LMFDB - Embedded newform 560.3.p.g.209.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [560,3,Mod(209,560)] 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("560.209"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	560.p (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$15.2588948042$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.211319484596224.6
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 2x^{6} - 48x^{5} - 23x^{4} - 48x^{3} + 1226x^{2} - 7512x + 24408 $ x^8 + 2x^6 - 48x^5 - 23x^4 - 48x^3 + 1226x^2 - 7512x + 24408
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.1
Root			$-3.29232 - 1.99575i$ of defining polynomial
Character	$\chi$	$=$	560.209
Dual form			560.3.p.g.209.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.76222 q^{3} +(-1.82242 - 4.65605i) q^{5} +(-1.46990 - 6.84393i) q^{7} +13.6788 q^{9} +16.6788 q^{11} +14.9917 q^{13} +(8.67878 + 22.1731i) q^{15} -11.3469 q^{17} -3.16043i q^{19} +(7.00000 + 32.5923i) q^{21} -22.1731i q^{23} +(-18.3576 + 16.9706i) q^{25} -22.2814 q^{27} +16.0363 q^{29} -21.7846i q^{31} -79.4281 q^{33} +(-29.1869 + 19.3165i) q^{35} -35.8610i q^{37} -71.3939 q^{39} -74.6660i q^{41} +11.7680i q^{43} +(-24.9285 - 63.6891i) q^{45} +11.2272 q^{47} +(-44.6788 + 20.1198i) q^{49} +54.0363 q^{51} +20.8103i q^{53} +(-30.3958 - 77.6572i) q^{55} +15.0507i q^{57} +93.1210i q^{59} -43.4000i q^{61} +(-20.1065 - 93.6166i) q^{63} +(-27.3212 - 69.8021i) q^{65} +13.6879i q^{67} +105.593i q^{69} -81.3576 q^{71} +40.0933 q^{73} +(87.4228 - 80.8176i) q^{75} +(-24.5162 - 114.148i) q^{77} -40.6061 q^{79} -17.0000 q^{81} -42.4477 q^{83} +(20.6788 + 52.8316i) q^{85} -76.3686 q^{87} +52.7121i q^{89} +(-22.0363 - 102.602i) q^{91} +103.743i q^{93} +(-14.7151 + 5.75964i) q^{95} -116.701 q^{97} +228.145 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 36 q^{9} + 60 q^{11} - 4 q^{15} + 56 q^{21} - 92 q^{29} - 52 q^{35} - 204 q^{39} - 284 q^{49} + 212 q^{51} - 292 q^{65} - 504 q^{71} - 692 q^{79} - 136 q^{81} + 92 q^{85} + 44 q^{91} + 176 q^{95}+ \cdots + 944 q^{99}+O(q^{100}) $ 8 * q + 36 * q^9 + 60 * q^11 - 4 * q^15 + 56 * q^21 - 92 * q^29 - 52 * q^35 - 204 * q^39 - 284 * q^49 + 212 * q^51 - 292 * q^65 - 504 * q^71 - 692 * q^79 - 136 * q^81 + 92 * q^85 + 44 * q^91 + 176 * q^95 + 944 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−4.76222			−1.58741			−0.793704	−	0.608304i	$-0.791850\pi$
$3$	−4.76222			−1.58741			−0.793704	+	0.608304i	$0.791850\pi$
$4$		0			0
$5$	−1.82242	−	4.65605i	−0.364484	−	0.931210i
$5$	−1.82242	−	4.65605i	−0.364484	−	0.931210i
$6$		0			0
$7$	−1.46990	−	6.84393i	−0.209986	−	0.977704i
$7$	−1.46990	−	6.84393i	−0.209986	−	0.977704i
$8$		0			0
$9$	13.6788			1.51986
$10$		0			0
$11$	16.6788			1.51625			0.758126	−	0.652108i	$-0.226115\pi$
$11$	16.6788			1.51625			0.758126	+	0.652108i	$0.226115\pi$
$12$		0			0
$13$	14.9917			1.15321			0.576604	−	0.817024i	$-0.304377\pi$
$13$	14.9917			1.15321			0.576604	+	0.817024i	$0.304377\pi$
$14$		0			0
$15$	8.67878	+	22.1731i	0.578585	+	1.47821i
$16$		0			0
$17$	−11.3469			−0.667463			−0.333731	−	0.942668i	$-0.608308\pi$
$17$	−11.3469			−0.667463			−0.333731	+	0.942668i	$0.608308\pi$
$18$		0			0
$19$		−	3.16043i		−	0.166338i	−0.996535	−	0.0831692i	$-0.973496\pi$
$19$		−	3.16043i		−	0.166338i	0.996535	−	0.0831692i	$-0.0265042\pi$
$20$		0			0
$21$	7.00000	+	32.5923i	0.333333	+	1.55202i
$22$		0			0
$23$		−	22.1731i		−	0.964050i	−0.876158	−	0.482025i	$-0.839901\pi$
$23$		−	22.1731i		−	0.964050i	0.876158	−	0.482025i	$-0.160099\pi$
$24$		0			0
$25$	−18.3576	+	16.9706i	−0.734302	+	0.678823i
$26$		0			0
$27$	−22.2814			−0.825237
$28$		0			0
$29$	16.0363			0.552977			0.276489	−	0.961017i	$-0.410829\pi$
$29$	16.0363			0.552977			0.276489	+	0.961017i	$0.410829\pi$
$30$		0			0
$31$		−	21.7846i		−	0.702730i	−0.936239	−	0.351365i	$-0.885718\pi$
$31$		−	21.7846i		−	0.702730i	0.936239	−	0.351365i	$-0.114282\pi$
$32$		0			0
$33$	−79.4281			−2.40691
$34$		0			0
$35$	−29.1869	+	19.3165i	−0.833911	+	0.551899i
$36$		0			0
$37$		−	35.8610i		−	0.969216i	−0.874731	−	0.484608i	$-0.838962\pi$
$37$		−	35.8610i		−	0.969216i	0.874731	−	0.484608i	$-0.161038\pi$
$38$		0			0
$39$	−71.3939			−1.83061
$40$		0			0
$41$		−	74.6660i		−	1.82112i	−0.413376	−	0.910561i	$-0.635650\pi$
$41$		−	74.6660i		−	1.82112i	0.413376	−	0.910561i	$-0.364350\pi$
$42$		0			0
$43$			11.7680i			0.273674i	0.990594	+	0.136837i	$0.0436937\pi$
$43$			11.7680i			0.273674i	−0.990594	+	0.136837i	$0.956306\pi$
$44$		0			0
$45$	−24.9285	−	63.6891i	−0.553967	−	1.41531i
$46$		0			0
$47$	11.2272			0.238877			0.119439	−	0.992842i	$-0.461891\pi$
$47$	11.2272			0.238877			0.119439	+	0.992842i	$0.461891\pi$
$48$		0			0
$49$	−44.6788	+	20.1198i	−0.911812	+	0.410608i
$50$		0			0
$51$	54.0363			1.05954
$52$		0			0
$53$			20.8103i			0.392648i	0.980539	+	0.196324i	$0.0629004\pi$
$53$			20.8103i			0.392648i	−0.980539	+	0.196324i	$0.937100\pi$
$54$		0			0
$55$	−30.3958	−	77.6572i	−0.552650	−	1.41195i
$56$		0			0
$57$			15.0507i			0.264047i
$58$		0			0
$59$			93.1210i			1.57832i	0.614187	+	0.789161i	$0.289484\pi$
$59$			93.1210i			1.57832i	−0.614187	+	0.789161i	$0.710516\pi$
$60$		0			0
$61$		−	43.4000i		−	0.711476i	−0.934586	−	0.355738i	$-0.884230\pi$
$61$		−	43.4000i		−	0.711476i	0.934586	−	0.355738i	$-0.115770\pi$
$62$		0			0
$63$	−20.1065	−	93.6166i	−0.319150	−	1.48598i
$64$		0			0
$65$	−27.3212	−	69.8021i	−0.420326	−	1.07388i
$66$		0			0
$67$			13.6879i			0.204296i	0.994769	+	0.102148i	$0.0325716\pi$
$67$			13.6879i			0.204296i	−0.994769	+	0.102148i	$0.967428\pi$
$68$		0			0
$69$			105.593i			1.53034i
$70$		0			0
$71$	−81.3576			−1.14588			−0.572941	−	0.819597i	$-0.694197\pi$
$71$	−81.3576			−1.14588			−0.572941	+	0.819597i	$0.694197\pi$
$72$		0			0
