LMFDB - Embedded newform 240.12.f.b.49.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [240,12,Mod(49,240)] 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("240.49"); S:= CuspForms(chi, 12); N := Newforms(S);

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, 12, names="a")

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	240.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$184.402363334$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 350078x^{4} + 30638651521x^{2} + 173683668788100 $ x^6 + 350078x^4 + 30638651521x^2 + 173683668788100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$-451.888i$ of defining polynomial
Character	$\chi$	$=$	240.49
Dual form			240.12.f.b.49.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-243.000i q^{3} +(6755.14 + 1787.80i) q^{5} -41527.6i q^{7} -59049.0 q^{9} -839520. q^{11} +1.66178e6i q^{13} +(434436. - 1.64150e6i) q^{15} +3.23669e6i q^{17} +5.44307e6 q^{19} -1.00912e7 q^{21} +2.97481e7i q^{23} +(4.24356e7 + 2.41537e7i) q^{25} +1.43489e7i q^{27} -1.55089e8 q^{29} +2.76977e8 q^{31} +2.04003e8i q^{33} +(7.42432e7 - 2.80525e8i) q^{35} -4.58098e8i q^{37} +4.03812e8 q^{39} +1.42131e8 q^{41} -1.09829e9i q^{43} +(-3.98884e8 - 1.05568e8i) q^{45} -1.89330e9i q^{47} +2.52786e8 q^{49} +7.86516e8 q^{51} -1.28322e9i q^{53} +(-5.67107e9 - 1.50090e9i) q^{55} -1.32267e9i q^{57} -5.01298e9 q^{59} +7.45699e9 q^{61} +2.45216e9i q^{63} +(-2.97093e9 + 1.12255e10i) q^{65} -1.14069e9i q^{67} +7.22879e9 q^{69} -8.23753e9 q^{71} -1.90626e10i q^{73} +(5.86935e9 - 1.03119e10i) q^{75} +3.48632e10i q^{77} +2.39605e10 q^{79} +3.48678e9 q^{81} -6.55948e10i q^{83} +(-5.78657e9 + 2.18643e10i) q^{85} +3.76867e10i q^{87} -3.18476e10 q^{89} +6.90096e10 q^{91} -6.73053e10i q^{93} +(3.67687e10 + 9.73114e9i) q^{95} +4.42047e10i q^{97} +4.95728e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 9926 q^{5} - 354294 q^{9} - 1753400 q^{11} - 551124 q^{15} + 4069824 q^{19} - 32748624 q^{21} + 169978326 q^{25} + 71715564 q^{29} + 122243352 q^{31} - 825571904 q^{35} - 153183312 q^{39} + 2006768564 q^{41}+ \cdots + 103536516600 q^{99}+O(q^{100}) $ 6 * q - 9926 * q^5 - 354294 * q^9 - 1753400 * q^11 - 551124 * q^15 + 4069824 * q^19 - 32748624 * q^21 + 169978326 * q^25 + 71715564 * q^29 + 122243352 * q^31 - 825571904 * q^35 - 153183312 * q^39 + 2006768564 * q^41 + 586120374 * q^45 - 9590905446 * q^49 + 1396921464 * q^51 - 9162341400 * q^55 + 4069460200 * q^59 - 12386466660 * q^61 - 30310342152 * q^65 + 24235021800 * q^69 - 13703505600 * q^71 + 22735606824 * q^75 + 13438327752 * q^79 + 20920706406 * q^81 + 50595843144 * q^85 - 127436589092 * q^89 - 271843383552 * q^91 + 87931077696 * q^95 + 103536516600 * 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
$3$		−	243.000i		−	0.577350i
$3$		−	243.000i		−	0.577350i
$4$		0			0
$5$	6755.14	+	1787.80i	0.966717	+	0.255850i
$5$	6755.14	+	1787.80i	0.966717	+	0.255850i
$6$		0			0
$7$		−	41527.6i		−	0.933894i	−0.884285	−	0.466947i	$-0.845354\pi$
$7$		−	41527.6i		−	0.933894i	0.884285	−	0.466947i	$-0.154646\pi$
$8$		0			0
$9$	−59049.0			−0.333333
$10$		0			0
$11$	−839520.			−1.57171			−0.785853	−	0.618414i	$-0.787776\pi$
$11$	−839520.			−1.57171			−0.785853	+	0.618414i	$0.787776\pi$
$12$		0			0
$13$			1.66178e6i			1.24132i	0.784079	+	0.620661i	$0.213136\pi$
$13$			1.66178e6i			1.24132i	−0.784079	+	0.620661i	$0.786864\pi$
$14$		0			0
$15$	434436.	−	1.64150e6i	0.147715	−	0.558134i
$16$		0			0
$17$			3.23669e6i			0.552882i	0.961031	+	0.276441i	$0.0891550\pi$
$17$			3.23669e6i			0.552882i	−0.961031	+	0.276441i	$0.910845\pi$
$18$		0			0
$19$	5.44307e6			0.504312			0.252156	−	0.967687i	$-0.418860\pi$
$19$	5.44307e6			0.504312			0.252156	+	0.967687i	$0.418860\pi$
$20$		0			0
$21$	−1.00912e7			−0.539184
$22$		0			0
$23$			2.97481e7i			0.963732i	0.876245	+	0.481866i	$0.160041\pi$
$23$			2.97481e7i			0.963732i	−0.876245	+	0.481866i	$0.839959\pi$
$24$		0			0
$25$	4.24356e7	+	2.41537e7i	0.869082	+	0.494668i
$26$		0			0
$27$			1.43489e7i			0.192450i
$28$		0			0
$29$	−1.55089e8			−1.40408			−0.702042	−	0.712136i	$-0.747728\pi$
$29$	−1.55089e8			−1.40408			−0.702042	+	0.712136i	$0.747728\pi$
$30$		0			0
$31$	2.76977e8			1.73762			0.868808	−	0.495150i	$-0.164887\pi$
$31$	2.76977e8			1.73762			0.868808	+	0.495150i	$0.164887\pi$
$32$		0			0
$33$			2.04003e8i			0.907425i
$34$		0			0
$35$	7.42432e7	−	2.80525e8i	0.238936	−	0.902811i
$36$		0			0
$37$		−	4.58098e8i		−	1.08605i	−0.839718	−	0.543023i	$-0.817280\pi$
$37$		−	4.58098e8i		−	1.08605i	0.839718	−	0.543023i	$-0.182720\pi$
$38$		0			0
$39$	4.03812e8			0.716678
$40$		0			0
$41$	1.42131e8			0.191592			0.0957958	−	0.995401i	$-0.469460\pi$
$41$	1.42131e8			0.191592			0.0957958	+	0.995401i	$0.469460\pi$
$42$		0			0
$43$		−	1.09829e9i		−	1.13930i	−0.821886	−	0.569652i	$-0.807078\pi$
$43$		−	1.09829e9i		−	1.13930i	0.821886	−	0.569652i	$-0.192922\pi$
$44$		0			0
$45$	−3.98884e8	−	1.05568e8i	−0.322239	−	0.0852832i
$46$		0			0
$47$		−	1.89330e9i		−	1.20415i	−0.798440	−	0.602075i	$-0.794341\pi$
$47$		−	1.89330e9i		−	1.20415i	0.798440	−	0.602075i	$-0.205659\pi$
$48$		0			0
$49$	2.52786e8			0.127842
$50$		0			0
$51$	7.86516e8			0.319206
$52$		0			0
$53$		−	1.28322e9i		−	0.421487i	−0.977541	−	0.210743i	$-0.932412\pi$
$53$		−	1.28322e9i		−	0.421487i	0.977541	−	0.210743i	$-0.0675884\pi$
$54$		0			0
$55$	−5.67107e9	−	1.50090e9i	−1.51939	−	0.402120i
$56$		0			0
$57$		−	1.32267e9i		−	0.291165i
$58$		0			0
$59$	−5.01298e9			−0.912872			−0.456436	−	0.889756i	$-0.650874\pi$
$59$	−5.01298e9			−0.912872			−0.456436	+	0.889756i	$0.650874\pi$
$60$		0			0
$61$	7.45699e9			1.13045			0.565223	−	0.824938i	$-0.308790\pi$
$61$	7.45699e9			1.13045			0.565223	+	0.824938i	$0.308790\pi$
$62$		0			0
$63$			2.45216e9i			0.311298i
$64$		0			0
$65$	−2.97093e9	+	1.12255e10i	−0.317592	+	1.20001i
$66$		0			0
$67$		−	1.14069e9i		−	0.103219i	−0.998667	−	0.0516093i	$-0.983565\pi$
