LMFDB - Embedded newform 1392.2.c.c.1391.39

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1392,2,Mod(1391,1392)] 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("1392.1391"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1392, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1392 = 2^{4} \cdot 3 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1392.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$11.1151759614$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1391.39
Character	$\chi$	$=$	1392.1391
Dual form			1392.2.c.c.1391.38

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.68301 + 0.409236i) q^{3} +3.05264i q^{5} -2.64216i q^{7} +(2.66505 + 1.37750i) q^{9} -0.324090i q^{11} +4.33763 q^{13} +(-1.24925 + 5.13763i) q^{15} +5.30412 q^{17} -1.65101 q^{19} +(1.08126 - 4.44678i) q^{21} -0.308926 q^{23} -4.31862 q^{25} +(3.92159 + 3.40898i) q^{27} +(-5.38460 + 0.0781992i) q^{29} -0.989329 q^{31} +(0.132629 - 0.545446i) q^{33} +8.06555 q^{35} +6.03392i q^{37} +(7.30028 + 1.77512i) q^{39} +2.28529 q^{41} -2.64034 q^{43} +(-4.20500 + 8.13545i) q^{45} -8.46146i q^{47} +0.0190147 q^{49} +(8.92689 + 2.17064i) q^{51} +11.7335i q^{53} +0.989329 q^{55} +(-2.77867 - 0.675652i) q^{57} +10.3680 q^{59} -1.34986i q^{61} +(3.63956 - 7.04148i) q^{63} +13.2412i q^{65} +12.3379i q^{67} +(-0.519925 - 0.126423i) q^{69} -6.97020 q^{71} -14.9347i q^{73} +(-7.26828 - 1.76733i) q^{75} -0.856295 q^{77} -11.1127 q^{79} +(5.20500 + 7.34220i) q^{81} +7.55001 q^{83} +16.1916i q^{85} +(-9.09434 - 2.07196i) q^{87} -8.79022 q^{89} -11.4607i q^{91} +(-1.66505 - 0.404869i) q^{93} -5.03994i q^{95} +6.04110i q^{97} +(0.446432 - 0.863716i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 8 q^{9} - 8 q^{13} - 48 q^{25} + 8 q^{45} - 56 q^{49} - 48 q^{57} + 32 q^{81} + 48 q^{93}+O(q^{100}) $ 40 * q - 8 * q^9 - 8 * q^13 - 48 * q^25 + 8 * q^45 - 56 * q^49 - 48 * q^57 + 32 * q^81 + 48 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1392\mathbb{Z}\right)^\times$.

$n$	$175$	$929$	$1045$	$1249$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	1.68301	+	0.409236i	0.971687	+	0.236272i
$3$	1.68301	+	0.409236i	0.971687	+	0.236272i
$4$		0			0
$5$			3.05264i			1.36518i	0.730800	+	0.682591i	$0.239147\pi$
$5$			3.05264i			1.36518i	−0.730800	+	0.682591i	$0.760853\pi$
$6$		0			0
$7$		−	2.64216i		−	0.998641i	−0.866417	−	0.499320i	$-0.833583\pi$
$7$		−	2.64216i		−	0.998641i	0.866417	−	0.499320i	$-0.166417\pi$
$8$		0			0
$9$	2.66505	+	1.37750i	0.888351	+	0.459166i
$10$		0			0
$11$		−	0.324090i		−	0.0977167i	−0.998806	−	0.0488583i	$-0.984442\pi$
$11$		−	0.324090i		−	0.0977167i	0.998806	−	0.0488583i	$-0.0155583\pi$
$12$		0			0
$13$	4.33763			1.20304			0.601521	−	0.798857i	$-0.294561\pi$
$13$	4.33763			1.20304			0.601521	+	0.798857i	$0.294561\pi$
$14$		0			0
$15$	−1.24925	+	5.13763i	−0.322555	+	1.32653i
$16$		0			0
$17$	5.30412			1.28644			0.643219	−	0.765682i	$-0.277599\pi$
$17$	5.30412			1.28644			0.643219	+	0.765682i	$0.277599\pi$
$18$		0			0
$19$	−1.65101			−0.378767			−0.189384	−	0.981903i	$-0.560649\pi$
$19$	−1.65101			−0.378767			−0.189384	+	0.981903i	$0.560649\pi$
$20$		0			0
$21$	1.08126	−	4.44678i	0.235951	−	0.970366i
$22$		0			0
$23$	−0.308926			−0.0644154			−0.0322077	−	0.999481i	$-0.510254\pi$
$23$	−0.308926			−0.0644154			−0.0322077	+	0.999481i	$0.510254\pi$
$24$		0			0
$25$	−4.31862			−0.863724
$26$		0			0
$27$	3.92159	+	3.40898i	0.754710	+	0.656058i
$28$		0			0
$29$	−5.38460	+	0.0781992i	−0.999895	+	0.0145212i
$29$	−5.38460	+	0.0781992i	−0.999895	+	0.0145212i
$30$		0			0
$31$	−0.989329			−0.177689			−0.0888444	−	0.996046i	$-0.528317\pi$
$31$	−0.989329			−0.177689			−0.0888444	+	0.996046i	$0.528317\pi$
$32$		0			0
$33$	0.132629	−	0.545446i	0.0230878	−	0.0949500i
$34$		0			0
$35$	8.06555			1.36333
$36$		0			0
$37$			6.03392i			0.991971i	0.868331	+	0.495985i	$0.165193\pi$
$37$			6.03392i			0.991971i	−0.868331	+	0.495985i	$0.834807\pi$
$38$		0			0
$39$	7.30028	+	1.77512i	1.16898	+	0.284246i
$40$		0			0
$41$	2.28529			0.356903			0.178452	−	0.983949i	$-0.442891\pi$
$41$	2.28529			0.356903			0.178452	+	0.983949i	$0.442891\pi$
$42$		0			0
$43$	−2.64034			−0.402648			−0.201324	−	0.979525i	$-0.564524\pi$
$43$	−2.64034			−0.402648			−0.201324	+	0.979525i	$0.564524\pi$
$44$		0			0
$45$	−4.20500	+	8.13545i	−0.626845	+	1.21276i
$46$		0			0
$47$		−	8.46146i		−	1.23423i	−0.786873	−	0.617115i	$-0.788301\pi$
$47$		−	8.46146i		−	1.23423i	0.786873	−	0.617115i	$-0.211699\pi$
$48$		0			0
$49$	0.0190147			0.00271638
$50$		0			0
$51$	8.92689	+	2.17064i	1.25001	+	0.303950i
$52$		0			0
$53$			11.7335i			1.61173i	0.592103	+	0.805863i	$0.298298\pi$
$53$			11.7335i			1.61173i	−0.592103	+	0.805863i	$0.701702\pi$
$54$		0			0
$55$	0.989329			0.133401
$56$		0			0
$57$	−2.77867	−	0.675652i	−0.368043	−	0.0894923i
$58$		0			0
$59$	10.3680			1.34980			0.674902	−	0.737907i	$-0.264186\pi$
$59$	10.3680			1.34980			0.674902	+	0.737907i	$0.264186\pi$
$60$		0			0
$61$		−	1.34986i		−	0.172832i	−0.996259	−	0.0864162i	$-0.972459\pi$
$61$		−	1.34986i		−	0.172832i	0.996259	−	0.0864162i	$-0.0275415\pi$
$62$		0			0
$63$	3.63956	−	7.04148i	0.458542	−	0.887143i
$64$		0			0
$65$			13.2412i			1.64237i
$66$		0			0
$67$			12.3379i			1.50731i	0.657270	+	0.753655i	$0.271711\pi$
$67$			12.3379i			1.50731i	−0.657270	+	0.753655i	$0.728289\pi$
$68$		0			0
$69$	−0.519925	−	0.126423i	−0.0625916	−	0.0152196i
$70$		0			0
$71$	−6.97020			−0.827210			−0.413605	−	0.910456i	$-0.635731\pi$
$71$	−6.97020			−0.827210			−0.413605	+	0.910456i	$0.635731\pi$
$72$		0			0
$73$		−	14.9347i		−	1.74797i	−0.485954	−	0.873984i	$-0.661528\pi$
$73$		−	14.9347i		−	1.74797i	0.485954	−	0.873984i	$-0.338472\pi$
$74$		0			0
