LMFDB - Embedded newform 240.3.l.c.161.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [240,3,Mod(161,240)] 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("240.161"); S:= CuspForms(chi, 3); N := Newforms(S);

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, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [4,0,-4] 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:	$6.53952634465$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	240.161
Dual form			240.3.l.c.161.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.58114 - 1.52896i) q^{3} -2.23607i q^{5} +7.48683 q^{7} +(4.32456 + 7.89292i) q^{9} -8.48528i q^{11} -10.0000 q^{13} +(-3.41886 + 5.77160i) q^{15} -30.3870i q^{17} -26.9737 q^{19} +(-19.3246 - 11.4471i) q^{21} -9.17377i q^{23} -5.00000 q^{25} +(0.905694 - 26.9848i) q^{27} -26.8328i q^{29} -8.00000 q^{31} +(-12.9737 + 21.9017i) q^{33} -16.7411i q^{35} +15.9473 q^{37} +(25.8114 + 15.2896i) q^{39} +47.3575i q^{41} +14.4605 q^{43} +(17.6491 - 9.67000i) q^{45} -45.8688i q^{47} +7.05267 q^{49} +(-46.4605 + 78.4330i) q^{51} -30.3870i q^{53} -18.9737 q^{55} +(69.6228 + 41.2417i) q^{57} +24.0789i q^{59} -53.9473 q^{61} +(32.3772 + 59.0930i) q^{63} +22.3607i q^{65} +110.460 q^{67} +(-14.0263 + 23.6788i) q^{69} -15.5936i q^{71} +87.9473 q^{73} +(12.9057 + 7.64481i) q^{75} -63.5279i q^{77} +46.9737 q^{79} +(-43.5964 + 68.2668i) q^{81} -26.1443i q^{83} -67.9473 q^{85} +(-41.0263 + 69.2592i) q^{87} +60.7739i q^{89} -74.8683 q^{91} +(20.6491 + 12.2317i) q^{93} +60.3150i q^{95} +36.0527 q^{97} +(66.9737 - 36.6951i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{7} - 8 q^{9} - 40 q^{13} - 20 q^{15} - 32 q^{19} - 52 q^{21} - 20 q^{25} - 28 q^{27} - 32 q^{31} + 24 q^{33} - 88 q^{37} + 40 q^{39} - 56 q^{43} + 20 q^{45} + 180 q^{49} - 72 q^{51}+ \cdots + 192 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^7 - 8 * q^9 - 40 * q^13 - 20 * q^15 - 32 * q^19 - 52 * q^21 - 20 * q^25 - 28 * q^27 - 32 * q^31 + 24 * q^33 - 88 * q^37 + 40 * q^39 - 56 * q^43 + 20 * q^45 + 180 * q^49 - 72 * q^51 + 152 * q^57 - 64 * q^61 + 256 * q^63 + 328 * q^67 - 132 * q^69 + 200 * q^73 + 20 * q^75 + 112 * q^79 + 28 * q^81 - 120 * q^85 - 240 * q^87 + 80 * q^91 + 32 * q^93 + 296 * q^97 + 192 * 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
$3$	−2.58114	−	1.52896i	−0.860380	−	0.509654i
$3$	−2.58114	−	1.52896i	−0.860380	−	0.509654i
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$	7.48683			1.06955			0.534774	−	0.844995i	$-0.320397\pi$
$7$	7.48683			1.06955			0.534774	+	0.844995i	$0.320397\pi$
$8$		0			0
$9$	4.32456	+	7.89292i	0.480506	+	0.876991i
$10$		0			0
$11$		−	8.48528i		−	0.771389i	−0.922627	−	0.385695i	$-0.873962\pi$
$11$		−	8.48528i		−	0.771389i	0.922627	−	0.385695i	$-0.126038\pi$
$12$		0			0
$13$	−10.0000			−0.769231			−0.384615	−	0.923077i	$-0.625666\pi$
$13$	−10.0000			−0.769231			−0.384615	+	0.923077i	$0.625666\pi$
$14$		0			0
$15$	−3.41886	+	5.77160i	−0.227924	+	0.384773i
$16$		0			0
$17$		−	30.3870i		−	1.78747i	−0.448596	−	0.893734i	$-0.648076\pi$
$17$		−	30.3870i		−	1.78747i	0.448596	−	0.893734i	$-0.351924\pi$
$18$		0			0
$19$	−26.9737			−1.41967			−0.709833	−	0.704370i	$-0.751230\pi$
$19$	−26.9737			−1.41967			−0.709833	+	0.704370i	$0.751230\pi$
$20$		0			0
$21$	−19.3246	−	11.4471i	−0.920217	−	0.545099i
$22$		0			0
$23$		−	9.17377i		−	0.398859i	−0.979912	−	0.199430i	$-0.936091\pi$
$23$		−	9.17377i		−	0.398859i	0.979912	−	0.199430i	$-0.0639090\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	0.905694	−	26.9848i	0.0335442	−	0.999437i
$28$		0			0
$29$		−	26.8328i		−	0.925270i	−0.886549	−	0.462635i	$-0.846904\pi$
$29$		−	26.8328i		−	0.925270i	0.886549	−	0.462635i	$-0.153096\pi$
$30$		0			0
$31$	−8.00000			−0.258065			−0.129032	−	0.991640i	$-0.541187\pi$
$31$	−8.00000			−0.258065			−0.129032	+	0.991640i	$0.541187\pi$
$32$		0			0
$33$	−12.9737	+	21.9017i	−0.393141	+	0.663688i
$34$		0			0
$35$		−	16.7411i		−	0.478316i
$36$		0			0
$37$	15.9473			0.431009			0.215504	−	0.976503i	$-0.430860\pi$
$37$	15.9473			0.431009			0.215504	+	0.976503i	$0.430860\pi$
$38$		0			0
$39$	25.8114	+	15.2896i	0.661830	+	0.392041i
$40$		0			0
$41$			47.3575i			1.15506i	0.816369	+	0.577531i	$0.195984\pi$
$41$			47.3575i			1.15506i	−0.816369	+	0.577531i	$0.804016\pi$
$42$		0			0
$43$	14.4605			0.336291			0.168145	−	0.985762i	$-0.446222\pi$
$43$	14.4605			0.336291			0.168145	+	0.985762i	$0.446222\pi$
$44$		0			0
$45$	17.6491	−	9.67000i	0.392202	−	0.214889i
$46$		0			0
$47$		−	45.8688i		−	0.975933i	−0.872862	−	0.487966i	$-0.837739\pi$
$47$		−	45.8688i		−	0.975933i	0.872862	−	0.487966i	$-0.162261\pi$
$48$		0			0
$49$	7.05267			0.143932
$50$		0			0
$51$	−46.4605	+	78.4330i	−0.910990	+	1.53790i
$52$		0			0
$53$		−	30.3870i		−	0.573339i	−0.958030	−	0.286670i	$-0.907452\pi$
$53$		−	30.3870i		−	0.573339i	0.958030	−	0.286670i	$-0.0925482\pi$
$54$		0			0
$55$	−18.9737			−0.344976
$56$		0			0
$57$	69.6228	+	41.2417i	1.22145	+	0.723538i
$58$		0			0
$59$			24.0789i			0.408116i	0.978959	+	0.204058i	$0.0654132\pi$
$59$			24.0789i			0.408116i	−0.978959	+	0.204058i	$0.934587\pi$
$60$		0			0
$61$	−53.9473			−0.884382			−0.442191	−	0.896921i	$-0.645799\pi$
$61$	−53.9473			−0.884382			−0.442191	+	0.896921i	$0.645799\pi$
$62$		0			0
$63$	32.3772	+	59.0930i	0.513924	+	0.937984i
$64$		0			0
$65$			22.3607i			0.344010i
$66$		0			0
$67$	110.460			1.64866			0.824332	−	0.566107i	$-0.191551\pi$
$67$	110.460			1.64866			0.824332	+	0.566107i	$0.191551\pi$
$68$		0			0
$69$	−14.0263	+	23.6788i	−0.203280	+	0.343171i
$70$		0			0
$71$		−	15.5936i		−	0.219628i	−0.993952	−	0.109814i	$-0.964974\pi$
$71$		−	15.5936i		−	0.219628i	0.993952	−	0.109814i	$-0.0350255\pi$
$72$		0			0
$73$	87.9473			1.20476			0.602379	−	0.798210i	$-0.294220\pi$
$73$	87.9473			1.20476			0.602379	+	0.798210i	$0.294220\pi$
