LMFDB - Embedded newform 192.9.e.j.65.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [192,9,Mod(65,192)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("192.65"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	192.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,56,0,0,0,-1584,0,328] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$78.2166931317$
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} - 4x^{7} - 78x^{6} + 144x^{5} + 2079x^{4} + 936x^{3} - 658x^{2} + 2884x + 30633 $ x^8 - 4x^7 - 78x^6 + 144x^5 + 2079x^4 + 936x^3 - 658x^2 + 2884*x + 30633
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{42}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.8
Root			$-1.50551 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	192.65
Dual form			192.9.e.j.65.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(80.7366 + 6.52720i) q^{3} -920.542i q^{5} +2012.50 q^{7} +(6475.79 + 1053.97i) q^{9} +9968.11i q^{11} +11295.6 q^{13} +(6008.56 - 74321.4i) q^{15} +151773. i q^{17} +186183. q^{19} +(162482. + 13136.0i) q^{21} +224838. i q^{23} -456773. q^{25} +(515954. + 127363. i) q^{27} +1.17495e6i q^{29} -300789. q^{31} +(-65063.8 + 804791. i) q^{33} -1.85259e6i q^{35} +1.47981e6 q^{37} +(911966. + 73728.5i) q^{39} -2.69503e6i q^{41} +5.09749e6 q^{43} +(970221. - 5.96124e6i) q^{45} -2.64836e6i q^{47} -1.71466e6 q^{49} +(-990655. + 1.22537e7i) q^{51} -3.07581e6i q^{53} +9.17606e6 q^{55} +(1.50318e7 + 1.21525e6i) q^{57} -1.78203e7i q^{59} -8.90915e6 q^{61} +(1.30325e7 + 2.12111e6i) q^{63} -1.03981e7i q^{65} -1.07217e7 q^{67} +(-1.46756e6 + 1.81527e7i) q^{69} -1.05459e6i q^{71} +1.03895e7 q^{73} +(-3.68783e7 - 2.98145e6i) q^{75} +2.00608e7i q^{77} -6.99649e7 q^{79} +(4.08250e7 + 1.36505e7i) q^{81} +2.34028e7i q^{83} +1.39714e8 q^{85} +(-7.66913e6 + 9.48615e7i) q^{87} +7.62936e7i q^{89} +2.27323e7 q^{91} +(-2.42847e7 - 1.96331e6i) q^{93} -1.71389e8i q^{95} +4.70613e7 q^{97} +(-1.05061e7 + 6.45514e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 56 q^{3} - 1584 q^{7} + 328 q^{9} - 25232 q^{13} - 38336 q^{15} + 157936 q^{19} - 30480 q^{21} - 579704 q^{25} - 276040 q^{27} - 805552 q^{31} + 102848 q^{33} + 3985008 q^{37} + 2297104 q^{39} + 6962672 q^{43}+ \cdots - 369701504 q^{99}+O(q^{100}) $ 8 * q + 56 * q^3 - 1584 * q^7 + 328 * q^9 - 25232 * q^13 - 38336 * q^15 + 157936 * q^19 - 30480 * q^21 - 579704 * q^25 - 276040 * q^27 - 805552 * q^31 + 102848 * q^33 + 3985008 * q^37 + 2297104 * q^39 + 6962672 * q^43 - 8670592 * q^45 - 5884520 * q^49 - 15590144 * q^51 - 27101312 * q^55 + 36756688 * q^57 + 51583600 * q^61 + 69759312 * q^63 + 58200688 * q^67 - 94226048 * q^69 - 116854768 * q^73 - 143181896 * q^75 - 172454576 * q^79 + 194700040 * q^81 + 264333824 * q^85 + 242851008 * q^87 + 382128480 * q^91 - 313470352 * q^93 - 337326704 * q^97 - 369701504 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	80.7366	+	6.52720i	0.996748	+	0.0805827i
$3$	80.7366	+	6.52720i	0.996748	+	0.0805827i
$4$		0			0
$5$		−	920.542i		−	1.47287i	−0.676510	−	0.736434i	$-0.736508\pi$
$5$		−	920.542i		−	1.47287i	0.676510	−	0.736434i	$-0.263492\pi$
$6$		0			0
$7$	2012.50			0.838191			0.419096	−	0.907942i	$-0.362347\pi$
$7$	2012.50			0.838191			0.419096	+	0.907942i	$0.362347\pi$
$8$		0			0
$9$	6475.79	+	1053.97i	0.987013	+	0.160641i
$10$		0			0
$11$			9968.11i			0.680835i	0.940274	+	0.340418i	$0.110568\pi$
$11$			9968.11i			0.680835i	−0.940274	+	0.340418i	$0.889432\pi$
$12$		0			0
$13$	11295.6			0.395489			0.197745	−	0.980254i	$-0.436638\pi$
$13$	11295.6			0.395489			0.197745	+	0.980254i	$0.436638\pi$
$14$		0			0
$15$	6008.56	−	74321.4i	0.118688	−	1.46808i
$16$		0			0
$17$			151773.i			1.81719i	0.417680	+	0.908594i	$0.362843\pi$
$17$			151773.i			1.81719i	−0.417680	+	0.908594i	$0.637157\pi$
$18$		0			0
$19$	186183.			1.42865			0.714325	−	0.699814i	$-0.246734\pi$
$19$	186183.			1.42865			0.714325	+	0.699814i	$0.246734\pi$
$20$		0			0
$21$	162482.	+	13136.0i	0.835465	+	0.0675437i
$22$		0			0
$23$			224838.i			0.803450i	0.915760	+	0.401725i	$0.131589\pi$
$23$			224838.i			0.803450i	−0.915760	+	0.401725i	$0.868411\pi$
$24$		0			0
$25$	−456773.			−1.16934
$26$		0			0
$27$	515954.	+	127363.i	0.970858	+	0.239655i
$28$		0			0
$29$			1.17495e6i			1.66122i	0.556854	+	0.830611i	$0.312008\pi$
$29$			1.17495e6i			1.66122i	−0.556854	+	0.830611i	$0.687992\pi$
$30$		0			0
$31$	−300789.			−0.325698			−0.162849	−	0.986651i	$-0.552068\pi$
$31$	−300789.			−0.325698			−0.162849	+	0.986651i	$0.552068\pi$
$32$		0			0
$33$	−65063.8	+	804791.i	−0.0548635	+	0.678621i
$34$		0			0
$35$		−	1.85259e6i		−	1.23454i
$36$		0			0
$37$	1.47981e6			0.789585			0.394792	−	0.918770i	$-0.370817\pi$
$37$	1.47981e6			0.789585			0.394792	+	0.918770i	$0.370817\pi$
$38$		0			0
$39$	911966.	+	73728.5i	0.394203	+	0.0318696i
$40$		0			0
$41$		−	2.69503e6i		−	0.953735i	−0.878975	−	0.476868i	$-0.841772\pi$
$41$		−	2.69503e6i		−	0.953735i	0.878975	−	0.476868i	$-0.158228\pi$
$42$		0			0
$43$	5.09749e6			1.49102			0.745508	−	0.666496i	$-0.232207\pi$
$43$	5.09749e6			1.49102			0.745508	+	0.666496i	$0.232207\pi$
$44$		0			0
$45$	970221.	−	5.96124e6i	0.236603	−	1.45374i
$46$		0			0
$47$		−	2.64836e6i		−	0.542732i	−0.962476	−	0.271366i	$-0.912525\pi$
$47$		−	2.64836e6i		−	0.542732i	0.962476	−	0.271366i	$-0.0874753\pi$
$48$		0			0
$49$	−1.71466e6			−0.297436
$50$		0			0
$51$	−990655.	+	1.22537e7i	−0.146434	+	1.81128i
$52$		0			0
$53$		−	3.07581e6i		−	0.389812i	−0.980822	−	0.194906i	$-0.937560\pi$
$53$		−	3.07581e6i		−	0.389812i	0.980822	−	0.194906i	$-0.0624402\pi$
$54$		0			0
$55$	9.17606e6			1.00278
$56$		0			0
$57$	1.50318e7	+	1.21525e6i	1.42400	+	0.115125i
$58$		0			0
$59$		−	1.78203e7i		−	1.47064i	−0.677720	−	0.735320i	$-0.737032\pi$
$59$		−	1.78203e7i		−	1.47064i	0.677720	−	0.735320i	$-0.262968\pi$
$60$		0			0
$61$	−8.90915e6			−0.643453			−0.321727	−	0.946833i	$-0.604263\pi$
$61$	−8.90915e6			−0.643453			−0.321727	+	0.946833i	$0.604263\pi$
$62$		0			0
$63$	1.30325e7	+	2.12111e6i	0.827305	+	0.134648i
$64$		0			0
$65$		−	1.03981e7i		−	0.582504i
$66$		0			0
