LMFDB - Embedded newform 648.4.i.c.217.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(217,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("648.217");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$38.2332376837$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	648.217
Dual form			648.4.i.c.433.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 - 4.33013i) q^{5} +(-18.0000 + 31.1769i) q^{7} +O(q^{10})$	$q+(-2.50000 - 4.33013i) q^{5} +(-18.0000 + 31.1769i) q^{7} +(32.0000 - 55.4256i) q^{11} +(32.5000 + 56.2917i) q^{13} -59.0000 q^{17} -28.0000 q^{19} +(80.0000 + 138.564i) q^{23} +(50.0000 - 86.6025i) q^{25} +(-28.5000 + 49.3634i) q^{29} +(-82.0000 - 142.028i) q^{31} +180.000 q^{35} -321.000 q^{37} +(-123.000 - 213.042i) q^{41} +(4.00000 - 6.92820i) q^{43} +(42.0000 - 72.7461i) q^{47} +(-476.500 - 825.322i) q^{49} -478.000 q^{53} -320.000 q^{55} +(-16.0000 - 27.7128i) q^{59} +(-207.500 + 359.401i) q^{61} +(162.500 - 281.458i) q^{65} +(110.000 + 190.526i) q^{67} -884.000 q^{71} -77.0000 q^{73} +(1152.00 + 1995.32i) q^{77} +(40.0000 - 69.2820i) q^{79} +(634.000 - 1098.12i) q^{83} +(147.500 + 255.477i) q^{85} -123.000 q^{89} -2340.00 q^{91} +(70.0000 + 121.244i) q^{95} +(-673.000 + 1165.67i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{5} - 36 q^{7}+O(q^{10}) $ 2 * q - 5 * q^5 - 36 * q^7	$ 2 q - 5 q^{5} - 36 q^{7} + 64 q^{11} + 65 q^{13} - 118 q^{17} - 56 q^{19} + 160 q^{23} + 100 q^{25} - 57 q^{29} - 164 q^{31} + 360 q^{35} - 642 q^{37} - 246 q^{41} + 8 q^{43} + 84 q^{47} - 953 q^{49} - 956 q^{53} - 640 q^{55} - 32 q^{59} - 415 q^{61} + 325 q^{65} + 220 q^{67} - 1768 q^{71} - 154 q^{73} + 2304 q^{77} + 80 q^{79} + 1268 q^{83} + 295 q^{85} - 246 q^{89} - 4680 q^{91} + 140 q^{95} - 1346 q^{97}+O(q^{100}) $ 2 * q - 5 * q^5 - 36 * q^7 + 64 * q^11 + 65 * q^13 - 118 * q^17 - 56 * q^19 + 160 * q^23 + 100 * q^25 - 57 * q^29 - 164 * q^31 + 360 * q^35 - 642 * q^37 - 246 * q^41 + 8 * q^43 + 84 * q^47 - 953 * q^49 - 956 * q^53 - 640 * q^55 - 32 * q^59 - 415 * q^61 + 325 * q^65 + 220 * q^67 - 1768 * q^71 - 154 * q^73 + 2304 * q^77 + 80 * q^79 + 1268 * q^83 + 295 * q^85 - 246 * q^89 - 4680 * q^91 + 140 * q^95 - 1346 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−2.50000	−	4.33013i	−0.223607	−	0.387298i	0.732294	−	0.680989i	$-0.238450\pi$
$5$	−2.50000	−	4.33013i	−0.223607	−	0.387298i	−0.955901	+	0.293691i	$0.905116\pi$
$6$		0			0
$7$	−18.0000	+	31.1769i	−0.971909	+	1.68340i	−0.282128	+	0.959377i	$0.591040\pi$
$7$	−18.0000	+	31.1769i	−0.971909	+	1.68340i	−0.689781	+	0.724018i	$0.742293\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	32.0000	−	55.4256i	0.877124	−	1.51922i	0.0226410	−	0.999744i	$-0.492793\pi$
$11$	32.0000	−	55.4256i	0.877124	−	1.51922i	0.854483	−	0.519480i	$-0.173874\pi$
$12$		0			0
$13$	32.5000	+	56.2917i	0.693375	+	1.20096i	0.970725	+	0.240192i	$0.0772105\pi$
$13$	32.5000	+	56.2917i	0.693375	+	1.20096i	−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−59.0000			−0.841741			−0.420871	−	0.907121i	$-0.638275\pi$
$17$	−59.0000			−0.841741			−0.420871	+	0.907121i	$0.638275\pi$
$18$		0			0
$19$	−28.0000			−0.338086			−0.169043	−	0.985609i	$-0.554068\pi$
$19$	−28.0000			−0.338086			−0.169043	+	0.985609i	$0.554068\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	80.0000	+	138.564i	0.725268	+	1.25620i	0.958864	+	0.283867i	$0.0916173\pi$
$23$	80.0000	+	138.564i	0.725268	+	1.25620i	−0.233596	+	0.972334i	$0.575049\pi$
$24$		0			0
$25$	50.0000	−	86.6025i	0.400000	−	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−28.5000	+	49.3634i	−0.182494	+	0.316088i	−0.942729	−	0.333559i	$-0.891750\pi$
$29$	−28.5000	+	49.3634i	−0.182494	+	0.316088i	0.760235	+	0.649648i	$0.225084\pi$
$30$		0			0
$31$	−82.0000	−	142.028i	−0.475085	−	0.822871i	0.524508	−	0.851406i	$-0.324249\pi$
$31$	−82.0000	−	142.028i	−0.475085	−	0.822871i	−0.999593	+	0.0285343i	$0.990916\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	180.000			0.869302
$36$		0			0
$37$	−321.000			−1.42627			−0.713136	−	0.701026i	$-0.752726\pi$
$37$	−321.000			−1.42627			−0.713136	+	0.701026i	$0.752726\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−123.000	−	213.042i	−0.468521	−	0.811503i	0.530831	−	0.847477i	$-0.321880\pi$
$41$	−123.000	−	213.042i	−0.468521	−	0.811503i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$		0			0
$43$	4.00000	−	6.92820i	0.0141859	−	0.0245707i	−0.858845	−	0.512235i	$-0.828818\pi$
$43$	4.00000	−	6.92820i	0.0141859	−	0.0245707i	0.873031	+	0.487664i	$0.162151\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	42.0000	−	72.7461i	0.130347	−	0.225768i	−0.793463	−	0.608618i	$-0.791724\pi$
$47$	42.0000	−	72.7461i	0.130347	−	0.225768i	0.923811	+	0.382850i	$0.125057\pi$
$48$		0			0
$49$	−476.500	−	825.322i	−1.38921	−	2.40619i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−478.000			−1.23884			−0.619418	−	0.785061i	$-0.712632\pi$
$53$	−478.000			−1.23884			−0.619418	+	0.785061i	$0.712632\pi$
$54$		0			0
$55$	−320.000			−0.784523
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−16.0000	−	27.7128i	−0.0353055	−	0.0611509i	0.847833	−	0.530264i	$-0.177907\pi$
$59$	−16.0000	−	27.7128i	−0.0353055	−	0.0611509i	−0.883138	+	0.469113i	$0.844574\pi$
$60$		0			0
$61$	−207.500	+	359.401i	−0.435535	+	0.754369i	−0.997339	−	0.0729012i	$-0.976774\pi$
$61$	−207.500	+	359.401i	−0.435535	+	0.754369i	0.561804	+	0.827270i	$0.310108\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	162.500	−	281.458i	0.310087	−	0.537086i
$66$		0			0
$67$	110.000	+	190.526i	0.200577	+	0.347409i	0.948714	−	0.316134i	$-0.102385\pi$
$67$	110.000	+	190.526i	0.200577	+	0.347409i	−0.748138	+	0.663544i	$0.769052\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−884.000			−1.47763			−0.738813	−	0.673910i	$-0.764614\pi$
$71$	−884.000			−1.47763			−0.738813	+	0.673910i	$0.764614\pi$
