LMFDB - Embedded newform 280.6.g.b.169.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,6,Mod(169,280)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("280.169");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 280 = 2^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	280.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$44.9074695476$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.12
Character	$\chi$	$=$	280.169
Dual form			280.6.g.b.169.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.32294i q^{3} +(-6.05478 + 55.5728i) q^{5} +49.0000i q^{7} +237.604 q^{9} +O(q^{10})$	$q-2.32294i q^{3} +(-6.05478 + 55.5728i) q^{5} +49.0000i q^{7} +237.604 q^{9} -24.5475 q^{11} -522.496i q^{13} +(129.092 + 14.0649i) q^{15} +1046.48i q^{17} +2876.21 q^{19} +113.824 q^{21} -685.167i q^{23} +(-3051.68 - 672.963i) q^{25} -1116.41i q^{27} +931.643 q^{29} -6369.55 q^{31} +57.0223i q^{33} +(-2723.07 - 296.684i) q^{35} +15978.1i q^{37} -1213.73 q^{39} +7308.29 q^{41} +3729.69i q^{43} +(-1438.64 + 13204.3i) q^{45} +4316.07i q^{47} -2401.00 q^{49} +2430.92 q^{51} +342.571i q^{53} +(148.630 - 1364.17i) q^{55} -6681.26i q^{57} +10806.1 q^{59} -30920.3 q^{61} +11642.6i q^{63} +(29036.6 + 3163.60i) q^{65} +39564.9i q^{67} -1591.60 q^{69} +32418.6 q^{71} +70423.5i q^{73} +(-1563.25 + 7088.86i) q^{75} -1202.83i q^{77} +78053.7 q^{79} +55144.4 q^{81} +41312.2i q^{83} +(-58156.1 - 6336.24i) q^{85} -2164.15i q^{87} -60986.9 q^{89} +25602.3 q^{91} +14796.1i q^{93} +(-17414.8 + 159839. i) q^{95} +112719. i q^{97} -5832.58 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 30 q^{5} - 2106 q^{9}+O(q^{10}) $ 24 * q - 30 * q^5 - 2106 * q^9	$ 24 q - 30 q^{5} - 2106 q^{9} + 630 q^{11} + 1302 q^{15} - 3000 q^{19} - 882 q^{21} + 3400 q^{25} - 10210 q^{29} + 13372 q^{31} + 2058 q^{35} - 62478 q^{39} + 4216 q^{41} + 58642 q^{45} - 57624 q^{49} + 112914 q^{51} - 19492 q^{55} - 22916 q^{59} - 11268 q^{61} + 89258 q^{65} + 50276 q^{69} + 134240 q^{71} - 98520 q^{75} - 78834 q^{79} + 85504 q^{81} + 5414 q^{85} + 54988 q^{89} + 37730 q^{91} - 242844 q^{95} + 112988 q^{99}+O(q^{100}) $ 24 * q - 30 * q^5 - 2106 * q^9 + 630 * q^11 + 1302 * q^15 - 3000 * q^19 - 882 * q^21 + 3400 * q^25 - 10210 * q^29 + 13372 * q^31 + 2058 * q^35 - 62478 * q^39 + 4216 * q^41 + 58642 * q^45 - 57624 * q^49 + 112914 * q^51 - 19492 * q^55 - 22916 * q^59 - 11268 * q^61 + 89258 * q^65 + 50276 * q^69 + 134240 * q^71 - 98520 * q^75 - 78834 * q^79 + 85504 * q^81 + 5414 * q^85 + 54988 * q^89 + 37730 * q^91 - 242844 * q^95 + 112988 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$141$	$241$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		−	2.32294i		−	0.149017i	−0.997220	−	0.0745083i	$-0.976261\pi$
$3$		−	2.32294i		−	0.149017i	0.997220	−	0.0745083i	$-0.0237387\pi$
$4$		0			0
$5$	−6.05478	+	55.5728i	−0.108311	+	0.994117i
$5$	−6.05478	+	55.5728i	−0.108311	+	0.994117i
$6$		0			0
$7$			49.0000i			0.377964i
$7$			49.0000i			0.377964i
$8$		0			0
$9$	237.604			0.977794
$10$		0			0
$11$	−24.5475			−0.0611681			−0.0305841	−	0.999532i	$-0.509737\pi$
$11$	−24.5475			−0.0611681			−0.0305841	+	0.999532i	$0.509737\pi$
$12$		0			0
$13$		−	522.496i		−	0.857481i	−0.903428	−	0.428740i	$-0.858958\pi$
$13$		−	522.496i		−	0.857481i	0.903428	−	0.428740i	$-0.141042\pi$
$14$		0			0
$15$	129.092	+	14.0649i	0.148140	+	0.0161402i
$16$		0			0
$17$			1046.48i			0.878235i	0.898430	+	0.439117i	$0.144709\pi$
$17$			1046.48i			0.878235i	−0.898430	+	0.439117i	$0.855291\pi$
$18$		0			0
$19$	2876.21			1.82783			0.913917	−	0.405901i	$-0.133042\pi$
$19$	2876.21			1.82783			0.913917	+	0.405901i	$0.133042\pi$
$20$		0			0
$21$	113.824			0.0563230
$22$		0			0
$23$		−	685.167i		−	0.270070i	−0.990841	−	0.135035i	$-0.956885\pi$
$23$		−	685.167i		−	0.270070i	0.990841	−	0.135035i	$-0.0431147\pi$
$24$		0			0
$25$	−3051.68	−	672.963i	−0.976537	−	0.215348i
$26$		0			0
$27$		−	1116.41i		−	0.294724i
$28$		0			0
$29$	931.643			0.205710			0.102855	−	0.994696i	$-0.467202\pi$
$29$	931.643			0.205710			0.102855	+	0.994696i	$0.467202\pi$
$30$		0			0
$31$	−6369.55			−1.19043			−0.595215	−	0.803566i	$-0.702933\pi$
$31$	−6369.55			−1.19043			−0.595215	+	0.803566i	$0.702933\pi$
$32$		0			0
$33$			57.0223i			0.00911506i
$34$		0			0
$35$	−2723.07	−	296.684i	−0.375741	−	0.0409378i
$36$		0			0
$37$			15978.1i			1.91876i	0.282111	+	0.959382i	$0.408965\pi$
$37$			15978.1i			1.91876i	−0.282111	+	0.959382i	$0.591035\pi$
$38$		0			0
$39$	−1213.73			−0.127779
$40$		0			0
$41$	7308.29			0.678979			0.339489	−	0.940610i	$-0.389746\pi$
$41$	7308.29			0.678979			0.339489	+	0.940610i	$0.389746\pi$
$42$		0			0
$43$			3729.69i			0.307611i	0.988101	+	0.153805i	$0.0491529\pi$
$43$			3729.69i			0.307611i	−0.988101	+	0.153805i	$0.950847\pi$
$44$		0			0
$45$	−1438.64	+	13204.3i	−0.105906	+	0.972042i
$46$		0			0
$47$			4316.07i			0.284999i	0.989795	+	0.142500i	$0.0455139\pi$
$47$			4316.07i			0.284999i	−0.989795	+	0.142500i	$0.954486\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	2430.92			0.130871
$52$		0			0
$53$			342.571i			0.0167518i	0.999965	+	0.00837589i	$0.00266616\pi$
$53$			342.571i			0.0167518i	−0.999965	+	0.00837589i	$0.997334\pi$
$54$		0			0
$55$	148.630	−	1364.17i	0.00662519	−	0.0608083i
$56$		0			0
$57$		−	6681.26i		−	0.272377i
$58$		0			0
$59$	10806.1			0.404147			0.202073	−	0.979370i	$-0.435232\pi$
$59$	10806.1			0.404147			0.202073	+	0.979370i	$0.435232\pi$
$60$		0			0
$61$	−30920.3			−1.06394			−0.531972	−	0.846762i	$-0.678549\pi$
$61$	−30920.3			−1.06394			−0.531972	+	0.846762i	$0.678549\pi$
$62$		0			0
$63$			11642.6i			0.369571i
$64$		0			0
$65$	29036.6	+	3163.60i	0.852436	+	0.0928748i
$66$		0			0
$67$			39564.9i			1.07677i	0.842699	+	0.538386i	$0.180966\pi$