$75$	−7.26828	−	1.76733i	−0.839269	−	0.204074i
$76$		0			0
$77$	−0.856295			−0.0975839
$78$		0			0
$79$	−11.1127			−1.25028			−0.625140	−	0.780512i	$-0.714958\pi$
$79$	−11.1127			−1.25028			−0.625140	+	0.780512i	$0.714958\pi$
$80$		0			0
$81$	5.20500	+	7.34220i	0.578334	+	0.815800i
$82$		0			0
$83$	7.55001			0.828722			0.414361	−	0.910113i	$-0.364005\pi$
$83$	7.55001			0.828722			0.414361	+	0.910113i	$0.364005\pi$
$84$		0			0
$85$			16.1916i			1.75622i
$86$		0			0
$87$	−9.09434	−	2.07196i	−0.975015	−	0.222137i
$88$		0			0
$89$	−8.79022			−0.931761			−0.465881	−	0.884848i	$-0.654262\pi$
$89$	−8.79022			−0.931761			−0.465881	+	0.884848i	$0.654262\pi$
$90$		0			0
$91$		−	11.4607i		−	1.20141i
$92$		0			0
$93$	−1.66505	−	0.404869i	−0.172658	−	0.0419830i
$94$		0			0
$95$		−	5.03994i		−	0.517087i
$96$		0			0
$97$			6.04110i			0.613381i	0.951809	+	0.306690i	$0.0992216\pi$
$97$			6.04110i			0.613381i	−0.951809	+	0.306690i	$0.900778\pi$
$98$		0			0
$99$	0.446432	−	0.863716i	0.0448681	−	0.0868067i
$100$		0			0
$101$	−17.2359			−1.71504			−0.857519	−	0.514451i	$-0.827996\pi$
$101$	−17.2359			−1.71504			−0.857519	+	0.514451i	$0.827996\pi$
$102$		0			0
$103$			0.923146i			0.0909602i	0.998965	+	0.0454801i	$0.0144818\pi$
$103$			0.923146i			0.0909602i	−0.998965	+	0.0454801i	$0.985518\pi$
$104$		0			0
$105$	13.5744	+	3.30071i	1.32473	+	0.322117i
$106$		0			0
$107$	−7.79466			−0.753538			−0.376769	−	0.926307i	$-0.622965\pi$
$107$	−7.79466			−0.753538			−0.376769	+	0.926307i	$0.622965\pi$
$108$		0			0
$109$	−0.742271			−0.0710967			−0.0355483	−	0.999368i	$-0.511318\pi$
$109$	−0.742271			−0.0710967			−0.0355483	+	0.999368i	$0.511318\pi$
$110$		0			0
$111$	−2.46930	+	10.1552i	−0.234375	+	0.963885i
$112$		0			0
$113$	6.34397			0.596790			0.298395	−	0.954442i	$-0.403549\pi$
$113$	6.34397			0.596790			0.298395	+	0.954442i	$0.403549\pi$
$114$		0			0
$115$		−	0.943039i		−	0.0879388i
$116$		0			0
$117$	11.5600	+	5.97508i	1.06872	+	0.552396i
$118$		0			0
$119$		−	14.0143i		−	1.28469i
$120$		0			0
$121$	10.8950			0.990451
$122$		0			0
$123$	3.84618	+	0.935225i	0.346798	+	0.0843264i
$124$		0			0
$125$			2.08001i			0.186042i
$126$		0			0
$127$	15.4041			1.36689			0.683446	−	0.730001i	$-0.260481\pi$
$127$	15.4041			1.36689			0.683446	+	0.730001i	$0.260481\pi$
$128$		0			0
$129$	−4.44372	−	1.08052i	−0.391248	−	0.0951346i
$130$		0			0
$131$		−	17.9726i		−	1.57027i	−0.619325	−	0.785135i	$-0.712594\pi$
$131$		−	17.9726i		−	1.57027i	0.619325	−	0.785135i	$-0.287406\pi$
$132$		0			0
$133$			4.36222i			0.378253i
$134$		0			0
$135$	−10.4064	+	11.9712i	−0.895639	+	1.03032i
$136$		0			0
$137$	−18.1751			−1.55280			−0.776400	−	0.630241i	$-0.782956\pi$
$137$	−18.1751			−1.55280			−0.776400	+	0.630241i	$0.782956\pi$
$138$		0			0
$139$		−	7.02940i		−	0.596226i	−0.954531	−	0.298113i	$-0.903643\pi$
$139$		−	7.02940i		−	0.596226i	0.954531	−	0.298113i	$-0.0963573\pi$
$140$		0			0
$141$	3.46273	−	14.2407i	0.291615	−	1.19929i
$142$		0			0
$143$		−	1.40578i		−	0.117557i
$144$		0			0
$145$	−0.238714	−	16.4372i	−0.0198241	−	1.36504i
$146$		0			0
$147$	0.0320019	+	0.00778150i	0.00263948	+	0.000641807i
$148$		0			0
$149$		−	3.58772i		−	0.293917i	−0.989143	−	0.146959i	$-0.953052\pi$
$149$		−	3.58772i		−	0.293917i	0.989143	−	0.146959i	$-0.0469484\pi$
$150$		0			0
$151$		−	18.4441i		−	1.50096i	−0.660892	−	0.750481i	$-0.729822\pi$
$151$		−	18.4441i		−	1.50096i	0.660892	−	0.750481i	$-0.270178\pi$
$152$		0			0
$153$	14.1358	+	7.30641i	1.14281	+	0.590688i
$154$		0			0
$155$		−	3.02007i		−	0.242578i
$156$		0			0
$157$		−	11.6674i		−	0.931161i	−0.885005	−	0.465581i	$-0.845845\pi$
$157$		−	11.6674i		−	0.931161i	0.885005	−	0.465581i	$-0.154155\pi$
$158$		0			0
$159$	−4.80178	+	19.7477i	−0.380806	+	1.56609i
$160$		0			0
$161$			0.816229i			0.0643279i
$162$		0			0
$163$	14.7424			1.15471			0.577357	−	0.816492i	$-0.304084\pi$
$163$	14.7424			1.15471			0.577357	+	0.816492i	$0.304084\pi$
$164$		0			0
$165$	1.66505	+	0.404869i	0.129624	+	0.0315190i
$166$		0			0
$167$	−10.5747			−0.818292			−0.409146	−	0.912469i	$-0.634173\pi$
$167$	−10.5747			−0.818292			−0.409146	+	0.912469i	$0.634173\pi$
$168$		0			0
$169$	5.81506			0.447312
$170$		0			0
$171$	−4.40002	−	2.27426i	−0.336478	−	0.173917i
$172$		0			0
$173$		−	13.2785i		−	1.00954i	−0.863253	−	0.504772i	$-0.831577\pi$
$173$		−	13.2785i		−	1.00954i	0.863253	−	0.504772i	$-0.168423\pi$
$174$		0			0
$175$			11.4105i			0.862550i
$176$		0			0
$177$	17.4495	+	4.24298i	1.31159	+	0.318922i
$178$		0			0
$179$	17.8157			1.33161			0.665806	−	0.746125i	$-0.268088\pi$
$179$	17.8157			1.33161			0.665806	+	0.746125i	$0.268088\pi$
$180$		0			0
$181$	−7.93300			−0.589655			−0.294827	−	0.955551i	$-0.595262\pi$
$181$	−7.93300			−0.589655			−0.294827	+	0.955551i	$0.595262\pi$
$182$		0			0
$183$	0.552413	−	2.27184i	0.0408355	−	0.167939i
$184$		0			0
$185$	−18.4194			−1.35422
$186$		0			0
$187$		−	1.71901i		−	0.125706i
$188$		0			0
$189$	9.00705	−	10.3615i	0.655166	−	0.753685i
$190$		0			0
$191$		−	6.12163i		−	0.442946i	−0.975167	−	0.221473i	$-0.928914\pi$
$191$		−	6.12163i		−	0.442946i	0.975167	−	0.221473i	$-0.0710864\pi$
$192$		0			0
$193$		−	2.76669i		−	0.199151i	−0.995030	−	0.0995753i	$-0.968252\pi$
$193$		−	2.76669i		−	0.199151i	0.995030	−	0.0995753i	$-0.0317484\pi$
$194$		0			0
$195$	−5.41879	+	22.2851i	−0.388048	+	1.59587i
$196$		0			0
$197$		−	15.4776i		−	1.10274i	−0.834262	−	0.551368i	$-0.814106\pi$
$197$		−	15.4776i		−	1.10274i	0.834262	−	0.551368i	$-0.185894\pi$
$198$		0			0
$199$			10.4674i			0.742016i	0.928630	+	0.371008i	$0.120988\pi$
$199$			10.4674i			0.742016i	−0.928630	+	0.371008i	$0.879012\pi$
$200$		0			0
$201$	−5.04910	+	20.7648i	−0.356136	+	1.46463i
$202$		0			0