$74$		0			0
$75$	12.9057	+	7.64481i	0.172076	+	0.101931i
$76$		0			0
$77$		−	63.5279i		−	0.825037i
$78$		0			0
$79$	46.9737			0.594603			0.297302	−	0.954784i	$-0.403913\pi$
$79$	46.9737			0.594603			0.297302	+	0.954784i	$0.403913\pi$
$80$		0			0
$81$	−43.5964	+	68.2668i	−0.538228	+	0.842799i
$82$		0			0
$83$		−	26.1443i		−	0.314992i	−0.987520	−	0.157496i	$-0.949658\pi$
$83$		−	26.1443i		−	0.314992i	0.987520	−	0.157496i	$-0.0503421\pi$
$84$		0			0
$85$	−67.9473			−0.799380
$86$		0			0
$87$	−41.0263	+	69.2592i	−0.471567	+	0.796083i
$88$		0			0
$89$			60.7739i			0.682853i	0.939908	+	0.341427i	$0.110910\pi$
$89$			60.7739i			0.682853i	−0.939908	+	0.341427i	$0.889090\pi$
$90$		0			0
$91$	−74.8683			−0.822729
$92$		0			0
$93$	20.6491	+	12.2317i	0.222033	+	0.131524i
$94$		0			0
$95$			60.3150i			0.634894i
$96$		0			0
$97$	36.0527			0.371677			0.185838	−	0.982580i	$-0.440500\pi$
$97$	36.0527			0.371677			0.185838	+	0.982580i	$0.440500\pi$
$98$		0			0
$99$	66.9737	−	36.6951i	0.676502	−	0.370657i
$100$		0			0
$101$		−	48.1577i		−	0.476809i	−0.971166	−	0.238405i	$-0.923376\pi$
$101$		−	48.1577i		−	0.476809i	0.971166	−	0.238405i	$-0.0766245\pi$
$102$		0			0
$103$	−140.408			−1.36318			−0.681591	−	0.731733i	$-0.738712\pi$
$103$	−140.408			−1.36318			−0.681591	+	0.731733i	$0.738712\pi$
$104$		0			0
$105$	−25.5964	+	43.2110i	−0.243776	+	0.411534i
$106$		0			0
$107$			43.1149i			0.402943i	0.979494	+	0.201471i	$0.0645723\pi$
$107$			43.1149i			0.402943i	−0.979494	+	0.201471i	$0.935428\pi$
$108$		0			0
$109$	133.842			1.22791			0.613954	−	0.789342i	$-0.289578\pi$
$109$	133.842			1.22791			0.613954	+	0.789342i	$0.289578\pi$
$110$		0			0
$111$	−41.1623	−	24.3829i	−0.370831	−	0.219665i
$112$		0			0
$113$			7.90852i			0.0699869i	0.999388	+	0.0349935i	$0.0111410\pi$
$113$			7.90852i			0.0699869i	−0.999388	+	0.0349935i	$0.988859\pi$
$114$		0			0
$115$	−20.5132			−0.178375
$116$		0			0
$117$	−43.2456	−	78.9292i	−0.369620	−	0.674609i
$118$		0			0
$119$		−	227.502i		−	1.91178i
$120$		0			0
$121$	49.0000			0.404959
$122$		0			0
$123$	72.4078	−	122.236i	0.588682	−	0.993792i
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	134.460			1.05874			0.529372	−	0.848390i	$-0.322428\pi$
$127$	134.460			1.05874			0.529372	+	0.848390i	$0.322428\pi$
$128$		0			0
$129$	−37.3246	−	22.1095i	−0.289338	−	0.171392i
$130$		0			0
$131$			220.394i			1.68240i	0.540727	+	0.841198i	$0.318149\pi$
$131$			220.394i			1.68240i	−0.540727	+	0.841198i	$0.681851\pi$
$132$		0			0
$133$	−201.947			−1.51840
$134$		0			0
$135$	−60.3399	−	2.02519i	−0.446962	−	0.0150014i
$136$		0			0
$137$			95.5153i			0.697192i	0.937273	+	0.348596i	$0.113341\pi$
$137$			95.5153i			0.697192i	−0.937273	+	0.348596i	$0.886659\pi$
$138$		0			0
$139$	76.8157			0.552631			0.276315	−	0.961067i	$-0.410887\pi$
$139$	76.8157			0.552631			0.276315	+	0.961067i	$0.410887\pi$
$140$		0			0
$141$	−70.1317	+	118.394i	−0.497388	+	0.839673i
$142$		0			0
$143$			84.8528i			0.593376i
$144$		0			0
$145$	−60.0000			−0.413793
$146$		0			0
$147$	−18.2039	−	10.7833i	−0.123836	−	0.0733555i
$148$		0			0
$149$		−	276.237i		−	1.85394i	−0.375139	−	0.926969i	$-0.622405\pi$
$149$		−	276.237i		−	1.85394i	0.375139	−	0.926969i	$-0.377595\pi$
$150$		0			0
$151$	−18.0527			−0.119554			−0.0597770	−	0.998212i	$-0.519039\pi$
$151$	−18.0527			−0.119554			−0.0597770	+	0.998212i	$0.519039\pi$
$152$		0			0
$153$	239.842	−	131.410i	1.56759	−	0.858890i
$154$		0			0
$155$			17.8885i			0.115410i
$156$		0			0
$157$	103.842			0.661414			0.330707	−	0.943733i	$-0.392713\pi$
$157$	103.842			0.661414			0.330707	+	0.943733i	$0.392713\pi$
$158$		0			0
$159$	−46.4605	+	78.4330i	−0.292204	+	0.493289i
$160$		0			0
$161$		−	68.6825i		−	0.426599i
$162$		0			0
$163$	11.3815			0.0698251			0.0349126	−	0.999390i	$-0.488885\pi$
$163$	11.3815			0.0698251			0.0349126	+	0.999390i	$0.488885\pi$
$164$		0			0
$165$	48.9737	+	29.0100i	0.296810	+	0.175818i
$166$		0			0
$167$		−	252.270i		−	1.51060i	−0.655382	−	0.755298i	$-0.727492\pi$
$167$		−	252.270i		−	1.51060i	0.655382	−	0.755298i	$-0.272508\pi$
$168$		0			0
$169$	−69.0000			−0.408284
$170$		0			0
$171$	−116.649	−	212.901i	−0.682159	−	1.24504i
$172$		0			0
$173$			11.8160i			0.0683005i	0.999417	+	0.0341502i	$0.0108725\pi$
$173$			11.8160i			0.0683005i	−0.999417	+	0.0341502i	$0.989128\pi$
$174$		0			0
$175$	−37.4342			−0.213910
$176$		0			0
$177$	36.8157	−	62.1509i	0.207998	−	0.351135i
$178$		0			0
$179$			69.0358i			0.385675i	0.981231	+	0.192837i	$0.0617690\pi$
$179$			69.0358i			0.385675i	−0.981231	+	0.192837i	$0.938231\pi$
$180$		0			0
$181$	−189.684			−1.04798			−0.523989	−	0.851725i	$-0.675557\pi$
$181$	−189.684			−1.04798			−0.523989	+	0.851725i	$0.675557\pi$
$182$		0			0
$183$	139.246	+	82.4834i	0.760905	+	0.450729i
$184$		0			0
$185$		−	35.6593i		−	0.192753i
$186$		0			0
$187$	−257.842			−1.37883
$188$		0			0
$189$	6.78078	−	202.031i	0.0358771	−	1.06895i
$190$		0			0
$191$		−	108.708i		−	0.569153i	−0.958653	−	0.284577i	$-0.908147\pi$
$191$		−	108.708i		−	0.569153i	0.958653	−	0.284577i	$-0.0918530\pi$
$192$		0			0
$193$	−167.947			−0.870193			−0.435097	−	0.900384i	$-0.643286\pi$
$193$	−167.947			−0.870193			−0.435097	+	0.900384i	$0.643286\pi$
$194$		0			0
$195$	34.1886	−	57.7160i	0.175326	−	0.295980i
$196$		0			0
$197$			171.659i			0.871367i	0.900100	+	0.435684i	$0.143493\pi$
$197$			171.659i			0.871367i	−0.900100	+	0.435684i	$0.856507\pi$
$198$		0			0
$199$	−35.0790			−0.176276			−0.0881382	−	0.996108i	$-0.528092\pi$
$199$	−35.0790			−0.176276			−0.0881382	+	0.996108i	$0.528092\pi$
$200$		0			0
$201$	−285.114	−	168.890i	−1.41848	−	0.840248i
$202$		0			0
$203$		−	200.893i		−	0.989620i
$204$		0			0
$205$	105.895			0.516559
$206$		0			0
$207$	72.4078	−	39.6725i	0.349796	−	0.191654i
$208$		0			0
$209$			228.879i			1.09512i
$210$		0			0
