LMFDB - Embedded newform 240.8.f.e.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [240,8,Mod(49,240)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("240.49"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	240.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-444] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$74.9724061162$
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} + 162x^{6} + 7361x^{4} + 87300x^{2} + 160000 $ x^8 + 162x^6 + 7361x^4 + 87300*x^2 + 160000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{12}\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-7.26440i$ of defining polynomial
Character	$\chi$	$=$	240.49
Dual form			240.8.f.e.49.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000i q^{3} +(-272.800 + 60.8703i) q^{5} -1505.28i q^{7} -729.000 q^{9} -1596.37 q^{11} -956.100i q^{13} +(1643.50 + 7365.60i) q^{15} -32470.6i q^{17} -39168.1 q^{19} -40642.5 q^{21} -59361.5i q^{23} +(70714.6 - 33210.8i) q^{25} +19683.0i q^{27} -66150.5 q^{29} +19664.1 q^{31} +43102.1i q^{33} +(91626.6 + 410640. i) q^{35} -376045. i q^{37} -25814.7 q^{39} +385003. q^{41} -466410. i q^{43} +(198871. - 44374.4i) q^{45} +468903. i q^{47} -1.44232e6 q^{49} -876706. q^{51} +1.60516e6i q^{53} +(435491. - 97171.8i) q^{55} +1.05754e6i q^{57} -2.04044e6 q^{59} -378667. q^{61} +1.09735e6i q^{63} +(58198.1 + 260824. i) q^{65} -4644.40i q^{67} -1.60276e6 q^{69} +2.79333e6 q^{71} -2.01174e6i q^{73} +(-896692. - 1.90929e6i) q^{75} +2.40299e6i q^{77} +1.76767e6 q^{79} +531441. q^{81} +3.06625e6i q^{83} +(1.97649e6 + 8.85797e6i) q^{85} +1.78606e6i q^{87} +6.14397e6 q^{89} -1.43920e6 q^{91} -530930. i q^{93} +(1.06851e7 - 2.38417e6i) q^{95} -3.02969e6i q^{97} +1.16376e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 444 q^{5} - 5832 q^{9} - 10752 q^{11} + 17496 q^{15} + 30464 q^{19} + 64152 q^{21} + 127616 q^{25} + 240072 q^{29} - 233728 q^{31} - 593520 q^{35} + 454896 q^{39} + 507648 q^{41} + 323676 q^{45} - 3267160 q^{49}+ \cdots + 7838208 q^{99}+O(q^{100}) $ 8 * q - 444 * q^5 - 5832 * q^9 - 10752 * q^11 + 17496 * q^15 + 30464 * q^19 + 64152 * q^21 + 127616 * q^25 + 240072 * q^29 - 233728 * q^31 - 593520 * q^35 + 454896 * q^39 + 507648 * q^41 + 323676 * q^45 - 3267160 * q^49 - 264384 * q^51 - 3525696 * q^55 - 1091424 * q^59 - 6433520 * q^61 - 2555592 * q^65 - 5940864 * q^69 + 1381824 * q^71 - 7768224 * q^75 + 14380160 * q^79 + 4251528 * q^81 - 1452008 * q^85 + 45778896 * q^89 - 24075648 * q^91 + 25774656 * q^95 + 7838208 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		−	27.0000i		−	0.577350i
$3$		−	27.0000i		−	0.577350i
$4$		0			0
$5$	−272.800	+	60.8703i	−0.975999	+	0.217776i
$5$	−272.800	+	60.8703i	−0.975999	+	0.217776i
$6$		0			0
$7$		−	1505.28i		−	1.65872i	−0.558714	−	0.829361i	$-0.688705\pi$
$7$		−	1505.28i		−	1.65872i	0.558714	−	0.829361i	$-0.311295\pi$
$8$		0			0
$9$	−729.000			−0.333333
$10$		0			0
$11$	−1596.37			−0.361627			−0.180813	−	0.983517i	$-0.557873\pi$
$11$	−1596.37			−0.361627			−0.180813	+	0.983517i	$0.557873\pi$
$12$		0			0
$13$		−	956.100i		−	0.120698i	−0.998177	−	0.0603492i	$-0.980779\pi$
$13$		−	956.100i		−	0.120698i	0.998177	−	0.0603492i	$-0.0192214\pi$
$14$		0			0
$15$	1643.50	+	7365.60i	0.125733	+	0.563493i
$16$		0			0
$17$		−	32470.6i		−	1.60295i	−0.598031	−	0.801473i	$-0.704050\pi$
$17$		−	32470.6i		−	1.60295i	0.598031	−	0.801473i	$-0.295950\pi$
$18$		0			0
$19$	−39168.1			−1.31007			−0.655036	−	0.755598i	$-0.727347\pi$
$19$	−39168.1			−1.31007			−0.655036	+	0.755598i	$0.727347\pi$
$20$		0			0
$21$	−40642.5			−0.957663
$22$		0			0
$23$		−	59361.5i		−	1.01732i	−0.860968	−	0.508660i	$-0.830141\pi$
$23$		−	59361.5i		−	1.01732i	0.860968	−	0.508660i	$-0.169859\pi$
$24$		0			0
$25$	70714.6	−	33210.8i	0.905147	−	0.425098i
$26$		0			0
$27$			19683.0i			0.192450i
$28$		0			0
$29$	−66150.5			−0.503663			−0.251831	−	0.967771i	$-0.581033\pi$
$29$	−66150.5			−0.503663			−0.251831	+	0.967771i	$0.581033\pi$
$30$		0			0
$31$	19664.1			0.118552			0.0592758	−	0.998242i	$-0.481121\pi$
$31$	19664.1			0.118552			0.0592758	+	0.998242i	$0.481121\pi$
$32$		0			0
$33$			43102.1i			0.208785i
$34$		0			0
$35$	91626.6	+	410640.i	0.361230	+	1.61891i
$36$		0			0
$37$		−	376045.i		−	1.22049i	−0.792213	−	0.610245i	$-0.791071\pi$
$37$		−	376045.i		−	1.22049i	0.792213	−	0.610245i	$-0.208929\pi$
$38$		0			0
$39$	−25814.7			−0.0696853
$40$		0			0
$41$	385003.			0.872409			0.436205	−	0.899848i	$-0.356322\pi$
$41$	385003.			0.872409			0.436205	+	0.899848i	$0.356322\pi$
$42$		0			0
$43$		−	466410.i		−	0.894600i	−0.894384	−	0.447300i	$-0.852386\pi$
$43$		−	466410.i		−	0.894600i	0.894384	−	0.447300i	$-0.147614\pi$
$44$		0			0
$45$	198871.	−	44374.4i	0.325333	−	0.0725920i
$46$		0			0
$47$			468903.i			0.658779i	0.944194	+	0.329390i	$0.106843\pi$
$47$			468903.i			0.658779i	−0.944194	+	0.329390i	$0.893157\pi$
$48$		0			0
$49$	−1.44232e6			−1.75136
$50$		0			0
$51$	−876706.			−0.925461
$52$		0			0
$53$			1.60516e6i			1.48100i	0.672058	+	0.740498i	$0.265410\pi$
$53$			1.60516e6i			1.48100i	−0.672058	+	0.740498i	$0.734590\pi$
$54$		0			0
$55$	435491.	−	97171.8i	0.352947	−	0.0787536i
$56$		0			0
$57$			1.05754e6i			0.756371i
$58$		0			0
$59$	−2.04044e6			−1.29342			−0.646712	−	0.762734i	$-0.723856\pi$
$59$	−2.04044e6			−1.29342			−0.646712	+	0.762734i	$0.723856\pi$
$60$		0			0
$61$	−378667.			−0.213601			−0.106800	−	0.994280i	$-0.534061\pi$
$61$	−378667.			−0.213601			−0.106800	+	0.994280i	$0.534061\pi$
$62$		0			0
$63$			1.09735e6i			0.552907i
$64$		0			0
$65$	58198.1	+	260824.i	0.0262852	+	0.117801i
$66$		0			0
$67$		−	4644.40i		−	0.00188655i	−1.00000		0.000943274i	$-0.999700\pi$
$67$		−	4644.40i		−	0.00188655i	1.00000		0.000943274i	$-0.000300253\pi$
$68$		0			0
$69$	−1.60276e6			−0.587350
$70$		0			0
