LMFDB - Embedded newform 648.4.i.j.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	648.217
Dual form			648.4.i.j.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	0.955901	−	0.293691i	$-0.0948835\pi$
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i	−0.732294	+	0.680989i	$0.761550\pi$
$6$		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	$0.507207\pi$
$11$	−32.0000	+	55.4256i	−0.877124	+	1.51922i	−0.854483	+	0.519480i	$0.826126\pi$
$12$		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	$-0.361725\pi$
$17$	59.0000			0.841741			0.420871	+	0.907121i	$0.361725\pi$
$18$		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	$-0.908383\pi$
$23$	−80.0000	−	138.564i	−0.725268	−	1.25620i	0.233596	−	0.972334i	$-0.424951\pi$
$24$		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	−0.760235	−	0.649648i	$-0.774916\pi$
$29$	28.5000	−	49.3634i	0.182494	−	0.316088i	0.942729	+	0.333559i	$0.108250\pi$
$30$		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	0.999353	−	0.0359748i	$-0.0114536\pi$
$41$	123.000	+	213.042i	0.468521	+	0.811503i	−0.530831	+	0.847477i	$0.678120\pi$
$42$		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	−0.923811	−	0.382850i	$-0.874943\pi$
$47$	−42.0000	+	72.7461i	−0.130347	+	0.225768i	0.793463	+	0.608618i	$0.208276\pi$
$48$		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	$-0.287368\pi$
$53$	478.000			1.23884			0.619418	+	0.785061i	$0.287368\pi$
$54$		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	0.883138	−	0.469113i	$-0.155426\pi$
$59$	16.0000	+	27.7128i	0.0353055	+	0.0611509i	−0.847833	+	0.530264i	$0.822093\pi$
$60$		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	$-0.235386\pi$
$71$	884.000			1.47763			0.738813	+	0.673910i	$0.235386\pi$
$72$		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	$0.483199\pi$
$83$	−634.000	+	1098.12i	−0.838440	+	1.45222i	−0.891198	+	0.453614i	$0.850135\pi$
$84$		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	$-0.476664\pi$
$89$	123.000			0.146494			0.0732470	+	0.997314i	$0.476664\pi$
$90$		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	−0.584158	−	0.811640i	$-0.698575\pi$
$101$	417.000	−	722.265i	0.410822	−	0.711565i	0.994980	+	0.100075i	$0.0319083\pi$
$102$		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	$-0.456728\pi$
$107$	300.000			0.271048			0.135524	+	0.990774i	$0.456728\pi$
$108$		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	$0.0683595\pi$
$113$	808.500	+	1400.36i	0.673073	+	1.16580i	−0.303955	+	0.952686i	$0.598307\pi$
$114$		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	0.846036	−	0.533126i	$-0.178983\pi$
$131$	−58.0000	−	100.459i	−0.0386831	−	0.0670011i	−0.884719	+	0.466125i	$0.845650\pi$
$132$		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	−0.945834	−	0.324649i	$-0.894754\pi$
$137$	−307.500	+	532.606i	−0.191763	+	0.332143i	0.754072	+	0.656792i	$0.228087\pi$
$138$		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	0.727350	−	0.686267i	$-0.240752\pi$
$149$	−419.500	−	726.595i	−0.230650	−	0.399497i	−0.957999	+	0.286770i	$0.907418\pi$
$150$		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	0.507863	−	0.861438i	$-0.330436\pi$
$167$	−1062.00	−	1839.44i	−0.492096	−	0.852335i	−0.999959	+	0.00910286i	$0.997102\pi$
$168$		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	$-0.490848\pi$
$173$	2002.50	−	3468.43i	0.880042	−	1.52428i	0.851293	−	0.524691i	$-0.175819\pi$
$174$		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	$-0.596366\pi$
$179$	−1428.00			−0.596278			−0.298139	+	0.954522i	$0.596366\pi$
$180$		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	$0.418629\pi$
