LMFDB - Embedded newform 4320.2.f.k.1729.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4320,2,Mod(1729,4320)] 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("4320.1729"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,-6,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 3x^{10} - 21x^{8} - 174x^{6} - 525x^{4} + 1875x^{2} + 15625 $ x^12 + 3x^10 - 21x^8 - 174x^6 - 525x^4 + 1875*x^2 + 15625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1729.4
Root			$-1.28187 + 1.83216i$ of defining polynomial
Character	$\chi$	$=$	4320.1729
Dual form			4320.2.f.k.1729.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.28187 + 1.83216i) q^{5} -3.41531i q^{7} +1.00351 q^{11} +1.56023i q^{13} +2.54580i q^{17} +4.54580 q^{19} +2.66433i q^{23} +(-1.71364 - 4.69717i) q^{25} -3.97203 q^{29} -1.11852 q^{31} +(6.25740 + 4.37797i) q^{35} +6.83062i q^{37} -5.97904 q^{41} -9.09949i q^{43} +2.23704i q^{47} -4.66433 q^{49} +0.881478i q^{53} +(-1.28636 + 1.83859i) q^{55} +6.68769 q^{59} +9.75593 q^{61} +(-2.85859 - 2.00000i) q^{65} -2.00701i q^{67} -0.142926 q^{71} -8.94756i q^{73} -3.42728i q^{77} -3.87445 q^{79} +17.4203i q^{83} +(-4.66433 - 3.26338i) q^{85} +14.8167 q^{89} +5.32865 q^{91} +(-5.82711 + 8.32865i) q^{95} +11.0975i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{19} - 6 q^{25} - 4 q^{31} - 8 q^{49} - 30 q^{55} - 8 q^{61} + 88 q^{79} - 8 q^{85} - 32 q^{91}+O(q^{100}) $ 12 * q + 16 * q^19 - 6 * q^25 - 4 * q^31 - 8 * q^49 - 30 * q^55 - 8 * q^61 + 88 * q^79 - 8 * q^85 - 32 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$1$	$1$	$-1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	−1.28187	+	1.83216i	−0.573268	+	0.819368i
$5$	−1.28187	+	1.83216i	−0.573268	+	0.819368i
$6$		0			0
$7$		−	3.41531i		−	1.29086i	−0.763817	−	0.645432i	$-0.776677\pi$
$7$		−	3.41531i		−	1.29086i	0.763817	−	0.645432i	$-0.223323\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00351			0.302568			0.151284	−	0.988490i	$-0.451659\pi$
$11$	1.00351			0.302568			0.151284	+	0.988490i	$0.451659\pi$
$12$		0			0
$13$			1.56023i			0.432729i	0.976313	+	0.216364i	$0.0694199\pi$
$13$			1.56023i			0.432729i	−0.976313	+	0.216364i	$0.930580\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.54580i			0.617448i	0.951152	+	0.308724i	$0.0999019\pi$
$17$			2.54580i			0.617448i	−0.951152	+	0.308724i	$0.900098\pi$
$18$		0			0
$19$	4.54580			1.04288			0.521439	−	0.853288i	$-0.325395\pi$
$19$	4.54580			1.04288			0.521439	+	0.853288i	$0.325395\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.66433i			0.555550i	0.960646	+	0.277775i	$0.0895970\pi$
$23$			2.66433i			0.555550i	−0.960646	+	0.277775i	$0.910403\pi$
$24$		0			0
$25$	−1.71364	−	4.69717i	−0.342728	−	0.939435i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.97203			−0.737587			−0.368793	−	0.929511i	$-0.620229\pi$
$29$	−3.97203			−0.737587			−0.368793	+	0.929511i	$0.620229\pi$
$30$		0			0
$31$	−1.11852			−0.200893			−0.100446	−	0.994942i	$-0.532027\pi$
$31$	−1.11852			−0.200893			−0.100446	+	0.994942i	$0.532027\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.25740	+	4.37797i	1.05769	+	0.740011i
$36$		0			0
$37$			6.83062i			1.12295i	0.827495	+	0.561473i	$0.189765\pi$
$37$			6.83062i			1.12295i	−0.827495	+	0.561473i	$0.810235\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.97904			−0.933769			−0.466885	−	0.884318i	$-0.654624\pi$
$41$	−5.97904			−0.933769			−0.466885	+	0.884318i	$0.654624\pi$
$42$		0			0
$43$		−	9.09949i		−	1.38766i	−0.720139	−	0.693830i	$-0.755922\pi$
$43$		−	9.09949i		−	1.38766i	0.720139	−	0.693830i	$-0.244078\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.23704i			0.326307i	0.986601	+	0.163153i	$0.0521665\pi$
$47$			2.23704i			0.326307i	−0.986601	+	0.163153i	$0.947834\pi$
$48$		0			0
$49$	−4.66433			−0.666332
$50$		0			0
$51$		0			0
$52$		0			0
$53$			0.881478i			0.121080i	0.998166	+	0.0605401i	$0.0192823\pi$
$53$			0.881478i			0.121080i	−0.998166	+	0.0605401i	$0.980718\pi$
$54$		0			0
$55$	−1.28636	+	1.83859i	−0.173453	+	0.247915i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.68769			0.870663			0.435331	−	0.900270i	$-0.356631\pi$
$59$	6.68769			0.870663			0.435331	+	0.900270i	$0.356631\pi$
$60$		0			0
$61$	9.75593			1.24912			0.624560	−	0.780977i	$-0.285279\pi$
$61$	9.75593			1.24912			0.624560	+	0.780977i	$0.285279\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.85859	−	2.00000i	−0.354564	−	0.248069i
$66$		0			0
$67$		−	2.00701i		−	0.245195i	−0.992456	−	0.122598i	$-0.960878\pi$
$67$		−	2.00701i		−	0.245195i	0.992456	−	0.122598i	$-0.0391225\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.142926			−0.0169622			−0.00848111	−	0.999964i	$-0.502700\pi$
$71$	−0.142926			−0.0169622			−0.00848111	+	0.999964i	$0.502700\pi$
$72$		0			0
$73$		−	8.94756i		−	1.04723i	−0.851954	−	0.523616i	$-0.824583\pi$
$73$		−	8.94756i		−	1.04723i	0.851954	−	0.523616i	$-0.175417\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.42728i		−	0.390575i
$78$		0			0
$79$	−3.87445			−0.435910			−0.217955	−	0.975959i	$-0.569939\pi$
