LMFDB - Embedded newform 240.5.l.d.161.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [240,5,Mod(161,240)] 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, 1, 0])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("240.161"); S:= CuspForms(chi, 5); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [6,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$24.8087911401$
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} + 73x^{4} + 1096x^{2} + 180 $ x^6 + 73x^4 + 1096x^2 + 180
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$7.20990i$ of defining polynomial
Character	$\chi$	$=$	240.161
Dual form			240.5.l.d.161.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-8.77108 + 2.01697i) q^{3} -11.1803i q^{5} -23.3388 q^{7} +(72.8637 - 35.3820i) q^{9} -33.9875i q^{11} -125.965 q^{13} +(22.5504 + 98.0636i) q^{15} -308.095i q^{17} +363.724 q^{19} +(204.707 - 47.0738i) q^{21} -102.256i q^{23} -125.000 q^{25} +(-567.728 + 457.302i) q^{27} +1565.10i q^{29} -326.954 q^{31} +(68.5518 + 298.107i) q^{33} +260.936i q^{35} +72.4778 q^{37} +(1104.85 - 254.068i) q^{39} +2149.01i q^{41} -1501.27 q^{43} +(-395.583 - 814.640i) q^{45} +1958.33i q^{47} -1856.30 q^{49} +(621.418 + 2702.32i) q^{51} +2965.17i q^{53} -379.992 q^{55} +(-3190.25 + 733.621i) q^{57} +5045.29i q^{59} +736.716 q^{61} +(-1700.55 + 825.775i) q^{63} +1408.34i q^{65} -4254.63 q^{67} +(206.248 + 896.898i) q^{69} -1303.84i q^{71} +6768.14 q^{73} +(1096.38 - 252.121i) q^{75} +793.229i q^{77} +6367.07 q^{79} +(4057.23 - 5156.13i) q^{81} +6892.17i q^{83} -3444.60 q^{85} +(-3156.76 - 13727.6i) q^{87} -6401.82i q^{89} +2939.88 q^{91} +(2867.74 - 659.456i) q^{93} -4066.56i q^{95} -9879.94 q^{97} +(-1202.55 - 2476.45i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{3} - 76 q^{7} + 118 q^{9} - 424 q^{13} - 50 q^{15} + 244 q^{19} - 876 q^{21} - 750 q^{25} + 352 q^{27} - 3772 q^{31} + 4420 q^{33} + 1896 q^{37} + 1336 q^{39} + 7384 q^{43} + 1900 q^{45} - 1318 q^{49}+ \cdots - 9680 q^{99}+O(q^{100}) $ 6 * q - 8 * q^3 - 76 * q^7 + 118 * q^9 - 424 * q^13 - 50 * q^15 + 244 * q^19 - 876 * q^21 - 750 * q^25 + 352 * q^27 - 3772 * q^31 + 4420 * q^33 + 1896 * q^37 + 1336 * q^39 + 7384 * q^43 + 1900 * q^45 - 1318 * q^49 + 8492 * q^51 + 1300 * q^55 - 11584 * q^57 + 6452 * q^61 - 14796 * q^63 - 13816 * q^67 + 5472 * q^69 + 596 * q^73 + 1000 * q^75 + 16124 * q^79 + 5086 * q^81 - 3100 * q^85 + 4900 * q^87 + 11632 * q^91 - 8184 * q^93 + 9756 * q^97 - 9680 * 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
$3$	−8.77108	+	2.01697i	−0.974564	+	0.224108i
$3$	−8.77108	+	2.01697i	−0.974564	+	0.224108i
$4$		0			0
$5$		−	11.1803i		−	0.447214i
$5$		−	11.1803i		−	0.447214i
$6$		0			0
$7$	−23.3388			−0.476303			−0.238151	−	0.971228i	$-0.576541\pi$
$7$	−23.3388			−0.476303			−0.238151	+	0.971228i	$0.576541\pi$
$8$		0			0
$9$	72.8637	−	35.3820i	0.899551	−	0.436815i
$10$		0			0
$11$		−	33.9875i		−	0.280888i	−0.990089	−	0.140444i	$-0.955147\pi$
$11$		−	33.9875i		−	0.280888i	0.990089	−	0.140444i	$-0.0448531\pi$
$12$		0			0
$13$	−125.965			−0.745357			−0.372678	−	0.927961i	$-0.621561\pi$
$13$	−125.965			−0.745357			−0.372678	+	0.927961i	$0.621561\pi$
$14$		0			0
$15$	22.5504	+	98.0636i	0.100224	+	0.435838i
$16$		0			0
$17$		−	308.095i		−	1.06607i	−0.846093	−	0.533036i	$-0.821051\pi$
$17$		−	308.095i		−	1.06607i	0.846093	−	0.533036i	$-0.178949\pi$
$18$		0			0
$19$	363.724			1.00755			0.503773	−	0.863836i	$-0.331945\pi$
$19$	363.724			1.00755			0.503773	+	0.863836i	$0.331945\pi$
$20$		0			0
$21$	204.707	−	47.0738i	0.464188	−	0.106743i
$22$		0			0
$23$		−	102.256i		−	0.193301i	−0.995318	−	0.0966506i	$-0.969187\pi$
$23$		−	102.256i		−	0.193301i	0.995318	−	0.0966506i	$-0.0308129\pi$
$24$		0			0
$25$	−125.000			−0.200000
$26$		0			0
$27$	−567.728	+	457.302i	−0.778777	+	0.627301i
$28$		0			0
$29$			1565.10i			1.86100i	0.366296	+	0.930498i	$0.380626\pi$
$29$			1565.10i			1.86100i	−0.366296	+	0.930498i	$0.619374\pi$
$30$		0			0
$31$	−326.954			−0.340222			−0.170111	−	0.985425i	$-0.554413\pi$
$31$	−326.954			−0.340222			−0.170111	+	0.985425i	$0.554413\pi$
$32$		0			0
$33$	68.5518	+	298.107i	0.0629493	+	0.273744i
$34$		0			0
$35$			260.936i			0.213009i
$36$		0			0
$37$	72.4778			0.0529421			0.0264711	−	0.999650i	$-0.491573\pi$
$37$	72.4778			0.0529421			0.0264711	+	0.999650i	$0.491573\pi$
$38$		0			0
$39$	1104.85	−	254.068i	0.726398	−	0.167040i
$40$		0			0
$41$			2149.01i			1.27841i	0.769036	+	0.639205i	$0.220736\pi$
$41$			2149.01i			1.27841i	−0.769036	+	0.639205i	$0.779264\pi$
$42$		0			0
$43$	−1501.27			−0.811934			−0.405967	−	0.913888i	$-0.633065\pi$
$43$	−1501.27			−0.811934			−0.405967	+	0.913888i	$0.633065\pi$
$44$		0			0
$45$	−395.583	−	814.640i	−0.195350	−	0.402292i
$46$		0			0
$47$			1958.33i			0.886522i	0.896393	+	0.443261i	$0.146178\pi$
$47$			1958.33i			0.886522i	−0.896393	+	0.443261i	$0.853822\pi$
$48$		0			0
$49$	−1856.30			−0.773136
$50$		0			0
$51$	621.418	+	2702.32i	0.238915	+	1.03896i
$52$		0			0
$53$			2965.17i			1.05560i	0.849370	+	0.527798i	$0.176982\pi$
$53$			2965.17i			1.05560i	−0.849370	+	0.527798i	$0.823018\pi$
$54$		0			0
$55$	−379.992			−0.125617
$56$		0			0
$57$	−3190.25	+	733.621i	−0.981919	+	0.225799i
$58$		0			0
$59$			5045.29i			1.44938i	0.689075	+	0.724690i	$0.258017\pi$
$59$			5045.29i			1.44938i	−0.689075	+	0.724690i	$0.741983\pi$
$60$		0			0
$61$	736.716			0.197989			0.0989944	−	0.995088i	$-0.468437\pi$
$61$	736.716			0.197989			0.0989944	+	0.995088i	$0.468437\pi$
$62$		0			0
$63$	−1700.55	+	825.775i	−0.428459	+	0.208056i
$64$		0			0
$65$			1408.34i			0.333334i
$66$		0			0
$67$	−4254.63			−0.947791			−0.473895	−	0.880581i	$-0.657153\pi$
$67$	−4254.63			−0.947791			−0.473895	+	0.880581i	$0.657153\pi$
$68$		0			0