$67$		−	1.14069e9i		−	0.103219i	0.998667	−	0.0516093i	$-0.0164351\pi$
$68$		0			0
$69$	7.22879e9			0.556411
$70$		0			0
$71$	−8.23753e9			−0.541847			−0.270923	−	0.962601i	$-0.587329\pi$
$71$	−8.23753e9			−0.541847			−0.270923	+	0.962601i	$0.587329\pi$
$72$		0			0
$73$		−	1.90626e10i		−	1.07623i	−0.842871	−	0.538116i	$-0.819136\pi$
$73$		−	1.90626e10i		−	1.07623i	0.842871	−	0.538116i	$-0.180864\pi$
$74$		0			0
$75$	5.86935e9	−	1.03119e10i	0.285597	−	0.501765i
$76$		0			0
$77$			3.48632e10i			1.46781i
$78$		0			0
$79$	2.39605e10			0.876085			0.438042	−	0.898954i	$-0.355672\pi$
$79$	2.39605e10			0.876085			0.438042	+	0.898954i	$0.355672\pi$
$80$		0			0
$81$	3.48678e9			0.111111
$82$		0			0
$83$		−	6.55948e10i		−	1.82785i	−0.405885	−	0.913924i	$-0.633037\pi$
$83$		−	6.55948e10i		−	1.82785i	0.405885	−	0.913924i	$-0.366963\pi$
$84$		0			0
$85$	−5.78657e9	+	2.18643e10i	−0.141455	+	0.534480i
$86$		0			0
$87$			3.76867e10i			0.810648i
$88$		0			0
$89$	−3.18476e10			−0.604549			−0.302274	−	0.953221i	$-0.597746\pi$
$89$	−3.18476e10			−0.604549			−0.302274	+	0.953221i	$0.597746\pi$
$90$		0			0
$91$	6.90096e10			1.15926
$92$		0			0
$93$		−	6.73053e10i		−	1.00321i
$94$		0			0
$95$	3.67687e10	+	9.73114e9i	0.487527	+	0.129028i
$96$		0			0
$97$			4.42047e10i			0.522665i	0.965249	+	0.261333i	$0.0841620\pi$
$97$			4.42047e10i			0.522665i	−0.965249	+	0.261333i	$0.915838\pi$
$98$		0			0
$99$	4.95728e10			0.523902
$100$		0			0
$101$	1.88396e11			1.78362			0.891812	−	0.452407i	$-0.149435\pi$
$101$	1.88396e11			1.78362			0.891812	+	0.452407i	$0.149435\pi$
$102$		0			0
$103$			1.74933e11i			1.48685i	0.668820	+	0.743424i	$0.266800\pi$
$103$			1.74933e11i			1.48685i	−0.668820	+	0.743424i	$0.733200\pi$
$104$		0			0
$105$	−6.81675e10	−	1.80411e10i	−0.521238	−	0.137950i
$106$		0			0
$107$		−	1.75156e11i		−	1.20730i	−0.797249	−	0.603650i	$-0.793712\pi$
$107$		−	1.75156e11i		−	1.20730i	0.797249	−	0.603650i	$-0.206288\pi$
$108$		0			0
$109$	−5.67974e10			−0.353576			−0.176788	−	0.984249i	$-0.556571\pi$
$109$	−5.67974e10			−0.353576			−0.176788	+	0.984249i	$0.556571\pi$
$110$		0			0
$111$	−1.11318e11			−0.627029
$112$		0			0
$113$		−	7.76853e10i		−	0.396650i	−0.980136	−	0.198325i	$-0.936450\pi$
$113$		−	7.76853e10i		−	0.396650i	0.980136	−	0.198325i	$-0.0635501\pi$
$114$		0			0
$115$	−5.31838e10	+	2.00953e11i	−0.246570	+	0.931655i
$116$		0			0
$117$		−	9.81263e10i		−	0.413774i
$118$		0			0
$119$	1.34412e11			0.516333
$120$		0			0
$121$	4.19482e11			1.47026
$122$		0			0
$123$		−	3.45377e10i		−	0.110615i
$124$		0			0
$125$	2.43477e11	+	2.39028e11i	0.713595	+	0.700558i
$126$		0			0
$127$			2.44393e11i			0.656401i	0.944608	+	0.328200i	$0.106442\pi$
$127$			2.44393e11i			0.656401i	−0.944608	+	0.328200i	$0.893558\pi$
$128$		0			0
$129$	−2.66884e11			−0.657778
$130$		0			0
$131$	−3.21349e10			−0.0727755			−0.0363877	−	0.999338i	$-0.511585\pi$
$131$	−3.21349e10			−0.0727755			−0.0363877	+	0.999338i	$0.511585\pi$
$132$		0			0
$133$		−	2.26038e11i		−	0.470974i
$134$		0			0
$135$	−2.56530e10	+	9.69288e10i	−0.0492383	+	0.186045i
$136$		0			0
$137$		−	1.03419e12i		−	1.83079i	−0.402557	−	0.915395i	$-0.631878\pi$
$137$		−	1.03419e12i		−	1.83079i	0.402557	−	0.915395i	$-0.368122\pi$
$138$		0			0
$139$	−6.51000e11			−1.06414			−0.532071	−	0.846700i	$-0.678586\pi$
$139$	−6.51000e11			−1.06414			−0.532071	+	0.846700i	$0.678586\pi$
$140$		0			0
$141$	−4.60071e11			−0.695216
$142$		0			0
$143$		−	1.39510e12i		−	1.95099i
$144$		0			0
$145$	−1.04765e12	−	2.77269e11i	−1.35735	−	0.359234i
$146$		0			0
$147$		−	6.14269e10i		−	0.0738097i
$148$		0			0
$149$	−1.48776e12			−1.65962			−0.829808	−	0.558049i	$-0.811550\pi$
$149$	−1.48776e12			−1.65962			−0.829808	+	0.558049i	$0.811550\pi$
$150$		0			0
$151$	−1.69992e12			−1.76220			−0.881100	−	0.472929i	$-0.843197\pi$
$151$	−1.69992e12			−1.76220			−0.881100	+	0.472929i	$0.843197\pi$
$152$		0			0
$153$		−	1.91123e11i		−	0.184294i
$154$		0			0
$155$	1.87101e12	+	4.95180e11i	1.67978	+	0.444568i
$156$		0			0
$157$			3.75740e11i			0.314369i	0.987569	+	0.157184i	$0.0502417\pi$
$157$			3.75740e11i			0.314369i	−0.987569	+	0.157184i	$0.949758\pi$
$158$		0			0
$159$	−3.11822e11			−0.243345
$160$		0			0
$161$	1.23537e12			0.900023
$162$		0			0
$163$		−	1.19182e12i		−	0.811296i	−0.914029	−	0.405648i	$-0.867046\pi$
$163$		−	1.19182e12i		−	0.811296i	0.914029	−	0.405648i	$-0.132954\pi$
$164$		0			0
$165$	−3.64718e11	+	1.37807e12i	−0.232164	+	0.877222i
$166$		0			0
$167$			7.86209e11i			0.468379i	0.972191	+	0.234190i	$0.0752436\pi$
$167$			7.86209e11i			0.468379i	−0.972191	+	0.234190i	$0.924756\pi$
$168$		0			0
$169$	−9.69345e11			−0.540880
$170$		0			0
$171$	−3.21408e11			−0.168104
$172$		0			0
$173$			2.61407e11i			0.128252i	0.997942	+	0.0641258i	$0.0204259\pi$
$173$			2.61407e11i			0.128252i	−0.997942	+	0.0641258i	$0.979574\pi$
$174$		0			0
$175$	1.00305e12	−	1.76225e12i	0.461968	−	0.811630i
$176$		0			0
$177$			1.21815e12i			0.527047i
$178$		0			0
$179$	1.08084e12			0.439614			0.219807	−	0.975543i	$-0.429457\pi$
$179$	1.08084e12			0.439614			0.219807	+	0.975543i	$0.429457\pi$
$180$		0			0
$181$	3.87701e12			1.48342			0.741711	−	0.670720i	$-0.234015\pi$
$181$	3.87701e12			1.48342			0.741711	+	0.670720i	$0.234015\pi$
$182$		0			0
$183$		−	1.81205e12i		−	0.652663i
$184$		0			0
$185$	8.18988e11	−	3.09451e12i	0.277865	−	1.04990i
$186$		0			0
$187$		−	2.71727e12i		−	0.868967i
$188$		0			0
$189$	5.95876e11			0.179728
$190$		0			0
$191$	−4.34227e12			−1.23604			−0.618021	−	0.786161i	$-0.712065\pi$
$191$	−4.34227e12			−1.23604			−0.618021	+	0.786161i	$0.712065\pi$
$192$		0			0
$193$			4.45221e11i			0.119677i	0.998208	+	0.0598384i	$0.0190585\pi$
$193$			4.45221e11i			0.119677i	−0.998208	+	0.0598384i	$0.980941\pi$
$194$		0			0
$195$	2.72781e12	+	7.21936e11i	0.692824	+	0.183362i
$196$		0			0
$197$		−	1.92250e12i		−	0.461640i	−0.972997	−	0.230820i	$-0.925859\pi$