$73$	40.0933			0.549223			0.274611	−	0.961555i	$-0.411451\pi$
$73$	40.0933			0.549223			0.274611	+	0.961555i	$0.411451\pi$
$74$		0			0
$75$	87.4228	−	80.8176i	1.16564	−	1.07757i
$76$		0			0
$77$	−24.5162	−	114.148i	−0.318392	−	1.48245i
$78$		0			0
$79$	−40.6061			−0.514001			−0.257001	−	0.966411i	$-0.582734\pi$
$79$	−40.6061			−0.514001			−0.257001	+	0.966411i	$0.582734\pi$
$80$		0			0
$81$	−17.0000			−0.209877
$82$		0			0
$83$	−42.4477			−0.511418			−0.255709	−	0.966754i	$-0.582309\pi$
$83$	−42.4477			−0.511418			−0.255709	+	0.966754i	$0.582309\pi$
$84$		0			0
$85$	20.6788	+	52.8316i	0.243280	+	0.621548i
$86$		0			0
$87$	−76.3686			−0.877801
$88$		0			0
$89$			52.7121i			0.592271i	0.955146	+	0.296136i	$0.0956980\pi$
$89$			52.7121i			0.592271i	−0.955146	+	0.296136i	$0.904302\pi$
$90$		0			0
$91$	−22.0363	−	102.602i	−0.242158	−	1.12750i
$92$		0			0
$93$			103.743i			1.11552i
$94$		0			0
$95$	−14.7151	+	5.75964i	−0.154896	+	0.0606278i
$96$		0			0
$97$	−116.701			−1.20311			−0.601553	−	0.798833i	$-0.705451\pi$
$97$	−116.701			−1.20311			−0.601553	+	0.798833i	$0.705451\pi$
$98$		0			0
$99$	228.145			2.30450
$100$		0			0
$101$		−	9.14290i		−	0.0905237i	−0.998975	−	0.0452619i	$-0.985588\pi$
$101$		−	9.14290i		−	0.0905237i	0.998975	−	0.0452619i	$-0.0144122\pi$
$102$		0			0
$103$	−41.4499			−0.402427			−0.201213	−	0.979547i	$-0.564488\pi$
$103$	−41.4499			−0.402427			−0.201213	+	0.979547i	$0.564488\pi$
$104$		0			0
$105$	138.995	−	91.9893i	1.32376	−	0.876089i
$106$		0			0
$107$		−	123.191i		−	1.15132i	−0.817691	−	0.575658i	$-0.804746\pi$
$107$		−	123.191i		−	1.15132i	0.817691	−	0.575658i	$-0.195254\pi$
$108$		0			0
$109$	−134.182			−1.23102			−0.615512	−	0.788127i	$-0.711051\pi$
$109$	−134.182			−1.23102			−0.615512	+	0.788127i	$0.711051\pi$
$110$		0			0
$111$			170.778i			1.53854i
$112$		0			0
$113$			73.6419i			0.651698i	0.945422	+	0.325849i	$0.105650\pi$
$113$			73.6419i			0.651698i	−0.945422	+	0.325849i	$0.894350\pi$
$114$		0			0
$115$	−103.239	+	40.4088i	−0.897732	+	0.351381i
$116$		0			0
$117$	205.068			1.75272
$118$		0			0
$119$	16.6788	+	77.6572i	0.140158	+	0.652581i
$120$		0			0
$121$	157.182			1.29902
$122$		0			0
$123$			355.576i			2.89086i
$124$		0			0
$125$	112.471	+	54.5462i	0.899768	+	0.436369i
$126$		0			0
$127$		−	51.4687i		−	0.405266i	−0.979255	−	0.202633i	$-0.935050\pi$
$127$		−	51.4687i		−	0.405266i	0.979255	−	0.202633i	$-0.0649498\pi$
$128$		0			0
$129$		−	56.0418i		−	0.434432i
$130$		0			0
$131$		−	170.440i		−	1.30107i	−0.759478	−	0.650533i	$-0.774545\pi$
$131$		−	170.440i		−	1.30107i	0.759478	−	0.650533i	$-0.225455\pi$
$132$		0			0
$133$	−21.6298	+	4.64552i	−0.162630	+	0.0349287i
$134$		0			0
$135$	40.6061	+	103.743i	0.300786	+	0.768469i
$136$		0			0
$137$			99.9035i			0.729223i	0.931160	+	0.364611i	$0.118798\pi$
$137$			99.9035i			0.729223i	−0.931160	+	0.364611i	$0.881202\pi$
$138$		0			0
$139$			204.866i			1.47386i	0.675971	+	0.736928i	$0.263725\pi$
$139$			204.866i			1.47386i	−0.675971	+	0.736928i	$0.736275\pi$
$140$		0			0
$141$	−53.4666			−0.379196
$142$		0			0
$143$	250.043			1.74856
$144$		0			0
$145$	−29.2250	−	74.6660i	−0.201552	−	0.514938i
$146$		0			0
$147$	212.770	−	95.8150i	1.44742	−	0.651803i
$148$		0			0
$149$	54.6424			0.366728			0.183364	−	0.983045i	$-0.441301\pi$
$149$	54.6424			0.366728			0.183364	+	0.983045i	$0.441301\pi$
$150$		0			0
$151$	−27.4666			−0.181898			−0.0909489	−	0.995856i	$-0.528990\pi$
$151$	−27.4666			−0.181898			−0.0909489	+	0.995856i	$0.528990\pi$
$152$		0			0
$153$	−155.211			−1.01445
$154$		0			0
$155$	−101.430	+	39.7008i	−0.654389	+	0.256134i
$156$		0			0
$157$	163.326			1.04029			0.520146	−	0.854078i	$-0.325878\pi$
$157$	163.326			1.04029			0.520146	+	0.854078i	$0.325878\pi$
$158$		0			0
$159$		−	99.1034i		−	0.623292i
$160$		0			0
$161$	−151.751	+	32.5923i	−0.942556	+	0.202437i
$162$		0			0
$163$		−	213.495i		−	1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$163$		−	213.495i		−	1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$164$		0			0
$165$	144.751	+	369.821i	0.877282	+	2.24134i
$166$		0			0
$167$	270.861			1.62192			0.810962	−	0.585099i	$-0.198944\pi$
$167$	270.861			1.62192			0.810962	+	0.585099i	$0.198944\pi$
$168$		0			0
$169$	55.7515			0.329890
$170$		0			0
$171$		−	43.2308i		−	0.252812i
$172$		0			0
$173$	190.423			1.10071			0.550355	−	0.834931i	$-0.314492\pi$
$173$	190.423			1.10071			0.550355	+	0.834931i	$0.314492\pi$
$174$		0			0
$175$	143.129	+	100.693i	0.817881	+	0.575388i
$176$		0			0
$177$		−	443.463i		−	2.50544i
$178$		0			0
$179$	−96.0727			−0.536719			−0.268359	−	0.963319i	$-0.586481\pi$
$179$	−96.0727			−0.536719			−0.268359	+	0.963319i	$0.586481\pi$
$180$		0			0
$181$			248.774i			1.37444i	0.726449	+	0.687220i	$0.241169\pi$
$181$			248.774i			1.37444i	−0.726449	+	0.687220i	$0.758831\pi$
$182$		0			0
$183$			206.681i			1.12940i
$184$		0			0
$185$	−166.971	+	65.3539i	−0.902544	+	0.353264i
$186$		0			0
$187$	−189.252			−1.01204
$188$		0			0
$189$	32.7515	+	152.492i	0.173288	+	0.806838i
$190$		0			0
$191$	234.254			1.22646			0.613231	−	0.789903i	$-0.289869\pi$
$191$	234.254			1.22646			0.613231	+	0.789903i	$0.289869\pi$
$192$		0			0
$193$		−	180.917i		−	0.937391i	−0.883360	−	0.468696i	$-0.844724\pi$
$193$		−	180.917i		−	0.937391i	0.883360	−	0.468696i	$-0.155276\pi$
$194$		0			0
$195$	130.110	+	332.413i	0.667230	+	1.70468i
$196$		0			0
$197$		−	158.495i		−	0.804542i	−0.915521	−	0.402271i	$-0.868221\pi$
$197$		−	158.495i		−	0.804542i	0.915521	−	0.402271i	$-0.131779\pi$
$198$		0			0
$199$		−	223.490i		−	1.12307i	−0.827454	−	0.561533i	$-0.810212\pi$
$199$		−	223.490i		−	1.12307i	0.827454	−	0.561533i	$-0.189788\pi$
$200$		0			0
$201$		−	65.1847i		−	0.324302i
$202$		0			0
$203$	−23.5718	−	109.752i	−0.116117	−	0.540648i
$204$		0			0
$205$	−347.648	+	136.073i	−1.69585	+	0.663770i
