LMFDB - Embedded newform 384.7.h.f.65.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,7,Mod(65,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("384.65");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	384.h (of order $2$, degree $1$, 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:	$88.3407681100$
Analytic rank:	$0$
Dimension:	$8$
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} - 172x^{6} + 13179x^{4} - 522628x^{2} + 8755681 $ x^8 - 172x^6 + 13179x^4 - 522628*x^2 + 8755681
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.5
Root			$-6.86097 + 3.28347i$ of defining polynomial
Character	$\chi$	$=$	384.65
Dual form			384.7.h.f.65.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(22.4164 - 15.0501i) q^{3} -122.733 q^{5} +206.594 q^{7} +(275.991 - 674.737i) q^{9} +O(q^{10})$	\(q+(22.4164 - 15.0501i) q^{3} -122.733 q^{5} +206.594 q^{7} +(275.991 - 674.737i) q^{9} -142.827 q^{11} +3499.27i q^{13} +(-2751.23 + 1847.14i) q^{15} -3382.86i q^{17} -755.858i q^{19} +(4631.09 - 3109.25i) q^{21} -16911.3i q^{23} -561.656 q^{25} +(-3968.12 - 19278.9i) q^{27} +41131.3 q^{29} -22916.1 q^{31} +(-3201.66 + 2149.55i) q^{33} -25355.8 q^{35} +26835.0i q^{37} +(52664.2 + 78441.0i) q^{39} +40460.6i q^{41} -128319. i q^{43} +(-33873.1 + 82812.4i) q^{45} -60668.5i q^{47} -74968.0 q^{49} +(-50912.4 - 75831.7i) q^{51} -5295.60 q^{53} +17529.5 q^{55} +(-11375.7 - 16943.6i) q^{57} -97392.6 q^{59} -213477. i q^{61} +(57018.0 - 139397. i) q^{63} -429475. i q^{65} -385320. i q^{67} +(-254517. - 379091. i) q^{69} +356373. i q^{71} -141507. q^{73} +(-12590.3 + 8452.97i) q^{75} -29507.1 q^{77} -722207. q^{79} +(-379099. - 372442. i) q^{81} -8040.09 q^{83} +415188. i q^{85} +(922016. - 619029. i) q^{87} +410420. i q^{89} +722927. i q^{91} +(-513697. + 344889. i) q^{93} +92768.6i q^{95} -1.52876e6 q^{97} +(-39418.8 + 96370.4i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 72 q^{3} - 1656 q^{9}+O(q^{10}) $ 8 * q + 72 * q^3 - 1656 * q^9	$ 8 q + 72 q^{3} - 1656 q^{9} + 1648 q^{11} - 200 q^{25} + 21384 q^{27} - 22608 q^{33} + 136320 q^{35} + 105208 q^{49} + 275328 q^{51} - 391104 q^{57} + 836624 q^{59} - 1964944 q^{73} - 59400 q^{75} + 166536 q^{81} - 587024 q^{83} - 1477232 q^{97} - 1688976 q^{99}+O(q^{100}) $ 8 * q + 72 * q^3 - 1656 * q^9 + 1648 * q^11 - 200 * q^25 + 21384 * q^27 - 22608 * q^33 + 136320 * q^35 + 105208 * q^49 + 275328 * q^51 - 391104 * q^57 + 836624 * q^59 - 1964944 * q^73 - 59400 * q^75 + 166536 * q^81 - 587024 * q^83 - 1477232 * q^97 - 1688976 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	22.4164	−	15.0501i	0.830237	−	0.557410i
$3$	22.4164	−	15.0501i	0.830237	−	0.557410i
$4$		0			0
$5$	−122.733			−0.981863			−0.490931	−	0.871198i	$-0.663343\pi$
$5$	−122.733			−0.981863			−0.490931	+	0.871198i	$0.663343\pi$
$6$		0			0
$7$	206.594			0.602315			0.301157	−	0.953574i	$-0.402627\pi$
$7$	206.594			0.602315			0.301157	+	0.953574i	$0.402627\pi$
$8$		0			0
$9$	275.991	−	674.737i	0.378588	−	0.925565i
$10$		0			0
$11$	−142.827			−0.107308			−0.0536539	−	0.998560i	$-0.517087\pi$
$11$	−142.827			−0.107308			−0.0536539	+	0.998560i	$0.517087\pi$
$12$		0			0
$13$			3499.27i			1.59275i	0.604804	+	0.796374i	$0.293251\pi$
$13$			3499.27i			1.59275i	−0.604804	+	0.796374i	$0.706749\pi$
$14$		0			0
$15$	−2751.23	+	1847.14i	−0.815179	+	0.547300i
$16$		0			0
$17$		−	3382.86i		−	0.688554i	−0.938868	−	0.344277i	$-0.888124\pi$
$17$		−	3382.86i		−	0.688554i	0.938868	−	0.344277i	$-0.111876\pi$
$18$		0			0
$19$		−	755.858i		−	0.110200i	−0.998481	−	0.0550998i	$-0.982452\pi$
$19$		−	755.858i		−	0.110200i	0.998481	−	0.0550998i	$-0.0175477\pi$
$20$		0			0
$21$	4631.09	−	3109.25i	0.500064	−	0.335736i
$22$		0			0
$23$		−	16911.3i		−	1.38993i	−0.719042	−	0.694967i	$-0.755419\pi$
$23$		−	16911.3i		−	1.38993i	0.719042	−	0.694967i	$-0.244581\pi$
$24$		0			0
$25$	−561.656			−0.0359460
$26$		0			0
$27$	−3968.12	−	19278.9i	−0.201601	−	0.979468i
$28$		0			0
$29$	41131.3			1.68647			0.843235	−	0.537546i	$-0.180648\pi$
$29$	41131.3			1.68647			0.843235	+	0.537546i	$0.180648\pi$
$30$		0			0
$31$	−22916.1			−0.769229			−0.384615	−	0.923077i	$-0.625666\pi$
$31$	−22916.1			−0.769229			−0.384615	+	0.923077i	$0.625666\pi$
$32$		0			0
$33$	−3201.66	+	2149.55i	−0.0890909	+	0.0598144i
$34$		0			0
$35$	−25355.8			−0.591390
$36$		0			0
$37$			26835.0i			0.529780i	0.964279	+	0.264890i	$0.0853357\pi$
$37$			26835.0i			0.529780i	−0.964279	+	0.264890i	$0.914664\pi$
$38$		0			0
$39$	52664.2	+	78441.0i	0.887814	+	1.32236i
$40$		0			0
$41$			40460.6i			0.587057i	0.955950	+	0.293528i	$0.0948296\pi$
$41$			40460.6i			0.587057i	−0.955950	+	0.293528i	$0.905170\pi$
$42$		0			0
$43$		−	128319.i		−	1.61393i	−0.590599	−	0.806965i	$-0.701108\pi$
$43$		−	128319.i		−	1.61393i	0.590599	−	0.806965i	$-0.298892\pi$
$44$		0			0
$45$	−33873.1	+	82812.4i	−0.371721	+	0.908778i
$46$		0			0
$47$		−	60668.5i		−	0.584345i	−0.956366	−	0.292173i	$-0.905622\pi$
$47$		−	60668.5i		−	0.584345i	0.956366	−	0.292173i	$-0.0943782\pi$
$48$		0			0
$49$	−74968.0			−0.637217
$50$		0			0
$51$	−50912.4	−	75831.7i	−0.383807	−	0.571663i
$52$		0			0
$53$	−5295.60			−0.0355703			−0.0177852	−	0.999842i	$-0.505661\pi$
$53$	−5295.60			−0.0355703			−0.0177852	+	0.999842i	$0.505661\pi$
$54$		0			0
$55$	17529.5			0.105361
$56$		0			0
$57$	−11375.7	−	16943.6i	−0.0614263	−	0.0914917i
$58$		0			0
$59$	−97392.6			−0.474209			−0.237105	−	0.971484i	$-0.576198\pi$
$59$	−97392.6			−0.474209			−0.237105	+	0.971484i	$0.576198\pi$
$60$		0			0
$61$		−	213477.i		−	0.940506i	−0.882532	−	0.470253i	$-0.844163\pi$
$61$		−	213477.i		−	0.940506i	0.882532	−	0.470253i	$-0.155837\pi$
$62$		0			0
$63$	57018.0	−	139397.i	0.228029	−	0.557481i
$64$		0			0
$65$		−	429475.i		−	1.56386i
$66$		0			0
$67$		−	385320.i		−	1.28114i	−0.767898	−	0.640572i	$-0.778697\pi$
$67$		−	385320.i		−	1.28114i	0.767898	−	0.640572i	$-0.221303\pi$
$68$		0			0
$69$	−254517.	−	379091.i	−0.774763	−	1.15397i