$67$	−1.07217e7			−0.532063			−0.266032	−	0.963964i	$-0.585713\pi$
$67$	−1.07217e7			−0.532063			−0.266032	+	0.963964i	$0.585713\pi$
$68$		0			0
$69$	−1.46756e6	+	1.81527e7i	−0.0647442	+	0.800837i
$70$		0			0
$71$		−	1.05459e6i		−	0.0415001i	−0.999785	−	0.0207501i	$-0.993395\pi$
$71$		−	1.05459e6i		−	0.0415001i	0.999785	−	0.0207501i	$-0.00660542\pi$
$72$		0			0
$73$	1.03895e7			0.365850			0.182925	−	0.983127i	$-0.441443\pi$
$73$	1.03895e7			0.365850			0.182925	+	0.983127i	$0.441443\pi$
$74$		0			0
$75$	−3.68783e7	−	2.98145e6i	−1.16554	−	0.0942285i
$76$		0			0
$77$			2.00608e7i			0.570670i
$78$		0			0
$79$	−6.99649e7			−1.79627			−0.898135	−	0.439720i	$-0.855078\pi$
$79$	−6.99649e7			−1.79627			−0.898135	+	0.439720i	$0.855078\pi$
$80$		0			0
$81$	4.08250e7	+	1.36505e7i	0.948389	+	0.317110i
$82$		0			0
$83$			2.34028e7i			0.493124i	0.969127	+	0.246562i	$0.0793009\pi$
$83$			2.34028e7i			0.493124i	−0.969127	+	0.246562i	$0.920699\pi$
$84$		0			0
$85$	1.39714e8			2.67648
$86$		0			0
$87$	−7.66913e6	+	9.48615e7i	−0.133866	+	1.65582i
$88$		0			0
$89$			7.62936e7i			1.21598i	0.793943	+	0.607992i	$0.208025\pi$
$89$			7.62936e7i			1.21598i	−0.793943	+	0.607992i	$0.791975\pi$
$90$		0			0
$91$	2.27323e7			0.331496
$92$		0			0
$93$	−2.42847e7	−	1.96331e6i	−0.324639	−	0.0262456i
$94$		0			0
$95$		−	1.71389e8i		−	2.10421i
$96$		0			0
$97$	4.70613e7			0.531590			0.265795	−	0.964030i	$-0.414366\pi$
$97$	4.70613e7			0.531590			0.265795	+	0.964030i	$0.414366\pi$
$98$		0			0
$99$	−1.05061e7	+	6.45514e7i	−0.109370	+	0.671993i
$100$		0			0
$101$		−	1.33013e8i		−	1.27823i	−0.769110	−	0.639116i	$-0.779300\pi$
$101$		−	1.33013e8i		−	1.27823i	0.769110	−	0.639116i	$-0.220700\pi$
$102$		0			0
$103$	−1.03845e8			−0.922649			−0.461324	−	0.887231i	$-0.652626\pi$
$103$	−1.03845e8			−0.922649			−0.461324	+	0.887231i	$0.652626\pi$
$104$		0			0
$105$	1.20922e7	−	1.49572e8i	0.0994829	−	1.23053i
$106$		0			0
$107$		−	8.24148e7i		−	0.628739i	−0.949301	−	0.314369i	$-0.898207\pi$
$107$		−	8.24148e7i		−	0.628739i	0.949301	−	0.314369i	$-0.101793\pi$
$108$		0			0
$109$	2.22099e8			1.57340			0.786701	−	0.617334i	$-0.211787\pi$
$109$	2.22099e8			1.57340			0.786701	+	0.617334i	$0.211787\pi$
$110$		0			0
$111$	1.19475e8	+	9.65901e6i	0.787017	+	0.0636269i
$112$		0			0
$113$		−	9.17665e7i		−	0.562821i	−0.959587	−	0.281411i	$-0.909198\pi$
$113$		−	9.17665e7i		−	0.562821i	0.959587	−	0.281411i	$-0.0908023\pi$
$114$		0			0
$115$	2.06973e8			1.18337
$116$		0			0
$117$	7.31478e7	+	1.19052e7i	0.390353	+	0.0635319i
$118$		0			0
$119$			3.05443e8i			1.52315i
$120$		0			0
$121$	1.14996e8			0.536463
$122$		0			0
$123$	1.75910e7	−	2.17587e8i	0.0768545	−	0.950633i
$124$		0			0
$125$			6.08919e7i			0.249413i
$126$		0			0
$127$	1.87904e8			0.722305			0.361153	−	0.932507i	$-0.382383\pi$
$127$	1.87904e8			0.722305			0.361153	+	0.932507i	$0.382383\pi$
$128$		0			0
$129$	4.11554e8	+	3.32723e7i	1.48617	+	0.120150i
$130$		0			0
$131$			2.06519e8i			0.701253i	0.936515	+	0.350626i	$0.114031\pi$
$131$			2.06519e8i			0.701253i	−0.936515	+	0.350626i	$0.885969\pi$
$132$		0			0
$133$	3.74693e8			1.19748
$134$		0			0
$135$	1.17243e8	−	4.74957e8i	0.352980	−	1.42995i
$136$		0			0
$137$		−	2.15371e8i		−	0.611371i	−0.952133	−	0.305686i	$-0.901114\pi$
$137$		−	2.15371e8i		−	0.611371i	0.952133	−	0.305686i	$-0.0988856\pi$
$138$		0			0
$139$	−2.49471e8			−0.668285			−0.334142	−	0.942523i	$-0.608447\pi$
$139$	−2.49471e8			−0.668285			−0.334142	+	0.942523i	$0.608447\pi$
$140$		0			0
$141$	1.72864e7	−	2.13819e8i	0.0437348	−	0.540967i
$142$		0			0
$143$			1.12596e8i			0.269263i
$144$		0			0
$145$	1.08159e9			2.44676
$146$		0			0
$147$	−1.38436e8	−	1.11919e7i	−0.296469	−	0.0239682i
$148$		0			0
$149$			3.89834e8i			0.790923i	0.918483	+	0.395461i	$0.129415\pi$
$149$			3.89834e8i			0.790923i	−0.918483	+	0.395461i	$0.870585\pi$
$150$		0			0
$151$	3.94946e8			0.759679			0.379840	−	0.925052i	$-0.375979\pi$
$151$	3.94946e8			0.759679			0.379840	+	0.925052i	$0.375979\pi$
$152$		0			0
$153$	−1.59964e8	+	9.82853e8i	−0.291915	+	1.79359i
$154$		0			0
$155$			2.76889e8i			0.479710i
$156$		0			0
$157$	−4.13244e8			−0.680155			−0.340077	−	0.940397i	$-0.610453\pi$
$157$	−4.13244e8			−0.680155			−0.340077	+	0.940397i	$0.610453\pi$
$158$		0			0
$159$	2.00764e7	−	2.48330e8i	0.0314121	−	0.388544i
$160$		0			0
$161$			4.52486e8i			0.673444i
$162$		0			0
$163$	−1.52612e8			−0.216191			−0.108095	−	0.994141i	$-0.534475\pi$
$163$	−1.52612e8			−0.216191			−0.108095	+	0.994141i	$0.534475\pi$
$164$		0			0
$165$	7.40844e8	+	5.98940e7i	0.999519	+	0.0808067i
$166$		0			0
$167$			2.84359e8i			0.365596i	0.983151	+	0.182798i	$0.0585154\pi$
$167$			2.84359e8i			0.365596i	−0.983151	+	0.182798i	$0.941485\pi$
$168$		0			0
$169$	−6.88141e8			−0.843588
$170$		0			0
$171$	1.20568e9	+	1.96231e8i	1.41010	+	0.229500i
$172$		0			0
$173$			5.67130e7i			0.0633138i	0.999499	+	0.0316569i	$0.0100784\pi$
$173$			5.67130e7i			0.0633138i	−0.999499	+	0.0316569i	$0.989922\pi$
$174$		0			0
$175$	−9.19254e8			−0.980129
$176$		0			0
$177$	1.16317e8	−	1.43875e9i	0.118508	−	1.46586i
$178$		0			0
$179$		−	1.02066e9i		−	0.994185i	−0.867697	−	0.497093i	$-0.834401\pi$
$179$		−	1.02066e9i		−	0.994185i	0.867697	−	0.497093i	$-0.165599\pi$
$180$		0			0
$181$	6.37279e8			0.593766			0.296883	−	0.954914i	$-0.404053\pi$
$181$	6.37279e8			0.593766			0.296883	+	0.954914i	$0.404053\pi$
$182$		0			0
$183$	−7.19294e8	−	5.81518e7i	−0.641361	−	0.0518512i
$184$		0			0
$185$		−	1.36223e9i		−	1.16295i
$186$		0			0
$187$	−1.51289e9			−1.23721
$188$		0			0
$189$	1.03836e9	+	2.56317e8i	0.813765	+	0.200877i
$190$		0			0
$191$			1.25733e9i			0.944750i	0.881398	+	0.472375i	$0.156603\pi$
$191$			1.25733e9i			0.944750i	−0.881398	+	0.472375i	$0.843397\pi$
$192$		0			0
$193$	2.98104e7			0.0214851			0.0107426	−	0.999942i	$-0.496580\pi$
$193$	2.98104e7			0.0214851			0.0107426	+	0.999942i	$0.496580\pi$
$194$		0			0
$195$	6.78702e7	−	8.39503e8i	0.0469397	−	0.580609i
$196$		0			0
$197$		−	2.19401e9i		−	1.45671i	−0.685199	−	0.728356i	$-0.740285\pi$
$197$		−	2.19401e9i		−	1.45671i	0.685199	−	0.728356i	$-0.259715\pi$