$79$	−3.87445			−0.435910			−0.217955	+	0.975959i	$0.569939\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			17.4203i			1.91212i	0.293168	+	0.956061i	$0.405290\pi$
$83$			17.4203i			1.91212i	−0.293168	+	0.956061i	$0.594710\pi$
$84$		0			0
$85$	−4.66433	−	3.26338i	−0.505917	−	0.353963i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.8167			1.57056			0.785282	−	0.619139i	$-0.212518\pi$
$89$	14.8167			1.57056			0.785282	+	0.619139i	$0.212518\pi$
$90$		0			0
$91$	5.32865			0.558594
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.82711	+	8.32865i	−0.597849	+	0.854502i
$96$		0			0
$97$			11.0975i			1.12678i	0.826191	+	0.563390i	$0.190503\pi$
$97$			11.0975i			1.12678i	−0.826191	+	0.563390i	$0.809497\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.38734			0.735067			0.367534	−	0.930010i	$-0.380202\pi$
$101$	7.38734			0.735067			0.367534	+	0.930010i	$0.380202\pi$
$102$		0			0
$103$			2.00701i			0.197757i	0.995100	+	0.0988784i	$0.0315255\pi$
$103$			2.00701i			0.197757i	−0.995100	+	0.0988784i	$0.968475\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			10.9930i			1.06273i	0.847143	+	0.531366i	$0.178321\pi$
$107$			10.9930i			1.06273i	−0.847143	+	0.531366i	$0.821679\pi$
$108$		0			0
$109$	2.66433			0.255196			0.127598	−	0.991826i	$-0.459273\pi$
$109$	2.66433			0.255196			0.127598	+	0.991826i	$0.459273\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	7.09161i		−	0.667122i	−0.942728	−	0.333561i	$-0.891750\pi$
$113$		−	7.09161i		−	0.667122i	0.942728	−	0.333561i	$-0.108250\pi$
$114$		0			0
$115$	−4.88148	−	3.41531i	−0.455200	−	0.318479i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.69470			0.797042
$120$		0			0
$121$	−9.99298			−0.908452
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.8026	+	2.88148i	0.966218	+	0.257727i
$126$		0			0
$127$			16.5288i			1.46670i	0.679854	+	0.733348i	$0.262043\pi$
$127$			16.5288i			1.46670i	−0.679854	+	0.733348i	$0.737957\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	21.3524			1.86557			0.932785	−	0.360433i	$-0.117371\pi$
$131$	21.3524			1.86557			0.932785	+	0.360433i	$0.117371\pi$
$132$		0			0
$133$		−	15.5253i		−	1.34622i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			3.45420i			0.295112i	0.989054	+	0.147556i	$0.0471407\pi$
$137$			3.45420i			0.295112i	−0.989054	+	0.147556i	$0.952859\pi$
$138$		0			0
$139$	22.6573			1.92177			0.960884	−	0.276952i	$-0.0893245\pi$
$139$	22.6573			1.92177			0.960884	+	0.276952i	$0.0893245\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1.56570i			0.130930i
$144$		0			0
$145$	5.09161	−	7.27740i	0.422835	−	0.604355i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.99801			−0.163683			−0.0818416	−	0.996645i	$-0.526080\pi$
$149$	−1.99801			−0.163683			−0.0818416	+	0.996645i	$0.526080\pi$
$150$		0			0
$151$	−14.7559			−1.20082			−0.600410	−	0.799692i	$-0.704996\pi$
$151$	−14.7559			−1.20082			−0.600410	+	0.799692i	$0.704996\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.43380	−	2.04931i	0.115165	−	0.164605i
$156$		0			0
$157$			22.3559i			1.78420i	0.451841	+	0.892099i	$0.350768\pi$
$157$			22.3559i			1.78420i	−0.451841	+	0.892099i	$0.649232\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.09949			0.717140
$162$		0			0
$163$			4.82360i			0.377814i	0.981995	+	0.188907i	$0.0604944\pi$
$163$			4.82360i			0.377814i	−0.981995	+	0.188907i	$0.939506\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			17.0846i			1.32204i	0.750366	+	0.661022i	$0.229877\pi$
$167$			17.0846i			1.32204i	−0.750366	+	0.661022i	$0.770123\pi$
$168$		0			0
$169$	10.5657			0.812746
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	2.88148i		−	0.219075i	−0.993983	−	0.109537i	$-0.965063\pi$
$173$		−	2.88148i		−	0.219075i	0.993983	−	0.109537i	$-0.0349369\pi$
$174$		0			0
$175$	−16.0423	+	5.85261i	−1.21268	+	0.442416i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	16.5288			1.23542			0.617711	−	0.786405i	$-0.288060\pi$
$179$	16.5288			1.23542			0.617711	+	0.786405i	$0.288060\pi$
$180$		0			0
$181$	−9.51889			−0.707533			−0.353767	−	0.935334i	$-0.615099\pi$
$181$	−9.51889			−0.707533			−0.353767	+	0.935334i	$0.615099\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.5148	−	8.75593i	−0.920106	−	0.643749i
$186$		0			0
$187$			2.55473i			0.186820i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	18.1990			1.31683			0.658416	−	0.752654i	$-0.271227\pi$
$191$	18.1990			1.31683			0.658416	+	0.752654i	$0.271227\pi$
$192$		0			0
$193$			14.3789i			1.03501i	0.855679	+	0.517507i	$0.173140\pi$
$193$			14.3789i			1.03501i	−0.855679	+	0.517507i	$0.826860\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	14.4472i		−	1.02932i	−0.857395	−	0.514659i	$-0.827918\pi$
$197$		−	14.4472i		−	1.02932i	0.857395	−	0.514659i	$-0.172082\pi$
$198$		0			0
$199$	3.24407			0.229966			0.114983	−	0.993367i	$-0.463319\pi$
$199$	3.24407			0.229966			0.114983	+	0.993367i	$0.463319\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			13.5657i			0.952125i
$204$		0			0
$205$	7.66433	−	10.9546i	0.535300	−	0.765101i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.56174			0.315542
