LMFDB - Embedded newform 336.6.h.b.239.29

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(239,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("336.239");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.29
Character	$\chi$	$=$	336.239
Dual form			336.6.h.b.239.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(11.3960 - 10.6363i) q^{3} -1.64589i q^{5} +49.0000i q^{7} +(16.7365 - 242.423i) q^{9} +O(q^{10})$	\(q+(11.3960 - 10.6363i) q^{3} -1.64589i q^{5} +49.0000i q^{7} +(16.7365 - 242.423i) q^{9} -338.070 q^{11} -419.299 q^{13} +(-17.5062 - 18.7565i) q^{15} +1091.66i q^{17} +1783.13i q^{19} +(521.181 + 558.403i) q^{21} +1082.02 q^{23} +3122.29 q^{25} +(-2387.76 - 2940.66i) q^{27} +6526.88i q^{29} -2259.29i q^{31} +(-3852.64 + 3595.83i) q^{33} +80.6484 q^{35} -5723.56 q^{37} +(-4778.32 + 4459.81i) q^{39} +5959.06i q^{41} +9145.68i q^{43} +(-399.000 - 27.5463i) q^{45} -2181.89 q^{47} -2401.00 q^{49} +(11611.2 + 12440.5i) q^{51} +142.738i q^{53} +556.425i q^{55} +(18966.0 + 20320.6i) q^{57} +22221.3 q^{59} -38659.1 q^{61} +(11878.7 + 820.088i) q^{63} +690.118i q^{65} +49837.7i q^{67} +(12330.7 - 11508.7i) q^{69} +156.642 q^{71} +14656.6 q^{73} +(35581.6 - 33209.8i) q^{75} -16565.4i q^{77} +8355.00i q^{79} +(-58488.8 - 8114.62i) q^{81} +122707. q^{83} +1796.74 q^{85} +(69422.1 + 74380.2i) q^{87} -39034.7i q^{89} -20545.7i q^{91} +(-24030.6 - 25746.8i) q^{93} +2934.83 q^{95} -146768. q^{97} +(-5658.10 + 81956.0i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 8 q^{9}+O(q^{10}) $ 40 * q + 8 * q^9	$ 40 q + 8 q^{9} - 1048 q^{13} + 980 q^{21} - 43416 q^{25} + 20296 q^{33} - 16192 q^{37} + 56488 q^{45} - 96040 q^{49} + 31088 q^{57} + 173112 q^{61} - 114176 q^{69} - 267488 q^{73} + 64888 q^{81} + 508112 q^{85} - 224544 q^{93} - 276400 q^{97}+O(q^{100}) $ 40 * q + 8 * q^9 - 1048 * q^13 + 980 * q^21 - 43416 * q^25 + 20296 * q^33 - 16192 * q^37 + 56488 * q^45 - 96040 * q^49 + 31088 * q^57 + 173112 * q^61 - 114176 * q^69 - 267488 * q^73 + 64888 * q^81 + 508112 * q^85 - 224544 * q^93 - 276400 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	11.3960	−	10.6363i	0.731052	−	0.682322i
$3$	11.3960	−	10.6363i	0.731052	−	0.682322i
$4$		0			0
$5$		−	1.64589i		−	0.0294425i	−0.999892	−	0.0147212i	$-0.995314\pi$
$5$		−	1.64589i		−	0.0294425i	0.999892	−	0.0147212i	$-0.00468609\pi$
$6$		0			0
$7$			49.0000i			0.377964i
$7$			49.0000i			0.377964i
$8$		0			0
$9$	16.7365	−	242.423i	0.0688744	−	0.997625i
$10$		0			0
$11$	−338.070			−0.842413			−0.421207	−	0.906965i	$-0.638393\pi$
$11$	−338.070			−0.842413			−0.421207	+	0.906965i	$0.638393\pi$
$12$		0			0
$13$	−419.299			−0.688123			−0.344061	−	0.938947i	$-0.611803\pi$
$13$	−419.299			−0.688123			−0.344061	+	0.938947i	$0.611803\pi$
$14$		0			0
$15$	−17.5062	−	18.7565i	−0.0200892	−	0.0215240i
$16$		0			0
$17$			1091.66i			0.916144i	0.888915	+	0.458072i	$0.151460\pi$
$17$			1091.66i			0.916144i	−0.888915	+	0.458072i	$0.848540\pi$
$18$		0			0
$19$			1783.13i			1.13318i	0.823999	+	0.566591i	$0.191738\pi$
$19$			1783.13i			1.13318i	−0.823999	+	0.566591i	$0.808262\pi$
$20$		0			0
$21$	521.181	+	558.403i	0.257893	+	0.276312i
$22$		0			0
$23$	1082.02			0.426497			0.213248	−	0.976998i	$-0.431596\pi$
$23$	1082.02			0.426497			0.213248	+	0.976998i	$0.431596\pi$
$24$		0			0
$25$	3122.29			0.999133
$26$		0			0
$27$	−2387.76	−	2940.66i	−0.630351	−	0.776311i
$28$		0			0
$29$			6526.88i			1.44116i	0.693374	+	0.720578i	$0.256123\pi$
$29$			6526.88i			1.44116i	−0.693374	+	0.720578i	$0.743877\pi$
$30$		0			0
$31$		−	2259.29i		−	0.422249i	−0.977459	−	0.211124i	$-0.932288\pi$
$31$		−	2259.29i		−	0.422249i	0.977459	−	0.211124i	$-0.0677125\pi$
$32$		0			0
$33$	−3852.64	+	3595.83i	−0.615848	+	0.574797i
$34$		0			0
$35$	80.6484			0.0111282
$36$		0			0
$37$	−5723.56			−0.687325			−0.343662	−	0.939093i	$-0.611668\pi$
$37$	−5723.56			−0.687325			−0.343662	+	0.939093i	$0.611668\pi$
$38$		0			0
$39$	−4778.32	+	4459.81i	−0.503053	+	0.469521i
$40$		0			0
$41$			5959.06i			0.553628i	0.960923	+	0.276814i	$0.0892786\pi$
$41$			5959.06i			0.553628i	−0.960923	+	0.276814i	$0.910721\pi$
$42$		0			0
$43$			9145.68i			0.754301i	0.926152	+	0.377151i	$0.123096\pi$
$43$			9145.68i			0.754301i	−0.926152	+	0.377151i	$0.876904\pi$
$44$		0			0
$45$	−399.000	−	27.5463i	−0.0293726	−	0.00202783i
$46$		0			0
$47$	−2181.89			−0.144075			−0.0720375	−	0.997402i	$-0.522950\pi$
$47$	−2181.89			−0.144075			−0.0720375	+	0.997402i	$0.522950\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	11611.2	+	12440.5i	0.625105	+	0.669749i
$52$		0			0
$53$			142.738i			0.00697990i	0.999994	+	0.00348995i	$0.00111089\pi$
$53$			142.738i			0.00697990i	−0.999994	+	0.00348995i	$0.998889\pi$
$54$		0			0
$55$			556.425i			0.0248027i
$56$		0			0
$57$	18966.0	+	20320.6i	0.773195	+	0.828416i
$58$		0			0
$59$	22221.3			0.831072			0.415536	−	0.909577i	$-0.363594\pi$
$59$	22221.3			0.831072			0.415536	+	0.909577i	$0.363594\pi$
$60$		0			0
$61$	−38659.1			−1.33023			−0.665115	−	0.746741i	$-0.731617\pi$
$61$	−38659.1			−1.33023			−0.665115	+	0.746741i	$0.731617\pi$
$62$		0			0
$63$	11878.7	+	820.088i	0.377067	+	0.0260321i
$64$		0			0
$65$			690.118i			0.0202600i
$66$		0			0
$67$			49837.7i			1.35635i	0.734902	+	0.678173i	$0.237228\pi$
$67$			49837.7i			1.35635i	−0.734902	+	0.678173i	$0.762772\pi$
$68$		0			0
$69$	12330.7	−	11508.7i	0.311791	−	0.291008i
$70$		0			0
$71$	156.642			0.00368775			0.00184387	−	0.999998i	$-0.499413\pi$
$71$	156.642			0.00368775			0.00184387	+	0.999998i	$0.499413\pi$
$72$		0			0
$73$	14656.6			0.321905			0.160952	−	0.986962i	$-0.448543\pi$
$73$	14656.6			0.321905			0.160952	+	0.986962i	$0.448543\pi$
$74$		0			0
$75$	35581.6	−	33209.8i	0.730418	−	0.681730i
$76$		0			0
$77$		−	16565.4i		−	0.318402i
$78$		0			0