$198$		0			0
$199$	−1.23376e9			−0.786714			−0.393357	−	0.919386i	$-0.628686\pi$
$199$	−1.23376e9			−0.786714			−0.393357	+	0.919386i	$0.628686\pi$
$200$		0			0
$201$	−8.65631e8	−	6.99825e7i	−0.530333	−	0.0428751i
$202$		0			0
$203$			2.36458e9i			1.39242i
$204$		0			0
$205$	−2.48089e9			−1.40473
$206$		0			0
$207$	−2.36972e8	+	1.45601e9i	−0.129067	+	0.793015i
$208$		0			0
$209$			1.85589e9i			0.972675i
$210$		0			0
$211$	−5.81426e8			−0.293335			−0.146668	−	0.989186i	$-0.546855\pi$
$211$	−5.81426e8			−0.293335			−0.146668	+	0.989186i	$0.546855\pi$
$212$		0			0
$213$	6.88350e6	−	8.51438e7i	0.00334419	−	0.0413652i
$214$		0			0
$215$		−	4.69245e9i		−	2.19607i
$216$		0			0
$217$	−6.05337e8			−0.272997
$218$		0			0
$219$	8.38813e8	+	6.78143e7i	0.364660	+	0.0294812i
$220$		0			0
$221$			1.71437e9i			0.718679i
$222$		0			0
$223$	1.29312e9			0.522901			0.261450	−	0.965217i	$-0.415799\pi$
$223$	1.29312e9			0.522901			0.261450	+	0.965217i	$0.415799\pi$
$224$		0			0
$225$	−2.95797e9	−	4.81424e8i	−1.15415	−	0.187844i
$226$		0			0
$227$		−	4.25456e8i		−	0.160233i	−0.996786	−	0.0801163i	$-0.974471\pi$
$227$		−	4.25456e8i		−	0.160233i	0.996786	−	0.0801163i	$-0.0255292\pi$
$228$		0			0
$229$	−1.34280e9			−0.488280			−0.244140	−	0.969740i	$-0.578506\pi$
$229$	−1.34280e9			−0.488280			−0.244140	+	0.969740i	$0.578506\pi$
$230$		0			0
$231$	−1.30941e8	+	1.61964e9i	−0.0459861	+	0.568814i
$232$		0			0
$233$			2.62002e9i			0.888956i	0.895790	+	0.444478i	$0.146611\pi$
$233$			2.62002e9i			0.888956i	−0.895790	+	0.444478i	$0.853389\pi$
$234$		0			0
$235$	−2.43792e9			−0.799372
$236$		0			0
$237$	−5.64872e9	−	4.56675e8i	−1.79043	−	0.144748i
$238$		0			0
$239$		−	6.00705e8i		−	0.184107i	−0.995754	−	0.0920534i	$-0.970657\pi$
$239$		−	6.00705e8i		−	0.184107i	0.995754	−	0.0920534i	$-0.0293430\pi$
$240$		0			0
$241$	4.91474e7			0.0145691			0.00728454	−	0.999973i	$-0.497681\pi$
$241$	4.91474e7			0.0145691			0.00728454	+	0.999973i	$0.497681\pi$
$242$		0			0
$243$	3.20697e9	+	1.36857e9i	0.919751	+	0.392502i
$244$		0			0
$245$			1.57842e9i			0.438084i
$246$		0			0
$247$	2.10305e9			0.565016
$248$		0			0
$249$	−1.52755e8	+	1.88946e9i	−0.0397372	+	0.491520i
$250$		0			0
$251$		−	5.49890e9i		−	1.38542i	−0.721217	−	0.692709i	$-0.756417\pi$
$251$		−	5.49890e9i		−	1.38542i	0.721217	−	0.692709i	$-0.243583\pi$
$252$		0			0
$253$	−2.24121e9			−0.547017
$254$		0			0
$255$	1.12800e10	+	9.11940e8i	2.66777	+	0.215678i
$256$		0			0
$257$		−	1.62748e9i		−	0.373065i	−0.982449	−	0.186532i	$-0.940275\pi$
$257$		−	1.62748e9i		−	0.373065i	0.982449	−	0.186532i	$-0.0597249\pi$
$258$		0			0
$259$	2.97811e9			0.661823
$260$		0			0
$261$	−1.23836e9	+	7.60873e9i	−0.266861	+	1.63965i
$262$		0			0
$263$			3.55235e9i			0.742494i	0.928534	+	0.371247i	$0.121070\pi$
$263$			3.55235e9i			0.742494i	−0.928534	+	0.371247i	$0.878930\pi$
$264$		0			0
$265$	−2.83141e9			−0.574142
$266$		0			0
$267$	−4.97984e8	+	6.15968e9i	−0.0979873	+	1.21203i
$268$		0			0
$269$			2.16296e9i			0.413086i	0.978438	+	0.206543i	$0.0662213\pi$
$269$			2.16296e9i			0.413086i	−0.978438	+	0.206543i	$0.933779\pi$
$270$		0			0
$271$	−4.12387e9			−0.764589			−0.382294	−	0.924041i	$-0.624866\pi$
$271$	−4.12387e9			−0.764589			−0.382294	+	0.924041i	$0.624866\pi$
$272$		0			0
$273$	1.83533e9	+	1.48378e8i	0.330418	+	0.0267128i
$274$		0			0
$275$		−	4.55316e9i		−	0.796127i
$276$		0			0
$277$	−5.01959e9			−0.852608			−0.426304	−	0.904580i	$-0.640185\pi$
$277$	−5.01959e9			−0.852608			−0.426304	+	0.904580i	$0.640185\pi$
$278$		0			0
$279$	−1.94785e9	−	3.17022e8i	−0.321468	−	0.0523205i
$280$		0			0
$281$		−	9.64527e9i		−	1.54700i	−0.633799	−	0.773498i	$-0.718505\pi$
$281$		−	9.64527e9i		−	1.54700i	0.633799	−	0.773498i	$-0.281495\pi$
$282$		0			0
$283$	−3.72563e9			−0.580836			−0.290418	−	0.956900i	$-0.593794\pi$
$283$	−3.72563e9			−0.580836			−0.290418	+	0.956900i	$0.593794\pi$
$284$		0			0
$285$	1.11869e9	−	1.38374e10i	0.169563	−	2.09737i
$286$		0			0
$287$		−	5.42373e9i		−	0.799412i
$288$		0			0
$289$	−1.60594e10			−2.30217
$290$		0			0
$291$	3.79957e9	+	3.07178e8i	0.529861	+	0.0428370i
$292$		0			0
$293$		−	1.12558e10i		−	1.52724i	−0.645667	−	0.763619i	$-0.723421\pi$
$293$		−	1.12558e10i		−	1.52724i	0.645667	−	0.763619i	$-0.276579\pi$
$294$		0			0
$295$	−1.64043e10			−2.16606
$296$		0			0
$297$	−1.26956e9	+	5.14308e9i	−0.163166	+	0.660994i
$298$		0			0
$299$			2.53968e9i			0.317756i
$300$		0			0
$301$	1.02587e10			1.24976
$302$		0			0
$303$	8.68205e8	−	1.07390e10i	0.103003	−	1.27408i
$304$		0			0
$305$			8.20125e9i			0.947721i
$306$		0			0
$307$	9.70242e8			0.109226			0.0546131	−	0.998508i	$-0.482607\pi$
$307$	9.70242e8			0.109226			0.0546131	+	0.998508i	$0.482607\pi$
$308$		0			0
$309$	−8.38409e9	−	6.77817e8i	−0.919648	−	0.0743495i
$310$		0			0
$311$			1.23353e10i			1.31859i	0.751886	+	0.659294i	$0.229145\pi$
$311$			1.23353e10i			1.31859i	−0.751886	+	0.659294i	$0.770855\pi$
$312$		0			0
$313$	−2.47417e9			−0.257781			−0.128891	−	0.991659i	$-0.541142\pi$
$313$	−2.47417e9			−0.257781			−0.128891	+	0.991659i	$0.541142\pi$
$314$		0			0
$315$	1.95257e9	−	1.19970e10i	0.198319	−	1.21851i
$316$		0			0
$317$			1.69441e10i			1.67796i	0.544162	+	0.838980i	$0.316848\pi$
$317$			1.69441e10i			1.67796i	−0.544162	+	0.838980i	$0.683152\pi$
$318$		0			0
$319$	−1.17120e10			−1.13102
$320$		0			0
$321$	5.37938e8	−	6.65389e9i	0.0506655	−	0.626694i
$322$		0			0
$323$			2.82576e10i			2.59613i
$324$		0			0
$325$	−5.15951e9			−0.462461
$326$		0			0
$327$	1.79315e10	+	1.44968e9i	1.56829	+	0.126789i
$328$		0			0
$329$		−	5.32981e9i		−	0.454913i
$330$		0			0
$331$	−1.73248e10			−1.44330			−0.721651	−	0.692257i	$-0.756617\pi$
$331$	−1.73248e10			−1.44330			−0.721651	+	0.692257i	$0.756617\pi$
$332$		0			0
$333$	9.58293e9	+	1.55967e9i	0.779330	+	0.126840i
$334$		0			0
$335$			9.86975e9i			0.783659i
$336$		0			0
$337$	7.54415e9			0.584912			0.292456	−	0.956279i	$-0.405528\pi$
$337$	7.54415e9			0.584912			0.292456	+	0.956279i	$0.405528\pi$
$338$		0			0
$339$	5.98978e8	−	7.40891e9i	0.0453536	−	0.560991i
$340$		0			0
