LMFDB - Embedded newform 2320.2.j.e.289.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2320,2,Mod(289,2320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2320.289"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,4,0,-1,0,0,0,8,0,0,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.75200995984.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 10x^{6} + 30x^{4} + 27x^{2} + 4 $ x^8 + 10x^6 + 30x^4 + 27*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 290)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.3
Root			$1.83507i$ of defining polynomial
Character	$\chi$	$=$	2320.289
Dual form			2320.2.j.e.289.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.367473 q^{3} +(1.82635 - 1.29013i) q^{5} +2.99580i q^{7} -2.86496 q^{9} -1.08988i q^{11} +1.37471i q^{13} +(-0.671136 + 0.474088i) q^{15} -0.212256 q^{17} -6.70297i q^{19} -1.10088i q^{21} -1.06384i q^{23} +(1.67114 - 4.71246i) q^{25} +2.15522 q^{27} +(2.49749 + 4.77101i) q^{29} -4.22738i q^{31} +0.400501i q^{33} +(3.86496 + 5.47139i) q^{35} +5.72993 q^{37} -0.505170i q^{39} -10.3731i q^{41} +10.8599 q^{43} +(-5.23244 + 3.69617i) q^{45} +6.38765 q^{47} -1.97480 q^{49} +0.0779983 q^{51} +10.0993i q^{53} +(-1.40608 - 1.99050i) q^{55} +2.46316i q^{57} +5.28523 q^{59} -3.38530i q^{61} -8.58285i q^{63} +(1.77356 + 2.51071i) q^{65} +10.2314i q^{67} +0.390934i q^{69} +5.04538 q^{71} -3.51767 q^{73} +(-0.614098 + 1.73170i) q^{75} +3.26505 q^{77} +8.85669i q^{79} +7.80291 q^{81} -12.4112i q^{83} +(-0.387654 + 0.273837i) q^{85} +(-0.917761 - 1.75322i) q^{87} -2.95876i q^{89} -4.11836 q^{91} +1.55345i q^{93} +(-8.64769 - 12.2420i) q^{95} +16.0605 q^{97} +3.12246i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - q^{5} + 8 q^{9} + 3 q^{15} - 2 q^{17} + 5 q^{25} + 10 q^{27} - 4 q^{29} - 16 q^{37} + 8 q^{43} - 4 q^{45} + 6 q^{47} - 6 q^{49} + 24 q^{51} + 11 q^{55} + 18 q^{59} - 15 q^{65} + 12 q^{71}+ \cdots + 14 q^{97}+O(q^{100}) $ 8 * q + 4 * q^3 - q^5 + 8 * q^9 + 3 * q^15 - 2 * q^17 + 5 * q^25 + 10 * q^27 - 4 * q^29 - 16 * q^37 + 8 * q^43 - 4 * q^45 + 6 * q^47 - 6 * q^49 + 24 * q^51 + 11 * q^55 + 18 * q^59 - 15 * q^65 + 12 * q^71 + 34 * q^73 + 11 * q^75 + 40 * q^77 - 24 * q^81 + 42 * q^85 + 10 * q^87 + 20 * q^91 + 10 * q^95 + 14 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$1857$	$2031$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−0.367473			−0.212161			−0.106080	−	0.994358i	$-0.533830\pi$
$3$	−0.367473			−0.212161			−0.106080	+	0.994358i	$0.533830\pi$
$4$		0			0
$5$	1.82635	−	1.29013i	0.816770	−	0.576963i
$5$	1.82635	−	1.29013i	0.816770	−	0.576963i
$6$		0			0
$7$			2.99580i			1.13230i	0.824301	+	0.566152i	$0.191569\pi$
$7$			2.99580i			1.13230i	−0.824301	+	0.566152i	$0.808431\pi$
$8$		0			0
$9$	−2.86496			−0.954988
$10$		0			0
$11$		−	1.08988i		−	0.328611i	−0.986410	−	0.164305i	$-0.947462\pi$
$11$		−	1.08988i		−	0.328611i	0.986410	−	0.164305i	$-0.0525382\pi$
$12$		0			0
$13$			1.37471i			0.381277i	0.981660	+	0.190638i	$0.0610558\pi$
$13$			1.37471i			0.381277i	−0.981660	+	0.190638i	$0.938944\pi$
$14$		0			0
$15$	−0.671136	+	0.474088i	−0.173287	+	0.122409i
$16$		0			0
$17$	−0.212256			−0.0514796			−0.0257398	−	0.999669i	$-0.508194\pi$
$17$	−0.212256			−0.0514796			−0.0257398	+	0.999669i	$0.508194\pi$
$18$		0			0
$19$		−	6.70297i		−	1.53777i	−0.639390	−	0.768883i	$-0.720813\pi$
$19$		−	6.70297i		−	1.53777i	0.639390	−	0.768883i	$-0.279187\pi$
$20$		0			0
$21$		−	1.10088i		−	0.240231i
$22$		0			0
$23$		−	1.06384i		−	0.221826i	−0.993830	−	0.110913i	$-0.964622\pi$
$23$		−	1.06384i		−	0.221826i	0.993830	−	0.110913i	$-0.0353776\pi$
$24$		0			0
$25$	1.67114	−	4.71246i	0.334227	−	0.942493i
$26$		0			0
$27$	2.15522			0.414772
$28$		0			0
$29$	2.49749	+	4.77101i	0.463772	+	0.885954i
$29$	2.49749	+	4.77101i	0.463772	+	0.885954i
$30$		0			0
$31$		−	4.22738i		−	0.759259i	−0.925139	−	0.379630i	$-0.876051\pi$
$31$		−	4.22738i		−	0.759259i	0.925139	−	0.379630i	$-0.123949\pi$
$32$		0			0
$33$			0.400501i			0.0697183i
$34$		0			0
$35$	3.86496	+	5.47139i	0.653298	+	0.924833i
$36$		0			0
$37$	5.72993			0.941994			0.470997	−	0.882135i	$-0.343894\pi$
$37$	5.72993			0.941994			0.470997	+	0.882135i	$0.343894\pi$
$38$		0			0
$39$		−	0.505170i		−	0.0808920i
$40$		0			0
$41$		−	10.3731i		−	1.62001i	−0.586426	−	0.810003i	$-0.699465\pi$
$41$		−	10.3731i		−	1.62001i	0.586426	−	0.810003i	$-0.300535\pi$
$42$		0			0
$43$	10.8599			1.65613			0.828063	−	0.560635i	$-0.189443\pi$
$43$	10.8599			1.65613			0.828063	+	0.560635i	$0.189443\pi$
$44$		0			0
$45$	−5.23244	+	3.69617i	−0.780006	+	0.550993i
$46$		0			0
$47$	6.38765			0.931735			0.465868	−	0.884854i	$-0.345742\pi$
$47$	6.38765			0.931735			0.465868	+	0.884854i	$0.345742\pi$
$48$		0			0
$49$	−1.97480			−0.282114
$50$		0			0
$51$	0.0779983			0.0109220
$52$		0			0
$53$			10.0993i			1.38724i	0.720341	+	0.693620i	$0.243985\pi$
$53$			10.0993i			1.38724i	−0.720341	+	0.693620i	$0.756015\pi$
$54$		0			0
$55$	−1.40608	−	1.99050i	−0.189596	−	0.268399i
$56$		0			0
$57$			2.46316i			0.326254i
$58$		0			0
$59$	5.28523			0.688079			0.344039	−	0.938955i	$-0.388205\pi$
$59$	5.28523			0.688079			0.344039	+	0.938955i	$0.388205\pi$
$60$		0			0
$61$		−	3.38530i		−	0.433443i	−0.976233	−	0.216722i	$-0.930464\pi$
$61$		−	3.38530i		−	0.433443i	0.976233	−	0.216722i	$-0.0695364\pi$
$62$		0			0
$63$		−	8.58285i		−	1.08134i
$64$		0			0
$65$	1.77356	+	2.51071i	0.219983	+	0.311415i
$66$		0			0
$67$			10.2314i			1.24996i	0.780639	+	0.624982i	$0.214894\pi$
$67$			10.2314i			1.24996i	−0.780639	+	0.624982i	$0.785106\pi$
$68$		0			0
$69$			0.390934i			0.0470629i
$70$		0			0
$71$	5.04538			0.598777			0.299388	−	0.954131i	$-0.403217\pi$