$206$		0			0
$207$		−	303.302i		−	1.46522i
$208$		0			0
$209$		−	52.7121i		−	0.252211i
$210$		0			0
$211$	−407.612			−1.93181			−0.965905	−	0.258897i	$-0.916641\pi$
$211$	−407.612			−1.93181			−0.965905	+	0.258897i	$0.916641\pi$
$212$		0			0
$213$	387.443			1.81898
$214$		0			0
$215$	54.7923	−	21.4462i	0.254848	−	0.0997499i
$216$		0			0
$217$	−149.092	+	32.0212i	−0.687062	+	0.147563i
$218$		0			0
$219$	−190.933			−0.871841
$220$		0			0
$221$	−170.109			−0.769724
$222$		0			0
$223$	−322.714			−1.44715			−0.723574	−	0.690247i	$-0.757502\pi$
$223$	−322.714			−1.44715			−0.723574	+	0.690247i	$0.757502\pi$
$224$		0			0
$225$	−251.109	+	232.137i	−1.11604	+	1.03172i
$226$		0			0
$227$	169.259			0.745633			0.372817	−	0.927905i	$-0.378392\pi$
$227$	169.259			0.745633			0.372817	+	0.927905i	$0.378392\pi$
$228$		0			0
$229$		−	307.130i		−	1.34118i	−0.741829	−	0.670589i	$-0.766041\pi$
$229$		−	307.130i		−	1.34118i	0.741829	−	0.670589i	$-0.233959\pi$
$230$		0			0
$231$	116.751	+	543.600i	0.505418	+	2.35325i
$232$		0			0
$233$			9.29105i			0.0398757i	0.999801	+	0.0199379i	$0.00634684\pi$
$233$			9.29105i			0.0398757i	−0.999801	+	0.0199379i	$0.993653\pi$
$234$		0			0
$235$	−20.4607	−	52.2745i	−0.0870670	−	0.222445i
$236$		0			0
$237$	193.375			0.815930
$238$		0			0
$239$	−32.8968			−0.137644			−0.0688218	−	0.997629i	$-0.521924\pi$
$239$	−32.8968			−0.137644			−0.0688218	+	0.997629i	$0.521924\pi$
$240$		0			0
$241$		−	186.750i		−	0.774894i	−0.921892	−	0.387447i	$-0.873357\pi$
$241$		−	186.750i		−	0.774894i	0.921892	−	0.387447i	$-0.126643\pi$
$242$		0			0
$243$	281.490			1.15840
$244$		0			0
$245$	175.102	+	171.360i	0.714703	+	0.699428i
$246$		0			0
$247$		−	47.3803i		−	0.191823i
$248$		0			0
$249$	202.145			0.811829
$250$		0			0
$251$		−	298.325i		−	1.18855i	−0.804263	−	0.594274i	$-0.797440\pi$
$251$		−	298.325i		−	1.18855i	0.804263	−	0.594274i	$-0.202560\pi$
$252$		0			0
$253$		−	369.821i		−	1.46174i
$254$		0			0
$255$	−98.4770	−	251.596i	−0.386184	−	0.986650i
$256$		0			0
$257$	224.463			0.873398			0.436699	−	0.899608i	$-0.356147\pi$
$257$	224.463			0.873398			0.436699	+	0.899608i	$0.356147\pi$
$258$		0			0
$259$	−245.430	+	52.7121i	−0.947607	+	0.203522i
$260$		0			0
$261$	219.358			0.840450
$262$		0			0
$263$			333.403i			1.26769i	0.773459	+	0.633846i	$0.218525\pi$
$263$			333.403i			1.26769i	−0.773459	+	0.633846i	$0.781475\pi$
$264$		0			0
$265$	96.8938	−	37.9252i	0.365637	−	0.143114i
$266$		0			0
$267$		−	251.027i		−	0.940176i
$268$		0			0
$269$			72.1823i			0.268336i	0.990959	+	0.134168i	$0.0428361\pi$
$269$			72.1823i			0.268336i	−0.990959	+	0.134168i	$0.957164\pi$
$270$		0			0
$271$			21.1078i			0.0778887i	0.999241	+	0.0389443i	$0.0123995\pi$
$271$			21.1078i			0.0778887i	−0.999241	+	0.0389443i	$0.987600\pi$
$272$		0			0
$273$	104.942	+	488.615i	0.384403	+	1.78980i
$274$		0			0
$275$	−306.182	+	283.048i	−1.11339	+	1.02927i
$276$		0			0
$277$			22.4218i			0.0809453i	0.999181	+	0.0404727i	$0.0128864\pi$
$277$			22.4218i			0.0809453i	−0.999181	+	0.0404727i	$0.987114\pi$
$278$		0			0
$279$		−	297.987i		−	1.06805i
$280$		0			0
$281$	−43.6119			−0.155203			−0.0776013	−	0.996984i	$-0.524726\pi$
$281$	−43.6119			−0.155203			−0.0776013	+	0.996984i	$0.524726\pi$
$282$		0			0
$283$	59.1956			0.209172			0.104586	−	0.994516i	$-0.466648\pi$
$283$	59.1956			0.209172			0.104586	+	0.994516i	$0.466648\pi$
$284$		0			0
$285$	70.0767	−	27.4287i	0.245883	−	0.0962410i
$286$		0			0
$287$	−511.009	+	109.752i	−1.78052	+	0.382410i
$288$		0			0
$289$	−160.249			−0.554493
$290$		0			0
$291$	555.757			1.90982
$292$		0			0
$293$	−371.148			−1.26672			−0.633358	−	0.773859i	$-0.718324\pi$
$293$	−371.148			−1.26672			−0.633358	+	0.773859i	$0.718324\pi$
$294$		0			0
$295$	433.576	−	169.706i	1.46975	−	0.575273i
$296$		0			0
$297$	−371.627			−1.25127
$298$		0			0
$299$		−	332.413i		−	1.11175i
$300$		0			0
$301$	80.5393	−	17.2978i	0.267572	−	0.0574677i
$302$		0			0
$303$			43.5405i			0.143698i
$304$		0			0
$305$	−202.073	+	79.0932i	−0.662533	+	0.259322i
$306$		0			0
$307$	−49.4319			−0.161016			−0.0805079	−	0.996754i	$-0.525654\pi$
$307$	−49.4319			−0.161016			−0.0805079	+	0.996754i	$0.525654\pi$
$308$		0			0
$309$	197.394			0.638815
$310$		0			0
$311$			425.196i			1.36719i	0.729862	+	0.683595i	$0.239584\pi$
$311$			425.196i			1.36719i	−0.729862	+	0.683595i	$0.760416\pi$
$312$		0			0
$313$	−407.383			−1.30154			−0.650771	−	0.759274i	$-0.725554\pi$
$313$	−407.383			−1.30154			−0.650771	+	0.759274i	$0.725554\pi$
$314$		0			0
$315$	−399.241	+	264.226i	−1.26743	+	0.838811i
$316$		0			0
$317$			515.185i			1.62519i	0.582830	+	0.812594i	$0.301946\pi$
$317$			515.185i			1.62519i	−0.582830	+	0.812594i	$0.698054\pi$
$318$		0			0
$319$	267.467			0.838453
$320$		0			0
$321$			586.662i			1.82761i
$322$		0			0
$323$			35.8610i			0.111025i
$324$		0			0
$325$	−275.211	+	254.418i	−0.846804	+	0.782824i
$326$		0			0
$327$	639.003			1.95414
$328$		0			0
$329$	−16.5029	−	76.8384i	−0.0501608	−	0.233551i
$330$		0			0
$331$	−72.0727			−0.217742			−0.108871	−	0.994056i	$-0.534724\pi$
$331$	−72.0727			−0.217742			−0.108871	+	0.994056i	$0.534724\pi$
$332$		0			0
$333$		−	490.535i		−	1.47308i
$334$		0			0
$335$	63.7313	−	24.9451i	0.190243	−	0.0744628i
$336$		0			0
$337$		−	362.450i		−	1.07552i	−0.843098	−	0.537759i	$-0.819271\pi$
$337$		−	362.450i		−	1.07552i	0.843098	−	0.537759i	$-0.180729\pi$
$338$		0			0
$339$		−	350.699i		−	1.03451i
$340$		0			0
$341$		−	363.341i		−	1.06552i
$342$		0			0
$343$	203.372	+	276.204i	0.592921	+	0.805260i
$344$		0			0
$345$	491.648	−	192.436i	1.42507	−	0.557785i
$346$		0			0
$347$		−	310.175i		−	0.893877i	−0.894565	−	0.446938i	$-0.852514\pi$
$347$		−	310.175i		−	0.893877i	0.894565	−	0.446938i	$-0.147486\pi$
$348$		0			0