$69$	206.248	+	896.898i	0.0433203	+	0.188384i
$70$		0			0
$71$		−	1303.84i		−	0.258647i	−0.991602	−	0.129323i	$-0.958719\pi$
$71$		−	1303.84i		−	0.258647i	0.991602	−	0.129323i	$-0.0412805\pi$
$72$		0			0
$73$	6768.14			1.27006			0.635029	−	0.772488i	$-0.280988\pi$
$73$	6768.14			1.27006			0.635029	+	0.772488i	$0.280988\pi$
$74$		0			0
$75$	1096.38	−	252.121i	0.194913	−	0.0448216i
$76$		0			0
$77$			793.229i			0.133788i
$78$		0			0
$79$	6367.07			1.02020			0.510100	−	0.860115i	$-0.329608\pi$
$79$	6367.07			1.02020			0.510100	+	0.860115i	$0.329608\pi$
$80$		0			0
$81$	4057.23	−	5156.13i	0.618385	−	0.785875i
$82$		0			0
$83$			6892.17i			1.00046i	0.865893	+	0.500230i	$0.166751\pi$
$83$			6892.17i			1.00046i	−0.865893	+	0.500230i	$0.833249\pi$
$84$		0			0
$85$	−3444.60			−0.476762
$86$		0			0
$87$	−3156.76	−	13727.6i	−0.417064	−	1.81366i
$88$		0			0
$89$		−	6401.82i		−	0.808208i	−0.914713	−	0.404104i	$-0.867583\pi$
$89$		−	6401.82i		−	0.808208i	0.914713	−	0.404104i	$-0.132417\pi$
$90$		0			0
$91$	2939.88			0.355016
$92$		0			0
$93$	2867.74	−	659.456i	0.331568	−	0.0762465i
$94$		0			0
$95$		−	4066.56i		−	0.450588i
$96$		0			0
$97$	−9879.94			−1.05005			−0.525026	−	0.851086i	$-0.675944\pi$
$97$	−9879.94			−1.05005			−0.525026	+	0.851086i	$0.675944\pi$
$98$		0			0
$99$	−1202.55	−	2476.45i	−0.122696	−	0.252674i
$100$		0			0
$101$			11119.7i			1.09006i	0.838415	+	0.545032i	$0.183482\pi$
$101$			11119.7i			1.09006i	−0.838415	+	0.545032i	$0.816518\pi$
$102$		0			0
$103$	12245.5			1.15426			0.577128	−	0.816653i	$-0.304173\pi$
$103$	12245.5			1.15426			0.577128	+	0.816653i	$0.304173\pi$
$104$		0			0
$105$	−526.301	−	2288.69i	−0.0477370	−	0.207591i
$106$		0			0
$107$		−	708.790i		−	0.0619085i	−0.999521	−	0.0309542i	$-0.990145\pi$
$107$		−	708.790i		−	0.0619085i	0.999521	−	0.0309542i	$-0.00985462\pi$
$108$		0			0
$109$	−908.407			−0.0764588			−0.0382294	−	0.999269i	$-0.512172\pi$
$109$	−908.407			−0.0764588			−0.0382294	+	0.999269i	$0.512172\pi$
$110$		0			0
$111$	−635.708	+	146.186i	−0.0515955	+	0.0118647i
$112$		0			0
$113$		−	3441.26i		−	0.269501i	−0.990880	−	0.134751i	$-0.956977\pi$
$113$		−	3441.26i		−	0.269501i	0.990880	−	0.134751i	$-0.0430233\pi$
$114$		0			0
$115$	−1143.26			−0.0864469
$116$		0			0
$117$	−9178.29	+	4456.91i	−0.670487	+	0.325583i
$118$		0			0
$119$			7190.57i			0.507773i
$120$		0			0
$121$	13485.8			0.921102
$122$		0			0
$123$	−4334.48	−	18849.1i	−0.286502	−	1.24589i
$124$		0			0
$125$			1397.54i			0.0894427i
$126$		0			0
$127$	−28648.5			−1.77621			−0.888106	−	0.459638i	$-0.847979\pi$
$127$	−28648.5			−1.77621			−0.888106	+	0.459638i	$0.847979\pi$
$128$		0			0
$129$	13167.7	−	3028.01i	0.791282	−	0.181961i
$130$		0			0
$131$		−	400.763i		−	0.0233531i	−0.999932	−	0.0116766i	$-0.996283\pi$
$131$		−	400.763i		−	0.0233531i	0.999932	−	0.0116766i	$-0.00371685\pi$
$132$		0			0
$133$	−8488.90			−0.479897
$134$		0			0
$135$	5112.80	+	6347.40i	0.280538	+	0.348280i
$136$		0			0
$137$		−	29277.7i		−	1.55989i	−0.625845	−	0.779947i	$-0.715246\pi$
$137$		−	29277.7i		−	1.55989i	0.625845	−	0.779947i	$-0.284754\pi$
$138$		0			0
$139$	−1852.04			−0.0958565			−0.0479282	−	0.998851i	$-0.515262\pi$
$139$	−1852.04			−0.0958565			−0.0479282	+	0.998851i	$0.515262\pi$
$140$		0			0
$141$	−3949.89	−	17176.6i	−0.198677	−	0.863973i
$142$		0			0
$143$			4281.25i			0.209362i
$144$		0			0
$145$	17498.3			0.832263
$146$		0			0
$147$	16281.7	−	3744.10i	0.753470	−	0.173266i
$148$		0			0
$149$			19905.0i			0.896582i	0.893888	+	0.448291i	$0.147967\pi$
$149$			19905.0i			0.896582i	−0.893888	+	0.448291i	$0.852033\pi$
$150$		0			0
$151$	24648.2			1.08101			0.540506	−	0.841340i	$-0.318233\pi$
$151$	24648.2			1.08101			0.540506	+	0.841340i	$0.318233\pi$
$152$		0			0
$153$	−10901.0	−	22448.9i	−0.465676	−	0.958986i
$154$		0			0
$155$			3655.45i			0.152152i
$156$		0			0
$157$	1399.18			0.0567642			0.0283821	−	0.999597i	$-0.490964\pi$
$157$	1399.18			0.0567642			0.0283821	+	0.999597i	$0.490964\pi$
$158$		0			0
$159$	−5980.66	−	26007.8i	−0.236568	−	1.02875i
$160$		0			0
$161$			2386.54i			0.0920699i
$162$		0			0
$163$	−8023.34			−0.301981			−0.150991	−	0.988535i	$-0.548246\pi$
$163$	−8023.34			−0.301981			−0.150991	+	0.988535i	$0.548246\pi$
$164$		0			0
$165$	3332.94	−	766.432i	0.122422	−	0.0281518i
$166$		0			0
$167$			20948.5i			0.751138i	0.926794	+	0.375569i	$0.122553\pi$
$167$			20948.5i			0.751138i	−0.926794	+	0.375569i	$0.877447\pi$
$168$		0			0
$169$	−12693.7			−0.444443
$170$		0			0
$171$	26502.3	−	12869.3i	0.906339	−	0.440111i
$172$		0			0
$173$			20564.6i			0.687112i	0.939132	+	0.343556i	$0.111632\pi$
$173$			20564.6i			0.687112i	−0.939132	+	0.343556i	$0.888368\pi$
$174$		0			0
$175$	2917.36			0.0952606
$176$		0			0
$177$	−10176.2	−	44252.6i	−0.324817	−	1.41251i
$178$		0			0
$179$		−	32395.6i		−	1.01107i	−0.862807	−	0.505533i	$-0.831296\pi$
$179$		−	32395.6i		−	1.01107i	0.862807	−	0.505533i	$-0.168704\pi$
$180$		0			0
$181$	18055.4			0.551125			0.275562	−	0.961283i	$-0.411136\pi$
$181$	18055.4			0.551125			0.275562	+	0.961283i	$0.411136\pi$
$182$		0			0
$183$	−6461.79	+	1485.93i	−0.192953	+	0.0443708i
$184$		0			0
$185$		−	810.326i		−	0.0236764i
$186$		0			0
$187$	−10471.4			−0.299447
$188$		0			0
$189$	13250.1	−	10672.9i	0.370934	−	0.298785i
$190$		0			0
$191$			11720.8i			0.321285i	0.987013	+	0.160642i	$0.0513566\pi$
$191$			11720.8i			0.321285i	−0.987013	+	0.160642i	$0.948643\pi$
$192$		0			0
$193$	−28253.2			−0.758496			−0.379248	−	0.925295i	$-0.623817\pi$
$193$	−28253.2			−0.758496			−0.379248	+	0.925295i	$0.623817\pi$
$194$		0			0
$195$	−2840.57	−	12352.6i	−0.0747027	−	0.324855i
$196$		0			0
$197$			53312.1i			1.37370i	0.726797	+	0.686852i	$0.241008\pi$
$197$			53312.1i			1.37370i	−0.726797	+	0.686852i	$0.758992\pi$
$198$		0			0