$197$		−	1.92250e12i		−	0.461640i	0.972997	−	0.230820i	$-0.0741408\pi$
$198$		0			0
$199$	−2.52317e12			−0.573131			−0.286566	−	0.958061i	$-0.592514\pi$
$199$	−2.52317e12			−0.573131			−0.286566	+	0.958061i	$0.592514\pi$
$200$		0			0
$201$	−2.77189e11			−0.0595933
$202$		0			0
$203$			6.44048e12i			1.31126i
$204$		0			0
$205$	9.60112e11	+	2.54102e11i	0.185215	+	0.0490186i
$206$		0			0
$207$		−	1.75660e12i		−	0.321244i
$208$		0			0
$209$	−4.56957e12			−0.792630
$210$		0			0
$211$	−7.60254e12			−1.25143			−0.625713	−	0.780053i	$-0.715192\pi$
$211$	−7.60254e12			−1.25143			−0.625713	+	0.780053i	$0.715192\pi$
$212$		0			0
$213$			2.00172e12i			0.312835i
$214$		0			0
$215$	1.96352e12	−	7.41909e12i	0.291491	−	1.10138i
$216$		0			0
$217$		−	1.15022e13i		−	1.62275i
$218$		0			0
$219$	−4.63221e12			−0.621363
$220$		0			0
$221$	−5.37866e12			−0.686304
$222$		0			0
$223$			2.46671e12i			0.299531i	0.988722	+	0.149765i	$0.0478518\pi$
$223$			2.46671e12i			0.299531i	−0.988722	+	0.149765i	$0.952148\pi$
$224$		0			0
$225$	−2.50578e12	−	1.42625e12i	−0.289694	−	0.164889i
$226$		0			0
$227$		−	1.11438e13i		−	1.22713i	−0.789642	−	0.613567i	$-0.789734\pi$
$227$		−	1.11438e13i		−	1.22713i	0.789642	−	0.613567i	$-0.210266\pi$
$228$		0			0
$229$	1.12840e13			1.18404			0.592021	−	0.805923i	$-0.298330\pi$
$229$	1.12840e13			1.18404			0.592021	+	0.805923i	$0.298330\pi$
$230$		0			0
$231$	8.47177e12			0.847438
$232$		0			0
$233$		−	3.66685e12i		−	0.349813i	−0.984585	−	0.174906i	$-0.944038\pi$
$233$		−	3.66685e12i		−	0.349813i	0.984585	−	0.174906i	$-0.0559623\pi$
$234$		0			0
$235$	3.38484e12	−	1.27895e13i	0.308081	−	1.16407i
$236$		0			0
$237$		−	5.82239e12i		−	0.505808i
$238$		0			0
$239$	1.40189e13			1.16285			0.581427	−	0.813598i	$-0.302494\pi$
$239$	1.40189e13			1.16285			0.581427	+	0.813598i	$0.302494\pi$
$240$		0			0
$241$	−1.19865e13			−0.949723			−0.474862	−	0.880060i	$-0.657502\pi$
$241$	−1.19865e13			−0.949723			−0.474862	+	0.880060i	$0.657502\pi$
$242$		0			0
$243$		−	8.47289e11i		−	0.0641500i
$244$		0			0
$245$	1.70760e12	+	4.51931e11i	0.123587	+	0.0327083i
$246$		0			0
$247$			9.04517e12i			0.626014i
$248$		0			0
$249$	−1.59395e13			−1.05531
$250$		0			0
$251$	4.71793e12			0.298914			0.149457	−	0.988768i	$-0.452247\pi$
$251$	4.71793e12			0.298914			0.149457	+	0.988768i	$0.452247\pi$
$252$		0			0
$253$		−	2.49741e13i		−	1.51470i
$254$		0			0
$255$	5.31302e12	+	1.40614e12i	0.308582	+	0.0816688i
$256$		0			0
$257$		−	2.26178e13i		−	1.25840i	−0.777244	−	0.629199i	$-0.783383\pi$
$257$		−	2.26178e13i		−	1.25840i	0.777244	−	0.629199i	$-0.216617\pi$
$258$		0			0
$259$	−1.90237e13			−1.01425
$260$		0			0
$261$	9.15787e12			0.468028
$262$		0			0
$263$		−	2.63896e13i		−	1.29323i	−0.762815	−	0.646617i	$-0.776183\pi$
$263$		−	2.63896e13i		−	1.29323i	0.762815	−	0.646617i	$-0.223817\pi$
$264$		0			0
$265$	2.29414e12	−	8.66832e12i	0.107837	−	0.407458i
$266$		0			0
$267$			7.73896e12i			0.349036i
$268$		0			0
$269$	−1.82859e13			−0.791550			−0.395775	−	0.918347i	$-0.629524\pi$
$269$	−1.82859e13			−0.791550			−0.395775	+	0.918347i	$0.629524\pi$
$270$		0			0
$271$	2.37685e13			0.987803			0.493902	−	0.869518i	$-0.335570\pi$
$271$	2.37685e13			0.987803			0.493902	+	0.869518i	$0.335570\pi$
$272$		0			0
$273$		−	1.67693e13i		−	0.669301i
$274$		0			0
$275$	−3.56256e13	−	2.02775e13i	−1.36594	−	0.777473i
$276$		0			0
$277$		−	2.00012e13i		−	0.736914i	−0.929645	−	0.368457i	$-0.879886\pi$
$277$		−	2.00012e13i		−	0.736914i	0.929645	−	0.368457i	$-0.120114\pi$
$278$		0			0
$279$	−1.63552e13			−0.579205
$280$		0			0
$281$	2.16062e13			0.735689			0.367845	−	0.929887i	$-0.380096\pi$
$281$	2.16062e13			0.735689			0.367845	+	0.929887i	$0.380096\pi$
$282$		0			0
$283$		−	4.22652e13i		−	1.38407i	−0.721865	−	0.692034i	$-0.756715\pi$
$283$		−	4.22652e13i		−	1.38407i	0.721865	−	0.692034i	$-0.243285\pi$
$284$		0			0
$285$	2.36467e12	−	8.93479e12i	0.0744944	−	0.281474i
$286$		0			0
$287$		−	5.90234e12i		−	0.178926i
$288$		0			0
$289$	2.37957e13			0.694322
$290$		0			0
$291$	1.07417e13			0.301761
$292$		0			0
$293$			5.66640e13i			1.53298i	0.642259	+	0.766488i	$0.277997\pi$
$293$			5.66640e13i			1.53298i	−0.642259	+	0.766488i	$0.722003\pi$
$294$		0			0
$295$	−3.38634e13	−	8.96222e12i	−0.882488	−	0.233558i
$296$		0			0
$297$		−	1.20462e13i		−	0.302475i
$298$		0			0
$299$	−4.94347e13			−1.19630
$300$		0			0
$301$	−4.56093e13			−1.06399
$302$		0			0
$303$		−	4.57801e13i		−	1.02978i
$304$		0			0
$305$	5.03730e13	+	1.33316e13i	1.09282	+	0.289224i
$306$		0			0
$307$			1.99364e13i			0.417239i	0.977997	+	0.208619i	$0.0668970\pi$
$307$			1.99364e13i			0.417239i	−0.977997	+	0.208619i	$0.933103\pi$
$308$		0			0
$309$	4.25087e13			0.858433
$310$		0			0
$311$	−1.05339e13			−0.205309			−0.102655	−	0.994717i	$-0.532734\pi$
$311$	−1.05339e13			−0.205309			−0.102655	+	0.994717i	$0.532734\pi$
$312$		0			0
$313$		−	5.27355e13i		−	0.992224i	−0.868259	−	0.496112i	$-0.834761\pi$
$313$		−	5.27355e13i		−	0.992224i	0.868259	−	0.496112i	$-0.165239\pi$
$314$		0			0
$315$	−4.38399e12	+	1.65647e13i	−0.0796455	+	0.300937i
$316$		0			0
$317$		−	6.68311e13i		−	1.17261i	−0.810091	−	0.586304i	$-0.800583\pi$
$317$		−	6.68311e13i		−	1.17261i	0.810091	−	0.586304i	$-0.199417\pi$
$318$		0			0
$319$	1.30201e14			2.20681
$320$		0			0
$321$	−4.25630e13			−0.697035
$322$		0			0
$323$			1.76175e13i			0.278825i
$324$		0			0
$325$	−4.01381e13	+	7.05186e13i	−0.614042	+	1.07881i
$326$		0			0
$327$			1.38018e13i			0.204137i
$328$		0			0
$329$	−7.86241e13			−1.12455
$330$		0			0
$331$	−2.79590e13			−0.386783			−0.193392	−	0.981122i	$-0.561949\pi$
$331$	−2.79590e13			−0.386783			−0.193392	+	0.981122i	$0.561949\pi$
$332$		0			0
$333$			2.70502e13i			0.362016i
$334$		0			0
$335$	2.03934e12	−	7.70555e12i	0.0264084	−	0.0997831i
$336$		0			0
$337$			1.16982e14i			1.46606i	0.680194	+	0.733032i	$0.261896\pi$