$349$		−	190.418i		−	0.545609i	−0.962069	−	0.272804i	$-0.912049\pi$
$349$		−	190.418i		−	0.545609i	0.962069	−	0.272804i	$-0.0879512\pi$
$350$		0			0
$351$	−334.036			−0.951670
$352$		0			0
$353$	−177.839			−0.503793			−0.251896	−	0.967754i	$-0.581054\pi$
$353$	−177.839			−0.503793			−0.251896	+	0.967754i	$0.581054\pi$
$354$		0			0
$355$	148.268	+	378.805i	0.417656	+	1.06706i
$356$		0			0
$357$	−79.4281	−	369.821i	−0.222488	−	1.03591i
$358$		0			0
$359$	366.933			1.02210			0.511049	−	0.859552i	$-0.329257\pi$
$359$	366.933			1.02210			0.511049	+	0.859552i	$0.329257\pi$
$360$		0			0
$361$	351.012			0.972332
$362$		0			0
$363$	−748.534			−2.06208
$364$		0			0
$365$	−73.0668	−	186.676i	−0.200183	−	0.511442i
$366$		0			0
$367$	−213.129			−0.580734			−0.290367	−	0.956915i	$-0.593777\pi$
$367$	−213.129			−0.580734			−0.290367	+	0.956915i	$0.593777\pi$
$368$		0			0
$369$		−	1021.34i		−	2.76786i
$370$		0			0
$371$	142.424	−	30.5891i	0.383893	−	0.0824505i
$372$		0			0
$373$			594.278i			1.59324i	0.604481	+	0.796619i	$0.293380\pi$
$373$			594.278i			1.59324i	−0.604481	+	0.796619i	$0.706620\pi$
$374$		0			0
$375$	−535.612	−	259.761i	−1.42830	−	0.692696i
$376$		0			0
$377$	240.412			0.637698
$378$		0			0
$379$	397.939			1.04997			0.524985	−	0.851111i	$-0.324071\pi$
$379$	397.939			1.04997			0.524985	+	0.851111i	$0.324071\pi$
$380$		0			0
$381$			245.106i			0.643322i
$382$		0			0
$383$	−464.822			−1.21363			−0.606817	−	0.794841i	$-0.707554\pi$
$383$	−464.822			−1.21363			−0.606817	+	0.794841i	$0.707554\pi$
$384$		0			0
$385$	−486.802	+	322.175i	−1.26442	+	0.836818i
$386$		0			0
$387$			160.972i			0.415947i
$388$		0			0
$389$	564.969			1.45236			0.726182	−	0.687503i	$-0.241293\pi$
$389$	564.969			1.45236			0.726182	+	0.687503i	$0.241293\pi$
$390$		0			0
$391$			251.596i			0.643467i
$392$		0			0
$393$			811.672i			2.06532i
$394$		0			0
$395$	74.0014	+	189.064i	0.187345	+	0.478643i
$396$		0			0
$397$	−460.060			−1.15884			−0.579421	−	0.815029i	$-0.696721\pi$
$397$	−460.060			−1.15884			−0.579421	+	0.815029i	$0.696721\pi$
$398$		0			0
$399$	103.006	−	22.1230i	0.258160	−	0.0554462i
$400$		0			0
$401$	−124.109			−0.309499			−0.154749	−	0.987954i	$-0.549457\pi$
$401$	−124.109			−0.309499			−0.154749	+	0.987954i	$0.549457\pi$
$402$		0			0
$403$		−	326.589i		−	0.810394i
$404$		0			0
$405$	30.9812	+	79.1528i	0.0764967	+	0.195439i
$406$		0			0
$407$		−	598.118i		−	1.46958i
$408$		0			0
$409$		−	59.8790i		−	0.146403i	−0.997317	−	0.0732017i	$-0.976678\pi$
$409$		−	59.8790i		−	0.146403i	0.997317	−	0.0732017i	$-0.0233217\pi$
$410$		0			0
$411$		−	475.763i		−	1.15757i
$412$		0			0
$413$	637.313	−	136.879i	1.54313	−	0.331425i
$414$		0			0
$415$	77.3576	+	197.638i	0.186404	+	0.476237i
$416$		0			0
$417$		−	975.618i		−	2.33961i
$418$		0			0
$419$			552.743i			1.31920i	0.751618	+	0.659598i	$0.229274\pi$
$419$			552.743i			1.31920i	−0.751618	+	0.659598i	$0.770726\pi$
$420$		0			0
$421$	378.400			0.898812			0.449406	−	0.893328i	$-0.351636\pi$
$421$	378.400			0.898812			0.449406	+	0.893328i	$0.351636\pi$
$422$		0			0
$423$	153.575			0.363061
$424$		0			0
$425$	208.301	−	192.563i	0.490120	−	0.453089i
$426$		0			0
$427$	−297.027	+	63.7938i	−0.695613	+	0.149400i
$428$		0			0
$429$	−1190.76			−2.77567
$430$		0			0
$431$	−310.036			−0.719342			−0.359671	−	0.933079i	$-0.617111\pi$
$431$	−310.036			−0.719342			−0.359671	+	0.933079i	$0.617111\pi$
$432$		0			0
$433$	−672.484			−1.55308			−0.776541	−	0.630067i	$-0.783027\pi$
$433$	−672.484			−1.55308			−0.776541	+	0.630067i	$0.783027\pi$
$434$		0			0
$435$	139.176	+	355.576i	0.319945	+	0.817416i
$436$		0			0
$437$	−70.0767			−0.160359
$438$		0			0
$439$		−	503.192i		−	1.14622i	−0.819478	−	0.573111i	$-0.805736\pi$
$439$		−	503.192i		−	1.14622i	0.819478	−	0.573111i	$-0.194264\pi$
$440$		0			0
$441$	−611.151	+	275.214i	−1.38583	+	0.624069i
$442$		0			0
$443$			143.693i			0.324363i	0.986761	+	0.162181i	$0.0518530\pi$
$443$			143.693i			0.324363i	−0.986761	+	0.162181i	$0.948147\pi$
$444$		0			0
$445$	245.430	−	96.0637i	0.551529	−	0.215874i
$446$		0			0
$447$	−260.220			−0.582147
$448$		0			0
$449$	−337.042			−0.750651			−0.375325	−	0.926893i	$-0.622469\pi$
$449$	−337.042			−0.750651			−0.375325	+	0.926893i	$0.622469\pi$
$450$		0			0
$451$		−	1245.34i		−	2.76128i
$452$		0			0
$453$	130.802			0.288746
$454$		0			0
$455$	−437.561	+	289.587i	−0.961674	+	0.636454i
$456$		0			0
$457$		−	719.140i		−	1.57361i	−0.617201	−	0.786805i	$-0.711734\pi$
$457$		−	719.140i		−	1.57361i	0.617201	−	0.786805i	$-0.288266\pi$
$458$		0			0
$459$	252.824			0.550815
$460$		0			0
$461$		−	310.629i		−	0.673815i	−0.941538	−	0.336908i	$-0.890619\pi$
$461$		−	310.629i		−	0.673815i	0.941538	−	0.336908i	$-0.109381\pi$
$462$		0			0
$463$		−	278.960i		−	0.602505i	−0.953544	−	0.301253i	$-0.902595\pi$
$463$		−	278.960i		−	0.602505i	0.953544	−	0.301253i	$-0.0974047\pi$
$464$		0			0
$465$	483.034	−	189.064i	1.03878	−	0.406589i
$466$		0			0
$467$	60.1527			0.128807			0.0644033	−	0.997924i	$-0.479486\pi$
$467$	60.1527			0.128807			0.0644033	+	0.997924i	$0.479486\pi$
$468$		0			0
$469$	93.6788	−	20.1198i	0.199742	−	0.0428994i
$470$		0			0
$471$	−777.794			−1.65137
$472$		0			0
$473$			196.276i			0.414959i
$474$		0			0
$475$	53.6343	+	58.0178i	0.112914	+	0.122143i
$476$		0			0
$477$			284.660i			0.596771i
$478$		0			0
$479$			344.378i			0.718953i	0.933154	+	0.359476i	$0.117045\pi$
$479$			344.378i			0.718953i	−0.933154	+	0.359476i	$0.882955\pi$
$480$		0			0
$481$		−	537.618i		−	1.11771i
$482$		0			0
$483$	722.674	−	155.212i	1.49622	−	0.321350i
$484$		0			0
$485$	212.679	+	543.366i	0.438513	+	1.12034i
$486$		0			0
$487$		−	213.495i		−	0.438388i	−0.975681	−	0.219194i	$-0.929657\pi$
$487$		−	213.495i		−	0.438388i	0.975681	−	0.219194i	$-0.0703427\pi$
$488$		0			0
$489$			1016.71i			2.07916i