$67$			39564.9i			1.07677i	−0.842699	+	0.538386i	$0.819034\pi$
$68$		0			0
$69$	−1591.60			−0.0402449
$70$		0			0
$71$	32418.6			0.763219			0.381609	−	0.924324i	$-0.375370\pi$
$71$	32418.6			0.763219			0.381609	+	0.924324i	$0.375370\pi$
$72$		0			0
$73$			70423.5i			1.54672i	0.633970	+	0.773358i	$0.281424\pi$
$73$			70423.5i			1.54672i	−0.633970	+	0.773358i	$0.718576\pi$
$74$		0			0
$75$	−1563.25	+	7088.86i	−0.0320904	+	0.145520i
$76$		0			0
$77$		−	1202.83i		−	0.0231194i
$78$		0			0
$79$	78053.7			1.40710			0.703551	−	0.710645i	$-0.251597\pi$
$79$	78053.7			1.40710			0.703551	+	0.710645i	$0.251597\pi$
$80$		0			0
$81$	55144.4			0.933875
$82$		0			0
$83$			41312.2i			0.658238i	0.944288	+	0.329119i	$0.106752\pi$
$83$			41312.2i			0.658238i	−0.944288	+	0.329119i	$0.893248\pi$
$84$		0			0
$85$	−58156.1	−	6336.24i	−0.873068	−	0.0951227i
$86$		0			0
$87$		−	2164.15i		−	0.0306541i
$88$		0			0
$89$	−60986.9			−0.816135			−0.408067	−	0.912952i	$-0.633797\pi$
$89$	−60986.9			−0.816135			−0.408067	+	0.912952i	$0.633797\pi$
$90$		0			0
$91$	25602.3			0.324097
$92$		0			0
$93$			14796.1i			0.177394i
$94$		0			0
$95$	−17414.8	+	159839.i	−0.197975	+	1.81708i
$96$		0			0
$97$			112719.i			1.21637i	0.793794	+	0.608187i	$0.208103\pi$
$97$			112719.i			1.21637i	−0.793794	+	0.608187i	$0.791897\pi$
$98$		0			0
$99$	−5832.58			−0.0598098
$100$		0			0
$101$	−70404.9			−0.686751			−0.343375	−	0.939198i	$-0.611570\pi$
$101$	−70404.9			−0.686751			−0.343375	+	0.939198i	$0.611570\pi$
$102$		0			0
$103$		−	107510.i		−	0.998520i	−0.866452	−	0.499260i	$-0.833605\pi$
$103$		−	107510.i		−	0.998520i	0.866452	−	0.499260i	$-0.166395\pi$
$104$		0			0
$105$	−689.179	+	6325.52i	−0.00610041	+	0.0559916i
$106$		0			0
$107$			166698.i			1.40757i	0.710411	+	0.703787i	$0.248509\pi$
$107$			166698.i			1.40757i	−0.710411	+	0.703787i	$0.751491\pi$
$108$		0			0
$109$	−44142.8			−0.355872			−0.177936	−	0.984042i	$-0.556942\pi$
$109$	−44142.8			−0.355872			−0.177936	+	0.984042i	$0.556942\pi$
$110$		0			0
$111$	37116.2			0.285928
$112$		0			0
$113$			87317.4i			0.643287i	0.946861	+	0.321644i	$0.104235\pi$
$113$			87317.4i			0.643287i	−0.946861	+	0.321644i	$0.895765\pi$
$114$		0			0
$115$	38076.7	+	4148.53i	0.268481	+	0.0292516i
$116$		0			0
$117$		−	124147.i		−	0.838440i
$118$		0			0
$119$	−51277.7			−0.331941
$120$		0			0
$121$	−160448.			−0.996258
$122$		0			0
$123$		−	16976.7i		−	0.101179i
$124$		0			0
$125$	55875.7	−	165516.i	0.319851	−	0.947468i
$126$		0			0
$127$			2231.60i			0.0122774i	0.999981	+	0.00613870i	$0.00195402\pi$
$127$			2231.60i			0.0122774i	−0.999981	+	0.00613870i	$0.998046\pi$
$128$		0			0
$129$	8663.84			0.0458391
$130$		0			0
$131$	253184.			1.28901			0.644507	−	0.764598i	$-0.277063\pi$
$131$	253184.			1.28901			0.644507	+	0.764598i	$0.277063\pi$
$132$		0			0
$133$			140934.i			0.690856i
$134$		0			0
$135$	62042.2	+	6759.64i	0.292990	+	0.0319219i
$136$		0			0
$137$			187500.i			0.853492i	0.904372	+	0.426746i	$0.140340\pi$
$137$			187500.i			0.853492i	−0.904372	+	0.426746i	$0.859660\pi$
$138$		0			0
$139$	−155638.			−0.683247			−0.341624	−	0.939837i	$-0.610977\pi$
$139$	−155638.			−0.683247			−0.341624	+	0.939837i	$0.610977\pi$
$140$		0			0
$141$	10026.0			0.0424696
$142$		0			0
$143$			12825.9i			0.0524505i
$144$		0			0
$145$	−5640.90	+	51774.0i	−0.0222807	+	0.204499i
$146$		0			0
$147$			5577.37i			0.0212881i
$148$		0			0
$149$	−274452.			−1.01274			−0.506372	−	0.862315i	$-0.669014\pi$
$149$	−274452.			−1.01274			−0.506372	+	0.862315i	$0.669014\pi$
$150$		0			0
$151$	152868.			0.545599			0.272799	−	0.962071i	$-0.412050\pi$
$151$	152868.			0.545599			0.272799	+	0.962071i	$0.412050\pi$
$152$		0			0
$153$			248649.i			0.858733i
$154$		0			0
$155$	38566.2	−	353974.i	0.128937	−	1.18343i
$156$		0			0
$157$			100062.i			0.323982i	0.986792	+	0.161991i	$0.0517915\pi$
$157$			100062.i			0.323982i	−0.986792	+	0.161991i	$0.948209\pi$
$158$		0			0
$159$	795.772			0.00249629
$160$		0			0
$161$	33573.2			0.102077
$162$		0			0
$163$		−	106073.i		−	0.312706i	−0.987701	−	0.156353i	$-0.950026\pi$
$163$		−	106073.i		−	0.312706i	0.987701	−	0.156353i	$-0.0499737\pi$
$164$		0			0
$165$	−3168.89	−	345.257i	−0.00906144	−	0.000987263i
$166$		0			0
$167$		−	497781.i		−	1.38117i	−0.723252	−	0.690585i	$-0.757353\pi$
$167$		−	497781.i		−	1.38117i	0.723252	−	0.690585i	$-0.242647\pi$
$168$		0			0
$169$	98291.2			0.264727
$170$		0			0
$171$	683399.			1.78725
$172$		0			0
$173$		−	520694.i		−	1.32272i	−0.750069	−	0.661359i	$-0.769980\pi$
$173$		−	520694.i		−	1.32272i	0.750069	−	0.661359i	$-0.230020\pi$
$174$		0			0
$175$	32975.2	−	149532.i	0.0813939	−	0.369096i
$176$		0			0
$177$		−	25101.9i		−	0.0602245i
$178$		0			0
$179$	414464.			0.966840			0.483420	−	0.875388i	$-0.339394\pi$
$179$	414464.			0.966840			0.483420	+	0.875388i	$0.339394\pi$
$180$		0			0
$181$	−570429.			−1.29421			−0.647106	−	0.762400i	$-0.724021\pi$
$181$	−570429.			−1.29421			−0.647106	+	0.762400i	$0.724021\pi$
$182$		0			0
$183$			71825.9i			0.158545i
$184$		0			0
$185$	−887950.	−	96744.0i	−1.90748	−	0.207824i
$186$		0			0
$187$		−	25688.5i		−	0.0537199i
$188$		0			0
$189$	54704.2			0.111395
$190$		0			0
$191$	259086.			0.513878			0.256939	−	0.966428i	$-0.417286\pi$
$191$	259086.			0.513878			0.256939	+	0.966428i	$0.417286\pi$
$192$		0			0
$193$		−	715882.i		−	1.38340i	−0.722184	−	0.691701i	$-0.756862\pi$
$193$		−	715882.i		−	1.38340i	0.722184	−	0.691701i	$-0.243138\pi$
$194$		0			0
$195$	7348.84	−	67450.2i	0.0138399	−	0.127027i
$196$		0			0
$197$		−	518715.i		−	0.952278i	−0.879370	−	0.476139i	$-0.842036\pi$
$197$		−	518715.i		−	0.952278i	0.879370	−	0.476139i	$-0.157964\pi$
$198$		0			0
$199$	220161.			0.394101			0.197050	−	0.980393i	$-0.436864\pi$
