LMFDB - Embedded newform 432.3.q.d.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,3,Mod(17,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("432.17");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	432.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$11.7711474204$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	432.17
Dual form			432.3.q.d.305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 + 2.59808i) q^{5} +(-4.17423 - 7.22999i) q^{7} +O(q^{10})$	$q+(4.50000 + 2.59808i) q^{5} +(-4.17423 - 7.22999i) q^{7} +(0.825765 - 0.476756i) q^{11} +(4.84847 - 8.39780i) q^{13} -18.8776i q^{17} +24.6969 q^{19} +(0.825765 + 0.476756i) q^{23} +(1.00000 + 1.73205i) q^{25} +(-11.8485 + 6.84072i) q^{29} +(1.52270 - 2.63740i) q^{31} -43.3799i q^{35} +46.6969 q^{37} +(9.45459 + 5.45861i) q^{41} +(22.5227 + 39.0105i) q^{43} +(39.2196 - 22.6435i) q^{47} +(-10.3485 + 17.9241i) q^{49} -94.3879i q^{53} +4.95459 q^{55} +(-16.2650 - 9.39063i) q^{59} +(-6.54541 - 11.3370i) q^{61} +(43.6362 - 25.1934i) q^{65} +(37.5227 - 64.9912i) q^{67} -18.0204i q^{71} -7.90918 q^{73} +(-6.89388 - 3.98018i) q^{77} +(-21.8712 - 37.8820i) q^{79} +(-112.871 + 65.1662i) q^{83} +(49.0454 - 84.9491i) q^{85} +145.300i q^{89} -80.9546 q^{91} +(111.136 + 64.1645i) q^{95} +(54.9393 + 95.1576i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 18 q^{5} - 2 q^{7}+O(q^{10}) $ 4 * q + 18 * q^5 - 2 * q^7	$ 4 q + 18 q^{5} - 2 q^{7} + 18 q^{11} - 10 q^{13} + 40 q^{19} + 18 q^{23} + 4 q^{25} - 18 q^{29} - 38 q^{31} + 128 q^{37} + 126 q^{41} + 46 q^{43} + 54 q^{47} - 12 q^{49} + 108 q^{55} + 126 q^{59} + 62 q^{61} - 90 q^{65} + 106 q^{67} - 208 q^{73} + 90 q^{77} - 14 q^{79} - 378 q^{83} + 108 q^{85} - 412 q^{91} + 180 q^{95} + 14 q^{97}+O(q^{100}) $ 4 * q + 18 * q^5 - 2 * q^7 + 18 * q^11 - 10 * q^13 + 40 * q^19 + 18 * q^23 + 4 * q^25 - 18 * q^29 - 38 * q^31 + 128 * q^37 + 126 * q^41 + 46 * q^43 + 54 * q^47 - 12 * q^49 + 108 * q^55 + 126 * q^59 + 62 * q^61 - 90 * q^65 + 106 * q^67 - 208 * q^73 + 90 * q^77 - 14 * q^79 - 378 * q^83 + 108 * q^85 - 412 * q^91 + 180 * q^95 + 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	4.50000	+	2.59808i	0.900000	+	0.519615i	0.877200	−	0.480125i	$-0.159409\pi$
$5$	4.50000	+	2.59808i	0.900000	+	0.519615i	0.0227998	+	0.999740i	$0.492742\pi$
$6$		0			0
$7$	−4.17423	−	7.22999i	−0.596319	−	1.03286i	−0.993359	−	0.115054i	$-0.963296\pi$
$7$	−4.17423	−	7.22999i	−0.596319	−	1.03286i	0.397040	−	0.917801i	$-0.370037\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.825765	−	0.476756i	0.0750696	−	0.0433414i	−0.461995	−	0.886882i	$-0.652866\pi$
$11$	0.825765	−	0.476756i	0.0750696	−	0.0433414i	0.537065	+	0.843541i	$0.319533\pi$
$12$		0			0
$13$	4.84847	−	8.39780i	0.372959	−	0.645984i	−0.617060	−	0.786916i	$-0.711676\pi$
$13$	4.84847	−	8.39780i	0.372959	−	0.645984i	0.990019	+	0.140932i	$0.0450098\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	18.8776i		−	1.11045i	−0.831701	−	0.555223i	$-0.812633\pi$
$17$		−	18.8776i		−	1.11045i	0.831701	−	0.555223i	$-0.187367\pi$
$18$		0			0
$19$	24.6969			1.29984			0.649919	−	0.760003i	$-0.274803\pi$
$19$	24.6969			1.29984			0.649919	+	0.760003i	$0.274803\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.825765	+	0.476756i	0.0359028	+	0.0207285i	0.517844	−	0.855475i	$-0.326735\pi$
$23$	0.825765	+	0.476756i	0.0359028	+	0.0207285i	−0.481941	+	0.876204i	$0.660068\pi$
$24$		0			0
$25$	1.00000	+	1.73205i	0.0400000	+	0.0692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−11.8485	+	6.84072i	−0.408568	+	0.235887i	−0.690174	−	0.723643i	$-0.742466\pi$
$29$	−11.8485	+	6.84072i	−0.408568	+	0.235887i	0.281606	+	0.959530i	$0.409133\pi$
$30$		0			0
$31$	1.52270	−	2.63740i	0.0491195	−	0.0850774i	−0.840420	−	0.541935i	$-0.817692\pi$
$31$	1.52270	−	2.63740i	0.0491195	−	0.0850774i	0.889540	+	0.456858i	$0.151025\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	43.3799i		−	1.23943i
$36$		0			0
$37$	46.6969			1.26208			0.631040	−	0.775751i	$-0.282628\pi$
$37$	46.6969			1.26208			0.631040	+	0.775751i	$0.282628\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	9.45459	+	5.45861i	0.230600	+	0.133137i	0.610849	−	0.791747i	$-0.290828\pi$
$41$	9.45459	+	5.45861i	0.230600	+	0.133137i	−0.380249	+	0.924884i	$0.624162\pi$
$42$		0			0
$43$	22.5227	+	39.0105i	0.523784	+	0.907220i	0.999617	+	0.0276845i	$0.00881337\pi$
$43$	22.5227	+	39.0105i	0.523784	+	0.907220i	−0.475833	+	0.879536i	$0.657853\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	39.2196	−	22.6435i	0.834460	−	0.481776i	−0.0209170	−	0.999781i	$-0.506659\pi$
$47$	39.2196	−	22.6435i	0.834460	−	0.481776i	0.855377	+	0.518005i	$0.173325\pi$
$48$		0			0
$49$	−10.3485	+	17.9241i	−0.211193	+	0.365797i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	94.3879i		−	1.78090i	−0.455077	−	0.890452i	$-0.650388\pi$
$53$		−	94.3879i		−	1.78090i	0.455077	−	0.890452i	$-0.349612\pi$
$54$		0			0
$55$	4.95459			0.0900835
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−16.2650	−	9.39063i	−0.275679	−	0.159163i	0.355787	−	0.934567i	$-0.384213\pi$
$59$	−16.2650	−	9.39063i	−0.275679	−	0.159163i	−0.631466	+	0.775404i	$0.717546\pi$
$60$		0			0
$61$	−6.54541	−	11.3370i	−0.107302	−	0.185852i	0.807375	−	0.590039i	$-0.200888\pi$
$61$	−6.54541	−	11.3370i	−0.107302	−	0.185852i	−0.914676	+	0.404187i	$0.867554\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	43.6362	−	25.1934i	0.671327	−	0.387591i
$66$		0			0
$67$	37.5227	−	64.9912i	0.560040	−	0.970018i	−0.437452	−	0.899242i	$-0.644119\pi$
$67$	37.5227	−	64.9912i	0.560040	−	0.970018i	0.997492	−	0.0707765i	$-0.0225477\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	18.0204i		−	0.253808i	−0.991915	−	0.126904i	$-0.959496\pi$
$71$		−	18.0204i		−	0.253808i	0.991915	−	0.126904i	$-0.0405041\pi$
$72$		0			0
$73$	−7.90918			−0.108345			−0.0541725	−	0.998532i	$-0.517252\pi$
$73$	−7.90918			−0.108345			−0.0541725	+	0.998532i	$0.517252\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.89388	−	3.98018i	−0.0895309	−	0.0516907i
$78$		0			0