$72$		0			0
$73$	−77.0000			−0.123454			−0.0617272	−	0.998093i	$-0.519661\pi$
$73$	−77.0000			−0.123454			−0.0617272	+	0.998093i	$0.519661\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1152.00	+	1995.32i	1.70497	+	2.95309i
$78$		0			0
$79$	40.0000	−	69.2820i	0.0569665	−	0.0986688i	−0.836136	−	0.548522i	$-0.815191\pi$
$79$	40.0000	−	69.2820i	0.0569665	−	0.0986688i	0.893102	+	0.449854i	$0.148524\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	634.000	−	1098.12i	0.838440	−	1.45222i	−0.0527581	−	0.998607i	$-0.516801\pi$
$83$	634.000	−	1098.12i	0.838440	−	1.45222i	0.891198	−	0.453614i	$-0.149865\pi$
$84$		0			0
$85$	147.500	+	255.477i	0.188219	+	0.326005i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−123.000			−0.146494			−0.0732470	−	0.997314i	$-0.523336\pi$
$89$	−123.000			−0.146494			−0.0732470	+	0.997314i	$0.523336\pi$
$90$		0			0
$91$	−2340.00			−2.69559
$92$		0			0
$93$		0			0
$94$		0			0
$95$	70.0000	+	121.244i	0.0755984	+	0.130940i
$96$		0			0
$97$	−673.000	+	1165.67i	−0.704462	+	1.22016i	0.262424	+	0.964953i	$0.415478\pi$
$97$	−673.000	+	1165.67i	−0.704462	+	1.22016i	−0.966885	+	0.255211i	$0.917855\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−417.000	+	722.265i	−0.410822	+	0.711565i	−0.994980	−	0.100075i	$-0.968092\pi$
$101$	−417.000	+	722.265i	−0.410822	+	0.711565i	0.584158	+	0.811640i	$0.301425\pi$
$102$		0			0
$103$	−530.000	−	917.987i	−0.507014	−	0.878174i	−0.999967	−	0.00811820i	$-0.997416\pi$
$103$	−530.000	−	917.987i	−0.507014	−	0.878174i	0.492953	−	0.870056i	$-0.335917\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−300.000			−0.271048			−0.135524	−	0.990774i	$-0.543272\pi$
$107$	−300.000			−0.271048			−0.135524	+	0.990774i	$0.543272\pi$
$108$		0			0
$109$	−557.000			−0.489458			−0.244729	−	0.969592i	$-0.578699\pi$
$109$	−557.000			−0.489458			−0.244729	+	0.969592i	$0.578699\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−808.500	−	1400.36i	−0.673073	−	1.16580i	−0.977028	−	0.213111i	$-0.931641\pi$
$113$	−808.500	−	1400.36i	−0.673073	−	1.16580i	0.303955	−	0.952686i	$-0.401693\pi$
$114$		0			0
$115$	400.000	−	692.820i	0.324349	−	0.561790i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1062.00	−	1839.44i	0.818096	−	1.41698i
$120$		0			0
$121$	−1382.50	−	2394.56i	−1.03869	−	1.79907i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1125.00			−0.804984
$126$		0			0
$127$	−1136.00			−0.793730			−0.396865	−	0.917877i	$-0.629902\pi$
$127$	−1136.00			−0.793730			−0.396865	+	0.917877i	$0.629902\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	58.0000	+	100.459i	0.0386831	+	0.0670011i	0.884719	−	0.466125i	$-0.154350\pi$
$131$	58.0000	+	100.459i	0.0386831	+	0.0670011i	−0.846036	+	0.533126i	$0.821017\pi$
$132$		0			0
$133$	504.000	−	872.954i	0.328589	−	0.569133i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	307.500	−	532.606i	0.191763	−	0.332143i	−0.754072	−	0.656792i	$-0.771913\pi$
$137$	307.500	−	532.606i	0.191763	−	0.332143i	0.945834	+	0.324649i	$0.105246\pi$
$138$		0			0
$139$	−726.000	−	1257.47i	−0.443011	−	0.767317i	0.554900	−	0.831917i	$-0.312756\pi$
$139$	−726.000	−	1257.47i	−0.443011	−	0.767317i	−0.997911	+	0.0645994i	$0.979423\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4160.00			2.43270
$144$		0			0
$145$	285.000			0.163227
$146$		0			0
$147$		0			0
$148$		0			0
$149$	419.500	+	726.595i	0.230650	+	0.399497i	0.957999	−	0.286770i	$-0.0925816\pi$
$149$	419.500	+	726.595i	0.230650	+	0.399497i	−0.727350	+	0.686267i	$0.759248\pi$
$150$		0			0
$151$	1028.00	−	1780.55i	0.554023	−	0.959596i	−0.443956	−	0.896049i	$-0.646425\pi$
$151$	1028.00	−	1780.55i	0.554023	−	0.959596i	0.997979	−	0.0635472i	$-0.0202413\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−410.000	+	710.141i	−0.212464	+	0.367999i
$156$		0			0
$157$	−549.500	−	951.762i	−0.279330	−	0.483814i	0.691888	−	0.722005i	$-0.256779\pi$
$157$	−549.500	−	951.762i	−0.279330	−	0.483814i	−0.971219	+	0.238190i	$0.923446\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−5760.00			−2.81958
$162$		0			0
$163$	−72.0000			−0.0345980			−0.0172990	−	0.999850i	$-0.505507\pi$
$163$	−72.0000			−0.0345980			−0.0172990	+	0.999850i	$0.505507\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1062.00	+	1839.44i	0.492096	+	0.852335i	0.999959	−	0.00910286i	$-0.00289757\pi$
$167$	1062.00	+	1839.44i	0.492096	+	0.852335i	−0.507863	+	0.861438i	$0.669564\pi$
$168$		0			0
$169$	−1014.00	+	1756.30i	−0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2002.50	+	3468.43i	−0.880042	+	1.52428i	−0.0287490	+	0.999587i	$0.509152\pi$
$173$	−2002.50	+	3468.43i	−0.880042	+	1.52428i	−0.851293	+	0.524691i	$0.824181\pi$
$174$		0			0
$175$	1800.00	+	3117.69i	0.777527	+	1.34672i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1428.00			0.596278			0.298139	−	0.954522i	$-0.403634\pi$
$179$	1428.00			0.596278			0.298139	+	0.954522i	$0.403634\pi$
$180$		0			0
$181$	−2226.00			−0.914129			−0.457064	−	0.889434i	$-0.651099\pi$
$181$	−2226.00			−0.914129			−0.457064	+	0.889434i	$0.651099\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	802.500	+	1389.97i	0.318924	+	0.552393i
$186$		0			0
$187$	−1888.00	+	3270.11i	−0.738311	+	1.27879i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1878.00	−	3252.79i	0.711452	−	1.23227i	−0.252860	−	0.967503i	$-0.581371\pi$
$191$	1878.00	−	3252.79i	0.711452	−	1.23227i	0.964312	−	0.264768i	$-0.0852954\pi$
$192$		0			0
$193$	1818.50	+	3149.73i	0.678231	+	1.17473i	0.975513	+	0.219940i	$0.0705863\pi$
$193$	1818.50	+	3149.73i	0.678231	+	1.17473i	−0.297283	+	0.954789i	$0.596080\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−795.000			−0.287520			−0.143760	−	0.989613i	$-0.545919\pi$
$197$	−795.000			−0.287520			−0.143760	+	0.989613i	$0.545919\pi$
$198$		0			0
$199$	2500.00			0.890554			0.445277	−	0.895393i	$-0.353105\pi$