$199$	220161.			0.394101			0.197050	+	0.980393i	$0.436864\pi$
$200$		0			0
$201$	91906.9			0.160457
$202$		0			0
$203$			45650.5i			0.0777509i
$204$		0			0
$205$	−44250.1	+	406143.i	−0.0735410	+	0.674985i
$206$		0			0
$207$		−	162798.i		−	0.264073i
$208$		0			0
$209$	−70603.7			−0.111805
$210$		0			0
$211$	−1.12935e6			−1.74632			−0.873161	−	0.487432i	$-0.837934\pi$
$211$	−1.12935e6			−1.74632			−0.873161	+	0.487432i	$0.837934\pi$
$212$		0			0
$213$		−	75306.5i		−	0.113732i
$214$		0			0
$215$	−207269.	−	22582.5i	−0.305801	−	0.0333177i
$216$		0			0
$217$		−	312108.i		−	0.449941i
$218$		0			0
$219$	163589.			0.230486
$220$		0			0
$221$	546784.			0.753069
$222$		0			0
$223$			524793.i			0.706686i	0.935494	+	0.353343i	$0.114955\pi$
$223$			524793.i			0.706686i	−0.935494	+	0.353343i	$0.885045\pi$
$224$		0			0
$225$	−725091.	−	159899.i	−0.954852	−	0.210566i
$226$		0			0
$227$		−	14762.6i		−	0.0190151i	−0.999955	−	0.00950755i	$-0.996974\pi$
$227$		−	14762.6i		−	0.0190151i	0.999955	−	0.00950755i	$-0.00302639\pi$
$228$		0			0
$229$	−599491.			−0.755429			−0.377715	−	0.925922i	$-0.623290\pi$
$229$	−599491.			−0.755429			−0.377715	+	0.925922i	$0.623290\pi$
$230$		0			0
$231$	−2794.09			−0.00344517
$232$		0			0
$233$		−	1.05756e6i		−	1.27619i	−0.769958	−	0.638095i	$-0.779723\pi$
$233$		−	1.05756e6i		−	1.27619i	0.769958	−	0.638095i	$-0.220277\pi$
$234$		0			0
$235$	−239856.	−	26132.8i	−0.283322	−	0.0308686i
$236$		0			0
$237$		−	181314.i		−	0.209681i
$238$		0			0
$239$	779537.			0.882758			0.441379	−	0.897321i	$-0.354489\pi$
$239$	779537.			0.882758			0.441379	+	0.897321i	$0.354489\pi$
$240$		0			0
$241$	87213.2			0.0967252			0.0483626	−	0.998830i	$-0.484600\pi$
$241$	87213.2			0.0967252			0.0483626	+	0.998830i	$0.484600\pi$
$242$		0			0
$243$		−	399385.i		−	0.433887i
$244$		0			0
$245$	14537.5	−	133430.i	0.0154730	−	0.142017i
$246$		0			0
$247$		−	1.50281e6i		−	1.56733i
$248$		0			0
$249$	95965.7			0.0980884
$250$		0			0
$251$	1.22334e6			1.22564			0.612822	−	0.790221i	$-0.290034\pi$
$251$	1.22334e6			1.22564			0.612822	+	0.790221i	$0.290034\pi$
$252$		0			0
$253$			16819.1i			0.0165197i
$254$		0			0
$255$	−14718.7	+	135093.i	−0.0141749	+	0.130102i
$256$		0			0
$257$			1.18806e6i			1.12203i	0.827806	+	0.561014i	$0.189589\pi$
$257$			1.18806e6i			1.12203i	−0.827806	+	0.561014i	$0.810411\pi$
$258$		0			0
$259$	−782928.			−0.725224
$260$		0			0
$261$	221362.			0.201142
$262$		0			0
$263$			41965.5i			0.0374113i	0.999825	+	0.0187057i	$0.00595454\pi$
$263$			41965.5i			0.0374113i	−0.999825	+	0.0187057i	$0.994045\pi$
$264$		0			0
$265$	−19037.6	−	2074.19i	−0.0166532	−	0.00181441i
$266$		0			0
$267$			141669.i			0.121618i
$268$		0			0
$269$	−1.86599e6			−1.57228			−0.786138	−	0.618051i	$-0.787922\pi$
$269$	−1.86599e6			−1.57228			−0.786138	+	0.618051i	$0.787922\pi$
$270$		0			0
$271$	2.39583e6			1.98168			0.990840	−	0.135042i	$-0.0431171\pi$
$271$	2.39583e6			1.98168			0.990840	+	0.135042i	$0.0431171\pi$
$272$		0			0
$273$		−	59472.5i		−	0.0482959i
$274$		0			0
$275$	74911.0	+	16519.5i	0.0597329	+	0.0131724i
$276$		0			0
$277$		−	158159.i		−	0.123850i	−0.998081	−	0.0619248i	$-0.980276\pi$
$277$		−	158159.i		−	0.123850i	0.998081	−	0.0619248i	$-0.0197239\pi$
$278$		0			0
$279$	−1.51343e6			−1.16400
$280$		0			0
$281$	2.11447e6			1.59748			0.798741	−	0.601675i	$-0.205500\pi$
$281$	2.11447e6			1.59748			0.798741	+	0.601675i	$0.205500\pi$
$282$		0			0
$283$		−	2.26078e6i		−	1.67800i	−0.544133	−	0.838999i	$-0.683141\pi$
$283$		−	2.26078e6i		−	1.67800i	0.544133	−	0.838999i	$-0.316859\pi$
$284$		0			0
$285$	371297.	+	40453.6i	0.270775	+	0.0295015i
$286$		0			0
$287$			358106.i			0.256630i
$288$		0			0
$289$	324727.			0.228704
$290$		0			0
$291$	261839.			0.181260
$292$		0			0
$293$			35687.5i			0.0242855i	0.999926	+	0.0121428i	$0.00386526\pi$
$293$			35687.5i			0.0242855i	−0.999926	+	0.0121428i	$0.996135\pi$
$294$		0			0
$295$	−65428.5	+	600525.i	−0.0437736	+	0.401769i
$296$		0			0
$297$			27405.1i			0.0180277i
$298$		0			0
$299$	−357997.			−0.231580
$300$		0			0
$301$	−182755.			−0.116266
$302$		0			0
$303$			163546.i			0.102337i
$304$		0			0
$305$	187215.	−	1.71833e6i	0.115237	−	1.05768i
$306$		0			0
$307$			1.66987e6i			1.01120i	0.862768	+	0.505601i	$0.168729\pi$
$307$			1.66987e6i			1.01120i	−0.862768	+	0.505601i	$0.831271\pi$
$308$		0			0
$309$	−249740.			−0.148796
$310$		0			0
$311$	−1.79587e6			−1.05287			−0.526434	−	0.850216i	$-0.676471\pi$
$311$	−1.79587e6			−1.05287			−0.526434	+	0.850216i	$0.676471\pi$
$312$		0			0
$313$			2.26887e6i			1.30903i	0.756050	+	0.654514i	$0.227127\pi$
$313$			2.26887e6i			1.30903i	−0.756050	+	0.654514i	$0.772873\pi$
$314$		0			0
$315$	−647012.	−	70493.4i	−0.367397	−	0.0400287i
$316$		0			0
$317$			1.35249e6i			0.755937i	0.925819	+	0.377968i	$0.123377\pi$
$317$			1.35249e6i			0.755937i	−0.925819	+	0.377968i	$0.876623\pi$
$318$		0			0
$319$	−22869.5			−0.0125829
$320$		0			0
$321$	387230.			0.209752
$322$		0			0
$323$			3.00991e6i			1.60527i
$324$		0			0
$325$	−351620.	+	1.59449e6i	−0.184657	+	0.837362i
$326$		0			0
$327$			102541.i			0.0530308i
$328$		0			0
$329$	−211487.			−0.107720
$330$		0			0
$331$	2.22523e6			1.11636			0.558180	−	0.829720i	$-0.311500\pi$
$331$	2.22523e6			1.11636			0.558180	+	0.829720i	$0.311500\pi$
$332$		0			0
$333$			3.79647e6i			1.87616i
$334$		0			0
$335$	−2.19874e6	−	239557.i	−1.07044	−	0.116626i
$336$		0			0
$337$			1.14308e6i			0.548279i	0.961690	+	0.274140i	$0.0883931\pi$
$337$			1.14308e6i			0.548279i	−0.961690	+	0.274140i	$0.911607\pi$
$338$		0			0
$339$	202833.			0.0958604
$340$		0			0
$341$	156356.			0.0728164
$342$		0			0
$343$		−	117649.i		−	0.0539949i
$344$		0			0