$71$	2.79333e6			0.926229			0.463115	−	0.886298i	$-0.346732\pi$
$71$	2.79333e6			0.926229			0.463115	+	0.886298i	$0.346732\pi$
$72$		0			0
$73$		−	2.01174e6i		−	0.605259i	−0.953108	−	0.302630i	$-0.902136\pi$
$73$		−	2.01174e6i		−	0.605259i	0.953108	−	0.302630i	$-0.0978645\pi$
$74$		0			0
$75$	−896692.	−	1.90929e6i	−0.245431	−	0.522587i
$76$		0			0
$77$			2.40299e6i			0.599838i
$78$		0			0
$79$	1.76767e6			0.403372			0.201686	−	0.979450i	$-0.435358\pi$
$79$	1.76767e6			0.403372			0.201686	+	0.979450i	$0.435358\pi$
$80$		0			0
$81$	531441.			0.111111
$82$		0			0
$83$			3.06625e6i			0.588619i	0.955710	+	0.294310i	$0.0950897\pi$
$83$			3.06625e6i			0.588619i	−0.955710	+	0.294310i	$0.904910\pi$
$84$		0			0
$85$	1.97649e6	+	8.85797e6i	0.349083	+	1.56447i
$86$		0			0
$87$			1.78606e6i			0.290790i
$88$		0			0
$89$	6.14397e6			0.923813			0.461907	−	0.886929i	$-0.347166\pi$
$89$	6.14397e6			0.923813			0.461907	+	0.886929i	$0.347166\pi$
$90$		0			0
$91$	−1.43920e6			−0.200205
$92$		0			0
$93$		−	530930.i		−	0.0684459i
$94$		0			0
$95$	1.06851e7	−	2.38417e6i	1.27863	−	0.285302i
$96$		0			0
$97$		−	3.02969e6i		−	0.337053i	−0.985697	−	0.168526i	$-0.946099\pi$
$97$		−	3.02969e6i		−	0.337053i	0.985697	−	0.168526i	$-0.0539008\pi$
$98$		0			0
$99$	1.16376e6			0.120542
$100$		0			0
$101$	7.17509e6			0.692951			0.346475	−	0.938059i	$-0.387378\pi$
$101$	7.17509e6			0.692951			0.346475	+	0.938059i	$0.387378\pi$
$102$		0			0
$103$			1.45694e7i			1.31374i	0.754002	+	0.656872i	$0.228121\pi$
$103$			1.45694e7i			1.31374i	−0.754002	+	0.656872i	$0.771879\pi$
$104$		0			0
$105$	1.10873e7	−	2.47392e6i	0.934678	−	0.208556i
$106$		0			0
$107$			1.64442e7i			1.29769i	0.760923	+	0.648843i	$0.224747\pi$
$107$			1.64442e7i			1.29769i	−0.760923	+	0.648843i	$0.775253\pi$
$108$		0			0
$109$	−2.34832e7			−1.73686			−0.868430	−	0.495812i	$-0.834871\pi$
$109$	−2.34832e7			−1.73686			−0.868430	+	0.495812i	$0.834871\pi$
$110$		0			0
$111$	−1.01532e7			−0.704650
$112$		0			0
$113$			1.25747e7i			0.819826i	0.912125	+	0.409913i	$0.134441\pi$
$113$			1.25747e7i			0.819826i	−0.912125	+	0.409913i	$0.865559\pi$
$114$		0			0
$115$	3.61335e6	+	1.61938e7i	0.221548	+	0.992903i
$116$		0			0
$117$			696997.i			0.0402328i
$118$		0			0
$119$	−4.88772e7			−2.65884
$120$		0			0
$121$	−1.69388e7			−0.869226
$122$		0			0
$123$		−	1.03951e7i		−	0.503686i
$124$		0			0
$125$	−1.72694e7	+	1.33643e7i	−0.790846	+	0.612015i
$126$		0			0
$127$			2.41070e7i			1.04431i	0.852850	+	0.522155i	$0.174872\pi$
$127$			2.41070e7i			1.04431i	−0.852850	+	0.522155i	$0.825128\pi$
$128$		0			0
$129$	−1.25931e7			−0.516497
$130$		0			0
$131$	2.08040e7			0.808533			0.404266	−	0.914641i	$-0.367527\pi$
$131$	2.08040e7			0.808533			0.404266	+	0.914641i	$0.367527\pi$
$132$		0			0
$133$			5.89589e7i			2.17304i
$134$		0			0
$135$	−1.19811e6	−	5.36952e6i	−0.0419110	−	0.187831i
$136$		0			0
$137$		−	9.94308e6i		−	0.330369i	−0.986263	−	0.165184i	$-0.947178\pi$
$137$		−	9.94308e6i		−	0.330369i	0.986263	−	0.165184i	$-0.0528219\pi$
$138$		0			0
$139$	1.47414e7			0.465571			0.232785	−	0.972528i	$-0.425216\pi$
$139$	1.47414e7			0.465571			0.232785	+	0.972528i	$0.425216\pi$
$140$		0			0
$141$	1.26604e7			0.380347
$142$		0			0
$143$			1.52629e6i			0.0436478i
$144$		0			0
$145$	1.80458e7	−	4.02660e6i	0.491574	−	0.109686i
$146$		0			0
$147$			3.89425e7i			1.01115i
$148$		0			0
$149$	2.21899e7			0.549545			0.274772	−	0.961509i	$-0.411398\pi$
$149$	2.21899e7			0.549545			0.274772	+	0.961509i	$0.411398\pi$
$150$		0			0
$151$	−4.90759e7			−1.15998			−0.579988	−	0.814625i	$-0.696943\pi$
$151$	−4.90759e7			−1.15998			−0.579988	+	0.814625i	$0.696943\pi$
$152$		0			0
$153$			2.36711e7i			0.534315i
$154$		0			0
$155$	−5.36436e6	+	1.19696e6i	−0.115706	+	0.0258177i
$156$		0			0
$157$			6.21368e6i			0.128145i	0.997945	+	0.0640723i	$0.0204088\pi$
$157$			6.21368e6i			0.128145i	−0.997945	+	0.0640723i	$0.979591\pi$
$158$		0			0
$159$	4.33394e7			0.855054
$160$		0			0
$161$	−8.93555e7			−1.68745
$162$		0			0
$163$		−	1.54205e7i		−	0.278895i	−0.990229	−	0.139448i	$-0.955467\pi$
$163$		−	1.54205e7i		−	0.278895i	0.990229	−	0.139448i	$-0.0445327\pi$
$164$		0			0
$165$	−2.62364e6	−	1.17583e7i	−0.0454684	−	0.203774i
$166$		0			0
$167$			7.74611e6i			0.128699i	0.997927	+	0.0643496i	$0.0204973\pi$
$167$			7.74611e6i			0.128699i	−0.997927	+	0.0643496i	$0.979503\pi$
$168$		0			0
$169$	6.18344e7			0.985432
$170$		0			0
$171$	2.85536e7			0.436691
$172$		0			0
$173$			1.00086e7i			0.146965i	0.997297	+	0.0734824i	$0.0234113\pi$
$173$			1.00086e7i			0.146965i	−0.997297	+	0.0734824i	$0.976589\pi$
$174$		0			0
$175$	−4.99915e7	−	1.06445e8i	−0.705120	−	1.50139i
$176$		0			0
$177$			5.50918e7i			0.746758i
$178$		0			0
$179$	1.09231e8			1.42350			0.711751	−	0.702431i	$-0.247902\pi$
$179$	1.09231e8			1.42350			0.711751	+	0.702431i	$0.247902\pi$
$180$		0			0
$181$	−1.27824e7			−0.160228			−0.0801138	−	0.996786i	$-0.525528\pi$
$181$	−1.27824e7			−0.160228			−0.0801138	+	0.996786i	$0.525528\pi$
$182$		0			0
$183$			1.02240e7i			0.123322i
$184$		0			0
$185$	2.28900e7	+	1.02585e8i	0.265794	+	1.19120i
$186$		0			0
$187$			5.18352e7i			0.579668i
$188$		0			0
$189$	2.96284e7			0.319221
$190$		0			0
$191$	−1.54482e8			−1.60421			−0.802103	−	0.597185i	$-0.796286\pi$
$191$	−1.54482e8			−1.60421			−0.802103	+	0.597185i	$0.796286\pi$
$192$		0			0
$193$		−	1.51365e8i		−	1.51556i	−0.652507	−	0.757782i	$-0.726283\pi$
$193$		−	1.51365e8i		−	1.51556i	0.652507	−	0.757782i	$-0.273717\pi$
$194$		0			0
$195$	7.04225e6	−	1.57135e6i	0.0680127	−	0.0151758i
$196$		0			0
$197$		−	1.23829e8i		−	1.15396i	−0.816758	−	0.576981i	$-0.804231\pi$
$197$		−	1.23829e8i		−	1.15396i	0.816758	−	0.576981i	$-0.195769\pi$
$198$		0			0
$199$	4.77920e7			0.429902			0.214951	−	0.976625i	$-0.431041\pi$