$79$	−21.8712	−	37.8820i	−0.276850	−	0.479519i	0.693750	−	0.720216i	$-0.255957\pi$
$79$	−21.8712	−	37.8820i	−0.276850	−	0.479519i	−0.970600	+	0.240697i	$0.922624\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−112.871	+	65.1662i	−1.35989	+	0.785135i	−0.989609	−	0.143783i	$-0.954073\pi$
$83$	−112.871	+	65.1662i	−1.35989	+	0.785135i	−0.370284	+	0.928918i	$0.620740\pi$
$84$		0			0
$85$	49.0454	−	84.9491i	0.577005	−	0.999402i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			145.300i			1.63258i	0.577642	+	0.816290i	$0.303973\pi$
$89$			145.300i			1.63258i	−0.577642	+	0.816290i	$0.696027\pi$
$90$		0			0
$91$	−80.9546			−0.889611
$92$		0			0
$93$		0			0
$94$		0			0
$95$	111.136	+	64.1645i	1.16985	+	0.675416i
$96$		0			0
$97$	54.9393	+	95.1576i	0.566384	+	0.981007i	0.996919	+	0.0784327i	$0.0249916\pi$
$97$	54.9393	+	95.1576i	0.566384	+	0.981007i	−0.430535	+	0.902574i	$0.641675\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−127.772	+	73.7695i	−1.26507	+	0.730391i	−0.974052	−	0.226326i	$-0.927329\pi$
$101$	−127.772	+	73.7695i	−1.26507	+	0.730391i	−0.291022	+	0.956716i	$0.593995\pi$
$102$		0			0
$103$	−51.5681	+	89.3186i	−0.500661	+	0.867171i	0.499338	+	0.866407i	$0.333576\pi$
$103$	−51.5681	+	89.3186i	−0.500661	+	0.867171i	−1.00000		0.000763745i	$0.999757\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	36.0408i		−	0.336830i	−0.985716	−	0.168415i	$-0.946135\pi$
$107$		−	36.0408i		−	0.336830i	0.985716	−	0.168415i	$-0.0538649\pi$
$108$		0			0
$109$	−148.272			−1.36030			−0.680149	−	0.733074i	$-0.738085\pi$
$109$	−148.272			−1.36030			−0.680149	+	0.733074i	$0.738085\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	148.166	+	85.5439i	1.31121	+	0.757025i	0.982296	−	0.187336i	$-0.0599852\pi$
$113$	148.166	+	85.5439i	1.31121	+	0.757025i	0.328910	+	0.944361i	$0.393319\pi$
$114$		0			0
$115$	2.47730	+	4.29080i	0.0215417	+	0.0373113i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−136.485	+	78.7995i	−1.14693	+	0.662180i
$120$		0			0
$121$	−60.0454	+	104.002i	−0.496243	+	0.859518i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	119.512i		−	0.956092i
$126$		0			0
$127$	78.0908			0.614888			0.307444	−	0.951566i	$-0.400526\pi$
$127$	78.0908			0.614888			0.307444	+	0.951566i	$0.400526\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	202.704	+	117.031i	1.54736	+	0.893369i	0.998342	+	0.0575598i	$0.0183320\pi$
$131$	202.704	+	117.031i	1.54736	+	0.893369i	0.549019	+	0.835810i	$0.315001\pi$
$132$		0			0
$133$	−103.091	−	178.559i	−0.775119	−	1.34255i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−129.758	+	74.9156i	−0.947136	+	0.546829i	−0.892190	−	0.451660i	$-0.850832\pi$
$137$	−129.758	+	74.9156i	−0.947136	+	0.546829i	−0.0549460	+	0.998489i	$0.517499\pi$
$138$		0			0
$139$	−42.2650	+	73.2052i	−0.304065	+	0.526656i	−0.977053	−	0.212998i	$-0.931677\pi$
$139$	−42.2650	+	73.2052i	−0.304065	+	0.526656i	0.672988	+	0.739654i	$0.265011\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	9.24614i		−	0.0646584i
$144$		0			0
$145$	−71.0908			−0.490281
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−100.030	−	57.7524i	−0.671343	−	0.387600i	0.125242	−	0.992126i	$-0.460029\pi$
$149$	−100.030	−	57.7524i	−0.671343	−	0.387600i	−0.796585	+	0.604526i	$0.793362\pi$
$150$		0			0
$151$	−32.3865	−	56.0950i	−0.214480	−	0.371490i	0.738632	−	0.674109i	$-0.235472\pi$
$151$	−32.3865	−	56.0950i	−0.214480	−	0.371490i	−0.953112	+	0.302619i	$0.902139\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	13.7043	−	7.91220i	0.0884151	−	0.0510465i
$156$		0			0
$157$	10.4092	−	18.0292i	0.0663005	−	0.114836i	−0.830970	−	0.556318i	$-0.812214\pi$
$157$	10.4092	−	18.0292i	0.0663005	−	0.114836i	0.897270	+	0.441482i	$0.145547\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	7.96036i		−	0.0494433i
$162$		0			0
$163$	−133.060			−0.816320			−0.408160	−	0.912910i	$-0.633829\pi$
$163$	−133.060			−0.816320			−0.408160	+	0.912910i	$0.633829\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−255.053	−	147.255i	−1.52726	−	0.881765i	−0.999475	−	0.0323885i	$-0.989689\pi$
$167$	−255.053	−	147.255i	−1.52726	−	0.881765i	−0.527787	−	0.849377i	$-0.676978\pi$
$168$		0			0
$169$	37.4847	+	64.9254i	0.221803	+	0.384174i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	59.9847	−	34.6322i	0.346732	−	0.200186i	−0.316513	−	0.948588i	$-0.602512\pi$
$173$	59.9847	−	34.6322i	0.346732	−	0.200186i	0.663245	+	0.748402i	$0.269179\pi$
$174$		0			0
$175$	8.34847	−	14.4600i	0.0477055	−	0.0826284i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	47.4829i		−	0.265268i	−0.991165	−	0.132634i	$-0.957657\pi$
$179$		−	47.4829i		−	0.265268i	0.991165	−	0.132634i	$-0.0423435\pi$
$180$		0			0
$181$	242.879			1.34187			0.670935	−	0.741516i	$-0.265893\pi$
$181$	242.879			1.34187			0.670935	+	0.741516i	$0.265893\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	210.136	+	121.322i	1.13587	+	0.655796i
$186$		0			0
$187$	−9.00000	−	15.5885i	−0.0481283	−	0.0833607i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.52270	−	3.76588i	0.0341503	−	0.0197167i	−0.482828	−	0.875715i	$-0.660390\pi$
$191$	6.52270	−	3.76588i	0.0341503	−	0.0197167i	0.516978	+	0.855999i	$0.327057\pi$
$192$		0			0
$193$	−172.727	+	299.172i	−0.894959	+	1.55011i	−0.0611031	+	0.998131i	$0.519462\pi$
$193$	−172.727	+	299.172i	−0.894959	+	1.55011i	−0.833856	+	0.551983i	$0.813871\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			77.2247i			0.392004i	0.980604	+	0.196002i	$0.0627959\pi$
$197$			77.2247i			0.392004i	−0.980604	+	0.196002i	$0.937204\pi$
$198$		0			0
$199$	−153.485			−0.771280			−0.385640	−	0.922649i	$-0.626019\pi$
$199$	−153.485			−0.771280			−0.385640	+	0.922649i	$0.626019\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	98.9166	+	57.1095i	0.487274	+	0.281328i
$204$		0			0
$205$	28.3638	+	49.1275i	0.138360	+	0.239646i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	20.3939	−	11.7744i	0.0975784	−	0.0563369i
$210$		0			0
$211$	−25.7804	+	44.6529i	−0.122182	+	0.211625i	−0.920628	−	0.390441i	$-0.872322\pi$