$70$		0			0
$71$			356373.i			0.995703i	0.867262	+	0.497851i	$0.165877\pi$
$71$			356373.i			0.995703i	−0.867262	+	0.497851i	$0.834123\pi$
$72$		0			0
$73$	−141507.			−0.363754			−0.181877	−	0.983321i	$-0.558217\pi$
$73$	−141507.			−0.363754			−0.181877	+	0.983321i	$0.558217\pi$
$74$		0			0
$75$	−12590.3	+	8452.97i	−0.0298437	+	0.0200367i
$76$		0			0
$77$	−29507.1			−0.0646330
$78$		0			0
$79$	−722207.			−1.46481			−0.732403	−	0.680871i	$-0.761601\pi$
$79$	−722207.			−1.46481			−0.732403	+	0.680871i	$0.761601\pi$
$80$		0			0
$81$	−379099.	−	372442.i	−0.713342	−	0.700816i
$82$		0			0
$83$	−8040.09			−0.0140613			−0.00703067	−	0.999975i	$-0.502238\pi$
$83$	−8040.09			−0.0140613			−0.00703067	+	0.999975i	$0.502238\pi$
$84$		0			0
$85$			415188.i			0.676065i
$86$		0			0
$87$	922016.	−	619029.i	1.40017	−	0.940055i
$88$		0			0
$89$			410420.i			0.582181i	0.956695	+	0.291091i	$0.0940182\pi$
$89$			410420.i			0.582181i	−0.956695	+	0.291091i	$0.905982\pi$
$90$		0			0
$91$			722927.i			0.959335i
$92$		0			0
$93$	−513697.	+	344889.i	−0.638643	+	0.428776i
$94$		0			0
$95$			92768.6i			0.108201i
$96$		0			0
$97$	−1.52876e6			−1.67504			−0.837520	−	0.546407i	$-0.815995\pi$
$97$	−1.52876e6			−1.67504			−0.837520	+	0.546407i	$0.815995\pi$
$98$		0			0
$99$	−39418.8	+	96370.4i	−0.0406254	+	0.0993203i
$100$		0			0
$101$	175149.			0.169998			0.0849992	−	0.996381i	$-0.472911\pi$
$101$	175149.			0.169998			0.0849992	+	0.996381i	$0.472911\pi$
$102$		0			0
$103$	−1.33846e6			−1.22488			−0.612440	−	0.790517i	$-0.709812\pi$
$103$	−1.33846e6			−1.22488			−0.612440	+	0.790517i	$0.709812\pi$
$104$		0			0
$105$	−568387.	+	381607.i	−0.490994	+	0.329647i
$106$		0			0
$107$	1.06224e6			0.867103			0.433552	−	0.901129i	$-0.357260\pi$
$107$	1.06224e6			0.867103			0.433552	+	0.901129i	$0.357260\pi$
$108$		0			0
$109$		−	390541.i		−	0.301570i	−0.988567	−	0.150785i	$-0.951820\pi$
$109$		−	390541.i		−	0.301570i	0.988567	−	0.150785i	$-0.0481801\pi$
$110$		0			0
$111$	403868.	+	601544.i	0.295305	+	0.439843i
$112$		0			0
$113$		−	2.53393e6i		−	1.75614i	−0.478533	−	0.878070i	$-0.658831\pi$
$113$		−	2.53393e6i		−	1.75614i	0.478533	−	0.878070i	$-0.341169\pi$
$114$		0			0
$115$			2.07557e6i			1.36472i
$116$		0			0
$117$	2.36109e6	+	965765.i	1.47419	+	0.602995i
$118$		0			0
$119$		−	698879.i		−	0.414726i
$120$		0			0
$121$	−1.75116e6			−0.988485
$122$		0			0
$123$	608934.	+	906980.i	0.327231	+	0.487397i
$124$		0			0
$125$	1.98663e6			1.01716
$126$		0			0
$127$	2.59182e6			1.26530			0.632649	−	0.774439i	$-0.281968\pi$
$127$	2.59182e6			1.26530			0.632649	+	0.774439i	$0.281968\pi$
$128$		0			0
$129$	−1.93121e6	−	2.87645e6i	−0.899621	−	1.33995i
$130$		0			0
$131$	91522.7			0.0407113			0.0203557	−	0.999793i	$-0.493520\pi$
$131$	91522.7			0.0407113			0.0203557	+	0.999793i	$0.493520\pi$
$132$		0			0
$133$		−	156156.i		−	0.0663748i
$134$		0			0
$135$	487019.	+	2.36615e6i	0.197945	+	0.961703i
$136$		0			0
$137$		−	2.50257e6i		−	0.973249i	−0.873611	−	0.486624i	$-0.838228\pi$
$137$		−	2.50257e6i		−	0.973249i	0.873611	−	0.486624i	$-0.161772\pi$
$138$		0			0
$139$			988914.i			0.368226i	0.982905	+	0.184113i	$0.0589412\pi$
$139$			988914.i			0.368226i	−0.982905	+	0.184113i	$0.941059\pi$
$140$		0			0
$141$	−913065.	−	1.35997e6i	−0.325720	−	0.485145i
$142$		0			0
$143$		−	499789.i		−	0.170914i
$144$		0			0
$145$	−5.04816e6			−1.65588
$146$		0			0
$147$	−1.68051e6	+	1.12827e6i	−0.529042	+	0.355191i
$148$		0			0
$149$	−4.84935e6			−1.46597			−0.732985	−	0.680245i	$-0.761873\pi$
$149$	−4.84935e6			−1.46597			−0.732985	+	0.680245i	$0.761873\pi$
$150$		0			0
$151$	−3.63349e6			−1.05534			−0.527671	−	0.849449i	$-0.676935\pi$
$151$	−3.63349e6			−1.05534			−0.527671	+	0.849449i	$0.676935\pi$
$152$		0			0
$153$	−2.28254e6	−	933639.i	−0.637301	−	0.260678i
$154$		0			0
$155$	2.81256e6			0.755278
$156$		0			0
$157$			4.51450e6i			1.16657i	0.812268	+	0.583285i	$0.198233\pi$
$157$			4.51450e6i			1.16657i	−0.812268	+	0.583285i	$0.801767\pi$
$158$		0			0
$159$	−118708.	+	79699.2i	−0.0295318	+	0.0198273i
$160$		0			0
$161$		−	3.49377e6i		−	0.837177i
$162$		0			0
$163$		−	5.64963e6i		−	1.30454i	−0.757987	−	0.652269i	$-0.773817\pi$
$163$		−	5.64963e6i		−	1.30454i	0.757987	−	0.652269i	$-0.226183\pi$
$164$		0			0
$165$	392949.	−	263820.i	0.0874750	−	0.0587295i
$166$		0			0
$167$		−	4.42913e6i		−	0.950975i	−0.879723	−	0.475487i	$-0.842272\pi$
$167$		−	4.42913e6i		−	0.950975i	0.879723	−	0.475487i	$-0.157728\pi$
$168$		0			0
$169$	−7.41807e6			−1.53685
$170$		0			0
$171$	−510006.	−	208610.i	−0.101997	−	0.0417202i
$172$		0			0
$173$	−1.66599e6			−0.321762			−0.160881	−	0.986974i	$-0.551434\pi$
$173$	−1.66599e6			−0.321762			−0.160881	+	0.986974i	$0.551434\pi$
$174$		0			0
$175$	−116035.			−0.0216508
$176$		0			0
$177$	−2.18319e6	+	1.46577e6i	−0.393706	+	0.264329i
$178$		0			0
$179$	−4.85953e6			−0.847296			−0.423648	−	0.905827i	$-0.639251\pi$
$179$	−4.85953e6			−0.847296			−0.423648	+	0.905827i	$0.639251\pi$
$180$		0			0
$181$		−	8.92904e6i		−	1.50581i	−0.658132	−	0.752903i	$-0.728653\pi$
$181$		−	8.92904e6i		−	1.50581i	0.658132	−	0.752903i	$-0.271347\pi$
$182$		0			0
$183$	−3.21285e6	−	4.78539e6i	−0.524248	−	0.780843i
$184$		0			0
$185$		−	3.29353e6i		−	0.520171i
$186$		0			0
$187$			483163.i			0.0738871i
$188$		0			0
$189$	−819790.	−	3.98289e6i	−0.121427	−	0.589948i
$190$		0			0
$191$			7.88180e6i			1.13116i	0.824692	+	0.565582i	$0.191348\pi$
$191$			7.88180e6i			1.13116i	−0.824692	+	0.565582i	$0.808652\pi$
$192$		0			0
$193$	−5.97521e6			−0.831154			−0.415577	−	0.909558i	$-0.636420\pi$
$193$	−5.97521e6			−0.831154			−0.415577	+	0.909558i	$0.636420\pi$
$194$		0			0
$195$	−6.46363e6	−	9.62729e6i	−0.871711	−	1.29837i
$196$		0			0
$197$	−4.16081e6			−0.544226			−0.272113	−	0.962265i	$-0.587722\pi$
$197$	−4.16081e6			−0.544226			−0.272113	+	0.962265i	$0.587722\pi$
$198$		0			0
$199$	1.21801e7			1.54558			0.772792	−	0.634660i	$-0.218860\pi$