$203$	0.206615	+	14.2269i	0.0145015	+	0.998536i
$204$		0			0
$205$			6.97618i			0.487238i
$206$		0			0
$207$	−0.823303	−	0.425544i	−0.0572235	−	0.0295774i
$208$		0			0
$209$			0.535075i			0.0370119i
$210$		0			0
$211$	−16.0658			−1.10601			−0.553006	−	0.833177i	$-0.686520\pi$
$211$	−16.0658			−1.10601			−0.553006	+	0.833177i	$0.686520\pi$
$212$		0			0
$213$	−11.7309	−	2.85245i	−0.803789	−	0.195447i
$214$		0			0
$215$		−	8.06001i		−	0.549688i
$216$		0			0
$217$			2.61396i			0.177447i
$218$		0			0
$219$	6.11180	−	25.1352i	0.412997	−	1.69848i
$220$		0			0
$221$	23.0073			1.54764
$222$		0			0
$223$			21.1522i			1.41645i	0.705985	+	0.708226i	$0.250504\pi$
$223$			21.1522i			1.41645i	−0.705985	+	0.708226i	$0.749496\pi$
$224$		0			0
$225$	−11.5093	−	5.94888i	−0.767290	−	0.396592i
$226$		0			0
$227$	−7.48574			−0.496846			−0.248423	−	0.968652i	$-0.579912\pi$
$227$	−7.48574			−0.496846			−0.248423	+	0.968652i	$0.579912\pi$
$228$		0			0
$229$		−	11.5003i		−	0.759964i	−0.924994	−	0.379982i	$-0.875930\pi$
$229$		−	11.5003i		−	0.759964i	0.924994	−	0.379982i	$-0.124070\pi$
$230$		0			0
$231$	−1.44115	−	0.350427i	−0.0948210	−	0.0230564i
$232$		0			0
$233$		−	13.0190i		−	0.852900i	−0.904511	−	0.426450i	$-0.859764\pi$
$233$		−	13.0190i		−	0.852900i	0.904511	−	0.426450i	$-0.140236\pi$
$234$		0			0
$235$	25.8298			1.68495
$236$		0			0
$237$	−18.7029	−	4.54773i	−1.21488	−	0.295407i
$238$		0			0
$239$	−20.4272			−1.32132			−0.660662	−	0.750684i	$-0.729724\pi$
$239$	−20.4272			−1.32132			−0.660662	+	0.750684i	$0.729724\pi$
$240$		0			0
$241$	6.89851			0.444372			0.222186	−	0.975004i	$-0.428681\pi$
$241$	6.89851			0.444372			0.222186	+	0.975004i	$0.428681\pi$
$242$		0			0
$243$	5.75539	+	14.4871i	0.369208	+	0.929347i
$244$		0			0
$245$			0.0580450i			0.00370836i
$246$		0			0
$247$	−7.16147			−0.455674
$248$		0			0
$249$	12.7068	+	3.08974i	0.805258	+	0.195804i
$250$		0			0
$251$		−	13.5014i		−	0.852201i	−0.904676	−	0.426100i	$-0.859887\pi$
$251$		−	13.5014i		−	0.852201i	0.904676	−	0.426100i	$-0.140113\pi$
$252$		0			0
$253$			0.100120i			0.00629446i
$254$		0			0
$255$	−6.62617	+	27.2506i	−0.414947	+	1.70650i
$256$		0			0
$257$		−	2.15970i		−	0.134718i	−0.997729	−	0.0673592i	$-0.978543\pi$
$257$		−	2.15970i		−	0.134718i	0.997729	−	0.0673592i	$-0.0214573\pi$
$258$		0			0
$259$	15.9426			0.990622
$260$		0			0
$261$	−14.4580	−	7.20886i	−0.894925	−	0.446217i
$262$		0			0
$263$			23.4078i			1.44339i	0.692213	+	0.721693i	$0.256636\pi$
$263$			23.4078i			1.44339i	−0.692213	+	0.721693i	$0.743364\pi$
$264$		0			0
$265$	−35.8183			−2.20030
$266$		0			0
$267$	−14.7940	−	3.59727i	−0.905380	−	0.220150i
$268$		0			0
$269$	19.3985			1.18274			0.591372	−	0.806399i	$-0.298586\pi$
$269$	19.3985			1.18274			0.591372	+	0.806399i	$0.298586\pi$
$270$		0			0
$271$	28.1803			1.71183			0.855915	−	0.517117i	$-0.172995\pi$
$271$	28.1803			1.71183			0.855915	+	0.517117i	$0.172995\pi$
$272$		0			0
$273$	4.69013	−	19.2885i	0.283860	−	1.16739i
$274$		0			0
$275$			1.39962i			0.0844002i
$276$		0			0
$277$	−8.57277			−0.515088			−0.257544	−	0.966267i	$-0.582913\pi$
$277$	−8.57277			−0.515088			−0.257544	+	0.966267i	$0.582913\pi$
$278$		0			0
$279$	−2.63661	−	1.36280i	−0.157850	−	0.0815886i
$280$		0			0
$281$		−	14.0891i		−	0.840486i	−0.907412	−	0.420243i	$-0.861945\pi$
$281$		−	14.0891i		−	0.840486i	0.907412	−	0.420243i	$-0.138055\pi$
$282$		0			0
$283$		−	13.0897i		−	0.778101i	−0.921216	−	0.389050i	$-0.872803\pi$
$283$		−	13.0897i		−	0.778101i	0.921216	−	0.389050i	$-0.127197\pi$
$284$		0			0
$285$	2.06252	−	8.48227i	0.122173	−	0.502446i
$286$		0			0
$287$		−	6.03810i		−	0.356418i
$288$		0			0
$289$	11.1337			0.654922
$290$		0			0
$291$	−2.47223	+	10.1672i	−0.144925	+	0.596014i
$292$		0			0
$293$	12.2982			0.718470			0.359235	−	0.933247i	$-0.383038\pi$
$293$	12.2982			0.718470			0.359235	+	0.933247i	$0.383038\pi$
$294$		0			0
$295$			31.6499i			1.84273i
$296$		0			0
$297$	1.10481	−	1.27095i	0.0641078	−	0.0737478i
$298$		0			0
$299$	−1.34001			−0.0774945
$300$		0			0
$301$			6.97618i			0.402101i
$302$		0			0
$303$	−29.0083	−	7.05356i	−1.66648	−	0.405216i
$304$		0			0
$305$	4.12065			0.235948
$306$		0			0
$307$	−25.8929			−1.47779			−0.738893	−	0.673822i	$-0.764651\pi$
$307$	−25.8929			−1.47779			−0.738893	+	0.673822i	$0.764651\pi$
$308$		0			0
$309$	−0.377784	+	1.55366i	−0.0214914	+	0.0883849i
$310$		0			0
$311$		−	7.24247i		−	0.410683i	−0.978690	−	0.205341i	$-0.934170\pi$
$311$		−	7.24247i		−	0.410683i	0.978690	−	0.205341i	$-0.0658304\pi$
$312$		0			0
$313$	2.29603			0.129779			0.0648897	−	0.997892i	$-0.479330\pi$
$313$	2.29603			0.129779			0.0648897	+	0.997892i	$0.479330\pi$
$314$		0			0
$315$	21.4951	+	11.1103i	1.21111	+	0.625993i
$316$		0			0
$317$	−22.2565			−1.25005			−0.625024	−	0.780606i	$-0.714911\pi$
$317$	−22.2565			−1.25005			−0.625024	+	0.780606i	$0.714911\pi$
$318$		0			0
$319$	0.0253436	+	1.74509i	0.00141897	+	0.0977064i
$320$		0			0
$321$	−13.1185	−	3.18986i	−0.732203	−	0.178040i
$322$		0			0
$323$	−8.75715			−0.487261
$324$		0			0
$325$	−18.7326			−1.03910
$326$		0			0
$327$	−1.24925	−	0.303764i	−0.0690837	−	0.0167982i
$328$		0			0
$329$	−22.3565			−1.23255
$330$		0			0
$331$	−28.0193			−1.54008			−0.770040	−	0.637996i	$-0.779764\pi$
$331$	−28.0193			−1.54008			−0.770040	+	0.637996i	$0.779764\pi$
$332$		0			0
$333$	−8.31171	+	16.0807i	−0.455479	+	0.881218i
$334$		0			0
$335$	−37.6631			−2.05775
$336$		0			0
$337$		−	8.55494i		−	0.466017i	−0.972475	−	0.233009i	$-0.925143\pi$
$337$		−	8.55494i		−	0.466017i	0.972475	−	0.233009i	$-0.0748570\pi$
$338$		0			0
$339$	10.6770	+	2.59618i	0.579893	+	0.141005i
$340$		0			0
$341$			0.320631i			0.0173632i
$342$		0			0
$343$		−	18.5453i		−	1.00135i