$211$	−25.7804	+	44.6529i	−0.122182	+	0.211625i	0.798446	+	0.602066i	$0.205656\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			234.063i			1.08866i
$216$		0			0
$217$	−25.4245			−0.117164
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−158.530	−	91.5274i	−0.717331	−	0.414151i
$222$		0			0
$223$	156.614	+	271.263i	0.702303	+	1.21642i	0.967656	+	0.252273i	$0.0811781\pi$
$223$	156.614	+	271.263i	0.702303	+	1.21642i	−0.265353	+	0.964151i	$0.585489\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−66.0528	+	38.1356i	−0.290982	+	0.167998i	−0.638384	−	0.769718i	$-0.720397\pi$
$227$	−66.0528	+	38.1356i	−0.290982	+	0.167998i	0.347403	+	0.937716i	$0.387064\pi$
$228$		0			0
$229$	60.7724	−	105.261i	0.265382	−	0.459655i	−0.702282	−	0.711899i	$-0.747835\pi$
$229$	60.7724	−	105.261i	0.265382	−	0.459655i	0.967664	+	0.252244i	$0.0811686\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	151.021i		−	0.648157i	−0.946030	−	0.324079i	$-0.894946\pi$
$233$		−	151.021i		−	0.648157i	0.946030	−	0.324079i	$-0.105054\pi$
$234$		0			0
$235$	235.318			1.00135
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−75.9620	−	43.8567i	−0.317833	−	0.183501i	0.332593	−	0.943070i	$-0.392076\pi$
$239$	−75.9620	−	43.8567i	−0.317833	−	0.183501i	−0.650426	+	0.759570i	$0.725410\pi$
$240$		0			0
$241$	−100.894	−	174.753i	−0.418647	−	0.725118i	0.577157	−	0.816633i	$-0.304162\pi$
$241$	−100.894	−	174.753i	−0.418647	−	0.725118i	−0.995804	+	0.0915158i	$0.970829\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−93.1362	+	53.7722i	−0.380148	+	0.219478i
$246$		0			0
$247$	119.742	−	207.400i	0.484787	−	0.839675i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	52.6261i		−	0.209666i	−0.994490	−	0.104833i	$-0.966569\pi$
$251$		−	52.6261i		−	0.209666i	0.994490	−	0.104833i	$-0.0334307\pi$
$252$		0			0
$253$	0.909185			0.00359362
$254$		0			0
$255$		0			0
$256$		0			0
$257$	69.8939	+	40.3532i	0.271961	+	0.157017i	0.629778	−	0.776775i	$-0.283146\pi$
$257$	69.8939	+	40.3532i	0.271961	+	0.157017i	−0.357818	+	0.933791i	$0.616479\pi$
$258$		0			0
$259$	−194.924	−	337.618i	−0.752602	−	1.30355i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	401.614	−	231.872i	1.52705	−	0.881641i	0.527564	−	0.849515i	$-0.323106\pi$
$263$	401.614	−	231.872i	1.52705	−	0.881641i	0.999484	−	0.0321259i	$-0.0102278\pi$
$264$		0			0
$265$	245.227	−	424.746i	0.925385	−	1.60281i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	43.4762i		−	0.161622i	−0.996729	−	0.0808109i	$-0.974249\pi$
$269$		−	43.4762i		−	0.161622i	0.996729	−	0.0808109i	$-0.0257510\pi$
$270$		0			0
$271$	342.636			1.26434			0.632169	−	0.774830i	$-0.282165\pi$
$271$	342.636			1.26434			0.632169	+	0.774830i	$0.282165\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.65153	+	0.953512i	0.00600557	+	0.00346732i
$276$		0			0
$277$	24.5000	+	42.4352i	0.0884477	+	0.153196i	0.906855	−	0.421442i	$-0.138476\pi$
$277$	24.5000	+	42.4352i	0.0884477	+	0.153196i	−0.818407	+	0.574638i	$0.805143\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	17.8791	−	10.3225i	0.0636266	−	0.0367349i	−0.467849	−	0.883808i	$-0.654971\pi$
$281$	17.8791	−	10.3225i	0.0636266	−	0.0367349i	0.531476	+	0.847073i	$0.321638\pi$
$282$		0			0
$283$	26.7043	−	46.2533i	0.0943616	−	0.163439i	−0.814980	−	0.579489i	$-0.803252\pi$
$283$	26.7043	−	46.2533i	0.0943616	−	0.163439i	0.909342	+	0.416049i	$0.136586\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	91.1421i		−	0.317568i
$288$		0			0
$289$	−67.3633			−0.233091
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.9245	+	7.46196i	0.0441109	+	0.0254674i	0.521893	−	0.853011i	$-0.325226\pi$
$293$	12.9245	+	7.46196i	0.0441109	+	0.0254674i	−0.477782	+	0.878478i	$0.658559\pi$
$294$		0			0
$295$	−48.7951	−	84.5157i	−0.165407	−	0.286494i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.00740	−	4.62307i	0.0267806	−	0.0154618i
$300$		0			0
$301$	188.030	−	325.678i	0.624685	−	1.08199i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	68.0219i		−	0.223023i
$306$		0			0
$307$	−65.9092			−0.214688			−0.107344	−	0.994222i	$-0.534235\pi$
$307$	−65.9092			−0.214688			−0.107344	+	0.994222i	$0.534235\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−216.659	−	125.088i	−0.696652	−	0.402213i	0.109447	−	0.993993i	$-0.465092\pi$
$311$	−216.659	−	125.088i	−0.696652	−	0.402213i	−0.806099	+	0.591780i	$0.798425\pi$
$312$		0			0
$313$	213.197	+	369.268i	0.681140	+	1.17977i	0.974633	+	0.223808i	$0.0718490\pi$
$313$	213.197	+	369.268i	0.681140	+	1.17977i	−0.293493	+	0.955961i	$0.594818\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	401.818	−	231.990i	1.26756	−	0.731829i	0.293038	−	0.956101i	$-0.405334\pi$
$317$	401.818	−	231.990i	1.26756	−	0.731829i	0.974527	+	0.224272i	$0.0720005\pi$
$318$		0			0
$319$	−6.52270	+	11.2977i	−0.0204473	+	0.0354158i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	466.219i		−	1.44340i
$324$		0			0
$325$	19.3939			0.0596735
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−327.424	−	189.038i	−0.995210	−	0.574585i
$330$		0			0
$331$	236.401	+	409.459i	0.714203	+	1.23704i	0.963266	+	0.268549i	$0.0865441\pi$
$331$	236.401	+	409.459i	0.714203	+	1.23704i	−0.249063	+	0.968487i	$0.580123\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	337.704	−	194.974i	1.00807	−	0.582011i
$336$		0			0
$337$	−152.803	+	264.663i	−0.453422	+	0.785349i	−0.998596	−	0.0529735i	$-0.983130\pi$
$337$	−152.803	+	264.663i	−0.453422	+	0.785349i	0.545174	+	0.838323i	$0.316463\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	2.90383i		−	0.00851564i
$342$		0			0
$343$	−236.287			−0.688884
$344$		0			0
$345$		0			0
$346$		0			0
$347$	115.766	+	66.8373i	0.333618	+	0.192615i	0.657446	−	0.753501i	$-0.271637\pi$
$347$	115.766	+	66.8373i	0.333618	+	0.192615i	−0.323828	+	0.946116i	$0.604970\pi$
$348$		0			0
$349$	49.3786	+	85.5262i	0.141486	+	0.245061i	0.928056	−	0.372440i	$-0.121479\pi$
$349$	49.3786	+	85.5262i	0.141486	+	0.245061i	−0.786570	+	0.617500i	$0.788145\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	282.424	−	163.058i	0.800068	−	0.461919i	−0.0434270	−	0.999057i	$-0.513828\pi$