$337$			1.16982e14i			1.46606i	−0.680194	+	0.733032i	$0.738104\pi$
$338$		0			0
$339$	−1.88775e13			−0.229006
$340$		0			0
$341$	−2.32527e14			−2.73102
$342$		0			0
$343$		−	9.26112e13i		−	1.05328i
$344$		0			0
$345$	4.88315e13	+	1.29237e13i	0.537891	+	0.142357i
$346$		0			0
$347$			1.62553e12i			0.0173453i	0.999962	+	0.00867265i	$0.00276062\pi$
$347$			1.62553e12i			0.0173453i	−0.999962	+	0.00867265i	$0.997239\pi$
$348$		0			0
$349$	1.40028e14			1.44769			0.723843	−	0.689965i	$-0.242374\pi$
$349$	1.40028e14			1.44769			0.723843	+	0.689965i	$0.242374\pi$
$350$		0			0
$351$	−2.38447e13			−0.238893
$352$		0			0
$353$			1.07894e13i			0.104770i	0.998627	+	0.0523850i	$0.0166823\pi$
$353$			1.07894e13i			0.104770i	−0.998627	+	0.0523850i	$0.983318\pi$
$354$		0			0
$355$	−5.56457e13	−	1.47271e13i	−0.523812	−	0.138631i
$356$		0			0
$357$		−	3.26621e13i		−	0.298105i
$358$		0			0
$359$	−8.36216e13			−0.740114			−0.370057	−	0.929009i	$-0.620662\pi$
$359$	−8.36216e13			−0.740114			−0.370057	+	0.929009i	$0.620662\pi$
$360$		0			0
$361$	−8.68632e13			−0.745669
$362$		0			0
$363$		−	1.01934e14i		−	0.848854i
$364$		0			0
$365$	3.40801e13	−	1.28770e14i	0.275353	−	1.04041i
$366$		0			0
$367$		−	1.00049e13i		−	0.0784420i	−0.999231	−	0.0392210i	$-0.987512\pi$
$367$		−	1.00049e13i		−	0.0784420i	0.999231	−	0.0392210i	$-0.0124876\pi$
$368$		0			0
$369$	−8.39267e12			−0.0638639
$370$		0			0
$371$	−5.32890e13			−0.393624
$372$		0			0
$373$		−	2.32365e14i		−	1.66637i	−0.552993	−	0.833186i	$-0.686514\pi$
$373$		−	2.32365e14i		−	1.66637i	0.552993	−	0.833186i	$-0.313486\pi$
$374$		0			0
$375$	5.80839e13	−	5.91648e13i	0.404467	−	0.411994i
$376$		0			0
$377$		−	2.57724e14i		−	1.74292i
$378$		0			0
$379$	1.60902e14			1.05693			0.528465	−	0.848955i	$-0.322768\pi$
$379$	1.60902e14			1.05693			0.528465	+	0.848955i	$0.322768\pi$
$380$		0			0
$381$	5.93876e13			0.378973
$382$		0			0
$383$		−	3.41226e12i		−	0.0211568i	−0.999944	−	0.0105784i	$-0.996633\pi$
$383$		−	3.41226e12i		−	0.0211568i	0.999944	−	0.0105784i	$-0.00336727\pi$
$384$		0			0
$385$	−6.23286e13	+	2.35506e14i	−0.375538	+	1.41895i
$386$		0			0
$387$			6.48528e13i			0.379768i
$388$		0			0
$389$	2.49915e14			1.42255			0.711277	−	0.702912i	$-0.248117\pi$
$389$	2.49915e14			1.42255			0.711277	+	0.702912i	$0.248117\pi$
$390$		0			0
$391$	−9.62854e13			−0.532830
$392$		0			0
$393$			7.80879e12i			0.0420169i
$394$		0			0
$395$	1.61856e14	+	4.28366e13i	0.846926	+	0.224146i
$396$		0			0
$397$		−	2.50976e14i		−	1.27727i	−0.769508	−	0.638637i	$-0.779498\pi$
$397$		−	2.50976e14i		−	1.27727i	0.769508	−	0.638637i	$-0.220502\pi$
$398$		0			0
$399$	−5.49272e13			−0.271917
$400$		0			0
$401$	3.56396e14			1.71648			0.858240	−	0.513248i	$-0.171558\pi$
$401$	3.56396e14			1.71648			0.858240	+	0.513248i	$0.171558\pi$
$402$		0			0
$403$			4.60273e14i			2.15694i
$404$		0			0
$405$	2.35537e13	+	6.23369e12i	0.107413	+	0.0284277i
$406$		0			0
$407$			3.84582e14i			1.70695i
$408$		0			0
$409$	4.76849e13			0.206017			0.103008	−	0.994680i	$-0.467153\pi$
$409$	4.76849e13			0.206017			0.103008	+	0.994680i	$0.467153\pi$
$410$		0			0
$411$	−2.51309e14			−1.05701
$412$		0			0
$413$			2.08177e14i			0.852525i
$414$		0			0
$415$	1.17271e14	−	4.43102e14i	0.467654	−	1.76701i
$416$		0			0
$417$			1.58193e14i			0.614382i
$418$		0			0
$419$	2.26851e14			0.858150			0.429075	−	0.903269i	$-0.358840\pi$
$419$	2.26851e14			0.858150			0.429075	+	0.903269i	$0.358840\pi$
$420$		0			0
$421$	−1.36580e14			−0.503309			−0.251654	−	0.967817i	$-0.580975\pi$
$421$	−1.36580e14			−0.503309			−0.251654	+	0.967817i	$0.580975\pi$
$422$		0			0
$423$			1.11797e14i			0.401383i
$424$		0			0
$425$	−7.81781e13	+	1.37351e14i	−0.273493	+	0.480500i
$426$		0			0
$427$		−	3.09671e14i		−	1.05572i
$428$		0			0
$429$	−3.39008e14			−1.12641
$430$		0			0
$431$	−1.21003e13			−0.0391896			−0.0195948	−	0.999808i	$-0.506238\pi$
$431$	−1.21003e13			−0.0391896			−0.0195948	+	0.999808i	$0.506238\pi$
$432$		0			0
$433$			3.05977e14i			0.966064i	0.875603	+	0.483032i	$0.160464\pi$
$433$			3.05977e14i			0.966064i	−0.875603	+	0.483032i	$0.839536\pi$
$434$		0			0
$435$	−6.73764e13	+	2.54579e14i	−0.207404	+	0.783667i
$436$		0			0
$437$			1.61921e14i			0.486021i
$438$		0			0
$439$	3.68147e14			1.07762			0.538811	−	0.842427i	$-0.318874\pi$
$439$	3.68147e14			1.07762			0.538811	+	0.842427i	$0.318874\pi$
$440$		0			0
$441$	−1.49267e13			−0.0426140
$442$		0			0
$443$		−	1.38978e13i		−	0.0387012i	−0.999813	−	0.0193506i	$-0.993840\pi$
$443$		−	1.38978e13i		−	0.0387012i	0.999813	−	0.0193506i	$-0.00615987\pi$
$444$		0			0
$445$	−2.15135e14	−	5.69372e13i	−0.584427	−	0.154674i
$446$		0			0
$447$			3.61525e14i			0.958179i
$448$		0			0
$449$	−3.92376e13			−0.101472			−0.0507362	−	0.998712i	$-0.516157\pi$
$449$	−3.92376e13			−0.101472			−0.0507362	+	0.998712i	$0.516157\pi$
$450$		0			0
$451$	−1.19321e14			−0.301126
$452$		0			0
$453$			4.13081e14i			1.01741i
$454$		0			0
$455$	4.66170e14	+	1.23376e14i	1.12068	+	0.296597i
$456$		0			0
$457$			1.37201e14i			0.321973i	0.986957	+	0.160986i	$0.0514675\pi$
$457$			1.37201e14i			0.321973i	−0.986957	+	0.160986i	$0.948532\pi$
$458$		0			0
$459$	−4.64430e13			−0.106402
$460$		0			0
$461$	2.48659e14			0.556223			0.278111	−	0.960549i	$-0.410292\pi$
$461$	2.48659e14			0.556223			0.278111	+	0.960549i	$0.410292\pi$
$462$		0			0
$463$			6.31268e14i			1.37885i	0.724355	+	0.689427i	$0.242138\pi$
$463$			6.31268e14i			1.37885i	−0.724355	+	0.689427i	$0.757862\pi$
$464$		0			0
$465$	1.20329e14	−	4.54657e14i	0.256672	−	0.969822i
$466$		0			0
$467$		−	8.60246e14i		−	1.79217i	−0.443880	−	0.896086i	$-0.646398\pi$
$467$		−	8.60246e14i		−	1.79217i	0.443880	−	0.896086i	$-0.353602\pi$
$468$		0			0
$469$	−4.73703e13			−0.0963952
$470$		0			0
$471$	9.13048e13			0.181501
$472$		0			0
$473$			9.22035e14i			1.79065i
$474$		0			0
$475$	2.30980e14	+	1.31470e14i	0.438288	+	0.249467i
$476$		0			0
$477$			7.57728e13i			0.140496i
