LMFDB - Embedded newform 504.6.s.f.289.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,6,Mod(289,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("504.289");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	504.s (of order $3$, degree $2$, 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:	$80.8334451857$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 2041 x^{10} - 63452 x^{9} + 3932036 x^{8} - 70117724 x^{7} + 1560078988 x^{6} + \cdots + 472919482810944 $ x^12 - x^11 + 2041x^10 - 63452x^9 + 3932036x^8 - 70117724x^7 + 1560078988x^6 - 7057906348x^5 + 150204387604x^4 + 416833058064x^3 + 20453850998736x^2 + 83518334422848x + 472919482810944
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{18}\cdot 3\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.5
Root			$10.9316 + 18.9342i$ of defining polynomial
Character	$\chi$	$=$	504.289
Dual form			504.6.s.f.361.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(30.2849 - 52.4550i) q^{5} +(76.1691 + 104.906i) q^{7} +(-152.194 - 263.607i) q^{11} +600.507 q^{13} +(674.856 + 1168.88i) q^{17} +(-1162.12 + 2012.85i) q^{19} +(-438.629 + 759.727i) q^{23} +(-271.855 - 470.866i) q^{25} +3249.80 q^{29} +(-1804.67 - 3125.78i) q^{31} +(7809.63 - 818.380i) q^{35} +(-3467.78 + 6006.37i) q^{37} +2865.20 q^{41} +2487.91 q^{43} +(-3300.58 + 5716.77i) q^{47} +(-5203.55 + 15981.2i) q^{49} +(12516.5 + 21679.2i) q^{53} -18436.7 q^{55} +(-14446.5 - 25022.1i) q^{59} +(-21953.7 + 38025.0i) q^{61} +(18186.3 - 31499.6i) q^{65} +(-25162.3 - 43582.4i) q^{67} +64705.5 q^{71} +(11306.9 + 19584.1i) q^{73} +(16061.5 - 36044.8i) q^{77} +(12983.3 - 22487.8i) q^{79} -55740.6 q^{83} +81751.9 q^{85} +(39958.5 - 69210.2i) q^{89} +(45740.1 + 62996.8i) q^{91} +(70389.6 + 121918. i) q^{95} +116989. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 17 q^{5} + 144 q^{7} + 565 q^{11} - 842 q^{13} - 52 q^{17} + 107 q^{19} - 700 q^{23} - 4881 q^{25} - 12958 q^{29} - 3552 q^{31} + 938 q^{35} - 3453 q^{37} + 15148 q^{41} + 38278 q^{43} - 12136 q^{47}+ \cdots + 28934 q^{97}+O(q^{100}) $ 12 * q - 17 * q^5 + 144 * q^7 + 565 * q^11 - 842 * q^13 - 52 * q^17 + 107 * q^19 - 700 * q^23 - 4881 * q^25 - 12958 * q^29 - 3552 * q^31 + 938 * q^35 - 3453 * q^37 + 15148 * q^41 + 38278 * q^43 - 12136 * q^47 - 39926 * q^49 + 37591 * q^53 - 68670 * q^55 - 3905 * q^59 + 13296 * q^61 - 45638 * q^65 - 2889 * q^67 - 132992 * q^71 + 127931 * q^73 + 306817 * q^77 + 100226 * q^79 + 146246 * q^83 - 181328 * q^85 + 3418 * q^89 - 275865 * q^91 - 108810 * q^95 + 28934 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	30.2849	−	52.4550i	0.541753	−	0.938344i	−0.457050	−	0.889441i	$-0.651094\pi$
$5$	30.2849	−	52.4550i	0.541753	−	0.938344i	0.998804	−	0.0489035i	$-0.0155727\pi$
$6$		0			0
$7$	76.1691	+	104.906i	0.587535	+	0.809199i
$7$	76.1691	+	104.906i	0.587535	+	0.809199i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−152.194	−	263.607i	−0.379241	−	0.656865i	0.611711	−	0.791081i	$-0.290482\pi$
$11$	−152.194	−	263.607i	−0.379241	−	0.656865i	−0.990952	+	0.134217i	$0.957148\pi$
$12$		0			0
$13$	600.507			0.985507			0.492754	−	0.870169i	$-0.335990\pi$
$13$	600.507			0.985507			0.492754	+	0.870169i	$0.335990\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	674.856	+	1168.88i	0.566355	+	0.980956i	0.996922	+	0.0783972i	$0.0249802\pi$
$17$	674.856	+	1168.88i	0.566355	+	0.980956i	−0.430567	+	0.902559i	$0.641686\pi$
$18$		0			0
$19$	−1162.12	+	2012.85i	−0.738529	+	1.27917i	0.214628	+	0.976696i	$0.431146\pi$
$19$	−1162.12	+	2012.85i	−0.738529	+	1.27917i	−0.953157	+	0.302475i	$0.902187\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−438.629	+	759.727i	−0.172893	+	0.299460i	−0.939430	−	0.342741i	$-0.888645\pi$
$23$	−438.629	+	759.727i	−0.172893	+	0.299460i	0.766537	+	0.642200i	$0.221978\pi$
$24$		0			0
$25$	−271.855	−	470.866i	−0.0869935	−	0.150677i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3249.80			0.717565			0.358782	−	0.933421i	$-0.383192\pi$
$29$	3249.80			0.717565			0.358782	+	0.933421i	$0.383192\pi$
$30$		0			0
$31$	−1804.67	−	3125.78i	−0.337282	−	0.584190i	0.646639	−	0.762797i	$-0.276174\pi$
$31$	−1804.67	−	3125.78i	−0.337282	−	0.584190i	−0.983920	+	0.178607i	$0.942841\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	7809.63	−	818.380i	1.07761	−	0.112924i
$36$		0			0
$37$	−3467.78	+	6006.37i	−0.416435	+	0.721287i	−0.995578	−	0.0939393i	$-0.970054\pi$
$37$	−3467.78	+	6006.37i	−0.416435	+	0.721287i	0.579143	+	0.815226i	$0.303387\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2865.20			0.266193			0.133096	−	0.991103i	$-0.457508\pi$
$41$	2865.20			0.266193			0.133096	+	0.991103i	$0.457508\pi$
$42$		0			0
$43$	2487.91			0.205194			0.102597	−	0.994723i	$-0.467285\pi$
$43$	2487.91			0.205194			0.102597	+	0.994723i	$0.467285\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3300.58	+	5716.77i	−0.217944	+	0.377491i	−0.954179	−	0.299235i	$-0.903268\pi$
$47$	−3300.58	+	5716.77i	−0.217944	+	0.377491i	0.736235	+	0.676726i	$0.236602\pi$
$48$		0			0
$49$	−5203.55	+	15981.2i	−0.309606	+	0.950865i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12516.5	+	21679.2i	0.612059	+	1.06012i	0.990893	+	0.134653i	$0.0429919\pi$
$53$	12516.5	+	21679.2i	0.612059	+	1.06012i	−0.378834	+	0.925465i	$0.623675\pi$
$54$		0			0
$55$	−18436.7			−0.821820
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−14446.5	−	25022.1i	−0.540297	−	0.935822i	−0.998887	−	0.0471739i	$-0.984979\pi$
$59$	−14446.5	−	25022.1i	−0.540297	−	0.935822i	0.458590	−	0.888648i	$-0.348355\pi$
$60$		0			0
$61$	−21953.7	+	38025.0i	−0.755412	+	1.30841i	0.189758	+	0.981831i	$0.439230\pi$
$61$	−21953.7	+	38025.0i	−0.755412	+	1.30841i	−0.945169	+	0.326581i	$0.894104\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	18186.3	−	31499.6i	0.533902	−	0.924745i
$66$		0			0
$67$	−25162.3	−	43582.4i	−0.684800	−	1.18611i	−0.973500	−	0.228689i	$-0.926556\pi$
$67$	−25162.3	−	43582.4i	−0.684800	−	1.18611i	0.288699	−	0.957420i	$-0.406777\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	64705.5			1.52333			0.761667	−	0.647969i	$-0.224381\pi$