$353$	282.424	−	163.058i	0.800068	−	0.461919i	0.843495	+	0.537137i	$0.180494\pi$
$354$		0			0
$355$	46.8184	−	81.0918i	0.131883	−	0.228428i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			418.736i			1.16639i	0.812331	+	0.583197i	$0.198199\pi$
$359$			418.736i			1.16639i	−0.812331	+	0.583197i	$0.801801\pi$
$360$		0			0
$361$	248.939			0.689581
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−35.5913	−	20.5487i	−0.0975105	−	0.0562977i
$366$		0			0
$367$	93.6135	+	162.143i	0.255078	+	0.441808i	0.964917	−	0.262557i	$-0.0845656\pi$
$367$	93.6135	+	162.143i	0.255078	+	0.441808i	−0.709839	+	0.704364i	$0.751232\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−682.423	+	393.997i	−1.83942	+	1.06199i
$372$		0			0
$373$	−225.515	+	390.603i	−0.604597	+	1.04719i	0.387518	+	0.921862i	$0.373333\pi$
$373$	−225.515	+	390.603i	−0.604597	+	1.04719i	−0.992115	+	0.125331i	$0.960001\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			132.668i			0.351905i
$378$		0			0
$379$	489.666			1.29200			0.645998	−	0.763339i	$-0.276442\pi$
$379$	489.666			1.29200			0.645998	+	0.763339i	$0.276442\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	89.2492	+	51.5281i	0.233027	+	0.134538i	0.611968	−	0.790883i	$-0.290378\pi$
$383$	89.2492	+	51.5281i	0.233027	+	0.134538i	−0.378941	+	0.925421i	$0.623712\pi$
$384$		0			0
$385$	−20.6816	−	35.8216i	−0.0537185	−	0.0930432i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−29.6816	+	17.1367i	−0.0763024	+	0.0440532i	−0.537666	−	0.843158i	$-0.680694\pi$
$389$	−29.6816	+	17.1367i	−0.0763024	+	0.0440532i	0.461363	+	0.887211i	$0.347360\pi$
$390$		0			0
$391$	9.00000	−	15.5885i	0.0230179	−	0.0398682i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	227.292i		−	0.575423i
$396$		0			0
$397$	8.27245			0.0208374			0.0104187	−	0.999946i	$-0.496684\pi$
$397$	8.27245			0.0208374			0.0104187	+	0.999946i	$0.496684\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−358.636	−	207.059i	−0.894355	−	0.516356i	−0.0189903	−	0.999820i	$-0.506045\pi$
$401$	−358.636	−	207.059i	−0.894355	−	0.516356i	−0.875364	+	0.483464i	$0.839378\pi$
$402$		0			0
$403$	−14.7656	−	25.5747i	−0.0366391	−	0.0634608i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	38.5607	−	22.2630i	0.0947438	−	0.0547003i
$408$		0			0
$409$	163.106	−	282.508i	0.398792	−	0.690729i	−0.594785	−	0.803885i	$-0.702763\pi$
$409$	163.106	−	282.508i	0.398792	−	0.690729i	0.993577	+	0.113156i	$0.0360960\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			156.795i			0.379648i
$414$		0			0
$415$	−677.227			−1.63187
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−468.325	−	270.388i	−1.11772	−	0.645317i	−0.176903	−	0.984228i	$-0.556608\pi$
$419$	−468.325	−	270.388i	−1.11772	−	0.645317i	−0.940818	+	0.338912i	$0.889941\pi$
$420$		0			0
$421$	−141.848	−	245.689i	−0.336932	−	0.583584i	0.646922	−	0.762556i	$-0.276056\pi$
$421$	−141.848	−	245.689i	−0.336932	−	0.583584i	−0.983854	+	0.178973i	$0.942723\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	32.6969	−	18.8776i	0.0769340	−	0.0444178i
$426$		0			0
$427$	−54.6441	+	94.6464i	−0.127972	+	0.221654i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			257.429i			0.597282i	0.954365	+	0.298641i	$0.0965334\pi$
$431$			257.429i			0.597282i	−0.954365	+	0.298641i	$0.903467\pi$
$432$		0			0
$433$	476.272			1.09994			0.549968	−	0.835186i	$-0.314640\pi$
$433$	476.272			1.09994			0.549968	+	0.835186i	$0.314640\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	20.3939	+	11.7744i	0.0466679	+	0.0269437i
$438$		0			0
$439$	−278.931	−	483.123i	−0.635379	−	1.10051i	−0.986435	−	0.164154i	$-0.947511\pi$
$439$	−278.931	−	483.123i	−0.635379	−	1.10051i	0.351056	−	0.936355i	$-0.385823\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	720.400	−	415.923i	1.62619	−	0.938879i	0.640969	−	0.767567i	$-0.278533\pi$
$443$	720.400	−	415.923i	1.62619	−	0.938879i	0.985217	−	0.171312i	$-0.0548005\pi$
$444$		0			0
$445$	−377.499	+	653.848i	−0.848313	+	1.46932i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			729.927i			1.62567i	0.582492	+	0.812836i	$0.302078\pi$
$449$			729.927i			1.62567i	−0.582492	+	0.812836i	$0.697922\pi$
$450$		0			0
$451$	10.4097			0.0230814
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−364.296	−	210.326i	−0.800650	−	0.462255i
$456$		0			0
$457$	−354.818	−	614.563i	−0.776407	−	1.34478i	−0.934000	−	0.357272i	$-0.883707\pi$
$457$	−354.818	−	614.563i	−0.776407	−	1.34478i	0.157594	−	0.987504i	$-0.449626\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.96990	−	4.60142i	0.0172883	−	0.00998140i	−0.491331	−	0.870973i	$-0.663489\pi$
$461$	7.96990	−	4.60142i	0.0172883	−	0.00998140i	0.508619	+	0.860992i	$0.330156\pi$
$462$		0			0
$463$	−27.5987	+	47.8024i	−0.0596085	+	0.103245i	−0.894290	−	0.447488i	$-0.852319\pi$
$463$	−27.5987	+	47.8024i	−0.0596085	+	0.103245i	0.834681	+	0.550733i	$0.185652\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			625.811i			1.34007i	0.742331	+	0.670033i	$0.233720\pi$
$467$			625.811i			1.34007i	−0.742331	+	0.670033i	$0.766280\pi$
$468$		0			0
$469$	−626.514			−1.33585
$470$		0			0
$471$		0			0
$472$		0			0
$473$	37.1969	+	21.4757i	0.0786405	+	0.0454031i
$474$		0			0
$475$	24.6969	+	42.7764i	0.0519936	+	0.0900555i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	267.856	−	154.647i	0.559199	−	0.322854i	−0.193625	−	0.981076i	$-0.562024\pi$
$479$	267.856	−	154.647i	0.559199	−	0.322854i	0.752824	+	0.658222i	$0.228691\pi$
$480$		0			0
$481$	226.409	−	392.151i	0.470704	−	0.815283i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			570.946i			1.17721i
$486$		0			0
$487$	28.3337			0.0581800			0.0290900	−	0.999577i	$-0.490739\pi$
$487$	28.3337			0.0581800			0.0290900	+	0.999577i	$0.490739\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	822.461	+	474.848i	1.67507	+	0.967105i	0.964727	+	0.263254i	$0.0847956\pi$
$491$	822.461	+	474.848i	1.67507	+	0.967105i	0.710348	+	0.703851i	$0.248538\pi$
$492$		0			0
$493$	129.136	+	223.670i	0.261940	+	0.453693i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−130.287	+	75.2214i	−0.262147	+	0.151351i