$199$	−50464.4			−1.27432			−0.637161	−	0.770731i	$-0.719891\pi$
$199$	−50464.4			−1.27432			−0.637161	+	0.770731i	$0.719891\pi$
$200$		0			0
$201$	37317.7	−	8581.47i	0.923683	−	0.212407i
$202$		0			0
$203$		−	36527.6i		−	0.886398i
$204$		0			0
$205$	24026.6			0.571722
$206$		0			0
$207$	−3618.03	−	7450.77i	−0.0844368	−	0.173884i
$208$		0			0
$209$		−	12362.1i		−	0.283008i
$210$		0			0
$211$	67329.0			1.51230			0.756148	−	0.654400i	$-0.227079\pi$
$211$	67329.0			1.51230			0.756148	+	0.654400i	$0.227079\pi$
$212$		0			0
$213$	2629.80	+	11436.1i	0.0579648	+	0.252068i
$214$		0			0
$215$			16784.7i			0.363108i
$216$		0			0
$217$	7630.72			0.162049
$218$		0			0
$219$	−59363.9	+	13651.1i	−1.23775	+	0.284630i
$220$		0			0
$221$			38809.2i			0.794604i
$222$		0			0
$223$	56592.7			1.13802			0.569011	−	0.822330i	$-0.307326\pi$
$223$	56592.7			1.13802			0.569011	+	0.822330i	$0.307326\pi$
$224$		0			0
$225$	−9107.96	+	4422.75i	−0.179910	+	0.0873630i
$226$		0			0
$227$		−	19015.9i		−	0.369033i	−0.982829	−	0.184516i	$-0.940928\pi$
$227$		−	19015.9i		−	0.369033i	0.982829	−	0.184516i	$-0.0590719\pi$
$228$		0			0
$229$	−79079.9			−1.50798			−0.753989	−	0.656887i	$-0.771873\pi$
$229$	−79079.9			−1.50798			−0.753989	+	0.656887i	$0.771873\pi$
$230$		0			0
$231$	−1599.92	−	6957.47i	−0.0299829	−	0.130385i
$232$		0			0
$233$		−	36312.6i		−	0.668876i	−0.942418	−	0.334438i	$-0.891454\pi$
$233$		−	36312.6i		−	0.668876i	0.942418	−	0.334438i	$-0.108546\pi$
$234$		0			0
$235$	21894.8			0.396465
$236$		0			0
$237$	−55846.0	+	12842.2i	−0.994250	+	0.228635i
$238$		0			0
$239$			46783.7i			0.819028i	0.912304	+	0.409514i	$0.134302\pi$
$239$			46783.7i			0.819028i	−0.912304	+	0.409514i	$0.865698\pi$
$240$		0			0
$241$	30646.4			0.527649			0.263824	−	0.964571i	$-0.415016\pi$
$241$	30646.4			0.527649			0.263824	+	0.964571i	$0.415016\pi$
$242$		0			0
$243$	−25186.5	+	53408.1i	−0.426535	+	0.904471i
$244$		0			0
$245$			20754.0i			0.345757i
$246$		0			0
$247$	−45816.6			−0.750981
$248$		0			0
$249$	−13901.3	−	60451.8i	−0.224211	−	0.975013i
$250$		0			0
$251$			114804.i			1.82226i	0.412120	+	0.911130i	$0.364788\pi$
$251$			114804.i			1.82226i	−0.412120	+	0.911130i	$0.635212\pi$
$252$		0			0
$253$	−3475.44			−0.0542960
$254$		0			0
$255$	30212.9	−	6947.66i	0.464635	−	0.106846i
$256$		0			0
$257$			89602.7i			1.35661i	0.734780	+	0.678305i	$0.237285\pi$
$257$			89602.7i			1.35661i	−0.734780	+	0.678305i	$0.762715\pi$
$258$		0			0
$259$	−1691.55			−0.0252165
$260$		0			0
$261$	55376.3	+	114039.i	0.812911	+	1.67406i
$262$		0			0
$263$			113892.i			1.64658i	0.567620	+	0.823291i	$0.307864\pi$
$263$			113892.i			1.64658i	−0.567620	+	0.823291i	$0.692136\pi$
$264$		0			0
$265$	33151.6			0.472077
$266$		0			0
$267$	12912.3	+	56150.9i	0.181126	+	0.787651i
$268$		0			0
$269$			39232.3i			0.542174i	0.962555	+	0.271087i	$0.0873832\pi$
$269$			39232.3i			0.542174i	−0.962555	+	0.271087i	$0.912617\pi$
$270$		0			0
$271$	−52549.5			−0.715533			−0.357767	−	0.933811i	$-0.616462\pi$
$271$	−52549.5			−0.715533			−0.357767	+	0.933811i	$0.616462\pi$
$272$		0			0
$273$	−25786.0	+	5929.66i	−0.345986	+	0.0795618i
$274$		0			0
$275$			4248.44i			0.0561777i
$276$		0			0
$277$	−28707.3			−0.374139			−0.187069	−	0.982347i	$-0.559899\pi$
$277$	−28707.3			−0.374139			−0.187069	+	0.982347i	$0.559899\pi$
$278$		0			0
$279$	−23823.0	+	11568.3i	−0.306047	+	0.148614i
$280$		0			0
$281$		−	83537.2i		−	1.05796i	−0.848636	−	0.528978i	$-0.822576\pi$
$281$		−	83537.2i		−	1.05796i	0.848636	−	0.528978i	$-0.177424\pi$
$282$		0			0
$283$	−126719.			−1.58223			−0.791115	−	0.611667i	$-0.790499\pi$
$283$	−126719.			−1.58223			−0.791115	+	0.611667i	$0.790499\pi$
$284$		0			0
$285$	8202.13	+	35668.1i	0.100980	+	0.439127i
$286$		0			0
$287$		−	50155.3i		−	0.608910i
$288$		0			0
$289$	−11401.3			−0.136508
$290$		0			0
$291$	86657.7	−	19927.5i	1.02334	−	0.235325i
$292$		0			0
$293$			69385.0i			0.808222i	0.914710	+	0.404111i	$0.132419\pi$
$293$			69385.0i			0.808222i	−0.914710	+	0.404111i	$0.867581\pi$
$294$		0			0
$295$	56408.1			0.648182
$296$		0			0
$297$	15542.6	+	19295.7i	0.176202	+	0.218749i
$298$		0			0
$299$			12880.7i			0.144078i
$300$		0			0
$301$	35037.8			0.386727
$302$		0			0
$303$	−22428.2	−	97532.1i	−0.244292	−	1.06234i
$304$		0			0
$305$		−	8236.74i		−	0.0885432i
$306$		0			0
$307$	−63952.5			−0.678549			−0.339274	−	0.940687i	$-0.610182\pi$
$307$	−63952.5			−0.678549			−0.339274	+	0.940687i	$0.610182\pi$
$308$		0			0
$309$	−107406.	+	24698.8i	−1.12490	+	0.258678i
$310$		0			0
$311$		−	18806.5i		−	0.194441i	−0.995263	−	0.0972203i	$-0.969005\pi$
$311$		−	18806.5i		−	0.194441i	0.995263	−	0.0972203i	$-0.0309951\pi$
$312$		0			0
$313$	62746.1			0.640469			0.320235	−	0.947338i	$-0.396238\pi$
$313$	62746.1			0.640469			0.320235	+	0.947338i	$0.396238\pi$
$314$		0			0
$315$	9232.45	+	19012.8i	0.0930456	+	0.191613i
$316$		0			0
$317$		−	93427.4i		−	0.929728i	−0.885382	−	0.464864i	$-0.846103\pi$
$317$		−	93427.4i		−	0.929728i	0.885382	−	0.464864i	$-0.153897\pi$
$318$		0			0
$319$	53193.8			0.522732
$320$		0			0
$321$	1429.61	+	6216.86i	0.0138742	+	0.0603338i
$322$		0			0
$323$		−	112061.i		−	1.07412i
$324$		0			0
$325$	15745.7			0.149071
$326$		0			0
$327$	7967.71	−	1832.23i	0.0745140	−	0.0171350i
$328$		0			0
$329$		−	45705.1i		−	0.422253i
$330$		0			0
$331$	71675.0			0.654202			0.327101	−	0.944989i	$-0.393928\pi$
$331$	71675.0			0.654202			0.327101	+	0.944989i	$0.393928\pi$
$332$		0			0
$333$	5281.00	−	2564.41i	0.0476242	−	0.0231259i
$334$		0			0
$335$			47568.3i			0.423865i
$336$		0			0
$337$	−87920.2			−0.774157			−0.387078	−	0.922047i	$-0.626516\pi$
$337$	−87920.2			−0.774157			−0.387078	+	0.922047i	$0.626516\pi$
$338$		0			0
$339$	6940.92	+	30183.6i	0.0603973	+	0.262646i