$71$	64705.5			1.52333			0.761667	+	0.647969i	$0.224381\pi$
$72$		0			0
$73$	11306.9	+	19584.1i	0.248334	+	0.430127i	0.963064	−	0.269274i	$-0.0867837\pi$
$73$	11306.9	+	19584.1i	0.248334	+	0.430127i	−0.714730	+	0.699401i	$0.753450\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	16061.5	−	36044.8i	0.308717	−	0.692812i
$78$		0			0
$79$	12983.3	−	22487.8i	0.234055	−	0.405395i	−0.724943	−	0.688809i	$-0.758134\pi$
$79$	12983.3	−	22487.8i	0.234055	−	0.405395i	0.958998	+	0.283414i	$0.0914671\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−55740.6			−0.888129			−0.444065	−	0.895995i	$-0.646464\pi$
$83$	−55740.6			−0.888129			−0.444065	+	0.895995i	$0.646464\pi$
$84$		0			0
$85$	81751.9			1.22730
$86$		0			0
$87$		0			0
$88$		0			0
$89$	39958.5	−	69210.2i	0.534730	−	0.926179i	−0.464447	−	0.885601i	$-0.653747\pi$
$89$	39958.5	−	69210.2i	0.534730	−	0.926179i	0.999176	−	0.0405781i	$-0.0129199\pi$
$90$		0			0
$91$	45740.1	+	62996.8i	0.579020	+	0.797472i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	70389.6	+	121918.i	0.800202	+	1.38599i
$96$		0			0
$97$	116989.			1.26246			0.631228	−	0.775597i	$-0.282551\pi$
$97$	116989.			1.26246			0.631228	+	0.775597i	$0.282551\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−14237.2	−	24659.6i	−0.138874	−	0.240537i	0.788197	−	0.615424i	$-0.211015\pi$
$101$	−14237.2	−	24659.6i	−0.138874	−	0.240537i	−0.927071	+	0.374886i	$0.877682\pi$
$102$		0			0
$103$	41737.4	−	72291.4i	0.387644	−	0.671419i	−0.604488	−	0.796614i	$-0.706622\pi$
$103$	41737.4	−	72291.4i	0.387644	−	0.671419i	0.992132	+	0.125195i	$0.0399558\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−65743.6	+	113871.i	−0.555129	+	0.961511i	0.442765	+	0.896638i	$0.353998\pi$
$107$	−65743.6	+	113871.i	−0.555129	+	0.961511i	−0.997894	+	0.0648734i	$0.979336\pi$
$108$		0			0
$109$	107070.	+	185450.i	0.863179	+	1.49507i	0.868844	+	0.495085i	$0.164863\pi$
$109$	107070.	+	185450.i	0.863179	+	1.49507i	−0.00566575	+	0.999984i	$0.501803\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−94500.0			−0.696203			−0.348101	−	0.937457i	$-0.613174\pi$
$113$	−94500.0			−0.696203			−0.348101	+	0.937457i	$0.613174\pi$
$114$		0			0
$115$	26567.7	+	46016.6i	0.187331	+	0.324466i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−71219.9	+	159829.i	−0.461035	+	1.03464i
$120$		0			0
$121$	34199.6	−	59235.5i	0.212353	−	0.367806i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	156348.			0.894991
$126$		0			0
$127$	214404.			1.17957			0.589784	−	0.807561i	$-0.299213\pi$
$127$	214404.			1.17957			0.589784	+	0.807561i	$0.299213\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−171480.	+	297012.i	−0.873042	+	1.51215i	−0.0142087	+	0.999899i	$0.504523\pi$
$131$	−171480.	+	297012.i	−0.873042	+	1.51215i	−0.858834	+	0.512255i	$0.828810\pi$
$132$		0			0
$133$	−299678.	+	31403.6i	−1.46902	+	0.153940i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	88485.9	+	153262.i	0.402784	+	0.697643i	0.994061	−	0.108826i	$-0.0347091\pi$
$137$	88485.9	+	153262.i	0.402784	+	0.697643i	−0.591276	+	0.806469i	$0.701376\pi$
$138$		0			0
$139$	267589.			1.17471			0.587357	−	0.809328i	$-0.300169\pi$
$139$	267589.			1.17471			0.587357	+	0.809328i	$0.300169\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−91393.5	−	158298.i	−0.373745	−	0.647345i
$144$		0			0
$145$	98419.9	−	170468.i	0.388743	−	0.673323i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−29535.6	+	51157.1i	−0.108988	+	0.188773i	−0.915361	−	0.402635i	$-0.868094\pi$
$149$	−29535.6	+	51157.1i	−0.108988	+	0.188773i	0.806372	+	0.591408i	$0.201428\pi$
$150$		0			0
$151$	179646.	+	311156.i	0.641173	+	1.11054i	0.985171	+	0.171574i	$0.0548852\pi$
$151$	179646.	+	311156.i	0.641173	+	1.11054i	−0.343998	+	0.938970i	$0.611781\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−218617.			−0.730895
$156$		0			0
$157$	287924.	+	498698.i	0.932241	+	1.61469i	0.779481	+	0.626426i	$0.215483\pi$
$157$	287924.	+	498698.i	0.932241	+	1.61469i	0.152760	+	0.988263i	$0.451184\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−113110.	+	11852.9i	−0.343903	+	0.0360380i
$162$		0			0
$163$	−226635.	+	392543.i	−0.668125	+	1.15723i	0.310303	+	0.950638i	$0.399569\pi$
$163$	−226635.	+	392543.i	−0.668125	+	1.15723i	−0.978428	+	0.206588i	$0.933764\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	405992.			1.12649			0.563243	−	0.826291i	$-0.309553\pi$
$167$	405992.			1.12649			0.563243	+	0.826291i	$0.309553\pi$
$168$		0			0
$169$	−10684.0			−0.0287751
$170$		0			0
$171$		0			0
$172$		0			0
$173$	256454.	−	444191.i	0.651469	−	1.12838i	−0.331298	−	0.943526i	$-0.607486\pi$
$173$	256454.	−	444191.i	0.651469	−	1.12838i	0.982767	−	0.184851i	$-0.0591802\pi$
$174$		0			0
$175$	28689.8	−	64384.6i	0.0708161	−	0.158923i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	140325.	+	243050.i	0.327343	+	0.566974i	0.981984	−	0.188966i	$-0.0605135\pi$
$179$	140325.	+	243050.i	0.327343	+	0.566974i	−0.654641	+	0.755940i	$0.727180\pi$
$180$		0			0
$181$	551836.			1.25203			0.626013	−	0.779813i	$-0.284686\pi$
$181$	551836.			1.25203			0.626013	+	0.779813i	$0.284686\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	210043.	+	363805.i	0.451210	+	0.781519i
$186$		0			0
$187$	205418.	−	355794.i	0.429570	−	0.744037i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	410784.	−	711499.i	0.814762	−	1.41121i	−0.0947374	−	0.995502i	$-0.530201\pi$
$191$	410784.	−	711499.i	0.814762	−	1.41121i	0.909499	−	0.415706i	$-0.136466\pi$
$192$		0			0
$193$	−46835.6	−	81121.7i	−0.0905072	−	0.156763i	0.817217	−	0.576329i	$-0.195515\pi$
$193$	−46835.6	−	81121.7i	−0.0905072	−	0.156763i	−0.907725	+	0.419566i	$0.862182\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−76717.0			−0.140840			−0.0704200	−	0.997517i	$-0.522434\pi$
$197$	−76717.0			−0.140840			−0.0704200	+	0.997517i	$0.522434\pi$
$198$		0			0
$199$	152265.	+	263730.i	0.272563	+	0.472093i	0.969517	−	0.245023i	$-0.0787954\pi$