$490$		0			0
$491$	289.260			0.589125			0.294562	−	0.955632i	$-0.404826\pi$
$491$	289.260			0.589125			0.294562	+	0.955632i	$0.404826\pi$
$492$		0			0
$493$	−181.962			−0.369092
$494$		0			0
$495$	−415.777	−	1062.26i	−0.839954	−	2.14597i
$496$		0			0
$497$	119.588	+	556.806i	0.240619	+	1.12033i
$498$		0			0
$499$	−31.3212			−0.0627680			−0.0313840	−	0.999507i	$-0.509991\pi$
$499$	−31.3212			−0.0627680			−0.0313840	+	0.999507i	$0.509991\pi$
$500$		0			0
$501$	−1289.90			−2.57466
$502$		0			0
$503$	−11.3597			−0.0225839			−0.0112919	−	0.999936i	$-0.503594\pi$
$503$	−11.3597			−0.0225839			−0.0112919	+	0.999936i	$0.503594\pi$
$504$		0			0
$505$	−42.5698	+	16.6622i	−0.0842966	+	0.0329945i
$506$		0			0
$507$	−265.501			−0.523671
$508$		0			0
$509$			732.326i			1.43875i	0.694620	+	0.719377i	$0.255573\pi$
$509$			732.326i			1.43875i	−0.694620	+	0.719377i	$0.744427\pi$
$510$		0			0
$511$	−58.9332	−	274.396i	−0.115329	−	0.536978i
$512$		0			0
$513$			70.4188i			0.137269i
$514$		0			0
$515$	75.5393	+	192.993i	0.146678	+	0.374743i
$516$		0			0
$517$	187.256			0.362198
$518$		0			0
$519$	−906.836			−1.74728
$520$		0			0
$521$			10.4965i			0.0201468i	0.999949	+	0.0100734i	$0.00320652\pi$
$521$			10.4965i			0.0201468i	−0.999949	+	0.0100734i	$0.996793\pi$
$522$		0			0
$523$	−359.468			−0.687319			−0.343659	−	0.939094i	$-0.611667\pi$
$523$	−359.468			−0.687319			−0.343659	+	0.939094i	$0.611667\pi$
$524$		0			0
$525$	−681.613	−	479.522i	−1.29831	−	0.913375i
$526$		0			0
$527$			247.187i			0.469046i
$528$		0			0
$529$	37.3517			0.0706082
$530$		0			0
$531$			1273.78i			2.39883i
$532$		0			0
$533$		−	1119.37i		−	2.10013i
$534$		0			0
$535$	−573.582	+	224.505i	−1.07212	+	0.419636i
$536$		0			0
$537$	457.520			0.851992
$538$		0			0
$539$	−745.188	+	335.574i	−1.38254	+	0.622586i
$540$		0			0
$541$	383.746			0.709326			0.354663	−	0.934994i	$-0.384596\pi$
$541$	383.746			0.709326			0.354663	+	0.934994i	$0.384596\pi$
$542$		0			0
$543$		−	1184.72i		−	2.18180i
$544$		0			0
$545$	244.536	+	624.756i	0.448689	+	1.14634i
$546$		0			0
$547$			788.077i			1.44073i	0.693598	+	0.720363i	$0.256025\pi$
$547$			788.077i			1.44073i	−0.693598	+	0.720363i	$0.743975\pi$
$548$		0			0
$549$		−	593.660i		−	1.08135i
$550$		0			0
$551$		−	50.6818i		−	0.0919814i
$552$		0			0
$553$	59.6870	+	277.905i	0.107933	+	0.502541i
$554$		0			0
$555$	795.151	−	311.230i	1.43270	−	0.560774i
$556$		0			0
$557$			64.6592i			0.116085i	0.998314	+	0.0580424i	$0.0184858\pi$
$557$			64.6592i			0.116085i	−0.998314	+	0.0580424i	$0.981514\pi$
$558$		0			0
$559$			176.422i			0.315603i
$560$		0			0
$561$	901.260			1.60652
$562$		0			0
$563$	656.960			1.16689			0.583446	−	0.812152i	$-0.301704\pi$
$563$	656.960			1.16689			0.583446	+	0.812152i	$0.301704\pi$
$564$		0			0
$565$	342.880	−	134.207i	0.606868	−	0.237534i
$566$		0			0
$567$	24.9883	+	116.347i	0.0440711	+	0.205197i
$568$		0			0
$569$	834.375			1.46639			0.733194	−	0.680019i	$-0.238029\pi$
$569$	834.375			1.46639			0.733194	+	0.680019i	$0.238029\pi$
$570$		0			0
$571$	66.4971			0.116457			0.0582286	−	0.998303i	$-0.481455\pi$
$571$	66.4971			0.116457			0.0582286	+	0.998303i	$0.481455\pi$
$572$		0			0
$573$	−1115.57			−1.94690
$574$		0			0
$575$	376.291	+	407.045i	0.654419	+	0.707904i
$576$		0			0
$577$	31.9127			0.0553079			0.0276540	−	0.999618i	$-0.491196\pi$
$577$	31.9127			0.0553079			0.0276540	+	0.999618i	$0.491196\pi$
$578$		0			0
$579$			861.565i			1.48802i
$580$		0			0
$581$	62.3939	+	290.509i	0.107391	+	0.500015i
$582$		0			0
$583$			347.091i			0.595353i
$584$		0			0
$585$	−373.721	−	954.808i	−0.638839	−	1.63215i
$586$		0			0
$587$	783.466			1.33470			0.667348	−	0.744746i	$-0.267430\pi$
$587$	783.466			1.33470			0.667348	+	0.744746i	$0.267430\pi$
$588$		0			0
$589$	−68.8488			−0.116891
$590$		0			0
$591$			754.787i			1.27714i
$592$		0			0
$593$	1024.80			1.72816			0.864078	−	0.503357i	$-0.167902\pi$
$593$	1024.80			1.72816			0.864078	+	0.503357i	$0.167902\pi$
$594$		0			0
$595$	331.180	−	219.181i	0.556605	−	0.368372i
$596$		0			0
$597$			1064.31i			1.78277i
$598$		0			0
$599$	1109.11			1.85161			0.925805	−	0.378001i	$-0.123388\pi$
$599$	1109.11			1.85161			0.925805	+	0.378001i	$0.123388\pi$
$600$		0			0
$601$		−	1028.00i		−	1.71048i	−0.518232	−	0.855240i	$-0.673410\pi$
$601$		−	1028.00i		−	1.71048i	0.518232	−	0.855240i	$-0.326590\pi$
$602$		0			0
$603$			187.233i			0.310503i
$604$		0			0
$605$	−286.451	−	731.845i	−0.473473	−	1.20966i
$606$		0			0
$607$	581.737			0.958381			0.479190	−	0.877711i	$-0.340930\pi$
$607$	581.737			0.958381			0.479190	+	0.877711i	$0.340930\pi$
$608$		0			0
$609$	112.254	+	522.662i	0.184326	+	0.858229i
$610$		0			0
$611$	168.315			0.275475
$612$		0			0
$613$			303.242i			0.494685i	0.968928	+	0.247342i	$0.0795573\pi$
$613$			303.242i			0.494685i	−0.968928	+	0.247342i	$0.920443\pi$
$614$		0			0
$615$	1655.58	−	648.009i	2.69200	−	1.05367i
$616$		0			0
$617$			530.544i			0.859877i	0.902858	+	0.429938i	$0.141465\pi$
$617$			530.544i			0.859877i	−0.902858	+	0.429938i	$0.858535\pi$
$618$		0			0
$619$		−	171.117i		−	0.276440i	−0.990402	−	0.138220i	$-0.955862\pi$
$619$		−	171.117i		−	0.276440i	0.990402	−	0.138220i	$-0.0441382\pi$
$620$		0			0
$621$			494.049i			0.795569i
$622$		0			0
$623$	360.758	−	77.4816i	0.579066	−	0.124369i
$624$		0			0
$625$	49.0000	−	623.076i	0.0784000	−	0.996922i
$626$		0			0
$627$			251.027i			0.400362i
$628$		0			0
$629$			406.910i			0.646916i
$630$		0			0
$631$	1021.70			1.61917			0.809585	−	0.587003i	$-0.199692\pi$
$631$	1021.70			1.61917			0.809585	+	0.587003i	$0.199692\pi$
$632$		0			0
$633$	1941.14			3.06657
$634$		0			0
$635$	−239.641	+	93.7977i	−0.377387	+	0.147713i
$636$		0			0
$637$	−669.811	+	301.630i	−1.05151	+	0.473517i
$638$		0			0
$639$	−1112.87			−1.74158