$341$		−	2.99830e9i		−	0.221747i
$342$		0			0
$343$	−1.50524e10			−1.08750
$344$		0			0
$345$	1.67103e10	+	1.35095e9i	1.17953	+	0.0953596i
$346$		0			0
$347$		−	2.20011e10i		−	1.51749i	−0.651387	−	0.758746i	$-0.725812\pi$
$347$		−	2.20011e10i		−	1.51749i	0.651387	−	0.758746i	$-0.274188\pi$
$348$		0			0
$349$	1.10710e10			0.746251			0.373126	−	0.927781i	$-0.378286\pi$
$349$	1.10710e10			0.746251			0.373126	+	0.927781i	$0.378286\pi$
$350$		0			0
$351$	5.82799e9	+	1.43863e9i	0.383964	+	0.0947810i
$352$		0			0
$353$			1.54479e10i			0.994880i	0.867498	+	0.497440i	$0.165727\pi$
$353$			1.54479e10i			0.994880i	−0.867498	+	0.497440i	$0.834273\pi$
$354$		0			0
$355$	−9.70792e8			−0.0611242
$356$		0			0
$357$	−1.99369e9	+	2.46605e10i	−0.122740	+	1.51820i
$358$		0			0
$359$		−	1.17345e10i		−	0.706458i	−0.935537	−	0.353229i	$-0.885084\pi$
$359$		−	1.17345e10i		−	0.706458i	0.935537	−	0.353229i	$-0.114916\pi$
$360$		0			0
$361$	1.76806e10			1.04104
$362$		0			0
$363$	9.28436e9	+	7.50600e8i	0.534719	+	0.0432297i
$364$		0			0
$365$		−	9.56397e9i		−	0.538849i
$366$		0			0
$367$	3.18806e10			1.75737			0.878684	−	0.477404i	$-0.158422\pi$
$367$	3.18806e10			1.75737			0.878684	+	0.477404i	$0.158422\pi$
$368$		0			0
$369$	2.84047e9	−	1.74524e10i	0.153209	−	0.941349i
$370$		0			0
$371$		−	6.19005e9i		−	0.326737i
$372$		0			0
$373$	4.99978e9			0.258295			0.129147	−	0.991625i	$-0.458776\pi$
$373$	4.99978e9			0.258295			0.129147	+	0.991625i	$0.458776\pi$
$374$		0			0
$375$	−3.97453e8	+	4.91620e9i	−0.0200984	+	0.248602i
$376$		0			0
$377$			1.32717e10i			0.656996i
$378$		0			0
$379$	−3.07635e10			−1.49101			−0.745503	−	0.666502i	$-0.767791\pi$
$379$	−3.07635e10			−1.49101			−0.745503	+	0.666502i	$0.767791\pi$
$380$		0			0
$381$	1.51707e10	+	1.22649e9i	0.719956	+	0.0582053i
$382$		0			0
$383$		−	1.94789e10i		−	0.905251i	−0.891701	−	0.452625i	$-0.850488\pi$
$383$		−	1.94789e10i		−	0.905251i	0.891701	−	0.452625i	$-0.149512\pi$
$384$		0			0
$385$	1.84668e10			0.840521
$386$		0			0
$387$	3.30103e10	+	5.37259e9i	1.47165	+	0.239519i
$388$		0			0
$389$		−	4.42619e9i		−	0.193300i	−0.995318	−	0.0966499i	$-0.969187\pi$
$389$		−	4.42619e9i		−	0.193300i	0.995318	−	0.0966499i	$-0.0308127\pi$
$390$		0			0
$391$	−3.41244e10			−1.46002
$392$		0			0
$393$	−1.34799e9	+	1.66736e10i	−0.0565089	+	0.698972i
$394$		0			0
$395$			6.44056e10i			2.64567i
$396$		0			0
$397$	−1.25488e10			−0.505172			−0.252586	−	0.967574i	$-0.581281\pi$
$397$	−1.25488e10			−0.505172			−0.252586	+	0.967574i	$0.581281\pi$
$398$		0			0
$399$	3.02514e10	+	2.44570e9i	1.19359	+	0.0964963i
$400$		0			0
$401$		−	1.35459e10i		−	0.523879i	−0.965084	−	0.261939i	$-0.915638\pi$
$401$		−	1.35459e10i		−	0.523879i	0.965084	−	0.261939i	$-0.0843621\pi$
$402$		0			0
$403$	−3.39758e9			−0.128810
$404$		0			0
$405$	1.25659e10	−	3.75812e10i	0.467061	−	1.39685i
$406$		0			0
$407$			1.47509e10i			0.537577i
$408$		0			0
$409$	4.31951e10			1.54362			0.771812	−	0.635850i	$-0.219350\pi$
$409$	4.31951e10			1.54362			0.771812	+	0.635850i	$0.219350\pi$
$410$		0			0
$411$	1.40577e9	−	1.73883e10i	0.0492659	−	0.609383i
$412$		0			0
$413$		−	3.58633e10i		−	1.23268i
$414$		0			0
$415$	2.15433e10			0.726306
$416$		0			0
$417$	−2.01415e10	−	1.62835e9i	−0.666111	−	0.0538522i
$418$		0			0
$419$		−	5.59494e10i		−	1.81526i	−0.419770	−	0.907631i	$-0.637889\pi$
$419$		−	5.59494e10i		−	1.81526i	0.419770	−	0.907631i	$-0.362111\pi$
$420$		0			0
$421$	3.82532e10			1.21770			0.608848	−	0.793287i	$-0.291632\pi$
$421$	3.82532e10			1.21770			0.608848	+	0.793287i	$0.291632\pi$
$422$		0			0
$423$	2.79128e9	−	1.71502e10i	0.0871851	−	0.535683i
$424$		0			0
$425$		−	6.93260e10i		−	2.12491i
$426$		0			0
$427$	−1.79296e10			−0.539337
$428$		0			0
$429$	−7.34933e8	+	9.09058e9i	−0.0216980	+	0.268387i
$430$		0			0
$431$			4.42900e9i			0.128350i	0.997939	+	0.0641751i	$0.0204416\pi$
$431$			4.42900e9i			0.128350i	−0.997939	+	0.0641751i	$0.979558\pi$
$432$		0			0
$433$	−1.29438e8			−0.00368222			−0.00184111	−	0.999998i	$-0.500586\pi$
$433$	−1.29438e8			−0.00368222			−0.00184111	+	0.999998i	$0.500586\pi$
$434$		0			0
$435$	8.73240e10	+	7.05976e9i	2.43880	+	0.197166i
$436$		0			0
$437$			4.18611e10i			1.14785i
$438$		0			0
$439$	−4.59331e10			−1.23671			−0.618355	−	0.785899i	$-0.712200\pi$
$439$	−4.59331e10			−1.23671			−0.618355	+	0.785899i	$0.712200\pi$
$440$		0			0
$441$	−1.11038e10	−	1.80719e9i	−0.293573	−	0.0477805i
$442$		0			0
$443$		−	3.70397e10i		−	0.961728i	−0.876795	−	0.480864i	$-0.840323\pi$
$443$		−	3.70397e10i		−	0.961728i	0.876795	−	0.480864i	$-0.159677\pi$
$444$		0			0
$445$	7.02315e10			1.79098
$446$		0			0
$447$	−2.54452e9	+	3.14738e10i	−0.0637347	+	0.788351i
$448$		0			0
$449$			2.63633e10i			0.648657i	0.945945	+	0.324328i	$0.105138\pi$
$449$			2.63633e10i			0.648657i	−0.945945	+	0.324328i	$0.894862\pi$
$450$		0			0
$451$	2.68643e10			0.649336
$452$		0			0
$453$	3.18866e10	+	2.57789e9i	0.757209	+	0.0612170i
$454$		0			0
$455$		−	2.09260e10i		−	0.488249i
$456$		0			0
$457$	1.51289e10			0.346850			0.173425	−	0.984847i	$-0.444517\pi$
$457$	1.51289e10			0.346850			0.173425	+	0.984847i	$0.444517\pi$
$458$		0			0
$459$	−1.93302e10	+	7.83081e10i	−0.435498	+	1.76423i
$460$		0			0
$461$			5.13740e10i			1.13747i	0.822521	+	0.568735i	$0.192567\pi$
$461$			5.13740e10i			1.13747i	−0.822521	+	0.568735i	$0.807433\pi$
$462$		0			0
$463$	1.82665e10			0.397495			0.198747	−	0.980051i	$-0.436313\pi$
$463$	1.82665e10			0.397495			0.198747	+	0.980051i	$0.436313\pi$
$464$		0			0
$465$	−1.80731e9	+	2.23551e10i	−0.0386563	+	0.478150i
$466$		0			0
$467$		−	5.75031e10i		−	1.20899i	−0.796608	−	0.604496i	$-0.793374\pi$
$467$		−	5.75031e10i		−	1.20899i	0.796608	−	0.604496i	$-0.206626\pi$
$468$		0			0
$469$	−2.15773e10			−0.445971
$470$		0			0
$471$	−3.33639e10	−	2.69733e9i	−0.677943	−	0.0548087i
$472$		0			0
$473$			5.08123e10i			1.01514i
$474$		0			0
$475$	−8.50434e10			−1.67058
$476$		0			0
$477$	3.24180e9	−	1.99183e10i	0.0626199	−	0.384750i
$478$		0			0
$479$			4.64781e10i			0.882889i	0.897288	+	0.441445i	$0.145534\pi$