$71$	5.04538			0.598777			0.299388	+	0.954131i	$0.403217\pi$
$72$		0			0
$73$	−3.51767			−0.411712			−0.205856	−	0.978582i	$-0.565998\pi$
$73$	−3.51767			−0.411712			−0.205856	+	0.978582i	$0.565998\pi$
$74$		0			0
$75$	−0.614098	+	1.73170i	−0.0709099	+	0.199960i
$76$		0			0
$77$	3.26505			0.372087
$78$		0			0
$79$			8.85669i			0.996455i	0.867046	+	0.498227i	$0.166016\pi$
$79$			8.85669i			0.996455i	−0.867046	+	0.498227i	$0.833984\pi$
$80$		0			0
$81$	7.80291			0.866989
$82$		0			0
$83$		−	12.4112i		−	1.36230i	−0.732143	−	0.681151i	$-0.761480\pi$
$83$		−	12.4112i		−	1.36230i	0.732143	−	0.681151i	$-0.238520\pi$
$84$		0			0
$85$	−0.387654	+	0.273837i	−0.0420470	+	0.0297018i
$86$		0			0
$87$	−0.917761	−	1.75322i	−0.0983943	−	0.187965i
$88$		0			0
$89$		−	2.95876i		−	0.313628i	−0.987628	−	0.156814i	$-0.949878\pi$
$89$		−	2.95876i		−	0.313628i	0.987628	−	0.156814i	$-0.0501224\pi$
$90$		0			0
$91$	−4.11836			−0.431721
$92$		0			0
$93$			1.55345i			0.161085i
$94$		0			0
$95$	−8.64769	−	12.2420i	−0.887234	−	1.25600i
$96$		0			0
$97$	16.0605			1.63070			0.815351	−	0.578968i	$-0.196544\pi$
$97$	16.0605			1.63070			0.815351	+	0.578968i	$0.196544\pi$
$98$		0			0
$99$			3.12246i			0.313819i
$100$		0			0
$101$		−	13.7584i		−	1.36901i	−0.729007	−	0.684506i	$-0.760018\pi$
$101$		−	13.7584i		−	1.36901i	0.729007	−	0.684506i	$-0.239982\pi$
$102$		0			0
$103$			9.16613i			0.903165i	0.892229	+	0.451583i	$0.149140\pi$
$103$			9.16613i			0.903165i	−0.892229	+	0.451583i	$0.850860\pi$
$104$		0			0
$105$	−1.42027	−	2.01059i	−0.138604	−	0.196213i
$106$		0			0
$107$		−	2.34345i		−	0.226550i	−0.993564	−	0.113275i	$-0.963866\pi$
$107$		−	2.34345i		−	0.226550i	0.993564	−	0.113275i	$-0.0361341\pi$
$108$		0			0
$109$	−1.12260			−0.107526			−0.0537628	−	0.998554i	$-0.517121\pi$
$109$	−1.12260			−0.107526			−0.0537628	+	0.998554i	$0.517121\pi$
$110$		0			0
$111$	−2.10560			−0.199854
$112$		0			0
$113$	−14.7451			−1.38710			−0.693551	−	0.720408i	$-0.743955\pi$
$113$	−14.7451			−1.38710			−0.693551	+	0.720408i	$0.743955\pi$
$114$		0			0
$115$	−1.37249	−	1.94295i	−0.127986	−	0.181181i
$116$		0			0
$117$		−	3.93850i		−	0.364115i
$118$		0			0
$119$		−	0.635875i		−	0.0582906i
$120$		0			0
$121$	9.81217			0.892015
$122$		0			0
$123$			3.81184i			0.343702i
$124$		0			0
$125$	−3.02760	−	10.7626i	−0.270796	−	0.962637i
$126$		0			0
$127$	−2.57047			−0.228092			−0.114046	−	0.993475i	$-0.536381\pi$
$127$	−2.57047			−0.228092			−0.114046	+	0.993475i	$0.536381\pi$
$128$		0			0
$129$	−3.99074			−0.351365
$130$		0			0
$131$			1.20697i			0.105454i	0.998609	+	0.0527269i	$0.0167913\pi$
$131$			1.20697i			0.105454i	−0.998609	+	0.0527269i	$0.983209\pi$
$132$		0			0
$133$	20.0807			1.74122
$134$		0			0
$135$	3.93619	−	2.78051i	0.338773	−	0.239308i
$136$		0			0
$137$	−5.81534			−0.496838			−0.248419	−	0.968653i	$-0.579911\pi$
$137$	−5.81534			−0.496838			−0.248419	+	0.968653i	$0.579911\pi$
$138$		0			0
$139$	−15.0428			−1.27591			−0.637955	−	0.770074i	$-0.720219\pi$
$139$	−15.0428			−1.27591			−0.637955	+	0.770074i	$0.720219\pi$
$140$		0			0
$141$	−2.34729			−0.197678
$142$		0			0
$143$	1.49827			0.125292
$144$		0			0
$145$	10.7165	+	5.49147i	0.889958	+	0.456042i
$146$		0			0
$147$	0.725686			0.0598536
$148$		0			0
$149$	0.968162			0.0793150			0.0396575	−	0.999213i	$-0.487373\pi$
$149$	0.968162			0.0793150			0.0396575	+	0.999213i	$0.487373\pi$
$150$		0			0
$151$	10.3004			0.838234			0.419117	−	0.907932i	$-0.362340\pi$
$151$	10.3004			0.838234			0.419117	+	0.907932i	$0.362340\pi$
$152$		0			0
$153$	0.608105			0.0491624
$154$		0			0
$155$	−5.45386	−	7.72069i	−0.438065	−	0.620140i
$156$		0			0
$157$	−11.5654			−0.923023			−0.461512	−	0.887134i	$-0.652693\pi$
$157$	−11.5654			−0.923023			−0.461512	+	0.887134i	$0.652693\pi$
$158$		0			0
$159$		−	3.71121i		−	0.294318i
$160$		0			0
$161$	3.18706			0.251175
$162$		0			0
$163$	−3.50675			−0.274670			−0.137335	−	0.990525i	$-0.543854\pi$
$163$	−3.50675			−0.274670			−0.137335	+	0.990525i	$0.543854\pi$
$164$		0			0
$165$	0.516698	+	0.731457i	0.0402249	+	0.0569438i
$166$		0			0
$167$			6.80763i			0.526791i	0.964688	+	0.263395i	$0.0848423\pi$
$167$			6.80763i			0.526791i	−0.964688	+	0.263395i	$0.915158\pi$
$168$		0			0
$169$	11.1102			0.854628
$170$		0			0
$171$			19.2038i			1.46855i
$172$		0			0
$173$		−	12.6960i		−	0.965258i	−0.875825	−	0.482629i	$-0.839682\pi$
$173$		−	12.6960i		−	0.965258i	0.875825	−	0.482629i	$-0.160318\pi$
$174$		0			0
$175$	14.1176	+	5.00638i	1.06719	+	0.378447i
$176$		0			0
$177$	−1.94218			−0.145983
$178$		0			0
$179$	3.44971			0.257844			0.128922	−	0.991655i	$-0.458848\pi$
$179$	3.44971			0.257844			0.128922	+	0.991655i	$0.458848\pi$
$180$		0			0
$181$	19.3826			1.44070			0.720350	−	0.693611i	$-0.243981\pi$
$181$	19.3826			1.44070			0.720350	+	0.693611i	$0.243981\pi$
$182$		0			0
$183$			1.24401i			0.0919597i
$184$		0			0
$185$	10.4649	−	7.39234i	0.769393	−	0.543496i
$186$		0			0
$187$			0.231333i			0.0169167i
$188$		0			0
$189$			6.45659i			0.469648i
$190$		0			0
$191$		−	24.5210i		−	1.77428i	−0.461503	−	0.887139i	$-0.652689\pi$
$191$		−	24.5210i		−	1.77428i	0.461503	−	0.887139i	$-0.347311\pi$
$192$		0			0
$193$	10.0101			0.720546			0.360273	−	0.932847i	$-0.382684\pi$
$193$	10.0101			0.720546			0.360273	+	0.932847i	$0.382684\pi$
$194$		0			0
$195$	−0.651735	−	0.922620i	−0.0466717	−	0.0660702i
$196$		0			0
$197$			19.1667i			1.36557i	0.730618	+	0.682786i	$0.239232\pi$
$197$			19.1667i			1.36557i	−0.730618	+	0.682786i	$0.760768\pi$
$198$		0			0
$199$	−14.7249			−1.04382			−0.521910	−	0.853000i	$-0.674781\pi$
$199$	−14.7249			−1.04382			−0.521910	+	0.853000i	$0.674781\pi$
