LMFDB - Embedded newform 432.6.i.c.145.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [432,6,Mod(145,432)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("432.145"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")

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

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$69.2858101592$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 40x^{6} + 568x^{4} + 3363x^{2} + 7056 $ x^8 + 40x^6 + 568x^4 + 3363*x^2 + 7056
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			145.2
Root			$2.56934i$ of defining polynomial
Character	$\chi$	$=$	432.145
Dual form			432.6.i.c.289.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-23.6560 + 40.9735i) q^{5} +(-1.01546 - 1.75882i) q^{7} +(-91.8836 - 159.147i) q^{11} +(-364.672 + 631.630i) q^{13} -1212.88 q^{17} +473.092 q^{19} +(1806.42 - 3128.81i) q^{23} +(443.283 + 767.788i) q^{25} +(-663.121 - 1148.56i) q^{29} +(-2590.41 + 4486.72i) q^{31} +96.0867 q^{35} -14715.2 q^{37} +(3159.83 - 5472.98i) q^{41} +(3067.89 + 5313.75i) q^{43} +(-1584.17 - 2743.86i) q^{47} +(8401.44 - 14551.7i) q^{49} +12265.0 q^{53} +8694.41 q^{55} +(14773.5 - 25588.5i) q^{59} +(20153.6 + 34907.1i) q^{61} +(-17253.4 - 29883.8i) q^{65} +(11748.9 - 20349.7i) q^{67} -123.786 q^{71} +35217.6 q^{73} +(-186.607 + 323.214i) q^{77} +(-24072.8 - 41695.3i) q^{79} +(-5167.97 - 8951.18i) q^{83} +(28692.0 - 49696.1i) q^{85} +42585.7 q^{89} +1481.23 q^{91} +(-11191.5 + 19384.2i) q^{95} +(-49347.3 - 85472.1i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 78 q^{5} - 28 q^{7} + 444 q^{11} - 182 q^{13} + 4356 q^{17} - 952 q^{19} + 8844 q^{23} - 1654 q^{25} - 12018 q^{29} - 1132 q^{31} - 16224 q^{35} - 15176 q^{37} - 1248 q^{41} + 6092 q^{43} - 60 q^{47}+ \cdots + 33976 q^{97}+O(q^{100}) $ 8 * q - 78 * q^5 - 28 * q^7 + 444 * q^11 - 182 * q^13 + 4356 * q^17 - 952 * q^19 + 8844 * q^23 - 1654 * q^25 - 12018 * q^29 - 1132 * q^31 - 16224 * q^35 - 15176 * q^37 - 1248 * q^41 + 6092 * q^43 - 60 * q^47 + 9090 * q^49 - 20952 * q^53 + 36120 * q^55 + 2076 * q^59 + 48142 * q^61 + 13146 * q^65 + 7148 * q^67 - 71856 * q^71 + 122452 * q^73 - 39534 * q^77 + 59516 * q^79 + 117696 * q^83 + 28836 * q^85 + 451728 * q^89 - 111392 * q^91 + 294888 * q^95 + 33976 * q^97

Character values

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

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−23.6560	+	40.9735i	−0.423172	+	0.732956i	−0.996248	−	0.0865467i	$-0.972417\pi$
$5$	−23.6560	+	40.9735i	−0.423172	+	0.732956i	0.573076	+	0.819503i	$0.305750\pi$
$6$		0			0
$7$	−1.01546	−	1.75882i	−0.00783278	−	0.0135668i	0.862082	−	0.506768i	$-0.169160\pi$
$7$	−1.01546	−	1.75882i	−0.00783278	−	0.0135668i	−0.869915	+	0.493201i	$0.835827\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−91.8836	−	159.147i	−0.228958	−	0.396567i	0.728541	−	0.685002i	$-0.240199\pi$
$11$	−91.8836	−	159.147i	−0.228958	−	0.396567i	−0.957500	+	0.288435i	$0.906865\pi$
$12$		0			0
$13$	−364.672	+	631.630i	−0.598472	+	1.03658i	0.394574	+	0.918864i	$0.370892\pi$
$13$	−364.672	+	631.630i	−0.598472	+	1.03658i	−0.993047	+	0.117720i	$0.962441\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1212.88			−1.01788			−0.508940	−	0.860802i	$-0.669963\pi$
$17$	−1212.88			−1.01788			−0.508940	+	0.860802i	$0.669963\pi$
$18$		0			0
$19$	473.092			0.300650			0.150325	−	0.988637i	$-0.451968\pi$
$19$	473.092			0.300650			0.150325	+	0.988637i	$0.451968\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1806.42	−	3128.81i	0.712031	−	1.23327i	−0.252063	−	0.967711i	$-0.581109\pi$
$23$	1806.42	−	3128.81i	0.712031	−	1.23327i	0.964094	−	0.265563i	$-0.0855578\pi$
$24$		0			0
$25$	443.283	+	767.788i	0.141850	+	0.245692i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−663.121	−	1148.56i	−0.146419	−	0.253605i	0.783482	−	0.621414i	$-0.213441\pi$
$29$	−663.121	−	1148.56i	−0.146419	−	0.253605i	−0.929901	+	0.367809i	$0.880108\pi$
$30$		0			0
$31$	−2590.41	+	4486.72i	−0.484133	+	0.838542i	−0.999834	−	0.0182261i	$-0.994198\pi$
$31$	−2590.41	+	4486.72i	−0.484133	+	0.838542i	0.515701	+	0.856769i	$0.327531\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	96.0867			0.0132585
$36$		0			0
$37$	−14715.2			−1.76710			−0.883552	−	0.468334i	$-0.844854\pi$
$37$	−14715.2			−1.76710			−0.883552	+	0.468334i	$0.844854\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3159.83	−	5472.98i	0.293564	−	0.508469i	−0.681085	−	0.732204i	$-0.738492\pi$
$41$	3159.83	−	5472.98i	0.293564	−	0.508469i	0.974650	+	0.223735i	$0.0718251\pi$
$42$		0			0
$43$	3067.89	+	5313.75i	0.253028	+	0.438258i	0.964358	−	0.264600i	$-0.0852401\pi$
$43$	3067.89	+	5313.75i	0.253028	+	0.438258i	−0.711330	+	0.702858i	$0.751907\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1584.17	−	2743.86i	−0.104606	−	0.181183i	0.808971	−	0.587848i	$-0.200025\pi$
$47$	−1584.17	−	2743.86i	−0.104606	−	0.181183i	−0.913577	+	0.406665i	$0.866692\pi$
$48$		0			0
$49$	8401.44	−	14551.7i	0.499877	−	0.865813i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12265.0			0.599763			0.299881	−	0.953977i	$-0.403053\pi$
$53$	12265.0			0.599763			0.299881	+	0.953977i	$0.403053\pi$
$54$		0			0
$55$	8694.41			0.387555
$56$		0			0
$57$		0			0
$58$		0			0
$59$	14773.5	−	25588.5i	0.552528	−	0.957006i	−0.445563	−	0.895250i	$-0.646997\pi$
$59$	14773.5	−	25588.5i	0.552528	−	0.957006i	0.998091	−	0.0617560i	$-0.0196701\pi$
$60$		0			0
$61$	20153.6	+	34907.1i	0.693472	+	1.20113i	0.970693	+	0.240323i	$0.0772534\pi$
$61$	20153.6	+	34907.1i	0.693472	+	1.20113i	−0.277221	+	0.960806i	$0.589413\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−17253.4	−	29883.8i	−0.506514	−	0.877308i
$66$		0			0
$67$	11748.9	−	20349.7i	0.319750	−	0.553823i	−0.660686	−	0.750662i	$-0.729734\pi$
$67$	11748.9	−	20349.7i	0.319750	−	0.553823i	0.980436	+	0.196840i	$0.0630678\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−123.786			−0.00291425			−0.00145713	−	0.999999i	$-0.500464\pi$