$344$		0			0
$345$	0.385925	−	1.58715i	0.0207775	−	0.0854490i
$346$		0			0
$347$	−10.0591			−0.540002			−0.270001	−	0.962860i	$-0.587024\pi$
$347$	−10.0591			−0.540002			−0.270001	+	0.962860i	$0.587024\pi$
$348$		0			0
$349$	−7.39498			−0.395844			−0.197922	−	0.980218i	$-0.563419\pi$
$349$	−7.39498			−0.395844			−0.197922	+	0.980218i	$0.563419\pi$
$350$		0			0
$351$	17.0104	+	14.7869i	0.907949	+	0.789266i
$352$		0			0
$353$		−	20.5150i		−	1.09190i	−0.837816	−	0.545952i	$-0.816168\pi$
$353$		−	20.5150i		−	1.09190i	0.837816	−	0.545952i	$-0.183832\pi$
$354$		0			0
$355$		−	21.2775i		−	1.12929i
$356$		0			0
$357$	5.73516	−	23.5862i	0.303537	−	1.24832i
$358$		0			0
$359$		−	13.6902i		−	0.722542i	−0.932461	−	0.361271i	$-0.882343\pi$
$359$		−	13.6902i		−	0.722542i	0.932461	−	0.361271i	$-0.117657\pi$
$360$		0			0
$361$	−16.2742			−0.856535
$362$		0			0
$363$	18.3363	+	4.45861i	0.962409	+	0.234016i
$364$		0			0
$365$	45.5901			2.38630
$366$		0			0
$367$	−32.2982			−1.68595			−0.842976	−	0.537951i	$-0.819199\pi$
$367$	−32.2982			−1.68595			−0.842976	+	0.537951i	$0.819199\pi$
$368$		0			0
$369$	6.09043	+	3.14799i	0.317055	+	0.163878i
$370$		0			0
$371$	31.0018			1.60953
$372$		0			0
$373$	19.4368			1.00640			0.503200	−	0.864170i	$-0.332156\pi$
$373$	19.4368			1.00640			0.503200	+	0.864170i	$0.332156\pi$
$374$		0			0
$375$	−0.851216	+	3.50069i	−0.0439566	+	0.180775i
$376$		0			0
$377$	−23.3564	+	0.339200i	−1.20292	+	0.0174697i
$378$		0			0
$379$	4.26600			0.219130			0.109565	−	0.993980i	$-0.465054\pi$
$379$	4.26600			0.219130			0.109565	+	0.993980i	$0.465054\pi$
$380$		0			0
$381$	25.9252	+	6.30390i	1.32819	+	0.322959i
$382$		0			0
$383$	−14.7911			−0.755790			−0.377895	−	0.925848i	$-0.623352\pi$
$383$	−14.7911			−0.755790			−0.377895	+	0.925848i	$0.623352\pi$
$384$		0			0
$385$		−	2.61396i		−	0.133220i
$386$		0			0
$387$	−7.03664	−	3.63706i	−0.357692	−	0.184882i
$388$		0			0
$389$	12.6881			0.643311			0.321655	−	0.946857i	$-0.395761\pi$
$389$	12.6881			0.643311			0.321655	+	0.946857i	$0.395761\pi$
$390$		0			0
$391$	−1.63858			−0.0828665
$392$		0			0
$393$	7.35501	−	30.2480i	0.371011	−	1.52581i
$394$		0			0
$395$		−	33.9232i		−	1.70686i
$396$		0			0
$397$	26.6275			1.33640			0.668199	−	0.743982i	$-0.267065\pi$
$397$	26.6275			1.33640			0.668199	+	0.743982i	$0.267065\pi$
$398$		0			0
$399$	−1.78518	+	7.34167i	−0.0893707	+	0.367543i
$400$		0			0
$401$		−	26.1618i		−	1.30646i	−0.757159	−	0.653230i	$-0.773413\pi$
$401$		−	26.1618i		−	1.30646i	0.757159	−	0.653230i	$-0.226587\pi$
$402$		0			0
$403$	−4.29135			−0.213767
$404$		0			0
$405$	−22.4131	+	15.8890i	−1.11372	+	0.789531i
$406$		0			0
$407$	1.95553			0.0969321
$408$		0			0
$409$			35.3883i			1.74984i	0.484267	+	0.874920i	$0.339086\pi$
$409$			35.3883i			1.74984i	−0.484267	+	0.874920i	$0.660914\pi$
$410$		0			0
$411$	−30.5888	−	7.43788i	−1.50883	−	0.366884i
$412$		0			0
$413$		−	27.3940i		−	1.34797i
$414$		0			0
$415$			23.0475i			1.13136i
$416$		0			0
$417$	2.87668	−	11.8306i	0.140872	−	0.579345i
$418$		0			0
$419$	36.2231			1.76961			0.884807	−	0.465957i	$-0.154290\pi$
$419$	36.2231			1.76961			0.884807	+	0.465957i	$0.154290\pi$
$420$		0			0
$421$			19.2897i			0.940122i	0.882634	+	0.470061i	$0.155768\pi$
$421$			19.2897i			0.940122i	−0.882634	+	0.470061i	$0.844232\pi$
$422$		0			0
$423$	11.6556	−	22.5502i	0.566716	−	1.09643i
$424$		0			0
$425$	−22.9065			−1.11113
$426$		0			0
$427$	−3.56655			−0.172597
$428$		0			0
$429$	0.575296	−	2.36595i	0.0277756	−	0.114229i
$430$		0			0
$431$	−35.0160			−1.68666			−0.843330	−	0.537396i	$-0.819408\pi$
$431$	−35.0160			−1.68666			−0.843330	+	0.537396i	$0.819408\pi$
$432$		0			0
$433$			26.1873i			1.25848i	0.777211	+	0.629240i	$0.216634\pi$
$433$			26.1873i			1.25848i	−0.777211	+	0.629240i	$0.783366\pi$
$434$		0			0
$435$	6.32495	−	27.7618i	0.303258	−	1.33107i
$436$		0			0
$437$	0.510039			0.0243985
$438$		0			0
$439$		−	28.0344i		−	1.33801i	−0.743259	−	0.669004i	$-0.766721\pi$
$439$		−	28.0344i		−	1.33801i	0.743259	−	0.669004i	$-0.233279\pi$
$440$		0			0
$441$	0.0506751	+	0.0261927i	0.00241310	+	0.00124727i
$442$		0			0
$443$			2.34599i			0.111461i	0.998446	+	0.0557307i	$0.0177488\pi$
$443$			2.34599i			0.111461i	−0.998446	+	0.0557307i	$0.982251\pi$
$444$		0			0
$445$		−	26.8334i		−	1.27202i
$446$		0			0
$447$	1.46822	−	6.03817i	0.0694445	−	0.285595i
$448$		0			0
$449$	−31.3303			−1.47857			−0.739283	−	0.673394i	$-0.764836\pi$
$449$	−31.3303			−1.47857			−0.739283	+	0.673394i	$0.764836\pi$
$450$		0			0
$451$		−	0.740640i		−	0.0348754i
$452$		0			0
$453$	7.54800	−	31.0417i	0.354636	−	1.45846i
$454$		0			0
$455$	34.9854			1.64014
$456$		0			0
$457$	30.1214			1.40902			0.704511	−	0.709693i	$-0.251166\pi$
$457$	30.1214			1.40902			0.704511	+	0.709693i	$0.251166\pi$
$458$		0			0
$459$	20.8006	+	18.0816i	0.970888	+	0.843978i
$460$		0			0
$461$	3.60887			0.168082			0.0840408	−	0.996462i	$-0.473217\pi$
$461$	3.60887			0.168082			0.0840408	+	0.996462i	$0.473217\pi$
$462$		0			0
$463$		−	20.3196i		−	0.944330i	−0.881510	−	0.472165i	$-0.843473\pi$
$463$		−	20.3196i		−	0.944330i	0.881510	−	0.472165i	$-0.156527\pi$
$464$		0			0
$465$	1.23592	−	5.08281i	0.0573144	−	0.235709i
$466$		0			0
$467$		−	35.9513i		−	1.66363i	−0.555056	−	0.831813i	$-0.687303\pi$
$467$		−	35.9513i		−	1.66363i	0.555056	−	0.831813i	$-0.312697\pi$
$468$		0			0
$469$	32.5986			1.50526
$470$		0			0
$471$	4.77473	−	19.6364i	0.220008	−	0.904797i
$472$		0			0
$473$			0.855706i			0.0393454i
$474$		0			0
$475$	7.13008			0.327150
$476$		0			0
$477$	−16.1629	+	31.2705i	−0.740049	+	1.43178i
$478$		0			0
$479$		−	20.1812i		−	0.922105i	−0.887373	−	0.461052i	$-0.847472\pi$
$479$		−	20.1812i		−	0.922105i	0.887373	−	0.461052i	$-0.152528\pi$