$200$		0			0
$201$		−	3.75977i		−	0.265193i
$202$		0			0
$203$	−14.2930	+	7.48197i	−1.00317	+	0.525132i
$204$		0			0
$205$	−13.3826	−	18.9450i	−0.934684	−	1.32317i
$206$		0			0
$207$			3.04787i			0.211842i
$208$		0			0
$209$	−7.30542			−0.505326
$210$		0			0
$211$			8.19596i			0.564233i	0.959380	+	0.282116i	$0.0910365\pi$
$211$			8.19596i			0.564233i	−0.959380	+	0.282116i	$0.908964\pi$
$212$		0			0
$213$	−1.85404			−0.127037
$214$		0			0
$215$	19.8341	−	14.0107i	1.35267	−	0.955523i
$216$		0			0
$217$	12.6644			0.859713
$218$		0			0
$219$	1.29265			0.0873492
$220$		0			0
$221$		−	0.291791i		−	0.0196280i
$222$		0			0
$223$		−	6.32850i		−	0.423788i	−0.977293	−	0.211894i	$-0.932037\pi$
$223$		−	6.32850i		−	0.423788i	0.977293	−	0.211894i	$-0.0679631\pi$
$224$		0			0
$225$	−4.78774	+	13.5010i	−0.319183	+	0.900069i
$226$		0			0
$227$		−	20.8358i		−	1.38292i	−0.722413	−	0.691461i	$-0.756967\pi$
$227$		−	20.8358i		−	1.38292i	0.722413	−	0.691461i	$-0.243033\pi$
$228$		0			0
$229$		−	14.2925i		−	0.944473i	−0.881472	−	0.472236i	$-0.843447\pi$
$229$		−	14.2925i		−	0.944473i	0.881472	−	0.472236i	$-0.156553\pi$
$230$		0			0
$231$	−1.19982			−0.0789424
$232$		0			0
$233$		−	13.5230i		−	0.885923i	−0.896541	−	0.442961i	$-0.853928\pi$
$233$		−	13.5230i		−	0.885923i	0.896541	−	0.442961i	$-0.146072\pi$
$234$		0			0
$235$	11.6661	−	8.24090i	0.761014	−	0.537577i
$236$		0			0
$237$		−	3.25460i		−	0.211409i
$238$		0			0
$239$	18.6697			1.20764			0.603822	−	0.797119i	$-0.293644\pi$
$239$	18.6697			1.20764			0.603822	+	0.797119i	$0.293644\pi$
$240$		0			0
$241$	−19.6628			−1.26660			−0.633298	−	0.773908i	$-0.718299\pi$
$241$	−19.6628			−1.26660			−0.633298	+	0.773908i	$0.718299\pi$
$242$		0			0
$243$	−9.33301			−0.598713
$244$		0			0
$245$	−3.60668	+	2.54775i	−0.230422	+	0.162769i
$246$		0			0
$247$	9.21465			0.586314
$248$		0			0
$249$			4.56077i			0.289027i
$250$		0			0
$251$			11.5370i			0.728212i	0.931358	+	0.364106i	$0.118625\pi$
$251$			11.5370i			0.728212i	−0.931358	+	0.364106i	$0.881375\pi$
$252$		0			0
$253$	−1.15946			−0.0728945
$254$		0			0
$255$	0.142453	−	0.100628i	0.00892073	−	0.00630156i
$256$		0			0
$257$		−	7.27525i		−	0.453817i	−0.973916	−	0.226909i	$-0.927138\pi$
$257$		−	7.27525i		−	0.453817i	0.973916	−	0.226909i	$-0.0728619\pi$
$258$		0			0
$259$			17.1657i			1.06662i
$260$		0			0
$261$	−7.15522	−	13.6688i	−0.442897	−	0.846076i
$262$		0			0
$263$	29.7334			1.83344			0.916721	−	0.399528i	$-0.130826\pi$
$263$	29.7334			1.83344			0.916721	+	0.399528i	$0.130826\pi$
$264$		0			0
$265$	13.0293	+	18.4448i	0.800387	+	1.13306i
$266$		0			0
$267$			1.08727i			0.0665396i
$268$		0			0
$269$			26.5290i			1.61750i	0.588152	+	0.808750i	$0.299856\pi$
$269$			26.5290i			1.61750i	−0.588152	+	0.808750i	$0.700144\pi$
$270$		0			0
$271$		−	19.2989i		−	1.17232i	−0.810195	−	0.586161i	$-0.800639\pi$
$271$		−	19.2989i		−	1.17232i	0.810195	−	0.586161i	$-0.199361\pi$
$272$		0			0
$273$	1.51339			0.0915944
$274$		0			0
$275$	−5.13601	−	1.82133i	−0.309713	−	0.109831i
$276$		0			0
$277$			11.2042i			0.673194i	0.941649	+	0.336597i	$0.109276\pi$
$277$			11.2042i			0.673194i	−0.941649	+	0.336597i	$0.890724\pi$
$278$		0			0
$279$			12.1113i			0.725083i
$280$		0			0
$281$	−7.60493			−0.453672			−0.226836	−	0.973933i	$-0.572838\pi$
$281$	−7.60493			−0.453672			−0.226836	+	0.973933i	$0.572838\pi$
$282$		0			0
$283$			4.75740i			0.282798i	0.989953	+	0.141399i	$0.0451601\pi$
$283$			4.75740i			0.282798i	−0.989953	+	0.141399i	$0.954840\pi$
$284$		0			0
$285$	3.17779	+	4.49860i	0.188236	+	0.266474i
$286$		0			0
$287$	31.0757			1.83434
$288$		0			0
$289$	−16.9549			−0.997350
$290$		0			0
$291$	−5.90182			−0.345971
$292$		0			0
$293$	22.2904			1.30222			0.651108	−	0.758985i	$-0.274304\pi$
$293$	22.2904			1.30222			0.651108	+	0.758985i	$0.274304\pi$
$294$		0			0
$295$	9.65271	−	6.81863i	0.562002	−	0.396996i
$296$		0			0
$297$		−	2.34892i		−	0.136298i
$298$		0			0
$299$	1.46248			0.0845772
$300$		0			0
$301$			32.5342i			1.87524i
$302$		0			0
$303$			5.05585i			0.290451i
$304$		0			0
$305$	−4.36747	−	6.18276i	−0.250081	−	0.354024i
$306$		0			0
$307$	−19.9531			−1.13878			−0.569392	−	0.822066i	$-0.692821\pi$
$307$	−19.9531			−1.13878			−0.569392	+	0.822066i	$0.692821\pi$
$308$		0			0
$309$		−	3.36831i		−	0.191616i
$310$		0			0
$311$		−	15.8156i		−	0.896820i	−0.893828	−	0.448410i	$-0.851990\pi$
$311$		−	15.8156i		−	0.896820i	0.893828	−	0.448410i	$-0.148010\pi$
$312$		0			0
$313$			9.71119i			0.548909i	0.961600	+	0.274454i	$0.0884973\pi$
$313$			9.71119i			0.548909i	−0.961600	+	0.274454i	$0.911503\pi$
$314$		0			0
$315$	−11.0730	−	15.6753i	−0.623892	−	0.883204i
$316$		0			0
$317$	−31.6058			−1.77516			−0.887580	−	0.460654i	$-0.847615\pi$
$317$	−31.6058			−1.77516			−0.887580	+	0.460654i	$0.847615\pi$
$318$		0			0
$319$	5.19982	−	2.72196i	0.291134	−	0.152400i
$320$		0			0
$321$			0.861157i			0.0480651i
$322$		0			0
$323$			1.42274i			0.0791636i
$324$		0			0
$325$	6.47828	+	2.29733i	0.359350	+	0.127433i
$326$		0			0
$327$	0.412526			0.0228127
$328$		0			0
$329$			19.1361i			1.05501i
$330$		0			0
$331$			18.4649i			1.01492i	0.861674	+	0.507462i	$0.169416\pi$
$331$			18.4649i			1.01492i	−0.861674	+	0.507462i	$0.830584\pi$
$332$		0			0
$333$	−16.4160			−0.899593
$334$		0			0
$335$	13.1998	+	18.6862i	0.721183	+	1.02093i
$336$		0			0
$337$	−11.7729			−0.641311			−0.320656	−	0.947196i	$-0.603903\pi$
$337$	−11.7729			−0.641311			−0.320656	+	0.947196i	$0.603903\pi$
$338$		0			0
$339$	5.41843			0.294289
$340$		0			0
$341$	−4.60733			−0.249501
$342$		0			0
$343$			15.0545i			0.812866i
$344$		0			0
$345$	0.504355	+	0.713983i	0.0271535	+	0.0384396i
$346$		0			0