$71$	−123.786			−0.00291425			−0.00145713	+	0.999999i	$0.500464\pi$
$72$		0			0
$73$	35217.6			0.773486			0.386743	−	0.922188i	$-0.373600\pi$
$73$	35217.6			0.773486			0.386743	+	0.922188i	$0.373600\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−186.607	+	323.214i	−0.00358676	+	0.00621245i
$78$		0			0
$79$	−24072.8	−	41695.3i	−0.433969	−	0.751656i	0.563242	−	0.826292i	$-0.309554\pi$
$79$	−24072.8	−	41695.3i	−0.433969	−	0.751656i	−0.997211	+	0.0746359i	$0.976221\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5167.97	−	8951.18i	−0.0823426	−	0.142622i	0.821913	−	0.569613i	$-0.192907\pi$
$83$	−5167.97	−	8951.18i	−0.0823426	−	0.142622i	−0.904256	+	0.426991i	$0.859574\pi$
$84$		0			0
$85$	28692.0	−	49696.1i	0.430739	−	0.746062i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	42585.7			0.569887			0.284944	−	0.958544i	$-0.408025\pi$
$89$	42585.7			0.569887			0.284944	+	0.958544i	$0.408025\pi$
$90$		0			0
$91$	1481.23			0.0187508
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11191.5	+	19384.2i	−0.127227	+	0.220363i
$96$		0			0
$97$	−49347.3	−	85472.1i	−0.532518	−	0.922348i	−0.999279	−	0.0379646i	$-0.987913\pi$
$97$	−49347.3	−	85472.1i	−0.532518	−	0.922348i	0.466761	−	0.884383i	$-0.345421\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−37818.9	−	65504.3i	−0.368898	−	0.638949i	0.620496	−	0.784210i	$-0.286931\pi$
$101$	−37818.9	−	65504.3i	−0.368898	−	0.638949i	−0.989393	+	0.145260i	$0.953598\pi$
$102$		0			0
$103$	73869.1	−	127945.i	0.686072	−	1.18831i	−0.287026	−	0.957923i	$-0.592667\pi$
$103$	73869.1	−	127945.i	0.686072	−	1.18831i	0.973099	−	0.230389i	$-0.0739999\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−209139.			−1.76594			−0.882969	−	0.469430i	$-0.844459\pi$
$107$	−209139.			−1.76594			−0.882969	+	0.469430i	$0.844459\pi$
$108$		0			0
$109$	115605.			0.931988			0.465994	−	0.884788i	$-0.345697\pi$
$109$	115605.			0.931988			0.465994	+	0.884788i	$0.345697\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	23415.8	−	40557.3i	0.172509	−	0.298795i	−0.766787	−	0.641901i	$-0.778146\pi$
$113$	23415.8	−	40557.3i	0.172509	−	0.298795i	0.939296	+	0.343107i	$0.111479\pi$
$114$		0			0
$115$	85465.5	+	148031.i	0.602623	+	1.04377i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1231.63	+	2133.25i	0.00797283	+	0.0138094i
$120$		0			0
$121$	63640.3	−	110228.i	0.395156	−	0.684431i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−189796.			−1.08645
$126$		0			0
$127$	93444.5			0.514096			0.257048	−	0.966399i	$-0.417250\pi$
$127$	93444.5			0.514096			0.257048	+	0.966399i	$0.417250\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	183869.	−	318470.i	0.936115	−	1.62140i	0.163481	−	0.986546i	$-0.447728\pi$
$131$	183869.	−	318470.i	0.936115	−	1.62140i	0.772634	−	0.634852i	$-0.218939\pi$
$132$		0			0
$133$	−480.404	−	832.083i	−0.00235493	−	0.00407885i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−76809.2	−	133037.i	−0.349632	−	0.605581i	0.636552	−	0.771234i	$-0.280360\pi$
$137$	−76809.2	−	133037.i	−0.349632	−	0.605581i	−0.986184	+	0.165653i	$0.947027\pi$
$138$		0			0
$139$	122213.	−	211679.i	0.536513	−	0.929268i	−0.462575	−	0.886580i	$-0.653075\pi$
$139$	122213.	−	211679.i	0.536513	−	0.929268i	0.999088	−	0.0426878i	$-0.0135921\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	134029.			0.548101
$144$		0			0
$145$	62747.3			0.247842
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4924.61	+	8529.67i	−0.0181721	+	0.0314751i	−0.874968	−	0.484180i	$-0.839118\pi$
$149$	−4924.61	+	8529.67i	−0.0181721	+	0.0314751i	0.856796	+	0.515655i	$0.172451\pi$
$150$		0			0
$151$	−109892.	−	190339.i	−0.392216	−	0.679338i	0.600526	−	0.799606i	$-0.294958\pi$
$151$	−109892.	−	190339.i	−0.392216	−	0.679338i	−0.992741	+	0.120268i	$0.961625\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−122558.	−	212276.i	−0.409743	−	0.709696i
$156$		0			0
$157$	−15290.5	+	26483.9i	−0.0495077	+	0.0857498i	−0.889717	−	0.456512i	$-0.849099\pi$
$157$	−15290.5	+	26483.9i	−0.0495077	+	0.0857498i	0.840210	+	0.542262i	$0.182432\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7337.35			−0.0223087
$162$		0			0
$163$	272367.			0.802945			0.401472	−	0.915871i	$-0.368499\pi$
$163$	272367.			0.802945			0.401472	+	0.915871i	$0.368499\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	99337.8	−	172058.i	0.275628	−	0.477402i	−0.694665	−	0.719333i	$-0.744448\pi$
$167$	99337.8	−	172058.i	0.275628	−	0.477402i	0.970293	+	0.241931i	$0.0777809\pi$
$168$		0			0
$169$	−80324.9	−	139127.i	−0.216338	−	0.374709i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	33895.9	+	58709.5i	0.0861058	+	0.149140i	0.905862	−	0.423573i	$-0.139224\pi$
$173$	33895.9	+	58709.5i	0.0861058	+	0.149140i	−0.819756	+	0.572713i	$0.805891\pi$
$174$		0			0
$175$	900.268	−	1559.31i	0.00222217	−	0.00384891i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	21849.0			0.0509681			0.0254840	−	0.999675i	$-0.491887\pi$
$179$	21849.0			0.0509681			0.0254840	+	0.999675i	$0.491887\pi$
$180$		0			0
$181$	188080.			0.426722			0.213361	−	0.976973i	$-0.431559\pi$
$181$	188080.			0.426722			0.213361	+	0.976973i	$0.431559\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	348104.	−	602933.i	0.747789	−	1.29521i
$186$		0			0
$187$	111444.	+	193027.i	0.233052	+	0.403658i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	123200.	+	213389.i	0.244359	+	0.423241i	0.961951	−	0.273222i	$-0.0880893\pi$
$191$	123200.	+	213389.i	0.244359	+	0.423241i	−0.717592	+	0.696463i	$0.754756\pi$
$192$		0			0
$193$	113565.	−	196701.i	0.219459	−	0.380114i	−0.735184	−	0.677868i	$-0.762904\pi$
$193$	113565.	−	196701.i	0.219459	−	0.380114i	0.954643	+	0.297754i	$0.0962375\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	503086.			0.923584			0.461792	−	0.886988i	$-0.347207\pi$
$197$	503086.			0.923584			0.461792	+	0.886988i	$0.347207\pi$
$198$		0			0
$199$	−4126.65			−0.00738694			−0.00369347	−	0.999993i	$-0.501176\pi$