$191$	−1878.00	+	3252.79i	−0.711452	+	1.23227i	−0.964312	+	0.264768i	$0.914705\pi$
$192$		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	$-0.454081\pi$
$197$	795.000			0.287520			0.143760	+	0.989613i	$0.454081\pi$
$198$		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	$0.361146\pi$
$227$	−1962.00	+	3398.28i	−0.573667	+	0.993621i	−0.996185	+	0.0872664i	$0.972187\pi$
$228$		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	$-0.717620\pi$
$233$	−4493.00			−1.26329			−0.631644	+	0.775258i	$0.717620\pi$
$234$		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	0.996207	−	0.0870103i	$-0.0277313\pi$
$239$	1562.00	+	2705.46i	0.422751	+	0.732225i	−0.573457	+	0.819236i	$0.694398\pi$
$240$		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	$-0.445625\pi$
$251$	1352.00			0.339990			0.169995	+	0.985445i	$0.445625\pi$
$252$		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	0.549864	−	0.835254i	$-0.314680\pi$
$257$	−1847.50	−	3199.96i	−0.448420	−	0.776686i	−0.998283	+	0.0585686i	$0.981346\pi$
$258$		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	−0.802849	−	0.596183i	$-0.796683\pi$
$263$	490.000	−	848.705i	0.114885	−	0.198986i	0.917734	+	0.397196i	$0.130017\pi$
$264$		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	$-0.704774\pi$
$269$	−5293.00			−1.19970			−0.599851	+	0.800112i	$0.704774\pi$
$270$		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	$-0.558264\pi$
$281$	3582.50	−	6205.07i	0.760548	−	1.31731i	0.942568	−	0.334013i	$-0.108403\pi$
$282$		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	$0.0736169\pi$
$293$	3436.50	+	5952.19i	0.685196	+	1.18679i	−0.288179	+	0.957577i	$0.593050\pi$
$294$		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	0.619762	−	0.784789i	$-0.287229\pi$
$311$	−2028.00	−	3512.60i	−0.369766	−	0.640454i	−0.989529	+	0.144335i	$0.953896\pi$
$312$		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	$0.540947\pi$
$317$	−5209.50	+	9023.12i	−0.923012	+	1.59870i	−0.794728	+	0.606965i	$0.792387\pi$
$318$		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	0.992797	−	0.119807i	$-0.0382276\pi$
$347$	2538.00	+	4395.94i	0.392643	+	0.680077i	−0.600155	+	0.799884i	$0.704894\pi$
$348$		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	$-0.587294\pi$
$353$	4631.00	−	8021.13i	0.698253	−	1.20941i	0.969072	−	0.246779i	$-0.0793722\pi$
$354$		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	$-0.470758\pi$
$359$	1248.00			0.183473			0.0917367	+	0.995783i	$0.470758\pi$
$360$		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	0.806921	−	0.590659i	$-0.201132\pi$
$383$	−810.000	−	1402.96i	−0.108065	−	0.187175i	−0.914986	+	0.403484i	$0.867799\pi$
$384$		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	−0.998406	−	0.0564423i	$-0.982024\pi$
$389$	−3455.00	+	5984.24i	−0.450323	+	0.779981i	0.548083	+	0.836424i	$0.315358\pi$
$390$		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	0.963353	−	0.268237i	$-0.0864409\pi$
$401$	2002.50	+	3468.43i	0.249377	+	0.431933i	−0.713976	+	0.700170i	$0.753108\pi$
$402$		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	0.653476	−	0.756947i	$-0.273310\pi$
$419$	−2820.00	−	4884.38i	−0.328797	−	0.569493i	−0.982273	+	0.187454i	$0.939977\pi$
$420$		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	$-0.724386\pi$
$431$	−11596.0			−1.29596			−0.647981	+	0.761656i	$0.724386\pi$
$432$		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	$-0.604792\pi$
$443$	6134.00	−	10624.4i	0.657867	−	1.13946i	0.981167	−	0.193163i	$-0.0618745\pi$
$444$		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	$-0.685622\pi$