$347$		−	12.1497i		−	0.652232i	−0.945330	−	0.326116i	$-0.894260\pi$
$347$		−	12.1497i		−	0.652232i	0.945330	−	0.326116i	$-0.105740\pi$
$348$		0			0
$349$	−8.13241			−0.435318			−0.217659	−	0.976025i	$-0.569842\pi$
$349$	−8.13241			−0.435318			−0.217659	+	0.976025i	$0.569842\pi$
$350$		0			0
$351$			2.96280i			0.158143i
$352$		0			0
$353$		−	17.7354i		−	0.943958i	−0.881610	−	0.471979i	$-0.843540\pi$
$353$		−	17.7354i		−	0.943958i	0.881610	−	0.471979i	$-0.156460\pi$
$354$		0			0
$355$	9.21465	−	6.50919i	0.489063	−	0.345472i
$356$		0			0
$357$			0.233667i			0.0123670i
$358$		0			0
$359$		−	9.14009i		−	0.482396i	−0.970476	−	0.241198i	$-0.922460\pi$
$359$		−	9.14009i		−	0.482396i	0.970476	−	0.241198i	$-0.0775402\pi$
$360$		0			0
$361$	−25.9297			−1.36472
$362$		0			0
$363$	−3.60571			−0.189251
$364$		0			0
$365$	−6.42451	+	4.53825i	−0.336274	+	0.237543i
$366$		0			0
$367$	−25.5654			−1.33451			−0.667253	−	0.744831i	$-0.732530\pi$
$367$	−25.5654			−1.33451			−0.667253	+	0.744831i	$0.732530\pi$
$368$		0			0
$369$			29.7186i			1.54709i
$370$		0			0
$371$	−30.2553			−1.57078
$372$		0			0
$373$		−	21.8731i		−	1.13255i	−0.824217	−	0.566273i	$-0.808385\pi$
$373$		−	21.8731i		−	1.13255i	0.824217	−	0.566273i	$-0.191615\pi$
$374$		0			0
$375$	1.11256	+	3.95497i	0.0574524	+	0.204234i
$376$		0			0
$377$	−6.55877	+	3.43333i	−0.337794	+	0.176826i
$378$		0			0
$379$			32.8410i			1.68693i	0.537184	+	0.843465i	$0.319488\pi$
$379$			32.8410i			1.68693i	−0.537184	+	0.843465i	$0.680512\pi$
$380$		0			0
$381$	0.944579			0.0483922
$382$		0			0
$383$			34.6298i			1.76950i	0.466064	+	0.884751i	$0.345672\pi$
$383$			34.6298i			1.76950i	−0.466064	+	0.884751i	$0.654328\pi$
$384$		0			0
$385$	5.96314	−	4.21234i	0.303910	−	0.214681i
$386$		0			0
$387$	−31.1133			−1.58158
$388$		0			0
$389$			12.9103i			0.654580i	0.944924	+	0.327290i	$0.106135\pi$
$389$			12.9103i			0.654580i	−0.944924	+	0.327290i	$0.893865\pi$
$390$		0			0
$391$			0.225807i			0.0114195i
$392$		0			0
$393$		−	0.443531i		−	0.0223732i
$394$		0			0
$395$	11.4263	+	16.1754i	0.574918	+	0.813875i
$396$		0			0
$397$		−	28.7398i		−	1.44241i	−0.692723	−	0.721204i	$-0.743589\pi$
$397$		−	28.7398i		−	1.44241i	0.692723	−	0.721204i	$-0.256411\pi$
$398$		0			0
$399$	−7.37913			−0.369419
$400$		0			0
$401$	24.2030			1.20864			0.604320	−	0.796742i	$-0.293445\pi$
$401$	24.2030			1.20864			0.604320	+	0.796742i	$0.293445\pi$
$402$		0			0
$403$	5.81143			0.289488
$404$		0			0
$405$	14.2509	−	10.0668i	0.708131	−	0.500221i
$406$		0			0
$407$		−	6.24492i		−	0.309549i
$408$		0			0
$409$			17.4820i			0.864431i	0.901770	+	0.432216i	$0.142268\pi$
$409$			17.4820i			0.864431i	−0.901770	+	0.432216i	$0.857732\pi$
$410$		0			0
$411$	2.13698			0.105410
$412$		0			0
$413$			15.8335i			0.779115i
$414$		0			0
$415$	−16.0120	−	22.6672i	−0.785997	−	1.11269i
$416$		0			0
$417$	5.52781			0.270698
$418$		0			0
$419$	−26.2378			−1.28180			−0.640900	−	0.767625i	$-0.721439\pi$
$419$	−26.2378			−1.28180			−0.640900	+	0.767625i	$0.721439\pi$
$420$		0			0
$421$		−	13.6673i		−	0.666105i	−0.942908	−	0.333053i	$-0.891921\pi$
$421$		−	13.6673i		−	0.666105i	0.942908	−	0.333053i	$-0.108079\pi$
$422$		0			0
$423$	−18.3004			−0.889796
$424$		0			0
$425$	−0.354708	+	1.00025i	−0.0172059	+	0.0485191i
$426$		0			0
$427$	10.1417			0.490790
$428$		0			0
$429$	−0.550574			−0.0265820
$430$		0			0
$431$	20.1544			0.970805			0.485403	−	0.874291i	$-0.338673\pi$
$431$	20.1544			0.970805			0.485403	+	0.874291i	$0.338673\pi$
$432$		0			0
$433$	−9.30542			−0.447190			−0.223595	−	0.974682i	$-0.571779\pi$
$433$	−9.30542			−0.447190			−0.223595	+	0.974682i	$0.571779\pi$
$434$		0			0
$435$	−3.93803	−	2.01797i	−0.188814	−	0.0967542i
$436$		0			0
$437$	−7.13090			−0.341117
$438$		0			0
$439$	−38.2655			−1.82631			−0.913156	−	0.407610i	$-0.866362\pi$
$439$	−38.2655			−1.82631			−0.913156	+	0.407610i	$0.866362\pi$
$440$		0			0
$441$	5.65773			0.269416
$442$		0			0
$443$	−34.1561			−1.62281			−0.811403	−	0.584488i	$-0.801296\pi$
$443$	−34.1561			−1.62281			−0.811403	+	0.584488i	$0.801296\pi$
$444$		0			0
$445$	−3.81719	−	5.40375i	−0.180952	−	0.256162i
$446$		0			0
$447$	−0.355774			−0.0168275
$448$		0			0
$449$			2.90669i			0.137175i	0.997645	+	0.0685876i	$0.0218493\pi$
$449$			2.90669i			0.137175i	−0.997645	+	0.0685876i	$0.978151\pi$
$450$		0			0
$451$	−11.3054			−0.532351
$452$		0			0
$453$	−3.78512			−0.177840
$454$		0			0
$455$	−7.52158	+	5.31321i	−0.352617	+	0.249087i
$456$		0			0
$457$		−	2.65336i		−	0.124119i	−0.998072	−	0.0620596i	$-0.980233\pi$
$457$		−	2.65336i		−	0.124119i	0.998072	−	0.0620596i	$-0.0197669\pi$
$458$		0			0
$459$	−0.457457			−0.0213523
$460$		0			0
$461$		−	6.53979i		−	0.304589i	−0.988335	−	0.152294i	$-0.951334\pi$
$461$		−	6.53979i		−	0.304589i	0.988335	−	0.152294i	$-0.0486662\pi$
$462$		0			0
$463$		−	27.1658i		−	1.26250i	−0.775579	−	0.631250i	$-0.782542\pi$
$463$		−	27.1658i		−	1.26250i	0.775579	−	0.631250i	$-0.217458\pi$
$464$		0			0
$465$	2.00415	+	2.83715i	0.0929401	+	0.131570i
$466$		0			0
$467$	−32.2561			−1.49264			−0.746318	−	0.665590i	$-0.768180\pi$
$467$	−32.2561			−1.49264			−0.746318	+	0.665590i	$0.768180\pi$
$468$		0			0
$469$	−30.6512			−1.41534
$470$		0			0
$471$	4.24999			0.195829
$472$		0			0
$473$		−	11.8360i		−	0.544221i
$474$		0			0
$475$	−31.5875	−	11.2016i	−1.44933	−	0.513963i
$476$		0			0
$477$		−	28.9340i		−	1.32480i
$478$		0			0
$479$			25.8295i			1.18018i	0.807337	+	0.590090i	$0.200908\pi$
$479$			25.8295i			1.18018i	−0.807337	+	0.590090i	$0.799092\pi$
$480$		0			0
$481$			7.87700i			0.359160i
$482$		0			0
$483$	−1.17116			−0.0532895
$484$		0			0
$485$	29.3322	−	20.7202i	1.33191	−	0.940854i