$480$		0			0
$481$			26.1729i			1.19338i
$482$		0			0
$483$	−0.334030	+	1.37372i	−0.0151989	+	0.0625066i
$484$		0			0
$485$	−18.4413			−0.837377
$486$		0			0
$487$			1.77423i			0.0803982i	0.999192	+	0.0401991i	$0.0127992\pi$
$487$			1.77423i			0.0803982i	−0.999192	+	0.0401991i	$0.987201\pi$
$488$		0			0
$489$	24.8116	+	6.03312i	1.12202	+	0.272827i
$490$		0			0
$491$			29.5894i			1.33535i	0.744453	+	0.667674i	$0.232710\pi$
$491$			29.5894i			1.33535i	−0.744453	+	0.667674i	$0.767290\pi$
$492$		0			0
$493$	−28.5605	+	0.414778i	−1.28630	+	0.0186807i
$494$		0			0
$495$	2.63661	+	1.36280i	0.118507	+	0.0612532i
$496$		0			0
$497$			18.4163i			0.826086i
$498$		0			0
$499$			6.31292i			0.282605i	0.989966	+	0.141303i	$0.0451290\pi$
$499$			6.31292i			0.282605i	−0.989966	+	0.141303i	$0.954871\pi$
$500$		0			0
$501$	−17.7973	−	4.32753i	−0.795123	−	0.193340i
$502$		0			0
$503$			38.7164i			1.72628i	0.504965	+	0.863140i	$0.331505\pi$
$503$			38.7164i			1.72628i	−0.504965	+	0.863140i	$0.668495\pi$
$504$		0			0
$505$		−	52.6151i		−	2.34134i
$506$		0			0
$507$	9.78681	+	2.37973i	0.434648	+	0.105688i
$508$		0			0
$509$			29.6256i			1.31313i	0.754268	+	0.656567i	$0.227992\pi$
$509$			29.6256i			1.31313i	−0.754268	+	0.656567i	$0.772008\pi$
$510$		0			0
$511$	−39.4597			−1.74559
$512$		0			0
$513$	−6.47458	−	5.62825i	−0.285860	−	0.248493i
$514$		0			0
$515$	−2.81803			−0.124177
$516$		0			0
$517$	−2.74227			−0.120605
$518$		0			0
$519$	5.43403	−	22.3478i	0.238527	−	0.980960i
$520$		0			0
$521$			38.6692i			1.69413i	0.531492	+	0.847063i	$0.321632\pi$
$521$			38.6692i			1.69413i	−0.531492	+	0.847063i	$0.678368\pi$
$522$		0			0
$523$			37.7301i			1.64982i	0.565263	+	0.824911i	$0.308775\pi$
$523$			37.7301i			1.64982i	−0.565263	+	0.824911i	$0.691225\pi$
$524$		0			0
$525$	−4.66957	+	19.2039i	−0.203797	+	0.838128i
$526$		0			0
$527$	−5.24752			−0.228586
$528$		0			0
$529$	−22.9046			−0.995851
$530$		0			0
$531$	27.6314	+	14.2820i	1.19910	+	0.619784i
$532$		0			0
$533$	9.91277			0.429370
$534$		0			0
$535$		−	23.7943i		−	1.02872i
$536$		0			0
$537$	29.9841	+	7.29084i	1.29391	+	0.314623i
$538$		0			0
$539$		−	0.00616246i		−	0.000265436i
$540$		0			0
$541$		−	29.3544i		−	1.26204i	−0.775765	−	0.631022i	$-0.782636\pi$
$541$		−	29.3544i		−	1.26204i	0.775765	−	0.631022i	$-0.217364\pi$
$542$		0			0
$543$	−13.3513	−	3.24647i	−0.572960	−	0.139319i
$544$		0			0
$545$		−	2.26589i		−	0.0970599i
$546$		0			0
$547$			18.3628i			0.785138i	0.919723	+	0.392569i	$0.128414\pi$
$547$			18.3628i			0.785138i	−0.919723	+	0.392569i	$0.871586\pi$
$548$		0			0
$549$	1.85943	−	3.59746i	0.0793587	−	0.153536i
$550$		0			0
$551$	8.89002	−	0.129108i	0.378728	−	0.00550017i
$552$		0			0
$553$			29.3616i			1.24858i
$554$		0			0
$555$	−31.0000	−	7.53788i	−1.31588	−	0.319965i
$556$		0			0
$557$			11.4926i			0.486957i	0.969906	+	0.243478i	$0.0782885\pi$
$557$			11.4926i			0.486957i	−0.969906	+	0.243478i	$0.921712\pi$
$558$		0			0
$559$	−11.4528			−0.484403
$560$		0			0
$561$	0.703480	−	2.89311i	0.0297010	−	0.122147i
$562$		0			0
$563$			11.1439i			0.469660i	0.972036	+	0.234830i	$0.0754534\pi$
$563$			11.1439i			0.469660i	−0.972036	+	0.234830i	$0.924547\pi$
$564$		0			0
$565$			19.3659i			0.814728i
$566$		0			0
$567$	19.3992	−	13.7524i	0.814691	−	0.577548i
$568$		0			0
$569$	12.4043			0.520017			0.260009	−	0.965606i	$-0.416275\pi$
$569$	12.4043			0.520017			0.260009	+	0.965606i	$0.416275\pi$
$570$		0			0
$571$			21.9281i			0.917663i	0.888523	+	0.458832i	$0.151732\pi$
$571$			21.9281i			0.917663i	−0.888523	+	0.458832i	$0.848268\pi$
$572$		0			0
$573$	2.50519	−	10.3028i	0.104656	−	0.430404i
$574$		0			0
$575$	1.33413			0.0556371
$576$		0			0
$577$			17.9419i			0.746929i	0.927644	+	0.373465i	$0.121830\pi$
$577$			17.9419i			0.746929i	−0.927644	+	0.373465i	$0.878170\pi$
$578$		0			0
$579$	1.13223	−	4.65637i	0.0470538	−	0.193512i
$580$		0			0
$581$		−	19.9483i		−	0.827595i
$582$		0			0
$583$	3.80272			0.157492
$584$		0			0
$585$	−18.2398	+	35.2886i	−0.754121	+	1.45900i
$586$		0			0
$587$	−19.7070			−0.813395			−0.406698	−	0.913563i	$-0.633320\pi$
$587$	−19.7070			−0.813395			−0.406698	+	0.913563i	$0.633320\pi$
$588$		0			0
$589$	1.63339			0.0673027
$590$		0			0
$591$	6.33401	−	26.0491i	0.260546	−	1.07151i
$592$		0			0
$593$		−	14.3859i		−	0.590756i	−0.955380	−	0.295378i	$-0.904554\pi$
$593$		−	14.3859i		−	0.590756i	0.955380	−	0.295378i	$-0.0954457\pi$
$594$		0			0
$595$	42.7807			1.75384
$596$		0			0
$597$	−4.28364	+	17.6168i	−0.175318	+	0.721007i
$598$		0			0
$599$			40.5960i			1.65871i	0.558725	+	0.829353i	$0.311291\pi$
$599$			40.5960i			1.65871i	−0.558725	+	0.829353i	$0.688709\pi$
$600$		0			0
$601$		−	8.63868i		−	0.352379i	−0.984356	−	0.176190i	$-0.943623\pi$
$601$		−	8.63868i		−	0.352379i	0.984356	−	0.176190i	$-0.0563772\pi$
$602$		0			0
$603$	−16.9954	+	32.8811i	−0.692105	+	1.33902i
$604$		0			0
$605$			33.2584i			1.35215i
$606$		0			0
$607$	−0.724798			−0.0294186			−0.0147093	−	0.999892i	$-0.504682\pi$
$607$	−0.724798			−0.0294186			−0.0147093	+	0.999892i	$0.504682\pi$
$608$		0			0
$609$	−5.47444	+	24.0287i	−0.221836	+	0.973690i
$610$		0			0
$611$		−	36.7027i		−	1.48483i
$612$		0			0
$613$	−19.0477			−0.769328			−0.384664	−	0.923057i	$-0.625683\pi$
$613$	−19.0477			−0.769328			−0.384664	+	0.923057i	$0.625683\pi$
$614$		0			0
$615$	−2.85490	+	11.7410i	−0.115121	+	0.473443i
$616$		0			0
$617$	3.32514			0.133865			0.0669326	−	0.997757i	$-0.478679\pi$
$617$	3.32514			0.133865			0.0669326	+	0.997757i	$0.478679\pi$
$618$		0			0
$619$	−34.6233			−1.39163			−0.695814	−	0.718222i	$-0.744956\pi$
$619$	−34.6233			−1.39163			−0.695814	+	0.718222i	$0.744956\pi$
$620$		0			0
$621$	−1.21148	−	1.05312i	−0.0486150	−	0.0422603i