$199$	152265.	+	263730.i	0.272563	+	0.472093i	−0.696955	+	0.717115i	$0.745462\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	247534.	+	340923.i	0.421594	+	0.580653i
$204$		0			0
$205$	86772.5	−	150294.i	0.144211	−	0.249780i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	707471.			1.12032
$210$		0			0
$211$	−391659.			−0.605622			−0.302811	−	0.953051i	$-0.597925\pi$
$211$	−391659.			−0.605622			−0.302811	+	0.953051i	$0.597925\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	75346.3	−	130504.i	0.111164	−	0.192543i
$216$		0			0
$217$	190453.	−	427408.i	0.274561	−	0.616160i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	405256.	+	701924.i	0.558147	+	0.966739i
$222$		0			0
$223$	−732879.			−0.986893			−0.493447	−	0.869776i	$-0.664263\pi$
$223$	−732879.			−0.986893			−0.493447	+	0.869776i	$0.664263\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−425958.	−	737782.i	−0.548659	−	0.950305i	−0.998367	−	0.0571296i	$-0.981805\pi$
$227$	−425958.	−	737782.i	−0.548659	−	0.950305i	0.449708	−	0.893176i	$-0.351528\pi$
$228$		0			0
$229$	−342010.	+	592379.i	−0.430973	+	0.746468i	−0.996957	−	0.0779485i	$-0.975163\pi$
$229$	−342010.	+	592379.i	−0.430973	+	0.746468i	0.565984	+	0.824416i	$0.308496\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	579124.	−	1.00307e6i	0.698846	−	1.21044i	−0.270021	−	0.962855i	$-0.587031\pi$
$233$	579124.	−	1.00307e6i	0.698846	−	1.21044i	0.968867	−	0.247582i	$-0.0796361\pi$
$234$		0			0
$235$	199916.	+	346264.i	0.236144	+	0.409014i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−638377.			−0.722908			−0.361454	−	0.932390i	$-0.617719\pi$
$239$	−638377.			−0.722908			−0.361454	+	0.932390i	$0.617719\pi$
$240$		0			0
$241$	79265.3	+	137291.i	0.0879104	+	0.152265i	0.906628	−	0.421932i	$-0.138648\pi$
$241$	79265.3	+	137291.i	0.0879104	+	0.152265i	−0.818717	+	0.574197i	$0.805314\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	680705.	+	756942.i	0.724509	+	0.805651i
$246$		0			0
$247$	−697863.	+	1.20873e6i	−0.727826	+	1.26063i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.62185e6			−1.62490			−0.812451	−	0.583029i	$-0.801867\pi$
$251$	−1.62185e6			−1.62490			−0.812451	+	0.583029i	$0.801867\pi$
$252$		0			0
$253$	267026.			0.262272
$254$		0			0
$255$		0			0
$256$		0			0
$257$	389207.	−	674127.i	0.367577	−	0.636662i	−0.621609	−	0.783327i	$-0.713521\pi$
$257$	389207.	−	674127.i	0.367577	−	0.636662i	0.989186	+	0.146666i	$0.0468542\pi$
$258$		0			0
$259$	−894242.	+	93708.7i	−0.828335	+	0.0868022i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	294945.	+	510860.i	0.262937	+	0.455420i	0.967021	−	0.254696i	$-0.0819755\pi$
$263$	294945.	+	510860.i	0.262937	+	0.455420i	−0.704084	+	0.710117i	$0.748642\pi$
$264$		0			0
$265$	1.51625e6			1.32634
$266$		0			0
$267$		0			0
$268$		0			0
$269$	230565.	+	399351.i	0.194273	+	0.336492i	0.946662	−	0.322228i	$-0.104432\pi$
$269$	230565.	+	399351.i	0.194273	+	0.336492i	−0.752389	+	0.658719i	$0.771098\pi$
$270$		0			0
$271$	−456680.	+	790993.i	−0.377736	+	0.654258i	−0.990733	−	0.135827i	$-0.956631\pi$
$271$	−456680.	+	790993.i	−0.377736	+	0.654258i	0.612996	+	0.790086i	$0.289964\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−82749.2	+	143326.i	−0.0659830	+	0.114286i
$276$		0			0
$277$	−1.03563e6	−	1.79376e6i	−0.810969	−	1.40464i	−0.912187	−	0.409775i	$-0.865607\pi$
$277$	−1.03563e6	−	1.79376e6i	−0.810969	−	1.40464i	0.101218	−	0.994864i	$-0.467726\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	493074.			0.372517			0.186259	−	0.982501i	$-0.440364\pi$
$281$	493074.			0.372517			0.186259	+	0.982501i	$0.440364\pi$
$282$		0			0
$283$	−264091.	−	457418.i	−0.196014	−	0.339506i	0.751218	−	0.660054i	$-0.229466\pi$
$283$	−264091.	−	457418.i	−0.196014	−	0.339506i	−0.947232	+	0.320547i	$0.896133\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	218240.	+	300577.i	0.156397	+	0.215403i
$288$		0			0
$289$	−200933.	+	348025.i	−0.141516	+	0.245113i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.49808e6			−1.69995			−0.849977	−	0.526821i	$-0.823384\pi$
$293$	−2.49808e6			−1.69995			−0.849977	+	0.526821i	$0.823384\pi$
$294$		0			0
$295$	−1.75005e6			−1.17083
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−263400.	+	456222.i	−0.170387	+	0.295120i
$300$		0			0
$301$	189502.	+	260997.i	0.120559	+	0.166043i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.32973e6	+	2.30317e6i	0.818494	+	1.41767i
$306$		0			0
$307$	3.02907e6			1.83427			0.917137	−	0.398573i	$-0.130494\pi$
$307$	3.02907e6			1.83427			0.917137	+	0.398573i	$0.130494\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	278275.	+	481986.i	0.163145	+	0.282575i	0.935995	−	0.352014i	$-0.114503\pi$
$311$	278275.	+	481986.i	0.163145	+	0.282575i	−0.772850	+	0.634589i	$0.781170\pi$
$312$		0			0
$313$	1.41584e6	−	2.45230e6i	0.816868	−	1.41486i	−0.0911102	−	0.995841i	$-0.529042\pi$
$313$	1.41584e6	−	2.45230e6i	0.816868	−	1.41486i	0.907979	−	0.419017i	$-0.137625\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.07815e6	−	1.86741e6i	0.602604	−	1.04374i	−0.389822	−	0.920890i	$-0.627463\pi$
$317$	1.07815e6	−	1.86741e6i	0.602604	−	1.04374i	0.992425	−	0.122850i	$-0.0392033\pi$
$318$		0			0
$319$	−494599.	−	856670.i	−0.272130	−	0.471343i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.13706e6			−1.67308
$324$		0			0
$325$	−163251.	−	282759.i	−0.0857327	−	0.148493i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−851126.	+	89190.5i	−0.433515	+	0.0454285i
$330$		0			0
$331$	−532428.	+	922193.i	−0.267111	+	0.462649i	−0.968114	−	0.250508i	$-0.919402\pi$
$331$	−532428.	+	922193.i	−0.267111	+	0.462649i	0.701004	+	0.713157i	$0.252736\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.04816e6			−1.48397
$336$		0			0
$337$	−223972.			−0.107428			−0.0537141	−	0.998556i	$-0.517106\pi$
$337$	−223972.			−0.107428			−0.0537141	+	0.998556i	$0.517106\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−549319.	+	951448.i	−0.255822	+	0.443097i
$342$		0			0