$79$			8355.00i			0.150619i	0.997160	+	0.0753093i	$0.0239944\pi$
$79$			8355.00i			0.150619i	−0.997160	+	0.0753093i	$0.976006\pi$
$80$		0			0
$81$	−58488.8	−	8114.62i	−0.990513	−	0.137422i
$82$		0			0
$83$	122707.			1.95513			0.977563	−	0.210643i	$-0.0675556\pi$
$83$	122707.			1.95513			0.977563	+	0.210643i	$0.0675556\pi$
$84$		0			0
$85$	1796.74			0.0269736
$86$		0			0
$87$	69422.1	+	74380.2i	0.983331	+	1.05356i
$88$		0			0
$89$		−	39034.7i		−	0.522367i	−0.965289	−	0.261183i	$-0.915887\pi$
$89$		−	39034.7i		−	0.522367i	0.965289	−	0.261183i	$-0.0841127\pi$
$90$		0			0
$91$		−	20545.7i		−	0.260086i
$92$		0			0
$93$	−24030.6	−	25746.8i	−0.288109	−	0.308686i
$94$		0			0
$95$	2934.83			0.0333637
$96$		0			0
$97$	−146768.			−1.58381			−0.791906	−	0.610643i	$-0.790911\pi$
$97$	−146768.			−1.58381			−0.791906	+	0.610643i	$0.790911\pi$
$98$		0			0
$99$	−5658.10	+	81956.0i	−0.0580207	+	0.840413i
$100$		0			0
$101$			159536.i			1.55616i	0.628164	+	0.778081i	$0.283807\pi$
$101$			159536.i			1.55616i	−0.628164	+	0.778081i	$0.716193\pi$
$102$		0			0
$103$			140555.i			1.30543i	0.757604	+	0.652714i	$0.226370\pi$
$103$			140555.i			1.30543i	−0.757604	+	0.652714i	$0.773630\pi$
$104$		0			0
$105$	919.067	−	857.804i	0.00813530	−	0.00759302i
$106$		0			0
$107$	145462.			1.22826			0.614132	−	0.789204i	$-0.289506\pi$
$107$	145462.			1.22826			0.614132	+	0.789204i	$0.289506\pi$
$108$		0			0
$109$	40512.9			0.326608			0.163304	−	0.986576i	$-0.447785\pi$
$109$	40512.9			0.326608			0.163304	+	0.986576i	$0.447785\pi$
$110$		0			0
$111$	−65225.6	+	60877.7i	−0.502470	+	0.468977i
$112$		0			0
$113$		−	25294.8i		−	0.186352i	−0.995650	−	0.0931762i	$-0.970298\pi$
$113$		−	25294.8i		−	0.186352i	0.995650	−	0.0931762i	$-0.0297020\pi$
$114$		0			0
$115$		−	1780.88i		−	0.0125571i
$116$		0			0
$117$	−7017.60	+	101648.i	−0.0473940	+	0.686488i
$118$		0			0
$119$	−53491.2			−0.346270
$120$		0			0
$121$	−46759.6			−0.290340
$122$		0			0
$123$	63382.6	+	67909.3i	0.377753	+	0.404731i
$124$		0			0
$125$		−	10282.3i		−	0.0588595i
$126$		0			0
$127$			1363.36i			0.00750071i	0.999993	+	0.00375035i	$0.00119378\pi$
$127$			1363.36i			0.00750071i	−0.999993	+	0.00375035i	$0.998806\pi$
$128$		0			0
$129$	97276.6	+	104224.i	0.514676	+	0.551434i
$130$		0			0
$131$	206877.			1.05326			0.526628	−	0.850096i	$-0.323456\pi$
$131$	206877.			1.05326			0.526628	+	0.850096i	$0.323456\pi$
$132$		0			0
$133$	−87373.6			−0.428303
$134$		0			0
$135$	−4839.99	+	3929.99i	−0.0228565	+	0.0185591i
$136$		0			0
$137$			93138.4i			0.423962i	0.977274	+	0.211981i	$0.0679916\pi$
$137$			93138.4i			0.423962i	−0.977274	+	0.211981i	$0.932008\pi$
$138$		0			0
$139$			104434.i			0.458464i	0.973372	+	0.229232i	$0.0736214\pi$
$139$			104434.i			0.458464i	−0.973372	+	0.229232i	$0.926379\pi$
$140$		0			0
$141$	−24864.8	+	23207.4i	−0.105326	+	0.0983055i
$142$		0			0
$143$	141753.			0.579683
$144$		0			0
$145$	10742.5			0.0424312
$146$		0			0
$147$	−27361.7	+	25537.9i	−0.104436	+	0.0974745i
$148$		0			0
$149$			475761.i			1.75559i	0.479036	+	0.877795i	$0.340986\pi$
$149$			475761.i			1.75559i	−0.479036	+	0.877795i	$0.659014\pi$
$150$		0			0
$151$		−	196604.i		−	0.701699i	−0.936432	−	0.350849i	$-0.885893\pi$
$151$		−	196604.i		−	0.701699i	0.936432	−	0.350849i	$-0.114107\pi$
$152$		0			0
$153$	264643.	+	18270.5i	0.913969	+	0.0630989i
$154$		0			0
$155$	−3718.54			−0.0124320
$156$		0			0
$157$	−454423.			−1.47133			−0.735667	−	0.677344i	$-0.763131\pi$
$157$	−454423.			−1.47133			−0.735667	+	0.677344i	$0.763131\pi$
$158$		0			0
$159$	1518.21	+	1626.64i	0.00476254	+	0.00510267i
$160$		0			0
$161$			53019.0i			0.161201i
$162$		0			0
$163$			183238.i			0.540190i	0.962834	+	0.270095i	$0.0870551\pi$
$163$			183238.i			0.540190i	−0.962834	+	0.270095i	$0.912945\pi$
$164$		0			0
$165$	5918.32	+	6341.00i	0.0169234	+	0.0181321i
$166$		0			0
$167$	−542867.			−1.50627			−0.753134	−	0.657867i	$-0.771459\pi$
$167$	−542867.			−1.50627			−0.753134	+	0.657867i	$0.771459\pi$
$168$		0			0
$169$	−195481.			−0.526487
$170$		0			0
$171$	432273.	+	29843.4i	1.13049	+	0.0780473i
$172$		0			0
$173$		−	3143.85i		−	0.00798633i	−0.999992	−	0.00399316i	$-0.998729\pi$
$173$		−	3143.85i		−	0.00798633i	0.999992	−	0.00399316i	$-0.00127107\pi$
$174$		0			0
$175$			152992.i			0.377637i
$176$		0			0
$177$	253233.	−	236353.i	0.607557	−	0.567059i
$178$		0			0
$179$	−594264.			−1.38627			−0.693133	−	0.720809i	$-0.743770\pi$
$179$	−594264.			−1.38627			−0.693133	+	0.720809i	$0.743770\pi$
$180$		0			0
$181$	103750.			0.235393			0.117696	−	0.993050i	$-0.462449\pi$
$181$	103750.			0.235393			0.117696	+	0.993050i	$0.462449\pi$
$182$		0			0
$183$	−440558.	+	411191.i	−0.972467	+	0.907645i
$184$		0			0
$185$			9420.32i			0.0202365i
$186$		0			0
$187$		−	369056.i		−	0.771772i
$188$		0			0
$189$	144092.	−	117000.i	0.293418	−	0.238250i
$190$		0			0
$191$	578338.			1.14709			0.573546	−	0.819173i	$-0.305567\pi$
$191$	578338.			1.14709			0.573546	+	0.819173i	$0.305567\pi$
$192$		0			0
$193$	−226522.			−0.437742			−0.218871	−	0.975754i	$-0.570237\pi$
$193$	−226522.			−0.437742			−0.218871	+	0.975754i	$0.570237\pi$
$194$		0			0
$195$	7340.34	+	7864.57i	0.0138239	+	0.0148111i
$196$		0			0
$197$		−	724593.i		−	1.33024i	−0.746738	−	0.665118i	$-0.768381\pi$
$197$		−	724593.i		−	1.33024i	0.746738	−	0.665118i	$-0.231619\pi$
$198$		0			0
$199$		−	455160.i		−	0.814764i	−0.913258	−	0.407382i	$-0.866442\pi$
$199$		−	455160.i		−	0.814764i	0.913258	−	0.407382i	$-0.133558\pi$
$200$		0			0
$201$	530090.	+	567949.i	0.925465	+	0.991560i
$202$		0			0
$203$	−319817.			−0.544705
$204$		0			0
$205$	9807.93			0.0163002
$206$		0			0
$207$	18109.2	−	262306.i	0.0293747	−	0.425484i
$208$		0			0
$209$		−	602824.i		−	0.954608i
$210$		0			0
$211$		−	964129.i		−	1.49083i	−0.666599	−	0.745416i	$-0.732251\pi$