$622$		0			0
$623$			23.2251i			0.930495i
$624$		0			0
$625$	−27.9426			−1.11771
$626$		0			0
$627$	−0.218972	+	0.900537i	−0.00874489	+	0.0359640i
$628$		0			0
$629$			32.0046i			1.27611i
$630$		0			0
$631$		−	7.69991i		−	0.306529i	−0.988185	−	0.153264i	$-0.951021\pi$
$631$		−	7.69991i		−	0.306529i	0.988185	−	0.153264i	$-0.0489786\pi$
$632$		0			0
$633$	−27.0389	−	6.57469i	−1.07470	−	0.261320i
$634$		0			0
$635$			47.0231i			1.86606i
$636$		0			0
$637$	0.0824788			0.00326793
$638$		0			0
$639$	−18.5759	−	9.60142i	−0.734853	−	0.379826i
$640$		0			0
$641$	13.2200			0.522159			0.261080	−	0.965317i	$-0.415921\pi$
$641$	13.2200			0.522159			0.261080	+	0.965317i	$0.415921\pi$
$642$		0			0
$643$			16.5687i			0.653406i	0.945127	+	0.326703i	$0.105938\pi$
$643$			16.5687i			0.653406i	−0.945127	+	0.326703i	$0.894062\pi$
$644$		0			0
$645$	3.29844	−	13.5651i	0.129876	−	0.534124i
$646$		0			0
$647$	36.1450			1.42101			0.710504	−	0.703693i	$-0.248467\pi$
$647$	36.1450			1.42101			0.710504	+	0.703693i	$0.248467\pi$
$648$		0			0
$649$		−	3.36018i		−	0.131898i
$650$		0			0
$651$	−1.06973	+	4.39933i	−0.0419259	+	0.172423i
$652$		0			0
$653$	34.2866			1.34174			0.670868	−	0.741576i	$-0.265922\pi$
$653$	34.2866			1.34174			0.670868	+	0.741576i	$0.265922\pi$
$654$		0			0
$655$	54.8638			2.14370
$656$		0			0
$657$	20.5724	−	39.8016i	0.802607	−	1.55281i
$658$		0			0
$659$			27.5734i			1.07411i	0.843548	+	0.537053i	$0.180463\pi$
$659$			27.5734i			1.07411i	−0.843548	+	0.537053i	$0.819537\pi$
$660$		0			0
$661$	−11.0644			−0.430356			−0.215178	−	0.976575i	$-0.569033\pi$
$661$	−11.0644			−0.430356			−0.215178	+	0.976575i	$0.569033\pi$
$662$		0			0
$663$	38.7216	+	9.41542i	1.50382	+	0.365665i
$664$		0			0
$665$	−13.3163			−0.516384
$666$		0			0
$667$	1.66344	−	0.0241578i	0.0644086	−	0.000935392i
$668$		0			0
$669$	−8.65622	+	35.5993i	−0.334669	+	1.37635i
$670$		0			0
$671$	−0.437477			−0.0168886
$672$		0			0
$673$	14.0892			0.543100			0.271550	−	0.962424i	$-0.412464\pi$
$673$	14.0892			0.543100			0.271550	+	0.962424i	$0.412464\pi$
$674$		0			0
$675$	−16.9359	−	14.7221i	−0.651861	−	0.566653i
$676$		0			0
$677$	−8.09475			−0.311107			−0.155553	−	0.987828i	$-0.549716\pi$
$677$	−8.09475			−0.311107			−0.155553	+	0.987828i	$0.549716\pi$
$678$		0			0
$679$	15.9615			0.612547
$680$		0			0
$681$	−12.5986	−	3.06343i	−0.482779	−	0.117391i
$682$		0			0
$683$	−23.8250			−0.911639			−0.455819	−	0.890072i	$-0.650654\pi$
$683$	−23.8250			−0.911639			−0.455819	+	0.890072i	$0.650654\pi$
$684$		0			0
$685$		−	55.4819i		−	2.11985i
$686$		0			0
$687$	4.70635	−	19.3552i	0.179558	−	0.738447i
$688$		0			0
$689$			50.8958i			1.93897i
$690$		0			0
$691$		−	28.2828i		−	1.07593i	−0.842968	−	0.537963i	$-0.819194\pi$
$691$		−	28.2828i		−	1.07593i	0.842968	−	0.537963i	$-0.180806\pi$
$692$		0			0
$693$	−2.28207	−	1.17954i	−0.0866887	−	0.0448072i
$694$		0			0
$695$	21.4582			0.813958
$696$		0			0
$697$	12.1215			0.459134
$698$		0			0
$699$	5.32783	−	21.9111i	0.201517	−	0.828752i
$700$		0			0
$701$		−	14.5903i		−	0.551068i	−0.961291	−	0.275534i	$-0.911145\pi$
$701$		−	14.5903i		−	0.551068i	0.961291	−	0.275534i	$-0.0888547\pi$
$702$		0			0
$703$		−	9.96206i		−	0.375726i
$704$		0			0
$705$	43.4718	+	10.5705i	1.63724	+	0.398107i
$706$		0			0
$707$			45.5400i			1.71271i
$708$		0			0
$709$	38.8982			1.46085			0.730426	−	0.682992i	$-0.239322\pi$
$709$	38.8982			1.46085			0.730426	+	0.682992i	$0.239322\pi$
$710$		0			0
$711$	−29.6160	−	15.3078i	−1.11069	−	0.574086i
$712$		0			0
$713$	0.305629			0.0114459
$714$		0			0
$715$	4.29135			0.160487
$716$		0			0
$717$	−34.3791	−	8.35953i	−1.28391	−	0.312192i
$718$		0			0
$719$	30.6929			1.14465			0.572326	−	0.820026i	$-0.306041\pi$
$719$	30.6929			1.14465			0.572326	+	0.820026i	$0.306041\pi$
$720$		0			0
$721$	2.43909			0.0908366
$722$		0			0
$723$	11.6103	+	2.82312i	0.431790	+	0.104993i
$724$		0			0
$725$	23.2540	−	0.337713i	0.863633	−	0.0125423i
$726$		0			0
$727$	45.6258			1.69217			0.846083	−	0.533051i	$-0.178955\pi$
$727$	45.6258			1.69217			0.846083	+	0.533051i	$0.178955\pi$
$728$		0			0
$729$	3.75775	+	26.7372i	0.139176	+	0.990268i
$730$		0			0
$731$	−14.0047			−0.517981
$732$		0			0
$733$		−	34.5462i		−	1.27599i	−0.770039	−	0.637996i	$-0.779764\pi$
$733$		−	34.5462i		−	1.27599i	0.770039	−	0.637996i	$-0.220236\pi$
$734$		0			0
$735$	−0.0237541	+	0.0976904i	−0.000876184	+	0.00360337i
$736$		0			0
$737$	3.99858			0.147289
$738$		0			0
$739$	6.45597			0.237487			0.118743	−	0.992925i	$-0.462113\pi$
$739$	6.45597			0.237487			0.118743	+	0.992925i	$0.462113\pi$
$740$		0			0
$741$	−12.0528	−	2.93073i	−0.442772	−	0.107663i
$742$		0			0
$743$			33.8323i			1.24119i	0.784132	+	0.620594i	$0.213108\pi$
$743$			33.8323i			1.24119i	−0.784132	+	0.620594i	$0.786892\pi$
$744$		0			0
$745$	10.9520			0.401251
$746$		0			0
$747$	20.1212	+	10.4001i	0.736195	+	0.380520i
$748$		0			0
$749$			20.5947i			0.752514i
$750$		0			0
$751$	−1.49003			−0.0543719			−0.0271859	−	0.999630i	$-0.508655\pi$
$751$	−1.49003			−0.0543719			−0.0271859	+	0.999630i	$0.508655\pi$
$752$		0			0
$753$	5.52526	−	22.7230i	0.201352	−	0.828072i
$754$		0			0
$755$	56.3033			2.04909
$756$		0			0
$757$			1.69146i			0.0614773i	0.999527	+	0.0307386i	$0.00978595\pi$
$757$			1.69146i			0.0614773i	−0.999527	+	0.0307386i	$0.990214\pi$
$758$		0			0
$759$	−0.0409725	+	0.168502i	−0.00148721	+	0.00611625i
$760$		0			0
$761$			43.9031i			1.59149i	0.605633	+	0.795744i	$0.292920\pi$
$761$			43.9031i			1.59149i	−0.605633	+	0.795744i	$0.707080\pi$
$762$		0			0
$763$			1.96120i			0.0710000i
$764$		0			0
$765$	−22.3038	+	43.1514i	−0.806397	+	1.56014i
$766$		0			0
$767$	44.9728			1.62387
$768$		0			0