$199$	2500.00			0.890554			0.445277	+	0.895393i	$0.353105\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1026.00	−	1777.08i	−0.354734	−	0.614418i
$204$		0			0
$205$	−615.000	+	1065.21i	−0.209529	+	0.362915i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−896.000	+	1551.92i	−0.296544	+	0.513629i
$210$		0			0
$211$	1972.00	+	3415.60i	0.643403	+	1.11441i	0.984668	+	0.174440i	$0.0558114\pi$
$211$	1972.00	+	3415.60i	0.643403	+	1.11441i	−0.341265	+	0.939967i	$0.610855\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−40.0000			−0.0126883
$216$		0			0
$217$	5904.00			1.84696
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1917.50	−	3321.21i	−0.583643	−	1.01090i
$222$		0			0
$223$	340.000	−	588.897i	0.102099	−	0.176841i	−0.810450	−	0.585807i	$-0.800777\pi$
$223$	340.000	−	588.897i	0.102099	−	0.176841i	0.912549	+	0.408967i	$0.134111\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1962.00	−	3398.28i	0.573667	−	0.993621i	−0.422518	−	0.906355i	$-0.638854\pi$
$227$	1962.00	−	3398.28i	0.573667	−	0.993621i	0.996185	−	0.0872664i	$-0.0278132\pi$
$228$		0			0
$229$	−2307.50	−	3996.71i	−0.665869	−	1.15332i	−0.979049	−	0.203625i	$-0.934728\pi$
$229$	−2307.50	−	3996.71i	−0.665869	−	1.15332i	0.313181	−	0.949694i	$-0.398606\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4493.00			1.26329			0.631644	−	0.775258i	$-0.282380\pi$
$233$	4493.00			1.26329			0.631644	+	0.775258i	$0.282380\pi$
$234$		0			0
$235$	−420.000			−0.116586
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1562.00	−	2705.46i	−0.422751	−	0.732225i	0.573457	−	0.819236i	$-0.305602\pi$
$239$	−1562.00	−	2705.46i	−0.422751	−	0.732225i	−0.996207	+	0.0870103i	$0.972269\pi$
$240$		0			0
$241$	154.500	−	267.602i	0.0412955	−	0.0715259i	−0.844639	−	0.535336i	$-0.820185\pi$
$241$	154.500	−	267.602i	0.0412955	−	0.0715259i	0.885934	+	0.463811i	$0.153518\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2382.50	+	4126.61i	−0.621275	+	1.07608i
$246$		0			0
$247$	−910.000	−	1576.17i	−0.234421	−	0.406029i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1352.00			−0.339990			−0.169995	−	0.985445i	$-0.554375\pi$
$251$	−1352.00			−0.339990			−0.169995	+	0.985445i	$0.554375\pi$
$252$		0			0
$253$	10240.0			2.54460
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1847.50	+	3199.96i	0.448420	+	0.776686i	0.998283	−	0.0585686i	$-0.0186536\pi$
$257$	1847.50	+	3199.96i	0.448420	+	0.776686i	−0.549864	+	0.835254i	$0.685320\pi$
$258$		0			0
$259$	5778.00	−	10007.8i	1.38621	−	2.40098i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−490.000	+	848.705i	−0.114885	+	0.198986i	−0.917734	−	0.397196i	$-0.869983\pi$
$263$	−490.000	+	848.705i	−0.114885	+	0.198986i	0.802849	+	0.596183i	$0.203317\pi$
$264$		0			0
$265$	1195.00	+	2069.80i	0.277012	+	0.479799i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5293.00			1.19970			0.599851	−	0.800112i	$-0.295226\pi$
$269$	5293.00			1.19970			0.599851	+	0.800112i	$0.295226\pi$
$270$		0			0
$271$	−4912.00			−1.10104			−0.550522	−	0.834821i	$-0.685571\pi$
$271$	−4912.00			−1.10104			−0.550522	+	0.834821i	$0.685571\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3200.00	−	5542.56i	−0.701699	−	1.21538i
$276$		0			0
$277$	697.000	−	1207.24i	0.151187	−	0.261863i	−0.780477	−	0.625184i	$-0.785024\pi$
$277$	697.000	−	1207.24i	0.151187	−	0.261863i	0.931664	+	0.363321i	$0.118357\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3582.50	+	6205.07i	−0.760548	+	1.31731i	0.182020	+	0.983295i	$0.441736\pi$
$281$	−3582.50	+	6205.07i	−0.760548	+	1.31731i	−0.942568	+	0.334013i	$0.891597\pi$
$282$		0			0
$283$	4670.00	+	8088.68i	0.980928	+	1.69902i	0.658794	+	0.752324i	$0.271067\pi$
$283$	4670.00	+	8088.68i	0.980928	+	1.69902i	0.322134	+	0.946694i	$0.395600\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8856.00			1.82144
$288$		0			0
$289$	−1432.00			−0.291472
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3436.50	−	5952.19i	−0.685196	−	1.18679i	−0.973375	−	0.229218i	$-0.926383\pi$
$293$	−3436.50	−	5952.19i	−0.685196	−	1.18679i	0.288179	−	0.957577i	$-0.406950\pi$
$294$		0			0
$295$	−80.0000	+	138.564i	−0.0157891	+	0.0273475i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5200.00	+	9006.66i	−1.00577	+	1.74204i
$300$		0			0
$301$	144.000	+	249.415i	0.0275748	+	0.0477610i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2075.00			0.389555
$306$		0			0
$307$	204.000			0.0379247			0.0189624	−	0.999820i	$-0.493964\pi$
$307$	204.000			0.0379247			0.0189624	+	0.999820i	$0.493964\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2028.00	+	3512.60i	0.369766	+	0.640454i	0.989529	−	0.144335i	$-0.0461044\pi$
$311$	2028.00	+	3512.60i	0.369766	+	0.640454i	−0.619762	+	0.784789i	$0.712771\pi$
$312$		0			0
$313$	−547.500	+	948.298i	−0.0988707	+	0.171249i	−0.911217	−	0.411926i	$-0.864856\pi$
$313$	−547.500	+	948.298i	−0.0988707	+	0.171249i	0.812347	+	0.583175i	$0.198190\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5209.50	−	9023.12i	0.923012	−	1.59870i	0.128283	−	0.991738i	$-0.459053\pi$
$317$	5209.50	−	9023.12i	0.923012	−	1.59870i	0.794728	−	0.606965i	$-0.207613\pi$
$318$		0			0
$319$	1824.00	+	3159.26i	0.320139	+	0.554497i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1652.00			0.284581
$324$		0			0
$325$	6500.00			1.10940
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1512.00	+	2618.86i	0.253372	+	0.438852i
$330$		0			0
$331$	−1798.00	+	3114.23i	−0.298571	+	0.517140i	−0.975809	−	0.218624i	$-0.929843\pi$
$331$	−1798.00	+	3114.23i	−0.298571	+	0.517140i	0.677238	+	0.735764i	$0.263177\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	550.000	−	952.628i	0.0897006	−	0.155366i
$336$		0			0
$337$	1455.00	+	2520.13i	0.235190	+	0.407360i	0.959328	−	0.282295i	$-0.0910956\pi$
$337$	1455.00	+	2520.13i	0.235190	+	0.407360i	−0.724138	+	0.689655i	$0.757762\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10496.0			−1.66683