$640$		0			0
$641$	337.212			0.526072			0.263036	−	0.964786i	$-0.415276\pi$
$641$	337.212			0.526072			0.263036	+	0.964786i	$0.415276\pi$
$642$		0			0
$643$	−844.298			−1.31306			−0.656530	−	0.754300i	$-0.727977\pi$
$643$	−844.298			−1.31306			−0.656530	+	0.754300i	$0.727977\pi$
$644$		0			0
$645$	−260.933	+	102.132i	−0.404548	+	0.158344i
$646$		0			0
$647$	279.281			0.431656			0.215828	−	0.976431i	$-0.430755\pi$
$647$	279.281			0.431656			0.215828	+	0.976431i	$0.430755\pi$
$648$		0			0
$649$			1553.14i			2.39313i
$650$		0			0
$651$	710.012	−	152.492i	1.09065	−	0.234243i
$652$		0			0
$653$		−	807.833i		−	1.23711i	−0.785742	−	0.618555i	$-0.787718\pi$
$653$		−	807.833i		−	1.23711i	0.785742	−	0.618555i	$-0.212282\pi$
$654$		0			0
$655$	−793.576	+	310.613i	−1.21157	+	0.474218i
$656$		0			0
$657$	548.427			0.834744
$658$		0			0
$659$	114.958			0.174443			0.0872214	−	0.996189i	$-0.472201\pi$
$659$	114.958			0.174443			0.0872214	+	0.996189i	$0.472201\pi$
$660$		0			0
$661$		−	419.383i		−	0.634467i	−0.948347	−	0.317234i	$-0.897246\pi$
$661$		−	419.383i		−	0.634467i	0.948347	−	0.317234i	$-0.102754\pi$
$662$		0			0
$663$	810.097			1.22187
$664$		0			0
$665$	61.0483	+	92.2432i	0.0918020	+	0.138712i
$666$		0			0
$667$		−	355.576i		−	0.533098i
$668$		0			0
$669$	1536.84			2.29721
$670$		0			0
$671$		−	723.860i		−	1.07878i
$672$		0			0
$673$			1071.68i			1.59240i	0.605037	+	0.796198i	$0.293158\pi$
$673$			1071.68i			1.59240i	−0.605037	+	0.796198i	$0.706842\pi$
$674$		0			0
$675$	409.032	−	378.128i	0.605974	−	0.560189i
$676$		0			0
$677$	−592.231			−0.874788			−0.437394	−	0.899270i	$-0.644098\pi$
$677$	−592.231			−0.874788			−0.437394	+	0.899270i	$0.644098\pi$
$678$		0			0
$679$	171.539	+	798.695i	0.252635	+	1.17628i
$680$		0			0
$681$	−806.048			−1.18362
$682$		0			0
$683$		−	93.2784i		−	0.136572i	−0.997666	−	0.0682858i	$-0.978247\pi$
$683$		−	93.2784i		−	0.136572i	0.997666	−	0.0682858i	$-0.0217530\pi$
$684$		0			0
$685$	465.155	−	182.066i	0.679059	−	0.265790i
$686$		0			0
$687$			1462.62i			2.12900i
$688$		0			0
$689$			311.982i			0.452805i
$690$		0			0
$691$		−	48.8749i		−	0.0707307i	−0.999374	−	0.0353653i	$-0.988741\pi$
$691$		−	48.8749i		−	0.0707307i	0.999374	−	0.0353653i	$-0.0112595\pi$
$692$		0			0
$693$	−335.351	−	1561.41i	−0.483912	−	2.25312i
$694$		0			0
$695$	953.866	−	373.352i	1.37247	−	0.537198i
$696$		0			0
$697$			847.225i			1.21553i
$698$		0			0
$699$		−	44.2460i		−	0.0632991i
$700$		0			0
$701$	1151.83			1.64312			0.821562	−	0.570119i	$-0.193103\pi$
$701$	1151.83			1.64312			0.821562	+	0.570119i	$0.193103\pi$
$702$		0			0
$703$	−113.336			−0.161218
$704$		0			0
$705$	97.4386	+	248.943i	0.138211	+	0.353111i
$706$		0			0
$707$	−62.5734	+	13.4392i	−0.0885054	+	0.0190087i
$708$		0			0
$709$	−113.249			−0.159730			−0.0798650	−	0.996806i	$-0.525449\pi$
$709$	−113.249			−0.159730			−0.0798650	+	0.996806i	$0.525449\pi$
$710$		0			0
$711$	−555.442			−0.781212
$712$		0			0
$713$	−483.034			−0.677466
$714$		0			0
$715$	−455.685	−	1164.21i	−0.637321	−	1.62827i
$716$		0			0
$717$	156.662			0.218497
$718$		0			0
$719$		−	575.543i		−	0.800477i	−0.916411	−	0.400239i	$-0.868927\pi$
$719$		−	575.543i		−	0.800477i	0.916411	−	0.400239i	$-0.131073\pi$
$720$		0			0
$721$	60.9273	+	283.681i	0.0845039	+	0.393454i
$722$		0			0
$723$			889.343i			1.23007i
$724$		0			0
$725$	−294.388	+	272.146i	−0.406053	+	0.375373i
$726$		0			0
$727$	−802.036			−1.10321			−0.551607	−	0.834104i	$-0.685985\pi$
$727$	−802.036			−1.10321			−0.551607	+	0.834104i	$0.685985\pi$
$728$		0			0
$729$	−1187.52			−1.62897
$730$		0			0
$731$		−	133.530i		−	0.182667i
$732$		0			0
$733$	−123.726			−0.168794			−0.0843970	−	0.996432i	$-0.526896\pi$
$733$	−123.726			−0.168794			−0.0843970	+	0.996432i	$0.526896\pi$
$734$		0			0
$735$	−833.877	−	816.054i	−1.13453	−	1.11028i
$736$		0			0
$737$			228.297i			0.309765i
$738$		0			0
$739$	−632.193			−0.855471			−0.427736	−	0.903904i	$-0.640689\pi$
$739$	−632.193			−0.855471			−0.427736	+	0.903904i	$0.640689\pi$
$740$		0			0
$741$			225.636i			0.304501i
$742$		0			0
$743$		−	895.660i		−	1.20546i	−0.797944	−	0.602732i	$-0.794079\pi$
$743$		−	895.660i		−	1.20546i	0.797944	−	0.602732i	$-0.205921\pi$
$744$		0			0
$745$	−99.5816	−	254.418i	−0.133667	−	0.341500i
$746$		0			0
$747$	−580.632			−0.777286
$748$		0			0
$749$	−843.109	+	181.078i	−1.12565	+	0.241760i
$750$		0			0
$751$	461.115			0.614001			0.307001	−	0.951709i	$-0.400675\pi$
$751$	461.115			0.614001			0.307001	+	0.951709i	$0.400675\pi$
$752$		0			0
$753$			1420.69i			1.88671i
$754$		0			0
$755$	50.0557	+	127.886i	0.0662989	+	0.169385i
$756$		0			0
$757$		−	833.169i		−	1.10062i	−0.834961	−	0.550310i	$-0.814510\pi$
$757$		−	833.169i		−	1.10062i	0.834961	−	0.550310i	$-0.185490\pi$
$758$		0			0
$759$			1761.17i			2.32038i
$760$		0			0
$761$			228.343i			0.300056i	0.988682	+	0.150028i	$0.0479364\pi$
$761$			228.343i			0.300056i	−0.988682	+	0.150028i	$0.952064\pi$
$762$		0			0
$763$	197.234	+	918.330i	0.258498	+	1.20358i
$764$		0			0
$765$	282.860	+	722.671i	0.369752	+	0.944668i
$766$		0			0
$767$			1396.04i			1.82013i
$768$		0			0
$769$		−	1211.59i		−	1.57554i	−0.615972	−	0.787768i	$-0.711236\pi$
$769$		−	1211.59i		−	1.57554i	0.615972	−	0.787768i	$-0.288764\pi$
$770$		0			0
$771$	−1068.94			−1.38644
$772$		0			0
$773$	139.181			0.180053			0.0900267	−	0.995939i	$-0.471305\pi$
$773$	139.181			0.180053			0.0900267	+	0.995939i	$0.471305\pi$
$774$		0			0
$775$	369.697	+	399.913i	0.477029	+	0.516016i
$776$		0			0
$777$	1168.79	−	251.027i	1.50424	−	0.323072i
$778$		0			0
$779$	−235.977			−0.302923
$780$		0			0
$781$	−1356.94			−1.73745
$782$		0			0
$783$	−357.312			−0.456337
$784$		0			0
$785$	−297.648	−	760.452i	−0.379170	−	0.968729i
$786$		0			0
$787$	149.466			0.189919			0.0949596	−	0.995481i	$-0.469728\pi$