$498$		0			0
$499$	−280.113	+	485.170i	−0.561349	+	0.972284i	0.436030	+	0.899932i	$0.356384\pi$
$499$	−280.113	+	485.170i	−0.561349	+	0.972284i	−0.997379	+	0.0723525i	$0.976949\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			897.832i			1.78495i	0.451094	+	0.892477i	$0.351034\pi$
$503$			897.832i			1.78495i	−0.451094	+	0.892477i	$0.648966\pi$
$504$		0			0
$505$	−766.635			−1.51809
$506$		0			0
$507$		0			0
$508$		0			0
$509$	170.454	+	98.4114i	0.334879	+	0.193343i	0.658005	−	0.753013i	$-0.271400\pi$
$509$	170.454	+	98.4114i	0.334879	+	0.193343i	−0.323126	+	0.946356i	$0.604734\pi$
$510$		0			0
$511$	33.0148	+	57.1833i	0.0646082	+	0.111905i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−464.113	+	267.956i	−0.901190	+	0.520302i
$516$		0			0
$517$	21.5908	−	37.3964i	0.0417617	−	0.0723334i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	375.837i		−	0.721377i	−0.932686	−	0.360688i	$-0.882542\pi$
$521$		−	375.837i		−	0.721377i	0.932686	−	0.360688i	$-0.117458\pi$
$522$		0			0
$523$	−91.1827			−0.174345			−0.0871727	−	0.996193i	$-0.527783\pi$
$523$	−91.1827			−0.174345			−0.0871727	+	0.996193i	$0.527783\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−49.7878	−	28.7450i	−0.0944739	−	0.0545445i
$528$		0			0
$529$	−264.045	−	457.340i	−0.499141	−	0.864537i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	91.6806	−	52.9318i	0.172009	−	0.0993092i
$534$		0			0
$535$	93.6367	−	162.184i	0.175022	−	0.303147i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			19.7348i			0.0366137i
$540$		0			0
$541$	−38.8490			−0.0718096			−0.0359048	−	0.999355i	$-0.511431\pi$
$541$	−38.8490			−0.0718096			−0.0359048	+	0.999355i	$0.511431\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−667.226	−	385.223i	−1.22427	−	0.706831i
$546$		0			0
$547$	−233.022	−	403.606i	−0.426000	−	0.737854i	0.570513	−	0.821289i	$-0.306744\pi$
$547$	−233.022	−	403.606i	−0.426000	−	0.737854i	−0.996513	+	0.0834344i	$0.973411\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−292.621	+	168.945i	−0.531072	+	0.306615i
$552$		0			0
$553$	−182.591	+	316.257i	−0.330182	+	0.571893i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			695.042i			1.24783i	0.781492	+	0.623916i	$0.214459\pi$
$557$			695.042i			1.24783i	−0.781492	+	0.623916i	$0.785541\pi$
$558$		0			0
$559$	436.803			0.781400
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−473.780	−	273.537i	−0.841528	−	0.485857i	0.0162552	−	0.999868i	$-0.494826\pi$
$563$	−473.780	−	273.537i	−0.841528	−	0.485857i	−0.857783	+	0.514011i	$0.828159\pi$
$564$		0			0
$565$	444.499	+	769.895i	0.786724	+	1.36265i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−215.954	+	124.681i	−0.379533	+	0.219123i	−0.677615	−	0.735417i	$-0.736986\pi$
$569$	−215.954	+	124.681i	−0.379533	+	0.219123i	0.298082	+	0.954540i	$0.403653\pi$
$570$		0			0
$571$	36.9166	−	63.9414i	0.0646525	−	0.111981i	−0.831887	−	0.554945i	$-0.812739\pi$
$571$	36.9166	−	63.9414i	0.0646525	−	0.111981i	0.896540	+	0.442963i	$0.146073\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1.90702i			0.00331656i
$576$		0			0
$577$	−43.9092			−0.0760991			−0.0380496	−	0.999276i	$-0.512114\pi$
$577$	−43.9092			−0.0760991			−0.0380496	+	0.999276i	$0.512114\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	942.302	+	544.038i	1.62186	+	0.936382i
$582$		0			0
$583$	−45.0000	−	77.9423i	−0.0771870	−	0.133692i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−381.386	+	220.194i	−0.649721	+	0.375117i	−0.788349	−	0.615228i	$-0.789064\pi$
$587$	−381.386	+	220.194i	−0.649721	+	0.375117i	0.138628	+	0.990345i	$0.455731\pi$
$588$		0			0
$589$	37.6061	−	65.1357i	0.0638474	−	0.110587i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	347.232i		−	0.585551i	−0.956181	−	0.292776i	$-0.905421\pi$
$593$		−	347.232i		−	0.585551i	0.956181	−	0.292776i	$-0.0945789\pi$
$594$		0			0
$595$	−818.908			−1.37632
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−684.083	−	394.956i	−1.14204	−	0.659359i	−0.195107	−	0.980782i	$-0.562505\pi$
$599$	−684.083	−	394.956i	−1.14204	−	0.659359i	−0.946936	+	0.321423i	$0.895839\pi$
$600$		0			0
$601$	353.455	+	612.201i	0.588111	+	1.01864i	0.994480	+	0.104929i	$0.0334614\pi$
$601$	353.455	+	612.201i	0.588111	+	1.01864i	−0.406369	+	0.913709i	$0.633205\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−540.409	+	312.005i	−0.893237	+	0.515711i
$606$		0			0
$607$	−596.628	+	1033.39i	−0.982913	+	1.70246i	−0.332048	+	0.943263i	$0.607739\pi$
$607$	−596.628	+	1033.39i	−0.982913	+	1.70246i	−0.650866	+	0.759193i	$0.725594\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	439.145i		−	0.718731i
$612$		0			0
$613$	629.181			1.02640			0.513198	−	0.858270i	$-0.328461\pi$
$613$	629.181			1.02640			0.513198	+	0.858270i	$0.328461\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−166.909	−	96.3648i	−0.270516	−	0.156183i	0.358606	−	0.933489i	$-0.383252\pi$
$617$	−166.909	−	96.3648i	−0.270516	−	0.156183i	−0.629122	+	0.777306i	$0.716586\pi$
$618$		0			0
$619$	−76.4773	−	132.463i	−0.123550	−	0.213994i	0.797615	−	0.603166i	$-0.206095\pi$
$619$	−76.4773	−	132.463i	−0.123550	−	0.213994i	−0.921165	+	0.389172i	$0.872761\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1050.51	−	606.515i	1.68622	−	0.973539i
$624$		0			0
$625$	335.500	−	581.103i	0.536800	−	0.929765i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	881.525i		−	1.40147i
$630$		0			0
$631$	−44.8786			−0.0711229			−0.0355615	−	0.999367i	$-0.511322\pi$
$631$	−44.8786			−0.0711229			−0.0355615	+	0.999367i	$0.511322\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	351.409	+	202.886i	0.553399	+	0.319505i
$636$		0			0
$637$	100.348	+	173.809i	0.157533	+	0.272855i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	209.106	−	120.727i	0.326219	−	0.188342i	−0.327942	−	0.944698i	$-0.606355\pi$
$641$	209.106	−	120.727i	0.326219	−	0.188342i	0.654161	+	0.756355i	$0.273022\pi$
$642$		0			0
$643$	395.704	−	685.380i	0.615403	−	1.06591i	−0.374910	−	0.927061i	$-0.622326\pi$