$210$		0			0
$211$	−8.30876			−0.571999			−0.285999	−	0.958230i	$-0.592325\pi$
$211$	−8.30876			−0.571999			−0.285999	+	0.958230i	$0.592325\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	16.6717	+	11.6643i	1.13700	+	0.795500i
$216$		0			0
$217$			3.82010i			0.259325i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.97203			−0.267187
$222$		0			0
$223$			20.2060i			1.35309i	0.736400	+	0.676547i	$0.236524\pi$
$223$			20.2060i			1.35309i	−0.736400	+	0.676547i	$0.763476\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			11.0846i			0.735710i	0.929883	+	0.367855i	$0.119908\pi$
$227$			11.0846i			0.735710i	−0.929883	+	0.367855i	$0.880092\pi$
$228$		0			0
$229$	−13.3216			−0.880318			−0.440159	−	0.897920i	$-0.645078\pi$
$229$	−13.3216			−0.880318			−0.440159	+	0.897920i	$0.645078\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.4203i			1.07573i	0.843032	+	0.537863i	$0.180768\pi$
$233$			16.4203i			1.07573i	−0.843032	+	0.537863i	$0.819232\pi$
$234$		0			0
$235$	−4.09863	−	2.86759i	−0.267365	−	0.187061i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.6542			−0.753849			−0.376924	−	0.926244i	$-0.623018\pi$
$239$	−11.6542			−0.753849			−0.376924	+	0.926244i	$0.623018\pi$
$240$		0			0
$241$	13.3216			0.858121			0.429061	−	0.903276i	$-0.358845\pi$
$241$	13.3216			0.858121			0.429061	+	0.903276i	$0.358845\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.97904	−	8.54580i	0.381987	−	0.545971i
$246$		0			0
$247$			7.09248i			0.451284i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−25.3154			−1.59790			−0.798948	−	0.601399i	$-0.794610\pi$
$251$	−25.3154			−1.59790			−0.798948	+	0.601399i	$0.794610\pi$
$252$		0			0
$253$			2.67367i			0.168092i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	1.69124i		−	0.105497i	−0.998608	−	0.0527484i	$-0.983202\pi$
$257$		−	1.69124i		−	0.105497i	0.998608	−	0.0527484i	$-0.0167981\pi$
$258$		0			0
$259$	23.3287			1.44957
$260$		0			0
$261$		0			0
$262$		0			0
$263$			8.42026i			0.519215i	0.965714	+	0.259608i	$0.0835932\pi$
$263$			8.42026i			0.519215i	−0.965714	+	0.259608i	$0.916407\pi$
$264$		0			0
$265$	−1.61501	−	1.12994i	−0.0992093	−	0.0694114i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−23.9162			−1.45819			−0.729097	−	0.684410i	$-0.760060\pi$
$269$	−23.9162			−1.45819			−0.729097	+	0.684410i	$0.760060\pi$
$270$		0			0
$271$	−23.7758			−1.44428			−0.722139	−	0.691748i	$-0.756841\pi$
$271$	−23.7758			−1.44428			−0.722139	+	0.691748i	$0.756841\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.71965	−	4.71364i	−0.103699	−	0.284243i
$276$		0			0
$277$		−	23.7732i		−	1.42839i	−0.699944	−	0.714197i	$-0.746792\pi$
$277$		−	23.7732i		−	1.42839i	0.699944	−	0.714197i	$-0.253208\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−17.6333			−1.05191			−0.525956	−	0.850512i	$-0.676292\pi$
$281$	−17.6333			−1.05191			−0.525956	+	0.850512i	$0.676292\pi$
$282$		0			0
$283$		−	4.27589i		−	0.254175i	−0.991892	−	0.127088i	$-0.959437\pi$
$283$		−	4.27589i		−	0.254175i	0.991892	−	0.127088i	$-0.0405629\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			20.4203i			1.20537i
$288$		0			0
$289$	10.5189			0.618758
$290$		0			0
$291$		0			0
$292$		0			0
$293$			19.2031i			1.12186i	0.827864	+	0.560929i	$0.189556\pi$
$293$			19.2031i			1.12186i	−0.827864	+	0.560929i	$0.810444\pi$
$294$		0			0
$295$	−8.57272	+	12.2529i	−0.499123	+	0.713393i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.15695			−0.240403
$300$		0			0
$301$	−31.0776			−1.79128
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.5058	+	17.8745i	−0.716080	+	1.02349i
$306$		0			0
$307$		−	27.8462i		−	1.58927i	−0.607090	−	0.794633i	$-0.707663\pi$
$307$		−	27.8462i		−	1.58927i	0.607090	−	0.794633i	$-0.292337\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	29.1865			1.65502			0.827509	−	0.561453i	$-0.189757\pi$
$311$	29.1865			1.65502			0.827509	+	0.561453i	$0.189757\pi$
$312$		0			0
$313$			1.14643i			0.0648002i	0.999475	+	0.0324001i	$0.0103151\pi$
$313$			1.14643i			0.0648002i	−0.999475	+	0.0324001i	$0.989685\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			27.3017i			1.53342i	0.641995	+	0.766709i	$0.278107\pi$
$317$			27.3017i			1.53342i	−0.641995	+	0.766709i	$0.721893\pi$
$318$		0			0
$319$	−3.98595			−0.223170
$320$		0			0
$321$		0			0
$322$		0			0
$323$			11.5727i			0.643923i
$324$		0			0
$325$	7.32865	−	2.67367i	0.406520	−	0.148308i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.64020			0.421218
$330$		0			0
$331$	28.1692			1.54832			0.774159	−	0.632992i	$-0.218173\pi$
$331$	28.1692			1.54832			0.774159	+	0.632992i	$0.218173\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.67717	+	2.57272i	0.200905	+	0.140563i
$336$		0			0
$337$			18.9496i			1.03225i	0.856513	+	0.516126i	$0.172626\pi$
$337$			18.9496i			1.03225i	−0.856513	+	0.516126i	$0.827374\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.12244			−0.0607837
$342$		0			0
$343$		−	7.97705i		−	0.430720i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			1.90137i			0.102071i	0.998697	+	0.0510354i	$0.0162521\pi$