$787$	149.466			0.189919			0.0949596	+	0.995481i	$0.469728\pi$
$788$		0			0
$789$		−	1587.74i		−	2.01234i
$790$		0			0
$791$	504.000	−	108.246i	0.637168	−	0.136847i
$792$		0			0
$793$		−	650.641i		−	0.820480i
$794$		0			0
$795$	−461.430	+	180.608i	−0.580415	+	0.227180i
$796$		0			0
$797$	435.452			0.546364			0.273182	−	0.961962i	$-0.411924\pi$
$797$	435.452			0.546364			0.273182	+	0.961962i	$0.411924\pi$
$798$		0			0
$799$	−127.394			−0.159442
$800$		0			0
$801$			721.038i			0.900172i
$802$		0			0
$803$	668.707			0.832761
$804$		0			0
$805$	428.307	+	647.165i	0.532058	+	0.803932i
$806$		0			0
$807$		−	343.748i		−	0.425958i
$808$		0			0
$809$	−455.551			−0.563104			−0.281552	−	0.959546i	$-0.590849\pi$
$809$	−455.551			−0.563104			−0.281552	+	0.959546i	$0.590849\pi$
$810$		0			0
$811$			11.6265i			0.0143360i	0.999974	+	0.00716802i	$0.00228167\pi$
$811$			11.6265i			0.0143360i	−0.999974	+	0.00716802i	$0.997718\pi$
$812$		0			0
$813$		−	100.520i		−	0.123641i
$814$		0			0
$815$	−994.042	+	389.078i	−1.21968	+	0.477396i
$816$		0			0
$817$	37.1919			0.0455225
$818$		0			0
$819$	−301.430	−	1403.47i	−0.368047	−	1.71364i
$820$		0			0
$821$	−485.600			−0.591474			−0.295737	−	0.955269i	$-0.595565\pi$
$821$	−485.600			−0.591474			−0.295737	+	0.955269i	$0.595565\pi$
$822$		0			0
$823$			1249.13i			1.51777i	0.651223	+	0.758886i	$0.274256\pi$
$823$			1249.13i			1.51777i	−0.651223	+	0.758886i	$0.725744\pi$
$824$		0			0
$825$	1458.11	−	1347.94i	1.76740	−	1.63387i
$826$		0			0
$827$		−	1368.11i		−	1.65430i	−0.561978	−	0.827152i	$-0.689959\pi$
$827$		−	1368.11i		−	1.65430i	0.561978	−	0.827152i	$-0.310041\pi$
$828$		0			0
$829$			1111.64i			1.34094i	0.741937	+	0.670469i	$0.233907\pi$
$829$			1111.64i			1.34094i	−0.741937	+	0.670469i	$0.766093\pi$
$830$		0			0
$831$		−	106.778i		−	0.128493i
$832$		0			0
$833$	506.964	−	228.297i	0.608601	−	0.274066i
$834$		0			0
$835$	−493.624	−	1261.14i	−0.591166	−	1.51035i
$836$		0			0
$837$			485.392i			0.579919i
$838$		0			0
$839$		−	423.166i		−	0.504369i	−0.967679	−	0.252184i	$-0.918851\pi$
$839$		−	423.166i		−	0.504369i	0.967679	−	0.252184i	$-0.0811490\pi$
$840$		0			0
$841$	−583.836			−0.694216
$842$		0			0
$843$	207.690			0.246370
$844$		0			0
$845$	−101.603	−	259.581i	−0.120240	−	0.307197i
$846$		0			0
$847$	−231.042	−	1075.74i	−0.272776	−	1.27006i
$848$		0			0
$849$	−281.903			−0.332041
$850$		0			0
$851$	−795.151			−0.934373
$852$		0			0
$853$	723.461			0.848137			0.424068	−	0.905630i	$-0.360602\pi$
$853$	723.461			0.848137			0.424068	+	0.905630i	$0.360602\pi$
$854$		0			0
$855$	−201.285	+	78.7848i	−0.235421	+	0.0921460i
$856$		0			0
$857$	−1014.89			−1.18423			−0.592115	−	0.805853i	$-0.701707\pi$
$857$	−1014.89			−1.18423			−0.592115	+	0.805853i	$0.701707\pi$
$858$		0			0
$859$			97.4114i			0.113401i	0.998391	+	0.0567005i	$0.0180580\pi$
$859$			97.4114i			0.113401i	−0.998391	+	0.0567005i	$0.981942\pi$
$860$		0			0
$861$	2433.54	−	522.662i	2.82641	−	0.607040i
$862$		0			0
$863$		−	480.498i		−	0.556776i	−0.960469	−	0.278388i	$-0.910200\pi$
$863$		−	480.498i		−	0.556776i	0.960469	−	0.278388i	$-0.0898001\pi$
$864$		0			0
$865$	−347.031	−	886.617i	−0.401191	−	1.02499i
$866$		0			0
$867$	763.139			0.880207
$868$		0			0
$869$	−677.260			−0.779356
$870$		0			0
$871$			205.204i			0.235596i
$872$		0			0
$873$	−1596.33			−1.82856
$874$		0			0
$875$	207.989	−	849.921i	0.237702	−	0.971338i
$876$		0			0
$877$		−	525.779i		−	0.599520i	−0.954015	−	0.299760i	$-0.903093\pi$
$877$		−	525.779i		−	0.599520i	0.954015	−	0.299760i	$-0.0969066\pi$
$878$		0			0
$879$	1767.49			2.01080
$880$		0			0
$881$			1289.75i			1.46396i	0.681324	+	0.731982i	$0.261405\pi$
$881$			1289.75i			1.46396i	−0.681324	+	0.731982i	$0.738595\pi$
$882$		0			0
$883$		−	51.6578i		−	0.0585026i	−0.999572	−	0.0292513i	$-0.990688\pi$
$883$		−	51.6578i		−	0.0585026i	0.999572	−	0.0292513i	$-0.00931231\pi$
$884$		0			0
$885$	−2064.78	+	808.176i	−2.33309	+	0.913194i
$886$		0			0
$887$	194.121			0.218851			0.109426	−	0.993995i	$-0.465099\pi$
$887$	194.121			0.218851			0.109426	+	0.993995i	$0.465099\pi$
$888$		0			0
$889$	−352.249	+	75.6540i	−0.396230	+	0.0851001i
$890$		0			0
$891$	−283.539			−0.318226
$892$		0			0
$893$		−	35.4829i		−	0.0397345i
$894$		0			0
$895$	175.085	+	447.319i	0.195626	+	0.499798i
$896$		0			0
$897$			1583.03i			1.76480i
$898$		0			0
$899$		−	349.346i		−	0.388594i
$900$		0			0
$901$		−	236.132i		−	0.262078i
$902$		0			0
$903$	−383.546	+	82.3759i	−0.424746	+	0.0912247i
$904$		0			0
$905$	1158.30	−	453.371i	1.27989	−	0.500962i
$906$		0			0
$907$			1386.01i			1.52812i	0.645145	+	0.764060i	$0.276797\pi$
$907$			1386.01i			1.52812i	−0.645145	+	0.764060i	$0.723203\pi$
$908$		0			0
$909$		−	125.064i		−	0.137584i
$910$		0			0
$911$	−449.939			−0.493896			−0.246948	−	0.969029i	$-0.579428\pi$
$911$	−449.939			−0.493896			−0.246948	+	0.969029i	$0.579428\pi$
$912$		0			0
$913$	−707.975			−0.775439
$914$		0			0
$915$	962.315	−	376.659i	1.05171	−	0.411650i
$916$		0			0
$917$	−1166.48	+	250.530i	−1.27206	+	0.273206i
$918$		0			0
$919$	−310.036			−0.337363			−0.168681	−	0.985671i	$-0.553951\pi$
$919$	−310.036			−0.337363			−0.168681	+	0.985671i	$0.553951\pi$
$920$		0			0
$921$	235.406			0.255598
$922$		0			0
$923$	−1219.69			−1.32144
$924$		0			0
$925$	608.581	+	658.321i	0.657926	+	0.711698i
$926$		0			0
$927$	−566.985			−0.611634
$928$		0			0
$929$		−	568.376i		−	0.611815i	−0.952061	−	0.305908i	$-0.901040\pi$
$929$		−	568.376i		−	0.611815i	0.952061	−	0.305908i	$-0.0989598\pi$
$930$		0			0
$931$	63.5873	+	141.204i	0.0683000	+	0.151669i
$932$		0			0
$933$		−	2024.88i		−	2.17029i
$934$		0			0
$935$	344.897	+	881.166i	0.368874	+	0.942424i
$936$		0			0
$937$	−610.588			−0.651641			−0.325821	−	0.945432i	$-0.605641\pi$