$199$	−4126.65			−0.00738694			−0.00369347	+	0.999993i	$0.501176\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1346.74	+	2332.62i	−0.00229374	+	0.00397287i
$204$		0			0
$205$	149498.	+	258938.i	0.248457	+	0.430340i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−43469.3	−	75291.1i	−0.0688363	−	0.119228i
$210$		0			0
$211$	−494954.	+	857285.i	−0.765347	+	1.32562i	0.174716	+	0.984619i	$0.444099\pi$
$211$	−494954.	+	857285.i	−0.765347	+	1.32562i	−0.940063	+	0.341001i	$0.889234\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−290297.			−0.428298
$216$		0			0
$217$	10521.8			0.0151684
$218$		0			0
$219$		0			0
$220$		0			0
$221$	442305.	−	766094.i	0.609173	−	1.05512i
$222$		0			0
$223$	−523910.	−	907438.i	−0.705496	−	1.22195i	−0.966512	−	0.256620i	$-0.917391\pi$
$223$	−523910.	−	907438.i	−0.705496	−	1.22195i	0.261017	−	0.965334i	$-0.415942\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−355537.	−	615808.i	−0.457952	−	0.793196i	0.540900	−	0.841087i	$-0.318084\pi$
$227$	−355537.	−	615808.i	−0.457952	−	0.793196i	−0.998853	+	0.0478902i	$0.984750\pi$
$228$		0			0
$229$	−147071.	+	254734.i	−0.185327	+	0.320995i	−0.943687	−	0.330841i	$-0.892668\pi$
$229$	−147071.	+	254734.i	−0.185327	+	0.320995i	0.758360	+	0.651836i	$0.226001\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.56670e6			−1.89059			−0.945294	−	0.326218i	$-0.894226\pi$
$233$	−1.56670e6			−1.89059			−0.945294	+	0.326218i	$0.894226\pi$
$234$		0			0
$235$	149901.			0.177066
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−260961.	+	451998.i	−0.295516	+	0.511849i	−0.975105	−	0.221744i	$-0.928825\pi$
$239$	−260961.	+	451998.i	−0.295516	+	0.511849i	0.679589	+	0.733593i	$0.262158\pi$
$240$		0			0
$241$	213855.	+	370407.i	0.237179	+	0.410806i	0.959904	−	0.280330i	$-0.0904439\pi$
$241$	213855.	+	370407.i	0.237179	+	0.410806i	−0.722725	+	0.691136i	$0.757111\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	397490.	+	688472.i	0.423068	+	0.732776i
$246$		0			0
$247$	−172523.	+	298819.i	−0.179931	+	0.311649i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−231914.			−0.232350			−0.116175	−	0.993229i	$-0.537063\pi$
$251$	−231914.			−0.232350			−0.116175	+	0.993229i	$0.537063\pi$
$252$		0			0
$253$	−663921.			−0.652101
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−707127.	+	1.22478e6i	−0.667828	+	1.15671i	0.310682	+	0.950514i	$0.399443\pi$
$257$	−707127.	+	1.22478e6i	−0.667828	+	1.15671i	−0.978510	+	0.206199i	$0.933891\pi$
$258$		0			0
$259$	14942.6	+	25881.4i	0.0138413	+	0.0239739i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	615838.	+	1.06666e6i	0.549006	+	0.950907i	0.998343	+	0.0575445i	$0.0183271\pi$
$263$	615838.	+	1.06666e6i	0.549006	+	0.950907i	−0.449336	+	0.893363i	$0.648340\pi$
$264$		0			0
$265$	−290143.	+	502542.i	−0.253803	+	0.439600i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−119245.			−0.100475			−0.0502377	−	0.998737i	$-0.515998\pi$
$269$	−119245.			−0.100475			−0.0502377	+	0.998737i	$0.515998\pi$
$270$		0			0
$271$	−50141.4			−0.0414738			−0.0207369	−	0.999785i	$-0.506601\pi$
$271$	−50141.4			−0.0414738			−0.0207369	+	0.999785i	$0.506601\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	81460.8	−	141094.i	0.0649557	−	0.112506i
$276$		0			0
$277$	−235820.	−	408452.i	−0.184663	−	0.319846i	0.758800	−	0.651324i	$-0.225786\pi$
$277$	−235820.	−	408452.i	−0.184663	−	0.319846i	−0.943463	+	0.331478i	$0.892453\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.03652e6	+	1.79530e6i	0.783087	+	1.35635i	0.930135	+	0.367217i	$0.119689\pi$
$281$	1.03652e6	+	1.79530e6i	0.783087	+	1.35635i	−0.147048	+	0.989129i	$0.546977\pi$
$282$		0			0
$283$	758046.	−	1.31297e6i	0.562639	−	0.974519i	−0.434626	−	0.900611i	$-0.643120\pi$
$283$	758046.	−	1.31297e6i	0.562639	−	0.974519i	0.997265	−	0.0739079i	$-0.0235471\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12834.7			−0.00919770
$288$		0			0
$289$	51229.6			0.0360808
$290$		0			0
$291$		0			0
$292$		0			0
$293$	385100.	−	667013.i	0.262062	−	0.453905i	−0.704727	−	0.709478i	$-0.748931\pi$
$293$	385100.	−	667013.i	0.262062	−	0.453905i	0.966790	+	0.255573i	$0.0822641\pi$
$294$		0			0
$295$	698967.	+	1.21065e6i	0.467629	+	0.809957i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.31750e6	+	2.28198e6i	0.852262	+	1.47616i
$300$		0			0
$301$	6230.62	−	10791.8i	0.00396383	−	0.00686556i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.90702e6			−1.17383
$306$		0			0
$307$	−2.13559e6			−1.29322			−0.646610	−	0.762821i	$-0.723814\pi$
$307$	−2.13559e6			−1.29322			−0.646610	+	0.762821i	$0.723814\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	301713.	−	522582.i	0.176886	−	0.306375i	−0.763927	−	0.645303i	$-0.776731\pi$
$311$	301713.	−	522582.i	0.176886	−	0.306375i	0.940812	+	0.338928i	$0.110064\pi$
$312$		0			0
$313$	−1.08619e6	−	1.88133e6i	−0.626677	−	1.08544i	−0.988214	−	0.153079i	$-0.951081\pi$
$313$	−1.08619e6	−	1.88133e6i	−0.626677	−	1.08544i	0.361537	−	0.932358i	$-0.382252\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.60049e6	+	2.77213e6i	0.894550	+	1.54941i	0.834361	+	0.551218i	$0.185837\pi$
$317$	1.60049e6	+	2.77213e6i	0.894550	+	1.54941i	0.0601886	+	0.998187i	$0.480830\pi$
$318$		0			0
$319$	−121860.	+	211067.i	−0.0670477	+	0.116130i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−573805.			−0.306026
$324$		0			0
$325$	−646611.			−0.339574
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3217.31	+	5572.54i	−0.00163871	+	0.00283834i
$330$		0			0
$331$	1.30413e6	+	2.25883e6i	0.654263	+	1.13322i	0.982078	+	0.188475i	$0.0603545\pi$
$331$	1.30413e6	+	2.25883e6i	0.654263	+	1.13322i	−0.327815	+	0.944742i	$0.606312\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	555865.	+	962787.i	0.270618	+	0.468725i
$336$		0			0
$337$	414934.	−	718687.i	0.199024	−	0.344719i	−0.749189	−	0.662357i	$-0.769556\pi$
$337$	414934.	−	718687.i	0.199024	−	0.344719i	0.948212	+	0.317638i	$0.102890\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	952065.			0.443385
$342$		0			0
$343$	−68258.7			−0.0313273