$449$	−10478.0			−1.10131			−0.550654	+	0.834734i	$0.685622\pi$
$450$		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	−0.838727	−	0.544552i	$-0.816700\pi$
$461$	517.000	−	895.470i	0.0522323	−	0.0904690i	0.890959	+	0.454083i	$0.150033\pi$
$462$		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	$-0.130787\pi$
$467$	18504.0			1.83354			0.916770	+	0.399416i	$0.130787\pi$
$468$		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	$0.467948\pi$
$479$	−8506.00	+	14732.8i	−0.811376	+	1.40534i	−0.911901	+	0.410410i	$0.865386\pi$
$480$		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	$0.00108494\pi$
$491$	5472.00	+	9477.78i	0.502949	+	0.871133i	−0.497045	+	0.867725i	$0.665582\pi$
$492$		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	$-0.479671\pi$
$503$	1440.00			0.127647			0.0638235	+	0.997961i	$0.479671\pi$
$504$		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	0.811345	−	0.584568i	$-0.198736\pi$
$509$	−1155.00	−	2000.52i	−0.100579	−	0.174207i	−0.911923	+	0.410361i	$0.865403\pi$
$510$		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	$-0.636801\pi$
$521$	−9910.00			−0.833330			−0.416665	+	0.909060i	$0.636801\pi$
$522$		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	$-0.546719\pi$
$557$	−3845.00			−0.292492			−0.146246	+	0.989248i	$0.546719\pi$
$558$		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	0.919500	−	0.393091i	$-0.128594\pi$
$563$	1594.00	+	2760.89i	0.119323	+	0.206674i	−0.800176	+	0.599765i	$0.795261\pi$
$564$		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	$0.386055\pi$
$569$	−8631.50	+	14950.2i	−0.635943	+	1.10148i	−0.986315	+	0.164874i	$0.947278\pi$
$570$		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	$-0.620625\pi$
$587$	8812.00	−	15262.8i	0.619608	−	1.07319i	0.989557	−	0.144141i	$-0.0460419\pi$
$588$		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	$-0.290089\pi$
$593$	17695.0			1.22537			0.612687	+	0.790326i	$0.290089\pi$
$594$		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	0.815352	−	0.578966i	$-0.196543\pi$
$599$	−1374.00	−	2379.84i	−0.0937231	−	0.162333i	−0.909075	+	0.416633i	$0.863210\pi$
$600$		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	0.835977	−	0.548765i	$-0.184902\pi$
$617$	−877.500	−	1519.87i	−0.0572558	−	0.0991699i	−0.893233	+	0.449595i	$0.851568\pi$
$618$		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	−0.977476	−	0.211045i	$-0.932313\pi$
$641$	−4965.50	+	8600.50i	−0.305968	+	0.529952i	0.671508	+	0.740997i	$0.265647\pi$
$642$		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	$-0.321103\pi$
$647$	17540.0			1.06579			0.532897	+	0.846180i	$0.321103\pi$
$648$		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	$0.190953\pi$
$653$	15045.0	+	26058.7i	0.901618	+	1.56165i	0.0762235	+	0.997091i	$0.475714\pi$
$654$		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	−0.705742	−	0.708469i	$-0.749386\pi$
$659$	4410.00	−	7638.34i	0.260682	−	0.451514i	0.966423	+	0.256955i	$0.0827194\pi$
$660$		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	−0.701010	−	0.713151i	$-0.747267\pi$
$677$	4705.00	−	8149.30i	0.267102	−	0.462634i	0.968112	+	0.250517i	$0.0806007\pi$
$678$		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	$-0.490547\pi$
$683$	1060.00			0.0593847			0.0296924	+	0.999559i	$0.490547\pi$
$684$		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	$-0.346071\pi$
$701$	17259.0			0.929905			0.464953	+	0.885336i	$0.346071\pi$
$702$		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	$-0.652662\pi$
$719$	−17792.0			−0.922851			−0.461425	+	0.887179i	$0.652662\pi$