$211$	58.1580			0.275630			0.137815	−	0.990458i	$-0.455992\pi$
$211$	58.1580			0.275630			0.137815	+	0.990458i	$0.455992\pi$
$212$		0			0
$213$	−23.8420	+	40.2492i	−0.111934	+	0.188963i
$214$		0			0
$215$		−	32.3347i		−	0.150394i
$216$		0			0
$217$	−59.8947			−0.276012
$218$		0			0
$219$	−227.004	−	134.468i	−1.03655	−	0.614009i
$220$		0			0
$221$			303.870i			1.37498i
$222$		0			0
$223$	−99.3815			−0.445657			−0.222828	−	0.974858i	$-0.571529\pi$
$223$	−99.3815			−0.445657			−0.222828	+	0.974858i	$0.571529\pi$
$224$		0			0
$225$	−21.6228	−	39.4646i	−0.0961012	−	0.175398i
$226$		0			0
$227$			216.951i			0.955733i	0.878432	+	0.477867i	$0.158590\pi$
$227$			216.951i			0.955733i	−0.878432	+	0.477867i	$0.841410\pi$
$228$		0			0
$229$	325.684			1.42220			0.711100	−	0.703090i	$-0.248197\pi$
$229$	325.684			1.42220			0.711100	+	0.703090i	$0.248197\pi$
$230$		0			0
$231$	−97.1317	+	163.974i	−0.420483	+	0.709845i
$232$		0			0
$233$		−	51.7119i		−	0.221939i	−0.993824	−	0.110970i	$-0.964604\pi$
$233$		−	51.7119i		−	0.221939i	0.993824	−	0.110970i	$-0.0353957\pi$
$234$		0			0
$235$	−102.566			−0.436450
$236$		0			0
$237$	−121.246	−	71.8209i	−0.511585	−	0.303042i
$238$		0			0
$239$			410.047i			1.71568i	0.513917	+	0.857840i	$0.328194\pi$
$239$			410.047i			1.71568i	−0.513917	+	0.857840i	$0.671806\pi$
$240$		0			0
$241$	445.526			1.84866			0.924328	−	0.381599i	$-0.124627\pi$
$241$	445.526			1.84866			0.924328	+	0.381599i	$0.124627\pi$
$242$		0			0
$243$	216.906	−	109.549i	0.892616	−	0.450818i
$244$		0			0
$245$		−	15.7702i		−	0.0643683i
$246$		0			0
$247$	269.737			1.09205
$248$		0			0
$249$	−39.9737	+	67.4821i	−0.160537	+	0.271013i
$250$		0			0
$251$			237.364i			0.945675i	0.881150	+	0.472838i	$0.156770\pi$
$251$			237.364i			0.945675i	−0.881150	+	0.472838i	$0.843230\pi$
$252$		0			0
$253$	−77.8420			−0.307676
$254$		0			0
$255$	175.381	+	103.889i	0.687771	+	0.407407i
$256$		0			0
$257$		−	318.887i		−	1.24080i	−0.784284	−	0.620402i	$-0.786970\pi$
$257$		−	318.887i		−	1.24080i	0.784284	−	0.620402i	$-0.213030\pi$
$258$		0			0
$259$	119.395			0.460985
$260$		0			0
$261$	211.789	−	116.040i	0.811453	−	0.444598i
$262$		0			0
$263$			36.2300i			0.137757i	0.997625	+	0.0688784i	$0.0219420\pi$
$263$			36.2300i			0.137757i	−0.997625	+	0.0688784i	$0.978058\pi$
$264$		0			0
$265$	−67.9473			−0.256405
$266$		0			0
$267$	92.9210	−	156.866i	0.348019	−	0.587513i
$268$		0			0
$269$			528.041i			1.96298i	0.191518	+	0.981489i	$0.438659\pi$
$269$			528.041i			1.96298i	−0.191518	+	0.981489i	$0.561341\pi$
$270$		0			0
$271$	475.895			1.75607			0.878034	−	0.478597i	$-0.158855\pi$
$271$	475.895			1.75607			0.878034	+	0.478597i	$0.158855\pi$
$272$		0			0
$273$	193.246	+	114.471i	0.707859	+	0.419307i
$274$		0			0
$275$			42.4264i			0.154278i
$276$		0			0
$277$	188.158			0.679271			0.339635	−	0.940557i	$-0.389696\pi$
$277$	188.158			0.679271			0.339635	+	0.940557i	$0.389696\pi$
$278$		0			0
$279$	−34.5964	−	63.1434i	−0.124002	−	0.226320i
$280$		0			0
$281$		−	24.4322i		−	0.0869473i	−0.999055	−	0.0434736i	$-0.986158\pi$
$281$		−	24.4322i		−	0.0869473i	0.999055	−	0.0434736i	$-0.0138424\pi$
$282$		0			0
$283$	−198.460			−0.701274			−0.350637	−	0.936511i	$-0.614035\pi$
$283$	−198.460			−0.701274			−0.350637	+	0.936511i	$0.614035\pi$
$284$		0			0
$285$	92.2192	−	155.681i	0.323576	−	0.546250i
$286$		0			0
$287$			354.558i			1.23539i
$288$		0			0
$289$	−634.368			−2.19504
$290$		0			0
$291$	−93.0569	−	55.1231i	−0.319783	−	0.189427i
$292$		0			0
$293$			513.825i			1.75367i	0.480794	+	0.876834i	$0.340349\pi$
$293$			513.825i			1.75367i	−0.480794	+	0.876834i	$0.659651\pi$
$294$		0			0
$295$	53.8420			0.182515
$296$		0			0
$297$	−228.974	−	7.68507i	−0.770955	−	0.0258757i
$298$		0			0
$299$			91.7377i			0.306815i
$300$		0			0
$301$	108.263			0.359679
$302$		0			0
$303$	−73.6313	+	124.302i	−0.243008	+	0.410237i
$304$		0			0
$305$			120.630i			0.395508i
$306$		0			0
$307$	11.3815			0.0370733			0.0185366	−	0.999828i	$-0.494099\pi$
$307$	11.3815			0.0370733			0.0185366	+	0.999828i	$0.494099\pi$
$308$		0			0
$309$	362.412	+	214.678i	1.17285	+	0.694751i
$310$		0			0
$311$		−	518.756i		−	1.66802i	−0.551746	−	0.834012i	$-0.686038\pi$
$311$		−	518.756i		−	1.66802i	0.551746	−	0.834012i	$-0.313962\pi$
$312$		0			0
$313$	−46.3160			−0.147974			−0.0739872	−	0.997259i	$-0.523572\pi$
$313$	−46.3160			−0.147974			−0.0739872	+	0.997259i	$0.523572\pi$
$314$		0			0
$315$	132.136	−	72.3977i	0.419479	−	0.229834i
$316$		0			0
$317$		−	39.0957i		−	0.123330i	−0.998097	−	0.0616651i	$-0.980359\pi$
$317$		−	39.0957i		−	0.123330i	0.998097	−	0.0616651i	$-0.0196411\pi$
$318$		0			0
$319$	−227.684			−0.713743
$320$		0			0
$321$	65.9210	−	111.286i	0.205361	−	0.346684i
$322$		0			0
$323$			819.648i			2.53761i
$324$		0			0
$325$	50.0000			0.153846
$326$		0			0
$327$	−345.465	−	204.639i	−1.05647	−	0.625808i
$328$		0			0
$329$		−	343.412i		−	1.04381i
$330$		0			0
$331$	445.421			1.34568			0.672841	−	0.739787i	$-0.265074\pi$
$331$	445.421			1.34568			0.672841	+	0.739787i	$0.265074\pi$
$332$		0			0
$333$	68.9651	+	125.871i	0.207102	+	0.377991i
$334$		0			0
$335$		−	246.997i		−	0.737305i
$336$		0			0
$337$	325.684			0.966421			0.483211	−	0.875504i	$-0.339471\pi$
$337$	325.684			0.966421			0.483211	+	0.875504i	$0.339471\pi$
$338$		0			0
$339$	12.0918	−	20.4130i	0.0356691	−	0.0602153i
$340$		0			0
$341$			67.8823i			0.199068i
$342$		0			0
$343$	−314.053			−0.915605
$344$		0			0
$345$	52.9473	+	31.3638i	0.153471	+	0.0909097i
$346$		0			0
$347$		−	51.8236i		−	0.149348i	−0.997208	−	0.0746738i	$-0.976208\pi$
$347$		−	51.8236i		−	0.149348i	0.997208	−	0.0746738i	$-0.0237916\pi$
$348$		0			0
$349$	−97.5787			−0.279595			−0.139798	−	0.990180i	$-0.544645\pi$
$349$	−97.5787			−0.279595			−0.139798	+	0.990180i	$0.544645\pi$
$350$		0			0
$351$	−9.05694	+	269.848i	−0.0258033	+	0.768798i