$479$			4.64781e10i			0.882889i	−0.897288	+	0.441445i	$0.854466\pi$
$480$		0			0
$481$	1.67153e10			0.312272
$482$		0			0
$483$	−2.95347e9	+	3.65322e10i	−0.0542680	+	0.671254i
$484$		0			0
$485$		−	4.33219e10i		−	0.782961i
$486$		0			0
$487$	−7.02986e10			−1.24977			−0.624886	−	0.780716i	$-0.714855\pi$
$487$	−7.02986e10			−1.24977			−0.624886	+	0.780716i	$0.714855\pi$
$488$		0			0
$489$	−1.23213e10	−	9.96126e8i	−0.215488	−	0.0174212i
$490$		0			0
$491$			1.95781e10i			0.336857i	0.985714	+	0.168428i	$0.0538692\pi$
$491$			1.95781e10i			0.336857i	−0.985714	+	0.168428i	$0.946131\pi$
$492$		0			0
$493$	−1.78326e11			−3.01875
$494$		0			0
$495$	5.94223e10	+	9.67127e9i	0.989757	+	0.161088i
$496$		0			0
$497$		−	2.12235e9i		−	0.0347850i
$498$		0			0
$499$	−3.94522e10			−0.636311			−0.318155	−	0.948039i	$-0.603063\pi$
$499$	−3.94522e10			−0.636311			−0.318155	+	0.948039i	$0.603063\pi$
$500$		0			0
$501$	−1.85607e9	+	2.29582e10i	−0.0294607	+	0.364407i
$502$		0			0
$503$		−	9.64015e10i		−	1.50595i	−0.658047	−	0.752977i	$-0.728617\pi$
$503$		−	9.64015e10i		−	1.50595i	0.658047	−	0.752977i	$-0.271383\pi$
$504$		0			0
$505$	−1.22444e11			−1.88267
$506$		0			0
$507$	−5.55581e10	−	4.49163e9i	−0.840845	−	0.0679786i
$508$		0			0
$509$			4.54849e10i			0.677635i	0.940852	+	0.338818i	$0.110027\pi$
$509$			4.54849e10i			0.677635i	−0.940852	+	0.338818i	$0.889973\pi$
$510$		0			0
$511$	2.09088e10			0.306652
$512$		0			0
$513$	9.60619e10	+	2.37128e10i	1.38702	+	0.342383i
$514$		0			0
$515$			9.55937e10i			1.35894i
$516$		0			0
$517$	2.63991e10			0.369511
$518$		0			0
$519$	−3.70177e8	+	4.57882e9i	−0.00510200	+	0.0631079i
$520$		0			0
$521$			8.36756e10i			1.13566i	0.823146	+	0.567829i	$0.192217\pi$
$521$			8.36756e10i			1.13566i	−0.823146	+	0.567829i	$0.807783\pi$
$522$		0			0
$523$	−5.10989e10			−0.682975			−0.341487	−	0.939886i	$-0.610931\pi$
$523$	−5.10989e10			−0.682975			−0.341487	+	0.939886i	$0.610931\pi$
$524$		0			0
$525$	−7.42174e10	−	6.00015e9i	−0.976942	−	0.0789814i
$526$		0			0
$527$		−	4.56517e10i		−	0.591854i
$528$		0			0
$529$	2.77588e10			0.354469
$530$		0			0
$531$	1.87820e10	−	1.15400e11i	0.236246	−	1.45154i
$532$		0			0
$533$		−	3.04419e10i		−	0.377192i
$534$		0			0
$535$	−7.58663e10			−0.926049
$536$		0			0
$537$	6.66203e9	−	8.24043e10i	0.0801141	−	0.990952i
$538$		0			0
$539$		−	1.70919e10i		−	0.202505i
$540$		0			0
$541$	4.61169e10			0.538358			0.269179	−	0.963090i	$-0.413248\pi$
$541$	4.61169e10			0.538358			0.269179	+	0.963090i	$0.413248\pi$
$542$		0			0
$543$	5.14517e10	+	4.15965e9i	0.591835	+	0.0478473i
$544$		0			0
$545$		−	2.04451e11i		−	2.31741i
$546$		0			0
$547$	6.61213e10			0.738571			0.369285	−	0.929316i	$-0.379603\pi$
$547$	6.61213e10			0.738571			0.369285	+	0.929316i	$0.379603\pi$
$548$		0			0
$549$	−5.76938e10	−	9.38996e9i	−0.635097	−	0.103365i
$550$		0			0
$551$			2.18756e11i			2.37330i
$552$		0			0
$553$	−1.40804e11			−1.50562
$554$		0			0
$555$	8.89152e9	−	1.09982e11i	0.0937140	−	1.15917i
$556$		0			0
$557$		−	1.26433e11i		−	1.31353i	−0.754095	−	0.656766i	$-0.771924\pi$
$557$		−	1.26433e11i		−	1.31353i	0.754095	−	0.656766i	$-0.228076\pi$
$558$		0			0
$559$	5.75791e10			0.589681
$560$		0			0
$561$	−1.22146e11	−	9.87496e9i	−1.23318	−	0.0996974i
$562$		0			0
$563$			6.46239e10i			0.643220i	0.946872	+	0.321610i	$0.104224\pi$
$563$			6.46239e10i			0.643220i	−0.946872	+	0.321610i	$0.895776\pi$
$564$		0			0
$565$	−8.44749e10			−0.828961
$566$		0			0
$567$	8.21602e10	+	2.74717e10i	0.794931	+	0.265799i
$568$		0			0
$569$			3.65015e10i			0.348226i	0.984726	+	0.174113i	$0.0557059\pi$
$569$			3.65015e10i			0.348226i	−0.984726	+	0.174113i	$0.944294\pi$
$570$		0			0
$571$	1.26958e11			1.19431			0.597153	−	0.802127i	$-0.296298\pi$
$571$	1.26958e11			1.19431			0.597153	+	0.802127i	$0.296298\pi$
$572$		0			0
$573$	−8.20686e9	+	1.01513e11i	−0.0761305	+	0.941678i
$574$		0			0
$575$		−	1.02700e11i		−	0.939505i
$576$		0			0
$577$	−1.91393e10			−0.172673			−0.0863363	−	0.996266i	$-0.527516\pi$
$577$	−1.91393e10			−0.172673			−0.0863363	+	0.996266i	$0.527516\pi$
$578$		0			0
$579$	2.40679e9	+	1.94578e8i	0.0214153	+	0.00173133i
$580$		0			0
$581$			4.70981e10i			0.413332i
$582$		0			0
$583$	3.06600e10			0.265398
$584$		0			0
$585$	1.09592e10	−	6.73356e10i	0.0935741	−	0.574938i
$586$		0			0
$587$			8.68383e10i			0.731406i	0.930732	+	0.365703i	$0.119171\pi$
$587$			8.68383e10i			0.731406i	−0.930732	+	0.365703i	$0.880829\pi$
$588$		0			0
$589$	−5.60018e10			−0.465308
$590$		0			0
$591$	1.43207e10	−	1.77137e11i	0.117386	−	1.45198i
$592$		0			0
$593$			8.91939e10i			0.721301i	0.932701	+	0.360650i	$0.117445\pi$
$593$			8.91939e10i			0.721301i	−0.932701	+	0.360650i	$0.882555\pi$
$594$		0			0
$595$	2.81174e11			2.24340
$596$		0			0
$597$	−9.96093e10	−	8.05297e9i	−0.784156	−	0.0633956i
$598$		0			0
$599$		−	9.31780e9i		−	0.0723779i	−0.999345	−	0.0361889i	$-0.988478\pi$
$599$		−	9.31780e9i		−	0.0723779i	0.999345	−	0.0361889i	$-0.0115218\pi$
$600$		0			0
$601$	−1.87743e11			−1.43901			−0.719507	−	0.694485i	$-0.755632\pi$
$601$	−1.87743e11			−1.43901			−0.719507	+	0.694485i	$0.755632\pi$
$602$		0			0
$603$	−6.94313e10	−	1.13003e10i	−0.525153	−	0.0854713i
$604$		0			0
$605$		−	1.05858e11i		−	0.790140i
$606$		0			0
$607$	6.11762e10			0.450638			0.225319	−	0.974285i	$-0.427658\pi$
$607$	6.11762e10			0.450638			0.225319	+	0.974285i	$0.427658\pi$
$608$		0			0
$609$	−1.54341e10	+	1.90908e11i	−0.112205	+	1.38789i
$610$		0			0
$611$		−	2.99147e10i		−	0.214645i
$612$		0			0
$613$	1.31009e11			0.927814			0.463907	−	0.885884i	$-0.346447\pi$
$613$	1.31009e11			0.927814			0.463907	+	0.885884i	$0.346447\pi$
$614$		0			0
$615$	−2.00298e11	−	1.61932e10i	−1.40016	−	0.113197i
$616$		0			0
$617$			2.05077e11i			1.41506i	0.706682	+	0.707531i	$0.250191\pi$
$617$			2.05077e11i			1.41506i	−0.706682	+	0.707531i	$0.749809\pi$
$618$		0			0
$619$	−2.25693e11			−1.53729			−0.768643	−	0.639678i	$-0.779068\pi$
$619$	−2.25693e11			−1.53729			−0.768643	+	0.639678i	$0.779068\pi$
$620$		0			0
$621$	−2.86360e10	+	1.16006e11i	−0.192551	+	0.780036i
$622$		0			0
$623$			1.53541e11i			1.01923i