$211$		−	964129.i		−	1.49083i	0.666599	−	0.745416i	$-0.267749\pi$
$212$		0			0
$213$	1785.08	−	1666.09i	0.00269594	−	0.00251623i
$214$		0			0
$215$	15052.7			0.0222085
$216$		0			0
$217$	110705.			0.159595
$218$		0			0
$219$	167027.	−	155893.i	0.235329	−	0.219642i
$220$		0			0
$221$		−	457731.i		−	0.630419i
$222$		0			0
$223$		−	63217.5i		−	0.0851286i	−0.999094	−	0.0425643i	$-0.986447\pi$
$223$		−	63217.5i		−	0.0851286i	0.999094	−	0.0425643i	$-0.0135527\pi$
$224$		0			0
$225$	52256.2	−	756915.i	0.0688147	−	0.996761i
$226$		0			0
$227$	1.10400e6			1.42202			0.711009	−	0.703183i	$-0.248239\pi$
$227$	1.10400e6			1.42202			0.711009	+	0.703183i	$0.248239\pi$
$228$		0			0
$229$	−574557.			−0.724010			−0.362005	−	0.932176i	$-0.617908\pi$
$229$	−574557.			−0.724010			−0.362005	+	0.932176i	$0.617908\pi$
$230$		0			0
$231$	−176196.	−	188779.i	−0.217253	−	0.232769i
$232$		0			0
$233$		−	908368.i		−	1.09616i	−0.836427	−	0.548078i	$-0.815360\pi$
$233$		−	908368.i		−	1.09616i	0.836427	−	0.548078i	$-0.184640\pi$
$234$		0			0
$235$			3591.14i			0.00424193i
$236$		0			0
$237$	88866.7	+	95213.4i	0.102770	+	0.110110i
$238$		0			0
$239$	−933752.			−1.05739			−0.528697	−	0.848811i	$-0.677319\pi$
$239$	−933752.			−1.05739			−0.528697	+	0.848811i	$0.677319\pi$
$240$		0			0
$241$	477979.			0.530110			0.265055	−	0.964233i	$-0.414610\pi$
$241$	477979.			0.530110			0.265055	+	0.964233i	$0.414610\pi$
$242$		0			0
$243$	−752846.	+	529633.i	−0.817882	+	0.575386i
$244$		0			0
$245$			3951.77i			0.00420607i
$246$		0			0
$247$		−	747667.i		−	0.779769i
$248$		0			0
$249$	1.39837e6	−	1.30516e6i	1.42930	−	1.33402i
$250$		0			0
$251$	−204288.			−0.204672			−0.102336	−	0.994750i	$-0.532632\pi$
$251$	−204288.			−0.204672			−0.102336	+	0.994750i	$0.532632\pi$
$252$		0			0
$253$	−365799.			−0.359286
$254$		0			0
$255$	20475.6	−	19110.8i	0.0197191	−	0.0184046i
$256$		0			0
$257$		−	124182.i		−	0.117280i	−0.998279	−	0.0586400i	$-0.981324\pi$
$257$		−	124182.i		−	0.117280i	0.998279	−	0.0586400i	$-0.0186764\pi$
$258$		0			0
$259$		−	280454.i		−	0.259784i
$260$		0			0
$261$	1.58227e6	+	109237.i	1.43773	+	0.0992587i
$262$		0			0
$263$	−1.16004e6			−1.03415			−0.517074	−	0.855940i	$-0.672979\pi$
$263$	−1.16004e6			−1.03415			−0.517074	+	0.855940i	$0.672979\pi$
$264$		0			0
$265$	234.930			0.000205506
$266$		0			0
$267$	−415186.	−	444838.i	−0.356422	−	0.381877i
$268$		0			0
$269$			2.06129e6i			1.73684i	0.495833	+	0.868418i	$0.334863\pi$
$269$			2.06129e6i			1.73684i	−0.495833	+	0.868418i	$0.665137\pi$
$270$		0			0
$271$		−	1.23369e6i		−	1.02043i	−0.860048	−	0.510214i	$-0.829566\pi$
$271$		−	1.23369e6i		−	1.02043i	0.860048	−	0.510214i	$-0.170434\pi$
$272$		0			0
$273$	−218531.	−	234138.i	−0.177462	−	0.190136i
$274$		0			0
$275$	−1.05555e6			−0.841683
$276$		0			0
$277$	−1.94351e6			−1.52191			−0.760953	−	0.648807i	$-0.775268\pi$
$277$	−1.94351e6			−1.52191			−0.760953	+	0.648807i	$0.775268\pi$
$278$		0			0
$279$	−547704.	−	37812.6i	−0.421246	−	0.0290821i
$280$		0			0
$281$		−	703887.i		−	0.531786i	−0.964002	−	0.265893i	$-0.914333\pi$
$281$		−	703887.i		−	0.531786i	0.964002	−	0.265893i	$-0.0856669\pi$
$282$		0			0
$283$		−	1.32443e6i		−	0.983022i	−0.870871	−	0.491511i	$-0.836445\pi$
$283$		−	1.32443e6i		−	0.983022i	0.870871	−	0.491511i	$-0.163555\pi$
$284$		0			0
$285$	33445.3	−	31215.9i	0.0243906	−	0.0227648i
$286$		0			0
$287$	−291994.			−0.209252
$288$		0			0
$289$	228143.			0.160680
$290$		0			0
$291$	−1.67257e6	+	1.56108e6i	−1.15785	+	1.08067i
$292$		0			0
$293$		−	193285.i		−	0.131531i	−0.997835	−	0.0657656i	$-0.979051\pi$
$293$		−	193285.i		−	0.131531i	0.997835	−	0.0657656i	$-0.0209490\pi$
$294$		0			0
$295$		−	36573.6i		−	0.0244688i
$296$		0			0
$297$	807232.	+	994150.i	0.531016	+	0.653974i
$298$		0			0
$299$	−453690.			−0.293482
$300$		0			0
$301$	−448138.			−0.285099
$302$		0			0
$303$	1.69688e6	+	1.81807e6i	1.06180	+	1.13764i
$304$		0			0
$305$			63628.4i			0.0391653i
$306$		0			0
$307$		−	1.26426e6i		−	0.765579i	−0.923836	−	0.382789i	$-0.874963\pi$
$307$		−	1.26426e6i		−	0.765579i	0.923836	−	0.382789i	$-0.125037\pi$
$308$		0			0
$309$	1.49499e6	+	1.60176e6i	0.890722	+	0.954336i
$310$		0			0
$311$	929022.			0.544659			0.272330	−	0.962204i	$-0.412206\pi$
$311$	929022.			0.544659			0.272330	+	0.962204i	$0.412206\pi$
$312$		0			0
$313$	−1.16482e6			−0.672043			−0.336022	−	0.941854i	$-0.609081\pi$
$313$	−1.16482e6			−0.672043			−0.336022	+	0.941854i	$0.609081\pi$
$314$		0			0
$315$	1349.77	−	19551.0i	0.000766449	−	0.0111018i
$316$		0			0
$317$			245723.i			0.137340i	0.997639	+	0.0686702i	$0.0218756\pi$
$317$			245723.i			0.137340i	−0.997639	+	0.0686702i	$0.978124\pi$
$318$		0			0
$319$		−	2.20654e6i		−	1.21405i
$320$		0			0
$321$	1.65769e6	−	1.54719e6i	0.897924	−	0.838071i
$322$		0			0
$323$	−1.94657e6			−1.03816
$324$		0			0
$325$	−1.30917e6			−0.687526
$326$		0			0
$327$	461684.	−	430909.i	0.238768	−	0.222852i
$328$		0			0
$329$		−	106913.i		−	0.0544552i
$330$		0			0
$331$			115530.i			0.0579597i	0.999580	+	0.0289799i	$0.00922587\pi$
$331$			115530.i			0.0579597i	−0.999580	+	0.0289799i	$0.990774\pi$
$332$		0			0
$333$	−95792.3	+	1.38752e6i	−0.0473391	+	0.685693i
$334$		0			0
$335$	82027.1			0.0399342
$336$		0			0
$337$	1.86018e6			0.892237			0.446119	−	0.894974i	$-0.352806\pi$
$337$	1.86018e6			0.892237			0.446119	+	0.894974i	$0.352806\pi$
$338$		0			0
$339$	−269044.	−	288259.i	−0.127152	−	0.136233i
$340$		0			0
$341$			763799.i			0.355708i
$342$		0			0
$343$		−	117649.i		−	0.0539949i
$344$		0			0
$345$	−18942.0	−	20294.9i	−0.00856799	−	0.00917991i
$346$		0			0
$347$	−1.30222e6			−0.580579			−0.290290	−	0.956939i	$-0.593752\pi$
$347$	−1.30222e6			−0.580579			−0.290290	+	0.956939i	$0.593752\pi$
$348$		0			0
$349$	−973221.			−0.427708			−0.213854	−	0.976866i	$-0.568602\pi$