$347$			1.90137i			0.102071i	−0.998697	+	0.0510354i	$0.983748\pi$
$348$		0			0
$349$	18.4132			0.985638			0.492819	−	0.870132i	$-0.335967\pi$
$349$	18.4132			0.985638			0.492819	+	0.870132i	$0.335967\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	10.2370i		−	0.544863i	−0.962175	−	0.272431i	$-0.912172\pi$
$353$		−	10.2370i		−	0.544863i	0.962175	−	0.272431i	$-0.0878278\pi$
$354$		0			0
$355$	0.183212	−	0.261864i	0.00972389	−	0.0138983i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	29.8532			1.57559			0.787796	−	0.615936i	$-0.211222\pi$
$359$	29.8532			1.57559			0.787796	+	0.615936i	$0.211222\pi$
$360$		0			0
$361$	1.66433			0.0875961
$362$		0			0
$363$		0			0
$364$		0			0
$365$	16.3934	+	11.4696i	0.858069	+	0.600345i
$366$		0			0
$367$		−	3.70116i		−	0.193199i	−0.995323	−	0.0965995i	$-0.969203\pi$
$367$		−	3.70116i		−	0.193199i	0.995323	−	0.0965995i	$-0.0307966\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.01052			0.156298
$372$		0			0
$373$		−	0.809581i		−	0.0419185i	−0.999780	−	0.0209592i	$-0.993328\pi$
$373$		−	0.809581i		−	0.0419185i	0.999780	−	0.0209592i	$-0.00667202\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	6.19726i		−	0.319175i
$378$		0			0
$379$	36.1115			1.85492			0.927462	−	0.373919i	$-0.121986\pi$
$379$	36.1115			1.85492			0.927462	+	0.373919i	$0.121986\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	11.5727i		−	0.591338i	−0.955290	−	0.295669i	$-0.904457\pi$
$383$		−	11.5727i		−	0.591338i	0.955290	−	0.295669i	$-0.0955426\pi$
$384$		0			0
$385$	6.27934	+	4.39331i	0.320025	+	0.223904i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−22.2040			−1.12579			−0.562894	−	0.826529i	$-0.690312\pi$
$389$	−22.2040			−1.12579			−0.562894	+	0.826529i	$0.690312\pi$
$390$		0			0
$391$	−6.78285			−0.343023
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.96653	−	7.09863i	0.249893	−	0.357171i
$396$		0			0
$397$		−	9.50428i		−	0.477006i	−0.971142	−	0.238503i	$-0.923343\pi$
$397$		−	9.50428i		−	0.477006i	0.971142	−	0.238503i	$-0.0766567\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.41730			−0.0707766			−0.0353883	−	0.999374i	$-0.511267\pi$
$401$	−1.41730			−0.0707766			−0.0353883	+	0.999374i	$0.511267\pi$
$402$		0			0
$403$		−	1.74515i		−	0.0869320i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6.85456i			0.339768i
$408$		0			0
$409$	−33.2230			−1.64277			−0.821386	−	0.570373i	$-0.806799\pi$
$409$	−33.2230			−1.64277			−0.821386	+	0.570373i	$0.806799\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	22.8405i		−	1.12391i
$414$		0			0
$415$	−31.9167	−	22.3304i	−1.56673	−	1.09616i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.5113			−0.562363			−0.281182	−	0.959655i	$-0.590726\pi$
$419$	−11.5113			−0.562363			−0.281182	+	0.959655i	$0.590726\pi$
$420$		0			0
$421$	23.0846			1.12507			0.562537	−	0.826772i	$-0.309825\pi$
$421$	23.0846			1.12507			0.562537	+	0.826772i	$0.309825\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11.9581	−	4.36259i	0.580052	−	0.211617i
$426$		0			0
$427$		−	33.3195i		−	1.61244i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−29.9961			−1.44486			−0.722431	−	0.691443i	$-0.756975\pi$
$431$	−29.9961			−1.44486			−0.722431	+	0.691443i	$0.756975\pi$
$432$		0			0
$433$		−	7.54827i		−	0.362747i	−0.983414	−	0.181373i	$-0.941946\pi$
$433$		−	7.54827i		−	0.362747i	0.983414	−	0.181373i	$-0.0580542\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			12.1115i			0.579372i
$438$		0			0
$439$	38.1563			1.82110			0.910550	−	0.413398i	$-0.135658\pi$
$439$	38.1563			1.82110			0.910550	+	0.413398i	$0.135658\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			2.48111i			0.117881i	0.998261	+	0.0589406i	$0.0187723\pi$
$443$			2.48111i			0.117881i	−0.998261	+	0.0589406i	$0.981228\pi$
$444$		0			0
$445$	−18.9930	+	27.1465i	−0.900353	+	1.28687i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.38232			0.159621			0.0798107	−	0.996810i	$-0.474568\pi$
$449$	3.38232			0.159621			0.0798107	+	0.996810i	$0.474568\pi$
$450$		0			0
$451$	−6.00000			−0.282529
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.83062	+	9.76296i	−0.320224	+	0.457694i
$456$		0			0
$457$		−	28.4869i		−	1.33256i	−0.745701	−	0.666280i	$-0.767885\pi$
$457$		−	28.4869i		−	1.33256i	0.745701	−	0.666280i	$-0.232115\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.98103			0.185415			0.0927076	−	0.995693i	$-0.470448\pi$
$461$	3.98103			0.185415			0.0927076	+	0.995693i	$0.470448\pi$
$462$		0			0
$463$			12.2289i			0.568327i	0.958776	+	0.284164i	$0.0917159\pi$
$463$			12.2289i			0.568327i	−0.958776	+	0.284164i	$0.908284\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	20.3287i		−	0.940698i	−0.882481	−	0.470349i	$-0.844128\pi$
$467$		−	20.3287i		−	0.940698i	0.882481	−	0.470349i	$-0.155872\pi$
$468$		0			0
$469$	−6.85456			−0.316514
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	9.13139i		−	0.419862i
$474$		0			0
$475$	−7.78987	−	21.3524i	−0.357424	−	0.979716i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.16396			0.281639			0.140819	−	0.990035i	$-0.455026\pi$
