−	0.995415i	$-0.469506\pi$
$971$	−677500.	−	391155.i	−0.718573	−	0.414868i	−0.814227	+	0.580546i	$0.802839\pi$
$972$		0			0
$973$	−125776.	−	217062.i	−0.132853	−	0.229275i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	638714.	+	1.10629e6i	0.669141	+	1.15899i	0.978145	+	0.207925i	$0.0666711\pi$
$977$	638714.	+	1.10629e6i	0.669141	+	1.15899i	−0.309004	+	0.951061i	$0.599996\pi$
$978$		0			0
$979$		−	218557.i		−	0.228033i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.36250e6	+	786638.i	−1.41003	+	0.814081i	−0.995391	−	0.0959049i	$-0.969426\pi$
$983$	−1.36250e6	+	786638.i	−1.41003	+	0.814081i	−0.414639	+	0.909986i	$0.636092\pi$
$984$		0			0
$985$	912930.	+	527080.i	0.940947	+	0.543256i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	987661.	−	1.71068e6i	1.00975	−	1.74894i
$990$		0			0
$991$	−162173.	−	280892.i	−0.165132	−	0.286017i	0.771570	−	0.636144i	$-0.219472\pi$
$991$	−162173.	−	280892.i	−0.165132	−	0.286017i	−0.936702	+	0.350127i	$0.886138\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.16099e6			2.18276
$996$		0			0
$997$	−1.57129e6	+	907185.i	−1.58076	+	0.912652i	−0.586012	+	0.810303i	$0.699303\pi$
$997$	−1.57129e6	+	907185.i	−1.58076	+	0.912652i	−0.994748	+	0.102350i	$0.967364\pi$
$998$		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	252.5.z.f.145.1		6
3.2	odd	2		28.5.h.a.5.2	✓	6
7.3	odd	6	inner	252.5.z.f.73.1		6
12.11	even	2		112.5.s.c.33.2		6
15.2	even	4		700.5.o.a.649.4		12
15.8	even	4		700.5.o.a.649.3		12
15.14	odd	2		700.5.s.a.201.2		6
21.2	odd	6		196.5.b.a.97.3		6
21.5	even	6		196.5.b.a.97.4		6
21.11	odd	6		196.5.h.c.129.2		6
21.17	even	6		28.5.h.a.17.2	yes	6
21.20	even	2		196.5.h.c.117.2		6
84.23	even	6		784.5.c.e.97.4		6
84.47	odd	6		784.5.c.e.97.3		6
84.59	odd	6		112.5.s.c.17.2		6
105.17	odd	12		700.5.o.a.549.3		12
105.38	odd	12		700.5.o.a.549.4		12
105.59	even	6		700.5.s.a.101.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.5.h.a.5.2	✓	6	3.2	odd	2
28.5.h.a.17.2	yes	6	21.17	even	6
112.5.s.c.17.2		6	84.59	odd	6
112.5.s.c.33.2		6	12.11	even	2
196.5.b.a.97.3		6	21.2	odd	6
196.5.b.a.97.4		6	21.5	even	6
196.5.h.c.117.2		6	21.20	even	2
196.5.h.c.129.2		6	21.11	odd	6
252.5.z.f.73.1		6	7.3	odd	6	inner
252.5.z.f.145.1		6	1.1	even	1	trivial
700.5.o.a.549.3		12	105.17	odd	12
700.5.o.a.549.4		12	105.38	odd	12
700.5.o.a.649.3		12	15.8	even	4
700.5.o.a.649.4		12	15.2	even	4
700.5.s.a.101.2		6	105.59	even	6
700.5.s.a.201.2		6	15.14	odd	2
784.5.c.e.97.3		6	84.47	odd	6
784.5.c.e.97.4		6	84.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

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	252.5.z.f.145.1

Level	$252$
Weight	$5$
Character	252.145
Analytic conductor	$26.049$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$