$349$	−973221.			−0.427708			−0.213854	+	0.976866i	$0.568602\pi$
$350$		0			0
$351$	1.00119e6	+	1.23302e6i	0.433758	+	0.534197i
$352$		0			0
$353$			1.31546e6i			0.561877i	0.959726	+	0.280938i	$0.0906457\pi$
$353$			1.31546e6i			0.561877i	−0.959726	+	0.280938i	$0.909354\pi$
$354$		0			0
$355$		−	257.814i		−	0.000108576i
$356$		0			0
$357$	−609584.	+	568950.i	−0.253141	+	0.236267i
$358$		0			0
$359$	2.20797e6			0.904183			0.452091	−	0.891972i	$-0.350678\pi$
$359$	2.20797e6			0.904183			0.452091	+	0.891972i	$0.350678\pi$
$360$		0			0
$361$	−703468.			−0.284103
$362$		0			0
$363$	−532871.	+	497351.i	−0.212254	+	0.198105i
$364$		0			0
$365$		−	24123.1i		−	0.00947767i
$366$		0			0
$367$			1.38578e6i			0.537068i	0.963270	+	0.268534i	$0.0865392\pi$
$367$			1.38578e6i			0.537068i	−0.963270	+	0.268534i	$0.913461\pi$
$368$		0			0
$369$	1.44461e6	+	99733.7i	0.552314	+	0.0381308i
$370$		0			0
$371$	−6994.15			−0.00263816
$372$		0			0
$373$	−834777.			−0.310669			−0.155335	−	0.987862i	$-0.549646\pi$
$373$	−834777.			−0.310669			−0.155335	+	0.987862i	$0.549646\pi$
$374$		0			0
$375$	−109366.	−	117177.i	−0.0401611	−	0.0430293i
$376$		0			0
$377$		−	2.73672e6i		−	0.991691i
$378$		0			0
$379$			4.32010e6i			1.54488i	0.635085	+	0.772442i	$0.280965\pi$
$379$			4.32010e6i			1.54488i	−0.635085	+	0.772442i	$0.719035\pi$
$380$		0			0
$381$	14501.2	+	15536.8i	0.00511789	+	0.00548341i
$382$		0			0
$383$	2.52528e6			0.879654			0.439827	−	0.898083i	$-0.355040\pi$
$383$	2.52528e6			0.879654			0.439827	+	0.898083i	$0.355040\pi$
$384$		0			0
$385$	−27264.8			−0.00937455
$386$		0			0
$387$	2.21712e6	+	153067.i	0.752510	+	0.0519521i
$388$		0			0
$389$			2.85190e6i			0.955567i	0.878478	+	0.477783i	$0.158560\pi$
$389$			2.85190e6i			0.955567i	−0.878478	+	0.477783i	$0.841440\pi$
$390$		0			0
$391$			1.18119e6i			0.390732i
$392$		0			0
$393$	2.35757e6	−	2.20041e6i	0.769985	−	0.718659i
$394$		0			0
$395$	13751.4			0.00443459
$396$		0			0
$397$	−4.63481e6			−1.47590			−0.737948	−	0.674858i	$-0.764205\pi$
$397$	−4.63481e6			−1.47590			−0.737948	+	0.674858i	$0.764205\pi$
$398$		0			0
$399$	−995707.	+	929335.i	−0.313112	+	0.292240i
$400$		0			0
$401$		−	6.19417e6i		−	1.92363i	−0.273697	−	0.961816i	$-0.588247\pi$
$401$		−	6.19417e6i		−	1.92363i	0.273697	−	0.961816i	$-0.411753\pi$
$402$		0			0
$403$			947320.i			0.290559i
$404$		0			0
$405$	−13355.7	+	96265.8i	−0.00404604	+	0.0291632i
$406$		0			0
$407$	1.93496e6			0.579011
$408$		0			0
$409$	−4.85132e6			−1.43401			−0.717004	−	0.697069i	$-0.754487\pi$
$409$	−4.85132e6			−1.43401			−0.717004	+	0.697069i	$0.754487\pi$
$410$		0			0
$411$	990652.	+	1.06140e6i	0.289279	+	0.309939i
$412$		0			0
$413$			1.08884e6i			0.314116i
$414$		0			0
$415$		−	201962.i		−	0.0575638i
$416$		0			0
$417$	1.11080e6	+	1.19013e6i	0.312820	+	0.335161i
$418$		0			0
$419$	3.14837e6			0.876093			0.438047	−	0.898952i	$-0.355670\pi$
$419$	3.14837e6			0.876093			0.438047	+	0.898952i	$0.355670\pi$
$420$		0			0
$421$	−5.27959e6			−1.45176			−0.725880	−	0.687821i	$-0.758567\pi$
$421$	−5.27959e6			−1.45176			−0.725880	+	0.687821i	$0.758567\pi$
$422$		0			0
$423$	−36517.2	+	528941.i	−0.00992308	+	0.143733i
$424$		0			0
$425$			3.40847e6i			0.915350i
$426$		0			0
$427$		−	1.89429e6i		−	0.502780i
$428$		0			0
$429$	1.61541e6	−	1.50773e6i	0.423779	−	0.395531i
$430$		0			0
$431$	5.93285e6			1.53840			0.769201	−	0.639006i	$-0.220654\pi$
$431$	5.93285e6			1.53840			0.769201	+	0.639006i	$0.220654\pi$
$432$		0			0
$433$	23551.4			0.00603667			0.00301834	−	0.999995i	$-0.499039\pi$
$433$	23551.4			0.00603667			0.00301834	+	0.999995i	$0.499039\pi$
$434$		0			0
$435$	122421.	−	114261.i	0.0310194	−	0.0289517i
$436$		0			0
$437$			1.92939e6i			0.483299i
$438$		0			0
$439$		−	780660.i		−	0.193331i	−0.995317	−	0.0966653i	$-0.969182\pi$
$439$		−	780660.i		−	0.193331i	0.995317	−	0.0966653i	$-0.0308176\pi$
$440$		0			0
$441$	−40184.3	+	582058.i	−0.00983920	+	0.142518i
$442$		0			0
$443$	5.11778e6			1.23900			0.619501	−	0.784996i	$-0.287335\pi$
$443$	5.11778e6			1.23900			0.619501	+	0.784996i	$0.287335\pi$
$444$		0			0
$445$	−64246.6			−0.0153798
$446$		0			0
$447$	5.06036e6	+	5.42176e6i	1.19788	+	1.28343i
$448$		0			0
$449$			2.13133e6i			0.498926i	0.968384	+	0.249463i	$0.0802540\pi$
$449$			2.13133e6i			0.498926i	−0.968384	+	0.249463i	$0.919746\pi$
$450$		0			0
$451$		−	2.01458e6i		−	0.466384i
$452$		0			0
$453$	−2.09115e6	−	2.24050e6i	−0.478784	−	0.512979i
$454$		0			0
$455$	−33815.8			−0.00765757
$456$		0			0
$457$	7.54577e6			1.69010			0.845051	−	0.534686i	$-0.179570\pi$
$457$	7.54577e6			1.69010			0.845051	+	0.534686i	$0.179570\pi$
$458$		0			0
$459$	3.21019e6	−	2.60662e6i	0.711212	−	0.577492i
$460$		0			0
$461$			1.93363e6i			0.423761i	0.977296	+	0.211880i	$0.0679587\pi$
$461$			1.93363e6i			0.423761i	−0.977296	+	0.211880i	$0.932041\pi$
$462$		0			0
$463$		−	2.86201e6i		−	0.620467i	−0.950660	−	0.310234i	$-0.899593\pi$
$463$		−	2.86201e6i		−	0.620467i	0.950660	−	0.310234i	$-0.100407\pi$
$464$		0			0
$465$	−42376.3	+	39551.6i	−0.00908848	+	0.00848266i
$466$		0			0
$467$	−5.10191e6			−1.08253			−0.541266	−	0.840851i	$-0.682055\pi$
$467$	−5.10191e6			−1.08253			−0.541266	+	0.840851i	$0.682055\pi$
$468$		0			0
$469$	−2.44205e6			−0.512651
$470$		0			0
$471$	−5.17859e6	+	4.83340e6i	−1.07562	+	1.00392i
$472$		0			0
$473$		−	3.09188e6i		−	0.635433i
$474$		0			0
$475$			5.56746e6i			1.13220i
$476$		0			0
$477$	34602.9	+	2388.93i	0.00696333	+	0.000480737i
$478$		0			0
$479$	−6.21679e6			−1.23802			−0.619010	−	0.785383i	$-0.712466\pi$
$479$	−6.21679e6			−1.23802			−0.619010	+	0.785383i	$0.712466\pi$
$480$		0			0
$481$	2.39989e6			0.472964
$482$		0			0
$483$	563928.	+	604203.i	0.109991	+	0.117846i
$484$		0			0
$485$			241564.i			0.0466314i