$643$	395.704	−	685.380i	0.615403	−	1.06591i	0.990314	−	0.138849i	$-0.0443402\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	294.028i		−	0.454448i	−0.973842	−	0.227224i	$-0.927035\pi$
$647$		−	294.028i		−	0.454448i	0.973842	−	0.227224i	$-0.0729650\pi$
$648$		0			0
$649$	−17.9082			−0.0275935
$650$		0			0
$651$		0			0
$652$		0			0
$653$	665.379	+	384.156i	1.01896	+	0.588295i	0.913802	−	0.406161i	$-0.133133\pi$
$653$	665.379	+	384.156i	1.01896	+	0.588295i	0.105155	+	0.994456i	$0.466466\pi$
$654$		0			0
$655$	608.113	+	1053.28i	0.928417	+	1.60807i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−373.204	+	215.469i	−0.566318	+	0.326964i	−0.755678	−	0.654944i	$-0.772692\pi$
$659$	−373.204	+	215.469i	−0.566318	+	0.326964i	0.189359	+	0.981908i	$0.439359\pi$
$660$		0			0
$661$	−506.136	+	876.653i	−0.765712	+	1.32625i	0.174157	+	0.984718i	$0.444280\pi$
$661$	−506.136	+	876.653i	−0.765712	+	1.32625i	−0.939869	+	0.341534i	$0.889053\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1071.35i		−	1.61105i
$666$		0			0
$667$	−13.0454			−0.0195583
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−10.8099	−	6.24112i	−0.0161102	−	0.00930123i
$672$		0			0
$673$	−281.606	−	487.755i	−0.418433	−	0.724748i	0.577349	−	0.816498i	$-0.304087\pi$
$673$	−281.606	−	487.755i	−0.418433	−	0.724748i	−0.995782	+	0.0917499i	$0.970754\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	303.227	−	175.068i	0.447897	−	0.258594i	−0.259044	−	0.965865i	$-0.583408\pi$
$677$	303.227	−	175.068i	0.447897	−	0.258594i	0.706942	+	0.707272i	$0.250074\pi$
$678$		0			0
$679$	458.659	−	794.421i	0.675492	−	1.16999i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	502.818i		−	0.736190i	−0.929788	−	0.368095i	$-0.880010\pi$
$683$		−	502.818i		−	0.736190i	0.929788	−	0.368095i	$-0.119990\pi$
$684$		0			0
$685$	−778.546			−1.13656
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−792.650	−	457.637i	−1.15044	−	0.664205i
$690$		0			0
$691$	188.159	+	325.902i	0.272300	+	0.471638i	0.969450	−	0.245287i	$-0.0788823\pi$
$691$	188.159	+	325.902i	0.272300	+	0.471638i	−0.697150	+	0.716925i	$0.745549\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−380.385	+	219.616i	−0.547317	+	0.315994i
$696$		0			0
$697$	103.045	−	178.480i	0.147841	−	0.256069i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			489.681i			0.698546i	0.937021	+	0.349273i	$0.113571\pi$
$701$			489.681i			0.698546i	−0.937021	+	0.349273i	$0.886429\pi$
$702$		0			0
$703$	1153.27			1.64050
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1066.70	+	615.862i	1.50878	+	0.871092i
$708$		0			0
$709$	−237.014	−	410.521i	−0.334294	−	0.579014i	0.649055	−	0.760741i	$-0.275164\pi$
$709$	−237.014	−	410.521i	−0.334294	−	0.579014i	−0.983349	+	0.181728i	$0.941831\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.51479	−	1.45192i	0.00352706	−	0.00203635i
$714$		0			0
$715$	24.0222	−	41.6077i	0.0335975	−	0.0581925i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			108.122i			0.150379i	0.997169	+	0.0751894i	$0.0239561\pi$
$719$			108.122i			0.150379i	−0.997169	+	0.0751894i	$0.976044\pi$
$720$		0			0
$721$	861.030			1.19422
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−23.6969	−	13.6814i	−0.0326854	−	0.0188709i
$726$		0			0
$727$	−222.296	−	385.027i	−0.305771	−	0.529611i	0.671662	−	0.740858i	$-0.265581\pi$
$727$	−222.296	−	385.027i	−0.305771	−	0.529611i	−0.977433	+	0.211247i	$0.932248\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	736.423	−	425.174i	1.00742	−	0.581634i
$732$		0			0
$733$	358.181	−	620.388i	0.488651	−	0.846368i	−0.511264	−	0.859424i	$-0.670823\pi$
$733$	358.181	−	620.388i	0.488651	−	0.846368i	0.999915	+	0.0130556i	$0.00415584\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	71.5567i		−	0.0970918i
$738$		0			0
$739$	−933.362			−1.26301			−0.631504	−	0.775373i	$-0.717562\pi$
$739$	−933.362			−1.26301			−0.631504	+	0.775373i	$0.717562\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−13.7793	−	7.95550i	−0.0185455	−	0.0107073i	0.490699	−	0.871329i	$-0.336742\pi$
$743$	−13.7793	−	7.95550i	−0.0185455	−	0.0107073i	−0.509244	+	0.860622i	$0.670075\pi$
$744$		0			0
$745$	−300.090	−	519.772i	−0.402806	−	0.697680i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−260.574	+	150.443i	−0.347896	+	0.200858i
$750$		0			0
$751$	404.916	−	701.334i	0.539169	−	0.933867i	−0.459781	−	0.888033i	$-0.652072\pi$
$751$	404.916	−	701.334i	0.539169	−	0.933867i	0.998949	−	0.0458347i	$-0.0145947\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	336.570i		−	0.445788i
$756$		0			0
$757$	689.637			0.911013			0.455506	−	0.890232i	$-0.349458\pi$
$757$	689.637			0.911013			0.455506	+	0.890232i	$0.349458\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	825.393	+	476.541i	1.08462	+	0.626204i	0.932138	−	0.362103i	$-0.117941\pi$
$761$	825.393	+	476.541i	1.08462	+	0.626204i	0.152479	+	0.988307i	$0.451274\pi$
$762$		0			0
$763$	618.924	+	1072.01i	0.811172	+	1.40499i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−157.721	+	91.0604i	−0.205634	+	0.118723i
$768$		0			0
$769$	328.348	−	568.715i	0.426980	−	0.739552i	−0.569623	−	0.821906i	$-0.692911\pi$
$769$	328.348	−	568.715i	0.426980	−	0.739552i	0.996603	+	0.0823545i	$0.0262440\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	278.021i		−	0.359665i	−0.983697	−	0.179832i	$-0.942445\pi$
$773$		−	278.021i		−	0.359665i	0.983697	−	0.179832i	$-0.0575555\pi$
$774$		0			0
$775$	6.09082			0.00785912
$776$		0			0
$777$		0			0
$778$		0			0
$779$	233.499	+	134.811i	0.299743	+	0.173056i
$780$		0			0
$781$	−8.59133	−	14.8806i	−0.0110004	−	0.0190533i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	93.6827	−	54.0877i	0.119341	−	0.0689015i
$786$		0			0
$787$	410.977	−	711.833i	0.522207	−	0.904489i	−0.477459	−	0.878654i	$-0.658442\pi$
$787$	410.977	−	711.833i	0.522207	−	0.904489i	0.999666	−	0.0258350i	$-0.00822444\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1428.32i		−	1.80572i
$792$		0			0
$793$	−126.941			−0.160077
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1145.33	−	661.257i	−1.43705	−	0.829683i	−0.439409	−	0.898287i	$-0.644812\pi$