$352$		0			0
$353$		−	569.797i		−	1.61416i	−0.590445	−	0.807078i	$-0.701048\pi$
$353$		−	569.797i		−	1.61416i	0.590445	−	0.807078i	$-0.298952\pi$
$354$		0			0
$355$	−34.8683			−0.0982206
$356$		0			0
$357$	−347.842	+	587.215i	−0.974347	+	1.64486i
$358$		0			0
$359$		−	274.283i		−	0.764019i	−0.924158	−	0.382010i	$-0.875232\pi$
$359$		−	274.283i		−	0.764019i	0.924158	−	0.382010i	$-0.124768\pi$
$360$		0			0
$361$	366.579			1.01545
$362$		0			0
$363$	−126.476	−	74.9191i	−0.348418	−	0.206389i
$364$		0			0
$365$		−	196.656i		−	0.538784i
$366$		0			0
$367$	−461.828			−1.25839			−0.629194	−	0.777248i	$-0.716615\pi$
$367$	−461.828			−1.25839			−0.629194	+	0.777248i	$0.716615\pi$
$368$		0			0
$369$	−373.789	+	204.800i	−1.01298	+	0.555014i
$370$		0			0
$371$		−	227.502i		−	0.613213i
$372$		0			0
$373$	−491.947			−1.31889			−0.659447	−	0.751751i	$-0.729209\pi$
$373$	−491.947			−1.31889			−0.659447	+	0.751751i	$0.729209\pi$
$374$		0			0
$375$	17.0943	−	28.8580i	0.0455848	−	0.0769547i
$376$		0			0
$377$			268.328i			0.711746i
$378$		0			0
$379$	258.763			0.682752			0.341376	−	0.939927i	$-0.389107\pi$
$379$	258.763			0.682752			0.341376	+	0.939927i	$0.389107\pi$
$380$		0			0
$381$	−347.061	−	205.585i	−0.910922	−	0.539593i
$382$		0			0
$383$		−	522.422i		−	1.36402i	−0.731341	−	0.682012i	$-0.761105\pi$
$383$		−	522.422i		−	1.36402i	0.731341	−	0.682012i	$-0.238895\pi$
$384$		0			0
$385$	−142.053			−0.368968
$386$		0			0
$387$	62.5352	+	114.136i	0.161590	+	0.294924i
$388$		0			0
$389$		−	610.847i		−	1.57030i	−0.619306	−	0.785150i	$-0.712586\pi$
$389$		−	610.847i		−	1.57030i	0.619306	−	0.785150i	$-0.287414\pi$
$390$		0			0
$391$	−278.763			−0.712949
$392$		0			0
$393$	336.974	−	568.867i	0.857439	−	1.44750i
$394$		0			0
$395$		−	105.036i		−	0.265915i
$396$		0			0
$397$	−214.000			−0.539043			−0.269521	−	0.962994i	$-0.586865\pi$
$397$	−214.000			−0.539043			−0.269521	+	0.962994i	$0.586865\pi$
$398$		0			0
$399$	521.254	+	308.770i	1.30640	+	0.773859i
$400$		0			0
$401$		−	454.557i		−	1.13356i	−0.823869	−	0.566780i	$-0.808189\pi$
$401$		−	454.557i		−	1.13356i	0.823869	−	0.566780i	$-0.191811\pi$
$402$		0			0
$403$	80.0000			0.198511
$404$		0			0
$405$	152.649	+	97.4846i	0.376911	+	0.240703i
$406$		0			0
$407$		−	135.318i		−	0.332476i
$408$		0			0
$409$	−573.842			−1.40304			−0.701518	−	0.712651i	$-0.747494\pi$
$409$	−573.842			−1.40304			−0.701518	+	0.712651i	$0.747494\pi$
$410$		0			0
$411$	146.039	−	246.538i	0.355326	−	0.599850i
$412$		0			0
$413$			180.274i			0.436500i
$414$		0			0
$415$	−58.4605			−0.140869
$416$		0			0
$417$	−198.272	−	117.448i	−0.475472	−	0.281650i
$418$		0			0
$419$			97.9159i			0.233690i	0.993150	+	0.116845i	$0.0372780\pi$
$419$			97.9159i			0.233690i	−0.993150	+	0.116845i	$0.962722\pi$
$420$		0			0
$421$	717.315			1.70384			0.851918	−	0.523675i	$-0.175439\pi$
$421$	717.315			1.70384			0.851918	+	0.523675i	$0.175439\pi$
$422$		0			0
$423$	362.039	−	198.362i	0.855885	−	0.468942i
$424$		0			0
$425$			151.935i			0.357494i
$426$		0			0
$427$	−403.895			−0.945889
$428$		0			0
$429$	129.737	−	219.017i	0.302416	−	0.510529i
$430$		0			0
$431$			293.077i			0.679994i	0.940427	+	0.339997i	$0.110426\pi$
$431$			293.077i			0.679994i	−0.940427	+	0.339997i	$0.889574\pi$
$432$		0			0
$433$	487.526			1.12593			0.562963	−	0.826482i	$-0.309661\pi$
$433$	487.526			1.12593			0.562963	+	0.826482i	$0.309661\pi$
$434$		0			0
$435$	154.868	+	91.7377i	0.356019	+	0.210891i
$436$		0			0
$437$			247.450i			0.566247i
$438$		0			0
$439$	−257.237			−0.585961			−0.292981	−	0.956118i	$-0.594647\pi$
$439$	−257.237			−0.585961			−0.292981	+	0.956118i	$0.594647\pi$
$440$		0			0
$441$	30.4997	+	55.6662i	0.0691602	+	0.126227i
$442$		0			0
$443$		−	293.096i		−	0.661615i	−0.943698	−	0.330808i	$-0.892679\pi$
$443$		−	293.096i		−	0.661615i	0.943698	−	0.330808i	$-0.107321\pi$
$444$		0			0
$445$	135.895			0.305381
$446$		0			0
$447$	−422.355	+	713.005i	−0.944866	+	1.59509i
$448$		0			0
$449$			585.614i			1.30426i	0.758106	+	0.652132i	$0.226125\pi$
$449$			585.614i			1.30426i	−0.758106	+	0.652132i	$0.773875\pi$
$450$		0			0
$451$	401.842			0.891002
$452$		0			0
$453$	46.5964	+	27.6018i	0.102862	+	0.0609312i
$454$		0			0
$455$			167.411i			0.367936i
$456$		0			0
$457$	−813.052			−1.77911			−0.889554	−	0.456831i	$-0.848984\pi$
$457$	−813.052			−1.77911			−0.889554	+	0.456831i	$0.848984\pi$
$458$		0			0
$459$	−819.986	−	27.5213i	−1.78646	−	0.0599593i
$460$		0			0
$461$		−	554.074i		−	1.20190i	−0.799288	−	0.600948i	$-0.794790\pi$
$461$		−	554.074i		−	1.20190i	0.799288	−	0.600948i	$-0.205210\pi$
$462$		0			0
$463$	449.723			0.971324			0.485662	−	0.874147i	$-0.338579\pi$
$463$	449.723			0.971324			0.485662	+	0.874147i	$0.338579\pi$
$464$		0			0
$465$	27.3509	−	46.1728i	0.0588191	−	0.0992964i
$466$		0			0
$467$		−	30.7221i		−	0.0657862i	−0.999459	−	0.0328931i	$-0.989528\pi$
$467$		−	30.7221i		−	0.0657862i	0.999459	−	0.0328931i	$-0.0104721\pi$
$468$		0			0
$469$	826.999			1.76332
$470$		0			0
$471$	−268.031	−	158.770i	−0.569067	−	0.337092i
$472$		0			0
$473$		−	122.701i		−	0.259411i
$474$		0			0
$475$	134.868			0.283933
$476$		0			0
$477$	239.842	−	131.410i	0.502813	−	0.275493i
$478$		0			0
$479$		−	735.242i		−	1.53495i	−0.641078	−	0.767476i	$-0.721512\pi$
$479$		−	735.242i		−	1.53495i	0.641078	−	0.767476i	$-0.278488\pi$
$480$		0			0
$481$	−159.473			−0.331545
$482$		0			0
$483$	−105.013	+	177.279i	−0.217418	+	0.367037i
$484$		0			0
$485$		−	80.6162i		−	0.166219i
$486$		0			0
$487$	92.6185			0.190182			0.0950909	−	0.995469i	$-0.469686\pi$
$487$	92.6185			0.190182			0.0950909	+	0.995469i	$0.469686\pi$
$488$		0			0
$489$	−29.3772	−	17.4019i	−0.0600761	−	0.0355866i
$490$		0			0
$491$		−	898.323i		−	1.82958i	−0.403933	−	0.914789i	$-0.632357\pi$
$491$		−	898.323i		−	1.82958i	0.403933	−	0.914789i	$-0.367643\pi$