$478$		0			0
$479$	−5.71890e14			−1.03626			−0.518129	−	0.855303i	$-0.673371\pi$
$479$	−5.71890e14			−1.03626			−0.518129	+	0.855303i	$0.673371\pi$
$480$		0			0
$481$	7.61256e14			1.34813
$482$		0			0
$483$		−	3.00194e14i		−	0.519629i
$484$		0			0
$485$	−7.90293e13	+	2.98609e14i	−0.133724	+	0.505269i
$486$		0			0
$487$			7.95200e14i			1.31543i	0.753268	+	0.657714i	$0.228476\pi$
$487$			7.95200e14i			1.31543i	−0.753268	+	0.657714i	$0.771524\pi$
$488$		0			0
$489$	−2.89612e14			−0.468402
$490$		0			0
$491$	5.04124e13			0.0797240			0.0398620	−	0.999205i	$-0.487308\pi$
$491$	5.04124e13			0.0797240			0.0398620	+	0.999205i	$0.487308\pi$
$492$		0			0
$493$		−	5.01976e14i		−	0.776292i
$494$		0			0
$495$	3.34871e14	+	8.86264e13i	0.506465	+	0.134040i
$496$		0			0
$497$			3.42085e14i			0.506027i
$498$		0			0
$499$	−2.91835e14			−0.422264			−0.211132	−	0.977458i	$-0.567715\pi$
$499$	−2.91835e14			−0.422264			−0.211132	+	0.977458i	$0.567715\pi$
$500$		0			0
$501$	1.91049e14			0.270419
$502$		0			0
$503$		−	7.29427e14i		−	1.01008i	−0.863095	−	0.505042i	$-0.831477\pi$
$503$		−	7.29427e14i		−	1.01008i	0.863095	−	0.505042i	$-0.168523\pi$
$504$		0			0
$505$	1.27264e15	+	3.36814e14i	1.72426	+	0.456339i
$506$		0			0
$507$			2.35551e14i			0.312277i
$508$		0			0
$509$	3.75076e14			0.486600			0.243300	−	0.969951i	$-0.421770\pi$
$509$	3.75076e14			0.486600			0.243300	+	0.969951i	$0.421770\pi$
$510$		0			0
$511$	−7.91623e14			−1.00509
$512$		0			0
$513$			7.81021e13i			0.0970549i
$514$		0			0
$515$	−3.12746e14	+	1.18170e15i	−0.380410	+	1.43736i
$516$		0			0
$517$			1.58946e15i			1.89257i
$518$		0			0
$519$	6.35218e13			0.0740461
$520$		0			0
$521$	−1.27745e15			−1.45793			−0.728967	−	0.684549i	$-0.759999\pi$
$521$	−1.27745e15			−1.45793			−0.728967	+	0.684549i	$0.759999\pi$
$522$		0			0
$523$			1.16147e15i			1.29792i	0.760824	+	0.648959i	$0.224795\pi$
$523$			1.16147e15i			1.29792i	−0.760824	+	0.648959i	$0.775205\pi$
$524$		0			0
$525$	−4.28227e14	−	2.43740e14i	−0.468595	−	0.266717i
$526$		0			0
$527$			8.96487e14i			0.960696i
$528$		0			0
$529$	6.78604e13			0.0712213
$530$		0			0
$531$	2.96011e14			0.304291
$532$		0			0
$533$			2.36190e14i			0.237827i
$534$		0			0
$535$	3.13145e14	−	1.18321e15i	0.308887	−	1.16712i
$536$		0			0
$537$		−	2.62645e14i		−	0.253811i
$538$		0			0
$539$	−2.12219e14			−0.200930
$540$		0			0
$541$	1.14931e15			1.06623			0.533115	−	0.846043i	$-0.321021\pi$
$541$	1.14931e15			1.06623			0.533115	+	0.846043i	$0.321021\pi$
$542$		0			0
$543$		−	9.42113e14i		−	0.856454i
$544$		0			0
$545$	−3.83675e14	−	1.01543e14i	−0.341808	−	0.0904623i
$546$		0			0
$547$			5.91724e13i			0.0516641i	0.999666	+	0.0258321i	$0.00822352\pi$
$547$			5.91724e13i			0.0516641i	−0.999666	+	0.0258321i	$0.991776\pi$
$548$		0			0
$549$	−4.40328e14			−0.376815
$550$		0			0
$551$	−8.44162e14			−0.708096
$552$		0			0
$553$		−	9.95020e14i		−	0.818170i
$554$		0			0
$555$	−7.51966e14	−	1.99014e14i	−0.606160	−	0.160425i
$556$		0			0
$557$		−	8.98701e14i		−	0.710251i	−0.934819	−	0.355125i	$-0.884438\pi$
$557$		−	8.98701e14i		−	0.710251i	0.934819	−	0.355125i	$-0.115562\pi$
$558$		0			0
$559$	1.82511e15			1.41424
$560$		0			0
$561$	−6.60296e14			−0.501699
$562$		0			0
$563$			1.00499e15i			0.748799i	0.927268	+	0.374399i	$0.122151\pi$
$563$			1.00499e15i			0.748799i	−0.927268	+	0.374399i	$0.877849\pi$
$564$		0			0
$565$	1.38886e14	−	5.24775e14i	0.101483	−	0.383448i
$566$		0			0
$567$		−	1.44798e14i		−	0.103766i
$568$		0			0
$569$	−2.14457e15			−1.50738			−0.753691	−	0.657228i	$-0.771729\pi$
$569$	−2.14457e15			−1.50738			−0.753691	+	0.657228i	$0.771729\pi$
$570$		0			0
$571$	1.36068e15			0.938119			0.469060	−	0.883167i	$-0.344593\pi$
$571$	1.36068e15			0.938119			0.469060	+	0.883167i	$0.344593\pi$
$572$		0			0
$573$			1.05517e15i			0.713629i
$574$		0			0
$575$	−7.18527e14	+	1.26238e15i	−0.476727	+	0.837562i
$576$		0			0
$577$			9.41505e14i			0.612852i	0.951894	+	0.306426i	$0.0991332\pi$
$577$			9.41505e14i			0.612852i	−0.951894	+	0.306426i	$0.900867\pi$
$578$		0			0
$579$	1.08189e14			0.0690955
$580$		0			0
$581$	−2.72400e15			−1.70702
$582$		0			0
$583$			1.07729e15i			0.662453i
$584$		0			0
$585$	1.75431e14	−	6.62857e14i	0.105864	−	0.400002i
$586$		0			0
$587$		−	7.11947e14i		−	0.421636i	−0.977525	−	0.210818i	$-0.932387\pi$
$587$		−	7.11947e14i		−	0.421636i	0.977525	−	0.210818i	$-0.0676128\pi$
$588$		0			0
$589$	1.50760e15			0.876300
$590$		0			0
$591$	−4.67169e14			−0.266528
$592$		0			0
$593$		−	2.44357e15i		−	1.36844i	−0.729278	−	0.684218i	$-0.760144\pi$
$593$		−	2.44357e15i		−	1.36844i	0.729278	−	0.684218i	$-0.239856\pi$
$594$		0			0
$595$	9.07972e14	+	2.40302e14i	0.499148	+	0.132104i
$596$		0			0
$597$			6.13130e14i			0.330898i
$598$		0			0
$599$	−6.81013e14			−0.360835			−0.180417	−	0.983590i	$-0.557745\pi$
$599$	−6.81013e14			−0.360835			−0.180417	+	0.983590i	$0.557745\pi$
$600$		0			0
$601$	−1.35124e15			−0.702947			−0.351474	−	0.936198i	$-0.614319\pi$
$601$	−1.35124e15			−0.702947			−0.351474	+	0.936198i	$0.614319\pi$
$602$		0			0
$603$			6.73569e13i			0.0344062i
$604$		0			0
$605$	2.83366e15	+	7.49951e14i	1.42132	+	0.376165i
$606$		0			0
$607$		−	2.98246e15i		−	1.46905i	−0.678582	−	0.734524i	$-0.737405\pi$
$607$		−	2.98246e15i		−	1.46905i	0.678582	−	0.734524i	$-0.262595\pi$
$608$		0			0
$609$	1.56504e15			0.757059
$610$		0			0
$611$	3.14624e15			1.49474
$612$		0			0
$613$		−	2.62361e15i		−	1.22424i	−0.790766	−	0.612119i	$-0.790317\pi$
$613$		−	2.62361e15i		−	1.22424i	0.790766	−	0.612119i	$-0.209683\pi$
$614$		0			0
$615$	6.17467e13	−	2.33307e14i	0.0283009	−	0.106934i
$616$		0			0
$617$			1.36222e15i			0.613310i	0.951821	+	0.306655i	$0.0992098\pi$
$617$			1.36222e15i			0.613310i	−0.951821	+	0.306655i	$0.900790\pi$
$618$		0			0
$619$	−4.10602e15			−1.81603			−0.908013	−	0.418942i	$-0.862401\pi$
$619$	−4.10602e15			−1.81603			−0.908013	+	0.418942i	$0.862401\pi$