$199$	1.21801e7			1.54558			0.772792	+	0.634660i	$0.218860\pi$
$200$		0			0
$201$	−5.79910e6	−	8.63750e6i	−0.714122	−	1.06365i
$202$		0			0
$203$	8.49747e6			1.01578
$204$		0			0
$205$		−	4.96584e6i		−	0.576409i
$206$		0			0
$207$	−1.14107e7	−	4.66737e6i	−1.28647	−	0.526212i
$208$		0			0
$209$			107957.i			0.0118253i
$210$		0			0
$211$		−	1.27411e7i		−	1.35631i	−0.734919	−	0.678154i	$-0.762780\pi$
$211$		−	1.27411e7i		−	1.35631i	0.734919	−	0.678154i	$-0.237220\pi$
$212$		0			0
$213$	5.36344e6	+	7.98860e6i	0.555015	+	0.826669i
$214$		0			0
$215$			1.57489e7i			1.58466i
$216$		0			0
$217$	−4.73433e6			−0.463318
$218$		0			0
$219$	−3.17207e6	+	2.12969e6i	−0.302003	+	0.202760i
$220$		0			0
$221$	1.18375e7			1.09669
$222$		0			0
$223$	8.46685e6			0.763497			0.381749	−	0.924266i	$-0.375322\pi$
$223$	8.46685e6			0.763497			0.381749	+	0.924266i	$0.375322\pi$
$224$		0			0
$225$	−155012.	+	378970.i	−0.0136087	+	0.0332704i
$226$		0			0
$227$	−7.29952e6			−0.624046			−0.312023	−	0.950075i	$-0.601007\pi$
$227$	−7.29952e6			−0.624046			−0.312023	+	0.950075i	$0.601007\pi$
$228$		0			0
$229$			1.83981e7i			1.53202i	0.642826	+	0.766012i	$0.277762\pi$
$229$			1.83981e7i			1.53202i	−0.642826	+	0.766012i	$0.722238\pi$
$230$		0			0
$231$	−661443.	+	444084.i	−0.0536607	+	0.0360271i
$232$		0			0
$233$		−	2.00421e7i		−	1.58444i	−0.610237	−	0.792219i	$-0.708926\pi$
$233$		−	2.00421e7i		−	1.58444i	0.610237	−	0.792219i	$-0.291074\pi$
$234$		0			0
$235$			7.44601e6i			0.573747i
$236$		0			0
$237$	−1.61893e7	+	1.08693e7i	−1.21614	+	0.816498i
$238$		0			0
$239$		−	2.39694e7i		−	1.75575i	−0.478887	−	0.877877i	$-0.658960\pi$
$239$		−	2.39694e7i		−	1.75575i	0.478887	−	0.877877i	$-0.341040\pi$
$240$		0			0
$241$	5.88680e6			0.420560			0.210280	−	0.977641i	$-0.432562\pi$
$241$	5.88680e6			0.420560			0.210280	+	0.977641i	$0.432562\pi$
$242$		0			0
$243$	−1.41033e7	−	2.64335e6i	−0.982885	−	0.184219i
$244$		0			0
$245$	9.20103e6			0.625660
$246$		0			0
$247$	2.64495e6			0.175520
$248$		0			0
$249$	−180230.	+	121004.i	−0.0116743	+	0.00783793i
$250$		0			0
$251$	2.80782e7			1.77561			0.887807	−	0.460217i	$-0.152228\pi$
$251$	2.80782e7			1.77561			0.887807	+	0.460217i	$0.152228\pi$
$252$		0			0
$253$			2.41539e6i			0.149151i
$254$		0			0
$255$	6.24862e6	+	9.30703e6i	0.376845	+	0.561294i
$256$		0			0
$257$			1.62429e7i			0.956894i	0.878116	+	0.478447i	$0.158800\pi$
$257$			1.62429e7i			0.956894i	−0.878116	+	0.478447i	$0.841200\pi$
$258$		0			0
$259$			5.54394e6i			0.319094i
$260$		0			0
$261$	1.13519e7	−	2.77528e7i	0.638477	−	1.56094i
$262$		0			0
$263$		−	572672.i		−	0.0314803i	−0.999876	−	0.0157401i	$-0.994990\pi$
$263$		−	572672.i		−	0.0314803i	0.999876	−	0.0157401i	$-0.00501045\pi$
$264$		0			0
$265$	649944.			0.0349252
$266$		0			0
$267$	6.17685e6	+	9.20014e6i	0.324514	+	0.483349i
$268$		0			0
$269$	1.60268e7			0.823358			0.411679	−	0.911329i	$-0.364943\pi$
$269$	1.60268e7			0.823358			0.411679	+	0.911329i	$0.364943\pi$
$270$		0			0
$271$	−2.21330e7			−1.11207			−0.556035	−	0.831159i	$-0.687678\pi$
$271$	−2.21330e7			−1.11207			−0.556035	+	0.831159i	$0.687678\pi$
$272$		0			0
$273$	1.08801e7	+	1.62054e7i	0.534743	+	0.796476i
$274$		0			0
$275$	80219.5			0.00385728
$276$		0			0
$277$			329177.i			0.0154878i	0.999970	+	0.00774392i	$0.00246499\pi$
$277$			329177.i			0.0154878i	−0.999970	+	0.00774392i	$0.997535\pi$
$278$		0			0
$279$	−6.32463e6	+	1.54624e7i	−0.291221	+	0.711972i
$280$		0			0
$281$			2.42415e7i			1.09255i	0.837607	+	0.546274i	$0.183954\pi$
$281$			2.42415e7i			1.09255i	−0.837607	+	0.546274i	$0.816046\pi$
$282$		0			0
$283$			2.32930e7i			1.02770i	0.857880	+	0.513850i	$0.171781\pi$
$283$			2.32930e7i			1.02770i	−0.857880	+	0.513850i	$0.828219\pi$
$284$		0			0
$285$	1.39617e6	+	2.07954e6i	0.0603122	+	0.0898323i
$286$		0			0
$287$			8.35890e6i			0.353593i
$288$		0			0
$289$	1.26938e7			0.525894
$290$		0			0
$291$	−3.42694e7	+	2.30080e7i	−1.39068	+	0.933684i
$292$		0			0
$293$	−1.21239e7			−0.481993			−0.240996	−	0.970526i	$-0.577474\pi$
$293$	−1.21239e7			−0.481993			−0.240996	+	0.970526i	$0.577474\pi$
$294$		0			0
$295$	1.19533e7			0.465608
$296$		0			0
$297$	566753.	+	2.75353e6i	0.0216334	+	0.105104i
$298$		0			0
$299$	5.91772e7			2.21381
$300$		0			0
$301$		−	2.65099e7i		−	0.972094i
$302$		0			0
$303$	3.92622e6	−	2.63601e6i	0.141139	−	0.0947588i
$304$		0			0
$305$			2.62006e7i			0.923448i
$306$		0			0
$307$		−	1.04782e7i		−	0.362136i	−0.983471	−	0.181068i	$-0.942045\pi$
$307$		−	1.04782e7i		−	0.362136i	0.983471	−	0.181068i	$-0.0579555\pi$
$308$		0			0
$309$	−3.00035e7	+	2.01439e7i	−1.01694	+	0.682761i
$310$		0			0
$311$		−	5.87497e7i		−	1.95310i	−0.215292	−	0.976550i	$-0.569070\pi$
$311$		−	5.87497e7i		−	1.95310i	0.215292	−	0.976550i	$-0.430930\pi$
$312$		0			0
$313$	5.12380e7			1.67093			0.835467	−	0.549540i	$-0.185197\pi$
$313$	5.12380e7			1.67093			0.835467	+	0.549540i	$0.185197\pi$
$314$		0			0
$315$	−6.99798e6	+	1.71085e7i	−0.223893	+	0.547370i
$316$		0			0
$317$	−5.13280e7			−1.61130			−0.805651	−	0.592390i	$-0.798184\pi$
$317$	−5.13280e7			−1.61130			−0.805651	+	0.592390i	$0.798184\pi$
$318$		0			0
$319$	−5.87464e6			−0.180971
$320$		0			0
$321$	2.38116e7	−	1.59868e7i	0.719901	−	0.483332i
$322$		0			0
$323$	−2.55697e6			−0.0758783
$324$		0			0
$325$		−	1.96539e6i		−	0.0572529i
$326$		0			0
$327$	−5.87768e6	−	8.75454e6i	−0.168098	−	0.250374i
$328$		0			0
$329$		−	1.25337e7i		−	0.351960i
$330$		0			0
$331$			5.26211e7i			1.45103i	0.688207	+	0.725515i	$0.258398\pi$
$331$			5.26211e7i			1.45103i	−0.688207	+	0.725515i	$0.741602\pi$
$332$		0			0
$333$	1.81065e7	+	7.40620e6i	0.490346	+	0.200569i
$334$		0			0
$335$			4.72915e7i			1.25791i
$336$		0			0
$337$	3.13406e7			0.818876			0.409438	−	0.912338i	$-0.365725\pi$
$337$	3.13406e7			0.818876			0.409438	+	0.912338i	$0.365725\pi$
$338$		0			0
$339$	−3.81358e7	−	5.68016e7i	−0.978890	−	1.45801i
$340$		0			0
$341$	3.27303e6			0.0825443
$342$		0			0
$343$	−3.97935e7			−0.986120