$343$	−2.07287e6	+	671389.i	−0.951343	+	0.308133i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.37069e6	−	2.37411e6i	−0.611106	−	1.05847i	−0.991054	−	0.133459i	$-0.957391\pi$
$347$	−1.37069e6	−	2.37411e6i	−0.611106	−	1.05847i	0.379948	−	0.925008i	$-0.375942\pi$
$348$		0			0
$349$	2.12214e6			0.932631			0.466316	−	0.884618i	$-0.345581\pi$
$349$	2.12214e6			0.932631			0.466316	+	0.884618i	$0.345581\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.22077e6	−	3.84649e6i	−0.948566	−	1.64297i	−0.748448	−	0.663193i	$-0.769201\pi$
$353$	−2.22077e6	−	3.84649e6i	−0.948566	−	1.64297i	−0.200118	−	0.979772i	$-0.564133\pi$
$354$		0			0
$355$	1.95960e6	−	3.39413e6i	0.825271	−	1.42941i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	641834.	−	1.11169e6i	0.262837	−	0.455247i	−0.704158	−	0.710044i	$-0.748675\pi$
$359$	641834.	−	1.11169e6i	0.262837	−	0.455247i	0.966995	+	0.254797i	$0.0820085\pi$
$360$		0			0
$361$	−1.46301e6	−	2.53400e6i	−0.590851	−	1.02338i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.36971e6			0.538143
$366$		0			0
$367$	−2.49672e6	−	4.32444e6i	−0.967618	−	1.67596i	−0.702410	−	0.711773i	$-0.747893\pi$
$367$	−2.49672e6	−	4.32444e6i	−0.967618	−	1.67596i	−0.265208	−	0.964191i	$-0.585441\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.32091e6	+	2.96434e6i	−0.498240	+	1.11813i
$372$		0			0
$373$	242108.	−	419344.i	0.0901026	−	0.156062i	−0.817452	−	0.575997i	$-0.804614\pi$
$373$	242108.	−	419344.i	0.0901026	−	0.156062i	0.907554	+	0.419935i	$0.137947\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.95153e6			0.707165
$378$		0			0
$379$	4.13495e6			1.47867			0.739336	−	0.673337i	$-0.235140\pi$
$379$	4.13495e6			1.47867			0.739336	+	0.673337i	$0.235140\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−327620.	+	567455.i	−0.114123	+	0.197667i	−0.917429	−	0.397900i	$-0.869739\pi$
$383$	−327620.	+	567455.i	−0.114123	+	0.197667i	0.803306	+	0.595567i	$0.203073\pi$
$384$		0			0
$385$	−1.40431e6	−	1.93412e6i	−0.482848	−	0.665016i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	512840.	+	888265.i	0.171834	+	0.297624i	0.939061	−	0.343751i	$-0.111698\pi$
$389$	512840.	+	888265.i	0.171834	+	0.297624i	−0.767227	+	0.641375i	$0.778364\pi$
$390$		0			0
$391$	−1.18404e6			−0.391675
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−786397.	−	1.36208e6i	−0.253600	−	0.439248i
$396$		0			0
$397$	−2.38535e6	+	4.13154e6i	−0.759583	+	1.31564i	0.183481	+	0.983023i	$0.441263\pi$
$397$	−2.38535e6	+	4.13154e6i	−0.759583	+	1.31564i	−0.943064	+	0.332612i	$0.892070\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.32830e6	−	4.03274e6i	0.723067	−	1.25239i	−0.236697	−	0.971583i	$-0.576065\pi$
$401$	2.32830e6	−	4.03274e6i	0.723067	−	1.25239i	0.959765	−	0.280806i	$-0.0906017\pi$
$402$		0			0
$403$	−1.08372e6	−	1.87705e6i	−0.332394	−	0.575723i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.11110e6			0.631717
$408$		0			0
$409$	1.27969e6	+	2.21648e6i	0.378265	+	0.655173i	0.990810	−	0.135262i	$-0.0431876\pi$
$409$	1.27969e6	+	2.21648e6i	0.378265	+	0.655173i	−0.612545	+	0.790435i	$0.709854\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.52459e6	−	3.42143e6i	0.439823	−	0.987036i
$414$		0			0
$415$	−1.68810e6	+	2.92387e6i	−0.481147	+	0.833371i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	342145.			0.0952083			0.0476042	−	0.998866i	$-0.484841\pi$
$419$	342145.			0.0952083			0.0476042	+	0.998866i	$0.484841\pi$
$420$		0			0
$421$	−3.61976e6			−0.995348			−0.497674	−	0.867364i	$-0.665812\pi$
$421$	−3.61976e6			−0.995348			−0.497674	+	0.867364i	$0.665812\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	366925.	−	635534.i	0.0985384	−	0.170674i
$426$		0			0
$427$	−5.66124e6	+	593248.i	−1.50260	+	0.157459i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−629229.	−	1.08986e6i	−0.163161	−	0.282602i	0.772840	−	0.634601i	$-0.218836\pi$
$431$	−629229.	−	1.08986e6i	−0.163161	−	0.282602i	−0.936001	+	0.351999i	$0.885502\pi$
$432$		0			0
$433$	−2.22735e6			−0.570912			−0.285456	−	0.958392i	$-0.592145\pi$
$433$	−2.22735e6			−0.570912			−0.285456	+	0.958392i	$0.592145\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.01948e6	−	1.76579e6i	−0.255373	−	0.442319i
$438$		0			0
$439$	8124.26	−	14071.6i	0.00201197	−	0.00348484i	−0.865018	−	0.501741i	$-0.832693\pi$
$439$	8124.26	−	14071.6i	0.00201197	−	0.00348484i	0.867030	+	0.498257i	$0.166026\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.08982e6	+	5.35173e6i	−0.748039	+	1.29564i	0.200722	+	0.979648i	$0.435671\pi$
$443$	−3.08982e6	+	5.35173e6i	−0.748039	+	1.29564i	−0.948761	+	0.315994i	$0.897662\pi$
$444$		0			0
$445$	−2.42028e6	−	4.19205e6i	−0.579383	−	1.00352i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.39476e6			−1.49695			−0.748477	−	0.663161i	$-0.769214\pi$
$449$	−6.39476e6			−1.49695			−0.748477	+	0.663161i	$0.769214\pi$
$450$		0			0
$451$	−436066.	−	755289.i	−0.100951	−	0.174853i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.68974e6	−	491443.i	1.06199	−	0.111287i
$456$		0			0
$457$	−2.74197e6	+	4.74923e6i	−0.614147	+	1.06373i	0.376387	+	0.926463i	$0.377166\pi$
$457$	−2.74197e6	+	4.74923e6i	−0.614147	+	1.06373i	−0.990534	+	0.137271i	$0.956167\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−317176.			−0.0695101			−0.0347551	−	0.999396i	$-0.511065\pi$
$461$	−317176.			−0.0695101			−0.0347551	+	0.999396i	$0.511065\pi$
$462$		0			0
$463$	−2.27371e6			−0.492926			−0.246463	−	0.969152i	$-0.579268\pi$
$463$	−2.27371e6			−0.492926			−0.246463	+	0.969152i	$0.579268\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.99318e6	+	6.91638e6i	−0.847279	+	1.46753i	0.0363490	+	0.999339i	$0.488427\pi$
$467$	−3.99318e6	+	6.91638e6i	−0.847279	+	1.46753i	−0.883628	+	0.468190i	$0.844906\pi$
$468$		0			0
$469$	2.65547e6	−	5.95931e6i	0.557454	−	1.25102i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−378645.	−	655833.i	−0.0778179	−	0.134785i
$474$		0			0
$475$	1.26371e6			0.256989
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−57787.1	−	100090.i	−0.0115078	−	0.0199321i	0.860214	−	0.509933i	$-0.170330\pi$