$486$		0			0
$487$			1.63351i			0.0740215i	0.999315	+	0.0370107i	$0.0117836\pi$
$487$			1.63351i			0.0740215i	−0.999315	+	0.0370107i	$0.988216\pi$
$488$		0			0
$489$	1.28864			0.0582742
$490$		0			0
$491$			10.3456i			0.466892i	0.972370	+	0.233446i	$0.0750002\pi$
$491$			10.3456i			0.466892i	−0.972370	+	0.233446i	$0.925000\pi$
$492$		0			0
$493$	−0.530107	−	1.01267i	−0.0238748	−	0.0456086i
$494$		0			0
$495$	4.02838	+	5.70272i	0.181062	+	0.256318i
$496$		0			0
$497$			15.1149i			0.677998i
$498$		0			0
$499$	−0.704726			−0.0315479			−0.0157739	−	0.999876i	$-0.505021\pi$
$499$	−0.704726			−0.0315479			−0.0157739	+	0.999876i	$0.505021\pi$
$500$		0			0
$501$		−	2.50162i		−	0.111764i
$502$		0			0
$503$	6.19285			0.276126			0.138063	−	0.990423i	$-0.455912\pi$
$503$	6.19285			0.276126			0.138063	+	0.990423i	$0.455912\pi$
$504$		0			0
$505$	−17.7501	−	25.1277i	−0.789869	−	1.11817i
$506$		0			0
$507$	−4.08269			−0.181319
$508$		0			0
$509$	−14.6292			−0.648426			−0.324213	−	0.945984i	$-0.605100\pi$
$509$	−14.6292			−0.648426			−0.324213	+	0.945984i	$0.605100\pi$
$510$		0			0
$511$		−	10.5382i		−	0.466184i
$512$		0			0
$513$		−	14.4463i		−	0.637822i
$514$		0			0
$515$	11.8255	+	16.7406i	0.521093	+	0.737679i
$516$		0			0
$517$		−	6.96176i		−	0.306178i
$518$		0			0
$519$			4.66544i			0.204790i
$520$		0			0
$521$	−6.40511			−0.280613			−0.140306	−	0.990108i	$-0.544809\pi$
$521$	−6.40511			−0.280613			−0.140306	+	0.990108i	$0.544809\pi$
$522$		0			0
$523$		−	16.5805i		−	0.725013i	−0.931981	−	0.362507i	$-0.881921\pi$
$523$		−	16.5805i		−	0.725013i	0.931981	−	0.362507i	$-0.118079\pi$
$524$		0			0
$525$	−5.18783	−	1.83971i	−0.226416	−	0.0802917i
$526$		0			0
$527$			0.897285i			0.0390864i
$528$		0			0
$529$	21.8682			0.950793
$530$		0			0
$531$	−15.1420			−0.657107
$532$		0			0
$533$	14.2600			0.617671
$534$		0			0
$535$	−3.02336	−	4.27997i	−0.130711	−	0.185039i
$536$		0			0
$537$	−1.26768			−0.0547043
$538$		0			0
$539$			2.15229i			0.0927057i
$540$		0			0
$541$			40.5567i			1.74367i	0.489803	+	0.871833i	$0.337069\pi$
$541$			40.5567i			1.74367i	−0.489803	+	0.871833i	$0.662931\pi$
$542$		0			0
$543$	−7.12260			−0.305660
$544$		0			0
$545$	−2.05027	+	1.44830i	−0.0878238	+	0.0620383i
$546$		0			0
$547$		−	43.0100i		−	1.83897i	−0.393119	−	0.919487i	$-0.628604\pi$
$547$		−	43.0100i		−	1.83897i	0.393119	−	0.919487i	$-0.371396\pi$
$548$		0			0
$549$			9.69876i			0.413933i
$550$		0			0
$551$	31.9799	−	16.7406i	1.36239	−	0.713173i
$552$		0			0
$553$	−26.5328			−1.12829
$554$		0			0
$555$	−3.84556	+	2.71649i	−0.163235	+	0.115309i
$556$		0			0
$557$			35.0002i			1.48301i	0.670949	+	0.741503i	$0.265887\pi$
$557$			35.0002i			1.48301i	−0.670949	+	0.741503i	$0.734113\pi$
$558$		0			0
$559$			14.9293i			0.631442i
$560$		0			0
$561$		−	0.0850087i		−	0.00358907i
$562$		0			0
$563$	7.82733			0.329882			0.164941	−	0.986303i	$-0.447257\pi$
$563$	7.82733			0.329882			0.164941	+	0.986303i	$0.447257\pi$
$564$		0			0
$565$	−26.9297	+	19.0231i	−1.13294	+	0.800306i
$566$		0			0
$567$			23.3759i			0.981696i
$568$		0			0
$569$		−	16.3098i		−	0.683741i	−0.939747	−	0.341870i	$-0.888940\pi$
$569$		−	16.3098i		−	0.683741i	0.939747	−	0.341870i	$-0.111060\pi$
$570$		0			0
$571$	26.6814			1.11658			0.558291	−	0.829645i	$-0.311457\pi$
$571$	26.6814			1.11658			0.558291	+	0.829645i	$0.311457\pi$
$572$		0			0
$573$			9.01082i			0.376432i
$574$		0			0
$575$	−5.01332	−	1.77783i	−0.209070	−	0.0741404i
$576$		0			0
$577$	−34.3986			−1.43203			−0.716016	−	0.698084i	$-0.754036\pi$
$577$	−34.3986			−1.43203			−0.716016	+	0.698084i	$0.754036\pi$
$578$		0			0
$579$	−3.67846			−0.152872
$580$		0			0
$581$	37.1813			1.54254
$582$		0			0
$583$	11.0070			0.455862
$584$		0			0
$585$	−5.08117	−	7.19310i	−0.210081	−	0.297398i
$586$		0			0
$587$			25.7402i			1.06241i	0.847243	+	0.531205i	$0.178261\pi$
$587$			25.7402i			1.06241i	−0.847243	+	0.531205i	$0.821739\pi$
$588$		0			0
$589$	−28.3360			−1.16756
$590$		0			0
$591$		−	7.04326i		−	0.289721i
$592$		0			0
$593$			8.07625i			0.331652i	0.986155	+	0.165826i	$0.0530290\pi$
$593$			8.07625i			0.331652i	−0.986155	+	0.165826i	$0.946971\pi$
$594$		0			0
$595$	−0.820361	−	1.16133i	−0.0336315	−	0.0476100i
$596$		0			0
$597$	5.41101			0.221458
$598$		0			0
$599$		−	6.48120i		−	0.264815i	−0.991195	−	0.132407i	$-0.957729\pi$
$599$		−	6.48120i		−	0.264815i	0.991195	−	0.132407i	$-0.0422707\pi$
$600$		0			0
$601$		−	9.59123i		−	0.391235i	−0.980680	−	0.195617i	$-0.937329\pi$
$601$		−	9.59123i		−	0.391235i	0.980680	−	0.195617i	$-0.0626710\pi$
$602$		0			0
$603$		−	29.3126i		−	1.19370i
$604$		0			0
$605$	17.9205	−	12.6590i	0.728571	−	0.514660i
$606$		0			0
$607$	28.1680			1.14330			0.571652	−	0.820496i	$-0.306303\pi$
$607$	28.1680			1.14330			0.571652	+	0.820496i	$0.306303\pi$
$608$		0			0
$609$	5.25229	−	2.74943i	0.212833	−	0.111412i
$610$		0			0
$611$			8.78119i			0.355249i
$612$		0			0
$613$			41.3758i			1.67115i	0.549373	+	0.835577i	$0.314867\pi$
$613$			41.3758i			1.67115i	−0.549373	+	0.835577i	$0.685133\pi$
$614$		0			0
$615$	4.91776	+	6.96176i	0.198303	+	0.280725i
$616$		0			0
$617$	9.70399			0.390668			0.195334	−	0.980737i	$-0.437421\pi$
$617$	9.70399			0.390668			0.195334	+	0.980737i	$0.437421\pi$
$618$		0			0
$619$			29.7952i			1.19757i	0.800909	+	0.598786i	$0.204350\pi$
$619$			29.7952i			1.19757i	−0.800909	+	0.598786i	$0.795650\pi$
$620$		0			0
$621$		−	2.29281i		−	0.0920073i
$622$		0			0
$623$	8.86386			0.355123
$624$		0			0
$625$	−19.4146	−	15.7503i	−0.776584	−	0.630013i
$626$		0			0
$627$	2.68455			0.107210
$628$		0			0
$629$	−1.21621			−0.0484935
$630$		0			0
$631$	35.9799			1.43234			0.716169	−	0.697927i	$-0.245894\pi$