$769$			15.0016i			0.540972i	0.962724	+	0.270486i	$0.0871844\pi$
$769$			15.0016i			0.540972i	−0.962724	+	0.270486i	$0.912816\pi$
$770$		0			0
$771$	0.883827	−	3.63480i	0.0318302	−	0.130904i
$772$		0			0
$773$	15.0342			0.540741			0.270370	−	0.962756i	$-0.412854\pi$
$773$	15.0342			0.540741			0.270370	+	0.962756i	$0.412854\pi$
$774$		0			0
$775$	4.27254			0.153474
$776$		0			0
$777$	26.8315	+	6.52427i	0.962575	+	0.234057i
$778$		0			0
$779$	−3.77304			−0.135183
$780$		0			0
$781$			2.25897i			0.0808322i
$782$		0			0
$783$	−21.3828	−	18.0493i	−0.764158	−	0.645030i
$784$		0			0
$785$	35.6164			1.27121
$786$		0			0
$787$		−	28.0294i		−	0.999140i	−0.866274	−	0.499570i	$-0.833491\pi$
$787$		−	28.0294i		−	0.999140i	0.866274	−	0.499570i	$-0.166509\pi$
$788$		0			0
$789$	−9.57931	+	39.3956i	−0.341032	+	1.40252i
$790$		0			0
$791$		−	16.7618i		−	0.595979i
$792$		0			0
$793$		−	5.85521i		−	0.207925i
$794$		0			0
$795$	−60.2825	−	14.6581i	−2.13800	−	0.519870i
$796$		0			0
$797$	35.5928			1.26076			0.630381	−	0.776286i	$-0.282899\pi$
$797$	35.5928			1.26076			0.630381	+	0.776286i	$0.282899\pi$
$798$		0			0
$799$		−	44.8806i		−	1.58776i
$800$		0			0
$801$	−23.4264	−	12.1085i	−0.827731	−	0.427833i
$802$		0			0
$803$	−4.84017			−0.170806
$804$		0			0
$805$	−2.49166			−0.0878193
$806$		0			0
$807$	32.6478	+	7.93855i	1.14926	+	0.279450i
$808$		0			0
$809$	5.71587			0.200959			0.100480	−	0.994939i	$-0.467962\pi$
$809$	5.71587			0.200959			0.100480	+	0.994939i	$0.467962\pi$
$810$		0			0
$811$			15.6015i			0.547843i	0.961752	+	0.273921i	$0.0883208\pi$
$811$			15.6015i			0.547843i	−0.961752	+	0.273921i	$0.911679\pi$
$812$		0			0
$813$	47.4277	+	11.5324i	1.66336	+	0.404458i
$814$		0			0
$815$			45.0033i			1.57640i
$816$		0			0
$817$	4.35922			0.152510
$818$		0			0
$819$	15.7871	−	30.5434i	0.551645	−	1.06727i
$820$		0			0
$821$		−	35.6929i		−	1.24569i	−0.782346	−	0.622845i	$-0.785977\pi$
$821$		−	35.6929i		−	1.24569i	0.782346	−	0.622845i	$-0.214023\pi$
$822$		0			0
$823$	−13.4761			−0.469748			−0.234874	−	0.972026i	$-0.575468\pi$
$823$	−13.4761			−0.469748			−0.234874	+	0.972026i	$0.575468\pi$
$824$		0			0
$825$	−0.572774	+	2.35557i	−0.0199414	+	0.0820106i
$826$		0			0
$827$			18.0179i			0.626542i	0.949664	+	0.313271i	$0.101425\pi$
$827$			18.0179i			0.626542i	−0.949664	+	0.313271i	$0.898575\pi$
$828$		0			0
$829$			53.4043i			1.85481i	0.374060	+	0.927404i	$0.377965\pi$
$829$			53.4043i			1.85481i	−0.374060	+	0.927404i	$0.622035\pi$
$830$		0			0
$831$	−14.4281	−	3.50829i	−0.500504	−	0.121701i
$832$		0			0
$833$	0.100856			0.00349446
$834$		0			0
$835$		−	32.2806i		−	1.11712i
$836$		0			0
$837$	−3.87974	−	3.37260i	−0.134104	−	0.116574i
$838$		0			0
$839$		−	24.0362i		−	0.829823i	−0.909862	−	0.414911i	$-0.863813\pi$
$839$		−	24.0362i		−	0.829823i	0.909862	−	0.414911i	$-0.136187\pi$
$840$		0			0
$841$	28.9878	−	0.842143i	0.999578	−	0.0290394i
$842$		0			0
$843$	5.76577	−	23.7121i	0.198584	−	0.816689i
$844$		0			0
$845$			17.7513i			0.610663i
$846$		0			0
$847$		−	28.7862i		−	0.989105i
$848$		0			0
$849$	5.35677	−	22.0301i	0.183844	−	0.756070i
$850$		0			0
$851$		−	1.86403i		−	0.0638982i
$852$		0			0
$853$		−	20.3607i		−	0.697138i	−0.937283	−	0.348569i	$-0.886668\pi$
$853$		−	20.3607i		−	0.697138i	0.937283	−	0.348569i	$-0.113332\pi$
$854$		0			0
$855$	6.94250	−	13.4317i	0.237428	−	0.459354i
$856$		0			0
$857$			16.4101i			0.560557i	0.959919	+	0.280279i	$0.0904269\pi$
$857$			16.4101i			0.560557i	−0.959919	+	0.280279i	$0.909573\pi$
$858$		0			0
$859$	−20.9085			−0.713389			−0.356694	−	0.934221i	$-0.616096\pi$
$859$	−20.9085			−0.713389			−0.356694	+	0.934221i	$0.616096\pi$
$860$		0			0
$861$	2.47101	−	10.1622i	0.0842117	−	0.346327i
$862$		0			0
$863$	−54.2074			−1.84524			−0.922621	−	0.385707i	$-0.873957\pi$
$863$	−54.2074			−1.84524			−0.922621	+	0.385707i	$0.873957\pi$
$864$		0			0
$865$	40.5344			1.37821
$866$		0			0
$867$	18.7381	+	4.55630i	0.636379	+	0.154740i
$868$		0			0
$869$			3.60152i			0.122173i
$870$		0			0
$871$			53.5172i			1.81336i
$872$		0			0
$873$	−8.32160	+	16.0998i	−0.281643	+	0.544897i
$874$		0			0
$875$	5.49572			0.185789
$876$		0			0
$877$	14.8343			0.500920			0.250460	−	0.968127i	$-0.419418\pi$
$877$	14.8343			0.500920			0.250460	+	0.968127i	$0.419418\pi$
$878$		0			0
$879$	20.6981	+	5.03288i	0.698128	+	0.169755i
$880$		0			0
$881$	14.3779			0.484405			0.242202	−	0.970226i	$-0.422130\pi$
$881$	14.3779			0.484405			0.242202	+	0.970226i	$0.422130\pi$
$882$		0			0
$883$		−	33.4000i		−	1.12400i	−0.827137	−	0.562000i	$-0.810032\pi$
$883$		−	33.4000i		−	1.12400i	0.827137	−	0.562000i	$-0.189968\pi$
$884$		0			0
$885$	−12.9523	+	53.2672i	−0.435386	+	1.79056i
$886$		0			0
$887$		−	4.87261i		−	0.163606i	−0.996649	−	0.0818031i	$-0.973932\pi$
$887$		−	4.87261i		−	0.163606i	0.996649	−	0.0818031i	$-0.0260679\pi$
$888$		0			0
$889$		−	40.7000i		−	1.36503i
$890$		0			0
$891$	2.37953	−	1.68689i	0.0797173	−	0.0565129i
$892$		0			0
$893$			13.9699i			0.467486i
$894$		0			0
$895$			54.3851i			1.81789i
$896$		0			0
$897$	−2.25524	−	0.548379i	−0.0753004	−	0.0183098i
$898$		0			0
$899$	5.32714	−	0.0773648i	0.177670	−	0.00258026i
$900$		0			0
$901$			62.2361i			2.07338i
$902$		0			0
$903$	−2.85490	+	11.7410i	−0.0950053	+	0.390716i
$904$		0			0
$905$		−	24.2166i		−	0.804987i
$906$		0			0
$907$	34.1416			1.13365			0.566826	−	0.823837i	$-0.308171\pi$
$907$	34.1416			1.13365			0.566826	+	0.823837i	$0.308171\pi$
$908$		0			0
$909$	−45.9346	−	23.7424i	−1.52356	−	0.787487i
$910$		0			0
$911$		−	2.12516i		−	0.0704098i	−0.999380	−	0.0352049i	$-0.988792\pi$
$911$		−	2.12516i		−	0.0704098i	0.999380	−	0.0352049i	$-0.0112084\pi$
$912$		0			0
$913$		−	2.44688i		−	0.0809799i
$914$		0			0