$345$	9636.79	−	88449.7i	0.00435898	−	0.0400082i
$346$		0			0
$347$		−	3.43866e6i		−	1.53308i	−0.642195	−	0.766541i	$-0.721976\pi$
$347$		−	3.43866e6i		−	1.53308i	0.642195	−	0.766541i	$-0.278024\pi$
$348$		0			0
$349$	−510173.			−0.224209			−0.112105	−	0.993696i	$-0.535759\pi$
$349$	−510173.			−0.224209			−0.112105	+	0.993696i	$0.535759\pi$
$350$		0			0
$351$	−583321.			−0.252720
$352$		0			0
$353$		−	2.48028e6i		−	1.05941i	−0.848182	−	0.529705i	$-0.822302\pi$
$353$		−	2.48028e6i		−	1.05941i	0.848182	−	0.529705i	$-0.177698\pi$
$354$		0			0
$355$	−196288.	+	1.80160e6i	−0.0826651	+	0.758729i
$356$		0			0
$357$			119115.i			0.0494648i
$358$		0			0
$359$	−1.42523e6			−0.583646			−0.291823	−	0.956472i	$-0.594262\pi$
$359$	−1.42523e6			−0.583646			−0.291823	+	0.956472i	$0.594262\pi$
$360$		0			0
$361$	5.79649e6			2.34098
$362$		0			0
$363$			372712.i			0.148459i
$364$		0			0
$365$	−3.91363e6	−	426399.i	−1.53762	−	0.167527i
$366$		0			0
$367$			3.21259e6i			1.24506i	0.782596	+	0.622530i	$0.213895\pi$
$367$			3.21259e6i			1.24506i	−0.782596	+	0.622530i	$0.786105\pi$
$368$		0			0
$369$	1.73648e6			0.663902
$370$		0			0
$371$	−16786.0			−0.00633158
$372$		0			0
$373$		−	890810.i		−	0.331522i	−0.986166	−	0.165761i	$-0.946992\pi$
$373$		−	890810.i		−	0.331522i	0.986166	−	0.165761i	$-0.0530081\pi$
$374$		0			0
$375$	−384483.	−	129796.i	−0.141188	−	0.0476631i
$376$		0			0
$377$		−	486780.i		−	0.176392i
$378$		0			0
$379$	−2.61527e6			−0.935231			−0.467616	−	0.883932i	$-0.654887\pi$
$379$	−2.61527e6			−0.935231			−0.467616	+	0.883932i	$0.654887\pi$
$380$		0			0
$381$	5183.86			0.00182954
$382$		0			0
$383$			951960.i			0.331606i	0.986159	+	0.165803i	$0.0530215\pi$
$383$			951960.i			0.331606i	−0.986159	+	0.165803i	$0.946978\pi$
$384$		0			0
$385$	66844.4	+	7282.85i	0.0229834	+	0.00250409i
$386$		0			0
$387$			886189.i			0.300780i
$388$		0			0
$389$	4.71548e6			1.57998			0.789991	−	0.613119i	$-0.210085\pi$
$389$	4.71548e6			1.57998			0.789991	+	0.613119i	$0.210085\pi$
$390$		0			0
$391$	717017.			0.237185
$392$		0			0
$393$		−	588130.i		−	0.192084i
$394$		0			0
$395$	−472598.	+	4.33766e6i	−0.152405	+	1.39882i
$396$		0			0
$397$			976212.i			0.310862i	0.987847	+	0.155431i	$0.0496767\pi$
$397$			976212.i			0.310862i	−0.987847	+	0.155431i	$0.950323\pi$
$398$		0			0
$399$	327382.			0.102949
$400$		0			0
$401$	3.59551e6			1.11661			0.558303	−	0.829637i	$-0.311453\pi$
$401$	3.59551e6			1.11661			0.558303	+	0.829637i	$0.311453\pi$
$402$		0			0
$403$			3.32806e6i			1.02077i
$404$		0			0
$405$	−333887.	+	3.06453e6i	−0.101149	+	0.928381i
$406$		0			0
$407$		−	392223.i		−	0.117367i
$408$		0			0
$409$	691170.			0.204304			0.102152	−	0.994769i	$-0.467427\pi$
$409$	691170.			0.204304			0.102152	+	0.994769i	$0.467427\pi$
$410$		0			0
$411$	435550.			0.127184
$412$		0			0
$413$			529499.i			0.152753i
$414$		0			0
$415$	−2.29584e6	−	250136.i	−0.654366	−	0.0712946i
$416$		0			0
$417$			361537.i			0.101815i
$418$		0			0
$419$	−4.98420e6			−1.38695			−0.693474	−	0.720482i	$-0.743921\pi$
$419$	−4.98420e6			−1.38695			−0.693474	+	0.720482i	$0.743921\pi$
$420$		0			0
$421$	3.58138e6			0.984794			0.492397	−	0.870371i	$-0.336121\pi$
$421$	3.58138e6			0.984794			0.492397	+	0.870371i	$0.336121\pi$
$422$		0			0
$423$			1.02551e6i			0.278670i
$424$		0			0
$425$	704245.	−	3.19354e6i	0.189126	−	0.857629i
$426$		0			0
$427$		−	1.51509e6i		−	0.402133i
$428$		0			0
$429$	29793.9			0.00781599
$430$		0			0
$431$	6.43823e6			1.66945			0.834725	−	0.550668i	$-0.185627\pi$
$431$	6.43823e6			1.66945			0.834725	+	0.550668i	$0.185627\pi$
$432$		0			0
$433$		−	6345.19i		−	0.00162639i	−1.00000		0.000813195i	$-0.999741\pi$
$433$		−	6345.19i		−	0.00162639i	1.00000		0.000813195i	$-0.000258848\pi$
$434$		0			0
$435$	120268.	+	13103.5i	0.0304738	+	0.00332019i
$436$		0			0
$437$		−	1.97068e6i		−	0.493644i
$438$		0			0
$439$	1.88283e6			0.466283			0.233142	−	0.972443i	$-0.425099\pi$
$439$	1.88283e6			0.466283			0.233142	+	0.972443i	$0.425099\pi$
$440$		0			0
$441$	−570487.			−0.139685
$442$		0			0
$443$			3.27757e6i			0.793493i	0.917928	+	0.396747i	$0.129861\pi$
$443$			3.27757e6i			0.793493i	−0.917928	+	0.396747i	$0.870139\pi$
$444$		0			0
$445$	369263.	−	3.38922e6i	0.0883965	−	0.811333i
$446$		0			0
$447$			637534.i			0.150916i
$448$		0			0
$449$	−3.48358e6			−0.815475			−0.407737	−	0.913099i	$-0.633682\pi$
$449$	−3.48358e6			−0.815475			−0.407737	+	0.913099i	$0.633682\pi$
$450$		0			0
$451$	−179400.			−0.0415319
$452$		0			0
$453$		−	355102.i		−	0.0813032i
$454$		0			0
$455$	−155016.	+	1.42279e6i	−0.0351034	+	0.322191i
$456$		0			0
$457$			5.46472e6i			1.22399i	0.790862	+	0.611994i	$0.209632\pi$
$457$			5.46472e6i			1.22399i	−0.790862	+	0.611994i	$0.790368\pi$
$458$		0			0
$459$	1.16831e6			0.258837
$460$		0			0
$461$	−3.88786e6			−0.852036			−0.426018	−	0.904715i	$-0.640084\pi$
$461$	−3.88786e6			−0.852036			−0.426018	+	0.904715i	$0.640084\pi$
$462$		0			0
$463$		−	2.00091e6i		−	0.433787i	−0.976195	−	0.216893i	$-0.930408\pi$
$463$		−	2.00091e6i		−	0.433787i	0.976195	−	0.216893i	$-0.0695924\pi$
$464$		0			0
$465$	−822259.	−	89586.9i	−0.176350	−	0.0192137i
$466$		0			0
$467$		−	6.93164e6i		−	1.47077i	−0.677652	−	0.735383i	$-0.737002\pi$
$467$		−	6.93164e6i		−	1.47077i	0.677652	−	0.735383i	$-0.262998\pi$
$468$		0			0
$469$	−1.93868e6			−0.406981
$470$		0			0
$471$	232438.			0.0482786
$472$		0			0
$473$		−	91554.4i		−	0.0188160i
$474$		0			0
$475$	−8.77727e6	−	1.93558e6i	−1.78495	−	0.393620i
$476$		0			0
$477$			81396.3i			0.0163798i
$478$		0			0
$479$	−7.93363e6			−1.57991			−0.789957	−	0.613163i	$-0.789897\pi$
$479$	−7.93363e6			−1.57991			−0.789957	+	0.613163i	$0.789897\pi$
$480$		0			0
$481$	8.34850e6			1.64530
$482$		0			0