$344$		0			0
$345$	3.12375e7	+	4.65269e7i	0.760710	+	1.13304i
$346$		0			0
$347$	5.67725e7			1.35878			0.679391	−	0.733777i	$-0.262244\pi$
$347$	5.67725e7			1.35878			0.679391	+	0.733777i	$0.262244\pi$
$348$		0			0
$349$			9.45512e6i			0.222429i	0.993796	+	0.111214i	$0.0354740\pi$
$349$			9.45512e6i			0.222429i	−0.993796	+	0.111214i	$0.964526\pi$
$350$		0			0
$351$	6.74619e7	−	1.38855e7i	1.56005	−	0.321100i
$352$		0			0
$353$		−	1.84722e7i		−	0.419948i	−0.977707	−	0.209974i	$-0.932662\pi$
$353$		−	1.84722e7i		−	0.419948i	0.977707	−	0.209974i	$-0.0673379\pi$
$354$		0			0
$355$		−	4.37386e7i		−	0.977643i
$356$		0			0
$357$	−1.05182e7	−	1.56664e7i	−0.231172	−	0.344321i
$358$		0			0
$359$			8.02076e7i			1.73353i	0.498714	+	0.866767i	$0.333806\pi$
$359$			8.02076e7i			1.73353i	−0.498714	+	0.866767i	$0.666194\pi$
$360$		0			0
$361$	4.64746e7			0.987856
$362$		0			0
$363$	−3.92548e7	+	2.63551e7i	−0.820677	+	0.550992i
$364$		0			0
$365$	1.73675e7			0.357157
$366$		0			0
$367$	6.88440e6			0.139273			0.0696366	−	0.997572i	$-0.477816\pi$
$367$	6.88440e6			0.139273			0.0696366	+	0.997572i	$0.477816\pi$
$368$		0			0
$369$	2.73002e7	+	1.11667e7i	0.543360	+	0.222253i
$370$		0			0
$371$	−1.09404e6			−0.0214245
$372$		0			0
$373$		−	4.53815e7i		−	0.874485i	−0.899344	−	0.437243i	$-0.855955\pi$
$373$		−	4.53815e7i		−	0.874485i	0.899344	−	0.437243i	$-0.144045\pi$
$374$		0			0
$375$	4.45332e7	−	2.98990e7i	0.844481	−	0.566973i
$376$		0			0
$377$			1.43929e8i			2.68612i
$378$		0			0
$379$		−	1.51053e7i		−	0.277467i	−0.990330	−	0.138734i	$-0.955697\pi$
$379$		−	1.51053e7i		−	0.277467i	0.990330	−	0.138734i	$-0.0443032\pi$
$380$		0			0
$381$	5.80992e7	−	3.90070e7i	1.05050	−	0.705290i
$382$		0			0
$383$		−	2.37036e7i		−	0.421908i	−0.977496	−	0.210954i	$-0.932343\pi$
$383$		−	2.37036e7i		−	0.421908i	0.977496	−	0.210954i	$-0.0676570\pi$
$384$		0			0
$385$	3.62149e6			0.0634607
$386$		0			0
$387$	−8.65814e7	−	3.54148e7i	−1.49380	−	0.611015i
$388$		0			0
$389$	3.21550e7			0.546261			0.273130	−	0.961977i	$-0.411941\pi$
$389$	3.21550e7			0.546261			0.273130	+	0.961977i	$0.411941\pi$
$390$		0			0
$391$	−5.72087e7			−0.957044
$392$		0			0
$393$	2.05161e6	−	1.37742e6i	0.0338000	−	0.0226929i
$394$		0			0
$395$	8.86385e7			1.43824
$396$		0			0
$397$			1.01939e8i			1.62918i	0.580035	+	0.814591i	$0.303039\pi$
$397$			1.01939e8i			1.62918i	−0.580035	+	0.814591i	$0.696961\pi$
$398$		0			0
$399$	−2.35015e6	−	3.50045e6i	−0.0369980	−	0.0551068i
$400$		0			0
$401$			2.78698e7i			0.432215i	0.976370	+	0.216108i	$0.0693362\pi$
$401$			2.78698e7i			0.432215i	−0.976370	+	0.216108i	$0.930664\pi$
$402$		0			0
$403$		−	8.01896e7i		−	1.22519i
$404$		0			0
$405$	4.65279e7	+	4.57109e7i	0.700404	+	0.688105i
$406$		0			0
$407$		−	3.83275e6i		−	0.0568495i
$408$		0			0
$409$	1.07187e8			1.56665			0.783326	−	0.621611i	$-0.213522\pi$
$409$	1.07187e8			1.56665			0.783326	+	0.621611i	$0.213522\pi$
$410$		0			0
$411$	−3.76638e7	−	5.60986e7i	−0.542499	−	0.808028i
$412$		0			0
$413$	−2.01207e7			−0.285623
$414$		0			0
$415$	986783.			0.0138063
$416$		0			0
$417$	1.48832e7	+	2.21679e7i	0.205253	+	0.305715i
$418$		0			0
$419$	6.69022e6			0.0909491			0.0454745	−	0.998965i	$-0.485520\pi$
$419$	6.69022e6			0.0909491			0.0454745	+	0.998965i	$0.485520\pi$
$420$		0			0
$421$			5.45684e7i			0.731299i	0.930753	+	0.365649i	$0.119153\pi$
$421$			5.45684e7i			0.731299i	−0.930753	+	0.365649i	$0.880847\pi$
$422$		0			0
$423$	−4.09353e7	−	1.67439e7i	−0.540850	−	0.221226i
$424$		0			0
$425$			1.90001e6i			0.0247508i
$426$		0			0
$427$		−	4.41031e7i		−	0.566481i
$428$		0			0
$429$	−7.52185e6	−	1.12035e7i	−0.0952693	−	0.141899i
$430$		0			0
$431$		−	4.58145e7i		−	0.572231i	−0.958195	−	0.286115i	$-0.907636\pi$
$431$		−	4.58145e7i		−	0.572231i	0.958195	−	0.286115i	$-0.0923641\pi$
$432$		0			0
$433$	−6.27032e6			−0.0772372			−0.0386186	−	0.999254i	$-0.512296\pi$
$433$	−6.27032e6			−0.0772372			−0.0386186	+	0.999254i	$0.512296\pi$
$434$		0			0
$435$	−1.13162e8	+	7.59752e7i	−1.37477	+	0.923005i
$436$		0			0
$437$	−1.27826e7			−0.153170
$438$		0			0
$439$	4.02491e7			0.475732			0.237866	−	0.971298i	$-0.423552\pi$
$439$	4.02491e7			0.475732			0.237866	+	0.971298i	$0.423552\pi$
$440$		0			0
$441$	−2.06905e7	+	5.05837e7i	−0.241243	+	0.589786i
$442$		0			0
$443$	7.03746e7			0.809477			0.404739	−	0.914432i	$-0.367363\pi$
$443$	7.03746e7			0.809477			0.404739	+	0.914432i	$0.367363\pi$
$444$		0			0
$445$		−	5.03720e7i		−	0.571622i
$446$		0			0
$447$	−1.08705e8	+	7.29831e7i	−1.21710	+	0.817146i
$448$		0			0
$449$		−	1.22328e8i		−	1.35141i	−0.737170	−	0.675707i	$-0.763838\pi$
$449$		−	1.22328e8i		−	1.35141i	0.737170	−	0.675707i	$-0.236162\pi$
$450$		0			0
$451$		−	5.77884e6i		−	0.0629958i
$452$		0			0
$453$	−8.14498e7	+	5.46843e7i	−0.876184	+	0.588258i
$454$		0			0
$455$		−	8.87269e7i		−	0.941935i
$456$		0			0
$457$	−1.10853e6			−0.0116144			−0.00580722	−	0.999983i	$-0.501849\pi$
$457$	−1.10853e6			−0.0116144			−0.00580722	+	0.999983i	$0.501849\pi$
$458$		0			0
$459$	−6.52178e7	+	1.34236e7i	−0.674416	+	0.138813i
$460$		0			0
$461$	−4.08136e7			−0.416583			−0.208292	−	0.978067i	$-0.566790\pi$
$461$	−4.08136e7			−0.416583			−0.208292	+	0.978067i	$0.566790\pi$
$462$		0			0
$463$	−3.22650e7			−0.325079			−0.162540	−	0.986702i	$-0.551969\pi$
$463$	−3.22650e7			−0.325079			−0.162540	+	0.986702i	$0.551969\pi$
$464$		0			0
$465$	6.30475e7	−	4.23292e7i	0.627060	−	0.420999i
$466$		0			0
$467$	9.16957e7			0.900323			0.450162	−	0.892947i	$-0.351366\pi$
$467$	9.16957e7			0.900323			0.450162	+	0.892947i	$0.351366\pi$
$468$		0			0
$469$		−	7.96049e7i		−	0.771651i
$470$		0			0
$471$	6.79435e7	+	1.01199e8i	0.650257	+	0.968529i
$472$		0			0
$473$			1.83273e7i			0.173187i
$474$		0			0
$475$			424533.i			0.00396123i
$476$		0			0
$477$	−1.46154e6	+	3.57314e6i	−0.0134665	+	0.0329227i
$478$		0			0
$479$		−	3.72123e7i		−	0.338594i	−0.985565	−	0.169297i	$-0.945850\pi$