$479$	6.16396			0.281639			0.140819	+	0.990035i	$0.455026\pi$
$480$		0			0
$481$	−10.6573			−0.485931
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−20.3324	−	14.2255i	−0.923248	−	0.645947i
$486$		0			0
$487$			13.3994i			0.607183i	0.952802	+	0.303592i	$0.0981858\pi$
$487$			13.3994i			0.607183i	−0.952802	+	0.303592i	$0.901814\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.96302			0.178849			0.0894244	−	0.995994i	$-0.471497\pi$
$491$	3.96302			0.178849			0.0894244	+	0.995994i	$0.471497\pi$
$492$		0			0
$493$		−	10.1120i		−	0.455422i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			0.488137i			0.0218959i
$498$		0			0
$499$	23.2172			1.03934			0.519671	−	0.854366i	$-0.326054\pi$
$499$	23.2172			1.03934			0.519671	+	0.854366i	$0.326054\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	13.8098i		−	0.615747i	−0.951427	−	0.307873i	$-0.900383\pi$
$503$		−	13.8098i		−	0.615747i	0.951427	−	0.307873i	$-0.0996173\pi$
$504$		0			0
$505$	−9.46957	+	13.5348i	−0.421390	+	0.602291i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.6352			0.693020			0.346510	−	0.938046i	$-0.387367\pi$
$509$	15.6352			0.693020			0.346510	+	0.938046i	$0.387367\pi$
$510$		0			0
$511$	−30.5587			−1.35184
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.67717	−	2.57272i	−0.162036	−	0.113368i
$516$		0			0
$517$			2.24489i			0.0987300i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4.25788			−0.186541			−0.0932706	−	0.995641i	$-0.529732\pi$
$521$	−4.25788			−0.186541			−0.0932706	+	0.995641i	$0.529732\pi$
$522$		0			0
$523$		−	33.3195i		−	1.45696i	−0.685067	−	0.728480i	$-0.740227\pi$
$523$		−	33.3195i		−	1.45696i	0.685067	−	0.728480i	$-0.259773\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	2.84754i		−	0.124041i
$528$		0			0
$529$	15.9014			0.691364
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	9.32865i		−	0.404069i
$534$		0			0
$535$	−20.1409	−	14.0915i	−0.870768	−	0.609230i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.68068			−0.201611
$540$		0			0
$541$	32.6035			1.40173			0.700866	−	0.713293i	$-0.252797\pi$
$541$	32.6035			1.40173			0.700866	+	0.713293i	$0.252797\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.41531	+	4.88148i	−0.146296	+	0.209100i
$546$		0			0
$547$			34.4149i			1.47148i	0.677266	+	0.735738i	$0.263165\pi$
$547$			34.4149i			1.47148i	−0.677266	+	0.735738i	$0.736835\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−18.0561			−0.769214
$552$		0			0
$553$			13.2325i			0.562701i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	30.9930i		−	1.31321i	−0.754233	−	0.656607i	$-0.771991\pi$
$557$		−	30.9930i		−	1.31321i	0.754233	−	0.656607i	$-0.228009\pi$
$558$		0			0
$559$	14.1973			0.600480
$560$		0			0
$561$		0			0
$562$		0			0
$563$			19.1902i			0.808772i	0.914588	+	0.404386i	$0.132515\pi$
$563$			19.1902i			0.808772i	−0.914588	+	0.404386i	$0.867485\pi$
$564$		0			0
$565$	12.9930	+	9.09049i	0.546619	+	0.382440i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.6333			0.739225			0.369612	−	0.929186i	$-0.379491\pi$
$569$	17.6333			0.739225			0.369612	+	0.929186i	$0.379491\pi$
$570$		0			0
$571$	6.29471			0.263425			0.131713	−	0.991288i	$-0.457952\pi$
$571$	6.29471			0.263425			0.131713	+	0.991288i	$0.457952\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	12.5148	−	4.56570i	0.521903	−	0.190403i
$576$		0			0
$577$		−	36.8267i		−	1.53312i	−0.642174	−	0.766559i	$-0.721967\pi$
$577$		−	36.8267i		−	1.53312i	0.642174	−	0.766559i	$-0.278033\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	59.4955			2.46829
$582$		0			0
$583$			0.884568i			0.0366351i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	28.9321i		−	1.19416i	−0.802183	−	0.597078i	$-0.796328\pi$
$587$		−	28.9321i		−	1.19416i	0.802183	−	0.597078i	$-0.203672\pi$
$588$		0			0
$589$	−5.08458			−0.209507
$590$		0			0
$591$		0			0
$592$		0			0
$593$			17.4940i			0.718392i	0.933262	+	0.359196i	$0.116949\pi$
$593$			17.4940i			0.718392i	−0.933262	+	0.359196i	$0.883051\pi$
$594$		0			0
$595$	−11.1454	+	15.9301i	−0.456918	+	0.653071i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.49727			0.306330			0.153165	−	0.988201i	$-0.451053\pi$
$599$	7.49727			0.306330			0.153165	+	0.988201i	$0.451053\pi$
$600$		0			0
$601$	17.8943			0.729926			0.364963	−	0.931022i	$-0.381082\pi$
$601$	17.8943			0.729926			0.364963	+	0.931022i	$0.381082\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	12.8097	−	18.3088i	0.520786	−	0.744357i
$606$		0			0
$607$			13.1135i			0.532261i	0.963937	+	0.266131i	$0.0857452\pi$
$607$			13.1135i			0.532261i	−0.963937	+	0.266131i	$0.914255\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.49029			−0.141202
$612$		0			0
$613$			38.6908i			1.56271i	0.624088	+	0.781354i	$0.285471\pi$
$613$			38.6908i			1.56271i	−0.624088	+	0.781354i	$0.714529\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	32.7688i		−	1.31922i	−0.751607	−	0.659611i	$-0.770721\pi$
$617$		−	32.7688i		−	1.31922i	0.751607	−	0.659611i	$-0.229279\pi$
$618$		0			0