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.15930e6	+	2.00796e6i	−0.516858	+	0.895224i	0.482951	+	0.875648i	$0.339565\pi$
$347$	−1.15930e6	+	2.00796e6i	−0.516858	+	0.895224i	−0.999808	+	0.0195761i	$0.993768\pi$
$348$		0			0
$349$	1.56680e6	+	2.71377e6i	0.688572	+	1.19264i	0.972300	+	0.233737i	$0.0750955\pi$
$349$	1.56680e6	+	2.71377e6i	0.688572	+	1.19264i	−0.283728	+	0.958905i	$0.591571\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.33508e6	−	2.31243e6i	−0.570259	−	0.987717i	−0.996539	−	0.0831261i	$-0.973510\pi$
$353$	−1.33508e6	−	2.31243e6i	−0.570259	−	0.987717i	0.426280	−	0.904591i	$-0.359824\pi$
$354$		0			0
$355$	2928.30	−	5071.96i	0.00123323	−	0.00213602i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.05907e6			−0.843210			−0.421605	−	0.906780i	$-0.638533\pi$
$359$	−2.05907e6			−0.843210			−0.421605	+	0.906780i	$0.638533\pi$
$360$		0			0
$361$	−2.25228e6			−0.909610
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−833109.	+	1.44299e6i	−0.327318	+	0.566931i
$366$		0			0
$367$	−509190.	−	881943.i	−0.197340	−	0.341803i	0.750325	−	0.661069i	$-0.229897\pi$
$367$	−509190.	−	881943.i	−0.197340	−	0.341803i	−0.947665	+	0.319266i	$0.896564\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12454.6	−	21572.0i	−0.00469781	−	0.00813685i
$372$		0			0
$373$	−2.61768e6	+	4.53395e6i	−0.974191	+	1.68735i	−0.291612	+	0.956537i	$0.594192\pi$
$373$	−2.61768e6	+	4.53395e6i	−0.974191	+	1.68735i	−0.682579	+	0.730812i	$0.739142\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	967286.			0.350511
$378$		0			0
$379$	−2.73883e6			−0.979417			−0.489708	−	0.871886i	$-0.662897\pi$
$379$	−2.73883e6			−0.979417			−0.489708	+	0.871886i	$0.662897\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.17454e6	−	3.76641e6i	0.757478	−	1.31199i	−0.186655	−	0.982425i	$-0.559765\pi$
$383$	2.17454e6	−	3.76641e6i	0.757478	−	1.31199i	0.944133	−	0.329564i	$-0.106902\pi$
$384$		0			0
$385$	−8828.79	−	15291.9i	−0.00303563	−	0.00525787i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.50378e6	+	2.60462e6i	0.503860	+	0.872712i	0.999990	+	0.00446327i	$0.00142071\pi$
$389$	1.50378e6	+	2.60462e6i	0.503860	+	0.872712i	−0.496130	+	0.868248i	$0.665246\pi$
$390$		0			0
$391$	−2.19098e6	+	3.79488e6i	−0.724762	+	1.25533i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.27787e6			0.734574
$396$		0			0
$397$	2.16602e6			0.689740			0.344870	−	0.938650i	$-0.387923\pi$
$397$	2.16602e6			0.689740			0.344870	+	0.938650i	$0.387923\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.76996e6	−	3.06566e6i	0.549670	−	0.952057i	−0.448626	−	0.893719i	$-0.648087\pi$
$401$	1.76996e6	−	3.06566e6i	0.549670	−	0.952057i	0.998297	−	0.0583377i	$-0.0185800\pi$
$402$		0			0
$403$	−1.88930e6	−	3.27236e6i	−0.579480	−	1.00369i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.35209e6	+	2.34188e6i	0.404593	+	0.700775i
$408$		0			0
$409$	783644.	−	1.35731e6i	0.231638	−	0.401210i	−0.726652	−	0.687006i	$-0.758925\pi$
$409$	783644.	−	1.35731e6i	0.231638	−	0.401210i	0.958290	+	0.285796i	$0.0922580\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−60007.5			−0.0173113
$414$		0			0
$415$	489015.			0.139380
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.18321e6	−	2.04937e6i	0.329250	−	0.570277i	−0.653114	−	0.757260i	$-0.726538\pi$
$419$	1.18321e6	−	2.04937e6i	0.329250	−	0.570277i	0.982363	+	0.186983i	$0.0598709\pi$
$420$		0			0
$421$	1.84288e6	+	3.19196e6i	0.506747	+	0.877712i	0.999970	+	0.00780885i	$0.00248566\pi$
$421$	1.84288e6	+	3.19196e6i	0.506747	+	0.877712i	−0.493222	+	0.869903i	$0.664181\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−537650.	−	931238.i	−0.144387	−	0.250085i
$426$		0			0
$427$	40930.3	−	70893.3i	0.0108636	−	0.0188164i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.78152e6			−1.75847			−0.879233	−	0.476391i	$-0.841945\pi$
$431$	−6.78152e6			−1.75847			−0.879233	+	0.476391i	$0.841945\pi$
$432$		0			0
$433$	−3.20434e6			−0.821332			−0.410666	−	0.911786i	$-0.634704\pi$
$433$	−3.20434e6			−0.821332			−0.410666	+	0.911786i	$0.634704\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	854601.	−	1.48021e6i	0.214072	−	0.370784i
$438$		0			0
$439$	698435.	+	1.20972e6i	0.172967	+	0.299588i	0.939456	−	0.342670i	$-0.111331\pi$
$439$	698435.	+	1.20972e6i	0.172967	+	0.299588i	−0.766489	+	0.642258i	$0.777998\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.48597e6	+	4.30583e6i	0.601847	+	1.04243i	0.992541	+	0.121910i	$0.0389018\pi$
$443$	2.48597e6	+	4.30583e6i	0.601847	+	1.04243i	−0.390694	+	0.920521i	$0.627765\pi$
$444$		0			0
$445$	−1.00741e6	+	1.74488e6i	−0.241160	+	0.417702i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4.85114e6			−1.13561			−0.567804	−	0.823164i	$-0.692207\pi$
$449$	−4.85114e6			−1.13561			−0.567804	+	0.823164i	$0.692207\pi$
$450$		0			0
$451$	−1.16134e6			−0.268856
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−35040.1	+	60691.3i	−0.00793482	+	0.0137435i
$456$		0			0
$457$	−2.14244e6	−	3.71081e6i	−0.479864	−	0.831149i	0.519869	−	0.854246i	$-0.325981\pi$
$457$	−2.14244e6	−	3.71081e6i	−0.479864	−	0.831149i	−0.999733	+	0.0230972i	$0.992647\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	11973.0	+	20737.8i	0.00262392	+	0.00454476i	0.867334	−	0.497726i	$-0.165831\pi$
$461$	11973.0	+	20737.8i	0.00262392	+	0.00454476i	−0.864710	+	0.502271i	$0.832498\pi$
$462$		0			0
$463$	−1.76420e6	+	3.05569e6i	−0.382469	+	0.662456i	−0.991415	−	0.130756i	$-0.958259\pi$
$463$	−1.76420e6	+	3.05569e6i	−0.382469	+	0.662456i	0.608945	+	0.793212i	$0.291593\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.66263e6			−0.989324			−0.494662	−	0.869086i	$-0.664708\pi$
$467$	−4.66263e6			−0.989324			−0.494662	+	0.869086i	$0.664708\pi$
$468$		0			0
$469$	−47722.0			−0.0100181
$470$		0			0
$471$		0			0
$472$		0			0
$473$	563778.	−	976492.i	0.115866	−	0.200686i
$474$		0			0
$475$	209713.	+	363234.i	0.0426473	+	0.0738673i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.27492e6	−	5.67232e6i	−0.652171	−	1.12959i	−0.982595	−	0.185760i	$-0.940525\pi$