$199$	4.77920e7			0.429902			0.214951	+	0.976625i	$0.431041\pi$
$200$		0			0
$201$	−125399.			−0.00108920
$202$		0			0
$203$			9.95748e7i			0.835436i
$204$		0			0
$205$	−1.05029e8	+	2.34352e7i	−0.851470	+	0.189990i
$206$		0			0
$207$			4.32745e7i			0.339107i
$208$		0			0
$209$	6.25270e7			0.473757
$210$		0			0
$211$	−6.28256e7			−0.460413			−0.230207	−	0.973142i	$-0.573940\pi$
$211$	−6.28256e7			−0.460413			−0.230207	+	0.973142i	$0.573940\pi$
$212$		0			0
$213$		−	7.54200e7i		−	0.534759i
$214$		0			0
$215$	2.83905e7	+	1.27237e8i	0.194822	+	0.873128i
$216$		0			0
$217$		−	2.95999e7i		−	0.196644i
$218$		0			0
$219$	−5.43169e7			−0.349447
$220$		0			0
$221$	−3.10451e7			−0.193473
$222$		0			0
$223$		−	5.17882e6i		−	0.0312726i	−0.999878	−	0.0156363i	$-0.995023\pi$
$223$		−	5.17882e6i		−	0.0312726i	0.999878	−	0.0156363i	$-0.00497739\pi$
$224$		0			0
$225$	−5.15510e7	+	2.42107e7i	−0.301716	+	0.141699i
$226$		0			0
$227$		−	2.43478e8i		−	1.38156i	−0.723067	−	0.690778i	$-0.757268\pi$
$227$		−	2.43478e8i		−	1.38156i	0.723067	−	0.690778i	$-0.242732\pi$
$228$		0			0
$229$	1.09734e8			0.603832			0.301916	−	0.953335i	$-0.402374\pi$
$229$	1.09734e8			0.603832			0.301916	+	0.953335i	$0.402374\pi$
$230$		0			0
$231$	6.48806e7			0.346317
$232$		0			0
$233$			1.85771e8i			0.962129i	0.876685	+	0.481064i	$0.159750\pi$
$233$			1.85771e8i			0.962129i	−0.876685	+	0.481064i	$0.840250\pi$
$234$		0			0
$235$	−2.85422e7	−	1.27917e8i	−0.143466	−	0.642968i
$236$		0			0
$237$		−	4.77270e7i		−	0.232887i
$238$		0			0
$239$	3.06316e8			1.45137			0.725684	−	0.688028i	$-0.241523\pi$
$239$	3.06316e8			1.45137			0.725684	+	0.688028i	$0.241523\pi$
$240$		0			0
$241$	−5.78177e7			−0.266073			−0.133037	−	0.991111i	$-0.542473\pi$
$241$	−5.78177e7			−0.266073			−0.133037	+	0.991111i	$0.542473\pi$
$242$		0			0
$243$		−	1.43489e7i		−	0.0641500i
$244$		0			0
$245$	3.93464e8	−	8.77942e7i	1.70932	−	0.381403i
$246$		0			0
$247$			3.74486e7i			0.158124i
$248$		0			0
$249$	8.27888e7			0.339840
$250$		0			0
$251$	3.90109e8			1.55714			0.778571	−	0.627557i	$-0.215945\pi$
$251$	3.90109e8			1.55714			0.778571	+	0.627557i	$0.215945\pi$
$252$		0			0
$253$			9.47632e7i			0.367890i
$254$		0			0
$255$	2.39165e8	−	5.33653e7i	0.903249	−	0.201543i
$256$		0			0
$257$		−	5.32200e8i		−	1.95573i	−0.209237	−	0.977865i	$-0.567098\pi$
$257$		−	5.32200e8i		−	1.95573i	0.209237	−	0.977865i	$-0.432902\pi$
$258$		0			0
$259$	−5.66053e8			−2.02445
$260$		0			0
$261$	4.82237e7			0.167888
$262$		0			0
$263$		−	3.12888e8i		−	1.06058i	−0.847816	−	0.530291i	$-0.822083\pi$
$263$		−	3.12888e8i		−	1.06058i	0.847816	−	0.530291i	$-0.177917\pi$
$264$		0			0
$265$	−9.77068e7	−	4.37889e8i	−0.322526	−	1.44545i
$266$		0			0
$267$		−	1.65887e8i		−	0.533364i
$268$		0			0
$269$	−2.95534e8			−0.925710			−0.462855	−	0.886434i	$-0.653175\pi$
$269$	−2.95534e8			−0.925710			−0.462855	+	0.886434i	$0.653175\pi$
$270$		0			0
$271$	−3.96274e8			−1.20949			−0.604746	−	0.796419i	$-0.706725\pi$
$271$	−3.96274e8			−1.20949			−0.604746	+	0.796419i	$0.706725\pi$
$272$		0			0
$273$			3.88583e7i			0.115588i
$274$		0			0
$275$	−1.12887e8	+	5.30169e7i	−0.327325	+	0.153727i
$276$		0			0
$277$			4.00402e8i			1.13192i	0.824432	+	0.565961i	$0.191495\pi$
$277$			4.00402e8i			1.13192i	−0.824432	+	0.565961i	$0.808505\pi$
$278$		0			0
$279$	−1.43351e7			−0.0395172
$280$		0			0
$281$	3.58146e8			0.962916			0.481458	−	0.876469i	$-0.340107\pi$
$281$	3.58146e8			0.962916			0.481458	+	0.876469i	$0.340107\pi$
$282$		0			0
$283$		−	3.95688e8i		−	1.03777i	−0.854845	−	0.518884i	$-0.826348\pi$
$283$		−	3.95688e8i		−	1.03777i	0.854845	−	0.518884i	$-0.173652\pi$
$284$		0			0
$285$	−6.43727e7	−	2.88497e8i	−0.164719	−	0.738217i
$286$		0			0
$287$		−	5.79536e8i		−	1.44708i
$288$		0			0
$289$	−6.44000e8			−1.56943
$290$		0			0
$291$	−8.18017e7			−0.194597
$292$		0			0
$293$			1.93280e8i			0.448900i	0.974486	+	0.224450i	$0.0720586\pi$
$293$			1.93280e8i			0.448900i	−0.974486	+	0.224450i	$0.927941\pi$
$294$		0			0
$295$	5.56631e8	−	1.24202e8i	1.26238	−	0.281677i
$296$		0			0
$297$		−	3.14214e7i		−	0.0695951i
$298$		0			0
$299$	−5.67555e7			−0.122789
$300$		0			0
$301$	−7.02077e8			−1.48389
$302$		0			0
$303$		−	1.93727e8i		−	0.400075i
$304$		0			0
$305$	1.03300e8	−	2.30496e7i	0.208474	−	0.0465171i
$306$		0			0
$307$		−	2.11267e8i		−	0.416722i	−0.978052	−	0.208361i	$-0.933187\pi$
$307$		−	2.11267e8i		−	0.416722i	0.978052	−	0.208361i	$-0.0668129\pi$
$308$		0			0
$309$	3.93373e8			0.758490
$310$		0			0
$311$	−2.59442e8			−0.489079			−0.244540	−	0.969639i	$-0.578637\pi$
$311$	−2.59442e8			−0.489079			−0.244540	+	0.969639i	$0.578637\pi$
$312$		0			0
$313$		−	9.29772e8i		−	1.71384i	−0.515446	−	0.856922i	$-0.672374\pi$
$313$		−	9.29772e8i		−	1.71384i	0.515446	−	0.856922i	$-0.327626\pi$
$314$		0			0
$315$	−6.67958e7	−	2.99356e8i	−0.120410	−	0.539637i
$316$		0			0
$317$			1.63510e8i			0.288295i	0.989556	+	0.144147i	$0.0460439\pi$
$317$			1.63510e8i			0.288295i	−0.989556	+	0.144147i	$0.953956\pi$
$318$		0			0
$319$	1.05601e8			0.182138
$320$		0			0
$321$	4.43993e8			0.749219
$322$		0			0
$323$			1.27181e9i			2.09997i
$324$		0			0
$325$	−3.17529e7	−	6.76102e7i	−0.0513087	−	0.109250i
$326$		0			0
$327$			6.34047e8i			1.00278i
$328$		0			0
$329$	7.05828e8			1.09273
$330$		0			0
$331$	2.57948e8			0.390962			0.195481	−	0.980707i	$-0.437373\pi$
$331$	2.57948e8			0.390962			0.195481	+	0.980707i	$0.437373\pi$
$332$		0			0
$333$			2.74137e8i			0.406830i
$334$		0			0
$335$	282706.	+	1.26699e6i	0.000410845	+	0.00184127i
$336$		0			0
$337$		−	2.34282e8i		−	0.333452i	−0.986003	−	0.166726i	$-0.946680\pi$
$337$		−	2.34282e8i		−	0.333452i	0.986003	−	0.166726i	$-0.0533196\pi$
$338$		0			0
$339$	3.39516e8			0.473327
$340$		0			0
$341$	−3.13912e7			−0.0428715
$342$		0			0