$486$		0			0
$487$			2.26782e6i			0.433297i	0.976250	+	0.216649i	$0.0695126\pi$
$487$			2.26782e6i			0.433297i	−0.976250	+	0.216649i	$0.930487\pi$
$488$		0			0
$489$	1.94898e6	+	2.08817e6i	0.368583	+	0.394907i
$490$		0			0
$491$	8.99591e6			1.68400			0.841999	−	0.539479i	$-0.181379\pi$
$491$	8.99591e6			1.68400			0.841999	+	0.539479i	$0.181379\pi$
$492$		0			0
$493$	−7.12511e6			−1.32031
$494$		0			0
$495$	134890.	+	9312.59i	0.0247438	+	0.00170827i
$496$		0			0
$497$			7675.44i			0.00139384i
$498$		0			0
$499$			9.36173e6i			1.68308i	0.540195	+	0.841540i	$0.318350\pi$
$499$			9.36173e6i			1.68308i	−0.540195	+	0.841540i	$0.681650\pi$
$500$		0			0
$501$	−6.18650e6	+	5.77412e6i	−1.10116	+	1.02776i
$502$		0			0
$503$	−1.05570e6			−0.186046			−0.0930230	−	0.995664i	$-0.529653\pi$
$503$	−1.05570e6			−0.186046			−0.0930230	+	0.995664i	$0.529653\pi$
$504$		0			0
$505$	262578.			0.0458173
$506$		0			0
$507$	−2.22770e6	+	2.07920e6i	−0.384890	+	0.359234i
$508$		0			0
$509$			9.97613e6i			1.70674i	0.521304	+	0.853371i	$0.325446\pi$
$509$			9.97613e6i			1.70674i	−0.521304	+	0.853371i	$0.674554\pi$
$510$		0			0
$511$			718175.i			0.121668i
$512$		0			0
$513$	5.24359e6	−	4.25770e6i	0.879702	−	0.714302i
$514$		0			0
$515$	231337.			0.0384351
$516$		0			0
$517$	737633.			0.121371
$518$		0			0
$519$	−33439.1	−	35827.3i	−0.00544924	−	0.00583842i
$520$		0			0
$521$		−	3.86656e6i		−	0.624065i	−0.950071	−	0.312033i	$-0.898990\pi$
$521$		−	3.86656e6i		−	0.624065i	0.950071	−	0.312033i	$-0.101010\pi$
$522$		0			0
$523$		−	3.98061e6i		−	0.636349i	−0.948032	−	0.318174i	$-0.896930\pi$
$523$		−	3.98061e6i		−	0.636349i	0.948032	−	0.318174i	$-0.103070\pi$
$524$		0			0
$525$	1.62728e6	+	1.74350e6i	0.257670	+	0.276072i
$526$		0			0
$527$	2.46637e6			0.386841
$528$		0			0
$529$	−5.26558e6			−0.818101
$530$		0			0
$531$	371906.	−	5.38694e6i	0.0572396	−	0.829099i
$532$		0			0
$533$		−	2.49863e6i		−	0.380964i
$534$		0			0
$535$		−	239414.i		−	0.0361631i
$536$		0			0
$537$	−6.77222e6	+	6.32079e6i	−1.01343	+	0.945880i
$538$		0			0
$539$	811706.			0.120345
$540$		0			0
$541$	7.73175e6			1.13576			0.567878	−	0.823113i	$-0.307765\pi$
$541$	7.73175e6			1.13576			0.567878	+	0.823113i	$0.307765\pi$
$542$		0			0
$543$	1.18234e6	−	1.10352e6i	0.172084	−	0.160614i
$544$		0			0
$545$		−	66679.6i		−	0.00961616i
$546$		0			0
$547$		−	1.75053e6i		−	0.250150i	−0.992147	−	0.125075i	$-0.960083\pi$
$547$		−	1.75053e6i		−	0.250150i	0.992147	−	0.125075i	$-0.0399172\pi$
$548$		0			0
$549$	−647017.	+	9.37184e6i	−0.0916188	+	1.32707i
$550$		0			0
$551$	−1.16383e7			−1.63309
$552$		0			0
$553$	−409395.			−0.0569285
$554$		0			0
$555$	100198.	+	107354.i	0.0138078	+	0.0147940i
$556$		0			0
$557$			8.11205e6i			1.10788i	0.832557	+	0.553940i	$0.186876\pi$
$557$			8.11205e6i			1.10788i	−0.832557	+	0.553940i	$0.813124\pi$
$558$		0			0
$559$		−	3.83478e6i		−	0.519052i
$560$		0			0
$561$	−3.92541e6	−	4.20576e6i	−0.526596	−	0.564205i
$562$		0			0
$563$	−1.07935e7			−1.43513			−0.717565	−	0.696491i	$-0.754744\pi$
$563$	−1.07935e7			−1.43513			−0.717565	+	0.696491i	$0.754744\pi$
$564$		0			0
$565$	−41632.3			−0.00548668
$566$		0			0
$567$	397616.	−	2.86595e6i	0.0519405	−	0.374379i
$568$		0			0
$569$		−	1.31525e7i		−	1.70305i	−0.524313	−	0.851526i	$-0.675678\pi$
$569$		−	1.31525e7i		−	1.70305i	0.524313	−	0.851526i	$-0.324322\pi$
$570$		0			0
$571$		−	1.20772e6i		−	0.155015i	−0.996992	−	0.0775077i	$-0.975304\pi$
$571$		−	1.20772e6i		−	0.155015i	0.996992	−	0.0775077i	$-0.0246962\pi$
$572$		0			0
$573$	6.59073e6	−	6.15140e6i	0.838585	−	0.782686i
$574$		0			0
$575$	3.37838e6			0.426127
$576$		0			0
$577$	3.91715e6			0.489814			0.244907	−	0.969547i	$-0.421243\pi$
$577$	3.91715e6			0.489814			0.244907	+	0.969547i	$0.421243\pi$
$578$		0			0
$579$	−2.58144e6	+	2.40937e6i	−0.320012	+	0.298681i
$580$		0			0
$581$			6.01265e6i			0.738968i
$582$		0			0
$583$		−	48255.4i		−	0.00587996i
$584$		0			0
$585$	167301.	+	11550.2i	0.0202119	+	0.00139540i
$586$		0			0
$587$	7.18456e6			0.860607			0.430303	−	0.902684i	$-0.358407\pi$
$587$	7.18456e6			0.860607			0.430303	+	0.902684i	$0.358407\pi$
$588$		0			0
$589$	4.02862e6			0.478485
$590$		0			0
$591$	−7.70702e6	−	8.25745e6i	−0.907649	−	0.972472i
$592$		0			0
$593$			1.81897e6i			0.212417i	0.994344	+	0.106208i	$0.0338710\pi$
$593$			1.81897e6i			0.212417i	−0.994344	+	0.106208i	$0.966129\pi$
$594$		0			0
$595$			88040.3i			0.0101950i
$596$		0			0
$597$	−4.84124e6	−	5.18700e6i	−0.555931	−	0.595635i
$598$		0			0
$599$	−1.14164e7			−1.30005			−0.650026	−	0.759912i	$-0.725242\pi$
$599$	−1.14164e7			−1.30005			−0.650026	+	0.759912i	$0.725242\pi$
$600$		0			0
$601$	1.40657e7			1.58846			0.794230	−	0.607618i	$-0.207875\pi$
$601$	1.40657e7			1.58846			0.794230	+	0.607618i	$0.207875\pi$
$602$		0			0
$603$	1.20818e7	+	834107.i	1.35313	+	0.0934176i
$604$		0			0
$605$			76960.9i			0.00854834i
$606$		0			0
$607$		−	1.39505e6i		−	0.153680i	−0.997043	−	0.0768399i	$-0.975517\pi$
$607$		−	1.39505e6i		−	0.153680i	0.997043	−	0.0768399i	$-0.0244830\pi$
$608$		0			0
$609$	−3.64463e6	+	3.40169e6i	−0.398208	+	0.371664i
$610$		0			0
$611$	914866.			0.0991413
$612$		0			0
$613$	3.63840e6			0.391074			0.195537	−	0.980696i	$-0.437355\pi$
$613$	3.63840e6			0.391074			0.195537	+	0.980696i	$0.437355\pi$
$614$		0			0
$615$	111771.	−	104321.i	0.0119163	−	0.0111220i
$616$		0			0
$617$			7.09314e6i			0.750111i	0.927002	+	0.375056i	$0.122376\pi$
$617$			7.09314e6i			0.750111i	−0.927002	+	0.375056i	$0.877624\pi$
$618$		0			0
$619$		−	6.19094e6i		−	0.649427i	−0.945813	−	0.324713i	$-0.894732\pi$
$619$		−	6.19094e6i		−	0.649427i	0.945813	−	0.324713i	$-0.105268\pi$
$620$		0			0
$621$	−2.58361e6	−	3.18185e6i	−0.268842	−	0.331094i
$622$		0			0
$623$	1.91270e6			0.197436
$624$		0			0
$625$	9.74024e6			0.997400