$340$		0			0
$341$			11112.3i			0.0955645i
$342$		0			0
$343$	99360.4			0.844550
$344$		0			0
$345$	10027.6	−	2305.92i	0.0842481	−	0.0193734i
$346$		0			0
$347$		−	5941.38i		−	0.0493433i	−0.999696	−	0.0246717i	$-0.992146\pi$
$347$		−	5941.38i		−	0.0493433i	0.999696	−	0.0246717i	$-0.00785403\pi$
$348$		0			0
$349$	−65214.3			−0.535417			−0.267709	−	0.963500i	$-0.586266\pi$
$349$	−65214.3			−0.535417			−0.267709	+	0.963500i	$0.586266\pi$
$350$		0			0
$351$	71514.1	−	57604.2i	0.580467	−	0.467563i
$352$		0			0
$353$		−	36220.5i		−	0.290673i	−0.989382	−	0.145337i	$-0.953573\pi$
$353$		−	36220.5i		−	0.290673i	0.989382	−	0.145337i	$-0.0464265\pi$
$354$		0			0
$355$	−14577.4			−0.115670
$356$		0			0
$357$	−14503.2	−	63069.1i	−0.113796	−	0.494857i
$358$		0			0
$359$		−	152925.i		−	1.18656i	−0.804997	−	0.593279i	$-0.797833\pi$
$359$		−	152925.i		−	1.18656i	0.804997	−	0.593279i	$-0.202167\pi$
$360$		0			0
$361$	1974.27			0.0151493
$362$		0			0
$363$	−118285.	+	27200.6i	−0.897673	+	0.206426i
$364$		0			0
$365$		−	75670.1i		−	0.567987i
$366$		0			0
$367$	−143778.			−1.06748			−0.533740	−	0.845649i	$-0.679214\pi$
$367$	−143778.			−1.06748			−0.533740	+	0.845649i	$0.679214\pi$
$368$		0			0
$369$	76036.2	+	156584.i	0.558429	+	1.15000i
$370$		0			0
$371$		−	69203.7i		−	0.502784i
$372$		0			0
$373$	102685.			0.738056			0.369028	−	0.929418i	$-0.379691\pi$
$373$	102685.			0.738056			0.369028	+	0.929418i	$0.379691\pi$
$374$		0			0
$375$	−2818.80	−	12258.0i	−0.0200448	−	0.0871677i
$376$		0			0
$377$		−	197148.i		−	1.38711i
$378$		0			0
$379$	148649.			1.03486			0.517432	−	0.855724i	$-0.326888\pi$
$379$	148649.			1.03486			0.517432	+	0.855724i	$0.326888\pi$
$380$		0			0
$381$	251279.	−	57783.3i	1.73103	−	0.398063i
$382$		0			0
$383$		−	88390.0i		−	0.602568i	−0.953535	−	0.301284i	$-0.902585\pi$
$383$		−	88390.0i		−	0.602568i	0.953535	−	0.301284i	$-0.0974152\pi$
$384$		0			0
$385$	8868.57			0.0598318
$386$		0			0
$387$	−109388.	+	53117.8i	−0.730376	+	0.354665i
$388$		0			0
$389$		−	107542.i		−	0.710687i	−0.934736	−	0.355344i	$-0.884364\pi$
$389$		−	107542.i		−	0.710687i	0.934736	−	0.355344i	$-0.115636\pi$
$390$		0			0
$391$	−31504.6			−0.206073
$392$		0			0
$393$	808.328	+	3515.13i	0.00523362	+	0.0227591i
$394$		0			0
$395$		−	71186.0i		−	0.456247i
$396$		0			0
$397$	−131243.			−0.832710			−0.416355	−	0.909202i	$-0.636693\pi$
$397$	−131243.			−0.832710			−0.416355	+	0.909202i	$0.636693\pi$
$398$		0			0
$399$	74456.8	−	17121.9i	0.467691	−	0.107549i
$400$		0			0
$401$			21480.0i			0.133581i	0.997767	+	0.0667907i	$0.0212760\pi$
$401$			21480.0i			0.133581i	−0.997767	+	0.0667907i	$0.978724\pi$
$402$		0			0
$403$	41184.8			0.253587
$404$		0			0
$405$	−57647.3	−	45361.2i	−0.351454	−	0.276550i
$406$		0			0
$407$		−	2463.34i		−	0.0148708i
$408$		0			0
$409$	118822.			0.710313			0.355156	−	0.934807i	$-0.384428\pi$
$409$	118822.			0.710313			0.355156	+	0.934807i	$0.384428\pi$
$410$		0			0
$411$	59052.2	+	256797.i	0.349585	+	1.52022i
$412$		0			0
$413$		−	117751.i		−	0.690344i
$414$		0			0
$415$	77056.8			0.447419
$416$		0			0
$417$	16244.4	−	3735.52i	0.0934183	−	0.0214822i
$418$		0			0
$419$		−	95945.4i		−	0.546507i	−0.961942	−	0.273254i	$-0.911900\pi$
$419$		−	95945.4i		−	0.546507i	0.961942	−	0.273254i	$-0.0880998\pi$
$420$		0			0
$421$	73818.2			0.416485			0.208242	−	0.978077i	$-0.433226\pi$
$421$	73818.2			0.416485			0.208242	+	0.978077i	$0.433226\pi$
$422$		0			0
$423$	69289.6	+	142691.i	0.387246	+	0.797472i
$424$		0			0
$425$			38511.8i			0.213214i
$426$		0			0
$427$	−17194.1			−0.0943026
$428$		0			0
$429$	−8635.15	−	37551.1i	−0.0469197	−	0.204037i
$430$		0			0
$431$		−	107733.i		−	0.579955i	−0.957033	−	0.289977i	$-0.906352\pi$
$431$		−	107733.i		−	0.579955i	0.957033	−	0.289977i	$-0.0936478\pi$
$432$		0			0
$433$	−114738.			−0.611974			−0.305987	−	0.952036i	$-0.598986\pi$
$433$	−114738.			−0.611974			−0.305987	+	0.952036i	$0.598986\pi$
$434$		0			0
$435$	−153479.	+	35293.6i	−0.811094	+	0.186517i
$436$		0			0
$437$		−	37193.1i		−	0.194760i
$438$		0			0
$439$	−257591.			−1.33660			−0.668300	−	0.743892i	$-0.732978\pi$
$439$	−257591.			−1.33660			−0.668300	+	0.743892i	$0.732978\pi$
$440$		0			0
$441$	−135257.	+	65679.6i	−0.695475	+	0.337717i
$442$		0			0
$443$		−	35360.6i		−	0.180182i	−0.995934	−	0.0900911i	$-0.971284\pi$
$443$		−	35360.6i		−	0.180182i	0.995934	−	0.0900911i	$-0.0287158\pi$
$444$		0			0
$445$	−71574.5			−0.361442
$446$		0			0
$447$	−40147.8	−	174588.i	−0.200931	−	0.873777i
$448$		0			0
$449$			8230.74i			0.0408269i	0.999792	+	0.0204134i	$0.00649825\pi$
$449$			8230.74i			0.0408269i	−0.999792	+	0.0204134i	$0.993502\pi$
$450$		0			0
$451$	73039.4			0.359091
$452$		0			0
$453$	−216191.	+	49714.6i	−1.05352	+	0.242263i
$454$		0			0
$455$		−	32868.9i		−	0.158768i
$456$		0			0
$457$	−180660.			−0.865027			−0.432514	−	0.901627i	$-0.642373\pi$
$457$	−180660.			−0.865027			−0.432514	+	0.901627i	$0.642373\pi$
$458$		0			0
$459$	140892.	+	174914.i	0.668748	+	0.830232i
$460$		0			0
$461$		−	34875.9i		−	0.164106i	−0.996628	−	0.0820529i	$-0.973852\pi$
$461$		−	34875.9i		−	0.164106i	0.996628	−	0.0820529i	$-0.0261476\pi$
$462$		0			0
$463$	206020.			0.961054			0.480527	−	0.876980i	$-0.340445\pi$
$463$	206020.			0.961054			0.480527	+	0.876980i	$0.340445\pi$
$464$		0			0
$465$	−7372.94	−	32062.3i	−0.0340985	−	0.148282i
$466$		0			0
$467$		−	274766.i		−	1.25988i	−0.776644	−	0.629940i	$-0.783079\pi$
$467$		−	274766.i		−	1.25988i	0.776644	−	0.629940i	$-0.216921\pi$
$468$		0			0
$469$	99298.2			0.451436
$470$		0			0
$471$	−12272.3	+	2822.11i	−0.0553204	+	0.0127213i
$472$		0			0
$473$			51024.3i			0.228063i
$474$		0			0
$475$	−45465.5			−0.201509
$476$		0			0
$477$	104914.	+	216053.i	0.461101	+	0.949564i
$478$		0			0