$479$	−3.27492e6	−	5.67232e6i	−0.652171	−	1.12959i	0.330424	−	0.943832i	$-0.392808\pi$
$480$		0			0
$481$	5.36622e6	−	9.29457e6i	1.05756	−	1.83175i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.66945e6			0.901387
$486$		0			0
$487$	9.23457e6			1.76439			0.882194	−	0.470886i	$-0.156066\pi$
$487$	9.23457e6			1.76439			0.882194	+	0.470886i	$0.156066\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.62130e6	−	4.54023e6i	0.490697	−	0.849912i	−0.509246	−	0.860621i	$-0.670075\pi$
$491$	2.62130e6	−	4.54023e6i	0.490697	−	0.849912i	0.999943	+	0.0107089i	$0.00340882\pi$
$492$		0			0
$493$	804288.	+	1.39307e6i	0.149037	+	0.258140i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	125.700	+	217.718i	2.28267e−5	+	3.95370e-5i
$498$		0			0
$499$	3.23974e6	−	5.61140e6i	0.582451	−	1.00883i	−0.412737	−	0.910850i	$-0.635427\pi$
$499$	3.23974e6	−	5.61140e6i	0.582451	−	1.00883i	0.995188	−	0.0979844i	$-0.0312395\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4.42908e6			0.780538			0.390269	−	0.920701i	$-0.372382\pi$
$503$	4.42908e6			0.780538			0.390269	+	0.920701i	$0.372382\pi$
$504$		0			0
$505$	3.57859e6			0.624429
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.14026e6	−	3.70704e6i	0.366162	−	0.634210i	−0.622800	−	0.782381i	$-0.714005\pi$
$509$	2.14026e6	−	3.70704e6i	0.366162	−	0.634210i	0.988962	+	0.148170i	$0.0473384\pi$
$510$		0			0
$511$	−35761.9	−	61941.4i	−0.00605855	−	0.0104937i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.49490e6	+	6.05335e6i	0.580653	+	1.00572i
$516$		0			0
$517$	−291118.	+	504232.i	−0.0479009	+	0.0829667i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8.64335e6			−1.39504			−0.697522	−	0.716564i	$-0.745714\pi$
$521$	−8.64335e6			−1.39504			−0.697522	+	0.716564i	$0.745714\pi$
$522$		0			0
$523$	9.11201e6			1.45667			0.728333	−	0.685224i	$-0.240296\pi$
$523$	9.11201e6			1.45667			0.728333	+	0.685224i	$0.240296\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.14187e6	−	5.44187e6i	0.492789	−	0.853536i
$528$		0			0
$529$	−3.30813e6	−	5.72984e6i	−0.513976	−	0.890233i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.30460e6	+	3.99168e6i	0.351380	+	0.608609i
$534$		0			0
$535$	4.94740e6	−	8.56915e6i	0.747296	−	1.29436i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.08782e6			−0.457804
$540$		0			0
$541$	5.76360e6			0.846643			0.423322	−	0.905979i	$-0.360864\pi$
$541$	5.76360e6			0.846643			0.423322	+	0.905979i	$0.360864\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.73476e6	+	4.73674e6i	−0.394391	+	0.683106i
$546$		0			0
$547$	−2.80744e6	−	4.86263e6i	−0.401183	−	0.694869i	0.592686	−	0.805434i	$-0.298067\pi$
$547$	−2.80744e6	−	4.86263e6i	−0.401183	−	0.694869i	−0.993869	+	0.110564i	$0.964734\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−313717.	−	543374.i	−0.0440209	−	0.0762464i
$552$		0			0
$553$	−48889.7	+	84679.4i	−0.00679837	+	0.0117751i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.18422e6			−0.298303			−0.149152	−	0.988814i	$-0.547654\pi$
$557$	−2.18422e6			−0.298303			−0.149152	+	0.988814i	$0.547654\pi$
$558$		0			0
$559$	−4.47510e6			−0.605722
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−339673.	+	588330.i	−0.0451637	+	0.0782258i	−0.887724	−	0.460377i	$-0.847714\pi$
$563$	−339673.	+	588330.i	−0.0451637	+	0.0782258i	0.842560	+	0.538603i	$0.181048\pi$
$564$		0			0
$565$	1.10785e6	+	1.91885e6i	0.146002	+	0.252883i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.28195e6	−	5.68451e6i	−0.424964	−	0.736059i	0.571453	−	0.820635i	$-0.306380\pi$
$569$	−3.28195e6	−	5.68451e6i	−0.424964	−	0.736059i	−0.996417	+	0.0845759i	$0.973046\pi$
$570$		0			0
$571$	−952119.	+	1.64912e6i	−0.122208	+	0.211671i	−0.920638	−	0.390417i	$-0.872331\pi$
$571$	−952119.	+	1.64912e6i	−0.122208	+	0.211671i	0.798430	+	0.602088i	$0.205664\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.20302e6			0.404008
$576$		0			0
$577$	1.13367e7			1.41758			0.708792	−	0.705417i	$-0.249240\pi$
$577$	1.13367e7			1.41758			0.708792	+	0.705417i	$0.249240\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10495.7	+	18179.1i	−0.00128994	+	0.00223425i
$582$		0			0
$583$	−1.12696e6	−	1.95195e6i	−0.137321	−	0.237846i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.88666e6	−	1.19280e7i	−0.824923	−	1.42881i	−0.901978	−	0.431781i	$-0.857885\pi$
$587$	−6.88666e6	−	1.19280e7i	−0.824923	−	1.42881i	0.0770557	−	0.997027i	$-0.475448\pi$
$588$		0			0
$589$	−1.22550e6	+	2.12263e6i	−0.145554	+	0.252108i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9.94674e6			−1.16157			−0.580783	−	0.814058i	$-0.697254\pi$
$593$	−9.94674e6			−1.16157			−0.580783	+	0.814058i	$0.697254\pi$
$594$		0			0
$595$	−116542.			−0.0134955
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.77125e6	+	4.79995e6i	−0.315579	+	0.546600i	−0.979561	−	0.201150i	$-0.935532\pi$
$599$	−2.77125e6	+	4.79995e6i	−0.315579	+	0.546600i	0.663981	+	0.747749i	$0.268866\pi$
$600$		0			0
$601$	−6.63325e6	−	1.14891e7i	−0.749101	−	1.29748i	−0.948254	−	0.317513i	$-0.897152\pi$
$601$	−6.63325e6	−	1.14891e7i	−0.749101	−	1.29748i	0.199153	−	0.979968i	$-0.436181\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.01096e6	+	5.21513e6i	0.334438	+	0.579264i
$606$		0			0
$607$	3.04326e6	−	5.27108e6i	0.335249	−	0.580668i	−0.648284	−	0.761399i	$-0.724513\pi$
$607$	3.04326e6	−	5.27108e6i	0.335249	−	0.580668i	0.983533	+	0.180731i	$0.0578462\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.31081e6			0.250416
$612$		0			0
$613$	−4.59806e6			−0.494223			−0.247112	−	0.968987i	$-0.579481\pi$
$613$	−4.59806e6			−0.494223			−0.247112	+	0.968987i	$0.579481\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.01311e6	−	1.38791e7i	0.847399	−	1.46774i	−0.0361221	−	0.999347i	$-0.511501\pi$
$617$	8.01311e6	−	1.38791e7i	0.847399	−	1.46774i	0.883521	−	0.468391i	$-0.155166\pi$
$618$		0			0
$619$	1.19878e6	+	2.07634e6i	0.125751	+	0.217807i	0.922026	−	0.387127i	$-0.126533\pi$