$620$		0			0
$621$	−4.26853e14			−0.185470
$622$		0			0
$623$			1.32255e15i			0.564585i
$624$		0			0
$625$	1.21738e15	+	2.04996e15i	0.510607	+	0.859814i
$626$		0			0
$627$			1.11040e15i			0.457625i
$628$		0			0
$629$	1.48272e15			0.600455
$630$		0			0
$631$	3.07651e15			1.22433			0.612163	−	0.790731i	$-0.290300\pi$
$631$	3.07651e15			1.22433			0.612163	+	0.790731i	$0.290300\pi$
$632$		0			0
$633$			1.84742e15i			0.722511i
$634$		0			0
$635$	−4.36928e14	+	1.65091e15i	−0.167940	+	0.634554i
$636$		0			0
$637$			4.20073e14i			0.158693i
$638$		0			0
$639$	4.86418e14			0.180616
$640$		0			0
$641$	−2.43456e14			−0.0888590			−0.0444295	−	0.999013i	$-0.514147\pi$
$641$	−2.43456e14			−0.0888590			−0.0444295	+	0.999013i	$0.514147\pi$
$642$		0			0
$643$		−	6.07705e14i		−	0.218038i	−0.994040	−	0.109019i	$-0.965229\pi$
$643$		−	6.07705e14i		−	0.218038i	0.994040	−	0.109019i	$-0.0347710\pi$
$644$		0			0
$645$	−1.80284e15	−	4.77136e14i	−0.635885	−	0.168292i
$646$		0			0
$647$		−	3.75432e14i		−	0.130184i	−0.997879	−	0.0650919i	$-0.979266\pi$
$647$		−	3.75432e14i		−	0.130184i	0.997879	−	0.0650919i	$-0.0207341\pi$
$648$		0			0
$649$	4.20849e15			1.43477
$650$		0			0
$651$	−2.79503e15			−0.936894
$652$		0			0
$653$			2.76536e15i			0.911442i	0.890123	+	0.455721i	$0.150618\pi$
$653$			2.76536e15i			0.911442i	−0.890123	+	0.455721i	$0.849382\pi$
$654$		0			0
$655$	−2.17076e14	−	5.74509e13i	−0.0703533	−	0.0186196i
$656$		0			0
$657$			1.12563e15i			0.358744i
$658$		0			0
$659$	3.26634e15			1.02374			0.511872	−	0.859062i	$-0.328952\pi$
$659$	3.26634e15			1.02374			0.511872	+	0.859062i	$0.328952\pi$
$660$		0			0
$661$	1.72112e15			0.530521			0.265260	−	0.964177i	$-0.414542\pi$
$661$	1.72112e15			0.530521			0.265260	+	0.964177i	$0.414542\pi$
$662$		0			0
$663$			1.30701e15i			0.396238i
$664$		0			0
$665$	4.04111e14	−	1.52692e15i	0.120498	−	0.455298i
$666$		0			0
$667$		−	4.61361e15i		−	1.35316i
$668$		0			0
$669$	5.99410e14			0.172934
$670$		0			0
$671$	−6.26029e15			−1.77673
$672$		0			0
$673$		−	4.18650e15i		−	1.16888i	−0.811438	−	0.584438i	$-0.801315\pi$
$673$		−	4.18650e15i		−	1.16888i	0.811438	−	0.584438i	$-0.198685\pi$
$674$		0			0
$675$	−3.46579e14	+	6.08905e14i	−0.0951989	+	0.167255i
$676$		0			0
$677$		−	4.21054e15i		−	1.13789i	−0.822376	−	0.568945i	$-0.807352\pi$
$677$		−	4.21054e15i		−	1.13789i	0.822376	−	0.568945i	$-0.192648\pi$
$678$		0			0
$679$	1.83571e15			0.488114
$680$		0			0
$681$	−2.70795e15			−0.708486
$682$		0			0
$683$		−	6.85825e15i		−	1.76563i	−0.469722	−	0.882814i	$-0.655646\pi$
$683$		−	6.85825e15i		−	1.76563i	0.469722	−	0.882814i	$-0.344354\pi$
$684$		0			0
$685$	1.84893e15	−	6.98612e15i	0.468407	−	1.76986i
$686$		0			0
$687$		−	2.74201e15i		−	0.683607i
$688$		0			0
$689$	2.13243e15			0.523201
$690$		0			0
$691$	7.01392e15			1.69368			0.846841	−	0.531847i	$-0.178502\pi$
$691$	7.01392e15			1.69368			0.846841	+	0.531847i	$0.178502\pi$
$692$		0			0
$693$		−	2.05864e15i		−	0.489269i
$694$		0			0
$695$	−4.39759e15	−	1.16386e15i	−1.02872	−	0.272260i
$696$		0			0
$697$			4.60033e14i			0.105927i
$698$		0			0
$699$	−8.91045e14			−0.201965
$700$		0			0
$701$	−3.59916e15			−0.803067			−0.401533	−	0.915844i	$-0.631523\pi$
$701$	−3.59916e15			−0.803067			−0.401533	+	0.915844i	$0.631523\pi$
$702$		0			0
$703$		−	2.49346e15i		−	0.547706i
$704$		0			0
$705$	−3.10784e15	−	8.22517e14i	−0.672077	−	0.177871i
$706$		0			0
$707$		−	7.82361e15i		−	1.66571i
$708$		0			0
$709$	−9.13743e15			−1.91544			−0.957722	−	0.287695i	$-0.907111\pi$
$709$	−9.13743e15			−1.91544			−0.957722	+	0.287695i	$0.907111\pi$
$710$		0			0
$711$	−1.41484e15			−0.292028
$712$		0			0
$713$			8.23952e15i			1.67459i
$714$		0			0
$715$	2.49416e15	−	9.42406e15i	0.499161	−	1.88606i
$716$		0			0
$717$		−	3.40659e15i		−	0.671375i
$718$		0			0
$719$	−4.61453e15			−0.895609			−0.447805	−	0.894131i	$-0.647794\pi$
$719$	−4.61453e15			−0.895609			−0.447805	+	0.894131i	$0.647794\pi$
$720$		0			0
$721$	7.26454e15			1.38856
$722$		0			0
$723$			2.91271e15i			0.548323i
$724$		0			0
$725$	−6.58131e15	−	3.74598e15i	−1.22026	−	0.694555i
$726$		0			0
$727$			4.47421e15i			0.817104i	0.912735	+	0.408552i	$0.133966\pi$
$727$			4.47421e15i			0.817104i	−0.912735	+	0.408552i	$0.866034\pi$
$728$		0			0
$729$	−2.05891e14			−0.0370370
$730$		0			0
$731$	3.55482e15			0.629901
$732$		0			0
$733$			4.68168e15i			0.817203i	0.912713	+	0.408601i	$0.133983\pi$
$733$			4.68168e15i			0.817203i	−0.912713	+	0.408601i	$0.866017\pi$
$734$		0			0
$735$	1.09819e14	−	4.14947e14i	0.0188842	−	0.0713530i
$736$		0			0
$737$			9.57636e14i			0.162229i
$738$		0			0
$739$	5.58278e15			0.931764			0.465882	−	0.884847i	$-0.345737\pi$
$739$	5.58278e15			0.931764			0.465882	+	0.884847i	$0.345737\pi$
$740$		0			0
$741$	2.19798e15			0.361429
$742$		0			0
$743$			5.84984e15i			0.947776i	0.880585	+	0.473888i	$0.157150\pi$
$743$			5.84984e15i			0.947776i	−0.880585	+	0.473888i	$0.842850\pi$
$744$		0			0
$745$	−1.00500e16	−	2.65982e15i	−1.60438	−	0.424612i
$746$		0			0
$747$			3.87331e15i			0.609283i
$748$		0			0
$749$	−7.27383e15			−1.12749
$750$		0			0
$751$	−6.23385e15			−0.952218			−0.476109	−	0.879386i	$-0.657953\pi$
$751$	−6.23385e15			−0.952218			−0.476109	+	0.879386i	$0.657953\pi$
$752$		0			0
$753$		−	1.14646e15i		−	0.172578i
$754$		0			0
$755$	−1.14832e16	−	3.03912e15i	−1.70355	−	0.450858i
$756$		0			0
$757$		−	1.20579e16i		−	1.76297i	−0.472210	−	0.881486i	$-0.656543\pi$
$757$		−	1.20579e16i		−	1.76297i	0.472210	−	0.881486i	$-0.343457\pi$
$758$		0			0
$759$	−6.06871e15			−0.874514
$760$		0			0
$761$	5.06900e15			0.719957			0.359979	−	0.932961i	$-0.382784\pi$
$761$	5.06900e15			0.719957			0.359979	+	0.932961i	$0.382784\pi$
$762$		0			0
$763$			2.35866e15i			0.330203i
$764$		0			0
$765$	3.41691e14	−	1.29106e15i	0.0471515	−	0.178160i
$766$		0			0
$767$		−	8.33045e15i		−	1.13317i
$768$		0			0
$769$	−1.35439e15			−0.181614			−0.0908072	−	0.995868i	$-0.528945\pi$