$483$		−	77988.4i		−	0.0152112i
$484$		0			0
$485$	−6.26410e6	−	682487.i	−1.20922	−	0.131747i
$486$		0			0
$487$			678105.i			0.129561i	0.997900	+	0.0647806i	$0.0206347\pi$
$487$			678105.i			0.129561i	−0.997900	+	0.0647806i	$0.979365\pi$
$488$		0			0
$489$	−246401.			−0.0465984
$490$		0			0
$491$	−3.21431e6			−0.601706			−0.300853	−	0.953671i	$-0.597271\pi$
$491$	−3.21431e6			−0.601706			−0.300853	+	0.953671i	$0.597271\pi$
$492$		0			0
$493$			974950.i			0.180661i
$494$		0			0
$495$	35315.0	−	324133.i	0.00647807	−	0.0594579i
$496$		0			0
$497$			1.58851e6i			0.288470i
$498$		0			0
$499$	6119.98			0.00110027			0.000550134	−	1.00000i	$-0.499825\pi$
$499$	6119.98			0.00110027			0.000550134		1.00000i	$0.499825\pi$
$500$		0			0
$501$	−1.15631e6			−0.205817
$502$		0			0
$503$		−	6.01661e6i		−	1.06031i	−0.847901	−	0.530154i	$-0.822134\pi$
$503$		−	6.01661e6i		−	1.06031i	0.847901	−	0.530154i	$-0.177866\pi$
$504$		0			0
$505$	426286.	−	3.91260e6i	0.0743828	−	0.682711i
$506$		0			0
$507$		−	228324.i		−	0.0394487i
$508$		0			0
$509$	3.71667e6			0.635858			0.317929	−	0.948115i	$-0.397013\pi$
$509$	3.71667e6			0.635858			0.317929	+	0.948115i	$0.397013\pi$
$510$		0			0
$511$	−3.45075e6			−0.584604
$512$		0			0
$513$		−	3.21104e6i		−	0.538707i
$514$		0			0
$515$	5.97465e6	+	650951.i	0.992646	+	0.108151i
$516$		0			0
$517$		−	105949.i		−	0.0174329i
$518$		0			0
$519$	−1.20954e6			−0.197107
$520$		0			0
$521$	5.81741e6			0.938935			0.469467	−	0.882950i	$-0.344446\pi$
$521$	5.81741e6			0.938935			0.469467	+	0.882950i	$0.344446\pi$
$522$		0			0
$523$		−	6.50390e6i		−	1.03973i	−0.854249	−	0.519864i	$-0.825983\pi$
$523$		−	6.50390e6i		−	1.03973i	0.854249	−	0.519864i	$-0.174017\pi$
$524$		0			0
$525$	−347354.	−	76599.3i	−0.0550015	−	0.0121290i
$526$		0			0
$527$		−	6.66563e6i		−	1.04548i
$528$		0			0
$529$	5.96689e6			0.927062
$530$		0			0
$531$	2.56757e6			0.395172
$532$		0			0
$533$		−	3.81855e6i		−	0.582211i
$534$		0			0
$535$	−9.26389e6	−	1.00932e6i	−1.39929	−	0.152456i
$536$		0			0
$537$		−	962775.i		−	0.144075i
$538$		0			0
$539$	58938.5			0.00873830
$540$		0			0
$541$	1.13075e7			1.66102			0.830509	−	0.557005i	$-0.188050\pi$
$541$	1.13075e7			1.66102			0.830509	+	0.557005i	$0.188050\pi$
$542$		0			0
$543$			1.32507e6i			0.192859i
$544$		0			0
$545$	267275.	−	2.45314e6i	0.0385449	−	0.353778i
$546$		0			0
$547$			1.23034e6i			0.175815i	0.996129	+	0.0879076i	$0.0280180\pi$
$547$			1.23034e6i			0.175815i	−0.996129	+	0.0879076i	$0.971982\pi$
$548$		0			0
$549$	−7.34678e6			−1.04032
$550$		0			0
$551$	2.67960e6			0.376003
$552$		0			0
$553$			3.82463e6i			0.531835i
$554$		0			0
$555$	−224730.	+	2.06265e6i	−0.0309692	+	0.284245i
$556$		0			0
$557$		−	753742.i		−	0.102940i	−0.998675	−	0.0514700i	$-0.983609\pi$
$557$		−	753742.i		−	0.102940i	0.998675	−	0.0514700i	$-0.0163907\pi$
$558$		0			0
$559$	1.94875e6			0.263770
$560$		0			0
$561$	−59672.9			−0.00800516
$562$		0			0
$563$		−	9.32019e6i		−	1.23924i	−0.784904	−	0.619618i	$-0.787288\pi$
$563$		−	9.32019e6i		−	1.23924i	0.784904	−	0.619618i	$-0.212712\pi$
$564$		0			0
$565$	−4.85248e6	−	528688.i	−0.639503	−	0.0696752i
$566$		0			0
$567$			2.70208e6i			0.352972i
$568$		0			0
$569$	8.03692e6			1.04066			0.520330	−	0.853965i	$-0.325809\pi$
$569$	8.03692e6			1.04066			0.520330	+	0.853965i	$0.325809\pi$
$570$		0			0
$571$	1.45113e7			1.86259			0.931293	−	0.364270i	$-0.118681\pi$
$571$	1.45113e7			1.86259			0.931293	+	0.364270i	$0.118681\pi$
$572$		0			0
$573$		−	601840.i		−	0.0765764i
$574$		0			0
$575$	−461092.	+	2.09091e6i	−0.0581591	+	0.263734i
$576$		0			0
$577$			7.41897e6i			0.927692i	0.885916	+	0.463846i	$0.153531\pi$
$577$			7.41897e6i			0.927692i	−0.885916	+	0.463846i	$0.846469\pi$
$578$		0			0
$579$	−1.66295e6			−0.206150
$580$		0			0
$581$	−2.02430e6			−0.248791
$582$		0			0
$583$		−	8409.25i		−	0.00102467i
$584$		0			0
$585$	6.89920e6	+	751683.i	0.833507	+	0.0908124i
$586$		0			0
$587$		−	1.57362e7i		−	1.88497i	−0.334255	−	0.942483i	$-0.608485\pi$
$587$		−	1.57362e7i		−	1.88497i	0.334255	−	0.942483i	$-0.391515\pi$
$588$		0			0
$589$	−1.83202e7			−2.17591
$590$		0			0
$591$	−1.20494e6			−0.141905
$592$		0			0
$593$		−	8.95641e6i		−	1.04592i	−0.852358	−	0.522959i	$-0.824828\pi$
$593$		−	8.95641e6i		−	1.04592i	0.852358	−	0.522959i	$-0.175172\pi$
$594$		0			0
$595$	310476.	−	2.84965e6i	0.0359530	−	0.329989i
$596$		0			0
$597$		−	511420.i		−	0.0587275i
$598$		0			0
$599$	4.65323e6			0.529892			0.264946	−	0.964263i	$-0.414646\pi$
$599$	4.65323e6			0.529892			0.264946	+	0.964263i	$0.414646\pi$
$600$		0			0
$601$	−1.11044e7			−1.25403			−0.627014	−	0.779008i	$-0.715723\pi$
$601$	−1.11044e7			−1.25403			−0.627014	+	0.779008i	$0.715723\pi$
$602$		0			0
$603$			9.40078e6i			1.05286i
$604$		0			0
$605$	971480.	−	8.91657e6i	0.107906	−	0.990398i
$606$		0			0
$607$		−	2.36299e6i		−	0.260309i	−0.991494	−	0.130155i	$-0.958453\pi$
$607$		−	2.36299e6i		−	0.260309i	0.991494	−	0.130155i	$-0.0415474\pi$
$608$		0			0
$609$	106043.			0.0115862
$610$		0			0
$611$	2.25513e6			0.244381
$612$		0			0
$613$		−	1.72952e7i		−	1.85898i	−0.368849	−	0.929489i	$-0.620248\pi$
$613$		−	1.72952e7i		−	1.85898i	0.368849	−	0.929489i	$-0.379752\pi$
$614$		0			0
$615$	943444.	+	102790.i	0.100584	+	0.0109588i
$616$		0			0
$617$		−	8.21073e6i		−	0.868298i	−0.900841	−	0.434149i	$-0.857049\pi$
$617$		−	8.21073e6i		−	0.868298i	0.900841	−	0.434149i	$-0.142951\pi$
$618$		0			0
$619$	−9.00518e6			−0.944639			−0.472320	−	0.881427i	$-0.656583\pi$
$619$	−9.00518e6			−0.944639			−0.472320	+	0.881427i	$0.656583\pi$
$620$		0			0
$621$	−764929.			−0.0795962
$622$		0			0
$623$		−	2.98836e6i		−	0.308470i
$624$		0			0
$625$	8.85987e6	+	4.10733e6i	0.907250	+	0.420591i
$626$		0			0