$479$	−57787.1	−	100090.i	−0.0115078	−	0.0199321i	−0.871722	+	0.490001i	$0.836996\pi$
$480$		0			0
$481$	−2.08243e6	+	3.60687e6i	−0.410400	+	0.710834i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.54301e6	−	6.13667e6i	0.683940	−	1.18462i
$486$		0			0
$487$	−2.71618e6	−	4.70456e6i	−0.518962	−	0.898869i	−0.999757	−	0.0220360i	$-0.992985\pi$
$487$	−2.71618e6	−	4.70456e6i	−0.518962	−	0.898869i	0.480795	−	0.876833i	$-0.340348\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8.62648e6			1.61484			0.807420	−	0.589977i	$-0.200863\pi$
$491$	8.62648e6			1.61484			0.807420	+	0.589977i	$0.200863\pi$
$492$		0			0
$493$	2.19314e6	+	3.79864e6i	0.406396	+	0.703899i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.92855e6	+	6.78799e6i	0.895011	+	1.23268i
$498$		0			0
$499$	375738.	−	650798.i	0.0675514	−	0.117002i	−0.830272	−	0.557359i	$-0.811815\pi$
$499$	375738.	−	650798.i	0.0675514	−	0.117002i	0.897823	+	0.440357i	$0.145148\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.12863e6			−0.551358			−0.275679	−	0.961250i	$-0.588903\pi$
$503$	−3.12863e6			−0.551358			−0.275679	+	0.961250i	$0.588903\pi$
$504$		0			0
$505$	−1.72469e6			−0.300942
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.21487e6	+	3.83627e6i	−0.378926	+	0.656319i	−0.990906	−	0.134554i	$-0.957040\pi$
$509$	−2.21487e6	+	3.83627e6i	−0.378926	+	0.656319i	0.611980	+	0.790873i	$0.290373\pi$
$510$		0			0
$511$	−1.19326e6	+	2.67786e6i	−0.202153	+	0.453666i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.52803e6	−	4.37868e6i	−0.420015	−	0.727487i
$516$		0			0
$517$	2.00931e6			0.330614
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.33826e6	+	4.04999e6i	0.377397	+	0.653672i	0.990683	−	0.136190i	$-0.0434857\pi$
$521$	2.33826e6	+	4.04999e6i	0.377397	+	0.653672i	−0.613285	+	0.789861i	$0.710152\pi$
$522$		0			0
$523$	741533.	−	1.28437e6i	0.118543	−	0.205323i	−0.800647	−	0.599136i	$-0.795511\pi$
$523$	741533.	−	1.28437e6i	0.118543	−	0.205323i	0.919191	+	0.393813i	$0.128844\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.43578e6	−	4.21890e6i	0.382043	−	0.661717i
$528$		0			0
$529$	2.83338e6	+	4.90756e6i	0.440216	+	0.762476i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.72058e6			0.262335
$534$		0			0
$535$	3.98208e6	+	6.89716e6i	0.601486	+	1.04180i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.00471e6	−	1.06054e6i	0.742005	−	0.157238i
$540$		0			0
$541$	2.09764e6	−	3.63323e6i	0.308133	−	0.533703i	−0.669821	−	0.742523i	$-0.733629\pi$
$541$	2.09764e6	−	3.63323e6i	0.308133	−	0.533703i	0.977954	+	0.208820i	$0.0669624\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.29704e7			1.87052
$546$		0			0
$547$	5.83053e6			0.833181			0.416591	−	0.909094i	$-0.363225\pi$
$547$	5.83053e6			0.833181			0.416591	+	0.909094i	$0.363225\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.77666e6	+	6.54137e6i	−0.529943	+	0.917888i
$552$		0			0
$553$	3.34803e6	−	350844.i	0.465560	−	0.0487866i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.06141e6	−	3.57047e6i	−0.281531	−	0.487627i	0.690231	−	0.723589i	$-0.257509\pi$
$557$	−2.06141e6	−	3.57047e6i	−0.281531	−	0.487627i	−0.971762	+	0.235963i	$0.924176\pi$
$558$		0			0
$559$	1.49401e6			0.202220
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−995072.	−	1.72352e6i	−0.132307	−	0.229163i	0.792258	−	0.610186i	$-0.208905\pi$
$563$	−995072.	−	1.72352e6i	−0.132307	−	0.229163i	−0.924566	+	0.381023i	$0.875572\pi$
$564$		0			0
$565$	−2.86193e6	+	4.95700e6i	−0.377170	+	0.653278i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	746894.	−	1.29366e6i	0.0967115	−	0.167509i	−0.813610	−	0.581411i	$-0.802501\pi$
$569$	746894.	−	1.29366e6i	0.0967115	−	0.167509i	0.910322	+	0.413902i	$0.135834\pi$
$570$		0			0
$571$	−5.19490e6	−	8.99783e6i	−0.666787	−	1.15491i	−0.978798	−	0.204830i	$-0.934336\pi$
$571$	−5.19490e6	−	8.99783e6i	−0.666787	−	1.15491i	0.312011	−	0.950079i	$-0.398997\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	476973.			0.0601623
$576$		0			0
$577$	3.30161e6	+	5.71856e6i	0.412844	+	0.715068i	0.995200	−	0.0978670i	$-0.0312020\pi$
$577$	3.30161e6	+	5.71856e6i	0.412844	+	0.715068i	−0.582355	+	0.812935i	$0.697869\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.24571e6	−	5.84752e6i	−0.521807	−	0.718673i
$582$		0			0
$583$	3.80987e6	−	6.59889e6i	0.464236	−	0.804080i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.23858e7			−1.48364			−0.741820	−	0.670599i	$-0.766037\pi$
$587$	−1.23858e7			−1.48364			−0.741820	+	0.670599i	$0.766037\pi$
$588$		0			0
$589$	8.38898e6			0.996371
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.69859e6	+	4.67409e6i	−0.315137	+	0.545833i	−0.979467	−	0.201606i	$-0.935384\pi$
$593$	−2.69859e6	+	4.67409e6i	−0.315137	+	0.545833i	0.664330	+	0.747440i	$0.268717\pi$
$594$		0			0
$595$	6.22696e6	+	8.57626e6i	0.721081	+	0.993129i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.00340e6	−	6.93409e6i	−0.455892	−	0.789628i	0.542847	−	0.839831i	$-0.317346\pi$
$599$	−4.00340e6	−	6.93409e6i	−0.455892	−	0.789628i	−0.998739	+	0.0502038i	$0.984013\pi$
$600$		0			0
$601$	1.02285e7			1.15511			0.577557	−	0.816351i	$-0.304006\pi$
$601$	1.02285e7			1.15511			0.577557	+	0.816351i	$0.304006\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.07147e6	−	3.58788e6i	−0.230086	−	0.398520i
$606$		0			0
$607$	−2.66714e6	+	4.61963e6i	−0.293816	+	0.508903i	−0.974709	−	0.223479i	$-0.928258\pi$
$607$	−2.66714e6	+	4.61963e6i	−0.293816	+	0.508903i	0.680893	+	0.732383i	$0.261592\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.98202e6	+	3.43296e6i	−0.214786	+	0.372020i
$612$		0			0
$613$	−8.06888e6	−	1.39757e7i	−0.867286	−	1.50218i	−0.864759	−	0.502186i	$-0.832529\pi$
$613$	−8.06888e6	−	1.39757e7i	−0.867286	−	1.50218i	−0.00252631	−	0.999997i	$-0.500804\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.66405e7			−1.75976			−0.879879	−	0.475197i	$-0.842377\pi$
$617$	−1.66405e7			−1.75976			−0.879879	+	0.475197i	$0.842377\pi$