$626$		0			0
$627$	−6.41185e6	−	6.86977e6i	−0.651350	−	0.697868i
$628$		0			0
$629$		−	6.24816e6i		−	0.629688i
$630$		0			0
$631$		−	969152.i		−	0.0968988i	−0.998826	−	0.0484494i	$-0.984572\pi$
$631$		−	969152.i		−	0.0968988i	0.998826	−	0.0484494i	$-0.0154280\pi$
$632$		0			0
$633$	−1.02548e7	−	1.09872e7i	−1.01723	−	1.08988i
$634$		0			0
$635$	2243.94			0.000220839
$636$		0			0
$637$	1.00674e6			0.0983032
$638$		0			0
$639$	2621.63	−	37973.5i	0.000253991	−	0.00367899i
$640$		0			0
$641$		−	1.73952e6i		−	0.167219i	−0.996499	−	0.0836094i	$-0.973355\pi$
$641$		−	1.73952e6i		−	0.167219i	0.996499	−	0.0836094i	$-0.0266448\pi$
$642$		0			0
$643$			4.13653e6i			0.394556i	0.980348	+	0.197278i	$0.0632102\pi$
$643$			4.13653e6i			0.394556i	−0.980348	+	0.197278i	$0.936790\pi$
$644$		0			0
$645$	171541.	−	160106.i	0.0162356	−	0.0151533i
$646$		0			0
$647$	−1.45403e7			−1.36557			−0.682784	−	0.730620i	$-0.739231\pi$
$647$	−1.45403e7			−1.36557			−0.682784	+	0.730620i	$0.739231\pi$
$648$		0			0
$649$	−7.51235e6			−0.700106
$650$		0			0
$651$	1.26160e6	−	1.17750e6i	0.116672	−	0.108895i
$652$		0			0
$653$		−	2.16386e7i		−	1.98585i	−0.118761	−	0.992923i	$-0.537892\pi$
$653$		−	2.16386e7i		−	1.98585i	0.118761	−	0.992923i	$-0.462108\pi$
$654$		0			0
$655$		−	340496.i		−	0.0310105i
$656$		0			0
$657$	245301.	−	3.55310e6i	0.0221710	−	0.321140i
$658$		0			0
$659$	−3.79096e6			−0.340044			−0.170022	−	0.985440i	$-0.554384\pi$
$659$	−3.79096e6			−0.340044			−0.170022	+	0.985440i	$0.554384\pi$
$660$		0			0
$661$	−6.93084e6			−0.616996			−0.308498	−	0.951225i	$-0.599826\pi$
$661$	−6.93084e6			−0.616996			−0.308498	+	0.951225i	$0.599826\pi$
$662$		0			0
$663$	−4.86858e6	−	5.21629e6i	−0.430149	−	0.460869i
$664$		0			0
$665$			143807.i			0.0126103i
$666$		0			0
$667$			7.06221e6i			0.614648i
$668$		0			0
$669$	−672403.	−	720425.i	−0.0580851	−	0.0622334i
$670$		0			0
$671$	1.30695e7			1.12060
$672$		0			0
$673$	2.22164e7			1.89076			0.945379	−	0.325974i	$-0.105692\pi$
$673$	2.22164e7			1.89076			0.945379	+	0.325974i	$0.105692\pi$
$674$		0			0
$675$	−7.45530e6	−	9.18160e6i	−0.629804	−	0.775638i
$676$		0			0
$677$		−	1.95190e7i		−	1.63676i	−0.574678	−	0.818380i	$-0.694873\pi$
$677$		−	1.95190e7i		−	1.63676i	0.574678	−	0.818380i	$-0.305127\pi$
$678$		0			0
$679$		−	7.19166e6i		−	0.598624i
$680$		0			0
$681$	1.25812e7	−	1.17425e7i	1.03957	−	0.970273i
$682$		0			0
$683$	1.34295e7			1.10156			0.550781	−	0.834650i	$-0.314330\pi$
$683$	1.34295e7			1.10156			0.550781	+	0.834650i	$0.314330\pi$
$684$		0			0
$685$	153295.			0.0124825
$686$		0			0
$687$	−6.54764e6	+	6.11119e6i	−0.529289	+	0.494008i
$688$		0			0
$689$		−	59849.9i		−	0.00480303i
$690$		0			0
$691$		−	1.91529e6i		−	0.152595i	−0.997085	−	0.0762974i	$-0.975690\pi$
$691$		−	1.91529e6i		−	0.152595i	0.997085	−	0.0762974i	$-0.0243098\pi$
$692$		0			0
$693$	−4.01584e6	−	277247.i	−0.317646	−	0.0219298i
$694$		0			0
$695$	171886.			0.0134983
$696$		0			0
$697$	−6.50525e6			−0.507203
$698$		0			0
$699$	−9.66171e6	−	1.03517e7i	−0.747931	−	0.801347i
$700$		0			0
$701$		−	1.49746e7i		−	1.15096i	−0.817815	−	0.575481i	$-0.804815\pi$
$701$		−	1.49746e7i		−	1.15096i	0.817815	−	0.575481i	$-0.195185\pi$
$702$		0			0
$703$		−	1.02059e7i		−	0.778865i
$704$		0			0
$705$	38196.6	+	40924.6i	0.00289436	+	0.00310107i
$706$		0			0
$707$	−7.81725e6			−0.588174
$708$		0			0
$709$	1.88950e7			1.41166			0.705831	−	0.708380i	$-0.250574\pi$
$709$	1.88950e7			1.41166			0.705831	+	0.708380i	$0.250574\pi$
$710$		0			0
$711$	2.02544e6	+	139833.i	0.150261	+	0.0103738i
$712$		0			0
$713$		−	2.44460e6i		−	0.180088i
$714$		0			0
$715$		−	233308.i		−	0.0170673i
$716$		0			0
$717$	−1.06410e7	+	9.93171e6i	−0.773010	+	0.721483i
$718$		0			0
$719$	−7.05630e6			−0.509044			−0.254522	−	0.967067i	$-0.581918\pi$
$719$	−7.05630e6			−0.509044			−0.254522	+	0.967067i	$0.581918\pi$
$720$		0			0
$721$	−6.88719e6			−0.493406
$722$		0			0
$723$	5.44703e6	−	5.08394e6i	0.387538	−	0.361705i
$724$		0			0
$725$			2.03788e7i			1.43991i
$726$		0			0
$727$			1.64554e7i			1.15471i	0.816494	+	0.577353i	$0.195914\pi$
$727$			1.64554e7i			1.15471i	−0.816494	+	0.577353i	$0.804086\pi$
$728$		0			0
$729$	−2.94607e6	+	1.40432e7i	−0.205316	+	0.978696i
$730$		0			0
$731$	−9.98394e6			−0.691049
$732$		0			0
$733$	−1.24028e6			−0.0852631			−0.0426315	−	0.999091i	$-0.513574\pi$
$733$	−1.24028e6			−0.0852631			−0.0426315	+	0.999091i	$0.513574\pi$
$734$		0			0
$735$	42032.4	+	45034.3i	0.00286989	+	0.00307486i
$736$		0			0
$737$		−	1.68486e7i		−	1.14260i
$738$		0			0
$739$		−	8.86966e6i		−	0.597442i	−0.954340	−	0.298721i	$-0.903440\pi$
$739$		−	8.86966e6i		−	0.597442i	0.954340	−	0.298721i	$-0.0965600\pi$
$740$		0			0
$741$	−7.95244e6	−	8.52039e6i	−0.532053	−	0.570052i
$742$		0			0
$743$	993571.			0.0660278			0.0330139	−	0.999455i	$-0.489489\pi$
$743$	993571.			0.0660278			0.0330139	+	0.999455i	$0.489489\pi$
$744$		0			0
$745$	783048.			0.0516889
$746$		0			0
$747$	2.05369e6	−	2.97470e7i	0.134658	−	1.95048i
$748$		0			0
$749$			7.12766e6i			0.464240i
$750$		0			0
$751$			3.78698e6i			0.245015i	0.992468	+	0.122508i	$0.0390936\pi$
$751$			3.78698e6i			0.245015i	−0.992468	+	0.122508i	$0.960906\pi$
$752$		0			0
$753$	−2.32806e6	+	2.17288e6i	−0.149626	+	0.139652i
$754$		0			0
$755$	−323588.			−0.0206598
$756$		0			0
$757$	3.08256e7			1.95512			0.977558	−	0.210666i	$-0.0675634\pi$
$757$	3.08256e7			1.95512			0.977558	+	0.210666i	$0.0675634\pi$
$758$		0			0
$759$	−4.16863e6	+	3.89076e6i	−0.262657	+	0.245149i
$760$		0			0
$761$			2.63831e7i			1.65145i	0.564076	+	0.825723i	$0.309233\pi$
$761$			2.63831e7i			1.65145i	−0.564076	+	0.825723i	$0.690767\pi$
$762$		0			0
$763$			1.98513e6i			0.123446i
$764$		0			0
$765$	30071.1	−	435571.i	0.00185779	−	0.0269095i
$766$		0			0