$342$		0			0
$343$	21960.0			3.45693
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2538.00	−	4395.94i	−0.392643	−	0.680077i	0.600155	−	0.799884i	$-0.295106\pi$
$347$	−2538.00	−	4395.94i	−0.392643	−	0.680077i	−0.992797	+	0.119807i	$0.961772\pi$
$348$		0			0
$349$	−4059.00	+	7030.39i	−0.622560	+	1.07830i	0.366448	+	0.930439i	$0.380574\pi$
$349$	−4059.00	+	7030.39i	−0.622560	+	1.07830i	−0.989007	+	0.147866i	$0.952760\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4631.00	+	8021.13i	−0.698253	+	1.20941i	0.270819	+	0.962630i	$0.412706\pi$
$353$	−4631.00	+	8021.13i	−0.698253	+	1.20941i	−0.969072	+	0.246779i	$0.920628\pi$
$354$		0			0
$355$	2210.00	+	3827.83i	0.330407	+	0.572282i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1248.00			−0.183473			−0.0917367	−	0.995783i	$-0.529242\pi$
$359$	−1248.00			−0.183473			−0.0917367	+	0.995783i	$0.529242\pi$
$360$		0			0
$361$	−6075.00			−0.885698
$362$		0			0
$363$		0			0
$364$		0			0
$365$	192.500	+	333.420i	0.0276052	+	0.0478137i
$366$		0			0
$367$	−4940.00	+	8556.33i	−0.702632	+	1.21699i	0.264907	+	0.964274i	$0.414659\pi$
$367$	−4940.00	+	8556.33i	−0.702632	+	1.21699i	−0.967539	+	0.252720i	$0.918675\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8604.00	−	14902.6i	1.20404	−	2.08545i
$372$		0			0
$373$	4889.00	+	8468.00i	0.678667	+	1.17549i	0.975382	+	0.220520i	$0.0707754\pi$
$373$	4889.00	+	8468.00i	0.678667	+	1.17549i	−0.296715	+	0.954966i	$0.595891\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3705.00			−0.506146
$378$		0			0
$379$	3260.00			0.441834			0.220917	−	0.975293i	$-0.429095\pi$
$379$	3260.00			0.441834			0.220917	+	0.975293i	$0.429095\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	810.000	+	1402.96i	0.108065	+	0.187175i	0.914986	−	0.403484i	$-0.132201\pi$
$383$	810.000	+	1402.96i	0.108065	+	0.187175i	−0.806921	+	0.590659i	$0.798868\pi$
$384$		0			0
$385$	5760.00	−	9976.61i	0.762485	−	1.32066i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3455.00	−	5984.24i	0.450323	−	0.779981i	−0.548083	−	0.836424i	$-0.684642\pi$
$389$	3455.00	−	5984.24i	0.450323	−	0.779981i	0.998406	+	0.0564423i	$0.0179757\pi$
$390$		0			0
$391$	−4720.00	−	8175.28i	−0.610488	−	1.05740i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−400.000			−0.0509524
$396$		0			0
$397$	−1705.00			−0.215545			−0.107773	−	0.994176i	$-0.534372\pi$
$397$	−1705.00			−0.215545			−0.107773	+	0.994176i	$0.534372\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2002.50	−	3468.43i	−0.249377	−	0.431933i	0.713976	−	0.700170i	$-0.246892\pi$
$401$	−2002.50	−	3468.43i	−0.249377	−	0.431933i	−0.963353	+	0.268237i	$0.913559\pi$
$402$		0			0
$403$	5330.00	−	9231.83i	0.658824	−	1.14112i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10272.0	+	17791.6i	−1.25102	+	2.16683i
$408$		0			0
$409$	2692.50	+	4663.55i	0.325515	+	0.563808i	0.981616	−	0.190864i	$-0.0611291\pi$
$409$	2692.50	+	4663.55i	0.325515	+	0.563808i	−0.656102	+	0.754673i	$0.727796\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1152.00			0.137255
$414$		0			0
$415$	−6340.00			−0.749924
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2820.00	+	4884.38i	0.328797	+	0.569493i	0.982273	−	0.187454i	$-0.0600234\pi$
$419$	2820.00	+	4884.38i	0.328797	+	0.569493i	−0.653476	+	0.756947i	$0.726690\pi$
$420$		0			0
$421$	−2165.50	+	3750.76i	−0.250689	+	0.434206i	−0.963716	−	0.266931i	$-0.913990\pi$
$421$	−2165.50	+	3750.76i	−0.250689	+	0.434206i	0.713027	+	0.701137i	$0.247324\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2950.00	+	5109.55i	−0.336697	+	0.583175i
$426$		0			0
$427$	−7470.00	−	12938.4i	−0.846601	−	1.46636i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	11596.0			1.29596			0.647981	−	0.761656i	$-0.275614\pi$
$431$	11596.0			1.29596			0.647981	+	0.761656i	$0.275614\pi$
$432$		0			0
$433$	−2765.00			−0.306876			−0.153438	−	0.988158i	$-0.549035\pi$
$433$	−2765.00			−0.306876			−0.153438	+	0.988158i	$0.549035\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2240.00	−	3879.79i	−0.245203	−	0.424704i
$438$		0			0
$439$	−3966.00	+	6869.31i	−0.431177	+	0.746821i	−0.996975	−	0.0777231i	$-0.975235\pi$
$439$	−3966.00	+	6869.31i	−0.431177	+	0.746821i	0.565798	+	0.824544i	$0.308568\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6134.00	+	10624.4i	−0.657867	+	1.13946i	0.323300	+	0.946297i	$0.395208\pi$
$443$	−6134.00	+	10624.4i	−0.657867	+	1.13946i	−0.981167	+	0.193163i	$0.938126\pi$
$444$		0			0
$445$	307.500	+	532.606i	0.0327571	+	0.0567369i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10478.0			1.10131			0.550654	−	0.834734i	$-0.314378\pi$
$449$	10478.0			1.10131			0.550654	+	0.834734i	$0.314378\pi$
$450$		0			0
$451$	−15744.0			−1.64380
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5850.00	+	10132.5i	0.602752	+	1.04400i
$456$		0			0
$457$	8064.50	−	13968.1i	0.825474	−	1.42976i	−0.0760834	−	0.997101i	$-0.524242\pi$
$457$	8064.50	−	13968.1i	0.825474	−	1.42976i	0.901557	−	0.432661i	$-0.142425\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−517.000	+	895.470i	−0.0522323	+	0.0904690i	−0.890959	−	0.454083i	$-0.849967\pi$
$461$	−517.000	+	895.470i	−0.0522323	+	0.0904690i	0.838727	+	0.544552i	$0.183300\pi$
$462$		0			0
$463$	−4540.00	−	7863.51i	−0.455706	−	0.789305i	0.543023	−	0.839718i	$-0.317280\pi$
$463$	−4540.00	−	7863.51i	−0.455706	−	0.789305i	−0.998728	+	0.0504127i	$0.983946\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18504.0			−1.83354			−0.916770	−	0.399416i	$-0.869213\pi$
$467$	−18504.0			−1.83354			−0.916770	+	0.399416i	$0.869213\pi$
$468$		0			0
$469$	−7920.00			−0.779769
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−256.000	−	443.405i	−0.0248856	−	0.0431031i
$474$		0			0