$343$			9.31426e8i			1.24629i
$344$		0			0
$345$	4.37233e8	−	9.75605e7i	0.573253	−	0.127911i
$346$		0			0
$347$		−	6.73974e8i		−	0.865944i	−0.901407	−	0.432972i	$-0.857465\pi$
$347$		−	6.73974e8i		−	0.865944i	0.901407	−	0.432972i	$-0.142535\pi$
$348$		0			0
$349$	−8.77464e8			−1.10494			−0.552472	−	0.833531i	$-0.686315\pi$
$349$	−8.77464e8			−1.10494			−0.552472	+	0.833531i	$0.686315\pi$
$350$		0			0
$351$	1.88189e7			0.0232284
$352$		0			0
$353$		−	6.75088e8i		−	0.816863i	−0.912789	−	0.408431i	$-0.866076\pi$
$353$		−	6.75088e8i		−	0.816863i	0.912789	−	0.408431i	$-0.133924\pi$
$354$		0			0
$355$	−7.62021e8	+	1.70031e8i	−0.903999	+	0.201711i
$356$		0			0
$357$			1.31969e9i			1.53508i
$358$		0			0
$359$	−7.40244e8			−0.844393			−0.422196	−	0.906504i	$-0.638741\pi$
$359$	−7.40244e8			−0.844393			−0.422196	+	0.906504i	$0.638741\pi$
$360$		0			0
$361$	6.40271e8			0.716290
$362$		0			0
$363$			4.57346e8i			0.501848i
$364$		0			0
$365$	1.22455e8	+	5.48802e8i	0.131811	+	0.590732i
$366$		0			0
$367$		−	1.05986e9i		−	1.11922i	−0.828755	−	0.559611i	$-0.810951\pi$
$367$		−	1.05986e9i		−	1.11922i	0.828755	−	0.559611i	$-0.189049\pi$
$368$		0			0
$369$	−2.80667e8			−0.290803
$370$		0			0
$371$	2.41622e9			2.45656
$372$		0			0
$373$			1.28239e9i			1.27950i	0.768585	+	0.639748i	$0.220961\pi$
$373$			1.28239e9i			1.27950i	−0.768585	+	0.639748i	$0.779039\pi$
$374$		0			0
$375$	3.60837e8	+	4.66274e8i	0.353347	+	0.456595i
$376$		0			0
$377$			6.32465e7i			0.0607913i
$378$		0			0
$379$	−2.97355e8			−0.280568			−0.140284	−	0.990111i	$-0.544802\pi$
$379$	−2.97355e8			−0.280568			−0.140284	+	0.990111i	$0.544802\pi$
$380$		0			0
$381$	6.50888e8			0.602933
$382$		0			0
$383$			8.10508e8i			0.737160i	0.929596	+	0.368580i	$0.120156\pi$
$383$			8.10508e8i			0.737160i	−0.929596	+	0.368580i	$0.879844\pi$
$384$		0			0
$385$	−1.46270e8	−	6.55535e8i	−0.130630	−	0.585441i
$386$		0			0
$387$			3.40013e8i			0.298200i
$388$		0			0
$389$	3.38899e8			0.291908			0.145954	−	0.989291i	$-0.453375\pi$
$389$	3.38899e8			0.291908			0.145954	+	0.989291i	$0.453375\pi$
$390$		0			0
$391$	−1.92750e9			−1.63071
$392$		0			0
$393$		−	5.61708e8i		−	0.466807i
$394$		0			0
$395$	−4.82220e8	+	1.07598e8i	−0.393691	+	0.0878448i
$396$		0			0
$397$		−	1.00265e9i		−	0.804235i	−0.915588	−	0.402117i	$-0.868274\pi$
$397$		−	1.00265e9i		−	0.804235i	0.915588	−	0.402117i	$-0.131726\pi$
$398$		0			0
$399$	1.59189e9			1.25461
$400$		0			0
$401$	−1.09336e9			−0.846755			−0.423377	−	0.905953i	$-0.639156\pi$
$401$	−1.09336e9			−0.846755			−0.423377	+	0.905953i	$0.639156\pi$
$402$		0			0
$403$		−	1.88008e7i		−	0.0143090i
$404$		0			0
$405$	−1.44977e8	+	3.23490e7i	−0.108444	+	0.0241973i
$406$		0			0
$407$			6.00309e8i			0.441362i
$408$		0			0
$409$	−1.09511e9			−0.791453			−0.395726	−	0.918368i	$-0.629507\pi$
$409$	−1.09511e9			−0.791453			−0.395726	+	0.918368i	$0.629507\pi$
$410$		0			0
$411$	−2.68463e8			−0.190738
$412$		0			0
$413$			3.07142e9i			2.14543i
$414$		0			0
$415$	−1.86644e8	−	8.36473e8i	−0.128187	−	0.574492i
$416$		0			0
$417$		−	3.98016e8i		−	0.268797i
$418$		0			0
$419$	−1.75196e9			−1.16352			−0.581762	−	0.813359i	$-0.697636\pi$
$419$	−1.75196e9			−1.16352			−0.581762	+	0.813359i	$0.697636\pi$
$420$		0			0
$421$	−4.58026e8			−0.299160			−0.149580	−	0.988750i	$-0.547792\pi$
$421$	−4.58026e8			−0.299160			−0.149580	+	0.988750i	$0.547792\pi$
$422$		0			0
$423$		−	3.41830e8i		−	0.219593i
$424$		0			0
$425$	−1.07837e9	−	2.29614e9i	−0.681410	−	1.45090i
$426$		0			0
$427$			5.69999e8i			0.354304i
$428$		0			0
$429$	4.12099e7			0.0252000
$430$		0			0
$431$	7.60129e8			0.457317			0.228658	−	0.973507i	$-0.426566\pi$
$431$	7.60129e8			0.457317			0.228658	+	0.973507i	$0.426566\pi$
$432$		0			0
$433$			2.48461e9i			1.47079i	0.677638	+	0.735396i	$0.263004\pi$
$433$			2.48461e9i			1.47079i	−0.677638	+	0.735396i	$0.736996\pi$
$434$		0			0
$435$	−1.08718e8	−	4.87238e8i	−0.0633271	−	0.283811i
$436$		0			0
$437$			2.32508e9i			1.33276i
$438$		0			0
$439$	1.69934e8			0.0958637			0.0479318	−	0.998851i	$-0.484737\pi$
$439$	1.69934e8			0.0958637			0.0479318	+	0.998851i	$0.484737\pi$
$440$		0			0
$441$	1.05145e9			0.583785
$442$		0			0
$443$		−	1.53246e9i		−	0.837481i	−0.908106	−	0.418741i	$-0.862472\pi$
$443$		−	1.53246e9i		−	0.837481i	0.908106	−	0.418741i	$-0.137528\pi$
$444$		0			0
$445$	−1.67607e9	+	3.73985e8i	−0.901640	+	0.201184i
$446$		0			0
$447$		−	5.99127e8i		−	0.317280i
$448$		0			0
$449$	3.45325e9			1.80039			0.900194	−	0.435489i	$-0.143425\pi$
$449$	3.45325e9			1.80039			0.900194	+	0.435489i	$0.143425\pi$
$450$		0			0
$451$	−6.14609e8			−0.315486
$452$		0			0
$453$			1.32505e9i			0.669713i
$454$		0			0
$455$	3.92612e8	−	8.76042e7i	0.195400	−	0.0435999i
$456$		0			0
$457$			7.56279e8i			0.370659i	0.982676	+	0.185330i	$0.0593353\pi$
$457$			7.56279e8i			0.370659i	−0.982676	+	0.185330i	$0.940665\pi$
$458$		0			0
$459$	6.39118e8			0.308487
$460$		0			0
$461$	−8.78272e8			−0.417518			−0.208759	−	0.977967i	$-0.566942\pi$
$461$	−8.78272e8			−0.417518			−0.208759	+	0.977967i	$0.566942\pi$
$462$		0			0
$463$			3.54973e9i			1.66212i	0.556185	+	0.831058i	$0.312264\pi$
$463$			3.54973e9i			1.66212i	−0.556185	+	0.831058i	$0.687736\pi$
$464$		0			0
$465$	3.23179e7	+	1.44838e8i	0.0149059	+	0.0668031i
$466$		0			0
$467$			3.03363e9i			1.37833i	0.724605	+	0.689165i	$0.242022\pi$
$467$			3.03363e9i			1.37833i	−0.724605	+	0.689165i	$0.757978\pi$
$468$		0			0
$469$	−6.99111e6			−0.00312926
$470$		0			0
$471$	1.67769e8			0.0739843
$472$		0			0
$473$			7.44566e8i			0.323511i
$474$		0			0
$475$	−2.76976e9	+	1.30081e9i	−1.18581	+	0.556910i
$476$		0			0
$477$		−	1.17017e9i		−	0.493666i
$478$		0			0
$479$	3.00343e9			1.24866			0.624328	−	0.781162i	$-0.285373\pi$
$479$	3.00343e9			1.24866			0.624328	+	0.781162i	$0.285373\pi$