$631$	35.9799			1.43234			0.716169	+	0.697927i	$0.245894\pi$
$632$		0			0
$633$		−	3.01180i		−	0.119708i
$634$		0			0
$635$	−4.69458	+	3.31623i	−0.186299	+	0.131601i
$636$		0			0
$637$		−	2.71478i		−	0.107564i
$638$		0			0
$639$	−14.4548			−0.571824
$640$		0			0
$641$			7.21413i			0.284941i	0.989799	+	0.142471i	$0.0455046\pi$
$641$			7.21413i			0.284941i	−0.989799	+	0.142471i	$0.954495\pi$
$642$		0			0
$643$			16.4543i			0.648895i	0.945904	+	0.324448i	$0.105178\pi$
$643$			16.4543i			0.648895i	−0.945904	+	0.324448i	$0.894822\pi$
$644$		0			0
$645$	−7.28850	+	5.14857i	−0.286984	+	0.202725i
$646$		0			0
$647$		−	30.8486i		−	1.21278i	−0.795166	−	0.606392i	$-0.792616\pi$
$647$		−	30.8486i		−	1.21278i	0.795166	−	0.606392i	$-0.207384\pi$
$648$		0			0
$649$		−	5.76026i		−	0.226110i
$650$		0			0
$651$	−4.65382			−0.182397
$652$		0			0
$653$	22.3089			0.873014			0.436507	−	0.899701i	$-0.356215\pi$
$653$	22.3089			0.873014			0.436507	+	0.899701i	$0.356215\pi$
$654$		0			0
$655$	1.55715	+	2.20436i	0.0608430	+	0.0861316i
$656$		0			0
$657$	10.0780			0.393180
$658$		0			0
$659$			27.0186i			1.05249i	0.850332	+	0.526247i	$0.176401\pi$
$659$			27.0186i			1.05249i	−0.850332	+	0.526247i	$0.823599\pi$
$660$		0			0
$661$	−9.63068			−0.374590			−0.187295	−	0.982304i	$-0.559972\pi$
$661$	−9.63068			−0.374590			−0.187295	+	0.982304i	$0.559972\pi$
$662$		0			0
$663$			0.107225i			0.00416429i
$664$		0			0
$665$	36.6745	−	25.9067i	1.42218	−	1.00462i
$666$		0			0
$667$	5.07560	−	2.65694i	0.196528	−	0.102877i
$668$		0			0
$669$			2.32555i			0.0899111i
$670$		0			0
$671$	−3.68956			−0.142434
$672$		0			0
$673$			46.9574i			1.81007i	0.425334	+	0.905037i	$0.360157\pi$
$673$			46.9574i			1.81007i	−0.425334	+	0.905037i	$0.639843\pi$
$674$		0			0
$675$	3.60166	−	10.1564i	0.138628	−	0.390919i
$676$		0			0
$677$	28.1645			1.08245			0.541224	−	0.840878i	$-0.317961\pi$
$677$	28.1645			1.08245			0.541224	+	0.840878i	$0.317961\pi$
$678$		0			0
$679$			48.1141i			1.84645i
$680$		0			0
$681$			7.65661i			0.293402i
$682$		0			0
$683$		−	19.8006i		−	0.757650i	−0.925468	−	0.378825i	$-0.876328\pi$
$683$		−	19.8006i		−	0.757650i	0.925468	−	0.378825i	$-0.123672\pi$
$684$		0			0
$685$	−10.6209	+	7.50254i	−0.405803	+	0.286657i
$686$		0			0
$687$			5.25210i			0.200380i
$688$		0			0
$689$	−13.8836			−0.528922
$690$		0			0
$691$	−3.64179			−0.138540			−0.0692701	−	0.997598i	$-0.522067\pi$
$691$	−3.64179			−0.138540			−0.0692701	+	0.997598i	$0.522067\pi$
$692$		0			0
$693$	−9.35426			−0.355339
$694$		0			0
$695$	−27.4734	+	19.4071i	−1.04213	+	0.736153i
$696$		0			0
$697$			2.20175i			0.0833973i
$698$		0			0
$699$			4.96935i			0.187958i
$700$		0			0
$701$	−24.5639			−0.927767			−0.463883	−	0.885896i	$-0.653544\pi$
$701$	−24.5639			−0.927767			−0.463883	+	0.885896i	$0.653544\pi$
$702$		0			0
$703$		−	38.4075i		−	1.44857i
$704$		0			0
$705$	−4.28699	+	3.02831i	−0.161457	+	0.114053i
$706$		0			0
$707$	41.2174			1.55014
$708$		0			0
$709$	36.5676			1.37333			0.686663	−	0.726976i	$-0.259075\pi$
$709$	36.5676			1.37333			0.686663	+	0.726976i	$0.259075\pi$
$710$		0			0
$711$		−	25.3741i		−	0.951602i
$712$		0			0
$713$	−4.49726			−0.168424
$714$		0			0
$715$	2.73637	−	1.93296i	0.102334	−	0.0722886i
$716$		0			0
$717$	−6.86062			−0.256215
$718$		0			0
$719$	4.40968			0.164453			0.0822267	−	0.996614i	$-0.473797\pi$
$719$	4.40968			0.164453			0.0822267	+	0.996614i	$0.473797\pi$
$720$		0			0
$721$	−27.4599			−1.02266
$722$		0			0
$723$	7.22557			0.268722
$724$		0			0
$725$	26.6569	−	3.79632i	0.990011	−	0.140992i
$726$		0			0
$727$	−37.9900			−1.40897			−0.704485	−	0.709719i	$-0.748822\pi$
$727$	−37.9900			−1.40897			−0.704485	+	0.709719i	$0.748822\pi$
$728$		0			0
$729$	−19.9791			−0.739966
$730$		0			0
$731$	−2.30509			−0.0852567
$732$		0			0
$733$	6.57895			0.242999			0.121500	−	0.992591i	$-0.461230\pi$
$733$	6.57895			0.242999			0.121500	+	0.992591i	$0.461230\pi$
$734$		0			0
$735$	1.32536	−	0.936228i	0.0488866	−	0.0345333i
$736$		0			0
$737$	11.1510			0.410752
$738$		0			0
$739$			26.6171i			0.979127i	0.871968	+	0.489563i	$0.162844\pi$
$739$			26.6171i			0.979127i	−0.871968	+	0.489563i	$0.837156\pi$
$740$		0			0
$741$	−3.38614			−0.124393
$742$		0			0
$743$	4.67606			0.171548			0.0857741	−	0.996315i	$-0.472664\pi$
$743$	4.67606			0.171548			0.0857741	+	0.996315i	$0.472664\pi$
$744$		0			0
$745$	1.76821	−	1.24905i	0.0647821	−	0.0457618i
$746$		0			0
$747$			35.5575i			1.30098i
$748$		0			0
$749$	7.02051			0.256524
$750$		0			0
$751$		−	12.3371i		−	0.450187i	−0.974337	−	0.225093i	$-0.927731\pi$
$751$		−	12.3371i		−	0.450187i	0.974337	−	0.225093i	$-0.0722687\pi$
$752$		0			0
$753$		−	4.23956i		−	0.154498i
$754$		0			0
$755$	18.8122	−	13.2888i	0.684645	−	0.483630i
$756$		0			0
$757$	10.2685			0.373216			0.186608	−	0.982434i	$-0.440251\pi$
$757$	10.2685			0.373216			0.186608	+	0.982434i	$0.440251\pi$
$758$		0			0
$759$	0.426070			0.0154654
$760$		0			0
$761$	49.1824			1.78286			0.891430	−	0.453159i	$-0.149703\pi$
$761$	49.1824			1.78286			0.891430	+	0.453159i	$0.149703\pi$
$762$		0			0
$763$		−	3.36308i		−	0.121752i
$764$		0			0
$765$	1.11062	−	0.784534i	0.0401544	−	0.0283649i
$766$		0			0
$767$			7.26568i			0.262348i
$768$		0			0
$769$			36.4972i			1.31612i	0.752964	+	0.658062i	$0.228624\pi$
$769$			36.4972i			1.31612i	−0.752964	+	0.658062i	$0.771376\pi$
$770$		0			0
$771$			2.67346i			0.0962823i
$772$		0			0
$773$	−29.4854			−1.06052			−0.530258	−	0.847836i	$-0.677905\pi$
$773$	−29.4854			−1.06052			−0.530258	+	0.847836i	$0.677905\pi$
$774$		0			0
$775$	−19.9214	−	7.06452i	−0.715596	−	0.253765i
$776$		0			0
$777$		−	6.30794i		−	0.226296i