$627$			164008.i			0.0166608i
$628$		0			0
$629$	−1.67209e7			−1.68512
$630$		0			0
$631$	8.64584e6			0.864438			0.432219	−	0.901769i	$-0.357731\pi$
$631$	8.64584e6			0.864438			0.432219	+	0.901769i	$0.357731\pi$
$632$		0			0
$633$			2.62342e6i			0.260231i
$634$		0			0
$635$	−124016.	−	13511.8i	−0.0122052	−	0.00132978i
$636$		0			0
$637$			1.25451e6i			0.122497i
$638$		0			0
$639$	7.70280e6			0.746271
$640$		0			0
$641$	9.67494e6			0.930043			0.465022	−	0.885299i	$-0.346047\pi$
$641$	9.67494e6			0.930043			0.465022	+	0.885299i	$0.346047\pi$
$642$		0			0
$643$			1.76923e7i			1.68755i	0.536699	+	0.843774i	$0.319671\pi$
$643$			1.76923e7i			1.68755i	−0.536699	+	0.843774i	$0.680329\pi$
$644$		0			0
$645$	−52457.6	+	481474.i	−0.00496489	+	0.0455694i
$646$		0			0
$647$		−	1.17040e7i		−	1.09919i	−0.835432	−	0.549594i	$-0.814782\pi$
$647$		−	1.17040e7i		−	1.09919i	0.835432	−	0.549594i	$-0.185218\pi$
$648$		0			0
$649$	−265262.			−0.0247209
$650$		0			0
$651$	−725007.			−0.0670486
$652$		0			0
$653$		−	3.02668e6i		−	0.277769i	−0.990309	−	0.138884i	$-0.955648\pi$
$653$		−	3.02668e6i		−	0.277769i	0.990309	−	0.138884i	$-0.0443517\pi$
$654$		0			0
$655$	−1.53297e6	+	1.40701e7i	−0.139615	+	1.28143i
$656$		0			0
$657$			1.67329e7i			1.51237i
$658$		0			0
$659$	−92261.4			−0.00827573			−0.00413787	−	0.999991i	$-0.501317\pi$
$659$	−92261.4			−0.00827573			−0.00413787	+	0.999991i	$0.501317\pi$
$660$		0			0
$661$	−1.41297e7			−1.25785			−0.628924	−	0.777467i	$-0.716504\pi$
$661$	−1.41297e7			−1.25785			−0.628924	+	0.777467i	$0.716504\pi$
$662$		0			0
$663$		−	1.27014e6i		−	0.112220i
$664$		0			0
$665$	−7.83212e6	−	853326.i	−0.686792	−	0.0748275i
$666$		0			0
$667$		−	638331.i		−	0.0555560i
$668$		0			0
$669$	1.21906e6			0.105308
$670$		0			0
$671$	759014.			0.0650794
$672$		0			0
$673$		−	7.96708e6i		−	0.678050i	−0.940777	−	0.339025i	$-0.889903\pi$
$673$		−	7.96708e6i		−	0.678050i	0.940777	−	0.339025i	$-0.110097\pi$
$674$		0			0
$675$	−751304.	+	3.40694e6i	−0.0634682	+	0.287809i
$676$		0			0
$677$			1.55789e7i			1.30636i	0.757201	+	0.653182i	$0.226566\pi$
$677$			1.55789e7i			1.30636i	−0.757201	+	0.653182i	$0.773434\pi$
$678$		0			0
$679$	−5.52322e6			−0.459746
$680$		0			0
$681$	−34292.7			−0.00283357
$682$		0			0
$683$		−	1.29198e7i		−	1.05975i	−0.848075	−	0.529876i	$-0.822238\pi$
$683$		−	1.29198e7i		−	1.05975i	0.848075	−	0.529876i	$-0.177762\pi$
$684$		0			0
$685$	−1.04199e7	−	1.13527e6i	−0.848471	−	0.0924427i
$686$		0			0
$687$			1.39258e6i			0.112571i
$688$		0			0
$689$	178992.			0.0143643
$690$		0			0
$691$	469320.			0.0373916			0.0186958	−	0.999825i	$-0.494049\pi$
$691$	469320.			0.0373916			0.0186958	+	0.999825i	$0.494049\pi$
$692$		0			0
$693$		−	285796.i		−	0.0226060i
$694$		0			0
$695$	942353.	−	8.64923e6i	0.0740034	−	0.679228i
$696$		0			0
$697$			7.64802e6i			0.596303i
$698$		0			0
$699$	−2.45665e6			−0.190173
$700$		0			0
$701$	−1.65314e7			−1.27062			−0.635310	−	0.772257i	$-0.719128\pi$
$701$	−1.65314e7			−1.27062			−0.635310	+	0.772257i	$0.719128\pi$
$702$		0			0
$703$			4.59565e7i			3.50718i
$704$		0			0
$705$	−60705.0	+	557171.i	−0.00459993	+	0.0422197i
$706$		0			0
$707$		−	3.44984e6i		−	0.259567i
$708$		0			0
$709$	−6.68599e6			−0.499517			−0.249758	−	0.968308i	$-0.580351\pi$
$709$	−6.68599e6			−0.499517			−0.249758	+	0.968308i	$0.580351\pi$
$710$		0			0
$711$	1.85459e7			1.37586
$712$		0			0
$713$			4.36420e6i			0.321500i
$714$		0			0
$715$	−712774.	−	77658.3i	−0.0521419	−	0.00568097i
$716$		0			0
$717$		−	1.81082e6i		−	0.131546i
$718$		0			0
$719$	2.43419e7			1.75603			0.878015	−	0.478633i	$-0.158868\pi$
$719$	2.43419e7			1.75603			0.878015	+	0.478633i	$0.158868\pi$
$720$		0			0
$721$	5.26800e6			0.377405
$722$		0			0
$723$		−	202591.i		−	0.0144137i
$724$		0			0
$725$	−2.84308e6	−	626961.i	−0.200883	−	0.0442992i
$726$		0			0
$727$		−	1.93114e7i		−	1.35512i	−0.735469	−	0.677558i	$-0.763038\pi$
$727$		−	1.93114e7i		−	1.35512i	0.735469	−	0.677558i	$-0.236962\pi$
$728$		0			0
$729$	1.24723e7			0.869219
$730$		0			0
$731$	−3.90306e6			−0.270154
$732$		0			0
$733$			1.95802e7i			1.34604i	0.739625	+	0.673019i	$0.235003\pi$
$733$			1.95802e7i			1.34604i	−0.739625	+	0.673019i	$0.764997\pi$
$734$		0			0
$735$	−309950.	−	33769.8i	−0.0211628	−	0.00230574i
$736$		0			0
$737$		−	971219.i		−	0.0658641i
$738$		0			0
$739$	2.20510e7			1.48531			0.742656	−	0.669673i	$-0.233566\pi$
$739$	2.20510e7			1.48531			0.742656	+	0.669673i	$0.233566\pi$
$740$		0			0
$741$	−3.49093e6			−0.233558
$742$		0			0
$743$			7.70392e6i			0.511965i	0.966681	+	0.255982i	$0.0823989\pi$
$743$			7.70392e6i			0.511965i	−0.966681	+	0.255982i	$0.917601\pi$
$744$		0			0
$745$	1.66174e6	−	1.52520e7i	0.109692	−	1.00679i
$746$		0			0
$747$			9.81594e6i			0.643622i
$748$		0			0
$749$	−8.16821e6			−0.532013
$750$		0			0
$751$	−2.64964e7			−1.71430			−0.857150	−	0.515067i	$-0.827767\pi$
$751$	−2.64964e7			−1.71430			−0.857150	+	0.515067i	$0.827767\pi$
$752$		0			0
$753$		−	2.84175e6i		−	0.182641i
$754$		0			0
$755$	−925580.	+	8.49529e6i	−0.0590945	+	0.542389i
$756$		0			0
$757$			1.50006e7i			0.951415i	0.879604	+	0.475707i	$0.157808\pi$
$757$			1.50006e7i			0.951415i	−0.879604	+	0.475707i	$0.842192\pi$
$758$		0			0
$759$	39069.8			0.00246171
$760$		0			0
$761$	2.85896e7			1.78956			0.894780	−	0.446507i	$-0.147332\pi$
$761$	2.85896e7			1.78956			0.894780	+	0.446507i	$0.147332\pi$
$762$		0			0
$763$		−	2.16300e6i		−	0.134507i
$764$		0			0
$765$	−1.38181e7	−	1.50551e6i	−0.853681	−	0.0930104i
$766$		0			0
$767$		−	5.64614e6i		−	0.346548i
$768$		0			0
$769$	1.99376e7			1.21579			0.607893	−	0.794019i	$-0.292015\pi$
$769$	1.99376e7			1.21579			0.607893	+	0.794019i	$0.292015\pi$
$770$		0			0
$771$	2.75978e6			0.167201