$618$		0			0
$619$	1.93849e6	+	3.35757e6i	0.203347	+	0.352207i	0.949605	−	0.313450i	$-0.101485\pi$
$619$	1.93849e6	+	3.35757e6i	0.203347	+	0.352207i	−0.746258	+	0.665657i	$0.768151\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.03042e7	−	1.07979e6i	1.06364	−	0.111460i
$624$		0			0
$625$	5.58455e6	−	9.67272e6i	0.571858	−	0.990487i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9.36101e6			−0.943401
$630$		0			0
$631$	−8.87637e6			−0.887487			−0.443744	−	0.896154i	$-0.646350\pi$
$631$	−8.87637e6			−0.887487			−0.443744	+	0.896154i	$0.646350\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.49321e6	−	1.12466e7i	0.639035	−	1.10684i
$636$		0			0
$637$	−3.12477e6	+	9.59682e6i	−0.305119	+	0.937084i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.95914e6	+	1.37856e7i	0.765105	+	1.32520i	0.940191	+	0.340647i	$0.110646\pi$
$641$	7.95914e6	+	1.37856e7i	0.765105	+	1.32520i	−0.175087	+	0.984553i	$0.556021\pi$
$642$		0			0
$643$	1.80650e7			1.72310			0.861551	−	0.507670i	$-0.169493\pi$
$643$	1.80650e7			1.72310			0.861551	+	0.507670i	$0.169493\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.82101e6	+	8.35023e6i	0.452769	+	0.784220i	0.998557	−	0.0537040i	$-0.0171027\pi$
$647$	4.82101e6	+	8.35023e6i	0.452769	+	0.784220i	−0.545787	+	0.837924i	$0.683769\pi$
$648$		0			0
$649$	−4.39734e6	+	7.61641e6i	−0.409806	+	0.709804i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.01681e7	+	1.76117e7i	−0.933166	+	1.61629i	−0.155293	+	0.987868i	$0.549632\pi$
$653$	−1.01681e7	+	1.76117e7i	−0.933166	+	1.61629i	−0.777872	+	0.628422i	$0.783701\pi$
$654$		0			0
$655$	1.03865e7	+	1.79900e7i	0.945947	+	1.63843i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	8.68194e6			0.778759			0.389380	−	0.921077i	$-0.372689\pi$
$659$	8.68194e6			0.778759			0.389380	+	0.921077i	$0.372689\pi$
$660$		0			0
$661$	−4.75543e6	−	8.23665e6i	−0.423337	−	0.733241i	0.572926	−	0.819607i	$-0.305808\pi$
$661$	−4.75543e6	−	8.23665e6i	−0.423337	−	0.733241i	−0.996264	+	0.0863655i	$0.972475\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−7.42846e6	+	1.66707e7i	−0.651395	+	1.46184i
$666$		0			0
$667$	−1.42545e6	+	2.46896e6i	−0.124062	+	0.214882i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.33649e7			1.14593
$672$		0			0
$673$	7.89064e6			0.671544			0.335772	−	0.941943i	$-0.391003\pi$
$673$	7.89064e6			0.671544			0.335772	+	0.941943i	$0.391003\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.50722e6	−	7.80674e6i	0.377952	−	0.654633i	−0.612812	−	0.790229i	$-0.709962\pi$
$677$	4.50722e6	−	7.80674e6i	0.377952	−	0.654633i	0.990764	+	0.135596i	$0.0432949\pi$
$678$		0			0
$679$	8.91095e6	+	1.22729e7i	0.741737	+	1.02158i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−529513.	−	917144.i	−0.0434335	−	0.0752291i	0.843491	−	0.537143i	$-0.180496\pi$
$683$	−529513.	−	917144.i	−0.0434335	−	0.0752291i	−0.886925	+	0.461914i	$0.847163\pi$
$684$		0			0
$685$	1.07192e7			0.872839
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.51625e6	+	1.30185e7i	0.603189	+	1.04475i
$690$		0			0
$691$	6.16510e6	−	1.06783e7i	0.491185	−	0.850758i	−0.508763	−	0.860906i	$-0.669897\pi$
$691$	6.16510e6	−	1.06783e7i	0.491185	−	0.850758i	0.999949	+	0.0101487i	$0.00323047\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.10393e6	−	1.40364e7i	0.636405	−	1.10229i
$696$		0			0
$697$	1.93360e6	+	3.34909e6i	0.150760	+	0.261123i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.18317e7			−0.909392			−0.454696	−	0.890647i	$-0.650252\pi$
$701$	−1.18317e7			−0.909392			−0.454696	+	0.890647i	$0.650252\pi$
$702$		0			0
$703$	−8.05997e6	−	1.39603e7i	−0.615099	−	1.06538i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.50250e6	−	3.37187e6i	0.113049	−	0.253701i
$708$		0			0
$709$	1.47513e6	−	2.55500e6i	0.110209	−	0.190887i	−0.805646	−	0.592398i	$-0.798181\pi$
$709$	1.47513e6	−	2.55500e6i	0.110209	−	0.190887i	0.915854	+	0.401511i	$0.131515\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.16632e6			0.233255
$714$		0			0
$715$	−1.10714e7			−0.809910
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.54286e6	+	7.86847e6i	−0.327723	+	0.567634i	−0.982060	−	0.188570i	$-0.939615\pi$
$719$	−4.54286e6	+	7.86847e6i	−0.327723	+	0.567634i	0.654336	+	0.756204i	$0.272948\pi$
$720$		0			0
$721$	1.07629e7	−	1.12786e6i	0.771065	−	0.0808008i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−883472.	−	1.53022e6i	−0.0624235	−	0.108121i
$726$		0			0
$727$	2.38405e7			1.67293			0.836467	−	0.548018i	$-0.184617\pi$
$727$	2.38405e7			1.67293			0.836467	+	0.548018i	$0.184617\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.67898e6	+	2.90809e6i	0.116213	+	0.201286i
$732$		0			0
$733$	−1.01233e7	+	1.75340e7i	−0.695922	+	1.20537i	0.273947	+	0.961745i	$0.411671\pi$
$733$	−1.01233e7	+	1.75340e7i	−0.695922	+	1.20537i	−0.969869	+	0.243627i	$0.921663\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7.65910e6	+	1.32660e7i	−0.519408	+	0.899642i
$738$		0			0
$739$	1.04773e7	+	1.81472e7i	0.705729	+	1.22236i	0.966428	+	0.256938i	$0.0827136\pi$
$739$	1.04773e7	+	1.81472e7i	0.705729	+	1.22236i	−0.260699	+	0.965420i	$0.583953\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.06903e7			1.37497			0.687486	−	0.726198i	$-0.258714\pi$
$743$	2.06903e7			1.37497			0.687486	+	0.726198i	$0.258714\pi$
$744$		0			0
$745$	1.78896e6	+	3.09858e6i	0.118089	+	0.204537i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.69534e7	+	1.77657e6i	−1.10421	+	0.115712i
$750$		0			0
$751$	4.50787e6	−	7.80786e6i	0.291656	−	0.505164i	−0.682545	−	0.730843i	$-0.739127\pi$
$751$	4.50787e6	−	7.80786e6i	0.291656	−	0.505164i	0.974202	+	0.225680i	$0.0724603\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.17623e7			1.38943
$756$		0			0
$757$	428276.			0.0271634			0.0135817	−	0.999908i	$-0.495677\pi$
$757$	428276.			0.0271634			0.0135817	+	0.999908i	$0.495677\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.87239e6	−	3.24308e6i	0.117202	−	0.203000i	−0.801456	−	0.598054i	$-0.795941\pi$