$797$	−1145.33	−	661.257i	−1.43705	−	0.829683i	−0.997644	+	0.0686043i	$0.978145\pi$
$798$		0			0
$799$	−427.454	−	740.372i	−0.534986	−	0.926624i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.53113	+	3.77075i	−0.00813341	+	0.00469583i
$804$		0			0
$805$	20.6816	−	35.8216i	0.0256915	−	0.0444989i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			235.681i			0.291324i	0.989334	+	0.145662i	$0.0465311\pi$
$809$			235.681i			0.291324i	−0.989334	+	0.145662i	$0.953469\pi$
$810$		0			0
$811$	587.362			0.724244			0.362122	−	0.932131i	$-0.382052\pi$
$811$	587.362			0.724244			0.362122	+	0.932131i	$0.382052\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−598.771	−	345.701i	−0.734688	−	0.424172i
$816$		0			0
$817$	556.242	+	963.439i	0.680835	+	1.17924i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	817.453	−	471.956i	0.995679	−	0.574856i	0.0887121	−	0.996057i	$-0.471725\pi$
$821$	817.453	−	471.956i	0.995679	−	0.574856i	0.906967	+	0.421202i	$0.138392\pi$
$822$		0			0
$823$	−807.871	+	1399.27i	−0.981617	+	1.70021i	−0.325520	+	0.945535i	$0.605539\pi$
$823$	−807.871	+	1399.27i	−0.981617	+	1.70021i	−0.656097	+	0.754676i	$0.727794\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	582.354i		−	0.704177i	−0.935967	−	0.352088i	$-0.885472\pi$
$827$		−	582.354i		−	0.704177i	0.935967	−	0.352088i	$-0.114528\pi$
$828$		0			0
$829$	877.121			1.05805			0.529024	−	0.848607i	$-0.322558\pi$
$829$	877.121			1.05805			0.529024	+	0.848607i	$0.322558\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	338.363	+	195.354i	0.406198	+	0.234519i
$834$		0			0
$835$	−765.158	−	1325.29i	−0.916357	−	1.58718i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−984.778	+	568.562i	−1.17375	+	0.677666i	−0.954561	−	0.298016i	$-0.903675\pi$
$839$	−984.778	+	568.562i	−1.17375	+	0.677666i	−0.219191	+	0.975682i	$0.570342\pi$
$840$		0			0
$841$	−326.909	+	566.223i	−0.388715	+	0.673274i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			389.552i			0.461009i
$846$		0			0
$847$	1002.57			1.18368
$848$		0			0
$849$		0			0
$850$		0			0
$851$	38.5607	+	22.2630i	0.0453122	+	0.0261610i
$852$		0			0
$853$	159.909	+	276.970i	0.187466	+	0.324701i	0.944405	−	0.328785i	$-0.106639\pi$
$853$	159.909	+	276.970i	0.187466	+	0.324701i	−0.756939	+	0.653486i	$0.773306\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−691.061	+	398.984i	−0.806372	+	0.465559i	−0.845694	−	0.533668i	$-0.820813\pi$
$857$	−691.061	+	398.984i	−0.806372	+	0.465559i	0.0393225	+	0.999227i	$0.487480\pi$
$858$		0			0
$859$	−233.901	+	405.128i	−0.272294	+	0.471627i	−0.969449	−	0.245293i	$-0.921116\pi$
$859$	−233.901	+	405.128i	−0.272294	+	0.471627i	0.697155	+	0.716921i	$0.254449\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1304.85i			1.51199i	0.654578	+	0.755994i	$0.272846\pi$
$863$			1304.85i			1.51199i	−0.654578	+	0.755994i	$0.727154\pi$
$864$		0			0
$865$	359.908			0.416079
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−36.1209	−	20.8544i	−0.0415661	−	0.0239982i
$870$		0			0
$871$	−363.855	−	630.216i	−0.417744	−	0.723554i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−864.067	+	498.869i	−0.987505	+	0.570136i
$876$		0			0
$877$	186.878	−	323.682i	0.213088	−	0.369079i	−0.739592	−	0.673056i	$-0.764981\pi$
$877$	186.878	−	323.682i	0.213088	−	0.369079i	0.952679	+	0.303977i	$0.0983146\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			229.979i			0.261043i	0.991445	+	0.130522i	$0.0416652\pi$
$881$			229.979i			0.261043i	−0.991445	+	0.130522i	$0.958335\pi$
$882$		0			0
$883$	1381.79			1.56488			0.782439	−	0.622728i	$-0.213976\pi$
$883$	1381.79			1.56488			0.782439	+	0.622728i	$0.213976\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	758.794	+	438.090i	0.855461	+	0.493901i	0.862490	−	0.506075i	$-0.168904\pi$
$887$	758.794	+	438.090i	0.855461	+	0.493901i	−0.00702852	+	0.999975i	$0.502237\pi$
$888$		0			0
$889$	−325.969	−	564.596i	−0.366670	−	0.635091i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	968.605	−	559.224i	1.08466	−	0.626231i
$894$		0			0
$895$	123.364	−	213.673i	0.137837	−	0.238741i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			41.6655i			0.0463465i
$900$		0			0
$901$	−1781.82			−1.97760
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1092.95	+	631.017i	1.20768	+	0.697256i
$906$		0			0
$907$	590.037	+	1021.97i	0.650537	+	1.12676i	0.982993	+	0.183644i	$0.0587894\pi$
$907$	590.037	+	1021.97i	0.650537	+	1.12676i	−0.332456	+	0.943119i	$0.607877\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1100.13	−	635.158i	1.20760	−	0.697210i	0.245368	−	0.969430i	$-0.421091\pi$
$911$	1100.13	−	635.158i	1.20760	−	0.697210i	0.962235	+	0.272220i	$0.0877578\pi$
$912$		0			0
$913$	−62.1367	+	107.624i	−0.0680578	+	0.117880i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1954.07i		−	2.13093i
$918$		0			0
$919$	−1316.63			−1.43268			−0.716340	−	0.697751i	$-0.754184\pi$
$919$	−1316.63			−1.43268			−0.716340	+	0.697751i	$0.754184\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−151.332	−	87.3713i	−0.163956	−	0.0946602i
$924$		0			0
$925$	46.6969	+	80.8815i	0.0504832	+	0.0874394i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−543.424	+	313.746i	−0.584956	+	0.337724i	−0.763100	−	0.646280i	$-0.776324\pi$
$929$	−543.424	+	313.746i	−0.584956	+	0.337724i	0.178145	+	0.984004i	$0.442991\pi$
$930$		0			0
$931$	−255.576	+	442.670i	−0.274517	+	0.475478i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	93.5307i		−	0.100033i
$936$		0			0
$937$	469.789			0.501375			0.250688	−	0.968068i	$-0.419343\pi$
$937$	469.789			0.501375			0.250688	+	0.968068i	$0.419343\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−805.984	−	465.335i	−0.856518	−	0.494511i	0.00632656	−	0.999980i	$-0.497986\pi$
$941$	−805.984	−	465.335i	−0.856518	−	0.494511i	−0.862845	+	0.505469i	$0.831320\pi$
$942$		0			0
$943$	5.20485	+	9.01506i	0.00551946	+	0.00955998i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.14465	+	1.81556i	−0.00332064	+	0.00191717i	−0.501659	−	0.865065i	$-0.667277\pi$
$947$	−3.14465	+	1.81556i	−0.00332064	+	0.00191717i	0.498339	+	0.866982i	$0.333944\pi$