$492$		0			0
$493$	−815.368			−1.65389
$494$		0			0
$495$	−82.0527	−	149.758i	−0.165763	−	0.302541i
$496$		0			0
$497$		−	116.747i		−	0.234903i
$498$		0			0
$499$	−136.921			−0.274391			−0.137195	−	0.990544i	$-0.543809\pi$
$499$	−136.921			−0.274391			−0.137195	+	0.990544i	$0.543809\pi$
$500$		0			0
$501$	−385.710	+	651.143i	−0.769881	+	1.29969i
$502$		0			0
$503$		−	443.077i		−	0.880868i	−0.897785	−	0.440434i	$-0.854825\pi$
$503$		−	443.077i		−	0.880868i	0.897785	−	0.440434i	$-0.145175\pi$
$504$		0			0
$505$	−107.684			−0.213236
$506$		0			0
$507$	178.099	+	105.498i	0.351279	+	0.208083i
$508$		0			0
$509$		−	213.062i		−	0.418590i	−0.977853	−	0.209295i	$-0.932883\pi$
$509$		−	213.062i		−	0.418590i	0.977853	−	0.209295i	$-0.0671168\pi$
$510$		0			0
$511$	658.447			1.28855
$512$		0			0
$513$	−24.4299	+	727.879i	−0.0476216	+	1.41887i
$514$		0			0
$515$			313.961i			0.609634i
$516$		0			0
$517$	−389.210			−0.752824
$518$		0			0
$519$	18.0662	−	30.4987i	0.0348096	−	0.0587643i
$520$		0			0
$521$		−	3.20085i		−	0.00614366i	−0.999995	−	0.00307183i	$-0.999022\pi$
$521$		−	3.20085i		−	0.00614366i	0.999995	−	0.00307183i	$-0.000977795\pi$
$522$		0			0
$523$	−966.644			−1.84827			−0.924134	−	0.382069i	$-0.875212\pi$
$523$	−966.644			−1.84827			−0.924134	+	0.382069i	$0.875212\pi$
$524$		0			0
$525$	96.6228	+	57.2354i	0.184043	+	0.109020i
$526$		0			0
$527$			243.096i			0.461282i
$528$		0			0
$529$	444.842			0.840911
$530$		0			0
$531$	−190.053	+	104.130i	−0.357915	+	0.196102i
$532$		0			0
$533$		−	473.575i		−	0.888509i
$534$		0			0
$535$	96.4078			0.180202
$536$		0			0
$537$	105.553	−	178.191i	0.196561	−	0.331827i
$538$		0			0
$539$		−	59.8439i		−	0.111028i
$540$		0			0
$541$	−186.105			−0.344002			−0.172001	−	0.985097i	$-0.555023\pi$
$541$	−186.105			−0.344002			−0.172001	+	0.985097i	$0.555023\pi$
$542$		0			0
$543$	489.601	+	290.019i	0.901659	+	0.534106i
$544$		0			0
$545$		−	299.280i		−	0.549137i
$546$		0			0
$547$	309.434			0.565693			0.282847	−	0.959165i	$-0.408721\pi$
$547$	309.434			0.565693			0.282847	+	0.959165i	$0.408721\pi$
$548$		0			0
$549$	−233.298	−	425.802i	−0.424951	−	0.775596i
$550$		0			0
$551$			723.779i			1.31357i
$552$		0			0
$553$	351.684			0.635957
$554$		0			0
$555$	−54.5217	+	92.0417i	−0.0982373	+	0.165841i
$556$		0			0
$557$		−	4.00106i		−	0.00718323i	−0.999994	−	0.00359161i	$-0.998857\pi$
$557$		−	4.00106i		−	0.00718323i	0.999994	−	0.00359161i	$-0.00114325\pi$
$558$		0			0
$559$	−144.605			−0.258685
$560$		0			0
$561$	665.526	+	394.230i	1.18632	+	0.702728i
$562$		0			0
$563$		−	166.970i		−	0.296572i	−0.988945	−	0.148286i	$-0.952624\pi$
$563$		−	166.970i		−	0.296572i	0.988945	−	0.148286i	$-0.0473756\pi$
$564$		0			0
$565$	17.6840			0.0312991
$566$		0			0
$567$	−326.399	+	511.102i	−0.575660	+	0.901414i
$568$		0			0
$569$			156.289i			0.274673i	0.990524	+	0.137337i	$0.0438542\pi$
$569$			156.289i			0.274673i	−0.990524	+	0.137337i	$0.956146\pi$
$570$		0			0
$571$	144.105			0.252374			0.126187	−	0.992006i	$-0.459726\pi$
$571$	144.105			0.252374			0.126187	+	0.992006i	$0.459726\pi$
$572$		0			0
$573$	−166.211	+	280.591i	−0.290071	+	0.489688i
$574$		0			0
$575$			45.8688i			0.0797719i
$576$		0			0
$577$	532.947			0.923651			0.461826	−	0.886971i	$-0.347195\pi$
$577$	532.947			0.923651			0.461826	+	0.886971i	$0.347195\pi$
$578$		0			0
$579$	433.495	+	256.785i	0.748697	+	0.443497i
$580$		0			0
$581$		−	195.738i		−	0.336899i
$582$		0			0
$583$	−257.842			−0.442268
$584$		0			0
$585$	−176.491	+	96.7000i	−0.301694	+	0.165299i
$586$		0			0
$587$		−	190.342i		−	0.324262i	−0.986769	−	0.162131i	$-0.948163\pi$
$587$		−	190.342i		−	0.324262i	0.986769	−	0.162131i	$-0.0518368\pi$
$588$		0			0
$589$	215.789			0.366366
$590$		0			0
$591$	262.460	−	443.077i	0.444096	−	0.749707i
$592$		0			0
$593$			345.719i			0.583001i	0.956571	+	0.291500i	$0.0941544\pi$
$593$			345.719i			0.583001i	−0.956571	+	0.291500i	$0.905846\pi$
$594$		0			0
$595$	−508.710			−0.854975
$596$		0			0
$597$	90.5438	+	53.6344i	0.151665	+	0.0898399i
$598$		0			0
$599$			704.055i			1.17538i	0.809085	+	0.587692i	$0.199963\pi$
$599$			704.055i			1.17538i	−0.809085	+	0.587692i	$0.800037\pi$
$600$		0			0
$601$	338.474			0.563185			0.281592	−	0.959534i	$-0.409137\pi$
$601$	338.474			0.563185			0.281592	+	0.959534i	$0.409137\pi$
$602$		0			0
$603$	477.693	+	871.856i	0.792193	+	1.44586i
$604$		0			0
$605$		−	109.567i		−	0.181103i
$606$		0			0
$607$	816.513			1.34516			0.672581	−	0.740024i	$-0.265186\pi$
$607$	816.513			1.34516			0.672581	+	0.740024i	$0.265186\pi$
$608$		0			0
$609$	−307.157	+	518.532i	−0.504363	+	0.851449i
$610$		0			0
$611$			458.688i			0.750717i
$612$		0			0
$613$	−229.263			−0.374001			−0.187001	−	0.982360i	$-0.559877\pi$
$613$	−229.263			−0.374001			−0.187001	+	0.982360i	$0.559877\pi$
$614$		0			0
$615$	−273.329	−	161.909i	−0.444437	−	0.263266i
$616$		0			0
$617$			1072.25i			1.73785i	0.494945	+	0.868924i	$0.335188\pi$
$617$			1072.25i			1.73785i	−0.494945	+	0.868924i	$0.664812\pi$
$618$		0			0
$619$	80.7103			0.130388			0.0651941	−	0.997873i	$-0.479233\pi$
$619$	80.7103			0.130388			0.0651941	+	0.997873i	$0.479233\pi$
$620$		0			0
$621$	−247.552	−	8.30863i	−0.398635	−	0.0133794i
$622$		0			0
$623$			455.004i			0.730344i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	349.947	−	590.769i	0.558130	−	0.942215i
$628$		0			0
$629$		−	484.591i		−	0.770415i
$630$		0			0
$631$	−492.894			−0.781131			−0.390566	−	0.920575i	$-0.627721\pi$
$631$	−492.894			−0.781131			−0.390566	+	0.920575i	$0.627721\pi$
$632$		0			0
$633$	−150.114	−	88.9213i	−0.237147	−	0.140476i
$634$		0			0
$635$		−	300.663i		−	0.473485i
$636$		0			0
$637$	−70.5267			−0.110717
$638$		0			0
$639$	123.079	−	67.4353i	0.192612	−	0.105533i
$640$		0			0
$641$		−	65.4816i		−	0.102155i	−0.998695	−	0.0510777i	$-0.983734\pi$