$778$		0			0
$779$	−69.5305			−2.49119
$780$		0			0
$781$		−	5.49885i		−	0.196764i
$782$		0			0
$783$	5.38263	+	10.2826i	0.192360	+	0.367469i
$784$		0			0
$785$	−21.1226	+	14.9209i	−0.753898	+	0.532550i
$786$		0			0
$787$			38.4961i			1.37224i	0.727490	+	0.686119i	$0.240687\pi$
$787$			38.4961i			1.37224i	−0.727490	+	0.686119i	$0.759313\pi$
$788$		0			0
$789$	−10.9262			−0.388985
$790$		0			0
$791$		−	44.1733i		−	1.57062i
$792$		0			0
$793$	4.65382			0.165262
$794$		0			0
$795$	−4.78794	−	6.77798i	−0.169811	−	0.240390i
$796$		0			0
$797$	44.1311			1.56320			0.781602	−	0.623777i	$-0.214403\pi$
$797$	44.1311			1.56320			0.781602	+	0.623777i	$0.214403\pi$
$798$		0			0
$799$	−1.35582			−0.0479653
$800$		0			0
$801$			8.47675i			0.299511i
$802$		0			0
$803$			3.83383i			0.135293i
$804$		0			0
$805$	5.82069	−	4.11171i	0.205152	−	0.144919i
$806$		0			0
$807$		−	9.74869i		−	0.343170i
$808$		0			0
$809$		−	8.79023i		−	0.309048i	−0.987989	−	0.154524i	$-0.950616\pi$
$809$		−	8.79023i		−	0.309048i	0.987989	−	0.154524i	$-0.0493844\pi$
$810$		0			0
$811$	3.91186			0.137364			0.0686820	−	0.997639i	$-0.478121\pi$
$811$	3.91186			0.137364			0.0686820	+	0.997639i	$0.478121\pi$
$812$		0			0
$813$			7.09181i			0.248721i
$814$		0			0
$815$	−6.40457	+	4.52416i	−0.224342	+	0.158474i
$816$		0			0
$817$		−	72.7938i		−	2.54673i
$818$		0			0
$819$	11.7990			0.412289
$820$		0			0
$821$	−8.59729			−0.300047			−0.150024	−	0.988682i	$-0.547935\pi$
$821$	−8.59729			−0.300047			−0.150024	+	0.988682i	$0.547935\pi$
$822$		0			0
$823$	2.31892			0.0808323			0.0404162	−	0.999183i	$-0.487132\pi$
$823$	2.31892			0.0808323			0.0404162	+	0.999183i	$0.487132\pi$
$824$		0			0
$825$	1.88735	+	0.669292i	0.0657090	+	0.0233018i
$826$		0			0
$827$	13.5864			0.472446			0.236223	−	0.971699i	$-0.424090\pi$
$827$	13.5864			0.472446			0.236223	+	0.971699i	$0.424090\pi$
$828$		0			0
$829$			24.1451i			0.838594i	0.907849	+	0.419297i	$0.137723\pi$
$829$			24.1451i			0.838594i	−0.907849	+	0.419297i	$0.862277\pi$
$830$		0			0
$831$		−	4.11724i		−	0.142825i
$832$		0			0
$833$	0.419163			0.0145231
$834$		0			0
$835$	8.78272	+	12.4331i	0.303939	+	0.430267i
$836$		0			0
$837$		−	9.11092i		−	0.314919i
$838$		0			0
$839$		−	19.9698i		−	0.689434i	−0.938707	−	0.344717i	$-0.887975\pi$
$839$		−	19.9698i		−	0.689434i	0.938707	−	0.344717i	$-0.112025\pi$
$840$		0			0
$841$	−16.5251	+	23.8311i	−0.569831	+	0.821762i
$842$		0			0
$843$	2.79461			0.0962514
$844$		0			0
$845$	20.2911	−	14.3335i	0.698035	−	0.493089i
$846$		0			0
$847$			29.3953i			1.01003i
$848$		0			0
$849$		−	1.74822i		−	0.0599987i
$850$		0			0
$851$		−	6.09574i		−	0.208959i
$852$		0			0
$853$	−36.0840			−1.23549			−0.617746	−	0.786378i	$-0.711954\pi$
$853$	−36.0840			−1.23549			−0.617746	+	0.786378i	$0.711954\pi$
$854$		0			0
$855$	24.7753	+	35.0728i	0.847298	+	1.19947i
$856$		0			0
$857$			19.0491i			0.650705i	0.945593	+	0.325352i	$0.105483\pi$
$857$			19.0491i			0.650705i	−0.945593	+	0.325352i	$0.894517\pi$
$858$		0			0
$859$		−	43.1051i		−	1.47073i	−0.677673	−	0.735364i	$-0.737011\pi$
$859$		−	43.1051i		−	1.47073i	0.677673	−	0.735364i	$-0.262989\pi$
$860$		0			0
$861$	−11.4195			−0.389175
$862$		0			0
$863$			25.9382i			0.882947i	0.897274	+	0.441474i	$0.145544\pi$
$863$			25.9382i			0.882947i	−0.897274	+	0.441474i	$0.854456\pi$
$864$		0			0
$865$	−16.3795	−	23.1874i	−0.556918	−	0.788394i
$866$		0			0
$867$	6.23049			0.211599
$868$		0			0
$869$	9.65271			0.327446
$870$		0			0
$871$	−14.0652			−0.476582
$872$		0			0
$873$	−46.0129			−1.55730
$874$		0			0
$875$	32.2426	−	9.07007i	1.09000	−	0.306624i
$876$		0			0
$877$			43.0492i			1.45367i	0.686812	+	0.726835i	$0.259009\pi$
$877$			43.0492i			1.45367i	−0.686812	+	0.726835i	$0.740991\pi$
$878$		0			0
$879$	−8.19111			−0.276279
$880$		0			0
$881$		−	49.8210i		−	1.67851i	−0.543736	−	0.839256i	$-0.682991\pi$
$881$		−	49.8210i		−	1.67851i	0.543736	−	0.839256i	$-0.317009\pi$
$882$		0			0
$883$		−	9.29618i		−	0.312841i	−0.987691	−	0.156421i	$-0.950004\pi$
$883$		−	9.29618i		−	0.312841i	0.987691	−	0.156421i	$-0.0499955\pi$
$884$		0			0
$885$	−3.54711	+	2.50567i	−0.119235	+	0.0842270i
$886$		0			0
$887$	1.49827			0.0503070			0.0251535	−	0.999684i	$-0.491993\pi$
$887$	1.49827			0.0503070			0.0251535	+	0.999684i	$0.491993\pi$
$888$		0			0
$889$		−	7.70060i		−	0.258270i
$890$		0			0
$891$		−	8.50422i		−	0.284902i
$892$		0			0
$893$		−	42.8162i		−	1.43279i
$894$		0			0
$895$	6.30040	−	4.45057i	0.210599	−	0.148766i
$896$		0			0
$897$	−0.537421			−0.0179440
$898$		0			0
$899$	20.1689	−	10.5578i	0.672669	−	0.352123i
$900$		0			0
$901$		−	2.14363i		−	0.0714146i
$902$		0			0
$903$		−	11.9554i		−	0.397852i
$904$		0			0
$905$	35.3995	−	25.0061i	1.17672	−	0.831231i
$906$		0			0
$907$	32.7075			1.08603			0.543017	−	0.839722i	$-0.317282\pi$
$907$	32.7075			1.08603			0.543017	+	0.839722i	$0.317282\pi$
$908$		0			0
$909$			39.4173i			1.30739i
$910$		0			0
$911$		−	38.5452i		−	1.27706i	−0.769598	−	0.638529i	$-0.779543\pi$
$911$		−	38.5452i		−	1.27706i	0.769598	−	0.638529i	$-0.220457\pi$
$912$		0			0
$913$	−13.5266			−0.447667
$914$		0			0
$915$	1.60493	+	2.27200i	0.0530573	+	0.0751099i
$916$		0			0
$917$	−3.61585			−0.119406
$918$		0			0
$919$	36.8375			1.21516			0.607579	−	0.794259i	$-0.292141\pi$
$919$	36.8375			1.21516			0.607579	+	0.794259i	$0.292141\pi$
$920$		0			0
$921$	7.33223			0.241605
$922$		0			0
$923$			6.93595i			0.228300i
$924$		0			0
$925$	9.57549	−	27.0021i	0.314840	−	0.887822i
$926$		0			0
$927$		−	26.2606i		−	0.862512i
$928$		0			0
$929$	29.3787			0.963885			0.481942	−	0.876203i	$-0.339931\pi$
$929$	29.3787			0.963885			0.481942	+	0.876203i	$0.339931\pi$