$767$	−9.31736e6			−0.571880
$768$		0			0
$769$	1.29795e7			0.791485			0.395743	−	0.918362i	$-0.370487\pi$
$769$	1.29795e7			0.791485			0.395743	+	0.918362i	$0.370487\pi$
$770$		0			0
$771$	−1.32084e6	−	1.41517e6i	−0.0800227	−	0.0857378i
$772$		0			0
$773$			2.48016e7i			1.49290i	0.665439	+	0.746452i	$0.268244\pi$
$773$			2.48016e7i			1.49290i	−0.665439	+	0.746452i	$0.731756\pi$
$774$		0			0
$775$		−	7.05417e6i		−	0.421883i
$776$		0			0
$777$	−2.98301e6	−	3.19605e6i	−0.177256	−	0.189916i
$778$		0			0
$779$	−1.06258e7			−0.627362
$780$		0			0
$781$	−52955.8			−0.00310661
$782$		0			0
$783$	1.91933e7	−	1.55847e7i	1.11878	−	0.908433i
$784$		0			0
$785$			747928.i			0.0433197i
$786$		0			0
$787$			2.80000e7i			1.61147i	0.592277	+	0.805734i	$0.298229\pi$
$787$			2.80000e7i			1.61147i	−0.592277	+	0.805734i	$0.701771\pi$
$788$		0			0
$789$	−1.32198e7	+	1.23386e7i	−0.756017	+	0.705622i
$790$		0			0
$791$	1.23944e6			0.0704346
$792$		0			0
$793$	1.62097e7			0.915361
$794$		0			0
$795$	2677.26	−	2498.80i	0.000150235	−	0.000140221i
$796$		0			0
$797$		−	2.18479e7i		−	1.21833i	−0.793045	−	0.609164i	$-0.791505\pi$
$797$		−	2.18479e7i		−	1.21833i	0.793045	−	0.609164i	$-0.208495\pi$
$798$		0			0
$799$		−	2.38188e6i		−	0.131993i
$800$		0			0
$801$	−9.46290e6	−	653303.i	−0.521126	−	0.0359777i
$802$		0			0
$803$	−4.95497e6			−0.271177
$804$		0			0
$805$	87263.1			0.00474615
$806$		0			0
$807$	2.19246e7	+	2.34904e7i	1.18508	+	1.26972i
$808$		0			0
$809$			1.20561e7i			0.647643i	0.946118	+	0.323821i	$0.104968\pi$
$809$			1.20561e7i			0.647643i	−0.946118	+	0.323821i	$0.895032\pi$
$810$		0			0
$811$			3.24274e7i			1.73125i	0.500692	+	0.865625i	$0.333079\pi$
$811$			3.24274e7i			1.73125i	−0.500692	+	0.865625i	$0.666921\pi$
$812$		0			0
$813$	−1.31219e7	−	1.40591e7i	−0.696260	−	0.745985i
$814$		0			0
$815$	301588.			0.0159045
$816$		0			0
$817$	−1.63080e7			−0.854761
$818$		0			0
$819$	−4.98074e6	−	343862.i	−0.259468	−	0.0179133i
$820$		0			0
$821$		−	5.93027e6i		−	0.307055i	−0.988144	−	0.153528i	$-0.950937\pi$
$821$		−	5.93027e6i		−	0.307055i	0.988144	−	0.153528i	$-0.0490634\pi$
$822$		0			0
$823$		−	1.61840e7i		−	0.832886i	−0.909162	−	0.416443i	$-0.863276\pi$
$823$		−	1.61840e7i		−	0.832886i	0.909162	−	0.416443i	$-0.136724\pi$
$824$		0			0
$825$	−1.20291e7	+	1.12272e7i	−0.615314	+	0.574298i
$826$		0			0
$827$	−4.71859e6			−0.239910			−0.119955	−	0.992779i	$-0.538275\pi$
$827$	−4.71859e6			−0.239910			−0.119955	+	0.992779i	$0.538275\pi$
$828$		0			0
$829$	1.91120e7			0.965873			0.482937	−	0.875655i	$-0.339570\pi$
$829$	1.91120e7			0.965873			0.482937	+	0.875655i	$0.339570\pi$
$830$		0			0
$831$	−2.21482e7	+	2.06719e7i	−1.11259	+	1.03843i
$832$		0			0
$833$		−	2.62107e6i		−	0.130878i
$834$		0			0
$835$			893497.i			0.0443483i
$836$		0			0
$837$	−6.64381e6	+	5.39466e6i	−0.327796	+	0.266165i
$838$		0			0
$839$	−1.38363e7			−0.678602			−0.339301	−	0.940678i	$-0.610191\pi$
$839$	−1.38363e7			−0.678602			−0.339301	+	0.940678i	$0.610191\pi$
$840$		0			0
$841$	−2.20890e7			−1.07693
$842$		0			0
$843$	−7.48678e6	−	8.02148e6i	−0.362849	−	0.388764i
$844$		0			0
$845$			321739.i			0.0155011i
$846$		0			0
$847$		−	2.29122e6i		−	0.109738i
$848$		0			0
$849$	−1.40871e7	−	1.50932e7i	−0.670737	−	0.718640i
$850$		0			0
$851$	−6.19300e6			−0.293142
$852$		0			0
$853$	1.04104e7			0.489885			0.244942	−	0.969538i	$-0.421231\pi$
$853$	1.04104e7			0.489885			0.244942	+	0.969538i	$0.421231\pi$
$854$		0			0
$855$	49118.8	−	711471.i	0.00229791	−	0.0332845i
$856$		0			0
$857$			3.94441e7i			1.83455i	0.398250	+	0.917277i	$0.369617\pi$
$857$			3.94441e7i			1.83455i	−0.398250	+	0.917277i	$0.630383\pi$
$858$		0			0
$859$		−	443047.i		−	0.0204865i	−0.999948	−	0.0102432i	$-0.996739\pi$
$859$		−	443047.i		−	0.0204865i	0.999948	−	0.0102432i	$-0.00326058\pi$
$860$		0			0
$861$	−3.32756e6	+	3.10575e6i	−0.152974	+	0.142777i
$862$		0			0
$863$	3.73239e7			1.70593			0.852963	−	0.521972i	$-0.174803\pi$
$863$	3.73239e7			1.70593			0.852963	+	0.521972i	$0.174803\pi$
$864$		0			0
$865$	−5174.42			−0.000235137
$866$		0			0
$867$	2.59991e6	−	2.42660e6i	0.117466	−	0.109636i
$868$		0			0
$869$		−	2.82458e6i		−	0.126883i
$870$		0			0
$871$		−	2.08969e7i		−	0.933333i
$872$		0			0
$873$	−2.45639e6	+	3.55801e7i	−0.109084	+	1.58005i
$874$		0			0
$875$	503834.			0.0222468
$876$		0			0
$877$	3.94244e7			1.73087			0.865437	−	0.501017i	$-0.167041\pi$
$877$	3.94244e7			1.73087			0.865437	+	0.501017i	$0.167041\pi$
$878$		0			0
$879$	−2.05584e6	−	2.20267e6i	−0.0897466	−	0.0961562i
$880$		0			0
$881$		−	1.95878e7i		−	0.850251i	−0.905134	−	0.425125i	$-0.860230\pi$
$881$		−	1.95878e7i		−	0.850251i	0.905134	−	0.425125i	$-0.139770\pi$
$882$		0			0
$883$			3.75521e7i			1.62081i	0.585869	+	0.810406i	$0.300753\pi$
$883$			3.75521e7i			1.62081i	−0.585869	+	0.810406i	$0.699247\pi$
$884$		0			0
$885$	−389010.	−	416792.i	−0.0166956	−	0.0178880i
$886$		0			0
$887$	3.72169e7			1.58830			0.794148	−	0.607724i	$-0.207917\pi$
$887$	3.72169e7			1.58830			0.794148	+	0.607724i	$0.207917\pi$
$888$		0			0
$889$	−66804.8			−0.00283500
$890$		0			0
$891$	1.97733e7	+	2.74331e6i	0.834421	+	0.115766i
$892$		0			0
$893$		−	3.89061e6i		−	0.163263i
$894$		0			0
$895$			978090.i			0.0408151i
$896$		0			0
$897$	−5.17024e6	+	4.82560e6i	−0.214551	+	0.200249i
$898$		0			0
$899$	1.47461e7			0.608526
$900$		0			0
$901$	−155821.			−0.00639460
$902$		0			0
$903$	−5.10697e6	+	4.76655e6i	−0.208422	+	0.194529i
$904$		0			0
$905$		−	170761.i		−	0.00693055i
$906$		0			0
$907$			2.55703e7i			1.03209i	0.856561	+	0.516046i	$0.172597\pi$
$907$			2.55703e7i			1.03209i	−0.856561	+	0.516046i	$0.827403\pi$
$908$		0			0
$909$	3.86751e7	+	2.67007e6i	1.55247	+	0.107180i
$910$		0			0
$911$	2.72611e7			1.08830			0.544149	−	0.838988i	$-0.316852\pi$