$475$	−1400.00	+	2424.87i	−0.135235	+	0.234233i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8506.00	−	14732.8i	0.811376	−	1.40534i	−0.100525	−	0.994935i	$-0.532052\pi$
$479$	8506.00	−	14732.8i	0.811376	−	1.40534i	0.911901	−	0.410410i	$-0.134614\pi$
$480$		0			0
$481$	−10432.5	−	18069.6i	−0.988942	−	1.71290i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6730.00			0.630090
$486$		0			0
$487$	−6140.00			−0.571314			−0.285657	−	0.958332i	$-0.592212\pi$
$487$	−6140.00			−0.571314			−0.285657	+	0.958332i	$0.592212\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5472.00	−	9477.78i	−0.502949	−	0.871133i	−0.999994	−	0.00340843i	$-0.998915\pi$
$491$	−5472.00	−	9477.78i	−0.502949	−	0.871133i	0.497045	−	0.867725i	$-0.334418\pi$
$492$		0			0
$493$	1681.50	−	2912.44i	0.153612	−	0.266065i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15912.0	−	27560.4i	1.43612	−	2.48743i
$498$		0			0
$499$	2604.00	+	4510.26i	0.233609	+	0.404623i	0.958868	−	0.283854i	$-0.0916130\pi$
$499$	2604.00	+	4510.26i	0.233609	+	0.404623i	−0.725258	+	0.688477i	$0.758280\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1440.00			−0.127647			−0.0638235	−	0.997961i	$-0.520329\pi$
$503$	−1440.00			−0.127647			−0.0638235	+	0.997961i	$0.520329\pi$
$504$		0			0
$505$	4170.00			0.367451
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1155.00	+	2000.52i	0.100579	+	0.174207i	0.911923	−	0.410361i	$-0.134597\pi$
$509$	1155.00	+	2000.52i	0.100579	+	0.174207i	−0.811345	+	0.584568i	$0.801264\pi$
$510$		0			0
$511$	1386.00	−	2400.62i	0.119986	−	0.207822i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2650.00	+	4589.93i	−0.226744	+	0.392731i
$516$		0			0
$517$	−2688.00	−	4655.75i	−0.228662	−	0.396054i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9910.00			0.833330			0.416665	−	0.909060i	$-0.363199\pi$
$521$	9910.00			0.833330			0.416665	+	0.909060i	$0.363199\pi$
$522$		0			0
$523$	6640.00			0.555157			0.277578	−	0.960703i	$-0.410468\pi$
$523$	6640.00			0.555157			0.277578	+	0.960703i	$0.410468\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4838.00	+	8379.66i	0.399899	+	0.692645i
$528$		0			0
$529$	−6716.50	+	11633.3i	−0.552026	+	0.956137i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	7995.00	−	13847.7i	0.649722	−	1.12535i
$534$		0			0
$535$	750.000	+	1299.04i	0.0606081	+	0.104976i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−60992.0			−4.87405
$540$		0			0
$541$	−15969.0			−1.26906			−0.634530	−	0.772899i	$-0.718806\pi$
$541$	−15969.0			−1.26906			−0.634530	+	0.772899i	$0.718806\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1392.50	+	2411.88i	0.109446	+	0.189566i
$546$		0			0
$547$	−250.000	+	433.013i	−0.0195416	+	0.0338470i	−0.875631	−	0.482981i	$-0.839554\pi$
$547$	−250.000	+	433.013i	−0.0195416	+	0.0338470i	0.856089	+	0.516828i	$0.172887\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	798.000	−	1382.18i	0.0616986	−	0.106865i
$552$		0			0
$553$	1440.00	+	2494.15i	0.110732	+	0.191794i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3845.00			0.292492			0.146246	−	0.989248i	$-0.453281\pi$
$557$	3845.00			0.292492			0.146246	+	0.989248i	$0.453281\pi$
$558$		0			0
$559$	520.000			0.0393446
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1594.00	−	2760.89i	−0.119323	−	0.206674i	0.800176	−	0.599765i	$-0.204739\pi$
$563$	−1594.00	−	2760.89i	−0.119323	−	0.206674i	−0.919500	+	0.393091i	$0.871406\pi$
$564$		0			0
$565$	−4042.50	+	7001.82i	−0.301008	+	0.521360i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8631.50	−	14950.2i	0.635943	−	1.10148i	−0.350372	−	0.936611i	$-0.613945\pi$
$569$	8631.50	−	14950.2i	0.635943	−	1.10148i	0.986315	−	0.164874i	$-0.0527218\pi$
$570$		0			0
$571$	2328.00	+	4032.21i	0.170620	+	0.295522i	0.938637	−	0.344908i	$-0.112090\pi$
$571$	2328.00	+	4032.21i	0.170620	+	0.295522i	−0.768017	+	0.640429i	$0.778756\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	16000.0			1.16043
$576$		0			0
$577$	−23209.0			−1.67453			−0.837265	−	0.546798i	$-0.815847\pi$
$577$	−23209.0			−1.67453			−0.837265	+	0.546798i	$0.815847\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	22824.0	+	39532.3i	1.62977	+	2.82285i
$582$		0			0
$583$	−15296.0	+	26493.4i	−1.08661	+	1.88207i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8812.00	+	15262.8i	−0.619608	+	1.07319i	0.369949	+	0.929052i	$0.379375\pi$
$587$	−8812.00	+	15262.8i	−0.619608	+	1.07319i	−0.989557	+	0.144141i	$0.953958\pi$
$588$		0			0
$589$	2296.00	+	3976.79i	0.160620	+	0.278202i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17695.0			−1.22537			−0.612687	−	0.790326i	$-0.709911\pi$
$593$	−17695.0			−1.22537			−0.612687	+	0.790326i	$0.709911\pi$
$594$		0			0
$595$	−10620.0			−0.731727
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1374.00	+	2379.84i	0.0937231	+	0.162333i	0.909075	−	0.416633i	$-0.136790\pi$
$599$	1374.00	+	2379.84i	0.0937231	+	0.162333i	−0.815352	+	0.578966i	$0.803457\pi$
$600$		0			0
$601$	−9789.50	+	16955.9i	−0.664429	+	1.15083i	0.315010	+	0.949088i	$0.397992\pi$
$601$	−9789.50	+	16955.9i	−0.664429	+	1.15083i	−0.979440	+	0.201737i	$0.935341\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6912.50	+	11972.8i	−0.464518	+	0.804568i
$606$		0			0
$607$	−8902.00	−	15418.7i	−0.595257	−	1.03102i	−0.993511	−	0.113740i	$-0.963717\pi$
$607$	−8902.00	−	15418.7i	−0.595257	−	1.03102i	0.398253	−	0.917275i	$-0.369616\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5460.00			0.361519
$612$		0			0
$613$	−22690.0			−1.49501			−0.747504	−	0.664257i	$-0.768748\pi$
$613$	−22690.0			−1.49501			−0.747504	+	0.664257i	$0.768748\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	877.500	+	1519.87i	0.0572558	+	0.0991699i	0.893233	−	0.449595i	$-0.148432\pi$
$617$	877.500	+	1519.87i	0.0572558	+	0.0991699i	−0.835977	+	0.548765i	$0.815098\pi$
$618$		0			0
$619$	3460.00	−	5992.90i	0.224667	−	0.389135i	−0.731552	−	0.681785i	$-0.761204\pi$