$641$		−	65.4816i		−	0.102155i	0.998695	−	0.0510777i	$-0.0162656\pi$
$642$		0			0
$643$	428.619			0.666592			0.333296	−	0.942822i	$-0.391839\pi$
$643$	428.619			0.666592			0.333296	+	0.942822i	$0.391839\pi$
$644$		0			0
$645$	−49.4384	+	83.4602i	−0.0766487	+	0.129396i
$646$		0			0
$647$		−	462.801i		−	0.715303i	−0.933855	−	0.357652i	$-0.883578\pi$
$647$		−	462.801i		−	0.715303i	0.933855	−	0.357652i	$-0.116422\pi$
$648$		0			0
$649$	204.316			0.314817
$650$		0			0
$651$	154.596	+	91.5766i	0.237475	+	0.140671i
$652$		0			0
$653$			425.064i			0.650941i	0.945552	+	0.325470i	$0.105523\pi$
$653$			425.064i			0.650941i	−0.945552	+	0.325470i	$0.894477\pi$
$654$		0			0
$655$	492.816			0.752390
$656$		0			0
$657$	380.333	+	694.161i	0.578894	+	1.05656i
$658$		0			0
$659$			182.769i			0.277343i	0.990338	+	0.138671i	$0.0442831\pi$
$659$			182.769i			0.277343i	−0.990338	+	0.138671i	$0.955717\pi$
$660$		0			0
$661$	−482.053			−0.729278			−0.364639	−	0.931149i	$-0.618808\pi$
$661$	−482.053			−0.729278			−0.364639	+	0.931149i	$0.618808\pi$
$662$		0			0
$663$	464.605	−	784.330i	0.700762	−	1.18300i
$664$		0			0
$665$			451.568i			0.679050i
$666$		0			0
$667$	−246.158			−0.369052
$668$		0			0
$669$	256.517	+	151.950i	0.383434	+	0.227131i
$670$		0			0
$671$			457.758i			0.682203i
$672$		0			0
$673$	184.579			0.274264			0.137132	−	0.990553i	$-0.456212\pi$
$673$	184.579			0.274264			0.137132	+	0.990553i	$0.456212\pi$
$674$		0			0
$675$	−4.52847	+	134.924i	−0.00670885	+	0.199887i
$676$		0			0
$677$		−	1065.85i		−	1.57437i	−0.616715	−	0.787187i	$-0.711537\pi$
$677$		−	1065.85i		−	1.57437i	0.616715	−	0.787187i	$-0.288463\pi$
$678$		0			0
$679$	269.920			0.397526
$680$		0			0
$681$	331.710	−	559.982i	0.487093	−	0.822293i
$682$		0			0
$683$			788.926i			1.15509i	0.816359	+	0.577545i	$0.195989\pi$
$683$			788.926i			1.15509i	−0.816359	+	0.577545i	$0.804011\pi$
$684$		0			0
$685$	213.579			0.311794
$686$		0			0
$687$	−840.636	−	497.958i	−1.22363	−	0.724830i
$688$		0			0
$689$			303.870i			0.441030i
$690$		0			0
$691$	−932.000			−1.34877			−0.674385	−	0.738380i	$-0.735591\pi$
$691$	−932.000			−1.34877			−0.674385	+	0.738380i	$0.735591\pi$
$692$		0			0
$693$	501.421	−	274.730i	0.723551	−	0.396436i
$694$		0			0
$695$		−	171.765i		−	0.247144i
$696$		0			0
$697$	1439.05			2.06464
$698$		0			0
$699$	−79.0655	+	133.476i	−0.113112	+	0.190952i
$700$		0			0
$701$			1352.75i			1.92974i	0.262721	+	0.964872i	$0.415380\pi$
$701$			1352.75i			1.92974i	−0.262721	+	0.964872i	$0.584620\pi$
$702$		0			0
$703$	−430.158			−0.611889
$704$		0			0
$705$	264.737	+	156.819i	0.375513	+	0.222439i
$706$		0			0
$707$		−	360.549i		−	0.509970i
$708$		0			0
$709$	−269.473			−0.380075			−0.190038	−	0.981777i	$-0.560861\pi$
$709$	−269.473			−0.380075			−0.190038	+	0.981777i	$0.560861\pi$
$710$		0			0
$711$	203.140	+	370.759i	0.285711	+	0.521462i
$712$		0			0
$713$			73.3901i			0.102931i
$714$		0			0
$715$	189.737			0.265366
$716$		0			0
$717$	626.947	−	1058.39i	0.874403	−	1.47614i
$718$		0			0
$719$			537.103i			0.747014i	0.927627	+	0.373507i	$0.121845\pi$
$719$			537.103i			0.747014i	−0.927627	+	0.373507i	$0.878155\pi$
$720$		0			0
$721$	−1051.21			−1.45799
$722$		0			0
$723$	−1149.96	−	681.192i	−1.59055	−	0.942174i
$724$		0			0
$725$			134.164i			0.185054i
$726$		0			0
$727$	1117.83			1.53759			0.768795	−	0.639495i	$-0.220856\pi$
$727$	1117.83			1.53759			0.768795	+	0.639495i	$0.220856\pi$
$728$		0			0
$729$	−727.359	−	48.8800i	−0.997750	−	0.0670507i
$730$		0			0
$731$		−	439.411i		−	0.601109i
$732$		0			0
$733$	7.52599			0.0102674			0.00513369	−	0.999987i	$-0.498366\pi$
$733$	7.52599			0.0102674			0.00513369	+	0.999987i	$0.498366\pi$
$734$		0			0
$735$	−24.1121	+	40.7052i	−0.0328056	+	0.0553812i
$736$		0			0
$737$		−	937.288i		−	1.27176i
$738$		0			0
$739$	823.079			1.11377			0.556887	−	0.830588i	$-0.311996\pi$
$739$	823.079			1.11377			0.556887	+	0.830588i	$0.311996\pi$
$740$		0			0
$741$	−696.228	−	412.417i	−0.939579	−	0.556568i
$742$		0			0
$743$			3.21898i			0.00433241i	0.999998	+	0.00216620i	$0.000689524\pi$
$743$			3.21898i			0.00433241i	−0.999998	+	0.00216620i	$0.999310\pi$
$744$		0			0
$745$	−617.684			−0.829106
$746$		0			0
$747$	206.355	−	113.063i	0.276245	−	0.151356i
$748$		0			0
$749$			322.794i			0.430967i
$750$		0			0
$751$	−1185.63			−1.57874			−0.789368	−	0.613920i	$-0.789592\pi$
$751$	−1185.63			−1.57874			−0.789368	+	0.613920i	$0.789592\pi$
$752$		0			0
$753$	362.921	−	612.671i	0.481967	−	0.813639i
$754$		0			0
$755$			40.3670i			0.0534662i
$756$		0			0
$757$	−863.315			−1.14044			−0.570221	−	0.821491i	$-0.693143\pi$
$757$	−863.315			−1.14044			−0.570221	+	0.821491i	$0.693143\pi$
$758$		0			0
$759$	200.921	+	119.017i	0.264718	+	0.156808i
$760$		0			0
$761$			570.597i			0.749800i	0.927065	+	0.374900i	$0.122323\pi$
$761$			570.597i			0.749800i	−0.927065	+	0.374900i	$0.877677\pi$
$762$		0			0
$763$	1002.05			1.31331
$764$		0			0
$765$	−293.842	−	536.303i	−0.384107	−	0.701050i
$766$		0			0
$767$		−	240.789i		−	0.313936i
$768$		0			0
$769$	−741.684			−0.964479			−0.482239	−	0.876040i	$-0.660176\pi$
$769$	−741.684			−0.964479			−0.482239	+	0.876040i	$0.660176\pi$
$770$		0			0
$771$	−487.565	+	823.090i	−0.632380	+	1.06756i
$772$		0			0
$773$		−	623.203i		−	0.806214i	−0.915153	−	0.403107i	$-0.867930\pi$
$773$		−	623.203i		−	0.806214i	0.915153	−	0.403107i	$-0.132070\pi$
$774$		0			0
$775$	40.0000			0.0516129
$776$		0			0
$777$	−308.175	−	182.550i	−0.396622	−	0.234943i
$778$		0			0
$779$		−	1277.41i		−	1.63980i
$780$		0			0
$781$	−132.316			−0.169419
$782$		0			0
$783$	−724.078	−	24.3023i	−0.924749	−	0.0310375i
$784$		0			0
$785$		−	232.198i		−	0.295793i
$786$		0			0
$787$	−335.303			−0.426052			−0.213026	−	0.977046i	$-0.568332\pi$
$787$	−335.303			−0.426052			−0.213026	+	0.977046i	$0.568332\pi$
$788$		0			0