$769$	−1.35439e15			−0.181614			−0.0908072	+	0.995868i	$0.528945\pi$
$770$		0			0
$771$	−5.49612e15			−0.726537
$772$		0			0
$773$		−	8.61815e15i		−	1.12312i	−0.827435	−	0.561561i	$-0.810201\pi$
$773$		−	8.61815e15i		−	1.12312i	0.827435	−	0.561561i	$-0.189799\pi$
$774$		0			0
$775$	1.17537e16	+	6.69001e15i	1.51013	+	0.859543i
$776$		0			0
$777$			4.62276e15i			0.585579i
$778$		0			0
$779$	7.73627e14			0.0966219
$780$		0			0
$781$	6.91557e15			0.851623
$782$		0			0
$783$		−	2.22536e15i		−	0.270216i
$784$		0			0
$785$	−6.71749e14	+	2.53817e15i	−0.0804311	+	0.303905i
$786$		0			0
$787$			4.95922e15i			0.585534i	0.956184	+	0.292767i	$0.0945761\pi$
$787$			4.95922e15i			0.585534i	−0.956184	+	0.292767i	$0.905424\pi$
$788$		0			0
$789$	−6.41268e15			−0.746649
$790$		0			0
$791$	−3.22608e15			−0.370429
$792$		0			0
$793$			1.23919e16i			1.40325i
$794$		0			0
$795$	−2.10640e15	−	5.57477e14i	−0.235246	−	0.0622598i
$796$		0			0
$797$		−	1.48779e16i		−	1.63878i	−0.573235	−	0.819391i	$-0.694312\pi$
$797$		−	1.48779e16i		−	1.63878i	0.573235	−	0.819391i	$-0.305688\pi$
$798$		0			0
$799$	6.12802e15			0.665752
$800$		0			0
$801$	1.88057e15			0.201516
$802$		0			0
$803$			1.60034e16i			1.69152i
$804$		0			0
$805$	8.34507e15	+	2.20859e15i	0.870067	+	0.230271i
$806$		0			0
$807$			4.44347e15i			0.457002i
$808$		0			0
$809$	1.18817e16			1.20549			0.602744	−	0.797935i	$-0.294074\pi$
$809$	1.18817e16			1.20549			0.602744	+	0.797935i	$0.294074\pi$
$810$		0			0
$811$	7.07941e14			0.0708569			0.0354284	−	0.999372i	$-0.488720\pi$
$811$	7.07941e14			0.0708569			0.0354284	+	0.999372i	$0.488720\pi$
$812$		0			0
$813$		−	5.77574e15i		−	0.570309i
$814$		0			0
$815$	2.13074e15	−	8.05091e15i	0.207570	−	0.784293i
$816$		0			0
$817$		−	5.97806e15i		−	0.574565i
$818$		0			0
$819$	−4.07495e15			−0.386421
$820$		0			0
$821$	4.20085e15			0.393052			0.196526	−	0.980499i	$-0.437034\pi$
$821$	4.20085e15			0.393052			0.196526	+	0.980499i	$0.437034\pi$
$822$		0			0
$823$		−	1.76608e16i		−	1.63046i	−0.579135	−	0.815232i	$-0.696609\pi$
$823$		−	1.76608e16i		−	1.63046i	0.579135	−	0.815232i	$-0.303391\pi$
$824$		0			0
$825$	−4.92744e15	+	8.65701e15i	−0.448874	+	0.788626i
$826$		0			0
$827$			7.42500e15i			0.667446i	0.942671	+	0.333723i	$0.108305\pi$
$827$			7.42500e15i			0.667446i	−0.942671	+	0.333723i	$0.891695\pi$
$828$		0			0
$829$	1.02610e16			0.910208			0.455104	−	0.890438i	$-0.349602\pi$
$829$	1.02610e16			0.910208			0.455104	+	0.890438i	$0.349602\pi$
$830$		0			0
$831$	−4.86028e15			−0.425457
$832$		0			0
$833$			8.18189e14i			0.0706816i
$834$		0			0
$835$	−1.40559e15	+	5.31095e15i	−0.119835	+	0.452790i
$836$		0			0
$837$			3.97431e15i			0.334404i
$838$		0			0
$839$	5.46116e15			0.453518			0.226759	−	0.973951i	$-0.427187\pi$
$839$	5.46116e15			0.453518			0.226759	+	0.973951i	$0.427187\pi$
$840$		0			0
$841$	1.18522e16			0.971449
$842$		0			0
$843$		−	5.25032e15i		−	0.424750i
$844$		0			0
$845$	−6.54806e15	−	1.73300e15i	−0.522878	−	0.138384i
$846$		0			0
$847$		−	1.74201e16i		−	1.37307i
$848$		0			0
$849$	−1.02704e16			−0.799092
$850$		0			0
$851$	1.36275e16			1.04666
$852$		0			0
$853$		−	1.38989e16i		−	1.05380i	−0.849926	−	0.526902i	$-0.823353\pi$
$853$		−	1.38989e16i		−	1.05380i	0.849926	−	0.526902i	$-0.176647\pi$
$854$		0			0
$855$	−2.17115e15	−	5.74614e14i	−0.162509	−	0.0430093i
$856$		0			0
$857$			1.35605e16i			1.00203i	0.865437	+	0.501017i	$0.167041\pi$
$857$			1.35605e16i			1.00203i	−0.865437	+	0.501017i	$0.832959\pi$
$858$		0			0
$859$	4.45328e15			0.324876			0.162438	−	0.986719i	$-0.448064\pi$
$859$	4.45328e15			0.324876			0.162438	+	0.986719i	$0.448064\pi$
$860$		0			0
$861$	−1.43427e15			−0.103303
$862$		0			0
$863$			1.15395e16i			0.820593i	0.911952	+	0.410297i	$0.134575\pi$
$863$			1.15395e16i			0.820593i	−0.911952	+	0.410297i	$0.865425\pi$
$864$		0			0
$865$	−4.67344e14	+	1.76584e15i	−0.0328131	+	0.123983i
$866$		0			0
$867$		−	5.78236e15i		−	0.400867i
$868$		0			0
$869$	−2.01153e16			−1.37695
$870$		0			0
$871$	1.89558e15			0.128128
$872$		0			0
$873$		−	2.61024e15i		−	0.174222i
$874$		0			0
$875$	9.92627e15	−	1.01110e16i	0.654247	−	0.666422i
$876$		0			0
$877$			1.61714e16i			1.05257i	0.850308	+	0.526285i	$0.176416\pi$
$877$			1.61714e16i			1.05257i	−0.850308	+	0.526285i	$0.823584\pi$
$878$		0			0
$879$	1.37693e16			0.885064
$880$		0			0
$881$	−2.06661e16			−1.31187			−0.655936	−	0.754817i	$-0.727726\pi$
$881$	−2.06661e16			−1.31187			−0.655936	+	0.754817i	$0.727726\pi$
$882$		0			0
$883$		−	2.69113e15i		−	0.168714i	−0.996436	−	0.0843569i	$-0.973116\pi$
$883$		−	2.69113e15i		−	0.168714i	0.996436	−	0.0843569i	$-0.0268836\pi$
$884$		0			0
$885$	−2.17782e15	+	8.22880e15i	−0.134845	+	0.509505i
$886$		0			0
$887$			2.20727e16i			1.34982i	0.737900	+	0.674911i	$0.235818\pi$
$887$			2.20727e16i			1.34982i	−0.737900	+	0.674911i	$0.764182\pi$
$888$		0			0
$889$	1.01491e16			0.613009
$890$		0			0
$891$	−2.92722e15			−0.174634
$892$		0			0
$893$		−	1.03053e16i		−	0.607267i
$894$		0			0
$895$	7.30125e15	+	1.93234e15i	0.424982	+	0.112475i
$896$		0			0
$897$			1.20126e16i			0.690685i
$898$		0			0
$899$	−4.29561e16			−2.43976
$900$		0			0
$901$	4.15339e15			0.233032
$902$		0			0
$903$			1.10831e16i			0.614295i
$904$		0			0
$905$	2.61897e16	+	6.93133e15i	1.43405	+	0.379533i
$906$		0			0
$907$			2.50362e16i			1.35434i	0.735827	+	0.677170i	$0.236794\pi$
$907$			2.50362e16i			1.35434i	−0.735827	+	0.677170i	$0.763206\pi$
$908$		0			0
$909$	−1.11246e16			−0.594541
$910$		0			0
$911$	1.34918e16			0.712393			0.356196	−	0.934411i	$-0.384073\pi$
$911$	1.34918e16			0.712393			0.356196	+	0.934411i	$0.384073\pi$
$912$		0			0
$913$			5.50682e16i			2.87284i
$914$		0			0
$915$	3.23959e15	−	1.22406e16i	0.166984	−	0.630940i
$916$		0			0
$917$			1.33449e15i			0.0679646i
$918$		0			0
$919$	1.86406e16			0.938048			0.469024	−	0.883185i	$-0.344606\pi$