$930$		0			0
$931$			13.2370i			0.433826i
$932$		0			0
$933$			5.81181i			0.190270i
$934$		0			0
$935$	0.298449	+	0.422496i	0.00976033	+	0.0138171i
$936$		0			0
$937$		−	29.1188i		−	0.951270i	−0.879643	−	0.475635i	$-0.842218\pi$
$937$		−	29.1188i		−	0.951270i	0.879643	−	0.475635i	$-0.157782\pi$
$938$		0			0
$939$		−	3.56860i		−	0.116457i
$940$		0			0
$941$	−34.4784			−1.12396			−0.561982	−	0.827149i	$-0.689961\pi$
$941$	−34.4784			−1.12396			−0.561982	+	0.827149i	$0.689961\pi$
$942$		0			0
$943$	−11.0353			−0.359360
$944$		0			0
$945$	8.32984	+	11.7920i	0.270970	+	0.383595i
$946$		0			0
$947$	−16.7368			−0.543873			−0.271936	−	0.962315i	$-0.587664\pi$
$947$	−16.7368			−0.543873			−0.271936	+	0.962315i	$0.587664\pi$
$948$		0			0
$949$		−	4.83579i		−	0.156976i
$950$		0			0
$951$	11.6143			0.376619
$952$		0			0
$953$			49.5247i			1.60426i	0.597149	+	0.802131i	$0.296300\pi$
$953$			49.5247i			1.60426i	−0.597149	+	0.802131i	$0.703700\pi$
$954$		0			0
$955$	−31.6353	−	44.7840i	−1.02369	−	1.44918i
$956$		0			0
$957$	−1.91080	+	1.00025i	−0.0617672	+	0.0323334i
$958$		0			0
$959$		−	17.4216i		−	0.562572i
$960$		0			0
$961$	13.1293			0.423525
$962$		0			0
$963$			6.71391i			0.216353i
$964$		0			0
$965$	18.2821	−	12.9144i	0.588520	−	0.415728i
$966$		0			0
$967$	−28.3924			−0.913040			−0.456520	−	0.889713i	$-0.650904\pi$
$967$	−28.3924			−0.913040			−0.456520	+	0.889713i	$0.650904\pi$
$968$		0			0
$969$		−	0.522820i		−	0.0167954i
$970$		0			0
$971$			4.86155i			0.156014i	0.996953	+	0.0780072i	$0.0248557\pi$
$971$			4.86155i			0.156014i	−0.996953	+	0.0780072i	$0.975144\pi$
$972$		0			0
$973$		−	45.0650i		−	1.44472i
$974$		0			0
$975$	−2.38060	−	0.844208i	−0.0762401	−	0.0270363i
$976$		0			0
$977$			34.0078i			1.08801i	0.839083	+	0.544003i	$0.183092\pi$
$977$			34.0078i			1.08801i	−0.839083	+	0.544003i	$0.816908\pi$
$978$		0			0
$979$	−3.22469			−0.103062
$980$		0			0
$981$	3.21621			0.102686
$982$		0			0
$983$	−38.6176			−1.23171			−0.615855	−	0.787860i	$-0.711189\pi$
$983$	−38.6176			−1.23171			−0.615855	+	0.787860i	$0.711189\pi$
$984$		0			0
$985$	24.7275	+	35.0052i	0.787885	+	1.11536i
$986$		0			0
$987$		−	7.03201i		−	0.223831i
$988$		0			0
$989$		−	11.5533i		−	0.367372i
$990$		0			0
$991$	−26.3189			−0.836048			−0.418024	−	0.908436i	$-0.637277\pi$
$991$	−26.3189			−0.836048			−0.418024	+	0.908436i	$0.637277\pi$
$992$		0			0
$993$		−	6.78536i		−	0.215327i
$994$		0			0
$995$	−26.8929	+	18.9970i	−0.852562	+	0.602246i
$996$		0			0
$997$	−34.2096			−1.08343			−0.541715	−	0.840562i	$-0.682225\pi$
$997$	−34.2096			−1.08343			−0.541715	+	0.840562i	$0.682225\pi$
$998$		0			0
$999$	12.3492			0.390713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.j.e.289.3		8
4.3	odd	2		290.2.d.b.289.5	yes	8
5.4	even	2		2320.2.j.d.289.6		8
12.11	even	2		2610.2.b.d.289.2		8
20.3	even	4		1450.2.c.g.1101.5		16
20.7	even	4		1450.2.c.g.1101.12		16
20.19	odd	2		290.2.d.a.289.4	yes	8
29.28	even	2		2320.2.j.d.289.5		8
60.59	even	2		2610.2.b.f.289.1		8
116.115	odd	2		290.2.d.a.289.3	✓	8
145.144	even	2	inner	2320.2.j.e.289.4		8
348.347	even	2		2610.2.b.f.289.2		8
580.347	even	4		1450.2.c.g.1101.6		16
580.463	even	4		1450.2.c.g.1101.11		16
580.579	odd	2		290.2.d.b.289.6	yes	8
1740.1739	even	2		2610.2.b.d.289.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
290.2.d.a.289.3	✓	8	116.115	odd	2
290.2.d.a.289.4	yes	8	20.19	odd	2
290.2.d.b.289.5	yes	8	4.3	odd	2
290.2.d.b.289.6	yes	8	580.579	odd	2
1450.2.c.g.1101.5		16	20.3	even	4
1450.2.c.g.1101.6		16	580.347	even	4
1450.2.c.g.1101.11		16	580.463	even	4
1450.2.c.g.1101.12		16	20.7	even	4
2320.2.j.d.289.5		8	29.28	even	2
2320.2.j.d.289.6		8	5.4	even	2
2320.2.j.e.289.3		8	1.1	even	1	trivial
2320.2.j.e.289.4		8	145.144	even	2	inner
2610.2.b.d.289.1		8	1740.1739	even	2
2610.2.b.d.289.2		8	12.11	even	2
2610.2.b.f.289.1		8	60.59	even	2
2610.2.b.f.289.2		8	348.347	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:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.367473 q^{3} +(1.82635 - 1.29013i) q^{5} +2.99580i q^{7} -2.86496 q^{9} -1.08988i q^{11} +1.37471i q^{13} +(-0.671136 + 0.474088i) q^{15} -0.212256 q^{17} -6.70297i q^{19} -1.10088i q^{21} -1.06384i q^{23} +(1.67114 - 4.71246i) q^{25} +2.15522 q^{27} +(2.49749 + 4.77101i) q^{29} -4.22738i q^{31} +0.400501i q^{33} +(3.86496 + 5.47139i) q^{35} +5.72993 q^{37} -0.505170i q^{39} -10.3731i q^{41} +10.8599 q^{43} +(-5.23244 + 3.69617i) q^{45} +6.38765 q^{47} -1.97480 q^{49} +0.0779983 q^{51} +10.0993i q^{53} +(-1.40608 - 1.99050i) q^{55} +2.46316i q^{57} +5.28523 q^{59} -3.38530i q^{61} -8.58285i q^{63} +(1.77356 + 2.51071i) q^{65} +10.2314i q^{67} +0.390934i q^{69} +5.04538 q^{71} -3.51767 q^{73} +(-0.614098 + 1.73170i) q^{75} +3.26505 q^{77} +8.85669i q^{79} +7.80291 q^{81} -12.4112i q^{83} +(-0.387654 + 0.273837i) q^{85} +(-0.917761 - 1.75322i) q^{87} -2.95876i q^{89} -4.11836 q^{91} +1.55345i q^{93} +(-8.64769 - 12.2420i) q^{95} +16.0605 q^{97} +3.12246i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - q^{5} + 8 q^{9} + 3 q^{15} - 2 q^{17} + 5 q^{25} + 10 q^{27} - 4 q^{29} - 16 q^{37} + 8 q^{43} - 4 q^{45} + 6 q^{47} - 6 q^{49} + 24 q^{51} + 11 q^{55} + 18 q^{59} - 15 q^{65} + 12 q^{71}+ \cdots + 14 q^{97}+O(q^{100}) \) 8 * q + 4 * q^3 - q^5 + 8 * q^9 + 3 * q^15 - 2 * q^17 + 5 * q^25 + 10 * q^27 - 4 * q^29 - 16 * q^37 + 8 * q^43 - 4 * q^45 + 6 * q^47 - 6 * q^49 + 24 * q^51 + 11 * q^55 + 18 * q^59 - 15 * q^65 + 12 * q^71 + 34 * q^73 + 11 * q^75 + 40 * q^77 - 24 * q^81 + 42 * q^85 + 10 * q^87 + 20 * q^91 + 10 * q^95 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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