$789$	55.3943	−	93.5147i	0.0702083	−	0.118523i
$790$		0			0
$791$			59.2098i			0.0748543i
$792$		0			0
$793$	539.473			0.680294
$794$		0			0
$795$	175.381	+	103.889i	0.220606	+	0.130678i
$796$		0			0
$797$		−	550.520i		−	0.690740i	−0.938467	−	0.345370i	$-0.887753\pi$
$797$		−	550.520i		−	0.690740i	0.938467	−	0.345370i	$-0.112247\pi$
$798$		0			0
$799$	−1393.81			−1.74445
$800$		0			0
$801$	−479.684	+	262.820i	−0.598856	+	0.328115i
$802$		0			0
$803$		−	746.258i		−	0.929337i
$804$		0			0
$805$	−153.579			−0.190781
$806$		0			0
$807$	807.354	−	1362.95i	1.00044	−	1.68891i
$808$		0			0
$809$			560.288i			0.692569i	0.938130	+	0.346284i	$0.112557\pi$
$809$			560.288i			0.692569i	−0.938130	+	0.346284i	$0.887443\pi$
$810$		0			0
$811$	237.842			0.293270			0.146635	−	0.989191i	$-0.453156\pi$
$811$	237.842			0.293270			0.146635	+	0.989191i	$0.453156\pi$
$812$		0			0
$813$	−1228.35	−	727.624i	−1.51089	−	0.894987i
$814$		0			0
$815$		−	25.4498i		−	0.0312267i
$816$		0			0
$817$	−390.053			−0.477421
$818$		0			0
$819$	−323.772	−	590.930i	−0.395326	−	0.721526i
$820$		0			0
$821$		−	65.4816i		−	0.0797584i	−0.999205	−	0.0398792i	$-0.987303\pi$
$821$		−	65.4816i		−	0.0797584i	0.999205	−	0.0398792i	$-0.0126973\pi$
$822$		0			0
$823$	−521.512			−0.633673			−0.316836	−	0.948480i	$-0.602621\pi$
$823$	−521.512			−0.633673			−0.316836	+	0.948480i	$0.602621\pi$
$824$		0			0
$825$	64.8683	−	109.508i	0.0786283	−	0.132738i
$826$		0			0
$827$			987.512i			1.19409i	0.802208	+	0.597045i	$0.203658\pi$
$827$			987.512i			1.19409i	−0.802208	+	0.597045i	$0.796342\pi$
$828$		0			0
$829$	333.631			0.402450			0.201225	−	0.979545i	$-0.435508\pi$
$829$	333.631			0.402450			0.201225	+	0.979545i	$0.435508\pi$
$830$		0			0
$831$	−485.662	−	287.686i	−0.584431	−	0.346193i
$832$		0			0
$833$		−	214.309i		−	0.257274i
$834$		0			0
$835$	−564.092			−0.675559
$836$		0			0
$837$	−7.24555	+	215.878i	−0.00865657	+	0.257919i
$838$		0			0
$839$		−	129.363i		−	0.154187i	−0.997024	−	0.0770934i	$-0.975436\pi$
$839$		−	129.363i		−	0.154187i	0.997024	−	0.0770934i	$-0.0245640\pi$
$840$		0			0
$841$	121.000			0.143876
$842$		0			0
$843$	−37.3559	+	63.0629i	−0.0443130	+	0.0748077i
$844$		0			0
$845$			154.289i			0.182590i
$846$		0			0
$847$	366.855			0.433123
$848$		0			0
$849$	512.254	+	303.438i	0.603362	+	0.357407i
$850$		0			0
$851$		−	146.297i		−	0.171912i
$852$		0			0
$853$	1080.42			1.26661			0.633306	−	0.773902i	$-0.281697\pi$
$853$	1080.42			1.26661			0.633306	+	0.773902i	$0.281697\pi$
$854$		0			0
$855$	−476.061	+	260.835i	−0.556797	+	0.305071i
$856$		0			0
$857$		−	702.548i		−	0.819776i	−0.912136	−	0.409888i	$-0.865568\pi$
$857$		−	702.548i		−	0.819776i	0.912136	−	0.409888i	$-0.134432\pi$
$858$		0			0
$859$	281.132			0.327278			0.163639	−	0.986520i	$-0.447677\pi$
$859$	281.132			0.327278			0.163639	+	0.986520i	$0.447677\pi$
$860$		0			0
$861$	542.105	−	915.163i	0.629623	−	1.06291i
$862$		0			0
$863$			419.221i			0.485772i	0.970055	+	0.242886i	$0.0780941\pi$
$863$			419.221i			0.485772i	−0.970055	+	0.242886i	$0.921906\pi$
$864$		0			0
$865$	26.4213			0.0305449
$866$		0			0
$867$	1637.39	+	969.924i	1.88857	+	1.11871i
$868$		0			0
$869$		−	398.585i		−	0.458671i
$870$		0			0
$871$	−1104.60			−1.26820
$872$		0			0
$873$	155.912	+	284.561i	0.178593	+	0.325958i
$874$		0			0
$875$			83.7053i			0.0956632i
$876$		0			0
$877$	1079.42			1.23081			0.615405	−	0.788211i	$-0.288992\pi$
$877$	1079.42			1.23081			0.615405	+	0.788211i	$0.288992\pi$
$878$		0			0
$879$	785.618	−	1326.25i	0.893763	−	1.50882i
$880$		0			0
$881$			748.212i			0.849275i	0.905363	+	0.424638i	$0.139599\pi$
$881$			748.212i			0.849275i	−0.905363	+	0.424638i	$0.860401\pi$
$882$		0			0
$883$	875.749			0.991788			0.495894	−	0.868383i	$-0.334840\pi$
$883$	875.749			0.991788			0.495894	+	0.868383i	$0.334840\pi$
$884$		0			0
$885$	−138.974	−	82.3223i	−0.157032	−	0.0930196i
$886$		0			0
$887$			1015.05i			1.14436i	0.820127	+	0.572182i	$0.193903\pi$
$887$			1015.05i			1.14436i	−0.820127	+	0.572182i	$0.806097\pi$
$888$		0			0
$889$	1006.68			1.13238
$890$		0			0
$891$	579.263	+	369.928i	0.650126	+	0.415183i
$892$		0			0
$893$			1237.25i			1.38550i
$894$		0			0
$895$	154.369			0.172479
$896$		0			0
$897$	140.263	−	236.788i	0.156369	−	0.263977i
$898$		0			0
$899$			214.663i			0.238779i
$900$		0			0
$901$	−923.368			−1.02483
$902$		0			0
$903$	−279.443	−	165.530i	−0.309460	−	0.183312i
$904$		0			0
$905$			424.146i			0.468670i
$906$		0			0
$907$	−1504.70			−1.65898			−0.829491	−	0.558520i	$-0.811369\pi$
$907$	−1504.70			−1.65898			−0.829491	+	0.558520i	$0.811369\pi$
$908$		0			0
$909$	380.105	−	208.261i	0.418158	−	0.229110i
$910$		0			0
$911$		−	1002.19i		−	1.10010i	−0.835131	−	0.550051i	$-0.814608\pi$
$911$		−	1002.19i		−	1.10010i	0.835131	−	0.550051i	$-0.185392\pi$
$912$		0			0
$913$	−221.842			−0.242981
$914$		0			0
$915$	184.438	−	311.363i	0.201572	−	0.340287i
$916$		0			0
$917$			1650.05i			1.79940i
$918$		0			0
$919$	780.289			0.849063			0.424532	−	0.905413i	$-0.360439\pi$
$919$	780.289			0.849063			0.424532	+	0.905413i	$0.360439\pi$
$920$		0			0
$921$	−29.3772	−	17.4019i	−0.0318971	−	0.0188945i
$922$		0			0
$923$			155.936i			0.168945i
$924$		0			0
$925$	−79.7367			−0.0862018
$926$		0			0
$927$	−607.201	−	1108.23i	−0.655018	−	1.19550i
$928$		0			0
$929$		−	1093.84i		−	1.17744i	−0.808339	−	0.588718i	$-0.799633\pi$
$929$		−	1093.84i		−	1.17744i	0.808339	−	0.588718i	$-0.200367\pi$
$930$		0			0
$931$	−190.236			−0.204335
$932$		0			0
$933$	−793.157	+	1338.98i	−0.850115	+	1.43513i
$934$		0			0
$935$			576.552i			0.616633i
$936$		0			0
$937$	−407.947			−0.435376			−0.217688	−	0.976018i	$-0.569852\pi$
$937$	−407.947			−0.435376			−0.217688	+	0.976018i	$0.569852\pi$
$938$		0			0
$939$	119.548	+	70.8154i	0.127314	+	0.0754157i
$940$		0			0