$479$			160960.i			0.701533i	0.936463	+	0.350766i	$0.114079\pi$
$479$			160960.i			0.701533i	−0.936463	+	0.350766i	$0.885921\pi$
$480$		0			0
$481$	−9129.68			−0.0394608
$482$		0			0
$483$	−4813.59	−	20932.6i	−0.0206336	−	0.0897280i
$484$		0			0
$485$			110461.i			0.469597i
$486$		0			0
$487$	69827.5			0.294421			0.147210	−	0.989105i	$-0.452971\pi$
$487$	69827.5			0.294421			0.147210	+	0.989105i	$0.452971\pi$
$488$		0			0
$489$	70373.4	−	16182.9i	0.294300	−	0.0676764i
$490$		0			0
$491$		−	430893.i		−	1.78734i	−0.448727	−	0.893669i	$-0.648122\pi$
$491$		−	430893.i		−	1.78734i	0.448727	−	0.893669i	$-0.351878\pi$
$492$		0			0
$493$	482198.			1.98396
$494$		0			0
$495$	−27687.6	+	13444.9i	−0.112999	+	0.0548715i
$496$		0			0
$497$			30430.1i			0.123194i
$498$		0			0
$499$	−282872.			−1.13603			−0.568013	−	0.823019i	$-0.692288\pi$
$499$	−282872.			−1.13603			−0.568013	+	0.823019i	$0.692288\pi$
$500$		0			0
$501$	−42252.5	−	183741.i	−0.168336	−	0.732032i
$502$		0			0
$503$			240734.i			0.951483i	0.879585	+	0.475742i	$0.157820\pi$
$503$			240734.i			0.951483i	−0.879585	+	0.475742i	$0.842180\pi$
$504$		0			0
$505$	124322.			0.487491
$506$		0			0
$507$	111338.	−	25602.9i	0.433138	−	0.0996032i
$508$		0			0
$509$		−	225427.i		−	0.870102i	−0.900406	−	0.435051i	$-0.856730\pi$
$509$		−	225427.i		−	0.870102i	0.900406	−	0.435051i	$-0.143270\pi$
$510$		0			0
$511$	−157961.			−0.604933
$512$		0			0
$513$	−206497.	+	166332.i	−0.784654	+	0.632035i
$514$		0			0
$515$		−	136909.i		−	0.516199i
$516$		0			0
$517$	66558.6			0.249014
$518$		0			0
$519$	−41478.2	−	180374.i	−0.153987	−	0.669635i
$520$		0			0
$521$			154960.i			0.570880i	0.958397	+	0.285440i	$0.0921397\pi$
$521$			154960.i			0.570880i	−0.958397	+	0.285440i	$0.907860\pi$
$522$		0			0
$523$	128163.			0.468553			0.234277	−	0.972170i	$-0.424728\pi$
$523$	128163.			0.468553			0.234277	+	0.972170i	$0.424728\pi$
$524$		0			0
$525$	−25588.4	+	5884.22i	−0.0928376	+	0.0213486i
$526$		0			0
$527$			100733.i			0.362701i
$528$		0			0
$529$	269385.			0.962635
$530$		0			0
$531$	178513.	+	367618.i	0.633111	+	1.30379i
$532$		0			0
$533$		−	270700.i		−	0.952872i
$534$		0			0
$535$	−7924.52			−0.0276863
$536$		0			0
$537$	65340.9	+	284144.i	0.226588	+	0.985349i
$538$		0			0
$539$			63090.9i			0.217165i
$540$		0			0
$541$	−499660.			−1.70718			−0.853591	−	0.520943i	$-0.825580\pi$
$541$	−499660.			−1.70718			−0.853591	+	0.520943i	$0.825580\pi$
$542$		0			0
$543$	−158365.	+	36417.2i	−0.537107	+	0.123511i
$544$		0			0
$545$			10156.3i			0.0341934i
$546$		0			0
$547$	244301.			0.816490			0.408245	−	0.912872i	$-0.366141\pi$
$547$	244301.			0.816490			0.408245	+	0.912872i	$0.366141\pi$
$548$		0			0
$549$	53679.8	−	26066.5i	0.178101	−	0.0864845i
$550$		0			0
$551$			569264.i			1.87504i
$552$		0			0
$553$	−148600.			−0.485924
$554$		0			0
$555$	1634.40	+	7107.43i	0.00530608	+	0.0230742i
$556$		0			0
$557$		−	247875.i		−	0.798954i	−0.916743	−	0.399477i	$-0.869192\pi$
$557$		−	247875.i		−	0.798954i	0.916743	−	0.399477i	$-0.130808\pi$
$558$		0			0
$559$	189107.			0.605181
$560$		0			0
$561$	91845.2	−	21120.4i	0.291830	−	0.0671085i
$562$		0			0
$563$		−	265392.i		−	0.837279i	−0.908152	−	0.418640i	$-0.862507\pi$
$563$		−	265392.i		−	0.837279i	0.908152	−	0.418640i	$-0.137493\pi$
$564$		0			0
$565$	−38474.5			−0.120525
$566$		0			0
$567$	−94690.9	+	120338.i	−0.294539	+	0.374315i
$568$		0			0
$569$			551671.i			1.70394i	0.523587	+	0.851972i	$0.324593\pi$
$569$			551671.i			1.70394i	−0.523587	+	0.851972i	$0.675407\pi$
$570$		0			0
$571$	−325456.			−0.998205			−0.499102	−	0.866543i	$-0.666337\pi$
$571$	−325456.			−0.998205			−0.499102	+	0.866543i	$0.666337\pi$
$572$		0			0
$573$	−23640.5	−	102804.i	−0.0720024	−	0.313113i
$574$		0			0
$575$			12782.0i			0.0386602i
$576$		0			0
$577$	−262408.			−0.788181			−0.394091	−	0.919072i	$-0.628940\pi$
$577$	−262408.			−0.788181			−0.394091	+	0.919072i	$0.628940\pi$
$578$		0			0
$579$	247811.	−	56985.9i	0.739203	−	0.169985i
$580$		0			0
$581$		−	160855.i		−	0.476522i
$582$		0			0
$583$	100779.			0.296505
$584$		0			0
$585$	49829.7	+	102616.i	0.145605	+	0.299851i
$586$		0			0
$587$			427538.i			1.24079i	0.784290	+	0.620395i	$0.213028\pi$
$587$			427538.i			1.24079i	−0.784290	+	0.620395i	$0.786972\pi$
$588$		0			0
$589$	−118921.			−0.342790
$590$		0			0
$591$	−107529.	−	467605.i	−0.307858	−	1.33876i
$592$		0			0
$593$			63018.6i			0.179209i	0.995977	+	0.0896043i	$0.0285603\pi$
$593$			63018.6i			0.179209i	−0.995977	+	0.0896043i	$0.971440\pi$
$594$		0			0
$595$	80393.0			0.227083
$596$		0			0
$597$	442628.	−	101785.i	1.24191	−	0.285586i
$598$		0			0
$599$			634561.i			1.76856i	0.466959	+	0.884279i	$0.345350\pi$
$599$			634561.i			1.76856i	−0.466959	+	0.884279i	$0.654650\pi$
$600$		0			0
$601$	−474806.			−1.31452			−0.657260	−	0.753664i	$-0.728285\pi$
$601$	−474806.			−1.31452			−0.657260	+	0.753664i	$0.728285\pi$
$602$		0			0
$603$	−310008.	+	150538.i	−0.852587	+	0.414009i
$604$		0			0
$605$		−	150776.i		−	0.411929i
$606$		0			0
$607$	163861.			0.444731			0.222365	−	0.974963i	$-0.428622\pi$
$607$	163861.			0.444731			0.222365	+	0.974963i	$0.428622\pi$
$608$		0			0
$609$	73675.1	+	320386.i	0.198649	+	0.863852i
$610$		0			0
$611$		−	246681.i		−	0.660775i
$612$		0			0
$613$	328099.			0.873139			0.436570	−	0.899670i	$-0.356193\pi$
$613$	328099.			0.873139			0.436570	+	0.899670i	$0.356193\pi$
$614$		0			0
$615$	−210739.	+	48461.0i	−0.557180	+	0.128127i
$616$		0			0
$617$			405778.i			1.06590i	0.846146	+	0.532952i	$0.178917\pi$
$617$			405778.i			1.06590i	−0.846146	+	0.532952i	$0.821083\pi$
$618$		0			0
$619$	−184587.			−0.481747			−0.240874	−	0.970557i	$-0.577434\pi$
$619$	−184587.			−0.481747			−0.240874	+	0.970557i	$0.577434\pi$
$620$		0			0
$621$	46762.0	+	58053.8i	0.121258	+	0.150538i