$619$	1.19878e6	+	2.07634e6i	0.125751	+	0.217807i	−0.796275	+	0.604934i	$0.793199\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−43243.9	−	74900.6i	−0.00446380	−	0.00773153i
$624$		0			0
$625$	3.10456e6	−	5.37725e6i	0.317906	−	0.550630i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.78478e7			1.79870
$630$		0			0
$631$	−6.25228e6			−0.625122			−0.312561	−	0.949898i	$-0.601187\pi$
$631$	−6.25228e6			−0.625122			−0.312561	+	0.949898i	$0.601187\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.21053e6	+	3.82875e6i	−0.217551	+	0.376810i
$636$		0			0
$637$	6.12754e6	+	1.06132e7i	0.598325	+	1.03633i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.48142e6	−	1.64223e7i	−0.911440	−	1.57866i	−0.812031	−	0.583614i	$-0.801638\pi$
$641$	−9.48142e6	−	1.64223e7i	−0.911440	−	1.57866i	−0.0994090	−	0.995047i	$-0.531695\pi$
$642$		0			0
$643$	−1.64962e6	+	2.85722e6i	−0.157346	+	0.272532i	−0.933911	−	0.357506i	$-0.883627\pi$
$643$	−1.64962e6	+	2.85722e6i	−0.157346	+	0.272532i	0.776565	+	0.630038i	$0.216961\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.56481e7			−1.46961			−0.734805	−	0.678278i	$-0.762726\pi$
$647$	−1.56481e7			−1.46961			−0.734805	+	0.678278i	$0.762726\pi$
$648$		0			0
$649$	−5.42978e6			−0.506023
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.78514e6	−	1.34843e7i	0.714469	−	1.23750i	−0.248695	−	0.968582i	$-0.580002\pi$
$653$	7.78514e6	−	1.34843e7i	0.714469	−	1.23750i	0.963164	−	0.268915i	$-0.0866650\pi$
$654$		0			0
$655$	8.69921e6	+	1.50675e7i	0.792276	+	1.37226i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.40513e6	−	1.10940e7i	−0.574532	−	0.995119i	−0.996092	−	0.0883184i	$-0.971851\pi$
$659$	−6.40513e6	−	1.10940e7i	−0.574532	−	0.995119i	0.421560	−	0.906800i	$-0.361483\pi$
$660$		0			0
$661$	−230248.	+	398801.i	−0.0204971	+	0.0355020i	−0.876092	−	0.482144i	$-0.839858\pi$
$661$	−230248.	+	398801.i	−0.0204971	+	0.0355020i	0.855595	+	0.517646i	$0.173192\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	45457.8			0.00398616
$666$		0			0
$667$	−4.79150e6			−0.417020
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3.70358e6	−	6.41479e6i	0.317552	−	0.550017i
$672$		0			0
$673$	4.03608e6	+	6.99069e6i	0.343496	+	0.594952i	0.985079	−	0.172101i	$-0.0550555\pi$
$673$	4.03608e6	+	6.99069e6i	0.343496	+	0.594952i	−0.641583	+	0.767053i	$0.721722\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.82149e6	+	3.15491e6i	0.152741	+	0.264554i	0.932234	−	0.361856i	$-0.117857\pi$
$677$	1.82149e6	+	3.15491e6i	0.152741	+	0.264554i	−0.779493	+	0.626410i	$0.784523\pi$
$678$		0			0
$679$	−100220.	+	173586.i	−0.00834219	+	0.0144491i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.57475e6			0.785373			0.392686	−	0.919672i	$-0.371546\pi$
$683$	9.57475e6			0.785373			0.392686	+	0.919672i	$0.371546\pi$
$684$		0			0
$685$	7.26801e6			0.591819
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.47272e6	+	7.74698e6i	−0.358941	+	0.621705i
$690$		0			0
$691$	3.01871e6	+	5.22857e6i	0.240506	+	0.416569i	0.960859	−	0.277039i	$-0.0893531\pi$
$691$	3.01871e6	+	5.22857e6i	0.240506	+	0.416569i	−0.720352	+	0.693608i	$0.756020\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.78215e6	+	1.00150e7i	0.454075	+	0.786481i
$696$		0			0
$697$	−3.83250e6	+	6.63809e6i	−0.298814	+	0.517560i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.37603e7			−1.05763			−0.528813	−	0.848739i	$-0.677363\pi$
$701$	−1.37603e7			−1.05763			−0.528813	+	0.848739i	$0.677363\pi$
$702$		0			0
$703$	−6.96164e6			−0.531279
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−76806.9	+	133033.i	−0.00577899	+	0.0100095i
$708$		0			0
$709$	809892.	+	1.40277e6i	0.0605078	+	0.104803i	0.894693	−	0.446683i	$-0.147395\pi$
$709$	809892.	+	1.40277e6i	0.0605078	+	0.104803i	−0.834185	+	0.551485i	$0.814061\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	9.35873e6	+	1.62098e7i	0.689435	+	1.19414i
$714$		0			0
$715$	−3.17061e6	+	5.49165e6i	−0.231941	+	0.401734i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.42931e7			1.03111			0.515555	−	0.856856i	$-0.327586\pi$
$719$	1.42931e7			1.03111			0.515555	+	0.856856i	$0.327586\pi$
$720$		0			0
$721$	−300043.			−0.0214954
$722$		0			0
$723$		0			0
$724$		0			0
$725$	587900.	−	1.01827e6i	0.0415392	−	0.0719481i
$726$		0			0
$727$	−1.37208e7	−	2.37652e7i	−0.962818	−	1.66765i	−0.715367	−	0.698749i	$-0.753741\pi$
$727$	−1.37208e7	−	2.37652e7i	−0.962818	−	1.66765i	−0.247450	−	0.968901i	$-0.579593\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.72100e6	−	6.44496e6i	−0.257553	−	0.446094i
$732$		0			0
$733$	2.55310e6	−	4.42209e6i	0.175512	−	0.303996i	−0.764826	−	0.644237i	$-0.777175\pi$
$733$	2.55310e6	−	4.42209e6i	0.175512	−	0.303996i	0.940338	+	0.340241i	$0.110509\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.31812e6			−0.292837
$738$		0			0
$739$	1.34356e7			0.904997			0.452498	−	0.891765i	$-0.350533\pi$
$739$	1.34356e7			0.904997			0.452498	+	0.891765i	$0.350533\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	343246.	−	594519.i	0.0228104	−	0.0395088i	−0.854395	−	0.519624i	$-0.826072\pi$
$743$	343246.	−	594519.i	0.0228104	−	0.0395088i	0.877205	+	0.480116i	$0.159405\pi$
$744$		0			0
$745$	−232994.	−	403557.i	−0.0153799	−	0.0266387i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	212371.	+	367838.i	0.0138322	+	0.0239581i
$750$		0			0
$751$	5.46542e6	−	9.46639e6i	0.353610	−	0.612470i	−0.633269	−	0.773931i	$-0.718288\pi$
$751$	5.46542e6	−	9.46639e6i	0.353610	−	0.612470i	0.986879	+	0.161462i	$0.0516208\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.03985e7			0.663900
$756$		0			0
$757$	−2.31128e7			−1.46593			−0.732966	−	0.680265i	$-0.761865\pi$
$757$	−2.31128e7			−1.46593			−0.732966	+	0.680265i	$0.761865\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5.94528e6	+	1.02975e7i	−0.372144	+	0.644572i	−0.989895	−	0.141801i	$-0.954711\pi$
$761$	−5.94528e6	+	1.02975e7i	−0.372144	+	0.644572i	0.617751	+	0.786374i	$0.288044\pi$