$720$		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	$-0.851477\pi$
$743$	−16934.0	−	29330.5i	−0.836135	−	1.44823i	0.0569682	−	0.998376i	$-0.481857\pi$
$744$		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	$0.125637\pi$
$761$	16680.5	+	28891.5i	0.794570	+	1.37624i	−0.128542	+	0.991704i	$0.541030\pi$
$762$		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	$-0.640259\pi$
$773$	−18333.0			−0.853030			−0.426515	+	0.904480i	$0.640259\pi$
$774$		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	0.630277	−	0.776370i	$-0.282941\pi$
$797$	−8037.50	−	13921.4i	−0.357218	−	0.618720i	−0.987495	+	0.157651i	$0.949608\pi$
$798$		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	$-0.253488\pi$
$809$	32183.0			1.39863			0.699316	+	0.714812i	$0.253488\pi$
$810$		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	$-0.639094\pi$
$821$	13480.5	−	23348.9i	0.573048	−	0.992549i	0.996251	−	0.0865136i	$-0.0275726\pi$
$822$		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	$-0.235602\pi$
$827$	35120.0			1.47671			0.738357	+	0.674410i	$0.235602\pi$
$828$		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	$-0.652987\pi$
$839$	13044.0	−	22592.9i	0.536745	−	0.929669i	0.999077	−	0.0429623i	$-0.0136795\pi$
$840$		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	−0.834747	−	0.550634i	$-0.814386\pi$
$857$	1492.50	−	2585.09i	0.0594899	−	0.103039i	0.894237	+	0.447595i	$0.147719\pi$
$858$		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	$-0.359312\pi$
$863$	21688.0			0.855467			0.427734	+	0.903905i	$0.359312\pi$
$864$		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	$-0.778371\pi$
$881$	−40126.0			−1.53448			−0.767241	+	0.641358i	$0.778371\pi$
$882$		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	$0.00755799\pi$
$887$	13748.0	+	23812.2i	0.520420	+	0.901394i	−0.479298	+	0.877652i	$0.659109\pi$
$888$		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	−0.797288	−	0.603599i	$-0.793733\pi$
$911$	3412.00	−	5909.76i	0.124088	−	0.214928i	0.921376	+	0.388672i	$0.127066\pi$
$912$		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	−0.594583	−	0.804034i	$-0.702683\pi$
$929$	11298.5	−	19569.6i	0.399022	−	0.691127i	0.993606	+	0.112907i	$0.0360163\pi$
$930$		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	$0.104836\pi$
$941$	21742.5	+	37659.1i	0.753226	+	1.30462i	−0.193026	+	0.981194i	$0.561830\pi$
$942$		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	$-0.548920\pi$
$947$	22710.0	−	39334.9i	0.779278	−	1.34975i	0.932359	−	0.361535i	$-0.117747\pi$
$948$		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	$-0.686301\pi$
$953$	−32505.0			−1.10487			−0.552435	+	0.833556i	$0.686301\pi$
$954$		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	$-0.701789\pi$
$971$	−35844.0			−1.18464			−0.592322	+	0.805702i	$0.701789\pi$
$972$		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	0.993401	−	0.114694i	$-0.0365887\pi$
$977$	12135.0	+	21018.4i	0.397373	+	0.688270i	−0.596028	+	0.802963i	$0.703255\pi$
$978$		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	−0.978516	−	0.206171i	$-0.933900\pi$
$983$	−9576.00	+	16586.1i	−0.310709	+	0.538164i	0.667807	+	0.744334i	$0.267233\pi$
$984$		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.j.217.1		2
3.2	odd	2		648.4.i.c.217.1		2
9.2	odd	6		648.4.a.b.1.1	yes	1
9.4	even	3	inner	648.4.i.j.433.1		2
9.5	odd	6		648.4.i.c.433.1		2
9.7	even	3		648.4.a.a.1.1	✓	1
36.7	odd	6		1296.4.a.c.1.1		1
36.11	even	6		1296.4.a.f.1.1		1

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

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

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