$479$		−	3.72123e7i		−	0.338594i	0.985565	−	0.169297i	$-0.0541498\pi$
$480$		0			0
$481$	−9.39027e7			−0.843807
$482$		0			0
$483$	−5.25816e7	−	7.83179e7i	−0.466651	−	0.695056i
$484$		0			0
$485$	1.87629e8			1.64466
$486$		0			0
$487$	1.89081e8			1.63705			0.818525	−	0.574471i	$-0.194792\pi$
$487$	1.89081e8			1.63705			0.818525	+	0.574471i	$0.194792\pi$
$488$		0			0
$489$	−8.50273e7	−	1.26644e8i	−0.727163	−	1.08308i
$490$		0			0
$491$	1.81333e8			1.53190			0.765952	−	0.642898i	$-0.222268\pi$
$491$	1.81333e8			1.53190			0.765952	+	0.642898i	$0.222268\pi$
$492$		0			0
$493$		−	1.39142e8i		−	1.16122i
$494$		0			0
$495$	4.83798e6	−	1.18278e7i	0.0398886	−	0.0975189i
$496$		0			0
$497$			7.36245e7i			0.599726i
$498$		0			0
$499$			1.73596e8i			1.39713i	0.715546	+	0.698566i	$0.246178\pi$
$499$			1.73596e8i			1.39713i	−0.715546	+	0.698566i	$0.753822\pi$
$500$		0			0
$501$	−6.66587e7	−	9.92852e7i	−0.530083	−	0.789535i
$502$		0			0
$503$			8.79058e7i			0.690739i	0.938467	+	0.345369i	$0.112246\pi$
$503$			8.79058e7i			0.690739i	−0.938467	+	0.345369i	$0.887754\pi$
$504$		0			0
$505$	−2.14966e7			−0.166915
$506$		0			0
$507$	−1.66286e8	+	1.11642e8i	−1.27595	+	0.856654i
$508$		0			0
$509$	−1.69584e8			−1.28597			−0.642987	−	0.765877i	$-0.722305\pi$
$509$	−1.69584e8			−1.28597			−0.642987	+	0.765877i	$0.722305\pi$
$510$		0			0
$511$	−2.92344e7			−0.219095
$512$		0			0
$513$	−1.45721e7	+	2.99934e6i	−0.107937	+	0.0222164i
$514$		0			0
$515$	1.64273e8			1.20266
$516$		0			0
$517$			8.66507e6i			0.0627048i
$518$		0			0
$519$	−3.73456e7	+	2.50733e7i	−0.267139	+	0.179354i
$520$		0			0
$521$		−	1.96877e8i		−	1.39213i	−0.717977	−	0.696067i	$-0.754932\pi$
$521$		−	1.96877e8i		−	1.39213i	0.717977	−	0.696067i	$-0.245068\pi$
$522$		0			0
$523$			1.70811e8i			1.19402i	0.802235	+	0.597008i	$0.203644\pi$
$523$			1.70811e8i			1.19402i	−0.802235	+	0.597008i	$0.796356\pi$
$524$		0			0
$525$	−2.60108e6	+	1.74633e6i	−0.0179753	+	0.0120684i
$526$		0			0
$527$			7.75221e7i			0.529656i
$528$		0			0
$529$	−1.37957e8			−0.931914
$530$		0			0
$531$	−2.68795e7	+	6.57144e7i	−0.179530	+	0.438912i
$532$		0			0
$533$	−1.41582e8			−0.935034
$534$		0			0
$535$	−1.30372e8			−0.851376
$536$		0			0
$537$	−1.08933e8	+	7.31363e7i	−0.703457	+	0.472291i
$538$		0			0
$539$	1.07074e7			0.0683783
$540$		0			0
$541$			1.09184e8i			0.689555i	0.938684	+	0.344778i	$0.112046\pi$
$541$			1.09184e8i			0.689555i	−0.938684	+	0.344778i	$0.887954\pi$
$542$		0			0
$543$	−1.34383e8	−	2.00157e8i	−0.839351	−	1.25018i
$544$		0			0
$545$			4.79323e7i			0.296100i
$546$		0			0
$547$		−	2.10962e7i		−	0.128897i	−0.997921	−	0.0644483i	$-0.979471\pi$
$547$		−	2.10962e7i		−	0.128897i	0.997921	−	0.0644483i	$-0.0205287\pi$
$548$		0			0
$549$	−1.44041e8	−	5.89177e7i	−0.870500	−	0.356064i
$550$		0			0
$551$		−	3.10894e7i		−	0.185848i
$552$		0			0
$553$	−1.49203e8			−0.882274
$554$		0			0
$555$	−4.95679e7	−	7.38291e7i	−0.289949	−	0.431866i
$556$		0			0
$557$	2.31350e8			1.33876			0.669381	−	0.742919i	$-0.266559\pi$
$557$	2.31350e8			1.33876			0.669381	+	0.742919i	$0.266559\pi$
$558$		0			0
$559$	4.49022e8			2.57058
$560$		0			0
$561$	7.27164e6	+	1.08308e7i	0.0411854	+	0.0613439i
$562$		0			0
$563$	2.49317e8			1.39709			0.698547	−	0.715564i	$-0.253830\pi$
$563$	2.49317e8			1.39709			0.698547	+	0.715564i	$0.253830\pi$
$564$		0			0
$565$			3.10996e8i			1.72429i
$566$		0			0
$567$	−7.83196e7	−	7.69443e7i	−0.429656	−	0.422112i
$568$		0			0
$569$		−	1.88477e8i		−	1.02311i	−0.859252	−	0.511553i	$-0.829070\pi$
$569$		−	1.88477e8i		−	1.02311i	0.859252	−	0.511553i	$-0.170930\pi$
$570$		0			0
$571$			6.38897e7i			0.343181i	0.985168	+	0.171590i	$0.0548905\pi$
$571$			6.38897e7i			0.343181i	−0.985168	+	0.171590i	$0.945109\pi$
$572$		0			0
$573$	1.18622e8	+	1.76682e8i	0.630522	+	0.939134i
$574$		0			0
$575$			9.49835e6i			0.0499625i
$576$		0			0
$577$	−2.64150e8			−1.37506			−0.687531	−	0.726155i	$-0.741306\pi$
$577$	−2.64150e8			−1.37506			−0.687531	+	0.726155i	$0.741306\pi$
$578$		0			0
$579$	−1.33943e8	+	8.99274e7i	−0.690055	+	0.463294i
$580$		0			0
$581$	−1.66103e6			−0.00846935
$582$		0			0
$583$	756353.			0.00381697
$584$		0			0
$585$	−2.89783e8	−	1.18531e8i	−1.44745	−	0.592059i
$586$		0			0
$587$	6.84017e7			0.338184			0.169092	−	0.985600i	$-0.445917\pi$
$587$	6.84017e7			0.338184			0.169092	+	0.985600i	$0.445917\pi$
$588$		0			0
$589$			1.73213e7i			0.0847687i
$590$		0			0
$591$	−9.32704e7	+	6.26205e7i	−0.451837	+	0.303357i
$592$		0			0
$593$		−	2.30466e7i		−	0.110521i	−0.998472	−	0.0552604i	$-0.982401\pi$
$593$		−	2.30466e7i		−	0.110521i	0.998472	−	0.0552604i	$-0.0175989\pi$
$594$		0			0
$595$			8.57754e7i			0.407204i
$596$		0			0
$597$	2.73035e8	−	1.83312e8i	1.28320	−	0.861524i
$598$		0			0
$599$			2.25012e8i			1.04695i	0.852042	+	0.523474i	$0.175364\pi$
$599$			2.25012e8i			1.04695i	−0.852042	+	0.523474i	$0.824636\pi$
$600$		0			0
$601$	−8.08740e7			−0.372551			−0.186275	−	0.982498i	$-0.559642\pi$
$601$	−8.08740e7			−0.372551			−0.186275	+	0.982498i	$0.559642\pi$
$602$		0			0
$603$	−2.59990e8	−	1.06345e8i	−1.18578	−	0.485026i
$604$		0			0
$605$	2.14925e8			0.970556
$606$		0			0
$607$	−2.86741e8			−1.28211			−0.641054	−	0.767496i	$-0.721502\pi$
$607$	−2.86741e8			−1.28211			−0.641054	+	0.767496i	$0.721502\pi$
$608$		0			0
$609$	1.90483e8	−	1.27888e8i	0.843343	−	0.566209i
$610$		0			0
$611$	2.12295e8			0.930715
$612$		0			0
$613$		−	3.51391e8i		−	1.52549i	−0.646699	−	0.762745i	$-0.723851\pi$
$613$		−	3.51391e8i		−	1.52549i	0.646699	−	0.762745i	$-0.276149\pi$
$614$		0			0
$615$	−7.47362e7	−	1.11316e8i	−0.321296	−	0.478556i
$616$		0			0
$617$		−	3.46355e8i		−	1.47457i	−0.675580	−	0.737287i	$-0.736107\pi$
$617$		−	3.46355e8i		−	1.47457i	0.675580	−	0.737287i	$-0.263893\pi$
$618$		0			0
$619$		−	3.14225e8i		−	1.32486i	−0.749126	−	0.662428i	$-0.769526\pi$
$619$		−	3.14225e8i		−	1.32486i	0.749126	−	0.662428i	$-0.230474\pi$
$620$		0			0
$621$	−3.26031e8	+	6.71062e7i	−1.36139	+	0.280213i