$762$		0			0
$763$	−117392.	−	203328.i	−0.00730005	−	0.0126441i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.07750e7	+	1.86628e7i	0.661345	+	1.14548i
$768$		0			0
$769$	−1.32545e6	+	2.29575e6i	−0.0808253	+	0.139994i	−0.903605	−	0.428367i	$-0.859089\pi$
$769$	−1.32545e6	+	2.29575e6i	−0.0808253	+	0.139994i	0.822779	+	0.568361i	$0.192422\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.12893e7			1.28148			0.640742	−	0.767757i	$-0.278627\pi$
$773$	2.12893e7			1.28148			0.640742	+	0.767757i	$0.278627\pi$
$774$		0			0
$775$	−4.59314e6			−0.274698
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.49489e6	−	2.58922e6i	0.0882601	−	0.152871i
$780$		0			0
$781$	11373.9	+	19700.3i	0.000667242	+	0.00115570i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−723426.	−	1.25301e6i	−0.0419006	−	0.0725739i
$786$		0			0
$787$	−1.14439e7	+	1.98214e7i	−0.658623	+	1.14077i	0.322349	+	0.946621i	$0.395528\pi$
$787$	−1.14439e7	+	1.98214e7i	−0.658623	+	1.14077i	−0.980972	+	0.194148i	$0.937806\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−95110.7			−0.00540491
$792$		0			0
$793$	−2.93979e7			−1.66010
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.28424e7	+	2.22437e7i	−0.716146	+	1.24040i	0.246370	+	0.969176i	$0.420762\pi$
$797$	−1.28424e7	+	2.22437e7i	−0.716146	+	1.24040i	−0.962516	+	0.271225i	$0.912571\pi$
$798$		0			0
$799$	1.92141e6	+	3.32799e6i	0.106477	+	0.184423i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.23592e6	−	5.60478e6i	−0.177096	−	0.306739i
$804$		0			0
$805$	173573.	−	300637.i	0.00944043	−	0.0163513i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.88061e6			0.315901			0.157951	−	0.987447i	$-0.449511\pi$
$809$	5.88061e6			0.315901			0.157951	+	0.987447i	$0.449511\pi$
$810$		0			0
$811$	2.04938e7			1.09414			0.547068	−	0.837088i	$-0.315744\pi$
$811$	2.04938e7			1.09414			0.547068	+	0.837088i	$0.315744\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.44313e6	+	1.11598e7i	−0.339784	+	0.588523i
$816$		0			0
$817$	1.45139e6	+	2.51389e6i	0.0760730	+	0.131762i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.81711e6	−	3.14732e6i	−0.0940855	−	0.162961i	0.815141	−	0.579263i	$-0.196659\pi$
$821$	−1.81711e6	−	3.14732e6i	−0.0940855	−	0.162961i	−0.909227	+	0.416302i	$0.863326\pi$
$822$		0			0
$823$	−7.90337e6	+	1.36890e7i	−0.406736	+	0.704488i	−0.994522	−	0.104529i	$-0.966666\pi$
$823$	−7.90337e6	+	1.36890e7i	−0.406736	+	0.704488i	0.587786	+	0.809017i	$0.300000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.98096e6			−0.100719			−0.0503597	−	0.998731i	$-0.516037\pi$
$827$	−1.98096e6			−0.100719			−0.0503597	+	0.998731i	$0.516037\pi$
$828$		0			0
$829$	2.56811e7			1.29786			0.648929	−	0.760849i	$-0.275217\pi$
$829$	2.56811e7			1.29786			0.648929	+	0.760849i	$0.275217\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.01900e7	+	1.76495e7i	−0.508815	+	0.881294i
$834$		0			0
$835$	4.69988e6	+	8.14043e6i	0.233276	+	0.404046i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.27342e6	−	3.93767e6i	−0.111500	−	0.193123i	0.804875	−	0.593444i	$-0.202232\pi$
$839$	−2.27342e6	−	3.93767e6i	−0.111500	−	0.193123i	−0.916375	+	0.400321i	$0.868899\pi$
$840$		0			0
$841$	9.37612e6	−	1.62399e7i	0.457123	−	0.791760i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.60068e6			0.366193
$846$		0			0
$847$	−258496.			−0.0123807
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.65818e7	+	4.60411e7i	−1.25823	+	2.17932i
$852$		0			0
$853$	−708511.	−	1.22718e6i	−0.0333407	−	0.0577477i	0.848874	−	0.528596i	$-0.177281\pi$
$853$	−708511.	−	1.22718e6i	−0.0333407	−	0.0577477i	−0.882214	+	0.470848i	$0.843948\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.07200e6	−	7.05290e6i	−0.189389	−	0.328032i	0.755658	−	0.654967i	$-0.227317\pi$
$857$	−4.07200e6	−	7.05290e6i	−0.189389	−	0.328032i	−0.945047	+	0.326935i	$0.893984\pi$
$858$		0			0
$859$	5.33500e6	−	9.24050e6i	0.246690	−	0.427280i	−0.715915	−	0.698187i	$-0.753990\pi$
$859$	5.33500e6	−	9.24050e6i	0.246690	−	0.427280i	0.962605	+	0.270907i	$0.0873237\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.21144e7			−0.553701			−0.276851	−	0.960913i	$-0.589291\pi$
$863$	−1.21144e7			−0.553701			−0.276851	+	0.960913i	$0.589291\pi$
$864$		0			0
$865$	−3.20737e6			−0.145750
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.42379e6	+	7.66222e6i	−0.198721	+	0.344196i
$870$		0			0
$871$	8.56899e6	+	1.48419e7i	0.382723	+	0.662895i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	192729.	+	333816.i	0.00850995	+	0.0147397i
$876$		0			0
$877$	−1.65673e7	+	2.86954e7i	−0.727366	+	1.25983i	0.230627	+	0.973042i	$0.425922\pi$
$877$	−1.65673e7	+	2.86954e7i	−0.727366	+	1.25983i	−0.957993	+	0.286792i	$0.907411\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−49573.7			−0.00215185			−0.00107592	−	0.999999i	$-0.500342\pi$
$881$	−49573.7			−0.00215185			−0.00107592	+	0.999999i	$0.500342\pi$
$882$		0			0
$883$	−4.21109e7			−1.81758			−0.908788	−	0.417259i	$-0.862991\pi$
$883$	−4.21109e7			−1.81758			−0.908788	+	0.417259i	$0.862991\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−550680.	+	953806.i	−0.0235012	+	0.0407053i	−0.877537	−	0.479509i	$-0.840815\pi$
$887$	−550680.	+	953806.i	−0.0235012	+	0.0407053i	0.854036	+	0.520215i	$0.174148\pi$
$888$		0			0
$889$	−94888.8	−	164352.i	−0.00402680	−	0.00697463i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−749458.	−	1.29810e6i	−0.0314498	−	0.0544727i
$894$		0			0
$895$	−516860.	+	895228.i	−0.0215683	+	0.0373574i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6.87102e6			0.283545
$900$		0			0
$901$	−1.48761e7			−0.610487
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.44922e6	+	7.70627e6i	−0.180577	+	0.312769i
$906$		0			0
$907$	−3.72157e6	−	6.44595e6i	−0.150213	−	0.260177i	0.781093	−	0.624415i	$-0.214663\pi$
$907$	−3.72157e6	−	6.44595e6i	−0.150213	−	0.260177i	−0.931306	+	0.364238i	$0.881329\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.61672e7	+	2.80024e7i	0.645414	+	1.11789i	0.984206	+	0.177028i	$0.0566483\pi$