$772$		0			0
$773$		−	3.26440e6i		−	0.196497i	−0.995162	−	0.0982483i	$-0.968676\pi$
$773$		−	3.26440e6i		−	0.196497i	0.995162	−	0.0982483i	$-0.0313239\pi$
$774$		0			0
$775$	1.94378e7	+	4.28647e6i	1.16250	+	0.256357i
$776$		0			0
$777$			1.81869e6i			0.108070i
$778$		0			0
$779$	2.10202e7			1.24106
$780$		0			0
$781$	−795796.			−0.0466846
$782$		0			0
$783$		−	1.04010e6i		−	0.0606276i
$784$		0			0
$785$	−5.56073e6	−	605854.i	−0.322076	−	0.0350908i
$786$		0			0
$787$			8.35276e6i			0.480721i	0.970684	+	0.240361i	$0.0772657\pi$
$787$			8.35276e6i			0.480721i	−0.970684	+	0.240361i	$0.922734\pi$
$788$		0			0
$789$	97483.3			0.00557491
$790$		0			0
$791$	−4.27855e6			−0.243140
$792$		0			0
$793$			1.61557e7i			0.912311i
$794$		0			0
$795$	−4818.22	+	44223.3i	−0.000270377	+	0.00248161i
$796$		0			0
$797$			2.72699e7i			1.52068i	0.649525	+	0.760340i	$0.274968\pi$
$797$			2.72699e7i			1.52068i	−0.649525	+	0.760340i	$0.725032\pi$
$798$		0			0
$799$	−4.51670e6			−0.250296
$800$		0			0
$801$	−1.44907e7			−0.798012
$802$		0			0
$803$		−	1.72872e6i		−	0.0946097i
$804$		0			0
$805$	−203278.	+	1.86576e6i	−0.0110561	+	0.101476i
$806$		0			0
$807$			4.33458e6i			0.234295i
$808$		0			0
$809$	−2.15748e6			−0.115898			−0.0579489	−	0.998320i	$-0.518456\pi$
$809$	−2.15748e6			−0.115898			−0.0579489	+	0.998320i	$0.518456\pi$
$810$		0			0
$811$	−2.01132e7			−1.07381			−0.536906	−	0.843642i	$-0.680407\pi$
$811$	−2.01132e7			−1.07381			−0.536906	+	0.843642i	$0.680407\pi$
$812$		0			0
$813$		−	5.56537e6i		−	0.295303i
$814$		0			0
$815$	5.89478e6	+	642249.i	0.310866	+	0.0338696i
$816$		0			0
$817$			1.07274e7i			0.562261i
$818$		0			0
$819$	6.08321e6			0.316900
$820$		0			0
$821$	−1.08889e7			−0.563803			−0.281902	−	0.959443i	$-0.590965\pi$
$821$	−1.08889e7			−0.563803			−0.281902	+	0.959443i	$0.590965\pi$
$822$		0			0
$823$		−	1.34320e7i		−	0.691261i	−0.938371	−	0.345630i	$-0.887665\pi$
$823$		−	1.34320e7i		−	0.691261i	0.938371	−	0.345630i	$-0.112335\pi$
$824$		0			0
$825$	38373.8	−	174014.i	0.00196291	−	0.00890120i
$826$		0			0
$827$			1.57603e7i			0.801308i	0.916229	+	0.400654i	$0.131217\pi$
$827$			1.57603e7i			0.801308i	−0.916229	+	0.400654i	$0.868783\pi$
$828$		0			0
$829$	−7.22449e6			−0.365108			−0.182554	−	0.983196i	$-0.558436\pi$
$829$	−7.22449e6			−0.365108			−0.182554	+	0.983196i	$0.558436\pi$
$830$		0			0
$831$	−367394.			−0.0184556
$832$		0			0
$833$		−	2.51261e6i		−	0.125462i
$834$		0			0
$835$	2.76631e7	+	3.01395e6i	1.37304	+	0.149596i
$836$		0			0
$837$			7.11104e6i			0.350849i
$838$		0			0
$839$	−1.70208e7			−0.834785			−0.417392	−	0.908726i	$-0.637056\pi$
$839$	−1.70208e7			−0.834785			−0.417392	+	0.908726i	$0.637056\pi$
$840$		0			0
$841$	−1.96432e7			−0.957684
$842$		0			0
$843$		−	4.91179e6i		−	0.238051i
$844$		0			0
$845$	−595132.	+	5.46232e6i	−0.0286729	+	0.263169i
$846$		0			0
$847$		−	7.86197e6i		−	0.376550i
$848$		0			0
$849$	−5.25164e6			−0.250049
$850$		0			0
$851$	1.09477e7			0.518201
$852$		0			0
$853$			4.57133e6i			0.215115i	0.994199	+	0.107557i	$0.0343029\pi$
$853$			4.57133e6i			0.215115i	−0.994199	+	0.107557i	$0.965697\pi$
$854$		0			0
$855$	−4.13783e6	+	3.79784e7i	−0.193579	+	1.77673i
$856$		0			0
$857$		−	1.85107e7i		−	0.860937i	−0.902606	−	0.430468i	$-0.858348\pi$
$857$		−	1.85107e7i		−	0.860937i	0.902606	−	0.430468i	$-0.141652\pi$
$858$		0			0
$859$	−1.38323e7			−0.639606			−0.319803	−	0.947484i	$-0.603617\pi$
$859$	−1.38323e7			−0.639606			−0.319803	+	0.947484i	$0.603617\pi$
$860$		0			0
$861$	831859.			0.0382421
$862$		0			0
$863$		−	1.98922e7i		−	0.909193i	−0.890698	−	0.454596i	$-0.849783\pi$
$863$		−	1.98922e7i		−	0.909193i	0.890698	−	0.454596i	$-0.150217\pi$
$864$		0			0
$865$	2.89364e7	+	3.15269e6i	1.31494	+	0.143265i
$866$		0			0
$867$		−	754321.i		−	0.0340807i
$868$		0			0
$869$	−1.91602e6			−0.0860698
$870$		0			0
$871$	2.06725e7			0.923311
$872$		0			0
$873$			2.67824e7i			1.18936i
$874$		0			0
$875$	8.11028e6	+	2.73791e6i	0.358109	+	0.120892i
$876$		0			0
$877$			2.90439e7i			1.27514i	0.770394	+	0.637568i	$0.220059\pi$
$877$			2.90439e7i			1.27514i	−0.770394	+	0.637568i	$0.779941\pi$
$878$		0			0
$879$	82900.0			0.00361895
$880$		0			0
$881$	−2.01498e7			−0.874642			−0.437321	−	0.899306i	$-0.644073\pi$
$881$	−2.01498e7			−0.874642			−0.437321	+	0.899306i	$0.644073\pi$
$882$		0			0
$883$			1.46721e7i			0.633270i	0.948547	+	0.316635i	$0.102553\pi$
$883$			1.46721e7i			0.633270i	−0.948547	+	0.316635i	$0.897447\pi$
$884$		0			0
$885$	1.39498e6	+	151986.i	0.0598702	+	0.00652299i
$886$		0			0
$887$		−	3.51096e7i		−	1.49836i	−0.662365	−	0.749181i	$-0.730447\pi$
$887$		−	3.51096e7i		−	1.49836i	0.662365	−	0.749181i	$-0.269553\pi$
$888$		0			0
$889$	−109348.			−0.00464042
$890$		0			0
$891$	−1.35366e6			−0.0571234
$892$		0			0
$893$			1.24139e7i			0.520931i
$894$		0			0
$895$	−2.50949e6	+	2.30330e7i	−0.104720	+	0.961152i
$896$		0			0
$897$			831604.i			0.0345093i
$898$		0			0
$899$	−5.93414e6			−0.244883
$900$		0			0
$901$	−358495.			−0.0147120
$902$		0			0
$903$			424528.i			0.0173256i
$904$		0			0
$905$	3.45382e6	−	3.17004e7i	0.140178	−	1.28660i
$906$		0			0
$907$		−	6.47843e6i		−	0.261488i	−0.991416	−	0.130744i	$-0.958263\pi$
$907$		−	6.47843e6i		−	0.261488i	0.991416	−	0.130744i	$-0.0417366\pi$
$908$		0			0
$909$	−1.67285e7			−0.671501
$910$		0			0
$911$	−5.94759e6			−0.237435			−0.118718	−	0.992928i	$-0.537878\pi$
$911$	−5.94759e6			−0.237435			−0.118718	+	0.992928i	$0.537878\pi$
$912$		0			0
$913$		−	1.01411e6i		−	0.0402632i
$914$		0			0
$915$	−3.99157e6	−	434890.i	−0.157613	−	0.0171722i
$916$		0			0
$917$			1.24060e7i			0.487202i
$918$		0			0
$919$	−7.15761e6			−0.279563			−0.139781	−	0.990182i	$-0.544640\pi$