$622$		0			0
$623$			8.47902e7i			0.350656i
$624$		0			0
$625$	−2.35049e8			−0.962762
$626$		0			0
$627$	1.62476e6	+	2.42000e6i	0.00659152	+	0.00981777i
$628$		0			0
$629$	9.07791e7			0.364782
$630$		0			0
$631$	−5.41176e7			−0.215402			−0.107701	−	0.994183i	$-0.534349\pi$
$631$	−5.41176e7			−0.215402			−0.107701	+	0.994183i	$0.534349\pi$
$632$		0			0
$633$	−1.91754e8	−	2.85609e8i	−0.756020	−	1.12606i
$634$		0			0
$635$	−3.18101e8			−1.24235
$636$		0			0
$637$		−	2.62333e8i		−	1.01493i
$638$		0			0
$639$	2.40458e8	+	9.83556e7i	0.921588	+	0.376961i
$640$		0			0
$641$		−	1.42947e8i		−	0.542751i	−0.962474	−	0.271375i	$-0.912522\pi$
$641$		−	1.42947e8i		−	0.542751i	0.962474	−	0.271375i	$-0.0874785\pi$
$642$		0			0
$643$		−	5.21812e7i		−	0.196282i	−0.995173	−	0.0981411i	$-0.968710\pi$
$643$		−	5.21812e7i		−	0.196282i	0.995173	−	0.0981411i	$-0.0312897\pi$
$644$		0			0
$645$	2.37022e8	+	3.53034e8i	0.883304	+	1.31564i
$646$		0			0
$647$		−	4.37963e8i		−	1.61705i	−0.588460	−	0.808526i	$-0.700266\pi$
$647$		−	4.37963e8i		−	1.61705i	0.588460	−	0.808526i	$-0.299734\pi$
$648$		0			0
$649$	1.39103e7			0.0508863
$650$		0			0
$651$	−1.06127e8	+	7.12520e7i	−0.384664	+	0.258258i
$652$		0			0
$653$	2.41321e7			0.0866674			0.0433337	−	0.999061i	$-0.486202\pi$
$653$	2.41321e7			0.0866674			0.0433337	+	0.999061i	$0.486202\pi$
$654$		0			0
$655$	−1.12328e7			−0.0399729
$656$		0			0
$657$	−3.90545e7	+	9.54798e7i	−0.137713	+	0.336679i
$658$		0			0
$659$	2.76788e8			0.967142			0.483571	−	0.875305i	$-0.339339\pi$
$659$	2.76788e8			0.967142			0.483571	+	0.875305i	$0.339339\pi$
$660$		0			0
$661$			3.29605e8i			1.14127i	0.821203	+	0.570636i	$0.193303\pi$
$661$			3.29605e8i			1.14127i	−0.821203	+	0.570636i	$0.806697\pi$
$662$		0			0
$663$	2.65355e8	−	1.78156e8i	0.910515	−	0.611308i
$664$		0			0
$665$			1.91654e7i			0.0651709i
$666$		0			0
$667$		−	6.95584e8i		−	2.34408i
$668$		0			0
$669$	1.89796e8	−	1.27427e8i	0.633884	−	0.425581i
$670$		0			0
$671$			3.04902e7i			0.100924i
$672$		0			0
$673$	−2.21755e8			−0.727493			−0.363746	−	0.931498i	$-0.618503\pi$
$673$	−2.21755e8			−0.727493			−0.363746	+	0.931498i	$0.618503\pi$
$674$		0			0
$675$	2.22872e6	+	1.08281e7i	0.00724677	+	0.0352079i
$676$		0			0
$677$	−5.76656e8			−1.85845			−0.929225	−	0.369513i	$-0.879524\pi$
$677$	−5.76656e8			−1.85845			−0.929225	+	0.369513i	$0.879524\pi$
$678$		0			0
$679$	−3.15833e8			−1.00890
$680$		0			0
$681$	−1.63629e8	+	1.09858e8i	−0.518106	+	0.347850i
$682$		0			0
$683$	−8.03596e7			−0.252218			−0.126109	−	0.992016i	$-0.540249\pi$
$683$	−8.03596e7			−0.252218			−0.126109	+	0.992016i	$0.540249\pi$
$684$		0			0
$685$			3.07147e8i			0.955597i
$686$		0			0
$687$	2.76892e8	+	4.12418e8i	0.853965	+	1.27194i
$688$		0			0
$689$		−	1.85307e7i		−	0.0566546i
$690$		0			0
$691$			2.40157e8i			0.727884i	0.931422	+	0.363942i	$0.118569\pi$
$691$			2.40157e8i			0.727884i	−0.931422	+	0.363942i	$0.881431\pi$
$692$		0			0
$693$	−8.14369e6	+	1.99095e7i	−0.0244693	+	0.0598221i
$694$		0			0
$695$		−	1.21372e8i		−	0.361547i
$696$		0			0
$697$	1.36873e8			0.404220
$698$		0			0
$699$	−3.01635e8	−	4.49271e8i	−0.883181	−	1.31546i
$700$		0			0
$701$	−4.71751e8			−1.36949			−0.684744	−	0.728784i	$-0.740086\pi$
$701$	−4.71751e8			−1.36949			−0.684744	+	0.728784i	$0.740086\pi$
$702$		0			0
$703$	2.02834e7			0.0583815
$704$		0			0
$705$	1.12063e8	+	1.66913e8i	0.319812	+	0.476346i
$706$		0			0
$707$	3.61848e7			0.102392
$708$		0			0
$709$		−	3.03570e8i		−	0.851765i	−0.904779	−	0.425882i	$-0.859964\pi$
$709$		−	3.03570e8i		−	0.851765i	0.904779	−	0.425882i	$-0.140036\pi$
$710$		0			0
$711$	−1.99322e8	+	4.87300e8i	−0.554558	+	1.35577i
$712$		0			0
$713$			3.87542e8i			1.06918i
$714$		0			0
$715$			6.13405e7i			0.167814i
$716$		0			0
$717$	−3.60741e8	−	5.37308e8i	−0.978675	−	1.45769i
$718$		0			0
$719$		−	2.70537e8i		−	0.727846i	−0.931429	−	0.363923i	$-0.881437\pi$
$719$		−	2.70537e8i		−	0.727846i	0.931429	−	0.363923i	$-0.118563\pi$
$720$		0			0
$721$	−2.76518e8			−0.737763
$722$		0			0
$723$	1.31961e8	−	8.85967e7i	0.349164	−	0.234424i
$724$		0			0
$725$	−2.31017e7			−0.0606218
$726$		0			0
$727$	3.96563e8			1.03207			0.516035	−	0.856567i	$-0.327407\pi$
$727$	3.96563e8			1.03207			0.516035	+	0.856567i	$0.327407\pi$
$728$		0			0
$729$	−3.55929e8	+	1.53002e8i	−0.918714	+	0.394924i
$730$		0			0
$731$	−4.34085e8			−1.11128
$732$		0			0
$733$			1.12080e8i			0.284589i	0.989824	+	0.142295i	$0.0454480\pi$
$733$			1.12080e8i			0.284589i	−0.989824	+	0.142295i	$0.954552\pi$
$734$		0			0
$735$	2.06254e8	−	1.38476e8i	0.519446	−	0.348749i
$736$		0			0
$737$			5.50340e7i			0.137477i
$738$		0			0
$739$			2.33774e8i			0.579245i	0.957141	+	0.289622i	$0.0935297\pi$
$739$			2.33774e8i			0.579245i	−0.957141	+	0.289622i	$0.906470\pi$
$740$		0			0
$741$	5.92903e7	−	3.98067e7i	0.145723	−	0.0978367i
$742$		0			0
$743$			6.99167e8i			1.70457i	0.523078	+	0.852285i	$0.324783\pi$
$743$			6.99167e8i			1.70457i	−0.523078	+	0.852285i	$0.675217\pi$
$744$		0			0
$745$	5.95175e8			1.43938
$746$		0			0
$747$	−2.21899e6	+	5.42495e6i	−0.00532346	+	0.0130147i
$748$		0			0
$749$	2.19452e8			0.522269
$750$		0			0
$751$	−5.11140e8			−1.20676			−0.603379	−	0.797455i	$-0.706179\pi$
$751$	−5.11140e8			−1.20676			−0.603379	+	0.797455i	$0.706179\pi$
$752$		0			0
$753$	6.29413e8	−	4.22579e8i	1.47418	−	0.989745i
$754$		0			0
$755$	4.45948e8			1.03620
$756$		0			0
$757$		−	5.13325e8i		−	1.18333i	−0.806185	−	0.591664i	$-0.798471\pi$
$757$		−	5.13325e8i		−	1.18333i	0.806185	−	0.591664i	$-0.201529\pi$
$758$		0			0
$759$	3.63517e7	+	5.41443e7i	0.0831380	+	0.123830i
$760$		0			0
$761$			9.53837e7i			0.216431i	0.994127	+	0.108216i	$0.0345137\pi$
$761$			9.53837e7i			0.216431i	−0.994127	+	0.108216i	$0.965486\pi$
$762$		0			0
$763$		−	8.06835e7i		−	0.181640i
$764$		0			0
$765$	2.80143e8	+	1.14588e8i	0.625742	+	0.255950i
$766$		0			0
$767$		−	3.40803e8i		−	0.755296i
$768$		0			0
$769$	−1.74508e8			−0.383740			−0.191870	−	0.981420i	$-0.561455\pi$