$915$	6.93510	+	1.68632i	0.229267	+	0.0557479i
$916$		0			0
$917$	−47.4863			−1.56814
$918$		0			0
$919$			26.3749i			0.870026i	0.900424	+	0.435013i	$0.143256\pi$
$919$			26.3749i			0.870026i	−0.900424	+	0.435013i	$0.856744\pi$
$920$		0			0
$921$	−43.5781	−	10.5963i	−1.43595	−	0.349160i
$922$		0			0
$923$	−30.2342			−0.995169
$924$		0			0
$925$		−	26.0582i		−	0.856789i
$926$		0			0
$927$	−1.27163	+	2.46023i	−0.0417658	+	0.0808046i
$928$		0			0
$929$			14.2303i			0.466881i	0.972371	+	0.233440i	$0.0749984\pi$
$929$			14.2303i			0.466881i	−0.972371	+	0.233440i	$0.925002\pi$
$930$		0			0
$931$	−0.0313934			−0.00102888
$932$		0			0
$933$	2.96388	−	12.1892i	0.0970330	−	0.399055i
$934$		0			0
$935$	5.24752			0.171612
$936$		0			0
$937$	−55.2741			−1.80573			−0.902864	−	0.429927i	$-0.858539\pi$
$937$	−55.2741			−1.80573			−0.902864	+	0.429927i	$0.858539\pi$
$938$		0			0
$939$	3.86425	+	0.939618i	0.126105	+	0.0306633i
$940$		0			0
$941$		−	33.2871i		−	1.08513i	−0.840015	−	0.542564i	$-0.817454\pi$
$941$		−	33.2871i		−	1.08513i	0.840015	−	0.542564i	$-0.182546\pi$
$942$		0			0
$943$	−0.705986			−0.0229901
$944$		0			0
$945$	31.6298	+	27.4953i	1.02892	+	0.894422i
$946$		0			0
$947$		−	39.0684i		−	1.26955i	−0.772696	−	0.634777i	$-0.781092\pi$
$947$		−	39.0684i		−	1.26955i	0.772696	−	0.634777i	$-0.218908\pi$
$948$		0			0
$949$		−	64.7810i		−	2.10288i
$950$		0			0
$951$	−37.4579	−	9.10814i	−1.21465	−	0.295352i
$952$		0			0
$953$		−	11.3865i		−	0.368845i	−0.982847	−	0.184422i	$-0.940959\pi$
$953$		−	11.3865i		−	0.368845i	0.982847	−	0.184422i	$-0.0590415\pi$
$954$		0			0
$955$	18.6871			0.604702
$956$		0			0
$957$	−0.671501	+	2.94738i	−0.0217065	+	0.0952753i
$958$		0			0
$959$			48.0213i			1.55069i
$960$		0			0
$961$	−30.0212			−0.968427
$962$		0			0
$963$	−20.7732	−	10.7371i	−0.669406	−	0.345999i
$964$		0			0
$965$	8.44571			0.271877
$966$		0			0
$967$	−26.9548			−0.866807			−0.433403	−	0.901200i	$-0.642687\pi$
$967$	−26.9548			−0.866807			−0.433403	+	0.901200i	$0.642687\pi$
$968$		0			0
$969$	−14.7384	−	3.58374i	−0.473465	−	0.115126i
$970$		0			0
$971$		−	33.2812i		−	1.06804i	−0.845471	−	0.534022i	$-0.820680\pi$
$971$		−	33.2812i		−	1.06804i	0.845471	−	0.534022i	$-0.179320\pi$
$972$		0			0
$973$	−18.5728			−0.595416
$974$		0			0
$975$	−31.5271	−	7.66605i	−1.00968	−	0.245510i
$976$		0			0
$977$		−	14.9482i		−	0.478234i	−0.970991	−	0.239117i	$-0.923142\pi$
$977$		−	14.9482i		−	0.478234i	0.970991	−	0.239117i	$-0.0768580\pi$
$978$		0			0
$979$			2.84882i			0.0910486i
$980$		0			0
$981$	−1.97819	−	1.02248i	−0.0631588	−	0.0326452i
$982$		0			0
$983$		−	36.3642i		−	1.15984i	−0.814674	−	0.579919i	$-0.803084\pi$
$983$		−	36.3642i		−	1.15984i	0.814674	−	0.579919i	$-0.196916\pi$
$984$		0			0
$985$	47.2477			1.50544
$986$		0			0
$987$	−37.6262	−	9.14908i	−1.19766	−	0.291218i
$988$		0			0
$989$	0.815668			0.0259367
$990$		0			0
$991$		−	16.0086i		−	0.508531i	−0.967134	−	0.254266i	$-0.918166\pi$
$991$		−	16.0086i		−	0.508531i	0.967134	−	0.254266i	$-0.0818337\pi$
$992$		0			0
$993$	−47.1568	−	11.4665i	−1.49647	−	0.363878i
$994$		0			0
$995$	−31.9533			−1.01299
$996$		0			0
$997$			59.3764i			1.88047i	0.340525	+	0.940235i	$0.389395\pi$
$997$			59.3764i			1.88047i	−0.340525	+	0.940235i	$0.610605\pi$
$998$		0			0
$999$	−20.5695	+	23.6626i	−0.650790	+	0.748651i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1392.2.c.c.1391.39	yes	40
3.2	odd	2	inner	1392.2.c.c.1391.37	yes	40
4.3	odd	2	inner	1392.2.c.c.1391.2	yes	40
12.11	even	2	inner	1392.2.c.c.1391.4	yes	40
29.28	even	2	inner	1392.2.c.c.1391.1	✓	40
87.86	odd	2	inner	1392.2.c.c.1391.3	yes	40
116.115	odd	2	inner	1392.2.c.c.1391.40	yes	40
348.347	even	2	inner	1392.2.c.c.1391.38	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1392.2.c.c.1391.1	✓	40	29.28	even	2	inner
1392.2.c.c.1391.2	yes	40	4.3	odd	2	inner
1392.2.c.c.1391.3	yes	40	87.86	odd	2	inner
1392.2.c.c.1391.4	yes	40	12.11	even	2	inner
1392.2.c.c.1391.37	yes	40	3.2	odd	2	inner
1392.2.c.c.1391.38	yes	40	348.347	even	2	inner
1392.2.c.c.1391.39	yes	40	1.1	even	1	trivial
1392.2.c.c.1391.40	yes	40	116.115	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:	\( N \)	\(=\)	\( 1392 = 2^{4} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1392.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.68301 + 0.409236i) q^{3} +3.05264i q^{5} -2.64216i q^{7} +(2.66505 + 1.37750i) q^{9} -0.324090i q^{11} +4.33763 q^{13} +(-1.24925 + 5.13763i) q^{15} +5.30412 q^{17} -1.65101 q^{19} +(1.08126 - 4.44678i) q^{21} -0.308926 q^{23} -4.31862 q^{25} +(3.92159 + 3.40898i) q^{27} +(-5.38460 + 0.0781992i) q^{29} -0.989329 q^{31} +(0.132629 - 0.545446i) q^{33} +8.06555 q^{35} +6.03392i q^{37} +(7.30028 + 1.77512i) q^{39} +2.28529 q^{41} -2.64034 q^{43} +(-4.20500 + 8.13545i) q^{45} -8.46146i q^{47} +0.0190147 q^{49} +(8.92689 + 2.17064i) q^{51} +11.7335i q^{53} +0.989329 q^{55} +(-2.77867 - 0.675652i) q^{57} +10.3680 q^{59} -1.34986i q^{61} +(3.63956 - 7.04148i) q^{63} +13.2412i q^{65} +12.3379i q^{67} +(-0.519925 - 0.126423i) q^{69} -6.97020 q^{71} -14.9347i q^{73} +(-7.26828 - 1.76733i) q^{75} -0.856295 q^{77} -11.1127 q^{79} +(5.20500 + 7.34220i) q^{81} +7.55001 q^{83} +16.1916i q^{85} +(-9.09434 - 2.07196i) q^{87} -8.79022 q^{89} -11.4607i q^{91} +(-1.66505 - 0.404869i) q^{93} -5.03994i q^{95} +6.04110i q^{97} +(0.446432 - 0.863716i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 8 q^{9} - 8 q^{13} - 48 q^{25} + 8 q^{45} - 56 q^{49} - 48 q^{57} + 32 q^{81} + 48 q^{93}+O(q^{100}) \) 40 * q - 8 * q^9 - 8 * q^13 - 48 * q^25 + 8 * q^45 - 56 * q^49 - 48 * q^57 + 32 * q^81 + 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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