$624$		0			0
$625$	−1.22373e11			−0.801986
$626$		0			0
$627$	−1.21138e10	+	1.49839e11i	−0.0783808	+	0.969512i
$628$		0			0
$629$			2.24596e11i			1.43482i
$630$		0			0
$631$	−2.05261e11			−1.29476			−0.647380	−	0.762167i	$-0.724135\pi$
$631$	−2.05261e11			−1.29476			−0.647380	+	0.762167i	$0.724135\pi$
$632$		0			0
$633$	−4.69423e10	−	3.79508e9i	−0.292381	−	0.0236377i
$634$		0			0
$635$		−	1.72973e11i		−	1.06386i
$636$		0			0
$637$	−1.93681e10			−0.117633
$638$		0			0
$639$	1.11150e9	−	6.82929e9i	0.00666663	−	0.0409611i
$640$		0			0
$641$		−	2.63885e10i		−	0.156308i	−0.996941	−	0.0781541i	$-0.975097\pi$
$641$		−	2.63885e10i		−	0.156308i	0.996941	−	0.0781541i	$-0.0249026\pi$
$642$		0			0
$643$	2.90994e11			1.70232			0.851158	−	0.524910i	$-0.175901\pi$
$643$	2.90994e11			1.70232			0.851158	+	0.524910i	$0.175901\pi$
$644$		0			0
$645$	3.06286e10	−	3.78853e11i	0.176965	−	2.18893i
$646$		0			0
$647$		−	7.95056e10i		−	0.453712i	−0.973928	−	0.226856i	$-0.927155\pi$
$647$		−	7.95056e10i		−	0.453712i	0.973928	−	0.226856i	$-0.0728447\pi$
$648$		0			0
$649$	1.77635e11			1.00126
$650$		0			0
$651$	−4.88728e10	−	3.95115e9i	−0.272109	−	0.0219988i
$652$		0			0
$653$			1.44238e11i			0.793282i	0.917974	+	0.396641i	$0.129824\pi$
$653$			1.44238e11i			0.793282i	−0.917974	+	0.396641i	$0.870176\pi$
$654$		0			0
$655$	1.90109e11			1.03285
$656$		0			0
$657$	6.72802e10	+	1.09502e10i	0.361099	+	0.0587706i
$658$		0			0
$659$			1.94035e11i			1.02882i	0.857545	+	0.514410i	$0.171989\pi$
$659$			1.94035e11i			1.02882i	−0.857545	+	0.514410i	$0.828011\pi$
$660$		0			0
$661$	7.71719e10			0.404253			0.202127	−	0.979359i	$-0.435215\pi$
$661$	7.71719e10			0.404253			0.202127	+	0.979359i	$0.435215\pi$
$662$		0			0
$663$	−1.11900e10	+	1.38412e11i	−0.0579131	+	0.716342i
$664$		0			0
$665$		−	3.44921e11i		−	1.76373i
$666$		0			0
$667$	−2.64174e11			−1.33471
$668$		0			0
$669$	1.04402e11	+	8.44045e9i	0.521200	+	0.0421367i
$670$		0			0
$671$		−	8.88074e10i		−	0.438086i
$672$		0			0
$673$	2.72425e11			1.32796			0.663982	−	0.747749i	$-0.268865\pi$
$673$	2.72425e11			1.32796			0.663982	+	0.747749i	$0.268865\pi$
$674$		0			0
$675$	−2.35674e11	−	5.81757e10i	−1.13526	−	0.280238i
$676$		0			0
$677$			2.44635e11i			1.16457i	0.812986	+	0.582283i	$0.197841\pi$
$677$			2.44635e11i			1.16457i	−0.812986	+	0.582283i	$0.802159\pi$
$678$		0			0
$679$	9.47107e10			0.445574
$680$		0			0
$681$	2.77703e9	−	3.43499e10i	0.0129120	−	0.159712i
$682$		0			0
$683$		−	3.20799e11i		−	1.47418i	−0.675794	−	0.737090i	$-0.736199\pi$
$683$		−	3.20799e11i		−	1.47418i	0.675794	−	0.737090i	$-0.263801\pi$
$684$		0			0
$685$	−1.98258e11			−0.900468
$686$		0			0
$687$	−1.08413e11	−	8.76472e9i	−0.486692	−	0.0393469i
$688$		0			0
$689$		−	3.47430e10i		−	0.154167i
$690$		0			0
$691$	−2.33613e11			−1.02467			−0.512335	−	0.858785i	$-0.671220\pi$
$691$	−2.33613e11			−1.02467			−0.512335	+	0.858785i	$0.671220\pi$
$692$		0			0
$693$	−2.11434e10	+	1.29909e11i	−0.0916732	+	0.563259i
$694$		0			0
$695$			2.29649e11i			0.984295i
$696$		0			0
$697$	4.09033e11			1.73312
$698$		0			0
$699$	−1.71014e10	+	2.11531e11i	−0.0716345	+	0.886065i
$700$		0			0
$701$		−	2.88649e11i		−	1.19536i	−0.801736	−	0.597679i	$-0.796090\pi$
$701$		−	2.88649e11i		−	1.19536i	0.801736	−	0.597679i	$-0.203910\pi$
$702$		0			0
$703$	2.75515e11			1.12804
$704$		0			0
$705$	−1.96830e11	−	1.59128e10i	−0.796772	−	0.0644155i
$706$		0			0
$707$		−	2.67689e11i		−	1.07140i
$708$		0			0
$709$	−1.36081e11			−0.538533			−0.269266	−	0.963066i	$-0.586781\pi$
$709$	−1.36081e11			−0.538533			−0.269266	+	0.963066i	$0.586781\pi$
$710$		0			0
$711$	−4.53078e11	−	7.37407e10i	−1.77294	−	0.288555i
$712$		0			0
$713$		−	6.76288e10i		−	0.261682i
$714$		0			0
$715$	1.03649e11			0.396589
$716$		0			0
$717$	3.92092e9	−	4.84989e10i	0.0148358	−	0.183508i
$718$		0			0
$719$			5.50687e10i			0.206058i	0.994678	+	0.103029i	$0.0328534\pi$
$719$			5.50687e10i			0.206058i	−0.994678	+	0.103029i	$0.967147\pi$
$720$		0			0
$721$	−2.08988e11			−0.773356
$722$		0			0
$723$	3.96799e9	+	3.20795e8i	0.0145217	+	0.00117402i
$724$		0			0
$725$		−	5.36685e11i		−	1.94253i
$726$		0			0
$727$	5.15315e11			1.84474			0.922371	−	0.386306i	$-0.126249\pi$
$727$	5.15315e11			1.84474			0.922371	+	0.386306i	$0.126249\pi$
$728$		0			0
$729$	2.49987e11	+	1.31426e11i	0.885131	+	0.465342i
$730$		0			0
$731$			7.73663e11i			2.70946i
$732$		0			0
$733$	4.66106e11			1.61462			0.807308	−	0.590131i	$-0.200924\pi$
$733$	4.66106e11			1.61462			0.807308	+	0.590131i	$0.200924\pi$
$734$		0			0
$735$	−1.03026e10	+	1.27436e11i	−0.0353020	+	0.436659i
$736$		0			0
$737$		−	1.06875e11i		−	0.362247i
$738$		0			0
$739$	1.67363e11			0.561154			0.280577	−	0.959831i	$-0.409474\pi$
$739$	1.67363e11			0.561154			0.280577	+	0.959831i	$0.409474\pi$
$740$		0			0
$741$	1.69793e11	+	1.37270e10i	0.563179	+	0.0455305i
$742$		0			0
$743$		−	5.57723e11i		−	1.83005i	−0.403395	−	0.915026i	$-0.632170\pi$
$743$		−	5.57723e11i		−	1.83005i	0.403395	−	0.915026i	$-0.367830\pi$
$744$		0			0
$745$	3.58858e11			1.16492
$746$		0			0
$747$	−2.46658e10	+	1.51552e11i	−0.0792160	+	0.486720i
$748$		0			0
$749$		−	1.65860e11i		−	0.527003i
$750$		0			0
$751$	−5.22604e11			−1.64291			−0.821454	−	0.570275i	$-0.806836\pi$
$751$	−5.22604e11			−1.64291			−0.821454	+	0.570275i	$0.806836\pi$
$752$		0			0
$753$	3.58924e10	−	4.43962e11i	0.111641	−	1.38091i
$754$		0			0
$755$		−	3.63565e11i		−	1.11891i
$756$		0			0
$757$	−6.71867e10			−0.204597			−0.102299	−	0.994754i	$-0.532620\pi$
$757$	−6.71867e10			−0.204597			−0.102299	+	0.994754i	$0.532620\pi$
$758$		0			0
$759$	−1.80948e11	−	1.46288e10i	−0.545238	−	0.0440801i
$760$		0			0
$761$			6.17807e11i			1.84211i	0.389438	+	0.921053i	$0.372669\pi$
$761$			6.17807e11i			1.84211i	−0.389438	+	0.921053i	$0.627331\pi$
$762$		0			0
$763$	4.46973e11			1.31881
$764$		0			0
$765$	9.04757e11	+	1.47254e11i	2.64172	+	0.429953i
$766$		0			0
$767$		−	2.01290e11i		−	0.581623i
$768$		0			0
$769$	−1.52656e11			−0.436524			−0.218262	−	0.975890i	$-0.570039\pi$
$769$	−1.52656e11			−0.436524			−0.218262	+	0.975890i	$0.570039\pi$