$619$	3460.00	−	5992.90i	0.224667	−	0.389135i	0.956220	+	0.292650i	$0.0945371\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2214.00	−	3834.76i	0.142379	−	0.246607i
$624$		0			0
$625$	−3437.50	−	5953.92i	−0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	18939.0			1.20055
$630$		0			0
$631$	10744.0			0.677832			0.338916	−	0.940817i	$-0.389940\pi$
$631$	10744.0			0.677832			0.338916	+	0.940817i	$0.389940\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2840.00	+	4919.02i	0.177483	+	0.307410i
$636$		0			0
$637$	30972.5	−	53645.9i	1.92649	−	3.33678i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4965.50	−	8600.50i	0.305968	−	0.529952i	−0.671508	−	0.740997i	$-0.734353\pi$
$641$	4965.50	−	8600.50i	0.305968	−	0.529952i	0.977476	+	0.211045i	$0.0676866\pi$
$642$		0			0
$643$	−2034.00	−	3522.99i	−0.124748	−	0.216070i	0.796886	−	0.604129i	$-0.206479\pi$
$643$	−2034.00	−	3522.99i	−0.124748	−	0.216070i	−0.921635	+	0.388059i	$0.873146\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−17540.0			−1.06579			−0.532897	−	0.846180i	$-0.678897\pi$
$647$	−17540.0			−1.06579			−0.532897	+	0.846180i	$0.678897\pi$
$648$		0			0
$649$	−2048.00			−0.123869
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15045.0	−	26058.7i	−0.901618	−	1.56165i	−0.825394	−	0.564557i	$-0.809047\pi$
$653$	−15045.0	−	26058.7i	−0.901618	−	1.56165i	−0.0762235	−	0.997091i	$-0.524286\pi$
$654$		0			0
$655$	290.000	−	502.295i	0.0172996	−	0.0299638i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4410.00	+	7638.34i	−0.260682	+	0.451514i	−0.966423	−	0.256955i	$-0.917281\pi$
$659$	−4410.00	+	7638.34i	−0.260682	+	0.451514i	0.705742	+	0.708469i	$0.250614\pi$
$660$		0			0
$661$	−6659.50	−	11534.6i	−0.391868	−	0.678735i	0.600828	−	0.799378i	$-0.294838\pi$
$661$	−6659.50	−	11534.6i	−0.391868	−	0.678735i	−0.992696	+	0.120643i	$0.961504\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5040.00			−0.293899
$666$		0			0
$667$	−9120.00			−0.529427
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13280.0	+	23001.6i	0.764037	+	1.32335i
$672$		0			0
$673$	−7017.50	+	12154.7i	−0.401939	+	0.696178i	−0.993960	−	0.109744i	$-0.964997\pi$
$673$	−7017.50	+	12154.7i	−0.401939	+	0.696178i	0.592021	+	0.805922i	$0.298330\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4705.00	+	8149.30i	−0.267102	+	0.462634i	−0.968112	−	0.250517i	$-0.919399\pi$
$677$	−4705.00	+	8149.30i	−0.267102	+	0.462634i	0.701010	+	0.713151i	$0.252733\pi$
$678$		0			0
$679$	−24228.0	−	41964.1i	−1.36935	−	2.37178i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1060.00			−0.0593847			−0.0296924	−	0.999559i	$-0.509453\pi$
$683$	−1060.00			−0.0593847			−0.0296924	+	0.999559i	$0.509453\pi$
$684$		0			0
$685$	−3075.00			−0.171518
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15535.0	−	26907.4i	−0.858979	−	1.48779i
$690$		0			0
$691$	−4270.00	+	7395.86i	−0.235077	+	0.407166i	−0.959295	−	0.282405i	$-0.908868\pi$
$691$	−4270.00	+	7395.86i	−0.235077	+	0.407166i	0.724218	+	0.689571i	$0.242201\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3630.00	+	6287.34i	−0.198121	+	0.343155i
$696$		0			0
$697$	7257.00	+	12569.5i	0.394374	+	0.683075i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−17259.0			−0.929905			−0.464953	−	0.885336i	$-0.653929\pi$
$701$	−17259.0			−0.929905			−0.464953	+	0.885336i	$0.653929\pi$
$702$		0			0
$703$	8988.00			0.482203
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−15012.0	−	26001.5i	−0.798563	−	1.38315i
$708$		0			0
$709$	12072.5	−	20910.2i	0.639481	−	1.10761i	−0.346065	−	0.938210i	$-0.612482\pi$
$709$	12072.5	−	20910.2i	0.639481	−	1.10761i	0.985547	−	0.169404i	$-0.0541842\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13120.0	−	22724.5i	0.689127	−	1.19360i
$714$		0			0
$715$	−10400.0	−	18013.3i	−0.543969	−	0.942182i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17792.0			0.922851			0.461425	−	0.887179i	$-0.347338\pi$
$719$	17792.0			0.922851			0.461425	+	0.887179i	$0.347338\pi$
$720$		0			0
$721$	38160.0			1.97109
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2850.00	+	4936.34i	0.145995	+	0.252871i
$726$		0			0
$727$	−5310.00	+	9197.19i	−0.270890	+	0.469195i	−0.969090	−	0.246708i	$-0.920651\pi$
$727$	−5310.00	+	9197.19i	−0.270890	+	0.469195i	0.698200	+	0.715903i	$0.253985\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−236.000	+	408.764i	−0.0119409	+	0.0206822i
$732$		0			0
$733$	4325.00	+	7491.12i	0.217937	+	0.377477i	0.954177	−	0.299243i	$-0.0967341\pi$
$733$	4325.00	+	7491.12i	0.217937	+	0.377477i	−0.736240	+	0.676720i	$0.763401\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14080.0			0.703722
$738$		0			0
$739$	4652.00			0.231565			0.115783	−	0.993275i	$-0.463062\pi$
$739$	4652.00			0.231565			0.115783	+	0.993275i	$0.463062\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	16934.0	+	29330.5i	0.836135	+	1.44823i	0.893103	+	0.449852i	$0.148523\pi$
$743$	16934.0	+	29330.5i	0.836135	+	1.44823i	−0.0569682	+	0.998376i	$0.518143\pi$
$744$		0			0
$745$	2097.50	−	3632.98i	0.103150	−	0.178660i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5400.00	−	9353.07i	0.263434	−	0.456280i
$750$		0			0
$751$	6426.00	+	11130.2i	0.312234	+	0.540806i	0.978846	−	0.204600i	$-0.0655892\pi$
$751$	6426.00	+	11130.2i	0.312234	+	0.540806i	−0.666611	+	0.745406i	$0.732256\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10280.0			−0.495533
$756$		0			0
$757$	−9730.00			−0.467164			−0.233582	−	0.972337i	$-0.575045\pi$
$757$	−9730.00			−0.467164			−0.233582	+	0.972337i	$0.575045\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16680.5	−	28891.5i	−0.794570	−	1.37624i	−0.923112	−	0.384531i	$-0.874363\pi$
$761$	−16680.5	−	28891.5i	−0.794570	−	1.37624i	0.128542	−	0.991704i	$-0.458970\pi$
$762$		0			0