$769$	−1.74508e8			−0.383740			−0.191870	+	0.981420i	$0.561455\pi$
$770$		0			0
$771$	2.44457e8	+	3.64107e8i	0.533382	+	0.794449i
$772$		0			0
$773$	4.01318e8			0.868860			0.434430	−	0.900706i	$-0.356950\pi$
$773$	4.01318e8			0.868860			0.434430	+	0.900706i	$0.356950\pi$
$774$		0			0
$775$	1.28710e7			0.0276507
$776$		0			0
$777$	8.34367e7	+	1.24275e8i	0.177866	+	0.264924i
$778$		0			0
$779$	3.05825e7			0.0646934
$780$		0			0
$781$		−	5.08995e7i		−	0.106847i
$782$		0			0
$783$	−1.63214e8	−	7.92965e8i	−0.339995	−	1.65184i
$784$		0			0
$785$		−	5.54077e8i		−	1.14541i
$786$		0			0
$787$		−	7.87876e7i		−	0.161634i	−0.996729	−	0.0808172i	$-0.974247\pi$
$787$		−	7.87876e7i		−	0.161634i	0.996729	−	0.0808172i	$-0.0257530\pi$
$788$		0			0
$789$	−8.61875e6	−	1.28372e7i	−0.0175474	−	0.0261361i
$790$		0			0
$791$		−	5.23494e8i		−	1.05775i
$792$		0			0
$793$	7.47013e8			1.49799
$794$		0			0
$795$	1.45694e7	−	9.78171e6i	0.0289962	−	0.0194676i
$796$		0			0
$797$	1.55228e8			0.306616			0.153308	−	0.988178i	$-0.451007\pi$
$797$	1.55228e8			0.306616			0.153308	+	0.988178i	$0.451007\pi$
$798$		0			0
$799$	−2.05233e8			−0.402353
$800$		0			0
$801$	2.76926e8	+	1.13272e8i	0.538847	+	0.220407i
$802$		0			0
$803$	2.02109e7			0.0390337
$804$		0			0
$805$			4.28801e8i			0.821993i
$806$		0			0
$807$	3.59262e8	−	2.41204e8i	0.683582	−	0.458948i
$808$		0			0
$809$			6.91494e8i			1.30600i	0.757358	+	0.653000i	$0.226490\pi$
$809$			6.91494e8i			1.30600i	−0.757358	+	0.653000i	$0.773510\pi$
$810$		0			0
$811$			9.19813e8i			1.72440i	0.506572	+	0.862198i	$0.330912\pi$
$811$			9.19813e8i			1.72440i	−0.506572	+	0.862198i	$0.669088\pi$
$812$		0			0
$813$	−4.96142e8	+	3.33103e8i	−0.923281	+	0.619879i
$814$		0			0
$815$			6.93394e8i			1.28088i
$816$		0			0
$817$	−9.69908e7			−0.177854
$818$		0			0
$819$	4.87786e8	+	1.99521e8i	0.887928	+	0.363193i
$820$		0			0
$821$	−3.54626e8			−0.640827			−0.320414	−	0.947278i	$-0.603822\pi$
$821$	−3.54626e8			−0.640827			−0.320414	+	0.947278i	$0.603822\pi$
$822$		0			0
$823$	−6.29337e8			−1.12897			−0.564487	−	0.825442i	$-0.690926\pi$
$823$	−6.29337e8			−1.12897			−0.564487	+	0.825442i	$0.690926\pi$
$824$		0			0
$825$	1.79823e6	−	1.20731e6i	0.00320246	−	0.00215009i
$826$		0			0
$827$	−8.77055e8			−1.55064			−0.775318	−	0.631571i	$-0.782411\pi$
$827$	−8.77055e8			−1.55064			−0.775318	+	0.631571i	$0.782411\pi$
$828$		0			0
$829$			2.31492e8i			0.406323i	0.979145	+	0.203162i	$0.0651217\pi$
$829$			2.31492e8i			0.406323i	−0.979145	+	0.203162i	$0.934878\pi$
$830$		0			0
$831$	4.95414e6	+	7.37898e6i	0.00863308	+	0.0128586i
$832$		0			0
$833$			2.53606e8i			0.438758i
$834$		0			0
$835$			5.43600e8i			0.933726i
$836$		0			0
$837$	9.09339e7	+	4.41797e8i	0.155078	+	0.753435i
$838$		0			0
$839$			2.86832e8i			0.485671i	0.970068	+	0.242835i	$0.0780775\pi$
$839$			2.86832e8i			0.485671i	−0.970068	+	0.242835i	$0.921923\pi$
$840$		0			0
$841$	1.09696e9			1.84418
$842$		0			0
$843$	3.64836e8	+	5.43407e8i	0.608997	+	0.907073i
$844$		0			0
$845$	9.10440e8			1.50897
$846$		0			0
$847$	−3.61779e8			−0.595379
$848$		0			0
$849$	3.50561e8	+	5.22145e8i	0.572850	+	0.853234i
$850$		0			0
$851$	4.53815e8			0.736359
$852$		0			0
$853$			3.87271e8i			0.623976i	0.950086	+	0.311988i	$0.100995\pi$
$853$			3.87271e8i			0.623976i	−0.950086	+	0.311988i	$0.899005\pi$
$854$		0			0
$855$	6.25944e7	+	2.56033e7i	0.100147	+	0.0409635i
$856$		0			0
$857$			1.83015e8i			0.290767i	0.989375	+	0.145383i	$0.0464415\pi$
$857$			1.83015e8i			0.290767i	−0.989375	+	0.145383i	$0.953558\pi$
$858$		0			0
$859$		−	6.26953e8i		−	0.989135i	−0.869139	−	0.494567i	$-0.835327\pi$
$859$		−	6.26953e8i		−	0.989135i	0.869139	−	0.494567i	$-0.164673\pi$
$860$		0			0
$861$	1.25802e8	+	1.87377e8i	0.197096	+	0.293566i
$862$		0			0
$863$			6.24437e8i			0.971530i	0.874089	+	0.485765i	$0.161459\pi$
$863$			6.24437e8i			0.971530i	−0.874089	+	0.485765i	$0.838541\pi$
$864$		0			0
$865$	2.04472e8			0.315926
$866$		0			0
$867$	2.84549e8	−	1.91043e8i	0.436617	−	0.293138i
$868$		0			0
$869$	1.03150e8			0.157185
$870$		0			0
$871$	1.34834e9			2.04054
$872$		0			0
$873$	−4.21924e8	+	1.03151e9i	−0.634150	+	1.55036i
$874$		0			0
$875$	4.10426e8			0.612648
$876$		0			0
$877$		−	4.74884e8i		−	0.704026i	−0.935995	−	0.352013i	$-0.885497\pi$
$877$		−	4.74884e8i		−	0.704026i	0.935995	−	0.352013i	$-0.114503\pi$
$878$		0			0
$879$	−2.71775e8	+	1.82466e8i	−0.400168	+	0.268668i
$880$		0			0
$881$			4.64654e8i			0.679520i	0.940512	+	0.339760i	$0.110346\pi$
$881$			4.64654e8i			0.679520i	−0.940512	+	0.339760i	$0.889654\pi$
$882$		0			0
$883$			2.30656e8i			0.335028i	0.985870	+	0.167514i	$0.0535740\pi$
$883$			2.30656e8i			0.335028i	−0.985870	+	0.167514i	$0.946426\pi$
$884$		0			0
$885$	2.67949e8	−	1.79898e8i	0.386565	−	0.259535i
$886$		0			0
$887$			1.94025e8i			0.278027i	0.990290	+	0.139014i	$0.0443932\pi$
$887$			1.94025e8i			0.278027i	−0.990290	+	0.139014i	$0.955607\pi$
$888$		0			0
$889$	5.35453e8			0.762108
$890$		0			0
$891$	5.41455e7	+	5.31947e7i	0.0765471	+	0.0752030i
$892$		0			0
$893$	−4.58568e7			−0.0643945
$894$		0			0
$895$	5.96424e8			0.831928
$896$		0			0
$897$	1.32654e9	−	8.90621e8i	1.83799	−	1.23400i
$898$		0			0
$899$	−9.42570e8			−1.29728
$900$		0			0
$901$			1.79143e7i			0.0244921i
$902$		0			0
$903$	−3.98975e8	−	5.94256e8i	−0.541855	−	0.807069i
$904$		0			0
$905$			1.09589e9i			1.47849i
$906$		0			0
$907$		−	2.19565e7i		−	0.0294267i	−0.999892	−	0.0147133i	$-0.995316\pi$
$907$		−	2.19565e7i		−	0.0294267i	0.999892	−	0.0147133i	$-0.00468357\pi$
$908$		0			0
$909$	4.83396e7	−	1.18180e8i	0.0643593	−	0.157345i
$910$		0			0
$911$		−	1.15098e9i		−	1.52235i	−0.648547	−	0.761175i	$-0.724623\pi$
$911$		−	1.15098e9i		−	1.52235i	0.648547	−	0.761175i	$-0.275377\pi$
$912$		0			0
$913$	1.14834e6			0.00150889
$914$		0			0
$915$	3.94322e8	+	5.87324e8i	0.514739	+	0.766681i
$916$		0			0
$917$	1.89080e7			0.0245210
$918$		0			0