$622$		0			0
$623$			149411.i			0.384952i
$624$		0			0
$625$	15625.0			0.0400000
$626$		0			0
$627$	24933.9	+	108429.i	0.0634243	+	0.275810i
$628$		0			0
$629$		−	22330.0i		−	0.0564401i
$630$		0			0
$631$	−712031.			−1.78830			−0.894149	−	0.447769i	$-0.852219\pi$
$631$	−712031.			−1.78830			−0.894149	+	0.447769i	$0.852219\pi$
$632$		0			0
$633$	−590548.	+	135801.i	−1.47383	+	0.338918i
$634$		0			0
$635$			320300.i			0.794346i
$636$		0			0
$637$	233829.			0.576262
$638$		0			0
$639$	−46132.4	−	95002.4i	−0.112981	−	0.232666i
$640$		0			0
$641$			388033.i			0.944393i	0.881493	+	0.472196i	$0.156539\pi$
$641$			388033.i			0.944393i	−0.881493	+	0.472196i	$0.843461\pi$
$642$		0			0
$643$	276131.			0.667872			0.333936	−	0.942596i	$-0.391623\pi$
$643$	276131.			0.667872			0.333936	+	0.942596i	$0.391623\pi$
$644$		0			0
$645$	−33854.2	−	147220.i	−0.0813753	−	0.353872i
$646$		0			0
$647$			603278.i			1.44115i	0.693378	+	0.720574i	$0.256122\pi$
$647$			603278.i			1.44115i	−0.693378	+	0.720574i	$0.743878\pi$
$648$		0			0
$649$	171477.			0.407114
$650$		0			0
$651$	−66929.6	+	15390.9i	−0.157927	+	0.0363164i
$652$		0			0
$653$			625614.i			1.46717i	0.679598	+	0.733584i	$0.262154\pi$
$653$			625614.i			1.46717i	−0.679598	+	0.733584i	$0.737846\pi$
$654$		0			0
$655$	−4480.67			−0.0104438
$656$		0			0
$657$	493152.	−	239471.i	1.14248	−	0.554781i
$658$		0			0
$659$		−	706201.i		−	1.62614i	−0.582167	−	0.813069i	$-0.697795\pi$
$659$		−	706201.i		−	1.62614i	0.582167	−	0.813069i	$-0.302205\pi$
$660$		0			0
$661$	−296423.			−0.678437			−0.339219	−	0.940708i	$-0.610163\pi$
$661$	−296423.			−0.678437			−0.339219	+	0.940708i	$0.610163\pi$
$662$		0			0
$663$	−78277.1	−	340399.i	−0.178077	−	0.774392i
$664$		0			0
$665$			94908.8i			0.214617i
$666$		0			0
$667$	160041.			0.359733
$668$		0			0
$669$	−496379.	+	114146.i	−1.10908	+	0.255040i
$670$		0			0
$671$		−	25039.1i		−	0.0556127i
$672$		0			0
$673$	52622.9			0.116184			0.0580918	−	0.998311i	$-0.481498\pi$
$673$	52622.9			0.116184			0.0580918	+	0.998311i	$0.481498\pi$
$674$		0			0
$675$	70966.0	−	57162.8i	0.155755	−	0.125460i
$676$		0			0
$677$			341720.i			0.745578i	0.927916	+	0.372789i	$0.121599\pi$
$677$			341720.i			0.745578i	−0.927916	+	0.372789i	$0.878401\pi$
$678$		0			0
$679$	230586.			0.500143
$680$		0			0
$681$	38354.5	+	166790.i	0.0827032	+	0.359646i
$682$		0			0
$683$			247237.i			0.529996i	0.964249	+	0.264998i	$0.0853714\pi$
$683$			247237.i			0.529996i	−0.964249	+	0.264998i	$0.914629\pi$
$684$		0			0
$685$	−327334.			−0.697606
$686$		0			0
$687$	693616.	−	159502.i	1.46962	−	0.337950i
$688$		0			0
$689$		−	373509.i		−	0.786796i
$690$		0			0
$691$	−471510.			−0.987496			−0.493748	−	0.869605i	$-0.664373\pi$
$691$	−471510.			−0.987496			−0.493748	+	0.869605i	$0.664373\pi$
$692$		0			0
$693$	28066.0	+	57797.6i	0.0584406	+	0.120349i
$694$		0			0
$695$			20706.5i			0.0428683i
$696$		0			0
$697$	662097.			1.36288
$698$		0			0
$699$	73241.5	+	318501.i	0.149900	+	0.651863i
$700$		0			0
$701$		−	530831.i		−	1.08024i	−0.841588	−	0.540120i	$-0.818379\pi$
$701$		−	530831.i		−	1.08024i	0.841588	−	0.540120i	$-0.181621\pi$
$702$		0			0
$703$	26361.9			0.0533416
$704$		0			0
$705$	−192041.	+	44161.1i	−0.386380	+	0.0888508i
$706$		0			0
$707$		−	259522.i		−	0.519200i
$708$		0			0
$709$	−938943.			−1.86787			−0.933936	−	0.357441i	$-0.883649\pi$
$709$	−938943.			−1.86787			−0.933936	+	0.357441i	$0.883649\pi$
$710$		0			0
$711$	463928.	−	225280.i	0.917722	−	0.445639i
$712$		0			0
$713$			33433.1i			0.0657653i
$714$		0			0
$715$	47865.8			0.0936296
$716$		0			0
$717$	−94361.3	−	410343.i	−0.183551	−	0.798195i
$718$		0			0
$719$		−	918433.i		−	1.77660i	−0.459263	−	0.888300i	$-0.651886\pi$
$719$		−	918433.i		−	1.77660i	0.459263	−	0.888300i	$-0.348114\pi$
$720$		0			0
$721$	−285796.			−0.549776
$722$		0			0
$723$	−268802.	+	61812.8i	−0.514228	+	0.118250i
$724$		0			0
$725$		−	195637.i		−	0.372199i
$726$		0			0
$727$	314965.			0.595928			0.297964	−	0.954577i	$-0.403692\pi$
$727$	314965.			0.595928			0.297964	+	0.954577i	$0.403692\pi$
$728$		0			0
$729$	113190.	−	519247.i	0.212987	−	0.977055i
$730$		0			0
$731$			462532.i			0.865580i
$732$		0			0
$733$	958567.			1.78408			0.892040	−	0.451956i	$-0.149274\pi$
$733$	958567.			1.78408			0.892040	+	0.451956i	$0.149274\pi$
$734$		0			0
$735$	−41860.3	−	182035.i	−0.0774868	−	0.336962i
$736$		0			0
$737$			144604.i			0.266224i
$738$		0			0
$739$	−197682.			−0.361976			−0.180988	−	0.983485i	$-0.557929\pi$
$739$	−197682.			−0.361976			−0.180988	+	0.983485i	$0.557929\pi$
$740$		0			0
$741$	401861.	−	92410.8i	0.731880	−	0.168301i
$742$		0			0
$743$		−	81590.4i		−	0.147796i	−0.997266	−	0.0738978i	$-0.976456\pi$
$743$		−	81590.4i		−	0.147796i	0.997266	−	0.0738978i	$-0.0235439\pi$
$744$		0			0
$745$	222545.			0.400964
$746$		0			0
$747$	243859.	+	502189.i	0.437016	+	0.899965i
$748$		0			0
$749$			16542.3i			0.0294872i
$750$		0			0
$751$	−132505.			−0.234938			−0.117469	−	0.993077i	$-0.537478\pi$
$751$	−132505.			−0.234938			−0.117469	+	0.993077i	$0.537478\pi$
$752$		0			0
$753$	−231557.	−	1.00696e6i	−0.408383	−	1.77591i
$754$		0			0
$755$		−	275575.i		−	0.483443i
$756$		0			0
$757$	997153.			1.74008			0.870042	−	0.492978i	$-0.164092\pi$
$757$	997153.			1.74008			0.870042	+	0.492978i	$0.164092\pi$
$758$		0			0
$759$	30483.3	−	7009.85i	0.0529150	−	0.0121682i
$760$		0			0
$761$		−	316363.i		−	0.546281i	−0.961974	−	0.273140i	$-0.911938\pi$
$761$		−	316363.i		−	0.546281i	0.961974	−	0.273140i	$-0.0880623\pi$
$762$		0			0
$763$	21201.2			0.0364175
$764$		0			0
$765$	−250986.	+	121877.i	−0.428872	+	0.208257i
$766$		0			0
$767$		−	635532.i		−	1.08031i
$768$		0			0
$769$	584632.			0.988621			0.494311	−	0.869285i	$-0.335421\pi$