$619$	39.1314			1.57282			0.786412	−	0.617703i	$-0.211936\pi$
$619$	39.1314			1.57282			0.786412	+	0.617703i	$0.211936\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	50.6035i		−	2.02739i
$624$		0			0
$625$	−19.1269	+	16.0985i	−0.765075	+	0.643941i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−17.3894			−0.693361
$630$		0			0
$631$	−11.7360			−0.467204			−0.233602	−	0.972332i	$-0.575051\pi$
$631$	−11.7360			−0.467204			−0.233602	+	0.972332i	$0.575051\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−30.2835	−	21.1877i	−1.20176	−	0.840809i
$636$		0			0
$637$		−	7.27740i		−	0.288341i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	22.1950			0.876650			0.438325	−	0.898817i	$-0.355572\pi$
$641$	22.1950			0.876650			0.438325	+	0.898817i	$0.355572\pi$
$642$		0			0
$643$		−	11.9161i		−	0.469924i	−0.972005	−	0.234962i	$-0.924503\pi$
$643$		−	11.9161i		−	0.469924i	0.972005	−	0.234962i	$-0.0754966\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	29.2678i		−	1.15064i	−0.817930	−	0.575318i	$-0.804878\pi$
$647$		−	29.2678i		−	1.15064i	0.817930	−	0.575318i	$-0.195122\pi$
$648$		0			0
$649$	6.71113			0.263435
$650$		0			0
$651$		0			0
$652$		0			0
$653$			2.21013i			0.0864890i	0.999065	+	0.0432445i	$0.0137694\pi$
$653$			2.21013i			0.0864890i	−0.999065	+	0.0432445i	$0.986231\pi$
$654$		0			0
$655$	−27.3709	+	39.1211i	−1.06947	+	1.52859i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16.3859			−0.638304			−0.319152	−	0.947704i	$-0.603398\pi$
$659$	−16.3859			−0.638304			−0.319152	+	0.947704i	$0.603398\pi$
$660$		0			0
$661$	−16.0398			−0.623875			−0.311938	−	0.950103i	$-0.600978\pi$
$661$	−16.0398			−0.623875			−0.311938	+	0.950103i	$0.600978\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	28.4449	+	19.9014i	1.10305	+	0.771742i
$666$		0			0
$667$		−	10.5828i		−	0.409767i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	9.79013			0.377944
$672$		0			0
$673$		−	27.2305i		−	1.04966i	−0.851207	−	0.524830i	$-0.824129\pi$
$673$		−	27.2305i		−	1.04966i	0.851207	−	0.524830i	$-0.175871\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			35.0916i			1.34868i	0.738421	+	0.674340i	$0.235572\pi$
$677$			35.0916i			1.34868i	−0.738421	+	0.674340i	$0.764428\pi$
$678$		0			0
$679$	37.9014			1.45452
$680$		0			0
$681$		0			0
$682$		0			0
$683$			25.2818i			0.967383i	0.875239	+	0.483691i	$0.160704\pi$
$683$			25.2818i			0.967383i	−0.875239	+	0.483691i	$0.839296\pi$
$684$		0			0
$685$	−6.32865	−	4.42782i	−0.241805	−	0.169178i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.37530			−0.0523949
$690$		0			0
$691$	−30.2947			−1.15247			−0.576233	−	0.817286i	$-0.695478\pi$
$691$	−30.2947			−1.15247			−0.576233	+	0.817286i	$0.695478\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−29.0436	+	41.5119i	−1.10169	+	1.57463i
$696$		0			0
$697$		−	15.2215i		−	0.576554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−46.1622			−1.74352			−0.871760	−	0.489933i	$-0.837021\pi$
$701$	−46.1622			−1.74352			−0.871760	+	0.489933i	$0.837021\pi$
$702$		0			0
$703$			31.0506i			1.17110i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	25.2300i		−	0.948873i
$708$		0			0
$709$	13.0776			0.491138			0.245569	−	0.969379i	$-0.421025\pi$
$709$	13.0776			0.491138			0.245569	+	0.969379i	$0.421025\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	2.98011i		−	0.111606i
$714$		0			0
$715$	−2.86861	−	2.00701i	−0.107280	−	0.0750580i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−44.4260			−1.65681			−0.828405	−	0.560129i	$-0.810751\pi$
$719$	−44.4260			−1.65681			−0.828405	+	0.560129i	$0.810751\pi$
$720$		0			0
$721$	6.85456			0.255277
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.80663	+	18.6573i	0.252792	+	0.692915i
$726$		0			0
$727$		−	30.9996i		−	1.14971i	−0.818254	−	0.574856i	$-0.805058\pi$
$727$		−	30.9996i		−	1.14971i	0.818254	−	0.574856i	$-0.194942\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	23.1655			0.856807
$732$		0			0
$733$			45.0157i			1.66269i	0.555754	+	0.831347i	$0.312430\pi$
$733$			45.0157i			1.66269i	−0.555754	+	0.831347i	$0.687570\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	2.01405i		−	0.0741884i
$738$		0			0
$739$	7.27098			0.267467			0.133734	−	0.991017i	$-0.457303\pi$
$739$	7.27098			0.267467			0.133734	+	0.991017i	$0.457303\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	40.8405i		−	1.49829i	−0.662404	−	0.749147i	$-0.730464\pi$
$743$		−	40.8405i		−	1.49829i	0.662404	−	0.749147i	$-0.269536\pi$
$744$		0			0
$745$	2.56118	−	3.66068i	0.0938343	−	0.134117i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	37.5444			1.37184
$750$		0			0
$751$	−27.5388			−1.00490			−0.502452	−	0.864605i	$-0.667569\pi$
$751$	−27.5388			−1.00490			−0.502452	+	0.864605i	$0.667569\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	18.9151	−	27.0353i	0.688392	−	0.983914i
$756$		0			0
$757$		−	17.9791i		−	0.653462i	−0.945117	−	0.326731i	$-0.894053\pi$
$757$		−	17.9791i		−	0.653462i	0.945117	−	0.326731i	$-0.105947\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−31.9022			−1.15645			−0.578227	−	0.815876i	$-0.696255\pi$