$919$	1.86406e16			0.938048			0.469024	+	0.883185i	$0.344606\pi$
$920$		0			0
$921$	4.84454e15			0.240893
$922$		0			0
$923$		−	1.36889e16i		−	0.672606i
$924$		0			0
$925$	1.10648e16	−	1.94397e16i	0.537233	−	0.943864i
$926$		0			0
$927$		−	1.03296e16i		−	0.495616i
$928$		0			0
$929$	−1.26618e16			−0.600358			−0.300179	−	0.953883i	$-0.597046\pi$
$929$	−1.26618e16			−0.600358			−0.300179	+	0.953883i	$0.597046\pi$
$930$		0			0
$931$	1.37593e15			0.0644723
$932$		0			0
$933$			2.55975e15i			0.118535i
$934$		0			0
$935$	4.85794e15	−	1.83555e16i	0.222325	−	0.840045i
$936$		0			0
$937$		−	3.98788e16i		−	1.80374i	−0.432006	−	0.901871i	$-0.642194\pi$
$937$		−	3.98788e16i		−	1.80374i	0.432006	−	0.901871i	$-0.357806\pi$
$938$		0			0
$939$	−1.28147e16			−0.572861
$940$		0			0
$941$	−2.35800e16			−1.04184			−0.520920	−	0.853606i	$-0.674411\pi$
$941$	−2.35800e16			−1.04184			−0.520920	+	0.853606i	$0.674411\pi$
$942$		0			0
$943$			4.22812e15i			0.184643i
$944$		0			0
$945$	4.02522e15	+	1.06531e15i	0.173746	+	0.0459833i
$946$		0			0
$947$			2.50128e16i			1.06718i	0.845743	+	0.533591i	$0.179158\pi$
$947$			2.50128e16i			1.06718i	−0.845743	+	0.533591i	$0.820842\pi$
$948$		0			0
$949$	3.16778e16			1.33595
$950$		0			0
$951$	−1.62400e16			−0.677005
$952$		0			0
$953$			3.78726e16i			1.56068i	0.625355	+	0.780340i	$0.284954\pi$
$953$			3.78726e16i			1.56068i	−0.625355	+	0.780340i	$0.715046\pi$
$954$		0			0
$955$	−2.93326e16	−	7.76313e15i	−1.19490	−	0.316241i
$956$		0			0
$957$		−	3.16387e16i		−	1.27410i
$958$		0			0
$959$	−4.29476e16			−1.70976
$960$		0			0
$961$	5.13075e16			2.01931
$962$		0			0
$963$			1.03428e16i			0.402433i
$964$		0			0
$965$	−7.95967e14	+	3.00753e15i	−0.0306193	+	0.115694i
$966$		0			0
$967$		−	1.01035e16i		−	0.384262i	−0.981369	−	0.192131i	$-0.938460\pi$
$967$		−	1.01035e16i		−	0.384262i	0.981369	−	0.192131i	$-0.0615398\pi$
$968$		0			0
$969$	4.28106e15			0.160980
$970$		0			0
$971$	1.09869e16			0.408478			0.204239	−	0.978921i	$-0.434528\pi$
$971$	1.09869e16			0.408478			0.204239	+	0.978921i	$0.434528\pi$
$972$		0			0
$973$			2.70344e16i			0.993795i
$974$		0			0
$975$	1.71360e16	+	9.75356e15i	0.622852	+	0.354518i
$976$		0			0
$977$			2.52822e16i			0.908644i	0.890837	+	0.454322i	$0.150119\pi$
$977$			2.52822e16i			0.908644i	−0.890837	+	0.454322i	$0.849881\pi$
$978$		0			0
$979$	2.67367e16			0.950173
$980$		0			0
$981$	3.35383e15			0.117859
$982$		0			0
$983$			4.29168e16i			1.49136i	0.666304	+	0.745680i	$0.267875\pi$
$983$			4.29168e16i			1.49136i	−0.666304	+	0.745680i	$0.732125\pi$
$984$		0			0
$985$	3.43706e15	−	1.29868e16i	0.118110	−	0.446275i
$986$		0			0
$987$			1.91056e16i			0.649258i
$988$		0			0
$989$	3.26720e16			1.09798
$990$		0			0
$991$	5.20325e16			1.72930			0.864648	−	0.502377i	$-0.167541\pi$
$991$	5.20325e16			1.72930			0.864648	+	0.502377i	$0.167541\pi$
$992$		0			0
$993$			6.79404e15i			0.223309i
$994$		0			0
$995$	−1.70443e16	−	4.51093e15i	−0.554056	−	0.146635i
$996$		0			0
$997$		−	5.42887e16i		−	1.74537i	−0.488288	−	0.872683i	$-0.662378\pi$
$997$		−	5.42887e16i		−	1.74537i	0.488288	−	0.872683i	$-0.337622\pi$
$998$		0			0
$999$	6.57320e15			0.209010

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.12.f.b.49.3		6
4.3	odd	2		30.12.c.b.19.6	yes	6
5.4	even	2	inner	240.12.f.b.49.6		6
12.11	even	2		90.12.c.c.19.1		6
20.3	even	4		150.12.a.u.1.2		3
20.7	even	4		150.12.a.t.1.2		3
20.19	odd	2		30.12.c.b.19.3	✓	6
60.59	even	2		90.12.c.c.19.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.12.c.b.19.3	✓	6	20.19	odd	2
30.12.c.b.19.6	yes	6	4.3	odd	2
90.12.c.c.19.1		6	12.11	even	2
90.12.c.c.19.4		6	60.59	even	2
150.12.a.t.1.2		3	20.7	even	4
150.12.a.u.1.2		3	20.3	even	4
240.12.f.b.49.3		6	1.1	even	1	trivial
240.12.f.b.49.6		6	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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	240.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-243.000i q^{3} +(6755.14 + 1787.80i) q^{5} -41527.6i q^{7} -59049.0 q^{9} -839520. q^{11} +1.66178e6i q^{13} +(434436. - 1.64150e6i) q^{15} +3.23669e6i q^{17} +5.44307e6 q^{19} -1.00912e7 q^{21} +2.97481e7i q^{23} +(4.24356e7 + 2.41537e7i) q^{25} +1.43489e7i q^{27} -1.55089e8 q^{29} +2.76977e8 q^{31} +2.04003e8i q^{33} +(7.42432e7 - 2.80525e8i) q^{35} -4.58098e8i q^{37} +4.03812e8 q^{39} +1.42131e8 q^{41} -1.09829e9i q^{43} +(-3.98884e8 - 1.05568e8i) q^{45} -1.89330e9i q^{47} +2.52786e8 q^{49} +7.86516e8 q^{51} -1.28322e9i q^{53} +(-5.67107e9 - 1.50090e9i) q^{55} -1.32267e9i q^{57} -5.01298e9 q^{59} +7.45699e9 q^{61} +2.45216e9i q^{63} +(-2.97093e9 + 1.12255e10i) q^{65} -1.14069e9i q^{67} +7.22879e9 q^{69} -8.23753e9 q^{71} -1.90626e10i q^{73} +(5.86935e9 - 1.03119e10i) q^{75} +3.48632e10i q^{77} +2.39605e10 q^{79} +3.48678e9 q^{81} -6.55948e10i q^{83} +(-5.78657e9 + 2.18643e10i) q^{85} +3.76867e10i q^{87} -3.18476e10 q^{89} +6.90096e10 q^{91} -6.73053e10i q^{93} +(3.67687e10 + 9.73114e9i) q^{95} +4.42047e10i q^{97} +4.95728e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 9926 q^{5} - 354294 q^{9} - 1753400 q^{11} - 551124 q^{15} + 4069824 q^{19} - 32748624 q^{21} + 169978326 q^{25} + 71715564 q^{29} + 122243352 q^{31} - 825571904 q^{35} - 153183312 q^{39} + 2006768564 q^{41}+ \cdots + 103536516600 q^{99}+O(q^{100}) \) 6 * q - 9926 * q^5 - 354294 * q^9 - 1753400 * q^11 - 551124 * q^15 + 4069824 * q^19 - 32748624 * q^21 + 169978326 * q^25 + 71715564 * q^29 + 122243352 * q^31 - 825571904 * q^35 - 153183312 * q^39 + 2006768564 * q^41 + 586120374 * q^45 - 9590905446 * q^49 + 1396921464 * q^51 - 9162341400 * q^55 + 4069460200 * q^59 - 12386466660 * q^61 - 30310342152 * q^65 + 24235021800 * q^69 - 13703505600 * q^71 + 22735606824 * q^75 + 13438327752 * q^79 + 20920706406 * q^81 + 50595843144 * q^85 - 127436589092 * q^89 - 271843383552 * q^91 + 87931077696 * q^95 + 103536516600 * q^99

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