$480$		0			0
$481$	−3.59537e8			−0.147311
$482$		0			0
$483$			2.41260e9i			0.974249i
$484$		0			0
$485$	1.84418e8	+	8.26500e8i	0.0734020	+	0.328963i
$486$		0			0
$487$			3.28649e8i			0.128938i	0.997920	+	0.0644690i	$0.0205354\pi$
$487$			3.28649e8i			0.128938i	−0.997920	+	0.0644690i	$0.979465\pi$
$488$		0			0
$489$	−4.16353e8			−0.161020
$490$		0			0
$491$	6.97275e8			0.265839			0.132919	−	0.991127i	$-0.457565\pi$
$491$	6.97275e8			0.265839			0.132919	+	0.991127i	$0.457565\pi$
$492$		0			0
$493$			2.14794e9i			0.807344i
$494$		0			0
$495$	−3.17473e8	+	7.08382e7i	−0.117649	+	0.0262512i
$496$		0			0
$497$		−	4.20474e9i		−	1.53636i
$498$		0			0
$499$	−3.18226e9			−1.14652			−0.573262	−	0.819372i	$-0.694322\pi$
$499$	−3.18226e9			−1.14652			−0.573262	+	0.819372i	$0.694322\pi$
$500$		0			0
$501$	2.09145e8			0.0743046
$502$		0			0
$503$			4.07550e9i			1.42788i	0.700205	+	0.713942i	$0.253092\pi$
$503$			4.07550e9i			1.42788i	−0.700205	+	0.713942i	$0.746908\pi$
$504$		0			0
$505$	−1.95736e9	+	4.36749e8i	−0.676319	+	0.150908i
$506$		0			0
$507$		−	1.66953e9i		−	0.568939i
$508$		0			0
$509$	−4.18725e8			−0.140740			−0.0703699	−	0.997521i	$-0.522418\pi$
$509$	−4.18725e8			−0.140740			−0.0703699	+	0.997521i	$0.522418\pi$
$510$		0			0
$511$	−3.02822e9			−1.00396
$512$		0			0
$513$		−	7.70946e8i		−	0.252124i
$514$		0			0
$515$	−8.86842e8	−	3.97452e9i	−0.286102	−	1.28221i
$516$		0			0
$517$		−	7.48544e8i		−	0.238232i
$518$		0			0
$519$	2.70233e8			0.0848502
$520$		0			0
$521$	−2.02201e9			−0.626400			−0.313200	−	0.949687i	$-0.601401\pi$
$521$	−2.02201e9			−0.626400			−0.313200	+	0.949687i	$0.601401\pi$
$522$		0			0
$523$			1.87087e8i			0.0571857i	0.999591	+	0.0285928i	$0.00910262\pi$
$523$			1.87087e8i			0.0571857i	−0.999591	+	0.0285928i	$0.990897\pi$
$524$		0			0
$525$	−2.87402e9	+	1.34977e9i	−0.866826	+	0.407101i
$526$		0			0
$527$		−	6.38504e8i		−	0.190032i
$528$		0			0
$529$	−1.18962e8			−0.0349392
$530$		0			0
$531$	1.48748e9			0.431141
$532$		0			0
$533$		−	3.68101e8i		−	0.105298i
$534$		0			0
$535$	−1.00096e9	−	4.48598e9i	−0.282605	−	1.26654i
$536$		0			0
$537$		−	2.94922e9i		−	0.821860i
$538$		0			0
$539$	2.30248e9			0.633337
$540$		0			0
$541$	3.94897e9			1.07224			0.536122	−	0.844140i	$-0.319889\pi$
$541$	3.94897e9			1.07224			0.536122	+	0.844140i	$0.319889\pi$
$542$		0			0
$543$			3.45125e8i			0.0925075i
$544$		0			0
$545$	6.40622e9	−	1.42943e9i	1.69517	−	0.378247i
$546$		0			0
$547$		−	1.71845e9i		−	0.448932i	−0.974482	−	0.224466i	$-0.927936\pi$
$547$		−	1.71845e9i		−	0.448932i	0.974482	−	0.224466i	$-0.0720638\pi$
$548$		0			0
$549$	2.76048e8			0.0712003
$550$		0			0
$551$	2.59099e9			0.659835
$552$		0			0
$553$		−	2.66083e9i		−	0.669082i
$554$		0			0
$555$	2.76980e9	−	6.18030e8i	0.687738	−	0.153456i
$556$		0			0
$557$			3.22406e9i			0.790516i	0.918570	+	0.395258i	$0.129345\pi$
$557$			3.22406e9i			0.790516i	−0.918570	+	0.395258i	$0.870655\pi$
$558$		0			0
$559$	−4.45935e8			−0.107977
$560$		0			0
$561$	1.39955e9			0.334671
$562$		0			0
$563$			5.69907e9i			1.34594i	0.739671	+	0.672968i	$0.234981\pi$
$563$			5.69907e9i			1.34594i	−0.739671	+	0.672968i	$0.765019\pi$
$564$		0			0
$565$	−7.65423e8	−	3.43037e9i	−0.178539	−	0.800149i
$566$		0			0
$567$		−	7.99966e8i		−	0.184302i
$568$		0			0
$569$	−3.70804e9			−0.843823			−0.421912	−	0.906637i	$-0.638641\pi$
$569$	−3.70804e9			−0.843823			−0.421912	+	0.906637i	$0.638641\pi$
$570$		0			0
$571$	3.34811e9			0.752615			0.376308	−	0.926495i	$-0.377194\pi$
$571$	3.34811e9			0.752615			0.376308	+	0.926495i	$0.377194\pi$
$572$		0			0
$573$			4.17101e9i			0.926189i
$574$		0			0
$575$	−1.97144e9	−	4.19773e9i	−0.432461	−	0.920824i
$576$		0			0
$577$		−	8.37348e9i		−	1.81464i	−0.420441	−	0.907320i	$-0.638124\pi$
$577$		−	8.37348e9i		−	1.81464i	0.420441	−	0.907320i	$-0.361876\pi$
$578$		0			0
$579$	−4.08685e9			−0.875012
$580$		0			0
$581$	4.61556e9			0.976355
$582$		0			0
$583$		−	2.56244e9i		−	0.535568i
$584$		0			0
$585$	−4.24264e7	−	1.90141e8i	−0.00876174	−	0.0392672i
$586$		0			0
$587$		−	9.32230e8i		−	0.190235i	−0.995466	−	0.0951174i	$-0.969677\pi$
$587$		−	9.32230e8i		−	0.190235i	0.995466	−	0.0951174i	$-0.0303226\pi$
$588$		0			0
$589$	−7.70205e8			−0.155311
$590$		0			0
$591$	−3.34339e9			−0.666240
$592$		0			0
$593$			4.69858e9i			0.925283i	0.886545	+	0.462642i	$0.153098\pi$
$593$			4.69858e9i			0.925283i	−0.886545	+	0.462642i	$0.846902\pi$
$594$		0			0
$595$	1.33337e10	−	2.97517e9i	2.59502	−	0.579032i
$596$		0			0
$597$		−	1.29038e9i		−	0.248204i
$598$		0			0
$599$	−2.66487e9			−0.506620			−0.253310	−	0.967385i	$-0.581519\pi$
$599$	−2.66487e9			−0.506620			−0.253310	+	0.967385i	$0.581519\pi$
$600$		0			0
$601$	−1.21738e9			−0.228753			−0.114376	−	0.993437i	$-0.536487\pi$
$601$	−1.21738e9			−0.228753			−0.114376	+	0.993437i	$0.536487\pi$
$602$		0			0
$603$			3.38577e6i			0.000628849i
$604$		0			0
$605$	4.62089e9	−	1.03107e9i	0.848364	−	0.189297i
$606$		0			0
$607$		−	2.74917e9i		−	0.498932i	−0.968384	−	0.249466i	$-0.919745\pi$
$607$		−	2.74917e9i		−	0.498932i	0.968384	−	0.249466i	$-0.0802551\pi$
$608$		0			0
$609$	2.68852e9			0.482339
$610$		0			0
$611$	4.48318e8			0.0795136
$612$		0			0
$613$			7.89654e9i			1.38460i	0.721609	+	0.692301i	$0.243403\pi$
$613$			7.89654e9i			1.38460i	−0.721609	+	0.692301i	$0.756597\pi$
$614$		0			0
$615$	6.32751e8	+	2.83578e9i	0.109691	+	0.491597i
$616$		0			0
$617$		−	3.64303e9i		−	0.624403i	−0.950016	−	0.312202i	$-0.898934\pi$
$617$		−	3.64303e9i		−	0.624403i	0.950016	−	0.312202i	$-0.101066\pi$
$618$		0			0
$619$	7.22253e9			1.22397			0.611987	−	0.790868i	$-0.290370\pi$
$619$	7.22253e9			1.22397			0.611987	+	0.790868i	$0.290370\pi$
$620$		0			0
$621$	1.16841e9			0.195783
$622$		0			0
$623$		−	9.24838e9i		−	1.53235i
$624$		0			0