$761$	1.87239e6	−	3.24308e6i	0.117202	−	0.203000i	0.918658	+	0.395054i	$0.129274\pi$
$762$		0			0
$763$	−1.12994e7	+	2.53578e7i	−0.702661	+	1.57689i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.67523e6	−	1.50259e7i	−0.532467	−	0.922260i
$768$		0			0
$769$	−2.40091e7			−1.46406			−0.732032	−	0.681270i	$-0.761428\pi$
$769$	−2.40091e7			−1.46406			−0.732032	+	0.681270i	$0.761428\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.65473e6	−	1.15263e7i	−0.400573	−	0.693813i	0.593222	−	0.805039i	$-0.297856\pi$
$773$	−6.65473e6	−	1.15263e7i	−0.400573	−	0.693813i	−0.993795	+	0.111226i	$0.964522\pi$
$774$		0			0
$775$	−981215.	+	1.69951e6i	−0.0586827	+	0.101641i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.32972e6	+	5.76724e6i	−0.196591	+	0.340506i
$780$		0			0
$781$	−9.84777e6	−	1.70568e7i	−0.577710	−	1.00062i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3.48790e7			2.02018
$786$		0			0
$787$	3.49008e6	+	6.04499e6i	0.200862	+	0.347903i	0.948806	−	0.315858i	$-0.102292\pi$
$787$	3.49008e6	+	6.04499e6i	0.200862	+	0.347903i	−0.747944	+	0.663761i	$0.768959\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−7.19798e6	−	9.91362e6i	−0.409043	−	0.563367i
$792$		0			0
$793$	−1.31834e7	+	2.28343e7i	−0.744464	+	1.28945i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.91181e7			1.62374			0.811872	−	0.583836i	$-0.198449\pi$
$797$	2.91181e7			1.62374			0.811872	+	0.583836i	$0.198449\pi$
$798$		0			0
$799$	−8.90966e6			−0.493735
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.44168e6	−	5.96116e6i	0.188357	−	0.326243i
$804$		0			0
$805$	−2.80378e6	+	6.29215e6i	−0.152495	+	0.342223i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−8.90045e6	−	1.54160e7i	−0.478124	−	0.828135i	0.521561	−	0.853214i	$-0.325350\pi$
$809$	−8.90045e6	−	1.54160e7i	−0.478124	−	0.828135i	−0.999685	+	0.0250786i	$0.992016\pi$
$810$		0			0
$811$	−2.86322e6			−0.152863			−0.0764315	−	0.997075i	$-0.524353\pi$
$811$	−2.86322e6			−0.152863			−0.0764315	+	0.997075i	$0.524353\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.37272e7	+	2.37763e7i	0.723918	+	1.25386i
$816$		0			0
$817$	−2.89126e6	+	5.00781e6i	−0.151542	+	0.262478i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	367651.	−	636790.i	0.0190361	−	0.0329715i	−0.856350	−	0.516395i	$-0.827274\pi$
$821$	367651.	−	636790.i	0.0190361	−	0.0329715i	0.875387	+	0.483424i	$0.160607\pi$
$822$		0			0
$823$	−2.96647e6	−	5.13808e6i	−0.152665	−	0.264424i	0.779541	−	0.626351i	$-0.215452\pi$
$823$	−2.96647e6	−	5.13808e6i	−0.152665	−	0.264424i	−0.932207	+	0.361927i	$0.882119\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.57974e7			−0.803197			−0.401598	−	0.915816i	$-0.631545\pi$
$827$	−1.57974e7			−0.803197			−0.401598	+	0.915816i	$0.631545\pi$
$828$		0			0
$829$	−1.22213e7	−	2.11679e7i	−0.617634	−	1.06977i	−0.989916	−	0.141653i	$-0.954758\pi$
$829$	−1.22213e7	−	2.11679e7i	−0.617634	−	1.06977i	0.372283	−	0.928119i	$-0.378575\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.21918e7	+	4.70265e6i	−1.10810	+	0.234817i
$834$		0			0
$835$	1.22954e7	−	2.12963e7i	0.610278	−	1.05703i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.17577e7			1.06711			0.533555	−	0.845766i	$-0.320856\pi$
$839$	2.17577e7			1.06711			0.533555	+	0.845766i	$0.320856\pi$
$840$		0			0
$841$	−9.94997e6			−0.485101
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−323564.	+	560430.i	−0.0155890	+	0.0270010i
$846$		0			0
$847$	8.81911e6	−	924165.i	0.422392	−	0.0442630i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.04214e6	−	5.26914e6i	−0.143997	−	0.249411i
$852$		0			0
$853$	−1.45057e7			−0.682600			−0.341300	−	0.939954i	$-0.610867\pi$
$853$	−1.45057e7			−0.682600			−0.341300	+	0.939954i	$0.610867\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.08316e7	−	1.87609e7i	−0.503779	−	0.872571i	−0.999990	−	0.00436899i	$-0.998609\pi$
$857$	−1.08316e7	−	1.87609e7i	−0.503779	−	0.872571i	0.496212	−	0.868202i	$-0.334724\pi$
$858$		0			0
$859$	−6.16479e6	+	1.06777e7i	−0.285060	+	0.493738i	−0.972624	−	0.232386i	$-0.925347\pi$
$859$	−6.16479e6	+	1.06777e7i	−0.285060	+	0.493738i	0.687564	+	0.726124i	$0.258680\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−5.19670e6	+	9.00095e6i	−0.237520	+	0.411397i	−0.960002	−	0.279993i	$-0.909668\pi$
$863$	−5.19670e6	+	9.00095e6i	−0.237520	+	0.411397i	0.722482	+	0.691390i	$0.243001\pi$
$864$		0			0
$865$	−1.55334e7	−	2.69046e7i	−0.705871	−	1.22260i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.90392e6			−0.355053
$870$		0			0
$871$	−1.51102e7	−	2.61716e7i	−0.674876	−	1.16892i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.19089e7	+	1.64019e7i	0.525838	+	0.724226i
$876$		0			0
$877$	1.47654e7	−	2.55745e7i	0.648256	−	1.12281i	−0.335283	−	0.942118i	$-0.608832\pi$
$877$	1.47654e7	−	2.55745e7i	0.648256	−	1.12281i	0.983539	−	0.180695i	$-0.0578348\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.55937e7			0.676875			0.338437	−	0.940989i	$-0.390102\pi$
$881$	1.55937e7			0.676875			0.338437	+	0.940989i	$0.390102\pi$
$882$		0			0
$883$	−2.59236e7			−1.11890			−0.559452	−	0.828863i	$-0.688988\pi$
$883$	−2.59236e7			−1.11890			−0.559452	+	0.828863i	$0.688988\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7.50215e6	−	1.29941e7i	0.320167	−	0.554545i	−0.660355	−	0.750953i	$-0.729594\pi$
$887$	7.50215e6	−	1.29941e7i	0.320167	−	0.554545i	0.980522	+	0.196408i	$0.0629277\pi$
$888$		0			0
$889$	1.63309e7	+	2.24923e7i	0.693038	+	0.954506i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.67135e6	−	1.32872e7i	−0.321917	−	0.557576i
$894$		0			0
$895$	1.69989e7			0.709356
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5.86480e6	−	1.01581e7i	−0.242022	−	0.419194i
$900$		0			0
$901$	−1.68937e7	+	2.92607e7i	−0.693286	+	1.20081i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.67123e7	−	2.89466e7i	0.678289	−	1.17483i
$906$		0			0
$907$	4.82185e6	+	8.35169e6i	0.194624	+	0.337098i	0.946777	−	0.321890i	$-0.104318\pi$