$761$	−31.9022			−1.15645			−0.578227	+	0.815876i	$0.696255\pi$
$762$		0			0
$763$		−	9.09949i		−	0.329424i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			10.4343i			0.376761i
$768$		0			0
$769$	−50.9321			−1.83666			−0.918330	−	0.395817i	$-0.870462\pi$
$769$	−50.9321			−1.83666			−0.918330	+	0.395817i	$0.870462\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	6.34854i		−	0.228341i	−0.993461	−	0.114171i	$-0.963579\pi$
$773$		−	6.34854i		−	0.228341i	0.993461	−	0.114171i	$-0.0364211\pi$
$774$		0			0
$775$	1.91675	+	5.25389i	0.0688515	+	0.188725i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.1795			−0.973808
$780$		0			0
$781$	−0.143427			−0.00513223
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−40.9597	−	28.6573i	−1.46191	−	1.02282i
$786$		0			0
$787$			33.3195i			1.18771i	0.804571	+	0.593856i	$0.202395\pi$
$787$			33.3195i			1.18771i	−0.804571	+	0.593856i	$0.797605\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−24.2200			−0.861165
$792$		0			0
$793$			15.2215i			0.540530i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			14.3933i			0.509838i	0.966962	+	0.254919i	$0.0820489\pi$
$797$			14.3933i			0.509838i	−0.966962	+	0.254919i	$0.917951\pi$
$798$		0			0
$799$	−5.69508			−0.201477
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	8.97893i		−	0.316860i
$804$		0			0
$805$	−11.6643	+	16.6717i	−0.411113	+	0.587602i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13.3154			−0.468144			−0.234072	−	0.972219i	$-0.575205\pi$
$809$	−13.3154			−0.468144			−0.234072	+	0.972219i	$0.575205\pi$
$810$		0			0
$811$	15.3146			0.537768			0.268884	−	0.963173i	$-0.413345\pi$
$811$	15.3146			0.537768			0.268884	+	0.963173i	$0.413345\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8.83763	−	6.18321i	−0.309569	−	0.216588i
$816$		0			0
$817$		−	41.3645i		−	1.44716i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10.7787			−0.376178			−0.188089	−	0.982152i	$-0.560229\pi$
$821$	−10.7787			−0.376178			−0.188089	+	0.982152i	$0.560229\pi$
$822$		0			0
$823$			1.69415i			0.0590543i	0.999564	+	0.0295272i	$0.00940015\pi$
$823$			1.69415i			0.0590543i	−0.999564	+	0.0295272i	$0.990600\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	15.5587i		−	0.541028i	−0.962716	−	0.270514i	$-0.912806\pi$
$827$		−	15.5587i		−	0.541028i	0.962716	−	0.270514i	$-0.0871936\pi$
$828$		0			0
$829$	−1.38248			−0.0480156			−0.0240078	−	0.999712i	$-0.507643\pi$
$829$	−1.38248			−0.0480156			−0.0240078	+	0.999712i	$0.507643\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	11.8745i		−	0.411425i
$834$		0			0
$835$	−31.3017	−	21.9001i	−1.08324	−	0.757886i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−40.6978			−1.40505			−0.702523	−	0.711661i	$-0.747943\pi$
$839$	−40.6978			−1.40505			−0.702523	+	0.711661i	$0.747943\pi$
$840$		0			0
$841$	−13.2230			−0.455965
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.5438	+	19.3581i	−0.465921	+	0.665938i
$846$		0			0
$847$			34.1291i			1.17269i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−18.1990			−0.623853
$852$		0			0
$853$			6.02103i			0.206156i	0.994673	+	0.103078i	$0.0328692\pi$
$853$			6.02103i			0.206156i	−0.994673	+	0.103078i	$0.967131\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			48.7290i			1.66455i	0.554362	+	0.832276i	$0.312962\pi$
$857$			48.7290i			1.66455i	−0.554362	+	0.832276i	$0.687038\pi$
$858$		0			0
$859$	25.4542			0.868486			0.434243	−	0.900796i	$-0.357016\pi$
$859$	25.4542			0.868486			0.434243	+	0.900796i	$0.357016\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	25.1384i		−	0.855722i	−0.903845	−	0.427861i	$-0.859267\pi$
$863$		−	25.1384i		−	0.855722i	0.903845	−	0.427861i	$-0.140733\pi$
$864$		0			0
$865$	5.27934	+	3.69367i	0.179503	+	0.125588i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.88804			−0.131893
$870$		0			0
$871$	3.13139			0.106103
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.84113	−	36.8943i	0.332691	−	1.24726i
$876$		0			0
$877$		−	30.8057i		−	1.04023i	−0.854095	−	0.520117i	$-0.825888\pi$
$877$		−	30.8057i		−	1.04023i	0.854095	−	0.520117i	$-0.174112\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	20.8377			0.702040			0.351020	−	0.936368i	$-0.385835\pi$
$881$	20.8377			0.702040			0.351020	+	0.936368i	$0.385835\pi$
$882$		0			0
$883$			53.2876i			1.79327i	0.442769	+	0.896636i	$0.353996\pi$
$883$			53.2876i			1.79327i	−0.442769	+	0.896636i	$0.646004\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	42.8335i		−	1.43821i	−0.694903	−	0.719104i	$-0.744553\pi$
$887$		−	42.8335i		−	1.43821i	0.694903	−	0.719104i	$-0.255447\pi$
$888$		0			0
$889$	56.4510			1.89331
$890$		0			0
$891$		0			0
$892$		0			0
$893$			10.1692i			0.340298i
$894$		0			0
$895$	−21.1877	+	30.2835i	−0.708228	+	1.01227i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.44280			0.148176
$900$		0			0
$901$	−2.24407			−0.0747608
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.2019	−	17.4401i	0.405606	−	0.579730i
$906$		0			0
$907$		−	11.0825i		−	0.367989i	−0.982927	−	0.183994i	$-0.941097\pi$