$625$	3.89760e9	−	4.69698e9i	0.638583	−	0.769553i
$626$		0			0
$627$		−	1.68823e9i		−	0.273524i
$628$		0			0
$629$	−1.22104e10			−1.95638
$630$		0			0
$631$	1.55879e9			0.246993			0.123496	−	0.992345i	$-0.460589\pi$
$631$	1.55879e9			0.246993			0.123496	+	0.992345i	$0.460589\pi$
$632$		0			0
$633$			1.69629e9i			0.265820i
$634$		0			0
$635$	−1.46740e9	−	6.57638e9i	−0.227426	−	1.01925i
$636$		0			0
$637$			1.37900e9i			0.211386i
$638$		0			0
$639$	−2.03634e9			−0.308743
$640$		0			0
$641$	3.18649e9			0.477869			0.238935	−	0.971036i	$-0.423202\pi$
$641$	3.18649e9			0.477869			0.238935	+	0.971036i	$0.423202\pi$
$642$		0			0
$643$		−	1.65144e9i		−	0.244976i	−0.992470	−	0.122488i	$-0.960913\pi$
$643$		−	1.65144e9i		−	0.244976i	0.992470	−	0.122488i	$-0.0390873\pi$
$644$		0			0
$645$	3.43539e9	−	7.66544e8i	0.504101	−	0.112481i
$646$		0			0
$647$			3.95797e9i			0.574523i	0.957852	+	0.287262i	$0.0927449\pi$
$647$			3.95797e9i			0.574523i	−0.957852	+	0.287262i	$0.907255\pi$
$648$		0			0
$649$	3.25730e9			0.467736
$650$		0			0
$651$	−7.99197e8			−0.113533
$652$		0			0
$653$		−	2.09998e9i		−	0.295134i	−0.989052	−	0.147567i	$-0.952856\pi$
$653$		−	2.09998e9i		−	0.295134i	0.989052	−	0.147567i	$-0.0471443\pi$
$654$		0			0
$655$	−5.67533e9	+	1.26635e9i	−0.789127	+	0.176079i
$656$		0			0
$657$			1.46656e9i			0.201753i
$658$		0			0
$659$	−1.18536e10			−1.61344			−0.806718	−	0.590937i	$-0.798758\pi$
$659$	−1.18536e10			−1.61344			−0.806718	+	0.590937i	$0.798758\pi$
$660$		0			0
$661$	−1.14873e10			−1.54709			−0.773543	−	0.633744i	$-0.781517\pi$
$661$	−1.14873e10			−1.54709			−0.773543	+	0.633744i	$0.781517\pi$
$662$		0			0
$663$			8.38218e8i			0.111702i
$664$		0			0
$665$	−3.58884e9	−	1.60840e10i	−0.473237	−	2.12089i
$666$		0			0
$667$			3.92679e9i			0.512386i
$668$		0			0
$669$	−1.39828e8			−0.0180552
$670$		0			0
$671$	6.04494e8			0.0772437
$672$		0			0
$673$		−	1.99478e9i		−	0.252257i	−0.992014	−	0.126128i	$-0.959745\pi$
$673$		−	1.99478e9i		−	0.252257i	0.992014	−	0.126128i	$-0.0402551\pi$
$674$		0			0
$675$	6.53688e8	+	1.39188e9i	0.0818102	+	0.174196i
$676$		0			0
$677$			3.21860e9i			0.398664i	0.979932	+	0.199332i	$0.0638772\pi$
$677$			3.21860e9i			0.398664i	−0.979932	+	0.199332i	$0.936123\pi$
$678$		0			0
$679$	−4.56053e9			−0.559076
$680$		0			0
$681$	−6.57389e9			−0.797642
$682$		0			0
$683$			1.87259e9i			0.224890i	0.993658	+	0.112445i	$0.0358683\pi$
$683$			1.87259e9i			0.224890i	−0.993658	+	0.112445i	$0.964132\pi$
$684$		0			0
$685$	6.05238e8	+	2.71247e9i	0.0719464	+	0.322439i
$686$		0			0
$687$		−	2.96281e9i		−	0.348622i
$688$		0			0
$689$	1.53470e9			0.178754
$690$		0			0
$691$	−3.08223e8			−0.0355380			−0.0177690	−	0.999842i	$-0.505656\pi$
$691$	−3.08223e8			−0.0355380			−0.0177690	+	0.999842i	$0.505656\pi$
$692$		0			0
$693$		−	1.75178e9i		−	0.199946i
$694$		0			0
$695$	−4.02144e9	+	8.97310e8i	−0.454396	+	0.101390i
$696$		0			0
$697$		−	1.25013e10i		−	1.39842i
$698$		0			0
$699$	5.01583e9			0.555485
$700$		0			0
$701$	−1.39953e10			−1.53450			−0.767251	−	0.641346i	$-0.778376\pi$
$701$	−1.39953e10			−1.53450			−0.767251	+	0.641346i	$0.778376\pi$
$702$		0			0
$703$			1.47290e10i			1.59893i
$704$		0			0
$705$	−3.45375e9	+	7.70640e8i	−0.371218	+	0.0828304i
$706$		0			0
$707$		−	1.08005e10i		−	1.14941i
$708$		0			0
$709$	1.02356e10			1.07858			0.539290	−	0.842120i	$-0.318693\pi$
$709$	1.02356e10			1.07858			0.539290	+	0.842120i	$0.318693\pi$
$710$		0			0
$711$	−1.28863e9			−0.134457
$712$		0			0
$713$		−	1.16729e9i		−	0.120605i
$714$		0			0
$715$	−9.29059e7	−	4.16373e8i	−0.00950544	−	0.0426002i
$716$		0			0
$717$		−	8.27054e9i		−	0.837948i
$718$		0			0
$719$	9.63078e9			0.966297			0.483148	−	0.875538i	$-0.339493\pi$
$719$	9.63078e9			0.966297			0.483148	+	0.875538i	$0.339493\pi$
$720$		0			0
$721$	2.19309e10			2.17913
$722$		0			0
$723$			1.56108e9i			0.153617i
$724$		0			0
$725$	−4.67781e9	+	2.19691e9i	−0.455889	+	0.214106i
$726$		0			0
$727$		−	1.62740e10i		−	1.57081i	−0.618980	−	0.785407i	$-0.712454\pi$
$727$		−	1.62740e10i		−	1.57081i	0.618980	−	0.785407i	$-0.287546\pi$
$728$		0			0
$729$	−3.87420e8			−0.0370370
$730$		0			0
$731$	−1.51446e10			−1.43399
$732$		0			0
$733$			1.62212e10i			1.52132i	0.649152	+	0.760659i	$0.275124\pi$
$733$			1.62212e10i			1.52132i	−0.649152	+	0.760659i	$0.724876\pi$
$734$		0			0
$735$	−2.37044e9	−	1.06235e10i	−0.220203	−	0.986877i
$736$		0			0
$737$			7.41420e6i			0.000682226i
$738$		0			0
$739$	1.78796e10			1.62968			0.814841	−	0.579684i	$-0.196824\pi$
$739$	1.78796e10			1.62968			0.814841	+	0.579684i	$0.196824\pi$
$740$		0			0
$741$	1.01111e9			0.0912927
$742$		0			0
$743$			1.10072e10i			0.984499i	0.870454	+	0.492249i	$0.163825\pi$
$743$			1.10072e10i			0.984499i	−0.870454	+	0.492249i	$0.836175\pi$
$744$		0			0
$745$	−6.05340e9	+	1.35070e9i	−0.536355	+	0.119678i
$746$		0			0
$747$		−	2.23530e9i		−	0.196206i
$748$		0			0
$749$	2.47531e10			2.15250
$750$		0			0
$751$	8.82495e9			0.760278			0.380139	−	0.924929i	$-0.375876\pi$
$751$	8.82495e9			0.760278			0.380139	+	0.924929i	$0.375876\pi$
$752$		0			0
$753$		−	1.05329e10i		−	0.899016i
$754$		0			0
$755$	1.33879e10	−	2.98727e9i	1.13214	−	0.252615i
$756$		0			0
$757$			1.26747e10i			1.06194i	0.847389	+	0.530972i	$0.178173\pi$
$757$			1.26747e10i			1.06194i	−0.847389	+	0.530972i	$0.821827\pi$
$758$		0			0
$759$	2.55861e9			0.212401
$760$		0			0
$761$	−4.76538e9			−0.391969			−0.195984	−	0.980607i	$-0.562790\pi$
$761$	−4.76538e9			−0.391969			−0.195984	+	0.980607i	$0.562790\pi$
$762$		0			0
$763$			3.53487e10i			2.88097i
$764$		0			0
$765$	−1.44086e9	−	6.45746e9i	−0.116361	−	0.521491i
$766$		0			0
$767$			1.95086e9i			0.156114i
$768$		0			0
$769$	3.33620e9			0.264552			0.132276	−	0.991213i	$-0.457772\pi$
$769$	3.33620e9			0.264552			0.132276	+	0.991213i	$0.457772\pi$