$770$		0			0
$771$	1.06229e10	−	1.31397e11i	0.0300626	−	0.371851i
$772$		0			0
$773$			4.38021e11i			1.22681i	0.789769	+	0.613404i	$0.210200\pi$
$773$			4.38021e11i			1.22681i	−0.789769	+	0.613404i	$0.789800\pi$
$774$		0			0
$775$	1.37392e11			0.380851
$776$		0			0
$777$	2.40442e11	+	1.94387e10i	0.659671	+	0.0533315i
$778$		0			0
$779$		−	5.01769e11i		−	1.36255i
$780$		0			0
$781$	1.05122e10			0.0282547
$782$		0			0
$783$	−1.49645e11	+	6.06220e11i	−0.398120	+	1.61281i
$784$		0			0
$785$			3.80408e11i			1.00178i
$786$		0			0
$787$	−1.77780e11			−0.463429			−0.231714	−	0.972784i	$-0.574433\pi$
$787$	−1.77780e11			−0.463429			−0.231714	+	0.972784i	$0.574433\pi$
$788$		0			0
$789$	−2.31869e10	+	2.86805e11i	−0.0598322	+	0.740079i
$790$		0			0
$791$		−	1.84680e11i		−	0.471752i
$792$		0			0
$793$	−1.00634e11			−0.254479
$794$		0			0
$795$	−2.28598e11	−	1.84812e10i	−0.572275	−	0.0462659i
$796$		0			0
$797$			2.28725e11i			0.566867i	0.958992	+	0.283433i	$0.0914734\pi$
$797$			2.28725e11i			0.566867i	−0.958992	+	0.283433i	$0.908527\pi$
$798$		0			0
$799$	4.01950e11			0.986245
$800$		0			0
$801$	−8.04110e10	+	4.94061e11i	−0.195337	+	1.20019i
$802$		0			0
$803$			1.03564e11i			0.249084i
$804$		0			0
$805$	4.16532e11			0.991894
$806$		0			0
$807$	−1.41181e10	+	1.74630e11i	−0.0332876	+	0.411742i
$808$		0			0
$809$		−	6.42295e11i		−	1.49948i	−0.661733	−	0.749739i	$-0.730179\pi$
$809$		−	6.42295e11i		−	1.49948i	0.661733	−	0.749739i	$-0.269821\pi$
$810$		0			0
$811$	6.90320e11			1.59576			0.797879	−	0.602817i	$-0.205955\pi$
$811$	6.90320e11			1.59576			0.797879	+	0.602817i	$0.205955\pi$
$812$		0			0
$813$	−3.32947e11	−	2.69173e10i	−0.762102	−	0.0616126i
$814$		0			0
$815$			1.40485e11i			0.318420i
$816$		0			0
$817$	9.49067e11			2.13014
$818$		0			0
$819$	1.47210e11	+	2.39591e10i	0.327191	+	0.0532519i
$820$		0			0
$821$		−	2.41129e11i		−	0.530735i	−0.964147	−	0.265367i	$-0.914507\pi$
$821$		−	2.41129e11i		−	0.530735i	0.964147	−	0.265367i	$-0.0854932\pi$
$822$		0			0
$823$	6.33487e11			1.38082			0.690412	−	0.723416i	$-0.257429\pi$
$823$	6.33487e11			1.38082			0.690412	+	0.723416i	$0.257429\pi$
$824$		0			0
$825$	2.97194e10	−	3.67607e11i	0.0641540	−	0.793538i
$826$		0			0
$827$			5.42766e11i			1.16035i	0.814491	+	0.580177i	$0.197017\pi$
$827$			5.42766e11i			1.16035i	−0.814491	+	0.580177i	$0.802983\pi$
$828$		0			0
$829$	4.78621e11			1.01338			0.506691	−	0.862128i	$-0.330868\pi$
$829$	4.78621e11			1.01338			0.506691	+	0.862128i	$0.330868\pi$
$830$		0			0
$831$	−4.05265e11	−	3.27639e10i	−0.849836	−	0.0687055i
$832$		0			0
$833$		−	2.60239e11i		−	0.540497i
$834$		0			0
$835$	2.61765e11			0.538474
$836$		0			0
$837$	−1.55193e11	−	3.83092e10i	−0.316206	−	0.0780552i
$838$		0			0
$839$			3.51190e11i			0.708753i	0.935103	+	0.354376i	$0.115307\pi$
$839$			3.51190e11i			0.708753i	−0.935103	+	0.354376i	$0.884693\pi$
$840$		0			0
$841$	−8.80262e11			−1.75966
$842$		0			0
$843$	6.29566e10	−	7.78726e11i	0.124661	−	1.54196i
$844$		0			0
$845$			6.33463e11i			1.24249i
$846$		0			0
$847$	2.31428e11			0.449659
$848$		0			0
$849$	−3.00794e11	−	2.43179e10i	−0.578947	−	0.0468054i
$850$		0			0
$851$			3.32718e11i			0.634392i
$852$		0			0
$853$	−2.13313e11			−0.402922			−0.201461	−	0.979497i	$-0.564569\pi$
$853$	−2.13313e11			−0.402922			−0.201461	+	0.979497i	$0.564569\pi$
$854$		0			0
$855$	1.80639e11	−	1.10988e12i	0.338023	−	2.07688i
$856$		0			0
$857$			3.64415e11i			0.675575i	0.941222	+	0.337787i	$0.109678\pi$
$857$			3.64415e11i			0.675575i	−0.941222	+	0.337787i	$0.890322\pi$
$858$		0			0
$859$	5.33725e11			0.980269			0.490135	−	0.871647i	$-0.336948\pi$
$859$	5.33725e11			0.980269			0.490135	+	0.871647i	$0.336948\pi$
$860$		0			0
$861$	3.54018e10	−	4.37894e11i	0.0644188	−	0.796812i
$862$		0			0
$863$			7.88098e11i			1.42081i	0.703792	+	0.710406i	$0.251489\pi$
$863$			7.88098e11i			1.42081i	−0.703792	+	0.710406i	$0.748511\pi$
$864$		0			0
$865$	5.22067e10			0.0932528
$866$		0			0
$867$	−1.29658e12	−	1.04823e11i	−2.29469	−	0.185515i
$868$		0			0
$869$		−	6.97417e11i		−	1.22296i
$870$		0			0
$871$	−1.21107e11			−0.210425
$872$		0			0
$873$	3.04759e11	+	4.96011e10i	0.524686	+	0.0853953i
$874$		0			0
$875$			1.22545e11i			0.209056i
$876$		0			0
$877$	−6.10907e11			−1.03271			−0.516353	−	0.856376i	$-0.672711\pi$
$877$	−6.10907e11			−1.03271			−0.516353	+	0.856376i	$0.672711\pi$
$878$		0			0
$879$	7.34690e10	−	9.08757e11i	0.123069	−	1.52227i
$880$		0			0
$881$			2.52172e11i			0.418595i	0.977852	+	0.209297i	$0.0671177\pi$
$881$			2.52172e11i			0.418595i	−0.977852	+	0.209297i	$0.932882\pi$
$882$		0			0
$883$	−1.10902e11			−0.182431			−0.0912154	−	0.995831i	$-0.529075\pi$
$883$	−1.10902e11			−0.182431			−0.0912154	+	0.995831i	$0.529075\pi$
$884$		0			0
$885$	−1.32443e12	−	1.07074e11i	−2.15901	−	0.174547i
$886$		0			0
$887$			1.58698e11i			0.256377i	0.991750	+	0.128188i	$0.0409162\pi$
$887$			1.58698e11i			0.256377i	−0.991750	+	0.128188i	$0.959084\pi$
$888$		0			0
$889$	3.78156e11			0.605430
$890$		0			0
$891$	−1.36070e11	+	4.06948e11i	−0.215900	+	0.645696i
$892$		0			0
$893$		−	4.93079e11i		−	0.775374i
$894$		0			0
$895$	−9.39557e11			−1.46430
$896$		0			0
$897$	−1.65770e10	+	2.05045e11i	−0.0256056	+	0.316723i
$898$		0			0
$899$		−	3.53412e11i		−	0.541056i
$900$		0			0
$901$	4.66825e11			0.708362
$902$		0			0
$903$	8.28251e11	+	6.69605e10i	1.24569	+	0.100709i
$904$		0			0
$905$		−	5.86642e11i		−	0.874538i
$906$		0			0
$907$	−1.28126e12			−1.89326			−0.946629	−	0.322325i	$-0.895536\pi$
$907$	−1.28126e12			−1.89326			−0.946629	+	0.322325i	$0.895536\pi$
$908$		0			0
$909$	1.40192e11	−	8.61367e11i	0.205337	−	1.26163i
$910$		0			0
$911$		−	5.55958e11i		−	0.807176i	−0.914941	−	0.403588i	$-0.867763\pi$
$911$		−	5.55958e11i		−	0.807176i	0.914941	−	0.403588i	$-0.132237\pi$
$912$		0			0
$913$	−2.33282e11			−0.335736
$914$		0			0
$915$	−5.35312e10	+	6.62141e11i	−0.0763699	+	0.944639i
$916$		0			0
$917$			4.15619e11i			0.587784i
$918$		0			0
$919$	−2.94315e11			−0.412619			−0.206310	−	0.978487i	$-0.566145\pi$