$919$	5.98945e8			0.771685			0.385843	−	0.922565i	$-0.373911\pi$
$919$	5.98945e8			0.771685			0.385843	+	0.922565i	$0.373911\pi$
$920$		0			0
$921$	−1.57698e8	−	2.34884e8i	−0.201858	−	0.300659i
$922$		0			0
$923$	−1.24704e9			−1.58590
$924$		0			0
$925$		−	1.50720e7i		−	0.0190435i
$926$		0			0
$927$	−3.69402e8	+	9.03109e8i	−0.463725	+	1.13371i
$928$		0			0
$929$			1.34334e9i			1.67548i	0.546066	+	0.837742i	$0.316125\pi$
$929$			1.34334e9i			1.67548i	−0.546066	+	0.837742i	$0.683875\pi$
$930$		0			0
$931$			5.66652e7i			0.0702210i
$932$		0			0
$933$	−8.84187e8	−	1.31696e9i	−1.08868	−	1.62154i
$934$		0			0
$935$		−	5.93000e7i		−	0.0725470i
$936$		0			0
$937$	1.14989e9			1.39777			0.698887	−	0.715232i	$-0.253679\pi$
$937$	1.14989e9			1.39777			0.698887	+	0.715232i	$0.253679\pi$
$938$		0			0
$939$	1.14857e9	−	7.71136e8i	1.38727	−	0.931396i
$940$		0			0
$941$	−8.14092e7			−0.0977023			−0.0488511	−	0.998806i	$-0.515556\pi$
$941$	−8.14092e7			−0.0977023			−0.0488511	+	0.998806i	$0.515556\pi$
$942$		0			0
$943$	6.84241e8			0.815970
$944$		0			0
$945$	1.00615e8	+	4.88832e8i	0.119225	+	0.579247i
$946$		0			0
$947$	−8.31324e8			−0.978860			−0.489430	−	0.872043i	$-0.662795\pi$
$947$	−8.31324e8			−0.978860			−0.489430	+	0.872043i	$0.662795\pi$
$948$		0			0
$949$		−	4.95170e8i		−	0.579369i
$950$		0			0
$951$	−1.15059e9	+	7.72491e8i	−1.33776	+	0.898156i
$952$		0			0
$953$		−	1.24911e9i		−	1.44318i	−0.692319	−	0.721592i	$-0.743411\pi$
$953$		−	1.24911e9i		−	1.44318i	0.692319	−	0.721592i	$-0.256589\pi$
$954$		0			0
$955$		−	9.67356e8i		−	1.11065i
$956$		0			0
$957$	−1.31688e8	+	8.84138e7i	−0.150249	+	0.100875i
$958$		0			0
$959$		−	5.17015e8i		−	0.586202i
$960$		0			0
$961$	−3.62355e8			−0.408286
$962$		0			0
$963$	2.93168e8	−	7.16732e8i	0.328275	−	0.802560i
$964$		0			0
$965$	7.33355e8			0.816079
$966$		0			0
$967$	−1.42091e9			−1.57140			−0.785700	−	0.618607i	$-0.787697\pi$
$967$	−1.42091e9			−1.57140			−0.785700	+	0.618607i	$0.787697\pi$
$968$		0			0
$969$	−5.73180e7	+	3.84825e7i	−0.0629970	+	0.0422953i
$970$		0			0
$971$	7.12527e8			0.778294			0.389147	−	0.921176i	$-0.372770\pi$
$971$	7.12527e8			0.778294			0.389147	+	0.921176i	$0.372770\pi$
$972$		0			0
$973$			2.04304e8i			0.221788i
$974$		0			0
$975$	−2.95792e7	−	4.40569e7i	−0.0319134	−	0.0475335i
$976$		0			0
$977$		−	8.18474e8i		−	0.877650i	−0.898573	−	0.438825i	$-0.855395\pi$
$977$		−	8.18474e8i		−	0.877650i	0.898573	−	0.438825i	$-0.144605\pi$
$978$		0			0
$979$		−	5.86189e7i		−	0.0624726i
$980$		0			0
$981$	−2.63513e8	−	1.07786e8i	−0.279122	−	0.114171i
$982$		0			0
$983$			1.48038e7i			0.0155852i	0.999970	+	0.00779260i	$0.00248049\pi$
$983$			1.48038e7i			0.0155852i	−0.999970	+	0.00779260i	$0.997520\pi$
$984$		0			0
$985$	5.10668e8			0.534355
$986$		0			0
$987$	−1.88634e8	−	2.80961e8i	−0.196186	−	0.292210i
$988$		0			0
$989$	−2.17004e9			−2.24326
$990$		0			0
$991$	1.84976e9			1.90062			0.950308	−	0.311310i	$-0.100768\pi$
$991$	1.84976e9			1.90062			0.950308	+	0.311310i	$0.100768\pi$
$992$		0			0
$993$	7.91952e8	+	1.17958e9i	0.808819	+	1.20470i
$994$		0			0
$995$	−1.49490e9			−1.51755
$996$		0			0
$997$			1.00022e9i			1.00928i	0.863330	+	0.504640i	$0.168375\pi$
$997$			1.00022e9i			1.00928i	−0.863330	+	0.504640i	$0.831625\pi$
$998$		0			0
$999$	5.17348e8	−	1.06484e8i	0.518903	−	0.106805i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.7.h.f.65.5	yes	8
3.2	odd	2		384.7.h.e.65.2	yes	8
4.3	odd	2		384.7.h.e.65.3	yes	8
8.3	odd	2	inner	384.7.h.f.65.6	yes	8
8.5	even	2		384.7.h.e.65.4	yes	8
12.11	even	2	inner	384.7.h.f.65.8	yes	8
24.5	odd	2	inner	384.7.h.f.65.7	yes	8
24.11	even	2		384.7.h.e.65.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.7.h.e.65.1	✓	8	24.11	even	2
384.7.h.e.65.2	yes	8	3.2	odd	2
384.7.h.e.65.3	yes	8	4.3	odd	2
384.7.h.e.65.4	yes	8	8.5	even	2
384.7.h.f.65.5	yes	8	1.1	even	1	trivial
384.7.h.f.65.6	yes	8	8.3	odd	2	inner
384.7.h.f.65.7	yes	8	24.5	odd	2	inner
384.7.h.f.65.8	yes	8	12.11	even	2	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	384.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(22.4164 - 15.0501i) q^{3} -122.733 q^{5} +206.594 q^{7} +(275.991 - 674.737i) q^{9} +O(q^{10})\)	\(q+(22.4164 - 15.0501i) q^{3} -122.733 q^{5} +206.594 q^{7} +(275.991 - 674.737i) q^{9} -142.827 q^{11} +3499.27i q^{13} +(-2751.23 + 1847.14i) q^{15} -3382.86i q^{17} -755.858i q^{19} +(4631.09 - 3109.25i) q^{21} -16911.3i q^{23} -561.656 q^{25} +(-3968.12 - 19278.9i) q^{27} +41131.3 q^{29} -22916.1 q^{31} +(-3201.66 + 2149.55i) q^{33} -25355.8 q^{35} +26835.0i q^{37} +(52664.2 + 78441.0i) q^{39} +40460.6i q^{41} -128319. i q^{43} +(-33873.1 + 82812.4i) q^{45} -60668.5i q^{47} -74968.0 q^{49} +(-50912.4 - 75831.7i) q^{51} -5295.60 q^{53} +17529.5 q^{55} +(-11375.7 - 16943.6i) q^{57} -97392.6 q^{59} -213477. i q^{61} +(57018.0 - 139397. i) q^{63} -429475. i q^{65} -385320. i q^{67} +(-254517. - 379091. i) q^{69} +356373. i q^{71} -141507. q^{73} +(-12590.3 + 8452.97i) q^{75} -29507.1 q^{77} -722207. q^{79} +(-379099. - 372442. i) q^{81} -8040.09 q^{83} +415188. i q^{85} +(922016. - 619029. i) q^{87} +410420. i q^{89} +722927. i q^{91} +(-513697. + 344889. i) q^{93} +92768.6i q^{95} -1.52876e6 q^{97} +(-39418.8 + 96370.4i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 72 q^{3} - 1656 q^{9}+O(q^{10}) \) 8 * q + 72 * q^3 - 1656 * q^9	\( 8 q + 72 q^{3} - 1656 q^{9} + 1648 q^{11} - 200 q^{25} + 21384 q^{27} - 22608 q^{33} + 136320 q^{35} + 105208 q^{49} + 275328 q^{51} - 391104 q^{57} + 836624 q^{59} - 1964944 q^{73} - 59400 q^{75} + 166536 q^{81} - 587024 q^{83} - 1477232 q^{97} - 1688976 q^{99}+O(q^{100}) \) 8 * q + 72 * q^3 - 1656 * q^9 + 1648 * q^11 - 200 * q^25 + 21384 * q^27 - 22608 * q^33 + 136320 * q^35 + 105208 * q^49 + 275328 * q^51 - 391104 * q^57 + 836624 * q^59 - 1964944 * q^73 - 59400 * q^75 + 166536 * q^81 - 587024 * q^83 - 1477232 * q^97 - 1688976 * q^99

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