$907$		−	11.0825i		−	0.367989i	0.982927	−	0.183994i	$-0.0589028\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−37.7383			−1.25033			−0.625163	−	0.780494i	$-0.714968\pi$
$911$	−37.7383			−1.25033			−0.625163	+	0.780494i	$0.714968\pi$
$912$		0			0
$913$			17.4813i			0.578548i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	72.9251i		−	2.40820i
$918$		0			0
$919$	43.4132			1.43207			0.716035	−	0.698064i	$-0.245955\pi$
$919$	43.4132			1.43207			0.716035	+	0.698064i	$0.245955\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	0.222997i		−	0.00734004i
$924$		0			0
$925$	32.0846	−	11.7052i	1.05493	−	0.384865i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−50.6729			−1.66252			−0.831262	−	0.555881i	$-0.812381\pi$
$929$	−50.6729			−1.66252			−0.831262	+	0.555881i	$0.812381\pi$
$930$		0			0
$931$	−21.2031			−0.694904
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.68068	−	3.27482i	−0.153075	−	0.107098i
$936$		0			0
$937$			30.7138i			1.00338i	0.865049	+	0.501688i	$0.167287\pi$
$937$			30.7138i			1.00338i	−0.865049	+	0.501688i	$0.832713\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−43.5474			−1.41961			−0.709803	−	0.704401i	$-0.751216\pi$
$941$	−43.5474			−1.41961			−0.709803	+	0.704401i	$0.751216\pi$
$942$		0			0
$943$		−	15.9301i		−	0.518756i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	23.2370i		−	0.755102i	−0.925989	−	0.377551i	$-0.876766\pi$
$947$		−	23.2370i		−	0.755102i	0.925989	−	0.377551i	$-0.123234\pi$
$948$		0			0
$949$	13.9602			0.453168
$950$		0			0
$951$		0			0
$952$		0			0
$953$			52.5894i			1.70354i	0.523917	+	0.851769i	$0.324470\pi$
$953$			52.5894i			1.70354i	−0.523917	+	0.851769i	$0.675530\pi$
$954$		0			0
$955$	−23.3287	+	33.3435i	−0.754897	+	1.07897i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	11.7971			0.380950
$960$		0			0
$961$	−29.7489			−0.959642
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−26.3445	−	18.4318i	−0.848058	−	0.593341i
$966$		0			0
$967$		−	5.68418i		−	0.182791i	−0.995815	−	0.0913955i	$-0.970867\pi$
$967$		−	5.68418i		−	0.182791i	0.995815	−	0.0913955i	$-0.0291327\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.38431			−0.0444245			−0.0222123	−	0.999753i	$-0.507071\pi$
$971$	−1.38431			−0.0444245			−0.0222123	+	0.999753i	$0.507071\pi$
$972$		0			0
$973$		−	77.3817i		−	2.48074i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.76296i		−	0.0564019i	−0.999602	−	0.0282010i	$-0.991022\pi$
$977$		−	1.76296i		−	0.0564019i	0.999602	−	0.0282010i	$-0.00897784\pi$
$978$		0			0
$979$	14.8686			0.475203
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	43.9930i		−	1.40316i	−0.712592	−	0.701579i	$-0.752479\pi$
$983$		−	43.9930i		−	1.40316i	0.712592	−	0.701579i	$-0.247521\pi$
$984$		0			0
$985$	26.4696	+	18.5193i	0.843391	+	0.590075i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.2440			0.770914
$990$		0			0
$991$	−9.35557			−0.297189			−0.148595	−	0.988898i	$-0.547475\pi$
$991$	−9.35557			−0.297189			−0.148595	+	0.988898i	$0.547475\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4.15846	+	5.94366i	−0.131832	+	0.188427i
$996$		0			0
$997$			18.6278i			0.589947i	0.955505	+	0.294974i	$0.0953108\pi$
$997$			18.6278i			0.589947i	−0.955505	+	0.294974i	$0.904689\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.f.k.1729.4	yes	12
3.2	odd	2	inner	4320.2.f.k.1729.9	yes	12
4.3	odd	2		4320.2.f.h.1729.4	yes	12
5.4	even	2	inner	4320.2.f.k.1729.3	yes	12
12.11	even	2		4320.2.f.h.1729.9	yes	12
15.14	odd	2	inner	4320.2.f.k.1729.10	yes	12
20.19	odd	2		4320.2.f.h.1729.3	✓	12
60.59	even	2		4320.2.f.h.1729.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4320.2.f.h.1729.3	✓	12	20.19	odd	2
4320.2.f.h.1729.4	yes	12	4.3	odd	2
4320.2.f.h.1729.9	yes	12	12.11	even	2
4320.2.f.h.1729.10	yes	12	60.59	even	2
4320.2.f.k.1729.3	yes	12	5.4	even	2	inner
4320.2.f.k.1729.4	yes	12	1.1	even	1	trivial
4320.2.f.k.1729.9	yes	12	3.2	odd	2	inner
4320.2.f.k.1729.10	yes	12	15.14	odd	2	inner

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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.28187 + 1.83216i) q^{5} -3.41531i q^{7} +1.00351 q^{11} +1.56023i q^{13} +2.54580i q^{17} +4.54580 q^{19} +2.66433i q^{23} +(-1.71364 - 4.69717i) q^{25} -3.97203 q^{29} -1.11852 q^{31} +(6.25740 + 4.37797i) q^{35} +6.83062i q^{37} -5.97904 q^{41} -9.09949i q^{43} +2.23704i q^{47} -4.66433 q^{49} +0.881478i q^{53} +(-1.28636 + 1.83859i) q^{55} +6.68769 q^{59} +9.75593 q^{61} +(-2.85859 - 2.00000i) q^{65} -2.00701i q^{67} -0.142926 q^{71} -8.94756i q^{73} -3.42728i q^{77} -3.87445 q^{79} +17.4203i q^{83} +(-4.66433 - 3.26338i) q^{85} +14.8167 q^{89} +5.32865 q^{91} +(-5.82711 + 8.32865i) q^{95} +11.0975i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{19} - 6 q^{25} - 4 q^{31} - 8 q^{49} - 30 q^{55} - 8 q^{61} + 88 q^{79} - 8 q^{85} - 32 q^{91}+O(q^{100}) \) 12 * q + 16 * q^19 - 6 * q^25 - 4 * q^31 - 8 * q^49 - 30 * q^55 - 8 * q^61 + 88 * q^79 - 8 * q^85 - 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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