$919$	−2.94315e11			−0.412619			−0.206310	+	0.978487i	$0.566145\pi$
$920$		0			0
$921$	7.83340e10	+	6.33296e9i	0.108871	+	0.00880174i
$922$		0			0
$923$		−	1.19122e10i		−	0.0164129i
$924$		0			0
$925$	−6.75937e11			−0.923292
$926$		0			0
$927$	−6.72478e11	−	1.09449e11i	−0.910666	−	0.148216i
$928$		0			0
$929$		−	6.48271e11i		−	0.870350i	−0.900346	−	0.435175i	$-0.856687\pi$
$929$		−	6.48271e11i		−	0.870350i	0.900346	−	0.435175i	$-0.143313\pi$
$930$		0			0
$931$	−3.19240e11			−0.424932
$932$		0			0
$933$	−8.05151e10	+	9.95912e11i	−0.106255	+	1.31430i
$934$		0			0
$935$			1.39268e12i			1.82224i
$936$		0			0
$937$	−9.26935e11			−1.20252			−0.601258	−	0.799055i	$-0.705334\pi$
$937$	−9.26935e11			−1.20252			−0.601258	+	0.799055i	$0.705334\pi$
$938$		0			0
$939$	−1.99756e11	−	1.61494e10i	−0.256943	−	0.0207727i
$940$		0			0
$941$			7.98701e11i			1.01865i	0.860574	+	0.509326i	$0.170105\pi$
$941$			7.98701e11i			1.01865i	−0.860574	+	0.509326i	$0.829895\pi$
$942$		0			0
$943$	6.05945e11			0.766278
$944$		0			0
$945$	2.35950e11	−	9.55850e11i	0.295865	−	1.19857i
$946$		0			0
$947$			3.06154e11i			0.380663i	0.981720	+	0.190331i	$0.0609562\pi$
$947$			3.06154e11i			0.380663i	−0.981720	+	0.190331i	$0.939044\pi$
$948$		0			0
$949$	1.17355e11			0.144690
$950$		0			0
$951$	−1.10598e11	+	1.36801e12i	−0.135215	+	1.67250i
$952$		0			0
$953$		−	1.70654e11i		−	0.206892i	−0.994635	−	0.103446i	$-0.967013\pi$
$953$		−	1.70654e11i		−	0.206892i	0.994635	−	0.103446i	$-0.0329869\pi$
$954$		0			0
$955$	1.15743e12			1.39149
$956$		0			0
$957$	−9.45589e11	−	7.64468e10i	−1.12734	−	0.0911405i
$958$		0			0
$959$		−	4.33433e11i		−	0.512446i
$960$		0			0
$961$	−7.62417e11			−0.893921
$962$		0			0
$963$	8.68626e10	−	5.33701e11i	0.101001	−	0.620573i
$964$		0			0
$965$		−	2.74417e10i		−	0.0316448i
$966$		0			0
$967$	−8.15354e11			−0.932482			−0.466241	−	0.884658i	$-0.654392\pi$
$967$	−8.15354e11			−0.932482			−0.466241	+	0.884658i	$0.654392\pi$
$968$		0			0
$969$	−1.84443e11	+	2.28143e12i	−0.209203	+	2.58768i
$970$		0			0
$971$		−	1.27970e12i		−	1.43956i	−0.694200	−	0.719782i	$-0.744242\pi$
$971$		−	1.27970e12i		−	1.43956i	0.694200	−	0.719782i	$-0.255758\pi$
$972$		0			0
$973$	−5.02060e11			−0.560150
$974$		0			0
$975$	−4.16561e11	−	3.36772e10i	−0.460957	−	0.0372664i
$976$		0			0
$977$			1.07767e12i			1.18279i	0.806382	+	0.591395i	$0.201423\pi$
$977$			1.07767e12i			1.18279i	−0.806382	+	0.591395i	$0.798577\pi$
$978$		0			0
$979$	−7.60503e11			−0.827885
$980$		0			0
$981$	1.43826e12	+	2.34085e11i	1.55297	+	0.252753i
$982$		0			0
$983$		−	8.12637e11i		−	0.870328i	−0.900351	−	0.435164i	$-0.856690\pi$
$983$		−	8.12637e11i		−	0.870328i	0.900351	−	0.435164i	$-0.143310\pi$
$984$		0			0
$985$	−2.01968e12			−2.14554
$986$		0			0
$987$	3.47887e10	−	4.30311e11i	0.0366581	−	0.453433i
$988$		0			0
$989$			1.14611e12i			1.19796i
$990$		0			0
$991$	−1.61365e12			−1.67307			−0.836537	−	0.547911i	$-0.815423\pi$
$991$	−1.61365e12			−1.67307			−0.836537	+	0.547911i	$0.815423\pi$
$992$		0			0
$993$	−1.39875e12	−	1.13083e11i	−1.43861	−	0.116305i
$994$		0			0
$995$			1.13572e12i			1.15873i
$996$		0			0
$997$	−1.63148e11			−0.165121			−0.0825604	−	0.996586i	$-0.526310\pi$
$997$	−1.63148e11			−0.165121			−0.0825604	+	0.996586i	$0.526310\pi$
$998$		0			0
$999$	7.63513e11	+	1.88472e11i	0.766575	+	0.189228i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.9.e.j.65.8		8
3.2	odd	2	inner	192.9.e.j.65.7		8
4.3	odd	2		192.9.e.i.65.1		8
8.3	odd	2		24.9.e.a.17.8	yes	8
8.5	even	2		48.9.e.e.17.1		8
12.11	even	2		192.9.e.i.65.2		8
24.5	odd	2		48.9.e.e.17.2		8
24.11	even	2		24.9.e.a.17.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.9.e.a.17.7	✓	8	24.11	even	2
24.9.e.a.17.8	yes	8	8.3	odd	2
48.9.e.e.17.1		8	8.5	even	2
48.9.e.e.17.2		8	24.5	odd	2
192.9.e.i.65.1		8	4.3	odd	2
192.9.e.i.65.2		8	12.11	even	2
192.9.e.j.65.7		8	3.2	odd	2	inner
192.9.e.j.65.8		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	192.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(80.7366 + 6.52720i) q^{3} -920.542i q^{5} +2012.50 q^{7} +(6475.79 + 1053.97i) q^{9} +9968.11i q^{11} +11295.6 q^{13} +(6008.56 - 74321.4i) q^{15} +151773. i q^{17} +186183. q^{19} +(162482. + 13136.0i) q^{21} +224838. i q^{23} -456773. q^{25} +(515954. + 127363. i) q^{27} +1.17495e6i q^{29} -300789. q^{31} +(-65063.8 + 804791. i) q^{33} -1.85259e6i q^{35} +1.47981e6 q^{37} +(911966. + 73728.5i) q^{39} -2.69503e6i q^{41} +5.09749e6 q^{43} +(970221. - 5.96124e6i) q^{45} -2.64836e6i q^{47} -1.71466e6 q^{49} +(-990655. + 1.22537e7i) q^{51} -3.07581e6i q^{53} +9.17606e6 q^{55} +(1.50318e7 + 1.21525e6i) q^{57} -1.78203e7i q^{59} -8.90915e6 q^{61} +(1.30325e7 + 2.12111e6i) q^{63} -1.03981e7i q^{65} -1.07217e7 q^{67} +(-1.46756e6 + 1.81527e7i) q^{69} -1.05459e6i q^{71} +1.03895e7 q^{73} +(-3.68783e7 - 2.98145e6i) q^{75} +2.00608e7i q^{77} -6.99649e7 q^{79} +(4.08250e7 + 1.36505e7i) q^{81} +2.34028e7i q^{83} +1.39714e8 q^{85} +(-7.66913e6 + 9.48615e7i) q^{87} +7.62936e7i q^{89} +2.27323e7 q^{91} +(-2.42847e7 - 1.96331e6i) q^{93} -1.71389e8i q^{95} +4.70613e7 q^{97} +(-1.05061e7 + 6.45514e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 56 q^{3} - 1584 q^{7} + 328 q^{9} - 25232 q^{13} - 38336 q^{15} + 157936 q^{19} - 30480 q^{21} - 579704 q^{25} - 276040 q^{27} - 805552 q^{31} + 102848 q^{33} + 3985008 q^{37} + 2297104 q^{39} + 6962672 q^{43}+ \cdots - 369701504 q^{99}+O(q^{100}) \) 8 * q + 56 * q^3 - 1584 * q^7 + 328 * q^9 - 25232 * q^13 - 38336 * q^15 + 157936 * q^19 - 30480 * q^21 - 579704 * q^25 - 276040 * q^27 - 805552 * q^31 + 102848 * q^33 + 3985008 * q^37 + 2297104 * q^39 + 6962672 * q^43 - 8670592 * q^45 - 5884520 * q^49 - 15590144 * q^51 - 27101312 * q^55 + 36756688 * q^57 + 51583600 * q^61 + 69759312 * q^63 + 58200688 * q^67 - 94226048 * q^69 - 116854768 * q^73 - 143181896 * q^75 - 172454576 * q^79 + 194700040 * q^81 + 264333824 * q^85 + 242851008 * q^87 + 382128480 * q^91 - 313470352 * q^93 - 337326704 * q^97 - 369701504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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