$919$	−7.15761e6			−0.279563			−0.139781	+	0.990182i	$0.544640\pi$
$920$		0			0
$921$	3.87901e6			0.150686
$922$		0			0
$923$		−	1.69386e7i		−	0.654445i
$924$		0			0
$925$	1.07527e7	−	4.87601e7i	0.413202	−	1.87374i
$926$		0			0
$927$		−	2.55449e7i		−	0.976347i
$928$		0			0
$929$	−1.97090e7			−0.749246			−0.374623	−	0.927177i	$-0.622228\pi$
$929$	−1.97090e7			−0.749246			−0.374623	+	0.927177i	$0.622228\pi$
$930$		0			0
$931$	−6.90578e6			−0.261119
$932$		0			0
$933$			4.17170e6i			0.156895i
$934$		0			0
$935$	1.42759e6	+	155539.i	0.0534039	+	0.00581847i
$936$		0			0
$937$		−	1.75240e7i		−	0.652055i	−0.945360	−	0.326028i	$-0.894290\pi$
$937$		−	1.75240e7i		−	0.652055i	0.945360	−	0.326028i	$-0.105710\pi$
$938$		0			0
$939$	5.27044e6			0.195067
$940$		0			0
$941$	−2.42171e7			−0.891556			−0.445778	−	0.895144i	$-0.647073\pi$
$941$	−2.42171e7			−0.891556			−0.445778	+	0.895144i	$0.647073\pi$
$942$		0			0
$943$		−	5.00740e6i		−	0.183372i
$944$		0			0
$945$	−331222.	+	3.04007e6i	−0.0120654	+	0.110740i
$946$		0			0
$947$		−	4.71428e7i		−	1.70821i	−0.520105	−	0.854103i	$-0.674107\pi$
$947$		−	4.71428e7i		−	1.70821i	0.520105	−	0.854103i	$-0.325893\pi$
$948$		0			0
$949$	3.67960e7			1.32628
$950$		0			0
$951$	3.14175e6			0.112647
$952$		0			0
$953$		−	5.54215e7i		−	1.97672i	−0.152121	−	0.988362i	$-0.548610\pi$
$953$		−	5.54215e7i		−	1.97672i	0.152121	−	0.988362i	$-0.451390\pi$
$954$		0			0
$955$	−1.56871e6	+	1.43981e7i	−0.0556588	+	0.510855i
$956$		0			0
$957$			53124.4i			0.00187506i
$958$		0			0
$959$	−9.18749e6			−0.322590
$960$		0			0
$961$	1.19420e7			0.417126
$962$		0			0
$963$			3.96081e7i			1.37632i
$964$		0			0
$965$	3.97836e7	+	4.33451e6i	1.37526	+	0.149838i
$966$		0			0
$967$			1.97491e7i			0.679175i	0.940574	+	0.339588i	$0.110288\pi$
$967$			1.97491e7i			0.679175i	−0.940574	+	0.339588i	$0.889712\pi$
$968$		0			0
$969$	6.99184e6			0.239211
$970$		0			0
$971$	−4.19853e7			−1.42906			−0.714528	−	0.699607i	$-0.753358\pi$
$971$	−4.19853e7			−1.42906			−0.714528	+	0.699607i	$0.753358\pi$
$972$		0			0
$973$		−	7.62625e6i		−	0.258243i
$974$		0			0
$975$	3.70390e6	+	816792.i	0.124781	+	0.0275169i
$976$		0			0
$977$		−	4.38468e7i		−	1.46961i	−0.678280	−	0.734804i	$-0.737274\pi$
$977$		−	4.38468e7i		−	1.46961i	0.678280	−	0.734804i	$-0.262726\pi$
$978$		0			0
$979$	1.49708e6			0.0499214
$980$		0			0
$981$	−1.04885e7			−0.347970
$982$		0			0
$983$			6.38237e6i			0.210668i	0.994437	+	0.105334i	$0.0335912\pi$
$983$			6.38237e6i			0.210668i	−0.994437	+	0.105334i	$0.966409\pi$
$984$		0			0
$985$	2.88265e7	+	3.14071e6i	0.946676	+	0.103142i
$986$		0			0
$987$			491272.i			0.0160520i
$988$		0			0
$989$	2.55546e6			0.0830765
$990$		0			0
$991$	−6.01041e7			−1.94411			−0.972053	−	0.234761i	$-0.924569\pi$
$991$	−6.01041e7			−1.94411			−0.972053	+	0.234761i	$0.924569\pi$
$992$		0			0
$993$		−	5.16907e6i		−	0.166356i
$994$		0			0
$995$	−1.33303e6	+	1.22350e7i	−0.0426855	+	0.391782i
$996$		0			0
$997$			2.82105e7i			0.898821i	0.893325	+	0.449410i	$0.148366\pi$
$997$			2.82105e7i			0.898821i	−0.893325	+	0.449410i	$0.851634\pi$
$998$		0			0
$999$	1.78382e7			0.565506

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.g.b.169.12	✓	24
4.3	odd	2		560.6.g.h.449.13		24
5.4	even	2	inner	280.6.g.b.169.13	yes	24
20.19	odd	2		560.6.g.h.449.12		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.g.b.169.12	✓	24	1.1	even	1	trivial
280.6.g.b.169.13	yes	24	5.4	even	2	inner
560.6.g.h.449.12		24	20.19	odd	2
560.6.g.h.449.13		24	4.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	280.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.32294i q^{3} +(-6.05478 + 55.5728i) q^{5} +49.0000i q^{7} +237.604 q^{9} +O(q^{10})\)	\(q-2.32294i q^{3} +(-6.05478 + 55.5728i) q^{5} +49.0000i q^{7} +237.604 q^{9} -24.5475 q^{11} -522.496i q^{13} +(129.092 + 14.0649i) q^{15} +1046.48i q^{17} +2876.21 q^{19} +113.824 q^{21} -685.167i q^{23} +(-3051.68 - 672.963i) q^{25} -1116.41i q^{27} +931.643 q^{29} -6369.55 q^{31} +57.0223i q^{33} +(-2723.07 - 296.684i) q^{35} +15978.1i q^{37} -1213.73 q^{39} +7308.29 q^{41} +3729.69i q^{43} +(-1438.64 + 13204.3i) q^{45} +4316.07i q^{47} -2401.00 q^{49} +2430.92 q^{51} +342.571i q^{53} +(148.630 - 1364.17i) q^{55} -6681.26i q^{57} +10806.1 q^{59} -30920.3 q^{61} +11642.6i q^{63} +(29036.6 + 3163.60i) q^{65} +39564.9i q^{67} -1591.60 q^{69} +32418.6 q^{71} +70423.5i q^{73} +(-1563.25 + 7088.86i) q^{75} -1202.83i q^{77} +78053.7 q^{79} +55144.4 q^{81} +41312.2i q^{83} +(-58156.1 - 6336.24i) q^{85} -2164.15i q^{87} -60986.9 q^{89} +25602.3 q^{91} +14796.1i q^{93} +(-17414.8 + 159839. i) q^{95} +112719. i q^{97} -5832.58 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 30 q^{5} - 2106 q^{9}+O(q^{10}) \) 24 * q - 30 * q^5 - 2106 * q^9	\( 24 q - 30 q^{5} - 2106 q^{9} + 630 q^{11} + 1302 q^{15} - 3000 q^{19} - 882 q^{21} + 3400 q^{25} - 10210 q^{29} + 13372 q^{31} + 2058 q^{35} - 62478 q^{39} + 4216 q^{41} + 58642 q^{45} - 57624 q^{49} + 112914 q^{51} - 19492 q^{55} - 22916 q^{59} - 11268 q^{61} + 89258 q^{65} + 50276 q^{69} + 134240 q^{71} - 98520 q^{75} - 78834 q^{79} + 85504 q^{81} + 5414 q^{85} + 54988 q^{89} + 37730 q^{91} - 242844 q^{95} + 112988 q^{99}+O(q^{100}) \) 24 * q - 30 * q^5 - 2106 * q^9 + 630 * q^11 + 1302 * q^15 - 3000 * q^19 - 882 * q^21 + 3400 * q^25 - 10210 * q^29 + 13372 * q^31 + 2058 * q^35 - 62478 * q^39 + 4216 * q^41 + 58642 * q^45 - 57624 * q^49 + 112914 * q^51 - 19492 * q^55 - 22916 * q^59 - 11268 * q^61 + 89258 * q^65 + 50276 * q^69 + 134240 * q^71 - 98520 * q^75 - 78834 * q^79 + 85504 * q^81 + 5414 * q^85 + 54988 * q^89 + 37730 * q^91 - 242844 * q^95 + 112988 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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