$763$	10026.0	−	17365.5i	0.475708	−	0.823951i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1040.00	−	1801.33i	0.0489599	−	0.0848010i
$768$		0			0
$769$	13278.5	+	22999.0i	0.622672	+	1.07850i	0.988986	+	0.148009i	$0.0472863\pi$
$769$	13278.5	+	22999.0i	0.622672	+	1.07850i	−0.366314	+	0.930491i	$0.619380\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18333.0			0.853030			0.426515	−	0.904480i	$-0.359741\pi$
$773$	18333.0			0.853030			0.426515	+	0.904480i	$0.359741\pi$
$774$		0			0
$775$	−16400.0			−0.760136
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3444.00	+	5965.18i	0.158401	+	0.274358i
$780$		0			0
$781$	−28288.0	+	48996.3i	−1.29606	+	2.24484i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2747.50	+	4758.81i	−0.124920	+	0.216368i
$786$		0			0
$787$	18688.0	+	32368.6i	0.846449	+	1.46609i	0.884357	+	0.466811i	$0.154597\pi$
$787$	18688.0	+	32368.6i	0.846449	+	1.46609i	−0.0379081	+	0.999281i	$0.512069\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	58212.0			2.61666
$792$		0			0
$793$	−26975.0			−1.20796
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8037.50	+	13921.4i	0.357218	+	0.618720i	0.987495	−	0.157651i	$-0.0503919\pi$
$797$	8037.50	+	13921.4i	0.357218	+	0.618720i	−0.630277	+	0.776370i	$0.717059\pi$
$798$		0			0
$799$	−2478.00	+	4292.02i	−0.109719	+	0.190039i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2464.00	+	4267.77i	−0.108285	+	0.187555i
$804$		0			0
$805$	14400.0	+	24941.5i	0.630476	+	1.09202i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−32183.0			−1.39863			−0.699316	−	0.714812i	$-0.746512\pi$
$809$	−32183.0			−1.39863			−0.699316	+	0.714812i	$0.746512\pi$
$810$		0			0
$811$	−1424.00			−0.0616565			−0.0308282	−	0.999525i	$-0.509814\pi$
$811$	−1424.00			−0.0616565			−0.0308282	+	0.999525i	$0.509814\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	180.000	+	311.769i	0.00773635	+	0.0133998i
$816$		0			0
$817$	−112.000	+	193.990i	−0.00479606	+	0.00830703i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13480.5	+	23348.9i	−0.573048	+	0.992549i	0.423202	+	0.906035i	$0.360906\pi$
$821$	−13480.5	+	23348.9i	−0.573048	+	0.992549i	−0.996251	+	0.0865136i	$0.972427\pi$
$822$		0			0
$823$	10550.0	+	18273.1i	0.446841	+	0.773951i	0.998178	−	0.0603314i	$-0.0192157\pi$
$823$	10550.0	+	18273.1i	0.446841	+	0.773951i	−0.551338	+	0.834282i	$0.685882\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35120.0			−1.47671			−0.738357	−	0.674410i	$-0.764398\pi$
$827$	−35120.0			−1.47671			−0.738357	+	0.674410i	$0.764398\pi$
$828$		0			0
$829$	21238.0			0.889778			0.444889	−	0.895586i	$-0.353243\pi$
$829$	21238.0			0.889778			0.444889	+	0.895586i	$0.353243\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	28113.5	+	48694.0i	1.16936	+	2.02539i
$834$		0			0
$835$	5310.00	−	9197.19i	0.220072	−	0.381176i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13044.0	+	22592.9i	−0.536745	+	0.929669i	0.462332	+	0.886707i	$0.347013\pi$
$839$	−13044.0	+	22592.9i	−0.536745	+	0.929669i	−0.999077	+	0.0429623i	$0.986320\pi$
$840$		0			0
$841$	10570.0	+	18307.8i	0.433392	+	0.750657i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10140.0			0.412813
$846$		0			0
$847$	99540.0			4.03806
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−25680.0	−	44479.1i	−1.03443	−	1.79168i
$852$		0			0
$853$	5561.00	−	9631.93i	0.223218	−	0.386625i	−0.732565	−	0.680697i	$-0.761677\pi$
$853$	5561.00	−	9631.93i	0.223218	−	0.386625i	0.955783	+	0.294072i	$0.0950105\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1492.50	+	2585.09i	−0.0594899	+	0.103039i	−0.894237	−	0.447595i	$-0.852281\pi$
$857$	−1492.50	+	2585.09i	−0.0594899	+	0.103039i	0.834747	+	0.550634i	$0.185614\pi$
$858$		0			0
$859$	24774.0	+	42909.8i	0.984026	+	1.70438i	0.646189	+	0.763178i	$0.276362\pi$
$859$	24774.0	+	42909.8i	0.984026	+	1.70438i	0.337837	+	0.941205i	$0.390305\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−21688.0			−0.855467			−0.427734	−	0.903905i	$-0.640688\pi$
$863$	−21688.0			−0.855467			−0.427734	+	0.903905i	$0.640688\pi$
$864$		0			0
$865$	20025.0			0.787133
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2560.00	−	4434.05i	−0.0999333	−	0.173090i
$870$		0			0
$871$	−7150.00	+	12384.2i	−0.278150	+	0.481770i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	20250.0	−	35074.0i	0.782371	−	1.35511i
$876$		0			0
$877$	−5467.50	−	9469.99i	−0.210518	−	0.364628i	0.741359	−	0.671109i	$-0.234182\pi$
$877$	−5467.50	−	9469.99i	−0.210518	−	0.364628i	−0.951877	+	0.306481i	$0.900848\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	40126.0			1.53448			0.767241	−	0.641358i	$-0.221629\pi$
$881$	40126.0			1.53448			0.767241	+	0.641358i	$0.221629\pi$
$882$		0			0
$883$	−42748.0			−1.62920			−0.814601	−	0.580022i	$-0.803044\pi$
$883$	−42748.0			−1.62920			−0.814601	+	0.580022i	$0.803044\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−13748.0	−	23812.2i	−0.520420	−	0.901394i	−0.999718	−	0.0237419i	$-0.992442\pi$
$887$	−13748.0	−	23812.2i	−0.520420	−	0.901394i	0.479298	−	0.877652i	$-0.340891\pi$
$888$		0			0
$889$	20448.0	−	35417.0i	0.771433	−	1.33616i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1176.00	+	2036.89i	−0.0440687	+	0.0763292i
$894$		0			0
$895$	−3570.00	−	6183.42i	−0.133332	−	0.230937i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9348.00			0.346800
$900$		0			0
$901$	28202.0			1.04278
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5565.00	+	9638.86i	0.204405	+	0.354040i
$906$		0			0
$907$	10840.0	−	18775.4i	0.396843	−	0.687352i	−0.596492	−	0.802619i	$-0.703439\pi$
$907$	10840.0	−	18775.4i	0.396843	−	0.687352i	0.993334	+	0.115268i	$0.0367725\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3412.00	+	5909.76i	−0.124088	+	0.214928i	−0.921376	−	0.388672i	$-0.872934\pi$
$911$	−3412.00	+	5909.76i	−0.124088	+	0.214928i	0.797288	+	0.603599i	$0.206267\pi$
$912$		0			0