$769$	584632.			0.988621			0.494311	+	0.869285i	$0.335421\pi$
$770$		0			0
$771$	−180726.	−	785913.i	−0.304027	−	1.32210i
$772$		0			0
$773$		−	662736.i		−	1.10913i	−0.832141	−	0.554564i	$-0.812885\pi$
$773$		−	662736.i		−	1.10913i	0.832141	−	0.554564i	$-0.187115\pi$
$774$		0			0
$775$	40869.2			0.0680444
$776$		0			0
$777$	14836.7	−	3411.80i	0.0245751	−	0.00565121i
$778$		0			0
$779$			781646.i			1.28806i
$780$		0			0
$781$	−44314.2			−0.0726509
$782$		0			0
$783$	−715723.	−	888551.i	−1.16740	−	1.44930i
$784$		0			0
$785$		−	15643.3i		−	0.0253857i
$786$		0			0
$787$	773806.			1.24935			0.624673	−	0.780886i	$-0.285232\pi$
$787$	773806.			1.24935			0.624673	+	0.780886i	$0.285232\pi$
$788$		0			0
$789$	−229718.	−	998959.i	−0.369012	−	1.60470i
$790$		0			0
$791$			80315.0i			0.128364i
$792$		0			0
$793$	−92800.7			−0.147572
$794$		0			0
$795$	−290776.	+	66865.9i	−0.460070	+	0.105796i
$796$		0			0
$797$			481401.i			0.757863i	0.925425	+	0.378932i	$0.123708\pi$
$797$			481401.i			0.757863i	−0.925425	+	0.378932i	$0.876292\pi$
$798$		0			0
$799$	603350.			0.945096
$800$		0			0
$801$	−226509.	−	466460.i	−0.353038	−	0.727025i
$802$		0			0
$803$		−	230032.i		−	0.356745i
$804$		0			0
$805$	26682.4			0.0411749
$806$		0			0
$807$	−79130.4	−	344109.i	−0.121506	−	0.528384i
$808$		0			0
$809$			391432.i			0.598081i	0.954240	+	0.299040i	$0.0966665\pi$
$809$			391432.i			0.598081i	−0.954240	+	0.299040i	$0.903333\pi$
$810$		0			0
$811$	−1.15536e6			−1.75661			−0.878305	−	0.478101i	$-0.841325\pi$
$811$	−1.15536e6			−1.75661			−0.878305	+	0.478101i	$0.841325\pi$
$812$		0			0
$813$	460916.	−	105991.i	0.697333	−	0.160357i
$814$		0			0
$815$			89703.7i			0.135050i
$816$		0			0
$817$	−546047.			−0.818061
$818$		0			0
$819$	214211.	−	104019.i	0.319355	−	0.155076i
$820$		0			0
$821$			299979.i			0.445045i	0.974928	+	0.222522i	$0.0714291\pi$
$821$			299979.i			0.445045i	−0.974928	+	0.222522i	$0.928571\pi$
$822$		0			0
$823$	−32530.4			−0.0480275			−0.0240138	−	0.999712i	$-0.507645\pi$
$823$	−32530.4			−0.0480275			−0.0240138	+	0.999712i	$0.507645\pi$
$824$		0			0
$825$	−8568.97	−	37263.4i	−0.0125899	−	0.0547488i
$826$		0			0
$827$		−	764819.i		−	1.11827i	−0.829076	−	0.559136i	$-0.811133\pi$
$827$		−	764819.i		−	1.11827i	0.829076	−	0.559136i	$-0.188867\pi$
$828$		0			0
$829$	−137003.			−0.199352			−0.0996758	−	0.995020i	$-0.531781\pi$
$829$	−137003.			−0.199352			−0.0996758	+	0.995020i	$0.531781\pi$
$830$		0			0
$831$	251794.	−	57901.8i	0.364622	−	0.0838475i
$832$		0			0
$833$			571916.i			0.824218i
$834$		0			0
$835$	234211.			0.335919
$836$		0			0
$837$	185621.	−	149517.i	0.264957	−	0.213422i
$838$		0			0
$839$			1.12228e6i			1.59433i	0.603759	+	0.797167i	$0.293669\pi$
$839$			1.12228e6i			1.59433i	−0.603759	+	0.797167i	$0.706331\pi$
$840$		0			0
$841$	−1.74225e6			−2.46331
$842$		0			0
$843$	168492.	+	732711.i	0.237096	+	1.03105i
$844$		0			0
$845$			141920.i			0.198761i
$846$		0			0
$847$	−314744.			−0.438723
$848$		0			0
$849$	1.11146e6	−	255589.i	1.54199	−	0.354590i
$850$		0			0
$851$		−	7411.31i		−	0.0102338i
$852$		0			0
$853$	43550.6			0.0598544			0.0299272	−	0.999552i	$-0.490472\pi$
$853$	43550.6			0.0598544			0.0299272	+	0.999552i	$0.490472\pi$
$854$		0			0
$855$	−143883.	−	296304.i	−0.196824	−	0.405327i
$856$		0			0
$857$			300075.i			0.408572i	0.978911	+	0.204286i	$0.0654872\pi$
$857$			300075.i			0.408572i	−0.978911	+	0.204286i	$0.934513\pi$
$858$		0			0
$859$	498647.			0.675782			0.337891	−	0.941185i	$-0.390287\pi$
$859$	498647.			0.675782			0.337891	+	0.941185i	$0.390287\pi$
$860$		0			0
$861$	101162.	+	439916.i	0.136462	+	0.593422i
$862$		0			0
$863$			13487.9i			0.0181102i	0.999959	+	0.00905512i	$0.00288237\pi$
$863$			13487.9i			0.0181102i	−0.999959	+	0.00905512i	$0.997118\pi$
$864$		0			0
$865$	229919.			0.307286
$866$		0			0
$867$	100002.	−	22996.1i	0.133036	−	0.0305926i
$868$		0			0
$869$		−	216401.i		−	0.286562i
$870$		0			0
$871$	535936.			0.706443
$872$		0			0
$873$	−719888.	+	349572.i	−0.944575	+	0.458678i
$874$		0			0
$875$		−	32617.0i		−	0.0426018i
$876$		0			0
$877$	−944732.			−1.22831			−0.614157	−	0.789184i	$-0.710504\pi$
$877$	−944732.			−1.22831			−0.614157	+	0.789184i	$0.710504\pi$
$878$		0			0
$879$	−139948.	−	608582.i	−0.181129	−	0.787664i
$880$		0			0
$881$		−	1.28495e6i		−	1.65552i	−0.561082	−	0.827760i	$-0.689615\pi$
$881$		−	1.28495e6i		−	1.65552i	0.561082	−	0.827760i	$-0.310385\pi$
$882$		0			0
$883$	1.22382e6			1.56963			0.784815	−	0.619731i	$-0.212758\pi$
$883$	1.22382e6			1.56963			0.784815	+	0.619731i	$0.212758\pi$
$884$		0			0
$885$	−494760.	+	113773.i	−0.631695	+	0.145263i
$886$		0			0
$887$			755435.i			0.960173i	0.877221	+	0.480087i	$0.159395\pi$
$887$			755435.i			0.960173i	−0.877221	+	0.480087i	$0.840605\pi$
$888$		0			0
$889$	668624.			0.846015
$890$		0			0
$891$	−175244.	−	137895.i	−0.220743	−	0.173697i
$892$		0			0
$893$			712291.i			0.893212i
$894$		0			0
$895$	−362193.			−0.452162
$896$		0			0
$897$	−25980.1	−	112978.i	−0.0322891	−	0.140414i
$898$		0			0
$899$		−	511714.i		−	0.633152i
$900$		0			0
$901$	913553.			1.12534
$902$		0			0
$903$	−307319.	+	70670.2i	−0.376890	+	0.0866685i
$904$		0			0
$905$		−	201866.i		−	0.246471i
$906$		0			0
$907$	−1.28653e6			−1.56389			−0.781946	−	0.623346i	$-0.785773\pi$
$907$	−1.28653e6			−1.56389			−0.781946	+	0.623346i	$0.785773\pi$
$908$		0			0
$909$	393439.	+	810225.i	0.476156	+	0.980568i
$910$		0			0
$911$		−	501962.i		−	0.604831i	−0.953176	−	0.302416i	$-0.902207\pi$
$911$		−	501962.i		−	0.604831i	0.953176	−	0.302416i	$-0.0977931\pi$
$912$		0			0
$913$	234248.			0.281018
$914$		0			0
$915$	16613.3	+	72245.1i	0.0198432	+	0.0862911i
$916$		0			0
$917$			9353.35i			0.0111232i
$918$		0			0
$919$	252142.			0.298548			0.149274	−	0.988796i	$-0.452306\pi$