$937$	−610.588			−0.651641			−0.325821	+	0.945432i	$0.605641\pi$
$938$		0			0
$939$	1940.05			2.06608
$940$		0			0
$941$		−	1588.92i		−	1.68855i	−0.535912	−	0.844274i	$-0.680032\pi$
$941$		−	1588.92i		−	1.68855i	0.535912	−	0.844274i	$-0.319968\pi$
$942$		0			0
$943$	−1655.58			−1.75565
$944$		0			0
$945$	650.325	−	430.398i	0.688174	−	0.455447i
$946$		0			0
$947$			595.949i			0.629302i	0.949207	+	0.314651i	$0.101888\pi$
$947$			595.949i			0.629302i	−0.949207	+	0.314651i	$0.898112\pi$
$948$		0			0
$949$	601.067			0.633369
$950$		0			0
$951$		−	2453.43i		−	2.57984i
$952$		0			0
$953$		−	707.004i		−	0.741872i	−0.928658	−	0.370936i	$-0.879037\pi$
$953$		−	707.004i		−	0.741872i	0.928658	−	0.370936i	$-0.120963\pi$
$954$		0			0
$955$	−426.910	−	1090.70i	−0.447026	−	1.14209i
$956$		0			0
$957$	−1273.74			−1.33097
$958$		0			0
$959$	683.733	−	146.848i	0.712964	−	0.153126i
$960$		0			0
$961$	486.430			0.506171
$962$		0			0
$963$		−	1685.10i		−	1.74984i
$964$		0			0
$965$	−842.356	+	329.706i	−0.872908	+	0.341664i
$966$		0			0
$967$			91.3586i			0.0944763i	0.998884	+	0.0472381i	$0.0150420\pi$
$967$			91.3586i			0.0944763i	−0.998884	+	0.0472381i	$0.984958\pi$
$968$		0			0
$969$		−	170.778i		−	0.176242i
$970$		0			0
$971$			1137.54i			1.17152i	0.810485	+	0.585759i	$0.199204\pi$
$971$			1137.54i			1.17152i	−0.810485	+	0.585759i	$0.800796\pi$
$972$		0			0
$973$	1402.09	−	301.133i	1.44100	−	0.309489i
$974$		0			0
$975$	1310.62	−	1211.59i	1.34422	−	1.24266i
$976$		0			0
$977$			476.101i			0.487309i	0.969862	+	0.243654i	$0.0783463\pi$
$977$			476.101i			0.487309i	−0.969862	+	0.243654i	$0.921654\pi$
$978$		0			0
$979$			879.174i			0.898033i
$980$		0			0
$981$	−1835.44			−1.87099
$982$		0			0
$983$	777.439			0.790884			0.395442	−	0.918491i	$-0.370591\pi$
$983$	777.439			0.790884			0.395442	+	0.918491i	$0.370591\pi$
$984$		0			0
$985$	−737.959	+	288.844i	−0.749197	+	0.293243i
$986$		0			0
$987$	78.5906	+	365.922i	0.0796257	+	0.370741i
$988$		0			0
$989$	260.933			0.263835
$990$		0			0
$991$	1152.10			1.16256			0.581280	−	0.813704i	$-0.302552\pi$
$991$	1152.10			1.16256			0.581280	+	0.813704i	$0.302552\pi$
$992$		0			0
$993$	343.226			0.345646
$994$		0			0
$995$	−1040.58	+	407.294i	−1.04581	+	0.409340i
$996$		0			0
$997$	990.955			0.993937			0.496968	−	0.867769i	$-0.334446\pi$
$997$	990.955			0.993937			0.496968	+	0.867769i	$0.334446\pi$
$998$		0			0
$999$			799.033i			0.799833i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.3.p.g.209.1		8
4.3	odd	2		70.3.d.a.69.4	yes	8
5.4	even	2	inner	560.3.p.g.209.7		8
7.6	odd	2	inner	560.3.p.g.209.8		8
12.11	even	2		630.3.h.d.559.7		8
20.3	even	4		350.3.b.c.251.4		8
20.7	even	4		350.3.b.c.251.5		8
20.19	odd	2		70.3.d.a.69.5	yes	8
28.3	even	6		490.3.h.a.19.8		16
28.11	odd	6		490.3.h.a.19.5		16
28.19	even	6		490.3.h.a.129.4		16
28.23	odd	6		490.3.h.a.129.1		16
28.27	even	2		70.3.d.a.69.1	✓	8
35.34	odd	2	inner	560.3.p.g.209.2		8
60.59	even	2		630.3.h.d.559.2		8
84.83	odd	2		630.3.h.d.559.6		8
140.19	even	6		490.3.h.a.129.5		16
140.27	odd	4		350.3.b.c.251.8		8
140.39	odd	6		490.3.h.a.19.4		16
140.59	even	6		490.3.h.a.19.1		16
140.79	odd	6		490.3.h.a.129.8		16
140.83	odd	4		350.3.b.c.251.1		8
140.139	even	2		70.3.d.a.69.8	yes	8
420.419	odd	2		630.3.h.d.559.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.3.d.a.69.1	✓	8	28.27	even	2
70.3.d.a.69.4	yes	8	4.3	odd	2
70.3.d.a.69.5	yes	8	20.19	odd	2
70.3.d.a.69.8	yes	8	140.139	even	2
350.3.b.c.251.1		8	140.83	odd	4
350.3.b.c.251.4		8	20.3	even	4
350.3.b.c.251.5		8	20.7	even	4
350.3.b.c.251.8		8	140.27	odd	4
490.3.h.a.19.1		16	140.59	even	6
490.3.h.a.19.4		16	140.39	odd	6
490.3.h.a.19.5		16	28.11	odd	6
490.3.h.a.19.8		16	28.3	even	6
490.3.h.a.129.1		16	28.23	odd	6
490.3.h.a.129.4		16	28.19	even	6
490.3.h.a.129.5		16	140.19	even	6
490.3.h.a.129.8		16	140.79	odd	6
560.3.p.g.209.1		8	1.1	even	1	trivial
560.3.p.g.209.2		8	35.34	odd	2	inner
560.3.p.g.209.7		8	5.4	even	2	inner
560.3.p.g.209.8		8	7.6	odd	2	inner
630.3.h.d.559.2		8	60.59	even	2
630.3.h.d.559.3		8	420.419	odd	2
630.3.h.d.559.6		8	84.83	odd	2
630.3.h.d.559.7		8	12.11	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	560.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.76222 q^{3} +(-1.82242 - 4.65605i) q^{5} +(-1.46990 - 6.84393i) q^{7} +13.6788 q^{9} +16.6788 q^{11} +14.9917 q^{13} +(8.67878 + 22.1731i) q^{15} -11.3469 q^{17} -3.16043i q^{19} +(7.00000 + 32.5923i) q^{21} -22.1731i q^{23} +(-18.3576 + 16.9706i) q^{25} -22.2814 q^{27} +16.0363 q^{29} -21.7846i q^{31} -79.4281 q^{33} +(-29.1869 + 19.3165i) q^{35} -35.8610i q^{37} -71.3939 q^{39} -74.6660i q^{41} +11.7680i q^{43} +(-24.9285 - 63.6891i) q^{45} +11.2272 q^{47} +(-44.6788 + 20.1198i) q^{49} +54.0363 q^{51} +20.8103i q^{53} +(-30.3958 - 77.6572i) q^{55} +15.0507i q^{57} +93.1210i q^{59} -43.4000i q^{61} +(-20.1065 - 93.6166i) q^{63} +(-27.3212 - 69.8021i) q^{65} +13.6879i q^{67} +105.593i q^{69} -81.3576 q^{71} +40.0933 q^{73} +(87.4228 - 80.8176i) q^{75} +(-24.5162 - 114.148i) q^{77} -40.6061 q^{79} -17.0000 q^{81} -42.4477 q^{83} +(20.6788 + 52.8316i) q^{85} -76.3686 q^{87} +52.7121i q^{89} +(-22.0363 - 102.602i) q^{91} +103.743i q^{93} +(-14.7151 + 5.75964i) q^{95} -116.701 q^{97} +228.145 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 36 q^{9} + 60 q^{11} - 4 q^{15} + 56 q^{21} - 92 q^{29} - 52 q^{35} - 204 q^{39} - 284 q^{49} + 212 q^{51} - 292 q^{65} - 504 q^{71} - 692 q^{79} - 136 q^{81} + 92 q^{85} + 44 q^{91} + 176 q^{95}+ \cdots + 944 q^{99}+O(q^{100}) \) 8 * q + 36 * q^9 + 60 * q^11 - 4 * q^15 + 56 * q^21 - 92 * q^29 - 52 * q^35 - 204 * q^39 - 284 * q^49 + 212 * q^51 - 292 * q^65 - 504 * q^71 - 692 * q^79 - 136 * q^81 + 92 * q^85 + 44 * q^91 + 176 * q^95 + 944 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more