$907$	4.82185e6	+	8.35169e6i	0.194624	+	0.337098i	−0.752153	+	0.658988i	$0.770985\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.37570e7			−1.74683			−0.873417	−	0.486973i	$-0.838101\pi$
$911$	−4.37570e7			−1.74683			−0.873417	+	0.486973i	$0.838101\pi$
$912$		0			0
$913$	8.48337e6	+	1.46936e7i	0.336815	+	0.583380i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.42198e7	+	4.63385e6i	−1.73658	+	0.181978i
$918$		0			0
$919$	1.73458e7	−	3.00439e7i	0.677495	−	1.17346i	−0.298238	−	0.954492i	$-0.596399\pi$
$919$	1.73458e7	−	3.00439e7i	0.677495	−	1.17346i	0.975733	−	0.218965i	$-0.0702679\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3.88561e7			1.50126
$924$		0			0
$925$	3.77093e6			0.144909
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7.91456e6	+	1.37084e7i	−0.300876	+	0.521132i	−0.976335	−	0.216265i	$-0.930612\pi$
$929$	−7.91456e6	+	1.37084e7i	−0.300876	+	0.521132i	0.675459	+	0.737398i	$0.263946\pi$
$930$		0			0
$931$	−2.61207e7	−	2.90461e7i	−0.987665	−	1.09828i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.24421e7	−	2.15504e7i	−0.465442	−	0.806169i
$936$		0			0
$937$	1.64558e7			0.612307			0.306153	−	0.951982i	$-0.400958\pi$
$937$	1.64558e7			0.612307			0.306153	+	0.951982i	$0.400958\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.24672e6	−	1.25517e7i	−0.266789	−	0.462091i	0.701242	−	0.712923i	$-0.252629\pi$
$941$	−7.24672e6	−	1.25517e7i	−0.266789	−	0.462091i	−0.968031	+	0.250832i	$0.919296\pi$
$942$		0			0
$943$	−1.25676e6	+	2.17677e6i	−0.0460229	+	0.0797139i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8.07707e6	−	1.39899e7i	0.292670	−	0.506920i	−0.681770	−	0.731567i	$-0.738789\pi$
$947$	8.07707e6	−	1.39899e7i	0.292670	−	0.506920i	0.974440	+	0.224647i	$0.0721228\pi$
$948$		0			0
$949$	6.78987e6	+	1.17604e7i	0.244735	+	0.423893i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.86700e7			−0.665904			−0.332952	−	0.942944i	$-0.608045\pi$
$953$	−1.86700e7			−0.665904			−0.332952	+	0.942944i	$0.608045\pi$
$954$		0			0
$955$	−2.48812e7	−	4.30954e7i	−0.882800	−	1.52905i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9.33823e6	+	2.09565e7i	−0.327882	+	0.735822i
$960$		0			0
$961$	7.80092e6	−	1.35116e7i	0.272482	−	0.471952i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.67366e6			−0.196130
$966$		0			0
$967$	661254.			0.0227406			0.0113703	−	0.999935i	$-0.496381\pi$
$967$	661254.			0.0227406			0.0113703	+	0.999935i	$0.496381\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−8.67867e6	+	1.50319e7i	−0.295396	+	0.511641i	−0.975077	−	0.221867i	$-0.928785\pi$
$971$	−8.67867e6	+	1.50319e7i	−0.295396	+	0.511641i	0.679681	+	0.733508i	$0.262118\pi$
$972$		0			0
$973$	2.03820e7	+	2.80717e7i	0.690185	+	0.950577i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.36720e7	+	2.36807e7i	0.458244	+	0.793702i	0.998868	−	0.0475625i	$-0.0151453\pi$
$977$	1.36720e7	+	2.36807e7i	0.458244	+	0.793702i	−0.540624	+	0.841264i	$0.681812\pi$
$978$		0			0
$979$	−2.43258e7			−0.811166
$980$		0			0
$981$		0			0
$982$		0			0
$983$	7.19703e6	+	1.24656e7i	0.237558	+	0.411462i	0.960013	−	0.279956i	$-0.0903198\pi$
$983$	7.19703e6	+	1.24656e7i	0.237558	+	0.411462i	−0.722455	+	0.691418i	$0.756986\pi$
$984$		0			0
$985$	−2.32337e6	+	4.02420e6i	−0.0763006	+	0.132157i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.09127e6	+	1.89014e6i	−0.0354766	+	0.0614473i
$990$		0			0
$991$	−6.36599e6	−	1.10262e7i	−0.205912	−	0.356650i	0.744511	−	0.667610i	$-0.232683\pi$
$991$	−6.36599e6	−	1.10262e7i	−0.205912	−	0.356650i	−0.950423	+	0.310960i	$0.899349\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.84453e7			0.590647
$996$		0			0
$997$	−1.19987e7	−	2.07823e7i	−0.382293	−	0.662150i	0.609097	−	0.793096i	$-0.291532\pi$
$997$	−1.19987e7	−	2.07823e7i	−0.382293	−	0.662150i	−0.991390	+	0.130945i	$0.958199\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.6.s.f.289.5		12
3.2	odd	2		168.6.q.d.121.2	yes	12
7.4	even	3	inner	504.6.s.f.361.5		12
12.11	even	2		336.6.q.n.289.2		12
21.11	odd	6		168.6.q.d.25.2	✓	12
84.11	even	6		336.6.q.n.193.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.6.q.d.25.2	✓	12	21.11	odd	6
168.6.q.d.121.2	yes	12	3.2	odd	2
336.6.q.n.193.2		12	84.11	even	6
336.6.q.n.289.2		12	12.11	even	2
504.6.s.f.289.5		12	1.1	even	1	trivial
504.6.s.f.361.5		12	7.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	504.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(30.2849 - 52.4550i) q^{5} +(76.1691 + 104.906i) q^{7} +(-152.194 - 263.607i) q^{11} +600.507 q^{13} +(674.856 + 1168.88i) q^{17} +(-1162.12 + 2012.85i) q^{19} +(-438.629 + 759.727i) q^{23} +(-271.855 - 470.866i) q^{25} +3249.80 q^{29} +(-1804.67 - 3125.78i) q^{31} +(7809.63 - 818.380i) q^{35} +(-3467.78 + 6006.37i) q^{37} +2865.20 q^{41} +2487.91 q^{43} +(-3300.58 + 5716.77i) q^{47} +(-5203.55 + 15981.2i) q^{49} +(12516.5 + 21679.2i) q^{53} -18436.7 q^{55} +(-14446.5 - 25022.1i) q^{59} +(-21953.7 + 38025.0i) q^{61} +(18186.3 - 31499.6i) q^{65} +(-25162.3 - 43582.4i) q^{67} +64705.5 q^{71} +(11306.9 + 19584.1i) q^{73} +(16061.5 - 36044.8i) q^{77} +(12983.3 - 22487.8i) q^{79} -55740.6 q^{83} +81751.9 q^{85} +(39958.5 - 69210.2i) q^{89} +(45740.1 + 62996.8i) q^{91} +(70389.6 + 121918. i) q^{95} +116989. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 17 q^{5} + 144 q^{7} + 565 q^{11} - 842 q^{13} - 52 q^{17} + 107 q^{19} - 700 q^{23} - 4881 q^{25} - 12958 q^{29} - 3552 q^{31} + 938 q^{35} - 3453 q^{37} + 15148 q^{41} + 38278 q^{43} - 12136 q^{47}+ \cdots + 28934 q^{97}+O(q^{100}) \) 12 * q - 17 * q^5 + 144 * q^7 + 565 * q^11 - 842 * q^13 - 52 * q^17 + 107 * q^19 - 700 * q^23 - 4881 * q^25 - 12958 * q^29 - 3552 * q^31 + 938 * q^35 - 3453 * q^37 + 15148 * q^41 + 38278 * q^43 - 12136 * q^47 - 39926 * q^49 + 37591 * q^53 - 68670 * q^55 - 3905 * q^59 + 13296 * q^61 - 45638 * q^65 - 2889 * q^67 - 132992 * q^71 + 127931 * q^73 + 306817 * q^77 + 100226 * q^79 + 146246 * q^83 - 181328 * q^85 + 3418 * q^89 - 275865 * q^91 - 108810 * q^95 + 28934 * q^97

Introduction

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