$911$	2.72611e7			1.08830			0.544149	+	0.838988i	$0.316852\pi$
$912$		0			0
$913$	−4.14836e7			−1.64702
$914$		0			0
$915$	676773.	+	725107.i	0.0267233	+	0.0286319i
$916$		0			0
$917$			1.01370e7i			0.398093i
$918$		0			0
$919$			2.60245e7i			1.01647i	0.861219	+	0.508235i	$0.169702\pi$
$919$			2.60245e7i			1.01647i	−0.861219	+	0.508235i	$0.830298\pi$
$920$		0			0
$921$	−1.34471e7	−	1.44075e7i	−0.522371	−	0.559678i
$922$		0			0
$923$	−65679.7			−0.00253762
$924$		0			0
$925$	−1.78706e7			−0.686729
$926$		0			0
$927$	3.40738e7	+	2.35240e6i	1.30233	+	0.0899106i
$928$		0			0
$929$			4.54664e7i			1.72843i	0.503125	+	0.864213i	$0.332183\pi$
$929$			4.54664e7i			1.72843i	−0.503125	+	0.864213i	$0.667817\pi$
$930$		0			0
$931$		−	4.28130e6i		−	0.161883i
$932$		0			0
$933$	1.05871e7	−	9.88139e6i	0.398174	−	0.371633i
$934$		0			0
$935$	−607425.			−0.0227229
$936$		0			0
$937$	−1.41416e7			−0.526198			−0.263099	−	0.964769i	$-0.584745\pi$
$937$	−1.41416e7			−0.526198			−0.263099	+	0.964769i	$0.584745\pi$
$938$		0			0
$939$	−1.32742e7	+	1.23894e7i	−0.491299	+	0.458550i
$940$		0			0
$941$		−	9.68022e6i		−	0.356378i	−0.983996	−	0.178189i	$-0.942976\pi$
$941$		−	9.68022e6i		−	0.356378i	0.983996	−	0.178189i	$-0.0570239\pi$
$942$		0			0
$943$			6.44782e6i			0.236121i
$944$		0			0
$945$	−192569.	−	237159.i	−0.00701468	−	0.00863895i
$946$		0			0
$947$	2.97694e7			1.07869			0.539343	−	0.842086i	$-0.318673\pi$
$947$	2.97694e7			1.07869			0.539343	+	0.842086i	$0.318673\pi$
$948$		0			0
$949$	−6.14552e6			−0.221510
$950$		0			0
$951$	2.61360e6	+	2.80026e6i	0.0937103	+	0.100403i
$952$		0			0
$953$		−	2.27703e7i		−	0.812151i	−0.913840	−	0.406076i	$-0.866897\pi$
$953$		−	2.27703e7i		−	0.812151i	0.913840	−	0.406076i	$-0.133103\pi$
$954$		0			0
$955$		−	951878.i		−	0.0337733i
$956$		0			0
$957$	−2.34696e7	−	2.51457e7i	−0.828371	−	0.887532i
$958$		0			0
$959$	−4.56378e6			−0.160243
$960$		0			0
$961$	2.35247e7			0.821706
$962$		0			0
$963$	2.43453e6	−	3.52634e7i	0.0845959	−	1.22535i
$964$		0			0
$965$			372830.i			0.0128882i
$966$		0			0
$967$		−	3.19631e7i		−	1.09921i	−0.835423	−	0.549607i	$-0.814778\pi$
$967$		−	3.19631e7i		−	1.09921i	0.835423	−	0.549607i	$-0.185222\pi$
$968$		0			0
$969$	−2.21831e7	+	2.07044e7i	−0.758948	+	0.708358i
$970$		0			0
$971$	−6.18007e6			−0.210351			−0.105176	−	0.994454i	$-0.533541\pi$
$971$	−6.18007e6			−0.210351			−0.105176	+	0.994454i	$0.533541\pi$
$972$		0			0
$973$	−5.11727e6			−0.173283
$974$		0			0
$975$	−1.49193e7	+	1.39248e7i	−0.502617	+	0.469114i
$976$		0			0
$977$		−	2.28930e7i		−	0.767303i	−0.923478	−	0.383651i	$-0.874666\pi$
$977$		−	2.28930e7i		−	0.767303i	0.923478	−	0.383651i	$-0.125334\pi$
$978$		0			0
$979$			1.31965e7i			0.440048i
$980$		0			0
$981$	678043.	−	9.82125e6i	0.0224949	−	0.325833i
$982$		0			0
$983$	4.53673e7			1.49747			0.748737	−	0.662867i	$-0.230661\pi$
$983$	4.53673e7			1.49747			0.748737	+	0.662867i	$0.230661\pi$
$984$		0			0
$985$	−1.19260e6			−0.0391655
$986$		0			0
$987$	−1.13716e6	−	1.21837e6i	−0.0371560	−	0.0398096i
$988$		0			0
$989$			9.89581e6i			0.321707i
$990$		0			0
$991$			4.98408e7i			1.61213i	0.591824	+	0.806067i	$0.298408\pi$
$991$			4.98408e7i			1.61213i	−0.591824	+	0.806067i	$0.701592\pi$
$992$		0			0
$993$	1.22882e6	+	1.31658e6i	0.0395472	+	0.0423716i
$994$		0			0
$995$	−749142.			−0.0239887
$996$		0			0
$997$	−3.86590e7			−1.23172			−0.615862	−	0.787854i	$-0.711192\pi$
$997$	−3.86590e7			−1.23172			−0.615862	+	0.787854i	$0.711192\pi$
$998$		0			0
$999$	1.36665e7	+	1.68311e7i	0.433256	+	0.533577i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.h.b.239.29	yes	40
3.2	odd	2	inner	336.6.h.b.239.11	✓	40
4.3	odd	2	inner	336.6.h.b.239.12	yes	40
12.11	even	2	inner	336.6.h.b.239.30	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.6.h.b.239.11	✓	40	3.2	odd	2	inner
336.6.h.b.239.12	yes	40	4.3	odd	2	inner
336.6.h.b.239.29	yes	40	1.1	even	1	trivial
336.6.h.b.239.30	yes	40	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(11.3960 - 10.6363i) q^{3} -1.64589i q^{5} +49.0000i q^{7} +(16.7365 - 242.423i) q^{9} +O(q^{10})\)	\(q+(11.3960 - 10.6363i) q^{3} -1.64589i q^{5} +49.0000i q^{7} +(16.7365 - 242.423i) q^{9} -338.070 q^{11} -419.299 q^{13} +(-17.5062 - 18.7565i) q^{15} +1091.66i q^{17} +1783.13i q^{19} +(521.181 + 558.403i) q^{21} +1082.02 q^{23} +3122.29 q^{25} +(-2387.76 - 2940.66i) q^{27} +6526.88i q^{29} -2259.29i q^{31} +(-3852.64 + 3595.83i) q^{33} +80.6484 q^{35} -5723.56 q^{37} +(-4778.32 + 4459.81i) q^{39} +5959.06i q^{41} +9145.68i q^{43} +(-399.000 - 27.5463i) q^{45} -2181.89 q^{47} -2401.00 q^{49} +(11611.2 + 12440.5i) q^{51} +142.738i q^{53} +556.425i q^{55} +(18966.0 + 20320.6i) q^{57} +22221.3 q^{59} -38659.1 q^{61} +(11878.7 + 820.088i) q^{63} +690.118i q^{65} +49837.7i q^{67} +(12330.7 - 11508.7i) q^{69} +156.642 q^{71} +14656.6 q^{73} +(35581.6 - 33209.8i) q^{75} -16565.4i q^{77} +8355.00i q^{79} +(-58488.8 - 8114.62i) q^{81} +122707. q^{83} +1796.74 q^{85} +(69422.1 + 74380.2i) q^{87} -39034.7i q^{89} -20545.7i q^{91} +(-24030.6 - 25746.8i) q^{93} +2934.83 q^{95} -146768. q^{97} +(-5658.10 + 81956.0i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{9}+O(q^{10}) \) 40 * q + 8 * q^9	\( 40 q + 8 q^{9} - 1048 q^{13} + 980 q^{21} - 43416 q^{25} + 20296 q^{33} - 16192 q^{37} + 56488 q^{45} - 96040 q^{49} + 31088 q^{57} + 173112 q^{61} - 114176 q^{69} - 267488 q^{73} + 64888 q^{81} + 508112 q^{85} - 224544 q^{93} - 276400 q^{97}+O(q^{100}) \) 40 * q + 8 * q^9 - 1048 * q^13 + 980 * q^21 - 43416 * q^25 + 20296 * q^33 - 16192 * q^37 + 56488 * q^45 - 96040 * q^49 + 31088 * q^57 + 173112 * q^61 - 114176 * q^69 - 267488 * q^73 + 64888 * q^81 + 508112 * q^85 - 224544 * q^93 - 276400 * q^97

Introduction

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