$770$		0			0
$771$	−1.43694e10			−1.12914
$772$		0			0
$773$			1.22441e9i			0.0953453i	0.998863	+	0.0476726i	$0.0151804\pi$
$773$			1.22441e9i			0.0953453i	−0.998863	+	0.0476726i	$0.984820\pi$
$774$		0			0
$775$	1.39054e9	−	6.53060e8i	0.107307	−	0.0503961i
$776$		0			0
$777$			1.52834e10i			1.16882i
$778$		0			0
$779$	−1.50798e10			−1.14292
$780$		0			0
$781$	−4.45921e9			−0.334949
$782$		0			0
$783$		−	1.30204e9i		−	0.0969300i
$784$		0			0
$785$	−3.78228e8	−	1.69509e9i	−0.0279068	−	0.125069i
$786$		0			0
$787$		−	1.72256e10i		−	1.25969i	−0.776721	−	0.629844i	$-0.783119\pi$
$787$		−	1.72256e10i		−	1.25969i	0.776721	−	0.629844i	$-0.216881\pi$
$788$		0			0
$789$	−8.44799e9			−0.612327
$790$		0			0
$791$	1.89284e10			1.35986
$792$		0			0
$793$			3.62043e8i			0.0257813i
$794$		0			0
$795$	−1.18230e10	+	2.63808e9i	−0.834531	+	0.186210i
$796$		0			0
$797$		−	1.01065e10i		−	0.707127i	−0.935411	−	0.353564i	$-0.884970\pi$
$797$		−	1.01065e10i		−	0.707127i	0.935411	−	0.353564i	$-0.115030\pi$
$798$		0			0
$799$	1.52255e10			1.05599
$800$		0			0
$801$	−4.47895e9			−0.307938
$802$		0			0
$803$			3.21149e9i			0.218878i
$804$		0			0
$805$	2.43762e10	−	5.43909e9i	1.64695	−	0.367486i
$806$		0			0
$807$			7.97943e9i			0.534459i
$808$		0			0
$809$	−9.46505e9			−0.628497			−0.314248	−	0.949341i	$-0.601753\pi$
$809$	−9.46505e9			−0.628497			−0.314248	+	0.949341i	$0.601753\pi$
$810$		0			0
$811$	−1.67478e10			−1.10251			−0.551257	−	0.834335i	$-0.685852\pi$
$811$	−1.67478e10			−1.10251			−0.551257	+	0.834335i	$0.685852\pi$
$812$		0			0
$813$			1.06994e10i			0.698300i
$814$		0			0
$815$	9.38649e8	+	4.20671e9i	0.0607367	+	0.272202i
$816$		0			0
$817$			1.82684e10i			1.17199i
$818$		0			0
$819$	1.04917e9			0.0667350
$820$		0			0
$821$	−3.34978e9			−0.211259			−0.105630	−	0.994406i	$-0.533686\pi$
$821$	−3.34978e9			−0.211259			−0.105630	+	0.994406i	$0.533686\pi$
$822$		0			0
$823$		−	6.69632e9i		−	0.418732i	−0.977837	−	0.209366i	$-0.932860\pi$
$823$		−	6.69632e9i		−	0.418732i	0.977837	−	0.209366i	$-0.0671401\pi$
$824$		0			0
$825$	1.43146e9	+	3.04795e9i	0.0887543	+	0.188981i
$826$		0			0
$827$		−	5.81000e9i		−	0.357196i	−0.983922	−	0.178598i	$-0.942844\pi$
$827$		−	5.81000e9i		−	0.357196i	0.983922	−	0.178598i	$-0.0571562\pi$
$828$		0			0
$829$	−6.84048e9			−0.417009			−0.208505	−	0.978021i	$-0.566860\pi$
$829$	−6.84048e9			−0.417009			−0.208505	+	0.978021i	$0.566860\pi$
$830$		0			0
$831$	1.08108e10			0.653516
$832$		0			0
$833$			4.68329e10i			2.80733i
$834$		0			0
$835$	−4.71508e8	−	2.11314e9i	−0.0280276	−	0.125610i
$836$		0			0
$837$			3.87048e8i			0.0228153i
$838$		0			0
$839$	−1.04085e8			−0.00608445			−0.00304223	−	0.999995i	$-0.500968\pi$
$839$	−1.04085e8			−0.00608445			−0.00304223	+	0.999995i	$0.500968\pi$
$840$		0			0
$841$	−1.28740e10			−0.746324
$842$		0			0
$843$		−	9.66995e9i		−	0.555940i
$844$		0			0
$845$	−1.68684e10	+	3.76388e9i	−0.961780	+	0.214604i
$846$		0			0
$847$			2.54975e10i			1.44180i
$848$		0			0
$849$	−1.06836e10			−0.599155
$850$		0			0
$851$	−2.23226e10			−1.24163
$852$		0			0
$853$			6.47246e9i			0.357066i	0.983934	+	0.178533i	$0.0571351\pi$
$853$			6.47246e9i			0.357066i	−0.983934	+	0.178533i	$0.942865\pi$
$854$		0			0
$855$	−7.78941e9	+	1.73806e9i	−0.426210	+	0.0951008i
$856$		0			0
$857$			2.27261e10i			1.23337i	0.787211	+	0.616684i	$0.211524\pi$
$857$			2.27261e10i			1.23337i	−0.787211	+	0.616684i	$0.788476\pi$
$858$		0			0
$859$	−1.08363e10			−0.583315			−0.291657	−	0.956523i	$-0.594207\pi$
$859$	−1.08363e10			−0.583315			−0.291657	+	0.956523i	$0.594207\pi$
$860$		0			0
$861$	−1.56475e10			−0.835474
$862$		0			0
$863$			3.23218e10i			1.71182i	0.517125	+	0.855910i	$0.327002\pi$
$863$			3.23218e10i			1.71182i	−0.517125	+	0.855910i	$0.672998\pi$
$864$		0			0
$865$	−6.09228e8	−	2.73035e9i	−0.0320054	−	0.143437i
$866$		0			0
$867$			1.73880e10i			0.906114i
$868$		0			0
$869$	−2.82186e9			−0.145870
$870$		0			0
$871$	−4.44051e6			−0.000227703
$872$		0			0
$873$			2.20865e9i			0.112351i
$874$		0			0
$875$	2.01170e10	+	2.59952e10i	1.01516	+	1.31179i
$876$		0			0
$877$		−	4.21367e9i		−	0.210941i	−0.994422	−	0.105471i	$-0.966365\pi$
$877$		−	4.21367e9i		−	0.210941i	0.994422	−	0.105471i	$-0.0336349\pi$
$878$		0			0
$879$	5.21856e9			0.259173
$880$		0			0
$881$	5.03021e9			0.247840			0.123920	−	0.992292i	$-0.460453\pi$
$881$	5.03021e9			0.247840			0.123920	+	0.992292i	$0.460453\pi$
$882$		0			0
$883$		−	2.81606e10i		−	1.37651i	−0.725469	−	0.688255i	$-0.758377\pi$
$883$		−	2.81606e10i		−	1.37651i	0.725469	−	0.688255i	$-0.241623\pi$
$884$		0			0
$885$	−3.35345e9	−	1.50290e10i	−0.162626	−	0.728835i
$886$		0			0
$887$		−	1.59422e10i		−	0.767038i	−0.923533	−	0.383519i	$-0.874712\pi$
$887$		−	1.59422e10i		−	0.767038i	0.923533	−	0.383519i	$-0.125288\pi$
$888$		0			0
$889$	3.62877e10			1.73222
$890$		0			0
$891$	−8.48379e8			−0.0401807
$892$		0			0
$893$		−	1.83660e10i		−	0.863049i
$894$		0			0
$895$	−2.97981e10	+	6.64889e9i	−1.38934	+	0.310005i
$896$		0			0
$897$			1.53240e9i			0.0708922i
$898$		0			0
$899$	−1.30079e9			−0.0597101
$900$		0			0
$901$	5.21206e10			2.37396
$902$		0			0
$903$			1.89561e10i			0.856725i
$904$		0			0
$905$	3.48704e9	−	7.78068e8i	0.156382	−	0.0348937i
$906$		0			0
$907$			2.20381e10i			0.980729i	0.871517	+	0.490364i	$0.163136\pi$
$907$			2.20381e10i			0.980729i	−0.871517	+	0.490364i	$0.836864\pi$
$908$		0			0
$909$	−5.23064e9			−0.230984
$910$		0			0
$911$	−1.50904e10			−0.661283			−0.330641	−	0.943756i	$-0.607265\pi$
$911$	−1.50904e10			−0.661283			−0.330641	+	0.943756i	$0.607265\pi$
$912$		0			0
$913$		−	4.89489e9i		−	0.212860i
$914$		0			0
$915$	−6.22338e8	−	2.78911e9i	−0.0268567	−	0.120363i
$916$		0			0
$917$		−	3.13158e10i		−	1.34113i
$918$		0			0
$919$	9.21105e9			0.391476			0.195738	−	0.980656i	$-0.437290\pi$