$948$		0			0
$949$	−38.3474	+	66.4197i	−0.0404083	+	0.0699892i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	719.641i		−	0.755132i	−0.925983	−	0.377566i	$-0.876761\pi$
$953$		−	719.641i		−	0.755132i	0.925983	−	0.377566i	$-0.123239\pi$
$954$		0			0
$955$	39.1362			0.0409803
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1083.28	+	625.431i	1.12959	+	0.652170i
$960$		0			0
$961$	475.863	+	824.218i	0.495175	+	0.857667i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1554.54	+	897.516i	−1.61093	+	0.930068i
$966$		0			0
$967$	16.8870	−	29.2491i	0.0174633	−	0.0302473i	−0.857162	−	0.515047i	$-0.827774\pi$
$967$	16.8870	−	29.2491i	0.0174633	−	0.0302473i	0.874625	+	0.484800i	$0.161108\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	970.472i		−	0.999456i	−0.866182	−	0.499728i	$-0.833433\pi$
$971$		−	970.472i		−	0.999456i	0.866182	−	0.499728i	$-0.166567\pi$
$972$		0			0
$973$	705.697			0.725279
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1359.92	+	785.151i	1.39194	+	0.803635i	0.993529	−	0.113574i	$-0.0362300\pi$
$977$	1359.92	+	785.151i	1.39194	+	0.803635i	0.398406	+	0.917209i	$0.369563\pi$
$978$		0			0
$979$	69.2724	+	119.983i	0.0707584	+	0.122557i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−671.930	+	387.939i	−0.683551	+	0.394648i	−0.801192	−	0.598408i	$-0.795800\pi$
$983$	−671.930	+	387.939i	−0.683551	+	0.394648i	0.117641	+	0.993056i	$0.462467\pi$
$984$		0			0
$985$	−200.636	+	347.511i	−0.203691	+	0.352803i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			42.9513i			0.0434290i
$990$		0			0
$991$	−870.454			−0.878359			−0.439180	−	0.898399i	$-0.644731\pi$
$991$	−870.454			−0.878359			−0.439180	+	0.898399i	$0.644731\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−690.681	−	398.765i	−0.694152	−	0.400769i
$996$		0			0
$997$	−622.499	−	1078.20i	−0.624372	−	1.08144i	−0.988662	−	0.150159i	$-0.952022\pi$
$997$	−622.499	−	1078.20i	−0.624372	−	1.08144i	0.364290	−	0.931286i	$-0.381312\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.3.q.d.17.1		4
3.2	odd	2		144.3.q.c.113.2		4
4.3	odd	2		54.3.d.a.17.1		4
8.3	odd	2		1728.3.q.d.449.2		4
8.5	even	2		1728.3.q.c.449.1		4
9.2	odd	6	inner	432.3.q.d.305.1		4
9.4	even	3		1296.3.e.g.161.2		4
9.5	odd	6		1296.3.e.g.161.4		4
9.7	even	3		144.3.q.c.65.2		4
12.11	even	2		18.3.d.a.5.2	✓	4
20.3	even	4		1350.3.k.a.449.4		8
20.7	even	4		1350.3.k.a.449.1		8
20.19	odd	2		1350.3.i.b.1151.2		4
24.5	odd	2		576.3.q.e.257.1		4
24.11	even	2		576.3.q.f.257.2		4
36.7	odd	6		18.3.d.a.11.2	yes	4
36.11	even	6		54.3.d.a.35.1		4
36.23	even	6		162.3.b.a.161.4		4
36.31	odd	6		162.3.b.a.161.1		4
60.23	odd	4		450.3.k.a.149.1		8
60.47	odd	4		450.3.k.a.149.4		8
60.59	even	2		450.3.i.b.401.1		4
72.11	even	6		1728.3.q.d.1601.2		4
72.29	odd	6		1728.3.q.c.1601.1		4
72.43	odd	6		576.3.q.f.65.2		4
72.61	even	6		576.3.q.e.65.1		4
180.7	even	12		450.3.k.a.299.1		8
180.43	even	12		450.3.k.a.299.4		8
180.47	odd	12		1350.3.k.a.899.4		8
180.79	odd	6		450.3.i.b.101.1		4
180.83	odd	12		1350.3.k.a.899.1		8
180.119	even	6		1350.3.i.b.251.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.3.d.a.5.2	✓	4	12.11	even	2
18.3.d.a.11.2	yes	4	36.7	odd	6
54.3.d.a.17.1		4	4.3	odd	2
54.3.d.a.35.1		4	36.11	even	6
144.3.q.c.65.2		4	9.7	even	3
144.3.q.c.113.2		4	3.2	odd	2
162.3.b.a.161.1		4	36.31	odd	6
162.3.b.a.161.4		4	36.23	even	6
432.3.q.d.17.1		4	1.1	even	1	trivial
432.3.q.d.305.1		4	9.2	odd	6	inner
450.3.i.b.101.1		4	180.79	odd	6
450.3.i.b.401.1		4	60.59	even	2
450.3.k.a.149.1		8	60.23	odd	4
450.3.k.a.149.4		8	60.47	odd	4
450.3.k.a.299.1		8	180.7	even	12
450.3.k.a.299.4		8	180.43	even	12
576.3.q.e.65.1		4	72.61	even	6
576.3.q.e.257.1		4	24.5	odd	2
576.3.q.f.65.2		4	72.43	odd	6
576.3.q.f.257.2		4	24.11	even	2
1296.3.e.g.161.2		4	9.4	even	3
1296.3.e.g.161.4		4	9.5	odd	6
1350.3.i.b.251.2		4	180.119	even	6
1350.3.i.b.1151.2		4	20.19	odd	2
1350.3.k.a.449.1		8	20.7	even	4
1350.3.k.a.449.4		8	20.3	even	4
1350.3.k.a.899.1		8	180.83	odd	12
1350.3.k.a.899.4		8	180.47	odd	12
1728.3.q.c.449.1		4	8.5	even	2
1728.3.q.c.1601.1		4	72.29	odd	6
1728.3.q.d.449.2		4	8.3	odd	2
1728.3.q.d.1601.2		4	72.11	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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	432.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 + 2.59808i) q^{5} +(-4.17423 - 7.22999i) q^{7} +O(q^{10})\)	\(q+(4.50000 + 2.59808i) q^{5} +(-4.17423 - 7.22999i) q^{7} +(0.825765 - 0.476756i) q^{11} +(4.84847 - 8.39780i) q^{13} -18.8776i q^{17} +24.6969 q^{19} +(0.825765 + 0.476756i) q^{23} +(1.00000 + 1.73205i) q^{25} +(-11.8485 + 6.84072i) q^{29} +(1.52270 - 2.63740i) q^{31} -43.3799i q^{35} +46.6969 q^{37} +(9.45459 + 5.45861i) q^{41} +(22.5227 + 39.0105i) q^{43} +(39.2196 - 22.6435i) q^{47} +(-10.3485 + 17.9241i) q^{49} -94.3879i q^{53} +4.95459 q^{55} +(-16.2650 - 9.39063i) q^{59} +(-6.54541 - 11.3370i) q^{61} +(43.6362 - 25.1934i) q^{65} +(37.5227 - 64.9912i) q^{67} -18.0204i q^{71} -7.90918 q^{73} +(-6.89388 - 3.98018i) q^{77} +(-21.8712 - 37.8820i) q^{79} +(-112.871 + 65.1662i) q^{83} +(49.0454 - 84.9491i) q^{85} +145.300i q^{89} -80.9546 q^{91} +(111.136 + 64.1645i) q^{95} +(54.9393 + 95.1576i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 18 q^{5} - 2 q^{7}+O(q^{10}) \) 4 * q + 18 * q^5 - 2 * q^7	\( 4 q + 18 q^{5} - 2 q^{7} + 18 q^{11} - 10 q^{13} + 40 q^{19} + 18 q^{23} + 4 q^{25} - 18 q^{29} - 38 q^{31} + 128 q^{37} + 126 q^{41} + 46 q^{43} + 54 q^{47} - 12 q^{49} + 108 q^{55} + 126 q^{59} + 62 q^{61} - 90 q^{65} + 106 q^{67} - 208 q^{73} + 90 q^{77} - 14 q^{79} - 378 q^{83} + 108 q^{85} - 412 q^{91} + 180 q^{95} + 14 q^{97}+O(q^{100}) \) 4 * q + 18 * q^5 - 2 * q^7 + 18 * q^11 - 10 * q^13 + 40 * q^19 + 18 * q^23 + 4 * q^25 - 18 * q^29 - 38 * q^31 + 128 * q^37 + 126 * q^41 + 46 * q^43 + 54 * q^47 - 12 * q^49 + 108 * q^55 + 126 * q^59 + 62 * q^61 - 90 * q^65 + 106 * q^67 - 208 * q^73 + 90 * q^77 - 14 * q^79 - 378 * q^83 + 108 * q^85 - 412 * q^91 + 180 * q^95 + 14 * q^97

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