$911$	1.61672e7	+	2.80024e7i	0.645414	+	1.11789i	−0.338792	+	0.940861i	$0.610018\pi$
$912$		0			0
$913$	−949703.	+	1.64493e6i	−0.0377060	+	0.0653087i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−746841.			−0.0293295
$918$		0			0
$919$	−3.18905e7			−1.24558			−0.622792	−	0.782388i	$-0.714002\pi$
$919$	−3.18905e7			−1.24558			−0.622792	+	0.782388i	$0.714002\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	45141.5	−	78187.3i	0.00174410	−	0.00302087i
$924$		0			0
$925$	−6.52299e6	−	1.12982e7i	−0.250664	−	0.434163i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7.20450e6	−	1.24786e7i	−0.273883	−	0.474379i	0.695970	−	0.718071i	$-0.254975\pi$
$929$	−7.20450e6	−	1.24786e7i	−0.273883	−	0.474379i	−0.969853	+	0.243692i	$0.921641\pi$
$930$		0			0
$931$	3.97465e6	−	6.88429e6i	0.150288	−	0.260307i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.05453e7			−0.394485
$936$		0			0
$937$	2.33199e7			0.867718			0.433859	−	0.900981i	$-0.357152\pi$
$937$	2.33199e7			0.867718			0.433859	+	0.900981i	$0.357152\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2.35788e7	+	4.08396e7i	−0.868054	+	1.50351i	−0.00407185	+	0.999992i	$0.501296\pi$
$941$	−2.35788e7	+	4.08396e7i	−0.868054	+	1.50351i	−0.863982	+	0.503522i	$0.832037\pi$
$942$		0			0
$943$	−1.14159e7	−	1.97730e7i	−0.418054	−	0.724091i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.21868e7	−	2.11081e7i	−0.441585	−	0.764847i	0.556223	−	0.831033i	$-0.312250\pi$
$947$	−1.21868e7	−	2.11081e7i	−0.441585	−	0.764847i	−0.997807	+	0.0661864i	$0.978917\pi$
$948$		0			0
$949$	−1.28429e7	+	2.22445e7i	−0.462910	+	0.801784i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.33987e7			1.54790			0.773952	−	0.633244i	$-0.218277\pi$
$953$	4.33987e7			1.54790			0.773952	+	0.633244i	$0.218277\pi$
$954$		0			0
$955$	−1.16577e7			−0.413623
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−155993.	+	270187.i	−0.00547719	+	0.00948677i
$960$		0			0
$961$	894119.	+	1.54866e6i	0.0312311	+	0.0540938i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.37302e6	+	9.30634e6i	0.185738	+	0.321707i
$966$		0			0
$967$	137300.	−	237810.i	0.00472176	−	0.00817832i	−0.863655	−	0.504084i	$-0.831830\pi$
$967$	137300.	−	237810.i	0.00472176	−	0.00817832i	0.868377	+	0.495905i	$0.165164\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3.36920e7			1.14678			0.573388	−	0.819284i	$-0.305629\pi$
$971$	3.36920e7			1.14678			0.573388	+	0.819284i	$0.305629\pi$
$972$		0			0
$973$	−496407.			−0.0168096
$974$		0			0
$975$		0			0
$976$		0			0
$977$	6.93084e6	−	1.20046e7i	0.232300	−	0.402356i	−0.726184	−	0.687500i	$-0.758708\pi$
$977$	6.93084e6	−	1.20046e7i	0.232300	−	0.402356i	0.958485	+	0.285144i	$0.0920415\pi$
$978$		0			0
$979$	−3.91293e6	−	6.77739e6i	−0.130480	−	0.225999i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	4.34952e6	+	7.53359e6i	0.143568	+	0.248667i	0.928838	−	0.370487i	$-0.120809\pi$
$983$	4.34952e6	+	7.53359e6i	0.143568	+	0.248667i	−0.785270	+	0.619154i	$0.787476\pi$
$984$		0			0
$985$	−1.19010e7	+	2.06132e7i	−0.390835	+	0.676947i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.21676e7			0.720656
$990$		0			0
$991$	4.44501e7			1.43777			0.718884	−	0.695130i	$-0.244653\pi$
$991$	4.44501e7			1.43777			0.718884	+	0.695130i	$0.244653\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	97620.2	−	169083.i	0.00312595	−	0.00541430i
$996$		0			0
$997$	8.21072e6	+	1.42214e7i	0.261603	+	0.453110i	0.966668	−	0.256033i	$-0.0824154\pi$
$997$	8.21072e6	+	1.42214e7i	0.261603	+	0.453110i	−0.705065	+	0.709143i	$0.749082\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.6.i.c.145.2		8
3.2	odd	2		144.6.i.c.49.3		8
4.3	odd	2		27.6.c.a.10.1		8
9.2	odd	6		144.6.i.c.97.3		8
9.7	even	3	inner	432.6.i.c.289.2		8
12.11	even	2		9.6.c.a.4.4	✓	8
36.7	odd	6		27.6.c.a.19.1		8
36.11	even	6		9.6.c.a.7.4	yes	8
36.23	even	6		81.6.a.c.1.1		4
36.31	odd	6		81.6.a.d.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.6.c.a.4.4	✓	8	12.11	even	2
9.6.c.a.7.4	yes	8	36.11	even	6
27.6.c.a.10.1		8	4.3	odd	2
27.6.c.a.19.1		8	36.7	odd	6
81.6.a.c.1.1		4	36.23	even	6
81.6.a.d.1.4		4	36.31	odd	6
144.6.i.c.49.3		8	3.2	odd	2
144.6.i.c.97.3		8	9.2	odd	6
432.6.i.c.145.2		8	1.1	even	1	trivial
432.6.i.c.289.2		8	9.7	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-23.6560 + 40.9735i) q^{5} +(-1.01546 - 1.75882i) q^{7} +(-91.8836 - 159.147i) q^{11} +(-364.672 + 631.630i) q^{13} -1212.88 q^{17} +473.092 q^{19} +(1806.42 - 3128.81i) q^{23} +(443.283 + 767.788i) q^{25} +(-663.121 - 1148.56i) q^{29} +(-2590.41 + 4486.72i) q^{31} +96.0867 q^{35} -14715.2 q^{37} +(3159.83 - 5472.98i) q^{41} +(3067.89 + 5313.75i) q^{43} +(-1584.17 - 2743.86i) q^{47} +(8401.44 - 14551.7i) q^{49} +12265.0 q^{53} +8694.41 q^{55} +(14773.5 - 25588.5i) q^{59} +(20153.6 + 34907.1i) q^{61} +(-17253.4 - 29883.8i) q^{65} +(11748.9 - 20349.7i) q^{67} -123.786 q^{71} +35217.6 q^{73} +(-186.607 + 323.214i) q^{77} +(-24072.8 - 41695.3i) q^{79} +(-5167.97 - 8951.18i) q^{83} +(28692.0 - 49696.1i) q^{85} +42585.7 q^{89} +1481.23 q^{91} +(-11191.5 + 19384.2i) q^{95} +(-49347.3 - 85472.1i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 78 q^{5} - 28 q^{7} + 444 q^{11} - 182 q^{13} + 4356 q^{17} - 952 q^{19} + 8844 q^{23} - 1654 q^{25} - 12018 q^{29} - 1132 q^{31} - 16224 q^{35} - 15176 q^{37} - 1248 q^{41} + 6092 q^{43} - 60 q^{47}+ \cdots + 33976 q^{97}+O(q^{100}) \) 8 * q - 78 * q^5 - 28 * q^7 + 444 * q^11 - 182 * q^13 + 4356 * q^17 - 952 * q^19 + 8844 * q^23 - 1654 * q^25 - 12018 * q^29 - 1132 * q^31 - 16224 * q^35 - 15176 * q^37 - 1248 * q^41 + 6092 * q^43 - 60 * q^47 + 9090 * q^49 - 20952 * q^53 + 36120 * q^55 + 2076 * q^59 + 48142 * q^61 + 13146 * q^65 + 7148 * q^67 - 71856 * q^71 + 122452 * q^73 - 39534 * q^77 + 59516 * q^79 + 117696 * q^83 + 28836 * q^85 + 451728 * q^89 - 111392 * q^91 + 294888 * q^95 + 33976 * q^97

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