$941$		−	671.008i		−	0.713079i	−0.934280	−	0.356540i	$-0.883956\pi$
$941$		−	671.008i		−	0.713079i	0.934280	−	0.356540i	$-0.116044\pi$
$942$		0			0
$943$	434.447			0.460707
$944$		0			0
$945$	−451.754	−	15.1623i	−0.478047	−	0.0160447i
$946$		0			0
$947$		−	1608.13i		−	1.69813i	−0.528290	−	0.849064i	$-0.677167\pi$
$947$		−	1608.13i		−	1.69813i	0.528290	−	0.849064i	$-0.322833\pi$
$948$		0			0
$949$	−879.473			−0.926737
$950$		0			0
$951$	−59.7758	+	100.911i	−0.0628557	+	0.106111i
$952$		0			0
$953$			695.440i			0.729737i	0.931059	+	0.364869i	$0.118886\pi$
$953$			695.440i			0.729737i	−0.931059	+	0.364869i	$0.881114\pi$
$954$		0			0
$955$	−243.079			−0.254533
$956$		0			0
$957$	587.684	+	348.120i	0.614090	+	0.363762i
$958$		0			0
$959$			715.107i			0.745680i
$960$		0			0
$961$	−897.000			−0.933403
$962$		0			0
$963$	−340.302	+	186.453i	−0.353377	+	0.193617i
$964$		0			0
$965$			375.542i			0.389162i
$966$		0			0
$967$	1030.07			1.06522			0.532609	−	0.846361i	$-0.321212\pi$
$967$	1030.07			1.06522			0.532609	+	0.846361i	$0.321212\pi$
$968$		0			0
$969$	1253.21	−	2115.63i	1.29330	−	2.18331i
$970$		0			0
$971$			1165.24i			1.20004i	0.799986	+	0.600019i	$0.204840\pi$
$971$			1165.24i			1.20004i	−0.799986	+	0.600019i	$0.795160\pi$
$972$		0			0
$973$	575.106			0.591065
$974$		0			0
$975$	−129.057	−	76.4481i	−0.132366	−	0.0784083i
$976$		0			0
$977$		−	726.440i		−	0.743541i	−0.928325	−	0.371771i	$-0.878751\pi$
$977$		−	726.440i		−	0.743541i	0.928325	−	0.371771i	$-0.121249\pi$
$978$		0			0
$979$	515.684			0.526746
$980$		0			0
$981$	578.807	+	1056.40i	0.590017	+	1.07686i
$982$		0			0
$983$			1024.21i			1.04192i	0.853581	+	0.520960i	$0.174426\pi$
$983$			1024.21i			1.04192i	−0.853581	+	0.520960i	$0.825574\pi$
$984$		0			0
$985$	383.842			0.389687
$986$		0			0
$987$	−525.064	+	886.395i	−0.531980	+	0.898070i
$988$		0			0
$989$		−	132.657i		−	0.134133i
$990$		0			0
$991$	1797.89			1.81422			0.907111	−	0.420892i	$-0.138283\pi$
$991$	1797.89			1.81422			0.907111	+	0.420892i	$0.138283\pi$
$992$		0			0
$993$	−1149.69	−	681.031i	−1.15780	−	0.685832i
$994$		0			0
$995$			78.4390i			0.0788332i
$996$		0			0
$997$	901.368			0.904080			0.452040	−	0.891998i	$-0.350696\pi$
$997$	901.368			0.904080			0.452040	+	0.891998i	$0.350696\pi$
$998$		0			0
$999$	14.4434	−	430.336i	0.0144579	−	0.430766i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.l.c.161.1		4
3.2	odd	2	inner	240.3.l.c.161.2		4
4.3	odd	2		30.3.d.a.11.2	✓	4
5.2	odd	4		1200.3.c.k.449.4		8
5.3	odd	4		1200.3.c.k.449.5		8
5.4	even	2		1200.3.l.u.401.4		4
8.3	odd	2		960.3.l.e.641.1		4
8.5	even	2		960.3.l.f.641.4		4
12.11	even	2		30.3.d.a.11.4	yes	4
15.2	even	4		1200.3.c.k.449.6		8
15.8	even	4		1200.3.c.k.449.3		8
15.14	odd	2		1200.3.l.u.401.3		4
20.3	even	4		150.3.b.b.149.2		8
20.7	even	4		150.3.b.b.149.7		8
20.19	odd	2		150.3.d.c.101.3		4
24.5	odd	2		960.3.l.f.641.3		4
24.11	even	2		960.3.l.e.641.2		4
36.7	odd	6		810.3.h.a.701.2		8
36.11	even	6		810.3.h.a.701.3		8
36.23	even	6		810.3.h.a.431.2		8
36.31	odd	6		810.3.h.a.431.3		8
60.23	odd	4		150.3.b.b.149.8		8
60.47	odd	4		150.3.b.b.149.1		8
60.59	even	2		150.3.d.c.101.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.3.d.a.11.2	✓	4	4.3	odd	2
30.3.d.a.11.4	yes	4	12.11	even	2
150.3.b.b.149.1		8	60.47	odd	4
150.3.b.b.149.2		8	20.3	even	4
150.3.b.b.149.7		8	20.7	even	4
150.3.b.b.149.8		8	60.23	odd	4
150.3.d.c.101.1		4	60.59	even	2
150.3.d.c.101.3		4	20.19	odd	2
240.3.l.c.161.1		4	1.1	even	1	trivial
240.3.l.c.161.2		4	3.2	odd	2	inner
810.3.h.a.431.2		8	36.23	even	6
810.3.h.a.431.3		8	36.31	odd	6
810.3.h.a.701.2		8	36.7	odd	6
810.3.h.a.701.3		8	36.11	even	6
960.3.l.e.641.1		4	8.3	odd	2
960.3.l.e.641.2		4	24.11	even	2
960.3.l.f.641.3		4	24.5	odd	2
960.3.l.f.641.4		4	8.5	even	2
1200.3.c.k.449.3		8	15.8	even	4
1200.3.c.k.449.4		8	5.2	odd	4
1200.3.c.k.449.5		8	5.3	odd	4
1200.3.c.k.449.6		8	15.2	even	4
1200.3.l.u.401.3		4	15.14	odd	2
1200.3.l.u.401.4		4	5.4	even	2

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	240.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.58114 - 1.52896i) q^{3} -2.23607i q^{5} +7.48683 q^{7} +(4.32456 + 7.89292i) q^{9} -8.48528i q^{11} -10.0000 q^{13} +(-3.41886 + 5.77160i) q^{15} -30.3870i q^{17} -26.9737 q^{19} +(-19.3246 - 11.4471i) q^{21} -9.17377i q^{23} -5.00000 q^{25} +(0.905694 - 26.9848i) q^{27} -26.8328i q^{29} -8.00000 q^{31} +(-12.9737 + 21.9017i) q^{33} -16.7411i q^{35} +15.9473 q^{37} +(25.8114 + 15.2896i) q^{39} +47.3575i q^{41} +14.4605 q^{43} +(17.6491 - 9.67000i) q^{45} -45.8688i q^{47} +7.05267 q^{49} +(-46.4605 + 78.4330i) q^{51} -30.3870i q^{53} -18.9737 q^{55} +(69.6228 + 41.2417i) q^{57} +24.0789i q^{59} -53.9473 q^{61} +(32.3772 + 59.0930i) q^{63} +22.3607i q^{65} +110.460 q^{67} +(-14.0263 + 23.6788i) q^{69} -15.5936i q^{71} +87.9473 q^{73} +(12.9057 + 7.64481i) q^{75} -63.5279i q^{77} +46.9737 q^{79} +(-43.5964 + 68.2668i) q^{81} -26.1443i q^{83} -67.9473 q^{85} +(-41.0263 + 69.2592i) q^{87} +60.7739i q^{89} -74.8683 q^{91} +(20.6491 + 12.2317i) q^{93} +60.3150i q^{95} +36.0527 q^{97} +(66.9737 - 36.6951i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{7} - 8 q^{9} - 40 q^{13} - 20 q^{15} - 32 q^{19} - 52 q^{21} - 20 q^{25} - 28 q^{27} - 32 q^{31} + 24 q^{33} - 88 q^{37} + 40 q^{39} - 56 q^{43} + 20 q^{45} + 180 q^{49} - 72 q^{51}+ \cdots + 192 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^7 - 8 * q^9 - 40 * q^13 - 20 * q^15 - 32 * q^19 - 52 * q^21 - 20 * q^25 - 28 * q^27 - 32 * q^31 + 24 * q^33 - 88 * q^37 + 40 * q^39 - 56 * q^43 + 20 * q^45 + 180 * q^49 - 72 * q^51 + 152 * q^57 - 64 * q^61 + 256 * q^63 + 328 * q^67 - 132 * q^69 + 200 * q^73 + 20 * q^75 + 112 * q^79 + 28 * q^81 - 120 * q^85 - 240 * q^87 + 80 * q^91 + 32 * q^93 + 296 * q^97 + 192 * q^99

Introduction

L-functions

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