$919$	9.21105e9			0.391476			0.195738	+	0.980656i	$0.437290\pi$
$920$		0			0
$921$	−5.70420e9			−0.240595
$922$		0			0
$923$		−	2.67071e9i		−	0.111794i
$924$		0			0
$925$	−1.24888e10	−	2.65919e10i	−0.518828	−	1.10472i
$926$		0			0
$927$		−	1.06211e10i		−	0.437915i
$928$		0			0
$929$	−4.08831e10			−1.67297			−0.836487	−	0.547987i	$-0.815394\pi$
$929$	−4.08831e10			−1.67297			−0.836487	+	0.547987i	$0.815394\pi$
$930$		0			0
$931$	5.64929e10			2.29440
$932$		0			0
$933$			7.00494e9i			0.282370i
$934$		0			0
$935$	−3.15522e9	−	1.41406e10i	−0.126238	−	0.565755i
$936$		0			0
$937$			3.38015e10i			1.34229i	0.741324	+	0.671147i	$0.234198\pi$
$937$			3.38015e10i			1.34229i	−0.741324	+	0.671147i	$0.765802\pi$
$938$		0			0
$939$	−2.51038e10			−0.989488
$940$		0			0
$941$	2.16815e10			0.848252			0.424126	−	0.905603i	$-0.360581\pi$
$941$	2.16815e10			0.848252			0.424126	+	0.905603i	$0.360581\pi$
$942$		0			0
$943$		−	2.28543e10i		−	0.887519i
$944$		0			0
$945$	−8.08262e9	+	1.80349e9i	−0.311559	+	0.0695187i
$946$		0			0
$947$			1.47060e10i			0.562692i	0.959606	+	0.281346i	$0.0907809\pi$
$947$			1.47060e10i			0.562692i	−0.959606	+	0.281346i	$0.909219\pi$
$948$		0			0
$949$	−1.92342e9			−0.0730538
$950$		0			0
$951$	4.41476e9			0.166447
$952$		0			0
$953$			1.26663e10i			0.474050i	0.971504	+	0.237025i	$0.0761724\pi$
$953$			1.26663e10i			0.474050i	−0.971504	+	0.237025i	$0.923828\pi$
$954$		0			0
$955$	4.21426e10	−	9.40335e9i	1.56570	−	0.349358i
$956$		0			0
$957$		−	2.85123e9i		−	0.105157i
$958$		0			0
$959$	−1.49671e10			−0.547989
$960$		0			0
$961$	−2.71259e10			−0.985945
$962$		0			0
$963$		−	1.19878e10i		−	0.432562i
$964$		0			0
$965$	9.21362e9	+	4.12923e10i	0.330054	+	1.47919i
$966$		0			0
$967$			2.68713e10i			0.955644i	0.878457	+	0.477822i	$0.158574\pi$
$967$			2.68713e10i			0.955644i	−0.878457	+	0.477822i	$0.841426\pi$
$968$		0			0
$969$	3.43389e10			1.21242
$970$		0			0
$971$	−1.85571e9			−0.0650494			−0.0325247	−	0.999471i	$-0.510355\pi$
$971$	−1.85571e9			−0.0650494			−0.0325247	+	0.999471i	$0.510355\pi$
$972$		0			0
$973$		−	2.21898e10i		−	0.772252i
$974$		0			0
$975$	−1.82548e9	+	8.57327e8i	−0.0630754	+	0.0296231i
$976$		0			0
$977$		−	3.13360e10i		−	1.07501i	−0.843260	−	0.537506i	$-0.819367\pi$
$977$		−	3.13360e10i		−	1.07501i	0.843260	−	0.537506i	$-0.180633\pi$
$978$		0			0
$979$	−9.80808e9			−0.334075
$980$		0			0
$981$	1.71193e10			0.578953
$982$		0			0
$983$			2.78355e9i			0.0934679i	0.998907	+	0.0467339i	$0.0148813\pi$
$983$			2.78355e9i			0.0934679i	−0.998907	+	0.0467339i	$0.985119\pi$
$984$		0			0
$985$	7.53752e9	+	3.37806e10i	0.251305	+	1.12626i
$986$		0			0
$987$		−	1.90574e10i		−	0.630889i
$988$		0			0
$989$	−2.76868e10			−0.910094
$990$		0			0
$991$	−2.20283e10			−0.718991			−0.359495	−	0.933147i	$-0.617051\pi$
$991$	−2.20283e10			−0.718991			−0.359495	+	0.933147i	$0.617051\pi$
$992$		0			0
$993$		−	6.96461e9i		−	0.225722i
$994$		0			0
$995$	−1.30376e10	+	2.90911e9i	−0.419584	+	0.0936223i
$996$		0			0
$997$		−	5.02880e10i		−	1.60706i	−0.595266	−	0.803528i	$-0.702953\pi$
$997$		−	5.02880e10i		−	1.60706i	0.595266	−	0.803528i	$-0.297047\pi$
$998$		0			0
$999$	7.40170e9			0.234883

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.f.e.49.1		8
4.3	odd	2		15.8.b.a.4.5	yes	8
5.4	even	2	inner	240.8.f.e.49.5		8
12.11	even	2		45.8.b.d.19.4		8
20.3	even	4		75.8.a.i.1.3		4
20.7	even	4		75.8.a.j.1.2		4
20.19	odd	2		15.8.b.a.4.4	✓	8
60.23	odd	4		225.8.a.bb.1.2		4
60.47	odd	4		225.8.a.z.1.3		4
60.59	even	2		45.8.b.d.19.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.b.a.4.4	✓	8	20.19	odd	2
15.8.b.a.4.5	yes	8	4.3	odd	2
45.8.b.d.19.4		8	12.11	even	2
45.8.b.d.19.5		8	60.59	even	2
75.8.a.i.1.3		4	20.3	even	4
75.8.a.j.1.2		4	20.7	even	4
225.8.a.z.1.3		4	60.47	odd	4
225.8.a.bb.1.2		4	60.23	odd	4
240.8.f.e.49.1		8	1.1	even	1	trivial
240.8.f.e.49.5		8	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-27.0000i q^{3} +(-272.800 + 60.8703i) q^{5} -1505.28i q^{7} -729.000 q^{9} -1596.37 q^{11} -956.100i q^{13} +(1643.50 + 7365.60i) q^{15} -32470.6i q^{17} -39168.1 q^{19} -40642.5 q^{21} -59361.5i q^{23} +(70714.6 - 33210.8i) q^{25} +19683.0i q^{27} -66150.5 q^{29} +19664.1 q^{31} +43102.1i q^{33} +(91626.6 + 410640. i) q^{35} -376045. i q^{37} -25814.7 q^{39} +385003. q^{41} -466410. i q^{43} +(198871. - 44374.4i) q^{45} +468903. i q^{47} -1.44232e6 q^{49} -876706. q^{51} +1.60516e6i q^{53} +(435491. - 97171.8i) q^{55} +1.05754e6i q^{57} -2.04044e6 q^{59} -378667. q^{61} +1.09735e6i q^{63} +(58198.1 + 260824. i) q^{65} -4644.40i q^{67} -1.60276e6 q^{69} +2.79333e6 q^{71} -2.01174e6i q^{73} +(-896692. - 1.90929e6i) q^{75} +2.40299e6i q^{77} +1.76767e6 q^{79} +531441. q^{81} +3.06625e6i q^{83} +(1.97649e6 + 8.85797e6i) q^{85} +1.78606e6i q^{87} +6.14397e6 q^{89} -1.43920e6 q^{91} -530930. i q^{93} +(1.06851e7 - 2.38417e6i) q^{95} -3.02969e6i q^{97} +1.16376e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 444 q^{5} - 5832 q^{9} - 10752 q^{11} + 17496 q^{15} + 30464 q^{19} + 64152 q^{21} + 127616 q^{25} + 240072 q^{29} - 233728 q^{31} - 593520 q^{35} + 454896 q^{39} + 507648 q^{41} + 323676 q^{45} - 3267160 q^{49}+ \cdots + 7838208 q^{99}+O(q^{100}) \) 8 * q - 444 * q^5 - 5832 * q^9 - 10752 * q^11 + 17496 * q^15 + 30464 * q^19 + 64152 * q^21 + 127616 * q^25 + 240072 * q^29 - 233728 * q^31 - 593520 * q^35 + 454896 * q^39 + 507648 * q^41 + 323676 * q^45 - 3267160 * q^49 - 264384 * q^51 - 3525696 * q^55 - 1091424 * q^59 - 6433520 * q^61 - 2555592 * q^65 - 5940864 * q^69 + 1381824 * q^71 - 7768224 * q^75 + 14380160 * q^79 + 4251528 * q^81 - 1452008 * q^85 + 45778896 * q^89 - 24075648 * q^91 + 25774656 * q^95 + 7838208 * q^99

Introduction

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