$913$	−40576.0	−	70279.7i	−1.47083	−	2.54756i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4176.00			−0.150386
$918$		0			0
$919$	26512.0			0.951632			0.475816	−	0.879545i	$-0.342153\pi$
$919$	26512.0			0.951632			0.475816	+	0.879545i	$0.342153\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−28730.0	−	49761.8i	−1.02455	−	1.77457i
$924$		0			0
$925$	−16050.0	+	27799.4i	−0.570509	+	0.988150i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−11298.5	+	19569.6i	−0.399022	+	0.691127i	−0.993606	−	0.112907i	$-0.963984\pi$
$929$	−11298.5	+	19569.6i	−0.399022	+	0.691127i	0.594583	+	0.804034i	$0.297317\pi$
$930$		0			0
$931$	13342.0	+	23109.0i	0.469674	+	0.813499i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	18880.0			0.660366
$936$		0			0
$937$	38115.0			1.32888			0.664441	−	0.747341i	$-0.268670\pi$
$937$	38115.0			1.32888			0.664441	+	0.747341i	$0.268670\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−21742.5	−	37659.1i	−0.753226	−	1.30462i	−0.946252	−	0.323431i	$-0.895164\pi$
$941$	−21742.5	−	37659.1i	−0.753226	−	1.30462i	0.193026	−	0.981194i	$-0.438170\pi$
$942$		0			0
$943$	19680.0	−	34086.8i	0.679607	−	1.17711i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22710.0	+	39334.9i	−0.779278	+	1.34975i	0.153081	+	0.988214i	$0.451080\pi$
$947$	−22710.0	+	39334.9i	−0.779278	+	1.34975i	−0.932359	+	0.361535i	$0.882253\pi$
$948$		0			0
$949$	−2502.50	−	4334.46i	−0.0856002	−	0.148264i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	32505.0			1.10487			0.552435	−	0.833556i	$-0.313699\pi$
$953$	32505.0			1.10487			0.552435	+	0.833556i	$0.313699\pi$
$954$		0			0
$955$	−18780.0			−0.636342
$956$		0			0
$957$		0			0
$958$		0			0
$959$	11070.0	+	19173.8i	0.372752	+	0.645625i
$960$		0			0
$961$	1447.50	−	2507.14i	0.0485885	−	0.0841578i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9092.50	−	15748.7i	0.303314	−	0.525355i
$966$		0			0
$967$	−7138.00	−	12363.4i	−0.237376	−	0.411147i	0.722585	−	0.691283i	$-0.242954\pi$
$967$	−7138.00	−	12363.4i	−0.237376	−	0.411147i	−0.959961	+	0.280135i	$0.909621\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	35844.0			1.18464			0.592322	−	0.805702i	$-0.298211\pi$
$971$	35844.0			1.18464			0.592322	+	0.805702i	$0.298211\pi$
$972$		0			0
$973$	52272.0			1.72226
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−12135.0	−	21018.4i	−0.397373	−	0.688270i	0.596028	−	0.802963i	$-0.296745\pi$
$977$	−12135.0	−	21018.4i	−0.397373	−	0.688270i	−0.993401	+	0.114694i	$0.963411\pi$
$978$		0			0
$979$	−3936.00	+	6817.35i	−0.128493	+	0.222557i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9576.00	−	16586.1i	0.310709	−	0.538164i	−0.667807	−	0.744334i	$-0.732767\pi$
$983$	9576.00	−	16586.1i	0.310709	−	0.538164i	0.978516	+	0.206171i	$0.0661002\pi$
$984$		0			0
$985$	1987.50	+	3442.45i	0.0642914	+	0.111356i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1280.00			0.0411543
$990$		0			0
$991$	−9164.00			−0.293748			−0.146874	−	0.989155i	$-0.546921\pi$
$991$	−9164.00			−0.293748			−0.146874	+	0.989155i	$0.546921\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6250.00	−	10825.3i	−0.199134	−	0.344910i
$996$		0			0
$997$	−18129.5	+	31401.2i	−0.575895	+	0.997479i	0.420049	+	0.907501i	$0.362013\pi$
$997$	−18129.5	+	31401.2i	−0.575895	+	0.997479i	−0.995944	+	0.0899775i	$0.971320\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.i.c.217.1		2
3.2	odd	2		648.4.i.j.217.1		2
9.2	odd	6		648.4.a.a.1.1	✓	1
9.4	even	3	inner	648.4.i.c.433.1		2
9.5	odd	6		648.4.i.j.433.1		2
9.7	even	3		648.4.a.b.1.1	yes	1
36.7	odd	6		1296.4.a.f.1.1		1
36.11	even	6		1296.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.a.1.1	✓	1	9.2	odd	6
648.4.a.b.1.1	yes	1	9.7	even	3
648.4.i.c.217.1		2	1.1	even	1	trivial
648.4.i.c.433.1		2	9.4	even	3	inner
648.4.i.j.217.1		2	3.2	odd	2
648.4.i.j.433.1		2	9.5	odd	6
1296.4.a.c.1.1		1	36.11	even	6
1296.4.a.f.1.1		1	36.7	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.50000 - 4.33013i) q^{5} +(-18.0000 + 31.1769i) q^{7} +O(q^{10})\)	\(q+(-2.50000 - 4.33013i) q^{5} +(-18.0000 + 31.1769i) q^{7} +(32.0000 - 55.4256i) q^{11} +(32.5000 + 56.2917i) q^{13} -59.0000 q^{17} -28.0000 q^{19} +(80.0000 + 138.564i) q^{23} +(50.0000 - 86.6025i) q^{25} +(-28.5000 + 49.3634i) q^{29} +(-82.0000 - 142.028i) q^{31} +180.000 q^{35} -321.000 q^{37} +(-123.000 - 213.042i) q^{41} +(4.00000 - 6.92820i) q^{43} +(42.0000 - 72.7461i) q^{47} +(-476.500 - 825.322i) q^{49} -478.000 q^{53} -320.000 q^{55} +(-16.0000 - 27.7128i) q^{59} +(-207.500 + 359.401i) q^{61} +(162.500 - 281.458i) q^{65} +(110.000 + 190.526i) q^{67} -884.000 q^{71} -77.0000 q^{73} +(1152.00 + 1995.32i) q^{77} +(40.0000 - 69.2820i) q^{79} +(634.000 - 1098.12i) q^{83} +(147.500 + 255.477i) q^{85} -123.000 q^{89} -2340.00 q^{91} +(70.0000 + 121.244i) q^{95} +(-673.000 + 1165.67i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{5} - 36 q^{7}+O(q^{10}) \) 2 * q - 5 * q^5 - 36 * q^7	\( 2 q - 5 q^{5} - 36 q^{7} + 64 q^{11} + 65 q^{13} - 118 q^{17} - 56 q^{19} + 160 q^{23} + 100 q^{25} - 57 q^{29} - 164 q^{31} + 360 q^{35} - 642 q^{37} - 246 q^{41} + 8 q^{43} + 84 q^{47} - 953 q^{49} - 956 q^{53} - 640 q^{55} - 32 q^{59} - 415 q^{61} + 325 q^{65} + 220 q^{67} - 1768 q^{71} - 154 q^{73} + 2304 q^{77} + 80 q^{79} + 1268 q^{83} + 295 q^{85} - 246 q^{89} - 4680 q^{91} + 140 q^{95} - 1346 q^{97}+O(q^{100}) \) 2 * q - 5 * q^5 - 36 * q^7 + 64 * q^11 + 65 * q^13 - 118 * q^17 - 56 * q^19 + 160 * q^23 + 100 * q^25 - 57 * q^29 - 164 * q^31 + 360 * q^35 - 642 * q^37 - 246 * q^41 + 8 * q^43 + 84 * q^47 - 953 * q^49 - 956 * q^53 - 640 * q^55 - 32 * q^59 - 415 * q^61 + 325 * q^65 + 220 * q^67 - 1768 * q^71 - 154 * q^73 + 2304 * q^77 + 80 * q^79 + 1268 * q^83 + 295 * q^85 - 246 * q^89 - 4680 * q^91 + 140 * q^95 - 1346 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more