$919$	252142.			0.298548			0.149274	+	0.988796i	$0.452306\pi$
$920$		0			0
$921$	560933.	−	128990.i	0.661289	−	0.152068i
$922$		0			0
$923$			164238.i			0.192784i
$924$		0			0
$925$	−9059.72			−0.0105884
$926$		0			0
$927$	892253.	−	433271.i	1.03831	−	0.504197i
$928$		0			0
$929$			1.28900e6i			1.49356i	0.665074	+	0.746778i	$0.268400\pi$
$929$			1.28900e6i			1.49356i	−0.665074	+	0.746778i	$0.731600\pi$
$930$		0			0
$931$	−675181.			−0.778970
$932$		0			0
$933$	37932.1	+	164953.i	0.0435757	+	0.189495i
$934$		0			0
$935$			117073.i			0.133917i
$936$		0			0
$937$	596815.			0.679768			0.339884	−	0.940467i	$-0.389612\pi$
$937$	596815.			0.679768			0.339884	+	0.940467i	$0.389612\pi$
$938$		0			0
$939$	−550351.	+	126557.i	−0.624178	+	0.143534i
$940$		0			0
$941$			1.55891e6i			1.76052i	0.474493	+	0.880259i	$0.342631\pi$
$941$			1.55891e6i			1.76052i	−0.474493	+	0.880259i	$0.657369\pi$
$942$		0			0
$943$	219749.			0.247118
$944$		0			0
$945$	−119327.	−	148141.i	−0.133621	−	0.165887i
$946$		0			0
$947$			226143.i			0.252164i	0.992020	+	0.126082i	$0.0402402\pi$
$947$			226143.i			0.252164i	−0.992020	+	0.126082i	$0.959760\pi$
$948$		0			0
$949$	−852551.			−0.946647
$950$		0			0
$951$	188440.	+	819459.i	0.208359	+	0.906080i
$952$		0			0
$953$		−	1.71407e6i		−	1.88731i	−0.330927	−	0.943656i	$-0.607361\pi$
$953$		−	1.71407e6i		−	1.88731i	0.330927	−	0.943656i	$-0.392639\pi$
$954$		0			0
$955$	131042.			0.143683
$956$		0			0
$957$	−466567.	+	107290.i	−0.509436	+	0.117148i
$958$		0			0
$959$			683307.i			0.742982i
$960$		0			0
$961$	−816622.			−0.884249
$962$		0			0
$963$	−25078.4	−	51645.1i	−0.0270426	−	0.0556899i
$964$		0			0
$965$			315880.i			0.339210i
$966$		0			0
$967$	−1.48328e6			−1.58625			−0.793123	−	0.609061i	$-0.791546\pi$
$967$	−1.48328e6			−1.58625			−0.793123	+	0.609061i	$0.791546\pi$
$968$		0			0
$969$	226025.	+	982900.i	0.240718	+	1.04680i
$970$		0			0
$971$		−	25773.5i		−	0.0273360i	−0.999907	−	0.0136680i	$-0.995649\pi$
$971$		−	25773.5i		−	0.0273360i	0.999907	−	0.0136680i	$-0.00435080\pi$
$972$		0			0
$973$	43224.5			0.0456567
$974$		0			0
$975$	−138106.	+	31758.5i	−0.145280	+	0.0334081i
$976$		0			0
$977$		−	1.63957e6i		−	1.71767i	−0.512249	−	0.858837i	$-0.671187\pi$
$977$		−	1.63957e6i		−	1.71767i	0.512249	−	0.858837i	$-0.328813\pi$
$978$		0			0
$979$	−217582.			−0.227016
$980$		0			0
$981$	−66189.9	+	32141.3i	−0.0687786	+	0.0333984i
$982$		0			0
$983$			1.02737e6i			1.06322i	0.846991	+	0.531608i	$0.178412\pi$
$983$			1.02737e6i			1.06322i	−0.846991	+	0.531608i	$0.821588\pi$
$984$		0			0
$985$	596047.			0.614339
$986$		0			0
$987$	92185.8	+	400883.i	0.0946302	+	0.411513i
$988$		0			0
$989$			153514.i			0.156948i
$990$		0			0
$991$	−101778.			−0.103635			−0.0518174	−	0.998657i	$-0.516501\pi$
$991$	−101778.			−0.103635			−0.0518174	+	0.998657i	$0.516501\pi$
$992$		0			0
$993$	−628667.	+	144566.i	−0.637562	+	0.146612i
$994$		0			0
$995$			564210.i			0.569894i
$996$		0			0
$997$	−466189.			−0.468999			−0.234499	−	0.972116i	$-0.575345\pi$
$997$	−466189.			−0.468999			−0.234499	+	0.972116i	$0.575345\pi$
$998$		0			0
$999$	−41147.7	+	33144.3i	−0.0412301	+	0.0332106i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.5.l.d.161.2		6
3.2	odd	2	inner	240.5.l.d.161.1		6
4.3	odd	2		15.5.c.a.11.6	yes	6
12.11	even	2		15.5.c.a.11.1	✓	6
20.3	even	4		75.5.d.d.74.12		12
20.7	even	4		75.5.d.d.74.1		12
20.19	odd	2		75.5.c.i.26.1		6
60.23	odd	4		75.5.d.d.74.2		12
60.47	odd	4		75.5.d.d.74.11		12
60.59	even	2		75.5.c.i.26.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.5.c.a.11.1	✓	6	12.11	even	2
15.5.c.a.11.6	yes	6	4.3	odd	2
75.5.c.i.26.1		6	20.19	odd	2
75.5.c.i.26.6		6	60.59	even	2
75.5.d.d.74.1		12	20.7	even	4
75.5.d.d.74.2		12	60.23	odd	4
75.5.d.d.74.11		12	60.47	odd	4
75.5.d.d.74.12		12	20.3	even	4
240.5.l.d.161.1		6	3.2	odd	2	inner
240.5.l.d.161.2		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-8.77108 + 2.01697i) q^{3} -11.1803i q^{5} -23.3388 q^{7} +(72.8637 - 35.3820i) q^{9} -33.9875i q^{11} -125.965 q^{13} +(22.5504 + 98.0636i) q^{15} -308.095i q^{17} +363.724 q^{19} +(204.707 - 47.0738i) q^{21} -102.256i q^{23} -125.000 q^{25} +(-567.728 + 457.302i) q^{27} +1565.10i q^{29} -326.954 q^{31} +(68.5518 + 298.107i) q^{33} +260.936i q^{35} +72.4778 q^{37} +(1104.85 - 254.068i) q^{39} +2149.01i q^{41} -1501.27 q^{43} +(-395.583 - 814.640i) q^{45} +1958.33i q^{47} -1856.30 q^{49} +(621.418 + 2702.32i) q^{51} +2965.17i q^{53} -379.992 q^{55} +(-3190.25 + 733.621i) q^{57} +5045.29i q^{59} +736.716 q^{61} +(-1700.55 + 825.775i) q^{63} +1408.34i q^{65} -4254.63 q^{67} +(206.248 + 896.898i) q^{69} -1303.84i q^{71} +6768.14 q^{73} +(1096.38 - 252.121i) q^{75} +793.229i q^{77} +6367.07 q^{79} +(4057.23 - 5156.13i) q^{81} +6892.17i q^{83} -3444.60 q^{85} +(-3156.76 - 13727.6i) q^{87} -6401.82i q^{89} +2939.88 q^{91} +(2867.74 - 659.456i) q^{93} -4066.56i q^{95} -9879.94 q^{97} +(-1202.55 - 2476.45i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{3} - 76 q^{7} + 118 q^{9} - 424 q^{13} - 50 q^{15} + 244 q^{19} - 876 q^{21} - 750 q^{25} + 352 q^{27} - 3772 q^{31} + 4420 q^{33} + 1896 q^{37} + 1336 q^{39} + 7384 q^{43} + 1900 q^{45} - 1318 q^{49}+ \cdots - 9680 q^{99}+O(q^{100}) \) 6 * q - 8 * q^3 - 76 * q^7 + 118 * q^9 - 424 * q^13 - 50 * q^15 + 244 * q^19 - 876 * q^21 - 750 * q^25 + 352 * q^27 - 3772 * q^31 + 4420 * q^33 + 1896 * q^37 + 1336 * q^39 + 7384 * q^43 + 1900 * q^45 - 1318 * q^49 + 8492 * q^51 + 1300 * q^55 - 11584 * q^57 + 6452 * q^61 - 14796 * q^63 - 13816 * q^67 + 5472 * q^69 + 596 * q^73 + 1000 * q^75 + 16124 * q^79 + 5086 * q^81 - 3100 * q^85 + 4900 * q^87 + 11632 * q^91 - 8184 * q^93 + 9756 * q^97 - 9680 * q^99

Introduction

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